Neutral Kaons (Springer Tracts in Modern Physics)

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Springer Tracts in Modern Physics Volume 153 Managing Editor: G. H6hler, Karlsruhe Editors: J. K/ihn, Karlsruhe Th. Miiller, Karlsruhe R. D. Peccei, Los Angeles F. Steiner, Ulm J. Tr/imper, Garching P. W61fle, Karlsruhe Honorary Editor: E. A. Niekisch, Jfilich

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Radoje Belu]evi~

Neutral Kaons With 67 Figures

Springer

Dr. Radoje Belu~evi~ High Energy Accelerator Research OrganizationKEK Department of Physics 1-10ho, Tsukuba-shi 3o5-o8ol Ibaraki-ken, Japan Email: [email protected]

Physics and Astronomy Classification Scheme (PACS): 14.4o.A, o3.65.-w, 13.2o.Eb, 13.25.Es, ll.3o.Er, 12.15.Ff, 12.15.Ji

ISSN oo81-3869 ISBN 3-54o-65645-6 Springer-Verlag Berlin Heidelberg New York Library of Congress Cataloging-in-Publication Data applied for. Die Deutsche Bibliothek - CIP Einheitsaufnahme Belu~evi~, Radoje: Neutral kaons/Radoje Belugevi~. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1999 (Springer tracts in modern physics; Vol. 153) ISBN 3-54o-65645-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by EDV-Beratung E Herweg, Hirschberg Cover design: design &production GmbH, Heidelberg SPIN: lO7O9232

56/3144- 5 4 3 21 o - Printed on acid-free paper

Dedicated to Jack S t e i n b e r g e r

Foreword

Science Museum, London. Science & Society Picture Library

Among the five thousand stereoscopic photographs of cosmic ray showers obtained by George Rochester and Clifford Butler at Manchester University, using a cloud chamber placed in a magnetic field, there was a picture containig "forked tracks of a very striking character". In the lower right-hand side of the picture, just below a 3-cm lead plate mounted across the centre of the chamber, they observed, on 15th October 1946, a pair of tracks forming a two-pronged fork (an inverted V) with the apex in the gas (see the reprinted image). The direction of the magnetic field was such that a positively charged particle moving downward is deflected in an anticlockwise direction. They determined that the particle corresponding to the upper track had positive charge and a momentum of 340 + 100 MeV/c; the lower particle had negative charge and a momentum of 350 + 150 MeV/c. The ionization and curvature

VIII

Foreword

of the tracks showed that they were due to particles much less massive than the proton. If the tracks were associated with a collision process, one would have expected several hundred times as m a n y of these interactions in the lead plate as in the gas. Since very few events similar to this were observed in the plate, they argued that the fork "must be due to some type of spontaneous process for which the probability depends on the distance travelled and not on the amount of m a t t e r traversed". This conclusion is supported by the following argument: if the fork were due to a deflection of a backscattered charged particle by a nucleus, the m o m e n t u m transfer would be so large as to produce a visible recoiling nucleus at the apex. Based on their past experience, the electron pair production by a highenergy photon in the Coulomb field of the nucleus was excluded because the two tracks would have to be much closer together if they were an electronpositron pair. They also excluded the possibility of this picture representing the decay of a charged pion or muon coming up from below the chamber, since in that case conservation of energy and m o m e n t u m would require the incident particle to have a minimum mass of 1280me (me is the electron

mass).

Rochester and Butler therefore concluded that this had to be a photographic image of the decay of a new type of uncharged elementary particle into two lighter charged particles. For the case where the incident particle decays into two particles of equal mass, they determined the mass of the parent particle to be 870 + 200 M e V / c 2, for an assumed secondary particle mass of 200me.

Preface

Enormous progress has been made in the field of high-energy, or elementary particle, physics over the past three decades. The existence of a subnuclear world of quarks and leptons, whose dynamics can be described by quantum field theories possesing local gauge symmetry (gauge theories), has been firmly established. The cosmological and astrophysical implications of experimental results and theoretical ideas from particle physics have become essential to our understanding of the formation of the universe. For example, a tiny violation of CP symmetry, which has been observed so far only in the K ° system, is believed to have played an important role in the early stages of cosmic evolution. The main purpose of this book is to convey the unique beauty of a quantum-mechanical system that contains so many of the aspects of modern physics. Inevitably, this imposes considerable constraints on the content and nature of the presentation. In outlining the basic formalism necessary to describe the K ° system and its time evolution in both vacuum and matter, effort was made to keep the presentation as clear as possible and to justify the main steps in the derivations. To highlight their quantum-mechanical origin, extraordinary properties of neutral kaons are illustrated through analogous experiments with polarized light and atomic beams. A formal theory of the discrete symmetry operations C (charge conjugation), P (parity transformation) and T (time reversal) is presented. These subtle concepts are discussed in the context of parity violation, time reversal asymmetry and CP noninvariance in kaon decays. In order to emphasize the complementary roles of theory and measurement, a number of "classic" experiments with neutral K mesons are described and some major current projects and proposals are reviewed. A detailed and pedagogical discussion of the K ° physics within the framework of gauge theories of the electroweak interactions is also provided. Athough this book was written primarily for graduate students and researchers in high-energy physics, I have endeavored to make its content accessible to curious undergraduates and physicists not specializing in the field.

Acknowledgements I would like to thank Bruce Winstein and Italo Mannelli for valuable comments regarding the experiments E731 at Fermilab and NA31 at CERN.

X

Preface

I have benefitted from discussions with Robert Sachs about the K ° phenomenology in the presence of T and C P T violation, and with Kaoru Hagiwara, Makoto Kobayashi, Yasuhiro Okada and Yasuhiro Shimizu concerning K°-/~ ° mixing and rare kaon decays in the Standard Model. Helpful comments and suggestions by Volker Hepp, Martin Wunsch and Sher Alam are appreciated. I am particularly indebted to Asish Satpathy and Bruce Winstein for their interest, help and advice. For permission to reprint various plots and drawings I am grateful to Bill Carithers, Val Fitch, Erwin Gabathuler, Jack Ritchie, Jack Steinberger and Bruce Winstein. I wish to express my special gratitude to Hans KSlsch, Victoria Wicks and the production team at Springer for their help in preparing the manuscript for publication. Support from Prof. Sakue Yamada, Head of the Institute for Particle and Nuclear Studies at KEK, and the Japanese Ministry of Education, Science and Culture (Monbusho) is gratefully acknowledged. Tsukuba-shi Februar~ 1999

R. Belu~evid

Contents

1.

.

Introduction .............................................. 1.1 K ° a n d / ~ 0 as E i g e n s t a t e s of S t r a n g e n e s s . . . . . . . . . . . . . . . . . . 1.2 C P E i g e n s t a t e s of N e u t r a l K a o n s : K ° a n d K ° . . . . . . . . . . . . . 1.3 D u a l i t y of N e u t r a l K a o n s : ( K ° , / ~ 0 ) vs. ( g °, K °) . . . . . . . . . . . 1.4 T h e E i n s t e i n - P o d o l s k y R o s e n P a r a d o x in t h e K ° S y s t e m . . . . 1.5 S t r a n g e n e s s Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 T h e K 10 - K 20 M a s s Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 6 8 9 10 13

of Neutral Kaons in Matter .................. The K ° Regeneration ................................... Coherent Regeneration Amplitude ........................ K 01 - K 1o Interference a n d t h e Sign of A m k . . . . . . . . . . . . . . . . . .

17 17 20 28

Propagation

2.1 2.2 2.3

CP Violation

3.

3.1 3.2 3.3 3.4 3.5

in K ° Decays ............................... Discovery of C P V i o l a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P h e n o m e n o l o g i c a l I m p l i c a t i o n s of K ° --+ 27r . . . . . . . . . . . . . . . . Unitarity, C P T Invariance and T Violation ................ Isospin A n a l y s i s of K°,L -~ 21r . . . . . . . . . . . . . . . . . . . . . . . . . . . K L0- K s0 I n t e r f e r e n c e as E v i d e n c e for CP V i o l a t i o n . . . . . . . . . . .

1

33 33 35 39 43 49

4.

Interference in Semileptonic and Pionic Decay Modes .... 4.1 S e m i l e p t o n i c D e c a y s of N e u t r a l K a o n s . . . . . . . . . . . . . . . . . . . . 4.2 K ° - K ° Interference in 7r+Tr- a n d giTr=F~ D e c a y s . . . . . . . . . . . . 4.3 K ° - K ° I n t e r f e r e n c e W i t h o u t R e g e n e r a t o r . . . . . . . . . . . . . . . . . .

57 57 62 65

5.

Precision Measurements of 0oo, ¢+- and e'/e ............ 5.1 T h e E x p e r i m e n t NA31 a t C E R N . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 T h e E x p e r i m e n t E 7 3 1 / E 7 7 3 a t F e r m i l a b . . . . . . . . . . . . . . . . . . 5.3 C o m p a r i s o n of NA31 a n d E731 E x p e r i m e n t a l Techniques . . . .

71 72 75 80

Neutral Kaons in Proton-Antiproton Annihilations ....... 6.1 T h e C P L E A R E x p e r i m e n t a t C E R N . . . . . . . . . . . . . . . . . . . . . . 6.2 Is CP V i o l a t i o n C o m p e n s a t e d by T i m e - R e v e r s a l A s y m m e t r y ?

81 81 87

.

XII

Contents

7.

Neutral Kaons in Electron-Positron Annihilations 7.1 T h e D A C N E P r o j e c t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.

Neutral Kaons in Fixed-Target Experiments .............. 8.1 T h e E x p e r i m e n t s K T e V a n d NA48 . . . . . . . . . . . . . . . . . . . . . . .

.

The 9.1 9.2 9.3

9.4

K ° System in the Standard Model ................... C a l c u l a t i o n of A m k a n d % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B°-/~ ° M i x i n g a n d C o n s t r a i n t s on C K M P a r a m e t e r s . . . . . . . . Rare K a o n Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 KL° --+ 7r°v~ a n d K + -+ 7r+v~ . . . . . . . . . . . . . . . . . . . . . . 9.3.2 K ° - + p + # - a n d K ° - + e + e - . . . . . . . . . . . . . . . . . . . . . . Direct CP V i o l a t i o n ( # ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendices

A B C D E F

........

89 89 93 93 99 99 109 116 117 125 139

...................................................

145 145 Forward S c a t t e r i n g A m p l i t u d e a n d the O p t i c a l T h e o r e m . . . . 147 W a t s o n ' s T h e o r e m a n d the Decay A m p l i t u d e s K ° , / ~ o __+ 27r. 150 T i m e Reversal a n d CPT Violation . . . . . . . . . . . . . . . . . . . . . . . . 153 T r a n s f o r m a t i o n P r o p e r t i e s of Dirac Fields U n d e r C, P a n d T 156 The Vacuum Insertion Approximation . . . . . . . . . . . . . . . . . . . . . 169

CP P r o p e r t i e s of K --+ 27r a n d K --+ 37r . . . . . . . . . . . . . . . . . . . .

References ....................................................

171

Name Index

175

..................................................

Subject Index

................................................

Subject Index (Decays) .......................................

177 183

1. I n t r o d u c t i o n

"This is one of the greatest achievements of theoretical physics. It is not based on an elegant mathematical hocus-pocus such as the general theory of relativity yet the predictions are just as important as, say, the prediction of positrons."

Richard Feynman, The Theory of Fundamental Processes The neutral K meson (neutral kaon), K ~ and its antiparticle, /7/~ form a remarkable quantum-mechanical two-state system that has played an important role in the history of elementary particle physics. Indeed, ever since the discovery of K ~ half a century ago, neutral kaons have been a rich source of unique and fascinating phenomena associated with their production, decay and propagation in both vacuum and matter. What makes the K ~ system so special is that K ~ and /7/~ which have the same charge, mass, spin and parity, but different strangeness quantum number, S, cannot always be distinguished from one another. Whereas in strangeness-conserving strong interactions K ~ (S = +1) a n d / ~ 0 (S = - 1 ) are as distinct as the neutron and antineutron, this distinction is erased in strangeness-violating weak interactions, thus allowing K ~ ++/7/o transitions. 1 As a consequence, an initially pure IK ~> or I/( ~> state will gradually evolve into a state of mixed strangeness, ]K~

> a(t)]K ~ + b(t)l/~~

in accordance with the principle of superposition of amplitudes in quantum mechanics. This strangeness oscillation effect has a nice optical analogy: right-circularly polarized light rapidly acquires a large left-circularly polarized component while passing through a crystal that absorbs predominantly x-polarized light. The K ~ and /~0 mesons are two unconnected, degenerate (mao = m~o) states in the absence of the weak interaction. As is well known from quantum mechanics, the mixing of two degenerate levels in vacuum must result in level splitting (this splitting shows up in the hydrogen molecular ion and in the inversion spectrum of ammonia). The application of ordinary perturbation theory to the K~ ~ system produces the following result: the weak interaction, /t/w, slightly shifts the value of the kaon mass, mko, and splits the degenerate levels by a tiny amount:

Amk --

Iml - m21 --
+
where Amk is the mass difference of the two states, K ~ and K ~ created by strangeness-changing K ~ <-+/~0 transitions. These new states are the correct I The neutron and antineutron do not mix becauseof baryon number conservation.

2

1. Introduction

linear superpositions of K ~ a n d / ~ o , which diagonalize the perturbation. In turn, K ~ and KO are linear superpositions of KL~ and K ~ Since both K ~ and its antiparticle decay to two or more pions, the initial state in a pionic decay of a neutral kaon tmtst be some linear combination of K a and /~a states. To see what this implies, suppose that weak interactions do not make an arbitrary distinction between particles and antiparticles, i.e, that they are invariant under the combined operation of charge conjugation and parity transformation (space inversion), ~ / 5 The wavefunction of a 27r final state does not change its sign under C P (it is "even" under this symmetry transformation). Consequently, one linear combination of neutral kaon states, K ~ = K ~ + / ~ 0 can decay into a pair of pious with no CP violation. The other, equally probable, combination K2~ - K ~ cannot, because it is "odd" under ~/5. This state is thus forced to find other CP-conserving ways to decay, such as into three plans, in which case the relatavely small threebody phase space makes the lifetime of K2~ much longer than that of K1~ This remarkable prediction was made in 1955 by M. Gell-Mann and A. Pals. The unique beauty of the K ~ system stems from the quantum-mechanical interplay between the two related sets of particles

(/
The importance of the K ~ system transcends its quantum-mechanical intricacy. Many of the theoretical ideas that form the basis of our current understanding of particle physics are intimately associated with K mesons, in particular with neutral kaons. In the early 1950s, the multitude of observed K-meson decays led to the realization that parity may be violated in weak interactions. Central to this development was a brilliant analysis by R. Dalitz who showed, in 1953, that the 21r and 31r decay modes of the charged kaon required the parent particles

1.1 K ~ and/~o as Eigenstates of Strangeness

3

to have opposite intrinsic parities. This created a serious dilema: either the K -+ 2~ and K -+ 3~ decays were due to different initial states, or parity was not conserved. In 1956 T. D. Lee and C. N. Yang questioned the experimental basis for the assumption of parity conservation in weak interactions. Soon thereafter experiments suggested by them observed that parity was indeed violated. The dilema was thus resolved: the parent particles were the same, but parity was not conserved in the decays. The prevailing sentiment among physicists prior to this discovery was quaintly expressed by W. Pauli: "What God hath put asunder no man shall ever join." Subsequent investigations showed that parity violation was compensated by a failure of charge conjugation. This possibility was first suggested by L. Landau in 1957 before the observation of parity noninvariance! CP was consequently considered to be an exact symmetry of nature until "these same particles, in effect, dropped the other shoe". In 1964, J. Cronin and V. Fitch with their collaborators detected one 27r event among 500 or so common decays of the long-lived neutral l~on. This tiny violation of CP symmetry, which has been observed so far only in the K ~ system, remains a great mystery to this day, especially because it may have played an essential role in the early formation of the universe. The concept of strangeness, introduced in 1953 by M. Gell-Mann and, independently, T. Nakano and K. Nishijima to explain the anomalously long lifetimes of K-mesons and hyperons, was crucial for the development of the quark model of particles (M. Gell-Mann, 1964). Quark flavors, such as strangeness, are conserved in strong interactions but not in weak decays, resulting in the long lifetimes of strange particles. The smallness of the observed branching ratio for the decay K ~ --+ # + # - , implying the absence of strangeness-changing neutral weak currents, led to the prediction of a fourth quark, the charm quark. This prediction is based on quark mixing, which lies at the heart of the assumed universality of the weak interactions of quarks and leptons, and thus of the highly successful Standard Model of elementary particles. CP violation was introduced in the Standard Model by increasing the number of quark and lepton families to at least three (M. Kobayashi and T. Maskawa, 1973). This idea became very attractve with the subsequent discovery of the bottom quark, which forms, together with the recently detected top quark, a third family of quarks.

1.1 K ~ a n d / ~ o

as E i g e n s t a t e s o f S t r a n g e n e s s

The neutral kaon 2 was discovered in 1946 by G. Rochester and C. Butler [1] in a cloud chamber exposed to cosmic rays. They observed a pair of charged 2 Kaons behave in some respects like heavy pions and so they are included in the family of mesons.

4

1. Introduction

particles, later identified as pions, that could be associated with the decay of a neutral particle about 900 times heavier than the electron. 3 With the benefit of hindsight we can say that the particle they detected was either a K ~ or a/~0. But how do we know that there is an antiparticle to the neutral kaon, i.e., how can we distinguish a K ~ from a/7/o? The answer is provided by the empirical fact that some strong-interaction processes, e.g.,

7r- + p - c K ~ + n,

K- + p-+ K~ + A~

(1.1)

have not been observed, although they violate neither the charge nor the baryon number conservation. This prompted M. Gell-Mann and, independently from him, T. Nakano and K. Nishijima to introduce, in 1953, a new quantum number called strangeness [2], which is conserved in strong but not in weak interactions. If we assign the quantum numbers I (isospin) and S (strangeness) according to Table 1.1 to the particles participating in (1.1), then those reactions are clearly not allowed. On the other hand, the strong interactions

K + + n -+ K ~ + p,

zc- + p - + A~ + K ~

K - + p--+ fV~ + n

(1.2)

are allowed and have indeed been observed. From (1.2) we see that K + is associated with K ~ and K - with /~0, thus forming two K doublets (see Table 1.1).

Table 1.1. Isospin and strangeness assignments.

/3 S I

-1

1

0 ~ -1 0 0 1

-2

1

~

-3 0 0 0

0

+~1

zr+

K o

K +

K-

/~o

&-

+1

p A 7r~

7r-

2

1

-~ n

+1 1 -1 1 -1

1

&0 _~-

~+ ~0

~

~2-

s The K ~ was discovered in 1946 and the K ~ in 1947, both by G. Rochester and C. Butler. The discovery of rJ: by C. Lattes, G. Occhialini and C. Powell was reported a day after the first observation of the K~:! The first direct detection of the 7f~ was made in 1950 by J. Steinberger, W. Panofsky and J. Steller.

1.1 K ~ and/~0 as Eigenstates of Strangeness

5

Now, to distinguish a K ~ from a/~o we just have to let them interact with matter. For example, a/7/~ can produce a A particle (the so-called associated

production), /~o + p _+ A o + ~+,

(1.3)

whereas a K ~ cannot. Therefore, K0r

unlike 7, ~o, ~o, wo, etc.

As already mentioned, strangeness is not conserved in weak interactions. As a consequence, K ~ and/7/0 can decay weakly into the same final states. If we observe their common decay modes only, the two neutral kaons look like the same particle; they can be distinguished only through their production in strong interactions. The fact that K ~ and KO have common decay modes, K~

~ --+ 7c+~r- , 7c~ ~ 7c+~r-Tr~ 37r~ etc.

(1.4)

suggests that they are connected through strangeness-violating (IASI = 2), second-order weak transitions (see Fig. 1.1). Therefore, the particles we observe in experiments are not K ~ and/~0, but rather a linear superposition of the two. This situation is (almost) unique to the K ~ system (only D ~ ~ and B~ ~ share this property), because the only quantum number that distinguishes them, S, is not conserved in weak interactions. This is not the case for other particle-antiparticle pairs, since their quantum numbers (charge, lepton number, baryon number) are conserved in all interactions.

Ko

x/

.,.z •+

x /"

Ko Fig.

1.1.

The K~

~ transition via an

intermediate 7r+Tr- pair

Describing the K ~ system as a linear superposition of K ~ a n d / ~ o states is analogous to representing photons as a linear superposition of right- and left-circularly polarized states. We shall thus express the states of neutral kaons by two-dimensional (complex) vectors. Let ]K ~ denote the state in which the meson is a K ~ and I/7/~ the state in which it is a/~o. The general state of a neutral kaon is then given by

I~) = alK ~ + b]/~~

lal: + ]b]2 = 1.

(1.5)

The strangeness operator, S, is defined by

~1K~ = +lK~

~ _-

(1.6)

6

1. Introduction

1.2 C P E i g e n s t a t e s of N e u t r a l Kaons: K1~ and K ~ Weak interactions are not invariant under space inversion (parity transformation)/5. Indeed, all neutrinos have negative helicity (they are said to be lefthanded) and all antineutrinos have positive helicity (they are right-handed), and so in this sense parity is maximally violated in weak interactions. As a consequence, charge conjugation C is also not conserved in weak interactions: applying C to a left-handed neutrino turns it into a left-handed antineutrino, a state which does not exist. In this section it will be assumed, however, that the combined operation of charge conjugation and parity transformation, ~ / 5 is conserved in weak interactions. 4 We define the effect of ~/5 on K ~ a n d / ~ o to be 5

~PlK~ = Ik~

~Plk~ = Ig~

(1.7)

i.e., IK~

I/~ ~ # C P eigenstates

In accordance with the above assumption, we form the following linear, orthonormal combinations of IK~ and [/7/0), which are CP eigenstates: 1

IK~ _-- ~ [IK~ + IK~

CPlK~ = +IK~ (1.s)

1

IKO)=- ~ [[K~ -[K~

C/3[K~ _- _IKO).

Thus by assuming that CP is conserved in weak interactions, we infer that there is a particle that can decay only into a state with CP = +1 ( K ~ and a particle which decays only into a CP = - 1 state (K~ According to (1.4), neutral kaons typically decay into two or three pions, i.e. into states with parity +1 and - 1 , respectively (both states have C = +1; see Appendix A). Therefore, K ~ is expected to decay only into two pions, whereas K ~ must decay only into three pions: K ~ -+ 2~,

K ~ ~ 3~r.

(1.9)

In the 27~ decay, there is 215 MeV of kinetic energy available for the pions (mk ,-~ 500 MeV, m . ~ 140 MeV); in the 37r decay only 78 MeV. The small 37r phase space makes the lifetime of K ~ much longer than that of K ~ (see (1.12) and [21]). 4 C and P are strictly conserved in strong and electromagnetic interactions. CP violation in weak decays of neutral kaons will be discussed later on. 5 There is an arbitrary phase associated with this definition: ~ P l K ~ - ei~176 ) ~

&Plk ~

_- ~ P (e-i~176

= e-i~176 ).

We choose Ocp =O.The relative phase of K ~ and /~o is not fixed by the strong and electromagnetic interactions, since these states do not couple except through very feeble second-order weak transitions.

1.2 C P Eigenstates of Neutral Kaons: K ~ and K ~ We can invert equations (1.8) to express [K ~ and positions of [K ~ and [K ~ states:

I/~~

7

as linear super-

1

IK~ - - ~ fig~ + Ig~ Ig~ _=

1

(1.1o)

[IKl~ -IK~

Now, if a b e a m of K ~ particles is produced at t = 0, then at a later time t > T1 (T1 iS the lifetime of g l ~ only K ~ mesons will survive:

0 IK~ c~ Ig~ + IK~ --+ IK 25,

(1.11)

t >> ~1.

Near the beam production point one would thus see mainly 27r decays, and farther down the beamline only 3~ decays. There is a nice optical analogy to (1.11): if we were to shine a linearly polarized b e a m of light on a crystal t h a t absorbs preferentially left-circularly polarized light, the beam would become increasingly right-circularly polarized as it passes through the crystal, just as a K ~ b e a m decays into a K ~ beam. The remarkable prediction (1.11) was originally made in 1955 by M. GellMann and A. Pals in a paper famed for the sheer beauty of their reasoning [3].6 The first experimental confirmation was swift: in 1956, L. Lederman and his collaborators discovered the K ~ meson at Brookhaven in a cloud chamber 7 placed sufficiently far from the beam production point to allow all K ~ mesons and A ~ particles to decay before reaching the chamber [4] (see Fig. 1.2). Subsequent experiments, which could detect both the 7r+Tr- and the 27r~ decays, showed that about a half of all originally produced K ~ mesons decayed by these two modes.

& !////1 Collimator ~//.,~ CU Target

Magnetic field

~ ~ X 7#

Protons/ ~

xX

/ ~/ -~--~

/

/

CloudChamber

Fig. 1.2. The experimental setup used by K. Lande et al. [4, 5] 6 This work was published before the discovery of parity violation. Nevertheless, the essence of their argument remains the same when C is replaced by C/5 7 This was one of the last cloud chamber experiments.

8

1. Introduction The measured lifetimes of the two CP eigenstates K ~ and K ~ are 7-1 = 0 . 8 9 •

-l~

T2=5.17•

-ss,

CT1----2.7cm

(1.12)

c~-2=15.5m

The data for K ~ a n d / ~ o decays are mutually consistent in the sense that the lifetimes and decay modes for the K ~ and K ~ components are the same.

1.3 Duality

of Neutral

Kaons:

(K~

~

vs. (K~

~

Unlike K ~ and /~o, the CP eigenstates K ~ and K ~ are not a particleantiparticle pair: each is its own antiparticle and each has a unique lifetime; hence their masses may also differ. In contrast, K ~ and /~o have the same mass but do not have a unique lifetime, for they decay faster into two pious than into three pious. According to (1.8) and (1.10), 9

K~

~ is an equal mixture of K ~ a n d K ~

9 K~

~ is an equal mixture of K ~ a n d / ~ o

The K ~ system, therefore, is described by two related sets of particles: (K ~176

>(K ~176

Which set of particles is observed depends on the nature of the measurement: o K1, K 2o are detected through their decays into states with definite CP parity (2~ or 3~), and K ~ through mutually distinct semileptonic decays K ~ -+ e+Tr-u~ a n d / ~ o __+ e - T r + ~ , which obey the empirical rule AStrangeness = ACharge. To summarize:

9 are distinguished through production (in strong interactions) and through semileptonic weak decays, 9 are eigenstates of strangeness, K o, ~7o { 9 have no unique lifetime, 9 have the same mass, 9 are linear superpositions of K~ and K~

~176

K1, K2

9 are distinguished through pionic weak decays, 9 are eigenstates of CP, 9 each has a unique lifetime, 9 have different mass, 9 are linear superpositions of K ~ and fifo.

1.4 The Einstein-Podolsky-Rosen Paradox in the K ~ System 1.4 The in the

Einstein-Podolsky-Rosen

9

Paradox

K ~ System

The decays of two-kaon states provide a unique insight into the dynamic behavior of q u a n t u m systems over macroscopic distances, and thus into the difference between classical and q u a n t u m concepts of measurement. This difference lies at the heart of the famous Einstein Podolsky Rosen "paradox". The classical concept is outlined in the following remark by Albert Einstein: "The real factual situation of the system $2 is independent of what is done with the system $1, which is spatially separated from the former." This is in sharp contrast with the quantum-mechanical interpretation, according to which a measurement on what appears to be a part of the system is in fact a measurement on the whole system. To understand the meaning of the latter statement, suppose that two neutral kaons are emitted in the + z and - z directions from the decay at rest of an odd eigenstate of C (the ~5 meson):

14) )

) [KO[~O),

y~C = 1--.

The two kaons emitted simultaneously are always in opposite states: one is K ~ and the other/~0, or one is K ~ and the other K ~ The reason is t h a t the initial state has odd parity ( P = - 1 ) , while Bose statistics forbids odd parity states for two identical spinless bosons. Note also that ~5 has zero strangeness, and t h a t strangeness is conserved in the strong decay ~5 -+ K ~ ~ Now, consider a coincidence measurement of the decays of kaons emitted in opposite directions. If we detect the pionic decay K ~ -+ 27r of the kaon emitted in the + z direction, then its companion in the - z direction must be a K2~ However, if we detect the semileptonic decay K ~ -+ e+Tr-v~ at +z, then the kaon at - z is definitely a / ~ 0 . Even when the two kaons are very far apart with no possibility of interacting, we can still "select" the state of the kaon at - z based on what we choose to measure at +z. In q u a n t u m mechanics, all relevant information about a system is contained in its wave function. When two systems have wave functions which differ at most by a constant phase factor, they are considered to be in the same quantum state, which is a linear superposition of the individual state vectors. We can thus write 14~) = ~ :

1

{ [K~176

- [fi[~176

}

1

for the two-kaon state before any decays have taken place. Since the initial state has C = - 1 , charge conjugation invariance in the strong decay of the r meson requires the negative sign between the two terms in the above expression. This description clearly includes quantum-mechanical interference between states which are spatially separated (see Sect. 7.1). Note that the

10

1. Introduction

above state vector is formally analogous to that of a spin-singlet state composed of two s p i n - l / 2 particles: IS ----0> = [ ( ~ ) - ( $ t ) ] / v %

1.5 S t r a n g e n e s s

Oscillations

"Especially interesting is the fact that we have taken the principle of superposition to its ultimately logical conclusion."

Richard Feynman, The Theory of Fundamental Processes Suppose t h a t at time t = 0 a pure K ~ b e a m is generated and then allowed to propagate in vacuum. After a sufficiently long period of time only the K ~ component will survive. This component is a superposition of the strangeness eigenstates K ~ a n d / ~ o :

IK~

> IK~ =

1 [ig0 ) _ iK0)],

t >> T,.

(1.13)

Thus starting out as a pure K ~ beam, the neutral kaon will evolve into a state of mixed strangeness. This is known as the K ~ ~ or strangeness, oscillation. The first evidence for this effect was reported by K. Lande et al. in 1957 [5], who employed the same experimental set-up as was used to discover K ~ (see Fig. 1.2). T h e y observed the process KO + 7-le --+ Z - p p n T r +

by using a b e a m of neutral kaons produced predominantly through reactions p+n--+p+

A~ + K ~

Since the energy threshold for the above interaction is much lower than for p + n ---+p + n + K ~ + [(o

the b e a m overwhelmingly contained K ~ mesons. The strangeness oscillation is a purely quantum-mechanical effect that enables one to test the principle of superposition of amplitudes in the most direct way. To learn more about this effect we first have to determine how the states of neutral kaons change in time. The C P eigenstates K ~ and K ~ have different lifetimes and decay modes, and hence different weak couplings. Consequently, their masses also ought to differ, just as the mass difference betwen proton and neutron can be attributed to their different electromagnetic couplings. A quantum-mechanical state (a particle) is described by a Schr5dinger wave function, which is in general a complex number. The interference between two q u a n t u m states is determined by the difference in the phases of 0 their wave functions. As we will see shortly, a K 1-K 20 mass difference causes the relative phase of K ~ and K ~ states to vary in time. It is precisely this

1.5 Strangeness Oscillations

11

time variation of the relative phase of the two-state s y s t e m /\ K ~i ~ K 27 ~ that we want to study. Suppose that at time t -- 0 the state I~} of a neutral knon is pure K ~ A stable particle propagating in vacuum can be described by a plane wave (~ = c = 1): I ~ ( z , t)) = e i ( k z - E t ) I~(0)),

(1.14)

where E is the energy of the particle and k its wave number. Since K ~ is not stable, we expect that at a later time the probability of finding the particle in the state I~'(t = 0)} should decrease by a factor e -t/~ because of the exponential decay law for K1~ -+ ~-. This probability is given by Probability(t) = IAmplitude(t)[ 2 -- I{K~ I ~P(t)}]2 = e -t/~L.

(1.15)

Therefore, Amplitude(t) = (K ~ I ~P(t)} c( e -t/2~-~

(1.16)

and ] ~ ( t ) > ~- e - i E l t

e -t/2T'

[K~ = 0)}.

(1.17)

There is a similar expression for K ~ If t is measured in the paricle rest frame, then E = m, ~- =- 1/F is the proper lifetime and

[~,,2(t))

= e-(iml'2+r"2/~)tpIK~,2(t = 0)),

where tp is the proper time. An initially pure K~ ]~'(t)}---- ~

1

(1.18)

propagating in vacuum is thus described by

[e-(iml+r'/2)t'lK~ } + e-(im2+r2/2)t'lK~

(1.19)

The amplitude of the probability that the K-meson state ]~(t)) is a K ~ at time t reads A = {K~ I ~P(t)) = ~1[er

+ er

,

(1.20)

where

r = - ( i r a + F/2)tp

(1.21)

and F is the decay rate. Similarly, the probability amplitude for/~o is

= (/~~ I ~,(t)) = 1 [er _ er

(1.22)

The corresponding probabilities are P = ~1 [ e - F ' t + e_F2 t + 2e -(F~+F2)t/2 cos(Amkt) ]

(1.23)

p : 41 [e_r, , + e -r2t - 2e -
(1.24)

and

12

1. Introduction

where A m k = Ira1 - m21

(1.25)

is the K 1-K~ 0 0 mass difference. The K ~ and/7/0 intensities oscillate with the frequency Amk, as shown in Fig. 1.3. When a K ~ meson is created, the probability that it is a/720 is zero: P(to) -- 1 and P(to) = 0. As the K ~ component decays away, the original K ~ evolves into a state of mixed strangeness (see (1.13)). Each curve in Fig. 1.3 is the result of strangeness oscillation, associated with the cosine-term in (1.23), (1.24), superposed over exponential damping due to F ~ 0. The cosine term reflects quantum-mechanical interference between the K ~ and K ~ amplitudes of the particle over macroscopic distances! l.O

0.8

Ko 0.6

0.4

0.2

I

I

I

I

I

2

4

8

10

12

• 10 -1~ sec

Fig. 1.3. Oscillations of K ~ and k ~ intensities for an initially pure K ~ beam assuming Amk = 1"1/2 0 Pioneering work on strangeness oscillations and the K1-K 20 mass difference [6] reported a result in qualitative agreement with Amk = h/c2T1. This experiment produced K ~ mesons by a low-energy 7r- beam, and then detected /~0 particles via hyperon production in a cloud chamber. Again there is a nice optical analogy to this phenomenon: if we were to shine right-circularly polarized light through a crystal that absorbs predominantly x-polarized light, after a short distance there would be a large left-circularly polarized component. Both phenomena are based on the principle of superposition of amplitudes, which in q u a n t u m mechanics holds for "probability waves" of particles; these (complex) probability amplitudes can interfere constructively or destructively, just like waves do.

1.6 The K1-K2 o o Mass Difference 1.6 The

13

K 1o- K 2o Mass Difference

In the absence of the weak interaction, K ~ and /7/~ are two unconnected, degenerate (mko = mTr ) states:

=
(1.26)

In deriving the above result we used the fact that the hamiltonian /:/s+em commutes with S (strangeness is conserved in strong and electromagnetic interactions). In the particle rest frame, the masses of K ~ and/7/0 can be expressed as expectation values of/~s+em over the states IK ~ and IK~ mk~ = ( KO I /-~/s+em I K ~ = (/~o I /~s+em

I /~0) : m~o

(1.27)

The weak interaction, /:/w, is very feeble compared with /:/~+~m, and can thus be treated as a small perturbation. As before, we will assume that/:/w is strictly invariant under the combined operation of charge conjugation and parity transformation. The weak interaction connects K ~ and/7/0 (they are not eigenstates of the perturbed problem), but not the CP eigenstates K ~ and K~ (K~ I/:/w [ g ~ = 0 = (K ~ I/:/w I K~ 9

(1.28)

Note also that when a degeneracy exists, any linear combination of degenerate eigenstates is itself an eigenstate:

/~/s+eml~} = f/s+em [aIK ~

+ bl/~~

= mkol~'>,

where I~> is the unperturbed state. We can consequently apply ordinary degenerate perturbation theory to the K ~ system. S{nce neutral lmons decay through a number of channels, our Hilbert space should, in principle, be expanded to include all possible transitions. However, we keep the analysis simple by restricting ourselves to the two-dimensional Hilbert space spanned by [K ~ and 1/~o>, in which case the effect of decays is incorporated into an effective hamiltonian

He/= Hw+ Z

/:/w In> (n I/:/w

+

(1.29)

n ^ eft" H~ is determined by the virtual transitions to all intermediate states n outside the two-particle subspace. Now,

[f/sq-em -~-/:/ewff]I~> ~-[mko ~-m'] Ik~>

(1.30)

14

1. Introduction

where a/:/~fr [ g ~ + b/:/ef t/f ~ = m '

[alK ~ + bl/;;~

(1.31)

is a small p e r t u r b a t i o n due to the weak interaction. Taking the inner p r o d u c t of (1.31) with (K~ and then (/<~ we obtain a s y s t e m of two coupled linear equations:

a(KO I H,~ A f I K~ + b(K~ I ^e~

I K~

(1.32)

a(k~ I H$ ]K~ + b(k~ ]/:/~f [ K~ = bm'. Since/?/ewffand ~/5 commute,

I K~

H ~I dP/rf I k ~ k~ wfdP I k ~ =
(1.33)

and ( K0

I ^ eft

H~ I g ~

=

^ eft C. P H~,

.

I R~.

=. ( K. o I C P H weft

= (/~o I/:/,~ I/~o) = m.

I /~0) (1.34)

Expressed in m a t r i x form, (1.32) now reads Am

\(m-m'm-m')

(1.35)

To find nontrivial solutions of (1.35) we set the d e t e r m i n a n t of the coefficients of a and b equal to zero, with the result m ' = m :i: A m . Inserting each of the values of m ' into (1,35) yields

a/b = 1, a/b = - 1 ,

m' = m + A m , m' = m - A m .

(1.36)

To ensure t h a t the new ]~P) is normalized (]al 2 + ]bl2 = 1), we set a = 1 / v ~ a n d b -- • Therefore,

,k~)new = a,K~ + b,K~ = ( 'K~

,

(1.37)

where K1~ and K ~ are the correct linear superpositions of the original eigenfunctions which diagonalize the perturbation. T h e "weak" masses of the two nondegenerate levels K ~ and K ~ are given by s ( K 0[ H,~ ^ee~ I K~

-- ~ 1 [(K~ + (/(~ = m + Am

^ e f t IlK 0 ) + [ K 0 ) ] Hw

(1.38)

s Note that ml = m - A m and m2 = m + A m if we choose 0cp -- zr (see footnote 5). Experimentally, m2 > ml, as shown in Sect. 2.3.

1.6 The K1-K2 o o Mass Difference

J

m+ z~m

"~"

m- txm

f.1

m'~

Hw

J

K~

Hs + Hem

15

The KO_/~ o mixing results in level splitting

Fig. 1 .4.

and (KOl

Ae~ H7 IK~ =

[
----m - Am

(1.39)

The above perturbation calculation can be summarized as shown in Fig. 1.4. The weak interaction shifts mko by m and splits the degenerate levels by an amount 2Am - Amk. The two new levels with definite CP parity, K ~ and K ~ differ in mass:

Amk--Iml-m21 = .

(1.40)

This mass difference is due to strangeness-changing K ~ ~ transitions. We can m a k e a rather accurate e s t i m a t e of A m k based on the Heisenberg uncertainty relation, A E A t = 1 (h = c = 1), by setting At ---- 1/(F~ + F~) ~ 1 / 5 and A E = A m k (cf. Sect. 9.1): A m k ~ F1 ~ 10 l~ s -1 ~ 7 • 10 -6 eV.

(1.41)

T h e K Io- K 2o mass difference can be o b t a i n e d by counting the n u m b e r of /~~ events as a function of their distance from the K ~ p r o d u c t i o n point (see (1.24)). In the e x p e r i m e n t [7], for example, a p r o p a n e bubble chamber was used to detect the following sequence of events: charge--exchange

"K+ + n-+ p + K ~, K ~ --+ KO -_ K ~ + [~~ /7/o + p _+ A 0 + 7r+.

(1.42)

T h e m e a s u r e d value of A m k is indeed tiny:

Amk=3.5x10

-6eV~F1/2

or

Amk mk

--0.7•

10 -14.

(1.43)

T h e smallness of the observed K Io- K 2o mass difference indicates t h a t ( K ~ I /2/w I /~o> = (/7(o I /2/w I K~ = 0, thus confirming t h a t first-order weak interactions obey the empirical rule IASI <_ 1. A much more precise way of measuring A m k is based on the K ~ regeneration phenomenon, a n o t h e r spectacular q u a n t u m - m e c h a n i c a l effect associated with the K ~ s y s t e m t h a t we describe next in some detail.

2. Propagation of N e u t r a l Kaons in Matter

2.1 The

K ~ Regeneration

"I believe that this concept of probability amplitude is perhaps the most fundamental concept of quantum theory." P.A.M. Dirac, Relativity and Quantum Theory This phenomenon - - the conversion of K ~ mesons into K ~ mesons is of the same nature as the K ~ ~ oscillation. Its existence was predicted by A. Pals and O. Piccioni in 1955 [8], a year before the K ~ particle was discovered! A detailed analysis of the process was presented in the seminal paper by M.L. Good [8]. Suppose t h a t a pure K ~ b e a m is allowed to decay in vacuum until we are left with only the K ~ component. If we then pass this K ~ beam through a thin slab of material where it can interact, the strong interactions will select both eigenstates of strangeness present in the beam. T h e / ~ o component will be more strongly absorbed than the K ~ component because K ~ mesons can only scatter elastically or through charge-exchange reactions, whereas /~o mesons can be both scattered and absorbed (see (1.2),(1.3)). After the slab, the b e a m composition can be expressed as

1 [alKO) _ biKO>] _ a + b []KO ) _ i/~0)] + a - b []KO ) + ]/~o)] 1 (a + b)lK ~ + 1 (a - b)lK~

(2.1)

Since a r b, we conclude that after all the K ~ mesons in the initial b e a m have decayed away, some can be regenerated by passing the pure K ~ b e a m through matter. Figure 2.1 contains a schematic drawing of such an experiment. Like the K ~ ~ oscillation, the K ~ regeneration phenomenon is a direct consequence of the principle of superposition of amplitudes in quantum mechanics. The analogous experiment with polarized light is shown in Fig. 2.2. This analogy follows from the wave aspect of q u a n t u m theory. There is another close analogy to the K ~ regeneration which beautifully illustrates the concept of quantization. It is based on the Stern-Gerlach atomic

18

2. Propagation of Neutral Kaons in Matter

~_Kt + K2 o

Negative pion [-] beam

K2

o

Proton target

O

0

K2 + K1

Regenerator

Fig. 2.1. Schematic drawing of a K ~ regeneration experiment

K0absorbed Re! enerator Polarized

f~Rotator

~r~

K0

K2

K~I decays

Rotator

o

Ko

K01 decays

K20

Fig. 2.2. An experiment with polarized light analogous to K ~ regeneration beam experiment, i.e., on the fact that it is impossible to quantize the spin components of the beam along two orthogonal axis simultaneously. As shown in Fig. 2.3, an unpolarized atomic beam of spin 1/2 propagating in the z direction enters an inhomogeneous magnetic field that points along the y axis (Hy). This causes the beam to split equally into two components, one deflected upward and the other downward. The two components correspond to the atoms quantized by the field in the spin eigenstates ay = + 1 / 2 and O'y ~ - 1 / 2 , respectively. If the "lower" component is then sent through an inhomogeneous magnetic field pointing in the direction perpendicular to the y axis (the x direction), it will again split into two equal components, one deflected in the x direction (a= = + 1 / 2 ) and the other in the opposite

unpolarized r spin 1/2 / a t o ~

r~ absorbed

. protontarget.~f ~ ~ 1 7 6 I //~/ .~,~

/.x

"y

If' ~ Kl~decays f ~

~,oabsorbed

K~~,,

Hy

regenerator,,

Fig. 2.3. Analogy between the Stern-Cerlach and K ~ regeneration experiments

2.1 The K ~ Regeneration

]9

direction (ax = - 1 / 2 ) . We now repeat the above two steps starting with the ax -- - 1 / 2 component, and assume that after each beam-splitting the a~,y = + 1 / 2 component is absorbed in matter. The analogy between this and the K1~ regeneration experiment is as follows. Each time the atomic beam passes through Hy corresponds to the selection of strangeness eigenstates K ~ a n d / ~ 0 via strong interactions in the proton target or the regenerator. The beam splitting in /Ix is analogous to the selection of CP eigenstates K ~ and K ~ through weak decays. Note that once the beam passes through an inhomogeneous magnetic field in, say, the y direction, all preexisting information about quantization along the x axis is lost (the axis singled out in space is defined by the magnetic field). This means that it is impossible to quantize the spin components of the beam along two orthogonal axes; in other words, the spin operators &x and ~y do not commute. Similarly, the operators S and C/~ also do not commute; hence K ~ states are eigenstates of either S or C/~, but not both. The first experimental confirmation of the regeneration phenomenon, and o 20 mass difference, was that by O. Picalso the first measurement of the K 1-K cioni and his collaborators [9]. To produce K ~ mesons, they passed negative pions through a liquid-hydrogen target (see (1.1)), and then allowed the K1~ component of the K ~ beam to decay away. About 200 K1~ mesons were regenerated in a 30-inch propane bubble chamber fitted with lead and iron plates (see Fig. 2.4), from a beam of approximately 105 K~) particles. A particularly elegant way of measuring Amk is based on the variable gap method. Refer to Fig. 2.5. A K ~ beam passes through two thin slabs of material separated by distance g, which can be varied. Let us assume for the sake of simplicity that t h e / ~ 0 component is totally absorbed in each of the two slabs. After exiting from the first slab (t = 0), the beam is pure K ~

Incident Kz beam

lllllll -W.,~~

30"

J I

,,^~, K,meson !/

", ,~+

I \/ i /

T 19"

Fig. 2.4. A regenerated K ~ decays in the experiment by R. Good et al. [9]

20

2. Propagation of Neutral Kaons in Matter ~-

g

Absorber

Regenerator

Fig. 2.5. A sketch of the variable gap method Just before it enters the second slab (regenerator), the beam is described by (1.19), where tp = t 9 = t / 7 = g/e'~ = g / [ ( p k / E k ) ( E k / m k ) ] = g ( m k / p k ) is the proper time to traverse g. Immediately after the regenerator (t~ >> tslab), where t h e / ~ 0 component is totally absorbed, the K ~ amplitude reads (see (1.20)) AKo(t~) = (K~ I ~P(tg)) = ~1( K ~ with r

I ~(t~)) = ~ 1

[e~l + e ~2]

(2.2)

given by (1.21). The corresponding decay intensity is (F1 >>/"2)

I(g) _= AK o~ ~-4-1(0) [1 + e -rltg + 2e -rltg/2 cos(Amktg)] ,

(2.3)

where I(0) = I < K ~

2 =

1

(2.4)

The mass difference Amk can be obtained by measuring the K1~ -+ 7r+Trdecay intensity after the regenerator as a function of g. This is, of course, a very simplified description of the method (see [10] regarding the original variable gap experiment).

2.2 Coherent

Regeneration

Amplitude

We have seen that regeneration occurs because K ~ and/~0 mesons interact differently with matter. Let us rewrite (2.1) as ]k~after)

_

_

a + b []KO ) + t~lK10)] 2

'

t) ~

a-b a+b'

(2.5)

where 0, the regeneration parameter, is a complex number that can be related to the physical properties of the regenerator, as will be shown in what follows. The regenerated K ~ mesons will coincide in direction and momentum with the incident K ~ beam whenever the forward scattering amplitudes for

2.2 Coherent Regeneration Amplitude

21

K ~ and /~0 are different. This situation is called coherent regeneration, or regeneration by transmission, and is a result of the constructive contribution from all scatterers in the target. The coherence is preserved in a region that is typically contained within a micro-radian. As in the case of forward scattering of neutrons, light, X-rays, etc., the forward scattering of neutral kaons is always coherent, and its interference with the incoming beam gives rise to the refractive index of the scattering material (see Appendix B): 2rN n = 1 + - - ~ - f ( w , 0),

(2.6)

where N is the number of scatterers per unit volume, k the wave number of the incident particles and f(w, 0) the scattering amplitude in the forward direction (0 = 0). The forward amplitude is related to the total cross-section by the optical theorem 47r O'tot = T ]m ~(w, 0).

(2.7)

A particle traversing a slab of thickness z picks up an extra phase, which is proportional to the refractive index n: = k(n - 1)z.

(2.8)

Not only are the K ~ a n d / ~ 0 mesons absorbed differently in matter, but their elastic scattering amplitudes also differ, just as those for K + p --4 K + p and K - p --4 K - p do. Since K ~ and /2(0 have different total cross-sections, they must also have different indices of refraction; hence they acquire unequal phase shifts while propagating through matter. To see what this implies, we first write down the time development of K ~ a n d / ~ 0 states in vacuum (see (1.19)): = ~1 [ IK~

Ig~

=

1

er

-~-[K0) er 2] ,

[igO ) er 1 _

[KO) er

'

(2.9)

where r

=

-

(i- 1,2 + rl,2/2) t,

(2.10)

and tp = t / 7 = t x / 1 - v 2 is the proper time: z = vt = t p V / V ~ - v 2. The two-state SchrSdinger equation describing the system (2.9) reads 9 1

-

-

dtp

- -

~m - i6F/2

rh

iF~2 ] ~Pvac,

(2.tl)

9 The K ~ system is described by two coupled differential equations because of the K 0 ++ /~0 mixing. If neutral kaons interacted only through strong and electromagnetic interactions, both of which conserve S, there would be no transitions between the K ~ and/~o states, and each state would be separately described by a SchrSdinger equation. In fact, (2.11) is not a "real" SchrSdinger equation.

22

2. Propagation of Neutral Kaons in Matter

where ~va~ -- (IK~

IR~

~

(2.12)

)

and

_ ml + m2 /~ _ I"1 + ['2 ~m -- ml -- m2 5F -- F I _ _ -F2. - (2.13) 2 ' 2 ' 2 ' 2 While traversing a distance z in matter, the K~ extra phase: IK~

~ IK~

ir

IK~

~ meson acquires an

--+ I/7/~

(2.14)

i•,

where = k(n

27cN v

-

1)z- k~tpf(0), (2.15) 27rN v

r = k ( f i - 1)z - k lx/]_:~_ v2 tpf(0). It is straightforward to show that (2.11) now becomes . d~matt 1 - -

dtp

( ~ t - i/~/2 - k ~ l ( 0 ) =

5m-iSF/2

5m - i ~ r / 2

rh - i/~/2

27rgv

-

k~_~f(0) ]

k~mat t.

(2.16)

The matrix equation (2.16) may be expressed as 'matt a9d ~dt-----~

__

(M - iF)

2~rNv

~matt

k 1~/~_ v 2

(~(:)

0 ) k~matt

7(0)

(2.17)

or

i dk~matt -~p - ( M ' - i F ' )

(2.18)

~matt,

with

ir

ir

,2,9,

and -

2~Nv

(2.20)

kx/1 - v 2"

The time development of the K ~ system in vacuum and in matter is therefore described by the SchrSdinger equations 9d~vac 1 - H~wc dtp

and

dk~mat t

i--

dtp

'

- - H ~[tmatt ,

(2.21)

2.2 Coherent Regeneration Amplitude

23

respectively, with H_--M-iF,

H'__M'-iF',

(2.22)

where M and F are 2 x 2 matrices. 1~ M is called the mass matrix and F the decay matri3:, both are hermitian (M~k = M k j , F~k = Fkj) because they represent observable quantities. However, H and H' are not hermitian, otherwise K ~ and/~0 would not decay. To understand this statement note that the decay process is the equivalent of an absorption, and that an absorbing medium can be described in terms of a complex index of refraction, the imaginary part of which is associated with attenuation (see Appendix B). We may now proceed with our main task, which is to obtain the intensity of the regenerated K ~ component at z. To this end we express ~ m a t t in (2.16) in terms of the C P eigenstates K ~ and K ~ and write the result down as a system of two coupled differential equations (IK~ = K~ i d

:

(,h

-

i/~/2

-

~f)

+ (Sin - i 6 r / 2 )

i ~dp [K~

-

K o] =

K2o]

+

[K ~ -

K~

(2.23)

(sin -i~r/2) [K ~ + K ~ + (~h - i / ~ / 2 - ~ ) [K1~ - K ~

Adding and subtracting equations (2.23) yields .dP1 2 dtp

0)

( m l - iF1/2 m2-iF2/2 0

(

f+

P1,2 - z

~'~-f--i)~t12'(2.24)

with P1,2 = ~ KO ] .

(2.25)

We see that d'Pl,2/dtp has two components: the first one describes the propagation of free particles, and the second one the scattering with nuclei inside the regenerator. The change of P1,2 with respect to z as the beam passes through matter may thus be expressed as

d 12 _ [d 12]

dz

L-h-~z'-Jvac+L dz

J~c,

(2.26)

The second term in (2.26) is readily obtained from (2.24) by setting dz = v2:

v d t = v'ydtp = v d t p / V ~ -

lo Since we have restricted ourselves to a two-dimensional Hilbert space, H and H' are effective hamiltonians: H = Hstrong + Helect...... gnetic +/-/weak. From (2.22) it follows that M = (H + Ht)/2 and F = i(H - Ht).

24

2. Propagation of Neutral Kaons in Matter

- ~ z ].uc ~

k

\f-~

f+

'

Regarding the first term, recall that an unstable particle propagating in vacuum may be described by a plane wave

~P(z) = eikz-zl2Ao(o), where we used Therefore,

(2.28)

Ftp/2 = z/2/37v = z/2A (A - 137T is the decay length).

--~-z'-J vac =

0

ik2 -

1/2A2 ~1,2.

Expressions (2.27) and (2.29) form a system of two coupled differential equations:

dK~ [ikl _ l/2Al + ilrN ] dz = --~1 (f+~) K~

iTcN

(f-~)K~

- CK~ + :DK~

(2.30)

d K ~ _ [ ik2 - 1/2A2 + i~rN (f + ~) 1 n ~ + brN (~ - ~) K ~ dz ~ - . 4 K o + t~K ~

The coefficients A, B, C and :D in (2.30) represent the rates of change of the K ~ and K ~ amplitudes due to propagation, decay and (coherent) scattering. The coupled differential equations (2.30) can be formally solved for the z dependence of the amplitudes. However, the solution is simplified, and underlying physics made more transparent, by noting that K2~ >> K ~ and/"1 >> F2. We can thus neglect the contribution of K ~ to the change in K ~ and write the second equation in (2.30) as

~ ik2-1/2A2+~(f+~)

dz.

(2.31)

This yields lnKO(z)

inN

A2_.~zik2z + ~2 (~ +9) z + In K~

(2.32)

i.e.

KO(z) = eia~z e - N f .... /2 KO(0),

(2.33)

where we defined 11 11 Writing r

0) ---- drtot(K O) = [(Ttot(K 0) + atot ( / ~ ~ ---- 47r[Im I + Im~]/2k = -4~_i[iImJ + iIm~]/2k we could identify ftot with O'tot provided iImf = f and iIm I = I, i.e. the forward scattering amplitudes were purely imaginary (the beam was only attenuated). In general this is not the case because of refraction.

2.2 Coherent Regeneration Amplitude 47ri ~+~ k 2 Inserting (2.33) into the first equation (2.30) gives

(2.34)

ftot ~

dKl~ ~zz

[ -- i k l - 1 / 2 A l + iTrN ~- ~ - 1 (~

-

25

brN ] ~-1 ( f + ~ ) K~

~) eik2z e-gft~

g~

(2.35)

dz

Fig. 2.6. Schematic drawing of a solid regenerator of thickness l A K ~ travels as a K ~ with a wave number k2 until it is regenerated, and after that as a K ~ with a wave number kj. Since Amk is so tiny, kl ~ k2. Note also that Avac ~ Amatt. The first term on the right-hand side of (2.35) describes the behavior of K ~ mesons, created before z, in the interval z, z+dz. The second term gives the amplitude of K ~ particles created between z and z + dz (see Fig. 2.6). To see how this amplitude is modified at the end of the regenerator due to propagation, decay and coherent scattering, we integrate the first term in the interval l - z, where l is the thickness of the regenerator, with the result (2.36)

(~(l -- Z) = e ikl (l-z) e-(l-z)/2A~ e - N f t o t ( l - z ) / 2 .

Since we are dealing with coherent scattering, the amplitudes of K1~ mesons created in all dz intervals have to be added. This gives the following K ~ amplitude at the end of the regenerator: /q0q) = --

cq - z)dV(z) brN

X

(f-f)/q~176

f

e ikl(/-z) e -(l-z)/2A~ e -NIt~

f

e i(kl-k2)(l-z) e -(l-z)/2A1 d z

e ik2z e -N/t~

dz

brN

•

-

-

i~N kl ( ~ - ~ ) K 2 (o/ ) fo z e(l-z)[i(k*-k~)-l/2A1]dz

(2.37)

26

2. Propagation of Neutral Kaons in Matter

with g~

=- e -Nft~

e ik2/g~

= 0).

(2.38)

The simple integration above results in KO(l ) _

iNA1Alfr [1 --e -(-ihm+l/2)~] K ~ - i h m + 1/2

-

ocK~

(2.39)

where fr = ( ~ - 7 ) / 2 ,

A1 - 27r/kl,

~ =-l/A1,

(2.40)

and

6m-- (m2

ml)c2

h/'Q

-- (kl - k2) A1

(2.41)

is the K ~ ~ mass difference expressed in units of the K1~ mass width h/T1. Expression (2.41) requires a few words of explanation. As discussed in Appendix B, for coherent scattering the regenerator as a whole absorbs the difference in m o m e n t u m Pl - P~ = h(kt - k2) = h A k and recoils with the m o m e n t u m - h A k , taking away from the meson the energy (hAk)2/2Mreg. Since Mreg is very large, this energy is negligible compared with m l - m2. Hence E2 = El, which we express as (h, c ~ 1) h

-

:

-

c 2

i.e. kt - k2 m2 - m l . . . . kh 2, me 2 where k - (kl + ka)/2 and m - (ml + m 2 ) / 2 can be taken as the m o m e n t u m and mass of either kaon. Expression (2.41) follows from the above result if we note that k (like [) is determined in the rest-frame of the scatterers: k = p/h = flE/h = fl~m/h. Equation (2.39) defines the coherent regeneration amplitude Oc. The intensity of K ~ mesons emerging from the regenerator is thus given by

IK~

-

(NA1AI)252 + 1/4 Ifrl2 [1 + e -(

--

loci 2

tKO(1).

-- 2e -~/2 COS(~m~)] IK~

-ms o ' (2.42)

Note that expression (2.42) has the same oscillatory form as that for the K~ ~ intensity (see (1.23), (1.24)). This is to be expected since in both cases oscillations arise from the interference of the two eigenstates of the hamiltonian, which have the same masses and lifetimes in the two cases. The above results are summarized in Fig. 2.7. The magnitude of 6m can be determined by measuring 0c as a flmction of regenerator thickness. This method is relatively simple and does not require a knowledge of ft. Using an iron regenerator, a spark chamber experiment

2.2 Coherent Regeneration Amplitude

27

IK2> + .~1 KO 1>

I K2>

Fig. 2.7. K ~ beam composition before and immediately after a regenerator

[11] obtained 5 m = 0.82 • 0.12 by plotting the K ~ intensity as a function of the iron thickness I (see Fig. 2.8). Since IKo(/) ~ e -N/t~

(2.43)

----e - l / t L

they measured the nuclear mean path, #, in a separate attenuation experiment.

I 900

I

I

I

I

8,1.5

9 9

800

9

RUN I RUN ~ RUN m

700 =0.8

600 0

m

500

Z

400

~=o

300

ZOO

100

o~

i

t

I

I

0

5

I0

15

20

THICKNESS OF IRON, r

Fig. 2.8. The magnitude of Amk measured by [11]

25

|

28

2. Propagation of Neutral Kaons in Matter

2 . 3 K x -0K

10 I n t e r f e r e n c e

and the

Sign of Amk

We next describe a regeneration experiment which determined the magnitude, as well as the sign, of the K ~ ~ mass difference A m k . The measurement was based on the observation of the interference between the regenerated (reg) and originally produced (orig) K ~ mesons, which proves unequivocally that the states Kl~ and K~ are quantum-mechanically identical [12]. Consider a pure K ~ beam impinging on a regenerator of thickness I placed at a distance d from the production point. Both l and d can be so adjusted that the two interfering waves, g ~(orig) and K ~(reg), have comparable amplitudes, thus maximizing the interference effect. At the exit of the regenerator

~after :/C~

+ K:l~

(2.44)

Since {K~ = ~

1

[tK ~ § IK~

(2.45)

> K:~ -- 0) = K:~ -- 0),

the above two amplitudes read (see (2.36), (2.39) and (2.43)) K:~

= e (ik~-l/2A~)(d+O e -1/2"/C~

-- 0)

K:~

- - t [1 - e [i(kl-k2)-l/2A1]l] e ik:(d+l) e -l/2t~ ~ ~

(2.46)

and = O)

: teikl(d+l) [e-i(kl-k2)(d+l) _ e-i(kl-k~)d-l/2A1] x e -t/2" 1C~

= O)

:e-ihmd/Al r, [e-ih~l/A1 _e-l/2A1] • e ikl(d+l) e -I/2~ K;~ = 0),

(2.47)

where iNA1AI~r - i h m + 1/2"

(2.48)

Hence,

)i~after ~ [e-(d+l)/2A1 q-e-ihmd/Axt(e-ihml/A1 --e-l/2A1)] • e ikm(d+l)-l/2tz K:~ -- 0).

(2.49)

Defining A(I) ~-r [e -ihm//A1 _ e-//2A,] -- {A(/)I e i arg A(l)

(2.50)

means the (normalized) intensity of K ~ mesons right after the regenerator can be expressed as

2.3 K1-K1 0 o Interference and the Sign of Am~

I~afterl2

I~o(d + z)

29

- e -(d+t)/A' -4- IA(Z)I 2 + 2 IA(Z)I

• e -(a+l)/2nl cos [arg A(1) -

5md/A1].

(2.51)

T h e last term in (2.51) describes the K o1 - K 1o interference. T h e difference in phase between/C~ and/C~ depends on the proper time elapsed between p r o d u c t i o n and regeneration because of Amk. By altering d, this phase difference can be changed to maximize the destructive interference between the two waves. Tile K ~ intensity will exibit a pronounced m i n i m u m when cos [argA(/) - 5md/A1] = - 1

i.e.

a r g A ( / ) - ~ m d / a l = 180 ~

(2.52)

Since the m a x i m u m interference occurs when the amplitudes of the two waves are a b o u t equal, l can be p r o p o r t i o n a t e l y reduced as d is increased to keep the ratio of intensities ]/C~ ]]CO(orig)l 2

IA(I)I2 e-(d+l)/AI

(2.53)

close to unity. In the experiment [12], a K + b e a m of 900 M e V / c at Berkeley was used to p r o d u c e K ~ mesons via K + n --+ K ~ in a copper target (see Fig. 2.9). S p a r k - c h a m b e r pictures of K ~ -4 7r+Tr- events were taken behind an iron regenerator. B y measuring the K ~ intensity as a function of (d + l)/A1 one can deduce b o t h the m a g n i t u d e and sign of A m k , provided the m a g n i t u d e and phase of [r arc known. T h e latter were determined by the authors in a separate scattering experiment with charged kaons on iron nuclei. 12 T h e phase of A(1) is the sum of the four nonzero phases in 13

A(1) = i N )~l Alfr e - i h m l / A1 -- r - i h m + 1/2

(2.54)

Based on reference [12], we obtain 14 arg(i) =

arg [e i~r/2] L J

=

71"/2 =

90 ~

arg([r) --

~ ~ 50~

~

12 The imaginary part of ~r can be determined from the K i cross-sections on nucleons: a ( K + n) = a ( K o p), a ( K - p ) = a(fi[ 0 n), etc. The real part of [ is obtained from the interference between Coulomb and nuclear scattering ([ is usually assumed to be purely imaginary). To calculate [r for iron nuclei, optical-model fits to nucleon data are used. 13 The phase of a complex function f ( z ) = fl(z)f2(z)f3(z) ... is the algebraic sum of the phases of its individual factors: arg f ( z ) ----arg f l ( z ) + arg f2(z) + arg f3 (z) + . . . . Recall also that arg[fl (z)/f2 (z)] = arg fl(z) - arg f2 (z). 14 The average momentum of the K ~ beam was approximately 760 MeV/c, which corresponds to A1 = 3.98 cm (l/A1 = 1.9).

30

2. Propagation of Neutral Kaons in Matter

Coil

Magnet

Coil

_

_

~

~2???ers.Y_./_~ L__

~'Cutargets ~ Foil sparkchamber K+ Beam . z,.,X,~,~I ~,~. r gapsof89 9so.eV/cc.:ll,, ,ml]l~,^ ~ , lO~'/pulse

Smal"'l " spark chambers

]1

~ I~, A

u

9

./

- Cz

IJ

////, Magnet

~"//~/

Coil

L .~

Coil

Fig. 2.9. The experimental set-up of W. Mehlhop et al. [12]

arg [1/2 -- iSm]-1 ---=- arg [1/2 - i6m] = arcsin

26m

+45 , 6m = +0.5, =

arg [e-i~m//A1 - e

- - 4 5 0 , ~m = - - 0 . 5 ,

-U2A'] = arctan [

1 ~ T~=,/2,~,j

sin(-Sml/A1)

Lcos(-~

_ ~ - 7 6 ~ (~m -~ + 0 . 5 , - - [ + 7 6 ~ 5m = - 0 . 5 .

Hence, arg A(l) = - 6 0 ~ =]=31~ for ~rn = "4-0.5 (see Fig. 2.10). The minimum is given by (2.52), which yields d/A1 = 3.1(6m = +0.5) and 7.3(6m = --0.5). The K1~ --+ 7r+Tr- decay intensity measured as a function of (d + l)/A1 = d/A1 + 1.9 is shown in Fig. 2.11. This distribution has a minimum at 5.5 (in units of ~h), corresponding to d/A1 = 5 . 5 - 1.9 = 3.6, which clearly favor8 the value of 3.1 predicted for 6m = +0.5. We thus conclude that K ~ is heavier than K ~ If one assumes a value for 6m, the experiment can provide the magnitude and phase of ~r. Excellent agreement was found between I~rl and Or obtained this way for 6m = 0.46 and the corresponding values from the scattering experiment with charged kaons mentioned above.

2.3 K ~

~ Interference and the Sign of A m k

. ~0 for 8 < 0

for

S>0

A for S>0

F i g . 2.10. Phase relation between Kl~ and g ~ [12]

10CO--

-

10

! ~.5

,t 3.5

5,5

I, 75 Ks LIFETIHES

I cl5

! II,S

f 135

F i g . 2.11. The rate of 7r+rr - decays as a function of distance measured by W. Mehlhop et al. [12]

31

3. C P V i o l a t i o n in K ~ D e c a y s

"But then in 1964 these same particles, in effect, dropped the other shoe."

Val Fitch, Nobel Prize lecture (1980) We have assumed up to now that the combined operation of charge conjugation and parity transformation, C P , which turns a particle state into an antiparticle state, is conserved in weak interactions. Considering that parity violation is such a large effect (as we mentioned earlier, all neutrinos are left-handed and all antineutrinos are right-handed), and that both P and C are not conserved in weak interactions (applying C to a left-handed neutrino changes it into a left-handed antineutrino), one may wonder if this assumption is justified. It turns out that it is not, as we will now explain. ^

^

3.1 Discovery of C P Violation In 1964, J. Christenson, J. Cronin, V. Fitch and R. Turlay 15 detected one 27r event among 500 or so common decays of the long-lived neutral kaon clear evidence of C P violation. Subsequent studies of semileptonic decays of neutral kaons have confirmed this finding (see Sect. 4.1). Unlike parity violation, which is maximal in weak interactions, C P is violated only infinitesimally (at a rate of about 10-3). Moreover, C P violation has been observed so far only in the K ~ system. While the nonconservation of parity was readily incorporated into the theory of weak interactions, primarily because the neutrino is adequately described by the Dirac equation for massless particles (the W e y l equation), a "natural" way to accomodate C P violation has yet to be found. Invariance under C P implies a particle antiparticle symmetry in nature. As it happens, there is practically no animatter in the universe. From measurements of galactic masses and nucleosynthesis calculations, and from the temperature of the microwave background radiation, the ratio of baryon to photon densities at the present time is found to be nb -- ~tb / ] o b s e r v e d

- -

-

-

n~

- -

10-7/cm 3 ~ 10_9. 400/cm 3

Since the universe is electrically neutral (n~ - ne+ = rip), there is also an excess of electrons over positrons of about 10-7/cm 3. On the other hand the 15 A Princeton University group.

3. CP Violation in K ~ Decays

34

theoretical prediction, based on the assumption of baryon number conservation and initial symmetry between matter and antimatter 16, is ~lth ~ 10-1s As shown in Sect. 4.1, the long-lived neutral kaon decays more often (about 3 • 10 -3 times) into a positron than into an electron. If the K2~ meson were a pure CP eigenstate, and if CP were strictly conserved, the two decay modes (4.5), which transform into one another under CP, would be equally probable. The nonconservation of CP therefore permits particle-antiparticle "discrimination" and thus may be responsible for the observed asymmetry between matter and antimatter in the universe. The charge asymmetry in the decay of neutral kaons not only distinguishes between matter and antimatter, but also provides an unambigous definition of positive charge: it is the electric charge carried by the lepton preferentially emitted in the decay of K ~ In essence, CP violation implies that the laws of nature do make an arbitrary distinction between left and right and between particles and antiparticles. So far, CP violation has been observed via the CP forbidden decays K ~ -+ Ir+Tr- [13a], 7r%r~ [135] and 7r+Ir-7 [13c], and in the form of charge asymmetry in the semileptonic decays of K ~ [26, 27]. In the celebrated CP-violation experiment of J. Christenson et al. [13a], a beryllium target was placed in the circulating 30 GeV proton beam of the Brookhaven A.G. Synchrotron. Neutral beams of approximately 1 GeV/c, emitted at 30 ~ to the proton direction, pass through two collimators and a sweeping magnet before entering a plastic "bag" filled with helium gas at atmospheric pressure, placed 17 m from the target. At this point the K ~ component has decayed away leaving a pure K ~ beam. Pairs of charged particles originating from the (cross-hatched) area inside the helium bag (see Fig. 3.1) are analyzed by two spectrometers consisting of bending magnets and spark chambers, triggered on a coincidence between water Cerenkov and scintillator counters. The helium bag serves to minimize secondary interactions in the decay region ("cheap vacuum"). Spark-chamber photographs were measured on machines equiped with digitized angular encoders, The rare 27r decays are distinguished from the c o m m o n semileptonic and 37r decays on the basis of their invariant mass ]7 and the direction 0 of their resultant m o m e n t u m vector relative to the incident b e a m (0 ~ 0 for K ~ --+ ~r+~r-). The apparatus was calibrated for 21r events by measuring K ~ -+ ~r+~r- decays produced by coherent regeneration in a tungsten regenerator successively placed at intervals of 28 c m along the sensitive decay region. Since the regenerated K ~ mesons have the same m o m e n t u m and direction as the K ~ beam, their decays simulate the CP-violating K ~ -+ Ir+Tr- decay. For these measurements a thin anticoincidence counter was placed immediately 16 In the context of the cosmological Big Bang model. 17

Tytlr+,r

-

=

/ytk 0

=

i=1

--

(Y]~L1p~)2]

1/2

3.2 Phenomenological Implications of K ~ -+ 27r Water Scintillator ~ e ~ o v

Plan view

.

.

.

35

>parkcham/r

.

57 ft to < internal target

Scintillat ,r -'..I Water Cerenkov

Fig. 3.1. The experimental arrangement of J. Christenson et al. [13a]

behind the regenerator to ensure that K ~ mesons decay downstream from it, thereby eliminating neutron-induced background events. Taking into account the relative detection efficiency for two- and threebody decays, they retained 45 + 9 events from the helium gas that appeared indentical with those from the coherent regeneration in tungsten in both mass and angular distribution. The total corrected sample of K ~ decays was 22700. They concluded that K ~ decays to two pions with a branching ratio of T~ - K ~ --+ 7r+Tr- - (2.0 + 0.4) • 10 -3. K ~ --+ all

3.2 Phenomenological

Implications

(3.1)

o f K ~ --+ 27r

Let us assume - - and this is generally believed to be the case - - t h a t the weak hamiltonian is not invariant under CP. As a consequence, its eigenstates are not CP eigenstates, but linear superpositions of CP-odd and CP-even components. Since CP violation is so small, we expect the particle eigenstates to differ only slightly from the CP eigenstates K ~ and K ~ We therefore write the short- and long-lived eigenstates of the weak hamiltonian, /:/w, as Ig~> - IK~ + el IK~ (3.2) IK~ ~ IK~ -4- e21K~ i.e. IK~ = ICP = +1) + (10 -3) x ICP : - 1 ) , (3.3)

IK~ = ICP = - 1 ) + (10 -3) x ICP : +1>,

36

3. CP Violation in K ~ Decays

where el,2 is generally a complex number. In the above expressions we neglected the normalization factors l / v / 1 + lel2, which are second order in c. The new states K ~ and K ~ are clearly not CP eigenstates. Moreover, they are not even orthogonah

( K~ I K~ = el + ~ .

(3.4)

This lack of orthogonality is to be expected since K~ and K ~ have the same decay modes. From (1.8), (1.10) and (3.2) it follows (neglecting terms proportional to c1~2) that

1

IK~ = ~

1

IK~ = ~

[(1 + el)lK~ + (1 - el)l/<~ (3.5) [(1+ ~ 2 ) [ K ~

(1 - ~2)lk~

and [K ~ = ~ IK~ = ~

1

1

[(1 - c2)[K ~ + (1 - el)lK~ (3.6) [(1 +

~2)lK~>-

(1 + c1)lK~

.

If we assume that weak interactions are invariant under the combined operation of charge conjugation, C, parity transformation,/5, and time reversal, 7~, then CPT invariance means ci = e2 -- e,

(3.7)

where e is a measure of nature's deviation from perfect CP invariance. The above assumption is in accordance with the CPT theorem, which states that any quantum theory that is based on relativistic invariance and locality is automatically invariant under ~/5~. An important consequence of this theorem is that a particle and its antiparticle must have the same mass, decay lifetime and magnetic moment. To prove (3.7), consider a transition 2[(Pi, si) --+ .T(pi, si) , where 2[ stands for one or more particles in the initial state and $v for particles in the final state; p and s are the corresponding momenta and spins, respectively. This transition is described by the matrix element

.A.fi = (.T" I A [ Z}.

(3.8)

The operation of C flips the signs of internal charges, such as the electric charge, baryon number, etc., but spins and momenta are not affected; /5 reverses 3-momenta; under T, initial and final states are interchanged and spins and momenta are reversed. The effect of the combined C/ST operation is thus

I A I z>

,

(3.9)

3.2 Phenomenological Implications of K ~ -+ 27r

37

where the overbar denotes antiparticles and the prime means that spins are reversed. Therefore, assuming C P T invariance means

<J: I A I z) = (z' I A I Y>.

(3.10)

For spinless kaons this implies ( C P T invariance)

(KO i/:/[ KO) = (/~o i/:/i ko).

(3.11)

If we define

~--~--' ~-=--&~' ~--~-e2

1 + et

1

~--v~'

-

•1

-

1 + e2

1 -

(3.12)

expressions (3.5) and (3.6) can be written as (IK~) = g~, etc.) K~=aK

~

~

g~

~

~

(3.13) K ~ = S g ~ + Z K o,

~:~ = .~KO _ ~ K ~

By using (2.9) and (2.10), the time development of the K ~ state is given by g ~ = ,~K~ e ~S + Z K ~ e *L = ~ ( , ~ g ~

~ er + ~ ('~g ~ - , ~ k ~ e %

i.e. K ~ = (6ae Cs + fl'),eeL) g ~ + (6fie r - flSe r

k ~

(3.14)

Similarly, /~o = (3~aeCs _ a,),eCL) g o + (~fleCs + a6eCL)/~o.

(3.15)

From (3.11) and (3.14) and (3.15) it follows that 6a e Cs + air eel = 7fl e Cs + a6 e r

---+ 6oe = ~/fl.

Therefore, 1 - e2

l+e2

1 e1 l+el -

--

-

-

) ~1

z~2

~s

By assuming C P T invariance, expressions (3.5) and (3.6) become

IK~) = f(e) [(] + e)lK~ + (1 - e)lR~ IK ~ =/(,)

(3.16)

[(1 + e)lK ~ - (1 - e)IR~

and

IK~

= f(e) [(1 -

e)lK~>

+ (1 - ~)IK~

Ig~

= f(e) [(1 +

e)lK~)

- (1 + e)lK~

(3.17)

where we included the normalization factor 1

f(e) - V/2( 1 +

lel2).

(3.18)

38

3. C P Violation in K ~ Decays CP

Found in nature

Not found in nature

/ Fig. 3.2. K ~ is a superposion of K ~ and /~o, with a slightly larger amplitude for K ~ This violates CP symmetry

/ / /

T h e C P n o n s y m m e t r i c state IK ~ is illustrated in Fig. 3.2. Expression (3.4) now reads (K~ ] K~ -

e + e* 2Re c _ ( K o ] KO), C P T i n v a r i a n c e . (3.19) 1 + ]~l2 - --------5 1 + [el

To relate a possible C P T - v i o l a t i n g mass difference mko - mko to the measurable q u a n t i t y AmL,S, we consider (3.14), (3.15) in a small time interval At. In this case e--(ims,L+Fs,L/2)t =

e_i~elS,Lt t~O 1 -- iA/Is,LAt.

(3.20)

Hence, K ~ -+ K ~ - i A t [(Sa.MIs +/33~ML) K ~ +/35 ( M s - J~L) NO] ,

(3.21) /~o _+ KO _ iAt [Ta ( M s - .A/[L) K 0 -f- ("//3MS A- C~6ML) R 0] , where we used (Sa+/37) K~

~

~

etc.

We can express (3.21) as . d~vac

1

dtp

(3.22)

( 1 (.A/IS-q-./~L) q- (MS --.~L)~ 1(.s

h4L) ( 1 _ 2~)

1 (MS--.A4L)(1 q-2~ ) 89(A4S +.A4L)_ ( M S _ M L ) g ] k~vac,

i.e.

9dl/fvac 1 dtp

{/J~ 11J~ 12 "~ -- \.A,~21

M22 ] ~vac

(3.23)

with _ _el _- e2 2 '

~ = -s --~- e2 2

(3.24)

Now, M22 - Mll

= (ML

- Ms)

(e, - e2)

= [(m~ - ms) + i (Fs - FL)/2]

(mL -- m s ) (el -- e2) (1 + i)

(e, -- e2)

3.3 Unitarity, CPT Invariance and T Violation

39

since ZlrnL,S ~ Fs/2 (see (1.43)) and Fs >> FL. Therefore, Re (M22 - M u ) -= mko - m~o = (mL -- ms) (cl -- e2) _< 3.5 • 10 -9 eV, which sets the upper limit on a possible CPT-violating K~ ence. Experimentally [32, 36], mko

-

m~o

mko

(3.25)

~ mass differ-

< 1.3 x 10 -18.

(3.26)

The above results can be summarized in the following way. According to (3.24), 2~=el+e2,

2~=el-e2

>el=~+~,

*

e2 ~

~*

-- ~-*,

which we use to express (3.4) as (3.27)

{KL~ 1 7 6

If CPT is conserved (and consequently T is violated), then according to (3.7) and (3.11) J~ll

=-M22, (K ~ I K ~ = 2 R e e (el =c2), C P T invariance. (3.28)

If, on the other hand, T is conserved (which means that CPT is violated), then ~ ( e 1 = - c 2 ) , T invariance.

A/ll2 =A/I21, (K ~ t K ~

(3.29)

because time reversal interchanges indices of the initial and final states. Ignoring the small CP violation, we find that m L -- m s

= -- (.hd12 + .hd~2 ) = - 2 R e

M12.

(3.30)

Recall that the mass and decay matrices are both hermitian: M21 = M{2,

3.3 U n i t a r i t y ,

F21 =/'1"2.

CPT

(3.31)

Invariance

and

T Violation

The observed CP violation in the neutral kaon decay implies that either T is violated but CPT is conserved, or CPT is violated but T is conserved, or both T and CPT are violated. To test these possibilities we will rely on the unitarity relations, first derived by J. Bell and J. Steinberger [14a]. These relations are direct consequences of probability conservation, as shown in what follows. The time development of the (K ~ K ~ system is given by -- ~s e-i'MStlK~ + as

e-i'MLtlK~

(3.32)

40

3. CP Violation in K ~ Decays

with A4S,L as defined in (3.20). The states IK~ due to CP violation. We can thus write

IK~ are not orthogonal

<~'s,L I ~S,L> = lasl2e -Fst + ]aLI2e -rLt + a~aL ei(A~[-A%)t
+ a*Lasei( -Mslt
(3.33)

At t --+ 0, the decrease in the norm of I~S,L> can be expressed as d ~

~< S,L I ~SL> = lasI2Fs + laLI2rI~ - i(M;

-i(M~

_

ML)

a;aL
o I K~ 9 Ms)aLas~/K L

(3.34)

*

Since the kaon decay results in the appearance of the decay product, conservation of probability requires (3.34) to be compensated by the total transition rate

(f [/:/w I ~S,L> 2

=

lasl2 ~ IAsI2 + laLI2 ~ IALI2 y

f

f

+a~aL E A~AL +aLas E ALAs' Y Y

(3.35)

where the transition amplitudes are AS,L -- (f I /:/w I KS~

(3.36)

Equating coefficients in (3.34) and (3.35) yields the Bell-Steinberger unitarity

relations Fs = E I(f I/:/w [K~ 2 - ~ IAsl 2, f y

rL = ~ I
(3.37)

[ Ks~

[KL~

=- E A~AL, f - i (M~ - A4s) (K ~ I K~ = E ( f Y

I/:/w I K~ * (f I/:/w I K~}

=- ~ At As. f Note that in deriving (3.37) we set the same transition rate for different final states. This point will be addressed again shortly.

3.3 Unitarity, CPT Invariance and T Violation

41

By using the Schwartz inequality A*L A s 2 - <- E

E

(3.38)

]ALl2 IAs[2

it follows from expressions (3.37) that

2

+ (mL -- ms) 2 I(K ~ I K~ 2 < rSVL,

(3.39)

i.e. (K ~ [ K ~ < 0.06,

(3.40)

which confirms that IK~ and IK~ are nearly orthogonal. To test C P T and T invariance we will employ the fourth unitarity relation --i(mL - ms) + FL 5 +

= ~(fl/:/w /

t KD o.

rs] (Ko i gO) .

(3.41)

Both sides of this equation violate CP symmetry since (a) (KL~ ] K~} # 0, and (b) the final states of K ~ and KL~ could not otherwise be the same. To simplify (3.41), we define the following CP-violating amplitude ratios (f

/?/w

(f & r/i -

(f

//w

(f /Iw

K~ (CP = K o)

+1), (3.42)

K~} (CP = - 1 ) . K o)

Expression (3.41) now reads 3 FL + --iZlmL,S + Fs]

(Ko [ K~)

= E

7/}F/,

(3.43)

where F / is the transition rate to a specific final state, or a group of states. This reflects the fact that various decay modes of K ~ and K ~ have different decay rates. Experimentally, the main three-body decay modes have small decay rates compared with Fs (see, for example, [15, 21]): Fs(semileptonic) ~ FL(semileptonic) ~ FL(37r) --~ 10-3Fs.

(3.44)

The K ~ --~ 3~r decays are not only forbidden by CP but also suppressed by the phase-space volumelS; hence Fs(37r) << 10-3Fs. The radiative decay mode K ~ ~ ~+~-V can also be ignored compared with K ~ -+ 21r [13c]. 18 As shown in Appendix A, the ~+Tr-~ ~ final state is not a CP eigenstate. The K~ may decay into the kinematics-suppressed but CP-allowed final state with g = 1 and C P = +1, or into the kinematics-favored but CP-forbidden state with g = 0 and C P - -1, where g is the relative angular momentum between the ~r+ and rt-.

42

3. C P Violation in K ~ Decays

Therefore, the right-hand side of (3.43) is d o m i n a t e d by K ~ --+ 7r+~r- , 7r%r~ decays, i.e,, it is sufficiently accurate to write

*

.

r/}Ff ~ ~?+_ F+_ + 7/oo/"o0,

(3.45)

$ where F + _ and Fo0 are the partial decay rates for K ~ ~ 7r+Tr- and K ~ -+ 7r%r~ repectively. T h e complex a m p l i t u d e ratios r/+_ and r/0o are defined as

<~+~- I/:/w I K~

@+Tr- [ ~---~w[ K~ - I t / + _ [e ir

,

@ %r~ i/2/w [ K~ _--[f]0ol e i4)~

,

(3.46)

In the next section it will be shown t h a t 2 IF, F + _ + Foo ~ g F s + g s = Fs.

(3.47)

F u r t h e r m o r e , the most accurate m e a s u r e m e n t s of the CP-violating p a r a m e ters r/+_ and r/o0 are m u t u a l l y consistent and yield [z/+-I ~-- [r/oo[ ~ 2.3 x 10 -3,

r

"~ 000 ~ 44 o.

(3.48)

Hence, * * F,s, ~ +* _ F + _ + rlooFoo - ~,~

(3,49)

a n d (3.43) simplifies to (FL << Fs) i ~

+

;1

(KL~ [ K ~ ) * = ~ . , .

Using (3.28), we can rewrite this expression as

1 + i ~-s

] Re ~ = IW~] cos r

(1 + i t a n r

(3.51)

and similarly for (KL~ I K~ = 2Jim ~. These results are s u m m a r i z e d in Table 3.1 (see [14b]). It is obvious t h a t experiments strongly favour C P T conservation, whereas T is clearly violated. In w h a t follows we shall thus set ~-0.

T h e foregoing analysis represents one of the most stringent tests of C P T invariance in physics. We hasten to point out t h a t it relies solely on some of

T a b l e 3.1. CPT conserved

T conserved

el = e ~ = e ( ~ = 0 ) (KL ~ I K~} = 2Re e

~1 ---- - - s

- i] Im f = ~ , , [2amL.s rs

. 2AmL,S ]

r

,~, arctan \

(~ = 0)

(KL~ ] K ~ = 2 J i m

rs

] ~ 43"7~

~b.. ~ arctan (\ 2~mL,s -r~ ~] ~ 133.7 ~

3.4 Isospin Analysis of g~, L --~ 27r

43

the basic principles of quantum mechanics: the principle of superposition of amplitudes and conservation of probability. The analysis also reveals that, for the first time, we have a physical system which behaves asymmetrically in time as a result of an interaction (the weak decay). It is well known that entropy in a closed system increases with time, but this effect is due to boundary conditions, not specific interactions. Moreover, as Stiickelberg showed in 1952 [16], the second law of thermodynamics does not depend on the validity of microscopic reversibility, and is thus deeoupled from the question of time invarianee. A direct test of time-reversal asymmetry in the K ~ system is discussed in Sect. 7.1.

3.4 Isospin

Analysis

o f K S,L ~ --+ 27r

"Anyone who has played with these invarianees knows that it is an orgy of relative phases."

Abraham Pais, Inward Bound Before presenting experimental results on CP violation in the K ~ system, it is important to make sure that the reader is familiar with the isospin analysis of KSO L --~ 27l" decays [14@ Recall that pions and kaons are pseudoscalars (spin-zero bosons), whose isospin assignments are given in Table 1.1. The pions are identified as the three states of an isotopic spin triplet, i.e., the pion has isospin I~ = 1. The neutral kaons belong to an isotopic spin doublet: Ik = 1/2. Since pions and kaons are spinless, angular momentum conservation requires the two pions in the K~, L -+ 2n decay to be in an S state. By Bose statistics, their total wavefunction must be symmetric under particle interchange. Consequently, the isospin function of the 2n-state must also be symmetric. The isospin states of 7r+, 7r- and 7r~ read In + } - [ 1 , 1 ) ,

In-)_--]l,-1),

]To)-=]1,0),

(3.52)

and so the possible total isotopic spins of the combined states [n+~r -) and [n~ ~ are I - 0, 1 or 2, with I3 = 0. In terms of Clebsch-Gordan coefficients, we have 1

+

1

+

[1,0) = ~ [ n ,

1

+

n2) - ~ l T r ~ - n l ),

(3.53)

44

3. C P Violation in K ~ Decays

where lTr+~r-) -I~+>1~->, e t c The I1, 0) state is not symmetric under particle interchange because it changes sign for 7r+ ~ ~r-. This leaves us with 12, 0) and 10, 0) as the only allowed isospin states. To express the experimentally observed pion states 17r+Tr- ) and 17r%r~ in terms of the allowed isospin states 12, 0) and }0, 0), note that 17r+Tr- ) is the symmetric combination of the two states 17r+Tr~-) and I~r~-~r+): I~+~r- ) = ~

1

[Ir+rr2) + ]~2~+)].

(3.54)

Inverting the two remaining expressions (3.53) gives

I~+~-) =

d

,Tr~176 = ~ , 2 ,

12, o) +

d 1

0) - ~ , 0 ,

Io, o), (3.55) 0).

In the isotopic spin space, the K ~ --+ 27r decay is specified by the following four transition amplitudes (Trr I = 0 I f/w I K~

(.., s = O l S:/w I sT~

(==, z

= 2 I/:/w

(3.56)

I sT~

The weak decay amplitudes K ~ -+ f and /~o ~ invariance (see (3.9)):

f are related by C P T

~ T - * =
(3.57)

because/:/~ is hermitian (/:/tw =/:/w) in first-order perturbation theory (see Appendix C). pion

Fig. 3.3. Final-state interaction (Hs) in the decay pion

K ~ --+ 2~r

So far we have considered only the weak interaction responsible for the kaon decay. In reality, the final state pions interact strongly with each other (see Fig. 3.3). By resorting again to unitarity and C P T conservation, we can generalize (3.57) to include final state interactions (Watson's theorem; see Appendix C), with the result

(]' I Hw I K0) = ei2~
(3.5S)

where 5 is the scattering phase shift for the states If) and If p) at v ~ = mko. In the limit of vanishing final-state interactions (5 = 0) this reverts to the old

3.4 Isospin Analysis of KOL ~ 2~r

45

relation between particle and antiparticle transition amplitudes guaranteed by C P T . The above expression can be rewritten as [e -i6

(f']~IwIK~

*=

e -i6 (f ] /2/w ] /s176

(3.59)

In view of this result we define the following decay amplitudes Ao - e -i~~ <-~, / = 0 1 / ~ w I K~

(3.60)

A2 - e -i~2 <7rTr, I = 2 ] /;/w I K~

where 5o and 52 are S-wave scattering phases (phase shifts) for ]0, 0) and 12, 0), respectively. 19 These phase shifts are known from the analysis of pion nucleon scattering, e.g., (3.61)

7r- p ---+ n Tr+ ~r- , pTr - Tr~ .

From (3.59) and (3.60) it follows that A~ - e-i5~
(3.62) A~ = e - i ~ <~., S = 21 Hw I R~

Taken together, expressions (3.60) and (3.62) read Ac~e i6~ A ~ e i6~

=
(3.63)

We have now established the basic formalism needed to analyze C P violation in K ~ -+ 2~r decays. Using (3.16), (3.55) and (3.63), we obtain @+Tr-[/;/w I K ~ = ~ :

<0, 0 I/:/w

IK~>+

~(2,0

I(, +

I /?/w I K ~

+

+ (1 + e)A2e i62 + (1 - e)A~e i6~ ~, )

i.e. @+~-I/;/w ] K~ = f(e)V~-{v/2[Re Ao + ieIm Ao]ei6~ + [Re A2 + idm A2] ei52}.

(3.64)

19 Because pions are present only in the final state they acquire half the usual strong-interaction phase 25.

46

3. CP Violation in K ~ Decays Similarly, Qr+Tr- [/~/w [ K ~ = f(e) +

v ~ [eRe Ao + i Im Ao] e i5~

[~Re A2 +

<~~176I Hw I K~ : / ( ~ / ~ {

i Im A2] e i62 },

-[~e Ao + ir

(3.65)

Aol ei~~

+ V/2 [Re A 2 + ieIm A2] e i52 },

I

:

.o +

(3.66)

Aoi eiSo

+ v ~ [eRe A2 + Jim A2] e i62 }.

(3.67)

These results can be simplified by observing that the relative contribution of the I = 2 decay mode to the K ~ --+ 27r transition amplitude is small compared with that of the I = 0 mode, as shown below. Since Ik = 1/2, the decay K ~ ~ --+ 27r can be regarded as a mixture of A I = 1/2 and 3/2 transitions. Experimentally, the branching ratio

I s , + - _ F ( K ~ --+ 7r+Tr-) _ 68.6 _ 2.18 Fs,oo -- F ( K ~ -+ 7r~ ~ 31.3

(3.68)

is close to that expected for a pure [0,0) final state (see (3.55)):

0r+cr-;I = 0 [/:/w I K ~ 22 2. ]<~o~o ;i = 01 ~qw i KO>

"~I=0 =

(3.69)

This is in agreement with the empirical " A I = 1/2 rule", which states that weak amplitudes for processes with A I = 1/2 are strongly enhanced compared with those with ,51 > 1/2. For example, the decay K + ~ 7r+Tr~ (,51 = 3/2) is about 600 times slower than K~ --+ ~r+Tr- (,51 = 1/2). Neglecting a small phase-space correction factor (see (9.156)) and the terms containing e, we have from (3.64) and (3.66) that + ~22Re (A2/Ao) e i(6~-6~ 2 (3.70) F8,+- _ Qr+zr- if/w [K~) 2 .~2 1 -rs,oo

-

(Tr%ro I Hw IK~>

2

]1

v/2Re (A2/Ao) e i(*.-*~

which yields

Re Aoo ~0.04 for the mr scattering phase 7r/2 + 52 - 6o = (37.5 + 5) ~

(3.71)

3.4 Isospin Analysis

of

KO,L --+ 27r

47

In what follows we will neglect the terms proportional to eRe (A2/Ao) and e(ImA2/ReAo). The experimentally observed amplitude ratios (3.46) are then given by

e + iImAo/ReAo + --~2(iImA2/ReAo)ei(~2-5~ 1 + ~2Re (A2/Ao)ei('~2-~o)

~+-

_ ~ + ~_~22~ c + e', 1+~

(3.72)

e + i Im Ao/Re Ao - v~(i Im A2/Re Ao)e i(~2-~~ 1 - x/2Re (n2/Ao)ei(~2-~o)

qoo

C -- 2r

: - -

1 - 2~

~- c - 2~ I

(3.73)

in any phase convention for which le(Im Ao/Re Ao)l << 1. In (3.72) and (3.73) we defined z - e + i-Im- Ao ReAo ' i Im A2 ei(~~_8o)

(3.74)

e2 - v ~ Re Ao

~ ~ ---~Re (-~-~) e i(82-8~ and iImAo

~'-~2

i ReA2 [ImA2

neA----~~- v~neAo ~A2

ImAo] ei(~2_~o).

(3.75)

R-~oJ

Using expressions (3.64)-(3.67) and <0, 01 = ~ ( ~ + ~ r - [

- ~

@~176 (3.76)

(2,o, = ff~ (~+~-I + ff~5@%~ it is straightforward to show that the parameters z, e2 and ~ can be expressed as ratios of K ~ and K ~ transition amplitudes to the I = 0 and I = 2 states: c~

(o,o I & IK~ (o,o LHw I K~ ' 1 (2,01HwlKL~ 1 ( 2 , 0 1 / : / w [ K s~

v~ (0,01HwIK~

(3.77)

3. C P Violation in K ~ Decays

48

The phase of e' is determined from (3.75) (note that i ei~r/2). Based on (3.81) and (3.82), the phase of c = e + i I m A o / R e A o =- e + i40 is, to a good approximation, r ~ r ~ zr/4 (see (9.150)): =

r

~ arctan

'~ J ,

(2ArnL,S ~ss

r

= zr/2 + ~2 -- (~o.

(3.78)

We see that % _ and 700 do not vanish even without K ~ - K ~ mixing (e = 0), if either A0 or A2 has a nonzero imaginary part. In other words, Ao and A2 are real if CP is conserved. The parameter r ~, which affects the 7r+Tr- and 7r%r~ channels differently, determines the magnitude of direct CP violation in [ASI = 1 decays. If CP violation is associated with decay only, then K ~ ~ 27r is possible. Note t h a t e t is expressed as a difference in CP violation in the I = 0 and I = 2 decay amplitudes. It vanishes only if A0 and A2 have the same phase (see Appendix C and Sect. 9.3). Adopting the phase convention I m A0 -- 0 eliminates direct CP violation in the I -- 0 decay mode. The p a r a m e t e r e arises primarily from mixing in the mass matrix, and is thus associated with indirect CP violation that mixes CP-even states into K ~ (JASJ = 2). To show this, we invert expressions for the off-diagonal matrix elements in (3.22), assuming C P T conservation (~ = e), with the result -h/J12 -- J ~ 2 1 2(Ms -Ada)

(/~12 --/";2) § i(M12 - Mr2) (Fs - FL) + 2i(ms -- mL)

i Im F12 - Im M12 Amk(1 - i 2 A m k / F s ) '

(3.79)

where we used M i j = Mij - iFij, F21 = 1"{2, M21 = M~2 and 2Amk ~ Fs. According to Table 3.1 and (3.72)-(3.74), Re e =

7/~ 1 + i 2Amk/Fs 1 +i

~ 1~ 2 - - ~ k na

-~- \

2-~mkk +

'

(3.80)

because 3 r / ~ ~ 2r1+- + rloo ~ 3e and e = e + i Im A o / R e Ao = e + i 40- Since Re e is a real quantity, ImF12 ~-4o, Amk

_

_

2Rec~

-

Im M12 +4oAmk

(3.81)

Now, we can always define the relative phase of K ~ a n d / ~ 0 by choosing I m A o = 0 or I m A 2 = 0 (see footnote 5). In either case the experimental result e'/e ~ 10 -3 (see Sect. 5.1) implies that 140,2[ << leJ - This shows that e arises primarily from mixing in the mass matrix. CP violation, as measured by the parameter e, is an effect of the order of 10 -3 relative to weak transitions: I m M12 ~ 10 - s eV compared with A m k 3.5 x 10 -6 eV. Using (3.79) and (3.30), we obtain

3.5 KL-K 0 so Interference as Evidence for CP Violation ilm/"12 - Im M12 Amk(1 - - i t a n r

- I m M12 eir v/-2 Amk

1 I m M12 eir 2v/2 Re M12

49 (3.82)

If M12 and/'12 were both real, e would be zero (see (3.79)). Since H is so small, I m M12 << Re M12, I m / ' 1 2 << Re/"12 As promised in the preceding section, we now derive expression (3.47). First note that

{I = 0 ]/:/w I K~ = ~ _

(I = 0 {/:/w { K ~ + {I = 0 I/:/w I/~~ 2 ReA0e i6~

(3.83)

We can use the fact that Re A2 << Re Ao to write Fs ~

{I -- 0 I/:/w [ K~) 2 = 2 (Re Ao) 2

(3.84)

and (neglecting the terms with e and A2 in (3.64) and (3.66))

rs,+_ -

(Tr%r- I/:/w I g ~

2 ~ 54 (Re Ao) 2 ,

(3.85) Fs,oo =

<7r% ~ I/:/w I K~ 2 ~ 52 (Re Ao)2

(3.86) 3.5

K ~o- K

so I n t e r f e r e n c e

as Evidence

for CP

Violation

After the detection of the K ~ ~ ~+Tr- decay by the Princeton group in 1964, most physicists were not keen to forsake yet another cherished symmetry principle. They still believed that CP was an exact s y m m e t r y of nature which successfully replaced the defunct idea of parity conservation. Various a t t e m p t s were made to explain the effect without abandoning CP invariance. Among other suggestions, it was proposed t h a t either the observed pions, or their long-lived parent particles, were not the usual mesons; or that, similar to Pauli's neutrino hypothesis, the decay was accompanied by a third particle with small mass and energy. 2~ One can confront these hypotheses by proving that: (a) the mass and lifetime of the particle responsible for the 27r-decay are the same as those of the K~ (b) the rate of K ~ --+ 27r decays does not depend on how the kaons are produced; and (c) the decay products are similar to pions in every respect. 20 It was even suggested that the Princeton result might be due to regeneration of K1~ mesons in a fly trapped in the helium bag! It turns out that if that had been the case the fly would have to be more dense than uranium [92].

50

3.

CP

Violation in K ~ Decays

Such tests are, in fact, not necessary, for there is a direct proof that the effect is indeed due to CP violation. In 1965, V. Fitch and his collaborators observed constructive interference from a coherent beam of K ~ and K~ mesons by comparing the decay rates of KL~ -+ 27r in vacuum and in the presence of a diffuse regenerator [17]. In addition to providing conclusive evidence for CP violation, the results of their experiment also demonstrated that it is possible to make clear empirical distinction between a world made of matter and that composed of antimatter, as we will now explain. Suppose that the regenerator in Fitch's experiment is composed of antimatter. In this case the forward scattering amplitudes f and [ would be interchanged because strong interactions conserve C invariance. As a consequence, the regeneration amplitude would change sign (see (2.39) and (2.40)): coo(matter) ---+ -~0c(antimatter)

(3.87)

resulting in destructive interference. Before presenting their results, we will outline the basic idea behind the experiment. Immediately after the regenerator (t = 0), an initially pure K ~ beam contains a K ~ component (see Fig. 2.7): ]~P(t = 0)) =

[g~

= 0)> +

Qc]g~

= 0)>.

Here we set the KL~ amplitude to unity, in which case the amplitude of K~ is just the regeneration amplitude Qc = I~odeir

(3.88)

At a later time t, I+(t)) = e-iMLt]KL0(t = 0)) +

Oce-iMStlK~

= 0)).

(3.89)

The ~P -+ 7r%r- amplitude reads A~+=-

I#w

= (7r+1r I +(t)) = Atcso+~+ ~_ [/]+_e -i2vtLt § ~Oce-i'A4st]

(3.90)

and the corresponding decay rate (per K ~ meson) is

I.+.- =

Fs,+- {l~/+-12e -rLt + le~12e -r~' + 21,1+_ll0cle-(rs+rL)~/= cos [5.# -- (r

- r

(3.91)

where t is the proper time of decay. The value of 17/+_12 was obtained from the rate of K ~ -+ 7r+Tr- decays without regenerator; 21 that of ]0~]2 was determined by measuring the ~ + u decay rate immediately behind a dense regenerator (~c >> ~7+-), in which case the interference is small (for this measurement a 7.6 cm solid piece of berillium was employed). ~1 The result was in excellent agreement with that of Christenson et al. [13a].

3.5

o so Interference as KL-K

Evidence for

CP Violation

51

The maximum interference occurs when the KL~ and K ~ amplitudes are about equal. For a berillium regenerator this corresponds to a density of roughly 0.1 g/cm 3, or N ~ 7 • 1021 nuclei/cm 3. To attain low density, they used berillium plates 0.5 mm thick, separated by 1 cm; the whole assembly was 1 m long. Note that granularity effects in this case are negligible, since the element spacing is small compared with the K ~ decay length As = (pk/mk)/Fs (1.3GeV/0.5GeV) • 10-1~ ~ 2.6 • 10 - l ~ • 3 • 101~ ~ 7.5cm and the oscillation length lo~c ~ 12As ~ 90 cm. This arrangement is, therefore, equivalent to a uniform distribution of the same amount of berillium over lo~c. Since the length of the diffuse regenerator is considerably larger than As, the coherent regeneration amplitude is independent of ~ ~ I/As at distances sufficiently far from the face of the regenerator:

Qc l>>As iNoAsAsfr -iSm + 1/2 --: Co-

(3.92)

T h a t is to say, at large l the number of regenerated K ~ mesons is equal to the number of those which decay (AKo = c o n s t ) . The interference experiments are always performed over a time scale t << TL, i.e., over a decay length ldec << AL, which means that the KL~ decays can be neglected (kKo = c o n s t ) . Expression (3.91) for the ~+~-- decay rate thus simplifies to Id - r s , + _

+ IQ012 + 21

+-IIQ01 cos ( r

-

(3.93)

From (3.92) and (2.42) it folows that I c] = I 01

( N ) 2 [l + e-r - 2e-r

cos((~m~r)] ,

(3.94)

where N and No are the nuclear densities of the solid and the diffuse regenerator, respectively. The experiment was performed at the Brookhaven AGS. The K ~ beam was similar to that used in the original CP violation experiment [13a]. The experimental set-up of Fitch et al. is shown in Fig. 3.4. A particularly useful feature of the detector is that the pion trajectories intersect at a point which lies approximately on the extrapolated line of flight of the kaons. Spark chambers were used as tracking devices. To retain the applicability of (3.92), they considered only those events that originated in the last 45 cm of the diffuse regenerator. The regeneration amplitude Q0 was determined from (3.94) by measuring It~cl2 as described above. The intensity ]d Was measured over the same kaon decay length /dec as the free-space intensity ]~ = 17+-]2. Using (3.93), the phase angle between ~?+_ and t~0 is given by cos(C+_

with

-

=

Id 2--I~ --Ip

(3.95)

3. C P Violation in K ~ Decays

52

Cz

~/////////~..~'," ~/////////////~ ~//////A c,

...."..~Z!~

Fig. 3.4. The experimental set-up used by V. Fitch et al. [17]

1"6 l"k

/fT

--

ff

/~ffJ J/ / / / //f~

- -

1-Z 1-0

/ / J'~j,/J'

0"8 "

0.4

~/

//

/ //// ~

]'

/

1/

-0-2~ - 0.4

i

i

0-5

I-0

t4oss difference161

Ip ~ ]~0] 2

Idec

As '

Fig. 3.5. cos(C+_ - r versus Amk calculated from the [17] data

(3.96)

where lae~/As is the decay length scale factor. T h e d a t a were corrected for a t t e n u a t i o n in berillium and for interference in the regeneration in solid Be. T h e rate of 7r+rc - decays was found to be a b o u t four times the rate without regenerator a clear indication of constructive interference between the KL~ --+ 7r+~r- and K s~ -+ 7r+Tr- decays. According to (3.92), LOo depends on the value of ~m ~ AmL,s. Consequently, c o s ( C + - - r calculated from the d a t a also is a function of 6m, as shown in Fig. 3.5. Clearly, the d a t a strongly rejects values of 16ml > 0.8. 22 In their second paper, Fitch et al. reported the 7r+~r- yield as a function of the diffuse regenerator amplitude Ar - O0. Measurements were made at densities of 0, 0.05, 0.1 and 0.4 g / c m 3. As explained earlier, in a world m a d e 22 The correction for interference in solid Be causes the curve in Fig. 3.5 to be double valued.

3.5 K L0- K s0 Interference as Evidence for C P Violation 12

'

I

'

I

-

- COSe =

'

r

53

' /n

/!I, .// .... ----M

Ic

/'~

+

cos~ :o coso =-I

///I ///

L

/ / // 6

4

/

//

./~// 2

,.~../

,,.J/ 0

// //

/// 12

2.0 3.0 A r (Arbitrory Units )

4.0

Fig. 3.6. Rate of 7f+~- decays as a function of the regenerator amplitude A~ [17]

of antimatter an experimenter would observe destructive interference, similar to the dashed line in Fig. 3.6. Note that the result shown in the figure is independent of the sign and magnitude of AmL,S. Interference effects between KL~ and K ~ mesons can also be studied by measuring the time dependence of the 7r+Tr- decay intensity following a thick regenerator. Several such experiments have been performed (e.g., [18, 19]). The diffuse and solid regenerator techniques are nicely summarized by J.M. Gaillard in [20]. Each group referenced above used different regenerator thicknesses, resulting in increased sensitivity for interference effects at different distances from the regenerator. Both experiments were performed at the CERN Proton Synchrotron (PS). Figure 3.7 shows the measurement of C. Alff-Steinberger et al., who used copper as the regenerating material (M. Bott-Bodenhausen et al. employed carbon plates). The theoretical expectations, expression (3.91), for different regenerator densities were fitted to the data to obtain the best values of l0cl, I?]+-J, ~mL,s and r -- r - r (Fs was known to an adequate precision). The distributions corresponding to the best fit solutions are also presented in Fig. 3.7. No acceptable fit was obtained without the interference term. The latter was extracted from the data for each distribution in Fig. 3.7 by subtracting the quadratic terms according to the fitted parameters J0cJ and J~+_J, and then dividing by 21pcI]~+-Jexp -(rs +rL)t/2. The results from both groups are shown in Fig. 3.8. The oscillatory behavior of the interference term provides a precise measurement of the mass difference IAmL,S] and the phase difference r (remember, m L - - m s > 0, as demonstrated by W. Mehlhop et al. [12]). Since we will describe shortly an experiment that measured the phase difference

54

3. C P Violation in K ~ Decays

9

L

ID

w Z v

r~

~

o

9 a s ~ . O i I N . L N 3 A ~J

..o

hill

J J l

(2 ~$

R c - O i l SJ, N 3 A 3

-

//

IN

:Das,~.O~ I S J . N 3 A 3

Fig. 3.7. Event rates measured by C. AlffSteinberger et al. [18]

~ K S~ Interference as Evidence for 3.5 K L-

CP Violation

55

I'(

0'"

-0,

-0" -b

-1.{

0

~

/

)[

~ 'lP(Z IO"~)

0

,

Z

3

4

,

.

,

.

1. ( s NO'l~

F i g . 3.8. Interference terms measured by C. Alff-Steinberger et al. Bott-Bodenhausen et al. (right)

(left) and M.

r = r - r a n d , concurrently, t h e r e g e n e r a t i o n p h a s e r m u c h m o r e precisely t h a n t h e C E R N g r o u p s did, we do n o t p r e s e n t t h e values of t h e i r fitted parameters.

4. Interference in Semileptonic and Pionic Decay Modes

4.1 Semileptonic Decays of N e u t r a l Kaons In our discussion of the K ~ system so far we have associated the (K ~ doublet with strong interactions and the (K ~ K ~ doublet with weak decays. This classification is, in fact, not entirely correct. The reason is that the semileptonic weak decays of neutral kaons (see [21]), Neutral kaon --+ e+rrmve,

#+Trmv~,

(4.1)

have also been observed. These final states are clearly not CP eigenstates, for they transform into one another under CP. Consequently, they cannot be described in terms of the CP eigenstates K ~ and K~ instead they are decay modes of the strangeness eigenstates K ~ and/~o. The semileptonic decays of strange particles obey the selection rule AStrangeness = ACharge

(AS = AQ)

(4.2)

first postulated by R. Feynman and M. Gell-Mann in 1958. As an example of this rule, the decay ~- -+n+e-

(4.3)

+~

is observed (branching ratio ~ 10-3), whereas 57+ --~ n + e + + ve

(4.4)

is not (branching ratio < 5 • 10-6). In the case of neutral kaons, the AS = AQ rule implies K ~ -~ e%r-v~,

R ~ ~ e-Tr+~.

(4.5)

A test of (4.2) is shown in Fig. 4.1, where the measured distribution of K ~ ~ e+Tr-ve and /7/0 __+ e-rr+#e events from an initially pure K ~ state is plotted as a function of the K ~ decay time [22]. The result is in good agreement with the strangeness oscillation plot of Fig. 1.3, thus confirming the AS = AQ rule (to about 2%). A related measurement is that of J. Steinberger and his collaborators [23], who obtained the time dependence of the charge asymmetry: N ( K ~ ~ e+r-v~) - N ( f ( ~ --+ e-Tr+p~) N + - NA ( t ) = N ( K o --+ e+~r_ue) ~ N - - ~ - - ~ e-zc+o~) =-- N + ~ N-

(4.6)

58

4. Interference in Semileptonic and Pionic Decay Modes

1.0

9 N+ o N"

--x,O 0.8

0.6

0.4

0.2

I

.2

t

.4

I .6

I

I T

"

98"10"9s

eigentime

Fig. 4.1. A test of the AS A Q rule usin K ~ -+ e+Tc-uc and K w --+ e-~+~c decays [22])

(see Fig. 4.2). From expressions for the K ~ a n d / ~ 0 probabilities (1.23) and (1.24), it follows 23 that .A(t) = 2e-(Fs+FL)t/2 COS((~mt) = 2COS((~mt) e-FS t + e--FL t e - A F t ~- e+AFt

(4.7)

for a pure K ~ at t = 0 ( A F -- (Fs - FL)/2 and t is the proper time). Again, the measurement is in good agreement with (4.7) in the strangeness oscillation region. Note the apparent decay rate a s y m m e t r y between the K ~ --+ e+~r-L,~ a n d / ~ 0 ~ e - ~ r + ~ decays for large values of the K ~ decay time in Fig. 4.2. Assuming the validity of the A S = A Q rule, this a s y m m e t r y represents the CP-violating effect mentioned in Sect. 3.1. 23 For a beam of neutral kaons, the probabilities can be replaced by the number densities of particles and antiparticles in the beam.

4.1 Semileptonic Decays of Neutral Kaons

59

0.075 7" 0.05 0.025

z+

§

0

~-§

.

I

i J

"

I

-0.025

'

'

j

*~'

~

-

'

:'

2"

' r (lO-l%er 0 10* 'Decaytime

-0.05 -0.075 Fig. 4.2. Charge asymmetry as a function of the Kl3 decay time [23] In order to study the semileptonic decays Ks ~ K 0 --+ ~4-71"T/2 in more detail, we define the following transition amplitudes f = (t+~r-v [/:/w [K~

A S = AQ,

g - (e+~-~ I//w I K~

AS = - A Q .

(4.8)

Assuming CPT invariance, we have f* ---- (e-~+~ I ~qw I K~

as = aq, AS = -AQ.

g* - (e-~+~ I/tw I K~

(4.9)

The operation of ~/5 transforms particles into antiparticles. If CP is conserved, both f and g are real. The amplitudes g and g* violate the AS = AQ rule. This violation is small (g/f <_ 10-2), as mentioned above. We therefore adjust the relative phase between K ~ and /~0 so that the AS = AQ amplitudes are real: f = f* (see footnote 5). The time development of an initially pure K ~ (/~0) state can he written as (see (3.17) and (3.32))

I~,(t)) -- ~1 {(1 :F e)e-iA~stIK~ • (1 T e)e-iA%tlK~

(4.10)

where the upper (lower) signs refer to K ~ (/~o). Expressing K ~ and K ~ in terms of K ~ a n d / ~ o (see (3.16)), we obtain

1 ~'e_i.Adst

I~V(t)) = 2 t

[iKO> + (1 - 2e)[/(~

+ e--iA4Lt[IK0

-

(1 -

2e)[/~~

"~, (4.11)

1 {e-i.Mst [[/~0) + (1 + 2e)[K~ Ig,(t)) = I

+ e -i~aLt [I/f"~ - (1 + 2e)lK~

}.

60

4. Interference in Semileptonic and Pionic Decay Modes

The transition amplitudes for ~P(~) --+ t~=LTrT=uread

A+ = ~/ [ e - '"~ s t ( 1 + x - 2ex) + e - i ~ L t ( 1 A_

x+2ex)],

= L [e-i2~4St(x * + 1 -- 2e) + e-iMLt(x * -- 1 + 2e)] 2

/~+ ---- ~f

(4.12)

[e--i'AASt(x+ 1 + 2e) + e - - i ~ L t ( x - 1-- 2e)] ,

A_ ---- f [e-i2cist(1 + x* + 2ex*) + e-iJv[Lt(1 - x* - 2ex*)] 2 where we defined x =- g / f . A nonzero value of x implies a violation of the A S = AQ rule; Im x =/= 0 indicates C P violation. Squaring the above amplitudes gives the corresponding decay rates (we omit the factor lfl2/4):

s

[ T _ ] o~ {11 + xl = :F 4lxl2Re ~} e -r~t + { l l - xl = :F 4lxl2Re ~}

e -rLt

+ 2(1 - Ixl2)e -F' cos(Star) :]: 4Im x e - r t sin(Smt), (4.13) F_ [ F + ] o ( { [ l + xl2~:4Re <}e - r s t + { 1 1 - xl2~: 4Re e } e - r L t

- 2{(1 - I x l =) :F4Re e} x e -s

cos((~mt) T 4Im x e - s

sin(Smt),

where the upper (lower) signs refer to K ~ (/~0) and F --- (Fs + FL)/2. Note that the above decay rates depend on the real part of e. Neglecting the small C P violation, F•

= 0) ~ }1 + xt2e -rs~ + I1 - xl2e - r L t 4- 2(1 - lx}2)e - r t cos(Smt) 4Im x e - r t sin(Smt).

-

(4.14)

In the experiment by Niebergall et al. [22], the initial K ~ state was produced in the inelastic charge-exchange reaction K + p -+ K~

(4.15)

+

using a 2.4 G e V / c K + beam. From a sample of 4724 K ~ --+ e+Tr:F~ decays, containing less than 8 background events, they obtained 5m --= AmL,S = (0.53 + 0.04) x 10 l~ S-1

(4.16)

by comparing their data with the theoretical prediction for the time dependence of charge asymmetry in Ke3 decays: F+ - F_ o< (1 - [xl 2)

e -Ft

COS((~mt).

(4.17)

The time distribution of events, together with the theoretical expectation for x = 0, is shown in Fig. 4.1. The best fit to the data yielded Re x = 0.04+0.03 and Im x = -0.06 • 0.05. Combining this result with other measurements referenced in their paper, they found Re x -- 0.023 • 0.02,

Im x -- -0.0015 • 0.025,

(4.18)

4.1 Semileptonic Decays of Neutral Kaons

61

which is consistent with x = 0, i.e., they saw no evidence for the AS = - A Q transitions in semileptonic decays of the neutral kaon. To determine the time dependence of charge asymmetry

.A(t) = F+(t) - F_(t)

r+(t) u r_ (t)'

-fi+(t) - F _ ( t )

A(t) = r--+ (t) + r--_ (t)

(4.19)

(F+(t) and F_(t) are the decay rates to positive and negative leptons, respectively) for an initially pure K~ ~ beam, consider neutral kaon decays in vacuum close to the beam production point. From (4.13) it follows that A ( t ) [A(t)] =

(4.20)

2(1 - Ixl 2) {Re e [e - r s t + e -rLt] + e - r t cos(Smt) }

I1 + xl2e - r s t + I1 - xl2e -rLt + 4e - r t [Re e cos(Smt) - I m x sin(Star)] ' where the upper (lower) signs refer to K ~ (/~o). For x, c -+ 0 this expression becomes (4.7), as it should. A neutral kaon beam produced by protons hitting a stationary target is a mixture of K ~ and /~0 mesons. In fact, it is an incoherent mixture because K ~ and its antiparticle are produced differently. Consequently, the interference terms in (4.20) have to be multiplied by the dilution factor/)(p): +COS(Smt) -+ /)(p)COS(Star), T sin(Smt) -+ --/)(p)sin(Smt) (see (4.42)); T)(p) is defined by

I)(p) - S(p) - S(p)

(4.21)

where S(p) and S(p) are momentum-dependent production intensities of K ~ a n d / ~ 0 mesons, respectively. Note that for S(p) = S(p) there would be no interference at all. However, because of the lower production threshold for K ~ (Sect. 1.5), S(p) >> S(p) in general. In the interference region e -rLt ~ 1 and e -Fst (( 1, and so (4.20) simplifies to A(t) - 2(1 - I x l 2) {7)(p)e_(rs_rL)t/2 eos(Smt) + Re ~}.

I1 xl:

(4.22)

The apparatus employed by J. Steinberger and his coworkers in the experiment mentioned above is described in Sect. 4.3. Semileptonic decays were selected from the data by an unambiguous lepton signature on one of the two charged decay products of K ~ The reconstructed momenta of charged tracks were restricted to (a) Pe,P,~ < 8GeV/c in order to eliminate pions above the Cerenkov threshold (8.4 GeV/e) from the Ke3 sample, and (b) p~ > 1.6 GeV/c, since the minimum momentum to reach muon counters was 1.45 GeV/c. In total, 6 million K~3 and 2 million K~3 decays with lifetimes t ~ 12.75 • 10-1~ remained after these and other cuts to remove possible backround events. The mass difference A?TtL, S Was determined from a comparison of the measured ~4(t) distribution (see Fig. 4.2) with the theoretical prediction (4.20),

62

4. Interference in Semileptonic and Pionic Decay Modes

(4.22), taking into account radiative corrections, experimental resolution and acceptance, etc., with the result AmL,S = (0.5334 3- 0.004) • 101~s -1.

(4.23)

From (4.22) we see that ,4(t) ---+

>>2Ree,

Fst ~ l,

2Ree,

Fst >> l.

(4.24)

Therefore, at Fst >> 1 the charge asymmetry is given by the second term in (4.22): F ( K ~ --+ g+Tr-v) - F ( K ~ --+ g-zr+v) 1 -Ixl 2 AL(t) = F(KO -~ *+7r-v)+ F ( K ~ -+ •-zr+v) ~ 2Re e F1 L-~-~" (4.25)

Assuming that x = 0, a measurement of Aa(t) at Fst >> 1 yields 2Re e = (K~ ] K~>. To keep the effects of K ~ ~ interference as low as possible, they selected events according to t~3 > 12.75 • 10-1~ and t#3 > 14.75 • 10 - l ~ s. Based on a total of 34 million Kr and 15 million K~3 events, their measurement yielded (see [24]) Re r = (1.67 + 0.08) x 10 -3.

(4.26)

Using (3.80) Re c =

17/+- I v/1 + (2AmL,s/Fs) 2

(4.27)

and the measured values of ZimL,S, ]U+-[ and Fs (see Sect. 4.3), they computed Ir/+-I = (1.66 J: 0.03) • 10 -3, v/1 + (2n,nL,s/rs) 2

(4.28)

in good agreement with the above result.

4.2

o o Interference K~-K~

in ~-+~-- a n d s

Decays

We now describe a high-precision K ~ ~ interference experiment which measured the phase difference r -r using the pionic decay modes K ~ --+ 7r+Tr- and, concurrently, the regeneration phase r from the timedependent charge asymmetry in K~ --+ e+zrT=v~ and #+Tr~=v~ decays [25]. In this experiment, a KL~ beam of 4 to 10 GeV/c momentum from the Brookhaven AGS traversed an 81-cm-long block of carbon. Multiwire proportional chambers were used to measure time distributions of 7r+Tr- and semileptolliC decays behind the regenerator.

K L0 - K s0 Interference in 7r+~r- and g•

4.2

Decays

63

The semileptonic decay rates F+ and F_ behind a regenerator are obtained from (3.89) by repeating the steps which led to (4.13), with the result F+(0c) c( +2Re e(1 - I x [

2) {Iocl2e -rSt + e -rL+ }

+ I1 + xl21ocl2e -rSt + IX - x[2e - r L t

+ 2 {2Re e • (1 - I x 1 2 ) }

loci

-- 410olin x e - r t sin(6mt + r

x e - r t COS0mt + r

(4.29)

In the interference region, e -rLt ~ 1 and e - r s t << 1. Observing also that 10cl2 << 1, (4.29) yields

A(t)~ 2(1-1xl2) {Igcle-(rs-lu162 I1-~

(4.30)

which is the expression used by W. Carithers et al. [25] The measured charge asymmetry of K~ - K ~ ~ e• and /403 - K ~ -+ #+lr+v u decays as a function of proper time is shown in Fig. 4.3. Prom this asymmetry they extracted the nuclear regeneration phase, r defined by r

(4.31)

= Cf + r

where e-(-i6m+l/2)< ] -- arg /r 1 (4.32) Ti? ] L Namely, they split the phase of the coherent regeneration amplitude (see (2.39)) r

r

-- arg [ i ( [ - ~ ) ] ,

% 8

'

1

[

I

f

0

2

4.

6

(a)

S

uO

I

[

r ' x IO-tOsec

%

I

'

I

l

,

I

I

I

8

4,,

0

2

~

I

4 r' x

~

I

g 10"L

6

I0-~~sec

: ~ + I

8

,

~, ~ : I

iO

J

Fig. 4.3. (a) Kc3 charge asymmetry versus proper time; (b) Ku3 charge asymmetry [25]

64

4. Interference in Semileptonic and Pionic Decay Modes

N.~sAs [ i ( f - 9 ) ] 1 - e--(--i~m+l/2)~ --i~m + 1/2

2

pc = IQcle ir176 -

(4.33)

into two parts, one of which, r can be easily computed. The data were simultaneously fitted for XIL~cland r with AmL,S, FS and AL fixed; X and AL are defined as (see (4.25)) 2(1 - I = 1 2 ) Ii-xl

X =

A L ( t ) - 2xRe

2

~ = x(K ~ I K~ 9

(4.34)

'

AL(t) is the charge asymmetry in the decays K ~ --+ g+or~=u, first observed by J. Steinberger and his colleagues at Brookhaven [26], and by D. Dorfan et al. at SLAC [27]. The value of .AL(t) used by Carithers et al. is from [24]. The combined g e 3 and K#3 data yielded Cf = -41 ~ + 2.6%

(4.35)

The 7r+Tr- intensity in the forward direction behind a regenerator placed in a K~ is given by (3.91). Similar to the first part of their analysis, the ;r+;r - data (see Fig. 4.4) were fitted simultaneously for IQr r -r and _Fs, with AmL,S and EL fixed. Combining the result for r - Cf with the measurement of Cf (see Fig. 4.5), the CP violating phase r/+_ was found to be r

= 45.5 ~ + 2.8 ~

(4.36)

iO 6

2 ~ INTENSITY ( 4 - 1 0 G E V ) [ EXPERIMENT~,L POINTS -NO INTERFERENCE

1os

io 4

9

\ I

I0 z

0

I

2

3

4

!!

.1111{i i I

5 6 7 8 9 I0 II PROPER TIME

12 13 14 x I0-10 =er

Fig. 4.4. Rate of fr+Trevents as a function of proper time downstream from a carbon regenerator (T. Modis, Columbia Univ. thesis, 1973; and [25])

4.3 KL-K 0 s0 Interference Without Regenerator I

I

-2.0

I

I

I

0

(~f-~B+_)2 ~ DATA

9

~f K/.L5 DATA

9

~f Ke5 DATA

65

I

-I.6 C

.o - 1 . 2 "O O

~- - o . 8

-

-0.4

-

oL

1 4

0

I 6

I

12

I0

8

PK (GeV/c)

Fig. 4.5. Results of the combined fit for 6+- [25]

Their analysis can be summarized as follows: ~+TrTy, [Smt + ( e l + r 7r+Tr- , r

>r

-41 ~

[Smt+(OS + r 1 6 2 1 6 2 1 6 2 1 6 2

~-87~

(4.37)

~ 46 ~

4 . 3 K ~o - K so I n t e r f e r e n c e

Without

Regenerator

The CP violating phase r can be measured independently of the regeneration phase Ce by observing K L0- K s0 interference in vacuum close to the K ~ 1 6 3~ production point. This so-called vacuum interference method requires the mass difference A~mL,S to be known very accurately: a 1% error in z~mL,s corresponds to an uncertainty of 3 ~ in 0+_. Here we describe an experiment by J. Steinberger and his collaborators at the CERN Proton Synchrotron, 24 who used results from two high precision measurements of Ama,s in the same detector [23, 28] to obtain r They were the first to employ multiwire proportional chambers (MWPC), invented by G. Charpak. MWPCs can handle event rates that are hundreds of times higher than those possible with spark chambers. The detector, described below, was also used in the charge asymmetry measurement discussed earlier. 24 A CERN-Heidelberg collaboration.

66

4. Interference in Semileptonic and Pionic Decay Modes

The apparatus is sketched in Fig. 4.6. The neutral kaons, produced by 24 GeV/c protons hitting a platinum target, were selected by a collimator at an angle of roughly 75 mrad. Protons with such momenta produce at small angles about three times as many K~ a s / ( ~ The kaon momenta were in the range 3-15 GeV/c. The collimator was followed by a 9-m-long decay volume filled with helium ("cheap vacuum"). A 6-m-long threshold Cerenkov counter, containing hydrogen gas at atmospheric pressure, was used to identify electrons. Muons were identified by two counters behind a concrete absorber at the far end of the detector. The decay region, extending 2.2 m to 11.6 m after the target, permitted detection in the proper time interval 3.5 • ]0 -10 S < tp < 30 z 10 -10 S. The momenta of charged decay products of neutral kaons were measured in a spectrometer consisiting of four MWPCs and a bending magnet. A total of 109 events was registered, with an average rate of about 1000 events per machine cycle. To select K~ --+ 7c+7r- events only inward bending pairs of charged particles were retained. They also required that: (a) there must be no signal in the Cerenkov counter and no coincidence between the two muon counters; and (b) the momenta of both particles must lie in the interval 1.5 GeV/c to 8.5 GeV/c, i.e., above the minimum momentum to traverse the muon absorber (1.45 GeV/c) and below the threshold for pion detection in the Cerenkov counter (8.4 GeV/c). To derive the 7r~rdecay distribution from the interfering K ~ and K~ states, let us assume that at t = 0 a pure K~ ~ beam is produced. At a later time t (see (4.10)) [~P(t)) = ~ 1 {(1 =7 e)e -iA4st lKs) 0 :h (1 ~

e)e-i~LtlK~

(4.38)

where the upper (lower) signs refer to a K ~ (/~0). Note that the phase between KL~ and K ~ at t = 0 is 0 ~ (180 ~ if the original state is a K ~ (/~0). The decay amplitudes read 1 A~r = AKo._+Tr~-~ e -irnst X {(1 ~ e ) e -rst/2 + (1Te)lrll e-i(~mt-r and the corresponding decay rates (we omit the factor Fs,+_/2, i.e.,

(4.39)

Fs,oo/2)

F,r. o( (1 T 2Re e) • {[r/12e-FLt + e - r s t

-t-2[rile -(rs+f'L)t/2 COS(Smt--r

(4.40)

The interference term changes sign when K ~ is replaced by/~o, resulting in different ~rTr decay distributions for the two states, as shown in Fig. 4.7. This illustrates nicely the violation of CP symmetry. The distribution (4.40) is practically identical to that behind a regenerator (see (3.91)), with the regeneration amplitude 0c = 1.

4.3 K Lo- K so Interference Without Regenerator

67

rr IJJ

_k

LJ~ rr laJ

r W ~D O re

a: r

ot tLJ r

Z

i r

t~J r~

~w~N

~7

tlJ m

{2.

Fig. 4.6. The apparatus used by J. Steinberger and his collaborators at CERN

68

4. Interference in Semileptonic and Pionic Decay Modes 10-3,

10 - 4 -

"

K 0 "-* 7rTr

r~

10-s

~o..~

~..

:. :

: : 9

10 -6

0

i

i

5

l0

'"

o

i

i

i

I5

20

25

30

['st

Fig. 4.7. Difference in ;rrr decay distributions between initially pure K ~ and /2/o beams, illustrating the violation of CP symmetry

T h e magnitude of rj+_ was obtained by measuring the ;r+rr - decay rate at Fst >> 1, FLt << 1 (t _> 15 x 10 -1~ T h e phase of 77+_ was measured in the interference region (5 x 10-1~ < t < 15 x 10 - l ~ s) by isolating the cosine-term. The results were [29, 30] r

= 45-9~ • 1-6~

I~+-I = (2.3 4- 0.035) x 10 -3.

(4.41)

To obtain r they used AmL,S = (0.5338 4-0.00215) x 101~ -1, the combined value of the two measurements of AmL,S in the same detector referenced earlier (one measurement was based on the variable gap regeneration method, and the other one on charge a s y m m e t r y in semileptonic K ~ decays). In fact, their analysis was slightly more complicated than this simplified description because the neutral kaon b e a m is an incoherent mixture of K ~ a n d / ~ 0 particles, as explained in Sect. 4.1. The rrTr decay intensity is therefore a linear combination of the two distributions (4.40):

x

{ I.12e-rL ~+ e - r s t + 2V(p)lule-(r~-cL)~/2cos(Star -

r } (4.42)

4.3 KL-K o so Interference Without Regenerator

69

(see (4.21)). Expression (4.42) was fitted to the ~ + ~ - data to obt&in Fs, I~+-], r S(p) &nd S(p), assuming that AmL,s and FL are known (see Fig. 4.8). The fit yielded Fs = (1.119 • 0.006) x 101~ s - ] .

A'~ ~ ~'~"

(a)

event

(4.43)

rate

10 5

-- ~ ~

I O"

With interference interfelnce

10 3

10 2

~

+I

L

l

L

(b)

Ai

g I

o u

5

.L 10 ro 10" io sec

j 15

.L__

Fig. 4.8. (a) The measured K ~ -+ 7r+Tr- event rate and (b) the extracted interference term [29]

5. P r e c i s i o n and #/e

Measurements

of r

r

The decay modes KL~ --+ 7r~ ~ are considerably more difficult to investigate experimentally than their charge counterparts K~ --+ 7r+Tr- . The reason is that neutral pions cannot be observed directly: a 7r~ decays within 10-16s into two photons. Instead, one has to detect and measure electromagnetic "showers" associated with the reaction shown in Fig. 5.1. The difficulty lies in measuring accurately the direction and energy of the final state photons. Y

K~

_~ " ' ~

Y

F i g . 5 . 1 . K ~ --4 lr%r~ --4 47

An additional complication arises from the fact that the C P conserving decay K ~ --+ 37r~ --+ 67 is two hundred times more frequent than the CP violating decay K ~ --+ 27r~ --4 47. Since 37r~ decays can simulate 47 events if two photons are not detected, one has to rely on distinct kinematical features of the 27r~ decay mode to eliminate this background. As the photon energy increases, its measurement precision improves. The two experiments described below used intense fluxes of neutral kaons with energies of about 100 GeV. High beam intensities are essential to achieve adequate statistical accuracy, especially in the K ~ -+ 27r~ channel. Both experiments measured the CP-violating parameter st/s, which can be related to the K~ -+ 7r~ ~ 7r+Tr- decay intensities in the following way. Using

170o 12 - rL,oo

rs,oo '

17+-I2 - /'L,+-rs,+_

(5.1)

(see (3.46)) we form the double ratio of decay intensities

]~7oo ]2 _ F ( K o _+ 7r%O)/F(KO __+ ~-o~-o) I~+_l 2 r ( K o ~ ~ + ~ - ) / V ( K O ~ ~§

(5.2)

72

5. Precision Measurements of r

r

and

~'/~

From expressions (3.72) and (3.73) it follows that ~?oo ~ c ( 1 - 2 ~ )

,

r/+_ ~ e ( 1 + ~ ) .

(5.3)

Hence,

] and Re

(@)

1[ ~ ~ 1

(5.6)

[ rio012 ] 1~+_12j .

By observing all four K L,S ~ ~ 7r%r~ 7r+~r- decay modes simultaneously, or at least two at a time, beam intensities and detection efficieneies cancel in the double ratio (5.2), thus minimizing systematic uncertainties in the measurement of e'/c.

5.1 The

Experiment

HA31

at CERN

We first describe an experiment by J. Steinberger and his coworkers at the CERN Super Proton Synchrotron (SPS). 25 Intense beams of KL~ and K ~ mesons with energies around 100 GeV were produced alternately by 450 GeV protons (1011 and 107 protons per pulse, respectively) at two different targets (see Fig. 5.2). The ~r%r~ and 7r+Tr- decay modes were detected concurrently, however. The K ~ data were taken with the corresponding target displaced in steps of 1.2 m, which resulted in uniform K ~ and K ~ decay distributions over a 48-m-long decay region, despite the short Ks~ decay length (6 m on the average). The decay region was evacuated and the space between two tracking wire chambers, set 25 m apart, was filled with helium. Photons from 7r~ decays were measured in a liquid-argon/lead calorimeter which was also used, together with an iron/scintillator calorimeter, to measure the energy of charged pions. There was no magnetic spectrometer. The KL~ --~ ~r+Tr- decays were reconstructed from hits in the two wire chambers, and the K ~ ~ 7r%r~ decays from the measured positions and energies of the photons. The energy spectra of accepted 7r%r~ and 7r+Tr- events are shown in Fig. 5.3. After corrections for various systematic uncertainties, the analysis, based on their 1986 data, yielded Re ( - ~ ) = ( 3 . 3 + 1 . 1 ) x 1 0 -3, 25 The HA31 Collaboration.

(5.7)

5.1 The Experiment NA31 at CERN

73

anficount er -ring!;

V 'q

] .....

Ilffll

beam dump

/

i , , .... ~176 ,

| k

'

II

I' M

co,,ma*0~s \

~-----~1

tllll

lifl-....

veto counters

I Ill~" h,d~o. calorimeter phofon calorimeter

wire chamber 1

wire chamber 2

Fig. 5.2. The NA31 experimental set-up at CERN which was interpreted as the first evidence of direct CP violation in the KL~ -+ 2~r decay [31@ Using data collected in 1988 and 1989, they reported Re (e'/g) = (2.0 + 0.7) • 10 -3 [315]. The NA31 collaboration also determined the phases of the CP-violating parameters U00 and 77+_ fi'om the time dependence of the 27r decay rates by using (4.42). F ~ is most sensitive to r in the region where K ~ and K ~ decay rates are about equal ( ~ 12K ~ lifetimes). The original beam layout was therefore modified to obtain the maximum acceptance in the interference region. The phases r and 0o0 were determined from the ratio of decay distributions for two different target positions (one near to the detector, ]Cnear, and one far from the detector, ]Cf~r), which renders the acceptance correction negligible. The maximum sensitivity is obtained when the interference patterns from the two targets are displaced by ~/2; at 100 G e V / c this corresponds to a distance of about 15 m. The measured decay rates from the combined /Cnear and Kfar data are shown in Fig. 5.4. The interference term was extracted by subtracting the fitted lifetime distribution without interference from the data. The phases r and r were obtained in the following way. They took the ratio of events observed from the two targets (see Fig. 5.5). A simultaneous fit to (~Cnear/K:far)00 and (K~near/]~far) +_ was made using bins of 5 GeV/c momentum and 0.5~'s (rs was computed for each event from the mid-point between the two targets). The phases r and r and the dilution factor :D(p) were varied in the fit, while TS, TL, I~+--I, I~lool and AmL,s were fixed. Taking into account various systematic uncertainties, they obtained [32] r r

= (46.9 ! 2.2) ~ = (47.1 4- 2.8) ~

(5.8) )r

- r

= (0.2 4- 2.9) ~

74

5. Precision Measurements of r

r

and r

K~--> 7T+n .~, 105

K~---> ~"TT ~

K~'--> ~*TT-

10 4

K~--'>~o~o 103

102

60

80

120

100

160

160

180

F i g . 5.3. Energy spectra of ~r~ ~ and Tc+Tr- decays a n d the corresponding event statistics [31a]

200 GEV

KAON ENERGY

K~--> .ri.Orf ~

K ~ --~ ~ * ~ . . . .

1.6 ~

:i~',"

K~--> n'+/r-

1.2

1 t

/tt t

E 107

106 -1.6~

6

8

10

12

6

14 16 18 Ks Lifetimes

8

10

12

14 16 18 Ks Lifetimes

105

oD m

104

i 5

7.5

10

t2.5

15

17~

20 2z5 25 K s Lifetime~

5

,

,

i

7.5

,

,

i

10

,

,

i , ,

12.~

i

15

,

,

]

,,

17.$

i

,

-1 i

,

L

20 22.5 2~ K, Lifetimes

F i g . 5.4. Acceptance-corrected lifetime distributions. Insets show the difference between a fit without the interference term and the data, averaged over energy [32]

5.2 The Experiment E731/E773 at Fermilab

75

KO > ~+~14 __ 12

_

,

IO

With int*r~=rence _

9

~thout ;hterference

12

P = I O0 GeV

~o

__

With interference

_ _

W~thout i n t e r f e r e n c e 9

P=

IOOGeV.

@ 6

2

$

7.5

10

12.5

I$

t7.5

2

22.$

25

5

7.5

I0

t2.$

15

17.$

20

22.5

25

Ks L i f e t i m e s

Ks L i f e t i m e s

Fig. 5.5. Ratio of decay distributions for two target positions [32]

5.2 The Experiment E731/E773 at Fermilab The second experiment was performed at Fermilab by B. Winstein and his collaborators, 26 who used 800 GeV protons incident on a berillium target to produce two parallel kaon beams, one pure K ~ and one with coherently regenerated K ~ mesons (see Fig. 5.6). In this experiment IPr ~ 10 • I~l, and so the 2~r-decays from the regenerator b e a m were mostly K s~ mesons. This way they obtained K ~ and K ~ beams with almost identical m o m e n t u m and spatial distributions. The regenerator alternated between the beams once

Photon Veto..~ .........

PhotonVeto

Muon Veto Lead Glass "...

/i~.,,~:i ....................li

i~

' iiil.ii.1, I' :i I ......i.

~ch~,s-?-":~!i::il I.......i...........j........... I "" = 2,omI Tngger Planes

Magnet

, 10 m

Fig. 5.6. Detector layout of the experiment E731 at Fermilab every minute, thus essentially eliminating any small difference in b e a m intensity or detector acceptance for decays from the two beams. However, since TL >~> TS, the detector acceptance as a function of decay vertex must be 26 The E731/E773 Collaborations.

76

5. Precision Measurements of r

o.•'"•,S'•~,,

10 3 --

KS

r

and c'/e

I OATA ,' MONTECARLO

-

-

~, ~t

0

10 2

0 tn

r

~

Z tJJ Q

bJ

10

tt4 t j ,: I 120

I , 130

VERTEX ( m )

rl 110

150

Fig. 5.7.7r~ ~ decay positions

[331

precisely known. This acceptance was determined by using a highly detailed Monte Carlo simulation which relied on K~3 and 37r~ decays. To minimize systematic uncertainties, K ~ and K~ decays to 7r~ ~ and 7r+:r - final states were detected simultaneously. A drift chamber spectrometer was employed to determine the 7r+Tr- momenta, mass and decay vertex. The energies and positions of the four photons from the :r%r~ decays were measured with a lead-glass calorimeter. The KL~ --+ ~r%r~ decay position and the 7r%r~ effective mass were obtained from the best pairing of photons into two pions (see Figs. 5.7 and 5.8 [33]). Semileptonic events were removed from the 7r+:r - sample using the ratio of shower energy to track momentum (for Ke3 decays) and a muon "hodoscope" (for Ku3 decays). The 37r~ backround to the 7r~ ~ data was estimated by Monte Carlo calculations (Fig. 5.8). After background subtraction, the full E731 data set contained (3.27 • 1057r+~r- , 4.1 x 1057r%r~ vacuum events and (1.06 • 10%r+Tr- , 8.0 • 105:r~ ~ regenerator events. The data were collected in 1987 and 1988 at the Fermilab "Tevatron" accelerator. To obtain Re (s'/g), the ratio of vacuum to regenerator events was fitted in momentum (p) and decay position (z) bins by using the following expression for the event rate downstream of the regenerator:

dpdzdN o( ~c(p)e_Z(Fs/2_iz~mL.s)//3. w ~- T/e_ZFL/2/3.yc 2 ,

(5.9)

5.2 The Experiment E731/E773 at Fermilab I

i

i

i

77

i

10 3 9 3~T" BkCKGROUNO

f OkTX

>

[0 2

O4

W

tO o 9 1 4 99

w.

1

420

[

I

]

I

I

440

460

480

500

520

HAS$ ( HeY

540

Fig. 5.8. 7r~ ~ effective mass

]

where c is the speed of light in vacuum, -), = Ek/rnk, 77+_ = s(1 + s'/e) and ~1oo = e(1 - 2el/e). In the fit they used (a) their own values of AmL,S and Fs, (b) the world average of Ir/+_l for H , (c) r = arctan(2AmL,s/Fs), (d) r = (43 + 6) ~ and (e) the empirical power-law parametrization of the regeneration amplitude

~0c(p) (2( p--C~e--i(2--c~)Tr/2

(5.10)

(see [34] regarding the above parametrization). Fits were first done for each decay mode separately, setting s / = 0, to extract a and Pc. The 7c+7r- and 7c~ ~ results were found to be mutually consistent, which points to a small value of el/s. A grand fit was next made to both modes simultaneously for the value of Re (s'/s), allowing the regeneration parameters to vary. They obtained [35] Re(~)=(0.74•215

-3,

(5.11)

a value not significantly different from zero. The full E731 data set was also used to measure the neutral kaon parameters ArnL,s, TS, r and Ar =-- r - r To extract AmL,S and ~-s they fixed r

~ r

~ arctan \

~

= 43.7 ~

(5.12)

78

5. Precision Measurements of r ....

I ....

r

I ....

i ....

I ....

I ....

L ....

I'''

....

i ....

I ....

I ....

I ....

I ....

i'''

ill,

,,It

,,,,

,,,i

,,,,

JlJJ

,ll

and e'/e

1.2 ~

0.8

o~

0.4

"

0

-04 -0.8 ~

-1.2

[-. eo

M

10"~ 10 4

10: i,i,

2 3 Time a (21 0 "l&s) Proper

,

'

I

'

I

'

I

'

I

""

Fig. 5.9a,b. Distributions in proper time for lr+lr - decays. The lines are the best fit results described in [35]

, I

'

1.2

[-..

0.8

O

0.4

~ ~

0

r/

-0,4

N -0.s *'* -1.2 l

(b)

10.1 O

r

f~'b,,

t 0 "J 10.4

0

2

4

6

8

I0

Proper Time (•176

12

Fig. 5.10a, b. Distributions in proper time for 2~r~ decays. The lines are the best fit results described in [35]

and simultaneously varied rs, ArnL,S, a and Oc(P = 70 G e V / c ) in expressions (5.9) and (5.10)). T h e extracted interference and exponential terms, together with the superposed best fits, are shown in Figs. 5.9a,b and 5.10a,b [35]. Combining the values for ~ m L , s and TS obtained from the two decay modes, they found

AmL,S =

(0.5286 + 0.0028) x 101~ -1,

'rs = (0.8929 -t- 0.0016) x 10 -1~ s.

(5.13)

5.2 The Experiment E731/E773 at Fermilab

79

The AmL,S result was lower than the existing world average by about two standard deviations. Based on the reported dependences upon AmL,S, they corrected the best previous measurements of r by using their value of Z~mL,S. The corrected values were found to be in excellent agreement with each other and with (5.12), as expected from CPT symmetry. To extract Ar a simultaneous fit to the charged and neutral data was made, allowing r Ar and s//g to vary, with the result 27 r

- r

= ( - 1 . 6 + 1.2) ~

(5.14)

A similar fit with AmL,S floating yielded r

= (42.2 =t= 1.4) ~

(5.15)

in agreement with (5.12), which is based on the world-average values for AmL,S and Fs. The apparatus of experiment E773, which took d a t a in 1991, was essentially the same as that of experiment E731, the main difference being that K ~ mesons this time impinged on two different regenerators, one placed 117m and the other one 128 m from the target. For this run a new "active" regenerator made of plastic scintillator was used, thereby reducing inelastic regeneration by a factor of 10 (kaons scattered inelastically may be assigned to the wrong beam). Downstream of its regenerator each b e a m is a coherent superposition of K ~ and K s~ mesons. The 27r decay rate is given by (5.9). The phases CQc - r and A r _ r -- 0 + - were extracted from the measured decay rates into b o t h neutral and charged pions [36] (see Fig. 5.11). From the fits, performed simultaneously to both regenerators, they found 102

102

~+~"

l i_.......... .... , IllJlllJ

t,lJ~]ntlllll[lll

120

130

140

Z decay (m)

150

120

130

IIII

140

I JI

150

Z decay (m)

Fig. 5.11. Measured rates for decay into 7r+Tr- and ~%r ~ The predictions from the fits with (solid line) and without (dotted line) the interference term are also shown [36]

27 Since they used their own values for Ts and AmL,S, derived assuming (5.12), they could not report 4)+- in the same fit.

80

5. Precision Measurements of r r

= (43.53 =t=0.97) ~

As with the cos [tAmL,s -- (r for 7r~ ~ events.

r

- r

r

and c'/~ = (0.62 =t= 1.03) ~

(5.16)

E731 data, the interference terms were fitted by -- Cec)] for 7r+~r- and cos [tAmL,s -- (r -- r + Ar

5.3 Comparison of NA31 and E731 Experimental Techniques We conclude this chapter with a few brief comments regarding the experiments NA31 and E731/E773 (see also [37]). The presence of a regenerator in E731 leads to the quantum-mechanical interference between the K ~ --+ 27r and K ~ --+ 27r amplitudes, the measurement of which can provide independent confirmation of an cl/c signal. The NA31 experiment had to be concerned with possible shifts in the overall detection efficiency, since the K ~ -+ 27r and K ~ --~ 27r decays were collected at different times under different rate conditions. The K ~ decay distribution was not uniform in E731, resulting in large relative acceptance corrections. Because of the shorter decay region, the residual background from 37r~ decays in the K ~ --+ 27r~ sample in E731 is considerably smaller than in NA31. The energy and position resolutions of the NA31 electromagnetic calorimeter are superior to those of E731. However, the plane resolution of its tracking chambers is much worse. As a consequence, the background in the K ~ -+ 7c+7r- sample is significantly smaller in E731 (NA31 was forced to discard about 40% of its ~r+Tr- d a t a in order to keep the eTrv background low). Concerning backgrounds, it should be noted that a 1% shift in the double ratio (5.2) corresponds to 1.6 • 10 -3 in e//e. The largest backgrounds in E731 and NA31 were at a few-percent level. In the Fermilab experiment the regenerator b e a m flux was significantly reduced by a 66 cm carbon absorber. Consequently, lack of statistics prevented t h e m from extracting r from the time-dependent charge a s y m m e t r y in semileptonic decays. The presence of D(p) in (4.42) is a fundamental deficiency in this class of experiments, as a source of uncertainty in the NA31 data analysis.

6. N e u t r a l K a o n s in P r o t o n - A n t i p r o t o n A n n i h i l a t i o n s

Proton antiproton annihilations were first used as a source of neutral kaons in early 1960s (see, e.g., Armenteros et al. [38]). In a typical experiment of this kind, J. Steinberger and his collaborators produced K ~ a n d / ~ 0 mesons in equal, but relatively small, numbers in the annihilation of low-energy antiprotons in a liquid-hydrogen chamber at Brookhaven: ~ + p -+/~~

+ pions,

D+p--+K~

(6.1)

Due to strangeness conservation in strong interactions, the K ~ (/~o) is "tagged" by the charge sign of the accompanying kaon. 2s Figure 6.1 shows the time distribution of the semileptonic decays K~ ~ --+ g• measured by F~anzini et al. [38].

6.1 The

CPLEAR

Experiment at

CERN

High-precision studies of CP violation based on this idea began a quarter of a century later at CERN, following the construction of the Low Energy Antiproton Ring (LEAR) and a dedicated detector. The C P L E A R experiment produces intense fluxes of tagged K ~ and /7/0 mesons by stopping low-energy antiprotons (200 MeV/c, 106 antiprotons/second) from LEAR in a low-density hydrogen target:

(PP)rest -+ /~~

(PP)rest --+ K ~

(6.2)

The branching ratio for each of the above two processes is about 0.2%, which means that K ~ and K ~ mesons are produced in equal numbers. However, the tagging efficiencies for K ~ and /7/0 are not identical because of different cross-sections for interactions of K + and K - mesons in the detector material. Note also that K ~ and/7/o undergo coherent regeneration in the detector, which must be taken into account. Tagged K ~ a n d / ~ 0 beams offer the possibility of observing directly K ~ K ~ interference. The K ~ 1 7 6 decay rate to any final state f reads (see (4.39), (4.40)) 28 Reactions (6.1) have also been used to test charge conjugation invariance in strong interactions.

82

6. Neutral Kaons in Proton Antiproton Annihilations |

t

I

I

I

!

i

I

I

I w

36109 EVENTS 32

Z8- 1 m 24Z

--

> 20tlJ

o e,-

167

W

--

~E I 2 Z

84-

q

o 2

4

6

8 I0 12 t x I0 "l~ sec

14

16

18

20

Fig. 6.1. The result from Franzini et al. [38] on the time-distribution of the leptonic decays of an equal mixture of K ~ and/s The solid curve is the prediction of the AS = AIQ rule

FKO R o o f = ~ ( 1 T 2 R e

e)FKo__+f{l~fl2e -rLt + e - r s t

+ 21wle-(rs+rL)t/2 eos(Smt -- e l ) } , where

Ft
(6.3)

as before, the upper (lower) signs refer to

K ~ (R~ As for (4.6), the time-dependent decay a s y m m e t r y is defined by r (/~o _+ f ) _ r ( K ~ --+ f )

(6.4)

As(t) -- r (KO _+ f ) + r (KO -+ f ) Using (6.3),

.Af(t) can be expressed as

Af(t) ,~ 2Re ~ - 21zlfle(rS-rL)t/2 cos(Smt -- Of) 1 + I?~fl2e(Fs-FL)t thereby isolating the

K~

(6.5)

~ interference term in (6.3). We see that:

(a) since K ~ --+ f a n d / ~ o __+ f are CP-conjugate processes, any difference in their rates is a clear sign of C P violation;

6.1 The CPLEAR Experiment at CERN

83

(b) by measuring A/(t), one can determine both the modulus and the phase of the CP-violating parameter Wf; (c) the acceptances common to K ~ a n d / ~ o decays cancel in ~4/(t). The CPLEAR detector is shown in Fig. 6.2. It consists of: a spherical gaseous hydrogen target (16 bar pressure); cylindrical tracking chambers; a threshold Cerenkov counter (filled with liquid freon and sandwiched between two layers of scintillator) to identify charged kaons; and an electromagnetic calorimeter consisting of lead plates interspaced with streamer tubes. All components are assembled inside a solenoidal magnet, which provides a uniform 0.44 tesla field neeeded to measure the momenta of charged particles. The acceptance-corrected decay rates of initially tagged K ~ and /~0 mesons into ~+7~- are shown in Fig. 6.3. The CP violating parameter W+_ was determined from the time-dependent asymmetry in the decay rates: expression (6.5) was fitted to the observed asymmetry in Fig. 6.4, keeping IW+-I and r as free parameters, with the result [39a] (in the fit they used their own value of Ama,s obtained from semileptonic decays of neutral kaons): I~+-I -- (2.312 • 0.054) • 10 -3,

r

= (42.7 • 1.8) ~

(6.6)

See also [39b] for a correlation analysis of ~7+_ and Ama,s based on the results from different experiments, which yielded (r = 43.82 ~ • 0.63 ~ Unlike in most other experiments, the measurement of AmL,S by CPLEAR is independent of r The K ~ ~ mass difference can be extracted from the decay rate asymmetry Drift chambers Calorimeter

S

Streamer tubes

~erenkov

Fig. 6.2. The layout of the CPLEAR detector at CERN

84 i

6. Neutral Kaons in Proton Antiproton Annihilations

1o, I 0s

10"

10~

10=

10

. . . . . . . . . . .

2

4

. . . . . . . . . . . . .

6

8

10

12

14

16

18 vJ'~s

Fig. 6.3. Acceptancecorrected decay rate of K ~ (El) and /~o (.) into 7r+Tr- . The lines are the expected rates

[39a]

A

*" 0.6

0.4 0.2 0 -0.2

I -0.4 -0.6 -0.8 -1.0

0

2

4

6

8

10

12

14

16

18

Fig. 6.4. Decay rate asymmetry versus the proper decay time. The solid line is the result of the fit described in [39a]. The inset displays the data at the short decay times with a refined b~nning

6.1 The CPLEAR Experiment at CERN

85

Azure(t) -- [F+(t) - F_(t)] - [ F + ( t ) - F _ ( t ) ]

(6.7)

[ r + ( t ) + F_(t)] + [T+(t) + F _ ( t ) ] ' where the K ~ a n d / ~ 0 decay rates F+(t) = P ( K ~ --~ g + ~ - v ) ,

F_(t) - F ( K ~ -+ g-~r+L,) ,

T+(t) - r (K0 _~ ~ + ~ - . ) ,

T _ ( t ) - r (~:0 ~ ~ - ~ + v )

(6.8)

and t is the decay eigentime. T h e expression for ,A,~m(t) follows readily from (4.20)

Aam(t) -

4 (1 - Ixl 2) e - r t cosC~mt) 211 + x l 2 e - r s t + 211 - x]2e-rLt 2e - F t cos(6mt) (1 + 2Re x) e - r s t + (1 - 2Re x) e-rL,'

(6.9)

where we neglected the t e r m s proportional to Ixl 2 and used x + x* = 2Re x. A possible violation of the A S = A Q rule is taken into account by the p a r a m e t e r Re x. T h e m e a s u r e d a s y m m e t r y is shown in Fig. 6.5. Leaving ~mL,S and Re x as free p a r a m e t e r s , they fitted (6.9) to the d a t a in Fig. 6.5 and o b t a i n e d [40] Am[,,s = (0.5274 + 0.0029) • 101~ s -1,

(6.10)

thus confirming the low value of AmL,S m e a s u r e d by E731.

~,~ 0 . 8

~

0.7

~>.~0.8 0

0.5 0.4

0.2 0.1 0

-0.1

2

4-

6

8

10

12

14

16

18

20

Fig. 6.5. The asymmetry .Azure(t) as a function of the decay time (in units of Ts). The solid line represents the result of the fit [40]

86

6. Neutral Kaons in Proton-Antiproton Annihilations Their most recent results on A m L s and Re x (full statistics) are [41a] AmL,s = (0.5295 + 0.00202) x 101~ s -1, (6.11) Re x = [-1.8 + 4.1(stat) + 4.5(syst)] • 10 -a.

As explained in Appendix A, the K ~ meson may decay into the kinematicssupressed and CP-allowed final state 7r+Tr-Tr~ with L = 1 = 1 and CP = +1, or into the kinematics-favored and CP-forbidden state :r+Tr-zr~ with L = l = 0 and CP = - 1 . This results in a Dalitz plot distribution which is symmetric with respect to the 7r+ and :r- for the CP-violating amplitude and antisymmetric for the CP-conserving amplitude. Thus by integrating the decay amplitude over the entire phase space of the K ~ -+ 7r+Tr-Tr~ decay, the CP-allowed contribution can be eliminated. From (4.38), and defining T/+- 0 ~

AKg-~+~- ~~

the decay amplitude for K ~

A+-o =

(6.12)

AKo-+Tr+;r- ~o

AKo~r+Tr-Tro

~-+Tr-~ ~ can

~ -+

be expressed as

1 e_imLt

x {(1Te)e-FLt/2+(l:Fe)rl+_oe(ia=-vs/2)t},

(6.13)

where the upper (lower) signs refer to K ~ (KO). The corresponding decay rates read F + - o c( (1 T 2Re e){e -rLt +

I +_olZe-r

-t-e-(rs+rL)t/2 [,~__oe-iamt + ,+_o ei~''t] }.

(6.14)

The time-dependent decay rate asymmetry, which is a direct measure of the KL-K o so interference, is given by

A+-o(t) =

r+_o(t) F+-o(t)

-

r+_o(t)

+ V+-o(t)

2Re e - 2e -(rs-FL)t/2 x

{Re ~7+-o cos(Smt) -- [m U+-o sin(e~mt)},

(6.15)

where F and P are the KO and K ~ decay rates, respectively. A recent result on the CP-violating p a r a m e t e r ~+-0, based on the full statistics of CPLEAR, is [42a] Re 7/+-o = [ - 2 + 7(stat)+4(syst)] • 10 -3,

(6.16)

I m ~/+-o = [ - 2 + 9(stat)+_2(syst)] x 10 -3. Additional information about their measurement of ~?+-o can be found elsewhere [42b].

6.2 Is CP Violation Compensated by Time-Reversal Asymmetry?

87

The Fermilab experiment E621 has also published a result on Im ~+-o by fixing Re rl+-o = Re c and assuming C P T invariance [43]: Im rl+_o = [-15 + 30] x 10 -3.

(6.17)

6.2 Is C P Violation Compensated by Time-Reversal Asymmetry? If C P T is conserved, the observed CP violation demonstrates the failure of time-reversal invariance. A direct test of T asymmetry in the K ~ system was suggested by Aharony and Kabir in 1970 [44]. As shown in Appendix D, the operation of time reversal gives the identity

(K0 i e-ira i/?0 > = (f;o i e-~HTt i K0 >

(6,1S)

where /2/T is the time-reverse of the hamiltonian /2/. If /2/T ---- /2/, the two amplitudes are equal. A nonzero value of the ratio of transition intensities F(I~O _+ K o) _ F ( K o _+ [(o) A T ( t ) -- r ( K o -~ g o ) + F ( K o ~ fi2o),

(6.19)

where

F(K ~ ~ K ~ - (KO e_i/:/t [KO ) 2 (6.20) r ( g o _+ [(o) == (KO e_i/:/t ] Ko ) 2,

would thus imply /2/T r /2/, 1.e., a violation of time-reversal invariance. The first evidence for time-reversal noninvariance has been reported by the C P L E A R collaboration based on semileptonic decays of tagged neutral kaons [41b].They extracted A T ( t ) = -F+(t) - F_(t) m 4Re e + 2Ira sin((~mt) (6.21) F+ (t) + F_ (t) X c o s h ( A F t ) - cos((~mt)

from the measured decay rates F + ( t ) - F [/~~ = 0) ~ g+Tr-u] and F_(t) F [K~ = 0) --9 f-lr+O] (see Fig. 6.6). Expression (6.21), which was derived assuming C P T invariance, follows readily from (4.13), with A F -- ( F s - F L ) / 2 (see also Appendix D). The C P L E A R measurement yielded AT(t) = [6.6 + 1.3(star) + 1.0(syst)] x 10 -3,

(6.22)

which should be compared with 4Re e ~ 6.6 x 10 -a (see (6.21), (4.26), and (4.28)), assuming Im x = 0.

88

6. Neutral Kaons in P r o t o n - A n t i p r o t o n Annihilations

t.. 0.04 0.05 0.02

._~ .

0.01

-0.01 -0.02

2

4

6

8

10 12 14 16 18 20 Neutral-kaon decay time ["rs]

F i g . 6.6. The asymmetry .AT, indicating a violation of T invariance in semileptonic decays of neutral kaons. The solid line represents the fitted average (.AT(t)) [41b]

7. N e u t r a l K a o n s in E l e c t r o n - P o s i t r o n

7.1 The

DA(I)NE

Annihilations

Project

The DA(I)NE project at Frascati (Italy) will study the process e+e - -+ virtual photon --+ ~5 -+ K~

~

jPC _- 1 - - ,

(7.1)

by producing at rest about 5000 r mesons per second at the centre-of-mass energy x/~ -- 1020 MeV. As explained in Sect. 1.4, the neutral kaon pair in (7.1) is in a pure C = - 1 quantum state: 1

{[K~176

- IK~176

= --~ { I K ~ 1 7 6

[K~176

[4~> = ~

1

} (7.2)

0 S0 - K s0K L. 0 With an i.e., the final state is either K ~ ~ - / ( ~ 1 7 6 or KLK expected collider luminosity of 5 • 1032 c m -2 s -1, about 10 l~ coherently produced K~ pairs per year will provide a particularly beautiful method of quantum interferometry. To study the time evolution of the state (7.2), we denote by f l ( t l , +z) and f2(t2,-z) any two final states in the neutral-kaon decay, and define the amplitude ratios

~ _=
(7.3)

:e-(i"s'L+Fs'L/2)t[K0,L(t : 0)>--e-iMs'Lt]K~

(7.4)

and (7.2), the decay amplitude for 4~ ~ fir2 reads (see also [46a]) 1 I"

e-i(.Mst2+.MLt~)

- As--,f, AL--+f2 e -i(Mst'+A4Lt2) } __ As~$1 A s ~ f 2 e _ i y ~ t / 2

where

{7/1 e iA2vIAt/2 - ~/2 e -iA'MAt/2 }

(7.5)

90

7. Neutral Kaons in Electron-Positron Annihilations

AS,L~f ~
A ~ f l I ~ ~
(7.6)

and At=t2-tl,

t=tl+t2,

M=ML+MS,

AM=ML-MS.

(7.7)

For equal times (tl = t2) the amplitude A~_~/1/2 vanishes for identical final states of the two knons (fl = f2 = 7r+Tr-, 7r~176 9 This result follows from Bose statistics: since the kaon is spinless, the total angular m o m e n t u m in a 27r state must be zero. Two identical 2~r states thus behave like two identical spinless bosons, and therefore cannot be in an antisymmetric spatial state (recall t h a t Po = - 1 ) . However, if the two final states are not identical at equal times, e.g., f l = 7r+~r- and f2 = ~r%r~ then Ar

o( r/+_ - ~o0 = 3r

for

At = 0.

(7.8)

Simultaneous decay to distinct CP eigenstates, such as ~r+~r- and ~r%r~ is therefore an unambiguous signature of direct CP violation. T h e decay rate corresponding to (7.5) is given by Fs~/1Fs--+$2 e_Ft • {[r]ll2e -z~rz~t + Ir]212e~rz~t - 2lr]l[lr]2l e o s O }

(7.9)

with O ~-~ (~rnAt -t- r

- q~2,

F ~ (Fs -t- FL)/2,

z:~F ~- (/'s - FL)/2. (7.10)

From an experimental point of view, it is convenient to consider the probability distribution in the relative time At :

F ( f l , f2; At) -- ~1

/?

t d(tl + t2) F~o/~ h.

(7.11)

In the above expression we integrate over all experimentally "accessible" times t = tl + t2, keeping At constant. T h e factor 1/2 is the Jacobian for the transformation from (tl, t2) to (t, At). For At _> 0 a straightforward calculation yields

F ( A t > O) -- Fs~/~Fs-~I2 4F x { I r / l l 2 e - r ~ t + [rl2lee - r ~ t

(7.12) -21r~lll,721e - r ~ t c o s O } .

For At < 0 we obtain an expression analogous to this, with At ~ [Atl and with indices 1 and 2 interchanged. If we set f l = 7r+rr- and f2 = lr%r ~ in (7.5), the quadratic terms in IA,/~__~fIj.212 are proportional to

Iv+_12 ~ lel2 [1 + 2Re (J/e)],

Ir/o012~ lel2 [1 - 4Re (J/e)]

(see (5.4) and (5.5)) and the mixed terms to

(7.13)

7.1 The DA~NE Project

91

Interference ~ - ~+_ (rl00)*e iSmAt - (/]+_)*r]00 e-iSmAt -

21~12{[1 -

Re (s'/e)]

• cos 5 m A t -- 3Im (S'/s) sin 5 m A t } .

(7.14)

Expression (7.12) now reads

r ( n t > 0) = c kl 2 {(1 + 2~)e -rs~* + (1 - 49~)e - r L n t - 2e - r n t [(1 - 9~)cos0 - 3 3 s i n 0 ] } ,

(7.15)

where C --= Fs,ooFs,+4F '

9~_= R e ( ~ ' / c ) ,

3 -- Im (sl/e),

0 ----5mAt.

(7.16)

For At < 0 there is an expression analogous to this with A t -+ Intl, rs FL and I m ( s ' / c ) -~ - I m (el/e). A nonzero value of r would thus result in an a s y m m e t r y between the A t > 0 and A t < 0 distributions. T h e interference p a t t e r n s for different combinations of final states shown in Fig. 7.1 can be used to extract Re (e'/e), I m (el/e), z~mL,S, [T/TrTrI, r etc., as discussed in [46b, c]. T h e a s y m m e t r y in K ~ --+ g+lrT=v decays provides tests o f T and C P T invariance. 1.2 3.5

b

3 2.5

0.8

'~ f '2 '~

2

0.6

1.5

0.4 1 0.2

o .,s

\

\

\

0.5

:iO . ?

" ~ .... ; 0 i s

0 -30

'-i0

:i0

~,'oio-)o

-15-10 -$

0

$

10 15 20 25 30

~l=(tFt2)l~ s

Fig. 7.1. Calculated interference patterns for the following final states in two-kaon decays: (a) fl : wTTr-, f2 = w~176 (b) fl = l+Tv-v, f2 : /-Tr+v; (c) 11 = 2~, f2 = lzcv

Measuring el/r at DAgPNE requires very accurate reconstruction of t + _ and too. T h e high-statistics d a t a from K ~ --+ g• 3~r decays will be used to m a p detector acceptance and reconstruction efficiency. A b o u t 107 7r+Tr-Tr~ 0 decays are needed for a statistical error of 10 -4 on Re (r the CP-violating K ~ --+ 27r decays being the modes with the limiting statistics. In this context, the K ~ and K ~ decay lengths from ~5 --+ K ~ K ~ are As = 7/3c7"s = 0.592 cm and AL = 7~CTL = 343 cm, respectively. A b o u t one quarter of K ~ mesons are expected to decay within the tracking volume of the K L O E detector at DA(I)NE.

92

7. Neutral Kaons in Electron Positron Annihilations

The DAONE project is, undoubtedly, very versatile. It provides a novel method of q u a n t u m interferometry and precision tests of the discrete symmetries C, P and T not readily achieved in other experiments. The neutral kaons in the reaction (7.1) change their identity continually and in a completely correlated way. This can be used to test q u a n t u m mechanics by studying correlations of the Einstein-Podolsky-Rosen type (see Sect. 1.4 and [47, 48]). Note also t h a t a 4~ factory is characterized by low background, since about a third of the 4~-decay final states are neutral-kaon pairs ( K + K - : K ~ ~ : 0~r = 0.49 : 0.34 : 0.13).

8. Neutral Kaons in Fixed-Target Experiments

8.1 The

Experiments

KTeV

and

NA48

Presently there are two major fixed-target experiments with neutral kaons: NA48 at C E R N and KTeV at Fermilab. The elegance and sophistication of these experiments reflect years of experience with K ~ beams, especially that gained with their predecessors, NA31 and E731, respectively. The main aim of each of the two groups is to determine Re (e//e) with a precision of ~ 10 -4. In order to achieve this goal it is necessary to collect roghly 4 million KL~ --+ 27r~ decays, an increase of about an order of magnitude compared with NA31 or E731. The KTeV apparatus is shown in Fig. 8.1. The experiment retains the basic features of E731: it uses two beams, which are side by side and identical in shape, and records all four decay modes simultaneously. To reduce inelastic regeneration, an "active" regenerator made of plastic scintillator is employed. The most significant improvement with respect to E731 is the use of an Muon

Analysis Magnet Photon Veto Detectors i

i .................. Ku

beams

[email protected] ...........t,I............ F::::II IH 1 1.............. lq.. Vacuum DecayRegion

IT

V

,

.......... : ..... acuum Window l l ~ ; ~ : T .

Drift ;5' .

.

li

~'~"' ;' ' ) ....... -.- ] ,

| IllIf i

]~egenerat~ [' I I /il~~ilTr,, i

Muon

Filter Veto i" ii

.

.

g

/

Hadron Veto with Lead Wall I

120

I

I

140

I

I

I

160 Distance from Target (m)

Fig. 8.1. A schematic drawing of the KTeV detector

180

i

94

8. Neutral Kaons in Fixed-Target Experiments

~t

E Regenerator beam data Prediction without interference > -,

40-50 GeV kaon energy

lO

10 4 .....~

10 3

125

130

135

140

145

150 155 Distance from target (m)

Fig. 8.2. Interference in K ~ -+ 7r+~r- decays measured by KTeV undoped CsI electromagnetic calorimeter. This should result in a much better energy resolution and thus more efficient background rejection in the 2~r~ decay channel. KTeV expects to achieve an error of 0.5 ~ in the determination of r and r and also of r using semileptonic decays. Fig. 8.2 shows their preliminary result on kaon interference. The NA48 detector is sketched in Fig. 8.3. An important element of the new design is the concurrent presence of (almost) collinear K~ and K ~ beams in the detector, produced by protons hitting two different targets. A channeling crystal is used to simultaneously attenuate and deflect protons emerging from the KL~ target, which are then sent toward the K ~ target located close to the detector (see Fig. 8.4 [49]). The protons producing the K~ component are tagged in order to distinguish between the K~ and KL~ beams. A major improvement compared with NA31 is a liquid-krypton calorimeter with superior energy and position resolution. A magnetic spectrometer has been added for charged decay modes.

8.1 The Experiments KTeV and NA48 lavetocounters, hadroncalorimeter

"x

li filter (iron) triggerhodoscope anticounter ~

~

~

homogeneousLKr calorimeter

magnet

/

/

~

ch r am s be/

beampipe Fig. 8.3. The NA48detector at CERN Detectors 1012ppp "~-

KS t a r g e ~ ....~

Z

~

~

target ~ j~" tagging ~ . 107.ppp/ counter Channeling crystal ~ Fig. 8.4. Simultaneous K~ and K~ beams in NA48

"

95

96

8. Neutral Kaons in Fixed-Target Experiments

Supplement:

Kaon

Beams

As a supplement to the preceding section, we will describe (a) the production of K mesons by a high-energy proton beam striking a stationary target, and (b) the techniques of electrostatic and RF separation of K + beams. Our discussion is partly based on the lecture notes by B. Winstein [104]. Other sources of K-mesons are discussed in Sects. 6.1 (proton antiproton annihilations) and 7.1 (e+e - collisions). An experiment that uses K + "beams" at rest is described in Sect. 9.3.1. The production of charged kaons by incident protons on a Be target can be expressed as a yield per incident proton per GeV/c kaon momentum per steradian of the solid angle ~2. An approximate parametrization of the yield at angle 0 is given by d2a [( mb r] P(~I - ~-~)a (1 + 5e-20x) dpdf2 G e ~ / c ) s t o( j -=pf(x, Pt)

(S8.1)

with x = P/Pbeam and Pt = Op. Using d3p = p2dpd~, we can write d d2a p d ~ -p2d3a~pp

d3cr'~_ ,~,p ( E--~p] -= p O'inv,

(88.2)

where "inv" denotes Lorentz invariance. The k~on yield per incident proton therefore reads d2N P O'inv Yield-= d p d ~ - ~rinel '

($8.3)

where ainel is the inelastic cross-section. From ($8.1) we see that the kaon production is most copious in the forward direction (Pt = 0). The most significant difference in the production of K + and K - mesons is in the x dependence: K + (x: (1 - x) 3,

K-

(x (1 -- X) 6.

(S8.4)

By simple quark-counting, one can show that the relative production rates of charged and neutral kaons are given by K ~ ,-~ ( K + + K - ) / 2 ,

[(o ~ K - ,

($8.5)

i.e., K o _/~o K 0 +/7/o

K + - KK + + 3K-"

(s8.6)

The asymmetry in the K ~ and/~o production spectra is due to an interplay between associated production and baryon number conservation. Expressions ($8.5) and (88.6) are corroborated by a measurement of the "dilution factor" KO(p) _ [gO(p) D(p) - KO(p ) T fiO(p ) at the CERN SPS (the NA31 collaboration [32]; see Sect. 5.1).

($8.7)

8.1 The Experiments KTeV and NA48

97

Therefore, a proton b e a m produces a mixture of K ~ a n d / ~ o mesons that depends upon energy and angle. At high energies (large x), the K ~ production dominates. As was mentioned in Sect. 4.1, if K ~ and /~0 mesons were produced with equal spectra (D(p) = 0), the interference t e r m in (4.42) could not be observed in a target experiment! Sufficiently far from the production target, where only the KL~ component can survive, the initial composition of the kaon b e a m becomes irrelevant. In the forward direction (0 = 0~ the neutral b e a m is dominated by photons and neutrons. Photons, which originate mainly from rr~ decays, are easily removed with a lead converter near the production target. It is sufficient to have a converter t h a t is about 10 radiation lengths thick ( ~ 6cm), since the probability of photon non-conversion is e - ( 7 / 9 ) x 1 0 ~ 4 • 10 - 4 . The charged products of the resulting electromagnetic shower are swept away by a magnetic field. The neutrons are produced with the invariant cross-section 25 mb Oinv(n ) ~ (GeM/c)2 e -5pt,

($8.8)

which does not depend on x = P/Pbeam" The neutron flux is m a n y times the kaon flux at 0 = 0 ~ and thus presents a much more serious problem t h a n the photon beam. Fortunately, the kaon to neutron ratio can be enhanced very efficiently by targeting away from 0 = 0% It follows from the above differential distributions t h a t an enhancement factor of over 100 can be obtained with less than a factor of 2 loss in the kaon flux at 0 ~ 5 m r (see [104]). In the design of a K ~ beam, one also has to consider: (a) the "soft" component of neutral particles in and around the b e a m ("beam halo"), (b) the forward hadronic "jet" and (c) the b e a m of noninteracting particles. The first component can be reduced considerably by collimators that subtend little solid angle compared with ~2beam- To dispose of the other two components, a "beam dump" is used t h a t is thick enough to completely absorb the hadronic shower, and is also sufficiently close to the target to reduce the muon flux originating from ~ and K decays. As an example, we outline the main features of the KTeV beamline. The kaons are produced by 800 GeV protons striking a BeO target. The proton b e a m is delivered every minute in spills lasting 23 s. About half of the 3 • 1012 protons per spill interact in the target, while the other half are absorbed in a b e a m dump. Neutral kaons produced in the horizontal plane at an angle of 4.8 mrad relative to the proton direction point toward the detector. Two identical sets of collimators produce two secondary neutral beams at • mrad in the horizontal plane. The size of the beams is about 8 cm • 8 cm at 180 m from the target. A large sweeping magnet downstream of the b e a m d u m p removes muons from both the primary target and the dump. The photon flux is reduced using a 7.6-cm-long lead absorber t h a t transmits 55% of kaons and neutrons.

98

8. Neutral Kaons in Fixed-Target Experiments

To further enhance the kaon to neutron ratio, a 52-cm-long Be absorber is used. To see how this enhancement works, note that the neutron and kaon total cross-sections have the following atomic number dependences: O'tot(/t ) ~,~ 49mb • A ~

5rtot(g )

~

24rob • A ~

A > 7.

($8.9)

The cross-section ratio is thus greatest for small A. The number of interaction lengths, Xint, in the Be absorber (A = 9, L -- 52 cm) is Xint ~- (Tt~ ( - ~ )

L -~ { 0.97, 1"74'

kaons, neutrons,

($8.10)

where No ~ 6 • 1023 is Avogadro's number and ~) = 1.85 g/cm 3 is the density of Be. The respective transmission coefficients are e -Xint -- 0.18 (neutrons) and 0.38 (kaons), resulting in an enhancement factor of about 2. As mentioned in Sect. 9.3, the purity of K + beams at Brookhaven (experiment E787) has been considerably improved through electrostatic (DC) separation. This method of particle separation works in the following way. An almost parallel beam enters a separator which has a vertical electric field and a horizontal magnetic field. The separator is tuned in such a way that the action of the magnetic field cancels the action of the electric field for the kaons, whereas the pions are deflected. At a vertical focus downstream of the seperator, the pion beam is displaced from the axis and then stopped. The separation is given by Separation (in rad) .-~

,

($8.11)

P where, in the case of E787, E = 60 K V / c m and L = 220 cm (p is the beam momentum). The method of RF separation employs, in general, two cavities and a system of quadrupole magnets between them. The first RF cavity imposes a transverse momentum kick on the beam of a few MeV/c per meter of cavity length. The second cavity, located downstream of the first one, is so tuned that the Tr+ mesons arrive with the same phase that they had in the first cavity. Since the 7r+ momenta have been reversed in the quadrupoles, the positive pions end up with no net kick. If the RF frequency and the distance between the cavities are such that the ~+s and protons are 360 ~ out of phase at the second cavity, the protons will also receive no net kick. The K + mesons, on the other hand, are 90 ~ out of phase with respect to the 7r+s, and thus get a net transverse kick corresponding to ~ 1 mrad. A beam plug downstram of the second cavity stops the pions and protons. RF-separated K + beams can be used, for example, to search for the rare decay K + --+ 7r+vO (see Sect. 9.3). Alternatively, they can be focused on a target to produce K ~ mesons by charge exchange.

9. T h e K ~ S y s t e m in t h e S t a n d a r d M o d e l

9.1 C a l c u l a t i o n

of Am k and ek

So far our description of the K ~ system has been restricted to its quantummechanical properties and phenomenological implications of the observed CP violation. In this section we consider K~ ~ mixing within the framework of gauge theories of the electroweak and strong interactions (the Standard Model (SM)). We will outline the calculation of the off-diagonal elements of the K~176 mass matrix that generate the tiny K~ mass difference and the CP-violating parameter e (see (3.30) and (3.82)). This is the forerunner of all the calculations that are forbidden in the lowest order of the electroweak theory. The smallness of the K ~ ~ mass difference (Amk = 3.5 • 10-6 eV) enables the quantum-mechanical interference effects between the K~ and K ~ components of a K ~ beam to spread out over macroscopic distances (see Sect. 1.5). Phenomenologically, it played an important role in the establishment of the charm hypothesis, as explained in what follows. There is overwhelming experimental evidence that strangeness-changing neutral currents are heavily suppressed. For example, F ( K ~ -+ #+#-) ~ 10_8, r ( K ~ ~ all)

F ( K • --+ ~• F ( K • --4 all)

.~ 10_10.

(9.1)

The suppression of K ~ --+ # + # - was very surprising because the analogous charged decay K + --+ #+v~ has a branching ratio of 63% (see Fig. 9.1, where the mesons are represented, in accordance with SM, as bound states of quark-antiquark pairs). To explain this puzzle, Glashow, Iliopoulos and Maiani (GIM) incorporated the charm quark into the hadronic weak current originally proposed by Cabibbo. The weak charged current in the GIM model reads [50] j a CC -

(UC)L?~

S'

(9.2)

'

L

where

=

{' cosOr sin 0r "~ (d) k,-sinOr cosOr s L'

r

s')= V r

s),

(9.3)

100

9. The K ~ System in the Standard Model

Zo 9 s9 $ U

K+

r

g+

,

sinOr

) o

KL

(b)

(a) e+

w+{

I

sinOr

V:o

K+ U

11

(c) Fig. 9.1. Allowed decays K + -+ #+v, (a) and K + --+ rc~ first-order contribution to K ~ --~ # + # - (b)

(c); forbidden

g is the SU(2) weak coupling, y~ are Dirac matrices and 0c is the Cabibbo angle. The subscript L denotes "left-handed" spinors: qL = ~1 (1 --~5)q. The quarks have charges Q~,c,t = + 2 / 3 and Qd,s,b = --1/3. Since the flavor quantum numbers are not conserved in weak interactions, the weak eigenstates d~ and s ~ are mixtures of the mass eigenstates d and 8. 29 The mixing matrix V, which also couples the up (u) as well as the charm (e) quark to a linear combination of the down (d) and strange (s) quarks, is unitary: vW

= v-~v

(9.4)

= 11

that is to say, the inverse of V is its hermitian conjugate (ll is the unit matrix). The unitarity of V ensures the absence of strangeness-changing neutral currents. Indeed, ~0r

= CV t O r e

= ~OV t Yr = r162

(9.5)

for any arbitrary operator 0 . From (9.5) we conclude that the neutral component of the hadronic weak current does not contain terms that mix quark flavours. Changesof flavor alwaysinvolve change of charge (hence A S -- AQ). This explains why the amplitude in Fig. 9.1b is zero. 29 It is sufficient to mix ("rotate") either

(ds) or (uc).

9.1 Calculation of Am k and ek cosO,=

d

W +

[ l i i ui i i i i l l sinOc W-slnOc

g co~O~

101

,

~'-

W+

W-

g+

Fig. 9.2. Second-order contributions to K ~ --~ #+#-

The decay K ~ -+ # + # - can also proceed through the diagrams in Fig. 9.2. The amplitudes corresponding to these diagrams are proportional to cos 0c sin 0c and - sin 0c cos 0c, respectively. The second amplitude cancels most of the first. If the u and c quarks had the same mass, the cancellation would be exact (see (9.13)). The foregoing discussion demonstrates the importance of the Cabibbo angle 0c, which not only suppresses the weak transitions that are proportional to 0c ~ 13~ but also removes strangeness-changing neutral currents via the charm quark. The origin of 0c, however, is not explained in the Standard Model. It is instructive to calculate the amplitude that corresponds to the lowestorder diagrams contributing to Amk (see Fig. 9.3a,b). The lowest order for which the transition sd ++ ,~d is possible is fourth order in the weak hamiltonian/:/w: the two W bosons in Fig. 9.3a,b are emitted and reabsorbed, so that a total of four weak "vertices" are involved. Of the two sets of diagrams in Fig. 9.3a,b it is sufficient to compute the first one: it can be readily verified that, in the limit of vanishing external momenta, the amplitude corresponding to the second set is identical. We ignore, for the moment, the contribution of the top (t) quark, which is justified when computing A m k (but not e), as explained later on. Neglecting the external quark momenta, 3~ the electroweak Feynman rules applied to the box diagrams in Fig. 9.3a yield, in the so-called t'Hooft-Feynman gauge, 31

30 In the K ~ rest frame, their components are of the order of mk and thus can be neglected compared with the W boson and heavy quark masses. 31 The presence of unphysical scalars in this gauge can be ignored when the top quark is not included, since their coupling to fermions (f) is proportional to

m//m~

~ I.

102

9. The K ~ System in the Standard Model Vid i i i

d

.~--S m

w+:' l k 1 W"

K~

o t i

v.~, ,.~.~

v~

(a) v~ d

K~

~

w +

v~ ~s

. . . . . . . . .

qi

v.~

~<

~o

~j

w-

v~

a Fig. 9.3a,b. Feynman diagrams for the AS ----2 transition $d --+ sd

(b)

2

-ig

+

2

i(]r + /

•

~onvd \k2

_

m ~ / \k2 - m~

(]r -- "~'k,). The terms in u and c have opposite signs because of the GIM mechanism (see Fig. 9.2). We can simplify the above expression by using % L = R % , L 2 = L, R 2 - - R a n d R L = L R = O (L,R(1=t= ~5)/2). The quark masses in the numerators thus drop out, with the result i M = g4 sin2 Pc4c~ wherea-k

Pc (m 2 _ m2)2

2-m 2,b_=k 2-m~,c-k

d4k k , k , (2~r)4 a2b2c2 T t'',

(9.7)

2-mw2 and

T t'" - ~ts~/A~/t~"/eLUdVs~/ey'~/ALv d.

(9.8)

To calculate the integral I,~ = f dak kt'k" J ( 2 . ) 4 (k: - , n ~ ) 2 ( k ~ - m ~ ) : ( k ~ - m ~ ) 2

(9.9)

9.1 Calculation of Am k and ~k

103

we use Feynman's "parameter formula"

fl

aSg2c

-12o3/0

fl-x xdXJo

( 1 - x - y )y dY [ a + ( b _ a ) x + ( c _ a ) y ] 6

(9.10)

and

kt, k.

F(n - 3) g~. - i~r2 2F(n) t n-3

dak (k: ~ +t)

n_> 4.

(9.11)

Both expressions are frequently employed in the evaluation of loop integrals. Writing a + ( b - a)x + ( c - a)y ~ k 2 - [ m ~2, y + ( m c2 - r n ~2) x + m ~ ] =_ k S + t , we have

k k t , (k s +kvt) 6 I,v = 120 ~ooI x dx ~o 1-~: dy ( 1 - x - y )y / d 4(27r)4 -igu~, fol x d x fo 1-x dy [ y + x (( m 1 -~x - y- ) m~)mj~2] y - 16-6---~-K~m6 3 -igt'v 647r2(m2 _ ?Tt u2 ) rrt w4

(9.12)

since rn w2>7 mcS >> mu2 [rn~ = (80.3+0.05) GeV/c 2, rnc = (1.3+0.3) GeV/cS]. Therefore, i M --

--ig4

2

28 2m

2

mc -- mu sin20c cos20c

• [~'7~'~.TQLUdO~'ye'~ ~/~Lvd ].

(9.13)

The factor (m~ - m u s) / m ~ s represents the typical GIM suppression mentioned above (it contains m~ 2 because the loop integration is cut off by m~). A single box diagram would yield A4 o( g4mj,2. The Dirac algebra in (9.13) can be simplified by virtue of

7~'~,'~e = ( g~,~,g~ + g~,~,g,~Q - g:~g~,o + ie~o'~5 ) 7 ~, .~e~.,,/~ = ( ga~'g~,~ + ge.g,Z~ _ ge~g,~ + iEQ~-~/5 ) ~/a (Co1~3 = + 1, e ~ 7 7v7 (1 -

(9.14)

= - 1 ). It is then straightforward to show that |

-

= 4

"(1 - 75) | 7 (1 - 75),

(9.15)

where the symbol | separates the matrices from two different fermion lines. Expressing the SU(2) weak coupling g in terms of the Fermi coupling constant GF (g2/8m 2 = GF/V~), it follows (mc2 >> m 2) that M - - G2 m 2 sinS0r c0s20r [ ~ 7 ~ ( 1 - "/5)Ud$)s'~a(1 8~s

-

-

~5)Vd ]

(9.16)

The Feynman amplitude (9.16) describes the AS = 2 transitions sd ++ Sd, with quark-antiquark pairs in the intermediate states. What we are actually concerned with are the hadronic transitions K ~ ++/~o mediated by genuine physical states. We thus express the above amplitude (obtained in the limit

104

9. The K ~ System in the Standard Model

of vanishing external quark momenta) as the matrix element of an equivalent four-fermion operator between the K ~ and/22o states (normalized to unity in a volume V):

M12- (K~176

(9.17)

2mk 2

_ - a ~ m_z~ sin20r cos20r 16~ 2 2rnk

~"(1

- ~)d~.(1

- ~ ) d IK~

In deriving this expression the spinors in (9.16) were replaced by the field operators ~s,d -- s, d. Also, a factor of 1/2 was included to compensate for the fact that /2/eft contains four terms which contribute to sd ++ sd, two corresponding to Fig. 9.3a and two to Fig. 9.3b, as can be readily verified by writing the fields in terms of creation and annihilation operators. This brings us to the most obscure part in the calculation of M12, the evaluation of the hadronic matrix element. Following the early calculations of Area based on the box diagrams in Fig. 9.3a,b [51a], it is customary to insert the vacuum intermediate state in the middle of the four-fermion operator, which amounts to neglecting strong-interaction effects. Since the renormalized operator /?/eft cannot really be treated as a product of two factors, the whole procedure is dubious, to say the least. These uncertainties are embodied in the parameter

~ : _ < o I ~-yo(1 - 75)d

IK~

(9.18)

s'Ta(1 - 75)d gTa(1 - ~5)d I K ~

Using the definition of the

(OIg~,75dlK~

K2t decay constant fk 32,

=- fkqo,, f~xp ~ 160MeV,

(9.19)

we obtain, in the K ~ rest frame, (/;;~ ~ ( 1

- 7 5 ) d I 0)(01 g~,,~(1- 75)d IK ~ --

~(fkmk) 2.

(9.20)

The presence of the additional factor of 8/3 is explained in Appendix .F (see also [98]). Therefore,

Amk = - 2 Re M12 = ~G2 f~ BKmkm2c sin 2 0c cos 20c,

(9.21)

which is in good agreement with the measured value of Amk, provided BK 1 and m~ ~ 1.5 G e V / c 2. Historically, a correct upper limit was set on the charm mass in this way [51b] before chacmonium (the bound state of a charmanticharm pair) was observed. In the light of what we said earlier, however, this must be viewed as a fortuitous coincidence. 32 In the matrix element of a vector-axial vector current between the vacuum and a pseudoscalar state only the axial current contributes (see (9.105)).

9.1 Calculation of Am k and ek

105

Since the K~ ~ mass splitting is caused by weak interactions, one would expect Amk to be comparable to the K ~ decay width, which is indeed the c a s e (see (1.41)). The quarks inside the K mesons are "glued" together by the strong force, resulting in gluonic corrections to the box diagrams of Fig. 9.3a,b. It is beyond the scope of this book to discuss strong-interaction effects on A m k (see [54] for an extensive review of the subject), except to mention that among various nonperturbative calculations of the denominator in (9.18), the lattice gauge theory yields the most accurate value: BK = 0.8 • 0.2. We also note that the real part of 71//12 is dominated by low momenta (k < m~) in the loop diagram. In this region the effect of "virtual" low-energy transitions K ~ --+ ~r~, 7r, ~, 6, 77' --~/~0 is important, yet difficult to estimate reliably. We now turn to the evalulation of the CP-violating parameter e based on the box diagrams in Fig. 9.3a,b. According to (3.82), e - - - v /Amk ~ Im M12.

(9.22)

With its two weak isospin doublets (ud') and (cs'), the GIM model cannot account for an imaginary part of the matrix element M12 (see below). This motivated Kobayashi and Maskawa (KM) to introduce, in 1973, a third quark doublet. Their proposal is not a trivial extension of the four-quark model because it allows for the existence of CT' violation within SM. The KM model [52] was proposed before the discovery of even the charm quark (see Chap. 1). In this context, recall that a third quark generation is not required to explain K~ ~ mixing. The GIM model can be extended to include the two additional quarks b (bottom, or beauty) and t (top) by defining, in analogy with (9.2) and (9.3),

jcc=

g, - ~ ( gv~ t ) / 3 ~~V ( ~ ) b \

vt

=v-l'

(9.23)

/L

where V is the unitary, 3 x 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix. The matrix elements of V can be expressed in terms of a certain number of independent parameters. A unitary n x n matrix for n quark generations is characterized by no = (n - 1)n/2 rotation angles and n~ = (n - 1)(n 2)/2 physical phases. For n = 2 we have n5 = 0 and no = 1, the only parameter being the Cabibbo angle 0r For three generations no = 3 and n~ = 1, i.e., in addition to three mixing angles there is also a nonvanishing phase 5. Therefore, the CIM matrix is real and has only one parameter, whereas the CKM matrix is complex and contains four independent parameters. There is a number of ways (three dozen, in fact) to express the elements of V in terms of three rotation angles and one phase. The parametrization suggested by Wolfenstein [53] is particularly convenient because it emphasizes the observed strong hierarchy of the CKM matrix elements:

106

9. The K ~ System in the Standard Model

V-

(

Yu~Yus Yub) Ycd Vcs Ycb Ytd Vts Vtb 1 - A2/2

A

AA3re -i~

-A(1 + A2A4re i~) 1 - A2/2 - A2A6re i~ AA3(1 - re i~)

-AA~(1 + A2re i~)

AA2

) ,

(9.24)

1

where A and r are of the order of unity and A - sin 0c = 0.22 • 0.002. The notation for V~j on the left in (9.24) may seem peculiar: the matrix element Vud, for example, does not refer to the mixing of u and d quarks (they cannot mix because of charge conservation), but rather to that of the d' and d states (see (9.3)). Recall, however, that V may also be viewed as containing the transition amplitudes for weak processes, in which case Vii represents the relative strength of the transition i ++ j. The rows and columns of V must satisfy E j ]Viii 2 = E i [Viii 2 = 1. From (9.24) we infer that (a) the quarks of one generation are coupled to those of the successive generations with decreasing strength: Vub << Vus < Vud, etc.; (b) V is almost diagonal, i.e., it is practically the unit matrix; (e) Iv~jl ~ Iv~l; and (d) the third generation contributes marginally to the 2 • 2 GIM submatrix (otherwise the Cabibbo model would not have been so successful). Before outlining the calculation of c, we would like to stress that the Standard Model does not reveal the origin of CP violation - - it merely allows for its existence. Indeed, by offering no insight into the CKM matrix, the model betrays one of its most conspicuous shortcomings. The hope is that CP violation will contribute to a more profound understanding of flavor dynamics, this intricate component of SM that involves spontaneous breaking of electroweak symmetry and contains most of the free parameters of the model. Neglecting the external quark momenta, the amplitude corresponding to the box diagram in Fig. 9.3a reads 33 g4 i M = - - ~ - f d4k E ~ i ~ j '3

k2--

2 7e Lud mi

~s'~L 'k2

2 yaLvd -- m j

• _ i ( g ~ - - k ~ k o / m ~2) - i ( g ~e _ k ~ k ~ / m 2) k2 _ m 2 k2 _ m 2 ,

(9.25)

33 This time we employ the "unitary gauge", in which there are no diagrams with unphysical scalars. Since m t > row, the terms k~ka/m2~ and k~ke/m2w in (9.25) cannot be neglected.

9.1 Calculation of Am k and %

107

where ~i ---- Y/;Y/d.

(9.26)

As explained earlier, the quark masses in the numerators drop out. ~5~rthermore, the unitarity of the CKM matrix (which implies that any pair of rows or columns are orthogonal) ( v t v ) i 3 = 5,j = ( v v t ) # ++ ~ Vs k

= 6,j = ~

EkV3*k

(9.27)

k

gives E ~k = ~u -t- ~c -~ ~t ~ E Ys k=u,a,t k=u,c,t

= 0

(9.28)

(Y]~k ~k ---- 0 is an off-diagonal element of the unit matrix), thus simplifying the calculation considerably. Writing

1

m i2 mj2

1

(ks _ m~)(k~ - m~)

kn(k2 _ m~)(k~ - . @

k4

1 1 + k2(k ~ - my) + k2(k ~ - , @

(9.29)

we see that (a) the last three terms do not contribute because of (9.28), and (b) if the quark masses were equal, the GIM cancellation would result in 3/[ = 0. The remaining term yields a convergent integral that is straightforward to evaluate9 For mu = 0 it follows that g4 f iAd= ~a x •

r.

d4k 1 (2~r)4 k2(k 2 - m 2 ) 2 2

r.L(k2 -

(

4

2 2 "1 2~c~tmcmt ~t mt m~)~ + (k~ - m~)~ + (ks - - ~ m~)

1 - m--~ + ~

l

~(1

- ~5)~(1

- ~)v~,

(9.3o)

where we used kUk ~ = gU~'k2/4 (which holds because of symmetric integration in d4k), Ir162= k s and expression (9.15). Replacing the spinors by the corresponding field operators, equation (9.30) can be rewritten as (see the text between (9.16) and (9.21)) 9

2

2

M12 - --1GFfkmk

127r4 2 : ~t2mt4 f dak [ _~m~ 2~c~tmcmt • ~- L(k2-.~)~ + (k~-m~)~ + (k~-~--.~) [1 3k4 X 4(k2 _--m~)2 ] .

] (9.31)

We will ignore, for the moment, the term 3k4/4(k 2 - m2) 2 in the second square bracket (it gives corrections that are of the order of mq/m~ 2 2 and

108

9. The K ~ System in the Standard Model

therefore negligible except for the top quark: m t = (180 4- 15) GeV/c2). The remaining integrals are easy to solve, resulting in M12 ~ - G ~ f~ mk 2 2 127r2 BK [~cmc + ~t2 mt2 +

(9.32)

The first, second and third terms correspond to box diagrams with c~, t{ and c{ + t~ loops, respectively. The CKM matrix elements given in (9.24) yield (re i6 - co+ it/) ~2 ~ ,~2 ( 1 - 2iAZA4r/), ~2 ~ A4)~10 [(1 - co)2 _ r]2 + 2it/(1 _ co)],

(9.33)

2 ~ t ~ 2A2A6(1 - co+ it/). Clearly, the real part of M12 is dominated by ~ ~ sin 2 0c, despite the fact that mt :>> me. This justifies our earlier claim that the top quark can be ignored as far as Amk is concerned. Regarding the imaginary part of M12, note that the contributions from the three terms in (9.32), which are of the same order, are suppressed by the common factor A~)~6 sinS. This "explains" why the CP-violating parameter e is so small, and shows explicitely that e can be attributed to the presence of a complex phase factor in the CKM matrix. From (9.22), (9.32) and (9.33) it follows that

G~f~ mk BKA2A6~? 6v/-2 7r2Amk X {m2c [ln(mt2/rn2c)- 1]-f-mt2A2A4(1- co)}.

s ~ eir

(9.34)

The result of a more detailed calculation of e, which includes stronginteraction (QCD) corrections to the lowest-order electroweak amplitude, can be expressed as [54] (the original calculation is due to Inami and Lim [55]) 2

2

2

GFfk m k m w BK ] e ] - 12v~Tr2Amk x {~XcIm~ 2 + 2VctE(xc, xt)Im~c~t + VtE(xt)Im~ 2 }

= C~I3KA2)~6rl {[rktE(xc, xt) - VcXc] + A2A%?tE(xt)( 1 - co)}- (9.35) Here xi = m 2i / m ~2, the coefficients rh are perturbative QCD corrections ( ~ = 1.38, ~?t = 0.57, ~?~t= 0.47) and 2

2

2

GFI~ rnkmw -- 3.8 x 104 Cr =- 6v/~ rr2 Amk

(GF = 1.17 x 10 -5 G e V - 2 ) .

(9.36)

The functions E(xt) and E(xr obtained after the loop integration in (9.31), depend weakly on the top quark mass: 34 34 When next-to-leading-order QCD corrections are included, the "current-quark" mass Iilt(mt) rn;[ 1 - (4/3)a~(mt)/Tr] should be used (rn~" is the pole mass measured in collider experiments). =

9.2 B~

~ Mixing and Constraints on CKM Parameters

[4 - llxt_+_xt2 E ( x , ) = x, L 4(1 - xt) 2

E(xc, xt)=xr

[

109

3xt2 in xt ] 20:x-~t)3J

3x, 4(1-x,)

3x2!nxt 4(1-x,)2]

(9.37)

] "

Since ImM12 is dominated by large loop momenta (k > me), the lowenergy mesonic transitions K ~ --~ 7rzr,~, 7/, 0, ~# --+ /~0, while significant for Amk, do not affect e. As explained in [54], 7/~ depends strongly on the QCD parameter A, a number that is not fixed by the theory (one can think of A as the energy at which quasi-free quarks and gluons bind themselves together to become hadrons). Luckily, ~kxc is smaller than either of the other two terms in (9.35).

9.2 B~

~ Mixing and Constraints on CKM Parameters

The computed value of e depends on the CKM parameters A, ~/and Q (A is known to a high precision). From (9.24),

A- IVcb[ 2,

?]2 = ;1 GVub

(9.38)

Combining measurements of inclusive semileptonic B decays to charmed mesons with those of the exclusive decays B ~ --+ ( D * + ) D + g - G it is possible to deduce [56] [V~bl = 0.038 4- 0.002 --+ A = 0.79 4- 0.04.

(9.39)

The ratio [Vub/Vcb[can be obtained from semileptonic decays of B mesons, produced on the T(4S) resonance, by measuring the lepton energy spectrum above the endpoint allowed for the predominant B --~ Dgu transitions. There are large theoretical uncertainties in the predicted lepton spectrum used to extract Vub. We quote [56]

Vub

~r

= 0.08 4- 0.015 --+ r b _----V/~ + ~/2 = 0.36 + 0.07.

(9.40)

The magnitude of Vtd places an additional constraint on ~oand ~1. [Vtd] can be determined from "virtual" processes involving t ~ d transitions. The only available process is B~ ~ mixing, identified by the presence of "same-sign" leptons in semileptonic decays of B~ ~ pairs: e+e - -+ T(4S) --+ B~ ~ --~ ~+~+ + anything.

(9.41)

This reaction is possible if one of the neutral B-mesons can transform to its antiparticle before decaying. Otherwise, the flavor-specific transitions B ~ --+ ~ - ~ X and/~o __+ g+ PX would result in opposite-sign dilepton events in (9.41). It does not require much imagination, nor effort, to extend our description of K~ ~ mixing to the B ~ system. The Feynman diagrams responsible for

110

9. The K ~ System in the Standard Model Vtd

d

~

Bo

vg

t

= b

l I I

W +,J

d

~o

W-

v~

w*

v,;

vd

w-

~d

"-

Bo

I I

V~ (a)

(b)

Fig. 9.4a,b. Box diagrams for B~

~ mixing

B ~ ~ oscillation are shown in Fig. 9.43,b. Whereas the top quark contribution to kaon mixing is suppressed by [YtdYt*l2 ~/~10 in the case of neutral n mesons 35 .2 Am(B~)t c<mt2 I~dVtbJ 2

m2A2A 6 [(1 - Q)2 +~]2],

* 2

(9.42)

m t 2 A 2 A4.

2 4 , the diagrams with t quarks Since Am(B~ (x m2A 6 and Am(B~ o( m~A in b o t h internal fermion lines dominate the B~ ~ mass difference. The expressions for Am(B~,s) follow directly from (9.31) and (9.37):

Am(BO=d,s) = -~2[ C~ m BfBBB]d,~?B 2 x

f

G2 -

67r2

IV~qV~;12m~[ 1

d4k k2--~:~-~t2) 2 2

2

3k4 1

4(k2~n2)2 ] 9 2

mw[mBfBBBld,~lVtqVtbl r]BE(Xt),

(9.43)

where msd = 5 . 2 8 G e V / c 2, roSs = 5.38GeV/& and the QCD correction factor rlB = 0.55 + 0.01. The parameter BB, analogous to •K, is related to the probability of (rib) or (sb) quarks forming a B ~ meson. The product f2BBe contains all the uncertainties associated with the hadronic matrix element. Prom lattice QCD calculations, fBe BV/~d = (200+40) MeV (see [54]). Solving (9.43) for IVtd~;I yields

IV~d%l ~ IV~dl= (8.3 • 1.85)

(9.44)

x 10 - 3 ,

where we used the world-average value [57]

Am(B~) = (3.0 • 0.13) x 10 -4 eV -+ Am(B~ Amk -- 85.8 • 3.6.

(9.45)

Since 1 Vtd

V~b = V/(o -- 1)2 + ~/2,

35 In contrast to the K ~ system, there are two neutral B mesons: the the B~

(9.46)

B~

and

9.2 B~

~ Mixing and Constraints on CKM Parameters

111

it follows that ~t -= V/(Y - 1) 2 + r/2 = 0.99 • 0.22.

(9.47)

The constraints on COand r/ given by (9.40) and (9.47) have the form of rings centred at (0, 0) and (1, 0) in the (co,rl) plane, with radii % and t~, respectively. Another constraint on these parameters comes from CPviolating decays of KL~ mesons, as given by (9.35). The allowed values of 0 and rI are shown in Fig. 9.5.

0.6 0.5 0.4

~-o.3 0.2 0.1 0 -0.5

-0.4

-0.2

0

0.2

0.4

0.6

P Fig. 9.5. Allowed region in the ~

plane

As mentioned earlier, the CKM matrix can be parametrized in many different ways. We will now demonstrate, however, that there exists a quantity that "measures" the amount of CP violation in a parametrizationindependent manner, and in doing so will provide a simple geometrical interpretation of the unitarity equation (vvt)ij = 5~j. Expressions (9.27) lead to six relations that can be represented as triangles in the complex (CO,r/) plane. Consider, for example, the unitarity relation vudv

Since

(9.48)

+ v sv * + Y bS; -- 0.

V,,d ,-~ Vtb ~ 1, Vt*, ,~ - Vcb and V=s = ~, we can write (9.48) as vub

_

= 0.

(9.49)

+Y-7

Equation (9.48) requires the sum of three complex quantities to vanish. Geometrically, it defines a triangle in the (& q) plane, with sides

-~1 ~VubI = V / ~ + r l 2 = % '

xllVtd~cb : k//(aO-- 1)2-}-r/2 -----~t t '

1.(9.50)

112

9. The K ~ System in the Standard Model

The six triangles that represent the unitarity relations (9.27) have very different shapes, yet they all contain the same area. To show this we multiply each term in (9.48) by the phase factor V~dVtd/lV~dVtd I =-- a/tal, which leaves the shape and the area of the triangle intact, with the result la I + a VusVt____~* ~ + a VubVt*b _ O. lal lal

(9.51)

From (9.51) we infer that 1 Area(triangle) = ~ Im Vu*dYtdYt*sVus

:

- -

1 -~ Im Vu*dYtdVt*bVub .

(9.52)

Multiplying (9.48) by V~sVts/IV~sVts I leads to yet another expression for the same area: 1 Area(triangle) = ~ Im Vu*sYtsYt*bVub . (9.53) Repeating this analysis for other pairs of rows or columns, we obtain the following result [59] 2 • Area(each triangle) = Im V*jV~kV~kVzj =-- J.

(9.54)

In the Wolfenstein parametrization, J

= A2/~ 6

sin 5.

(9.55)

All C'P-violating observables within the Standard Model are proportinal to this quantity (see, for example, (9.34)). Note that each of the subscripts in (9.54) appears twice, once with V and once with V*. The quantity J is thus invariant under redefinitions of the phases of quark fields: qL --+ eiCqqL" In other words, unlike the mixing matrix itself, J is parametrization independent. We conclude this section with a brief review of some aspects of B~ ~ mixing that are relevant to our discussion of the CKM parameters ~ and 6. Ignoring C P violation, the time evolution of "flavor eigenstates" (K ~ or (B 0'/~0) is given by (D.19): IM~

= f+(t)lM~

+ f-(t)lM~

IM~

=/+(t)l ~~

+ f_(t)lM~

(9.56)

The functions f+(t) and f _ ( t ) , defined in (D.20), can also be written as f +(t) = e-imte -Ft/2

COS

[(Am/2 - iAF/4)t] , (9.57)

f _ ( t ) --~ ie-imte -Ft/2 sin [(Am/2 - i n r / 4 ) t ] , where F = (F1 +/'2)/2 is the average width of M ~ and M ~ m = (ml +m2)/2, Am = m2 - rnl and A F = F2 - F1 (we define 1 and 2 such that A m > 0). The probabilities of finding IM ~ or I-~/~ at time t are (see also (1.23) and (1.24))

9.2 B~

~ Mixing and Constraints on CKM Parameters

113

p ( M o _). M o) = p(]~,/o _> AT/o) = if+(Oi =

e-Pt 2

[ c o s h ( A F / 2 ) t + cos

p ( M o _.)./~/-o) = p(gT/-o _> M o) = -

Amt] (9.58)

[f_(t)]2

e-Ft 2 [cosh(AF/2)t-cosAmt].

The above expressions describe how an initially pure ]M ~ or ]1~/o) state evolves with time into a state of mixed flavor. As it decays, the system oscillates between M ~ and _~/0 with frequency A m . This deviation from a simple exponential time evolution is an unambiguous sign of mixing. For oscillations to be detected, the system must not decay away too fast. The magnitudes of Am and A F relative to F are therefore crucial parameters: Am F

lifetime mixing time"

(9.59)

Although (9.56) (9.58) apply equally well to both ( K ~ and (B ~ there are significant differences in the behavior of the two systems. Because of the light l~on mass, the dominant decay mode is K ~ --+ 7r~ (the CP-odd kaon state decays into the phase-space suppressed 3~ mode); hence Fs >> FL. In contrast, B ~ and B ~ have a number of important decay modes. However, the channels that are common to both B ~ a n d / ~ (and are thus responsible for a width difference AF) have branching ratios _< 10 -3, leading to A F / F < 0.01 for B ~ and < 0.2 for/~o. We can therefore neglect A F / F in the case of B~ ~ mixing. On the other hand, A m ( B ~ >> Amk (see (9.45) and the text below). Experimentally [57, 58], xk :

Z~mk --

-- 0.473 + 0.0018,

Fs

n.~(B ~

Xd -- ~

xs=

nm(B ~ -

-

Fs

-- 0.728 • 0.025,

(9.60)

> 10.5.

The parameter x expresses the oscillation frequency in terms of the average lifetime. Using (9.58), we obtain the following ratio of time-integrated probabilities: f o P( B~ -+/}~ dt r -= f o P( B~ -+ B~ dt (rim) 2 + (nv/2) 2 2r2 + (n.~)2 - ( n r / 2 ) ~

x2

- -

2 + x~'

0 < r < 1.

Another useful parameter is the oscillation probability:

(9.61)

114

9. The K ~ System in the Standard Model

fo~ P(B ~ -->/3~ X =- fo~ p(Bo --> BO)d t + J o P( B~ ~ / 3 ~ r 1 -

0 < X< -5"

l+r'

(9.62)

The measurement of B~ ~ mixing requires the flavour quantum number, B, of the neutral meson to be identified at both its production and decay. Since B (like S) is conserved in strong and electromagnetic interactions, B mesons are produced in pairs. The flavour of B ~ (/~o) can be traced by observing semileptonic decays B ~ -+ g - v X and/~0 _~ g+~X. One thus expects

B0 mi~ B0 d_~ ~+~X.

(9.63)

Experimentally, the amount of mixing is determined through

Tg = N(BB) + N(JBB) = U(g-g-) + N(g+g +) N(B[~) + N(BB) N(g+g -) '

(9.64)

where N(BB) is the number of B B final states in a sample of events from a process where a B/3 pair is initially produced, etc. Note that N(BB) and N(BB) are experimentally indistinguishable. When a B/3 pair is produced incoherently, 36 which occurs at energies well above the bb threshold, the time evolution of one meson is independent of the other. In this case, 7~-

2X(1-X) (1 - X) 2 + X 2

_

2____f__r incoherent production, r 2'

(9.65)

1+

since P ( B B ) = P(/3/3) = P(Ba oscillates) x P(Bb remains the same) = X ( 1 X), P(B/3) = (1 - ~()2 (neither oscillates) and P(/3B) = ?(2 (both oscillate). At L E P and hadron colliders, where both the B ~ and B ~ are produced, one measures the sum of their mixing probabilities, weighted by the corresponding production fractions: ;~ = fdXd + fsXs. The situation is quite different on the Y(4S) resonance, or at the B/3* threshold, where the two mesons are produced coherently (i.e., they form a quantum-mechanical state of definite orbital angular momentum, g, and parity). The Y(4S) resonance is a P wave bb bound state with C = - 1 and P = - 1 t h a t lies just above the Ba[~a threshold (its mass is less than 2 m(B~ The Y(4S) state decays strongly into B+B - or B~ ~ (see (9.41)). Since it is produced via a "virtual" photon, the B~ ~ pair is in a pure C = - 1 q u a n t u m state (see Sects. 1.4 and 7.1): 1

IB~ ~ = ~

{IB~176

-IB~176

(9.66)

The two B mesons are strongly correlated: at no time can the original B~ ~ system evolve into two identical states in the Y(4S) rest frame. As 36 That is, the angular momentum and parity of the pair are different for each event, i.e., the final state is a superposition of many angular-momentum states.

9.2 B~

~ Mixing and Constraints on CKM Parameters

115

explained in Sect. 7.1, if the mesons were to decay at the same time to the same final state, there would be two identical J = 0 bosonic systems in an overall P wave. But this would violate the rule that two identical spinless bosons cannot be in an antisymmetric spatial state. The B ~ and/~o propagate coherently until one of them decays. Only then will the state of the second particle be uniquely defined: it will have the flavor quantum number opposite to that of the first B meson. Suppose that the two decays occur at times tl and t2. Using expressions (9.56), which describe the time evolution of flavor eigenstates IB~ and IB~ we obtain (with f~_ ~ f + ( t l ) , etc.) 1 2 _ fl_f2_) ]BOBO) ]BOBO(t)) ~ (f~_f2_ _ f~_f2) iBOBO I + (f~_fr

1 2 +(f_f__ I 2 IB~176 + if_f+ 1 2 /;f+)

(9.67)

From (9.67) we see that P ( B ~ ~ = p(/~o/~o) and P ( B ~ ~ = P(/~~176 hence,

Tr = f o I f + ( t l ) f - ( t 2 ) - f - ( t l ) f + ( t 2 ) l 2 dtldt2 _ Af

(9.68)

f o If+(tl)f+(t2) - f - ( t l ) f - ( t 2 ) ] 2dtldt2 - :D" Writing

f•

= l e - i m t e - C t / 2 [eiZ~mt/2e z~Ft/4 • e-iZ~mt/2e -z~Ft/4]

(9.69)

2

it follows that .M"

T)

=

=

dtldt2e -rt

/j

e z~Fz~t/2 + e -zlcz~t/2 T 2 R e e - i A m A t

dtldt2 [e-&tle -Fit2 -t- e-lht~e -F2t~

:~22 R e e - F t l eiAmtl e - F t ~ e -iAmt2 ]

2 -

FIF2

:F

2 : r2

_

(

2Re

1

1

F - iAm F + iAm 2

r/2)2

+

(9.70)

Therefore 7~=r-

1 -XX '

coherent production (g odd).

(9.71)

The T(4S) resonance decays to B~ ~ or B + B - , and so the observed number of N(t~+/-) events has to be corrected for leptons coming from charged B mesons, a procedure that is not entirely unambiguous. The first evidence for B/~ mixing was provided in 1987 by the UA1 experiment a7 at the CERN pp collider [60]. Soon thereafter the ARGUS collaboration at the DORIS e+e - storage ring observed large B d0- B- 0d mixing 37 In 1983, the UA1 collaboration, led by C. Rubbia, discovered the intermediate vector bosons W J: and Z ~

116

9. The K ~ System in the Standard Model

(r -- 0.21 4- 0.08) among B mesons produced in T(4S) decays [61]. Their result strongly suggested that the top quark was much heavier than expected. Until recently, all measurements of B / ) mixing were time integrated. These studies are insensitive to x when mixing is maximal because x --+ ec as 0 -0 X ~ 0.5 (see (9.60)-(9.62)). To measure Bs-B s transitions one thus needs to determine the time evolution of B ~ mesons, which is not an easy task given their rapid oscillation rate. The oscillation period gives a direct measurement of the mass difference between the CP eigenstates B1~ and B ~ provided the proper time of the Bmeson decay, tp, is known with sufficient accuracy: tp = L/~'~ = L(m/p), where L is the measured decay length; m and p are are the mass and momentum of the meson, respectively. The typical experimental resolution of L E P experiments is 2.5 ps in tp. The value of A m is found from the fraction of "mixed" or "unmixed" events as a function of tp by using (9.58). Based on data collected between 1991 and 1994, the DELPHI collaboration at LEP has reported [58]

Am(B ~ > 6.5ps -1 (4.3 x 10-3eV)

at 95% CL

(9.72)

corresponding to xs > 10.5, where xs = Am(B~ ~ and ~-(B ~ = (1.61 40.1) ps. As we mentioned earlier, the measurement of IVtd[ suffers from large theoretical uncertainties associated with fB V/BBB.This uncertainty can be reduced by measuring

Am(B ~ m.~ Vts 2 mB ~ ~s2 Am(B ~ - robe Vtd ~ - mBd A2 [ ( Q - 1 ) 2+r/21 '

(9.73)

where is is the ratio of hadronic matrix elements for the B ~ and B ~ [54]: fB~ ~

_ 1.16 4- 0.05.

(9.74)

Unfortunately, it is much more difficult to determine Arn(B ~ than Arn(B~) because (a) the fraction of B ~ mesons produced in b decays is considerably smaller than that of B ~ particles, and (b) the large value of A m ( B ~ (experimentally, Am(B~ ~ > 14) leads to rapid oscillations that complicate the measurement.

9.3 Rare Kaon Decays Over the past forty years, studies of rare meson decays have contributed significantly to our present understanding of weak interactions. As explained in Chap. 1, the observation of both K -+ 27r and K --~ 37c decays led to the discovery of parity violation. Parity is maximally violated in weak interactions

9.3 Rare Kaon Decays

117

in the sense t h a t all neutrinos are left-handed and all antineutrinos are righthanded. This motivated Feynman, Gell-Mann and others to formulate the weak interaction in terms of vector-axial vector (V-A) currents. The helicity suppressed decay 3s ~r --~ cue provided crucial support for the "V-A theory" ,39 which has been very successfifl in explaining most of the low-energy weakinteraction phenomena. The observed violation of C P s y m m e t r y in K ~ decays (at a rate of about 10 -3) may be a fundamental property of nature, with important implications for the early evolution of the universe. A deeper insight into C P violation is expected to be gained from precision measurements of theoretically "clean" rare kaon decays, such as K ~ --e 7r~ The suppression of the KL~ -+ # + p decay (at the level of 10-s), discussed in Sect. 9.1, suggested the existence of the charm quark, and thus played an important role in the development of the Standard Model (the GIM mechanism). In the same section we described how the sensitivity of K ~ ~ mixing to energies higher than the kaon mass scale was used to predict the mass of the charm quark. Similarly, rare kaon decays t h a t are dominated by one-loop Feynman diagrams with top quark exchange can yield valuable measurements of the C K M matrix elements Vtd and Vts. Since the branching ratio for t -+ d + W is very small, it is difficult to determine the coupling Vtd directly from t decays. Rare kaon decays are an important source of information on higher-order effects in electroweak interactions, and can therefore serve as a probe into physics beyond the Standard Model. Experiments at the highest-energy particle colliders, and those studying the rarest of K-meson decays at low energies, are pursuing different aspects of the same physics. In what follows we will concentrate on those processes that are theoretically best understood. Our exposition is meant to be pedagogical, rather than comprehensive, in order to highlight the underlying physics. We will not discuss decays t h a t violate lepton number conservation.

9.3.1 K ~ --+ 7r~

and K + -+ n'+v0

Within the Standard Model, these transitions are loop-induced semileptonic decays of the type s -+ d + g + g. They are entirely due to second-order weak processes determined by Z~ and W-box diagrams: since photons do not couple to neutrinos, there is no electromagnetic contribution. 3s Since the pion has spin zero and, according to the V-A law, the neutrino is left-handed, the lepton in 7r -+ gv~ must have negative helicity (~ = -1). The probability for a lepton of velocity v to be left-handed is P()~ = - 1 ) = 1 v/c. This probability is much smaller for the light electron than for the muon ( m u / m ~ ~ 200). The electronic decay mode is even more suppressed in the K -+ gu~ case because the electron is more relativistic than in the pion decay. The phase space can do little to improve its odds against the muonic decay mode. 39 In fact, the suppression is proportional to ( m ~ / m u ) z for any arbitrary mixing of V and A couplings: ,.7~e = ftt",/~,(Cv + CA",/5)yv --+ F(71" --4' g//g) (X 4(C~ + C~)m~.

118

9. The K ~ System in the Standard Model

Both decays are theoretically "clean" because the hadronic transition amplitudes are matrix elements of quark currents between mesonic states, which can be extracted from the leading semileptonic decays by using isospin symmetry. The process K ~ --4 7r~ offers the most transparent window into the origin of CP violation proposed so far. It proceeds almost entirely through direct CP violation [62], and is completely dominated by "short-distance" loop diagrams with top quark exchange. Although this decay has a miniscule branching ratio (about 10 -11 ) and is experimentally very challenging, its measurement, which is complementary to those planned in the B ~ system, is feasible and certainly worth the effort. The main features of the decay K --4 ~rup, summarized above, can be discerned from the relatively simple calculation of the box diagram in Fig. 9.6a. Neglecting the charged lepton mass and external quark momenta, the corresponding amplitude reads (the contribution of unphysical scalars in the t'Hooft-Feynman gauge can be ignored because their coupling to leptons is proportional to me/m~ << 1) Vid

d

W

,~

",r

~'

d

r,

I g

-,~

~d

ql

.

] l'k l

(a)

_ s ~

(b)

,

gv(fa-)

d

r

w-.r d' ~

v~

~i w+,

vi; w+,,+

d ~

W-,r

.w. ~.~

\ "~ (~)

~

// v 0~')

~

v0r)

d

~

g

U,C,t

(~+)

v(U-)

(~+) (c)

Fig. 9.6. Electroweak diagrams for K --+ 7ru~ (a,c) and K~ -~ lt+p - (b,c)

9.3 Rare Kaon Decays

iM =

Z

i=u,c,t x ua

V/dV/* ~ ' ~

( -igv,,

~s

[ig

yr2 3~ L

~_

-igue

" s ---~2

u,

k2 -igo

--7

2

"y'L

- m~ L

]

119

1 (9.75)

i

[-ig

o ]

"-,

i.e. -94 i M - (27r)4 Z

VidVi*~ [f&7zLud] [fivTzLv.]

i=u,C~t

d4k •

(9.76)

(k: - m~)(k:

- m~):'

where we used k~k~ = g ~ k 2 / 4 and 3'~'Y~3'~(1 - 3/5) | 3~u~W'Yv(1- ~5) = 16"y~(1 - %) | "y~(1 - "Ys) (9.77) (note that the order of "y-matrix indices in this equation is not the same as in (9.15); hence 4 ~-+ 16). Using Feynman's parameter formula, 1 _ fl - Jo

2xdx

(9.78)

l a x + b(1 - x)] 3'

the well-known expression d4k F ( n - 2) 1 (k 2 + t ) n - i 7 r 2 F ( n ) t n-2'

n>3,

(9.79)

and

VudV&=

- vc~E~

- ~dE~

(9.8o)

(see (9.28)), it is straightforward to show that V/dV/s I

.A/[ -- (47r)2m~ E

~-~ :~-)2-

i=-c,t

) (9.81)

• [~sT~LUd] [~tv%Lvv]

with xi =- m i2/ r n w2, xu ,-~ 0, x~ ~ 2.6 • 10 -4 and xt ~-- 5. As for (9.17), we can express this amplitude as the matrix element of an equivalent operator between the states IK) and [Tr):

i:c,t

x (Tr I $ ' ~ d

[ K)[P~/~(1

- ~5)u],

(9.82)

where [] stands for "box diagram", GF ~ CK =- V ~ 27rsin20w ' -

"TD(xi) =

Xi(X i -- ln xi -- 1) ( x i - 1) 2

(9.83)

120

9. The K ~ System in the Standard Model

and 0w is the weak mixing, or Weinberg, angle: e = gsin0w, a - e2/47r. Since Ir and K have the same parity, only the vector current contributes to The amplitudes @l.i7~d]K) are much simpler objects than the matrix element of the four-quark operator in (9.17). In the limit of exact isospin symmetry, which is a very good approximation, @+l$7~d[K +) = v/2 @~176 ). Moreover, the matrix element of the weak current Sy~d between K + and ~+ is related by isospin to the known matrix element of the operator gT~u between K + and 7r~ @+l$7"d]K +) = v/2@~ I ~7"u ] K + ) .

(9.84)

The operator $7~u is measured in the decay K + --~ ~~ for this transition is given by (see Fig. 9.1c)

A( K+ --+ 7~~

The amplitude

= ~22V~( :r176I s7 ~u I K+) [~Ta(1 - 75)e] 9

(9.85)

Neglecting the positron mass, the branching ration for K + -+ 7c+vi per neutrino flavor reads (the decays K + --+ ~r and K + -+ 7:~ have essentially the same phase space)

B(K+--+zc+u~')=B(K+--+Tc~ = B ( K + ~ 7r~

27r sin 2a 0w 7)/-~2 Vus]2 (x/2) 2

a2 [7)[2 sin40w A2 "

(9.86)

The complex coefficient 7) depends on the charm and top quark masses: 7) = E

V/dV~; ~-(x,).

(9.87)

i=c,t

To show that the decay K ~ -+ 7r~ is CP violating, consider the behavior of the corresponding interaction lagrangian under CP: _- qSKLO,q5 o~7~( 1 _ 75)V c P [_4~KL] OU [--4i~o][-- PTt'(1 -- 75)V] Using (3.2), (1.8) and (9.82), we can write [62]

.A(K ~ ~ T:Ov~,) = ~ A(K~ --+ 7c~

+ A ( K ~ -+ 7r~

(9.88)

where 4~

40 Note that (Tr~ (see Appendix E).

"~ = (~~

~ = (.~176

9.3 Rare Kaon Decays

A(K o ~ .o~)=

121

1 [~4(KO --~ 7rOmp) + ~4(k o -+ 7ro~) ] = Re A ( K + -+ 7r+~,~),

A(Ko ~ ~o.v) :

(9.89)

1 [A(K o -~ ~ o ~ ) _ A ( k o _~ ~o.v)]

= iIm A ( K + --+ 7r+v~,). Summed over three neutrino flavors, the branching ratios for the indirect and direct CP-violating contributions are, respectively, B ( K L0 --> 71"0VV)indirect ~,~ 3 I~l2 ~KL • 2.8 • 10 -~

TK+

• [br(Xc) + A2A4(1

-

Lo).~(xt)]

2

(9.90) U(KL0 --~ 71"0/-'P)direct ~ 3 TKL • 2.8 • 10 -6 [A2A4~.T(xt)] 2 TK +

for B ( K + -+ 7r~ = 0.0482, sin 2 0w = 0.23 and c~(mw) = 1/128. The small value of e renders the contribution from indirect C P violation (and hence from the charm quark) insignificant. Therefore, B ( K ~ --4 T:~ ') ~ B ( K ~ ~ 7r0up)direct = 8 • 10 -11 [~$'(Xt)] 2

(9.91)

based on TKL = 4.18~-K+ and A = 0.8. To complete the calculation of B ( K -~ 7r~D), we consider the remaining diagrams in Fig. 9.6. The presence of unphysical scalars in Fig. 9.6c cannot be ignored because of the large top quark mass. The result of a somewhat lengthy calculation yields Az(K

V~dV~* $'z(X~) (~')V-A,

~ Try,F,) = r

(9.92)

i:c,t

where jZz(xi ) -_ xi [x~ + xi(31nxi - 7) + 21nxi + 6] 8(xi - 1) 2

(9.93)

Combining this result with (9.83), it follows that xi [ 3xi - 6 2 + x~ ] 9~ ( x i ) = .,~z(Xi) + 9rD(xi) = ~- [(~-i :-1~ 2 lnxi + xi - l J"

(9.94)

We thus finally obtain, for ~] = 0.36 and mt = 180 GeV/c 2, B ( K ~ -+ ~r~

,,~ 2.8 • 10 -11.

(9.95)

Isospin-violating quark mass effects and electroweak radiative corrections reduce this branching ratio by 5.6% [63]. Next-to-leading-order QCD effects are known to within +1% [54]. The overall theoretical ambiguity in the calculation of B ( K ~ --+ 7r~ is below 2%. This uncertainty does not include

122

9. The K ~ System in the Standard Model

the error on the CKM p a r a m e t e r ~, as given by the (correlated) constraints (9.40) and (9.47). The detection of K ~ --+ 7r~ presents a formidable challenge. The experimental signature of this decay is a single unbalanced 7r~ which makes background rejection very difficult. The direction of photons from the decay 7r~ --+ 2"y can be determined through their conversion to e+e - pairs. In general, the most important backgrounds are K ~ --+ 2"y (B ~ 5 • 10-4), K ~ --+ 27r~ (/~ .~ 10-3), neutron interactions at residual gas atoms in the decay region t h a t produce 7r~ A --+ n~r~ decays, etc. The K ~ --+ 2"~ decay, for example, can be discriminated by using both the transverse m o m e n t u m balance of the two-gamma system and the position of the detected photons with respect to the b e a m axis. Alternatively, the final state can be defined by selecting those events in which the ~r~ undergoes the Dalitz decay 7r~ ~ e + e - % In this case it is possible to reconstruct the vertex of the decay and hence the invariant mass of the Ir~ Another advantage over the 2"y final state is t h a t a relatively wide b e a m can be used. However, this method has the disadvantage that (a) the 7r~ --~ e + e - ' y decay has a small branching ratio (about 1%) and (b) the final state in the radiative decay K ~ --+ 7r+eT'yp looks like e + e - 7 + "nothing" if the 7r+ is misidentified as e +. All a t t e m p t s to detect K ~ --+ ~r~ thus far have relied on the Dalitz decay mode. The best published limit to date is B ( K ~ --+ 7r~ < 5.8 z 10 -5 (90% CL) from Fermilab experiment E731/799 [64]. There are several proposals to measure B ( K ~ -+ ~r~ The KTeV experiment, described in Sect. 8.1, is expected to reach a sensitivity of 10 - s by identifying ~r~ through the Dalitz decay. The K A M I collaboration [65] has proposed to use the Main Injector at Fermilab as a source of very highintensity and high-energy neutral kaons, and to detect 7r~ -+ 27 decays in the pure CsI crystals of the KTeV apparatus. They aim at a sensitivity of better than 10 -12. An experiment at the K E K laboratory [66] intends to employ an array of CeF3 crystals to measure the energy and position of the two g a m m a s from the ~r~ --+ "y'~ decay, and a lead/scintillator barrel calorimeter to select two-photon events. The experiment E926 at Brookhaven [67] would exploit high b e a m intensities of the AGS proton synchrotron, which will be able to provide, by the year 2000, over 1014 protons/pulse. The Brookhaven group proposes to obtain low-momentum kaons ((Pk) =700 MeV/e) from a microbunched proton beam. This would allow them to determine the m o m e n t u m of the decaying KL~ using time of flight measurements. The expression for 13(K + -+ 7r+v~) with three quark and lepton families was originally derived by T. Inami and C. Lim [55]:

9.3 Rare Kaon Decays B(I( + ~ ~ + ~ ) =

123

a 2 ( m w ) B ( K + --+ 7rOe+u~)

2zr2 sin 4 0w

x }]

E~=~,~v,~v~;7(x~,y~)

(9.96)

(cf. (9.86) and (9.87)), with J:(x,y)-

lay xy 16 x - y

(y-4

+~--

~

\~]

7

x nx[x x4 x

+8

3x

\~L-~_l] + 1 +

(

16(x--l)

----------7 (x 1)

]

3) l+y_l

"

(9.97)

In the above two equations, xi - m i2/ m w2, i = c, t (quarks) and Yt =- m e2/ m w2, = e, tt, T (leptons). For y --+ 0, (9.97) reduces to (9.94). Experimentally, xt ~ 5, xc, y~- ..~ 10 -4 (mr ~ 1.78GeV/c2), y~ -~ 10 -6 and yr ~ 10 -11. We can thus write lnxr - 1] . % ( x t , y t ) ~ : F ( x t ) , .%(x~,yt),~xc. x~lnz--~- y--elnye F - (9.98) t Ye - x~ 4

[

and B ( K + --~ it+uP) ~-. 2.8 x 10-6{3 [A2 A4r] 2F(xt)] 2

+ ~

f(~c,y~) + A~a4(1 - o)f(~t)

, (9.99)

#.=e,,u,Tk

where, to a very good approximation, .T(xe, y~) ~ .T(Xc, y,) -~ ~'(xc, 0). In contrast to the CP-violating decay K ~ --+ 7r~ the charm and top quark contributions to K + --+ 7r+u0 are of comparable size: the smallness of ~-(x~, Ye) in comparison with .T(xt) is compensated by the strong CKM suppression of the t contribution. Isospin-violating quark-mass effects and electroweak radiative corrections (which do not affect the short-distance structure of K -+ zruO) result in a decrease of the branching ratio by 10% [63]. Possible long-distance contributions are estimated to be negligibly small [68]. Short-distance QCD effects represent the most important class of radiative corrections to this process. The inclusion of next-to-leading-order QCD corrections yields [69, 54] 0.88 x 10 - l ~ <_B(K+--+zr+ug,) <_1.02 x 10 -m, QCD uncertainty.(9.100) for rAc = 1.3GeV/c 2, rSt = 170GeV/c 2, 0 = 0 and 7/ = 0.36. The above theoretical ambiguity is associated mainly with the charm quark. Based on

124

9. The K ~ System in the Standard Model

our present knowledge of Standard Model parameters, Buchalla et al. predict [54] 0.6 • 10 - l ~ ___B ( K + --~ 7r+~)theo~ < 1.5 • 10 10.

(9.101)

The determination of IVtdl from /3(K + -+ 7 r + ~ ) and of (~,~) by using both B ( K + -+ 7r+vS) and 13(K ~ -+ 7r~ is discussed in [69, 54] (see Fig. 9.7). The expected accuracy is comparable to that one can achieve by studying CP asymmetries in B decays [67]. = 7(1 -

2/2)

~--K /

(0 3)

+

/

~=0(1-~2/2)

(1,0)

Fig. 9.7. Determination of p and ~/ from K -+ 7rvP

The decay K + -+ 7r+v# occurs at a very low rate in the presence of large background sources of pions and muons, and has only the weak kinematic constraint P~ < (m 2 - m2)/2rnk =227 MeV/c. Fortunately, each of the two major K + decay modes K + --+ n+Tr ~ and K + --+ p+uu produces a single charged track of unique center-of-mass momentum (205 MeV/c and 236 MeV/c, respectively), i.e., it yields a kinematic peak that can be avoided with sufficiently good momentum resolution. Unambiguous ~-# identification and efficient photon detection are also required to reject these and other background decays. The only other important background sources are scattering of beam pions and K + "charge-exchange" reactions (see (1.42)) resulting in K ~ --+ n + g - ~ decays, where t~ = e, # is missed. In the experiment E787 at Brookhaven [70], K + mesons are stopped in a target at the center of a hermetic detector (see Fig. 9.8). The momentum, kinetic energy and range of the charged decay products are measured to distinguish pions from muons. In addition, the pulse shape in a stopping counter is used to identify the decay sequence n+ --4 p+ -+ e +. For their 1995 datataking, kaons of 790 MeV/c were delivered at a rate of 7 • 108 per 1.6 s spill of the AGS proton synchrotron. The pion contamination in the beam was reduced to about 25% by using a particle separator (see Sect. 8.1). The kaons are identified by (~erenkov, tracking and energy-loss counters. Before being brought to rest in a highly segmented target composed of scintillating fibers, approximately 20% of the kaons pass through a BeO cylinder, where their momentum is reduced to about 300 MeV/c. Beam protons are completely ranged out and many of the remaining pions are absorbed or scattered in this

9.3 Rare Kaon Decays Magnet

125

~nr~,an~

Stack bes

upport

Fig. 9.8. Schematic side view of the E787 detector

"degrader". The energy and pathlength information from the target is used to correct the total energy and range determination for charged decay products. Charged particles emerging from the target are m o m e n t u m analyzed as they traverse a cylindrical drift chamber. They are subsequently stopped in an array of plastic scintillation counters and straw tube tracking chambers ("range stack") to measure their range and kinetic energy. The pion detector is completely surrounded by a lead/scintillator photon detection system. The whole apparatus is contained in a solenoidal field of 1 T. In the pion m o m e n t u m region 211 M e V / c < P~ <230 MeV/c, E787 observed one event consistent with the decay K + --4 7r+uP. The background was estimated to contribute 0.08 =h 0.03 events. Assuming that the observed event is due to K + --4 7r+u#, they obtained [71]

B( K+ -4 lr+uP)exp = ~'"-3.5/I 9 + 9.7 • 10-10. 9.3.2

(9.102)

K ~ --+ i t + i t - a n d K ~ --+ e + e -

The short-distance contributions to K ~ --+ # + # - and K -+ ~rup come from one-loop diagrams in Fig. 9.6, and are therefore very similar (the only difference is due to the reversed lepton line in the box diagram). There is another similarity between these two flavor-changing neutral-current processes: their amplitudes can be related in a simple way to the matrix elements for the charged-current transitions K + --+ # + % and K + --+ ~r~ respectively

126

9. The K ~ System in the Standard Model

(see (9.108) and (9.84)). Neglecting the mass of the "virtual" lepton, one readily infers from the box diagrams in Figs. 9.6a,b and expressions (9.15) and (9.77):

Jk413 d--+

=

CKEVidVi*s~[](xi)($d)v_A(~g)V_ A (9.103) i=c,t

lip

(cf. (9.81) and (9.82)), where e = #, v and ~'D(xi) is given by (9.83). The penguin diagrams in Fig. 9.6c yield (cf. (9.92))

"~Z 8d-+

= C K E YidYi*s.~z(Xi)(sd)v_A ~ (~12)V_A/ i=c,t

l/l/

with JZz(x ) given by (9.93). Note that the term sin20wpV~p is absent from the expression for gd --+ ## due to vector-current conservation (photonic penguin diagrams do not contribute to K ~ --+ # + # - for the same reason). To show this, we write A ( K ~ -+ #fz) c< (01ff~lK~ Since the K ~ meson is a pseudoscalar, the only kinematical quantity available to construct the hadronic matrix element is the four-momentum transfer q~: (0 IJ~l K~ -= fkq~.

(9.105)

This implies that .7~ is an axial current because q~ is a polar vector (see footnote 32). Now, ~q~v~g - ~Ot?= ~(15e -/Sg)t? = 0, where we used the Dirac equation (15 - m)g = 0. The gd --+ #fz annihilation amplitude thus reads

A4(gd -+ ##) = --CK E VidVi*G(xi)(sV~vsd)(#V~75#)' i=c,t

(9.106)

G(x) = g ~

(9.107)

where l n x + 8(1 - x~---~"

We take A,t + .M t to be an effective hamiltonian for the decay K ~ ~ #+#-.41 As a consequence of isospin invariance, (01 s 7 ~ 5 u I K+> = (0l g%75dl K~ =- fkqa.

(9.108)

Therefore,

A ( K ~ ~ #p) = - CK~-~ 6(x~)( 0 IV~dV~s(ST~d) i=c,t + Vi~V~ (dT~ 75 s ) I g ~ (#7~75#)

=

-

x/2Cgae E VidVis* 6( X i)(O ]gTa75u]K+>(fi'7'~75#) i=c,t aGFfkq~ Re E V~dVi;6( i)(#7 75#), 27r sin20w i=c,t

(9.109)

al Since K ~ is (mainly) CP-odd, only the CP-odd combination of axial currents gT~v5d + dV~75s contributes to the transition K ~ -4 vacuum.

9.3 Rare Kaon Decays

where we used IK~ ~ [IK~ -/<~ ( 0 157~75dlK ~ = ( 0 = -

127

and (see Appendix E)

ICP)tCP($'7,~'75d)(Cp)-1CPlK~ <0 ]dT~75slR~ - p )),

p ))

i.e.,

(0 ]$7~75dIK~

p )) = - (0 IdT~75sl/~~

(9.110)

p )).

Expression (9.109) can be written as A ( K ~ -+ ##) - 21r sin20~,

aa~fkb

-- ~sin20----~

(9.111)

m.ft(k)'y5v(p)

with qa = (k - p ) ~ and /~ -= Re E

VidVi*~6(xi).

(9.112)

i=c,t

Hence,

]A] 2 = 4(aFfk) 2 \~sin20w ) m~ (kp + m~).

(9.113)

In the center-of-mass system, the decay rate is given by dF-

(2n)4 d3p d3k 2ink 1"412 (27r)a2E u (2~r)32E, 5(ink - 2E,) 3(3)(k + p). (9.114)

Using p = - k --+ kp = E~ + p 2 and

/d3k=/,k,2d,k,d~=47r/~E2-m2E.dE.

(9.115)

it can easily be shown that

F(K ~

--}

##)

__

zlm,u

8~

\ ~ ~i--Jew ] m , ,1 - ~ ]

.

(9.116)

The K + --+ #+vt, amplitude reads (see Fig. 9.1a)

A ( K + -+ #+L,t, )

-

GFfk V,s(k - p)~u~,(k)'7 ( - 0'5)v,(p)

-

- GFSk v~~ Vu,mtfi(k)(1 +75)v(p)

-

'~

1

(9.117)

since fi(k)k = 0 and (15 - mt,)v(p ) = 0. Therefore,

1.,412 = 4( GF f l )21V~sl2m~ (kp).

(9.118)

128

9. The K ~ System in the Standard Model = 47rf E .2d E . , it follows readily

Ifwenotethatkp=E.(Eu+E.)andfdak from (9.114) and (9.118) that

F ( K + -4 #+u,) =

1 - m#

(GFfk)2[Wusl2mk 2 87r m,

(9.119)

The branching ratio for K ~ --* # + # - can thus be expressed as [55]

( NK~

(,

~ ~sin20w

-

2 TKL

-

Iv~sl 2 (1

2

2 2

x B ( K + -4 #+vu).

(9.120)

The Standard Model expectation for the short-distance contribution to the KL~ -4 # + # - branching ratio is [54] 0.6 x 10 -9 < B ( K ~ -4 #+#-)SD --< 2.0 X 10 -9.

(9.121)

In passing we note that the K decay constant J'k is obtained from K -4 #u using

(GFfk)elv~sl~mk

F ( K -4 #vt, ) =

87r

x ( 1 + 2c~lnm--2 7r rnk

2 m~,

l _ rnt*

3~lnmk" ~ , 7r m~ /

(9.122)

which includes the leading radiative corrections [72]. Experimentally, 2 2 =l.23x10-3GeV f~lV~l

7

or

fk=159=t=l.6MeV.

(9.123)

g

~t

Fig. 9.9. The two-photon absorptive contribution to K o __+#+/~-

It turns out that the most important contribution to K ~ -4 # + # - comes from the real two-photon intermediate state shown in Fig. 9.9. The corresponding absorptive (imaginary) part of the amplitude .A(K ~ -4 #+#-)2~ can be estimated reliably: it is simply the product of the experimentally known decay amplitude A(KL~ -4 77) and the pure QED scattering amplitude ~4(77 -4 # + # - ) . The decay rate for this process is given by [73]42 42 The factor (me~ink) 2 in (9.124) suppresses g~ --~ e+e - relative to K~ ~ #+p-. This is the familiar helicity suppression of the decay of a pseudoscalar meson into a fermion-antifermion pair through the axial vector coupling.

9.3 Rare Kaon Decays

F(K~

=-2 \mk/

~ ln\~/j

129

=1.2• 10 -5, (9.124)

where 3 = (1 - A~,2/~,2~1/2 .... u/,,okj . Based on the experimental value [21]

B( K~ --+ 77)exp -- (5.92 + 0.15) • 10 -4,

(9.125)

this leads to B(KL~ --+ # + # - ) a b , = (7.1 • 0.18) • 10 -9,

(9.126)

which pretty much accounts for the observed decay rate [21]

I3(K~ --+ #+#-)e~v = (7.2 + 0.5) x 10 -9.

(9.127)

The dispersive (real) part of the amplitude .A(K~ --+ # + # - ) is the sum of long- and short-distance contributions. The long-distance part is mainly due to the virtual two-photon intermediate state ( K ~ -+ 7"7" --+ # + # - ) , and is difficult to calculate reliably (see, e.g., [107]). This is unfortunate because, as we have seen, the short-distance part is known rather well. The experiment E871 at Brookhaven has detected approximately 6400 KL~ --+ # + # - events [74] (previous experiments had accumulated about 900 K ~ --+ # + i t - decays [75]). As shown in Fig. 9.10, protons of energy 24 GeV from the AGS produce a neutral beam, in which about 2 x 107 K 0 mesons decay per beam spill within the vacuum decay region of the E871 detector. The neutral beam is stopped inside the spectrometer by a hadronic "beam plug". Tracking is provided by straw chambers in the upstream section of the spectrometer (where rates are the highest), and by drift chambers in the section downstream of the beam plug. The beam volume within the spectrometer is filled with helium to reduce multiple scattering and particle interactions. Trajectories of two-body charged decay products are nearly parallel to the initial beam direction downstream of two dipole magnets. A threshold Cerenkov counter and a lead-glass array are used for electron identification (the experiment also intends to search for the lepton-number-violating decay K ~ -+ #e at the 10 -12 level). Muons are identified and stopped in a segmented absorber stack containing proportional wire chambers and scintillator hodoscopes. The experiment is expected to measure the K ~ --+ p+p- branching ratio much more precisely than its predecessor, the E791. It remains to be seen, however, if the anticipated improvement in experimental precision can be matched by an improvement in the theoretical estimate of the long-distance dispersive contribution to this decay. Should E871 confirm that the measured branching ratio is indeed saturated by •( K ~ --4 #+#-)abs, it would be difficult to understand why the two apparently unrelated dispersive contributions cancel each other. Since the kaon is spinless, the muon pair in the decay K ~ -+ # + # - has the total angular momentum J = 0. This implies that s = 2, where s and g are the spin and orbital angular momentum of the muonic system, respectively. The replacement #+ ++ # - means not only an interchange of spins, but also

130

9. The K ~ System in the Standard Model

Slraw Cerenkov i Magnets Lead GlassMuonRangefinder | A _~, ~ ,.~...

Chambers Sweeping

Target Magr~fs Pro,on

N 9=-

Straw Chambers

f

Orilt C h a m b e r s

!

!

0

10

l Instrumentediron TriggerCounters(x andy) TriggerCounters(x only)

I

I

I

I

20

30

40

5O

m

Fig. 9.10. Plan view of the E871 spectrometer space inversion r --+ - r. The combined operation of charge conjugation and parity transformation results in ~/5r

= (_1)s+1r

(9.128)

Thus there are two possible states for this system, which are eigenstates of 5P:

CD](#+#-)3P~ = + 1(#+#-)3P~

(9.129)

= _ I(,+,-),So>.

The most general amplitude for K ~ --+ p + p - has the following form (cf. (9.111))

A 4 ( K ~ --+ #+#-) = ~t(p_, s_ )(a% + ib)v(p+, s+ ).

(9.130)

The amplitude a is CP conserving (the # pair is in the 1So state), whereas b is CP violating (#+#- is in the 3p0 state). The interference between the two amplitudes results in longitudinal polarization of the leptons, a CP violating observable. If we sum over the possible spin states of both muons, IA4]: = Tr [(15- + mt,)(a% + ib)(f)_ - m , ) ( - a * % - ib*)] = 4 {p+. p_(lal 2 + ]bl2) + rn2(laJ 2 -Ib]2)}.

(9.131)

Using (9.114) and (9.115), it can easily be shown that the total K ~ --+ # + p decay rate is given by

F-

mk~ (la12 + ~21bl2) 87r

where, as before, ~ = (1 - A~,2/..2~1/2 ~,,~tt/,,~k) 9

(9.132)

9.3 Rare Kaon Decays

131

Since we are interested in polarized muons, we must insert a spin projection operator (1 § in front of the muon spinor. In the muon rest frame, where s --- (0, s), the operator projects onto a spin direction parallel to the unit vector s. After the insertion, the probability for the decay into polarized muons can be calculated using the standard trace techniques. Summing, for example, over the spin states of the #+, we obtain 43 [M, 2 = T R I 0 _ + m ~ ) ( 1 + ' ~2 5 ~ - ) (a'y5 + ib)(O_ - m . ) ( - a *Z5 - ib* )] = 2{p+ .p_(la, 2 + Ibl2) +m~(la, 2 - , b l 2) + imt~ab*(p + - p _ ) s _

-im,ba*(p+ +p_)s_}.

(9.133)

The spin 4-vector s_ can be expressed in terms of the unit vector s_ in the muon rest frame by means of the Lorentz transformation s=

[p.s (p.s)p ] mu ,S + m~,(Eu + mu ) .

By virtue of p+ - p _ becomes

(9.134)

= (0,2p_) and p+ + p _ = (2E, 0), expression (9.133)

IMI 2 = 2 { p + . p _ ( l a l ~ + Ibl~) + m2(lal 2 - I b l 2) + 4 E I m (ba*)(p_. s_)}.

(9.135)

The corresponding decay rate in the K ~ rest frame reads [76] d W - mki3 (]al 2 +/32]b]2) (1 + Pa n . s ) d ~ , 64~r~ where n = P/IP], n . s = +lnl and T,L(KO --+ # + # _ ) _= 2flIm (ba*) _ mk/32 Im (be*) [a[ 2 q- ~21b[2 4zrF

(9.136)

(9.137)

PL is the degree of longitudinal polarization of the muon: NR --NL PL -- N a + NL'

(9.138)

where NR(NL) is the number of # - s emerging with positive (negative) helicity. The longitudinal polarization violates CP because p. s is invariant under charge conjugation, but changes sign under parity transformation. We now take into account that the KL~ state has a small CP-even admixture: KL~ ~ g ~ + cK ~ The amplitude fox" KL~ --~ # + # - is given by (9.130), with a=a2+eal,

b=b2+ebl.

(9.139)

43 TO avoid depolarization effects associated with the atomic capture of the #-, it is better to measure the longitudinal polarization of the p+.

132

9. The K ~ System in the Standard Model

To first order in CP-violating quantities, Im (ba*) ~ Im [a~(b2 + ebl)] 9

(9.140)

The amplitudes a2 and bx are CP conserving, whereas b2 is CP violating. In the Standard Model, b2 is due to the $d --+ p# transition mediated by the Higgs scalar H ~ This contribution is negligible (PL(Higgs) < 10 -4 for m H > 10 GeV/c 2) in view of the LEP result m H > 58.4GeV/c 2 [21]. Writing =

lel e i r

~

lel e i ~ / 4

= ~Icl, • r + i),

(9.141)

expression (9.137) and the second term in (9.140) thus give PL(K ~

v 91 ] {Re a2(ae bl + Im bl) + Im a2 (Im bl - Re bl) }.

(9.142)

Recall that Im a2 is completely dominated by the real two-photon intermediate state:

lira

a2127

-- 4/3mkarn~in _/jjl---~:,l'1+/3'[64~rF(K~ \ 1 l/2mk

(9.143)

An upper bound on IRe a2l follows from (9.126) and (9.127). Based on a leading-order calculation of the decay K ~ ~ K ~ -+ 7*7* -+ e + e - in chiral perturbation theory, which yields the amplitude bl, G. Ecker and A. Pich obtained [77] P L ( K ~ -+ # + # - ) ~ 2 x 10 -3.

(9.144)

Taking into account theoretical uncertainties in their calculation, they conclude that an observation of PL > 5 x 1 0 - 3 would indicate the existence of a CP-violating mechanism beyond the Standard Model. Under the assumption of #-e universality, the decays KL~ --+ # + p - and KL~ --+ e+e - are induced by the same physical processes. The absorptive (imaginary) part of the amplitude ~4(K ~ --+ e+e - ) is thus expected to be dominated by the real two-photon intermediate state. From (9.124) it follows that l+fle

2

B( K~ -+e+e-)2~ = (m___Z~2 ~3it,[ln (1--:-&~)] B(K ~ -+ #+#-)2-y \ m s / /3, [ln (l+--~a~~] 2' L

(9.145)

\l-ft. ]J

i.e.,

B(K ~ ~ e+e-)2~ ~ 3.1 x 10 -12.

(9.146)

The short-distance contribution to the real (dispersive) part of A(KL~ --+ e+e - ) can be expressed as (see (9.119) and (9.120))

9.3 Rare Kaon Decays

B(K ~ --+ #+#-)SD ~ B ( K + ~ #+u u) ~ \m~/--

"

133

(9.147)

We see that ( K ~ -+ e+e-)sD is even more suppressed with respect to (KL~ --+ mU+#-)SD than ( K ~ -+ e+e-)2~ is with respect to ( K ~ ~ # + # - ) z ~ . The decay K ~ --+ e+e - is, therefore, unlikely to contribute much to our understanding of the short-distance physics. The first observation of the KL~ --+ e+e - decay mode has recently been made by the E871 collaboration [74]. They observed four K ~ --~ e+e - candidate events, with a predicted background of 0.17 =t=0.10 K ~ --+ e + e - ? and K ~ ~ e+e-e+e - events. Their observation translates into a branching fraction of (8.7_+5:17) • 10 -12.

Supplement:

K ~ -+ 7r-s

and

K + - + 7r~163

The amplitude for the semileptonic decay K --+ 7cgue, where g is #+ or e +, has the form

GF

I JR I K>

-

($9.1)

(c.f. (9.85)). Measurements of the ~~ angular distribution in the decay K + ~ 7r~ reveal that the scalar and tensor contributions to Jh~ are very small [80] (see Fig. 9.11). Taking also into account that 7r and K have the same parity, we infer that Jh~ contains only the vector current. The hadronic amplitude must be formed from the available 4-vectors. It is convenient to write

(~ I

aft I K )

: f+(q2)kZ + f_(q2)qZ,

($9.2)

where

k - k K + k~,

q2 = (k K _ k~)2 = (p€ + pe)2.

($9.3)

By virtue of the Dirac equation, 5(p~)(~(1 - ?5)v(pe) = me~(p~)(1 + ?5)v(pe). The amplitude ($9.1) can thus be written as

GF

.54 : ~

{f+kZ~(p,),'/Z(1-,.fh)v(pe) + f_mefi(p~)(1 +?5)v(pe) }.

(S9.4)

Summing over the spin states of the leptons, we obtain _~_

Tr

2^

^ ^

2

2^

^

+ f + f - m e p ~ k ( p e + me) + f _ f + m e p , ( p e + me)k

= 4ag{f

[2(kp.)(kpe) - k2(p~,pe)]

+ f~-2m2e(P~,Pe) + 2 R e ( f + f _ ) m ~ ( k p ~ ) } .

($9.5)

134

9. The K ~ System in the Standard Model

200 Z

100

-1

-0.5

0 r

0.5

l

Ot

Fig. 9.11. Cosine of the angle between 7r~ and L~ in the dilepton CM system f<JrK,a decay. Predictions for vector (V), scalar (S) and tensor (T) couplings are plotted against data [80]

To derive ($9.5) we used

which reflects the fact that there are ~a[y three tadepeedeat 4 - m a m e n ~ (s is the aamp[ete~y ~n~isyn]raet~iv L~vi-Civita ~e~so;: ~ = +I, In the ka~n rest frame, m k = E~ + E F + E V and p~ 4- p . -~ k , = 0 t ~ e d ~n this~ the 4-vector products in ($9.5) can be expressed in terms ~f the center-~f-mass (CM) energies of the lepton and pion as Iellows k 2 = m 2 + m,~2 + 2mkErr ,

p~.p~

=

{m~ + . ~ - 2

m~- 2m~E~}/2,

($9.7)

+ m~ - 2 . ~ ( 2 E ~ + E ~ ) } / 9 , k.p~ = {2m~(2E~ + E~) - m~ - m~ In the CM system, the decay rate is given by d r = I~1--~ d~3, 2m k where d ~ 3 is the differential element of the three-particle phase space

1 f Ca--(27r) 5 J

dak~ d3pe d3p, (~(3)(krr -[- Pt + Pv) 2E~ 2Ee 2 E ,

(S9.8)

9.3 Rare Kaon Decays x

_

135

5(ink - E . - E~ - ~:.)

1 f k~ dlk.l d n . m ~dlpl d ~ 8(27r) 5 J E~ Ee x

5(m k - E~ - Ee - E . ) Eu

(S9.9)

Since pd]p I = E d E and d~2 = d(cos0)dr this becomes

27:

[ d~2~dE~dEe Ik~]lP~ d(cos0~)

~3 -- 8(27F)5 j

E,,

x 5(m k - E~ - Ee - E~).

(S9.10)

To simplify ($9.10), note that

E~ = p2 -_ k~ + p~ + 21k=flps cos0=~,

($9.11)

which yields

E, dE, = Ik=llp~d(cos0~),

(S9.12)

after differentiation with respect to 0.~, while keeping Ik.I and IP~ fixed. Substituting ($9.12) in ($9.10) and integrating out the (f-function results in d~ 3 = ~

1

dE~dE,~.

($9.13)

When ($9.5) and ($9.7) are substituted in the expression for the decay rate, it reads

G2mk ~ 2

d2F

dEedE~ --

8~3 l f + [ P 2 - ( m

k_E~_2E~) 2

m2

+ ~@2(SmkE~ + 6mkE~ + m 2 -~,,%

+#_ m~

_

3m2)]

[m2k + m ~ - m ~ - 2rnkE~]

($9.14) If the charged lepton is a positron, we can neglect the terms proportional to rn~, in which case d2F-

G2Fmk 8~r3 f~ {p~ -- (mk -- E~ - 2Er 2 } dE~dE~.

($9.15)

To determine the kinematic limits of phase-space integration, consider p~ = _ ( p . + kTr) ~

Hence,

p2e = k ,2 + p . 2 + 2lk~l]p~ cos0,,.

(S9.16)

136

9. The K ~ System in the Standard Model ax Pe m min

2

=

([p~[-4-[k.[) 2

(Ev • [kTr[)2 ($9.17)

= (ink - E . - E~ + Ik.I) 2,

i.e. E~

-m~

= (ink - E= • Ik.I) ~ + E~ - 2(ink - E . • Ik.I)E~.

m a x or m i n

Therefore, (E~) mien x : (mk - E , • Ik.I) 2 + m~

($9.18)

2 ( m k - E= • Ik=l)

and we find that the positron energy at a fixed E , varies within the range m k - E , - Ik,I < 2E~ ~ mk - E . + Ik.I.

($9.19)

Analogously, the pion energy at a given E~ is constrained by max

(E.)mi

n __

(mk - E~ • [p~) 2 + m .

2

(S9.20)

2(m~ - E~ + IP~) If the positron mass is neglected, 2

(ink - 2E~) 2 + m 2 < E , < m~ + m , 2(ink - 2Ee) 2m k

(S9.21)

Expressions ($9.18) and ($9.20) define the contour of a Dalitz plot. Assuming that f+ (q2) ~ constant, the distribution ($9.15) can be readily integrated over d e e from E m i n t o E m a x , with the result dF-

G~mk f~ (E~ 127r3

-m2)3/2 d E , .

($9.22)

A measurement of the pion energy spectrum gives the q2 dependence of f+. Using ($9.21) and rewriting expression ($9.15) in a slightly different form, we have dF dEC-

G2Fmk 8~ 3 f2 • /E ~m~ { ( ~

- 2E~)[2E.

- (.~

- 2E~)] - ~n~} d E . .

(S9.23)

A simple integration yields the positron energy spectrum:

dF G2mk E2(W~ - E~)2 dE--~ - -------527~ f2 m k - 2E~ '

($9.24)

where We is the maximum energy of the positron in K~3 decay (see ($9.29) and ($9.30)). The neutrino spectrum in the rest-frame of the kaon can be constructed from the measured momentum distributions of the pion and electron in the laboratory frame:

9.3 Rare Kaon Decays 2 =

-

pe-k~)Lab

2(E~E~ -

137

-

(S9.25)

2m k

This expression follows from

(kK -

2 = (k~ + P~)aab 2 9 P-)CM

($9.26)

To find the m a x i m u m energy of a particle in a t h r e e - b o d y decay M -+ m I + m 2 + m 3 , let M12 be the invariant mass of particles 1 and 2. In the rest s y s t e m of M , M122 = (El + E2) 2 - (Pl -4- p2) 2 = ( M - E3) 2 - (p3) 2.

(S9.27)

Since P3 = E32 - m32, we have M22 = M 2 - 2 M E 3 + m 2

> E3 =

M r + m2 - M~2

($9.28)

2M

E3 has the m a x i m u m value for M A in = rn 1 + m : . Therefore, W3 = g ~ nax --

M 2 + m~ - ( m 1 + m2) 2 2M

($9.29)

In our case one of the particles is a massless neutrino. We thus obtain the following kinematic limits 2

2

m,~ < E . < m~ + m,~ - m~ _

_

2m k

me < Ee < m~ + m~ - r n , '

_

_

($9.30)

2m k

T h e Ke3 decay rate is therefore given by F-

G2mk

127r3

2 fmk(1Ta)/2 3 / f~_ [ E,~tl-

.,m,~

,

m2

2

dE,~ ,

~2

2 2k. Setting 1 - m ~2 / E ~ 2 = where a =- mTr/rn obtain

F-

,3/2 X2

($9.31)

and integrating by parts, we

5

GFmk

7687r 3 f 2 {(1 -- a)3(1 + a) -- 6a(1 -- a)(1 + a) -- 12a 2 l n a } .($9.32)

If only the leading orders in a are kept, this simplifies to F-

G F2 m ks

(

7681r a f 2

1-

Sm~'~ m2 j .

($9.33)

Of special interest is the m u o n p o l a r i z a t i o n in the decay K + -+ 7r~ S u m m i n g over the spin states of the neutrino (see the text preceding expressions (9.133) and (9.134) in Sect. 9.3), we obtain

IMI 2 = - ~ Tr[(f+k~

+ f _ q ~ ) ( f ~ _ k ~' + if_q)')

Sv

x/5.3,Z(1 - "/5) (15u - m , ) ( 1 + 75g,)'),x(1 - 75)] = G.__}_~T r [ ( f + k Z + f _ q Z ) ( f ~ _ k ~ + f_* qa)(1 - ~'s) 2 x ~,'),~(~, + rnu~u)7)~].

($9.34)

138

9. The K ~ System in the Standard Model

To derive ($9.34) we used pt*. st* = 0 (see (9.134)). A straightforward evaluation of the traces yields

E JMI = Sv

1

IMI 2 + 2c , t* {

- k2(v -st*)]

+ f2_ [2(q.p~)(q.st*) - q2(p~-su) ] +2Re ( f + f * ) [(k.p~)(q.st*) + ( q . p ~ ) ( k . s u ) -

(k.q)(p,.st*)]

or } +2Im ( f + f *_) e , X ~ kZa)~_a ._ = V,%

($9.35)

with ]~]2 given by ($9.5). In the kaon rest frame,

q.st* = p~.st* = E , Pu'St* (PdSt*)(Pu'P~) rot* P~'Su - mt*(Et* + mt*) ' ($9.36)

k.st*

(2k g

--

pt* - p , ) . s u

=

2m k PdSt* - -

_

p

.st,

mt*

(st* is the spin unit vector of the muon in its rest frame) and el3Xar

kt3a,~na.qr "~ "* r v - l *

=

al~Aar(2kK

--

a r pt* - p.)~(pt* + p~) .X p.st*

= 2 gl3.~ar kJ3nAna'qr i kst* = 2m k (pt*• p,)" s u. = 2mkeioyk pill2,

($9.37)

Each term in ($9.35), with the exception of the last one, contains components of the muon polarization in the decay plane. The last term gives rise to a polarization component normal to that plane. Since this term changes sign under time reversal, its presence would violate T invariance. Indeed, the time-reversed amplitude of ($9.1) and ($9.2) GF

.h/[ T) - ~ [f~_(q2)k ~ + f._(q2)qz] ~(V.)~,Z(1 _ %)v(Pe)

(S9.38)

also leads to ($9.35), except that in this case the last term is ~ Im ( f ~ f _ ) = - I m ( f + f _ * ) . If the decay is caused by an interaction that is T invariant, Im (f+f*_) = 0 (f+ and f_ have the same phase). To obtain ($9.38) we used Table E.2 in Appendix E and the fact that the operation of time reversal implies charge conjugation. We now define ( ~ = ~m-t * [ 1 and rewrite

(S9.35)

f~+] = ~mt* - [1-~(q2)],

9~=2Rec~, 3 ~ - 2 I m ~ ,

(S9.39)

as

= 8 G F f + [A + B . s ~ ] ,

($9.40)

Sv

where A = [2(kK.p,) - fftmt*] (kK.P~) -- (m2k -- a2)(pdp~),

($9.41)

9.4 Direct CP Violation (e')

139

B = [2m,(kK.pu ) - fJt(p,.pu)]k ~ - [rn,(rn~ + a 2 ) _ N(kK.pt,)]p~ ar + ~:Je ~ , k ~ q, x pus,.

(S9.42)

It can be readily verified, using ($9.36) and ($9.37), that in the kaon rest frame

B'%:{a(()k~+[b(~)+a(()(?

~-:p-~ + r n k - - E ~ ) ] P , \ L', + rn,

+ 3 rnk(k,~x p~) }. s,.

($9.43)

In the above expression,

mtt

l

($9.44)

b(() = m : [ 2 E , -

~R ( W . - E~)]. mtt (W,~ is the maximum energy of the pion; see ($9.29) and ($9.30)). From ($9.43) it follows immediately that the muon polarization vector is given by [81] ~ ' ( ~ ) - I~t(~)l'

($9.45)

[

.,4 = a(~)k,~ + b(~) + a(~) \ E, + mr, mk(k~ x p,). Since the pion and K meson are spinless and the neutrino has a definite helicity, the muon is completely polarized. The direction of its polarization is fixed by the kinematics of the decay. The nonvanishing mass of the muon gives rise to a polarization component along the pion momentum. As we explained earlier, if the decay is T invariant, ( is real and the polarization vector lies entirely in the decay plane.

9.4 Direct C P Violation (e') The parameter e', defined by (3.75), determines the amount of direct CP violation in AS : 1 transitions: e'-- x/~ A 0ReA2 Re 1 [IIA2ReA2

Re~00ImA~

ei(Tr/2+52_5o )

(9.148)

Any CP-violating observable must involve an interference of two amplitudes. In the case of e/, the interfering amplitudes are

140

9. The K ~ System in the Standard Model

(rTr, I = al/:/w]K ~ = A~e i~~

(~ = 0, 2,

(9.149)

where [Trlr,I = 0, 2) are the I = 0 and I = 2 isospin states of the 27r system. Remarkably, the phase factor in (9.148) is very close to 7r//4, i.e., the phase of c ~ is almost the same as that of ~. Using (3.74), (3.81) and (3.82), we can express the parameter a, which measures the interference between K ~ --+ ~Tr and K ~ -+/7/0 _+ 7rTr, as follows: e ~ ~22 eir

[ ImM12 ImAo] [2ReM12 + ReAoJ "

(9.150)

The expressions for e ~ and c are independent of phase convention. To show this we redefine the phases of the strangeness eigenstates K ~ and K~ ]K ~ -+ e-i;~lK~ Ig~ ~ e~lk~ For Iml << 1, Im
~ ~ Im

[e2i~'
(K~ I/:ZwIZ~~ + 2ARe (K~176

(9.151)

i.e., Im M12 Re M12

Im M12 - + 2A. Re M12

(9.152)

Similarly, Im A~ Im Am § Re Am Re Am

A,

(9.153)

which proves the above assertion. In the absence of A I = 3//2 transitions, r c< Im A0//Re A0. Since this expression is not phase-convention invariant, c ' must vanish for A2 = 0. Therefore, to determine ~', one has to calculate the phases of both A0 and A2. Using the experimental ratio of A I = 3//2 and A I = 1/2 amplitudes

/F(K + --+7r+Tr~

ReA2

1

(9.154)

R-- o V and the measured values (7r/2 + 52 - 50) ~ ~r/4, r ~'

w

limA2

LReA2

ImAo]

-w

~ ~/4, we obtain

ImAo(1 - 1-2),

(9.155)

where f2 = w-lImA2//ImAo. For reasons unrelated to CP violation, the phase Im A2//ReA~ is enhanced due to the smallness of Re A2. This means that any interaction that gives rise to A2 is 22 times more effective in producing E~ than the one which contributes only to Ao. From r s = (I = 01HwIK~) + (I -- 21//wl K~ 2 v/1 _ 4m~/m~ 167rrn k it follows (see (9.149)) that

(9.156)

9.4 Direct CP Violation (e')

141

W

q

q (a)

W

U,C,t

I ~/,Z

~/,Z q

q

q

q

(b)

s

it.--

W ........

U,r [

,,-q

W (c)

Fig. 9.12. Gluonic (a) and electroweak (b,c) "penguins" giving rise to e'

ReA0 .-~ 3.34 • 10-7 GeV.

(9.157)

As first noted by F. Gilman and M. Wise [78], a nonzero value of e' could arise from the diagram in Fig. 9.12a, whimsically named "gluonic penguin". The diagram, which is the main source of CP violation in AS = 1 transitions, describes the A I = 1/2 decay s -+ d + g, where g is a gluon. Since this graph turns an s quark into a d quark, it can only change isospin by a unit of 1/2, and thus cannot lead to an I = 2 final state. When g is replaced by a photon (or a Z~ the resulting "electroweak penguin" diagram (Fig. 9.12b) generates a phase in As. To see this, note that the electromagnetic current _ a1+75.-. (17aQqq = q'7 - - - ~ q q

, _ ,~1-~/5~, -v q7 ~ - ~ 4 q q .

(9.158)

(Qq is the charge of quark q) is a mixture of I = 1 and I = 0 components: (1Qqq =

2~u - d d - ~s ~u - d d (tu + d d - 2~s 3 2 + 6

(9.159)

142

9. The K ~ System in the Standard Model

The I = 1 component of the electromagnetic current and the s -+ d transition current, which changes isospin by a unit of 1/2, can induce a AI = 3/2 transition and thus create an I = 2 final state (recall that Ik = 1/2). At lowest order, the diagrams in Fig. 9.12a,b yield the following effective hamiltonians [78, 79]:

GF as {

rn2 m2} v ~ 127r V~'*aV~sln~-~ + Vt~Vts ln'~'--~rnr d%(1 - 3'5)A%

/2/w(g) ~ x

E

q3'"~q

(9.160)

q=u,d,s

and 2

Hw( )

G m 2a

x

m2

* rnc + VtdVts * in mc~ .)d % ( 1 - ~ 5 ) s V],dVus ln~-ff

E vT~/"Qqq"

(9.161)

q=u,d,s

Here we show only the logarithmic mc,t dependences. In (9.160) and (9.161), /s is a typical hadronic mass scale, /~a are color SU(3) matrices and as is the strong coupling constant (as ~ 10-1). Since V~*dVus is purely real, the contribution to c ~ comes only from Im (Vt~Vts) = - A2A57/: ImAo ~ -~2A2~5~/@Tr, I

= 01Cp0

(9.162)

p IK ~

with Ols

fftt2

Cp = 127r lnrn-~'

0p - d%(1 - ~/5)Aas E

q')'#)~aq

(9.163)

q=u,d,s

and similarly for A2. The photonic penguin is suppressed with respect to its gluonic counterpart by the factor of a/as. However, the phase Im A2/Re A2 is enhanced due to the smallness of Re A2, as mentioned above. A full gauge-invariant calculation includes Z~ and box diagrams (Fig. 9.12c). The detailed calculations are very complicated. In essence, one evaluates the relevant Feynman diagrams to obtain an effective hamiltonian for K ~ ~ Ir~r (as was discussed for K~ ~ mixing), extracts its operator structure and then tries to estimate the hadronic matrix elements of these operators. The amplitudes Im A0,2 are calculated from the effective hamiltonian for AS = 1 transitions 8

f.IeAS---1 __

ff

GF , v/~ Im (VtaVts) E Ci(#)Oi

(9.164)

i=1

derived using renormalization group (RG) equations. The coefficient functions Ci(#), which are governed by the short-distance structure of QCD (# < E _< rn~), have been calculated at next-to-leading order in perturbation theory.

9.4 Direct CP Violation (e')

143

The main difficulty lies in the evaluation of the hadronic matrix elements 0,2 -- < ~ , I = 0, 2 I 0~ I K~

(9.165)

which describe the long-distance physics (0 _< E _< /t), and therefore cannot be treated perturbatively. The dominant contribution comes from two operators, 66 (gluonic penguin) and Os (electroweak penguin), which are responsible for the A I = 1/2 and A I = 3/2 transitions, respectively. Expression (9.155) thus simplifies to c'e _ 21~lRe AoWG F Im(Vt*dVts)C6(O6} 0 [ 1 - - W

-1]C8~(Oj8.}2

(9.166)

We should also mention the small breaking of isospin invariance that gives rise to 7r-r/and 7r-q' mixing. As a consequence, K ~ --+ 7r~ ~ will receive contributions from diagrams in Fig. 9.13. The amplitudes A ( K ~ -+ 7r~ and A ( K ~ --+ 7r~ ') get imaginary parts from the gluonic penguin in Fig. 9.12a. This induces, via the processes in Fig. 9.13, an imaginary part in A ( K ~ -+ 7r~ ~ and thus in both the I = 0 and I = 2 components of the amplitude A ( K ~ --~ 2~r) (see (3.55) and (3.76)).

~o

Fig.

9.13.

Isospin-breaking contribution to K ~ --+ 7c~ ~

The electroweak-penguin (ewp) and isospin-breaking (IB) effects modify the purely gluonic contribution by (see (9.155)) ~- ~ewp -{- ~vl+~',

~,+rl, = W_ 1 ( I m A 2 ) I B

(9.167)

Im Ao The Standard Model prediction for ~//c suffers from large hadronic uncertainties, aggravated by substantial cancellations between the I = 0 and I = 2 contributions (the 66 contribution to s ' / c is positive and only weakly mr-dependent, whereas that of 0 s is negative and shows a strong m t dependence). This renders a precise calculation of this quantity impossible at present. A recent comprehensive review of the subject that provides access to the theoretical literature can be found in [54]. Here we quote their own result, -

2.1 x 10 -4 _< cl/~ < 13.2 x 10 -4,

(9.168)

but stress that some other authors obtain a much wider range of values for

CI/s

Appendices

A CP

Properties

of K

--+ 27r a n d

K

- + 3~-

The pion has spin zero and negative intrinsic parity (it is a pseudoscalar): J R = 0 - . Conservation of angular m o m e n t u m therefore requires the spin of the kaon to be equal to the relative orbital angular momentum, f, of the two pions in the decay K --4 27r: Jk = J2~ = g. The parity of the 2~r state is P2~ = ( P ~ ) 2 ( - 1 ) t = ( - 1 ) 2 ( - 1 ) t = ( - 1 ) ~.

(A.1)

To find the allowed values of g, note that the identity of the final state pions in the K ~ --+ 2~r~ decay implies that their wavefunction is symmetric, which means t h a t g must be even: g = 0, 2, 4, . . . . Hence, Jff~ = 0 +, 2 + , 4+, . . . .

(A.2)

The 3~r system in the charged kaon decay can be regarded as consisting of two parts: a 27r state containing the pions of like charge (~r+Tr+ or 7r-Tr-), plus the remaining pion. If we denote the relative orbital angular m o m e n t u m in the 27r state by g, and that of the third pion relative to 27r state by L, then the spin of the 3~r system is constrained by It - g[ < J37r _~ ]t 4- L I.

(A.3)

The parity of the final state is P3~ = (P~)3(--1)t(--1)L = (--1) TM,

(A.4)

because g must be even for two indentical bosons. Since the sum of the masses of the three pions is close to the kaon mass, the decays into 37r states with g > 0 are supressed by kinematics. We can thus write JP. = L (-1)L+~ -- 0 - , 1+, 2 - , . . . .

(A.5)

From (A.2) and (A.5) we infer that J3P,~ = 0 - , 2 - , 4 - , . . . .

(A.6)

Measurements of the charged 37r decay modes favor Jk = 0. This result is corroborated by the absence of the decay K • --+ 7r+ 4- 7, which is forbidden if the K + spin is zero. Furthermore, the muon polarization in the decay

146

Appendices

K + --+ #+ + v~ is the same as in rr + --+ #• + vt,, thus indicating t h a t Jk = J~ = 0. Therefore, g P ~ = 0 +, JP~ = 0 - .

(A.7)

T h e kaon parity can be determined from the reaction K - + 4He--+ 4H A 4- rr0 ( K - capture from an a t o m i c S state), where the hypernucleus 4H A contains a b o u n d A h y p e r o n in place of a neutron. Measurements of 4H 4 weak decay modes suggest t h a t J = t? = 0 on b o t h sides of the reaction. Hence j R __ 0 - , if the parity of the A-particle is defined to be positive, like t h a t of the nucleon. Application of the charge conjugation o p e r a t o r 0 interchanges the rr + and 7r- in the decay K ~ --+ rr+r~ - . In this case the operation is equivalent to space inversion, 0g%+,~- = Pg%+~- = (-1)e~P~+.-,

(A.8)

and thus 0/SkV=+rr - = (-1)2g~P~+~ - = + ~P~+~r-"

(A.9)

For the 7r~ ~ final state 0/5g%o~o =/3~V~o,o = (-1)~,P.o.o = +~P.o,~o

(A.10)

because two identical bosons must be in an overall s y m m e t r i c state. If we also take into account t h a t the kaon is spinless, conservation of angular m o m e n t u m requires ~ = 0 in the 27r state. Hence, 0~2~ = / 3 ~ 2 , = + ~P2,,

(i.ll)

Regarding the decay K ~ --+ 3 r ~ we have shown t h a t for a system of three pions, at least two of which are identical, P = - 1 : 0/3g'3~o = - g'3~o.

(A.12)

As for the charged 3rr decay mode, for K ~ --+ rt+rr-rr ~ we take the relative orbital angular m o m e n t u m in the r~+r~- state to be s and of the rr ~ relative to the lr+lr - centre of mass to be L. Now,

OPO~+~-.o = 015 {~+~- (s = (--1)L+I~P~+~_~o.

= + ~+~- P ~ o ( n ) (A.13)

We see that, in contrast to the 7r+Tr- , 7r~ ~ and 39r~ final states, the r~+rr-rr ~ system does not have a well-defined C P eigenvalue: C P = ( - 1 ) L+I. T h e kaon is spinless, and so the total angular m o m e n t u m in the 37r-system must be zero, which means t h a t L must be m a t c h e d by g = L. If L = 0 or an even integer, then C P = - 1 ; if L is odd, C P = +1. As explained above, states with ~ > 0 are supressed by kinematics. Therefore, the decay K ~ -+ rr+rr-rc ~ is expected to be d o m i n a t e d by the CP-allowed decay K ~ --+ rr+rc-rr ~ with L = t? = 0 and C P = - 1 . T h e K ~ m a y decay into the kinematics-supressed

B Forward Scattering Amplitude and the Optical Theorem

147

and CP-allowed final state with L = g = 1 and C P = +1, or into the kinematics-favored and CP-forbidden state with L = g = 0 and C P -- - 1 . The charge conjugation operator C changes the sign of all charges, including the charge of the sources and thus of the electromagnetic fields they produce: Au(x)

~ - A t ( x ),

under C.

(i.14)

If we consider an electromagnetic field to be a collection of photons, then the electromagnetic potential A u ( x ) can be taken to represent the photon wavefunction. From (A.14) we infer that the photon has odd charge-conjugation parity C)l')') = -17},

(A.15)

i.e., 1"),}is an eigenstate of 0 with eigenvalue C = -1. Since C is multiplicative, 0In'),) = Ch,1)l-Y2)... [')'n} ~---(--1)nln3'}.

(A.16)

The rr~ decays predominantly to two photons. Hence, C)l~r~ = + Ire~

0/51rr ~ = -Irl~

(A.17)

B Forward Scattering Amplitude and the Optical Theorem Suppose in the z (see Fig. outward

an incident beam of particles, represented by a plane wave traveling direction, ~i = eikZ, impinges normally on a thin slab of material B.1). The scattered beam at large r is a spherical wave propagating from each of the scatterers: eikr ~Psc = ~(0) , (n.1) r

dx a X

z

\

p

Fig. B.1. A plane wave impinges on a thin slab of material

148

Appendices

where ~(0) is the scattering amplitude. Note that in (B.1) there is no dependence on azimuthal angle r because the z component of the angular mom e n t u m is zero. We have also ignored the factor e -iEt, since for coherent scattering the slab as a whole takes up the recoil, and therefore very little energy is exchanged in the process: E1 - E2 -

P~a~

2Mslab

<< P s l a b - - P1 - P 2 .

The scattered wave must have a 1/r dependence to ensure t h a t the rate of scattered particles t h a t pass through a spherical surface centred around r = 0 is independent of r: ~ . ~ I~'scl2Vrel r 2 d . Q =

Vrel]~(O)12d~'~

(B.2)

7~ is the number of scattered particles per unit time through the surface r 2 d ~ and Vre1 the relative velocity of outgoing particles and the target. According to the definition of a cross-section, (B.3)

Tr ~ flux. dr = Vre I d a , for a scattering centre per unit area and the incident flux of particles

(B.4)

flUX -----Vi~'il~"; = Vi ~ Vre 1.

Therefore, the differential cross-section for elastic scattering reads da

(B.5)

h-5 = lI(~

For a slab of material of thickness l, containing N scatterers per unit volume, the amplitude fl,sc(Z) of the scattered wave at a distance z is given

by Asc(Z) =

~0~

[27rxdxNl] f(w,0) ~eikr --

( B .6)

The expression in the square bracket represents the number of scatterers in a ring at radius x. Since r = v / ~ + z 2, we can write

Asc(z) = 27rN1

dr f(w, 0)e ikr.

(B.7)

Defining u = f(w, 0), dv = eikrdr and integrating by parts, results in ik But e ikr

f(w,0) eikrlz --

eikrd[(w,0)

}

.

(B.8)

r --~oo

) 0, and so the first term in (B.8) gives

i27rNl k ~(o),O)eikz. For the scattering in the forward direction (0 --+ 0) the second t e r m does not contribute. The scattering amplitude at 0 = 0 is, therefore, determined by the first term:

B Forward Scattering Amplitude and the Optical Theorem i2~rN l ~:(02, 0)e ikz. k The total amplitude at large z is then ~sc(Z) --

~ ( Z ) = ~Ain(Z ) @ .Asc(Z ) =

e ikz Jr-

149

(B.9)

i2~N1 - ~(cO,0)e ikz,

(B.IO)

i.e. A(z) =

i27vNl ] 1+ ~ f(w,0) e ikz.

(B.11)

Let us assume that the slab is a homogeneous material of refractive index n(w). The wave number in the material is nk, whereas in vacuum it is k. The above amplitude can thus be expressed as (B.12)

.A( z ) = e ik( z-l) +inkl = eikZ e ik(n-1)l,

which shows that the particle picks up an extra phase Cext ~ k(n - 1)l. For Cext << 1, a Taylor expansion of the second exponential in (B.12) gives .A(z) ~ e ikz [1 + ik(n - 1)l].

(B.13)

Comparing (B.11) and (B.13) we infer that 2 7rN n(w)=l+-~[(w,O)=l+

2 7rN c 2 w~ [(w,0),

(8.14)

where w = kc. Expression (B.14) relates the forward scattering amplitude [(w, 0) with the index of refraction n(w). We next derive the optical theorem O'tot

=

- ~ I m f(w, 0),

(8.15)

which states that if there is any scattering at all, there must be some scattering in the forward direction. Consider the attenuation (through scattering, nuclear reactions, etc.) of a beam of particles as it passes through a slab of material. An absorbing medium can be regarded as having a complex index of refraction, the imaginary part of which is associated with attenuation. We thus write n=nsc+nabs=

l+~5-~sc(W,0 ) +a~-~abs(w,0).

(B.16)

Similar to (B.12), the amplitude in this case reads A ( z) = Aoe inkz = .Aoein~r

(B.17)

e -n~b~kz.

The normalized beam intensity is I

l0

e_2nabsk z

e -47rNf~b~(w'O)/k ~

e -zN~176

(B.18)

150

Appendices

so that 4rr 4~r atot = ~ - labs(W, 0) = %- Im f(w, 0),

(B.19)

which is the optical theorem (B.15). Therefore, the imaginary part of n describes the attenuation of the beam as governed by the total cross-section - not by the absorption cross-section. The real part of n gives the phase shift of the wave relative to its propagation in free space.

C Watson's Theorem and the Decay Amplitudes K ~ R ~ --+ 27r The unitarity condition

SS* = 1

(C.1)

is imposed on the scattering matrix S to ensure probability conservation. In matrix form this condition reads

(ss*)s

=

: 6s,,

(c.2)

n

where ~-~.f5Si = 1 and n runs over all possible intermediate states. The diagonal elements satisfy E

]S'~i]2 = 1,

(C.3)

n

which expresses the fact that every intermediate state must disintigrate into one of the available final states, i.e., that the sum of the transition probabilities from a given initial state to all final states is unity. The transition matrix Tli is defined by

SSi = a f i - i(2rr)4($ (4) (pf - pi)Tfi.

(C.4)

Substituting (C.4) in (C.2) yields

i(Tfi - T;~) = (27r)4 E (~(4)(Pi pn)TfnTi*n. -

-

(C.5)

n

If we take f -- i (forward elastic scattering), this expression leads to the optical theorem (B.15). The unitarity condition (C.1) should hold true whether or not weak interactions are included. Let aS denote the small change in S due to the weak interaction: S = So + (~S, with

5S = -i(2rr)a5 (4) (pf - pi)T.

(C.6)

C Watson's Theorem and the Decay Amplitudes K ~ /~0 _+ 27r

151

Using (C.1) we have, to first-order,

s t s = (S~o + ~ s t ) ( s o + 5s) = StoSo +~StSo + s*o~s + ~ s t ~ s = 1, 1

neglect

i.e.

~st so + Sto~S = 0.

(c.7)

Expressions (C.6) and (C.7) give

TtSo - StoT = 0,

(C.8)

or in terms of transition amplitudes

(f l Tt So l i) = (f IS toT I i). Since

(C.9)

(f [ 0 Wi) = (i[0t If)*, it follows that (i IS loT If) * = (f l StoT ] i).

(C.10)

If we ignore the effect of strong and electromagnetic interactions, we can set So = 1 in (C10). From the resulting relation we infer that T is hermitian:

(i l T [ f)* = (f [T I i)

) Z = T t.

Now, recall that CPT invariance implies

(i l T l f) = (f' l T l i'),

CPT.

(C.11)

(see (3.10)) (C.12)

Combining this result with (C.11), which is based on the unitarity condition (C.1), we obtain

(f IT I i) = (]' IT I[')*,

CPT,

(C.13)

in agreement with (3.57). In vacuum, the initial state IK ~ would be stable in the absence of weak interactions:

Soil) = li),

(ilSto = (i I.

(C.14)

Since kaons are not stable, we must take into account strong interaction effects among their decay products, as discussed in Sect. 3.4. In the decay K ~ --+ 27r, for example, the two final state pions undergo elastic scattering. Consequently, the phase of their wave function changes by an angle (called the phase shift), which is conventionally denoted by 25I, where 6t=0 and 5i=2 are S-wave phase shifts for [0, 0) and 12, 0), respectively. We thus write Sol f ) = ei2~'lf),

(fl st = e-i2~'(fl,

(C.15)

which ensures that So is diagonal. Based on (C.14) and (C.15), equation (C.10) becomes (i I r l f)* = e-i2a~(f IT I i).

(C.16)

152

Appendices

Expression (C.12) now reads

(C.17)

which is Watson's theorem (3.58) (see [14c]). Note that, due to the strong interaction phase shift in the two-pion wave function, the particle and antiparticle decay amplitudes are not complex conjugates of each other. Using (3.63) and (3.55), the transition amplitudes A(K ~ -+ 270 and A(/~ ~ -+ 21r) can be written as A ( K ~ 1 6 3~ --+ ~-oTro) : ~/f~lA2le+ir v t)

i~2 -

A(K~176 --+ 7r+Tr-) = V ~ IA21e•162

/
•

e

~i6o ~ ,

(c.18) + V/-2'A ~ l o ~^•162eih~,

where the upper (lower) signs refer to K ~ (/~o) and A~ = lAdle ir with a = 0 or 2. The expressions for the corresponding transition rates, which include the interference between the I = 0 and I = 2 modes, are

F(K~176

:r~176 = ~,Ao,2 + ~,A212 3 IAoIIA21cos [+(r

- r

+ (60 - 52)],

(c.19)

F(K~

2 12 + ~lA212 1 ~ -+ 7r+Tr-) = ~lAo + - ~ l A o l l A 2 [ cos [+(r

- Cz) + (60 - 52)].

From (C.19) we see that: (a) F(K ~ -+ 27r;I = 0 or 2) = r ( / ~ ~ -+ 27v;I = 0 or 2) ~ Iaol 2 or IA212; (b) F(K ~ -+ :r~ ~ or 7r+~r-) = F ( / ( ~ --+ 7r%r~ or 7r+~ -) only if 6o = 62; (c) F ( g ~ K ~ --+ 7r%r~ + F ( g ~ K ~ -+ 7r+Tr-) = IAo[2 + IA212. The difference in F(K ~ --+ 7r% ~ or 7r+Tr- ) and F ( K ~ --+ 7r%r~ or 7r+Tr- ) is due to r - r and is therefore associated with the parameter e' (see (3.75)), which determines the magnitude of direct CP violation. For reasons unrelated to CP invariance, ReAo ~ 20 x ReA2, as shown in (3.70), (3.71). The intrinsic scale for direct CP violation in weak decays of neutral kaons is thus O(10 -5 - 10 -4) (see (5.7) and (5.11)).

D Time Reversal and CPT Violation D Time

Reversal

and

153

Violation

CPT

To illustrate the effect of time reversal on a quantum-mechanical state, consider the Schr6dinger equation d

i ~(x, t) =/:/~P(x, t).

(D.1)

If the hamiltonian /:/ is invariant under time reversal, the new wavefunction obtained under this transformation must also satisfy (D.1). As can be readily verified by complex conjugation of the above equation, the function kV*(x,-t) is a solution of (D.1) provided /7/ = /:/*. We thus conclude that time reversal has something to do with complex conjugation, and that a quantum-mechanical state is invariant under time reversal if its hamiltonian is real. In the above elementary example the time-reversed state was obtained simply by complex conjugation, which is not the case in general. For wavefunctions in momentum space, time reversal implies 4~(p) --+ 45"(-p), that is, in addition to complex conjugation there is also the transformation ~(p) -+ ~ ( - p ) . We can generalize our discussion by introducing the time-independent operator T, which has the effect of creating a new wavefunction and a new hamiltonian (see, e.g., [45] and [97]): = :rH7

J

= Hr

7'7

(D.2)

= 1.

We require that kVT satisfy the SchrSdinger equation with t --+ - t and hamiltonian [/7", as does if' with hamiltonian H. I f / : / i s invariant under time reversal (/2/= I/T), the operator T commutes with the hamiltonian: T / : / = / / 7 ~. If (D.1) is multiplied by ]b from the left, we obtain 7

(D.3)

=

By comparing (D.3) with the Schr6dinger equation for the time-reversed wavefunction kVT d~T

i d(-t)

_

(D.4)

d~T .~- i~iT~] T

i ~--

we infer that T i T~ - ' = - i .

(D.5)

This is a special case of the property /4(e~P)=c*'P* ~ R = / ~ - I

or

R 2=1,

(D.6)

which defines an antilinear operator. For comparison, the corresponding equation for a linear operator is

154

Appendices

O(cgQ = c()O.

(D.7)

Therefore, unlike other transformation operators in quantum mechanics, 2P is not a linear operator. Instead, 2~ can be expressed as the product of two operators 0 and K, where 0 is a unitary operator ( 0 t 0 = 0 - 1 0 = 1) and K changes a quantity into its complex conjugate. Such an operator is called antilinear-unitary, or antiunitary. To show that (D,8)

OR,

we substitute (D.8) in (D.4) and obtain 9 d0

" _ 0(R/

~d ~

K_l)0_10e. = 0 H , 0 _ 1 0 e .

'

(D.9)

since (0/~) -1 = / ~ - 1 0 - 1 and/:/*~* =/~(/:/kP) = / ~ / : / / ~ - l / ~ p . Taking the complex conjugate of (D.9) and factoring out the time-independent operator 0, yields the unreversed SchrSdinger equation (D.1). According to (D.2) and (D.8), OT = TO = 0~'*,

(D.10)

which we use to express the inner product of two time-reversed states as

(AT I BT} =--/ dV(g~AT)tOBr = f dV(O~P~)?(OqJ~) : f dV(O~4/rOtOg"~= {/dVOtA~B}" =(AIB}*={BIA

I.

(D.11)

Taking the point of view that the operator T is to act on Dirac's kets, we write the time-reversed states

IAT} -- T IA}, IBT} = T IB). Let us further define I C } ~ O ttB)++ {C I = {B I0, where 0 is a linear operator (see (D.7). Now,

{B I O I A } = {VIA ) = {AT ICT}= {AT ITOtIB) = {AT I T0tT-12b I B} = {AT I:FOtT -1 I BT}, i.e.

(BI()IA} = {AT I T 0 t Z -1 I BT}.

(D.12)

The time evolution of state vectors in the Schr6dinger representation can be expressed as =

e

(D.13)

D Time Reversal and

CPT Violation

155

Indeed, if this expression is differentiated with respect to t, we obtain d i~-~lk~(t)> =

[-te-it:I(t-t')[~(t')) = [-IlO(t)) ,

which is SchrSdinger's equation for fl'(t)), provided/2/ is the total hamiltonian. Setting A =- KO, B = K ~ and 0 = e -i/:/t in (D.12) gives 44

(D.14)

where rei/C/tT -1 --

e -i[fTt.

Consider the possibility of (3.24), we define q=~+~,

(D.15)

CPT violation in the K ~ system. Based on

e2-~-~e+q-e2=2~

(D.16)

and use equations (3.6) to write the time evolution of an initially pure K ~ or R ~ state as

IK~

= ~ "

[(1 - e + ~ ) l K ~

- e - ~-)lgL~

, (D.17)

l

IKa(t)) = ~2 [(I + e - ~)IK~)e-~S"~t-(I + e + ~)IK~ Substituting in (D.17) expressions (3.5) for IK~ and IK~):

IK~> ~1 IKL~ = ~

1

[(1 + e + ~)IK~ + (1 - e - ~)1/~o}] ,

(D.lS) [(1 + e-- ~)lK~ - (1 - e § f ) l R ~

we obtain

Ig ~ +

]g~

= [f+(t) + 2~/_(t)]

(1 - 2e)f_(t)l/q~

Ik~

= [f+(t) - 2~f_(t)] IK~ + (1 +

f•

[ e _ i ~ s t 4- e - L M L t ] .

(D.20)

r ( K ~ -~ k ~ = If_(t)12(1 - 4Re e),

(D.21)

(D.19)

2e)f_(t)lK~

with

Hence~

r ( k ~ ~ K ~ = If_(t)12(1 + 4Re e). 44 This result is valid only in the two-dimensional Hilbert space of the K ~ system, with/2/defined by (2.21).

156

Appendices

resulting in the time-reversal asymmetry (see (6.19) and (6.20))

F([(o _+ K o) _ F ( K o __+/~o) .AT(t) -- F([4 o ~ KO ) --~ F(KO ~ / ~ o ) = 4Re e.

(D.22)

Note that the transition rates (D.21), which were obtained without any reference to the method of detecting the two K ~ states, do not depend on the C P T violating, T invariant parameter ~ (see [44]). In contrast,

r ( K ~ -+ K ~ ~ I f + ( t ) l 2 + 2 [f+f_* C + f _ f ~ ] ,

(D.23)

F ( K ~ -+ R ~ ~ I/+(t)l 2 - 2 [f+f_* C + f_f~_~], which yields Ac.r(t)

-

F([(o _+ [fo) _ F ( K o __+ K o) = _ 4Re r ( R 0 -~ KO) u r ( K O ~ KO)

for t >> Ts.

(D.24)

E Transformation P r o p e r t i e s of Dirac Fields U n d e r C, P and T Using Maxwell's equations Ot, Ft'~ = j~, where F t'€ - O~A ~ - O ' A t' is the electromegnatic field tensor, and the transformation properties of 0" - O/Ox ~ and j~, one can determine how the electromagnetic potential A" = (A ~ A) transforms under parity, P, charge conjugation, C, and time reversal, T. Similarly, the transformation properties of Dirac spinors follow from his relativistic equation {i3'~0~ - m } r

"cgr

: 0

or

- H~p(x)= {-ic~Jo-~ + flm}r

(E.1)

In the Dirac-Pauli representation, a j=

,

/3=3, ~

,

aj 0

3'J=

0-11

,

(E.2a)

-a j 0

where aJ are the Pauli spin matrices O "1 =

(0110) (0 )(100 ) ,

~r2 =

,a3 =

0

(E.2b)

-1

and 11 is the unit 2 x 2 matrix. The results are shown in Table E.1. The transpose matrix ~T = 3"o^r = 3 ' o / ~ is expressed in terms of the the complex-conjugation operator K defined in (D.6). In the representation (E.2b), the spinor transformation matrices read C = i3'23'o and 7~ = i3"23'5, where 3'5 --- i')'~

E Transformation Properties of Dirac Fields Under C, P and T

157

Table E.1. P

x" = (t, x) (t, - x )

C

(t,x)

T

(-t, x)

0" 0r 0r -0"

j" j~ _jr j.

A, A~

~p .yo~b

~o

-A.

C~-T

--~)Td--i

A"

7~ T

~DT~%-1

The gauge (or phase) invariance of the Dirac theory is ensured through the replacement iO~ -+ iOt, + eA,: the transformation ~b(x) --+ ei~(x)r introduces a vector (gauge) field A~, which couples to the electron via ~ i n t ----

-j~' A, = er

A~.

In the following we will outline a formal theory of the discrete symmetry operations P, C and T. Although most of the ensuing discussion can be carried out without recourse to second quantization, the invariance under charge conjugation is an important exception. We thus seek to establish a formalism that would allow us to include also the more elementary symmetry operations, such as space reflections, in theories with second quantization.

Parity Operation We begin with the parity operation, or space inversion, P: x -+ x ~ = - x , t -+ t t = t. This is a subtle concept whose true raison d'etre lies in elementary particle physics9 If we compare (E.1) with the parity-transformed Dirac equation

"Or

1

Ot

- H'r

-- {i(~J ~---~ + /3m}~b(t,-x)

(E.3)

and note that 7~ ~ -- -c,J, (70) 2 = 1, we see that the hamiltonian has the following symmetry property: H ~= 7 ~

~

(E.4)

Equation (E.3) can thus be written as 9o r -x) 1 Ot

- 7 ~ 1 7 6 ~p(t, - x ) .

(n.5)

We now define a new function Cp(t, x) = 7~

- x ) = PC(t, - x )

(E.aa)

and multiply (E.5) by 7 ~ from the left, with the result

1"OCp(t,o______[___x) _ H~pp(t, x). The above equation is formally identical with (E.1). The Dirac theory is, therefore, invariant under space reflections provided the parity-transformed

158

Appendices

spinor is defined according to (E.6a). The operator /5, which connects the original and parity-transformed spinors, is linear-unitary: 15= t5-1

i.e.

/52 = 1.

(E.7)

This expresses the fact that the relation between the two states is reciprocal. The adjoint spinor has the following transformation property Cp(t, x) = Opt(t, x)7 ~ = [7~162 - x ) ] t 7 ~ = ~(t, - x ) 7 ~

(E.6b)

If r is regarded as a field operator, it can be expanded in terms of plane waves: d3p r x) = f (2~) 3/2

x E

[d~(p)v'(p)eipx + b'(p)u~(p)e-ip~]

(E.8)

s = d=l/2

with the expansion coefficients as operators. The operator dts(p) creates a positron with the z component of spin s = +1/2: d~(p)[ 0) = le+(p, s)), whereas b~ (p) annihilates an electron. The quantum fields @p(t,x) and ~b(t, x) satisfy the same equation of motion. Furthermore, it can be readily verified that they obey the same anticommutation rules {r

x), r

y)} = {~.(t, ~), ~.(t, y)} = 0,

(E.9a)

{Ca(t, x), Ca(t, y)} --- (.yo).z 5(3)(x - y). Indeed,

{r

~), ~ ( t , y)} = ( , ~ 1 7 6 1 6 2 = (~0)c~fl (~(3)(x - y),

~.(t,-y)} (E.9b)

and similarly

= {r

~(t,~)}

We thus expect the field operators means of a unitary transformation:

U(P)~(t, x)Ut(P) ? r

= 0.

@p(t,x)

(E.gc)

and @(t, x) to be related by

x) - "~%(t, - x ) .

(E.10)

When the Fourier expansion (E.8) is substituted into this relation, it reads

v(p)r

_/d,. (2.)3/.

[v(P)d~(p)vt(P)"'(p)ei'"

+ U(P)bs (p)U*(P)u8(p)e -ipx] --=/ (27r)3/2d3py~s[d~(_p)~/Ovs(p,)eipX

+bs(-p)7~

(E.11)

E Transformation Properties of Dirac Fields Under C, P and T

159

with p' = (E, - p ) . By using the explicit form of the Dirac spinors U8

V 8 ~---

t,

L>o'

(E.12a)

IEl+m /~s Xs

E<0

where

(E.12b) the following symmetry properties of u~ and v~ can be easily derived = -vs(p).

Prom (E.11) and (E.13) we infer that

U ( P)b~ (p)Ui ( P )

=

(E.14)

U ( P)d~ (p)U t ( P) = - d~ (-p), where the first expression is the hermitian conjugate of U(P)bs(p)Ut(p) = bs(-p) (recall that u t u = U - 1 U = 1 for a unitary operator U). If we further assume tha,t U(P)I O } = I 0 }, i.e., that the vacuum is invariant under P, then

U(P)I e- (p, s)) -- le-(-p, s)),

U(P)le+(p, s))

= - le+(-p,

(E.15)

s)).

The unitarity of U ensures that the state vectors are orthonormal:

(e-(p,s) [ V t U l e-(p,s)) = (e-(-p,s) l e-(-p,s)).

(E.16)

Expressions (E.14) define a unitary operator in the space of electron and positron state vectors 45 that satisfies (E.10). Based on the above results we conclude that for every state of free electrons and positrons there exists a parity-transformed state in which all momenta are reversed, but the spins are not affected. To test this prediction, it is necessary to set up a physical situation which cannot revert back into itself under space inversion. By aligning the spins of 13-emitting 6~ nuclei in a strong magnetic field and then measuring the relative electron intensities along and against the field, C. S. Wu and her collaborators were able to demonstrate, in 1957, that the left right symmetry of free Dirac fields is violated in weak interactions. Of particular interest is the second equality in (E.15). It shows that the intrinsic parity of the electron is opposite to that of the positron. This prediction has been verified experimentally, for example, in the decay of positro-

nium, 45 The inversion operators U(P, C) and V(T) act in Hilbert space, whereas /5, and T (which are composed of 2 matrices) operate in spinor space.

160

Appendices

Charge Conjugation We next consider the operation of operator

charge conjugation. The

adjoint Dirac field

dap

= f (27r)3/2 V~E •

E

[b~(p)~s(p)eipx + ds(p)vs(p)e-ipx]

(E.17)

s= rkl/2

annihilates positrons and creates electrons, whereas r does exactly the opposite. It is, therefore, plausible to expect that the Dirac equation is invariant under the replacement r ++ r To see if this assumption is correct, we take the complex conjugate of (E.1) and obtain [i(--'~')T0, -- role T = 0,

(E.18)

where we used 70(~f).~o = (7,)W,

~-W ~ (r

= 70r

(E.19)

If we multiply (E.18) by C from the left and compare the result with^(E.1), we find that two equations are equivalent if there exists an operator C such that ~(_~/~)W __ 7 / ~ , i.e., ~--l~/z~

= ( ,~#)T.

(E.20)

A suitable choice for C is = i7270

> ~ = _ ~ - 1 = _ ~t.

(E.21)

Under space inversion, the charge-conjugated spinor Co(x) transforms as r

_ ~w

= ~(r

~ ~[(70r

= ~(~,~0)T = ~,~0~-I~-T

= __ ~/0r

Therefore, r

~ 7~162

r

~ - 7~162

(E.22)

which shows that the intrinsic parity of a particle is indeed opposite to that of its antiparticle (cf. (E.15)). To demonstrate the equivalence of electrons and positrons in the Dirac theory, we look for a unitary operator U(C) that satisfies the following relation among field operators:

V(C)r i.e.,

s r

- C C r ( x ) = C [r

(E.23)

E Transformation Properties of Dirac Fields Under C, P and T

U(C)~(x)Ut(C)__ I (271-)3//2 dap V ~~ E

s

161

[U ( C)d~ (V) Ut (C)vs (P)eipx

+ V(C)bs(p)Vf(C)us(p)e -ipx] -J

d3p (2r~)3/2 ~

Ibm(p)i2/2u**(pleipx $ 9 2v s*(p)e -ipx ]. + ds (p) 17

E

(E.24)

From (E.24) and using i72u:(p) =

.2*

,7

(v) =

(E.25)

we obtain

U(C)b~(p)U t (C) = d~(p), U(C)d~ (p)U t (C) = bt~(p).

(E.26)

To prove (E.25), consider the following equations for the Dirac spinors

(i7~0~ - m)u = O, ~(i7~0~ + m ) = 0.

(E.27)

If the second equation is transposed and multiplied by C from the left, it yields, based on (E.20),

(i7~0~ - m)i72V * = O. The relation on the right of (E.25) is obtained by comparing this result with the first equation (E.27). The relation on the left of (E.25) can be proved in a similar manner. Now, assuming that U(C)I 0} = 10), expressions (E.26) lead to

U(C)[e-(p, s)} =

le+(p, s)),

(E.28)

V(C)le+(p, s)> = le-(p, s)). Under charge conjugation, the roles of creation and annihilation operators are interchanged: b~(p) ~ bt~(p) and ds(p) ++ d~(p). This symmetry operation flips the signs of internal charges, such as the electric charge, baryon number etc., but spins and momenta are not affected. For example, C turns a left-handed neutrino into a left-handed antineutrino, a state which does not exist. The operation of charge conjugation, therefore, does not transform a particle into its antiparticle; this can be accomplished through the combined CPT operation. Like parity conservation, charge conjugation invariance too is violated in weak interactions. Direct evidence for this violation is provided, for example, by the fact that the positive and negative electrons in the decay #+ --+ ed=~P have opposite longitudinal polarization. This effect was first observed in 1957 by measuring the circular polarization of bremsstrahlung photons emitted by e + and e- in the muon decay (the total transmission cross-section for photons propagating through magnetized iron depends on their helicity).

162

Appendices

Having defined C conjugation for free Dirac fields, we will now examine the effect of this symmetry transformation on the interaction parts of the lagrangian, ~int. For two Dirac fields r and r interacting with a vector field V~, ~int = g (~1")'/~r

+ w2! Wl /~, )

(E.29)

where g is a real coupling constant. The second term is the hermitian conjugate of the first one, thus ensuring the hermiticity of the lagrangian. It should be remembered that every quantized field theory that obeys commutation or anticommutation rules must be properly symmetrized or antisymmetrized. Thus all the bilinear forms of the Dirac field must be antisymmetrized. We will encounter shortly an important consequence of this rule. The adjoint field transforms as

: r

: __eTa-l,

(E.30)

where we used (E.20) and (E.21). Hence,

--~ --~2 ^/ttr

(E.31)

(the superscript "T" can be omitted because ~ 7 u r is a number, i.e., a oneby-one "matrix"). The origin of the minus sign in (E.31) is both subtle and important; it is related to the connection between spin and statistics. Since the fermion fields are antisymmetric, a minus sign must be introduced when we move one spinor past another. For the electron current this implies

jU(x) =- - e ~,.,/ur ~ _ jU(x).

(E.32)

The above result was anticipated: C conjugation flips the signs of all charges, including that associated with the current operator. We see that the method of second quantization is indeed essential for a self-consistent formulation of charge conjugation invariance. From (E.31) it follows that the lagrangian (E.29) is invariant under charge conjugation provided Vu -+ - V t in which case this symmetry transformation merely turns each term in the lagrangian into its hermitian conjugate. The electromagnetic potential Au(x ) and the current jU(x) are related through a simple differential operator (see the beginning of this appendix). Hence they must have the same transformation properties under C, which means that the combination jUA u is invariant under C conjugation. If we associate a photon with the field A , ( x ) , then C~ = - 1 . As shown in Appendix A, for a state with n photons, C = ( - 1 ) % In the decay of positronium, C invariance implies that the 1S0 singlet state decays to two photons and the 3S0 triplet state to three photons.

E Transformation Properties of Dirac Fields Under C, P and T

163

We conclude our discussion of charge conjugation by showing that ~r satisfies the anticommutation rules (E.9a). In terms of field components, a

~

Lo'

where we dropped the superscript T since it pertains to the complete field operator, not to its components. Now,

= (7~

(~(3)(x - y),

(E.33)

and similarly {r

x), r

y)} = {--c r

r

_- 0.

(E.34)

T i m e Reversal Turning next to the time-reversal transformation (t, x) ~ ( - t , x), we will demonstrate that the free Dirae field does not possess a unique time direction, i.e., that it is invariant under this symmetry operation. As explained in Appendix D, the time-reversal operator T is antilinearunitary (or antiuuitary): = unitary transformation (U) • complex conjugation (/4).

(E.35)

Since complex conjugation is involved in time reflections, the Dirac equation for the time-reversed state reads (cf. (E.18)) {-iT~

+i(--TJ)T~xj

-- m}-~T(--t,x)=0.

(E.36)

From (E.1) and (E.36 it follows that x) -

(E.37)

x)

and ~-17o~ = 7~ = -~-17o~ '

for the Dirac equation to remain invariant under time reversal. Clearly, the operators T and C are of similar nature. In the Dirac-Pauli representation, = i7275 = 717370

) ~b = -- ~ - 1 = _ 5hi, ~2 = _ 1

(E.39)

and

~%~'T(__t,x) = .,f173,.~0 [r where

T = 7 17 3~ 9 ( - t , x ) ,

(E.40)

164

Appendices

"Y5~ i70717273 =

(0:) ]1

/End/

"

The adjoint of the time-reversed spinor is given by

-~t(t,~) : [~/%~Z(--t,X)]t'),0 : [~'(~3'(--t,x)70)Z]'~ 0 = [r

x)70] * T*3~0 : ~pT(--t, x ) T -1

since ~t = ~b-1 and T-17~ = ~/o~-1. Therefore,

~(t,x) ~ -~t(t,x) = cT(--t,x)T -1.

(E.42)

Using (E.40) and (E.42), it can be readily shown that the quantization conditions remain invariant under time reversal: =

(~/l~/3)aa(~O~/3"yl)o~3{~)~(--t

,

X), r

= (71~3)~. (~0).~(70)~(70~37~)~,

y)}

~(3)(~ _ y)

= (../0)c~~ (~(3)(X -- y).

(E.43)

Similarly, {r

x), r

y)} = {r

(t, x), r

y)} = 0.

(E.44)

In analogy with our treatment of parity and charge conjugation, we seek an antiunitary transformation in Hilbert space that transforms ~b(x) into Ct(x). A clue is provided by expression (D.12) from Appendix D:

(BIOIA)

= (At 10t I Ut),

(E.45)

where

Ot -- ~b0tT -1

(E.46)

and 0 is a linear operator. Note that, due to complex conjugation, the time reversal transformation exchanges "bra" and "ket" vectors. This represents the exchange of the initial and final states in an interaction. In view of (E.46), we postulate the existence of an antiunitary operator V ( T ) -- U ( T ) K that satisfies

V (T)Ot (t, x) V - I (T) = ~Pt(t, x) =- T~T(--t, x).

(E.47)

Expressed in terms of field components, equation (E.47) reads V(T)r

(t, ~ ) V -1 (T) = (71737~ z [r ( - t , x)7~

w

= (713~3)a,O~(-t, x),

(E.48a)

U ( T ) ~ ( t , x ) U -1 (T) = (713~3)~p~(-t, x).

(E.4Sb)

i.e.,

E Transformation Properties of Dirac Fields Under C, P and T

165

By Fourier-transforming this expression into momentum space, we obtain

U(T)r

(21r)3/2

+ U(T)bs (p)U-l(T)Us (p)e -ipx]

=-i (,.)','d'" + bs(p)')'l"y3U*(p)e-i(Et+p':)].

(E.49)

Now, (E.50)

7 ~/ u,(p) =io "2 tr*. Xs and -Xs=-l/2

io'2Xs=+l/2 =

Xs=A_I/2 (E.51)

i 2 tr*

{-a'(--P))G=-I/2

O" (

,, O. (__p)Xs=+l/2

"P)~s=•

Equation (E.50) can thus be written as

s) =

', -s),

(E.52a)

with p' = ( E , - p ) . Similarly

3'l"y3v*(p, S) ---- (--1)s+l/2v(ff,--S).

(E.52b)

When (E.52a) is substituted in (E.49) and p changed to - p in the second integral, we find that

U(T)bt(p, s)U-I(T) = ( - 1 ) s-1/2 bt(-p,-s), (E.53)

V(T)di(p, s)U-I(T) = ( - 1 ) s-l/2 dt(-p, -s),

where the first relation (E.53) is the hermitian conjugate of Ub(p, s)U -1 = ( - 1 ) s-l/2 b(-p, -s). The phase factor in (E.53) implies that the result of two time reversal transformations performed on the Dirac field is the original field multiplied by a minus sign. Indeed, from (E.48a) it follows that

V2(T)r

x)V-2(T) = U(T) [-~173@(-t, x)]* U-I(T) = (ffl")'3)(~1~3)~b(t, x) ---- - r

and from (E.53),

x),

(E.54)

166

Appendices

U2(T)bt(p, s)U-2(T) = ( - 1 ) s - l / 2 V ( T ) b t ( - p , - s ) U - I ( T ) = - b t (p, s).

(E.55)

Assuming that U(T)I O) = 10), expressions (E.53) yield

U(T) l e•

-- (-1)s-1/21e+(-p,-s)).

(E.56)

This symmetry transformation, therefore, reverses the momentum and spin of an electron (positron) with respect to the original orientation along the z axis. This is to be expected, since p is the time derivative of x and the spin transforms as angular momentum (x x p). According to (E.45), the expectation values of the observables () and Or, which are constructed from the field operators r and Ct, respectively, satisfy

(A I O I A ) = ( At l Ot l At),

(E.57)

where

IAt)=- V(T)IA),

vtv=

v v t = l.

(E.58)

With the existence of an antiunitary operator that transforms r into Ct previously established, expression (E.57) demonstrates time-reversal invariance for a free Dirac field. Transformation P r o p e r t i e s o f Dirac Bilinears By virtue of Lorentz invariance, the quark and lepton spinors appear in bilinear forms in the lagrangians of quantum field theories. The transformation properties of the Dirac bilinears r 1 6 2(scalar), ~'Y5r (pseudoscalar), r162 (vector), ~'~'Y5r (axial vector) and r162 (tensor) under the discrete symmetry operations CP, T and CPT are given in Table E.2. Table E.2.

CP T

(t, ~) (t, -x) (--t, x)

r162

~_~~ Cb

~Jb~Ja -~a~Jb

--~Jb75~Ja --~bTiz~Ja --~b~fDTS~Ja --~JbO'l~r -~a75~)b ~aT~Jb ~aTi~75~Jb --~aO'tzv~Jb

~o~"~

~o~"~r

~_~cr'~r

The lagrangian of a local field theory must be hermitian 46 and behave either as a scalar or a pseudoscalar under Lorentz transformations. Based on this one can show, referring to Table E.2, that CPT is a good symmetry. For example, a term in the lagrangian s that includes only scalars and/or pseudoscalars transforms under CPT as 46 Hermiticity of the lagrangian ensures probability conservation (unitarity condition). A "local" lagrangian is composed only of terms containing products of fields at the same space-time point.

E Transformation Properties of Dirac Fields Under C, P and T

s

x) = g (-~aCb) (ir162

~

s

167 (E.59)

where we used the fact that T implies charge conjugation: c number --+ (c number)*. For ~ to be hermitian, it must also contain a term ~ . The sum s + s is evidently CPT invariant. The same holds true for a combination of vector and/or axial vector fields, e.g.,

s

x) = g VU(t, x)Au(t, x) + h.c. CPT s

--x),

(E.60)

where h.c. denotes hermitian conjugate. Since tensors transform as products of vectors and/or axial vectors, we conclude that

[CPT] s

[CPT] -1 = ~ ( - x ) .

(E.61)

Ignoring irrelevant phases, transformation relations (E.14), (E.26) and (E.53) amount to

bt(p, s) P bt(-p, s) c dt(_p, s) T dt(p ' -s).

(E.62)

The combined CPT operation converts particles to antiparticles and exchanges kets and bras. Under this symmetry transformation, the momentum of the particle is unchanged because both space and time are reflected, but the sign of the spin state is reversed. We will round off our discussion of the discrete symmetry transformations C, P and T by deriving the entries in Table E.2. Consider first the transformation properties of the Dirac bilinears CaF~Cb under CP:

r

r~ r

= V(C) r U - ' ( C ) ~U(C) r U-~(C) = ~ ( t , - x ) F~Cpr

-x),

(E.63)

where i - ~ (V~')y - 7"7 ~)

F~ = ll, 75, 7 ~, V~V5, r

(E.64)

( II is the unit 4 • 4 matrix ) and F~cp -- "~~(C-1FiC)TT~

(E.65)

Expressions (E.63) and (E.65) were derived by using

v(c) r

-l(C) = ~0 v ( c ) r = - C~ ~ ~ ( t ,

V(C) ~ f ( x ) V -1 (C)

-1(6)

-~)

=

g(c)-~a(t, - x ) V

-_

[v(c)r

(E.66)

-1(C)~ '0 =

CdT ( ,t- ~ ) 7

~

168

Appendices

To evaluate Ficp, note that &-iT5&

= 75 = ( 7 5 ) T

c-l,-)fl'*TvC = C - 1 7 t t C C - 1 7 v C ~-- (7/z)T(Tv) T -------(TttTv) T,

(E.67)

C-1,-),tt75 ~ = (---,//z)Tc-175 ~ _ (--7tt)T(7t) T = (7#7t) T,

based on (E.20). It is then straightforward to show that F cp = ]1,

-75,

-7~,

-7~75,

(E.68)

-a~v,

where we set 7~ ~ = 7g. Under the time-reversal transformation,

g,~Fi ~,~ = V(T) -~a(x)V-l(T) Fi* V(T) ~ ( x ) V - I ( T ) = ~a(--t, x) Fit r X),

(E.69)

Fit = (7173)t Fi,7,7 3

(n.70)

--t

where and (see (E.48a)) a7

V(T) r

= 7173r

V(T) r

= V(T) ~ / ) ~ ' ( x ) V - 1 7 = [V(T)r =

x),

(E.71) 0

t 70

~(-t, .) (7173)*

From (E.70) we obtain ~t = 11,75, 7,, 7~75, -(Y~.

(E.72)

Finally, under the combined CPT transformation, ~.cpt Fi CbPt ~_ V ( T ) ~cp V-I(T)/~/* V ( T ) r cp V - I ( T ) = ~--~b(--x)., FcPt Ca(-x),

(E.73)

where ~cvt ~ (7570 Fi,7075)W : 7570 Fit7o75.

(E.74)

To derive (E.73) and (E.74), we used V ( T ) ~ P V - I ( T ) -- 7173~/)bP(--t, x ) ----')'173 (-- C70 ~bb) = 7%~(-x)

V(T)-~:pv-I(T)

:

(E.75) (~/):P)t (?'73)t 70 = (__ ~70~-~ra) t (_ 71737 ~

= _ r

a7 Note that v r

-1 --- u c T u -1 -- [Ur

~

t = [V~bV-1] t.

F The Vacuum Insertion Approximation

169

Upon substituting (7") t : "),07"70

(E.76)

in expression (E.74), it yields

~iicpt ~-~ ]]-,

--75,

--7",

--7"75,

(E.77)

attu"

In general, the transformation properties of a free Dirac field under the discrete symmetry operations P, C and T can be expressed as U(P)r

%r162 -x), = ~; ~(t,-x)7 ~ = ~cS~(t, x), =

U(P)-~(x)U-~(P) u(c)r u(c)~(x)U-'(c) V(T)r V(T)-~(x)V-I(T)

=

-- ?~c ~2T(x) ~ - 1 '

,1,~r = ~, r

=

,--

(E.78)

~), ^

x)'P,

where Op, ~c and 7/t are arbitrary phases. One can easily show that these phases do not enter into the transformation laws for the Dirac bilinears.

F The

Vacuum

Insertion

Approximation

If we define O , =- 7u (1 -75), the matrix element of the four-fermion operator in (9.17) reads

M =- (K~ g O u d g O " d l K ~ = o~joke
(F.1)

where Greek letters denote the color indices (1,2,3) of quark creation and annihilation operators, and gOd = ~ o g'~Od'~" Like weak quark currents, physical hadrons are colorless; that is, they are singlets in color space. In order to estimate the magnitude of M , it is customary to insert the vacuum state between the two ~Od currents. In this approximation, the operators are treated as free fields (i.e., strong interactions are neglected) and the K ~ (/~0) is considered as made up of only ~d (ds). Taking into account that quark fields anticommute, one can form the following combinations of two-fermion operators:

0 = OijOke {siadJa 9 gk'~dt7 + gk'~dt'~ 9 siadJa _ gi~5~de~, gk767rdJa - gk'~67~_dJa, gia6aader } .

(F.2)

To simplify the calculation, we use the Fierz identities 1

3

1 a

a

(F.3)

170

Appendices

where Aa are the 3 • 3 color generator matrices (a -- 1, ... , 8). If the twofermion operators are inserted between the vacuum and the colour singlet K ~ state, the o c t e t ~ c t e t term cannot contribute. Therefore,

.M = 2 (R~ ~O~d l O)i O l ~O~dl K~ + ~1 (K~ ~i~O it dea I 0}(0 I 8k'YOkjdJ'~ [K~ 1

+ ~ (/~~ I ~k'~OkjdJ'~ [0)(0

~i~Oitde'~ IK~

= 2 ( l + 3) l(R~176 = 8(fkmk)2.

(F.4)

References

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174

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Name Index

Aharony, A., 87, 156 Alff-Steinberger, C., 53 Altarelli, G., 104 Armenteros, R., 81 Bell, J., 39 Bott-Bodenhausen, M., 53 Buchalla, G., 124, 143 Butler, C., 3, 4 Carithers, W., 63, 64 Charpak, G., 65 Christenson, J., 33, 34 Cronin, J., 3, 33 Dalitz, R., 2 Dirac, P.A.M., 17 Dorfan, D., 64 Einstein, A., 9 Feynman, R., 1, 10, 57, 117 Fitch, V., 3, 33, 50-52 Franzini, P., 81 Gaillard, M., 104 Gaillard, J.-M., 53 Gell-Mann, M., 2-4, 7, 57, 117 Gilman, F., 141 Good, M.L., 2, 17

Lattes, C., 4 Lederman, L., 7 Lee, B., 104 Lee, T.D., 3 Lim, C., 108, 122 Maskawa, T., 3, 105 Mehlhop, W., 53 Nakano, T., 3, 4 Niebergall, F., 60 Nishijima, K., 3, 4 Occhialini, G., 4 Pais, A., 2, 7, 17 Panofsky, W., 4 Pauli, W., 3 Piccioni, O., 17, 19 Powell, C., 4 Ritchie, J., 129 Rochester, G., 3, 4 Rubbia, C., 115 Sachs, R., 153 Steinberger, J., 4, 39, 42, 57, 61, 64, 65, 72, 81 Steller, J., 4 Stiickelberg, E., 43

Kabir, P., 87, 156 Kobayashi, M., 3, 105

Winstein, B., 75 Wise, M., 141 Wojcicki, S., 129 Wolfenstein, L., 105 Wu C.S., 159

Landau, L., 3

Yang, C.N., 3

Inami, T., 108, 122

Subject Index

absorptive (imaginary) amplitude, 128 active regenerator, 93 adjoint field, 162 adjoint of time-reversed spinor, 164 angular momentum, 90, 114, 129, 145, 146, 148, 166 annihilation, 81 annihilation operators, 104 anticommutation rules, 158, 162, 163 antimatter, 34, 50, 53 antineutrinos, 6, 33, 117 antiparticles, 34, 160, 167 antiunitary operator, 163, 164, 166 antiunitary transformation, 164 arbitrary phases, 169 associated production, 5, 96 Avogadro's number, 98 axial current, 104, 126 axial vector coupling, 128 axial vector fields, 167 B decays, 109 B°-JB ° mass difference, 110 B°-B ° mixing, 109, 112, 114 B°-/~ ° oscillation, 110 B°_/~ B ! - B !° mixing, 115 transition, 116 B mesons exclusive decays, 109 - inclusive decays, 109 baryon and photon densities, 33 baryon number, 5, 34, 36 conservation, 1, 96 beam halo, 97 beam-splitting, 19 bilinear forms, 166 Bose statistics, 9, 43, 90 box diagram, 101, 103, 105, 119, 126, 142 bubble chamber, 19

Cabibbo angle, 100, 101, 105 Cabibbo-Kobayashi-Maskawa (CKM) matrix, 105 calorimeter, 72, 80, 83 (~erenkov counter, 34, 66, 83, 124, 129 (~erenkov threshold, 61 channeling crystal, 94 charge asymmetry, 34, 57, 62, 63, 65, 68, 80 charge conjugation, 2, 3, 6, 13, 33, 36, 130, 131, 138, 146, 147, 156, 160, 161 invariance, 9 charge conservation, 106 charge-conjugated spinor, 160 charge-exchange reactions, 17, 60, 98, 124 charm hypothesis, 99 charm quark, 3, 99, 101, 105, 117, 121, 123 mass, 103 charmonium, 104 CKM parameters, 109, 112 matrix elements, 105-108, 117 Clebsch-Gordan coefficients, 43 cloud chamber, 3, 7, 12 coherent production, 114, 115 coherent regeneration, 34, 35, 81 - amplitude, 26, 63 coherent scattering, 24-26, 148 coherent superposition, 79 coincidence measurement, 9 collimator, 97 color indices, 169 color space, 169 color SU(3) matrices, 142, 170 commutation rules, 162 complex conjugate, 160 complex conjugation, 153, 163 - operator, 156 complex phase factor, 105, 108 -

-

178

Subject Index

connection between spin and statistics, 162 constructive interference, 52 coupled linear equations, 14 coupling electromagnetic, 10 - gravitational, 2 - weak, 10 C P eigenstates K1° and K °, 2, 6, 8, 19, 35, 57 C P invariance, 36, 152 C P parity, 8, 15 C P symmetry, 3 C P violation, 3, 33, 34, 45, 50, 106, 118, 140 CP-violating parameter, 42 e, 99, 105, 108 -

~'/~,

71

86 ~+_, 47, 83 ~?00, 47 CP-violating phase, 65 C P T conservation, 42, 44, 48 C P T invariance, 37, 39, 151 C P T operation, 36, 161, 167 C P T theorem, 36 C P T transformation, 168 C P T violation, 155 mass difference, 38, 39 creation and annihilation operators, 104, 158 cross-section - absorption, 148, 150 - differential, 148 - total, 148, 150 current axial, 104, 126 scalar, 133 - tensor, 133 - vector, 133 -

~+-0,

A I = 1/2 rule, 46 A I ----1/2 transitions, 143 A I ----3/2 transitions, 140, 142, 143 AS = AQ rule, 8, 57-60, 85, 100 Dalitz decay, 122 Dalitz plot, 86, 136 decay matrix, 23, 39 degenerate eigenstates, 13 - levels, 1, 15 - perturbation theory, 13 states, 1

depolarization effects, 131 destructive interference, 29, 50, 53 diffuse regenerator, 50-52 dilution factor, 61, 96 Dirac algebra, 103 Dirac bilinears, 166, 167, 169 Dirac equation, 33, 126, 133, 157, 160, 163 Dirac field, 160, 162, 165, 169 Dirac matrices, 100 Dirac spinors, 104, 156, 159, 161 Dirac's kets, 154 Dirac-Pauli representation, 156, 163 direct C P violation, 47, 48, 73, 90, 139, 152 discovery of K °, 1 discrete symmetries C, P and T, 92, 157, 166, 167, 169 dispersive (real) amplitude, 129 drift chamber, 76, 125, 129 duality of neutral kaons, 8 effective hamiltonian, 13, 23, 126, 142 eigenfunctions, 14 Einstein-Podolsky-Rosen "paradox", 9, 92 elastic scattering, 21 electromagnetic calorimeter, 94 electromagnetic current, 141 electromagnetic field tensor, 156 electromagnetic potential, 147, 156, 162 electron positron annihilations, 89 electrons, 159, 160, 166 electrostatic separator, 98, 124 electroweak penguin, 141, 143 electroweak radiative corrections, 121, 123 electroweak theory, 99 enhancement factor, 98 entropy, 43 equation of motion, 158 expectation value, 13 exponential decay law, 11 45 decay, 92 meson, 9, 89 Fermi coupling constant, 103 fermion fields, 162 fermion lines, 103 Feynman amplitude, 103 Feynman diagrams, 117 Feynman rules, 101 Feynman's parameter formula, 103, 119

Subject Index field operators, 104 Fierz identities, 169 final state interactions, 44 fixed-target experiments, 93 flavor dynamics, 106 flavor eigenstates, 112 flavor quantum numbers, 100 forward elastic scattering, 150 forward scattering, 21 amplitude, 20, 24, 50, 149 four-fermion operator, 104, 169 four-quark operator, 120 Fourier transformation, 165 gauge (or phase) invariance, 157 gauge theories, 99 GIM matrix, 105 GIM mechanism, 99, 102, 117 GIM model, 105 GIM suppression, 103 gluon, 141 gluonic corrections, 105 gluonic penguin, 141, 143 hadronic amplitude, 133 hadronic matrix element, 104, 110, 142 hamiltonian, 13, 26, 87, 153 Heisenberg uncertainty relation, 15 helicity, 6, 117, 161 suppression, 128 hermitian conjugate, 159, 162, 167 Higgs scalar, 132 Hilbert space, 13, 23, 155, 159, 164 t'Hooft-Feynman gauge, 101, 118 hydrogen target, 83 incoherent mixture, 61 incoherent production, 114 indirect C P violation, 47, 48, 121 inelastic regeneration, 79 interaction lagrangian, 120 interference patterns, 73 interference region, 61 intermediate states, 103, 150 internal charges, 36, 161 intrinsic parity, 3, 145, 159, 160 inversion operators, 159 iron regenerator, 26, 29 lsospin, 4 analysis, 43 assignments, 43 invariance, 126, 143 symmetry, 118, 120 isospin-breaking effects, 143 -

179

isotopic spin doublet, 43 isotopic spin triplet, 43 K + beams at rest, 96, 98 K°-/~ ° mixing, 21, 109, 117, 142 K ° regeneration, 2, 15, 17, 19 K ° - K y interference, 29 K ° - K ° mass difference, 10, 12, 13, 15, 19, 26, 28 K2~ decay constant fk, 104 K L0 - K s0 interference, 49, 53, 62, 65, 81 K L0 - K OS mass difference, 83, 99, 104 K s0- K Oa mixing, 48 K~3 decay rate, 137 K doublets, 4 kaon beams, 96 kaon mixing, 110 kaon parity, 146 kaon to neutron ratio, 97, 98 kaon yield, 96 kinematic limits, 135, 137 Kobayashi-Maskawa model, 105 lagrangian, 162, 166 lattice QCD, 105, 110 left-handed spinors, 100 leptons, 3 families, 3 number, 5 spinors, 166 level splitting, 1 Levi Civita tensor, 103, 134 lifetime, 2, 6, 8, 10, 36, 113 distribution, 73 - proper, 11 linear operator, 164 linear superposition, 2, 5, 7, 9, 14 linear-unitary operator, 158 local field theory, 166 longitudinal polarization, 130, 131, 161 loop diagrams, 118 loop integrals, 103, 108 Lorentz invariance, 96, 166 Lorentz transformation, 131, 166 -

~-e universality, 132 magnetic moment, 36 magnetic spectrometer, 72, 94 mass difference, 39, 65 mass eigenstates, 100 mass matrix, 23, 39, 48, 99 mass splitting, 105 matrix element, 99, 104 imaginary part, 105, 108

180

Subject Index

real part, 105, 108 maximum interference, 29 Maxwell's equations, 156 micro-bunched proton beam, 122 microwave background radiation, 33 mixed strangeness, 1 mixing matrix, 100 mixing time, 113 momentum space, 153 multiwire proportional chamber, 65 muon polarization, 137-139, 145

diagram, 126 pion energy spectrum, 136 plane wave, 147, 158 polar vector, 126 polarized beam, 7 polarized light, 1, 12 polarized states, 5 porportional chamber, 62 positronium, 162 positrons, 159, 160, 166 - energy spectrum, 136 probability amplitudes, 11, 12, 17 probability conservation, 39, 40, 43, 150, 166 probability waves, 12 production intensities, 61 production of K + and K - , 96 production of K °, 1 propagation of K °, 1 propane bubble chamber, 15 proper time, 11, 63 proper time interval, 66 proton antiproton annihilations, 81, 96 pseudoscalar, 43, 104, 126, 128, 145, 166 -

neutrino, 6, 33, 117 - flavors, 121 - spectrum, 136 neutron flux, 97 nondegenerate levels, 14 nuclear regeneration phase, 63 nucleosynthesis calculations, 33 operator structure, 142 operators annihilation, 161, 169 antilinear, 153 - creation, 161, 169 linear, 153 optical theorem, 21, 149, 150 orbital angular momentum, 114, 129, 145 orthogonality of states, 36, 40, 41 oscillation frequency, 113 -

-

-

parity, 156 conservation, 3, 49 operation, 157 transformation, 2, 6, 13, 33, 36, 130, 131 violation, 7, 33, 116 particle-antiparticle discrimination, 34 particle-antiparticle symmetry, 33 Pauli spin matrices, 156 penguin diagram, 126 perturbation, 13, 14 theory, 1, 44 phase associated, 6 convention, 48, 140 difference, 53, 62 of wave function, 10 - relative, 6 - shift, 151 phase space, 2, 6, 41, 113, 117, 134 photon beam, 97 photonic penguin, 142 -

-

-

-

-

QCD correction factor, 110 rk, 108 108 108 QCD parameter A, 109 quantization conditions, 164 quantized field theory, 162 quantum fields, 158 quantum interferometry, 89, 92 quantum-mechanical interference, 9, 12, 80, 99 quarks, 3, 100 bottom, 105 charm, 100 - counting, 96 - currents, 169 - doublet, 105 down, 100 families, 3 - fields, 169 - flavors, 3 masses, 101, 102 mixing, 3 - model, 3 spinors, 166 strange, 100 top, 105 -

-

-

z ~ t ,

-

~ c t ,

Subject Index

-

u p ,

1 0 0

rare kaon decays, 116 refraction, 24 refractive index, 21, 149 regeneration, 19 amplitude, 50, 66, 77 - parameter, 20 - phase CQ, 62, 65 regenerator, 19, 20, 23, 25, 26, 28, 34, 50, 53, 62, 64, 75 relative phase, 11, 48, 59 relativistic invariance and locality, 36 renormalization group equations, 142 RF separation, 98 -

scalar, 166 scattering amplitude, 21 scattering matrix, 150 scattering phase shift, 44 SchrSdinger equation, 21, 22, 153, 154 Schr5dinger representation, 154 SchrSdinger wave function, 10 Schwartz inequality, 41 second law of thermodynamics, 43 second quantization, 157, 162 semileptonic decays, 8, 9, 33, 34, 57, 61, 62, 68, 76, 80, 81, 83, 94 separation of K + beams - electrostatic, 96 RF, 96 solid regenerator, 51 spark chamber, 26, 34, 51, 65 - pictures, 29 spin projection operator, 131 spinor space, 159 spinor transformation matrices, 156 Standard Model, 3, 99, 101, 105, 106, 112, 117, 124, 128, 132, 143 state vectors, 9, 10, 159 StermGerlach experiment, 18 strangeness, 3, 4, 9, 13 conservation, 81 eigenstates of, 10, 17, 19, 57, 140 mixed, 10, 12 - operator, S, 5 - oscillation, 1, 10, 12, 57 oscillation region, 58 quantum number, 1 strangeness-changing neutral currents, 3, 100, 101 strangeness-changing transition, 1, 15 strangeness-conserving strong interactions, 1 -

-

-

-

181

strangeness-violating weak interactions, 1 straw chamber, 129 strong coupling constant (~, 142 SU(2) weak coupling, 100 superposition of amplitudes, 1, 10, 12, 17, 43 T invariance, 39 T violation, 39 tagged beams, 94 - K ° and/~0, 81 tagging efficiences, 81 time evolution of states, 2, 89 time invariance, 43 time reversal, 36, 39, 138, 153, 156, 163 - amplitude, 138 - asymmetry, 87 invariance, 166 top quark, 3, 101, 110 mass, 108, 121 trace techniques, 131 transition matrix, 150 transmission coefficient, 98 transpose matrix, 156 two-fermion operators, 169 two-kaon states, 9 two-photon intermediate state, 129, 132 two-state system, 1, 11 Y(4S) resonance, 109, 114, 115 unit matrix, 100 unitarity, 39, 44, 100 condition, 150, 151 relations, 39, 40, 111 unitary operator, 154, 159, 160 unitary transformation, 158 universality of weak interactions, 3 unphysical scalars, 101, 118, 121

-

V - A theory, 117 vacuum, 159 vacuum interference method, 65 vacuum state, 169 variable gap method, 19, 68 vector, 167 vector current, 120 vector-axial vector current, 104 vectors - bra, 164 - ket, 164 W bosons, 101 mass, 103

182

Subject Index

W-box diagram, 117 Watson's theorem, 44, 152 wave function, 9 weak eigenstates, 100 weak hamiltonian, 35, 101 weak isospin doublets, 105 weak transitions, 5 Weinberg angle 0w, 120 Weyl equation, 33 Wolfenstein parametrization, 105, 112 Z ° penguin, 142 - diagram, 117

Subject Index (Decays)

B ° --~ ( D * + ) D + g ~ , 109 B ° -+ g - v X , 109, 114 [3 ° -+ g+F,X, 109, 114

~5 ~ KO/~ o, 9 K -+ 27r, 3, 116, 145 K -+ 37r, 3, 116 K + -+ # + v . , 99, 124, 125, 127 K + -+ 7ru5, 123 K + -+ 7r+uP, 117, 120, 122, 124, 125 K + --~ r+Tr °, 124 K + --~ 7r°g+ve, 133, 137 K + -+ 7r°e+v~, 125 K ° -+ 37r°, 146 K ° -+ 7r+Tr- , 34 K ° -+ 7r+Tr-Tr°, 146 K ° ~ 7r-g+v~, 133 K ° --+ e+Tr-u~, 57, 58 K °, /7/o -+ ~'+~--, 152 K o, /.~o _+ ro~ro, 152 K ° ~ #+#-, 3 K ± -+ p± + v~,, 146 K ° ~ 7r+Tr- , 34 K ° --+ 7r+Tr-7, 34 K ° --+ 7r°Tr°, 34 K ° ~ 37r, 41

Computer to plate: Mercedes Druck, Berlin Binding: Buchbinderei Liideritz & Bauer, Berlin

K°3 = K ° -+ #±~rt:uu, 63 K°,s --+ 7r+Tr- , 71 K°,s --+ 7r°Tr°, 71, 76 K°,s -+ e±TrTu~, 62 K ° -+ 27r, 49 K ° -+ 27r° -+ 4% 71 KL~ -+ 37r° -+ 6% 71 K ° -+ "yT, 129 K ° -~ # + i t - , 99, 101, 117, 125, 128, 129, 132 K ° --+ # + # - , 131 K~ -+ 7r+Tr- , 50 K L -+ 7r°vD, 117, 122, 123 K~ -+ e+e - , 125, 132 K~, L -+ 27r, 43 K ~° =-- K ° -~ e±Tr~=ue, 63 /~o _+ e-Tr+~e, 57, 58 #± --+ e+u~, 161 7r ~ ep~, 117

Z '+ -+ n + e + + u~, 57 Z - --~ n + e - + ~ , 57 T(4S) --+ B°/} °, 109, 115

Springer Tracts in Modern Physics 135 From Coherent Tunneling to Relaxation Dissipative Quantum Dynamics of Interacting Defects By A. Wiirger 1996. 51 figs. VIII, 216 pages 136 Optical Properties of Semiconductor Quantum Dots By 13. Woggon 1997.126 figs. VIII, 252 pages 137 The Mott Metal-Insulator Transition Models and Methods By F. Gebhard 1997. 38 figs. XVI, 322 pages 138 The Partonic Structure of the Photon Photoproduction at the Lepton-Proton Collider HERA By M. Erdmann 1997. 54 figs. X, 118 pages 139 Aharonov-Bohm and Other Cyclic Phenomena By J. Hamilton 1997. 34 figs. X, 186 pages 140 Exclusive Production of Neutral Vector Mesons at the Electron-Proton Collider HERA ByJ. A. Crittenden 1997. 34 figs. VIII, lO8 pages 141 Disordered Alloys Diffusive Scattering and Monte Carlo Simulations By W. Schweika 1998.48 figs. X, 126 pages 142 Phonon Raman Scattering in Semiconductors, Quantum Wells and Superlattices Basic Results and Applications By T. Ruf 1998.143 figs. VIII, 252 pages 143 Femtosecond Real-Time Spectroscopy of Small Molecules and Clusters By E. Schreiber 1998. 131 figs. XII, 212 pages 144 New Aspects of Electromagnetic and Acoustic Wave Diffusion By POAN Research Group 1998.31 figs. IX, 117 pages 145 Handbook of Feynman Path Integrals By C. Grosche and F. Steiner 1998. X, 449 pages 146 Low-Energy Ion Irradiation of Solid Surfaces By H. Gnaser 1999.93 figs. VIII, 293 pages 147 Dispersion, Complex Analysis and Optical Spectroscopy By K.-E. Peiponen, E.M. Vartiainen, and T. Asakura 1999.46 figs. VIII, 13o pages 148 X-Ray Scattering from Soft-Matter Thin Films Materials Science and Basic Research By M. Tolan 1999.98 figs. IX, 197 pages 149 High-Resolution X-Ray Scattering from Thin Films and Multilayers By V. Hol~r, U. Pietsch, and T. Baumbach 1999. 148 figs. XI, 256 pages 150 QCDat HERA The Hadronic Final State in Deep Inelastic Scattering By M. Kuhlen 1999. 99 figs. X, 172 pages 151 Atomic Simulation of Electrooptic and Magnetooptic Oxide Materials By H. Donnerberg 1999.45 figs. VIII, 205 pages 152 ThermocapiUary Convection in Modds of Crystal Growth By H. Kuhlmann 1999. lol figs. XVIII, 224 pages 153 Neutral Kaons By R. Belu~evi~ 1999.67 figs. XII, 183 pages 154 Applied RHEED Reflection High-Energy Electron Diffraction During Crystal Growth By W. Braun 1999.15o figs. IX, 222 pages

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Heavy Quark Effective Theory (Springer Tracts in Modern Physics)

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Coverings of Discrete Quasiperiodic Sets (Springer Tracts in Modern Physics)

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Heavy Quark Effective Theory (Springer Tracts in Modern Physics)

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Semiconductor Cavity Quantum Electrodynamics (Springer Tracts in Modern Physics)

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High Power Laser-Matter Interaction (Springer Tracts in Modern Physics)

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Jets of Hadrons (Springer Tracts in Modern Physics 90)

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Handbook of Feynman Path Integrals (Springer Tracts in Modern Physics)

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High Power Laser-Matter Interaction (Springer Tracts in Modern Physics)

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Coverings of Discrete Quasiperiodic Sets (Springer Tracts in Modern Physics)

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Flavor Physics and the TeV Scale (Springer Tracts in Modern Physics, 233)

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Flavor Physics and the TeV Scale (Springer Tracts in Modern Physics, 233)

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High Energy Polarized Proton Beams: A Modern View (Springer Tracts in Modern Physics)

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Electroweak Physics at LEP and LHC (Springer Tracts in Modern Physics)

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Electroweak Physics at LEP and LHC (Springer Tracts in Modern Physics)

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Precision Electroweak Physics at Electron-Positron Colliders (Springer Tracts in Modern Physics)

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Magnetic Heterostructures: Advances and Perspectives in Spinstructures and Spintransport (Springer Tracts in Modern Physics, 227)

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Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics)

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Capillary Surfaces: Shape - Stability - Dynamics, in Particular Under Weightlessness (Springer Tracts in Modern Physics)

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Electronic Quantum Transport in Mesoscopic Semiconductor Structures (Springer Tracts in Modern Physics)

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Electronic Quantum Transport in Mesoscopic Semiconductor Structures (Springer Tracts in Modern Physics)

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Magnetic Heterostructures: Advances and Perspectives in Spinstructures and Spintransport (Springer Tracts in Modern Physics, 227)

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Symmetries in Atomic Nuclei: From Isospin to Supersymmetry (Springer Tracts in Modern Physics)

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Capillary Surfaces: Shape - Stability - Dynamics, in Particular Under Weightlessness (Springer Tracts in Modern Physics)

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Symmetries in Atomic Nuclei: From Isospin to Supersymmetry (Springer Tracts in Modern Physics)

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Current-induced Nonequilibrium Phenomena in Quasi One-dimensional Superconductors (Springer Tracts in Modern Physics)

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The Anomalous Magnetic Moment of the Muon (Springer Tracts in Modern Physics, 226)

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