Particle Physics And Cosmology: The Quest For Physics Beyond The Standard Model(s) : Tasi 2002 Boulder, Colorado, USA3 2- 28 June 2002

PARTICLE PHYSICS AND COSMOLOGY THE QUEST FOR PHYSICS BEYOND THE STANDARD MODEL(S) Editors

Howard E. Haber Ann E. Nelson

PARTICLE PHYSICS AND COSMOLOGY THE QUEST FOR PHYSICS BEYOND THE STANDARD MODEL(S)

TASI 2002

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PARTICLE PHYSICS AND COSMOLOGY THE QUEST FOR PHYSICS BEYOND THE STANDARD MODEL(S)

TASI2002 Boulder, Colorado, USA

3 - 2 8 June 2002

Editors

Howard E. Haber University of California, Santa Cruz, USA

Ann E. Nelson University of Washington, USA

\^p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401^02, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

The editors and the publisher would like to thank Professor John Terning for permission to reproduce his lectures in this book.

PARTICLE PHYSICS AND COSMOLOGY The Quest for Physics Beyond the Standard Model(s) (TASI 2002) Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-892-3

Printed in Singapore by World Scientific Printers (S) Pte Ltd

PREFACE

The Theoretical Advanced Study Institute (TASI) in elementary particle physics has been held each summer since 1984. TASI was first organized in Ann Arbor, and was held in subsequent years in New Haven, Santa Cruz, Santa Fe and Providence. Since 1989, TASI has been permanently hosted by the University of Colorado in Boulder. Each year TASI attracts about fifty of the most promising advanced theory graduate students in the United States, along with some graduate students from abroad, and a few experimental graduate students and beginning postdoctoral fellows. The emphasis of the topics treated at TASI has shifted from year to year, but typically there have been courses of lectures in phenomenology, field theory, string theory, mathematical physics, and particle-astrophysics, as well as seminars on related theoretical and experimental topics. TASI has been highly successful, especially in introducing the students to a much broader range of ideas than they normally experience at their home institutions while working on their (often narrow) dissertation topics. The title of TASI-2002 was Particle Physics and Cosmology: The Quest for Physics Beyond the Standard Model(s). There were three parallel lecture series given. The Phenomenology lecture series aimed to cover a broad spectrum of standard research techniques used to interpret current and future collider data. This series included the following lecture courses: • • • • • • •

Neutrino Mixing and Masses, by Yuval Grossman Precision Electroweak Physics, by Konstantin Matchev Higgs Theory and Phenomenology, by Marcela Carena Techniques in Effective Field Theory, by Ira Rothstein Bottom Quark Physics, by Michael Luke Top Quark Physics, by Sally Dawson Tevatron Physics, by John Womersley

Complementing the phenomenology lecture series, the TeV-Scale Physics lecture series focused on modern speculations about physics beyond the Standard Model, with an emphasis on supersymmetry and extraV

vi

dimensional theories. This series included the following lecture courses: • • • • • • • •

Introduction to Super symmetry, by Carlos Wagner Non-perturbative Techniques in Supersymmetry, by John Terning The $upersymmetric Flavor Problem, by Ann Nelson Kaluza-Klein Theory and Large Extra Dimensions, by Keith Dienes New Ideas in Symmetry Breaking, by Mariano Quiros Extra Dimensions and Branes, by Csaba Csaki Deconstruction, by Andrew Cohen Strings for Pedestrians, by Mirjam Cvetic

Finally, in recognition of the increasingly important role of astrophysics and cosmology in modern particle physics as we seek to find a single description of physics that can be used at all distance and energy scales, the traditional particle physics lecture series was supplemented by an Astroparticle Physics lecture series. In this series of courses, after an introduction to cosmology, we learned about the current dark matter paradigms; the recent exciting discovery of dark energy and its implications; the implications of the observation of the cosmic microwave background, its acoustic peaks and polarization; and the upcoming era of gravitational wave astronomy. The Astroparticle Physics lecture series included the following lecture courses: • • • • •

Introduction to Cosmology, by Sean Carroll Dark Matter, by Keith Olive Dark Energy, by Joshua Frieman The Cosmic Microwave Background, by Mathias Zaldariagga Gravitational Waves, by Alessandra Buonanno

The Scientific Program was organized by Howard Haber and Ann Nelson, with invaluable assistance provided by the local director, K.T. Mahanthappa. In addition, we would especially like to acknowledge Kathy Oliver for efficient secretarial help; Mu-Chun Chen and Alex Flournoy for help in the day-to-day running of TASI; Tom DeGrand for assisting participants with computer related matters and also leading the weekend hikes; Dan Hooper and Bjorn O. Lange for organizing student seminars; and the National Science Foundation, U.S. Department of Energy and University of Colorado for providing the facilities and financial support. Howard E. Haber and Ann Nelson TASI-02 Program Co-directors January 30, 2004

CONTENTS Preface

v

Lecturers, local organizing committee and directors Student seminars

ix xii

P A R T I: P h e n o m e n o l o g y L e c t u r e S e r i e s Neutrinos Yuval

5 Grossman

Precision Electroweak Physics Konstantin

Matchev

Effective Field Theories Ira Z.

Michael

193

Luke

T h e Top Quark, QCD and New Physics

245

Dawson

Tevatron Physics John

101

Rothstein

B o t t o m Quark Physics and the Heavy Quark Expansion

Sally

51

303

Womersley

P A R T II: T e V - S c a l e P h y s i c s L e c t u r e S e r i e s Non-perturbative Supersymmetry John

343

Terning

New Directions for New Dimensions: Kaluza-Klein Theory, Large E x t r a Dimensions and the Brane World • Keith R.

Dienes

447

viii

New Ideas in Symmetry Breaking

549

Mariano Quiros Extra Dimensions and Branes

605

Csaba Csaki PART III: Astroparticle Physics Lecture Series Introduction to Cosmology

703

Mark Trodden and Sean M. Carroll Dark Matter

797

Keith A. Olive Gravitational Waves from the Early Universe Alessandra Buonanno

855

Lecturers

Name

(Institution)

Alessandra Buonanno Marcela Carena Sean Carroll Andrew Cohen Csaba Csaki Mirjam Cvetic Sally Dawson Keith Dienes Joshua Prieman Yuval Grossman Michael Luke Konstantin Matchev Ann Nelson Keith Olive Mariano Quiros Ira Rothstein John Terning Carlos Wagner John Womersley Matias Zaldarriaga

California Institute of Technology Fermilab Enrico Fermi Institute Boston University Cornell University University of Pennsylvania Brookhaven National Laboratory University of Arizona Fermilab Israel Institute of Technology, Technion University of Toronto University of Florida University of Washington University of Minnesota Universidad Autonoma de Barcelona Carnegie Mellon University Los Alamos National Laboratory Argonne National Laboratory Fermilab New York University

Local Organizing Committee

Name

(Institution)

Shanta DeAlwis Tom DeGrand Anna Hasenfratz K.T. Mahanthappa

University University University University

of of of of

Colorado Colorado Colorado Colorado

Directors Name

(Institution)

Howard Haber K.T. Mahanthappa A n n Nelson

University of California, Santa Cruz University of Colorado University of Washington

Participants Name

(Institution)

Christopher Aubin Sandor Benczik HuBo Elena Brewer Qinghong Cao Spencer Chang Jennifer Chen Mu-Chun Chen Ana Maria Curiel Samik Dasgupta Rob Fardon Bryan Field Terrance Figy Alex Flournoy Michael Forbes Hock-Seng Goh Roni Harnik Daniel Hooper Kyuwan Hwang Alberto Iglesias Christopher Jackson Abu K h a n Vishesh Khemani Savvas Koushiappas Tadas Krupovnickas Andrei Kryjevski

Washington University Virginia Tech Texas A&M University SUNY at Buffalo Michigan State University Harvard University University of Chicago University of Colorado CSIC, Madrid University of Colorado University of Washington SUNY at Stony Brook University of Wisconsin University of Colorado MIT University of Maryland UC Berkeley University of Wisconsin Seoul National University SUNY at Stony Brook Florida State University University of Florida MIT Ohio State University Florida State University University of Washington

Joydip Kundu Herry Kwee Bjorn Lange Christopher Lee Jason Lennon Nuno Leonardo Eugene Lim Zhaofeng Liu Mattew Martin Michael May David Maybury Robert McElrath Stavros Mouslopoulos Brandon Murakami Siew-Phang Ng Takemichi Okui Steven Oliver Vakif Onemli John Phillips Martin Puchwein Saswat Sarangi Liguo Song Marc Soussa Keith Thomas Manuel Toharia Dionysis Triantafyllopoulos Michael Trott Mithat Unsal Haibin Wang Ho-Ung Yee Andrew Zentner Xiaomin Zu

MIP College of William & Mary Cornell University Caltech University of Notre Dame Fermilab University of Chicago University of Colorado Carnegie Mellon University Johns Hopkins University University of Alberta University of Wisconsin UC Berkeley UC Davis University of Maryland Lawrence Berkeley National Lab UC Berkeley University of Florida Ohio State University ITP, ETH-Zurich Cornell University Vanderbilt University University of Florida University of Wisconsin UC Davis Columbia University University of Toronto University of Washington University of Michigan Yale University Ohio State University Pennsylvania State University

xii

Student Seminars* (In Alphabetic Order)

1. Sandor Benczik, "The classical limit of the minimal-length uncertainty relation." 2. Spencer Chang, "SU(3) Symmetry at a TeV and the Weak Mixing Angle." 3. Jennifer Chen, "Quintessential time variation of the fine structure constant?" 4. Ana Maria Curiel, "Optimal Observables to search for indirect SUSY-QCD signals." 5. Rob Fardon, "The littlest Higgs." 6. Michael McNeil Forbes, "Topological Defects in QCD." 7. Roni Harnik, "Viable Anomaly Mediation in 4-D." 8. Dan Hooper, "Microscopic Black Holes and TeV Gravity in Neutrino Telescopes." 9. Bo Hu, "Yukawa texture, neutrinos, \i —> e + 7 and M-theory." 10. Kyuwan Hwang, "Orbifolded SU(n) and Unification of families." 11. Vishesh Khemani, "Decoupling Fermion doublets from the SM." 12. Savvas Koushiappas, "Gamma rays from neutralino annihilations." 13. Joydip Kundu, "Crystaline Color Superconductivity." 14. Bjorn O. Lange, "QCD Sum Rules (and the heavy meson decay constant / B ) . " 15. Jason E. Lennon, "Higgs mediated Bs —> /Lt+/x~ in SUSY models." 16. Eugene Lim, "How to make Stuff (or: Generating cosmological perturbations)." 17. Matthew R. Martin, "Radion Induced Brane Preheating." 18. Mike May, "Brane running in RS 1." 19. David Maybury, "Model building with SO(10)." 20. Robert McElrath, "Black Holes at Colliders." 21. Stavros Mouslopoulos, "Multigravity from Brane worlds." 22. Brandon Murakami, "Lepton Flavor Violation through Z' Bosons." 23. Takemichi Okui, "Gauge Coupling Unification from Particle in a Box Physics." 24. John Phillips, "Halo substructure from lensing." 25. Saswat Sarangi, "Brane-Antibrane Inflation." 'Organized by Dan Hooper and Bjorn O. Lange.

Xlll

26. 27. 28. 29. 30.

Manuel Toharia, "On the Radion." Mithat Unsal, "Supersymmetry on a spatial lattice." Ho-Ung Yee, "K K contributions to S,T,U." Andrew Zentner, "Small-scale Cold Dark Matter and Inflation. Xiaomin Zu, "Factorization Theorems and perturbative QCD."

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PART I: Phenomenology Lecture Series

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illpiiMMlii

YUVAL GROSSMAN

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NEUTRINOS

YUVAL GROSSMAN Department of Physics, Technion-Israel Institute of Technology, Technion City, 32000 Haifa, Israel We present a pedagogical review of neutrino physics. In the first lecture we describe the theoretical motivation for neutrino masses, and explain how neutrino flavor oscillation experiments can probe neutrino masses. In the second lecture we review the experimental data, and show that it is best explained if neutrinos are massive. In the third lecture we explain what are the theoretical implications of the data, in particular, what are the challenges they impose on models of physics beyond the SM. We give examples of theoretical models that cope with some of these challenges.

Contents 1

Introduction

6

2

Neutrino masses 2.1 Fermion masses 2.2 Neutrino masses in the SM 2.3 Neutrino masses beyond the SM 2.4 Neutrino mixing

7 7 9 10 13

3

M e t h o d s for probing neutrino masses 3.1 Kinematic tests 3.2 Neutrinoless double (3 decay 3.3 Neutrino vacuum oscillation 3.4 Matter effects 3.5 Non-uniform density 3.6 Neutrino oscillation experiments

14 15 15 16 18 19 22

4

Solar neutrinos 4.1 Solar neutrinos production 4.2 Solar neutrinos propagation 4.3 Solar neutrinos detections

23 23 24 25 5

6

4.4 4.5 4.6 4.7 4.8

Chlorine: Homestake Gallium: SAGE, GALLEX and GNO Water Cerenkov: Kamiokande and SuperKamiokande Heavy Water: SNO Solar neutrinos: fits

26 27 27 28 29

5

Atmospheric neutrinos 5.1 Atmospheric neutrino production 5.2 Atmospheric neutrino propagation 5.3 Atmospheric neutrino detection 5.4 Atmospheric neutrino data 5.5 The atmospheric neutrino problem

32 32 32 33 33 35

6

Terrestrial neutrinos 6.1 K2K 6.2 KamLAND 6.3 LSND

35 36 36 37

7

Theoretical implications 7.1 Two neutrino mixing 7.2 Three neutrino mixing 7.3 Four neutrino mixing

38 38 40 41

8

Models for neutrino masses 8.1 Grand Unified Theories (GUTs) 8.2 Flavor physics

42 42 43

9

Conclusions

45

1. I n t r o d u c t i o n T h e success of the S t a n d a r d Model (SM) can be seen as a proof t h a t it is an effective low energy description of Nature. We are therefore interested in probing the more fundamental theory. One way to go is to search for new particles t h a t can be produced in yet unreached energies. Another way to look for new physics is to search for indirect effects of heavy unknown particles. In this set of lectures we explain how neutrino physics is used to probe such indirect signals of physics beyond the SM. In the SM the neutrinos are exactly massless. This prediction, however, is rather specific to the SM. In almost all of the SM extensions the neutrinos are massive and they mix. T h e search for neutrino flavor oscillation, a phenomenon which is possible only for massive neutrinos, is a search for new physics beyond the SM. T h e recent experimental indications for neutrino oscillations are indirect evidences for new physics, most likely, at distances much shorter t h a n the weak scale.

7

In the first lecture the basic mechanisms for generating neutrino masses are described and the ingredients of the SM that ensure massless neutrinos are explained. Then, the neutrino oscillation formalism is developed. In the second lecture the current experimental situation is summarized. In particular, we describe the oscillation signals observed by solar neutrino experiments, atmospheric neutrino experiments and long baseline terrestrial neutrino experiments. Each of these results separately can be accounted for by a rather simple modification to the SM. Trying to accommodate all of them simultaneously, however, is not trivial. In the third lecture we explain what are the theoretical challenges in trying to combine all the experimental indications for neutrino masses, and give several examples of models that cope with some of these challenges. These lecture notes are aimed to provide an introduction to the topic of neutrino physics. They are not meant to be a review. Therefore, many details are not given and many references are omitted. There are many textbooks [1] and reviews [2-4] about neutrinos. There is also a lot of information about neutrinos on the web [5,6]. All these sources provide more detailed discussions with complete set of references on the topics covered in these lectures. Moreover, they also cover many subjects that are not mentioned here. In preparing the lectures I used mainly the recent review by GonzalezGarcia and Nir [4]. This review is a very good starting point to anyone who wants to learn more about neutrino physics. 2. Neutrino masses 2.1. Fermion

masses

In general, there are two possible mass terms for fermions: Dirac and Majorana mass terms. All fermions can have Dirac mass terms, but only neutral fermions can have Majorana mass terms. Indeed, all the massive fermions in the SM, the quarks and charged leptons, have Dirac mass terms. The neutrinos, however, while massless in the SM, can have both Dirac and Majorana mass terms. Dirac masses couple left and right handed fields mo^L^R

+ h.c,

(1)

where m_o is the Dirac mass and ipL and ipn are left and right handed Weyl spinor fields, respectively. Note the following points regarding eq. (1): • Consider a theory with one or more exact U(l) symmetries. The

charges of 4>L and ipR under these symmetries must be opposite. In particular, the two fields can carry electric charge as long as

QhPi) = QWR)• Since ipL and ipR are different fields, there are four degrees of freedom with the same mass, mo• When there are several fields with the same quantum numbers, we define the Dirac mass matrix, (rriD)ij ( " i D l . j f f L J . W j + /l.C,

(2)

where i(j) runs from one to the number of left (right) handed fields with the same quantum numbers. In the SM, the fermion fields are present in three copies, and the Dirac mass matrices are 3 x 3 matrices. In general, however, mo does not have to be a square matrix. A Majorana mass couples a left handed or a right handed field to itself. Consider ipR, a SM singlet right handed field. Its Majorana mass term is mMipcRipR,

ipc = Cip ,

(3)

where TUM is the Majorana mass and C is the charge conjugation matrix [7]. A similar expression holds for left handed fields. Note the following points regarding eq. (3): • Since only one Weyl fermion field is needed in order to generate a Majorana mass term, there are only two degrees of freedom that have the same mass, TOM• When there are several neutral fields, TOM is promoted to be a Majorana mass matrix

(mMkW^OM,-.

(4)

Here, i and j runs from one to the number of neutral fields. A Majorana mass matrix is symmetric. • The additive quantum numbers of ipR and tpn are the same. Thus, a fermion field can have a Majorana mass only if it is neutral under all unbroken local and global U(l) symmetries. In particular, fields that carry electric charges cannot acquire Majorana masses. • We usually work with theories where only local symmetries are imposed. In such theories global symmetries can only be accidental. A Majorana mass term for a fermion field ip breaks all the global U(l) symmetries under which the fermion is charged.

9

2.2. Neutrino

masses

in the SM

The SM is a renormalizable four dimensional quantum field theory where • The gauge group is SU(3)C x SU(2)L x £7(l)y. • There are three generations of fermions QL(3,2)+1/6,

UR(3, l)+2/3,

LL(1,2)_1/2)

E*(1,1)-I.

DR(3,

1)_I/3, (5)

We use the (qc,qL)qY notation, such that qc is the representation under SU(3)c, 1L is the representation under SU{2)iJ and qy is the U(l)y charge. • The vev of the scalar Higgs field, H(l, 2) + 1 / 2 , leads to the Spontaneous Symmetry Breaking (SSB) SU(2)LXU(1)Y^U(1)EM-

(6)

The SM has four accidental global symmetries: baryon number (B) and lepton flavor numbers (Le, L^ and LT). It is also convenient to define total lepton number as L = Le + LM + LT, which is also conserved in the SM. An accidental symmetry is a symmetry that is not imposed on the action. It is there only due to the field content of the theory and by the requirement of renormalizability. If we would not require the theory to be renormalizable, accidental symmetries would not be present. For example, in the SM dimension five and six operators break lepton and baryon numbers. The SM implies that the neutrinos are exactly massless. There are several ingredients that combine to ensure it: • The SM does not include fields that are singlets under the gauge group, NR(1, 1)O- This implies that there are no Dirac mass terms {H)uZvR.a (We define H = ir2H*.) • There are no scalar triplets, A(l,3)i, in the SM. Therefore, Majorana mass terms of the form ( A ) I / £ I / £ cannot be written. • The SM is renormalizable. This implies that no dimension five Majorana mass terms of the form {H)(H)V1VL are possible. a

O u r notation is such that we use capital letters to denote fields before SSB and lowercase Greek or Roman letters to denote the fields after SSB. While such differentiation is meaningful only to field that are charged under the SM gauge group, we use this convention also for SM singlets.

10

•

is an accidental non-anomalous global symmetry of the SM. Thus, quantum corrections cannot generate Majorana (i?)(ii")^£^L mass terms. b U(1)B-L

Both the neutrinos and the 5i7(3)c x U(1)EM gauge bosons are massless in the SM. There is, however, a fundamental difference between these two cases. The symmetry that protects the neutrinos from acquiring masses is lepton number, which is an accidental symmetry. Namely, one does not impose it on the SM. In contrast, the photon and gluon are massless due to local gauge invariance, which is a symmetry that one imposes on the theory. This difference is significant. An imposed symmetry is exact also when one considers possible new physics at very short distances. Accidental symmetries, however, are likely to be broken by new heavy fields. One can understand the result that the neutrinos are massless in the SM in simple terms as follows. For a massive particle one can always find a reference frame where the particle is left handed and another reference frame where it is right handed. Thus, in order to have massive neutrinos, the SM left handed neutrino fields should couple to right handed fields. There are two options for such couplings. First, the SM left handed neutrinos can couple to right handed SM singlet fermions. This is not allowed in the SM just because there are no such fields in the SM. The second option is to have couplings between the left handed neutrinos and the right handed anti-neutrinos. Such couplings break lepton number. The fact that lepton number is an accidental symmetry of the SM forbids also this possibility. 2.3. Neutrino

masses

beyond the SM

Once the SM is embedded into a more fundamental theory, some of its properties mentioned above are lost. Therefore, in many SM extensions the neutrinos are massive. Neutrino masses can be generated by adding light or heavy fields to the SM. (By light we refer to fields that have weak scale or smaller masses, while heavy means much above the weak scale.) When light fields are added the resulting neutrino masses are generally too large and some mechanism is needed in order to suppress them. New heavy fields, on the contrary, generate very small masses to the neutrinos. We are now going to elaborate on these two possibilities. b

Note that U(1)B-L has gravitational anomalies. Namely, once gravity effects are included the symmetry is broken and neutrino masses can be generated. The SM, however, does not include gravity. What we learn is that it is likely that the neutrinos are massive in any SM extension that include gravity, in particular, in the real world.

11

There are two kinds of light fields that can be used to generate neutrino masses. First, right handed neutrino fields, NR(1, 1) 0 , couple to the SM left handed lepton fields via the Yukawa couplings and generate Dirac masses for the neutrinos yNHLLNR

+ h.c.

=>

mDvlvR,

(7)

where %JN is a dimensionless Yukawa coupling and the symbol "=>" indicates electroweak SSB. This mechanism offers no explanation why the neutrinos are so light and why the right handed neutrinos do not acquire large Majorana masses. Another way to generate neutrino masses is to add to the SM a scalar triplet, A(l, 3)i. Assuming that the neutral component of this triplet acquires a vev, a Majorana mass term is generated XNALCLLL

=>

mNvcLvL,

(8)

where \M is a dimensionless coupling. In that case it remains to be explained why the triplet vev is much smaller than the SM Higgs doublet vev. Note that the triplet vev is required to be small not only to ensure light neutrinos but also from electroweak precision data, in particular, from the measurement of the p parameter [8]. The other way to generate neutrino masses is to add new heavy fields to the SM. In that case the low energy theory is the SM, but the full high energy theory includes new fields. Using an effective field theory approach, the effects of these heavy fields are described by adding non-renormalizable terms to the SM action. All such terms are suppressed by powers of the small parameter v/M. Here v is the SM Higgs vev, which characterizes the weak scale, and M is the unknown scale of the new physics, which can be, for example, the Planck scale or the GUT scale. Neutrino masses are generated by the following dimension five operator ^HHLILL

=*

m„ = A ^ ,

(9)

where A is a dimensionless coupling. Note the following points regarding eq. (9): • Taking the SM to be an effective field theory implies mu ^ 0. Since we know that new physics must exist at, or below, the Planck scale, it is rather likely that the neutrinos are massive. • m„ is small since it arises from non-renormalizable terms. • Neutrino masses probe the high energy physics. • The neutrino acquires Majorana mass. Thus, total lepton number is broken by two units.

12

Figure 1. The seesaw mechanism acquire its name from analogy with a children seesaw. The lower one child takes its arm of the seesaw, the higher the other child goes. For neutrinos, the larger MN is, the lighter the neutrino becomes. (I Thank Oleg Khasanov for the figure.)

• The situation can be generalized to the case of several generations. Then, generically, both total lepton number and family lepton numbers are broken and lepton mixing and CP violation are expected. A famous example of a realization of this effective field theory description is the seesaw mechanism, see fig. 1. (Historically, it was invented within 50(10) GUT theory, and later implemented in many other models.) Consider one generation SM with an additional fermion singlet NR(1, 1)OThe following two terms in the Lagrangian are relevant for neutrino masses ^MNN%NR

+ YvHTENR + h.c,

(10)

where MN ~3> v is the Majorana mass of the right handed neutrino. After electroweak symmetry breaking the second term leads to a Dirac mass of the neutrino. In the (VL,VR) basis the neutrino mass matrix is

\mD

MNJ

where mp = Yvv. Using Tr(m„) = MN ,

| det(m„)| = m2D ,

(12)

13

to first order in V/MN we find 2

mUR =MN

mPL=^-.

(13)

Comparing this result with eq. (9) one identifies the new physics scale M with MM and the new physics coupling A with Y^. We learn that the seesaw mechanism is indeed a realization of the effective field theory approach for neutrino masses. 2.4.

Neutrino

mixing

Massive neutrinos generally mix. The neutrino mass terms break the accidental family lepton number symmetries. (Total lepton number is also violated if the neutrinos have Majorana masses.) This phenomenon is very similar to quark mixing in the SM. It is therefore instructive to describe both lepton and quark mixing in parallel. For quarks, the Cabibbo-Kobayashi-Maskawa (CKM) matrix, V, corresponds to non-diagonal charged current interactions between quark mass eigenstates

^=(ul)iVl3r(dL)JW+

••u,c,t,

j = d,s,b.

(14)

For leptons, it is common to use two different bases. The flavor basis is defined to be the one where the charged lepton mass matrix and the W interactions are diagonal. In the mass basis both the charged lepton and the neutrino mass matrices are diagonal, but the W interaction is not. In that basis the leptonic mixing matrix U is the analogue of the CKM matrix. 0 Namely, it shows up in the charged current interactions A=TlUtirf{vL)iW-,

£ = e,n,T,

i = 1,2,3.

(15)

When working in the mass basis, the formalism of quark and lepton flavor mixings are very similar. The difference between these two phenomena arises due to the way neutrino experiments are done. While quarks and c

Recently, in many papers the matrix U is called the Maki-Nakagawa-Sakata (MNS) matrix, UMNSI o r the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, L^PMNSI since these authors were the first to discuss leptonic flavor mixing. Others call it the CKM or KM analogue for the lepton sector since Kobayashi and Maskawa were the first to discuss CP violation from such matrices. Here we adopt the notation of [4] and call U the leptonic mixing matrix. The fact that the CKM matrix is denoted by V helps in avoiding confusion.

14

charged leptons are identified as mass eigenstates, neutrinos are identified as flavor eigenstates. Indeed, there are two equivalent ways to think about fermion mixing. For quarks, mixing is best understood as the fact that the W interaction is not diagonal in the mass basis. For leptons, we usually refer to the rotation between the neutrino mass and flavor bases as neutrino mixing. Note that the indices of the two matrices are of different order. In V the first index corresponds to the up type component of the doublet and the second one to the down type. In U it is the other way around. In general, U is described by three mixing angles and one Dirac CP violating phase. If the neutrinos are Majorana particles there are in addition two more CP violating phases called Majorana phases. The Majorana phases, however, do not affect neutrino flavor oscillations, and we do not discuss them any further. SM singlet states can mix with the three SM neutrinos. When the masses of the singlet states are small they can participate in neutrino oscillation. Then, the singlet states are usually called sterile neutrinos while the standard model neutrinos are called active neutrinos. d The mixings between all neutrinos are taken into account by enlarging the mixing matrix U to a 3 x (n + 3) matrix, where n is the number of sterile states. When the SM singlet states are very heavy (with mass M 3> v), they cannot participate in neutrino oscillations. Their couplings to the light neutrino is important since it gives them their masses. Thus, the heavy states are usually referred to as right handed neutrinos. The mixings relevant for neutrino oscillations is described by the matrix U that refers only to the 3 x 3 sub-matrix that corresponds to the mixing between the mostly active states. The presence of the heavy states affects the theory since in that case the matrix U does not describe the full mixing and, therefore, it is not unitary. This deviation from unitarity, however, is very small, of order v/M, and it is usually neglected. 3. Methods for probing neutrino masses The conclusion of the last section is that it is very likely that the neutrinos have very small masses and that they mix. There are several ways to probe Oscillation occurs only between left handed states, that is, both active and sterile states are left handed. One should not get confused by the fact that the sterile states are left handed; the identification of left handed states as SM doublets and right handed states as SM singlets does not have to be true in theories that extend the SM.

15

neutrino masses and mixing angles. Discussions about astrophysical and cosmological probes of neutrino masses can b e found, for example, in [1,3]. Here we briefly discuss kinematic tests for neutrino masses and neutrinoless double b e t a decay. Then we develop the formalism of neutrino oscillation, which is the most promising way t o probe the neutrino masses and mixing angles. 3 . 1 . Kinematic

tests

In decays t h a t produce neutrinos the decay spectra are sensitive to neutrino masses. For example, in TT —> \iv the muon m o m e n t u m is fixed (up to tiny width effects), and it is determined only by t h e masses of the pion, t h e muon and the neutrino. To first order in m%/m%, the muon m o m e n t u m in the pion rest frame is given by l

I-I \P = a

( r

2

2

m

\ K- u,

m

l + ml 5

M

2m7r \

2\

/1fix

T mt

ml - ml

•

v

J

(16) v

'

Since the correction to the massless neutrino limit is proportional to m ^ , the kinematic tests are not very sensitive to small neutrino masses. T h e current best bounds obtained using kinematic tests are [8] m „ < 18.2 MeV

from r -> 5TT + v ,

mu < 190 KeV

from TT -^ fit/,

mv < 2.2 eV from 3 H ^ 3 H e + e + 17. (17) T h e sensitivities of neutrino oscillation experiments are much better t h a n these bounds. Note t h a t while oscillation experiments are sensitive to the neutrino mass-squared differences (see below) the kinematic tests are sensitive to the neutrino masses themselves. 3 . 2 . Neutrinoless

double

f3

decay

Neutrino Majorana masses violate lepton number by two units. Therefore, if neutrinos have Majorana masses we expect there there are also AL = 2 processes. T h e smallness of the neutrino masses indicates t h a t such processes have very small rates. Therefore, the only practical way to look for AL = 2 processes is in places where the lepton number conserving ones are forbidden or highly suppressed. Neutrinoless double (3 decay where the single b e t a decay is forbidden is such a process. An example for such processes is ?§Ge - •

3

7t

Se + 2 e - .

(18)

16

The only physical background to neutrinoless double (3 decay is from double (3 decay with two neutrinos. Note the following points: • Since neutrinoless double (3 decay is a AL = 2 process, it is sensitive only to Majorana neutrino masses. Dirac masses conserve lepton number and do not contribute to this decay. • The neutrinoless double (3 decay rate is related to the neutrino mass-squared. It is also proportional to some nuclear matrix elements. Those matrix elements introduce theoretical uncertainties in extracting the neutrino mass from the signal, or in deriving a bound if no signal is seen. • Neutrinoless double (3 decay is sensitive to any AL = 2 operator, not only to the neutrino Majorana masses. Thus, the relation between the Neutrinoless double (3 decay rate and the neutrino mass is model dependent. • The best bound derived from neutrinoless double (3 decay is mv < 0.34 eV [9]. 3.3. Neutrino

vacuum

oscillation

We now turn to the derivation of the neutrino oscillation formalism. We use several assumptions in this derivation. We emphasize, however, that in all practical situations it is correct to use these assumptions and that more sophisticated derivations give the same results (see, for example, [10]). In an ideal neutrino oscillation experiment, a neutrino beam is generated with known flavor and energy spectrum. The flavor and the energy spectrum is then measured at some distance away. If the flavor composition of the beam changed during the propagation it is a signal of neutrino oscillation, which indicates neutrino masses and mixings. The flavor of the neutrino is identified via its charged current interaction. It is convenient to express a flavor state in terms of mass states

ka)=E^>*)>

(19)

i

where Greek indices runs over the flavor states (a = e, /i, r) and Roman ones over the mass states (i = 1,2,3). An initially produced flavor state \va) evolves in time according to

Wa(t))=^2e~^tU*mWi),

(20)

17

where we assume that all the components in the neutrino wave packet have the same momenta (pi = p). For ultra relativistic neutrinos we can use the following approximation o '171-

Bk*P+-£.

(21)

Then, the probability to observe oscillation between flavor a and /3 is given by Pa/3 = \{vp\Va(t))\2

= 5a0 ~ ^^UaiUfjiUajUpj

s i n 2 Xtj,

(22)

where in the last step we assumed CP conservation and Xij

_ Am?,-* ~~ 4p '

'

In many cases the three flavor oscillation is well approximated by considering only two flavors. Then, °

/ cos0 \-sm6

sin£\ costfy*'

[Z

'

and for a ^ (3 we obtain Pap = sin2 261 sin2 x .

(25)

Since the neutrinos are relativistic we usually use L = t, E = p and write _ TTL __ AnE . x = 7 > -^osc — -7 5" j (2o) L osc Am 2 such that L osc is the oscillation length. The following formula is also convenient

»-(£)GI)(^)The oscillation master formula, eq. (25), teaches us the following: • Longer baseline, L, or smaller energy, E, is needed in order to probe smaller Am 2 . • When L <€. Losc we can approximate sin x ~ x2. In that case the signal is usually too small to be detected. • When L » Losc the energy spread of the beam and decoherence effects cause the oscillations to average out. That is, in our formula we should take the average value: (sin x) = 0.5. In that case only a lower bound on the mass-squared difference between the two neutrinos can be obtained.

18

3.4. Matter

effects

When neutrinos travel through matter the oscillation formalism is modified. While neutrinos can scatter off the medium, in almost all relevant cases the scattering cross sections are very small, and the effect of scattering is negligible. The effect we are concentrating on is that of the forward scattering of the neutrinos. Like photons, when neutrinos travel inside a medium they acquire effective masses. There is no energy or momentum exchange between the neutrinos and the medium. The effect of the medium is that it induces effective masses for the neutrinos. First we discuss the case of a medium with constant density. Moreover, we consider only matter at low temperature, which implies that it consists of electrons, protons and neutrons. Note that there are no muons, taus and anti-leptons in that case. The charged current interaction between the electron neutrinos and the electrons in the medium induces an effective potential for the neutrinos [11] Vc = V2GFNe « 7.6 Ye ( ^ — j )

eV,

(28)

where Ne (Np, Nn) is the number density of electrons (protons, neutrons) and Ye = Ne/(Np + Nn) is the relative electron number density. To get a feeling for the size of this potential, note that at the Earth core p ~ 10 g/cm 3 , which gives rise to Vc ~ 1 0 - 1 3 eV. As we discuss below, the current data indicates that m„ > 10~ 3 eV, and thus mu ^> Vc- This seems to suggest that matter effects are irrelevant. This is, however, not the case since the matter effects arise from vector interactions while masses are scalar operators. The right comparison to make is between m2, and EVc where E is the neutrino energy. Since E » mv matter effects are enhanced and can be important. To explain this enhancement, we consider a uniform, unpolarized medium at rest. (Discussion about the general case which includes all types of interactions and arbitrary polarization can be found, for example, in [12].) In that case the neutrino feels the four-vector interaction VM = (Vc, 0,0,0). Due to Vc the vacuum dispersion relation of the neutrino, p^p1* = m2, is modified as follows (pM-^)(PM-^)=m2

777

=»

E*p

2

+ Vc + —,

(29)

where the approximation holds for ultra relativistic neutrinos. We learn that the effective mass-squared in matter, m ^ , is given by m2m=m2+A,

A = 2EVC,

(30)

19

where we used p w E. It is the vector nature of the weak interaction that makes the matter effects practically relevant. Interactions that are flavor universal only shift the neutrino energy by a negligibly small amount and do not affect the oscillation. Non-universal interactions, however, are important since in their presence the values of the effective masses and mixing angles are different from their vacuum ones. While the weak neutral current is flavor universal, the charged current is not. In normal matter, where there are electrons but not muons and taus, only electron neutrinos interact via the charged current with the medium. The mixing matrix is modified by this non-universal matter effect such that the effective squared mass difference and mixing angle are given by AmJ, = ^ / ( A T O 2 cos 29 - A)2 + (Am2 sin 2<9)2, (31) 2 _ Am sin 29 t&nWm ~ Am2cos20-A' where the subindex m stand for matter. The oscillation probability is then obtained from (25) by replacing the vacuum masses and mixing angles by the corresponding parameters in matter Pap = sin2 29m sin2 xm,

xm =

™ .

(32)

The following points are worth mentioning regarding eq. (31): • The vacuum result is reproduced for A — 0, as it should. • Vacuum mixing is needed in order to get mixing in matter. • To first order in xm the matter effects cancel in the oscillation probability. To see this note that xsin28 = xmsm.29m + 0(x^n). Therefore, when approximating smxm ~ xm eq. (32) reduces to the oscillation probability in vacuum, eq. (25). • For Am 2 cos 29 3> \A\, the matter effect is a small perturbation to the vacuum result. • For Am 2 cos 29 <€. \A\, the neutrino mass is a small perturbation to the matter effect. In that case the oscillations are highly suppressed since the effective mixing angle is very small. • For Am 2 cos 29 = A, the mixing is maximal, namely it is on resonance.

3.5. Non-uniform

density

When the matter density is not constant there are further modifications to the oscillation formalism. Density variation results in changing the effective

20

neutrino masses and their mixing angles. Then, the flavor composition of the neutrinos along their path is a function of the medium density profile. At any point r on the neutrino path we define the derivative of the density

^ W - ^ l .

(33)

For constant density, A' = 0, the flavor conversion probability is controlled by the effective masses and mixing angles. For varying density, A' ^ 0, there are extra parameters that affect the flavor conversion probability. Most important is the adiabatic parameter Q ( r ) =

Am 2 sin2 26 A(r) £ cos 20 1 ^ ) -

(34)

In the adiabatic limit, Q ^> 1, the density variation is slow. In this case the transition between effective mass eigenstates is highly suppressed, and the constant density formalism can be applied locally. In the non-adiabatic limit, Q < 1, the density variation is fast. Then, transition between effective mass eigenstates is possible, and the constant density formalism cannot be used. Both limits can be of interest in reality. The best known example of the effect of the density variation is the MSW effect [11,f 3]. It can account for a very efficient flavor conversion in the case of small mixing angle. Consider a two generation model where m,2 > rrii with small vacuum mixing angle, 6
zve(r = 0) « z,2m,

v^r = 0) « v™.

(35)

(By writing va « v™ we mean that a very small rotation in flavor space is needed in order to move from the flavor eigenstate va to the effective mass eigenstate v™.) In particular, the produced electron neutrino is almost a pure V2- Also at the edge of the Sun, r = RQ, the flavor eigenstates are almost pure mass eigenstates 9m(r = RQ) -> 9 < 1,

ve{r = R®) & Vl,

Vy.{r

=

RQ)

W V2-

(36)

When the adiabatic limit applies, neutrinos propagate in the Sun as effective mass eigenstates. Since the neutrinos are produced as almost pure V2, they

21

e

sin2 2i?=0.001:

80 60 40 20

A Figure 2. A demonstration of an MSW resonance conversion. 9m is the effective mixing angle in matter, A is the effective mass due to the matter effect defined in (30) and AR is the resonance point. The neutrinos are produced at a region where A > An. They propagate as effective mass eigenstates, which is shown in the figure as solid lines. In the plot the effect is demonstrated for two different values of the vacuum mixing angles. (The plot is taken from [4].)

leave the Sun as v-i- Since in vacuum J/2 is almost a pure i/M, we learn that there is almost full conversion from ve to v^. This phenomenon, which is called MSW resonance conversion, is demonstrated in fig. 2. In the more general case, corrections to the adiabatic limit are taken into account. In particular, transitions between the effective mass eigenstates are important in the vicinity of the resonance. Such level crossing is described by the Landau-Zener probability, PLZ- Then, the flavor conversion probability is given by the Parke formula [14] Pee = \ [1 + (1 " 2 P L Z ) COS 20m

COS 26?] ,

(37)

where 9m is the effective mixing angle at the production point. In this general case, the flavor transition probabilities become a rather involved function of the neutrino and medium parameters. It is this richness that makes the flavor composition of the solar neutrinos a complicated and not a monotonic function of their energies. As it is shown below this is important in order to be able to explain the solar neutrino data.

22

3.6. Neutrino

oscillation

experiments

Ideally, neutrino oscillation experiments measure the neutrino flavor transition probabilities as a function of the neutrino energies and their path. In practice, however, there are many experimental complications. The basic setup of any neutrino oscillation experiment is as follows: Neutrinos are produced and then propagate until few of them are detected at another location, far away from their production point. Therefore, in order to be able to probe the fundamental physics parameters we would like to know and control the following: • The total flux, the flavor composition and the energy spectrum of the beam. • The distance traveled by the neutrinos and the matter profile that they passed. • The total flux, the flavor composition and the energy spectrum of the detected neutrinos. In reality, it is impossible to know and control all of these parameters. Therefore, a lot of work is involved in order to get the maximum out of each experimental setup. In general we distinguish between two kinds of experiments. Disappearance experiments search for reduction in the flux of a specific neutrino flavor N[va(L)} < JV[i/a(0)],

(38)

where N[i/a(L)] is the number of neutrinos of flavor a in the neutrino beam at distance L. This would imply Paa(L)
(39)

Appearance experiments, on the other hand, look for enhancement of a flux of a specific flavor N[vf,(L)] > N[Vf3(0)},

(40)

Pap(L) > 0.

(41)

which would imply

In particular, in many cases -/V[z/g(0)] = 0. Then, any observation of neutrinos with flavor (3 is an appearance signal. Oscillation experiments are usually named after their production place and mechanism. In the following we discuss solar neutrinos, atmospheric neutrinos and terrestrial neutrinos.

23

4. Solar neutrinos Solar neutrinos receive a lot of attention since there are indications for disappearance of electron neutrinos. These indications are collectively referred to as the solar neutrino problem or the solar neutrino anomaly. The best explanation for the solar neutrino problem is neutrino flavor oscillation. The use of the words "problem" or "anomaly" should not worry us. They are used only to indicate the fact that solar neutrinos pose a problem to the SM. We think, however, that the SM is an effective theory of Nature, and thus, the fact that it faces problems is not surprising. Actually, we expect that neutrinos have masses and that they mix, and consequently, that the solar neutrino data cannot be explained by the SM. 4.1. Solar neutrinos

production

Based on our understanding of stellar evolution we know that nuclear reactions fuel the Sun. Nuclear processes involve only the first generation of fermions and thus only electron neutrinos are generated in the Sun. Moreover, since the Sun is burn by fusion reactions it produces only neutrinos and not anti-neutrinos. The neutrino energies are relatively small, Ev < 10 MeV. Therefore, only disappearance experiments can be done; the neutrinos energies are too small to produce muons or taus in any target at rest. There are many processes that produce solar neutrinos. The main reaction chain, which produces about 98.5% of the solar energy (and most of the solar neutrinos) is the pp cycle Ap - • 4 He + 2e+ + 2ve + 2 7 .

(42)

The neutrinos emitted in this cycle have continuum spectrum which goes up to Eu < 0.42 MeV. Assuming that the Sun is in equilibrium, one can use its current luminosity to calculate the neutrino flux from the pp reaction chain. The other nuclear reactions in the Sun are not very important for its luminosity, but they are important for neutrino physics. The difference originates from the fact that photons are thermalized almost immediately after they are produced. Neutrinos, on the other hand, do not interact in the Sun, and leave the Sun with their original energies. Several subdominant reactions produce neutrinos with energies above the threshold of the pp chain. Since it is easier to detect high energy neutrinos, in many cases the experimental signals consist only of neutrinos that were produced by sub-dominant reactions.

24

SuperK, SNO

X

JE! E o c 2

0.3

1

3

10

Neutrino Energy (MeV) Figure 3. The Bahcall and Pinsonneau.lt 2000 solar model prediction of the neutrino flux as function of the neutrino energy. This figure was taken from [15], where many more details on solar neutrinos can be found.

While the spectra of the different reactions are known, the sub-dominant reaction chains fluxes are not known in a model independent way. Solar models, namely, theoretical arguments, are used to obtain the total neutrino spectrum. As an example, the Bahcall and Pinsonneault 2000 (BP00) solar model prediction for the neutrino flux is given in fig. 3. Since solar models involve theoretical inputs, one would like to perform experiments where the interpretation of the data can be done with minimum dependence on a specific solar model. 4.2. Solar neutrinos

propagation

Solar neutrinos are produced, in the vicinity 'of the solar core. Then, they freely propagate out of the Sun and some of them travel to our detectors. During the night, neutrinos reach the detector after they have passed

25

through the Earth. It is convenient to separate the solar neutrino propagation into three regions: inside the Sun, between the Sun and the Earth, and inside the Earth. We discuss each of these below. As already mentioned before, matter effects in the Sun can be important for flavor conversion. To calculate these effects we need to know the matter profile in the Sun and the rates of the different reactions in the different parts of the Sun. While these parameters are roughly known, there are uncertainties in their values. Therefore, we have to count on solar models to provide the needed inputs. The distance to the Sun is known to high precision. It varies by about 7% during the year. If vacuum oscillation is relevant to solar neutrinos then this distance variation can be used in order to get better control on the neutrino parameters. During the night neutrinos pass through the Earth on their way to the detector, while during the day they do not. Therefore, the Earth matter effect, which shows up as a day-night difference of the detected neutrino flux, provides further sensitivity to the neutrino parameters. Moreover, depending on the location of the detector and the time of the year the neutrino may pass through both the Earth mantle and its core or only through the mantle. This is important since the matter density in these two regions is different, and therefore can affect the oscillations [16]. 4.3. Solar neutrinos

detections

There are three reactions that are used to detect solar neutrinos. Charged Current (CC) interaction ve + n —> p + e~ ,

The (43)

where n is a neutron, can be used to detect only electron neutrinos. All the neutrinos can undergo Elastic Scattering (ES) vi + e~ —+ vt + e~ .

(44)

This reaction is mediated by neutral current Z exchange for all neutrino flavors. For electron neutrinos there is also contribution from a W exchange amplitude. Due to this difference, the elastic scattering cross section for electron neutrinos is about six times larger than for muon and tau neutrinos. Finally, the cross section for the Neutral Current (NC) interaction vi + n —> vi + n , is the same for all neutrino flavors (here n is a neutron).

(45)

26

When presenting results of neutrino oscillation experiment it is convenient to define R = Nobs/NMc

,

(46)

such that N0bs is the number of detected neutrinos and NMC is the number of events predicted from a solar model based Monte Carlo (MC) simulation assuming no flavor oscillation. For detectors that use neutral current reactions R should be one regardless if there are active neutrino flavor oscillations or not. For charged current and elastic scattering based detectors, however, R = 1 is expected only if there are no neutrino flavor oscillations, and R < 1 is expected if there are oscillations. There are several solar neutrino experiments that use the above mentioned basic reactions. They have different experimental setups that us various reactions with different sensitivities and thresholds. Next we describe these experiments. 4.4. Chlorine:

Homestake

The first detection of solar neutrinos was announced 35 years ago. The neutrinos were detected in the Homestake mine in South Dakota [17]. The experiment uses the ue + 37C1 -> 37Ar + e~ ,

(47)

charged current interaction in order to capture the electron neutrinos. We note the following points: • The signal is extracted off-line. Namely, only integrated rates are measured. Therefore, no day-night and almost no spectral information are available. (The only information about the neutrino spectrum is that the energy of the neutrino is above the threshold.) • The energy threshold is 0.814 MeV. Therefore, the experiment is sensitive mainly to the 7 Be and 8 B neutrinos, pp neutrinos cannot be detected in this experiment. • After many years of data collection the published result is [18] i?«0.3.

(48)

Of course, in order to perform a fit to the neutrino parameters the above crude approximation should not be used. Yet, quoting the order of magnitude, as we did in eq. (48) is sufficient to understand the main ingredients of the solar neutrino problem.

27

The fact that the Homestake experiment found R < 1 was the first indication for solar electron neutrino disappearance. This result gave the motivation to build different types of detectors in order to further study the solar neutrinos. 4.5. Gallium:

SAGE,

GALLEX

and

GNO

The Homestake experiment had relatively high threshold. The SAGE, GALLEX [19] and GNO [20] Gallium experiments were built in order to detect neutrinos with much smaller energies. In particular, their threshold is low enough to detect pp neutrinos. In the Gallium experiments the neutrinos are detected using the i/e +

71

G a ^ 7 1 G e + e-

(49)

charged current reaction. We note the following points: • Like the Chlorine experiment, the signal is extracted off-line. Thus, also here no day-night and spectral information are available. • The energy threshold is 0.233 MeV. Thus, Gallium detectors are sensitive to pp neutrinos. The 7 Be and 8 B neutrinos also contribute significantly to the neutrino signal. Therefore, the total predicted flux is not solar model independent. • Combining the results of the three experiments, one obtains R w 0.6.

(50)

We learn that like in the Chlorine experiment, also here, the number of observed electron neutrinos is less than predicted. Moreover, the measured R is different for the two type of experiments. Since these two types are sensitive to different energy ranges, this fact indicates that the electron neutrino flux reduction must be energy dependent. 4.6. Water

Cerenkov:

Kamiokande

and

SuperKamiokande

Both the Chlorine and Gallium experiments extract their signal off-line. The Kamiokande and SuperKamiokande [21] water Cerenkov detectors can extract their signal on-line. Therefore, in these experiments day-night and spectral information can be extracted. The price to pay is that the energy threshold is much higher compared to the thresholds of the Chlorine and Gallium experiments.

28

In the water Cerenkov detectors the neutrinos are detected by elastic scattering of the neutrinos off the electrons in the water va + e~ —> va + e~ .

(51)

We note the following points: • Since the detection is done on-line, the time of each event is known, and also some information about the direction and energy of the neutrino in each event can be extracted. These features enable the experiments to clearly show that the detected neutrinos are coming from the Sun. • The water Cerenkov detectors threshold is about 6 MeV. Thus, the Kamiokande and SuperKamiokande experiments are sensitive mainly to 8 B neutrinos. • Since elastic scattering is used for the detection, all neutrino flavors contribute to the signal. This fact is important if there are active flavor oscillations. Then, the converted neutrinos also contribute to the signal. This is in contrast to detectors that use charged current interactions where only electron neutrinos generate the signal. • The integrated suppression is found to be R « 0.5.

(52)

Spectrum and day-night information were also reported. We learn again that there is an energy dependent suppression of the electron neutrino flux. 4.7. Heavy

Water:

SNO

We already mentioned that neutrino flavor conversion is the most plausible explanation to the observed energy dependent electron neutrino flux suppression. Thus, it is interesting to measure solar neutrinos using neutral current reactions since in that case all neutrino flavors contribute with the same strength. Namely, neutral current based experiments measure the solar neutrino flux independent of whether there are active flavor oscillations or not. The SNO detector [22] was built for this purpose. Its apparatus contains an inner tank filled with heavy water and an outer tank filled with regular water. Neutrino can be detected by all the three basic reaction types CC(Eth = 2.23 MeV):

ve + 2 H -> p + p + e~ ,

29

ES(Eth ~ 6 MeV) : NC(£ t / l ~ 6MeV) :

Va

+e~ -+va +e~ ,

^ a + 2 H - • n + p + va ,

(53)

where Eth is the threshold energy. Note that the presence of Deuterium (2H) in the heavy water is crucial for the charged current and neutral current measurements. Note the following points: • The signal is extracted on-line. Thus day-night and spectrum information is available. • We define the ratios

RB = %£,

^f52.

-KES

( 54 )

-KNC

where RQC {RES, -RNC) is the ratio between the observed number of charged current (elastic scattering, neutral current) events and the predicted number assuming no oscillation. Extracting RE and RN has several advantages. First, many systematic uncertainties cancel in their measurements. Second, their predicted values are almost solar model independent and depend only on the neutrino parameters. In particular, RE ^ 1 or RN J^ 1 can occur only if electron neutrinos are converted to muon or tau neutrinos. • SNO found RE ^ 1 and i?j\r 7^ 1- This result confirms the existence of Up or vT component in the solar neutrino flux at 5.3a. In addition, the BPOO solar model prediction is confirmed by the neutral current measurement. The SNO result [23] is illustrated in fig. 4.

4.8. Solar neutrinos:

fits

The general picture emerging by combining all the solar neutrino data is as follows. All charged current and elastic scattering based measurements found less events than predicted for massless neutrinos. This suppression is energy dependent. The BPOO solar model prediction for the neutral current based measurement, which holds also if there are active neutrino flavor conversions, was confirmed by SNO. The first and most robust conclusion is that the SM, which predicts massless neutrinos, fails to accommodate the data. While there are several exotic explanations that may be able to explain the data [24], the simplest and most plausible one is that of active neutrino flavor oscillation ve —> i/x,

x = fj,,r.

(55)

30

E

.SNO

,SNO

Figure 4. The SNO data is plotted as three independent measurements of the muon or tau neutrino flux (<^MT) as a function of the electron neutrino i u x (e)- The three bands represent the neutral current, charged current and elastic scattering based measurements. The dotted line is the BP00 solar model prediction for the neutrino flux. It is plotted only in conjunction with the neutral current measurement since only this measurement is insensitive to electron to muon or tau flavor conversion. (This plot is taken from [23].)

In that case, experiments that use reactions that are flavor dependent, namely charged current interactions and elastic scattering, are expected to detect fewer events than anticipated. On the other hand, experiments that use neutral current interactions, which are flavor blind, should not observe such reduction. Indeed, this is what the data shows. Assuming that the neutrinos are massive, the data can be used to determine the neutrino masses and mixing angles. While three flavor oscillation fits are available [25], it is a good approximation to fit the data assuming only two neutrino mixing. An example of such fit is shown in fig. 5. (Recent fits can be found in [25]. Note, however, that whenever new data is published, new fits are performed and published usually within days.) The regions where a good fit is found are indicated in the figure. The best fit is obtained in the Large Mixing Angle (LMA) region. The recent SNO data actually exclude the Small Mixing Angle (SMA) region. The LOW and the

31

10""

•fUH'I'l

t i l l i>M;t|;

i H Hill

I -I-,' t i \nW'"~

1^

i I Ml±

io4fe ID*

SMA 10

S

LOW ,'i'rV', 10'

E <

io r

I 1 i \

10

vo 10

10

10

li I

10

•'*»* i

i i MIII

10"

i

I i 11 m l

i

i

i l mil

I 11 n i l

-i

10"

.-J3T =

10

10

10

tan"© Figure 5. An example of a neutrino parameters fit to some solar neutrino data. The names of the regions where good fit is found are indicated in the figure. Note that by now this fit is outdated, and it is shown here for illustration only. (This plot is based on a plot taken from [26].)

Vacuum Oscillation (VO) regions, are still allowed by all solar data. They are excluded, however, by the recent KamLAND result, see below. To conclude, there are strong indications for active ue —> vx oscillation. This conclusion is robust since it has practically no dependence on the solar model. The best fit to the neutrino parameters is in the LMA region with A m 2 = 6 x 10" 5 eV 2 ,

tan 2 0 = 0.4.

(56)

Future data and new experiments will teach us more about solar neutrinos.

32

Besides having more statistics, future experiments will measure observables where we have only very little information at present. In particular, we expect to have more information about the day-night effect and the low energy neutrino spectrum. 5. Atmospheric neutrinos In the reaction chain initiated by cosmic ray collisions off nuclei in the upper atmosphere neutrinos are produced. These neutrinos are searched for by atmospheric neutrinos experiments. There are indications for muon neutrinos disappearance. These indications are called the atmospheric neutrino problem or the atmospheric neutrino anomaly. The likely explanation of this phenomenon is active neutrino oscillation. 5.1. Atmospheric

neutrino

production

Cosmic ray collisions off nuclei in the upper atmosphere produce hadrons, mainly pions. In their decay chain / i + -> e+veVli,

Tr+^/i+z^,

(57)

(and the charge conjugated decay chain) neutrinos are produced. Note that almost exclusively electron and muon neutrinos are produced. Based on (57) we expect NiyJ/Niye)

« 2,

(58)

where N{vz) is the number of neutrinos of flavor I. There are corrections to this prediction. Other mesons, mainly kaons, are also produced by cosmic rays. Their decay chains produce different neutrino flavor ratio. Another effect is that at high energies the muon lifetime in the lab frame is much longer than in its rest frame. In fact, it is long enough such that not all the muons decay before they arrive to the detector. All in all, the neutrino production rates and spectra are known to an accuracy of about 20%. The ratio N(y!j)/N(ve) is known better, to an accuracy of about 5%. The atmospheric neutrino energies are relatively large, Eu > 100 MeV. Therefore, both disappearance and appearance experiments are possible. In particular, since the initial tau neutrino flux is tiny, searching for tau appearance is an attractive option. 5.2. Atmospheric

neutrino

propagation

Since neutrinos are produced isotropically in the upper atmosphere, they reach the detector from all directions. The neutrino zenith angle in the

33

detector is correlated with the distance it traveled. Neutrinos with small zenith angle, namely those coming from above, travel short distances (of order 102 km). Neutrinos with large zenith angle, namely those coming from below, are neutrinos that cross the Earth and travel much longer distances (of order 104 km). Since the production is isotropic, the neutrino flux in the detector is expected to be isotropic in the absence of oscillation. (There are small known corrections due to geo-magnetic effects.) Matter effects can be important for neutrinos that cross the Earth. Oscillations that involve electron neutrinos or oscillations between active and sterile neutrinos are modified in the presence of matter. In contrast, the matter has almost no effect effect on fM —> vT oscillation. 5.3. Atmospheric

neutrino

detection

There are two types of atmospheric neutrino detectors. The water Cerenkov detectors, 1MB, Kamiokande and SuperKamiokande [21], use large tanks of water as targets. The water is surrounded by photomultipliers which detect the Cerenkov light emitted by charged leptons that are produced in charged current interactions between the neutrinos and the water. Compared to electrons, muons are likely to create sharper Cerenkov rings. Thus, the shape of the ring helps in determining the flavor of the incoming neutrino. Iron calorimeter detectors are based on a set of layers of iron which act as a target, and some tracing elements that are used to reconstruct muon tracks or showers produced by electrons. Both water Cerenkov and Iron calorimeter detectors identify the neutrino flavor via its charged current interactions. They have some ability to look for neutral current interactions. Information on the neutrino energy and direction can also be extracted by both methods. 5.4. Atmospheric

neutrino

data

All atmospheric neutrino experiments measure the double ratio TVt INe, r> _

ly

obs/obs

JV

MC7JVMC

(KQ\

where NlMC is the expected number of flavor I type events and N*bs are the observed ones. The advantage of using this double ratio is that many systematic and theoretical errors cancel in this ratio. Almost all of the experiments found R < 1; see fig. 6. The observation of R < 1 can be explained by muon neutrino disappearance, electron neutrino appearance, or both. The SuperKamiokande

34

1.5

a:

~

FREJUS

- " t 1

.

1MB

t

0.5 - NUSEX

S0UDAN2

\

KAM KAM multi sub

+

•

SK SK multi sub

Figure 6. Summary of the experimental measurements of R defined in eq. (59). (This plot is taken from [4].)

450 Snh-fipV e-like 400 350 300 *-3 Z±&**J±^+_ 250 200 150 100 50 0 -0.5 0.5 cosG

.+

i

i i i i i i i i i i i i i

i

450 fiuh-GpV[i-likP 400 350 300 ^y¥ i ^ t o * * 250 * • 200 150 100 50 0 111 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-0.5

0rt

0.5

f*

i

1

cosG

Figure 7. The low energy electron and muon neutrino fluxes compared to the MC predictions as observed by SuperKamiokande. The solid lines are the expected fluxes assuming no oscillations. The dashed lines are the best fits to neutrino oscillation. The solid circles are the data points. (This plot is taken from [27].)

experiment also measured the zenith angle dependence of the neutrino flux. The data indicate that the preferred explanation is that of muon neutrino disappearance. For example, fig. 7 shows that the low energy ve flux agrees while the v^ flux disagrees with the MC prediction.

35

5.5. The atmospheric

neutrino

problem

The zenith angle dependence of the muon neutrino flux as well as R < 1 cannot be explained with SM massless neutrinos. This is called the atmospheric neutrino anomaly. In order to explain it disappearance of muon neutrinos is needed. While there are several exotic explanations [24], the most attractive explanation is active neutrino flavor oscillation. In particular, the best fit is achieved for v^ —> vT oscillation with Am 2 = 2.6 x 10" 3 eV 2 ,

sin2 29 = 0.97.

(60)

The hypothesis of neutrino oscillation can be further tested. First, terrestrial long baseline experiments probe the same region of parameter space as the atmospheric neutrino experiments. Indeed, as discussed in more detail below, the recent K2K long baseline experiment results agree with the atmospheric neutrino data. Second, search for tau appearance in the atmospheric neutrino data is also possible. The SuperKamiokande experiment found such indications [28], but they are not yet convincing. 6. Terrestrial neutrinos One disadvantage of solar and atmospheric neutrino measurements is that since the neutrinos are not produced on Earth, there is no control over the neutrino source. Neutrinos from accelerators and nuclear reactors have the advantage that their source is known. That is, we know to high accuracy the initial neutrino flux, the energy spectrum and the flavor composition of the neutrino source. Moreover, in some cases these parameters can be tuned in order to maximize the experimental sensitivity. The main advantage of solar and atmospheric neutrinos is that they are sensitive to very small Am 2 . That is, their sensitivity is much better compared to most terrestrial neutrino experiments. Therefore, it is not surprising that most terrestrial neutrino experiments did not find disappearance or appearance signals. Their data was used to set bounds on the neutrino parameters. There are, however, some exceptions. The sensitivity of the K2K [29] and KamLAND [30] experiments are similar to those of, respectively, the atmospheric and solar neutrino experiments. Recently, positive disappearance signals were found in these two terrestrial experiments. The situation is different with regard to the LSND [31] experiment. In that case there is a claim for an appearance signal. This signal can only be explained with neutrino masses that are much larger compared to the masses deduced from the solar and atmospheric neutrino data. Next we

36

describe these three experiments and their results. 6.1.

K2K

In order for accelerator neutrinos to be sensitive to the same range of parameter space as the atmospheric neutrino experiments, long baseline, of order 103 km, is needed. In such long baseline experiments it is possible to search for v^ disappearance or vT appearance. In these experiments a neutrino beam from an accelerator is aimed at a detector located far away. The first operating long baseline experiment is the K2K experiment. It uses an almost pure z/M beam that is generated at KEK and is detected at SuperKamiokande, which is about 250 km away. Recently, the K2K experiment announced their first result [32]. They observed 56 muon neutrino events with an expectation of about 80. The v^ disappearance can be explained by v^ —> vx flavor oscillation. The best fit is found for the following values Am 2 = 2.8 x 10" 3 eV 2 ,

sin2 26 = 1.0.

(61)

These values of the neutrino parameters are very close to those indicated by the atmospheric neutrino data, see eq. (60). The statistical significance of the K2K oscillation signal is smaller than the atmospheric neutrino ones. The importance of the K2K result lies in the fact that it provides an independent test of the atmospheric neutrino parameter space. 6.2.

KamLAND

Reactor neutrinos have much smaller energies compared to neutrinos produced at accelerator. Thus, long baseline reactor experiments are sensitive to smaller values of Am 2 . Another important difference is that reactors generate only electron anti-neutrinos, while accelerators produce mainly muon neutrinos. Thus, reactor experiments can be used to provide independent probes of the solar neutrino parameter space. The KamLAND experiment is placed in the Kamioka mine in Japan. This site is located at an average distance of about 180 km from several large Japanese nuclear power stations. The initial fluxes and spectra of the ve emitted in each reactor are known to a good accuracy, since they are related to the power that is generated in each reactor. Thus, measurement of the total flux and energy spectrum of the Ve at KamLAND can be used to search for disappearance. This is particularly interesting since the KamLAND setup is such that it is sensitive to the region of the parameter space indicated by the LMA solution to the solar neutrino anomaly.

37

Recently, the KamLAND experiment announced their first result [33]. They found - r ^ = 0.611 ± 0.085 ± 0.041

(62)

NMC

where N0bs is the number of observed events and NMc is the expected number. This result can be explained by Ve —> Vx neutrino flavor oscillation with the best fit at Am2 = 6.9 x 10" 5 eV 2 ,

sin2 29 = 1.0.

(63)

These values are very close to the ones indicated by the solar neutrino data, see eq. (56). The importance of this result is twofold. First, it provides a completely solar model independent test for the neutrino oscillation solutions of the solar neutrino problem. Moreover, it discriminates between the different possible solutions, pointing at the LMA as the favorable one. 6.3.

LSND

The set up of the LSND experiment [31] is as follows. Neutrinos are produced by sending protons on a fixed target that generates pions. Almost all the negatively charged pions are absorbed in the target before they decay. The neutrinos are produced via 7T+ - > / / + ^ ,

fj,+ - > e+veVIJi.

(64)

Then, the neutrinos travel for about 30 meter to the detector. Since the beam contains u^, ve and F M , with negligible fraction of Ve, it best to search for Vfj, —» Ve appearance. The LSND collaboration announced a 3.3<7 indication for Ve appearance P(Pn -» ve) = (2-64 ± 0.67 ± 0.45) x K T 3 .

(65)

This result can be explained with v^ —> ve oscillation where the best fit is achieved for Am 2 = 1.2 eV 2 , 2

sin2 2(9 = 0.003.

(66)

Note that the value of Am in (66) is much larger than the values indicated by the solar and atmospheric neutrino data. The LSND indications for neutrino masses are not as strong as the solar and atmospheric neutrino ones. First, the statistical significance of the LSND signal is rather low (traditionally, only a 5a effect is called a discovery). Second, the LSND signal still needs an independent confirmation.

38

The Karmen experiment [34], which has a setting similar to LSND, but with a shorter baseline, L (=s 17 meter, did not find an appearance signal. Soon, the MiniBooNE experiment [35] will be able to clarify the situation.

7. Theoretical implications As we have seen, there are solid experimental indications for neutrino masses. The most important implication of these results is also the most simple one: the SM is not the complete picture of Nature. Namely, there are experimental indications that the SM is only an effective low energy description of Nature. There are basically two ways to extend the SM. One way is to assume that there is new physics only at a very high scale, much above the weak scale. Then, neutrino masses can be accounted for by the seesaw mechanism. Alternatively, one can think about weak scale new physics. Neutrino masses that are generated by such new physics are, in general, too large and some mechanism is required in order to explain the smallness of the neutrino masses. In the following we concentrate on the former option.

7.1. Two neutrino

mixing

We start by considering only one type of results at a time. Namely, we draw conclusions from the solar and KamLAND data or from the atmospheric and K2K data. In that case there is one theoretical challenge imposed by the data: how to generate neutrino masses at the right scale? The atmospheric neutrinos and K2K results indicate z/M —> vT oscillation with Am^ N ~ few x 10" 3 eV 2 .

(67)

The solar neutrinos and KamLAND results indicate ve —> i/^tT oscillation with Am!N~l(r4eV2.

(68)

We use (9) in order to obtain the scale of the new physics that is responsible for neutrino masses

mu = -§M

=*

M=-^-. vav

69

39

In order to find the high energy scale, M, we need to know m„ and moThe most likely situation is that the neutrinos are not degenerate Am.2, 13 2 m - • 1.

(70)

Then, we use mv ~ v Am 2 . In general, the Dirac masses are of the order of the weak scale times dimensionless couplings. The problem is that we do not know the value of these couplings. In the following we make two plausible choices for their values. Motivated by S'O(IO) GUT theories, we assume that these couplings are of order of the up type quark Yukawa couplings. In particular, for the heaviest neutrino, the coupling is of the order of the top Yukawa coupling. Another plausible choice is to use the charged lepton Yukawa couplings as a guide to the neutrino ones. Considering atmospheric neutrino and K2K data we use mv ~ few x !CT 2 eV and get mD~mt mD~mT If instead we use mv ~ 10 LAND data, we obtain mD~mt mD ~ mT

M ~ 1016 GeV, M ~ 10 11 GeV.

=> => 2

(71)

eV as indicated by solar neutrinos and Kam-

=> =^

M ~ few x 1016 GeV, M ~ 1012 GeV.

, ^ '

We emphasize the following points: • Both eqs. (71) and (72) imply that there exists a new scale between the weak scale and Mpi ~ 10 19 GeV. • The values obtained assuming mo ~ mt in both eqs. (71) and (72) are very close to the GUT scale, M G U T ~ 3 X 10 16 GeV. Therefore, we can say that neutrino masses are indications in favor of unification. • In several new physics models, a new scale, usually called the intermediate scale, Mi n t ~ \fm~wMp\ ~ 10 11 GeV, is introduced. For example, supersymmetry breaking has to occur at this scale if it is mediated via Planck scale physics to the observable sector. The values obtained assuming mn ~ mT in both eqs. (71) and (72) are very close to Mint- This could be an indication that neutrino masses are also generated by such models.

40

7.2. Three neutrino

mixing

When we combine the solar, KamLAND, atmospheric and K2K data, we have to consider the full three flavor mixing. There are three mass differences that control the oscillations. They are subject to one constraint Am 2 2 + Aml3 + A m ^ = 0.

(73)

The data indicates that the mass-squared differences are hierarchical, |Am 2 2 | "C |Am 2 3 |. This implies that |Am| 3 | RS (ATO^I, and therefore, that there are only two different oscillation periods. There are three mixing angles. Solar neutrino oscillations depend mainly on 0i2 and atmospheric neutrino oscillations depend mainly on 0 23 . The third angle, #13, is known to be small, mainly from terrestrial experiments (see, for example, [4] for details). Thus, the oscillation phenomena are described by two mass-squared differences and three mixing angles Am? 2 - 1(T 4 eV 2 , 012 ~ 1,

Am^ 3 - few x 1(T 3 eV 2 ,

#23 ~ 1,

(74)

013 < 0.2.

As before, we assume that the neutrinos are not degenerate. Then, we learn from (74) that the neutrino masses are somewhat hierarchical with two large and one small mixing angles. The theoretical challenge imposed by considering three neutrino flavor mixing is to explain the flavor structure of (74). There are basically two approaches to do so. One is called neutrino anarchy [36]. It assumes that there are no parametrically small numbers. The two apparently small numbers in the neutrino sector m2 m3

rs^

1 5

#13~^r,

(75)

are assumed to be accidentally small. That is, numbers of order five are considered natural. Therefore, this mechanism predicts that 0i3 is close to its current upper bound. Consequently, it also predicts that CP violating observables, which are proportional to the product of all the mixing angles and the neutrino mass-squared differences, should also be close to their current upper bounds. The other option to explain (74) is to assume that there is an underlying broken flavor symmetry that controls the neutrino sector parameters. Such a symmetry is often assumed in order to explain the quark sector flavor parameters. Within this framework it is assumed that there is one small

41

parameter, e, such that all the small parameters of the theory are functions of it. In particular, — ~ en « 1, 913 ~ em « 1, (76) m3 where n and m are some positive integers. In that case we often refer to observables like 1112/1713 and #13 as parametrically small numbers. While flavor symmetries can explain the observed hierarchies in the quark sector, it is not straightforward to extend the treatment to the neutrinos. The main reason is that in general large mixing angles come without mass hierarchies and vice versa. Explicitly, we consider the two generation case. The neutrino mass matrix is

- - ( ; : ) .

P7)

and we distinguish the following three cases • a^>b,c ==>• sin 2 0 <^i 1, mi/m2 < 1. • a, 6 , c ~ l , det(m) ~ 1 = > s i n 2 # ~ l , mi/m2 ~ 1. • a, b, c ~ 1, det(m) s i n 2 0 ~ l , mi/m2 < 1. We learn that in order to have large mixing with mass hierarchy the determinant of the mass matrix must be much smaller than the typical value of its entries (to the appropriate power). This is not a common situation in flavor models. In particular, this is not the case in the quark sector. 7.3. Four neutrino

mixing

While the LSND data cannot be considered as a solid indication for neutrino masses, one may like to try to accommodate it as well. In that case the situation become much more complicated. The reason is that three different mass-squared differences are needed A m | N ~ 10~ 4 eV 2 ,

Am2AN ~ few x 1CT3 eV 2 ,

Am£ S N D ~ 1 eV 2 . (78)

With three neutrinos, however, the. constraint (73) implies that there are at most two different scales. Therefore, at least four neutrinos are needed in order to accommodate all of these results. Experimentally we know that there are only three active neutrinos, and thus we need at least one additional light sterile neutrino field. There are two theoretical challenges: First, a sterile neutrino, by definition, is a fermion that is a singlet under the SM gauge group. As such,

42

it can acquire very large mass since there is no gauge symmetry that forbids it. Thus, a sterile neutrino naturally has very large mass. There are, however, several ideas that produce naturally light sterile states [37]. Even if there are light sterile states, complicated patterns of masses and mixing angles are required in order to accommodate all the data. In fact, even adding a sterile state does not help in obtaining a good fit to the data [38]. In the following we do not address the four neutrino mixing any further. 8. Models for neutrino masses As we saw there are three different kinds of theoretical challenges imposed by the neutrino data. Considering separately the solar and the atmospheric neutrino data, the challenge is to generate neutrino masses at the right scale. Combining them, we would like to find ways to generate the flavor structure, in particular, to explain mass hierarchy with large mixing. In order to accommodate also the LSND result, we should find a mechanism that generates light sterile neutrinos, and construct even more complicated flavor structure. The simplest way to generate neutrino masses is the seesaw mechanism. It is particularly attractive when it appears in models that are well motivated because they address various theoretical puzzles. Below we describe two models where neutrino masses are generated as one aspect of a model. (See [2,4] for more examples.) 8.1. Grand Unified Theories

(GUTs)

There are several supporting evidences for supersymmetric grand unification. First, in the minimal supersymmetric SM (MSSM) the three gauge couplings unify at one point at a scale of AGUT ~ 3 x 10 16 GeV.

(79)

Second, one generation of SM fermions fits into two 5*17(5) multiplets and into one multiplet for higher rank GUT groups. Neutrino physics provide further support for GUT, particularly, for GUTs that are based on 50(10) (or higher rank) gauge group. Next we describe two of the facts that make 50(10) GUTs attractive from the point of view of neutrino physics. The 15 degrees of freedom of one SM fermion generation fall into a 16 of 50(10). The one extra degree of freedom is a SM singlet. Thus, it can serve as a right handed neutrino field. Once 50(10) is broken, this singlet

43

becomes massive. Consequently, the SM active neutrinos acquire small masses via the seesaw mechanism. Namely, neutrino masses are predictions of 50(10) GUT theories. 50(10) also predicts the rough scale of the neutrino masses. To see it recall that 50(10) relates the up type quark masses to the Dirac masses of the neutrinos. In particular, the Dirac mass of the heaviest neutrino is of the order of the mass of the top quark. The Majorana mass of the singlet is expected to be of the order of the GUT breaking scale, Mjy = A AGUT where A is an unknown dimensionless number. Then, the seesaw generated mass of the heaviest active neutrino is of the order of rrii

10" 3 eV

,

,

For A ~ O(10 _ 1 ) it is the mass scale that was found by solar neutrinos. Somewhat smaller A is needed in order to get the scale that is relevant for atmospheric neutrino oscillation. Generally, it is fair to say that 5O(10) based theories predict neutrino masses in accordance with the experimental findings. 8.2. Flavor

physics

The most puzzling features of the SM fermion parameters are their smallness and the hierarchies between them. These features suggest that there is a more fundamental theory where the hierarchies are generated in a natural way. Generally, a broken horizontal symmetry is invoked. (By horizontal symmetry we refer to a symmetry that distinguishes between the three SM generations.) We demonstrate this broken flavor symmetry idea by working in the framework of a supersymmetric U(1)H horizontal symmetry [39]. We assume that the low energy spectrum consists of the fields of the MSSM. Each of the supermultiplets carries a charge under U{1)H- The horizontal symmetry is explicitly broken by a small parameter A to which we attribute a charge —1. Then, the following selection rules apply: Terms in the superpotential that carry a charge n > 0 under H are suppressed by 0(A n ), while those with n < 0 are forbidden due to the holomorphy of the superpotential. Explicitly, the lepton parameters arise from the Yukawa terms Z YijLiEjHd + ——LiLjHuHu. (81) Here Li (Ei) are the lepton doublet (singlet) superfields, Hu {Hj) is the up type (down type) Higgs superfields, Yy is a generic complex dimensionless

44

3 x 3 matrix that gives masses to the charged leptons, Zij is a symmetric complex dimensionless 3 x 3 matrix that gives Majorana masses to the neutrinos and M is a high energy scale. The selection rule implies H(Li) + H{Ej) + H(Hd) =n

=»

Ytj = 0(Xn)

H(Li) + H{Lj) + 2H(HU) =m

=>

Zi5 = 0(Xm)

(82)

where we assume that the horizontal charges of all the superfields are non-negative integers. We learn that the dimensionless couplings that are naively of order unity are suppressed by powers of a small parameter. Next we demonstrate a realization of the neutrino anarchy scenario using the above framework. Consider the following set of U(l)u charge assignments H{HU) = 0,

H{Hd) = 0,

(83)

H(Li) = 0,

H(L2) = 0,

H{L3) = 0,

H{E{) = 8,

H(E2) = 5,

H(E3) = 3.

Then, the lepton mass matrices have the following forms / A 8 A5 A 3 \ M ~ (Hd) A8 A5 A3 , \ A 8 A5 A 3 / l

/ l 1 1\ 1 1 1 . \1 1 1 / 2

M" ~ ^ M

(84)

We emphasize that the sign "~" implies that we only give the order of magnitude of the various entries; there is an unknown (complex) coefficient of 0(1) in each entry that we do not write explicitly. Equation (84) predicts large mixing angles sin #23 ~ 1,

sin #12 ~ 1,

sin #13 ~ 1,

(85)

non-hierarchical neutrino masses (Hu) (Hu)2 v 2 M M and hierarchical charged lepton masses v 1

me~(Hd)\8,

m^(Hd)X5,

v 3

(Hu) M

mr~(Hd)\3.

y

'

(87)

We see that this model predicts neutrino masses and mixing angles that are in accordance with the neutrino anarchy scenario. When we assume that both (Hd) and (Hu) are at the weak scale and that A ~ 0.2, this model also reproduces the order of magnitude of the observed charged lepton masses. Other, more complicated, flavor structure can also be generated using a similar framework. See [2,4] for more details.

45

9. Conclusions Neutrino physics is important since it is a tool for probing unknown physics at very short distances. Since we know that the SM has to be extended, it is of great interest to search for such new physics. In general, new physics at high scale predicts massive neutrinos. Therefore, there is a strong theoretical motivation to look for neutrino masses. In recent years various neutrino oscillation experiments found strong evidences for neutrino masses and mixing • Atmospheric neutrinos show deviation from the expected ratio between the fluxes of muon neutrinos and electron neutrinos. Moreover, the muon neutrino flux has strong zenith angle dependence. The simplest interpretation of these results is that there are v^ — vr oscillations. • The electron neutrino flux from the Sun is smaller than the theoretical expectation. Furthermore, the suppression is energy dependent. The total neutrino flux, as measured using neutral current reaction by the SNO experiment, agrees with the theoretical expectation. The simplest interpretation of these results is that there are ve — vx oscillations, where vx can be any combination of v^ and vT. • The K2K long baseline experiments found indications for muon neutrino disappearance. Moreover, the same neutrino parameters that account for the atmospheric neutrino disappearance also explain the K2K data. • The KamLAND experiment found evidences for F e disappearance. The data provide an independent test of the same parameter space that is probed by solar neutrinos. The theoretical expectation and the experimental data fit comfortably. Yet, there are many unsolved problems associated with neutrino physics. In the future we expect to have more data and thus we will be able to learn more about neutrinos, and eventually, about the short distance new physics. Acknowledgments I thank Yossi Nir, Subhendu Rakshit, Sourov Roy, Ze'ev Surujon and Jure Zupan for comments on the manuscript. Y.G. is supported by the United States-Israel Binational Science Foundation through Grant No. 2000133, by a Grant from the G.I.F., the German-Israeli Foundation for Scientific

46

Research and Development, and by the Israel Science Foundation under Grant No. 2 3 7 / 0 1 . References 1. A partial list of books includes: B. Kayser, "The physics of massive neutrinos", 1989, World Scientific; J. N. Bahcall, "Neutrino astrophysics", 1989, Cambridge Univ. Press; R. N. Mohapatra and P. B. Pal, "Massive neutrinos in physics and astrophysics", 1991, World Scientific; C. W. Kim and A. Pevsner, "Neutrinos in physics and astrophysics", 1993 Harwood Academic; G. G. Raffelt, "Stars as laboratories for fundamental physics", 1996, Chicago Univ. Press. 2. For some recent reviews see: S. Goswami, hep-ph/0303075; S. Pakvasa and J. W. Valle, hep-ph/0301061; R. N. Mohapatra, hep-ph/0211252; E. K. Akhmedov, hep-ph/0001264. 3. A. D. Dolgov, hep-ph/0208222; Phys. Rept. 370, 333 (2002) [hepph/0202122]. 4. M. C. Gonzalez-Garcia and Y. Nir, Rev. Mod. Phys. 75, 345 (2003) [hepph/0202058]. 5. The neutrino oscillation industry web page: "neutrinooscillation.org"; Juha Peltoniemi's ultimate neutrino page: "cupp.oulu.fi/neutrino". 6. Maury Goodman runs the long-baseline neutrino oscillation newsletters. These well written short monthly letters, are very nice summaries of neutrino news. It is highly recommended to sign up for the free monthly email service for anyone who likes to keep up with the developments in neutrino physics, "www.hep.anl.gov/ndk/longbnews/index.html" 7. See, e.g., M. E. Peskin and D. V. Schroeder, "Introduction to quantum field theory", 1995, Addison-Wesley. 8. K. Hagiwara et al. [Particle Data Group Collaboration], Phys. Rev. D 66, 010001 (2002). 9. H. V. Klapdor-Kleingrothaus et al, Eur. Phys. J. A 12 147 (2001). 10. Y. Grossman and H. J. Lipkin, Phys. Rev. D 55, 2760 (1997) [hepph/9607201]. 11. L. Wolfenstein, Phys. Rev. D 17, 2369 (1978). 12. See, e.g., H. Nunokawa, V. B. Semikoz, A. Y. Smirnov and J. W. Valle, Nucl. Phys. B 501, 17 (1997) [hep-ph/9701420]; S. Bergmann, Y. Grossman and E. Nardi, Phys. Rev. D 60, 093003 (1999) [hep-ph/9903517]. 13. S. P. Mikheev and A. Y. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985) [Yad. Fiz. 42, 1441 (1985)]. 14. S. J. Parke, Phys. Rev. Lett. 57, 1275 (1986). 15. J. Bahcall web site, "www.sns.ias.edu/~jnb"

47

16. See, for example, E. K. Akhmedov, Pramana 54, 47 (2000) [hep-ph/9907435]; S. T. Petcov, Phys. Lett. B 434, 321 (1998) [hep-ph/9805262]. 17. R. Jr. Davis, D. S. Harmer and K. C. Hoffman, Phys. Rev. Lett. 20, 1205 (1968). 18. B. T. Cleveland et al., Astrophys. J. 496, 505 (1998); K. Lande et al., Nucl. Phys. Proc. Suppl. 77, 13 (1999). 19. GALLEX home page: "kosmopc.mpi-hd.mpg.de/gallex/gallex.htm" 20. GNO home page: "www.lngs.infn.it/site/exppro/gno/Gno_home.htm" 21. SuperKamiokande home page: "www-sk.icrr.u-tokyo.ac.jp/doc/sk/index.html" 22. SNO home page: "www.sno.phy.queensu.ca" 23. Q. R. Ahmad et al. [SNO Collaboration], Phys. Rev. Lett. 89 011302 (2002) [nucl-ex/0204009]. 24. See, for example, S. Pakvasa, Pramana 54, 65 (2000) [hep-ph/9910246]. 25. P. C. de Holanda and A. Y. Smirnov, hep-ph/0212270; J. N. Bahcall, M. C. Gonzalez-Garcia and C. Pena-Garay, hep-ph/0212147; A. Bandyopadhyay, S. Choubey, R. Gandhi, S. Goswami and D. P. Roy, hepph/0212146; M. Maltoni, T. Schwetz and J. W. Valle, Phys. Rev. D 67, 093003 (2003) [arXiv:hep-ph/0212129]; V. Barger and D. Marfatia, Phys. Lett. B 555, 144 (2003) [hep-ph/0212126]. 26. A. Bandyopadhyay, S. Choubey, S. Goswami and K. Kar, Phys. Lett. B 519, 83 (2001) [hep-ph/0106264]. 27. T. Toshito [Super-Kamiokande Collaboration], hep-ex/0105023. 28. S. Fukuda et al. [Super-Kamiokande Collaboration], Phys. Rev. Lett. 85, 3999 (2000) [hep-ex/0009001]. 29. K2K home page: "neutrino.kek.jp" 30. KamLAND home page: "www.awa.tohoku.ac.jp/html/KamLAND/index.html" 31. LSND home page: "www.neutrino.lanl.gov/LSND" 32. M. H. Ahn et al. [K2K Collaboration], Phys. Rev. Lett. 90, 041801 (2003) [hep-ex/0212007]. 33. K. Eguchi et al. [KamLAND Collaboration], Phys. Rev. Lett. 90, 021802 (2003) [hep-ex/0212021]. 34. Karmen home page: "www-ik.fzk.de/~karmen/karmen_e.html" 35. MiniBooNE home page: "www-boone.fnal.gov/publicpages/index.html" 36. A. de Gouvea and H. Murayama, hep-ph/0301050; N. Haba and H. Murayama, Phys. Rev. D 63, 053010 (2001) [hepph/0009174]; L. J. Hall, H. Murayama and N. Weiner, Phys. Rev. Lett. 84, 2572 (2000) [hep-ph/9911341]. 37. For a review see, for example, R. N. Mohapatra, hep-ph/9910365. 38. For a recent paper see, for example, C. Giunti, hep-ph/0302173.

39. M. Leurer, Y. Nir and N. Seiberg, Nucl. Phys. B 398, 319 (1993) [hepph/9212278]; M. Leurer, Y. Nir and N. Seiberg, Nucl. Phys. B 420, 468 (1994) [hepph/9310320]; Y. Grossman and Y. Nir, Nucl. Phys. B 448, 30 (1995) [hep-ph/9502418].

It

KONSTANTIN MATCHEV

This page is intentionally left blank

PRECISION ELECTROWEAK PHYSICS

KONSTANTIN MATCHEV Department of Physics, University of Florida, Gainesville, FL 32611, USA and Institute for High Energy Phenomenology, Newman Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA E-mail: [email protected]

These notes are a written version of a set of lectures given at TASI-02 on the topic of precision electroweak physics.

Contents

1

Preliminary Remarks 1.1 What is it all about? 1.2 Useful references for further reading

53 53 54

2

The 2.1 2.2 2.3

55 55 57 58

3

Precision Measurements at the Z Pole 3.1 Z resonance parameters 3.2 Branching ratios and partial widths 3.3 Unpolarized forward-backward asymmetry 3.4 Left-right asymmetry 3.5 Left-right forward-backward asymmetry 3.6 Tau polarization

61 63 63 65 66 66 66

4

Precision Measurements at LEP-II

67

Tools of the Trade Theory Fundamental parameters, input parameters and observables . . . . Experimental facilities

51

52

5

Precision Measurements at the Tevatron 5.1 Top mass measurement 5.2 W mass measurement

71 71 73

6

Theoretical Interpretation of the Precision Electroweak D a t a 6.1 Testing the Standard Model 6.2 Fixed parameters 6.3 Floating parameters 6.4 Comparing the values for the electroweak observables 6.5 The Higgs mass prediction 6.6 Impact of the My/ measurement on m^ 6.7 Impact of the asymmetry measurements on m^ 6.8 A Higgs puzzle?

74 76 77 78 79 80 80 86 89

7

Testing for N e w Physics 7.1 S, T, U parameters 7.2 Constraining new physics scenarios

90 92 92

8

Concluding Remarks

94

53

1. Preliminary Remarks Precision experiments have been crucial in both validating the Standard Model (SM) of particle physics and in providing directions in searching for new physics. The precision program, which started with the discovery of the weak neutral currents in 1973, has been extremely successful': it confirmed the gauge principle in the Standard Model, established the gauge groups and representations and tested the one-loop structure of the SM, validating the basic principles of renormalization, which in turn allowed for a prediction of the top quark and Higgs boson masses. In relation to new physics models, precision measurements have severely constrained new physics at the TeV scale and provided a hint of a possible (supersymmetric) gauge coupling unification at high energies. 1.1. What is it all about? On the one hand, the term Precision Electroweak Physics (PEW) refers to quantities which are very well measured. How well? Let us say at the % level or better. However, not every well-measured quantity is of interest here. For example, the amount of cold dark matter in the Universe £ICDM, Newton's constant GN and the strong coupling constant as are all very well known, yet they do not belong to the electroweak sector of the Standard Model: SU(S) x SU(2)W QCD

x U(1)Y

Electroweak

The observables which are typically included in the precision electroweak data set are conveniently summarized in Tables 1 and 2. Our goal in these lectures will be the following: (1) we shall learn the meaning of some of the quantities in Tables 1 and 2; (2) we shall find out how these quantities are measured by experiment; (3) we shall discuss the prospects for improving these measurements in the near future; (4) we shall find out their implications about the validity of the Standard Model; (5) we shall learn how they can be used to predict or constrain as yet undiscovered physics - for example, the mass of the Higgs boson, or the existence/lack of new physics beyond the Standard Model near the TeV scale.

54 Table 1. Principal Z-pole observables, their experimental values, theoretical predictions using the SM parameters from the global best fit as of 1/03, and pull (difference from the prediction divided by the theoretical uncertainty). T(had), T(inv), and r{(.+(.~) are not independent, but are included for completeness. From Ref. 1. Quantity Mz [GeV] Tz [GeV] T(had) [GeV] T(inv) [MeV] F(e+£-) [MeV] Thad [nb] Re Rfj. RT

AFB(e) AFB(P) AFB(T)

Rb Re Rs,d/R(d+u+s) AFB(b) AFB(C)

AFB(s) Ab Ac As ALR (hadrons) ALR (leptons) A» Ar Ae{QLR) AT(VT) Ae{Vr) QFB

Group(s)

Value

Standard Model

pull

LEP LEP LEP LEP LEP LEP LEP LEP LEP

91.1876 ±0.0021 2.4952 ± 0.0023 1.7444 ± 0.0020 499.0 ± 1 . 5 83.984 ± 0.086 41.541 ± 0 . 0 3 7 20.804 ±0.050 20.785 ± 0 . 0 3 3 20.764 ± 0.045

91.1874 ±0.0021 2.4972 ± 0.0011 1.7436 ±0.0011 501.74 ± 0 . 1 5 84.019 ± 0.027 41.470 ± 0.010 20.753 ± 0.012 20.753 ± 0.012 20.799 ± 0.012

0.1 -0.9

LEP LEP LEP LEP + S L D LEP + S L D OPAL LEP LEP DELPHI.OPAL SLD SLD SLD

0.0145 ± 0.0025 0.0169 ±0.0013 0.0188 ± 0.0017 0.21664 ± 0.00065 0.1718 ±0.0031 0.371 ± 0.023 0.0995 ± 0.0017 0.0713 ± 0.0036 0.0976 ±0.0114 0.922 ± 0.020 0.670 ± 0.026 0.895 ± 0.091

0.01639 ± 0.00026

-0.8 0.4 1.4 1.1 -0.2 0.5 -2.4 -0.8 -0.5 -0.6 0.1 -0.4

SLD SLD SLD SLD SLD LEP LEP LEP

1.2. Useful references

0.15138 0.1544 0.142 0.136 0.162 0.1439 0.1498 0.0403

for further

±0.00216 ±0.0060 ±0.015 ±0.015 ±0.043 ±0.0043 ±0.0048 ± 0.0026

0.21572 0.17231 0.35918 0.1036 0.0741 0.1037 0.93476 0.6681 0.93571

± 0.00015 ± 0.00006 ± 0.00004 ±0.0008 ± 0.0007 ±0.0008 ± 0.00012 ± 0.0005 ± 0.00010

0.1478 ±0.0012

0.0424 ± 0.0003

1.9 1.0 1.0 -0.8

1.7 1.1 -0.4 -0.8 0.3 -0.9 0.4 -0.8

reading

For the current status of the precision electroweak observables, one should consult the "Electroweak model and constraints on New Physics" review [2] in the Particle Data Book [3]. There are nice lecture summaries from previous schools [4-11], as well as book collections of relevant review articles [12-14]. The long-term prospects for improvement in the precision measurements have been discussed in several workshop reports, e.g. the Tevatron Run II Workshop [15] and Snowmass 2001 [16-18]. The website of the LEP Electroweak Working Group is another excellent source of information, with its continuously updated electroweak summary notes [20].

55 Table 2.

The recent status of non-Z-pole observables, as of 1/03. From Ref. 1.

Quantity mt Mw Mw

[GeV] [GeV] [GeV]

Group (s)

Value

Standard Model

pull

Tevatron LEP Tevatron,UA2

174.3 ± 5 . 1 80.447 ± 0.042 80.454 ± 0.059

174.4 ± 4.4 80.391 ± 0 . 0 1 8

0.0 1.3 1.1

NuTeV NuTeV CCFR CDHS CHARM CDHS CHARM CDHS 1979

9l Rv R" Rv R? RD R"

CHARM II all CHARM II all

97 97 97 9A

Qw (Cs) Qw(Tl) 1 0 3 r(fc-.. 7 )

-0.035 -0.041 -0.503 -0.507

± 0.017 ± 0.015 ±0.017 ± 0.014

-2.9 0.6 -0.3 0.1 -1.7 -0.1 1.0 -1.0

-0.0398 ± 0.0003 -0.1 -0.5065 ± 0.0001 0.0

Boulder Oxford/Seattle

-72.69 ± 0.44 -116.6 ± 3 . 7

-73.10 ± 0 . 0 4 -116.7 ± 0 . 1

0.8 0.0

BaBar/Belle/CLEO

3.48i°;«|

3.20 ± 0 . 0 9

0.5

direct/Be /B^ e + e ~ / r decays BNL/CERN

290.96 ± 0.59 ± 5.66 56.53 ± 0.83 ± 0.64 4510.64 ± 0.79 ± 0.51

291.90 ± 1 . 8 1 57.52 ± 1 . 3 1 4508.30 ± 0.33

TT [fs] 10 9 (a„ -

0.30396 ± 0.00023 0.30005 ± 0.00137 0.03005 ± 0.00004 0.03076 ± 0.00110 0.5820 ± 0.0027 ± 0.0031 0.5833 ± 0.0004 0.3096 ± 0.0033 ± 0.0028 0.3092 ± 0.0002 0.3021 ± 0.0031 ± 0.0026 0.3862 ± 0.0002 0.384 ± 0 . 0 1 6 ± 0 . 0 0 7 0.403 ± 0.014 ± 0.007 0.3816 ± 0.0002 0.365 ± 0.015 ± 0.007

£)

-0.4 -0.9 2.5

2. The Tools of the Trade As usual, we need to have a theory and experiments to test it. Before we go on to discuss the observables from Tables 1 and 2 in the next few sections, we first need to introduce the necessary theoretical ingredients and describe the relevant type of experiments. In Section 2.1 we review the electroweak Lagrangian and in Section 2.2 we discuss its input parameters. Sections 2.3 provides a general review of some of the types of experiments which have contributed to the results in Tables 1 and 2.

2.1.

Theory

Achieving the remarkable precision displayed in Tables 1 and 2 is only possible because the Standard Model provides a well-defined theoretical framework for computing the different electroweak observables in terms of a few input parameters and thus predicting different relations among them.

56

The SM electroweak Lagrangian for fermions is given by

CF = J2i;i[iP-mi-^T7-)^i 2MW e^Q^a^iAp

(!)

(QED)

(2)

i

-^^^7M(i_75)(r+w+

2cos#i^

• E ^ " ( 0 V - 0 A 7

5

+

T-W-)^

) ^

(CC)

(^C)

(3) (4)

i

where the vector and axial vector couplings are g^=Tl-2Qism20w,

(5)

9A = It, (6) Q is the electric charge of fermion i, TJ is its weak isospin, 9w is the Weinberg angle ( t a n # ^ = g'/g), g (g') is the SU(2)w (U(l)y) gauge coupling constant, and e = gsiu.9w is the electromagnetic coupling constant. The photon (A/j) and Z-boson (Z^) fields are given in terms of the hypercharge gauge boson B^ and neutral W-boson W? as A^ = B^ cos 9W + Wl sin 6W, (7) l

Z„ = -Bp sin 9W + W* cos 6W.

(8)

Often the neutral current interactions are equivalently written as - ^ ^ E ^ ( f f i ( l - 7

5

) + s k ( l + 7

5

) ) ^

(JVC)

(9)

i

with the redefinition 9v = 9L+

9R

9A=gL~gR

(10) (11)

Exercise: Use (5), (6), (7) and (8) to derive (1-4) from the SU(2)w x U(l)y gauge theory lagrangian supplemented with the Yukawa terms for the fermions. Hint: you will need to know the Biggs vacuum expectation value v, which is computed by minimizing the tree-level Higgs potential VH = -

1

- ^

2

+ ^

(12)

57

2.2. Fundamental observables

parameters,

input parameters

and

Fundamental parameters are the parameters appearing in the original gauge theory Lagrangian. In the case of the electroweak Lagrangian, the fundamental parameters are the gauge couplings g' and g. The Higgs sector of the theory (12) adds two more fundamental parameters: the Higgs mass parameter /i and self-coupling XH • The remaining fundamental parameters are the Yukawa couplings in the Yukawa sector (1). Putting all together, the set of fundamental parameters is {
(14)

Often, as was the case above in Eq. (1-4), it is convenient to rewrite the Lagrangian in terms of derived quantities, i.e. input parameters. Through a simple redefinition we can trade one fundamental parameter for a combination of a new (input) parameter and the remaining fundamental parameters, e.g. g' = gtan0w

(15)

eliminates g' from the Lagrangian and replaces it with 9w- Another example is 2MW v= , (16) 9

which eliminates the combination v = [ij \J\JI in favor of the V7-boson mass Mw- The latter is directly accessible experimentally, unlike either A# or fi. A similar trick allows to go from Yukawa couplings to fermion masses: gnu K

-

JiMw

(

}

which is very convenient, as most fermion masses are well-known experimentally while in contrast the Yukawa couplings are small (with the exception of the top Yukawa) and thus difficult to measure. Of course, one should make sure the total number of parameters stays the same for both the fundamental and input parameters. For the precision electroweak tests of the SM the following set of wellmeasured input parameters is typically used: {a,GF,Mz,mh,mi}, 2

(18)

where a = e /47r is the fine structure constant, GF is the Fermi constant, which is measured from the muon lifetime, and Mz (m,h) is the Z-boson (Higgs boson) mass.

58

Exercise Find the relation between the input parameter set (18) and the fundamental parameter set (14). Finally, observables are the quantities which are measured by experiment. An example is the set of electroweak observables listed in Tables 1 and 2. If the number of observables is larger than the number of input parameters, we can test the model. 2.3. Experimental

facilities

As can be seen from Tables 1 and 2, the electroweak observables have been measured at a variety of experimental facilities. We shall now briefly describe each type. Lepton colliders Lepton (e + e~) colliders have played a very important role in the precision program. Lepton colliders have numerous advantages, perhaps best summarized in the following aphorism: "At a lepton collider, every event is signal. At a hadron collider, every event is background." The main attractive features are: • Fixed center of mass energy in each event. This provides an additional kinematic constraint, useful for eliminating the backgrounds as well as extracting properties of new particles, allowing e.g. a missing mass measurement. • All of the available beam energy is used (minus beamstrahlung), i.e. we have an efficient use of particle acceleration. • Clean environment: easier identification of the underlying physics in the event, easier heavy flavor tagging, less backgrounds. • Polarizability of the colliding beams (SLC). The polarization of the colliding beams can be used to reweight the contribution of the different diagrams involving W's, Z's and 7's. Naturally, lepton colliders also have certain disadvantages in comparison to hadron colliders (see below), which makes both types of facilities equally valuable and to a large extent complementary. Figure 1 summarizes some of the history of e+e~ colliders. In reverse chronological order, they are: • The Next Linear Collider (NLC) with an initial CM energy of 500 GeV is now under active discussion as the next large international

59

10

-a

10

§ o

a u

10

10

10

Figure 1.

0

20 40 60 80 100 120 Center of Mass Energy [GeV]

Total e + e - cross section from CESR/DORIS to LBP energies.

particle physics facility. Its timescale is still uncertain, but the NLC was ranked as the top priority among mid-term science facilities in the Department of Energy's Office of Science 20-year science facility plan [21]. LEP-II at CERN (1996-2000). It had the highest CM energy so far among lepton colliders. LEP-II took data at several yfs before shutting down in November 2000 amidst much speculation and heated discussions. There were four experiments (detectors) at diametrically opposite sites: ALEPH, DELPHI, L3 and OPAL. LEP-I at CERN (1989-1995), which took data at several energies around s/s = MzSLC at SLAC (1989-1998). It reached energies up to 100 GeV and also took data around the Z-pole. Unlike LEP, it had an important advantage: the availability of beam polarization (80%). B-factories: PEP-II at SLAC (1999-present), 9 x 3.1 GeV; KERB at

60

KEK (Japan) (1999-present) 8 x 3.5 GeV; CESR at Cornell (1979present). Exercise: Why are the B-factories asymmetric? • Others (y/s < 10 GeV) (Novosibirsk, Bejing, Frascati).

Hadron colliders In turn, hadron colliders have their own advantages: • Protons are much heavier than electrons, which leads to a reduction in the radiation losses. As a result, for a given CM energy, the ringsize of a hadron collider is smaller. Equivalently, for a given ringsize (and fixed magnetic field), a hadron collider allows to reach higher CM energies. • Even though the CM energy of the colliding beams is large, the typical parton CM energy is much smaller, but usually still beyond the CM energy of LC competitors. The main hadron collider facilities are the following: • LHC (CERN) is currently under construction and is projected to begin operation in 2007-8 as a 7 x 7 TeV pp collider. There will be four experiments - ATLAS, CMS, LHCB and ALICE. The initial data taking rate will be 10 fb~ per year at low luminosity and will subsequently increase to 100 fb~ per year. • The Tevatron Run II at Fermilab is now operating as 0.98 x 0.98 pp TeV collider. There are two experiments: CDF and DO, and the hope nowadays is for 8 fb~ per experiment before the LHC. • The Tevatron Run I (1987-1996): discovered the top quark, operated in a 0.9 x 0.9 pp mode and delivered 110 pb~ of data per experiment. • SppS at CERN (1981-1990). 300 x 300 GeV pp collider, discovered W and Z. It is interesting that the last three SM particles (W, Z and t) were discovered at hadron colliders and chances are that the last one (the Higgs boson) will follow suit. Other facilities In addition to the collider experiments mentioned above, there are numer-

61

cms fixed target experiments, whose main advantage is the lower cost and large luminosity (because of the dense target). Their main disadvantage is the low center of mass energy, which scales with the beam energy Ef, only as ECM ~ \fE~b- Another class of experiments on muon dipole moments are done at muon storage rings at CERN, and more recently, at BNL. In conclusion of this section, we should comment on the experimental identification of heavy particles, which decay promptly and are detected only through their decay products. For example, a W-boson decays either to a pair of quarks, which later materialize into QCD jets, or to a lepton and the corresponding neutrino. The W branching fractions are B(W —> qq') ~ 2/3, and B(W ->• ivt) ~ 1/9 for £=e,n, r. The jets (or the lepton and its neutrino) may have had a different source, and the only indication that they may have come from a W decay is that their invariant mass is close to MwSimilarly, a Z-boson can decay to pairs of quarks, leptons or neutrinos, with branching fractions B(Z —> qq) ~ 0.7, B(Z —> vv) ~ 0.2 and B{Z —> £+£~) ~ 0.1 (summed over the three generations). When both decay products are visible, their invariant mass is again required to be near Mz. A word of caution: r is not always a lepton! A r-lepton can decay leptonically to e or \i with a branching ratio 0.18 for each, or hadronically, with a branching ratio 0.64. In the latter case it appears as a "tau jet", which is similara to an ordinary QCD jet. To an experimentalist, a "lepton" is either an e or a /i, which may or may not have come from a tau decay. By T'S, experimentalists often mean "tau jets".

3. Precision Measurements at the Z Pole At the Z pole the cross-section for e+e~ —> ff diagram. For / / e w e have

dafz _ 9 dn

sr e e r / 7 /M|

±{s-Ml)2

+ s2T2z/M2z

is dominated by the Z

(l + c o s ^ X l - i V y L

2cos9Af{-Pe

B

+ Ae)

Although not quite - a tau jet is more narrow and has lower track multiplicity.

(19)

62

where P e is the polarization of the electron beam (relevant at SLC), s is the square of the CM energy, Y z is the total Z width b , r e e and Y ff are the partial widths for Z —> ee and Z —> / / , correspondingly. (The partial widths are related to the Z branching fractions as B(Z —> / / ) = Tjj/Tz-) In (19) 9 is the angle between the incident electron and the outgoing fermion and Af is the left-right coupling constant asymmetry: 2gjgA

{g[f - {gfRf

(9fv)2 + (gfA)2

(gfL)2 + (9fR)2

A f

where in the second equation we have used (5) and (6). Notice that Af < 1. Since gv and gA only depend on the quantum numbers of the particles, it follows that Af is the same for all charged leptons, all up-type quarks and all down-type quarks. For example, for charged leptons gev = - i - 2 ( - l ) sin2 6W ~ -0.50 + 0.462 = -0.038 9A = ~ \ =

-0.50

(21) (22)

We see that because of an accidental cancellation g\?
_

2(-0.038)(-0.5) _, Q _ 15 (-0.038) 2 + ( - 0 . 5 ) 2

(23)

(compare to the values measured in Table 1). This small value of An makes it particularly sensitive to electroweak vacuum polarization corrections (which have an impact on sin2 6w)- In terms of sin2 9w> we have _ Ai

2(1 - 4 sin2 9w)

' l + (l-4sin2M2-

(

}

Therefore small changes in sin2 6w are amplified by a factor of 8 in the leptonic asymmetry Ag. Exercise: Confirm roughly the numerical values for A^ and Ac in Table 1. b

T h e result (19) already incorporates the s-dependent width

rz(s) = ^rz(s

= M2z),

which in turn accounts for the effect of the so called "non-photonic", or oblique, one-loop corrections, i.e. the corrections to the Z propagator. For details, see 22.

63

Let us now discuss the various measurements on the Z-pole. We shall need the following integrals for the forward and backward regions: I-K/2

F : B :

/ Jo J

,TT/2

4

(l + cos2(9)sm6W0 = - ; 3 (1 + cos 2 (9) sin 6W6> = - ;

JTT/2

3.1. Z resonance

3

/ J0

2cos0sin0d0 = 1; (25)

f

2 c o s 0 s i n 0 d 0 = - 1 . (26)

JTT/2

parameters

Scanning the Z peak and fitting to a'z yields measurements of Mz, Tz and the peak hadronic (/ = q) total cross-section (obtained for s = M\ by integrating (19) over the full solid angle):

'" = m! Sff

(27

»

Recall that the LEP beams are not polarized, so Pe = 0. Figure 2 shows the hadronic cross-section (Jhad measured by the four collaborations as a function of the center-of-mass energy (solid line). Also shown as a dashed line is the cross-section after unfolding all effects due to photon radiation. The radiative corrections are large but very well known. At the peak the QED deconvoluted cross-section is 36% larger and the peak position is shifted by ~ 100 MeV. 3.2. Branching

ratios and partial

widths

Looking at various exclusive final states one can determine the following ratios rhnd 0.7 21; (28) Ri — 0.0333 Te Rb = Rr =

1

had

(29)

)

•*- cc

(30)

J- had

with £ = {e,/x, r } . Basically this amounts to measuring the Z branching fractions. Having measured the total width from the peak scan and all visible partial widths, one can determine the invisible partial width (for Z —* vv): •L inv

=

-L Z

-L had

*• ee

-L /x/x

-L r r

l^-V

64 —

i

|

i

,

i

|

i

i

i

|

,

i

,

i

i

i

°° yy

40 ALEPH DELPHI L3 OPAL

. .

30

/ / / /

kj

20 - • measurements, error bars / increased by factor 10

10

- _ — c from fit

/

/

. . -

/

—

vti

-

/

•

\V -

/ / / /

... QED unfolded

N:

•

ass

i . . .

86

i . ,y¥z,

i . . .

88

90

, .

•

1

92 94 E [GeV] cm

Figure 2. The final hadronic cross-section averaged over the four experiments at LEP-I, as a function of the center-of-mass energy, as measured (solid line) and QED deconvoluted (dashed line).

and count the number of neutrino species Nv coupling to the Z. Assuming that Yi N T where Tu is the Z partial width to pairs of a single neutrino species, we have Nv

•L inv / *- £

{YU/Y()SM

(32)

The ratio of Tinv and the Z partial width into charged leptons is used in order to minimize the uncertainties due to the electroweak corrections which are common to both partial widths and cancel out in their ratio. The Z lineshape in the hadronic channel is shown in Fig. 3 along with the SM prediction for two, three or four neutrinos species. The result confirms that there are three light neutrino flavors with SM-like couplings to the Z.

65

ALEPH

C

o 1.05

b

w

1 0.95

88

89

-4-, 90

91

92

93

94

95

Energy (GeV) Figure 3. The Z lineshape in the hadronic channel along with the SM prediction for two, three or four neutrinos species [23]. The lower plot shows the ratio of the measured points to the best-fit values.

3.3. Unpolarized

forward-backward

asymmetry

The unpolarized forward-backward asymmetry is determined by a simple counting method in e + e _ —> / / scattering Af

—a =

-AeAi

(33)

A

FB

where aF (aB) is the total cross-section for forward (backward) scattering of / with respect to the incident e~ direction. Exercise: Use (25-26) in order to derive the second equality in (33) and then check the numerical values for ApB from Table 1 for f = £,b, c.

66

3.4. Left-right

asymmetry

The left-right asymmetry is defined as ,

_

iff/(-|Pe|)-g/(+|Pe|) Pe(Jf{-\Pe\) + af{+\Pe\) ~Ae>

ALR

~

^

where o-f(Pe) is the total (integrated over all angles) cross-section for producing / / pairs with an electron beam of polarisation Pe. Exercise: Derive the second equality. 3.5. Left-right

forward-backward

asymmetry

The left-right forward-backward asymmetry is defined as Af FB

_ 4(~\Pe\) ~ 4(~\Pe\) ~ 4(+\Pe\) + 4(+l^l) _ 3 p ^ 4(-|p e |) + 4(-|p e |) + 4(+|p e |) + 4(+|p e |) 4 e '•

It allows a direct determination of the A/ quantities which are presented in Table 1. Exercise: Derive the second equality. To summarize or discussion so far, there are three observable asymmetries, measuring either Ae, Af or their product. 3.6. Tau

polarization

The tau lepton is the only fundamental fermion whose polarization is experimentally accessible at LEP and SLC. The average polarization of r leptons is defined by 6

Ms) =

+

(s) - o * ° * ( s )

" ' , , \

(36)

where atot(s) is the r production cross-section via Z exchange, integrated over all angles, and the subscripts + and — refer to the r helicity states +1 and — 1. The dependence of PT at the Z pole on the polar angle 6 has the form [24] P(cozC) i^costfj -

Ml 1 +

+ ™s20) + 2Aecose • c o s 2 e + 2AAe cosg

W

The r polarization is measured using five exclusive decay modes which comprise about 80% of all r decays: ei/P, fii/i>, TV(K)I/, pv and a\v. The single 7r and K modes are not normally distinguished. The different channels do not all have the same sensitivity to the r polarization, TT(K)V being the

67

in

°. 1000 C 750 M

T^JTV

\

VI ' ' ' '

J

\ X2/dof= 60.4/ 39 !

/

1It

T-»pV

j

J

- -_

[t

:

500

'250

^L=r5^MO<X>8D?^KXXTCk7S*7c^

MCFit

•

Data

R551 tBkg

• • Pos. Helicity T

^ B Non-x Bkg

—

Neg. Helicity z

OPAL

0

Figure 4. Observed distributions of the polarization estimators for the different channels studied by OPAL [26], compared to the SM expectation for positive or negative r helicity.

best in that respect (see Fig. 4). The energy and/or angular distributions of the decay products can be used as r polarization analysers [25]. Figure 4 shows an example of the different distributions used and their sensitivity to the T polarization. Fitting the experimental data for P T (cos0) (an example from the ALEPH experiment [27] is shown in Figure 5) to Eq. (37) allows an extraction of AT and Ae which enter Table 1 as AT(VT) and Ae(VT), correspondingly. 4. Precision Measurements at LEP-II LEP-II was able to pair produce W's and perform measurements of Mw, Tw and the W branching fractions. There are three diagrams contributing to W-pair production: a i-channel v exchange, and an s-channel 7*

68

-0.1

-0.2

-0.3

No-Universality

i

-1

-0.8

-0.6

-0.4

-0.2

i

0

i

i

i

0.2

i

i

,

i

0.4

,

,

,

i

0.6

,

i

,

i

0.8

i

,

i

1

COS0

Figure 5. Tau polarization as a function of cos 9. The dashed (dotted) line is the result from a global fit to the data with (without) the universality assumption Ae = AT-

and Z exchange. The latter two diagrams contain triple gauge boson couplings, therefore LEP also tested the nonabelian nature of the SM gauge interactions. The W mass measurement involves reconstruction of the invariant mass of the W decay products. For WW, there are three possible final states: 4g, qqlvg and 2l2vi. The LEP measurements used only the first two channels. In the case of qqlvi, the energy and momentum of the missing neutrino are easily deduced from energy and momentum conservation. Then the four objects are paired up and the invariant mass of each pair is computed. The resulting invariant mass distribution in the qq^v^, channel is shown in Fig. 6. The mass peak is very pronounced, with virtually no background. The position of the peak allows an extraction of Mw, while its width is

69

cs 100 % 90 O 7^ 80

1 70

ALEPH Preliminary jxvqq selection •

Vs = 191.6,195.5, 199.5,201.6 GeV

Data (Luminosity = 237 pb"1)

—

MC ( m w = 80.60 GeV/c 2 )

H

Non-WW background

W 60

M w (GeV/c 2 )

Figure 6. Reconstructed W mass distribution from the ALEPH experiment for the qqiiv^ channel. The data are compared to the Monte Carlo prediction for M\y = 80.60 GeV.

indicative of Tw, where in addition one has to worry about detector resolution effects (smearing). All four experiments are consistent and so far the LEP Mw measurement is slightly better than the Mw measurement from the Tevatron (see below). The W mass measurement is of extreme importance — as we shall see later, Mw is one of the observables most sensitive to the Higgs mass TO/,. The W branching ratios were measured in WW production and the results displayed in Table 3 nicely demonstrate lepton universality. The measurement of the rise in the VF-pair cross-section near threshold provides an alternative method for determining Mw • The result for energies up to 207 GeV is shown in Figure 7. The method is statistically limited and yields inferior precision compared to direct Mw reconstruction. However,

70 Table 3. LEP measurements of the W branching fractions derived from WW production cross-section measurements [20]. Decay mode W -> eue W —> fiVit W -> r i / T

B r a n c h i n g f r a c t i o n (%)

10.59 10.55 11.20 67.77

W -^gg'

±0.17 ±0.16 ±0.22 ± 0.28

it clearly demonstrates the need for all triple gauge boson couplings in order to understand the data. LEP-II also measured the ZZ cross-section. However, the electron couplings to the Z are smaller, and in addition Mz > Mw- Thus the sample of ZZ pairs was much smaller and the measurement of the ZZ cross-section was not as good as for the WW case. Nevertheless, a comparison of the combined data to theory (Fig. 8) reveals good agreement within the large errors of the data.

.Q Q.

30 -1

11/07/2003

'

'LEP

/

',--'

PRELIMINARY

20/

/

_A—-*—-t-t—jr-"-* -

//y

10-

/// :!/ / ,/

lltlYFSWW/RacoonWW ... no ZWW vertex (Gentle) ... only ve exchange (Gentle)

r\

160

180

200

Vs (GeV) Figure 7. The measured WW cross-section at LEP as a function of the center-of-mass energy, compared to the predictions of the event generators RacoonWW and YFSWW. The magenta (upper dotted) line is the theoretical result ignoring both WW~y and WWZ couplings, while the red (lower dotted) line neglects WWZ only.

71 11/07/2003

.o

L E P PRELIMINARY ''.. ZZTOandYFSZZ

N N

1-

0.5

o

180

190

200

Vs (GeV) Figure 8. The measured e + e ~ —» ZZ cross-section at LEP as a function of the centerof-mass energy. The solid line is the SM prediction and the band indicates its ± 2 % uncertainty.

5. Precision Measurements at the Tevatron 5.1. Top mass

measurement

Since the top quark is so heavy, so far it can only be produced at the Tevatron. The dominant mechanism for pair-production is qq —> ti through a virtual s-channel gluon. Single top production is also possible. The top quark decays as t —> Wb almost 100% of the time. Top quark pair-production then gives W+W~bb events and there are three possible discovery channels, depending on the W decays: • Both W's decaying hadronically — 6 jet final state (2b4j). This is the largest signal cross-section, but with large QCD backgrounds. • One W decays hadronically, the other leptonically: £2b2jtfT (lepton plus jet sample). This was the best channel for Run I, because there was a sufficient number of events, yet the background was under control.

72

• Both W's decaying leptonically: 2£2bIfT (dilepton sample). This channel has the largest S/VB ratio, however it is limited by statistics. Both experiments (CDF and DO) have conducted measurements of mt in Run I and the results from the two experiments as well as from different channels are consistent with each other (see Fig. 9). The direct mt measurements from the Teavtron are in very good agreement with the best fit value for mt extracted from the electroweak precision fits (see Section 6.6). Right now mt is the best measured of all quark masses. In Run II one can anticipate significant improvements — the expected precision on the mt measurements is shown in Table 4. In fact, there are already some preliminary top results from Run II [29].

Tevatron Top Quark Mass Measurements 168.4 ± 12.8 GeV/c 2

I

I

w

I

Dilepton

173.3 ±

7.8 GeV/c2

Lepton+jets

172.1 ±

7.1 GeV/c2

Combined

f D0 "i'CDF 167.4 ± 11.4 GeV/c2 I I

'm* w

I I

176.1 ±

I

7.4 GeV/c 2

186.0 ± 1 1 . 5 GeV/c 2 176.1 ±

6.6 GeV/c 2

174.3 ±

5.1

Dilepton Lepton+jets All-Hadronic Combined

Tevatron combined

i

150

160

170

GeV/c 2

i i i i i

180

190

200

Mtop (GeV/c 2 ) Figure 9.

Run I Tevatron results for mt and the global average. (From Ref. 28.)

73 Table 4. Expected precision (in GeV) on the rat measurements in Run II for the lepton plus jet and for the dilepton channel.

fCdt bbijj$T

bb2e$T

5.2. W mass

(ft- 1 ) Statistical Systematic Total Statistical Systematic Total

2 1.7 2.1 2.7 2.4 1.4 2.8

15 0.63 1.2 1.3 0.87 1.0 1.3

30 0.44 1.1 1.2 0.62 1.0 1.2

measurement

W bosons can be singly produced at the Tevatron with a relatively large cross-section ~ 1 nb. The W is subsequently detected through its leptonic decay mode to a charged lepton and the corresponding neutrino. (The hadronic decays cannot be used because of the enormous dijet background from QCD.) The signature is l.I$T + X. Unfortunately, we cannot reconstruct the invariant mass of the W, because of the unknown longitudinal component of the neutrino momentum. Hence we must extract Mw from transverse quantities only. There are several possible methods [30]: looking at the transverse mass (M^Y), the transverse momentum of the charged lepton (pj,) o r the missing transverse energy I$T. Since different methods have different systematic errors, it is desirable to have as many independent measurements as possible. In Run I the transverse mass method was used. (The p^T method was limited by the number of leptonic Z events - see below.) The transverse mass c is defined as M^

= ^2plTpuT{l-cos<j>)

(38)

where (j> is the angle between p\ and p\ in the transverse plane. p\ is nothing but the missing transverse energy and is measured as

FT = - 0 + fT)

(39)

where U is total transverse momentum of the remaining hadronic activity in the event (typically a jet from initial state radiation). The transverse mass distribution exhibits a characteristic drop-off around Mw, as shown in Fig. 10. The Mw measurement is done by fitting Monte Carlo predictions c T h e name is appropriate since when the W has zero longitudinal momentum p™, the transverse mass M^f coincides with the invariant mass Mw- If, however, \p^\ > 0, then M™ < Mw.

74

^500 a

CDF(IB) Preliminary

> *400

X7df = 158/139 (50 < MT < 120) X2/df = 62/69 (65 < M T < 100) Mw = 80.430 +/- 0.100 (stat) GeV

300 200

90

100 110 120 Transverse Mass (GeV)

Figure 10. Transverse mass distribution of W —^ \iv events from the CDF Run IB data, with the best fit. The shaded distribution is the background.

for different values of My/ to the observed Mjf distribution. In addition, at large Mjf the shape of the distribution is sensitive to the intrinsic W width TV, as illustrated in Fig. 11, which allows for a direct measurement

of rV: Tw = 2.055 ± 0.125 GeV.

(40)

An alternative method for measuring Mw employs the lepton pr distribution, which cuts off around M^y/2. However, unlike M^f, the pj. distribution depends on the W boost, and therefore on the transverse momentum p^ with which the W was produced. It is difficult to compute p^ by theoretical means, especially in the low px region, and it is best to extract it from data. For this purpose the Tevatron experiments have used the observed pij, distribution, which has a similar shape, but fewer events. Hence, in Run I this method was statistically limited, but offers good prospects for Run II. The expected precision of the Mw measurements at the Tevatron by the two methods is shown in Table 5. 6. Theoretical Interpretation of the Precision Electroweak Data We can make use of the precision electroweak data in three different ways: • Test the Standard Model.

75

>

10'

O

• Tw=L6GeV • Tw = 2.1GeV *rw = 2.6GeV

in +->

in J 10

>

10

10

100

120

140

160

180 200

mp (GeV) Figure 11. Monte Carlo simulations of the transverse mass spectrum for different W-boson widths. The normalization is arbitrary. (From Ref. 31.)

Table 5. Expected precision (in MeV) on the Tevatron Myy measurements in Run II.

JCdt (fb" 1 )

Mf PT

Statistical Systematic Total Statistical Systematic

Total

2 19 19 27 44 10 44

15 7 16 17 16 4 16

30 5 15 16 11 3 12

• Predict the preferred mass range of yet undiscovered particles: top quark in the past, nowadays the Higgs boson. • Point towards specific new physics models (in case of some discrepancy) or constrain new physics models (if agreement is found).

76

We shall discuss each one in turn. 6.1. Testing the Standard

Model

Having made a variety of measurements for different observables, we can test the SM by comparing theory to experiment. For this purpose we need to compute the theoretical prediction (within the SM) for each observable. How do we do it? Recall that in the SM we start with the input parameters {p} = {a,GF,Mz,g3,mh,mt,mb,mc,ms,...}

,

(41)

where the first three are the parameters of the electroweak sector. Any other electroweak quantity, e.g. sin2 9w, Mw etc. can be expressed in terms of these and thus is predicted by the model. At tree level the expressions are simple and only involve the electroweak inputs a,Gp,Mz, e.g. the Weinberg angle can be found from

Similarly, the W mass is Mw = Mz cos 6W

(43)

with 6w given by (42). The radiative corrections, however, modify the tree level relations and introduce the remaining (non-electroweak) parameters into the game, so that each electroweak observable depends in principle on the full set of input parameters (41): Of e°ry({p})

= Oree(a,GF,Mz)

[1 + A,({p})].

(44)

By measuring enough electroweak observables (with sufficient precision so that we are sensitive to the radiative corrections) we can extract information about the values of the other parameters outside the electroweak sector. We can then test the model for consistency by comparing the values deduced indirectly with direct measurements of those parameters. So the strategy is as follows. First compute the corrections in a certain renormalization scheme, treating some parameters as fixed inputs (usually the best measured ones). Then perform a global fit to the electroweak data. This will result in "best fit" values for the remaining (floating) input parameters. Then

77

(1) Compare the "best fit" values of the floating parameters to their direct measurements (if available). For example, the fit will choose "best" values for Mw, rnt, a3, which have been measured by other means directly. On the other hand, the best fit value for rrih cannot be checked against experiment yet, but is perhaps the most valuable piece of information from the global fit. (2) For the best fit values found above, compute and quote the theoretical prediction of the SM for each observable. This is the number quoted under "Standard Model" in Tables 1 and 2. Then compare the experimental and theoretical values of the observables themselves. A large discrepancy in a certain place may signal new physics which affects that particular measurement but not the others... There are two popular programs on the market which would accomplish this program: • ZFITTER [32], which uses the on-shell scheme; • GAPP [33] (global analysis of particle properties), which utilizes the MS scheme and is used for the PDG review. 6.2. Fixed

parameters

The parameters which are held fixed in the global fits are the following. Fine structure constant a. It is measured at low energies, so in order to obtain g'(Mz) and g(Mz) we need to evolve it up to the Z scale: a Mz

^ ^

= i

f^(M

Y

( 45 )

1 — i\a{Mz) Aa has QED and (two-loop) QCD contributions. The latter are denoted as Aa;)J d (for five active quark flavors below Mz)- The global fit produces a "best fit" value for it [16] Aa

hld(Mz)

= 0-02778 ± 0.00020,

(46)

which can again be compared to theoretical estimates. The uncertainty arises from higher order perturbative and nonperturbative corrections, from the uncertainty in the light quark masses (most notably mc) and from insufficient e + e~ data below 1.8 GeV. Sometimes, however, the theoretical estimates are used as a constraint in the fit and predetermine the best fit value. Note that a " 1 = 137.03599976 ±0.00000050 is much better known than l a~ (Mz) = 127.922 ± 0.020, which explains why it is preferred as an input parameter.

78

Fermi constant GF = 1.16637(1) 10~ 5 GeV 2 . It is determined from the muon lifetime formula .

25

o - p ill,,,

n2\ airrifj)

1927T3

'»

a 2 (m M )' G2 " (47)

where F{x) = l-8x

+ 8x3-x4-12x2lnx

(48)

and Ci is a known number. The remaining uncertainty in GF is almost entirely from the experimental input. Fermion masses. For simplicity, the fermion masses are held fixed, with the exception of mt and mc. 6.3. Floating

parameters

The remaining parameters from the set (41), which were not mentioned in Sec. 6.2, are floating and their values are determined from the global fit. • Mz • It is measured very well, and since the experimental measurement is included in the fit, the "best fit" value for Mz is very close to the experimental one (see Table 1). • as. Its best fit value [1] as{Mz)

= 0.1210 ±0.0018

(49)

is in very good agreement with direct determinations from tau decays, charmonium and upsilon spectra, jet properties etc. • mt. The best fit value, including the Tevatron data, is [1] mt = 174.2 ± 4.4 GeV.

(50)

However, even if we take the Tevatron data out, the prediction of precision data alone [1] mt = 174.0l?;2 GeV.

(51)

is in impressive agreement with the direct Tevatron value 174.3 ±5.1 GeV. • Higgs boson mass m/,. The best fit value is [1] mh = 86+^ GeV.

(52)

The central value is well below the LEP exclusion limit from direct searches, but is consistent at la. The result for the Higgs mass is often shown as "the blue band plot" (Fig. 12). However, there are

79

two caveats which we shall discuss in more detail below. First, the blue band hides a potential discrepancy between individual contradictory measurements. Second, the dependence on rrih is only logarithmic, and new physics contributions at the same level may have an impact on the Higgs mass prediction.

6.4. Comparing

the values for the electroweak

observables

Once we have the best fit values for the floating parameters, we can predict the theoretical central values Oth for the remaining observables in Tables 1

CM

<

2H

0 ExcludedT ^ n ^ - r ^ 20 100

Preliminary

400

mH [GeV] Figure 12. Of particular interest is the constraint on the mass rrih of the Higgs boson, because this fundamental ingredient of the Standard Model has not been observed yet. The figure shows the Ax versus rrih curve derived from precision elecXji troweak measurements, assuming the Standard Model to be the correct theory of nature. The solid line is the result of the fit using all data while the band represents an estimate of the theoretical error due to missing higher order corrections. The vertical band shows the 95% CL exclusion limit on rrih from the direct Higgs search at LEP. (Prom Ref. 20.)

80

and 2. We can then compare theory (Oth) to experiment (Oexp). The results are usually presented as "pulls", which in Tables 1 and 2 are defined as /nth P"

11

=

/nexp atH

(53)

»

where ath is the error on Oth. All in all, the fit is successful, as evidenced from Fig. 13. One is tempted to conclude that the SM works pretty well, having been tested at the level of 1% and less. No major discrepancies are observed, and, with so many measurements, a few 2a — 3a deviations are simply inevitable. 6.5. The Higgs mass

prediction

Let us now understand better where the Higgs mass prediction comes from. Ideally, we should concentrate on those observables which exhibit the strongest dependence on the Higgs boson mass rrih- For this purpose, it is useful to plot the theoretical prediction for each observable as a function of rrih and contrast it with the experimental measurement (see Figs. 1417). We see that the observables most sensitive to m^ are the W mass Mw and the asymmetries. We will discuss these in more detail in the next two subsections. 6.6. Impact

of the Mw

measurement

on irih

An approximate formula for Mw (in the MS scheme), which exhibits the dependence on the relevant electroweak parameters, is [34] MW = 80.3827 - 0.0579!n ( j ^ )

- 0.008h,"

( ^ ^ )

1

-H^-fH(i^)'- ) 1 -H^- )-

(54)

The experimentally measured value for Mw seems to be a bit high (Table 2). As we can see from (54), a largish Mw can be explained by either a small rrih, a larger mt, a smaller as, or some combination of the above. The correlation between Mw, rnt and rrih from the precision data is shown in Fig. 18, where the solid contour delineates the 68% CL region resulting from a global fit omitting the Mw, Tw and mt measurements. Also shown is the

81

Summer 2003 Measurement

Fit

I^meas /-Jiti / meas

(}

1

2

J

r

m 7 [GeVl

r 7 (GeVJ °hari R:

^

91.1875 + 0.0021 2.4952 + 0.0023

91.1875 2.4960 •

41,540 ± 0,037

41.478 M M

20.767 ± 0.025

20.742

ii^^^^i§i§ii!i;l

s


0.01714 + 0.00095 0.01836

R

0.21638 ±0.00066 0.21579 _

b D rt c

0.1720 ±0.0030

™"

0.0997 ±0.0016

0.1723 ' 0.1036 —

0.0706 ± 0.0035

0.0740

Ab

0.925 ± 0.020

0.935

A

0.670 ± 0.026

0

A * M

f bc

A°' n

fb

A,(SLD)

0.1513 + 0.0021

0.668 0.1477

:T)-.,, \(.«'.:.y \

00.423 + 0.034

80.385

i .... iCoVj

2.139 ±0,089 174.3 ±5.1

c

mt [GeV]

-•••I

••WWl SHU I i:iilliiiii^

~

•

2.093 174.3 0.2223 — 1

0

1 2

3

Figure 13. Agreement between the measured values for some of the precision' electro-weak observables and their SM predictions, as of the Summer of 2003. (From the L E P EWWG[19].)

82

10

;\ v \ i \ \

10'

; •

\x\>

>

>

\v\ % K \

CD

o 10

•

1

2_

V.

i

i

i

i

CD

O

-

10 2H 41.6

0.02

0.02761 ± 0.00036 as= 0.118 ±0.002 m,= 174.3+ 5.1 GeV

1.992

2

2.008

o?ep [nb] Figure 14. of m/,. The SM band is of the band

Comparison of LEP-I measurements with the SM prediction as a function measurement with its error is shown as the vertical band. The width of the due to the uncertainties in Aah(^d(Mz), as(Mz) and mt- The total width is the linear sum of these effects. (From the LEP EWWG [20].)

SM prediction for this correlation, for mh = 114,300,1000 GeV. We first see that the indirect determination of Mw and mt from precision data alone is in very good agreement with the direct Mw and mt measurements shown

83

Preliminary

0.105

0.14

0.08

0.15

A,(P T ) 10' Measurement ^ AajfJd= 0.02761 + 0.00036

>

a = 0.118 + 0.002 jm,= 174.3 ±5.1 GeV 10' 0.14

0.15

A, (SLD) Figure 15. Comparison of LEP-I measurements with the SM prediction as a function of m^. The measurement with its error is shown as the vertical band. The width of the SM band is due to the uncertainties in A a ^ d ( M z ) , aa(Mz) and mt- The total width of the band is the linear sum of these effects. Also shown is the comparison of the SLD measurement of At, dominated by A°LR, with the SM. (From the LEP EWWG [20].)

84

Preliminary 10?

10'

>

> CD

CD

o

O

I

E

X

E 10*-

10' 0.216

0.218

Rg

m3

1

> CD

O

10 2-

: •

^

-

^

0.173

1

i

'

0.179

0.72

Measurement x(5)-

a s = 0.118 ±0.002 U m t = 174.3 ±5.1 GeV

0.234

Figure 16. Comparison of LEP-I and SLD heavy flavor measurements with the SM prediction as a function of mh. The measurement with its error is shown as the vertical band. The width of the SM band is due to the uncertainties in Aa^d(Mz), as(Mz) and m±. The total width of the band is the linear sum of these effects. Also shown is the comparison of the LEP-I measurement of the inclusive hadronic charge asymmetry Q1^ with the SM. (From the LEP EWWG [20].)

85

Preliminary

sirf9w(vN) Measurement W,\ A a f f ^ 0.02761 ± 0.00036 as= 0.118 ±0.002 | | | m t = 174.3 ±5.1 GeV Figure 17. Comparison of Mw and Yw measured at LEP-II and pp colliders, of sin 2 d\y measured by NuTeV and of atomic parity violation in caesium with the SM prediction as a function of m^. The measurement with its error is shown as the vertical band. The width of the SM band is due to the uncertainties in Aorh^d(Mz), cts{Mz) and mt. The total width of the band is the linear sum of these effects. (From the LEP EWWG [20].)

with the dashed contour. This fact may not look that impressive, now that the top quark and the W boson have been discovered, but nevertheless it should be considered as a triumphant success of the precision program. We also see that both the direct and indirect measurements of Mw and mt prefer a light Higgs boson.

86

6.7. Impact of the asymmetry

measurements

on m/,

2

The asymmetries (or equivalently, sin # e //) also exhibit significant sensitivity to nih- Examining Fig. 15, we notice a couple of things. First, At *b,c and ApB place contradictory demands on the Higgs mass: Ag prefers a \b,c very small mh while ApB prefer a heavier Higgs boson. There are a couple of At measurements — one from LEP and the other from SLD, and they seem to be in agreement. Second, since the best fit value for mh is low, this means that AFB will be off from its "SM prediction'"1. Finally, the

1

OU.D

1

1—[ —i

1

1

1

r 1 - i —1

i

'

'

'/

— LEP1, SLD Data -

80.5-

LEP2, ppLJSXB. 68% CL

> CD

—~^y /'

1

JL.

£ 80.4E

/

a

/ x&

80.3m H [GeV] 114/30QA on o OKJ.C. 150 1 C 30

000/ 170

/

Preliminary"

190

210

mt [GeV] Figure 18. A comparison of the indirect measurements of Myy and raj (LEP-I+SLD data) (solid contour) and the direct measurements (pp colliders and LEP-II data) (dashed contour). In both cases the 68% CL contours are plotted. Also shown is the SM relationship for the masses as a function of m ^ . The arrow labelled A Q shows the variation of this relation if a(Mz) is changed by \a. This variation leads to an additional uncertainty to the SM band shown in the figure. From Ref. 20.

d

Of course, if we compute the AjfB prediction with a large raj,, then AjfB will be OK, but a number of other well measured observables will deviate, most notably Ag and Mw< and the fit will become worse.

87

NuTeV measurement of sin2 9w (Figure 17) also seems to prefer a rather heavy Higgs boson. The results for the various asymmetries are often conveniently summarized as measurements of sin2 6^ (Fig. 19).

Final 0,1

0.23099 ± 0.00053 0,23098 ± 0.00026 0.23159 ±0.00041

A.(P_ had

-

-*-

Q fb

0.2324 ±0.0012 Preliminary

0,b fb i0,c A,fb

0.23212 ±0.00029 0.23223 ± 0.00081

Average 10

0.23150 ±0.00016

3

X2/d.o.f.: 10.5/5

>

CD E T ^ A a ^ 0.02761 ±0.00036

10 2i

m z = 91.1875 ±0.0021 GeV

ES23mt= 174.3 + 5.1 GeV

-i

0.23 •

0.232

p-

T

'

'

0.234

sin 2 e eff = (1 - g v /g A i) / 4 Figure 19. Comparison of several determinations of sin 2 # g? from asymmetries. Also shown is the prediction of the SM as a function of m/,. The width of the SM band is due to the uncertainties in Aa^d(Mz), Mz and roj. The total width of the band is the linear sum of these effects. (From Ref. 20.)

Summer 2003

r z [GeV] z > had [nb]

f|™

1

A|(P.) -,0

C

M^^p«^aa«m»««»»i«J

A,(SLD) sin 2 9^(Q fb ) m w [GeV] r w [GeV] sin29w(vN) Qw(Cs) •

f

y

y

^

.

^

^

,

^

j

B

i||i,»,||»,„||,„r,t„»|,

ggyBgMg^KHmg^iMMgMWgaWigg^p

2

10

10

10

M H [GeV] Figure 20. Preferred range for the SM Higgs mass m ^ from various electroweak observables. The shaded band denotes the overall constraint on the mass of the Higgs boson derived from the full data set. (Erom Ref. 20.)

89

6.8. A Higgs

puzzle?

Tables 1 and 2 contain a large number of observables, with different sensitivity to the Higgs mass rrih • Let us concentrate on a few selected observables with sufficient resolving power with respect to rrih and look at the range of rrih preferred by each one (Figure 20). We see that, as discussed previously, Mw and the leptonic asymmetries prefer a very light Higgs boson, while ApCB and NuTeV prefer a heavy Higgs boson. The current best fit value of rrih of just below the LEP limit arises from the combination of these somewhat contradictory measurements. Is this a problem? Chanowitz has made the argument that this discrepancy presents a "no-lose" case for new physics [35,36]: • One possible alternative is that ApCB is already indicative of new physics, and the discrepancy is real. However, any such new physics effect should not contribute too much to -R&, which seems to be in agreement with the SM. Proposed solutions to the puzzle include a heavy Z' boson with nonuniversal couplings to the third family [37-39] and mirror vector-like fermions mixing with the bji quark [40]. • Alternatively, the ApCB (and possibly the NuTeV) measurement could be wrong due to a statistical fluctuation or some unknown systematics. In that case, we should throw out the suspect measurements from the fit altogether. This significantly improves the goodness of fit, but at the expense of a much lower best fit value of rrih, whose upper bound from the fit becomes rrih < 120 GeV at 95% CL, leading to some tension with the direct LEP bound rrih > 114 GeV. In this scenario new physics is again needed in order to modify the radiative corrections and thus allow an acceptable rrih [41,42]. However, one should keep in mind that the tension between the indirect determination of rrih and the direct lower bound from LEP is statistically rather weak at the moment, and may be alleviated significantly, if, for example, the top quark turned out to be a little heavier, as suggested by the recent DO analysis of their Run I data [43] (which gives rrit = 180.1 ± 5.4 GeV). This is illustrated in Figure 21, which shows the effect of a l a upward fluctuation in mt on the indirect Higgs mass determination [44].

90

(5) Acc h a d -Aa

•0.0276110.00036 0.02768±0.00036

4-

MT = 179.4±5.1 GeV

CM

<

0

Excluded

Preliminary

mH Figure 21. Ref. 44.)

10 [GeV]

The effect of a la change in mt on the Higgs mass constraints.

(From

7. Testing for N e w Physics The precision data can be used to constrain or point towards new physics. To this end, there are different approaches. • Model-independent. If there is new physics, then upon integrating out the new heavy degrees of freedom we will end up with a set of higher dimensional operators in the low-energy lagrangian. The most model-independent approach, therefore, will be to write down all possible higher-dimensional operators consistent with the symmetries of the Standard Model. These operators will be suppressed by some scale A (presumably the scale where the new physics will appear) to the appropriate power. The size of these operators can

91

be parameterized by dimensionless coefficients of order unity. We can then use the precision electroweak data in order to constrain the dimensionless coefficients of the relevant operators [45-47]. The advantage of this approach is that it is completely model-independent. Unfortunately it is too complicated and time-consuming, and is rarely used for the case of more than a few operators. • Model-dependent. Alternatively, one can completely specify the new physics model of interest, e.g. minimal supergravity [48], minimal gauge mediation [48], minimal universal extra dimensions [49,50], little Higgs models [51-56], etc. One can then compute the contributions from new physics to the precision observables in terms of (41) supplemented by the new physics model parameters

&heory

=

&ree

^

G

^

M z )

[X

M

+

Af

({P})

+ A?P({p,

q})] ,

'(55) redo the fits and derive best fit values and constraints on {q} in a similar way as was done for {p}. In full generality (i.e. full leading order corrections to all precision observables), the method is again very time-consuming, so it is usually applied for a single (or a few) observables. • Oblique parameters. A third method, which sits somewhere in between, is the method of the so called S, T, U parameters (also called Peskin-Takeuchi parameters [57,58]). It amounts to making the approximation that the dominant new physics effects reside in the gauge boson propagators (self-energies). Notice that almost every electroweak observable involves some gauge boson propagator. So once we compute the new physics effects on the gauge boson propagators, we have essentially accounted for a whole class of corrections which appear in every observable, i.e. they are universal. In this method one neglects the process-specific corrections, i.e. the vertex and box corrections and the fermion (and Higgs) self-energies. Apriori we don't know whether this approximation is justified within a specific new physics model, however, there are many classes of models where it works. In scenarios with many new particles, there is a simple argument as to why the oblique corrections are most of the story - the gauge bosons couple to all particles charged under the corresponding gauge group, hence their self-energy corrections are enhanced by the multiplicity of the new particles. In contrast, the

92

flavor of the loop particles in the process-specific corrections is fixed by the flavor on the external legs. To summarize, the advantages of the method are: 1) simple calculations (there aren't too many relevant diagrams); 2) universality - i.e. the parameters are computed once and for all and affect all observables; 3) if S, T, U are added as free parameters to (41), their best fit values can be computed ahead of time by the fitting experts, and then supplied to model-builders, who in turn only need to learn to calculate S, T, U within a specific model and need not worry about the fitting procedure. 7.1. S, T, U

parameters

The S, T, U parameters are defined as [2] S

e

( M

_ ) nff%,(o) _ WzW T = ™ ^ - ^ ^ ,

z

Mir

(56)

M|

ae(Mz) _ U^(Ml) - H^*(0) 4?|c| * Ml 3e(Mz)fo ,m _ „w, (S + U)=

nwwy ^ ( Mw^ ) - nww ^(0) ' ~ ^>.

(57) •

(58)

where sfj ls defined in terms of the MS running couplings as s"| = 5 / 2 ( M z ) / ( 5 / 2 ( M z ) +
(59)

T = -0.15 ± 0.12(+0.09),

(60)

C/ = +0.32±0.12(+0.01)

(61)

for mh = 115.6 GeV (the parentheses show the change in going to m/, = 300 GeV). Best fit contours in the S — T plane (for U = 0) are shown in Fig. 22. We see that S = T = 0 is in the region of a light Higgs boson, i.e. in the absence of new physics, the fits prefer small mh, as we saw in the previous Section. However, a heavy Higgs is not out of the question, provided there are positive new physics contributions to T. 7.2. Constraining

new physics

scenarios

Previously we saw that precision electroweak data generally prefers a light SM Higgs boson (Figure 12), in which case any new physics effects

93 1.5

0.5

-0.5

—

r„ a „,, R., R Z*

had'

P

q

— asymmetries — —. - y scattering —

- Mw

—

Qw

all:M H =: 115 GeV all: M u = 1000 GeV -1.5 -1.5

0.5

1.5

Figure 22. la constraints on S and T from various inputs. The contours assume wih = 115 GeV except for the central and upper 90% CL regions (shaded, from all data) which are for m^ = 340 GeV and m^ = 1000 GeV, correspondingly. All fits assume U = 0. (Prom Ref. 2.)

appearing through 5, T and U should be small. However, this does not rule out a possible "conspiracy" [59,60], where the undesired contributions from a heavy Higgs boson in the SM are fortuitously cancelled by new physics effects. While a "conspiracy" is still a valid option in principle, the current thinking is that it would involve a large degree of fine-tuning and is therefore theoretically disfavored. As a result, we tend to like new physics models where no major deviations in the precision observables are expected. A simple and effective way to guarantee this is to have the new physics contribute only at the loop level, for example due to a conserved symmetry which distinguishes the SM particles from the rest. Some well

94

known examples are: supersymmetry with conserved i?-parity, Universal Extra Dimensions with conserved i^X-parity [49], and little Higgs models with conserved T-parity [61]. Conversely, models with tree-level contributions to electroweak observables (e.g. generic little Higgs theories) tend to have trouble with the data. 8. Concluding Remarks Overall, the SM is in good shape, and the agreement between theory and experiment has in fact improved since the time of TASI-2002. The results in Tables 1 and 2 are from 1/03 [1], while the electroweak data shown at the TASI school was from 7/01 [16]. Some notable changes and recent updates are the following: • Table 2 lists the Summer 2002 value for Mw (80.447± 0.042 GeV), but in the Winter of 2003 a revised ALEPH analysis lowered the LEP average to 80.412 ± 0.042 GeV, which is closer to the SM best fit prediction and will lead to a small increase in the best fit value for rah• New measurements of Mw and m-t are expected soon from Run II at the Tevatron. A new preliminary DO analysis [43] of Run I data gives rrit = 180.1 ± 5.4 GeV, which again will raise the best fit value for rrih• The NuTeV results on deep inelastic scattering in terms of sin2 0w (0.2277 ± 0.0016) are 3.0cr above the global fit value of 0.2228 ± 0.0004. Possible explanations of the NuTeV anomaly include an unexpectedly large violation of isospin in the quark sea [62,63], the effect of an asymmetric strange sea [62-64], nuclear shadowing [65,66] or NLO QCD corrections [67,68]. • The status of the g^ — 1 anomaly after the 2002 result from BNL was still somewhat uncertain, depending on the particular theoretical analysis used to extract the hadronic contribution to the Standard Model prediction for g^ — 2. In addition, last year the CMD-2 Collaboration at Novosibirsk discovered that part of the radiative treatment was incorrectly applied to their e + e~ data (a lepton vacuum polarization diagram was omitted) which prompted a reanalysis of the e+e~ data [69]. This subsequently brought the theory prediction in better agreement with experiment: the deviation was 1.9cr (0.7er) for the e + e~ (r-based) estimates [70,71]. The final result based on negative muons has just been announced [72].

95 It is consistent with the previous measurements and brings the world average to a^exp) = 11659208(6) x 1 0 " 1 0 (0.5 p p m ) . T h e difference between a^exp) and the SM theoretical prediction based on the e+e~ or r d a t a is now 2.7a and 1.4CT, respectively. New results from K L O E and B A B A R may help sort out the theoretical predictions. If the discrepancy stands, a n obvious candidate for a new physics explanation would be supersymmetry with relatively low superpartner masses and large t a n / 3 [73,74].

Acknowledgments It is a pleasure to t h a n k the organizers of TASI-2002 (Howard Haber and Ann Nelson) for creating a very stimulating atmosphere at the school, as well as the participants for their enthusiasm and insight. I would like to t h a n k Andreas Birkedal for helpful comments on the manuscript. This work is supported in part by the US DoE under grant DE-FG02-97ER41029.

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IRA Z. ROTHSTEIN

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EFFECTIVE FIELD THEORIES

IRA Z. ROTHSTEIN* Carnegie Mellon University Dept. of Physics, Pittsburgh PA 15213, USA

These notes are a written version of a set of lectures given at TASI-02 on the topic of effective field theories. They are meant as an introduction to some of the latest techniques and applications in the field.

Contents 1

Lecture I: The Big Picture 1.1 A Toy Model 1.2 Matching On or Off the Mass Shell? 1.3 Non-Decoupling and Wess-Zumino Terms 1.4 Types of EFTs 1.5 EFT and Summing Logs 1.6 Heavy Quark Effective Theory 1.7 The Method of Regions 1.8 A Caveat About Scaleless Integrals 1.9 Power Counting 1.10 The End of Our Calculation 1.11 EFT and WS 1.12 The Force Law in Gauge Theories

105 105 115 116 120 121 125 128 130 131 131 132 133

2

Lecture II: Non-Relativistic Effective Theories 2.1 The Relevant Scales and the Free Action 2.2 Interactions 2.3 The Multipole Expansion and Ultra-Soft Modes 2.4 Soft Modes 2.5 Calculating Loops in NRQCD

135 135 138 139 142 145

*Work supported by DOE contracts DOE-ER-40682-143 and DEAC02-6CH03000. 101

102

2.6 2.7 2.8 2.9 2.10 2.11

Matching at Subleading Orders in D Matching at Subleading Orders in a Summing Logs: The Velocity Renormalization Group (VRG) The Lamb Shift in QED States in the Effective Theory Summary

Lecture III: Effective Theories and Extra Dimensions 3.1 UV Fixed Points: A Theory Can Be More Than Effective 3.2 Naive Dimensional Analysis 3.3 The Force Law in d Dimensional Yang Mills Theory 3.4 Grand Unification 3.5 Orbifolding 3.6 Classical Divergences as UV Ignorance 3.7 Effective Field Theory in Curved Space 3.8 Unification in AdS5

147 149 . . . 153 157 160 161 162 163 165 166 170 173 177 178 181

It is a little known fact t h a t the Pioneer spacecraft launched in the mid-seventies contains a d a t a repository with a s u m m a r y of h u m a n scientific knowledge. It is interesting to ask what would h a p p e n if an alien civilization found this satellite and studied its contents? Let us furthermore suppose t h a t while these aliens were/are greatly advanced technologically, they are not terribly well versed in basic research, mainly because of the alien government's short-sightedness 8 , and are thus very interested in the free knowledge t h a t we are offering them. One of the pieces of information in the d a t a repository is a copy of the particle d a t a group booklet (the hand sized one, not the tome). T h e booklet is handed over to the relevant scientists, and a massive project is launched to decipher the information. After a couple of years of going over the d a t a and a copy of Bjorken and Drell, which was also on board, the aliens are up to speed and ready to test our theories. They've come to the conclusion t h a t we claim to have a well defined theory of strong and electro-weak interactions u p to the TeV scale (let's not quibble over anachronisms), and t h a t while we don't know exactly what happens at the TeV scale there is something t h a t unitarizes the theory above t h a t scale. They also conclude t h a t our understanding of gravity is very limited. In particular, while we know how to calculate q u a n t u m corrections at energies which are small compared to the Planck scale, above the Planck scale we have no idea what is going on. a

We know that without basic research they probably would never have achieved space travel, but I'll invoke suspension of disbelief at this point.

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After some period of intense investigations, the aliens decide to test the theoretical predictions based on what they learned from us. A proposal is put forward by a prominent experimentalist for a high precision measurement of the level splittings in Hydrogen. The proposal is summarily rejected based on a review written by a top alien theorist. The rejection was based on the referee's argument that it is silly to try to calculate loop corrections in the human's field theory. The referee's justification for this statement is that given that gravitons run through the loops and the fact the integrals include momenta above the Planck scale, where we know the theory breaks down, there is no way that one can expect that the results will be correct. Moreover, even if gravitational couplings are suppressed by the Planck scale, there are power law divergences which can compensate for these suppressions, so it seems that any attempt at calculating loop corrections would be fruitless. Fortunately, for us, there were some wise aliens working at the funding agency, and they said, "let's for the moment ignore gravitational effects, and see if the data agrees with what we get using just electro-magnetic effects". So the experiment was done and the theory did indeed meet with great success. So where did the referee go wrong? Well, to be fair, the referee was not completely wrong. There is a kernel of truth to its claims. It is true that the theory breaks down and that we should indeed pause to think about the results of any calculation which involves loop momenta above the Planck scale (the scale where the theory is no longer reliable). Since we know we are not correctly portraying the physics at the high scale, whatever the result of the loop integration may be, it will have to be treated in some way so as to remove the dependence on the UV physics. Indeed, the loop integrals will, in general, be divergent and need to be regulated. But there is no preferred regulation procedure b and, therefore, the numerical value of the integral is ambiguous. However, the point the referee missed is that if we are not too ambitious, this is not a problem. The short distance physics, which is not properly accounted for in the loop integrals, looks local at low energies. Another way of saying this is, that if we think of the loop integral in coordinate space, the part of the integral from which we know we are getting nonsense can be contracted to a point and therefore looks like an insertion of a local operator into our Feynman diagram. This means that all of the effects of the unknown short distance physics can be

b

However, a prudent calculator would choose to use a regulator which preserves the symmetries of the action, though this may not always be possible.

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mimicked by adding some new operators into our Lagrangian (as we will see these operators may be of arbitrarily high dimensions). That is to say, all the effects of the short distance physics can be accounted for by either shifting the parameters in our renormalizable Lagrangian, or by adding higher dimensional operators which are suppressed by the UV scale, Muv • Since we needed to perform measurements to fix those parameters anyway we still have predictivity as long as we truncate the Lagrangian at some order in powers of 1/Muv- So we see now in what way the referee was right. It's not that the unknown effects of the short distance physics are parametrically small in any way, indeed these effects are very important as they contribute at order one to the parameters in the renormalizable part of the Lagrangian. It's that all the effects go into making up the electron mass and all the other low energy parameters of the theory.0 If you changed the short distance physics it could, and would, change the values of the low energy parameters. Thus, as long as we are not too ambitious, we don't need to worry about the fact that our integrals diverge! Who cares if they diverge, we didn't expect to get the answer right anyway. So we should really call our field theories "effective" field theories. This moniker warns the consumer that it should be used with caution, as at some scale the theory no longer makes any sense. What that scale is depends upon the interactions one is trying to describe. Let's for the moment consider a universe where the effects of short distance physics could not be absorbed into low energy parameters. We couldn't calculate anything without knowing the complete theory of quantum gravity. While, this line of reasoning seems almost obvious now, thinking in terms of effective field theories (EFT) only became generally accepted in the last few decades. Indeed, from an EFT point of view, renormalizability, which was a standard benchmark for acceptable quantum field theories not that long ago, d is no longer relevant, unless we're interested in theories which are supposed to correctly describe all the short distance physics. Does this mean that field theory as a mathematical construct is incapable of being a "complete" theory? Certainly not. There are many field theories which are valid to arbitrarily high scales. The classic examples being asymptotically free theories such as QCD. These theories possess trivial UV c As we will discuss at length in the text, the number of such low energy parameters is dictated by the desired accuracy. d A s far as I know the only text books prior to those by Peskin and Schroeder [2] and Weinberg[3], which espoused the point of view described above was Georgi's book on weak interactions [4].

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fixed points, which allow for the cut-off to be taken to infinity. In fact, the necessary criteria for completeness is the existence of a UV fixed point, and taking the cut-off to infinity is called taking the "continuum limit", following the lattice terminology. The reasoning behind this criteria is that at the fixed point, the correlation length £, (read inverse mass) diverges, and thus one can study physics at arbitrarily large distances compared to the cut-off (the inverse of the lattice spacing a), which is equivalent to saying that the cut-off can be taken to infinity since lattice artifacts will be suppressed by powers of £/
1. Lecture I: The Big Picture This lecture is meant to give a global overview of effective field theories. The first part of the lecture in an introduction to the basic idea and a tutorial on the matching procedure, all discussed within the context of a simple toy scalar model. I then categorize the various types of effective field theories into two main headings depending upon whether the theory admits a perturbative matching procedure. Most of the rest of the lecture is dedicated to some sample theories where we can match. I will assume throughout these lectures that the reader has a basic understanding of renormalization.

1.1. A Toy

Model

Let us now try to be more quantitative by starting with a simple toy model. What we would like to illustrate is that all the effects of ultra-violet (UV) physics may be absorbed into local counter-terms and furthermore, that all of the physical effects of the UV physics are suppressed by powers of the UV scale.6 To do so we will work with a model of two scalar fields, <&L and e

This idea is usually referred to as "decoupling" as first codified in [5].

106 &H, with mji S>TOL,in four dimensions. The interaction Lagrangian will be given by

~Lint = ±*l + | M | + ^l&H

+ |$ 4 H-

(1)

It is typical, in most pedagogical treatises, to impose a discrete symmetry to forbid tri-linear interactions, as such terms introduce dimensionful couplings, which complicate the systematics. However, such interactions will illuminate certain ideas which otherwise would not arise until higher orders in the loop expansion. The usual problems with tri-linear theories arise from issues of vacuum stability, but for our purposes these issues will not be an obstruction. I will assume that all the couplings, except for mi, take on their "natural values". Here I use the term "natural" in both the sense of Dirac and t'Hooft. That is, dimensionless couplings should be of order one (Dirac) and dimensionful couplings should be of order of the largest scale in the problem, unless a symmetry arises in the limit when the coupling vanishes (t'Hooft). Thus, we take A2 ~ 1 and Ai ~ m # . Furthermore, since there is no symmetry protecting the light scalar mass, choosing m^
= Full Theory Result - Effective Theory Result.

(2)

In a sense what the left hand side of this equation is telling you is exactly what the effective theory is missing. Since the effective theory is designed to reproduce the IR physics of the full theory, SLEFT will have a well defined

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Taylor expansion around p — 0. Thus the holes in the effective theory can be plugged by adding local operators. If this seems a little obscure at this point don't worry as we will discuss many concrete examples of this matching procedure below. Notice that matching each n-point function will generate distinct local operators in the effective theory, with higher n-point functions obviously generating higher dimensional operators. Thus the number of n-point function one needs to calculate depends upon the desired accuracy. The more accuracy we want the more operators we must keep, as each operator of successively higher dimension will be suppressed by additional powers of the UV scale. Let's see how this works in our toy theory. Consider the amputated four point function for the light fields. At tree level, there are four diagrams which contribute to this process. One is the usual contribution involving only the light fields gives — i\o, and the other three have intermediate heavy scalars as shown in figure 1. These latter contributions are given by

Figl =

-i\f

+s

{t,u).

(3)

°H

Figure 1. Tree level contributions to the full theory four point functions involving the heavy scalar (thick lines).

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After expanding (3) we can see t h a t the four point function can be reproduced, u p to corrections of order p2 /m2H by using SLM

= C { ^ .

(4)

Here I have established notation for the coefficients (called "Wilson coefficients" ), the subscript counts the number of fields while the superscripts denote the order in the expansion in coupling and derivatives (more generally powers of 1/mjj), respectively. We can now see clearly what was emphasized in the introduction. As far as a low energy observer is concerned this looks like a simple <&4 theory with a coupling t h a t we had to fix via experiment anyway. T h e low energy observer would be completely oblivious to the UV physics unless the experiment were sensitive enough to see the deviations from $ 4 theory due to p/mn corrections. If we wanted to, we could add additional operators to ensure t h a t we correctly reproduce the higher n point functions as well. Suppose the low energy observer upgrades the machine, either by raising the energy or luminosity, to the point where the d a t a can no longer be fit by pure 0 4 theory. We know exactly what to do to correctly reproduce the data, simply add the higher derivative interaction SLint

=

J 77 • (5) mzH 4! Other operators with two derivatives and four fields can all be p u t into the form above via integration by p a r t s and use of the equation of motion. T h a t this latter manipulation is allowed will be discussed later. As the energy of the experiment is further raised one needs to include higher and higher derivative interactions to reproduce the data. Notice t h a t until we get to high enough energies to actually produce the heavy scalar we will be in the dark as to the true underlying theory. This is because in principle there will be an equivalence class (called a "universality class") of theories which look the same in the infra-red. As we fit more and more of the couplings of the higher dimension operators we would be able to eliminate theories, but removing all ambiguities would be difficult without including some other selection criteria. So far, we've shown t h a t to tree level we can reproduce the full theory by adjusting coefficients in an effective Lagrangian, this is called "matching" . This matching procedure can be carried out to arbitrary order in the couplings and p/ran- Indeed, suppose we wish to match at higher orders in the couplings. T h e matching procedure now takes on a new twist, but the

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idea is the same. We Taylor expand the full theory result and compare it to the result in the effective theory. We then modify the effective theory so as to get the answer right. I should mention at this point that in most theories the loop and coupling expansions coincide. But in our toy model, because of the Ai coupling this is no longer true. Nonetheless we can overcome this issue by assuming that (Ai/m# = Ai) is of order A/A~O- In this way all tree level contributions are of order Ao — A2 — A3. This is why I labelled the (i 2) Wilson coefficient above C\ . Returning to our calculation, let us see if we can reproduce the full theory at the next order in the couplings, beginning with the two point function. There are two leading order (dimension four) operators which get renormalized when calculating the two point function. The mass m\<&\, and the kinetic energy $ i D $ i . The one loop contributions to the two point function are shown in figure 2. Diagram 2a is the usual contribution found in $ 4 theory and is given by ^. „

Pig2a

i\0m2r

(\

, ,m? .

- = i^U-

ln(

#

A

)+1

,„.

(6)

)

where I will use the notation i = i - 7 + ln(47r). e e

O 2a.

_Q 2b.

(7)

Q 2c.

_2_ 2d.

Figure 2.

One loop contributions to the full theory two point function.

110

The diagram in 2c. gives the same result as 2a., but with the replacement (mi, Ao) —> (mu, A2). Both of these contributions are momentum independent and thus only renormalize the mass. Figure 2b, on the other hand, will have non-trivial momentum dependence. We expect to have a cut beginning at (ran + rriL)2, and thus this contribution will be analytic in the neighborhood p2 ~ m2L. Doing the one-loop integration gives

^•=M\-^t)+i)

(8)

where I have expanded in I/ma, and kept only the leading term. Finally, there is a contribution from the tadpole diagram 2d which is given by

Fig2d. = - J ^ 4 ( ' - m ( 4 ) + 0-

0)

Now let's calculate in the effective theory. We must note that to be consistent we must use the 0(A) effective Lagrangian which we generated at tree level when we calculate on the effective theory side (the aforementioned twist). This is in contrast to the tree level matching where there is nothing to calculate on the effective theory side. At leading order in l / m # , the only relevant operator is the <&\ term and so we find that the effective theory gives the same result as Fig 2a. except now the coupling is not Ai but Cf , 0 ) . To match we take the renormalized (here in MS) full theory result expanded around small {rh2L,p2 = p2 /m2H), and subtract the renormalized effective theory result, finding C (i,o)

1 32TT 2

A2mk(l-ln(^))

2

2 2

ta +i ,2

^( >"<^)- - 5< ^ » (10) To insure that perturbation theory is well behaved, we should choose [i to minimize the large logs. This scale is called the "matching scale". Intuitively we would think that this scale should be the high scale m # . Given that the effective theory is supposed to reproduce the infrared physics, we would expect that all the dependence on the low energy parameter m^ should be identical on both sides of the matching equation, thus leaving all logs in the matching independent of mi- However a glance at our results shows that choosing \i = m # does not get rid of all the large logs. Clearly we have done something wrong. The problem stems from the fact that we

Ill

have kept some terms which we had no business keeping. In particular, diagram 2d is power suppressed relative (remember that Ai scales as run) to the other diagrams. Thus, to be consistent we need to expand our full theory result for diagram 2b. to one more order. So we should make the replacement

A? m\ ,_ m\

F i g 2 b . ^ ( 8 ) + i - ^2 (( - f2 l n - f ) . '16TT rn H m\{

(11) 1,0)

The log term will exactly kill off all of the large logs in Cm'

«0, = ii« 1 + 5 ) + A^-

, leaving

(12)

Notice that this correction wants to pull up the light scalar mass to the scale mn since Ai c± mjj. This is just what we called "unnatural" in the sense of t'Hooft, since as m^ is taken to zero we do not enhance the symmetry. Keeping mi,
=

^MS

2

+ 0(\/(32n

))(m2H),

(13)

and we will need to delicately choose m2LMS to cancel the large 0{m2H) piece. The momentum dependent terms will simply shift the field normalization in the effective theory with Cp = ,» J* . If we wish to go to higher O^TT Tit TT

orders in the l / m # expansion we will generate the operator

02=cf2)$L^UL.

(14)

"H

(1 2)

Exercise 1.1: Calculate C\ ' Now let us consider the four point function. A partial list of the relevant full theory diagrams are shown in figure 3. The diagrams with external leg corrections will automatically be reproduced by the effective theory, since we already matched the two point function. In the effective theory, there will be only one diagram, and it is identical to diagram 3c except that at each vertex we use the coupling C\ . By considering the combination of

112

3a.

3b.

3d

3c.

-

3e.

Figure 3. functions.

A partial list of the one loop contributions to the full theory four point

couplings coming from (C\ ' ')2, we can begin to see that we will indeed reproduce diagrams 3a, 3b and 3c. However, we can also see that this diagram will not be able to reproduce the log of m^ from diagram 3e. To match this log we will have to first reproduce the six point function at tree level. Then diagram 3e will be reproduced by tying together two external legs. You can see how this will happen by noting that heavy field propagators can be contracted to a point in the full theory graph. Diagram 3d will have no partner in the effective theory and will contribute ( in the

113

s channel), after renormalization and expansion in s = -%-,

We see that since we are matching at the scale /x = m # , this diagram will not contribute to C\ ' , at a least in the MS1 scheme. The other channels give similar contributions.

Exercise 1.2: First draw all the remaining diagrams which contribute to the four point function in the full theory. Then show that all the dependence on ln(mi) cancels in the matching for the four point function. Hint: By working at zero external momentum the calculation greatly simplifies. What happens if we go to first order in p2/m%, staying at one loop? At this order when we calculate in the effective theory, we will need to include the graph depicted in figure 4, where the circle represents an insertion of the operator (5). The integral for this Feynman diagram will have a contribution which is quadratically divergent d v / (2TT)4

D

(p2 - ml + ie)((p! + p2 + p)2 ~ m% + ie)'

(16)

Figure 4. One loop contributions to the effective theory four point functions, with the dark circle representing an insertion of the derivative operator (5).

Now, suppose we regulate this divergence with a cut-off, then this contribution will scale as A2 /m2H. What order is this in our momentum expansion? If it's order one, it would seem that we have lost all hope of calculating in a systematic fashion, since no matter how many powers of ran may suppress a given operator, it may give an order one contribution

114

if we stick it into a multi-loop diagram. This would be a disaster. However, this reasoning is misleading. The fact that something can't be right can be immediately seen by working with a different regulator. Suppose we work in dimensional regularization. In this scheme, there are no power divergence as they are automatically set to zero [6]. It had better not be that our answer depends on our choice of regulator, otherwise we're clearly lost. So what's going on? Which regulator is giving us the correct result? The answer is, both. The difference between the two is "pure counter-term", which means the discrepancy in the two results can be accounted for by a shift in the couplings of local operators. Since the couplings are measured, and not predicted, there is no physical distinction between the two results. The only difference is in the bookkeeping, that is, the counter-terms in the two schemes differ but that's it. The nice thing about dimensional regularization is that we don't have to worry about mixing orders in l/rnn- We can see that if we use a cut-off and take it to be of order m # , then operators which are formally suppressed will "mix up" with operators of lower dimension.f But if we stick to dimensional regularization, then powers of IjmH are explicit, as they only appear in coefficients of operators (modulo the caveat about Ai in our toy theory), so we can read off the order of a loop diagram immediately. Thus, in general it is wiser to always work in "dim. reg." when doing effective field theory calculations. This is not to say that the cut-off doesn't have its time and place. In particular the cut-off makes quadratic divergences explicit, whereas in dim. reg. they show up as poles in lower dimensions. Another interesting point which arises from the calculation of the diagrams in figure 4 is the fact that there are divergences which can only be subtracted using counter-terms for operators with dimensions greater than four. The effective theory is non-renormalizable, an anathema in ancient times. That is, there are new UV divergences in the effective theory that arise in each order of perturbation theory. These divergences were not there in the full theory. This should not surprise us, since the effective theory is only built to mock up the IR of the full theory, not the UV. Thus the UV divergences will not cancel in the matching. This isn't a problem, we simply choose a scheme, and subtract. In the effective field theory approach we should be careful with out choice of scheme. If we're not careful we could mess up our power counting by introducing a scale. So we should choose a nice simple mass independent scheme like MS. I will have more I am using term "mix" in the technical sense of operator mixing.

115

to say about this later on in these lectures. Notice that the matching coefficient will clearly be prescription dependent, but any such prescription dependence will drop out in low energy predictions. Exercise 1.3: Calculate the anomalous dimensions of the dimension six operator $ | D $ | to one loop. The additional UV divergences which arise in the effective theory correspond to IR divergences in the full theory. This is easily understood from the fact that we have taken m # to infinity in the effective theory. So the "IR" divergence ln(m_ff) now looks UV. Thus effective field theory allows us to resum IR logs in a simple way using standard renormalization group techniques. We will see an example of this in the coming sections.

1.2. Matching

On or Off the Mass

Shell?

Before moving on I'd like to discuss one more technical aspect of matching. In particular, it's important to understand that it doesn't matter whether or not one chooses to match with external states on-shell or off-shell. The reason is that putting external lines off-shell only changes the IR behavior of the theory, and since the effective theory is designed to reproduce the non-analytic structure of the full theory, all the information regarding the virtuality of the external lines cancels in the matching. This can greatly reduce the amount of work one has to do to match. For instance, if one is not interested in matching operators with derivatives, then one could choose the external momenta to vanish identically, greatly simplifying the necessary integrals. Also if one matches off-shell one can induce operators in the effective theory which vanish by the leading order equations of motion. For example, in our toy model we could induce $ ^ 0 $ ^ . However, it is quite simple to show that we can eliminate this operator via a field redefinition, which are known to leave S-matrix elements unchanged due to the equivalence theorem [7]. The proof [9] is inductive. Suppose we have set all operators of order {p/rriH)n which vanish by the leading order equations of motion (by leading order I'm referring to the momentum expansion) to zero. If there exists an operator at order (p 2 / TO ir)™ +1 of the form F ( $ L , <J?#)n£, then we make the field redefinition 5$L = -F($L,QH)

(17)

so that variation of the leading order Lagrangian 8LLO = {5§L)U§L

(18)

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exactly cancels the offending term. Notice that since F is order n +1 in the derivative expansion the lower order operators remain unchanged but we do induce a change in the subleading set of operators, i.e. at order n + 2. However, since in an effective theory we generate all possible operators consistent with the symmetries, all this redefinition will do is to shift the value of some matching coefficients. 1.3. Non-Decoupling

and Wess-Zumino

Terms

Let us consider the case of a chiral gauge theory. These are theories for which a fermion mass term is disallowed by gauge invariance. We know that if the theory is to be unitary and gauge invariant, the particle content must be such that all the gauge anomalies arising from triangle diagrams sum to zero. The classic example of such a theory is the standard model itself, where the fermions get their masses from the Higgs Yukawa couplings. Taking the fermion masses large while holding the symmetry breaking scale fixed corresponds to the limit of a large Yukawa coupling A. Since the coupling is proportional to the mass, we might expect that low energy observables (below the scale of symmetry breaking) could be enhanced by a large mass. The classic example of this "non-decoupling" arises in K — K mixing. After integrating out the top quark one generates a term in the AS = 2 Hamiltonian Ht=m2t^r,204,

(19)

where O4 is a four quark operator. A similar effect shows up in the p parameter, which is a measure of the ratio of charged to neutral currents. These effects would disappear in the limit where the bottom and top quark become degenerate, the former vanishes as a consequence of the GIM mechanism. Both of these effects arise due to SU{2) breaking. If we took the limit where both the bottom and the top quark masses become equal and large, then we would expect that the effects of this heavy pair could all be absorbed into SU{2) symmetric operators. The thoughtful reader might be confused by this last statement if they are familiar with anomalies. Suppose that one of the SU(2)L doublets is much heavier than the weak scale, and we wish to integrate it out. The low energy theory would then have anomalous matter content and would apparently no longer be gauge invariant. Actually things can look even worse. If we start with an even number of SU(2)L doublets and integrate out one such pair, then naively, the partition function would vanish due to

117

the non-perturbative anomaly [12].g Now it should not come as no surprise that at "low energies" we do not have manifest gauge invariance. Nor should we be worried about this fact, since the lack of gauge invariance will only manifest itself in a pernicious way (a breakdown of unitarity) at energies near the masses of the gauge bosons. Nonetheless we should be able to add a tower of higher dimensional operators which will allow us to calculate sensibly at energies above the masses of the gauge bosons. The question is, what scale suppresses the effects of the fermions in these higher dimensional operators? The answer is the symmetry breaking scale v, and not the mass of the fermion. Integrating out a heavy fermion will generate a set of higher dimensional operators such as those arising from the graph in figure 5. This graph will generate a whole string of operators once

Figure 5. by 1/v2.

A diagram which contributes to a dimension six operator which is suppressed

we expand it in momentum. Consider the contribution to the dimension 6 operators, which are suppressed by l/MyV. Just from counting vertices and dimensional analysis we see that it must scale as 1/My-V oc A 2 /m 2 = 1/v2. Thus we can't write down a local effective theory which is valid in the region v < E < Xv, at least with these variables. The trouble all starts from the fact that fermions are getting their relatively large masses from dimensionless couplings. Even so, we should be able to calculate in a reliable fashion up to the scale v even with our low energy theory which appears anomalous. Imagine we're taking the limit g
This anomaly arises as a consequence of the fact that TT4(SU(2)) = Z2 and a topologically non-trivial gauge transformation multiply the partition function by a factor of -1 for each SU(2)L doublet.

118

I will only briefly summarize. The serious student may wish to spend some time looking at the original work [13]. If we allow for A S> 1, then the heavy fermions will be strongly coupled, but this should not stop us from matching onto an effective theory. We know that we will not be able to match perturbatively. However, the large Xv limit lends itself naturally to a derivative expansion which will keep our matching under control. We will simplify the problem by considering a toy Yukawa theory with SU(N)i x SU(N)R symmetry. The field content is a left (right) handed fermion in the fundamental and a Higgs in the bi-fundamental, which transform as fa -> 9L^L,

i)R - • 9RfpR,

$ - • gL$gR,

(20)

LW>L + ^RIWR

-

ML^R

-

WI&HL,

(21)

where the Higgs doublet, $ is written as $ = vU, with U being a unitary matrix. That is we have frozen the modulus of the Higgs field to simplify the problem. Allowing the modulus to fluctuate wont change our conclusions. Now we would like to match on to an effective theory without the heavy fermions. The matching involves only calculations in the full theory since the are no graphs in the effective theory. If we were working in the standard model and wanted only to integrate out the top quark, this would no longer be true. Thus matching corresponds to formally do the path integral over the fermions then expanding the effective action W(U) = -iIn

f d[ipL}d[i)R]exp fi f d4xL)

(22)

in powers of l/(Xv), keeping only the terms which do not vanish in the A —> oo limit. Here I will only sketch how this calculation is done. The trick is to perform a change of variables to absorb the Higgs field into a gauge field. This is accomplished by the redefinition NL = U^L,

NR = Vfl,

(23)

which results in L = NL{i$ - 4L)NL + NRi$NR

- Xv(NLNR

+ NRNL),

(24)

where AL = U'd^U. The advantage of using this form is that a derivative expansion corresponds to an expansion in the number of external fields.

119

We may write the new effective action denned in terms of an integral over the N fields, as W with W = W - U(U)

(25)

[ # L ] = J(U)[dNL].

(26)

since

Where J{U) is the Jacobian for the change of variables (23). W can be calculated using the usual Feynman diagram techniques, and, in general, contains many interesting pieces, such as the famous "Goldstone-Wilzcek" current [14] which can endow solitons with fermion number. However, in this model, all of these terms will be invariant under the anomalous symmetry, so we will not concern ourselves with them at present. We will be mostly concerned with the Wess-Zumino term which resides in the Jacobian [10]. This term can not be written as the four dimensional integral of an 5(7(2) x SU(2) invariant density but can be written as an invariant five dimensional integral [11] wz =

240TT 2

d4x

dx5eabcdeTr

&dauifidbUu*dc&uddUutdeu . (27)

U is an arbitrary (differentiate) extension of U into the x 5 direction which need only to satisfy U(x5 = 0, a;") = 1 ; U(x5 = 1, x?) = U{x»).

(28)

The reason that the interpolation is arbitrary is that the integrand is invariant under general coordinate transformations. Moreover, under a small variation of U, the integral is a total derivative whose value is fixed in terms of U(x><). Exercise 1.4: Prove that under small but arbitrary variations in U, the Wess-Zumino term changes by a total derivative, and thus depends only on the vale of U in four dimensional space-time. If we were to include the gauge fields, then a variation of the WZ term would exactly reproduce the usual ^vfXTF^Fpa contribution to the anomaly. It would also account for the non-perturbative anomaly [12] in the case where the gauge group is SU(2). The result for the gauged theory can be found in [13], and is slightly more complicated then what I have

120

described above. The non-local WZ term can be expanded in powers of v and the resulting theory will give well denned predictions in this standard model-like theory up to the scale v. 1.4. Types of EFTs I will separate EFTs into two types. Those for which the underlying UV physics is known and the matching can be done perturbatively, which I call "Type I", and those for which it is not possible to match, either because the UV physics is unknown, or because matching is non-perturbative, which I will call "Type II". Some examples of type II theories for which the underlying theory is unknown are • The Standard Model • Einstein Gravity • Any higher dimensional gauge theory11 whereas for the following effective theories • Chiral perturbation theory: Describes low energy pion interactions. • Nucleon Effective theory: Describes the low energy interaction of nucleons [15]. the underlying theory is just QCD, but the matching coefficients are not calculable, at least perturbatively. Examples of Type I theories are • Four Fermi Theory: For low energy weak interactions. In this case the underlying theory is the standard model, which in itself is a type II theory. • Heavy Quark Effective Theory: Describes mesons with one heavy quark [18]. • Non-Relativistic Effective Theories (NRQED, NRQCD): For nonrelativistic bound states such as onia or atomic physics [20,21]. • Soft-Collinear Effective Theory (SCET): For large light-like interactions such as those in heavy to light quark transitions [24], high energy jet processes [25] or quarkonia decays [26]. h

There are some higher dimensional SUSY field theories which are conjectured to have UV fixed points [16]. In these cases the theory is not effective, in that it is valid up to arbitrarily large momenta.

121

For type I theories, one may ask the question: "Why bother with an effective theory when we know the complete theory?". The answer is that in general the full theory can be quite complicated and going to an effective theory simplifies matters greatly. In particular, going to an effective theory can manifest approximate symmetries that are hidden in the full theory and, as we know, increased symmetry means increased predictive power. Furthermore, when the full theory contains several disparate scales (mi,mh) perturbation theory can be poorly behaved as typically one generates terms of the form a\n(mi/mh). When these logs become large, they need to be resummed in a systematic fashion in order to keep perturbation theory under control. Working within an effective theory simplifies the summation of these logs. The classic example of such a resummation arises in B meson decays, which I will now explore. 1.5. EFT and Summing

Logs

Let us consider the decay of a free b quark. This calculation is discussed in the text by Peskin and Schroeder [2] using slightly different language, but I will sketch it here for the sake of completeness . I will also simplify it some by ignoring operator mixing. At first sight there is no reason to believe that free quark decay has anything to do with the actual meson decay, but we'll come back to that point later. The tree level decay rate is proportional to the usual muon decay calculation, while at one loop we have diagrams such as those in figure 6a. This correction is finite and contains a large log of the form Diagram 6a = Ai—

m(mL/?7i|),

(29)

47T

The diagrams which do not traverse the W propagator do not contribute large logs and will not be of interest to us. If we are to save perturbation theory we must resum these logs. Indeed, the perturbative series for a generic Greens function of a theory which contains a large disparity of scales whose ratio is <5, will contain the series oo

n

n

C ln m

Giog = J2^ T, ^ ^ n—1

(30)

m—0

If we take (f* ln(<5)) to be order one, then we must resum all terms of the form (§£ ln(6))n. This is called the "leading log (LL)". approximation. This resummation as well as the "next to leading log (NLL) ( ( | £ \n(S))n~1)" and so forth are easily performed once we work within the framework of EFT.

122

6b. Figure 6.

One loop correction to heavy quark decay.

The way EFT makes the resummation of the logs simple is by converting the problem to one with which we are more familiar. We know how to sum logs arising from UV divergences using the renormalization group (RG), so all we need to do is to figure out how to relate the logs in (30) to RG logs. The logs in (30) can be thought of as IR logs in the rrib goes to zero limit, or UV logs if we take Mw to infinity. But this second limit is exactly the limit we take in constructing the EFT. In the four-Fermi theory Mw is taken formally to infinity, usually keeping only the first non-trivial term in the 1/Mw expansion. Let us see how the four Fermi theory reproduces the log in (29). The loop correction in the effective theory corresponding to those in diagram 6a is shown in diagram 6b, and is given by Diagram 6b = \

1

^ - ( -

47T

\e

Hml/v2)

(31)

Note that without doing any calculation we could have guessed the coefficient of the log in the effective theory from the full theory result. Since, as we discussed in the context of our toy model, the effective theory must reproduce the IR physics of the full theory, so the logs of m,b must be identical in both theories. Using this result we may match onto the effective theory. The Wilson coefficient of the four-Fermi operator is CiF(jj) = GF[l

+ X i ^ - (\n(Mw/m2b) A-K

+ l n K / M 2 ) + p)

(32)

Notice that we work in a mass independent prescription (usually MS) where

123

we subtract only the pole and possibly some dimensionless constants. Thus, the constant in (32) is scheme dependent. However, this scheme dependence is order 0(a), which is sub-leading in the LL approximation. To cancel this 0(a) scheme dependence we would need to go to the NLL, where the shift in the two loop anomalous dimension, generated by shifting schemes, will account for the change in the matching coefficient. Details of higher order calculations can be found in [17].

Exercise 1.5: Prove that the anomalous dimension is scheme independent only up to one loop. Also show that the beta function is scheme independent up to two loops. Assume that the coupling in the new scheme is analytic in the coupling in the old scheme, that is gl(g) = a±g + a2g2 + . . . Now in order to match we must make sure that series for C^F is converging. Thus we should choose fi to be order Mw, m which case dp — Gj?(l + A i " ^ ' p ) . In making this choice we have removed the large log from the matching, but if we are not careful the logs can still reappear. To see this we must consider the observable of interest. Using the optical theorem we may write the B meson decay rate as [18] T = Im [d4x— J

| ClF{n) I2 (B | T(O{F(x)OiF{0)

\ B),

(33)

TOB

Any fi dependence in the Wilson coefficient must be cancelled by the \i dependence of the matrix element. If we choose [i to be order Mw, then the matrix element will also be renormalized at this high scale. This would clearly be a blunder, since the physics responsible for the decay is all taking place at the scale irn,. This blunder will reprise the log, since it will now show up in the matrix element as ln(mt//x). To truly vanquish the log we want to renormalize the four Fermi operator at the lower scale m.(,. But we know how to do this. We use the fact that the bare operator, OB = ZOR, where Z includes both the operator and wave function renormalization, is fi independent ^ then since Z =

+ Pi9)§-g + 7 o 4 F ) OfF = 0,

(34)

Z^/ZoiF

loiF = ^g- i-S0iF

+ 4 x -S2J

(35)

124

where So4F = — |^if^ is the counter-term necessary to renormalize the four Fermi operator, and 82 is the fermionic wave function counter-term •

**

=

—

-

•

W

3 47T e

The solution to the RG equation for O^f is 0

»

4

= O 4 > 0 ) e x p [f™

^

^

2

) 1

#

),

07)

leaving r,

,

04F(mw)

^

r,

,

= 04F(mb)

J l n ( m ^ / A

ln(m2/A2)

(Al/(26<,))

(38)

where bo is the leading order beta function coefficient 11 — 2/3n/, and 7o 4 F = f^O^i = 2Ai — ^ ) - We are now free to evaluate the time order product, renormalized at the low scale mj without fear of generating large logs. Note that we can not claim to be accurate at order as. We can achieve this level of accuracy only if we run at two loops. Since at order as we must keep all terms of the form jj™ ln(<J)™_1. Thus keeping the constant in the matching really bought us nothing. The systematics follow the rule that to achieve accuracy at order an, you must match at order an and run at order n + 1. We have successfully removed that large and irrelevant scale Mw from our calculation. Doing so enabled us to easily sum the large logs. But we are not quite done removing scales. In the decay process itself there are two relevant scales, namely rriQ and AQCD, the scale of hadronic physics. So we would expect that in our matrix elements we could induce logs of the ratio of these scales. But this is not a large log, so should we bother integrating out the scale TUQI The answer is, of course, yes. Because if you recall in the introduction to this section we said that effective field theories do much more than just sum logs. They make approximate symmetries manifest. Just as the scale Mw was irrelevant for the decay process, so is TTIQ. Since rriQ 3> AQCD, the quark effectively acts as a static color source. As far as the light degrees of freedom are concerned, the heavy quark is a brick wall. If you increased its mass, the effects on the bound state would be essentially nil. Thus in the limit where the quark mass is taken to infinity, a new symmetry, "flavor symmetry", emerges. If we have more than one quark with TUQ 3> A, then to leading order in A/mg, the quark decay is flavor independent. Furthermore, the spin of the heavy quark should

125

decouple since the color magnetic dipole moment scales inversely with the mass, so we would expect the emergence of a spin symmetry as well in the large mass limit. 1.6. Heavy Quark Effective

Theory

In this section I will briefly introduce HQET. I wont go into detail as this is now textbook material [18]. But some of the concepts inherent in this theory will be necessary for the second lecture, so I will briefly recap the highlights and refer the reader to [18] for more details. What we would like to do is eliminate the scale ITIQ from the theory. However, we obviously need to keep the heavy quark field in the theory, so how do we keep the field and lose the mass? We begin by studying the dynamics of a heavy meson decay. As I said above the heavy quark should be static up to corrections of order A/TTIQ . We may write the heavy quark momentum as PQ

= "»««" +fcM,

(39)

where v^ is the quarks' 4-velocity ((1, 0) in the rest frame) and k^ is what's known as the "residual momentum" which is of order A. By studying the propagation of a heavy quark in a hadronic environment, It is easy to illustrate that the heavy quarks only role is to be a static color source Consider a heavy quark sitting inside a hadron being constantly bombarded by "soft" gluons(i.e. gluons with momentum of order A), as shown in figure (7). In the large mass limit we may simplify the intermediate propagators as follows Umrf + t + mchu w 7„(l + fl7. (rriQV + k)1 — m^ 2v • k

+

Q{k/m

}

(4Q)

so that the dependence on the heavy quark mass has disappeared. We expect the mass only to appear at sub-leading order in the mass expansion, and in phase space integrals. Furthermore, we also have

Mb) = «(*>) + 0{k/mQ)

(41)

so we may make the replacement 7^ —> v^ and the spin structure becomes trivial. In the real world where we have effectively two heavy quarks with mQ 3> A, charm and bottom, 1 we can see that in the large mass limit we 'The top quark decays too quickly to be relevant to this discussion.

126

mv

Figure 7. Scattering of a heavy quark off of gluons with momenta of order A.

should generate an SU(2)fi avor x SJJ{^)Spin symmetry, and this symmetry should be manifest in our effective theory. We can determine the action for the effective field theory by matching. We consider the full theory propagator, again splitting the momentum into a large and small pieces, and expand in the residual momentum

G™(p) = v

*

«

i{l+i,)

.

(42)

' mQi> + ft-mQ+ie 2v • k + ie Now we need to find an action which has the proper quadratic piece to reproduce this two point function. We do this in two steps: First we notice that the numerator of the effective propagator is a projector onto the large components of the heavy quark spinor. So we should split up the heavy quark spinor into "large" and "small" components. Then, to remove the heavy quark mass yet retain the field itself, we simply extract the mass dependence via a field redefinition [19], q(x) = e~imv-x (hv(x) + Hv(x)) ,

(43)

where

hv(x)=ei™-xl(l

+
Hv{x)=eimv-X\{l~i)q{x).

(44)

Here is where HQET differs from the usual effective theories where we remove entire fields from the action. The fields have labels v. A field with a label v, destroys a particle-* whose four momentum is fixed to be within J

If we were interested in anti-particles we would chose an opposite sign in the re-phasing.

127

A of mvp. That is, the field itself only has Fourier components less than A. This is in accordance with our physical expectations. Once we integrated out the hard modes, the only modes left have momenta of order A whose effect is only to jiggle the heavy quark. So we write the momentum of the heavy quark as p = TTIQV + k, where k is called the "residual momentum" and is of order A. Thus, derivatives acting on this rescaled field are of order A. We may now substitute this parameterization of the full QCD field into the Lagrangian to arrive at LQCD = q(il? ~ mQ)q

= V hv(iv • D)hv „

+ 0[

).

(45)

\mQJ

Notice the sum over velocities. It is needed because we should allow for arbitrary four velocities of the heavy quark, and remember that once we fix a label the momentum is fixed up to some small an amount of 0(A). Technically, this means that there is a super-selection rule, which is defined by the statement that states with four-velocities which differ by more than A exist in different Hilbert spaces. So to allow for differing velocities we have to "integrate in" the different sectors. If we are only interested in processes with no weak decays this point becomes moot since we can go to a frame where the heavy quark is static. However, if we want to look at for instance a b to c transition then a large "external" momentum is injected which drastically changes the velocity of the heavy quark. In the effective theory this is described by the flavor changing current, whose labels reflect the connection between different Hilbert spaces

J(x)=J2 ~KYhbv,{x).

(46)

v,v'

These type of label changing operators will play an important role in the next lecture. We performed a Taylor expansion of our tree level propagator, should we expect this replacement to work even in loop diagrams? The answer is of course, yes, and the reason is (not to flog a dead horse) that the part of the loop integral for which the Taylor expansion is ill suited comes from UV physics, which is local and can always be absorbed into counter terms. Technically, this can be see in a simple fashion. Take any full theory integral, / and write it as I = (I — IEFT) + I

EFT

(47)

128

If / is finite, then Taylor expanding and integrating commute and there is no work to be done, the EFT will reproduce the full theory to any desired order in the inverse mass. If the integral diverges, then all we need to do is differentiate it enough times with respect to the external momentum to make it finite and then perform the Taylor expansion. The value of the full integral will differ from the expanded integral by some polynomial in the external momentum, i.e. a counter-term. Using this type of argument inductively an all orders proof has been provided both for four Fermi theory [27] and HQET [28]. HQET is particularly simple because there is only one relevant IR region, k ~ AQCD • So we expect that Taylor expanding around small loop momenta will correctly reproduce the IR physics. Other theories may have more relevant IR momentum regions as will be discussed later on in these lectures. 1.7. The Method

of

Regions

This line of reasoning can greatly simplify matching calculations. Suppose we wish to match onto to HQET at one loop. The standard recipe is, take the full theory result, expanded in powers of k/rriQ, and subtract the value of the corresponding effective theory result. k But there is actually a way to make life a lot less complicated by utilizing some of the magic of dimensional regularization and asymptotic expansions [29]. Instead of first calculating in the full theory and then expanding, one can Taylor expand at the level of the integrand, assuming that the loop integrals are dominated by momenta much larger than the external momenta. By working in this way we will miss out on non-analytic pieces, but that's OK since we know that they cancel in the matching. How do we know that this method correctly reproduces the matching coefficient? One way to think about it is in terms of the "method of regions" [30]. The idea is that in dimensional regularization the integral will receive contributions from momenta of order the heavy quark mass ("hard modes") as well as momenta of order the residual momenta ("soft modes"). This is true even for divergent integrals since the contribution from modes much larger than the quark mass lead to scaleless integrals which vanish. Let us see how this works in a simple one loop example. Consider the two point function integral dnq /

1

(2TT)" (q2 + ie) ((pQ

1 + k - q)* - m2Q + ie)'

(48)

In addition one needs to worry about how the states are normalized in the two theories.

129

k is the residual momentum of order AQCD- TO extract the hard contribution we assume q is of order TUQ and Taylor expand the integrand up to order I/TUQ dnq x 2 g (i, n 2 ' 2 2 fr- >-*n. (27r) (q + ie) (q - 2p • q + ie) \ q - 2p • q + ie /

(49)

Whereas the soft part of the integral is found by assuming the loop momentum is order k. Keeping only the leading order piece in this region we find f dnq 1 1 Js = J (27r)ra (q2 + ie) 2p-(k~ q) + Ye' ^ which is exactly the HQET contribution. So if we can show that the sum of these two contributions is equal to I, then we've shown that the hard piece is exactly what we would call the matching. The full integral can be performed using text-book methods /

=I^(l-ln^ + 2+(^-1)ln(1^2))'

where p2 = 1 +

PQ 2

m

(51)

. The hard piece is given by

Q

The soft integral has denominators of differing dimensions, so it's helpful to use the identity

i -2^4

anbm

ry ^

I — F ,™-, JI T[n]T[m] 0 1

H /„ + i 26A)" O L \ \ n.-\-rr> (a

_

. ^

'

Then following standard techniques we arrive at k 2

-P«(i_+2_2HZ^)].

16-7T TTIQ \e

(54)

M

Expanding the full theory result (51) and comparing it to the sum of (54) and (52), we find agreement between the two. Notice that both 1^ and Is have spurious divergences which cancel each other. For more complicated theories, such as NRQCD, there are more regions to worry about, and one must be sure that they are all accounted for in order to correctly reproduce the full theory result. Furthermore, one must be careful when matching at two loops using the above trick. Doing so would necessitate keeping e dependence in the effective theory action [31,41].

130

Exercise 1.6: In the last example I have expanded in e before expanding in l/mq. Show that the equality holds true even if you don't expand in e by calculating the full result exactly and then expanding in I/TTOQ. The method of "Asymptotic Expansions" has been proven to work to all orders for both the large momentum and large mass limits for Euclidean momenta. However, there are no known counter-examples to two loops in Minkowski space.

1.8. A Caveat About Scaleless

Integrals

Suppose we chose to match exactly at zero external momentum. We are free to do this if we are not interested in corrections which are sub-leading p. It is often the case that at this point the integrals in the effective theory will be scaleless and thus vanish (this is true in HQET as well as NRQCD). However, this does not mean that we can conclude that the effective theory operator under consideration has no anomalous dimension. To calculate the running in the effective theory we need to account for the fact that the scaleless integral IEFT is giving zero because of a cancellation IEFT

= C (—

- )

.

(55)

Given any logarithmically divergent scaleless integral we can always perform this split algebraically [ ddk J J2^¥

1 _ f ddk ( 1 ~ J (2^)2 \(k2){k2-m2) 167T2 \euv

m2 \ ~ (fc )(fc 2 -m 2 )J 4

eIRJ

However, if and only if, we know that the full theory has no IR poles, can we correctly conclude that the anomalous dimension in the effective theory vanishes when we have scaleless integrals. When the full theory has no IR poles, the effective theory has no IR poles and since in the effective theory (for scaleless integrals) there is also a one to one correspondence of UV and IR poles we may conclude that the effective theory has no UV poles. This is an important point: We can't ignore dimensionless integrals in dimensional regularization in the effective theory unless we are sure that we have correctly accounted for all of the IR poles in the full theory.

131

1.9. Power

Counting

In most effective field theories the power counting is as simple as keeping tr.ack of powers of the mass. However, this will no longer be true in more complicated EFTs. Power counting is trivial in HQET, but I want to go through it pedantically, since such an analysis will bear fruit later. Once we've removed the heavy quark mass, there is only one relevant scale, A. All of the fields have support only over regions of order 1/A so we should expect that dimensional analysis will yield the correct power counting. Let's see if this is true. The leading order action, SHQET

Id4x

= v

'

J2hv(iv

• D)hv,

(57)

v

should scale as 0{1). Each field scales as A 3 ' 2 while the derivative scales as A. Finally, since the fields only have support over region of size A, d4x oc A~ 4 , the leading order action does indeed scale properly. In general, there may be more than one scale involved and the scaling of the fields is instead fixed according to the leading order action. All other operators in the action should scale as positive powers of A, since if there were operators with negative powers, power counting will be jeopardized.

1.10. The End of Our

Calculation

We now have all the ingredients to finish our inclusive B decay calculation. We started with a theory with three scales, Mw, vn-b and A. We eliminated Mw, ran down to nib and are now have the power, via HQET, to remove 77i{,. Removing this scale is crucial, even if we are not interested in the enhanced symmetry. To see this, we have to figure out how to evaluate the time ordered product (33). I will only sketch this final part of the calculation since it's is explained in details in [18]. The main point is that the non-local operator product can be simplified using an operator product expansion (OPE) much as in deep inelastic scattering (DIS) [8] (though the use of the OPE in the present context is not rigorously justified as it is in DIS). The TOP can be expanded into a set of local operators, schematically

/

d^xe-^xO{x)0{Q)

* &(
+ ... (58)

132

where q here is the large scale in the process (for us q ~ m^). The utility of the expansion is predicated on our ability to truncate it.1 As long as when we take the matrix elements of the RHS, we don't introduce a scale of order q upstairs, we can safely ignore terms of higher order in 1/q2. Now we can see that in the case of B decays we will run into exactly this problem since we will be taking matrix elements between B states. So that while each term in the expansion may be down by powers of q, i.e.TOj,,taking the matrix element will generate expectation values which scale with positive powers of 7715. The solution to this problem is simply to go to the effective theory, where the scale m,b is gone and all matrix elements scale only as A. How the OPE is performed in HQET is discussed in [18].

1.11. EFT and MS I want to emphasize the fact that using a mass independent prescription is crucial for this, as well as most other, EFT calculations. In fact, EFT and mass independent prescriptions go usually go hand in hand (lattice effective theories are exceptions to this rule). To see the need for this marriage, suppose we're trying to calculate the running of a gauge coupling. Our prescription can not affect any physical prediction, but a poor choice of prescription can lead to an ill behaved perturbative series. Let's compare and contrast the contributions of a massive fermion to the QED vacuum polarization in the MS and momentum subtraction (ms) schemes.

n

r/=^(^-p2v)(rf"(i-^)in[m2

n ™ = -^{V»VV

- P29^)

/ dx x(l - x) In

p2x{1 x), M2

•p2x(l

M2x(l~x)

(59) (60)

where M is the scale at which we have chosen to define the coupling, so that in the ms scheme IL(—M2) = 0. For p2
133

difference is that the ms scheme is a "decoupling scheme" while MS is not. That is, in the MS massive particles do not decouple in the low energy limit. That is exactly why we should work in an EFT, where we remove the heavy particles by hand. This is the small price we have to pay in order the easily sum the logs. Indeed, in the ms scheme we will need to calculate as{—M2), given its value at some other momenta at which it was measured MQ, and solving the RG equations in this scheme is more difficult. I should also say that even if you work in a mass independent scheme you don't have to use an EFT, it's just that without EFT calculations become more cumbersome since the low energy observables will receive contributions from all the heavy fields which re-introduce large logs. But in the end if you are careful you should always get the same result. Using an EFT will just make your life a lot easier. 1.12. The Force Law in Gauge

Theories

When working in an effective theory, or any theory for that matter, it is easy to get confused if one does not work with physical quantities. Coupling are in general scheme dependent and have no physical meaning, thus it is always best to have in mind some measurable observable when doing a calculation. Often in these lectures instead of talking about measuring a coupling, I will instead talk about measuring the force between static sources. In this way we can be assured that our result is in no way ambiguous. This force or potential is intimately related to what we normally call the coupling, but is a measurable quantity and thus scheme independent. We begin by showing that the static potential can be extracted by calculating the expectation value Z[J] = (0 | T{e-%e$ •^•"(xM/.c*) | o)

(61)

for a stationary charge distribution which turns off adiabatically at ±oo. then using standard arguments (see [2]) lim Z[J] oc eiE^T,

(62)

T—>oo

where E(J) is the vacuum energy in the presence of the source J. It is left to the reader to show that Z[J] is nothing but the expectation value of the gauge invariant time-like Wilson loop W(r, T) = (0 | Tr (p{e$ A»d^)^j | 0) where P stands for path ordering.

(63)

134

For a free Abelian theory, i.e. no matter, things simplify tremendously since the action is quadratic in the gauge field, and hence the functional integral can be done exactly, resulting in f dixd4yJ^(x)A^(x,y)J,'(y)

Z\J] OC e~^

/g^

where AM1/ is the photon propagator. We will be interested in the energy of two static sources, of opposite charge, separated by a spatial distance r. So that Mx)

= e5^5\x

- r) - e^0<53(x)

(65)

then it is straightforward to show that ^

(66)

= 4^-

Diagrammatically, one can arrive at result by using Wicks' theorem on (61). At leading order in the coupling there is one diagram corresponding to one photon exchange, resulting in the integral h

1 f ~ ^ J

(il

dtxdti _i2)2_r2-

. , (67)

At next order there are two Wick contractions, but these sum to give the square of Ii with a combinatoric factor of 1/2. One can then see that subsequent Wick contractions will just fill out the power series expansion of the exponential [32]. Now when we include matter, the only new diagrams which arise correspond to corrections to the photon propagator. Thus all the logarithmic corrections to Coulomb's law will arise from loops in the photonic two point function. Resumming the vacuum polarization graphs results in the corrected form of the potential o2

E{r) =

e fy

/

'

(68)

47IT

where e2(r)e// =

n+ ^>w/tt-

(69)

i 1 + "6^ ln ( r /W) When we go to QCD, things get slightly more complicated but, at least up to two loops, the same reasoning goes through [32,33]. The potential is given by the logarithmically corrected Coulomb potential with the appropriate beta function. However, at three loops one encounters an infrared divergence [34] arising from the fact that there can be intermediate

135

color octet states that live for long times. These colored states can interact with long wavelength gluons since they have non-vanishing dipole moments. However, this should not bother us. The vacuum matrix element contains long distance physics, and thus it need not be infra-red finite in perturbation theory.

2. Lecture II: Non-Relativistic Effective Theories 2.1. The Relevant

Scales and the Free

Action

In the previous lecture we displayed the utility of encapsulating the UV physics in an effective field theory. In our toy model we completely removed a heavy field, while in HQET we kept the field but removed the mass. In both of these cases the non-analytic momentum dependence came from rather simple kinematic configurations. In the toy model all the cuts were generated from intermediate light particles going on shell. In HQET the cuts come from either gluons or heavy quark lines going on shell with kfj, c± A. But there are certain kinematic situations where the momenta configurations which determine the cuts of a Greens function can be more complicated. Consider for instance a non-relativistic bound state such as Hydrogen or quarkonia. m In this case the on-shell external lines in our Greens functions will obey a non-relativistic dispersion relation, leading to the propagator

G^(E,p)=

I

,,

(70) -2

and the cuts will come from a momenta region where E ~ ^ . Notice that, as opposed to the case of our toy model and HQET where there was only one relevant low energy scale, TUL and A, respectively, a non-relativistic theory contains two scales namely E ~ mv2 and p ~ mv. That is, the typical size of the bound state is the Bohr radius TB — l/(mv) and the typical time scale is the Rydberg R ~ 1/mv2. The existence of these two scales complicates matters. As in our previous cases we'd like to eliminate all the large scales from the theory, leaving a nice simple theory with only one low energy scale. Unfortunately, as we shall see this is not possible due m For the quarkonia case we will assume that the bound state is Coulombic such that mv2 > > A. This hierarchy most probably does not hold for the case of the J/tp and may not for the T either. Attempts at alternate power countings which arise when this criteria are not met can be found in [35].

136

to the fact that the two scales mv and mv2 are correlated, and must exist simultaneously. Suppose we try to treat the scale mv in the same way that we treated the mass in HQET, i.e. by rescaling the quark fields. This would indeed remove it from the theory, and all derivatives would scale homogeneously as mv2. However, since spatial fluctuations on the scale mv are crucial to the dynamics of the bound state, we know that this can't be the end of the story as far as this scale is concerned. In fact, as we shall see, the Coulomb potential is built from the exchange of gluons with three momentum of order mv. So if we eliminate the modes with momenta of order mv by re-phasing the field we had better figure out how to put (integrate) them back in. At first, this seems like a silly thing to do, but we will see that this formal manipulation will simplify the construction of the effective theory since all spatial derivatives will scale homogeneously. If we did not do this re-phasing we would not have manifest power counting, as we shall see. But we are jumping the gun a little. Let us treat one scale at a time from top down. We begin by first eliminating the mass from theory as we did in HQET by a simple rescaling of the field Q = e - i m « t ( ^ + x),

(71)

,& = e ' m « t i ( l + 7 o ) Q ,

(72)

X= eim«^(l-7o)Q,

(73)

where

are the large and small two spinors of the full Dirac quark spinor, respectively. As opposed to HQET, there is now a preferred frame, namely the center of mass frame. So we will assume that we are working in this frame from now on. Thus when we rescale the field we won't sum over all four velocities, as we did in HQET. Also I won't bother labelling the fields by their four velocity, as there is only one relevant label velocity (1,0). If we wanted to allow for weak decays we would have to include a sum over four-momenta labels as we did in HQET. Now we can separate the next largest scale, mv, from the other scales in the theory by writing

V = Ee_i?'^W

(74)

p

where the label on the field fixes the three momentum to be order mv. Spatial derivatives acting on ipp(x) are of order the residual three momenta

137

raw2. This way of splitting up momenta into a large label piece and a residual momentum is depicted in figure 8.

<

mv

*•

P

>-

** 2

mv Figure 8. This diagram depicts how the momenta are divided by the labelling process. The largest label Vy. has been frozen to be (1, 0, 0, 0). The three momenta are split into a large label p oi order mv, and a small residual momenta of order mv2 depicted by the smallest boxes.

The free piece of the action can then be read off by substituting the expansions (71) and (74) into the full QCD action, solving for the small component x m terms of the large component ip, and keeping only the leading order terms in an expansion in the small parameter v, L

° = E <4 (ido - ~)

^5 + (^ - 0-

(75)

p

Notice that we have dropped all spatial derivative terms, such as p • <9, as they are higher order in v. Remember that the p which appears in the action is a C number and not an operator. £ is the large component of the

138

anti-quark field, which arises when we re-phase with the opposite sign in (60), and its action on the Fock space is to destroy an anti-quark.

(E.p + k)

•

(E.-p-k)

_ ^

,

"V

fr

^

(E.p'+k')

(E.-p'-k')

Figure 9. Full theory matching which generates the Coulomb interaction. The momenta are split up into a large piece (p) which is order mv and a residual piece k which is of order mv2. The large piece corresponds to the label of the field.

2.2.

Interactions

The interactions can be fixed by a matching calculations, though some interactions are more simply read off by gauge invariance, as we shall see. Let's start by considering the t-channel exchange of a gluon between a quark and anti-quark in the full theory as shown in figure 9. Since the initial and final state quarks are on shell, the gluon momentum must scale as (mn 2 , mv). We split the external three momenta into a piece of order mv, corresponding to the label of the field, and a residual momenta fc, which if of order mv2. We then expand the full theory result in v and write the full theory spinors in terms of the two spinors (72). The leading order piece generates the operator

Lc = 5 2 E 4 ' T ^ P ( ^ 7 ) l C l p , n - p .

(76)

This operator is nothing but the Coulomb potential. It is called a "potential" because the interaction is non-local in space but instantaneous in time. Notice that Lc includes a sum over all possible labels. This is in stark contrast to HQET where an external current is needed to change labels. Here in NRQCD, there are terms in the action which themselves change labels. What we have done seems a little perverse at first. We have integrated out

139

modes (via a field rescaling) only to integrate them back in by summing over labels. However, we will see that there is a method to the madness. Now let us power count to see if the operator (76) we matched onto should appear in the leading order Lagrangian. To do so we first need to determine how the fields scale. The easiest way to do this is to use the fact that the kinetic piece should be order one in our power counting. So, following the line of reasoning developed in section 1.9, we first power count the measure in the action. The quark field has support over 1/v2 in the time direction11 and order 1/v in the spatial directions so we conclude that d4x ~ v~5 and ip ~ v3/2. In this way the kinetic piece (75) scales as v°. Now let us consider the operator (76). The measure again scales the same way as in the kinetic piece since the operator contains only heavy quarks, so that Oc~v-5{v3/2)\r2~v-\

(77)

Thus Oc certainly should be treated as a leading order piece in the Lagrangian. Indeed, the fact that it actually is enhanced relative to kinetic piece is worrisome as it could lead us to the conclusion that there is no sensible v —> 0 limit, which is the limit in which the leading order action is exact. We will come back to this issue later. For now we will conclude that the Coulomb potential will have to be treated non-perturbatively in our theory. The operator can be interpreted in terms of the exchange of a non-propagating Coulomb gluon. If we wish we can re-write this operator in terms of an auxiliary field (i.e. a field without a kinetic term) [37]

Lc = E -9(4^P> + 4'V) A V' + ^ o , ) 2 •

(78)

Writing things in this way will make our relevant degrees of freedom more explicit when we discuss NRQCD in terms of the method of regions as discussed in (1.7). But this is just a formal manipulation. In general, "off shell" modes will not appear in the Lagrangian. 2.3. The Multipole

Expansion

and Ultra-Soft

Modes

What interactions are there in the one quark sector? We should allow for all interactions which leave the fields on-shell to within order mv2, which is the scale of their residual momentum, as these modes will certainly generate non-analyticity in our amplitudes. One obvious interaction we can have is n

All units are scaled to m.

140

Figure 10. Self energy diagram is NRQCD. The labelling technique automatically implements the dipole expansion.

of the usual form ip^Aip, if A is a gluon which carries momentum scaling as (mv2,mv2). In fact, gauge invariance seems to lead us to this type of interaction, since we would expect that the do in (75) should be elevated to a covariant derivative. Let's see if this new interaction does indeed to correspond to (mv2,mv2) gluons. The self energy diagram in figure 10 is given by

i£ = g2CF I ,f9,. . - 1 . .

^

.

(79)

Now we will show that the gluon running through the loop does have momentum which scales as (mv2,mv2). The important point to note here is that the fermion propagator does not carry any of the three momentum of the gluon. The p 2 piece is just the label of the external line. This is an implementation of a dipole expansion [38,39] Aa0(t, x) = Aa0(t, 0) + x • VAa0(t, 6) + • • • .

(80)

Exercise 2.1: Working with a coordinate space dipole expanded Lagrangian reproduce equation (79) for the self-energy. It is interesting that the act of separating the scales via the re-labelling technique has automatically generated a dipole expansion. This demonstrates a persistent fact in effective field theories. If you have properly separated the scales, then each diagram in the effective theory should scale homogeneously in the expansion parameter. In order for this to be true any given integral should be dominated by just one region of momenta. Let us see how this happens in our self-energy diagram. Doing the integral by contours we see that the go ~ | g | ~ mv2, there are no other relevant regions. Note that if we had not dipole expanded then the integral would have received a contribution where g was of order m, and the integral would not scale homogeneously in v. This would spoil the power counting in the

141

theory [39]. Gluons which have momenta of order (mv2,mv2) are called "Ultra-Soft (US)", and are on shell-modes which have the usual F2 kinetic term in the Lagrangian. These modes are sometimes also referred to as "radiation" or "dynamical" gluons which can be thought of as contributing to higher Fock states in the onia. Note, however, that the A0 US mode can be eliminated from the leading order action by redefining the quark field in the following fashion

tf(x) = Pe19^

Ao x+ dx

^ ^ i;(x),

(81)

where P stands for path ordering and n^ is the time like unit vector (1,0). The AQ US gluons will then reappear in the action at higher order in v. Thus without doing any calculations we can conclude that the interactions due to AQ US modes must vanish at leading order. There are additional US spatially polarized gluons (A) which play an important role at sub-leading order in v. We can determine the coupling of these gluons in a very simple way using what is known as reparameterization invariance [42] (RPI). The idea behind RPI is quite simple, yet its ramifications are powerful. The point is that when we pulled out the three momenta from the heavy quark field, there was an ambiguity.0 We could just as easily rescaled by e(imv + k) where k is order mv2. Thus our action should be independent under small shifts in the label. This just means that every time we see a p it should be accompanied by a d. Then by gauge invariance we know that we should make the replacement p - • p + it)

(82)

everywhere we see p. Notice that this form has to hold to all order in perturbation theory. This is quite a useful piece of information, as it tells us that many operators in our action will have vanishing anomalous dimension. The net effect of this substitution is the generation of subleading terms in the dipole expansion (80). For instance, the kinetic term p 2 /(2m) will generate the subleading dipole interaction V ^ ( P ' A/m)^>. Note that D = d — igA, scales as v2, since when the derivative acts on the heavy quark field it brings down a factor of the residual momentum which is of order v2, and, as you can prove for yourself, the ultra-soft gluon field scales as v2 as well.

°A similar ambiguity exists in the H Q E T rescaling. See [18] for a discussion.

142

f

\

Figure 11. A graph which one might have hoped would induce a running potential, which does not exist in the effective theory.

2.4. Soft

Modes

So now we have shown that there are two types of gluonic modes which contribute to the non-analytic behavior of the low energy theory: potential gluons (which are not dynamical), whose momenta scale as (mv2,mv), and "ultra-soft" (US) gluons whose momenta scale as (mv2, mv2). But this can not be the end of the story. In particular, we know that the Coulomb potential should run even once we are below the scale m, due to the non-Abelian nature of the theory. We might expect that a diagram such as figure 11 could account for the running of the coupling/potential. However, this diagram does not exist in the effective theory, since the Coulomb gluon is not a propagating degree of freedom (it is simple to show that this diagram vanishes). The Coulomb gluon sometimes called an "off-shell" mode because it can never satisfy the gluon dispersion relation g\ = g 2 , and thus never appears as an external state. To generate a running Coulomb potential we must include a new set of propagating degrees of freedom [30,40], namely, "soft gluons", whose momenta scale as (mv,mv). How could such degrees of freedom interact with the heavy quarks? We notice that because their energy scales as mv, any such interaction would throw the heavy quark off-shell by an amount of order mv. So we know that we need at least two interactions. The simplest such interactions are shown in figure 12. Now since these modes have a "large" component, in both their energy and momentum, we need to rescale the field so that all derivatives acting on the field scale as the lowest scale in the theory, mv2. So as in the case of the heavy quark field we write

Wx) = T,e~i9'XAMx)9

(83)

143

Figure 12. Leading order matching graphs between heavy quarks and "soft" gluonic modes with momenta of order (mv, mo). Not shown are the similar diagrams with ghost fields,

Notice the field is now labelled by a four momentum. Now we match the Compton graphs in figure 12 onto the effective theory. To do so we split the external momenta of the gluon fields into a label piece and a small residual momentum QP,

= q„ + k„ ,

(84)

and expand the diagram in powers of v. Keeping only the leading order result we find r int soft

L

4

™* £

\^4>[<>,<]^

q,q'v,p' At

gv0(q'-P+P'T-

SM0 ( « - P +P'r + ^iq-q')^ (p' - p )

" (p' - 0 p)2 ^ ^ '

Cq

^

rA,^A^^

2

f

+

^ ^

X

'

T

^

f

i

(85)

where I have also included the contribution from the ghosts cq, which are not shown in the figure. As we will see, this interaction will be responsible for the running of the Coulomb potential. Note that it vanishes for the

144

Abelian case, which is consistent with the fact that there is no change in the force law below the scale of the electron mass in QED. This cancellation would not arise if we did not drop the ie in the denominator. Keeping the ie would mean including a contribution where the gluons are potential and not soft. To see this write the intermediate fermion propagator as

- 4 - 7 - = P ( - ) - in5{q0) •

(86)

The delta function forces the gluon to have vanishing energy, i.e. it becomes a potential gluon [30]. In the effective theory this potential contribution is accounted for by operators which describe the scattering of soft modes off of a four fermion potential (see [45] for details). Furthermore, to include the effects of the "massless" quarks we will need to include a soft fermion field for each such flavor. So now we have two propagating gluonic degrees of freedom, the soft (mv,mv) and ultra-soft (US) (mv2,mv2). The kinetic terms for these fields are LKB

= -\F»VF^

+ J2

I P^P

~P"A$

I2 •

(87)

p

Here the fields strengths are composed of purely US gluons igF^ = [D^jD"]. While the second term in (87) generates the propagator for the soft gluons. We may treat the US field as a background gauge field. Given the relative time scales of these two modes, the US modes are frozen, as far as the soft gluons are concerned Thus in a background field formalism (see [2] for a review), we may gauge fix the US fields and the soft fields independently. The US gauge invariance is manifest while the soft gauge invariance is only seen when combined with re-parameterization invariance. Note that there will also be soft (labelled) ghost fields as well. The full soft Lagrangian, including ghosts, can be found in [41]. We are now prepared to see if the soft interaction Lagrangian is leading order in v. Each soft gluon field scales as v, as can be read off from (87), while each fermion field gives a factor of v 3 / 2 . How does the measure scale? The fermion has (temporal,spatial) support on the scale (l/(i> 2 ), l/(v)), whereas the gauge field has support only on scales of order 1/v. Thus, the measure scales as l/(i>) 4 , since outside this volume the integrand vanishes as depicted in figure 13. Thus the soft interaction Lagrangian I j " ' , t scales as v° and is indeed leading order. However, since this interaction is also order as we need not treat it non-perturbatively.

145

Figure 13. The soft gluons and heavy quarks have differing spatio-temporal extent, The overlap regions, which determines the scaling of the measure in the action is of order (y, v)

2.5. Calculating

Loops in

NRQCD

Now let us take stock in what we have built. So far we have a leading order action which contains the kinetic pieces for the fermions and gluons (75) and (87), as well as two leading order interactions (78) and (85). If we want to match beyond leading order in a, we need to know how to calculate using our effective theory. Calculating loop graphs seems rather complicated given that we will have loops with sums over labels. However, it will turn out in the end to be quite simple. As a first example suppose we're interested in a loop consisting of two insertions of Lc, as shown in figure 14. The external fermions have energy E and momenta ± p , r. In the effective theory, the external fermions have soft momentum (labels) ± p , ±r, and the ultra-soft momentum (E, 0). The intermediate fermions in the effective theory have soft momentum ±q, and ultra-soft momentum (E, 0) ± k. The graph in the effective theory is proportional to the integral

l

Coul

rf

1 ( p - q ) 2 ( ? - q ) 2 k° + E-q2/2m

?/ V

-k° + E-(i2/2m

+ ie'

+ ie (88)

146

(E.O)+k+(0.q) (E.O)+(0.p)

(E.O)+(0.r)

(E.O)+(0.-r)

(E.O)+(0.-p) (E.O)-k+(0,-q)

Figure 14. A one loop graph in the effective theory with two insertions of the coulomb interaction (solid dot).

There is an integral over the ultra-soft energy k°, the ultra-soft momentum k, and a sum over the soft momentum q. Summing over q and integrating over k is equivalent to integrating over the entire momentum space. Thus one can replace Eq. (88) by 1 1 1 Icoui = Id q 2 2 ( p - q ) ( r - q ) q° + E - q 2 /2m + ie /• X

1 -q° + E-q2/2m

(89)

+ ie'

where

dk° ^dq°,

Yl [ dk~+ f dq.

(90)

Soft loops are treated in a similar fashion. Consider the diagram with the insertion of two soft interactions as shown in figure 15. It gives a

(E.0)+(0.p)

(E.0)+(0.r)

(E.0)+(0.-p)

(E.0M0.-T)

Figure 15. A one loop graph in the effective theory with two insertions of the soft interaction (solid dot).

147

contribution of the form

/wt-E/d4*^^

(j5_f)2 (g0)2_(|2 ( g 0 ) 2 _ W +

^_ f ) 2 ' ( 9 1 )

where I have used the last interaction in Eq. (85) for the soft vertices. The sum on q is over a four-vector. As in the case of the potential loops, we make the replacement

J dAk-> f d4q,

(92)

Q

where the replacement must be done for all four components of q, since soft gluons carry a four-vector label q. There is a much simpler way to arrive at the effective theory integrand. Just take the full theory graphs, assume the proper scaling for the gluon loop momentum, i.e. soft, potential or US, and just keep the leading order term in each propagator. 2.6. Matching

at Subleading

Orders in v

Let us now consider to matching at higher orders in v, but still at tree level. We can guess the first correction to the kinetic energy from the dispersion relation

Formally this piece can be derived simply by keeping the sub-leading terms in the expansion of the QCD action in terms of the non-relativistic fields (74). The more relevant correction comes from higher order terms in the multipole expansion,

The ultra-soft gluon field inside D scales as v2, as does the derivative since it brings down a factor of the residual momentum of the heavy quark. Now let's consider the corrections to the quark/anti-quark potential. These are derived by expanding the full theory graph in figure 9 to higher orders in v.

Exercise 2.2: Show that, at order asv the subleading potentials generated by T-channel gluon exchange are given by

148

V{1)

= {TA ® TA)

^ C + ^2 S 2 TO2k2

m

+

^A(p',p) 2 v m

V,

'

T(k)

(95)

where k = p — p ' and • S-(p'xp)

A(p',p)

T(k)

=3l-S2

1

=;

k2

3 k • CTik • $2

and
V h f •• Vt

4:Tras 3 iras

(96)

(E.P1)

(E.P2)

(E.-P1)

(E-- P2)

Figure 16.

Tree level full theory diagram contributing to the potential.

T h e annihilation diagram in figure 16 generates terms of order asv in the potential. Using Fierz identities and charge conjugation, the contribution of this diagram can be transformed into the basis in Eq. (95) and give additional contributions to t h e matching. Only Vhf receives a non-zero contributions at tree level from annihilation: Vhf,a = —iras,

(97)

149

where the subscript a stands for annihilation. In addition this diagram generates the a hyper-fine interaction with a different color structure v(i)

jy« = (lxl)^fs2 with V^f = OL-K ^ 2.7. Matching

(98)

. The color factor arises from the Fierz rearrangement.

at Subleading

Orders in a

Let's now consider matching beyond tree level. But before we plunge into a calculation we must address an intricacy of NRQCD which I touched upon previously. In particular, we noticed that the Coulomb potential scales as 1/v. On the face of it, this presents a serious problem to the theory for the following reason. Suppose we wish to calculate the matrix element of some operator in the effective theory, (ip \ O \ tp). at order in vk. In this case we have (iP | O | V) = / f i i • • • (Txn x(iP\T(OH1(x2)...Hn(xn)exp(i

I dAxLG{x))

| if,) (99)

where Hi are terms in the sub-leading Hamiltonian such that their net order in v is vk. Given that the contribution from the Coulomb potential is of order 1/v and must be treated non-perturbatively, it appears that the time ordered product will not have a definite scaling in v. However, we are saved by the fact that while the Coulomb potential is of order 1/v it is also of order a. Furthermore, in a Coulombic state the Virial theorem tells us that n aAl/r mv2 ~ -^-^

~ mv) r

/„„„^ (100)

so that as(mv) — v.p So when we calculate matrix elements we should treat the Coulomb potential to be order one. However, when we match, we must account for the fact that the Coulomb potential can enhance a sub-leading operator, at the cost of a power of as. For instance, if we want to match at order a2s, then we have to include contributions of the form (L^LD^al p

(101)

If we are not in a Coulombic state, for instance we could be looking at threshold production, then we must have a < v.

150

where l/ 2 ) is an order v2a°s piece of the Lagrangian. As a practical example let us consider the matching of a production current at order v°. This matching calculation is relevant to top-quark threshold production [36]. At tree level we match by expanding the full QCD current in terms of the NRQCD fields, 97*9 = E ^liaiX-P

(102)

+

This operator produces a quark/anti-quark pair in a relative 3S\ state. Exercise 2.3: Show that at order v2 one generates the operators

(103)

O^E^P-^W-P-

(104)

°2 = E 4 ( P V X V

a.

b.

Figure 17. One loop matching diagrams in the full and effective theory. In the effective theory there are additional diagrams involving temporal US gluons which vanish.

At one loop we must calculate the graphs shown in figure 17. Since the full theory current is conserved, it has vanishing anomalous dimension once the wave function renormalization is included. We can simplify the matching calculation by working at threshold. In this way the effective

151

theory diagram vanishes, since it is scaleless. Calculating the full theory diagram gives Figa = f i r ' x

—

" 1 euv

2

31n(m 2 /V)-9

CIR

(105)

where I have explicitly separated the UV from the IR divergences. Adding the contribution from the wave function renormalization, all the UV and IR poles cancel. We then find the matching coefficient to be C=(1-2CF-).

(106)

IT

You should be able to show for yourself that this short cut to calculating the matching coefficient is identical to calculating the hard piece of the graph using the method of regions. Notice that at one loop the current has no anomalous dimension. We can conclude this from the fact that all of the IR poles in the full theory cancelled (see Section 1.8.2) In general we would not expect this since in the effective theory the current does not generate a symmetry. In fact we will see that at two loops we will generate an anomalous dimension. At two loops one must account for the possible Coulomb enhancement discussed previously in this section. So if we concerned with contributions of order a^v°, we must include all possible operator insertion such that the net power is v°. This means we must also include any order v°a2s potentials which might be generated. At this order the full theory generates the v° potential V<°> = Vk^m |k |

(107)

where Vk=o?{^--CACF),

(108)

and I have projected the potential onto the color singlet channel since that is all we need when calculating the production current. We must include one insertion of this potential in addition to insertions of the order v potentials already discussed in the previous section. The relevant diagrams are shown in figure 18. In addition there are corrections which contain Coulomb singularities 1/v, i.e. those contributions whose net scaling has inverse powers of v. However, these won't contribute to the matching since they will cancel in the full and effective theory. The purpose of this example was just

152

Figure 18. Diagrams contributing to the matching of the current at order a2v°. The large hatched circle is the current insertion. The small filled circle is a Coulomb potential while the square represents insertions of order v1 potential. The cross corresponds to the insertion of the p 4 / ( 8 m 3 ) operator. The triangle is the order v° potential.

to illustrate that when you match you must remember that sub-leading operators can be enhanced due to the 1/v nature of the Coulomb potential. To my knowledge, as of the time of this writing, this subtlety is unique to NRQCD. It is interesting to note that had we done the two loop calculation using HQET propagators we would get a vanishing anomalous dimension. This is most easily seen in AQ = 0 gauge.

Exercise 2.4: Prove that in the MS scheme the anomalous dimensions of the production current is gauge invariant. To prove this write down two equations for the current. One imposing its independence from fi and the other its independence of the gauge fixing parameter a. Then commute the two equations to derive a set of relations which allows you to complete the proof. If you get stuck consult the original reference [43].

The reason HQET gets it wrong lies in the theory's inability to properly describe fermion-fermion propagation. What happens is that HQET misses

153

out on the Coulomb singularity. This fact emphasizes an important point about NRQCD. You can't treat the scales mv and mv2 independently. Usually we would argue that between the scales m and mv, mv2 should be treatable as a perturbation, much as we would treat a light quark mass when working at a scale E 3> mq. But the fact that we need to keep the kinetic piece of the action at leading order is telling us that both mv and mv2 must exist simultaneously in the theory. After a brief diversion we will see the consequences of this correlation of scales.

2.8. Summing (VRG)

Logs: The Velocity Renormalization

Group

You have probably noticed that I have been careful not to specify the scale at which to evaluate the running couplings as(u). Such a specification implies we have control of the logarithmic corrections, which I will only now discuss. As shown in the previous lecture the recipe for summing large logs is typically straightforward. Suppose we have a theory with two "large" (relative to the scale at which experiments are to be performed E) scales mi > mi. In a general diagram we would expect to generate logs of the form Inimi/E), ln(mi/E), ln(mi/m2). We would sum these large logs in steps: 1) Match at the scale mi onto a theory where (p± has been integrated out. 2) Use the RG to run down to the scale m^. 3) Match onto a theory where $2 has been integrated out. 4) Run down to the scale of the experiment. We will call this procedure "two stage running". In the case of NRQCD, where we have two large scales m and mv and a low energy scale of mv2, we will get logs of the form \a(m/mv2) and \s\{m/mv). So we might be tempted to follow the standard machinery and turn the crank. Match at m, run to mv integrated modes of momenta mv and then run down to the scale mv2. However, NRQCD is a lot more interesting than that. The basic point is that the two scales mv and mv2 are correlated. They shouldn't be treated independently, since they coexist in the fermionic propagator. If we were to follow the two stage running in NRQCD, we would first match at m onto an HQET like theory by rescaling the fields only by their masses. In this case we should allow for quark virtualities q2 < m2Q and one might wish to treat ^ as a perturbation leaving the HQET form of the propagators. But as we discussed above, we know that this will give the wrong anomalous dimension for the current [21]. Instead, in NRQCD what we must do is match directly, at the scale m, onto the theory in which E ~ %—. So we can see immediately that the two stage

154

running will not be appropriate. Instead we do what is known as "one stage running". Given that we have to match directly onto a theory in which two scales exist simultaneously, how can we sum the large logs? In general, potential and soft loops will generate ln(VmE/n), while US loops will introduce logs of the form \n(E/fi). So in one-stage running we introduce two different H parameters, /is and nus- However, we are not free to choose /xs and pus independently, since /x| = m/ius- Instead we should write us = mv and nus = mis2, and think of a subtraction velocity v instead of a subtraction scale /x. This is the basis of what is known as the "velocity renormalization group" (VRG) [21]. The renormalization group equation is then derived using the fact that bare quantities are independent of the subtraction velocity, i.e. V-^-GB

= 0.

(109)

dv Then by solving these equations and running from v = 1 to v = v, we can sum all the large logs. How does the VRG differ from the standard RG [44]? In VRG we have d .dus d dfius d d d 1/ T~=l/^^r~ 1— + ^ — 1 '=^s^—+2/2US-, . (110) dv dv d/j,s dv d^us dfis d^us Note, the crucial factor of 2. In the two-stage method one would first integrate from m to mv using, 7s + "/us, then continue down to the scale mv2, using only jus • Whereas in one-stage one integrates 75 + 2"/us from v — 1 to v = v. If we assume that the anomalous dimensions are constant and don't run, then in the two stage running we would find for the running of some generic Wilson coefficient C(mv2) — C(m) = (75 + jus) \r\.{mv/m) + jus m(mi; 2 /'(mv)), (111) while in one-stage we have C(mv2) - C(m) = (7s + 2luS)

ln(v).

(112)

So we have agreement. However, in general the anomalous dimensions depend upon some couplings which are scale dependent. This scale dependence will lead to a discrepancy between the two methods at higher orders. Suppose we are interested in the anomalous dimensions ji of some set of Wilson coefficients [/j.q These anomalous dimensions will, in general, q

Operators and Wilson coefficients have opposite anomalous dimension since their product is RG invariant.

155

depend on t/j themselves. 7* = li{Ui(lA)-

(113)

7i will be some function of t = \n(/j,) = ln(w), which we can expand1"

*<'>-*<°>+<(KL+= 7.(0) + ' ( ^ )

M

•

+ -

("4)

Suppose the [/$ have both soft and ultra-soft anomalous dimension (this typically will occur at two loops). Then in one-stage running, integrating the anomalous dimensions will give S , n„,US\

Jo

J2 (ihfs

, &y?

us

d(21rs)..s , dW)

us

(115) whereas two-stage running yields ln(u)

jls'us\t')dt'+

0

H7

/-ln(w 2 )

^s\t')dt'

=

Jln(u)

*

+

7i

J +

2 ^(7 f c 7 f c + 5C/ f e l 7 f e

J +

5?7 fc

lk

+

3Uk

{lk

V (116)

and we see that the cross terms differ by a factor of two. Let us now return to the question of the running of the Coulomb potential. This running is particularly simple because the renormalization only involves soft loops. The matching coefficient is simply 4iras(m), and the running is generated by figure (15), with each vertex giving a factor of Vc

«, 1 5 _g ( W ) (H C j (i + h (g)) + ^ ) . r

(117)

This expansion is only for the purposes of illustrating the difference between one-stage and two-stage running.

156

There are also corrections stemming from US temporal gluon loops, but these vanish for reasons which were previously discussed. Calling the coefficient of the Coulomb operator in the action Vc, and using the fact that its bare value (VCB) is independent of v, we have

and we are left with v±Vc

= -2(30a2s{v),

(119)

where I've used /?o = ^ C U to make it clear that Vc obeys the same equation as the gauge coupling. Using our matching result for an initial condition, along with the explicit form for a3(u), we find Vc{mv) =A-Kas(mv).

(120)

At this order the running of the potential is identical to the running of the beta function. The large logs can be avoided by choosing a(\ k |) in the potential. This agrees with our intuitive picture of the force law (Wilson loop) getting quantum corrections due to the running of the coupling. However, this correspondence does not persist at higher orders as the running of the Coulomb potential includes effects from higher order potentials which vanish in the static limit [48,47].

Exercise 2.5: Show that the ^-CA gets replaced by the the full beta function (3Q = ^-CA — | n / i once nj soft light quark fields, q, are introduced. To begin this calculation first show that the soft Lagrangian gets an additional piece given by L's = -92

£

(7g73Bp(4'rA^)(^'7oT%))+(^^X)-

(121)

There is a simple way to determine when, where and how the fis a n d Hus show up. We just make sure that in 4 — 2e dimensions, the operators have the same v scalings as they would in 4 dimensions. For instance, in the Coulomb potential the matching comes with a factor of g2, which arises from an exchange of a Coulomb gluon. In 4 —2e dimensions, should we write this as g2^2g or g2fJ>2jS'? The coupling should still have units (mass)e but the v scaling appears ambiguous until we impose the fact that the operator should scale as v~x. The fermion fields now scale as v3^2~e and we have a factor of 1/v2 from the Wilson (label) coefficient. But now the measure

157

scales as v~2v~3+2c. So to insure that the operator scales as v'1 we should write g2 —> g2^\f. If we consider an operator where a US gluon attaches to the potential [41], the scaling of the measure won't change (see section 2.4) but since the US gluon scales as v2~2e then we should include a factor

2.9. The Lamb Shift in QED I would now like to utilize this formalism in a simple setting, namely, the Lamb shift in QED. This is a beautiful example because, not only does the formalism simplify the calculation, but it also provides one with immediate information about sub-leading orders, as we shall see. Let me remind you that the Lamb shift is the splitting between the 2SJ=1/2 and 2PJ=1/2 levels of Hydrogen. Recall that these levels are "accidentally" degenerate up to a2 corrections. Their splitting comes in at relative order a 3 , corresponding to a over all energy shift of order a 5 . This splitting is dominated by the so-called "Bethe log", a5 In v. Prom an effective field theory point of view, all logs are RG logs. If they look like "long distance" logs, its only because the scales have not been properly separated, i.e. you haven't written down the proper effective field theory. Thus, we should be able to also resum these logs if there are more of them. In the case of the "Bethe log" one can ask the question, is this log just a first term in a series? i.e. AE « a5 In v(l + a In v • a2 In2 v + ..

•

)

•

(122)

Figure 19. The QED diagram responsible for the Lamb shift. The photon over the top is non-instantaneous, i.e. ultra-soft, while the ladder gluons are Coulombic and must be resummed.

158

Typically the Lamb shift comes about from a low energy matrix element involving diagrams as shown in figure 19. But in NRQCD we know that there should be no large logs in any low energy matrix element. The log should arise from running. Let's see how this comes about. Recall that the leading order energy levels scale as v2, or using the Coulombic counting, a oc v, a2. So one would expect at order v4 there would be a splitting due to the subleading contribution to the Hamiltonian (123) and (107). However, we must account for the fact that in Hydrogen the masses of the constituents are not equal. In fact, we may take the mass of the proton to infinity.

Exercises 2.6: Show that for hydrogen we have 'V„r

yd)

•A(p',p) + ^ 2

(123)

where V2 is a contact delta function potential which vanishes in the equal mass case (considered in (95)). Matching at v = 1 we find ira I 1 1 \ not V2(y = !) = —-[ w —r. (124) K ' 2 \me mpj 2ml In addition, at this order, we should include corrections from the operator

4-i = £ 4 ( | ^ P + W-x)-

(125)

p

Thus the mass splitting is given by AE = ( 2 5 1 / 2 I L%>n + V2 12Sl/2) - ( 2 P 1/2 I L™n + ^ f A(p', p) 12P1/2).(126) Calculating these matrix elements is nothing more than solving the Coulomb problem to first order in perturbation theory, which we can solve quantum mechanically [46]. That is why it is simple to see that V2 does not contribute to the energy shift in the P wave since its wave function vanishes at the origin. The degeneracy is only split once we consider the (2) running of V2 and V\. L\(n is not renormalized due to reparameterization invariance but V2 gets an anomalous dimensions from the diagram shown in figure 20. A short calculation yields the result 2a ^ = 3 W

, N (12?)

159

Figure 20. The one loop diagram which generates an anomalous dimension for V2. The heavy lines are protons while the photon line with a line through it is ultra-soft. In the limit where the proton mass goes to infinity there is no soft anomalous dimension for V2 • The filled circle is a Coulomb vertex, while the US gluon couples to the electron via the order v dipole operator (94).

where Vc — —Aita. Because we are looking at a one loop diagram involving only US loops the VRG reduces to the usual RG (except of course for the factor of two arising from the fact that nus = mi/2) and we have v ^V2 = 7 2 (128) dv since in QED the coupling doesn't run below the scale me, the solution to the RG equation is trivial V2(v = v)=j2Vc\n{v) + V2{l). (129) We thus see that the series does indeed terminate. QED has a hard time generating logs because the coupling freezes below the scale me. At leading order the spin orbit potential has a vanishing anomalous dimension, so the energy splitting can be determined by taking the matrix element (126), but this time renormalized at the scale v = v, using =72Khu»|V(0)|2

(130)

and

| 2 = (i^2!

(13i)

AE = ~^-\n{v).

(132)

i m

we find the famous result

Using effective field theory we can go a step further and ask questions about the next series of logs, a5(aln(v)+a2ln2{v)

+ ...).

(133) s

It turns out [49], that this series terminates as well at order a In (v).

160

2.10. States

in the Effective

Theory

Before closing out this lecture I would like to emphasize an important point about the states in any effective theory. If we wish to keep our ducks in a row we should be sure to treat the states of the effective theory as eigenstates of the lowest order piece of the action. Power corrections to the states come from taking time ordered products in the low energy matrix elements. This is exactly what we do in quantum mechanics. We perturb around the lowest order wave functions. In this way we can keep track of various power law corrections. It is a simple exercise to see exactly how this works generically. Suppose we are considering a correction to the expectation value for some transition operator T = =

J A(*i(Pi) I T(O(0)ifi(y)) | *2(p2)>

(134)

where $i are eigenstates of HQ. The two time orderings correspond to corrections to $ i and $2- Let's consider the contribution from yo < 0. Using the representation of the theta function 0(-yo) = ~ / dr—— in J-oo T + ie and inserting a complete set of states yields

T = Yl I d4y— f

(135)

dTeil")(pa-1'»-T)(*i(pi) | O(0) | n'0') ;>(°>|ffi(0)|S2(p2))_

E2 — En (136) Then recalling from basic time independent perturbation theory that the first correction (ip^) to the zeroth order eigenstate (ip°) of the full Hamiltonian H = Ho + Hi is given by

,„,.») = E|»..»)'-7'f'

,137)

we explicitly see that the time ordered product accounts for corrections to the wave function. Let me emphasize that there is nothing wrong with treating the physical states as eigenstates of the exact Hamiltonian. This is often done in the literature. One often reads about "higher Fock states" of a bound state. For instance in the case of NRQCD one often hears the term "dynamical

161

gluons" as being part of a higher Fock state of the onium. Again, there is nothing wrong with this, it just goes against the spirit of effective field theory as espoused in these lectures. But there are advantages to treating the states as lowest order eigenstates. For instance, in the case of HQET the leading power corrections to the meson masses coming from the operators Oi=hv-

D2 2mQ

hv,

- acr 02 = h v ^

G^

4TOQ

hv

(138)

are cleanly separated from higher order power corrections. This would not be true if we took the meson to be an eigenstate of the full Hamiltonian since the expectation values of 0\ and O2 would contain contributions of order 1/m 2 . Of course this would only be an issue if we were interested in 1/m2 corrections.

2.11.

Summary

In this second lecture I have introduced some new concepts into the game. In particular, the idea that one can integrate out massless particles by breaking up the field into various components. In the case of NRQCD we split up the gluon field into four types of modes: hard (m,m), though I didn't use this term previously. These are just the gluons which we integrate out when we match at the scale m. Potential (TOW2, mv), soft (mv, mv) and ultra-soft (mv2,mv2). These last 3 modes are believed to be sufficient to correctly reproduce all the non-analytic structure of non-relativistic Greens function. To date this has only been proven up to two loops. Once we determined the modes, we fixed their couplings by matching, although gauge invariance and RPI fix many such couplings without doing any work. The way we arrived at the proper mode decomposition was far from systematic. Indeed, finding the proper modes is more of an art form at this point. The potential and US modes fell out directly from matching, while we deduced the need for the soft modes from physical argument. Of course had we carried out the matching in the two quark sector we would have immediately seen the need for a soft mode. In particular, if you calculate the full QCD box diagram you will see that just including the potential and the US modes is not sufficient to reproduce the full result [30]. There is no general rule for determining what modes are necessary. However, there are some guiding principles. First we must determine the scales involved in the external states of the theory. In the case of NRQCD, those would be mv and mv2. Then there are some obvious modes that one can pick out that are necessary. Those are modes which leave the

162

external lines on shell.s So for a fermion with momentum (mv,mv2) we know that US and potential gluons should contribute to the non-analytic behavior of the Greens functions. After that, finding more modes starts to get harder and you have to do some matching calculations to be sure. But its pretty clear that the other modes should also be formed from these scales. So the soft (mv,mv) modes are an obvious choice. Since these modes throw quarks off-shell, its clear that you will need more than one interaction vertex to generate an operator in the effective theory. Then what about (mv,mv2) modes? Physically, these modes would mediate interactions which are non-local in time but local in space, which is clearly unphysical, and in fact a simple calculation shows that such modes can't contribute. What about modes with higher powers of v such as mv3? It is simple to see that these modes would give zero since this scale can be dropped in all propagator, and so loop integrals would be independent of the loop variable itself. The interested reader should study SCET [24,25] where similar mode deconstructions are performed.

3. Lecture I I I : Effective Theories and Extra Dimensions I would now like to discuss the application of effective field theories to the speculative topic of extra dimensions. The basic idea of using extra dimensions to unify forces goes back to the early part of the twentieth century with the work of Kaluza and Klein. But while extra dimensions have always played a role in string theory, from a phenomenological stand point the topic was dormant until a few years back. The reason for this dormancy was that the size of the extra dimensions was always assumed to be of order the Planck scale, and therefore, the measurable effects of the extra dimensions are so suppressed that they are irrelevant for all intents and purposes. However, with the advent of D-brane ideas, it was realized that the extra dimensions can be of arbitrarily large size (up to the experimental limits for gravity at short distances [52]) since string theory provides a mechanism for field localization. That is, the usual problems involving observable effects of large extra-dimensions are eliminated because the standard model fields live on a fixed three dimensional space-like hyper-surface (the D-brane) and are thus impervious to the extra dimensions. This led to a flurry of model building, including new solutions to the hierarchy problem. Many of these models were discussed by other lecturers at this school [50,51]. My purpose s

By "on-shell" I mean its virtuality is on the same scale at the residual momentum.

163

here will be to treat these models from the point of view of effective field theories. Please note that I will be abusing the term D — brane throughout the rest of these lectures. You many substitute the term D — brane with a "domain wall of Planck scale thickness" for the purposes of these lectures. Let us consider a generic theory on a manifold M which can be written as a direct product M = M4xMd

(139)

where Md is some d dimensional compact manifold. We will assume for simplicity that there is only one relevant scale which characterizes Md, namely JR. So we would expect that for energies less than l/R, the effects of the compact extra dimensions should be suppressed by ER
(140)

Furthermore, naturalness tells us that g ~ Rd/2. Since for d > 0 the coupling is dimensionful, we would call this theory "non-renormalizable" in the old fashioned language. In our parlance we would simply say that its only an effective field theory, with limited predictive power. We would expect that this description of the theory would only correctly describe the physics up to a scale of order A = R-1. Typically the theory should contain a whole tower of higher dimensional operators which are suppressed by powers of A, which all become equally important when we have external momenta on this scale. 3.1. UV Fixed Points: Effective

A Theory Can Be More

Than

Now before going on I should say that the above conclusions need not necessarily hold. In particular, there could be a strongly coupled fixed point, where our power counting would break down. Indeed, as I previously mentioned, it is believed that there are five and six dimension supersymmetric Yang-Mills theories with 16 supercharges which posses non-trivial fixed points, although it is not known how to give Lagrangian descriptions of these theories. Our argument in the previous paragraph seems to exclude this possibility. So where is the loophole? The answer, is that our power counting arguments were all based upon reasoning which only necessarily holds at weak coupling. At strong coupling the power counting can change.

164

While its not clear how this happens in the higher dimensional SUSY theories, there is a nice example in three dimensions [53] which illustrates a mechanism by which naive power counting breaks down. Consider the theory of N fermions in three dimensions interacting via a four Fermi interaction L = $iifil>i + 2-$i1>i)2.

(141)

In three dimensions the coupling g2/N has mass dimensions — 1. So naively this theory is only an effective theory and breaks down at the scale N/g2. However, it has been shown that in the large N limit this is not the case. The conclusion that the theory is non-renormalizable was deduced from a weak coupling analysis. We know that quantum corrections can lead to scaling modifications. Typically, we study weakly coupled theory, where the quantum corrections only change the power counting by logs. But in strong coupling it can happen that this naive power counting breaks down, and a theory which we thought was non-renormalizable can indeed be renormalizable. This is exactly what happens if you study (141) in the large N limit. The way this comes about is seen by re-writing (141) in terms of an auxiliary scalar field L = Tpiipi/ji + a4>iipi-—a2. 2g-s

(142)

Now naively, the a propagator is constant. However, in the large N limit we are forced to resum sets of diagrams, and in doing so the a propagator scales as 1/p, if the bare coupling satisfies a certain bound. Given this scaling it is not hard, using textbook arguments, to show that all the divergences can be absorbed using only the operators shown in (142). This means that we can naturally set all the higher dimension operators in the action to zero without worrying that we will regenerate them via quantum corrections. The point of this little diversion was just to let the reader know that sometimes field theories can actually be more than just effective, they may be complete. The trouble is that the strong coupling, which is necessary to generate large anomalous dimensions, makes it difficult to perform controlled approximations. So finding strongly coupled fixed points is a difficult task. I should also point out, that even without such a fixed point, there could well be a field theoretic description of the theory above the cut-off in terms of a different set of variables. For instance, the underlying theory could consist of a quiver gauge theory [54].

165

3.2. Naive Dimensional

Analysis

Returning now to our non-renormalizable effective gauge theory, it actually turns out that there are simple arguments which lead to the conclusion that the scale at which the effective theory breaks down is numerically enhanced relative to the compactification scale by a geometrical factor. These arguments are based on what is called "naive dimensional analysis" which was first discussed in the context of chiral perturbation theory [55]. A discussion of NDA within the context of super-symmetric theories can be found in [56]. The basic argument goes as follows. Suppose we wish to calculate some n-point function (G\ ) in a non-renormalizable field theory. Generically there will be a tree level contribution from some unknown counterterm (Cj), as well as some calculable (in terms of lower order coefficients) non-analytic terms which arise from loop corrections. Both contributions will be suppressed by powers of the UV scale A. Furthermore, dimensional analysis tells us that there is a correlation between the order in the loop expansion at which the observable is generated and the dimensionality of the operator which contributes to the observable at tree level. Schematically the amplitude M can be written as M = (p2n/A2n)(CM

+ {C{\+

ln(p 2 /^ 2 )) + «)) •

(143)

Here the piece proportional to Ci is the contribution from some local operator, while the log piece comes from a loop integral involving lower order interactions whose couplings are related to C by a geometric factor generated by the loop integration (p). That is, if Co is the lower order coupling then C ~ CQP. The divergence will get absorbed into C,. The momentum p represents some generic set of external momenta. Varying the renormalization scale on a interval of order one generates an effective change in result of order C(p 2 n /A 2 n ). This is roughly equivalent to the statement that even if we set d to zero at tree level, it will be generated at the loop level. Thus, we would expect that the size of the counter-term is thus naturally of order Cp2n JA2n. Now given that the log arises from a loop, p is just a geometric factor which comes about from the angular integration and is of the form l/(77r r ) where 7 and r are a function of the number of dimensions. Thus NDA tells us that the cut-off is more like p _ 1 A instead of A. While this does not seem like a huge distinction, the chiral Lagrangian description of low energy QCD would be far less powerful if it were no for this factor of p — 47T. It is important to realize that NDA is a naturalness argument and assumes that couplings are all of order one. As such, it should only be taken as a guiding principle.

166

Let us now see how this works for a 4 + d dimensional Yang-Mills theory. The leading interaction is suppressed by k~d/2, so that any loop contribution to the effective action will be suppressed by Ad. Following the argument in the previous paragraph, let us compare the tree level contribution to the photon two point function arising from the higher dimensional operator O z = ^FABDFAB

(144)

to the contribution arising from loop corrections. The one loop correction to the two point function arising from interaction with a massless field is just the usual vacuum polarization graph, which in 4+d dimensions is given by -(«2) <* ^f^i-^d/2n-d/2}.

(145)

where, as usual -K^u(q2) = (q2g^v — q^qu)ir{q2). First we notice that the result only diverges in even dimensions. We could have guessed that since in odd dimensions there are no allowed (i.e. analytic) counter-terms which could absorb the divergence. So in odd dimensions we cannot use this particular calculation to learn anything from NDA. In even dimensions, e.g. d — 2, we would conclude that the true cut-off is not g^1 but (g(47r)3'/2)~1. Notice that I didn't worry about the overall constants in (144). The full result could include group theory factors or gamma functions which might arise from doing Feynman parameter integrals. However, as I said before, NDA is only supposed to give you a rough guess of the true cut-off. I would not wager much on its exactness, but it does make qualitative sense that the cut-off will in general be enhanced by a geometric factor, which grows with the dimensionality.

Exercise 3.1: Using a physical argument determine whether one would expect the NDA guess for the strong coupling scale to change due to compactification.

3.3. The Force Law in d Dimensional

Yang Mills

Theory

It is interesting to try to understand the force law in a compactified version of a d dimensional Yang-Mills theory. Let us calculate V(r) a la Exercise 1.6 for a five dimensional gauge theory compactified on an S 1 with radius R. If we KK reduce the theory (I am assuming here that the reader is familiar

167

with KK reduction, otherwise see [51]), then the force will get contributions from zero mode as well as KK mode exchange between our heavy sources. For distances much larger than R we would expect that only the zero mode force is non-negligible since the KK modes, whose masses scale with n/R, mediate a Yukawa force which dies off exponentially fast e~rn/R. Thus, in this regime, we reproduce the four dimensional result, with a rescaled coupling g/VR. The rescaling arises from the the norm of the zero mode. On the other hand, for r R: r<^R:

V(r) oc | - , Rr a2 y V(r) oc — .

(146)

How will this result be modified by quantum corrections? At large r 3> R we know that we should be dominated by external zero modes as well as zero modes in the loop. Whereas at small r < i J w e will get contributions not only from corrections to zero mode exchange but also from corrections to the exchange of external KK modes as well. At large r the dominant correction will arise from the pure zero mode vacuum polarization diagram which generates a correction of the form ^(r)K^ln(r),

(147)

whereas dimensional analysis tells us that the correction to the full five dimensional potential, at small r, will be of the form 8V(r) oc ^-,

(148)

so that at small r the quantum corrections dominate the classical result as per our previous discussion. Now let's do an honest calculation to see what the exact result looks like. Instead of calculating the full five dimensional potential, I will concentrate on the potential generated by external zero modes and its correction. The reason being that this will be the relevant object to calculate when we discuss grand unification. I will do the calculation in two ways. First by working exactly in five dimensions, and then by matching and running, as discussed in the first lecture.

168 I will specialize to the case of scalar QED for simplicity. In this case there are two contributions to the vacuum polarization. The result for a massless scalar can be written as [69] TI>(P2)

R

v f

(2fc + p)M(2/c + p)„ 1 ) ( / c - m 2 ) _| (p+k)2 4

25,flis

2

= (P>,PV-P29^MP2)2

nip )

•L R

256

If1 —-r2 /

(149) /— xdxyl x 2 ln(l

8TT JO

')

(150)

Exercise 3.2: Prove equation (150). A few points are worth noticing here. First off the result is finite. This is because dimensional regularization sets the power divergence to zero. This finiteness may seem strange given that each mode gives a divergent contribution. What happens is that the sum of the poles cancels. Also, we would expect that at long distances we should exactly reproduce the four dimensional non-analytic behavior. So if we expand for small values of the momentum we should get a log and nothing else non-analytic. Defining 6 = R^ and expanding the log we find ln(l

-+ln{S)+a152

• a254

(151)

where the rest of the series contains only even powers of S. The first term in the expansion cancels the power dependence, the second term is the 4-D log and the higher order terms are analytic. At pR 3> 1, the result is dominated by the power dependence and taking the Fourier transform of the first term in the Dyson series we get back the radius independent result we would expect from dimensional analysis (148). We can also see from dimensional analysis that in the regime pR 3> 1 the theory becomes strongly coupled as all loops will be of the same order. We should be able to reproduce the force law p dependence by the running and matching procedure. For long distances p « i ? _ 1 , all of the effects of the KK modes can be absorbed into local counter-terms and we simply reproduce the four dimensional result from the zero mode contribution. The higher KK modes just go into making up the low energy (measured) coupling. It is interesting though to see how we can reproduce

169

the short distance behavior as well. If we are interested in the two point function at the scale p, then we should sum over all the KK modes with masses less than that scale. Each mode will contribute a log(p/fj,) along with some scheme dependent constant. It seems clear how the linear dependence on p will arise from summing over the constant pieces, but how does the log(p) dependence becomes power-like? Suppose we have measured the coupling at some scale below the first KK mass Mi. To predict the coupling at P ~ M\, we run the coupling to the scale M\ using the four dimensional beta function, then we match, at the scale M\, onto a theory which contains the zero mode as well as the first KK mode. In fact, let's generalize this procedure. Suppose we know the value of the coupling at the scale Mn and we wish to predict its value at some higher scale. We just match in steps as we discussed in the first lecture. Since we are only interested in the log piece let us ignore additive constants in the matching (even though in this particular case the constants are of the same order as the logs, so this calculation does not follow the standard systematics). Thus, at the order we are working we may use the relation a(/x) = . R a ^ \ , , y (152) 1 - /30raa(/Jo) ln(/V/z0) where /3o is the beta function for one massless particle, and n counts the number of massless particles in the theory.* So the value of the coupling in a theory with n KK modes, an, at the scale Mn+\ is given by n+1 a„(M„ + 1 ) = an(Mn) + n(30a2n(Mn) H~^~) (153) where I have used the fact that Mn oc n and dropped higher order terms in the couplings since ln( IL ^ L ) is not a large log. Now we match onto a theory with n + 1 KK modes with a coupling we label an+\. To determine an+\, we equate the potentials of the two theories Vn(Mn+1)

= Vn+1(Mn+1)

(154)

Note that, in general, this relation does not mean we equate the couplings. Now we may use this result to derive the value of the coupling at the scale Mn in terms of a(l/R). a{n/R) = ail/R)

+ ^

a 2 (l/i?)n/? 0 m ( ^ ^ ) ,

(155)

'Remember once we are above threshold for a given particle it is treated as massless in the running. The mass effects are power suppressed.

170

taking A ^ > l w e have a{n/R) « a{l/R)

+ a2(l/R)f30N

(156)

Then using N ~ pR

(157)

we see that we have reproduced the power behavior we found when doing the full five dimensional calculation (150). But we also notice that the result is linearly sensitive to the matching scale. That is, an order one variation in the matching scale (157) changes the result by an amount of order one. Usually such a variation would lead to a subleading correction, but in this case the power law behavior leads to linear sensitivity to the UV physics. 3.4. Grand

Unification

One of the first uses of effective field theories was in the context of grand unification [57,58]. In the standard model the couplings approach each other near the scale 1 0 1 5 - 1 6 GeV, and in the minimal super-symmetric standard model (MSSM) the coupling are within errors of unifying. What unification of the couplings really means is that in the context of a GUT we get the correct post-diction for the weak mixing angle sin 2 9w • This postdiction is a compelling reason to believe that there is a desert of physics between the weak scale and the GUT scale." This desert would seem to exclude the existence of any extra dimensions whose scale is less than the usual (10 16 GeV) GUT scale. It is thus interesting see whether or not we can retain the grand unified prediction for sin2 9w within the confines of a theory of "large" (R_1 ~ TeV) extra dimensions [59]. Let us see what a compactified five dimensional GUT would predict for sin 2 9\y • I will discuss this calculation in the context of a simplified Abelian model. Suppose we have a unified 5-D theory with gauge group [/(l)ixC7(l) 2 x f7(l) 3 , which is compactified on a circle of radius R. We will furthermore impose a permutation symmetry to insure that the bare couplings for the three gauge groups are equal. We will assume that each gauge group has a matter content composed of two scalars. We will choose the first gauge group to have all its scalars massless. The second group will have one "Another hint that this might be the correct scenario is the long life time of the proton. Since higher dimensional baryon number violating operators need to be sufficiently suppressed. This is an additional challenge to extra dimensional model builders.

171

massless and one of mass M2, while the third group will have all its matter content of mass M3. All the masses for the scalars will be of the same order which we will call MQUT- These masses will mimic the effect of GUT symmetry breaking. For instance, they could be generated by choosing a Higgs potential in an appropriate way. Well above the scale of the masses, the values of the two point functions will be identical. Let us first review how the prediction for sin2 9w comes about in four dimensions in this model. Suppose that we've measured the low energy (say at the scale Mz) values of two of the couplings, gi(Mz) and g2{Mz)- For each gauge group we may write down a relation between the MS coupling, <ji, and the physical coupling, g(q)i, (the object sitting in front of the F2 piece of the effective action) -27T = ^27-T + "T MQ/H) » 9i(l) SI (M) ^ 1 1 +Ain(g//l)+*in( gl{q) glbx) ' 2 ^ " ^ ' ^ ' 2-K* ^7T = ^ n

aim

(158) M

+ ~2 l n ( M 3 / ^ ) -

aiw

^),

(159) (160)

^

In writing these results I have ignored constants of order one, as they are subleading in the leading log approximation. Furthermore, I have dropped the q dependence in the contributions from massive scalars, assuming that q < MGUT- When q is order MGUT this log will be small and sub-leading, so the q dependence will be irrelevant. We have at our disposal two measurements as well as the relation gi(q) = 92(9) = 53(9) for q > MGUTNaively, this is not enough information to make any predictions given the number of unknowns. However, it is the magic of the log that will save the day. First we solve for g~\ and §2 in terms of the measured couplings 1 1 , 0o H9/Mz) (161)

m-#wz)'**

1 1 gl{q) g\(Mz) Using this result we may solve for 2

(^ MGUT = Mz

exp

—-

'

, A) \n{q/Mz) 2n*

•

(162)

"

(163)

MGUT

\

l

1 2

—-2-

\ fa L51 ( M | ) 5i(M|)_ Now 33 is frozen at the GUT scale, so we can determine its value by solving for gi{MGuT) HMGUT/MZ). (164) g\1 g\(Mz) ^ 1 . A)

172

Notice that in making this prediction it is crucial that M3 ~ MQUT, otherwise #3 would run between those two scales. Perhaps more importantly the prediction relies upon the fact that the sensitivity to the threshold mass M3 is only logarithmic, so that even if M 3 differs from MQUT by a number of order one, the running between these two scales is negligible. If we only consider flat manifolds), then this property is unique to four dimensions. Exercise 3.3: This calculation was done in the context of a non-decoupling scheme. Redo the calculation in terms of an effective theory calculation by matching and running to each threshold. Let us now see how this prediction runs into trouble in five dimensions. Notice that the couplings do not "run" in the five dimensional theory since there are no log divergences. That is, there is no dimensional transmutation, the theory was not conformal classically. Thus it is always true that g\ = 92 = 5 3 , but is not true that the force between two charges will be the same for all the gauge groups at all distances. Thus instead of a running coupling I will define an "effective coupling", ge^ which will represent the strength of the force. Calculationally g^* is sum of the bare coupling gi plus radiative corrections. We will assume that we have compactified on a circle of radius R, and that the symmetry breaking masses are such that MiR > 1. For smaller masses we would expect that we would just reproduce the four dimensional log running which we know gets the answer for the weak angle wrong when the GUT scale is near a TeV. Consider the prediction for sin2 9w in this model. Assume we have measured the force between two charges at some low scale r _ 1 = Mz for g\ and g2- For the group with only massless fields in the loops the strength of the force at the scale E = Mz is

^ =ff|+ S l n ( 2 7 r M ^ ) -

(165)

This result is what we would expect. At low energies the corrections to the potential are governed by loops of zero modes of massless fields and we get back the four dimensional result. For the second gauge group we get 1 e// 2

G?2 )

1

< P0.H2irMzR) 9l W

+C

2

^ .

(166)

The term proportional to /3o is the usual massless result. The second term arises from the massive scalar field. Using the same logic as in the four

173

dimensional case, we would like to predict

.), (.9a

f/f)2

9l

+ C2

2M3R —.

via )

(167)

In these equations I have assumed that the power like piece dominates in (150), otherwise we would just reproduce the four dimensional result. We can see that when we try to solve for R using the first two equations, we won't be able to drop the term which goes as M2R, since it's not small compared to the log term. There are more unknowns than equations and we have no predictive power. Notice that this shortcoming is not related to calculability, but to the UV sensitivity of the result. Even if you knew the "beta function" exactly it would not matter, you still would need to know the values of the masses of the GUT particles to accurately predict the low energy value of sin2 9w •

3.5.

Orbifolding

As I mentioned at the beginning of this lecture, the phenomenological interest in extra dimensional theories was resurrected when it was realized that the D-brane techniques discovered in string theory [60] could be useful for model building. The world volume theory of a D-brane can be described by a gauge theory which lives on the brane. Thus, D-branes allow for a natural way of "trapping" fields onto a surface (these fields won't have KK modes). In this way one can make the space as large as one wants without worrying about introducing phenomenologically meddlesome light KK modes. However, we are not free to restrict gravity to the brane, since by general covariance, all stress-energy couples to the graviton, and thus gravity must propagate in the "bulk" of space-time. Let us explore the effective field theories of theories with D-branes. If all the fields of our theory are trapped to the D-brane, then there really isn't anything interesting going on, since the theory is just a normal four dimensional theory up to the scale of quantum gravity.v So to make things interesting we will allow at' least some of our fields to traverse the bulk, and discuss the description of the theory in terms of an effective field theory [61]. Let us assume, for the moment, we are working in a flat V

I will assume here that we are talking about D3 branes which have four dimensional world volumes.

174

spacetime. w Then in order to keep the KK modes sufficiently massive to conform to accelerator limits, we must also compactify the space with a radius at least as big as a TeV. The scenario I shall consider here is a compactification on the five dimensional manifold M = MAx

S1/Z2

(168)

where the Z2 defines an equivalence class via a reflection across the S1. Note that Z2 acts (K) non-trivially on both the coordinates and the fields K:

(x,y)^(x,-y)

(169)

K :

$ - • R{k)&

(170)

where R(k) furnishes a representation of the Z2. I have chosen the circle to have radius R and coordinatized it in the two dimensional plane (x, y). The points y = 0 and y = TTR are fixed points meaning they are left invariant under the action of the Z2 and are singular. The Einstein equations require that we place a singular sources of stress energy at those points, so that is where we must put a pair of D-branes. Such singular spaces are technically not manifolds, and are instead called orbifolds. The thickness of the Dbranes are of order the string scale, so as far as our effective field theory description is concerned, they are infinitely thin in the limit where E/Ms goes to zero. How would the force law in this space differ from the case of the pure S1 we previously discussed? We would expect that at long distances the force laws should also be identical, as long as the massless spectra are the same. Suppose we allow for our charged scalar to propagate into the bulk, as in the previous case. There is a simple way to use our result for S1 to calculate the vacuum polarization on S1/Z2. First we must decide how the bulk fields transform under the Z2. If we want the massless spectrum to be the same then we must choose the field to be even under the Z2 otherwise the theory would posses no zero mode. We then KK decompose the field in the usual way

* = £ *™w-jm> n— — oo w

v

<171>

I n general the brane tension itself will curve the space, but we will assume that the tension is small enough so as not to significantly perturb the theory away from Minkowski space.

175

and 0 > z < n coordinatizes the fifth dimension. We then further decompose the field as $ = $+ + ; $ _

(172)

where

*-' = 7^1 ( $ ( n ) ( a ; ) " $ ( - ) ^ ) s i n ( f ) •

(173)

Then on S\ the action can be written as

s=o^r 5X5

f

r

2

a

£ y ** (»„*S°) + ( ^ v - ^ (($in))2 + ($^)2)

• (174)

n=l'

Orbifolding by Zi eliminates the $ _ component so that we may write 7T51 = 7 T S i / Z 2 + 7T ( _ ) ,

(175)

where 7P ) is the contribution from $_ . Recall that in the case of Si, we found a finite result, and we concluded that the UV divergence from the zero mode must have cancelled with the pole from the KK sum. But since 7r(~) = 7f(+) (where the bar indicates that we don't include the zero mode) we can see that for the orbifold we no longer expect a finite result since 7TSV.Z2 = nsi - ^ ( ^ ( + ) +

7r(

~ ) ) = 2 ^ 1 + 271"0'

(176)

and we know that the KK mode sum (W^+' + n^) is divergent, since its divergent piece is minus the zero mode divergence. At first this may seems rather mysterious given that we would not expect that a non-local operation like a change in boundary conditions could affect a local structure like a UV counter-term. How can the local bulk physics be sensitive to what's going on at the boundaries? The answer is, it is not, the divergence corresponds to a local divergence on the boundary. Introducing the boundary allows for the existence of new counter-terms, since we can now write down localized interactions on the boundary, SB

f d5x(5(z)L0

+ S(z - R)LR)

(177)

where La and Ln are the most general actions allowed by the symmetries of the theory.

176

Returning to our calculation, we find

**'*"¥* + ^4

+C

>-ijt^rt-

(178)

As I said previously we would expect that at long distances one cannot distinguish between S1 and S1 /Z2, so we can immediately read off the q dependence of irs1/z2 from (178), when qR /z3

= \<12)s^

- 9 ^ 2 ln(g2/M2) = - ~

m(
(179)

and in fact this agrees with the expansion of the result we found previously in (150). Note again that' we do not expect constants to agree as these are local short distance contributions. The \ divergence can be subtracted by putting a FF counter-term on the boundary. Before closing this section I would like to discuss a common mistake you might find in the literature. Suppose we have a theory on S1/Z2, where we have a certain set of fields ($0) which are localized to a boundary at one boundary at z = 0, another set $ # on the second boundary which we will place at z = R, and finally yet another set which propagate in the bulk $ 3 . At low energies we may write down an effective field theory consisting of the low energy modes of $0 and QR and a few {NKK) of the low lying KK modes of the bulk fields,x such that dAxLEFT

= I d*x Y, (M*o) + mR) •^

+ L($5)).

(180)

n=0

At this point I have not allowed for interactions between the various fields.y Whose effective theory is this? An observer in the bulk? Or on one of the boundaries? The answer is that the question is not well posed. At low energies we cannot probe distances short enough to talk about positions in the fifth dimension. The action correctly describes the physics for all the observers. The Lagrangian will correctly describe all low energy experiments. However, the possible experiments which can be performed will depend upon the location of the observer, because an observer on the brane at z = R will not have access to a beam of particles which are interpolated x

T h e number of modes we choose to keep is fixed by the maximal energy allowed in the theory. y Strictly speaking this isn't possible since they must interact at least gravitationally given that the fields are sources of stress energy. Such interactions terms will be described in the effective theory via higher dimensional (i.e. d > 4) operators which are suppressed by the Planck scale.

177

by $ 0 and visa versa. In fact, matter on the brane at z = R will just look like dark matter for an observer on the brane at z = 0. 3.6. Classical

Divergences

as UV

Ignorance

Let me now utilize an extra dimensional theory to emphasize one of the key ideas that I have tried to convey in these lectures. The UV divergences we find in field theory calculations are not exotic, nor do they imply the theory is sick. They just tell us that our theory has limited predictive power, because they do not correctly describe the UV. To hammer this point home, I would now like to give an example where the UV divergences show up at the classical level [63,62]. Suppose I have a six dimensional theory with a 3-brane located at the origin, around which I impose a Z% symmetry, so that the space is M 4 x R2 jZi. Strictly speaking for arbitrary brane tension, the true geometry is conical [64] and not a flat orbifold, but I can always fine tune the tension so as to get a solution to Einstein's equation which yields a Z^ flat orbifold. To simplify matters I will consider a simple theory of a real scalar field in Euclidean space such that 5 = |ci6xQ(^)

2

+ i m V + A04 + ^ ( x ) i m ^ .

. (181)

where for simplicity I have only included a mass term on the brane. Now let's calculate the two point function for . Ignoring for the moment the brane mass term we find d4k

GoM

f

feikq-(y-y')

d2

= I T( 2HT TZ) 7/ T ^ *~ ' - * ' ' (2TT) 4

2

_|_

eikq-(y+y')\

T^—2q

k2 +

2

"

(182)

Exercise 3.4: Prove equation (182). To calculate the complete two point function we include insertions of the brane mass so that G(x,x')

= G0(x,x1) -ml +m4b f d6zd6z'G(x,

f

d6zG0(x,z)52(z)G0(z,x') z)52(z)G(z, z')52(z')G{z',

x') + ...

(183)

Since we have four dimensional Poincare symmetry it is simpler to work in the mixed momentum configuration space representation of the Greens

178

function Gk(w), where w are the transverse coordinates. Then Gk(w, w') = G°k(w, w') - m2G°k(w, 0)G°k(0, w') +m^(»,0)G2(0,0)G»(0,»') + ... .

(184)

Notice that G°(0,0) is divergent. This divergence arises as a consequence of the fact that we have approximated the brane as being infinitely thin. But we know this is not a problem, we simply parameterize our ignorance of the correct brane structure in a local counter-term.

Exercise 3.5: Show that in the MS scheme the beta function for mi, is given by ^ <

= <.

d/i

TT

(185)

Suppose we knew the true structure of the brane, and that it can be described by some profile F(w), which in the limit where the UV scale (which well call the Planck mass) goes to infinity approaches a delta function, lim

F(w) = \m2b82(w).

(186)

This information can then be encoded in the action S=

fd6x

Q (dA
.

(187)

Then the divergent integrals gets damped out by F and the result is finite. We have one less counter-term to fix and thus increased predictive power, but only because we claim to know the underlying theory. Of course, there is an even simpler example of UV divergences arising classically. The total energy in the Coulomb field of a charge particle diverges when the electron is treated as point-like. 3.7. Effective

Field Theory in Curved

Space

Adding curvature to the space-time manifold forces one to carefully inspect many of our starting assumptions in building a quantum field theory. For instance, even the notion of what is a particle must be reconsidered [65] since if the space-time does not admit a time-like killing vector there is no notion of a conserved energy. However, things simplify considerably as the space-time becomes more symmetric. So we will limit our discussion to a

179

specific set of theories which posses four dimensional Poincare symmetry. These theories have preferred vacua around which to quantize. Such spacetimes have metrics which can be written as ds2 = dz2 + g{z)-qtl,l/dx'1dxv,

(188)

for simplicity I've chosen to work in five dimensions, generalization to higher dimensional spaces is straightforward. Such space-times are often called "warped". We can use quantum field theory on this, as well as any other, manifold as long as the curvature scale is small compared to the Planck scale. Much, though not all, of our flat space effective field theory intuition carries over into curved space. Most importantly, the counter-term structure of the theory does not change since for momentum much shorter than the curvature the space looks flat.z So, as in flat space, we write down all possible counter-terms which are consistent with the symmetries of the theory under consideration. However, since momentum scales are now dependent upon the coordinate in the fifth direction, the momentum at which the loop expansion breaks down will depend upon the choice of observable. Let me specialize to a particular space which has received much attention in the past few years. Namely, five dimensional anti-deSitter space compactified on S1 jZ-i. anti-deSitter space is a maximally symmetric space with negative cosmological constant [66], and its compactification on Sl/Z2 was used in [67] to solve the hierarchy problem. The warped metric can be written in the conformally flat form ds2 = — ^ {dz2 + •nia/dx'idxv)

(189)

where k is the inverse radius of curvature. The two branes necessary to account for curvature singularities are positioned at z = 1/T (the "TeV brane") and z = 1/k (the "Planck brane"). T is taken to be near the TeV scale while the bulk cosmological constant (A) is related to the curvature scale via fc2 =

~ (190) V ; 24M 3 where M is the five dimensional fundamental scale, i.e. it is the scale in front of the Einstein term in the action. Note, that A will not be the four dimensional cosmological constant, nor will M be the effective Planck z

Here I'm ignoring counter-terms involving powers of the curvature which will in general be generated.

180

mass. This theory, in conjunction with a Goldberger-Wise stabilization mechanism [75], solves the hierarchy problem, and is sometimes referred to as RSI. A simple way to see this is to note that if the standard model fields are localized to the TeV brane, then the masses of the fields are naturally of order MT/k (see [51]), while gravity remains weak. Let us study the low energy effective field theory derived from this model. Now we would like to ask, what observables are calculable in an effective field theory? If we perform a KK decomposition, the action looks like

S= fd*xY,Hp,n), ^

(191)

n

where the fields 4>T (4>P) are fields localized to the TeV (Planck) brane, and 4>n are the KK modes of the bulk fields. The KK masses correspond to zeros of Bessel functions (see [51]), and the lowest lying masses are near a TeV (note that there is a vanishing effective cosmological constant. This only comes about after fine tuning a linear combination of the brane tensions.). Prom a low energy view point this looks like a theory with an effective compactification scale of a TeV, since the first KK mode has a mass of that order. So if we dimensionally reduce to a four dimensional effective theory, we will have, in general a set of zero modes and a tower of KK modes. Additionally we could have zero mass (m
S = ^ J d5x^FMNDFMN

(192)

where A is a dimensionful quantity of order 1/M in our normalization where the gauge field kinetic term is accompanied by a factor of 1/g2. The contribution of this term to the zero mode gauge field two point function will

181

be of the form U(q2) = Xk^,

(193)

so we can see explicitly that we have no hope of calculating the gauge field zero mode two point function for <j,2much larger than T2.

Exercise 3.6: Prove equation (193). Be sure to remember that the gauge potentials are the components of a one-form, so its index is naturally down.

At first this result is rather disheartening, since it appears to mean that extra dimensions with size of order the TeV scale have no hope of reproducing the four dimensional prediction for the weak mixing angle. Is there no way to have higher dimensional physics which is accessible at the LHC without the loss of this beautiful prediction [76]? The answer is yes, and it is through the magic of AdS. 3.8. Unification

in AdS 5

Suppose we calculate the low energy (q
<
l 48TT 2

VH&)

—r / dxx\Jl

2 16TT >TT2 Jo

— x2\a.N

where

\nN(p) = -\n\iv{jplT)kv{p/k)-iv{p/k)kv{p/T)\, iv(z) = (2-v)lv(z)

+

kv{z) = {2-v)Kv{z)-zKlJ_l{z),

(195)

zlv-1(z), (196)

and v = \Jk + m2/k2. Let's study this result in some detail. We first notice the UV pole A. This pole is identical to the pole we found in flat space orbifold, and can be subtracted away by a boundary counter-term. This fact also tells us that the ji dependence in (195) determines the running

182

of the boundary couplings, one of which lives at z = 1/T (AT) the other at z = l/k (Afc). In the MS scheme, the effective dimensionally reduced coupling looks like g2(q2

= - 2 +A f c (/i)95

+ 16TT

2

1

A T (/i)

(jl_\

1

96TT2

n

\4kTj

/ dxx v 1 — x2 In N Jo

(197)

where R = ln(fc/T)/fc, is the effective compactification radius. The first three terms result from the tree level bulk gauge coupling and boundary couplings, respectively. The remaining terms encode all the information about the KK modes. In flat space this term would be linearly sensitive to the bulk mass so in AdS we would expect that for m larger than the curvature scale fc we would again have linearly sensitivity. The interesting case is m < k. Using the expansions 2+u

Iz\u

r

„,

0N

-,

(198)

i > n ) ( 2 ) [i+ °(*') + - ] , r("), 2 "©""[ l + 0(z ) + ---}

(199)

for p ^C T, we have InJV(p)~-ln

k

i

(200)

T

and thus, for m ^ 0 1 g2(q2)

- -J2 + A fe(M)

9l

XT(n)

+analytic in q2.

1 96TT2

i (-£-] n

\4kTJ

48?r2

In

y2

(201)

This is a remarkable result, in that the five dimensional vacuum polarization looks just like what we would expect from a four dimensional calculation. The result (201) is only logarithmically sensitive to the masses, so there is hope that a GUT theory with a large mass scale (by large I mean large compared to the TeV scale, but still smaller than k) will still lead to the correct value of the weak mixing angle.

Exercise 3.7: Show that in the limit where m 3> k the inverse coupling is linearly sensitive to the mass.

183

Let us try to get a better understanding of how the logarithmic sensitivity to the symmetry breaking scale arises. It seems rather strange that we can make a prediction for a lower energy observable in terms of high energy parameters (the symmetry breaking scale m) when the observable under consideration is not well defined at scales above T. Clearly we can't get the result (201) from an effective field theory calculation where we run the zero mode correlator down from the scale m. Instead of tracking the zero mode correlator we will calculate another observable, calculable at higher scales, which matches onto the zero mode correlator at the scale T [68]. This other observable is the Planck brane gauge field correlator, which is defined as the gauge field two point function with its end point on the Planck brane. This correlator is calculable up to the scale k. In fact, the cut-off scale for any correlator restricted to a hyper-surface of fixed z is 1/z. The tree level Planck brane correlator, in mixed momentum coordinate space representation, can be written in the Az = 0 gauge, as (here r)^ is the flat Euclidean metric) £>£"(*, z') = rTDq(z,

z') + ^Hq{z,

z'),

(202)

where we won't need the form of Hq. If we take both end points on the Planck brane (z = 1/fc) one finds (for q> T) 'xe^iA^x,

l/k)Av(0,

l/k)) cx -^rrV^

+ ••-,

(203)

whereas in the case where only one end-point is on the Planck brane (this is needed for the one loop calculation) we find [69] D(zl/k)_kz

K,{qz)I0{qIT)+KQ{qlT)I0{qz)

At q
184

since it depends only upon the geometry. So that is a first step in understanding the result (201), but the real challenge in understanding the logarithmic mass dependence. From a four dimensional perspective this term looks like it's coming from a heavy scalar which decouples at its mass scale (compare to the result in section 3.4). In terms of the Planck brane correlator we can understand this from the following remarkable fact. All of the KK modes with masses much less than k, save for one, are localized to the TeV brane while the external gauge boson lines in the self energy die off exponentially fast as one moves away from the Planck brane. So the net contribution of the KK modes to the symmetry breaking effects is suppressed relative to contribution of a single mode. The modes with masses larger than k contribute in a symmetric way and thus only affect the unmeasurable counter-term. The one special mode has a mass near the value of the bulk mass. In the massless case this mode is the zero mode, which is flat, while in the massive case there is a "magic mode" which has support near the Planck brane [77]. Let us see how this happens quantitatively. In the limit T < A / ^ < k

Dg(z,l/k)

K^f'-

<205>

we see the exponential suppression of the gauge field propagator. This suppression will kill the contribution of the TeV brane localized KK modes. However, one must be careful to note that there is regulator dependence in this statement. A more precise statement would be that the non-analytic momentum dependence generated by the KK modes is exponentially small. I will refer the reader to reference [68,69,74] for more details. For the case of gauge fields and fermions, the suppression of the KK modes does not follow this argument, as they are not in general TeV brane localized. In fact, the contributions from these modes are only logarithmically suppressed. That is, they look like they contribute to the two loop beta function. This can be seen from studying the non-decoupling calculation for gauge field [73]. But given that these effects are of the same order as the usual unknown threshold effects, they won't change the prediction for the weak mixing angle [74]. To understand this behavior from a nondecoupling argument one can use arguments based on AdS/CFT, but these are beyond the scope of these lectures. At this point I should also mention that the gauge fields differ from scalar fields in another important way. The one loop calculation for a gauge field with mass mx gives a contribution of

185

the form 777

ir(q2) oc ln(^ff-) + . . .

(206)

so that the effective mass is parametrically enhanced [74], though numerically, this effect is small unless we are willing to tolerate a large hierarchy between k and rrixLet's now consider the effects of the higher dimensional operators on our prediction for the weak mixing angle, as they could perhaps ruin our predictive power. Any higher dimensional operator which is insensitive to the symmetry breaking masses will obviously not affect the prediction. These higher dimensional operators would all correspond to some universal contribution to the couplings and thus be irrelevant for the difference which determines the mixing angle. It is also possible that there are higher dimensional operators such as got an expectation value. However, since m
Acknowledgments I would like to thank the TASI organizing committee, Howie Haber, Ann Nelson and especially K.T. Manhanthappa for organizing yet another fruitful TASI meeting. I would also like to thank the UCSD theory group for its hospitality, as well as Iain Stewart and especially Walter Goldberger for comments on the manuscript.

186

References 1. The basic ideas behind effective field theory go back to Wilson's original work. The intuition gained by understanding the renormalization group in position space in terms of block spins makes it well worth the time to study this topic. I suggest the following review to the interested reader. K. G. Wilson and J. B. Kogut, Phys. Rept. 12, 75 (1974). 2. M. E. Peskin and D. V. Schroeder, "An Introduction To Quantum Field Theory,", Reading, USA: Addison-Wesley (1995) 3. S. Weinberg, "The Quantum Theory Of Fields. Vol. 2: Modern Applications", Cambridge, UK: Univ. Pr. (1996) 4. H. Georgi, "Weak Interactions And Modern Particle Theory," Menlo Park, USA: Benjamin/Cummings (1984) 5. T. Appelquist and J. Carazzone, Phys. Rev. D 11, 2856 (1975). 6. For a discussion see G. Sterman, "An Introduction To Quantum Field Theory,", Cambridge, UK: Univ. Pr. (1993) 7. S. Kamefuchi, L. O'Raifeartaigh and A. Salam, Nucl. Phys. 28, 529 (1961); R. E. Kallosh and I. V. Tyutin, Yad. Fiz. 17, 190 (1973) [Sov. J. Nucl. Phys. 17, 98 (1973)]. 8. The use of the OPE in DIS is discussed in [2]. The B decay calculation is discussed in [18], 9. H. Georgi, Nucl. Phys. B 361 (1991) 339. 10. J. Wess and B. Zumino, Phys. Lett. B 37, 95 (1971). 11. E. Witten, Nucl. Phys. B 223, 422 (1983). 12. E. Witten, Phys. Lett. B 117, 324 (1982). 13. E. D'Hoker and E. Farhi, Nucl. Phys. B 248, 59 (1984), Nucl. Phys. B 248, 77 (1984). 14. J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 (1981). 15. For a review and references, see P. F. Bedaque and U. van Kolck, arXiv:nuclth/0203055. 16. N. Seiberg, Phys. Lett. B 390, 169 (1997) [arXiv:hep-th/9609161]. N. Seiberg, Phys. Lett. B 388, 753 (1996) [arXiv:hep-th/9608111]. 17. A. J. Buras, arXiv:hep-ph/9806471. 18. A. V. Manohar and M. B. Wise, Cambridge Monogr. Part. Phys. Nucl. Phys. Cosmol. 10, 1 (2000). Also see M. Luke, these proceedings, as well as M. Neubert, Phys. Rept. 245, 259 (1994) [arXiv:hep-ph/9306320]. 19. H. Georgi, Phys. Lett. B 240, 447 (1990). 20. The theory was originally formulated in, W. E. Caswell and G. P. Lepage, Phys. Lett. B 167 (1986) 437. The formulation discussed in these lectures was given in [21]. For a review see [22]. NRQCD in the context of production and decay was discussed in [23]. 21. M. E. Luke, A. V. Manohar and I. Z. Rothstein, Phys. Rev. D 61, 074025 (2000) [arXiv:hep-ph/9910209]. 22. A. H. Hoang, arXiv:hep-ph/0204299. I. Z. Rothstein, arXiv:hep-ph/9911276. 23. G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51, 1125 (1995) [Erratum-ibid. D 55, 5853 (1997)] [arXiv:hep-ph/9407339].

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189 66. S.W. Hawking and G.F.R. Ellis, "The Large Scale Structure of Space-Time", Cambridge, Uk: Univ. pr. (1975). 67. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hepph/9905221]. 68. W. D. Goldberger and I. Z. Rothstein, Phys. Rev. Lett. 89, 131601 (2002) [arXiv:hep-th/0204160]. 69. A technique for calculating loop diagrams in higher dimensional spaces, including warped spaces, can be found in the appendix of, W. D. Goldberger and I. Z. Rothstein, arXiv:hep-th/0208060. For results including higher spin fields see [73]. A method of calculating vacuum bubbles is given in [70]. 70. W. D. Goldberger and I. Z. Rothstein, Phys. Lett. B 491, 339 (2000) [arXiv:hep-th/0007065]. 71. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150]. 72. E. Witten, remarks at ITP Santa Barbra conference, "New Dimensions in Field Theory and String Theory". S. S. Gubser, Phys. Rev. D 63, 084017 (2001) [arXiv:hep-th/9912001]. N. Arkani-Hamed, M. Porrati and L. Randall, JHEP 0108, 017 (2001) [arXiv:hep-th/0012148]. R. Rattazzi and A. Zaffaroni, JHEP 0104, 021 (2001) [arXiv:hep-th/0012248]. 73. K. w. Choi and I. W. Kim, Phys. Rev. D 67, 045005 (2003) [arXiv:hepth/0208071]. 74. W. D. Goldberger and I. Z. Rothstein, arXiv:hep-ph/0303158. 75. W. D. Goldberger and M. B. Wise, Phys. Rev. Lett. 83, 4922 (1999) [arXiv:hep-ph/9907447]. 76. A. Pomarol, Phys. Rev. Lett. 85, 4004 (2000) [arXiv:hep-ph/0005293]. L. Randall and M. D. Schwartz, Phys. Rev. Lett. 88, 081801 (2002) [arXiv:hep-th/0108115]. 77. K. Agashe, A. Delgado and R. Sundrum, Nucl. Phys. B 643, 172 (2002) [arXiv:hep-ph/0206099]. 78. K. Agashe, A. Delgado and R. Sundrum, Annals Phys. 304, 145 (2003) [arXiv:hep-ph/0212028]. W. D. Goldberger, Y. Nomura and D. R. Smith, Phys. Rev. D 67, 075021 (2003) [arXiv:hep-ph/0209158]. M. Carena, A. Delgado, E. Ponton, T. M. Tait and C. E. Wagner, arXiv:hep-ph/0305188. L. J. Hall, Y. Nomura, T. Okui and S. J. Oliver, arXiv:hep-th/0302192. Z. Chacko and E. Ponton, arXiv:hep-ph/0301171.

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MICHAEL LUKE

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BOTTOM QUARK PHYSICS A N D THE HEAVY QUARK EXPANSION

MICHAEL LUKE Department of Physics University of Toronto 60 St. George Street Toronto, Ontario Canada M5S1A7 E-mail: [email protected]

These lectures were given at TASI-02 on the subject of b quark physics. Contents 1

Introduction

194

2

Precision Measurements and Flavour Physics 2.1 b Quarks as Probes of New Physics

194 198

3

From Quarks t o Hadrons 3.1 Effective Field Theory 3.2 m\y > fi > rrn,: Four-Fermi Theory Generalized 3.2.1 Semileptonic decay 3.2.2 Nonleptonic b -> c decay 3.2.3 b -> S7 decay 3.3 n < mb: HQET 3.4 Inclusive Decays and the OPE

203 207 208 213 214 214 216 221

4

Applications 4.1 Inclusive Decays: Corrections to the Parton Model 4.1.1 Total decay widths 4.1.2 Differential widths and spectra 4.1.3 Vuf, and the shape function 4.2 Exclusive Decays and Symmetries 4.2.1 B -> D*tv and Vcb 4.2.2 B -> J/iiKs and sin 2/3

227 227 228 229 231 234 234 237

5

Conclusions

239 193

194

1. Introduction In these lectures I hope to give you an introduction to bottom quark physics, and a feel for the theoretical issues which are involved. Because of time constraints, I cannot cover all aspects of the field, so I have chosen to focus on a few key ideas. The most important is the idea of "factorization". This is a word which is used to mean many different things in particle physics, but here I use it in the very general sense of separating long and short-distance physics. This is important for B decays, because much of the theoretical progress in the subject relies on being able to separate the short-distance physics at scales 1/mw and 1/mj, which determines the quark-level process (and for which perturbation theory is a useful tool) from the complex longdistance physics of hadronization. Effective field theory will be introduced as a tool which makes this factorization automatic. The outline of these lectures is as follows: First, I will discuss the use of precision measurements of b quark transitions to probe physics at or above the weak scale. I will then discuss the extent to which 6-flavoured hadrons may be used to probe the interactions of b quarks, and then look at some important applications. Finally, I will briefly discuss some current directions of research. Throughout the lectures I will use the decay b —> sj to illustrate the concepts, since a complete understanding of this decay requires almost all of the theoretical machinery I will discuss. 2. Precision Measurements and Flavour Physics As you have heard in many lectures at this school, there are many good reasons to believe that the standard model is not the correct description of nature at scales above a few hundred GeV or so. A strongly-inter acting symmetry-breaking sector and supersymmetry in one of its many forms are perennial favourites for new physics, as are the more recent ideas of large extra dimensions and Little Higgs models. All of these theories conjecture new particles at the 100 GeV-few TeV scale. Unfortunately, to date no experimental evidence for such particles has been found. Historically, there have been two ways to discover new particles experimentally. The first is to build ever-larger colliders which can directly create more and more massive particles. The second is to perform precision lowenergy measurements, which corresponds to looking for new physics via virtual effects. While the first is usually preferable, because it's easiest to study particles if you can actually make them, important complementary information can be found by following the second path. This is the course

195

being pursued at the B factories at SLAC and KEK, as well as by many other precision low energy experiments, and has has a number of important successes in the past. Here are a few examples: (1) Virtual c quark loops contribute to K°—K° mixing via the diagrams in Fig. 1. From the measured size of the mixing, Gaillard and Lee [1] were able to predict that mc had to be less than a few GeV.

Kc

>u, c, r>

d Figure 1.

\ K°

s

K°\

*u,c,t

d

Kc

+

s

Quark-level diagrams contributing to K° — K° mixing.

(2) Similarly, virtual t quark loops contribute to B° — B° mixing. In 1986, the ARGUS and UA1 experiments measured a rate for this process that was much larger than expected [2], implying a top quark mass greater than 50 — 100 GeV. This was the first evidence for a heavy top quark. (3) Precision electroweak measurements (mostly at LEP) determined rrit = 169l 1 8 1 2 GeV [3], making it fractionally one of the best known quark masses, before it was even discovered. Indeed, the constraints from precision measurements puts strong constraints on a variety of new particles at the few TeV scale, and has even been dubbed the "little hierarchy" problem - the fact that there appears to be a sizeable hierarchy between the Higgs mass in the Standard Model and the scale of new four-fermi operators. It is straightforward to calculate the Standard Model contribution to flavour transitions at the quark level. Since there are many more possible transitions than free parameters, one could therefore in principle compare with experiment and either discover discrepancies or place constraints on models of new physics. However, there is a problem: when comparing with experiment the theoretical predictions often have large theoretical errors arising from the difficulties in relating the underlying quark process to the observed hadronic process (denoted by the black blobs in Fig. 1). Even purely leptonic interactions can be plagued by hadronic uncertainties

196

through loop effects, as the next example illustrates. Dealing with this issue will occupy much of these lectures. Example: g — 2 of the muon An instructive example of the importance of understanding low-energy QCD is provided by the recent measurement of the anomalous magnetic moment of the muon by the E821 experiment at Brookhaven [4]. As is well known, there was initially great excitement in the community due to a reported 2.6 a discrepancy between theory and experiment: (af - alxp) x 10 11 = 426 ± 150 exp ± 67 t h .

(1)

where a^ = \ (g — 2). This was seen as a possible signal of new physics, such as the supersymmetric contributions such as those in Fig. 2. Unfortunately, the discrepancy was reduced to a little over one standard deviation when a sign error was found in the theoretical calculation [5]. Nevertheless, it is worth considering what the theoretical limitations on this prediction are, to determine what level of discrepancy would be required to be a convincing signal of new physics.

Figure 2.

A supersymmetric contribution to g — 2 of the muon.

The bulk of the contribution to g — 2 arises from QED diagrams, which have been calculated to a quoted precision of ± 3 x 1 0 - 1 1 . Electroweak corrections give a much smaller contribution, but with a comparable error. These are both reliable because they are purely perturbative calculations. Unfortunately, even with muons, low-energy hadronic physics creeps in, via the vacuum polarization graph in Fig. 3(a). Fortunately, while this graph is nonperturbative, the hadronic contribution to the photon vacuum polarization may be measured experimentally via e+e~ annihilation to hadrons, as well as hadronic r decay, via a dispersion relation [6]. (This is discussed in more detail in a different context in Section 3.4.) Unfortunately, there is also a "light-by-light" scattering contribution, shown in Fig. 3(b), which cannot be extracted from data. This contribution is therefore typically estimated using models of hadrons, not from QCD [7].

197

A recent estimate [5] of this contribution to a p is (80 ±40) x 10 - 1 1 , but the size of this error is controversial. The only completely model independent analysis is based on chiral perturbation theory, where there is a contribution from the unknown coefficient of an operator which could easily be several times larger than this uncertainty: a£ L (had) = (l3±gg + 31C) x H T 1 1

(2)

where C is a nonperturbative parameter which is expected, on dimensional grounds, to be of order 1. However, a value of 2 or 5 is equally natural; even the original discrepancy could be fit with C ~ 10. Fitting the experimental result gives C = 7± 6

(3)

(where we have added the uncertainties in Ref. [8] in quadrature).. Most

>

\

^ (a)

>

>

^

^

>

(b)

Figure 3. Contributions of (a) vacuum polarization and (b) light-by-light scattering to the muon anomalous magnetic moment. The blobs contains low-energy quarks and gluons and so are strongly interacting.

importantly, the contribution of C is roughly the size of the proposed new physics contributions. What this means is that it is extremely hard to convincingly extract signs of new physics from this measurement. Since we are currently unable to calculate C from first principles, we are forced to rely on models of low-energy hadronic physics. Certainly, some models are demonstrably better than others [5], but the agreement between theory and experiment is good enough that without a model-independent treatment of the hadronic physics it is very difficult to convincingly argue that a discrepancy of this order could be a sign of new physics, rather than a failure in our understanding of low-energy QCD. Furthermore, it is difficult to see how this theoretical error could be reduced in the future without a fully nonperturbative calculation. The moral of this story is that, when searching for new physics in precision experiments, we need precise and model-independent predictions with

198

quantifiable error estimates. If hadrons are involved anywhere, this necessarily involves being able to systematically deal with low-energy QCD. 2.1. b Quarks

as Probes of New

Physics

Before looking for deviations from the Standard Model, we need to understand it. As indicated in Fig. 4, we observe a complicated pattern of flavour transitions, which follow a well-defined hierarchy. In the Standard Model, this pattern arises because of the hierarchy of the CKM matrix,

(

Vud Vus Vub Vcd Vcs Vcb Vtd Vt. Vtb

+ h.c.

(4)

where

O(l) O(X)

o(x32) o(\ ) FCNC Figure 4. Model.

The relative magnitudes of quark transition amplitudes in the Standard

(I A A3 Vud Vus Vuf>\ Vcd Vcs Vcb r.- A 1 A2 3 \A A2 1 Vtd Vu Vtb J

^ C KM

(5)

and A = sin^c = 0.2205 ± 0.0018 is the Cabibbo angle. Since A is small, it is useful to treat is as an expansion parameter. This allows us to write a particularly useful parameterization of the CKM matrix due to Wolfenstein [9]: 1 VcKM KA\\l

¥

-A -p-ir))

A 1 . £ -AX

AXi(p-ir1y AX2 | + 0(A4).

(6)

199

A is known at the ~ 1% level, A = 0.83 ± 0.03 is known (due to measurements I will later discuss) at the ~ 5% level, and p and r\ have larger uncertainties. A huge number of quark transitions is therefore determined by these four parameters (in addition to the quark masses, gauge couplings, and gauge boson masses), a few of which are shown in Fig. 5. The standard model is therefore extremely predictive, and the goal of the B factories and other precision low-energy experiments is to test these predictions as redundantly as possible.

s

Figure 5.

*

Some flavour-changing transitions.

Transitions between quarks of different charge, such as b —> civg. and b —> cud, are dominated by tree-level graphs in the SM and many of its extensions, and therefore simply reflect the CKM hierarchy. At one loop, however, things get more complicated. The decays denoted "FCNC" are flavour-changing-neutral current decays, in which the transition is between quarks of the same charge. These decays are forbidden at tree level in the SM, but can arise through loop diagrams such as those in Fig. 6, and so are related in a complicated way to the various SM parameters. (In addition, the GIM mechanism [10] ensures that they are also forbidden in the limit of degenerate intermediate quarks.) Hence, the amplitudes for FCNC processes are suppressed in the SM, and so new physics at the few hundred GeV scale may contribute at a comparable level to these decays, as illustrated schematically in Fig. 7. Thus, rare b decays provide a sensitive probe of the structure of flavour physics at high energy scales. Since there are many such transitions determined by a small number of parameters in the standard model, one can place strong constraints on (or discover!) new physics by studying as many transitions as possible.

200

fY U

(a)

(b)

(c)

Figure 6. FCNC processes in the standard model. Note that some four-fermion transitions, such as b —+ uud (which in turn determines the hadronic process B —> TTTT) receive comparable contributions from tree-level and one-loop "penguin" amplitudes, because the tree level amplitude is suppressed by the small CKM element Vub. This "penguin pollution" is a serious impediment to the theoretical prediction for such decays, as will be discussed in Section 4.2.2.

IT Figure 7.

A contribution to b —> S7 from new physics.

The preceding arguments hold for transitions involving any of the quarks; however, b quarks are particularly good probes of short-distance physics. There are a number of reasons for this, both theoretical and experimental, but one of the most important is that there are a number of observables that can be cleanly predicted in the SM, despite the complications of hadronization. As we will discuss in these lectures, many of these arise because the b quark is heavy: rrn, ~^> AQCD, SO b decays are shortdistance compared with the QCD scale; However, clean observables are not restricted to the b system; for example, some rare kaon decays (such as K ^ -KVV) are also very clean theoretically, and are also being intensely studied [11]. Let me make a couple of additional comments on the CKM matrix: (1) CP Violation: VCKM has a single nontrivial phase (i.e. a phase that cannot be absorbed in a field redefinition), which in the Wolfenstein parameterization is put into Vub = \Vui>\e~'17 and Vtd = \Vtd\e~l/3 (these two parameters are not independent in the SM). This phase violates charge-parity (CP) invariance of the theory, and in the SM is responsible for the observed CP violation in the K and B systems. In the presence of CP violation, the rate for a

201

(p.v)

(0,1)

Figure 8.

The unitarity relation (7) in the complex plane.

decay and its CP conjugate may differ: T(B —> / ) ^ T(B —> / ) , and so measurement of CP conjugate rates may be used to determine the phase. CP violation is interesting for a number of reasons. First of all, it is interesting in its own right, in particular because the observed universe is highly CP-asymmetric: there is far more matter than antimatter. Hence, CP violation is necessary to explain the observed state of the universe [12]. For our purposes, however, it is interesting because (i) the SM is highly predictive (there is only one phase in the CKM matrix, and so all CP violation depends on the same phase) and (ii) the CP invariance of QCD may be used to get clean hadronic measurements of this phase, and hence to test the CKM picture (as will be discussed in Section 4.2.2). Note that the presence of a single nontrivial phase is a general feature of a 3 x 3 unitary matrix, so the weak interactions with three generations naturally violate CP (and hence CP violation is not a mystery, as is sometimes stated). (2) The Unitarity Triangle: It is customary to express all of this graphically. The unitarity of VCKM leads to the relation

vudv:b + vcdv;b + vtdv;b = o.

(7)

Since each CKM element is a complex number, this relation may be drawn as a triangle the complex plane, as illustrated in Fig. 8. The length of the base of the triangle has been scaled to 1 by dividing each side by |VcdKfc| = sin^clKbl, and the apex is at the point (p, rj) as defined in Eq. (6). The angles (3 and 7 are the CKM phases denned in Eq. (6), and a = TT — j3 — 7. This construction is known as the "unitarity triangle" (UT). There are two other such triangles, arising from the two other unitarity relations, but the relation in

202 1.5

* * * « & U 0.5

sin 2 ^

ivjv, -0.5

-0.5

0.5

P

Figure 9. The experimental constraints on the unitarity triangle from different processes, and the best fit (as of summer, 2002).

Eq. (7) has the further property that each term is of order A3, so the sides of the unitarity triangle are of comparable length, while the other triangles have two sides much longer than the third and so are almost flat. This in turn means that the sides and angles of the UT should be measured with comparable precision to obtain useful constraints. Note that in the absence of CP violation, the UT would collapse to a line. Hence, by performing only CP conserving measurements of the sides of the triangle, one may determine the CP violating phase in VCKMThe UT provides a convenient way to plot constraints on the SM from different experiments, as illustrated in Fig. 9. Note, however, that it can be misleading as a guide to new physics: two measurements of a given parameter may be redundant in the SM, but in the presence of new physics are as interesting as measurements of two different parameters. For example, the CP violation in both B —* J/i^Kg and B —>• (f)Ks is proportional to sin 2/9 in the SM. However, in the presence of new physics which contributes to the b —> sss transition but not to b —* ccs this need no longer be true,

203

and so both measurements are important. Thus, the goal is not simply to measure the sides and/or angles of the unitarity triangle, but rather to measure as many different quark-level transitions as possible, to look for deviations from the SM predictions. Currently, all determinations of sides and angles of the UT are consistent with one another within the errors (see Fig. 9). 3. From Quarks to Hadrons In the last lecture we discussed how precision measurements of b quark transitions provide a sensitive probe of New Physics. Unfortunately, we can only do experiments on hadrons, not free quarks and gluons, and a simple quark transition like b —> cud can give rise to many complicated hadronic process, such as B —> DTT, B —> Dp or B —> Dinnrir, etc., all of which are completely incalculable analytically. The problem, of course, is that even though the decay of a 6 quark is a short-distance process, occurring over a typical distance scale r ~ 1/rrib, hadronization is by its very nature long-distance (r ~ 1/AQCD) and nonperturbative.

SJmn

^WD Figure 10.

An incalculable mess.

Fortunately, a Feynman diagram like that in Fig. 10 is in fact a terrible representation of the physics of B decay. If you were to actually calculate this diagram, you would find loop integrals containing wildly disparate scales: mw (the scale of the short-distance physics underlying the decay), mi, (the typical energy release of the decay) and AQCD (the scale of hadronization). Some of the physics would be perturbative, and some nonperturbative. The short-distance physics would be independent of the complications of hadronization, whereas the long-distance physics would be insensitive to many of the details of the short-distance physics (just as fluid mechanics is insensitive to the structure of the proton). Thus, a naive dia-

204

grammatic analysis obscures much of the important physics of the process, and the corresponding simplifications. Example: b —> sj We can illustrate this with an instructive example. Suppose we were particularly sheltered theorists who had never heard of hadronization, but wanted to calculate the rare FCNC process b —> sj in perturbation theory. As discussed in the previous section, because this decay arises at the loop level in the SM, new physics at the weak scale can be competitive with the SM rate, and so it provides a sensitive probe of weak-scale physics. We therefore need to be able to calculate the rate in the SM, where it gets contributions from the diagrams in Fig. 11.

b

,^%

^ y ^ v w ^ », ^ y (b)

(a)

(d)

(c)

Figure 11. Feynman diagrams contributing to b —> S7 decay in the SM at one loop. The contribution from intermediate u quarks is negligible.

Since we know about perturbation theory and infrared divergences we know we have to include graphs with both real and virtual gluons to have an infrared safe quantity. The decay is dominated by the t quark loop, and we find at 0(as) [13] TFmb

r(6 -+ x3l)

1287T

4

4a«

VM

b

w

+3

log-

(8)

7T

where [14] F2{x) = -:

+

4rr

4 0 - l) 2 1

2z-l

+ 4 (x - l ) 2

2{x 1 3 2(x-l)3

2 (x - l ) 4 2 (x - 1)

logs,

log a; (9)

Xs denotes s or s+gluon, c is a non-logarithmic term, and I have neglected small terms suppressed by powers of m^/m 2 ,. When we tried to compare with experiment, we would discover that nobody had measured the decay rate for the quark level process b —> Xsj. However, we could argue that the final state quarks and gluons have to

205

hadronize to something, so the appropriate thing to compare this to would be the sum of the decay rates to all final states containing a strange quark and a hard photon (K*j, K**j, Knj, etc.). Such decays are called inclusive decays, to be distinguished from exclusive decays to a specific hadronic state. The measured inclusive branching fraction is [15] Br(B - • Xsj)

= (3.3 ± 0.4) x 10" 4

(10)

Comparing this with the theoretical prediction in Eq. (8) is, however, problematic. At tree level, Eq. (8) gives to a branching fraction of about 1.3 x 10~ 4 , or almost a factor of three smaller than observation. Before we get too excited about having seen evidence for new physics, we notice that the one-loop correction to this result is huge, because it contains a term proportional to logm1/m\. This provides an enhancement by about a factor of seven, which counteracts the as/-7r suppression from the loop. Indeed, the oneloop correction is more than a factor of two larger than the tree level result! There is also a large scale uncertainty in this result: evaluating as(fj,) at fi = m,b increases the predicted branching fraction by almost an order of magnitude to 1.2 x 10~ 3 , or about four times the observed rate, while choosing fj, = raw lowers the prediction to about 7 x 1 0 - 4 . Both raw andTOf,are relevant scales in the problem, so there is no obvious way to choose between them. The difference is formally a two-loop effect, but the corresponding uncertainty in the branching fraction is huge. Perturbation theory is therefore showing no signs of converging: two-loop, one-loop and tree level effects are all about the same size. Clearly, we have a problem if we want to do precision physics. At the moment we have no idea whether the experimental result is in accord with the SM. Despite the fact that as(mb)/ir is less than 0.1, perturbation theory is failing badly because of these logarithmic enhancements. Nor will this problem stop at two loops - at n'th order in perturbation theory we will find terms proportional to a™ log"{m%/m\). It looks like we can only predict the decay rate to within an order of magnitude, much less the ~ 10% precision required to compete with the experimental result, and there is no way we can use this mode to search for new physics. Physically, this makes no sense. The scale m\, is large enough for perturbation theory to be valid, and the scales raw and rat are completely irrelevant to a decaying b quark - a b quark can't resolve anything smaller than its Compton wavelength, so as far as the b quark is concerned, it is decaying via a point interaction. The poor behaviour of perturbation the-

206

ory must just be an artifact of the fact that we are not calculating things sensibly. A sensible theory of b decay should never contain logarithms of mt/mi, or mw/'mb in matrix elements. Furthermore, we have been rather glib in comparing the quark-level process with the inclusive hadronic process. The decay of a b quark to free quarks and gluons is not the same as the decay of a B meson to all possible hadrons:

(1) The initial states are different. A b quark bound in a hadron is not the same as a free b quark - it is interacting with a complicated state of strongly-interacting quarks and gluons, and so is not at rest. In addition, there are dynamical light quarks and gluons in the initial state. As a practical matter, the parton-level rate in Eq. (8) is proportional to m\. Since this factor arises from phase space, we might be tempted to replace this with m% for B meson decay, which changes the prediction by about 50%. With no good reason as yet to choose one over the other, this is a large source of uncertainty. (2) The final states are different. If we were to measure the inclusive hadronic invariant mass spectrum, we would find broad peaks at the masses of all the possible kaon resonances superimposed on a continuum from multiparticle hadronic states, while the parton level prediction has a peak at ms followed by a perturbative tail from gluon and light-quark bremsstrahlung. Since the partons must hadronize to something, we may argue that if we average both distributions over a large number of resonances the two results should be equal (this idea is called "quark-hadron duality"), but how many resonances is enough? Should this argument also work for semileptonic D decay, which is dominated by only two resonances (the K and K*)l

Both of these problems arise because of the interplay of physics at the decay scale rrn, with the hadronization scale AQCD- Furthermore, in the limit mi, —> oo (or, more precisely, AQco/mb —• 0) both should vanish: an infinitely massive b quark is unaffected by collisions with light degrees of freedom, the difference between rrib and TUB is of order AQCD and so a subleading effect, and in the limit m;, —> oo there is a always a large number

207

of hadronic resonances to average over. Thus, we expect

r(l?->

J2

Xsl)=r(b^

Yl

Xsl)+0(AQCD/mb).

(11)

S99,sqq,...

However, on dimensional grounds these corrections should be at the 10% level, and as our phase space argument above indicates, may be much larger. If we really want to use B decays to look for new physics, we need a systematic way of calculating AQCD/TO& corrections. 3.1. Effective

Field

Theory

The two problems raised in the previous examples (AQCD/""ib corrections and large logarithms in perturbation theory) look different, but both have a common origin: the presence of multiple scales in the problem mw,

rnt >

mb »

AQCD-

(12)

To make progress, we need a tool which systematically disentangles the scales. This is the whole purpose of Effective Field Theory (EFT). Since Ira Rothstein has lectured extensively on EFT's at this school [16], I will be brief in my treatment and just stress the essential features [17]. The idea behind EFT is that the theory you use to calculate physics at a given scale /n should only contain degrees of freedom relevant at that scale. Anything happening at shorter distance scales — virtual heavy particle exchange, high momenta in loops — can't be resolved by external particles with momenta < fi, and so is replaced in the EFT by local interactions. Short-distance physics is therefore removed from the dynamics and simply enters in the values of the coefficients of the local operators in the effective Lagrangian. This is the same reasoning behind the multipole expansion in electrostatics: if we have a distribution of charges of size r^ and are interested in physics at distances of order r ~ ro away, we need to know the precise charge distribution. On the other hand, at distances r » ro, it is much more efficient to expand the theory in powers of TQJT. At leading order, we only need to know the total charge to determine the potential. If we wish to be more accurate, we can include the dipole moment, quadrupole moment, etc, systematically expanding the true result in powers of ro/r. It is important to stress that EFT's are useful whether or not we know the "full" (short-distance) theory. Even if we had a theory of quantum gravity, we wouldn't use it to calculate projectile motion. Similarly, the

208

fact t h a t we have a renormalizable theory of the weak interactions doesn't mean t h a t this is the appropriate theory to calculate low energy (/i
3 . 2 . mw

> fj, > nib:

Four-Fermi

Theory

Generalized

A familiar example of at E F T is four-fermi theory, the E F T of the weak interactions at scales much less t h a n mwSuppose we were interested in nonleptonic b —> c decay. At leading order this is determined by the diagram in Fig. 12(a). However, all external momenta in this graph are of order nib, while the mass of the W boson is mw 3> nib, so according to the rules of E F T it should not b e present in t h e theory. This is easy enough t o implement: since the m o m e n t u m transfer q through the W boson is
(13)

where ipL = | ( 1 — 75 )V'- This amplitude may be reproduced to arbitrary order in q/mw in an E F T with no W boson, but a series of higher-dimensional operators: 4GFycb £ S M —• -Ceff = £ Q C D + Q E D

1 m

2

w

-cLj^bLD2

V2 (dL^^uL)

cL-yfJ,bLdLJtj,uL (14)

This procedure of determining the coefficients of the operators in the E F T

Figure 12. For momentum transfers q <S mw> the W boson may be integrated out of the effective theory and replaced by a series of nonrenormalizable operators.

209

by demanding that the full and effective theories give the same physics is called matching, and the W (along with the Z, higgs and t quark) is said to have been "integrated out" of the theory. In principle the EFT has an infinite number of terms, but for q2
c

(a) Figure 13.

\c

(b)

Radiative correction to b decay in the full and effective theories.

calculate (because it contains four propagators, two of which are massive), and is a complicated function of mi, and mw- By power-counting, it is UV convergent. The effective theory graph is quite a bit simpler to calculate, but by power counting is UV divergent. This extra divergence looks worrisome, but it shouldn't be: the EFT is designed to get the physics right only at scales \i mw the W propagator softens the high-momentum behaviour of the loop integral. Since the EFT isn't supposed to get the physics right at momenta > mw, it is not surprising that it is divergent. (Both graphs are also IR divergent.) The EFT graph therefore requires additional renormalization. The result is then scheme-dependent, so it is not clear that it is telling us anything useful. However, the full theory graph will contain logarithms of the form V ^ l n ^ -

(15)

where the pj's are the external momenta in the graph, in addition to infrared divergent logarithms of the gluon mass (if we choose to regulate the theory this way). These are the large logarithms which we encountered in

210

the previous section, and which can spoil the convergence of perturbation theory. Since the EFT gets the physics of the external states right, it must reproduce these logarithms, and so must contain terms of the form

J^ln-^-

(16)

Pi • Pi

where c^ • = Cij + 0 ( 1 / T O ^ ) , and the renormalization scale /J, must appear in this way on dimensional grounds. In addition, the infrared divergences arise from low-momentum physics, so are identical (at leading order in 1/m^) in the two theories. Thus, as advertised, at one loop the EFT correctly reproduces the lowenergy physics (the logarithms) of the full theory. Of course, there are other important, non-logarithmic terms, but in the EFT these do not arise from loop graphs, but rather are contained in the coefficients of operators in the effective Lagrangian. These must be included for consistency, because before we can consistency calculate radiative corrections in the EFT, we must also calculate the radiative corrections to the matching procedure in Fig. 14; in other words, we must adjust the coefficients of the four-fermi operators in the EFT to ensure that matrix elements are the same in the two theories up to 0(as, l / m | ^ ) . This is illustrated schematically in Fig. 14. In this case, we find that we must add an additional operator to the effective Lagrangian,

Figure 14.

One-loop matching.

02 = cLTa^bLdLTalfluL.

(17)

whose coefficient is just given by the difference of the one-loop calculations in the full and effective theories. Thus, the difference in the finite pieces of the graphs goes into the coefficient C2(M)- Furthermore, to work consistently to this order in a3, matrix elements of O2 in the effective theory are only evaluated at tree level.

211

In general, then, the effective Lagrangian may be written £eff = £QCD+QED + J2

C

^)°i

(18)

i

where the 0;'s denote operators renormalized at the scale /J,, and the Cj(/Lt)'s are determined by matching to a given order i n a s . Note that in order for this whole procedure to work, it is crucial that logarithms of external momenta (as well as infrared divergences) are identical in the full and effective theories; otherwise, the difference between the graphs in the two theories could not be written as the matrix element of local operators. Tree-level matrix elements of local operators can only contain powers of external momenta (from derivatives), not logarithms. At this point it may seem that we've gone through a long song-anddance for no apparent reason. The EFT graphs may be simpler than the full theory graphs, but we have to calculate both to do the matching conditions. So what have we gained? (1) We have explicitly factorized the long- and short-distance physics, with /i serving as the factorization scale. The long-distance physics (the logs of fj?/pi • pj) is contained in the matrix elements of the operators Oi, while the short-distance logs of mw/fJ- are contained in the coefficients in the effective Lagrangian. Note that the longdistance physics is process-dependent (differing for different external states) whereas the short-distance physics is incorporated into the parameters of the theory, and so is independent of the particular process being studied. (2) Because the matching conditions are independent of the external states, we may calculate them using any external states we choose. In particular, if we are working at leading order in 1/m^, we may calculate the matrix elements in both theories between external states with zero momentum, which will greatly simplify the calculations. Both theories will give infrared divergent results, but the infrared divergences will cancel in the matching. (3) Finally, since the renormalization scale [J, is arbitrary, we can eliminate large logarithms in matrix elements by simply choosing to renormalize the theory at /x ~ ra^. Note that there appears to be a tradeoff here: choosing /z ~ rrn, makes perturbation theory work well for matrix elements, but poorly for the matching conditions, since the coefficients Cj(/i) in Eq. (18) will contain the large logarithms. However, the renormalization group gives us the best of

212

both worlds: we can calculate the matching conditions in the EFT renormalized at /J, ~ mw, where perturbation works well for the Cj's, and then use the renormalization group equations (RGE's) to lower the renormalization scale to \i = m,b, where we calculate matrix elements. Let me elaborate on that last point. Matrix elements of the effective Hamiltonian are physical, and so independent of the unphysical renormalization scale fji. Thus, /i-f

{CiMOi)

=0

(19)

and so

A ^ M = I>iCiG")

(20)

j

where d/d/i is the total derivative, including the change in the coupling

SO): d

d

_. . d

«" = ^ = A i S i

+

<21>

-

and /3o = —(11 — 2n//3) for n / light quark flavours. The anomalous dimension matrix is defined by

(22)

l,i = Zjk ( ^ ^ ) where Zij relates the bare and renormalized operators, 0^are = ZijOj.

(23)

The RGE is solved by diagonalizing the anomalous dimension matrix; for the simple case where there is only one operator 0(/z), the solution is 70

where the anomalous dimension of the operator O is 7oas(/u)/47r. Expanding this out gives C{fi2) =

1

-.

4TT

7ou In — + '

">

2

'

32^

70(70 + 2/30) In — + . . . >^'»

' - ™ -

C(m).

M2

(25)

213

and we see that the complete series of leading logs of ^2/^1 has been summed by the RGE and put into the coefficient function C{^2)The procedure is then straightforward. We calculate the matching conditions at the high scale fi = mw, where perturbation theory works well. Then, using the RGE, we evolve the coefficients from fs = mw to \i ~ mb. The large logarithms are summed to all orders in the Cj(/i)'s, and matrix elements at /i — mb contain no logarithms of m\/mw. Note that the leading logs in Eq. (25) correspond to matching at tree level, running at one loop and taking matrix elements at tree level. Including the one-loop matching corrections and one-loop matrix elements corresponds to including terms of order a " + 1 In™ [ixjiii- This is the same order as the terms summed using the two-loop running, and so both must be included to work consistently at this order. Thus, as a general rule, the matching conditions and matrix elements must be evaluated at one lower order in as than the anomalous dimensions of the operators. 3.2.1. Semileptonic decay Let me illustrate this with a few examples. In the SM, there are many flavour changing operators in the effective theory below mw, but for a given process one only deals with the appropriate subset of operators. Semileptonic b —> c (or b —> u) decay is the simplest: at leading order in 1/mw, the only operator is 0(/i) = cL^bLPe^eL.

(26)

M

Since C7 (1 — 75)6 is a partially conserved current in QCD (because chiral symmetry is only softly broken by the quark masses mc and mi,), this operator requires no additional renormalization, and so does not run below the scale mw- Thus, C(mb) = C(mw)

= -^Vcb

and there are no large logarithms of mb/mw to sum. The final result is r(6 ^ XJLv) = ^^f"

(27)

in the semileptonic b decays

f(mc/mb)

+ 0(as)

(28)

where f(x) = l-8x2 and the 0(as)

+ 8x4-...

terms do not contain any logarithms of mw-

(29)

214

3.2.2. Nonleptonic b —> c decay Nonleptonic decays are considerably more complicated. The simplest nonleptonic decay, the Afe = 1, Ac = 1 (i.e. b —> cqq) transition, is mediated at tree level by the operator Oj = cL^bLdLltluL

(30)

where

C K ) = -~^vcbv:d.

(31)

However, under renormalization, diagrams such as that in Fig. 13 (b) require a counterterm with a different colour structure: On = CL^^dLjT^ul

(32)

where i and j are explicit colour indices. (On is just a convenient linear combination of Oi and O2 from the previous discussion.) The corresponding RGE is therefore a matrix equation M

d ( din) \ dM \Cnfjj))

=

a ^ /711 7 i 2 \ ( C 7 (/i) \ 4TT U i 722J [Cuifj,))

(33)

where, at one loop, 7 n = 722 = - 2 , 712 = 721 = 6.

(34)

Since 712 and 721 are nonzero, 0 / and On mix under renormalization. The solution to Eq. (33) is

CiMn) = \

as(mw)\ as(fi) J

23

,

fas{mw)\ V «*(M) /

(35)

Thus, even though Cn{n~iw) = 0 at tree level, it is induced for \i < mwUsing as(m\v) = 0.12 and as(mb) = 0.22, we find Ci{mb) = 1.11, Cn{mb)

= -0.26.

(36)

3.2.3. b —> S7 decay The Ab = 1, As = 1 FCNC decay 6—> S7 is more complicated still. The diagrams in Fig. 11 (a, b) match onto an operator in the EFT of the form 07 ~ SL^baF^

(37)

whose matrix element determines the decay rate at tree level. However, Or mixes under renormalization with a whole slew of other operators. For

215

Figure 15. O2 and Or mix under renormalization. This is a two-loop graph, but it contains only one strong loop and so contributes to the leading order anomalous dimension matrix.

example, the operator 6L7' 1 CLCL7 M SL requires Or as a counterterm from the graph in Fig. 15. It turns out that a total of eight operators are required [18]: AC

HeS =

(38)

—lvtbv;sJ2cl(^)oi

where 0 1 = {CLf3l^bLa)

(SLaJ^Cip)

,

q=u..b °5=

^2 (sLotJ^bLa) (<7fl/37/^ii/3) . q=u..b

Or = ^ n i t i I a / ' W F ,

02 = {cLal^bLa)

(s L/3 7 M C L/3 )

q—u..b °& = J Z i^Lal^Lp) q=u..b

08 = -^mhsLa<j»vT^bR!iGlv

(QRPl^QRa)

(39)

and the en's and /3's are colour indices. At leading order in as only C2(mw), Cr(mw) and Cg(mw) are nonzero, but the other operators mix with these under renormalization, so one needs an 8 x 8 anomalous dimension matrix to obtain C-j{rrn,)- Furthermore, the calculation is complicated considerably by the fact that the mixing of O2 — 0% with Or and 08 first occurs at the two-loop level, through diagrams like that in Fig. 15. Since there is only one QCD loop in these diagrams, they contribute to the anomalous dimension matrix at leading order. The leading order (LO) RGE's for b —> Xsj were first solved by Grinstein, Springer and Wise [18] in the simplified approximation in which the mixing of O3 — Oe with Or and 08 was neglected. With this simplification,

216

they found as{mw) as(mb) _

C7(mw)

-

232

+ 513

-C8(mw)

_ / as(mb) \ '' as(mw)J

us(jnb) V as(mw)J

(40)

The complete anomalous dimension matrix was later completed by Misiak [19]; the results differ from this by about 10%. Resumming the leading logs in this way increases the tree level theoretical prediction in Eq. (8) by roughly a factor of three, bringing the prediction into agreement with the measurement, Eq. (10). However, there is still about a 25% theoretical uncertainty in this prediction, just by varying the renormalization scale by about a factor of two about /i = mb- For precision physics, this uncertainty is still disturbingly large, and to reduce it the next-to-leading-order (NLO) calculation is required. The complete NLO calculation of b —> Xsj is probably the most involved QCD calculation ever performed. There have been many papers over the past ten years to get to the current state of the art [20]. The complicating factor is that, since the leading order RGE equations involve two-loop diagrams, the NLO' calculation requires three-loop anomalous dimensions, as well as two-loop matching conditions and matrix elements. The current state of the art gives [20] Br(& -+ Xsl)f™l6

GeV

= (3.57 ± 0.30) x UT 4 .

(41)

Comparing this with the experimental result, Eq. (10), we see that as long as the l/m& corrections relating this to the branching fraction for B —» Xsj are small, there is little room for discovering new physics in this decay. Since the NNLO calculation requires four-loop calculations, it is unlikely that the theoretical error in this prediction will be able to be signficantly reduced anytime soon. 3.3. n <mb:

HQET

In the parton model calculation of inclusive B widths no scales below mj appear in the calculation, so it is sufficient to work with the effective fourfermi theory discussed in the previous section, renormalized at /J, = nibThis is a consequence of the fact that at leading order in l/nib the decay is entirely determined by short (r < l/mb) distance physics. On the other hand, decays to specific final states (exclusive decays) do get contributions

217

from long distance physics, since that determines the details of hadronization. For example, to calculate the exclusive process B —> if* 7 instead of the quark level process b —> X s 7 , we need to know the matrix elements of all eight operators (39) in the effective Hamiltonian (38) between a B meson and a if*, which are all determined by long-distance physics. Thus, in keeping with the EFT philosophy, we would like to work in an EFT which factorizes the physics of the parton-level decay at [i = rrib from the physics at lower scales. Since the long-distance contribution to exclusive decays is nonperturbative, it's not clear that switching to an EFT will be particularly useful, since EFT doesn't give us any new tools to solve strongly coupled theories. However, there are still reasons to switch to an EFT description: (1) The EFT allows us to identify simplications (in particular, approximate symmetries) which (in some cases) still allow us to say something about the hadronic physics. (2) The only way to calculate nonperturbative physics is to use lattice gauge theory. It is extremely inefficient to attempt to simulate a wide range of energy scales on the lattice, since the lattice spacing must be very fine (to simulate short distances correctly) and the lattice must be large (to fit the long-distance physics in). It is much more efficient to take care of the short-distance physics analytically, and only simulate the long-distance physics on the lattice. EFT does this automatically for us: an EFT renormalized at fj, can be matched smoothly onto a theory with a lattice spacing a ~ l//i. I won't have much to say about lattice gauge theory in these lectures, so I will instead focus on the first point. In general, factorizing physics at TOh from lower scales in B decay is a much trickier procedure than simply integrating heavy particles out of the theory, since in a given decay there are partons carrying large (~ TO;,) momenta (such as the constituents of the if* in B —> if *7 decay), so there may be both hard (~ mj,) and soft (~ AQCD) momentum transfers in the same process. This should be contrasted with four-fermi theory, where no external momenta are of order mw • It is not obvious how to unravel the physics at these different scales, making it complicated to construct the appropriate EFT. I will say a little bit at the end of these lectures on the general problem, but in this section I will discuss an important case where we do not have to worry about such subtleties. "Heavy Quark Effective Theory" [21-23] (HQET) is the EFT for a single

218

heavy quark in which all hadronic momentum transfers are much smaller than mi,. This is much simpler than the general case because the scale mj never enters the dynamics, so there is no complicated dynamical unraveling of scales required. This is the appropriate theory for semileptonic B to D decay, for example, at least in the region of phase space in which the D recoils slowly. However, it is not appropriate for B —> K*~f decay, since in the limit mj, —> oo the recoiling K* carries energy of order mb/2, which is not
p£ = mbv" + k»

(42)

where v^ is a fixed four-velocity (usually chosen to be the four-velocity of the meson, although any velocity differing from this by at most 0(AQCD/mb) will do). Since all momentum transfers are soft, scattering in the EFT can only change the residual momentum k^. The EFT can then be obtained by expanding amplitudes in powers of k^/nibThis is easy to do: for example, expanding the b quark propagator in powers of k^/mb gives j) + mb _ mbf + mb p2 — m2 + it 2m,bV-k + ie

| Q /fc"\ = TO

\

b/

/ ! +A 1 | \ 2 Jv-k + ie

which can then be inverted to give the kinetic term in the EFT: ZHQET

= bviD-vbv + o(-^-j.

(44)

Note that the propagator includes the projector \(l+f), which projects out the upper 2 (particle) components on the b field, so the EFT is a theory of 2 component spinors bv. The bv fields have also been defined with a large phase exp(imbv • x) relative to the QCD field, so that derivatives acting on the bv's pull down factors offc^instead of p%. The Feynman rules for HQET are shown in Fig. 16. Let me make some comments: a

Although we do integrate out b quark loops.

219

1

12VV K£>

XL

v-k Figure 16.

Feynman rules for HQET at leading order.

(1) Working for simplicity in the rest frame of the meson, v^ = (1,0,0,0), we see that the propagator is just the nonrelativistic energy k° of a static quark, and that it couples only to the electric field (via the scalar potential A0). Such objects had actually been extensively studied before the advent of HQET - they are called Wilson lines, and are simply static colour charges. Indeed, this is exactly what a heavy quark looks like in the rriQ —> oo limit: since all external momenta are negligible compared withTOQ,interactions can't change the 4-velocity of the heavy quark, so in its rest frame it is just a static source of colour. (2) The leading order HQET Lagrangian has no reference either to mj, or to the spin of the quark. Physically, this is because of the previous comment: as long as TUQ is much greater than any external momentum, the mass of the heavy quark is irrelevant to its dynamics - it's just a static source. Furthermore, the magnetic moment of a particle is inversely proportional to its mass, so the colour field of an infinitely massive quark only depends on its total charge, not its spin. This simple observation has profound implications, because it means that HQET has an enhanced symmetry compared with QCD: for nh heavy flavours, long-distance QCD is symmetric under the interchange of either spin state of any of the heavy flavours with the same four-velocity. Hence, the Lagrangian for HQET with n/i heavy flavours with the same four-velocity, nh

£ = ^

'

hiiViD • vhiiV

(45)

i=l

has an SU(2n/») spin-flavour symmetry. This will prove very useful shortly. Note that the symmetry exchanges quarks of the same velocity, not momentum. (3) We can easily go to higher orders in 1/mb by matching Green's functions [24,25]. For example, the three-point quark-quark-photon

220

vertex can be expanded using the Gordon decomposition , .q.v°'a'' ^ 2rrib

'{p + p'Y 2rrib = v-Au'u+

—

2mq

(k + k') -Au'u + i-^-u'tr^A^u. 2rrib

(46)

The first term arises from the leading order kinetic term in the Lagrangian; the second two arise from dimension five operators which can be added: £HQET

= bviD -vbv + -— (bv{iD)2bv + CV(/x)-6„cr • GbvJ

+0(l/ml).

(47) 2

The first correction term is just the k /2m kinetic energy term, while the second corresponds to the chromomagnetic moment of the b quark, and Cpirrib) = 1. This term is the leading operator which breaks the heavy quark spin symmetry. Weak Currents In addition, we can match weak currents between the full and effective theories. At leading order in as and l/m&, this is trivial. For example, the weak b —> c current matches onto the current CM6„7"(1 -

7 5 )c,

C(mb) = 1 + 0(as)

(48)

in HQET. However, unlike in the full theory, this operator requires additional renormalization in the EFT. From the diagrams in Fig. 17, the current has an anomalous dimension [21,22]

(49)

™=-6 and so

CM

/•

/

NN

=(w)

-6/25

c(mt)

(50)

But now we must be careful - this gives the correct running between \i = mb and fx = mc. But below // = mc, we now restrict all momentum transfers to be < mc, which means we must match onto a new EFT where the c quark is treated as heavy as well, £ = bviD • vbv + cv'iD • v'cv> + 0(l/m(, C)

(51)

221

Figure 17.

Diagrams contributing to the anomalous dimension of a heavy-light current

in HQET.

and the current is C(v • v', Ai)6„7/i(1 - 75)
(52)

The anomalous dimension for this current is [26] lT=ih(y-

v'r(v • v') - 1)

3TT2

(53)

where In (w + \/w2 - 1)

r(w) = 2

y/w

—1

V

(54)

/

and so nf i \ C(vv,(i)=[

(as{mb)\ — r \as(mc)J

fas(mc)\ C(mb), rT\ aa(n) J

n<mc.

(55)

Note that at when t> = v' (v • v' = 1), the anomalous dimension JT vanishes, and the heavy-heavy current doesn't run. This is because of the enhanced symmetry I mentioned earlier: when v = v', the Lagrangian (51) has an 577(4) spin-flavour symmetry which interchanges the spin components of both heavy flavours. The current bvTcv (for any T) is a generator of this symmetry. It is therefore related to a conserved charge, and so its anomalous dimension in HQET vanishes. 3.4. Inclusive

Decays and the

OPE

For general inclusive decays, we can't naively use HQET. Because the final state particles typically carry large (0(rrib)) momentum, it is not true that all the hadronic momentum transfers are soft. However, as we have argued on physical grounds, at leading order in A Q C D / ^ 6 , B meson decay is identical to free b quark decay. The argument depended on scale separation, so it is not surprising that we should be able to formulate the problem in the EFT language. The trick is that because the final state particles are highly

222

energetic, through a bit of cleverness in the complex plane we can relate the physical decay amplitude to an amplitude in which the decay products off their mass shells by 0(mb). In this region they may be integrated out of the theory just like a massive W boson, and we may treat the rest of the problem in HQET. This trick is called an operator product expansion (OPE), and its use long predates b physics. e+e~ —> hadrons and r decays In a classic 1975 paper, Poggio, Quinn and Weinberg [27] showed that the ratio = w

g(e+e--»

hadrons + X)(s) a{e+e~ -^n+n~ + X){s)

(where X denotes additional photons and fermi-antifermion pairs) was calculable in perturbation theory. It is instructive to go through their argument to see how it applies to the case of B decays. The optical theorem tells us that R(s) is proportional to the imaginary part of the photon self-energy (see Fig. 18) R(s) =

12TT Im

n ( s + ie)

(57)

where I P V ) =i|rf4xe^-(0|T^M(x)^M(0)|0).

(58)

Independent of perturbation theory, we know the analytic structure of

Im||vW\J

|/W\/]«£w rPvWNC^" 1 X

Figure 18.

The optical theorem.

H(q2): it is analytic everywhere except along the real axis for q2 > 4m^, where there is a branch cut due to real intermediate states, as shown in Fig. 19(a). (There are also isolated poles due to positronium bound states, but these are not important for our discussion.) Near the branch cut there are resonances which are sensitive to the details of long-distance QCD, so perturbation theory breaks down; however, in unphysical regions far from the cut we expect perturbation theory to be valid.

223

-*-x A WWWWAIV

(a) Figure 19. The analytic structure of n ( q 2 ) in the q2 plane. The wavy line denotes the branch cut due to real external states, (a) The smeared ratio for e+e~ —> hadrons is determined by II(q 2 ) in the unphysical region, (b) The phase space integral along the path C\ for hadronic r decay may be deformed to the path C2, in the unphysical region.

Of course, we are interested in the physical region. However, Poggio et. al. pointed out that if we average the value of R over a region of size A, Cauchy's theorem allows us to relate this to U(q2) in the unphysical region, where there are no hadrons: R(s') , , 1 A r (59) R(s,A) 2 2 da' = — [U(s + iA) - U(s - iA)}. (s' - s) + A 2i 7T JO This shouldn't come as a surprise. For q2 » A Q C D annihilation is a shortdistance process. For a fixed physical value of q2 the cross-section might be sensitive to long-distance physics (since q2 could be sitting just above or below a threshold), but if we average over a number of states these effects should cancel out, leaving us with a purely short-distance process. This idea is called "global duality", and it suggests that perturbation theory should be appropriate for sufficiently smeared observables. (The definition of "sufficient" here can be problematic, but in general it means that many resonances are averaged over.) This argument only strictly holds in the q2 —> 00 limit. To see how corrections arise, let's study this in a bit more detail. In the physical region, n(q 2 ) is nonlocal: the intermediate hadronic state is on shell, and so can travel a long distance. On the other hand, in the unphysical region the final-state quarks, like the virtual W in four-fermi theory, are far offshell, and so can be integrated out of the EFT and replaced by a series of local operators, as illustrated in Fig. 20. This procedure is called an operator product expansion (OPE) — morally, it's the same as integrating heavy particles out of the EFT, but in this case the intermediate particle isn't necessarily massive, just constrained by the kinematics to be far offshell.

224

Im v/vf

Vv^ + vA/^<|)jiA^+...

= 'vxgyv

Im (^ <<§•

££

%

Figure 20. The operator product expansion (OPE) for IT(g 2 ). The sum is over all operators with the quantum numbers of the vacuum. Oo is the identity operator, and Oi =Ga^G^va.

Just as in an EFT, the matrix elements of operators of increasing dimension are suppressed by powers of AQCD/? 2 > and so for q2

(60)

i

where the Oi's include all operators with the quantum numbers of the vacuum, and as in an EFT, the C^s are calculated order by order in perturbation theory. The leading order operator (denoted Oo in the figure) is the unit operator. Since its matrix element receives no radiative corrections, its coefficient is given by the imaginary part of the perturbative vacuum polarization, which is just the parton-level cross-section. Thus, as advertised, at leading order the OPE simply reproduces the parton model. The leading correction arises from the operator 0\ = G'\vG^va, whose coefficient is obtained by calculating the T-product of two currents in which two soft gluons are also emitted, and expanding the result in powers of the external gluon momenta. By dimensional analysis, the matrix element of 0\ is proportional to A Q C D , so the leading corrections to the parton model are suppressed by A Q C D / S 2 . This technique has many applications. In addition to e + e~ annihilation, it can also be easily extended to hadronic r decays [28], T —> vT + hadrons

(61)

which has been used to extract as(mT) with high precision. In this case the smearing is automatically done by the phase space integral, as illustrated in Fig. 19(b). The phase space integral of the imaginary part of II(s) from

225

s = 0 to s = m2 (weighted by the appropriate kinematic functions) is equivalent to the weighted contour integral of II(s) along the curve C±. C± may be deformed to C2, which lies almost entirely in the unphysical region, and so the OPE may be applied to the integrated rate. OPE for B decays For B decays similar arguments hold, although in this case the initial state contains a B meson rather than the vacuum [29]. This is in fact not much of a complication, since the QCD vacuum is also nonperturbative and complicated, and in both cases the hadronic physics enters in the matrix elements of local operators. In addition, in both cases the matrix element of the leading operator is normalized, so that the leading order calculation reproduces the parton model. As with r decays, the phase space integral over the lepton momentum in B decays performs the smearing automatically. As in Eq. (58), the decay rate is proportional to the imaginary part of the T-product of two weak currents. We can then decompose this into different Lorentz structures: T V

"

=

j{B\Tjt{x)jU^\B)e-^x^x

" 2 ^

= -g^T^q2, v

v»v"T2{q2, q • v) - ie^^vaq0T3(q27q

q-v)+ 2

+q»q Ti(q ,q

v

2

• v) + {v»q + v"q»)T5(q ,q

• v).

• v) (62)

Because the heavy meson rest frame defines a vector v^, the TVs are functions of q2 and q • v instead of just q2. For fixed q2, the analytic structure of the Tj's in the q • v plane is shown in Fig. 21. The left-hand branch cut corresponds to physical intermediate states for the decay process, while the right-hand cut corresponds to intermediate states containing a q quark and two b quarks (which is related to scattering, not decay). The differential decay rate is given by the integral over the left-hand cut,

dT~ / Jo ~ /

Im Im T^L^

T^L^diP.S.) d(RS.)

JCi

-
226

are free to deform the contour C\ into C2, which lies far from the resonances. We then perform an OPE for the intermediate state, as illustrated in Fig. 21. In addition to integrating out the intermediate hadronic state, in performing

q-v\ W M ws/VQ/ywyst <st-

\

VWVWSr

C2

Figure 21. The analytic structure of T^v. may be deformed to the path Ci.

The phase space integral on the path C\

the OPE for B decay we also switch to HQET, since the matrix elements of the resulting operators are to be evaluated at a renormalization scale below rrib. Explicitly, for semileptonic ( ) - » « decay, the forward matrix element of T^v between free quarks with momenta pb = m^v + k is (rribV — q + k)2 + ie

= {mbva - qa), + (mbv - q)2 + ie pHV

hvPhhv (iribV — q)2 + ie

0(hP/mb)

0{k^/mb) (64)

where P^ = \{mbva - qa){v"gatx + v»gav - vag^ + iea^xvx)Note that in the expansion, the external current q is taken to be of order nib, while fcM ~ AQCD- Taking the imaginary part converts the denominator to a delta function, —Im 7T

{mbV

2

q)

5 ({mbv - q)2) .

(65)

ie

The axial piece of the matrix element of bvPjJ3v vanishes in B-mesons, so at leading order the imaginary part of the nonlocal T-product is given by the matrix element of the local operator Oo

(66)

with coefficient given by Eq. (64). This operator is a symmetry current of the theory — it just counts the number of b quarks in the state — so its

227

matrix element is fixed by symmetry. With the conventional normalization of states, {B\hvhv\B)

= 2mB.

(67)

Thus, just as for e+e~ annihilation and r decay, the leading term in the OPE simply reproduces the free quark process, as calculated in perturbation theory. At subleading order, the only dimension four operator is hvD^hv. By Lorentz invariance, its matrix element must be proportional to ^ , so (B\bvD»bv\B)

= v»{B\bvv • Dbv\B).

(68)

However, from Eq. (44), the leading order equation of motion in HQET b is v • Dhv = 0, so the matrix element vanishes at this order. Thus, there are no corrections to the parton model at 0{l/rrn,)\ We will discuss higher order corrections in the next section. Note that one immediate implication of the OPE is that inclusive decay rates for ^-flavoured mesons are determined by parton-level kinematics, not hadron kinematics. In particular, the rate for B —> Xs-f decay is therefore proportional to ml, not mB, since the latter would introduce a 1/mb correction to the parton rate, which we have seen is absent. 4. Applications In the last few sections I introduced the main theoretical tools that allow us to factorize long and short-distance physics. Now we proceed to some applications. 4.1. Inclusive

Decays:

Corrections

to the Parton

Model

We saw in the previous section that up to 0(l/rrib), inclusive B decay rates are the same as that of free b quarks to quarks and gluons. To obtain higher accuracy, we must include in the OPE operators of dimension five, of which there are only two. Their matrix elements are conventionally denoted as Ai =

2mB

-^(B\bv(iD)2bv\B)

A2 = -^—(B\gbva^G^bv\B). 6ms

(69)

b A n important, but sometimes unappreciated, feature of the equations of motion is that they may always be applied at the operator level, even though the quarks in hadronic matrix elements are not on shell. See Ref. [30].

228

(In another convention, these matrix elements are denoted fj% = — \\ + ... and p?G = 3A2 + . . . , where the ellipses denote terms suppressed by powers of l/mb.) These are the same operators which appear in the HQET Lagrangian Eq. (47), and correspond to the kinetic energy of the heavy quark in the meson, and the chromomagnetic energy. Since the latter is the leading operator which violates spin symmetry, it may be determined from the experimentally measured B — B* mass splitting, A2 - i ( m | . - m2B) + 0(AQCD/mb)

~ 0.12GeV 2

(70)

while Ai cannot be simply related to meson masses. Thus, up to 0 ( A Q C D / T O ^ ) , inclusive quantities may be expressed in terms of only two unknown parameters, mb and Ai. From the HQET Lagrangian (47), the meson and quark masses are related via Ai+j3A 2 —-2 \-... (71) 2m{ so mb may be exchanged for A, the energy of the light degrees of freedom in the meson. At the price of introducing even more parameters, one may continue to expand to higher orders in 1/m;,. At 0 ( l / m | ) there are two more operators, whose matrix elements are defined as 1_ • D){iD»)bv\B) Pl = 7^-(B\bv(iDfl)(iv 2mB mB = mb + A

Pi = -T-^-e a ^*(B|6„(i£> a )(t£> M )(i£> /3 )7 4 7 5 6„|B). (72) orris In addition, there are the T-products of l/mb corrections to the HQET Lagrangian with the operators (69), for a total of six unknown quantities. For most quantities, expressions are available up to 0(l/mjj), although uncertainties at this order mean that the l/m^ terms are typically used to estimate theoretical uncertainties. 4.1.1. Total decay widths For example, for b —> s-y we have, at tree level [31], T(B - Xsl)

=

^r^\VtbVtsf\C7(rnb)\2

1m\

2ml

(73) The A2 term, which is known, reduces the rate by about 2%, while the Ai term is less than a 1% effect for reasonable values of Ai. The l/mb

229

corrections are therefore very small for this decay, and do not appreciably shift the prediction (41). The semileptonic B —> Xciv decay width is proportional to the CKM matrix element Vcb- The result is [29] T(B - • Xcev) = r 0

1 as fo{rhq)+ ^—2/i(m„Ai,A 2 ) + A0(mq) —

(74)

where G%ml\Vcb\"

r

1927T3

fo(mq) = l-8rh2q

+ 8mq - m8q - 24mq l n m , ,

/i(m„Ai,A 2 ) = Xi (l - 8ml + 8mq - m8q - 24mqlnmq) 2

(75) 8

+A 2 ( - 9 + 24m - 72m* + 72mq - 15m q - 72m^lnm g ) and AQ contains the one-loop" radiative corrections [32]. Hence, with a precise determination of A and Ai, a precise determination of Vcb is possible. Finally, setting mc = 0 in this expression, we obtain the rate for B —> Xu£v, T(B - •

G2Fml\Vub\

Xuev)

1

, Ax -9A2 , / 2 5

2

2\

as

1927T3

(76) 4.1.2. Differential widths and spectra We can construct an OPE not only for total decay rates, but also for differential rates and therefore spectral moments. There has been much recent interest in such quantities because they allow the nonperturbative parameters A, Ai (and, in principle, higher order terms) to be determined experimentally. This not only allows the rates in the previous section to be predicted with high accuracy, but consistency between different observables provides a stringent test of the whole OPE picture for inclusive decays. Some quantities of particular recent interest include the first moment of the photon energy spectrum in B —> Xsj decay [31,33], the first moment of the hadronic energy spectrum in B —> XJLv decay [34], and various weighted spectral moments of the charged lepton spectrum in B —> Xc£p decay [35]. Working to 0(\jmff), each of these observables constrains a linear combination of A and Ai. For example, the results of several recent CLEO analyses are shown in Fig. 22. This analysis obtained [36] { xs) mh

= 4.82 ± 0 . 0 7 B ± 0.11 T GeV,

Ai = -0.25 ± 0.02,sT ± 0 . 0 5 s y ± 0.14 T GeV 2

(77)

230 1630802-010

0

rr-.ro, | i i r.i | i n I [ i M i [ I I i I | I I I I | I i I I | i i

0 t h Moment of Lepton Energy

%v \\\

-0.05 -0.10

St

-0.15 1 a i Moment ^ % . of Lepton Energy $-0.25 l St

1 Moment of Photon Energy (b—sy)

=£-0.20

1 s t Hadronic _ Mass Moment

-C-0.30 -0.35

1a Total Experimental Ellipse

-0.40 -0.45 -0.50

11

0

0.1

J

1 1 1 '

0.2

1 1 1 1 1 i>

0.3 0.4 0.5 A (GeV)

Figure 22. Experimental constraints on A and Ai, from between the different measurements.

~

r I I l W l ' 1

0.6

0.7

I I

0.8

[36]. Note the consistency

(IS)

where the "IS" mass myb ' is an example of a short-distance b quark mass, which is preferred for practical purposes to A because of the poor behaviour of perturbation theory when the quark pole mass is used in calculations. 0 Global fits to a variety of spectral moments have also been performed by various groups, using expressions up to 0(1/ml), which give comparable results [38]. Combining these values with the experimental measurement of the semileptonic B —> Xc width gives [36] |Vcb\ = (40.8 ± 0.5 ± 0.4 ± 0.9) x 10~ 3

(78)

where the first error is from the uncertainty in the total semileptonic width, the second from mb and Ai, and the third an estimate of higher order corrections. c This is a fascinating side issue which I have no time to discuss here. See Ref. [37] for further discussion.

231

4.1.3. Vut, and the shape function In principle, one could use Eq. (76) and the values of rrn, and Ai from the moment fits to determine Vub from the total inclusive semileptonic b —> u width, with a theoretical uncertainty at the ~ 5% level. However, because of the ~ 100 x background from b —> c decays, the inclusive b —> u rate can only be measured in restricted regions of phase space where the charm background is absent. In these restricted regions, the usual OPE breaks down. This is easiest to see by instead considering the analogous situation in inclusive b —> Xs^f decay. Suppose we were interested not in the total rate but instead in the differential rate dTjdE1 This is in fact an important experimental quantity, since one has to impose a hard cut on the photon energy E1 > 2.2 GeV to eliminate background from b —> c decay, and it is important to know how much of the b —• sj this cuts out. Unfortunately, the OPE gives us little information. At tree level the decay is purely twobody, so the photon energy spectrum is just a delta function at E7 = m&/2. At higher orders in the OPE it doesn't look much better - higher order terms are proportional to derivatives of delta functions at the same point, so this spectrum makes no sense unless we integrate it over some region of E1. Radiative corrections populate the region E < rrib/2 because of gluon bremmstrahlung, but the resulting theoretical spectrum does not have peaks due to the various hadronic resonances. This is an example of quark-hadron duality failing if the rate is not sufficiently inclusive. But how large a region must we integrate over? The OPE tells us. The denominator T-produce of the currents for b —> Xs-i is (mbV + k — q)2 +ie

(mbV — q)2 + 2k • (mbV — q) + k2 + ie

where fcM is the residual momentum of the b meson, defined in Eq. (42). Writing q^ = xm(,nIJ'/2, where nM is a light-like vector, and x = 1 corresponds to the maximum photon energy at the parton level, this becomes D=

;

;

;

(80) 2

1 — x + k • n+ (1 — x)k • n + k + ie (where k = k/rrib). There are three possible regions: (1) l - i > A Q C D / W 6 : This is the usual assumption for the OPE. In this region, the denominator in (80) may be expanded in powers of k^/rrib, and the result matched onto local operators.

232

(2) 1 — x < A Q C D / 7 ? T | : In this region, there is no sensible 1/mb expansion of the denominator, and the OPE breaks down. This is physically reasonable: in this region, the invariant mass of the final hadronic state is constrained to be of order AQCD, and so the decay is dominated by a small number of resonances, and one would expect an inclusive description to fail. Indeed, in this region the leading term of the OPE is the same size as all the other terms in the denominator, and so the expansion in powers of 1/rrn, doesn't converge. (3) 1 — x ~ AQCD/«"16: In this region, the maximum invariant mass of the final state is of order {h-QCD'mb)1'2 3> AQCD, and one would expect an inclusive description to be valid. However, in this region k • n and 1 — x are both of order AQCD , and one cannot treat the k • n term as a perturbation. This is often known as the shape function region. [39] In the shape function region, the usual OPE fails, but we may still expand the denominator in powers of AQCB/iribD =

; + • • • =^ ImD = -7rJ(l -x + k-n) + ... (81) 1 — x + k-n + ie A delta function containing the residual momentum is not related to the matrix element of a local operator; instead, it is given by the function f(u) = (B\b5(u> + ib-n)b\B).

(82)

This is a strange-looking object, but it looks less peculiar if we perform a Fourier transform in u> and look at it in position space, f(t) =

B\b(0)Pexp

n • A(t')dt' b(t)\B) (83) o where the P denotes path-ordering, and t measures the distance along the light-like direction nM: b(t) = b(tn/mb). This is a nonlocal operator where the b and b fields are separated by a light-like displacement, and the pathordered exponential is there to restore gauge invariance. (This is technically known as a light-like Wilson line.) In n-A = 0 gauge, the function f(u>) gives the amplitude for the light-cone component of the residual momentum fcM to be UJ. f(u>) is therefore the light-cone distribution function of the heavy quark residual momentum; it is often referred to as the "shape function". (The shape function was actually originally defined as an integral of f(u), but this nomenclature seems to have stuck.)

233

Thus, at leading order in AQCD/'^ib the nonperturbative contribution to the photon spectrum in b —> Xsj is given, in the region 1 — x ~ AQCD/W&: is determined by the light-cone distribution of the heavy quark residual momentum. [39] At this stage we don't seem to have gained anything. However, the function f(u) is universal. Consider, for example, the shape of the hadronic invariant mass spectrum in b —> Xu£p decay. In this case, D=

, ^ , s 0 + 2k • (v - q) + k2 + ie

(84)

where gM is the momentum transfer to the leptons and So = (v — q)2 is the parton-level hadronic invariant mass. d If SQ
; + ---=>lmD = -n6(s0 + k-n) + ... (85) so + k • n + ie Comparing this with Eq. (81), we see that the hadronic invariant mass spectrum is determined in the region s 0 ^ A Q C D " ^ by the same function that determines the shape of the photon spectrum near the endpoint of the b —> Xsj. A similar argument shows that the rate for B —> Xulv in the region Ei > rrib — AQCD is given by the same distribution function. More precisely, at leading order in 1/rrib the various spectra are determined by convoluting f(uj) with the appropriate kinematic functions: l^(B^Xsl)

= 2f(l-2E7)

^ ^ - (B - • Xulv) r dEi ~ ( B ^ r dsH

+ ...

=2 [ 8(1 - 2Ee - u>)f(w) dw + ... J

xutv) = f j

^ ^ - I 4B * ) w

(86)

_ % ) / ( w _ A) d, + . . .

This is useful phenomenologically for determining the CKM mixing angle \Vub\- As mentioned in Section 4.2.1, the semileptonic width for b —> u decay is only easily experimentally accessible in restricted regions of phase space, such as Eg > (m2B — m2D)/2mB, or SJJ < m2D [40], where background from b —> c decay is absent. Because m2/m2 ~ AQCD/«T-6, this d

T h i s differs from the physical hadronic invariant mass ( m j « — q)2; however, we won't worry about that complication here.

234

corresponds in both cases to the shape function region. The shape function f(ui) may then be measured in B —> Xsj decay and then used to predict either the charged lepton or hadronic invariant mass spectrum in semileptonic B —> Xu decay to obtain a theoretically clean measurement of \Vub\. At subleading order in 1/rrib the universality is broken, and new subleading shape functions are required, introducing theoretical uncertainty into the extraction of \Vub\. The expansion has been carried out systematically to 0(l/rrib), while large 0 ( 1 /ml) effects due to annihilation graphs has also been considered [41]. 4.2. Exclusive

Decays and

Symmetries

4.2.1. B - • D*£u and Vcb In Section 3.3 I discussed the fact that HQET has a spin-flavour symmetry which is not manifest in QCD. We made us of this in Section 3.4, where we used the fact that the operator bvbv is a generator of that symmetry, and therefore its matrix element is fixed (matrix elements of symmetry generators are related to conserved charges). However, heavy quark symmetry gives us much more information. Any operator of the form hi,vThjtV

(87)

for any Dirac structure T and heavy flavours i and j is a generator of the symmetry, and so has an absolutely normalized matrix element. Physically, this just corresponds to the fact that exchanging the heavy quark in a meson with another heavy quark of arbitrary spin has no effect on the light degrees of freedom in the meson. This is true because, as we have discussed, in the rriQ —> oo limit a heavy quark simply behaves as a static source of colour charge, and the so the light degrees of freedom are insensitive to the mass and spin of the quark. The most important phenomenological application of this symmetry is to semileptonic B —> D/D*lv decays. Isgur and Wise [22] showed that heavy quark symmetry implies that all matrix elements of the form (H} (v')\hitV>ThjiV\Hj(*)(v)) (where i?j(*) denotes either the pseudoscalar (£) or B) or vector (D* or B*) meson with a heavy quark of flavour i = c, b) could be related to a single universal function: {Hf\v')\hi,v>rhj,v\Hf\v))

~ £(v • v').

(88)

£(v • v') has been dubbed the "Isgur-Wise function". It has the additional

235

property that, because of the absolute normalization discussed in the previous paragraph, (89)

£(1) = 1-

This means that all form factors for semileptonic B —» D and B —> £)(*) decay are known at the kinematic point where the c quark moves with the same velocity as the c quark (i.e., the q2 transfer to the leptons is maximal.) We can use the EFT machinery to calculate the corrections to this prediction. Quite generally, we can write the differential decay rate dT (B dw

D^ePe)

48TT 3

\Vbc\2(mB

- mD.)2m3D.(w

Aw m2B — 2wm,BV(iD* • (mB - mD*)2 W+ l

1 m2 D*

+ 1)V™2 - 1 \F(w)

(90)

where F(w) = £(w) + 0(as) +

0(l/mCib)

(91)

and w = v • v'. The complicated w dependence just comes from the phase space integrals. Radiative Corrections: These arise from perturbative corrections to matching conditions, as well as running of the bTc current below fi = rrib, and so are completely calculable. The one-loop running calculation has already been discussed in Section 3.3. From our previous results, then, we get the leading perturbative correction to Eq. (90), C7M(1 - -y5)b

as(mb)\ Ois{mc)J

6/25

cv-fil

~ l5)bv

(92)

and so 6/25

\as(mc)J

0 «

+ 1

log" mc/mb)

+

0(l/mb,c).

(93)

At next order, the current must be matched to 0(as) accuracy, and the running performed to two loops (resumming all terms of order Q,n+i \ogn mc/mb). This calculation has been done [42]. For practical purposes, however, logmc/mb is not very large, and so it is not necessary to sum terms of order a™ logmc/mb to all orders. Instead, one typically uses a fixed order calculation. This therefore includes terms of order a2 which are not enhanced by powers of log mc/mb, but neglects terms of 0(al) and higher.

236

Power Corrections: Just as the Lagrangian has corrections corresponding to higher dimension operators, so the weak current matches in the EFT onto additional operators. For general v and v', the only operator at 0(l/mc) is

cv,il?^(l-"/5)bv and so at 0(1/mc)

(94)

we need the matrix element



(95)

There are three possible form factors required to describe the matrix elements of this operator in HQET. Furthermore, additional 1/m.Q corrections arise due to T-products of 1/TTIQ operators in the effective Lagrangian and the leading order current, (D(*\v')\T{cv,YK,Zi)\B{v)).

(96)

However, at zero recoil (v = v') things simplify dramatically: both of these corrections vanish, although each for a different reason. • The T-product in (96) vanishes by the Ademollo-Gatto theorem when v = v'. The theorem states that the symmetry breaking corrections to matrix elements of an approximate symmetry current are second order in the breaking terms in the Lagrangian. • The matrix element (95) can be related, when v = v', to the matrix element of the operators cvD • vbv, which vanishes by the equations of motion. Thus, we have the nice result that the leading nonperturbative effects to the absolute normalization of the Isgur-Wise function at zero recoil are of order AQCD/W;?, n ° t A-QCi>/mc- [25] This immediately raises this prediction to a reasonable level of precision, since corrections of order A Q C D / T O ^ are expected to be of order 5%. Putting everything together, we find cvi\

i , (i)as(mb)

. (2) (as(mb)\

,

F(l) = 1+??V—-—+VA ( —-—J = 0.960 ± 0.007 + S1/m2 +0(a3s,

3

3

,

+^+OK,l/<)

l/m^c)

(97)

where ryA and ryA' are perturbatively calculable, and <5i/m2 refers to the incalculable 0(l/m2c b) corrections. There have been a number of attempts to estimate the size of these corrections from quark models, sum rules and

237

other methods. Combining the results in the literature gives the rough estimate [43] h/m? = -5-5 ± 3%

(98)

F ( l ) = 0 . 9 1 ±0.04.

(99)

which gives the result

Combining this with the world averaged measured value of dT/dw at the endpoint gives the value [44] \Vcb\ = (41.9 ± l . l e x p t ± 1.8theory) x 10" 3 .

(100)

Note that this is not as precise as the result from inclusive decays, Eq. (78). However, it is based on a completely different approach, both theoretically and experimentally, and so the agreement between the two results within errors is a nontrivial check on the validity of the 1/rrn, expansion. 4.2.2. B -> J/ipKs

and sin 2/?

I have been focusing in these lectures on the heavy quark expansion and its application to decay rates, etc. in CP conserving quantities, but I must say something about CP violation. The B factories were designed, after all, to study the SM picture of CP violation and while I cannot give this subject its due, I would like to at least mention it [45]. It also provides a nice illustration of another situation where we can use symmetry considerations to make model-independent predictions for hadronic processes. In this case, the relevant symmetry of QCD is CP. Consider charged B decay to some final state / . The amplitude for this decay may be written as Af = (F\H\B+) = Y, AkeiSkeilfk.

(101)

k

The sum over k corresponds to the sum over different amplitudes (or operators) which contribute to the decay. In general, such amplitudes are complex, since there may be real intermediate states, as shown in Fig The phase ipk is the weak phase which is present in the weak Hamiltonian itself. Similarly, the amplitude for the CP-conjugate process B~ —> / may be written

Af = (f\H\B~) = Y,Akel5ke-^. fc

(102)

238

The important thing is that, because QCD is CP-conserving, the strong amplitude is unchanged in the CP-conjugate process. The only change in the amplitude arises because the CP-violating weak phase flips sign. This immediately can give rise to CP-violation in the rates, since the different amplitudes can interfere, and so we can have \Af{2 7^ \A%. The problem is that in general, this interference depends on the values of the strong amplitudes Aket5k, which in turn depend on long-distance QCD. For most decays, then, it is difficult to translate measured CP violation into a determination of the weak phases J/ipKs, which proceeds dominantly via the b —> ccs process. (There is also a "penguin" diagram which contributes to the decay, but it is proportional to the same weak phase, so the argument still holds.) Since it is dominated by one amplitude, it does not appear that we will get any CP violation, since CP violation requires the interference of at least two amplitudes. However, there is another amplitude hidden in the process: the B° can oscillate into a B°, which can then decay, and this mixing carries a relative phase (due to Vtd m the Wolfenstein parametrization) which can interfere with the decay. Thus, we really do have two interfering amplitudes, and the relative phase between them is /3. More precisely it is easy to show that

r(B»(t)^^s) + r(B«(t)^^,)

H

v >\ >

and thus this measurement provides a clean measurement of sin 2/3. The current world average, from Belle and Babar, is [15] sin 2/3 = 0.79 ±0.14.

(104)

This is shown in Fig. 8 along with the other experimental constraints, and is clearly in good agreement with the SM expectation. A similar analysis holds for the rarer decay B —> sss. Since in the SM in the Wolfenstein parameterization, neither b —-> ccs nor b —-> sss contains a weak phase, both measurements measure the phase /3 in mixing. However, since the second process occurs at the loop level in the SM, it is easy to cook up models of new physics which contribute a large phase to the b —> sss amplitude, so consistency of the measured /3 from the two processes provides an important test of the SM. Indeed, there has been recent excitement over a possible

239

discrepancy between these measurements [46]: sin 2/3(5 - • Ka) = - 0 . 9 6 ± 0 . 5 0 t ° ; ^ (BELLE) sin 2f3(B -> 4>KS) = —0.19±g;|g ± 0.09 (BABAR)

(105)

However, the experimental errors are high enough that the discrepancy with Eq. (104) cannot be considered more than a tantalizing hint at present. Unfortunately, it is not so easy to come up with decays which are dominated by a single amplitude like this. For example, we could try a similar trick with b —> uus to try to predict CP violation in B —> TTTT. In this case, the relative phase between the amplitudes is 7 + /3 = 7r — a, so one could hope to use this decay to determine sin 2a. However, as shown in Fig. 6(c), this process receives contributions from both the tree-level b —> uus amplitude and a penguin amplitude. The latter is a one-loop process, but since it is a strong loop there is no reason for it to be suppressed, and it contributes with different weak and strong phases. Thus, without knowing the relative amplitudes between the tree and penguin processes, we can no longer make a model-independent determination of a in this manner, and more complicated and/or less theoretically rigorous approaches must be used.

5. Conclusions In these lectures I have discussed some of the most important applications of the heavy quark expansion to B decays. This is a large and rapidly evolving field. The applications I have discussed are mostly "classic", in that the theory was developed a number of years ago and is well understood. Recently, there has been much work attempting to further exploit (generalized) factorization in B decays, and to make use of the resulting simplifications, in particular to better understand nonleptonic decays. As an important example, several years ago, Beneke, Buchalla, Neubert and Sachrajda [47] proposed a generalized form of factorization applicable to two-body nonleptonic decays. Within their framework, they showed that so-called naive factorization, in which matrix elements of two currents are written as the product of matrix elements of the individual currents, arises modelindependently from QCD leading order in l/mb and as. This claim has generated much interest in the field, both phenomenological and theoretical, as the validity of the arguments has been heatedly debated. More recently, an EFT description which encompasses the kinematics of these decays, dubbed Soft-Collinear Effective Theory (SCET) has been developed [48]. Since the

240

kinematics are much more complicated t h a n in H Q E T — with hard, soft and collinear momenta possible — the E F T is more complicated, and (I believe) not completely understood. However, much progress has been made in understanding the interplay of the various scales, and this is an active and rapidly-evolving field.

References This list of references is by no means exhaustive; I have only referenced the original literature for specific points made in the text, along with some useful reviews on the subject. 1. M. K. Gaillard and B. W. Lee, Phys. Rev. D 10, 897 (1974). 2. H. Albrecht et al. [ARGUS COLLABORATION Collaboration], Phys. Lett. B 192, 245 (1987); C. Albajar et al. [UA1 Collaboration], Phys. Lett. B 186, 247 (1987) [Erratum-ibid. 197B, 565 (1987)]. 3. L. Montanet et al. [Particle Data Group Collaboration], Phys. Rev. D 50, 1173 (1994). 4. H. N. Brown et al. [Muon g-2 Collaboration], Phys. Rev. Lett. 86, 2227 (2001). 5. M. Knecht and A. Nyffeler, Phys. Rev. D 65, 073034 (2002); M. Knecht, A. Nyffeler, M. Perrottet and E. De Rafael, Phys. Rev. Lett. 88, 071802 (2002). 6. S. Narison, Phys. Lett. B 513, 53 (2001) [Erratum-ibid. B 526, 414 (2002)] [arXiv:hep-ph/0103199]. 7. K. Melnikov, Int. J. Mod. Phys. A 16, 4591 (2001) [arXiv:hep-ph/0105267]. 8. M. Ramsey-Musolf and M. B. Wise, Phys. Rev. Lett. 89, 041601 (2002). 9. L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983). 10. S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2, 1285 (1970). 11. see, for example, G. Buchalla and A. J. Buras, Phys. Rev. D 54, 6782 (1996). 12. A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967) [JETP Lett. 5, 24 (1967 SOPUA,34,392-393.1991 UFNAA,161,61-64.1991)]. 13. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Phys. Rev. D 18, 2583 (1978) [Erratum-ibid. D 19, 2815 (1979)]; B. A. Campbell and P. J. O'Donnell, Phys. Rev. D 25, 1989 (1982); S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. G. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987). 14. T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981) [Erratum-ibid. 65, 1772 (1981)]. 15. K. Hagiwara et al. [Particle Data Group Collaboration], Phys. Rev. D 66, 010001 (2002). 16. I. Z. Rothstein, "TASI lectures on effective field theories," arXiv:hepph/0308266 (in these proceedings.) 17. For other reviews of effective field theory, see D. B. Kaplan, arXiv:nuclth/9506035; A. V. Manohar, arXiv:hep-ph/9606222.

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35. M. B. Voloshin, Phys. Rev. D 51, 4934 (1995); M. Gremm, A. Kapustin, Z. Ligeti and M. B. Wise, Phys. Rev. Lett. 77, 20 (1996); M. Gremm and I. Stewart, Phys. Rev. D 55, 1226 (1997); 36. R. A. Briere et al. [CLEO Collaboration], arXiv:hep-ex/0209024. 37. M. Beneke, Nucl. Phys. B 405, 424 (1993); M. E. Luke, A. V. Manohar and M. J. Savage, Phys. Rev. D 51, 4924 (1995); M. Neubert and C. T. Sachrajda, Nucl. Phys. B 438, 235 (1995). 38. C. W. Bauer, Z. Ligeti, M. Luke and A. V. Manohar, Phys. Rev. D 67, 054012 (2003); M. Battaglia et al, eConf C0304052, WG102 (2003) [Phys. Lett. B 556, 41 (2003)]. 39. M. Neubert, Phys. Rev. D49 (1994) 3392; D49 (1994) 4623; I.I. Bigi et al, Int. J. Mod. Phys. A9 (1994) 2467; T. Mannel and M. Neubert, Phys. Rev. D50 (1994) 2037. 40. V. Barger, C. S. Kim and R. J. Phillips, Phys. Lett. B251, (1990) 629; A.F. Falk, Z. Ligeti, and M.B. Wise, Phys. Lett. B406 (1997) 225; I. Bigi, R.D. Dikeman, and N. Uraltsev, Eur. Phys. J. C4 (1998) 453; R. D. Dikeman and N. Uraltsev, Nucl. Phys. B509 (1998) 378. 41. C. W. Bauer, M. E. Luke and T. Mannel, arXiv:hep-ph/0102089; Phys. Lett. B543 (2002) 261; M. Neubert, Phys. Lett. B 543 (2002) 269; A. K. Leibovich, Z. Ligeti and M. B. Wise, Phys. Lett. B539 (2002) 242. 42. D. J. Broadhurst and A. G. Grozin, Phys. Lett. B 267, 105 (1991). 43. P. F. . Harrison and H. R. . Quinn [BABAR Collaboration], "The BaBar physics book: Physics at an asymmetric B factory," SLAC-R-0504 . 44. http://www.slac.stanford.edu/xorg/hfag/semi/summer02/summer02.shtml 45. I. I. Y. Bigi and A. I. Sanda, Cambridge Monogr. Part. Phys. Nucl. Phys. Cosmol. 9, 1 (2000). 46. K. Abe et al. [Belle Collaboration], arXiv:hep-ex/0308035; B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 89, 201802 (2002). 47. M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); Nucl. Phys. B 591, 313 (2000); arXiv:hep-ph/0007256. 48. C. W. Bauer, S. Fleming and M. E. Luke, Phys. Rev. D 63, 014006 (2001); C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63, 114020 (2001); C. W. Bauer and I. W. Stewart, Phys. Lett. B 516, 134 (2001); C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. Lett. 87, 201806 (2001); C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D 65, 054022 (2002); M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, Nucl. Phys. B 643, 431 (2002);

^ _ ^

m

Y

^ ^ .

SALLY DAWSON

^

This page is intentionally left blank

THE TOP QUARK, QCD A N D N E W PHYSICS

S. DAWSON* Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA [email protected]

The role of the top quark in completing the Standard Model quark sector is reviewed, along with a discussion of production, decay, and theoretical restrictions on the top quark properties. Particular attention is paid to the top quark as a laboratory for perturbative QCD. As examples of the relevance of QCD corrections in the top quark sector, the calculation of e + e - —> it at next- to-leading-order QCD using the phase space slicing algorithm and the implications of a precision measurement of the top quark mass are discussed in detail. The associated production of a it pair and a Higgs boson in either e + e - or hadronic collisions is presented at next-to-leading-order QCD and its importance for a measurement of the top quark Yukawa coupling emphasized. Implications of the heavy top quark mass for model builders are briefly examined, with the minimal supersymmetric Standard Model and topcolor discussed as specific examples.

Contents 1

Introduction

246

2

W h o N e e d s a Top Quark? 2.1 Anomaly Cancellation 2.2 b Quark Properties 2.3 Precision Measurements

247 248 249 251

3

Top 3.1 3.2 3.3

253 253 258 259

Quark Properties Hadronic Production Weak Interactions of Top Top Quark Decay

"This work is supported by the U.S. Department of Energy under contract number DE-AC02-76-CH-00016. 245

246

3.4 3.5 3.6 3.7 3.8

W Helicity in Top Quark Decay Top Production and Decay Single Top Production Measurements of Mt Top Spin Correlations

261 263 264 264 266

4

T h e Top Quark as a Q C D Laboratory 4.1 NLO QCD Corrections to e+e~ -> it 4.2 0(as) Corrections to e+e~ -> it 4.3 Real Contributions : 4.4 Threshold Scan in e+e~ - • it 4.5 tth Production in e + e ~ and pp Collisions 4.6 tth at an e + e ~ Collider 4.7 pp -»• tth at NLO 4.8 Collinear Singularities and Phase Space Slicing 4.9 Mass Factorization

5

Heavy Top as Inspiration for Model Builders 5.1 EWSB and the Top Quark Mass 5.2 Supersymmetry and the Top Quark 5.3 mSUGRA 5.4 Charged Higgs Decays 5.5 The Top Squark 5.6 The Top Quark and Dynamical Symmetry Breaking

6

Conclusions

267 267 269 271 275 277 278 280 281 283 285 286 289 •. . . 291 294 295 295 296

1. I n t r o d u c t i o n Long before its discovery in 1995, 1 the top quark was regarded as an essential ingredient of the Standard Model (SM) of particle physics. Its existence, and many of its properties, are determined by requiring the consistency of the S t a n d a r d Model. These requirements (discussed in Section 2) specify the couplings of the top quark to the SU{2>) x SU{2)L X U(1)Y gauge bosons and many of the top quark properties: • O*

= - I e I

Hem

3 I °I

• Weak isospin p a r t n e r of b quark: T | = • Color triplet

• spin-1

|

247

The top quark mass, which is measured at CDF and DO to be Mt = 174.3 ± 5 . 1 GeV,2 is not predicted in the Standard Model, but is restricted by precision electroweak measurements. 3 ' 4 In order to confirm that the observed top quark is that predicted by the Standard Model, all of the properties listed above must be experimentally verified by direct observation. Section 3 contains a discussion of the measurements of top quark properties at the Tevatron and surveys the improvements expected at future colliders. Recent reviews of top quark physics can be found in Ref. 5. In Section 4, we discuss the top quark as a laboratory for perturbative QCD. At the top quark mass scale, the strong coupling constant is small, as(Mt) ~ 0.1, and so QCD effects involving the top quark are well behaved and we expect a perturbation series in as to converge rapidly. The process e+e~~ —• it provides an example of QCD effects in top quark production and the next-to-leading-order (NLO) QCD corrections are described in detail using the phase space slicing (PSS) algorithm. A knowledge of these higher order QCD corrections is vital for extracting a precise value of the top quark mass from the threshold behavior of the e+e~ —> it cross section. As a further example of the role of QCD effects in the top quark sector, we consider the associated production of tih and the implications for measuring the top quark Yukawa coupling. Finally, in Section 5, we discuss the importance of the top quark mass in model building. Since the top quark is heavy, it is expected to play a special role in elucidating the source of fermion masses. We begin by discussing the significance of the top quark in supersymmetric models, and finish with a brief discussion of the top quark and models with dynamical symmetry breaking.

2. W h o Needs a Top Quark? In this section we discuss some of the reasons why the top quark was believed to exist even before its experimental discovery. These considerations fall into three general categories: theoretical consistency of the Standard Model gauge theory (anomaly cancellation), consistency of b quark measurements with SM predictions, and consistency of precision measurements with the SM. We then turn to a discussion of top quark production and decay mechanisms at the Tevatron and the LHC. The particles of the first generation of fermions, along with the Higgs doublet, are shown in Table 1, with their gauge quantum numbers. The third generation is assumed to follow the same pattern, with the left-handed

248

top and bottom quarks forming an SU(2)i doublet with hypercharge, Y = | . The right-handed top and bottom quarks are SU(2)L singlets. Our normalization is such that: Qem =T3+Y

(1)

with T 3 = ±\. Table 1. Fermions in t h e first generation of t h e Standard Model and their SU(3) X SU(2)L X U(1)Y quantum numbers. Field {uL,dL) UR,

dn, (yL,eL) eR H

2.1. Anomaly

SU(S) 3 3 3 1 1 1

SU(2)L

U{1)Y

2 1 1 2 1 2

1

I 1 ! 2

-1

1 2

Cancellation

The requirement of gauge anomaly cancellation 6,7 puts restrictions on the couplings of the fermions to vector and axial gauge bosons, denoted here by V* and AM. The fermions of the Standard Model have couplings to the gauge bosons of the general form: L~g ^ T ^ ^ A ^

+ gv^T^^V^,

(2)

where Ta is the gauge generator in the adjunct representation. These fermion-gauge boson couplings contribute to triangle graphs of the form shown in Fig. 1. The triangle graphs diverge at high energy,

T^-Tr^iT^T^J-^^,

(3)

where r\i = ^ 1 for left- and right-handed fermions, ipL,R = | ( 1 T 75)V'This divergence is independent of the fermion mass and depends only on the fermion couplings to the gauge bosons. Such divergences cannot exist in a physical theory, and must somehow be cancelled. The theory can be anomaly free in a vector-like model where the left- and right-handed particles have identical couplings to gauge bosons and the contribution to Eq. (3) cancels for each pair of particles. From Table 1, however, it is

249

K"

Vr

'www

Figure 1. Generic Feynman diagram contributing to gauge anomalies. The fermion loop contains all fermions transforming under the gauge symmetry, a, b, c are gauge indices.

clear that the Standard Model is not vector-like. The anomaly, Tabc, must therefore be cancelled by a judicious choice of fermion representations under the various gauge groups. The only non-vanishing contribution to the anomaly in the Standard Model is from T,Tr[Y{Ta,Tb}},

(4)

a

where T are the SU(2)L generators and the sum is over all fermions in the theory. Equation (4) vanishes for the hypercharge assignments given in Table 1. Note that the anomaly cancels separately for each generations of fermions. The cancellation of the gauge anomalies in the Standard Model for the third generation therefore requires that the b quark have a T3 = | partner with electric charge Q\m = § | e |, and hypercharge Y* = Q\m — T|. The partner of the b quark is by definition the top quark. Since anomaly cancellation is independent of mass, a priori, the top quark mass could be anything. 2.2. b Quark

Properties

Many of the experimental properties of the b quark require that it be a T36 = - \ particle with Qbem = - § and Y 6 = \. The coupling of the b quark to the Z boson can be tested to check if it has these quantum numbers. The SM fermions couple to the Z boson as, L=-

9 T 4 cos 6w

Ri(l + J5) + HI - 75) AZ* ,

where Ri

-2QiSin

Ow

(5)

250

Li = 2T3i - 2Qi sin2 0W ,

(6)

and 6w is the electroweak mixing angle. The experimental value of Rhad is sensitive to Qlm, a(e+e Rhad

—> hadrons)

(7)

=

At a center-of-mass energy, y/s = 2m),, Rb depends sensitively on the b quark electric charge, SRhad(2mb) = Nc(Qb)2 + 0(as),

(8)

where Nc = 3 is the number of colors. The experimental measurement, 8 SRhad(2mb)

= 0.36 ±0.09 ±0.03

(9)

is in good agreement with the theoretical prediction from Qbem = — \ , verifying the SM electric charge assignment of the b quark. Similarly, the SU(2)L quantum numbers of the left- and right- handed b quarks are probed by the decay rate for Z to bb quark pairs. If the b did not have a top quark partner, it would be an isospin 0 particle ( T | = 0) and the decay width would be dramatically different from that of the SM. The decay width is given in terms of the left and right- handed couplings of the Z to the bb pair, 9 r(Z^bb) = ^^(Ll+Rl).

(10)

If the b quark were an isospin singlet, then the decay width would be changed, T(Z -> bb)T3 =~i _ 1 + 4Q b sin 2 0W + 8Q 2 sin 4 9W T(Z^bb)Ts=° ~ 8Q2bsm4ew

13.

The measurement of the Z —> bb decay width excludes the T36 — 0 hypothesis for the b quark. 3 The measured b couplings, combined with anomaly cancelation, require that the b quark have a T | = |2l ', color triplet, fermion partner: this is the top quark.

251

Figure 2. Feynman diagrams for gauge boson self-energies which give contributions proportional to M t 2 .

2.3. Precision

Measurements

Before the top quark was discovered, an approximate value for its mass was known from precision measurements, which depend sensitively on the top quark mass. At tree level, all electroweak measurements depend on just three parameters: the SU(2)L X U(1)Y gauge coupling constants and the Higgs vacuum expectation value, v = 246 GeV. These are typically traded for the precisely measured quantities, a, GF, and Mz- All electroweak measurements at lowest order can be expressed in terms of these three parameters. Beyond the lowest order, electroweak quantities depend on the masses of the top quark and the Higgs boson. A typical example of the role of the top quark mass in precision measurements is the calculation of the p parameter, M ._ w ~~ M§ cos2 Bw _ Aww{0) AzziO) M^ Ml '

H

( U [

'

where Ayv is defined by the gauge boson 2-point functions, Ol£V = Avv(p2)
+ Bvvp^p"

.

(12)

At tree level, the p parameter in the Standard Model is exactly one, but at one loop it receives contributions from gauge boson, Higgs boson, and fermion loops. The largest corrections are those involving the top quark loop. For simplicity, we compute only the corrections proportional to M t 2 , which are found from the diagrams of Fig. 2. The 2-point function of the Z boson is given by,

i""z =--N m
~

"c\4cwJ

W

J

(2n)n[den\

(is)

252

Winter 2003 Measurement

//-vmeas r > f i t w r r m e a s

Pull

-3-2-10123 -0.16

4«^ r! {m 7 ) mz [GeV]

0.02761 +0.00036 91.187510.0021

0.02

r z [GeV]

2.4952 ± 0.0023 41.540 ±0.037

-0.36 1.67

20.767 ± 0.025 0.01714 ±0.00095 0.1465 ±0.0032

1.01 0.79

° L [nb] R, A o,i M

fb

A(Pt) Rb Rc

0.21644 ±0.00065 0.1718 ±0.0031 0.0995 ±0.0017

c

A,°b C

0.0713 ±0.0036

Ab

0.922 ± 0.020

0.670 ± 0.026 Ac 0.1513 ±0.0021 A,(SLD) sin2e^tf(Q(b) 0.232410.0012

i

•

a

-0.42

.

0.99 -0.15

i

-2.43 -0.78 -0.64

•

0.07 1.67 0.82

^™ •i

m w [GeV]

80.426 ± 0.034

1.17

••

r w [GeV] mt [GeV] sin26w(vN)

2.139 ±0.069 174.3 ±5.1

0,67

•

0.227710.0016

(3w(Gs)

-72.83 + 0.49

0.05 2.94

— —

U.I £-.

1

-3-2-10123 Figure 3. Fits to electroweak data. The bars show the pull from the Standard Model global fit.3

where cw = cos 9w and , T"" = Tr[(k + Mt)Y(.RtP+ den={k2-M?}[(k+P)2-M?},

+ LtP-)(k-p

+ Mt)Y(RtP+

+ LtP-)] (14)

and P± = ^(1 ± 75). The coupling of the Z to the top quark is given in Eq. (5) and p is the external gauge boson momentum. Shifting momentum in the integral, k' —> k — px, and using Feynman parameters, the integral in n = 4 — 2e dimensions becomes, i^zz

... 2 f dnk' ( -ig \4cw [4k' 2 ( - l + l){R2t+Ll)

-N,

f1 ,

+ (k'2+p2x{l-x)-Mf)

l6LtRtM?}g^ 2^2

(15)

253

where we retain only those terms contributing to Azz{0)portional to M 2 is,

The result pro-

The analogous result for the W two-point function is,

A

™V

=

g2Nc(A*\\f2(i

,l

V-2^\M)

Mt

\t

+

V-

{17)

Combining Eqs. (16) and (17), the contribution to the p parameter which is proportional to M 2 is found from Eq. (II), 1 0 2 g

JVcM2

Sp

-64^M*;-^v^-

_GFNCM? ( 8 )

Experimentally 4 p = 1.00126±o;ooi44 and so an upper limit on the top quark mass can be obtained from Eq. (18). Many of the precision measurements shown in Fig. 3 are sensitive to M 2 and so a prediction for the top quark mass can be extracted quite precisely by combining many measurements. In fact, precision measurements were sensitive to the top quark mass before top was discovered at Fermilab! The agreement between the direct measurement of the top quark mass in the Fermilab collider experiments and the indirect prediction from the precision measurements (as shown in Fig. 4) is one of the triumphs of the Standard Model. It is interesting to note that a similar limit on the Higgs boson mass from precision measurements gives Mh < 193 GeV at the 95% confidence level.3 Precision measurements depend logarithmically on the Higgs mass, and so it is much more difficult to bound the Higgs mass in this manner than it is to restrict the top quark mass. Increases in the experimental precisions of the top quark mass and the W boson mass in Run II measurements at the Tevatron will provide an improved bound on the Higgs boson mass, SMh/Mh ~ 4 0 % . n In the next sections, we discuss the discovery of the top quark at Fermilab and the experimental exploration of the top quark properties, both at Fermilab and the LHC and at a future high energy e+e~ collider. 3. Top Quark Properties 3.1. Hadronic

Production

The top quark was discovered in 1995 at Fermilab in pp collisions at \/S = 1.8 TeV.1 This data set (called Run I) consists of an integrated luminosity of

254

Top-Quark Mass CDF

[GeV]

•—

176.1 ±6.6

D0

172.1 ±7.1

Average

174.3 ±5.1 170.7 ±10.3

LEP1/SLD —U—

LEP1/SLD/mw/rw 125

150

175

mt

177.5 ±9.3 200

[GeV]

Figure 4. Measurements of the top quark mass at Fermilab (CDF and DO) and indirect predictions from precision measurements (LEP1, SLD and M\y).3

L ~ 125 pb~x. Both CDF and DO are currently rediscovering the top quark in the Run II data set. Run II will produce roughly 500 clean top quark events for each inverse femtobarn of data and so precision measurements of many top quark properties will be possible. In hadronic interactions, the top quark is produced by gluon fusion and by qq annihilation as shown in Fig. 5, 99

tt

qq

it.

(19)

The hadronic top quark production cross section, CTH, at the Tevatron, pp —> it, (or pp —• it at the LHC) is found by convoluting the parton level cross section with the parton distribution functions (PDFs), <7H(S)

= Sjj

/ fi(Xi,

[l)fj(x2,

fjL)&ij(xiX2S,

fi),

(20)

where the hadronic center of mass energy is S, the partonic center of mass energy is s = xix2S, the parton distribution functions are fi(x, fi), and the parton level cross section is &ij(s,fi). The parameter fi is an unphysical

255

' ^smrnmc-fY-

t

+

+ /

rjirinrirwsOTir'Figure 5. liders.

8-

1

„/

'flaaaaasa.

g'

Feynman diagrams contributing to top quark pair production in hadron col-

renormalization/factorization scale. To 0(a2s), the sum in Eq. (20) is over the qq and gg initial states shown in Fig. 5. It is useful to parameterize the parton level cross section (to next-toleading order in as) as, Vij{s,n)

Mi

u °ij(p) + Hjas{lA + h2i:jas(n) log( -^2

h

At lowest order, 0(a2s), the parton level cross sections are

,0 99

h

=

*PP

192

-p{p2

(21)

12

+ 16p + 1 6 ) l o g ( [ ± | ) - 28 - 31p (22)

o,

where 4Mt2

/3 = y r

(23)

The threshold condition is p = 1 and so &ij vanishes at threshold. To any given order in as(p), the hadronic cross section an must be independent of p, daH d log p/

= 0.

(24)

256 Applying this restriction to Eq. (21) yields 0=

2-/(x 2 ,At) + / ( x i , / i i j — 2~ d log p? d log p? da(x1x2S,p) f(xi,p)f(x2,p)'9 log p2 -dx\dx2

}a(x1x2S,p,)dx1dx2 (25)

The scale dependence of the PDFs is governed by the Altarelli-Parisi evolution functions, Pij (z): dfi(x,n)

f1 dy

as

Ix .

(26)

y Similarly, the scale dependence of the strong coupling constant, a s (/z), is governed by the evolution of the QCD /3 function. das(n) d log p2

2

(27)

-60o;;

where 6o = -—12™lf , and nif is the number of light flavors. At nextto-leading order the scale dependence of the various terms must cancel, yielding a prediction for ft.2-,

* & - *

Anboh^ (p) - Sfc J

dzh°kj ( j j Pki (z) - Sfe J

dzh°k (A Pk M (28)

Using the explicit forms of the Altarelli-Parisi evolution functions, combined with Eq. (22). we find12 h2qq- = — 2TT

16?rP

81

^ ( m )

7T

h2gg

*{*(•

~~ 2n 192

1-/3

+ \hUP) (l27 - 6nlf + 4 8 1 o g ( ^

59p2 + 198/9 - 288 log

1 + /3 1-/9

+12p( p 2 + 16/0+16 )
h2qg

4 /?( 7449/02 - 3328/0+724 ) [> - 12h°gg(j>)]og[ -^ ~15 4p 32 1+ 0 = 3— 14/92 + 27/o - 136 log p{2-p)9l{(3) 8 4 1-/3 ~

(1319/32 - 3468/0 + 724

(29)

257

where we have defined,

»c»-*(^)-*(^) + ^)-™,(V) The quantities ft1 • can only be obtained by performing a complete nextto-leading order calculation 12 ' 13 and analytic results are not available, although a numerical parameterization is quite accurate. The strong coupling evaluated at the top quark mass is small, as(Mt) ~ 0.1 , and so a perturbative expansion converges rapidly. At energy scales significantly different from the top quark mass, there are also large logarithms of the form (a3 log(-pj5-))ra, which can be summed to all orders to obtain an improved prediction for the cross section. 14 The inclusion of next-to-leading-logarithm effects reduces the scale dependence of the cross section to roughly ±5%. 1 5 Figure 6 shows the Run I CDF and DO measured top production cross sections and top quark mass compared with two theoretical predictions. The Run I it cross sections are, 16 * « = 6.5il;Ip6 <7tI = 5 . 9 ± 1 . 7 p 6

CDF DO.

(31)

The curves labelled NLO+NLL include some of the logarithms of the form (as log(-gj-))™ and the agreement with the experimental results is clearly improved from the NLO prediction alone. An updated theoretical study including next-to-leading-logarithm resummation gives the prediction atl(VS

= 1.8 TeV) = (4.81 - 5.29) pb for Mt = 175 GeV,

(32)

with the range in the prediction corresponding to Mt/2 < fi < 2Mt-15 The agreement between the predicted and the experimental production rates implies that the top quark is a color triplet, since that rate would be significantly different for a different color representation. At the LHC, the subprocess gg —> it dominates the production rate with 90% of the total rate and the cross section is large, (JLHC ~ 800 pb, yielding about 108 it pairs/year. Because of the large rate, the LHC can search for new physics in top interactions. The large rate also has the implication that top production is often largest background to other new physics signals. A detailed understanding of the top quark signal is therefore a necessary ingredient of new physics searches.

258

S = 1.8 TeV, pp MRSR3

a. CDF NLO NLO + NLL, A=2

180

140

160

1B0

200

m t (GeV) Figure 6. Fermilab Tevatron Run I cross sections for pp —» it. The dashed line is the NLO prediction, while the solid lines incorporate the next-to-leading logarithms.14

3.2. Weak Interactions

of Top

T h e weak interaction eigenstates are not mass eigenstates, 9 S,=d,»,6*7"(l - l5)VtgqW+ 2 ^

+ h.c.

(33)

and so the interaction of the t o p quark with a W boson and a light quark is proportional to the C K M mixing element, Vtq. Measurements of the decay rates of the top quark into the lighter quarks therefore translate directly into measurements of the C K M mixing angles. T h e unitarity of the C K M m a t r i x and the restriction to 3 generations of fermions gives a limit on Vtb from the measured values of | Vub | and

I Vcb |:4 1 = | Vub

vcb |2 +1 vtb

(34)

giving, 0.9991 < | Vtb \< 0.9994

(35)

If there are more t h a n 3 generations, however, Eq. (34) is no longer valid and there is almost no limit from unitarity on VtbUnitarity in the three generation model can be tested by measuring the ratio of the rate for top decays to the b quark t o the rate for top decays to

259

Figure 7.

Feynman diagram for top decay to Wb.

lighter quarks, 17 R •tb

T(t - • Wb) T{t - Wq)

| Vu, | Vu | 2 + | Vu ? + I Vt6 | 2 = 0.94+°Ji

(CDF)

(36)

This quantity can be measured at Fermilab by counting the number of tagged 6's in a top quark event. Since the b quark lives approximately 1.5 ps, the b quark will travel 450 fim before decaying. The secondary b vertex can then be measured with a silicon vertex detector. Assuming unitarity of the CKM matrix, the denominator of Eq. (36) is 1, and the measurement can be interpreted as a measurement of Vtb, I Vtt |= 0.97±°;&

(37)

consistent with Eq. (35). If there is a fourth generation of quarks, (t',b'), the charge — | b' is experimentally restricted to be heavier than mt — Mw,4 and so the top quark cannot decay into the b'. Then the denominator of Rtb need not equal 1, and the measurement implies only | Vtb \»\ Vta |, | Vtd |The direct measurement of Vtd in single top production will be discussed in Section 3.6. 3.3. Top Quark

Decay

The top quark is the only quark which decays before it can form a bound state. This is due to the very short lifetime of the top quark, rt - 5 x 1CT25 sec ,

(38)

260

compared to the QCD time scale: TQCD ~ 3 x

10" 2 4

sec .

(39)

This is very different from the b quark system, where the b quark combines with the lighter quarks to form B mesons, which then decay. Since | Vtb | ~ 1, the dominant decay of the top quark is t(p)^W»(Pw)

+ b(P'),

(40)

The lowest order amplitude is shown in Fig. 7 and is, A0(t -> W»b) = --JL=utf)rf(l

-

7 5 )«(p)

.

(41)

Squaring the amplitude, and summing over the W polarization vectors, Ee"(p w )e*"(p w ) = - < T + ^ # , M w gives the amplitude-squared, M2 A0(t - Wb) | 2 =
(42)

(43)

The total decay width is

T0(t -+ Wb) ={^)~J\

Mt - Wb) |2 (dPS2) .

(44)

The factor of ^ is the average over the initial top quark spin, and the phase space factor is,

M? - Ml

{dFS

* = I 87rM sjf • 2

(45)

t

Including the CKM mixing, the final result is then

1 „ ^ „,^( -f) ( 1+ ^ ) . ( «, r (1 )H 1 Higher order QCD corrections can be calculated in a straightforward manner and yield a precise prediction for the decay width, 9,18 T(t - • bW+) = | Vtb | 2 1-42 GeV.

261

3.4.

W Helicity

in Top Quark

Decay

The helicity structure of the top quark decays is interesting because it yields information on the V — A nature of the tbW vertex as is clear from Eq. (41). The top quark is produced through the weak interactions as a left-handed fermion (neglecting the b quark mass), which has spin opposite from the direction of the top quark motion. If the W is produced as a longitudinal W (helicity 0), then its momentum and polarization vectors are (see Fig. 8):

Pw = (Ew,0,0,\pw eY =

|)

~(\Pw\,0,0,Ew)

Mw ^

K

\Mt J

'

The amplitude for a top decaying into a longitudinal W is then,

This gives the result that the decay of the top into longitudinal W s is, 777

r(t - bwL) ~ -j-.

(49) w

The decay of the top quark into a positive helicity W, h = +, is forbidden by angular momentum conservation since a massless b quark is always left-handed, while the heavy top can be either left- or right-handed. This is illustrated in Fig. 9. CDF has measured the decay of the top quark to a right handed W and finds a result consistent with 0, 19 BR(t -+bWR)=

0.11 ± 0.15 .

(50)

For a transversely polarized W, the polarization vector is given by, ef = -^=(0,l,±z,0).

b ** Figure 8.

t •

(51)

|

W

v A A / V W W W

Spin vectors for longitudinally polarized W's produced in the decay t —+ Wb.

262

t

W

•

v/VWWWW

forbidden Figure 9. Spin vectors for positive helicity (right handed) W's produced in the decay t —> Wb. Conservation of angular momentum forbids this decay.

b •* Figure 10.

t W • v/wwwvw Spin vectors for transverse helicity W's produced in the decay t —> Wb.

This gives a decay rate, T(t - • bWT) ~ g2mt .

(52)

Comparing Eqs. (49) and (52), it is clear that the rate for top decay to transverse W's is suppressed relative to that for decay to longitudinal W's,

-TO

—

r(t - bwL) T(t -> Wb)ToT 2M,2,

1+ = 70.1% .

(53)

Can we measure this? The W helicity is correlated with the momentum of the decay leptons: hw = + gives harder charged leptons than hw = -. A preliminary measurement of the polarization of the W's in top decay yields, 19 F 0 = 9 1 ± 3 7 ± 1 3 % CDF,

(54)

consistent with Eq. (53). With 2 / 6 " 1 of data from Run II at the Tevatron, a more definitive (~ 5%) measurement of F0 will be possible.20

263

3.5. Top Production

and

Decay

The Tevatron and the LHC produce predominantly it pairs, which decay almost entirely by t —> Wb (since | Vtb |~ l ) , a pp-+tt->

W+W~bb.

(55)

It is useful to classify top quark events by the decay pattern of the pair:

W+W~

• Di-lepton: Roughly 5% of the it events fall into this category,

W~ -> l~v .

(56)

Since Br(W -> ev) + BR(W - • \iv) ~ 22%, the 125 pb~x at Run I of the Tevatron produced ~ 625 it events which yielded ~ 0.22 x 0.22 x 625 ~ 30 di-lepton events. These events are clean, but suffer from the small statistics. • lepton + jets: This class of events comprises roughly 30% of the total, W± -> l±v,

WT - • jets .

(57)

Lepton plus jet events are fully reconstructable and have small backgrounds. Assuming at least 1 b quark is tagged, CDF observed 34 such events in Run I, with an expected background of 8.22 In Run II, with 2 / 6 " 1 , there should be 1000 lepton+jets events/experiment. This increase from Run I results from higher luminosity, the higher cross section at vS = 2 TeV compared to \/S = 1.8 TeV, and upgraded detectors with improved capabilities for 6-tagging. This channel produced the most precise measurement of the cross section for top pair production. The combined average from CDF and DO at Run I is, a(pp -+ it) \jg=1A

TeV=

5.9 ± 1.7 pb

(58)

• All jets: 44% of the events fall into this category. This class of events was observed by both CDF/DO at Run I, with large backgrounds. a

Recent reviews of the experimental issues involved in top quark physics can be found in Ref. 21.

264 Table 2. Contributions to single top production at the Tevatron and the LHC. &qb

(Tqq Oqb

Tevatron, Runll 2.1 pb .9pb .1 pb

LHC 238 pb 10 pb 56 pb

Figure 11. Feynman diagrams contributing to single top production at the Tevatron and the LHC.

3.6. Single Top

Production

Single top production can provide a precise measurement of Vtb- The dominant production processes at the Tevatron are the t channel W exchange and the s channel qq annihilation shown in Fig. 11. At the LHC, the subprocess gb —> tW is also important. The contributions (to NLO QCD) of the various subprocesses at the Tevatron and the LHC are given in Table 2 23,24 ^yj Qf these processes are proportional to | Vtb |2Single top production was not observed in Run I, and both experiments set limits on the production rate. At Run II of the Tevatron, and at the LHC, single top production can be observed with a rate roughly half that of the tt rate and will lead to a measurement at the Tevatron of,

with 30 fb 1 of data. With 2 / 6 accuracy of | ^ |~ 13%. 20 3.7.

Measurements

1

, the Tevatron Run II should achieve an

of Mt

Mt is a fundamental parameter of the Standard Model, but the value is not predicted (except indirectly from precision measurements such as the p parameter discussed earlier). In the Standard Model, a precise value of Mt is important primarily for its role in precision electroweak observ-

265

ables and for the prediction for the Higgs boson mass extracted from these measurements. 25 In extentions of the Standard Model, such as those discussed in Section 5, a precise knowledge of Mt plays a critical role in defining the parameters of the theory. The top quark mass was measured in Run I at the Tevatron and the most accurate value of Mt comes from the single lepton+jets channel, 2 Mt = 176.1 ± 4.8 (stat) ± 5.3 (syst)

CDF

= 173.3 ± 5.6 (stat) ± 5.5 (syst)

DO .

(59)

In the lepton plus jets channel there is one unknown parameter (the longitudinal momentum of the v coming from the W decay) and three constraints from the reconstruction of the W masses and the requirement that the reconstructed masses of the top and anti-top be identical, Mt = Mj. The best value of the top quark mass found from combining all channels at the Tevatron is, Mt = 174.3 ± 5 . 1 GeV .

(60)

The systematic error is dominated by the uncertainty on the jet energy scale, ±4.4 GeV

CDF

±4 GeV DO .

(61)

With increased data from Run II, the mass measurement at the Tevatron will be improved to SMt ~ 3 GeV with an integrated luminosity of 2 fb~l. Figure 12 shows the impact of a precise measurement of the top quark mass combined with a W mass measurement for the prediction of the Higgs boson mass. Clearly, for the precision measurements of the W and top masses to be useful, the experimental error on both measurements needs to be comparable. The Higgs mass prediction extracted from precision measurements can be compared with the direct measurement of the Higgs mass at the LHC and the consistency of the Standard Model tested. Tevatron Run II with 2 / 6 - 1 should achieve accuracies: 8MW ~ 27 MeV 5Mt ~ 3 GeV ,

(62)

yielding a prediction for the Higgs mass with an uncertainty of25 ^ ~ 4 0 % . Mh

(63)

266

80.6

—i

1

1

1

1

1

1

1

1

r

— LEP1,SLDData • - LEP2, pp Data

80.5-

68% CL

> CD

£ 80.4 5 E 80.3

mH [Ge

114 ' 130 150

80.2

170

190

mt [GeV] Figure 12. Implications of precision measurements of Mt and Myy on the indirect extraction of the Higgs boson mass. The Standard Model allows only the shaded region. 3

The LHC will have large statistics on top quark production due to the large rate and can use the lepton+jets channel with 10 fb^1 of data to obtain a measurement 26 SMt ~ 0.07 GeV .

3.8. Top Spin

(64)

Correlations

When pair produced in either e+e~ or qq collisions, the spins of the it pair are almost 100% correlated. 27 The top decays before the spin can flip, and so spin correlations between the t and t result in angular correlations among the decay products. At threshold, the it cross section is s-wave, and the e+e~ or qq spins are translated directly to the it spins. The incoming q, q have opposite helicities, since they are interacting through a gauge interaction and the it spins aligned along the beamline. At high energy, v ^ > > mt, the top mass is irrelevant and the t,i must have opposite helicities. Finding the correct basis for intermediate energies is subtle, however.

267

e~(p)

t^)

/ e+(q) Figure 13.

4.

1 , z

/

\ X

t(q>)

Lowest order Feynman diagram for e + e - —> it.

The Top Quark as a QCD Laboratory

The production and decay mechanisms for the top quark in both e + e~ and hadronic collisions are well understood and so the top quark system provides an ideal laboratory for studying perturbative QCD. This section contains a pedagogical treatment of the 0(as) next- to- leading order (NLO) QCD corrections to the process e + e~ —> ti.2S This process is particularly simple since the QCD corrections affect only the final state. The calculation is performed using the two cut-off phase space slicing (PSS) algorithm for evaluating the real gluon emission diagrams. 29 ' 30 The total rate for e+e~ —> it tests our understanding of perturbative QCD and additionally the threshold behavior of the total cross section can be used to obtain a precise measurement of Mt, along with information on as(Mt) and the iih Yukawa coupling. An understanding of the QCD corrections are crucial for these interpretations. As a second example of the role of QCD in top quark physics, we consider the associated production of a it pair along with a Higgs boson, h. The iih Yukawa coupling is a fundamental parameter of the theory and probes the mechanism of mass generation, since in the Standard Model, 9tth = —• The processes e+e~ —> iih and pp(pp) —> iih provide sensitive measurements of the top quark- Higgs coupling and can be used to verify the correctness of the Standard Model and to search for new physics in the top quark and Higgs sectors. The importance of the QCD NLO corrections to iih production is discussed at the end of this section. 4.1. NLO QCD Corrections +

to e+e~ —• it

The Born amplitude for e e~ —> it proceeds via s— channel 7 and Z exchange, as shown in Fig. 13. The dominant contribution at yfs = 500 GeV

268

is from 7 exchange and so for simplicity we will neglect the Z contribution here. (The complete 0{as) result, including the contribution from Z— exchange can be found in Refs. 28 and 30.) The electron mass can also be neglected. The lowest order amplitude is:

ALO{e+e~ - t t ) = ^ ^ ^ ) 7 ^ b W ) 7 ^ ( < / ) % , s where i, j are the color indices of the outgoing top quarks. It is simplest to work in the center-of-mass frame of the incoming e + e~ pair, where the 4— momenta of the particles are given by: p=

^(1,0,0,1)

q=

^(1,0,0,-1)

p> = ^ ( l , O , / ? s i n 0 , / 3 c o s 0 ) q' = ^ ( 1 , 0 , - / 3 sin 0,-/3 cos 0) with s = (p + q) 2

*-Vi-^

(«)

and 6 is the angle of the top quark emission with respect to the electron beam direction. The threshold condition is defined by j3 = 0. It is straightforward to find the tree level amplitude-squared in n = 4—2e dimensions, (summed over colors), ALO

\2= 64ir2a2Q2tNc 2(1 -e)+P2(cos2

0-1)

(66)

The color factor is Nc = 3. The corresponding total cross section is:

vLo = YsJ\ALO\2(dPS)2. Integrating over the two body phase space,

(67)

269

*(PO

Figure 14. Vertex correction to e+e —> tt. The 7*" denotes an off-shell photon which is connected to the initial e + e - state.

and including a factor of | for the average over the initial electron spins, gives a total rate, <JLo{e+e- - • it) = -Jj—\ OZ7TS 4

f

dcos9 \ ALO | 2 + 0(e)

J_i

The 0(e) terms which we have neglected will be important in computing the 0(as) QCD corrections. The cross section vanishes at threshold, (3 —> 0, and decreases with increasing energy, a ~ K This has the obvious implication that the higher energy an e + e~ collider, the larger the required luminosity. 4.2. 0(as) The 0(as) butions:

Corrections

to e+e — —• it

corrections to e+e~ —> it contain both real and virtual contri-

ONLO = CLO + ^virtual

+ Vreal-

(69)

The one-loop virtual contribution consists of a vertex correction and a wavefunction renormalization of the top quark leg. The vertex correction is shown in Fig. 14 and can be parameterized as: Averts = eQev(q)j„u(p) -V(p',

q')

(70)

with,

r„(p',g') = {~eQt)5i3g2sCF[N}u{p') ^

+ ^

v(q>)

(71)

270

where the normalization is, [N] =

1

fA-np?

16^2

\Mf

r(i + c)

(72)

and

N?

(73) 2Nr_ is the color factor for the vertex diagram. The parameter /x is an arbitrary renormalization scale. Explicit calculation gives the results, 28 ' 30 CF

1

A,

is A/3 = log

1-/3 1 + /9

B= - 1

3/SA^

+2Li2

- A | + 2A/3log

/ ^

2/9 1 + /3

2TT2

1 + /3

+

e The di-logarithm is denned by, (74) ! > ^ The 0{as) contribution to the virtual cross section is the interference of the vertex correction with the lowest order result. (The A term of Eq. (71) does not contribute to the interference with the lowest order result when the electron mass is neglected). Li2(x) =

^vertex

1 as /47r;r 2 2TT V Mt

J{dPS)2AL()Avertex CFT{1 + e)aLO

El

B fin

(75)

Note that the 0(e) terms from the lowest order amplitude and from the 2particle phase space combine with the — term to give a finite contribution 10 (Jyertex •

The wavefunction renormalization for the top quark must also be included in the virtual correction, as shown in Fig. 15. This diagram gives

271

t(P')

t(q') Figure 15. Feynman diagram for top quark wavefunction renormalization.

a multiplicative correction to the lowest order rate. Using the on-shell renormalization scheme, (76)

az = 26Ztt
where 5Zt,

dp2

\P

~Mt

4HMr;^ r ( 1 + e ) U + 4

(77)

The total virtual contribution is thus: 28 ' 30 0~VIRT = ^vertex + <J Z «LO^T(l '2n

4.3. Real

+ e)CF{

^

(-3 + Be)--4

+ Bfi.

(78)

Contributions

The virtual singularities of Eq. (78) are cancelled by those arising from real gluon emission, e + e~ —> tig. A sample diagram of this type is shown in Fig. 16. The real gluon emission diagram of Fig. 16 contains a fermion propagator on the outgoing top quark leg of the form: 1 (p'+pg)2-M?

_ 1 ~ 2p'-pg

~ 2E'Eg(l

1 - /3cos0)'

where E' and Eg are the energies of the outgoing top quark and gluon, 8 is the angle between them, and (3 = \/l — ^ 2 • Because of the massive top quark, /3 ^ 1, and this diagram has no collinear singularity as 6 —> 0. There is, however, a soft singularity as the energy of the gluon vanishes, Eg —> 0.

272

Figure 16.

Sample Feynman diagram for real gluon emission in the process e + e ~ —* tig.

The soft singularity is regulated using a technique called phase space slicing (PSS). 30 This method divides the gluon phase space into a soft gluon plus a hard gluon region: Cereal = CTsoft + &hard •

(79)

The soft region is defined by the energy of the emitted gluon, Eg<Ss^,

(80)

while the hard region is the remainder of the gluon phase space. Eg>6.^.

(81)

The separation into hard and soft regions depends on the arbitrary parameter 53. Individually, asoft and a hard depend on the cutoff, 5S, and the independence of the sum is a check on the calculation. S3 must be small enough that terms of 0(SS) can be neglected, while large enough to prevent numerical instabilities, a hard is finite and can be evaluated numerically using standard Monte Carlo techniques. In the soft gluon regime, the cross section can be evaluated analytically. The three body phase space is evaluated in the soft limit in the rest frame of the incoming e + e~: p + q = p' + q'+pg,

(82)

giving a gluon energy of TP

_

I

S

~

S

P'<,

(p' + q'f.

(83)

273

The 3-body phase space in the soft limit can be separated into a 2 body phase space factor times a soft phase space factor:31 dn~xt)'

f { PSh

"

dn~xa'

f

"/ ^

J 2 M S F

| !

"

T ( P

+ «"!'-«'-»>

(84)

x/^S^-

For the soft limit, we set the gluon momenta pg = 0 in the delta function and the phase space then factorizes: (dPS)3(soft) = (dPS)2 J

2

f^_i

•

(85)

The phase space is most easily evaluated by choosing an explicit representation for the gluon momentum, pg = Eg(l, ...,sin#i sin9 2 , sin61 cos#2, cos#1).

(86)

yielding,32 dn-1Pg=d\pg\\pg\n-2dSln-2 = dEgEg2'2 sin™"3 01d01 sin™"4 02d62dnn^4

(87)

where, 27r(n-3)/2

(88)

°"-4 = ~Yr^3y-

The integrals can be performed analytically to find the three body phase space in soft limit: 32 (dPS)s(soft) = (dPS)2(^y

(J^jI^±dPSaoft

%/J

dPSsoft = - ( ~ j

f '

2

dEgEg~2t

sin 1 " 26 QxdQx sin" 26 02d62 . (89)

Note that the soft phase space depends explicitely on the cut-off 6S through the limit on the Eg integral.

274

The soft phase space must be combined with the amplitude for e + e tig in pg —> 0 limit: i-soft

e2QeQt s

-9sTAv{q)llxu{p)

"{'£

x u(p

+ Pg + Mt) _M ^ ^ (-
F

9sTA

'}v(q>) (90)

_P -Pg

Q 'Pg.

where we have set pg = 0 in the numerator. The approximation of retaining only the most singular contributions is known as the eikonal (or double pole) approximation. 33 The soft limit of the amplitude always factorizes in this manner into a factor multiplying the lowest order amplitude. The soft amplitude-squared is A soft | 2 = g2sTr(TATA)

| ALO | 2 *«2 Mi

Mi .

2

- -2Mf 2

(P'-Pg)

(
+

(91)

P'-PgQ'-Pg.

giving the cross section in the soft limit, b

J | ASOft |2 (dPSHsoft)

(92)

The problem reduces to evaluating the soft integrals. A typical integral is 1

jsoft

;dPS3(Soft) .

(P' -Pg)2

(93)

Since the phase space and the amplitude squared are Lorentz invariant, we are free to chose any convenient frame. We use pg = Eg(l, ...sin6*i sin#2;sin#i cos62,cos9{) p' =

(94)

^(1,0,0,0)

and the soft integral is, jsoft

_

W^dPS'{soIt) s/l

- "

dEJnEl-2e 9^g

7T\4 b

As always, the factor of 1/4 is the spin average.

2e

e ™ iL--zee1d8 a 1JO sm sm-'„:„-2£. d02

275

sE\)) ( 1 - /? cos 6»02 Ye+^g(5s)

M2

+

^A0)+O(8s,e)

(95)

The complete set of soft integrals relevant for this process is given in the appendix of Ref. 34. The soft cross section is

/V

r(l-e)

Csoft

Us ^LO-^—CF

Cfi

~Uog(Ss)-^A0-2^+p

9i

(96) •Cfi e where B€ = —Ce so the sum of virtual plus soft contributions is finite. The finite contribution, C/j„, depends on 6S and is given by 30

'to'

Li2 The complete 0(as)

s

2(3

, 1 + /3,

J

r(l-2e)

A2

-f + log(5s)Ap

0(SS)

(97)

result for e+e~ —> it is then,

&NLO = &LO + 0'virtual + &soft + CT/iard •

(98)

The combination (Jsoft + Phard is independent of6s, while the \ poles cancel between cr so / t + (JvirtUaiThis example, e + e~ —> ii, is simple enough that it could have been done analytically without the approximation of phase space slicing.28 Phase space slicing, however, is useful for more complicated examples and also to construct Monte Carlo distributions beyond the leading order. 4.4. Threshold

Scan in e + e

tt

The energy dependence of the cross section for e + e~ —> it at threshold depends sensitively on the parameters of the top sector. The location of the peak depends on Mt, while the height of the peak depends on Tt- The problem is that the location of the peak as a function of center-of-mass energy shifts at higher orders and there is a large renormalization scale dependence to the prediction as seen in Fig. 17. 35 At threshold, (3 —> 0, the theory possesses two small parameters: as and j3 and the cross section diverges at 0(aa),9 (j(e4

tt)

1/3^0 = C r L O

1

2nas 3/?

(99)

276

/

/

/

/

N

v/

/ ' /

f ' ~

//

*-., "" - -

\C""-»-^L

f

/ / / /

s

=:/--.

<*, ^

—— ====

~^-^- .r-^^-"" Figure 17. Cross section for e + e _ —> it as a function of the center-of-mass energy in GeV using the pole mass definition for the top quark mass at lowest order ( dot-dashed), NLO (dashed), and NNLO (solid). The vertical axis shows tt)/avt, where Opt = 47ra 2 /3s. The three sets of curves at each order represent the variation with the renormalization scale /U, (jj, = 15, 30, and 60 GeV).37

This divergence as /? —> 0 is known as a Coulomb singularity and nonrelativistic quantum mechanics can be used to sum the leading contributions in powers of i . 3 6 Beyond lowest order in as, the predictions are quite sensitive to the precise definition of the top quark mass. The most straightforward definition is the pole mass appearing in the top quark propagator, D{p)

(100) Mt

\Pole - S ( p )

The pole mass, Mt \poiei is measured by reconstructing the four-momenta of top quark decay products. This definition of the top quark mass is, however, uncertain by QCD hadronization effects, O(AQCD)- These effects connect the top quark with its decay products. Even more disturbing is the fact that the location and height of the peak in the e+e~ —> it cross section change significantly between lowest order, NLO, and NNLO as seen in Fig. 17. A better definition of the top quark mass is the short distance 15 mass: 37 - 38 m i s = Mtt \pole

l

9 '

(101)

277

/'~'N

/

\

/

V

' /

4*—s.

' / ft /^

/ /

/ / //

/ // /

K*- - ^

"^

X^-*-^^ •

-

=

^-

/

: _ __ •

W '/

?s?= ^ ~~ Figure 18. Cross section for e + e - —> it as a function of the center-of-mass energy in GeV using the IS mass definition for the top quark mass at lowest order ( dot-dashed), NLO (dashed), and NNLO (solid). The vertical axis shows <x(e+e~ —> tt)/apt, where &pt = 47ra 2 /3s. The three sets of curves at each order represent the variation with the renormalization scale /i, (fi = 15, 30, and 60 GeV). 3 7

Using this mass definition, the cross section for e+e —> tt peaks at 2niis and the location of the peak does not shift at NLO or NNLO, as seen in Fig. 18. The NNLO result still has roughly a 20% scale uncertainty. It is estimated that by scanning the threshold dependence of the cross section for e+e~ —> tt a measurement with the accuracy 8Mt ~ 200 MeV can be made with an integrated luminosity of L = 50 fb^1. This is to be compared with the expected accuracy of the LHC measurement, SMt ~ 1 — 2 GeV.26 A precise measurement of the top quark mass is of interest primarily because of its implications for electroweak precision measurements. 25

4.5. tth Production in e+e

and pp

Collisions

In many models with physics beyond the Standard Model, the Higgs coupling to the top quark is significantly different from that in the Standard Model. In the Standard Model, the fermion-Higgs Yukawa couplings are, L

=

-9ffhipiph

(102)

278

where 9ffh = ~f

•

(103)

Since the top quark is the heaviest quark, it has the largest Yukawa coupling to the Higgs boson. It is important to measure this coupling accurately in order to verify that the Higgs mechanism is the source of fermion masses. The Higgs boson couplings to the lighter quarks can be tested by measuring the Higgs decays to fermion pairs. This is not possible, however, for the top-fermion coupling. Only for very heavy Higgs bosons (M/j > 2Mt) is the decay h —> it kinematically accessible, and the branching ratio of the Higgs into it pairs is significantly smaller than that into gauge boson pairs. It does not appear feasible to measure guh through this channel. The most promising prospect for measuring the top quark Yukawa coupling is ith production in e+e~ and pp (or pp) collisions. 4.6. iih at an e + e -

Collider

+

At an e e~ collider, iih associated production occurs through s-channel •y and Z exchange, as shown in Fig. 19. The contribution from Z exchange is a few percent of the total rate and will be neglected here. With this assumption, the rate is directly proportional to gfth. In order to interpret the cross section as a measurement of guh, however, it is necessary to include the QCD corrections which are potentially of the same order of magnitude as new physics effects which may change gtth from the SM prediction. The NLO QCD result includes virtual and real gluon corrections on the outgoing t, t legs. The calculation follows that of the previous section for e + e~ —> it, although in this case the virtual

Figure 19.

Contributions to e + e

tth at lowest order.

279

e+e - > tth

Mh=110GeV Mh=130GeV

500

600

700

800

900

1000

Vs (GeV) Figure 20.

NLO rate for e + e

—> tth as a function of center of mass energy.'

calculation also includes box diagrams. These corrections have been found in Refs. 39 and 40 and can be parameterized as K

&NLO <JLO

l + as(n)F(s,Mt2,M£)

.

(104)

The parameter \i is an unphysical renormalization scale and, at this order, the only dependence on JJL is in the running of as(fi). For Mh = 120 GeV, and /j, ranging from Mt to y/s, K(sfs = 500 GeV) ~ 1.4 - 1.5 K(y/s = 1 TeV) ~ 0.8 - 0.9 .

(105)

The NLO results for tth production in e+e~ collisions are shown in Fig. 20. The rate is quite small and has a maximum value for ^fs ~ 700 — 800 GeV for Mh in the 120 GeV region. In order to measure the top quark Yukawa coupling in e+e~ collisions, we consider the final state tth -» W+W~bbbb

(106)

The events are then classified according to the W decays, in an analogous manner to that discussed in Section 3.5 for top quark decays. Since

280

the event rate is quite small, it is advantageous to combine hadronic and semi-leptonic W decay channels. 41 ' 42 At \/s = 800 GeV , an integrated luminosity of L = 1000 / 6 _ 1 , can measure the coupling to an accuracy of41 ^

~ 5.5% .

(107)

9tth

At a lower energy, i / i = 500 GeV, the rate is considerably smaller and even with an integrated luminosity of L = 1000 / 6 - 1 , the precision is quite 42

poor, Sgtth

20% ,

(108)

gtth

for Mh = 120 GeV.

4.7. pp —• tth at NLO The QCD corrections to pp —> tt/i (or pp —> iift) can be calculated in a similar fashion to those for e + e~ —> ii/i. In the hadronic case, however, the calculation is complicated by the existence of strong interactions in the initial state. In this section, we discuss the treatment of initial state singularities in hadron collisions. The ingredients of an NLO calculation for pp or pp —> tih are: 34,43 • &virtual'- This contains all of the one-loop diagrams contributing to the process. Finite diagrams, or diagrams which contain only ultraviolet singularities, can be evaluated numerically.44 Diagrams which contain infrared singularities must be evaluated analytically. • asoft: This contains the soft singularities arising from both the initial and final state soft gluon radiation in the limit pg —> 0. The calculation is analogous to the discussion of Section 4.3. • Chard'- This is the contribution from real gluon radiation when the gluon has an energy above the soft cut-off. In contrast to the e+e~ example of Section 4.3, this contribution has a singularity when the initial state quark or gluon is parallel to the radiated final state gluon. The initial state collinear singularities are absorbed into the parton distribution functions (PDFs). The combination <jsoft + ahard + crvirtual is finite.

281 m,=174 GeV , mH=120 GeV , n=m t , 8C=0.001

Figure 21. NLO QCD corrections to pp —> tth at \ / S = 2 TeV. This figure shows the independence of the total cross section on the soft cutoff, Ss.34

4.8. Collinear

Singularities

and Phase Space

Slicing

To illustrate the effects of collinear singularities, we consider the parton level subprocess, q(p) + Q(Q) - • t(pt) + t(pt<) + h(ph) + g{pg) .

(109)

As in the e+e~ example, we divide the gluon phase space into a soft and a hard region. We then further divide dhard into a region with collinear divergences and a finite region, &hard — &hard \coLl \^hard

\not—coll

(110)

The separation depends on a new cut-off, 5C, although a hard must be independent of Sc. If the emitted gluon is from the initial state quark line, then there is a quark propagator, 1

1

(P ~ PgY

2p • Pg

1 y/sEg{l

— COS t

(111)

282

(1 - z)p

Figure 22.

Diagram contributing to collinear singularity for the process qq —> tthg.

which becomes singular when p and pg are parallel (cos# —> 1). hard/not-collinear region is defined by, 2p-Pg >5C EgV~s

2q-pg EgV~s

>

The

(112)

$c

In this region the quark propagator of Eq. ( I l l ) is finite and the cross section can be computed numerically. In the collinear region, 2p-Pg < Sc EgV~s

Eg^S

(113)

<Sc,

and the matrix element factorizes into a parton i splitting into a hard parton i' plus a collinear gluon with kinematics Pi' - zpi Pg = (1~ z)Pi 2pi • Pg

°ig

These kinematics are illustrated in Fig. 22. In the current example, q —> qg and the matrix element factorizes,34 Aiij -> tth + g) \2-+couinear

Y,i>g2s \ ALO(i'j

-> tth)

2Pu,(z)

(114)

zs 1-9 where PiV is the probability that parton i splits into parton i', with fraction z of the initial momentum. In this example, Pu>(z) = Pqg(z) = CF

l + z2 1-z

s(l-z)

(115)

283

The phase space also factorizes in the collinear limit: 31

(ps)A(ij -+ tth+g)collinear - (psui'j -v m ^ n ' - V ^ l 1 (1 — 2e) 167T2 Z xz dz dsig[(l-z)sig}-ee([ 'p^ -pjSc-Sig) (116) Combining Eqs. (115) and (116), the hard-collinear parton level cross section becomes:45

_as r ( l - e ) (AitfXf Vhard/coll

~ ^

r ( 1

x S^ •( /rl-6.

_

2 e

) \

?

)

\

r{l__z)2pi,.p.

dz

+ (i"j)j.

M

\\s.

zM?

e)

'

Pu'{z)(JLo{i'j

-^tth)

-

(117)

This result has a dependence on both Sc and 5S. The 1 — 6S limit on the z integration excludes soft gluon region. The remaining singularities in Eq. (117) are absorbed into the mass factorization as described in the next section. 4.9. Mass

Factorization

The hadronic cross section depends on the PDFs, fi(x): °H(S)

— Sy / fi{xi)fj(x2)<Tij(xix2S)dxidx2

,

where a is the parton level cross section. At lowest order, there is no dependence on the renormalization/ factorization scale, /i. Since the only physical quantity is the hadronic cross section, <JH(S), we can define scale dependent PDFs to absorb the hard/collinear singularity of Eq. (117), 30,31

\[5S TPlAz)fi' (l)

+CF 21

( °e^) + I)} • (118)

The upper limit on the z integration excludes the soft region. After convoluting the parton cross section with the renormalized quark distribution function of Eq. (118), the infrared singular counterterm of Eq. (118) exactly cancels the remaining infrared poles of &Virt + asoft + & hard/coll-

284

6.5

£

5.5 O _l Z O _i

0 4.5

3.5 150

250

350

450

Figure 23. Leading order and next-to-leading order rate for pp —> tth at the Tevatron for M/, = 120 GeV and '/S = 2 TeV as a function of the unphysical factorization/renormalization scale. 34

The cancellation of the cutoff dependence at the level of the total NLO cross section is a very delicate issue, since it involves both analytical and numerical contributions. It is crucial to study the behavior of <JNLO m a region where the cutoff(s) are small enough to justify the approximations used in the analytical calculation of the infra-red divergent soft and collinear limits, but not so small to give origin to numerical instabilities. This effect is illustrated in Fig. 21. The NLO result for pp —> tth at the Tevatron is shown in Fig. 23 as a function of the unphysical factorization/ renormalization scale. The complete calculation includes not only the qq initial state described here, but also the gg and qg initial states. Note the reduced \i dependence of the NLO rate from the lowest order prediction. The cross section is very small, making this a challenging measurement. At the LHC, the process pp —> tth has a significant rate for Mh < 130 GeV and we can look for h —> 66, giving the final state of W+W~bbbb. This is an important discovery channel for a Higgs with Mh < 130 GeV. The ATLAS experiment, with L = 100 / 6 ~ \ and assuming Mh = 120 GeV can measure ^ ^ ~ 16%. This is illustrated in Fig. 24. 26 The signal and the background have similar shapes at the LHC, increasing the difficulty of this measurement.

285

> O 150

m 100

50

mbb(GeV) Figure 24. Invariant mass for the bb pair from the Higgs decay in the process pp —* tih at the LHC with M^ = 120 GeV (data points). The dashed line is the background. The error bars correspond to the statistical uncertainty with an integrated luminosity of L = 100 fb-1.26

5.

Heavy Top as Inspiration for Model Builders

We turn now to a discussion of the top quark in models where there is new physics involving the top quark beyond that of the Standard Model. The fact that the top quark is much heavier than the other quarks gives it a special role in electroweak symmetry breaking. In this section, we examine the role of the top quark in the minimal supersymmetric model and in models with dynamical symmetry breaking. We pay particular attention to how data from the Tevatron and the LHC can help us to test these ideas. The large value of the top quark mass has important consequences in a supersymmetric (SUSY) model. 46 In the SM, the Higgs boson mass is a free parameter, while in a SUSY model there is an upper limit on the Higgs mass. At tree level in the minimal SUSY model, Mh < Mz, but the large top quark mass, and the subsequent large coupling to the Higgs boson, mt

V2

1,

(119)

lead to large radiative corrections to the prediction for the Higgs boson

286

mass, raising the lower limit on the Higgs mass significantly. In addition, corrections due to the large top quark mass typically imply that the scalar partner of the top, the stop, is the lightest squark. A further difference between the SM and a SUSY model which is due to the large top quark mass is the understanding of the mechanism of electroweak symmetry breaking in a SUSY model. If the supersymmetric model is embedded in a grand unified model, then the simplest version (mSUGRA) has the interesting property that the evolution of the parameters from the GUT scale to the weak scale drives the Higgs mass-squared negative as a consequence of the large value of Mt, and therefore explains electroweak symmetry breaking. In fact, this mechanism requires that the top quark have a mass near its observed value. In models with dynamical symmetry breaking, the large value of the top quark mass can imply that the top quark is different from the lighter quarks, perhaps having different gauge interactions in an extended gauge sector. In Section 5.6, we discuss topcolor as an example of a model of dynamical symmetry breaking where the top quark plays a special role. Many possibilities for dynamical symmetry breaking in electroweak interactions are detailed in the review by Hill and Simmons. 47 5.1. EWSB

and the Top Quark

Mass

We begin by considering the importance of the top quark mass in the SM. In the SM, electroweak symmetry breaking occurs through the interactions of an SU(2)L doublet scalar field, , with a scalar potential, 48 V = A * 2 * ^ + A($i- f ) 2

.

(120)

If y? < 0, the neutral component of the Higgs doublet gets a vacuum expectation value, <*o> - ^=

(121)

and the scalar potential has the shape shown in Fig. 25. Electroweak symmetry breaking (EWSB) occurs and masses are generated for the W and Z gauge bosons. After EWSB, there remains in the spectrum a physical particle, the Higgs boson, h, and at tree level the quartic self- coupling of the Higgs boson, A, is related to the Higgs mass by Ml = 2Xv2.

(122)

A measurement of this relationship would test the mechanism of EWSB.

287

Figure 25.

Higgs potential in the Standard Model with p? < 0.

Equation (120) is the scalar potential at the electroweak scale, A ~ v. All parameters of the electroweak theory change with the energy scale A in a precisely calculable way, and so will be different at higher energies. The most interesting parameters for us are the Higgs quartic self-coupling, A, and the top quark Yukawa coupling, gUh = — . To one loop accuracy,0 A — - —— 24A2 - (A d\

A

dgtth dA

~ 16TT2

1 f dtth 167T 2 V 1 2

Wtth )X + B

54flL

ugtth

17g'2 - 27g'2 - 96g2s

(123)

where, A = B

^g2+g12) 3 2g* + 8

/2\2 (g2+g")

(124)

We have omitted the equations for the running of the gauge coupling constants. They can be found in Ref. 48. In order for electroweak symmetry breaking to occur, the quartic coupling must remain finite at all scales, 1 >0. A(A)

(125)

As A is evolved from the weak scale to higher energy, it grows and eventually becomes infinite at some large scale. When A becomes large, the scaling of c

g and g' are the S[/(2)x, and U(V)Y gauge coupling constants.

288

Eq. (124) can be approximated, A ^ = ^

2TT2

d\

.

(126)

Equation (126) is easily solved,

AM = A(k)"ilog(^:)-

(127)

The position where A(A) —> oo is known as the Landau Pole. Since A is related to the Higgs mass, M% = 2v2X, the requirement that the theory not approach the Landau pole gives an upper bound on the Higgs boson mass, Ml <

4 v2

f x 31og[A/M h

.

(128)

This means that at the scale A there must be some new physics beyond that of the SM which enters. The SM simply makes no sense at energy scales larger than A since in this region the Higgs quartic self-coupling is infinite. Assuming no new physics before the GUT scale, A ~ 10 16 GeV, gives a bound on the Higgs mass of49'50 Mh < 160 GeV .

'

(129)

This bound corresponds to the upper curve in Fig. 26. Non-perturbative studies using lattice gauge theory confirm the validity of this limit. 51 Conversely, we require that the potential be bounded from below, which is equivalent to the restriction that A > 0 at large $. The scaling of the quartic coupling near A = 0 is, dX

1

~dA

16TT2

B - I2gfth

(130)

which can be solved, A(A) = A(M,) + ^ — l ^2 M l o g ( A ) . 167T \Mh

(131)

The requirement that A (A) > 0 gives a lower bound on the Higgs mass,

A complete approach must include the full set of renormalization group equations, including the running of the coupling constants for large values of $.

289 800.0

600.0

>~ CD CD 400.0 x

200.0

""10"

10°

10*

10"

10°

A (GeV) Figure 26. Limits on the Higgs boson mass as a function of the cutoff scale A. The theoretically consistent region lies between the two curves. 4 9

The restrictions on the Higgs mass from consistency of the theory are shown in Fig. 26, where the allowed region for a scale A is the area between the curves, known as the "chimney". At a given scale A, the Higgs mass has a lower limit from bounding the potential from below and an upper limit from the requirement that Higgs quartic coupling, A, be finite. The exact shape of the allowed region depends sensitively on the value of M 4 . 49 ' 50 The SM with a Higgs mass in the 130 GeV region is consistent at energy scales all the way to the Planck scale. The theory is still unsatisfactory however, because it provides no explanation of why /i 2 < 0, as required for EWSB. In addition, the Higgs mass is a free parameter in the SM and there is no understanding of why it should have any particular value. In the next section, we will see that low energy supersymmetry can potentially solve both these problems. 5.2. Supersymmetry

and the Top

Quark

TeV scale supersymmetric models explain several mysteries about the SM: • A heavy top quark and mSUGRA (minimal supergravity) explain why fi1 < 0. • A heavy top quark and the MSSM (minimal supersymmetric Standard Model) explain why the Higgs mass may light, as suggested by precision measurements.

290

In this section, we give a brief review of phenomenological models of supersymmetry and discuss the relationship between the top quark mass and the requirement that supersymmetry exist below the TeV scale. Supersymmetry is a symmetry which relates the masses and couplings of particles of differing spin. To each SM chiral fermion is associated a complex scalar and each of the SU(3) x SU(2)L X U(1)Y gauge bosons similarly acquires a Majorana fermion partner. Supersymmetry connects particles of differing spin, but with all other characteristics the same. 52 In an unbroken supersymmetric model, particles and their superpartners have identical masses. It is clear, then, that supersymmetry must be a broken symmetry if it is to be a theory of low energy interactions. There is no scalar particle, for example, with the mass and quantum numbers of the electron. In fact, there are no candidate supersymmetric scalar partners for any of the fermions in the experimentally observed spectrum. The most general soft breaking of supersymmetry introduces masses and mixing parameters and the theory loses much of its capabilities for specific predictions. The Higgs sector in a SUSY model always contains at least two SU(2)L doublets. In the supersymmetric extension of the Standard Model, the Higgs doublet acquires a SUSY partner which is an SU(2)L doublet of Majorana fermion fields, hi (the Higgsinos), which contribute to the triangle SU(2)L and U(1)Y gauge anomalies as described in Section 2.1. Since the fermions of the Standard Model have exactly the right quantum numbers to cancel these anomalies, it follows that the contribution from the fermionic partner of the Higgs doublet remains uncancelled. 6 ' 7 This contribution must be cancelled by adding a second Higgs doublet (and its Majorana fermion partner) with precisely the opposite U(l)y quantum numbers from the first Higgs doublet. All supersymmetric models thus have at least 2-Higgs doublets and so the minimal supersymmetric model has 5 physical Higgs bosons: h°, H0^0,!!^. The ratio of the vacuum expectation values of the neutral Higgs bosons, Vo

tan/3=—

(133)

Vi

is a fundamental parameter of the theory. At tree level, the neutral Higgs boson masses are predicted in terms of two parameters, which are usually taken to be tan/3 and the mass of the pseudoscalar Higgs boson, MA- The neutral Higgs boson masses are then predicted, KH

= \{M\ + Ml)

T

\y/(MA

+ Mlf-

4M2ZM2A cos2 2/3

(134)

291

which gives an upper bound on the lightest Higgs mass: Mh<Mz

| cos 2/3 | <MZ

(135)

which is excluded experimentally. (In a SUSY model, the LEP and LEPII results require Mh > 89.8 GeV.4) The Higgs mass prediction in a supersymmetric model receives large radiative corrections from top and stop squark loops. At 1-loop, and neglecting possible soft SUSY breaking tri-linear scalar mixing, the lightest Higgs mass becomes: „.r9

•

Ml =

Ml+Ml+e A

z 2

If,

,

9

,9

9 9 9

1 (M 2 + M\ + e)2 - 4M 2 M 2 COS2 2/3 >. 1/2

- 4 e ( M l sin2 /3 + M§ cos2 /3) I

(136)

where, 3GF

Ml

lo

(M?

sh^ h

(137)

and Mt is the average stop squark mass. For unbroken supersymmetry, the top and stop have the same masses, Mt = M t , and the bound of Eq. (135) is recovered. The mass splitting between the top and stop in a broken SUSY theory changes the upper bound on the Higgs mass to, Ml < M § cos2 2/3 + e sin2 /?.

(138)

The large Mf corrections raise the Higgs mass bound above the experimental limit, as shown in Fig. 27. There are many analyses 53 which include a variety of two-loop effects, renormalization group effects, etc. but the important point is that for given values of tan/3, the scalar mixing parameters, and the squark masses, there is an upper bound on the lightest neutral Higgs boson mass. For large values of tan /3 the limit is relatively insensitive to the value of tan (3 and with a squark mass less than about 1 TeV, the upper limit on the Higgs mass is about 110 GeV if mixing in the top squark sector is negligible. For large mixing, this limit is raised to around 130 GeV.

5.3.

mSUGRA

The minimal supersymmetric model (MSSM) is constructed by adding to the Lagrangian all of the soft supersymmetry breaking terms allowed by the gauge symmetries. These include masses for all the new SUSY particles

292

Maximum Higgs Mass (No mixing) 120.0

'

110.0

1

tan (3 =

100.0 • . - - - r ^

tan p=10

> O

90.0

80.0

-

^ a n p =2 70.0

60.0 200.0

I

400.0

.

600.0

.

800.0

msquark (GeV) Figure 27. mixing. 5 3

I

1000.0

Upper bound on Higgs mass in MSSM at one loop and with no soft trilinear

and tri-linear scalar mixing terms. T h e most general Lagrangian of this type has 105 new parameters beyond those of the S t a n d a r d Model. In order to make definite predictions, the simplifying assumption t h a t the soft parameters unify at MQUT is often made. (The resulting model is usually called m S U G R A or the CMSSM). In this framework the model is completely specified by five parameters: • • • • •

Common scalar mass at MQUT, mo Common gaugino mass at MQUT, Afi/2 1 Higgs mass-squared, m\2 1 tri-linear coupling at MQUT, AoXp sign(/t)

T h e parameter ft describes the mixing between the two Higgs doublets. At the G U T scale, the Higgs masses are, MlH(MGUT)

mn

•M

(139)

while the squark and slepton masses are M?(MGUT)

=

MKMGUT)

(140)

293

The theory must explain why the lightest neutral Higgs mass-squared becomes negative at the weak scale and breaks the electroweak symmetry, while the scalar fermion masses remain positive so as to not break color and electromagnetism. In an mSUGRA model, the masses and parameters are evolved using the renormalization group equations (RGE) to scale from MQUT down to the weak scale in order to make predictions for the masses at the weak scale. The large top quark mass plays an important role here. We can understand the basic scaling by considering only the Higgs doublet, i?2, d the scalar partners of the third generation SU(2)L fermion doublet, (£L, &L) = Qs, and the scalar partner of the right-handed top quark, in. The renormalization group scaling is given by: 54 dM2H2 = ^L(3glhXt dlog(A) dM? tR dlog(A)

3g2M2 - g'2M2)

- i {2g^Xt - f^ M °" f9'2M?

dM"k Qs = J L (&hXt dlog(A)

- fg2M2

- 1g2M2 - -\g'2M2

where,

Xt = M2H2+M2Q3+M2R+A2

.

(141)

-^3,2,1 a r e the gaugino masses associated with the SU(3) x SU(2)L X U(1)Y gauge sector, g^ is the coupling of the Higgs to the top quark in the MSSM, and At is the soft SUSY mixing term in the top sector. The QCD term (proportional to g2) makes M2R,MQ3 increase faster than MH2 a s the scale is reduced from MQUT- The Yukawa term (proportional to gtth) drives Mfj < 0, while the other scalar mass-squared terms remain positive. This behavior is a consequence of the large Mt value. Neglecting the gauge couplings, we have:

(142)

dlog(A)

i

H2 is the SU(2)L

doublet with isospin Y -

+\2 '

294 Evolution of sparticle masses

700 600

> o

500 400 300 200 100 n

io 6

io 10

io 17

Q (GeV) Figure 28.

Typical pattern of masses in an mSUGRA model. 5 5

which can be easily solved, Ml{K) = Ml{MGUT) - | j f Xt log ( ^ p )

.

(143)

For heavy Mt, M\ becomes negative and triggers EWSB. It is interesting that this behavior requires Mt ~ 175 GeV.55 The RGE scaling from the GUT scale to the weak scale in mSUGRA type models yields a a typical pattern of masses, as seen in Fig. 28. The gluino is the heaviest gaugino, while the squarks are significantly heavier than the sleptons. 5.4. Charged Higgs

Decays

The pattern of top quark decays is changed in the MSSM from that of the SM, due to the extended Higgs sector. The presence of a charged Higgs boson allows the decay, t —* H+b if the mass of the charged Higgs is less than Mt— m^. This decay looks similar to t —> Wb and can have a significant rate. The coupling of the charged Higgs to the t b system is, 9tH+b ~ mt cot f3P+ + nib tan/3P_

(144)

295

and so this decay is relevant both for very large and for very small tan/?. For small tan/3, H+ —> cs and H+ -> t*b, while for large tan/3, H+ —> ri/. These decays would suppress the rate for top decays to leptons plus jets in the SM top search. In Run I, the Tevatron experiments placed a limit on MH+ of roughly MH+ > 100 GeV for tan/3 < 1 or tan/3 > 50. 56 5.5. The Top

Squark

The mass-squared matrix for the scalar partners of the left and right handed top quarks (ti,, IR) includes mixing between the partners of the left- and right- handed top: M2=( M2Q3 + M? + M\Lt cos 2/3 Mt{At-ii * V Mt{At-(i cot (3) M?R+M? +

cot /3) \ Ml.Rtcos2p)

The mixing is proportional to the top quark mass and so the large value of Mt can cause one of the top squarks to be relatively light. If the scalar mass scale, MQ3 is much larger than Mz, Mt,At, then the stop masses are roughly degenerate ~ MQ3. On the other hand if MQ3 ~ Mz, Mt,At, then the large mixing effects drive the stop mass to a small value and the stop squark becomes the lightest squark. It is possible to search for a light stop in the decays of the top, t —> tx°, where x° is the lightest neutral SUSY particle. The limits then depend on the assumed branching ratio for the top into stop decays. The Run I data was sensitive to stop and chargino masses in the 100 GeV range. 57 5.6. The Top Quark and Dynamical

Symmetry

Breaking

The top quark has a special role in technicolor models where a ti condensate can play the part of the Higgs boson in generating EWSB. This can happen in models where there is a new strong interaction felt by the top quark, but not by the lighter quarks. The proto-type model of this type is called topcolor. A review of this class of models can be found in Ref. 47. Topcolor models are constructed by expanding the strong gauge group from the SU(3) of QCD to, SU{3)tc x SU{3) - •

SU(3)QCD

(145)

SU(3)tc couples only to the 3rd generation with a coupling gtc, while the second SU(3) gauge symmetry couples only to the first two generations. The symmetry is broken to SU(3)QCD at a scale M. Below M, the gauge

296 bosons of the SU(3)tc group, the "top-gluons", are massive and lead to effective 4-Fermi interactions between the top quarks: L ~ | |

( Q

3

L 7 ^ Q

3

L )

( W ^ * * )

,

(146)

where TA is the SU{2>) generator in the adjoint representation and Q^,L is the left-handed (tL,bL) doublet. If the SU(3)tc coupling, gtc, is larger t h a n a critical value, a top condensate forms, (tt), which breaks the electroweak symmetry and generates a mass for the top quark. One must arrange t h a t {tt) ^ 0 and (bb) = 0. This is usually done by adding an additional U(l)h gauge symmetry under which the t and b quarks transform differently. This class of models can be tested at the Tevatron and the LHC by searching for effects of the top gluons and the new Z', and through B decays. 5 9 Since the Z' boson corresponding to the U(l)h and the top gluons of SU(3)tc couple differently to the third generation and to the lighter quarks, they lead to flavor changing neutral currents. Present limits from B meson mixing require t h a t the topgluon be heavier t h a n 3 — 5 TeV and Mz, > 7 - 10 GeV.58

6.

Conclusions

T h e story of the top quark is far from over. Its properties and its role in the SM and in physics beyond the SM will be further elucidated in the emerging R u n II d a t a at the Tevatron. T h e increased statistics of R u n II will enable more precise measurements of the mass, couplings, and decay properties of the top. T h e LHC will provide even more insight into the role of the top quark. It is possible t h a t surprises are just around the corner! Acknowledgements I would like to t h a n k K.T. Mohanthapa, H. Haber, and A. Nelson for organizing such an enjoyable and successful school.

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.JOHN WOMERSLEY

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TEVATRON PHYSICS* JOHN WOMERSLEY Fermi National Accelerator Laboratory Batavia, IL 60510, USA E-mail: [email protected] These lectures form a personal, and not necessarily comprehensive, survey of physics at the Fermilab Tevatron proton-antiproton collider. They cover detectors, analysis issues, and physics prospects for the current Tevatron run.

1.

Introduction

1.1. Hadron

Colliders

The term "hadron colliders" is used to describe accelerators where a beam of protons is collided with a contra-rotating beam of protons or antiprotons. The next decade belongs to these machines: first the Fermilab Tevatron, and (starting around 2008) the LHC at CERN. Historically, there has been a complementarity of capability between hadron colliders and electron-positron colliders. Hadron machines emphasize maximum energy and hence maximum physics reach for new discoveries, while the electron-positron machines tend to have lower center-of-mass energy but cleaner events and better capabilities for precision measurements. Hadron Colliders have the advantage that protons can be easily accelerated to very high energies and stored in circular rings. They have the disadvantage that antiprotons, if used, must be collected from the results of colliding a lower energy, high intensity beam on a target. This can of course be avoided by building a proton-proton collider, at the cost of needing a second ring of magnets. They also have the disadvantage that protons are made of quarks and gluons, so the whole of the beam energy is not concentrated in a single point-like collision. Quarks and gluons are also strongly interacting particles, so the collisions are messy. Despite these problems, hadron colliders are the best way to explore the highest mass scales for new physics.

* Lectures presented at the TASI 2002 Summer School, University of Colorado, Boulder, CO, June 2-28, 2002. 303

304

1.2. The Tevatron The overarching question for the world high energy physics program is "what sets the mass scale of the weak interaction to be -100 GeV?" This question is addressed solely by colliders operating at the energy frontier. In the 1990's, there were four such machines: the Tevatron collider (Run I), LEP at CERN, the SLC at SLAC, and HERA. In contrast, between 2002 and roughly 2008, the Tevatron is the only machine that can address the central problems of the field: LEP and SLC have closed, and HERA's reach is limited. Increased energy and a greatly increased luminosity at the Tevatron make possible a new round of experimentation. The Tevatron physics program involves: • Precise measurements of the known quanta of the standard model, to search for indirect hints (or constraints) on new particles or forces; • Direct searches for new physics i.e. beyond the known standard model particles and forces. The Tevatron program has the potential for a discovery that would change the direction of particle physics. Between 1992 and 1995 (Run I), the Tevatron delivered about 120 pb~' to each experiment at a center-of-mass energy of 1.8 TeV. Run II started in 2001, with an increased center-of-mass energy of 1.96 TeV, using the new Fermilab Main Injector as part of the accelerator chain. We expect 2-4 ftf1 in the first phase of Run II, after which detector upgrades will be needed; the goal is to have delivered 10-15 ftf1 before the LHC becomes competitive. 2.

Detectors

A typical detector surrounds the interaction point with concentric layers of particle measurement devices, as shown in Fig. 1. The innermost parts of the detector measure charged particle trajectories in a magnetic field. Particle direction and momentum can be inferred. The innermost layers need to make the most precise measurements and use silicon wafers; these tend to be expensive, and so the outer layers use less precise but less costly technologies. Surrounding the tracker is a calorimeter. The calorimeter induces particle showers in dense material and measures the particle energies. It also distinguishes electrons and photons from hadrons by the shower topology. Finally, the outermost layers of the detector measure and identify muons. Combining all these measurements, it is also possible to infer the

305

presence of non-interacting particles like neutrinos, since they leave unbalanced momentum in the transverse plane ("missing transverse energy", ET™SS). Interaction point

Magnetized volume Tracking system \

Innermost tracking layers use silicon

Calorimeter induces shower j n dense material J,

EM layers fine sampling

Hadronic layers

Absorber material

Muon detector

Experimental signature of a quark or gluon

"Missing transverse energy" Signature of a non-interacting (or weakly interacting) particle like a neutrino

Figure 1: schematic of an azimuthal section of a tyrpical collider detector, and the particle signatures seen in it. There are two large multi-purpose detectors at the Tevatron: CDF and D 0 (D Zero). The detectors are complementary in that CDF detector places more emphasis on tracking measurements, while D 0 emphasizes calorimetry. The physics reach of the detectors is not dramatically different: it tends to either be driven by cross sections (as in searches for SUSY) or the respective strengths of the detectors tends to balance (as for the top quark in Run I).

2.1. Tracking The ability to identify b-quarks is very important at hadron colliders, as a signal for top, supersymmetry and Higgs. Experimentally, we can exploit the fact that a b-quark forms a B-meson which travels ~ 1mm from the collision point before decaying. To reconstruct this decay, we need to reconstruct tracks with a precision at the 10 urn level. We can tag the B-decay either by asking for two or three tracks having a significant "impact parameter" (distance of closest approach to the fitted primary vertex), or by explicitly reconstructing a

306

secondary decay vertex separated from the primary. Figure 2 shows these two methods schematically. Both CDF and D 0 use silicon detectors close to the beam pipe to achieve the necessary tracking precision. Silicon detectors use wafers of silicon of order 2 cm x 12 cm with -50 Jim pitch between readout strips and on-board amplifier-discriminator chips. These "ladders" are arrayed in cylindrical layers around the beam pipe and supported on structures that provide rigidity and cooling. Both the CDF and D 0 silicon detectors were assembled at Fermilab's Silicon Detector Facility. They are commissioned and working well in the collider. In CDF, the silicon detector feeds an impact parameter trigger which allows B-decay candidates to be selected online: a first at a hadron collider.

Figure 2: Impact parameter and secondary vertex techniques for b-tagging.

The detectors use separate outer tracker systems to cover the tracking region beyond the silicon. The CDF Central Outer Tracker is a large drift chamber with 96 wire planes and 30,000 sense wires. The D 0 Central Fiber Tracker uses 8 layers of 1mm diameter scintillating fibers, read out with cryogenic photodetector devices. Both the COT and CFT have the capability to provide triggers on high-transverse-momentum charged tracks.

2.2. Calorimetry The basic tool for detection of electrons, photons and hadronic jets (the signatures of quarks and gluons) is a segmented calorimeter surrounding the tracking system. The basic idea is to induce a shower of interactions between the incident particles and a dense material, and to measure the ionization energy deposited in the shower. From this, the particle energy is deduced. For electromagnetically interacting particles (e*, y) the situation is straightforward. Above -100 MeV, pair production and bremsstrahlung dominate the energy loss mechanisms. Shower development involves the processes y —> e+e~ and e* —> e*y and it scales with the radiation length Xo- It is easy to show that the total charged track length in the shower, and hence the sum of ionization, is proportional to the incident energy E, and that the RMS is proportional to VE.

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Strongly interacting particles (hadrons) also cascade in matter, but many more processes are involved: as well as inelastic hadronic collisions, energy is lost to ionization, electromagnetic cascades from produced n° mesons that decay to photons, nuclear binding energy, neutrino production, and nuclear excitation. Hadronic showers scale with the nuclear interaction length X: the showers are longer, wider, start later and have more fluctuations than an EM shower of the same energy. Because not all of the above processes are detectable, the response to a hadron is usually lower than to an electron of the same energy, the so-called e/7t ratio. For reasons of cost and compactness, collider calorimeters are "sampling calorimeters" that alternate dense but inert absorber with layers of sensitive medium. A fixed fraction of the shower energy is sampled in the sensitive medium. In the case of CDF, the sampler is scintillator tiles. New forward calorimeters were added for Run II and these exploit wavelength-shifting fibers to give a compact and flexible way to route the light signals to photomultiplier tubes. D 0 uses liquid argon as the sampling medium: the calorimeter consists of an absorber structure with electrodes, all of which is immersed in a vessel containing liquid argon, which serves as the ionization collector. Calorimeter energy resolution is usually dominated by statistical fluctuations in the number of shower particles, which scale as VE. The fractional resolution is therefore often quoted as "x%/VE" (where E is assumed to be measured in GeV): typical real-world values are 15%/VE for electrons, 50%/VE for single hadrons and 80%/VE for jets. Other terms contribute in quadrature: a "noise term", independent of E, which dominates at low E, and a "constant term", a constant fraction of E, which dominates at high energies and can come from calibration uncertainties, nonlinear response, and unequal response to hadrons and electrons. At high energies, jets become very clear in hadron-hadron collisions, and the calorimeter gives a very visual picture of the event topology. It also allows electrons to be clearly distinguished from jets.

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2.3. Muon System The outermost part of the detectors is the muon system. Since all other types of charged particles are absorbed, muons can be identified as charged tracks that traverse absorber outside the calorimeter. The outside of the detector is therefore covered with large-area detector planes to identify and trigger on muons. In the case of D0, an approximate stand-alone muon momentum measurement can be made using magnetized iron, but in both D 0 and CDF the central tracker provides a precise measurement once the muon track has been linked to matching trace in the central tracker. 2.4. Trigger The accelerator luminosity is set by the physics goals. For example, to find the Higgs we need -lOfb"1 of data; to do this in a reasonable time requires a collision rate of 2 x 1032 cm'2 s"1. The total inelastic proton-antiproton cross section is huge compared to the processes we are interested in studying. Even with bunch crossings every 396 ns at the Tevatron, there is more than one inleastic collision each time the bunches cross. The triggering challenge is then the real-time selection of perhaps 50 events per second (the maximum we can store to tape) out of this collision rate of 2.5 million per second. We do this by rapid identification of high energy particles, and of comparatively rare objects (electrons, muons...). Triggering is perhaps the greatest challenge to experiments at hadron colliders. Both CDF and D 0 use a three-level trigger system to reduce the rate progressively. The first level selects 10-40 kHz of collisions based on fast information from specialized detectors. The second level reduces this to a few hundred Hz using microprocessors that perform fast calculations on a subset of the data and reject most of the level 1 candidate events. The third level uses a farm of computers, with access to the full event readout; again most of the candidates from the previous level are rejected and 50 Hz is stored to tape. These events are passed to a computer farm where the reconstruction program is run on them within a few days of data taking, followed by storage in a tape robot system. The experiments operate with a "trigger list" of order 100 different trigger requirements at each trigger level, optimized for particular physics processes or final states. For many relatively high-rate processes, like jet production, we cannot run the trigger without restriction, because the cross section at low energy would exceed our capabilities. Such triggers are "prescaled", and only a known, fixed fraction of the events satisfying the trigger are saved.

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2.5. Simulation Tools A "Monte Carlo" is a Fortran or C++ program that generates events. Events vary from one to the next (random numbers are used), so we can expect to reproduce both the average behavior and the fluctuations of real data. Event generators may be parton level, which include just the parton distribution functions and the hard matrix element, or they may additionally include initial state radiation, final state radiation, hadronization of quarks and gluons, decays of unstable particles, and generation of the underlying event. Separate programs are used for simulation of the detector response to events. These are written by the detector collaborations but based on standard toolkits (GEANT being the best known and most widely used). 2.6. Computing Computing for data processing and analysis is a challenge for modern experiments both because of the quantity of data and because of the size and distributed geography of the collaborations. In the past, this led to the invention of the World-Wide Web as a way to share information. Now the challenge is to share data and computing power. There is a natural synergy between the needs of our experiments and current ideas about "Grid" computing. The Tevatron experiments are already making something like a Grid a reality and are distributing their data for analysis using a Fermilab-developed system called SAM. They are also exploring ways for remote collaborators to assist in monitoring detector operations using web tools over the internet. 3.

Hadron-Hadron Collisions

Hadron-hadron collisions are complicated by the fact that a hadron collider is really a broad-band quark and gluon collider. The incident particles to the parton-level hard scattering are selected from parton distributions at some factorization scale. The hard scattering is calculated at a given order in as, and at some renormalization scale. There may be initial state radiation before the hard scattering, or final state radiation after it. Outgoing quarks and gluons fragment into jets of hadrons, and the remaining quark/gluon content of the interacting beam particles forms a soft "underlying event." Since the incoming parton momentum fractions x, and x2 are a priori unknown, and usually beam particle fragments escape down the beampipe, the longitudinal motion of the parton-parton center of mass cannot be reconstructed. We therefore focus on transverse variables. We use the transverse energy and

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momentum, ET = E sinG and p T = p sinG, where G is the angle between a particle direction and the beam. We also use longitudinally boost-invariant quantities, the primary one being the pseudorapidity T| = -In tan(G/2). This maps on to angles in the detector: r\ = 0, 1, 2 being G = 90°, and roughly 40° and 15° respectively. For massless particles, pseudorapidity and true rapidity coincide. Particle production usually scales per unit of (pseudo)rapidity. 4.

Quantum Chromodynamics

No one doubts that Quantum Chromodynamics (QCD) describes the strong interaction between quarks and gluons. Its effects are all around us: it is the origin of the masses of hadrons, and thus of the mass of stars and planets. This doesn't mean it is an easy theory to work with. As well as using hadron colliders to test QCD itself, we find that it is so central to the calculation of new physics processes, and their backgrounds, that we need to make sure we can have confidence in our ability to make predictions in this framework. QCD is a gauge theory describing fermions (quarks) which carry an SU(3) color charge and interact through the exchange of vector bosons (gluons). It has the interesting features that the gluons are themselves colored, and that (as a consequence) the coupling constant runs rapidly and becomes weak at momentum transfers greater than a few (GeV)2. These features lead to a picture where the quarks and gluons are bound inside hadrons if left to themselves, but behave like free particles if probed at high momentum transfer. This is exactly what was seen in the deep inelastic scattering experiments at SLAC in the late 1960's which led to the genesis of QCD. The short-timescale, hard scattering involves the electron scattering elastically off a single, pointlike constituent inside the nucleon; the other quarks do not participate. Afterwards (over a longer timescale) the scattered quark, knocked out of the proton, radiates lots of gluons and quark-antiquark pairs which combine with each other and the colored remnant of the proton to form colorless hadrons. This is called "fragmentation" or "hadronization." At high momentum transfers these hadrons form a jet of roughly collinear particles whose energy and direction correspond to that of the quark or gluon. This picture becomes very intuitive when compared with highenergy events from LEP. One can see the topologies of two jets corresponding to e+e~ —» qq and three jets from e+e" —> qqg. High energy events at the Tevatron are also simple to recognize (see Fig. 3). Event generators model the hadronization process by dividing in into two phases: a perturbative phase, wherein a shower of quarks and radiated gluons

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develops and Q2 evolves downwards; and a nonperturbative phase, once Q2 ~ 1 GeV2 and the coupling constant becomes strong, where those quarks and gluons are grouped into hadrons (which may then decay). The PYTHIA and HERWIG programs contain two popular Monte Carlo treatments of this process.

Figure 3: High energy dijet event in Run II, as recorded by D0. 4.1. Jets Experimentally, we cannot make the same identification between hadrons and their "parent" parton. We need a jet algorithm to identify jets and define their properties rigorously. We would like an algorithm that can be run both on partons and on final state particles, and on detector measurements, and yield comparable results in all cases: in other words, one that does a good job of integrating over the non-perturbative and hard-to-calculate hadronization process. It is important to define jets properly even at the parton level: a naive one parton = one jet identification makes no sense for soft or collinear gluon radiation, for example.

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The traditional choice at hadron colliders has been the "cone algorithm." Jets are defined as the sum of energy within a cone of radius R2 = Ar)2+A
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balance in two jet events. For two jets, we can define an asymmetry A = (E-n-E-rc )/(ET1+Ex2); then the E T resolution oET/ET = ^2 oA- Since real events have additional soft jets and are not perfectly balanced in ET, it is necessary to progressively tighten cuts and extrapolate to the limit of no soft radiation. Typical jet resolutions are in the range 75-100%/VE(GeV).

4.2. Jet Cross Sections at the Tevatron In Run I, both CDF and D 0 measured jet cross sections at 1.8 TeV using the cone algorithm (R = 0.7). The cross section falls by seven orders of magnitude from 50 to 400 GeV. Agreement with NLO QCD is generally good, except perhaps at the highest energy end of the spectrum. When these results first appeared, there was some excitement that the CDF data, especially, were significantly in excess of QCD for ET > 250 GeV. This sparked a renewed investigation of uncertainies in the prediction arising from parton distributions, and it was found that the gluon distribution is not in fact well determined in this kinematic regime; a small enhancement in the gluon content is allowed and can model the CDF data very well. The latest PDF fits from CTEQ and MRST now make use of the Run I jet cross sections from D 0 to pin down the gluon distribution at high x. Both CDF and D 0 have now started measuring jet energy distributions from Run II. CDF are making use of their new forward calorimetry to cover the whole range of pseudorapidity (see Fig. 4). Jet calibrations are not yet final, but already we see events with transverse energies beyond 400 GeV. With the full Run II dataset this will reach as far as 600 GeV, allowing us to pin down the high-energy behavior of the cross section and thus better determine the gluon content of the proton.

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Figure 4: Preliminary Run II jet distributions from CDF. D 0 also measured the jet cross section in Run I data using the k x algorithm. The two algorithms yield different cross sections for collider data. Qualitatively this is what is expected, since the kx algorithm tends to gather more energy into jets. Quantitatively, however, it is not yet clear whether the differences are really as expected. We will try to address this question with early Run II data.

4.3. Heavy Flavor Production Run I left many unanswered questions about heavy flavour (charm and bottom) production. Resolving these is important because many new particles result in heavy flavour signatures. The inclusive b-jet and B-meson production cross sections lie significantly above the QCD prediction (see Fig. 5), though it can be made to fit better using resummation and retuned fragmentation functions (from LEP data). For charmonium, the measured cross section requires a large colour-

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octet component but that is not consistent with the observed JA|/ polarization. The CDF secondary vertex trigger in Run II is working beautifully, and the resulting huge charm and bottom samples will allow these puzzles to be explored in much more detail. D 0 now has preliminary Run II JA|/ and muon+jet cross sections which are the first steps in measuring the charmonium polarization (and thus production process) and the b-jet cross section.

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4.4. Hard Diffraction Another QCD-related puzzle is hard diffraction. In this kind of event, a highmomentum-transfer collision occurs but one of the incoming beam particles appears to leave the collision intact, instead of being destroyed in the process. This observation is rather surprising and needs to be pinned down better, and related quantitatively with similar phenomena observed at HERA. Both CDF and D 0 have new instrumentation for diffractive physics in Run II. This will allow us to test some of the basic assumptions on how to tag hard diffraction that have been used in earlier studies and will provide a sanity check for ideas of Higgs production through this mechanism at the LHC.

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5.

B-physics

The mixing between the three generations of quarks results in subtle violations of the CP symmetry relating particles and antiparticles. Understanding this symmetry will help explain why the universe is filled with matter, not antimatter. In the decays of B-mesons, these symmetry violations can be large, and so Bhadrons have become an important laboratory to explore the "unitarity triangle," which relates the elements of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. In Run II we want to confront the CKM matrix in ways that are complementary to the electron-positron B-factories. CP violation is now established in the B system through the decay Ba —> JAj/ Ks. The measured mixing angle is consistent with the standard model but, by itself, cannot exclude new physics. The BaBar and BELLE experiments can and will do much more with their data, but the Tevatron can uniquely access the B s meson, which is not produced at the B-factories and has therefore been called the "el Dorado" for hadron collider B-physics. By measuring the mixing rate between B s and B s , we can determine the length of one of the sides of the unitarity-triangle and complement the B-factories' measurements of its angles. CDF expect to be sensitive to Standard Model mixing with a few hundred inverse picobarns. It will also be interesting to see if there is sizeable CP violation in B s —» JA|/ <>| (it is expected to be small); while the decay Bs —> KK at the Tevatron complements Bd —» Tin that is measured at the B-factories. Together they can pin down the unitarity triangle angle Y- There are many other opportunities, such as Ab properties and searches for rare decays. CDF already have most impressive results from Run II, building on their Run I experience together with new detector capabilities (silicon vertex trigger and time of flight detector). Lepton-

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triggered signals for B* -> JA|/ K*, B° -> JA|/ K*°, and B s -> JA|/ 0 are seen; while using the Silicon Vertex Trigger, the purely hadronic modes B* —> D°7t —> Krai and B —> hadron hadron are being recorded (see Fig 6). We can also look forward to CDF exploiting an enormous sample of charm mesons. In D 0 the tools are being put in place for a B-physics program. The inclusive B lifetime has been measured and B mesons are being reconstructed. D 0 does not exploit purely hadronic triggers but benefits from its large muon acceptance, forward tracking coverage, and ability to exploit JA|/ —> e+e". 6.

W and Z Bosons

In Run II, we will complement our direct searches for new phenomena with indirect probes. New particles and forces can be seen indirectly through their effects on electro weak observables. The tightest constraints will come from improved determination of the masses of the W and the top quark. Until the LHC starts operations, the Tevatron is the world's only source of W and Z bosons. Production is dominated by quark-antiquark annihilation. The typical boson p T is a few GeV, but about 10% of the W/Z bosons are produced with one or more jets of significant E T (>25 GeV). Though the dominant decays of W's and Z's are into jets, they are swamped by the enormous QCD jet cross section. This means that experimentally we focus on the leptonic decay modes of the W and Z.

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Center of Mass Energy (TeV) Figure 8: W and Z cross section measurements at proton-antiproton colliders. W bosons are identified in the detector as a high-pT lepton together with a neutrino, identified as unbalanced transverse energy. We cannot reconstruct the longitudinal momentum of the neutrino but we can construct a transverse mass rriT :

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the maximum transverse mass is equal to the the W mass. Transverse mass distributions from Run II are shown in Fig. 7. The W and Z cross sections can be measured extremely well and are in excellent agreement with the QCD prediction (Fig. 8). In fact it is likely we shall start to use the W cross section as an absolute luminosity normalization at some point in Run II, as it is easier both to measure and to predict than the total inelastic proton-antiproton cross section which is used at present. One of the major goals of the Tevatron program is the W mass measurement. The simplest method is to fit the transverse mass distribution; recently we have started to fit the lepton p T and the ETmss distributions as well, and combine the three results. The whole key is to understand the systematics: one must keep beating them down as the data demand more precision. One can use the Z to constrain many effects, such as the lepton energy scale, the boson production process and boson pT. Currently, the W mass is known to be m w = 80 451 ± 33 MeV; the measurement is dominated by LEP data. The Tevatron Run I results fixed the W mass at the 60 MeV level, but it will take a Run II dataset of order 1 ftf before we can significantly improve the world knowledge of m w — not a

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short-term prospect. Given 2 fb"1 we will be able to drive the uncertainty down to the 25 MeV level per experiment, with an ultimate capability of 15 MeV per experiment. Both experiments now have preliminary results from their Run II samples of W and Z candidates. They have measured the cross sections at the Tevatron's new centre of mass energy of 1.96 TeV (Fig. 8) and used the ratio of the W to the Z to indirectly extract the W width. CDF have also taken a first look at the forward-backward asymmetry in e+e~ production in Run II. 6.1. QCD Aspects of W and Z Production To describe the production of bosons with large p T (> 30 GeV), we can use perturbative QCD; NLO calculations exist. For small p T (< 10 GeV), the fixed order calculation fails, and one needs to resum large logarithms of mw2/pT2. At intermediate p T , the two regions must be matched. This approach works well, but one needs to use data to extract a couple of non-perturbative parameters for the resummation calculation. We can use Z data to predict the W p T in this way. W and Z production with jets is an important background to the top quark, to the Higgs, and to many new physics searches. CDF have measured the W+jet cross section as a function of jet E T and find good agreement with NLO QCD. For 2, 3, 4 or more jets, only leading order calculations exist, and the CDF data agree reasonably well as long as the renormalization scale is chosen appropriately. D 0 has used their W+jet sample to study color coherence. By comparing the pattern of energy flow around the (colorless) W to that around the jet, evidence for the QCD-predicted color interference effects in soft gluon emission is seen in the data. Because leptons can be measured well, and the production process is wellunderstood, Drell-Yan lepton pair production at dilepton masses above the Z pole provides a sensitive test for new physics. Examples are searches for compositeness and for indirect effects of extra dimensions.

7.

The Top Quark

The top quark is unique among the elementary constituents of matter because of its high mass (175 times the proton mass, compared to 5 times the proton mass for the next heaviest quark). In the standard model this means that it alone has a strong (non-perturbative) coupling to the Higgs boson. Is nature giving us a hint

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here? Whether the Higgs or something else turns out to be the origin of the electroweak symmetry breaking, the top quark seems to be uniquely connected to the mechanism of mass generation, and the Tevatron collider is the world's only source of top quarks. The search for the top was pursued directly at e+e~ and pp colliders for many years, with mass limits increasing from m, >23 GeV in 1984 to mt > 131 GeV by 1994. In parallel, the increasingly precise electroweak parameter measurements made at LEP and SLC permitted indirect estimates of the top mass to be made, which by the early 1990's favored the mass range around 150-200 GeV. Direct and indirect measurements converged in 1995 when the top quark was discovered by CDF and D 0 with a mass around 175 GeV. Top quarks are produced in tt pairs at the Tevatron (electoweak single-top or tb production has not yet been observed, and will be discussed later). In the Standard Model, the top decays almost 100% to a W and a b quark. The most accessible signatures are those where one or both W's decays to an electron or muon, i.e. lepton + 4 jets + ETmiss, or 2 leptons + 2 jets + ETmiss. The all-hadronic (6 jets) channel has also been investigated but is challenging because of the large QCD backgrounds. Tagging the jets from the b-quarks, either using soft lepton tags or the silicon vertex detector, helps to pull the top signal out of the W+jets background. Both D 0 and CDF are on the road to "rediscovering" top for the spring 2003 conferences, and both experiments have candidate events. We can look forward to significant improvements in the short to medium term because the Run I dataset was so statistically limited—of order 20 clean events per detector. In Run II, we expect roughly 500 such clean events for every inverse femtobarn recorded. 7.1. Top Mass The top mass is a fundamental parameter of the Standard Model and critical input to electroweak fits. It affects the Standard Model prediction of the W and Higgs mass through radiative corrections. In the lepton + jets channel, there is one unknown in reconstructing the event kinematics (the longitudinal momentum of the neutrino). There are three constraints (the lepton+neutrino and two of the jets should reconstruct to the W mass, and the top mass should be equal to the antitop). This allows a twoconstraint kinematic fit, but there are up to 24 combinatoric ambiguities. The

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latter can be greatly reduced if one, or even better, two jets are b-tagged. In addition to the combinatorics, gluon radiation can add extra jets which worsens the mass resolution significantly. The basic fitting procedure is to evaluate a function /which is an estimator of the top mass (such as the result of a kinematic fit) for each event in the sample. The distribution of/is then compared to what is expected, given a hypothesised mt, for a sample of simulated signal and background events. The value of mt giving the best fit t o / i s then the estimated top mass. In Run I, CDF and D 0 obtained: mt = 176.1 ± 5.1 (stat.) + 5.3 (sys.) GeV (CDF) 173.3 ± 5.6 (stat.) ± 5.5 (sys.) GeV (D0) The largest systematic, for both experiments, is the uncertainty on the jet energy scale (~ 4 GeV). CDF also measured the top mass in the 6-jets sample. Here there is a threeconstraint fit, but the signal-to-background ratio is much worse. The result is mt = 186.0 ± 10.0 (stat.) ± 5.7 (sys.) GeV (CDF) Again, the largest systematic is the uncertainty on the jet energy scale (4.4 GeV). In the dilepton sample, things are harder, because two neutrinos are produced. Only four particles (two leptons, two jets) and two components of ETmss are observed. There remain three constraints, so the system is underconstrained. A dynamical likelihood analysis is therefore performed. For each assumed value of the top mass, the kinematics can be reconstructed. An event weight is then calculated which parametrizes how probable it is that this event originated from a top-antitop of this given mass. The event weight distributions, as a function of mt, are combined for all events, and compared with Monte Carlo samples as before. In Run I, CDF and D 0 obtained: mt = 168.4 ± 12.3 (stat.) ± 3.6 (sys.) GeV (D0) 167.4 ± 10.3 (stat.) ± 4.8 (sys.) GeV (CDF) The largest systematic, for both experiments, is once again the uncertainty on the jet energy scale, but the systematics in this method are reduced compared to the lepton+jets channel.

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The overall Tevatron average top mass is mt = 174.3 ± 5.1 GeV (CDF+D0). In Run II, the data samples will be much larger. The systematic errors will need to be reduced to match, and this will require improved knowledge of the jet energy scale using data. This can come from Z+jet events, from photon+jet events, by using the W —> jj decays within the top events, and by using Z —> bb as a calibration sample. Gluon radiation effects will also need to be constrained using data, or reduced with harder cuts given a sufficient number of top events to afford the loss in statistics. Using double b-tagged events, again with a loss of statistics, we can greatly reduce the combinatorics. New mass extraction techniques are also being developed: D 0 has reported a new, preliminary determination of the top mass using existing Run I lepton+jets data. The new technique makes use of more information per event, giving better discrimination between signal and background than the published 1998 analysis, and improves the statistical error equivalently to a factor 2.4 increase in the number of events. With 2 ftf1 of Run II data we will be able to drive the uncertainty on mt down to about 3 GeV/c2 per experiment, and on mw down to about 25 MeV/c per experiment. Measurements at these sensitivities will present a powerful test of the consistency of the standard model. With an ultimate capability of 15 MeV/c2 for m w and 1-2 GeV/c2 for mt per experiment, the indirect evidence for physics beyond the standard model would be compelling.

7.2. Production Cross Section and Kinematics The top cross section is a test of QCD, and any discrepancy could possibly indicate new physics. In Run I, the measurement was performed in the lepton+jets, dilepton and all-jets final states. The results (5.9 ± 1.4 pb for D0, 6.5 +1.7/-1.4 pb for CDF) are well within the range of QCD predictions (5-7 pb). The top quark transverse momentum is another test of the QCD prediction; in Run I, CDF found good agreement between data and expectations. In Run II we will also test for resonances in the top-antitop invariant mass that would signal the production of new particles decaying into top. The Standard Model predicts that the top and antitop spins should be correlated. This is because their production is predominantly through quark-antiquark annihilation with a spin-1 s-channel gluon. Since the top lifetime (4 x 10~25 s) is less than the timescale of QCD hadronization (1/AQCD ~ few x 10-24 s), the top decays before the spin information is lost. The spin correlation can therefore be reconstructed from the decay products. The motivation to do this is to test the top quark's spin (is it really a spin-Vi object as it should be?) and to check these

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assumptions about production and decay. The optimal spin-quantization basis is off-diagonal and has been derived by Mahlon and Parke [1]. Only like-spin combinations are produced in this reference frame. The analysis uses dilepton events, so in Run I it was very statistically limited; nevertheless, D 0 carried the analysis through on the six available events. The result is consistent with expectations; with 2 ftf1, it should be possible to distinguish between uncorrelated spins and the standard model expected correlation at better than the 2o level. The top is expected to decay predominantly (70%) into a longitudinally polarized W, with a 30% admixture of left-handed W's. The lepton p T distrribution can distinguish the various states. CDF find the Run I data are consistent with the standard model expectation, and the fraction of right-handed W's is consistent with zero. 7.3. Single-Top Production Single top (top+bottom) production tests the electroweak properties of the top quark, and allows the Wtb coupling and the CKM element |Vtb|2 to be extracted.

Figure 9: Single top production. The cross section is about 0.7 pb for the s-channel process in Fig. 9 (a) and 1.7 pb for the t-channel process (b). The signature consists of a W (lepton and missing ET) together with two b-jets, one coming from top decay. In the s-channel process both b-jets are high-pT and central, while in the t-channel case the jet from top decay is still hard and central but the other is much softer, there also being a high-pT light quark jet (shown as a d-quark in Fig. 9(b)). It is desirable to separate the two processes based on these kinematic differences since they have different systematic errors and different sensitivities to new physics. The backgrounds for both are significantly higher than they are for top-antitop production (the W + 2 jet cross section is much higher than the W + 4 jet cross section).

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In Run I, both experiments could only set limits (at several times the standard model level) on single top production. With 2 ftf1 in Run II, we expect to see a clear signal. We will use it to measure the cross section to ~ 20%, to indirectly extract the width for t -> Wb to ~ 25%, and thus |Vtb| to ~ 12%. It is important to measure single top production since it is a background to many new physics searches; it is also sensitive to new dynamics like topcolor. 8.

The Higgs Boson

In the standard model, the weak force is weak because the W and Z bosons interact with a scalar field (called the Higgs field) that permeates the universe with a finite vacuum expectation value of 246 GeV. If this same field couples to the fundamental fermions, it can explain their masses too. It should be possible to excite this field and observe its quanta—the long sought Higgs boson. It is the last piece of the standard model, and also a key to understanding any beyondthe-standard-model physics like supersymmetry. Finding it is a very high priority, but will require large datasets because the production cross section is low and the irreducible backgrounds are large. All of the properties of the Higgs are fixed in the standard model with the exception of its own mass: its couplings and decays are all determined. There is no doubt that electroweak symmetry breaking occurs, so we know that something is coupling to the W and Z. We should remember, however, that the Higgs field need not result from a single, elementary scalar boson: there could be more than one particle (as occurs in SUSY), or composite particles could play this role (as in technicolor or topcolor). Whatever the field is, precision electroweak measurements tell us that it looks very much like a Standard Model Higgs (though its fermion couplings are less constrained). We also know that the WW cross section would violate unitarity at Vs ~ 1 TeV without the existence of a Higgs; this is a real experiment accessible to the LHC, so there is no doubt that whatever plays the role of the Higgs will eventually be seen. Over the last decade, the focus was on experiments at LEP. Direct searches excluded masses below 114 GeV, while indirect electroweak measurements combined with Tevatron top mass data in global fits to exclude Higgs masses above about 200 GeV. In mid 2000, there was excitement that hints of a signal for a Higgs with a mass of 115 GeV, right at the limit of sensitivity, had been seen at LEP. In the event, CERN management decided to shut off LEP operations in order to expedite construction of the LHC, and the resolution of the puzzle is now left to the Tevatron.

325

A relatively light Higgs, with a mass in the range between the current lower bound of 114 GeV and about 140 GeV, is produced with a cross section of about 1 pb at the Tevatron and decays mainly to bb quark pairs. This gives a signature of two b-jets, which is almost impossible to extract from the huge background of two b-jets from QCD processes. Instead, we will search for the production of a Higgs together with a W or Z boson. The cross section is a factor of 5 lower, but the W or Z decays a significant fraction of the time into an electron or a muon, and a high energy electron or muon is relatively easy to trigger on and isolate. We then have the simpler, but still very challenging, task of separating the signal of a vector boson plus a Higgs, from the backgrounds: vector bosons plus two bjets, vector bosons plus a Z, with Z —» bb, top pairs and single top. The bb mass resolution directly influences the signal significance; Z —» bb will serve as a calibration for H —> bb. As an example, consider mH = 115 GeV (interesting because of the LEP "hint"). In 15 fb"1, we expect[2] 92 WH -» lv bb events (with a background of 450); 90 ZH —> vv bb events (with a background of 880); and 10 ZH —» 11 bb events (with a background of 44). These event rates would allow a 95% exclusion (if no signal is seen) with 2fb_1 per experiment, and three and five standard deviation signals to be established with 5 and 15fb_1 respectively. If we do see something, we will want to test whether it is really a Standard Model Higgs by measuring the production cross section (which is predicted for any given ma), and searching for other decay modes. The modes H —> xx and H —> WW are expected to have branching ratios around 8-9% for mH = 115 GeV and might just be visible. H —> YY should not be visible for a SM Higgs at the Tevatron. It will also be interesting to look for the process where the Higgs is produced in association with a top-antitop pair. The cross section is very low (a few fb) but the signal to background ratio is good, and one might see a signal at the few event level. This process tests the top quark Yukawa coupling. At higher Higgs masses, above about 140 GeV, the dominant Higgs decay mode becomes W pairs. Here, the challenge is not to reconstruect b-jets, but to separate the H —> WW signal from the other sources of W pairs. Angular cuts can be used to extract the Higgs signal from the WW background (the Higgs is spin-0 while a virtual Z is spin-l). With tight cuts one can obtain a signal to background of order 1 to 3. All of these Higgs strategies and estimated sensitivities are taken from the Higgs and Supersymmetry workshop held at Fermilab in 1999 [2]. Right now, the

326

experiments are developing the foundations needed to do this analysis for real in Run II: good jet resolution, high b-identification and trigger efficiencies, and a good understanding of all the backgrounds. These will also enable us to firm up these earlier estimates of Higgs sensitivity as a function of luminosity. In the most popular extension to the Standard Model, supersymmetry (described below), there is still a Higgs boson: in fact the existence of a light Higgs is a very basic prediction of SUSY. These models contain an extended suite of Higgs particles: two neutral scalars h and H, a pseudoscalar A and a charged pair H*. At tree level, the Higgs sector is decsribed by two parameters (usually taken to be mA and tan(3); on top of this, there are radiative corrections that depend on sparticle masses and the top mass. LEP has excluded mh < 91 GeV, mA < 92 GeV and mH± < 79 GeV, and tan|3 < 2.4. Over much of the remaining allowed parameter space, the h is "light" (115-130 GeV) and the H, A and H* are much more massive. In this "decoupling limit", the h looks very much like the standard model Higgs and decays similarly, while the heavy Higgses tend to have enhanced decays to b's and taus. Tevatron searches for the standard Higgs therefore apply to its supersymmetric cousin h as well; if we exclude the existence of such a Higgs, we will have gone a long way towards excluding minimal supersymmetry at the electroweak scale. One area of Higgs physics that can be attacked with relatively modest luminosities (already feasible in 2003) is to search for the other supersymmetric Higgs bosons. Associated production of a neutral SUSY Higgs (h/H/A) together with a b b quark pair is enhanced for high but plausible values of tanp\ and tighter limits than those from LEP will already be reachable with a few hundred inverse picobarns. The Tevatron can also search for the production of charged Higgses in top decays: the decay to H ^ competes with the standard model Wb decay, and would result in an enhanced production of tau leptons in top decays at large tan(3. As long as we have adequate sensitivity, exclusion of a Higgs would still be a very important discovery for the Tevatron. In the Standard Model, we can only exclude the lower part of the mass range (up to 175 GeV with -10 fb"1), but this is the region favored by electroweak fits. In minimal SUSY, we can potentially exclude all of the allowed parameter space with ~5 fb"1. There is also the possibility of discovering something other than a Higgs. In dynamical models like technicolor and topcolor, the role of the Higgs is played by a composite particle. Such models predict many other new particles in the mass range 1001000 GeV, with strong couplings, large cross sections, and clear signatures at the Tevatron.

327

It is useful to remember what the Higgs search will and will not tell us. It will tell us what is the source of mass of the W and Z, and therefore why the weak force is weak. It will tell us the source of mass of the fundamental fermions. It will tell us whether there are fundamental scalars, and give a strong indication whether there is weak-scale SUSY (since a light Higgs is a basic feature of such models). It could tell us that there is other new physics at the weak scale, should we discover something like technicolor, or should we find a Higgs, but one whose mass, together with the top and W masses, is not consistent with precision electroweak fits. It will tell us the mass scale of new physics and point towards the machine we want to build after the linear collider. The search will not reveal why fermion masses have the values they do, nor what is the origin of flavor (what distinguishes a top quark from an up quark). It will not reveal what is the origin of mass in the universe, since the baryon masses are almost all due to QCD, while that of dark matter, if it turns out to be the lightest supersymmetric neutralino, is due to some unknown SUSY breaking mechanism. We don't even know what two thirds of the universes's mass—dark energy—is made of.

9.

Searches for New Physics

As the world's highest energy collider, the Tevatron is the most likely place to directly discover a new particle or force. We know the standard model is incomplete; theoretically the most popular extension is to make it a part of a larger picture called supersymmetry (which is a basic prediction of superstring models). Here each known particle has a so-far unobserved and more-massive partner, to which it is related through a change of spin. If it exists, the lightest supersymmetric particle would be stable, and vast numbers of them would pervade the universe, explaining the astronomers' observations of dark matter. LEP has excluded sparticle masses below the 80-100 GeV range (and lightest neutralinos below 36 GeV); the Tevatron is now the only place to directly search for supersymmetry. There are a number of search strategies. The most copiously produced SUSY particles at a hadron collider would be the colored squarks and gluinos. These have cascade decays giving final states with missing energy ^miss-j p j u s j e t s ^ an( j 0ften iepton(s) from charginos and neutralinos in the cascade). In Run I, squark and gluino masses below about 200 GeV were exluded (Fig. 10); with 2 fb"1, this mass reach can be roughly doubled.

328

Ms > 300 GeV/c2 Mr, a Mg

Figure 10: Squark and gluino mass limits from Run I (CDF). Another attractive search mode is for charginos and neutralinos through multilepton final states. Associated production of a chargino and a neutralino can give a trilepton signature for which the standard model background is very small. In Run I, this search was not competitive with LEP, but in Run II it will become increasingly important as squark and gluino production reaches its kinematic limits. The chargino mass reach is 150-180 GeV. We will also search for SUSY signatures with heavy flavor. Quite often the lightest squarks are the sbottom and stop, which can decay to b or c-jets. Gauge mediated SUSY models result in signatures with ETmiss + photon(s). Searches for other new phenomena include leptoquarks, dijet resonances, new heavy W and Z' bosons, massive stable particles, and monopoles. The Tevatron allows us to experimentally test the new and exciting idea that gravity may propagate in more than four dimensions of spacetime. If there are extra dimensions that are open to gravity, but not to the other particles and forces of the standard model, then we could not perceive them in our everyday lives. But particle physics experiments at the TeV scale could see signatures such as a quark or gluon jet recoiling against a graviton, or indirect indications like an increase in high energy electron-pair production. These studies use the Tevatron to literally measure the shape and structure of space-time.

329

While it is good to be guided by theory, one should also remain open to the unexpected. Therefore both experiments have .developed quasi-modelindependent (signature-based) searches, which look for significant deviations from the Standard Model. In the Run I dateset, no significant evidence for new physics was found. Perhaps revealing different psychologies, D 0 has quantified its agreement with the Standard Model at the 89% confidence level, while CDF has preferred to highlight some potential anomalies as worth pursuing early in Run II.

Figure lis Run II search for virtual effects of extra dimensions in the pair production of photons and electrons (D0). The experiments have already embarked on a number of searches using Run II data. Work has started on understanding the Bfm$s distribution in multijet events as a prelude to squark and gluino searches; trilepton candidates are also being accumulated. At D0, a gauge-mediated SUSY search has set a limit on the cross section for pp —» Ejmss + YY- Also at D0, virtual effects of extra dimensions are being sought in e+e", p+f4.~ and YY fina^ states, and limits on the scale of new dimensions at the 0.9 TeV level can already be set. A search for leptoquarks decaying to electron+jet has been carried out. None of the cross sections or mass limits is better, yet, than published Run I results, but serves as a demonstration that the pieces are all in place.

330

10. Physics Prospects for Run II At the same time as Fermilab's accelerator experts have been working to improve Tevatron operations, they have been trying to incorporate the lessons learned into a solid plan for the future. The planning for the accelerator complex is in two phases. The first focuses on US Fiscal Year 2003, which ends in September 2003. A full plan and schedule are in place. By the summer of 2003, each experiment should have recorded around 200 pb_1 of Run II data (almost twice the Run I dataset). The centrepiece will be a greatly increased top quark sample, thanks to the higher beam energy and the much improved b-tagging capabilities of the detectors. A first look at B s mixing will be possible, together with lifetimes and branching ratio measurements from the B, B s , Ab and charm samples. Jet distributions at the highest energies will constrain proton structure, and searches will follow up on Run I anomalies and extend the Run I reach for many extensions to the standard model. The second phase covers 2004 and beyond. It is now clear that it will take somewhat longer than had been anticipated to accumulate the large datasets ultimately foreseen for Run II: such is the price of realism. As long as the Tevatron remains the world's highest energy collider, it is a unique facility that must be exploited to the fullest; this will remain true until the LHC experiments start producing competitive physics results. If the LHC delivers first beam to ATLAS and CMS in 2008, we might expect a year to be spent commissioning the detectors and accelerator, with the first physics data by 2009. One year of concurrent running (2009-10) between LHC and Tevatron would not be unreasonable, since the low-mass Higgs searches at the two machines are complementary in several ways. With this in mind, we are preparing to run the Tevatron until the end of the decade in order to fully realize its physics potential. The Run II physics program is a broad and deep one and will answer crucial questions about the universe. As Fig. 12 shows, there is no threshold at which this starts. There is compelling physics to be done each year and with each doubling of luminosity, starting now with a few hundred inverse picobarns and to the end of the decade with multi femtobarn data sets. To explore the 5 ftf1 to 15 ftf1 domain calls for upgrades to the CDF and D 0 detectors. Primarily, these involve new trigger systems to handle more than ten interactions per crossing at the expected luminosity, and new silicon detectors that make use of LHC R&D to sustain the high radiation doses. These upgrades were successfully reviewed by the Department of Energy in September and are now moving towards approval with installation planned for 2005-6.

331

Run I I Physics Program iiiiiii

* 5
* 30 Higgs signal « m H - 115-125, 155-170 GeV * Exclude Higgs over whole range of 115-180 GeV * Possible discovery of supersymmetry in a larger fraction of parameter space

* • • *

3?j Higgs signal @ m^ = 115 GeV Exclude SM Higgs 115-130, 155-170 GeV Exclude much of SUSY Higgs parameter space Possible discovery of supersymmetry in a significant fraction of minimal SUSY parameter space (the source of cosmic dark matter?)

* * * * *

Measure top mass ± 3 GeV and W mass ± 25 MeV Directly exclude m^ =* 115 GeV Significant SUSY and SUSY Higgs searches Probe extra dimensions at the 2 TeV (10- 19 m) sea If B physics: constrain the CKM matrix

* * * * *

Improved top mass measurement High p j jets constrain proton structure Start to explore B§ mixing and B physics SUSY Higgs search <§ large tan p Searches beyond Run I sensitivity

Figure 12; Summary of Run II physics reach for various integrated luminosities.

332

Run II is a marathon and not a sprint. The combination of high accelerator energy, excellent detectors, enthusiastic collaborations and data samples that are doubling every year guarantees interesting new physics results at each step. Each step answers important questions. Each leads on to the next. This is how we will lay the foundations for a successful LHC physics program—and hopefully a linear collider to follow. References 1. G. Mahlon and S. Parke, Phys. Lett. B 411, 173 (1997). 2. M. Carena et ah, Report of the Higgs Working Group of the Tevatron Run II SUSY/Higgs Workshop, hep-ph/0010338.

333

Appendix: Student Exercise for TASI 2002

A Search for New Physics using D 0 Run 1 Data

John Womersley, Fermilab TASI Summer School, June 2002 Goal of this exercise: explore the limits that can be placed on the production of technicolor particles using the data already taken at the Tevatron in 1992-95. Extrapolate to Run 2. We will use the public interface to D 0 Run 1 data, which is called Quaero. 1. First, look at the PRL* article describing Quaero: http://www-dO.fnal.gov/www buffer/pub/pub 228.pdf or http://www-dO.fnal.gov/www buffer/pub/pub 228.ps You'll see (in Table III of the paper) a number of channels where limits were set, including pp —> WH —» evjj. The paper did not, however, set limits on the technicolor process with a similar final state, pp —» W% —> evjj. It is suggested that this process may have a much higher cross section than the Higgs process, so it's interesting to see what can be done with the relatively low luminosity accumulated in Run 1. This process, and a suggested search strategy, are described in: http://arxiv.org/abs/hep-ph/9704455 2. Go to the Quaero website: http://quaero.fnal.gov Look at the examples to see what one can do with this interface. You can ask Quaero to generate the signal events itself, using Pythia, or you can feed it an input file. In this case we'll ask it to use Pythia because Technicolor is implemented in Pythia. The Pythia web page is http://www.thep.lu.se/~torbjorn/Pvthia.html but a more useful short guide from LAPP can be found at http://wwwlapp.in2p3.fr/Pythia/. Page 28 and 29 of the short guide (PDF version) describe the technicolor implementation. 3. Now let's set up our search.

334

Under "signal:" • Select the final state corresponding to evjj (e met 2j (nj)) • Select "smear" (we want the detector resolutions to be modelled) • Select "Pythia input" since we want Quaero to generate the technicolor events itself. • Enter the Pythia commands to generate W% events: The easiest way is MSEL=50 which turns on all the technicolor processes. • Click on "search" (that's what we want to do) • Check the boxes corresponding to the backgrounds to be considered Now under "variables:" • If you like, enter a constraint (e.g. you could constrain the e and the missing E T to be a W). This is optional. • Enter up to 3 variables in which Quaero will search for signal events: the hep-ph paper suggests the p T of the W, the p T of the dijet system, A^Qet], jet2), and the mass of the dijet systemUnder "Requestor," enter your name and e-mail, and a short note describing what you are doing (please mention TASI). 4. Now click "submit"! Stay logged in - if there are errors in the submission, Quaero will bounce an email right back to you. 5. Check your e-mail tomorrow for the results. See the examples on the following pages for typical output. 6. What limit did you get? How do this limit, and the Pythia cross section reported by Quaero, compare with the expected theoretical cross section from the hep-ph paper? Remember that Quaero quotes limits on (cross section x branching ratio), so you'll have to multiply by 9 to remove B(W —> ev). How much more luminosity would it take to observe a signal in Run 2? 7. If you have time, try optimizing the constraints or variables to see if you can get a better limit on the same process. Try constraining the e and Missing E T to a W mass, for example, if you didn't already. Try other variables which might be better than mine.

335

Here is an example Quaero submission: httpv' /^na«m. ttwlgtw / quAvm;

Q».*rev

Quaero

Ttm *«b £3*$® presses a trtfwftet to $m mmm ^ m m mpm^m «i m* F««W n#w ^ w ^ f t s n a af ift« stsfe of # * * » # *w#*as <3*V. Th* ft «t* links t * t a * . founts in *&&towsf * « » *MSH<W • * » * • • r * t a m * i by *

^d^ll'f T '^ ; r -^33 pr^w^r"

** £!gafiL£B£i

jBon:fP

,

^rw"

.__^_^JXL^^^

* t&fcaty w w * i M » day

336

Example of the e-mail response: i!hrip:A^w«im^toy®tiiM|^^i2J'^

Date; Morv 20 May 2002 1557:54 -ffiPQ To: wcirassteieastosfcgafc' Dearitfiri Vtonarsfciy: Therea]i!ofycuet^£Kraatm^&aKa*iad.. R«asefn£hidfi*eaoai«5{BfffWgre(«patioeti «vpitecaw!aeftatus«s.^nire5*it, Regard^ The DZeta aastxYattan

V.-J«jasKw {\sl at af}., \#241 Man Kay 33 203Z

Stfijaa; Qjaem R&^iest #24).

news

mass£|lfj2) DZero cuts: j t e n d a ^ t « l ) M t e j ^ 2 a ' ) 8 & C ^ t ^

us* (mas^syme^>3aj &&

MaddOeserts&ori Tsttt fcr sttfimar street e«SMise

P>ora oo»* sectftn xfcrarKJiHgfatio = &3S fife, ujapw ***& afi trie trass, sadg-wt m ttfe process at cmSOensEtewfeof 59%, 93%, and 95%. ar« fcund to &e 8 5 fife I $ pb, and i.a jp^, jespsrttwsy. Madras* sensiStfty (fl,635 f* *-1) fs =K*)feed ai a aegSMi of vatfatfe SJB<E v * * 2.9 sfigpas «**]ES a p « X a ^ 23.4 *- 5,4 SBdapsatid events esfift^cKKi, se\d 14 (wtrts abs«8"w>dfatfe d£&< Piots Plots af$K> va^aHe* Sis*: you used aneavaiMIe f » ^gwtiq ^ t i g ^ j / ^ a r g j ^ a f l ^ ^ u ^ ^ / f ^ ^ i t ^ ^ f e i S i f ; ^ ! p& "TO red «jtvefs»eaKpecj«i »iactapa*«dj efts grseri curve rsym* sscjcaiflxiKpled by a feoaraf 50; ate ESack daisare D&dasa," Pieass &»p tie Mtfti&tg caveats £i mtid vtfietftHKtirasintjtrt"s result: '* Your Sgrtai tiasfeem&m tlraagh a fast eisteaaF sfiTV-iatof. '• Your Sgnai ha* tjeen generated wiSi PyU«a. Any inadequacytoP>»is*ii nxtfattrig cT trie <jg«3l praeefci w^ prepagasa tfto a oifrasgorelsig &i adequacy fet •* a amrrpmQ mim$e ariaysesti*tfte same ssgnat, you shouH qjote asthe inai aaswsf tte resufeeorresptadMg to-tfte greatest sensiSyty.. *No&&^h2£be^in^e&a!*^tesEa^^

337

Example of the plots that are produced:

I

I 'lint)

|a>

l| ' ' | I ' ' I |"|'Enfcfai ' | '

Uif

. j - l . , , | ,.,|,..! I l h.

0

0

1 2 deltaphi(j'1,j2)

3

100 200 300 (e+met)_pt

100 2 0 0 3 0 0 4 0 0 mass(j1 ,j2)

The plots show the three variables I specified in my search. The black points are the data, the red curve is the background, and the green is the signal expected multiplied by 50. You can see that the deltaphi and mass variables give quite good distinction between signal and backgound, but the px variable isn't so great. Try to do better!

Quaero FAQ: 1. Yes, this is real collider data. 2. Yes, if you find a signal you can publish it.

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PART II: TeV-Scale Physics Lecture Series

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JOHN TERNING

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NON-PERTURBATIVE

SUPERSYMMETRY*

JOHN TERNING Theory Division T-8, Los Alamos National Laboratory Los Alamos, NM 87545 E-mail: [email protected]

These lectures contain a pedagogical review of non-perturbative results from holomorphy and Seiberg duality with applications to dynamical SUSY breaking. Background material on anomalies, instantons, unitarity bounds from superconformal symmetry, and gauge mediation are also included.

Contents

1

Holomorphy 1.1 Non-Renormalization Theorems 1.2 Wavefunction Renormalization 1.3 Integrating Out 1.4 The Holomorphic Gauge Coupling

345 345 347 347 349

2

R e v i e w of Anomalies and Instantons 2.1 Anomalies in the Path Integral 2.2 Gauge Anomalies 2.3 Review of Instantons 2.4 Instantons in Broken Gauge Theories

352 353 356 358 359

3

Gaugino Condensation

361

4

The 4.1 4.2 4.3 4.4 4.5 4.6

363 363 366 368 371 373 374

Affleck-Dine-Seiberg Super-potential Symmetry and Holomorphy Consistency of W A D S : Moduli Space Consistency of W A D S : Mass Perturbations Generating W A D S from Instantons Generating W A D S from Gaugino Condensation Vacuum Structure

'Printed with permission from the author. The copyright of these lectures is retained by the author. 343

344

5

't Hooft's A n o m a l y Matching

374

6

Duality for S U S Y Q C D . 6.1 The Classical Moduli Space for F > N 6.2 The Quantum Moduli Space for F > N 6.3 Infrared Fixed Points 6.4 An Aside on Superconformal Symmetry 6.5 Duality 6.6 Integrating out a flavor 6.7 Consistency

375 376 379 380 383 388 392 393

7

Confinement in S U S Y Q C D 7.1 F = N: Confinement with Chiral Symmetry Breaking 7.2 F = N: Consistency Checks

394 394 397

8

S-Confinement in S U S Y Q C D 400 8.1 F = N + 1: Confinement without Chiral Symmetry Breaking . . . 400 8.2 Connection to theories with F > N + 1 403

9

Duality for SO(N) 9.1 The SO{N) Theories and Their Classical Moduli Spaces 9.2 The Dynamical Superpotential for F < N - 2 9.3 Duality 9.4 Some Special Cases

407 407 409 409 410

10 Sp{2N) and Chiral Theories 10.1 Duality for Sp(2N) 10.2 Why Chiral Gauge Theories are Interesting

412 412 414

11 S-Confinement

414

12 Deconfinement

417

13 Gauge Mediation 13.1 Messengers of SUSY breaking 13.2 RG Calculation of Soft Masses 13.3 Gauge Mediation and the /j, Problem

419 419 421 423

14 Dynamical S U S Y Breaking 14.1 A Rule of Thumb for SUSY Breaking 14.2 The 3-2 Model 14.3 The SU(5) Model 14.4 SUSY Breaking and Deformed Moduli Spaces 14.5 SUSY Breaking from Baryon Runaways 14.6 Direct Gauge Mediation 14.7 Single Sector Models

425 425 425 428 430 432 435 437

345

1. Holomorphy Two of the early exciting observations about J\f = 1 SUSY [1] gauge theories were the absence of quadratic loop corrections to scalar masses and the absence of renormalization for many superpotentials [2-5]. While the former property was an immediately obvious consequence of Bose-Fermi symmetry, the latter was only much later realized to follow in a very simple way from the holomorphy [6,7] of the superpotential, that is from the fact that superpotentials are functions of chiral superfields but not of the complex conjugates of these fields. Holomorphy arguments laid the groundwork for the later development of Seiberg duality [8,9]. 1.1. Non-Renormalization

Theorems

Couplings in the superpotential can always be regarded as background fields, and so superpotentials are also holomorphic functions of coupling constants. The fact that superpotentials are holomorphic can be used to prove powerful non-renormalization theorems [6]. If we integrate out physics above a scale [i (i.e. calculate the Wilsonian effective action 1 ) then the effective superpotential must also be a holomorphic function of the couplings. Consider a theory renormalized at some scale A with a superpotential: Wtree = ^

+^ .

(l.l)

Here 0 is a chiral superfield. I will also refer to the scalar component as

(1-2)

so we have R[tp] = R{(j>] — 1, R[8] = 1. Since the Lagrangian in our toy model has Yukawa couplings, £ D ±4>U , 1

(1.3)

As opposed to the 1PI effective action, see ref. [10,11] for a related discussion. For k = 2 there is an SU(2)R R symmetry and for M = 4 there is an S(7(4) H ~ SO(6)R R symmetry. 3 The existence of a discrete symmetry often referred to a -R-parity is completely independent of the continuous R symmetry discussed here [12]. 2

346

which must have zero R charge, we have 3R[<j>] - 2 = 0 ,

(1.4)

and therefore R[W] = 2. More generally we could get the same result by noting: d29W .

/

(1.5)

By convention superfields are labeled by the R charge of the lowest component, which in our example of a chiral superfield is the scalar component. By convention the gaugino assigned R charge 1. In our toy model we can make the following charge assignments: U(l)

U(1)R

m -2 A -3

(L6)

0 -1

where we are treating the mass and coupling as background spurion fields in order to keep track of all the symmetry information. Note that nonzero values for m and A explicitly break both £/(l) symmetries, but these symmetries still lead to selection rules. If we consider the effective superpotential generated by integrating out modes from A down to some scale fi, then the symmetries and holomorphy of the effective superpotential restrict it to be of the form Weff = mcf)2 h, .

/ ^

TO

^a^TO1"".^2 ,

(1.7)

since m 0, and the massless limitTO—> 0 restricts n < 1 so

Weff = jtf

+ ^

= Wtree .

Thus we have shown that the superpotential is not renormalized.

(l.g

347

1.2. Wavefunction

Renormalization

Next consider the kinetic terms: Acin. = Zd^*d^ + iZ^d^

,

(1.9)

where the wavefunction renormalization factor is a non-holomorphic function Z = Z(m, A, m\ \ \ /x, A) .

(1.10)

If we integrate out modes down to /J > ra we have (at one-loop order) Z = l + c\tf\n.(^\ ,

(1.11)

where c is a constant determined by the perturbative calculation. If we integrate out modes down to scales below m we have Z = l + +cAA t ln( -^—r ) (1.12) \mm< ) So there is wavefunction renormalization, and the couplings of canonically normalized fields run. In our example the running mass and running coupling are given by m A Z'Z?' 1.3. Integrating

(1.13)

Out

Consider a model with two different chiral superfields: W = \M
+ -CPH^2

(1-14)

This model has three global (7(1) symmetries: U(1)A U(1)B 0 4>H 1 1 0 -2 0 M -2 A -1

U(1)R

1 l 2

(1.15)

0

where (7(1)A and (7(1)B are spurious symmetries for M, A / 0. If we want to integrate out modes down to \x < M, we must integrate out 4>H- An arbitrary term in the effective superpotential has the form 4PMk\p .

(1.16)

348

To preserve the symmetries we must have j = 4, p = 2, and k = — 1. By comparing with tree-level perturbation theory we can determine the coefficient:

*.-££.

(1-17)

We could also derive this exact result using the algebraic equation of motion ^

= Mfe + ^

2

= 0.

(1.18)

We can simply solve this equation for <J>H and plug the result back into the superpotential (1.14), which yields the perturbative effective superpotential (1.17). Another interesting example is w

=

\MH4>2 + 1^>3H •

. (1.19)

The <J>H equation of motion yields

Note that as y —> 0, the two vacua approach (f>H = —A0/(2M) (as in the previous example) and H = oo. Integrating out CJ>H yields ^

f f

M^ 3Xy£ f Xyf\ \y2 - ^ 2 ( l - ^ r ±2 ( l - ^ - 2) \ / l - i ^ 2 |

3y 1

2M

V

M J V

M

•

(1.21)

The singularities in Weff indicate points in the parameter space and the space of # becomes massless and we shouldn't have integrated it out. The mass of <\>H can be found by calculating d2W -^r=M

+ y^H,

(1.22)

and substituting in the solution of the equation of motion: d2W b2H

A#2 M l ^ \ -^W' '" V * M 2

(1-23)

We could also apply the holomorphy analysis to this problem. First, we can maintain the symmetries of the previous example by assigning charges (-3,0,-1) under U(1)A x U(1)B x U{1)R to the coupling y. Then requiring

349

that the effective superpotential has no dependence on u and maintains the three symmetries we find that it must have the form

for some function / , just as we found from integrating out #. 1.4. The Holomorphic

Gauge 4

Coupling M

Using the superspace notation y = x^ — ido^B (where 9 is an anticommuting Grassmannian variable) we can write a chiral supermultiplet consisting of a scalar (f>, a fermion ijj and an auxiliary field F as a chiral superfield $ = cf,(y) + V26 i/>(y) + 92F(y) .

(1.25)

We can also use this notation to represent an SU(N) gauge super-multiplet as a chiral superfield W% = -iK(v)

+ eaDa{y) - (a^0)aF^(y)

- (06)
,(1.26)

where the index a labels an element of the adjoint representation, running from 1 to N2 — 1, Aa is the gaugino field, F " is the usual gauge field strength,and Da is the auxiliary field. We have used the notation that the a% are the usual Pauli matrices and *" = (/,*') afM = (I,-al)

(1-27) (1.28)

a"" = - ( c r ' V - o - ^ ) .

(1.29)

The field strength can be written as F% = d^Al - dvA% + gthcAlAcv

,

(1.30)

where fabc is the structure constant of the gauge group determined through the adjoint generators Ta by [Ta,Tb]=ifabc

.

Finally, using the notation

*For an excellent review see ref. [12].

6»YM

4?n

2TT

gz

(1.31)

350

we can write the SUSY Yang-Mills action as a superpotential term

— [d*xId'28TWX

16iri >7T« J

(1.33)

+ h.c.

J

= / (fx /

^ Y M j?ap,v

~4g2

327T2

""

+ -Xa^D^Xa g2

1

+

pa

""

-^DaDa 2

2g

where 1 Aivu0

pa

pa ap •

(1.34)

Note that the gauge coupling g only appears only in r which is a holomorphic parameter, but the gauge fields are not canonically normalized. To go to a canonically normalized basis 5 we would rescale the fields by (A«,\aa,Da)^g(Al,\aa,Da)

(1.35)

Recall that the one-loop running of the gauge coupling g is given by the renormalization group equation: dg where for an SU(N) gauge theory with F flavors and N = 1 supersymmetry, b = 3N - F .

(1.37)

The solution for the running coupling is 1

•Xf

2

9M

(1.38)

where |A| is the intrinsic scale of the non-Abelian gauge theory that enters through dimensional transmutation. We can then write the one-loop running version of our holomorphic parameter r as 4iri Ti-loop — —

2TT

h

_1_ In 2ni

(1.39)


A* J

(1.40)

Due to subtleties with the measure in the path integral there is a non-trivial relation between the running of the holomorphic gauge coupling and the running of the canonical gauge coupling [3,4,13,14].

351

We can then define a holomorphic intrinsic scale A = \k\ei9™'b = fie27riT/b

(1.41) ,

(1.42)

or equivalently r1_loop = ^ l n ( ^ ) .

(1.43)

In order to take account of non-perturbative effects we need to understand the term in the action proportional to #YM- This FF term violates a discrete symmetry: CP (the product of charge conjugation and parity). The CP violating term can be rewritten as Fa^F^

= 4e^ C T a M Tr [AudpAa

+ -A„ApAa

) .

(1.44)

Thus the CP violating term is a total derivative and can have no effect in perturbation theory since it integrates to terms at the boundary of spacetime. Nevertheless it is well known that it can have non-perturbative effects. To see this consider the following semi-classical instanton configuration of the gauge field:

where ft" describes how the instanton is oriented in the gauge space and spacetime. Equation (1.45) represents an instanton configuration of size p centered about the point XQ. Such instantons have a non-trivial, topological winding number, n, which takes integer values. The CP violating term measures the winding number:

Jd4xF"""f£,

^

=n9YU.

(1.46)

eiS ,

(1.47)

Since the path integral has the form /

VAaVXaVDa

and the action S depends on #YM only through a term which is an integer times #YM it follows that #YM -> 0YM + 2TT ,

(1.48)

is a symmetry of the theory since it has no effect on the path integral.

352

The Euclidean action of a instanton configuration can be bounded, since 0 < I dAxTr {F^ ± F^Y

. 2

= fd4xTr

(2F2 ± 2 F F ) ,

(1.49)

so we have f d4xTrF2

> I Jd4xTrFF\

= 32ir2\n\ .

(1.50)

Thus one instanton effects are suppressed by e -&ot = e P & + t f ™ =

^ Y

(1.51)

If we integrate down to the scale JJ we have the effective superpotential WeS = TJ^WZWS

.

(1.52)

Since the physics must be periodic in #YM, A _> e27ri/bA

,

(1.53)

is a symmetry. To allow for non-perturbative corrections we can write the most general form of r as:

T(A;M) = l n

i (j) + / ( A ; A t ) '

(L54)

where / is a holomorphic function of A. Since A —> 0 corresponds to weak coupling where we must recover the perturbative result (1.43), / must have Taylor series representation in positive powers of A. Since plugging A -> e27ri/feA

(1.55)

into the perturbative term already shifts #YM by 2TT, f must be invariant under this transformation, so the Taylor series must be in positive powers of A6. Thus in general we can write: h

/ A\

.°°.

T(As )

/ A\

" -2s ta u) + S""W

bn

(L56)

So the holomorphic gauge coupling only receives one-loop corrections and non-perturbative n-instanton corrections, or in other words in perturbation theory there is no additional running beyond one-loop. 2. Review of Anomalies and Instantons The reader should feel free to skip over this section if they already have a basic familiarity with anomalies and instantons.

353

2.1. Anomalies

in the Path

Integral

Consider some fermions V coupled to a gauge field Aa: Sferndon = / dtxitfa"^

+ iA»T0)^

(2.1)

Under a position dependent chiral rotation i> - • eia{x)ip

(2.2)

Sfermion - • SWmion ~ / d^Xa^O^i^a**^)

(2.3)

Thus, since a = constant is a symmetry of the action, classically we find

c^W)

= dfifA =

o

(2.4)

Quantum mechanically this is not true: the global current has an anomaly. One way to see this is to calculate the diagram with a fermion triangle and the global current and and two gauge currents at the three vertices, as in Figure 1.

Figure 1. The fermion triangle which contributes to the anomaly. One also adds the crossed graph where the gauge bosons are interchanged.

One finds directly from this calculation that the global current is not conserved: d^fA oc Fa^F^

.

(2.5)

Another way to see this is from Fujikawa's path integral derivation [15] of the anomaly. First define p^i^B^d^ + iA^)

(2-6)

0 = i o ^ - iAJ

(2.7)

354

where a% are the usual Pauli matrices and a% = {U,<Ti),

(2.8)


(2.9)

^ =- ( ^ / - 4 a / ) .

(2.10)

f d4xtp'!ptjj= Jdtxipjp^

(2.11)

Then

and we can define orthonormal eigenfunctions fn and gn D2fn=flPfn 2

D gn = 00gn

= -\2nfn,

(2.12)

2

(2.13)

= -X ngn ,

V>fn = K9n ,

(2.14)

P9n = -Kfn ,

(2.15)

with the usual completeness properties

Y.ti{*)Uy)=5{x-y),

(2.16)

n

Y/9*n(x)gn(y)

= 5(x-y),

(2.17)

Tr Jd4xf:(x)fm(x)

= Snm ,

(2.18)

Tr / dixg*n{x)gm(x)

= 5nm .

(2.19)

n

In general D2 and £> have different numbers of zero eigenvalues. We can expand the fermion fields in this basis: ^(x)=J^affn(x) ,

(2.20)

n

^{x)=^bn9n(x)

.

(2.21)

n

The partition function in the background gauge field A^ is given by a path integral over the fermion fields which in this basis reduces to ipV^ e

S

=

Rnmdandbme

s

.

(2.22)

Under a chiral rotation (Xn &n

> (Xn = {^nmQ"m ?

(2.23)

>

(2.24)

On — Onm^m ?

355

where the transformation matrices are defined by Cnm = TIJd4xeia^f*(x)fm(x)

,

Cnm = Tvjdixe-ia^g*m(x)gn(x)

(2.25) .

(2.26)

So the path integral measure changes by —> ( d e t C d e t C ) - 1 ] ! ^ ^ ^ ™ , .

Unmdandbm

(2.27)

where (detCCy1

= exp (-i

[ d4xa{x)A(x)\

A(x) =TrJ2(fn(x)fn(x)

- g*n(x)gn(x))

,

(2.28)

.

(2.29)

n

Since the partition function, Z, is independent of the chiral rotation parameter a, if we take a functional derivative with respect to it we find: 0 = ^ | | a = 0 = i(d^(x)

- A(x)) .

(2.30)

To evaluate A, we use a smooth regulator function R(z) to suppress the effects of the highest eigenmodes. R(z) is chosen such that R(0) = 1, R(oo) — 0, R'(oo) = 0, R"(oo) = 0, . . . For example, e~z satisfies these conditions. Inserting the regulator we have: A(x) = lim Tr J2^l/M2)

(f*n{x)fn{x)

A

M—»oo

- g*n{x)gn{x)) ,

(2.31)

—'

= lim T r £ (f*(x)R(-D2/M2)fn(x)-g*n(x)R(-D2/M2)gn(x))

,

n

= lim

lim Tr (R(-D2/M2)

y—*x M—>oo

- R{-T>2/M2))

\

5{y - x) .

(2.32)

I

To simplify this first note that D2 = d2 - A2 - ci^{F^

- 2A{lldv]) = d2 - A2 - F+ + 0(d • A),

D2 = d2-A2

- 2A[lidv]) = d2-A2+F'+

+ u^iF^

0(d • A). (2.33)

For simplicity we will chose a gauge where d • A = 0. Using the Fourier transform of 5(y — x), one finds by Taylor expanding R(z) around (p2 + A2)/M2 that (2?r)4 ^

n\

356

The n = 0 in the series vanishes trivially, the Dirac trace of the n = 1 term vanishes, and terms with n > 2 vanish in the large M limit. Defining 2 2 z p /M we are left with 1 f°° A{x) = — ^ /

zdzRW^e^TrF^F^

3271-* J 0 1

=

^

2

^

/

^

(2-35)

•

So the anomaly is given by d»f{xE)

= ^Fa^F;u

.

(2.36)

Now consider integrating this equation over spacetime. The left-hand side is given by integrating Eq. (2.31) which yields [d4xA

y/R(X2jM2)rIt[dix(f:(x)fn(x)-g*n(x)gn(x)),

= lim - n^ - n^

,

(2.37)

which is just the number of fermion zero-modes minus the number of antifermion zero-modes, since all other modes occur in pairs and cancel in the difference. So we have 1 n n ip i/)t — QQ_22 / " X F jiV?\IV 32^ n , (2.38) where n is the winding number (Pontryagin number) of the gauge field configuration. This result is known as the Atiyah-Singer index theorem. 2.2. Gauge

Anomalies

There are also potentially anomalies for three-point functions of gauge bosons which are proportional to Aabc

= Tr[T°{T 6 ,T c }] .

(2.39)

These anomalies are potentially non-vanishing for example in the case with one U(l) gauge boson and and two SU(N) gauge bosons, or with three SU(N) gauge bosons for N > 3. The anomaly for two fermions in two different representations R i and R-2 simply add A a 6 c (R! © R 2 ) = A a6c (Rx) + Aabc(R2)

,

(2.40)

357

while for a fermion that is in two different representations R x and R 2 of two different groups one finds: A a 6 c (Ri ® R 2 ) = dim(R 1 )A a b c (R 2 ) + dim(R 2 )A a b c (R 1 ) .

(2.41)

For the fundamental (defining) representation we define dabc

_ ^ [ ^ { 7 ^ ^ j « j] _

(2.42)

It is then convenient to define an anomaly factor A(R), which measures the anomaly relative to the to the fundamental representation: Aabc(K)

= A(R)d ibc

(2.43)

So the gauge anomaly for a theory vanishes if

X>(R0=0

(2.44)

The dimension, index, and anomaly coefficient of the smallest representations are listed below. Irrep

dim(r) N N2-l

2T(r) 1 2iV

A(v) 1 0

N(N-l) 2 N(N+1) 2 N(N-l)(N-2)

7V-2 N +2

iV-4 iV + 4

(JV-3)(iV-2)

(JV-3)(JV-6)

N{N+l)(N+2)

(N+2)(AT+3)

(JV+3)(iV+6)

iV2-3

N2 - 9

•

Ad

y m 1 U

:n

JV(JV-1)(JV+1)

SU(N)

(2.45)

2

]V (JV+1)(JV-1) AT(iV-2)(iV+2) iV(JV-4)(iV+4) 12 (W+2)(JV+3)(JV+4) (JV+3)(N+4)(iV+8) 1 | 1 1 1W(JV+l)(W+2)(W+3) 6 6 24 i(Af-2)(JV 2 -iV-4) (JV-4)(W 2 -JV-8) N(N+l)(N-l)(N-2) 2 8 2 =

:n

If the gauge anomaly does not vanish, then the theory can only make sense as a spontaneously broken theory, since two triangle graphs back to back generate a mass for the gauge bosons. Alternatively if we start with an anomaly free gauge theory and give masses to some subset of the fields so that the anomaly no longer cancels, then the low energy effective theory has an anomaly, but we can only produce such masses if the gauge symmetry is spontaneously broken.

358

2.3. Review

of

Instantons

Recall that instantons 6 are Euclidean solutions of D»F% = 0 ,

(2.46)

characterized by a size p, which, as |x| —* oo, approach pure gauge: - • iU{x)dllU(x)i

A^x)

.

(2.47)

In general, instantons break one linear combination of axial U(l) symmetries. Consider the axial symmetry that has charge + 1 for all (left-handed) fermions. We have oc Fa^F«„

«

(2.48)

Integrating this current in the instanton background with winding number n, one finds: /

d^xd^

=n

Y2 nr 2T(r)

(2.49)

where nr is the number of fermions in the representation r. Thus instantons can create or annihilate fermions. Also an axial rotation of the fermions ip -> eiai\) ,

(2.50)

eYM^0YM-aJ2^r2T(r) .

(2.51)

is equivalent to a shift of #YM r

In the instanton background the gauge covariant derivative can be diagonalized: WD^i

= Xiipi .

(2.52)

For a fermion in representation r one finds 2T(r) zero eigenvalues corresponding to 2T(r) "zero-modes". (In the anti-instanton background ip^ has 2T(r) zero eigenvalues.) Consider a fundamental of SU(N) with T(D) = \. We can write tpi in terms of a Grassmann variable and a complex eigenfunction ipi = £,ifi(x). We then have

For an excellent review see ref. [16].

359

where /o corresponds to the zero eigenvalue. The path integration over this particular fermion is then

(2-55)

J V^V^ = J d£o J I I <&#] • So

= / ^° / n d ^ exp (- E x^n j = / #° / n <&#* IK1 - A^^«) /< n

= 0,

(2.56)

since integrating a constant over a Grassmann variable vanishes: d£0 = 0 .

(2.57)

/ •

However if we insert the fermion field into the path integral we find f TtyThfi exp (-

j ^a»D^\

^{x) = f0(x) J\

\n .

(2.58)

Thus for an instanton amplitude to be non-vanishing it must be accompanied by one external fermion leg for each zero-mode. At distances much larger than the instanton size 't Hooft [17] showed that instantons produce effective interactions for a quark Qnj with color index n and flavor index j : £inst = cdet H?nQnj + h.c.

(2.59)

This interaction respects the non-Abelian SU(F) x SU(F) flavor symmetry but breaks the U(1)A symmetry. 2.4. Instantons

in Broken

Gauge

Theories

In a theory with scalars that carry gauge quantum numbers, VEVs of the scalars prevent us from finding solutions of the classical Euclidean equation

360

of motion. However we can find approximate solutions [18] when we drop the scalar contribution to the gauge current: D»F% = 0 D»D^

+ ^ ^

= 0 ,

(2.60) (2.61)

As \x\ —> oo we want the gauge field to go to pure gauge and the scalar field to go to its VEV (up to a gauge transformation): A„{x) - iU{x)dlxU{x)^ <jp -> U(x){<jt) ,

(2.62) (2.63)

where {(j>>) is a vacuum solution. For small (p < l/(gv)) instantons with a completely broken gauge symmetry we find Euclidean actions: Sinst = S S^ = 8ir2p2v2 .

(2.64) (2.65)

Integrating over instanton locations and sizes we find d^xa I

-^e~s''nst~s*

\b = Jdix0J%(pA)°e-^-^-,

(2.66)

which is dominated at p2

= 1 6 ^ •

^

Thus the integration is convergent: breaking the gauge symmetry provides an exponential infrared cutoff. Note that since A^ is related to an element of the gauge group at \x\ —> oo, the topological character of the instanton relies on U : S3 - • SU{2) c G

(2.68)

If the scalar fields break the gauge group G down to H, then there will still be pure instanton in the H gauge theory if SU(2) C H. If the instantons in G/H can be gauge rotated into SU(2) C H, then all G instanton effects can be accounted for by the effective theory through H instantons. If not, we must add new interactions in the effective theory in order to match the

361

physics properly. Examples of when this is necessary [18] include SU(N) breaks completely SU(N) - • £7(1) SU(N) x SU(N) ->

SU(N)diag

SU(N) -> SO(N) This obviously happens whenever there are a different number of zero modes for G and H instantons. In general new interactions must be included in the effective theory when n^G/H) is non-trivial. 3. Gaugino Condensation We now turn to the chiral symmetry transformations and condensation of gauginos. Note that in pure SU(N) SUSY Yang-Mills (that is with F = 0 flavors, so the only fermion is a gaugino) the U(1)R symmetry is broken by instantons. We can see this by calculating the mixed triangle anomaly between one (7(1).R current and two gluons (see Figure 1). The anomaly is proportional to a group theoretical factor (the anomaly index) which arises by summing (tracing) over all possible fermions in the loop taking in to account the Abelian and non-Abelian charges. In this example the anomaly index is given by the i?-charge of the gaugino (which is 1) times the index of the adjoint representation (T(Ad) = N), so the anomaly is non-vanishing. Because of the anomaly, the chiral rotation Xa

_> eia>a

j

(3 ^

is equivalent to a shift in the coefficient of the CP violating term in the the action, Fa^vF^v, OYM

- • OYM - 2Na .

(3.2)

The factor 27V arises because the gaugino Aa is in the adjoint representation of the gauge group and thus has 2N zero-modes in a one-instanton background. Thus the chiral rotation is only a symmetry when

where k is an integer, so the U(1)R symmetry is explicitly broken down to a discrete Z2N subgroup. Treating the holomorphic gauge coupling T as a

362

background (spurion) chiral superfield we can define a spurious symmetry given by Xa

_^ eiaxa

j

T-+T+

,

(3.4)

7T

which leaves the p a t h integral invariant. Assuming t h a t SUSY Yang-Mills has no massless particles, just massive color-singlet composites, t h e n holomorphy a n d symmetries determine t h e effective superpotential t o be: T T r

Q

2-KJT

Weff = a/Te = aA3 .

N

(3.5)

This is the unique form because under the spurious U(1)R rotation (3.4) the superpotential (which has J?-charge 2) transforms as WeS

- • e2iaWeS

.

(3.6)

Since the holomorphic gauge coupling parameter r transforms linearly under the spurious U(1)R rotation, it must appear an exponential with the coefficient given in (3.5). Again treating r as a background chiral superfield, t h e n in the SUSY Yang Mills action (1.34) the F component of T (which we will refer to as FT) acts as a source for A a A a . W i t h our assumption t h a t there are no massless degrees of freedom, the effective action at low energies is given by just the effective superpotential (3.5). T h u s t h e gaugino condensate is given by (AaAa> = 1671-1^-1112 oFT

'wj

167TJ—- / =

d26WeS

d lQni-pr-Wes OT

= 167TI-—a/i 3 e w .

(3.7)

Dropping the non-perturbative corrections 7 to the running of r and plugging b = 3AT into (1.56) we find
Which only contribute a phase.

3

,

(3.8)

363

thus there is a non-zero gaugino condensate 8 . The presence of this condensate means that the vacuum does not respect the discrete ZN symmetry since (A a A a )-*e 2 i a (A a A a ) ,

(3.9)

is not invariant for all possible values of a = kir/N. In fact it is only invariant for k = 0 or k = N. So we see that Z^JV symmetry is spontaneously broken to Z2, and that there should be TV degenerate but distinct vacua. It is easy to check that the symmetry transformation OYM —• $YM + 2-7T sweeps out N different values for (AaA°). It should be kept in mind that at this point we have not justified the assumption of no massless degrees of freedom which is crucial to the calculation. 4. The Affleck-Dine-Seiberg Superpotential 4.1. Symmetry

and

Holomorphy

Consider SU(N) SUSY QCD with F flavors (that is there are 2NF chiral supermultiplets) where F < N. We will denote the quarks and their superpartner squarks that transform in the SU(N) fundamental (defining) representation by Q and $ respectively, and use Q and $ for the quarks and squarks in the anti-fundamental representation. The theory has an SU(F) x SU(F) x U{1) x U(1)R global symmetry. The quantum numbers of the chiral supermultiplets are summarized in the following table 9 where • denotes the fundamental representation of the group. SU{N)

SU(F)

SU(F)

17(1)

U{1)R

•

•

1

1

1

D

-1

F-N F F-N F

(4.1)

The SU{F) x SU{F) global symmetry is the analog of the SU(3)L x SU{3)R chiral symmetry of non-supersymmetric QCD with 3 flavors, while the U(l) is the analog 10 of baryon number since quarks (fermions in the fundamental representation of the gauge group) and anti-quarks (fermions in the anti-fundamental representation of the gauge group) have opposite charges. There is an additional ^ ( 1 ) ^ relative to non-supersymmetric QCD since in the supersymmetric theory there is also a gaugino. 8

For related work using instantons, see ref. [19-21]. As usual only the .R-charge of the squark is given, and R[Q] = R[&] — 110 U p to a factor of N.

9

364

Recall that the auxiliary Da fields for this theory are given by (4.2)

where j is a flavor index that runs from 1 to F, m and n are color indices that run from 1 to iV, the index a labels an element of the adjoint representation, running from 1 to N2 — 1, and Ta is a gauge group generator. The D-term potential is given by: V

1 a a -D D 2g2

(4.3)

Classically there is a (D-flat) moduli space (space of VEVs where the potential V vanishes) which is given by \

vi

Vp

($ ) = ($)

0

(4.4)

\0 ... 0 J where ($) is a matrix with iV rows and F columns and we have used global and gauge symmetries to rotate (<&) to a simple form. At a generic point in the moduli space the SU(N) gauge symmetry is broken to SU(N — F). There are N2 -1-({N-

F)2 - 1) = 2NF - F2

(4.5)

broken generators, so of the original 2NF chiral supermultiplets only F2 singlets are left massless. This is because in the supersymmetric Higgs mechanism a massless vector supermultiplet "eats" an entire chiral supermultiplet to form a massive vector supermultiplet. We can describe the remaining F2 light degrees of freedom in a gauge invariant way by an F x F matrix field M/

Wr%r,

(4.6)

where we sum over the color index n. Since M can be written as a chiral superfield which is a product of of chiral superfields11, then, because of 11

That is there is a fermionic superpartner of M (the "mesino") given by M^, = Q

<£„

365

holomorphy, the only renormalization of M is the product of wavefunction renormalizations for $ and $. The axial U(1)A symmetry which assigns each quark a charge 1 is explicitly broken by instantons, while the U(1)R symmetry remains unbroken. To check this we can calculate the corresponding mixed anomalies between the global current and two gluons. For U(1)R we multiply the i?-charge by the index of the representation, and sum over fermions. The gaugino contributes 1 • N while each of the 2F quarks contributes ^(F~^N — 1). Adding these together we find that the mixed anomaly for (7(1) .R vanishes: ARgg

\ 1

(F-N

= N+{—___ij-=o.

(4.7)

For the (7(1)^ anomaly, the gauginos do not contribute since they have no (7(1)A charge and we find the anomaly coefficient is non-vanishing: AAgg

= l-2F-1-.

(4.8)

To keep track of selection rules arising from the broken U(1)A we can define a spurious symmetry in the usual way. The transformations Q -* eiaQ , Q - • eiaQ , GYM

-+ GYM + 2Fa ,

(4.9)

leave the path integral invariant. Under this transformation the holomorphic intrinsic scale (1-42) transforms as Ab

_^

ei2FaAb

_

(410)

To construct the effective superpotential we can make use of the following chiral superfields: Wa, A, and M. Their charges under the U(1)R and spurious U{1)A symmetries are given in the following table. U(1)A U(1)R 0 2 ww b 2F 0 A 2F 2{F - N) detM a

a

Note that detM is the only SU(F) x SU(F) invariant we can make out of M. To be invariant, a general non-perturbative term in the Wilsonian superpotential must have the form Abn(WaWa)m(detM)p

.

(4.12)

366

As usual to preserve the periodicity of 8YM we can only have powers of A6 (for m = 1 we can still have the perturbative term b log AW a W a because the change in the path integral measure due to the anomaly). Since the superpotential is neutral under U(1)A and has charge 2 under U(1)R, the two symmetries require: 0 = n2F + p2F 2 = 2m + p2(F-

.

N) .

(4.13) (4.14)

The solution of these equations is n

(4 15)

= -P=W^F-

'

Since b = 3N — F > 0 we can only have a sensible weak-coupling limit (A —• 0) if n > 0, which implies p < 0 and (because N > F) m < 1. Since WaWa contains derivative terms, locality requires m > 0 and that m is integer valued. In other words, since we trying to find a Wilsonian effective action (which corresponds to performing the path integral over field modes with momenta larger than a scale /i) which is valid at lowenergies (momenta below n) it must have a sensible derivative expansion. So there are only two possible terms in the effective superpotential: m = 0 and m = 1. The m = 1 term is just the tree-level field strength term. The coefficient of this term is restricted by the periodicity of #YM to be proportional to b In A. So we see that the gauge coupling receives no nonperturbative renormalizations. The other term (m = 0) is the Affleck-DineSeiberg superpotential 12 : / \3N-F\

W

=C

Trh?

F

™ »' {li*M)

'

(4 16)

-

where CN,F is in general renormalization scheme dependent. 4.2. Consistency

of

WADS-'

Moduli

Space

We can check whether the Affleck-Dine-Seiberg superpotential is consistent by constructing effective theories with fewer colors or flavors by going out in the classical moduli space or by adding mass terms for some of the flavors. Consider giving a large vacuum expectation value (VEV), v, to one flavor. This breaks the gauge symmetry to SU(N - 1) and one flavor is partially 12

First discussed by Davis, Dine, and Seiberg [22] and explored in more detail by Affleck, Dine, and Seiberg in [23]

367

"eaten" by the Higgs mechanism (since there are 2N — 1 broken generators) so the effective theory has F—l flavors. There are 2F—1 additional gauge singlet chiral supermultiplets left over as well since 2NF - (2JV - 1) - (2F - 1) = 2(N - 1)(F - 1) .

(4.17)

We can write an effective theory for the SU(N — 1) gauge theory with F — l flavors (since the gauge singlets only interact with the fields in the effective gauge theory by the exchange of heavy gauge bosons they must decouple from the gauge theory at low energies, i.e. they interact only through irrelevant operators with dimension greater than 4). The running holomorphic gauge coupling, g^, in the low-energy theory is given by 871-2

u

( A*

i

where bi is the standard /3-mnction coefficient of the low-energy theory bL = 3(N-l)-(F-l)

= 3N-F-2

,

(4.19)

and AL is the holomorphic intrinsic scale of the low-energy effective theory \AL\eie™/bL

AL =

= iie2viTL/bL

,

(4.20)

This coupling should match onto the running coupling of the high-energy theory 8TT2

(5) '

•6 In

52(M)

at the scale of the heavy gauge boson threshold 2

8TT

gi(v)-

_

^ 13

v.

2

8TT

gl(v)+C>

(4.22)

where c represents scheme dependent corrections, which vanish in the DR renormalization scheme 14 . So we have A\ f e

(KL^H

13 The fact that the heavy gauge boson mass is v rather than gv is a result of having an unconventional normalization for the quark and squark fields, this is necessary to maintain gL and A^ as holomorphic parameters. For related discussions on this point see [3,4,13]. 14 DR uses dimensional regularization through dimensional reduction with modified minimal subtraction [24,25].

368 A3iV-F

li

^2

-

A3N-F-2 iV-l,F-l '

— A

/ 4 2 ON ^.Z«3J

where we have started labeling (for later convenience) the intrinsic scale of the low-energy effective theory with a subscript showing the number of colors and flavors in the gauge theory that it corresponds to: A J V - I , F - I = A^. If we represent the light (F — l ) 2 degrees of freedom (corresponding to the gauge invariant combinations of the chiral superfields that are fundamentals under the remaining gauge symmetry) as an (F — 1) x (F — 1) matrix M then we have detM = v 2 detM + . . . ,

(4.24)

where . . . represents terms involving the decoupled gauge singlet fields. Plugging these results into the Affieck-Dine-Seiberg superpotential for N colors and F flavors (which we denote by WADS(N, F)) and using

(which follows from equation (4.23)) we reproduce WADS(N — 1,F — 1) provided that CJV,F is only a function oi N — F. Giving equal VEVs to all flavors we break the gauge symmetry from SU(N) down to SU(N - F), and all the flavors are "eaten". Through the same method of matching running couplings,

we then have

ir-w A3JV-F V2F

l(N-F) '~F)

(4.26)

,3(JV-F) iV-F,0 '

A

(4.27)

So the effective superpotential is given by Weff — CN,F-^-N-F,0

i

(4.28)

which agrees with the result (3.5) derived from holomorphy arguments for gaugino condensation in pure SUSY Yang-Mills, as described in section 3. 4.3.

Consistency

O / W A D S - ' Mass

Perturbations

Now consider giving a mass, m, to one flavor. Below this mass threshold the low-energy effective theory is an SU(N) gauge theory with F-1 flavors.

369

Matching the holomorphic gauge coupling of the effective theory to that of the underlying theory at the scale m gives: (^ ) b \m. )

V J

(AL

\m

F+l

A 3 7 V " -F _ A3N-

(4.29)

-1

Using holomorphy the superpotential must have the form / A3N~F\ Wexact = ( - ^ )

wzr¥

/(*) ,

(4.30)

where

and /(£) is an as yet undetermined function. Note that since mMf is actually a mass term in the underlying superpotential, it has U(1)A charge 0, and i?-charge 2, so t has i?-charge 0. Taking the weak coupling, small mass limit A —• 0, m —> 0, we must recover our previous results with the addition of a small mass term, hence f(t) = CN,F+t.

(4.32)

However in this double limit t is still arbitrary so this is the exact form of f(t). Thus we find / \3N-F\

W^F

Wexact = CN,F (-^f)

+ mMf

.

(4.33)

The equations of motion for Mf and MF -8Mr dWexa, dMF (where coi(Mf)

=

CN F

{-N^F)+m

' {-detM)

, -1

s^-ry^

}

C

°

= °>

(4 34)

W )

(4 351

is the cofactor of the matrix element Mf)

- 0

'

imply that

F

and that the cofactor of Mf is zero. Thus M has the block diagonal form M=(™

°F)

,

I 0 MfJ '

(4.37) K

!

370

where M is an (F — 1) x (F — 1) matrix. Plugging the solution (4.36) into the exact superpotential (4.33) we find We^N,F-l) = (N-F+l)^)

( ^ ^ ^ j

(4.38)

which is just WADS up to an overall constant. Thus,for consistency, we have a recursion relation: C W . F - 1 = (N-F

+ 1) ( j ^ j

""'+1

•

(4-39)

An instanton calculation is reliable for F = N — 1 since the non-Abelian gauge group is completely broken, and such a calculation [20,25] determines CN,N-I = 1 in the DR scheme, and hence CNtF = N -F

.

(4.40)

We can also consider adding masses for all the flavors. Holomorphy determines the superpotential to be Wexact = CN,F

( -

^

detM

)

+ T^M?

,

(4.41)

where mj is the quark (and squark) mass matrix. The equation of motion for M gives / \3N-F\

Mt = (m-1n

W=F

-r-TT) V detM

•

(4-42)

Taking the determinant and plugging the result back in to (4.42) gives &$i = M{ = (m-l){

(Aetmh3N-FY

.

(4.43)

Note that since the result involves the Ar-th root there are N distinct vacua that differ by the phase of M. This is in precise accord with the Witten index argument [26]. Matching the holomorphic gauge coupling at the mass thresholds gives A 3 i V - i ? detm = A ^ r 0 .

(4.44)

So Weff = NA3Nfi

,

(4.45)

371

which agrees with our holomorphy result 15 , equation (3.5), for pure SUSY Yang-Mills and determines the parameter a = N up to a phase. So the gaugino condensate (3.8) is given by (AaA°) = -32n2e2*ik'N

A%fi ,

(4.46)

where k — 1...N. Thus starting with F = N — 1 flavors we can consistently derive the correct Affleck-Dine-Seiberg effective superpotential for 0 < F < N — 1, and gaugino condensation for F = 0. This retroactively justifies the assumption that there was a mass gap in SUSY Yang Mills. 4.4. Generating

W A D S from

Instantons

Recall that the Affleck-Dine-Seiberg superpotential WADS OC AT^F ,

(4.47)

while instanton effects are suppressed by 16 e-sinst

^ Ab _

(448)

Sc for F = N — 1 is is possible that instantons can generate WADS- Since SU{N) can be completely broken in this case, we can do a reliable instanton calculation. When all VEVs are equal the ADS superpotential predicts quark masses of order d2WADS

A2N+l

(4.49)

v2N

d^idtf and a vacuum energy density of order 8WADs d$i

2

A2^1 „2JV-1

2

(4.50)

Looking at a single instanton vertex we find 2N gaugino legs (corresponding to 2N zero-modes) and 2F = 2N - 2 quark legs, as shown in Figure 2. All the quark legs can be connected to gaugino legs by the insertion of a scalar VEV. The remaining two gaugino legs can be converted to two quark legs by the insertion of two more VEVs. Thus a fermion mass is generated.

15 16

Which assumed a mass gap. See Eq. (1.51).

372

2N-2

Figure 2. Instanton with 2N-2 quark legs (solid, straight lines) and 2N gaugino legs (wavy lines), connected by 2N squark VEVs (dashed lines with crosses).

Prom the instanton calculation we find the quark mass is given by m ~ e

_s^ 2 /„^„i /S'(P)

V

v2N

p2JV-l

b

V2N p2N-l

2^V+1

-)

V2N n2iV-l

(4.51)

PJ The dimensional analysis works because the only other scale in the problem is the instanton size p, and the integration over p is dominated by the region around b Forcing the quark legs to end at the same space time point generates the F component of M, and hence a vacuum energy of the right size. From our previous arguments we recall that we can derive the ADS superpotential for smaller values of F from the case F = N — 1, so in particular we can derive gaugino condensation for zero flavors from this instanton calculation with N — 1 flavors.

373

4.5. Generating

W A D S from

Gaugino

Condensation

For F < N — 1 instantons cannot generate WADS since at a generic point in the classical moduli space with detM ^ 0 the SU(N) gauge group breaks to SU(N — F) D SU(2). Matching the gauge coupling in the effective theory at a generic point in the classical moduli space gives A3N-F

= AHN-F)detM

(453)

In the far infrared the effective theory splits into and SU(N — F) gauge theory and F2 gauge singlets described by M. However these sectors can be coupled by irrelevant operators. Indeed they must be, since by themselves the SU(N-F) gauginos have an anomalous R symmetry. The R symmetry of the underlying theory was spontaneously broken by squark VEVs but it should not be anomalous. An analogous situation occurs in QCD where the SU(2)L x SU(2)R chiral symmetry is spontaneously broken and the axial anomaly of the quarks is reproduced in the low-energy theory by an irrelevant operator (the Wess-Zumino term [27]) which manifests itself in the anomalous decay ir° —> 77. In SUSY QCD the correct term is in fact present since the effective holomorphic coupling in the low-energy effective theory,

depends on In detM through the matching condition (4.53). The relevant term in the low-energy Lagrangian is (factoring out 3(iV — F)/(32ir2))

J d26 In detMWaWa + h.c. = [ r r ( F M M - 1 ) A a A a + A r g ( d e t M ) F a ^ F ; i / + . . . ] + h.c.

(4.55)

where FM is the auxiliary field (i.e. the coefficient of 92 in superspace notation) for M. The second term can be seen to arise through triangle diagrams involving the fermions in the massive gauge supermultiplets. Note Arg(detM) transforms under a chiral rotation in the correct manner to be the Goldstone boson of the spontaneously broken R symmetry: Arg(detM) -> Arg(detM) + 2Fa . The equation of motion for FM gives dW FM

~dM =

M-l(XaXa)

(4.56)

374

M-1A%_Ffi

oc

/ \3N-F\

7T=F

This gives a vacuum energy density that agrees with the Affleck-DineSeiberg calculation. This potential energy implies that a non-trivial superpotential was generated for M, and since the only superpotential consistent with holomorphy and symmetry is WADS we can conclude that for F < N — 1 flavors gaugino condensation generates WADS • 4.6. Vacuum

Structure

Now that we believe WADS is correct what does it tells us about the vacuum structure of the theory? It is easy to see that

y = Y,

dW dQi

dW

Wi = J2\Fi\ + \Fi\2, 2

2

(4.58)

i

is minimized as detM —> oo, so there is a "run-away vacuum", or more strictly speaking no vacuum. A loop-hole in this argument would seem to be that we have not included wavefunction renormalization effects, which could produce wiggles or even local minima in the potential, but it could not produce new vacua unless the renormalization factors were singular. It is usually assumed that this cannot happen unless there are particles that become massless at some point in the field space, which would also produce a singularity in the superpotential. This is just what happens at detM = 0, where the massive gauge supermultiplets become massless. So we do not yet understand the theory without VEVs. 5. 't Hooft's Anomaly Matching 't Hooft made one of the most important advances in understanding strongly coupled theories with composite degrees of freedom by pointing out that the anomalies of the constituents and the composites must match [28]. Consider an asymptotically free gauge theory, with a global symmetry group G. We can easily compute the anomaly for three global G currents in the ultraviolet by looking at triangle diagrams of the fermions. We will call the result Auv. Now imagine that we weakly gauge G with a

375

gauge coupling g < 1. If Auv ^ 0, then we can add some spectators that only have G gauge couplings, they can be chosen such that their G anomaly is As = -Auv, so the total G anomaly vanishes. Now construct the effective theory at a scale less than the strong interaction scale. If we compute the G anomaly at this scale we add up the triangle diagrams of light fermions, which consist of the spectators and strongly interacting or composite fermions. If G is not spontaneously broken by the strong interactions its anomaly must still vanish 17 , so 0 = AIR + As .

(5.1)

Thus we have AIR

= Auv

.

(5.2)

Taking g —> 0 decouples the weakly coupled gauge bosons but does not change the three point functions of currents. 6. Duality for S U S Y QCD Theoretical effort in the mid 1990s (mainly due to Seiberg [8,29]) led to a dramatic break-through in the understanding of strongly coupled Af = 1 SUSY gauge theories 18 . After this work we now have a detailed understanding of the infrared (IR) behavior of many strongly-coupled theories, including the phase structure such theories. The phase of a gauge theory can be understood by considering the potential V(R) between two static test charges a distance R apart 1 9 . Up to an additive constant we expect the functional form of the potential will fall into one of the following categories: Coulomb :

V(R) ~ ^

free electric :

V(R) ~

flln(RA)

free magnetic : V(R) ~ ]^)Higgs :

V(R) ~ constant

confining :

V(R) ~ crR .

(6.1)

17 If G is spontaneously broken the anomaly will still be reproduced by the interactions of the Goldstone bosons, this is the origin of the Wess-Zumino term [27]. 18 For other review of these developments see [9,30,31]. 19 Holding the charges fixed for a time T corresponds to a Wilson loop of area T • R.

376

The explanation of these functional forms is as follows. In a gauge theory where the coupling doesn't run (e.g. at an IR fixed point or in QED at energies below the electron mass) then we expect to simply have a Coulomb potential. In a gauge theory where the coupling runs to zero in the IR (e.g.. QED with massless electrons) there is an inverse logarithmic correction to squared gauge coupling and hence to the potential. Since electric and magnetic charges are inversely related by the Dirac quantization condition, the squared charge of a static magnetic monopole grows logarithmically in the IR due to the renormalization by loops of massless electrons. Using electricmagnetic duality to exchange electrons with monopoles one finds that the logarithmic correction to the potential for static electrons renormalized by massless monopole loops appears in the numerator since the coupling grows in the IR. In a Higgs phase the gauge bosons are massive so there are no long range forces. In a confining phase 20 we expect a tube of confined gauge flux between the charges, which, at large distances, acts like a string with constant mass per unit length, and thus gives rise to a linear potential. The familiar electric-magnetic duality exchanges electrons and magnetic monopoles and hence the free electric phase with the free magnetic phase. Mandelstam and 't Hooft [32] conjectured that duality could also exchange the Higgs and confining phases and that confinement can be thought of as a dual Meissner effect arising from a monopole condensate 21 . Electricmagnetic duality also exchanges an Abelian Coulomb phase with another Abelian Coulomb phase. Seiberg [9,29] conjectured that analogs of the first and last of these dualities actually occur in the IR of non-Abelian J\f = 1 SUSY gauge theories and demonstrated that his conjecture satisfies many non-trivial consistency checks.

6.1. The Classical Moduli Space for F > N Consider SU(N) SUSY QCD with F flavors and F>N. This theory has a global SU(F) x SU(F) x t/(l) x U(1)R symmetry. The quantum numbers 22

20 More precisely a confining phase with area law confinement. Note however that with dynamical quarks in the fundamental representation of the gauge group we can produce quark anti-quark pairs which screen the test charge, so there is no long range force, that is the flux tube breaks. 21 Seiberg and Witten later showed that this is actually the case in certain Af = 2 SUSY gauge theories [33]. 22 As usual only the .R-charge of the squark is given, and R[Q] = R[&] — 1.

377

of the squarks and quarks are summarized in table 6.2. SU{N)

SU(F)

SU(F)

C/(l)

U(1)R

*, Q

•

D

1

1

$, Q

3

1

D

-1

F-N F F-N F

(6.2)

The SU{F) x SU{F) global symmetry is the analog of the SI7(3) L x SU(3)R chiral symmetry of non-supersymmetric QCD with 3 flavors, while the U(l) is the analog 23 of baryon number since quarks (fermions in the fundamental representation of the gauge group) and anti-quarks (fermions in the anti-fundamental representation of the gauge group) have opposite charges. There is an additional U(1)R relative to non-supersymmetric QCD since in the supersymmetric theory there is also a gaugino. Recall that the .D-terms for this theory are given in terms of the squarks by Da = g(*jn{Ta)™<5>ri

•* 3 "n;i),

(6.3)

where j is a flavor index that runs from 1 to F, m and n are color indices that run from 1 to AT, the index a labels an element of the adjoint representation, running from 1 to N2 — 1, and Ta is a gauge group generator. The D-term potential is: V

1 a a -D D 2g2

(6.4)

where we sum over the index a. Define (6.5) (6.6) Dm and Dm are N x N positive semi-definite Hermitian matrices. In a SUSY vacuum state the vacuum energy vanishes and we must have: Da = gTZm(Dm

-Dm)=0

(6.7)

.Since Ta is a complete basis for traceless matrices, we must have that the second matrix is proportional to the identity: Dn Up to a factor of N.

-D

Pi

(6.8)

378

D^ can be diagonalized by an SU(N) gauge transformation (6.9)

D' = U^DU , so we can take -D" to have the form:

/hi

\ «2

D =

(6.10)

KIV

V

In this basis, because of Eq. (6.8), Dm must also be diagonal, with eigenvalues \vi\2. This tells us that

k l 2 = \vi\2 + p

(6.11)

Since D^ and Dm are invariant under flavor transformations, we can use SU(F) x SU(F) flavor transformations to put ($) and ($) in the form 0...0\

Vl

(6.12)

<*> VJV

0 ... 0 /

«i

($)

0 ... 0

3.13)

\ ° - °/ Thus we have a space of degenerate vacua, which is referred to as a moduli space of vacua. The vacua are physically distinct since, for example, different values of the VEVs correspond to different masses for the gauge bosons. With a VEV for a single flavor turned on we break the gauge symmetry down to SU(N — 1). At a generic point in the moduli space the SU(N) gauge symmetry is broken completely and there are 2NF— (N2 — 1) massless* chiral supermultiplets left over. We can describe these light degrees of freedom in a gauge invariant way by scalar "meson" and "baryon" fields and their superpartners: M? = Wn<$>ni ,

(6.14)

379

(6.15) = $

(6.16)

The fermion partners of these fields are the corresponding products of scalars and one fermion. There are constraints relating the fields M and B, since the M has F2 components, B and B each have

components, N and all three are constructed out of the same 2NF underlying squark fields. For example, at the classical level, there is a relationship between the product of the B and B eigenvalues and the product of the non-zero eigenvalues of M: Mj71 . . . M3"

,Bn'

Biu

(6.17)

where [] denotes antisymmetrization. Up to flavor transformations the moduli can be written as: \

fvivi

vNvN

{M) =

0/

\ (BI,...,N)

(B

(6.18)

=VI...VN

)=vi

VN

(6.19) (6.20)

with all other components set to zero. We also see that the rank of M is at most N. If it is less than N, then B or B (or both) vanish. If the rank of M is k, then SU(N) is broken to SU(N - k) with F -k massless flavors. 6.2. The Quantum

Moduli

Space for F > N

Recall that the ADS superpotential made no sense for F > N however the vacuum solution

Ml = {m~l)i (detmA3Ar-F)

3.21)

is still sensible. Giving large masses, m # , to flavors N through F and matching the gauge coupling at the mass thresholds gives

A3JV-^detmff = A 2 / + V

(6.22)

380

The low-energy effective theory has N — 1 flavors and an ADS superpotential. If we give small masses, mi, to the light flavors we have M( = {nq})\

(detmLA2NN^1)Jr

= (m^(detmLdetmHA3iv-F)* .

(6.23)

Since the masses are holomorphic parameters of the theory, this relationship can only break down at isolated singular points, so equation (6.21) is true for generic masses and VEVs. For F > N we can take m*- —> 0 with components of M finite or zero. So the vacuum degeneracy is not lifted and there is a quantum moduli space [34] for F > N, however the classical constraints between M, B and B may be modified. Thus we can parameterize the quantum moduli space 24 by M, B and B. When these fields have large values (with maximal rank) then the squark VEVs are large (compared to A) and the gauge theory is broken in the perturbative regime. As the meson and baryon fields approach zero (the origin of moduli space) then the gauge couplings become stronger, and at the origin we would naively expect a singularity since the gluons are becoming massless at this point. We shall see that this expectation is too naive.

6.3. Infrared Fixed

Points

For F > 3N we lose asymptotic freedom, so the theory can be understood as a weakly coupled low-energy effective theory. For F just below 3iV we have an infrared fixed point. This was pointed out by Banks and Zaks [36] as a general property of gauge theories. By considering the large N limit of SU(N) with F/N infmitesimally below the point where asymptotic freedom is lost they showed that the /? function has a perturbative fixed point. Here we will apply their argument to SUSY QCD. Novikov, Shifman, Vainshtein and Zakharov [37] have shown that the exact (3 function for the running canonical 25 gauge coupling is given by

24

There is a theorem which shows that in general a moduli space is parameterized by the gauge invariant operators, see [35]. 25 For discussions of how the holomorphic gauge coupling is related to the canonical gauge coupling see [3,4,13,14].

381

where 7 is the anomalous dimension of the quark mass term. In perturbation theory one finds

-< = -h—+0^a2

<6-25>

N2-l

So 2

16vr /3(g) = - 5

3

o5 / (3iV - F) - -^ (w2

+0(g7)

-FN-

N2 — 1 F ^ -

.

(6.26)

Now take the number of flavors to be infinitesimally close to the point where asymptotic freedom is lost. For F = 2>N — eN we have

L

167r2/3(ff) = -g\N

- f j (3(iV2 - 1) + 0(e)) 8TT 2 7

+0(g ) . (6.27) So there is an approximate solution of the condition (3 = 0 where there first two terms cancel. This solution corresponds to a perturbative infrared (IR) fixed point at

£ = *rw=i*>

(6 28)

-

7

and we can safely neglect the 0(g ) terms since they are higher order in e. Without any masses this fixed point gauge theory theory is scale invariant when the coupling is set to the fixed point value g = g*- A general result of field theory is that a scale invariant theory of fields with spin < 1 is actually conformally invariant [38]. In a conformal SUSY theory the SUSY algebra is extended to a superconformal algebra. A particular .R-charge enters the superconformal algebra in an important way, we will refer to this superconformal i?-charge as Rsc- Mack found [39] that in the case of conformal symmetry there are lower bounds on the dimensions of gauge invariant fields. In the superconformal case 26 it was shown for A/" = 1 by Flato and Fronsdal [40] (and for general M by Dobrev and Petkova [41]) that the dimensions of the scalar component of gauge invariant chiral and anti-chiral superfields are given by 3 d = -Rsc, for chiral super fields , 3 d = — R s c , foranti — chiral superfields . 'A brief review is given in the next section.

(6.29) (6.30)

382

Since the charge of a product of fields is the sum of the individual charges, RSc[0102]

= R^ld]

+ RSC[02] ,

(6.31)

we have the result that for chiral superfields dimensions simply add: Z?[0i<9 2 ] = D[Oi] + D[02] .

(6.32)

This is a highly non-trivial statement since in general the dimension of a product of fields is affected by renormalizations that are independent of the renormalizations of the individual fields. In general the R symmetry of a SUSY gauge theory seems ambiguous 27 , since we can always form linear combinations of any U(1)R with other U(l)'s, but for the fixed point of SUSY QCD i? sc is unique since we must have RSC[Q] = RSC[Q] .

(6.33)

Thus we can identify the R charge we have been using (as given in table 6.2) with Rsc. If we denote the anomalous dimension of the squarks at the fixed point by 7* then the dimension of the meson field at the IR fixed point is D[M] - D[*«] = 2 + T , =

\2{F~FN)

- 3 - ^ .

(e-34)

and the anomalous dimension of the mass operator at the fixed point is 37V 7* = 1 - — . (6.35) We can also check that the exact (3 function (6.24) vanishes: /3oc3A^-F(l-7»)=0 .

(6.36)

For a scalar field in a conformal theory we also have [39] D{4>) > 1 ,

(6.37)

with equality holding for a free field. Requiring that D[M] > 1 implies F > ^N .

(6.38)

Thus the IR fixed point is an interacting conformal theory for |iV < F < 3N. Such conformal theories have no particle interpretation, but anomalous dimensions are physical quantities. 27

A general procedure for determining the superconformal R symmetry was given by Intriligator and Wecht [42]

383

6.4. An Aside

on Swperconformal

Symmetry

The reader may skip this section on her first time through, and return to it later if she remains curious about how the results quoted above relating scaling dimensions to i?-charges were obtained. I will loosely follow the discussion of ref. [43]. The generators of the conformal group can be represented by Mpv = -iix^dy

-

xvd^)

Pp. = -id^, K^ - -i{x2dfj, D = ixada

2Xfj,xada)

,

(6.39)

where M^u are the ordinary Lorentz rotations/boosts, PM are the translation generators, K^ are referred to as the "special" conformal generators, and D is the dilation generator. (D is also know as the "dilatation" generator by those who like extra syllables.) In 4 spacetime dimensions, the conformal group is isomorphic to SO(A, 2) We would like to find the restrictions that can be placed on conformal field theory operators by constructing the unitary irreducible representations of the conformal group. The simplest method to perform this construction is not with the generators defined above but by a related set of generators that correspond to "radial quantization" in Euclidean space. These generators are defined with indices that runs from 1 to 4: % = Mjk

*u =\{Pj-Ki) D' = -\{P* + KQ) J =••^(Pj+K^

P

+ iM'jo

l Pi = -D- -(P0-K0)

.^ - > «'i = iPj+Kj) iM 0 l

K = :-D+ -(P0-K0),

(6.40)

where j = 1,2,3 and D' acts as the "Hamiltonian" of the "radial quantization" . The eigenstates of the "radial quantization" are in one to one correspondence with the operators of the conformal field theory. From these

384

definitions we see that D't =

-D'

P? = K'j .

(6.41)

An SO (4, 2) representation can be specified by its decomposition into irreducible representations of its maximal compact subgroup which is 50(4) x 50(2). The 50(4) « 5(7(2) x SU(2) representations are just the usual representations of the Lorentz group which can be specified by two half-integers (J,j) (Lorentz spins): (scalar) = (0,0) (spinor)= (vector) =

(l,o)

or (o,±)

(6.42)

1 1 2'2

The 50(2) representation is labeled by the eigenvalue of D' acting on the state (operator). I will denote this eigenvalue by —id where d is the scaling dimension of the operator. To complete the construction we need raising and lowering operators for the 50(2) group. We can choose a basis such that P' is a raising operator and therefore K' is a lowering operator. Then we can classify multiplets by the scaling dimension of the lowest weight state (the state annihilated by K'). The operator corresponding to the lowest weight state is also known as the primary operator. One can check (using the commutator [D',P'] = —iPL) that acting on the lowest weight operator with the raising operator P', gives a new operator (sometimes called a descendant operator) with scaling dimension d + 1. Unitarity requires that any linear combination of states have positive norm, so in particular the state aP'm\d,(ji,ji))+bP'n\d,U2,J2))

,

(6.43)

(with m ^ n) must have positive norm. Using the commutator [P'm, K'n] = -i(25mnD'

+ 2M'mn) ,

(6.44)

we find (\a\2 + \b\2)d + 2Re(a*b{d,(j1,j1)\iM^nn\d,(j2,j2))

>0 .

(6.45)

Given that iM'mn has real eigenvalues and restricting to \a\ = \b\ this condition can be written as d > ±(d,(ju~jl)\iM'mn\d,(J2,h))

•

(6.46)

385

To find the eigenvalues of iM'mn write (for fixed m, n) iM

Ln

= ^(Smadn0

~ 6mp5na)M'a0 ,

(6-47)

which can be re-written as M'mn = (V.M')mn

,

(6.48)

where V is the generator of 50(4) rotations in the vector representation: Va0mn = i(5ma5n/3 — Smf35na) ,

(6.49)

and A.B = ^AaPBa0

.

(6.50)

By a similarity transformation we can go to a Clebsch-Gordan basis where (V + M'), V, and M' are "simultaneous (commuting) quantum numbers" and use the relation 28 V.M = 1 [(V + M'f

- V2 - M'2]

(6.51)

From this one can show that that the most negative eigenvalue of V.M has a larger magnitude than the most positive eigenvalue, so the most restrictive bound from the inequality (6.46) is the case with the — sign. If our state |d, j'2, J2) corresponds to the representation r of 50(4), and r' is the representation with the smallest quadratic Casimir in the product rxV then we have d > \ [C2 (r) + C2 (V) - C2 (r')]

(6.52)

C 2 (V) = V.V

(6.53)

where

and so on. Using the fact that the 50(4) Casimir is twice the sum of the SU(2) Casimirs (i.e. C2(jJ) = 2(J 2 + J2) = 2(j(j + l)+](j + 1))), one can check that C2(scalar) = 0 3 C2{spinor) = -

(6.54)

C2 {vector) = 3 , 28 Readers of a certain age will recognize this technique from the calculation of the mostattractive channel in single gauge boson exchange [44], while all readers should recognize the calculation of the spin-orbit term.

386

and thus: d > 0 , (scalar) 3 d > — , (spinor)

(6.55)

d > 3 , (vector) . The first two bounds are perfectly reasonable since the identity operator is a scalar and has dimension 0, and a free (massless) spinor has dimension 3/2. The third bound may be a little surprising since a free massless vector (gauge) boson field has dimension 1, however such a field is not gauge invariant, and thus unitarity cannot be applied. A conserved current is a gauge-invariant vector field and does have dimension 3. Applying similar arguments to states with P[P'k acting on them, one finds for scalars d(d-l)>0,

(6.56)

which means that for scalar operators with d > 0 (i.e. operators other than the identity) d>1.

(6.57)

Turning to superconformal symmetry, first note that in addition to the usual (Euclidian) supersymmetry generators Q'ia (where a is a spinor index and i runs from 1 to M) there is also a superconformal generator S'^ such that Q ,f = S' .

(6.58)

Q' and S' can be chosen to be real Majorana (four component) spinors. There is also an iZ-symmetry which is 1/(1) for M = 1, U(2) for A/" = 2, and SU(4) for A/" = 4. In general we can take the matrix (Tij)pq = Sip5jq ,

(6.59)

as a generator of the full i?-symmetry, and R as the generator of the Abelian .R-symmetry (i.e. for M = 1 we have R = T n ; for A/" = 2, R = J^ Ty; while for M = 4 set R = 0). The /^-symmetry does not commute with the supersymmetry generators, to see this explicitly we need a few more definitions. Define the Euclidian T matrices in terms of the Lorentzian 7 matrices by r< = H ,

(6.60)

T 4 = -«7o ,

(6.61)

387

and choose a basis where Ta is real and hermitian Then we can define left-handed and right-handed projection operators by P- = i ( l - 7 5 ) ,

(6-62)

P+ = i ( l + 7 5 ) .

(6.63)

The convention is that (j,j) = (1/2, 0) corresponds to the fermion component of chiral supermultiplet which is projected to zero by P _ . Finally we can then write [Ta, Q'm\ = P+ Q[Sjm - P Q'3^m

(6.64)

[Ttj, S'm] = P+ S<8jm - P- S'jSjm

(6.65)

Most importantly for the unitarity bounds, the anticommutator of Q' and S"is: {Q'ia, S'jP} = i5-f[{M'mnTmTn)a0 -2{P+)aPTij

+ 2Sa0D'}

+ 2{P-)a0Tji

+-^(rfB)apR.

(6.66)

Now apply the requirement of unitarity to the state aQja|d,iJ)(jiJ1)>+6Q^|d,JR,(j2j2)> ,

(6-67)

(with a ^ /?, and for simplicity I will only consider TV = 1). The result is d>±(d,RJ(jl,jl)\l-(M^nTmTn)a0

- l(
-(6.68)

The operator inside the matrix element can be split into left and righthanded parts (i.e. proportional to P+ and P_): 4J.S - ^R\

+ P_ Uj.S

+ ^R\

,

(6.69)

where S is the rotation generator in the spin-half representation of the first SU(2) embedded in SO(4), and S is the corresponding generator of the second SU(2). The eigenvalues of 2J.S are —j — 1 and j for j > 0 and 0 for j = 0, so, as before, the most negative eigenvalues provide the strongest bounds. d > P+ Uj + 2 - 25j0

+ ^R\+P-(2]

+ 2 - 2S-J0

-

^R\

. (6.70)

388

Although P+ and P_ were denned in the superconformal algebra to act on spinors, they have been implicitly extended to act as projection operators on superpartners of spinors in the same way they act on the spinors themselves. The bound we have found above is not the whole story. The bound (6.70) is a necessary condition [43] but if either j or j are zero there are more restrictive conditions [40,41] that require: d > dmax = max (2j + 2 + ^R, 2j + 2 - ^Rj > 2 + j + j or d = | | i ? | .

(6.71)

Here dmax is the boundary of the continuous range, while the isolated point (which happen only in the supersymmetric case [40,41]) is achieved for chiral and anti-chiral supermultiplets. We can check the result (6.70) by considering a chiral supermultiplet with i?-charge Rsc. The primary field is the lowest component of the supermultiplet, which is just the scalar with (j,j) = (0,0). Equality is achieved in the necessary condition (6.70) on the scaling dimension of the scalar component at: d = \Rsc ,

(6.72)

in agreement with the results of [40,41]. It is this special case which is of interest for the chiral supermultiplet. The reason is that superconformal multiplets are generally much larger than supermultiplets (due to the existence of the S' generator), but the superconformal multiplets can degenerate ("shorten" in the language of ref. [45]) precisely at the point where some of the states have zero norm, which is the point where the bound is saturated. In other words, in a superconformal theory, chiral supermultiplets are "short" multiplets.

6.5.

Duality

In a conformal theory (even if it is strongly coupled) we don't expect any global symmetries to break, so 't Hooft anomaly matching should apply to any description of the low-energy degrees of freedom. The anomalies of the mesons and baryons described above do not match to those of the quarks and gaugino. However Seiberg [29] found a non-trivial solution to the anomaly matching using a "dual" SU(F — N) gauge theory with a "dual" gaugino, "dual" quarks and a gauge singlet "dual mesino" with the

389

following quantum numbers 29 : SU(F-

N)

SU{F)

SU{F)

C/(l)

U(1)R

N F-N N F-N

N F N F r> F-N Z F

n

1

q

n n

1

D

mesino

1

•

•

q

0

(6.73)

In the language of duality the dual quarks can be thought of as "magnetic" quarks, in analogy with the duality between electrons and magnetic monopoles. The anomalies of the two dual theories match as follows: SU{Ff

:-{F-N)

U(1)RSU(F)2 U{1)

N

N ^ < N-F/ri F -2N ArN l „ (F-N)+ — F F~ 2

U(1)SU{F)2

3

+F

1

AT 2

2F

:0

C/(l) : 0 U(1)U(1)2R : 0 N -F 2{F-N)FJ7(l) F = ~7V 2 N -F 2{F-N)FU(l) F 2N4 + N2 - 1 2

F

-2N F

F2 + (F-

F-2N

N)2 - 1

F2 + (F - N)2 - 1

F

U{l)2U{l)j

N ^F-NJ

N

-F F

2F(F - N) =

-2N2

(6.74)

This theory admits a unique superpotential: (6.75) where represents the "dual" squark (that is the scalar superpartner of the "dual" quark q) and M is the dual meson (the scalar superpartner of the "dual" mesino). This ensures that the two theories have the same number 29

As usual only the R-charge of the scalar component is given, and i?[fermion] R [scalar] — 1.

390

of degrees of freedom since the M equation of motion removes the color singlet (/>(/> degrees of freedom. The counting works because both theories have the same number of massless chiral superfields at a generic point in moduli space: 2FN - 2(N2 - 1) = 2FN - 2(N2 - 1) ,

(6.76)

where N = F — N is the number of colors in the dual theory. The dual theory also has baryon operators: b

i l , .•,IF-N

bn,.

• •AF

_

(

JJIF-NIF-N

Arllil

r n ^ i •

—N

,

.,TIF-N • •
ni

d> • ~rnF-NtF-N

e '-

(6.77) (6.78)

Thus the two moduli spaces have a simple mapping

Bilt. Bh>-

..,iN

•<-> C i i , . .•;iN,jl,-

...xN

^

e i u

,3FN

..JF-N'S

. •;iN,jl,-

•JF-NT.

•

JFN

(6.79)

The one-loop (3 function in the dual theory is (3(g) oc -53(37V - F) = -g3(2F

- 3JV) .

(6.80)

So the dual theory loses asymptotic freedom when F < 3N/2. In other words, the dual theory leaves the conformal regime to become infrared free at exactly the point where the meson of the original theory becomes a free field which implies very strong coupling for the underlying quarks and squarks. We can also apply the Banks-Zaks [36] argument to the dual theory [46]. When F = 37V - eiV

=

3

2 V

-(l+e-)N ) 6/

(6.81)

there is a perturbative! fixed point at 8TT2

=

A2

=

TV

3 ^ _ ! ( 16TT2 ~—f

3N

F

/

1

+

N

(6.82) (6.83)

At this fixed point D(M<j>4>) = 3, so the superpotential term is marginal. If the superpotential coupling A = 0, then M has no interactions (it is a free field) and therefore its dimension is 1. If the dual gauge coupling is set

391

close to the Banks-Zaks fixed point and A « 0 then we can calculate the dimension of (jxf) from the i? sc charge for F > 3N/2: D(H>) =

3(F - N) 3N K >= — < 2 . p

, (6.84)

So the superpotential is a relevant operator (not marginal) and thus there is an unstable fixed point at _a yq

8TT2

~2

N

= yqt = ———2 e * 3 N - 1 A2 = 0 .

(6.85) (6.86)

In other words the pure Banks-Zaks fixed point in the dual theory is unstable and the superpotential coupling flows toward the non-zero value given in (6.83). So we have found that not only does SUSY QCD have an interacting IR fixed point for 3N/2 < F < 3N there is a dual description that also has an interacting fixed point in the same region. The original theory is weakly coupled near F = 3N and moves to stronger coupling as F is reduced, while the dual theory is weakly coupled near F = 3N/2 and moves to stronger coupling as F is increased. For F < 3N/2 the IR fixed point of the dual theory is trivial (asymptotic freedom is lost in the dual): Qt = 0 2

A = 0.

(6.87) (6.88)

Since M has no interactions it has scaling dimension 1, and there is an accidental U(l) symmetry in the infrared. For this range of F, Rsc is a linear combination of R and this accidental U{1). This is consistent with the relation D{M) = (3/2)i? sc (M). Surprisingly in this range we find that the the IR is a theory of free massless composite gauge bosons, quarks, mesons, and their superpartners. We can lower the number of flavors to F = N + 2, but to go below this requires new considerations since there is no dual gauge group SU(F — N) when F = N + 1. We will examine what happens in this case in detail later on. To summarize, for 3./V > F > 3N/2 what we have found is two different theories that have IR fixed points that describe the same physics. For 3N/2 > F > N + 1 we have found that a strongly coupled theory and an IR free theory describe the same physics. Two theories having the same

392

IR physics are referred to as "being in the same universality class" by condensed matter physicists. This phenomenon also occurs in particle theory, a well known example of this is QCD and the chiral Lagrangian. Having two different description can be very useful if one theory is strongly coupled and the other is weakly coupled. (Two different theories could not both be weakly coupled and describe the same physics.) Then we can calculate non-perturbative effects in one theory by simple doing perturbative calculations in the other theory. Here we see that even when the dual theory cannot be thought of as being composites of the original degrees of freedom (since there is no particle interpretation of conformal theories it doesn't make sense to talk about composite particles) it still provides a weakly coupled description in the region where the original theory is strongly coupled. The name duality has been introduced because both theories are gauge theories and thus there is some resemblance to electric-magnetic duality and the Olive-Montonen duality [47] of Af = 4 SUSY gauge theories. However. Olive-Montonen duality is valid at all energy scales, while for these Af = 1 theories, as we go up in energy the infrared correspondence of Seiberg duality is lost.

6.6. Integrating

out a flavor

If we give a mass to one flavor in the original theory we have added a superpotential term Wmass = m ¥ F $ f .

(6.89)

In the dual theory we then have the superpotential Wd = XMffifa + mM£ ,

(6.90)

where we have made use of the mapping (6.79). Because of this mapping it is common (though somewhat confusing) to write ~ M XM = — ,

(6.91)

which trades the coupling A for a scale // and uses the same symbol, M, for fields in the two different theories that have the same quantum numbers. With this notation the dual superpotential is Wd = -Mffifa

+ mMf

.

(6.92)

393

The equation of motion for Mf is: dWd

dMf

1-rF

M

>F

+m =0

(6.93)

So the dual squarks have VEVs: -rF ,

(6.94)

-fj,m .

We saw earlier that along such a D-flat direction we have a theory with one less color, one less flavor, and some singlets. The spectrum of light degrees of freedom is: q' q' M' q" q"

s

Mf MF

Mf

SU(F -ND D 1 1 1 1 1 1 1

-1)

SU{FD 1 D D 1 1 D 1 1

1)

SU(F - 1) 1

•

•1 •

(6.95)

1 1

•1

The low-energy effective superpotential is:

WeS = - (tf)MF'; + (fcjMfJ* + Mfs) + -M'q

(6.96)

So we can integrate out M3F, ", Mf, 4> , Mf, and S since they all have mass terms in the superpotential. This leaves just the dual of SU(N) with F — 1 flavors which has a superpotential W = -M'4>'4>' .

(6.97)

Similarly one can check that there is a consistent mapping between the two dual theories when one flavor of the original squarks and anti-squarks have £>-flat VEVs, and the corresponding meson VEV gives a mass to the dual quarks and dual squarks, which can then be integrated out. The analysis of baryonic D-flat directions was done in ref. [48]. 6.7.

Consistency

We have seen that Seiberg's conjectured duality satisfies three non-trivial consistency checks:

394

• The global anomalies of the original quarks and gauginos match those of the dual quarks, dual gauginos, and "mesons". • The moduli spaces have the same dimensions and the gauge invariant operators match: M

^M t

tl,---,iN,3l,---3F-NLr

-Dll,...,8JV "

^ii,...,*,, ^

il €

--'iNdl-JF~Nbju...jFN

.

(6.98)

• Integrating out a flavor in the original theory results in an SU(N) theory with F — 1 flavors, which should have a dual with SU(F — N — 1) and F — 1 flavors. Starting with the dual of the original theory, the mapping of the mass term is a linear term for the "meson" which forces the dual squarks to have a VEV and Higgses the theory down to SU(F - N - 1) with F - 1 flavors. The duality exchanges weak and strong coupling and also classical and quantum effects. For example in the original theory M satisfies a classical constraint 30 rank(M) < N. In the dual theory there are F — rank(M) light dual quarks. If rank(M) > N then the number of light dual quarks is less than N = F — N, and an ADS superpotential is generated, so there is no vacuum with rank(M) > N. Thus in the dual, rank(M) < iV is enforced by quantum effects. 7. Confinement in SUSY QCD For certain special cases (particular values of the number of flavors) the description in terms of a dual gauge theory breaks down since there is no possible dual gauge group. Remarkably one finds that the low-energy effective theory simply contains mesons and baryons and that their properties can actually be derived from the case where there is a dual gauge group. 7.1. JF = N:

Confinement

with Chiral Symmetry

Breaking

For SUSY QCD with F = N flavors, Seiberg [34] found that all the 't Hooft anomaly matching conditions can be solved with just the color singlet meson and baryon fields, that is no dual quarks are required. Thus this theory is confining in the sense that all of the massless degrees of freedom are color singlet particles. Discussed at the end of Section 6.1.

395

Recall that the classical moduli space of SUSY QCD is parameterized by Ml = Wn
(7.1) nN

Bh,...,iN=^niil...^nNiNe '--' B

= $

...$

(7.2) £„!,..,„„ .

(7.3)

For F = N flavors the baryons are flavor singlets: B = ^-'ip

Bix_iF

(7.4)

B = eiu...tiFBiu-'iF

.

(7.5)

We also saw that classically these fields satisfy a constraint: detM = BB~ .

(7.6)

With quark masses turned on we have: (M/) = ( m - 1 ) ^ ( d e t m A 3 i V - F ) ^ .

(7.7)

Taking a determinant of this equation (and using F = N) we have det(M) = det (m" 1 ) det mA2N

= A2N ,

(7.8)

independent of the masses. However det m / 0 sets (B) = (B) = 0, since we can integrate out all the fields that have baryon number. Thus the classical constraint is violated! To understand what is going on it is helpful to use holomorphy and the symmetries of the theory. The flavor invariants are: detM B B A2JV

U(l)A 2N N N 2N

£7(1) 0 N -N 0

U(1)R_ 0 0 0 0

(7.9)

vanishes since F = N. Thus a o f „MV t h f > oHu^iivo, snnarlc: ,,«»v u„™ uiiv ii,-vu Ui.rP'e , 6 >, ^J. F -, vanishes generalized form of the constraint that has the correct A —> 0 and B,B —> 0 limits is

detM -BB = A™ (l + Y: Cpq ^ J j ^

V9) ,

(7-10)

with p, q > 0. For (BB) ;§> A2N the theory is perturbative, but with Cpq / O w e find solutions of the form d e t M « (BB)*^

,

(7.11)

396

which do not reproduce the weak coupling A —> 0 limit, thus we can conclude Cpq = 0. Therefore the quantum constraint is: detM - ~BB = A2N .

(7.12)

First note that this equation has the correct form to be an instanton effect since e -s l n .t

oc Ab = A2N .

(7.13)

Also note that we cannot take M = B = B = 0, that is we cannot go to the origin of moduli space (this situation is referred to as a "deformed" moduli space). This means that the global symmetries are at least partially broken everywhere in the quantum moduli space. For example consider the following special points with enhanced symmetry: Ml = A25{, B = ~B = 0 and M = 0, BB~ = -A2N. In the first case, with M = A2, the global SU(F) x SU{F) x U(l) x U(1)R symmetry is broken to SU(F)d x [7(1) x U(1)R, that is the theory undergoes chiral symmetry breaking and, as in non-supersymmetric QCD, the chiral symmetry is broken to the diagonal subgroup. For BB = —A2N the global symmetry is broken to SU(F) x SU(F) x U(1)R, that is baryon number is spontaneously broken. For large VEVs we can understand this symmetry breaking as a perturbative Higgs phase with squark VEVs giving masses to quarks and gauginos (See Figure 3). There is no point in the moduli space where gluons become light, so there are no singular points, and the moduli space is smooth. This is an example of a theory that exhibits "complementarity" [49,50] since we can go smoothly from a Higgs phase (large VEVs) to a confining phase (VEVs of 0(A)) without going through a phase transition. Complementarity holds in any theory where there are scalars (in our case they are squarks) in the fundamental representation 31 of the gauge group. In such a theory we can take any colored field and, by multiplying by enough scalar fields, we can form a color singlet operator 32 . In other words any color charge can be screened. In a theory where screening does not happen a Wilson loop 33 can obey an area law (which indicates confinement) or a •"•Also known as defining or faithful representations of the gauge group. I n the perturbative regime this color singlet operator is just the original colored field multiplied by VEVs, while in the regime of small VEVs it can be a complicated nonperturbative object. 33 A Wilson loop is a closed quark loop. If the expectation value of the loop goes like e —C1 , where A is the area enclosed by the Wilson loop it is said to obey an area law. If the expectation value of the loop goes like e~C2L, where L is the length of the perimeter of the Wilson loop it is said to obey a perimeter law. 32

397

perimeter law (which happens in a Higgs phase) and it can therefore act as a gauge invariant order parameter which can distinguish between the area law confinement phase and the Higgs phase. In a screening theory a Wilson loop will always obey a perimeter law (even if confinement occurs, since all the dynamics takes place along the perimeter of the loop where the screening occurs) as it does in a Higgs phase. Thus in screening theories there is no gauge invariant order parameter that can distinguish between a confining phase and a Higgs phase, so there can be no phase transition between the two regimes and the theory exhibits complementarity.

Figure 3. Squark VEV (dashed line with cross) gives a mass to a quark (solid lines) and a gaugino (wavy line).

7.2. F = N: Consistency

Checks

The constraint (7.12) can be put in the form of an equation of motion arising from a superpotential by introducing a Lagrange multiplier field, which we can refer to as X: Wconstraint = X(detM-BB-A2N)

.

(7.14)

We can now check the consistency of the confined picture by adding a mass for the Nth flavor. It is convenient to re- write the meson field as

M=

r * ;

,

a-is)

where M is an (N — 1) x (N — 1) matrix. We can then write the mass term for the iVth flavor in the confined description as Wmass = rnY, so that the full superpotential is W = X (detM - ~BB - A2N) + mY .

(7.16)

398

We then have the following equations of motion: dW — — = -IB = 0 dW ^L = -XB = 0 dB

(7.17) (7.18) '

K

dW —i=Xcoi{N>)=0 (7.19) dW — =Xcof(PJ)=0 (7.20) dW ~ —-=XdetM + m = 0 , (7.21) where cof(Mj) is the cofactor of the matrix element Mj. The solution of these equations is: X = -m (detM)

(7.22)

B = B~ = Nj = Pi = 0 .

(7.23)

Plugging this solution into the equation of motion for X (the constraint equation) gives ^ = YdetM - A2N = 0 . (7.24) oX So, putting this result into the full superpotential (7.16), we find the lowenergy effective superpotential is

WeS = "^Z

.

(7.25)

detM Using the usual matching relation for the holomorphic gauge coupling we find mA2" = A™+I1} (7.26) so A2N+1

WeS = ^ ^ . (7.27) detM This is just the Afneck-Dine-Seiberg superpotential for SU(N) with N — 1 flavors As a further consistency check let's consider in more detail the points in the moduli space with enhanced symmetry. When M/ = A25j, B = B = 0 the global symmetry is broken to SU(F)d x U{1) x U(1)R. In terms of the elementary fields, the $ and $ VEVs break SU(N) x SU(F) x

399

SU(F) to SU(F)d. The quarks transform as • x D = 1 + Ad under this diagonal group, while the gluino transforms as Ad. The composites have the following quantum numbers 34 :

M-TrM TrM B B

SU(F)d Ad 1 1 1

f/(l) 0 0 N -N

U(1)R _ 0 0 0 0

The field TrM gets a mass with the Lagrange multiplier field X. non-trivial anomalies match as follows (recalling that F = N):

U(1)2U(1)R : U(1)R: U(1)R: U(1)RSU(F)2:

elem. comp. -2FN -2N2 2 2 -2FN + N -1-(F -1)-1~1 -2FN + N2-1~{F2-1)-1~1 -2N + N -N .

(7.28)

The

(7.29)

At M = 0, B~B = -A 2 J V only the U{\) is broken. The composites transform as:

M B B

SU(F) D 1 1

SU(F) D 1 1

U(1)R 0 0 0

(7.30)

The linear combination B + B gets a mass with the Lagrange multiplier field X. The anomalies match as follows: SU(F)3 : U(1)RSU(F)2 U(1)R: U(1)3R:

elem. N : -N\ -2FN + N2-1 -2FN + N2-1

-(F2

comp. F -F\ -F2-l - 1) - 1 - 1 .

(7.31)

which, again, agree because F = N. 34

As usual only the R-charge of the scalar component is given, and it[fermion] R [scalar] — 1.

400

8. S-Conflnement in S U S Y QCD 8.1. F = N + 1; Confinement Breaking

without

Chiral

Symmetry

For SUSY QCD with F = N+l flavors Seiberg [34] again found that all the 't Hooft anomaly matching conditions can be satisfied with just the color singlet meson and baryon fields; thus this theory is also confining. This theory also exhibits complementarity [49,50] since screening can occur (as discussed at the end of section 7.1). However, the theory with F = N + 1 flavors does not require chiral symmetry breaking, that is we can go to the origin of moduli space. Furthermore the theory develops a dynamical superpotential. Confining theories that screen, do not spontaneously break global symmetries, and have dynamical superpotential have been dubbed "s-confming" [51]. To see that spontaneous chiral symmetry breaking does not occur we begin by recalling the classical constraints on the meson and baryon fields. For F = N + 1 flavors the baryons are flavor anti-fundamentals (and the anti-baryons are flavor fundamentals) since they are anti-symmetrized in N = F -1 colors: Bi

=

e^-^Bi,,...^ ,

Bi = eilt...,iN,iB

' '

.

(8.1) (8.2)

With this notation the classical constraints are: (Af-^jdetM = B % , M3iBi = M^Bj = 0 .

(8.3) (8.4)

With non-zero quark masses we have: (M/) = (m- 1 )^(det™A 2 J V - 1 )* , (B*) = (B3) = 0 .

(8.5) (8.6)

So taking a determinant of equation (8.5) gives

(Af-^jdetM = mJA 2 ^" 1 .

(8.7)

Thus we see that the classical constraint is satisfied as m*- —> 0. Taking this limit in different ways we can cover the classical moduli space, so the classical and quantum moduli spaces are the same. In particular chiral symmetry remains unbroken at M = B = B = 0.

401

The most general superpotential allowed for the mesons and baryons is: W •

1 A2JV-1

aB^lBj

+ f3detM + detM /

detM

&MiB<

where / is an as yet unknown function. Actually only / = 0 reproduces the classical constraints: dW _ ~ BM{ ~ A2N dW 1

[aB'Bj + ^(Af-^jdetM] - 0 ;_

(8.9)

n

(8.10)

^Z- = - — 2 A-r - a B ' M / = 0

(8.11)

dW dBj

1

A

-!

provided that (3 = — a. To determine a, consider adding a mass for one flavor so that we can compare with the results for F = N flavors.

a^ W = — 2W-1

BlM\Bj -- detM + mX , B^MlBj

A

(8.12)

where

-

(

^

)

B={U\B')

I

.

5.13)

,

U4)

'-(H

5.15)

For this theory with one massive flavor and F = N light flavors we have the following equations of motion: dW _ a - (B'U - cof (Y)) = 0 ~8Y ~ A™~~ dW dW dU dW dU

a A2N-1

a a

ZB'

(8.16) (8.17)

0

(8.18)

B'Y = 0

(8.19) (8.20) (8.21)

402

The solution of these equations is: Y =Z =U=U=0, detM' - B'B' =

(8.22)

mA2N 1

~ = 1A™ . (8.23) a a Equation (8.23) gives the correct quantum constraint (7.12) for F = N flavors if and only if a = 1. Plugging the solutions of the equations of motion into the superpotential (8.12) we find the effective superpotential is w

<* = X27TT (B'Z'

~

detM

' + mA2""1) •

(8-24)

With the usual matching relation for the holomorphic gauge coupling we find m A 2 " " 1 = A2NNN ,

(8.25)

so ^eff = ^ ^ n (B'B' - detM' + A2NNN)

.

(8.26)

Holding AJV,JV fixed as m —> oo implies that A —> 0, so X becomes a Lagrange multiplier field in this limit. Thus we can completely reproduce the superpotential (7.14) used to discuss F — N flavors in section 7.2. Thus to summarize we have determined that the correct superpotential for the confined description of SUSY QCD with F = N + 1 flavors is: 1 W = A ,2ni >Vr- , IB^fBj - detM 3.27) Since the point M = B = -B = 0 i n o n the quantum moduli space, we should worry about what singular behavior occurs there. Naively gluons and gluinos should become massless. What actually happens is that only the components of M, B, B become massless. That is we simply have confinement without chiral symmetry breaking. This is the type of theory that 't Hooft was searching for when he proposed his anomaly matching conditions [28]. The color singlet composites transform as 35 : M B B 35

SU(F)

SU(F)

•

•

•1

1

•

U{1) 0 N -N

U(1)R 2 F N F N F

3.28)

As usual only the .R-charge of the scalar component is given, and _R[fermion] = it[scalar] — 1.

403

Some of the anomaly matchings go as follows: elem. N

SU(F)3 : U(1)SU{F)2 : U{1)RSU{F)2 U(1)R : U(l)R :

comp. F-l

l y 2 __JV JV 2F 2

JV 2 2-F F _|_ F 2 ^

2

-^F2^

-N -l J

2

F

2(N - F) 2

- (f) 2NF + N -1 (^Y

3.29)

N-F

F + (^Y

2F

which agree because F = N + 1. 8.2. Connection

to theories

with F > iV + 1

We can also check that the confined descriptions of the theories with F = N and F = N + 1 flavors are consistent with dual descriptions of the theories with more flavors. Consider the dual theory for F = N + 2: SU(N + 2)

C/(l)

U(1)R

•

1

JV 2

D

1

•

JV JV+2 JV JV+2

1

•

D

SU(2)

SU{N + 2)

q

•

q M

JV 2

3.30)

4 JV+2

0

The dual theory has a superpotential .31)

W = -M~4>

As we have seen in section 6.6, giving a mass to one flavor in the corresponding SUSY QCD theory produces a dual squark vev —F

3.32)

{
SU(N + 1)

C/(l)

q'

•

1

A^

V

1

•

-N

M'

D

D

0

U{1)R JV JV+l JV JV+l 2 JV+l

That is after integrating out the gauge singlet fields which are massive.

,33)

404

Comparing with the confined spectrum (8.28) we see that we should identify qH = cBi ,

(8.34)

Wj = cBj ,

(8.35)

where c and c are some constant re-scalings. After integrating out the massive gauge singlet fields the remnant of the tree-level dual superpotential is Wtree = -BiM'iB3 . (8.36) A* Since we have completely broken the dual gauge group we expect that instantons will generate extra terms in the superpotential. Indeed one finds:

WW. = ^ ^ d e t (ML) ,

(ftp)

^

(8.37)

/

where A is the intrinsic holomorphic scale of the dual gauge theory. A simple way to verify that this superpotential is generated by instantons is to check that the corresponding interaction between two fermion components of M (the mesinos) and N — 1 mesons is actually generated. The instanton contribution to this interaction can be seen in Figure 4. So the effective superpotential agrees with the result for F — N + 1 flavors (8.27): W

<* = x i

[B'MfBj - detM'] ,

(8.39)

if and only if c

c=T^rT, \2N-1 '

Iy

NJV,iV+2 N4-2

(8.40)

1

fJiN+2m

A2N-1

3.41)

•

The second relation (8.41) actually follows from a more general relation 13N-FA3N-F

=

{_1}F-iVMF

_

(g

42)

To see why this relation is true consider generic values of (M) in the dual of SUSY QCD. All the dual quarks are massive, so we have a pure SU(F-N)

405

N-1 Figure 4. Instanton contribution to the interaction between two mesinos (external straight lines) and N — 1 mesons (dash-dot lines). The instanton has 4 gaugino legs (internal wavy lines) and N + 2 quark and anti-quark legs (internal straight lines). A squark VEV and an anti-squark VEV are shown (dashed line with cross) gives a mass to a quark and a gaugino or a anti-quark and gaugino, however they can be arbitrarily many insertions of these VEVs which must be re-summed.

gauge theory. The intrinsic scale of the dual low-energy effective theory is given by matching:

Af = A3^-Fdet

.43)

This theory undergoes gaugino condensation, and as we have seen the effective superpotential is: WL = NA3L

(8.44)

406

= (F - N) i

=

<

*

-

"

(

j

•

*

*

"

)

(8.45)

(

•

8

-

4

6

)

where we have used equation (8.42). Adding a mass term m^Mj gives: Mi = {m-1yi(detmh3N-F)j}

,

(8.47)

which we have already seen is the correct result. If we consider the dual of the dual 37 of SUSY QCD then (with the assumption that (A = A) equation (8.42) implies A3N-F13N-F

=

^f-N-F

(8>48)

So, since F — N = N, we must have for consistency ji=-li

.

(8.49)

If we write the composite meson of the dual quarks as: N] = tfa

,

(8.50)

and the dual-dual squarks as d, then the superpotential of the dual of the dual is Wdd = -4-1 ij + -^Nl M M The equations of motion give dW 1 ^L = -Ni=0 dW

l-i

.

(8.51)

(8.52) 1

•

— L = -7dj + -M\ = 0 (8.53) y dNl n 3 n J •So, since Jl = —fi, we can identify the original squarks with the dual-dual squarks: *i = dj •

(8-54)

Plugging the solution (8.52) back into the dual-dual superpotential (8.51) we find that it vanishes and we conclude that the dual of the dual of SUSY QCD is just SUSY QCD. 37

Which has F - N = N colors.

407

9. Duality for

SO(N)

Dualities for SO(N) theories have also been found [52], and they exhibit some interesting new features since they can be non-screening if there are no spinor representations. Also there are massless composites made of squarks and gluinos. 9.1. The SO(N) Spaces

Theories

and Their Classical

Moduli

An SO(N) gauge theory with F quarks in the vector representation has a global SU(F) x U(1)R symmetry as follows: SO{N)

SU(F)

•

i-|

U(1)R F+2-JV

LJ Q F Since there are no dynamical spinors in this theory, static spinor sources cannot be screened, so there is a distinction between area-law confining and Higgs phases. Recall that the adjoint of SO(N) is the two-index antisymmetric tensor. For odd N, there is one spinor representation, while for even N there are two in-equivalent spinors. For N = Ak the spinors are self-conjugate, while for TV = 4fc + 2 the two spinors are complex conjugates. The simplest irreducible representations are summarized in the following two tables.

SO(2N +1) d(r) 2N + 1

Irrep

•

2T(r) 2 2 JV-2

S

B m

N{2N+1) AN -2 (N + 1)(2N + 1) - 1 4/V + 6

Irrep

• S S,(S>)

B m

SO(2N) d(r) 2/V •

2T(r) 2

2N-1

2JV-3

2N-1

2 JV-3

N{2N - 1) N(2N+1)-1

AN -A AN + A

The one-loop /3 function coefficient for A^ > 4 is b = 3(A - 2) - F .

(9.3)

408

Solving the D-flatness conditions one finds that up to flavor transformations, the classical vacua for F < N are given by

hi

\ VF

<$)

(9.4)

0 \0

... 0 )

At a generic point in the classical moduli space the SO(N) gauge symmetry is broken broken to SO(N-F) and there are NF-N(N-1) + (N-F)(NF — 1) massless chiral supermultiplets. For F > N the vacua are: 0 ... (T

Vi

($)

(9.5) vN 0 . . . 0,

At a generic point in the moduli space the SO(N) gauge symmetry is broken completely and there are NF — N(N — 1) massless chiral supermultiplets. We can describe these light degrees of freedom in a gauge invariant way by scalar "meson" and (for F > N) "baryon" fields and their superpartners: Mji = Q^i

,

(9.6)

B[iu..,iN] = % • • • * « ] ,

(9-7)

where [] denotes antisymmetrization. Up to flavor transformations the moduli space is described by:

J

N

(M)

\ (BI,...,N)

Vl

• ••VN

(9.

°/

(9.9) (9.10)

with all other components set to zero. The rank of M is at most N. If the rank of M is N, then B = i v d e t ' M , where det' is the product of non-zero eigenvalues.

409

9.2. The Dynamical

Superpotential

for F < N — 2

To construct the effective superpotential we should look at how the chiral superfields transform under the anomalous axial U(1)AU(1)A a

W A6 detM

U(1)R

0 2F 2F

1 0 2(F + 2-

(9.11) N)

So we see it is possible to generate a dynamical superpotential (the analogue of the ADS superpotential for SU(N)): Wdy

CN,F

(9.12)

\detMJ

for F < N - 2. 9.3.

(—Y

Duality

For F > 3(N — 2) we lose asymptotic freedom, so the theory can be understood as a weakly coupled low-energy effective theory. For F just below 3iV we have an infrared fixed point. A solution to the anomaly matching for F > N — 2 is given by: SO(F - N + 4)

SU(F)

q

•

M

1

• m

U(1)R N-2 F 2(F+2-N) F

For N > N — 3 this theory admits a unique superpotential: (9.13) The dual theory also has baryon operators: _§[ii,.-,iff] =Ah

...^1 .

(9.14)

There are additional hybrid "baryon" operators in both theories since the adjoint is an antisymmetric tensor. In the original SO(N) theory we have:

Hu

W $r- . . $ ;*JV-lN_4]

(9.15)

410

While in the dual theory we have: £[
=

w ^

... 4,^-tl .

(9.16)

The two theories thus have a mapping of mesons, baryons, and hybrids: M

^M

h • •Hl,...,lN-4

< - > F-

• tl,---,iFJD

Ftii'--->iN

t

H[n,...,iN-2]

„

i?i 2]

eil,...,iFHt"-

-

•

(9.17)

- 3(N - 2)) .

(9.18)

The dual /3 function is 0(g) ex -g3(3(N

- 2) - F) = -g3(2F

So the dual theory loses asymptotic freedom when F < 3(N — 2)/2. When F = 3N -eN

,

(9.19)

there is a perturbative IR fixed point in the dual theory. One can check that the exact /? function vanishes in this range using the relation between dimensions and R charges in a superconformal theory. So we have found that SO(N) with F vectors has an interacting IR fixed point for 3(N — 2)/2
Cases

For F < N - 5, SO(N) breaks to SO(N - F) D SO(5), which undergoes gaugino condensation and produces the dynamical superpotential: /16A3(AT-2)-F

W d y n oc (AA) oc (^

- ^

j

.

(9.20)

For F = N - 4, SO(N) breaks to SO(4) ~ SU(2)L x SU(2)R, so there are two gaugino condensates /16A2JV_1V

1 ^condensate = 2 ( A A ) L + 2(XX)R

= - ( e L + eR)

(

^

^

j

, (9.21)

411

where es = ±1 .

(9.22)

So there are four vacua corresponding to two physically distinct branches: one with (ex, + eR) = ±2 and the other with (ex + eR) = 0. The first branch has runaway vacua, while the second has a quantum moduli space. At M = 0, the composite M satisfies the 't Hooft anomaly matching conditions. One can check that this only happens for F = N — 4. So we have another example of confinement without chiral symmetry breaking, this time without any baryons. Integrating out a flavor on the first branch gives the correct runaway vacua of the previous case (F + N — 5), while on the second branch we find no supersymmetric vacua after integrating out a flavor, which is a consistency check. For F = 7V-3, SO(N) breaks to SO(4) ~ SU{2)L x SU(2)R, which then breaks to SU(2)d ~ 50(3) so there are instanton effects (since H^G/H) = H^(SU(2)) = Z) and gaugino condensation y^2JV-3

W conde „ sat e = 4(1 + e) — — , detM

(9.23)

e = ±1 ,

(9.24)

where corresponding to the two phases of the gaugino condensate. So there are two physically distinct branches: one with e = 1 and the other with e = — 1. The first has runaway vacua, while the second has a quantum moduli space. Integrating out a flavor, we would need to find two branches again, so W ^ 0 even on the second branch. For this to be true we must have some other fields that interact with M. We also know that M does not match the anomalies by itself. The solution of the anomaly matching is given by: SU{F) q M

• m

U(1)R AT-2 F

2^+2-iV)

The most general superpotential is

where f(t) is an unspecified function. Adding a mass term gives qF = ±iv ,

(9.26)

412

which gives us the correct number of ground states. Note that the operator mapping must be: -.JV-

Q1

(9.27)

w„r

which is a hybrid operator. For N = 4 this is a gluino-ball. This is an example of confinement without chiral symmetry breaking with hybrids. Starting with the F = N dual which has an 50(4) gauge group, and integrating out a flavor there will be instanton effects when we break to 50(3) so the dual superpotential is modified in the case F = N — 1: W

M,

Vrh1 -

2/j, '

6

rdetM

4A2N-5

(9.28)

For F AT — 2 we see in the original theory that we can generically break to SO(2), while in the dual we also break so 5*0(2) ~ U(l). This case needs a separate treatment [52]. 10. Sp(2N) 10.1. Duality

and Chiral Theories for

Sp(2N)

An Sp(2N) gauge theory 38 with 2F quarks (F flavors) in the fundamental representation has a global SU(2F) x U(1)R symmetry as follows:

Q

Sp(2N)

SU(2F)

•

•

U(1)R F-l-N F

(10.1)

Recall that the adjoint of Sp(2N) is the two-index symmetric tensor. The simplest irreducible representations are summarized in the following table. Irrep

•

B

d(r) 2N

T(r)

N(2N-1)-1 N(2N+1)

2N-2 2N + 2

N(2N-l)(2N-2)

cm 3

_

2 N

N(2N+l)(2N+2) 2N(2N-1)(2N+1)

We'll use the notation that Sp(2) ~

_

SU(2).

2N

(2iV-3)(2JV-2) _ 2 (2JV+2)(2JV+3) 2

(2N)2 - 4

(10.2)

413

The one-loop /3 function coefficient for N > 4 is b = 3(2iV + 2) - 2F .

(10.3)

The moduli space is parameterized by a "meson" Mji = QjQi ,

(10.4)

which is antisymmetric in the flavor indices i,j. The holomorphic intrinsic scale (spurion) and the flavor singlet chiral superfield transform under the global symmetry as: U(1)A U{l)R 2F 0 PfM 2F 2(F - 1 N)

(10.5)

where the Pfaffian of a 2F x 2F matrix M is given by PfM

'M,„

(10.6)

• Mi,

So we see it is possible to generate a dynamical superpotential W d y n oc

At

(10.7)

PfM

for F < N + 1. For F = N + 1 one finds confinement with chiral symmetry breaking PfM = A2**"1"1) .

(10.8)

For F = N + 2 one finds confinement without chiral symmetry breaking with a superpotential: W = PfM .

(10.9)

A solution to the anomaly matching for F > N — 2 is given by: Sp(2(F - N -- 2 ) ) q

•

M

1

SU(2F)

B

U(1)R N+l F 2(F+2-N) F

This dual theory admits a unique superpotential: (10.10) For 3(AT + l ) / 2 < F < 3(N + 1) we have an IR point. For N + 3 < F < 3(N + l ) / 2 the dual is IR free and the low energy theory is made up of quarks, gluons, mesons, and their superpartners. A complete discussion of Sp(N) theories with fundamentals is given in [53].

414

10.2. Why Chiral Gauge Theories

are

Interesting

We would eventually like to use non-perturbative methods like duality for understanding SUSY gauge theories that dynamical break SUSY. Usually vector-like gauge theories don't break SUSY, while chiral gauge theories can. If a theory is vector-like we can give masses to all the matter fields. If these masses are large we have a pure gauge theory that has gaugino condensation but no SUSY breaking. By Witten's index argument [26] we can vary the mass but the number of bosonic minus fermionic vacua doesn't change. If taking the mass to zero does not move some vacua in from or out to infinity, then the massless theory has the same number of vacua, and SUSY is not broken. As our first example of a chiral gauge theory consider

Q T

SU(N)

SU(N + 4)

D

•

m

1

I will not write down the charges under the two global U(l)'s. This theory is dual to [54] SO(8)

SU(N + 4)

q

•

D

p

S

1

U ~ detT

1

1

M-QTQ

1

m

with a superpotential W = Mqq + Upp .

(10.11)

This theory is vector-like! The dual /3 function coefficient is: b = 3(8-2)-(N

+ 4)-l

= 13-N

.

(10.12)

So the dual is IR free for N > 13. 11. S- Confinement We need some semi-systematic way to survey chiral gauge theories. One way to do this is to generalize well understood dual descriptions. The simplest of these is confinement without chiral symmetry breaking in SU(N)

415

with N + 1 flavors. Recall (from section 8.1) the confined description had a superpotential W

= A ^ I (detM -

BM

B)

•

(11-1)

The crucial features of this description were that since there was no chiral symmetry breaking and that the meson-baryon description was valid over the whole moduli space. That is, there was a smooth description with no phase-transitions. This is because the theory obeyed complementarity [49,50], since every static source could be screened by squarks. To generalize this we will need to have fields that are fundamentals of SU or Sp and spinors of SO. We will only consider theories that have a superpotential in the confined description. This requirement gives us an index constraint. Theories that satisfy these conditions are called s-confining [51]. Consider a gauge theory with one gauge group and arbitrary matter fields. Choose an anomaly free U(1)R such that 4>i that transforms in the representation ri of the gauge group and has i?-charge q while all other fields have zero charge. The charge q is determined by anomaly cancellation: 0=(q-l)T(ri)+T(Ad)-Y/T(*i) = qpr(rl)+T(Ad)-YtT(ri).

(H-2) (11.3)

3

Since we can do this for any field, and for each choice the superpotential has jR-charge 2, we have 6

Woe A

*(t)

TW

2/(E J T(r J )-T(Ad))

(11.4)

There may in general be a sum of terms corresponding to different contractions of gauge indices. Requiring that this superpotential be holomorphic at the origin means there should be integer powers of the composite fields, which implies integer powers of the fundamental fields. Unless all the T(ri) have a common divisor we must have ^2T{TJ)

- T(Ad) = 1 or 2 for SO or Sp

3

2 ( ^ T ( r - j ) - T(Ad)) = 1 or 2 for SU .

(11.5)

3

The differing cases come from the different conventions for normalizing generators, for SO and Sp we have T(D) = 1, while for SU we have T(D) = 1/2.

416

Anomaly cancellation for SU and Sp require that the left hand side be even. This condition is necessary for s-confinement, but not sufficient. One has to check explicitly (by exploring the moduli space, as discussed below) that for SO none of the candidate theories where the sum is 2 turn out to be s-confining. Thus we have

Y^T{r0)-T{Ad) = {\2 for for

SU or SO Sp

(11.6)

This condition gives a finite list of candidate s-confining theories. We can check whether candidate theories that satisfy the index constraint really are s-confining by going out in moduli space. Generically we break to theories with smaller gauge groups and singlet fields that decouple in the infrared. If the smaller gauge theory is not s-confining the the original theory was not s-confining. Alternatively if we have an s-confining theory and we go out in moduli space we must end up with another s-confining theory. Using these checks one can go through the list of candidates. For SU one finds that the following theories are the only ones that satisfy the conditions for s-confinement. SU(N) (JV + 1)P + D) SU(N) SU(N)

L] + B + 3(n + n)

SU(5) (11.7)

f2D+4D

517(5) SU(6)

U+5D+D

SU(6)

-4(D + D)

SU{7)

B + 3D

Let's consider the special case: SU(2N + l)|5I/(4) SU(2N + 1)

¥D •

1 1 •

1 • 1

[/(!)!

U{1)2

0 2N + 5 4 -2JV + 1 -2iV - 1-2AT + 1

U(1)R 0 0 i

(11.

417

This theory has a confined description in terms of the following composite fields: SU(4:)SU(2N + 1) •

QQ

C7(l)! 3-27V

•

AQ2 ANQ AN-'Q3

1

U 1 1 1

• •

Q2N+1

1

U(l)2 -AN+ 2

U{l)R \

8 -2N + 7 - 2 7 V - 1 27V2 + 37V+1 - 6 7 V - 3 27V2-37V-2 4(27V+1) -47V 2 + 1

0 \ § 0

(11.9)

with a superpotential

W

1 A2N

(A»Q)(QQy(AQT-'

+

.TT2S

N~ir,3\ (A»-'Q*)(QQ){AQ')

~(Qz"+L)(ANQ)(AN-lQ3)

N

(11.10)

One can check that the equations of motion reproduce the classical constraints, and that integrating out a flavor gives confinement with chiral symmetry breaking.

12. Deconfinement Given that we can find confined descriptions of certain SUSY gauge theories, we can also reverse engineer a dual description of a given theory where some of the fields correspond to mesons or baryons of the dual description. Such a procedure is referred to as "deconfinement" [55]. Consider the SU(N) gauge theory with an antisymmetric tensor [56,57] for odd TV theory with F > 5 flavors: SU{N) SU(F) SU(N + F - 4) f/(l)i 1

•

• 1

1 1

•

U(l)2U(l)R -2F TV

-12 N

F

•£•

r

N

- F 2-

N+F-i

—

We can find a dual description of this theory by taking A to be a composite

418

meson of a s-confining Sp theory SU(N) Sp{N --3) SU(F) SU(N + F --4) E/(l)! c/(i)2 u{\)R 1 1 0 -F =$ • • 1 0 1 1 FN 8 • ~P 1 1 1 0 F(l-N)6-% 1 1 • 1 N-F 2-f Q -F 1 1 F -§• Q AT+F-4

DID Dl

Y Z

r

N

(12.1) with a superpotential (12.2)

W = yzP The SU(N) group has iV + F duality: to find another dual: SU(Fy p q q M L B

• • • •

1 1 1

3 flavors so we can using our standard

3) Sp(N - 3) SU(F) SU(N + 1 1 • 1 1 1 1 1 • 1 1 • 1 • • 1 • • 1 1 •

F-4)

(12.3)

with (12.4)

W = Mqq + Bqp + Lyq . But Sp(N - 3) with N + 2F -7 fundamentals has an Sp(2F • SU(Fy p q Q

M I B

• • • •

1 1 1

a

B

H

1

(Ly)

•

3) Sp(2F - 8) SU(F) SU(N + 1 1 • 1 1 1 1 1 • 1 1 • 1 • • 1 • • 1 1 • 1

1

1

1 1

1 1

B •

dual:

F-4)

(12.5)

419

with W = ayy + Hll + (Ly)ly + Mqq + Bqp + (Ly)q ,

(12.6)

which, after integrating out (Ly) and q becomes W = ayy + Hll + Mqly + Bqp .

(12.7)

With F = 5 we have a gauge group 5(7(2) x SU(2) and one can show (using the fact that gauge invariant scalar operators have dimensions larger than or equal to one) that for TV > 11 this theory has an IR fixed point [57]. One can also show that some of the fields are IR free. Integrating out one flavor completely breaks the gauge group and the light degrees of freedom are just the composites of the s-confining description discussed at the end of section 11. With the other dual descriptions we would have to discuss strong interaction effects to see that we get the correct confined description. 13. Gauge Mediation The basic idea of gauge mediation 39 as proposed 40 by Dine, Nelson, Nir, and Shirman [60] is that there are three sectors in the theory, a dynamical SUSY breaking sector, a messenger sector, and the Minimal Supersymmetric Standard Model (MSSM). SUSY breaking is communicated to the messenger sector so that the messengers have a SUSY breaking spectrum. They also have standard model gauge interactions, which then communicate SUSY breaking to the ordinary superpartners. This mechanism has the great advantage that since the gauge interactions are flavor blind it does not introduce flavor changing neutral currents which are an enormous problem for supergravity mediation models. 13.1. Messengers

of SUSY

breaking

We will first consider a model with Nf messengers i, 4>i and a Goldstino multiplet X with an expectation value: (X)=M

+ 02F ,

(13.1)

so the scale of SUSY breaking is set by \/F. The Goldstino is coupled to the messenger fields via a superpotential W = Xfafa . 'For an excellent review see [58]. 'For early work along these lines see ref. [59].

(13.2)

420

In order to preserve gauge unification, fa and fa should form complete Grand Unified Theory (GUT) multiplets. The existence of the messengers shifts the coupling at the GUT scale, /LiGUT, relative to that in the MSSM by ^GUT = - ^ l n ( ^ )

,

(13-3)

where N,

Nm = J22T(ri)-

( 13 -4)

i-l

For the unification to remain perturbative we need Nm < ln(

*50 x •

(13.5)

M^r)

^

>

The VEV of X gives each messenger fermion a mass M, and the scalars squared masses equal to M2 ± F. We will be interested in the case that F
At two-loop order one finds squared masses for squarks and sleptons of order

the sum indicates that there are contributions from all the gauge interactions that the scalar couples to. This is very nice since the squark and slepton masses turn out to be of the same order as the gaugino masses, however doing two-loop calculations is quite tedious. Fortunately since we are only interested in the finite parts of two-loop graphs, there is a simple method for doing these calculations using the renormalization group (RG) and holomorphy [58,61]. The super-partners of the the standard model gauge bosons.

421

13.2. RG Calculation

of Soft

Masses

Below the mass scale of the messengers we can integrate them out and write the pure gauge part of the Lagrangian as: CG = ~^fd20T(X,fi)WaWa,

(13.E

where the effects of the messengers are included through their effects on the one-loop running of the holomorphic gauge coupling. Taylor expanding in the F component of X we find a gaugino mass given by i 9T , MX = - — \X=MF j 91nr , F VI \x=M2dlnX^-' M

(13.9)

The holomorphic coupling is given by

^^-w+4kO+^ln(i)'

(13 10)

-

where, as usual, b' is the (3 function coefficient of the theory including the messenger fields and b is the f3 function coefficient in the effective theory (i.e. the MSSM) below the mass scale of the messengers. These two coefficients are related by: b' = b-Nm.

(13.11)

So the gaugino mass is simply given by

Note that in the MSSM, where we have three gauginos, the ratio of the gaugino mass to the gauge coupling is universal: M± a\

=

M ^ «2

=

M ^

c*3

=

F -iVl

This type of relation was originally found in supergravity mediation models and, at the time, was thought to be a signature of such models. Next consider the wavefunction renormalization for the matter fields of the MSSM: C=

I' dAeZ{X,X])Q^Q'

,

(13.14)

422

where Z must be real and the superscript ' is meant to indicate that we have not yet canonically normalized the field. Taylor expanding in the superspace coordinate 6 we have

c

z+

2+

n

d2z

= /*» ( If™ §T^ + a*§oFeW2

QrtQ' . X=M

(13.15) Canonically normalizing we have:

Zl 2 { l +

MF°2

'

\X=M

(13.16)

Q',

so £ " /

t2 J_IJ5L^Wi^ ZdXdXy

dlnZdln dX 9Xt


QfQ. X=M

(13.17) Thus we have a sfermion mass term: _ d2h\Z FFt « " "51nXainXtU=MMMt ' 2

m

(13.18)

Rescaling the matter fields also introduces an A term in the effective potential from Taylor expanding the superpotential: Z~1'2

dlnZ 8X

dW

(13.19)

FQ

which is suppressed by a Yukawa coupling. To calculate Z, we do a supersymmetric calculation and replace M by vXJO. At I loops an RG analysis gives In Z = a(fi0)

f(a(iJ,0)L0,

a(fi0)Lx)

(13.20)

where In

U Lx d2lnZ SlnXSlnXt

In

a{ii)l+1h(a{n)Lx)

(13.21) (13.22)

(13.23)

Thus the two-loop scalar masses are determined by a one-loop RG equation.

423

At one-loop we have dlnZ

_ C2(r)

dhi/j,

TT

<*(/*) ,

(13.24)

2Cj(r)

2C2(r)

where

a' 1 (X)=a- 1 (// 0 ) + ^ l n f — g - J ,

a-1(M)=^1W + A

l n

(13.26)

^j .

(13.27)

So we finally obtain -Q = 2 C

2

(r)^

m

(e

2 +

^ ( l - a ) ( ^ )

,

(13.28)

where {

-i

+ *°M-*-

(13 29)

-

What is particularly impressive about this calculation [58,61] is that a twoloop result is obtained from a one-loop calculation. 13.3.

Gauge Mediation

and the \i

Problem

Recall that in order to obtain a viable mass spectrum for the electroweak sector in the MSSM, the two Higgs doublets, Hu and Hd, needed two types of masses terms: a supersymmetric \x term: W = fiHuHd ,

(13.30)

and a soft SUSY-breaking b term: V = bHuHd ,

(13.31)

with a peculiar relation between these two seemingly unrelated parameters b ~ fj2 .

(13.32)

In gauge mediated models we need to be of the same order as the squark and slepton masses:

"~1^£-

< 13 - 33 >

424

If we introduce a coupling of the Higgses to the SUSY breaking field X, W = \XHuHd

,

(13.34)

we get (j, = XM ,

(13.35)

b = XF ~ 1 6 T T V ,

(13.36)

so b is much too large for this type of model to be viable. A more indirect coupling W = X ( A i 0 i ^ + A2^2^2)Afl-u0i^2 + XHafafa ,

(13.37)

yields a one-loop correction to the effective Lagrangian: d4e^^f(X1/X2)HuHd^J

.

(13.38)

This unfortunately still gives the same, non-viable, ratio for bjJJ?. The correct ratio can be arranged with the introduction of two additional singlet fields S and N: W = S(X1HuHd + X2N2 + XH> - M2N) + X(f4> ,

(13.39)

then /i = A i ( S ) ,

(13.40)

b = AjFs .

(13.41)

A VEV for S is generated at one loop 1

F\

<^> ~ i ^ J ^ 'WN • L

(13 42)

-

but Fs is only generated at two loops: Fs

~ ( 1 6 ^ W ~ 16^17" •

(13 43)

'

Thus we can obtain b ~ fj,2 provided that we arrange for M2N ~ Fx- So it seems that somewhat elaborate models are needed to solve the fj, problem in the gauge mediated scenario.

425

14. Dynamical SUSY Breaking Non-perturbative SUSY techniques can be used to understand how to dynamically break gauge symmetries (like the the GUT symmetry, of the electroweak gauge symmetry of the standard model), however the most important application is to break SUSY itself. We would like to have a theory of dynamical breaking so that the SUSY breaking scale can be naturally much smaller than the Planck or string scale without fine-tuning the ratio of these scales by hand. 42 This is what naturally happens in asymptotically free gauge theories since the coupling at high energies can be perturbative and slowly running and then get strong at some much lower scale. This is just what happens in QCD: the coupling is perturbative near the (putative) unification scale, but gets strong in the IR, producing a large but natural ratio between the GUT scale and the proton mass.

14.1. A Rule of Thumb for SUSY

Breaking

A theory that has no flat directions and spontaneously breaks a continuous global symmetry generally breaks SUSY [63,65]. This is because there must be a Goldstone boson (which has no interactions in the potential), and by SUSY it must have a scalar partner (a modulus), but if there are no flat directions this is impossible. (Unless the modulus is also a Goldstone boson.) In the early days people looked for theories that had no classical flat directions (assuming that quantum corrections would not cancel the classical potential) and tried to make them break global symmetries in the perturbative regime. This method produced a handful of dynamical SUSY breaking theories. With duality we can find many examples of dynamical SUSY breaking. An important twist is that we will find that non-perturbative quantum effects can lift flat directions both at the origin of moduli space as well as for large VEVs.

14.2. The 3-2

Model

In the mid 1980s, Affleck, Dine, and Seiberg [65] found the simplest known model of dynamical SUSY breaking. Their model has a gauge group SU(3) x SU(2) and two global £/(!) symmetries with the following

For an excellent review see [62].

426

chiral supermultiplets: Q L U D

SU(3) SU(2) U(l) U(1)R 1 D • 1/3 -3 1 • -1 • 1 -4/3 -8 • 1 2/3 4

(14.1)

I will write Q = (U,D), and denote the intrinsic scales of the two gauge groups by A3 and A2 respectively. For A 3 » A2 (that is when the SU(3) interactions are much stronger than the SU(2) interactions), instantons give the standard Affleck-Dine-Seiberg superpotential: Wdyn

A5

(14.2)

det(QQ)

which has a runaway vacuum. Adding a tree-level trilinear term to the superpotential (14.3)

— ^ L — + XQDL det(QQ)

W

removes the classical flat directions and produces a stable minimum for the potential. Since the vacuum is driven away from the point where the VEVs vanish by the dynamical ADS potential (14.2), the global C/(l) symmetries are broken and we expect (by the rule of thumb described above) that SUSY is broken. The L equation of motion dW 0Lo

\ea»QmaD

(14.4)

= 0,

tries to set detQQ to zero since detQQ

=

det

{DQiDQ2

= U QmaD Qnp€ a@

(14.5)

Thus the potential can't have a zero-energy minimum since the dynamical term blows up at detQQ=0. Therefore SUSY is indeed broken. We can crudely estimate the vacuum energy for by taking all the VEVs to be of order 2> A3 and A < 1 we are in a perturbative regime. The potential is then given by V

dW dQ

2

+

dW dU

2

+

dW 3D

2

+

dW dL

(14.6)

427

A3"

, .\ ^Al 3 UO "•" AA-f A3

+, y\2 ^ ,

(14.7)

where in the last line we have dropped the numerical factors since we are only interested in the scaling behavior of the solution. This potential has a minimum near

<> « H ,

(14.8)

A' so we see that this solution is self-consistent: the vacuum is weakly coupled for small A since this ensures that 3> A3. Plugging the solution back into the potential (14.7) we find the vacuum energy is of order V«A^A£,

(14.9)

which vanishes as A or A go to zero, as it must. Using duality Intriligator and Thomas [66] showed that we can also understand the case where A2 ^> A3 and supersymmetry is broken nonperturbatively. The SU(2) gauge group has 4 doublets which is equivalent to 2 flavors, so we have confinement with chiral symmetry breaking. The SU(3) gauge group has two flavors and is completely broken for generic VEVs. It is simpler to consider SU{2) as an SU group rather than an Sp group, so we write the gauge invariant composites as mesons and baryons:

M~(LQ'

LQ

* ci4in N i ' '

VQ3Q1Q3Q2/ B ~ Q1Q2 ,

{

B ~Q3L . In this notation the effective superpotential is

W = X(detM-BB-A*)+\(j2MliDi

+ BD3\

,

(14.11)

where X is a Lagrange multiplier field that imposes the constraint for confinement with chiral symmetry breaking. The D equations of motion try to force M\i and B to zero while the constraint means that at least one of M11, M12, or B is non-zero, so we see that SUSY is broken at tree-level in the dual (confined) description. We can estimate the vacuum energy as V « A2A! .

(14.12)

Comparing the vacuum energies in the two cases we see that the SU(3) interactions dominate when A3 3> A? A2.

428

Without making the approximation that one gauge group is much stronger than the other we should consider the full superpotential

W = A (detM - BB - K\) +

A7 J=— + XQDL , det(QQ)

(14.13)

which still breaks SUSY, although the analysis is more complicated.

14.3. The SU(5)

Model

Another simple model analyzed by Affleck, Dine, and Seiberg [63] as well as Meurice and Veneziano [64] is SU(5) with matter content • + Q, that is an antifundamental and an antisymmetric tensor representation. This chiral gauge theory has no classical flat directions, since there are no gauge invariant operators that we can write down. Affleck, Dine, and Seiberg tried to match the anomalies in order to find a confined description but found only "bizarre", "implausible" solutions. This lead to the belief that the at least one of the global U(l) symmetries was broken and that therefore SUSY was broken (using the rule of thumb described earlier). Adding extra flavors ( • + • ) with masses Murayama [67] showed that SUSY is broken, but taking the masses to oo takes the theory to a strongly coupled regime, so the possibility remained that non-perturbative effects induced some phase transition as the mass was varied. With duality arguments Pouliot [68] showed that SUSY is indeed broken at strong coupling. The simplest way to see SUSY breaking is to consider the case that where there are enough flavors that the theory s-confines. Csaki, Schmaltz, and Skiba [51] showed that with four flavors SU(5) 5J7(4) SU(5) tf(l)i U(l)2 A Q Q

D

•

1 1 •

1

0 • 1

4 -5

9 -3 -3

U(l)R 0 0 \

(14.14)

the theory s-confines. To simplify the notation, rather than introducing new symbols for each color-singlet composite I will simply label it by it's constituents between parenthesis. Thus the meson is denoted by (QQ).

429

The spectrum of massless composites is: SU(4) 577(5) C/(l)i t/(l) 2 (QQ) (AQ2) (A2Q) (AQ3)

•

D

1

B 1 1 1

•

D 1

(Q5)

U(1)R

-1

-6

I 2

8 -5 -15 20

3 15 0 -15

0

(14.15)

1 2 3 2

0

with a superpotential l r

Wdy

(A2Q)(QQ)3(AQZ)

+ (AQ3)(QQ)(AQ

-(Q*)(A2Q)(AQ3)

f

(14.16) (14.17)

Note that by examining the global symmetry index structure one can see that the first term in Wdyn is anti-symmetrized in both the SU(5) and SU(4:) indices, while the second term is anti-symmetrized in just the SU(5) indices. We can add mass terms and Yukawa couplings for the extra flavors: 4

AW = ] T mQiQi + ] T XijAQiQj ,

(14.18)

which lift all the flat directions. The equations of motion give dW

(A2Q)(AQ3)

5

= Q,

(14.19)

d(Q ) d(QQ)

=

3(A2Q)(QQ)2(AQ2)

(AQ3)(AQy

0 . (14.20)

Assuming that (A2Q) ^ 0 then the first equation of motion (14.19) requires (AQ3) = 0 and multiplying the second equation of motion (14.20) by (A2Q) we see that because of the antisymmetrizations the first term vanishes and therefore (AQ3)(AQZ)2

(14.21)

but this contradicts (AQ3) = 0, so there is no solution with our assumption

(A2Q) + 0.

Assuming that (AQ3) ^ 0 then the first equation of motion requires (A Q) = 0, and plugging this into the second equation of motion (14.20) 2

430

we find equation (14.21) directly. Multiplying equation (14.21) by (AQ3) we find that the left hand side vanishes again due to antisymmetrizations, so (AQ3) = 0 but this contradicts equation (14.21) and our assumption. Therefore the equations of motion cannot be satisfied, and SUSY is broken at tree level in the dual description. 14.4. SUSY

Breaking

and Deformed

Moduli

Spaces

The Intriligator-Thomas-Izawa-Yanagida [69,70] model is a vector-like theory which consists of an SU(2) SUSY gauge theory with two flavors43 and a gauge singlet:

Q S

SU(2) SU(4) D

•

1

(14.22)

B

with a superpotential W = XSijQiQj

.

(14.23)

The strong SU(2) dynamics enforces a constraint Pf (QQ) = A4 ,

(14.24)

where the Pfafnan44 of a 2F x 2F matrix M is given by Pf(M) = e*i- i2FMili2 ... Ml2F_li2F .

(14.25)

The equation of motion for for the gauge singlet S is dW — = \QiQj

=0.

(14.26)

Since this equation is incompatible with the constraint (14.24) we see that SUSY is broken. Another way to see this is that at least for large values of XS we can integrate out the quarks, leaving an SU(2) gauge theory with no flavors 43

Since doublets and anti-doublets of SU(2) are equivalent, an SU(2) theory with F flavors has a global SU(2F) symmetry rather than an SU(F) X SU(F) as one finds for a larger number of colors. If we artificially divided Q up into q and q then we could follow our previous notation M = qq, B = e^qtqj, B = e^q^ and write the constraint as Pf(QQ) = d e t M — BB = A4.

431

which has gaugino condensation: 3JV

A:•eff

2 A'3N-2 (XS) 2

Weff = 2A^ff = 2A XS , dWeS 2AA^

as*

(14.27) (14.28) (14.29)

so again we see that the vacuum energy is non-zero. Since this theory is vector-like (it admits mass terms for all the quarks and for S) one would naively expect that this model could not break SUSY. This is because the Witten index Tr(—1)F is non-zero with mass terms turned on so there is at least one supersymmetric vacuum. Since the index is topological, it does not change under variations of the mass. However Witten noted that there is a loop-hole in the index argument since the potential for large field values are very different with AW = msS2 from the theory with ms —• 0, since in this limit vacua can come in from or go out to co and thus change the index. At the level of analysis described above S appears to be a flat direction but a general feature of SUSY breaking theories is that flat directions become pseudo-flat. With a non-zero vacuum energy, flat directions can be modified by corrections from the Kahler potential. For large values of XS in this model there is a wavefunction renormalization for S [71,72]

2s =

H- c A A . l n ( ; » 2 ) .

(14.30)

So the vacuum energy is corrected to be

v = 4 ' A 'V \Zs\ w |A|2A4 1 + cAA1 In — 5 V Mo J.

(14.31)

So we see that the potential slopes towards the origin. This can be stabilized by gauging a subgroup of SU(4). Otherwise there is a calculable low-energy effective theory [73] near ASwO (essentially an O'Raifeartaigh model [74]) which has a local minimum at S = 0. The effective theory becomes noncalculable near XS « A. The behavior near this region is unknown.

432

14.5. SUSY

Breaking from Baryon

Runaways

Consider a generalization [75] of the 3-2 model: Sp(2N) SU(2N - 1) SU(2N-1) 1 1 D B B 1 1 l B

U(l)

U(l)i l

•

• •

-l

2JV-1 2AT+2 2JV-1

o 6

(14.32)

-4JV

with a tree-level superpotential (14.33)

W = XQLU.

If we turn off the SU(2N — 1) gauge coupling and the superpotential, Sp(2N) is in a non-Abelian Coulomb phase for N > 6, it has a weaklycoupled dual description for N = 4, 5, it s-confines for N = 3, and confines with a quantum-deformed moduli space for N = 2. If we turn off the Sp(2N) gauge coupling and the superpotential, SU(2N — 1) s-confines for any N > 2. Here we will consider the case that Asu ^> AspIncluding the effects of the tree-level superpotential, this theory has a classical moduli space that can be parameterized by the gauge-invariants: SU(2N -- l ) t f ( l ) M = (LL) B =

2

•

b = (TJ2N-1)

fi

°

4(JV -iV+l) 2N-1

(14.34)

1

subject to the constraints MjkBte

6

2

(U2N-2D)

klmi

17(1) R

0 ""12JV-3

2A^ + 2

_

(14.35)

These constraints split the moduli space into two = 0, Mjkbranches: b = 0. on one of them M = 0 and B, b ^ 0, and on the other M ^ 0 and B, b = 0. First consider the branch where M = 0. We will see that later that the true vacuum actually lies on this branch. In terms of the elementary fields, this corresponds to the VEVs (up to gauge and flavor transformations)

(U) =

/ v sin I \ 0

v cos 9

(14.36)

<£>> =

vl

2N-2

V oJ

433

For large VEVs, v > ASu, SU(2N — 1) is generically broken and the superpotential gives masses to Q and L or order Xv. The low-energy effective theory is pure Sp(2N), "which has gaugino condensation. The intrinsic scale of the low-energy theory is A 3(2iV+2) A

eff

__ .3(2iV+2)-2(2JV-l) — ASp

(xuf2N-1]

(14.37)

The effective superpotential is given by A3

Weff OC A 3 f f

(

XU

(14.38)

For N > 2 this forces (U) to zero. Now consider the case of small VEVs, v < Asu, then SU(2N — 1) s-confines so we have the following effective theory Sp(2N) SU(2N - 1)

• •

L (QU) (QD) (Q2"-1) B b

• 5

• • 1

1 1 •

1

1

(14.39)

with a superpotential W = X(QU)L 1 A4JV-3 SU

[(Q2N~l)(QU)B + (Q2N-x)(QD)b - det

(14.40)

The first term in the superpotential is a mass term, so (QU) and L can be integrated out with (QU) — 0, and we have a low-energy superpotential W

1 A4AT-3

(14.41)

(Q^-')(QD)b

On this branch of the moduli space (b) = (U ) ^ 0, and this VEV gives a mass to (Q 2 i v _ 1 ) and (QD) which leaves an pure Sp(2N) as the low energy effective theory. So we again find gaugino condensation with a scale given by A3(2N+2) =

A3(27V+2)-2(2N-l)

^ ^ 2 ( 2 ^ - 1 )

(_b_^

A SU

^

^

434

and a effective superpotential Wee oc Ae3ff ~ b ^

(2JV-2) ( A ^ A ^ ASU )

N

^ ,

(14.43)

which forces b —> oo, (this is a baryon runaway vacuum) but the effective theory is only valid for scales below Asu, since we assumed v < Asu- We have already seen that beyond this point the potential starts to rise again, so the vacuum is around 72JV-1,

(b) = (U'"~x)~AlT1.

(14-44)

With some more work [75] one can also see that SUSY is also broken when ASp » AsuAn especially interesting case is when N = 3, where the Sp(2N) sconfines and we have the following effective theory 5(7(5) S77(5)

(QQ)

B

(LL) (QL)

1 D

(Q2^-1)

•

u

n

D

1

B

(14.45)

D 1 D 1

D

with W = \{QL)U + Q21*-^2"-1

.

(14.46)

The non-Abelian global symmetry group in this case is 5(7(5) which is just large enough to embed the standard model gauge groups, and hence this is a candidate model for gauge mediation [60]. After integrating out (QL) and U we find 5C/(5) with an antisymmetric tensor, an antifundamental, and some gauge singlets, which we have already seen breaks SUSY in Section 14.3. Returning to general N and considering the other branch of the moduli space where M = (LL) ^ 0, one can see that the D-flat directions for L break Sp(2N) to 5/7(2), and the spectrum of the effective theory is 5(7(2) SU(2N-1) Q' V

V D

• •

1 1

•

1

• •

(14.47)

435

and some singlets with a superpotential W = XQ'V'L'

.

(14.48)

This is a generalized 3-2 model (see Section 14.2) with a dynamical superpotential. For (L) 3> Asu the vacuum energy is independent of the SU(2) scale and proportional to ^su(2N-i) which itself is proportional to a positive power of (L), thus the effective potential in this region drives (L) smaller. For (L) -C A$u w e can use the s-confmed description above, and find again that the baryon b runs away. For (L) w Asu, t n e vacuum energy is V ~ Asu

.

(14.49)

which is larger than the vacuum energy on the other branch, so we see that the global minimum is on the baryon branch with b = (U ) ^ 0. Further examples of dynamical SUSY breaking can be found in references [62,76-78]. 14.6.

Direct

Gauge

Mediation

A more elegant (and more difficult from the model building view point) approach to gauge mediation is direct gauge mediation where the fields that break SUSY have SM gauge couplings. These types of models only have two sectors (a SUSY breaking sector and an MSSM sector) rather than three. Consider the model [79] SI/(5)i SU{5)2 SU(5) Y

4>

1 •

•

• 1

•

• i

with a superpotential

W = \Y0i .

(14.50)

We can weakly gauge the global SU(5) with the SM gauge groups. For large Y S> Ai, A2, (p and <j> get a mass and, by the usual matching argument, the scale of the effective gauge theory is Ae3ff5 = A?" 5 - 5 (AX) 5 ,

(14.51)

1 5

where X = (dety) / . The effective gauge theory undergoes gaugino condensation, so the superpotential is given by W eff = A3eS - XXAj

,

(14.52)

436

so SUSY is broken a la the Intriligator-Thomas-Izawa-Yanagida model discussed in Section 14.4. The vacuum energy is given by |AAf|2 (14.53) ZX where Zx is the wavefunction renormalization for X. For large X the vacuum energy grows monotonically. A local minimum occurs at the point where the anomalous dimension 7 = 0. For (X) > 10 14 GeV, the Landau pole for A is above the Planck scale. The theoretical problem with this model is that for small values of X there is a supersymmetric minimum along a baryonic direction [80]. For this regime it is appropriate to look at the constrained mesons and baryons of SU(5)\. The superpotential is V:

W = A(detM -BB-

A}0) + XYM .

(14.54)

0

There is a supersymmetric minimum at BB = —A} , Y — 0, M = 0. This supersymmetric minimum would have to be removed, or the nonsupersymmetric minimum made sufficiently metastable (with a lifetime of the order of the age of the Universe) by adding appropriate terms to the superpotential that force BB = 0 The phenomenological problem with this model is that heavy gauge boson messengers can give negative contributions to squark and slepton squared masses. Consider the general case where a Goldstino VEV (X) = M + 62F,

(14.55)

breaks SUSY and breaks two gauge groups down to the SM gauge groups GxH

-f SU(3)C x SU(2)L x U(1)Y ,

(14.56)

with 1 a{M)

1 aG(M)

+

1 aH(M)

(14.57)

Analytic continuation in superspace gives MA

<*(M) 47T

(b-bH-

bG)

(14.58)

M

and m

Q

2 2C 2 (r) a(M)2 2

(16TT )

(b

(R2 ~ 2)bH

2bG)e +

b-bH

~bG

(i-e

(14.59)

437

where

R

=

a

- ^ (14-61) y a(M) ' This typically gives a negative mass squared for right-handed sleptons. Another danger for direct mediation models arises if not all the messengers are heavy. Then two-loop RG evolution gives: d 2 2 2 4 2F (14.62) ex - 5 M A + c S T r ( ( - l ) m- ) the one-loop term proportional to the gaugino mass squared drives the scalar mass positive as the renormalization scale is run down, while the twoloop term can drive the mass squared negative. This effect is maximized when the gaugino is light, so in the standard case where the gluino is heavy, then again it is the sleptons that receive dangerous negative contributions. This effect is also dangerous [81] in models where the squarks and sleptons of the first two generations are much heavier that 1 TeV. 14.7. Single Sector

Models

Another appealing approach to gauge mediation is to have the strong dynamics that break SUSY also produce composite MSSM particles [82]. Thus rather that having three sectors (one for the standard model, one hidden sector that dynamically breaks SUSY, and one "messenger" sector that couples to the SUSY breaking and has standard model gauge interactions) there is really just one sector. Consider a SUSY model with gauge group SU(k) x SO(10) with the following matter content 45 :

SU{k) 50(10) SU(W) SU(2) 1 1 Q • • L • 1 • 1 U 1 • • 1 1 16 1 • s

(14.63)

and a superpotential W = XQLU Note that the spinor representation of 5 0 ( 1 0 ) has dimension 16.

(14.64)

438

The 5(7(10) global symmetry is large enough to embed the standard model gauge group, or a complete Grand Unified Theory gauge group (SO(W) or SU(5)) can be embedded. This is a baryon runaway model (similar to the models described in Section 14.5). For large detU » Ai 0 WeS~UW/k,

(14.65)

while for small detU
~ [7 1 0 ( 1 -7 ) / f c ,

(14.66)

where 7 is the anomalous dimension of U which depends on the as yet not fully understood details of S'O(IO) gauge theories with spinors [83], but which seem to have infrared fixed points [57,83]. For 10 > k > 10(1 — 7) SUSY is broken. There are two composite generations corresponding to the spinor S. The composite squarks and sleptons have masses of order mcomp

F ?a = .

(14.67)

where F is the value of the F-component of U which breaks SUSY (i.e. the vacuum energy is |F| 2 ). This can be thought of as gauge mediation via the strong SO(10) interactions. The global SU{2) symmetry enforces a degeneracy that suppresses flavor changing neutral currents. The composite fermions can only get couplings to the Higgs fields from higher dimension operators, so they are light. The fundamental gauginos and third generation scalars get masses from gauge mediation. Since the superpartners of the the first two (composite) generations are much heavier than the superpartners of the third generation, the spectrum is very similar to that of the "More Minimal" Supersymmetric Standard Model [84]. Acknowledgments I would like to thank Howard Georgi and the Harvard and MIT students and post-docs who attended Physics 284 in the fall of 1999: they were the first guinea pigs for.these lectures. I would also like to thank the organizers, lecturers, and students from TASI 2002 who provided stimulating and fun environment for these lectures. Finally I must thank Csaba Csaki, Vladimir Dobrev, Josh Erlich, Shiraz Minwalla, Stavros Mouslopoulos, Ann Nelson, Ricardo Rattazzi, Yuri Shirman, and Witold Skiba for useful discussions and comments. J.T. is supported by in part by the US Department of Energy under contract W-7405-ENG-36, and during the original stages of

439

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KEITH R. DIENES

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N E W DIRECTIONS FOR N E W DIMENSIONS: A N I N T R O D U C T I O N TO KALUZA-KLEIN THEORY, LARGE EXTRA DIMENSIONS, A N D THE BRANE WORLD

KEITH R. DIENES Department of Physics, University of Arizona, Tucson, AZ 85721 E-mail: dienes@physics. arizona. edu This is the written version of an introductory self-contained course on theories with extra spacetime dimensions, delivered at the 2002 TASI summer school. No prior knowledge of such theories is assumed. The goal is to provide a short and practical introduction to this important field of modern research, focusing exclusively on the case of flat extra dimensions. Subsequent lectures at this school will focus on more advanced topics.

Contents 1

Lecture # 1 : Extra spacetime dimensions: General properties and bounds 451 1.1 Original Kaluza-Klein idea 451 1.2 The compactification procedure: Mode expansions, Kaluza-Klein reductions, and low-energy signatures 455 1.3 Orbifold compactifications 462 1.4 Types of extra dimensions 466 1.5 Bounds on extra dimensions 472 1.6 Embedding 4D theories into extra dimensions 478

2

Lecture # 2 : Large extra dimensions and the brane world 481 2.1 The Standard Paradigm for physics beyond the Standard Model: A quick review 482 2.2 Extra dimensions: Lowering the GUT scale 486 2.3 Extra dimensions: Lowering the Planck scale 491 2.4 Extra dimensions: Lowering the string scale 497 2.5 Extra dimensions: A unified picture for physics beyond the Standard Model? 498 447

448

2.6 Open questions

502

Lecture # 3 : Extra dimensions in action: Applications to particle phenomenology 503 3.1 Proton decay 504 3.2 The fermion mass hierarchy 508 3.3 Neutrino oscillations 511 3.4 QCD axion phenomenology 519 Lecture # 4 : Beyond the simple pictures: Lessons and warnings524 4.1 Shape, not just volume 524 4.2 Strings, not just effective field theories 534 4.3 Curved, not just flat 542

Introduction Within the past four years, the subject of theories with extra spacetime dimensions has undergone a profound transformation. What was once the almost exclusive province of practitioners of supergravity and string theory has exploded to become one of the dominant research topics for particle phenomenologists and even particle experimentalists. What began in the 1920's as an abstract mathematical attempt at unification near the Planck scale has become, in the short space of the past four years, one of the major contenders for possible physics at energy scales immediately beyond that of the Standard Model. Indeed, growing numbers of theorists and experimentalists are actively exploring the possibility that extra curled-up spacetime dimensions might soon be discovered, either in the next round of collider experiments, or in precision table-top experiments. One has only to glance at some of the headlines culled from the popular scientific press of the past few years (see Fig. 1) to appreciate the magnitude of the sudden interest in this subject. Looking further at these headlines, however, we can discern another theme at play. Certainly, many of the headlines focus on the possibility of large extra dimensions; for example, most the headlines in the upper half of Fig. 1 are in this category. But in the headlines in the lower half of Fig. 1, we instead perceive another theme entirely — the idea that fundamental ideas of grand unification and string theory may soon be directly testible! Since the energy scales of these fundamental theories are normally considered to be unreachably high (near the Planck scale ~ 1019 GeV), a direct examination of these theories can only happen if their characteristic energy

449

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450

scales are somehow significantly reduced, perhaps all the way to the TeV range. Of course, these two themes are related: theories with large extra spacetime dimensions tend to exhibit significantly reduced GUT, Planck, and string scales. The main goal of these lectures will be to explain this connection. We will begin, in Lecture # 1 , by discussing extra dimensions in and of themselves. How are such theories formulated? What are their lowenergy signatures? What bounds govern their allowed parameter ranges? Then, in Lecture # 2 , we shall discuss how and why such extra dimensions tend to reduce the characteristic energy scales of fundamental physics. Together, this picture of large extra dimensions and reduced energy scales gives rise to a paradigm commonly called the "brane world". In Lecture # 3 , we will then continue to see how extra dimensions can provide new mechanisms for addressing some long-standing phenomenological problems within the Standard Model or its extensions. To provide concrete examples, I have chosen such diverse topics as proton decay, fermion mass hierarchies, neutrino oscillations, and axion physics. Finally, in Lecture # 4 , we shall discuss some issues that are normally ignored in current treatments of the brane world, but which perhaps may play a role in more sophisticated developments of a complete higher-dimensional theory. I have chosen to raise such issues not only as a way of indicating caution concerning the standard treatments of higher-dimensional theories, but also to stimulate further thinking and research. These lectures were delivered at the 2002 TASI summer school. Since one of the main topics of this school was the physics of large extra dimensions, there were many lectures covering numerous advanced aspects of this subject. These lectures, by contrast, were requested by the TASI organizers to provide an overall, non-technical introduction to the subject. Consequently, there are clearly numerous topics which are omitted in these lectures, but which are as important as those which are included. Chief among these is the possibility of non-flat extra dimensions — i.e., extra dimensions which are warped or have non-trivial (e.g., non-factorizable) geometries. Such extra dimensions have played a huge role in recent discussions of higher-dimensional theories, but will not be covered here. By contrast, these lectures will focus primarily on the simpler case of flat, toroidally compactified extra dimensions. As we shall see, this will be sufficient in order to enable us to deduce most of the main properties that have excited the particle physics community in recent years.

451

1. Lecture # 1 : Extra spacetime dimensions: General properties and bounds We begin with a general discussion of the properties and constraints on extra spacetime dimensions. These properties and bounds will apply for all scenarios involving flat extra spacetime dimensions. 1.1. Original Kaluza-Klein

idea

Why consider physics in more than four dimensions? As indicated in the Introduction, one of the first application of extra spacetime dimensions to fundamental physics dates back to the 1920's, in the original work of T. Kaluza and 0 . Klein. Kaluza and Klein pointed out that if extra dimensions of suitable size exist, then they might provide a possible means of unifying the two fundamental forces which were known at that time to exist, namely gravity and electromagnetism. The Kaluza-Klein argument rests upon one of the important properties of extra dimensions, namely that they provide a way of unifying particles with different spins. To see this, we must first recognize that since the D-dimensional Lorentz group SO(D — 1,1) is larger than the fourdimensional Lorentz group SO(3,1) for all D > 4, each representation of the Z?-dimensional Lorentz group is larger than the corresponding representation of the four-dimensional Lorentz group. This implies that a single representation of the ZJ-dimensional Lorentz group decomposes into several different representations of the four-dimensional Lorentz group all at once. However, different representations of the four-dimensional Lorentz group correspond to different physical particles of different spins. Thus, we see that certain combinations of different particles with different spins in four dimensions can be viewed merely as different components of a single particle in higher dimensions, with a single spin associated with the higher-dimensional Lorentz group. Let us consider a trivial example of this phenomenon. A vector field in five dimensions has five components: V^ = (V0, V\, V2, V3, V4). If we identify the first four components as the directions corresponding to our usual four-dimensional spacetime, then we can decompose this five-dimensional vector inito representations of the four-dimensional Lorentz group, yielding a four-dimensional vector V^ = (Vo, Vi, V2, V3) and a four-dimensional scalar $ = V4. Thus a single Lorentz vector representation in five dimensions has yielded both a spin-1 particle and a spin-0 particle in four dimensions.

452

This is, of course, a trivial example. However, let us now turn to the case originally considered by Kaluza and Klein, concerning Einstein gravity in five dimensions. This situation is sketched in Fig. 2. In such a theory, the fundamental field is the five-dimensional metric gMN, with (M,N) = 0,1,2,3,4. As a symmetric tensor field, gMN has 15 independent components. Decomposed into representations of the four-dimensional Lorentz group, however, we find that gMN yields three different representations: 4

5

,( ) -= g\v „( ) with 0 < /z, v < 3; • a spin-two symmetric tensor field g^J •5) • a spin-one massless vector field Ajj, = g^JA* with 0 < [i < 3; and (5)

• a spin-zero massless scalar field <£> = g\4 . While a spin-two symmetric tensor field gjjj is interpreted as the fourdimensional graviton, a spin-one massless vector field Ajj is naturally interpreted as a photon. (The remaining scalar field combines with the graviton to produce a modified version of Einstein gravity called "Brans-Dicke gravity".) While this examination of the field content is certainly suggestive of these interpretations, we really must verify that these interpretations hold at the level of the actions (or at the level of the equations of motion of the fields). However, beginning with the five-dimensional Einstein-Hilbert (5")

action for the field gMN and decomposing this action into representations of the four-dimensional Lorentz group, we indeed obtain not only the Maxwell action for Ajj, but also the Brans-Dicke action involving gjjj and $. Indeed, the four-dimensional gauge coupling
732

I-'•••'• J

where K§ is the five-dimensional gravitational coupling and R is the radius of compactification. (We will see how to derive relations such as this shortly.) We see, then, that four-dimensional gravity can indeed be unified with electromagnetism into pure five-dimensional gravity! Or, phrased slightly differently, four-dimensional gauge invariance (the symmetry underlying electromagnetism) can be derived from five-dimensional general coordinate invariance (the symmetry underlying Einstein general relativity). While this setup succeeds in unifying both of the fundamental forces known in the 1920's, we now know that there are at least four fundamental forces: gravity (mediated by gravitons); electromagnetism (mediated by

453

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9jxv 4D Brans-Dicke gravity

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41~) photon (E&M)

Figure 2. The basic Kaluza-Klein observation: Four-dimensional Brans-Dicke gravity and four-dimensional electromagnetism can be unified within the framework of fivedimensional Einstein gravity.

photons); the weak nuclear force (mediated by the W^ and Z particles); and the strong nuclear force (mediated by gluons). Thus, unification along these lines a la Kaluza and Klein would require even more extra dimensions. Moreover, it turns out that generating non-Abelian gauge symmetries in this way would also require more complicated geometric compactifications than the simple circle-compactifications we will consider here. In general, the gauge group that is generated through a Kaluza-Klein (KK) reduction of this sort is the isometry group of the manifold on which the extra dimensions are compactified. Thus, if we have 5 extra spacetime dimensions, with each compactified on a circle, then the gauge group generated in four dimensions is U(l)s. (However, we shall see in Lecture # 4 that string theory provides an elegant way of generating non-abelian gauge symmetries even with simple toroidal compactifications.)

454

What is the current status of KK theory? Nowadays, the best modern candidate for unification of all forces and particles in nature is string theory. Needless to say, we will not have time in these lectures to provide a proper introduction to string theory. However, the basic premise of string theory is very simple: all elementary particles are really closed vibrating loops of energy called strings. The length scale of these loops of energy is on the order of 10~ 35 meters (corresponding to 10 19 GeV), so it is not possible to probe this stringy structure directly. This idea has great power, because it provides a way to unify all of the particles and forces in nature. Specifically, each different elementary particle can be viewed as corresponding to a different vibrational mode of the string. A pictorial representation of this idea is given in Fig. 3, where we are schematically associating higher vibrational string modes with string loops containing more "wiggles". From the point of view of a lowenergy observer who cannot make out this stringy structure, the different excitations each appear to be point particles. However, to such an observer, the states with more underlying "wiggles" appear to have higher spin. Thus, in this way we find that string theory predicts not only spin-1/2 and spin-1 states (which can be associated with the fermions and gauge bosons of the Standard Model respectively), but also a spin-2 state (which can naturally

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Figure 3. The basic hypothesis of string theory is that the different elementary particles correspond to the different vibrational modes of a single fundamental entity, a closed loop of energy called a string. In this way one obtains not only spin-1/2 and spin-1 states which can be associated with the matter and gauge bosons of the Standard Model, but also a spin-2 state which can be identified with the graviton. Thus, string theory provides a way of unifying the Standard Model with gravity.

455

be associated with the graviton). Thus, through string theory, we see that the gauge interactions, particles, and also gravity are unified into a common quantized description as corresponding to different excitation modes of a single fundamental entity, the string itself. This form of unification seems to be drastically different from the KK proposal. However, just like the original KK proposal, strings also require extra spacetime dimensions. Specifically, they require six extra dimensions (making a total spacetime dimensionality D = 10) if they are weakly coupled, and seven extra dimensions (making D = 11) if they are strongly coupled. Thus, in this respect, string theory is the modern incarnation of the original KK idea: instead of having particles vibrating in higher dimensions, we have strings vibrating in higher dimensions. Indeed, one of the ways in which string theories give rise to gauge symmetries is through a direct implementation of the KK idea. We shall discuss this in more detail in Lecture $ 4 . Thus, the original KK idea finds a natural home in string theory, and continues to play a role in modern theories of unification.

1.2. The compactification procedure: Mode expansions, Kaluza-Klein reductions, low-energy signatures

and

Given this motivation, let us begin our study of extra spacetime dimensions by understanding literally what an extra dimension entails. Clearly, in our everyday world, we are not familiar with any additional space dimensions beyond those of length, width, and height. These are dimensions of sufficient size that we can walk into them and experience them fully as spatial degrees of freedom. Therefore, if any extra space dimensions exist beyond those that we already know, they must be of sufficiently small size that they are not observable with the naked eye. (We will shortly define "naked eye" more scientifically.) In other words, they must be compactified down to a small length scale. What does it mean to compactify an extra dimension? The idea is simple: we imagine that our extra dimension takes some compact shape, such as a circle, such that by travelling along that extra dimension we quickly return to our original location. Thus, taking a circle compactification as our example, we imagine that at every point in spacetime, there exists an additional circle of radius R which is orthogonal to all of the known dimensions. This is illustrated in Fig. 4. In this example, spacetime has the topology A^4 x Si where M4 is four-dimensional Minkowski space and Si

456

is a circle of radius R. We then imagine that R is sufficiently small that that we cannot probe this additional structure in our spacetime. Thus, this extra dimensions is essentially unobservable.

0 I fi" 4D spacetime Figure 4. An extra dimension compactified on a circle of radius R. At every point in the four-dimensional spacetime (here collapsed onto a single line), there exists an "orthogonal" dimension compactified on a circle of radius R.

More generally, we can compactify 8 extra spacetime dimensions by formulating a theory on M.^ x K where M4 is our four-dimensional Minkowski spacetime and K is any (^-dimensional compact manifold. For example, for 5 = 2, we can take K = T^ (a two-torus) or K = S\ x S\ (two independent, orthogonal circles, one for each compactified dimension). Note that S\ x Si is a special case of Ti- However, we can also take K = S2 (a two-sphere). Many other topologies are possible, even for S = 2. For example, two-dimensional surfaces with boundaries include a rectangle and a cylinder. As we shall see later, consideration of manifolds with boundaries (more generally known as "orbifolds") will be necessary for phenomenological reasons. We should also mention that there are more general forms of compactification beyond the simple factorizable M4, x K structure. This product structure indicates that the properties of the extra spacetime dimension are the same no matter the location of this extra dimension in the physical four-dimensional Minkowski spacetime. However, this too is an assumption. Later, in Lecture # 4 , we shall see that the possibility of non-factorizable geometries leads to some extremely interesting physics. In general, the fact that K is compact means that we must impose boundary conditions appropriate for the specific choice of K. We shall generally let x^ indicate the coordinates on .M4, and let yl indicate the coordinates of K. While each of the a;M are valued along the infinite real line, our boundary conditions on the yl indicate the specific choice of compact-

457

ification manifold K. For example, in the case of a single extra dimension compactified on a circle, we have only a single y coordinate. Compactification on a circle then requires that we impose periodic boundary conditions under y —> y + 2irR. Indeed, it is this which defines the radius R of the circle. Note, in this connection, that the volume (i.e., the complete spatial extent) of the extra dimension is given by the circumference 2TTR. (By contrast, the quantity irR2 has no meaning here, since it ultimately depends on the manner in which this periodic extra dimension is embedded in some larger spacetime. Only the quantity 2irR has an intrinsic topological meaning.) Given the specific spacetime geometry, our next task is to construct suitable wavefunctions on this space. For example, in the circlecompactification case, all wavefunctions <&(x^,y) must be periodic under y —> y + 2irR, implying that ^(x,i,y) must have a mode expansion of the form oo

1

$(*",!/) = -/== V2irR

Y,

M^)eKP{iny/R).

(1.2)

n=_oo

This is the most general form for the wavefunction consistent with the compactification symmetries in this case. The normalization prefactor in Eq. (1.2) will be discussed shortly. In order to interpret the coefficients ^>n(a:'t), let us imagine that the five-dimensional field $(x M , y) is a complex Klein-Gordon field of mass mo. The five-dimensional action for such a field is given by TR

dy i(0 M *)*(0 M *)-^m§$*$

d4x

S5 =

(1.3)

where M is the five-dimensional spacetime index such that the range 0 < M < 3 corresponds to our ordinary four-dimensional Minkowski spacetime, with M = fi, and xM = y for M = 4. Our procedure is now to derive a corresponding four-dimensional action by substituting the mode expansion (1.2) into Eq. (1.3) and explicitly performing the integration over the compactified coordinate y. First, inserting the mode expansion (1.2) into Eq. (1.3), we obtain 1

S5 = ^

r

/-2TTR

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2TTR J

dy

J0

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458

"

\rnlJ2
.

(1.4)

ran

The first term comes from the kinetic term in Eq. (1.3) with M — fi < 3, while the second term arises from the kinetic term with M = 4. However, we now recall that exponential wavefunctions with different frequencies are orthogonal: dy

ei(n~m)y/R

27TRSmn

=

.

(1.5)

'0 Jo

Performing the y-integrations thus gives rise to the four-dimensional action

S4 = Jd^xl1- 5>„03^„) " \^ 2 ^

2

l' 2 1

n

0 „ l . (1.6)

Interpreting this action is straightforward, since this is nothing but the action of an infinite tower of Klein-Gordon fields
m2 = ml + jp.

(1.7)

The lightest field is thus 4>o, with mass mo, while 4>n and _„ are degenerate for each n. As evident from the wavefunction decomposition in Eq. (1.2), the index n is the (discrete) quantum number corresponding to (quantized) momentum in the (compactified) fifth dimension, with p$ = n/R; we thus see that our mass formula in Eq. (1.7) is nothing but the statement that the total mass is given by m§ + p | . Note that if we had compactified S extra dimensions on circles of radii Ri (i = 1,..., 5), we would have obtained the same action (1.6) with the masses (1.7) replaced by s

2

,2

ESi=l

l

^

This is an example of a Kaluza-Klein reduction, and the procedure we have just outlined is completely general. We begin with our £>-dimensional action SD, written in terms of our D-dimensional quantum fields. These fields may be scalars, spinors, vectors, or any bona-fide representation of the D-dimensional Lorentz group. The higher-dimensional action may also contain arbitrary interaction terms between these fields. In general, this action should respect all of the symmetries (such as Lorentz invariance and gauge invariance) appropriate for the fundamental higher-dimensional theory. Given this parent theory, we then compactify it by choosing an arbitrary ^-dimensional compactification manifold K, where S = D - 4.

459

First, we decompose the Lorentz structure of our D-dimensional fields as discussed in Sect. 1.1, writing these fields as sums of irreducible representations of the /cmr-dimensional Lorentz group. This will yield a spectrum of fields of different four-dimensional spins. Next, we perform a mode expansion for each field as in Eq. (1.2), where the exponentials are generally replaced by the eigenfunctions f{yi,y2,---,ys) of the Laplace operator on the manifold K. We then substitute these mode expansions into our higherdimensional action Sp, and integrate over the compactification volume to obtain the corresponding effective four-dimensional action S4: S4 =

f d6y SD .

(1.9)

JK

In performing this integral, the Laplace eigenfunctions will continue to exhibit an orthogonality relation analogous to that in Eq. (1.5) since they continue to be the eigenfunctions of a Hermitian operator. This will have the net effect of ensuring conservation of higher-dimensional momenta for each term in the resulting four-dimensional Lagrangian, so that any terms m4>n...4>p in the resulting four-dimensional Lagrangian will satisfy m + n+ ... +p = 0. Given our resulting four-dimensional Lagrangian, we then can read off the complete set of fields, effective masses, and interaction terms appropriate for the effective theory in four dimensions. Note that in general, the four-dimensional KK mass terms will emerge from the kinetic terms in the original higher-dimensional Lagrangian. The resulting KK mass spectrum is thus determined by the eigenvalues of the appropriate Laplace operator on the manifold K. Given this procedure, it is now also easy to understand why we chose the overall (volume) normalization given in Eq. (1.2). As evident from the action (1.3), the $ field has mass dimension +3/2 in five dimensions; in general, the mass dimensions of scalar, fermion, and gauge fields in D dimensions are given by m

= [A] = \{D-2),

[*] = \{D-l).

(1.10)

However, even though the $ field has mass dimension +3/2, we wish the individual KK modes n to have the mass dimension + 1 , as appropriate for-a four-dimensional interpretation. This explains the factor of VR in the prefactor in Eq. (1.2). The remaining factor of y/2n is inserted so that the kinetic terms for the resulting n fields in four dimensions will be appropriately normalized after the integration over y. In general, the appropriate prefactor for a KK mode expansion is [Vol(K)] - 1 / 2 where K is the compactification manifold.

460

The mass formula (1.7) simplifies in cases where the extra dimension is small compared toTOQ~. For R~1
mn

« m0 + —

(1.11)

where each excited mass level is doubly degenerate. We thus have an infinite tower of approximately equally spaced mass levels, with the ground state shifted by an overall constant mo- The lowest-lying state with n = 0 is called the Kaluza-Klein zero mode or ground state, whereas the <\>n for all n ^ 0 are the excited KK states. If the accessible energy in our experiment is much less than -R -1 , we will not have sufficient energy to excite the higher KK modes. In such cases, we would only observe the KK zero mode 4>o- This is therefore nothing but the the usual four-dimensional state, and TOO is nothing but the usual four-dimensional mass of the four-dimensional particle. The threshold energy for detecting extra dimensions is therefore ~ R~l, since this is the threshold for producing the first excited KK state. These results are in complete accordance with our expectations based on elementary quantum mechanics. An extra dimension compactified on a circle is mathematically equivalent to a particle placed into a periodic box of length L = 2TTR. In this analogy, the "compactification" is nothing but the imposition of the periodicity arising from the walls of the box. As we expect, the momenta inside the box are quantized in units of 1/L, and we do not expect to be able to probe the features of this box if our energy scale is much less than 1/L. Indeed, it precisely by mapping out the resonance pattern of momentum wavefunctions inside the box that we learn of the existence of the box, and eventually hope to determine its volume and shape. Thus, to summarize, we see that there are two primary signatures of extra spacetime dimensions: • the appearance of an infinite tower of Kaluza-Klein states for each known four-dimensional state; and • the appearance of new fields of different spins which combine with our known fields to fill out irreducible representations of the higherdimensional Lorentz group. The first feature implies that there will be an infinite tower of heavier KK excitations corresponding to each four-dimensional field. In other words, there will be a KK tower of electrons, a KK tower of quarks, a KK tower of photons, etc., with the ground state of each tower corresponding to our usual four-dimensional particle. Note that each state in the KK tower

461

will have exactly the same quantum numbers (same spins, same couplings, etc.) as those of their ground-state particle. These KK excitations are thus essentially "echoes" off the extra dimensions, with the higher excitations corresponding to the higher momentum modes in these extra dimensions and leading to a correspondingly higher KK mass in our effective four-dimensional theory. By contrast, the second feature implies we would naively also expect to see additional fields of varying spins, as necessary to fill out representations of our higher-dimensional Lorentz group. As expected, each of these new fields will also have its own KK tower of states. These conclusions hold in the general case where we compactify on a smooth manifold K. Thus, these conclusions apply for the simple case of circle-compactification that we have been discussing here. However, in many cases the second property (the existence of particles of new spins) causes phenomenological difficulties, since it tends to require the existence of light scalar particles associated with our usual four-dimensional fields. Unfortunately, the bounds on such scalar particles are very severe. Thus, for phenomenological reasons, it is often necessary to compactify in such a way that we evade certain aspects of these predictions. We shall discuss how this can be done in the next section. One might wonder why our original higher-dimensional action should respect D-dimensional Lorentz invariance. After all, the spacetime on which we are formulating our D-dimensional theory is not MD, but rather M4XK where K is presumably a compact space. This choice therefore already breaks the D-dimensional Lorentz invariance — certain directions of the D-dimensional geometry are being compactified while others are not. Why then should D-dimensional Lorentz invariance play such a critical role? For many applications, it turns out that higher-dimensional Lorentz invariance need not be imposed. For example, later in this lecture, we will consider different types of extra dimensions in the presence of branes; the presence of a brane will break the higher-dimensional Lorentz invariance explicitly. However, for the sorts of compactifications we are discussing here, a full D-dimensional Lorentz invariance should indeed emerge in the UV limit where our characteristic energy scales are large compared to all compactification radii (and also large compared any curvature scales inherent within K, if K is a non-flat manifold). In this UV limit, our particles have extremely small Compton wavelengths compared to the compactification and curvature radii, and thus experience K as though K were effectively uncompactified and flat. Phrased slightly differently, in the highenergy limit the KK spectrum appears relatively continuous, signalling the

462

effective emergence of an uncompactified extra dimension like those of .M4. In such cases, the full D-dimensional Lorentz invariance should be restored. Finally, there are other symmetries that our D-dimensional action might also exhibit. For example, if our four-dimensional action is to exhibit an N = 1 supersymmetry, then it turns out that our D-dimensional action will need to exhibit an extended N > 1 supersymmetry, with the precise value of N depending on D. We will see an explicit example of this in Sect. 1.6. In general, our D-dimensional action must exhibit whatever symmetries are necessary in order to yield a four-dimensional action of the sort that we desire; the particular choice of the D-dimensional action then describes how we choose to embed our four-dimensional theory into higher dimensions.

1.3. Orbifold

compactifications

In the previous section, we discussed compactifications where K is a manifold (i.e., a smooth surface with no special points or boundaries), and we concentrated on the explicit case of compactifications on circles. However, as we shall see later, another important class of compactifications are on manifolds with special points, such as endpoints or boundaries. For example, rather than compactify a single extra dimension on a circle, we might choose to compactify an extra dimension on a finite line segment of a given length. Since the two endpoints of the line segment are not identified, a line segment is topologically distinct from a circle; we shall see, moreover, that compactifications on line segments give rise to radically different low-energy phenomenologies than compactifications on circles. Likewise, if we have two extra dimensions, we could choose to compactify each extra dimension on a line segment (thereby taking the compactification space to be a bounded rectangle), or one extra dimension on a circle and the other on a line segment (thereby taking the compactification space to be a finite-length cylinder). In general, a manifold with special points such as endpoints or boundaries is called an orbifold. In this section, we shall discuss how the above Kaluza-Klein procedure is modified for compactifications on orbifolds. The proper mathematical construction of an orbifold is to begin with a manifold K, and then to "mod out" by a discrete symmetry T. Here the phrase "modding out" refers to imposing a further discrete identification between points on the manifold. The resulting quotient space K/T is the orbifold. As an example, let us consider our standard case of a circle compactification, where K is a circle of radius R. This implies the periodic boundary

463

condition y <-> y + 2irR. Given this boundary condition, we may continue to regard y as taking values along the whole real number line; alternatively, we can restrict our attention to a fundamental domain 0 < y < 2TTR, with the endpoints of this range identified. However, let us now impose the additional discrete Z2 identification Y : y <-> —y. (Note that as a result of the circle boundary condition, this is equivalent to Y : y <-> 2irR — y.) circle of radius R

line segment = circle/ Z, 2 fixed points

y=jrR y=7tR

y=0

Figure 5. The orbifolding process: the circle S1 defined by y <-> y + 2nR is subjected to a Z 2 identification y <-> —y, becoming a line segment S1 l~S-i with two fixed points at y = 0 and y = -KR.

What does this do to our original circle manifold K? Clearly, the resulting space K/Y no longer consists of the full range 0 < y < 2nR, since points in this range with y > -KR are now identified with points y < TTR. The fundamental domain is thus reduced to 0 < y < irR. Note that unlike the case of the circle, the endpoints of this smaller domain are not identified with each other by either the original circle identification or the new orbifold identification. Instead, they are "fixed points" with respect to the discrete symmetry Y : y —> — y: Y:

(y = 0) -» (y = 0) (y = nR) -> (y = -TTR)

<-> (y = nR)

(1-.12)

where we have used the circle identification in this last step. Thus, our resulting manifold K/Y has the topology of a line segment of length irR, formed by "folding" the circle in half on itself. This folding process is illustrated in Fig. 5. Since S1 is used to denote the circle, S1 /Z2 is often used to denote the line segment, where Z2 indicates any discrete Z2 symmetry Y. The presence of the special fixed-point boundaries is what prevents S1 )7Li from being a manifold in its own right. The above remarks are completely general. When we mod out a manifold K by a discrete symmetry T, there are usually a set of fixed points y,.

464

The resulting space K/T would be a manifold except for these special fixed points. The resulting space is thus an orbifold. The number of fixed points is often called the "order" of the orbifold, and the volume of the orbifold is always reduced compared to that of the manifold. How then do we compactify theories on an orbifold? Once again, let us concentrate on the case of a line segment S1 jTLi. We have already seen how to compactify on a circle. We thus must now consider the effects of introducing the discrete identification T : y —* —y. Clearly, since we are now interested in the Z2 action T, our first step should be to rewrite our KK mode expansions in Eq. (1.2) in terms of eigenfunctions which are even (denoted +) or odd (denoted —) under T: <S>(x»,y)

-.

1

OO

i=4 (^) + E^ +) (^) cos (f)

I-KR

v

+)

n— 1

OO

^

r

^rV)sin(f) " v" ' — \R.

(1.13)

Note that this is nothing but a rewriting of the original KK mode expansion. In each case, we have a single KK zero-mode (in this case 0 ), as well as an infinite tower of doubly degenerate excited KK modes. The explicit mapping between the original KK modes cpn in Eq. (1.2) and the modes 4>n ' in Eq. (1.13) is given by: £o +) = 4>o ,

^ 0 = -ft (<j>„ 0™ + + (/>_„)

4>\^ ^ i > o0 = = --=^( ( / > „ - -n) . (1.14)

Since this is just a reshuffling of our KK modes, we have not yet affected the physics in any way. Specifically, this is still compactification on a circle. Let us now turn to the original five-dimensional action. It is here that we will now impose our orbifold Z2 identification, requiring the five-dimensional action to be even under the discrete parity symmetry y —-> — y. If $ denotes a five-dimensional Lorentz scalar field, then the five-dimensional Klein-Gordon action (1.3) does not require a particular parity for $. However, if this parity along the fifth dimension is to be a good symmetry, then we must choose $ (which presumably corresponds to a physically observable quantum field) to have a definite parity. Depending on our choice, this then implies that we are forced to eliminate roughly half of our Kaluza-Klein modes: •t ih • / e v e n if $ is < , , >- o d d .

{ ] ^ for all n > 0 ,-, , ^ + + / 4> n = 0 then we must set < ^,, N (1.15) I 0 .

465

This "orbifold projection" thus removes roughly half of the degrees of freedom of the theory, signifying that we are cornpactifying on an orbifold rather than on a circle. The choice of whether $ is taken even or odd in this case is completely arbitrary, governed by our ultimate phenomenological goals for the field . We shall see explicit examples later in this lecture. However, in either case, our resulting Kaluza-Klein tower is now only singly degenerate. Moreover, if we choose $ odd, then we lose our zero-mode ground state as well. This orbifold projection also has another important effect on the Lorentz structure of our higher-dimensional fields. To see this, let us imagine that $ actually denotes a five-dimensional gauge vector field AM where 0 < M < 4. The kinetic term for such a field then takes the usual form d xdy where FMN = 8MAM

— 8MAM

1

771

TPMN

(1.16)

+ ••• is the five-dimensional field-strength

tensor. Let us consider the mixed component F^ = d^A^ — 84 A^ + .... Clearly, since c?M is even under our parity symmetry while 84 = 8y is odd, we see that A^ and A4 must be chosen to have opposite parities! Thus, even though A^ and A4 are part of the same Lorentz multiplet, either A^ can have zero modes or A4 can have a zero mode, but not both. Indeed, at the level of the zero modes, the higher-dimensional Lorentz symmetry is broken. Similar conclusions can be drawn for fields of higher spin as well. Note that the entire five-dimensional action must be invariant under the Z2 discrete symmetry. This includes not only kinetic terms, but also possible interaction terms. Thus, we see that imposition of an orbifold symmetry may be a powerful way of removing certain unwanted interaction terms from a higher-dimensional Lagrangian. Having chosen the parities of our fields, the rest of the KK reduction procedure follows as before. For example, we obtain exactly the same KK mass spectrum. The only subtlety is that the integral over the compactified dimensions is to be performed over the full circle 0 < y < 2-KR, and not over the reduced orbifold volume 0 < y < irR. The easiest way to see this is to realize that we require an integration over the full range 0 < y < 2TTR in order to maintain the orthogonality of our Kaluza-Klein modes:

466

for all rn, n € Z + . Note that these normalizations are also consistent with the normalizations in Eq. (1.13), guaranteeing that each of our KK modes ^n>o w ^ have a properly normalized kinetic term in the resulting effective four-dimensional action. Even though we have illustrated the orbifold procedure for the simple case of compactifications on a one-dimensional line segment S1 /~Z-2, these properties tend to be quite general and hold regardless of the manifold and the discrete symmetry. For example, if V corresponds to be a Z3 symmetry of the original manifold K, then compactification on the orbifold K/Y would tend to remove a third of the excited KK levels that would have arisen in a straightforward compactification on the manifold K. Moreover, the zero mode would survive for fields which are neutral {i.e., eigenvalue +1) under the action of V. Thus, to conclude, we see that compactification on an orbifold rather than a manifold generally has two important effects on the Kaluza-Klein spectrum: • a severe reduction in the total numbers of KK states; and • elimination of the zero mode for fields which transform non-trivially under the orbifold discrete action. As we shall see later, this elimination of the zero mode for certain fields is a powerful way of using compactification to break certain symmetries at the level of the zero modes (which, we recall, may be the only observable fields at low energies). Indeed, this procedure is often called "symmetry breaking by orbifolds", which stands in stark contrast to the Higgs mechanism but nevertheless retains many of the advantageous features of Higgs-induced spontaneous symmetry breaking. Moreover, as we shall see in subsequent lectures, the orbifold symmetry T can also be used to restrict the interaction terms which might otherwise exist in the original higher-dimensional Lagrangian. Thus, compactification on orbifolds rather than manifolds will play a large role in our attempts at realistic higher-dimensional modelbuilding.

1.4. Types of extra Thus far, we have sion, one in which of extra dimension forces and particles

dimensions

been considering the simplest type of extra dimenall forces and particles propagate. This is the type originally considered by Kaluza and Klein. Since all "feel" this extra dimension, we shall designate such an

467

extra dimension as "universal". For such an extra dimension, all particles accrue Kaluza-Klein excitations. There are, however, additional types of extra dimensions which may be considered. Specifically, recent developments in non-perturbative string theory predict the existence of solitonic membranes (or Dirichlet branes, abbreviated "D-branes") to which various gauge forces can be restricted. These branes may be imagined as dynamical, fluctuating hypersurfaces floating in the extended D-dimensional spacetime. These hypersurfaces may be of any dimensionality p < D - l ; when necessary, we shall specify the dimensionality of the brane by referring to a Dp-brane. (Here p traditionally refers to the number of space dimensions only, which is why p < D — 1 rather than p < D.) It is, of course, well beyond the scope of these lectures to discuss why and how such D-branes emerge in string theory, or how they manage to trap various forces on their surfaces (or "worldvolumes"). However, the important point is that such D-branes exist, and that they have this magic property. Given the existence of such D-branes, new types of extra dimensions become possible. For example, let us begin by considering what happens if there exists a D3-brane which is capable of trapping all of the gauge forces. In this case, all of the gauge bosons, as well as particles which carry gauge charges with respect to these gauge bosons, are restricted to the brane. They therefore fail to experience any extra dimensions beyond the four spacetime dimensions of the D3-brane; they are "trapped" on the brane, and cannot propagate off the brane into the so-called D-dimensional "bulk" of the full D-dimensional spacetime. Thus, in the effective four-dimensional theory, these gauge bosons and other charged particles do not accrue any infinite towers of KK excitations associated with these extra dimensions in the bulk. By contrast, states which are completely neutral with respect to these gauge forces, such as the graviton, are free to wander off the brane, and experience the full D-dimensional spacetime. For this reason, these sorts of extra dimensions which are perpendicular to a brane are often called "gravity-only" extra dimensions: the graviton accrues KK states from these extra dimensions, but the Standard-Model particles do not. These extra dimensions are illustrated in Fig. 6. Other possibilities exist if the D3-brane traps only some forces [e.g., the SU{2) color force associated with the strong interaction], while leaving other gauge forces [such as SU(2) x U(l) electroweak forces] free. Such a brane therefore traps gluons and quarks, but fails to trap leptons which are colorless. (Because mesons and baryons are composites of trapped quarks,

468

$

&

gauge forces and particles restricted to "brane"

only gravitational force can leave "brane" and propagate in "bulk'

Figure 6. A "gravity-only" extra dimension. Gauge forces and particles are restricted to the brane, while only the gravitational force can leave the brane and propagate in the bulk.

they are also presumably trapped.) However, gravitons would again be free to wander throughout the higher-dimensional space. It is also possible to imagine several D3-branes, one trapping SU(3) and another trapping SU(2) x U(l). Gluons would then be trapped to the SU(3) brane, while the W±, Z, and leptons would be trapped to the electroweak brane. Quarks would have to be simultaneously trapped on both branes — i.e., trapped at the location of their intersection. Indeed, having such branes intersect becomes a model-building requirement in order to give rise to light quarks. Thus far, we have only discussed D3-branes. In general, however, forces can be trapped on Dp-branes for arbitrary p < D — 1. In such cases, the particles trapped on the Dp-brane will accrue infinite towers of KK excitations associated with the p — 3 extra dimensions in the brane, but will not have any KK excitations associated with the D — p dimensions in the "bulk" perpendicular to the brane. (These are often called "longitudinal" and "transverse" extra dimensions respectively.) Of course, such a Dp-brane cannot be imagined as a flat, infinite hypersurface for p > 3, since ultimately the p — 3 extra dimensions in the brane will need to be compactified (just as the bulk extra dimensions also need to be

469

compactified). Thus, such a brane will typically take a "wrapped" shape {e.g., like a higher-dimensional cylinder, with the usual M4 running along the infinite direction of the compactified extra dimensions in the brane running around the circumference), or a truncated shape {e.g., like a semiinfinite rectangle where the infinite edge corresponds to M4 while the finite edge corresponds to a compactified space with a boundary). Note that since gravity is nothing but a reflection of the geometry of the spacetime, it is impossible to "trap" gravity except by curving the spacetime itself. We will discuss such theories briefly in Lecture # 4 . Such "Randall-Sundrum theories" give rise to very interesting phenomenologies as well as important theoretical insights, but are beyond the scope of these lectures. We see, then, that the existence of D-branes gives rise to a very rich set of geometric possibilities for higher-dimensional model-building. We can imagine branes of various types and dimensionalities, extending along different directions and wrapping and/or intersecting in a variety of different ways. By changing the sizes and compactification patterns of the overall spacetime and the branes that reside therein, we can construct models of incredible richness and variety. This, then, becomes the language of modern model-building in higher dimensions. We close this section with three important comments. First, it is important to realize that our D-branes are not infinitely thin objects; like any soliton, they have a characteristic thickness which is ultimately set by the underlying string scale. However, for many purposes it is legitimate to approximate such branes as being infinitely thin, and to incorporate the associated trapping of forces and particles into the higherdimensional Lagrangian by using Dirac (^-functions. For example, if our full spacetime is five-dimensional but there exists a D3-brane located at y = 0, we would write our action as S5 =

/ d4x / dy {£buik + CbTa,ne5{y)}

(1.18)

where £buik is the five-dimensional Lagrangian for the bulk fields £brane is the four-dimensional Lagrangian for the trapped brane fields. For example, if the D3-brane traps all of the gauge forces of the Standard Model, then £buik might be the Lagrangian for general relativity while £brane would be the Lagrangian for the Standard Model. This setup originally emerged in work of Hofava and Witten, and forms the basis of the so-called "ADD model" (due to Arkani-Hamed, Dimopoulos, and Dvali). Upon KaluzaKlein reduction as outlined above, the presence of the Dirac ^-function

470

then eliminates the KK excitations for the Standard-Model particles. Of course, in cases where the brane thickness is important (and we shall see some examples of this in Lecture #3), £brane would be replaced by a higherdimensional Lagrangian and the Dirac 5-function would be replaced by a brane profile function with the appropriate thickness. Our second comment is that D-branes are dynamical objects. While it is tempting to consider them as rigid and immovable, D-branes are actually fluctuating with degrees of freedom of their own which can be associated with their positions and momenta. Once again, the energy scale for such fluctuations is set by the underlying string scale, and thus may be sufficiently high that they the dynamics of the brane itself may be ignored for low-energy phenomenological applications. However, since the brane has its own stress energy, it warps the spacetime around it just as any massive object would. In these lectures, we shall ignore this effect and consider the extra dimensions perpendicular to brane to be flat (even though compactified); however, in some circumstances (such as the case where there is only one extra dimension perpendicular to the brane), this warping effect is actually quite significant and important. As indicated earlier, we shall discuss such Randall-Sundrum scenarios briefly in Lecture # 4 . Finally, even though we have presented D-branes as mysterious objects capable of trapping various kinds of particles, it is useful to have a rough sense of the underlying string-theoretic ideas that give rise to these pictures. In Fig. 3, we sketched the basic idea behind string theory, representing each fundamental particle as a different vibrational mode of a closed string. This picture is valid for closed-string theories (such as the Type II and heterotic strings), but there also exist string theories based on a mixture of open and closed strings. These are the so-called Type I strings. (It is not possible to formulate an interacting theory consisting only of open strings, since an open string theory always contains a string configuration in which the endpoints are joined together, forming a closed string.) In fact, as we shall discuss in Lecture # 2 , the most natural ways to realize our theories of extra dimensions in string theory are through Type I strings. In Type I string theory, particles which carry gauge charges are represented by open strings, and the actual charges themselves are associated with the endpoints of these open strings. This type of charge is often called a "Chan-Paton charge" in the string literature. Indeed, one can form an analogy with strong interactions by picturing a meson as an open string: the endpoints of the string correspond to the quark and antiquark, while the string itself corresponds to the gluonic flux tube. (It was, in fact, this

471

analogy that gave rise to string theory in the late 1960's as a theory of strong interactions.) By contrast, a closed string is one which contains no endpoints, and which must therefore be neutral with respect to these types of charges. For example, while all quarks and leptons and gauge bosons of the Standard Model carry gauge charges of one sort or another, and hence are typically represented within the Type I framework by open strings, gravitons are neutral and thus are represented by closed strings. a This contrast between open strings and closed strings is relevant for us because D-branes are the surfaces on which the endpoints of all open strings must end. In other words, open strings must be attached to (or "trapped" on) D-branes. Closed strings, by contrast, have no endpoints, and are thus free to wander off the D-brane into the bulk. Needless to say, there are many constraints which govern the types of D-branes and geometric configurations that can exist within a given Type I string model. These constraints govern all sorts of parameters such as the total number of D-branes, which gauge forces they may trap, the spectrum of Dp-brane dimensionalities (i.e., the allowed spectrum of values of p), and even which dimensions they may occupy, how they may overlap, and the angles, if any, of their intersections. All of these features must be chosen in order to satisfy numerous string-theoretic constraints, chief among them the cancellation of certain dangerous anomalies. Clearly, the art of Type I string model-building is a study unto itself which we cannot enter into here. Thus, in order to make progress, D > 4 model-builders can take either of two approaches. They may follow a "top-down" perspective, building a fully self-consistent Type I string model and then investigating its low-energy phenomenological properties. Such model-builders are typically a

Note that although the photon is neutral, it must also be represented by an open string. This arises because the photon, like all gauge bosons, actually transforms in the adjoint of the corresponding group; it is only an "accident" of the abelian nature of the C/(l) gauge group that the photon happens to be electrically neutral. Similarly, there may also be other particles (such as the right-handed neutrino) which are neutral with respect to all Standard Model gauge symmetries. Whether such particles are represented by open or closed strings depends on the particular Type I string model under study. If the Standard Model gauge symmetry is realized by breaking a grand unified (GUT) symmetry such as SO(10) or SU(5), the right-handed neutrino will carry appropriate gauge charges under this larger GUT symmetry and must therefore emerge as an open string state. However, if the Standard Model gauge symmetry is realized directly at the string scale without passing through an intermediate GUT symmetry, both options are possible and depend on the particular string construction. Similar comments also apply to other gauge-neutral particles such as axions. We shall discuss some of the phenomenological implications of these choices in Lecture # 3 .

472

string theorists or string phenomenologists. Alternatively, model-builders can follow a "bottom-up" perspective, studying various phenomenological properties of theories in extra dimensions, developing scenarios for solving phenomenological problems in low-energy particle physics using KaluzaKlein states, imagining various ad hoc brane and then determining whether there might exist a suitable brane configuration which might give rise to these hypothetical models. Such studies usually take place within the language of a higher-dimensional effective field theory, borrowing only those ideas from string theory (such as t h e notion of t h e brane and perhaps a few stringy constraints concerning allowed brane configurations) as necessary. Of course, one hopes t h a t these two approaches eventually meet, t h a t a given phenomenologically interesting brane configuration built'through a "bottom-up" analysis can ultimately be demonstrated to arise from a bonafide T y p e I string construction. Since higher-dimensional field theories are intrinsically non-renormalizable, one hopes t h a t such an embedding into a fully consistent Type I string model will be possible at some energy scale because string theory provides one of the few known finite theories in higher dimensions.

1.5. Bounds

on extra

dimensions

Having discussed general theoretical possibilities concerning the types and properties of extra dimensions, let us now t u r n to some experimental constraints. Perhaps the most important experimental fact concerning e x t r a dimensions is the following: at the present time, there exists absolutely no experimental evidence that any extra dimensions of any type exist. b Given our previous discussion in Sect. 1.2, what this means is t h a t the lightest Kaluza-Klein excitations have never been discovered for any particle. This refers not only to a direct discovery (e.g., through direct production of K K states), but also to an indirect discovery (e.g., through loop effects or new, effective interactions t h a t they may induce). However, turning this into a bound on extra dimensions depends on the type of extra dimension under study. "Universal" extra dimensions of the Kaluza-Klein type, felt by all forces and all particles, would in principle be accessible through collider experiments. Since colliders have probed the b

Of course, the same may also be said of supersymmetry, grand unification, string theory, or nearly any extension to the Standard Model. As a general rule, there is never any experimental evidence that something exists... until, of course, it is discovered!

473

structure of matter down to a length scale of approximately 10~ 18 meters, or equivalently probed the spectrum of particles up to an energy scale near the TeV range, our failure to have detected extra dimensions through collider experiments implies that any possible extra dimensions of this sort must have a radius R-1

> 0(1) TeV .

(1.19)

Here 0(1) denotes a factor of one to several (e.g., 3 or 4), depending on the particular model under study. If the Standard Model (or a subsector of the Standard Model) is trapped on a Dp-brane, this bound continues to apply for those p — 3 longitudinal dimensions in the brane. This is because those gauge bosons and charged particles trapped on the brane necessarily accrue Kaluza-Klein excitations corresponding to the extra dimensions in the brane, and thus could potentially lead to observable effects at colliders. Thus, the bound (1.19) applies quite generally for any longitudinal extra dimensions (i.e., extra dimensions in the brane). It is, however, sometimes possible to evade this constraint slightly. For example, there exist certain models in the literature in which there exists a conserved quantum number for KK states which implies that any excited KK states must be produced in pairs. In such cases, it therefore becomes twice as difficult to produce KK excitations, which implies that the experimental bound on the size of the corresponding extra dimensions becomes roughly half as severe. In such cases, one can then potentially tolerate i ? _ 1 > 500 GeV. However, the important point is that such extra dimensions are generally bounded in the TeV range. The remaining extra dimensions are of the "transverse" type, perpendicular to the brane(s) on which the Standard-Model gauge forces and particles are trapped. Since no Standard-Model forces or particles feel these extra dimensions, these extra dimensions are immune to collider probes. Instead, the primary effects of such extra dimensions are gravitational, since it is often only the graviton which can propagate in their bulk. We therefore must ask: down to what length scale have we tested gravity? Such tests could take the form of direct tests of the Newtonian gravitational inverse square law (e.g., through Cavendish-type experiments), or indirect tests through astrophysics or cosmology. In this case, not surprisingly, the experimental bounds are much weaker. Let us first consider Cavendish-type experiments. The normal four-dimensional graviton state is the particle responsible for ordinary

474

Newtonian gravity, with the inverse-square force law arising from graviton exchange. If the graviton is now considered to be merely the zero-mode of a potential KK tower of gravitons, we must consider how exchanges of the excited KK gravitons will affect the inverse-square law. In general, if there are gn graviton states with masses mn = n/R at the n t h KK level, the corresponding corrections to the Newtonian gravitational potential take the form

1

V(r)

+ XI 9ne~

(1.20)

where GN is the usual four-dimensional gravitational Newton constant. For length scales r S> R, the corrections from all but the first excited KK graviton will be exponentially suppressed. We therefore find

V(r) = -9lfMl

[i +

fllC-/«]

.

(1.21)

Precision laboratory Cavendish-type experiments have been performed, searching for deviations from Newtonian gravity of just this form. Although our main interest is in Kaluza-Klein gravitons as the origin of such deviations, there are many other effects which could potentially produce corrections of this form as well. Results are therefore usually quoted simply as excluded regions in the parameter space of (gi,R). The most recent plot summarizing the results of various experiments is shown in Fig. 7. Since our KK graviton theories have gi ~ 0(1) (depending on details to be discussed shortly), the excluded regions in Fig. 7 imply the constraint R < 0.1 millimeter

<=>

R'1

> O(10~ 4 eV) .

(1.22)

This may seem like a very weak constraint. Basically, this constraint indicates that despite years of intense, sophisticated experimental effort, we simply do not know much about the behavior of gravity at length scales shorter than fractions of a millimeter. However, it must be remembered that gravity is an extremely weak force. Indeed, by definition, the couplings of gravitons to matter are suppressed by powers of the Planck scale. It is an extremely subtle art to be able to design experiments capable of measuring such minute forces with such precision. Alternatively, it may seem that extra dimensions of this potentially large size could be ruled out by astrophysical or cosmological means. After all, if there exist excited KK gravitons which are really as light as 10 ~ 4 eV, it would seem that such light states could have a huge effect on processes such

475

Figure 7. Exclusion plots for deviations from Newtonian gravity. The heavy solid line denotes the most recent result from the so-called "Eot-Wash group" at the University of Washington; the entire range above this line is excluded at the 95% confidence level. Results from previous experiments, as well as important theoretical regions, are also shown. When comparing to Eq. (1.21), A denotes R and a denotes gi. The horizontal lines labeled as "two extra dimensions" indicate a = g\ = 3 and 4, corresponding to compactifications on a two-sphere and a two-torus respectively.

as stellar cooling (where extra KK gravitons could carry away energy and thereby increase stellar cooling rates beyond observational bounds). They could also potentially affect supernova dynamics, etc. However, detailed studies have been performed, and in all cases it is found that the constraints in Eq. (1.22) are not significantly altered. Discussions of these analysis can be found in the literature. The upshot, then, is that "gravity-only" extra dimensions could potentially be as large as fractions of a millimeter. Indeed, we might be living in a D3-brane universe, with an extra dimension of submillimeter size immediately adjacent, and yet have no experimental way of detecting this extra dimension. It is also possible to place constraints on the time-variation of the sizes of extra dimensions. To understand the procedure by which this is done,

476

let us consider a simplified picture in which the only forces are gravity and electromagnetism. The fundamental constants of nature thus appear to be GN (the four-dimensional gravitational Newton constant) with mass dimension (length) 2 , and a^4' (the four-dimensional electromagnetic finestructure constant) with mass dimension (length) 0 . In D > 4, however, these fundamental constants have different mass dimensions: [G™] = (length) 13 " 2 ,

[a] = (length) 0 " 4 .

(1.23)

As evident through a Kaluza-Klein reduction, the higher- and lowerdimensional constants are therefore related by powers of the compactification radius: /~i(D)

(D)

GN o* . G(4) _ (4) _ ( N D 4 D 4 { ~ R ' ~ R ~ ' ' But which are the true constants of nature? If our underlying theory is really D-dimensional, then it is the D-dimensional constants are the true constants of nature; the four-dimensional constants are only derived quantities once a particular compactification radius is specified. Therefore, if the compactification radius has a relative time rate of change R/R, then the apparent four-dimensional "constants" of nature will experience a relative rate of change ^ l - ^ G(4)

"

a (4)

= U-D)X ^

(125)

U

)

R

•

V-^)

The time variations of any possible compactification radii can thus be constrained by constraining the time variations of the fundamental constants of nature. Note that Eq. (1.25) holds for universal extra dimensions. In general, relations such as these hold for any gauge coupling as long as the corresponding gauge bosons experience the corresponding extra dimension^). The question then boils down to determining the constraints on time variations of the four-dimensional Newton constant GN and the fourdimensional fine structure constant a^4'. In general, such bounds come from many sources: • primordial nucleosynthesis and element abundances (which are sensitive to a through the proton/neutron mass differences and to az through nuclear stability bounds); • astrophysical observations {e.g., observations of high-redshift radio galaxies, whose signatures can provide information about a at earlier times); and

477

• terrestrial measurements (most notably, through evidence of a natural geochemical reactor in Oklo, West Africa, that went critical « 1.8 x 109 years ago and which thereby provides bounds on nuclear stability at that time). Putting all these constraints together, we find rather significant bounds on GN/GN: GN GN

< 10" 1 1 y " 1 ,

(1.26)

as well as even more stringent bounds on a/a: \a/a\ \a/a\

< 10_lr y_1 < 10

-16

y"

1

over 109 years over 10

10

years

(Oklo reactor) , (nucleosynthesis) . (1.27)

Together, these constraints imply that extra dimensions have been essentially fixed in size since nucleosynthesis (i.e., since the first three minutes after the big bang). There are, of course, methods by which these bounds can be evaded. For example, it is possible that R/R oscillates rather than changes monotonically; thus there would not be a discernable trend in the direction of evolution of these couplings. Similarly, there may be conspiracies: for example, even if the fine-structure constant changes, the strong coupling «3 might change in a compensatory way so that together, the couplings conspire to preserve nuclear properties. However, none of these scenarios are particularly likely. Is there any evidence for non-zero variations in the couplings? Using spectroscopic observations of the absorption lines of gas clouds against background quasars at redshifts in the range 1.0 < z % 1.6, there have recently been claims of a non-zero variation in the fine-structure constant of magnitude — = (-2.64 ± 0.35) x 10" 5 . (1.28) a Given the redshifts involved, this is just at the edge of the limits quoted above. However, these measurements tend to have large systematic calibration errors. Thus, further study is needed before a definitive conclusion can be drawn. Finally, let us consider whether there might be spatial variations in the fine-structure constant (i.e., variations in a observed across different

478

regions in the sky). The best bounds are currently — < 3 x 10" 6 for Ax « 3000 Mpc . (1.29) a Thus, once again we see that any possible variations are highly constrained. 1.6. Embedding

4D theories

into extra

dimensions

We now turn to a practical matter: given what we have learned about extra dimensions, how do we take a four-dimensional theory and embed it into higher dimensions? In this section, as an illustrative example, we shall describe the embedding of the Minimal Supersymmetric Standard Model (MSSM) into five dimensions. As we shall see, the issues that arise will be fairly typical for most models of interest. Recall that the MSSM matter content consists of gauge bosons (gluons, W^, Z, and photon); two Higgs fields (Hu and Hd); and three generations of chiral fermions. Moreover, because the MSSM has N = 1 supersymmetry, each of these fields comes in aa JV = 1 SUSY representation: the gauge bosons come in N = 1 vector supermultiplets (thereby including their gaugino superpartners); the Higgs fields come in N = 1 chiral supermultiplets (thereby including Higgsino fields); and the chiral fermions also come in N = 1 chiral supermultiplets (thereby including both the fermions and their sfermion superpartners). In order to embed this theory into five dimensions, at first glance it may seem that for each of these fields, we simply introduce a single KK tower of identical corresponding states. However, we can immediately see that this procedure runs into trouble. The first source of trouble concerns the fact that our four-dimensional theory (the MSSM itself) has N = 1 spacetime supersymmetry. Since our four-dimensional theory has N = 1 supersymmetry, this means that our five-dimensional theory must have at least this much supersymmetry as well. However, N = 1 supersymmetry in five dimensions is equivalent to N = 2 supersymmetry in four dimensions! The easiest way to see this is to realize that a single spin-3/2 gravitino in five dimensions decomposes into two spin-3/2 gravitino representations in four dimensions. Again, this is ultimately a reflection of the fact that our higher-dimensional fields typically have an expanded Lorentz (and thus also supersymmetry) structure compared with the structure in four dimensions. The second problem that we face concerns the chirality of the MSSM (and indeed, of the Standard Model itself). In five spacetime dimensions,

479

there is no such thing as chirality; chiral representations of the Lorentz algebra are possible only in even numbers of spacetime dimensions. Again, the reason for this is completely analogous to the reason for the expanded supersymmetry. A single fermion in five dimensions decomposes into two fermions in four dimensions — a chiral fermion, and its chiral "conjugate" which has opposite chirality. Thus, the five-dimensional theory can have no intrinsic chirality of its own. More mathematically, this follows from the fact that the (Euclidean) Lorentz (rotation) group in D dimensions is SO(D), yet only when D is even does this group contain chiral spinor representations. What this means, then, is that although our "zero-mode" theory (the MSSM) must be N — 1 supersymmetric and chiral, our field content at all excited KK modes must be N = 2 supersymmetric and chiral. What does this mean for embedding the MSSM into higher dimensions? Let us first consider the gauge fields. Rather than merely repeat the N = 1 supersymmetric vector supermultiplet at each excited KK level, we must also introduce a new N = 1 chiral supermultiplet at each excited level. Together, these representations combine to produce an TV = 2 vector supermultiplet at all excited KK levels. The corresponding situation is illustrated in Fig. 8(a). A similar situation exists for each Higgs field. Rather than merely repeat the N = 1 supersymmetric chiral supermultiplet at each excited KK level, we must also introduce an additional N = 1 chiral supermultiplet at each excited level. Together, these representations combine to produce an N = 2 hypermultiplet at each excited KK level, as illustrated in Fig. 8(b). Embedding each fermion field into five dimensions requires a similar situation. While we wish to retain only the original four-dimensional chiral supermultiplet fermion as our zero-mode field, we find that we must introduce an additional chiral supermultiplet (a so-called chiral-conjugate "mirror fermion") at each excited KK level in order to produce a non-chiral N = 2 hypermultiplet. In each case, we see that we are forced into a situation where the field content at each excited KK level is different from, and larger than, the field content of the zero modes. Clearly, there is only one way to achieve this. As discussed in Sect. 1.3, we must actually compactify on an orbifold (in this line segment) rather than on a circle. We then must choose this orbifold so that all of the MSSM fields are chosen even with respect to the Z2 orbifold symmetry, while all of the fields that we needed to introduce beyond the MSSM must be chosen odd. This guarantees that these extra

480

(a) gauge fields:

A, A,

(b) Higgs fields:

,V

H + ,v+

H_ .V-

A, X

H + .V+

H_ ,v_

A,X

H + .v+

H_ ,V-

A,X

H + .V+

(A, X): N=1 vector supermultiplet (0, V): N=1 chiral supermultiplet N=2 vector supermultiplet

(H + , V + ): N=1 chiral supermultiplet (H _, V_) : W= J c/i/'ra/ supermultiplet —^^-

N=2 hypermultiplet

Figure 8. Embedding the non-matter field content of the MSSM into five dimensions, (a) Gauge fields: The zero modes contain only the N = 1 vector supermultiplet, while each excited KK level contains this supermultiplet as well as an N = 1 chiral supermultiplet. Together, these representations combine to produce an N = 2 vector supermultiplet at each excited KK level, (b) Higgs fields: The zero modes contain only a single N = 1 chiral supermultiplet, while each excited KK levels contains two copies of such chiral multiplets. Together, these combine to produce an N = 2 hypermultiplet at each excited KK level.

fields beyond the MSSM will not have zero modes, which is in accordance with our original desire that our low-energy, observable, four-dimensional theory is merely the MSSM. The case of the MSSM is quite typical. Rather than compactify on higher-dimensional manifolds, we typically are forced to compactify on orbifolds for two reasons: • We must reduce the extended supersymmetry (N = 2 or N = 4 supersymmetry) that would arise higher dimensions, leaving us with only N — 1 supersymmetry at the zero-mode level; and • We must introduce chirality at the level of the zero modes. Once again, this is achieved through the (chiral) orbifold projection which removes the chiral-conjugate states.

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In other words, in the language of symmetry breaking, we would say that we must compactify on an orbifold rather than a manifold in order to (partially) break the supersymmetry as well as break the non-chiral spectrum into a chiral one. Having seen how to embed each individual MSSM field into five dimensions, we now must address the only remaining issue of deciding which fields should indeed feel the extra dimensions. As we discussed in Sect. 1.4, there are many different types of extra dimensions, some of which can be felt by some fields and not others. In general, assuming that we are not talking about "gravity-only" extra dimensions (which none of the MSSM fields would feel), we are left with various choices as to which gauge groups and which charged particles can propagate in which extra dimensions. If we are considering a five-dimensional theory, the most natural situation is to imagine that all of the gauge forces can feel this extra dimension. This means that all of the MSSM gauge bosons will propagate in the extra dimension. We then must decide whether our Higgs fields and/or fermion generations will feel this extra dimension as well. In general, we can let e = 0,1,2 denote the number of Higgs fields which feel the extra dimension; likewise, we can let r\ = 0,1,2, 3 denote the number of fermion generations which feel the extra dimension. (The remaining fields are imagined to be restricted to brane intersections or orbifold fixed points.) These parameters (e, r\) thus join the radius R as parameters which describe different possible embeddings of the MSSM into higher dimensions.

2. Lecture # 2 : Large extra dimensions and the brane world In the previous lecture, we discussed general properties of extra spacetime dimensions, and we noted that the experimental bounds on the sizes of such extra dimensions are actually quite generous. Extra "longitudinal" dimensions in the brane can potentially be as large as an inverse TeV, while extra "transverse" dimensions perpendicular to the brane can potentially be as large as tenths of a millimeter. Despite these bounds, extra dimensions were traditionally considered to be quite small, on the order of the Planck length. The reason for this was, indeed, naturalness: since extra dimensions emerge primarily in theories such as string theory which also contain gravity, their natural length scales are at the Planck length. Clearly, if these extra dimensions are so small, their corresponding Kaluza-Klein excitations will have masses ~ 10 19 GeV.

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These excitations will be essentially unobservable. For this reasons, it was not traditionally considered that such extra spacetime dimensions could ever have relevance for the low-energy world, or four our phenomenological ideas about physics beyond the Standard Model. In particular, it was felt that they could never alter our fundamental picture for physics beyond the Standard Model, which includes supersymmetry at the TeV scale; grand unification near 1016 GeV, and ultimately some sort of embedding into string theory near 10 1 8 " 1 9 GeV. However, all of this began to change dramatically in 1998. Specifically, high-energy theorists began to wonder what would happen if potential extra dimensions were not so small, or equivalently if their corresponding KK states were not so remotely heavy? If the KK states were to become sufficiently light, one could imagine that they could begin to have a direct effect on the physics beyond the Standard Model! What would these effects be? Surprisingly, it was found that large extra dimensions have the power to alter the fundamental high energy scales of physicsl In other words, supposedly "fixed" energy scales such as the GUT scale, the Planck scale, and the string scale are actually not fixed at all, but can be altered in theories with extra dimensions of suitable sizes. As we shall see, this has given rise to a dramatically new set of possibilities for physics beyond the Standard Model. In this lecture, I shall present this new picture and discuss how and why the presence of large extra dimensions can have these profound effects. Ultimately, taken together, these new possibilities have given rise to an alternative picture for physics beyond the Standard Model: this picture is often called the "brane world". It is our goal in this lecture to introduce this new paradigm.

2.1. The Standard Paradigm for physics Standard Model: A quick review

beyond

the

By way of contrast, let us first review the standard paradigm for physics beyond the Standard Model. This was the unique, prevailing picture which held sway until 1998, and it is still a compelling (and perhaps even correct) picture for physics at energy scales beyond those of present-day colliders. Our focus will be on the energy scales inherent in this picture. Our first energy scale of interest is the GUT scale. By measuring the strengths of the gauge couplings corresponding to the fundamental forces [strong SU(3), weak SU{2), and hypercharge U(l)] of the Standard Model and extrapolating them upwards within the framework of either the

483

Standard Model itself or its minimal N = 1 supersymmetric extension (the MSSM), one finds that they meet at approximately M GU T « 2 x 1016 GeV ,

(2.1)

with aouT ~ 0.04 = >
MPlanck = y / —

(2.2)

This is, of course, the Planck scale. Alternatively, we can view this as the energy scale at which the gravitational interaction becomes roughly as strong as the gauge interactions. If we view GJV as a "gauge coupling" for gravity, then the fact that it is dimensionful (with dimensions of inverse

60

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Figure 9. Gauge coupling unification within the framework of the MSSM. The three gauge couplings appear to unify at an energy scale MQTJT ~ 2 X 10 16 GeV. The thickness of the lines represents current experimental uncertainties in the low-energy measurements of these couplings.

484

mass squared) implies that GN "runs" with a classical scaling ~ y? where /j is the corresponding energy scale. In other words, the gravitational interaction gets dramatically stronger with increasing energy, ultimately becoming ~ 0(1) at the Planck scale. One therefore often associates the Planck scale with the scale at which quantum gravitational effects should become significant. The string scale is, strictly speaking, another independent energy scale governing physics beyond the Standard Model. In principle, this scale is set by the so-called "Regge slope" a' (which is in turn set by the string tension): one has

Mstring = - L = .

(2.3)

However, the relation between a' and other physical parameters such as GN depends on the particular string model under study. In the case of the weakly coupled heterotic string, we find that we must set ^string

where String is the string coupling (the strength of string interactions). The relation (2.4) must hold in order that the string states we wish to identify as gravitons actually induce gravitational interactions of the correct strength. Thus, combining Eqs. (2.2), (2.3), and (2.4), we find that -^string =

ffstringMpianck

•

(2.5)

Unfortunately, the value of String is not known; this value can only be determined by understanding the non-perturbative dynamics of a special string state known as the dilaton. However, based on the prejudice (coming from grand unified theories) that String should be approximately 0.7, we find that Mstring ~ 10 18 GeV. We emphasize, however, that this value for the string scale rests upon a number of important assumptions, including the choice of a particular string framework (namely, the weakly coupled heterotic string). For other types of string theories, the value of M s t r i n g may be quite different. We see, then, that we have a huge hierarchy of energy scales: the energy scales associated with "fundamental" physics (grand unification, quantum gravity, and string theory) are at least thirteen orders of magnitude beyond those that we currently probe through collider experiments. This is an important hierarchy, because the energy scales that we currently probe are associated with another important energy scale, namely the scale of

485

electroweak symmetry breaking. In order to stabilize this hierarchy against quantum corrections, the traditional approach is to introduce supersymmetry and imagine that this supersymmetry is broken at some energy scale beyond the scale of electroweak symmetry breaking. There are several considerations which go into the choice of this SUSYbreaking scale. Most importantly, since supersymmetry is meant to stabilize the gauge hierarchy, we must invoke this symmetry at an energy scale which is not too far beyond the scale of electroweak symmetry breaking. Otherwise, the protection that supersymmetry provides for quantities such as the radiative corrections to the Higgs mass will be lost. Moreover, it is found that supersymmetry just beyond the weak scale is also necessary in order to achieve the precise gauge coupling unification shown in Fig. 9. Thus, we are led to imagine that the scale for supersymmetry breaking (i.e., the energy scale at which the lightest superpartners will appear) should be somewhere not too far beyond the scale of electroweak symmetry breaking. This scale is typically taken to be in the TeV range.

The Standard Paradigm. (inverse) strengths of forces

Energy (GeV) 1000 SUSY? (TeV)

GUT scale

Planck (string) scale

fundamental physics

"The Desert" Figure 10.

The Standard Paradigm for physics beyond the Standard Model.

486

Thus, combining these energy scales, we are led to the "Standard Paradigm" for physics beyond the Standard Model. This picture is sketched in Fig. 10. As we have stated above, this picture is highly successful, and leads to a number of elegant solutions for many outstanding problems ranging from proton decay and neutrino physics to dark matter and cosmological evolution. However, our purpose in the rest of this lecture will be to present another, alternative picture. As we shall see, this new picture will be radically different from the standard one, providing new benefits but also simultaneously introducing new challenges. 2.2. Extra dimensions:

Lowering

the GUT

scale

Let us now see how the- standard paradigm may-be affected by the presence of large extra dimensions. We shall first focus on the case of the GUT scale.

Figure 11. One-loop wavefunction renormalization diagram. This diagram contributes to the one-loop running of the gauge couplings, where the entire field content of the theory runs in the loop.

In the usual MSSM calculation, one derives the one-loop running of the SU(3) x SU(2) x U(l) gauge couplings by evaluating the one-loop wavefunction renormalization diagram shown.in Fig. 11, where all MSSM states can propagate in the loop. This then leads to the logarithmic renormalization group equation (RGE)

ar\^) = a-\Mz)-^\n(j^}

,

(2.6)

where bi are the one-loop MSSM beta-function coefficients. Given the experimentally measured gauge couplings at the Z-scale, one can use this RGE to extrapolate the gauge couplings upwards in energy. This then

487

leads to the conventional unification near M Q U T ~ 2 X 10 16 GeV, as shown in Fig. 9. How do we extend this calculation into higher dimensions? First, let us imagine that there exist 5 = D — 4 "longitudinal" extra spacetime dimensions, each compactified on a circle of radius R. Here S = 1,2,... can take any integer value, and likewise .R - 1 , which sets the energy threshold for the extra dimensions, can range anywhere from the TeV scale all the way up to the usual GUT scale M Q U T ~ 2 X 10 16 GeV. Corresponding to each MSSM field, there will be an infinite tower of Kaluza-Klein states whose masses are separated by R~l. Of course, for the reasons discussed in Sect. 1.6, we actually must compactify on orbifolds rather than circles, and likewise our states at the excited levels of the Kaluza-Klein towers must fall into N = 2 supermultiplets (even though the ground-state zero-mode MSSM states are only N = 1 supersymmetric). It also turns out not all of the Standard Model particles need have Kaluza-Klein excitations. Thus, the particle content at each excited Kaluza-Klein level can be different from the content of the MSSM zero-mode ground state. Given this setup, let us now calculate the higher-dimensional "running" of the gauge couplings. To do this, we must simply re-evaluate the standard one-loop wavefunction renormalization diagram in Fig. 11. However, we now have not only the usual MSSM states, but also their their corresponding Kaluza-Klein excitations propagating in the loop. What is the effect of these Kaluza-Klein excitations? Evaluating the one-loop diagrams, we find the general result

(2.7) In this equation, the Jacobi theta-function $3(1-) = J^^L-oo exp(-7riTn2) reflects the sum over the Kaluza-Klein states. The b\ are the beta-function coefficients describing the matter content at the excited Kaluza-Klein levels (which depend on the particular embedding of the MSSM into higher dimensions, as discussed in Sect. 1.6). The numerical coefficient r is given by r = TT(XS)-2/5 where Xs = 2ns/2/6T(S/2); this factor is needed as an overall normalization, and corresponds to the volume of a <5-dimensional unit sphere. While the final term in Eq. (2.7) reflects the contributions from the Kaluza-Klein towers, the penultimate term reflects the fact that the zero-mode MSSM states and their Kaluza-Klein towers need not be identical.

While the result in Eq. (2.7) emerges from an exact summation over the Kaluza-Klein states, it is possible to simplify this expression in most cases of interest. For energy scales below R"1, we find that the excited KK states do not substantially contribute to Eq. (2.7); thus, at energy scales below i ? _ 1 , it is legitimate to use the standard four-dimensional RGE in Eq. (2.6). However, at energy scales above -ff-1, we can approximate A 3> R~x and evaluate the Jacobi-theta-function integral explicitly, obtaining the simplified RGE ar\A)

* a-\R-')

-

^

ln(Ai?) -

| J

[(AR)S - l] .

(2.8)

While this result is derived only in the limit A » -R -1 , it is easy to verify that it actually holds with extreme accuracy all the way down to AR « 1. We see, then, that the net effect of the Kaluza-Klein states running in the loop in Fig. (11) is to change our logarithmic evolution of the gauge couplings into a power-law evolution! There are several ways to interpret this result. From a four-dimensional perspective, we simply have a theory endowed with infinite towers of KK states. Thus, our gauge couplings continue to experience logarithmic evolution, but with a beta-function coefficient which is constantly changing as new KK thresholds are crossed at higher and higher energies. This combination of logarithmic running with a scale-dependent beta-function coefficient thus gives rise to an effective power-law evolution. But perhaps even more fundamentally, this is really a higherdimensional theory. As we have already seen in Eq. (1.23), gauge couplings in D — 4 + S spacetime dimensions are no longer dimensionless; they have mass dimensions [a] = S. Thus, we expect that they should accrue a power-law scaling with exponent S in addition to their quantum running. The Kaluza-Klein states thus manage to achieve, from a fourdimensional perspective, exactly what we expect to observe in our underlying five-dimensional theory. Note that the coefficient of the power-law scaling piece within Eq. (2.8) is indeed nothing but bi, which depends on the field content at the excited KK levels. So, we see that the presence of extra dimensions changes the evolution of the gauge couplings, introducing power-law evolution rather than the traditional (four-dimensional) logarithmic evolution. But what does this do the unification that exists in the four-dimensional case? To see this, let us first examine an extreme case. Let us choose R~l = 1 TeV, and take 6=1. Let us also imagine that only the gauge bosons

489

and a single Higgs gauge field propagate in the extra dimensions (i.e., we take e = 1 and rj = 0 in the language of Sect. 1.6). This implies the values (61,62,63) = (33/5,1,-3) and (61,62,63) = ( 3 / 5 , - 3 , - 6 ) . We then find the evolution of gauge couplings illustrated in Fig. 12. Note that although the running of the gauge couplings is profoundly altered, becoming power-law rather than logarithmic, the unification of the gauge couplings is nevertheless preserved! Thus, we see that extra dimensions are consistent with the emergence of a grand unified theory in the TeV range. In other words, the GUT scale has been lowered all the way to the TeV range! However, this is not the only possibility. For example, let us see what happens if we adjust the radius R of the extra dimensions. Clearly, taking smaller radii corresponds to increasing the energy scale associated with the KK states. We thus expect the power-law behavior to emerge later in the evolution of the gauge couplings. Nevertheless, regardless of the value of R, we see in Fig. 13 that we continue to have unification, with a GUT scale that depends on the value of R. Indeed, we now see that the usual unification in Fig. 9 is nothing but the special case of Fig. 13 when R-1

- • M G U T « 2 x 1016

GeV.

60

40

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2

3 log10(/x/GeV)

4

Figure 12. Unification of gauge couplings at the new unification scale MQ-^^ RJ 20 TeV, assuming the appearance of a single extra spacetime dimension of radius R~ = 1 TeV.

490

Similarly, one can ask whether this unification persists for different numbers of extra spacetime dimensions. However, the parameter 8 = D — 4 governs the power inherent in the power-law evolution. Thus, increasing 5 merely has the effect of increasing the rate at which the unification occurs. Nevertheless, the unification is preserved. This reduced-scale unification leads to many important quantitative questions. How predictive is this unification? How perturbative is it? How sensitive is it to unification-scale effects? What about higher-loop correc-

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491

tions? These issues are discussed in some detail in the original references. The upshot, however, is that this sort of unification scenario is predictive, perturbative, and not unreasonably sensitive to unification-scale effects. It is important to emphasize that not all embeddings of the MSSM into higher dimensions automatically lead to an acceptable gauge coupling unification. While increasing the value of r\ (the number of fermion generations which propagate in the extra dimensions) preserves the unification, this increase tends to drive the unified unified gauge coupling towards stronger and stronger values, thereby invalidating the accuracy of the above oneloop results. Moreover, changing the value of e (the number of Higgs fields experiencing the extra spacetime dimensions) also has a dramatic effect on the unification, and can spoil it entirely unless further new states are introduced. However, this is no different than the situation in the MSSM. As we know, gauge coupling unification in the usual MSSM is also extremely sensitive to particle content and shifts in energy thresholds. Nevertheless, we see that extra dimensions have broadened our perspective considerably. Indeed, through extra dimensions, it becomes possible for the first time to contemplate grand unification occurring at an intermediate scale — indeed, even as low as the TeV range. In other words, we now see that it becomes possible to replace a four-dimensional GUT at 1016 GeV with a higher-dimensional GUT at the TeV-scale. 2.3. Extra dimensions:

Lowering

the Planck

scale

Let us now turn to the fundamental scale associated with gravity, namely the Planck scale. As we shall see, a similar phenomenon occurs here as well. Recall that the Planck scale is defined through the value of Newton's constant GN, as in Eq. (2.2). However, as we have already seen in Eq. (1.24), if D — 4 extra spacetime dimensions exist, then the four-dimensional Newton's constant GN which defines the usual fourdimensional Planck scale Mpianck is ultimately related to a £>-dimensional Newton's constant GN : G = RD-*G%>,

(2.9)

or equivalently G™

= RD-AM^nck.

(2.10)

In other words, in the presence of D — 4 extra spacetime dimensions, it

492

is GN which is the fundamental constant of nature, not GN or Mpi anc k. Thus, depending on the value of R, we see that Mpi anc k can also be adjusted! In order to write this result in its more common form, note that GN has mass dimension ( m a s s ) 2 - 0 . Thus, we can define an effective higherdimensional Planck scale M pla ' nck M(D)

iW

Planck

__

\MD)

=

\ N

-,1/(2-13)

(2.11)

U

Inserting this into Eq. (2.10), we thus have m

Mh^JRD~A

(2-12)

= M2lanck/vn

(2.13)

Planck

or equivalently M{D) JW

n+2

Planck

where n = D — 4 and where Vn = Rn is (proportional to) the n-dimensional compactification volume. Since the D-dimensional Planck scale is the fundamental energy scale, we now have a new possible explanation of why M p l a n c k appears to be so large in four dimensions. Rather than imagine that Vn is small and M p l a n c k is also large, we can instead imagine that M p l a n c k is relatively small but that Mpi anc k appears large in four dimensions only because Vn is large! Thus, once again, we see that if we have large extra spacetime dimensions, our fundamental higher-dimensional Planck scale M p l a n c k can indeed be considerably lower than previously imagined. As an example, let us consider how large these extra dimensions would have to be if we were to attempt to take M p ] a n c k in the TeV range. With Mpi anc k « 1019 GeV, we find from Eq. (2.13) that R-1

« 10" 3 2 /" TeV

<=^

R P» io 32 /™- 19 meters .

(2.14)

Thus, we see that the required extra dimensions would have to have sizes R w 10 13 meters R w 1 millimeter i ? « 1 0 f e r m i « (10 MeV)- 1 .

(2.15)

Clearly, according to the discussion in Sect. 1.5, the case with n = 1 is ruled out; an extra dimension of this size would clearly disturb obvious phenomena such as planetary orbits. Indeed, the case with n = 2 is just at the edge of the bounds depicted in Fig. 7 coming from precision Cavendishtype experiments [note that we have not been careful with overall factors

493

of 0(1), which is why we cannot make a definitive claim about this case]. However, we see that the cases with n > 2 are comfortably within the range allowed by precision gravitational experiments, as well as all astronomical and cosmological data. Of course, this assumes that such extra dimensions are transverse to the brane on which all gauge forces are restricted, so that they are of the "gravity-only" type. This is a major result. We see that the Planck scale need not be unreachably high at all — indeed, the fundamental energy scale associated with quantum gravity may be accessible within the next round of collider experiments. Like our previous result concerning the lowering of the GUT scale, such a result thus calls for a drastic rethinking of the possibilities for physics beyond the Standard Model. Despite its importance, however, it is important to recognize that this result does not provide a solution to the gauge hierarchy problem. It may seem, at first glance, that since the fundamental energy scale M p l ^ k may be in the TeV range, the hierarchy between Mpi anc k (the effective fourdimensional Planck scale) and the electroweak symmetry breaking scale is eliminated. However, it is not eliminated: Mpi anc k continues to be at 1019 GeV. Or, phrased another way, the reason that Mpi anc k can differ so greatly from M p i a n c k is that we have an unusually large volume Vn = Rn. Thus, our previous hierarchy in energy scales has merely become a hierarchy in geometry: why are the extra dimensions so much larger than the fundamental energy scales in the theory? For example, if M p l k ~ O(TeV) with n = 2, then we require J ? - 1 « 1 0 - 4 eV. The hierarchy between the compactification scale R^1 of the extra dimensions and the electroweak scale is thus exactly as large as the hierarchy between the electroweak scale and the previous four-dimensional GUT scale. In other words, we have not eliminated our big number (the hierarchy); we have merely hidden it. Although this therefore not a solution to the hierarchy problem, this still represents a tremendous breakthrough in our attempts to solve this problem. We now have an additional perspective, one resting on geometry, which did not exist previously. Indeed, as will be discussed in subsequent lectures at this TASI school, there does exist a way of truly solving the hierarchy problem in the context of extra dimensions; the solution is to warp the geometry and thereby soak up the large number within a relatively small deformation of the compactification space. Because of the importance of Eq. (2.13), we conclude this subsection by providing three further ways of interpreting this result.

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gravitational flux lines "leak out" into the extra dimensions

Figure 14. If there exist extra spacetime dimensions perpendicular to the brane, then the gravitational flux lines between two masses on the brane will leak out into the extra dimensions, leading the gravity on the brane to appear weaker than it would otherwise have been.

First, geometry. Let us consider two masses located on a fourdimensional brane, as shown in Fig. 14. Without extra dimensions perpendicular to the brane, the gravitational flux lines all remain within the brane, giving rise to the familiar Newton force law F = GN MxMijr1 where r is the distance between the masses. However, if there exist n extra dimensions perpendicular to the brane, then the flux lines can leak off the brane into these extra dimensions as well. By Gauss' law, this changes the Newton force law to become F = G^ ' MiM2/r2+n. This leakage of the gravitational flux lines into the extra dimensions implies that the gravitational interaction will appear weaker than we would have naively expected in four dimensions. Thus, we can compensate by allowing gravity in higher dimensions to be intrinsically stronger than in four dimensions. In other words, the higher-dimensional Planck scale (where gravity gets strong) can be lower than the four-dimensional Planck scale. This geometric argument also explains why gravity appears to be fourdimensional until we probe distances on the order of the compactification radius R. Let us imagine that our extra dimension has only a finite "width" R perpendicular to the brane. If the'distance r between the point masses is much smaller than R, then our flux lines are effectively "spherical" in the full, higher-dimensional space. This gives rise to the higher-dimensional force law outlined above, which implies that our higher-dimensional gravity

495

is effectively stronger than expected. However, if the distance r between the point masses is much larger than R, then our flux lines, even though extending into the small width R, are still forced to travel along the constricted hypervolume parallel to the brane. They thus cannot effectively spread much beyond the four dimensions of the brane itself (or regions closely parallel to it), and consequently they reproduce the usual four-dimensional force law. Indeed, by matching the four- and higher-dimensional force laws smoothly in the region r « R, we can immediately derive our starting relation in Eq. (2.9). Our second way of understanding the emergence of this modified force law is through the exchange of Kaluza-Klein graviton states. Recall that the contributions from the exchange of KK graviton modes are already given in Eq. (1-20) where mn denote the corresponding KK masses. As we have seen in Sect. 1.4, for r » R the KK graviton modes are heavy, and produce only small corrections to the usual four-dimensional Newtonian inverse square law. However, in the opposite limit r < i { , many of the KK graviton modes become light and we must explicitly perform the sum over the KK states in Eq. (1.20). For D — 4 extra dimensions, and for r
J29n<

(2.16)

n=0

Thus, we find that the gravitational potential becomes

This is indeed the expected Gauss-law result in D dimensions, provided we make the identification given in Eq. (2.9). Finally, let us understand this result one final way, as a "running" of the gravitational coupling. This is closest in spirit to our discussion of the method by which extra dimensions can be used to lower the GUT scale. Since GN' is a dimensionful quantity, we really should define a dimensionless gravitational coupling GJV(M) = fi2GN . It is this quantity which is the closest analogue to the dimensionless four-dimensional gauge couplings cti(ii). However, because of the underlying mass dimension of GN , we see that the dimensionless gravitational coupling Gjv already experiences a power-law running, scaling like fx2 (or equivalently, with the inverse coupling GNl scaling like /x~2, as shown in Fig. 15). The energy scale at which GN ~ O(l) defines the Planck scale.

496

G N (M)= H 2 G N

AGJJ(p)

-, R-1

n=2:

(10~4eV)

i M'p|

(TeV)

i Mp,

•

log(u)

(lO'^GeV)

Figure 15. In higher dimensions, the effective, dimensionless gravitational coupling GJV(M) runs even more rapidly than in four dimensions, reaching order one (and thus denning MpiJ^ k) at a lower energy scale than expected.

However, if there are n extra dimensions which "open up" (i.e., become accessible) at a radius R, then the combined effect of the gravitational KK modes above this energy scale is to provide a further power-law growth for GN, turning the /i2 scaling into /z 2 + n . As illustrated in Fig. 15, this causes the gravitational interactions to get stronger at an even lower energy scale, thereby effectively reducing the (now higher-dimensional) Planck scale. Indeed, one can even deduce graphically that for n = 2, one requires R~x fa 1CP4 eV in order to bring the reduced Planck scale to the TeV range down from 10 19 GeV. The picture that emerges, then, is relatively simple. Ordinarily, the gravitational interaction is weak because it is mediated by a graviton whose couplings to ordinary matter are Planck-scale suppressed. (This is indeed the definition of the Planck scale.) However, in the presence of extra spacetime dimensions, the gravitational interaction is mediated not only by the single Planck-suppressed graviton, but also by its infinite towers of

497

Planck-suppressed KK gravitons as well. The combined effect of the exchange of this KK graviton tower thus leads to an enhancement in the strength of the resulting gravitational interaction, or equivalently a reduction in the Planck scale suppression with which the single higherdimensional graviton state couples to ordinary matter. 2.4. Extra dimensions:

Lowering

the string

scale

Finally, large extra dimensions can also be used to lower the string scale. This is important because higher-dimensional gauge theories are intrinsically non-renormalizable, and consequently they experience divergences which cannot be absorbed through shifts in the bare parameters in their effective Lagrangians. Consequently, if we construct a theory in which our GUT and Planck scales are reduced, we must somehow manage to embed this theory into a finite, higher-dimensional theory at an energy scale which is not too far above the reduced GUT or Planck scale. Since the best candidate for such a theory is string theory, we thus are faced with the need to find a way to lower the string scale at the same time we lower the GUT and Planck scales. Recall that in the perturbative heterotic string, the string scale is tied to the usual four-dimensional Planck scale as indicated in Eq. (2.3), and thus remains at the perturbative heterotic string value near 10 18 GeV. Therefore, unless the perturbative heterotic string coupling is extremely suppressed, with String <S 0(1), we cannot appreciably lower the string scale below the usual Planck scale. For heterotic strings at strong coupling, however, this behavior changes. Specifically, it turns out that various (closed) heterotic strings at strong coupling can be equivalently described as (open) -Type I strings at weak coupling. Thus, many non-perturbative features of heterotic string theory can be studied by analyzing weakly coupled Type I strings. Remarkably, Type I string theory offers the interesting possibility of lowering the fundamental string scale, for Eq. (2.3) no longer continues to apply. Instead, we find the relation M^ing ~ e* /2 ggaUge Mpi anck

(2-18)

where <\> represents the so-called ten-dimensional dilaton field andffgaugeis the Type I gauge coupling. Thus, simply by adjusting the VEV of the tendimensional dilaton, one can lower M str i n g relative to Mpi anc k. Of course, the value of the dilaton indirectly affects the values of the gauge and gravitational couplings, implying that 5gauge and Mpi anc k themselves change

498

their apparent values when the dilaton is changed. However, using other string relations it is possible to eliminate this dependence algebraically, and relate Mstring and Mpianck directly to each other without exhibiting the dependence on the dilaton. We then find the general relation String ~ J

^

V-1^

(2.19)

y agaugeJWpi a n c k

where agauge = ffgauge/^71") a n d where (2n)6V is the (normalized) sixdimensional volume of compactification. In writing Eq. (2.19), we have ignored (and will continue to ignore) all numerical factors of order one, since our goal will merely be to obtain order-of-magnitude estimates. The relation (2.19) thus represents the non-perturbative counterpart of Eq. (2.3). We see, then, that within the context of Type I string theory, we can again achieve a dramatic reduction in the value of M s t r i n g ; all that is necessary is to have a large compactification volume V. 2.5. Extra dimensions: A unified picture for beyond the Standard Model?

physics

Given these results, let us now attempt to construct a unified scenario for physics beyond the Standard Model utilizing extra dimensions. If all of the gauge forces of the Standard Model are restricted to a brane, we have already seen that extra "longitudinal" dimensions in the brane can lower the corresponding GUT scale while extra "transverse" dimensions perpendicular to the brane can lower the corresponding Planck scale. String theory, however, requires at least six (or seven) extra spacetime dimensions. Thus, in order to embed our reduced-scale theories into string theory, we should choose some of these extra dimensions to be longitudinal, and the remainder to be transverse. If we choose the sizes of these dimensions correctly according to Eq. (2.19), we can also lower simultaneously the string scale, and thereby produce a self-consistent Type I string theory without any high energy scales whatsoever. Let us give an explicit example of how this can be done. In order to embed our grand unification scenario into Type I string theory, we shall attempt to identify agaUge with the gauge coupling a'GXJT at the lowered GUT scale M Q U T . We shall also attempt to identify M str i n g with this lowered GUT scale MGVT, and seek to build a scenario where MGVT — 10 TeV. Moreover, let us assume that this is accomplished through the use of 8 extra longitudinal dimensions of common radius R. As we have

499

seen, these are the dimensions that cause the gauge couplings to experience power-law corrections. Given the relation (2.19), we can now easily determine the required common radius r for the remaining 6 — <5 compactified dimensions. Writing the normalized compactification volume as V ~ R5r6~5

,

(2.20)

we find

P ^ -

~ «'GUT {M'GmRf/2 ( M ^ r ) 8 - * / 2 .

(2.21)

AtPlanck

Let us begin by considering the case 5 = 1. Taking M Q U T = 10 TeV, we see from Fig. 12 that this requires MQVTR « 20 and a'GVT « 1/50. (Note that Fig. 12 is drawn for M Q U T RS 20 TeV rather than 10 TeV, but approximately the same results hold in both cases.) This in turn implies that MQVTr ss 10" 6 , which implies that the radius r of the five extra dimensions must be smaller than the string length scale! Rather than view this as an inconsistency, we instead recognize that this is actually a signal that we should pass to a slightly different description (the so-called Type I' description) of the physics. Technically, this procedure of passing from one description to the other is called a T-duality, and in general a Type I theory with a compactified radius r is the T-dual of an equivalent Type I' theory with a compactified radius r' = (M?tring r )~ 1: T-duality:

Mstlingr

<-> (Mstring/)^ 1 •

(2.22)

We therefore pass to a Type I' description by T-dualizing our five extra dimensions. We then find ( r ' ) " 1 ~ 10" 6 M Q U T ~ 10 MeV .

(2.23)

Note, however, that extra dimensions of this size transverse to the brane are precisely what are required to lower the Planck scale! Thus, putting the pieces together, we see this combination of extra spacetime dimensions longitudinal and transverse to the brane has managed to self-consistently lower the GUT, Planck, and string scales simultaneously. Indeed, we see the GUT-lowering scenario and the Planck-lowering scenario actually require each other when they are embedded within the contet of string theory. This makes sense, since string theory is a unified theory containing not only gauge forces but also gravity. Thus, to summarize our results, we see that we can naturally associate the scale of gauge coupling unification at 10 TeV with the string scale of

500

G"N^) Type 1' string

efft3CtlVe field thee>ry\

i

theory

V a

2

PaT1-

!

r

^string

«3^

10MeV 1/r'

gauge: gravity:

4D 4D

100 GeV Mz

4D 9D

0.5 TeV 1/R

5D 10D

10 TeV M GUT

log p,

10D 10D

Figure 16. The unification of gauge and gravitational couplings within the framework of a Type I' string theory, assuming a string scale at M s t r i n g = 10 TeV. Note that the running of Gj^(fi) is not shown to scale in this figure. This figure should be contrasted with Fig. 10.

a Type I' theory in which one dimension has radius R^1 ~ 0.5 TeV and the five remaining dimensions have radii r' ~ (10MeV) - 1 . The resulting scenario is sketched in Fig. 16. Above the string scale Mstring = 10 TeV, the physics is described in terms of a full Type I' string theory. Below 10 TeV, by contrast, the physics is described by a series of effective field theories in which the gauge and gravitational forces feel different numbers of spacetime dimensions. Together, everything is balanced so as to produce a self-consistent simultaneous lowering of GUT, Planck, and string scales. Needless to say, other configurations are also possible. However, the main point is that this picture provides a completely different alternative when compared with the usual paradigm sketched in Fig. 10. Rather than have fundamental energy scales which are very high and which are separated

501

from the electroweak symmetry breaking scale through a large "desert", we instead now have a scenario in which there are no high energy scales whatsoever. Instead, the desert in energy scales has been completely eliminated, and has been replaced by a large extra-dimensional volume. Thus is thus a completely different scenario for physics beyond the Standard Model.

X6,x7,x8,x9,x10

Figure 17. A D-brane configuration which can accommodate the scenario sketched in Fig. 16 within the context of Type I' string theory.

It is also interesting to understand the spacetime geometry associated with such a model. One such possibility is sketched in Fig. 17. At low energies, the Standard Model (or the MSSM) lives on the four-dimensional intersection of different "branes" (here shown as the intersection between the D4-brane and the D6-brane). At higher energies, however, the gauge forces feel the extra longitudinal dimension inherent in the D4-brane (here sketched as a cylinder with radius R), and the gauge bosons and Higgs fields propagate on this extra dimension as well. This is the dimension whose size is restricted to be no larger than an inverse TeV. Note that even though we have sketched this extra dimension as a cylinder, in order to break our extended supersymmetry down to N = 1 supersymmetry and introduce chirality, this circular extra dimension might be replaced by a line sigment of finite length. Also, depending on the value of 77, our chiral fermions may or may not be restricted to the intersection of the D4-brane and the D6brane. Thus, at this energy scale, the gauge world has effectively become a cylinder. As described in Sect. 2.2, this can lower the GUT scale and ensures gauge coupling unification.

502

Of course, there are also additional transverse directions which are felt only by the gravitational interactions. These directions are, for example, those that extend along the D6-brane. These extra dimensions lower the Planck scale, and thereby explain the weakness of gravity. Finally, as we have seen, choosing these combinations of extra dimensions correctly manages to simultaneously achieve a lowered string scale. This makes possible an embedding into string theory which ultimately ensures the consistency and calculability of such a theory. Loosely speaking, this type of scenario is called a "brane world". Of course, the various types of branes and geometric configurations is virtually limitless, and thus the potential for model-building in this new language is extremely rich as well. Ultimately, of course, we would like to be able to realize a given brane configuration as emerging from a bona-fide string model; work along these directions is much more difficult than the relatively simpler "bottom-up" constructions which we have described thus far. In any case, we see that the brane world provides a new and tantalizing alternative perspective in which to view the theoretical possibilities for physics beyond the Standard Model.

2.6. Open

questions

Needless to say, the brane world opens up whole new vistas that await exploration. Virtually all aspects of physics beyond the Standard Model must now be considered in a new light. For example, what is the role, if any, of supersymmetry and of supersymmetry-breaking? What are the consequences of low-scale GUT physics and string physics? What sorts of solutions do these problems provide for long-standing problems such as the fermion mass hierarchy problem, as well as general issues in flavor physics? How can we understand neutrino oscillations in this context? Indeed, for each of these questions, we now must consider the possible role of large extra dimensions, and the effects of fundamental energy scales which are now radically altered. There are also numerous questions of a more phenomenological, even experimental, nature. If the fundamental energy scales are really down in the TeV range, then many striking signals will be possible at future colliders. These include the direct detection of Kaluza-Klein states; decays of KK states into (s)fermions; Drell-Yan production of KK states in pp collisions; new effective four-fermi contact interactions that emerge via exchange of gauge-boson KK states; and even direct detection of GUT particles, Regge

503

states, and signatures of quantum gravity (such as TeV-scale black holes). Thus, it becomes possible possible, for the first time, to contemplate probing the properties of GUTs and strings experimentally, which would lead to a new experimental approach to string phenomenology! Of course, this new physics is not merely limited to elementary particles, for large extra dimensions would also have profound effects for astrophysics and cosmology. New issues that immediately emerge in this framework include the role of light stable KK states, possible new dark-matter candidates (such as stable low-lying KK states), reheating and thermal regeneration of KK states, new brane-related scenarios for inflation in higher dimensions, phase transitions in higher dimensions, topological defects, and density perturbations. One can also ponder whether there might be cosmological methods for generating the large radii that are required in the brane world, or whether there might even be cosmological explanations for the dimensionality of spacetime (i.e., a reason why only four dimensions became uncompactified and large, while the others remain compact). There may even be new brane-related approaches to the cosmological constant problem. Clearly, then, there is a lot to think about. In Lecture # 3 , we shall discuss several new approaches towards addressing some of these problems in more detail. Then, in Lecture # 4 , we shall raise some new issues which have not received much attention in the literature, but which should serve to broaden our perspective concerning some of the fuller implications of studying theories with extra spacetime dimensions.

3. Lecture # 3 : Extra dimensions in action: Applications to particle phenomenology In this lecture, we shall address some phenomenological issues that arise within the context of the brane world. As we shall see, most of these issues arise within the standard paradigm as well, but their resolutions within the brane world take on a wholly new character. The set of topics I have chosen to address in this lecture is, of necessity, quite limited. However, it is my hope that by examining these topics, it is possible to get a sense of how actual model-building in extra dimensions proceeds. Indeed, at this point, most of the major phenomenological issues facing the brane world scenario have been studied quite extensively; we therefore refer the interested reader to the original literature for more details.

504

3.1. Proton

decay

Perhaps the most immediate question that arises within any extension to the Standard Model is the issue of proton decay. In a nutshell, the proton is known to have a lifetime exceeding 10 33 years, yet most extensions to the Standard Model introduce new interactions which tend to provide baryon non-conserving channels for proton decay into leptons. In the standard paradigm, such new physics occurs at a high energy scale, however, and thus these theories may be able to maintain proton stability to a sufficient degree. By contrast, in the case of the brane world scenarios in which all fundamental high energy scales are eliminated, this issue becomes even more severe. However, in this lecture, we shall show that there exist various intrinsically higher-dimensional mechanisms which enable us to cancel the usual proton-decay diagrams to all orders in perturbation theory. Let us begin by considering the scenario outlined in the previous lecture wherein MQUT is lowered to a new smaller scale M'GVT. Any grand unified theories, in and of themselves, already tend to introduce severe complications for proton decay, for proton decay can be mediated by the X bosons of the GUT gauge group as well as new, colored Higgs triplets. The relevant diagrams are illustrated in Fig. 18.

¥.

Figure 18. Typical diagrams that can mediate proton decay. Here the external lines correspond to the MSSM (s)quarks and leptons, while the internal ^ fields correspond to X-bosons (as in the diagram on the left) and Higgsino fields (as in the diagram on the right).

In a traditional GUT, these diagrams are suppressed by the large GUT mass ~ 1016 GeV associated with these new particles; this mass appears in the intermediate propagators and can often suppress these amplitudes to an acceptable degree.

505

In the brane world scenarios, however, this situation is drastically altered: although the unified gauge coupling is typically weaker than the usual GUT gauge coupling, the unification scale is significantly lower! This means that the usual proton decay amplitudes are apparently increased by a factor

This would seem to indicate immediate trouble for these scenarios. The important point, however, is that these are actually grand unified theories in higher dimensions. Therefore, we should ask whether there might exist intrinsically higher-dimensional solutions to the proton decay problem. One possible mechanism proceeds as follows. Recall that all of the extra GUT states beyond the MSSM should not be observable below the unification scale. What this means, then, iis that such states should not have zero modes, or equivalently, such states should be chosen odd under the same Z2 orbifold projection that we used to break our extended supersymmetry to N = 1 supersymmetry and to introduce chirality at low energies. Indeed, such states must have wavefunctions which behave as sm(ny/R) in the higher-dimensional space. How does this help us? Recall that in bany scenarios, the three generations of chiral fermions are restricted to the orbifold fixed points located at y = 0 and y = irR. In such scenarios, then, we see that all GUT states beyond the MSSM have wavefunctions which necessarily vanish at the higher-dimensional fermion locations! These states therefore do not couple to the MSSM fermions, and consequently are not able to mediate proton decay along the lines of the diagrams in Fig. 18. This is actually a rather powerful mechanism. It completely suppresses the dangerous interactions not only for the lowest modes of the X-bosons and Higgs triplets, but also all of their infinite towers of KK excitations, as illustrated in Fig. 19. Thus, such proton decay diagrams vanish to all orders in perturbation theory. Although we have chosen to illustrate this mechanism as the failure of a wavefunction to reach a localized state, we should stress that this pictorial representation is only meant to be suggestive. In truth, we must remember that all such localizations are effective only down to a characteristic length scales associated with the string scale; indeed, as we know from elementary considerations of quantum-mechanical wavepackets, no perfect localizations

506

n=1 (A £

0)

o 0)

I I

CO CO

y=0

y=7TR

n=2 (0

c .2 E i_

//

/

s

\\

CO


s

•^

\

2

'

CO (0

\

N

^Ss / "-^^ ^,'

2

y==0

/

y= n:=3 ,-

w c o

I0)

1 1

k.

M—

:

s

(0 CO

s y=0

-v

/

/ \\ 1 \\ 1 \ 1

' \ / \ \

1

1

i

1

! \

\

\ '

1 1

/

/

' ^x

(0

CO

\

i i

I i

i

»/

x'

i

i

\

*x /

y=7i;R

Figure 19. The orbifold projection requires the GUT particles beyond the MSSM to have odd wavefunctions under y —> —y, which in turn prevents these particles from coupling to the MSSM fermions which reside at the orbifold fixed points. This applies not only for the lowest-lying KK mode (top figure), but all excited KK modes as well (lower figures).

are possible. Consequently there is always a little bit of overlap. However, our conclusion that these diagrams vanish to all orders in perturbation theory continues to hold. This is because our pictorial representations really

507

illustrate something deeper: the Z 2 orbifold projection that we have been discussing actually requires that all terms in our underlying Lagrangian must be Z 2 invariant, and it is this requirement that eliminates the direct couplings between these GUT states and the MSSM fermions. It is tempting to view this suppression mechanism as a Kaluza-Klein selection rule whereby an excited n > 1 KK state cannot couple directly to zero modes. Indeed, such KK selection rules would represent momentum conservation in the higher-dimensional space. However, it is important to realize that the presence of our orbifold fixed points (or branes) at y = 0 and y = irR actually breaks the higher-dimensional translation invariance that would have given rise to such a higher-dimensional momentum conservation. Thus, this orbifolding procedure breaks the KK selection rules, but (as we have seen) supplies a new discrete symmetry which ends up having the same effect. We see, then, that this is an intrinsically higher-dimensional solution to the proton decay problem. It is valid for all radii R and spacetime dimensions D, and instead makes use of a higher-dimensional (orbifold) symmetry to eliminate the coupling rather than a heavy mass scale to suppress a propagator. Such a mechanism therefore has no analogue for ordinary four-dimensional grand unified theories. There are, of course, many other contributions to proton decay. For example, there are still a plethora of additional, non-renormalizable interactions which can emerge, and which will require their own methods of suppression. In general, these can be handled by introducing further discrete symmetries which can eliminate whatever proton decay diagrams might destabilize the proton below experimental limits. There may also be non-perturbative effects leading to proton decay, such as instanton contributions which scale like ~ e x p ( - l / g 2 ) . These are generally small. If the scale of quantum gravity is also reduced, then there exist additional effects that can lead to observable proton decay. For example, black holes can typically mediate proton decay. Suppressing such contributions would therefore require not merely discrete symmetries, but gauged discrete symmetries. Finally, we should mention that there also exist other mechanisms for stabilizing the proton. For example, even though our MSSM fermions may be localized relative to the extra dimensions transverse to the brane, the brane itself may still have extra longitudinal dimensions. If so, then it may be possible to localize the quarks and leptons at different points within these longitudinal extra dimensions. (These scenarios are commonly referred to

508

as having "split fermions".) Thus, even though our GUT particles may couple to both the quarks and the leptons equally, the required three-point vertex will be suppressed by the exponentially small wavefunction overlap between the quarks and leptons within the brane. This mechanism works not only for quark/lepton localization along longitudinal extra dimensions within the brane, but also within any small (string-scale) transverse width that the brane itself may have (a so-called "fat brane"). This mechanism will be discussed in more detail by other lecturers. 3.2. The fermion

mass

hierarchy

We now address the question of understanding the fermion mass hierarchy within the context of a higher-dimensional theory. First, recall that the fermion mass hierarchy refers to the hierarchical pattern of masses for the fermions of the Standard Model. For example, our lightest first-generation fermions have masses me

« 0.5 MeV = 5 x 1CT4 GeV

mu

«

md «

5 MeV = 5 x 1CT3 GeV 10 MeV = 1 x 1 0 - 2 GeV ,

(3.2)

while our third-generation quarks have masses mb « mt

5 GeV = 5 x 10° GeV 180 GeV « 2 x 10+2 GeV .

(3.3)

Thus, spanning all of the fermions, we have an overall hierarchy ^ )

« 1012 ,

(3.4)

meJ which is quite substantial indeed. Note that we have not even included neutrinos in this discussion (we shall discuss the case of neutrinos in the Sect. 3.3). This hierarchy is exceedingly difficult to explain within the usual Standard Model or the MSSM. In these models, these masses arise from interactions with the Higgs fields parametrized by Yukawa couplings, yet these Yukawa couplings, like the gauge couplings, also run logarithmically. Can extra dimensions provide a solution? If we naively consider the MSSM in higher dimensions and evaluate the standard Yukawa wavefunction diagrams, then we have Kaluza-Klein states of our MSSM particles

509

running in the loops. We then find that our Yukawa couplings ctp for each fermion F runs according to

«F(MO)

iFjp \XsJ

\Moy

We thus see that the Yukawa couplings are driven rapidly towards weak values with power-law dependence. This might be useful for avoiding problems with Landau poles, even for the top quark, and might even be useful for string theory (where we can potentially avoid dilaton runaway problems with exceedingly weak couplings at the string scale). However, this cannot be used to explain the fermion mass hierarchy, sinice the power-law dependence in Eq.(3.5) is universal for all fermions. Another possibility is to introduce a new singlet S with a superpotential coupling W = \HuHdS. In this case, we then find that our Yukawa couplings become string rather than weak in the UV, with the couplings ultimately sharing a common Landau pole. However, while this scenario can explain certain pairwise hierarchies between up- and down-type fermions, this still cannot provide a satisfactory solution to the fermion mass hierarchy problem. Clearly, in order to explain the full hierarchy between all fermions, we require a flavor-dependent coupling. It is important to note, however, that this flavor dependence does not have to be large, since power-law running can amplify the effects of even small flavor-dependent couplings. To illustrate this, let us consider the a variant of the Frogatt-Nielsen scenario by imagining an extra scalar field $ with its own KK tower, coupling to the fermions F through a superpotential interaction term of the form W = Y, VF $™F FFHu,d .

(3.6)

F

Here rip depends on the specific flavor F (indeed, this is the input flavor dependence). However, unlike the Frogatt-Nielsen scenario, we shall treat $ as fully dynamical. There will also be no need for an arbitrarily small vacuum expectation value for <J>. To see the role that extra spacetime dimensions can now play, let us consider the Yukawa wavefunction diagram shown in Fig. 20. We see that the number of internal KK towers in the loop is flavor-dependent. Thus, each fermion feels a different effective number of spacetime dimensions: AF = S(l+nF)

+ 2nF .

(3.7)

510

t,b

J

/

H

u,d

\

t_

Figure 20. Yukawa wavefunction renormalization diagram corresponding to the interaction given in Eq. (3.6). This setup induces each fermion to feel a different effective number of spacetime dimensions, leading to different power-law scalings for the different fermions.

This can then be used to explain the fermion mass hierarchy. For example, for S = 1 (one extra dimension) and R~l = 1 TeV, the entire mass hierarchy between the electron and the top quark can be accommodated simply with rit = 0 a n d fit — 3. There is thus no need for the usual large Frogatt-Nielsen values of ripNeedless to say, this is only one scenario for addressing the fermion mass hierarchy. However, all higher-dimensional scenarios for addressing this issue generally revolve around the same basic idea: arranging our model in such a way that the different generations of fermions (and even the different flavors of fermion within a single generation) feel the higher-dimensional spacetime in a flavor-dependent way. Other scenarios by which this can be done include trapping different flavors on different branes with different transverse volumes, and/or trapping the different fermions on subspaces of different dimensionalities. One can then exploit volume effects and/or different power-law scalings in order to achieve a variety of effects. Of course, this is not a solution to the various flavor puzzles. In particular, the brane world scenario doees not explain the flavor-dependent positions or volumes that the different fermions experience; it merely can accommodate them. To provide an explanation would require a proper, "topdown" string-theoretic construction of a particular self-consistent Type I string model. In any case, the important point is that the brane world scenario has clearly given us a new language in which to think about these issues: the language of geometry. Moreover, thanks to power-law running

511

and/or exponentially small wavefunction overlaps, the brane world has also given us at least partial tools for addressing flavor hierarchies, since we now only require small input parameters at high energies in order to generate potentially significant hierarchies at lower energies.

3.3. Neutrino

oscillations

As we have seen, the lesson from the brane world has been that heavy mass scales in four dimensions can be replaced by lighter mass scales in higher dimensions. However, low-energy neutrino data seem to provide independent evidence for yet another heavy mass scale, namely the seesaw scale. Let us first recall how the usual seesaw mechanism works. GUT models based on SO (10) gauge symmetries or larger generically predict the existence of a right-handed neutrino N for each generation. Let us therefore assume a Yukawa coupling of the form yuULHuN, which leads to a Dirac massTO= y^(Hu) « O(10 2 ) GeV. However, let us also assume a Majorana mass for N (which might arise from GUT breaking via the 126 representation receiving a GUT-scale vacuum expectation value): M « O(10 16 ) GeV. The total mass terms for our left- and right-handed neutrinos then take the form

Diagonalizing to find the mass eigenvalues, we obtain 2 771

A_ «

- — ,

A+ * M .

(3.9)

Thus, we see that the lightest eigenvalue (which we associate with our observable neutrinos) "seesaws" against heavy mass scale. Indeed, if we wish the lightest eigenvalue to fall into the approximate range near 10~ 2 eV, this indicates that we must have M « 10 1 4 ^ 1 6 GeV. Thus, light neutrino masses seem to provide further evidence for a high fundamental GUT scale. The question then emerges as to whether it is possible to generate light neutrino masses without the introduction of a heavy mass scale, perhaps by some intrinsically higher-dimensional mechanism. To date, there have been a number of proposals concerning how this might be accomplished. The vast majority of these proposals, however, originate with the same observation: because the right-handed neutrino is a Standard-Model gauge singlet, it need not be restricted to a "brane"

512

with respect to the full higher-dimensional space. It is therefore possible for this field to experience extra spacetime dimensions and thereby accrue an infinite tower of Kahiza-Klein excitations. In other words, the right-handed neutrino state may emerge within the context of Type I string theory as an open, rather than closed, string state. This then leads to a number of higher-dimensional mechanisms for suppressing the resulting neutrino mass without a heavy mass scale. For example, if the right-handed neutrino exists in the bulk, then its Kaluza-Klein excitations can give rise to power-law running for the neutrino Yukawa coupling through the diagram in Fig. 21. We see, then, that the neutrino Yukawa coupling can be driven to very small values over short energy interval, implying that the neutrino mass will be power-law suppressed relative to masses of all other fermions!

N VL

(

\

\

Hu Figure 21. Neutrino Yukawa renormalization diagram leading to power-law suppression of the left-handed neutrino relative to all other fermions.

Another observation concerns a possible suppression by the so-called bulk "volume" factor. If the right-handed neutrino N is in the bulk, then its wavefunction must be normalized (suppressed) by the bulk volume factor (RM*)n where M* is the fundamental energy scale associated with the bulk physics (e.g., the Planck or string scale) and where n is the number of extra spacetime dimensions associated with the bulk. However, this volume factor in turn suppresses the Dirac coupling m between the left-handed neutrino on the brane; basically, the wavefunction of the right-handed neutrino is so spread out over the relatively large bulk that its overlap with the lefthanded neutrino on the brane is exceedingly small. This, too, can therefore lead to a small effective physical neutrino mass. Both of these mechanisms suppress the mass of the resulting light neutrino state by suppressing the Dirac coupling m. But is there a higherdimensional analogue of the seesaw mechanism?

513

In order to construct such a higher-dimensional seesaw mechanism, let us again begin by assuming that the right-handed neutrino feels extra dimensions, while the left-handed neutrino VL, does not. For concreteness, we consider a Dirac fermion <]/ in five dimensions, and work in the Weyl basis in which \& can be decomposed into two two-component spinors: vf = (^ 1) -i/)2)T. When the extra spacetime dimension is compactified on a Z2 orbifold, it is natural for one of the two-component Weyl spinors, e.g., ipi, to be taken to be even under the Z2 action y —> —y, while the other spinor -02 is taken to be odd. If the left-handed neutrino VL is restricted to a brane located at the orbifold fixed point y = 0, then 1^2 vanishes at this point and so the most natural coupling is between VL and if>i. This then results in a Lagrangian of the form

£ = fdtxdy Ms j ^ i i a ^ V i + V w / ^ 2 j

./*,{,

VLia^D^VL + (mi'Lip1\v=o + h.c.) I

(3.10)

where y is the coordinate of the extra compactified spacetime dimension and where Ms is the mass scale of the higher-dimensional fundamental theory (e.g., a reduced Type I string scale). Note that the last term represents the Dirac brane/bulk Yukawa coupling between VL and i/>i • Next, we compactify the Lagrangian (3.10) down to four dimensions by expanding the five-dimensional \& field in Kaluza-Klein modes. Imposing the orbifold relations Vi,2(—2/) — ±^1,2(2/) implies that our Kaluza-Klein decomposition takes the form

M*,y)

= ^=$>in)(*)cos(m//i?)

(3.11)

and a similar result for ^2 with cosine replaced by sine. For convenience, we shall also define the linear combinations N^ = (f/'i + C ) / V 2 and M(") = (ip[n) - ip^n))/V2 for all n > 0. Inserting this decomposition into Eq. (3.10) and integrating over the compactified dimension, we then obtain an effective four-dimensional Lagrangian in which the StandardModel neutrino v^ mixes with the entire tower of Kaluza-Klein states of

514

the higher-dimensional ^ field with a mass mixing matrix of the form

M

m m m

m M0 0 0

m 0 0

m 0

M0 + i

Mo

(3.12) R

In Eq. (3.12), we have defined the basis

{uL,1,[0),NW,MW,NW,M™,...).

(3.13)

Note that m = m/y/2irRMs is the Dirac coupling suppressed by the volume factor corresponding to the extra spacetime dimension. The parameter Mo in Eq. (3.12) represents an additional contribution that may arise if we implement a Scherk-Schwarz twist upon compactification. Moreover, it turns out that topological considerations permit only two values: Mo = 0 and Mo = (2R)"1. Let us here consider the non-trivial possibility Mo = (2.R) -1 . It is then possible to solve for the eigenvalues and eigenstates of the mass mixing matrix in Eq. (3.12). Remarkably, it turns out that for any value of mR, there exists an exactly zero eigenvalue, with a corresponding mass eigenstate given exactly by

\H)

1 V l + TT2m2R2

\vL)-mRj2

fc=i

|AT(^i)) _ |M(fc>)

fc-1/2 (3.14)

,.(0) where we have denned A^°) = tp^'. Even though this result is exact for all mR, in most realistic scenarios we have mR
515

higher-dimensional scenarios, neutrino oscillations do not necessarily require neutrino masses! It is easy to determine the shape of the resulting neutrino oscillations. In general, the neutrino survival probability is given by 2

Uuk\ exp

.(*) k

(3.15)

2p

where U is the matrix that diagonalizes the mass matrix M. in Eq. (3.12). Eq. (3.15) represents the cumulative effects of the pairwise oscillations between VL and each of the Kaluza-Klein states associated with the righthanded neutrino. However, since A4 is non-diagonal, U will also be nondiagonal. We thus obtain the non-trivial neutrino oscillations shown in Fig. 22.

10

20 t/(2R z p)

30

10

20 t/(2R z p)

30

40

Figure 22. Neutrino oscillations without neutrino masses in five dimensions. The lefthanded neutrino on the brane oscillates with each of the Kaluza-Klein modes of the righthanded neutrino in the bulk. The infinite numbers of resulting individual oscillations (the first two components of which are shown in the left plot) add constructively to produce the resulting oscillation pattern shown in the right plot.

It is evident from Fig. 22 that our resulting neutrino oscillations are quite sizable. Moreover, our neutrino oscillations are effectively periodic, just as in four dimensions. Interestingly, in this scenario, the neutrino regeneration is never total, but the deficits can be total. In general, the approximate oscillation "length" is set by i ? _ 1 , since this determines the mass spacings of the right-handed neutrino KK states in the bulk.

516

It is important to note t h a t these neutrino oscillations arise from the same left/right neutrino mixing which gives rise t o the neutrino mass matrix and thereby suppresses the physical neutrino mass. This is an important distinction relative to the usual four-dimensional seesaw mechanism, where the left/right mixing has such a small effective mixing angle 6 ss 1 0 - 1 6 t h a t no observable oscillations result. Just as in the usual four-dimensional case, however, this type of higherdimensional seesaw mechanism can easily be generalized to include the case of flavor oscillations. Most flavor models in the literature thus far introduce one bulk neutrino for each of the three brane neutrinos, thereby extending flavor into the bulk. However, in such cases, the b r a n e / b u l k couplings become arbitrary 3 x 3 mixing matrices whose parameters are undetermined. Moreover, the three bulk neutrinos can in principle correspond to different e x t r a spacetime dimensions with different radii. T h u s , one obtains a scenario with many undetermined parameters governing neutrino masses and mixing angles. Perhaps more interestingly, one can instead consider a "compact" model in which there is only one bulk neutrino \1/ for all three flavor brane neutrinos Vi (i = 1,2,3) on the brane. Moreover, let us imagine t h a t each of these brane neutrinos has a corresponding Majorana mass m;, but for simplicity we will take the brane theory (i.e., the S t a n d a r d Model) to be completely flavor-diagonal. Likewise, we shall also take our b r a n e / b u l k couplings to be completely flavor-neutral (parametrized by a single parameter m as before). W i t h Mo = 0 we then obtain a mass m a t r i x of the form / m i

0 0 m m m

M

0 m2 0 m m m

0 0 m3 m m m

m m m 0 0 0

m m m 0 l R

m m m 0 0

(3.16)

l R

0

V It t u r n s out t h a t the eigenvalues A of this matrix are given exactly as the solutions to the transcendental equation t a n 7rAi?

2

*E

i

(3.17)

For each solution A to Eq. (3.17), the corresponding mass eigenstate \i>.\) is

517

exactly given by \ ~l 3

l*A>=

\

3

-.

oo

,

l

/Nx

J= l

J

/

1=1

fc=

—OO

(3.18) where iV* is an overall normalization constant, and where we have defined f iV« #(*) EE i M « |^0)

if k> 0 if jfc < 0 if jfc = 0 .

(3.19)

Note that the different flavor eigenstates \v/) in Eq. (3.18) consist of different linear combinations of the different mass eigenstates \i/\} for each A. Thus, the different flavor eigenstates will experience a relative oscillation with each other. This oscillation is entirely "bulk-mediated" in the sense that there are no explicit flavor mixings on the brane; it is the presence of the higher-dimensional bulk which is completely responsible for inducing the flavor oscillations on the brane. As an explicit example, let us consider the case with mR = 0.01, 7TI3R S> 1, and rtii^R = 1 — dm/2 for various dm. We then obtain the neutrino oscillations shown in Fig. 23 as functions of a dimensionless time parameter i = (1/2R2)(L/E). Here the subscripts (1,2,3) correspond to (z/1,^2,^3) = (yeiV^v^) respectively. Note that these flavor oscillations arise even without flavor mixing angles on the brane. Although this special case assumes nearly degenerate neutrinos on the brane, it is also possible to achieve significant flavor oscillations even when the brane neutrinos are non-degenerate. This illustrates that in higher dimensions, it may not be necessary to have large flavor mixing angles on the brane in order to accommodate experimental data; small (or even vanishing) flavor mixing angles on the brane may suffice. Furthermore, even though this model contains only five free parameters, it yields a surprisingly rich neutrino oscillation phenomenology, and exploits the higher-dimensional nature of the bulk in an essential way. This model also exhibits another intriguing feature. When the brane/ bulk coupling m is large, one might have naively expected to lose all initial probability of brane neutrinos into bulk neutrinos due to the strong brane/bulk mixing. However, it turns out even as m —> 00, we lose only 1/3 of the initial probability into bulk neutrinos. In other words, 2/3 of the original probability always remains "on the brane" even when m —> 00.

518

Figure 23. Bulk-mediated neutrino flavor oscillations with mR.= 0.01 and m i ^ f i = l=F<5m/2, for various 5m. As discussed in the text, these flavor oscillations are essentially "maximal" even though all mixing angles on the brane vanish identically.

519

This property holds regardless of the values of R or m;, and is intimately connected with the structure of the model in which we have taken only a single bulk neutrino. (By contrast, if we had taken a separate bulk neutrino for each brane neutrino, we would have found that all initial neutrino probability would be lost into the bulk as m —> oo.) This surprising observation suggests that in this model, the large-m case may be able to evade various four-dimensional bounds (such as those from supernova cooling rates) which ordinarily restrict the sizes of mixings with sterile neutrinos. In this lecture, we have merely sketched a number of ideas which play a role when considering neutrino oscillations in higher dimensions. However, over the past few years this has become a rather large industry, and many detailed models have been constructed with a variety of realistic phenomenologies. In general, one seeks to find parameters governing the brane/bulk couplings and mixing angles in such a way as to yield neutrino flavor oscillations in accordance with observational data from solar neutrinos, atmospheric neutrinos, and even laboratory neutrinos (e.g., from the LSND experiment). One must also be careful that conversion into bulk neutrinos not affect observed supernova cooling rates. One important constraint which tends to limit the success of such models is the recent result from the SNO collaboration, which reports flavor conversion in the solar neutrino flux from electron neutrinos to muon and tau neutrinos, with apparently no losses into sterile neutrinos. Since the right-handed bulk Kaluza-Klein states we have been discussing act as sterile neutrinos from the four-dimensional point of view, these constraints require that any higher-dimensional treatment of the solar neutrino flux not leave too great a neutrino probability in the bulk neutrino states. While one can develop a higher-dimensional version of the MSW effect, one must be sure to choose parameters so that most, if not all, of the intermediate-stage bulk neutrino probability is ultimately returned to the active neutrino flavors on the brane. This is perhaps the most severe constraint currently facing models of neutrino oscillations in higher dimensions.

3.4. QCD axion

phenomenology

Many of the above ideas are completely general, and apply to other bulk fields as well. Towards this end, let us now discuss how extra spacetime dimensions may contribute to the invisibility of the QCD axion. The possibility of large extra dimensions may provide a new approach towards solving the strong CP problem. One of the standard approaches

520

to the strong CP problem is to introduce a Peccei-Quinn (PQ) symmetry with a corresponding axion. Unfortunately, the experimentally allowed parameter space for the mass and couplings of the axion have become very narrow, and it is not clear how to generate the high energy scale associated with the breaking of PQ symmetry or to explain the "invisibility" of the resulting QCD axion. Within the framework afforded by large extra dimensions, however, this situation may be radically altered. The basic idea is that since the QCD axion is a singlet under all Standard-Model symmetries, it is free to leave the D-brane to which the Standard-Model particles are restricted and propagate into the higher-dimensional bulk. In other words, in theories with extra dimensions, the QCD axion can accrue an infinite tower of Kaluza-Klein axion states. Can this be used to lower the fundamental PQ symmetry-breaking scale? As in the neutrino case, it is possible to exploit the large volume factor associated with the extra dimensions in order to realize a large effective four-dimensional PQ scale from a smaller, higher-dimensional fundamental PQ scale. Specifically, if / P Q is the underlying higher-dimensional PQ scale, the effective PQ scale / P Q in four dimensions is given by / P Q = (VMS)1'2 f?Q ^ / P Q - It is the volume-renormalized scale / P Q which parametrizes the couplings between the four-dimensional axion and the PQ-charged Standard-Model particles on the brane. Therefore, this volume-renormalization of the brane/bulk coupling can be used to obtain a sufficiently suppressed axion even if the underlying PQ scale is near the TeV range. This is therefore one method of generating an apparent high PQ symmetry-breaking scale in a natural way. However, the presence of the Kaluza-Klein axions can have important and unexpected effects on axion phenomenology. In theories with extra dimensions, it turns out that the four-dimensional axion no longer is a mass eigenstate; instead, this axion ao mixes with the infinite tower of Kaluza-Klein axions an, with a mass mixing matrix given by ^/2 ^/2 V2 2 2 V2 2 + y 2 2 2 V2 2+ V 2 2 2 + 9y2 V2

/ 1 AT

m

PQ

(3.20)

V where TOPQ ~ A Q C D / / P Q and y = (mpQR) of interesting consequences.

1

. This mixing has a number

521

First, under certain circumstances the mass of the axion essentially decouples from the PQ scale, and instead is set by the radius of the extra spacetime dimension. This occurs because the eigenvalues A of the matrix (3.20) are given as solutions to the transcendental equation 7ri?Acot(7riZA) = A2/mpQ where R is the radius of the extra compactified dimension. Note that the lightest eigenvalue necessarily satisfies A < l/(2i?); this result holds regardless of the value of TOPQ. This implies that axions in the 10~ 4 eV mass range are consistent with (sub-)millimeter extra dimensions. More interestingly, however, this implies that in higher dimensions, the size of the axion mass is set by the radius R and not by the Peccei-Quinn scale / P Q when mpQ > 1/(2R). This decoupling implies that it may be possible to adjust the mass of the axion independently of its couplings to matter. This is not possible in four dimensions. Second, since the usual four-dimensional axion is no longer a mass eigenstate, it should undergo laboratory oscillations as it propagates. Such oscillations are entirely analogous to neutrino oscillations. Moreover, because the axion is now a higher-dimensional field, Standard-Model particles couple not only to the usual four-dimensional axion zero mode, but rather to the entire linear superposition a' ~ Yln an of Kaluza-Klein modes. Therefore, the quantity of phenomenological interest is the probability Par^a>(t) that a' is preserved as a function of time. This probability is shown in Fig. 24. Unlike the case of neutrino oscillations, we see that this probability drops rapidly from 1 (at the initial time t = 0) to extremely small values (expected to be « 10~ 16 in realistic scenarios). At no future time does this probability regenerate. Thus, we see that in higher dimensions, the axion state a' rapidly "decoheres" and becomes invisible with respect to subsequent laboratory interactions! This decoherence is therefore a possible higher-dimensional mechanism contributing to an invisible axion. Finally, one can investigate the effects of Kaluza-Klein axions on cosmological relic axion oscillations. Recall that as the universe cools and passes through the QCD phase transition at T K AQCD, instanton effects suddenly establish a non-zero axion potential where none previously existed. In the usual four-dimensional situation, the axion can therefore find itself displaced relative to the newly-established minimum of the potential, and begin to oscillate around it with a "damping" term due to the expansion of the universe (the Hubble term). The equation governing this evolution is given by d2a +

TT da 9 3H-+mla

= 0

, (3.21)

522

where H is the Hubble constant and ma is the axion mass. Some of the most severe upper bounds on the PQ scale come from demanding that the energy trapped in these oscillations today be less than the critical energy density to overdose the universe. In higher dimensions, by contrast, all of the axion Kaluza-Klein modes are coupled together via the mixing matrix (3.20). This means that the usual equation (3.21) is now generalized to become d2ak

(3.22)

=0

• SH^+Mhat

~dt?~

for each individual axion mode a^, where the mass matrix MM is given in Eq. (3.20). Thus, even if we assume that only the zero-mode axion starts out initially displaced when the axion potential is generated, the coupling wj

1 1 1 1

0.8 --\

0.6

OH

1

1

!

1

i

,

i

|

i

1

(e)

i

i

|

i

i

-

Vb) \

\(c)

\

\(d)

0.2

i

(^T^-^

\

- 1 \

0.4 -

1 '

W

|\\.VVViW/iW

0

0.001

0.002

0.003

0.004

(m£Q/2p) t Figure 24. The axion preservation probability P 0 '_> a '(i) as a function of the number "max of axion Kaluza-Klein states which are included in the analysis. In this plot we show the results for (a) n m a x = 1; (b) n m a x = 2; (c) n m a x = 3; (d) n m a x = 5; and (e) "max = 30. As n m a x increases, the axion probability rapidly falls to zero as a result of the destructive interference of the Kaluza-Klein states, and remains suppressed without significant axion regeneration at any later times. This "decoherence" of the axion implies that there is negligible probability for subsequently detecting the original axion state at any future time.

523

inherent in the equations (3.22) is sufficient to trigger the excited KaluzaKlein modes into oscillation. This situation is illustrated in Fig. 25(a). It is therefore important to understand whether the infinite towers of Kaluza-Klein axion states accelerate or retard the dissipation of the cosmological energy density associated with these relic oscillations. Remarkably, one finds that the net effect of these coupled Kaluza-Klein axions is always either to preserve or to enhance the rate of energy dissipation. This behavior is shown in Fig. 25(b). Because these Kaluza-Klein states accelerate the rate at which this relic oscillation energy is dissipated, the usual relic oscillation bounds are loosened in higher dimensions. This suggests that it may be possible to raise the effective PQ symmetry-breaking scale beyond its usual four-dimensional value, thereby further contributing to axion invisibility.

time -.

y = (mp,R)-'

Figure 25. (a) Coupled Kaluza-Klein cosmological relic axion oscillations. Although the excited Kaluza-Klein modes have vanishing initial displacements, they are triggered into oscillation as a result of the initial displacement of the zero-mode. The superimposed dashed line shows the behavior of the usual four-dimensional axion zero-mode when no Kaluza-Klein modes are present, (b) The energy-dissipation ratio factor p(t)/p4D(*) where P4D (t) is the normalized energy density today in the usual four-dimensional case and p(t) is the analogous quantity in the higher-dimensional case. The different curves correspond to different input parameters. As y —» 0 (implying a large extra dimension), we see that the cosmological relic energy density is dissipated more rapidly than in four dimensions.

Together, these results suggest that it may be possible to develop a new, higher-dimensional approach to axion phenomenology. In this lecture, we have chosen to discuss only a small subset of the many phenomenological issues that have been examined within the context of the

524

brane world. For further details concerning these or other approaches, we refer the interested reader to the original literature. 4. Lecture # 4 : Beyond the simple pictures: Lessons and warnings In this final lecture, we shall discuss some issues which are normally neglected in current treatments of the brane world, but which perhaps may play a role in more sophisticated developments of a complete higherdimensional theory. I have chosen to raise such issues not only as a way of indicating caution concerning the standard treatments of higherdimensional theories, but also to stimulate further thinking and research. 4.1. Shape, not just

volume

Thus far in our discussions of large extra dimensions, we have seen that large extra dimensions have the potential to lower the fundamental energy scales of physics. Indeed, in all cases, the degree to which these scales may be lowered depends on the volume of the compactified dimensions. However, compactification geometry is not merely endowed with volume; to borrow a phrase from Shakespeare, it is also "beautified with goodly shape." 0 Indeed, both volume (or "Kahler") moduli and shape (or "complex") moduli are necessary in order to fully describe the geometry of compactification. This distinction has phenomenological relevance because the shape moduli also play a significant role in determining the experimental bounds on such scenarios. Unfortunately, in most discussions of large extra dimensions in the literature, relatively little attention has been paid to the implications of these moduli. In this lecture, we shall discuss some of the phenomenological implications of non-trivial shape moduli. First, as we shall see, shape moduli can have dramatic effects on the corresponding Kaluza-Klein (KK) spectrum: they can induce level-crossings and varying mass gaps, they can help to eliminate light KK states, and they can alter experimental constraints in such scenarios to the extent that the bounds on the largest extra dimension can be completely eliminated. In other words, we shall see that large extra dimensions can be essentially invisible (even if they are flat). Second, we shall see that shape casts "shadows". Specifically, we shall show that non-trivial shape moduli can distort our experimental percepC

W. Shakespeare, The Two Gentlemen

of Verona: IV, i.

525

tions of compactification geometry at low energies. Indeed, we shall find that spacetime geometry is essentially "renormalized" as a function of the energy scale with which it is probed. Finally, we shall briefly discuss the effects that non-trivial shape moduli can have on string winding modes. We shall show, in particular, that toroidal compactifications exist for which all KK states as well as all winding modes are heavier than the string scale. Thus, in the context of low-scale string theories, it is possible to cross the string scale without detecting any string states associated with spacetime compactification. In order to illustrate these points, let us begin by considering the case of compactification on a general two-torus, as illustrated in Fig. 26. Such a torus is realized as the space K subject to the two discrete identifications 2/1

2/1 + 27ri?!

2/1 -*

2/2

2/2

2 / 2 - ^ 2 / 2 + 2nR2 sin 9 .

2/i + 2KR2

COS 9

(4.23)

Thus, the flat two-torus is specified by three parameters: R\, R2, and 9. The volume of the torus is V = ATT2RIR2 sin 9, while the shape parameters are R2/R\ and 9. Most studies of large extra dimensions in the literature focus.on the effects of V and R2/Ri, essentially fixing 6 = TT/2. But what are the phenomenological implications of non-trivial 9?

27CR,

271R1

Yl

Figure 26. General two-dimensional torus with shift angle 9.

Clearly, we need to analyze the KK wavefunctions on this space. Demanding invariance under the torus periodicities in Eq. (4.23), we find that the KK wavefunctions take the form * r

exp lR-^Vl yi

( ~tan £e) 6J

.n2

y2

R2 sin 9

Applying the (mass) 2 operator —{d2/dy\ + d2/dy2),

n^TL

. (4.24)

we then find the KK

526

masses

Note that these masses are no longer invariant under n\ —> — ri\ or n2 —> —ri2 individually unless combined with 8 —> TT — 8. We can therefore restrict our attention to values of 8 in the range 0 < 8 < TT/2 without loss of generality. The case with 8 = TT/2 corresponds to the usual "rectangular" torus, but we are interested here in the behavior of the KK spectrum as 8 is varied while holding the radii (R\,R2) fixed. We then find dramatic changes in the KK spectrum, as illustrated in Fig. 27 (upper plots) for the cases with Rx = R2 and Ri = 4R2. We observe that in general, the mass gap /J, (defined as the mass of the first excited KK state) depends on 8, and tends to increase as 8 becomes smaller than TT/2. Note that in the case of KK gravitons, this increase in the mass gap implies that the deviations from Newtonian gravity will be exponentially suppressed as 8 becomes smaller, even though the radii of the extra dimensions are held fixed. Another possibility, of course, is to hold the volume of the torus fixed (as well as the ratio R2/Ri) while we vary 8. We then find the results illustrated in Fig. 27 (lower plots). In this case, the mass gap can vary, but need not necessarily grow as 8 gets smaller. It is important to note that modular symmetries imply a redundancy in whether we interpret changes in the shape moduli as changes in 8 or as changes in R2/R\. In other words, the effects of taking 8 —> 0 can equivalently be viewed as keeping 8 relatively large and instead varying R2/R1. However, the important point is that the Kaluza-Klein spectrum is indeed sensitive to changes in the shape moduli, and this feature may be used to evade experimental constraints even when the radii (or volume) of the extra dimensions may be large and/or fixed. In fact, there even exists a small window of parameter space in which, thanks to the shape moduli, the usual experimental constraints on the radii of the largest extra dimensions become, instead, constraints merely on the volume of the compactified space, regardless of the size of the largest radii. In other words, the experimental bounds becomes independent of the size of the largest extra dimension! Thus, under certain narrow circumstances, a large, flat extra dimension can be essentially invisible, even though the volume of compactification is held fixed.

527

\

A2-' ' ' //

7(2.-2)

,

|

1 ,

// I

-

:

/X y^^>^L /

(2.1)//

(2.0)/'

" ;

yC^s*^-^~e7^ •

(3.3)

L

r

/

/

'

(4,1)

/^Q.o)

j

L

s/<w)

\

^(1.0)

"" - R,=R " i

.

'

(2.2)

/ A\.-v>

i

\

-

^

-

2

,

1

i

I

i

.,

i

0.6

0.8

•

0.4

1

,

0.2

,

'.

-

R,=4R E

",,,!,,, 0

I

4

i

1= 3

/(2,-2)

^ /\ r/ ^^ L

/

i

1 i

/ '

/

,

•

I

51

' /'

'

AX

y / ( 1 . i)

/

'S

1 '

'

'

XK

4

f\

(2,0)

'"Si N(3,3)

"

\

( 2 . 2 ) \ ^ \

' R

1

1 ,

=

R

,

Z 1

0.8

,

1 ,

,

0.6

,

1 ,

0.4 sin0

_

0.2

__^

: 1

•

V

\«.i)\ vi

R, = 4R2 ,

j

\

(0.2)\ ~

\

^(1.0)

l

(UT^^^Mi , 1, , ,

/

/ \ .

B

a

a 1

•

-

\

C 2 1

/

yj

*

'

C!

\

^ 3 . ^^"(To)

2

^c-^.

1 ~

\ I'

\

,

0

' '/ ' '1 / \' \/l I -

t=

^"V

//(1.0)

0

'

0.2

0.4

(2.o) y

\ > *

,

sinf?

/(2.-1t

/

yS

1 i/ i

,

0.6

0.8

sin0

\

]

(1.0)

(1.1)

,

,

1

0.8

,

,

,

0.6

1

0.4

i , , ,, 0.2 0

sinfl

Figure 27. The masses of the lowest-lying KK states (tii, 712) as functions of the shape parameter d for Ri = R2 = R (upper left plot) and R\ = 4R2 (upper right plot), holding .Rl and R2 fixed. The lower plots represent the same scenarios where we vary 9 holding both the volume of the torus and the ratio R2/R1 fixed.

Let us now consider another surprising aspect of shape moduli: this is the phenomenon called "shadowing". To understand this, let us consider general tori in one, two, and three dimensions. These tori are illustrated in Fig. 28. The one-dimensional torus, of course, is nothing but a circle and has no corresponding shape moduli. By contrast, the two- and threedimensional tori are described not only by radii but also by the shift angles 6 and ctij which mix the periodicities associated with translations along the corresponding directions. Note that we shall now use the symbol p to indicate the radius of the one-torus; lowercase r\, r^ to indicate the radii of

528

(3)

Figure 28.

27Cp

General one-, two-, and three-dimensional tori with arbitrary shape angles.

the two-torus, with 8 serving as the shape angle; and uppercase i?i, R2, R3 to indicate the radii of the three-torus, with 0:12,011.3, «23 denoting the corresponding shape angles. Our goal is to study the extent to which various low-energy observers can determine the shapes of these tori by studying their associated KK spectra. Towards this end, let us assume that the "true" compactification geometry is given by the three-torus shown in Fig. 28(c). Furthermore, let us assume that there is a hierarchy of length scales such that R3 0(R^1), the KK spectrum will reveal the presence of all three dimensions of the torus. Such an observer can then determine all three radii Ri and shape angles Qy through a detailed spectral analysis of the KK states. However, for an observer with access to only intermediate energies 0(R^ ) the third dimension will be inaccessible; the compactification manifold would then appear to be a two-torus, as

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illustrated in Fig. 28(b). Finally, for the low-energy observer with access to energies 0(R{1) <S £ m a x
,

r2=R2

,

0 = a12 .

(4.26)

After all, at energies much below R^1, we expect all remnants of the third dimension to vanish, so that the two-torus experienced by the intermediateenergy observer is merely the base of the original three-torus in Fig. 28(c). Likewise, it is natural to expect that the lowest-energy observer would perceive a circle with radius p = n, which by Eq. (4.26) implies p = n

= Hi .

(4.27)

Once again, this would be the naive expectation, given that we have only sufficient energy to probe the largest dimension of the original three-torus. However, as we shall now demonstrate, the above relations in Eqs. (4.26) and (4.27) are incorrect, even in the presence of a large hierarchy R3
530

seen in E q . (4.25) t h a t t h e K K s p e c t r u m is given by TliTlj

M2 1=1

«

(4.28)

t^tj

where ra» G 2 . Finally, for t h e general three-torus shown in Fig. 28(c), t h e K K s p e c t r u m is given by

M2

1 ~K

3

En

2

'H2 2 J?2bjk

V ^ njn "» 7 Z^ P . RiR 1-Ll.J i^i

CikCjk)

(4.29)

where k ^ i,j, where Cij = c o s a ^ a n d s^- = s i n a ^ , a n d where K [the dimensionless squared volume of t h e parallelepiped in Fig. 28(c)] is given by

« =l-E4+2]!^ i<3

= E 4 + 2 n^' - ! i<3

i<3

(4.30)

\i<3

Given these results, we can now determine the appropriate correspondence relations. In deriving these relations, we shall assume a hierarchy i?3 < i?2 € R\ so that we can successively integrate out small extra dimensions when passing to larger length scales. Our procedure will be to disregard all KK states whose masses exceed the appropriate reference energy (either high, intermediate, or low) and are therefore inaccessible to the corresponding observer. The observer at highest energy clearly sees three dimensions worth of KK states, and deduces the "true" geometry of the compactified space by comparing the measured KK masses with Eq. (4.29). However, the observer at intermediate energy cannot perceive excitations in the R3 direction, since functionally R3 —> 0 for this observer. His attention is therefore restricted to states with 77,3 = 0, and he attempts a spectral analysis of the remaining states. This leads to the identifications 1

1

b

3k

(i = l , 2 ) sin C12 - C13C23 cos# 1 (4.31) sin 2 6>rir 2 KRiR2 This observer therefore deduces t h a t t h e compactified space is a two-torus parametrized by ( r i , r2, 0) given by KR\

(n

= si3Ri

\cos0

= (C12 - C13C23VS13S23 •

(4.32)

531

Note that both the radii r* and the shape angle 8 are affected, leading to apparent values for (ri,T2,0) which are not present in the original threetorus. The lowest-energy observer, by contrast, misses the i%i excitations as well. Upon comparing with the circle KK spectrum M2 = n\/p2, he therefore concludes that the compactified space is a circle of radius p = (sin0)n = —

i?i .

(4.33)

S23

Once again, this radius does not correspond to any periodicity in the original three-torus. Mathematically, these results reflect the geometric "shadows" that successive smaller extra dimensions cast onto the larger extra dimensions when they are integrated out. As such, they indicate that the low-energy observer can see only those "projections" of the compactification space which are perpendicular to the extinguished dimensions. But given the assumed large hierarchy of length scales, the physical implications of this shadowing effect are rather striking. A small extra spacetime dimension — even one no larger than the Planck length! — is able to cast a huge shadow over all other length scales and their associated dimensions, completely distorting our low-energy perception and interpretation of the compactification geometry. Indeed, the physics which we would normally associate with the Planck scale (such as the angles that parametrize the shape of the Planck-sized extra dimensions relative to the larger extra dimensions) fail to decouple at low energies. Of course, no observer at any energy scale can use this shadowing phenomenon in order to deduce the existence of an extra spacetime dimension beyond his own energy scale. Nevertheless, the observer's interpretation of that portion of the compactification geometry accessible to him is completely distorted, leading him to deduce geometric radii and shape angles that have no basis in reality. Since the existence of an even smaller extra dimension beyond those already perceived can never be ruled out, this shadowing effect implies that one can never know the "true" compactification geometry. Even when light KK states are detected and successful fits to the KK mass formulae are obtained via spectral analyses, the presence of further additional dimensions with appropriate shape moduli can always reveal the previous successes to have been illusory. We are not claiming that no "true" compactification geometry can ever exist. Indeed, if one takes the predictions of string theory seriously, then

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there is ultimately a true, maximum number of compactified dimensions, with associated radii and shape moduli. However, as an experimental question, one can never be satisfied concerning the true number of extra dimensions. Thus, these results imply that one can correspondingly never be certain of the nature of whatever compactification geometry is ultimately discovered. In this sense, the concept of a "true" compactification geometry does not exist. To make this point more explicit, let us consider a simple example. Suppose i?! _1 = 1 TeV, R^1 = 10 11 GeV, and R^1 = 10 19 GeV, with «12 = 90° and a±3 = a23 = 60°. How is this torus perceived at lower energies? Using the above results, we see that at intermediate energies this appears to be a two-torus with rj" 1 « 1.155 TeV, r^1 « 1.155 x 10 11 GeV, and 9 ~ 71°. Finally, at lowest energies we observe a single one-dimensional circle with radius p^1 ~ 1.225 TeV. Note that the distortions are not large in this example. However, this distortion for the size of the largest extra dimension persists over eight orders of magnitude in this example, and it does not disappear even as the smallest dimension(s) are taken to zero size! Let us now consider more dramatic examples of this "shadowing" phenomenon. Suppose the original three-torus has Ri = R2 and R% = 1019 GeV, with a12 = 90°, a13 = 60° + e, and a23 = 30° + e, where e < 1. In this configuration, the third periodicity associated with the Planck-sized extra dimension is nearly in the plane formed by the periodicities of the two larger extra dimensions. After integrating out the Planck-sized extra dimension, we apparently observe a two-torus with r\jr2 = \/3 and 6 ss e. Thus the "squashing" of the original Planck-sized extra dimension relative to the large dimensions is perceived by a low-energy observer as a squashing of the two large dimensions with respect to each other! Now let us suppose that the original three-torus has R\ — R2 and R-1 = 10 19 GeV, with a12 = 90° and a13 < a23. We then find that the effective two-torus at lower energies apparently has r\
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description of the compactification manifold changes. Our perception of quantities such as the radii and shape moduli associated with the largest (and experimentally accessible) extra dimensions continually evolves as a function of the energy with which we probe this manifold — even though the largest extra dimensions are already detected and their geometric properties are already presumed known. Thus, the apparent compactification geometry is not fixed at all, but rather undergoes renormalization much like other "constants" of nature. This may seem to be a rather unusual way to interpret these results. However, compactifications on these different tori constitute a series of different effective theories. As we cross the energy thresholds associated with the different extra dimensions, our effective theory changes. In this sense, then, our correspondence relations are "matching conditions" or "threshold effects" which describe the same physics at different energy scales. Indeed, such relations merely reflect the requirement that the physical KK spectrum remain invariant under change in our effective field theories with different cutoffs. However, this is nothing but renormalization. Indeed, we can imagine extrapolating our calculations to incorporate a continuing series of hierarchies corresponding to a continuing series of extra dimensions. The corresponding series of matching conditions would then constitute a renormalization group "flow." We stress that this is a purely classical effect. It is triggered entirely by non-trivial shape moduli, and thus is a property of geometry itself. However, as an experimental question, there can always be smaller which have not yet been discovered. We see, then, that there is no such thing as "true" compactification geometry. Indeed, even if we think we have discovered an extra dimension with a given size, the subsequent discovery of an additional extra dimension which is even smaller requires that we reevaluate the size of the extra dimension that had already been discovered and measured. Finally, if string theory is the ultimate theory, then spacetime compactification should produce not only KK states, but also winding states. Ordinarily, the masses of KK states and winding states play a reciprocal role: if the lightest KK states are lighter than the string scale, then the corresponding winding states are necessarily heavier than the string scale. Similarly, the reverse situation in which the lightest KK states are heavier than the string scale ordinarily results in winding states which are lighter than the string scale (and is equivalent to the previous configuration as a result of T-duality). The expectation, then, is that either the KK states

534

or the winding states must have masses at or below the string scale. Thus, it would seem to be impossible to cross the string scale without seeing at least some states (either KK or winding) associated with the spacetime compactification. However, this expectation also fails when non-trivial shape moduli are involved. Specifically, it can be shown that it is possible for the string scale to be simultaneously lighter than all the KK states as well as all the winding states. Thus, in such theories, it is possible to cross the string scale without seeing a single resonance associated with the spacetime compactification! Needless to say, this can therefore give rise to a low-energy phenomenology which is markedly different from that of the usual KK effective field theories. These results concerning shape moduli prompt a number of outstanding questions. What sets the values of these shape moduli? What are the effects of non-trivial shape moduli in more general toroidal compactifications with background torsion fields, or in more complicated non-toroidal geometries? How do modular symmetries affect these results, and how might these results be interpreted within the context of string theory when non-trivial vacuum energies and GSO projections are present? How can the degrees of freedom associated with these non-trivial shape moduli be exploited in order to produce phenomenologically interesting theories, and how do they affect current collider (and Cavendish) bounds on extra dimensions? How might theories with non-trivial shape moduli be deconstructed? If shape moduli can have such dramatic effects on KK masses, what are their effects on couplings and scattering amplitudes? How do quantum effects contribute to the renormalization of compactification geometry? And if compactification geometry is renormalized, to what extent is spacetime geometry knowable at all? Indeed, what are the analogues of the renormalization-group invariants? Finally, can we use the phenomenon of Planck-scale "shadowing" in order to develop a new approach for attacking the cosmological constant problem? Clearly, these and other questions await further study.

4.2. Strings,

not just effective field

theories

Most treatments of large extra dimensions in the literature rely on the use of an effective field theory. In other words, one writes a field-theoretic Lagrangian involving the Kaluza-Klein modes, and, if necessary, utilizes a cutoff of some sort in order to regulate possibly divergent Kaluza-Klein sums. While this may be a valid procedure in some circumstances, the results of such treatments are often highly sensitive to the value and form of the

535

cutoff. Moreover, such treatments may completely miss other important physics which can be as relevant as (or even more relevant than) the physics of only the Kaluza-Klein modes. To illustrate these points, let us consider what light string theory might shed on such issues. String theory may be especially relevant if, as discussed in Lecture # 2 , the string scale is significantly reduced, perhaps even to the TeV range. In string theory, there are actually three kinds of infinite towers of massive states. First, since strings are extended objects with an intrinsic tension (energy per unit length), there is a tower of states which correspond to the quantized oscillations of the string. As such, the fundamental energy scale of these states is set by the string tension T ~ 1/a' where a' is the Regge slope. Given the definition of the string scale in Eq. (2.3), we see that the masses of these so-called "oscillator states" are given by M£(osc) = V£M stri ng ,

<€Z.

(4.34)

Note that the masses of these oscillator states have no dependence on the compactification radius, and only depend on the string's internal structure. The quantity £ is the oscillation number. The second group of states are the Kaluza-Klein states arising from spacetime compactification. Just as for point particles, the masses of these states are given by MfK>

= ^,

n€Z

(4.35)

where R is the radius of compactification. In contrast with the oscillator states, the KK states depend on the geometry of the compactification but are completely insensitive to the string's internal structure (i.e., no dependence on a'). For closed strings, there is also a third group of states. When the spacetime is compactified with periodic boundary conditions, the string may wind around the compact direction. The mass or energy involved in this winding state therefore takes the form _^(wind) _ tension x wrapping length = (J^

x (wR)

= wM^ingR

,

weTL

(4.36)

where w represents the winding number. Unlike the previous groups of states, the masses of the winding states depend on both M s t r ; n g and R. Moreover, unlike the KK states, the masses actually grow linearly with R.

536

These states thus reflect both the spacetime compactification as well as the internal structure of the string. Thus, in string theory, we actually have three groups of infinite towers of states whose masses are governed by two mass scales (Mstring and l/R). We have taken the trouble to clearly identify these three separate groups of states because in string theory, they can actually conspire together in order to produce some novel effects. These are therefore effects which would not be apparent from considerations of just the KK states alone. For example, closed strings exhibit a symmetry called T-duality under which the tree-level spectrum is invariant under (Af s tring-R)

*

•

n

<

> W

(Af s t r ing i?)

KK states <—> winding states .

(4-37)

In fact, this T-duality symmetry holds for the mass spectrum not only at tree level, but also to all orders in perturbation theory and even nonperturbatively. It also holds for all string interactions. Thus, T-duality is an exact symmetry of string theory. T-duality has important implications for physics. For example, Tduality plays a crucial role amongst the complete set of non-perturbative string dualities, leading to a deeper understanding of the complete string moduli space. More generally, however, T-duality suggests a new geometric structure for spacetime near the string scale. In particular, T-duality indicates that our usual linear ordering of length scales and energy scales must somehow break down near the string scale, since strings cannot distinguish whether they are propagating on a space with a large or small compactification radius! This phenomenon has been the basis of a claim that there is actually a minimum possible length scale in string theory, i? m i n « M^a . It is interesting to ask whether these extra oscillator and winding states might also have implications for phenomenology. Of course, if R and Mstring are in the TeV range, all of these states might be observable at future colliders. However, if these scales are very high, these states will necessarily be very heavy. It might then seem that these states cannot have any effect on low-energy phenomenology. However, this conclusion may not always be correct. Since there are actually an infinite number of such states, they have the potential to combine to produce some important effects, even at low energies. After all, the decoupling theorem states that a single heavy state of mass M makes contributions to low-energy physics which are suppressed by powers of M, and

537

which disappear as M —> oo. However, this theorem may not apply when there is an infinite tower of such states, especially if (as in string theory) the number of such states grows exponentially as a function of mass. In order to see how this might be useful, let us consider how we might use these states in order to provide an alternative approach towards solving the gauge hierarchy problem without supersymmetry and without large extra dimensions. First, recall that in field theory, ultraviolet divergences are generic and lead to destabilization of light energy scales. This is the fundamental source of the gauge hierarchy problem. Moreover, the traditional routes to stabilizing the gauge hierarchy (such as supersymmetry or large extra dimensions) require the introduction of new physics (either superpartners or new Kaluza-Klein states) only slightly above the scale of electroweak symmetry breaking. This is because these mechanisms operate only scale-by-scale, and thus can protect the gauge hierarchy just below the TeV range only by introducing new physics just above the TeV range. However, in string theory, this need not be the case: the infinite towers of oscillator states, Kaluza-Klein states, and winding states together conspire to arrange themselves at all mass levels so as to cancel the divergences of the light (massless) states! Rather than rely on boson/fermion pairings or cancellations amongst states at the same energy levels, these cancellations occur across the entire infinite string spectrum. These cancellations thus represent conspiracies between physics at all energy scales simultaneously. Moreover, these cancellations have been shown to occur automatically in string theory, as a result of deeper string symmetries which would not be apparent from a purely low-energy perspective. These symmetries are ultimately at the root of the ultraviolet finiteness for which string theory is famous. Thus, in such a scenario, the divergences afflicting the Higgs mass calculation in the non-supersymmetric Standard Model would be cancelled not by the contributions of weak-scale superpartners, but rather through the contributions of infinite towers of string oscillator, KK, and winding states. Moreover, this can occur even if all of these states are at the Planck scale, and are hence unobservable at low energies. This would then represent yet another approach to the hierarchy problem. By not requiring any new physics at the TeV scale, this approach clearly leads to a markedly different phenomenology for physics beyond the Standard Model. More importantly, however, the existence of such an approach illustrates our broader point that approaches based exclusively on

538

low-energy effective field theories may not always be sufficient to address questions, such as the hierarchy problem, which necessarily involve physics at widely separated energy scales. As a final example of unexpected features emerging within the context of string theory, let us again consider the masses of Kaluza-Klein and winding states. We have already seen in Eq. (4.25) that the KK masses for a general two-torus depend on the shape angle 6. In general, for a flat n-dimensional torus, there are n different radii Ri as well as a variety of shape angles a^which parametrize the angle between the i t h and j t h periodicity directions, with 1 < i,j < n. The corresponding masses of the Kaluza-Klein and winding states are then given by M2ni.Wi = NTQ^N

(4.38)

where we have collected the corresponding KK momentum and winding numbers into a vector NT

=

fe.-,^;^,..,^) Rn a'

\Ri

,

(4-39)

a' J

and where O"1 =

(-SG-

G-BG-iB)

•

< 4 - 40)

In this expression, G is the dimensionless (n x n)-dimensional metric of the n-torus, now given by Gij = cos atij

(4-41)

(where an s 0), and B is the (n X n)-dimensional background antisymmetric tensor (representing a possible spacetime torsion). Thus, the mass formula (4.38) generalizes our previous results for both KK and winding states to the case of general n-dimensional tori. This much, again, is relatively standard. However, in string theory, this is not the only contribution to the mass of a given KK and/or winding state. Instead, the full mass spectrum of a toroidally compactified string is generally governed by two equations of the form a'M2tot = a'M2ni.Wi +2(N + N) + 2(a + a)

~N-N = Y2niwi + a-a .

(4.42)

i

While Mn^mi represents the mass contribution from the Kaluza-Klein and winding excitations given in Eq. (4.38), (N, N) represent the total energies

539

from the left- and right-moving string oscillator excitations. These energies are calculated as N = ^2nnNn and N = ^2nnNn, where iVn and JV„ count the number of excitations of frequency n of the underlying left- and right-moving string worldsheet fields. Likewise, (a, a) represent left- and right-moving vacuum-energy contributions. The first of these equations indicates that the total mass of a given string state receives contributions not only from Kaluza-Klein and winding excitations, but also from the oscillators and from the total vacuum energy. By contrast, the second of these equations is a level-matching constraint which is required for the self-consistency of the string. This constraint ensures that the total mass of a given state receives equal contributions from the left- and right-moving worldsheet excitations, possibly offset by a vacuum energy difference a — a. (The quantity ^ UiWi represents the energy offset due to the Kaluza-Klein/winding modes.) Note that the constraints listed in Eq. (4.42) are generic and appear as a minimal set for all toroidally compactified string theories. Given the constraints in Eq. (4.42), we immediately see two important differences relative to the traditional Kaluza-Klein picture which focuses only on Mni-Wi. First, we see that the total spacetime masses of our KaluzaKlein and winding modes are generally offset by a non-zero vacuum energy a + a. The value of a + a depends on the type of string in question (bosonic, superstring, etc), as well as on the type of sector (Neveu-Schwarz, Ramond, higher-order twists, etc) within a given string theory. The important point, however, is that a + a can have either sign, and is often negative in many string sectors. This implies that the lightest Kaluza-Klein states according to the mass formula in Eq. (4-38) need not be the lightest Kaluza-Klein states in the actual string spectrum. We shall see several dramatic examples of this below. The second important feature is the presence of a level-matching constraint which correlates the Kaluza-Klein/winding numbers (rij-, Wj) with the string oscillator energies (N, N). This implies that whether a given Kaluza-Klein or winding state is allowed in string theory depends not only on its total mass, but also on the particular combination J ^ riiWi. In order to see how these two features affect the mass spectrum, let us classify the various string states into sectors according to their values of N + 7f + a + a. In sectors with N + N + a + a > 0, the string states are already massive even before any Kaluza-Klein or winding modes are excited. Such states are therefore not likely to be among the lightest states in the full string spectrum.

540

Next, we turn our attention to string sectors with N + N + a + a = 0. As apparent from Eq. (4.42), these are the sectors in which our usual expectation A^tot = Mnnwi holds. Note that the special cases with rii = Wi = 0 correspond to the massless string states (such as the graviton) which already exist in the spectrum prior to compactification. The remaining cases with non-zero n, and Wi thus correspond to the usual Kaluza-Klein and winding excitations of these states, as well as the excitations of their Kaluza-Klein descendants [including the U(l) graviphoton gauge fields which emerge from the metric and antisymmetric tensor via the original Kaluza-Klein mechanism, as discussed in Lecture #1]. It is important to note, however, that the spectrum of Kaluza-Klein and winding excitations in such sectors is restricted to those states with a common value of ^ riiWi (typically zero since the lightest Kaluza-Klein states in such sectors have J2i n%wt = 0). Finally, let us consider the sectors with N + N + a + a < 0. In such sectors, the lightest Kaluza-Klein and winding excitations are actually tachyonic and must be removed from the string spectrum (typically through a so-called "GSO projection"). Depending on the specific string model in question, there are two possible results of such GSO projections. First, it may happen that this entire sector of the string theory is GSO-projected out of the spectrum. In such cases, no states from this sector will survive. On the other hand, it may happen that these GSO projections affect only the lightest states, preserving states with a'A42ot > 0. In such cases, then, we find that it is the multiply excited Kaluza-Klein/winding states in these sectors which are actually the lightest states appearing in the string spectrum! Clearly, this feature is completely beyond our elementary considerations based on considering only the direct masses of KK and winding states. Yet, as we shall now demonstrate, this can lead to some very important effects. As an explicit example, let us consider the case of the bosonic string (for which a = a = - 1 ) . If (N,N) = (1,0) or (0,1), then a'M2tot = a'M^..w. - 2 . Thus, Kaluza-Klein or winding states for which a!M2n..w. = 2 are actually massless. To see the implications of this, let us imagine that we are compactifying our bosonic string on a two-torus given by R1=R2

= Va' ,

cos<9 = 6 = l / 2 .

(4.43)

In such a case, the following twelve multiply excited Kaluza-Klein/winding states (n1,n2;wi,w2) have a'M%..w. = 2 and hence are massless: ±(1,1;1,0), ±(1,1;0,-1),

±(1,0:1,-1), ±(1,0; -1,0),

± ( 0 , 1 ; 0,1) ±(0,1; 1,-1).

(4.44)

541

Note that the first six states [top line of Eq. (4.44)] have ^2iriiWi = 1, while the second six states [bottom line of Eq. (4.44)] have J2iriiWi = — 1. Thus the first group of states can have (N,N) = (0,1), while the second group can have (N, N) = (1,0). If these oscillator excitations correspond to excitations of the spacetime coordinates of the uncompactified dimensions, then these string states will be spacetime vectors. Clearly, massless vectors must correspond to gauge bosons, and that's exactly what happens. These twelve states combine with the four U{1) graviphoton states which emerge from the Kaluza-Klein decomposition of the graviton and B^v field, thereby enhancing the total Kaluza-Klein gauge symmetry from £/(l) 4 to SU(3)L x SU(3)R\ This is remarkable result. At the beginning of Lecture # 1 (see Fig. 2), we showed that a Kaluza-Klein compactification of a single extra dimension gives rise to a U(l) gauge symmetry which emerges from the extra components of the metric; two extra dimensions would likewise produce C/(l) 2 . In a similar manner, an additional U(l)2 gauge symmetry emerges from decomposition of the anti-symmetric tensor field Bij. However, in a string context, we see that the appearance of the extra gauge boson states in Eq. (4.44) promotes this KK gauge symmetry from the abelian gauge symmetry U(l)4 to a non-abelian one! This is well beyond what is possible in ordinary Kaluza-Klein theory based on compactifications on flat tori. However, this mechanism is well known in string theory: this is nothing but the so-called "Narain mechanism" by which non-abelian gauge symmetries can be generated in string theory via toroidal compactifications. Through this mechanism, the Kaluza-Klein/winding quantum numbers become identified as the charges of a non-abelian gauge group. Indeed, by choosing our torus correctly, we can achieve any simply laced non-abelian gauge group we desire. These features illustrate that the simple Kaluza-Klein effective field theory picture becomes far richer, but also far more complex, when embedded within the full context of string theory. For example, if the SU(3) gauge symmetry of the strong interaction is realized in string theory as an enhanced gauge symmetry, as described above, then the massless gluons of the Standard Model are actually simultaneous combinations of KaluzaKlein, winding, and oscillator modes! In other words, such gluons correspond to string states which simultaneously resonate in extra dimensions (Kaluza-Klein), wrap around those extra dimensions (winding), and also vibrate along their length with a certain frequency (oscillators). Likewise, the masses of the W^ and Z gauge bosons, ordinarily generated through

542

the Higgs mechanism, can equivalently be generated by shifting various compactification radii away from the symmetric values which are needed to produce unbroken SU{2) x 17(1) gauge symmetry. This, then, opens up new avenues for higher-dimensional model-building which were not possible in the usual "field-theoretic" Kaluza-Klein treatments. 4.3. Curved, not just

flat

Finally, for the sake of completeness, we close this lecture with an important comment. Throughout these lectures, we have focused exclusively on the relatively simple case in which our extra dimensions are flat. Even though these extra dimensions may be compactified on circles or tori, these manifolds contain no intrinsic curvatures of their own. However, as we stressed in Lecture # 1 , the mere presence of massive branes implies that they must deform the spacetime around them, leading to "warping" effects which we have thus far neglected. It turns out that such warping effects can lead to a number of surprising and important phenomena, and ultimately give rise to wholly different theoretical approaches towards solving the gauge hierarchy problem and/or developing alternatives to standard methods of compactification. Models based on such warpings are generally referred to as "Randall-Sundrum models", and will be covered in subsequent lectures at this school.

A final thought Only experiment will decide if large extra spacetime dimensions actually exist in nature, and if the fundamental high-energy scales of physics are really as low as the TeV range. Nevertheless, what is remarkable about the recent developments we have presented is that they illustrate that the "fundamental" high energy scales are not immutable, and that the theoretical landscape for physics beyond the Standard Model may be significantly broader than had been previously thought. Moreover, it is equally remarkable and gratifying that ideas originally born in string theory are having such a profound effect in developing answers to primarily phenomenological questions, and that these ideas may be potentially testable in the not-toodistant future. These developments, then, suggest that an exciting future lies ahead of us. If nothing else, this may be the most valuable lesson that we may take with us from the brane world.

543 Acknowledgments First, I would like to t h a n k the organizers of the 2002 TASI school (Howie Haber, Ann Nelson, and K . T . M a h a n t h a p p a ) for giving me the opportunity to spend a week in Boulder, Colorado and deliver these lectures in such a pleasant and stimulating environment. I am also particularly grateful t o t h e TASI organizers — especially Howie Haber — for t h e infinite patience they exhibited while waiting for these lectures t o appear in written form. I would also like to t h a n k my research colleagues for collaborations on many of the results which have found their way into these lectures. Finally, and most importantly, I also wish to t h a n k the TASI students themselves for their questions and sustained interest in these lectures. There is no greater joy t h a n teaching t o an interested and intelligent audience. References 1. The field of large extra dimensions is huge and constantly growing. Therefore, rather than attempt to provide a large list of references, we shall limit these references to those papers which originally proposed the ideas we have discussed in these lectures. In Lecture # 1 , Fig. 7 is taken from: E. G. Adelberger [EOT-WASH Group Collaboration], arXiv:hep-ex/0202008. For recent claims of evidence implying a non-zero time variation of the fundamental constants of nature, see, e.g.: J. K. Webb, V. V. Flambaum, C. W. Churchill, M. J. Drinkwater and J. D. Barrow, Phys. Rev. Lett. 82 (1999) 884 [arXiv:astro-ph/9803165]; J. K. Webb et al., Phys. Rev. Lett. 87 (2001) 091301 [arXiv:astroph/0012539]. 2. In Lecture # 2 , the idea of lowering the Planck scale due to extra spacetime dimensions transverse to a brane originally appeared in: N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429 (1998) 263 [arXiv:hep-ph/9803315]; I. Antoniadis et al., Phys. Lett. B 436 (1998) 257 [arXiv:hep-ph/9804398]. The idea of lowering the GUT scale due to extra spacetime longitudinal dimensions within the brane originally appeared in: K. R. Dienes, E. Dudas and T. Gherghetta, Phys. Lett. B 436 (1998) 55 [arXiv:hep-ph/9803466]; Nucl. Phys. B 537 (1999) 47 [arXiv:hepph/9806292]; arXiv:hep-ph/9807522. The idea of lowering the string scale through the use of large extra dimensions originally appeared in: E. Witten, Nucl. Phys. B 471 (1996) 135 [arXiv:hep-th/9602070]; J. D. Lykken, Phys. Rev. D 54 (1996) 3693 [arXiv:hep-th/9603133]; G. Shiu and S. H. H. Tye, Phys. Rev. D 58 (1998) 106007 [arXiv:hepth/9805157];

544 Z. Kakushadze and S. H. H. Tye, Nucl. Phys. B 548 (1999) 180 [arXiv:hepth/9809147]. Note that important early work concerning the possibility of TeV-sized large extra spacetime dimensions appeared in: I. Antoniadis, Phys. Lett. B 246 (1990) 377; I. Antoniadis, K. Benakli and M. Quiros, Phys. Lett. B 331 (1994) 313 [arXiv:hep-ph/9403290]. 3. In Lecture # 3 , the ideas we presented concerning proton decay and the fermion mass hierarchy are discussed in: K. R. Dienes, E. Dudas and T. Gherghetta, Phys. Lett. B 436 (1998) 55 [arXiv:hep-ph/9803466]; Nucl. Phys. B 537 (1999) 47 [arXiv:hepph/9806292]. The proposal utilizing "split fermions" can be found in: N. Arkani-Hamed and M. Schmaltz, Phys. Rev. D 61 (2000) 033005 [arXiv:hep-ph/9903417]. The original papers proposing neutrino oscillations in extra dimensions are: N. Arkani-Hamed, S. Dimopoulos, G. R. Dvali and J. March-Russell, Phys. Rev. D 65 (2002) 024032 [arXiv:hep-ph/9811448]; K. R. Dienes, E. Dudas and T. Gherghetta, Nucl. Phys. B 557 (1999) 25 [arXiv:hep-ph/9811428]. The flavor model with only one bulk neutrino originally appeared in: K. R. Dienes and I. Sarcevic, Phys. Lett. B 500 (2001) 133 [arXiv:hepph/0008144]. Finally, the original proposals concerning axion physics in higher dimensions appeared in: K. R. Dienes, E. Dudas and T. Gherghetta, Phys. Rev. D 62 (2000) 105023 [arXiv:hep-ph/9912455]. 4. In Lecture # 4 , our discussion of the phenomenological implications of shape moduli is based on: K. R. Dienes, Phys. Rev. Lett. 88 (2002) 011601 [arXiv:hep-ph/0108115]; K. R. Dienes and A. Mafi, Phys. Rev. Lett. 88 (2002) 111602 [arXiv:hepth/0111264]; Phys. Rev. Lett. 89 (2002) 171602 [arXiv:hep-ph/0207009]. The ideas concerning string finiteness and its possible role in stabilizing the gauge hierarchy originally appeared in: K. R. Dienes, Nucl. Phys. B 429 (1994) 533 [arXiv:hep-th/9402006]; Nucl. Phys. B 611 (2001) 146 [arXiv:hep-ph/0104274]; K. R. Dienes, M. Moshe and R. C. Myers, Phys. Rev. Lett. 74 (1995) 4767 [arXiv:hep-th/9503055]. Finally, the use of "warped" extra dimensions in the development of alternative solutions to the hierarchy problem as well as alternatives to the whole idea of compactification can be found in: L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370 [arXiv:hepph/9905221]; Phys. Rev. Lett. 83 (1999) 4690 [arXiv:hep-th/9906064]. 5. Needless to say, there are many topics which we have not been able to cover in these lectures. The following is a list of reviews which might therefore be helpful as an initial approach to the literature in this subject.

545 For applications to collider signatures, see: I. Antoniadis and K. Benakli, Int. J. Mod. Phys. A 15 (2000) 4237 [arXiv:hepph/0007226]. For applications to cosmology, see: F. Quevedo, Class. Quant. Grav. 19 (2002) 5721 [arXiv:hep-th/0210292]. For an introduction to using D-branes in model-building, gravity, and cosmology, see: E. Kiritsis, Fortsch. Phys. 52 (2004) 200 [arXiv:hep-th/0310001]. For a general introduction to the brane world, see: G. Gabadadze, arXiv:hep-ph/0308112. For a review of precision searches for non-Newtonian gravity, see: E. G. Adelberger, B. R. Heckel and A. E. Nelson, arXiv:hep-ph/0307284. Finally, other TASI lectures at this school have covered various extensions of the ideas introduced here. In particular, for further details concerning orbifold compactifications, see: M. Quiros, TASI 2002 lectures, this volume. For further details concerning the properties of higher-dimensional effective field theories, see: I. Rothstein, TASI 2002 lectures, this volume. For further details concerning warped compactifications and orbifold GUT models, see: C. Csaki, TASI 2002 lectures, this volume.

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MARIANO QUIROS

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N E W IDEAS IN S Y M M E T R Y B R E A K I N G

M. QUIROS Department Theoretical Physics IFAE/UAB E-08193 Bellaterra Barcelona, Spain E-mail: quirosQifae. es Some old and new ideas on symmetry breaking, based on the presence of extra dimensions that have been the subject of a very fast development and intensive studies during the last years, will be presented in these lectures. Special attention will be devoted to the various compactification mechanisms, including toroidal and orbifold compactifications, and to non-trivial boundary conditions or ScherkSchwarzcompactification. Also symmetry breaking by Wilson lines, or Hosotani breaking characteristic of non-simply connected compact manifolds will be analyzed in some detail. The different mechanisms will be applied to the breaking of the most relevant symmetries in particle physics: supersymmetry and gauge symmetry. The required background for these lectures is Quantum Field Theory, Supersymmetry and some rudiments of Kaluza-Klein theory. The different sections "will be illustrated with examples.

Contents 1

INTRODUCTION

550

2

EXTRA DIMENSIONS A N D SYMMETRY BREAKING 2.1 Compactification 2.2 Scherk-Schwarz mechanism 2.3 Orbifolds 2.4 Scherk-Schwarz in Orbifolds 2.5 Scherk-Schwarz as Hosotani breaking

551 552 553 555 556 558

3

SUPERSYMMETRY BREAKING 3.1 Supersymmetry breaking by orbifolding 3.1.1 Vector multiplets 3.1.2 Hypermultiplets 3.2 Supersymmetry breaking by Scherk-Schwarz compactification . . .

560 562 562 565 566

549

550

3.3

3.2.1 Bulk breaking 567 3.2.2 Brane breaking 570 Supersymmetry breaking by Hosotani mechanism 573 3.3.1 Super-Higgs effect 576 3.3.2 Radiative determination of the Scherk-Schwarz parameter . 577 3.3.3 Brane assisted Scherk-Schwarz supersymmetry breaking . . 578

GAUGE SYMMETRY BREAKING 4.1 Gauge breaking by orbifolding 4.1.1 Rank preserving orbifold breaking 4.1.2 Rank lowering 4.2 Gauge breaking by the Hosotani mechanism 4.3 Top assisted electroweak breaking

1.

580 581 583 586 588 592

INTRODUCTION

Symmetry breaking is one of the main issues in contemporary particle physics. Its implementation in a perturbative q u a n t u m field theory has led to the notion of spontaneous symmetry breaking with the presence of a massless Goldstone particle for the case of a global symmetry (Goldstone theorem [1]) and a massive Higgs particle for a local symmetry (Higgs mechanism [2]). These ideas based on four dimensional field theories are nowadays thoroughly explained in many textbooks in q u a n t u m field theory [3]. T h e idea of unification of all interactions led to the introduction of extra dimensions starting from t h e pioneer work of Kaluza a n d Klein who a t t e m p t e d to unify gravity and electromagnetism through the presence of a fifth dimension [4]. More recently the a t t e m p t s to unify gravity with electroweak and strong interactions in a consistent q u a n t u m theory have led to the modern string theories t h a t incorporate supersymmetries and are described in ten (heterotic, type II and type I strings) or eleven (M-theory) space-time dimensions [5]. T h e recent string dualities relating all string constructions [6] as well as the presence in the spectrum of superstrings and supergravities of branes embedded in the higher dimensional space (as e.g. in type I strings Dp-branes with their world-sheet spanning a p + 1 dimensional space-time) opened the possibility t h a t the non-gravitational

551

sector could live in a p + 1 dimensional hypersurface. a Moreover the size of both the string scale and the compactification radius can be lowered from the Planck scale to the TeV range [7,8] thus making contact with the phenomenology of present and future colliders [9]. Therefore if the Standard Model lives in a Dp-brane with p > 3 this means that the Standard Model fields feel extra dimensions in its space-time propagation. This in turn opened a plethora of new possibilities for symmetry breaking associated with the different compactifications that extra dimensions can experience. These possibilities will be described in these lectures in some detail. Some of them are based on the possible compactification of extra dimensions. This compactification can break the higher-dimensional Lorentz invariance (as in toroidal compactifications) or the higher-dimensional Poincare invariance as in orbifolds where translational invariance is explicitly broken and four-dimensional fixed points can appear where localized (or twisted) states can propagate [10]. The compactification of extra dimensions can also introduce non-trivial boundary conditions, a mechanism known as Scherk-Schwarzbreaking [11]. Finally the extra dimensional components of gauge fields can acquire a constant background or vacuum expectation value and a symmetry can then be broken by a Wilson line, a mechanism known as Hosotani mechanism [12]. These three lectures will be organized in the following way. A general overview of all these mechanisms will be given in section 2 without explicit mention to the particular symmetry that is broken. Sections 3 and 4 will be devoted to the particularly interesting cases in particle physics where the symmetry is identified with supersymmetry and gauge symmetry, respectively. Since this is not an introduction on these general topics the reader is supposed to have a knowledge on supersymmetric and (non-abelian) gauge field theories as well as some notions on Kaluza-Klein theories. As we said above the required background on quantum field theories is provided by standard textbooks [3] while an introduction to supersymmetric theories can be found in Ref. [13]. As for Kaluza-Klein theories an introduction has been provided in these TASI lectures [14]. 2. E X T R A D I M E N S I O N S A N D S Y M M E T R Y BREAKING In this lecture we will review some general ideas dealing with symmetry breaking that can be realized in theories with extra dimensions. We will a

T h e gravitational sector must propagate in the whole higher-dimensional space time.

552

start by defining the compactification mechanisms on smooth manifolds (torii) with trivial and with twisted boundary conditions. The former being known as ordinary and the latter as Scherk-Schwarz compatification [11]. Then we will consider compactification on singular manifolds (orbifolds), with singularities concentrated on the fixed points [10]. In particular we will study the compatibility of orbifolds and Scherk-Schwarz compatification. Finally we will interpret the Scherk-Schwarz breaking as a Hosotani breaking [12] where the extra dimensional component of a gauge boson acquires a vacuum expectation value (VEV). In the present lecture we will only review general ideas on symmetry breaking by orbifold and/or ScherkSchwarz compatification. We will postpone the consideration of specific symmetries (as supersymmetry of gauge symmetry) to the second and third lectures.

2.1.

Compactification

We will consider a D-dimensional theory (D = 4 + d) with d extra dimensions and an action defined as SD=

fdDzCD[^(z)}.

(2.1)

We say that the theory is compactified on M.± x C, where M.4 is the Minkowski space-time and C a compact space if the coordinates of the _D-dimensional space can be split as zM = (x M ,j/ m ), (/j, = 0,1,2,3; m = 1 , . . . , d) and the coordinates ym describe the compact space C. The four dimensional (4D) Lagrangian is obtained after integration of the compact coordinates ym as

£4 = J'ddy CD[
(2.2)

The Lagrangian (2.2) contains propagation and interaction of massless and massive fields. For energies E M for g e G. M is the covering space of C. That G is acting freely on M means that only TZ has fixed points in M, where 1 is the identity in

553

G.h The operators rg constitute a representation of G, which means that T 9i92 = Tgi ' Tg2 • C is then constructed by the identification of points y and Tg(y) that belong to the same orbit V = rg(y).

(2.3)

After the identification (2.3) physics should not depend on individual points in M but only on orbits (points in C) as CD[4>{x,y)} = CD[cj){x,Tg{y))}.

(2.4)

A sufficient condition to fulfill Eq. (2.4) is 4>(x,Tg(y)) = {x,y)

(2.5)

which is known as ordinary compactification. However condition (2.5), if sufficient is normally not necessary. In fact a necessary and sufficient condition to fulfill Eq. (2.4) is provided by (x,Tg(y))=Tg(f>{x,y)

(2.6)

where Tg are the elements of a global symmetry group of the theory. Condition (2.6) is known as Scherk-Schwarz compactification and will be the subject of the next section. 2.2. Scherk-Schwarz

mechanism

As we described in the previous section the Scherk-Schwarz compactification mechanism occurs when some twist transformation corresponding to g € G, Tg, is different from the identity. The operators Tg are a representation of the group G acting on field space, i.e. they satisfy the property: Tgi92 = TgiTg2, g\,gi € G. The latter property is easily proven by using the group representation properties of operators Tg acting on the space M. A few comments are in order now: • The ordinary compactification corresponds to Tg = 1, \/g G G. • Scherk-Schwarz compactification corresponds to Tg ^ 1 for some g&G. • For ordinary and Scherk-Schwarz compactifications fields are functions on the covering space M. • For ordinary compactification fields are also functions on the compact space C. b

Trivially rt(y) = y, Vy e M.

554

• For Scherk-Schwarz compactification twisted fields are not singlevalues functions on C.

EXAMPLE

We will consider the simplest example of a compact space, the circle. We will take one extra dimension D = 5 (i.e. d = 1), M = R (the set of real numbers), G = Z (the set of integer numbers), and C = S1 (the circle). The n-th element of the group Z is represented by r„ with Tn(y) = y + 2TvnR, y eR

(2.7)

where R is the radius of the circle S1. The identification y = rn(y) leads to the fundamental domain of length 2TTR (the circle) as: [y, y + 2irR) or (y, y + 2nR\. The interval must be opened at one end because y and y+2nR describe the same point in S1 and should not be counted twice. Any choice for y leads to an equivalent fundamental domain in the covering space R. A convenient choice is y = — TTR which leads to the fundamental domain (-TTR,TTR\

.

(2.8)

The group Z has infinitely many elements but all of them can be obtained from just one generator, the translation 2TTR. Then only one independent twist can exist acting on the fields, as 4>(x,y + 27rR)=T(i>(x,y)

(2.9)

while twists corresponding to the other elements of Z are just given by Tn = Tn. As we said above, T must be an operator corresponding to a symmetry of the Lagrangian. As we will see later on in these lectures this symmetry can be a global, or local SU(2)R, when the theory is supersymmetric. Another candidate is a TLi symmetry associated to the invariance of the Lagrangian under the inversion — 4>.c The case of SU(2)R will be analyzed in the context of supersymmetric theories. Here we will analyze the simplest Z2 case. There are fields with untwisted T = 1 (bosons) and fields with twisted T = — 1 (fermions) boundary conditions. Untwisted fields can be described by real functions on S1 while twisted fields are not single-valued c

This symmetry is present in the fermionic sector and is associated to fermion number. It is used in field theory at finite temperature [15] where Euclidean time is compactified on S1.

555

functions on the circle. Of course one can define single-valued functions on the interval [—irR, nR] by the definition (j)(—irR) = T(f>(irR) — —4>{TTR) although still <j) is not a single-valued function on,the circle S1. The case we have just studied can be easily generalized to that of pextra dimensions, M = W, where G = 17 and C = Tp, the p-torus. In that case the torus periodicity is defined by a lattice vector v= (i>i,..., vp), where Vi = 2irRi and Ri are the different radii of Tp. Twisted boundary conditions are defined by p independent twists given by Tj (x,y*+6jiv>)=Ti4>(x,i/)

(2.10)

where rij = Sji is the unitary vector along the i-th dimension. 2.3.

Orbifolds

Orbifolding is a technique normally used to obtain chiral fermions from a (higher dimensional) vector-like theory [10]. Orbifold compactification can be defined in a similar way to ordinary or Scherk-Schwarz compactification. Let C be a compact manifold and H a discrete group represented by operators C,h • C —> C for h G H acting non freely on C. We mod out C by H by identifying points in C which differ by (h for some h G H and require that fields defined at these two points differ by some transformation Zh, a global or local symmetry of the theory, V = (h(y) 4>{x,Ch(v)) = Zh
(2.11)

The fact that H acts non-freely on C means that some transformations (h have fixed points in C. The resulting space O = C/H is not a smooth manifold but it has singularities at the fixed points: it is called an orbifold.

EXAMPLE

We will continue with the simple example of the previous section with d = 1 and C — S1. We can take in this case H = Z 2 and the orbifold is now O = S1/Z2. The action of the only non-trivial element of Z 2 (the inversion) is represented by £ where C(y) = -v

(2-12) 2

2

that obviously satisfies the condition Q {y) = ((—y) = y => C, = 1- For fields we can write as in (2.11) (j>(x,-y) = Z(Kx,y)

(2.13)

556

where using (2.12) and (2.13) one can easily prove that Z"1 = 1. This means that in field space Z is a matrix that can be diagonalized with eigenvalues ± 1 . The orbifold S' 1 /Zj is a manifold with boundaries: the fixed points are co-dimension one boundaries. Not all orbifolds possess boundaries: for example in d — 2, T 2 /Z2 is a "pillow" with four fixed points without boundaries. A detailed description of six-dimensional orbifolds can be found in Ref. [16]. 2.4. Scherk-Schwarz

in

Orbifolds

In this section we will consider the case of Scherk-Schwarz compactification in orbifolds. Remember that the starting point was a non-compact space M with a discrete group G acting freely (without fixed points) on the covering space M by operators rg, g G G and defining the compact space C = M/G. The elements g & G are represented on field space by operators Tg, Eq. (2.6). Subsequently we introduced another discrete group H acting non-freely (with fixed points) on C by operators Qu, h £ H and represented on field space by operators Zh, Eq. (2.11). We can always consider the group H as acting on elements y G M and then considering both G and H as subgroups of a larger discrete group J. Then in general Tg • £/, (y) j^ 0i' Ts (y) which means that g • h ^ h • g and J is not the direct product G ® H. Furthermore the twists Tg have to satisfy some consistency conditions. In fact from Eqs. (2.6) and (2.11) one can easily deduce a set of identities as TgZh cj)(x, y) = 4>(x, Tg • Ch(y)) = Zgh (x, y) = (f>(x, Ch • Tg(y)) = Zhg <j>{x, y) Tgi Zh Tg2 <j){x, y) = 4>(x, rgi • Ch • T92 (y)) = Zgihg2 <j>(x, y)

(2.14)

where gi,g2,h are considered as elements in the larger group J. The conditions (2.14) impose compatibility constraints in particular orbifold construction with twisted boundary conditions as we will explicitly illustrate next.

EXAMPLE

We will continue here by analyzing the simple case of the orbifold S1/^ with twisted boundary conditions. In this case there is only one independent group element for G — Z which is the translation r(y) = y + 2TTR while the orbifold group H = Z 2 contains only the inversion ((y) = —y.

557

First of all, notice that the translation and inversion do not commute to each other. In fact ( • r(y) = — y — 2TTR while r • ((y) = — y + 2irR. It follows then that (" • r • ( = T~1 and the group J is the semi-direct product Z K Z 2 [17]. Second, one can easily see that r • £ • r = C, which implies the consistency condition on the possible twist operators TZT

= Z <^=> ZTZ = T~1

(2.15)

as can be easily deduced from Eq. (2.14). Since we have seen that Z2 = 1, its eigenvalues are ± 1 and Z can be written in the diagonal basis as

where lp is the identity matrix in p-dimensional field space. This means that in the subspace spanned by some fields Z can be given either by Z = a3 or Z = ± 1 . On the other hand being T an operator corresponding to a global (or local) symmetry of the theory it can be written as T = e2™^

(2.17)

where Xa are the (hermitian) generators of the symmetry group acting on field space. Using the consistency condition (2.15) and (2.17) leads to the condition {/3-A,Z} = 0

(2.18)

for a generic T that does not commute with Z. There is however a singular solution for [T, Z] = 0 which corresponds to T = ± 1 . Using now (2.15) we can finally conclude that Z = a3 = • T = e^Hfii^+P^') Z = ±1 = > T = ±1

or

T

= e 2 *'*" 3

(2.19) (2.20)

where the parameters j3\t2 are real-valued. The previous equations deserve some explanation. In the case of Z = a3 we are implicitly considering the field subspace spanned by 5(7(2) doublets in which case X = a. This is typically the case where the global symmetry of the Lagrangian is SU(2). This case will appear, as we will see next, in supersymmetric theories where gauginos in vector superfields and scalar fields in hyperscalars transform as doublets under an SU(2)R symmetry of the theory [18]. In that case, using

558

the global residual invariance, we can rotate (/?i,/?2) ~~> (0,u>) and consider twists given by 2

T

=

e2-Kiu,a-

=

(

c o s

2 7 r w

s i n

2 7 r w

^ — sin 2ITUJ cos 2ircu

(2.21)

The twist (2.21) is a continuous function of w and so it is continuously connected with the identity that corresponds to the trivial no-twist solution (i.e. u> = 0). In this way Eq. (2.21) describes a continuous family of solutions to the consistency condition (2.15). There is also a discrete solution that is not connected with the identity and corresponds to w = 1/2 for T = exp(7ricr3) = - 1 2 as shown in Eq. (2.19). In this case [Z,T] = 0. In the case of Z = ± 1 we are considering a discrete global symmetry with even and odd fields under y —> —y. Now using Eq. (2.15) we obtain that boundary conditions can be either periodic or antiperiodic, i.e. T = ± 1 . For instance if the global symmetry can be associated to fermion number, bosons (fermions) have periodic (antiperiodic) boundary conditions. In particular this is the case of field theory at finite temperature [15].' Another example is the case of supersymmetric theories where we can use imparity of N = 1 four dimensional supersymmetry as the global symmetry. Here ordinary Standard Model fields are periodic while their supersymmetric partners are antiperiodic [19].

2.5. Scherk-Schwarz

as Hosotani

breaking

In theories compactified on a torus, or an orbifold, a symmetry can be broken by two mechanisms that are not present in simply-connected spaces: the Scherk-Schwarz and the Wilson/Hosotani mechanisms. As we have seen above the Scherk-Schwarz mechanism is based on twists Tg that represent the discrete group G defining the torus in field space by means of a global, or local symmetry of the Lagrangian. If the symmetry is a local one the Scherk-Schwarzbreaking is equivalent to a Hosotani breaking, where an extra dimensional component of the corresponding gauge field acquires a non-zero VEV. For simplicity we will consider the case of a five-dimensional, d = 1 theory compactified on the S1/Z2 orbifold. In this case the ScherkSchwarztwist is T = exp(27riojQ), i.e. (j)(x, y +

2TTR)

= e2viujQ

(x, y)

(2.22)

where Q corresponds to a given direction in the generator space and UJ is

559

the corresponding parameter. A trivial solution to Eq. (2.22) is <j>(x,y)=ei»Qv'R4{x,y)

(2.23)

where (f>(x, y + 2irR) = cj>(x, y) is a periodic single-valued function that can be expanded in Fourier modes. Obviously the symmetry generated by Q is broken by the five-dimensional kinetic term. Since the symmetry is a local one, there are associated gauge fields AM (M = /z, 5). If there is a VEV for A5 along the Q-direction, (X-A5) = Q(A%) all non-singlet fields will receive a mass-shift relative to their KK-values through their covariant derivatives oc (Ag ). In this representation all fields are periodic (no twist). We can switch to the Scherk-Schwarzpicture by allowing for gauge transformations with non-periodic parameters [20]. In particular the non-periodic gauge transformation U(y) =

2lxiQ e

^y

(2.24)

transforms away the VEV and ends up with non-periodic fields with ScherkSchwarzparameter LO given by <^> = |

(2-25)

EXAMPLE

As an introduction to the next section we will consider here a fivedimensional theory with N = 1 global supersymmetry where there is a global SU(2)R invariance acting on hyperscalars and gauginos. Hyperscalars *, tp) where ip is a Dirac spinor and (f>1 are complex scalars transforming as a doublet under SU(2)R [21]. Gauginos X1 (i = 1, 2) are contained in vector multiplets V = (j4.jvf,£, A1) where AM are five-dimensional gauge bosons and E is a real scalar. A1 are Symplectic-Majorana spinors transforming as doublets under SU(2)R, i.e. [22]

*=(4J' l

^-*2(A0* lJ

(2 26)

-

where \ L are Weyl fermions and e is the two-dimensional antisymmetric tensor with e12 = + 1 . To summarize, the objects transforming under SU{2)R

are the doublets

560

These fields are the only ones that can be given non-trivial boundary conditions based on the SU(2)R global symmetry. This property will be widely used in section 3. Suppose now that SU(2)R is realized locally (see section 3) and we define the orbifold breaking such that: (A^,A5' , S 1 ' 2 ) are even under the Z2 parity, while {A1*2, A%, S 3 ) are odd fields. This amounts to defining the parities in the language of section 2.4 as ZAs = Zv =-ZAlt

= (^l^

(2.28)

The five-dimensional kinetic terms for gauginos and hyperscalars can be written as Ckin =

|A7M2?MA

+

\VM4\2

(2.29)

where j M — (7 M ,7 5 ), with j 5 = —idiag(l, — 1), and the covariant derivative T>M = DM + i
(2.30)

generate a constant shift to the mass of all Kaluza-Klein modes as Am„ = {A\)

(2.31)

which is equivalent to the Scherk-Schwarzmechanism corresponding to the generator Q = a2 and with parameter UJ=(A\)R

(2.32)

3. S U P E R S Y M M E T R Y B R E A K I N G In this section we will study the different mechanisms of supersymmetry breaking based on the existence of extra dimensions. In order to simplify the analysis as much as possible we will concentrate on a five-dimensional theory, i.e. d = 1, where the extra dimension is compactified on the orbifold Sl /1i2- In particular we will set up the formalism for coupling the fivedimensional super-Yang-Mills multiplets and hypermultiplets to the 5 1 / Z 2 orbifold boundaries. On this issue we will follow the formalism introduced by Mirabelli and Peskin [22].

561

We will first consider a five-dimensional space-time with metric T)MN = diag(l, — 1, —1, —1, —1), M = fi, 5, and Dirac matrices 7 M = (7^,7 5 ) with

where
= ~eikejlxl7P

• • • 7MAfe

(3.2)

that includes a minus sign from fermion interchange. The five dimensional Yang-Mills on-shell multiplet [AM, S, A1) 3 is extended to an off-shell multiplet by adding an SU(2)R triplet of real-valued auxiliary fields Xa (a = 1, 2,3), i.e. / ' of f —shell

. -,\ Adj

= (AM,H,Xt,Xj

(3.3)

We will write the members of the multiplet as matrices in the adjoint representation of the gauge group with generators tA: V = YAtA. The N = 1 supersymmetric transformations are defined by a supersymmetric parameter £ l : a Symplectic-Majorana spinor. The supersymmetric transformations are given by: 5^AM

= iu^y

5^

= (jMNFMN

- jMDMZ)

5tXa = ^(aayijMDMX

C ~ i{X • S)ij?

- i [E, Si(
(3.4)

where D M S = <9MX - i[AM,T] and j M N = [ 7 M , 7 w ] / 4 . The five-dimensional on-shell hypermultiplet (A1, ip), where A% is an SU(2)R doublet and

*-(Z)

,3 5)

-

a Dirac spinor, is extended to the off-shell multiplet Moff-aheu

= (Ai,iP,Fi)

(3.6)

562

where Fl is an SU(2)R doublet of complex auxiliary fields. The components of the hypermultiplet transform under N = 1 supersymmetry as,

6tF* = -iV2^jMDM^

(3.7)

The previous formalism, and supersymmetric transformations (3.4) and (3.7), hold for a flat five-dimensional space and also in the case of toroidal compactifications, i.e. in this case compactification on the circle S1. For orbifold compactifications there appear four-dimensional fixed point branes where supersymmetry is reduced by a half. We will discuss in this section the four-dimensional supersymmetry that appears on the branes. 3.1. Supersymmetry

breaking by

orbifolding

In the orbifold S1 /Z2 there are four-dimensional branes at the fixed points y = 0,TTR where supersymmetry is reduced from jV = 2 t o i V = l. d In the rest of this section we will analyze this N = 1 supersymmetry at the four-dimensional fixed points of the orbifold. To project the bulk structure into the orbifold S1 /Z2 we must impose the boundary conditions on fields <j> as 4>(x,-y) = Z4,cf>(x,y)

(3.8)

where Z$ = ± 1 are the field intrinsic parities. The parities Z^ must be assigned such that they leave the bulk Lagrangian invariant. Fields (f> with Ztj, = — 1 vanish at the walls but have non-vanishing derivatives, 85(f), that can couple to them. We will separately consider the cases of vector and hypermultiplets. 3.1.1. Vector multiplets We will consider here a vector multiplet (AM, S, A1, Xa) and orbifold conditions that do not break the gauge structure. 6 The parity assignments are chosen to be those in table 1, where we have also included the parities of the supersymmetric parameters. Notice that £ is odd and so it does not N = 1 supersymmetry in five dimensions is like N = 2 supersymmetry in fourdimensions. e For a more general analysis where the gauge structure is broken by the orbifold projection, see section 4.

563 Table 1. Parities of the vector multiplet Z = +l AM

A^

s

Z = -1 A5 S

A*

A|

Xa

X3

e

eL

X1.2

f2

couple to the wall, while D5E = 9s E is then even and gauge-covariant on the wall. Prom table 1 we can see that £,\ is the parameter of the N = 1 supersymmetry on the wall. Then supersymmetric transformations (3.4) reduce on the wall y = 0 to the following transformations generated by £^ on the even-parity states:

<%Ai =

7

<%X3 =

- i ( * 3 - d 5 £)£i

^ « i fla'DnXl

<5e<95E = -iQDb\l

- i$D5\2L

+ h.c. (3.9)

+ h.c.

Gathering the last two equations in (3.9) yields, 5i(X3-d5E)

= eLi^D„X1L

(3.10)

which shows that the N = 1 vector multiplet on the brane in the WessZumino (WZ) gauge is given by (A1*, X\,D) where the auxiliary D-field is D = X3 -<9 5 £ [23]. In this way the five dimensional action can be written as

S = Jd*xl.£s+'£i6(y-yi)cA

(3.11)

where yi = 0, -KR in the present case. The bulk Lagrangian should be the standard one for a five dimensional super-Yang-Mills theory tr

1. -=FiMN

(DM?*)2

+ XijMDMX

+ X2-

A[£, A]

(3.12)

with txtAtB = SAB/2. The boundary Lagrangian should have the standard form corresponding to a four-dimensional chiral multiplet localized on the

564

brane at y = 0, (4>,ipi,F) and coupled to the gauge N = 1 multiplet (^4M,A^,X3 — c^E). The chiral multiplet is supposed to transform under the irreducible representation R of the gauge group and we will call t£ the generators of the gauge group in the corresponding representation. The brane Lagrangian is then written as A = tr [\D^\2

+ \F\2]

+ ^LI^D^L

- igV2(\1A<j>hAipL + ^LtifXl)

+ g4>HA4>(XA - <9 5 £ A ).

(3.13)

The Lagrangian involving the auxiliary fields XA and the scalar field <j> is jd5x

{\(Xf)2+g6(y)i4
- 95SA)|

(3.14)

Integrating out the auxiliary fields XA yields the boundary Lagrangian -g4>H£
l

-g2(4>UA4>)28{Q)

-

(3.15)

As we can see the formalism provides singular terms (5(0) on the boundary which arise naturally from integration of auxiliary fields. These singular terms are required by supesymmetry and they are necessary for cancellation of divergences in the supersymmetric limit. These terms can be formally understood as -.

OO

Using Eqs. (3.12) and (3.15) we can write the five dimensional Lagrangian for the Y,A fields as

£5 = -\{d^A? = -\[d^A

-5{y)gtftAd^A - \g2 (tftfo)2 52(y) + S(y)g
(3.17)

We can see that the Lagrangian (3.17) is a perfect square and the corresponding potential has a minimum at sA = -\ge(y)^tU

(3.18)

where e(y) is the sign function. We can see that if (f> acquires a VEV, also T,A acquires one breaking the gauge group. The function T,A(y) is an odd function and has jumps at the orbifold fixed points. This behaviour is typical of odd functions in orbifold backgrounds [24].

565

3.1.2. Hypermultiplets Hypermultiplets on the walls can be treated in the same way as we have just done with vector multiplets. A hypermultiplet is defined by (A1, ip, Fl), where F% is a doublet of complex auxiliary fields. A consistent set of assignments which yields N = 1 supersymmetry on the wall is given in table 2.

Table 2. Parities of the hypermultiplet Z = +l

Z = -1

A*

A1

A2

Fl

F1

F2

e

tl

f2

Similarly to vector multiplets, supersymmetry on the wall is generated by £^ and it acts on even-parity states as S^A1 = S^L S^A2

V ^ L

= iV2^d^AxeL*

~ V2d5A2&

+

y/2Flii

= V2$d5?pR

(3.19)

Putting together the last two equation of (3.19) leads to S^F1 - 05A2) = iV2$d»d^L

(3.20)

which shows that A = (A 1 , ip^, Fx — d$A2) transforms as an off-shell chiral multiplet on the boundary. Notice that, as it happened with the case of the vector multiplet, the auxiliary field of a chiral N = 1 multiplet on the brane does contain the 85 of an odd field [22]. We can now write the coupling of the bulk hypermultiplet to chiral superfields $0 = (^oiV'O; FQ) localized on the brane y = 0 through a superpotential W that depends on 4>o and the boundary value of the scalar field A\ W =

W($0,A)

(3.21)

566

The five dimensional action can then be written as in Eq. (3.11) with a bulk Lagrangian £5 = \8MAl\2

+ i^MdM^

+ \F'\2

(3.22)

and a brane Lagrangian dW

CA = {Fl-dbA2)^

+ h.c.

(3.23)

Integrating out the auxiliary field F1 yields "<%)

dW

(3.24)

Z41

and replacing it into the Lagrangian (3.22) and (3.23) gives an action

<%)

(d5^+/1.c.)+<%)

dW dA1

(3.25)

where we again find a singular coupling 6(0) as required by supersymmetry. Collecting in (3.25) the terms where A2 appears we get a potential

d5A2+5(y)

V

dW dA1

2

(3.26)

that is a perfect square and is then minimized for 1 . .dW (3.27) -2£{y)dAt Then if supersymmetry is spontaneously broken in the brane, i.e. if A2

then A2 acquires a VEV. This behaviour is reminiscent of a similar one in the Horava-Witten theory [25] in the presence of a gaugino condensation. 3.2. Supersymmetry compactification

breaking by

Scherk-Schwarz

In this section we will keep on considering the previous S1/Z2 orbifold and, in particular, a five dimensional generalization of the Supersymmetric Standard Model [18,26]. The gauge and Higgs sectors of the theory will be considered to live in the bulk while chiral matter (quarks and leptons)

567

will be supposed to be localized on the four-dimensional boundaries (fixed points) of the orbifold. As we have seen in the previous section the orbifold boundary conditions break the N = 2 supersymmetry in the bulk to N = 1 for the zero modes and on the branes. We will further break the residual N = 1 supersymmetry by using the Scherk-Schwarzboundary conditions and the global SU(2)ji symmetry. 3.2.1. Bulk breaking The on-shell gauge multiplet V = ( A M , S , A l ) A * belongs to the adjoint representation of the gauge group SU(3) (g> SU(2)L <8> U(l)y and % = 1,2 is an SU(2)R index. The Higgs boson fields belong to the hypermultiplets HP = (Hf, tpa), where a = 1,2 transforms as a doublet of the global group SU{2)H- The five-dimensional action can be written as in Eqs. (3.12) and (3.22) C5 = - ^ t r {-\F2MN

+ CD M £) 2 + a^DM*

- A4[E, A']

9 I ^ + \DMH?\2 + 4>a(ijMDM - £)° - (iV2H?%iJ>a + h-c.) - fffs2Ff - 9— (Hfa{TAH^

\

(3.28)

We now define the Z2 parity according to the symmetries of the bulk Lagrangian as in table 3. Columns in table 3 correspond to N = 1 D = 4 supersymmetric multiplets. Table 3.

Parities of bulk fields

Z--\

Z = +l

V»

Hi

H{

*L

*>l

^R

V5,S A

>l

Hi

Hi

*% * l

The Fourier expansion for Z = ± 1 fields ± is given by oo

0+ = ^ ) + v / 2 ^ c o s ^

)

oo n=l

We can see from the expansion (3.29) that the Z2 symmetry projects away half of the tower of KK-modes. In particular the zero modes are the chiral

568

N = 1 superfields constituting the columns of table 4 while the non-zero modes are arranged in TV = 2 multiplets, as shown in table 5. Table 4.

Zero modes

Vector

Chiral ff2(0)

Table 5.

Non-zero modes

Vector y(n) Al

(n)

^1(0)

Hypermultiplets Hl(n)

E(n)

^2 (n)

H

l(n)

H2(n)

^ 2 („) ij)2(n)

^,1 (n)

We will break supersymmetry by using SU(2)R as a global symmetry of the theory, as well as SU(2)H- Our definition of parity is then Z = ±o-3 ® ij5

(3.30)

where 0-3 is a matrix acting on both SU{2)R and SU(2)H indices while 7 5 is acting only on spinor indices. Similarly the twist is defined as T = e 2n iojcr

(3.31)

as we saw in the previous section, whereCT2is also acting on both SU(2)R and SU(2)H indices. Notice that Z and T satisfy the general condition (2.15). In particular we will introduce the o>twistf for gauginos, Higgsinos and Higgses as in Eq. (2.23): U{y)

ip1 ip2

m (| (3.32)

H{ H\

Hi Hi f

U(y)

(HUK'M. "<»)=

_ giwcr

y/R

In fact one could introduce different twists qn and qu for SU(2)R and SU(2)jf, respectively, as in Ref. [18,26]. Here we are taking for simplicity qn = qjj = w.

569

where tilded fields are periodic fields expandable in Fourier series as in Eq. (3.29). After replacing (3.32) in the Lagrangian (3.28) we obtain the following mass spectrum for n --£ 0 modes [18], £(«)

=

1 ^J(AKn) R

+

-±( 2

H{0n)

H{2n)

( #L

H[

n)

A 2(n))

J2(n)

(«)

(3.33)

Vh

ff.

(

»,y

R V

lnz 0 0 On2 0 0 0 n2 +4w2 \ 0 0 -4nw

/H\ (n) H:2

(n)

7

H^a^f and, to simplify where we have defined the fields H^ as Hf the notation, we skipped the tildes on the fields. Therefore, the nonzero mode mass eigenstates corresponding to the n-th level are two Majorana fermions [A^ ( n ) ± X2L(n)]/y/2 with masses |n±u>|, two Dirac fermions [iplL(n>ii/;^™']/\/2 with masses |n±w|, and four complex scalars H0(«) rr(n) and [H[n)±H^n)]/y/2 with masses n and |n±2w|, respectively. Of course the mass spectrum of the fields Vj; , V5 , S ^ is not modified by the ScherkSchwarztwist. Notice that only the Scherk-Schwarzmechanism with respect to the SU(2)R symmetry breaks supersymmetry while the twist with respect to the SU(2)H symmetry does just provide a supersymmetric mass. The mass Lagrangian for zero modes (A 1(0) , / 2 ( 0 ) , 1 ( 0 ) H$»,H<°>)*,

m

*°> = l

A i(o) A i(o)

•#<«H<°> + h.c\

R2^3

'

(3.34)

and the complex scalar HQ ' is massless (the Standard Model-like Higgs). From (3.34) we can see how the Scherk-Schwarzmechanism breaks supersymmetry in the zero mode sector of the five dimensional fields. In particular it provides a mass to gauginos; in the language of the Minimal Supersymmetric Standard Model (MSSM) we should write M-1/2

~R

(3.35)

On the other hand it gives a supersymmetric mass to Higgsinos thus providing an extra-dimensional solution to the MSSM /x-problem. Using again

570

the MSSM language we could also write (3.36)

»=R 3.2.2. Brane breaking

As we said above we are assuming left- and right-handed quark and lepton superfields localized on the boundary at y = 0.g The gauge superfield {y^X^X^ — d^Ti) coupling to the left-handed quark superfield (Q,QL) gives, after eliminating the auxiliary fields, the brane Lagrangian [see (3.11)], IAXQI 2

+ iQL^D^L

- iV2(Ql\lqL

+ h.c.) -

&d5ZQ

\ {QhAQ) + (Q^Q)(i£ttV 3 )?#;)

(3.37)

that can be easily obtained from the Lagrangian in Eq. (3.13). In the same way the interaction of all chiral fermions to the bulk gauge multiplet can be computed and integration J dy and mode decomposition yields a four dimensional Lagrangian for Kaluza-Klein modes. Similarly, the Yukawa couplings of the Higgs boson hypermultiplet (H%,ip2L,F2 — dsH2) to the quark superfields (Q>9L) a n d {U,UR) on the boundary can be easily computed using (3.23). It yields, A = fh \H2qLuR htHlQ

2

+ il>l(QuR + qLU) - (d5H*)QU + h.c. - htHJU

2

- htUQ * 6(y)

(3.38)

where ht is the top-quark Yukawa coupling. Similarly, integration J dy and mode decomposition yields a four dimensional Lagrangian for KaluzaKlein modes. A similar procedure could be followed for N = 1 localized superfields in the leptonic sector: (L,£L) and (E,en). The scalar fields on the boundary (squarks and sleptons) are massless at the tree level. Supersymmetry is broken in the bulk by gaugino masses, Eq. (3.35), and transmitted to the fields on the boundary by radiative corrections [26]. In particular the diagrams for gauge interactions contributing to squark masses are exhibited in Fig. 1, g

Of course situations where only part of matter fields are localized on the boundaries, and the rest propagating in the bulk, are easily considered following similar lines to those found in this section and the previous one. See section 4.3 and Ref. [27].

571

y(n)

Q__\^i_Q Q —cy—

Q _/_A_ Q Figure 1.

Q

Q___i^__. Q

Diagrams contributing to the Q mass from the gauge sector.

where the Kaluza-Klein gaugino mass eigenstates are indicated by A^±n^ = ^ i ( n ) _|_ A 2 ( n ))/\/2. The corresponding contribution to the squark masses is computed to be, 2 _ 171/=, =

g2C2(Q) 4TT4

[Am2(0)-Am2(w)]

(3.39)

where C2(Q) is the quadratic Casimir of the Q representation under the gauge group11 and 1 A m 2 H = — -2 [Li 3 {e2™) + h.c] 2~R where the polylogarithm functions are defined as

^ ) = E^-

(3.40)

(3-41)

fc=i

The diagrams from Yukawa interactions contributing to squark masses are given in Fig. 2, where the Kaluza-Klein mass eigenstates are defined as F(±") = [ ^ W ±iP2^]/V2, /i(M> = H(™\ fc(-M) = H£\ ff(±n) = n) [H[ ± H^}/s/2. The result is given by m

| = _M_ [Am2(2w) + Am 2 (0) - 2Am 2 (w)]

(3.42)

h We use the convention for the generators t r { t ^ tj^} = T(R)6AB and JZA^I^R *«) = Ci (R) II, where R is a representation of the gauge group. In particular if N is the fundamental representation of SU(N), T(N) = 1/2 and C2(N) = (N2 - 1)/2JV, while for the adjoint representation T(Adj) = C2(A.dj) = N.

572

h (n) ,H (n)

g (n)

Q-L^-Q Q—r v - Q U

hWR(n)

Q _/_A_ Q Figure 2. plings).

Q___i^__. Q

Diagrams contributing to the Q mass from the Higgs sector (Yukawa cou-

Notice that the radiative contributions to squark and slepton masses in Eqs. (3.39) and (3.42) are in all cases finite, a feature shared by thermal masses in field theories at finite temperature, known as Debye masses for the case of longitudinal gauge bosons [15]. The finiteness of radiative corrections to soft masses from the tower of Kaluza-Klein modes has been challenged in Refs. [28] where it was argued that introducing a sharp cut-off in the number of contributing Kaluza-Klein modes would restore the typical quadratic divergences of four-dimensional field theories. The finiteness of these radiative corrections, that was explicitly proven to hold at two-loop by explicit calculations [29], has finally been recognized to be a robust result when one introduces a regularization that preserves the symmetries of the five-dimensional theory, i.e. supersymmetry and Lorentz invariance. Furthermore explicit calculations with different consistent regularizations all yield the same finite result [30] therefore proving that the quadratic divergences obtained using a sharp cut-off were an artifact of the non-covariant regularization. Needless to say the finiteness of the previous results is due to supersymmetry. In fact there are indeed quadratic divergences in the one-loop calculation that cancel by supersymmetry. Finally one can compute the contribution of the Kaluza-Klein tower to the soft breaking trilinear coupling between two boundary and one bulk field, AtQUH%. The leading contribution to the parameter At is provided by the exchange of gluinos as in Fig. 3, and given by At = ^[iLz2(e2^)-i(e-^)]

(3.43)

573

I I I

qL/ \ y-

uR ^

?i
Q Figure 3.

U

Diagrams contributing to At-

Supersymmetry breaking is then gauge and Yukawa mediated to the bosonic fields of the chiral sector localized on the branes by radiative corrections. In this aspect the model shares common features with any gauge mediated supersymmetry breaking model but with a very characteristic spectrum. Electroweak symmetry breaking is also easily triggered by radiative corrections at one-loop if the top is living in the bulk [26]. In fact the tachyonic mass induced by the top Yukawa coupling is also finite [26]. The fact that the radiative contributions to soft masses are finite does not mean that they are not sensitive to the ultraviolet cut-off at any order of perturbation theory. It just means that there is no mass counterterm at any order in perturbation theory. An explicit dependence on the cut-off already appears in the two-loop correction to the soft masses [29]. However it comes exclusively from the wave function renormalization and can be absorbed in the redefinition the gauge and Yukawa couplings in the improved theory. In fact the wave function renormalization yields a renormalization of gauge and Yukawa couplings that contains a power-law dependence on the cut-off. This power law renormalization [31] has led to the possibility of power-law or accelerated unification [32] in the TeV range that parallels the ordinary logarithmic unification in the MSSM.

3.3. Supersymmetry

breaking by Hosotani

mechanism

In the five dimensional formulation of local supersymmetry the SU(2)R global supersymmetry is promoted to a local symmetry [33-35]. This subject is being extensively studied at present. We will use the formulation

574

of Ref. [33] where two multiplets are necessary to formulate off-shell five dimensional supergravity: the minimal supergravity multiplet (40B + 4 0 F ) and the tensor multiplet (8s + 8p). Their parities are given in tables 6 and 7, respectively. Fields in the upper panel of table 6 are physical fields Table 6.

Minimal supergravity multiplet

Z = +l

Field 9MN

graviton

i>M

gravitino

VM VMN

t

9p.v, 955

graviphoton

BM

B5

By.

3

h2

v, , v5

Sf7(2)H-gauge antisymmetric

y5 ,

y ^

t3

C

real scalar

C

c

S'C/(2) H -doublet

a

Tensor multiplet

Field

Z = +l

Z = -1

Y

y-1,2

Y3

Bpvp

Byvs

BMNP

3

vtiv

SE/(2)R-triplet

Table 7.

Z = -\

N

N

P

P\

Pi

while those in the lower panel are auxiliary fields. Fields in table 7 are all of them auxiliary fields. The SU(2)R gauge fixing is done by fixing the compensator field [33] Y = eu 1 1 1 that breaks ^minimal

~r

SU(2)R

—>

U(1)R

(3.44)

= {a2}. The invariant Lagrangian Cgrav =

^tensor

contains the term (1 — eu)C and then the equation of motion of C yields u = 0.

575

The auxiliary fields that are relevant for supersymmetry breaking are: V5' , i 1 ' 2 , that constitute the F-term of the radion superfield [36-39], n = [555 + iB5, V>52L, V5X + iV2 + 4i (i 1 + i t2)]

(3.45)

The relevant terms in Cgrav containing these fields are [33] Cgrav =

-\^P1PMNVM^N

-

Y2eMNPQRV2MdNBPQR

+ {V£)2 - 120 1 ) 2 - 48(t 2 ) 2 - \2Nt2 where j

P M N

- N2

(3.46)

is the normalized antisymmetric product of gamma matrices, VM = DM+ io2VM

(3.47)

and DM is the covariant derivative with respect to local Lorentz transformations. The field equations for the auxiliary fields yield Vg1 = N = t1 = t2 = 0 while the field equation for the 3-form tensor BMNP 1

rl _ n _ JVJ

T/2 _ «

v

(3.48) gives

_ _ J V? = 0 (Odd field) [ K5 = constant (even field)

where K is an odd field and the last implication is suggested by the simplest . choice

K = yji

( 3 - 5 °)

which leads to the background V? = |

(3.51)

and makes the connection between the Hosotani/Wilson picture and the Scherk-Schwarzone. In fact using the coupling of V2 to the gravitino field through the covariant derivative X>5 in (3.46) one obtains the gravitino mass eigenvalues for the Kaluza-Klein modes as m% = ^

(3.52)

An alternative choice to the odd function (3.50) has been proposed in Refs. [24,40] as K = A i % ) + A2S(y - TTR)

(3.53)

that leads, through the covariant derivative P 5 in (3.46), to a localized gravitino mass term. In this case the gravitino modes mass matrix has to

576

be diagonalized and mass eigenstates computed. The resulting spectrum is similar to that in Eq. (3.52) thus proving that supersymmetry breaking by a localized gravitino mass (that can arise e.g. from some non-perturbative dynamics) is equivalent to a global Scherk-Schwarzbreaking, as anticipated in Ref. [41]. The issue of supersymmetry breaking by a localized mass on the brane will the subject of section 3.3.3. Supersymmetry breaking is also manifest for gauginos and hyperscalars, SU(2)n doublets, that interact with VM through the covariant derivative, see Eq. (2.29) -matter

—^ 7

^AfA

(3.54)

VMA

where T>M = DM +ia -VM

(3.55)

After using the equations of motion V5 ' = 0 we obtain that T>M —• P M and the mass eigenvalues for gauginos and hyperscalars are, as for gravitinos,

m[% = r4n) = ^

(3.56)

which also shows the equivalence between the Hosotani/Wilson mechanism and the Scherk-Schwarzcompactification for the matter sector. Supersymmetry breaking is spontaneous provided there is a (massless) Goldstone fermion (Goldstino) "eaten" by the gravitino that becomes massive in the unitary gauge. This is known as the super-Higgs effect [42] and will be considered next. 3.3.1. Super-Higgs effect The Goldstino is provided by the fifth component of the gravitino: -05- This is obvious from the local supersymmetry transformation, S&s =V5£ + --- = ia2 V£t + •••

(3.57)

We will now analyze the corresponding super-Higgs effect [39]. The kinetic term for the gravitino can be decomposed in four-dimensional and extradimensional components as:

- ^7^75A^5 - ^57^7

5

^^

(3.58)

577

We can now make the redefinition ^

=

^ + ^ ( 2 ?

5

(3.59)

) - V B

which can be seen as a local supersymmetry transformation with parameter (Z?5)-1^>5 = £, gauging ip5 away. This defines a "super-unitary" gauge where ip5 n a s been "eaten" by the four dimensional gravitino ip^: ;^M7

MNP

+ •

•) €

(3-60)

The second term of (3.60) provides a mass term for the four dimensional gravitino as 1

r

R lP5 ~iP5 ^

-A^ILKL)^

*-"n

1>l

h.c

at

(3.61)

where ps = id$ = n/R for a flat (unwarped) extra dimension and ui is defined as a function of the background field V52 in Eq. (3.51). The eigenvalues of the mass matrix (3.61) are \n ± u>\/R in agreement with Eq. (3.52). 3.3.2. Radiative determination of the Scherk-Schwarz parameter The effective potential along the V52, i.e. u, direction is flat and hence the Scherk-Schwarzparameter is undetermined at the tree-level. The scale of supersymmetry breaking (i.e. the gravitino mass) also is undetermined at the tree-level, a situation which is typical of no-scale models in supergravity [43]. Since all mass eigenvalues, for gravitinos, gauginos and hyperscalars, are equal to (3.52) we can compute the Coleman-Weinberg [44] one-loop effective potential in the following way. For hyperscalars we have V0

2NH

V^

f

d

'P i

P

n + u> R

(3.62)

where 2NH — 2 ( # degrees of freedom of a complex scalar) XNH ( # of hypermultiplets) x 2 ( # scalars in one hypermultiplet)xl/2 (orbifold reduction of # degrees of freedom). Similarly for gauginos we get V,1/2

2iVv

£

dAP (2^

log

R

(3.63)

578

where the minus sign comes from the fermionic character of gauginos and Ny is the number of vector multiplets, while for the gravitino, in the five dimensional harmonic gauge lNipN = 0 the effective potential is easilyworked out to be [45] 2"

(3.64)

Using techniques from field theory at finite temperature one obtains [26]

where the polylogarithm functions are defined in Eq. (3.41), and Li5 has a power expansion as Li5 (e2™) + h.c. = 2C(5) - 47r2C(3)w2 + • • •

(3.66)

Moreover using the expansion (3.66) it is easy to see that the global minimum of the potential is LO = 0 (w = 1/2) for NH>2 + NV (NH < 2 + Nv). This value can be shifted if there is another source of supersymmetry breaking; for instance if the Scherk-Schwarzbreaking is "assisted" by brane effects, as we will see in the next section. 3.3.3. Brane assisted Scherk-Schwarz supersymmetry breaking The coupling of the minimal and tensor multiplets to the branes permits a total Lagrangian as £ = Cgrav + 2 W5(y)Cbrane

(3.67)

with [33] Cbrane = " 2 TV - 2 V£ - 12 t2 + ^ ^ , ( 7 V V V

(3-68)

where we can always assume that W comes from some non-perturbative dynamics that appears when some brane fields in the hidden sector are integrated out. This typically happens when supersymmetry is broken by gaugino condensation, as it is the case in M-theory [41,46]. Of course in general one could also introduce another term localized at the other brane, as 2 W'5(y — nR), but we will work out here the simplest case. In the presence of brane terms the field equations of the auxiliary fields N and V§ are modified to N = - 2 WS(y),

Vi = 2 W6(y).

(3.69)

579

The presence of these VEV's modify the mass terms [47] for gauginos, Mi/2 = h5V5 = ij5 (d5 + i^Vi

+ io2V£)

(3.70)

and hyperscalars, Ml = V5&

(3.71)

while the gravitino mass is directly modified by the term in Cbrane'2WS(y)^^a2^

(3.72)

The corresponding mass eigenvalues can be seen to be modified by the presence of W ^ 0 as [47] m („ )

=

n + A^,m >

a=

0,1/2,3/2

(3.73)

where the quantities Aa(tu, W) have been computed to be [24] / A3/2(LJ,W)

'

/ tan 2 w?r)+tanh 2 (W)

rxrx

y

'

= arctanW

'

y

-^

£—-

V l + tan 2 (w^) tanh 2 (W)

/ „ n / tan 2 (w7r)+tan 2 (W) v A1/20(u>,W) = arctanW„/ N 5—7v ; X t a n W7r tan

''

Y +

( )

/ 0 nA. (3.74)

W

The modified mass eigenvalues produce a modification of the effective potential that turns out to be [47] 3 Li 5 (e 2 - A 3/ 2 (-.wO) + h. c .' Veff e 6 4 367T i?

(3.75)

+M

S^h( e 2 7 r i A l / 2 ( ^') + H

We expect that the brane effects can modify continuously the location of the minimum of the effective potential away from its minima for W = 0 at u) = 0 and to = 1/2. We have studied numerically two extreme cases. First of all, the case where only the gravitational and gauge sectors are living in the bulk of the extra dimension, while matter and Higgs fields are localized in the observable brane. This case, that corresponds to N\r = 12 and Nfj = 0, is shown in Fig. 4. We can see that for W = 0 the minimum of the potential is ujmin = 1/2 while for W Z 0.82 it smoothly becomes wTO,„ = 0 with a smooth transition region around W ~ 0.815. Finally we have considered the other extreme case where all gravitational, gauge, matter and Higgs fields are propagating

580 0.813 0.5

0.815

0.817

0.819

0.4 0.3 0.2 0.1 0 0.813 0.815 0.817 0.819 Figure 4. Plot of the minimum of the effective potential ujmin as a function of W for Nv = 12 and NH = 0. 0.7737 0.5

0.774

0.7743

0.7746

0.7749

0.774

0.7743

0.7746

0.7749

0.4 0.3 0.2 0.1 0 0.7737

Figure 5. Plot of the minimum of the effective potential Lomin as a function of W for Nv = 12 and NH = 49.

in the bulk of the extra dimensions, which amounts to considering the case with Ny = 12 and NH = 49. This case is shown in Fig. 5. We can see from Fig. 5 that for W = 0 the minimum of the potential is uJmin = 0 while for W > 0.775 it becomes tomin = 1/2 with a smooth transition region around W ^ 0.774. 4. G A U G E S Y M M E T R Y B R E A K I N G In this section we will consider the different mechanisms of gauge symmetry breaking based on the existence of extra dimensions. As we did in section 3,

581

in order to simplify the analysis we will concentrate on a five-dimensional theory, i.e. d = 1, where the extra dimension is compactified on the orbifold S1 /Z2. As we will see we can use both the orbifold (section 4.1) and Seherk-Schwarz (section 4.2) boundary conditions to break the gauge symmetry on the boundaries. Moreover if bulk fermions strongly coupled to the Higgs (i.e. the top quark) propagate in the bulk of the extra dimensions they induce a spontaneous one-loop (Coleman-Weinberg) finite electroweak breaking as we will describe in some detail in section 4.3. 4.1. Gauge breaking by

orbifolding

In section 3.1 we have (almost) always been assuming that the orbifold action was commuting with the gauge structure and so the gauge group remained intact after orbifolding. This is clearly not the most general case for the orbifold action can break the gauge group Q in the bulk into its subgroup H on the branes. This breaking must be consistent with the orbifold action and so it is strongly constrained. Here we will consider the case of a general gauge group with breaking Q —> H by the Z2 projection. We consider gauge fields AM — AAiTA where TA are the Lie algebra generators of Q normalized such that tr {TATB}

= -5AB

(4.1)

with indices M = \i, 5 and A = 1, • • •, dim(Q). We couple to the gauge fields Dirac spinors tp that transform in the representation R [with dim(R) = dR\ of the group Q. The five dimensional action can be written as S 5 = j ' dbxTtl-l-FMNFMN

+i^MDMA

(4.2)

where FMN = FANTA, FAN = dMAA - ONA^ + gfABCAMA% and the gauge covariant derivative is DM = 9M — ^9-^MT^, where T# are Liealgebra valued matrices in the representation R of Q satisfying [TA,TB]=lfABCTg.

(4.3)

The Z2 parity assignment is defined as

AU^,-y)

=

aMAABAfI(x^,y)

V(^,-y) = AR®(n5)^(^,y)

(4.4)

582

where aM = + 1 and a5 = — 1, which just means that the parities of AA and AA are opposite, and \R = X'R = A^ is a matrix acting on the representation indices of rp. Finally A2 = 1 and so its eigenvalues are ± 1 . By imposing the requirement that under the Z2 action F^N —> M AB \ F^N, and that F^NF^N remains invariant it is straightforward a AB to check that A should satisfy the condition [48] sABC _ j^AA' ^BB'j^CC'

IA'B'C

u^\

which implies that the action of Z2 on the Lie-algebra of Q is a Lie-algebra automorphism.1 With no loss of generality we can consider the diagonal basis where AAA = rjASAA with JJA — ± 1 and the automorphism condition (4.5) takes the simpler form fABC

=

^B^CjABC

(nQ g u m )

(4g)

Using now the Z2 action (4.6) we can naturally split the adjoint index TA into an unbroken part Ta (rja = +1) and a broken part Ta (r)a = — 1) such that H = {Ta}, K = Q/H = {Ta}. In this way the parities of gauge bosons are given in table 8. Following table 8 only Aa and A§ contain Table 8. Parities gauge sector EVEN ODD

Aa

A%

Aa A V

A%

of

zero-modes and are non-vanishing on the branes. In particular Aa are the gauge bosons on the brane corresponding to the gauge group H and A^ are massless scalars on the brane. When the latter acquire a VEV they can spontaneously break the gauge group 7i to a subgroup, a mechanism known as the Hosotani mechanism [12]. This will be discussed later on in section 4.2. The automorphism constraint (4.5) implies that the only non-vanishing structure constants are fabc and fabc, which strongly constrains the possible breakings. A trivial example where we can see that the automorphism 'A Lie-algebra automorphism is a transformation TA —> AAA TA that preserves the structure constants, i.e. such that [TA ,TB ] = ifA B c Tc . It is easy to prove the latter equality using Eqs. (4.3) and (4.5).

583

constraint is a non-trivial one is the case Q = SU(2). Its only non-vanishing structure constant is / 1 2 3 = e 123 (and permutations thereof). There are thus only two possibilities: • The case a = 1,2,3, which corresponds to no breaking: TL = SU{2). • The case a = 3 and a = 1,2, which corresponds to the breaking SU(2) -* 1/(1). In particular we can easily check that the breaking SU(2) —> nothing is not allowed by the orbifold action. As for fermions ip in the representation R of the group Q, the requirement that the fermion-gauge boson coupling igAAIi[)^iMTA^p is invariant under the orbifold action implies that the matrix \R in (4.4) should satisfy the condition [49] \RT£\R

A A

r [\R,T%] = O

= r, T = • I 1{A R ,T!}=0

(4.7)

The final issue here is the gauge-fixing term and ghost Lagrangian Cg.f. + tghost = - y ^ A h 8MAAI

+ Tr dMcDMc

(4.8)

A quick glance at the ghost-gauge field interactions shows that ghosts cA have the same parity properties as the gauge fields AA in (4.4), i.e. cA(x»,-y)=AABcB(x^y)

(4.9)

Finally we would like to comment that the Lie-algebra automorphisms come in two classes that will be subsequently analyzed: • Inner automorphisms, that can be written as a group conjugation: TA —> gTAg~x, g G Q. Inner automorphisms preserve the rank => rank(H) = rank(Q). • Outer automorphisms, that can not be written as a group conjugation and therefore do not preserve the rank =>• rank(H) < rank(Q). 4.1.1. Rank preserving orbifold breaking The case rank{H) = rank(Q) implies that none of the generators T^ are diagonal and therefore XR can be chosen diagonal

584

where d\ and di are model dependent numbers. The general structure for rank preserving orbifolding is better presented in the Cartan-Weyl basis for the generators TA. In this basis the generators are organized into the Cartan subalgebra generators Hi, i = 1, • • •, rank(Q), and "raising" and "lowering" generators Ea, a = 1, • • •, dim(Q) -~rank(H) with [HuEa) = aiEa

(4.11)

where the rank (£)-dimensional vector a is the root associated to Ea [50]. The orbifold action on gauge fields can be written as the group conjugation TA->gTAg-1,

with

g = e"2™'5

(4.12)

The orbifold action is then fully specified by the rank (
(4.13)

1

gHig- =Hi

(4.14)

From the last equality it follows that the inner automorphism (4.12) preserves the rank. Moreover Ea generators, with roots a such that a • v = 0 belong to the unbroken subgroup. The problem of determining the unbroken subgroup is thus reduced to an algebraic problem, as we will see next in some examples.

SU(2)

- • U(l)

EXAMPLE

This is the simplest non-trivial example. We have already seen that the only non-trivial SU(2) breaking corresponds to SU(2) —> 1/(1) by the AAB matrix A=

0 - 1 0

(4.15)

In the Cartan-Weyl basis: # i = T 3 , E± = T1 ± iT2, with the non-zero ro'ots a± = ± 1 and the commutation rules [H1,E±] = ±E±

(4.16)

the unbroken U(l) corresponds to the generator Hi. The orbifold breaking (4.15) is triggered by the twist vector v = — | and the group element g = el7rl

3

(4.17)

585

Fermions in this example will be considered in the fundamental representation R = 2. One can easily see that

A

» =±(J-°i)'

<"8»

i.e. d\ = d,2 = 1 in the notation of (4.10). In fact A2 in (4.18) commutes with Ta =T3 = 5CT3 and anticommutes with T a = {T1 = \ax,T2 = ±a2} in agreement with the general orbifold condition (4.7). The surviving fermions on the brane are left-handed fermions with (/(l)-charge equal to + 1 and right-handed fermions with [/(l)-charge equal to —1, and so the fermion spectrum in the bulk is non-chiral.

5(7(3) -> SU{2) <8> (7(1) EXAMPLE The next to simplest example of orbifold breaking by an inner automorphism corresponds to SU(3) —> SU{2) ® (7(1). In order to achieve this breaking pattern we take the A matrix as

A=

/13 0 0 \ 0 -14 0

(4.19)

\0 0 +lj so that the 5(7(3) generators [see Eq. (4.24)] corresponding to 5(7(2)® (7(1) have positive parity and those corresponding to SU(3)/SU(2) <8> (7(1) negative parity. In the Cartan-Weyl basis we have four unbroken generators: the Cartan generators Hi, H2, and two other unbroken generators E±i corresponding to the roots a±i = (±1,0). The twist is v = (0, A/3) and so a±i-v = 0. For the rest of generators E±2, E±z we find exp(—2-KW-Q) = —1 as expected. Choosing fermions in the representation R = 3 we find A^ = ±diag(+l, + 1 , —1) that commutes with the generators of 5(7(2) <£> (7(1) and anticommute with those of 5(7(3)/5(7(2) (g) (7(1). Using now the decomposition 3 = 2 _ 1 © l _ + 2 we obtain on the brane a massless 5(7(2) doublet with a given chirality and a massless 5(7(2) singlet with opposite chirality. The spectrum is chiral and anomaly cancellation must be enforced in this model. A classification of breaking patterns by Z2 inner automorphisms is given in table 9. For more deatails see Ref. [48].

586 Table 9. phisms

Breaking pattern by inner automor-

H

Q

SU(p + q)

SU{p) SU(q) U{1)

SO(p + q)

SO(p) ® SO(q), p or q even

SO(2n)

SU{n) ® (7(1)

Ee

SU(6) ® St7(2), 5 0 ( 1 0 ) £7(1)

FINAL REMARKS

Rank preservation by inner automorphisms holds for arbitrary ZJV (abelian) orbifolds. For non-abelian orbifolds the group element g can not in general be written as exp{—2iriv • H} and therefore group conjugation can act non-trivially on some Hi leading to the projection of the corresponding gauge fields and thus to rank lowering. However considering non-abelian orbifolds is too complicated a way of achieving rank lowering for a simpler way of doing it does exist in the ZJV orbifolds when the discrete group is not realized as a group conjugation, as we will see in the next section.

4.1.2. Rank lowering If the discrete group in Eq. (4.4) can not be realized as the group conjugation (4.12) its action on the Lie algebra is called an outer automorphism and leads to rank reduction. In short, outer automorphisms are structure constant preserving linear transformations of the generators that can not be written as a group conjugation. For any given Lie algebra there is only a limited number of outer automorphisms depending on the symmetries of the Dynkin diagrams. Reduced rank implies that some of the T^ are diagonal and for them the identity {A_R, T^} = 0 can never be satisfied if XR is diagonal. The most interesting case is AABTB =

_ pAf

(4 2Q)

which of course preserves the structure constants. This is an outer automorphism that produces the breaking SU(N) —> SO(N). In this case we must find a non-diagonal (and unitary) matrix in the ^-representation, XR,

587

that solves Eqs. (4.7), i.e. \RT£\R

= - {T£f

(4.21)

This identity can only be satisfied if R is a real representation. In fact one can always choose R = 1Z © TZ where 72. is a non-real representation with generators

**=(1-
(422)

which implies that XR takes the block form

Mr?) and so it has d-n eigenstates (ej, it) with positive parity and dn eigenstates (ii, —e-i) with negative parity, where ej denotes the usual unit vector with a unit in the i-th entry and zero otherwise. The zero-mode spectrum resulting from this action will always be vector like and therefore anomaly-free. To produce a chiral theory the simplest possibility would be to add chiral, anomaly-free, matter of the Ti subgroup localized on the brane where Q —> H via the outer automorphism. Explicit examples will be presented next.

C/(l) -> nothing

EXAMPLE

The simplest example is U{\) in the bulk breaking down to nothing on the branes. The choice that performs this breaking is A = — 1, that is a simple realization of the equation A T = — TT, with T = 1. Charged fermions are necessarily accompanied by oppositely charged partners. The fermionic zero mode spectrum is a vector-like pair of Weyl fermions that can be .assembled into a Dirac fermion.

SU(3)

- • SO(3)

EXAMPLE

The second simplest example of an automorphism with rank breaking action is SU(3) —> SO(3). Solving equation (4.20) one must give a positive parity to antisymmetric generators and a negative parity to symmetric ones. In particular the generators for the SU(3) group are given by the Gell-Man

588

matrices:

A5 =

(4.24)

A7

Ab =

Ab =

This results in the choice A = diag(-l,

+1, - 1 , - 1 , +1, - 1 , +1, -1)

(4.25)

The positive parity generators (A2, A5, A7) form the fundamental representation of SO(3). Matter in the 3 0 3 representation transforms in the 3 © 3 of SO (3). The parity of these states is A303

0 h K

V o

(4.26)

which, after diagonalization yields 3 even and 3 odd states. The fermionic zero mode spectrum is then an SO(3) triplet of Weyl fermions plus their vector-like partners. A classification of breaking patterns by Z2 outer automorphisms is given in table 10. For more details see Ref. [48]. Table 10. Breaking pattern by outer automorphisms Q

H

SU(p)

SO(P)

SO(p+q) SU(2p) E6

SO{p) SO{q) SP(P)

Sp(4), F 4

4.2. Gauge breaking by the Hosotani

mechanism

In the previous section we have seen how a gauge symmetry Q can be broken down to a subgroup H by the orbifold action on the branes. The original

589

gauge bosons A^ split into gauge bosons on the brane Aa and coset "gauge bosons" AJJ that are parity-odd and have no zero modes. Similarly the extra-dimensional components A^ of the higher-dimensional gauge bosons split into odd scalars Ag and even scalars Ag. In particular the scalars Ag propagate in the brane and have massless zero modes that belong to non-trivial representations of H. For instance in the simple case presented in section 4.1, Q = SU(3) and Ji = SU(2) £7(1), we have that under 5/7(3) —> SU(2) £7(1) the branching ratio of the adjoint representation 8 is 8 = 3 0 0 lo © 2 r 0 2 _ F

(4.27)

and then the scalars A£ along the coset direction SU(3)/SU(2) U(l) are SU(2) doublets. The scalars Aj? do not have any tree-level mass since it is forbidden by the higher-dimensional gauge invariance. In flat infinite five-dimensional space-time the gauge invariance in the higher-dimensional space-time would keep on protecting the appearance of a mass for Ag at any order in perturbation theory. However in compact spaces, as it is the case of the Z2 orbifold, the masses of Ag are protected only up to finite terms of order 1/R2 (that go to zero in the "infinite" limit R —> 00). These mass corrections are similar to the thermal masses in field theory at finite temperature [15] that appear from radiative corrections ~ g T: the so-called Debye masses for longitudinal photons and gluons. At tree-level the gauge symmetry H on the brane is spontaneously broken in the presence of the background field Ag. However the VEV of Aj? is undetermined at the classical level. When radiative corrections are considered a tachyonic mass for some Ag fields can be generated, denoting an instability along the Aj? direction whose minimum can be found using effective potential techniques. In this case the VEV (Ag) is fixed .and some gauge bosons in Ti become massive. This spontaneous breaking of 7i to a subgroup by (Ag) is called Hosotani breaking [12] and it has been discussed several times along these lectures. A sufficient condition for Hosotani breaking is that A% acquires a tachyonic mass.J The requirements for this sufficient condition to hold can be established in a general theory with a gauge group Q in the bulk and an arbitrary fermionic content. Before showing them we will remind that such a condition (a tachyonic mass for A§) J

There are some exceptions to this rule when (Ag) = 1/2 and the gauge group TC is unbroken [51].

590

although sufficient is by no means necessary since the effective potential could have a minimum at the origin and another deeper (global) minimum located at some value A% ^ 0. In that case we must rely on the cosmological evolution leading the field towards the global minimum, or otherwise if the life-time of the transition from the local minimum at the origin to the global minimum is shorter than the age of the present universe. The diagrams contributing to the mass of A% are shown in Fig 6. AC,C

ivww A°

vwwi Figure 6. tion

The one-loop diagrams contributing to mass and wave function renormaliza-

This calculation has been done in Refs. [49,52] leading to the result for the A\ mass 3
m?

(4.28)

where C2{Q} is the quadratic Casimir of (the adjoint representation of) Q, e.g. C2[SU(N)] = N and T(R) is the Dynkin index of the representation R satisfying (4.29)

dRC2(R)=T(R)dg [See footnote h for more details.] From the expression (4.28) we see that for models with T{R)

3

f

(4.30)

m? < 0 and therefore H could be radiatively broken. To be more precise we have to look at the full effective potential Ve

"

=

128^4

Tr[V(rF)-V(rB)]

(4.31)

591

with TB,F = exp[2niqB,F(u)], QB,F being the shifts in boson and fermion Kaluza-Klein masses according to (n\ n + an p{w) BF = p »

m

n

e Z

(4.32)

where the parameter w measures the VEV of A§ as w = (Af )i?

(4.33)

which provides a Scherk-Schwarz-like breaking. The structure of Veff depends on the gauge and matter structure. A particularly simple case is when there are Nf fermions in the adjoint representation. Then T(Adj) = C2{Q) and the symmetry is fully determined by the factor 3 — 4 Nf in front of the effective potential. Notice that the radiative mass (4.28) is finite and we want to conclude this section with a few comments about this finiteness. The field A% is a scalar, in a given representation of the gauge group H, when it propagates on the brane. As such it seems that its radiatively corrected mass should not be protected from quadratic divergences (let us remind that the theory is not supersymmetric) by the H invariance and furthermore that radiative corrections localized on the brane should indeed be quadratically divergent. However the calculation leading to the result in (4.28) contains both bulk and brane effects and nevertheless it has been explicitly found to be finite. Is this fact reflecting some additional symmetry or is there an accidental cancellation of quadratic divergences? The answer to those questions were given in Refs. [53,54] where the invariance on the brane, the residue of the gauge invariance in the bulk, was fully identified. In fact it was proven that there is, apart from the Ti gauge invariance on the brane, an infinite set of symmetries that do not close in a group but prevent many explicit renormalization terms allowed by H invariance. In particular there is a residual shift symmetry [53] as Ag —> Ag + dsC", where £° is an odd gauge parameter, that prevents the appearance in a five-dimensional gauge theory of brane mass terms for A% at any order in perturbation theory. This striking result is a general feature of five-dimensional theories, where the non-appearance of quadratically divergent masses on the brane for the extra dimensional components of the five-dimensional gauge bosons is due to the original higher-dimensional gauge invariance. However this result can not be straightforwardly generalized to theories with more than one extra dimension, although some finite results have been found by explicit calculations in six-dimensional theories [55,56]. In fact we have proven [54] that in theories with six or more dimensions there can exist quadratic divergences

592

to the mass of extra dimensional gauge bosons localized on the branes only provided that there are U(l) subgroups in Ji that were not already present in Q. These quadratic divergences are associated to the possible existence of tadpoles for the even fields F^ a phenomenon similar, but not quite coincident, to that of localized anomalies on branes [57,16]. In particular models in six or more dimensions (unlike in five-dimensional theories) the cancellation of brane quadratic divergences has to be enforced, as the cancellation of global and local anomalies should be. Particular examples with finite renormalized masses [55,56] satisfy the condition for cancellation of quadratic divergences, as it should happen. To conclude this section, we have seen that supersymmetry is not the only symmetry that can protect the scalar masses from quadratic radiative corrections. If gauge symmetry breaking proceeds through the Hosotani breaking the higher-dimensional gauge invariance can do the job thus providing a possible non-supersymmetric alternative solution to the hierarchy problem. 4.3. Top assisted

electroweak

breaking

In this section we have been concentrating up to now in non-supersymmetric models where the finiteness of the Hosotani breaking was protected by the underlying higher-dimensional gauge theory. Another possibility for mass protection is when the higher (e.g. five) dimensional theory is supersymmetric, as in section 3. In that case, as we discussed there, supersymmetry can be broken by Scherk-Schwarzcompactification. In particular, in the model we discussed in section 3.2 the gauge and Higgs hypermultiplets were living in the bulk of the fifth dimension while quark and lepton chiral N = 1 supermultiplets were localized on the brane at y = 0. Since the brane is supersymmetric the contributions to the Higgs mass from the top quark Yukawa coupling, see Fig. 7, vanish at tree level, while the contributions to the squared Higgs mass from gauge interactions are positive =>• electroweak symmetry is unbroken at one-loop. A way of putting a remedy to this situation is by allowing the top quark to propagate in the bulk [26,27]. Now to have superpotential interactions on the brane as in the MSSM, W = [HtQLURH2 + hhQLDRBx

+ hTLLBxER

+ ^Hx • H2] 5{y)

(4.34)

where we are using standard MSSM notation by which superfields are denoted by capital Roman characters, we need to enforce orbifold invariance on it. Since the gauge group is unbroken by the orbifold action,

593

tL #2

Figure 7. TA

The one-loop renormalization of the Higgs mass

| T a | ; Ta = ^ ^ t h e c o n ditions (4.7) are trivially satisfied by the identity matrices, XR =1, and the orbifold conditions for bulk fermions are ip(x^, —y) = i75V'(a;A',2/), see Eq. (4.4), which means in Weyl components =

^L{x^,-y)

=

-ipL(x^,y)

i

rl>n(x' ,-y)=+il>R{x'l,y)

(4.35)

under which the theory is invariant. We can then consistently choose as fields propagating in the bulk: • V: the gauge sector multiplets • UR, DH, MR: "right-handed" hypermultiplets. Their massless zero modes surviving the Z2-projection are (uR ,uR), (dR ,dR ) and (£R,£R )• The localized fields are the SU(2)L doublets: i?i,2, QL = (<7L,9L), = (£L,£L)Their fermionic modes will cancel the global anomalies generated by uR, dR , £R. In other words qi, £L are the Z2-partners of are needed to cancel the anomalies, which means that they should have an intrisic odd parity [58]: LL

Z2 :

qL -^ -qL,

II -> -II

(4.36)

in the underlying theory that provides the localized states which are necessary to achieve anomaly cancellation Since only even (E) components of bulk fields couple to the brane [e.g. UR] and brane localized states are Z2-odd (O) [e.g. if2, QL] the only orbifold allowed brane couplings are EEE and EOO. Now while EEE couplings are suppressed in field theory, since no wave function of the involved fields is concentrated on the brane, the natural Yukawa couplings localized on the brane are EOO: for instance the coupling ^QLURHI that we are considering in (4.34). This justifies our choice of matter content. Of course there are other matter contents which are also consistent with the orbifold invariance and lead to somewhat different phenomenological behaviours.

594

The choice made here is just the most convenient one for phenomenological purposes. Given the previous matter content, the Higgs potential along the neutral components of Hi and H2 is, V(HUH2)

= mllHil2

+ m2\H2\2 + mlj(iJi • H2 + h.c.)

+ l±^(\Hl\2

- \H2\2)2 + Xt\H2\2 + A ^ l

2

(4.37)

where the tree-level relations Ml = m2, = n2,

m\=Q,

Xt = Xb = 0

(4.38)

are satisfied. Radiative corrections will alter these relations since s u p e r s y m m e t r y is broken by t h e Scherk-Schwarzmechanism as we described in section 3.2. To compute radiative corrections we first decompose even a n d o d d fields into modes as in Eq. (3.29) a n d change basis using (jr+ n> = <jy^' a n d <j>_ n' = —<j>_ which gives the Fourier decomposition for periodic fields as 7

v>


cos

(n + u))y

(n)

—#—^

n— — 00

*- = f>n^V>>

(4.39)

n= — 00

In particular one-loop corrections to m2 and m2 can be obtained using the results of section 3.2. They yield [Am 2 (0) - Am 2 (c)]

ml = ^ + ^ ~ ^

ml = 112 + 6 f e L + ? g 2 [Am2(0) - Am2 H I

(4-40)

where Am2(oj) was defined in Eq. (3.40) and we are neglecting O(g')corrections. The top and bottom Yukawa couplings are defined as mt /1 + tan' 3 v V tanz/3

,

mi / v v

~T~

where tan/3 = i^/^i, u = \Jv\ + v2 = 174 GeV. Obviously hb is only relevant for tan/3 3> 1. The mass term m2 is generated by the diagram of Fig. 8 and the resulting contribution is 2

Sg2u>/2 512TT2R

[iLi2 (e 2 ™ w ) +h.c] = IMB

(4.42)

595

( - \ Figure 8.

) /i

+ SUSY=0

J~

The one-loop renormalization of m.3

Prom (4.42) it is easy to check that m\ goes to zero both in the supersymmetric limit (w —> 0) and when /i = 0, as it should. It is also zero for the degenerate case u> = 1/2, reflecting the fact that in that case the gauginos X^ are Dirac fermions. The diagrams that contribute to At are given in Fig. 9, and similar diagrams changing t —> b would contribute to A;,. The result is given by [27] \b

?>Kh f &v ; = -^f 8TT2 {\ 1 - \ag(2izRMz* + (l +

1 4(r 2 -l)

(rz - 1) log I 2 - r - r

r2)(Li2(l/r)-Li2(r))

(4.43)

where r = exp(27rio;). Minimization of the one-loop effective potential, VJJ = VJJ = 0 yields the relations n mt 2 2 1 + tan 2 /3 L tan/3 ~ — , mA = -mi -r— A 3 m6 tan/3 and the subsequent Higgs mass spectrum is given by M2H = m2A

(4.44)

Ml = M | + 4Atv~ M2H± =m2A

+ M^

(4.45)

The masses, along with 1/R, turn out to be, as a consequence of the minimization conditions, functions of u and /i, which are the free parameters of the theory. In particular for p. < 1 TeV we obtain the bound 1/R $ 10 TeV. This range for values of 1/R is in agreement with present experimental bounds on the size of longitudinal extra dimensions. In particular in five-dimensional models the strongest bounds are provided by electroweak precision measurements that yield limits that are model dependent but generically in the range 1/R Z 2 — 5 TeV [59].

596

Hi

W,iL

, - - - - „

N

\

\

N N

s

N

'

•

\\

, '

N

•

/

/

#2

*S

31

•

•* K ,^

/

s/

4

\

\

/

/

s

N. *»

•

H.2 /

#2

4"^ tL

H2 _

/ H3

""y" /

T[n) i/

X

S

K /

\

N

#2

#2 - ' ' ' '

tL tL

# 2 -.^

) 4m)

An)

/^ „

„ X,, Hi , - ' #2

'

^

th

#2

tL

#2

"y

V \

/

Fw; \ . - ' "

'igure 9.

; tt] i

A_

^x #2

#2

th

#2

One-loop diagrams contributing to At

What is more interesting, the previous model can be tested by Higgs searches at the future LHC collider at CERN. In particular LEP searches in the MSSM Higgs sector settle a lower bound on VIA as niA ~ 95 GeV [60]. This bound translates into a parallel lower bound on \i which is w-dependent. The absolute lower bound turns out to be fi Z 350 GeV. On the other hand the bound m,A ~ 95 GeV also translates into a lower bound

597

on the S t a n d a r d Model-like Higgs mass Mh which is again w-dependent. T h e absolute lower limit on the Standard Model Higgs mass t u r n s out to be Mh ~ 145 GeV. This mass is probably too large for this Higgs to be discovered at Tevatron [61]. In particular if the Standard Model Higgs t u r n s out to have a mass ~ 115 GeV (i.e. in the final L E P region [62]) this model would be excluded. In fact, notice t h a t for not too large \x values, \x $ 600 GeV, MH < Mh- However in the large t a n / 3 region the state H is ~ H° with unconventional couplings to gauge bosons and fermions. In this way the coupling HZZ = (hZZ) cos/3 is strongly suppressed as ~ l / t a n / 3 and so is its direct H production at LEP. In short the Higgs predicted by the above model could not have been discovered at L E P and its discovery at Tevatron is only very marginal. Its discovery should only be done at LHC.

Acknowledgments I would like to t h a n k all my collaborators on this subject for so many fruitful discussions. In particular I. Antoniadis, K. Benakli, M. Carena, A. Delgado, S. Dimopoulos, G. v. Gersdorff, N. Irges, J. Lykken, A. Pomarol, S. Pokorski, A. Riotto and C. Wagner. This work supported in part by C I C Y T , Spain, under contracts F P A 2001-1806 and F P A 2002-00748, and by EU under contracts HPRN-CT-2000-00152 and HPRN-CT-2000-00148.

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CSABA CSAKI

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EXTRA DIMENSIONS A N D BRANES

CSABA CSAKI Institute for High Energy Phenomenology, Newman Laboratory of Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA This is a pedagogical introduction into theories with branes and extra dimensions. We first discuss the construction of such models from an effective field theory point of view, and then discuss large extra dimensions and some of their phenomenological consequences. Various possible phenomena (split fermions, mediation of supersymmetry breaking and orbifold breaking of symmetries) is discussed next. The second half of this review is entirely devoted to warped extra dimensions, including the construction of the Randall-Sundrum solution, intersecting branes, radius stabilization, KK phenomenology and bulk gauge bosons.

Contents

1

Introduction

606

2

Large Extra Dimensions 2.1 Matching the higher dimensional theory to the 4D effective theory 2.2 What is a brane and how to write an effective theory for it? . . . . 2.3 Coupling of SM fields to the various graviton components 2.4 Phenomenology with large extra dimensions

607 608 613 617 623

3

Various Models w i t h Plat Extra Dimensions 3.1 Split fermions, proton decay and flavor hierarchy 3.2 Mediation of supersymmetry breaking via extra dimensions (gaugino mediation) 3.3 Symmetry breaking via orbifolds :

627 628

Warped Extra Dimensions 4.1 The Randall-Sundrum background 4.2 Gravity in the RS model 4.3 Intersecting branes, hierarchies with infinite extra dimensions . . .

652 652 659 665

4

605

633 641

606

5

Phenomenology of Warped Extra Dimensions 5.1 The graviton spectrum and coupling in RSI 5.2 Radius stabilization 5.3 Localization of scalars and quasi-localization 5.4 SM Gauge fields in the bulk of RSI 5.5 AdS/CFT

670 670 673 682 684 687

6

Epilogue

690

1. I n t r o d u c t i o n Theories with extra dimensions have recently a t t r a c t e d enormous attention. These lectures a t t e m p t to give an introduction to these new models. First we a t t e m p t to motivate theories with branes in extra dimensions, and then explain how to write down an effective theory of branes, and what kind of 4D excitations one would expect to see in such models and how these modes couple to the s t a n d a r d model (SM) fields. T h e first lecture is closed with a discussion of the main phenomenological implications of models with large extra dimensions. In the second lecture we discuss various possible scenarios for theories with flat extra dimensions t h a t could be relevant to model building. First we discuss split fermions, which could give a new way of explaining the fermion mass hierarchy problem, and perhaps also explain proton stability. Next we discuss mediation of supersymmetry breaking via a flat extra dimension, and finally we close this lecture with a discussion of symmetry breaking via orbifolds. These topics were chosen such t h a t besides getting to know some of the most important directions in model building the reader will also be introduced to most of the relevant techniques used in this field. T h e second half of this review deals exclusively with warped extra dimensions: the Randall-Sundrum model and its variations. In the third lecture we first show in detail how to obtain the Randall-Sundrum solution, and how gravity would behave in such theories. Then localization of gravity to brane intersections is discussed, followed by a possible solution to the hierarchy problem in infinite extra dimensions. T h e final lecture discusses various issues in warped extra dimensions, including the graviton KK spectrum, radius stabilization and radion physics, quasi-localization of fields, bulk gauge fields and the A d S / C F T correspondence. Inevitably, many important topics have been left out from this review. These include the cosmology of extra dimensional models, running and unification in AdS space, universal extra dimensions and KK dark matter, black hole physics, brane induced gravity, electroweak symmetry breaking

607

with extra dimensions, and the list goes on. The purpose of these lectures is not to cover every imoprtant topic, but rather to provide the necessary tools for the reader to be able to delve deeper into some of the topics currently under investigation. I have tried to include a representative list of references for the topics covered, and also some list for the topics left out from the review. Clearly, there are many hundreds of references that are relevant to the topics covered here, and I apologize to everyone whose work was not quoted here. Other recent reviews of the subject can be found in Ref. 1.

2. Large Extra Dimensions Extra dimensions were first introduced in the 1920's by Kaluza and Klein,2 who were trying to unify electromagnetism with gravity, by assuming that the photon field originates from fifth component (g^) of a five dimensional metric tensor. The early 1980's lead to a revitalization of these ideas partly due to the realization that a consistent string theory will necessarily include extra dimensions. In order for such theories with extra dimensions to not flatly contradict with our observed four space-time dimensions, we need to be able to hide the existence of the extra dimensions in all observations that have been made to date. The most plausible way of achieving this is by assuming that the reason why we have not observed the extra dimensions yet is that contrary to the ordinary four space-time dimensions which are very large (or infinite), these hypothetical extra dimensions are finite, that is they are compactified. Then one would need to be able to probe length scales corresponding to the size of the extra dimensions to be able to detect them. If the size of the extra dimensions is small, then one would need extremely large energies to be able to see the consequences of the extra dimensions. Thus by making the size of the extra dimensions very small, one can effectively hide these dimensions. So the most important question that one needs to ask is how large could the size of the extra dimensions be without getting into conflict with observations. This is the first point we will address below. We will see that answering this question will naturally lead us towards theories with fields localized to branes. In order to be able to examine such theories, we will consider how to write down an effective Lagrangian for a theory with a brane, and find out how the different modes in such theories would couple to the particles of the SM of particle physics. We close this section by explaining how to calculate various processes including the Kaluza-Klein modes of the graviton, and by briefly sketching

608

ideas about black-hole production in theories with large extra dimensions. The very basics of Kaluza-Klein theories have been explained in the lectures by Keith Dienes. 3 Therefore I will assume that the reader is familiar with the concept of Kaluza-Klein decomposition of a higher dimensional field. Otherwise, the lectures should be self-contained.

2.1. Matching effective

the higher dimensional theory

theory to the 4.TD

The first question that we would like to answer is how large the extra dimensions could possibly be without us having them noticed until now. For this we need to understand how the effectively four dimensional world that we observe would be arising from the higher dimensional theory. In more formal terms, this procedure is called matching the effective theory to the fundamental higher dimensional theory. Thus what we would like to find out is how the observed gauge and gravitational couplings (which should be thought of as effective low-energy couplings) would be related to the "fundamental parameters" of the higher dimensional theory. Let us call the fundamental (higher dimensional) Planck scale of the theory M*, assume that there are n extra dimensions, and that the radii of the extra dimensions are given by r. In order to be able to perform the matching, we need to first write down the action for the higher dimensional gravitational theory, including the dimensionful constants, since these are the ones we are trying to match. For this it is very useful to examine the dimensions of the various quantities that will appear. The infinitesimal distance is related to the coordinates and the metric tensor by ds2 = gMNdxMdxN

(2.1)

Assuming that the coordinates carry proper dimensions (that is they are NOT angular variables) the metric tensor is dimensionless, [g] = 0. Since we can calculate the Christoffel symbols as ^MN ~ 9

9MgNB

(2.2)

we get that the Christoffel symbols carry dimension one, [r] = 1. Since RMN ~ T 2 , the Ricci tensor will carry dimension two, [RMN] = 2, and similarly the curvature scalar [R] = 2. The main point is that all of this is independent of the total number of dimensions, since these were based on local equations. In order to generalize the Einstein-Hilbert action to more

609

than four dimensions, we simply assume that the action will take the same form as in four dimensions: Ji+n

[d4+nxVg(i+n)R(i+n)-

(2.3)

In order to make the action dimensionless, we need to multiply by the appropriate power of the fundamental Planck scale M*. Since d4+n carries dimension —n —4, and i?( 4+n ) carries dimension 2, this has to be the power n + 2, thus we take Si+n = -M?+2

f d 4 + n aV
(2-4)

What we need to find out is how the usual four dimensional action S 4 = -M2Pl fd*xy/iJWRW.

(2.5)

is contained in this higher dimensional expression. Here Mpi is the observed 4D Planck scale ~ 10 18 GeV. For this we need to make some assumption about the geometry of the space-time. We will for now assume, that spacetime is flat, and that the n extra dimensions are compact. So the metric is given by ds2 = (j)^ + h^dxTdx"

+ r2dn\n),

(2.6)

where X ii IS 3. four dimensional coordinate, dfl2n-. corresponds to the line element of the flat extra dimensional space in some parametrization, rj^ is the flat (Minkowski) 4D metric, and /iM„ is the 4D fluctuation of the metric around its minimum. The reason why we have only put in 4D fluctuations is that our goal is to find out how the usual 4D action is contained in the higher dimensional one. For this the first thing to find out is how the 4D graviton is contained in the higher dimensional metric, this is precisely what is given in (2.6). This does not mean that there wouldn't be additional terms (and in fact there will be as we will see very soon). From this we can now calculate the expressions that appear in (2.4): / v

^4+^ = r

n

V^,

i?(4+n)=i?(4),

(2.7)

where these latter quantities are to be calculated from h. Therefore we get

s4+n = - M : + 2 f d 4 + n x v V 4 + n ) # ( 4 + n ) = -M™+ 2 fdn{n)rn

fcPxy/gWRW.

(2.8)

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The factor Jdfl^r11 is nothing but the volume of the extra dimensional space which we denote by V(n). For toroidal compactification it would simply be given by V(n) = {2irr)n. Comparing (2.8) with (2.5) we find the matching relation for the gravitational couplings that we have looked for: M2Pl = M:+2V(n)

= M:+2(2irr)n.

(2.9)

Let us now repeat the same matching procedure for the gauge couplings. Assume that the gauge fields live in the extra dimensions, and go into a normalization where the gauge fields are not canonically normalized: S(i+n) =

_ fd4+nxJ_FMNFMN^gi4+n)_ (2.10) J 4gi M, N denote indices that range from 1 to 4 + n, and g* denotes the higher dimensional ("fundamental") gauge coupling. Clearly, the four dimensional part of the field strength F^ is included the the full higher dimensional FMN • Again performing the integral over the extra dimension we find: 5 (4)

Vt,

- JdAx^F^F^Jg~W.

(2.11)

Thus the matching of the gauge couplings is given by 9eff 92 Note, that it is clear from this equation, that the coupling constant of a higher dimensional gauge theory is not dimensionless, but rather it has dimension [g*] = —n/2. As a consequence it is not a renormalizable theory, but can be thought of as the low-energy effective theory of some more fundamental theory at even higher energies. Now let us try to understand the consequences of (2.9) and (2.12). Since the gauge coupling is dimensionful in extra dimensions, one needs to ask what should be its natural size. The simplest assumption is that the same physics that sets the strength of gravitational couplings would also set the gauge coupling, and thus
s-. Mi Then we would have the following two equations:

(2.13)

±=V(n)M:~rnM: 9A

M2Pl = V(n)M:+2 - rnM?+2,

(2.14)

611

from which it follows that 1

"+2

This would imply that in a "natural" higher dimensional theory r ~ 1/Mpj! In this case there would be no hope of finding out about the existence of these tiny extra dimensions in the foreseeable future. This is what the prevailing view has been until the 90's about extra dimensions. However, we should note that these arguments crucially depended on the ASSUMPTION that every field propagates in all dimensions. The purpose of these lectures is to understand, what kind of physical phenomena one could expect if some or all the fields of the standard model were localized in the extra dimensions to a "brane". This possibility has been first raised in Refs. 4 and 5 (see also Refs. 6 and 7). Before jumping into the detailed description of theories with branes, we would like to first understand what the restrictions on the size of extra dimensions would be, if contrary to the previous assumption the SM fields were localized to 4 dimensions, and only gravity (or other yet unobserved fields) were to propagate into the extra dimension. In this case, new physics will only appear in the gravitational sector, and only when distances as short as the size of the extra dimension are actually reached. However, it is very hard to test gravity at very short distances. The reason is that gravity is a much weaker interaction than all the other forces. Over large distances gravity is dominant because there is only one type of gravitational charge, so it cannot be screened. However, as one starts going to shorter distances, inter-molecular van der Waals forces and eventually bare electromagnetic forces will be dominant, which will completely overwhelm the gravitational forces. This is the reason why the Newton-law of gravitational interactions has only been tested down to about a fraction of a millimeter using essentially Cavendish-type experiments. 8 Therefore, the real bound on the size of an extra dimension is r < 0.1 mm

(2.16)

if only gravity propagates in the extra dimension. How would a large value close to the experimental bound affect the fundamental Planck scale M*? Since we have the relation MPl ~ M™+2rn, if r > 1/Mpi, the fundamental Planck scale M* will be lowered from Mpi. How low could it possibly go down? If M* < 1 TeV, that would imply that quantum gravity should have already played a role in the collider experiments that have been performed until now. Since we have not seen a hint of that, one has to impose that

612

M* > 1 TeV. So the lowest possible value (and thus the largest possible size of the extra dimensions) would be for M* ~ 1 TeV. Such models are called theories with "Large extra dimensions", proposed by Arkani-Hamed, Dimopoulos and Dvali,9 see also Ref. 10. For earlier papers where the possibility of lowering the fundamental Planck scale has been mentioned see Refs. 11 and 12. Let us check, how large a radius one would need, if in fact M* was of the order of a TeV. Reversing the expression MpL ~ M™+2rn we would now get

'= M -(i£) i = (1TeV)10~f'

<2-17>

where we have used M* - 103 GeV and MPi ~ 1019 GeV. To convert into conventional length scales one should keep the conversion factor lGeV" 1 = 2 • l(T 1 4 cm

(2.18)

in mind. Using this we finally get r ~ 2 • 10" 1 7 10^ cm.

(2.19)

For n = 1 this would give the absurdly large value of r = 2 • 10 15 cm, which is grater than the astronomical unit of 1.51013 cm. This is clearly not possible: there can't be a single large extra dimension if one would like to lower M* all the way to the TeV scale. However, already for two extra dimensions one would get a much smaller number r ~ 2 mm. This is just borderline excluded by the latest gravitational experiments performed in Seattle. Conversely, one can set a bound on the size of two large extra dimensions from the Seattle experiments, which gave r < 0.2 mm= 10 12 1/GeV. This results in M* > 3 TeV. We will see that for two extra dimensions there are in fact more stringent bounds than the direct bound from gravitational measurements. For n > 2 the size of the extra dimensions is less than 10 ~ 6 cm, which is unlikely to be tested directly via gravitational measurements any time soon. Thus for n > 2 M* ~ 1 TeV is indeed a possibility that one has to carefully investigate. If M* was really of order the TeV scale, there would no longer be a large hierarchy between the the fundamental Planck scale M* and the scale of weak interactions Mw, thus this would resolve the hierarchy problem. In this case gravity would appear weaker than the other forces at long distances because it would get diluted by the large volume of the extra dimensions. However, this would only be an apparent hierarchy between the strength of the forces, as soon as one got below scales of order r one

613

would start seeing the fundamental gravitational force, and the hierarchy would disappear. However, as soon as one postulates the equality of the strength of the weak and gravitational interactions one needs to ask why this is not the scale that sets the size of the extra dimensions themselves. Thus by postulating a very large radius for the extra dimensions one would merely translate the hierarchy problem of the scales of interactions into the problem of why the size of the extra dimension is so large compared to its natural value.

2.2. What is a brane and how to write an effective for it?

theory

Above we have seen that theories where certain particles (especially the light SM particles) are localized to four dimensions, while other particles could propagate in more dimensions could be very interesting. In these lectures we would like to study theories of this sort. We will refer to the surface along which some of the particles are localized as "branes", which stands for a membrane that could have more spatial dimensions than the usual 2 dimensional membrane. A p-brane will mean that the brane has p spatial dimensions, so a 2-brane is just the usual membrane, a 1-brane is just a string, while the most important object for us in these lectures will be a 3-brane, which has 3 spatial dimensions just like our observed world, which could be embedded into more dimensions. What is really a brane? In field theory, it is best to think of it as a topological defect (like a soliton), which could have fields localized to its surface. For example, as we will see in the next lecture in detail, domain walls localize fermions to the location of the domain wall. String theories also contain objects called D-branes (short for Dirichlet banes). These are surfaces where open string can end on. These open strings will give rise to all kinds of fields localized to the brane, including gauge fields. In the supergravity approximation these D-branes will also appear as solitons of the supergravity equations of motion. In our approach, we will use a low-energy effective field theory description. We will (usually) not care very much where these branes come from, but simply assume that there is some consistent high-energy theory that would give rise to these objects. Therefore, this theory will be valid only up to some cutoff scale, above which the dynamics that actually generates the brane has to be taken into account. For our discussion we will follow the description and notation of Sundrum in Ref. 13.

614

To describe the branes, let us first set up some notation. The 3-brane will be assumed to be described by a flat four-dimensional space-time equivalent to R4, while the extra dimensions by Rn or if compactified by T™ (an n dimensional torus). The coordinates in the bulk (bulk stands for the all of spacetime) are denoted by XM, M — 0 , 1 , . . . , 3 + n, the coordinates on the brane are denoted by x^, fi = 0,1,2,3, while the coordinates along the extra dimensions only are denoted by xm, m = 4 , . . . , 3 + n. For now we will concentrate on the bosonic degrees of freedom. What are these degrees of freedom that we need to discuss a low-energy effective theory? Since we need to discuss the physics of higher dimensional gravity, this should include the metric in the 4 + n dimensions GMN(X), and also the position of the M brane in the extra dimensions Y (x). Note, that the metric is a function of the bulk coordinate XM, while the position of the brane is a function of the coordinate along the brane xM. In addition, we would like to take into account the fields that are localized to live along the brane. These could be some scalar fields $(x), gauge fields A/J,(x) or fermions ^>L(X). These fields are also functions of the coordinate along the brane x^.

Figure 1.

Parametrization of the position of the brane in the bulk.

The effective theory that we are trying to build up should describe small fluctuations of the the fields around the vacuum state. So we have to specify what we actually mean by the vacuum. We will assume that we have a flat brane embedded into flat space. The corresponding choice of vacua is then given by GMN(X) M

Y (x)

= rjMN = 6ffx».

(2.20)

Here and everywhere in the lectures we will use the metric convention

615

VMN = diag(+, —, —,..., - ) . The bulk action will then just be the usual higher dimensional Einstein-Hilbert action as discussed in the previous section, plus perhaps a term from a bulk cosmological constant A: Sbuik = - f di+nX^\G\{M?+2R^+^

+ A).

(2.21)

In order to be able to write down an effective action for the brane localized fields one has to first find the induced metric on the brane. This is the metric that should be used to contract Lorentz indices of the brane field. To get the induced metric, we need to find what the distance between two point on the branes is, x and x + dx: ds2 = GMNdY{x)MdY(x)N

PJYM

3YN

= GMN-R—dx^-^—dx".

(2.22)

From this the induced metric can be easily read off: 9v» = GMNiy^d^dvY".

(2.23)

This is the general expression for arbitrary background metric and for an arbitrary brane. For the flat vacuum that we have chosen one can easily see from (2.23) that the induced metric will fluctuate around the flat 4D Minkowski metric T]^u. After this we can discuss the basic principle for writing down the brane induced part of the action: it has to be invariant both under the general coordinate transformation of the bulk coordinates X and under the general coordinate transformations of a;. It is clear that the invariance under the general coordinate transformation of the bulk coordinates just corresponds to the usual general covariance of a higher dimensional gravitational theory. The additional requirement that the action also be invariant under the coordinate transformations of the brane coordinate x is an expression of the fact that x is just a possible parametrization of the surface (brane), which itself can not have a physical significance, and any different choice of parametrization has to give the same physics. This will ensure the usual 4D Lorentz invariance of the brane induced action. Thus there are two separate coordinate invariances that the action has to be invariant under, that corresponding to X and to x. Practically this means that one has to contract bulk indices with bulk indices, and brane indices with brane indices. For example d^Y1^ would be a vector under both the bulk and the brane coordinate invariance, and both indices have to be contracted to form a scalar that can appear in the action, while the induced metric g^v would be a tensor under the brane reparametrization, but a scalar under

616

bulk coordinate invariance, etc. Thus the general form of the brane action would be of the form Jbrane

/*„ 9»V9pa

_/4 _ Rd) F

+

EZD^D^

F

_ !/($) (2.24)

Here the possible constant piece / corresponds to the energy density of the brane, called the brane-tension. This brane tension has to be small (in the units of the fundamental Planck scale) in order to be able to neglect its backreaction on the gravitational background. In Chapter 4 we will investigate warped backgrounds where this back-reaction will be taken into account. As discussed above, in addition to the usual bulk coordinate invariance which everyone is familiar with, there is also a 4D reparametrization invariance, which is corresponding to the fact that a different parametrization of the surface describing the brane would yield the same physics, x —> x'(x) is an invariance of the Lagrangian. Thus one needs an additional gauge fixing condition, which will eliminate the non-physical components from YM. There are four coordinates, so one needs four conditions, which can be picked as Yfl(x)=x»,

(2.25)

which is a complete gauge fixing. Thus out of the 4 + n components of YM only the components along the extra dimension Ym(x), m = 4, 5 , . . . , 3 + n correspond to physical degrees of freedom. These n physical fields correspond to the position of the brane within the bulk. Let us now discuss how to normalize the fields in order to end up with canonically normalized 4D actions. The bulk metric GMN is dimensionless, and we expand it in fluctuations around the background which we assume to be flat space: GMN = VMN H

^xr^Miv2M*2 +

(2.26)

This way the graviton fluctuation HMN has dimension f + 1 which is the right one for a bosonic field in 4 + n dimensions. The prefactor was chosen such that the kinetic term reproduces the canonically normalized kinetic term when expanding the Einstein-Hilbert action in h. To get a canonically normalized field for the coordinates describing the position of the brane Ym{x) we expand the leading term in the brane-action

617

which is just the brane tension:

ydWJfffhf 4 + .-.],

(2.27)

where for the induced metric the leading dependence on Y is 9^ = GMNdIJYmdl,Ym

= n^ + dtlYmdvYm

+ ...,

(2.28)

and expanding the determinant of the metric in powers of Y we obtain detg = -d(lYm&iYm,

(2.29)

from which the leading term in the action is S = fdixf%Ymd"Ym.

(2.30)

Thus the canonically normalized field will be

Note, that it is the brane tension which sets the size of the kinetic term for the Ym fields. In particular, if the tension is negative, then one would have a field with negative kinetic energy, which is thus a physical ghost. This shows that a brane with negative tension is likely unstable, the brane wants to crumble, unless somehow these modes with negative kinetic energy are projected out, by not allowing the brane to move. This possibility will be encountered when we consider branes at orbifold fixed points. 2.3. Coupling of SM fields to the various components

graviton

In the previous section we have discussed how one should construct an action for fields localized on branes coupled to gravity. Next we would like to explicitly construct the generic interaction Lagrangians between the matter on the brane (which we will simply call the SM matter) and the various graviton modes. For our discussion we will follow the work of Giudice, Rattazzi and Wells.14 We have seen above that the SM fields feel only the induced metric 9^{x)

= GMN(X)d^YMdvYN.

(2.32)

It is quite clear from the previous subsection how to deal with the Ym fields, so we will concentrate on the modes of the bulk graviton, and for

618

now set the fluctuations of the brane to zero, that is set YM = Sjfx^, and thus Ym = 0. In this case 9v,(x) = Gl„,(xli,xm

= 0).

(2.33)

The action is given by

S = J'd*xLSMy/g(gilv,$,*,A,...)

(2.34)

The definition of the energy-momentum tensor is given by J-gT^

6

=

-p^-.

(2.35)

From this it is clear that at linear order the interaction between the SM matter and the graviton field is given by (expanding in the fluctuation around flat space again g^ — ??„„ H — a ^ i ) : Sint= [d*xT*v!^®.

(2.36)

Thus generically the graviton couples linearly to the energy-momentum tensor of the matter (this is in fact the definition of the stress-energy tensor, but it is a quantity that we know quite well and is easy to find). Note, that in the above expression what appears is the graviton field at the position of the brane, which is not a mass eigenstate field from the 4D point of view, but rather a superposition of all KK modes. Using the lecture by K. Dienes we write the graviton field in the KK expansion as

hMN{xty)= E kx = — oo

••• E fcn

^ p ^ ,

—— oo

(2.37)

n

where we have denoted the coordinates along the extra dimension by ym = xm, and assumed a toroidal compactification with volume Vn = (2TrR)n. Plugging the KK expansion back into (2.36) we find the coupling of the SM fields to the individual KK modes to be

y fdAxT^ * % = y f dUx^-T^h* k

(2.38)

k

where we have again used the relation between the fundamental Planck scale and the observed one. Thus we can see that an individual KK mode couples with strength 1/iWp; to the SM fields. However, since there are many of them, the total coupling in terms of the field at the brane sums up to a coupling proportional to 1/M*, as we have seen in (2.36).

619

Next let us discuss what the different modes contained in the bulk graviton field are. Clearly, the graviton is a D by D symmetric tensor, where D = n + 4 is the total number of dimensions. Therefore this tensor has in principle D{D-\-l)/2 components. However, we know that general relativity has a large gauge symmetry - D dimensional general coordinate invariance. Therefore we can impose D separate conditions to fix the gauge, for example using the harmonic gauge dMh% = \dNh%.

(2.39)

However, just like in the ordinary Lorentz gauge for gauge theories, this is not yet a complete gauge fixing. Gauge transformations which satisfy the equation OeM = 0 are still allowed (where the gauge transformation is hMN —> fiMN + 9M^N + 9N£M), and this means that another D conditions can be imposed. This means that generically a graviton has D(D + l ) / 2 — 2D = D(D — 3)/2 independent degrees of freedom. For D = 4 this gives the usual 2 helicity states for a massless spin two particle, however in D = 5 we get 5 components, in D = 6 we get 9 components, etc. This means that from the 4D point of view a higher dimensional graviton will contain particles other than just the ordinary 4D graviton. This is quite clear, since the higher dimensional graviton has more components, and thus will have to contain more fields. The question is what these fields are arid how many degrees of freedom they contain. We will go through these modes carefully in the following. • The 4D graviton and its KK modes These live in the upper left 4 by 4 block of the bulk graviton which is given b y a 4 + n b y 4 + n matrix: rik

k

These modes are labeled by the vector k, which is an n-component vector, specifying the KK numbers along the various extra dimensions. These

620

modes are generically massive, except for the zero mode. In the limit of no sources these modes satisfy the 4D equation (D+P)G^=0

(2.41)

which as usual would imply 10 components, however the gauge conditions d"G liv

0,

Gf,k = 0

(2.42)

eliminate 5 components, and we are left generically with 5 degrees of freedom, which is just the right number for a massive graviton in four dimensions. The reason is that a massive graviton contains a normal 4D massless graviton with two components, but also "eats" a massless gauge field and a massless scalar, as in the usual Higgs mechanism. Thus 5 = 2 + 2 + 1. • 4D vectors and their KK modes The off-diagonal blocks of the bulk graviton form vectors under the four dimensional Lorentz group. This is clear, since they come from the GMj components of the graviton: /

(2.43)

V

)

Naively one could think there there would be n such four dimensional vectors, however we have seen before that the 4D graviton has to eat one 4D vector to form a full massive KK tower, thus there are only n — 1 massive KK towers describing spin 1 particles in 4D. The missing of the last tower is expressed by the constraint

*X- = o.

(2.44)

The usual Lorentz gauge condition can also be imposed dnVk

0.

(2.45)

Each of these massive vectors absorbed a scalar via the Higgs mechanism. • 4D scalars and their KK modes

621

The remaining lower right n by n block of the graviton matrix clearly corresponds to 4D scalar fields: \

/

(2.46)

J

\ Originally there are n(n + l ) / 2 scalars, however as we discussed before the graviton eats one scalar, and the remaining n — 1 vectors eat one scalar each. In addition, there is a special scalar mode, whose zero mode sets the overall size of the internal manifold, and therefore this special scalar is usually called the radion. Thus there are n ( n + 1 ) / 2 — n — 1 = (n 2 —n — 2)/2 scalars left. The equations that express the fact that n fields were eaten are 0,

k Sjk

(2.47)

while the fact that we usually separate out the radion as a special field not included among the scalars is expressed as the additional condition kj

(2.48)

0.

S. kj

The radion is then given by h^ Let us now count the total number of degrees of freedom taken into account: 5 (graviton) +3(n — 1) (vectors) +(n 2)/2 (scalars) + 1 (radion) = (4 + n)(l + n)/2 = D(D - 3)/2. Thus we can see that all the modes of the graviton have been accounted for. The explicit expressions for the canonically normalized 4D fields are 3(w-l) n+2

given in unitary gauge by (using the notation K = radion

H

scalars &.

K

J

hk. »

)HS

^ _ {Vi. n-1

vectors V* = — hUgravitons Gk^v

j k

'

r(»?/i V

3

"-

+ •

k2

^)Hn.

(2.49)

The equation of motion in the presence of sources is given for the above fields by

622

(D+fc2)

0 0

ok

V

3MPi

(2.50)

/

M

We can see why the radion is special: besides setting the overall size of the compact dimension it also is the only field besides the 4D graviton that couples to brane sources. It couples to the trace of the energy-momentum tensor. All the other fields are not important when one tries to calculate the coupling of the matter fields on the brane to the modes of the bulk graviton. For example, if we take QED on the brane, the Lagrangian is given by

L = V9 ( W A , * - -AF^F^) ,

(2.51)

where the covariant derivative for fermions is given by Dn

M

dfj, -

ieQAf,

1

<J

(2.52)

e^djxe.iic

where e^f is the vielbein, and abc the spin-connection (we will not go into detail in these lectures on how to couple fermions to gravity, for those interested in more details we refer to Ref. 13). From this the energy-momentum tensor is given by L

liv

'M^dv + •-eQV^A,,

~1vdv)*--Ad^v1 )* + F M A F; +

X - v , P PF-Xp-

(2.53)

Typical Feynman diagrams involving the graviton which we would get from this T when coupled to gravity on the brane are given by:

623

The last one appears only in non-Abelian gauge theories. There are similar vertices involving the radion field. 2.4. Phenomenology

with large extra

dimensions

We have discussed before that the mass splitting of the KK modes in large extra dimensional theories is extremely small, 1

/ M* \ "

/ M* \ ^

12n-31

This implies that for a typical particle physics process with high energies there is an enormous number of KK modes available. This suggests that since the splitting of the KK modes is very small, it is useful to turn the sum over KK modes into an integral. If iV denotes the number of KK modes whose momentum along the extra dimension is less than k, then clearly dN = Sn-xk^dk,

(2.55)

where Sn = {2ir)%/T(n/2) is the surface of an n dimensional sphere with unit radius. The actual mass of a given KK mode is given by m = '-£, therefore we get that dN = Sn^1mn-1Rn~1dmR

Ml,

= Sn-lir^mn~1dm.

(2.56)

In order to calculate an inclusive cross-section for the production of graviton modes, what one needs to do is to first calculate the cross-section for the production of an individual mode with mass m, dam/dt, and then using the above formula get ^ -Sn n _ i-^m^. dtdm M,n+2

dt

(2.57)

Since the cross section for an individual KK mode is proportional to l/MPl, the inclusive cross section will have a behavior of the form

AfL^s^ZH.^.

(2.58)

n dtdm M*™+2 With this we have basically covered all elements of calculating processes with large extra dimensions. In the following we will briefly list some of the most interesting features/constraints on these models. Clearly, not everything will be covered here, for those interested in further details we refer to the original papers. A nice general overview of the phenomenology of large extra dimensions is given in Ref. 15. For collider signals see Refs. 1418. For the running of couplings and unification in extra dimensions see

624

Ref. 19. For consequences in electroweak precision physics see Ref. 20. For neutrino physics with large extra dimensions see Ref. 21. For topics related to inflation with flat extra dimensions see Ref. 22. Issues related to radius stabilization for large extra dimensions is discussed in Ref. 23. Connections to string theory model building can be found in for example in Refs. 24, 25 and 26. More detailed description of supernova cooling into extra dimensions is in Ref. 27, while of black hole production in Refs. 28, 29. • Graviton production in colliders. 15-16 Some of the most interesting processes in theories with large extra dimensions involve the production of a single graviton mode at the LHC or at a linear collider. Some of the typical Feynman diagrams for such a process are given by:

Note, that the lifetime of an individual graviton mode is of the order r ~ Tp2~i which means that each graviton produces is extremely long lived, M

PI

and once produced will not decay again within the detector. Therefore, it is like a stable particle, and takes away undetected energy and momentum. Thus in the above diagrams wherever we see a graviton, what one really observes is missing energy. This would lead to the spectacular events when a single photon recoils against missing energy in the linear collider, or a single jet against missing energy at the LHC. These are processes with no backgrounds, and also very different from the signals of supersymmetry, where one would usually have two jets, or two photons produced in the presence of missing energy. • Virtual graviton exchange. 17

625

Besides the direct production of gravitons, another interesting consequence of large extra dimensions is that the exchange of virtual gravitons can lead to enhancement of certain cross-sections above the SM values. For example, in a linear collider the e + e~ —> / / process would also get contributions from the diagram

The scattering amplitude for this process can be calculated using our previous rules: A-

rp ±_ rp MA ,^ ^L, ""s-m? a 22

where S(s) = ^ see Ref. 17.

£

n

, 0 +

K

3

7=Z? and r = T^T^

V^ s-m2

=

S(S)T,

- -^T^.

(2.59)

For more details

• Supernova cooling. 15 ' 27 Some of the strongest constraints on the large extra dimension scenarios come from astrophysics, in particular from the fact that just as axions could lead to a too fast cooling of supernovae, since gravitons are also weakly coupled particles they could also transport a significant fraction of the energy within a supernovae. These processes have been discussed in detail in Ref. 27, here we will just briefly mention the essence of the calculation. The production of axions in supernovae is proportional to the axion decay constant l / / 2 . The production of gravitons as we have seen is roughly proportional to 1 /'M Pl (T'/'8m) n ~ Tn/M™+2, where T is a typical temperature within the supernova. This means that the bounds obtained for the axion cooling calculation can be applied using the substitution l / / 2 -> Tn/M?+2. For a supernova T ~ 30 MeV, and the usual axion bound fa > 109 GeV implies a bound of order M* > 10 — 100 TeV for n = 2. For n > 2 one does not get a significant bound on M* from this process. • Cooling into the bulk. 15 Another strong constraint on the cosmological history of models with large extra dimensions comes from the fact that at large temperatures emissions of gravitons into the bulk would be a very likely process. This would empty

626

our brane from energy density, and move all the energy into the bulk in the form of gravitons. To find out at which temperature this would cease to be a problem, one has to compare the cooling rates of the brane energy density via the ordinary Hubble expansions and the cooling via the graviton emission. The two cooling rates are given by dp Ct6• expansion

~ - 3 f f p ~ - 3 - 5lvl - p ,

(2.60)

ivl rPl

dp

(2-61)

n+2 • Ml

dt evaporation

These two are equal at the so called "normalcy temperature" T„, below which the universe would expand as a normal 4D universe. By equating the above two rates we get T

* ~ ( ~M— )

=

^ " ^

MeV

'

< 2 62

-' ^

This suggests, that after inflation the reheat temperature of the universe should be such, that one ends up below the normalcy temperature, otherwise one would overpopulate the bulk with gravitons, and overdose the universe. This is in fact a very stringent constraint on these models, since for example for n = 2, T* ~ 10 MeV, so there is just barely enough space to reheat above the temperature of nucleosynthesis. However, this makes baryogenesis a tremendously difficult problem in these models. • Black hole production at colliders.28 One of the most amazing predictions of theories with large extra dimensions would be that since the scale of quantum gravity is lowered to the TeV scale, one could actually form black holes from particle collisions at the LHC. Black holes are formed when the mass of an object is within the horizon size corresponding to the mass of the object. For example, the horizon size corresponding to the mass of the Earth is 8 mm's, so since the radius of the Earth is 6000 km's most of the mass is outside the horizon of the Earth and it is not a black hole. What would be the characteristic size of the horizon in such models? This usually can be read off from the Schwarzschild solution which in 4D is given by , ds

2

/ = [

1

GM\ - — )

,2 dt2 ~

dr2 {1_GM}

, „ + r2d2n ,

, (2.63)

627

and the horizon is at the distance where the factor multiplying dt2 vanishes: rjf = GM. In 4 + n dimensions in the Schwarzschild solution the prefactor is replaced by 1 —GM ^- _, —>-, 1 — M2+n 1+n , from which the horizon size is given cpici^eu uy ± — —

—

by

The exact solution gives a similar expression except for a numerical prefactor in the above equation. Thus we know roughly what the horizon size would be, and a black hole will form if the impact parameter in the collision is smaller, than this horizon size. Then the particles that collided will form a black hole with mass MBH = y/s, and the cross section as we have seen is roughly the geometric cross section corresponding to the horizon size of a given collision energy

r2

1

(MBHy+1

» ~ MI, \TMT)

(2 65)

'

2

The cross section would thus be of order 1/TeV ~ 400 pb, and the LHC would produce about 107 black holes per year! These black holes would not be stable, but decay via Hawking radiation. This has the features that every particle would be produced with an equal probability in a spherical distribution. In the SM there are 60 particles, out of which there are 6 leptons, and one photon. Thus about 10 percent of the time the black hole would decay into leptons, 2 percent of time into photons, and 5 percent into neutrinos, which would be observed as missing energy. These would be very specific signatures of black hole production at the LHC. 3. Various Models with Flat Extra Dimensions In the previous section we have discussed theories with large extra dimensions: the motivation, basic idea, some calculational tools and some of the most interesting consequences. In this section we will consider some topics that are related to flat extra dimensions (that is theories where the gravitational background along the extra dimension is flat, as compared to the warped extra dimensional scenario discussed in the following two lectures). These models do not necessarily assume that the size of the extra dimension is as large as in the large extra dimension scenario discussed previously. The first model, theories with split fermions will still be closely related to large extra dimensions, while the other two examples: mediation

628

of supersymmetry breaking via extra dimensions and symmetry breaking via orbifold compactifications will be models of their own, usually in a supersymmetric context, and thus in those models we will not assume the presence of large extra dimensions at all. 3.1. Split fermions,

proton

decay and

flavor

hierarchy

If there are indeed large extra dimensions, and the scale of gravity is of order M* ~ TeV, then one has to confront the following issue. It is usually a well-accepted fact that quantum gravity generically breaks all global symmetries (but not the gauge symmetries), and therefore if one is assuming that a theory has a global symmetry, one can only do this up to symmetry breaking operators suppressed by the scale of quantum gravity. However, this would generically cause problems with proton decay. In the SM baryon number is an accidental global symmetry of the Lagrangian (this means that every renormalizable operator consistent with gauge invariance also conserves baryon number, without having to explicitly require that). However, one can easily write down non-renormalizable operators that do violate baryon number, and would give rise to proton decay. However, in the SM if there is no new physics up to some high scale (the GUT or the Planck scales) then these operators are expected to be suppressed by this large scale, and proton decay could be sufficiently suppressed. In principle, any new physics beyond the SM could give rise to new baryon number violating operators, suppressed by the scale of new physics. For example, in the supersymmetric standard model the exchange of superpartners could in principle lead to some baryon number violating operators, which would be disastrous, since they would only be suppressed by the scale of the mass of the superpartners. Therefore, in the MSSM one has to make sure that all the interactions with the superpartners also conserve baryon number, which can be achieved by imposing the so-called R-parity. Then again the remaining baryon number violating operators will be suppressed by the next (very high) scale of physics. However, in the large extra dimensional case the situation is slightly worse. The reason is that since we are assuming that the scale of quantum gravity itself is of order M* we can not really rely on a global symmetry to forbid the unwanted baryon number violating operators. For example the operator

would cause very rapid proton decay.

629

A very nice way out of this problem that relies specifically on the existence of the extra dimensions was proposed by Arkani-Hamed and Schmaltz. 30 Their idea is to make use of the extra dimensions in a way that can explain why the dangerous operators are very suppressed without having to worry about quantum gravity spoiling the suppression. This will be done by localizing the SM fermions at slightly different points along the extra dimension. This localization of fermions at different points is sometimes called the "split fermion" scenario. The reason why this could be interesting for the suppression of proton decay is that if the fermions are split, and their wave functions have a relatively narrow width compared to the distance of splitting, then operators in the effective 4D theory that involve fermions localized at different points along the extra dimensions could have a very large suppression due to the small overlap of the fermion wave functions. This way one can generate large suppression factors without the use of any symmetry, and this could be used both to highly suppress baryon number violating operators, and also to generate the observed fermion mass hierarchy of the SM fermions. In this section we will follow the discussion of Ref. 30 of split fermions. Until now we have not discussed the mechanism of localization of various fields. If we want to pursue the program of localizing fermions at slightly different points, we will have to go into the detail of the stabilization mechanism of fermions in extra dimensions. We will discuss the simplest case, namely a single extra dimension. For this, we have to first understand how fermions in 5D are different from fermions in 4D. Fermions are forming representations of the 5D Lorentz group, which is different from the 4D Lorentz group, since it is larger. In particular, the 5D Clifford algebra contains the 75 Dirac matrix as well. While in 4D the smallest representation of the Lorentz group is a two-component Weyl fermion, and one gets four component Dirac fermions only if one requires parity invariance, in 5D due to the fact the Clifford algebra contains 75 the smallest irreducible representation is the four component Dirac fermion. Since a Dirac fermion contains two two-component fermions with opposite chirality, whenever one talks about higher dimensional fermions one has to start out with an intrinsically non-chiral set of 4D fermions. Since the SM is a chiral gauge theory, non-chirality of the higher dimensional fermions has to be overcome somehow. We will see that the localization mechanism and (in the next two subsection) orbifold projections can achieve this. First we will concentrate on describing a viable localization mechanism that produces chiral fermions localized at different points along the extra dimension.

630

We will use the following representation for the 7 matrices:

The two type of 5D Lorentz invariants that can be formed from two fourcomponent 5D spinors ^ 1 and ^ 2 are the usual # 1 ^ 2 which corresponds to the usual 4D Dirac mass term, and ^^C^2, which corresponds to the Majorana mass term, and where C5 is the 5D charge conjugation matrix C5 = 7 °7 2 7 5 To describe the localization mechanism for fermions, we consider a 5D spinor in the background of a scalar field $ which forms a domain wall. This domain wall is a physical example of brane that has a finite width (a "fat brane"). The background configuration of this scalar is denoted by $(y). The action for a 5D spinor in this background is given by

S= f d4xdy^> [ t 7 X + ilhdy + $(j/)] *.

(3.3)

Note, that we have added a Yukawa type coupling between the scalar field and the fermion, which will be essential for the localization of the fermions on the domain wall. From this the 5D Dirac equation is given by [*7M^4 + i-Kdy + *(y)] * = 0

(3.4)

We will look for solutions to this which are left- or right-handed 4D modes: i75*L = * L ,

il5^R = - * f l -

(3.5)

To find the 4D eigenmodes of this system, we look for the eigensystem of the y-dependent piece of the above equation. To simplify the equation (as usual when solving a Dirac-type equation) we also multiply by the conjugate of the differential operator to get the equation:

[-il5dy + $(y)] [il5dy + $( y )] * g j = ift&LtR,

(3.6)

which gives (-d2y + $(y) 2 ± $(1/)) *LtR

= n2n*1>fl.

(3.7)

If $(y) had a linear profile, this would exactly give a harmonic oscillator equation. However, even for the generic background one can define creation and annihilation operators as for the usual harmonic oscillator using the definition a = dy + ${y), ai = -dv + ${y).

(3.8)

631

With these operators one can turn the above Schrodinger-like problem into a SUSY quantum mechanics problem, which means that one can define the operators Q and Q^ such, that {Q, Q^} = H, where H is the Hamiltonian of the system. For the case at hand Q = a^P^, Q^ = O1J°PR will give the right anti-commutation relation (here PL,R are the left and right-handed projectors. The SUSY quantum mechanics-like Schrodinger problems have very special properties. For example, the eigenvalues of the L and R modes always come in pairs, except possibly for the zero modes. The pairing of eigenmodes just corresponds to the expected vectorlike behavior of the bulk fermions: for every L mode there is an R-mode, since these are massive modes that is exactly what one would expect. The fact, that the zero modes need not be paired is the most important part of the statement, since these zero modes are the most interesting for us: they are the ones that could give chiral 4D fermions. Therefore let us examine the equation for the zero mode. Since {Q,Q^} = H, the solution to Hty = 0 are the solutions to Q\$> = 0 of Qtijr = o, that is [±a„ + $(i/)]* i ,fl(y) = 0.

(3.9)

From this we find the left and right-handed zero modes to be

tf^e+tf^J/W.

(3.10)

Since we have an exponential with two different signs for the exponent, clearly both of these solutions can not be normalizable at the same time. Therefore we get chiral zero modes localized to the domain wall. For example, if the profile of the domain wall is linear, $(y) ~ 2/x2j/, then the solution for the left-handed zero mode is: i (TT/2)*

This would be a zero mode localized at y = 0, which is exactly the point at which the domain wall profile switches sign, that is where <3>(y) = 0. Thus we have described a dynamical mechanism to localize a chiral fermion to a domain wall. These are sometimes called domain-wall fermions, and are extensively used for example in trying to put chiral fermions on a lattice. 31 Let us now consider the case when we have many fermions in the bulk, i&i, and their action is given by S = / d4xdy$i [i 7 X

+ i-Kdy + mv)

~ Ma *i-

( 3 - 12 )

632

There will now be chiral fermion zero modes centered at the zeroes of (A$(y) - m))ij.

(3.13)

If we again imagine that the domain wall is linear in the region where all these fermions will be localized, then the center of the fermion wave functions will be at <3-14>

»' - $ •

where the m,'s are the eigenvalues of the rriij matrix. Thus the different fermions will be localized at different positions! Let us now try to write down what the Yukawa couplings between such fermion zero modes would be in the presence of a bulk Higgs field that connects these fermions. The 5D action would be given by S = fd5xL[ijMdM

+ (y)]L + fd5xEc[i^MdM

+ f d5XKHLTC5Ec

+ Hy) -m]Ec

,

(3.15)

where the last term is the bulk Yukawa coupling written such that it is both gauge invariant and 5D Lorentz invariant. From this the effective 4D Yukawa coupling for the zero modes will be of the for fd4xKHlec [dy*L(y)*ec(y-r),

(3.16)

where I, e c are the zero modes of the bulk fields, and \PL(J/), \Pec(y — r) are the wave functions of these zero modes. These wave functions are assumed to be Gaussians centered around different points (y = 0 and y = r) in the linear domain wall approximation, and so this integral will just be a convolution of two Gaussians, which is also a Gaussian: J dy$L(y)<S>ec(y) = ^

J dye~^e~^-^

= e~^.

(3.17)

Thus the effective 4D Yukawa coupling between the zero modes is Ke-^,

(3.18)

which could be exponentially small even if the bulk Yukawa couplingreis of order one. The exponential suppression factor depends on the relative position of the localized zero modes. Thus split fermions could naturally generate the fermion mass hierarchy observed in the SM. Basically, this

633

method translates the issue of fermion masses into the geography of fermion localization along the extra dimension. To close this subsection, we get back to our original motivation of proton stability. Imagine, that the leptons and the quarks are localized on opposite ends of a fat brane. A dangerous proton decay operator generated by quantum gravity at the TeV scale would have the form

J d5x~(QTC5L)\UcTC5Dc).

(3.19)

Again calculating the wave function overlap of the zero modes as before we get that the effective 4D coupling is suppressed by the factor f d2/(e-^ 2 )3 e -M 2 (y-r) 2 „ e-y2r\

(3.20)

Here r is the separation between quarks and leptons, and we can see that arbitrarily large suppression of proton decay is possible in such models without invoking symmetries. What would a model with large extra dimensions then look like in this scenario? One would have the large extra dimensions of size T e V - 1 1 0 ^ , where the gravitational degrees of freedom propagate. Within that large extra dimension there is a "fat brane", which is basically the domain wall that we have discussed above. The fat brane has the gauge fields and the Higgs scalar living along its world volume. The width of this fat brane could be of the order TeV - 1 , such that the KK modes of the gauge fields arising from the "fatness" of the brane are sufficiently heavy. Within this fat brane fermions are localized at different positions of this brane, which will eliminate the problem with proton decay and the geography of the localized zero modes will generate the fermion mass hierarchy. The width of the localized fermions is much smaller than the width of the fat brane, of order 0.1-0.01 TeV - 1 . For more on split fermions see Ref. 32.

3.2. Mediation of supersymmetry breaking via dimensions (gaugino mediation)

extra

Until now we have considered non-supersymmetric theories, and were trying to solve the hierarchy problem using large extra dimensions. In the following two examples we will assume that the hierarchy problem is solved by supersymmetry, and ask the question whether extra dimensions and branes could still play an important role in particle phenomenology. First we will discuss the issue of mediation of supersymmetry breaking via an extra

634

dimension, and then discuss how to break symmetries via the geometry of the extra dimension. Assuming that the hierarchy problem is solved by supersymmetry, the most important issue would then be how supersymmetry is broken. In the MSSM (the minimal supersymmetric standard model — see the lectures by Carlos Wagner) 33 the soft supersymmetry breaking parameters are usually inputted by hand, parameterizing our ignorance of what exactly the supersymmetry breaking sector is. However, most of the arbitrary parameter space for the supersymmetry breaking parameters is excluded by experiments. For example, if the susy breaking masses for the scalar partners of the fermions form an arbitrary mass matrix, then there would be large flavor changing neutral currents generated at the one loop level, contributing for example to K — K mixing. In the SM the K — K is suppressed by the well-known GIM mechanism, namely the one loop diagram d

"•

^

"•

wS s

•>

^

3

"•

s

Sw »>

^

*>

d

is proportional to leading order to the off-diagonal element of (V^V)CKM, which vanishes due to the unitarity of the CKM matrix. However, in the MSSM due to the presence of the superpartners of both the quarks and the gluino there is an additional vertex of the form

which is proportional to VQKMM% and is therefore not suparksVcKM, pressed, unless the mass matrix of the squarks themselves is close to the unit matrix, in which case the ordinary GIM mechanism would operate here as well. The question of why should the squarks (at least for the first two generations) be almost degenerate is called the susy flavor problem (see much more on this in the lectures by Ann Nelson). 34 Here we will show that extra dimensions can be used to transmit supersymmetry breaking to the SM in a way that this susy flavor problem could be resolved. There are two prominent proposals for this: anomaly mediation 35 (proposed by Randall and Sundrum, simultaneously to a similar proposal by Guidice, Luty, Murayama and Rattazzi), and gaugino mediation 36 (proposed simultaneously by D.E.Kaplan, Kribs and Schmaltz and by Chacko, Luty, Nelson

635

and Ponton). Anomaly mediation involves supergravity in the bulk of extra dimensions, and therefore is technically more involved. Here we will only consider proposal for gaugino mediation of supersymmetry breaking following the paper by Kaplan, Kribs and Schmaltz in Ref. 36, and for anomaly mediation we refer the reader to the original papers (Refs. 35 and 37). In both anomaly mediated and gaugino mediated scenarios the main assumption is that the MSSM matter fields are localized to a brane along an extra dimension, on which supersymmetry is unbroken. Supersymmetry is only broken on another brane in the extra dimension (it is not broken in the bulk either), and is transmitted to the MSSM matter fields via fields that live in the bulk, and thus couple to both the susy breaking brane (the hidden brane) and the visible MSSM brane. The difference between the two scenarios is what those bulk fields are that transmit the supersymmetry breaking. In the anomaly mediated scenario one has pure supergravity in the bulk, while in the gaugino mediated scenario the MSSM gauge fields live in the bulk, and thus the gauginos of the MSSM' will directly feel supersymmetry breaking, and will get the leading supersymmetry breaking terms. The MSSM scalars, since they are localized on a different brane will only get susy breaking masses via loops in the extra dimension, and will be therefore suppressed compared to the gaugino masses. The arrangement of fields for the gaugino mediated scenario is given in the figure below:

MSSM gauge fields

MSSM matter

StTSY

As mentioned before, these models have been formulated in the presence of a single extra dimension. In order to be able to proceed with the discussion of the gaugino mediated model, we need to first formulate supersymmetry in 5D. The issue is similar to the discussion of fermions in 5D when we were considering the split fermion model: since in 5D the smallest spinor is the 4 component Dirac spinor, in 5D the smallest number of supercharges must be 8. (As a reminder, in 4D the smallest supersymmetry algebra corresponds to the situation when there is a single complex

636

two-component Weyl spinor supercharge, which means there would be four real components. Thus N = 1 susy in 4D corresponds to 4 supercharges.) Since in 5D the smallest spinor has 8 real components, N = 1 supersymmetry in 5D is twice as large as in 4D, and the dimensional reduction of the 5D theory would correspond to N = 2 in 4D. For example, the 5D vectormultiplet would have to contain a massless 5D vector, and also a Dirac spinor. Since the Dirac spinor has four components on-shell, and the 5D massless vector 3, there needs to be an additional massless real scalar in the vectormultiplet, which is then: (A M ,A,$), where AM is the 5D vector, A is a Dirac spinor, and $ is a real scalar. When reduced to 4D the vector will go into a 4D vector plus A5 which is a scalar, and the Dirac fermion into two Weyl fermions. The A5 together with $ will form a complex scalar, and we can see that from the 4D point of view the 5D vector multiplet is a 4D vector multiplet plus a 4D chiral superfield in the adjoint (or equivalently from the 4D point of view an N = 2 vector superfield, which is exactly a vector plus a chiral superfield). This would mean that we would get two gauginos in the 4D theory, if 5D Lorentz invariance is completely intact. However, when one has branes in the extra dimension, 5D Lorentz invariance is anyways broken. One of the simplest ways of implementing the breaking of the 5D Lorentz invariance is to compactify the extra dimension on an orbifold instead of a circle. The meaning of an orbifold is to geometrically identify certain points along the extra dimension, and then require that the bulk fields have a definite transformation property under this symmetry geometric symmetry. We will discuss such orbifolds in much more detail in the next subsection, and there is also a whole lecture about them in these volumes by Mariano Quiros. 38 For now we will simply compactify the extra dimension on an interval (an S1 /Z2 orbifold) rather than a circle, and require that the bulk fields are either even or odd under the Zi parities y —> —y, and L — y^L + y. This is basically a fancy way of saying that we require that the bulk fields satisfy either Dirichlet or Neumann boundary conditions. A convenient choice of these Z2 parities for the bulk 5D vector superfield discussed above is (All,XL)-*(A(t,XL), (A5,$,\R)^-(A5,$,\R).

(3.21)

This means that A^ and \L satisfy Neumann boundary conditions at

637

y = 0, L while the other fields Dirichlet boundary conditions. Therefore, the KK expansions for these fields will differ from the usual expansion on a circle. The modes for the various fields will be of the form: . . ny A^,\L

~COSTT— ,

A5,$,Afi~sin^.

(3.22)

Here the mass of the n-th mode is as usual m 2 = n 2 7r 2 /L 2 . We can see, that for n = 0 the wave functions of the odd modes are vanishing. Thus the orbifolding procedure effectively "projects out" the zero mode for the states that have odd parities (that is the ones that have a Dirichlet be). Therefore, the zero.mode spectrum is no longer necessarily vectorlike, but orbifold compactifications can lead to a chiral fermion zero mode spectrum. What we see is that with the above choice of parities the zero modes will only appear for the fields that also appear in the MSSM. Thus the boundary conditions take care of the unwanted modes as compared to the MSSM! The Lagrangian of the gaugino mediated model will then schematically be of the form: L —

dy[L5

+ S(y)LMSSMmatter

+ S(y — L)Lsu3ybreaking\

.

(3.23)

We will not specify the dynamics that leads to supersymmetry breaking on the hidden brane, but rather assume that there is a field S which has a supersymmetry breaking F-term: (S)=F62.

(3.24)

Since the gauginos live in the bulk, they can couple to this susy breaking field via the operator d20^WaWa + h.c. (3.25) / What should be the suppression scale M in the above operator be? We have assumed, that the gauge fields (contrary to the large extra dimensional scenario) live in the bulk. This means that we have a 5D gauge theory. However, the 5D gauge coupling is not dimensionless, and therefore the 5D gauge theory is not renormalizable. This simply means, that at some scale M the 5D gauge interactions will become strong, above which the perturbative definition of the theory should not be trusted. However, below that scale the theory is a perfectly well-defined effective field theory, which however has to be completed in the ultra-violet (above the scale M).

638

The real question is .how the scale M is related to the length of the extra dimension and the 4D gauge coupling. The scale at which the theory becomes strongly coupled is where the effective dimensionless coupling g\M becomes as large as a loop factor, ^

~ 1.

167T 2

(3-26) V

'

However, as we have seen already the effective 4D gauge coupling, which is of order one since these are the visible MSSM gauge couplings are given by

4 = 4 = 0(1)-

(3-27)

Combining these two equations we get that ML ~ 16ir2 ~ (9(100), which implies that the scale M i s a factor of 100 or so larger than 1/L, the scale that sets the mass of the first KK mode. This means, that there is a hierarchy between M and 1/L, and between these two scales it is justified to use an effective 5D gauge theory picture. Coming back to the size of the supersymmetry breaking gaugino mass, substituting the vev of S into (3.25) we get that the gaugino mass term is F (3.28) f AA|„ =L . ~M~r This is a 4D mass term (since it is on the surface of the susy breaking brane only) for a 5D bulk field A. In order to get the effective 4D mass term for the zero mode, we have to rescale the 5D field to have a canonically normalized kinetic term for the zero mode. Since the 5D kinetic term is just the usual Lkin

-I

= / Aid M 7 Acf x,

(3.29)

after integrating over the extra dimension we get for the zero mode that A(5)Za = A(4). Then the gaugino mass for the correctly normalized 4D field will be F 1 mgaugino

= ~J ] ~ ^ '

(3.30)

Note, that the physical mass is suppressed by the size of the extra dimension, and this is the consequence of the fact that the mass is localized on a brane while the field extends everywhere in the extra dimension. (3.30) implies that at the high scale M all gauginos have a common mass term. The most important question is what the magnitude of the masses of the scalar partners of the SM fermions will be, since as we have seen at the beginning of this section, this is what leads to the susy flavor problem.

639

Qualitatively we can already see its features: since we have assumed that the SM matter fields are on the visible brane, they will not directly couple to the susy breaking sector. Thus they will feel susy breaking effects only through loops that go across the extra dimension and connect the scalar fields on one brane with the susy breaking vev on the other brane. This means, that this operator should be a loop factor smaller, than the gaugino mass term. However, even this would be too large, if arbitrary flavor structure was allowed. The important point is that the loops in the extra dimensions involve the gaugino fields, which couple universally (as determined by the gauge couplings) to the MSSM matter fields. Therefore, just as in the so called gauge mediated models (which we have not discussed here since those are not relying on extra dimensions) the corrections to the soft scalar masses will be flavor universal, and thus this setup has a chance of solving the susy flavor problem. Even though we basically already roughly know the answer for what the size of the scalar masses is going to be, it is still illustrative to go through the calculation in detail, since this should also teach us how to handle diagrams that involve loops in the bulk. Before we go into the detail of the calculation, let us just make one comment: every extra dimensional loop which involves propagation from one brane to the other must be necessarily UV finite. The reason is that usually divergences appear when the size of the loops in a usual Feynman diagram shrinks to zero. However here the size of the extra dimension provides a UV cutoff, and the integral will be finite (the loop can not shrink to zero in coordinate space since the fields have to propagate from one brane to the other.) The diagram that one needs to calculate is illustrated below:

Since we know that the fields have to propagate from one brane to the other and back, it is useful to use the propagators in the y coordinate, however, for the 4D propagators we want to use the usual momentum

640

representation. Therefore we need the propagator in a mixed q, yi,yf representation. Even though what we really need is the propagator for fermions, since in the above loop it is the gaugino that is propagating through the bulk, it is instructive to first calculate the scalar propagator P(q2,a,b), which corresponds to a scalar with four momentum q2 propagating from y = a to y = b. Since the bulk scalars are odd under the Z%, their wavefunctions are <My) = y § s i n ^ .

(3.31)

The bulk propagator can then be written in terms of the wave functions as

P(q\a,b) = Y6* (a)4> (b) / , rn n r / r mm "

Smn

o .

9

-z? m (—) m hr)?T3-

(332)

Here pn = ^ , and one can convert the sum to an integral via An — dp L/ir to get

P(?,a,b) = -1- r

dpMbp)Map)

^ _ 1 -i*-.,,.

(333)

A similar calculation for the fermions yields P(g2, 0, L) ~

2PLq

^e-iL,

(3.34)

Evaluating the diagram above we get 5

' ( S O Y ^^^W^L)CPT{q,L,L)C^P{q,L,Q).

(3.35)

Plugging in the propagator and evaluating the integral we get for the scalar mass " l - ^ 2l ^ 2 l 16TT \M )

•h3 = ^ S 2 < L

16TT

(3.36)

Thus we find that as expected the scalar mass is flavor universal (it depends only on the gauge coupling) and a loop-factor suppressed compared to the gaugino mass. In the end, the free parameters of the gaugino mediated model are the size of the extra dimension L, the unified gaugino mass Mi, the ratio of Higgs VEVs tan/?, and the sign of the /x-term. With these input parameters one has to start running down the above susy breaking soft masses from the

641

scale M to the electroweak scale, and find the appropriate physical mass spectrum. This gives a phenomenologically acceptable mass spectrum, for details see Ref. 39. 3.3. Symmetry

breaking via

orbifolds

We have seen in the previous section, that compactification can break some of the symmetries of the higher dimensional theory. The particular example that we discussed was the breaking of supersymmetry from 8 supercharges (N = 2 in 4D) to just 4 supercharges (N = 1 in 4D). In this section we will consider examples both of gauge symmetry breaking and of supersymmetry breaking via boundary conditions in extra dimensions. Such symmetry breakings have been considered by the string theorists for a long time, see for example. 40 Here the first example we will discuss in detail will be the breaking of the grand unified gauge group as proposed by Hall and Nomura, 41 see also Refs. 42-45 while the discussion of the breaking of supersymmetry is based on the model by Barbieri, Hall and Nomura, 46 see also Ref. 47. We will use orbifold projections on a theory on a circle to obtain the symmetry breaking boundary conditions, even though these can be also formulated very conveniently just by considering a theory on an interval. Through out this section we will just consider a single extra dimension, and try to find what are the orbifold projections one can do to arrive at the final set of boundary conditions. By orbifold boundary projections we mean a set of identifications of a geometric manifold which will reduce the fundamental domain of the theory. Let us first start with an infinite extra dimension, an infinite line R, parametrized by y, —oo < y < oo. One can obtain a circle 5 1 from the line by the identification y —> y + 2TTR, which we can denote as "modding out the infinite line by the translation T, R —> S1 = R1 jr. This way we obtain the circle. Another discrete symmetry that we could use to mod out the line is a reflection Z2 which takes y —> —y. Clearly, under this reflection the line is modded out to the half-line R1 —> R1 /Z2If we apply both discrete projections at the same time, we get S1 /Z2, the orbifold that we have already used in the previous section when we discussed gaugino mediation. This orbifold is nothing else but the line segment between 0 and irR.

642

Let us now see how the fields
= T-V(y +

2-KR),

Z
(3.37) (3.38)

where T and Z are matrices in the field space corresponding to some symmetry transformation of the action. This means that we have made the field identifications v(y + 2TrR)=Ty>(y),

(3.39)


(3.40)

Again, Z and T have to be symmetries of the action. However, Z and T are not completely arbitrary, but they have to satisfy a consistency condition. We can easily find what this consistency condition is by considering an arbitrary point at location y within the fundamental domain 0 and 2irR, apply first a reflection around 0 Z(0), and then a translation by 2TTR, which will take y to 2irR — y. However, there is another way of connecting these two points using the translations and the reflections: we can first translate y backwards by 2TTR, which takes y —> y — 2TTR, and then reflect around y = 0, which will also take the point into 2irR — y. This means that the translation and reflection satisfy the relation: Z(0)T~1(2WR)

=

T(2TTR)Z(0).

(3.41)

When implemented on the fields

or ZTZ = T _1

(3.42)

which is the consistency condition that the field transformations Z and T have to satisfy. As we have seen, the reflection Z is a Z2 symmetry, and so Z2 = 1. T is not a Z2 transformation, so T 2 ^ 1. However, for non-trivial T's T ^ 1 (T is sometimes called the Scherk-Schwarz-twist) one can always introduce a combination of T and Z which together act like another Z2 reflection. We can take the combined transformation T(2TTR)Z(0). This

643

combined transformation takes any point y into 2irR — y. That means, that it is actually a reflection around TTR, since if y = irR — x, then the combined transformation takes it to TTR + x, so x —> —x. So this is a Z2 reflection. And using the consistency condition (3.42) we see that for the combined field transformation Z' = TZ Z'2 = (TZ)2 = (TZ)iZT-1)

= 1,

(3.43)

so indeed the action of the transformation on the fields is also acting as another Z2 symmetry. Thus we have seen that the description of a generic S1 jZ-2, orbifold with non-trivial Scherk-Schwarz twists can be given as two non-trivial Z
2JIR

3TTR Figure 2. The action of the two Z2 reflections in the extended circle picture. The fundamental domain of the S1 /Z2 orbifold is just the interval between 0 and TTR, and the theory can be equivalently formulated on this line segment as well.

Breaking of the grand unified gauge group via orbifolds in SUSY

GUT's

644

One of the main motivation for considering supersymmetric gauge theories (besides the resolution of the hierarchy problem) is the unification of gauge couplings. The SM matter fields seem to fall nicely into SU(5) representations 10 + 5. This has motivated people in the 70's to consider theories based on an SU(5) gauge symmetry, which is broken down at high scales to SU(3)xSU(2)x U(l). If one calculates the running of the SM gauge couplings in the non-supersymmetric SM, they get close to each other at a scale ~ 10 14 GeV, but don't really unify to the level required by the experimental precision of the three gauge couplings. However, in supersymmetric theories the precision of unification of the couplings is much better than in the non-supersymmetric case and the scale of unification is pushed to somewhat larger values, of order 2 • 10 16 GeV. This is a consequence of the modification of the beta functions that determine the running of the couplings due to the addition of the superpartners at some relatively low scale. The other important consequence of the higher unification scale is that proton decay mediated by the exchange of gauge bosons that transform both under SU(3) and SU(2) (the so-called X, Y gauge bosons) will be more suppressed, since the decay rate depends on the fourth power of the X, Y gauge boson mass, which is proportional to the unification scale. Thus it looks like a complete supersymmetric GUT model could be completely realistic without fine tuning. There is however one important conceptual problem in SUSY GUT's (and also in ordinary GUT's in general): the doublet-triplet splitting problem. In order to get the set of beta functions in a SUSY GUT which leads to a sufficiently accurate prediction of the strong coupling at low energies one needs to have the beta functions of the MSSM. As explained before, the SM matter fields fall into complete SU(5) multiplets, however the two Higgs doublets of the MSSM do not. Thus if one takes the GUT idea seriously one needs to embed the Higgses into some complete SU(5) multiplets, and then somehow make the extra fields much heavier than the doublet. However, this seems very difficult: one needs to give a mass to the extra fields that are 10 14 times larger than the doublets. For example in the simplest case the two Higgs doublets are embedded as Hu = 2 —> 5, #d = 2* —> 5. In this case we had to add an extra triplet and antitriplet of SU(3), which have to be very heavy, while the doublets from the same multiplet light. This is another naturalness problem that is specific to GUT's. If the mass of the triplet was too low, the beta functions would change, and unification of couplings would not occur. Even if one can somehow arrange naturally for the triplets to be heavy (there are some nice natural solutions to the

645

doublet-triplet splitting problem), they still contribute to proton decay at a rate t h a t is usually too large. T h e reason is t h a t due to supersymmetry and grand unification the fermionic partners of heavy color triplet Higgses s necessarily couple to the SM fermions and the scalar partners via a Yukawa coupling term. Since the triplets have to be massive there is necessarily a diagram of the form

\

-

-

' '

which is baryon number violating. Here the dotted line is a squark, and Hit2 are the color triplet Higgsinos. T h e squark-quark-color triplet Higgsino vertex must exist, since the ordinary Yukawa coupling must exist, and this is the grand unified extension of the superpartner of the ordinary Yukawa vertex. T h e cross denotes the mass t e r m between the two color triplet Higgsinos, which should also exist to lift the mass of these triplets to above the G U T scale. Now one can take this diagram and dress it u p with some more couplings t h a t t u r n the squarks back t o quarks to get a contribution to proton decay:

T h e additional vertices used in this diagram must exist due to supersymmetry, as well as the gaugino mass insertions since the gauginos have not been observed. In extra dimensional theories an S1 /Z? orbifold structure with a Z-i x Z'2 reflection structure could give a nice explanation for the absence of the light triplets. In order t o s t a r t building u p an e x t r a dimensional G U T , we take a 5D SU(5) gauge theory with the minimal amount of supersymmetry N = 1 in 5D (meaning eight superchages) in the bulk. T h e n by definition we need to have the 5D N = 1 vector superfields in the bulk, while for the m a t t e r

646

fields one can choose if they are brane fields or bulk fields. For now we will choose the fermionic matter fields to be brane fields, while the Higgs fields to be bulk fields. One of the two Zi reflections will act exactly as in the case of gaugino mediation: its role is simply to split the 4D N = 2 multiplets into a 4D N — 1 vector superfield which has a zero mode, while the 4D TV = 1 chiral superfield which does not have a zero mode:

In order to have a grand unified extra dimensional SUSY GUT, the Higgs doublets will live in 5D chiral superfields in the 5 representation of SU(5), which correspond to a 4D N = 2 hyperrnultiplet. Since we have two Higgses in the MSSM, we will effectively introduce four 4D chiral superfields in the 5 of SU(5):

if = ( 5 , 5 ) ,

(3.44)

F ' = (5',5')-

We will use the second Z2 reflection to break the SU(5) GUT symmetry down to the SU(3)xSU(2)xU(l) of the MSSM. We will do this by picking the representation matrix for Z'2 on the fundamental of SU(5) to be nontrivial: ( Z'2:

\ 5

V )

(

/

\ (3.45)

-

V

+7 V J

Once the action of the second reflection is fixed on the fundamental 5 of SU(5), we can also determine its action on the adjoint 24. Since an adjoint is a product of a fundamental and anti-fundamental, the action of Z2 will be such, that the various components of the adjoint matrix pick up the

647

following signs:

( Z'2:

\ I

(3.46)

24

V

/

+/

This implies, that the fields at the position of the X, Y gauge bosons will be odd under this second reflection, while the others will be even. The fields that are even under both Z2 reflections will have a zero mode, while the others will not. We can now list the parities of all the bulk fields and find out what kind of KK towers these fields will have. The bulk fields are the following. The 5D vector superfield of SU(5) consists of the 4D vector superfield and the 4D chiral superfield in the adjoint: /

(

\ V

SM

v^

a

\

x

x

V" y

ya V SM

(3.47)

xaJ

The bulk Higgs fields decompose as # = (3 + 2,3 + 2),

H' = (3' + 2', 3' + 2').

(3.48)

With this notation we can give the two Z2 charges of all fields, and the corresponding KK masses: mode KK mass wave function 2n os^f ^SM'2,2 (+,+) R (2ra+l)y a 2n+l (3.49) n = 0,l,2,... A ,3,3' (+.-) R R 2n+l (2n+l)y (-,+) x , y , 3 , 3 ' 2n+2 cos R R X,Y,2,2' (-,-) sin 2ny R We can see from this table that with the charge assignments as above we would get a zero mode only for the fields that are present in the MSSM as well, while the KK towers for the other fields would start at a non-zero value, which is determined by the radius of the extra dimension. The structure of the KK towers for the various fields is depicted in Fig. 3.3. From this figure we can see that by simply assigning different Z2 transformation properties to the doublet and the triplet in 5 one can achieve the doublet-triplet splitting, due to the fact that the boundary condition for the triplet will not allow the existence of a triplet zero mode, while there is one for the doublet. Also note, that since 3 and 3 have different wave functions, there can be no mass term of the form 33, which implies {Z2,Z'2)

648

that the dimension five proton decay operator discussed above is vanishing, since in that diagram the mass insertion (the cross) is vanishing. Instead the 3 and 3 get masses with 3' and 3' respectively, which will not give any contribution to proton decay. The vanishing of the 33 mass term is a consequence of N = 2 supersymmetry. However, at the brane where N = 2 supersymmetry is broken to N = 1, one could in principle add an operator that would include a mass term for these fields. These mass terms can nevertheless be forbidden by requiring that a (7(1)R global R-symmetry is obeyed, under which the charges of the fields are given by

U(1)R

Figure 3. Nomura.

EH H W H' Tw Ft, 0 0 0 2 2 1 1

The KK towers for the various modes of the orbifold GUT theory of Hall and

Since we are talking about a GUT model, it is important to make sure that the model would actually predict the unification of the gauge couplings. Since the GUT symmetry is broken at one of the endpoints of the interval, there can be a brane-induced kinetic term for the SM gauge fields on the GUT breaking brane which does not necessarily have to be SU(5) symmetric. Therefore the tree-level matching between the 4D effective SM couplings and the 5D gauge coupling is given by

649

where the first term on the right is the SU(5) symmetric bulk gauge coupling, while the second term is the (possibly) SU(5) violating brane coupling. How precisely the couplings will unify depends on the ratio of the brane induced term to bulk term. If the size of the extra dimension is reasonably large compared to the cutoff scale (for example 1/R ~ MQUT, Mcut<,ff ~ IOOMQUT) then the SU(5) violating brane term will be volume suppressed, and it is reasonable to expect unification to a good precision. Of course it is not enough to build an extra dimensional GUT model, it is also important to check that the appearance of the new states charged under the SM gauge groups (the KK modes of the various gauge and Higgs fields) do not ruin the actual unification of couplings of the MSSM. This has been shown to be the case in Ref. 41, and the reader is referred for the details of that calculation to the original paper. Super•symmetry breaking via orbifolds We have seen in the previous two models that we can use a Z^ projection under the y —> —y reflection symmetry to break N = 2 supersymmetry down to N = 1 SUSY. We have also seen that in the case of a single extra dimension there are two separate i^'s at our disposal at both ends of the interval. This yields the following nice possibility: one could use one Z^ to break one half of the supercharges, and the other Z2 to break the other half:

First Z ,

Second Z2

v\*j / The two Z2S leave a different combination of N = 1 supercharges unbroken. The two projections together completely break supersymmetry, but locally there is always some amount of supersymmetry unbroken. The full supersymmetry breaking is only felt globally. This will have the important implication that the properties which follow from local supersymmetry will be maintained in the non-supersymmetric theory. The next task is to also include the MSSM matter fields and Higgs fields into this theory. A generic matter field of the SM will be an N = 2 hypermultiplet (two fermions and two scalars), and we can assign the Z2 parities for the matter fields again such that only the one of the fermions will have a zero mode, while all the other fields from this hypermultiplet

650

will have no zero modes. For example for quarks this can be achieved by the following choice of Z 2 projections:

For the Higgs, one has two possibilities: introducing two separate hypermultiplets for the two MSSM Higgs fields, which is the more straightforward but less exciting possibility, or introduce just a single Higgs doublet hypermultiplet which contains both of the MSSM Higgs doublets. We will consider the latter possibility, and then use the projections

First Z 2

In this case, there will only be a single Higgs zero mode, contrary to the MSSM which has two Higgs doublets. So the zero mode spectrum really reproduces that of the SM, and not that of the ordinary particles of the MSSM. N = 2 supersymmetry odes not allow a Yukawa-type coupling between hypermultiplets. This means that in the bulk one can not write down the appropriate couplings necessary for generating the fermion masses from the Yukawa couplings to the Higgs field. However, since on the fixed points N = 2 supersymmetry is broken to N = 1, these couplings can be added on the fixed points. One may wonder how this could be possibly done in a supersymmetric theory with just one Higgs field. The point is that since it is a different set of supercharges that remains unbroken at the two fixed point, one needs to use a different splitting of the bulk Higgs hypermultiplet into N = 1 chiral multiplets on the two fixed points. Thus from the full bulk Higgs hypermultiplet

#ct

H #

(3.52)

651

we form the chiral multiplets H

Hu

H*

Hd

(3.53)

Note, that it is the same scalar component (the one with a zero mode) appearing in both chiral multiplets, while one has a different fermionic partner corresponding to them depending on which set of supercharges are unbroken. Thus we can write down the following superpotential terms at the fixed point yielding the required Yukawa couplings:

'—Xd H^D-t^H^E

2JIR

37CR

The tree-level Higgs potential will be fixed just as in ordinary supersymmetric theories. Without a bulk mass term the form of the potential is just determined from the D-terms, and is given by: Vtr

92+9'2

H\

(3.54)

This on its own determines that the Higgs mass has to be light, as in any supersymmetric theory. Electroweak symmetry breaking can then happen radiatively, due to the top quark loops. Since the top Yukawa coupling is localized at one of the fixed points, the resulting contribution to the Higgs potential can also only be a term at the fixed point. Since at least some of supersymmetry is locally unbroken everywhere, the contribution can not be quadratically divergent. In fact, the contribution to the potential from the top quark loop is actually finite. However, there can be localized U(l) Fayet-Iliopoulos terms at the fixed points which can be quadratically divergent, which will contribute to the Higgs potential. Also, one can add a bulk mass term for the Higgs hypermultiplet, which also obviously contributes to the Higgs mass. For further details on the prediction of the physical Higgs mass of this model see Refs. 46, 48, 43.

652

4. Warped Extra Dimensions Until now we have considered flat extra dimensions, that is assumed that whatever gravitational sources there are present, their effects will be negligible for determining the background metric, that is we ignored the backreaction of gravity to the presence of the branes themselves. If the tensions of the branes are small, then this is a good approximation. However, interesting situations may arise, if this is not the case. One can immediately see that taking the reaction of gravity to brane sources into account could be very important. The reason is that if one has a 4D theory with only 4D sources, these will necessarily lead to an expanding universe. However, if one has 4D sources in 5D, one can balance the effects of the 4D brane sources by a 5D bulk cosmological constant to get a theory where the effective 4D cosmological constant would be vanishing, that is the 4D Universe would still appear to be static and flat for an observer on a brane. 4 The price to pay for this is that the 5D background itself will be curved, which is clear from the fact that one had to introduce a bulk cosmological constant. Thus one can "off-load curvature" from the brane into the bulk, 4 ' 49 and keep the brane to be still flat by curving the extra dimension. Such curved extra dimensioned are commonly referred to as "warped extra dimensions". This possibility has been first pointed out by Rubakov and Shaposhnikov.4 This shows that for example the cosmological constant problem will be placed into a completely different light from an extra dimensional point of view: in 4D one had to explain why the vacuum energy is very small, while in higher dimensions one has to explain why the vacuum energy of the SM fields is exactly canceled by a bulk vacuum 49

energy. In the following we will discuss the best known and most concrete example of warped extra dimensions, the Randall-Sundrum model. 50 ' 51 First we discuss in detail how to solve the Einstein equations to find the RS background, and discuss the mass scales in the model. Then we discuss the phenomenology of various interesting models based on the RandallSundrum backgrounds. 4.1. The Randall-Sundrum

background

We will show how to derive the Randall-Sundrum solution (and the conditions under which a flat background exists). We will begin again with a finite length 50 for the extra dimension, that is we consider an S1 jZi orbifold, which by now is familiar to everyone. Since we are interested in

653

warped solutions, we will assume that there is a non-vanishing 5D cosmological constant A in the bulk. We are also interested in solutions that have the property mentioned above, that is even though the extra dimension is curved, the brane itself remains static and flat, that is it preserves 4D Lorentz invariance. This means that the induced metric at every point along the extra dimension has to be the ordinary flat 4D Minkowski metric, and the components of the 5D metric depend only on the fifth coordinate y. The ansatz for the most general metric satisfying these properties is given by: ds2 = e-A^ddidxvr)liV

- dy2.

(4.1)

The amount of curvature (warping) along the extra dimension depends on the function e~A^v\ which is therefore called the warp-factor. Our task is to find this function A(y). Note, that there are of course various ways of writing this metric, by doing various coordinate transformations. For example one could as well have gone into the coordinate system where there is an overall prefactor in front of all coordinates, which is called the conformally flat metric. This is the simplest coordinate system, and this is what we will use to find A. To go into the conformally flat frame, we need to make a coordinate transformation of the form z = z(y). The coordinate transformation should not depend on the 4D coordinates x, since those would induce off-diagonal terms in the metric. In order to ensure that the metric be conformally flat in the new frame, dy and dz have to be related by

such that the full metric in the z coordinates will be: ds2 = e~AW (dxlidxvTiliV - dz2).

(4.3)

This metric is conformally flat, so a conformal transformation (that is an overall rescaling of the metric) connects it to the flat metric: 9MN = e~A(z)
(4.4)

A very useful relation connects the Einstein tensor GMN = RMN — \gMNR calculated from g and g in any number of dimensions and for arbitrary function A (see for example): 52 GMN

=

GMN

H

— -VMAVNA

.*-*VKAVKA)

+ VMVNA

-

~gMN{<JKVKA (4.5)

654

Here the covariant derivatives V are with respect to the metric g, while on the left the Einstein tensor GMN is calculated from gMN- For us CJMN = VMN, and therefore the covariant derivative V M —> 8M, and d = 5. Using these expression we can now easily evaluate the non-vanishing components of the Einstein tensor: 3

G.55 '55 =

2

^A'2

G^ = - ^ ( - A "

iA'2) .

+

(4.6)

This takes care of the left hand side of the bulk Einstein equation GMN = K2TMN, where K2 is the higher dimensional Newton constant, in 5D it is related to the 5D Planck scale by

«2 = w

w

The 5D Einstein action for gravity with a bulk cosmological constant A is S=-

f d5x^g-(MlR

+ A).

(4.8)

One can then use the definition of the stress-energy tensor to find the Einstein equation: GMN = « TMN = ^/iTI^Miv-

(4.9)

The 55 component of the Einstein equation will then be:

The first thing that we can see is that a solution can only exist if the bulk cosmological constant is negative A < 0. This means that the important case for us will be considering anti-de Sitter spaces, that is spaces with a negative cosmological constant. Once a negative cosmological constant is fixed, we can take the root of the above equation

Al

A2

^ im'~ ' A 2

Introducing the new function e~ l - f ' / f

2

(4 n)

-

= / we get the equation

1 A =2 V o \ -3Mj? ^

(4-12)

655

from which we can read off the solution for the metric e-A&

=

1

——t, (kz + const.) 2

(4.13) v

;

where we have introduced

The constant in (4.13) can be fixed by fixing the value of the metric at some point (this is an irrelevant rescaling of the units), we choose e~A^ = 1, from which the constant is set to 1. So a simple form for writing.the metric

e

1

-A(z)

However since we are on a S1 /Z2 orbifold, the solution must be symmetric under z —> —z reflection, and therefore the final form of the RS solution can be written in the form 1_ ds2 = /;.!..! , ^2{n^dx^dxv - dz2). (4.16)

(k\z\Tiy

With this the 55 component of the Einstein equations is solved everywhere. However one needs to check whether the 4D components are also satisfied. We have seen above that these components are given by -3/2TI^V(-A" + A'2/2). This means that (unlike in the 55 component) A" will appear, which will imply that there are delta function contributions to the Einstein tensor at the fixed points. Thus (4.16) can be a solution only, if there are also localized energy densities on the brane that compensate these delta functions. This is exactly as we have expected at the beginning of this section: there can be a non-vanishing bulk cosmological constant and still have flat induced 4D space if there is some energy density on the brane that compensates. Note however, that the 55 component of the Einstein equation already determined (together with the S1 /Z2 assumption) the full solution. So the energy density on the brane necessary to have a static solution would be fine-tuned against the bulk cosmological constant. Evaluating A" we get

A

9k2

+

Ah

"=-mrw mTi^z)-6iz-z^

(4 17)

-

where z\ is the location of the negative tension brane, and so the \iv components of the Einstein tensor are 4fc2 4k(5(z) - 5{z - Zl))~ r --1 (4.18) _{k\z\ + l)2 fc|z| + l .

656

The first term (the term without the delta functions) exactly matches the bulk contribution to the energy momentum tensor from the bulk cosmological constant. Therefore the remaining delta function terms are the ones that need to be compensated by adding additional sources onto the branes. To match these sources we need to find out what the energy-momentum tensor of a brane tension term V (an energy density localized on the brane) would be. The action for this is given by [ dixV^ginduced= J

fd5xV-^L5{y) J V955

(4.19)

for a flat brane at y = 0. This implies that the energy momentum tensor is

Thus the final form of the energy-momentum tensor for a brane tension is: r

t ™

=

I ^ g ^ _Vf _V> _v> Q)e-A'H(y).

(4.21)

Therefore to satisfy the Einstein equations at the brane we need to have two brane tensions, one at each end of the interval (at the two fixed points), and so we need the equality 3

4k{5{z) ~ 5{z - zi)) k\z\ + l

2M3

V05(z) + V15(z-z1)) k\z\ + 1

(4.22)

This implies that the two brane tensions at the two fixed points will have to be opposite, and given by VQ = -Vi = 12/cM,3.

(4.23)

Plugging in the expression for k we obtain that the bulk cosmological constant and the brane tensions have to be related by

Thus there is a static flat solution only, if the above two fine tuning conditions are satisfied. At this point it is not clear why we ended up with two fine tuning conditions. We could have expected at least one fine tuning, related to the vanishing of the 4D cosmological constant. However, at this point the meaning of the second fine tuning condition is obscure, but we will find out the reason behind the second fine tuning shortly in Sec. 5.2.

657

Now that we have found that the RS solution in the z coordinates where the metric is conformally flat is ds2 =

*

(7fr„ds'W - dz2),

(4.25)

we can ask what the metric in the original y coordinates would be. The reason why this is interesting is that y is the physical distance along the extra dimension, since in that metric there is no warp factor in front of the dy2 term. Since the relation between z and y was given by

we get that (by choosing y — 0 to correspond to z = 0): 1 {k\z\ + l)2

-

P -2fc|y|

(4.27)

and so the RS metric in its more well-known form is finally given by: ds2 = e-2kWdx>ldxvriiaj

- dy2.

(4.28)

Let us now discuss what the physical scales in a theory like this would be, if the matter fields were localized on one of the fixed points. First we assume this to be the brane with the negative tension (the brane where the induced metric is exponentially small compared to the other fixed point). Consider thus a scalar field (for example the Higgs scalar) on the negative tension brane. Its action would be given by gHiggs = I
- V{H)],

= A[(fftH) - v2)2}.

V(H) (4.29)

If the size of the extra dimension is b, then the induced metric at the negative tension is given by

CV* = e"2feV-

(4-30)

Plugging this in for the above action we get that the action for the Higgs is given by SHiggs =

I d4xe-4kb^2kbr]^dnHdvH

_ A ( j f f t f f _ y2^

(43^

We can see that due to the non-trivial value of the induced metric on the negative tension brane the Higgs field will not be canonically normalized.

658

To get the action for the canonically normalized field one needs a field redefinition H = e~kh H. In terms of this field the action is sHi9gs =

J dAx[rtflvdtlHdl'H

- \[(H*H) - (e-kbv)2}2}.

(4.32)

This is exactly the action for a normal Higgs scalar, but with the VEV (which sets all the mass parameters) "warped down" to VHiggs = ^

k h

V.

(4.33)

This shows, that all mass scales are exponentially suppressed on the negative tension brane, but not on the positive tension brane. Therefore, the positive tension brane is often also referred to as the Planck-brane, since the fundamental mass scale there would be unsuppressed of the order of the Planck scale, while the negative tension brane is referred to as the TeVbrane since the relevant mass scale there is TeV. The curvature in the bulk leads to a redshifting of all energy scales away from the positive tension brane. This means that if one introduces a bare Higgs VEV (and thus also Higgs mass) of order the 5D Planck scale M* into the Lagrangian, the physical Higgs mass and VEV will be exponentially suppressed. In order to find out, whether this exponential suppression is interesting we need to also find out what the effective scale of gravity (the 4D Planck scale) would be in this model. For this we need to find out how the effective 4D gravitational action depends on the radius b of the extra dimension. To find this out quickly, for now we will just assume that the 4D graviton h^ is embedded into the full 5D metric as ds2 = e-2fclyl[Vlu/ + h^x^dx^dx"

- dy2.

(4.34)

The subject of the next section will be mostly to verify this and to analyze the consequences of this equation in more detail, but for now we just accept (5)

this result. In this case the 5D Ricci tensor R^ will contain the Ricci tensor calculated from h^: R^u C R^J. The reason behind this is that R^v is invariant under a constant rescaling of the metric. Plugging in the explicit metric factors that appear in the action we get that the relevant piece of the 5D Einstein-Hilbert action containing the 4D metric is S = -Ml

J (Pxy/gR&

D -Ml

J e-4fcW y/^)e2k^R^d5x.

(4.35)

From this we can read off the value of the 4D effective Planck scale: fV=b

M2Pi = Ml / Jy=—b

iw-3

e~2k^dy

= -f-{\%

e~2kb).

(4.36)

659

For moderately large values of b this expression barely depends on the size of the extra dimension, and it is completely different from the analogous expression (2.9) for large extra dimensions. It also shows, that if all bare parameters M*, A, VQ, VHiggs are of the same order and of the order of the Planck scale, then while the scale of the Higgs VEV (the weak scale) gets exponentially suppressed, the scale of gravity itself will remain of the order of the Planck scale. This means that with a moderately large b one can naturally introduce an exponential hierarchy between the weak and the Planck scales! This is exactly what one would like to achieve when solving the hierarchy problem. So we can see that the Randall-Sundrum model has the possibility of giving a completely new explanation to the hierarchy problem. We will have a much better understanding of how the RS model solves the hierarchy problem after the next section, where we consider the behavior of gravity at different locations along the extra dimension in this model.

4.2. Gravity

in the RS

model

To find out more why the RS model solves the hierarchy problem, we need to study the properties of gravity in this AdS background in more detail [51]. Thus we really need to find the KK decomposition of the graviton in this curved background. Normally, when one has a flat extra dimension on a circle, one expects that there would be a graviton zero mode, a scalar zero mode and a vector zero mode, making up for the five degree of freedom in the 5D massless graviton, while at the massive level there would just be massive 4D gravitons, which also have 5 degrees of freedom. However, due to the y —> — y orbifold projection the zero mode of the vector will be eliminated. The reason is that from the generic form of the metric ds2 = e-^Mg^dx^dx"

+ A^dx^dy - b2dy2,

(4.37)

where g^u parametrizes the graviton, A^ the vector and b the scalar fluctuations, one can see that since ds2 is symmetric under y -^ —y, AM has to be flipping sign, and thus can not have a zero mode. Therefore at the zero mode level one expects only a graviton and a scalar, while at the massive level only gravitons. First we will focus exclusively on the graviton modes setting the scalar fluctuation to zero at first, and then later we will discuss the relevance of the scalar mode. In order to find the KK expansion of the graviton modes, we will go to the conformal frame for the metric, and parametrize the graviton fluctua-

660

tion by ds2 = e~A{z) [(77^ + h^x, z))dx»dxv

- dz2] .

(4.38)

Clearly the most general graviton fluctuations can be parametrized like this. Now we use our formula (4.5) that relates the Einstein tensor of conformally related metrics. We take as the metric g the metric following from (4.38), while the conformally related metric as g — e~A^g. Then the Einstein tensors are related by -VMAVNA

GMN=GMNH

+ VMVATA -

CJMN^RVRA

(4.39)

-VRAWRA)

in general d dimensions. Next we impose the gauge choice for the perturbations fc£ = dM = 0,

(4.40)

which is usually called the RS gauge choice. We will get back to the possibility of always choosing such a gauge later. Since g is the fluctuation of a flat metric, the linearized Einstein tensor for this case G^„ is written in every book on general relativity. This is however not everything, since the covariant derivatives V is evaluated with respect to the perturbed metric r\^v + h^, and so the Christoffel symbols are not all vanishing (as they would be for the flat background). Taking these extra terms in the derivatives in (4.39) into account is a tedious but doable work, which we will leave to the reader as an exercise. Finally, we need the perturbation of the right hand side of the Einstein equation (the perturbation of the energy-momentum tensor TMN), which is quite trivial since the only change is due to gMN —• 9MN + 5gMN- The perturbed Einstein equation is of course SGMN = K?&TMNI which after putting all terms together yields the linearized Einstein equation in a warped background to be -\dRdRh^+'^dRAdRhtlv

= Q.

(4.41)

In order to get to an equation that we have more intuition about, we can transform it into the form of a conventional one-dimensional Schrodinger equation, by rescaling the perturbation as

V =e

d-

T

V-

(4-42)

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This field redefinition was chosen such, that the first derivative terms in the differential equations cancel, so one is really left with a second derivative (kinetic energy) term and a no derivative potential term. The form of the equation in the new basis is

-\dRdRh^

+

(d

2

32

d 2

H ) -d XRAX..A Ad A

H ~ xRt —-d d R A V , = 0.

R

(4.43)

The second derivative is given by dRdR — —\JX ~ V^, where the 4D box operator is defined as • = —f^d^dy. Finally, we separate the variables as hnV(x, z) = hIJlI/(x)iS(z), and require that the H e a four dimensional mass eigenstate mode Oh^ = m2h^u. Then the final Schrodinger type equation that we get for the KK modes is given by 51 ' 53,54 ' 55

-d2^ + (^A'2 - ^AA * = mH.

(4.44)

Thus the Schrodinger potential is given by V(z) = ^A'2 In our case e~A^ is

= 1/(1 + k\z\)2,A

- \A".

= 21og(fc|z| + 1), and so the potential

V{z)=*-^ 1 j

4 (1 +

(4.45)

fcM)2

« £ l l + jfelzl

( 4 .4 6 ) [

'

This potential has the shape of a volcano, since there is a peak in it due to the first term, but then there is a delta function which is like the crater of the volcano. This is why it is commonly referred to as the "volcano potential". Note, that for any warp factor A(z) one can always define the operators Q = dz + | A ' and Q^ = —dz + | A ' , such that the equation is written as Q^QW = TO2\[/.53>55.54 This means, that we are again facing a SUSY quantum mechanics problem, which can be solved as in the case when discussing the localization of fermions. Therefore, as we know there is always a zero mode solution to this equation given by

^ = e-iA^=

I

(4.47)

(l + k\z\)4 The existence of a zero mode is not very surprising, since we have not broken 4D Lorentz invariance, so we expect a massless 4D graviton to exist. The interesting question to ask is under what circumstances will this graviton

662

zero mode actually be normalizable. For this we have to evaluate the usual quantum mechanical norm fzo

rzo

rza

i

/ dz\*0\2 = / dze-*'2AW = / dz (4.48) Jo Jo Jo (l + k\z\)2 If one is wondering whether this is really the relevant normalization, one can simply follow all the field and coordinate redefinitions backwards and find that the coefficient of the 4D kinetic term of the graviton zero mode is indeed proportional to this factor. Above ZQ is the size of the extra dimension in the z coordinates. The most important comment about the above integral is that it is converging even in the limit when the size of the extra dimension becomes infinitely large, ZQ —> oo. This usually does not happen in extra dimensional theories: when there is a compact extra dimension there is usually a graviton zero mode and KK tower. As the size of the extra dimension gets infinitely large, the zero mode becomes more and more decoupled due to its huge spread in the extra dimension, however the KK modes become lighter and lighter, and eventually form a continuum that reproduces higher dimensional gravity. Here however one does not find that the zero mode would decouple in the infinitely large extra dimension limit, and the reason for that is that contrary to the flat extra dimensional case, the zero mode of the graviton has a non-trivial wave function that is peaked around the positive tension brane. This implies that in the RS model gravity itself becomes localized around the positive tension brane, and far away one only has a small tail for the graviton wave function. In the original y coordinates the wave function of the graviton zero mode is given by da2=e-2kM(ri^

+ hv,)-dv>.

(4.49)

This confirms the ansatz that we have used in (4.34) to calculate the effective 4D Planck scale, and also sheds much more light on the nature of the solution to the hierarchy problem in the RS model. Since gravity is confined to the positive tension brane, observers living far away from the positive tension brane (for example on the other end of the interval) will only feel the tail end of the graviton wave function, which is exponentially suppressed compared to the Planck brane. On the other hand particle physics interactions will have equal strength irrespectively of where one is along the extra dimension. So the solution of the hierarchy problem can be summarized by saying that gravity is weak compared to particle physics because we happen to live at a point in extra dimension that is far away

663

from where gravity is localized. As a comparison, in large extra dimensions the way that the hierarchy problem is resolved is by saying that gravity is much weaker than particle physics because its fundamental strength does get diluted by fluxes spreading out into the large dimensions, while since particle physics is localized on a brane it does not get diluted and we feel the fundamental strength of interactions. Getting back to the issue of the wave function of the graviton zero mode: since it is localized around the TeV brane, and remains normalizable even in the limit of infinite extra dimension, it opens the possibility of recovering 4D gravitational interactions even when the size of the extra dimension is very large, since the zero mode remains localized around the Planck brane. However, this on its own is not sufficient to really have a 4D gravity theory without compactification. As explained above in the infinite extra dimension limit there will be a continuum of KK modes for the graviton, and usually these continuum modes are responsible for turning gravity into higher dimensional gravity. There will be a continuum of KK modes in the RS model as well, however there is hope that these would not turn gravity over into a higher dimensional theory of gravity, since these continuum modes also feel the volcano potential of Fig. 4, and this means that in order to get to the TeV brane they have to tunnel through a large barrier, and thus their wave functions will be strongly suppressed at the Planck brane. 51 One can easily estimate the suppression of the wave function by a the WKB approximation for the tunneling rate. 5 3 Consider a continuum

V(z)

Figure 4.

The shape of the volcano potential that localizes gravity in the RS model.

664

mode with mass m, then the semiclassical tunneling amplitude is mt

- 2 f21/"1' dzJV(z)-m?

\

t A cr>\

T(m) ~ e Ja=o('") v *. ' (4.50) ( where zo(m) and zi(m) are the the classical turning points for a particle with mass m in the volcano potential. In order for the KK modes to not give a large contribution to the 4D gravitational potential one would need that lim m ^o T{m) = 0. The reason is that the potential between two test charges located at the Planck brane are given by the exchange of the zero mode and the KK modes: U(r)

GNMXM2 , „ — ' +

r

1

f°° J „ . M i M 2 e - m r , T 2 / d m - ^ *^(0),

Mi J0

(4.51)

r

where the first term comes from the exchange of the zero mode which would cause 4D gravitational interactions, but the second term is due to the exchange of the KK modes. If for small m —> 0 we would have \? m (0) —> const, we would get unsuppressed 1/r2 corrections due to the integral around m = 0. Thus we need that 4' m (0) —> 0 for small m, and this can only be due to T{m) —> 0. In the RS case V{z) ~ 1/z2 for large z, and so as m —> 0 we get that T(0) ~ e~ Jo°° >dz.

(4.52)

The integral in the exponent diverges, so that we expect the KK modes to indeed decouple in the RS model and to recover 4D gravity even in the infinitely large extra dimension limit. In order to really establish this rigorously, a slightly more precise analysis is needed. Assume that the volcano potential for large z behaves like a(a + l)/z2 (for the RS case a = 3/2). Then the asymptotic form of the Schrodinger equation

- ^ +^ J ^ *

m

= m2*m

(4.53)

has solutions in terms of the Bessel functions *m = amz?Ya+i{mz)

+ bmz^Ja+i(mz)

(4.54)

If one does a careful matching of the wave function inside and outside the potential we get that in the RS case

Therefore the correction to Newton's law is of the form

665

Since J*0°° e mrm potential to be

~ 1/r2 we get the final form of the corrected Newton

where C is a calculable constant of order one. This shows, that the full correction due to the KK modes is extremely small for distances larger than the AdS curvature 1/fc, which in our case is of the order of the Planck length. This means that for distances above the Planck length one would really get a 4D gravitational potential irrespectively of the compactification. Thus in the presence of localized gravity there is no need to compactify [51] the extra dimension! Also, due to the wave function suppression of the KK modes the production of the continuum KK modes would be suppressed on the Planck brane by p2/k2. Besides reproducing the ordinary 4D Newton potential, one should however go somewhat further before claiming that 4D gravity as we know it (that is 4D Einstein gravity, not just Newtonian gravity) is in fact reproduced on the brane. The reason is that as we mentioned several times before, there could be an extra massless scalar in the graviton. That would also give a 1/r potential, so would also lead to Newtonian gravity, however the tensor structure of the graviton propagator could be modified, which would lead to a so called scalar-tensor theory of gravity. It is somewhat subtle to show that this does not indeed happen, and one needs to take so called brane bending effects into account for that. The analysis that shows that RS reproduces 4D Einstein gravity completely on the Planck brane has been performed in Refs. 56 and 57, see also Ref. 58. 4.3. Intersecting dimensions

branes, hierarchies

with infinite

extra

We close this section by discussing two pressing issues. The first is what happens if we have more than a single extra dimensions: can we still localize gravity somehow? The second issue is whether if one has an infinite extra dimension, can one still solve the hierarchy problem (since now there is no negative tension brane)? Localization of gravity to brane intersections The easiest way to localize 4D gravity in more than one extra dimension is to consider the intersection of co-dimension one branes (that is intersection of d — 1 spatial dimensional branes in d spatial dimensions). The simplest

666

possibility for intersecting branes is for two 4-branes in 6 space-time dimensions to intersect orthogonally. 59 To find the solution we should first think about what the RS solution with just the Planck brane is (this situation is usually referred to as the RS2 model): it is nothing else but two copies of slices of AdSs spaces glued together at the Planck brane. This is what we need to do in the case of orthogonally intersecting 4-branes as well: we need to take four "quarters" (or quadrangles) of the AdS6 space and glue them together at the branes (of course in the general case of intersecting 2 + n branes one needs to glue 2™ copies). One quadrangle can just be given by the usual AdS6 metric. An AdS6 metric is exactly the same form as the AdSs metric in the conformally flat coordinates: ds2

1 —^ [dx^dx^r]^ - dy2 - dz2] , (kz + 1)

(4.58)

where we can bring this metric by the coordinate redefinition z — (z\ + Z2)/y/2, y = (z1- z2)/V2 to ds2

1

(k(Zl+z2)

— [dx^dx^rj^

+ l)

- dz2 - dzl] .

(4.59)

A simple way of patching these solutions together at two four-branes is by the metric ds2

1

— [dx^dx^r]^ - dz2 - dzl] .

(4.60)

(fc(ki| + N ) + i)

This metric will be the same AdS6 metric in all four quadrangles, and continuous at the intersections. Generically for the intersection of n branes in 4 + n space-time dimensions the metric can be written as ds2 = n2(z)(dxMdxN7lMN),

H(Z)=

., ^

n

\

.

(4.61)

L l V

To find out what the relation of the tensions of the intersecting branes to the bulk cosmological constant is we need to find the Einstein tensor for the above metric. One can again use the formula for a conformally flat metric that gives the Einstein tensor: GMN

= (n + 2) - V M log fiVjv log fl + V M VAT log tt •9MN[

V/jVfllogft

n+ 1— V H log f2V# log ft

,

(4-62)

667

and we find that /I

G

M

n(n + 2)(n + 3)

JV =

^

2 M

2(n + 2)fcg,

« %

^

0(^l)

1/ /I

2(n + 2)fc

(4.63)

5(zn)

V

0/

This needs to be equated with the energy momentum tensor from the bulk cosmological constant, which exactly matches the first term, and the brane tensions on the intersecting branes exactly have to match the individual subleading terms. From these we get matching relations analogous to that of the 5D RS model: n(n + 2)(n + 3)

2

fc =A«4+n, 2(n + 2)k =

(4.64)

VKl+n,

where A is the 4 + n dimensional bulk cosmological constant, while V is the (common) brane tension. One can again find the perturbed Einstein equation for the graviton, the result is given by:

-V 2

z

n(n + 2)(n + 4)fc2

tf

(n + 2)kfl

E^

2

(4.65) One can again show59 that there is a single bound state zero mode solution to this equation, and its wave function is proportional to £1 ~^~. The zero mode is localized to the intersection of the branes. The relevant length

668

scale for gravity is again the AdS curvature scale L ~ k^1, where this is given by k2 ~ AK2+U ~ A/M2+n, where Ki+n is the 4 + n dimensional Newton constant, while M* as always is the 4 + n dimensional Planck scale. Therefore L

/ M2+n\

~{

f

A J '

(4 66)

'

and so we get a relation between the AdS length, the 4 + n D Planck scale and the 4D Planck scale of the form [59] M2+nLn

= M2Pl.

(4.67)

For scales much larger than L gravity will behave like ordinary 4D gravity, while for distances smaller than that scale gravity will behave as 4 + n dimensional gravity. (4.67) implies that the scale L is analogous to the real radius of compact flat extra dimensions. Thus one can try to solve the hierarchy problem in the presence of infinite extra dimensions by lowering the fundamental Planck scale M* down to a TeV, which would of course also imply that the brane tensions and the bulk cosmological constants would be themselves much smaller than M*. One can go further with the intersecting brane picture, and consider brane junctions 60 — that is semi-infinite co-dimension one branes ending at the same point, the brane junction (see Fig. 5). One can show, that such static junctions may exist only, if the total force from the tensions acting on the junction vanishes:

J2^ = °-

( 4 - 68 )

i

In that case there will be a single remaining fine tuning relation between the bulk cosmological constant and the brane tensions, equivalent to the vanishing of the 4D cosmological constant. For more details see Ref. 60. The hierarchy problem in the infinite RS case Finally, one may ask whether a solution to the hierarchy problem really necessitates the compactification as discussed at the beginning of this lecture. One possible way to avoid this we have seen above, where we simply lower the fundamental Planck scale M* down to the TeV as in the large extra dimension scenarios. However, this would then raise the issue of why the AdS curvature L is so large compared to its natural value. However, one could just live in the infinite extra dimensional scenario on a brane (that has a very small tension) away from the Planck brane. 61 Due to the

669

Figure 5.

Intersecting branes in the RS model.

small curvature gravity would not be much affected, and due to the large warp factor one would still get that the particle physics scales on this brane would be warped down to an exponentially small scale. But would we still get 4D gravity on this brane? The corrections to Newton's law would be given by the form , »(*o)V V(r, = G ^ ( 1 + j r * . i f *(*o(zo) )

(4.69)

The important point is that it is the local ratio of the KK amplitudes vs. the zero mode that will appear in this expression. One can show that this ratio is still scaling the same way as at the Planck brane *m(zo) V0(z0)

=

^m(0) *o(0)

• m

2

,

(4.70)

for small values of m. Thus the corrections to Newton's potential will be similarly negligible far away from the Planck brane as on the Planck brane. Thus the conclusion is that one really does not need to compactify, one can recover 4D gravity and solve the hierarchy problem. In fact, one can show without too much calculation that this indeed needs to be the case: the zero mode contribution gives a potential of the form k/(M%r), while the KK modes at worst would give l / ( M * r 2 ) . For relatively large values of r the second term is always suppressed compared to the first and gravity will have to look four dimensional.

670

5. Phenomenology of Warped Extra Dimensions This lecture will cover some important topics about further developments in the RS model, and warped spaces in general. 5.1.

The graviton

spectrum

and coupling in RSI

The first issue that we would like to discuss is what the phenomenology of the graviton KK modes would be in the RS model with two branes, which is often also referred to as the RSI model. We have already seen that the graviton fluctuations obey the Schrodinger-type equation

-d2V + f^-A'2

- ^AA # = m 2 *,

(5.1)

where /iM„ = e3'4AhfJ/U(x)ib(z), where the h/Jil/(x) are the 4D modes that satisfy Oh^ = rn^h^. For the RS background A = 21og(/c|z| + 1). Until now we have not discussed much what the BC of these equations should be, since we were imagining them in the infinite RS2 case. For the case, with finite branes we need to find the appropriate set of BC's, which can be read off from the fact that we imposed an orbifold projection y —> — y, under which the graviton was symmetric. Therefore, in y coordinates dyh^ = 0 at the two fixed points, from which it follows that also dzh^v = 0. Since the graviton can be recovered from the Schrodinger frame wave function *(z) as h„v = e3/4Ah^(x)^(z), and A = 21og(fc|z| + 1) we can translate the BC on h^ to a BC on $ :

^=-i*^f+i*i—

(5 2)

-

where z\ is the location of the second fixed point (with negative tension) in the z coordinates, z\ = ekb/k. The equation between the two boundaries for the RS case is

Generically, the solution of this equation is given in terms of Bessel functions: *(z) = am(kz + l ) i y 2 ( m ( z + 1/fc)) + bm(kz + 1)*J 2 (m(z + 1/fc)). (5.4)

671

The two boundary conditions will completely determine the masses of the KK tower m's, where the nth solution is given by m = kxne~kb

= 0(TeV),

Ji(xn)

= 0.

(5.5)

Thus the spacing of the graviton zero modes is of order TeV, and not of order Mpi as one would have naively suspected from the fact that the proper length of the extra dimension is of order 1/Mpi. The fact that the KK modes are spaced by TeV suggests that it is better to think of this model with extra dimension of size 1/TeV, that is a relatively large extra dimension, rather than a really small one. The answer to any physical observable will convey this impression. For example, if one were to ask how long it takes for light to bounce back and fourth between the two branes, the time we get will be of order 1/TeV due to the warping, much longer than the naively expected 1/Mpi. Let us now ask how one would detect these KK graviton resonances if one were to live on the TeV brane (the negative tension brane, where the hierarchy problem is resolved). Ordinarily one would think that these states will be completely unobservable, since they are relatively heavy (of order TeV), and only gravitationally coupled. However, our naive intuition fails us again! The point is that even though these KK modes are indeed gravitationally coupled, since they are repelled from the Planck brane by the tunneling through the volcano potential, their wave function must be exponentially peaked on the TeV brane. Therefore their couplings will also be exponentially enhanced to matter on the TeV brane (compared to gravitational strength coupling). Thus the picture that emerges is the following: on the TeV brane one has the graviton zero mode, that is peaked on the other brane therefore its coupling to TeV brane matter is weak (this is what sets the scale of gravitational coupling), and one also has the KK modes of the graviton whose masses are of order TeV, but their couplings are enhanced to 1/TeV rather than 1/Mpi on the TeV brane:

LTeV leV

L =

—T^h{0l

MPI

a/3

—r-hTa0 Y h(nl

MPte-kb

*->

a/3

(5.6) y

'

The detailed coefficients in the equation above have been worked out in [62], but by now none of the features of this coupling are surprising: the graviton modes couple to the stress-energy tensor of matter on the TeV brane, and the scales of coupling come from the consideration of the wave functions of the zero and KK modes as discussed above.

672

This implies that in TeV scale colliders such as the LHC will be (and already at the Tevatron) one could study these KK modes as individual resonances in the scattering cross section. Some of these cross-sections have been worked out in detail by Davoudiasl, Hewett and Rizzo in Ref. 62. Some sample cross sections in the presence of the KK resonances for the LHC and a linear collider calculated in Ref. 62 are given in Fig. 6. Note, that the phenomenology of the RS model is very different from that of large extra dimensions. In the case of large extra dimensions the spacing of the KK modes is very small, in the extreme case of 2 large extra dimensions of order 1CP3 eV. This is tiny enough to allow energetics to produce it in all colliders, however since the couplings of the individual modes is extremely small, of gravitational strength 1/Mpi one can certainly not see the individual modes. Their collective effects will be visible as a raise in cross sections at the LHC or the Tevatron. However, for RS as we discussed the individual KK modes are much heavier and much more strongly coupled, leading to well-defined resonances in high energy scatterings.

i

i

i

i

i

i

i

r

io6

io5 & b

io4

io3

250

500

750

1000

1350

1500

Vs (GeV) Figure 6. Scattering cross section for e + e ~ —> /i+ fx~ at a linear collider in the presence of the tower of Randall-Sundrum KK gravitons, as calculated and plotted by Davoudiasl, Hewett and Rizzo in Ref. 62. The mass of the first KK resonance is fixed to be 600 GeV, and the various curves correspond to different values of k/MPi = 1,0.7,0.5,0.3,0.2,0.1 from top to bottom. For details and LHC processes see the original paper. 6 2

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5.2. Radius

stabilization

Until now we have treated the radius of the RSI model as a fixed given constant, and found that radius r (which we have denoted b until now) has to be r ~ 30/A; in order for the hierarchy problem to be resolved (since Mweak ~ 10~16MPI, from which rk ~ loglO 16 ~ 30). This raises several important questions, that need to be addressed: • Since the radius is not dynamically fixed at the moment, but rather just set to its desirable value, there will be a corresponding massless scalar field in the effective theory, which corresponds to the fluctuations of the radius of the extra dimensions, called the radion.63^66 One can understand the masslessness of this field by realizing, that the RS solution discussed until now did not make any reference to the size of the extra dimension, it was a solution for arbitrary values. This means that in the effective theory this parameter is also arbitrary, and thus has no potential, and so is a flat direction. Thus it can have no mass. This massless radion would contribute to Newton's law and result in violations of the equivalence principle (would cause a fifth force), which is phenomenologically unacceptable, and therefore it does need to obtain a mass — the radius has to be stabilized. • The radius has to be stabilized at values somewhat larger than its natural value (we need kr ~ 30, while one would expect r ~ 1/fc). Does this reintroduce the hierarchy problem? • We have seen that one needed two fine tunings to obtain the static RS solution, one of which was equivalent to the vanishing of the 4D cosmological constant, and is thus expected. But can we shed light on what the nature of the second fine tuning is and whether we can eliminate it somehow? A mechanism for radius stabilization will address all the above mentioned issues. The simplest and most elegant solution for stabilization of the size of the extra dimension was proposed by Goldberger and Wise, 63 and is known as the Goldberger-Wise (GW) mechanism. Here we will discuss the details of this mechanism and its effect on the radion mass and radion physics, however before plunging into the details and the formalism let us first summarize the main idea behind the GW mechanism. Radius stabilization at non-trivial values of the radius usually occurs dynamically, if there are different forces, some of which would like to drive the extra

674

dimension very large, and some very small. Then there is a hope that these forces may balance each other at some value and a stable non-trivial minimum for the radius could be found. A possible way to find such a tension between large and small radii is if there is a tension between a kinetic and a potential term, one which would want derivatives to be small (and thus large radii) and the other which would want small radii to minimize the potential. The proposal of Goldberger and Wise was to use exactly this scenario. Introduce a bulk scalar field into the RS model, and add a bulk mass term to this field. This will result in a non-trivial potential for the radius, since due to the bulk mass the radius wants to be as small as possible. However, if we somehow achieve that there is also a non-trivial profile (a VEV that is changing with the extra dimensional coordinate) for this scalar, then the kinetic term would want the radius to be as large as possible, so as to minimize the kinetic energy in the 5th direction. Then there would be a non-trivial minimum. To achieve the non-trivial profile for this scalar field GW suggested to add brane potentials for this scalar on both fixed points, which have minima at different values from each other. Then one is guaranteed the non-trivial profile, and thus radius stabilization as well. The main question is not whether this mechanism works, it is intuitively clear that it should, but rather whether one can naturally achieve the somewhat large radius using this mechanism without fine tuning, and if yes what will be the characteristic mass of the radion. To describe the GW mechanism in some detail one needs to set up some formalism for theories with extra scalar fields in the bulk. The reason is that as we will see the back-reaction of the metric to the presence of the scalar field in the bulk will be important, therefore it would be very nice to simultaneously solve the Einstein and the bulk scalar equations, to have the back-reaction exactly under control. One can get away without such an exact solution either by just calculating a 4D effective potential for the radion, 63 ' 64 or by calculating the back-reaction order-by-order. However, with some formalism we will be able to solve the coupled equations exactly for certain scalar potentials, and therefore we will discuss the issue of radius stabilization using this approach. 54 ' 53 ' 67 ' 68 Denote the scalar field in the bulk by $, and consider the action

J
(5.7)

675

where the first term is the usual 5D Einstein-Hilbert action and the bulk action for the scalar field, while the next two terms are the brane induced potentials for the scalar field on the Planck (P) and on the TeV (T) branes. We will denote the 5D Newton constant as always by K2 = 1/2M^, and look for an ansatz of the background metric again of the generic form as in the RS case to maintain 4D Lorentz invariance: ds2 = e-2A^\l/dxtldx',-dy2.

(5.8)

The Einstein equations are exactly as we have derived for the RS case, except here we have an energy-momentum tensor that is derived from the action of the scalar field. So everyone should be able to derive the following equations on their own using the tools presented until now: 2

9^2

4^'2-A" = - ^ - V ( $ o ) - y £

\^0)6(y-yt)

i=P,T

The first equation is the i,j component of the Einstein equations, while the second one is the 5, 5 component. 3>o denotes the solution of the scalar field, which (by the requirement of Lorentz invariance) is assumed to be only a function of y:
= ^VS-

(5-10)

By the substituting the scalar and metric ansatz into this equation we get

i

We can separate these equations into the bulk equations that do not contain the delta functions, and the boundary conditions which will be obtained by matching the coefficients of the delta functions at the fixed points. For example, the first Einstein equation contains the explicit delta function term - y E i = P T ^ ( ^ ° ) ^ f e ~~ Vi) a t * n e branes which naively does not seem to be balanced by anything. However, there is no requirement for the derivative of the metric to be continuous (the metric itself is a physical quantity which should better be continuous), and so there could be a jump in the derivative A' at the branes, which would imply that A" also contains

676

a term proportional to a delta function. If the derivative A' jumps from A'(0 — e) to A'(0 + e), this implies that locally A' contains a term of the form (A'(0 + e) — A'(0 — e))9(y), where 0(y) is the step function. Therefore, A" will contain the term (A'(0 + e)-A'(0 — e))<%), thus the delta function is proportional to the jump of the derivative of A, sometimes denoted by [A']o = A'(0 + e) — A'(0 — e). Therefore, the boundary conditions derived this way are sometimes also called the jump equations, which in our case will be given by

[A1], = yA,(*„),

The bulk equations (5.9-5.11) together with these boundary conditions form the equations of the coupled gravity-scalar system. These are coupled second order differential equations, which are quite hard to solve. However, for specific potentials it is possible to simplify the solution. Let us assume that we have found a solution to the system of equations above given by A(y), $o(j/)- We can define the function W(&) via the equations

A'

= £w($0), 1 dW

If we plug in these expressions for A' and $g into the Einstein and scalar equations we will find that all equations are satisfied, if the following consistency condition holds: i

2

i / paxr\ fdwy - i-2 YW^2 =

V(*) =

8 V 9$ J

(5.14)

and the jump conditions are given by = A<(*o), 9Ai($ 0 ) (5.15) <9$ ' If W were given we would reduce the equations from coupled second order differential equations to ordinary first order equations that are quite easy to solve. The price to pay is that it is very hard to find W for a given V($): one needs to solve a non-linear second order differential equation (5.14) to find the "superpotential" W. However, if our goal is not to solve the equations

677

for a very particular bulk potential V(&), but rather a bulk potential which has some properties, we can simply start with a superpotential W that will produce a V with the required properties, and we will be able to easily get the full solution of the equations. In our case we would like the bulk potential to include a cosmological constant term (independent of $) and a mass term (quadratic in $), but we will not care if there are some other terms as well if those make solving the equations simpler. So we choose [54] W{$>)

= ^-v®2.

(5.16)

The first term is just what one needs for a cosmological constant, while the second term will provide the mass term when taking the derivative. Let us first discuss the jump conditions. Since the metric is an even function of y, if we try to solve the equations on the S1 /Z% orbifold again as in the RS case A(y) must also be an even function, and therefore A' is odd. However, one the equations we have above is that A' ~ W, which means that W must be an odd function at the branes, it must change signs. To satisfy the jump conditions (5.15) I must choose the brane potentials to be of the form A($)± = ± W ( $ ± ) ± W ' ( $ ± ) ( * - $±) + 7 ± ( * - $ ± ) 2 ,

(5.17)

where $± are the values of the scalar field at the two branes, which we will also denote by $ + = $ p at the Planck brane, and $ _ = $ r at the TeV brane. Then the solution will be given by the solution of the equation

* , = 5 ^ = - w *'

(5 18)

-

which is simply My)

= $pe-vy.

(5.19)

From this the value of the scalar field at the TeV brane is determined to be $T = $pe-m\

(5.20)

This means that the radius is no longer arbitrary, but given by r=-ln|£.

(5.21)

The value of the radius is determined by the equations of motion, which is exactly what we were after. This is the GW mechanism. The metric background will then be obtained from the equation A' = ^W(M 6

= k - ^*%e-*>y 6

(5.22)

678

given by the solution A{y) = ky+'^e-2vy.

(5.23)

The first term is the usual RS warp factor (remember that A has to be exponentiated to obtain the metric), while the second term is the backreaction of the metric to the non-vanishing scalar field in the bulk. We will assume that the back-reaction is small, and thus that K 2 $p,K 2 $|, 0. The values of $ p and $ y are determined by the bulk and brane potentials, so $ P / $ T is a fixed value. Since we want to generate the right hierarchy between the Planck and weak scales we need to ensure that kr ~ 30,

(5.24)

from which we get that

(525)

H%)~*

-

which implies that v/k is somewhat small (but not exponentially small). This is the ratio that will set the hierarchy in the RS model, and we can see that indeed one can generate this hierarchy using the GW stabilization mechanism by a very modest (O(50)) tuning of the input parameters of the theory. Once we have established the mechanism for radius stabilization, we know that the radion is no longer massless. The next obvious question then is what will be the value of the radion mass: is it naturally sufficiently large to avoid the problems mentioned at the beginning of this section or not? We will answer this question by finding what the radion mode is in the GW stabilized RS model, and explicitly find its mass, 68,69 see also Refs. 64, 65 and 70. For this, we need to find the scalar fluctuations of the coupled gravity-scalar system. This can be parametrized in the following way: ds2 = e-2A-2F{-x'y^liVdxiidxv $(x,y)=$0(y)+tp(x,y).

- (1 +

G{x,y))2dy2, (5.26)

At this moment it looks like there would be three different scalar fluctuations, F, G and tp. However, if we plug this ansatz into the Einstein equation the 4D off-diagonal fiu components are satisfied only if G = 2F,

(5.27)

679

while the /n5 components imply the following further relation among the fluctuations:
(5.28)

This means, that in the end there is just a single independent scalar fluctuation in the coupled equation, which we can choose to be F. Requiring the above two relations we find that the rest of the Einstein equations are all satisfied if the following equation holds: F" - 2A'F' - 4A"F - 2^-F'

+ AA'^-F

= e2AUF

(5.29)

in the bulk and the following boundary condition: (F' - 2A'F)i = 0.

(5.30)

Let us first assume that there is no stabilization mechanism, that is the background is exactly the RS background given by A = k\y\, and $o = 0. In this case most of the terms in the above equation disappear, and we are left with F' - 2kF = e2kym2F,

(F'-2kF)l

= 0.

(5.31)

We can see that the only solution is for m 2 = 0, and the wave function of the un-stabilized radion will be given by F(y)=e2kM.

(5.32)

This wave-function has been first found by Charmousis, Gregory and Rubakov, and thus the metric corresponding to radion fluctuations in the RS model corresponds to ds2 = e-2kM-2e"MfWril]H,dx»dxv

- (1 + 2e2k^f(x))dy2.

(5.33)

This is a single scalar mode, that is exponentially peaked at the TeV brane, just like all the graviton KK modes. Since it is peaked on the TeV brane, it means that its coupling on the Planck brane will be strongly suppressed, while it will be of its natural size (suppressed by 1/TeV) on the TeV brane. The exponential peaking on the TeV brane also implies, that if we move the TeV brane to infinity (that is consider the RS2 model) then the radion will no longer be a normalizable mode and completely decouples from the theory. This implies that in that case one is really recovering 4D gravity with just the tensor couplings on the Planck brane in RS2. Generically, for the RSI case we now have a complete set of modes (the radion mode is the one that is missing if we are imposing the RS gauge

680

choice): without stabilization at the zero mode level there is a graviton and a radion mode (which does not have a KK tower), while at the massive level there are the graviton KK modes only. In the presence of the GW stabilization the scalar mode will acquire a KK tower: the lowest mass mode of this tower will behave like the radion, while the higher mass modes will be identified with what used to be the KK tower of the GW bulk scalar. To find the radion mass for the case with stabilization, we simply need to plug into (5.29) the full background for A and $o with stabilization: F" - 2A'F' - AA"F + 2vF' - AvA'F + m2e2AF

= 0.

(5.34)

We can see that if the back-reaction of the metric was neglected (A" = 0) we would still not get a mass for the radion zero mode. Thus to find the leading term for the radion mass we expand in terms of the back-reaction of the metric in the parameter I = K&P/\/2, and obtain the mass of the radion _ -±v K^-rv)U ~ 3fc

2

m

radion

2(v+k)r

(

'

, oc.\ {0-00)

As a reminder, the mass of the graviton KK modes is of the order trigraviton ~ ke~kr,

(5.36)

and so the radion/graviton mass ratio is radion 2

m

graviton

~ l2j^e~2vr.

(5.37)

"-

Thus we can see that the radion mass is smaller than the graviton mass, and the radion would be likely the lightest new particle in the RSI model. The other scalar KK modes that originate mostly from the GW bulk scalar will have masses of the same order as the graviton. In order to find the coupling of the radion to SM matter fields on the TeV brane we need to find the canonically normalized radion field. The wave function (not yet normalized) is of the form F(x, y) = e2kM (1 + corrections) R(x),

(5.38)

where the correction terms above are suppressed by the back-reaction, and since these are not the leading terms we can neglect them. R(x) is the 4D radion field. The induced metric will be gf,f = e~2Ar1^(l-2F(x,y)).

(5.39)

The coupling to matter on the TeV brane will be as usually given by Tfl„gfJl1', which will result in a coupling of the form TffR(x). The question is what will

681

be the coefficient in front of this coupling be? Once we go to the canonically normalized radion field r(x) we find (after calculating the normalization of the kinetic term for the radion) that the radion coupling is given by 1

V6MPle~

•kr

r(x)TrT.

(5.40)

Thus the coupling is suppressed by the TeV scale just like for the graviton KK modes, except the masses are somewhat smaller. One can also write this coupling in the form v

V6A

^)-TrT =

7

^ T r r ,

(5.41)

where we have denoted A = Mpie~kr as the "TeV scale" that one obtains from the RS hierarchy, 7 = u/\/6A, and v is the usual Higgs VEV v — 246 GeV. This coupling is exactly equivalent to the coupling of the SM Higgs boson. The reason is that both the radion and the Higgs will couple to the mass terms in the Lagrangian, the Higgs via H/v, the radion via r/v, the only difference is the extra suppression factor 7 in the strength of the radion coupling. Thus the phenomenology of the radion would be very similar to that of the SM Higgs boson. There is one more important possibility for the physics of the radion. There could be a brane induced operator on the TeV brane mixing the radion and the SM Higgs field H [71]:

J dAx^HR{gind)yf^.

(5.42)

Using our master-formula (4.5) for calculating curvature tensors for conformally flat metrics we get L € = R(tf(r)r]^)

= - 6 f r 2 ( D l n f t + (Vlnfi) 2 ).

(5.43)

The interaction term will then be given by .

L c = -6^f2 2 (Dlnrj + (Vlnf2) 2 )ii" t iJ.

(5.44)

Expanding both the Higgs H and the warp factor Q, in the uneaten physical Higgs and the radion respectively H = ( ^ \ ,

n(r)=l-1~

+ ...

(5.45)

we get L 4 = 6^hDr + 3£72(<9r)2.

(5.46)

682

Thus there will be kinetic mixing induced between the Higgs and the radion fields. One needs to diagonalize the full quadratic Lagrangian L = -hiUh

- ]rm2hh2 - hi

+ 6 £ 7 2 ) r O - ^m2r2 + Q^hUr.

(5.47)

After the diagonalization, we find that not only is the coupling of the radions affected, but the couplings of the SM Higgs will also be modified from their SM expressions, and thus the radion-Higgs mixing could significantly affect ordinary Higgs physics as well. 68 ' 72 5.3. Localization

of scalars and

quasi-localization

We have seen in previous sections that the zero mode graviton is strongly peaked at the Planck brane, that is gravity is localized by the RS background. An important question to discuss is whether fields with spins other than 2 would also be localized or not. In the following we will discuss two examples of that: a bulk scalar field and a bulk gauge field. We will not discuss the issue of bulk fermions, the reader is referred to Refs. 73-78 for that case. We will start by (again) considering a bulk scalar field (see for example Refs. 79 and 75). We have already discussed the equation of motion for the bulk scalar, in the absence of a bulk potential it is given by dM(V9gMNdN$)

= 0.

(5.48)

If we are interested in whether or not a zero mode exists, we simply have to assume that the 4D derivative on $ vanishes, so the equation that a zero mode has to satisfy is dye-4kvdy = 0.

(5.49)

The solution to this is simply $ = const = $o- However, this is not enough. We need to figure out whether or not this mode is normalizable in the limit when the extra dimension becomes infinitely large (that is we want to find out if the effective 4D kinetic term remains normalizable or not). We write the zero mode as $ = $0¥>(x)> where ip is the 4D wave function for a massless 4D mode O^ip = 0. The normalization of the kinetic term then comes from

J ^d^d^g^dyd4x

= J e-^e2ky$>\dy

f d^d^V^d4x.

(5.50)

We can see that the y-integral converges even when the extra dimension is infinitely large, so this means that the scalar field is localized in the RS

683

background, even though in y coordinates its wave function is constant (if we went to the z coordinates we would of course find a wave function peaked at the Planck brane). However, this analysis was done in the absence of a mass term (or bulk potential in general), which is quite unnatural for a scalar field. Let us ask how the solutions will be modified if there is a bulk potential (and still keep the extra dimension infinitely large). The equation of motion will now be dM(V99MNdN*)

= V9%-

(5-51)

Clearly, there will be no more zero mode solution. In order to find out what happened to the zero mode, we need to consider the full continuum spectrum. 80 The scalar equation for a 4D mode with 4D momentum p in the presence of a bulk mass termTO2<0>2in the bulk potential will be: <92$ - 4fc$' + p2e2ky$

-

TO2$

= 0.

(5.52)

2

As we mentioned before, p = 0 is not a solution to this equation, there is no zero mode. Naively we would have thought that the zero mode just picks up a mass from the bulk mass term and will sit at p2 = TO2, but p2 = TO2 is not a solution to the bulk equations either. So now we might be really puzzled what happened to our zero mode. The continuum spectrum can be determined at large y, where the extra TO2 term from the bulk mass is unimportant. Thus the continuum spectrum will be unchanged compared to the TO2 = 0 spectrum, except the wave functions will be distorted for small y. The general solution to the equation above will be in terms of Hankel functions, $({/) = const .e2fc»Jff<1)(|efc»),

is=^4+~.

(5.53)

Since we have not included a source on the Planck brane the Z^ symmetry will imply that the BC (jump equation) at the Planck brane will just be dy$\y=o = 0. This implies that (!) ptf^-V!)

2 - v = 0.

(5.54)

The solution to this equation for small values of the bulk mass m is p = mo-*r,

m0 = — ,

_

= _ (

x

j

.

(5.55)

Thus what we find is that rather than having a single localized mode with mass TO, the mass will be shifted toTOO,and pick up an imaginary part.

684

This implies that a discrete mode with a finite lifetime will exist. We call this mode a quasi-localized mode for the following reason: if the lifetime is sufficiently long, then this mode for relatively short times will behave like an ordinary localized 4D mode. However, after some long times it will decay into some continuum bulk KK modes due to its finite lifetime. This mode can also be interpreted as a resonance in the continuum KK spectrum, rather than an individual mode with complex mass. This interpretation could be understood by calculating the effective volcano-type potential for this system. What we find is that the tunneling out amplitude for the mode trapped in the volcano will now be finite, and therefore it will have a finite width and lifetime. Either way the result is the same: for short times one has an effectively localized 4D particles, which however decays into the bulk after a long time. If this field were to carry bulk gauge charges, then it would lead to apparent non-conservation of this charge from the 4D observers point of view, even though of course there is no real violation of charge conservation in the full 5D theory once the continuum KK modes are also taken into account (which are however suppressed on the brane). This case is merely the simplest example of quasi-localization in an infinite volume space-time. Several attempts have been made to construct a viable quasi-localized model for gravity, which would be very exciting, since that would tell that the graviton would not strictly be massless, and have a finite width. The examples include the GRS model, 81 and the DGP modes. 82 However, none of them are quite satisfactory, either since there is an inherent instability in the spectrum, or because 4D gravity is not reproduced at large distances (the latter issue is still subject to debate). 5.4. SM Gauge fields in the bulk of RSI The next example we will discuss is the theory with gauge fields in the bulk. 83 ' 75 In this case one has to solve Maxwell's equation in a curved background, which is given by d^Vgg^g^F^) = 0.

(5.56)

We will choose a gauge where d^A^ = A5 = 0. The equation for the zero mode (similar to the considerations for the scalar) is just given by A; = 0.

(5.57)

The solution would again be a constant wave-function along the fifth dimension, A^ = Aoafj,(x), however in this case (contrary to the case with

685

the bulk scalar) the wave function is not normalizable (the kinetic term for the 4D gauge zero mode diverges), as one can see from J d^Xy/gg^g^F^F^

= J dVe~^e^e^

A\ J d4xF$F^.

(5.58)

Due to the different power of the warp factor (which is simply due to the more indices in the gauge field) the y integral is no longer normalizable, and thus there will be no localized zero mode. In fact, as we will see below the zero mode wave function is flat also in the z coordinates, and is thus not localized. One can show similarly, that bulk fermions would also not be localized in RS2. Nevertheless, we can consider the RSI model with the two branes in the presence of bulk gauge fields (and also possibly bulk fermions). However, we will assume that the Higgs is still localized on the negative tension brane (there is good reason to do that - it turns out that if it was in the bulk one would need extreme fine tuning in the bulk mass parameter to lower the Higgs mass from TeV to ~ 100 GeV). One should ask, how in this case with Higgs on the brane and gauge fields in the bulk would electroweak symmetry breaking work. The Lagrangian for the gauge fields would be given by (after the Higgs on the TeV brane gets a VEV) 1GMPGNQ

/,

4

\BGMP(Wl,Wh

(1WSINW°

+

\9z

1^BMNBPQ

g's

,

+ W2MW2 + (W3M - 2&)(W$ _ B 3 ) } (5.59)

Here we have used the perhaps simplest form of the RS metric: ds2 = ( - )

(rj^dx^dx"

-dz2),

(5.60)

where R = 1/k (do not confuse this with the size of the extra dimension in proper y coordinates), and the variable z runs between R and R', R'/R = 16 ekr _ xo . g5 and g'5 are the 5D bulk SU(2)ixU(l)y gauge couplings. One would actually like to calculate the gauge boson masses and check if the prediction agrees with that of the SM [84]. If there was no Higgs VEV, we have seen that the gauge boson would have zero modes and completely flat wave functions. However, the Higgs VEV on the TeV brane will deform the gauge boson wave functions, and the expression for the gauge boson masses will not be as simple as in the SM. To find the Mw, Mz masses one

686

needs to actually solve the bulk equations in the presence of a non-trivial boundary condition set by the brane Higgs field, and find the lowest lying eigenstates which should be identified with the bulk gauge bosons. The bulk equation for an eigenmode is (dt - \dz + m2- \v2g25(z

- i ? ' ) f ) *(z) = 0.

(5.61)

The solutions are as always Bessel functions zJ\{mz) and zY\(mz), and the BC's are that \l/ is flat at the Planck brane and satisfies a mixed BC depending on the Higgs VEV on the TeV brane. The equation determining the W mass is then J0(MWR){4MWR'Y0(MWR')

+glv2RY1(MwR'))

Y0(MWR)(4MWR'J0(MWR')

+ g^RJ^MwR')).

= (5.62)

Expanding the solution of this equation in powers of R2v2 we get that r2

^ The term

fff R2v2 9JR2v4R'2 2 lnRf ' += Rln^r ^ 4 f 4:R' i S - §32R H,4^

(5 63)

-

^sRi can be identified with the tree-level 4D gauge coupling

(see next section on AdS/CFT), and the term ^ | - = v2e~2kr is just the warped down (physical) Higgs VEV. Thus the first term is just the usual SM expression for the W-mass, however there is a relatively large correction to this expression which is of order

This on its own is not yet very meaningful, since a shift in a single mass can always be absorbed by redefining the value of the Higgs VEV v slightly. What one needs to do is calculate the full set of electroweak precision observables (for example the Mw/Mz mass ratio, etc) and compare those to the experimental values. This has been done in Ref. 84, where it was found that generically all electroweak precision observables will get these type of corrections, and can be brought in agreement with experiment only if the value of the TeV scale 1/R' is raised to above 10 TeV. This would be quite bad for these models, since in a case like this both the graviton and the KK gauge boson masses would be raised to values above those observable at the LHC. However, it has been pointed out recently in Ref. 85 that the reason for these corrections is that custodial SU(2) of the SM is violated. They have proposed a model based on SU(2)L X SU(2)R bulk

687

gauge symmetry where the corrections to electroweak precision observables can be greatly reduced and the bound on R' relaxed (see also the discussion in the next section).

5.5.

AdS/CFT

A big advance of string theory has been recently the realization, that certain string theories in an AdS background are completely equivalent to some 4D supersymmetric gauge theories. In particular, Maldacena conjectured [86], that type IIB string theory on AdS 5 x S5 space is equivalent to N — 4 supersymmetric SU(N) Yang-Mills theory for large N. AdS 5 x S5 is a ten dimensional space (as superstring theories live in 10 space-time dimensions), and the AdSs space that appears here is the full, un-truncated AdS space, for example given by the metric (5.60) for 0 < z < 00. This supersymmetric gauge theory is also a conformal field theory in four dimensions. The meaning of this correspondence is that if you take any operator O in the N = 4 SU(N) theory, there will be a field <& in the string theory that couples to this operator as 0 $ on the boundary of the AdS space. Then all correlators in the field theory will be related to the boundary-toboundary propagator of this field $ in the AdS space. The important piece for us is the notion that the bulk of AdSs space is equivalent to a conformal field theory in 4D. By bulk here we really mean the KK modes of gravity in the AdS background as discussed in the RS model. That it is the AdS and not the 5-sphere that really gives the CFT can be seen from the fact that the conformal symmetry group is exactly equivalent to the isometries of the AdS space. However, in the above correspondence gravity does not appear on the 4D CFT side. Why is that? The reason is that we have considered the full AdS space, without the Planck and the TeV branes as in the RS model. Without these branes the graviton zero mode will not be normalizable, and decouples from the theory. So it is not so surprising that the 5D gravity theory on AdS space corresponds to a 4D theory without gravity. However, once the Planck brane is introduced, and AdS space is cut off, the zero mode will become normalizable. 87 ' 88 This means that gravity will be coupled to the CFT, and also due to the appearance of a brane at high energies the CFT will have a UV cutoff determined by the position of the Planck brane. If we were to also introduce matter on the Planck brane, this matter would couple with order one strength to the zero mode graviton, however as we have seen the KK modes are exponentially suppressed on the Planck brane,

688

and therefore will basically not couple to matter on the Planck brane. Since the KK modes represent correspond to the CFT, this means that in the dual CFT picture the matter fields on the Planck brane do not couple directly to the CFT, only through the fact that the graviton zero mode couples both to the matter fields and the CFT. This picture can be checked via a non-trivial calculation. If the above proposal is right, one should be able to predict the form of the corrections to Newton's law on the Planck brane due to the fact that these corrections come from the KK modes, which should also have a prediction in terms of a CFT. This has been done by Gubser, 87 and below we will briefly summarize his argument. In AdS/CFT language the correction to the graviton propagator is due to the interaction with the CFT. A graviton always couples to the stress-energy tensor, so in this case it would be g^T^j,. The correction will be of the form

1/P2 Thus we can see that the corrections to Newton's potential generically boil down to calculating the two-point function of the stress-energy tensor in a CFT, (TT)CFTIn conformal field theories this correlator is wellknow (and determined by conformal invariance). The result is given by (T(x)T(O)) ~ c/xs, since the dimension of the stress-energy tensor in 4D is 4. Converting this to momentum space we get (T(p)T(—p)) = pilogp. Thus the Newton potential from the figure above will pick up the terms 4 J - j2 +i ~~P logp2 P logp 2 -.

pA

pZ

(5.65)

p.

Fourier-transforming back to coordinate space we get that the first term is of course 1/x2, while the second is the Fourier transform of 1/x4, thus AdS/CFT predicts the form of the gravitational potential on the Planckbrane to be

This is in agreement with what we have seen in (4.57). What would be the interpretation of the negative tension (TeV) brane on the CFT side? (Refs. 89 and 90) Once the TeV brane is introduced, instead of the continuum spectrum of KK modes we will get the discrete

689

spectrum of KK gravitons strongly peaked on the brane. These have a welldefined mass scale, which implies that conformal invariance must have been broken on the CFT side in the IR. Thus we may think of the RSI scenario as an almost conformal field theory, that very slowly runs, but suddenly becomes strongly interacting at some scale around the TeV, spontaneously breaks the conformal invariance, confines and produces the bound state resonances that correspond to the KK gravitons. All the modes localized on the TeV brane should correspond to such CFT bound states, including the SM Higgs and SM matter and gauge fields. So the RSI model can be simply interpreted as a CFT that becomes strongly interacting and produces the composite SM matter and Higgs fields. The hierarchy problem is simply solved due to the compositeness of the Higgs field! What happens if we put gauge fields in the bulk of AdS? (Refs. 89 and 90) We have mentioned at the beginning of this section, that N = 4 SYM corresponds to the full AdSs x S5 type IIB string theory. The gauge theory has an SU(4) global symmetry, called the R-symmetry. This global symmetry is reflected by the isometries of the S5 internal space. Since an isometry implies gauge boson zero modes, which we can indeed find in the string theory side. We thus conclude, that the global symmetry of the CFT corresponds to a gauge symmetry in the bulk. 89 ' 90 This global symmetry may be weakly gauged, if there is a normalizable zero mode for the bulk gauge field (as in the case of RSI). But then the CFT modes couple directly to the gauge fields (since they are no longer on the Planck brane), and will give direct contribution to the running of the gauge coupling. In fact, we have seen before that the matching relation between the 4D and the 5D gauge couplings at tree level is

£ = mSm-

(5 67)

-

The extra log appearing in this matching relation exactly reflects this running due to the CFT modes first noted by Pomarol, which is a tree-level effect in the gravity side, however a quantum loop effect on the CFT side. The nicest interpretation of the RSI model is when the gauge fields and fermions are in the bulk, while the Higgs is on the brane. In this case there is a CFT that is slowly running, the gauge and matter fields are coupled to the CFT, but are not composites. At some scale the slow running of the CFT will suddenly turn into strong interactions breaking the conformal symmetry, and producing a composite Higgs particle, which then breaks the electroweak symmetry. Thus this should be interpreted as a walking

690

technicolor model with a composite Higgs. 89 Prom the AdS/CFT point of view we now also understand why we got the large corrections to the W and Z masses: in the SM the relation of these masses is protected by a global symmetry called custodial SU(2). However, we have seen that if there was a global symmetry like this, we would need to see bulk gauge fields. So the real problem is that the strong interaction (the CFT) does not obey the custodial SU(2) symmetry. The resolution is simple: 85 simply impose this additional global SU(2) by putting SU(2)i x SU(2)R gauge bosons in the bulk, and break the SU(2),R on the Planck brane to eliminate its zero mode. This way custodial SU(2) will be restored and the corrections to electroweak observables will be smaller. However, this model would still correspond to a walking technicolor with a composite Higgs. One may ask, what would real technicolor (without a Higgs) then correspond to? In that case it is the strong interactions themselves (the appearance of the negative tension brane) which should break electroweak symmetry, rather than a composite Higgs on the TeV brane. This means that to get real technicolor, one needs to break electroweak symmetries by BC's on the TeV brane. A model for this has been recently proposed in Ref. 91, (see also Ref. 92).

6. Epilogue In these lectures I was trying to summarize some of the important/interesting topics in theories with extra dimensions. These notes should be sufficient for an advanced graduate student to get started on research in this area. However, the topics covered here are by far not complete. Some examples of entire fields left out are brane cosmology,93'94 higher dimensional warped space, 95 the self-tuning approach to the cosmological constant problem, 49 1/TeV size extra dimensions, 96 dark matter from Kaluza-Klein modes, 97 unification in warped extra dimensions, 98 black holes in the RS models, 99 electroweak symmetry breaking using the fifth component of the gauge field as a Higgs, 100-102 supersymmetric RandallSundrum models, 74,103 deconstruction of extra dimensions, 104-106 and the list could go on and on. That the list of omitted topics is so extensive shows how much interest there is currently in this subject. Whether any of these ideas will become reality or remain speculation forever can only be decided by experiment. All of those in the field hope that in ten years some of the topics listed will no longer be speculation about the behavior of nature at high energies, but undeniable facts established by experiments at the LHC. Until then we will happily speculate on...

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PART III: Astroparticle Physics Lecture Series

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SEAN M. CARROLL

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I N T R O D U C T I O N TO COSMOLOGY

MARK TRODDEN Department of Physics, Syracuse University Syracuse, NY 13244-1130, USA SEAN M. CARROLL Enrico Fermi Institute, Department of Physics, and Center for Cosmological Physics University of Chicago 5640 S. Ellis Avenue, Chicago, IL 60637, USA These proceedings summarize lectures that were and 2003 Theoretical Advanced Study Institutes (TASI) at the University of Colorado at Boulder. a pedagogical introduction to cosmology aimed at particle physics and string theory.

delivered as part of the 2002 in elementary particle physics They are intended to provide advanced graduate students in

Contents 1

Introduction

705

2

Fundamentals of the Standard Cosmology 2.1 Homogeneity and Isotropy: The Robertson-Walker Metric 2.2 Dynamics: The Friedmann Equations 2.3 Flat Universes 2.4 Including Curvature 2.5 Horizons 2.6 Geometry, Destiny and Dark Energy

706 706 709 714 714 716 718

3

Our 3.1 3.2 3.3 3.4 3.5

719 719 722 725 730 736

Universe Today and Dark Energy Matter: Ordinary and Dark Supernovae and the Accelerating Universe The Cosmic Microwave Background The Cosmological Constant Problem(s) Dark Energy, or Worse? 703

704

4

Early Times in the Standard Cosmology 4.1 Describing Matter 4.2 Particles in Equilibrium 4.3 Thermal Relics 4.4 Vacuum displacement 4.5 Primordial Nucleosynthesis 4.6 Finite Temperature Phase Transitions 4.7 Topological Defects 4.8 Baryogenesis 4.9 Baryon Number Violation 4.9.1 B-violation in Grand Unified Theories 4.9.2 B-violation in the Electroweak Theory 4.9.3 CP Violation 4.9.4 Departure from Thermal Equilibrium 4.9.5 Baryogenesis via Leptogenesis 4.9.6 Affleck-Dine Baryogenesis

741 742 743 748 750 751 753 756 763 765 765 765 767 768 769 770

5

Inflation 5.1 The Flatness Problem 5.2 The Horizon Problem 5.3 Unwanted Relics 5.4 The General Idea of Inflation 5.5 Slowly-Rolling Scalar Fields 5.6 Attractor Solutions in Inflation 5.7 Solving the Problems of the Standard Cosmology 5.8 Vacuum Fluctuations and Perturbations 5.9 Reheating and Preheating 5.10 The Beginnings of Inflation

771 771 772 774 775 775 778 779 780 783 784

705

1. Introduction The last decade has seen an explosive increase in both the volume and the accuracy of data obtained from cosmological observations. The number of techniques available to probe and cross-check these data has similarly proliferated in recent years. Theoretical cosmologists have not been slouches during this time, either. However, it is fair to say that we have not made comparable progress in connecting the wonderful ideas we have to explain the early universe to concrete fundamental physics models. One of our hopes in these lectures is to encourage the dialogue between cosmology, particle physics, and string theory that will be needed to develop such a connection. In this paper, we have combined material from two sets of TASI lectures (given by SMC in 2002 and MT in 2003). We have taken the opportunity to add more detail than was originally presented, as well as to include some topics that were originally excluded for reasons of time. Our intent is to provide a concise introduction to the basics of modern cosmology as given by the standard "ACDM" Big-Bang model, as well as an overview of topics of current research interest. In Lecture 1 we present the fundamentals of the standard cosmology, introducing evidence for homogeneity and isotropy and the FriedmannRobertson-Walker models that these make possible. In Lecture 2 we consider the actual state of our current universe, which leads naturally to a discussion of its most surprising and problematic feature: the existence of dark energy. In Lecture 3 we consider the implications of the cosmological solutions obtained in Lecture 1 for early times in the universe. In particular, we discuss thermodynamics in the expanding universe, finite-temperature phase transitions, and baryogenesis. Finally, Lecture 4 contains a discussion of the problems of the standard cosmology and an introduction to our best-formulated approach to solving them - the inflationary universe. Our review is necessarily superficial, given the large number of topics relevant to modern cosmology. More detail can be found in several excellent textbooks [1-7]. Throughout the lectures we have borrowed liberally (and sometimes verbatim) from earlier reviews of our own [8-15]. Our metric signature is —h + + . We use units in which h = c = 1, and define the reduced Planck mass by MP = (87rG)~ 1/2 ~ 1018GeV.

706

2. Fundamentals of the Standard Cosmology 2.1. Homogeneity Metric

and Isotropy:

The

Robertson-Walker

Cosmology as the application of general relativity (GR) to the entire universe would seem a hopeless endeavor were it not for a remarkable fact the universe is spatially homogeneous and isotropic on the largest scales. "Isotropy" is the claim that the universe looks the same in all direction. Direct evidence comes from the smoothness of the temperature of the cosmic microwave background, as we will discuss later. "Homogeneity" is the claim that the universe looks the same at every point. It is harder to test directly, although some evidence comes from number counts of galaxies. More traditionally, we may invoke the "Copernican principle," that we do not live in a special place in the universe. Then it follows that, since the universe appears isotropic around us, it should be isotropic around every point; and a basic theorem of geometry states that isotropy around every point implies homogeneity We may therefore approximate the universe as a spatially homogeneous and isotropic three-dimensional space which may expand (or, in principle, contract) as a function of time. The metric on such a spacetime is necessarily of the Robertson-Walker (RW) form, as we now demonstrate. a Spatial isotropy implies spherical symmetry. Choosing a point as an origin, and using coordinates (r, 6, ) around this point, the spatial line element must take the form da2 =dr2+f(r)(d92

+sin2 ed<j>2) ,

(1)

where f(r) is a real function, which, if the metric is to be nonsingular at the origin, obeys f(r) ~ r as r —> 0. Now, consider figure 1 in the 8 = n/2 plane. In this figure DH = HE = r, both DE and 7 are small and HA — x. Note that the two angles labeled 7 are equal because of homogeneity and isotropy. Now, note that EF ~ EF' = / ( 2 r ) 7 = /(r)/J .

(2)

Also AC = 7 / ( r + x) = AB + BC = 7 / ( r - x) + pf(x) a

.

(3)

O n e of the authors has a sentimental attachment to the following argument, since he learned it in his first cosmology course [16].

707

Figure 1.

Geometry of a homogeneous and isotropic space.

Using (2) to eliminate 0/j, rearranging (3), dividing by 2x and taking the limit x —> oo yields

<1 = IM dr

2/(r) •

l

(4) '

We must solve this subject to / ( r ) ~ r as r —> 0. It is easy to check that if f(r) is a solution then f(r/a) is a solution for constant a. Also, r, sinr and sinhr are all solutions. Assuming analyticity and writing / ( r ) as a power series in r it is then easy to check that, up to scaling, these are the only three possible solutions. Therefore, the most general spacetime metric consistent with homogeneity and isotropy is ds2 = -dt2 + a2(t) [dp2 + f2(p) (d62 + sin2 9d(f>2)] ,

(5)

where the three possibilities for f(p) are f(p) = {sin(p), p,

sinh(p)} .

(6)

This is a purely geometric fact, independent of the details of general relativity. We have used spherical polar coordinates (p, 8, ), since spatial isotropy implies spherical symmetry about every point. The time coordinate t, which is the proper time as measured by a comoving observer (one at constant spatial coordinates), is referred to as cosmic time, and the function a(t) is called the scale factor.

708

There are two other useful forms for the RW metric, First, a simple change of variables in the radial coordinate yields dsz

-dt2 + al(t)

"

dr2 1 — kr2

(d62 + sin2 6dcj)2

(7)

where C +1 k=l 0 { -1

if /(p) = sin(p) if/(p) = p if f(p) = sinh(p)

(8)

Geometrically, k describes the curvature of the spatial sections (slices at constant cosmic time), k = +1 corresponds to positively curved spatial sections (locally isometric to 3-spheres); k = 0 corresponds to local flatness, and k = — 1 corresponds to negatively curved (locally hyperbolic) spatial sections. These are all local statements, which should be expected from a local theory such as GR. The global topology of the spatial sections may be that of the covering spaces - a 3-sphere, an infinite plane or a 3-hyperboloid - but it need not be, as topological identifications under freely-acting subgroups of the isometry group of each manifold are allowed. As a specific example, the k = 0 spatial geometry could apply just as well to a 3-torus as to an infinite plane. Note that we have not chosen a normalization such that CLQ = 1- We are not free to do this and to simultaneously normalize \k\ = 1, without including explicit factors of the current scale factor in the metric. In the flat case, where k = 0, we can safely choose ao = 1. A second change of variables, which may be applied to either (5) or (7), is to transform to conformal time, r, via rt

'(*)

dif a(t>)

(9)

Applying this to (7) yields ds2 = a2(r)

dr2 +

dr2 + r2 {dOi2 , - 2 + sm 1 — kr2

(10)

where we have written a(r) = a\t{r)] as is conventional. The conformal time does not measure the proper time for any particular observer, but it does simplify some calculations. A particularly useful quantity to define from the scale factor is the Hubble parameter (sometimes called the Hubble constant), given by H . a a

(11)

709

The Hubble parameter relates how fast the most distant galaxies are receding from us to their distance from us via Hubble's law, (12)

v~Hd.

This is the relationship that was discovered by Edwin Hubble, and has been verified to high accuracy by modern observational methods (see figure 2). i

1

1

1

30000

1

'-

1000

s.

* •

X E

. .

t 500 J?

,s 10000

* 0

-

20000

/•

r•

i

- 1 100

Distance [Mpc]

i

i

200 300 Distance [Mpc]

Figure 2. Hubble diagrams showing the relationship between recessional velocities of distant galaxies and their distances. The left plot shows the original data of Hubble (and a rather unconvincing straight-line fit through it). To reassure you, the right plot shows much more recent data, using significantly more distant galaxies (note difference in scale).

2.2. Dynamics:

The Friedmann

Equations

As mentioned, the RW metric is a purely kinematic consequence of requiring homogeneity and isotropy of our spatial sections. We next turn to dynamics, in the form of differential equations governing the evolution of the scale factor a(t). These will come from applying Einstein's equation, R IXV

2R9^

8TTGT.fll>

(13)

to the RW metric. Before diving right in, it is useful to consider the types of energymomentum tensors Ty.v we will typically encounter in cosmology. For simplicity,- and because it is consistent with much we have observed about the

710

universe, it is often useful to adopt the perfect fluid form for the energymomentum tensor of cosmological matter. This form is (14) M

where C/ is the fluid four-velocity, p is the energy density in the rest frame of the fluid and p is the pressure in that same frame. The pressure is necessarily isotropic, for consistency with the RW metric. Similarly, fluid elements will be comoving in the cosmological rest frame, so that the normalized four-velocity in the coordinates of (7) will be tf" = ( 1 , 0 , 0 , 0 ) .

(15)

The energy-momentum tensor thus takes the form (P T

\ (16)

= P9ij

V

J

where gij represents the spatial metric (including the factor of a 2 ). Armed with this simplified description for matter, we are now ready to apply Einstein's equation (13) to cosmology. Using (7) and (14), one obtains two equations. The first is known as the Friedmann equation,

where an overdot denotes a derivative with respect to cosmic time t and i indexes all different possible types of energy in the universe. This equation is a constraint equation, in the sense that we are not allowed to freely specify the time derivative a; it is determined in terms of the energy density and curvature. The second equation, which is an evolution equation, is

It is often useful to combine (17) and (18) to obtain the acceleration equation a

a

_

4TTG •

• £ > i + 3p0 •

(19)

In fact, if we know the magnitudes and evolutions of the different energy density components pi, the Friedmann equation (17) is sufficient to solve for the evolution uniquely. The acceleration equation is conceptually useful, but rarely invoked in calculations.

711

The Friedmann equation relates the rate of increase of the scale factor, as encoded by the Hubble parameter, to the total energy density of all matter in the universe. We may use the Friedmann equation to define, at any given time, a critical energy density, (20) " ^ for which the spatial sections must be precisely flat (k = 0). We then define the density parameter

fitotal =

, (21) Pc which allows us to relate the total energy density in the universe to its local geometry via ^total > 1 <=> k = +1 fttotai = 1 «• k = 0

(22)

fitotal < l « f c = - l . It is often convenient to define the fractions of the critical energy density in each different component by fii = ^

.

(23)

Pc

Energy conservation is expressed in GR by the vanishing of the covariant divergence of the energy-momentum tensor, V^T^

=0.

(24)

Applying this to our assumptions - the RW metric (7) and perfect-fluid energy-momentum tensor (14) - yields a single energy-conservation equation, p + 3H(p + p) = 0 .

(25)

This equation is actually not independent of the Friedmann and acceleration equations, but is required for consistency. It implies that the expansion of the universe (as specified by H) can lead to local changes in the energy •density. Note that there is no notion of conservation of "total energy," as energy can be interchanged between matter and the spacetime geometry. One final piece of information is required before we can think about solving our cosmological equations: how the pressure and energy density are related to each other. Within the fluid approximation used here, we may assume that the pressure is a single-valued function of the energy density

712

p = p(p). It is often convenient to define an equation of state parameter, w, by p = wp .

(26)

This should be thought of as the instantaneous definition of the parameter w; it need represent the full equation of state, which would be required to calculate the behavior of fluctuations. Nevertheless, many useful cosmological matter sources do obey this relation with a constant value of w. For example, w = 0 corresponds to pressureless matter, or dust - any collection of massive non-relativistic particles would qualify. Similarly, w = 1/3 corresponds to a gas of radiation, whether it be actual photons or other highly relativistic species. A constant w leads to a great simplification in solving our equations. In particular, using (25), we see that the energy density evolves with the scale factor according to

P{a) K

^fefSJ •

(2?)

Note that the behaviors of dust (w = 0) and radiation (w = 1/3) are consistent with what we would have obtained by more heuristic reasoning. Consider a fixed comoving volume of the universe - i.e. a volume specified by fixed values of the coordinates, from which one may obtain the physical volume at a given time t by multiplying by a(t)3. Given a fixed number of dust particles (of massTO)within this comoving volume, the energy density will then scale just as the physical volume, i.e. as a(£)~ 3 , in agreement with (27), with w = 0. To make a similar argument for radiation, first note that the expansion of the universe (the increase of a(t) with time) results in a shift to longer wavelength A, or a redshift, of photons propagating in this background. A photon emitted with wavelength Ae at a time te, at which the scale factor is ae = a(te) is observed today (t = to, with scale factor oto = a{to)) at wavelength A0, obeying ^ = ^ = l + z. (28)' Ae ae The redshift z is often used in place of the scale factor. Because of the redshift, the energy density in a fixed number of photons in a fixed comoving volume drops with the physical volume (as for dust) and by an extra factor of the scale factor as the expansion of the universe stretches the wavelengths

713

of light. Thus, the energy density of radiation will scale as a(£)~ 4 , once again in agreement with (27), with w = 1/3. Thus far, we have not included a cosmological constant A in the gravitational equations. This is because it is equivalent to treat any cosmological constant as a component of the energy density in the universe. In fact, adding a cosmological constant A to Einstein's equation is equivalent to including an energy-momentum tensor of the form

This is simply a perfect fluid with energy-momentum tensor (14) with A PA PA

8TTG

=

,

-PA

(30)

so that the equation-of-state parameter is wA = - 1 •

(31)

This implies that the energy density is constant, PA = constant .

(32)

Thus, this energy is constant throughout spacetime; we say that the cosmological constant is equivalent to vacuum energy. Similarly, it is sometimes useful to think of any nonzero spatial curvature as yet another component of the cosmological energy budget, obeying 3fc Pcurv —Pcurv

=

87rGa2 k 8^G^ '

(33)

so that Wcurv = - 1 / 3 .

(34)

It is not an energy density, of course; pCurv is simply a convenient way to keep track of how much energy density is lacking, in comparison to a flat universe.

714

Type of Energy Dust Radiation Cosmological Constant

2.3. Flat

p(a) a a-4 constant

a{t) -jSJT fl/2

Universes

It is much easier to find exact solutions to cosmological equations of motion when k = 0. Fortunately for us, nowadays we are able to appeal to more than mathematical simplicity to make this choice. Indeed, as we shall see in later lectures, modern cosmological observations, in particular precision measurements of the cosmic microwave background, show the universe today to be extremely spatially flat. In the case of flat spatial sections and a constant equation of state parameter w, we may exactly solve the Friedmann equation (27) to obtain /

t

\ 2/3(1+™)

a(t) = a0 I-J

,

(35)

where ao is the scale factor today, unless w = — 1, in which case one obtains a(t) oc eHt. Applying this result to some of our favorite energy density sources yields table 1. Note that the matter- and radiation-dominated flat universes begin with a = 0; this is a singularity, known as the Big Bang. We can easily calculate the age of such a universe: f1

da

(36) 3 ( 1 + «;)#(, ' Unless w is close to —1, it is often useful to approximate this answer by to

Jo aH{a) z)

t0 ~ Ho1 .

(37)

It is for this reason that the quantity HQ is known as the Hubble time, and provides a useful estimate of the time scale for which the universe has been around. 2.4. Including

Curvature

It is true that we know observationally that the universe today is flat to a high degree of accuracy. However, it is instructive, and useful when considering early cosmology, to consider how the solutions we have already identified change when curvature is included. Since we include this mainly for

715

illustration we will focus on the separate cases of dust-filled and radiationfilled FRW models with zero cosmological constant. This calculation is an example of one that is made much easier by working in terms of conformal time r. Let us first consider models in which the energy density is dominated by matter (w = 0). In terms of conformal time the Einstein equations become 3(fc + h2) = 8TrGpa2 k + h2 + 2ti = 0 ,

(38)

where a prime denotes a derivative with respect to conformal time and h(r) = a' J a. These equations are then easily solved for h{r) giving cot(r/2) k= 1 2/r k=0 . coth(r/2) fc = - 1

1 — COS(T) T2/2

fc cosh(r) — 1

(39)

k = 1

= 0 . k = —1

(40)

One may use this to derive the connection between cosmic time and conformal time, which here is

{

T — sin(r)

k=1

r3/6 fc = 0 . (41) sinh(r) — r k = —1 Next we consider models dominated by radiation (w = 1/3). In terms of conformal time the Einstein equations become 3(fc + h2) = 8nGpa2 k + h2 +

2h' = -^f^a2

.

(42)

Solving as we did above yields

fc<

h(r) = <

1/T

k=0

,

(43)

a(r) oc {

T

k=0

,

(44)

716

and

{

1 — COS(T)

k = 1

r2/2 k=0 . (45) cosh(r) - 1 k = -1 It is straightforward to interpret these solutions by examining the behavior of the scale factor a(r); the qualitative features are the same for matter- or radiation-domination. In both cases, the universes with positive curvature (k = +1) expand from an initial singularity with a = 0, and later recollapse again. The initial singularity is the Big Bang, while the final singularity is sometimes called the Big Crunch. The universes with zero or negative curvature begin at the Big Bang and expand forever. This behavior is not inevitable, however; we will see below how it can be altered by the presence of vacuum energy. 2.5. Horizons One of the most crucial concepts to master about FRW models is the existence of horizons. This concept will prove useful in a variety of places in these lectures, but most importantly in understanding the shortcomings of what we are terming the standard cosmology. Suppose an emitter, e, sends a light signal to an observer, o, who is at r = 0. Setting 9 = constant and = constant and working in conformal time, for such radial null rays we have T0 — T = r. In particular this means that T0~Te=re

.

(46)

Now suppose r e is bounded below by fe; for example, fe might represent the Big Bang singularity. Then there exists a maximum distance to which the observer can see, known as the particle horizon distance, given by ?"ph (T G ) =T0-fe

.

(47)

The physical meaning of this is illustrated in figure 3. Similarly, suppose T0 is bounded above by f0. Then there exists a limit to spacetime events which can be influenced by the emitter. This limit is known as the event horizon distance, given by reh(T0) = f0 - Te ,

with physical meaning illustrated in figure 4.

(48)

717

Particles already seen Particles not yet seen

T=T n

T=T„ r

Ph

(x

^

r=0

V^

Figure 3. Particle horizons arise when the past light cone of an observer o terminates at a finite conformal time. Then there will be worldlines of other particles which do not intersect the past of o, meaning that they were never in causal contact.

T=T„

Never receives message Receives message from emitter at x e Figure 4. Event horizons arise when the future light cone of an observer o terminates at a finite conformal time. Then there will be worldlines of other particles which do not intersect the future of o,- meaning that they cannot possibly influence each other.

These horizon distances may be converted to proper horizon distances at cosmic time t, for example f dH == a ( r ) r p h =
dt1

-—

(49)

Just as the Hubble time HQ1 provides a rough guide for the age of the universe, the Hubble distance CHQ1 provides a rough estimate of the horizon distance in a matter- or radiation-dominated universe.

718

2.6. Geometry,

Destiny

and Dark

Energy

In subsequent lectures we will use what we have learned here to extrapolate back to some of the earliest times in the universe. We will discuss the thermodynamics of the early universe, and the resulting interdependency between particle physics and cosmology. However, before that, we would like to explore some implications for the future of the universe. For a long time in cosmology, it was quite commonplace to refer to the three possible geometries consistent with homogeneity and isotropy as closed (fc = 1), open (k — —1) and flat (k = 0). There were two reasons for this. First, if one considered only the universal covering spaces, then a positively curved universe would be a 3-sphere, which has finite volume and hence is closed, while a negatively curved universe would be the hyperbolic 3-manifold H3, which has infinite volume and hence is open. Second, with dust and radiation as sources of energy density, universes with greater than the critical density would ultimately collapse, while those with less than the critical density would expand forever, with flat universes lying on the border between the two. for the case of pure dust-filled universes this is easily seen from (40) and (44). As we have already mentioned, GR is a local theory, so the first of these points was never really valid. For example, there exist perfectly good compact hyperbolic manifolds, of finite volume, which are consistent with all our cosmological assumptions. However, the connection between geometry and destiny implied by the second point above was quite reasonable as long as dust and radiation were the only types of energy density relevant in the late universe. In recent years it has become clear that the dominant component of energy density in the present universe is neither dust nor radiation, but rather is dark energy. This component is characterized by an equation of state parameter w < —1/3. We will have a lot more to say about this component (including the observational evidence for it) in the next lecture, but for now we would just like to focus on the way in which it has completely separated our concepts of geometry and destiny. For simplicity, let's focus on what happens if the only energy density in the universe is a cosmological constant, with w = — 1. In this case, the Friedmann equation may be solved for any value of the spatial curvature parameter k. If A > 0 then the solutions are

719

cosh (J%t\

k = +1

— = < exp (^t) k=0 , (50) a ° / r-\ [ sinh ( J f t ) A; = - 1 where we have encountered the k = 0 case earlier. It is immediately clear that, in the t —-> oo limit, all solutions expand exponentially, independently of the spatial curvature. In fact, these solutions are all exactly the same spacetime - de Sitter space - just in different coordinate systems. These features of de Sitter space will resurface crucially when we discuss inflation. However, the point here is that the universe clearly expands forever in these spacetimes, irrespective of the value of the spatial curvature. Note, however, that not all of the solutions in (50) actually cover all of de Sitter space; the k = 0 and k = — 1 solutions represent coordinate patches which only cover part of the manifold. For completeness, let us complete the description of spaces with a cosmological constant by considering the case A < 0. This spacetime is called Anti-de Sitter space (AdS) and it should be clear from the Friedmann equation that such a spacetime can only exist in a space with spatial curvature k = — 1. The corresponding solution for the scale factor is a(t) = ao sin I y — —t I .

(51)

Once again, this solution does not cover all of AdS; for a more complete discussion, see [17]. 3. Our Universe Today and Dark Energy In the previous lecture we set up the tools required to analyze the kinematics and dynamics of homogeneous and isotropic cosmologies in general relativity. In this lecture we turn to the actual universe in which we live, and discuss the remarkable properties cosmologists have discovered in the last ten years. Most remarkable among them is the fact that the universe is dominated by a uniformly-distributed and slowly-varying source of "dark energy," which may be a vacuum energy (cosmological constant), a dynamical field, or something even more dramatic. 3.1. Matter:

Ordinary

and

Dark

In the years before we knew that dark energy was an important constituent of the universe, and before observations of galaxy distributions and CMB

720

anisotropies had revolutionized the study of structure in the universe, observational cosmology sought to measure two numbers: the Hubble constant Ho and the matter density parameter QM- Both of these quantities remain undeniably important, even though we have greatly broadened the scope of what we hope to measure. The Hubble constant is often parameterized in terms of a dimensionless quantity ft as H0 = 100ft km/sec/Mpc .

(52)

After years of effort, determinations of this number seem to have zeroed in on a largely agreed-upon value; the Hubble Space Telescope Key Project on the extragalactic distance scale [18] finds ft = 0.71 ± 0 . 0 6 ,

(53)

which is consistent with other methods [19], and what we will assume henceforth. For years, determinations of CIM based on dynamics of galaxies and clusters have yielded values between approximately 0.1 and 0.4, noticeably smaller than the critical density. The last several years have witnessed a number of new methods being brought to bear on the question; here we sketch some of the most important ones. The traditional method to estimate the mass density of the universe is to "weigh" a cluster of galaxies, divide by its luminosity, and extrapolate the result to the universe as a whole. Although clusters are not representative samples of the universe, they are sufficiently large that such a procedure has a chance of working. Studies applying the virial theorem to cluster dynamics have typically obtained values QM = 0.2 ± 0.1 [20-22]. Although it is possible that the global value of M/L differs appreciably from its value in clusters, extrapolations from small scales do not seem to reach the critical density [23]. New techniques to weigh the clusters, including gravitational lensing of background galaxies [24] and temperature profiles of the X-ray gas [25], while not yet in perfect agreement with each other, reach essentially similar conclusions. Rather than measuring the mass relative to the luminosity density, which may be different inside and outside clusters, we can also measure it with respect to the baryon density [26], which is very likely to have the same value in clusters as elsewhere in the universe, simply because there is no way to segregate the baryons from the dark matter on such large scales. Most of the baryonic mass is in the hot intracluster gas [27], and the fraction / g a s of total mass in this form can be measured either by direct

721

observation of X-rays from the gas [28] or by distortions of the microwave background by scattering off hot electrons (the Sunyaev-Zeldovich effect) [29], typically yielding 0.1 < / g a s < 0.2. Since primordial nucleosynthesis provides a determination of OB ~ 0.04, these measurements imply n M = fiB//ga8 = 0 . 3 ± 0 . 1 ,

(54)

consistent with the value determined from mass to light ratios. Another handle on the density parameter in matter comes from properties of clusters at high redshift. The very existence of massive clusters has been used to argue in favor of D,M ~ 0.2 [30], and the lack of appreciable evolution of clusters from high redshifts to the present [31,32] provides additional evidence that OM < 1.0. On the other hand, a recent measurement of the relationship between the temperature and luminosity of X-ray clusters measured with the XMM-Newton satellite [33] has been interpreted as evidence for (1M near unity. This last result seems at odds with a variety of other determinations, so we should keep a careful watch for further developments in this kind of study. The story of large-scale motions is more ambiguous. The peculiar velocities of galaxies are sensitive to the underlying mass density, and thus to Q,M, but also to the "bias" describing the relative amplitude of fluctuations in galaxies and mass [21,34]. Nevertheless, recent advances in very large redshift surveys have led to relatively firm determinations of the mass density; the 2df survey, for example, finds 0.1 < ^ M < 0.4 [35]. Finally, the matter density parameter can be extracted from measurements of the power spectrum of density fluctuations (see for example [36]). As with the CMB, predicting the power spectrum requires both an assumption of the correct theory and a specification of a number of cosmological parameters. In simple models (e.g., with only cold dark matter and baryons, no massive neutrinos), the spectrum can be fit (once the amplitude is normalized) by a single "shape parameter", which is found to be equal to T = H M ^ . (For more complicated models see [37].) Observations then yield V ~ 0.25, or Q,M ~ 0.36. For a more careful comparison between models and observations, see [38-41]. Thus, we have a remarkable convergence on values for the density parameter in matter: 0.1 < ftM < 0.4 .

(55)

As we will see below, this value is in excellent agreement with that which we would determine indirectly from combinations of other measurements.

722

As you are undoubtedly aware, however, matter comes in different forms; the matter we infer from its gravitational influence need not be the same kind of ordinary matter we are familiar with from our experience on Earth. By "ordinary matter" we mean anything made from atoms and their constituents (protons, neutrons, and electrons); this would include all of the stars, planets, gas and dust in the universe, immediately visible or otherwise. Occasionally such matter is referred to as "baryonic matter", where "baryons" include protons, neutrons, and related particles (strongly interacting particles carrying a conserved quantum number known as "baryon number"). Of course electrons are conceptually an important part of ordinary matter, but by mass they are negligible compared to protons and neutrons; the mass of ordinary matter comes overwhelmingly from baryons. Ordinary baryonic matter, it turns out, is not nearly enough to account for the observed matter density. Our current best estimates for the baryon density [42,43] yield flb = 0.04 ± 0.02 ,

(56)

where these error bars are conservative by most standards. This determination comes from a variety of methods: direct counting of baryons (the least precise method), consistency with the CMB power spectrum (discussed later in this lecture), and agreement with the predictions of the abundances of light elements for Big-Bang nucleosynthesis (discussed in the next lecture). Most of the matter density must therefore be in the form of non-baryonic dark matter, which we will abbreviate to simply "dark matter". (Baryons can be dark, but it is increasingly common to reserve the terminology for the non-baryonic component.) Essentially every known particle in the Standard Model of particle physics has been ruled out as a candidate for this dark matter. One of the few things we know about the dark matter is that is must be "cold" — not only is it non-relativistic today, but it must have been that way for a very long time. If the dark matter were "hot", it would have free-streamed out of overdense regions, suppressing the formation of galaxies. The other thing we know about cold dark matter (CDM) is that it should interact very weakly with ordinary matter, so as to have escaped detection thus far. In the next lecture we will discuss some currently popular candidates for cold dark matter. 3.2. Supernovae

and the Accelerating

Universe

The great story of fin de siecle cosmology was the discovery that matter does not dominate the universe; we need some form of dark energy to explain

723

a variety of observations. The first direct evidence for this finding came from studies using Type la supernovae as "standardizable candles," which we now examine. For more detailed discussion of both the observational situation and the attendant theoretical problems, see [45,46,8,47,48,15]. Supernovae are rare — perhaps a few per century in a Milky-Way-sized galaxy — but modern telescopes allow observers to probe very deeply into small regions of the sky, covering a very large number of galaxies in a single observing run. Supernovae are also bright, and Type la's in particular all seem to be of nearly uniform intrinsic luminosity (absolute magnitude M ~ —19.5, typically comparable to the brightness of the entire host galaxy in which they appear) [49]. They can therefore be detected at high redshifts (z ~ 1), allowing in principle a good handle on cosmological effects [50,51]. The fact that all SNe la are of similar intrinsic luminosities fits well with our understanding of these events as explosions which occur when a white dwarf, onto which mass is gradually accreting from a companion star, crosses the Chandrasekhar limit and explodes. (It should be noted that our understanding of supernova explosions is in a state of development, and theoretical models are not yet able to accurately reproduce all of the important features of the observed events. See [52-54] for some recent work.) The Chandrasekhar limit is a nearly-universal quantity, so it is not a surprise that the resulting explosions are of nearly-constant luminosity. However, there is still a scatter of approximately 40% in the peak brightness observed in nearby supernovae, which can presumably be traced to differences in the composition of the white dwarf atmospheres. Even if we could collect enough data that statistical errors could be reduced to a minimum, the existence of such an uncertainty would cast doubt on any attempts to study cosmology using SNe la as standard candles. Fortunately, the observed differences in peak luminosities of SNe la are very closely correlated with observed differences in the shapes of their light curves: dimmer SNe decline more rapidly after maximum brightness, while brighter SNe decline more slowly [55-57]. There is thus a one-parameter family of events, and measuring the behavior of the light curve along with the apparent luminosity allows us to largely correct for the intrinsic differences in brightness, reducing the scatter from 40% to less than 15% — sufficient precision to distinguish between cosmological models. (It seems likely that the single parameter can be traced to the amount of 56 Ni produced in the supernova explosion; more nickel implies both a higher peak luminosity and a higher temperature and thus opacity, leading to a slower decline. It would be an exaggeration, however, to claim that this behavior is well-understood theoretically.)

724

SCP 2003

>-*

24

1-

jf-:"'*;.

22 -

f?

./

20

18

0.2:5,0.7? 0.25,0.0(1 1.00,0.00

18

_l_

14 1.0 c

0.5

sl 0.0

1 fetrz+Mtrfri ^ rn

p
Figure 5.

0.0

0.2

0.4

0.6

0.8

1.0

Hubble diagram from the Supernova Cosmology Project, as of 2003 [67].

Following pioneering work reported in [58], two independent groups undertook searches for distant supernovae in order to measure cosmological parameters: the High-Z Supernova Team [59-63], and the Supernova Cosmology Project [64-67]. A plot of redshift vs. corrected apparent magnitude from the original SCP data is shown in Figure 5. The data are much better fit by a universe dominated by a cosmological constant than by a flat matter-dominated model. In fact the supernova results alone allow a substantial range of possible values of ^ M and il\; however, if we think we

725

know something about one of these parameters, the other will be tightly constrained. In particular, if Clyi ~ 0.3, we obtain nA ~ 0.7 .

(57)

This corresponds to a vacuum energy density p A ~ 10" 8 erg/cm 3 ~ ( 1 0 - 3 eV) 4 .

(58)

Thus, the supernova studies have provided direct evidence for a nonzero value for Einstein's cosmological constant. Given the significance of these results, it is natural to ask what level of confidence we should have in them. There are a number of potential sources of systematic error which have been considered by the two teams; see the original papers [60,61,66] for a thorough discussion. Most impressively, the universe implied by combining the supernova results with direct determinations of the matter density is spectacularly confirmed by measurements of the cosmic microwave background, as we discuss in the next section. Needless to say, however, it would be very useful to have a better understanding of both the theoretical basis for Type la luminosities, and experimental constraints on possible systematic errors. Future experiments, including a proposed satellite dedicated to supernova cosmology [68], will both help us improve our understanding of the physics of supernovae and allow a determination of the distance/redshift relation to sufficient precision to distinguish between the effects of a cosmological constant and those of more mundane astrophysical phenomena. 3.3. The Cosmic

Microwave

Background

Most of the radiation we observe in the universe today is in the form of an almost isotropic blackbody spectrum, with temperature approximately 2.7K, known as the Cosmic Microwave Background (CMB). The small angular fluctuations in temperature of the CMB reveal a great deal about the constituents of the universe, as we now discuss. We have mentioned several times the way in which a radiation gas evolves in and sources the evolution of an expanding FRW universe. It should .be clear from the differing evolution laws for radiation and dust that as one considers earlier and earlier times in the universe, with smaller and smaller scale factors, the ratio of the energy density in radiation to that in matter grows proportionally to l/a(t). Furthermore, even particles which are now massive and contribute to matter used to be hotter, and at

726

sufficiently early times were relativistic, and thus contributed to radiation. Therefore, the early universe was dominated by radiation. At early times the CMB photons were easily energetic enough to ionize hydrogen atoms and therefore the universe was filled with a charged plasma (and hence was opaque). This phase lasted until the photons redshifted enough to allow protons and electrons to combine, during the era of recombination. Shortly after this time, the photons decoupled from the now-neutral plasma and free-streamed through the universe. In fact, the concept of an expanding universe provides us with a clear explanation of the origin of the CMB. Blackbody radiation is emitted by bodies in thermal equilibrium. The present universe is certainly not in this state, and so without an evolving spacetime we would have no explanation for the origin of this radiation. However, at early times, the density and energy densities in the universe were high enough that matter was in approximate thermal equilibrium at each point in space, yielding a blackbody spectrum at early times. We will have more to say about thermodynamics in the expanding universe in our next lecture. However, we should point out one crucial thermodynamic fact about the CMB. A blackbody distribution, such as that generated in the early universe, is such that at temperature T, the energy flux in the frequency range \v, v + dv\ is given by the Planck distribution

P(v,T)dv = 8«hQ3

ehJT_idv,

(59)

where h is Planck's constant and k is the Boltzmann constant. Under a rescaling v —• av, with a=constant, the shape of the spectrum is unaltered if T —• T/a. We have already seen that wavelengths are stretched with the cosmic expansion, and therefore that frequencies will scale inversely due to the same effect. We therefore conclude that the effect of cosmic expansion on an initial blackbody spectrum is to retain its blackbody nature, but just at lower and lower temperatures, T oc l/a .

(60)

This is what we mean when we refer to the universe cooling as it expands. (Note that this strict scaling may be altered if energy is dumped into the radiation background during a phase transition, as we discuss in the next lecture.) The CMB is not a perfectly isotropic radiation bath. Deviations from isotropy at the level of one part in 105 have developed over the last decade into one of our premier precision observational tools in cosmology. The

727

small temperature anisotropies on the sky are usually analyzed by decomposing the signal into spherical harmonics via AT

— =5>, m Y i m (M),

(61)

where aim are expansion coefficients and 9 and

(62)

it is conventional to plot the quantity 1(1 + 1)C; against I in a famous plot that is usually referred to as the CMB power spectrum. An example is shown in figure (6), which shows the measurements of the CMB anisotropy from the recent WMAP satellite, as well as a theoretical model (solid line) that fits the data rather well. These fluctuations in the microwave background are useful to cosmologists for many reasons. To understand why, we must comment briefly on why they occur in the first place. Matter today in the universe is clustered into stars, galaxies, clusters and superclusters of galaxies. Our understanding of how large scale structure developed is that initially small density perturbations in our otherwise homogeneous universe grew through gravitational instability into the objects we observe today. Such a picture requires that from place to place there were small variations in the density of matter at the time that the CMB first decoupled from the photon-baryon plasma. Subsequent to this epoch, CMB photons propagated freely through the universe, nearly unaffected by anything except the cosmic expansion itself. However, at the time of their decoupling, different photons were released from regions of space with slightly different gravitational potentials. Since photons redshift as they climb out of gravitational potentials, photons from some regions redshift slightly more than those from other regions, giving rise to a small temperature anisotropy in the CMB observed today. On smaller scales, the evolution of the plasma has led to intrinsic differences in the temperature from point to point. In this sense the CMB carries with it a fingerprint of the initial conditions that ultimately gave rise to structure in the universe. One very important piece of data that the CMB fluctuations give us is the value of fitotai- Consider an overdense region of size R, which therefore contracts under self-gravity over a timescale R (recall c = 1). If R ;§> HQMB then the region will not have had time to collapse over the lifetime of the universe at last scattering. If R
728

6000

Angular Scale 2°

90°

0.5°

0.2° TT Cross Power Spectrum

5000

— i t I

4000

A - CDM All Data WMAP CBI ACBAR

3 3000

2000

1000

TE Cross Power Spectrum

Reionization

3

10

40

100 200 Multipole moment (/)

400

800

1400

Figure 6. The CMB power spectrum from the W M A P satellite [69]. The error bars on this plot are \-a and the solid line represents the best-fit cosmological model [70]. Also shown is the correlation between the temperature anisotropies and the (B-mode) polarization.

underway at last scattering, matter will have had time to fall into the resulting potential well and cause a resulting rise in temperature which, in turn, gives rise to a restoring force from photon pressure, which acts to damps out the inhomogeneity. Clearly, therefore, the maximum anisotropy will be on a scale which has had just enough time to collapse, but not had enough time to equilibrate;

729

that is R ~ -f^cMB- This means that we expect to see a peak in the CMB power spectrum at an angular size corresponding to the horizon size at last scattering. Since we know the physical size of the horizon at last scattering, this provides us with a ruler on the sky. The corresponding angular scale will then depend on the spatial geometry of the universe. For a flat universe (k = 0, f^totai = 1) we expect a peak at I a 220 and, as can be seen in figure (6), this is in excellent agreement with observations. Beyond this simple heuristic description, careful analysis of all of the features of the CMB power spectrum (the positions and heights of each peak and trough) provide constraints on essentially all of the cosmological parameters. As an example we consider the results from WMAP [70]. For the total density of the universe they find 0.98 < fitotai < 1.08

(63)

at 95% confidence ~ as mentioned, strong evidence for a flat universe. Nevertheless, there is still some degeneracy in the parameters, and much tighter constraints on the remaining values can be derived by assuming either an exactly flat universe, or a reasonable value of the Hubble constant. When for example we assume a flat universe, we can derive values for the Hubble constant, matter density (which then implies the vacuum energy density), and baryon density: h = 0.72 ±0.05 ftM = 1 - ^ A = 0.29 ±0.07 QB = 0.047 ±0.006 . If we instead assume that the Hubble constant is given by the value determined by the HST key project (53), we can derive separate tight constraints on ^ M and ^ A ; these are shown graphically in Figure 7, along with constraints from the supernova experiments. Taking all of the data together, we obtain a remarkably consistent picture of the current constituents of our universe: Q B = 0.04 f^DM = 0.26

QA = 0.7 .

(64)

Our sense of accomplishment at having measured these numbers is substantial, although it is somewhat tempered by the realization that we don't understand any of them. The baryon density is mysterious due to the asymmetry between baryons and antibaryons; as far as dark matter goes,

730

1

0 o« 0.6! G

0.4 0.2 0.0 0.0

0.2

0.4

0.6

a

0.8

1.0

(Verde et al. '05)

Figure 7. Observational constraints in the O M - ^ A plane. The wide green contours represent constraints from supernovae, the vertical blue contours represent constraints from the 2dF galaxy survey, and the small orange contours represent constraints from W M A P observations of CMB anisotropics when a prior on the Hubble parameter is included. Courtesy of Licia Verde; see [71] for details.

of course, we have never detected it directly and only have promising ideas as to what it might be. Both of these issues will be discussed in the next lecture. The biggest mystery is the vacuum energy; we now turn to an exploration of why it is mysterious and what kinds of mechanisms might be responsible for its value. 3.4. The Cosmological

Constant

Problemfs)

In classical general relativity the cosmological constant A is a completely free parameter. It has dimensions of [length]""2 (while the energy density PA has units [energy/volume]), and hence defines a scale, while general relativity is otherwise scale-free. Indeed, from purely classical considerations, we can't even say whether a specific value of A is "large" or "small"; it is simply a constant of nature we should go out and determine through experiment. The introduction of quantum mechanics changes this story somewhat. For one thing, Planck's constant allows us to define the reduced Planck

731

mass Mp ~ 10 18 GeV, as well as the reduced Planck length LP =

~ 1CT32 cm .

(8TTG) 1 / 2

(65)

Hence, there is a natural expectation for the scale of the cosmological constant, namely A (guess)

^

L-2

(gg)

)

or, phrased as an energy density, p feuess)

_

M

4 _ ^1Ql8

Ge

y^4 _

1 Q 112 e r g

/cm3 _

(g?)

We can partially justify this guess by thinking about quantum fluctuations in the vacuum. At all energies probed by experiment to date, the world is accurately described as a set of quantum fields (at higher energies it may become strings or something else). If we take the Fourier transform of a free quantum field, each mode of fixed wavelength behaves like a simple harmonic oscillator. ("Free" means "noninteracting"; for our purposes this is a very good approximation.) As we know from elementary quantum mechanics, the ground-state or zero-point energy of an harmonic oscillator with potential V(x) = | w 2 x 2 is EQ = \hw. Thus, each mode of a quantum field contributes to the vacuum energy, and the net result should be an integral over all of the modes. Unfortunately this integral diverges, so the vacuum energy appears to be infinite. However, the infinity arises from the contribution of modes with very small wavelengths; perhaps it was a mistake to include such modes, since we don't really know what might happen at such scales. To account for our ignorance, we could introduce a cutoff energy, above which ignore any potential contributions, and hope that a more complete theory will eventually provide a physical justification for doing so. If this cutoff is at the Planck scale, we recover the estimate (67). The strategy of decomposing a free field into individual modes and assigning a zero-point energy to each one really only makes sense in a flat spacetime background. In curved spacetime we can still "renormalize" the vacuum energy, relating the classical parameter to the quantum value by an infinite constant. After renormalization, the vacuum energy is completely arbitrary, just as it was in the original classical theory. But when we use general relativity we are really using an effective field theory to describe a certain limit of quantum gravity. In the context of effective field theory, if a parameter has dimensions [mass]", we expect the corresponding

732

mass parameter to be driven up to the scale at which the effective description breaks down. Hence, if we believe classical general relativity up to the Planck scale, we would expect the vacuum energy to be given by our original guess (67). However, we claim to have measured the vacuum energy (58). The observed value is somewhat discrepant with our theoretical estimate: /><£"> ~ 10- 1 2 0 p(.fr s ) .

(68)

This is the famous 120-orders-of-magnitude discrepancy that makes the cosmological constant problem such a glaring embarrassment. Of course, it is a little unfair to emphasize the factor of 10 120 , which depends on the fact that energy density has units of [energy]4. We can express the vacuum energy in terms of a mass scale, pvac = M4ac ,

(69)

so our observational result is M^°cbs) ~ 1(T 3 eV .

(70)

Ml2cs) ~ l(T 30 Mte c uess ) .

(71)

The discrepancy is thus

We should think of the cosmological constant problem as a discrepancy of 30 orders of magnitude in energy scale. In addition to the fact that it is very small compared to its natural value, the vacuum energy presents an additional puzzle: the coincidence between the observed vacuum energy and the current matter density. Our best-fit universe (64) features vacuum and matter densities of the same order of magnitude, but the ratio of these quantities changes rapidly as the universe expands: "M

PM

As a consequence, at early times the vacuum energy was negligible in comparison to matter and radiation, while at late times matter and radiation are negligible. There is only a brief epoch of the universe's history during which it would be possible to witness the transition from domination by one type of component to another. To date, there are not any especially promising approaches to calculating the vacuum energy and getting the right answer; it is nevertheless instructive to consider the example of supersymmetry, which relates to the

733

cosmological constant problem in an interesting way. Supersymmetry posits that for each fermionic degree of freedom there is a matching bosonic degree of freedom, and vice-versa. By "matching" we mean, for example, that the spin-1/2 electron must be accompanied by a spin-0 "selectron" with the same mass and charge. The good news is that, while bosonic fields contribute a positive vacuum energy, for fermions the contribution is negative. Hence, if degrees of freedom exactly match, the net vacuum energy sums to zero. Supersymmetry is thus an example of a theory, other than gravity, where the absolute zero-point of energy is a meaningful concept. (This can be traced to the fact that supersymmetry is a spacetime symmetry, relating particles of different spins.) We do not, however, live in a supersymmetric state; there is no selectron with the same mass and charge as an electron, or we' would have noticed it long ago. If supersymmetry exists in nature, it must be broken at some scale M s u s y . In a theory with broken supersymmetry, the vacuum energy is not expected to vanish, but to be of order Mvac ~ Msusy ,

(theory)

(73)

with p v a c = M^ac- What should M s u s y be? One nice feature of supersymmetry is that it helps us understand the hierarchy problem - why the scale of electroweak symmetry breaking is so much smaller than the scales of quantum gravity or grand unification. For supersymmetry to be relevant to the hierarchy problem, we need the supersymmetry-breaking scale to be just above the electroweak scale, or M s u s y ~ 103 GeV .

(74)

In fact, this is very close to the experimental bound, and there is good reason to believe that supersymmetry will be discovered soon at Fermilab or CERN, if it is connected to electroweak physics. Unfortunately, we are left with a sizable discrepancy between theory and observation: il4°J?s) ~ l(T 1 5 M s u s y .

(experiment)

(75)

Compared to (71), we find that supersymmetry has, in some sense, solved the problem halfway (on a logarithmic scale). This is encouraging, as it at least represents a step in the right direction. Unfortunately, it is ultimately discouraging, since (71) was simply a guess, while (75) is actually a reliable result in this context; supersymmetry renders the vacuum energy finite and calculable, but the answer is still far away from what we need. (Subtleties

734

in supergravity and string theory allow us to add a negative contribution to the vacuum energy, with which we could conceivably tune the answer to zero or some other small number; but there is no reason for this tuning to actually happen.) But perhaps there is something deep about supersymmetry which we don't understand, and our estimate M v a c ~ M s u s y is simply incorrect. What if instead the correct formula were

Mvac ~ (¥j^j

Msusy ?

(76)

In other words, we are guessing that the supersymmetry-breaking scale is actually the geometric mean of the vacuum scale and the Planck scale. Because Mp is fifteen orders of magnitude larger than M s u s y , and M s u s y is fifteen orders of magnitude larger than M v a c , this guess gives us the correct answer! Unfortunately this is simply optimistic numerology; there is no theory that actually yields this answer (although there are speculations in this direction [72]). Still, the simplicity with which we can write down the formula allows us to dream that an improved understanding of supersymmetry might eventually yield the correct result. As an alternative to searching for some formula that gives the vacuum energy in terms of other measurable parameters, it may be that the vacuum energy is not a fundamental quantity, but simply our feature of our local environment. We don't turn to fundamental theory for an explanation of the average temperature of the Earth's atmosphere, nor are we surprised that this temperature is noticeably larger than in most places in the universe; perhaps the cosmological constant is on the same footing. This is the idea commonly known as the "anthropic principle." To make this idea work, we need to imagine that there are many different regions of the universe in which the vacuum energy takes on different values; then we would expect to find ourselves in a region which was hospitable to our own existence. Although most humans don't think of the vacuum energy as playing any role in their lives, a substantially larger value than we presently observe would either have led to a rapid recollapse of the universe (if p v a c were negative) or an inability to form galaxies (if p v a c were positive). Depending on the distribution of possible values of p v a c , one can argue that the observed value is in excellent agreement with what we should expect [73-79]. The idea of environmental selection only works under certain special circumstances, and we are far from understanding whether those conditions hold in our universe. In particular, we need to show that there can be

735

a huge number of different domains with slightly different values of the vacuum energy, and that the domains can be big enough that our entire observable universe is a single domain, and that the possible variation of other physical quantities from domain to domain is consistent with what we observe in ours. Recent work in string theory has lent some support to the idea that there are a wide variety of possible vacuum states rather than a unique one [80-85]. String theorists have been investigating novel ways to compactify extra dimensions, in which crucial roles are played by branes and gauge fields. By taking different combinations of extra-dimensional geometries, brane configurations, and gauge-field fluxes, it seems plausible that a wide variety of states may be constructed, with different local values of the vacuum energy and other physical parameters. An obstacle to understanding these purported solutions is the role of supersymmetry, which is an important part of string theory but needs to be broken to obtain a realistic universe. Prom the point of view of a four-dimensional observer, the compactifications that have small values of the cosmological constant would appear to be exactly the states alluded to earlier, where one begins with a supersymmetric state with a negative vacuum energy, to which supersymmetry breaking adds just the right amount of positive vacuum energy to give a small overall value. The necessary fine-tuning is accomplished simply by imagining that there are many (more than 10 100 ) such states, so that even very unlikely things will sometimes occur. We still have a long way to go before we understand this possibility; in particular, it is not clear that the many states obtained have all the desired properties [86]. Even if such states are allowed, it is necessary to imagine a universe in which a large number of them actually exist in local regions widely separated from each other. As is well known, inflation works to take a small region of space and expand it to a size larger than the observable universe; it is not much of a stretch to imagine that a multitude of different domains may be separately inflated, each with different vacuum energies. Indeed, models of inflation generally tend to be eternal, in the sense that the universe continues to inflate in some regions even after inflation has ended in others [87,88]. Thus, our observable universe may be separated by inflating regions from other "universes" which have landed in different vacuum states; this is precisely what is needed to empower the idea of environmental selection. Nevertheless, it seems extravagant to imagine a fantastic number of separate regions of the universe, outside the boundary of what we can ever

736

possibly observe, just so that we may understand the value of the vacuum energy in our region. But again, this doesn't mean it isn't true. To decide once and for all will be extremely difficult, and will at the least require a much better understanding of how both string theory (or some alternative) and inflation operate - an understanding that we will undoubtedly require a great deal of experimental input to achieve.

3.5. Dark Energy,

or

Worse?

If general relativity is correct, cosmic acceleration implies there must be a dark energy density which diminishes relatively slowly as the universe expands. This can be seen directly from the Eriedmann equation (17), which implies a2 oc a2p + constant .

(77)

Prom this relation, it is clear that the only way to get acceleration (a increasing) in an expanding universe is if p falls off more slowly than a~2; neither matter (PM oc a - 3 ) nor radiation (/9R OC a - 4 ) will do the trick. Vacuum energy is, of course, strictly constant; but the data are consistent with smoothly-distributed sources of dark energy that vary slowly with time. There are good reasons to consider dynamical dark energy as an alternative to an honest cosmological constant. First, a dynamical energy density can be evolving slowly to zero, allowing for a solution to the cosmological constant problem which makes the ultimate vacuum energy vanish exactly. Second, it poses an interesting and challenging observational problem to study the evolution of the dark energy, from which we might learn something about the underlying physical mechanism. Perhaps most intriguingly, allowing the dark energy to evolve opens the possibility of finding a dynamical solution to the coincidence problem, if the dynamics are such as to trigger a recent takeover by the dark energy (independently of, or at least for a wide range of, the parameters in the theory). To date this hope has not quite been met, but dynamical mechanisms at least allow for the possibility (unlike a true cosmological constant). The simplest possibility along these lines involves the same kind of source typically invoked in models of inflation in the very early universe: a scalar field
737

kinetic, gradient, and potential energies, P*=l)2+V() .

(78)

For a homogeneous field (V ~ 0), the equation of motion in an expanding universe is 4> + 3H4> + ~ = 0 . (79) acp If the slope of the potential V is quite flat, we will have solutions for which cf> is nearly constant throughout space and only evolving very gradually with time; the energy density in such a configuration is p^ ss V(e/>) ~ constant .

(80)

Thus, a slowly-rolling scalar field is an appropriate candidate for dark energy. However, introducing dynamics opens up the possibility of introducing new problems, the form and severity of which will depend on the specific kind of model being considered. Most quintessence models feature scalar fields 4> with masses of order the current Hubble scale, m 0 ~ H0 ~ 10" 3 3 eV .

(81)

(Fields with larger masses would typically have already rolled to the minimum of their potentials.) In quantum field theory, light scalar fields are unnatural; renormalization effects tend to drive scalar masses up to the scale of new physics. The well-known hierarchy problem of particle physics amounts to asking why the Higgs mass, thought to be of order 10 11 eV, should be so much smaller than the grand unification/Planck scale, 10 25 10 27 eV. Masses of 1 0 - 3 3 eV are correspondingly harder to understand. On top of that, light scalar fields give rise to long-range forces and timedependent coupling constants that should be observable even if couplings to ordinary matter are suppressed by the Planck scale [95,96]; we therefore need to invoke additional fine-tunings to explain why the quintessence field has not already been experimentally detected. Nevertheless, these apparent fine-tunings might be worth the price, if we were somehow able to explain the coincidence problem. To date, many investigations have considered scalar fields with potentials that asymptote gradually to zero, of the form e1^ or l/. These can have cosmologically interesting properties, including "tracking" behavior that makes the current energy density largely independent of the initial conditions [97]. They do not, however, provide a solution to the coincidence problem, as the era

738

in which the scalar field begins to dominate is still set by finely-tuned parameters in the theory. One way to address the coincidence problem is to take advantage of the fact that matter/radiation equality was a relatively recent occurrence (at least on a logarithmic scale); if a scalar field has dynamics which are sensitive to the difference between matter- and radiation-dominated universes, we might hope that its energy density becomes constant only after matter/radiation equality. An approach which takes this route is fc-essence [98], which modifies the form of the kinetic energy for the scalar field. Instead of a conventional kinetic energy K = \{<j))2, in fc-essence we posit a form K = f(4>)g(<j>2) ,

(82)

where / and g are functions specified by the model. For certain choices of these functions, the fc-essence field naturally tracks the evolution of the total radiation energy density during radiation domination, but switches to being almost constant once matter begins to dominate. Unfortunately, it seems necessary to choose a finely-tuned kinetic term to get the desired behavior [99]. An alternative possibility is that there is nothing special about the present era; rather, acceleration is just something that happens from time to time. This can be accomplished by oscillating dark energy [100]. In these models the potential takes the form of a decaying exponential (which by itself would give scaling behavior, so that the dark energy remained proportional to the background density) with small perturbations superimposed: V(0) = e - *[l + acos(0)] .

(83)

On average, the dark energy in such a model will track that of the dominant matter/radiation component; however, there will be gradual oscillations from a negligible density to a dominant density and back, on a timescale set by the Hubble parameter, leading to occasional periods of acceleration. Unfortunately, in neither the fc-essence models nor the oscillating models do we have a compelling particle-physics motivation for the chosen dynamics, and in both cases the behavior still depends sensitively on the precise form of parameters and interactions chosen. Nevertheless, these theories stand as interesting attempts to address the coincidence problem by dynamical means. One of the interesting features of dynamical dark energy is that it is experimentally testable. In principle, different dark energy models can yield different cosmic histories, and, in particular, a different value for the

739

-0.5

i

99%

i i i | i r °r CMB+WMAP+HST+SN-Ia+ LSS+BBN

/

\ \ 9 5 » I

•1.5

•

-3.5

«

0.2

0.4

6»% i

0.6

95% 99% * > • I

•

0.8

Sa ^matter Figure 8. Constraints on the dark-energy equation-of-state parameter, as a function of Q M , assuming a flat universe. These limits are derived from studies of supernovae, CMB anisotropics, measurements of the Hubble constant, large-scale structure, and primordial nucleosynthesis. Prom [101].

equation of state parameter, both today and its redshift-dependenee. Since the CMB strongly constrains the total density to be near the critical value, It Is sensible to assume a perfectly flat universe and determine constraints on the matter density and dark energy equation of state; see figure (8) for some recent limits. As can be seen In (8), one possibility that Is consistent with the data is that w < — 1. Such a possibility violates the dominant energy condition, but possible models have been proposed [102]. However, such models run into serious problems when one takes them seriously as a particle physics theory [103,104]. Even if one restricts one's attention to more conventional matter sources, making dark energy compatible with sensible particle physics has proven tremendously difficult.

740

Given the challenge of this problem, it is worthwhile considering the possibility that cosmic acceleration is not due to some kind of stuff, but rather arises from new gravitational physics, there are a number of different approaches to this [105-110] and we will not review them all here. Instead we will provide an example drawn from our own proposal [109]. As a first attempt, consider the simplest correction to the EinsteinHilbert action,

S=~

J dAx^-9(R-^j+

J d'x^-gCM •

(84)

Here /x is a new parameter with units of [mass] and LM is the Lagrangian density for matter. The fourth-order equations arising from this action are complicated and it is difficult to extract details about cosmological evolution from them. It is therefore convenient to transform from the frame used in 84, which we call the matter frame, to the Einstein frame, where the gravitational Lagrangian takes the Einstein-Hilbert form and the additional degrees of freedom (H and H) are represented by a fictitious scalar field . The details of this can be found in [109]. Here we just state that, performing a simultaneous redefinition of the time coordinate, in terms of the new metric g^u, our theory is that of a scalar field fax*1) minimally coupled to Einstein gravity, and non-minimally coupled to matter, with potential

«P

1

^

"I-

(85)

Now let us first focus on vacuum cosmological solutions. The beginning of the Universe corresponds to R —> oo and ^ —> 0. The initial conditions we must specify are the initial values of <j> and ', denoted as fa and fa i. For simplicity we take fa just reaches the maximum of the potential V(fa) and comes to rest. In this case the Universe asymptotically evolves to a de Sitter solution. This solution requires tuning and is unstable, since any perturbation will induce the field to roll away from the maximum of its potential. 2. Power-Law Acceleration. For fai > fac, the field overshoots the maximum of V{fa) and the Universe evolves to late-time power-law inflation, with observational consequences similar to dark energy with equation-ofstate parameter » D E = —2/3.

741

3. Future Singularity. For <j>' i < does not reach the maximum of its potential and rolls back down to = 0. This yields a future curvature singularity. In the more interesting case in which the Universe contains matter, it is possible to show that the three possible cosmic futures identified in the vacuum case remain in the presence of matter. By choosing (j, ~ 10~ 33 eV, the corrections to the standard cosmology only become important at the present epoch, making this theory a candidate to explain the observed acceleration of the Universe without recourse to dark energy. Since we have no particular reason for this choice, such a tuning appears no more attractive than the traditional choice of the cosmological constant. Clearly our choice of correction to the gravitational action can be generalized. Terms of the form — fi2(n+1^ /Rn, with n > 1, lead to similar late-time self acceleration, with behavior similar to a dark energy component with equation of state parameter 3(2n+ l)(n + 1) Clearly therefore, such modifications can easily accommodate current observational bounds [101,70] on the equation of state parameter —1.45 < WDE < —0.74 (95% confidence level). In the asymptotic regime n = 1 is ruled out at this level, while n > 2 is allowed; even n = 1 is permitted if we are near the top of the potential. Finally, any modification of the Einstein-Hilbert action must, of course, be consistent with the classic solar system tests of gravity theory, as well as numerous other astrophysical dynamical tests. We have chosen the coupling constant fj, to be very small, but we have also introduced a new light degree of freedom. Chiba [111] has pointed out that the model with n = 1 is equivalent to Brans-Dicke theory with w = 0 in the approximation where the potential was neglected, and would therefore be inconsistent with experiment. It is not yet clear whether including the potential, or considering extensions of the original model, could alter this conclusion. 4. Early Times in the Standard Cosmology In the first lecture we described the kinematics and dynamics of homogeneous and isotropic cosmologies in general relativity, while in the second we discussed the situation in our current universe. In this lecture we wind the clock back, using what we know of the laws of physics and the universe to-

742

day to infer conditions in the early universe. Early times were characterized by very high temperatures and densities, with many particle species kept in (approximate) thermal equilibrium by rapid interactions. We will therefore have to move beyond a simple description of non-interacting "matter" and "radiation," and discuss how thermodynamics works in an expanding universe. 4 . 1 . Describing

Matter

In the first lecture we discussed how to describe matter as a perfect fluid, described by an energy-momentum tensor Tpu = {P + P)U„U„ + pg^ ,

(87)

M

where C/ is the fluid four-velocity, p is the energy density in the rest frame of the fluid and p is the pressure in that same frame. The energy-momentum tensor is covariantly conserved, V^T""

=

o.

(88)

In a more complete description, a fluid will be characterized by quantities in addition to the energy density and pressure. Many fluids have a conserved quantity associated with them and so we will also introduce a number flux density N^, which is also conserved V„A^ = 0 .

(89)

M

For non-tachyonic matter A?" is a timelike 4-vector and therefore we may decompose it as N>* = nU" .

(90)

We can also introduce an entropy flux density S^. This quantity is not conserved, but rather obeys a covariant version of the second law of thermodynamics V^S" > 0 .

(91)

Not all phenomena are successfully described in terms of such a local entropy vector (e.g., black holes); fortunately, it suffices for a wide variety of fluids relevant to cosmology. The conservation law for the energy-momentum tensor yields, most importantly, equation (25), which can be thought of as the first law of thermodynamics dU = TdS - pdV , with dS = 0.

(92)

743

It is useful to resolve S^ into components parallel and perpendicular to the fluid 4-velocity S" = sU" + a" ,

(93)

where s^U^ = 0. The scalar s is the rest-frame entropy density which, up to an additive constant (that we can consistently set to zero), can be written as s=P-^-

(94)

In addition to all these quantities, we must specify an equation of state, and we typically do this in such a way as to treat n and s as independent variables. 4.2. Particles

in

Equilibrium

The various particles inhabiting the early universe can be usefully characterized according to three criteria: in equilibrium vs. out of equilibrium (decoupled), bosonic vs. fermionic, and relativistic (velocities near c) vs. non-relativistic. In this section we consider species which are in equilibrium with the surrounding thermal bath. Let us begin by discussing the conditions under which a particle species will be in equilibrium with the surrounding thermal plasma. A given species remains in thermal equilibrium as long as its interaction rate is larger than the expansion rate of the universe. Roughly speaking, equilibrium requires it to be possible for the products of a given reaction have the opportunity to recombine in the reverse reaction and if the expansion of the universe is rapid enough this won't happen. A particle species for which the interaction rates have fallen below the expansion rate of the universe is said to have frozen out or decoupled. If the interaction rate of some particle with the background plasma is T, it will be decoupled whenever r
(95)

where the Hubble constant H sets the cosmological timescale. As a good rule of thumb, the expansion rate in the early universe is "slow," and particles tend to be in thermal equilibrium (unless they are very weakly coupled). This can be seen from the Priedmann equation when the energy density is dominated by a plasma with p ~ T 4 ; we then have

744

Thus, the Hubble parameter is suppressed with respect to the temperature by a factor of T/Mp. At extremely early times (near the Planck era, for example), the universe may be expanding so quickly that no species are in equilibrium; as the expansion rate slows, equilibrium becomes possible. However, the interaction rate F for a particle with cross-section a is typically of the form T = n{av) ,

(97)

where n is the number density and v a typical particle velocity. Since n ex a - 3 , the density of particles will eventually dip so low that equilibrium can once again no longer be maintained. In our current universe, no species are in equilibrium with the background plasma (represented by the CMB photons). Now let us focus on particles in equilibrium. For a gas of weaklyinteracting particles, we can describe the state in terms of a distribution function / ( p ) , where the three-momentum p satisfies £2(p)=m2 + |p|2.

(98)

The distribution function characterizes the density of particles in a given momentum bin. (In general it will also be a function of the spatial position x, but we suppress that here.) The number density, energy density, and pressure of some' species labeled i are given by

^

=

(^/

s ( p ) / i ( p ) d 3 p

* = ( ^ / ^

/ i ( p ) r f , p

'

<">

where gi is the number of spin states of the particles. For massless photons we have g1 = 2, while for a massive vector boson such as the Z we have gz = 3. In the usual accounting, particles and antiparticles are treated as separate species; thus, for spin-1/2 electrons and positrons we have ge- — ge+ = 2. In thermal equilibrium at a temperature T the particles will be in either Fermi-Dirac or Bose-Einstein distributions,

f<*) = ^ r ± I • where the plus sign is for fermions and the minus sign for bosons.

(10°)

745

We can do the integrals over the distribution functions in two opposite limits: particles which are highly relativistic (T ^>TO)or highly nonrelativistic (T
Relativistic Bosons

Pi

C

(!)

rii

^l±

30

gi{^f\^,r

(1) io^ 4

mini

I Pi

riiT
\pi

Pi

^T"

Non-relativistic (Either)

From this table we can extract several pieces of relevant information. Relativistic particles, whether bosons or fermions, remain in approximately equal abundances in equilibrium. Once they become non-relativistic, however, their abundance plummets, and becomes exponentially suppressed with respect to the relativistic species. This is simply because it becomes progressively harder for massive particle-antiparticle pairs to be produced in a plasma with T <^ m. It is interesting to note that, although matter is much more dominant than radiation in the universe today, since their energy densities scale differently the early universe was radiation-dominated. We can write the ratio of the density parameters in matter and radiation as «5£ "R

=

"MO

( ^

1'RO \

a

=

0/

£«>

-i .

(1Q1)

"R0

The redshift of matter-radiation equality is thus l + ,

e q =

^ « 3 x l 0

3

.

(102)

"R0

This expression assumes that the particles that are non-relativistic today were also non-relativistic at zeq; this should be a safe assumption, with the possible exception of massive neutrinos, which make a minority contribution to the total density. As we mentioned in our discussion of the CMB in the previous lecture, even decoupled photons maintain a thermal distribution; this is not because

746

they are in equilibrium, but simply because the distribution function redshifts into a similar distribution with a lower temperature proportional to 1/a. We can therefore speak of the "effective temperature" of a relativistic species that freezes out at a temperature Tf and scale factor a/: Tr\a)=Tf{^) .

(103)

For example, neutrinos decouple at a temperature around 1 MeV; shortly thereafter, electrons and positrons annihilate into photons, dumping energy (and entropy) into the plasma but leaving the neutrinos unaffected. Consequently, we expect a neutrino background in the current universe with a temperature of approximately 2K, while the photon temperature is 3K. A similar effect occurs for particles which are non-relativistic at decoupling, with one important difference. For non-relativistic particles the temperature is proportional to the kinetic energy |mi> 2 , which redshifts as 1/a2. We therefore have T/ l o n - r e l (a) = T

/

p )

2

.

(104)

In either case we are imagining that the species freezes out while relativistic/non-relativistic and stays that way afterward; if it freezes out while relativistic and subsequently becomes non-relativistic, the distribution function will be distorted away from a thermal spectrum. The notion of an effective temperature allows us to define a corresponding notion of an effective number of relativistic degrees of freedom, which in turn permits a compact expression for the total relativistic energy density. The effective number of relativistic degrees of freedom (as far as energy is concerned) can be defined as

**= £ 4 | ) 4 ^ £ 4|) 4 bosons

^

'

fermions

^

(105)

'

(The temperature T is the actual temperature of the background plasma, assumed to be in equilibrium.) Then the total energy density in all relativistic species comes from adding the contributions of each species, to obtain the simple formula P = ^9*T4

.

(106)

We can do the same thing for the entropy density. From (94), the entropy density in relativistic particles goes as T 3 rather than T 4 , so we define the

747

effective number of relativistic degrees of freedom for entropy as

g*s = £ ft (J) + g £ ft ( | ) bosons

^

/

fermions

^

•

(107)

'

The entropy density in relativistic species is then s = ^9*sT3

.

(108)

Numerically, g* and g*S will typically be very close to each other. In the Standard Model, we have (-100 g* « g*s ~ < 10 I3

T > 300 MeV 300 MeV > T > 1 MeV T < 1 MeV .

(109)

The events that change the effective number of relativistic degrees of freedom are the QCD phase transition at 300 MeV, and the annihilation of electron/positron pairs at 1 MeV. Because of the release of energy into the background plasma when species annihilate, it is only an approximation to say that the temperature goes a s T o c 1/a. A better approximation is to say that the comoving entropy density is conserved, s<xa~3 .

(110)

This will hold under all forms of adiabatic evolution; entropy will only be produced at a process like a first-order phase transition or an out-ofequilibrium decay. (In fact, we expect that the entropy production from such processes is very small compared to the total entropy, and adiabatic evolution is an excellent approximation for almost the entire early universe. One exception is inflation, discussed in the next lecture.) Combining entropy conservation with the expression (108) for the entropy density in relativistic species, we obtain a better expression for the evolution of the temperature, T ex ^ V

1

.

(Ill)

The temperature will consistently decrease under adiabatic evolution in an expanding universe, but it decreases more slowly when the effective number of relativistic degrees of freedom is diminished.

748

4.3. Thermal

Relics

As we have mentioned, particles typically do not stay in equilibrium forever; eventually the density becomes so low that interactions become infrequent, and the particles freeze out. Since essentially all of the particles in our current universe fall into this category, it is important to study the relic abundance of decoupled species. (Of course it is also possible to obtain a significant relic abundance for particles which were never in thermal equilibrium; examples might include baryons produced by GUT baryogenesis, or axions produced by vacuum misalignment.) In this section we will typically neglect factors of order unity. We have seen that relativistic, or hot, particles have a number density that is proportional to T 3 in equilibrium. Thus, a species X that freezes out while still relativistic will have a number density at freeze-out Tf given by nx{Tf)~TJ.

(112)

Since this is comparable to the number density of photons at that time, and after freeze-out both photons and our species X just have their number densities dilute by a factor a(t)~3 as the universe expands, it is simple to see that the abundance of X particles today should be comparable to the abundance of CMB photons, nxo ~ n-yo ~ 102 c m - 3 .

(113)

2

We express this number as 10 rather than 411 since the roughness of our estimate does not warrant such misleading precision. The leading correction to this value is typically due to the production of additional photons subsequent to the decoupling of X; in the Standard Model, the number density of photons increases by a factor of approximately 100 between the electroweak phase transition and today, and a species which decouples during this period will be diluted by a factor of between 1 and 100 depending on precisely when it freezes out. So, for example, neutrinos which are light (m„ < MeV) have a number density today of nv = 115 c m - 3 per species, and a corresponding contribution to the density parameter (if they are nevertheless heavy enough to be nonrelativistic today) of

(In this final expression we have secretly taken account of the missing numerical factors, so this is a reliable answer.) Thus, a neutrino with m„ ~ 10~ 2 eV would contribute Vtv ~ 2 x 10~ 4 . This is large enough to

749

be interesting without being large enough to make neutrinos be the dark matter. That's good news, since the large velocities of neutrinos make them free-stream out of overdense regions, diminishing primordial perturbations and leaving us with a universe which has much less structure on small scales than we actually observe. Now consider instead a species X which is nonrelativistic or cold at the time of decoupling. It is much harder to accurately calculate the relic abundance of a cold relic than a hot one, simply because the equilibrium abundance of a nonrelativistic species is changing rapidly with respect to the background plasma, and we have to be quite precise following the freezeout process to obtain a reliable answer. The accurate calculation typically involves numerical integration of the Boltzmann equation for a network of interacting particle species; here, we cut to the chase and simply provide a reasonable approximate expression. If o~o is the annihilation cross-section of the species X at a temperature T = mx, the final number density in terms of the photon density works out to be nx(T
—n7 .

(115)

Since the particles are nonrelativistic when they decouple, they will certainly be nonrelativistic today, and their energy density is Px = inxnx

•

(116)

We can plug in numbers for the Hubble parameter and photon density to obtain the density parameter, px n7 nx= (117)

Wr~^Mm-

Numerically, when ft = c = 1 we have 1 GeV~ 2 x 10~ 14 cm, so the photon density today is n 7 ~ 100 c m - 3 ~ 10~ 39 GeV - 3 . The Hubble constant is H0 ~ 10" 4 2 GeV, and the Planck mass is Mp ~ 10 18 GeV, so we obtain Vx ~

1 ,„a„.T2,

•

(H8)

It is interesting to note that this final expression is independent of the mass mx of our relic, and only depends on the annihilation cross-section; that's because more massive particles will have a lower relic abundance. Of course, this depends on how we choose to characterize our theory; we may use variables in which <JQ is a function of mx, in which case it is reasonable to say that the density parameter does depend on the mass.

750

T h e designation cold may ring a bell with many of you, for you will have heard it used in a cosmological context applied to cold dark matter (CDM). Let us see briefly why this is. One candidate for CDM is a Weakly Interacting Massive Particle ( W I M P ) . T h e annihilation cross-section of these particles, since they are weakly interacting, should be do ~ O^GF, where aw is the weak coupling constant and GF is the the Fermi constant. Using GF ~ ( 3 0 0 G e V ) - 2 and aw ~ 1CT 2 , we get a0 ~ a2wGF

~ 10"9 GeV"2 .

(119)

Thus, the density parameter in such particles would be fix ~ 1 .

(120)

In other words, a stable particle with a weak interaction cross section naturally produces a relic density of order the critical density today, and so provides a perfect candidate for cold dark matter. A paradigmatic example is provided by the lightest supersymmetric partner (LSP), if it is stable and supersymmetry is broken at the weak scale. Such a possibility is of great interest t o b o t h particle physicists and cosmologists, since it may be possible t o produce and detect such particles in colliders and to directly detect a W I M P background in cryogenic detectors in underground laboratories; this will be a major experimental effort over the next few years [13].

4.4.

Vacuum,

displacement

Another important possibility is the existence of relics which were never in thermal equilibrium. An example of these will be discussed later in this lecture: the production of topological defects at phase transitions. Let's discuss another kind of non-thermal relic, which derives from what we might call "vacuum displacement." Consider the action for a real scalar field in curved spacetime:

S = I dAx \f--g

g^d^dvcj>

- V{<j>)

(121)

If we assume t h a t = 0), its equation of motion in the Robertson-Walker metric (5) will be 4> + 3H4> + V'(4>)=0

,

(122)

where an overdot indicates a partial derivative with respect to time, and a prime indicates a derivative with respect to . For a free massive scalar

751

field, V(4>) = T}m2(f)2, and (122) describes a harmonic oscillator with a timedependent damping term. For H > m^ the field will be overdamped, and stay essentially constant at whatever point in the potential it finds itself. So let us imagine that at some time in the very early universe (when H was large) we had such an overdamped homogeneous scalar field, stuck at a value 4> = 4>* > the total energy density in the field is simply the potential energy |m|J. The Hubble parameter H will decrease to approximately rrirj, when the temperature reaches T* = ^/m^Mp, after which the field will be able to evolve and will begin to oscillate in its potential. The vacuum energy is converted to a combination of vacuum and kinetic energy which will redshift like matter, as p$ oc a - 3 ; in a particle interpretation, the field is a Bose condensate of zero-momentum particles. We will therefore have

M«)~^?(^)3 »

(123)

which leads to a density parameter today /2 "00- f ^ 5V 19 '* VlO- GeVV

(I24) V

;

A classic example of a non-thermal relic produced by vacuum displacement is the QCD axion, which has a typical primordial value (4>) ~ / P Q and a mass m^ ~ A Q C D / / P Q , where / P Q is the Peccei-Quinn symmetrybreaking scale and AQCD ~ 0.3 GeV is the QCD scale [1]. In this case, plugging in numbers reveals

""-(io^Sv)*"-

(I25)

The Peccei-Quinn scale is essentially a free parameter from a theoretical point of view, but experiments and astrophysical constraints have ruled out most values except for a small window around / P Q ~ 1012 GeV. The axion therefore remains a viable dark matter candidate [112,113]. Note that, even though dark matter axions are very light ( A Q C D / / P Q ~ 10~ 4 eV), they are extremely non-relativistic, which can be traced to the non-thermal nature of their production process. (Another important way to produce axions is through the decay of axion cosmic strings [1,114].) 4.5. Primordial

Nucleosynthesis

Given our time constraints, even some of the more important concepts in cosmology cannot be dealt with in significant detail. We have chosen

752

just to give a cursory treatment to primordial nucleosynthesis, although its importance as a crucial piece of evidence in favor of the big bang model and its usefulness in bounding any new physics of cosmological relevance cannot be overstated. Observations of primordial nebulae reveal abundances of the light elements unexplained by stellar nucleosynthesis. Although it does a great disservice to the analytic and numerical work required, not to mention the difficulties of measuring the abundances, we will just state that the study of nuclear processes in the background of an expanding cooling universe yields a remarkable concordance between theory and experiment. At temperatures below 1 MeV, the weak interactions are frozen out and neutrons and protons cease to interconvert. The equilibrium abundance of neutrons at this temperature is about 1/6 the abundance of protons (due to the slightly larger neutron mass). The neutrons have a finite lifetime (rn = 890 sec) that is somewhat larger than the age of the universe at this epoch, t(l MeV) « 1 sec, but they begin to gradually decay into protons and leptons. Soon thereafter, however, we reach a temperature somewhat below 100 keV, and Big-Bang nucleosynthesis (BBN) begins. (The nuclear binding energy per nucleon is typically of order 1 MeV, so you might expect that nucleosynthesis would occur earlier; however, the large number of photons per nucleon prevents nucleosynthesis from taking place until the temperature drops below 100 keV.) At that point the neutron/proton ratio is approximately 1/7. Of all the light nuclei, it is energetically favorable for the nucleons to reside in 4 He, and indeed that is what most of the free neutrons are converted into; for every two neutrons and fourteen protons, we end up with one helium nucleus and twelve protons. Thus, about 25% of the baryons by mass are converted to helium. In addition, there are trace amounts of deuterium (approximately 10~ 5 deuterons per proton), 3 He (also - 10~ 5 ), and 7 Li (~ 10" 1 0 ). Of course these numbers are predictions, which are borne out by observations of the primordial abundances of light elements. (Heavier elements are not synthesized in the Big Bang, but require supernova explosions in the later universe.) We have glossed over numerous crucial details, especially those which explain how the different abundances depend on the cosmological parameters. For example, imagine that we deviate from the Standard Model by introducing more than three light neutrino species. This would increase the radiation energy density at a fixed temperature, which in turn decreases the timescales associated with a given temperature (since t ~ H~x K ^ ' ). Nucleosynthesis would therefore happen

753

somewhat earlier, resulting in a higher abundance of neutrons, and hence in a larger abundance of 4 He. Observations of the primordial helium abundance, which are consistent with the Standard Model prediction, provided the first evidence that the number of light neutrinos is actually three. The most amazing fact about nucleosynthesis is that, given that the universe is radiation dominated during the relevant epoch (and the physics of general relativity and the Standard Model), the relative abundances of the light elements depend essentially on just one parameter, the baryon to entropy ratio

where TIB = n\, — n^ is the difference between the number of baryons and antibaryons per unit volume. The range of r\ consistent with the deuterium and 3 He primordial abundances is 2.6 x 1(T 10 < rj < 6.2 x l f r 1 0 .

(127)

Very recently this number has been independently determined to be r ^ e . l x l O ^ + ^ o -

(128)

from precise measurements of the relative heights of the first two microwave background (CMB) acoustic peaks by the WMAP satellite. This is illustrated in figure (9) and we will have a lot more to say about this quantity later when we discuss baryogenesis. 4.6. Finite

Temperature

Phase

Transitions

We have hinted at the need to go beyond perfect fluid sources for the Einstein equations if we are to unravel some of the mysteries left by the standard cosmology. Given our modern understanding of particle physics, it is natural to consider using field theory to model matter at early times in the universe. The effect of cosmic expansion and the associated thermodynamics yield some fascinating phenomena when combined with such a field theory approach. One significant example is provided by finite temperature phase transitions. Rather than performing a detailed calculation in finite-temperature field theory, we will illustrate this with a rough argument. Consider a theory of a single real scalar field <\> at zero temperature, interacting with a second real scalar field \- The Lagrangian density is £ = -^d^t

- \d»x^X - V(, T = 0) - | 0 V

,

(129)

754

0.01 fc » • i

d o 0.25 o

Fraction of critical density 0.02 0.05

«P

0.24

%4

^

TO 09

0.23 0.22,

0)

a

10~*4

o

-p

CP •

> j ^

^o

(d

10" 5 u m

&

f

icr

9

r

3 55

10"

••10

1 2 5 Baryon density (10" 3 1 g cm""3) Figure 9. Abundances of light elements produced by BBN, as a function of the baryon density. The vertical strip indicates the concordance region favored by observations of primordial abundances. Erom [44].

with potential V^T

= 0) =

P2 #

2^

X

r

(130)

where fi is a parameter with dimensions of mass and A and g are dimensionless coupling constants. Now, consider what the effective theory for looks like if we assume that x is i n thermal equilibrium. In this case, we may replace % by the

755

temperature T to obtain a Lagrangian for <j> only, with finite temperature effective -potential given by V(4,T)=1-(gT*-^)
+^ .

(131)

This simple example demonstrates a very significant result. At zero temperature, all your particle physics intuition tells you correctly that the theory is spontaneously broken, in this case the global Zi symmetry of the Lagrangian is broken by the ground state value {<j>) ^ 0 (we shall see more of this soon). However, at temperatures above the critical temperature T c given by

the full Zi symmetry of the Lagrangian is respected by the finite temperature ground state, which is now at (0) = 0. This behavior is know as finite temperature symmetry restoration, and its inverse, occurring as the universe cools, is called finite temperature spontaneous symmetry breaking. Zero temperature unification and symmetry breaking are fundamental features of modern particle physics models. For particle physicists this amounts to the statement that the chosen vacuum state of a gauge field theory is one which does not respect the underlying symmetry group of the Lagrangian. The melding of ideas from particle physics and cosmology described above leads us to speculate that matter in the early universe was described by a unified gauge field theory based on a simple continuous Lie group, G. As a result of the extreme temperatures of the early universe, the vacuum state of the theory respected the full symmetry of the Lagrangian. As the universe cooled, we hypothesize that the gauge theory underwent a series of spontaneous symmetry breakings (SSB) until matter was finally described by the unbroken gauge groups of QCD and QED. Schematically the symmetry breaking can be represented by G^H

-»

• SU(3)C x SU(2)L x U(1)Y -+ SU(3)C x U(l)em

. (133)

The group G is known as the grand unified gauge group and the initial breaking G —> H and is expected to take place at 1016GeV, as we discuss below. Let us briefly set up the mathematical description of SSB which will be useful later when we discuss topological properties of the theory. Consider a gauge field theory described by a continuous group G. Denote the vacuum state of the theory by |0). Then, given g(x) G G, the state

756

g(x)\0) is also a vacuum. Suppose all vacuum states are of the form g\0) but that the state |0) is invariant only under a subgroup H C G. Then the symmetry of the theory is said to have been spontaneously broken from G to H. Let us define two vacuum states |0}^ and |0)s to represent the same state under the broken group if g\0}A = \0}B for some g G G. We write \0)A ~ | 0 ) B and the distinguishable vacua of the theory are equivalence classes under ~ and are the cosets of H in G. Then the vacuum manifold, the space of all accessible vacua of the theory, is the coset space M = G/H .

(134)

Given that we believe the universe evolved through a sequence of such symmetry breakings, there are many questions we can ask about the cosmological implications of the scheme and many ways in which we can constrain and utilize the possible breaking patterns. 4.7. Topological

Defects

In a quantum field theory, small oscillations around the vacuum appear as particles. If the space of possible vacuum states is topologically nontrivial, however, there arises the possibility of another kind of solitonic object: a topological defect. Consider a field theory described by a continuous symmetry group G which is spontaneously broken to a subgroup H C G. Recall that the space of all accessible vacua of the theory, the vacuum manifold, is defined to be the space of cosets of H in G; M. = G/H. Whether the theory admits topological defects depends on whether the vacuum manifold has nontrivial homotopy groups. A homotopy group consists of equivalence classes of maps of spheres (with fixed base point) into the manifold, where two maps are equivalent if they can be smoothly deformed into each other. The homotopy groups defined in terms of n-spheres are denoted 7r„. In general, a field theory with vacuum manifold M. possesses a topological defect of some type if n{M)

± 1,

(135)

for some i = 0 , 1 , . . . . In particular, we can have a set of defects, as listed in table 3. In order to get an intuitive picture of the meaning of these topological criteria let us consider the first two. If TTQ(M) 7^ 1, the manifold M is disconnected. (A zero-sphere is the set of two points a fixed distance from the origin in R 1 . The set of topologically equivalent maps with fixed base point from such such a sphere into a manifold is simply the set of disconnected pieces into which the manifold

757

Homotopy Constraint

MM) ± 1 m(M)?l TT2(M) ± 1

MM) + 1

Topological Defect Domain Wall Cosmic String Monopole Texture

falls.) Let's assume for simplicity that the vacuum manifold consists of just two disconnected components M.\ and M.2 and restrict ourselves to one spatial dimension. Then, if we apply boundary conditions that the vacuum at —oo lies in M.\ and that at +oo lies in M.2, by continuity there must be a point somewhere where the order parameter does not lie in the vacuum manifold as the field interpolates between the two vacua. The region in which the field is out of the vacuum is known as a domain wall. Similarly, suppose n\(M) ^ 1. This implies that the vacuum manifold is not simply-connected: there are non-contractible loops in the manifold. The order parameter for such a situation may be considered complex and if, when traveling around a closed curve in space, the phase of the order parameter changes by a non-zero multiple of 2ir, by continuity there is a point within the curve where the field is out of the vacuum manifold. Continuity in the direction perpendicular to the plane of the curve implies that there exists a line of such points. This is an example of a cosmic string or line defect. Since we shall mostly concentrate on these defects let us now give a detailed analysis of the simplest example, the Nielsen-Olesen vortex in the Abelian Higgs model. Consider a complex scalar field theory based on the Abelian gauge group U(l). The Lagrangian density for this model is C = - i ( D / l ¥ > ) * D ' V - \F^F^

- V{
(136)

where „¥> = {&„ + ieA^cp , F^ = d^Au - dvA^ ,

(137) (138)

and the symmetry breaking (or "Mexican hat") potential is %)^(»)V-r,2)2.

(139)

758

Here e is the gauge coupling constant and r\ is a parameter that represents the scale of the symmetry breaking. In preparation for our discussion of the more general cosmological situation, let us note that at high temperatures the Lagrangian contains temperature-dependent corrections given by V(
(140)

where C is a constant. For T > Tc, where Tc is defined by T? = ^V2 2C

,

(141)

we see that

Vdp) = ^ »

+^

(^J - l) ^V + V

(142)

is minimized by (ip) = 0. However, for T < Tc, minimizing the above expression with respect to ip yields a minimum at 2 = » 7 2 ( l - ^ )

(143)

•

Thus, for T > Tc the full symmetry group G = U(l) is restored and the vacuum expectation value of the ip-field is zero. As the system cools, there is a phase transition at the critical temperature Tc and the vacuum symmetry is spontaneously broken, in this case entirely: G = U{\) —>1 = H .

(144)

The vacuum manifold is M = G/H = U(l)/1 = 17(1) = {

(145)

The group U(l) is topologically a circle, so M = S1 . The set of topologically equivalent ways to map is given by the integers. (We can wrap it any sense.) The first homotopy group is therefore homotopy groups in general is difficult, but for are trivial. We therefore have MS1)

= 1

1

(146) one circle to another circle number of times in either the integers. Calculating S"1 all of the other groups (147)

n1(S )=Z

(148)

1

= 1

(149)

1

= 1•

(150)

MS ) MS )

759

The Abelian Higgs model therefore allows for cosmic strings. To see how strings form in this model, consider what happens as T decreases through Tc. The symmetry breaks and the field acquires a VEV given by <¥>> = Veta ,

(151)

where a may be chosen differently in different regions of space, as implied by causality (we shall discuss this shortly). The requirement that (ip) be single valued implies that around any closed curve in space the change Act in a must satisfy Aa = 2irn ,

neZ

.

(152)

If for a given loop we have n j^ 0, then we can see that any 2-surface bounded by the loop must contain a singular point, for if not then we can continuously contract the loop to a point, implying that n = 0 which is a contradiction. At this singular point the phase, a, is undefined and (
,

A^ = -eY(r)d^9

.

(153)

The Lagrangian then yields the simplified equations of motion - X " - ^ + ^ z r r

+ ^ X ( X 2 - l ) = 0, 2 Y' 2e2r?X2Y

-Y" + — +

V

=°.

(154)

r A where a prime denotes differentiation with respect to r. It is not possible to solve these equations analytically, but asymptotically we have X ~ r , Y = 1 + 0(rl)2> ; as r -> 0 ,

760

/e-i1/V\ X ~ l - 0 \

) ,Y~

\

1/2

OiJ^e-2

er

"") ; as r -» 00 . (155)

V*F J

This solution corresponds to a string centered on the z-axis, with a central magnetic core of width ~ (ery) -1 carrying a total magnetic flux $= /

^ 2 ^ = - — .

(156)

e

ir=oo

The core region over which the Higgs fields are appreciably non-zero has width ~ (A 1 / 2 ??) -1 . Note that these properties depend crucially on the fact that we started with a gauged symmetry. A spontaneously broken global symmetry with nontrivial TTi(M) would still produce cosmic strings, but they would be much less localized, since there would be no gauge fields to cancel the scalar gradients at large distances. Strings are characterized by their tension, which is the energy per unit length. For the Nielsen-Olesen solution just discussed, the tension is approximately fi~r)2

•

(157)

Thus, the energy of the string is set by the expectation value of the order parameter responsible for the symmetry breaking. This behavior is similarly characteristic of other kinds of topological defects. Again, global defects' are quite different; the tension of a global string actually diverges, due to the slow fall-off of the energy density as we move away from the string core. It is often convenient to parameterize the tension by the dimensionless quantity Gfj,, so that a Planck-scale string would have G\i ~ 1. From the above discussion we can see that cosmology provides us with a unique opportunity to explore the rich and complex structure of particle physics theories. Although the topological solutions discussed above exist in the theory at zero temperature, there is no mechanism within the theory to produce these objects. The topological structures contribute a set of zero measure in the phase space of possible solutions to the theory and hence the probability of production in particle processes is exponentially suppressed. What the cosmological evolution of the vacuum supplies is a concrete causal mechanism for producing these long lived exotic solutions. At this point it is appropriate to discuss how many of these defects we expect to be produced at a cosmological phase transition. In the cosmological context, the mechanism for the production of defects is known as the Kibble mechanism. The guiding principle here is causality. As the phase transition takes place, the maximum causal distance imposed on the theory

761

by cosmology is simply the Hubble distance - the distance which light can have traveled since the big bang. As we remarked above, as the temperature of the universe falls well below the critical temperature of the phase transition, Tc, the expectation value of the order parameter takes on a definite value (~ r\e%a in the Abelian Higgs model) in each region of space. However, at temperatures around the critical temperature we expect that thermal fluctuations in () changes smoothly. If the phase changes by 2im for some n ^ 0 when traversing a loop in space, then any surface bounded by that loop intersects a cosmic string. These strings must be horizon-sized or closed loops. Numerical simulations of cosmic string formation indicate that the initial distribution of strings consists of 80% horizon-sized and 20% loops by mass. If we consider the temperature at which there is insufficient thermal energy to excite a correlation volume back into the unbroken state, the Ginsburg temperature, TQ, then, given the assumption of thermal equilibrium above the phase transition, a much improved estimate for the initial separation of the defects can be derived and is given by

where A is the self coupling of the order parameter. This separation is microscopic. Interesting bounds on the tension of cosmic strings produced by the Kibble mechanism arise from two sources: perturbations of the CMB, and gravitational waves. Both arise because the motions of heavy strings moving at relativistic velocities lead to time-dependent gravitational fields. The actual values of the bounds are controversial, simply because it is difficult to accurately model the nonlinear evolution of a string network, and the results can be sensitive to what assumptions are made. Nevertheless, the CMB bounds amount roughly to [115-118] Gil < 1 0 - 6 .

(158)

This corresponds roughly to strings at the GUT scale, 1016 GeV, which is certainly an interesting value. Bounds from gravitational waves come

762

from two different techniques: direct observation, and indirect measurement through accurate pulsar timings. Currently, pulsar timing measurements are more constraining, and give a bound similar to that from the CMB [119]. Unlike the CMB measurements, however, gravitational wave observatories will become dramatically better in the near future, through operations of ground-based observatories such as LIGO as well as satellites such as LISA. These experiments should be able to improve the bounds on G/J, by several orders of magnitude [120]. Now that we have established the criteria necessary for the production of topological defects in spontaneously broken theories, let us apply these conditions to the best understood physical example, the electroweak phase transition. As we remarked earlier, the GWS theory is based on the gauge group SU(2)L x U(1)Y- At the phase transition this breaks to pure electromagnetism, U(l)em. Thus, the vacuum manifold is the space of cosets MEW

= [SU(2) x U(l)]/U(l)

,

(159)

This looks complicated but in fact this space is topologically equivalent to the three-sphere, S3. The homotopy groups of the three-sphere are 7T0(S3) = 1 3

7n(5 ) = 1 3

MS ) 3

= 1

ir3(S ) = Z .

(160) (161) (162) (163)

(Don't be fooled into thinking that all homotopy groups of spheres vanish except in the dimensionality of the sphere itself; for example, 7T3(52) = Z.) Thus, the electroweak model does not lead to walls, strings, or monopoles. It does lead to what we called "texture," which deserves further comment. In a theory where irz{M) is nontrivial but the other groups vanish, we can always map three-dimensional space smoothly into the vacuum manifold; there will not be a defect where the field climbs out of MHowever, if we consider field configurations which approach a unique value at spatial infinity, they will fall into homotopy classes characterized by elements of 7T3(A/1); configurations with nonzero winding will be textures. If the symmetry is global, such configurations will necessarily contain gradient energies from the scalar fields. The energy perturbations caused by global textures were, along with cosmic strings, formerly popular as a possible origin of structure formation in the universe [121,122]; the predictions of these theories are inconsistent with the sharp acoustic peaks observed in the CMB, so such models are no longer considered viable.

763

In the standard model, however, the broken symmetry is gauged. In this case there is no need for gradient energies, since the gauge field can always be chosen to cancel them; equivalently, "texture" configurations can always be brought to the vacuum by a gauge transformation. However, transitions from one texture configuration to one with a different winding number are gauge invariant. These transitions will play a role in electroweak baryon number violation, discussed in the next section.

4.8.

Baryogenesis

The symmetry between particles and antiparticles [123,124], firmly established in collider physics, naturally leads to the question of why the observed universe is composed almost entirely of matter with little or no primordial antimatter. Outside of particle accelerators, antimatter can be seen in cosmic rays in the form of a few antiprotons, present at a level of around 10~ 4 in comparison with the number of protons (for example see [125]). However, this proportion is consistent with secondary antiproton production through accelerator-like processes, p+p —> ?>p+p, as the cosmic rays stream towards us. Thus there is no evidence for primordial antimatter in our galaxy. Also, if matter and antimatter galaxies were to coexist in clusters of galaxies, then we would expect there to be a detectable background of 7-radiation from nucleon-antinucleon annihilations within the clusters. This background is not observed and so we conclude that there is negligible antimatter on the scale of clusters (For a review of the evidence for a baryon asymmetry see [126].) More generally, if large domains of matter and antimatter exist, then annihilations would take place at the interfaces between them. If the typical size of such a domain was small enough, then the energy released by these annihilations would result in a diffuse 7-ray background and a distortion of the cosmic microwave radiation, neither of which is observed. While the above considerations put an experimental upper bound on the amount of antimatter in the universe, strict quantitative estimates of the relative abundances of baryonic matter and antimatter may also be obtained from the standard cosmology. The baryon number density does not remain constant during the evolution of the universe, instead scaling like a - 3 , where a is the cosmological scale factor [127]. It is therefore convenient to define the baryon asymmetry of the universe in terms of the

764

quantity

defined earlier. Recall that the range of r\ consistent with the deuterium and 3 He primordial abundances is 2.6 x 1CT10 < r] < 6.2 x 10" 1 0 .

(165)

Thus the natural question arises; as the universe cooled from early times to today, what processes, both particle physics and cosmological, were responsible for the generation of this very specific baryon asymmetry? (For reviews of mechanisms to generate the baryon asymmetry, see As pointed out by Sakharov [128], a small baryon asymmetry r\ may have been produced in the early universe if three necessary conditions are satisfied • baryon number (B) violation, • violation of C (charge conjugation symmetry) and CP (the composition of parity and C), • departure from thermal equilibrium. The first condition should be clear since, starting from a baryon symmetric universe with r\ = 0, baryon number violation must take place in order to evolve into a universe in which r\ does not vanish. The second Sakharov criterion is required because, if C and CP are exact symmetries, one can prove that the total rate for any process which produces an excess of baryons is equal to the rate of the complementary process which produces an excess of antibaryons and so no net baryon number can be created. That is to say that the thermal average of the baryon number operator B, which is odd under both C and CP, is zero unless those discrete symmetries are violated. CP violation is present either if there are complex phases in the Lagrangian which cannot be reabsorbed by field redefinitions (explicit breaking) or if some Higgs scalar field acquires a VEV which is not real (spontaneous breaking). We will discuss this in detail shortly. Finally, to explain the third criterion, one can calculate the equilibrium average of B at a temperature T = 1//3: (B)T = Tr (e~pHB) pH

= Tr [(CPT)(CPT)-1e->3H l

= Tr {e- (CPT)~ B{CPT)}

B)} fiH

= - T r (e~ B)

,

(166)

where we have used that the Hamiltonian H commutes with CPT. Thus (B)T ~ 0 in equilibrium and there is no generation of net baryon number.

765

Of the three Sakharov conditions, baryon number violation and C and CP violation may be investigated only within a given particle physics model, while the third condition - the departure from thermal equilibrium - may be discussed in a more general way, as we shall see (for baryogenesis reviews see [9, 10, 129-133]). Let us discuss the Sakharov criteria in more detail. 4.9. Baryon

Number

Violation

4.9.1. B-violation in Grand Unified Theories As discussed earlier, Grand Unified Theories (GUTs) [134] describe the fundamental interactions by means of a unique gauge group G which contains the Standard Model (SM) gauge group SU(3)C ® SU(2)L U{1)Y- The fundamental idea of GUTs is that at energies higher than a certain energy threshold MQUT the group symmetry is G and that, at lower energies, the symmetry is broken down to the SM gauge symmetry, possibly through a chain of symmetry breakings. The main motivation for this scenario is that, at least in supersymmetric models, the (running) gauge couplings of the SM unify [135-137] at the scale MQUT — 2 X 1016 GeV, hinting at the presence of a GUT involving a higher symmetry with a single gauge coupling. Baryon number violation seems very natural in GUTs. Indeed, a general property of these theories is that the same representation of G may contain both quarks and leptons, and therefore it is possible for scalar and gauge bosons to mediate gauge interactions among fermions having different baryon number. 4.9.2. B-violation in the Electroweak Theory It is well-known that the most general renormalizable Lagrangian invariant under the SM gauge group and containing only color singlet Higgs fields is automatically invariant under global abelian symmetries which may be identified with the baryonic and leptonic symmetries. These, therefore, are accidental symmetries and as a result it is not possible to violate B and L at tree-level or at any order of perturbation theory. Nevertheless, in many cases the perturbative expansion does not describe all the dynamics of the theory and, indeed, in 1976 't Hooft [138] realized that nonperturbative effects (instantons) may give rise to processes which violate the combination B + L, but not the orthogonal combination B — L. The probability of these processes occurring today is exponentially suppressed and probably irrelevant. However, in more extreme situations - like the primordial universe

766

at very high temperatures [139-142] - baryon and lepton number violating processes may be fast enough to play a significant role in baryogenesis. Let us have a closer look. At the quantum level, the baryon and the lepton symmetries are anomalous [143,144], so that their respective Noether currents jg and j£ are no longer conserved, but satisfy

«

= dtfZ = nf (j^W«vW^ -

JLF^F^

,

(167)

where g and g' are the gauge couplings of SU(2)L and U(l)y, respectively, nj is the number of families and W^v = (l/2)eIJ'vaf3Wap is the dual of the SU(2)i field strength tensor, with an analogous expression holding for F. To understand how the anomaly is closely related to the vacuum structure of the theory, we may compute the change in baryon number from time t — 0 to some arbitrary final time t — tf. For transitions between vacua, the average values of the field strengths are zero at the beginning and the end of the evolution. The change in baryon number may be written as AB = ANCS = nf[NCs(tf)

- NCS(0)] •

(168)

where the Chern-Simons number is defined to be Ncs(t) = J ^ J d3x j* Tr (A^AU + ^igA^A^

,

(169)

where At is the S U ( 2 ) L gauge field. Although the Chern-Simons number is not gauge invariant, the change ANcs is- Thus, changes in Chern-Simons number result in changes in baryon number which are integral multiples of the number of families rif (with n / = 3 in the real world). Gauge transformations U(x) which connects two degenerate vacua of the gauge theory may change the Chern-Simons number by an integer n, the winding number. If the system is able to perform a transition from the vacuum C?vac to the closest OH.6 ^vac , the Chern-Simons number is changed by unity and AB = AL = n / . Each transition creates 9 left-handed quarks (3 color states for each generation) and 3 left-handed leptons (one per generation). However, adjacent vacua of the electroweak theory are separated by a ridge of configurations with energies larger than that of the vacuum. The lowest energy point on this ridge is a saddle point solution to the equations of motion with a single negative eigenvalue, and is referred to as the sphaleron [140,141]. The probability of baryon number nonconserving

767

processes at zero temperature has been computed by 't Hooft [138] and is highly suppressed. The thermal rate of baryon number violation in the broken phase is

where \x is a dimensionless constant. Although the Boltzmann suppression in (170) appears large, it is to be expected that, when the electroweak symmetry becomes restored at a temperature of around 100 GeV, there will no longer be an exponential suppression factor. A simple estimate is that the rate per unit volume of sphaleron events is Tsp(T) = K{awTf

,

(171)

with K another dimensionless constant. The rate of sphaleron processes can be related to the diffusion constant for Chern-Simons number by a fluctuation-dissipation theorem [145] (for a good description of this see [131]). 4.9.3. CP Violation CP violation in GUTs arises in loop-diagram corrections to baryon number violating bosonic decays. Since it is necessary that the particles in the loop also undergo B-violating decays, the relevant particles are the X, Y, and Hs bosons in the case of 5/7(5). In the electroweak theory things are somewhat different. Since only the left-handed fermions are SU(2)L gauge coupled, C is maximally broken in the SM. Moreover, CP is known not to be an exact symmetry of the weak interactions. This is seen experimentally in the neutral kaon system through KQ, KQ mixing. Thus, CP violation is a natural feature of the standard electroweak model. While this is encouraging for baryogenesis, it turns out that this particular source of CP violation is not strong enough. The relevant effects are parameterized by a dimensionless constant which is no larger than 1 0 - 2 0 . This appears to be much too small to account for the observed BAU and, thus far, attempts to utilize this source of CP violation for electroweak baryogenesis have been unsuccessful. In light of this, it is usual to extend the SM in some fashion that increases the amount of CP violation in the theory while not leading to results that conflict with current experimental data. One concrete example of a well-motivated extension in the minimal supersymmetric standard model (MSSM).

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4.9.4. Departure from Thermal Equilibrium In some scenarios, such as GUT baryogenesis, the third Sakharov condition is satisfied due to the presence of superheavy decaying particles in a rapidly expanding universe. These generically fall under the name of out-of-equilibrium decay mechanisms. The underlying idea is fairly simple. If the decay rate Tx of the superheavy particles X at the time they become nonrelativistic (i.e. at the temperature T ~ Mx) is much smaller than the expansion rate of the universe, then the X particles cannot decay on the time scale of the expansion and so they remain as abundant as photons for T <, Mx- In other words, at some temperature T > Mx, the superheavy particles X are so weakly interacting that they cannot catch up with the expansion of the universe and they decouple from the thermal bath while they are still relativistic, so that nx ~ n 7 ~ T 3 at the time of decoupling. Therefore, at temperature T ~ Mx, they populate the universe with an abundance which is much larger than the equilibrium one. This overabundance is precisely the departure from thermal equilibrium needed to produce a final nonvanishing baryon asymmetry when the heavy states X undergo B and CP violating decays. The out-of-equilibrium condition requires very heavy states: Mx ^ ( 1 0 1 5 - 1 0 1 6 ) G e V a n d M x £ (10 1 0 -10 1 6 )GeV, for gauge and scalar bosons, respectively [146], if these heavy particles decay through renormalizable operators. A different implementation can be found in the electroweak theory. At temperatures around the electroweak scale, the expansion rate of the universe in thermal units is small compared to the rate of baryon number violating processes. This means that the equilibrium description of particle phenomena is extremely accurate at electroweak temperatures. Thus, baryogenesis cannot occur at such low scales without the aid of phase transitions and the question of the order of the electroweak phase transition becomes central. If the EWPT is second order or a continuous crossover, the associated departure from equilibrium is insufficient to lead to relevant baryon number production [142]. This means that for EWBG to succeed, we either need the EWPT to be strongly first order or other methods of destroying thermal equilibrium to be present at the phase transition. For a first order transition there is an extremum at ip = 0 which becomes separated from a second local minimum by an energy barrier. At the critical temperature T = Tc both

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phases are equally favored energetically and at later times the minimum at

4.9.5. Baryogenesis via Leptogenesis Since the linear combination B — L is left unchanged by sphaleron transitions, the baryon asymmetry may be generated from a lepton asymmetry [152-154] (see also [155,156].) Indeed, sphaleron transition will reprocess any lepton asymmetry and convert (a fraction of) it into baryon number. This is because B + L must be vanishing and the final baryon asymmetry results to be B ~ —L. In the SM as well as in its unified extension based on the group SU(5), B — L is conserved and no asymmetry in B — L can be generated. However,

770

adding right-handed Majorana neutrinos to the SM breaks B — L and the primordial lepton asymmetry may be generated by the out-of-equilibrium decay of heavy right-handed Majorana neutrinos 7V£ (in the supersymmetric version, heavy scalar neutrino decays are also relevant for leptogenesis). This simple extension of the SM can be embedded into GUTs with gauge groups containing 50(10). Heavy right-handed Majorana neutrinos can also explain the smallness of the light neutrino masses via the see-saw mechanism [154].

4.9.6. Affleck-Dine Baryogenesis Finally in this section, we mention briefly a mechanism introduced by Affleck and Dine [157] involving the cosmological evolution of scalar fields carrying baryonic charge. Consider a colorless, electrically neutral combination of quark and lepton fields. In a supersymmetric theory this object has a scalar superpartner, X, composed of the corresponding squark q and slepton I fields. An important feature of supersymmetric field theories is the existence of "flat directions" in field space, on which the scalar potential vanishes. Consider the case where some component of the field x h e s along a flat direction. By this we mean that there exist directions in the superpotential along which the relevant components of x can be considered as a free massless field. At the level of renormalizable terms, flat directions are generic, but supersymmetry breaking and nonrenormalizable operators lift the flat directions and sets the scale for their potential. During inflation it is likely that the x field is displaced from the position (x) = 0, establishing the initial conditions for the subsequent evolution of the field. An important role is played at this stage by baryon number violating operators in the potential V(x), which determine the initial phase of the field. When the Hubble rate becomes of the order of the curvature of the potential ~ TO3/2, the condensate starts oscillating around its present minimum. At this time, B-violating terms in the potential are of comparable importance to the mass term, thereby imparting a substantial baryon number to the condensate. After this time, the baryon number violating operators are negligible so that, when the baryonic charge of x is transferred to fermions through decays, the net baryon number of the universe is preserved by the subsequent cosmological evolution. The most recent implementations of the Affleck-Dine scenario have been in the context of the minimal supersymmetric standard model [158,159]

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in which, because there are large numbers of fields, flat directions occur because of accidental degeneracies in field space. 5. Inflation In our previous lectures we have described what is known as the standard cosmology. This framework is a towering achievement, describing to great accuracy the physical processes leading to the present day universe. However, there remain outstanding issues in cosmology. Many of these come under the heading of initial condition problems and require a more complete description of the sources of energy density in the universe. The most severe of these problems eventually led to a radical new picture of the physics of the early universe - cosmological inflation [160-162], which is the subject of this lecture. We will begin by describing some of the problems of the standard cosmology. 5.1. The Flatness

Problem

The Friedmann equation may be written as k (172) H2a2 ' where for brevity we are now writing Q, instead of fitotai- Differentiating this with respect to the scale factor, this implies

n-i

dVL , ,fi(fi-l) (173) — = (l + 3w)-± '. da a This equation is easily solved, but its most general properties are all that we shall need and they are qualitatively different depending on the sign of 1 + 3w. There are three fixed points of this differential equation, as given in table 4. Fixed Point fl = 1 $7 = oo

1 + 3w > 0 attractor repeller attractor

1 + 3w < 0 repeller attractor repeller

Observationally we know that f2 ~ 1 today - i.e., we are very close to the repeller of this differential equation for a universe dominated by

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ordinary matter and radiation (w > —1/3). Even if we only took account of the luminous matter in the universe, we would clearly live in a universe that was far from the attractor points of the equation. It is already quite puzzling that the universe has not reached one of its attractor points, given that the universe has evolved for such a long time. However, we may be more quantitative about this. If the only matter in the universe is radiation and dust, then in order to have tt in the range observed today requires (conservatively) 0 < 1 - 0 < 10" 6 0 .

(174)

This remarkable degree of fine tuning is the flatness problem. Within the context of the standard cosmology there is no known explanation of this fine-tuning. 5.2. The Horizon

Problem

The horizon problem stems from the existence of particle horizons in FRW cosmologies, as discussed in the first lecture. Horizons exist because there is only a finite amount of time since the Big Bang singularity, and thus only a finite distance that photons can travel within the age of the universe.

Figure 10. Past light cones in a universe expanding from a Big Bang singularity, illustrating particle horizons in cosmology. Points at recombination, observed today as parts of the cosmic microwave background on opposite sides of the sky, have non-overlapping past light cones (in conventional cosmology); no causal signal could have influenced them to have the same temperature.

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Consider a photon moving along a radial trajectory in a flat universe (the generalization to non-flat universes is straightforward). In a flat universe, we can normalize the scale factor to a0 = 1

(175)

without loss of generality. A radial null path obeys 0 = ds2 = -dt2 + a2dr2 ,

(176)

so the comoving (coordinate) distance traveled by such a photon between times t\ and ti is ft2 dt a - . Ar = J/tl — (t)

/iwx (177)

To get the physical distance as it would be measured by an observer at any time t, simply multiply by a(t). For simplicity let's imagine we are in a matter-dominated universe, for which , , to/ The Hubble parameter is therefore given by

(178)

H=2-t-i 3 = a^2H0 .

(179)

Then the photon travels a comoving distance Ar = 2HQ~1 ( > J - v/5I) •

(180)

The comoving horizon size when a = a* is the distance a photon travels since the Big Bang, rhor(a*)=2Jf0-1v^;.

(181)

The physical horizon size, as measured on the spatial hypersurface at a*, is therefore simply dhor(a*) = a*r ho r(a*) = 2H'1

.

(182)

Indeed, for any nearly-flat universe containing a mixture of matter and radiation, at any one epoch we will have dhorK) - HZ1 ,

(183)

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where H^1 is the Hubble distance at that particular epoch. This approximate equality leads to a strong temptation to use the terms "horizon distance" and "Hubble distance" interchangeably; this temptation should be resisted, since inflation can render the former much larger than the latter, as we will soon demonstrate. The horizon problem is simply the fact that the CMB is isotropic to a high degree of precision, even though widely separated points on the last scattering surface are completely outside each others' horizons. When we look at the CMB we were observing the universe at a scale factor OCMB ~ 1/1200; meanwhile, the comoving distance between a point on the CMB and an observer on Earth is Ar = 2H0l (1 « 2tf0"1 .

^O^MI)

(184)

However, the comoving horizon distance for such a point is fhor(aCMB) = %HQ

T/OCMB

sa 6 x l O - 2 ^ 1 .

(185)

Hence, if we observe two widely-separated parts of the CMB, they will have non-overlapping horizons; distinct patches of the CMB sky were causally disconnected at recombination. Nevertheless, they are observed to be at the same temperature to high precision. The question then is, how did they know ahead of time to coordinate their evolution in the right way, even though they were never in causal contact? We must somehow modify the causal structure of the conventional FRW cosmology. 5.3. Unwanted

Relics

We have spoken in the last lecture about grand unified theories (GUTs), and also about topological defects. If grand unification occurs with a simple gauge group G, any spontaneous breaking of G satisfies TT2(G/H) = TT\(H) for any simple subgroup H. In particular, breaking down to the standard model will lead to magnetic monopoles, since ir2(G/H) = TTI([SU(3) x SU(2) x U(l)]/Z 6 ) = Z .

(186)

(The gauge group of the standard model is, strictly speaking, [SU(3) x SU(2) x \J(1)]/Z6. The Z6 factor, with which you may not be familiar, only affects the global structure of the group and not the Lie algebra, and thus is usually ignored by particle physicists.)

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Using the Kibble mechanism, the expected relic abundance of monopoles works out to be tto,mo„o ~ 10 11 ( 1 0 1 4 G Q T e V )

( 10 76 m Ge°v) '

(18?)

This is far too big; the monopole abundance in GUTs is a serious problem for cosmology if GUTs have anything to do with reality. In addition to monopoles, there may be other model-dependent relics predicted by your favorite theory. If these are incompatible with current limits, it is necessary to find some way to dilute their density in the early universe. 5.4. The General Idea of

Inflation

The horizon problem especially is an extremely serious problem for the standard cosmology because at its heart is simply causality. Any solution to this problem is therefore almost certain to require an important modification to how information can propagate in the early universe. Cosmological inflation is such a mechanism. Before getting into the details of inflation we will just sketch the general idea here. The fundamental idea is that the universe undergoes a period of accelerated expansion, defined as a period when a > 0, at early times. The effect of this acceleration is to quickly expand a small region of space to a huge size, diminishing spatial curvature in the process, making the universe extremely close to flat. In addition, the horizon size is greatly increased, so that distant points on the CMB actually are in causal contact and unwanted relics are tremendously diluted, solving the monopole problem. As an unexpected bonus, quantum fluctuations make it impossible for inflation to smooth out the universe with perfect precision, so there is a spectrum of remnant density perturbations; this spectrum turns out to be approximately scale-free, in good agreement with observations of our current universe. 5.5. Slowly-Rolling

Scalar

Fields

If inflation is to solve the problems of the standard cosmology, then it must be active at extremely early times. Thus, we would like to address the earliest times in the universe amenable to a classical description. We expect this to be at or around the Planck time tp and since Planckian quantities arise often in inflation we will retain values of the Planck mass

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in the equations of this section. There are many models of inflation, but because of time constraints we will concentrate almost exclusively on the chaotic inflation model of Linde. We have borrowed heavily in places here from the excellent text of Liddle and Lyth [7]. Consider modeling matter in the early universe by a real scalar field 4>, with potential V((/>). The energy-momentum tensor for
a0 g^ lg (Va
+ V(
(188)

For simplicity we will specialize to the homogeneous case, in which all quantities depend only on cosmological time t and set k = 0. A homogeneous real scalar field behaves as a perfect fluid with P* = \i>2+V{) p = \i? - v{4>) •

(189) (wo)

The equation of motion for the scalar field is given by 0+ 3 - ^ + ^ = 0 , (191) a a

87rG =

\i2 + v(4>)

(192)

A very specific way in which accelerated expansion can occur is if the universe is dominated by an energy component that approximates a cosmological constant. In that case the associated expansion rate will be exponential, as we have already seen. Scalar fields can accomplish this in an interesting way. From (189) it is clear that if ) then the potential energy of the scalar field is the dominant contribution to both the energy density and the pressure, and the resulting equation of state is p ~ — p, approximately that of a cosmological constant, the resulting expansion is certainly accelerating. In a loose sense, this negligible kinetic energy is equivalent to the fields slowly rolling down its potential; an approximation which we will now make more formal. Technically, the slow-roll approximation for inflation involves neglecting the (/> term in (191) and neglecting the kinetic energy of compared to the

777

potential energy. The scalar field equation of motion and the Friedmann equation then become

1,,

V'M

(193)

3ff H2

^

8

-^V(4>)

,

(194)

where in this lecture a prime denotes a derivative with respect to <\>. These conditions will hold if the two slow-roll conditions are satisfied. These are |e| « 1 M « 1,

(195)

where the slow-roll parameters are given by

v

,

>

(196)

and

r\ = MP2Xr V

•

(197)

It is easy to see that the slow roll conditions yield inflation. Recall that inflation is defined by a/a > 0. We can write + H2,

(198)

§-2 > - 1 .

(199)

f2

(200)

-=H so that inflation occurs if

But in slow-roll - "« ,

which will be small. Smallness of the other parameter n helps to ensure that inflation will continue for a sufficient period. It is useful to have a general expression to describe how much inflation occurs, once it has begun. This is typically quantified by the number of e-folds, defined by

778

Usually we are interested in how many efolds occur between a given field value end) = 1. We also would like to express iV in terms of the potential. Fortunately this is simple to do via

The issue of initial conditions for inflation is one that is quite subtle and we will not get into a discussion of that here. Instead we will remain focused on chaotic inflation, in which we assume that the early universe emerges from the Planck epoch with the scalar field taking different values in different parts of the universe, with typically Planckian energies. There will then be some probability for inflation to begin in some places, and we shall focus on those. 5.6. Attractor

Solutions

in

Inflation

For simplicity, let us consider a particularly simple potential V{(t>) = \m24>2

,

(203)

where m has dimensions of mass. We shall also assume initial conditions that at the end of the quantum epoch, which we label as t = 0, p ~ M*. The slow-roll conditions give that, at t = 0, M2 M $ = fa „ _JL ^ ^ £ , m

(204)

£

where we have defined e = m/Mp ^C 1. We also have H = a4> ,

(205)

where a

£ ,

(206)

so that the scalar field equation of motion becomes 4> + 3 a # + m2(j) = 0 .

(207)

This is solved by =
(208)

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where /3 = m2/3a. that

Now, the slow-roll conditions remain satisfied provided

* » £ = ^ , m v/l27r and therefore, (208) is valid for a time

(209)

A * ~ | ~ £ .

(210)

Why is this important? This is important because (208) is an attractor solution! Let's see how this arises. Consider perturbing (208) by writing (t) - {t) + X(t) ,

(211)

where x(t) is a "small" perturbation, substituting in to the equation of motion and linearizing in x- One obtains X + 3ax = 0 .

(212)

This equation exhibits two solutions. The first is just a constant, which just serves to move along the trajectory. The second solution is a decaying mode with time constant tc = (3a)_1. Since tc
of the Standard

Cosmology

Let's stick with the simple model of the last section. It is rather easy to see that the Einstein equations are solved by 2TT

(213)

a(t) = a(0) exp M*^-*).

as we might expect in inflation. The period of time during which (208) is valid ends at t ~ At, at which a(At)~a(0)expQj)

,

(214)

Now, taking a typical value for m, for which s < 1 0 - 4 , we obtain a(At)~a(0)xl02-7xl°8 .

(215)

This has a remarkable consequence. A proper distance Lp at t = 0 will inflate to a size 1010 cm after a time At ~ 5 x 1 0 - 3 6 seconds. It is important at this point to know that the size of the observable universe

780

today is HQ1 ~ andhasnothadenoughl028cml Therefore, only a small fraction of the original Planck length comprises today's entire visible universe. Thus, homogeneity over a patch less than or of order the Planck length at the onset of inflation is all that is required to solve the horizon problem. Of course, if we wait sufficiently long we will start to see those inhomogeneities (originally sub-Planckian) that were inflated away. However, if inflation lasts long enough (typically about sixty e-folds or so) then this would not be apparent today. Similarly, any unwanted relics are diluted by the tremendous expansion; so long as the GUT phase transition happens before inflation, monopoles will have an extremely low density. Inflation also solves the flatness problem. There are a couple of ways to see this. The first is to note that, assuming inflation begins, the curvature term in the Friedmann equation very quickly becomes irrelevant, since it scales as a(t)~2. Of course, after inflation, when the universe is full of radiation (and later dust) the curvature term redshifts more slowly than both these components and will eventually become more important than both of them. However, if inflation lasts sufficiently long then even today the total energy density will be so close to unity that we will notice no curvature. A second way to see this is that, following a similar analysis to that leading to (173) leads to the conclusion that Q, = 1 is an attractor rather than a repeller, when the universe is dominated by energy with w < —1/3. Therefore, Q, is forced to be very close to one by inflation; if the inflationary period lasts sufficiently long, the density parameter will not have had time since to stray appreciably away from unity.

5.8. Vacuum Fluctuations

and

Perturbations

Recall that the structures - clusters and superclusters of galaxies - we see on the largest scales in the universe today, and hence the observed fluctuations in the CMB, form from the gravitational instability of initial perturbations in the matter density. The origin of these initial fluctuations is an important question of modern cosmology. Inflation provides us with a fascinating solution to this problem - in a nutshell, quantum fluctuations in the inflaton field during the inflationary epoch are stretched by inflation and ultimately become classical fluctuations. Let's sketch how this works. Since inflation dilutes away all matter fields, soon after its onset the universe is in a pure vacuum state. If we simplify to the case of

781

exponential inflation, this vacuum state is described by the GibbonsHawking temperature TGH =

2^ " Wp '

(216)

where we have used the Friedmann equation. Because of this temperature, the inflaton experiences fluctuations that are the same for each wavelength 8k = TGH- NOW, these fluctuations can be related to those in the density by

Inflation therefore produces density perturbations on every scale. The amplitude of the perturbations is nearly equal at each wavenumber, but there will be slight deviations due to the gradual change in V as the inflaton rolls. We can characterize the fluctuations in terms of their spectrum As{k), related to the potential via

Al(k)

V'3

W ) 2 k=aH '

(218)

where k = aH indicates that the quantity V3/(V')2 is to be evaluated at the moment when the physical scale of the perturbation A = a/k is equal to the Hubble radius H~x. Note that the actual normalization of (218) is convention-dependent, and should drop out of any physical answer. The spectrum is given the subscript "S" because it describes scalar fluctuations in the metric. These are tied to the energy-momentum distribution, and the density fluctuations produced by inflation are adiabatic — fluctuations in the density of all species are correlated. The fluctuations are also Gaussian, in the sense that the phases of the Fourier modes describing fluctuations at different scales are uncorrelated. These aspects of inflationary perturbations — a nearly scale-free spectrum of adiabatic density fluctuations with a Gaussian distribution — are all consistent with current observations of the CMB and large-scale structure, and have been confirmed to new precision by WMAP and other CMB measurements. It is not only the nearly-massless inflaton that is excited during inflation, but any nearly-massless particle. The other important example is the graviton, which corresponds to tensor perturbations in the metric (propagating excitations of the gravitational field). Tensor fluctuations have a spectrum

782

AUk) ~ ^ j P

•

(219)

k=aH

The existence of tensor perturbations is a crucial prediction of inflation which may in principle be verifiable through observations of the polarization of the CMB. Although CMB polarization has already been detected [163], this is only the .E-mode polarization induced by density perturbations; the B-mode polarization induced by gravitational waves is expected to be at a much lower level, and represents a significant observational challenge for the years to come. For purposes of understanding observations, it is useful to parameterize the perturbation spectra in terms of observable quantities. We therefore write Al(k) oc k713'1

(220)

A%{k) oc fcnT ,

(221)

and

where ns and « T are the "spectral indices". They are related to the slowroll parameters of the potential by n s = 1 - 6e + 2?7

(222)

nT = -2e .

(223)

and

Since the spectral indices are in principle observable, we can hope through relations such as these to glean some information about the inflaton potential itself. Our current knowledge of the amplitude of the perturbations already gives us important information about the energy scale of inflation. Note that the tensor perturbations depend on V alone (not its derivatives), so observations of tensor modes yields direct knowledge of the energy scale. If large-scale CMB anisotropies have an appreciable tensor component (possible, although unlikely), we can instantly derive Vjnflation ~ (1016 GeV) 4 . (Here, the value of V being constrained is that which was responsible for creating the observed fluctuations; namely, 60 e-folds before the end of inflation.) This is remarkably reminiscent of the grand unification scale, which is very encouraging. Even in the more likely case that the perturbations observed in the CMB are scalar in nature, we can still write N a t i o n - e ^ l O 1 6 GeV,

(224)

783

where e is the slow-roll parameter defined in (196). Although we expect e to be small, the 1/4 in the exponent means that the dependence on e is quite weak; unless this parameter is extraordinarily tiny, it is very likely that
and

Preheating

Clearly, one of the great strengths of inflation is its ability to redshift away all unwanted relics, such as topological defects. However, inflation is not discerning, and in doing so any trace of radiation or dust-like matter is similarly redshifted away to nothing. Thus, at the end of inflation the universe contains nothing but the inflationary scalar field condensate. How then does that matter of which we are made arise? How does the hot big bang phase of the universe commence? How is the universe reheated! For a decade or so reheating was thought to be a relatively wellunderstood phenomenon. However, over the last 5-10 years, it has been realized that the story may be quite complicated. We will not have time to go into details here. Rather, we will sketch the standard picture and mention briefly the new developments. Inflation ends when the slow-roll conditions are violated and, in most models, the field begins to fall towards the minimum of its potential. Initially, all energy density is in the inflaton, but this is now damped by two possible terms. First, the expansion of the universe naturally damps the energy density. More importantly, the inflaton may decay into other particles,such as radiation or massive particles, both fermionic and bosonic [185,186]. To take account of this one introduces a phenomenological decay term T^ into the scalar field equation. If we focus on fermions only, then a rough expression for how the energy density evolves is p 0 + (3H + T^P4> = 0 .

(225)

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The inflaton undergoes damped oscillations and decays into radiation which equilibrates rapidly at a temperature known as the reheat temperature TRH • In the case of bosons however, this ignores the fact that the inflaton oscillations may give rise to parametric resonance. This is signified by an extremely rapid decay, yielding a distribution of products that is far from equilibrium, and only much later settles down to an equilibrium distribution at energy TRH. Such a rapid decay due to parametric resonance is known as preheating [187,188]. One interesting outcome of preheating is that one can produce particles with energy far above the ultimate reheat temperature. Thus, one must beware of producing objects, such as topological relics, that inflation had rid us of. However, preheating can provide some useful effects, such as interesting new ways to generate dark matter candidates and topological relics and to generate the baryon asymmetry [189-194]. 5.10. The Beginnings

of

Inflation

If inflation begins, we have seen that it achieves remarkable results. It can be shown quite generally that the start of inflation in a previously noninflating region requires pre-existing homogeneity over a horizon volume [195]. It seems unlikely that classical, causal physics could cause this and we may interpret this as an indication that quantum physics may play a role in the process of inflation. Although it can be shown [196-200] that inflation cannot be past eternal, one consequence of quantum mechanics is that inflation may be future eternal [201-205]. To see this, consider inflation driven by a scalar field rolling down a simple potential as in figure (5.10). Imagine that, in a given space-like patch, inflation is occurring with the scalar field at a potential energy V\. Although the classical motion of the scalar field is towards the minimum of the potential, there is a nonzero probability that quantum fluctuations will cause the inflaton to jump up to a higher value of the potential V% > V\ over a sufficiently large part of the inflating patch. In the new region, the Friedmann equation tells us that the local Hubble parameter is now larger than in the inflating background

1

^/Sri/ir" '

<226)

and requiring the region to be sufficiently large means that its linear size is at least iTT , which may be considerably smaller than the size of the

785

V(4>)

Figure 11. Quantum fluctuations in a quasi-de Sitter spacetime can cause a scalar field to jump up its potential, increasing the expansion rate and leading to eternal inflation.

background inflationary horizon. Thus, inflation occurs in our new region with a larger Hubble parameter and, because of the exponential nature of inflation, the new patch very quickly contains a much larger volume of space than the old one. There are, of course, parts of space in which the classical motion of the inflaton to its minimum completes, and those regions drop out of the inflationary expansion and are free to become regions such of our own. However, it is interesting to note that, as long as the probability for inflation to begin at all is not precisely zero, any region like ours observed at a later time comes from a previously inflating region with probability essentially unity. Nevertheless, in this picture, most regions of the universe inflate forever and therefore this process is known as eternal inflation. If correct, eternal inflation provides us with an entirely new global view of the universe and erases many of our worries about the initial condition problems of inflation. However, some caution is in order when considering this model. Eternal inflation relies on the quantum backreaction of geometry to matter fluctuations over an extended spatial region. This is something we do not know how to treat fully at the present time. Perhaps string theory can help us with this. Acknowledgements We would each like the thank the TASI organizers (Howard Haber, K.T. Mahantappa, Juan Maldacena, and Ann Nelson) for inviting us to lecture and for creating such a vigorous and stimulating environment for

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students and lecturers alike. We would also like to t h a n k t h e Kavli Institute for Theoretical Physics for hospitality while these proceedings were being completed. T h e work of SMC is supported in p a r t by U.S. Dept. of Energy contract DE-FG02-90ER-40560, National Science Foundation Grant PHY01-14422 (CfCP) a n d t h e David and Lucile Packard Foundation. T h e work of M T is supported in part by the National Science Foundation (NSF) under grant PHY-0094122. Mark Trodden is a Cottrell Scholar of Research Corporation.

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KEITH A. OLIVE

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DARK MATTER*

KEITH A. OLIVEt William I. Fine Theoretical Physics Institute, School of Physics and University of Minnesota, Minneapolis, MN 55455 USA E-mail: [email protected]

Astronomy,

Observational evidence and theoretical motivation for dark matter are presented and connections to the CMB and BBN are made. Problems for baryonic and neutrino dark matter are summarized. Emphasis is placed on the prospects for supersymmetric dark matter.

Contents 1

Lecture 1 1.1 Observational Evidence 1.2 Theory

798 799 805

2

Lecture 2 2.1 Big Bang Nucleosynthesis 2.1.1 Theory 2.1.2 Abundances 2.2 Candidates 2.2.1 Baryons 2.2.2 Neutrinos 2.2.3 Axions

809 809 809 812 814 814 819 825

3

Lecture 3: Supersymmetric Dark Matter 3.1 Parameters 3.2 Neutralinos 3.3 The Relic Density 3.4 Phenomenological and Cosmological Constraints 3.5 Detection

826 827 829 830 832 839

* Summary of lectures given at the Theoretical Advanced Study Institute in Elementary Particle Physics at the University of Colorado at Boulder — June 2-28, 2002. + This work was supported in part by DOE grant DE-FG02-94ER40823 at Minnesota. 797

798

1. Lecture 1 The nature and identity of the dark matter of the Universe is one of the most challenging problems facing modern cosmology. The problem is a long-standing one, going back to early observations of mass-to-light ratios by Zwicky.1 Given the distribution (by number) of galaxies with total luminosity L, (f>(L), one can compute the mean luminosity density of galaxies £=

f L4>(L)dL

(1)

which is determined to be 2 £ ~ 2 ± 0 . 2 x 108hoLQMpc~3

(2)

where LQ — 3.8 x 10 33 erg s _ 1 is the solar luminosity. In the absence of a cosmological constant, one can define a critical energy density, pc = 3H2/8TTGN = 1-88 x 10~29ho2 g c m - 3 , such that p = pc for three-space curvature k = 0, where the present value of the Hubble parameter has been defined by H0 = 100ho km M p c - 1 s _ 1 . We can now define a critical mass-to-light ratio is given by (M/L)c

= Pc/C ~ 139O/i o (M o /L 0 )

(3)

which can be used to determine the cosmological density parameter nm = £• = (M/L)/(M/L)C (4) Pc Mass-to-light ratios are, however, strongly dependent on the distance scale on which they are determined. 3 In the solar neighborhood M/L ~ 2±1 (in solar units), yielding values of 57m of only ~ .001. In the bright central parts of galaxies, M/L ~ (10 — 20)ho so that f2m ~ 0.01. On larger scales, that of binaries and small groups of galaxies, M/L ~ (60 — 180)h0 and fim ~ 0.1. On even larger scales, that of clusters, M/L may be as large as (200 — 500)/io giving fim ~ 0.3. This progression in M/L seems to have halted, as even on the largest scales observed today, mass-to-light ratios imply values of ilm < 0.3 — 0.4. Thus when one considers the scale of galaxies (and their halos) and larger, the presence of dark matter (and as we shall see, non-baryonic dark matter) is required. Direct observational evidence for dark matter is found from a variety of sources. On the scale of galactic halos, the observed flatness of the rotation curves of spiral galaxies is a clear indicator for dark matter. There is also evidence for dark matter in elliptical galaxies, as well as clusters of galaxies coming from the X-ray observations of these objects. Also, direct

799

evidence has been obtained through the study of gravitational lenses. On the theoretical side, we predict the presence of dark matter (or dark energy) because (1) it is a strong prediction of most inflation models (and there is at present no good alternative to inflation), and (2) our current understanding of galaxy formation requires substantial amounts of dark matter to account for the growth of density fluctuations. One can also make a strong case for the existence of non-baryonic dark matter in particular. The recurrent problem with baryonic dark matter is that not only is it very difficult to hide baryons, but given the amount of dark matter required on large scales, there is a direct conflict with primordial nucleosynthesis if all of the dark matter is baryonic. In this first lecture, I will review the observational and theoretical evidence supporting the existence of dark matter. 1.1. Observational

Evidence

Assuming that galaxies are in virial equilibrium, one expects that one can relate the mass at a given distance r, from the center of a galaxy to its rotational velocity by M{r) oc v2r/GN 3,4

(5)

The rotational velocity, v, is measured by observing 21 cm emission lines in HI regions (neutral hydrogen) beyond the point where most of the light in the galaxy ceases. A subset of a compilation 5 of nearly 1000 rotation curves of spiral galaxies is shown in Fig. 1. The subset shown is restricted to a narrow range in brightness, but is characteristic for a wide range of spiral galaxies. Shown is the rotational velocity as a function of r in units of the optical radius. If the bulk of the mass is associated with light, then beyond the point where most of the light stops, M would be constant and i>2 oc 1/r. This is not the case, as the rotation curves appear to be flat, i.e., v ~ constant outside the core of the galaxy. This implies that M oc r beyond the point where the light stops. This is one of the strongest pieces of evidence for the existence of dark matter. Velocity measurements indicate dark matter in elliptical galaxies as well.6 Galactic rotation curves are not the only observational indication for the existence of dark matter. X-ray emitting hot gas in elliptical galaxies also provides an important piece of evidence for dark matter. A particularly striking example is that of the large elliptical M87. Detailed profiles of the temperature and density of the hot X-ray emitting gas have been mapped out. 7 Assuming hydrostatic equilibrium, these measurements allow one to determine the overall mass distribution in the galaxy necessary to bind

800 <M,> = - 2 1 . 2 i i i i i i i i i | i i i i | i i i i | i

-,

-miim-l

1

§•

ca ~>

g

.5 c> 1 1 1 ' ' ' • i ' ' ' ' i ' ' ' ' i

0 0

.5

1

1.5

2

R/Ropt Figure 1. Synthetic rotation curve 5 for galaxies with (M) = —21.2. The dotted curve shows the disk contribution, whereas the dashed curve shows the halo contribution.

the hot gas. Based on an isothermal model with temperature kT = 3keV (which leads to a conservative estimate of the total mass), Fabricant and Gorenstein 7 predicted that the total mass out to a radial distance of 392 kpc is 5.7 x 10 13 M Q , whereas the mass in the hot gas is only 2.8 x 1O 12 M 0 or only 5% of the total. The visible mass is expected to contribute only 1% of the total. M87 is not the only example of an elliptical galaxy in which X-ray emitting hot gas is observed to indicate the presence of dark matter. X-ray observations have shown that the total mass associated with elliptical galaxies is considerably larger than the luminous component in many examples of varying morphological types. 8 Mass-to-light ratios for these systems vary, with most being larger than 30/io a n d some ranging as high as ~ 200/irj. In addition, similar inferences pertaining to the existence of dark matter can be made from the X-ray emission from small groups of galaxies.9 On these scales, mass-to-light ratios are typically > 100/io and detailed studies have shown that the baryon faction in these systems is rather small. Furthermore, it was argued 10 that cluster baryon fractions should not differ from the Universal value given by f2s/f2 m . Using an estimate of QB = 0.04 from big bang nucleosynthesis (BBN, see below), and baryon fractions ranging from 0.1 to 0.3, one would obtain an estimate for the total matter density of 0.13-0.4. Another piece of evidence on large scales is available from gravitational lensing.11 The systematic lensing of the roughly 150,000 galaxies per deg 2 at redshifts between z = 1 — 3 into arcs and arclets allow one to trace

801

the matter distribution in a foreground cluster. Lensing observations can be categorized as either strong (multiple images) or weak (single images). Both require the presence of a dominant dark matter component. Strong lensing is particularly adept in testing the overall geometry of the Universe. 12 ' 13 While a cluster which provides multiple lenses of a single background galaxy (at known redshift) is useful for determining the total cluster mass, when several background galaxies are lensed, it is possible to constrain the values of Q,m and f^A-14 The recent results of Ref. 13 show a degeneracy in the flm — fl\ plane. Nevertheless, the allowed region is offset from similar types of degeneracies found in supernovae searches and the CMB (see below). Indeed, these lensing results are much more constraining in D,m than the other techniques, though a residual uncertainty of about 30 % persists. While in principle, these results find that any value (from 0 to 1) is possible for OA, ^m < 0.5 for low values of f^A and fim < 0.4 for higher values of Sl\ (^ 0.6). Weak lensing of galaxies by galaxies can (on a statistical basis) also probe the nature of galactic halos. Recent studies based on weak lensing data indicate that galactic halos may be far more extended than previously thought 15 (radii larger than 200 h^1 kpc). These results also imply a substantial contribution to Q m (of order 0.1-0.2) on this scale. On larger scales, using many cluster lenses enables one to estimate fi ~ 0.3. 16 Another use of weak lensing statistics is to determine the evolution of cosmic shear and hence an estimate of fi m . 1 7 Finally, there exist a number of examples of dark clusters, ie., lenses with no observable counterpart. 18 The contribution of these objects (if they are robust) to J7m is not clear. For a recent review of weak lensing see Ref. 19. Finally, on very large scales, it is possible to get an estimate of fim from the distribution of peculiar velocities of galaxies and clusters. On scales, A, where perturbations, 5, are still small, peculiar velocities can be expressed20 as v ~ HXSil^. On these scales, older measurements of the peculiar velocity field from the IRAS galaxy catalogue indicate that indeed f2 is close to unity. 21 Some of the new data indicates a lower value in the range 22 0.2-0.5, but does not conclusively exclude f2m = l. 23 The above discussion of observational evidence for dark matter pertains largely to the overall matter density of the Universe flm. (For a comprehensive review of determinations of the matter density, see Ref. 24.) However, the matter density and the overall curvature are not related one-to-one. The expansion rate of the Universe in the standard FRW model is expressed by

802

the Friedmann equation R2

2

H =

=

8irGNP

k

& -J--&

+

A

3

(6)

where R(t) is the cosmological scale factor, k is the three-space curvature constant (k = 0, + 1 , —1 for a spatially flat, closed or open Universe), and A is the cosmological constant. The Friedmann equation can be rewritten as

(f2-l)ff 2 = A

(7 )

so that k = 0,+l,—1 corresponds to fi = l,fi > 1 and Q < 1. However, the value of ft appearing in Eq. (7) represents the sum Q, = Q,m + £l\ of contributions from the matter density (fl m ) and the cosmological constant (n A = A/SH2). There has been a great deal of progress in the last several years concerning the determination of both fim and $1A- Cosmic Microwave Background (CMB) anisotropy experiments have been able to determine the curvature (i.e. the sum of fim and OA) to with in about 10%, while observations of type la supernovae at high redshift provide information on a (nearly) orthogonal combination of the two density parameters. The CMB is of course deeply rooted in the development and verification of the big bang model. 25 Indeed, it was the formulation of BBN that led to the prediction of the microwave background. The argument is rather simple. BBN requires temperatures greater than 100 keV, which according to the standard model time-temperature relation, i s T^[ eV = 2A/\/N, where iV is the number of relativistic degrees of freedom at temperature T, and corresponds to timescales less than about 200 s. The typical cross section for the first link in the nucleosynthetic chain is av(p + n^D

+ ^)~5x

10~ 20 cm 3 /s

(8)

This implies that it was necessary to achieve a density n

^

~ 10 17 cm~ 3

(9)

for nucleosynthesis to begin. The density in baryons today is known approximately from the density of visible matter to be UB0 ~ 10~ 7 c m - 3 and since we know that that the density n scales as R~3 ~ T 3 , the temperature today must be T0 = (n B o /n) 1 / 3 T B B N - 10K thus linking two of the most important tests of the Big Bang theory.

(10)

803

Microwave background anisotropy measurements have made tremendous advances in the last few years. The power spectrum 2 6 - 2 8 has been measured relatively accurately out to multipole moments corresponding to £ ~ 1000. The details of this spectrum enable one to make accurate predictions of a large number of fundamental cosmological parameters. 27 > 29 ~ 33 An example of these results as found by a recent frequentist analysis 34 is shown in Fig. 2. ts 4

\. X

/

|

A

i

2

n 40

. J . .

Ns _/ . 50

60

70 H„

80

0.6

0.8

1.0

1.2 n

1.4

A 0.00

«/.

0.05

0.10

0.15

QB

Figure 2. A%2 calculated with the MAXIMA-1 and COBE data as a function of parameter value. Solid blue circles show grid points in parameter space, and the green lines were obtained by interpolating between grid points. The parameter values where the green line intercepts the red dashed (dotted) line corresponds to the 68% (95%) frequentist confidence region. 34

Of particular interest to us here is the CMB determination of the total density, fitot, as well as the matter density fi m . The results of recent CMB anisotropy measurements are summarized in Table 1. As one can see, there is strong evidence that the Universe is flat or very close to it. Furthermore, the matter density is very consistent with the observational determinations discussed above and the baryon density, as we will see below, is consistent

804 Table 1.

Results from recent CMB anisotropy measurements.

BOOMERanG27 MAXIMA

MAXIMA (freq.) DASI

1.03 ± 0.06 o-9tr 0 9 8

29 34

30

o-^il

nBh2

hlfntl

fitot

5

1.04 ± 0.06

0 021+ 0 - 0 0 4

0.12 ± 0 . 0 5 0 U

1i 7 + 0 . 0 8

- '-0.04

0.0325 ± 0.0125

0 07 0 25+ - 0 . 0- 9 0.14 ± 0 . 0 4

0U.UZD_ 026+ 00 -000160

u,zo

0 022+ 0 ' 0 0 4 u

-u^-0.003

CBI 3 1

0.99 ± 0 . 1 2

0

17+0.08

0 022+ 0 - 0 1 5

VSA 3 2

1.03 ± 0 . 0 1 2

n

,0+0.08

0.029 ± 0.009

i liog + 0 . 2 4 - -0.20

Archeops 3 3

001O+0.006 u .uiy_0 007

-

1

with the BBN production of D/H and its abundance in quasar absorption systems. The discrepancy between the CMB value of Qra and O B is sign that non-baryonic matter (dark matter) is required. Furthermore, the apparent discrepancy between the CMB value of fitot a n d O m , though not conclusive on its own, is a sign that a contribution from the vacuum energy density or cosmological constant, is also required. The preferred region in the O m —OA plane as determined by a frequentist analysis of MAXIMA data is shown in Fig. 3.

1.0 0.8 0.6

< G

0.4 0.2 0.0

0 ,0

0.2

0.4

0.6

0.8

1.0

Figure 3. Two-dimensional frequentist confidence regions in the ( O M , ^ A ) plane. The red, orange and yellow regions correspond to the 68%, 95%, and 99% confidence regions respectively. The dashed black line corresponds to a flat universe, O = QTO ± O A = 1.

805

The presence or absence of a cosmological constant is a long standing problem in cosmology. To theorists, it is particularly offensive due to the necessary smallness of the constant. We know that the cosmological term is at most a factor of a few times larger than the current mass density. Thus from Eq. (6), we see that the dimensionless combination, GAT A < 10~ 121 . Nevertheless, even a small non-zero value for A could greatly affect the future history of the Universe: allowing open Universes to recollapse (if A < 0), or closed Universes to expand forever (if A > 0 and sufficiently large). Another exciting development has been the use of type la supernovae, which now allow measurement of relative distances with 5% precision. In combination with Cepheid data from the HST key project on the distance scale, SNe results are the dominant contributor to the best modern value for H0: 7 2 k m s - 1 M p c " 1 ± 10%. 35 Better still, the analysis of high-;? SNe has allowed the first meaningful test of cosmological geometry to be carried out, as shown in Fig. 4. These results can be contrasted with those from the CMB anisotropy measurements as in Fig. 5. We are led to a seemingly conclusive picture. The Universe is nearly flat with r2tot — 1- However the density in matter makes up only 20-50% of this total, with the remainder in a cosmological constant or some other form of dark energy. 1.2.

Theory

Theoretically, there is no lack of support for the dark matter hypothesis. The standard big bang model including inflation almost requires fitot = l. 3 8 This can be seen from the following simple solution to the curvature problem. The simple and unfortunate fact that at present we do not even know whether Q is larger or smaller than one, indicates that we do not know the sign of the curvature term further implying that it is subdominant in Eq. (6) (ll)

& < Tp

In an adiabatically expanding Universe, R ~ T^1 where T is the temperature of the thermal photon background. Therefore the quantity

*= 3^<W

<2X1

°"

<12)

is dimensionless and constant in the standard model. This is known as the curvature problem and can be resolved by a period of inflation. Before

806

0.01

0.02 0.05 redshift z

0.1

redshift z Figure 4. The type la supernova Hubble diagram 3 6 taken from Ref. 37. The first panel shows that, for z C 1, the large-scale Hubble flow is indeed linear and uniform; the second panel shows an expanded scale, with the linear trend divided out, and with the redshift range extended to show how the Hubble law becomes nonlinear. Comparison with the prediction of Friedmann-Lemaitre models appears to favor a vacuum-dominated universe.

inflation, let us write R — Ri, T = Ti and R ~ T~l. During inflation, R ~ T^1 ~ eHt, where H is constant. After inflation, R = Rf ^S> Ri but T = Tf = TR < Ti where TR is the temperature to which the Universe reheats. Thus R / T and k —* 0 is not constant. But from Eqs. (7) and (12) if k —> 0 then Q, —> 1, and since typical inflationary models contain much more expansion than is necessary, £1 becomes exponentially close to

807

Figure 5. Likelihood-based confidence contours 3 6 over the plane O A (i.e. Qv assuming w = — 1 vs O m . The SNe la results very nearly constrain Qv — O m , whereas the results of CMB anisotropics (from the Boomerang 98 data) favor a flat model with Qv + O T O ~ 1. The Intersection of these constraints Is the most direct (but far from the only) piece of evidence favoring a flat model with Qm ~ 0.3.

The inflationary prediction of O = Ids remarkably consistent with the CMB measurements discussed above. Furthermore, we know two things: Dark matter exists, since we don't see O = 1 in luminous objects, and most (about 90%) of the dark matter is not baryonic. The latter conclusion is a result of our forthcoming discussion on BBN which constrains the baryonto-photon ratio and hence O B - Thus 1 ~- O B is not only dark but also nonbaryonic. Furthermore, the matter density is surely composed of several contributions: O m = O B + OI/ + O X where the latter represents the dark matter contribution. Another important piece of theoretical evidence for dark matter comes from the simple fact that we are living in a galaxy. The type of perturbations produced by inflation39 are, in most models, adiabatic perturbations (Sp/p ex ST/T), and I will restrict my attention to these. Indeed, the perturbations produced by inflation also have the very nearly scalefree spectrum described by Harrison and Zeldovich.40 When produced,

808

scale-free perturbations fall off as -^ oc l~2 (increase as the square of the wave number). At early times Sp/p grows as t until the time when the horizon scale (which is proportional to the age of the Universe) is comparable to I. At later times, the growth halts (the mass contained within the volume I3 has become smaller than the Jean's mass) and -^ = 5 (roughly) independent of the scale I. When the Universe becomes matter dominated, the Jean's mass drops dramatically and growth continues as & ex R ~ 1/T. The transition to matter dominance is determined by setting the energy densities in radiation (photons and any massless neutrinos) equal to the energy density in matter (baryons and any dark matter). For three massless neutrinos and baryons (no dark matter), matter dominance begins at Tm = 0.22mBr)

(13)

and for 77 < 7 x 10~ 10 , this corresponds to Tm < 0.14 eV. Because we are considering adiabatic perturbations, there will be anisotropics produced in the microwave background radiation on the order of ST /T ~ 8. The value of S, the amplitude of the density fluctuations at horizon crossing, has been determined by COBE, 41 5 = (5.7±0.4) x 10~ 6 . Without the existence of dark matter, Sp/p in baryons could then achieve a maximum value of only Sp/p ~ A\5{Tm/T0) < 2 x 10~ 3 ^4A, where -4 T0 = 2.35 x 1 0 eV is the present temperature of the microwave background and A\ ~ 1 — 10 is a scale dependent growth factor. The overall growth in Sp/p is too small to argue that growth has entered a nonlinear regime needed to explain the large value (105) of Sp/p in galaxies. Dark matter easily remedies this dilemma in the following way. The transition to matter dominance is determined by setting equal to each other the energy densities in radiation (photons and any massless neutrinos) and matter (baryons and any dark matter). While if we suppose that there exists dark matter with an abundance Yx = nx/n1 (the ratio of the number density of x's to photons) then Tm = 0.22mxYx 2

(14) 7

Since we can write m x y x / G e V = O x /i /(4 x 10 ), we have Tm/T0 = 2.4 x 104f2x/i2 which is to be compared with 600 in the case of baryons alone. The baryons, although still bound to the radiation until decoupling, now see deep potential wells formed by the dark matter perturbations to fall into and are no longer required to grow at the rate Sp/p oc R. With regard to dark matter and galaxy formation, all forms of dark matter are not equal. They can be distinguished by their relative

809

temperature neutrinos or (due to free given by the

at T m . 4 2 Particles which are still largely relativistic at Tm (like other particles with mx < 100 eV) have the property 43 that streaming) they erase perturbations out to very large scales Jean's mass Mj

=3x 10

1 8

^VT

(15)

mx{ev)

Thus, very large scale structures form first and galaxies are expected to fragment out later. Particles with this property are termed hot dark matter particles. Cold particles (mx > 1 MeV) have the opposite behavior. Small scale structure forms first aggregating to form larger structures later. It is now well known that pure HDM cosmologies can not reproduce the observed large scale structure of the Universe. In contrast, CDM does much better. Current attention is focused on so-called ACDM cosmologies based on the OA — Q.m contribution to the curvature discussed above. 2. Lecture 2 2.1. Big Bang

Nucleosynthesis

The standard model 44 of big bang nucleosynthesis (BBN) is based on the relatively simple idea of including an extended nuclear network into a homogeneous and isotropic cosmology. Apart from the input nuclear cross sections, the theory contains only a single parameter, namely the baryonto-photon ratio, rj. Other factors, such as the uncertainties in reaction rates, and the neutron mean-life can be treated by standard statistical and Monte Carlo techniques. 45 ' 46 The theory then allows one to make predictions (with well-defined uncertainties) of the abundances of the light elements, D, 3 He, 4 He, and 7 Li. 2.1.1. Theory Conditions for the synthesis of the light elements were attained in the early Universe at temperatures T > 1 MeV. In the early Universe, the energy density was dominated by radiation with

'=isH + K) T4

<16)

from the contributions of photons, electrons and positrons, and Nv neutrino flavors (at higher temperatures, other particle degrees of freedom should be

810

included as well). At these temperatures, weak interaction rates were in equilibrium. In particular, the processes n + e + <-> p + Pe n + ve <-» p + e~ n <-> p + e~ + ve

(17)

fix the ratio of number densities of neutrons to protons. At T > 1 MeV, (n/p) ~ 1. The weak interactions do not remain in equilibrium at lower temperatures. Ereeze-out occurs when the weak interaction rate, r ^ ~ G%T5 falls below the expansion rate which is given by the Hubble parameter, H ~ VG^ ~ T2/MP, where MP = l/VG^ ~ 1.2 x 10 19 GeV. The /3-interactions in eq. (17) freeze-out at about 0.8 MeV. As the temperature falls and approaches the point where the weak interaction rates are no longer fast enough to maintain equilibrium, the neutron to proton ratio is given approximately by the Boltzmann factor, (n/p) ~ e _ A m / T ~ 1/6, where Am is the neutron-proton mass difference. After freeze-out, free neutron decays drop the ratio slightly to about 1/7 before nucleosynthesis begins. The nucleosynthesis chain begins with the formation of deuterium by the process, p + n —> D + 7. However, because of the large number of photons relative to nucleons, r^1 = n-y/ns ~ 10 10 , deuterium production is delayed past the point where the temperature has fallen below the deuterium binding energy, EB = 2.2 MeV (the average photon energy in a blackbody is E1 ~ 2.IT). This is because there are many photons in the exponential tail of the photon energy distribution with energies E > EB despite the fact that the temperature or E1 is less than EB- The degree to which deuterium production is delayed can be found by comparing the qualitative expressions for the deuterium production and destruction rates, T p RS UBOV

(18)

When the quantity ry _1 exp(—EB/T) ~ 1, the rate for deuterium destruction (D + 7 —> p + n) finally falls below the deuterium production rate and the nuclear chain begins at a temperature T ~ O.lMeV.

811 The dominant product of big bang nucleosynthesis is 4 He and its abundance is very sensitive to the (n/p) ratio

Y„

2(n/p)

(19)

0.25

[1 + (n/p)}

i.e., an abundance of close to 25% by mass. Lesser amounts of the other light elements are produced: D and 3 He at the level of about 1 0 - 5 by number, and 7 Li at the level of 10~~10 by number. The resulting abundances of the light elements 46 are shown in Fig. 6, over the range in 7710 = 1010n between 1 and 10. The left plot shows the abundance of 4 He by mass, Y, and the abundances of the other three isotopes by number. The curves indicate the central predictions from BBN, while the bands correspond to the uncertainty in the predicted abundances based primarily the uncertainty in the input nuclear reactions as computed by Monte Carlo in Ref. 46. This theoretical uncertainty is shown explicitly in the right panel as a function of T/IQ. The dark shaded boxes correspond to the observed abundances of 4 He and 7 Li and will be discussed below. The dashed boxes correspond to the ranges of the elements consistent with the systematic uncertainties in the observations. The broad band shows a 0.25 - ( b )

-

0.2 -

-

-X 0.15

\

/

0.1 -

3

0.05 -

10-9 Figure 6. of 7710-

10-

/

/

X"

He/H

"X

D/H// Y

10-

The light element abundances from big bang nucleosynthesis as a function

812 liberal range for 7710 consistent with the observations. At present, there is a general concordance between the theoretical predictions and the observational data.

2.1.2. Abundances In addition to it BBN production, 4 He is made in stars, and thus coproduced with heavy elements. Hence the best sites for determining the primordial 4 He abundance are in metal-poor regions of hot, ionized gas in nearby external galaxies (extragalactic HII regions). Helium indeed shows a linear correlation with metallicity in these systems, and the extrapolation to zero metallicity gives the primordial abundance (baryonic mass fraction) 47 Yp = 0.238 ± 0.002 ± 0.005.

(20)

Here, the first error is statistical and reflects the large sample of systems, whilst the second error is systematic and dominates. The systematic uncertainties in these observations have not been thoroughly explored to date. 48 In particular, there may be reason to suspect that the above primordial abundance will be increased due to effects such as underlying stellar absorption in the HII regions. We note that other analyses give similar results: Yp = 0.244 ± 0.002 ± 0.00549 and 0.239 ±0.002. 50 The primordial 7 Li abundance comes from measurements in the atmospheres of primitive (Population II) stars in the stellar halo of our Galaxy. The 7 Li/H abundance is found to be constant for stars with low metallicity, indicating a primordial component, and a recent determination gives -^

= (1.23 ± 0.06±g"H) x 10" 1 0 (95% CL),

(21)

H p

where the small statistical error is overshadowed by systematic uncertainties. 51 The range (21) may, however, be underestimated, as a recent determination 52 uses a different procedure to determine stellar atmosphere parameters, and gives 7 Li/H = (2.19 ±0.28) x 10~ 10 . At this stage, it is not possible to determine which method of analysis is more accurate, indicating the likelihood that the upper systematic uncertainty in (21) has been underestimated. Thus, in order to obtain a conservative bound from 7 Li, we take the lower bound from (21) and the upper bound from,52 giving 7 Li 9.0x10-" < — <2.8xl0-10. Hp

(22)

813

Deuterium is measured in high-redshift QSO absorption line systems via its isotopic shift from hydrogen. In several absorbers of moderate column density (Lyman-limit systems), D has been observed in multiple Lyman transitions. 53 ' 54 Restricting our attention to the three most reliable regions, 53 we find a weighted mean of - = (2.9 ±0.3) x 1(T S . rip

(23)

It should be noted, however, that the x2 P e r degree of freedom is rather poor ( ~ 3.4), and that the unweighted dispersion of these data is ~ 0.6 x 1 0 - 5 . This already points to the dominance of systematic effects. Observation of D in systems with higher column density (damped systems) find lower D/H, 5 5 at a level inconsistent with (23), further suggesting that systematic effects dominate the error budget. 56 If all five available observations are used, we would find D/H = (2.6 ± 0.3) x 1 0 - 5 with an even worse x 2 per degree of freedom (~ 4.3) and an unweighted dispersion of 0.8. Because there are no known astrophysical sites for the production of deuterium, all observed D is assumed to be primordial. 57 As a result, any firm determination of a deuterium abundance establishes an upper bound on rj which is robust. Thus, the recent measurements of D/H (Ref. 53) at least provide a lower bound on D/H, D/H > 2.1 x 10^ 5 (2a) and hence provide an upper bound to rj, 7710 < 7.3 and fisft,2 < 0.027. Helium-3 can be measured through its hyperfme emission in the radio band, and has been observed in HII regions in our Galaxy. These observations find58 that there are no obvious trends in 3 He with metallicity and location in the Galaxy. There is, however, considerable scatter in the data by a factor ~ 2, some of which may be real. Unfortunately, the stellar and Galactic evolution of 3 He is not yet sufficiently well understood to confirm whether 3 He is increasing or decreasing from its primordial value. 59 Consequently, it is unclear whether the observed 3 He abundance represents an upper or lower limit to the primordial value. Therefore, we can not use 3 He abundance as a constraint. By combining the predictions of BBN calculations with the abundances of D, 4 He, and 7 Li discussed above one can determine the the 95% CL region 4.9 < 7710 < 6.4, with the peak value occurring at rjio = 5.6. This range corresponds to values of Q,B between 0.018 < nBh2 with a central value of fi^/i 2 = 0.020.

< 0.023

(24)

814

If we were to use only the deuterium abundance from Eq. (23), one obtains the 95% CL range 5.3 < 7710 < 7.3, with the peak value occurring at 7710 — 5.9. This range corresponds to values of Q.B between 0.019 < flBh2 < 0.027

(25)

with a central value of fish2 — 0.021. As one can see from a comparison with Table 1, these values are in excellent agreement with determinations from the CMB. 2.2.

Candidates

2.2.1. Baryons Accepting the dark matter hypothesis, the first choice for a candidate should be something we know to exist, baryons. Though baryonic dark matter can not be the whole story if Vim > 0.1, the identity of the dark matter in galactic halos, which appear to contribute at the level of $7 ~ 0.05, remains an important question needing to be resolved. A baryon density of this magnitude is not excluded by nucleosynthesis. Indeed we know some of the baryons are dark since VI < 0.01 in the disk of the galaxy. It is interesting to note that until recently, there seemed to be some difficulty in reconciling the baryon budget of the Universe. By counting the visible contribution to fl in stellar populations and the X-ray producing hot gas, Persic and Salucci60 found only f2v;s ~ 0.003. A subsequent accounting by Fukugita, Hogan and Peebles 61 found slightly more (ft ~ 0.02) by including the contribution from plasmas in groups and clusters. At high redshift on the other hand, all of the baryons can be accounted for. The observed opacity of the Ly a forest in QSO absorption spectra requires a large baryon density consistent with the determinations by the CMB and BBN. 62 In galactic halos, however, it is quite difficult to hide large amounts of baryonic matter. 63 Sites for halo baryons that have been discussed include Hydrogen (frozen, cold or hot gas), low mass stars/Jupiters, remnants of massive stars such as white dwarfs, neutron stars or black holes. In almost every case, a serious theoretical or observational problem is encountered. 2.2.1.1 Hydrogen A halo predominately made of hydrogen (with a primordial admixture of 4 He) is perhaps the simplest possibility. Hydrogen may however be present in a condensed snow-ball like state or in the form of gas. Aside from the

815

obvious question of how do these snowballs get made, it is possible to show that their existence today requires them to be so large as to be gravitationally bound. 63 Assuming that these objects are electrostatically bound, the average density of solid hydrogen is ps = 0.07 g cm~ 3 and the binding energy per molecule is about 1 eV. Given that the average velocity of a snowball is v ~ 250 k m s - 1 , corresponding to a kinetic energy Ek ~ 600 eV, snowballs must be collisionless in order to survive. Requiring that the collision rate T c = nsav be less than tu~x (tu is the age of the halo ~ age of the Universe) with ns — PH/ITT'S and a = irrs2 one finds rs > 2 cm and ms > 1 g, assuming a halo density pn = 1.7 x 1 0 - 2 6 g c m - 3 (corresponding to 10 12 M Q in a radius of 100 kpc). However, collisionless snowballs also require that their formation occur when the overall density p = pn- In this case, snowballs could not have formed later than a redshift (1 + z) = 3 . 5 or when the microwave background temperature was 9.5K. At this temperature, there is no equilibrium between the gaseous and condensed state and the snowballs would sublimate. For a snowball to survive, rs > 1016 cm is required making this no longer an electrostatically bound object. If snowballs sublimate, then we can consider the possibility of a halo composed of cold hydrogen gas. Because the collapse time-scale for the halo (< 109 yrs) is much less than the age of the galaxy, the gas must be in hydrostatic equilibrium. Combining the equation of state P{r) = (2p(r)/mp)kT

(26)

where mp is the proton mass, with dP(r)/dr = ~

G M

^

r )

(27)

one can solve for the equilibrium temperature T =

GmpM{r)

c 1.3 x 10«K

(28)

This is hot gas. As discussed earlier, hot gas is observed through X-ray emission. It is easy to show that an entire halo of hot gas would conflict severely with observations. Cooling of course may occur, but at the expense of star formation. 2.2.1.2 Jupiter-like objects A very popular candidate for baryonic dark matter is a very low mass star or JLO. These are objects with a mass m < m0 — 0.08M©, the mass

816

1000

w

*

100.0

M/MQ Figure 7.

The IMF in the solar neighborhood. 6 5

necessary to commence nuclear burning. Presumably there is a minimum mass 64 m > mmin = (0.004 — 0.007)M Q based on fragmentation, but the exact value is very uncertain. The contribution of these objects to the dark matter in the halo depends on how much mass can one put between mmin and m0 and depends on the initial mass function (IMF) in the halo. An IMF is the number of stars formed per unit volume per unit mass and can be parameterized as Am-(1+x)

(29)

In this parametrization, the Salpeter mass function corresponds to x = 1.35. Because stars are not observed with m < m0, some assumptions about the IMF for low masses must be made. An example of the observed 65 IMF in the solar neighborhood is shown in Fig. 7. It is possible to use infrared observations 66 to place a lower limit on the slope, x, of the IMF in the halo of galaxies by comparing a mass-to-light ratio defined by Q = (Pm/PL)

LQ/MQ

(30)

817

where the total mass density in JLO's and low mass stars is given by mG

Pm = /

m
(31)

Jmmin

-lV

where VTIQ = 0.75M Q is the mass of a giant. The luminosity density given by such a stellar distribution is rma

PL=

L(j>dm + PG (32) Jm0 where L(m) is the luminosity of a star of mass m. PG is the contribution to the luminosity density due to giant stars. The observed66 lower limits on Q translate to a limit 63 on x x > 1.7

(33)

with a weak dependence of mmin. Unfortunately, one cannot use Eq. (33) to exclude JLO's since we do not observe an IMF in the halo and it may be different from that in the disk. One can however make a comparison with existing observations, none of which show such a steep slope at low masses. Indeed, most observations leading to a determination of the IMF (such as the one shown in Fig. 7) show a turn over (or negative slope). To fully answer the questions regarding JLO's in the halo, one needs a better understanding of star formation and the IMF. For now, postulating the existence of a large fraction of JLO's in halo is rather ad hoc. Despite the theoretical arguments against them, JLO's or massive compact halo objects (MACHOs) are candidates which are testable by the gravitational microlensing of stars in a neighboring galaxy such as the LMC. 67 By observing millions of stars and examining their intensity as a function of time, it is possible to determine the presence of dark objects in our halo. It is expected that during a lensing event, a star in the LMC will have its intensity rise in an achromatic fashion over a period 6t ~ 3 ^ M / . O O I M Q days. Indeed, microlensing candidates have been found.68 For low mass objects, those with M < O.1M 0 , it appears however that the halo fraction of MACHOs is very small. The relative amount of machos in the halo is typically expressed in terms of an optical depth. A halo consisting 100% of machos would have an optical depth of T ~ 5 x 1 0 - 7 . The most recent results of the MACHO collaboration 69 indicate that r = 12 j ^ x 10" 8 , corresponding to a macho halo fraction of about 20% with a 95% CL range of 8 - 50% based on 13 - 17 events. They also exclude a 100% macho halo at the 95% CL. The typical macho mass falls in the range 0.15 - 0.9

818

M©. The EROS collaboration has set even stronger limits having observed 5 events toward the LMC and 4 toward the SMC. 70 The observed optical depth from EROS1 is r = 4 ^ ° x 10" 8 and from EROS2 r = 6 + | x 1CT8. They have excluded low mass objects (M < 0.1M©) to make up less than 10% of the halo and objects with 2 x l C r 7 M 0 < M < 1M Q to be less than 25% of the halo at the 95% CL. 2.2.1.3 Remnants of massive stars Next one should possibility that the halo is made up of the dead stellar remnants of stars whose initial masses were M > 1M Q . Briefly, the problem which arises in this context is that since at least 40% of the stars initial mass is ejected, and most of this mass is in the form of heavy elements, a large population of these objects would contaminate the disk and prevent the existence of extremely low metallicity objects (Z ~ 10~5) which have been observed. Thus either dust (from ejecta) or dead remnants would be expected to produce too large a metallicity. 63 ' 71 Clearly star formation is a very inefficient mechanism for producing dark matter. Many generations of stars would would be required to cycle through their lifetimes to continually trap more matter in remnants. This question has been studied in more detail. 72 By allowing a variable star formation rate, and allowing a great deal of flexibility in the IMF, a search for a consistent set of parameters so that the halo could be described primarily in terms of dead remnants (in this case white dwarfs) was performed. While a consistent set of chemical evolution parameters can be found, there is no sensible theory to support this choice. In such a model however, the dark matter is in the form of white dwarfs and the remaining gas is is heavily contaminated by Carbon and Nitrogen. 73 Though it is not excluded, it is hard to understand this corner of parameter space as being realistic. 2.2.1.4 Black holes There are several possibilities for black holes as the dark matter in halos. If the black holes are primordial 74 which have presumably formed before nucleosynthesis, they should not be counted as baryonic dark matter and therefore do not enter into the present discussion. If the black holes are formed as the final stage of star's history and its formation was preceded by mass loss or a supernovae, then the previous discussion on remnants of massive stars applies here as well. However, it is also possible that the black halos were formed directly from very massive stars (m > 100M©?)

819

through gravitational instability with no mass loss. 75 Though there are limits due to overheating the disk 76 and stellar systems. 77 In this case I know of no argument preventing a sufficiently large population of massive black holes as baryonic dark matter (Of course, now the IMF must have mmin > 100M Q .) 2.2.2. Neutrinos Light neutrinos (m < 30eV) are a long-time standard when it comes to nonbaryonic dark matter. 78 Light neutrinos produce structure on large scales, and the natural (minimal) scale for structure clustering is given in Eq. (15). Hence neutrinos offer the natural possibility for large scale structures 79,80 including filaments and voids. Light neutrinos are, however, ruled out as a dominant form of dark matter because they produce too much large scale structure. 81 Because the smallest non-linear structures have mass scale Mj and the typical galactic mass scale is ~ 10 12 M Q , galaxies must fragment out of the larger pancake-like objects. The problem with such a scenario is that galaxies form late 80 ' 82 (z < 1) whereas quasars and galaxies are seen out to redshifts z > 6. In the standard model, the absence of a right-handed neutrino state precludes the existence of a neutrino mass (unless one includes nonrenormalizable lepton number violating interactions such HHLL). By adding a right-handed state I/R, it is possible to generate a Dirac mass for the neutrino, rav = huv/\/2, as is the case for the charged lepton masses, where hv is the neutrino Yukawa coupling constant, and v is the Higgs expectation value. It is also possible to generate a Majorana mass for the neutrino when in addition to the Dirac mass term, m^i/Ri'L, a term MVRVR is included. If M S> mu, the see-saw mechanism produces two mass eigenstates given by mVl ~ m^/M which is very light, and mV2 ~ M which is heavy. The state v\ is a potential hot dark matter candidate as V2 is in general not stable. The simplicity of the standard big bang model allows one to compute in a straightforward manner the relic density of any stable particle if that particle was once in thermal equilibrium with the thermal radiation bath. At early times, neutrinos were kept in thermal equilibrium by their weak interactions with electrons and positrons. As we saw in the case of the /3-interaction used in BBN, one can estimate the thermally averaged lowenergy weak interaction scattering cross section (av) ~ g^/m^

(34)

820 for T
F

(35)

/

Neutrinos will be in equilibrium when T w k > H or T 3 > A/8TT 3 A^/90 mtv/Mp

(36)

where MP = G~N1/2 = 1.22 x 10 1 9 GeV is t h e Planck mass. For N = 4 3 / 4 (accounting for photons, electrons, positrons and three neutrino flavors) we see t h a t equilibrium is maintained at temperatures greater t h a n 0 ( 1 ) MeV (for a more accurate calculation see Ref. 83). T h e decoupling scale of 0 ( 1 ) MeV has an important consequence on t h e final relic density of massive neutrinos. Neutrinos more massive t h a n 1 MeV will begin t o annihilate prior t o decoupling, a n d while in equilibrium, their number density will become exponentially suppressed. Lighter neutrinos decouple as radiation on t h e other hand, a n d hence do not experience t h e suppression due t o annihilation. Therefore, t h e calculations of t h e number density of light (mv < 1 MeV) and heavy (m„ > 1 MeV) neutrinos differ substantially. The number of density of light neutrinos with mv expressed a t late times as pv = m^Yjyriy

< 1 MeV can be

(37)

where Yv = nvjn1 is t h e density of v's relative t o t h e density of photons, which today is 411 photons per c m 3 . It is easy t o show t h a t in an adiabatically expanding universe Yv = 3 / 1 1 . This suppression is a result of the e + e ~ annihilation which occurs after neutrino decoupling a n d heats t h e photon b a t h relative t o t h e neutrinos. In order t o obtain an age of t h e Universe, t > 12 Gyr, one requires t h a t t h e m a t t e r component is constrained by Q,h2 < 0.3.

(38)

From this one finds t h e strong constraint (upper bound) on Majorana neutrino masses 8 4 : mtot

] T mv < 28eV.

(39)

821

4 2 >

0

g, "bfl

°

-2 -4

-6 KeV

MeV

GeV

TeV lily

Figure 8. Summary plot 8 5 of the relic density of Dirac neutrinos (solid) including a possible neutrino asymmetry of r\v = 5 X 10 (dotted).

where the sum runs over neutrino mass eigenstates. The limit for Dirac neutrinos depends on the interactions of the right-handed states (see discussion below). Given the discussion of the CMB results in the previous section, one could make a case that the limit on flh2 should be reduced by a factor of 2, which would translate in to a limit of 14 eV on the sum of the light neutrino masses. As one can see, even very small neutrino masses of order 1 eV, may contribute substantially to the overall relic density. The limit (39) and the corresponding initial rise in Q,vh? as a function of mv is displayed in the Fig. 8 (the low mass end with m„ < 1 MeV). Combining the rapidly improving data on key cosmological parameters with the better statistics from large redshift surveys has made it possible to go a step forward along this path. It is now possible to set stringent limits on the light neutrino mass density fi„/i2, and hence on the neutrino mass based on the power spectrum of the Ly a forest,86 mtot < 5.5 eV, and the limit is even stronger if the total matter density, Q.m is less than 0.5. Adding additional observation constraints from the CMB and galaxy clusters drops this limit 87 to 2.4 eV. This limit has recently been improved by the 2dF Galaxy redshift88 survey by comparing the derived power spectrum of fluctuations with structure formation models. Focussing on the the presently favoured ACDM model, the neutrino mass bound becomes TOtot < 1-8 eV for O m < 0.5. When even more constraints such as HST

822

Key project data, supernovae type la data, and BBN are included 89 the limit can be pushed to m,t0t < 0.3 eV. The calculation of the relic density for neutrinos more massive than ~ 1 MeV, is substantially more involved. The relic density is now determined by the freeze-out of neutrino annihilations which occur at T < m„, after annihilations have begun to seriously reduce their number density.90 The annihilation rate is given by Tann = {av)annnu

~ ^(rnvTfl2e-m»fT

(40)

where we have assumed, for example, that the annihilation cross section is dominated by vv —> / / via Z-boson exchange51 and (av)ann ~ m1/mAz. When the annihilation rate becomes slower than the expansion rate of the Universe the annihilations freeze out and the relative abundance of neutrinos becomes fixed. For particles which annihilate through approximate weak scale interactions, this occurs when T ~ m x / 2 0 . The number density of neutrinos is tracked by a Boltzmann-like equation, ^

= -

3

|n-M(n

2

-ng)

(41)

where UQ is the equilibrium number density of neutralinos. By defining the quantity / = n / T 3 , we can rewrite this equation in terms of x, as df

___ {8n: „

A M

2

The solution to this equation at late times (small x) yields a constant value of / , so that n oc T 3 . Roughly, the solution to the Boltzmann equation goes as Yv ~ / ~ (m(av)ann)~l and hence i}„h2 ~ {uv)ann , so that parametrically Q 1/m2,. As a result, the constraint (38) now leads to a lower bound 90 " 92 on the neutrino mass, of about mv > 3 — 7 GeV, depending on whether it is a Dirac or Majorana neutrino. This bound and the corresponding downward trend £lvlr? ~ 1/w2, can again be seen in Fig. 8. The result of a more detailed calculation is shown in Fig. 9 (Ref. 92) for the case of a Dirac neutrino. The two curves show the slight sensitivity on the temperature scale associated with the quark-hadron transition. The result a

While this is approximately true for Dirac neutrinos, the annihilation cross section of Majorana neutrinos is p-wave suppressed and is proportional of the final state fermion masses rather than m „ .

823

1

0

OS

o

-J

-2

-3 0

5

10

15

20

25

30

Itl v (GeV)

Figure 9. The relic density of heavy Dirac neutrinos due to annihilations. 9 2 The curves are labeled by the assumed quark-hadron phase transition temperature in MeV.

for a Majorana mass neutrino is qualitatively similar. Indeed, any particle with roughly weak scale cross-sections will tend to give an interesting value off)/i2~l. The deep drop in f2„/i2, visible in Fig. 8 at around mv = Mz/2, is due to a very strong annihilation cross section at Z-boson pole. For yet higher neutrino masses the Z-annihilation channel cross section drops as ~ l/?n 2 , leading to a brief period of an increasing trend in fl^h2. However, for mv > raw the cross section regains its parametric form (crv)ann ~ m 2 due to the opening up of a new annihilation channel to W-boson pairs, 93 and the density drops again as Vtuh2 ~ 1/m 2 . The tree level W-channel cross section breaks the unitarity at around 94 O(few) TeV however, and the full cross section must be bound by the unitarity limit. 95 This behaves again as 1/m.2, whereby Q,„h2 has to start increasing again, until it becomes too large again at 9 5 ' 9 4 200-400 TeV (or perhaps somewhat earlier as the weak interactions become strong at the unitarity breaking scale). If neutrinos are Dirac particles, and have a nonzero asymmetry the relic density could be governed by the asymmetry rather than by the annihilation cross section. Indeed, it is easy to see that the neutrino mass density corresponding to the asymmetry r\v = (n„ — n p ) / n 7 is given by 96 p = mvr\vn.

(43)

824

which implies n„h2 ~ 0.004 T)V10 (m„/GeV).

(44)

10

where 77^10 = lO /^. The behaviour of the energy density of neutrinos with an asymmetry is shown by the dotted line in the Fig. 8. At low m„, the mass density is dominated by the symmetric, relic abundance of both neutrinos and antineutrinos which have already frozen out. At higher values of mv, the annihilations suppress the symmetric part of the relic density until f2„/i2 eventually becomes dominated by the linearly increasing asymmetric contribution. In the figure, we have assumed an asymmetry of r\v ~ 5 x 1 0 - 1 1 for-neutrinos with standard weak interaction strength. In this case, fi„/i2 begins to rise when m„ > 20 GeV. Obviously, the bound (38) is saturated for m„ = 75GeV/^ioBased on the leptonic and invisible width of the Z boson, experiments at LEP have determined that the number of neutrinos is N„ = 2.9841 ±0.0083. 9 7 Conversely, any new physics must fit within these brackets, and thus LEP excludes additional neutrinos (with standard weak interactions) with masses mv < 45 GeV. Combined with the limits displayed in Figs. 8 and 9, we see that the mass density of ordinary heavy neutrinos is bound to be very small, f2„/i2 < 0.001 for masses mv > 45 GeV up to m„ ~ 0(100) TeV. Lab constraints for Dirac neutrinos are available, 98 excluding neutrinos with masses between 10 GeV and 4.7 TeV. This is significant, since it precludes the possibility of neutrino dark matter based on an asymmetry between v and P.96 Majorana neutrinos are excluded as dark matter since flvh0 < 0.001 for mv > 45 GeV and are thus cosmologically uninteresting. A bound on neutrino masses even stronger than Eq. (39) can be obtained from the recent observations of active-active mixing in both solar- and atmospheric neutrino experiments. The inferred evidence for v^ — vT and v& — Vfj,,r mixings are on the scales m 2 ~ 1 —lOxlO - 5 a n d m 2 ~ 2 —5xl0~ 3 . When combined with the upper bound on the electron-like neutrino mass vnv < 2.8 e V , " and the LEP-limit on the number of neutrino species, one finds the constraint on the sum of neutrino masses: 0.05 eV £ rntot £ 8.4 eV.

(45)

Conversely, the experimental and observational data then implies that the cosmological energy density of all light, weakly interacting neutrinos can be restricted to the range 0.0005 < 9.vh2 < 0.09.

(46)

825

Interestingly there is now also a lower bound due to the fact that at least one of the neutrino masses has to be larger than the scale m 2 ~ 10~ 3 eV2 set by the atmospheric neutrino data. Combined with the results on relic mass density of neutrinos and the LEP limits, the bound (46) implies that the ordinary weakly interacting neutrinos, once the standard dark matter candidate, 78 can be ruled out completely as a dominant component of the dark matter. If instead, we consider right-handed neutrinos, we have new possibilities. Right-handed interactions are necessarily weaker than standard left-handed interactions implying that right-handed neutrinos decouple early and today are at a reduced temperature relative to VL (Ref. 100)

Tx\3 Tj

_ 43 ~ 4N(Td)

As such, for TdR > 1 MeV, nVR/nVL

= {TUR/TVL)'i

(

}

< 1. Thus the abun-

dance of right-handed neutrinos can be written as

In this case, the previous bound (39) on neutrino masses is weakened. For a suitably large scale for the right-handed interactions, right-handed neutrino masses may be as large as a few keV. 101 Such neutrinos make excellent warm dark matter candidates, albeit the viable mass range for galaxy formation is quite restricted. 102 2.2.3. Axions Due to space limitations, the discussion of this candidate will be very brief. Axions are pseudo-Goldstone bosons which arise in solving the strong CP problem 103 ' 104 via a global U(l) Peccei-Quinn symmetry. The invisible axion 104 is associated with the flat direction of the spontaneously broken PQ symmetry. Because the PQ symmetry is also explicitly broken (the CP violating OFF coupling is not PQ invariant) the axion picks up a small mass similar to pion picking up a mass when chiral symmetry is broken. We can expect that ma ~ m„fn/fa where /„, the axion decay constant, is the vacuum expectation value of the PQ current and can be taken to be quite large. If we write the axion field as a = fa9, near the minimum, the potential produced by QCD instanton effects looks like V ~ m 2 # 2 / 2 . The axion equations of motion lead to a relatively stable oscillating solution.

826

The energy density stored in the oscillations exceeds the critical density 105 unless fa < 1012 GeV. Axions may also be emitted stars and supernova. 106 In supernovae, axions are produced via nucleon-nucleon bremsstrahlung with a coupling gAN oc rriN/fa- As was noted above the cosmological density limit requires fa Si 1012 GeV. Axion emission from red giants imply 107 fa > 1010 GeV (though this limit depends on an adjustable axion-electron coupling), the supernova limit requires 108 / „ ^ 2 x 10 11 GeV for naive quark model couplings of the axion to nucleons. Thus only a narrow window exists for the axion as a viable dark matter candidate. 3. Lecture 3: Supersymmetric Dark Matter Although there are many reasons for considering supersymmetry as a candidate extension to the standard model of strong, weak and electromagnetic interactions, 109 one of the most compelling is its role in understanding the hierarchy problem 110 namely, why/how is mw
(49)

where A is a cut-off representing the appearance of new physics, and the inequality in (49) applies if A ~ 103 TeV, and even more so if A ~ TTIGUT ~ 10 16 GeV or ~ Mp ~ 10 19 GeV. If the radiative corrections to a physical quantity are much larger than its measured values, obtaining the latter requires strong cancellations, which in general require fine tuning of the bare input parameters. However, the necessary cancellations are natural in supersymmetry, where one has equal numbers of bosons and fermions with equal couplings, so that (49) is replaced by 8mw=o(^)

\mB-m2F\.

(50)

The residual radiative correction is naturally small if \m2B —mF\ < 1 TeV 2 . In order to justify the absence of interactions which can be responsible for extremely rapid proton decay, it is common in the minimal supersymmetric standard model (MSSM) to assume the conservation of R-parity. If R-parity, which distinguishes between "normal" matter and the supersymmetric partners and can be defined in terms of baryon, lepton and spin as

827

R = (—i) 3B + L + 2S ) i s unbroken, there is at least one supersymmetric particle (the lightest supersymmetric particle or LSP) which must be stable. Thus, the minimal model contains the fewest number of new particles and interactions necessary to make a consistent theory. There are very strong constraints, however, forbidding the existence of stable or long lived particles which are not color and electrically neutral. 111 Strong and electromagnetically interacting LSPs would become bound with normal matter forming anomalously heavy isotopes. Indeed, there are very strong upper limits on the abundances, relative to hydrogen, of nuclear isotopes, 112 n/nH <> 1(T 15 to 1(T 29 for 1 GeV <m%l TeV. A strongly interacting stable relic is expected to have an abundance n/nn Ss 10~ 10 with a higher abundance for charged particles. There are relatively few supersymmetric candidates which are not colored and are electrically neutral. The sneutrino 113 is one possibility, but in the MSSM, it has been excluded as a dark matter candidate by direct 98 and indirect 114 searches. In fact, one can set an accelerator based limit on the sneutrino mass from neutrino counting, m^ > 44.7 GeV. 115 In this case, the direct relic searches in underground low-background experiments require mc, > 20 TeV. 98 Another possibility is the gravitino which is probably the most difficult to exclude. I will concentrate on the remaining possibility in the MSSM, namely the neutralinos.

3.1.

Parameters

The most general version of the MSSM, despite its minimality in particles and interactions contains well over a hundred new parameters. The study of such a model would be untenable were it not for some (well motivated) assumptions. These have to do with the parameters associated with supersymmetry breaking. It is often assumed that, at some unification scale, all of the gaugino masses receive a common mass, m^2- The gaugino masses at the weak scale are determined by running a set of renormalization group equations. Similarly, one often assumes that all scalars receive a common mass, mo, at the GUT scale. These too are run down to the weak scale. The remaining parameters of importance involve the Higgs sector. There is the Higgs mixing mass parameter, /x, and since there are two Higgs doublets in the MSSM, there are two vacuum expectation values. One combination of these is related to the Z mass, and therefore is not a free parameter, while the other combination, the ratio of the two vevs, tan/3, is free.

828

.

.

,

.

.

.

.

,

.

.

.

.

,

.

600 - s" "

>


'

%

.

•

400

to

S M

200

C

a

'

§

m0

i i i i i

( -200

Figure 10.

i

i

i

i

1

5

i

i

i

i

1

10 Log10(Q/GeV)

i

i

i

i

1

i

15

RG evolution of the mass parameters in the CMSSM.

If the supersymmetry breaking Higgs soft masses are also unified at the GUT scale (and take the common value mo), then /i and the physical Higgs masses at the weak scale are determined by electroweak vacuum conditions (fi is determined up to a sign). This scenario is often referred to as the constrained MSSM or CMSSM. Once these parameters are set, the entire spectrum of sparticle masses at the weak scale can be calculated13. In Fig. 10, an example of the running of the mass parameters in the CMSSM is shown. Here, we have chosen rrii/2 = 250 GeV, mo = 100 GeV, tan/3 = 3, AQ = 0, and fi < 0. Indeed, it is rather amazing that from so few input parameters, all of the masses of the supersymmetric particles can be determined. The characteristic features that one sees in the figure, are for example, that the colored sparticles are typically the heaviest in the spectrum. This is due to the large positive correction to the masses due

b There are in fact, additional parameters: the supersymmetry-breaking tri-linear masses A (also assumed to be unified at the GUT scale) as well as two CP violating phases 6^ and 6A-

829

to az in the RGE's. Also, one finds that the B (the partner of the U(l)y gauge boson), is typically the lightest sparticle. But most importantly, notice that one of the Higgs mass, 2 goes negative triggering electroweak symmetry breaking. 116 (The negative sign in the figure refers to the sign of the mass, 2 even though it is the mass of the sparticles which are depicted.) 3.2.

Neutralinos

There are four neutralinos, each of which is a linear combination of the R = — 1 neutral fermions:111 the wino W3, the partner of the 3rd component of the SU(2)L gauge boson; the bino, B; and the two neutral Higgsinos, Hi and H2. Assuming gaugino mass universality at the GUT scale, the identity and mass of the LSP are determined by the gaugino mass rrii/2, M, and tan j3. In general, neutralinos can be expressed as a linear combination X = aB + (3W3 + 1H1 + 8H2

(51)

The solution for the coefficients a, /3,7 and d for neutralinos that make up the LSP can be found by diagonalizing the mass matrix ( (W3,B,H°UH°)

M2

0

0

Ml

—g2"i

givi

V2

V2

V ^f ^ f

- ^

^

VT ^7T 0

-M

\

/w3\ B

-11

(52)

0 J \H%)

where Mi(M 2 ) is a soft supersymmetry breaking term giving mass to the U(l) (SU(2)) gaugino(s). In a unified theory Mi — M2 = TO1/2 at the unification scale (at the weak scale, Mi ~ ^^M2)As one can see, the coefficients a,/3,7, and S depend only on mi/2, A*-, and tan/3. In Fig. II, 1 1 7 regions in the M2,fi plane with tan/3 = 2 are shown in which the LSP is one of several nearly pure states, the photino, 7, the bino, B, a symmetric combination of the Higgsinos, ^(12)1 o r the Higgsino, S = sin (iHi + cos /3H2. The dashed lines show the LSP mass contours. The cross hatched regions correspond to parameters giving a chargino ( M ^ , H±) state with mass m^ < 45GeV and as such are excluded by LEP. 118 This constraint has been extended by LEP 1 1 9 and is shown by the light shaded region and corresponds to regions where the chargino mass is < 103.5 GeV. The newer limit does not extend deep into the Higgsino region because of the degeneracy between the chargino and neutralino. Notice that the parameter space is dominated by the B or Hi2 pure states and that the photino only occupies a small fraction of the parameter space, as does

830

M2 (GeV) Figure 11.

Mass contoursand composition of nearly pure LSP states in the MSSM. 1 1 7

the Higgsino combination S. Both of these light states are experimentally excluded.

3.3. The Relic

Density

The relic abundance of LSP's is determined by solving the Boltzmann equation for the LSP number density in an expanding Universe. The technique 92 used is similar to that for computing the relic abundance of massive neutrinos. 90 The relic density depends on additional parameters in the MSSM beyond m ^ i M i a n d tan/?. These include the sfermion masses, mj and the Higgs pseudo-scalar mass, TTIA, derived from mo (and m ^ ) . To determine the relic density it is necessary to obtain the general annihilation cross-section for neutralinos. In much of the parameter space of interest, the LSP is a bino and the annihilation proceeds mainly through sfermion exchange. Because of the p-wave suppression associated with Majorana fermions, the s-wave part of the annihilation cross-section is suppressed by

831

the outgoing fermion masses. This means that it is necessary to expand the cross-section to include p-wave corrections which can be expressed as a term proportional to the temperature if neutralinos are in equilibrium. Unless the neutralino mass happens to lie near near a pole, such as mx ~ mz/2 or m/,/2, in which case there are large contributions to the annihilation through direct s-channel resonance exchange, the dominant contribution to the BB annihilation cross section comes from crossed i-channel sfermion exchange. Annihilations in the early Universe continue until the annihilation rate T ~ avnx drops below the expansion rate. The calculation of the neutralino relic density proceeds in much the same way as discussed above for neutrinos with the appropriate substitution of the cross section. The final neutralino relic density expressed as a fraction of the critical energy density can be written as 1 1 1

where (Tx/T^)3 accounts for the subsequent reheating of the photon temperature with respect to x, due to the annihilations of particles with mass m < Xfmx.100 The subscript / refers to values at freeze-out, i.e., when annihilations cease. The coefficients a and b are related to the partial wave expansion of the cross-section, crv = a + bx +.... Equation (53) results in a very good approximation to the relic density expect near s-channel annihilation poles, thresholds and in regions where the LSP is nearly degenerate with the next lightest supersymmetric particle. 120 When there are several particle species i, which are nearly degenerate in mass, co-annihilations are important. In this case, 120 the rate equation (41) still applies, provided n is interpreted as the total number density,

n = '^2ni ,

(54)

i

no as the total equilibrium number density, n0 = ^ n o , i ,

(55)

i

and the effective annihilation cross section as ^ - ^ n o iUo j , . {CTeftVrel) = > . 2 (°X^rel) • 13

, , (56)

u

In Eq. (42), mx is now understood to be the mass of the lightest sparticle under consideration.

832

Note that this implies that the ratio of relic densities computed with and without coannihilations is, roughly,

where a = a + bx/2 and sub- and superscripts 0 denote quantities computed ignoring coannihilations. The ratio x°Jxf ~ 1 + X0, ln(geff<7efj/giao), where geff = J2igi(mi/mi)3/2e~(mi~mi^T. For the case of three degenerate slepton NLSPs, 121 ges = J2i9* = 8 a n d x°f/xf ~ 1-2- The effects of co-annihilations are discussed below. 3.4. Phenomenological

and Cosmological

Constraints

For the cosmological limits on the relic density I will assume 0.1 < flxh2

< 0.3.

(58)

The upper limit being a conservative bound based only on the lower limit to the age of the Universe of 12 Gyr. Indeed, most analyses indicate that ^matter 5s 0.4 — 0.5 and thus it is very likely that ftxh2 < 0.2 (cf. the CMB results in Table 1). One should note that smaller values of $lxh2 are allowed, since it is quite possible that some of the cold dark matter might not consist of LSPs. The calculated relic density is found to have a relevant cosmological density over a wide range of susy parameters. For all values of tan/3, there is a 'bulk' region with relatively low values of m.1/2 and mo where 0.1 < Q,xh2 < 0.3. However there are a number of regions at large values of rrii/2 and/or mo where the relic density is still compatible with the cosmological constraints. At large values of ra^, the lighter stau, becomes nearly degenerate with the neutralino and co-annihilations between these particles must be taken into account. 121 ' 122 For non-zero values of Ao, there are new regions for which x ~ t coannihilations are important. 123 At large tan/3, as one increases mi/2, the pseudo-scalar mass, ITIA begins to drop so that there is a wide funnel-like region (at all values of mo) such that 2m x « m,A and s-channel annihilations become important. 124 ' 125 Finally, there is a region at very high mo where the value of fi begins to fall and the LSP becomes more Higgsino-like. This is known as the 'focus point' region. • ° As an aid to the assessment of the prospects for detecting sparticles at different accelerators, benchmark sets of supersymmetric parameters have

833

pV,f4.FHW.|.v.„.tTl.,^

m

l/2

Figure 12. Schematic overview of the CMSSM benchmark points proposed in. 1 2 7 The points are intended to illustrate the range of available possibilities. The labels correspond to the approximate positions of the benchmark points in the (rrci/2> m o) plane. They also span values of tan (3 from 5 to 50 and include points with /x < 0.

often been found useful, since they provide a focus for concentrated discussion. A set of proposed post-LEP benchmark scenarios 127 in the CMSSM are illustrated schematically in Fig. 12. Five of the chosen points are in the c bulk? region at small mi/2 and mo, four are spread along the coannihilation 'tail' at larger TO1/2 for various values of tan/3. This tail runs along the shaded region in the lower right corner where the stau is the LSP and is therefore excluded by the constraints against charged dark matter. Two points are in rapid-annihilation 'funnels5 at large m i / 2 and mo. At large values of mo, the focus-point region runs along the boundary where electroweak symmetry no longer occurs (shown in Fig. 12 as the shaded region in the upper left corner). Two points were chosen in the focus-point region at large mo- The proposed points range over the allowed values of tan/3

834

200

300

400 m

600

i/2

Figure 13. The light-shaded 'bulk' area is the cosmologieally preferred region with 0.1 < Qh2 < 0.3. The light dashed lines show the location of the cosmologieally preferred region if one ignores coannihilations with the light sleptons. In the dark shaded region in the bottom right, the LSP is the r i , leading to an unacceptable abundance of charged dark matter. Also shown is the isomass contour mx± = 104 GeV and m,h = 110,114 GeV, as well as an indication of the slepton bound from LEP.

between 5 and 50. The light shaded region corresponds to the portion of parameter space where the relic density Qxh2 is between 0.1 and 0.3. The effect of coannihilations is to create an allowed band about 2550 GeV wide in mo forTO1/2^ 1400 GeV, which tracks above the m? = mx contour. Along the line m? = m x , R « 10, from ( 5 7 ) . m As mo increases, the mass difference increases and the slepton contribution to aeff falls, and the relic density rises abruptly. This effect is seen in Fig. 13. The light shaded region corresponds to 0.1 < fth2 < 0.3. The dark shaded region has rrir < rnx and is excluded. The light dashed contours indicate the corresponding region in Qh2 if one ignores the effect of coannihilations. Neglecting coannihilations, one would find an upper bound of ^ 450 GeV on rai/2> corresponding to an upper bound of roughly 200 GeV on rag. Instead, values of rn 1/2 U P to ~ 1400 GeV are allowed corresponding to an upper bound of ~ 600 GeV on m^.

835

The most important phenomenological constraints are also shown schematically in Fig. 12. These include the constraint provided by the LEP lower limit on the Higgs mass: mn > 114.4 GeV. 128 This holds in the Standard Model, for the lightest Higgs boson h in the general MSSM for tan/3 < 8, and almost always in the CMSSM for all tan/3. Since rrih is sensitive to sparticle masses, particularly m^, via loop corrections, the Higgs limit also imposes important constraints on the CMSSM parameters, principally m ^ as seen by the dashed curve in Fig. 12. The constraint imposed by measurements of b —> s^y (Ref. 129) also exclude small values of m ^ - These measurements agree with the Standard Model, and therefore provide bounds on MSSM particles, such as the chargino and charged Higgs masses, in particular. Typically, the b —» sj constraint is more important for fi < 0, but it is also relevant for \x > 0, particularly when tan/3 is large. The BNL E821 experiment reported last year a new measurement of aM = \{g^ — 2) which deviated by 2.6 standard deviations from the best Standard Model prediction available at that time. 130 However, it had been realized that the sign of the most important pseudoscalar-meson pole part of the light-bylight scattering contribution 131 to the Standard Model prediction should be reversed, which reduces the apparent experimental discrepancy to about 1.6 standard deviations (Sa^, x 1010 = 26 ± 16). The largest contribution to the errors in the comparison with theory was thought to be the statistical error of the experiment, which has been significantly reduced just recently.132 The world average of aM = §(<7M — 2) now deviates by (33.9 ± 11.2) x 1(T 10 from the Standard Model calculation of Davier et al. 133 using e + e~ data, and by (17 ± 11) x 10~ 10 from the Standard Model calculation of Davier et al. 133 based on r decay data. Other recent analyses of the e + e _ data yield similar results. On the subsequent plots, the formal 2-cr range 11.5 x 10" 1 0 < 8a^ < 56.3 x K T 1 0 is displayed. Following a previous analysis, 125 ' 134 in Fig. 14 the m\/2 — rno parameter space is shown for tan/3 = 10. The dark shaded region (in the lower right) corresponds to the parameters where the LSP is not a neutralino but rather a TR . The cosmologically interesting region at the left of the figure is due to the appearance of pole effects. There, the LSP can annihilate through s-channel Z and h (the light Higgs) exchange, thereby allowing a very large value of mo. However, this region is excluded by phenomenological constraints. Here one can see clearly the coannihilation tail which extends towards large values of m 1( / 2 . In addition to the phenomenological constraints discussed above, Fig. 14 also shows the current experimental

836

tan p = 10 , \i>0

800

100

200

300

400

500

600

700

800

900 1000

m1/2 (GeV) Figure 14. Compilation of phenomenological constraints on the CMSSM for tan 0 = 10,/x > 0, assuming Ao = 0,mt = 175 GeV and mb(mb)^ = 4.25 GeV. The nearvertical lines are the LEP limits m ± = 103.5 GeV (dashed and black), 1 1 9 and m^ = 114.1 GeV (dotted and red). 1 2 8 Also, in the lower left corner we show the mg = 99 GeV contour. 1 3 5 In the dark (brick red) shaded regions, the LSP is the charged f i , so this region is excluded. The light (turquoise) shaded areas are the cosmologically preferred regions with 0.1 < Oft,2 < 0.3. 1 2 5 The medium (dark green) shaded regions are excluded by b —* s'y. 129 The shaded (pink) region in the upper right delineates the 2 a range of g^ — 2. The dashed curves within this region correspond to the 1 — a bounds.*

constraints on the CMSSM parameter space due to the limit mx± ^ 103.5 GeV provided by chargino searches at LEP. 119 LEP has also provided lower limits on slepton masses, of which the strongest is frig ?> 99 GeV. 135 This is shown by dot-dashed curve in the lower left corner of Fig. 14. Similar results have been found by other analyses. 136 As one can see, one of the most important phenomenological constraint at this value of tan/? is due to the Higgs mass (shown by the nearly

837

100

1000

2000

3000

m1/2 (GeV) Figure 15.

As in Fig. 14 for tan/3 = 50.

vertical dot-dashed curve). The theoretical Higgs masses were evaluated using FeynHiggs, 137 which is estimated to have a residual uncertainty of a couple of GeV in nth- The region excluded by the b —> sj constraint is the dark shaded (green) region to the left of the plot. As many authors have pointed out ? 138 a discrepancy between theory and the BNL experiment could well be explained by supersymmetry. As seen in Fig. 14, this is par-* ticularly easy if /i > 0. The medium (pink) shaded region in the figure corresponds to the overall allowed region (2a) by the new experimental result. As discussed above, another mechanism for extending the allowed CMSSM region to large mx is rapid annihilation via a direct-channel pole when mx ~ ™mA-124'125 This may yield a 'funnel' extending to large rai/2 and mo at large tan/3, as seen in Fig. 15.

838

tan p = 10, m 1 / 2 = 300, m 0 = 100

\i (GeV) Figure 16. Compilations of phenomenological constraints on the MSSM with NUHM in the (/Z,771A) plane for t a n ^ = 10 and mo = 100 GeV, m1f2 = 300 GeV, assuming A0 = 05 mt = 175 GeV and mh(mb)g$ = 4.25 GeV. The shading is as described in Fig. 14. The (blue) solid line is the contour mx = m ^ / 2 , near which rapid direct-channel annihilation suppresses the relic density. The dark (black) dot-dashed line indicates when one or another Higgs mass-squared becomes negative at the GUT scale: only lower |/x| and larger VTIA values are allowed. The crosses denote the values of JJL and TTIA found in the CMSSM.

In principle,, the true input parameters in the CMSSM are: |i J mi ) m2 3 and B, where mi and m 2 are the Higgs soft masses (in the CMSSM ^ 1 = ^ 2 = ^ o and B is the susy breaking bilinear mass term). In this "case, the electroweak symmetry breaking conditions lead to a prediction of Mz,tan/3 ,and ra^. Since we are not really interested in predicting MZj it is more useful to assume instead the following CMSSM input parameters: Mzimi}m2J and tan/3 again with mi = m 2 = m 0 . In this case, one predicts fj,,B, and m^. However, one can generalize the CMSSM case to include non-universal Higgs masses (NUHM), 139 ' 140 in which case the input parameters become: MZ} ^ 5 m^, and tan(3 and one predicts m i , m 2 , and B.

839

The NUHM parameter space was recently analyzed 40 and a sample of the results found is shown in Fig. 16. While much of the cosmologically preferred area with fx < 0 is excluded, there is a significant enhancement in the allowed parameter space for fi > 0. 3.5.

Detection

Because the LSP as dark matter is present locally, there are many avenues for pursuing dark matter detection. Direct detection techniques rely on an ample neutralino-nucleon scattering cross-section. The effective fourfermion lagrangian can be written as

+

"5JXX*75*

+ aeiX7 5 X9i*

( 59 )

However, the terms involving aii,a4i,a^i, and aei lead to velocity dependent elastic cross sections. The remaining terms are: the spin dependent coefficient, a-2i and the scalar coefficient otzi- Contributions to a-n are predominantly through light squark exchange. This is the dominant channel for binos. Scattering also occurs through Z exchange but this channel requires a strong Higgsino component. Contributions to a%i are also dominated by light squark exchange but Higgs exchange is non-negligible in most cases. Figure 17 displays contours of the spin-independent cross section for the elastic scattering of the LSP x o n protons in the m i ^ ^ o planes for (a) tan/3 = 10,/i, < 0, (b) tan/3 = 10,/z > 0. 141 The double dot-dashed (orange) lines are contours of the spin-independent cross section, and the contours a si = 10~ 9 pb in panel (a) and a si = 10~ 12 pb in panel (b) are indicated. The LEP lower limits on rxih and mx±, as well as the experimental measurement of b —> sj for \x < 0, tend to bound the cross sections from above, as discussed in more detail below. Generally speaking, the spin-independent cross section is relatively large in the 'bulk' region, but falls off in the coannihilation 'tail'. Also, we note also that there is a strong cancellation in the spin-independent cross section when y, < o, 142 ' 143 as seen along strips in panel (a) of Fig. 17 where m^2 ~ 500 GeV. In the cancellation region, the cross section drops lower than 10~ 14 pb. All these possibilities for suppressed spin-independent cross sections are disfavoured by the data on g^ — 2, which favour values of m.1/2 andTOOthat are not very large, as well as fj. > 0, as seen in panel (b) of Fig. 17. Thus g^ — 2 tends to provide a lower bound on the spin-independent cross section.

840

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G SI =10- 12 pb

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1000100

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1 •V %

:' :

300-

: : m h = 114.1 GeV

•

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1 /''

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j

|-'

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tan P = 10» H > 0

tan p = 10 , ji< 0

800-1

< i

\ .

•\

0100

900 1000

l a SI =Hr* P b

j : ;^Ǥiilj

^^^^^^^^^^^B

200

300

400

500

600

700

800

900 1000

m^CGeV)

m^(GeV)

Figure 17. Spin-independent cross sections in the (mi/2>w*o) planes for (a) tan/9 = 10, fj, < 0, (b) tan /3 = 10, \i > 0. The double dot-dashed (orange) curves are contours of the spin-independent cross section, differing by factors of 10 (bolder) and interpolating factors of 3 (finer - when shown). For example, in (b), the curves to the right of the one marked 10~~9 pb correspond to 3 x 1Q~ 10 pb and 10~ 1 0 pb.

tanP=10, } i > 0

so- 8 -

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400

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700

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relic density *

a 20

>.

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50

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tanp Figure 18. Allowed ranges of the cross sections for tan f$ = 10 (a) fi > 0 for spinindependent elastic scattering. The solid (blue) lines indicate the relic density constraint, the dashed (black) lines the b -> 57 constraint, the dot-dashed (green) lines the mh constraint, and the dotted (red) lines the g^ - 2 constraint. The shaded (pale blue) region is allowed by all the constraints, (b) The allowed ranges of the spin-independent cross section for fi > 0. The darker solid (black) lines show the upper limits on the cross sections obtained from m ^ and b —• 57, and (where applicable) the lighter solid (red) lines show the lower limits suggested by g^ - 2 and the dotted (green) lines the lower limits from the relic density.

841

Figure 18(a) illustrates the effect on the cross sections of each of the principal phenomenological constraints, for the particular case tan/3 = 1 0 /x > 0. The solid (blue) lines mark the bounds on the cross sections allowed by the relic-density constraint 0.1 < £lxh2 < 0.3 alone. For any given value of mi/2, only a restricted range of m 0 is allowed. Therefore, only a limited range of mo, and hence only a limited range for the cross section, is allowed for any given value of mx. The thicknesses of the allowed regions are due in part to the assumed uncertainties in the nuclear inputs. These have been discussed at length in Refs. 142 and 143. On the other hand, a broad range of mx is allowed, when one takes into account the coannihilation 'tail' region at each tan/3 and the rapid-annihilation 'funnel' regions for tan/3 = 35,50. The dashed (black) line displays the range allowed by the b —> S7 constraint alone. In this broader range of mo and hence the spin-independent cross section is possible for any given value of mx. The impact of the constraint due to m/, is shown by the dot-dashed (green) line. Comparing with the previous constraints, we see that a region at low mx is excluded by m/,, strengthening significantly the previous upper limit on the spin-independent cross section. Finally, the dotted (red) lines in Fig. 18 show the impact of the gM — 2 constraint. This imposes an upper bound on mi/2 and hence mx, and correspondingly a lower limit on the spin-independent cross section. This analysis is extended in panel (b) of Fig. 18 to all the values 8 < tan/3 < 55 and we find overall that 1 4 1 2 x 10" 1 0 pb < asi < 6 x 10" 8 pb, 7

5

2 x 10~ pb < aSD < 10~ pb,

(60) (61)

for n > 0. (asD is the spin-dependent cross-section not shown in the figures presented here.) As we see in panel (b) of Fig. 18, m/, provides the most important upper limit on the cross sections for tan/3 < 23, and b —> S7 for larger tan /3, with gM — 2 always providing a more stringent lower limit than the relic-density constraint. The relic density constraint shown is evaluated at the endpoint of the coannihilation region. At large tan/3, the Higgs funnels or the focus-point regions have not been considered, as their locations are very sensitive to input parameters and calculational details. 144 The results from a CMSSM and MSSM analysis 142 ' 143 for tan/3 = 3 and 10 are compared with the most recent CDMS 145 and Edelweiss146 bonds in Fig. 19. These results have nearly entirely excluded the region purported by the DAMA 147 experiment. The CMSSM prediction 142 is shown by the dark shaded region, while the NUHM case 143 is shown by the larger lighter shaded region. Other CMSSM results 148 are also available.

842

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a

y

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]

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i

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10"

44

10"

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45

10"

10z

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10'

mx [GeV] Figure 19. Limits from the C D M S 1 4 5 and Edelweiss 146 experiments on the neutralinoproton elastic scattering cross section as a function of the neutralino mass. The Edelweiss limit is stronger at higher mx. These results nearly exclude the shaded region observed by DAM A. 1 4 7 The theoretical predictions lie at lower values of the cross section.

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| (a) 10" -14

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200

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300

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500

700

1000

m^; (GeV) Figure 20, Elastic spin-independent scattering of supersymmetric relics on protons calculated in benchmark scenarios, 1 4 9 compared with the projected sensitivities for CDMS I I 1 5 0 and C R E S S T 1 5 1 (solid) and GENIUS 1 5 2 (dashed). The predictions of our code (blue crosses) and l e u t d r i v e r 1 5 3 (red circles) for neutralino-nucleon scattering are compared. The labels A, B , . . . , L correspond to the benchmark points as shown in Fig. 12.

843 I conclude by showing the prospects for direct detection for the benchmark points discussed above. 1 4 9 Figure 20 shows rates for the elastic spin-independent scattering of supersymmetric relics, including the projected sensitivities for CDMS I I 1 5 0 and C R E S S T 1 5 1 (solid) and G E N I U S 1 5 2 (dashed). Also shown are the cross sections calculated in the proposed benchmark scenarios discussed in the previous section, which are considerably below the D A M A 1 4 7 range (10~ 5 — 10~ 6 pb). Indirect searches for supersymmetric dark m a t t e r via the products of annihilations in the galactic halo or inside the Sun also have prospects in some of the benchmark scenarios.

14Q a

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ALESSANDRA BUONANNO

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GRAVITATIONAL WAVES FROM THE EARLY UNIVERSE

ALESSANDRA BUONANNO groupe de Gravitation et Cosmologie (GReCO) Institut d'Astrophysique de Paris (CNRS) 98hls Boulevard Arago, 75014 Paris, France email: [email protected]

These lectures discuss how the direct detection of gravitational waves can be used to probe the very early Universe. We review the main cosmological mechanisms which could have produced relic gravitational waves, and compare theoretical predictions with capabilities and time scales of current and upcoming experiments.

Contents

1

Overview of gravitational-wave research

856

2

K e y ideas underlying G W detectors

857

3

Production of relic gravitational waves: general features 3.1 Gravitational-wave spectrum 3.2 When gravitons decoupled: thermal spectrum? 3.3 Gravitational waves produced by casual mechanisms: typical frequencies 3.4 Phenomenological bounds 3.4.1 Big-bang nucleosynthesis bound 3.4.2 COBE bound 3.4.3 msec pulsar bound

858 858 861

Amplification of quantum vacuum fluctuations 4.1 Standard inflation 4.1.1 de Sitter inflation 4.1.2 Slow-roll inflation 4.2 Superstring-motivated cosmology 4.2.1 Pre-big-bang scenario 4.2.2 Bouncing-Universe scenarios

865 869 870 871 874 874 877

4

855

863 863 863 864 865

856

4.3 Non-standard equations of state in post-big-bang eras 5 Gravitational waves from cosmic strings

879 880

6 Gravitational waves from other mechanisms occurring in early Universe

881

7 Gravitational waves in brane world scenarios

884

8 Extraction of cosmological parameters from detection of GWs 9 Summary

886 887

1. Overview of gravitational-wave research In 1916 Einstein realized the propagation effects at finite velocity in the gravitational equations and predicted the existence of wave-like solutions of the linearized vacuum field equations [1]. The work of Bondi [2] in the mid 50s, applied to self-gravitating systems like binaries made of neutron stars and/or black holes, proved that gravitational waves (GWs) carry off energy from those systems, as the 1974 discovery of the binary pulsar PSR 1913+16 by Hulse and Taylor [3] confirmed indisputably. This discovery is an indirect observation of GWs. In fact, the speed up of the two-body orbital period can be explained as due to GW emission. The experimental search for direct observation of gravitational waves begun only in the 60s with the pioneering work of Joseph Weber, and for almost three decades it has been pursued solely using meter-scale resonantbar detectors. These detectors are cylindrical-shape bars whose mechanical oscillations can be driven by GWs. Five cryogenic resonant-bar detectors are in operation since 1990: ALLEGRO, AURIGA, EXPLORER, NAUTILUS and NIOBE [4]. They are narrow-band detectors sensitive to GW frequency ~kHz. During the last years a world-wide network of kilometerscale ground-based laser-interferometer detectors has been built and has begun operation in Japan (TAMA 300), in the United States (LIGOs) and Europe (GEO 600 and VIRGO) [5,6]. The frequency band is ~ 1-103 Hz. Meantime the first detectors begin the search, development of the next generation kilometer-scale interferometers is already underway [7-10]. A laser-interferometer space antenna (LISA) [11,6] is also planned by the European Space Agency (ESA) and NASA, though not yet fully funded, and could fly in ~ 2011. It consists of three drag-free spacecrafts in a triangle configuration. The spacecraft tracks the distance to each other (~ 5 million

857

kms) using laser beams, searching G W s in the frequency range ~ 1 0 ~ 4 - 1 0 _ 1 Hz. T h a n k s to the theoretical and experimental progress made by the G W community during the last forty years, the direct detection of G W s is not far ahead of us and, hopefully, it will mark the next decade. T h e most promising sources for b o t h ground- and space-based detectors are astrophysical sources at relatively low red-shift [12], like binaries made of black holes (BHs) a n d / o r neutron stars (NSs), white-dwarf (WD) binaries, supermassive black holes, low-mass X-ray binaries, supernovae and pulsars. T h e large scale interferometers were indeed conceived and designed to detect G W s from those sources. These lectures will focus on cosmological sources at much higher red-shift z » 1. We shall discuss if and how the detection of GWs can be used to investigate physical processes occured around Universe's birth. Various excellent lectures and reviews were written on this subject like, e.g., the ones by Allen [13] and Maggiore [14] [see also Sections 2.9 and 3.6 by Cutler and Thorne [12]]. It is not possible to cover in two lectures all the literature on the subject, so we have to make a selection. In the first lecture, after briefly reviewing the key ideas underlying G W detectors [see Section 2], we discuss general features of relic G W spectra, their typical intensity and range of frequencies and phenomenological bounds [see Section 3]. T h e bulk of the lecture will be the analysis of the phenomenon of amplification of quantum-vacuum fluctuations, and the evaluation of stochastic G W backgrounds in standard, quintessential and superstring-motivated inflationary scenarios [see Section 4]. T h e second lecture will review, in Section 5, other physical mechanisms which could produce GWs, like first-order phase transitions, turbulence, preheating, etc., and cosmic strings [see Section 6]. Then, in Section 7. we discuss which differences in the relic G W spectrum we could expect from brane-world scenarios and, finally, in Section 8 we comment on the possibility of extracting the red-shift-luminosity curve and cosmological parameters by detecting G W s from binary systems. Section 9 summarizes the conclusions.

2. K e y i d e a s u n d e r l y i n g G W d e t e c t o r s T h e simplest G W detector we can imagine is a body of mass m at a distance L from a fiducial laboratory point, connected to it by a spring of resonant frequency u>o and quality factor Q. Einstein equation of geodesic deviation predicts t h a t the infinitesimal displacement A L of the mass along the line of separation from its equilibrium position satisfies the equation [15] (valid

858

for GW wavelengths 3> L and in local Lorentz frame of the observer at the fiducial laboratory point) AL(t) + 2 ^ AL(t) + ul AL(t) = |

[F+

h+(t) + Fx hx (t)] ,

(1)

where -F+,x a r e coefficients of order unity which depend on the direction of the source and the GW polarization angle; /i + j X are the two independent polarizations of the GW. Laser-interferometer GW detectors, such as GEO, LIGO, VIRGO and TAMA, are composed of two perpendicular kilometer-scale arm cavities with two test-mass mirrors hung by wires at the end of each cavity. The tiny displacements AL of the mirrors induced by a passing-by GW are monitored with very high accuracy by measuring the relative optical phase between the light paths in each interferometer arm. The mirrors are pendula with quality factor Q quite high and resonant frequency LOQ much lower (~ 1 Hz) than the typical GW frequency (~ 100 Hz). In this case Eq. (1), written in Fourier domain, reduces to AL/L ~ h. The typical amplitude, at 100 Hz, of •GWs emitted by binary systems in the VIRGO cluster of galaxies (15 Mpc distant), which is the largest distance the first-generation of ground-based interferometers can probe, is ~ 10~ 21 . This means AL ~ 10~ 18 m, a very tiny number! It seems rather discouraging, especially if we think to monitor the test-mass motion with light of wavelength nearly 1012 times larger. It took a long time, theoretically and experimentally to develop sophisticated technology to achieve measurements of such tiny displacements. The electromagnetic signal coming out the interferometer will contain the GW signal but also noise - for example thermal noise in the suspension system and in the mirror itself, can shake the mirror mimicking the effect of a GW. The root-mean-square of the noise is generally expressed in terms of the noise power per unit frequency Sn by the relation hrms ~ y/Sn(f) A / ~ AL/L, being AL the mirror displacement induced by noise and A / the frequency bandwidth. 3. Production of relic gravitational waves: general features 3.1. Gravitational-wave

spectrum

Since in the following we will focus mainly on the primordial stochastic background of GWs, in this section we relate its energy density to the spectral density Sh, which is the quantity of direct interest when we want to compare theoretical predictions to experimental sensitivities. [Henceforth, we pose h = 1 = c]

859

The intensity of a stochastic background of GWs at (present) frequency / is generally expressed in terms of

M / ) =^

,

(2)

Pc dlogf where pc = 3HQ / (8ITGN) and pew is the energy density of the GW stochastic backgrounds In the transverse-traceless (TT) gauge, denoting with to, the direction of propagation of the GW, with rh, n the unit vectors forming with to an orthonormal basis, and with ef, = rhirhj — hihj,e^ = m, hj+hi rhj the polarization tensors, i, j = 1, 2,3 [e^-(^) e Ql J(fi) = 2<5PQ], a stochastic, isotropic GW can be written as +oo

/ r=-t-,x

/•

dClhP(f,Cl)e-^iftefJ((l),

df "°°

(3)

J

where dto, = d(f)dcosO, hp(f,to) = hp(—f,Cl), and in writing Eq. (3) we have assumed, in full generality, that the wave is located at x = 0. If the stochastic background is also stationary and unpolarized, averaging the Fourier components over an ensemble gives [14] (h*P(f, Cl) hQ(f, to')) = 5(f - / ' ) S2(to, to!) i - 5PQ \ Sh(f), (4) with 52(to, Cl') = 5(cos 6 — cos 0') 8{<j> — <j>'). The above equation defines the (one-sided) spectral density Su [Sh(f) = Sh(~/)], which has the dimensions of H z - 1 / 2 . The energy density of GWs is given by the 00 component of the energy-momentum tensor averaged over several wavelengths [see e.g., Ref. [15]] PGW = / J0

dlogf

GW

dlogf

=

QO

hij

tii.

(5)

32TTG N

Inserting in the above equation Eq. (3) and using the property that for a stochastic background the spatial average can be traded with the ensemble average defined by Eq. (4), a straightforward calculation leads to: «GW(/)

36 = %L Jf SnyjJ h(f) ~ 1.25 x 10 ~ (JL) 3H$ ' hi V Hz ,

f Sh(f).

(6)

Note that if ficw ~ C fa with 0 < a < 3, due to the factor / 3 appearing in the RHS of Eq. (6), a GW detector operating at lower frequency needs a

T h e value of Ho is generally written as Ho = ho X 100 km/sec/Mpc where ho parameterizes the existing experimental uncertainty. Henceforth, all primordial spectrum will be expressed in terms of h^ Q Q W •

860

Lo^T„(GeV)] i 1 '

-2

5 ' 1 '

VIRGO

7 * ii 13 • Jl JtU '1 1 t ' | F 1 • • 1 '

I I k\f

o

/

is i 1 • ' .

LIGOI

ft

/1

BT-8

*W~

'

LIGOII LIGOIII/EURO

^

o -10 -

V

/ LISA

i

""-6

-4

.

.

i

-2

.

.

i

0

.

.

i

2

.

.

i

4

.

.

i

6

.

.

i

8

.

.

10

Loyf(Hz)] Figure 1. Sensitivities, expressed in terms of Q Q W , versus frequency of LISA and ground-based detectors of first, second and third generation. On the top axis, the estimated temperature of the Universe when GWs are produced by causal mechanisms during RD era [see Eq. (10)].

lower sensitivity than a GW detector operating at higher frequency. This explains why space-based detectors like LISA can have better performances than ground-based detectors, although the expected sensitivity of LISA is worse than LIGO and VIRGO. In Fig. 1 we plot the sensitivities for ground-based detectors of first [5], second (~ 2008) [7,8] and third (> 2011) [9,10] generation. More specifically, for the second generation we use the Advanced LIGO configuration or LIGOII [7,8], for which considerable research and development (R&D) have been gained during the last years, and for the third generation we use, as an example, the speed-meter configuration [9,10]. The R&D for third generation of interferometers as LIGOIII/EURO started only more recently. A stochastic background is a random process which is intrinsically indistinguishable from the detector noise. As we shall see in the following, the GW signal is expected to be far too low to exceed the noise level in any existing or planned single detector on the earth. Moreover, the instrumental noise level will not be known sufficiently well a priori to search for excess noise in each instrument. Therefore, the optimal strategy which has been proposed [16] is to perform a correlation between two or more detectors, possibly widely separated to minimize common noise sources. By

861

correlating two detectors the increase in sensitivity hTms is ~ ( A / T ) 1 ' 4 where A / is the bandwidth and T the total observation time. Henceforth, we shall show the sensitivities obtained correlating two groundbased detectors. Assuming a constant GW spectrum, and correlating for fourth months, we obtain at 90% confident level for two LIGOI: hlVlGW ~ 3.5 x 10" 6 , for two LIGOII: fc§ fiGw - 5.1 x 10" 9 and for two LIGOIII: hi ftGW ~ 3.7 x 10" 1 1 . In the case of LISA, since only one space-based detector is currently planned, the method of correlating two detectors cannot be used. [In Ref. [17] optimal orbital alignements for correlating a pair of future LISA-kind detectors have been investigated.] Armstrong, Estabrook and Tinto [18] realized that by combining the signals from the three spacecrafts which compose LISA, it is possible to measure the instrumental noise power. Then, any excess noise above this power will be due to GWs. However, besides the primordial stochastic GW background, there are various astrophysically generated stochastic GW background due to binaries present in our galaxy or outside it. These stochastic backgrounds are produced when the incoherent superposition of gravitational radiation emitted by a large number of astrophysical sources cannot be resolved individually. A possible, although rather challenging way of discriminating between the primordial and astrophysically generated backgrounds was suggested by Ungarelli and Vecchio [19]. As LISA rotates along its orbit, its sensitivity to different directions changes, so LISA could measure the isotropy of the stochastic background and use it to separate the primordial stochastic background, which is isotropic, from the galactic stochastic background, which is anisotropic, being concentrated mostly in the galactic plane (rather than in the halo). In Fig. 1 we plot the planned LISA sensitivity [40].

3.2. When gravitons

decoupled:

thermal

spectrum?

It is well known that particles which decoupled from primordial plasma at time tdec, when the Universe had temperature T^ec, carry information on the state of the Universe at Tdec- The weaker the interaction, the higher the energy scale when they drop out of thermal equilibrium. To estimate Tdec for gravitons, we proceed as follows [20,21]. We assume that gravitons are in thermal equilibrium in the primordial plasma through point-like fourbody interactions involving two gravitons. We denote with T = ncr\v\ the interaction rate per particle, being n the number density of particles in equilibrium, a the scattering cross section, and v the relative velocity.

862

Prom dimensional considerations we expect a ~ G2^ E2 ~ G2^ T2, where GN is the Newton constant and E2 is the average energy squared. For relativistic particles in equilibrium at temperature T, n ~ T3. We also assume v ~ 1. The interaction rate is then F ~ G2^ T5. Assuming the Universe evolves adiabatically, we have g$T3 a3 — const., where gs is the number of effectively massless degrees of freedom [see Eq. (3.73) of Ref. [21]]. Since for relativistic particles p ~ T 4 , we have T/T oc H oc T2/Mp\, thus the condition to maintain thermal equilibrium is F > H, i.e. the interaction time-scale must exceed the local Hubble time. This gives:

(£).-(&)"• So, as it could have been anticipated from dimensional considerations, gravitons decoupled at Tdec ~ Mp\. Let us derive at which temperature it corresponds today. The temperature of the Universe, i.e. of the particles in thermal equilibrium, scales as T oc gs ' a - 1 , while the temperature associated to particles which have decoupled from the thermal bath drops as T oc a - 1 . Applying these considerations to the thermal bath of gravitons, and using the fact that at T > 300 GeV the Standard Model (SM) of particle physics predicts #s — 106.75, while at present time, assuming three neutrino species, gs — 3.91, it is easily derived that the graviton temperature today is at most [21] Tg = (3.91/106.75) 1 / 3 2.75 K ~ 0.9 K [at most because the effective number of massless degrees of freedom could be higher then 106.75 at Planckian energies.]. Thus, the thermal spectrum of gravitons ranges in the GHz and is given by [13]

n (/)

« = ^ (£) Tc^ri-i^^'^

GQ45&8)

with fg = k Tg/h ~ 1.9 x 1010 Hz. [Note that for CMBR we have T 7 ~ 2.73 K and / 7 = 5.7 x 1010 Hz.] However, since gravitons interact very weakly with matter and radiation, the time required to reach thermal equilibrium could be longer than the Hubble time. So, it is quite unlikely that this radiation ever existed. Moreover, if the Universe underwent a period of inflation after the Planck era, the background should have been strongly diluted. More importantly, the theory of gravity at the Planck energy could differ significantly from Einstein theory; if so, the above analysis should be modified, accordingly.

863

3.3.

Gravitational waves produced typical frequencies

by casual

mechanisms:

Two features determine the typical frequency of GWs of cosmological origin produced by some casual mechanism [14]: (i) the dynamics, which is model dependent, and (ii) the kinematics, that is the red-shift from the production era. Let us assume that a graviton is produced with frequency /* at time t% during RD or MD era. We want to evaluate its frequency today / = /*a*/ao- Assuming that the Universe evolved adiabatically, that is gs(T*) T 3 al = gs(T0) T03 og, and using T0 = 2.73K and gs = 3.91 we get

Since the size of the Hubble radius is the length scale beyond which causal microphysics cannot operate, from causality considerations we expect the characteristic wavelength of gravitons produced at time £* is either ~ H~l or smaller. So, we pose [14] A* = eH^1 with e < l . If the GW signal is produced during RD era H* = 8TTGN p r a d / 3 = 8TT3S* T* /(90 M ^ ) and

'=-i(i£v)(£rOn the top axis of Fig. 1 we show the production temperature T* as obtained from Eq. (10) by expressing T* in terms of the frequency / . For the kind of mechanism analysed in this section, LISA could probe physics at TeV scale, ground-based interferometers in the range ~ 10 8 -10 10 GeV, while GUT and Planck scales could be probed in GHz region, where however no GW detectors have been planned so far, but electromagnetic (EM) microwave cavities have been proposed [22]. 3.4. Phenomenological

bounds

3.4.1. Big-bang nucleosynthesis bound The theory of Big-bang nucleosynthesis (BBN) [23] predicts rather successfully the primordial abundances of light elements like deuterium, 3 He, 4 He and 7 Li. If at nucleosynthesis time (T ~ MeV), the contribution of the primordial GWs to the total energy density is too large, then the expansion rate of the Universe H, and the freeze-out temperature which determines the relative abundance of neutrons and protons, will be too high. Thus, neutrons will be more available and 4 He will be overproduced, spoiling

/msec pulsar bound

a en o -J

BBN bound

\
-18 -16 -14 -12 -10 -8 -6 -4 -2 0 Log [f (Hz)]

2 4

6 8 10

10

Figure 2.

Summary of phenomenological bounds on energy density of relic GWs.

BBN predictions. Detailed calculations provides the following bound on the energy density in GWs integrated over frequency, rf=+oo

d\ogfh20nGW{f)

< 5.6 x 10" 6 (N„ - 3),

(11) /=o where N„ is the effective number of neutrino species. The bound (11) holds only for GWs that were already produced at time of nucleosynthesis. Since the integral cannot exceed the bound, also its positive definite integrand cannot do it, unless JIQW *C 1 0 - 6 for most of the frequency range and has a very narrow peak on the order of ~ 1 0 - 5 at some frequency. This possibility is however not very plausible. Thus, Eq. (11) gives for A / ~ / and N„ < 4 [24] the value h% f2Gw < 5.6 x 10~ 6 , which is plotted in Fig. 2. 3.4.2. COBE bound The COBE bound comes from the measurement of temperature fluctuations in the Cosmic Microwave Background Radiation (CMBR). Indeed, a background of GWs at very low frequencies can produce, if sufficiently strong, a stochastic red-shift of the CMBR frequencies, through the well known Sachs-Wolfe effect. Indicating by 5T the fluctuation induced in the CMBR temperature, we have

fiowCf) <

-Ho /

(5JL

10 i 3 x 10" l s Hz < / < KT16 Hz

(12)

865

where the range of frequencies is determined by the condition the GWs are inside the Hubble radius today (/ > Ho ~ 3 x 10^ 18 Hz) and they were outside the Hubble radius at the last scattering surface (LSS) (/ < V ^ L S S ^ O ~ l(T 1 6 Hz with zLSS ~ 103). The value of 6T/T induced by GWs on CMBR cannot exceed the observed value ST/T ~ 5 x 10" 6 . Detailed analyses give [25,13]: h20 n G w ( / ) < 7 x 10" u ( ^

J

3 x 1(T 18 Hz < / < 1(T 16 Hz .

(13)

GWs can saturate this bound if the contribution from scalar perturbations is subdominant, and this depends on the specific inflationary model. The COBE bound (13) is shown in Fig. 2. 3.4.3. msec pulsar bound The very accurate timing of msec pulsars constraints f^GW- In fact, if a GW passes between us and the pulsar the time of arrival of the pulse is (doppler) shifted. Many years of observations lead to the bound [26,14] K nGW(f)

< 4.8 x 1(T 9 Cpj

f > U = 4.4 x 1(T 9 Hz ,

(14)

where /* is derived from the total observation time of ~ 8 years [see Fig. 2]. 4. Amplification of quantum vacuum

fluctuations

The amplification of quantum vacuum fluctuations is a very general mechanism characterizing quantum field theory in curved space time, first discussed in cosmology by Grishchuk [27] and Starobinsky [28]. Let us first give a formal description of it. We assume that the background field dynamics is described by the action:

s

= Tek;Jd4x^n + s~>

(15)

where 1Z is the Ricci scalar, Sm refers to possible matter sources with ^JJc^T^v = 26Sm/dg>J'", being T^v the energy-momentum tensor. We restrict ourselves to an isotropic and spatially homogeneous FriedmannLemaitre-Robertson-Walker (FLRW) unperturbed metric, with scale factor a and ds2 = g^ dx^ dxv — —dt2 + a2(t) dx2. The linearized wave equation for the TT metric perturbations Sg^, = h^ (h^ = 0, V M /i^ = 0, h^ = 0)

866

can be obtained by perturbing Einstein equations and keeping all the sources fixed, 5T " = 0. A straightforward calculation gives: Dh?(t, x) = -j= d^y/blg1™

8V) V'(*, x) = 0.

(16)

Introducing the conformal time 77, with dr] = dt/a{t) and writing h?[r,,S) = y/^G^ Y, P=+,x

E^fa)^'*6?^)'

(17)

k

each polarization mode h^(rj) satisfies the equation h^V) + 2^h'n(v)+k2hl:(r,)=07

(18)

where we indicate with a prime the derivative with respect to conformal time r\. For simplicity let us consider two cosmological phases I and II, e.g., inflation and RD or MD phase. We denote with h^(r\) and Hz(r\) the solution of Eq. (18) when the scale factor of phase I and II is used, respectively. The mode expansion of h/ in phase I reads:

V = V^G^YJ

J[^f

[b\>(k)hn(rj)ep(Cl)erks + R.G] , (19

where H.C. stands for hermitian conjugate, bp(k) is the annihilation operator with respect to the vacuum in phase I, i.e. &p(fc)|0)i = 0 with P = x, + [classically, it can be traded with a random variable.] We refer to x and k as the comoving coordinate and momentum, related to the physical quantities by Xphys = a(t) x and fcphys = k/a(t). The mode expansion of H^ in the phase II can be obtained from Eq. (19) with the substitution h% —> Hz, bp(k) —> bp(k). The annihilation operator of phase II satisfies the equations bp(k)\0)n = 0 with P = x , + . The quantities {h^h*-} and {H^,H*} are complete bases and can be related to each other by the so-called Bogoliubov transformation [29] H

k =E t e

h

k> + P%w hi),

(20)

^ = EK^^-%'^)'

(21)

k'

k'

Note that the above equations express the positive-frequency components of the operator in phase I in terms of both the positive and negative components of the operators in phase II, and viceversa. The coefficients a^p

867

and (3rz, are called Bogoliubov coefficients and satisfy the equations [29] Z ^ \ak1kak2k

PkikPfk

Z ^ I aki k @k2 k

% k ak2

^fcifo '

(22) (23)

Inserting H^, given by Eq. (20), in the expression of H/ and equating to ft./, we find the well-known relation between annihilation and creation operators in the two phases:

bl(k) = £ [^k^(k')+P*MbUHk')} ,

(24)

k'

bU

(k) = E Hk' blCk') ~ % *WJ\ •

(25)

Thus, if (3j:p ^ 0, the vacuum of phase II differs from the vacuum of phase I. This means 6I(fc)|0)n 7^ 0 and the number of particles in the state |0)u is n(0\b\k)b^(k)\0)u

= J2\PU,\2^0.

(26)

This phenomenon is known as amplification of quantum-vacuum fluctuations in curved space time. It is due to the inevitable presence of positive and negative components in the basis I when expressed in terms of the basis II [see Eqs. (20)]. The existence of the two complete bases, which are associated to two independent Fock spaces, is a consequence of being the space-time curved. The physical idea underlying the quantum-vacuum production of particles can be also explained as follows [14]. [Henceforth, we assume an isotropic and spatially homogeneous metric, so we can pose a^, = ctk 5^, and /3g£, = /3k <%£/•] Suppose that over a time-scale H^1 the cosmological phase changes, e.g., from phase I to phase II. If £* is the time at which the transition occurs and /» is the physical frequency of the mode at that time, there exist two possible regimes: abrupt transition for which 2-7r/* H^1 1. Let us assume that the quantum state before the transition has number of particles Nj. = i(0|6j. bk \0)i- For modes for which the transition is sudden, i.e. 2nft H^1 H^1, that is their wavelengths are larger than the Hubble

868

radius or in jargon they are said to be outside the horizon),h the physical state does not have time to change during the transition. However, after the transition, the number of particles should be evaluated from annihilation and creation operators of phase II, Nl1 = 1(01^^10)1 = Nl(l

+ 2\/3k\2) + \pk\2 .

(27)

Therefore, if the number of particles when inflation starts is Nj., this number is amplified by the factor 1 + 2|/?fc|2 [33,14]. Moreover, due to the mixing between positive and negative frequencies even the vacuum state of phase I (Nj. = 0) is a multiparticle state when referred to the vacuum state of phase II. On the other hand, for modes for which the transition is adiabatic, i.e. 27r/* H^1 ^> 1 (for these modes A^C H~x, so their wavelengths are smaller than the Hubble radius or in jargon these modes are said to be inside the horizon), the quantum state has the time to follow the evolution of the scale factor. No particle production occurs in this case. In some sense the modes for which the transition is adiabatic, do not feel any effect of being the spacetime curved or changing in time. No mixing between positive and negative frequencies occurs, as it is in flat spacetime. [Practically, the relic GWs spectrum drops off exponentially for frequencies larger than jcutoff _ H*/(2ir), i.e. for modes which never went outside the Hubble radius.] We want now to give a more intuitive explanation of the phenomenon of amplification of quantum-vacuum fluctuations, which is very close to the original derivation due to Grishchuk [27] and Starobinsky [28]. It is based on a semiclassical approach. By introducing the variable ipk(v) = ahk(v)> being hk one of the two polarization modes, Eq. (18) can be recast in the form

M+[k?-U(r,)]

A=0,

U

W =^,

(28)

which is the equation of an harmonic oscillator in the time-dependent potential U(rj). [Note that to get unambiguous results, the change of variables ip(k) —> h(k) can be used only if the scale factor and its first derivative are continuous at the transition between phase I and II [30].] For simplicity, let us consider a de Sitter inflationary era, i.e. a = —1/(77 i?ds), being H^s the constant Hubble parameter during de Sitter phase. b

Note that the event horizon rfg, defined by the relation d&(t) = a(t) / t ' m a x dt'/a(t''), where i m ax is the maximal allowed time extension, exists and dp, ~ H^1 if a(t) ~ (t)a with a > 1 and t > 0, e.g., in power-law inflation or de Sitter inflation.

If k2
Ak + Bk

'

V

a2 (r/)

hk Ak + Bk

~

I Jt)'

Since during de Sitter era the scale factor gets larger and larger, the second term in the RHS of Eq. (29) for hk becomes more and more negligible, thus the tensorial perturbation hk remains (almost) constant while outside the Hubble radius. On the other hand, if k2 ^> |t/(??)|, we have kr\ » 1 or A < H^g1, which means the the mode is inside the Hubble radius (or in jargon over the potential barrier \U(r])\). The solution of Eq. (28) is a plane wave ipk ~ e±ikr>. Thus, in this case the tensorial perturbation hf. ~ e±%kv/a decreases in time while inside the Hubble radius. Therefore, in de Sitter case, the longer the tensorial-perturbation mode remains outside the Hubble radius, the more it gets amplified. Before ending this section we want to express the intensity of the stochastic GW background (2) in terms of the number of gravitons per cell of the phase space, which we indicate by n / with / = |fc|/(27r). In the case of GWs produced out of vacuum, n / = Nj1, thus it is the crucial quantity to evaluate. For an isotropic stochastic GW background PGW = 2/d 3 fc/(27r) 3 {knk), thus «Gw(/) = - 1 6 7 r 2 n / / 4 . (30) Pc From the above expression it is straightforward to deduce that if the planned space- and ground- based GW experiments will detect a stochastic GW background whose spectrum satisfies the BBN phenomenological bound discussed in Section 3.4.1, i.e. Q,QW < 10~ 6 , the occupation number rif ^> 1. This means the observed stochastic GW background is classical. 4 . 1 . Standard

inflation

In this section we shall first derive the GW spectrum in de Sitter inflation, which although is an ideal case (because it does not lead naturally to the transition from inflation to RD era), it has the advantage of providing analytical solutions for the tensorial perturbations. Secondly, we discuss how the GW spectrum is modified in more realistic inflationary scenarios, like slow-roll inflation.

^

870

4.1.1. de Sitter inflation The scale factor during de Sitter era is a(rf) = —l/(iJas??) with —oo < rj < 77*, while during the RD era we have a(rj) = (rj — 2rj*)/(Hds V*) with ?7* < r\ < rjeq, where r]eq is the time at which there is equality between radiation and matter era. [Henceforth, we refer to quantities evaluated at present time with the subscript 0.] Let us observe that during de Sitter phase any (classical) tensorial perturbation is de-amplified and their wavelength gets always stretched. Indeed, for a given comoving wavenumber k, it can be easily found [31] that the de-amplification coefficient is A(f) = h(f,r]o)/h(f-m,r)ia) = e~u (Has a*/(Ho a0)), where / = k/(2ira0), /in = k/(2ira-m) and Afia = log(a*/o; n ) is the number of e-foldings. [To solve the horizon problem [32] the minimal amount of e-foldings is Afm-m = log(iJas a*/(Ho ao))-] Thus, the longer the inflationary era, the smaller the coefficient A. Moreover, the stretching of the wavelength is A/A;n = e" (ao/a*). In de Sitter case Eq. (18) can be solved exactly and the solutions are rather simple,0 they read: hk(r,) = ± ( l - ± \

e-<*»}

-0o
hk (v) = I K e^kri+

0k eik v] ,

V* < V < »?eq •

(31) (32)

[Here, we have assumed that the initial state of the Universe is the BunchDavis vacuum [33], so in Eq. (27) we could unambiguously set TV4, = 0. See Ref. [34] where this assumption is relaxed.] The Bogoliubov coefficients a.k and j3k in Eq. (32) are obtained imposing the continuity of hk and h'k across the transition. A straightforward calculation [33] gives /?& = l/(2k2rjl), so the occupation number is Nk = |/3fc|2- Using the following relations: A;|^»| = 27r/a0|?7*| = 2TT f a 0 /(a* Hds) where a* = 1/Hds \v*\i ao/a* = (to/teq)2/s (i e q A*) 1 / 2 , we obtain fc|i7*| = / / / * with

where we used zeq ~ 104, teq ~ 1010 s and £* = Hds/2. The frequency /* is the cutoff frequency beyond which the GW spectrum falls down exponentially. By using Eq. (30) with nk — l/(4/c4?74), it is straightforward to c

These solutions can be obtained easily rewriting Eq. (18) in terms of the function uk, defined by h^ = ( a n ^ ) ' / a 2 .

871

find that the GW spectrum for fluctuations that re-enter the Hubble radius during the RD era is then

^w(/)^»10-»(6xl^5MJ.

(34)

For fluctuations that re-enter the Hubble radius during the matter era, a similar calculation gives [33]:

^ . ( f l ^ x ! . - ^ ) ^ ^ ) ' ,

(35)

where / e q = if eq /(27r)/(l + ze). [Inflation can occur if the slow-roll conditions are satisfied [32]: <> / ) and \\ |.] Differently from de Sitter inflation, in this case the Hubble parameter is not exactly constant during the inflation era, it slowly decreases in time, and as a consequence, the GW spectrum for fluctuations that go out of the Hubble radius during the inflation era and re-enter during RD era, is not fiat but acquires a tilt: /ijj^Gw(/) ~ y fnT ,d where we denote by V the value of the inflaton potential at the time the present Hubble scale crossed the Hubble radius during inflation. The spectral slope is [36] nr = —(V /V)2 Mp[/(87r) and because of the slow-roll condition |TIT| <^ 1. Quite nicely, the spectral slope can be expressed in terms of the scalar contribution S and tensorial contribution T = 0.61V/Mpi to the quadrupole CMB anisotropy [see, e.g., Eqs. (3), (4) in Ref. [36]]. The result is nT = -T/{7S). The explicit expression of the GW spectrum is given by Eqs. (5), (6) in Ref. [36]. In Fig. 3 we plot the spectrum for two values of T/S, taking also into account the first-correction for the variation of the spectral slope d

Whereas in de Sitter inflation the amplitude of the GW spectrum depends on the square of the (constant) Hubble parameter -Has [ see Eqs. (34), (35)], in potential-driven inflation, it depends on V. Indeed, by applying the slow-roll conditions to the FLRW equations it can be easily seen that if 2 ~ 8-n G N V/3.

872

Figure 3. In the left panel we sketch the evolution of the expansion rate of the Universe H and of some physical wavelengths during a period of standard inflation, followed by RD and MD eras. In the right panel we plot the stochastic GW background for slow-roll inflation obtained in Ref. [36] for T/S = 0.18 and dn-r/dlogk = 0 (continuous line), T/S = 0.18 and dnT/dlogk = - 1 0 " 3 (dot line), T/S = 0.0018 and dnT/dlogk = 0 (dash line) and T/S = 0.0018 and dnT/dlogk = - 1 0 - 3 (dot-dash line) The sensitivity of space-, (correlated) ground-based detectors and the stochastic background from WD binaries is also shown for comparison.

with scale, nr(fc), and compare it to the capabilities of LISA and (correlated 6 ) ground-based detectors. Figure 3 shows that if the predictions of standard inflation are correct, unfortunately there is no hope of observing the stochastic GW background from slow-roll inflation with current and planned GW detectors. As seen from Fig. 3, in the case of LISA, primordial GW backgrounds can be "covered" by the galactic WD binary background [37]. Even if two LISAs will be used, by correlating their data stream, the minimum achievable sensitivity is f2 GW > K T 1 3 [37]. Thus, to probe QGW ~ l ( r 1 6 - l ( r 1 5 from inflation with space-detectors of LISA-kind, the detectors have to operate outside the regime of mHz which is dominated by close WD binaries. Various scientists have already started discussing a follow-on mission of LISA. Given the current astrophysical understanding, the most promising region should be ~ 0.1-1 Hz [37-41], which can be obtained by shortening LISA arms by a factor of 100. In this higher frequency band the astrophysically generated background from unresolved NS binaries is present; however, because the number of sources per frequency bin is expected to e

T h e curves have been produced by multypling LIGOI, LIGOII and LIGOIII sensitivities curves shown in Fig. 1 by a constant factor obtained correlating Hanford (WA) and Louisiana (LO) LIGOs for four months at 90% confident level assuming a flat GW spectrum.

873

be not very high, it should be possible to remove it. To achieve sensitivities on the order of SIGW ~ 10~ 1 6 -10 - 1 5 , major, rather challenging improvements should be made, as increasing the laser power, the dimensions of the mirrors, etc.. The detection of relic GWs from inflation could be achieved also by measuring the polarization [42] in the CMBR with future CMB probes, as originally suggested in Ref. [43]. Indeed, the polarization tensor on the celestial sphere can be decomposed into its "gradient" and "curl" components. The scalar fluctuations do not have handness, so they do not contribute to the curl, while tensorial fluctuations do contribute to the curl. If the polarization measurement will give a curl different from zero, GW will be detected and it will be possible to discriminate between the plethora of inflationary models currently available. Indeed, some of those models predict that the tensorial component T is negligible. In Sections 4.2, 4.3 we shall discuss other inflationary models. Before doing it, we want to point out that quite generally the tensorial perturbations will satisfy an equation similar to Eq. (28) but with the scale factor a replaced by the so-called "pump" field a. The pump field depends on the dynamics of the scale factor and the other fields which are part of the background. Following Refs. [44,45,54,30], if we have two cosmological phases with a = (rj/rj^)11 for —oo < r\ < r\\ < 0 and a = \(jl — ^Vi)/Vi\12 f° r rj\ < ij < r?2, then the solutions of Eq. (28) (having replaced a with a) can be expressed in terms of Hankel functions of first and second species, as

Mr)) = VW\CHg>{k\r,\), Mv)

= y/\v-2Vl\[ak

•

(36)

H
- 2m\)},

(37)

where v\ = | 7 l — 1/2| and v% = | 7 2 — 1/2|, and we have normalized Eq. (36) allowing only positive frequencies at r\ —> — oo, so that ipk - • Ce~ik,1/Vk. It can be derived that the Bogoliubov coefficient /3k entering the GW spectrum is given by [30] |£/|2~/2eT

er = 1 - |-yil - l-yal for 7 l , 7 2 > l / 2 all other cases. £ T = _|^ x — 7 2 |

or

7 l

,72<-l/2, (38)

The spectral slope parameter ny is defined by dlog^cw . , „ ,Qn, nT = —r, 7— = 4 + 2e T , (39) dlogf and it is directly related to the kinematic behaviour of the background, that is to the evolution of the curvature scale and Hubble parameter. If

874

we apply the above equations to de Sitter inflation, we find that 71 = — 1, for RD era 72 = 1 and for MD era 72 = 2. Thus, the spectral slope for fluctuations which re-enter the Hubble radius during RD era is nr = 0, and for those which re-enter the Hubble radius during MD era is TIT = —2, in agreement with Eqs. (34), (35). Moreover, in the case of slow-roll or powerlaw inflation, 71 < — 1, and it can be easily found that for fluctuations that re-enter the Hubble radius during RD era UT = 2 + 271 < 0, as anticipated in Section 4.1.2. Finally, we note that this way of estimating the spectral slope can be easily generalized [45] to the case of n cosmological phases occurring at times 771, rj2, •.. if the pump fields follow a power-law evolution with coefficients 71,72, In this case the relic GW spectrum is characterized by various frequency regions or branches, and in each of them the spectral slope depends only on the kinematics of the phases in which the mode associated to a given frequency went out of the Hubble radius and re-entered the Hubble radius. On the other hand, the amplitude of the GW spectrum and special features in it, like possible oscillations, are determined by the dynamics and by physical assumptions, such as when and how the reheating process took place, whether entropy production is produced at some later stages, etc.. 4.2. Super string-motivated

cosmology

Potential-driven inflation of the kind discussed in Section 4.1 cannot be easily implemented in string theory if the dilaton field or a modulus field is simply identified with the inflaton field [46]. This result forced people to conceive new ways of reconciling inflation and string theory. Henceforth, we shall briefly discuss some of those attempts which use supergravity description of superstring theory, the so-called pre-big-bang (PBB) scenario [47] and bouncing-Universe scenarios [48,49]. Quite generally those scenarios predict a stochastic GW background anything but flat, which could be of interest for space- and ground-based detectors. 4.2.1. Pre-big-bang scenario Due to the presence of other fields, at low energy superstring theory does not give Einstein general relativity - for example heterotic string theory in four dimensions is described by the action

• S j " -

TZ + g^d^d^-^idB)2

- V»

(40)

875

where

l/a(t), ip(t) —> (p(t) — 61oga(i), f with a(t) ~ i 1 /^ 3 and ip(t) ~ — logt. Noticing this property, Veneziano [50] conceived the idea of implementing the inflationary phase at times before the would-be big-bang singularity. Indeed, it is easily shown that for t < 0, a > 0, a > 0, thus the Universe undergoes a (super) inflationary phase. Two different but physically equivalent descriptions of the PBB phase exist: either the string-frame picture described by Eq. (40), where the Universe undergoes an accelerated expansion (H > 0, H > 0, (p > 0), or the Einsteinframe picture, where the action (40) has the standard Hilbert-Einstein form and the evolution of the Universe is described by an accelerated contraction, or gravitational collapse (H < 0,H < 0,

0). This new kind of inflation, which can be shown solves the homogeneity and flatness conundra, is driven by the kinetic energy of the dilaton field and forces both the string coupling (g > 0) and the spacetime curvature to grow toward the future. As a consequence, at least in the homogeneous case, the inflationary stage lasts for ever (t —> — oo) and the initial state of the Universe is nearly flat, cold and decoupled: j C l , ^A^ ^C 1. The tensorial perturbations hk in the PBB model satisfy an equation similar to Eq. (18) but with the scale factor a replaced by the pump field a = e-^l2 a, that is [44] h'i(rl) + 2~h'j:(r]) + k2hi:(r1)=0.

(41)

Note that a coincides with the scale factor in the Einstein frame. [In the case o f D = d + n + 1 dimensions with n internal dimensions contracting and d external dimensions expanding, isotropically, if tensorial perturbations depend only on the external dimensions, then Eq. (41) is still valid but the pump field a = e - * / 2 a^ d - 1 ^ 2 bn^2, being b the scale factor of internal Here for convenience the origin of time has been fixed at t = 0.

876

1

-6 -8

IF

g^Oy

lIAGOs

/

SyLISA N. j

-10 : -12

-

-16

f f3/

-14

-

/"

7 "

/

!,f" , . . < Log [f(Hz)]

Figure 4. In the left panel we sketch the evolution of H and of some physical wavelengths during the superinflationary PBB phase, RD and MD eras. The shaded box refers to the spacetime region around the would-be big-bang singularity for which we do not have a complete description, yet. In the right panel we show two examples of the stochastic G W spectrum.

dimensions [44].] Let us consider the simplest scenario (henceforth minimal model) characterized by an isotropic superinflationary phase with static internal dimensions, followed by RD era. In this case, the solution of the background field equations is [47] a(rj) = (—7y)(1_v^)/2 and ip(rj) = (po — \/3 log(—77), and during RD era, a(rj) = (77 - 2r)*)/(Hs 77*) and ip(rj) =
[ -j-

(42)

Depending on the present value of the string coupling [51] go = 0.03-0.3, the GW spectrum amplitude at the end point frequency is ~ 10 _ 7 -10~ 5 [fi7(£o) ~ 10 - 4 ], thus rather close to the BBN bound discussed in Section 3.4.1. In the right panel of Fig. 4 we show two examples of the stochastic GW background in the range of frequencies of space- and ground-based

877

detectors. In the minimal model the spectrum has the peak in the GHz region and being elsewhere rather steep, it cannot be detected by current and planned GW detectors. References [52,53,30] showed that if the PBB superinflationary phase is followed by other cosmological eras where (partial) higher-order derivatives and/or quantum loops corrections in the effective action (40) are included, the GW spectrum can have branches where it is less steeper, even flat, and could have amplitudes detectable by space- and ground-based detectors. However, robust predictions can be made only when a complete description of the transition from pre- to the post-big-bang era will be available, including the understanding of the bigbang singularity. Indeed, the spectral slope for / > /t, in Fig. 4, depends crucially on the cosmological backgounds around the would-be big-bang singularity. 4.2.2. Bouncing-Universe

scenarios

New cosmologies motivated by string theory and influenced also by braneworld ideas have been proposed [48,49]. As in the PBB model, in those scenarios it is assumed that the Universe had a long existence prior to the big bang singularity, and the initial state is assumed to be very perturbative. Here, as an example, we shall focus on the bouncing-Universe model discussed in Ref. [49]. This model uses two four-dimensional boundary branes, one of which is the visible brane where our Universe lives, the other is the hidden brane. The two branes are originally separated by a finite distance which can be regarded, according to the M-theory conjecture [55], as the dilaton field in the strong-coupling regime. This scenario is built on the following four-dimensional effective action on the visible brane (in Einstein frame):

T

°« = lekld4"

K-^d^dvt-Vif)

(43)

The hidden boundary brane (or bulk brane) is assumed to be initially in a state with minimal energy, and it is flat, parallel to the visible brane and at rest. Non-perturbative effects, for example due to a potential of the kind V(4>) = — Vo e~c^ with c > 0, can generate interactions between the visible and bulk brane, causing the bulk brane to move toward the visible one and collide with it. In this way the kinetic energy of the bulk brane can be converted into radiation, and the hot big-bang era starts. Differently from the PBB scenario, in the model of Ref. [49], the Universe evolves from strong (g2 = e* » 1) to weak coupling (g2 = e* C 1), and both in the string- and Einstein-frame it undergoes an accelerated

878

contraction. Moreover, it is assumed that going toward the singularity (77 —> 0~) the potential V changes its shape and goes to zero. Thus the cosmological background in the Einstein frame can be roughly described by a phase with a^rf) ~ (—rj)£ with 0 < e < l (very slow contraction), followed by a phase with (almost) zero potential, i.e. a superinflationary phase of the PBB kind, with OE(^) ~ (—v)1 a n d then a RD era. The pump field a entering the equation of tensorial perturbations coincides with the scale factor in the Einstein frame (a = CLE), so for fluctuations which go out of the Hubble radius during the phase dominated by the potential and re-enter the Hubble radius during RD era, the discussion around Eqs. (37)-(38) gives: 71 = e > 0 and 72 = 1, thus n r ~ 2 + 2e, whereas for fluctuations that go out of the Hubble radius during the phase with (almost) zero potential and re-enter during RD era, we recover the PBB result UT = 3. Hence, both the PBB and bouncing-Universe scenarios predict very steep GW spectra at very low frequency, so no contribution of tensorial fluctuations to the CMBR inhomogeneities. Thus, if a tensorial component will be found in CMBR, then the current version of those scenarios should be rejected. Due to the collapse phase, in both scenarios discussed in this section the initial (classical) tensor fluctuations are not de-amplified [31], as occurs in potential-driven inflation (see discussion at the beginning of Section 4.1.1). This result, though paradoxical, does not imply that the initial value of tensor inhomogeneities must be fine tuned to unnaturally small value. Indeed, it can be derived [31] that the energy density of tensor waves is indeed de-amplified. However, to solve the homogeneity problem in those superstring-inspired models, we have to apply constraints more severe [31] than in potential-driven inflation models. It is worth to mention that some aspects of both the PBB and bouncingUniverse scenarios, have been questioned. Doubts on the naturalness of the initial conditions and on the actual solution of the homogeneity and flatness conundra were raised [56]. The formation of large-scale structures and CMBR inhomogeneities cannot be explained in those models as naturally as in the standard inflationary models (see Section 3.1) using adiabatic perturbations. [In the case of the PBB scenario, a way out has been recently suggested by converting isocurvature into adiabatic fluctuations [57] during the post-big-bang phase.] More importantly, a debate [58] on whether the scalar perturbations can be unambiguously calculated in those superstringmotivated models is still underway. Those issues can be clarified only when a complete and unique description of the transition from pre to post era will be available.

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4.3. Non-standard

equations

of state in post-big-bang

eras

In this section we want to give another example of non-flat relic GW spectrum due to the presence, soon after inflation, of a phase which differs from the RD era. The possibility of having an all variety of spectra by introducing non-standard, i.e. different from RD or MD, equations of state in post-big-bang eras was originally pointed out by Grishchuk [27]. An interesting example is the case in which the inflationary phase is followed by an expanding phase driven by an effective source whose equation of state is stiffer than radiation. Peebles and Vilenkin [59] discussed a model where the occurrence of a stiff post-inflationary phase is dynamically realized through a potential of the kind V(<j>) = A (4>4 + M 4 ) for <j> < 0 and V() = A M 8 / ( 0 4 + M 4 ) for 0 > 0, with A ~ l ( T 1 4 and M ~ 106 GeV. This model has been denoted quintessential inflation and was originally introduced to explain the dark-energy conundrum, that is the present accelerated expansion of the Universe. The inflaton field starts its evolution at <j> . The cosmological evolution is then driven by the kinetic energy of
880

-6 ^

-8 O

O

J

-14 -16 '1S-6

-4

-2

0

2

4

6

8

10

Logio[f(Hz)] Figure 5. We show an example of the stochastic GW spectrum originated in quintessential inflation and contrast it with the sensitivity of space- and (correlated) ground-based GW detectors. For a detail discussion of the GW spectra as function of the free parameters see Ref. [60].

5. Gravitational waves from cosmic strings Topological defects could have been generated during symmetry-breaking phase transitions in the early Universe. Since the beginning of the 80s they received significant attention [62] as possible candidates for seeding structure formation. Recently, more accurate observations of CMBR inhomogeneities on smaller angular scales and compatibility with densityfluctuation spectrum on scales of 100 h^1 Mpc, restrict the contribution of topological defects to at most ~ 10% [63]. In the following, we shall restrict our discussion to cosmic strings formed when a U(l) local gauge symmetry is broken [see Ref. [64] for the possibility of producing cosmic strings from brane collision at the end of brane inflation scenario [65]]. Cosmic strings are characterized by a single dimensional scale: the massper-unit-length \i. Since they do not have any ends, they are in the form of loops, smaller or larger than the Hubble length. [The length of a string is defined as the energy of the loop divided by fj,.] An important feature of those loops is that their tension equals their mass-per-unit-length, they oscillate relativistically under their own tension, emitting GWs and shrinking in size. It is generally assumed, though difficult to prove, that a network of cosmic strings did form at some time during the evolution of the Universe. In this network the only relevant scale is the Hubble length. Small loops (smaller than Hubble radius length) oscillate, emit GWs and

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disappear, but they are all the time replaced by small loops broken off very long loops (longer than Hubble radius length). The wavelength of the GW is determined by the length of the loop, and since in the network there are loops of all sizes, the GW spectrum is (almost) flat in a large frequency band, extending from / ~ 1 0 - 8 Hz to / ~ 1010 Hz. The GW spectrum is nearly Gaussian and has been evaluated in detail in Refs. [66]. If r is the characteristic radius of the loop, the quadrupole moment is Q ~ / i r 3 and by assuming an oscillation period of r ~ r, the energy emitted is ...2

d£/dt ~ P ~ GN Q ~ 7 GN A* , where 7 is the dimensionless radiation efficiency ~ 60. Numerical simulations of cosmic-string network would predict ftcw ~ P/Pc < 10~9-1CT8 for cosmic strings with GNM < 10~ 6 . Thus, the Gaussian GW background from cosmic strings could be detectable by second and third generation of ground-based interferometers, and by LISA as well, being above the galactic stochastic GW background of WD binaries. More recently, Damour and Vilenkin [67] found that strong beams of high-frequency GWs could be produced at cusps (where the string reaches a speed very close to light velocity) and at kinks along the string loop. [In time domain they are "spikes" or "bursts".] As a consequence of these beams the GW background discussed above becomes strongly nonGaussian. The most interesting feature of this non-Gaussian GW background is that it could be detectable for a large range of values of GN/J,, larger than the usually considered search for the Gaussian spectrum. For example, if the average number of cusps in a string oscillation is only 10%, the non-Gaussian GW background could be detectable by ground- and space-based detectors up to GNA* ~ 10 - 1 3 . If the number of cusps in a string oscillation is very small, kinks, which have smaller bursts but are ubiquitous, could generate a signal detectable by LISA for a large range of values of GNA* [ see Figs. 1, 2 in Ref. [67]]. The most stringent bound on the GW backround from cosmic strings comes from pulsar timing [see Section 3.4.3]. The analysis of Ref. [67] suggests that the GUT value GNA*GUT ~ 10~ 6 is still compatible with existing pulsar data, and if accuracy will be improved, pulsar timing could allow to detect the non-Gaussian GW background up to G^/j, ~ 1 0 - 1 1 . 6. Gravitational waves from other mechanisms occurring in early Universe During its history the Universe could have undergone phase transitions, corresponding to symmetry breaking of particle-physics fundamental interactions.

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In a first-order phase transition the Universe is initially trapped in a high-temperature metastable phase (unbroken-symmetry phase), where the false vacuum is separated from the true vacuum by a barrier in the potential V(<j>,T) of some scalar field driving the transition { is also called the order parameter) [see Fig. 6]. The transition from the metastable to the ground state takes place via quantum tunneling across the barrier with random nucleation of bubbles: true vacuum is inside the bubbles, false vacuum is outside them. If the temperature when bubbles form is too high, they are born small, the volume energy cannot overcome the shrinking effect of the surface tension and they disappear quickly; but when the temperature drops below a critical temperature, bubbles are born larger, and since the latent heat released during the transition is converted to bubble-wall kinetic energy, they expand, approach velocity close to the speed of light, collide and leave the Universe in the broken-symmetry phase. When the collision occurs, the spherical symmetry is broken and GWs, as well as other particles [68], are radiated away [69]. Since the Universe is expanding, the temperature T* at which the transition takes place can be obtained comparing the probability of bubble nucleation per unit time and volume with the expansion rate of the Universe at that temperature. The transition occurs when the probability for a single bubble to be nucleated within one horizon volume is on the order of one. Two parameters determine the stochastic GW spectrum [70]: the bubble nucleation rate per unit volume /3 (T = To e'3*) and the ratio a between the false vacuum energy density and the energy density of the radiation at the transition temperature T*. The bubble collision produces a GW spectrum which is strongly peaked at the frequency characteristic of the duration of the phase transition, i.e. 27r/peak — /?. A detailed analysis gives [70]:

W^XU-

(A)

(jt)^"^

<«>

where H* and g* are the Hubble parameter and the number of degrees of freedom at the time of transition. Typical values for an electro-weak phase transition (EWPT) are /?/#* ^ 10 2 -10 3 , T* = 0(100) GeV, for which /peak — 10~ 4 -5 x 10~ 3 Hz, and lies in the LISA frequency band. The GW spectrum was calculated in Ref. [70]; it reads

where K quantifies the fraction of latent heat that is transformed into bubble-wall kinetic energy and Vb is the bubble expansion velocity. At

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Figure 6. We show the typical temperature-dependence of a potential V(<j>, T) in a first-order phase transition being the order parameter. At the temperature T the true (< 4> >7^ 0) a n d false (conventionally < <j> > = 0) vacua are degenerate, and at T» the transition from the metastable state to the ground state occurs.

frequencies lower than / p e a k the GW spectrum ex / 2 8 [70], while at higher frequencies it drops off as oc / ~ 1 8 [70]. Non-perturbative calculations done using lattice field theory have shown that there is no first-order EWPT in the SM of particle physics for Higgs mass larger than W masses (current results predict an Higgs mass larger than W masses). In Minimal Supersymmetric Standard Models (MSSM), if the Higgs mass is in the range 110-115 GeV, there is the possibility of having a first-order phase transition, if the right-handed stop mass has a mass in the range 105-165 GeV [the stop particle is the scalar supersymmetric partner of the top quark]. However, in this case the GWs produced are too weak. For example, for a Higgs mass of 110 GeV and right-handed stop mass of 140 GeV, there exist regions in the parameter space where [71] a ~ 10~ 2 , K ~ 0.05 and H*/(3 ~ 10" 4 and thus hi flGW ~ KT 1 9 around 10 mHz. The strength of the transition can be enhanced in next-to-minimal supersymmetric standard models (NMSSM). Apreda et al. [71] investigated the NMSSM obtained adding a gauge singlet in the Higgs sector [72]. This model is rather attractive also because it can explain the observed baryon asymmetry. Apreda et al. found that there exist regions in the parameter space for which the GW spectrum is h\ J7GW ~ 10~ 1 5 -10 - 1 0 at / p e a k — 10 mHz. Note that for frequencies 10~ 4 -3 x 1 0 - 3 Hz the stochastic GW background from WD binaries could "cover" the GW spectrum from firstorder phase transitions. If this GW background will be ever detected it will be a signature of supersymmetry and, combined with experimental bounds from future particle colliders, it could allow to discriminate between various supersymmetric models.

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A stochastic GW background could be also produced during a firstorder phase transition from turbulent (anisotropic) eddies generated in the background fluid during the fast expansion and collision of the true-vacuum bubbles [70,71,73]. In the NMSSM [72] there exist regions of the parameter space where [71] /igficw ~ 1 0 - 1 0 with peak frequency in the mHz. Recently, the authors of Ref. [74] evaluated the stochastic GW background generated by cosmic turbulence before neutrino decoupling, i.e. much later than EWPT, and at the end of a first-order phase transition if magnetic fields also affect the turbulent energy spectrum. The observational perspectives of those scenarios are promising for LISA. Turner and Wilczek [75] pointed out that if inflation ends with bubble collisions, as in extended inflation, the GW spectrum produced has a peak in the frequency range of ground-based detectors. Subsequent analyses have shown that in two-field inflationary models where a field performs the first-order phase transition and a second field provides the inflationary slow rolling (so-called first-order or false vacuum inflation [76]), if the phase transition occurs well before the end of inflation, a GW spectrum peaked around 10-10 3 Hz, can be produced [77], with an amplitude large enough, depending on the number of e-foldings left after the phase transition, to be detectable by ground-based interferometers. A successful detection of such a spectrum will allow to distinguish between inflation and other cosmological phase transitions, like QCD or electroweak, which have a different peak frequency. Another mechanism that could have produced GWs in the early universe is parametric amplification after preheating [78]. During this phase classical fluctuations produced by the oscillations of the inflaton field (j> can interact back, via parametric resonance, on the oscillating background producing GWs. In the model where the inflaton potential contains also the interaction term ~ 2X2, being x a scalar field, the authors of Ref. [78] estimated OQW ~ 10~ 12 at / m i n ~ 105 Hz, while in pure chaotic inflation fiGw < 10" 1 1 at / m i n - 104 Hz. [See Fig. 3 in Ref. [78] for the GW spectrum in the range 10 6 -10 8 Hz.] Unfortunately, the predictions lie in frequency range where no GW detectors have been planned so far, although EM cavities to detect GWs were proposed [79].

7. Gravitational waves in brane world scenarios During the last years there have been feverish activities around brane-world models [80,81]. Those scenarios are based on the idea that large (> 1/Mpi)

885

spatial, extra dimensions might exist. They assume that ordinary matter is confined onto a three-dimensional subspace, called the (visible) brane, which is embedded in a larger space, called the bulk. Only gravitational interactions are allowed to propagate in the bulk. If n is the number of extra dimensions, then the four-dimensional Planck mass Mpi is a derived quantity, while the fundamental scale is determined by the gravitational mass Mfun(j in n + 4 dimensions. In the case of a flat bulk, being I the typical length of the extra dimensions, Mpj = M f "+j ln with Mfund ranging between TeV to Mp\. In the following, we shall mention results obtained in five-dimensional brane-world models (n = 1). By normalizing properly Eq. (31) and taking the limit kr\ 1, then the amplitude of GWs g can be modified with respect to the value predicted in standard four-dimensional theory for example [83-85,82] fc3/2 hk = f{Hl) H/MPi with some function / . h In the brane-world models analysed in Refs. [83,84,82], it was found that for HI 3> 1 the tensor amplitude is enhanced with respect to the standard result, i.e. f(l H) > 1. Moreover, the ratio between scalar and tensorial perturbations T/S may also differ from the value predicted by standard four-dimensional theory [83,82] and it could depend on specific features of the brane-inflation model. However, as seen in Section 3, the overall shape of the relic GW spectrum depends not only on the evolution of the tensormode amplitude during inflation but also on the post-big-bang phases, i.e. on the era at which the mode re-enters the Hubble radius. Only when those cosmological phases will be consistently described in the brane-world scenarios, we will have robust predictions for the GW spectrum. By implementing quintessential inflation on the visible brane, and evaluating the Bogoliubov coefHents all along the Universe evolution, the authors s

Here the GW refers to the homogeneous Kaluza-Klein massless mode. Massive KaluzaKlein gravitons decay very fastly [83,84,82], so their production is very suppressed during brane inflation. "In the models analysed by Frolov and Kofman [82], f(Hl)/Mpi = 1/Mpi infl where Mpiinfl is the Planck mass during inflation.

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of Ref. [86] found results similar to the ones discussed in Section 4.3: the GW spectrum has a branch where it increases linearly as function of the frequency. GWs can be also produced by excitations of the so-called radion field, which controls the size of the extra dimension, and by inhomogeneities in the displacement of the brane [87]. Those waves should peak at a frequency fixed by the scale of the extra dimension. LISA could observe excitations emitted at scales from millimeters to microns, while ground-based interferometers could observe rather small extra dimensions, up to ~ 10~ 12 mm. In some brane world scenarios the causal propagation of gravitational and luminous signals can be different [88,89]. In the case of an asymmetric warped spacetime of the kind [89], ds2 = -n2(l) dt2 + a2(l) ( dT1 , + r2 di}2) + b2(l) dl2 , \ 1 — k rJ /

(46)

where I is the coordinate along the extra dimension, and k = ±1,0 is the spatial curvature of the three-dimensional sections parallel to the brane, the local speed of light c(l) = n(l)/a(l) depends on the position I along the extra dimension. Since GWs can propagate in the bulk, a GW signal emitted at point A on the brane can take [if c(l) is increasing away from the brane] a short-cut in the bulk, and to an observer located at point B on the brane it appears quicker than a photon traveling along the brane from A to B. If true, this effect predicts a difference between gravitational and luminous speed. By detecting GWs and EM waves from 7-ray bursts and supernovae, ground-based detectors of second generation can put upper limits on 5c/c at most on the order of 1 0 - 1 7 [12] for 7-ray bursts, and 1 0 - 1 1 for a supernova in the Virgo cluster of galaxies.

8. Extraction of cosmological parameters from detection of G W s In this section we want to discuss very briefly another possible way GW detection can be used to probe cosmology, although not directly the physical mechanisms occurring in the very early Universe. It was realized long ago that binaries made of compact bodies, like BHs or NSs, which spiral in toward each other loosing energy because of the emission of GWs, are "standard candles" [90]. Indeed, once the binary masses and spins, position and orientation angles are specified, the GW signal passing-by the detector depends only on the luminosity distance

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dh(z, £}M,CIA, • • •) between binary and detector. By using three groundbased interferometers or LISA (and suitably high signal-to-noise ratio) it is possible to determine the location and orientation of the binary in the sky, extract the masses, the spins and the cosmological distance but not the source's cosmological red-shift. Indeed, the inspiral GW signal emitted from a binary with masses (777-1,7712) at red-shift z cannot be distinguished, except for an overall factor in the amplitude, from the one emitted from a local binary with masses [(1 + z) mi, (1 + z) rri2\. To break the degeneracy and obtain the distance-red-shift curve and the cosmological parameters, one could associate the binary-coalescence event to an EM event which had very clear emission or absorption lines, from which one could read z. [Because the errors in position determinations can be rather large, the existence of EM counterpart can also improve the accuracy in measuring the luminosity distance - for example by mapping massive BH binaries with LISA 5dL/dh decreases from 1-10% to 0.15-1% [91].] However, Holz and Hughes [91] recently pointed out that, practically, because of gravitationallensing effects (GWs are lensed as EM radiation are lensed) the precision with which the luminosity distance can be determined is.degraded, though still slightly better than what Type-la supernovae probes can reach, but unless the event rates is high enough the extraction of cosmological parameters cannot be done with good accuracy.

9. Summary As seen, the search for GWs from the very early Universe is rather challenging but the outcome is certainly worth the effort. Theoretically, so little we know about the very early evolution of our Universe, that the predictions for the stochastic GW background from standard inflation [27,28] and/or superstring-motivated models [47,48], including also brane-world scenarios [83-87], should be considered just as indications. What we learned from those models, is that it could well be the GW spectrum is not (almost) flat all the way from / ~ 10~ 16 Hz to / ~ 10 10 Hz. There could be frequency regions in which it increases or decreases [27] due to cosmological phases existed before the would-be big-bang singularity, and/or post-big-bang phases with equations of state different than radiation or matter ones, whose presence we cannot currently exclude. [Even an initial state different from vacuum could affect the relic GW spectrum [34], as seen from Eq. (27).] Future CMB polarization experiments [43] could detect the tensor contribution

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at very large wavelength and discriminate within plethora of inflationary models. The detection of GWs from bubble collision [69,71,76,77] and/or turbulent motion [69,71,73,74] in the primordial plasma, and especially the frequency at which the GW spectrum is peaked, will reveal if first-order phase transitions occured, and at which particle physics mechanism they are associated to: QCD, EW, last stages of inflation, etc. If a network of cosmic strings ever formed, it should have produced a Gaussian [66] or even strongly non-Gaussian [67] GW spectrum, which could be detectable by the second generation of ground-based interferometers and LISA. If not observed, we could put upper limits on few free parameters. An independent estimation of the cosmological parameters could be obtained by detecting GWs from coalescing binaries at fairly high red-shift [90,91]. Experimentally, since the GW signal from astrophysical sources is much more promising than the one from the early Universe, the ground- and space-based detectors were conceived and designed to detect the formers. Various scientists in the GW community have already started thinking at future detectors, like the follow-on LISA missions which could focus more on early-time cosmological signals [37-41]. The technological challenges for these missions are considerable and deserve very careful investigations. Finally, ground-based GW detectors in the highfrequency region of MHz [79] and GHz [22] have gained more attention [47,60,78]. So, even if the detection of GWs from primordial Universe is still somewhat far ahead of us, maybe, if nature is kind with us, we will not wait for a long time. There could be surprises.

Acknowledgments I wish to thank the organizers of the TASI school for having invited me to such a pleasant and stimulating school and all the students for their interesting questions. In preparing and writing these lectures I benefited from conversations with Yanbei Chen, Scott Hughes, Marc Kamionkowski, Arthur Kosowsky, David Langlois, Shane Larson, Albert Lazzarini, Michele Maggiore, Jerome Martin, Alberto Nicolis, Sterl Phinney, Patricia Purdue and Kip Thorne. This research was also supported by Caltech's Richard Chace Tolman Foundation.

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