Pentaquark 04: Proceedings of International Workshop, Spring-8, Japan, 20-23 July 2004 (Proceedings of the International Workshop)

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PENTAQUARK

04

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PENTAQUAR K Proceedings of the International Workshop Spring-8, Japan

20 - 23 July 2004

04

edited by

Atsushi Hosaka Tomoaki Hotta Osaka University, Japan

N E W JERSEY

*

LONDON

v

World Scientific

SINGAPORE

BElJlNG * SHANGHAI

HONG KONG * TAIPEI * C H E N N A I

Published by

World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224

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PENTAQUARK 04 Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Printed in Singaporeby B & JO Enterprise

V

PREFACE

The workshop PENTAQUARK04 hosted by RCNP of Osaka University and by JASRI was held at Spring-8 site (Nishiharima, Hyogo, Japan), from July 20 to 23, 2004, where the first significant evidence of the exotic baryon 8+ was observed by the LEPS group lead by T. Nakano (RCNP). This workshop follows, after 9 months, the previous workshop held in November 2003 at Jefferson Lab. This time, 126 people registered including 48 participants from abroad. We have heard as many as 64 talks covering almost exclusively on the pentaquark baryons and related exotic hadrons. Since the first report by LEPS group, we have seen rapid growth of both experimental and theoretical works on the pentaquark baryons. At the same time, the status of the pentaquark has been changing. In the latest issue of the Review of Paricle Physics (Physics Letters B, Vol. 592, 2004) the Q+ is nominated as a three-star state, meaning that “existence ranges from very likely to certain, but further confirmation is desirable”. Indeed, several counter-evidences have been now reported. As Ken Hicks mentioned in his summary talk as well as in these proceedings, the existence is certainly an experimental issue. However, the understanding is a theoretical issue. Despite many theoretical works with new ideas, so far the situation is not yet conclusive. For low lying states, we thought that we could understood them from QCD, or QCD oriented models and methods. However, the variety of different theoretical predictions to date suggests that we are still far from the final goal. The workshop started with two opening addresses by Prof. Kira, the director of SPring-8/JASFU and by Prof. Toki, the director of RCNP. The scientific sessions were then organized as plenary and parallel ones. Because parallel sessions with overlapping talks were necessary, we also arranged a poster session in one evening with continuous discussions till night, which was made possible in this workshop at the Spring-8 site. During the workshop, we heard many interesting reports from both theorists and experimentalists. Especially, it was our great pleasure that we could invite two pioneering theorists, H. Lipkin and D. Diakonov. One reason that made the workshop very interesting is that many people showed

vi

preliminary results, some of which might not be suited t o appear in a printed version. However, we have decided to publish the proceedings in order to compile the results and thoughts presented there, which we believe to be an important process not only to bring the workshop to a fruitful conclusion but also to make another step toward new developments. Finally, we would like to mention that this workshop was not able to be realized without support from Inoue Science Foundation and Ohio University, for which we would like to express our thanks. We also thank our secretaries and students from RCNP, and local people from Spring-8 for their support of the workshop.

Editors, Atsushi Hosaka and Tomoaki Hotta January, 2005

International Workshop PENTAQUARK04 July 20-23, 2004 at Spring-8, Japan URL: http://www.rcnp.osaka-u.ac.jp/penta04/ Email: [email protected] Organizers: Hiroshi Toki (Chair, RCNP, Osaka) Schin Date’ (Scientific Secretary, Spring-8) Atsushi Hosaka (Scientific Secretary, RCNP, Osaka) Kenneth Hicks (Ohio) Tomoaki Hotta (RCNP, Osaka) Ken-ichi Imai (Kyoto) Noritaka Kumagai (Spring-8) Takashi Nakano (RCNP, Osaka) Yuji Ohashi (Spring-8) Makoto Oka (Tokyo Inst. Tech.)

vii

Contents Welcome Addresses

A. Kira (Director of Spring-8)

H. Toki (Director of RCNP, Osaka University)

...

xlll

xv

Keynote Talks History and New Ideas for Exotic Particles H. J. Lipkin Chiral Symmetry and Pentaquarks D. Diakonov

1

11

Experiments Study of the Of at LEPS T. Nakano

23

X(3872) and Other Spectroscopy Results from Belle K. Abe

27

Search for Exotic Baryon Resonances in p p Collisions at the CERN SPS K. Kadija (for the NA49 Collaboration)

35

Preliminary Results from the GRAAL Collaboration C. Schaerf (for the Graal Collaboration)

43

Search for Pentaquark States on Proton Target at CLAS R. De Vita (for the CLAS Collaboration)

50

Evidence for Of Resonance from the COSY-TOF Experiment W. Eyrich (for the COSY-TOF Collaboration)

58

Pentaquark Search at HERMES W. Lorenzon ( o n behalf of the HERMES Collaboration)

66

...

Vlll

Study of Narrow Baryonic Pentaquark Candidates with the ZEUS Detector at HERA U. Karshon

75

Pentaquarks with Charm at H1 R. Stamen

83

Pentaquark Search Via ( T - , K - ) Reaction K. Miwa (for the E522 Collaboration)

87

Search for Pentaquarks at Belle R. Mizuk (for the Belle Collaboration)

91

Search for Strange Pentaquark Production in e+e- Annihilations at fi = 10.58 GeV and in T(45) Decays V. Halyo (Representing the BABAR Collaboration)

99

Pentaquark Results from CDF M. J. Wang (for the CDF Collaboration)

107

Search for the Pentaquark O+ in the y 3He 4 PA@+Reaction Measured at CLAS S. Niccolai (for the CLAS Collaboration)

115

Spectroscopy of Exotic Baryons with CLAS: Search for Ground and First Excited States M. Battaglieri

119

A Search for Neutral Baryon Resonances Below Pion Threshold X . Jiang

127

Time Projection Chamber for Photoproduction of Hyperon Resonances at SPring-8/LEPS H. Fujimura et al.

134

Theories Remarks on the Parity Determination of Narrow Resonances C. Hanhart, J. Haidenbauer, K. Nakayama and U.-G. Meissner

138

Pentaquark Baryon Production in Nuclear Reactions C.M. KO and W. Liu

149

ix

Photoproduction of @+ on the Nucleon and Deuteron T. Mart, A . Salam, K . Miyagawa and C. Bennhold

157

On the @+ Parity Determination in K K Photoproduction A.I. Titov, H. Ejiri, H. Haberzettl and K . Nakayama

165

Comment on the @+-Production at High Energy A.I. Titov, A . Hosaka, S. Date'and Y. Ohashi

171

Spin-Parity Measurements of O+- Some Considerations C. Rangacharyulu

174

Reflection Symmetry and Spin Parity of @+ H. Ejiri and A . Titov

178

The Use of the Scattering Phase Shift in Resonance Physics M. Nowakowski and N . G. Kelkar

182

Pentaquark Resonances from Collision Times N.G. Kelkar and M. Nowakowski

190

Photon and Nucleon Induced Production of @+ S.I. Nam, A . Hosaka and H. C. K i m

198

Determining the @+ Quantum Numbers Through a Kaon Induced Reaction T. Hyodo, A . Hosaka, E. Oset and M.J. Vicente Vacas

202

Exotic Challenges M. Praszatowict

206

Pentaquarks in a Breathing Mode Approach to Chiral Solitons H. Weigel

215

The Skyrme Model Revisited: An Effective Theory Approach and Application to the Pentaquarks K . Harada

223

Magnetic Moments of the Pentaquarks H.C. Kim, G.S. Yang, M. Praszatowicz and K. Goeke

231

X

Narrow Pentaquark States in a Quark Model with Antisymmetrized Molecular Dynamics Y. Kanada-En’yo, 0. Morimatsu and T. Nishikawa

239

Decay of O+ in a Quark Model A . Hosaka

247

Dynamical Study of the Pentaquark Antidecuplet in a Constituent Quark Model F1. Stancu

254

Pentaquark with Diquark Correlations in a Quark Model S. Takeuchi and K. Shimizu

262

Contribution of Instanton Induced Interaction for Pentaquarks in MIT Bag Model T. Shinozaki, M. Oka and S. Takeuchi

270

Five-body Calculation of Resonance and Continuum States of Pentaquark Baryons with Quark-Quark Correlation E. Hiyama, M. Kamimura, A . Hosaka, H. Toki and M. Yahiro

274

Flavor Structure of Pentaquark Baryons in Quark Model Y. Oh and H. Kim

282

Parity of the Pentaquark Baryon from the QCD Sum Rule S.H. Lee, H. Kim and Y. Kwon

290

Pentaquark Baryons from Lattice Calculations S. Sasaki

298

Excited Baryons and Pentaquarks on the Lattice F. X . Lee

306

Anisotropic Lattice QCD Studies of Pentaquark Anti-decuplet N . Ishii, T. Doi, H. Iida, M. Oka, F. Okiharu and H. Suganuma

316

Lattice QCD Study of the Pentaquark Baryons T.T. Takahashi, T. Umeda, T. Onogi and T. Kunihiro

324

xi

Signal of O+ in Quenched Lattice QCD with Exact Chiral Symmetry T. W. Chiu and T.H. Hsieh

331

The Static Pentaquark Potential in Lattice QCD F. Okiharu, H. Suganuma and T.T. Takahashi

339

QCD Sum Rules of Pentaquarks M. Oka

344

Pentaquark Baryon from the QCD Sum Rule with the Ideal Mixing J. Sugiyama, T. Doi and M. Oka

354

Mass and Parity of Pentaquark from Two-hadron-irreducible QCD Sum Rule T. Nishikawa

358

Three-quark Flavour-dependent Force in Pentaquarks V. Dmitras'inoviC

362

Interaction of the O+ with the Nuclear Medium M.J. Vicente Vacas, D. Cabrera, Q.B. Li, V.K. Magas and E. Oset

370

Production of O+ Hypernuclei with the ( K + ,K + ) Reaction H. Nagahiro, S. Hirenzaki, E. Oset and M.J. Vicente Vacas

378

Dynamics of Pentaquark in Color Molecular Dynamics Simulation Y. Maezawa, T. Maruyama, N . Itagaki and T. Hatsuda

386

Exotic Pentaquarks, Crypto-heptaquarks and Linear Three-hadronic Molecules P. Bicudo

390

Hadronic Aspects of Exotic Baryons E. Oset, S. Sarkar, M.J. Vicente Vacas, V. Mateu, T. Hyodo, A . Hosaka and F. J. Llanes-Estrada

398

Pentaquark Baryons in String Theory M. Bando, T. Kugo, A . Sugamoto and S. Terunuma

406

xii

Narrow Width of Pentaquark Baryons in QCD String Theory H. Suganuma, H. Ichie, F. Okiharu and T.T. Takahashi

414

Summary Talks Workshop Summary: Experiment K. Hicks

422

Pentaquarks: Theory Overview, and Some More about Quark Models C.E. Carlson

430

List of Participants

439

xiii

WELCOME ADDRESS

AKIRA KIRA Director General, JASRI/SPring-8

I am very pleased to welcome the participants of Pentaquark04 to the site of Spring-8. I feel it a great honor that Spring-8 is the place where a particle consisting of 5 quarks was f i s t confirmed. The discovery of the pentaqurk particle is one of the best news in last year for Spring-8. Hearing this news, I was very pleased. I did not understand the details but I felt that something very new took place: extraordinary science that provides a big breakthrough has been done. To my poor knowledge, the laser reverse Compton is the different from the synchrotron radiations. Spring-8 was constructed as a huge synchrotron radiation facility, and the reverse Compton is a sort of spin off. The main facility provides the beam of the world-No.1 quality: the Japanese society or Government appreciated the completion of the best in the world, and they are now expecting the brilliant results from the No.1 machine? The discovery of the pentaquark particle fulfills this expectation. The beamline, BLSSLEPS, is the unique peculiar beamline in the present existing 47 beamlines in Spring-8. In other big synchrotron facilities, ESRF is equipped with a laser Compton beamline but I heard that it is not used any more because of the undesirable influence to the ring beam. Probably affecting by this fact, APS users rejected the installation of the laser Compton beamline. At Spring-8, the machine people were predominant to the users and the Japanese economy was at so good conditions that any proposal could be funded. Of course, our staff paid enough care to avoid the influence to the main beam. I am pleased that such a beamline contributed to the present big discovery. The news of the discovery was reported extensively by Japanese mass media and the Government officials were impressed and pleased. It is probably a rare chance for you to appeal your beamline to the Government. At Spring-8, machine people told me that their technology would allow one more reverse Compton beamline without disturbing the main beam. Good luck!

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xv

WELCOME ADDRESS

HIROSHI TOKI Director of RCNP, Osaka University

On behalf of the organizing committee, I would like to welcome you to Spring-8. Spring-8 stands for Super Photon Ring and 8 is for 8 GeV electron. RCNP has a beamline at Spring-8, where the discovery of the pentaquark particle has been made. I am very happy to have so many friends coming from many places. In particular, it is our great pleasure to have Prof. Lipkin, who have been discussing the possibility of pentaquark for many years. We have also Prof. Diakonov, who was convinced with the idea of pentaquark from his experience with instanton and the chiral quark model and suggested Prof. Nakano to look for the state around 1500 MeV with small width. It was the year 2000 and in the occasion of the Adelaide meeting in Australia, when these two theorist and experimentalist discussed on the experiment to be done with the photon at LEPS in SPring8. A possible signature of pentaquark was announced by Prof. Nakano at PANIC02 held in Osaka in 2002. The paper was published in Phys. Rev. Lett. in July of 2003 and the announcement of the finding of the pentaquark was made to all the world. The meeting site is chosen here at Spring-8. This site is beautiful. At the same time, everybody can be accommodated here and hence the participants can discuss even after dinner. In this workshop, we are successful in getting many scientists who are involved in this interesting subject. We will have an occasion to see the LEPS facility and Spring-8 machine. We hope some of you have time to visit RCNP also, which is close from here; it takes about 2 hours by car. I sincerely hope that the symposium is fruitful to all the participants. It is also very important to have discussions individually. Let us enjoy a lively meeting. Thank you very much.

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1

HISTORY AND NEW IDEAS FOR EXOTIC PARTICLES

HARRY J. LIPKIN Department of Particle Physics, Weizmann Institute of Science, Rehovot, Israel E-mail: harry.lipkin0weizmann.ac.il and School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and High Energy Physics Division, Argonne National Laboratory Argonne, IL 60439-4815, USA Basic 1966 physics of Sakharov, Zeldovich and Nambu updated by QCD with constituent-quark quasiparticles having effective masses fits all masses and magnetic moments of ground state meson and baryons having no more than one strange or heavy quark Flavor antisymmetry explains absence of low-lying exotics and suggests diquark-triquarkmodel and twc-state model for 0+ pentaquark. Variational approach gives mass bounds for other pentaquarks.

-

1. Introduction What can QED teach us about QCD?

QCD is a Great Theory, but how do we connect it with experiment or find approximations I recall Yoshio Yamaguchi’s response in 1960 when asked whether there had been any thought at CERN about a possible breakdown of QED at small distances: “NO. . Many calculations. No thought.” What can we learn from QED; a Great Theory that everyone knows how to connect with experiment? We know how isolated free electrons behave and carry currents. But nobody could explain the fractional Hall effect until Robert Laughlin told us the Hall Current is not carried by single electrons but by quasiparticles related to electrons by a complicated transformation. Nobody has ever seen an isolated free quark. Experiments tell us that baryons are qqq and mesons are qQ but these are not the current quarks whose fields appear in the QCD Lagrangian. Are these quarks complic&ed quasiparticles related to current quarks by a complicated transformation? Nobody knows. Is Hadron Spectroscopy Waiting for Laughlin? Does QCD need another Laughlin to tell us what constituent quarks are?

2

2. The 1966 basic physics of hadron spectroscopy 2.1. The QCD-updated Sakharov-Zeldovich maa8 formula

A unified mass formula for both meson and baryon ground state massed updated by DeRujula, Georgi and Glashow2 (DGG) using QCD arguments relating hyperfine splittings to constituent quark effective masses3 and baryon magnetic moments showed that all are made of the same quarks’ and gave remarkable agreement with experiment including three magnetic moment predictions with no free parameters 495

M =

aa aj C mi + C m, .mj *

i

(ms - mu)Bar =

p~ = -0.61n.m.

pp

t@P

(1)

i> j

6

PP . mu k - ME = -0.61 n.m. = --- -3 m, 3 MA- MN

2Mp = 0.865 n.m. + pn = 0.88n.m. = MP = 3mu M N + M A

The same value f 3 % for m, - mu is obtained from four independent is obtained from meson and calculations. The same value f2.5% for baryon masses. The same approach for mb - m, gives

3

2.2. Two Hadron Spectrum puzzles -Why qqq and qa ?

(1) The Meson-Baryon Puzzle - The qq and ijq forces bind both mesons and baryons differently. A vector interaction gives equal and opposite forces; a scalar or tensor gives equal attractions for both. (2) Exotics Puzzle - No low-lying hadrons with exotic quantum numbers have been observed; e.g. no T+T+ or K + N bound states. Nambu solved both puzzlesa in 1966, related mesons and baryons and eliminated exotics by introducing color and a two-body non-abelian-gauge interaction with the color-factor of one-gluon exchange. A unified treatment of qq and qq interactions binds both mesons and baryons with the same forces. Only qqq and qq are stable in any singlecluster model with color space factorization. Any color singlet cluster that can break up into two color singlet clusters loses no color electric energy and gains kinetic energy. The Nambu color factor does not imply dynamics of onegluon exchange. Higher order diagrams can have same color factor Looking beyond bag or single-cluster models for possible molecular bound states Lipkin(1972) lowered the color-electric potential energy in potential models by introducing color-space correlations; e,g, qqqq at corners of a square, but not enough to compensate for the kinetic energy7 2.3. Important systematics in the experimental spectrum

A large spin-dependent interaction M 300 MeV but a very weak interaction M 2 MeV binding normal hadrons.

+

M ( A ) - M ( N ) M 300MeV >> M(n) M ( p ) - M ( d ) M 2MeV 2.4. Conclusions from basics

-

(7)

What we do know and don’t

We know the low-lying hadron spectrum is described by quasiparticles called quarks with a linear effective mass term and a hypefine interaction with a one-gluon exchange color factor. Only color singlet and 3* color factors arise in the (qq) and (qqq) states which behave like neutral atoms with a strong color electric field inside hadrons and none outside. No molecular bound states arise in the simplest cases. A strong spin-dependent interaction is crucial to understanding the spectrum We don’t know what these quarks are and the low-lying hadron spectrum provides no direct experimental information on (ijq)a and (qq)f3 interactions needed for multiquark exotic configurations.

4

3. QCD Guide t o the search for exotics 3.1. Words of Wisdom from Wigner and Bjorken Wigner said “With a few free parameters I ca.n fit an elephant. With a few more I can make him wiggle his trunk” His response to questions about a particular theory he did not like was: “I think that this theory is wrong. But the old Bohr - Sommerfeld quantum theory also wrong. It is hard to see how we could have reached the right theory without going through that stage’. In 1986 Bjorken noted how a qq created in e+e- annihilation fragments into hadrons. The quark can pick up an antiquark to make a meson. or a quark to make diquark. The diquark can pick up another quark to make a baryon but might pickup an antiquark to make a “triquark” bound in a color triplet state. Picking up two more quarks makes a pentaquark BJ asked: “Should such states be bound or live long enough to be observable as hadron resonances? What does quark model say? 3.2. What the quark model says about ezotics To consider the possible mass difference between the Q+ and a separated KN system, first put a K+ and a neutron close together and keep the US in the kaon and the udd in the neutron coupled to color singlets. Nothing happens because color singlet states behave like neutral atoms with negligible new interactions. Next change color-spin couplings while keeping an overall color singlet and search for the minimm energy. Use a variational approach with wave functions having the same spatial two-body density matrix elements as those in the observed mesons and baryons. Experimental hadron mass differences are then used to determine all parameters and look for possible bound states. This approach finds no possibility for a K+n bound state. But the same method shows that this trial wave function for the D;p system gives a lower hyperfine potential energy for the anticharmed strange pentaquark (Emud)over the separated D;p. Whether this is enough to compensate for the kinetic energy required to localize the state is unclear and highly model dependent with too many unknown parameters as soon as the requirements on the two-body density matrix are relaxed . This anticharmed strange pentaquarka and Jaffe’s H dibaryong became the subjects of experimental searches. Although Fermilab E791 did not find convincing evidencelo for the Emud pentaquark, the possibility is still open that this stable bound pentaquark exists and needs a better search. The existence of the Q+ showed that wave functions with the same two-

5

body density matrix for all pairs did not work and a two cluster model was needed to separate the uu and dd pairs that have a repulsive short-range hyperfme interaction. This led to the diquark-triquark mode14v5.. 3.3. Crucial role of wlor-magnetic interaction (1) QCD motivated models show same color-electric interaction for large multiquark states and separated hadrons and no binding. Only short-range color-magnetic interaction produces binding. (2) Jaffeg (1977) extended DGG with same color factor to multiquark sector in a single cluster or bag model, defined (fjq)8 and (qq)6 interactions, explained absence of lowlying exotics and suggested search for H dibaryon uuddss. (3) Jaffe's model extended to heavy quarks and flavor-antisymmetry principle" suggested exotic tetraquarks and anticharmed strange pentaquark8 (i%uds) (1987) 3.4. Flavor antisymmetry principle

- No leading exotics

The Pauli principle requires flavor-symmetricquark pairs to be antisymmetric in color and spin at short distances. Thus the short-range color-magnetic interaction is always repulsive between flavor-symmetric pairs. (1) Best candidates for multiquark binding have minimum number of same-flavor pairs

(a) Nucleon has only one same-flavor pair (b) A++(uuu)has three same-flavor pairs Costs 300 MeV relative to nucleon with only one. (c) Deuteron separates six same-flavor pairs into two nucleons Only two same-flavor pairs feel short range repulsion. (d) H(uuddss) has three same-flavor pairs. Optimum for light quark dibaryon (e) The (uud~E) pentaquark has only one same-flavor pair (2) Pentaquark search. (uudst?) pentaquark has same binding as H.

(a) Quark model calculations told experimenters to look for (uudsE)pentaquark; not the Of. (b) O+ (uudds) has two same-flavor pairs pairs. Too many for a single baryon. (c) Calculations motivating the (uudsE) pentaquark search found no reason to look for (uudds)

6

Ashery’s E791 search for &uds found events”; not convincing enough. Better searches for this pentaquark are needed; e.g. searches with good vertex detectors and particle ID8. Any proton emitted from secondary vertex is interesting. One goldplated event not a known baryon is enough; No statistical analysis needed. 4. The 8+ was found! What can it be?

Following Wigner’s guidance to understand QCD and the pentaquark, find a good wrong model that can teach us; stay away from free parameters 4.1. The s b r m i o n model

Experimental search motivated by another “wrong model”. Skyrmion12 has no simple connection with quarks except by another “wrong model”. The l/Nc expansion invented13 pre-QCD to explain absence of free quarks. -The binding Energy of qQ pairs into mesons EM M g2NcAt large N , the cross section for meson-meson scattering breaking up a meson into its constituent quarks is

But N. = 13 ’ JL N. M 1 This is NOT A SMALL PARAMETER! 4.2. How to ezplain O+ with quark

-

The two-state model

No bag or single cluster model with the same flavor-space correlation for all quarks can work. Keeping same-flavor pairs apart led to diquark-triquark model with (ud) diquark separated from remaining (uda) triquark with triquark color-spin coupling minimizing color-magnetic energy 4*5. Noting two different color-spin couplings for triquark with roughly equal color-magnetic energy leads naturally to a two-state model14. Let 101) and 1 0 2 ) denote an orthonormal basis for the two diquarktriquark states with different triquark color-spin couplings. The mass matrix eigenstates can be defined with a mixing angle 4

+

10)s z cos 4 - 101) sin 4 *

lo),

(02)

-sin4.101) -cos4.102)

(9)

Loop diagram via the K N intermediate state Oi the mass matrix and mass eigenstates

+ K N + Oj gives

Mij = A d o . (Oil T IKN) ( K N I T ISj)

(10)

7

lo), = C[(KNIT 101) 101)+ (KNIT 1 0 2 ) IOz)] lo), = C[(KNIT 1 0 2 ) . 101)- (KNIT 1%) . l 0 2 ) l *

*

(11)

where C is a normalization factor Then (KNIT 101) (KNIT 1 0 2 ) - (KNI T 1 0 2 ) (KNIT 101)= 0 Thus (KNIT lo}, = 0; the state 0, is decoupled from K N and its decay into K N is forbidden. The state 0 s with normal hadronic width can escape observation against continuum background. But there are no restrictions on couplings to K*N. Both 1 0,) and 10s) are produced without suppression by K* exchange. Advantages of the two-state model

-

-

(1) Explains narrow width and strong production (2) Arises naturally in a diquark-triquark model where two states have different color-spin couplings (3) Loop diagram mixing via K N decouples one state from K N (4) Broad state decaying to K N not seen ( 5 ) Narrow state coupled weakly to K N produced via K* exchange 4.3.

A variational approach for the Pentaquark Multiplet

Apply the QM Variational Principle to the exact (unknown) hamiltonian H and unknown exact wave function I@+) with three simple a s s ~ m p t i o n s ~ ~ : (1) Assume 0+ and Z-- are pentaquarks uuddg and ssddti (2) Assume 0+and E-- are degenerate in S U ( 3 ) f limit. (3) Assume SU(3) breaking changes only quark masses and leaves QCD color couplings unchanged in H.

(@+I TJ,,HT,,,

-H

lo+) M

m, - m u+

10.(

SV!!~+ BV:!~

10)'

(12)

where the S U ( 3 ) f transformation T,,, interchanges u and s flavors and ShyG and 6 V t ! $ denote the change in the hyperfine interaction under the transformations 3 -+ a and u -+ s respectively. Define a trial wave function

IEiG)

I

Tu*, * 0+) The variational Principle gives an upper bound for M(E--) M ( Z - - ) 5 (=;GI

)H; ;=I

M ( Z - ) -M ( W )

M(E--)

- M(O+)

= (ZiLl H ISV,,)

+ Ad(@')

(13)

- (0'1 H 10)'

5 (0+ITJH,HT,,, - H 10)'

5 m, - mu+ (@+ISV;!:

+ SV:!?

10)' (14)

8

where we have substituted eq. (12) for the SV(3)f breaking piece of H. From quark model hadron spectroscopy and simple assumptions about SU (3) breaking

5 M(A)-M ( N ) lo+)5 0

m, - m u

(@+ISv:!:

(O+l 6V)Z; 1 ) ' 0

5 2 - (~ds=ol6V;>$ I ~ d s = o ),

(15)

Experiment violates both bounds! Is an experiment or one of our assumptions wrong? (i) O+ and Z-- not pentaquarks uud& and ssddii? (ii) O+ and E-- not degenerate in the SV(3)f limit? (iii) Is our SU(3)-breaking model wrong? One possibility is the two-state model. The Of and E-- are not in the same SU(3)f multiplet if the two nearly degenerate diquark-triquark multiplets mix differently.

9

-

5. Heavy flavor pentaquaraks The 0, charmed pentaquark We now use the variataional approach to examine pentaquark states obtained by replacing the 3 by E or other heavy aniquarks in the exact '0 wave functionl6 and define a trial wave function

leyr)3 T,,, . I"+)

(19) We have the same light quark system and a different flavored antiquark. There is the same color electric field and a mass change. The variational principle gives an upper bound for M ( 0 , )

M ( % ) 5 M(@+)+ mc- m,

+ (Vhyp(E))e+- (V~,,(S))Q+

(20)

Hyperfine interaction inversely proportional to quark m a s product, m,

(vhyp(c))

= - * (Vhyp(3)) m,

(21)

Now examine the difference between the mass and the decay threshold AEDN(OJ

= M(O,) - M , - M o

AEKN(@+)

= M(O+) - M N - M K w lOOMeV

AEo,(OC) - AEKN(Q+)= M(O,) - M(0') - M D + MK

AEDN(O,)5 0.7. I (Vhyp(S))e+ I -100 MeV

(23) (24)

Thus if I (Vhyp(S))e+ 15 140 MeV the 0,is stable against strong decays. But the K* - K mass difference tells us that in the kaon

I ( V h y p ( a K ( u S I=

300 MeV

(25)

Is the hyperfine interaction of 3 with four quarks in a O+ comparable to Vh,,(B) with one quark in a kaon? This determines the stability of the 0,.Experiment will tell us about how QCD maka hadrons from quarks and gluons 6. Experimental contradictions about the O+

Some experiments see the pentaquark 17&hers definitely do not". No theoretical model addresses why certain experiments see it and others do not. Comprehensive reviewlg analyzes different models. Further analysis is needed to check presence of speci6c production mechanisms in experiments that see the O+ and their absence in those that do

10

not1*. One possibility is production and decay of a cryptoexotic N*(2400) with hidden strangeness20 fitting naturally into P-wave (ud) diquark-udJ triquark model for the Of. The N* is a (ds) diquark in the same flavor SU(3) multiplet as the (ud) diquark in the Of in a D-wave with the udS triquark. Its dominant decay would produce the O+ in K-O+ via the diquark transition ds + ud K - . Decays like AKand CK would be suppressed by the centrifugal barrier forbidding a quark in the triquark from joining the diquark.

+

Acknowledgments The original work reported in this talk was in collaboration with Marek Karliner. This work was partially supported by the US. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38

References 1. Ya.B. Zeldovich and A.D. Sakharov, Yad. Fiz 4(1966)395; Sov. J. Nucl. Phys. 4, 283 (1967). 2. A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12, 147 (1975) 3. A. D. Sakharov, private communication; H.J. Lipkin, Annals NY Academy of Sci. 452, 79 (1985), 4. M. Karliner and H. J. Lipkin, hep-ph/0307243. 5. M. Karliner and H.J. Lipkin, Phys. Lett. B575, 249 (2003). 6. Y. Nambu, in Preludes in Theoretical Physics, edited by A. de Shalit, H. Feshbach and L. Van Hove, (North-Holland Publishing Company, Amsterdam, 1966), p. 133 7. H.J. Lipkin, Phys. Lett. B45, 267 (1973) 8. Harry J. Lipkin, Nucl. Phys.A625, 207 (1997) 9. R. L. Jaffe, Phys. Rev. Lett. 38, 195 (1977) 10. E.M. Aitala et al.,FERMILAB-Pub-97/118-E, Phys. Lett. B448,303 (1996). 11. H.J. Lipkin, Phys. Lett. B70, 113 (1977) 12. D. Diakonov, V. Petrov and M. V. Polyakov, 2. Phys. A359, 305 (1997), hep-ph/9703373. 13. Harry J. Lipkin, in: ”Physique Nucleaire, Les Houches 1968,” edited by C. de Witt and V. Gillet, Gordon and Breach, New York (1969). p. 585 14. M. Karliner and H. J. Lipkin, Phys.Lett. B586, 303 (2004) hep-ph/0401072. 15. M. Karliner and H. J. Lipkin, hep-ph/0402008. 16. M. Karliner and H.J. Lipkin, hep-ph/0307343. 17. T. Nakano et al. [LEPS Coll.], Phys. Rev. Lett. 91, 012002 (2003), hepex/0301020. See experimental papers in these proceedings for updated list for and against. 18. Marek Karliner and Harry J. Lipkin. hep-ph/0405002 19. Byron K. Jennings and Kim Maltman, hep-ph/0308286. 20. L. G. Landsberg, Phys.Rept.320 223 (1999); hep-exf9910048.

11

CHIRAL SYMMETRY AND PENTAQUARKS

DMITRI DIAKONOV Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA NORDITA, Blegdamsvej 17,DK-2100 Copenhagen, Denmark St. Petersburg Nuclear Physics Institute, Gatchina, 188 300, St. Petersburg, Russia Spontaneous chiral symmetry breaking, mesons and baryons are illustrated in the language of the Dirac theory. It becomes clear why the naive quark models overestimate pentaquark masses by some 500 MeV and why in the Mean Field Approximation to baryons pentaquarks are light.

1. On confinement Confinement of color may be realized in a way that is more subtle than some people think. An example of a subtle confinement is provided by the exactly solvable Quantum Electrodynamics in 1 1 dimensions, also known as the Schwinger model. In the “pure glue” variant of the model, ie. with light “quarks” switched off, there is a trivial linear confining potential between static external charges, since the Coulomb potential is linear in one dimension. However, as one switches in massless or nearly massless “quarks”, the would-be linear confining potential of the imaginary pure-glue world is completely screened: it is energetically more favorable to produce “mesons” than t o pump an infinitely rising energy into the ever-expanding string between the sources. Nevertheless, “quarks” are not observable in the Schwinger model: they are confined despite the absence of gluonic strings or flux tubes between them. Only “mesons” are observable, built of an indefinite number of quark-antiquark (QQ) pairs l . Turning to Quantum Chromodynamics in 3 1 dimensions, there may be certain doubts whether there actually exists a linear rising potential between static quarks in the pure glue version of the theory (the systematic errors for that potential measured in lattice simulations may be underestimated, especially for large separations where it is most interesting 2 ) , however in the real world with light u , d , s quarks color strings or flux tubes between quarks undoubtedly do not exist. It is reassuring that the

+

+

12

screening of the rising potential has just started to be revealed in lattice simulations with light quarks ’. Unfortunately, so far the string breaking has been observed either at non-zero temperatures, or in 2 1 dimensions, or on very coarse lattices: such computations are very time-expensive. It implies that all lattice simulations for the “real” QCD are at present running with inherent strings between quarks, which do not exist in nature! It means that either wrong physics is miraculously cured in the process of the extrapolation of the present-day lattice results to small quark masses, or that the artifact strings are not too relevant for most of the observables.

+

2. Spontaneous Chiral Symmetry Breaking (SCSB) Besides confinement, the other crucial aspect of QCD is the spontaneous breaking of the chiral symmetry: as the result the nearly massless “bare” or “current” u , d , s quarks obtain a dynamical, momentum-dependent mass M ( p ) with M ( 0 ) M 350MeV for the u , d quarks and M 470MeV for the s quark. The microscopic origin of how light quarks become heavy, including the above numbers, can be understood as due to instantons - large fluctuations of the gluon field in the vacuum, needed to make the ~’(958) meson heavy Instantons are specific fluctuations of the gluon field that are capable of capturing light quarks. Quantum-mechanically, quarks can hop from one instanton to another each time flipping the helicity. When it happens many times quarks obtain the dynamical mass M ( p ) . This mass goes to zero at large momenta since quarks with very high virtuality are not affected by any background, even if it is a strong gluon field as in the case of instantons, see Fig. 1. Instantons may not be the only and the whole truth but the mechanism of the SCSB as due to the delocalization of the zero quark modes in the vacuum is probably here to stay.

’.

0.4

1

-0.1

0

I

1

2

3

4

(GeV)

Figure 1. Dynamical quark mass M ( p ) from a lattice simulation6 . Solid curve: obtained from instantons two decades before lattice measurements4 .

13

When chiral symmetry is spontaneously broken, the eight pseudoscalar mesons T ,K , q become light (quasi) Goldstone bosons. In the chiral limit (ie. when the bare quark masses mu M 4,m d M 7, m, M 150MeV are set to zero) the pseudoscalar mesons are exactly massless as they correspond to going along the “Mexican hat” valley, which costs zero energy. For the future discussion of pentaquarks it will be useful to understand chiral symmetry breaking in the language of the Dirac sea of quarks, see Figs. 2,3 a.

Figure 2. Dirac spectrum of quarks before spontaneous chiral symmetry breaking. Since quarks are massless or nearly massless, there is no gap between the positive and negativeenergy Dirac continua.

Figure 3. Spontaneous chiral symmetry breaking makes a mass gap of 2Mc2 in the Dirac spectrum. The vacuum state (no particles) corresponds to filling in all negative energy levels.

The appearance of the dynamical mass M ( p ) is instrumental in understanding the world of hadrons made of u , d, s quarks. Indeed, the normal lowest lying vector mesons have approximately twice this mass while the ground-state baryons have the mass of approximately thrice M . It does not mean that they are weakly bound: as usual in quantum mechanics, the gain in the potential energy of a bound system is to a big extent compensated by the loss in the kinetic energy, as a consequence of the uncertainty principle. Therefore, one should expect the size of light hadrons to be of the scale of 1/M M 0.7fm, which indeed they are. At the same time the size of the constituent quarks is roughly given by the slope of M ( p ) in Fig. 1 , aOne may wonder if the general Dirac theory is applicable for confined quarks. Of course, it is: quarks in the sea are not free but interacting. Mathematically, one can decompose any state in plane waves or any other complete basis. An example of the exact description of confined electrons in the Schwinger model in terms of the Dirac sea is given in the second paper under Ref. l. A more fresh example is provided in Ref.

’.

14

5

corresponding to about fm. Therefore, constituent quarks in hadrons are generally well separated, which is a highly non-trivial fact. It explains why the constituent quark idea has been a useful guideline for 40 years.

3. Mesons In the language of the Dirac spectrum for quarks, vector, axial and tensor mesons are the particlehole excitations of the vacuum, see Fig. 4. In the Dirac theory, a hole in the negative-energy continuum is the absence of a quark with negative energy, or the presence of an antiquark with positive energy. To create such an excitation, one has to knock out a quark from the sea and place it in the upper continuum: that costs minimum 2M in a non-interacting case, and gives the scale of the vector (as well as axial and tensor) meson masses in the interacting case as well.

- - - - - - _ _=_ _ _ _ _ _ _E_=_ -_M _ c_2 positive-energy antiquark

negativc-ei,arg)--quuk

-------------- E = - M c z more dense

__t__

=

Figure 4. Vector are particlehole excitations of the vacuum. They are made of a quark with positive energy and an antiquark with positive energy, hence their mass is roughly 2 M .

positl:ve-energy-

less dense

Figure 5. Pseudoscalar mesons are not particlehole excitations but a collective rearrangement of the vacuum. They are made of an antiquark with positive energy and a quark with negative energy, hence their mass is roughly zero.

For pions, this arithmetic miserably fails: their mass is zero by virtue of the Goldstone theorem. One can say that in pions twice the constituent quark mass is completely eaten up by a strong interaction (which is correct) but there is a more neat way to understand it. Pseudoscalar mesons are totally different in nature from, say, the vector mesons. They are Goldstone bosons associated with symmetry breaking. A chiral rotation costs zero energy: it is the same vacuum state. Pseudoscalar mesons are described by the same filled Dirac sea with negative energies as the vacuum state. They are not particle-hole excitations. If the Goldstone boson carries some energy, it corresponds to a slightly distorted spectral density of the Dirac sea (Fig. 5). The region of the Dirac sea where the

15

level density is lower than in the vacuum, is a hole and corresponds to an antiquark with positive energy. The region with higher density than in the vacuum corresponds to an extra quark with a negative energy, since there are now “more quarks” in the negative-energy Dirac sea. Therefore, the pseudoscalar mesons are “made of’’ a positive-energy antiquark and a negative-energy quark. The mass is hence ( M - M ) = 0. This explains why their mass is zero in the chiral limit, or close to zero if one recalls the small u, d, s bare masses which break explicitly chiral symmetry from the start. The most interesting mesons are the scalar ones: they are chiral partners of the pseudoscalar mesons and their quark organization depends much on the concrete mechanism by which chiral symmetry is broken, in particular on the stiffness of the “Mexican hat”. In the instanton model of the QCD vacuum, the QQ interaction in scalar mesons is much stronger than in vector, axial and tensor mesons. One can then expect the intermediate status of the scalar mesons, between Figs. 4 and 5. In addition, two pseudoscalar excitations from Fig. 5 may resonate forming a scalar resonance. Therefore, scalar mesons can be a mix of a tightly bound positive-energy quark with a positive-energy antiquark, and two positive-energy antiquarks with two negative-energy quarks (and vice versa). Which component prevails is very difficult to predict without a detailed dynamical theory but the data seem to indicate that the lowest nonet (a(SOO), n(800),ao(980), fo(980)) is predominantly a four-quark state (with two negative-energy quarks which make them unusually light!) whereas the second nonet (f0(1370),K,*(1430),ao(1450), fo(1530)) are predominantly “normal” particle-hole mesons, although its singlet member can be already mixed with the gluonium.

4. Baryons

Without spontaneous chiral symmetry breaking, the nucleon would be either nearly massless or degenerate with its chiral partner, N(1535, f-). Both alternatives are many hundreds of MeV away from reality, which serves as one of the most spectacular experimental indications that chiral symmetry is spontaneously broken. It also serves as a warning that if we disregard the effects of the SCSB we shall get nowhere in understanding baryons. Reducing the effects of the SCSB to ascribing quarks a dynamical mass of about 350MeV and verbally adding that pions are light, is, however,

16

insufficient. In fact it is inconsistent to stop here: one cannot say that quarks get a constituent mass but throw out their strong interaction with the pion field. Constituent quarks necessarily have to interact with pions, as a consequence of chiral symmetry, and actually very strongly. I have had an opportunity to talk about it recently and shall not repeat it here. Inside baryons, quarks experience various kinds of interactions: color Coulomb, color spin-spin (or hyperfine) and the interaction with the chiral field mentioned above. It is important to know which interaction is stronger and which one is weaker and can be disregarded in the first approximation. A simple estimate using the running a, at typical interquark separations shows that the chiral force is, numerically, the most strong one. There is also a theoretical argument in its favor. Taking, theoretically, the large N, (the number of colors) limit has been always considered as a helpful guideline in hadron physics. It is supposed that if some observable is stable in this academic limit, then in the real world with N, = 3 it does not differ strongly from its limiting value at N, -+ 00. There are many calculations, both analytical and on the lattice, supporting this view. Therefore, if a quantity is stable in the large-N, limit, one has to be able to get it from physics that survives at large N,. At arbitrary N,, baryons are made of N, constituent quarks sharing the same s-wave orbital but antisymmetrized in color. Baryons’ masses grow linearly with N, but their sizes are stable in It means that one has to be able to get the quark wave function in N, the large-N, limit, and that presumably it will not differ more than by a few percent from the true wave function at N, = 3. When the number of participants is large, one usually applies the mean field approximation for bound states, the examples being the ThomasFermi approximation to atoms and the shell model for nuclei. In these two examples the large number of participants are distributed in many orbitals or shells, whereas in the nucleon all participants are in one orbital. This difference is in favor of the nucleon as one expects smaller corrections from the fluctuations about the mean field in this case. [Indeed, corrections to the Thomas-Fermi approximation are known to die out as l / f i whereas for nucleons they die out faster as l/N,.] If the mean field is the color one, it has to point out in some direction in the color space. Hence the gluon field cannot serve as the mean field without breaking color symmetry. The mean field can be only a color-neutral one, leaving us with the meson field as the only candidate for the mean field in baryons. Given that the interaction of constituent quarks with the chiral field is very strong, one can hope that the baryons’ properties obtained in

17

the mean field approximation will not be too far away from reality. It does not say that color Coulomb or color hyperfine interactions are altogether absent but that they can be treated as a perturbation, once the nucleon skeleton is built from the mean chiral field. Historically, this model of baryons lo has been named the Chiral Quark Soliton Model, where the word “soliton” just stands for the self-consistent chiral field in the nucleon. The name bears associations with the XIX century English gentleman racing after a solitary wave going up the Thames, and is not too precise. Probably a more adequate title would be the Relativistic Mean Field Approximation to baryons. It should be stressed that this approximation supports full relativistic invariance and all symmetries following from QCD. mean field - - - _ _ _ _ . . __ __ _ ____ E =_ +A4

3 valence quarks

mean field

discrete level

Q Q Q

. - - -_._.--._.__

E = -A4

more dense

extra quark with negative energy

less dense

antiquark with positive energy

Figure 6. A schematic view of baryons in the Mean Field Approximation. There are three ‘valence’ quarks at a discrete energy level created by the mean field, and the negativeenergy Dirac continuum distorted by the mean field, as compared to the free one.

Q

Figure 7. Equivalent view of baryons in the same approximation, where the distorted Dirac sea is presented as quark-antiquark pairs. The number of QQ pairs is proportional to the square of the mean field.

If the trial pion field the nucleon is large enough (shown schematically by the solid curve in Fig. 6), there is a discrete bound-state level for three ‘valence’ quarks, Eval. One has also to fill in the negative-energy Dirac sea of quarks (in the absence of the trial pion field it corresponds to the vacuum). The continuous spectrum of the negative-energy levels is shifted in the trial pion field, its aggregate energy, as compared to the free case, being E,,,. The nucleon mass is the sum of the ‘valence’and ‘sea’energies, multiplied by three colors,

M N = 3 (%a~[x(x)] + Esea[x(z)]).

(1)

The self-consistent mean pion field binding quarks is the one minimizing the nucleon mass. If it happens to be weak, the valence-quark level is shallow

18

and hence the three valence quarks are non-relativistic. In this limit the Mean Field Approximation reproduces the old non-relativistic SU(6) wave functions of the octet and decuplet baryons, and there are few antiquarks 12. If the self-consistent field happens to be large and broad, the bound-state level with valence quarks is so deep that it joins the Dirac sea. In this limit the Mean Field Approximation becomes very close to the Skyrme model which should be understood as the approximate non-linear equation for the self-consistent chiral field. Interesting, the famous Wess-ZuminoWitten term which is added “by hands” in the Skyrme model l 3 appears automatically lo. The truth is in between these two limiting cases. The self-consistent pion field in the nucleon turns out to be strong enough to produce a deep relativistic bound state for valence quarks and a sufficient number of antiquarks, so that the departure from the non-relativistic quarks is considerable. At the same time the mean field is spatially not broad enough to justify the use of the Skyrme model which is just a crude approximation to the reality, although shares with reality some qualitative features. Being relativistic-invariant, this approach allows to compute all quark (and antiquark) distributions in the nucleon at low virtuality where they are not accessible in perturbative QCD. Important, all parton distributions are positive-definite and automatically satisfy all known sum rules 14. This is because the account of the Dirac sea of quarks makes the basis states complete. The Relativistic Mean Field Approximation has no difficulties in explaining the “spin crisis” l5 and the huge experimental value of the so-called nucleon a-term - the two stumbling blocks of the naive quark models. Nucleon spin is carried mainly not by valence quarks but by the orbital moment between valence and sea quarks, and inside the sea. The a-term is experimentally 4 times (!) bigger than it follows from valence quarks because, again, the main contribution arises from the Dirac sea to which the 0-term is particularly sensitive. On the whole, the picture of the nucleon emerging from the simple Eq.(l) is amazingly coherent and so far adequate.

5 . Pentaquarks

Based on this picture, Victor Petrov, Maxim Polyakov and I predicted in 1997 a relatively light and narrow antidecuplet of exotic baryons 17; this prediction largely motivated the first experiments. Both circumstances lightness and narrowness - are puzzles for naive quark models.

19

After the first announcements of the observation of the exotic 8+ signal in the y12C 18, K+Xe 19, yd 2o and yp 21 reactions, several theoretical proposals appeared on how to understand pentaquarks from a traditional constituent-quarks-only viewpoint 22. There are basically two constituent quark models of pentaquarks: one of them emphasizes the string confinement and color hyperfine interactions, the other, which I shall call the Glozman-Riska (GR) model 23, stresses the pseudoscalar exchanges as the main constituent quark interaction. Both approaches claim certain successes in explaining the properties of the groundstate baryons and of their excitation spectrum. It is interesting that in order to achieve it in the GR model one needs to reduce the string tension by a factor of 5 (!) as compared to that given by the lattice simulations in the pure glue world, which I find very natural - see the beginning. If one has a quark model at hand with the parameters fitted in the normal baryon sector, one can try to apply it to pentaquarks. This has started to be done, and the results are, to my mind, remarkable. The most clear calculation so far is by Florence Stancu 22 in the GR model. Having assumed a natural color-flavor-spin-space symmetry of the pentaquark, she has found the best variational wave function using the model parameters fixed from the 3Q baryons, and obtained the Q+ mass. It turns out to be 510MeV heavier than 1540 MeV. An evaluation of the 8+ mass in the JaffeWilczek model with extreme diquark correlations has been recently carried out in Ref. 24 assuming string dynamics between quarks, probed in the usual baryons. The authors also get 8 ’ s mass about 0.5GeV heavier than needed if one assumes massless diquarks and still heavier if diquarks are not exactly massless b. It is easy to understand this typical half-a-GeV overestimate of the Q+ mass in the constituent quark models. One sums up five quark masses each about 350 MeV, adds 150 MeV for strangeness and gets something around 1900 MeV. In addition there is some penalty for the p-wave, assuming the 8 has positive parity. It gives more than 2 GeV. This is the starting point. Then one switches in his or her favorite interaction between quarks which may reduce the starting mass, but has to pay back the kinetic energy. Owing to the uncertainty principle, these two usually cancel each other to

bA direct lattice measurement of the diquark propagator has shown that its mass is bigger than twice the constituent quark mass about 700 MeV and hence diquarks are not bound 2 5 . It would be important t o repeat this study with the current more powerful technique.

20

a great extent, even if the binding force is strong. Therefore, the @+ mass of about 2 GeV is a natural and expected result in any constituent quark calculation. The fundamental difference with our approach to pentaquarks is seen from Fig. 6,7. The fourth quark in the 8+ is a higher density state in the Dirac sea: it has a negative energy E = One does not sum five quark masses but rather (3M M - M ) = 3M to start with. This is because the extra QQ pair in the pentaquark is added not in the form of, say, a vector meson where one indeed adds 2M but in the form of a pseudoscalar Goldstone meson, which costs nearly zero energy. The energy penalty for making a pentaquark is exactly zero in the chiral limit, had the baryon been infinitely large. Both assumptions are wrong but it gives the idea why one has to expect light pentaquarks. In reality, to make the 8+ from the nucleon, one has to create a quasi-Goldstone K-meson and to confine it inside the baryon of the size 2 1/M. It costs roughly

+

m ( 8 ) - m ( N )M

d

-Jm.

G 5 Jw 606 MeV. =

(2)

Therefore, one should expect the lightest exotic pentaquark around 1546 MeV. In fact one also adds an indefinite number of light pions to cook up the O + . In the Dirac language of section 3, the naive quark models attempt to make a pentaquark by adding a particle-hole excitation or a vector meson to the nucleon whereas in the world with the spontaneous chiral symmetry breaking there is a cheaper possibility: to add a collective excitation of the vacuum, i. e. the pseudoscalar meson(s). Some analogy can be found in the 0++ mesons. There is definitely a large 4Q component, say, in the ao(980). Naively, that would imply a 4M = 1400MeV mass but it is 400 MeV lighter, actually close to 2 m ~ . This hints a resolution: the four quarks of the a0 meson are in the form of two quasi-Goldstone bosons where all four M’s are eaten up. Q+ is not a bound state of five good old constituent quarks: such bound states, if they exist, necessarily have a mass about 2 GeV. At the same time it is not a K N molecule - first, because its size is only about larger than that of the nucleon 26, second, because it is an excitation of the pion field as well, third, because its coupling to the K N state is very weak. It is a new kind of a state. What is the giant resonance or a rotational state in a nucleus made of? If one wants a bound-state description of the 8 at all cost, the closest concept I can think of is a superposition of

21

K N , K T N , KmrN ... (including the scalar I ~ Nbound ) states '. However, it is simpler to think of the 8+ as of a rotational excitation of the mean chiral field in the nucleon 17. It does not mean that one needs to abandon the quark language altogether: the 8+ has a definite 5Q-component wave function 12. I thank the organizers of the Pentaquarks-2004 for support and hospitality, and Victor Petrov for numerous discussions. This work has been supported in part by the US Department of Energy under contract DEAC05-84ER40150. References 1. A. Casher, J. Kogut and L. Susskind, Phys. Rev. D9, 232 (1973); G.S. Danilov, I.T. Dyatlov and V.Yu. Petrov, Nucl. Phys. B174,68 (1980).

2. D. Diakonov and V. Petrov, Phys. Scripta 61,536 (2000). 3. F. Karsch, E. Laermann and A. Peikert, Nucl. Phys. B605, 579 (2001), hep-lat/0012023; C.W. Bernard et al., Phys. Rev. D 64,054506, 074509 (2001), hep-lat/0104002; A. Duncan, E. Eichten and J. Yoo, Phys. Rev. D68,054505 (2003); H.D. Trottier and K.Y. Wong, hep-lat/0408028. 4. D. Diakonov and V. Petrov, Phys. Lett. B147,351 (1984); Nucl. Phys. B272, 457 (1986); for a recent review see D. Diakonov, Prog. Part. Nucl. Phys. 51 (2003) 173, hep-ph/0212026. 5. G. 't Hooft, Phys. Rev. Lett. 37,8 (1976). 6. P. Bowman, U. Heller, D. Leinweber, A. Williams and J. Zhang, Nucl. Phys. Proc. Suppl. 128,23 (2004), hep-lat/0403002. 7. W. Broniowski, B. Golli and G. Ripka, Nucl. Phys. A703,667-701 (2002), hep-ph/0107139. 8. N.N. Achasov, hep-ph/0410051. 9. D. Diakonov, hep-ph/0406043; to be published by World Scientific in: Continuous Advances in QCD-2004, Minneapolis, May 12-16, 2004, hep-ph/0408219. 10. D. Diakonov and V. Petrov, JETP Lett. 43,75 (1986) [Pisrna Zh. Eksp. Teor. Fir. 43,57 (1986)l; D. Diakonov, V. Petrov and P.V. Pobylitsa, Nucl. Phys. B306,809(1988); D. Diakonov and V. Petrov, in Handbook of QCD,M. Shifman, ed., World Scientific, Singapore (2001), vol. 1, p. 359, hep-ph/0009006. 11. E. Witten, Nucl. Phys. B156,269 (1979). 12. D. Diakonov and V. Petrov, to be published in Annalen der Physik, hep-ph/0409362. 13. E.Witten, Nucl. Phys. B160,433 (1983).

CTheidea of the KnN and n(800)N bound states has been put forward in Ref. pilot study shows that there is a mild attraction.

27;

a

22 14. D. Diakonov, V. Petrov, P. Pobylitsa, M. Polyakov and C. Weiss, Nucl. Phys. B480,341 (1996), hep-ph/9606314; Phys. Rev. D56,4069 (1997), hep-ph/9703420. 15. M. Wakamatsu and H. Yoshiki, Nucl. Phys. A524,561 (1991). 16. D. Diakonov, V. Petrov and M. Praszalowicz, Nucl. Phys. B323,53 (1989). 17. D. Diakonov, V. Petrov and M. Polyakov, 2. Phys. A359, 305 (1997), hep-ph/9703373;hep-ph/0404212. 18. T. Nakano (LEPS Collaboration), Talk at the PANIC 2002 (Oct. 3,2002,Osaka); T. Nakano et al., Phys. Rev. Lett. 91,012002 (2003), hep-ex/0301020. 19. V.A. Shebanov (DIANA Collaboration), Talk at the Session of the Nuclear Physics Division of the Russian Academy of Sciences (Dec. 3,2002,Moscow); V.V. Barmin, A.G. Dolgolenko et al., Phys. Atom. Nucl. 66,1715 (2003) [ Yad. Fzz. 66,1763 (2003)],hep-ex/0304040. 20. S. Stepanyan, K. Hicks et al. (CLAS Collaboration), Phys. Rev. Lett. 91, 252001 (2003),hep-ex/0307018. 21. V. Kubarovsky et al. (CLAS Collaboration), Phys. Rev. Lett. 92,032001 (2004), hep-ex/0311046. 22. F1. Stancu and D.-0. Riska, Phys. Lett. B575,242 (2003), hep-ph/0307010; F1. Stancu, Phys. Lett. B595, 269 (2004), hep-ph/0402044; M. Karliner and H. Lipkin, Phys. Lett. B575,249 (2003),hep-ph/0402260; R.L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003), hep-ph/0307341; L. Glozman, Phys. Lett. B575,18 (2003), hep-ph/0308232;B. Jennings and K. Maltman, Phys. Rev. D69,094020(2004),hep-ph/0308286;C.E. Carlson, C.D. Carone, H.J. Kwee and V. Nazaryan, Phys. Rev. D70,037501 (2004), hep-ph/0312325. 23. L. Glozman and D.-0. Riska, Phys. Rep. 268,263 (1996). 24. I.M. Narodetskii, C. Semay, B. Silvestre-Brac and Yu.A. Simonov, hep-ph/0409304. 25. F. Karsch et al., Phys. Rev. D58,111502 (1998), hep-lat/9804023. 26. M. Polyakov, in: Proceedings of Nstar-2004. 27. P. Bicudo and G.M. Marques, Phys. Rev. D69, 011503 (2004), hep-ph/0308073; F.J. Llanes-Estrada, E. Oset and V. Mateu, Phys. Rev. C69,055203 (2004), hep-ph/031120; see also the contributions by Bicudo and by Oset to these Proceedings.

23

STUDY OF THE O+ AT LEPS

T.NAKANO RCNP, Osaka University 10-1 Mihogaoka, Ibamki, Osaka, 567-0047, Japan E-mail: nakano Orcnp.osaka-u.ac.j p The photon beam at the LEPS facility is produced by backward-Compton scattering of laser photons from 8 GeV electrons at the SPring-8. The status and prospects of the experimental study on the O+ at LEPS are reported.

1. The first Q+ search experiment

A hadron with a combination of qqqqq is a pentaquark, and it is called exotic if the flavor of the antiquark is different from those of the other quarks. The 8+ is an exotic pentaquark with a quark configuration of uuddi?. In 1997, Diakonov, Petrov and Polyakov predicted the mass of the Of to be 1530 MeV with a narrow width of 15 MeV by using the chiral quark soliton model Motivated by this prediction, we searched for an evidence for the 8+ in an existing experimental data. The experiment was carried out by using a laser-electron photon (LEP) beam which was generated by Backward-Compton scattering of laser photons with the 8-GeV electrons. The charged particles produced from photo-nuclear interaction were then momentum analyzed by a forward angle spectometer and kaons were identified by a time-of-flight measurement '. In the Q+ search analysis, we selected K+K- pair events produced in a plastic scintillator located 9.5 cm downstream from the liquid-hydrogen (LH2) target. The main physics background events due to the photoproduction of the q5 meson were eliminated by removing the events in the q5 peak in the invariant K + K - mass distribution. The Fermi motion corrected missing mass of the N(y,K+K-)X reaction was calculated by assuming that the target nucleon has zero momentum. A prominent narrow peak at 1.54 GeV/2 is found. The estimated number of the events above the background level is 19.0, which corresponds to a Gaussian significance

-

'.

-

24

of 4.6 u. The narrow peak indicates the existence of an S = +1 resonance which may be attributed to the exotic 5-quark baryon proposed as the Q+. Soon after a preliminary result on the was announced by the LEPS, many experiments found an evidence for the Q+ by mainly analyzing old data. There are some inconsistencies in the measured masses, which are larger than the experimental resolutions. Although some fluctuation in the mass measurement has not been rare for a newly discovered particle, we should be cautious since it is a common characteristic of disappeared narrow resonances in the past.

2. Counter-evidences for the Q+

=--

HERA-B collaboration searched for both Q+ and in proton-induced reactions on C with a 920 GeV/c beam 3 . They found no signal of the pentaquarks although the A(1520) and So peaks were clearly seen in the invariant mass spectra. The upper limit of relative yield ratios were (Q+)/(A(l520)) < 0.02 at the 95 % confidence level. There are several other experiments which have searched for the pentaquark but found no Those experiments were carried out at high energy with evidences a high statistics and a good mass resolution, typically 2 3 MeV for the Q+ mass. Although the most of the experiments search for the Q+ in the K,-proton invariant mass inclusively, the Q+ peak should be identified as a very narrow peak in the mass distribution above a large number of combinatory background events. If the pentaquarks exist, their production at high energy must be heavily suppressed with respect to normal baryons. Clearly it has become the most important issue to confirm the existence or non-existence of the @+ experimentally.

-

4,576,798.

3. Further study at

LEPS

We performed a new experimental search for the O+ using a 15cm-long liquid deuterium target in 2003. The most essential cut for the signal event selection for both the Q+ and the A(1520) was a q5 exclusion cut. The photon-energy dependent cut point was determined by using a Monte Calro sample and the h(1520) events from a liquid hydrogen target, which were taken with the same detector setup. Contributions from coherent K+K- productions from a deuteron were removed by rejecting events with a K+K- missing mass consistent with a deuteron mass. Other cuts which were used in the previous analysis were either relaxed or removed. A preliminary analysis shows a peak at 1.53 GeV in the Fermi motion corrected

-

25

K - missing mass distribution. To check if the peak is not generated by the event selection artificially, the same selection cuts were applied to 1) phase space KKN Monte Calro events, 2) phi Monte Calro events which were generated with realistic spin density matrix elements, and 3) the LH2 data. No narrow peak nor strong enhancement was observed. To estimate the background under the peak, a mixed event analysis was carried out, where a K+, a K - , and a photon were picked up from different events. This event mixing technique works well for inclusive measurements with a high multiplicity but generally it is not applicable to an exclusive reaction since the momentum and energy are not conserved in a mixed event. However, in our case a cut on the K+K- missing mass to be consistent with a nucleon mass forces the momentum and energy conservations approximately satisfied. By analyzing the phase-space Monte Calro data, we have confirmed the event mixing reproduces the original missing mass spectrum only by using a small number ( m 1000 events) of the sample. The number of events in the peak was about 90 with the S/N ratio of 0.4 above the smooth background level which was estimated by using a mixed event technique. The peak structure was not seen in MC events and the LH2 events. Further analysis is in progress to check if the peak is not generated artificially by event selection cuts, detector acceptances, kinematical reflections, or their combinations. We plan to take more data with a time projection chamber which covers a large angle region around the target and the 3 GeV LEP beam in order to study the 8+ in a wider kinematic region. The new experiment will provide us with a better understanding of the background processes and may reveal a production mechanism of the Q+ if it exists. Together with high statistics experiments which are on going or scheduled at Jlab, KEK and other labs, the question on the existence of the O+ will be answered in the near future.

References 1. D. Diakonov, V. Petrov, and M. Polyakov, Z. Phys. A359,305 (1997). 2. T. Nakano et al. (LEPS Collaboration), Phys. Rev. Lett. 91, 012002 (2003). 3. K.T. Knopfle, M. Zavertyaev, and T. Zivko (HERA-B Collaboration), J. Phys. G30, S1363 (2004). 4. J.Z. Bai et al. (BES Collaboration), Phys. Rev. D 70, 012004 (2004). 5. C. Pinkenburg (for the PHENIX Collaboration), J. Phys. G30, S1201 (2004). 6. M. Longo et al. (Hyper-CP Collaboration), arXiv:hep-ex/0410027.

26

7. E. Gottschalk (FNAL-E690 Collaboration), in the presentation of this workshop. 8. M.J. Wang (CDF Collaboration), in these proceedings.

27

X(3872) AND OTHER SPECTROSCOPY RESULTS FROM BELLE

K. ABE KEK, Tsukuba 305-0801, Japan E-mail: kazuo. [email protected]

I report recent development on the measurements and interpretation of charmonium-like state X(3872) and D, J mesons. Properties of X(3872) that were measured up to now are inconsistent with expectations for charmonium states. Masses of the two new D,J states, which are significantly lower than potential model expectations, remain unresolved.

1. Charm Production at B factories The B factories, which operate at the e+e- center of mass energy corresponding to the T(4S) resonance and serve as an intensive source of B meson anti B meson pairs, are at the same time powerful charm factories. A large number of charmed mesons and baryons are produced simultaneously with B meson pairs through continuum e+e- +. cC process (see Fig.l(a)) where the cross section is roughly 30% larger than the B meson pair production, and through B meson decays where 99% of the time they decay into charmed particles (see Fig.l(b) and (c)).

Figure 1. Diagram for charm production in the e+e- + CC continuum process (a), B meson decay producing charmed mesons (b), and charmonium (c).

28

2. Charmonium-like state X(3872)

The X(3872) is produced in the B+ + XK+ decay, and decays into r+n-J/$~.Fig. 2 shows the mass difference MT+T-J/$- M J / $ distribution reported by Belle '. A small peak at 0.775 GeV/c2 in addition to 0.589 GeV/c2 peak corresponding to well known @'(3686) is clearly visible. N

0.40

0.80 1.oo M(z+x-J/v)- M(J/v) (GeV/c2)

0.60

1.20

Figure 2. Mass difference M=+=- Jl+ - MJ,+ distribution

From a simultaneous fit to M T + T - J / $ lMbc, and A E , the mass and width are determined as M X = 3872.0 f 0.6 f 0.5MeV and r < 2.3MeV (9O%CL). The Mbc and A E are commonly used kinematical variables for reconstructing the B meson decays, and defined as Mbc = and A E = Ebeam - EB, where p~ and EB are the vector sum and summed energy of the B meson decay products, and Ebeamis the beam energy, all in the e+e- c.m. system. This state was since confirmed by CDF 2 , DO 3 , and BaBar 4. A weighted average of the reported masses is 3871.9 f 0.5 MeV. The width is consistent with the detector resolution in all experiments, but only Belle quoted the upper limit. A weighted average of the product branching fraction of Belle and BaBar gives BT(B- -+ K-X) x BT(X + r+r-J/$) = (1.3 *0.3) x All four experiments observe that the rr invariant mass distribution tends to cluster near p, although not conclusive.

d-

29

Figure 3. Distributions for Mbc (a), M T + r - J , $ (b), and A E (c) for the X(3872) candidate events.

3. What is it? The X(3872) is charmonium-like since it decays to r+r- J/+. Its width is rather narrow. Although its mass is above DD threshold (3740 MeV), decays into DD are not seen 5 , indicating that the X(3872) -+ DD decay is either forbidden or suppressed. Being produced in exclusive B+ 4 K+X(3872) decays makes the asignment of high J values for the X(3872) not likely since the high J must cancel with corresponding high orbital angular momentum (large contrifugal barrier). Fig. 4 shows the charmonium spectrum based on the potential model of Godfrey and Isgur '. Dotted lines at 3740 MeV and 3871 MeV indicate DD and DD* mass threshold, respectively. As a candidate state for the X(3872), we reject the states with J p c = O++, 1--, 2++ since they are allowed to decay into DD. We also do not consider the states with orbital angular momentum equal or greater than 3 (except $3). Possibles states are then 2lP1 (h:, l+-),1 3 0 2 ($2, 2--), 1303 ($3, 3--) for C = -1, and I'D2 ( ~ ~2-+), 2 , 23P1 (xL1,1++), and 3lS0 ($, 0-+) for C = +l. Fig. 5 shows a distribution for cos tlJ,+, cosine of an angle of J / $ in the X(3872) rest frame with respect to the motion of X(3872). If X(3872) is h', (l+-), this distribution should be 1 - cos2OJ/+, which is inconsistent with the data. Wigner-Eckart theorem requires r(+n r+n-J/$), I'($J~ t r + r - J / $ > , and I'($"'(3770) + n+n-J/$~)are all equal. We conservatively set I?(+" + r + r - J / $ ~ >< 130keV based on CLEO result (< 55 keV 9O%CL) and BESS I1 result (80 f 35 keV). Potential model calculations give l?(& -+ yxcl) = 210 keV 7,8. Thus we expect l?(& 4 yxcl) to be a few times larger than r(&t r+r- J/$J).

30

4000 3800

!i 3600

Figure 4. Charmonium spectrum

Fig. 6 shows the Mbc and AE distributions for the B+ -+ K+yXcl candidates where yxCl invariant mass lies in the region of $' ((a) and (b)), and X(3872) ((c) and (d)). While a clear signal is seen for the $' case, no signal is seen for the X(3872) case. We set a limit B r ( X y X c l ) / B r ( X --f d 7 r - J / $ ) < 0.89 (9O%CL)which is inconsistent with the $2 asignment for X(3872). Similarly we obtain B r ( X -+ y x c 2 ) / B r ( X x+x-J/$) < 1.1(9O%CL) where the potential model expectations are 2 3, therefore disfavoring the $3 asignment for X(3872). A potential model calculation gives r p 3 P 1 -+ y J / $ ) 11keV 7. Since the xLl(l++) x+7rTTJ / $ decay is isospin violating, we expect its width is similar to r($' -+ x0J/$) 0.3keV, and therefore expect rp3P1 y J / $ ~ > / r ( 2 ~-+ P lx+x-J/$) 30. However, data shows this ratio to be less than 0.4(90%CL). disfavoring the xLl asignment for X(3872). -+

--f

N

--f

-

N

-

--f

31

.,

0.00

0.25

0.50

0.75

1.oo

J

IWS~J,

Figure 5. Helicity distribution for X(3872) ---t nnJ/$ decay. Solid line shows expectation for hh hypothesis. Dotted line is background estimation.

4. Possibility of

DD* molecular state

The absence of suitable charmonium candidate, together with the observation that the X(3872) mass is almost exactly equal to the sum of D and 0.masses, brings an exciting possibility of X(3872) being a loosely bound moleculelike state of 00.. The idea itself has beeen around since 1970’s. Here I mention only a few examples where the experimental consequences are clearly identifiable. Tornqvist points out that the inter-mesonic force mediated by single pion exchange can become attractive for Jpc = 1++,0-+ states ’. Voloshin suggests that if X(3872) = OD*f OD*, interference between D*O and D3* decays should show up in the DoDo7roand DoDoy decay rates lo. We must find some experimental clue for further investigation of this idea.

5. D,~(2317) and 0,~(2457) The 2003 discoveries of two narrow (cs) mesons brought excitement because the measured masses turned out to be considerably lower than the potential model calculations. However, subsequent measurements are consistent with

3 2

Figure 6. M b c and A E distributions for B+ -+ K+yxcl candidates. (a), (b) and (c), (d) are for -+ yxcl,X(3872) + yxcl candidates, respectively.

+'

these two states being the missing O+ and 1+ members of L = 1 multiplet. Their narrow widths can be well explained by the observed lower masses because the states lie below the D,~(2317)-+ D K and D,j(2457) -+D*K transition thresholds and the D,~(2317)+ Ds7ro,0 , ~ ( 2 4 5 7 )+ D,*.rrotransitions are isospin-violating which are known to be highly suppressed. While the discrepancy of the masses with the potential models remains as a puzzle, the mass splitting can be explained by the chiral symmetry model l l . Its prediction that the D,~(2317)and D,j(2457) are the (O+, 1+) parity multiplet partner of the ground state (0-, 1-) multiplet, and have mass splitting of A M m ~ / 3 is, in good agreement with the observation (Fig. 7). Recently, the SELEX experiment at Fermilab reported another D,J state with 2632 MeV mass and decays into DZv and D°K+ modes 13. We have searched such state in the B meson decay and found none. We set an upper limit as ( ~ ( D ~ ~ ( 2 6 3x2Br(2632 )) + DoK+)/a(Dsj(2573)) x Br(2573 + DoK+) < 1.1%(9O%CL) where the corresponding SELEX ratio is 0.56 f 0.27. Search for D,Qmode is under study. N

33

potential

spin-orbit

L= 1

&=3/2 j p 2

tensor-force 2+ 2590 +2560 _--+2550 ----__ o+ 2480 1+ 2457

\

m,

large small

0- 1969

m c finite

mq finite

mq Figure 7. Predicted masses for L = 1 states in the potential model, and the masssplitting calculation between (0-, 1-) and ( O + , I+) multiplets.

6. Ordinary D mesons

+

Unlike the D,J mesons, the (O+, 1+) -+ (O-] 1-) T transitions in the ordinary D mesons are no longer isospin-violating, and we expect wider (O+, 1+) states. These two states were observed by Belle 12. The masses and widths, MD;o = 2308 f 17 f 15 f 28, r D ; o = 76 f 21 f 18 f 60, M ~ , O = 2 4 2 7 f 2 6 f 2 0 f 1 5 , rD;O = 384t:0,7&24f70, are in good agreement with the potential model expectation. Here the units are in MeV and the third errors come from PDG errors. 7. Anomalous (A,+$ structure in B - -+

AZfirr-

Belle observes a resonance-like structure in the A$?j mass distrubution in the B- -+ AZm- decay. Fig. 7(a) shows the MiT- vs MitT- Dalitz plot for the B- + A:?~T- signal candidates. Here the regions labeled by the numbers are, 1 for B- -+ C:(2455)?j, 2 for B- + C:(2520)?j1 3 for B- -+ A:A--(1232), 4 for B- -+ A$A--(1600), 5 for B- + A:A--(2420). For this study, we use the region labeled as 6. The A:p mass distrubution is shown in Fig. 7(b). A resonance-like structure is clearly visible near 3.3 GeV/c2. Fits give M ( A $ p ) = 3.32 f 0.02 GeV which has 6.10 statistical significance and rBw = 0.15 f 0.05 GeV.

34

(a) Figure 8. (a) M:=-

vs M,’&

Dalitz plot for the B-

---t

AfW- signal candidates,

(b) The Afp invariant mass distribution.

8. Summary

We face a difficulty for asigning X(3872) to vacant charmonium states. Next steps should be determination of J p c by complete angular analysis and J p c of dipion system. Establishing missing charmonium states and improved measurements of radiative decays for X(3872) are also important. We must search for any clue for “molecule” ideas. New heavy D,J state reported by SELEX and a resonance-like structure in A Z p need further investigation.

References 1. Belle Collaboration, S.K. Choi, et al. Phys. Rev. Lett. 91,262001 (2003). 2. CDF Collaboration, G. Bauer, et al. hep-ex/0312021. 3. DO Collaboration, V.M. Abazov, et al. hepex/0405004. 4. BaBar Collaboration, B. Aubert, et al. hep-ex/0406022. 5. Belle Collaboration, R. Chistov, et al. hep-ex/0307061. 6. Godfrey and Isgur, PRD 32 (1985)189. 7. Barns, Godfrey, PRD 69, 054008 (2004). 8. Eichten, Lane, Quigg, PRD 69,094019 (2004). 9. N.A. Tornqvist, Phys. Lett. B590,209 (2004). 10. M.B. Voloshin, Phys. Lett. B579, 316 (2004). 11. Bardeen, Eichten, Hill, PRD 68, 054024 (2003). 12. Belle Collaboration, K. Abe, et al. PRD 69, 112002 (2004). 13. SELEX Collaboration, A.V. Evdokimov, et al. hep-ex/0406045.

35

SEARCH FOR EXOTIC BARYON RESONANCES IN PP COLLISIONS AT THE CERN SPS

K. KADIJA FOR THE NA49 COLLABORATION Rudjer Boskovic Institute, Bijenicka cesta 54, 10002 Zagreb, Croatia E-mail: Kreso. [email protected]

The results of resonance searches in Z-a-, 8-?r+, $?rand %?r+ invariant mass spectra in proton-proton collisions at 6 =17.2 GeV are presented. A narrow state was observed in S-n- spectra with mass of 1.862 f 0.002 GeV/c2 and width below the detector resolution of about 0.018 GeV/c2. This state is identified as a candidate for the hypothesized exotic 8;- baryon with S = -2, I = $ and a quark content of (dsdsii). =-a+ and the corresponding antiparticle spectra show an indication of enhancements at the same mass.

1. Introduction

Several experimental groups' have recently observed a narrow resonant state in the nK+ and pKg invariant mass spectra near 1540 GeV/c2. This strangeness S = +1 baryon has been identified as a candidate for the € the lightest member of an antidecuplet of pentaquark states. Various models have been put forward to explain this state and the structure of the multiplet that contains it (see' as an example for the chiral soliton models and3 as an example for the correlated quark models). The pentaquark antidecuplet also contains an isospin quartet of S = -2 baryons. This isospin multiplet contains two 25s with ordinary charge assignments (EE, Z;) in addition to the exotic states Ec (uussd) and E,5-(ddss.ii). The NA49 results of a search for the E;- and 2: states and their antiparticles in proton-proton collisions at &=17.2 GeV were published in the article4. In these proceedings we will address some issues that were not included in4.

4

36

2. The Experiment

NA4g5 is a fixed target, large acceptance hadron experiment at the CERN SPS. The central part of the detector are four large volume Time Projection Chambers (TPC) which provide precise tracking of charged particles and particle identification through a measurement of specific energy loss (dE/dx). After careful calibration, 3-6% dE/dx resolution was achieved depending on the reconstructed track length. Two of the TPCs (VTPC1 and VTPC2) are operated inside superconducting dipole magnets, allowing momentum determination from the track curvature. Typical values for the total momentum resolution are d p / p 2 = 0.3 - 7.0 (GeV/c)-l depending on the track length and topology. The interactions were produced with a beam of 158 GeV/c protons impinging upon a cylindrical liquid hydrogen target of 20 cm length and 2 cm transverse diameter. The measured trigger cross section was 28.2 mb of which 1 mb was estimated to be elastic scattering. Thus the detector was sensitive to most of the inelastic cross section of 31.8 mb.

-

3. Analysis and results

The data sample consists of about 6.5 M events. For each event the primary vertex was determined. Events in which no primary vertex was found were rejected. To remove non-target interactions the reconstructed primary vertex had to lie within f 9 cm in the longitudinal (2) and within f 1 cm in the transverse (z,g)direction from the center of the target. These cuts reduced the data sample to 3.75 M events. was searched for through its The exotic 8;- hyperon with I , = characteristic decay topology: strong decay to Z-T-, followed by two weak AT- and A PT-. Of the other three members of the decays 8predicted isospin quartet only the I , = state 8; is observable in the NA49 experiment via the Z-T+ decay channel. Also the corresponding antibaryon states and s;, are expected to be produced and can be detected via the G+T+ and Z+T- decay channels, respectively. The first step in the analysis is the reconstruction of V o and cascade candidates, by locating their decay vertices. The detailed description of the selection procedure can be found in4. To study the inclusive production of z5 , 8; and their antiparticles, the 8- and candidates were selected within f 0.015 GeV/c2 of their nominal masses. This reduces the data sample to 1640 events containing one 8- and 551 events containing one

-

-

4 4

s;+

--5+ Y

.

s+

37

2.4

2.6

28

M(EK)[G~V/C*] Figure 1. E-T- and E-n+ invariant mass spectra The cuts are explained in the text. The insert shows background subtracted =-a+ spectra with the result of Gauss fit to the E(153O)O.

To search for E;- (E:) the selected E- candidates were combined with primary R - (n+) tracks. To select ns from the primary vertex, their lbzl and [barlahad to be less than 1.5 cm and 0.5 cm, respectively, and their dE/dx had to be within 1.5 u of their nominal Bethe-Bloch value. Figure 1 shows the resulting F n - and S-n+ invariant mass spectra. The shaded histograms are the mixed-event background, obtained by combining the Eand the n candidates from different events. With these (loose) cuts a peak at M 1.86 GeV/c2 is visible in E-n- invariant mass spectrum. In the E-nf invariant mass spectrum the only clearly visible resonance is the E(153O)O. The mass from the Gauss fit (insert in Figure 1) agrees with the nominal (PDG) mass value for the E(1530)", suggesting a systematic error on the absolute mass scale below 0.001 GeV/c2. For further analysis, several additional cuts were applied. It was found "Extrapolated track impact position in the z (magnetic bending) and y (non-bending) directon at the main vertex.

38

-0 A0 d +d

M(Ex) [GeV/c2]

+

Figure 2. The combined 2-n- + 8+nf and =-a+ B+n- spectra after the final cut. The insert shows background subtracted spectra with the result of Gauss fit.

from simulation that the background below the assumed peak at M 1.86 GeV/c2 can be reduced by the restriction 8 > 4.5" (with 8 being the angle between the E and n direction calculated in the laboratory frame). In addition to this cut, a lower cut of 3 GeV/c was imposed on the n+ momenta to minimize the large proton contamination, and a lower cut on the dE/dx of the n+ at 0 . 5 below ~ the nominal Bethe-Bloch value to reduce K + contamination. Figure 2 shows the combined 8-n- ?n+ and E-n+ ?nspectra with these additional cuts. The enhancements around 1.86 GeV/c2 are now seen in all cases. Gauss fits t o the background subtracted spectra of the Ec- and its antiparticle and 8: and its antiparticle (shown as insert in Figure 2) yield peak position of 1.862 f0.002 GeV/c2 and 1.864 f0.005 GeV/c2, respectively. The robustness of the 8;- peak was investigated by changing the width of accepted regions around the nominal E- and A masses, by varing the dE/dx cut used for particle selection, by selecting tracks with different number of points, by using different b, and b, cuts, as well as by investigat-

+

+

39 Table 1. Rejected invariant mass ranges for combinations of the negative primary tracks with positive tracks.

A-

7r+

K+

P

497.6 f 10.0 MeV/c2

892.0 f30.0 MeV/c2

1115.7 f 10.0 MeV/c2 1520.0 f 15.0 MeV/c2

769.0 f 60.0 MeV/c2

K-

892.0 f 30.0 MeV/c2

1019.0 f 10.0 MeV12

p

1115.7 f 10.0 MeV/c2

1520.0 f 15.0 MeV/c2

ing events with different topologies. In all cases the peak at 1.86 GeV/c2 proved to be robust. Further, the influence of resonances, including the possibility of particle misidentification was carefully investigated. The events that contain E-T+ candidates within f 1 0 MeV/c2 around the nominal E(1530)" mass were rejected. The selected negative pions were combined with all positive particles, under various mass hypotheses. If the resulting invariant mass fell within the ranges indicated in Table 1 the particle was excluded from the analysis. Figure 3 shows the invariant mass spectra obtained after these additional cuts. The peak in E-s- spectra remains clearly visible. However, the E-T- spectrum is very sensitive to the quality of the Eand T - selection, and to see resonances these should have only a minimal amount of contamination. This is demonstrated in Figure 5, which shows the E-T- invariant mass for progresively higher 2- and T - purity: Figure 5a shows the results after selecting E- and A candidates within 15 MeV around their nominal invariant mass, and the proton and T - dE/dx within 3 (T of their nominal Bethe-Bloch values; Figure 5b shows the results when the E- is additionally purified with its (b,1<2 cm and Ib,l0.5 cm; Figure 5c shows the results after selecting the primary T - candidates with their Ib,l
40

Figure 3. 2-a- invariant mass distribution (a) events with Z(1530)" candidates were rejected (b) in addition, primary pions that can contribute (under different mass hypotheses) to the known remnances were excluded.

4. Other experiments

To date, the NA49 experiement is the only one which has reported evidence for the E-- resonance, with a number of collaborations publishing their null results (for the references see7 and8). Also, the 0+state itself is not found at several high statistics experiements (for the references see8). It is interesting to note that of the numerous experiments which unsuccessfully searched for the Fpentaquark only the ZEUS collaboration found the 0+state. Also, assuming the existence of a narrow 0+,all of the theoretical pentaquark models on the marked also require a narrow E-particle. It thus seems that, rather than conclude that the NA49 results is a statistical anomaly, one could instead argue that pentaquarks in general have a production mechanism which is poorly understood. In fact, there are reports of S resonances at 1860 MeV, albeit not in the exotic channel. As an example Figure 4a shows results from 14.3 GeV K - p reactionlo. In the E-r+ channel a number of resonances are visible, including a prominent state at 1860 MeV and a less significant one at 1760 MeV.

41

Figure 4b shows the preliminary CLAS results for E- resonancesg. Most of the known E states are visible, and labaled as such. However, there are also visible peaks at 1770 MeV and 1860 MeV, which are marked as possibly new states.

'7

7 2

74

16

18

2 22 m&+K'KIX)

1

(Gr

Figure 4. a) The 8-r+ inclusive mass spectrum from K - p reaction at 14.3 GeV b) 8-* in the K+K+ missing mass spectra from the 7 p d K + K f S - * reaction.

Even though the results from these two experiments are not proof of new

E states at 1860 and perhaps 1770 MeV, they still need to be adequately explained. While the NA49 published spectra only show a prominent peak at 1860 MeV, in the spectra without the t9 cut there is a slight excess visible at 1780 MeV (see Figure 5d) One possible interpretation of this spectra would be that at the 1780 MeV is a spin J=1/2 state, while at the 1860 MeV is a spin J=3/2 E-- state. This would make the results consistent with the above mentioned two experiments. All this is of course very speculative and much further work needs to be done before more firm conclusions are made. There is ongoing analysis at NA49 to try t o resolve this issue. 5. Conclusion

The evidence for the existence of a narrow state in E-T- invariant mass spectra with a mass of 1.862 f 0.002 GeV/c2 and width below the detector resolution of about 0.018 GeV/c2 is presented. This state is identified as a candidate for the hypothesized exotic E;- baryon with S = -2, I =

4

42

N

y

200

60

150

40

100 20

50

m o r; 100

0 40

\

3

3

15

30

50

20

25

10

0

1.5

2

2.5

0

1.5

2

2.5

M(E-n-)[GeV/c2] Figure 5. 8-a- NA49 invariant mass distribution for progresively better 8-a- selection. See text for details.

and a quark content of (dsdsfi). Z ? r + and the corresponding antiparticle spectra show indication of enhancements at the same mass.

References 1. T. Nakano et al, Phys. Rev. Lett. 91 012002 2003 V. V. Barmin et al, Phys. Atom. Nucl. 6 6 , 1715 2003. S . Stepanyan et al, Phys. Rev. Lett. 91, 252001 2003. J. Barth et al, Phys. Lett. B 572, 127 2003. V. Kubarovsky et al, Phys. Rev. Lett.92 032001 2004.

A. E. Asratyan, A. G. Dolgolenko and M. A. Kubantsev, Phys. Atom. Nucl. 67, 682 (2004). A. Airapetian et al, Phys. Lett. B 585, 213 2004. A. Aleev et al, hep-m/0401024 2004. 2. D. Diakonov, V. Petrov and M. V. Polyakov, 2. Phys. A 359, 305 1997. 3. R. L. Jaffe and F. Wilcze, Phys. Rev. Lett. 91 232003 2003 4. C.Alt et al, Phys. Rev. Lett. 92 042003 2004. 5. S. Afanasiev et al, Nucl. Instrum. Meth. A 430, 210 1999. 6. K. Kadija et al, J. Phys. G 30, 1539 2004 7. H.G. Fischer and S. Wenig, Eur. Phys. J. C 37, 133 2004 8. J. Pochodzalla, a7.;civ:hep-ex/O406077. U. Karshon, a7.;civ:hep-m/0410029. 9. J. W. Price, J. Ducote, J. Goetz, B.M.K. Nefkens, arxiv:nucl-e~/0402006. 10. R. J. Hemingway, CERN/EP/PHYS 76-50.

43

PRELIMINARY RESULTS FROM THE GRAAL COLLABORATION CARL0 SCHAERF For the Graal Collaboration' Universita di Roma "Tor Vergata",00133 Roma and INFN - Sezione di Roma II We present some very preliminary results on the beam polarization asymmetry in the incoherent photoproduction of q and no mesons on the deuteron. We compare the results obtained on a free proton in Hydrogen with those obtained on a quasi-free proton and quasi-free neutron in Deuterium. Up to gamma-ray energy of 1.1 GeV, the results for free and quasi-free protons' are similar and similar are those obtained for quasi-free proton and quasi-free neutron in q photoproduction. In no photoproduction the results for quasi-free proton and quasi-free neutron exhibit substantial differences. At energy around 1680 MeV an anomalous peak appears in the spectrum of the invariant mass of the eta-neutron system. Nothing similar is present in the eta-proton spectrum.

1.

Introduction

The backscattering of laser light against the high-energy electrons circulating in storage rings produces polarized gamma-ray beams useful for the study of photoreactions on nucleons and nuclei. Their main characteristics are the high degree of polarization, the possibility of changing the polarization of the gamma-ray beam by changing the polarization (linear and circular) of the laser light with simple optical components, the low background of lower energy gamma rays, and the reasonable intensity and energy resolution. After the successful operation in 1979 of the Ladon beam' on the storage ring Adone in Frascati, this technique has been applied in several other laboratories as indicated in Table I. Today Ladon beams cover the entire gamma-ray spectrum from a few MeV to 2.4 GeV. The Graal beam at the ESRF' covers the energy interval between 600 and 1500 MeV. The Graal detector3 consists of plastic scintillators, wire chambers and a BGO crystal ball made of 480 crystals. It covers a solid angle very close to 4n. It has been used to study the photoproduction of mesons in Hydrogen and Deuterium.

* O.Bartalini, V.Bellini, J.-P.Bocquet, M.Castoldi, A.D'Angelo, J.-P.Didelez,

R.Di Salvo, A Fantini, G.Gervino, F.Ghio, B.Girolami, A.Giusa, M.Guida1, E.Hourany, V.Kouznetsov, M.Kunne, A.Lapik, P.Levi Sandri, A.Lleres, D.Moricciani, V.Nedorezov, D.Rebreyend, G.Russo, N.Rudnev, C.Schaerf, M.Sperduto, M.Sutera, and A.Turinge.

45

2.

Preliminary results

We report here some very preliminary results on the beam-polarization asymmetry in the photoproduction of no and q mesons in Hydrogen and Deuterium and we compare the results for the photoproduction on free protons in Hydrogen and quasi-free protons and quasi-free neutrons in Deuterium. For q photoproduction we select events with two photons in the BGO ball and one neutral or charged particle in the central BGO ball or the forward walls. From the measurements of the angles and the energies of the two photons, we calculate their invariant mass. If the result is close to the mass of the q, we extract the angles and energy of the q and those of the nucleon. With the information on the gamma-ray energy provided by the tagger we verify that the energy and momentum conservation is compatible with one of the two reactions:

y +d

+ y + n + (p) + q + n + (p)

(2)

The particles in parentheses behave as spectators. The constraints that we impose on the conservation of energy and momentum require that the spectator carries out a negligible amount of momentum (and energy) and exclude the production of another particle like a no. Figure 1 shows the invariant mass of the two photons before and after kinematical correlation with the detected nucleon. The experimental result reproduces very well the result obtained with simulated events.

lo6

lo5

lo4 lo3

lo2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

G eV Figure 1. The invariant mass of two photons in the BGO ball. The upper curve is the distribution without any constraint. The lower curve requires a nucleon in the Graal apparatus. The two peaks clearly correspond to the masses of the no and q.

46

Figure 2. Coplanarity between the q and the nucleon versus the difference between the measured polar angle of the nucleon and that calculated from the q kinematics. 1

: 4

K/ndl

PI

.........

.,................

1...............

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

02

1

3.465 / I I : O.d.558 4 .........

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

&q

-...............

;

+. ..

....&.

100

1

m

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.

I

m

:

09

T........

..

. ......... I

4

~

0

.

... ;

i

€,.-..0.&62fi.Ge~i

oa

0,

;

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09

;

4

/n+

$1

im

2m

3m

0

im

2m

3m

I

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..... .& .........

~

0

03 011

0.7 06

05 0.4 0-1

................................. +................ + ............... 4................ 4 4

02 01

0 0

I

I

I

100

m

m

03

@fl

Figure 3. Azyrnuthal distribution of q with respect to the gamma-ray polarization for two different values of the gamma-ray energy and the q center of mass angle. Upper curves for reaction 1 (quasifree proton) and lower curves for reaction 2 (quasi-free neutron).

47

Figure 2 shows the coplanarity between the q and the nucleon and the difference between the measured polar angle of the nucleon and its value calculated from the momentum of the q . Similar bidimensional plots can be shown for the correlation between the missing mass from the q and that from the nucleon. Once we have defined the cuts that select the events belonging to reactions 1 and 2 we can plot the azimuthal distribution of the q with respect to the gamma-ray polarization direction. Two curves for selected intervals of gamma ray energy and q centerof-mass angle €), are reproduced in Figure 3. The beam-polarization asymmetry is obtained from a fit of these data and the calculated values of the gamma-ray beam polarization. Figure 4 indicates a comparison between the values of the asymmetry obtained for a free proton in Hydrogen and a quasi-free proton in Deuterium.

0.6

c

0.4 -

E,=0.86201 GeV

- *::

---1

% *’: @-

0.2 -

.* . I

.

,*# .*

, *

0 -4 &+‘

-

6=0.9 1578 GeV

.;*

%.* a., %

Figure 4. Comparison of the preliminary results obtained for the beam-polarization asymmetry for -q photoproduction on free (grey points) and quasi-free protons (black points).

Figure 5 indicates the comparison between a quasi-free proton (reaction 1) and a quasi-free neutron (reaction 2).

A similar procedure has been followed to analyze the no data. The comparison between quasi-fiee proton and quasi-free neutron is indicated in Figure 6 . Figure 7 represents some very preliminary results for the comparison of the invariant masses of the q-n and q-p systems in the photoproduction of q on the quasi-free nucleons in deuterium. These data are now being analyzed by three independent groups within the Graal collaboration and their results appear compatible.

48 0.8

E

0.6

-..............

~

-..................................

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

i5=0.97!$19 GeV - .......................... .

..L

,...........................

j

jEy=1.05$4 GeV

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

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

’........

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

I

0.4 0.2 ....................... ..... .......... .......I,

0 ,

0

/

I

I

,

50

,

,

,

,

100

,

,

I

I

150

I

,

,

I

,

,

,

, , , ,

100

50

0

0:

,

, ,

150

0:

Figure 5 . Comparison of the preliminary results obtained for the beam-polarization asymmetry for q photoproduction on quasi-free protons (grey points) and quasi-free neutrons (black points).

0

50

100

150

$‘’T 7c

0

50

150

100

$ ’!;

Figure 6 . Comparison of the preliminary results obtained for the beam-polarization asymmetry for no photoproduction on quasi-free protons (grey points) and quasi-free neutrons (black points).

3.

Conclusions

Preliminary results on the q photoproduction on quasi-free protons and neutrons in the deuteron show that the beam-polarization asymmetries are very close for free protons and quasi-free protons and are also very similar for quasi-free protons and quasi-free neutrons up to a gamma-ray energy around 1 GeV. The same is not true for no photoproduction where T=3/2 channels are present. An anomalous peak appears in the eta-neutron invariant-mass distribution at energy around 1.68 GeV. This peak is not visible in the eta-proton channel and

49

is not reproduced by the simulations of eta-nucleon photoproduction where we have used the experimental values of the cross sections for the eta free-proton reaction. More data are necessary to clarify this problem.

DATA

SIMULATION mvln

1000 -

N".

PUS

500

.

1116r

1.02 a109

800 -

LOO

I 600

300

-

200 100 0

1.5

1.6

1.7

1.8

1.5

1.9

W (eta-neutron) final state, data

1.6

1.7

1.8

1.9

W (eta-neutron) final state, simul

2250

2000 1750 H

1500

I

1250 1000

750

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500 250

0

1.5

1.6

1.7

1.8

1.9

W(et3-proton)final state, data W(eta-proton)final state. simuhtion Figure 7 . The q nucleon invariant mass in the final state of reaction 1 (lower curves) and reaction 2 (upper curves). The curves on the left are the data and those on the right the simulations. The interesting result is the presence of the narrow peak in the neutron data that is not apparent in the proton data and in the simulations. Both simulations have been performed using the q photoproduction cross sections measured for the free proton.

Refererences I

L. Federici, et al. I1 Nuovo Cimento, 59B, 247 (1980). R. Caloi, et al., Lettere a1 Nuovo Cimento, 27, 339 (1980). F. Ghio, et al., Nuc. Instr. and Meth. A404,71 (1998)

50

SEARCH FOR PENTAQUARK STATES ON PROTON TARGET AT CLAS

R. DE VITA, FOR THE CLAS COLLABORATION Istituto Nazionale di Fisica Nuclenre via Dodecaneso 33, 16146 Genova, Italy E-mail: [email protected]

Baryon states beyond the usual qqq configuration has been searched for many years but no evidence was found until recently. The findings about a possible pentaquark state have driven a rebirth of the experimental activity in this field. A broad experimental program for the search for pentaquark states is presently in progress at Jefferson Lab with the CLAS detector. In this proceedings the results and perspective for experiments using proton targets are discussed.

1. Physics Motivation

The existance of baryon states that cannot be described in terms of three valence quark is not prohibited by QCD. Search for such states started already several decades ago but no positive results were obtained until first evidence for a S=+l baryon, called O f , was reported by the LEPS/Spring8 Collaboration based on the analysis of photoproduction data on a carbon target.l This baryon can be interpreted as pentaquark state with minimal uudds configuration. Since the first announcement, evidence for the state has been claimed in more than 10 published papers from medium energy to very high energy experiments, using a variety of probes and target. In the last months negative results were also reported by several experiments, mostly at high energy. Presently, there is no definite answer on the existence or nonexistence of the This is clearly of utmost importance: if confirmed, this discovery would open a new chapter in hadron spectroscopy and would provide fundamental information on hadronic structure. In this contribution I report on the results obtained at Jefferson Lab with the CLAS detector using a photon beam and a proton target. In addition I will discuss the ongoing program of second generation experiments

e+.

51

Figure 1. The CLAS detector. Left: Longitudinal cut along the beam line shows the different detector components. Right: Transverse cut through CLAS.

aimed at improving the statistical accuracy of the measurements by at least one order of magnitude. 2. Experimental Setup

The core of Jefferson Lab is the CEBAF accelerator (Continuous Electron Beam Accelerator Facility). This consists of two superconducting linacs connected by recirculation arcs. The superconductive cavities operate at a frequency of 1.497 GHz, resulting in a 2.0 ns duty cycle beam for the three experimental halls (Hall A, B, and C). The injected electrons can circulate up to five times reaching a maximum energy of 5.75 GeV. In Hall B, the electron beam is converted into a bremsstrahlung photon beam using a gold radiator located 20 meters upstream of the target.2 The photon energy is measured by detecting the scattered electron using a tagging system that operates over a range from 20% to 95 % of the incident electron energy with a resolution of The tagged photons, with rates between lo6 and lo7 per second, interact with a liquid hydrogen target located inside the CLAS detector (CEBAF Large Acceptance Spectrometer). The CLAS detector is shown in Figure 1. It is a magnetic spectrometer divided into six independent sectors. Each sector is equipped with three layers of drift chambers for tracking reconstruction, threshold Cherenkov counter for pion/electron discrimination, scintillators for timeof-flight measurements, and electromagnetic calorimeters for electrons and

52 Table 1. Runs completed in Hall B at Jefferson Lab with the CLAS detector using photon beams and proton targets. The experimental conditions are summarized. Experiment g6a g6b g6c glc

Year 1998 1999 1999 2000

Beam Energy 4.1 GeV 5.5 GeV 1.9-3.1 GeV 5.7 GeV

Target LH2 LHz LH2 LH2

1.C(pb-l) N

N N N

1.0 1.0 4.0 2.7

neutrals identification. The magnetic field produced by a six-coil toroidal magnet is oriented in such a way as to maintain the azimuthal angle of the scattered particles constant while changing only their polar angle. The large acceptance of the detector allows the simultaneous measurement of several processes in a large kinematic domain. This capability is very important for the search of exotics, like the O+ and its partners. In fact the production cross section for such particles are expected to be small, and the exotics signals may be easily hidden by known baryon production. Such background processes may be significantly reduced by selecting particular kinematics accessible with the CLAS detector. In the last year, the existing CLAS data were reanalyzed to study possible evidence for pentaquark production. A summary of the different experiments with photon beam and proton target is given in Table 1. 3. Results on Proton Farget

3.1. Published Results The first evidence for the O+ on proton target at CLAS was found in the analysis of the reaction yp +-.Ir+K+K-n .4 The analysis was performed using the data recorded during the g6 run. This data sets was accumulated in three different periods starting in 1998, with electron beam energy from 4.1 to 5.7 GeV (see Table 1). The reaction was isolated detecting the three charged particles in the final state and selecting the unmeasured neutron using the missing mass technique. Production of known hyperons like the q5 meson was suppressed applying cuts on the corresponding mass spectrum. To understand the possible production mechanism for the O+ and for the background, different possibilities were explored. For example the O+ may be produced through an intermediated N* resonance as shown in the top left diagram of Figure 2. Following this assumption events with a forward going n-+ were selected, while events with a forward going K + were rejected to reduce t-channel contribution. The final nK+ mass spectrum

53

Y

n

+

.----?---------

P

n

b) t

xo

Y K-

/K

8'. I

.

"

P d)

Figure 2. Diagrams that may contribute to the process -yp -+ r + K - n K + . The left top diagram contributes to O+ production through intermediate N' excitation. The other diagrams represent background processes.

-

1.55 GeV is shown in the left panel of Figure 3. A peak at a mass of is observed over a smooth background. The spectrum was fitted with the sum of a gaussian function and of a background function obtained from simulations. A mean of 1550 f 10 MeV, I? = 26 f 7 MeV, and a statistical significance of 7.8 f 1.00 for the observed structure were found. To further study the background processes and to exclude systematic effects due to meson reflections in the nK+ mass spectrum, a full partial wave analysis of the K+K-.rr+ meson system was performed. This confirmed that the background is expected to be smooth and structureless. If the O+ production mechanism indeed involves the excitation of an intermediate N * resonance, this may appear in the nK+K- mass spectrum. This is shown in the right panel of Figure 3 for the events with effective nK+ mass between 1.54 and 1.58 GeV. An excess of events at a mass of 2.4 GeV is observed while the partial wave analysis results shown by the line confirms a structureless background contribution. This result is consistent with the assumption of an intermediate N * contribution, but the statistics is at present too poor to reach any definite conclusion and further studies are needed. N

3.2. Ongoing Data Analysis In addition to the 96 run period, other data sets have been analyzed looking for pentaquark states. The glc data set was used to study the reactions yp + K+Kon and yp + pK+K-, searching for evidence of S = +1

54 40 12

35 10

30

p

N

Y

Y

25

g

.

20

::

I

15

::

.

3

&

8

Y

6

A

4

I

10

5 0.

2

M(nK+) (GeV/cZ)

M(nKfK) (GeV/cz)

Figure 3. Left: Invariant mass distribution of M ( n K + ) when all cuts are applied. The inset shows the n K + mass distribution with only the cos P, > 0.8 cut applied. Right: Mass distribution M ( K - n K + ) for events selected in the peak region of the graph on the left. The inset shows the distribution for events outside of the Q+ region.

resonances in the K + n and K+p invariant masses. The first reaction was selected by detecting the final state charged particles, the K+ and the T+T- from the 2 0 decay, in CLAS. The neutron was not measured but was reconstructed using the missing mass technique, ensuring therefore the exclusivity of the final state. Figure 4 shows the quality of the channel identification: both Ko and n are reconstructed within 1-2 MeV of the nominal mass value with small background. The events laying in the shaded region were selected for further analysis. This event sample is still dominated by the production of known hyperons decaying into the same final state. These include the production of A excited states as for example yp 4 KfA*(1520),or the production of Cs in the reactions -p + K+C+n- and yp + K+C-n+. Figure 5 shows the A*(1520) and C+ peaks reconstructed as missing mass of K + and the K+n- system. Events associated with these reactions were excluded by cutting on the corresponding masses. After such cuts, the nK+ invariant mass spectrum was constructed. After selecting events in which the K0 is emitted at backward angles, two structures with masses near 1525 and 1575 MeV were seen. However, the low statistics of the final event sample did not allow us to draw definitive conclusion on such structures. The reaction yp + pK+K- was selected by detecting at least two of the three charged particles in CLAS and using the missing mass technique allows us to identify the third one. Two different topologies, ~p + pK+(K-) and ~p + (p)K+K-,were analyzed while the topology with pK- detected N

N

55

was dropped due to limited acceptance. Background contribution from known hyperons as the 4(1020) or A*(1520) were rejected with cuts on the corresponding masses. To further reduce the background contribution coming from other reactions and to maximize the signal to background ratio, angular and energy regions were selected where Monte Carlo simulations showed maximum sensitivity to the reactions of interest.

1

M(Ko) (GeV)

M(n) (GeV)

Figure 4. Final particle identification for the reaction 7 p + K + K o n . The left plot shows the K0 mass spectrum reconstructed as invariant mass the a+?r- system. The right plot shows the K+r+n- missing mass.

After these additional cuts, the pK+ invariant mass spectrum showed a structure in the mass region around 1.58 MeV. However also in this case, the limited statistics did not allow us to reach definitive conclusions. In either case, the much higher statistics of the new g l l experiment that is presently in the analysis stage, will allow more definite conclusions as to the existence and significance of these possibly new narrow structures. Table 2.

New experiments proposed in Hall B for the search of pentaquark states

Run

Beam

Energy

Target

gll

7

4.0 GeV

LH2

g12

7

5.7 GeV

LH2

Reaction 7p-+Q+Ko 7 p + Q+K-T+ yp+Q+K-?r+ 7 p + QfK0 7 p -+ K + K - ~ -

Status Data Taking Completed To bescheduled

.

56 v1 900 *

B

U

___

800

e,

700

250

I6sE

600 500 400 300 200 100 9.4

1.5

1.6

1.7

1.8

1.9

2

3

MM(K+) (GeV)

Figure 5. K + and K+T- missing masses after the cuts on the K + and KO, and n masses. The A'(1520) and C+ peaks are clearly visible. The highlighted areas correspond to the events selected for further analysis.

4. Perspectives and Future Plans

The results obtained from the analysis of the existing data demonstrate the CLAS capability of detecting the reactions of interest for the pentaquark search with good resolution and limited background. The large acceptance of the detector allowed for fully identified final states while explicitly rejecting known background sources. However the available statistics do not exclude that the observed signals may be affected by statistical fluctuations, kinematic reflections, or some artifact of the data analysis. A definitive answer about the existence of pentaquark states can be obtained only with new high statistics, high resolution experiments. For this purpose two new experiments on proton target have been recently approved for CLAS in Hall B at Jefferson Lab. The experimental conditions of this new runs are summarized in table 2. The first experiment (911)~ whose data taking was completed in July 2004 aims at establishing the phenomenology of the O+ spectrum, e.g. determining in what production channels the O+ is seen and what higher mass states are excited.6 The second experiment (912) that will be scheduled in the near future will exploit the maximum available energy to study the O+, 2, and other pentaquark production measuring cross section, angular distribution, and decay distributions. If the existence of the O+ is confirmed, these new experiments will

57 $900

3 800

: * I-

700

g

600 500 400

300 200 100 1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

M (pKo) GeV) Expected statistical accuracy of the mass spectra for the reactions -yp + with a+(@+*) decaying into K+n (left) and p K o (right) in the 911 run. The background was estimated based on the existing data and the signal was simulated assuming a production cross section of N 10 nb. Figure 6. @+(@+*)l?O,

provide a solid foundation for a long term plan in pentaquaxk spectroscopy.

Acknowledgements This work was supported by the Italian Istituto Nazionale di Fisica Nucleare, the French Centre National de la Research Scientifique, the French Commissariat B 1'Energie Atomique, the US. Department of Energy, the U.S. National Science Foundation, and the Korean Science and Engineering Foundation. The Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility for the United States Department of Energy under contract DE-AC05-84ER40150.

References T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003). D. Sober et al., Nucl. Instr. and Meth. A440, 263 (2000). B. Mecking et al., Nucl. Instr. and Meth. A503, 513 (2003). V. Koubarovsky et al., Phys.Rev.Lett. 92, 032001 (2004). M. Battaglieri, R. De Vita, V. Koubarovsky, et al. (CLAS), JLab experiment E04-021. 6. M. Battaglieri, these proceedings. 7. J. Price, D. Weygand, et al. (CLAS), Jlab experiment E04-017. 1. 2. 3. 4. 5.

58

EVIDENCE FOR 0' RESONANCE FROM THE COSY-TOF EXPERIMENT* W. EYRICH Physical Institur, University of Erlangen-Nuremberg, Erwin-Rommel-Str. 1, 0-91058 Erlangen, Germany

FOR THE COSY-TOF COLLABORATION Using the TOF detector at the COSY storage ring the hadronic reaction pp 4Z+Pp was measured exclusively at a beam momentum of 2.95 GeV/c. A narrow peak was observed in the invariant mass spectrum of the @p subsystem at 1530k 5 MeV/cZwith a significance of 4 - 6 standard deviations, depending on background assumptions. The upper limit of 18 & 4 MeV/c2 (FWHM) for its width is given by the experimental resolution. The corresponding total cross section is estimated to be about 0.4 -+ O.l(stat) O.l(syst) pb. Since a resonance in this subsystem must have strangeness S = + 1 we claim it to be the @+ state for which very recently evidence was found in various experiments.

+

1. Introduction As presently understood, QCD does not forbid the existence of states other than quark-antiquark and three-quark systems as long as they form colour singlets. In numerous theoretical publications the possible existence of exotic systems including pentaquark states has been worked out based on specific assumptions and production scenarios also. One of the most cited publications concerning pentaquark states by Diakonov, Petrov and Polyakov [l] is based on the soliton model assuming an antidecuplet as third rotational excitation in a three flavour system. The corners of this antidecuplet are occupied by exotic pentaquark states with the lightest state having a mass of -1530 MeV/c2, strangeness + l , spin 1/2 and isospin 0. This state, originally known as the z', has more recently been renamed 0 ' . In this model the mass of the 0 ' is fixed by the N* resonance at 1710 MeV/c2, which is assumed to be a member of the antidecuplet. The most striking property of the 0 ' resonance is the predicted narrow width of r < 15 MeV/c2, which according to ref. [l] is connected with a narrow width of the 1710 MeV/c2 N* resonance of 50 MeV or less. With the predicted quark content for the 0 ' of uuddS this pentaquark resonance is expected to decay into the channels K'n and K o p . ~~

* This work is supported by German BMBF and FZ Jiilich

59

The first report on the discovery of a narrow resonance in the expected mass region came from the LEPS collaboration at Spring8 [2] where in the yK-missing mass spectrum of the reaction yn +K’K-n on 12C a narrow resonance was observed at 1.54 k 0.01 GeVlc2with an upper limit for the width of r = 25 MeVlc2. In the meantime several other experiments have presented observations in the mass region between about 1525 and 1555 MeVIc’ [3 - 93. In this paper we report on the search for the 0’resonance using the COSY-TOF experiment. Within the framework of the hyperon production program at COSYTOF [lo, 111 the reaction p p + X f K O p has been measured exclusively. Data were taken predominantly at a beam momentum of Pbem = 2.95 GeVlc, corresponding to an excess energy of 126 MeV. This limits the invariant mass spectrum of the K o p system between the threshold value of 1436 MeVIc’ and an upper value of about 1562 MeVIc’. Accordingly an optimal ratio between a possible resonance signal around 1530 MeVlc2and the non resonant background is expected [12]. A deviation from a smooth invariant mass spectrum of the KO p system was already observed in a first measurement performed in 2000, but the extracted event sample was too small for a definitive statement [ l 11. To improve the statistical significance a second production run at the same beam momentum was performed in 2002.

2. Experimental setup and analysis In the production runs in 2000 and 2002 reported the time-of-flight spectrometer COSY-TOF was used in its 3 m version [ 131. The extracted proton beam (-lmm 0) hits a liquid hydrogen target with a length of 4 mm. The geometrical reconstruction of the related tracks and vertices is mainly realized by the startdetector system, a scheme of which is shown in Fig. 1 together with an event of the type p p + X C ’ K o p with a subsequent decay of the as a Ks into a z+z-pair and the delayed decay of the C’ into a nz+pair. The events of interest are identified by these delayed decays. The reconstruction of the K $ and its decay vertex occurs via the tracks of its decay products z+z- by two scintillating fibre hodoscopes. The decay kinematics and angular distributions allow a clear separation from the remaining background, which is dominated by the reaction p p + K + A p . The 2’ hyperon together with its delayed decay into nN is characterized by a track with a kink and is detected via a double sided silicon microstrip detector close to the target. The momenta of the reconstructed particles are calculated directly from the extracted directions (“geometry spectrometer”) using momentum and energy conservation. Since there are usually several possible geometrical combinations

60

intermediate fiber hodoscope

fiber ”starttorte”

scintillator 2x12 wedges

I

/

Si-p-strip scintillator scintillator 100 rings 2x96 fibers 2x192 fiber 128 segments

Figure 1: Scheme of the Start detector system together with an event of the reaction p p +Z’K0p

and hence kinematical solutions for each event, a missing mass analysis is applied for both the mass of the Et using the tracks of the primary reaction products and the mass of the K ; determined by using the information of the tracks of its decay products. This contains two overconstraints. To find the best solution, both masses are required to be best-fitted simultaneously. Events outside the phase space of the reactions of interest were rejected. Geometrical cuts on the tracks and decay vertices were used to suppress the background. By varying these cuts and performing Monte Car10 simulations in parallel it was carefully checked that the restrictions used do not influence the results concerning the observables of the reaction of interest. Both measurements show clear mass distributions peaking at the related corresponding masses of the X’ (“primary mass”) and KO (“secondary mass”), respectively, and they are identical. To get very clean samples for further investigations of the reaction of interest cuts on the resulting mass peaks have

61

been applied. This is demonstrated in Fig. 2 where the spectra of the two runs are summed up. In the upper part the cuts are indicated which lead to the spectra in the lower part. Finally these cuts shown on the mass and the X+ mass lead to two samples of 421 and 518 events for the two runs respectively, and accordingly 939 events for the total sample which is used for further analyses. Extensive Monte Car10 simulations were performed to control and to optimize the analysis chain. Moreover they were used to deduce the resolution in the various observables. The resolution of the X+ and the KO mass of the simulated data is in quantitative agreement with the real data. For the KO p invariant mass Entries 4728

IEntries 4728

" 800 600

400

200

n+n-~nvanantmass

c*missing mass m GeVIc'

~evi d

300 I

250

200

- 939 events

150

100 -

100

0

939 events

200

~

150 -

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,

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50 I

,

0.6

'

,

cI&_

L .

.

0.8

0.5 z*rnismg mass in GeVIc'

0.6

0.7

0I

i+i-invananl mass (n Gevlc'

Figure 4: Reconstructed masses of Z+and K! for the summed data (2000 + 2002). In the upper spectra the cuts are indicated which lead to the lower spectra. Further explanation see text.

which is relevant for the search for a possible narrow state an overall mass resolution of 18 3 MeV/c2 (FWHM) has been deduced from the simulations.

3.

Results

To search for a possible resonance the data of the two runs have been investigated both separately and in sum. In Fig. 5 the invariant mass spectra of the KO p system are shown. They cover the full kinematical range corresponding to the excitation energy of 126 MeV. The shape of all three spectra is very similar. Within statistical fluctuations the spectra from the 2000 (top figure) and 2002 (middle) runs are identical.

62

M(Kop) i n CeV/c'

Figure 5: Invariant mass spectrum of the @p subsystem obtained from the 2000 data (upper part), the 2002 data (middle part) and the sum of both (lower part) together with a fitted background.

There is an obvious deviation from a smooth distribution in the spectra of both runs and in the spectrum of the summed samples (bottom) around 1.53 GeV/c2 (indicated by the arrow in the summed spectrum). Assuming a smooth background as obtained by a polynomial fit excluding the region between 1.51 GeV/c2 and 1.54 GeV/c2 (dashed curves in Fig. 5) the significance of the signal can be deduced. Three different expressions for the significance of the peak in the summed spectrum (Fig. 5 bottom) have been considered. The first alternative is the naive estimation N , /& where N , is the number of events corresponding to the signal on top of the fitted background and N , is the number of events corresponding to the background in the chosen area. In the present case this leads to a significance of 5.9 B on the basis of an interval of f 1.5 B around the peak value of 1530 MeV/c2. This estimator

63

however neglects the statistical uncertainty of the background and therefore usually overestimates the significance of the peak. A more conservative method which is reliable for cases where the background is smooth and well fixed in its shape uses the estimator N S I J " , . In our case this method leads to a significance of 4.7 0. The third expression taking into account the full uncertainty of a statistically independent background which should underestimate the significance of the signal is given by N s / J ( N s + N , ) + N , . This leads to a value of 3.7 6. Because the measurement presented and the event sample extracted from it cover the full phase space of the reaction products, an investigation of the corresponding Dalitz plot is possible. In Fig. 6 the Dalitz plot based on the 939 events of the summed spectrum of Fig.5 is shown. The peak around 1.53 GeV/c2 *

2 5.4

1.532

I

M2(Kop)in GeV2/c4

3

4

M2(KoX+)in GeV2/c4

Figure 6: Dalitz plots for the full sample at a beam momentum of 2.95 GeV/c. The dotted lines show

the phase space limits. The arrows correspond to a mass for the F?p system of 1.53 MeV/c*.

identified in the KO p invariant mass spectrum should show up in the ideal case as a band in the Dalitz plot at the corresponding squared mass around 2.34 GeV2/c4as indicated by the arrows in both distributions. As expected due to the low number of events there is only a slight indication for a band. But more importantly in both distributions there is no indication of an artefact which could give rise to a faked signal in the K'pmass spectrum. It should also be recognized that according to the low excess energy of 126 MeV the influence of a possible excitation of f-resonances is excluded . To correct for the efficiency of the detector and the analysis, Monte Carlo simulations were used. The correction function is very smooth giving some enhancement at the edges of the phase space. In Fig. 7 the efficiency-corrected KO p invariant mass spectrum corresponding to the total sample is shown.

64

.S 140

e

* 120

40

l 201.4

t

L 1.6 M(Kop) GeV/cf

Figure 7: Efficiency corrected invariant mass spectrum of the p p subsystem for the full sample.

In comparison to the uncorrected spectrum shown in Fig. 5 there is no major difference. Again there is a significant peak around 1.53 GeV/c2 on top of a smooth background. For a more quantitative analysis a polynomial fit on the background and a Gaussian for the remaining signal are used (Fig.7 dotted lines). This yields a peak value of 1530 f 5 MeV/c2. The deduced width of 18 k 4 MeV/c2 (FWHM) is in agreement with the value of the Monte Car10 analysis and accordingly only an upper limit for the physical width of the observed peak. The cross section of the observed peak around 1530 MeV/c2 has been estimated by comparing with the measured total cross section of the reaction. The normalisation was deduced by comparison with the elastic p p scattering which was measured simultaneously. For the observed peak at 1530 MeV/c2 we deduce a cross section of 0.4 f 0.1 (stat.) k 0.1 (sys.)pb . This value is in rough agreement with theoretical estimations by Polyakov et al. [12] and Liu and KO [ 151, where a total cross section in the order of 0.1 - 1 p b is predicted for the 0 ' production in the threshold region in p p and p n induced reactions.

4.

Summary and Outlook

The COSY-TOF experiment provides evidence for a narrow resonance in the K o p system at a mass of 1530 f 5 MeV/c2 from the exclusively measured reaction p p - + Z + K o p [15]. The extracted width of about 18 MeVIc' reflects the experimental resolution, Since a resonance in this subsystem must have strangeness S = + 1 we claim it to be the @+ state. This is the first evidence on the 0 ' resonance from an elementary hadron hadron reaction. There is now evidence from several experiments on a narrow state in the systems @p and c n in the mass region between about 1525 and 1555 MeV/c2. But this

65

evidence comes from signals which contain about 50 events or even less. That means that none of these experiments has the statistical accuracy which is necessary to pin down the result with a precision which is required for a final proof of the existence. Especially in the high energy regime there are also experiments which do not see a signal. The next round of experiments has to produce much larger data samples to clarify this. COSY-TOF will have a run in the fall of this year to confirm the signal. In a second step we will use a deuterium target to investigate the reaction channel p n + A K o p for which our apparatus should be optimally suited. In the positive case of a confirmation of the signal plans exist to use a polarized beam in combination with a polarized target to deduce the parity of the 0 ' [ 161. Moreover we are investigating the reaction p p + A K ' p to look for a possible double charged partner of the 0 ' in the subsystem r p and to investigate the parameters of the N*(1710) resonance in the K f i system. Acknowledgments We want to thank very much the COSY accelerator team for the preparation of the excellent proton beam and the good cooperation during the beam time. We gratefully acknowledge support from the German BMBF and the FZ Julich. References 1. D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359,305 (1997) 2. LEPS Coll., T. Nakano et al. Phys. Rev. Lett. 91,012002 (2003) 3. DIANA Coll.,V.V. Barmin et al. Phys. Atom. Nucl. 66, 1715 (2003); Yad. Fiz. 66, 1763 (2003) 4. CLAS Coll., S. Stepanyan et al., Phys. Rev. Lett. 91,252001 (2003) 5. CLAS Coll., V. Kubarovsky et al., Phys. Rev. Lett. 92,032001 (2004) 6. SAPHIR Coll., J. Barth et al., Phys. Lett. B 572, 127 (2003) 7. A.E. Asratyan, A.G. Dolgolenko and M.A. Kubantsev, hep-ed0309042 8. HERMES Coll., A. Airapetian et al., hep-ed0312044 9. SVD Collab., A. Aleev et al., hep-ed0401024 10. COSY-TOF Coll., R. Bilger et al., Phys. Lett. B 420,217 (1998) 1 1. COSY-TOF Coll., W. Eyrich et al., International workshop on nuclear physics (Erice, Sept. 2002), Prog. Part. and Nucl. Phys. 50,547 (2003) 12. M. V. Polyakov et al., Eur. Phys. J. A 9, 115 (2000) 13. www.fz-juelich.de/ikp/COSY-TOF/detektor/index_e.html 14. W. Liu and C. M. KO, nucl-tN0309023 15. COSY-TOF Coll., Phys. Lett. B 595, 127 (2004) 16. C. Hanhart et al., hep-pN03 12236

66

PENTAQUARK SEARCH AT HERMES*

W. LORENZON (on behalf of the HERMES Collaboration) Randall Laboratory of Physics, University of Michgan, Ann Arbor, Michigan &3109-1120, USA E-mail: lorenzonbumich. edu

Evidence for a narrow baryon state at 1528f2.6(stat) f 2 . l ( s y s t ) MeV is presented in quasi-real photoproduction on a deuterium target through the decay channel pKg + p n f r - . The statistical significance of the peak in the p K g invariant mass spectrum is 4 standard deviations and its extracted intrinsic width r = 17 f 9(stat)f3(syst) MeV. This state may be interpreted as the predicted S=+l exotic O+(uuddB) pentaquark baryon.

1. Introduction One of the central mysteries of hadronic physics has been the failure t o observe baryon states beyond those whose quantum numbers can be explained in terms of three quark configurations. Exotic hadrons with manifestly more complex quark structures, in particular exotics consisting of five quarks, were proposed on the basis of quark and bag models1 in the early days of QCD. More recently, an exotic baryon of spin 1/2, isospin 0, and strangeness S=+l was discussed as a feature of the Chiral Quark Soliton model.2 In this approach3i4 the baryons are rotational states of the soliton nucleon in spin and isospin space, and the lightest exotic baryon lies a t the apex of an anti-decuplet with spin 1/2, which corresponds to the third rotational excitation in a three flavor system. Treating the known N(1710) resonance as a member of the anti-decuplet, Diakanov, Petrov, and Polyakov4 derived a mass of 1530 MeV and a width of less than 15 MeV for this exotic baryon, since named the O+. It corresponds to a uud& configuration, and decays through the channels O+ 4 pKo or nK+. However, measurements of K+ scattering on proton and deuteron targets showed no evidence5 for strange *This work is supported by the U.S. National Science Foundation under grant 0244842.

67 baryon resonances, and appear t o limit the width t o remarkably small values of order an MeV. Experimental evidence for the O+ first came from the observation of a narrow resonance‘ a t 1540&10(syst) MeV in the K - missing mass spectrum for the y n -+ K + K - n reaction on 12C. This result was confirmed since then by a series of experiments, with the observation of sharp peaks7-17 in the nK+ and pKg invariant mass spectra near 1530 MeV, in most cases with a width limited by the experimental resolution. There are also many unpublished reports of failures t o observe this signal.

2. Experiment

Presented here are the results of a search for the O+ in quasi-real photoproduction on deuterium.” The data were obtained by the HERMES experiment with the 27.6 GeV positron beam of the HERA storage ring at DESY. The HERMES spectrometer is described in detail in Ref. 18. The analysis searched for inclusive photoproduction of the O+ followed by the decay O+ 4 pKZ -+ p.rr+.rr-. Events selected contained at least three tracks: two oppositely charged pions in coincidence with one proton. Identification of charged pions and protons was accomplished with a Ring-Imaging Cerenkov (RICH) detector,lg which provides separation of pions, kmns and protons over most of the kinematic acceptance of the spectrometer. In order to keep the contaminations for pions and protons at negligible levels, protons were restricted t o a momentum range of 4-9GeV/c and pions to a range of 1-15 GeV/c. The event selection included constraints on the event topology to maximize the yield of the Kg peak in the M,+,- spectrum while minimizing its background. Based on the intrinsic tracking resolution, the required event topology included a minimum distance of approach between the two pion tracks less than l c m , a minimum distance of approach between the proton and reconstructed Kg tracks less than 6mm, a radial distance of the production vertex from the positron beam axis less than 4mm, a z coordinate of the production vertex within the f 2 0 cm long target cell of -18cm < z < +18cm along the beam direction, and a Kg decay length greater than 7cm. To suppress contamination from the R(1116) hyperon, events were rejected where the invariant mass Mp,- fell within 2a of the nominal A mass, where a = 2.6MeV is the apparent width of the A peak observed in this experiment.

68

3. Results The resulting invariant M,+,spectrum yields a K$ peak at 496.8 f 0.2 MeV, which is within 1MeV of the expected value of 497.7f0.03 MeV,'' To search for the Of, events were selected with a M,+,- invariant mass within f 2 (T about the centroid of the Kg peak. The resulting spectrum of the invariant mass of the p7rf7r- system is displayed in Fig. 1 (left panel).

1.45

1.5

1.55

1.6

1.65

1.7

M(x+x'p)[GeV]

1.45

1.5

1.55

1.6

1.65

1.7

M(~t+x-p) [GeVl

Figure 1. Distribution in invariant mass of the p7r+s- system subject to various constraints described in the text. Experimental data are represented by filled circles with statistical error bars, while fitted smooth curves result in indicated position and 0 width of the peak of interest. Left panel: a fit to the data of a Gaussian plus a third-order polynomial is shown. Right panel: the PYTHIA6 Monte Carlo simulation is represented by the gray shaded histogram, the mixed-event model normalized to the P Y T H I A 6 simulation by the fine-binned histogram, the known C*+ resonances by the dotted curves, and the narrow Gaussian for the peak of interest by the dashed curve.

A narrow peak is observed at 1528.0 f 2.6 f 2.1 MeV with a Gaussian width of (T = 8 f 2MeV and a statistical significance of N,/SN, = 3.7. Here, N , is the full area of the peak from a fit to the data of a Gaussian plus a third-order polynomial, and SN, is its fully correlated uncertainty. All correlated uncertainties from the fit, including those of the background parameters, are accounted for in SN,. There is no known positively charged strangeness-containing baryon in this mass region (other than the O+) that could account for the observed peak. In an attempt to better understand the signal and its background, two additional models for the background were explored.12 For the first model, a version of the P Y T H I A 6 code21 tuned for HERMES kinematics" is taken to

69

represent the non-resonant background, and the remaining strength in the spectrum is attributed to a combination of known broad resonances and a new structure near 1.53GeV. For the second model, the non-resonant background is simulated by combining from different events a kaon and proton that satisfy the same kinematical requirements as the tracks taken from single events in the main analysis. This procedure yields a shape that is very similar to that from the PYTHIA6 simulation, as shown in Fig. 1 (right panel). By fitting a polynomial to the mixed-event background normalized t o the PYTHIA6 simulation, a peak is obtained a t 1527.0 f 2.3 f 2.1 MeV with a Gaussian width of (T = 9.2 f 2 MeV and a statistical significance of Ns/SNs = 4.3. The resulting values from the two fits for the centroid are found to be consistent, while the width and significance depend on the method chosen to describe the remaining strength of the spectrum. Using a O+ “toy Monte Carlo” with I’0+=2 MeV, an instrumental width of 10-14.6MeV (FWHM) was derived. This is somewhat smaller than the observed 19-24MeV (FWHM) width of the peak.” Therefore, the peak of interest was re-fit with a Breit-Wigner form convoluted with a Gaussian whose width was fixed at the simulated resolution. The resulting value for the intrinsic width is r = 17 f 9(stat)f3(syst) MeV. In order t o study the isospin of the observed resonance, the possibility that the O++ partner is present in the M p ~ spectrum + was explored. Although Fig. 2 shows a clear peak for the R(1520) in the M p ~ invariant mass spectrum, there is no peak structure observed in the M p ~ invariant + mass distribution. The upper limit of zero counts is at the 91% confidence level. The failure t o observe a Of+ suggests that the observed 0 is not isotensor and is probably isoscalar. Estimates of the spectrometer acceptance times efficiency from the toy Monte Carlo simulation mentioned above were used t o estimate some cross sections. Taking the branching fraction of the O+ t o pKg t o be 1/4, the cross section for its photoproduction is found t o range from about 100 t o 220 nb f25%(stat), depending on the model for the background and the functional form fitted t o the peak. The cross section for photoproduction of the h(1520) is found to be 6 2 f l l ( s t a t ) nb. All of these estimates are subject to an additional factor of two uncertainty, t o account for the assumptions about the kinematic distribution of the parents used in the simulation. *The indicated range in width depends on the background model and on the mass reconstruction method used.12

70

~ 6 -t

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w2500

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WPKL M(pK+) [GeVl Figure 2. Spectra of invariant mass MpK- (top) and MpK+ (bottom). A clear peak is seen for the A(1520) in the M p K - invariant mass distribution. However, no peak structure is seen for the hypothetical @++ in the M p K + invariant mass distribution near 1.53 GeV.

A comparison of the mass values reported to date for the Q+ state by other experiments to the present results indicates that there are large variations in mass. However, this is not uncommon for new decaying particles. Nevertheless, there is clearly a need for better estimates of experimental uncertainties. By fitting the available mass values6-17 with a constant, the weighted average is 1532.5 f 2.4 MeV. The uncertainty of the average was scaled by the usual2' factor of square root of the reduced x2. It is important to note though, that the experimental status of pentaquark baryons is still controversial. The signal of the Q+ claimed by Refs. 6-17 fails t o appear in the data from a large number of other experiments, as shown in Table 1. Unfortunately, none of these eleven results have been published at the time of this workshop, and only three23 have appeared on the e-Print archive server.24 3.1. Systematic Studies The general experimental situation is quite unsettled at this point. There are many opcn questions that need to be answered: how real are the positive results, and equally, how real are the null results; what is the actual mass, intrinsic width, the spin and parity, etc., of this new particle. In particular, the positive results need t o be checked whether they were produced from fake peaks that can arise from kinematic reflections or from detector acceptance and kinematic constraints on the data. While the former

71 Table 1. Summary of null results for three possible pentaquark states. Evidence for the 2--(1862) and the 8,(3100) has been reported by Refs. 25 and 26, respectively. Only the null results from the experiments with a *-symbol have appeared on the e-print archive server at the time of this workshop. Experiment

HERA-B' E690 CDF HyperCP BaBar ZEUS ALEPH DELPHI PHENIX* FOCUS BES'

0+(1540)

2-- (1862)

8,(3100)

(uuddS)

(ddssS)

(uud&)

NO NO NO

NO NO NO

NO

NO NO NO

NO NO

NO NO Yes NO NO NO

NO NO

Reaction

p A -+ Q + X , 2 - - X p p -+ Q + X , E--X p p - + Q + X , Z - X , Q'X T , K , p -+ Q+X e+e- + Q+X, 8 - - X e p -+ Q + X , 8 - - X , Q'X e f e - -+Q+X e + e - -+ C- K o p

AuAu -+ Q + X yA-+QCX e + e - + J/* -+ 0+0-

check has not been entirely done for the present result, the PYTHIA6 and O+ Monte Carlos have not produced any fake peaks due t o acceptance or kinematic constraints. Because the present experiment does not precisely determine the strangeness of the observed peak, the question has arisen whether it is a true pentaquark (with strangeness S=+l)state or a previously unobserved C*+ resonance. Under the assumption that the peak is a C*+ resonance, there should also appear a peak in the MAT+ spectrum.27 However, as can be seen in Fig. 3, no peak appears in the MAn+ spectrum near 1.53GeV1 even though the well established C( 1385)+ baryon resonance is clearly seen. Furthermore, the MA=- spectrum clearly shows the well known C(1385)and z- states, demonstrating the ability of the current experiment to identify narrow resonances in AT invariant mass spectra near 1530MeV. This indicates that the observed peak in the M,,+,invariant mass spectrum cannot be a previously unobserved C*+ resonance.28 Although the present experiment has excellent particle identification for protons and kaons, the Mpa+Tinvariant mass spectrum exhibits a relatively large background under the peak of interest. Most of this background is from K s mesons that originate from processes other than O+ decay. A large fraction of such K s mesons from exclusive processes can be removed if an additional hadron is required in the event. Due to the limited acceptance of the HERMES spectrometer, however, this additional requirement reduces the number of events that passed all the kinematic constraints from

72

E*-

3-

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u

900 800 700

600

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500 300

200 100 n "

1.4

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1.8

2

M [ GeV]

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1.8

2

M[GeV]

Figure 3. Distribution in invariant mass of the (pr-)r- (left panel) and (pr-)r+ (right panel) system. There is no peak in the (pr-)r+ invariant mass spectrum near 1.53GeV. The shaded histogram represents the mixed-event background.

1203 to 395, in 86% of which the extra hadron was a pion. Requiring the additional hadron t o be a pion, here termed 7r4th, removes K S mesons from the process y p -+ 4 p -+ K:K:p, because it is rather unlikely t o detect a pion from K L decay in our detector. The additional pion further helps t o remove K s from p(K*)* -+ pK:7rZh by introducing a new veto constraint on the M p n invariant mass. Since the proton and the additional pion can come from a A -+ p7r& decay, an additional veto constraint is placed on the M - invariant mass. Figure 4 shows the O+ mass spectrum with an pT4th additional pion in the event after applying these two new constraints (on the K* and A) in addition to all the standard kinematic constraints. The mass and the width of the peak are in good agreement with the published results,12 however, the ratio of signal t o background improves from 1:3 (see Fig. 1) to 2:l. It was further investigated whether the fourth hadron could come from the following exclusive processes: ~p -+ F O + -+ (7r+7r-)(K;p -+ ~ + 7 r - p ) , or yn 4 K-O+ --+ K-(K:p + 7r+7r-p). Results from a Monte Carlo study revealed that the associated K - or K s from these exclusive processes go t o backward angles, and that even the pions from K s decay are inaccessible with the HERMES detector. This is due to the PID threshold on the proton, which requires that the momentum of the O+ must be larger than 7 GeV. Therefore, the tagged pion events cannot originate from these exclu-

73 eD + K p X

.

s

$15 V

& 10

5

1.45

1.5

1.55

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M(n*fp) [GeV]

Figure 4. Invariant mass distribution of Mprr+rr- with an additional pion, subject to the constraints in event topology discussed in the text.

sive processes, which implies that the production cross section has to be a t least partially inclusive. This is an interesting observation because it does not support a possible explanation for the discrepancy between the positive results from low energy experiments, which are mainly from exclusive reactions, and the null results at high energy experiments, which are primarily inclusive measurements. The idea was that the O+ would be produced in exclusive processes, and the cross sections for such processes typically decrease with increasing energy (with the exception of elastic scattering). Even though the cross sections for inclusive processes tend t o increase with energy, it was hypothesized that the Q+ would not be produced in inclusive processes, thus failing t o appear in high energy experiments. However, the present new data (as well as the observation of the Of by ZEUS13) appear to contradict this idea.

Acknowledgments

I wish to thank my colleagues in the HERMES collaboration. I acknowledge Avetik Airapetian and Andy Miller for critical reading of the manuscript. The author’s research is supported in part by the U.S. National Science Foundation, Intermediate Energy Nuclear Science Division under grant No. PHY-0244842.

74

References 1. R.L. Jaffe, Proc. Topical Conference on Baryon Resonances, Oxford, July 1976, SLAC-PUB-1774. 2. M. Chemtob, Nucl. Phys. B256,600 (1985). 3. H. Walliser, Nucl. Phys. A548,649 (1992). 4. D. Diakonov, V. Petrov, and M. Polyakov, Z. Phys. A359,305 (1997). 5. R. Arndt, I. Strakovsky, and R. Workman, Phys. Rev. C68, 042201 (2003); nucl-th/O311030. 6. LEPS Collaboration, T. Nakano et al., Phys. Rev. Lett. 91,012002 (2003). 7. DIANA Collaboration, V.V. Barmin et al., Phys. Atom. Nucl. 66,1715 (2003); Yad. Fiz. 66,1763 (2003). 8. CLAS Collaboration, S. Stepanyan et al., Phys. Rev. Lett. 91,252001 (2003). 9. SAPHIR Collaboration, J. Barth et al., Phys. Lett. B572, 127 (2003). 10. A.E. Asratyan, A.G. Dolgolenko, and M.A. Kubantsev Phys. Atom. Nucl. 67,682 (2004); Yad. Fiz. 67,704 (2004). 11. CLAS Collaboration, V. Kubarovsky et al., Phys. Rev. Lett. 92, 032001 (2004); erratum ibid. 92,049902 (2004). 12. HERMES Collaboration, A. Airapetian et al., Phys. Lett. B585,213 (2004). 13. ZEUS Collaboration, S. Chekanov et al., Phys. Lett. B591,7 (2004). 14. SVD Collaboration, A. Aleev et al., hep-ex/0401024. 15. COSY-TOF Collaboration, M. Abdel-Bary et al.,Phys. Lett. B595, 127 (2004). 16. P.Zh. Aslanyan, V.N. Emelyanenko and G.G. Rikhkvizkaya, hep-ex/0403044. 17. Yu.A. Troyan et al., hep-ex/0404003. 18. HERMES Collaboration, K. AckerstaE et al., Nucl. Instr. Meth. A417,230 (1998). 19. N. Akopov et al., Nucl. Instr. Meth. A479,511 (2002). 20. Particle Data Group, K. Hagiwara et al., Phys. Rev. D66,010001 (2002). 21. T. Sjostrand et al., Comput. Phys. Commun. 135,238 (2001). 22. E.-C. Aschenauer, P. Liebing, and T. Sjostrand, in preparation. 23. As of 18-July-2004, the following entries have appeared on ArXiv: K.T. Knopfle, M. Zavertyaev, and T. Zivko (HERA-B collaboration), hep-ex/0403020; C. Pinkenburg (PHENIX collaboration), nucl-ex/0404001; J.Z. Bai, et al., (BES Collaboration), hep-ex/0402012. 24. e-Print archives is accessible at “http://arxiv.org/” . 25. NA49 Collaboration, C. Alt et al., Phys. Fkv. Lett. 92,042003 (2004). 26. H1 Collaboration, A. Aktas et al., Phys. Lett. B588,17 (2004). 27. Maxim Polyakov, private communications. 28. Frank Close, private communications.

75

STUDY OF NARROW BARYONIC PENTAQUARK CANDIDATES WITH THE ZEUS DETECTOR AT HERA*

urn KARSHON~ Weizmann Institute of Science, Rehovot, Ismel E-mail: [email protected]

The three pentaquark candidates 0+(1530), E(1862) and 82(3100) have been studied in ep collisions a t a centre-of-mass energy 4 = 300 - 318 GeV using the full luminosity of the HEM-I data. Searches for narrow baryonic states in the decay channels K i p , K+p, E-n*, =+n* and D**pT are reported. The results support the existence of a narrow resonance decaying into K i p and KZp, consistent with the 8(1530) state. No signals are seen in the K + p , T n * , Z+n* and D**pT channels.

1. Introduction and experiment

The H E M e-p collider accelerates electrons (or positrons) and protons to energies of E, = 27.5 GeV and Ep = 920 GeV (820 GeV until 1997), respectively. The two collider experiments, H1 and ZEUS, are located at two collision points along the circulating beams. The incoming e* interacts with the proton by first radiating a virtual photon. The photon is either w 3 GeV2, where Q2 is quasi-real with Q2 < 1 GeV2 and Qkedian the negative squared four-momentum transferred between the electron and proton, or highly virtual (Q2 > 1 GeV2). In the former case no scattered electron is visible and this is the photoproduction (PHP) regime. In the latter case the scattered electron is measured in the main detector and this is the deep inelastic scattering (DIS) regime. The analysis was performed with ZEUS data taken between 1995 - 2000 (“HERA-I”), corresponding to an integrated luminosity of w 120 pb-l. Charged particles are tracked in the Central Tracking Detector (CTD) covering polar angles of 15” < 0 < 164”.

-

______

*This work is supported by the Israel Science Foundation and the U.S.-Israel Bi-national Science Foundation t o n behalf of the ZEUS Collaboration

76

The energy loss of particles in the CTD, d E / d z , is estimated from the truncated mean of the anode-wire pulse heights, after removing the lowest 10% and at least the highest 30% depending on the number of saturated hits. The d E / d z resolution for full-length tracks is about 9%. 2. Evidence for the strange pentaquark 0+(1530)

Fixed-target low-energy experiments saw a narrow exotic baryon with strangeness +1 around 1530 MeV decaying into Kfn. It was attributed to the O+ = uudd3 pentaquark candidate predicted by Diakonov et al. at the top of a SU(3) spin 1 / 2 anti-decuplet of baryons. Narrow peaks were also seen at a similar mass in the final state Kgp, which is not necessarily exotic. They were attributed to the Of as well. It is interesting to search for the O+ baryon in high-energy collider experiments. In particular it can be searched at the central rapidity region, which has little sensitivity to the proton remnant region. This region is dominated by parton fragmentation with no net baryon number, unlike low-energy experiments, where the pentaquark is mainly produced in the nucleon fragmentation region. The ZEUS search for the O+(1530) used the 1996 - 2000 H E W data (121 pb-l) and was performed in the DIS regime (Q2 > 1 GeV2). The search in the Kop mode was complicated due to a few unestablished resonances, such as C(1480) and C(1560), called “C bumps” 3. There are no such known bumps around the O+ mass range; however, it is difficult to describe the background under a 0 signal due to these C bumps. KZ particles were reconstructed from secondary-vertex CTD tracks with transverse momenta p~ > 0.15 GeV and pseudo-rapidities (q( < 1.75. The KO transverse momenta and pseudorapidities were required to have ~ T ( K O>) 0.3 GeV and lq(P)l< 1.5. A very clean KO + T+T- signal After requiring 0.483 < M(?r+n-) < 0.513 GeV, the was obtained number of Ki candidates was FJ 867, OM) with only w 6% background. Protons and antiprotons were selected from a wide d E / & proton band, motivated by the Bethe-Bloch equation, defined for primary-vertex tracks ’. Pion and kaon contamination was reduced by requiring the proton momentum to be less than 1.5 GeV and d E / d x to be above 1.15 minimum ionising particles (mips). The purity of the proton sample obtained was w 60%. The K;p@) mass spectrum is shown in Fig. l(a-f) with a minimum Q2 ranging from 1 to 50 GeV2, as well as with Q2 > 1 GeV2 for two separate bins of the photon-proton centre-of-mass energy, W. For Q2 > 10 GeV2 or Q2 > 1 GeV2 and W < 125 GeV, a peak is seen near 1.52 GeV. The histograms are the ARIADNE Monte Carlo (MC) simulation, normalised

’.

77

to the data above 1.65 GeV. The shape of the data distributions is not well described by the MC which does not simulate the E bumps. ZEUS

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Figure 1. M(K!&(p)) for (a-d) Q2> 1,10,30,50 GeV'; (e-f) W < 125 and > 125 GeV; (g) Q2 > 20 GeV2. The histograms are MC predictions normalised to the data above 1.65 GeV. The solid line in (g) is a fit to the data using a background function (dotted Line) plus two Gaussians (dashed lines). The inset shows the K i p (open circles) and K i p (black dots) combinations, compared to the combined sample fit scaled by 0.5.

In Fig. l(g) the K$ p @) mass spectrum is shown for Q2 > 20 GeV2 together with a fit to two Gaussians and a background of the form Pl(M - mp - m ~ o ) ~(1 2 + q ( M - mp - m ~ o ) )where , M is the K i p mass, mp ( m p ) is the proton (KO) mass and P1,2,3are free parameters. The fit X2/ndf (35/44)is significantly better than a one-Gaussian fit for the 0 only. The improvement is mainly in the low mass region, where the second resonance may correspond to the C(1480). The 0 peak position is M = 1521.5f l.Fj(stat.)f::s,(syst.) MeV, with a Gaussian width o = 6.1 f 1.6(stat.)f::~(syst.) MeV, which is above, but consistent with the resolution (B 2 MeV). The fit gives 221 f48 events above background, corresponding to 4.60. The probability of a fluctuation leading to the observed signal in the mass range 1.5 - 1.56 GeV is below 6-10-5.Fitting the 0 with a BreitWigner convoluted with a Gaussian fixed to the experimental resolution, the intrinsic full width of the signal is estimated to be r = 8 f 4(stat.) MeV.

-

78

The signal is seen for both proton charges (inset in Fig. lg). The fitted number of events in the Kgp channel is 96 f 34. If the signal originates from the 0,this is a first evidence for the anti-pentaquark 0-. The 0 production cross section was measured in the kinematic region Q2 > 20 G e V , 0.04 < y < 0.95, p~(0) > 0.5 GeV and 1q(0)1< 1.5 to be o ( e p + eO*X + e p p * X ) = 125 f 27(stat.)+~~(syst.)pb,where y is the lepton inelasticity. The acceptance was calculated using the RAPGAP MC, where C+ baryons were treated as 0+ with M = 1.522 GeV, forced to decay 100% to p S p ( f j ) . The 0 visible acceptance was w 4%. Fig. 2(a) values. The crossshows the cross section integrated above several section ratio to that of A(1116), R = o(Q+ + K o p ) / a ( A )(antiparticles are included), was measured in the same kinematic region. A baryons were measured in the decay mode A + p?r- and protons were selected by d E / d z with identical cuts as for the 0 . The A acceptance ( w 10%) ' was calculated using the ARIADNE MC. The result for Q2 > 20 GeV2 is R = (4.2 &0.9?;::)%. Fig. 2(b) shows R for these Q i i n values. It is not compatible with upper limits from HERA-B and ALEPH, where R < 0.5%.

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If the observed 1.52 GeV peak is due to a I = 1 state, a Of+ signal is expected in the K + p and K-fi spectrum. Selecting protons and charged kaons using dE/ds,no peak was seen in the above distribution. A clean 10a

79

A(1520) -+ K - p or K+p signal was seen with mass and width consistent with the PDG values 3. The number of A(1520) and h(1520) are similar. 3. Search for pentaquarks in the 8.rr channels

The p p fixed-target NA49 Collaboration (4= 17.2 GeV) reported observation of the E multiplet pentaquark candidates E;, and Z!$2 predicted at the bottom of the anti-decuplet of baryons. They found narrow peaks in these EA combinations at M m 1862 MeV with a width < 18 MeV. The significance of the signal for the sum of all 4 3~ channels is 5.80. ZEUS searched for such states in its DIS HEM-I data '. E-(z+) states were reconstructed via the A A - ( ~ T + ) decay channel, with A -+ p- (A + @+). Very clean A and 9 signals were obtained with m 130000 A A and w 2600 E candidates. In Fig. 3 the ZA invariant mass spectrum for Q2 > 1 GeV2 is shown. The left histograms show each charge combination separately. The right histogram is the sum of all EA combinations. A clean EO(1530) + ZA signal of m 4 . 8 ~ is seen in the combined plot. No evidence is seen for the NA49 signal around 1862 MeV in any of the ET mass plots. No signal is visible also with Q2 > 20 GeV2. The discrepancy may be due to the fact that the ZEUS results come from the central rapidity region, while NA49 also covers the forward region.

+

+

Figure 3. M(%n)for Qa > 1 GeVa for each charge combination (left) and for all charge combinations combined (right).

4. Search for a charmed pentaquark decaying to D**pF

The existence of the strange pentaquark @+ implies that charmed pentaquarks, @: = uuddE, should also exist. One type of model predicts M(@Z) w 2710 MeV, which is below the threshold to decay strongly

80

to D mesons. Another model predicts a C3: which decays mainly to D - p or Don (charge conjugate included) with M ( 0 : ) = 2985 MeV and r(0:) w 21 MeV. If Ad(@:) is above the s u m of the D* and p masses (2948 MeV), it can decay also to D**pF. The H1 Collaboration found a narrow signal in the D**pf invariant mass at 3.1 GeV with a width consistent with the detector resolution. The signal was seen in a DIS sample of w 3400 D** + Don* + (K%*)T* with a rate of w 1%of the visible D* production. A less clean signal of a comparable rate was seen also in the H1 PHP sample. The 0: search of ZEUS in the D**:pF mode was performed with the full H E M - I data lo. Clean D** signals were seen in the AM = M(D**) - M ( D o ) plots (Fig. 4 left). Two D** + Don* decay channels were used with Do + KFnh and Do + Krn*t,+~-. The 0: search was performed in the kinematic range 1q(D*)1 < 1.6 and pr (D*) > 1.35 (2.8) GeV and with A M values between ) In 0.144 - 0.147(0.1445 - 0.1465) GeV for the KTT ( K n n n ~channel. these shaded bands a total of w 62000 D*’s was obtained (Fig. 4a-b left) after subtracting wrong-charge combinations with charge f 2 for the Do candidate. Selecting DIS events with Q2 > 1 GeV2 yielded smaller, but cleaner D* signals with a total of w 13500 D*’s (Fig. 4c-d left). Protons were selected with p ~ ( p >) 0.15 GeV. To reduce the pion and kaon background, a parameterisation of the expected dE/dx as a function of P / m was obtained using tagged protons from A decays and tagged pions from K: decays. The x2 probability of the proton hypothesis was required to be above 0.15. Fig. 5 shows the M ( D * p ) = M ( K x n p ) - M ( K n n ) M ( D * ) P D Gdistributions for the KTT channel for the full (left) and the DIS (right) samples, where M ( D * ) ~ D is G the D** mass 3. In the lowP selection (Fig. Sb), a clean proton sample separated from the n and K dE/dx bands was obtained by taking only tracks with P < 1.35 GeV and d E / d z > 1.3 mips. In the high-P selection (Fig. 5c) only tracks with P (p ) > 2 GeV were used. The latter selection was prompted by the H1 observation of a better C3: signal-to-background ratio for high proton momenta. No narrow signal is seen in the Knn (Fig. 5) as well as in the K n m (Fig. ~ 4b,d right) channel. The Kmr analysis was repeated using very similar selection criteria as in the H1 analysis g. No indication of a narrow resonance was found in either the DIS or the PHP event sample lo. 95% C.L. upper limits on the fraction of D* mesons originating from C3: decays, R(0: + D * p / D * ) , were calculated in a signal window 3.07 < M ( D * p ) < 3.13 GeV for the Knn and K m r m channels. A visible

+

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Figure 4. Left: AM distributions (dots) for (a) D' + Knn and (b) D' + Kmrnr candidates. Events with Qa > 1 GeVa for the two channels, respectively, are shown in (c) and (d). The histograms are for wrong charge combinations. Right: M(D'*pF) distributions (dots) for the same samples. Solid curves are fits to a background function (see text). Shaded historgams are MC 0: signals, normalid to e : / D * = 1%, on top of the background fit.

rate of 1% for this fraction (Fig. 4 right), as claimed by H1 ', is excluded by 90 (50) for the full (DIS) combined sample. The M ( D * p )distributions were fitted to the form xae-bx+cx2, where x = M ( D * p )- M ( D * ) - m, (Fig. 4 right). The number of reconstructed 0: baryons was estimated by subtracting in the signal window the background function from the observed number of events, yielding R(O: + D*p/D*)< 0.23% and < 0.35% for the full and DIS combined two channels. The acceptance-corrected rates are, respectively, 0.37% and 0.51%. The 95% C.L. upper limit on the fraction of charm quarks fragmenting to C3: times the branching ratio 0: + D*p for the combined two channels is f(c + 0:)Beg-, D~~ < 0.16% (< 0.19%) for the full (DIS) sample. 5. summary

The ZEUS HERA-I data sample was used to search for narrow baryonic pentaquark candidates. For the inclusive DIS sample a 4.60 narrow signal was seen in the fragmentation region in the combined M ( K g p ) and M ( K @ plot at the Of mass range. If due to the 0 baryon, this is the first evidence

82 ZEUS

ZEUS

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3.3

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3.6

(GeV)

0 2.9

3

3.1 3.2 3.3 3.4 M(D'p) =AM'* + M(D*),

3.5

3.6

(GeV)

Figure 5. Left: M ( D * * p F ) distributions for the Kmr channel (dots) with (a)all proton candidates, (b) candidates with P ( p ) < 1.35 GeV and dE/& > 1.3, and (c) candidates with P(p) > 2 GeV. Histograms show the M(D"*p*) like-sign combinations. Right: Same for DIS events with Qa > 1 GeVa.

for the anti-pentaquaxk W.The cross-section ratio a(@+ + P p ) / a ( A ) for Q2 > 20 GeV2 is (4.2 f 0.9fA:;)%. No evidence is found for the NA49 %r signal at 1862 MeV in the inclusive DIS sample. No resonance structure is seen in M(D**pr) around 3.1 GeV. The 95% C.L. upper limit on the visible rate R(O: + D*p/D*)is 0.23% (0.35%for DIS). The ZEUS data are not compatible with the H1 result of = 1%of the above rate. Such a rate is excluded by 90 for the full data and by 50 for the ZEUS DIS data. References D. Diakonov, V. Petrov and M.V. Polyakov, 2.Phys. A369,305 (1997). ZEUS Coll., S. Chekanov et al., Phys. Lett. B691, 7 (2004). Particle Data Group, K. Hagiwara et al., Phys. Rev. D66,10001 (2002). ZEUS Coll., Abstract 273, XXXII Int. Conf. on High Energy Physics, ICHEP2004, August 2004, Beijing, China. 5. NA49 Coll., C. Alt et al., , Phys. Rev. Lett. 92, 42003 (2004). 6. ZEUS Coll., Abstract 293, XXXII Int. Conf. on High Energy Physics, ICHEP2004, August 2004, Beijing, China. 7. A. J d e and F. Wilczek, , P h p . Rev. Lett. 91,232003 (2003). 8. M. Karliner and H.J. Lipkin, Preprint hep-ph/0307343 (2003). 9. H1 Coll., C. Atkas et al., , Phys. Lett. B688, 17 (2004). 10. ZEUS Coll., S. Chekanov et al., hep-ex/0409033, Eur. Phys. J., in print. 1. 2. 3. 4.

83

PENTAQUARKS WITH CHARM AT H1

M I N E R STAMEN (DESY, NOW AT KEK) Physics Division 1 Oh0 1-1, Tsuhba, 305-0801, Japan E-mail: stamenObmail.kek. j p A narrow resonance in D*-p and D*+p invariant mass combinations is observed in inelastic electron-proton collisions at centreof-mass energies of 300 GeV and 320GeV at the HERA collider. The resonance has a mass of 3099 f 3(stat.) f 5 (syst.) MeV and a measured Gaussian width of 12 f 3 (stat.) MeV, compatible with the experimental resolution. The resonance is interpreted as a n anti-harmed baryon with thc minimal constitucnt quark composition of uuddE, togcthcr with its charge conjugate state.

1. Introduction

With the discovery of the strange pentaquark' and the subsequent confirmation of this new state by several experiments the question whether there also exist other pentaquarks with different quark content became apparent. Several theoretical predictions for a pentaquark containing a charm quark exist which range from 2700 MeV (see e.g. Jaf€e et al. 2, up to about 3000MeV (see e.g. Karliner et al. 3 ) . An analysis by the H1 experiment was performed which searches for a charmed pentaquark in the D*-p decay mode and its charge conjugate '. This analysis is only sensitive to masses starting at about 2950MeV due to the large mass of the D* meson. However, the search for charmed pentaquarks in the Dp decay mode is not performed due to large combinatorial background for the D meson reconstruction. At HERA 27.5 GeV electrons collide with protons of 820 GeV (920 GeV in recent years) yielding a centre of mass energy of 300 GeV (320 GeV). In ep interaction charm and anti-charm quarks are produced predominantly in boson-gluon-fusion processes. The kinematic variables which describe the process are the photon virtuality Q2 and the invariant mass of the photon

84

proton system W . Two distinct kinematic regimes are studied: the deep inelastic scattering (DIS) regime is characterised by Q2 > 1GeV2, while in the photoproduction domain Q2 is restricted to Q2 < 1GeV2. The analysis is carried out using data taken in the years 1996-2000 with an integrated luminosity of 75 pb-’.

2. Selection of D* and proton candidates

The identification of D* mesons proceeds via the decay chain: D*+ + Don$ with Do ---* K-n+ (including charge conjugated modes). r3denotes the slow pion from the primary D* decay which has typically low momentum. For the D* selection the mass difference technique is used, based on the variable A M D . = m(Knn,) - ~ ( K Twhere ) ~ ( K T T ,and ) m(Kn) are the invariant masses of the corresponding combinations. In figure 1 a typical A M p distribution is shown for the DIS data sample. A prominent signal is seen around the expected M ( D * )- M ( D o ) mass difference. The distribution is compared with “wrong charge D” background where the Do is replaced by fake “D-mesons” composed of like charge KT. Proton candidates are selected on the basis of the dE/dx measurement of the drift chamber with an average resolution of 8%.

200 0 0.13

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1. AM,* distribution for combinations. The combinatorial background determined from a wrong charge D sample is shown as shaded histogram. Figure

KF?r*?r,f

lo-’

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Figure 2. Specific ionisation energy loss relative to that of a minimally ionisiig particle, plotted against momentum.

85

3. Analysis of D * p combinations Candidate D* mesons having a A M p value in a window &2.5MeV around the nominal M ( D * ) - M ( D o ) mass difference are combined with proton candidates. The mass of the D*p state is calculated as M (D* p ) = m (Knn,p) - m ( K m , ) + m p D G (D*). A clear narrow peak is observed in the invariant mass distribution as shown in figure 3. The data are compared with the predictions from a D* Monte Carlo simulation and the “wrong charge D” background model which accounts for combinatorial background. No enhancement is seen, neither in the charm MC simulation nor in the non charm background estimated from data, while the shape of the background is well described. No significant enhancement is observed in likecharge D*p combinations. Possible kinematic reflections that could fake the signal have been ruled out by studying invariant mass distributions and correlations involving the K ; 71, T , and proton candidates under various mass hypotheses. All events in the M ( D * p )distribution have been scanned visually with no anomalies observed in the reconstruction of the candidate tracks. Extensive studies were performed to test the D*and proton content of the signal. It was shown that the D*p signal is enriched with D* in comparison to the sidebands. The signal is also visible for low momentum proton candidates where protons can be unambiguously identified. The signal is also visible in the independent photoproduction data sample as shown in figure 4. In this analysis the combinatorial background to the D* selection is significantly larger than for the DIS sample which necessitates tighter selection criteria for the proton and D* candidates.

2

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3.2

34 3.6 M (PP) [ GeV 1

Figure 4. M (D’p) distribution for the photoproduction data sample.

86

4. Signal assessment

The momentum distribution of the proton candidates without any dE/dz requirement shown in figure 5 reveals a significantly harder spectrum in the D*p signal region compared to the sidebands. This supports the expected change in the D*p kinematics. The fits to the M ( D * p ) distribution in DIS are shown in figure 6 . A Gaussian distribution is used for the signal shape yielding a r.m.s of 12 f 3 (stat.). The background is parametrised with a power law and the mass of the resonance is determined to be 3099 f 3 (stat.)f 5 (syst.). The probability that the background distribution fluctuates to produce the signal is calculated considering the backgroundmly hypotheses (seedashed line in figure 6 ) to be less than 4 x lo-' which corresponds to 5.4 0 in terms of Gaussian standard deviations. A state strongly decaying to D*-p must have baryon number +1 and charm -1 and has thus the minimal constituent quark composition of uu&C.

Figure 5. Momentum spectrum of the proton candidates.

Figure 6. Fits t o the DIS data sample. The line shows the fit result including a signal and a background component. The dashed line shows the background only fit.

References 1. T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91 (2003) 012002 [arXiv:hep-ex/0301020]. 2. R. L. J&e and F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003 [arXiv:hepph/0307341]. 3. M. Karliier and H. J. Lipkin, arXiv:hep-ph/0307343. 4. A. Aktas et &. [HlCollaboration], Phys. Lett. B 588 (2004) 17 [arXiv:hepex/M03017].

87

PENTAQUARK SEARCH VIA ( T - , K-)REACTION

K. MIWA FOR T H E E522 COLLABORATION Department of physics, Kyoto University, Kyoto 606-8502, Japan E-mail: [email protected] An experiment to search for 8+ via ( x - , K - ) reaction was carried out for the first time. The preliminary upper limit of the production cross section of 8+ was estimated t o be a few pb.

1. Introduction

The O+ particle was first observed by SPring-g/LEPS collaboration1. Since @+ has positive strangeness, its minimal quark content is (uudds) which means @+ is a manifestly exotic particle. This discovery stimulated many physicists. Much theoretical work has been made. Experimentally, this observation was immediately confirmed by other experiment^^?^?^. Most of these experimental data are from photo-production experiments and high energy experiments. On the other hand, the experimental result via mesonic reaction is only from the DAINA collaboration, where a K+ beam and Xe bubble chamber were used2. The physical properties such as spin, parity and width have not been determined yet experimentally. To determine them, an experiment with higher statistics is a must. In general, mesonic production should have higher cross section than photo-production. Therefore it is quite important to confirm @+ with high statistics in hadronic reactions using K+ and T - beam. We focused attention on ( T - , K - ) reaction. This reaction was investi. backgrounds gated in a bubble chamber experiment in 1 9 6 0 ' ~ ~Physical are phase space, production and A( 1520) production. The cross section for these reactions are about 25.2pb, 30.0f8.8 p b and 20.8f5.0 pb, respectively. This experiment also checked the exotic channels such as K+n and Kop. Although the statistics was very poor, the expected peak was not observed. Theoretically the cross section via ( T - , K - ) reaction is calculated to be a few to a few hundred pb by W. Liu et aL6 and Y . Oh et aL7.

88

7

Figure 1. Mass spectrum obtained by (n-,K - ) data. We selected negative particle at the 1st level trigger.

Figure 2. Missing mass spectrum of (T+,K+) reaction. The obtained peak position of C+ is 1.185f0.002GeV/c2.

2. Experiment and analysis

We have performed the E522 experiment at the K2 beamline of KEK 12GeV Proton Synchrotron in February 2004. The main objective of this experiment was to search for H-dibaryon resonance with ( K - , K + ) reaction. Besides this reaction, we optionally took ( T - , K - ) data (about 3days), because the @+ search via mesonic reaction was important and the K2 beamline is unique beamline which can provide a high-momentum T beam. We used T - beam extracted at 1.9 and 1.95 GeV/c. As a target, we used a scintillation fiber (SCIFI) target consisted of CH and a bulk target of CH2 to make the contribution from free protons larger. In this paper, we focus our work on the analysis of data with CH2 target at 1.95GeV/c. As the calibration data, we took carbon target data to investigate the contribution from carbon in CH2 target and (T+, K+) data to measure C+ peak position for the calibration of the missing mass. The experimental set up consists of two parts; one part is a beamline spectrometer to analyze momentum of each incident beam particle with the resolution of AP/P=O.5%, and the other part is forward spectrometer to detect scattered particles. It is similar to the one used for E3738. Figure 1 shows the obtained mass spectrum. The K - mesons are clearly identified. At first, we show the missing mass spectrum of (T+, K+) reaction which is just inverse reaction of ( T - , K - ) (see figure 2). We can clearly recognize the peak of C+. We fit this spectrum with two gaussian peaks assuming that the broad peak is attributed to quasi free protons in carbon and the narrow one is attributed to free protons. The obtained width is 12.8t1.9MeV/c2 which is consistent with the expected value of 12MeV/c2 from the simulation. Next, we will mention the analysis of ( T - , K - ) reaction. To select good

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(.-,IT-) events, we apply the following cuts; 1) K - selection using the relation between momentum and square of mass, 2) T - beam selection using the time-of-flight, 3) x2 cut of scatted particles, 4) x2 cut of beam particles, 5) vertex point cut, 6) distance of closest approach cut at vertex point. Figure 3 shows the preliminary missing mass spectrum of ( T - , K - ) reaction. It seems that there is a structure around 1.53GeV/c2. However there is a possibility that the structure is only statistical fluctuation. We fitted this histogram with the background of cubic function and gaussian peak. As the width of the gaussian peak, we used a=6.1 MeV/c2 expected from the simulation. The counts of this peak was obtained to be 154f62. The error is only statistical error, and systematic error is not estimated yet. The upper limit of the peak count is 256 at 90% confidence level (preliminary), and we use this count for the following calculations. We need more studies of background and systematic error. To investigate the contribution from the carbon in CH2 target, we analyzed carbon target data. In the figure 3, the missing mass spectrum of carbon target data normalized by beam counts and target thickness are also shown. The contribution from free proton and carbon are about 9,000 and 14,000 respectively. This ratio is consistent with that the effective nucleon number of carbon is about 3. Even if the structure around 1.53GeV/c2 is just statistical fluctuation, it is quite important to estimate the upper limit of the production cross section of O+ via the ( T - , K - ) reaction. We estimated the upper limit by two different ways.

90

The cross sections of the physical backgrounds are measured by the past experiment, and the sum of the cross section is about 77 pb. If the angular distribution of K - is equal for all reactions, the ratio of the counts from O+ and background is roughly equal to the ratio of the cross section of O+ and background. We used 256 counts as the number of the structure around 1.53GeV/2 and 9,000 counts as the background reacted with free proton. Thus, we obtained that the cross section is about 2.2pb. It is more straightforward to calculate the cross section covered by the spectrometer with the following equation.

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C O T ( U ~ U Z ~ S ~ S ) x cw(DAQ)

x Ttarget

Here Ne+ , Nbeam and TtaTget represent the number of Q+, the number of beam and the thickness of target respectively and cor(track), cor(decay), cor(ana1ysis) and cor(DAQ) are correction factors of the tracking efficiency, decay of K - , analysis efficiency and the deadtime of DAQ. The tracking efficiency and the analysis efficiency are not estimated precisely, but these efficiencies are greater than 0.75 and 0.56 respectively. By using these values, we obtain crsp 0.26pb. Assuming that K - is s-wave, about 10% of K - is accepted by the spectrometer. Then, if the K - is s-wave, the total cross section is estimated to be about 2.6pb. N

3. Summary

We studied ( T - , K - ) reaction to search for O+ with CH2 target at the KEK-PS K2 beamline. The upper limit of O+ production cross section via ( T - , K - ) reaction was estimated to be a few pb by two methods. Our analysis, however, is still ongoing and the data presented here are very preliminary.

References T. Nakano et al., Phys. Rev. Lett. 91,012002 (2003). V. V. Barmin e t al., Phys. Atom. Nucl. 66,1715 (2003). S. Stepanyan e t al., Phys. Rev. Lett 91,252001 (2003). J. Barth et al., Phys. Lett. B 572, 127 (2003). 0. I. Dahl et al., Phys. Rev. 163,1377 (1967). W. Liu and C. M. KO , Phys. Rev. C68,045203 (2003). Y. Oh, H. Kim, and S. H. Lee , Phys. Rev. D69,074016 (2003). 8. A. Ichikawa et al., Phys. Lett. B 500, 37 (2001).

1. 2. 3. 4. 5. 6. 7.

91

SEARCH FOR PENTAQUARKS AT BELLE

R. MIZUK * (BELLE COLLABORATION) Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya,25, 11 '7259 Moscow, Russia E-mail: mizuk Bitep .ru

We search for the strange pentaquark 0+ using kaon interactions in the material of the Belle detector. No signal is observed in the p K s final state, while in the p K final state we observe N 1.6. lo4 A(1520) -+ p K - decays. We set an upper limit on the ratio of 8+ to h(1520) yields a(Q+)/u(A(1520)) < 2% a t 90% CL, assuming that the 0+ is narrow. We also report on searches for strange and ch'kned pentaquarks in B meson decays. These results are obtained from a 155 fb-' data sample collected with the Belle detector near the T(45)resonance, at the KEKB asymmetric energy e+e- collider.

1. Introduction Until recently, all reported particles could be understood as bound states of three quarks or a quark and an antiquark. QCD predicts also more complicated configurations such as glueballs gg, molecules qqqQ and pentaquarks qqqqq. Recently, observations of the pentaquark Of = uuddi? have been reported in the decay channels Kfnl and pKs? Many experimental groups have confirmed this observation and the isospin 3/2 members of the same pentaquark multiplet have also been o b s e r ~ e d Evidence .~ for the charmed pentaquark O: = uuddE has also been seen.4 The topic attracts enormous theoretical interest. However the existence and properties of pentaquarks remain a mystery. Some experimental groups do not see the pentaquark signals. The non-observing experiments correspond to higher center-of-mass energies. It has been argued5 that pentaquark production is suppressed in the fragmentation regime at high energies. Charged and neutral kaons are copiously produced at Belle. We treat h n s as projectiles and the detector material as a target, and search for 'Work partially supported by Russian grant SS551722.2003.

92

strange pentaquark formation, KN + @+, and production, K N + W X . The kaon momentum spectrum is soft, with a most probable momentum of only 0.6GeV/c. Therefore we can search for CV formation in the low energy region. We also search for strange and charmed pentaquarks in the decays of B mesons, where the suppression of production observed in s channel e+ecollisions* may be absent. Studies of B meson decays have proved to be very useful for discoveries of new particles (such as P-wave cij states), therefore it is interesting to search for pentaquarks in B decays although no firm theoretical predictions for branching fractions exist. 2. Detector and data set

These studies are performed using a data sample of 140%-' collected at the "(45') resonance and 15 fb-' at an energy 60 MeV below the resonance. The data were collected with the Belle detector7 at the KEKB asymmetric energy e+e- storage rings.* The Belle detector is a large-solid-angle magnetic spectrometer that consists of a three layer silicon vertex detector (SVD), a 50-layer cylindrical drift chamber (CDC), a mosaic of aerogel threshold Cherenkov counters (ACC), a barrel-like array of time-of-flight scintillator counters (TOF), and an array of CsI(T1) crystals (ECL) located inside a superconducting solenoidal coil that produces a 1.5 T magnetic field. An iron flux return located outside the coil is instrumented to detect muons and K L mesons (KLM). The proton, kaon and charged pion candidates are identified based on the dE/dz, TOF and Cherenkov light yield information for each track. K s candidates are reconstructed via the R+T- decays and must have an invariant mass consistent with the nominal K s mass. The K s candidate is further required to have a displaced vertex and a momentum direction consistent with the direction from its production to decay vertices. 3. Search for O+ using kaon interactions in the detector material

The analysis is performed by selecting p K - , pK+ and pKs secondary vertices. The protons and kaons are required not to originate from the region around the run-averaged interaction point (IP). The proton and kam candidate are combined and the pK vertex is fitted. The xy distribution of the secondary pK- vertices is shown in Fig. 1 for the barrel part (left) and

93

E0 l o

10 2

10

0

-5 1 -10

x,cm

x,cm

Figure 1. The zy distribution of secondaryp K - vertices for the barrel (left) and endcap (right) parts of the detector.

for the endcap part (right) of the detector. The double wall beam pipe, three layers of SVD, the SVD cover and the two support cylinders of the CDC are clearly visible. The zy distributions for secondary pK+ and pKs vertices are similar. The mass spectra for p K - , pK+ and pKs secondary vertices are shown in Fig. 2. No significant structures are observed in the M ( p K + ) or M(pKs) spectra, while in the M ( p K - ) spectrum a A(1520) signal is clearly visible. We fit the pK- mass spectrum to a s u m of a A(1520) probability density function (p.d.f.) and a threshold function. The signal p.d.f. is a D-wave Breit-Wigner shape convolved with a detector resolution function (u 2MeV/c2). The A(1520) parameters obtained from the fit are consistent with the PDG values.Q The A(1520) yield, defined as the signal p.d.f. integral over the 1.48-1.56GeV/c2 mass interval (2.5I'), is 15519 f 412 events. The pKs mass spectrum is fitted to a sum of a Of signal p.d.f. and a third order polynomial. The O+ signal shape can be rather complicated because of possible rescattering of particles inside nuclei.'' In order to compare our result with other experiments we assume that the signal is narrow and its shape is determined by the detector resolution (N 2MeV/c2). For m = 1540MeV/c2 the fit result is 29 f 65 events. Using the FeldmanCousins method of upper limit evaluationll we obtain N < 94 events at the 90% CL. We set an upper limit on the ratio of O+ to A(1520) yields N

94

9.4

1.45

1.5

1.55

1.6

1.65 1.7

Figure 2. Maw spectra of p K + (left), p K - (right, points with error bars) and p K s (right, histogram) secondary pairs. The fit is described in the text.

corrected for the efficiency and branching fractions:

at the 90% CL. It is assumed that B(O+ + pKs) = 25%. We take B(A(1520) + p K - ) = $B(A(1520) + Nl?) = $(45 f l)%.’The ratio of efficiencies for Of + pKs and A(1520) 3 pK- of 37% is obtained from the Monte Car10 (MC) simulation. Our limit is much smaller than the results reported by many experiments which observe O+. For example it is two orders of magnitude smaller than the value reported by the HERMES Collaboration.12 We do not know any physical explanation for such a difference. The momentum spectrum of the produced A(1520) is shown in Fig. 3 (left). This spectrum is obtained from fitting M@K-) in momentum bins and correcting for the efficiency obtained from MC. The K - should have a 440 MeV/c momentum to produce A(1520) on a proton at rest. Even in the presence of Fermi motion with a typical momentum of 150MeV/c, A(1520) produced in the formation channel should be contained in the first momentum bin, 0.4 to O.SGeV/$. Therefore most of the A(1520) are produced in the production channel. The projectiles that can produce A(1520) are K - , Ks, K L , A. The momentum spectra of K- and K + are given in Fig. 3 (right). The spectra are corrected for efficiency and for contamination from other particle species. It is not likely that A(1520) production is dominated by interactions induced by A projectiles, because

95

>

\

5000

0

\

Z 3000 2000

1 0.5

I

+

+

1

1.5

2

2.5

p(A(1520)),GeV/c2

P(K'>t G e v P

Figure 3. Left: momentum spectrum of the h(1520). Right: momentum spectrum of K- (points with error bars) and K+ (solid histogram).

-

the A(1520) momentum spectrum is too soft. Even at the threshold of the AN + A(1520)preaction the h(1520) momentum is 1.1GeV/c. To demonstrate that non-strange particles do not produce A(l520) we study the pK- vertices accompanied by a K f tag. The distance from the pK- vertex to the nearest K f is plotted in Fig. 4 as a dashed histogram. For comparison the distance to any track is plotted as a solid histogram. The peak at zero corresponds to the vertices with additional tracks. The

Distance, cm Figure 4. Distance from pK- secondary vertex to the nearest track (solid histogram) and to the nearest K+ (dashed histogram).

96

much smaller peak at zero for the K + tagged vertices leads us to the conclusion that most A(1520) are produced by strange projectiles. 4. Search for pentaquarks in B meson decays

In this analysis we search for O+ and O*++ (an isovector pentaquark predicted in some mode1d3) in the decays Bo + O+g followed by Of -+ pKs, and B+ -+ O*++g followed by O*++ + pK+, respectively (inclusion of charge conjugated modes is implied throughout this section). We also search for @: in the decay Bo + O$r+ followed by Oz -+ D(*)-p, and O,*+ (the charmed analogue of O*++)in the decay Bo -+ OE+g followed by O,*+ + Dop. We reconstruct D mesons in the decay modes D*+ -+ DOT+, D o-+ K-r+ and D- -+ K - r f r + . The dominant background arises from the continuum e+e- + qij process. It is suppressed using event shape variables (the continuum events are jet-like, while the B8 events are spherically symmetric). The B decays are identified by their CM energy difference, AE = ( C i E i ) and the beam constrained mass, M h = JE;,,,,, where Ebem is the beam energy and $i and Ej are the momenta and energies of the decay products of the B meson in the CM frame. The AE distribution (with Mbc > 5.27GeV/2) and Mbc distribution (with IAEl < 0.05GeV/2) for the Bo -+ ppKs and B+ -+ ppK+ decays are shown in Fig. 5. The signal yields are extracted by performing unbinned maximum likelihood fits to the sum of signal and background distributions in the two dimensional ( k f b c , A E ) space. The signal distributions are determined from MC, whereas the background distributions are determined from the AE and Mbc sideband data samples. The fits give 28.6?!:5, and 216.5?:::: signal yields for the ppKs and ppK+ modes, respectively. For the region 1.53 GeVl2 < M p ~ <s 1.55 GeV/c2, corresponding to the reported Of mass, we find no signal. Since there is only a theoretical + 1.8 GeV/2 conjecture for the O*++, we check the 1.6 GeV/2 < M p ~ < region and fmd no signal. Assuming both states are narrow, we set the upper limits

The AE and corresponding M ( D ( * ) p )plots for the decays Bo

+

97

8 6 4

2 00.1

0 0.1 AE (GeV)

12 10 8 6 4 2 0 0.2 5.2 5.22 5.245.26 5.28 M , (GeV/c2)

80 70 60 50 40 30 20 10

50 40 30 20 10

n

00.1

0 0.1 AE (GeV)

0.2 "5.2 5.2255.255.275 5.3 M, (GeV/c2)

Figure 5. AE and Mbc distributions for ppKs (top), and p p K + (bottom) modes. The curves represent the fit results.

D-pfh+, Bo + D*-pjh+ and Bo + Do@ are shown in Fig. 6. From the fit to AE spectra the numbers of reconstructed B decays are 303 f 21, 60 f8 and 66 f 9 for the three modes, respectively. No signal of 0:or OE+ is found in the M ( D ( * ) p )spectra. We set the following upper limits on the fractions of the final state proceeding via O: and O,*+:

B(Bo + O:j%r+) x a(@:+ D-p) < 1.2% (90% CL), B(Bo + D-pj%r+)

We assume here that the charmed pentaquark mass is 3.099GeV/c2 and that the signal p.d.f. is determined by the detector resolution (- 3.5 M~V/C?).~Our limits can be compared with the H1 claim that about

98

AE (GOV)

AE (Gev)

3.5

8

2.5 2

1.O 1 0.5

28

3

3.2

3.4

1.6 1.8

4

M a p )

2.8

a

a2 a.4

3.6

a.8

HJD*-P)

4

2.8

I

31

3.4

3.6

3.8

4

h@'pbor~

Figure 6. A E and M ( D ( * ) p )distributions for Bo + D-p& (left), Bo + D*-p@r+ (middle) and Bo --+ Do@ (right) decays. The hatched histogram in AE distributions corresponds to the D meson sidebands, while the hatched histogram in M ( D ( * ) p )distributions corresponds to the AE sidebands (shown with vertical lines on the A E plots). The vertical line in the M ( D ( * ) p )distributions shows the H1 @ mas,3.099 GeV/ca.

1%of D*+mesons originate from 0;decays. The branching fraction for 0;decays to D mesons is expected to be even larger because of the larger

phase space. References 1. T. Nakano et al. (LEPS Collaboration), Phys. Rev. Lett. 91,012002 (2003). 2. V. Barmin et al. (DIANA Collaboration), Phys. Atom. Nucl. 66,1715 (2003). 3. C. Alt et al. (NA49 Collaboration), Phys. Rev. Lett. 92,042003 (2004). 4. A. Aktas et al. (H1Collaboration), Phys. Lett. B588, 17 (2004). 5. A. Titov, these Proceedings. 6. V. Halyo (BaBar Collaboration), these Proceedings. 7. A. Abashian et al. (Belle Collaboration), Nucl. Instr. Meth. A 479,117 (2002). 8. S. Kurokawa and E. Kikutani, Nucl. Instr. Meth. A 499,1 (2003). 9. S. Eidelman et al. (Particle Data Group), Phys. Lett. B592, 1 (2004). 10. A. Sibirtsev et al., nucl-th/0407011. 11. G.J.Feldman and R.D. Cousins, Phys. Rev. D57,3873 (1998). 12. A. Airapetian et al. (HERMES Collaboration), Phys. Lett. B585,213 (2004). 13. T. Browder, I. Klebanov and D. Marlow, Phys. Lett. B587,62 (2004).

99

SEARCH FOR STRANGE PENTAQUARK PRODUCTION IN e+e- ANNIHILATIONS AT f i = 10.58 GeV AND IN T(4S) DECAYS

VALERIE HALYO REPRESENTING THE BABAR COLLABORATION * 2575 Sand Hill rd. Menlo Park, CA 94025, USA E-mail: valeriehQslac.stanford.edu

A preliminary inclusive search for strange pentaquark production in e+e- interactions at a center-of-mass energy of 10.58 GeV using 123 ft-' of data collected with the BABAR detector is presented. We look for the 0+(1540), interpreted as a udud3 state; and the 2--(1860) and E0(1860), candidate dsdsfi and uss(uU+dd) states, respectively. In addition we search for other members of the antidecuplet and corresponding octet to which these states are thought t o belong. We find no evidence for the production of such states and set preliminary upper limits on their production cross sections as functions of c.m. momentum. The corresponding limits on the 0+(1540) and E--(1860) rates per e+e- + qq event are suppressed compare t o the rates measured for ordinary baryons of similar mass. Detailed discussion of the topics in this paper may be found in Ref l5

1. Introduction Recently several experimental groups have reported observations of a new, manifestly exotic (B=l, S=1) baryon resonance, called the Of(1540) 1-6, with an unusually narrow width (I'<8 MeV/c2) for a particle in this mass range that has open channels to strong decay. The NA49 experiment reported evidence for an additional narrow exotic (B=l, Q=S= -2) state, called E--, as well as corresponding Eo state, with masses close to 1862 MeV/c2 '. Shortly after the H1 collaboration reported a narrow (I? < 30 MeV/c2) exotic charmed (B=l, C= -1) resonance, Q!?, with a mass of 3099 MeV/c2. These results prompted a surge of pentaquark searches in experimental data of many kinds, mostly with negative results g . Several theoretical models have been proposed to describe possible pentaquark 1oi11912

*Work supported in part by Department of Energy contract de-ac03-76sf00515.

100

structure. They predict that the lowest mass states containing u, d and s quarks should occupy a spin-112 antidecuplet and octet. Predictions for the masses of the unobserved N and C states vary; MN might be anywhere between -150 MeV/c2 below M e and M s , and M E anywhere between M e and -150 MeV/c2 above M,-. In order to distinguish pentaquarks from ordinary baryons, we adopt a notation that has names corresponding to ordinary baryons with similar s quark content, plus a subscript 5. For example, the EL pentaquark (dss(uG d z ) ) corresponds to the ordinary E3"-(dss) baryon. Although experiments with a baryon in the beam or the target might have some advantage in pentaquark production, e+e- interactions are also known for democratic production of hadrons. Baryons with nonzero beauty, charm, strangeness (up to three units) and/or orbital angular momentum have been observed with production rates that appear to depend on the mass and spin, but not quark content. If pentaquarks are produced similarly, then one might expect a pentaquark rate as high as that for an ordinary baryon of the same mass and spin, i.e. about 8 x 0; and about 4 x Zc- per e+e- -+ hadrons event at f i = 10.58 GeV 14. Decays of B mesons are also known to produce a rather high rate of baryons and so provide another fertile hunting ground. Our program include an inclusive search for the reported states and also the other members of the antidecuplet and octet in data from the BABAFt experiment. After describing some common aspects of the analyses in section 2, we describe searches in two classes of decay modes in the next two sections. Upper limits on pentaquark production are discussed in section 5, and we summarize the results in section 6.

-

-

+

2. Data Sample

This analysis is performed using a 123fb-ldata sample collected on or near the T(4S) resonance with the BABAR detector at the PEP-I1 asymmetric-energy e+e- storage rings. The BABAFt detector, a generalpurpose, solenoidal, magnetic spectrometer, is described in detail in Ref. 13. Our invariant mass resolution for a given decay mode is better than that of any experiment reporting the observation of a pentaquark candidate with that mode available. Minimal hadronic event selection is performed, as we wish to be as inclusive as possible and maintain maximum signal efficiency. We simply require three reconstructed charged tracks in the event. The vast majority of the events are hadronic. The acceptance for a pentaquark

101

signal from any of these sources is well understood; if a signal is found, we can attempt to isolate the source with cuts on the event properties.

3. Search for the Oz(1540)

--f

pK;

We reconstruct S; candidates in the pK,"decay mode, where K," -+ 7r-7r+, with selection criteria based on geometrical and kinematic cuts designed for high efficiency and low bias against any production mechanism.

Wlm wM

~~~~

,

,~

4.5

5

Moo 4Mx)

2wo 0

1.5

2

2.5

3

3.5

4

1.8

2.2

2

( p g ) Mass [GeV/c2]

2.4

2.6

2.8

3

( p g ) Mass [GeV/c2]

m Moo 4Mx)

m 1.6

1.65

1.7

I75 1.75

1.8

(p@ Mass [GeV/cZ]

Figure 1. Distribution of the p K g invariant mass for combinations satisfying all the criteria described in the text. The same data are plotted four times in different pKZ mass regions.

The simulated signal reconstruction efficiency varies with p * , the pK; momentum in the e+e- c.m. frame, from 13% at low p* to 22% at high p * . The invariant mass distribution of pK," is shown in Fig. 1. There is a clear peak at 2285 MeV/c2 from A$ -+ pK; but no other sharp structure.

102

The A: peak (upper right plot) shows a mass resolution of better than 6 MeV/c2 and contains roughly 52,000 entries, demonstrating our sensitivity to the presence of a narrow resonance. The lower plots zoom in on the mass ranges 140&1600 and 1600-1800 MeV/c2. The @; has been reported at values between 1520 and 1540 MeV/c2. There is no enhancement in our data anywhere in this region. The pKE decay mode is also possible for the .E$ states, whose masses might be expected to be between that of the @; and about 2000 MeV/c2 depending on their strange quark content. There is no sign of a narrow resonance anywhere in this region. We consider several additional criteria that might enhance a pentaquark signal. Such as more stringent requirements on the flight distance of the Kg candidate or an additional antibaryon and K - in the event these do not reveal any additional structure. We quantify these null results for the O,: assuming a mass of 1540 MeV/c2. In order to reduce model dependence, we split the data into the ten p* bins of 500 MeV/c2 each and fit a signal plus background function to the pK; invariant mass distribution in each bin. The natural width of the 0: has not been measured; the best upper limit, r < 8 MeV/c2, is larger than our detector resolution, so we must consider the range of widths up to this value. We use a P-wave Breit-Wigner lineshape multiplied by a phasespace factor and convolved with a resolution function derived from the A: data and simulation. The latter is a sum of two Gaussian distributions with a common center and an overall rms ranging from 2.5 MeV/c2 at low p* to 1.8 MeV/c2 at high p*; this is narrower than for the A: due to the proximity of the OZ(1540) to threshold. For the natural width we consider two possiblities, 1 MeV/c2 and 8 MeV/c2, corresponding to a very narrow state and the upper limit, respectively. We consider systematic effects in the fitting procedure by varying the signal and background functions and fit range; changes in the fitted signal are negligible compared with the statistical uncertainties. We set upper limits on the production cross section in section 6.

4. Search for e",-+e"-rr-

and E: + e"-&

We next search for the reported c",--(1862) and Zt(l862) states decaying into a E- and a charged pion, where z"- 4 nor- and Ao 4 p-.The unique topology of this decay allows considerable background reduction by reconstructing the two displaced vertices and cutting around the A' and E'- masses. The simulated signal reconstruction efficiency varies from 6.5%

103

at low p* to 12% at high p * . The invariant mass distributions for E-T- and ?T+ are shown in Fig. 2. In Fig. 2 on the right there are clear peaks as expected for the 2*'(1530) and Z:(2470) baryons, but no other structure is visible. In Fig. 2 on the left there is no visible sharp structure at all.

IJ

2.1

1.7

4.6

J.5

J

(E'x - ) Inv M a s [OeVlc'l

Figure 2.

3-K-invariant mass distribution and =-a+ invariant mass distribution.

As in the preceding section, we divide the Z;- candidates into ten bins of p* and find no sign of a pentaquark signal in any bin. We fit a similar signal plus background function to the E-T- invariant mass distribution in each bin. The resolution function is derived in this case from the Z*'(1530) and 2:(2470) data and simulation, and is described by a Gaussian with an rms of 8 MeV/c2. For the Breit-Wigner width we consider two possiblities, 1 and 18 MeV/c2, corresponding to a very narrow state and the upper limit on the 2;- width, respectively. The fits are performed over the mass range from 1760 to 1960 MeV/c2, and the background function is a seventh-order polynomial. In all p* bins the fit quality is good across the full mass range and the signal is consistent with zero. Systematic uncertainties on the fitting procedure are again found to be negligible compared with the statistical uncertainties, and variations of the EF- mass and selection criteria give consistent results. 5 . Upper Limits

For the states reported by previous experiments, there exist specific masses at which to search and experimental upper limits on the decay widths. We can therefore calculate upper limits on their differential production cross sections, assuming a branching fraction B(8: 4 p K z ) = 114 for (3; and

104

B(Z;- + Z n - ) = 1/2 for E;-. We then derive a conservative upper limit on this cross section in each bin as can be seen in figures 3-4 for @: and EC- respectively. We consider only the physically allowed region, and scale the limit by a factor of (1 & / E ) , where & / E = 0.049 for @: is the sum in quadrature of the relative systematic uncertainties on the efficiency and luminosity. The total cross section upper limit is nearly independent of p * . Two sets of

+

5

zw

BABAR preliminary

. e

preliminary

-

P

E p* [GeV/c]

p* [GeV/c]

Figure 3. The 95% C.L. upper limit on the production cross-section for the @,f assuming a mass of 1540 MeV/c2 and a natural width r = 1 MeV/c2 (left) or r = 8 MeV/c2 (right), as a function of c.m. momentum p * .

limits are shown; one corresponds to a very narrow state, and the other to a width at the current experimental upper limit of 8 MeV/c2. The limits correspond to the mass of 1540 MeV/c2 used in the fits; repeating the analysis at several nearby mass values gives consistent limits. Similarly we derive an upper limit on production cross section for c"~-(1862)-+TT-. Here we assume a mass of 1862 MeV/c2 and present limits for both a very narrow hypothesis and for r = 18 MeV/c?. Its relative systematic uncertainty again varies slightly with p * , and its average value of & / c = 0.064 is larger than for the pKZ mode, as there are two displaced vertices and more tracks in the decay. The limits quote on the total production cross section of the 0; and E;- (times branching fraction) is given in table 1. 6. Summary

We have performed a preliminary high statistics search for the reported states Ql (1540), E;- (1862) in e+e- annihilations, and also for most of the other members of the pentaquark octet and anti-decuplet to which

105

Y

5

50

-

BABAR preliminary

40-

......._.

. m

30;....A

i

: j

c......

20 -

10 -

I , , , , I I I I I I I ,,,I-

'0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 4. T h e 95% C.L. upper limit on the production cross-section for E r - times its branching fraction into E-T-, assuming a natural width of l? = 1 (dashed) and = 18 MeV/c2 (solid), as a function of c.m. momentum p * .

r

Table 1. The upper limits at the C.L on the O+5 total production cross section times its particle

0; ---

=5

X-section U.L. (fb) r = 1 r = 8(i8) 182.8 11.0

363.1 16.85

qtj event U.L. (10-5/event)

r =1

r = 8(18)

5.39 0.32

10.71 0.5

T(4S) decay U.L. (10-5/event) r =qi8) r=1 17.90 34.98 1.06 1.60

they are postulated to belong. In all cases we observe clear signals for known baryon resonances that demonstrate sensitivity t o any new narrow resonances, with invariant mass resolution better than the reported upper limits on the widths of the respective states. We find no evidence for the production of 0; (1540), EL- (1862) or any other member of the multiplet decaying t o A°K or C°K final states in 123fb-'of BABAFi data Taking the upper limit widths, we calculate 95% C.L. upper limits on the total production rates of 1.1x 0; and 1.0 x Ec- per e+e- 4 qq event (preliminary); these are roughly a factor of eight and four below the typical values measured for ordinary octet and decuplet baryons of the same masses as illustrated in Fig. 5.

Acknowledgements

I thank the organizers for arranging a fruitful and enjoyable workshop.

106

!

Figure 5. Compilation of baryon production rates in e+e- annihilation from experiments at the Zo (circles) and & % 10 GeV (squares) as a function of baryon mass. The vertical scale accounts for the number of spin and particle+antiparticle states. The arrows indicate our preliminary upper limits on spin-l/2 e: and EF- pentaquark states, assuming the branching fractions shown, and are seen to lie below the solid line.

References 1. LEPS Collaboration, T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003). 2. CLAS Collaboration, V. Kubarovsky et al., Phys. Rev. Lett. 92 032001 (2004). Erratum; ibid, 049902. 3. SAPHIR Collaboration, J. Barth et al., Phys. Lett. B 572, 127 (2003). 4. DIANA Collaboration, V.V. Barmin et al., Phys. Atom. Nucl. 66,1715 (2003). 5. HERMES Collaboration, A. Airapetian et al., Phys. Lett. B 585, 213 (2004). 6. COSY-TOF Collaboration, M. Abdel-Bary et al., Phys. Lett. B 595, 127 (2004). 7. NA49 Collaboration, C. Alt et al., Phys. Rev. Lett. 92, 042003 (2004). 8. H1 Collaboration, A. Aktas et al., hep-ex/0403017 (2004). 9. See, e.g., J. Pochodzalla, hep-ex/0406077 (2004) and references therein. 10. D. Diakonov, V. Petrov and M.V. Polyakov, Z. Phys. A 359, 305 (1997). 11. M. Karliner and H. Lipkin, hepph/0402260 (2004). 12. R. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003). 13. The BABAR Collaboration, B. Aubert et al., Nucl. Instrum. Methods A 479,1(2002). 14. Particle Data Group, K. Hagiwara et al., Phys. Rev. D 66, 010001 (2002). 15. BABAR Collaboration, B. Aubert et al., hepex/0408064.

107

PENTAQUARK RESULTS FROM CDF

MING-JER WANG (FOR THE CDF COLLABORATION GROUP) Institute of Physics, Academia Sinica Taipei, Taiwan We have carried out three searches of pentaquarks at CDF. First, we will present the search for a neutral pentaquark 0, which decays into D* and proton. This search has been performed using 240 pb-l data samples of a displaced-two-track trigger. Second, we will present the search for a charged pentaquark Q+ which decays into K," and proton. This search has been performed using data samples of a minimum bias trigger and a jet trigger. Third, we will present the search for a doubly-charged pentaquark which decays into Z and A. This search has been performed using 220 pb-' data samples of a displaced two-track trigger and a jet triggerers.

1. Introduction One of the recent research excitements is the evidence of O+ particle which is postulated as one type of the pentaquarks. As shown in figure 1, ten experiments [l-91 around the world have reported evidences since the end of 2002. These findings have revived the search for pentaquark states. Many theoretical papers have been published in order to explain the observed 8+ mass and the peculiar narrow width. The NA49 experiment [lo] claimed to have found the other pentaquark state E(1860). In addition, H1 [ll]also claimed to have found the evidence of pentaquark 0, early this year. We at CDF collaboration group were motivated by these news to search for pentaquarks in our data samples. We started at 3 pentaquark search modes. We will report the CDF detector, search strategies, three searches for 8, + D*p, 0+ -+ p K , , and E(1860) -+ En,summary and conclusion in the following sections. 2. CDF Detector By thoughtful design, quality construction, and careful operation, CDFII detector as shown schematically in figure 2 has very good tracking components to record and reconstruct charged tracks from particle decays with high momentum resolution. Further more, we are able to reconstruct the

108 5 yiiark stute predicted by

Diakonov, Petrov, P o l p k o ~1997): j 4 quarks + 1 antiquark Q+ : u u d d s(bar) Mass 1530 MeVIc2 Width 15 MeVIc2 Decays equally to nK+ and pKO

-

*

Sfnope

*

SAPHIR

*

CIAS 1 CIAS 2

*

DIANA

*

HERMES

0

c-

sw 3

c

Asratpneial

COSY TOF

70

10 experiments reeport evidence 3 experiments report no observation Figure 1. The 8+ mass measurements. The solid circles are for nK+ and the open circles are for p K s results.

displaced vertex with high position precision in identifying heavy flavor decay. In addition, we also have particle identification capability using the TimeOf-Flight (TOF) and d E / d z techniques. We used the data collected from the tracking and particle identification components of the CDF detector t o do these three pentaquark search analyses. All the charged tracks are reconstructed with the drift chambers and silicon detector located in the center part of the whole detector. Precise vertex reconstruction was achieved by adding the information obtained from the silicon vertex detector. The particle identifications were carried out using both TOF information from the TOF counters which are located outside the tracking volume and d E / d s information from the drift chambers. 3. General Search Strategy

There are several unknown factors in searching for pentaquark at m- collider of b , = 1.96 TeV. First, we do not know the pentaquark production cross sections at this energy, therefore, we do not know which type of pentaquark is the best to look for. Second, we do not know the specific production mechanism at p p collider, therefore, we do not know which data set is the

109

Search Modes: oc

large radius tracking wire

time of flight

bD'%p(har) bD .n bK .n 0

Ks p

4.n .n I

_ I 1

4 E .n b A .n

4.n P detector (SVX) Figure 2. CDFII detector. The relationships between particle reconstruction of three pentaquark search modes and the detector components are indicated by red, blue, and green colors and arrows.

best t o look for. Third, we don't know the specific distinguished decay properties, therefore, we do not know which kinematic region is the best to look for. Additional difficulty is the large combinatorial background in high track multiplicity environment of p p collider. We do not know in prior whether we have the sensitivity t o search for. Despite of these uncertainties, we came out a working search strategy: First, we start with the claimed pentaquark candidates, i.e. 8,,O+, and E(1860). Second, we selected possible datasets for each search. Third, we assumed the production ratio of pentaquark t o that of known resonance is independent of c.m. energy, therefore, we don't need to use cuts which are pentaquark specific. As for the sensitivity issue, we selected known resonances with similar decay products and event multiplicity as the search reference signals. 4. Search for

0,

We search for 0,in the D*p decay mode. This is a hadronic decay and the Silicon Vertex Trigger (SVT) has the capability to trigger on 2 displaced tracks in an event using the silicon information. Therefore, this data set

110

is ideal for the D*p search. The reference mode chosen is the D1 and D2+ resonances in the D*n mass spectrum by replacing the third track with the pion mass. In total, we reconstructed 3M Do with half width of 8 MeV and 0.5 M of D* with half width of 1.2 MeV. Both the D1 and D2+peaks are clearly seen in the D*Tmass spectrum with a yield of 14,700. This indicates that we should be able to find 0, if its production yield is at the same order. The D*p mass resolution at 3.1 GeV was found t o be 2.5 MeV using zero width Monte Carlo data. With this excellent mass resolution, we should be able to resolve the 0, peak claimed by H1. The proton PT range was found to be from 1 to 8 GeV assuming 0, has the same PT spectrum as that of J / 9 . By studying the proton and pion separations in particle identification pull distributions, we decided t o apply TOF cuts for proton track PT < 2.75 GeV and dE/dx cuts for proton track PT > 2.75 GeV. Proton PT was required t o be larger than 500 MeV. The x2 of D*p vertex was required to be smaller than 30. The mass difference M ( D * )- M ( D o ) was required t o be between 142.5 and 148.5 MeV. The left plot of figure 3 shows the D*p mass spectrum with the TOF cuts on the proton track and the left plot of figure 4 shows the D*p mass spectrum with the dE/dx cuts on the proton track. In this search, we did not find any resonance structure around 3.1 GeV. These mass spectra were used t o set yield upper limits assuming the 0 and 12 MeV widths. The right plots of figure 3 and figure 4 show the yield upper limits as function of mass.

sl-

8

'

3s

tt

M(D*F]

$2

4

lavd

of 3.m

les'

'*is

I#@

pf

'

WVl4

Figure 3. D*p mass spectrum with TOF cuts is shown in the left plot and the 90% CL event yield upper limit as function of mass is shown in the right plot. The two curves on the right panel are of width 12 MeV/c2 and for zero width as indicated.

111

Figure 4. D'p mass spectrum with d E / d x cuts is shown in the left plot and the 90% CL event yield upper limit as function of mass is shown in the right plot. The two curves on the right panel are of width 12 MeV/c2 and for zero width as indicated.

5. Search for O+ We search for $+ in the pK, decay mode where K, decay into n+n-. Both minimum bias and jet 20 data samples were used to search for O+. In total, we reconstructed 667,000 K, in the minimum bias data set. Proton from A decay was removed. 4 4 K+K-,A(1520) -+ pK-,K*+-+ K,n+ were clearly seen as reference channels. The pK, mass resolution at 1.54 GeV was found to be 2.5 MeV using zero width Monte Carlo data. With this mass resolution, we should be able to resolve the O+ peak seen by other experiments [l-91. Since most of the proton PT range was found to be less than 3GeV, we used TOF as proton identification. The left plot of figure 5 shows the pK, mass spectrum of the minimum bias data with the TOF cuts on the proton track and the right plot is for the jet 20 data. In this search, we did not find any resonance structure around 1.54 GeV. 6. Search for E(1860) We search for E(1860) in both the E-n+ and E-T- decay modes where E- decays into A T - . Both displaced track and jet 20 data samples were used to search for E(1860). The E- peaks were clearly seen in both data sets. In total, we reconstructed 36,000 E- in the displaced track dataset. In figures 6 and 7, the E(1530) was clearly seen in the E-n+ mass spectra in both data sets and served as a reference channel. The En mass resolution was found to be 5.8 MeV using zero width Monte Carlo data. With this mass resolution, we should be able to resolve the E(1860) peak seen by the NA49 experiment[lO]. Figure 6 shows the En mass spectra of the minimum

112

r Figure 5 . p K , mass spectrum. The left plot is for the minimum bias data and the right plot is for the jet 20 data.

bias data and the figure 7 is for the jet 20 data. In this search, we did not find any resonance structure around 1.860 GeV in either Z:-.rrf or =-ITspectra.

s

*_-

x"

and

E-ZE track found InSnt

-_

Figure 6. The %T mass spectra of the displaced track sample. The left plot shows the = K+ mass spectrum and the right plot shows the E - K - mass spectrum.

113

r-Z-

Y

-4f18

28

33

Figure 7. The En mass spectra of the jet 20 sample. The left plot shows the 8-.rr+ mass spectrum and the right plot shows the E-n- mass spectrum.

7. Summary and Conclusion We have searched for O,, 8+,z(1860) pentaquark states. With no a p plication of knowledge for pentaquark production mechanisms and decay properties in all analyses, no evidences of these states have been found at CDF. We are vigorously searching for another pentaquark states.

Acknowledgements

I would like t o acknowledge the workshop organizers and Professor Takashi Nakano for their hospitality and for their invitation to this vivid and successful workshop. I also would like to thank the efforts and information provided by the CDF pentaquark task force members. I also would like t o thank Dr. J. Antos for bringing this interesting topic t o my attention. References 1. LEPS Collaboration, T. Nakano et al., Phys. Rev. Lett. 91, (2003) 012002. 2. SAPHIR Collaboration, J. B a t h et al., Phys. Lett. B 572, (2003) 127.

114 3. CLAS Collaboration, V. Kubarovsky et al., Phys. Rev. Lett. 91, (2003) 252001. 4. DIANA Collaboration, V. V. Barmin et al., Phys. Atom. Nucl. 66, (2003) 1715. 5. HERMES Collaboration, A. Airapetian et al., Phys. Lett. B 585, (2004) 213. 6. SVD Collaboration, A. Aleev et al., Preprint hep-ex/0401024, 2004. 7. A. E. Asratyan, A. G. Dolgolenko, and M. A. Kubantsev, hep-ex/0309042. 8. COSY-TOF Collaboration, M. Abdel-Bary, et al., Preprint hep-ex/0403011, 2004. 9. Zeus Collaboration, S. Chekanov, et al., DESY-04-056, hep-ex/0403051. 10. NA49 Collaboration, C. Alt et al., Phys. Rev. Lett. 92, (2004) 042003. 11. H1 Collaboration, A. Aktas, et al., Preprint hep-ex/0403017, 2004.

115

SEARCH FOR THE PENTAQUARK Q+ IN THE ySHe+ PA@+ REACTION MEASURED AT CLAS

S. NICCOLAI, for the CLAS Collaboration IPN Orsay, 15 rue Georges Clemenceau, 91406 Orsay h n c e E-mail: [email protected] Preliminary results of the analysis of the y3He+ pAe+ channel, a reaction not observed until now, are here summarized. The data have been obtained at Jefferson Lab with the CLAS detector for photon energies from threshold up to 1.55 GeV. The final state has been identified through the reconstruction of the decay modes A + p r - , ef + pKo, KO + T+R-, as well as €3++ nK+. Three independent analysis techniques have been used.

We are investigating the photoproduction of Q+ together with a A hyperon. The advantage of this reaction channel (see for theoretical predictions of the cross section) is that it allows to identify the final state without the need of cutting on competing channels, while at the same time excluding kinematical reflections in the N K invariant mass spectrum. Moreover, thanks to the presence of the A having strangeness S = -1, also the p K o decay mode must have S = + l . In particular, the reaction we have been studying is y3He-+ pAQ+, whose threshold is E.,11 800 MeV for a 8+ mass of 1.55 GeV/c2. The main reaction mechanism can be pictured as a two-step process (Fig. 1): the initial photon interacts with one of the protons of 3He and produces a A and a K+ (yp -, K + A ) . The A leaves the target nucleus, while the K+ reinteracts with the neutron in 3He to form a 8+ (K+n -, W). In this process, one of the 2 protons of 3He can either be a spectator, as pictured in Fig. 1, or rescatter and thus gain enough momentum t o be detected. The data used for this analysis were taken in December 1999 during the CLAS experiment E93-044. The electron beam energy was 1.645 GeV and its current was 10 nA. Photons in the energy range from 0.35 to 1.55 GeV were tagged. The data were obtained using a cylindrical cryogenic target,

116

Figure 1. Production mechanism for A 8 + in 3He. The decay modes A Kop and KO 4 T+T- are here shown.

-+

p ~ - ,0+4

18 cm long and 4 cm in diameter, filled with liquid 3He having density p = 0.0675 g/cm3.

In order to maximize the detection of this reaction with the CLAS detector, mainly efficient for charged particles, we have choosen the following decay channels :

The final state therefore is p r - p r + r - p for the 8+ ---t pKo decay mode, and ppr-nK+ for the 8 + nK+ decay mode. Having many particles in the final state (6 for the Q+ + pKo case, 5 for the 8+ + nK+ case), many different topologies of detected particles in the final state are possible. The most promising three analysis techniques are summarized in Table 1. Event selections the three analyses are shown in Figs. 2, 3, 4. In all three cases the final state is well identified, without the need of applying cuts to remove background channels. The preliminary analysis reveals an enhancement in the N K invariant mass spectrum, near 1.55 GeV/c2. While the statistical significance of the peak is limited, this analysis shows that a search for the AQ+ channel can contribute to resolve the issue of the existence of the Q+. The final state can be identified, unambigously, and no kinematical reflections can produce peaks in the N K invariant mass distribution. An analysis of this reaction using the high-statistics photoproduction data on the deuteron just taken

117 Table 1. Decay modes, combinations of detected particles in the final state and channel-identification techniques adopted for the analysis of the -y3He+ p A 8 + reaction. Decay modes 8+ + p K o , A + p r -

Final-state particles pp7r-s+x

Channel ID

mx = mA m ( r - a + ) = mKo

'

0.3

0.4

0.5

06

0.7 0. M(rr'n) (GeVlc')

Figure 2. Analysis of the p p r + r - topology. The cuts E-,> 1 GeV and p , < 0.8 GeV/c have been applied in order t o reduce the background under the A peak. Here is shown the missing mass of the ppr+n- system as a function of the invariant mass of x + r - , the lines represent the selection cuts applied to select the A (horizontal lines) and the KO (vertical lines)

a t CLAS will give a more definitive answer as to the existence and the properties of the Q+ exotic baryon.

References 1. V. Guzay, hepph/0402060 (2004). 2. A.R. Dzierba et al., Phys. Rev. D 69, 051901 (2004). 3. K. Hicks and S. Stepanyan, CEBAF Proposal E03-113.

118

U(&,

,Q."h'l

Figure 3. Results of the analysis of the m - s + n - topology. Top left: missing mass of the p p n - d n - system, showing a peak a the proton mass; top right: invariant mass of the pr- system, peaking at the A mass; bottom: invariant mass of the s+n- system, the peak is at the KO mass. The dashed lines represent the selection cuts applied.

Hu [GeVk')

Figure 4. Results of the analysis of the p p r - K + topology. Left: missing mass of the ps- K+ system, showing a peak at the neutron mass; right: invariant mass of the pssystem, peaking at the A mass. The dashed lines represent the selection cuts applied.

119

SPECTROSCOPY OF EXOTIC BARYONS WITH CLAS: SEARCH FOR GROUND AND FIRST EXCITED STATES

M. BATTAGLIERI Zstituto Nationale d i Fisica Nucleare, Via Dodecaneso 33 Genova, 16100, Italy E-mail: [email protected] In the las year many Collaborations reported about the evidence of a possible pentaquark state but so far the results are not yet conclusive. New dedicated experiments with higher statistics and precision are necessary t o confirm the pentaquark existence and its properties. In this contribution I will report about a new photoproduction experiment on a proton target, the so called ' G l l ' experiment, just performed in the Hall-B at Jefferson Lab that collected ten times the existing statistics.

1. Introduction

The first evidence of a narrow resonance with S = +1, named 8+,was reported by the LEPS Collaboration and then confirmed by other experimental groups including the CLAS Collaboration who found a signal in the reactions yD 4 pK+K-(n) and yp 4 .rr+K-K+(n)'. However, due to the limited statistics, the reported results are not yet conclusive and experiments with higher statistics are needed to confirm these findings and do extensive checks of the systematic dependencies. Jefferson Lab and, in particular, the CLAS Collaboration in the Hall-B started a comprehensive experimental program to establish a firm and consistent phenomenology of the 8+ spectra, to determine in which production and decay channels the 8+ is seen, and what the production mechanisms are. The G11 experiment run for 65 calendar days in June-July 2004 using the CLAS detector and the Hall-B Bremsstrahlung photon beam on a proton target. We measured two production channels, y p 4 ROO+ and y p 4 K*O+, as well as two decay modes of the O+, K+n and Kop hoping to answer the following questions: - are the present signals a statistical fluctuation? 293t47596

120

- in which production and decay channels does the 8+ show up? - what is the angular dependence of the associated K meson? - what are the production mechanisms in terms of hadron dynamics? - what are the relative production cross-sections and branching ratios? and, most importantly, -what are the masses and widths of pentaquarks produced in photoproduction and decaying to a K-nucleon final state? This measurement will provide a solid foundation for a long-term plan for the investigation of the pentaquark spectrum and properties.

2. Theoretical predictions

Theoretical predictions on the existence and properties of pentaquark baryons are based on a variety of models that treat the basic degrees of freedom quite differently. Due to the lack of a well-established phenomenology, at present different approaches result in different expectations for masses, widths, and quantum numbers of the Of and its companions. A discussion of different scenarios for pentaquark spectroscopy is found in the paper by Close lo, where it is remarked that a spin-312 partner of the O+ may exist. The predicted mass gap varies from 50 MeV, obtained in the Skyrmesoliton model by Borisuyk and collaborators 11, to 250 MeV, reported in Ref. 1 2 . Very recently, a general classification of all the possible pentaquark states based on symmetry considerations has been reported in 13. From this model-independent discussion, it emerges that there should be a pair of states with spin 112 and 312, respectively, with a splitting due to a spinorbit interaction in the positive parity case, or to a spin-spin interaction in the negative parity case. The ground state is expected to have 1=0 while the first excited should be an iso-triplet. Estimates of the total and differential cross sections for the reactions y p + RoQ+have been carried out in hadronic models with effective Lagrangians l 4 and in the Regge theory approach. At present, the production mechanisms are completely unknown and the available calculations include contributions from meson-exchange in the t-channel, baryon exchange in the s-channel, and pentaquark exchange in the u-channel. Varying the unknown coupling constants within reasonable ranges, such models are able to predict the total and differential cross sections according to different hypotheses for the O+ quantum numbers (parity, spin, isospin, etc.). The result of the G11 experiment will help in constraining the model predictions.

121

3. Present CLAS measurements

Several photon-induced reactions have been studied using the available data on a proton target collected by CLAS in previous years. In the energy range 1.8-3.0 GeV we analyzed the reaction -yp -+ KO@+ and subsequent @+ decay to K+n or KOp. We studied the two final states: -yp -+ KO@+ -+ .rr+.rr-K+(n) -yp -+ KO@+ 4 .rr+7r-p(Ko) The K o (KO) was identified by detecting its K," component decaying to .rr+.rr- (bx. 68.6%). For both channels three charged particles in final states were required: .rr+7r-K+ in the first case and .rr+7r-p in the second case. The neutron (or K 0 / K o )was identified using the missing mass technique. In the former reaction, the @+ should show up in the Ko missing mass while in the latter it should be seen both in the detected KO missing mass or in the (KOp)system invariant mass. After removing the background coming from hyperon production with the same final state (K+h*(1520), K+C+(-).rr-(+)),the ( K + n )and (KOp) invariant mass spectra showed two possible narrow structures that were enhanced by selecting the low energy region (1.8 < E-, < 2.3 GeV), close to the expected production threshold for the @+. Although this analysis is still in a preliminary stage and parallel analyses on the same data set show that statistical fluctuations using different selection cuts are not negligible, it calls for a new experiment with high statistics and precision, At higher energy, 3.0-5.0 GeV, we studied the reactions -yp -+ KO@+ --t K°K+n and ~p 4 K*@+-+ K-.rr+K+n. In all of these channels we found possible evidence of a narrow peak in the ( K + n ) invariant mass located in the range 1.55-1.57 GeV. The results of the -yp + K*@+channel have been published in PRL 8 . N

4. The experimental set-up

G11 experiment measures two production channels on the proton: KO@+ and K*@+,each using two decay modes of the @+: K+n and K'p, for a total of four final states. The primary goal of the experiment is to establish the mass spectra with a precise measurement of the masses, widths, and errors on any peaks observed. With the high statistics collected in the running time, a determination of the total and differential cross sections as well as the decay angular distribution will also be possible. The experiment uses the Hall B Bremsstrahlung tagged photon beam and the CLAS detector with a 40cm hydrogen target. The primary electron

122

Figure 1.

The new CLAS Start Counter used for the first time in G11 experiment.

beam has an energy of 4 GeV to obtain tagged photons from the 8+ production threshold (1.6 GeV) to the maximum energy of 3.8 GeV. The CLAS spectrometer l5 (CEBAF Large Acceptance Spectrometer), is built around six superconducting coils producing a toroidal magnetic field. The detector package consists of three layers of drift chambers for track reconstruction, one layer of scintillators for time-of-flight measurements and hadron identification, forward Cerenkov counters for electron-pion discrimination, and electromagnetic calorimeters to identify electrons and neutral particles. It is well suited for simultaneous multi-hadron detection as required by the pentaquark experiments. Using CLAS together the Hall-B tagger facility, we will be able to study simultaneously the quoted reactions with a high experimental resolution. The experimental set-up (polarity and strength of the CLAS magnetic field, beam energy, trigger condition) was optimized by performing full MC simulations and exploiting the experience gained analyzing the existing CLAS data. A new longer start counter was specifically built to trigger the data acquisition. The new CLAS start counter is made by 24 strips of 2mm thick plastic scintillator with a single side PMT-based read-out. A time resolution of few ns, reduced to 300ps in the off-line analysis was achieved.

123

The high azimuthal segmentation allows to work with an higher photon flux (around 6OMHz on the tagger focal plane) reducing the time necessary to collect ten times the existing statistics to a reasonable number of days (-30 PAC days). A drawing of the CLAS new start counter is shown in fig. 1

"1.4

I&

1.5

1.55

1.6

1.65

1.7

1.75

1.8

M (PW (GeV) Figure 2. The ( K f n ) (left) and p K o (right) invariant mass spectra for KO at backward angles in the center-of-mass system.

5. Expected statistical accuracy

The statistical accuracy of G11 experiment was estimated by extrapolating the results obtained from the low-energy CLAS data analysis. The projected yields reported below refer to the photon energy range E7 = 1.8-2.3 GeV close to the 8+ production threshold (where the production cross section should be maximum) therefore represent a lower limit of the achievable statistics. Results are shown for the reactions yp 4 Roe+-+ .rr+.rr-K+(n) and yp -+ kO@++ r+r-p(Ko) with the CLAS magnetic field to 50% of its maximum, assuming a production cross section of 16 nb, equal branching ratios for @+ decaying to nK+ and pKo, and an integrated luminosity corresponding to 25 days/run. We expect 160(700) @+(1526) and 300(500)

124

@+(1571) candidates for the nK+ (pKo)final state, cutting at backward angles. The expected nK+ and pKo invariant mass spectra are shown in Fig. 2. With the same assumptions, the expected statistical error on the differential cross section has been estimated. Fig. 3 shows the expected results for the yp 4 I?'@+ -+ r+n-K+(n) channel, according to different hypotheses on production mechanism: a flat distribution in the (KO - @+) center-of-mass, u-channel production, and t-channel production.

15 : 12.5 :

+

10 : 7.5 1 -

+

5 : 2.5 :

-1 -0.8 -0.6 -0.4 -0.2

0

0.2 0.4

0.6 0.8

1

cos(e)

Figure 3. The expected errors on the differential cross section according to different production mechanisms: flat (dots), u-channel production (triangles), and t-channel (squares).

Decay angular distribution (W(0)) reflects spin and parity of the Q+. Different combinations of the two quantum numbers are related to different

125

angular behavior (e.g. if Jp=1/2+ and J,=f1/2 then W(0) = constant; if Jp=3/2+ and J,=f1/2 then W(0) = 1 3cos28 while if J,=f3/2 then W(0) = sin20). If the Q+ is produced in some defined polarization state, measuring the decay angular distribution will provide some informations about them. The G11, will measure the decay angular distributions with the statistics accuracy shown in Fig. 4.

+

black =flat distribution red = sin2@] blue = 1 + %os2(6)

cos

1

=*+,,

Figure 4. The expected error on the 0+ decay distributions according to different combination of spin and parity.

6. Summary

Experimental evidence for a pentaquark state of baryonic matter is increasingly convincing. However, many small contradictions plague the comparison of experimental results. No single experiment has the statistical power to rule out the chance of a correlation between a statistical fluctuation and

126

an unknown systematic enhancement acting to produce the state. The G11 experiment at Jefferson Lab will firmly establishing the phenomenology of the C3+ spectrum. Data were taken in June-July 2004 and now they are under analysis. With ten times the statistics of our present data sample, this new experiment will enable us to pin down the masses and widths of any peaks in the spectrum and measure differential and total cross section helping to determine the relevance of pentaquark production mechanisms. References 1. T. Nakano et al., Phys. Rev. Lett. 91,012002 (2003). 2. V.V. Barmin et al., Phys. Atom. Nucl. 66,1715 (2003); Yad. Fiz. 66,1763 (2003). 3. J. Barth et al., Phys. Lett. B 572,127 (2003). 4. A.E. Asratyan, A.G. Dolgolenko, and M.A. Kubantsev, Phys. Atom. Nucl. 67, 682 (2004); Yad. Fiz. 67,704 (2004). 5. A. Airapetian et al., Phys. Lett. B585 213, (2004). 6. S. Chekanov et al., Phys. Lett. B591 7, (2004). 7. S. Stepanyan et al., Phys. Rev. Lett. 91,252001 (2003). 8. V. Kubarovsky et al.,Phys. Rev. Lett. 92,032001 (2003). 9. M. Battaglieri et al., JLab Experiment 04-021. 10. F. Close and J.J. Dudek , Phys. Lett. B586 75, 2004. 11. D. Borisyuk, M. Faber and A. Kobushkin, hep-ph/0307370. 12. B.K. Jennings and K. Maltman, Phys. Rev. D69, 094020, 2004. 13. R. Bijker, M. M. Giannini and E. Santopinto, hep-ph/0310281. 14. Y. Oh et al., Phys. Rev. D69,014009, 2004. 15. B. Mecking et al.,Nucl. Instrum. and Meth. A503,513 (2003).

127

A SEARCH FOR NEUTRAL BARYON RESONANCES BELOW PION THRESHOLD

XIAODONG JIANG Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854. E-mail: [email protected] The reaction p ( e , e ’ a + ) X o was studied with two high resolution magnetic spectrometers to search for narrow baryon resonances. A missing mam resolution of 2.0 MeV was achieved. A search for structures in the mass region of 0.97 < Mxo < 1.06 GeV yielded no significant signal. The yield ratio of p(e, e’n+)XO/p(e,e’a+)n at 1.004 GeV and (0.34 f0.42) x wa8 determined to be (-0.35 f 0.35) x at 1.044 GeV. This measurement clearly demonstrated the potential of high resolution missing mass searches in coincidence experiments.

In this talk, I reported the results from the first dedicated experimental search of narrow baryon resonances below pion threshold. Although the narrow structures we were searching for are a different kind of states than the pentaquark states, the lessons learned earlier in our field should certainly not be forgotten. Since the discovery of the A(1232) resonance in the 1950s,many baryon resonances have been discovered with m > mA. These baryon states are interpreted as net three-quark color-singlet objects with angular and radial excitations 2. The quark model explains the mass difference of ma -mN by quark spin-spin interactions 3 . Since all the low-lying quark-model states are accounted for, a baryon bound state with a mass between the nucleon and the A should not exist in the present theoretical framework. Indeed, there was no evidence for such a resonance prior to 1997,and several searches 4,5 for charge-two resonances yielded null results. In 1997, however, possible evidence of neutral baryon states at 1.004, 1.044,and 1.094 GeV was reported in the pp + p + X 0 reaction 6 . The two lower mass states are below pion threshold, and the only allowed decay channels are the radiative ones, which implies that their natural widths are of the order of a few keV, much narrower than the experimental resolution of a

128

few MeV. The authors suggested a possible explanation of these resonance states in terms of interacting colored quark clusters. These experimental results are most astounding when one considers the countless experiments carried out with many different probes over more than 50-years in which the claimed states were never observed. L’vov and Workman argued that the reported structures are “completely excluded” by the fact that no such structure was reported in the existing real Compton scattering data *. Furthermore, the existence of these states appears to be ruled out by their effects on the predicted composition of a neutron star which lead to a reduced maximal mass inconsistent with the observational limit g. As a counter argument, Kobushkin lo suggested that the claimed states could be members of a total anti-symmetric representation of a spinflavor group such that the one-photon excitation or decay channels are forbidden and only the 2yN channels are allowed. While there is no room for these new exotic baryon states within the many theoretical constituent quark models 11, a colorless Diquark Cluster Model l2 mass formula closely reproduced the observed masses. Recently, a model based on the excitation of quark condensates l 3 was suggested which interpreted the resonances as multiple production of a “genuine” Goldstone Boson with a mass of 20 MeV. The existence of baryon resonances below pion threshold, if established experimentally, could profoundly change our understanding of quark-quark interactions and strongly suggest new degrees of freedom in the quark model. However, the states claimed in Ref. were of limited statistics amid a rather significant background. The signals of the p p + p d X 0 peaks were of the order of compared to that of the pp + pr+n peak. Given the potential impact of these states, experimental verification in different reaction channels is highly desired. Recently, single baryon states of 0.966, 0.987, and 1.003 GeV were reported in the missing mass spectra of the pd + ppXo reaction 14, but a similar search in the same reaction channel has reported no resonance structure 15. This paper reports the first dedicated search in the p ( e ,e’n+)Xo channel in the mass region of 0.97 < M ~
129

Figure 1. The Feynman diagrams of p ( e ,e'r+)n and p(e, e ' r + ) X o reaction.

can be determined through the cross section ratio: 2

(E)

cp(e,e'r+)Xo

=

cp(e,e'r+)n

K,.

(1)

According to B. Tatischeff et al. the suppression factor K s is expected which falls close to an upper limit of to be in the order of N suggested by Azimov16. Therefore, a high-resolution, high-statistic and low background ( e , e ' r + ) measurement could be used to reveal the resonance states as abnormal structures above the radiative tail of the p(e,e'x+)n reaction. The measurement was conducted at the Thomas Jefferson National Accelerator Facility's experimental Hall A, taking advantage of the high resolution spectrometer pair and the high quality CW beam of the CEBAF accelerator. The data were collected during a 12 hour period. An electron beam of energy 1.722 GeV and average current 33 pA was scattered on a liquid hydrogen target. The target cell was an aluminum can 15 cm in length and 6.35 cm in diameter, and was oriented with its axis along the beam direction. Two magnetic spectrometers were used in coincidence. One spectrometer was set to a central momentum of 1.040 GeV/c to detect the scattered electrons at 19" on beam right, the other spectrometer was set to 41.6O on beam left to detect x+ particles first at 0.621 GeV/c (Kinematics-A) corresponding to the missing neutron peak as a calibration, then at 0.543 GeV/c (Kinematics-B) for the resonance search. The four-momentum transfer square is 0.2 GeV2, and the invariant mass of the &X0 system is W M 1.44 GeV. A combination of a threshold gas Cherenkov counter and a lead-glass calorimeter array at the focal plane provided clean e / r - separation in the

130

electron arm. In the hadron arm, the & / p separation was achieved by an Aerogel Cherenkov detector combined with the particle’s velocity and energy loss measured by the trigger scintillators. The resolution of the reconstructed two-arm vertex was 0; = 0.6 cm along the beam direction, and was mainly limited by the multiple scattering through the window material at the target chamber and the spectrometer entrance. The path length corrected coincidence time had a resolution of 2.0 ns (FWHM) which is dominated by the rise time of the photomultiplier tubes attached to the scintillators. After particle ID cuts and a two-arm vertex cut of (Azl < 1.0 cm, the ratio of real-to-random coincidences was about 1:l in Kinematics-B. Yields were corrected for event-reconstruction efficiencies (- 90%) and data acquisition dead times ( w 10%). The effect of pion decay was accounted for by weighting each event with a survival factor which was calculated based on the pion’s momentum and path length (-23.5 m). A window of 3.4 ns centered around the timing peak was used to select the (e,e’n+) events, and accidental events were sampled from a 34 ns time window on each side of the timing peak. No noticeable fine-structure was observed in any spectrum of accidental events. The phase space volumes were calculated through a Monte Car10 simulation which started by sampling a missing mass range uniformly. Spectrometer models, reconstruction resolutions and target material effects were considered in the simulation. The charge-normalized ( e , e ’ d ) yields were obtained by subtracting the accidental events from the coincident events and dividing the result by the phase space volume. The normalized yield, as a function of missing mass, is plotted in Fig. 2(a) in 1.0 MeV bins in arbitrary units. The missing mass resolution is 2.0 MeV, as demonstrated in Kinematics-A, and is mainly due to the energy loss in the target material. Due to the large size of the target cell, the incident electron, the scattered electron, and the outgoing pion passed through averaged material thickness of 0.5,0.6, and 0.4 g/cm2, respectively. The yield of Kinematics-B, which is the Bethe-Heitler radiation tail of the p(e,e‘r+)nreaction, has been amplified fifty times in Fig. 2(a). A thirdorder polynomial was fit to the Kinematics-B yield with a reduced x2 of 1.2 for 90 data points. The signature of an X o resonance would be an excess of yield above the smooth shape of the radiation tail. A line shape corre spondmg to such a signal with the strength of K, = 1.0 x at 1.004 and 1.044 GeV is illustrated in Fig. 2(a) by the curve shifted from the data. The deviations of the data from the polynomial fit are divided by the p ( e , e ’ d ) n peak-height and plotted in Fig. 2(b). To the level of K , M 1.0 x loh3, no

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resonance signal can be identified in the mass region of 0.97 < M X O< 1.06 GeV. Fitting these fluctuations with a Gaussian of FWHM = 2.0 MeV, the experimental resolution, leads to K , = (-0.35f0.35) x at 1.004 GeV at 1.044 GeV. For the reported states at and K , = (0.34 f 0.42) x 0.975 GeV and 0.986 GeV of Ref. I*, we found K , = (1.12 f 0.73) x respectively. The 0.966 GeV state of and K , = (-0.94 f 0.44) x Ref. I4 was covered in Kinematics-A with limited statistics and we found K , M (3 f 5) x The state at 1.094 GeV of Ref I3 was not covered in this measurement. The null result of this experiment does not necessarily contradict the claim of Ref.13. First, the hadronic reaction could have allowed more exotic channels for the production of X o , for example, through a reaction with a dibaryon type intermediate state. Second, due to the limited beam time and the unfavorable target geometry, the sensitivity of this measurement was not much improved compared to Ref.13. However, this measurement clearly demonstrated the potential of high resolution missing mass searches in coincidence experiments. A dedicated search17 in the future can set a tighter upper limit beyond K , = 1.0 x Indeed, a missing mass search conducted recently at MAMI of Mainz I8 already improved the upper limit to the level of With the new generation of high resolution spectrometers and the high intensity CW electron beam of Jefferson Lab, a carefully planned experiment could set a much tighter limit on or even discover narrow baryon structures which might not have been visible in earlier lower resolution experiments. Very recently, another dedicated search l9 using the same missing mass technique has been completed in Jefferson Lab Hall A in the reaction channel of p ( e ,e ' K - ) X + + to search for a possible pentaquark partner of O+. In conclusion, a high resolution missing mass search in the p ( e , e f n + ) X o reaction yielded no significant signal in the mass region of 0.97 < MXo < 1.06 GeV. The yield ratio of p ( e ,e ' r + ) X O / p ( e efn+)n , was determined to at 1.004 GeV and (0.34 f 0.42) x at 1.044 be (-0.35 f 0.35) x GeV, consistent with a null signal. We thank Dn. E. L. Lomon, B. Norum, B. Tatischeff and W. Turchinetz for many discussions. We also thank the Jefferson Lab Hall A collaboration, Hall A technical staff and the Jefferson Lab Accelerator Division for their support of this experiment. This work was supported by the US. Department of Energy (contract DEFG02-99ER41065, FIU) and the National Science Foundation (contract PHY00-98642, Rutgers). Southeastern

132

I

.

(a;

-

.-FQ,

800

+

1.044

I

i

a)

L

1.004

600

1

N .-

E

X50

400

0

z 200 0

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.-0 + U

2

x

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x

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0.96

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.

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Missing Mass (GeV) Figure 2. Normalized (e, e’lr+) yields in arbitrary units are plotted in 1.0 MeV missing maas bins in (a). A third power polynomial fits the data, the fit residues divided by the p(e, e ‘ d ) n peak-height are plotted in (b). Error bars are statistical only.

Universities &search Association manages Thomas Jefferson National Accelerator Facility under DOE contract DEAC05-84ER40150. References 1. X.jiang et al., Phys. Rev. C 67, 028201 (2003). 2. R.P.Feynman, Photon-Hadron Interaction, (Benjamin New York, 1972). 3. F. Close, An Introduction to Quarks and Partom, (Academic Press, N.Y.

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1979). 4. 5. Ram et al., Phys. Rev. D 49, 3120 (1994). 5. T. T. Nakamura et al., Phys. Rev. D 39, 1261 (1989). 6. B. Tatischeff et al., Phys. Rev. Lett. 79, 601 (1!?97j, B. Tatischeff et al., ibid, 51, 1347 (1998), and B. Tatischeff et al., nucl-ex/0207003. 7. A. I. L’vov and R. L. Workman, Phys. Rev. Lett. 81, 1346 (1998). 8. See for example F. J. Federspiel et al., Phys. Rev. Lett. 67, 1151 (1991); and B. E. MacGibbon et al., Phys. Rev. C 52, 2097 (1995). 9. E. E. Kolomeitsev and D. N. Voskresensky, nucl-th/0207091. 10. A. P. Kobushkin, nucl-th/9804069. 11. S. Capstick and W. Roberts, Prog. Part. Nucl. Phys. 45, 5241 (2000). 12. N. Konno, Nuovo Camento A 111, 1393 (1998). 13. T. Walcher, hep-ph/0111279. 14. L. V. Fil‘kov, et al. Eur. Phys. J . A 12,369 (2001). 15. A. Tamii et al. Phys. Rev. C 65, 047001 (2002) 16. Ya. I. Azimov, Phys. Lett. B 32, 499 (1970). 17. X. Jiang and R. Ransome co-spokespersons, Jefferson Lab experiment proposal PO2-015 (unpublished). 18. M. Kohl et al., Phys. Rev. C 67, 065204 (2003). 19. B. Wojtsekhowski and P. Reimer co-spokespersons, Jefferson Lab experiment EM-012.

134

TIME PROJECTION CHAMBER FOR PHOTOPRODUCTION OF HYPERON RESONANCES AT SPRING-8/LEPS

H. FUJIMURA~, K. I M A I ~ ,M. NIIYAMA~,H. FUN AH AS HI^, M. MIYABE', K. M I W A ~Y. , NAKATSUGAWA~,M. YOSOI~,J.K. A H N ~D.S. , AHN~, T. HOTTA3, K. KIN03, H. KOHR13, T. MIBE3, N. MURAMATSU3, T. NAKAN03, T. SAWADA3, S. AJIMURA4, Y. SUGAYA4, W.C. CHANG5, J.Y. CHEN5, D. OSHUEV5, M. NAKAMURA' 'Department of Physics, Kyoto University, Japan Department of Physics, Pusan University, Korea Research Center for Nuclear Physics, Osaka University, Japan Department of Physics, Osaka University, Japan Institute of Physics, Academia Sinica, Taiwan ti Wakayama Medical University, Japan We have constructed a Time Projection Chamber (TPC) to study photoproduction of hyperon resonances at SPring-8/LEPS. The effective angular coverage of this . target holder is mounted in TPC and enable us to TPC is close to 0 . 9 ~ 4 ~The detect tracks as close as 16 millimeters from the target center. The active volume of TPC is filled with a P10 gas of 1 atmosphere. The spatial resolution in the pad plane is 340 pm and that along to drift direction is 550 pm.

1. Introduction

The nature of A*(1405) is still remaining problem whether A*(1405) is L = 1 SU(3)-single three quark state or KN bound state. Recently some theorist predict the modification of mass spectrum of A*(1405) in nuclear matter Therefore, measurement of the A* (1405) invariant mass distribution is crucial to test these model predictions. To obtain clean invariant mass of A* (1405), it is important to distinguish A* (1405) and C*(1385) particles by its decay topology. The A'(1405) decays into CT,while C*(1385) decays mainly into AT. Therefore, detecting C from A*(1405) is essential to reconstruct the A*(1405) invariant mass. We have built a TPC to detect decay particles from A*(1405). The TPC has advantages to detect decay topology because of its multi-track 1329334.

135

separation capability. Moreover, this TPC is enable t o detect tracks as close as 16 mm from center of target and detect short life particles like C which decays in a few centimeters from the target. 2. Design of TPC

The chassis of TPC is a hexagonal shape as shown in Fig. 1. The active volume of the TPC is cylindrical shape of 700 mm in length and 350 mm in diameter. The TPC is operated under 2 T high magnetic field of the solenoid magnet. The gas filled in TPC consists of Ar(90%) and CH2(10%) of 1 atmosphere. All wires are strung along the same direction, while circular pad rows are placed radially. There are three wire layers. The top layer is a gating grid with 2mm wire-spacing. The gating grid is necessary for preventing ions from penetrating into the drift area. We operate the gating grid by changing the wire voltage. The second layer is a cathode grid at 0 V. The wire-spacing of the cathode grid is 2 mm. The third layer is an anode and field wire layer, where the two wires are placed alternatively. The anode wire-spacing is 4 mm. Signals from the TPC are read out through 1055 cathode pads which are arranged in 14 circular rows, as shown in Fig. 2. There are two different pad configurations. The inner section has a denser pad configuration, which consists of 6 pad rows with 4 x 7 mm2 pad size. The outer section has 8 pad rows of 8 x 13 mm2. The target cylinder is mounted in TPC and a nuclear target of 12.5 mm in radius is placed at the center of the active volume. The target cylinder

Figure 1. The downstream view of TPC. The target holder of 12.5 mm in radius is mounted at the center of the TPC.

136

Figure 2. Cathode pad plane. The beam goes inward.

and the denser pad configuration enable to detect the tracks as close as 16 millimeters from the the target center and a decay topology of C. 3. Performance study of the TPC

We have studied the intrinsic properties of the TPC. The radial coordinate ( r ) and the azimuthal coordinate (4) of a point on a track is determined by the center of radial coordinate of each pad row and center-of-gravity of induced charge on pads respectively. The axial coordinate (2) is obtained from the drift time by calculated centroid of a signal pulse. The spatial resolution of the pad plane, r 4 plane, was studied by using comic rays under magnetic field of 2 T to reduce transverse diffusion effect. The Gaussian fit to the data gives 340 pm. The position resolution along drift direction was studied by using bremsstrahlung photon without magnetic field. The obtained spatial resolution was 550 pm. The drift velocity of electrons was measured to be 50.4 mm/psec under a uniform electric field of 180 V/cm. 4. Experiment at SPring-8/LEPS

We have started hyperon photoproduction experiment from 2004 April. The energy range of the backward compton scattering photon is 1.5 2.4 GeV. The TPC is installed in solenoid magnet and operated under 2 T. The 6 plastic scintillation counters surrounds the TPC and 4 plastic scintillation counters are placed at the downstream of TPC for a timing and triggering purpose. We also use LEPS standard detector system to

-

137

detect forward going particles, which consists of a plastic scintillator start counter, tracking detectors, a dipole magnets, and a time-of-flight scintillator array 5. Fig. 3 shows event display of TPC. The target is installed in target cylinder and place at the center of TPC. We can see three tracks of particles generated from the target. This experiment will be ended at the end of December 2004.

Figure 3. Event display of TPC. Photon beam goes from left to right.

5. Summary

A Time Projection Chamber has been constructed as a 47r detector for hadron photoproduction experiments at SPring-8/LEPS. We finished performance study of the TPC. The position resolution along drift detection is 550 pm, and that of pad plane is 340 pm. The drift velocity of electrons is 50.4 mm/psec. The physics run have started from 2004 April and will be continued by the end of December. Detailed analysis is underway. References 1. V. Koch, Phys. Lett. B377 7 (1994). 2. J.C.Nacher, E. Oset, H. Toki, A. Ramos, Phys. Lett. B455 5 (1999). 3. M.F.M. Lutz and E.E. Kolomeitsev, Nucl. Phys. A700 193 (2002). 4. M. Kimura e t af. Phys. Rev. C62, 015206 (2000). 5. T. Nakano e t al., Nucl. Phys. A629 559c (1998).

138

REMARKS ON THE PARITY DETERMINATION OF NARROW RESONANCES*

C. HANHARTl, J. HAIDENBAUERl, K. NAKAYAMAlt2,U.-G. MEIf3NER173 Institut f i r Kernphysik, Forschungszentmm Julich GmbH, D-52425 Jiilich, Germany Dept. of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA Helmholtz-Institut f i r Strahlen- und Kernphysik (Theorie), Universitat Bonn NuJallee 14-16, 0-53115 Bonn, Germany

Recently several proposals were put forward to determine the parity of narrow baryonic resonances, in particular the 8+.In these proceedings we will briefly comment on the general p r o b l e y f n this task and then discuss in detail the potential of reactions of the type N N -+ 8+Y,where Y denotes a hyperon (either C or A) and the arrows indicate that a polarized initial state is required. Besides reiterating the model-independent properties of this class of reactions we discuss the physics content of some model calculations.

1. Generalities

The parity of a hadron contains significant information on its substructure. Unfortunately, especially for spin-l/2 particles, the determination of the parity is a non-trivial problem especially when we talk about narrow states. To illustrate the origin of the difficulty we observe that-in leading order of the outgoing cms momentum-the decay vertex of a spin-l/2 resonance into a spin-1/2 particle and a pseudoscalar (e.g. O+ + N K , if the O+ were indeed a spin-1/2 particle as suggested by almost all models) reads 0'. for a positive parity resonance and 1 for a negative parity resonance. Thus, as long as we do not measure the polarization of the decay products, all observables scale as (3. $)(a' . $) = or 1, accordingly, leaving no unique trace of the intrinsic parity of the decaying object. Unfortunately a measurement of the polarization of a nucleon in the final state is technically very demanding and of low efficiency. In addition, as

a. a

*This work is partly supported by COSY grant no. 41445282

139

was clearly demonstrated by Titov in this conference1, possible interference phenomena with the background amplitudes put into question, if such a measurement would indeed allow to determine the parity of the resonance unambiguously. Thus, the only straightforward option that remains is to extract the quantum numbers of this narrow resonance from a partial wave analysis. However, if the resonance of interest is very narrow also this program might be unsuccessful: in this case the partial wave analysis allows solely to put an upper limit on the width of the decaying particle5i4-note, also the more sophisticated work of Ref.6, where data on K+d scattering is used directlyhere the K N partial wave analysis was used to fix the parameters of the model for the background amplitudes. Does this mean that there is no way to determine the parity of a narrow resonance? The answer to this question is no, for one can use the stringent selection rules that are enforced by the Pauli Principle on the nucleonnucleon ( N N ) systems to manipulate the total parity of a system. It is well known that a two-nucleon state acquires a phase ( - ) L + S + T under permutation of the two particles, where L , S, and T denote the angular momentum, the total spin and the total isospin of the two nucleon system. The required antisymmetry of the N N wavefunction thus calls for L+S+T to be odd. For example, for a proton-proton state T = 1 and the parity is given by (-)L-thus each S = 1 state has odd parity and each S = 0 state has even parity. Therefore, preparing a pure spin state of a p p system means preparing a N N state of known parity. In case of a T = 0 state, the assignment of spin and parity needs to be reversed. A well known textbook example that exploits this method is the measurement of the parity of the pion2 in n- capture on deuterium from an atomic s-state (n-d -,nn); this transition would be forbidden if the parity of the spin 0 pion were positive. To see this we recall that an even parity nn state, characterized by an even value of L, is to be a spin singlet; consequently, in this case we have for the total angular momentum J = L. On the other hand, since for the deuteron j = 1 the initial state has J = 1. Since the parity of the whole system agrees to that of the pion, an s-wave production is allowed only for negative parity pions. It was essential in case of the reaction dn- -, nn that the deuteron is a j = 1 state and the pion is a scalar-this fixed the total angular momentum to J = 1 for the s-wave initial state. For the system N N YO+ the final state can be both spin singlet as well as spin triplet. Therefore in this case we need to manipulate the spin of the initial state to fix its parity. This ---f

140

observation was first made in Ref.7 and then further exploited in a series of publications378igi10~11~12i13. In addition, we will have to use the energy dependence of the spin cross sections to identify the leading partial wave, as was observed in R e f ~ . ~ t I ~ .

-0.5

-0.5

h

P

00

0

“ W

Q [MeV]

Q MeV1

Q MeV1

Figure 1. Energy dependence of the total cmss section and the angular integrated polarization observable A,, and 3uc for the reaction pp 4 C+Q+. Solid (dashed) lines correspond to a negative (positive) paricy Q+. Shown are results for three different models for the production operator: the left column shows the results for the model with only kaon exchange, the middle one those for the one with K* exchange and the *ht one those for the model including K* and K exchange. All results for uo and 3uc are normalized to 1 at an excess energy of 80 MeV and are divided b y the phase-space volume.

141

2. The ideal observable

In terms of the so-called Cartesian polarization observables, the spindependent cross section can be written ad7

a(<,8,gt, @f) = go(<) 1 + G ( ( P b ) i A i 0 ( 5 + ) (Pf)iDOi(5))

+ C ( ( P b ) Z ( P t ) j A i j ( <+) (Pb)i(Pf)jDij(E)) aj

+

c

(Pb)i (Pt j ( P f1k Aij,k (5)...] .

( 11

ijk

where ao(5) is the unpolarized differential cross section, the labels i,j and k can be either x,y or z , and Pb, Pt and P f denote the polarization vector of beam, target and one of the final state particles, respectively. All kinematic variables are collected in <. In Refs.15>17>18 it was shown, that a measurement of the spin correlation parameters A,,, A,,, A,, as well as the unpolarized cross section allows to project on the individual initial spin states. More precisely

- A,,) 3 a= ~ a0(1+ A,, + A,, - A,,) , la0 = ao(1 - A,, - A,,

3a1 = ao(1

+ A,,)

7

,

(2)

where the spin cross sections are labeled following the convention of Ref. l7 as 2 s + 1 a ~with s , S the total spin of the initial state and M s its projection; 00 denotes the unpolarized cross section. Unfortunately, longitudinal polarization (needed for A z z ) is not easy to prepare in a storage ring. However, the following linear combination projects on spin triplet initial states and no longitudinal polarization is needed8: 1 1 3aE = - ( 3 ~ 0+3 al) = -ao(2 + A , , A,,) . (3) 2 2

+

For @++ O+C+, only negative parity states contribute to 3 a ~thus ; only in case of a negative parity O+ s-waves are allowed in the final state. If we assume the O+ to be an isoscalar, the reaction 3 + @+A gives the same amount of information, however, positive and negative parity change their roles. It is well known that for large momentum transfer reactions in the near threshold regime the energy dependence of a partial wave characterized by angular momentum I is given by (p/A)l, where A denotes the intrinsic scale

142

of the production process-for reactions of the type N N typically given by the momentum transfer

+

B1B2 this is

~ ( M +BM ~B ~ ~ M N ) M .N For an extensive discussion of this type of reaction we refer to Ref.15. It is thus sufficient to measure the energy dependence of 302:for small excess energies to pin down the parity of the O+: in case of a positive parity in the pp channel it scales as phasespace times an odd polynomial in the excess energy Q = f i - 6, where SO denotes the threshold energy. On the other hand, if the resonance has negative parity the energy dependence should be that of phase-space times an even polynomial. It should be stressed that these considerations apply only for outgoing cms momenta significantly smaller than A-for larger energies no general statement on the energy dependence is possible in a model-independent way. This is quite obvious once it is recalled that there are rigorous energy-dependent bounds on the strength of the individual partial waves set by unitarity-thus there should not be an unlimited growth. The near threshold properties of 302:are shown in the lower line of Fig. 1, where the results for different models (for more details about these see next section and the appendix of Ref.13) for the unpolarized cross section 00, the spin correlation coefficient A,, and 302: are shown. Although the energy dependence of 00 as well as A,, is vastly different, reflecting the different admixture of partial waves in the different models, the figure clearly illustrates that the energy dependence of 3 ~iscan unambiguous signal for the parity of the Q+. In Ref.13 we also investigate if the spin transfer coefficient D,, can be used for a parity determination and we refer the interested reader to this paper. The remaining space available for these proceedings will now be used to discuss the reliability and features of model calculations for reactions of the type N N + B1B2.

P

N

3. Remarks on models As mentioned above the reactions N N -, BlB2 are characterized by a large momentum transfer. This has two consequences: first of all the energy dependence of the production process in the near threshold regime is fixed model-independently; secondly, it is very difficult to construct a reliable microscopic model for these reactions. Probably the most clear illustration of the latter point is the fact that there is not even a microscopic model available to hescribe the data on

143

Y

Figure 2.

Q+

Y

Diagrams considered in the model calculations.

p p + p p ~ ? , although this is the first inelasticity of the N N system and much is known about all the subsystems. Only recently it was observed, that the large momentum transfer as it occurs in inelastic N N reactions leads to a relative enhancement of pion loop contributions (see discussion in Ref. 15). Given this it seems inappropriate to construct a model for the O+Y that is quantitatively reliable, since here a lot less is reaction N N known about possible production mechanisms-there might even be quite complicated production mechanisms of relevance, like the decay of a heavier resonance as proposed in Ref.lG. In addition, it is well established that the initial state interaction can have a significant effect on observables. First of all it reduces the cross section typically by a factor of 2-3 and secondly it introduces an additional phase to the individual amplitudes; especially the latter effect can well change the shape especially of polarization observables, for they are quite sensitive to the relative phases of the contributing amplitudes. Note, in reactions of the type N N + O+Y, where the final state interaction is expected to be weak, the relative phase of the amplitudes is largely introduced by the N N interaction in the initial state. Through the Watson theoremlg this phase can be related to the N N scattering phaseshifts. Although this is true rigorously only in the elastic case, it is still reasonable to expect a properly adjusted formula to also work for the N N system at energies as high as relevant for the O+ production20. However, such a detailed work is beyond the scope of this paper--especially, the initial state interaction will not change significantly the energy dependence of observables, as long as only energies close to the production threshold are investigated. ---f

144

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-1 -0.5 0 0.5 cos(e)

1

Figure 3. Angular distributions for the K-exchange model calculated at an excess energy of 40 MeV. The meaning of the curves is as in Fig. 1 . The lower curve an the upper left panel is scaled by a factor of 10 (as indicated in the panel).

As a result of all this as in Ref.13 we here take a more pragmatic point of view: if all statements made above are indeed true model-independently, they have to apply to any (realistic) model. Since we do not trust the overall scale of the model results and for a better comparison of the results for the two different parities, the results for the integrated observables in

145

+ +

pp->x 0

8 6 4 2 0 0.5 0 -0.5

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

-0.5

0.5

2 -0.50

0.5 0

...\.............A.........

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

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

-0.5

0.5

0.5 I-

2 -0.5O I-

-I

- 0

--I

-0.5

-0.5

-0.5

-1 -0.5 0

0.5

1

COqe) Figure 4. Angular distributions for the K + K*-exchange model calculated at an excess energy of 40 MeV. The meaning of the curves is as in Fig. 1. The lower curve in the upper left panel is scaled by a factor of 10 (as indicated an the panel).

Fig. 1 are normalized to 1 at 20 MeV. We constructed a model where a scan through a wide range of parameters allowed to study a large class of different effects. To be more specific, we included the diagrams shown in Fig. 2, fixing the coupling strength of the K exchange and then varied the parameters for the K* exchange. The

146

following three models turned out to be representative for the many cases we studied, namely a model with purely K exchange, a model with purely K* exchange and a mixture of both. In the third case the relative strength of the two diagrams was adjusted such that for a positive parity @+ the swave contributions of the two diagrams canceled in the pp reaction". This lead to a quite drastic energy dependence of A,,. It turned out that, although the energy dependence of the cross section as well as A,, was very different for different models, the behavior of 3crc was always as described above. One question repeatedly asked on the conference was that about the possible production mechanisms. Given the problems mentioned above regarding the construction of the production operator for @+ production in N N collisions, it will most probably not be possible to unambiguously identify a particular production mechanism as the most significant one from data on these reactions directly. However, what the polarization observables serve for is to exclude particular production mechanisms. To be more concrete: as different production mechanisms are typically characterized by different spin and isospin dependencies, they will lead to quite different polarization observables for the pp and pn induced reaction. This is illustrated in Figs. 3 and 4, where the angular distributions of various spin observables are shown for two different sets of model parameters (pure K exchange and K K* exchange) for both channels. Clearly, the two models lead to very different angular dependencies of most of the observables shown. Thus, these data could well be used to exclude particular production mechanisms (once the initial state interaction is included as described above) and thus improve our understanding of the hadronic interactions of the @+ (... if it exits).

+

4. Summary and outlook

-

In summary we have argued that the most promising method to model independently determine the parity of narrow resonances is a measurement of I?$ BlB2: the energy dependence of the spin triplet cross section, given by 3 c r ~= iao(2 A,, A,,), is the ideal observable. There is also a chance that croD,, also allows to determine the parity-for details on this we refer to Ref.13.

+

+

also unsuccessfully tried to construct a model where the same happens for the negative parity.

147

In addition, especially the angular dependence of the large number of existing polarization observables will put strong constraints on the allowed production mechanisms. In this project theory did its part-now we have to wait for the experimental realization. We want to close with a few comments on the prospects of a measurement of A,, for N N -+ W Y . So far a measurement for the unpolarized cross section in the p p channel is completed by the TOF collaboration at the COSY accelerator21: with a statistical significance of about 4.5 u a total cross section of 400 nb was extracted from data on p p --t C + p K o . At COSY polarized beams are routinely available and a frozen spin target is currently being adopted to the COSY conditions. The first double polarized measurement is expected for summer 2005. Acknowledgments C.H. thanks the organizers for a very informative, inspiring, educating, enjoyable, and perfectly organized workshop.

References 1. A. Titov, these proceedings. 2. see e.g. D.H. Perkins, Introduction to high energy physics, Addison-Wesley Publishing Company, 1987. 3. W. Liu, C. M. KO, Phys. Rev. C 68,045203 (2003); T. Hyodo, A. Hosaka, E. Oset, Phys. Lett. B 579,290 (2004). 4. J. Haidenbauer and G. Krein, Phys. Rev. C 68 (2003) 052201. 5. R. A. Arndt, I. I. Strakovsky and R. L. Workman, Phys. Rev. C 68 (2003) 042201 [Erratum-ibid. C 69 (2004) 019901] 6. A. Sibirtsev, J. Haidenbauer, S. Krewald and U. G. Meissner, arXiv:hep ph/0405099. 7. A. W. Thomas, K. Hicks and A. Hosaka, Prog. Theor. Phys. 111, 291 (2004). 8. C. Hanhart et al., Phys. Lett. B 590,39 (2004). 9. S. I. Nam, A. Hosaka and H. C. Kim, arXiv:hepph/0401074. 10. S. I. Nam, A. Hosaka and H. C. Kim, arXiv:hepph/0402138. 11. Y . N. Uzikov, arXiv:hepph/0401150; M. P. Rekalo and E. Tomasi-Gustafsson, Phys. Lett. B 591,225 (2004). 12. Y . N. Uzikov, arXiv:hepph/0402216; M. P. Rekalo and E. Tomasi-Gustafsson, arXiv:hepph/0402277. 13. C. Hanhart, J. Haidenbauer, K. Nakayama and U. G. Meiflner, O n the determination of the parity of the 8+,arXiv:hepph/0407107. 14. M. Abdel-Bary et al., arXiv:hepex/0403011. 15. C. Hanhart, Phys. Rept. 397,155 (2004). 16. M. Karliner and H.J. Lipkin, arXiv:hepph/0405002; and H.J. Lipkin, these proceedings. D. Diakonov, arXiv:hepph/0406043.

148 17. 18. 19. 20. 21.

H. 0. Meyer et al., Phys. Rev. C 63 (2001) 064002. P. N. Deepak and G. Ramachandran, Phys. Rev. C 65(2002) 027601 . M. Goldberger and K.M. Watson, Collision Theory, Wilney, New York, 1964. C. Hanhart and K. Nakayama, Phys. Lett. B 454, 176 (1999). M. Abdel-Bary et al. [COSY-TOF Collaboration], “Evidence for a narrow resonance at 1530-MeV/c**2 in the KO p system of the Phys. Lett. B 595, 127 (2004); W. Eyrich, these proceedings.

149

PENTAQUARK BARYON PRODUCTION IN NUCLEAR REACTIONS *

C. M. KO AND W. LIU Cyclotron Institute and Physics Department Texas A U M University College Station, T X 77845-3366,USA E-mail: [email protected], [email protected]

Using a hadronic model with empirical coupling constants and form factors, we have evaluated the cross sections for the production of exotic pentaquark Ot and/or E : and 5.5- in reactions induced by photons, nucleons, pions, and kaons on nucleon targets. We have also predicted the O+ yield in relativistic heavy ion collisions using a kinetic model that takes into account both O+ production from the coalescence of quarks and antiquarks in the quark-gluon plasma and the effects due to subsequent hadronic absorption and regeneration.

1. Introduction One of the most exciting recent experimental results in hadron spectroscopy is the narrow baryon state that was inferred from the invariant mass spectrum of K + n or K o p in nuclear reactions induced by photons ', kaons 2 , and protons '. The extracted mass of about 1.54 GeV and width of less than 21-25 MeV are consistent with those of the pentaquark baryon O+ consisting of uudds quarks and predicted in the chiral soliton model '. However, the reported large width is limited by experimental resolutions as the actual width is expected to be much smaller '. Its existence has also been verified in the Skyrme model 6, the chiral quark model ', the constituent quark model ', the QCD sum rules ', and the lattice QCD l o . Although most models predict that O+ has spin 1/2 and isospin 0, their predictions on O+ parity vary widely. While the soliton model gives a positive parity and the lattice QCD studies favor a negative parity, the quark model can give either positive or negative parities, depending on whether quarks are *Supported in part by the US National Science Foundation under Grant No. PHY0098805 and the Welch Foundation under Grant No. A-1358.

150

correlated or not. To help determine the quantum numbers of O+, studies have been carried out to understand its production mechanism in these reactions Using a hadronic model with empirical coupling constants and form factors, we have evaluated the cross sections for O+ production from these reactions and also those for other multistrange exotic pentaquark baryons E$(uussdd) and E;-(ddssuii) 16. The latter has been observed in proton-proton collisions at center-of-mass energy of 17.2 GeV by the NA49 Collaboration at SPS 17, with mass 1.86 GeV and width about 18 MeV due to detector resolution. Moreover, the yield of O+ in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) has been studied using the quark coalescence model 18. In this talk, we report the results from our studies. 11i12.

13114115

2. @+ production in photonucleon reactions

Figure 1.

Diagrams for O+ production from the reaction yn --+ K-O+.

For O+ production from photonucleon reactions, possible processes are yp + KO@+, yn + A?-@+,yp + I?*'@+, and yn + K*-@+. As an illustration, we show in Fig.1 four possible diagrams for the reaction yn + I<-@+, involving the t-channel I<- and K*-exchange, the s-channel nucleon pole, and the u-channel O+ exchange. To evaluate their amplitudes, we need the photon coupling to O+ as well as the coupling constants g N K @ and g N K * 0, besides the well-known couplings of photon to I<-, I<*-, and neutron. Since the anomalous magnetic moment of O+ is not known empirically and theoretical estimates give a much smaller value than that of nucleons l 2 ? l 9 ,we have neglected its contribution and included only the coupling of photon to the charge of @+ . For the coupling constant g K N @ , it is proportional to the square root of O+ width and depends on the spin and parity of O+. Taking the O+ width to be 1 MeV, which is consistent with available I<+N and l i d data 5 , we then have g K N @ % 1 if @+ has spin 1/2 and positive parity. It is reduced to g K N @ % 0.14 if the @+ parity is negative due to the absence of a centrifugal barrier between nucleon and kaon.

151

For the coupling constant Q K * N @ ,it would have been g K * N @ M 0 . 6 g ~ if ~e N(1710), which has same partial decay width of 15 MeV to N7r and N p , were to belong to the same antidecuplet as O+. On the other hand, it is given by g K * N @ = & K N @ in the fall-apart decay model 'O. In our work, we have assumed that g K * N @ is zero or has the same magnitude as g K N @ .

30,

-10

,

05

00

05

10

-m

Figure 2. Total (upper panels) and differential (lower panels) cross sections for production from photo-nucleon reactions.

Ot

To take into account the finite sizes of hadrons, we have included for each amplitude a form factor of the form F ( x ) = A4/[A4 (x - m:)'], where x = s, t , and u with corresponding masses rn, = m N , m K or m K - , and me of the off-shell particles a t strong interaction vertices 'l. After restoring the gauge invariance of resulting total amplitude by adding a suitable contact term in the interaction Lagrangian, the cross section for the reaction yn + K-O+ has been evaluated in Ref.15 with a cutoff parameter A M 1.2 GeV, that is determined from fitting the measured cross section for charmed hadron production from photon-proton reactions with two-body final states ". The resulting total and differential cross sections for the reaction yn + I<-@+ are shown in the leftmost panels of Fig.2. The cross sections at photon energy E-, = 3 GeV are about 30 nb and 15 nb with and

+

152

without Ii* exchange, respectively, and the produced Of peaks at forward direction in the center-of-mass system. For negative parity @+, the cross section is reduced by about a factor of 10, although the coupling constant g K N @ is about a factor of 7 smaller than that for positive parity @+, and the angular distribution remains forward peaked. The cross sections for O+ production from other photon-nucleon reactions y p + I?'@+, y n + I<*-@+, and yp + I?*'@+ can be similarly evaluated, and their cross sections are shown in other panels of Fig.2. For the reaction yp + I?'@+, its value at ETy= 3 GeV is about 35 nb if I<* exchange is included, and this is comparable to the revised value of about 50 nb measured in SAPHIR experiment at Bonn Electron Stretcher Accelerator. 3. O+ production in proton-proton reactions z+

P

z+

Q+

P

P

Q'

P

z+

P

P

P

P

Figure 3. Diagrams (left panels) and cross sections (right panel) for O+ production from the reaction p p C+ 0 s .

-+

For @+ production in proton-proton reactions, possible diagrams are shown in the left panels of Fig.3 involving both t- and u-channel exchange of Ii' and I{*. The coupling constants Q K N C M -3.78 and Q K * N C = 3.25 are obtained from well-known glF" = 13.5 and g p N N = 3.25 by SU(3) relations. The form factors at interaction vertices are taken to be F(q2)= A2/(A2 + q2), with q being the three momentum of exchanged Ii' or I<*. Using a cutoff parameter A = 0.42 GeV, obtained from fitting the experimental cross sections for the reactions p p + Ii'+Ap and p p + D+A,p with similar hadronic models 23, the resulting cross section for the reaction p p + C+O+ is shown in the right panel of Fig.3 for g K N @ = 1 and different values of g K * N @ . The value of 0.3 p b obtained with g K * N @ = 1.0 at

153

center-of-mass energy f i = 2.75 GeV, corresponding to a beam momentum = 2.95 GeV/c, is comparable to the 0.4 p b measured in the COSY experiment 3.

pbeam

4. O+ production in pion- and kaon-nucleon reactions

Figure 4. Diagrams for O+ production from the reactions T N -+ KO+ (left two panels) and K N -+ TO+ (right two panels).

For O+ production from the reactions T N + I?@+ and I i N + TO+, the relevant diagrams are shown in Fig.4. Using additional known empirical coupling constant g r K K * = 3.28 and similar form factors as in the reaction p p + C+O+, their cross sections are shown in the left two panels of Fig.5 with values of 0.5-5 p b for T N + KO+ and 20-75 p b for Ii" -+ TO+ at center-of-mass energy of 0.2 GeV above their respective threshold energies. It will be interesting to compare the predicted cross sections with future data from experiments that are being carried out at KEK in Japan.

.,.." O

:

35

40

,../

/

45

50

L

55

s'"(GeV) Figure 5. Cross sectionsfor O+ production from the reactions T N -+ KO+ and K N -+ TO+ (left two panels) and and Z i from photon-nucleon reactions (right two panels).

a,-

5.

=$

and

production in photonucleon reactions

For =$ production in photonucleon reactions, the relevant diagrams are shown in Fig.6. The needed additional coupling constants g K ( K * ) E a s can

154

Figure 6.

Diagrams for 2; production from the reaction y p

+ K°Ko2;.

be obtained from the SU(3) relations g K ( K * ) x z 5= gK(K.)N@/fiif O+ and 2 5 belong to the same antidecuplet 24. Introducing appropriate form factors with empirical cutoff parameters l6, the resulting cross section for the reaction yp + K°KoEj' together with that for the reaction yn + K+K+EFare shown in the right two panels of Fig.5 for gKCEs = 0.71 but different values of gK*Czs.The cross sections are about 0.01-0.1 nb at photon energy E, = 4.5 GeV, and this information will be useful for planning experiments to verify these exotic multistrange pentaquark baryons in photonucleon reactions. 6. @+ production in relativistic heavy ion collisions

The quark-gluon plasma formed in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) 25 provides a promising environment for producing hadrons consisting of multiple quarks such as the pentaquark baryons. Based on the quark coalescence model for hadron production from the quark-gluon plasma 26, which has been shown to give a consistent description of the observed large baryon/meson ratio at intermediate transverse momenta of about 3 GeV/c and scaling of hadron elliptic flows according t o their constituent quark content, we have evaluated the yield of O+ in Au+Au collisions at = 200 GeV. With a plasma of quarks of constituent masses and massive gluons at temperature Tc = 175 MeV and in a volume 1000 fm3 as in Ref.26, the resulting O+ number is about 0.19 if O+ has a radius of 0.9 fm, and is somewhat smaller than that predicted by the statistical model 27. To take into account subsequent hadronic effects due to the reactions 0 c) K N , On H K N , and OR H n N , a schematic isentropic expanding fireball model has been used With the cross sections evaluated in previous sections, the time evolution of the O+ abundance has been studied for different values of @+ width or coupling constant g K N @ . The final @+ yield increases t o about 0.5 for I?@ = 20 MeV but remain about 0.2 for

-

-

155

= 1 MeV. Similar effects are seen if the number of @+ produced from the quark-gluon plasma is increased t o 0.94, resulting from a @+ radius that is 30% larger, or decreased to zero. With the expected small O+ width, the hadronic effect is thus small, and the abundance of O+ is then sensitive to its production from the quark-gluon plasma. Since the baryon chemical potential in the quark-gluon plasma is small, the yield of anti-O is only slightly less than that of @+. 7. Summary We have studied exotic pentaquark baryon production both in elementary reactions involving photons, protons, pions, and kaons on nucleon targets and in relativistic heavy ion collisions. With a phenomenological hadronic model, cross sections for the elementary reactions are evaluated by taking into account the coupling of @+ to both K N and K * N . Assuming that @+ has positive parity and a width of 1 MeV, corresponding to a coupling constant g K N @ M 1, and depending on the value of g K * N @ , the cross sections are 4-35 nb for yp + ROO+, 15-30 nb for y n + I(-@+, 4-15 nb for yp + I?*'@+, and 5-150 nb for y n + I(*-@+ at photon energy E, = 3 GeV; 0.1-2 p b for pp + C+@+, 0.5-5 p b for n N + K O + ,and 20-75 p b for K N + KO+ at center-of-mass energy of 0.2 GeV above their threshold. The produced @+ in these reactions all peaked at forward angles in the center-of-mass system. The cross sections are smaller by about an order of magnitude if @+ has negative parity. For multistrange pentaquark baryons Et and ZF-, the cross sections for their production in the reactions yp + K°KoE$ and yn + I(+K+E;- are 0.01-0.1 nb at photon energy E, = 4.5 GeV. In heavy ion collisions at RHIC, about 0.19 O+ is produced during hadronization of the quark-gluon plasma, but the final number after hadronic absorption and regeneration depends on the width of O+. These results are useful for understanding not only the experimental data but also the properties of pentaquark baryons. Acknowledgments

We would like to thank Lie-Wen Chen, Vincenzo Greco, and Su Houng Lee for discussions and collaboration on some of the reported works. References 1. T. Nakano et al., Phys. Reo. Lett. 91, 012002 (2003); S. Stepanyan et al. (CLAS Collaboration), ibid. 91, 252001 (2003); J. Barth et al. (SAPHIR Collaboration), Phys. Lett. B572,127 (2003).

156

2. V. V. Barmin et al., Phys. A t . Nucl. 66, 1715 (2003); hep-ex/0304040. 3. M. Abdel-Bary et al. (COSY Collaboration), Phys. Lett. B595, 127 (2004). 4. D. Diakonov, V. Petrov, and M. Poliakov, Z. Phys. A359, 305 (1997). 5. R. A. Arndt, I. I Strakovsky, and R. L. Workman, Phys. Rev. C68,042201(R) (2003); R. A. Arndt, Ya. I. Azimov, M. V. Polyakov, I. I. Strakovsky, and R. L. Workman, ibid. C69, 035208 (2004); J. Haidenbauer and G. Krien, ibid. C68, 052201(R) (2003); R. N. Cahn and G. H. Trilling, ibid. D69, 011501 (2004). 6. M. Praszalowicz, Phys. Lett. B575, 234 (2003); M. V. Polyakov and A. Rathke, . Eur. Phys. J . A18, 691 (2003); H. Walliser and V. B. Kopeliovich, J. Exp. Theor. Phys. 97,433 (2003); N. Itzhaki, I. R. Klebanov, P. Ouyang, and L. Rastelli, Nucl. Phys. B684, 264 (2004). 7. A. Hosaka, Phys. Lett. B517, (2003); L. Y. Glozman, ibid. B575, 18 (2003). 8. F1. Stancu and D. 0. Riska, Phys. Lett. B575, 242 (2003); M. Karliner and H. J. Lipkin, hep-ph/0307243; R. L. J d e and F. Wilceek, Phys. Rev. Lett. 91,232003 (2003). 9. S. L. Zhu, Phys. Rev. Lett. 91,232002 (2003); R. D. Matheus, F. S. Navarra, M. Nielsen, R. Rodrigues da Silva, and S. H. Lee, Phys. Lett. B 578, 323 (2004); F. Sugiyama, T. Doi, and M. Oka,ibid. B581, 167 (2004). 10. F. Csikor, Z. Fodor, S. D. Katz, and T. G. Kovbcs, JHEP 0311,070 (2003); hep-lat/0309090; S. Sakaki, Phys. Rev. Lett. 93,152001 (2004). 11. Y. Oh, H. Kim, and S. H. Lee, Phys. Rev. D69,014009 (2004); D69, 074016 (2004); Q. Zhao, ibid. D69, 053009 (2004); E. Hyoto, A. Hosaka, E. Oset, Phys. Lett. B579, 290 (2004); K. Nakayama and K. Tsushima, ibid. B583, 269 (2004); P. KO et al., hep-ph/0312147. 12. S. I. Nam, A. Hosaka, and H. C. Kim, Phys. Lett. B579, 43 (2004). 13. W. Liu and C. M. KO, Plays. Rev. C68, 045203 (2003). 14. W. Liu, C. M. KO, and V. Kubarovsky, Phys. Rev. C69, 025202 (2004). 15. W. Liu and C. M. KO, Nucl. Phys. A741, 215 (2004). 16. W. Liu and C. M. KO, Phys. Rev. C69, 045204 (2004). 17. C. Alt et al. (NA49 Collaboration), Phys. Rev. Lett. 92,042003 (2003). 18. L. W. Chen, V. Greco, C. M. KO, S. H. Lee, and W. Liu, Phys. Lett. 601, 34 (2004). 19. H. C. Kim and Praszalowicz, Phys. Lett. B585, 99 (2004). 20. C. E. Carlson, C. D. Carone, H. J. Kwee, and V. Nazaryan, Phys.Rev. D70, 037501 (2004). 21. R. M.Davidson and R. Workman, Plays. Rev. C63, 058201 (2001) 22. W. Liu, S. H. Lee, and C. M. KO, Nucl. Phys. A724, 375 (2003). 23. W. Liu, C. M. KO, and S. H. Lee, Nucl. Phys. A728, 457 (2003). 24. Y. Oh, H. Kim, and S. H. Lee, Phys. Rev. D69,094009 (2004). 25. Proceedings of Quark Matter 2004, J . Phys. G30, S1 (2004). 26. V. Greco, C. M. KO, and P. LCvai, Phys. Rev. Lett. 90,202302 (2003); Phys. Rev. C68,054904 (2003). 27. J. Randrup, Phys. Rev. C68, 031903 (2003); J. Letessier, G. Torrierie, S. Steinke, and J. Rafelski, ibid. C68,061901(R) (2003).

157

PHOTOPRODUCTION OF O+ ON THE NUCLEON AND DEUTERON

T. MART Departemen Fisika, FMIPA, Universitas Indonesia, Depok 16&& Indonesia

A. SALAM AND K. MIYAGAWA Department of Applied Physics, Okayama University of Science, 1-1 Ridai-cho, Okayama 700, Japan C. BENNHOLD Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, D . C. 20052, USA Photoproduction of the pentaquark particle 8+ on the nucleon has been studied by using an isobar and a Regge model. Using the isobar model, total cross sections around 100 nb for the p --t K-O+ channel and 400 nb for the y p + K08+ process are obtained. The inclusion of the K *intermediate state yields a substantially large effect, especially in the y p + ROB+ process. The Regge approach predicts smaller cross sections, i.e., less than 100 nb (20 nb) for the process on the neutron (proton). By using an elementary operator from the isobar model, cross sections for the process on a deuteron are predicted.

1. Introduction The observation of the pentaquark 0+baryon' has triggered a great number of investigations on the production process of this unconventional particle. In general, these efforts can be divided into two categories, i.e., investigations using hadronic and electromagnetic processes. The electromagnetic (photoproduction) process is, however, well known as a more "clean" process. Furthermore, photoproduction process provides an easier way to "see" the'0 which contains an antiquark, since all required constituents are already present in the initial state2. Other processes, such as e+e- and pp annihilations, would produce the strangeness-antistrangeness from gluons, which has a consequence of the suppressed cross section3. Several'0 photoproduction studies have been performed by using iso-

158

bar models with Born approximation4, where the obtained cross section spans from several nanobarns to almost one pbarn, depending on the O+ width, parity, hadronic form factor cut-off, and the exchanged particles used in the process. Those parameters are unfortunately still uncertain at present. In this paper, we calculate the photoproduction cross sections by utilizing an isobar model. Since the production threshold is already high we compare the results with those obtained from a Regge model. The comparison is also very important, since most input parameters in the isobar model are less known. 2. Formalism

The basic background amplitudes for the processes

are obtained from a series of treelevel Feynman diagrams shown in Fig. 1. They contain the n, Q+, K - , K*- and K1 intermediate states in the first process, whereas in the second process the K O exchange does not present since a real photon cannot interact with a neutral meson. The K* and K1 intermediate states are considered here, since previous studies on K A and KC photoproductions have shown that their roles are significant. Y

K

Figure 1. Feynman diagrams for 8+ photoproduction on neutron 7 + n t K (top) and on the proton 7 p -+ K O 8+ (bottom).

+

+

+ Q+

159

The transition matrix for both reactions can be decomposed into 4

Mfi = a@')

C Ai Mi

U(P)

(1)

7

i=l

where the gauge and Lorentz invariant matrices Mi are given in, e.g., In terms of Mandelstam variables s, u, and t , the functions Ai are given by

with ge = g K O N , Q e = 1, Q N = 1(0) for proton (neutron), KN and K e indicate the anomalous magnetic moments of the nucleon and O+, and M is taken to be 1 GeV in order to make the coupling constants GVyT = V,T gK * e N g K * K r dimensionless. The inclusion of hadronic form factors at hadronic vertices is performed by utilizing the Haberzettl prescriptionls. The form factors are taken as

A4 = A4 + (q2 - m3)2

with

q2 = s,u,t

,

(6)

with A the corresponding cut-off. The form factor for non-gauge-invariant terms F ( s ,u, t ) in Eq. (3) is extra constructed in order to satisfy crossing symmetry and to avoid a pole in the amplitudel9, i.e.,

160

Since O+ is an isoscalar particle, the coupling constants relations read QKBN

= gK-Q+n = gKOe+p

VT

7

gi*eN

VT

V,T

= g&-e+n = g K * o e + p

*

(8)

The coupling constant g K - @ + , , can be calculated from the decay width of the Q+ + K f n by using

+

The e .precise with p = [ { m i - (mK rn,J2}{mi - (m, - r n ~ ) ~ ) l / ~ ] / 2 m measurement of the decay width is still lacking due to the experimental resolution. The reported width1 is in the range of 6-25 MeV. Using the partial wave analysis of K + N data Arndt et a15 found I' 5 1 MeV, whereas the PDG6 announces r = 0.9 f0.3 MeV. Based on this information we use a width of 1 MeV in our calculation. Explicitly, we use gKeN/&

= 0-39 .

(10)

The magnetic moment of O+ is also not well known. A recent chiral soliton calculationg yields a value of pe = 0.82 p ~from , which we obtain I E = ~ 0.35. Note that in the second channel the Regge model does not depend on this coupling constant as well as the 0 '- magnetic moment. The coefficient CK*in Eqs. (2)-(5) is introduced since in '?I photoproduction the vector meson exchange in the t-channel is K*'. The coefficient reads'' CK*

= 1 for K-O+ [-1.53 for I7OO+] .

(11)

The coupling constants g & e N and & e N are also not well known. Therefore, we follow Refs.4J1, i.e., using g g * Q N = 1.32 and neglecting g & e N due to the lack of information on this coupling. By combining the electromagnetic and hadronic coupling constants we obtain

G L * Q N / ~=T8.72

X

.

(12)

Most previous calculations excluded the K1 exchange, mainly due t o the lack of information on the corresponding coupling constants. Reference4 used the vector dominance relation g K I K r = e g K , K p / f p to determine the electromagnetic coupling Q K I K r r where f , 2 / 4 ~= 2.9 and g K I K p = 12 is taken from the effective Lagrangian calculation of Ref. 14. As in the case of K * , the K 1 hadronic tensor coupling will be neglected in this calculation due to the same reason. Following Ref.4, the K1 axial vector coupling g z l e N is estimated from an isobar model for K+A photopr~duction~~ by using the extracted ratio G;.,,N/GKIAN= -8.26. We note that the Same

161

ratio is also obtained in Ref. l2 for the model without missing resonance 013(1895). Therefore, in our calculation we use GK1eN/47r = -7.64 x

.

(13)

The constant C K ~in Eqs.(3) and (4) is extracted from fitting an isobar model to the K+Co and K0X+ photoproduction data13, i.e.,

CK, = 1 for K-Q+ [-0.17 for I?'@+]

.

(14)

3. Regge Model

In Regge model one should only use the K- and K' (K" and K1) diagrams in Fig.1 for the yn + K-@+ ( y p KO@+) channel. Hence, the result from Regge model will not depend on the value of g K B N and Q+ magnetic moment in the second channel. The procedure is adopted from Ref. 16, i.e., by replacing the Feynman propagator with the Regge propagator

where Ka refers to K' and K1,and sponding trajectory16.

( Y K(~t ) = (YO

+ a' t denotes the corre-

4. Results and Discussion

The differential cross sections obtained from the isobar model in both channels are shown in Fig. 2. Obviously, both channels show a forward peaking differentialcross section which is due to the strong contribution from the K* intermediate state. Previous studies which use only Born terms4 obtained

da I dQ (nWsr)

4

Figure 2.

Differential cross sections obtained by using the isobar model.

162

a backward peaking cross section for the .yp -+ Roo+channel, since in this case no t-channel intermediate state is included. Figure 2 also demonstrates that the hadronic form factors are unable to suppress the cross sections at higher energies. The strong contribution of the K* in both channels can be observed in Fig. 3, where we can see that the inclusion of this state increases the total cross sections by more than one order of magnitude. In contrast to the K * , contribution from the K I vector meson is negligible. This fact can be traced back to the coupling constants given by Eqs. (13) and (14).

__._---.--. /----------

y p +KO@+ 0.01

Born K* +

+

K'

Born 2

2.5

3

3.5

4

4.5

5

5.5

2

2.5

3

W (Gv)

Figure 3. sections.

3.5

4

4.5

5

5.5

W (Gev)

Contribution of the Born terms, K*- and K1-exchange to the total cross

y n +K O+

6-

0.01

............... A = 0.8 GeV

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

-

0.m1

2

2.5

3

3.5

4

4.5

w (GV)

5

5.5

2

2.5

3

3.5

W

4

4.5

5

5.5

mv)

Figure 4. Total cross sections for 8+ photoproduction off a neutron (left) and a proton (right) a8 a function of the hadronic form factor cut-off A.

Figure 4 demonstrates the sensitivity of the total cross sections to the

163 1wo

. . . . . . yn+K-Q+

Irn

10

N -

>

dg ?

P

10

1

2

0.1

0.1

P 0.01

0.01

0.001

0.001

0 0.2 0.4 0.6 0.8

1

1.2 1 4

-r (G&)

-r (G~v')

Figure 5. Differential cross section for Q+ photoproduction obtained from the Ftegge calculation. The corresponding total c.m. energy W is shown in each panel.

40

15

2

2.5

3

3.5

4

4.5

5

5.5

E., (&v) Figure 6. Total cross section for the inclusive 8+ photoproduction on the deuteron.

choice of the hadronic form factor cut-off. Clearly, a right choice of the cut-off is very important in this case. For this purpose, we calculate also the cross sections by using a Regge model. The results are shown in Fig. 5. Obviously, the Regge approach predicts smaller cross sections than those obtained from the isobar model. In the case of K A and KZ= photoproductions, Ref.2o showed that Regge model works nicely at higher energies (up to W = 5 GeV) but overpredicts the K+A (underpredicts the K+tCo) data at the resonance region (W 5 2 GeV) by up to 50%. Thus, we would expect the same result for O+ photoproduction. By comparing with the result obtained from the isobar model, we can conclude that the isobar prc+ diction could overestimate the realistic cross section, especially at higher

164

energies, unless a softer hadronic form factor is chosen. This result can partly explain why the high energy experiments are unable to observe the existence of the Of. Using the elementary operator of the isobar model we predict the inclusive total cross section for @+ photoproduction on the deuteron. The results for both possible channels are given in Fig.6, where we show the inclusive total cross section obtained by using an isobar model with A = 0.8 GeV. The fact that the K-@+ cross section is smaller than the KO@+ one is originated from the elementary process (see Fig. 3). In conclusion, we have calculated cross sections of @+ photoproduction by using an isobar and a Regge models. The Regge model predicts smaller cross sections, especially at higher energies. The work of TM has been partly supported by the QUE project. References 1. T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003); J. Barth et al., Phys. Lett. B 572, 127 (2003); S. Stepanyan et al., Phys. Rev. Lett. 91, 252001 (2003); V. Kubarovsky et al., Phys. Rev. Lett. 92,032001 (2004); V.V. Barmin et al., Phys. Atom Nucl. 66, 1715 (2003); A. Airapetian et al., Phys. Lett. B 585,213 (2004); A. Aleev et al., hep-ex/0401024; S. Nussinov, hep-ph/0307357 (2003). 2. M. Karliner and H.J. Lipkin, Phys. Lett. B 597, 309 (2004) 3. A.I. Titov, A. Hosaka, S. Date and Y. Ohashi, nucl-th/0408001. 4. B.G. Yu, T.K. Choi, and C.-R. Ji, nucl-th/0312075 and references therein. 5. R.A. Arndt, 1.1. Strakovsky, and R.L. Workman, nucl-th/0311030 (2003). 6. Particle Data Group: S. Eidelman et al., Phys. Lett. B 592, 1 (2004). 7. W. Liu and C.M. KO, nucl-th/0308034. 8. S.I. Nam, A. Hosaka, and H-Ch Kim, hep-ph/0308313. 9. Hyun-Chul Kim, hep-ph/0308242. 10. T. Mart, C. Bennhold and C.E. Hyde-Wright, Phys. Rev. C 51, 1074 (1995). 11. W. Liu, C.M. KO, and V. Kubarovsky, Phys. Rev. C 69, 025202 (2004). 12. T. Mart and C. Bennhold, Phys. Rev. C 61, 012201 (2000). 13. T. Mart, Phys. Rev. C 62, 038201 (2000). 14. K. Haglin, Phys. Rev. C 50, 1688 (1994). 15. R.A. Williams, C.-R. Ji, and S.R. Cotanch, Phys. Rev. C 46, 1617 (1992). 16. M. Guidal, J.M. Laget, and M. Vanderhaeghen, Nucl. Phys. A627, 645 (1997). 17. F.X.Lee, T. Mart, C. Bennhold and L.E. Wright, Nucl. Phys. A695, 237 (2001). 18. H. Haberzettl, C. Bennhold, T. Mart, and T. Feuster, Phys. Rev. C 58, R40 (1998). 19. R.M. Davidson and R. Workman, Phys. Rev. C 63, 025210 (2001). 20. T. Mart and T. Wijaya, Acta Phys. Polon. B 34, 2651 (2003).

165

ON THE O+ PARITY DETERMINATION IN KK PHOTOPRODUCTION

A.I. TITOV Advanced Photon Research Center, Japan Atomic Energy Research Institute, Kizu, Kyoto, 619-0215, Japan Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia

H. EJIRI Natural Science, International Christian University, Osawa, Mitaka, Tokyo, 181-8585, Japan JASRI, Spring8 Mikazuki-chou, Hyougo, 679-5198, Japan H. HABERZETTL Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, DC 20052, USA Institut f u r Kernphysik (Theorie), Forschungzentrum Julich, D-52425 Julich, Germany

K. NAKAYAMA Department of Physics and Astronomy, University of Georgia, Athens, G A 30602, USA Institut f u r Kernphysik (Theorie), Forschungzentrum Julich, D-52425 Julich, Germany The problem of determining the parity of the O+ pentaquark in K K photoproduc+ N K K . We tion is discussed in terms of the two-step mechanism y N 4 I?@+ demonstrate that the contribution of the non-resonant background of the entire reaction y N 4 N K K cannot be neglected, and that suggestions t o determine the parity based solely on the initial-stage process y N + KO+ cannot be implemented cleanly. Nonetheless, we identify some spin observables which are sensitive mostly to the O+ parity rather than to the details of the production mechanism.

1. Problems The first evidence for the Q+ pentaquark discovered by the LEPS collaboration' at Spring-8 was subsequently confirmed in other experiments

166

at low energy (see, for example, the Workshop Summary talk by K. Hicks at this workshop). The pentaquark is produced as an intermediate state in the two-step reaction y N + KO+ -+ N K K . In view of this indirect mechanism, none of these experiments could determine the spin and the parity of the Q+. As has been pointed out, there are great ambiguities in calculating the unpolarized and polarized (spin) observables (see, for example, talks by T. Mart, C.M. KO, and S.I. Nam at this conference). In the effective Lagrangian formalism, the problems can be summarized as follows: (1) Dependence on the coupling operator for the @ + N K interaction, i.e., whether one chooses pseudoscalar (PS) or pseudovector (PV) couplings. (2) Ambiguity due to the choice of the coupling constants. At the simplest level, five unknown coupling constants and their phases enter the formalism: g e N K in the @ + N K interaction; the vector and tensor couplings g e N K - and K * , respectively, in the @ N K * interaction, and the tensor coupling K e in the electromagnetic yo+ interaction. (3) Dependence on the choice of the phenomenological form factors: form factors suppress the individual channels in different ways, and the form factors generate (modify) the contact terms for the PS (PV) coupling schemes which affect the theoretical predictions. A possible solution to these problems is to use more complicated (triple) spin observables, discussed by Ejiri,2 Nakayama and Love,3 and Rekalo and Tomassi-Gustaf~son.~ Assuming the @+ to be a spin-1/2 baryon, these spin observables for y N -+ KO+ (involving the linear polarization of the incoming photon, and the polarizations of the target nucleon and the outgoing O+) can be directly related to the parity of the pentaquark by employing some basic symmetry properties (such as the invariance of the transition amplitude under rotation and parity inversion, and the reflection symmetry with respect to the scattering plane). While in principle being completely model-independent, there are two difficulties with this method of parity determination. First, the final state being measured in the photoproduction experiment is the three-body state N K K (and not the two-body O + K state). The spin observables of the measured process depend on the actual initial and final states, but are independent of the parity of the intermediate @+. It is difficult, if not impossible, to find “model-independent” observables that would depend on the parity of the pentaquark. Second, the contribution of the non-resonant background to the reaction y N -+ N K K is large and, depending on the kinematical circumstances, this may modify the spin observables considerably. In our report, we discuss these important aspects and we try to identify the kinematic regions where the dependence on the “three-body nature”

167

N

Figure 1. Treelevel diagrams for the reaction y N -+ Q+I? -+ N K K .

N

N

N

(0

N

Figure 2. Diagrams for the background process for the y N -+ M N -+ N K K reaction, where M denotes the mesons p, w , 4, 0 , fz, and a z .

of the final state and non-resonant background is weak and where a clear difference is expected for different 8+ parities. The resonant amplitude consists of s-, u-, t-channel terms and the contact ( c ) term defined by the Q+NK interaction, as depicted in Figs. la-d. We also incorporate a tchannel K* exchange as shown in Fig. le. We consider the determination of the Q+ parity ( ~ eof)the isoscalar spin-l/2 O+. In fact, most theories predict the J p value of Q+ to be 1/2+ or 1/2-. Hereafter, we limit ourselves to the y n 4 n K + K - reaction. For a more detailed consideration of this problem, with a discussion of the yp --+ p K ° K o reaction, see Ref. 5. 2. Background

We found that the main contribution to the %on-resonant” background comes from the virtual vector-meson photoproduction yN -+ V N 4 N K K , depicted in Figs. 2a-c. We also include the excitation of the virtual scalar (g)and tensor (fi, az) mesons shown in Figs. 2d-f, respectively, and found that their contribution in the near-threshold region with E7 M 2 GeV is negligible. The K K invariant-mass distribution at a forward angle of K K photoproduction is shown in Fig. 3a. The photon energy was taken from the threshold to 2.35GeV, in accordance with the measurements of Ref. 1 shown in Fig. 3b. The shapes of the calculated and measured K + K invariant-mass distributions are similar to each other. The known mechanism of the vector-meson photoproduction allows us to fix the absolute value of the background. One can see a strong &meson photoproduction peak at M K K M M4 and a long tail dominated by the pmeson channel. , f~ mesons is much smaller and is The contribution from the u, a ~ and not shown here. In order to reduce the strong $-meson background in our

168

o

N -

102

0

9 10

I

B

ij

0.9

1.0

1.1

M, ( G W

1.2

1.3

1.0

1.1

1.2

1.3

Invariant K+Kmass (GeV/cz)

Figure 3. K K invariant-mass distribution in the yn --t K + K - reaction: (a) our calculation; (b) data from Ref. 1. Arrows indicate the &meson cut window.

calculation we eliminate the phase space with the K K invariant mass from 1.00 to 1.04 GeV following Ref. 1.

3. Resonance channels

As mentioned before, we describe the basic resonance process by considering the photoproduction of 8+,with a subsequent decay of 8+ into a nucleon and a kaon, as shown in Figs. la-d. All vertices are dressed by form factors, with current conservation being ensured by an appropriate choice of the contact terms.6 By using the relation between the @+ decay width re and the coupling constant g e N K , i.e., re 0: l g e N K I 2 , and the linear relation 9 9 ~ =~Q C. J ~ N Kone , finds that the cross section of the resonant channels does not depend on the O+ decay width at the resonance position. Instead, it depends on Q which provides a measure for the relative K* admixture to the process. The latter is defined via the ratio of the resonance-tobackground contribution at the resonance position by comparison with the experiment (for details of how the parameters are fixed, see Ref. 5). 4. Results We analyze the unpolarized and spin observables as a function of K+ polar angle in the 8+ rest frame (8)and at fixed angle of the K - photoproduction (0 = 55" in the center of mass). Our analysis of unpolarized, single and double spin observables shows that they are unable to determine the Of parity. Therefore, here we limit our consideration to the triple spin observables. Specifically, we consider the beam asymmetry for the linearly polarized photon beam at a fixed polarization of the target and the recoil nucleons. The nucleon polarizations are chosen along the normal to the

169

s

1.o

1.0

0.5

0.5

in-

(b)

0.0

+Wc

-0.5

-0.5 -1

.o

0 -0.5

0.0 0.5 cose

1.0

-1.0 -1.0

-0.5

0.0

0.5

0

cose

Figure 4. The triple spin asymmetry C y y ( f f )in y n ---t nK+K- as a function of the K+ decay angle for (a) positive and (b) negative r e . 112-

1E+

1.o

-

0.5

+ 0.0

Y

bf -0.5 -1.0 -1.0

-0.5 0.0

0.5

1.0

0.4 -1.0

-0.5

coso

0.0

0.5

1.0

COSQ

Figure 5 . The triple spin asymmetry C y y ( f f ) (a) and C y y ( f l ) (b) at different tensor coupling constant ( K * = 0, f0.5) of the Q + N K * interaction.

production ~ l a n e , ~ > ~

where o(tt) and ~ ( 7 1 correspond ) to the spin-conserving and spin-flip transitions between the initial and the final nucleons, respectively. We choose these asymmetries because for the initial-stage two-body process yN --+ @+I?, Bohr's theorem7 based on reflection symmetry in the scattering plane results in ~

Y -9N ' YY

K

(TT)

= +re

,

CyN+e+K YY

01) = -re .

(2)

This prediction is very strict, it does not depend on the production mechanism (in our case PV or PS coupling schemes, coupling constants, etc.) and therefore it is extremely attractive. But the realistic case is more complicated. Thus, in the case of the coplanar reaction where all three outgoing particles are in the production plane perpendicular to the nucleon polarization, Bohr's theorem predicts x y y ( t t ) = -xyy(Tl)

= rK =

independently of the intermediate @+ parity.

-1

,

(3)

170 The result for the non-coplanar case for E;,(tt) when we integrate over the azimuthal angle of the K+ meson direction of flight is shown in Fig. 4. The calculation for the different 8+ parity is done for PV coupling, positive a and a vanishing tensor coupling constant K* for the W N K * interaction. The case of cos 0 = f l corresponds t o the coplanar geometry where E;,(tt) = -1 in accordance with Eq. (3). For negative 8+ parity, the asymmetries due t o the only resonant channel (shown by the dashed curves) remains at E;,(tt) = -1 and at all cos 8 because of the s-wave 0+ decay, and therefore predictions for 2 2 and 2 -+3 processes for this case are the same. For positive parity, we have a p-wave decay which leads t o a fast increasing C$,(tT) from -1 up t o positive and large values and results in a specific bell-shape behavior. The shapes of C,, for different 7re are quite different from each other in the region of -0.8 5 c o s 0 5 0.8 and practically do not depend on the production mechanism. These ideal pictures are modified by the interference between the resonance and background channels. The coherent sum of these channels is shown by the solid curves. The final result is not sensitive to the production mechanism. Thus, Fig. 5a shows E;,(TT) calculated at different K* = 0, f 0 . 5 , and one can see rather a weak dependence of the asymmetries on K * . The “spin-flip” asymmetry C$,(TJ) shown in Fig. 5b decreases from +l when c o s 8 = f l t o some minimum values a t c o s 0 = 0. The depth of the minimum depends on the parameters of the resonant amplitude: it increases at large negative K * . In summary, we conclude that in the region of 0.5 5 c o s 0 5 0.8, the asymmetries E,’,( t T) for different parities are qualitatively different from each other. This feature suggests t o use this observable for the determination of re.

References 1. T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91, 012002 (2003). 2. H. Ejiri, http://www.spring8.or.jp/e/conference/appeal/proceedings /Theta+Spin.pdf. 3. K. Nakayama and W. G. Love, Phys. Rev. C 70, 012201 (2004). 4. M. P. Rekalo and E. Tomasi-Gustafsson, arXiv:hepph/0401050. 5. A.I. Titov, H. Ejiri, H. Haberzettl, and K. Nakayama, arXiv: nucl-th/0410098. 6. H. Haberzettl, Phys. Rev. C 56, 2041 (1997); H. Haberzettl, C. Bennhold, T. Mart, and T. Feuster, Phys. Rev. C 58, R40 (1998). 7. A. Bohr, Nucl. Phys. 10, 486 (1959).

171

COMMENT ON THE @+-PRODUCTION AT HIGH ENERGY

A.I. TITOV Advanced Photon Research Center, Japan Atomic Energy Research Institute, K i m , Kyoto, 619-0215, Japan Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia A. HOSAKA Research Center of Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan S. DATE AND Y . OHASHI

Japan Synchrotron Radiation Research Institute, Spring-8, 1-1-1 Kouto Mikazuki-cho Sayo-gun Hyogo 679-5198, Japan We show that the cross sections of the @+-pentquark production in different processes decrease with energy faster than the cross sections of production of the conventional three-quark hyperons. Therefore, the threshold region with the initial energy of a few GeV or less seems to be more favorable for the production and experimental study of @+-pentquark.

The evidence of the @+-pentaquarkdicovered by the LEPS at Spring-8 and its subsequent confirmation in a series of other experiments performed mainly at low energy poses a problem of the energy dependence of the 8+production. However, up to now the high energy experiments have found no clear peak of 8+ (for the overview of the present experimens at low and high energy see the Workshop Summary talk by K. Hicks at this conference). In our comments we analyze the high-energy limit of @+-production in inclusive processes in the fragmentation region and show that the @+ production cross section is suppressed compared to the production of the 77conventional”three-quark hyperons. For analysis of the @+ production in other kinematics regions see Ref. Consider the @+-production in inclusive reaction AB --f @+X together with the hyperon production: AB -+ Y X . The cross section of these reactions may be estimated on the base of fragmentation-recombination

172

model, which assumes the elementary sub-processes as depicted in Fig. 1 . Thus, it is assumed that at the first stage the colliding hadrons fragmentate recombination recombFation

h

Figure 1. Production of 8+ in inclusive reactions in the fragmentation region.

into partons (quark, gluon, di-quarks, etc). The probability to find i-th constituent (parton) is described by the "fragmentation" function F i / ~ ( x ) , where x = & / P A . At the second stage there is some soft (quasi-elastic) interaction between parton i and some other constituent from the hadron B. Finally, the parton i is recombined into the observable hadron h (@+ or Y).The probability of this process is defined by the "recombination" function R h / i ( Y ) , where y = p h / p i . Thus, the cross section of AB 4 8 + X reaction is controlled by the folding

where z = P h / P A . By making use of the scale behaviour of F i / ~ ( xand ) Rh/i(Y) Fi/A(x)

FO(x)(l- XIbX;

Rh/i(y)

& ( Y ) ( l - Y)",

where Fo(x) and Ro(x)are smooth functions of x,and keeping the dominant terms we get 1

0 0: /

( 1 - ")bX (x - z)""x.

(2)

z

The integral can be performed by an elementary method, and we can estimate the cross section as (b> 3 b,c: c)

b!c! ( b c)!b

+

+ c1 + 1( 1 -

Z)b+c

(3)

For further estimation, we have to specify the power b and c in the fragmentation and recombination functions. In the quark-parton picture 3 , these coefficients are related to the number of the constituent partons in A and h: b> = 2 n -~3 and c? = 2 n h - 3. Consider now two extreme variants. In

173

the quark-diquark picture of the hadrons A and h. When A is a nucleon and h is a hyperon or 8+ we have b = 1, c ( 8 + ) = 3 and c(Y) = 1. In the quark picture b = 3, c ( 8 + ) = 7 and c(Y) = 3. Then the corresponding ratio 8+ t o Y-production reads

RgY

N

9x 3x

[quarks] [diquarks]

(4)

which means that the Q+-production in the fragmentation region is strongly suppressed. Notice that (1 - 2)-power behaviour of the hadron production cross sections in the fragmentation region as a rule starts from z N 0.4 0.5 4. At z = 0.5, the accuracy of this estimation is 2 0 ~ 3 5 % and ~ for z N 0.7 it is 10-20%, and becomes better when z -+ 1. At z 5 0.4, we have t o specify the functions Fo(z) and &(z) in Eq. (l),which may be important for the central rapidity region. We also have t o include the dependence on the transverse momentum (for the finite PI) which is, however, beyond the scope of our present qualitative analysis. In summary, we have analyzed the high energy limit of the @+pentaquark production in the fragmentation region. We found the ratio of the 8+ production compared t o the background processes in diffractive processes is rather small. Our estimation is done on the base of the fragmentation-recombination model but it has a general character and is valid for any model (for example, the relativistic string model). Physically, the @+-pentaquark production in the fragmentation region is accompanied by creation of additional 2 quark-antiquark (diquark-antidiquark) pairs with subsequent pick quarks up by the outgoing hadron. It may be worthwhile to point out that there will be no suppression with increasing energy in the central rapidity regions in inclusive reactions. Nevertheless, the 8+ production at low energies seems t o be most suitable for the study of the properties of @+. References LEPS Collaboration. T.Nakano et al., [LEPSCollaboration],Phys. Rev. Lett. 91, 012002 (2003).

A.I. Titov, A. Hosaka, S. DatB, and Y. Ohashi, Phys. Rev. C 71, (2004). F.E. Close, An Introduction to quarks and partons. Academic press London and San Francisco, 1979. T. Kanki, K. Kinoshita, H. Sumiyoshi, and F. Takagi, Prog. Theor. Phys. Suppl. 97, 1 (1989).

174

SPIN-PARITY MEASUREMENTS OF O+ CONSIDERATIONS

- SOME

C . RANGACHARWLU. Department of Physics University of Saskatchewan Saskatoon, SK, Canada, S7N 5E2 E-mail: chary Osask. w ask.ca

I will briefly summarize a couple of early formalisms of spin-parity determinations of resonances and paticles from the angular distributions and polarization correlations of the decay products. I will also reason that these experiments on 8+ are beat performed at hadron facilities and/or electron beam facilities.

INTRODUCTION At this time, the questions concerning the Penta-quark are two-fold. First, we await confirmation of the existence of the resonance at 1540 MeV and we wonder if it is a single resonance or if there are two resonances separated by about 25 MeV. There are concerns if the signals seen so far are statistical fluctuations or some kinematical artifacts. These questions are being addressed and we may hope to get the answem towards the end of 2004. The next pressing question about the resonance is it indeed a Penta-quark or if it belongs to some other structures. The contenders are anti-decuplet scheme, lattice QCD, di-quark clusters etc l . The first and foremost attribute, besides the resonance energy, are its spin and parity. Needless to say, one should strive to deduce the spin and parity in as nearly model independent manner as possible. Recognizing that spin-parity determinations of levels and resonances have been an industry for nuclear and particle physicists for over 50 years, I will recapitulate a couple of well-known schemes, which are likely useful in the task of Of spin-parity assignment. Also, I will have a few comments on the relative merits of the probes in attempts to determine 'Work partially supported by Natural Sciences and Engineering Research Council of Canada

175

the spin-parity of the systems.

Some Formulations In early 1960s, several publications dedicated to the formulation of spinparity determinations of resonances have appeared. While the formalisms differ in details, the main thrust of all these works was to rely on basic principles of rotational and inversion invariancesand apply tensor algebras. The unique characteristics of the symmetry with respect to a select frames and reference axes are exploited. Noteworthy among these are works of Ademollo and Gatto2, Byers and Fenster Berman and Jacob4, and Zemach5. All these works attempt to define the observables sensitive to the spin and parities of resonances and particles, independent of the production and/or decay mechanisms. Of immediate interest to us is the work of Ademollo and Gatto[2], where they consider the production and decay of a baryonic resonance, in a process such as a + b + F + ....,F + f’ + c, where a and b are arbitrary, f’ is a spin 1/2 Fermion and c is a spin zero boson. The problem deals with the assignments of spin and parity of F. We can identify F with the Theta+ resonance, f’ with the decay nucleon and c as the K-meson. They choose n as unit vector along the normal to the production plane, which serves as the reference axis and v is the unit vector of momentum off’ in the rest frame of F, simply a unit vector along the flight direction of the decay fermion in the rest frame of resonance. In our case, it will be the direction of decay nucleon in the rest frame of 0’. Then they deduce a simple result linking the spin of the resonance (s), and the polarization (P) of the decay nucleon f’ (29

+ (P.az(nzw))2 + 1)2 = (P.nzw)2 ((P.w)(n.w))2 - (n.w)2

One may consider a few tests of this equation. One deduces that the angular distribution for a specific orbital angular momentum L, is given as

I l(s1/2LOls1/2)lzJmaz(~P,LOlSP) I < 71-21 > I 5 l(29 + 2) for L=l, a result

I(PL (n.v))l

(2)

which results in deduced earlier by Lee and Yang’. If Q+ is of negative parity, this result implies I < n.v > I 5 0.2,0.33 for spin 3/2 and 1/2, respectively. A stringent test requires that one measures the polarizations. Longitudinal polarization is zero for even panty. For odd-parity, we have

176

1 (P.vPL ( n.PI) ) I 5 I (Sl/2LOIs1/2) 12Imm(sp, LOlsp) (3) The transverse polarization is non-zero for L=O with the result I < P.nxv > I 5 (2s + 1)/(2s + 2). A conclusive test of spin-parity would entail mrrelation measurements which result in angular distribution, longitudinaland transverse polarization measurements. The nice thing is that they can all be measured simultaneously. If the' 0 is of jT = 1/2+, one would expect to see no longitudinal polarization along with a large transverse polarization. Also, Berman and Jacob * formulated the angular distributions and polarization distributions of isobar decays t o a nucleon and one or more pions. They consider the normal to the decay plane of the isobar as the reference axis. We can apply these results to the 0++ N + K decays. They also find that the longitudinal polarization for even 1-values is zero. For 1odd, ratio of the average values of longitudinal and transverse polarizations is found to be sensitive to both the 1-value and the spin (j) of the particle.

This ratio is 0.7 and 0.35 forjT = 1/2- and 3/2-, respectively. It does not warrant a high precision measurement to distinguish between the two possibilities. Thus, it seems that these measurements can fix the parity from the longitudinal polarization measurements. For the negative parity case, one can also determine the spin with a modest time investment.

Practical concerns The above formulations require that we have clean samples of the 0+decay products free from backgrounds- So far, the data from various laboratories shows the signal to noise ratio in the resonance region of about 1:l or worse. One has to find ways to reduce the background, if not completely eliminate it, by a clever choice of production channels and kinematical conditions. In this regard, production using real photons is not a good choice. One should remember that photons come with both parities and also have both isovector and isoscalar components. One might opt for production with hadron (K and pion) beams to take advantage of the fact that angular momentum transfer uniquely determines the parity of the transition. It is unlikely that the experiments at current hadron beam facilities have the potential to determine the spin-parity of this resonance, even if they succeed

177

in establishing the resonance. One may have to wait for next generation machines such as J-PARC 7 .

Electroproduction

Meanwhile, one might attempt the Q+ production by virtual photons viz., p(e,e'Ko)Q+ reaction at facilities such as J-LAB. A few well known facts about the versatility of virtual photon probes are worth mentioning. First of all, the polarization of the virtual photon is determined by the inelasticity and angle of scattered electron, which can be precisely controlled. Also, the flexibility that one can vary the momentum transfer at fixed energy transfers will allow one to find the kinematical region where the resonance/background ratio is maximum and select those settings for spinparity measurements. Another advantage is that one can selectively populate the transverse or longitudinal excitation, simply by varying the kinematics. Furthermore, coincidence measurements will allow one to determine not only the longitudinal and transverse form factors but also the interference terms *. For unpolarized electron beams and unpolarized targets, the coincidence cross sections are written as

In the above equation, U S carry the structure information and the rest are kinematical functions. Thus, one can vary the kinematics to determine the structure factors separately. These observables will provide stringent tests of the spin-parity assignments and they will also contribute as sensitive tests of structure models, described by several authors in these conference proceedings. References 1. 2. 3. 4. 5. 6. 7. 8.

See the other contributions of these proceedings. M. Ademollo and R. Gatto, Phys. Rev. 133B 531(1963). N. Byers and S.F'enster, Phys. Rev. Lett. 11 52(1963). S.M. Berman and M. Jacob, Phys. Rev. 139B 1023(1965). C. Zemach, Phys. Rev. 140B 109 (1965) T.D. Lee and C. N. Yang, Phys. Rev. 109 1755 (1959). http://j-parc.jp See for example, J. D. Walecka, Electron Scattering for Nuclear and Nucleon Structure, Cambridge University Press (2001)

178

REFLECTION SYMMETRY AND SPIN PARITY OF O+

H. EJIRI Natural Science, International Christian University Osawa, Mitaka, Tokyo, 181-8585, Japan JASRI, Spring8, Mikazuki-chou, Hyougo, 679-5198, Japan E-mail: [email protected] A. TITOV Advanced Photon Research Center, J A E R I Kizu, Kyoto, 619-0125, Japan Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, 141980 Russia E-mail: [email protected] p

T h e spin and parity of O+ produced in the photonuclear reaction of y N i + O+K are discussed in the framework of the reflection symmetry. Using polarized y and N i , the O+ parity is determined from the O+ spin polarization, which can be inferred model-insensitively from the s and p wave properties of K mesons involved in the decay of O+.

1. Reflection symmetry and spin parity of O+ Recently the evidence for the O+ pentaquark baryon has been reported by the LEPS collaboration Several experiments have supported the LEPS experiment, but high-energy experiments have found no clear O+ peak '. The spin and parity of the O+ are crucial for clarifying structures of the baryon and for disentangling different models 3 . The present letter discusses briefly the spin and parity of the particleunbound (resonant) Of in the framework of the reflection symmetry. The major collision processes involved in the production and decay of O+ are party-conserving strong processes. Then the reflection symmetry in nuclear reactions is used for identifying the spin and party of particles involved in the reactions. According to the Bohr's theorem of the reflection symmetry of the col-

'.

4151617.

179

lision, the reflection eigen value is conserved as

where Pi(Pf) is the intrinsic parity of the initial(fina1) state and Si(S,)is the sum of the spin components along the normal n to the reaction plane for the initial(fina1) state. Thus the Pi = Pf or - Pf in case of Si - Sf = even or odd. Let's discuss the photo-production on a nucleon Ni and the decay of

y + Ni

+ K-

+ o++ K - + Nf + K + ,

(2)

as first studied at Spring-8 by the LEPS group l. In case of photonuclear reactions by linearly polarized photons, the reflection eigen value is -1 for the y ( l ) with linear polarization perpendicular to the reaction plane (the electric vector parallel to n), and +1 for the y(l1) with the polarization parallel to the plane. On the basis of the reflection symmetry, the spin and parity of O+ are related t o the spins and parities of the photon and particles involved in the production and decay processes as shown in Table 1. Here we consider co-planar reactions through the Q+ spin 1/2 for simplicity. Table 1. Spins and parities in the photo-production and decay of the Of. s,(Ni) and s,(Nf) are the spin components of the target and residual nucleons along n (normal to the reaction plane), respectively. R, is the reflection eigen value of the y.

We note the following points. 1. The spin polarization of the final nucleon depends on the initial photon R,, i.e. on the linear polarization of the photon, and not on the intermediate O+ parity. The initial and final nucleon spins are anti-parallel in case of y ( l ) and parallel in case of ~ ( 1 1 ) .

180

2. The relations of the O+ parity to the sin polarizations of y,Ni,Q+, as shown in Table 1, are given alternatively in terms of the asymmetries]

where (TT) and ( T I ) stand for the spin polarizations of s,(Ni) and sz(O+). Then the Table 1 leads to %Jy(l)

'yy(TT)

=4O+)

Zyy(ll) = -7@+),

(5)

= ++)

Zyy(T1) = -7@+),

(6)

Note that O+ is a particle-unbound state, and decays by the strong process which is insensitive to the spin orientation. Then it is hard t o measure directly the spin polarization in contrast t o hyperon decays by the weak process. Therefore one needs angular momentum observables in order to determine the spin-polarization and thus parity of Of 435.

2. Angular momenta and polarization/spin observables The spin and parity of Of are related t o the orbital angular momentum(s/p wave) of the K+ from O+ and the spin flip/ nonflip in the production and the decay as given in Table 2. Actually it has been shown by Titov et al.5 that angular distributions of triple spin asymmetries depend mainly on the O+ parity but are not sensitive t o the reaction models(mechanisms). Table 2. Partial waves of K+ from O+ and spin flip/nonflip in the O+ production and decay.

J"(O)

y

Production spin

l(K+)

Decay spin

1/2 + 1/2 + 1/2 1/2 -

Y(l) y(l1) y(l)

Parallel Anti-parallel Anti-parallel Parallel

p p

Anti-parallel Anti-parallel Parallel Parallel

%ll)

S S

Here we consider the s and p-wave K+ decays from the O+.

181

Using the ?(/I) and the spin-up target nucleon, the 7r(@+) and the sz(O+) are related to the s and p wave K+ decays as 7r(O+) = spin-down if the K+ angular distribution is non-isotropic, 7r(O+) = - spin-up if the K+ angular distribution is isotropic. In case of the y ( l ) , the O+ spin polarization is reversed.

+

3. Concluding remarks The spin and parity of O+ produced in the photonuclear reacton are discussed by using the reflection symmetry. Using polarized y’s and polarized target nucleons, the 0 parity is determined in principle by measuring the spin polarization of O+. Since O+ decays by the strong process, the decay is insensitive to the spin polarization. The s and p wave properties of the K+ in the decay of O+ are used to get information of the O+ spin polarization, and thus t o determine the O+ parity. Using the ?(]I) and the up-spin Ni, the s-wave K+ isotropic distribution gives the up-spin and negative-parity O+ , while the p-wave distribution gives the down-spin positive-parity O+. It is noted that measurements should be made for the K+ emitted forward with respect t o the incoming target nucleon in the @+ rest frame in order to minimize the interference with the non-resonant process 5 . In general the present method can be applied for other reactions such as 7rN --+ O + K , K N -+ Of7r, and N N -+ O+A(C).

References 1. LEPS Collaboration, T. Nakano et al., it Phys. Rev. Lett. 91,012002 (2003). 2. K.Hicks, Summary of this workshop, and experimental reports in this workshop (2004). 3. D. Diakonov, V. Petrov and M.V. Polyakov, 2. Phys. A 359 305 (1997),and theoretical reports in this workshop. 4. H. Ejiri, Pentaquark Workshop, Nov. 2003 RCNP http://www.spring8.or.jp/e/conference/appeal/proceedings/Theta+Spin.pdf. 5. A.I. Titov, H. Ejiri, H. Haberzettl and K. Nakayama, Phys. Rev. (2004). 6. A. Bohr, Nuclear.Physics 10 486 (1959). 7. K.Nakayama and W.G. Love, Phys. Rev. C 70 012201 (2004).

182

THE USE OF THE SCATTERING PHASE SHIFT IN RESONANCE PHYSICS

M. NOWAKOWSKI AND N. G. KELKAR Departamento de Fisica, Universidad de 10s Andes, A . A . 4976 Santafe de Bogota, Colombia E-mail: mnowakosOuniandes.edu.co, nlcelkarOuniandes.edu.co

The scattering phase shift encodes a good amount of physical information which can be used to study resonances from scattering data. Among others, it can be used to calculate the continuum density of states and the collision time in a resonant process. Whereas the first information can be employed to examine the evolution of unstable states directly from scattering data, the second one serves as a tool to detect resonances and their properties. We demonstrate both methods concentrating in the latter case on 'exotic' resonances in nn and nK scattering.

1. Introduction The phase shift 6r is a convenient variable to parameterize the scattering amplitude as known from many textbooks on quantum mechanics. In principle, the phase shift can be extracted from the differential cross section dg/dR or other observables. The knowledge of the phase shift (or amplitude) can give us additional information on the dynamics of the scattering process, the properties of occurring resonances etc. We shall discuss two such applications of the phase shift in the subsequent sections. Both have to do with resonances as intermediate states in the scattering process from which 6l is determined.

2. Continuum level density

While calculating the correction factors B and C to the equation of states of an ideal gas, namely, pV = RT[1+ B / V C / V 2 ] Beth , and Uhlenbeckl found that the derivative of the phase shift is proportional to the difference of the density of states (of the outgoing particles) with and without

+

183

interaction. In case of the Zth partial wave we have then,

To appreciate this result, we briefly recall the Fock-Krylov method2 to study the time evolution of unstable states. It is based on the fact that unstable states cannot be eigenstates to the Hamiltonian and as a result we can expand the resonance states in terms of the energy eigenstates, i.e. IS)= J dE a(E)IE ) . The survival amplitude can be recast as a Fourier transform of the so-called spectral function pw = Ja(A)I2 (which is the probability density to find the state IE) between E and E dE) and is given as, Aq (t)= dEp, (E)e-jEt. It is used in many investigationson quantum time evolution3. Now the probability density and the continuum density of states are related by a constant. As long as there are no interfering resonances is positive and the above identification works without doubt. We can take the first resonance and neglect the subsequent contribution of the higher lying resonances in order to study only the large time behaviour of the time evolution because in this case due to the time-energy duality we need to know only the threshold behaviour of the phase shift. To analyze a realistic example5 we opted for Q + Q + 8Be(2+)+ Q Q (see Figure 1) *. The analysis of this data following the method outlined above reveals that the survival probability behaves as Pq(t) = IAq(t)12 N t-0.30 for large times5.

+

Jgh.

+

3. Time delay

In Figure 1 we can see that all resonances of 8Be with Z = 2 are nicely mapped through the peaks of the derivative of the phase shift and the positions of these peaks correspond to masses of the resonant states. We would expect this due to the interpretation of dS/dE as a continuum level density. Such a level density should have a complex pole, responsible for the exponential decay, at the resonance position which is reflected through a bump. If we want to map all resonances by this method, then it is more appealing to reinterpret &/dE as a collision time or, in case this is positive, as a time delay in a scattering experiment. Such an interpretation was pioneered by Wigner, Eisenbud and BohmOand is a topic of standard textbooks by now. For a wavepacket A(E',E ) centered around E the exact expression is7

1

M

At(E) = 8r2

dd

dE'IA(E', E)I22dE'

184

600

-

b s 00

400

200 0 150

0

Figure 1. D-wave phase shift (upper half) [4] and its derivative (lower half) in a-a elastic scattering as a function of E,, = Ec.,,,. - E8Be(groundstste). The figure on the right displays the region of the first 2+ level of *Be in detail. The i m t displays the accuracy of our fit near the threshold energy region which is crucial for the large time behaviour of the decay law.

which for a sufficiently narrow wave-packet A(E', E ) gives

At(E) = 2-

d6 dE

(3)

With this interpretation we can reinforce the expectation that the collision time &/dE peaks in the vicinity of a resonance (at the resonant energy to be exact). Certainly, a collision is delayed if an intermediate state becomes on-shell. We emphasize that the collision time (3) is strictly the difference between time spent with and without inetraction and not simply the time that a projectile spends in the scattering region of radius a. 4. Time delay and resonance physics

Having identified the derivative of the phase shift as continuum level density and as time delay in resonant scattering, we can proceed to apply this concept to realistic examples (one of them is already displayed in Figure 1). It is, however, instructive to dwell first on some theoretical connections, misconceptions and expectations. We note that we consider the usage of

185

time delay in resonance physics as a supplementary tool to the other established methods. In literature one often encounters the statement of the correspondence 'phase shift motion' c) resonance. Time delay is nothing else but the exact mathematical formulation of this correspondence. However, this correspondence often carries a misunderstanding as it is attached also to a ..-jump of the phase shift. We stress that this .Ir-jump is not a necessary condition for a resonance. In the spirit of time delay the condition is a peak around the resonance energy. Indeed, there are examples of prominent established resonances without the strong .Ir-jump like n Q + 5He(P1/2) + n + a which is purely elastic with a jump from Oo to 40° 'only'*. A simple Breit-Wigner parameterization of the amplitude i.e.

+

ri 2 ri E R - E T = corresponds to 6 = tan-1 ( E ~ - E ) ~ + I ' ~ / 4 1 r ; r /which 2] gives ( md6) E = E R = +&. This would mean that time delay is negative if Br < 1/2! An improvement can be reached by including a non-resonant background parametrized here by the diagonal phase cig and energy depenBr ER dent width. One then gets (g ) E = E R = Br(k)J/2 + which, in principle, can save the time delay from becoming negative near a resonance. However, we would not expect that when the resonant contribution is large. Let us now confront this with experiment. In Figure 2 we have plotted the phase shift for the Sll resonances, the inelasticity parameter (note that in case that there are several channels the S-matrix is written as qexp(2id)) and the time delay. First of all we find sharp peaks at 1.5 GeV and 1.65 GeV corresponding to the well known resonances (Particle Data estimate of the pole value of the first 4 1 resonance is 1.505). Secondly, we get these peaks in spite of the small branching ratio of Sll(l535) which is Br(nN) = 35 - 55% and Br(qN) = 30 - 55%. It is also clear that the time delay becomes negative when the inelasticity parameter is largest. This can be understood ils the loss of flux from the elastic channel due to the interpretation of At as density of stated0. In Figure 3 we have done a similar exercise for the P11 casell. This is interesting from several points of view. Again we find two established resonances, but the focus is here on the three star P11(1710). We find this resonance by the time delay method at the right position even if the nN branching ratio is as small as 10 - 20%. We find it by using the FAO2l2 amplitudes even if the group which has performed the FA02 partial wave analysis cannot find the pole corresponding to 41(1710). This resonance is important for the theoretical prediction13 of the Pentaquark 0(1540)14. Through the time delay method we find this resonance and also the Pentaquark15 at the right positions. In passing we

ER-ELir/2

4

[

&

($$$>,=,

186

xN elastic scattering

N(1535) --f

ZN (35-5596)

100

+ q N (30-55%) 4

PDG estimate of pole position: 1.505 GeV

1.5

1.5

1.7

Figure 2. Single energy values of (a) inelasticities (triangles), (b) phase shift (circles) and best fit curves to phase shift (solid lines) in the S11 partial wave of aN elastic scattering and (c) distribution of time delay as a function of energy available in the aN centre of ma98 system.

P,,

1.357

(1440)

1.386

(1710)

-

-70%

I

I

lC-20%

Figure 3. Time delay in the Pi1 partial wave of aN elastic scattering evaluated using the FA02 T-matrix solutions.

note that even resonances like P13(1585), Gl~(2190)and Hlg(2220) with nN branching ratios of 10 - 20% leave clear fingerprints in the time delay plots11. 5. Resonances in m r and .rrK scattering The previous sections showed that the time delay method is reliable in nuclear and baryon resonance physics. We now turn our attention to the mesonic case". To show how reliable the method indeed is and how sensitive it is to small phase shift motion, we first apply the method to the case

187

of the pmesons. This is depicted in Figure 4. Evidently, we find the ‘not-

,-

200 150

50

0.5

1

1.5

E,(GeV)

Figure 4. Time delay plot of the p resonances evaluated from a fit to the p-wave phase shifts in mr elastic scattering.

to-be-missed’ p(770), its first excitation p( 1450) and its second excitation ~(1700)which are all indisputable resonances. The peak at 1166 MeV corresponds to a small phase motion and one could be tempted to disregard it as a fluctuation. However, several other cases, among others the three star resonance 013(1700) and the two star F15(2000), show that small phase shift ‘motion’ can signify a resonance. This seems to be the case also here. Particle Data Group lists also several mesons between 1100 - 2200 MeV which by itself is not a remarkable fact. But at the recent Hadron 2001 conference in Protvino some authors have pointed out a growing evidence for a plike resonance at 1200 MeV which we think appears in our time delay plotl7. Our result in the pmeson sector is then an independent confirmation by the time delay method! In the last few years the scalar sector attracted lots of attention. One of the reasons is the ‘re-discovery’ of the famous a-meson and its ‘re-appearance’ in the Particle Data Book. The difficulty with this meson is reflected in the wide range of its possible mass, 400 - 1200 MeV. The time delay analysis for this sector is summarized in Figure 5. Of course, fo(980) is a dominant contribution here. We iden-

188

Figure 5. Time delay plot of the scalar meson resonances evaluated using two different sets of the s-wave phase shifts [18,19]as indicated on the figure for mr elatic scattering.

tify the peak around 1.23(1.34) GeV with fo(1370) for which Particle Data Group quotes the range of possible pole mass between 1.2 and 1.5 GeV. Similarly the peak at 1.42(1.47) GeV is attributed to fo(1500) (the PDG value is 1.4- 1.6 GeV). The analysis of both phase shifts reveals a resonance at 700 MeV. If, in addition, we take the information of the Kaminski phase shift we see also a peak at 350 MeV. Can we take this as an evidence for two resonances? Let us first note that in the region of 400 - 1200 where the a-meson is found, one can identify two accumulation points. One at 350 - 600 MeV and the other one at 700 - 850 MeV. The low lying case is supported also by unitarized chiral perturbation theory and by unitarized quark model, by the so-called ABC effect which is with us since 1961 and the recent decay J / 9 + bw + mrw20. The 700 MeV case finds its confirmation in Nambu-Jona-Lasinio models, Weinberg's mended symmetry and Bethe-Salpeter calculation2'. Hence, these two accumulation points are not artificial constructs. We have checkedls that the information we get from the time delay is the same whether we consider the channel mr + mr, T A + K K or K K + KK. In the strange scalar sector the controversy regarding the lightest scalar (called 6 meson) is even bigger. This analysis" also reveals the existence of two low lying resonances: one at 0.8 GeV and the other around 1 GeV which we identify with the putative 6-meson. We have applied the time delay method to many 'standard' cases, the established baryon resonances, the p mesons and the K*-as well as K,*mesons (discussed in") and found a good agreement with data. Some less

189

established resonances found by different methods get confirmed through the time delay method. By using the K N phase shift we found the recently discovered Pentaquark with a mass very close to the observed and predicted value15. We found the spin-orbit partners of this Pentaquark very close to the theoretical expectations21. Last but not least, our nuclear physics case discussed here in section two, shows also the virtues of the time delay method not only in finding nuclear levels, but also in studying the quantum evolution of unstable systems for large times. References 1. E. Beth and G. E. Uhlenbeck, Physic0 4 (1937), 915 (1937) 2. N. S. Krylov, and V. A. Fock, JETP 17, 93 (1947). 3. H. Nakazato, M. Namiki and M. Pascazio, Int. J. Mod. Phys. B10,247 (1996). 4. For an exhaustive list of phase shift data in the elastic a-ascattering via 'Be resonance see reference5. 5. N. G. Kelkar, M. Nowakomki and P. P. Khemchandani, Phys. Rev. C70, 024601 (2004), nucl-th/0405043. 6. L. Eisenbud, dissertation, Princeton, June 1948 (unpublished); D. Bohm, Quantum Theory (1951) pp. 257-261; E. P. Wigner, Phys. Rev. 98,145 (1955). 7. H. M. Nussenzveig, Causality and Dispersion Relations, Academic Press (1972) 8. G. L. Morgan and R. L. Walter, Phys. Rev. Dl68, 114 (1968). 9. C. Garcia-Recio, J. Nieves, E. Ruiz Arriola and M. J. Vicente Vacas, Phys. Rev. D67,076009 (2003). 10. N. G. Kelkar, J. Phys G: Nucl. Part. Phys. 29, L1 (2003), hep-ph/0205188. 11. N. G. Kelkar, M. Nowakomki, K. P. Khemchandani and S. R. Jain, Nucl. Phys. A 730, 1 2 1 (2004), hep-ph/0208197. 12. FA02 Partial Wave Analysis at http: // gwdac.phys.gwu.edu (We thank A. Arndt and I. I. Strakovsky for providing us the pole values of their analysis). 13. D. Diakonov, V. Petrov and M. Polyakov, 2. Pys. A359, 305 (1997). 14. T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003). 15. N. G. Kelkar, M. Nowakomki, and K. P. Khemchandani, J. Phys. G: Nucl. Part. Phys. 29, 1001 (2003), hep-ph/0307134. 16. N. G. Kelkar, M. Nowakowski and K. P. Khemchandani, Nucl. Phys. A 724, 357 (2003), hep-ph/0307184. 17. M. Achasov, hep-ex/0109035; B. Pick, Crystal Barrel Collaboration; A. Donnachie and Yu. s. Kalashnikova, hep-ph/0110191. 18. G. Grayer et al., Nucl. Phys. B75, 189 (1974). 19. R. Kaminski, L. Lesniak and K, Rybicki, 2.Phys. C74, 79 (1997). 20. For references see''. 21. N. G. Kelkar, M. Nowakowski and K. P. Khemchandani, Mod. Phys. Lett. A19, 2001 (2004), nucl-th/0405008. 22. R. Kaminski, L. Lesniak and B. Loiseau, Eur. Phys. J. C9, 141 (1999).

190

PENTAQUARK RESONANCES FROM COLLISION TIMES

N. G. KELKAR AND M. NOWAKOWSKI Departamento de FaSica, Universidad de 10s Andes, Cra. 1 No. 18A-10, Santafe de Bogota, Colombia E-mail: nkelkarOuniandes.edu.co Having successfully explored the existing relations between the S-matrix and collision times in scattering reactions to study the conventional baryon and meson resonances, the method is now extended t o the exotic sector. To be specific, the collision time in various partial waves of K + N elastic scattering is evaluated using phaw shifts extracted from the K + N + K + N data as well as from model dependent T-matrix solutions. We find several pentaquark resonances including some low-lying ones around 1.5 to 1.6 GeV in the Pol, PO3 and Do3 partial waves of K + N elastic scattering.

1. Introduction The discovery of the pion in 1947 followed by that of several other mesons and baryons, gave birth to a specialized branch in particle physics which involved the characterization of hadronic resonances. However, even after half a century's experience in analyzing experimental data to infer on the existence of resonances we still come across examples where a resonance is confirmed by one type of analysis and is reported to be absent by another and history shows that this is especially true in case of the pentaquark (2") resonances. It is therefore important to examine the limitations of the various theoretical definitions used to extract information from data and then comment on the existence of the resonance. The 0+(1540) found in several experiments1 which followed its theoretical prediction2, being one such recent example, in the present talk we try to shed some light on the controversy of its existence using a somewhat forgotten but welldocumented method of collision time or time delay in scattering. In fact, we identify several pentaquark resonances by evaluating the time delay in various partial waves of K + N elastic scattering using the available K+N + K S N data.

191

2. Collision time: From the fifties until now Intuitively, one would expect that if a resonance is formed as an intermediate state in a scattering process (say a b + R -+ a b), then the scattered particles in the final state would emerge (alone from the fact that the resonance has a finite lifetime) later than in a non-resonant process a + b + a + b. The resonant process would be “delayed” as compared to the non-resonant one. This relevance of the delay time or collision time in scattering processes to resonance physics was noticed back in the fifties by Eisenbud3, B o b 4 and Wigner5. Starting with a simple wave packet description, they showed that the amount of time by which an incoming particle in a scattering process got delayed due to interaction with the scattering centre is proportional to the energy derivative of the scattering phase shift, 6 ( E ) . Clearly then, d6ldE would be large and positive close to energies where resonances occur. Eisenbud3 also defined a delay time matrix, At, where a typical element of At,

+

+

3-

A t , = Re [ - ih(Sij)-’ dSij dE

,

gave the delay in the peak outgoing signal in the j t h channel when the signal is injected in the ith channel. For m elastic scattering reaction, i = j and one can easily see that using a phase shift formulation of the S-matrix, i.e. S = e2a6 in the purely elastic case and S = qe2i6for elastic scattering in the presence of inelasticities, the above relation reduces simply toSi7 db Atij = 2h-. dE

Henceforth for simplicity, we shall drop the subscripts ii and write At whenever we refer to time delay in elastic scattering. Later on, Smithg constructed a lifetime matrix Q , in terms of the scattering matrix, s. He defined collision time to be the limit as R + 00, of the difference between the time the particles spend within a distance R of each other (with interaction) and the time they would have spent there without interaction. He showed that the average time delay for a collision beginning in the ith channel calculated using Eq. (1)is indeed the matrix element Q i j of the lifetime matrix and concluded that when Qij ’s are positive and large, we have a criterion for the existence of metastable states. The interest in this concept continued in the sixties and Goldberger and Watsong, using the concept of time interval in S-matrix theory found that A t = -ihd[lnS(E)]/dE.Lippmann” even defined a time delay operator,

192

r = -itia/8E, the expectation value of which (using the phase shift formulation of the S-matrix) gave the time delay to be the same as in Eq. (2). In the seventies, the time delay concept finally found a place in most books on scattering theory and quantum mechanics”, where it is mentioned as a necessary condition for the existence of a resonance. However, inspite of being so well-known in literature as well as books, it was rarely used to characterize resonances until its recent application6y7to meson and baryon resonances. Instead, mathematical definitions of a resonance have been used over the decades for its identification and characterization. The simple physical concept of time delay was somehow always overlooked in practice. In what follows, we now analyse the shortcomings of the various definitions or tools used to locate resonances.

3. What is a resonance?

A resonance is theoretically clearly defined as an unstable state characterized by different quantum numbers. However, to identify such a state when it has been produced, one needs to define a resonance in terms of theoretical quantities which can be extracted from data. In principle, if an unstable state is formed for example in a scattering process, then the various definitions should simply serve as complementary tools for its confirmation. However, it does often happen that a resonance extracted using one definition appears to be “missing” within another. Before discarding the existence of such missing resonances, it is important to take into account the limitations of the various definitions of a resonance. We shall discuss these below. 3.1. S-mat& poles

The most conventional method of locating a resonance involves assuming that whenever an unstable particle is formed, there exists a corresponding pole of the S-matrix on the unphysical sheet of the complex energy plane lying close to the real axis”. The experimental data is usually fitted with a model dependent S-matrix and resonances are identified by locating the poles. However, Calucci and co-workers12took a different point of view. In the case of a resonance R formed in a two body elastic scattering process, a + b + R + a + b, a sharp peak in the cross section accompanied by a rapid variation of the phase shift through 7r/2 with positive derivative (essentially the condition €or large positive time delay) was taken as the signal for the existence of a resonance. The authors then constructed S-

193

matrices satisfying all requirements of analyticity, unitarity and threshold and asymptotic behaviour in energy such that a sharp isolated resonance is produced without an accompanying pole on the unphysical sheet. They also ensured the exponential decay of such a state. It is both interesting and relevant to note that while concluding that resonances can belong to a “no-pole category”13,the authors stressed the need for high accuracy data in the case of the Z*’s (the pentaquark resonances) whose dynamical origin might be questionable. 3.2. C r o s s section bumps, Argand diagrrrms and Speed Plots

Though the existence of a resonance usually produces a large bump in the cross sections, it was shown in a pedagogic article by Ohanian and Ginsburg14that a maximum of the scattering probability (i.e. cross section) cannot be taken as a sufficient condition for the existence of a resonance. Resonances can also be identified from anticlockwise loops in the Argand diagrams of the complex scattering amplitude; however, these alone cannot gaurantee the existence of a r e s o n a n ~ e ~Finally, ~. the speed plot peaks, i-e. peaks in S P ( E ) = IdT/dEl, where T is the complex scattering transition matrix, can in fact be ambiguous due to being positive definite by definitions. In the next section, we shall present the results of a time delay analysis of the K+N elastic scattering using the existing K+N + K + N data as well as the SP92lS model dependent T-matrix solutions. 4. Time delay in K + N elastic scattering 4.1. Energy dependent calculations

We shall first present the time delay distributions (as a function of energy) using model dependent solutions of the T-matrix. The expression for time delay in terms of the T-matrix17 is obtained by replacing S = 1 + 2iT in Eq. (1). As can be seen in Fig. 1, in addition to the resonances around 1.8 GeV, we find some low-lying ones around 1.5-1.6 GeV. Table I shows that the time delay peak positions around 1.8 GeV agree with the pole positions obtained from the same T-matrix. However, the low-lying ones do not correspond to any poles. These peaks could possibly be considered as realistic examples of the no-pole category of resonances13 mentioned in the previous section. At this point it is of historical importance to note that a speed plot peak at 1.54 GeV in the Pol partial wave of K + N elastic

194

scattering was already noted by Nakajima et all8. However, due to lack of support from Argand diagrams they did not mention it as a pentaquark resonance. 0.1

0

1.4

1.6

1.8

2

2.2

0

1.4

1.6

2

1.8

2.2

E (GeV) Figure 1. Time delay in various partial waves of K + N evaluated from the SP92 T-matrix solutions.

+ K+N

elastic scattering,

Table 1. Comparison of time delay peaks with pole values

Partial wave

SP92 pole position (GeV)

so1

1.85

1.831 - i95

1.57 1.83

1.811 - ill8

1.48 1.75

1.788 - i170

1.49 1.81

- i253

2.0

Po1

p13

DO3

D?K

Position of time delay peak

2.074

195 4.2. Pentaquark resonances from single energy values of K+N phase shifts

Being motivated by our earlier experience with the meson and unflavoured baryon resonances7, where small fluctuations in the single energy values of the phase shifts gave rise to time delay peaks corresponding to lesser established resonances, we decided to perform a time delay analysis of the phase shifts in K + N elastic scattering tool7. In Figs. 2 and 3 we show the time delay distributions obtained from fits to the single energy values of the phase shifts. It is interesting to note a peak at 1.545 GeV in the 0 0 3 partial wave which comes very close to the discovery of the O+ from recent cross section data. The peak at 1.64 GeV agrees with some of the predictions1' of a J p = 3/2+, Do3 partner of the Q(1540). In Figs. 3 and 4 we see that the resonances occur at exactly the same positions, namely, 1.6 and 1.8 GeV in the case of the Pol and Po3 partial waves which are J = 1/2,3/2 partners. The J = 3/2 partners of the Of have also been predicted21 to lie in the region from 1.4 to 1.7 GeV.

1

I

- - model solution

M cc)

K ' N elastic scattering

10

1.5

1.6

1.7

1.8

1.9

2

E (GeV) Figure 2. Time delay in the Do3 partial wave of K + N + K + N elastic scattering, evaluated from a fit (solid line in [a]) to the single energy values of the phase shift.

196

60 h

v

40

-- model solution

W

- fit

20

-

0.6 -

8"

0.4

2

3

0.2

-

Q

Figure 3.

Same as Fig. 2, but for the Pol and PO3 partial waves.

In closing, we note that the three peaks, namely, 1.545 in the Do3 and 1.6 and 1.8 GeV in the Poland Po3 partial waves are in very good agreement with the experimental d u e s a o , 1.545k.012,1.612~.01and 1.821f.11 GeV of the resonant structures in the p K t invariant mass spectrum. We can

197

then identify t h e time delay peak in the Do3 partial wave to be t h e

O+.

References 1. T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003); S. Stepanyan et al., Phys. Rev. Lett. 91, 252001 (2003); V. Kubarovsky et al., Phys. Rev. Lett. 92,032001 (2004) Erratum-ibid. 92,049902 (2004); V. V. Barmin et al., Phys. Atom. Nucl. 66,1715 (2003), Yad. Fiz. 66, 1763 (2003), hepex/0304040; J. Barth et al., Phys. Lett. B572, 127 (2003); A. E. Asratyan, A. G. Dolgolenko

and M. A. Kubantsev, hep-ex/0309042; A. Airapetian et al., Phys. Lett. B 585, 213 (2004); A. Aleev et al., hep-ex/0401024; S . V. Chekanov et al., hepex/0404007. 2. D. Diakonov, V. Petrov and M. Polyakov, Z. Phys. A 359, 305 (1997). 3. L. Eisenbud, Dbsertataon, Princeton, unpublished (June 1948). 4. D. Bohm, Quantum theory, New York Prentice Hall, pp. 257-261 (1951). 5. E. P. W i p e r , Phys. Rev. 98, 145 (1955). 6. N. G. Kelkar, J. Phys. G: Nucl. Part. Phys. 29, L1 (2003), hep-ph/0205188. 7. N. G. Kelkar, M. Nowakowski and K. P. Khemchandani, Nucl. Phys. A724, 357 (2003); N. G. Kelkar, M. Nowakowski, K. P. Khemchandani and S. R. Jain, Nucl. Phys. A730, 121 (2004). 8. F. T. Smith, Phys. Rev. 118, 349 (1960). 9. M. L. Goldberger and K. M. Watson, Phys. Rev. 127, 2284 (1962). 10. B. A. Lippmann, Phys. Rev. 151, 1023 (1966). 11. M. L. Goldberger and K. M. Watson, Collision theory, Wiley, New York (1964); C. J. Joachain, Quantum Collision theory, North Holland, Amsterdam (1975); J. R. Taylor, Scattering theory, Wiley, New York (1972); B. H. Bransden and R. G. Moorhouse, The Paon-Nucleon System, Princeton University Press, NJ (1973). 12. G. Calucci, L. Fonda and G. C. Ghirardi, Phys. Rev. 166, 1719 (1968); G. Calucci and G. C. Ghirardi, Phys. Rev. 169, 1339 (1968). 13. L. Fonda, G. C. Ghirardi and G. L. Shaw, Phys. Rev. D8, 353 (1973). 14. H. Ohanian and C. G. Ginsburg, Am. J. Phys. 42, 310 (1974). 15. N. Masuda, Phys. Rev. D1, 2565 (1970); P. D. B. Collins, R. C. Johnson and G. G. ROSS,Phys. Rev. 176, 1952 (1968). 16. J. S. Hyslop, R. A. Arndt, L. D. Roper and R. L. Workman, Phys. Rev. D46, 961 (1992). 17. N. G. Kelkar, M. Nowakowski and K. P. Khemchandani, J. Phys. G: Nucl. Part. Phys. 29, 1001 (2003), hep-ph/0307134; ibid, it Mod. Phys. Lett. A, (2004), nucl-th/0405008. 18. K. Nakajima et al., Phys. Lett. B112, 80 (1982). 19. B. K. Jennings and K. Maltman, Phys. Rev. D69, 094020 (2004); D. Akers, hep-ph/0403142. 20. P. Zh. Aslanyan, V. N. Emelyanenko and G. G. Rikhkvitzkaya, hepex/0403044. 21. L. Ya. Glozman, Phys. Lett. B575, 18 (2003); J. J. Dudek and F. E. Close, Phys. Lett. B583, 278 (2004).

198

PHOTON AND NUCLEON INDUCED PRODUCTION OF @+

SEUNG-IL N A M ~ TATSUSHI ~, HOSAKA~AND HYUN-CHUL KIM^ 1 . Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, Japan sinamOrcnp.osaka-u.ac.jp [email protected] and 2. Nuclear physics & Radiation technology Institute (NuRI), Pusan University, Keum-Jung Gu,Busan 609-735, Korea hchkimapusan. ac.kr We investigate Q+ production via photon and nucleon induced reactions. We observe that the positive parity 8+ production provides about ten times larger total cross sections than those of the negative parity one in both photon and nucleon induced reactions due to P-wave enhancement of the K N Q vertex. We also consider the model independent method in the nucleon induced reaction to determine the parity of Q+ and show clearly distinguishable signals for the two parities.

1. Introduction

'

After the observation of the evidence of O+ by LEPS collaboration motivated by Diakonov et al. 2 , physics of exotic pentaquark baryon state has been scrutinized by huge amount of research activities. In the present work, we investigate O+ production via photon and nucleon induced reactions using Born diagrams with a pseudoscalar K and vector K*+xchange included. For the nucleon induced reaction, we consider the model independent method to determine the parity of 0' which has not been confirmed yet by experiments. In calculations, we assume that O+ has the quantum numbers of spin 1/2, isospin 0 and the decay width r @ - + K N = 15 MeV is used to obtain K N O coupling constant We perform calculations for both parities of O+. '32.

2. Photon induced reactions: yN

+ I?@+

In this section, we study the total cross sections of y N + KO+ reactions. Results axe given in Fig 1. Two models are employed for the K N O cou-

199

pling schemes. One is the pseudo-scalar (PS, thick lines) and the other is pseudo-vector (PV, thin lines) to investigate theoretical ambiguity. AS for the anomalous magnetic moment of O+, 60, we employ -0.8 considering several model calculations 3,4. We set the unknown K * N O coupling constant to be 1gKNQ1/2 with positive (dashed line) and negative signs (dot-dashed line). In order to take into account the baryon structure, we employ a gauge invariant form factor which suppresses s- and u-channels 3 . In Fig.1 we plot total cross sections of the neutron (left) and proton (right) targets only for the positive parity '0 since we observe that the overall shapes and tendencies for the negative parity O+ are quite similar to the positive parity one. A major difference between them is that the total cross

Figure 1. The total cross section of y n the positive parity.

+ K-O+

(left) and y p

-+ Roo+(right)

for

sections are about ten times larger for the positive parity O+ than for the negative parity one due to the P-wave coupling nature of the K N O vertex. We also find that theoretical ambiguities due to the PV and PS schemes, 60 and K*-exchange contribution become small for the neutron target where the t-channel K-exchange dominates, whereas we find large model dependence for the proton case, where the K-exchange does not appear. 3. Nucleon induced reactions: np

+ Y O + and &T+

E+O+

In this section we investigate N N scattering for the production of O+. Here, we make use of the Nijmegen potential for the K N Y coupling constants. We also take into account K*-exchange contributions with vector and tensor K*NO(Y) couplings 6 . We consider only Y = A since overall behaviors of n p + COO+ are similar to np + A@+ with differences in the order of magnitudes of the total cross sections (OA N 5 x (TCO). We employ a monopole type form factor with a cutoff mass 1.0 GeV 6 . In Fig.2 we plot the total cross sections for the reaction with two different parities of W .

200

We observe that difference in the magnitudes of the total cross sections for the two parities is similar to the photoproduction. Furthermore, the results are not very sensitive to the signs of vector and tensor K*NO coupling constants. The labels in parenthesis denote (sgn(gg,NO),sgn(g&NO)). We note that if we consider initial state interaction, the order of magnitudes will be reduced by about factor three 6,7. As suggested by Thomas et al. 8 , tak-

Figure 2. Total cross sections of np parities of @+.

--f

A@+ for the positive (left) and negative (right)

ing into account the Pauli principle and parity conservation, &+ + C+O+ provides a clear method for the determination of the parity of @+. Spin 0 initial state allows non-zero production rate near the threshold (S-wave) for the positive parity Of, while spin 1 initial state does for the negative parity one. This selection rule should not be affected by any model dependences. We confirm that at the threshold region (- 2730 MeV), the reaction process is dominated by S-wave so that the selection rule is applicable '. We observe clear evidences of the selection rule in Fig. 3. K*-exchange 3.5

<-,+I <-,->

32 0 I

1 0

06 00 272

2.74

2.78

E,.

278

2.80

C-Vl

Figure 3. Total cross sections of R-t C+Q+ for the different spin states, spin 0 and spin 1 and for the positive (left) and negative (right) parities of @+.

contribution is not so sensitive to the various sign combinations. The spin

201

observable A x x , which was suggested by Hanhart et al. lo is plotted in Fig. 4. Up to about 100 MeV above the threshold, the results of the two different parities of O+ show clear difference due to the selection rule ’+lo. 1.0

1 0 ,

I

I 40-2 7 3

276

277 ’

Em CGeVl

278

281

-i 0

I

273

.

,

.

I

1

I 275

277

278

281

Em CGeVl

Figure 4. A x x for the positive (left) and negative (right) O+

4. Summary

We have investigated the photon and nucleon induced reactions for the O+ production in Born diagram calculations with appropriate form factors and with some phenomenologically determined coupling constants. Due to the different K N O vertex structure, we observed about ten times larger total cross sections for the positive parity O+. Though we still have theoretical ambiguities, this property is quite universal for O+ production reactions. The model independent method to determine the parity of @+ via polarized p p scattering seems quite promising. As announced by COSY-TOF collaboration, we hope to see more experimental results from $5 scattering in the near future. References 1. T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91,012002 (2003). 2. D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359,305 (1997). 3. S. I. Nam, A. Hosaka and H. -Ch. Kim, Phys. Lett. B 579, 43 (2004) 4. H. C. Kim and M. Praszalowicz, Phys. Lett. B 585, 99 (2004). 5. V. G. J. Stokes and Th. A. Rijken, Phys. Rev. C 59,3009 (1999). 6. S. I. Nam, A. Hosaka and H. -Ch. Kim, hep-ph/0402138 to appear in PRD. 7. C. Hanhart, Phys. Rept. 397,155 (2004). 8. A. W. Thomas, K. Hicks and A. Hosaka, Prog. Theor. Phys. 111,291 (2004). 9. S. I. Nam, A. Hosaka and H. C. Kim, Phys. Lett. B 602, 180 (2004). 10. C. Hanhart et al., Phys. Lett. B 590, 39 (2004). 11. W. Eyrich [COSY-TOF collaboration], “Evidence for @+ resonance from the COSY-TOF experiment” at PENTAQUARKOl workshop in Japan.

202

DETERMINING THE @+ QUANTUM NUMBERS THROUGH A KAON INDUCED REACTION

T. HYODO AND A. HOSAKA Research Center for Nuclear Physics (RCNP), Ibaraki, Osaka 567-0047, Japan E-mail: [email protected]

E. OSET AND M. J. VICENTE VACAS Departmento de Fa'sica Tedrica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigacidn de Paterna, Aptd. 22085, 46071 Valencia, Spain We study the K + p t T + K N reaction with kinematical condition suited to the production of the Of resonance. It is shown that in this reaction with the polarization experiment, a combined consideration of the strength a t the peak and the angular dependence of cross section can help determine the O+ quantum numbers.

In 2003, the LEPS collaboration reported a signal for a narrow S = +1 resonance around 1540 MeV in the experiment held at Spring-8.' It is important for further theoretical studies t o determine experimentally the quantum numbers of the Of resonance such as spin and parity. Here we study the K+p -+ n f K N reaction with different O+ quantum numbers, and see qualitative differences in observables depending on the quantum n u m b e r ~ . ~The > ~K+p ? ~ + r + K N reaction was also studied theoretically in Refs. 5, 6, but in the present study, we take the background contribution into account, which should be important for the information on the signal/background ratio and the possible interference between them. A successful model for the reaction was considered in Ref. 7, consisting of the mechanisms depicted in terms of Feynman diagrams in the upper panel of Fig. 1. The terms (a) (meson pole) and (b) (contact term) are derived from the effective chiral Lagrangians. Since the term (c) is proportional to the momentum of the final pion, it is negligible when the momentum is small. In the following, we calculate in the limit of the pion momentum set to zero. If there is a resonant state for K+n then this will be seen in

203

the final state interaction of this system. These processes are expressed as in the lower panel of Fig. 1, and they contribute to the present reaction in addition to the diagrams (a) and (b). For an s-wave K+n resonance we have J p = 1/2-, and for a pwave, J p = 1/2+,3/2+. We write the couplings of the resonance to K+n as gK+,, i j ~ + , and CK+, for s-wave and pwave with J p = 1/2+, 3/2+ respectively, and relate them to the @+ K~ r &+, = K~ r and &+, = .*~. K Mr width I? = 20 MeV via g:+, = +, A straightforward evaluation of the diagrams ( a ) and ( b ) leads to the K f n -+ d K N amplitudes

+

+

-iti = a i a . kin + b i z . $

,

(1)

where the explicit form of coefficients ai, bi are given in Ref. 2, i = 1 , 2 stands for the final state K+n,Kop respectively, and kin and q’ are the initial and final K+ momenta. With some coefficients fi, di,. . . , the resonance terms & take on the form +

-i i 9 = &.kin

,

-4

-iip’3/2)= fia.kin-gia.$.

-ii!p’1/2) = dia.$,

(2)

where the subscript i accounts for the intermediate state and the upper superscripts denote the partial wave and the spin of the O+. Finally the total amplitude for K f n final state is given by

-

-it = -it1

-

-

- it1 - i t 2

(3)

We calculate the cross sections with the initial three momentum of K+ in the Laboratory frame ki,(Lab) = 850 MeV/c (& = 1722 MeV). At this K+

?r+

K+

K+

U+

Figure 1. Upper panel : Feynman diagrams of the reaction K + p Lower panel : the Q+ resonance contribution.

+ a+K+n in

a model.

204 10 -

6

I

.--* 8

-

15w

-

I

1520

I

k,(Lab) = 850 MeVk €I = 0 deg

I,JP=O,1/T I,Jp=0,1/2'

1540

1560

1580

MI

Figure 2. The double differential cross sections at 0 = 0 deg (forward direction) for I = 0 , l and J p = 1/2-, 1/2+, 3 / 2 + .

energy, the final T + momentum is small enough with respect to lzin1. In Fig. 2, we show the invariant mass distribution d2a/dMrd cos 0 in the K+ forward direction (0 = 0). A resonance signal is always observed, independently of the quantum numbers of O+. The signals for the resonance are quite clear for the case of I , J p = 0,1/2+ and I , J p = 0,1/2-, while in the other cases the signals are weaker and the background is more important. Next we consider polarized amplitudes. It is seen in Eqs. (2) that if the O+ has the negative parity, the amplitude is proportional to 3 .Zin while if it has positive parity with spin 1/2, it is proportional t o 3 .4". We try to use this property in order t o distinguish the two cases. Let us consider that the initial proton polarization is 1/2 in the direction z and the final neutron polarization is -1/2 (the experiment can be equally done with Kop in the final state, which makes the nucleon detection easier). In this spin flip amplitude (-1/2ltl 1/2) the 3 .zin term vanishes, and therefore the resonance signal disappears for the s-wave case, while the 3.g operator of the pwave case would have a finite matrix element proportional to q' sin 0. This means, away from the forward direction of the final kaon, the appearance of a resonance peak in the mass distribution would indicate a pwave coupling and hence a positive parity resonance. In Fig. 3, we show the results for the polarized cross section measured at 90 degrees as a function of the invariant mass. The absence of the resonance term in s-wave results is clearly seen. The only sizeable resonance peak comes from the I , J p = 0,1/2+ case, and the other cross sections for spin 3/2 are quite reduced. A clear experimental signal of the resonance in this observable would indicate the quantum numbers as I , J p = 0,1/2+.

(Zin)

+

205

ISW

1520

IW

1560

1580

MI

Figure 3. The double differential cross sections at -9 = 90 deg for I = 0 , l and J p = 1/2-, 1/2+, 3/2f, with polarized initial photon.

In summary, we have studied the K+p -+ r + K N reaction and shown a method to determine the quantum numbers of O+ in the experiment, by using the polarized cross sections. For future perspective, the calculation for a finite momentum of final pion is desired. A combined study of ( K + ,d) and ( T - , K - ) reactions based on two-meson coupling' is also in p r o g r e ~ s . ~

Acknowledgments This work is supported by the Japan-Europe (Spain) Research Cooperation Program of Japan Society for the Promotion of Science (JSPS) and Spanish Council for Scientific Research (CSIC), which enabled T. H. and A. H. to visit IFIC, Valencia and E. 0. and M. J. V. V. visit RCNP, Osaka. This work is also supported in part by DGICYT projects BFM2000-1326, and the EU network EURIDICE contract HPRN-CT-2002-00311.

References 1. LEPS, T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003). 2. T . Hyodo, A. Hosaka, and E. Oset, Phys. Lett. B579, 290 (2004). 3. E. Oset, T . Hyodo, and A. Hosaka, nucl-th/0312014, Talk given at HYP 2003. 4. E. Oset, T . Hyodo, A. Hosaka, F. J. Llanes-Estrada, V. Mateu, S. Sarkar and M. J. Vicente Vacas, hep-ph/0405239, Talk given at NSTAR 2004. 5. W. Liu and C. M. KO, Phys. Rev. C68, 045203 (2003). 6. Y. Oh, H. Kim, and S. H. Lee, Phys. Rev. D69, 074016 (2004). 7. E. Oset and M. J. Vicente Vacas, Phys. Lett. B386, 39 (1996). 8. T . Hyodo, F. J. Llanes-Estrada, E. Oset, J. R. Pelbz, A. Hosaka, and M. J. Vicente Vacas, in preparation . 9. T . Hyodo, A. Hosaka, E. Oset, and M. J. Vicente Vacas, in preparation .

206

EXOTIC CHALLENGES*

M. PRASZALOWICZ M . Smoluchowski Institute of Physics, Jagellonian University, ul. Reymonta 4 , 30-059 Kmkdw, Poland E-mail: michalOif.uj.edu.pl

We list and discuss theoretical consequences of recent discovery of €I+.

1. Has O+ been really found? Let us start with a word of warning. No evidence of @+ has been found in HERA-B I, RHIC 2, BES 3 , LEP and Fermilab 5 . The reasons maybe either of experimental nature or a peculiar production mechanism 6*7. In contrast to the low energy almost fully exclusive experiments that reported @+, experiments which do not see exotics are mostly high energy inclusive ones It is difficult to produce exotic states in the high energy experiments which are dominated by the Pomeron exchanges g*lO. Note that experiments which do not see Oc put in fact an upper bound on the (yet unknown) production mechanism, rather than exclude its existence. Another piece of negative evidence comes from the old K N scattering data that have been recently reanalyzed 11J2. Here one can accommodate at most one resonance near 1545 MeV with very small width re+ < 2 MeV. K S d cross-section including the hypothesis of a narrow resonance recalculated in the Jiilich meson exchange model l3 yields re+ < 1MeV. However, "non-standard" analysis of the phase shifts allows for more exotics 14*15. All these facts call for a new high precision K N experiment in the interesting energy range. *This work is supported by the Polish State Committee for Scientific Research (KBN) under grant 2 P03B 043 24. Talk at International Workshop PENTAQUARKOI, Spring8, July 20-23, 2004

207

2. How many 8 ’ s ?

Since the first report on O+ by LEPS l6 many other experiments confirmed its existence 17. Reported masses are shown in Fig. 1. Some of these results were reported at this Workshop l8 together with new results from LEPS 19. In principle data in Fig. 1 should represent one state. However, if taken literary, ZEUS and CLAS data for example are not compatible.

Figure 1. Mass of O+ a8 reported by various experiments. Statistical and stysternatic errors have been added in quadrature. Squares refer to K + n final state and circles to JCP.

It is therefore legitimate to ask do all these experiments see the same state? Before this issue is decided experimentally let us examine predictions of different models. Chiral models predict a tower of exotic rotational states starting with m1/2, 273/2,1/2, Z5/2,3/2 (subscripts refer to spin) etc. The lowest excitation of O+ is an isospin triplet of spin 3/2 belonging to flavor 27. The mass 0 2 7 is only slightly larger than the mass of @+ and depends weakly on the value of pion nucleon C,N term (see Fig. 2). Note that theoretical uncertainty of the model 2o is approximately f30 MeV. In the correlated quark models additional states are also unavoidable. In the diquark model 21 the spin-orbit interaction splits spin 1/2 and 3/2 states by a tiny amount of a AE N 35 + 65 MeV 22. Similarly in the diqaurk-triquark scenario 23, the mass splitting would be of the order of 40 MeV. Hence a nearby isosinglet O* state of spin 3/2 is expected in these

208 2100,

1500

4

40

50

70

4 80

Figure 2. The spectra of m1/2baryons (solid lines) together with the masses of the 81 and S3/2 in the 27312 (dashed lines) as functions of C*N,using parameters fitted from the masses of the 8+ and non-exotic states.

models. This is a distinguishing feature, since the soliton models do not accommodate spin 3/2 antidecuplet. Although there are no more exotics in the minimal diquark model 211 the tensor diquarks in 6 of SU(3) flavor are almost unavoidable. They lead to further exotics like 27 which in the schematic model of Shuryak and Zahed 24 is even lighter than We see therefore the importance of experimental searches both for the isospin partners of Of and for another peak in the O+ channel. Preliminary CLAS results 25 indicate two states at 1523 and 1573 MeV, similarly bubble chamber experiment analyzed by the Yerevan group l7 reports 3 states at 1545, 1612 and 1821 MeV. Finaly, there is also report of a number of exotic resonanses from Dubna l7 and from the "non-standard" phase shift analysis 15. So far the searches for W +provided no evidence although some structures in K + p channel have been seen by CLAS 27 and STAR 29. There is no evidence for El++ in the old K + p scattering data ll.

m.

17y27928

3. Spin and parity of O+

Spin and parity of O+ are at present unknown. While almost all theorists agree that spin should be 1/2 the parity distinguishes between different models. Chiral models predict positive parity, similarly quark models with flavor dependent forces and correlated quark models predict P = +. In uncorrelated quark models and s u m rules P = -.

209

Unlike model calculations lattice simulations (summarized at this Workshop by s. Sasaki 30) should give clean theoretical answer whether pentaquarks exists and what their quantum numbers are. However, since pentaquarks are excited QCD states, lattice simulations are difficult and give ambiguous message: either there is no bound O+ state 31, or there is one but with negative parity 32. One simulation indicates 33 P = +. Let us stress that, unlike in the case of W whose spin and parity are not measured but assumed after the quark model 34, the parity of Of is of utmost importance to discriminate between various models and to understand how QCD binds quarks. 4. The width of O+

A key prediction of the seminal paper by Diakonov, Petrov and Polyakov 35 (DPP) was the observation that (in contrast to the naive expectations) in the chiral quark soliton model antidecuplet states should be very narrow. The decay width for B + B' cp is given by:

+

Here M and M' are baryon masses, p is meson momentum in the B rest frame, C denotes pertinent SU(3) Clebsch-Gordan coefficient and GR stands for a coupling constant for baryon B in the SU(3) representation R. It has been observed 35 that Gm 0 in the nonrelativistic limit of the soliton model which is very useful as a first approximation. This was a clear indication that baryons would be narrow. How narrow is of course a question of reliability of approximations employed to derive (1) and the phenomenological input used to determine Gm DPP 35 made a conservative estimate that re+ < 15 MeV. In a more recent analysis they have argued that re+ 3.6 + 11.2 MeV 36. In the diquark models O+ decay proceeds via diquark breakup and is therefore believed to be small. Recently it was shown 37 that the narrowness of O+ in the quark model with flavor-spin interactions follows from the group-theoretical structure of the wave function. Further suppression comes from the SU(3) breaking corrections due the mixing with other representations for rn, # 0 20*38. Therefore moderate admixtures of other representations for which the relevant couplings are not suppressed may substantially modify the decay width. In the case of O+ + K N the admixtures of and 27 in the wave function of the final nucleon affect the decay width. In the quark-soliton model they further

m

N

m

210

suppress re+by a factor of 0.2 20*38. In Fig. 3 we show modification factor R('"zz) for the width of O+ and for two partial widths of Zm coming from representation mixing in the c h i d quark-soliton model 20,38 as functions of the pion-nucleon E T term. ~ To conclude: the decay widths within the antidecuplet may substantially differ from the SU(3) symmetry values. On experimental side the results for O+ width are unclear. Most experiments quote upper limits, however there are a few which claim to have measured re+and quote error bars. ZEUS gives 17: re+= 6.1k1.6f::g MeV This result is consistent with the upper limit from DIANA ( K f + X e ) : re+ < 9 MeV 17. Results from a C3Ha bubble chamber in Dubna by the Yerevan group 17: re+ = 16.3 f 3.6 MeV, from COSY 17: re+= 18 f 4 MeV and Hermes 17: re+ = 19 f 5 (stat) f2 (syst) MeV are two times larger. As discussed in Sect. 1 old K scattering data put the lowest limit re+ < 1 + 2 MeV. In almost all theoretical models mechanisms were found which suppress O+ decay width. The question is now: how much? Therefore the measurement of the Of width is of utmost importance and will provide constraints on various theoretical scenarios.

0.0

40

50

60

70

80

Figure 3. Correction coefficients R(miX)for 8+ and Zm decays a8 functions of C r ~ . Large supression of 8+ together with a moderate enhancement of Zm leads to strong SU(3) violation for the decay widths.

5. Exotic cascades

So far only one experiment 39 reported the states which form the "base" of -

G-

10, namely I = 3/2 and Z& at 1862 MeV. This result needs confirmation, so far reports from other groups are negative.

21 1

In the original paper DPP 35 predicted the mass of the exotic 5 3 states above 2 GeV. This prediction, however, depends on the residual freedom of the model which is conveniently parameterized in terms C,N 40. They used C,N = 45 MeV, while present estimates 41 indicate a larger value of approximately 70 MeV. As seen from Fig. 2 larger values of C,N are compatible with the NA49 result. Original prediction of the diquark model 21 was 1750 MeV. Pure SU(3) arguments indicate that for ideal mixing scenario employed by Jaffe and Wilczek it is difficult accommodate exotic cascades at 1862 MeV without Similar conclusion has invoking new nucleon-like narrow resonances been reached for arbitrary mixing 44*45. Similarly to Q+, the decay widths of exotic cascades will be modified by additional mixing, as depicted in Fig. 3. 42943.

6. Cryptoexotic states and mixing

If Q+ mass is 1539 MeV and Em 1862 MeV then equal spacing within the antidecuplet requires additional cryptoexotic nucleon-like and C-like states with masses 1648 MeV and 1757 MeV respectively. These states should be in principle narrow with the decay widths related to re+ by the SU(3) symmetry. However, as discussed above, mixing will modify these relations. The nucleon-like and C-like states can mix with known (and unknown) resonances of the same panty and spin. Most of analysis in this direction was done for two nucleon-lie states IS1,2) assuming J p = 1/2+ for antidecuplet. Here three possible scenarios are discussed: 1) both states IS1,2) correspond to known resonances, 2) one state corresponds to the yet undiscovered resonance and 3) both have to be discovered. Mixing has been also discussed by Weigel 46 within the framework of the Skyrme model (with the dilaton field) . In this approach, apart from rotations, another mode, namely the "breathing" mode of the soliton, was quantized and a subsequent mixing with other states was investigated. Radially excited octet states were identified with known N * resonances (Roper or N*(1710), etc.), so that no novel states were predicted. Unfortunately little can be said about the decay widths within this approach. Cohen 42 made an important remark that not only masses but also decay widths are affected by mixing and any phenomenological analysis should discuss both simultaneously. He excluded ideal mixing scenario, unless new cryptoexotic nucleon-like resonances exist. The analysis of masses and decay widths of the N * states under the as-

212

sumption that they correspond to the Roper and N*(1710)indicates 45 that it is impossible to match the mass splittings with the observed branching ratios even for arbitrary mixing. It is shown that the mixing required for the decay N*(1710) + AT is not compatible with the mixing deduced from the masses. A possibility based on the nonideal mixing scenario advocated by Diakonov and Petrov 44 is that there should be a new N * resonance in the mass range of 1650 + 1680 MeV. Similar conclusion has been reached in the quark soliton model 47. Here already the ordinary nucleon state has a non-negligible admixture of which leads to the suppression of the decay width. Further decrease may be achieved by adding a mixing to another nucleon-like state as Roper and/or N*(1710) and by the admixture of 27 38. The same authors 47 claim that the improved phase shift analysis admits two candidates for the narrow nucleon-like resonances at 1680 and 1730 MeV and with widths smaller than 0.5 and 0.3 MeV, respectively. To conclude this Section let us note that physics of N* and C* states will be most probably dominated by extensive mixing between different nearby states which will affect both masses and decay widths. New, narrow resonances are theoretically expected. Experimental searchesfor such states have been recently performed with positive preliminary evidence

m

29148.

7. Summary

A convincing experiment confirming W is in our opinion still missing. If Of exists we have to understand why some experiments do not see it while the others do. Although yet unknown production mechanism might provide an explanation, it is really hard to understand why similar experiments like ZEUS and H1 give contradictory results. There is a common agreement that spin of Of is 1/2. However, there is no such consensus as far as parity is concerned. Measuring the parity will discriminate between different models. Even more importantly it will either strengthen our confidence in lattice QCD simulations or pinpoint some yet unknown weaknesses of this approach. Certainly the measurement of the width is badly needed. An intuitive explanation why Of is so narrow is still missing although in various models + formal arguments have been given. Since the leading decay mode 8 + 8 (where the second 8 refers to the outgoing meson) is very small even moderate admixtures of other SU(3) representations in the final state or in the initial state for cryptoexotic members of are going to modify

213

substantially the decay widths. Warning: SU(3) relations between different decay widths will not hold! Mixing will be very important for cryptoexotic nucleon-like and E-like states. Most probably, new narrow resonances are required for consistent theoretical picture. Also the confirmation of Z ~ ( 1 8 6 2 )is badly needed. Somewhat unexpectedly the discovery of @+ and possibly of Em has shaken our understanding of the QCD bound state. Simple quark model pictures must be modified and very likely soliton models might contain necessary ingredients to explain new exotics. References 1. K. T. Knopfle, M. Zavertyaev, and T. Zivko (HEM-B), hep-ex/0403020. 2. C. Pinkenburg (PHENIX), nucl-ex/0404001; S. Salur (STAR), nucl-

ex/0403009. 3. J. Z. Bai et al. (BES), hep-ex/0402012. 4. P. Hansen (ALEPH), talk at DIS 2004; T. Wengler (DELPHI), talk at Moriond 2004. 5. M. J. Long0 (HyperCP (E871)), talk at QNP2004; D. Christian (E690), talk at QNP2004; M.-J. Wang (CDF), talk at QNP2004 and this Workshop; E. Gottschalk (E690), this Workshop. 6. M. Karliner and H. J. Lipkin, hep-ph/0405002; 7. H. Lipkin, this Workshop. 8. K. Hicks, hep-ph/0408001. 9. D. Diakonov, hep-ph/0406043. 10. A. Titov, this Workshop. 11. R. A. Arndt, I. I. Strakovsky, and R. L. Workman, Phys. Rev. C68, 042201 (2003); R. L. Workman, R. A. Arndt, I. I. Strakovsky, D. M. Manley, and J. Tulpan, n~~l-th/0404061. 12. W. R. Gibbs, nucl-th/0405024. 13. A. Sibirtsev, J. Haidenbauer, S. Krewald, and U.-G.Meissner, hepph /0405099. 14. N. G. Kelkar, M. Nowakowski, and K. P. Khemchandani, nucl-th/0405008 and J. Phys. 6 2 9 , 1001 (2003). 15. N. Kelkar and M. Nowakowski, this Workshop. 16. T. Nakano et al. (LEPS), Phys. Rev. Lett. 91, 012002 (2003). 17. V. V. Barmin et al. (DIANA), Phys. Atom. Nucl. 66,1715 (2003); S. Stepanyan et al. (CLAS), Phys. Rev. Lett. 91, 252001 (2003); J. Barth et al. (SAPHIR), hep-ex/0307083; A. E. Asratyan, A. G. Dolgolenko, and M. A. Kubantsev, Phys. Atom. Nucl. 67, 682 (2004);V. Kubarovsky et al. (CLAS), Phys. Rev. Lett. 92, 032001 (2004); A. Airapetian et al. (HERMES), Phys. Lett. B585, 213 (2004); A. Aleev et 01. (SVD), hep-ex/0401024; M. AbdelBary e t al. (COSY-TOF), Phys. Lett. B595, 127 (2004); P. Z. Aslanyan, V. N. Emelyanenko, and G. G. Rikhkvitzkaya (Yerevan), hep-ex/0403044;

214

S. Chekanov et al. (ZEUS), Phys. Lett. B591, 7 (2004). 18. D.J. Tedeschi (CLAS), E. Eyrich (COSY-TOF), C. Schaerf (GRAAL), U. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41.

42. 43. 44. 45. 46. 47.

48.

Karshon (ZEUS), W. Lorenzon (HERMES), M. Battaglieri (CLAS) and R. DeVita (CLAS) at this Workshop. T. Nakano (LEPS), this Workshop. J. R. Ellis, M. Karliner, and M. Praszdowicz, JHEP 05,002 (2004). R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91,232003 (2003). J. J. Dudek and F. E. Close, Phys. Lett. B583, 278 (2004). M. Karliner and H. J. Lipkin, Phys. Lett. B575, 249 (2003). E. Shuryak and I. Zahed, Phys. Lett. B589, 21 (2004). D. K. Hong, Y. J. Sohn, and I. Zahed, hep-ph/0403205. R. De Vita (CLAS), talk at NSTAR 2004. Y. A. a o y a n et al., hep-ex/0404003. H. G. Juengst (CLAS), nucl-ex/0312019. S. Chekanov (ZEUS), hep-ex/0405013. S. Kabana (STAR), talk at RHIC and AGS Users Meeting, BNL, 2004. S. Sasaki, this Workshop. S. J. Dong et al., hep-ph/0306199 and this Workshop; N. Ishii et al., heplat/0409121 and this Workshop. F. Csikor, Z. Fodor, S. D. Katz, and T. G. Kovacs, JHEP 11,070 (2003); S. Sasaki, hep-lat/0310014 and this Workshop; T. T. Takahashi, this Workshop. T.-W. Chiu and T.-H. Hsieh, hepph/0403020 and this Workshop. S. Eidelman et al. (Particle Data Group), Phys. Lett. B592, 1 (2004). D. Diakonov, V. Petrov, and M. V. Polyakov, Z. Phys. A359, 305 (1997). D. Diakonov, V. Petrov, and M. V. Polyakov, hep-ph/0404212. C. E. Carlson, C. D. Carone, H. J. Kwee, and V. Nazaryan, hep-ph/0312325. M. Praszdowicz, A d a Phys. Polon. B35, 1625 (2004). C. Alt et al. (NA49), Phys. Rev. Lett. 92, 042003 (2004) and K. Kadija at this Workshop. M. Praszdowicz, Phys. Lett. B575, 234 (2003). M. M. Pavan, I. I. Strakovsky, R. L. Workman, and R. A. Arndt, PIN Newslett. 16, 110 (2002); T. Inoue, V. E. Lyubovitskij, T. Gutsche, and A. Faessler, Phys. Rev. C69, 035207 (2004). T. D. Cohen, hep-ph/0402056. M. Praszdowicz, hep-ph/0410086. D. Diakonov and V. Petrov, Phys. Rev. D69, 094011 (2004). S. Pakvasa and M. Suzuki, Phys. Rev. D70,036002 (2004). H. Weigel, hep-ph/0404173; AIP Conf. Proc. 549, 271 (2002); Eur. Phys. J. A2, 391 (1998) and this Workshop. R. A. Arndt, Y. I. Azimov, M. V. Polyakov, I. I. Strakovsky, and R. L. Workman, Phys. Rev. C69,035208 (2004). S. Kouznetsov (GRAAL), talk at NSTAR2004.

215

PENTAQUARKS IN A BREATHING MODE APPROACH TO CHIRAL SOLITONS*

H.WEIGEL Fachbereich Physik, Siegen University Walter-Flex-Strape 3, 0-57068 Siegen, Germany

In this talk I report on a computation of the spectra of exotic pentaquarks and and baryons in a chiral soliton model. radial excitations of the low-lying In addition I present model results for the transition magnetic moments between the N(1710) and the nucleon.

++

g'

1. Introduction

Although chiral soliton model predictions for the mass of the lightest exotic pentaquark, the'0 with zero isospin and unit strangeness, have been around for some time1, the study of pentaquarks as baryon resonances b e came popular only recently when experiment^^>^ indicated their existence. These experiments were stimulated by a c h i d soliton model estimate4 sug7 ~ it gesting that such exotic baryons might have a widtha so ~ m a l l ~that could have escaped earlier detection. These novel observations initiated exhaustive studies on the properties of pentaquarks. Comprehensive lists of such studies are, for example, collected in In chiral soliton models states with baryon quantum numbers are generated from the soliton by canonically quantizing the collective coordinates associated with (would-be) zero modes such as SU(3) flavor rotations. The and decuplet replowest states are members of the flavor octet (J" = !j+) resentations (JT = $+). Due to flavor symmetry breaking the physical states acquire admixtures from higher dimensional representations. For the J" = ++baryons those admixtures originate dominantly from the antidecuplet, 10,and the 27-plet12. They also contain states with quantum 'This work is supportedin parts by DFG under contract We-1254f9-1. aEstimates for pentaquark decays are obtained from axial current matrix e l e m e n t s * ~ ~ * From what is known about the A + a N transition8, such estimates may be questioned.

216

numbers that cannot be built as t h r w u a r k composites but contain additional quark-antiquark pairs. Hence the notion of exotic pentaquarks. So far, the Of and E3/2 with masses of 1537flOMeV and 1862f2MeV have been observed, although the single observation of E3/2 is not ~ndisputed'~. Soliton models predict the quantum numbers I ( J " ) = O(i+) for O+ and $(++) for 3 3 / 2 . These quantum numbers are yet to be codrmed experiment ally. Radial excitations14 of the octet nucleon and Z!, are expected to have masses similar N and E type baryons in the 10. Hence sizable mixing should occur between an octet of radial excitations and ths antidecuplet. Roughly, this corresponds to the picture that pentaquarks are members of the direct sum 8@10which is also obtained in a quark-diquark approach15. Some time ago a dynamical model was developed" to investigate such mixing effects and also to describe static properties of the low-lying J" = &' and J" = $+ baryons. Essentially that model has only a single free parameter, the Skyrme constant e which should be in the range e M 5.0.. .5.5. Later the mass of the recently discovered Of pentaquark was predicted with reasonable accuracy in the same model6. In this talk I will present predictions for masses of the E3p and additional exotic baryons that originate the 27-plet from exactly that model without any further modifications. The latter may be considered as partners of O+ and E.312 in the same way as the A is the partner of the nucleon. It is also interesting to see whether established nucleon resonances, such as the N(1710), qualify as flavor partners of the O+ pentaquark. To this end, I will consider transition magnetic matrix elements between the nucleon and its excitations predicted by the model. A more complete description of the material presented in this talk may be found in ref.17. 2. Collective Quantization of the Soliton

I consider a chiral Lagrangian in flavor SU(3). The basic variable is the chiral field U = exp(iXa@/2) that represents the pseudoscalar fields cp" (a = 0,. . .,8). Other fields may be included as well. For example, the specific model used later also contains a scalar meson. In general a chiral Lagrangian can be decomposed as a sum, t = t s LSB,of flavor symmetric and flavor symmetry breaking pieces. Denoting the (classical) soliton solution of this Lagrangian by Uo(8)states with baryon quantum numbers are constructed by quantizing the flavor rotations

+

U(F,t) = A(t)Uo(r')At(t),

A ( t ) E SU(3)

(1)

217

canonically. According to the above separation the Hamiltonian for the collective coordinates A ( t ) can be written as H = Hs HSB. For unit baryon number the eigenstates of HS are the members of SU(3) representations with the condition that the representation contains a state with identical spin and isospin quantum numbers. Radial excitations that potentially mix with states in higher dimensional SU(3) representations are described by an additional collective coordinate ( ( t ) 14J6

+

U(F,t ) = A(t)Uo(((t)r‘)At(t) .

(2)

Changing to z(t) = [ ( ( t ) ] - 3 /the 2 flavor symmetric piece of the collective Hamiltonian for a given SU(3) representation of dimension p reads

Hs =

-1

2&2px

-/Gz+V+ a a ($- $) ff3p4

J ( J + 1 ) + - C12 ( p ) + s , 28

(3) where J and G ( p ) are the spin and (quadratic) Casimir eigendues associated with the representation p. Note that m = m(z),(Y = a ( x ) ,. ..,s = ~ ( 2are ) functions of the scaliig variable to be computed in the specified soliton modelle. For a prescribed p there are discrete eigenvalues (E,,,,,) and eigenstates (lp,n,)) of Hs- The radial quantum number n, counts the number of nodes in the respective wave-functions. The eigenstates Ip, n,) serve to compute matrix elements of the full Hamiltonian H,,n,;,),n;, - &,np~,,P’4z,,n;, - (P,n 1 , f t r ( w b A + )s(x)lp‘,n;‘) * (4) This “matrix” is diagonlized exactly yielding the baryonic states IB , m) = C,,n, C,(,B,,F,) lp,n,). Here B refers to the specific baryon and m labels its excitations. I would like to stress that quantizing the radial degree of freedom is also demanded by observing that the proper description of baryon magnetic moments requires a substantial feedback of flavor symme try breaking on the soliton size18. 3. Results I divide the model results for the spectrum into three categories. First there are the low-lying J = f and J = baryons together with their monopole excitations. Without flavor symmetry breaking these would be pure octet and decuplet states. Second are the J = 3 states that are dominantly members of the antidecuplet. Those that are non-exotic mix with octet baryons and their monopole excitations. Third are the J = baryons that would dwell in the 27-plet if flavor symmetry held. The J = baryons

3

3

3

218 Table 1. Mass differences of the eigenstates of the Hamiltonian (4) with respect to the nucleon in MeV. Experimental datalg refer to four and three star reaonances, unless otherwise noted. For the Roper resonance [N(1440)] I list the Breit-Wigner (BW) mass and the pole position (PP) estimatelo. The states ”?” are potential isospin E candidates with yet undetermined spin-party. B

3*

L

m=l e=5.0 e=5.5 expt. 501 BW 413 445 426 PP 657 688 661 694 722 721 751 941 971 1011 (?) 640 680 661 841 878 901 1068 1036 1386 1343

m=O e=5.0 e=5.5 expt.

175 284

Input 173 284

177 254

382 258 445 604 730

380 276 460 617 745

379 293 446 591 733

from the 27-plet are heavier than those with J = studied here.

m=2 e=5.0 e=5.5 expt. 836 1081 1068

869 1129 1096

1515 974 1112 1232 1663

1324 1010 1148 1269 1719

771 871 831 (*) 941 (w) 981 1141 -

-

and will thus not be

3.1. Ordinary Bamons and their Monopole Excitations Table 1 shows the predictions for the mass differences with respect to the nucleon of the eigenstates of the full Hamiltonian (4) for two values of the Skyrme parameter e. The agreement with the experimental data is quite astonishing. Only the Roper resonance (IN,l)) is predicted a bit on the low side when compared to the empirical Breit-Wigner mass but agrm with the estimated pole position. This is common for the breathing mode approach in soliton models14. All other first excited states are quite well baryons the energy eigenvalues for the second reproduced. For the excitations overestimate the corresponding empirical data somewhat. In the nucleon channel the model predicts the m = 3 state only about 40MeV higher than the m = 2 state, i.e. still within the regime where the model is assumed to be applicable. This is interesting because empirically it is suggestive that there might exist more than only one resonance in that baryons with m = 2 the agreement with data energy region20. For the is on the 3%level. The particle data group lists two “three star” isospin$ S resonances at 751 and lOllMeV above the nucleon whose spin-parity is not yet determined. The present model suggests that the latter is J“ = $+, while the former seems to belong to a different channel. The present model gives fair agreement with available data and thus

4’

2’

219 Table 2. Masses of the eigenstates of the Hamiltonian (4) for the exotic baryons Q+ and 2312. Energies are given in GeV with the absolute energy scale set by the nucleon mass. Experimental data are the average of refs.2 for 9+ and the NA49 result for SSIa3. I also compare the predictions for the ground state ( m = 0) to the treatment of ref.21. B l m=O I m=1 I e=5.0 e=5.5 expt. I WK" I e=5.0 e=5.5 expt. e+ I 1.57 1.59 1.537f0.010 I 1.54 I 2.02 2.07 S3/2 1.89 1.91 1.862 f 0 . 0 0 2 1.78 2.29 2.33 -

I

I

I

supports the picture of coupled monopole and rotational modes. Most notably, the inclusion of higher dimensional s U ~ ( 3flavor ) representations in three flavor chiral models does not lead to the prediction of any novel states in the regime between 1 and 2GeV in the non-exotic channels. 3.2. Exotic Bargons from the Antidecuplet Table 2 compares the model prediction for the exotics CP and &3/2 to available data2~3 and to a chiral soliton model calculation21 that does not include a dynamical treatment of the monopole excitation. In that calculation parameters have been tuned to reproduce the mass of the lightest exotic pentaquark, O+. The inclusion of the monopole excitation increases the mass of the E3l2 slightly and brings it closer to the empirical value. Furthermore, the first prediction4 for the mass of the Z 3 p was based on identifying N(1710) with the nucleon like state in the antidecuplet and thus resulted in a far too large mass of 2070MeV. Other chiral soliton model studies either take ME^,^ as input22,adopt the assumptions of ref. or are less predictive because the model parameters vary considerablylO. Without any fine-tuning the model prediction is only about 30-50MeV higher than the data. 'In view of the approximative nature of the model this should be viewed as good agreement. Especially the mass difference between the two potentially observed exotics is reproduced within 1OMeV. 3.3. Baryons from the 27-plet

The 27-plet contains states with the quantum numbers of the baryons that are also contained in the decuplet of the low-lying J = baryons: A, C* and E*. Under flavor symmetry breaking these states mix with the radial excitations of decuplet baryons and are already discussed in table 1. Table 3 shows the model predictions for the J = baryons that emerge from the 27-plet but do not have partners in the decuplet. Again, the experimental

g

220

3

Table 3. Predicted masses of the eigenstates of the Hamiltonian (4) for the exotic J = baryons with m = 0 and m = 1 that originate from the 27-plet with hypercharge (Y) and isospin (I) quantum numbers listed. I also compare the m = 0 case t o treatments of refs.21322923.All numbers are in GeV.

-

-

eZ7 N27 A27

rZ7

n27 027

B Y

I

2 1 0 0 -1 -2

1 1/2 0 2 312 1

e=5.0 1.66 1.82 1.95 1.70 1.90 2.08

m=O WKzl

e=5.5 1.69 1.84 1.98 1.73 1.92 2.10

1.67 1.76 1.86 1.70 1.84 1.99

BFKZ2 1.60

--

--

1.70 1.88 2.06

m = l e=5.5 e=5.0 2.10 2.14 2.28 2.33 2.50 2.56 2.12 2.17 2.35 2.40 2.54 2.59

WMZ3 1.60 1.73 1.86 1.68 1.87 2.07

nucleon mass is used to set the mass scale. Let me remark that the particle data grouplQ lists two states with the quantum numbers of N27 and A27 at 1.72 and 1.89GeV, respectively, that fit reasonably well into the model calculation. In all channels the m = 1 states turn out to be about 500MeV heavier than the exotic ground states. 3.4. Magnetic Moment lh.xnsition Matria: Elementa

Table 4 shows the model prediction for magnetic moment transition matrix elements for states with nucleon quantum numbers. I expect the model to reliably predict these matrix element because it also gives a good account of the magnetic moments of the spin-; baryons, in particular with regard to deviations from flavor symmetric relationsl6. It is especially interesting .to compare them with the result originating from the assumption that the N(1710) be a pure antidecuplet state24. This assumption yields a proton channel transition matrix element much smaller than in the neutron channel. While I do confirm this result for the case of omitted configuration mixing (entry I i 6 , O ) + 18,O))it no longer holds true when the effects of flavor symmetry breaking are included. Then the transition matrix eleTable 4. Transition magnetic moments of excited nucleons in the proton and neutron channels. Results are given in nucleon magnetons (n.m.) and with rmpect to the proton magnetic moment, pp. p.

= 5.n

2 (N1710) 3 l8,1) --t I8,O)

I m o ) + 18,O)

I

I

nmtnn

-0.28 -0.24 -0.53 0.00

-0.13 -0.11 -0.24 0.00

neutrnn

-0.17 -0.19 0.40 -0.62

-0.08 -0.09 0.18 -0.28

221

ments in the proton and neutron channels for the N(1710) candidate state (rn = 2) are of similar magnitude. This difference to the pure 10 picture for the N(1710) should be large enough that data on electromagnetic properties could test the proposed mixing scheme. 4. Conclusion

In this talk I have discussed the interplay between rotational and monopole excitations for the spectrum of pentaquarks in a chiral soliton model. In this approach the scaling degree of freedom has been elevated to a dynamical quantity which has been quantized canonically at the same footing as the (flavor) rotational modes. Then not only the ground states in individual irreducible s U ~ ( 3representations ) are eigenstates of the (flavor-symmetric part of the) Hamiltonian but also all their radial excitations. I have treated flavor symmetry breaking exactly rather then only at first order. Thus, even though the chiral soliton approach initiates from a flavor-symmetric formulation, it is capable of accounting for large deviations thereof. The spectrum of the low-lying and baryons is reasonably well reproduced. Also, the model results for various static properties are in acceptable agreement with the empirical datale. This makes the model reliable to study the spectrum of the excited states. Indeed the model states can clearly be identified with observed baryon excitations; except maybe an additional P11 nucleon state although there exist analyses with such a resonance. Otherwise, this model calculation did not indicate the existence of yet unobserved baryon states with quantum numbers of three-quark composites. Here the mass difference between mainly octet and mainly antidecuplet baryons is a prediction while it is an input quantity in most other approaches4*21j22g23Jo and the computed masses for the exotic O+ and =3/2 baryons nicely agree with the recent observation for these pentaquarks. The present predictions for the masses of the spin-; pentaquarks should be sensible as well and are roughly expect between 1.6 and 2.1GeV.

i'

2'

Acknowledgments

I am grateful to the organizers for providing this pleasant and worthwhile workshop. References 1. A. V. Manohar, Nucl. Phys. B 248 (1984)19; L.C. Biedenharn, Y. Dothan, in &om SU(3) To Gmuity, E. Gotsman, G. Tauber, eds, p 15; M. Chemtob,

222

2.

3. 4. 5. 6. 7. 8.

9. 10. 11.

12.

13. 14.

15. 16. 17. 18. 19. 20. 21.

22. 23. 24.

Nucl. Phys. B 256 (1985) 600; M. Prasdowicz, in Skyrnaions and Anomalies, M. Jezabek, M. Prasdowicz, eds., World Scientific (1987), p. 112; H. Walliser, Nucl. Phys. A 548 (1992) 649; H.Walliser, in Baryons as Skyrnae Solatons, G. Holzwarth, ed., World Scientific (1994), p. 247 T. Nakano et aZ. [LEPS Coll.], Phys. Rev. Lett. 91 (2003) 012002; for further references and a discussion of the experimental situation, see T. Nakano and K. Hicks, Mod. Phys. Lett. A 19 (2004) 645. C. Alt et ol. “A49 Coll.], Phys. Rev. Lett. 92 (2004) 042003 D. Diakonov, V. Petrav, M. Polyakov, Z. Phys. A359 (1997) 305 R. L. JafFe, Eur. Phys. J. C 35 (2004) 221 H. Weigel, Eur. Phys. J. A 2 (1998) 391; AIP Conf. Proc. 549 (2002) 271 M. Praszalowicz, Phys. Lett. B 583 (2004) 96 H. Verschelde, Phys. Lett. B 209 (1988) 34; S. Saito, Prog. Theor. Phys. 78 (1987) 746; G. Holzwarth, A. Hayashi, B. Schwesinger, Phys. Lett. B 191 (1987) 27; G. Holzwarth, Phys. Lett. B 241 (1990) 165; A. Hayashi, S. Saito, M. Uehara, Phys. Lett. B 246 (1990) 15, Phys. Rev. D 46 (1992) 4856; N. Dorey, J. Hughes, M. Mattis, Phys. Rev. D 50 (1994) 5816 B. K. Jennings, K. Maltman, arXiv:hep-ph/0308286 J. R. Ellis, M. Karliner, M. Praszalowicz, JHEP 0405 (2004) 002 R. L. Jaffe, arXiv:hep-ph/0409065 N. W. Park, J. Schechter, H. Weigel, Phys. Lett. B 224 (1989) 171; For a ) models see H. Weigel, Int. J. review and more references on s U ~ ( 3soliton Mod. Phys. A 11 (1996) 2419 H. G. Fischer, S. Wenig, arXiv:hep-ex/0401014; M. I. Adamovich et al., [WA89 Coll.], arXiv:hep-e~/0405042. C. Hajduk, B. Schwesinger, Phys. Lett. B 140 (1984) 172; A. Hayashi, G. Holzwarth, Phys. Lett. B 140 (1984) 175; I. Zahed, Ulf-G. Meibner, U. Kaulfuss, Nucl. Phys. A 426 (1984) 525; J. Breit, C. R. Nappi, Phys. RRv. Lett. 53 (1984) 889; J. Zhang, G. Black, Phys. Rev. D 30 (1984) 2015 R. L. Jaffe, F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003 J. Schechter, H. Weigd, Phys. Rev. D 44 (1991) 2916; Phys. Lett. B 261 (1991) 235 H. Weigel, Eur. Phys. J. A 21 (2004) 133 B. Schwesinger, H. Weigel, Phys. Lett. B 267 (1991) 438, Nucl. Phys. A 540 (1992) 461 K. Hagiwara et al. [PDG], Phys. Rev. D 66 (2002) 010001 M. BatiniC, et al., Phys. Rev. C 51 (1995) 2310 [Erratum-ibid. C 57 (1998) 10041; D. G. Ireland, S. Janssen, J. Ryckebusch, arXiv:nucl-th/0312103 H. Walliir, V. B. Kopeliavich, J. Exp. Theor. Phys. 97 (2003) 433 [Zh. Eksp. Teor. Fiz. 124 (2003) 4831. D. Borisyuk, M. Faber, A. Kobushkin, arXiv:hep-ph/0312213 B. Wu, B. Q. Ma, Phys. Lett. B 586 (2004) 62 M. V. Polyakov, A. Rathke, Eur. Phys. J. A 18 (2003) 691

223

THE SKYRME MODEL REVISITED: AN EFFECTIVE THEORY APPROACH AND APPLICATION TO THE PENTAQUARKS

KOJI HARADA~ Department of Physics, Kyushu University &oka 810-8581 Japan E-mail: kojalscpOmbox.nc.kyushu-u.ac.jp

The Skyrme model is reconsidered from an effective theory point of view. Starting with the moist general Lagrangian up to including terms of order p4, Nc and 6m2 (am m, - m),we obtain new interactions, which have never been d i s c d in the literature. We obtain the parameter set best fitted to the low-lying baryon masses by taking into account the representation mixing up to 27. A prediction for the mainly anti-decuplet excited nucleon N’ and C’ is given.

1. Introduction and Summary The narrowness of the newly discovered exotic baryonic resonance C3+ 1*2*3i4 has been a mystery. The direct experimental upper bound is re < 9 MeV, while some reexaminations of older data suggest re < 1 MeV. At this moment, it is not very clear what makes the width so narrow. Interestingly, the mass and its narrow width had been predicted by Diakonov, Petrov, and PolyakovQ. Compare their predicted values, Me = 1530 MeV and I’ = 15 MeV (or 30 MeV10v11*12),with the experimental oned3, M e = 1539.2 f 1.6 MeV and I’ = 0.9 f 0.3 MeV. It is astonishing! What allows the authors to predict these numbers? It deserves a serious look. Their predictions are based on the “chiral quark-soliton model14,” (xQSM) which may be regarded as a version of the Skyrme model“ with 536*798

‘This talk is a preliminary version of the work in the collaboration with Y.Mitsunari and N. Yamashita, hepph/0410145. *Work partially supported by Grant-in-Aid for Scientific Research on Priority Area, Number of Area 763, “Dynamics of Strings and Fields,” from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

224

specific symmetry breaking interactionsa,

where @;(A) = (AtX,AXs), Y is the hypercharge operator, and J, is the spin operator. Is this a general form of the symmetry breaking? Is it possible to justify it without following their long way, just by relying on a more general argument? What is the most general Skyrme model? Is it possible to have a “model-independent” Skyrme model? This is our basic motivation. A long time ago, WittenI6 showed that a soliton picture of baryons emerges in the large-N, limit’’ of QCD. If the large-N, QCD has a close resemblance to the real QCD, we may consider an effective theory (not just a model) of baryons based on the soliton picture, which may be called as the “Skyrme-Witten large-Nc effective theory.” The question is in which theory the soliton appears. A natural candidate seems the chiral perturbation theory (xPT), because it represents a low-energy QCD at least in the meson sector. Note that it is different from the conventional Skyrme model, which contains only a few interactions. We have now an i n h i t e number of terms. We have to systematically treat these infinitely many interactions. Because we are interested in the low-energy region, we only keep the terms up to including 0(p4), where p stands for a typical energy/momentum scale. Because we consider the baryons as solitons, we keep only the leading order terms in N,. In this way, we arrive at the starting Lagrangian. We quantize the soliton by the collective coordinate quantization, where only the “rotational” modes are treated as dynamical. The resulting Hamiltonian contains a set of new interactions, which have never been considered in the literature. We calculate the matrix elements by using the orthogonality of the irreducible representation of SV(3) and the Clebsch-Gordan coefficients. By using these matrix elements, we calculate the baryon masses in perturbation theory with respect to the symmetry breaking parameter 6m m, - m, where m, is the strange quark mass and m stands for the mass for the up and down quarks. We ignore the isospin breaking in this work. ~

~

~~

“The xQSM has its own scenario based on chid symmetry breaking due t o instantons. But for our purpose, it is useful to regard it as a Skyrme model.

225

The calculated masses contain undetermined parameters. In the conventional Skyrme model calculations, they are determined by the profile function of the soliton and the xPT theory parameters. In our effective theory approach, however, they are just parameters to be fitted, because there are infinitely many contributions from higher order terms which we cannot calculate. After fitting the parameters, we make predictions.

2. The Hamiltonian

Let us start with the S U f ( 3 )xPT action which includes the terms up to 0(P41l 8 ,

where L4 = 8 LiOi is the terms of Oh4), M is the quark mass matrix, M = diag(m, m,ma), and is the WZW t e ~ n " 1 ~ ~ . The large-N , dependence of these low-energy coefficientsare :

As explained in the previous section, we keep only the terms of order N,. Furthermore, we assume that the constants L1, L2 and L3 have the ratio,

which is consistent with the experimental values, L1 = 0.4f0.3,2L1- L2 = -0.6 f 0.5, and L3 = -3.5 f 1.1 (times 10-3)22.It enables us to write the three terms in a single expression,

where we introduced L2 = l/(16e2).This term is nothing but the Skyrme

226

term. In this way, we end up with the action,

which is up to including U ( N c )and Oh4) terms. Note that there are tree level contributions to F, and M,, and so on. For example, 1

+ (2rn)Lb-

This action allows a topological soliton, called “Skyrmion.” The classical hedgehog ansatz,

has topological (baryon) number B = 1 and stable against fluctuations. We introduce the collective coordinate A(t),

U ( t ,). = A(t)Uc(.)At(t), and treat it as a quantum mechanical degree of freedom. By substituting Eq. (10) into Eq. (7), we obtain the following quantum mechanical Lagrangian,

where wQ is the “angular velocity,”

At(t)A(t)=

i2

a X,w*(t). ff=l

In the conventional Skyrme model, all the couplings are given in terms of the xPT parameters and the integrals involving the profile function F ( r ) , which is determined by minimizing the classical energy. In our effective theory approach, on the other hand, they are determined by fitting the physical quantities calculated by using them to the experimental values.

227

The most important feature of the Lagrangian (11) is that the “inertia tensor” &@(A)depends on A. It has the following form, I a P ( 4 = c p + I&(4,

(13)

(a = 8 or @ = 8)

where Z = {1,2,3}, J’ = {4,5,6,7), and d,p, is the usual symmetric tensor. The collective coordinate quantization procedure23*24*25~26*27 is wellknown, and leads to the following Hamiltonian,

+ + + H2, 2- P a l 2

H = Mci Ho Hi 1 Ho = (Fa)2+ 211

c

aEZ

Hi = .Dg)(A)

c

c

(Fa)2 +y

[c c] +

+ ;(1 - D g ( A ) ) , = 21 (1 -

c

&daP7F,~g)(A)Fp

~ E J , @ E Z 7=1

aEZ,PEJ

C FaDg)(A)Fa+ w aE,7

H2

(17)

7

212 a € J

aEZ

+

(16)

cC 8

dap7FaDg)(A)Fp

a,BEJ7=1

(18)

( D g ( A ) ) 2- (Di:)(A))

aEZ

where

and Fa ( a = l , . . ., 8 ) are the SU(3) generators,

c 8

[Fa, q31 = i

,=l

faP&,

(19)

228

where fupr is the totally anti-symmetric structure constant of SU(3). Note that they act on A j h m the right. 3. Fitting the parameters

We calculate the baryon masses (eigenvaluesof the Hamiltonian) in perturbation theory. The calculation of the matrix elements of these operators is a hard task and described in Ref. 28 in detail. We consider the mixings of representations among (8,m,27) for spin-; baryons and (10,27)for spin-$ baryons. The best fit set of parameters are obtained by the multidimensional exp 2 minimization of the evaluation function, x2 = (Mi- Mi ) /c:, where Mi stands for the calculated mass of baryon i, and M r p ,the corresponding experimental value. How accurately the experimental values should be considered is measured by ni. The sum is taken over the octet and decuplet baryons, as well as 0+(1540) and +(1860). The results are summarized in the following table.

xi

(MeV) MiCap US

Mi

N 939 0.6 941

E 1193 4.0 1218

E:

A

1318 3.2 1355

1116

0.01 1116

A 1232 2.0 1221

E* 1385 2.2 1396

S*

R

0

9

1533 1.6 1546

1672 0.3 1672

1539 1.6 1547

1862 2.0 1853

The best fit set of values is Mcl = 435MeV, I;' = 132MeV, IF1 = 408MeV, y = 1111MeV, x = 14.8MeV, y = -33.5MeV, z = -292MeV, w = 44.3MeV,

v = -69.8MeV, (22) with x2 = 3.5 x lo2. Note that they are quite reasonable, though we do not impose any constraint that the higher order (in 6m) parameters should be small. The parameter y is unexpectedly large (even though it is of leading order in N c ) , but considerably smaller than the value (7 = 1573 MeV) for the case (3) of Yabu and Ando. The parameter z seems also too large and we do not know the reason. Our guess is that this is because we do not consider the mixings among an enough number of representations. 4. Predictions and Discussions

We have determined our parameters and now ready to calculate other quantities. First of all, we make a prediction to the masses of the other members

229

of anti-decuplet, M N I= 1782 MeV,

M p = 1884 MeV.

(23)

Compare with the chiral quark-soliton model p r e d i ~ t i o n ~ ~ ,

MNI= 1646 MeV,

M E ) = 1754 MeV.

(24)

It is interesting to note that E' is heavier than 4. The decay widths are such quantities that can be calculated. The results are reported in Ref. 28. What should we do to improve the results? First of all, we should include more (arbitrarily many(?)) representations. The mixings with other representations are quite large, so that we expect large mixings with the representations we did not include. Second, we may have a better fitting procedure. In the present method, all of the couplings are treated equally. The orders of the couplings are not respected. Third, in order to understand the narrow width of @+, we might have to consider general N, m ~ l t i p l e t s Finally it seems interesting to include "radial" modes3'. Acknowledgments

The author would like to thank the organizers for providing this interesting workshop. References 1. T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91,012002 (2003) [arXiv:hep-ex/0301020]. 2. V. V. Barmin et al. [DIANA Collaboration], Phys. Atom. Nucl. 66, 1715 (2003) pad.Fiz. 66,1763 (2003)] [arXiv:hep-ex/0304040]. 3. S. Stepanyan et al. [CLAS Collaboration], Phys. Rev. Lett. 91,252001 (2003) [arXiv:hep-ex/0307018]. 4. J. Barth et al. [SAPHIR Collaboration], Phys. Lett. B 572, 123 (2003) [arXiv:hep-ex/0307083]. 5. S. Nussinov, arXiv:hep-ph/0307357. 6. R. A. Arndt, I. I. Strakovsky and R. L. Workman, Phys. Rev. C 68,042201 (2003) [Erratum-ibid. C 69,019901 (2004)] [arXiv:nucl-th/0308012]. 7. R. L. Workman, R. A. Arndt, I. I. Strakovsky, D. M. Manley and J. Tulpan, Phys. Rev. C 70, 028201 (2004) [arXiv:nucl-th/0404061]. 8. R. N. Cahn and G. H. Trilling, Phys. Rev. D 69,011501 (2004) [arXiv:hepph/0311245]. 9. D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359, 305 (1997) [arXiv:hep-ph/9703373]. 10. R. L. Jaffe, Eur. Phys. J. C 35, 221 (2004) [arXiv:hep-ph/0401187].

230

11. D. Diakonov, V. Petrov and M. Polyakov, arXiv:hep-ph/0404212. 12. R. L. JafFe, arXiv:hep-ph/0405268. 13. S. Eidelman et al. [Particle Data Group Collaboration], Phys. Lett. B 592, 1 (2004). 14. D. Diakonov and V. Y. Petrov, "Nucleons as chiral solitons," in At the f i n tier of Padicle Physics Vol 1, M. Shifman ed. [arXiv:hep-ph/0009006]. 15. T. H. R. Skyrme, Proc. Roy. SOC.Lond. A 260, 127 (1961). 16. E. Witten, Nucl. Phys. B 160,57 (1979). 17. G. 't Hooft, Nucl. Phys. B 72, 461 (1974). 18. J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 465 (1985). 19. J. Wess and B. Zumino, Phys. Lett. B 37, 95 (1971). 20. E. Witten, Nucl. Phys. B 223, 422 (1983). 21. S. Peris and E. de Rafael, Phys. Lett. B 348, 539 (1995) [arXiv:hepph/9412343]. 22. A. Pich, Rept. Prog. Phys. 58, 563 (1995) [arXiv:hep-ph/9502366]. 23. E. Witten, Nucl. Phys. B 223, 433 (1983). 24. G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B 228, 552 (1983). 25. E. Guadagnini, Nucl. Phys. B 236, 35 (1984). 26. P. 0. Mazur, M. A. Nowak and M. Praszaiowicz, Phys. Lett. B 147, 137 (1984). 27. S. Jain and S. R. Wadia, Nucl. Phys. B 258, 713 (1985). 28. K. Harada, Y. Mitsunari and N. Yamashita, arXiv:hep-ph/0410145. 29. J. R. Ellis, M. Karliner and M. Praszdowicz, JHEP 0405, 002 (2004) [arxiv:hep-ph/0401127]. 30. M. Praszdowicz, Phys. Lett. B 583, 96 (2004) [arXiv:hep-ph/0311230]. 31. H. Weigel, Eur. Phys. J. A 2, 391 (1998) [arXiv:hep-ph/9804260].

23 1

MAGNETIC MOMENTS OF THE PENTAQUARKS

HYUN-CHUL KIM', GHIL-SEOK YANG' , MICHAl PRASZA10WICZ2 AND KLAUS GOEKE3 1 . Department of Physics, and Nuclear Physics & Radiation Technology Institute (NuRI), Pusan National University, 609-735 Busan, Republic of Korea 2. M . Smoluchowski Institute of Physics, Jagellonian University, ul. Reymonta 4, 30-059 Krakdw, Poland 3. Znstitut fur Theoretische Physik II, Ruhr- Universitat Bochum, 0-44 780 Bochum, Germany We present in this talk a recent analysis for the magnetic moments of the baryon antidecuplet within the framework of the chiral quark-soliton model with linear m, corrections considered. We take into account the mixing of higher representations to the collective magnetic moment operator, which comes from the SU(3) symmetry breaking. Dynamical parameters of the model are fixed by experimental data for the magnetic moments of the baryon octet as well as by the masses of the octet, decuplet and of 8+. The magnetic moment of O+ is rather sensitive to the pionnucleon sigma term and ranges from -1.19n.m. to -0.33n.m. as the sigma term is varied from C I l = ~ 45 t o 75 MeV, respectively. On top of them, we obtained that the strange magnetic moment of the nucleon has the value of p $ ) = +0.39 n.m. within this scheme and turns out to be almost independent of the sigma term.

1. Introduction Exotic pentaquark baryons has been a hot issue, since the LEPS collaboration announced the new finding of the S = +1 baryon O+ which was soon confirmed by a number of other experiments 2 , together with an observation of exotic Ew states by the NA49 experiment at CERN 3 , though it is still under debate. Those experiments searching for the pentaquark states was stimulated by Diakonov et al. 4: Masses and decay widths of exotic baryon antidecuplet were predicted within the chiral quark-soliton model. The discoveries of the pentaquark baryon O+ and possibly of Em have triggered intensive theoretical investigations (see, for example, R e f ~ . ' > ~ The production mechanism of the O+ has been discussed in ref^.^?^?'^. In particular, it is of great interest to understand the photoproduction of the O+ theoretically, since the LEPS and CLAS collaborations used photons

'

232

as a probe to measure the O f . In order t o describe the mechanism of the pentaquark photoproduction, we have t o know the magnetic moment of the @+ and its strong coupling constants. However, information on the static properties such as antidecuplet magnetic moments and their strong coupling constants is absent t o date, so we need t o estimate them theoretically. Recently, two of the present authors calculated the magnetic moments of the exotic pentaquarks, within the framework of the chiral quark-soliton model l 1 in the chiral limit. Since we were not able t o fix all the parameters for the magnetic moments in the chiral limit, we had t o rely on the explicit model calculations 12,13. A very recent work l 4 extended the analysis for the magnetic moments of the baryons, taking into account the effect of SU(3) symmetry breaking so that the necessary parameters are fixed by the magnetic moments of the baryon octet. In the present talk, we would like t o present the main results of Ref. 14. 2. Constraints on parameters The collective Hamiltonian describing baryons in the ”(3) soliton model takes the following form 15:

+ J ( J211+ 1 )

Cz(SU(3)) - J ( J 212 with the symmetry breaking term given by:

H = Msol

+

+ 1) -

chiral quark-

+H,

(1)

where parameters a, p and y are of order O(m,) and are given as functions of the 7rN sigma term 16. @)(R) denote SU(3) Wigner rotation matrices and j is a collective spin operator. The Hamiltonian given in Eq.(2) acts on the space of baryon wave functions:

Here, R stands for the allowed irreducible representations of the SU(3) flavor group, i.e. R = 8,10, . . and Y,T ,T3 are the corresponding hypercharge, isospin, and its third component, respectively. Right hypercharge Y’ = 1 is constrained to be unity for the physical spin states for which J and J3 are spin and its third component. The model-independent approach consists now in using Eqs. (1) and (2) (and/or possibly analogous equations for other observables) and determining model parameters such as I 1 , 1 2 , a ,p, y from experimental data.

m,.

233

The symmetry-breaking term ( 2 ) of the collective Hamiltonian mixes different SU(3 ) representations as follows: l 3 IBS)

=

B ) f cm 1 1 0 1 / 2 , B ) + C27 1 2 7 1 / 2 , B ) Y

IBlo) =

l1o3/zl

1%)

Im1/21 B )

=

B

B -

181/2,

B)f

af7 1 2 7 3 / 2 1 B )

+d f

l81/2,

+4

5 1 3 5 3 , ~B~) 1

B ) + df7 1 2 7 1 / 2 , B ) + d&

B ) , (4)

1%1/2,

where [ B R )denotes the state which reduces t o the SU(3) representation R in the formal limit m, + 0 and the spin index J 3 has been suppressed. All relevant expressions for the mixing coefficients c g , u g , and d g can be found in Ref. 14. 3. Magnetic moments in the chiral quark-soliton model

The collective operator for the magnetic moments can be parameterized by six constants By definition in the model-independent approach they are treated as free 12113:

The parameters ~ 1 , 2 , 3are of order O ( m t ) ,while W 4 , 5 , 6 are of order O(m,), m, being regarded as a small parameter. The full expression for the magnetic moments can be decomposed as follows: p B = p!)

+ pp)+ pgf),

(6)

where the p g ) is given by the matrix element of the between the purely symmetric states IRJ,B , J 3 ) , and the p (z ) is given as the matrix element of the @ ( l )between the symmetry states as well. The wave function correction p g ’ ) is given as a sum of the interference matrix elements of the p g ) between purely symmetric states and admixtures displayed in Eq.(4). These matrix elements were calculated for octet and decuplet baryons in Ref.13. The measurement of the Q+ mass constrains the parameter space of the model. Recent phenomenological analyzes indicate that our previous assumption on y,i.e. y = 0, has t o be most likely abandoned. Therefore, our previous results for the magnetic moments of 8, 10 and have to

m

234

be reanalyzed. Now, we show that a model-independent analysis with this new phenomenological input yields w2 much larger than initially assumed, (0) which causes pe+ for realistic values of C,N t o be negative and rather small. Our previous results for the decuplet magnetic moments turn out t o hold within the accuracy of the model. The octet and decuplet - magnetic moments were calculated in Refs.l2yl3. For the antidecuplet p g ( O ) can be found in Ref.ll. In order to calculate the p g f ) , several off-diagonal matrix elements of the ,L(O) are required. These have been calculated in Ref.16 in the context of the hadronic decay widths of the baryon antidecuplet. Denoting the set of the model parameters by

w'= (w1,. . . , w6)

(7)

the model formulae for the set of the magnetic moments in representation R (of dimension R)

gR = (/*.El . .,P E R ) 1 .

(8)

can be conveniently cast into the form of the matrix equations:

FR

=A

R [ C x ~. w]',

(9)

where rectangular matrices Asl AlO,and A" can be found in Refs.12>13i14. Note their dependence on the pion-nucleon C,N term. 4. Results and discussion

In order t o find the set of parameters wi[CxN], we minimize the mean square deviation for the octet magnetic moments:

where the sum extends over all octet magnetic moments, but the Co. The value Ap8 N 0.01 is in practice independent of the C,N in the physically interesting range 45 - 75 MeV. The values of the &, th[CxN] are independent of C x N . Similarly, the value of the nucleon strange magnetic moment is independent of C,N and reads p g ) = 0.39 n.m. in fair agreement with our previous analysis of Ref.13. Parameters wi, however, do depend on C = N . This is shown in Table.1: Note that parameters ~ 2 , are 3 formally c3(1/Nc) with respect t o w1. For smaller C x ~this , N , counting is not borne by explicit

235 C x N [MeV]

45 60 75

w1 -8.564 -10.174 -11.783

w2

w3

w4

w5

206

14.983 11.764 8.545

7.574 7.574 7.574

-10.024 -9.359 -6.440

-3.742 -3.742 -3.742

-2.443 -2.443 -2.443

fits. The p B(0) can be parametrized by the following two parameters v and W:

w

=

+

(2pn - pP + 3 p ~ 0 ps- - 2p-j- - 3p-j+)/60 + 4pn + / L E O - 3pE- - 4p-j- - p - j + )/60

w = (3pp

= =

-0.268, 0.060.

(11)

which are free of linear m, corrections 13. This is a remarkable feature of the present fit, since when the m, corrections are included, the m,-independent parameters need not be refitted. This property will be used in the following when we restore the linear dependence of the p g on m,. The magnetic moments of the baryon decuplet and antidecuplet depend on the C x ~However, . the dependence of the decuplet is very weak, which as summarized in Table 2, where we also display the theoretical predictions

5.40 5.39 5.39

75

2.82 2.81

2.65 2.66 2.66

0.13 0.13

-0.09 -0.08 -0.07

-2.56 -2.55

-2.83 -2.82 -2.80

0.34 0.33

-2.30 -2.30

-2.05 -2.05

from Ref.12 for p = 0.25. Let us note that the m, corrections are not large for the decuplet and the approximate proportionality of the p g to the baryon charge Q B still holds. Finally, for antidecuplet we have a strong dependence on C T ~yielding , the numbers of Table 3. The results listed in Table 3 are further depicted in Fig.1. The wave function corrections cancel for the non-exotic baryons and add constructively for the baryon antidecuplet. In particular, for C n =~

236

II

1

O+ C n [MeV] ~ 45 -1.19 -0.78 6o 75 -0.33 E n [MeV] ~ 2& 45 -0.53

II I

0.48 1.51

P* -0.97 -0.36 0.28

n* -0.34 -0.41 -0.43

CL

co

CI

-0.75 0.06 0.90

-0.02 0.15 0.36

0.71

2- =-

=--

0.30 0.70 1.14

1.95 1.15 0.39

Y-

1.13 0.93 0.77

'. '

-0.19 0.23

I I

/

1.o 0.5

< 0.0 -0.5

-1 .o -1.5 ! 40

I

I

I

I

50

60

70

80

&N

Figure 1. Magnetic moments of antidecuplet as functions of C , N .

75 MeV we have large admixture coefficient of 27-plet: d& tends to dominate otherwise small magnetic moments of antidecuplet. At this point, the reliability of the perturbative expansion for the antidecuplet magnetic moments may be questioned. On the other hand, as remarked above, the N , counting for the wicoefficients works much better for large C x ~ One .

237 notices for reasonable values of C,N some interesting facts, which were partially reported already in Ref.": The magnetic moments of the antidecuplet baryons are rather small in absolute value. For @+ and p* one obtains negative values although the charges are positive. For Z+ and 7 3 one obtains positive values although the signs of the charges are negative.

---

5. Conclusion and summary Our present analysis shows that p ~ < + 0, although the magnitude depends strongly on the model parameters. The measurement of p ~ could + therefore discriminate between different models. This also may add t o reduce the ambiguities in the pion-nucleon sigma term C , N . In the present work, we determined the magnetic moments of the baryon antidecuplet in the model-independent analysis within the chiral quarksoliton model, i.e. using the rigid-rotor quantization with the linear m, corrections included. Starting from the collective operators with dynamical parameters fixed by experimental data, we obtained the magnetic moments of the baryon antidecuplet. The expression for the magnetic moments of the baryon antidecuplet is different from those of the baryon decuplet. We found that the magnetic moment pe+ is negative and rather strongly dependent on the value of the C r ~Indeed, . the pe+ ranges from -1.19n.m. t o -0.33n.m. for C,N = 45 and 75 MeV, respectively.

Acknowledgments H.-Ch.K and G.-S. Y are very much thankful to the organizers of the Workshop Pentaquark 04, in particular, T . Nakano and A. Hosaka for their hospitality. H.-Ch.K is grateful t o J.K. Ahn, S.I. Nam, M.V. Polyakov, and I.K. Yo0 for valuable discussions. The present work is supported by Korea Research Foundation Grant: KRF-2003-041-C20067 (H.-Ch.K.) and by the Polish State Committee for Scientific Research under grant 2 P03B 043 24 (M.P.) and by Korean-German and Polish-German grants of the Deutsche Forschungsgemeinschaft .

References 1. T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91,012002 (2003)

[arXiv:hep-ex/0301020]. 2. S. Stepanyan et al. [CLAS Collaboration],Phys. Rev. Lett. 91 (2003) 252001 [arXiv:hep-ex/0307018]; Phys. Rev. Lett. 92 (2004) 032001 [Erratum-ibid.

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3. 4. 5.

6. 7. 8.

9.

10. 11.

12. 13. 14. 15.

16.

92 (2004) 0499021 [arXiv:hep-ex/0311046]; V. V. Barmin et al. [DIANA Collaboration], Phys. Atom. Nucl. 66 (2003) 1715 [Yad. Fiz. 66 (2003) 17631 [arXiv:hep-ex/0304040]; J. Barth [SAPHIR Collaboration], hep-ex/0307083; A. E. Asratyan, A. G. Dolgolenko and M. A. Kubantsev, Phys. Atom. Nucl. 67 (2004) 682 [Yad. Fiz. 67 (2004) 7041 [arXiv:hep-ex/0309042]; A. Airapetian et al. [HERMES Collaboration], Phys. Lett. B 585 (2004) 213 (arXiv:hep-ex/0312044]; A. Aleev et al. [SVD Collaboration], arXiv:hepex/0401024; M. Abdel-Bary et al. [COSY-TOF Collaboration], arXiv:hepex/0403011; P. Z. Aslanyan, V. N. Emelyanenko and G. G. Rikhkvitzkaya, arXiv:hep-ex/0403044; S. Chekanov et al. [ZEUS Collaboration], Phys. Lett. B 591 (2004) 7 [arXiv:hep-ex/0403051]; C. Alt et al. “A49 Collaboration], Phys. Rev. Lett. 92 (2004) 042003 [arXiv:hep-ex/03 10014]; D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A359 (1997) 305 [hep-ph/9703373]. B. K. Jennings and K. Maltman, Phys. Rev. D 69,094020 (2004) [arXiv:hepph/0308286] and references therein. S. L. Zhu, arXiv:hep-ph/0406204 and references therein. K. Goeke, H.-Ch. Kim, M. Praszalowicz, and Gh.-S. Yang, to appear in Prog. Part. Nucl. Phys. (2004). Y. Oh, H. Kim and S. H. Lee, Phys. Rev. D 69,074016 (2004) [arXiv:hepph/0311054]; Phys. Rev. D 69,094009 (2004) [arXiv:hepph/0310117]; Phys. Rev. D 69,014009 (2004) [arXiv:hep-ph/0310019]. S. I. Nam, A. Hosaka and H.-Ch. Kim, arXiv:hep-ph/0405227; arXiv:hepph/0403009; arXiv:hepph/0402138; arXiv:hepph/0401074; Phys. Lett. B 579,43 (2004) [arXiv:hepph/0308313]. B. G. Yu, T. K. Choi and C. R. Ji, arXiv:nucl-th/0312075. H.-Ch. Kim and M. Praszalowicz, Phys. Lett. B 585 (2004) 99 [arXiv:hepph/0308242]; arXiv:hep-ph/0405171. H.-Ch. Kim, M. Praszalowicz and K. Goeke, Phys. Rev. D57 (1998) 2859 [hep-ph/9706531]. H.-Ch. Kim, M. Praszalowicz, M. V. Polyakov and K. Goeke, Phys. Rev. D58 (1998) 114027 [hep-ph/9801295]. G. S. Yang, H.- Ch. Kim, M. Praszalowicz and K. Goeke, arXiv:hep ph/0410042. A. Blotz, K. Goeke, N. W. Park, D. Diakonov, V. Petrov and P. V. Pobylitsa, Phys. Lett. B 287 (1992) 29; A. Blotz, D. Diakonov, K. Goeke, N. W. Park, V. Petrov and P. V. Pobylitsa, Nucl. Phys. A 555 (1993) 765. J. R. Ellis, M. Karliner and M. Praszalowicz, JHEP 0405 (2004) 002 [arXiv:hep-ph/0401127]; M. Praszalowicz, Acta Phys. Polon. B 35 (2004) 1625 [arXiv:hep-ph/0402038].

239

NARROW PENTAQUARK STATES IN A QUARK MODEL WITH ANTISYMMETRIZED MOLECULAR DYNAMICS *

Y. KANADA-EN'YO, 0. MORIMATSU AND T. NISHIKAWA Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaralci 305-0801, Japan

The exotic baryon 0+(uudda) is studied with microscopic calculations in a quark model by using a method of antisymmetrized molecular dynamics(AMD). We predict narrow states, J" = 1/2+(1 = 0), J" = 3/2+(1 = 0), and J" = 3 / 2 - ( 1 = l), which nearly degenerate in a low-energy region of the uudda system. We discuss NK decay widths and estimate them to be I' < 7 for the J" = {1/2+, 3/2+}, and r < 1MeV for the J" = 312- state.

The evidence of an exotic baryon O+ has recently been reported by several experimental groups. This discovery proved the existence of the multiquark hadron, whose minimal quark content is uudd3 as given by the decay modes. The study of pentaquarks has become a hot subject in hadron physics. A chiral soliton model predicted a narrow O+(J" = 1/2+) state whose parity contradicts the naive quark model expectation. Theoretical studies were done to describe O+ by many group^^*^. The spin parity of O+ is not only a open problem but also a key property to understand the dynamics of pentaquark systems. In this paper we would like to clarify the mechanism of the existence of narrow pentaquark states. We try to extract a simple picture for the pentaquark baryon with levels, width, spin-parity and structure from explicit calculation. In order to achieve this goal, we study the pentaquark with a flux-tube model6*' based on strong coupling QCD, by using a AMD method4g5. In the flux-tube model, the interaction energy of quarks and anti-quarks *The authors would like to thank t o T. Kunihiro, Y.Akaishi and H. En'yo for valuable discussions. This work is supported by Japan Society for the Promotion of Science and Grants-in-Aid for Scientific &search of the Japan Ministry of Education, Science Sports, Culture, and Technology.

240

is given by the energy of the string-like color-electric flux, which is proportional to the minimal length of the flux-tube connecting quarks and antiquarks at long distances supplemented by perturbative one-gluon-exchange (OGE) interaction at short distances. For the cfq system the flux-tube configuration has an exotic topology, Fig.l(c), in addition to an ordinary meson-baryon topology, Fig. l(d). An important point is that the transition between the different flux-tube topologies (c) and (d) is strongly suppressed because it takes place only in higher order. (In 1991, Carlson and Pandharipande studied exotic hadrons in the flux-tube model* and calculated a few q4q states with very limited quantum numbers.) We apply the AMD method to the flux-tube model and calculate the uud& system. The AMD is a variational method to solve a finite manyfermion system. One of the advantages of this method is that the spatial and spin degrees of freedom for all particles are independently treated. This method can successfully describe various types of structure such as shell-model-like structure and clustering (correlated nucleons) in nuclear physic^^,^. With the AMD method we calculate all the possible spin parity states of uudds system, and analyze the wave function to estimate the decay widths of the obtained states with a method of reduced width amplitudes. In the present calculation, the quarks are treated as non-relativistic spin-; Fermions. We use a Hamiltonian as H = HO H I + H f , where HO is the kinetic energy of the quarks, H I represents the short-range OGE interaction between the quarks and H f is the energy of the flux tubes. HO and H I are represented as follows;

+

where m,(the i-th quark mass) is m, for a u or d quark and m, for a B quark, and TOdenotes the kinetic energy of the center-of-mass motion. Here, we do not take into account the mass difference between ud and s in the second term of Ho, for simplicity. ac is the quark-gluon coupling constant, and F,' is the generator of color SU(3). In H I , we take only the dominant terms, Coulomb and color-magnetic terms, and omit other terms. In the flux-tube quark model 6, the confining potential is written as H f = g L f - M o , where a is the string tension, L f is the minimum length of the flux tubes, and M o is the zero-point energy. M o depends on the topology of the flux tubes and is necessary to fit the qq, q3 and q4q potential. In

24 1

(4

(b)

4*

2

3

2

2

3

3

Figure 1. Flux-tube configurations for confined states of qq (a), q3 (b), q4q (c), and disconnected flux-tube of q4q (d). Figures (e) and (f) represent the flux tubes in the color configurations, [udl[u@ and [uu][d~!lB, respectively. The string potentials given by the flux tubes (b) and (c) are supported by Lattice QCD

’.

the present calculation, we adjust the M o to fit the absolute masses for each of 3q-baryon and pentaquark. For the meson and 3q-baryon systems, the flux tube configurations are given as Fig.l(a) and (b). For the pentaquark system, the different types of flux-tube configurations appear as shown in Fig.l(e),(f), and (d), which correspond to the states, I@(e)) = I[udl[uCI]B), I@(,))= l[uu1[4$, and l @ ( d ) ) = I(QQQ)l(QQ)l), respectively “nql is defined by color anti-triplet of qq). In the present calculation of energy variation, we neglect the transitions among I@(e)), I@(f)) and I@(d)) and solve 5q wave functions within the model space (e) or (f), which corresponds to the conhecl states. It is reasonable because the transitions are suppressed as mentioned before. In the practical calculations of the string potential ( @ I H j l @ ) , the minimum length of the flux tubes L f is approximated by a linear combination of two-body distances as L f M i ( r 1 2 ~ 2 3 ~ 3 1 for ) a 3q-baryon, and L j M i(7-12 ~ 3 4 ) $ ( T i 3 T14 + ~ 2 3 T24) + a ( r i 1 + ~ i +2 r i 3 + r i 4 ) for @(el or @(f) of the pentaquark systems. We note that the confinement is reasonably realized by the approximation for @(,,,I as follows. The fluxtube configuration (e)(or (f)) consists of seven bonds and three junctions. In the limit that the length(R) of any one bond becomes much larger than other bonds, the approximated ( H f ) behaves as a linear potential oR. It means that all the quarks and anti-quarks are bounded by the linear poor @y) tential with the tension o. Therefore, the approximation for is a natural extension of the usual approximation for 3q-baryons. It is easily proved that the approximations are equivalent to (@IHjI@)M (@161 where 0 E -2o FrFyrij - Mo, within each of the flux-tube configurations. We solve the eigenstates of the Hamiltonian with a variational method in the AMD model space4t5. We take a base AMD wave function in a quark model as follows.

+

+

+

@(Z)= (1f P ) A [4z,$z2 -.‘q$z,,X] ,

+

+ +

4zi

0: e-*(r-4bzi)2

1

(3)

where 1f P is the parity projection operator, A is the anti-symmetrization

242

operator, and the spatial part $zi of the i-th single-particle wave function is written by a Gaussian with the center Z,(Zi is a complex parameter). X is the spin-isospin-color function. For the pentaquark(uudd3) system, X is expressed as

x=

Cml,m2,rn3,m4,m5 h m z m s m 4 m s lmlm2m3m4m5)S

@{ IududS) or Iuudds)) @ cabgEcehEghflakef)c,

(4)

where Iudud3) and luuddg) correspond to the configurations [ud[udaand [uu][d@ in Fig.1, respectively. Here, la)c(a = 1,2,3) denotes the color function, and Im)s(m is the intrinsic-spin function. Since we are interested in the confined states, we adopt those model space for the color configurations (qq)3 (qq)gQ, but do not use the meson-baryon configurations (qqq)l(qq)l. The variational parameters are Z = { & , Z 2 , - - - , Z 5 } and Cmlmzm3m4m5 which specify the spatial and spin configurations. The energy variation for Z is performed by a frictional cooling method, and the coefficientscml m2m3m4m5are determined by diagonalization of Hamiltonian and norm matrices. After the energy variation, the intrinsic-spin and parity S" eigen wave function @(Z)for the lowest state is obtained for each SR. In the numerical calculation, the linear and Coulomb potentials are approximated by seven-range Gaussians. We use the parameters, a, = 1.05, A = 0.13 fm, m, = 0.313 GeV, (T = 0.853 GeV/fm, and Ams = m, - m, = 0.2 GeV. The quark-gluon coupling constant a, is chosen so as to fit the N and A mass difference. The string tension Q is adopted to adjust the excitation energy of N*(1520). The width parameter b is chosen to be 0.5 = 972 MeV, the masses of N , N*(1520) and A fm. By choosing MO as are fitted l o , and the masses of A, C and C*1385 are well reproduced with these parameters. Now, we apply the AMD method to the uudd3 system. For each spin parity, we calculate energies of the [udl[ud]Sand [ u u ] [ 4 sstates and adopt the lower one. In table.1, the calculated results are shown. We adjust the zero-point energy of the string potential MO as M:4q = 2385 MeV to fit the absolute mass of the recently observed O+. This M:4q for pentaquark system is chosen independently of for 3q-baryon. If M:4q = 5M:3 is assumed as Ref.[8],the calculated mass of the pentaquark is around 2.2 GeV, which is consistent with the result of Ref.[8]. The most striking point in the results is that the S" = 3/2- and S" = 1/2+ states nearly degenerate with the S" = 1/2- states. The S" = 1/2+ correspond to J" = 1/2+ and 3/2+ with S = 1/2,L = 1, and the S" = 3/2- is J" = 3/2-(S = 3/2,L = 0). The lowest state J" = 1/2-(S" =

=r,J.)

243

1/2-, L = 0) exists just below the J" = 3/2- state, however, this state, as we discuss later, is expected to be much broader than other states. Other spin-parity states are much higher than these low-lying states. The LS-partners, J" = 1/2+ and 3/2+ exactly degenerate in the present Hamiltonian where the spin-orbit and tensor terms are omitted. If we introduce the spin-orbit force into the Hamiltonian the LS-splitting is small in the diquark structure because the effect of the spin-orbit force from the spin-zero diquarks is very weak as discussed in Ref.12. As shown later, since the present results show that the diquark structure is realized in the J" = 1/2+ and 3/2+ states, the LS-splitting should not be large in the uudds system. Table 1. Calculated masses(GeV) of the uud& system. The expectation values of the kinetic, string, Coulomb, color-magnetic terms, and that of the color-magnetic term in qQ pairs are listed. The S" = 3/2+ and S" = 5/2+ states are higher than the S" = 5/2- state. [uu][d49

S" Kinetic(H0) String(HF) Coulomb Color mag. q$olor mag. E

12

3.23 -0.67 -1.05 -0.01 -0.06 1.50

[ud][ud]a [uc.&&d]a 3&+ 7

7

3.22 -0.66 -1.04

3.36 -0.55 -0.99 -0.25 0.00 1.56

0.01 -0.01

1.53

[ud][ud]a

1I 3.19 -0.64 -1.03 0.04 0.02 1.56

[uu][dd]I 57

3.19 -0.64 -1.03 0.19 0.06 1.71

Next, we analyze the spin structure of these states, and found that the J" = {1/2+, 3/2+)(S = 1/2, L = 1) states consist of two spin-zero uddiquarks, while the J" = 3/2- consists of a spin-zero ud-diquark and a spin-one ud-diquark. Since the spin-zero ud-diquark has the isospin I = 0 and the spin-one ud-diquark has I = 1because of the color asymmetry, the isospin of the J" = 3/2- state is I = 1, while the even-parity states J" = 1/2+,3/2+ are I = 0. We consider that the J" = 1/2+ state corresponds to the Of(1530) in the flavor m-plet predicted by Diakonov et d.'. The odd-parity state, J" = 3/2- has I = 1, which means that this state is a member of the flavor 27-plet. We denote the J" = 1/2+,3/2+(I = 0) by QZ, and the J" = 3/2-(I = 1) by 63:. Although it is naively expected that unnatural spin parity states are much higher than the natural spin-parity 1/2- state, the results show the abnormal level structure of the (ududs) system, where the high spin state, J" = 3/2-, and the unnatural parity states, J" = {1/2+,3/2+}, nearly degenerate just above the J" = 1/2- state. By analysing the details of

244

these states, the abnormal level structure can be easily understood with a simple picture as follows. As shown in table.1, the J" = {1/2+,3/2+}(S = 1/2, L = 1) states have larger kinetic and string energies than the J" = 3/2-(S = 3/2,L = 0) and J" = 1/2-(S = 1/2,L = 0) states, while the former states gain the color-magnetic interaction. It indicates that the degeneracy of the even-parity states with the odd-parity states is realized by the balance of the loss of kinetic and string energies and the gain of the color-magnetic interaction. In the J" = {1/2+, 3/2+} and the 3/2states, the competition of the energy loss and gain can be simply understood from the point of view of the diquark structure as follows. As already mentioned by JafFe and Wilczek2, the relative motion between two spinzero diquarks must have the odd parity (L = 1) because of Pauli blocking between the two identical diquarks. In the J" = 312- state, one of the spinzero ud-diquarks is broken to be a spin-one ud-diquark to avoid the Pauli blocking, then, the L = 0 is allowed because diquarks are not identical. The L = 0 is energetically favored in the kinetic and string terms, and the energy gain cancels the color-magnetic energy loss of a spin-one ud-diquark. Although we can not describe the J" = 1/2- state by such a simple diquark picture, the competition of energy loss and gain in this state is similar to the J" = 3/2-. We remark that the existence of two spin-zero ud-diquarks in the J" = {1/2+, 3/2+} states predicted by Jaffe and Wilczek2 is actually confirmed in our ab iaitio calculations. We found that the component with two spin-zero ud-diquarks is 97% in the present J" = {1/2+,3/2+} state. In Fig.2, we show the quark and anti-quark density distributions in the J" = {1/2+, 3/2+} states. In the intrinsic state before parity projection, we found the spatial development of ud-uds clustering, which causes a parityasymmetric shape (Fig.2 (c)).

(b)

-

s-quark

Figure 2. q and q density distribution in the J" = 1/2+, 3 / 2 + ( S = 112, L = 1) states of Theta+. The u-quark density (a), density (b), and total quark-antiquark density (c) of the intrinsic state before parity projection are shown. The &quark density is same as the u-quark density. The root-mean-squareradius of q and Q is 0.63fm (the nucleon size is 0.5 fm).

245

We estimate the KN-decay widths of these states by using a method of reduced width amplitudesll. The decay width r is estimated by the product x Sf,,, where r",u, Eth) is given by the penetrability of the barrier", and Sfac(a) is the probability of the decaying particle at the channel radius a. In the following discussion, we use the channel radius a = 1 h and the threshold energy Eth = 100 MeV. We here estimate the maximum values of the widths, by taking into account only quark degrees of freedom. We omit the suppression of the transition between the confined state and the meson-baryon state due to the rearrangement of flux-tubes, which makes Sf,, small in general. In case of even parity J" = 1/2+, 3/2+ states, the K N decay modes are the P-wave, which gives w 100 MeV fm-l. We calculate the overlap between the obtained pentaquark wave function and the K + n state, and evaluate the probability as Sf,, = 0.034 fm-l. Roughly speaking, the main factors in this meson-baryon probability are the factor from the color configuration, the factor from the intrinsic spin part, and the other factor which arises from the spatial overlap. By using this value, the total width for K + n and Kop decays of the J" = 1/2+, 3/2+ states is estimated to be < 7 MeV. For more quantitative discussions, it is important to treat the coupling with the K N continuum states, where one must take into account the suppression due to the rearrangement of flux-tube topologies. It is interesting that the K N decay width of the J" = 3/2- state is w 30 extremely small due to the D-wave centrifugal barrier. In fact, MeV fm-' is much smaller than the P-wave case. Moreover, the J" = 3/2-(S" = 3/2-,L = 0) has no D-wave component, therefore, no overlap with the KN(L = 2) states in the present calculation. Even if we introduce the spin-orbit or tensor forces, the K N probability(Sf,,) in the J" = 3/2pentaquark state is expected to be minor. Consequently, the J" = 3/2state should be extremely narrow. If we assume the Sf,, in the J" = 3/2to be half of that in the Jn = 1/2+,3/2+ states, the KN decay width is estimated to be r < 1 MeV. Contrary to the narrow features of the J" = 3/2- state, in case of J" = 1/2-, S-wave(L = 0) decay is allowed and this state should be much broader. In conclusion, we proposed a quark model in the framework of the AMD method, and applied it to the uudd3 system. The level structure of the the uudds system and the properties of the low-lying states were studied. We predicted that thenarrow J n = {1/2+,3/2+)(@+,) and J" = 3/2states nearly degenerate. The widths of O$ and 0; are estimated to be I' < 7 MeV and r < 1 MeV, respectively. Two spin-zero diquarks are

rt

5

246

found in the @$,which confirms Jaf€e-Wilczek picture. The origin of the novel level structure is the 59 dynamics of the confined system bounded by the connected flux-tubes. We consider that the present results for the J" = { 1/2+, 3/2+}(Of,,) states correspond to the experimental observation of O+, while the OI=I is not observed yet. The existence of many narrow states, J" = 1/2+, 3/2+, and 3/2-, may give an light to further experimental observations. Concerning other pentaquarks, we give a comment on E(ddss.ii). The AMD calculations indicate that the diquark structure disappears in the ddss.ii(1/2+) due to the SU(3)-symmetry breaking in the color-magnetic interaction. As a result, the estimated width of the ddss.ii(l/2+) state is I? M 100 MeV, which is much broader than @+(1/2+). Also the 3/2- state is not so narrow because a S-wave decay channel E*(1530)~ is open. Finally, we would like to remind the readers that the absolute masses of the pentaquark in the present work are not predictions. We have an ambiguity of the zero-point energy of the string potential, which depends on the flux-tube topology in each of meson, 3q-baryon, pentaquark systems, We adjust that for the pentaquarks to reproduce the obseved Of mass. To c0nfh-m the zero-point energy, experimental information for other pentaquark states are desired.

References 1. 2. 3. 4.

5. 6. 7.

8. 9.

10. 11. 12.

D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A359 (1997) 305. R. J&e and F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003. M.Oka, Prog. Theor. Phys. 112 (2004) 1, and references therein. Y. Kanada-En'yo, H. Horiuchi and A. Ono, Phys. Rev. C 52 (1995) 628 ; Y. Kanada-En'yo and H. Horiuchi, Phys. Rev. C 52 (1995) 647 . Y. Kanada-En'yo, M. Kimura and H. Horiuchi, Comptes Rendus Physique vo1.4 (2003) 497. J. Carlson, J. B. Kogut and V. R. Pandharipande, Phys. Rev. D 2 7 (1982) 233; Phys. Rev. D28 (1983) 2807. 0. Morimatsu, Nucl. Phys. A505 (1989) 655; C. Alexandrou, T. Karapiperis and 0. Morimatsu, Nucl. Phys. A518 (1990) 723. J. Carlson and V. R. Pandharipande, Phys. Rev. D43 (1991) 1652. T. T. Takahashi, H. Matsufuru, Y. Nemoto and H. Suganuma, Phys. Rev. Lett. 86 (2001) 18; F. Okiharu, H. Suganuma and T. T. Takahashi, hep-lat/0407001. Y.Kanada-En'yo, M. Morimatsu and T. Nishikawa, hep-ph/0404144. H. Horiuchi and Y. Suzuki, Prog. Theor. Phys. 49 (1973) 1974, and references therein. By J.J. Dudek and F.E. Close, Phys. Lett. B583 (2004) 278.

247

DECAY OF O+ IN A QUARK MODEL

A. HOSAKA Research Center for Nuclear Physics ( R C N P ) , Ibaraki, Osaka 567-0047, Japan E-mail: [email protected] We study decay of the pentaquark O+ in a non-relativistic quark model for the cases of J p = 1/2*. The O+ + N matrix elements of the kaon source term are expressed as products of a spectroscopic factor and an interaction part. It is shown that a narrow width is realized for a positive parity state, while the negative parity state of ( 0 ~configuration ) ~ couples strongly to the continuum state resulting in a very broad width. It is also pointed out that a 3/2- state results in strong suppression of its width.

1. Introduction The observation of the evidence of the pentaquark particle @+ by the LEPS group at Spring-8 has triggered enormous amount of works both in experiments and theories An investigation may be required with new insights of the non-perturbative QCD dynamics for the description of the exotic particles 2 . It is a great challenge to understand in a unified way from the conventional hadronic matter which are made of three-quark baryons to the new type of matter of multi-quarks. Although the experimental situation for the existence of the pentaquarks requires further confirmation, some of the expected properties of the pentaquark baryons are very interesting to be explained from the underlying dynamics. Among them, the narrow width of @+ is particularly an important issue together with the determination of its spin and parity. As a physical quantity which reflects very sensitively the spin and parity, we investigate in this report the strong decay of @+. To the lowest order, fall-apart process dominates where a simple rearrangement of the five quarks in the pentaquark initial state going into a quark-antiquark kaon and a three-quark nucleon. Before going to details, we discuss briefly an important role of the centrifugal barrier for the decaying K N state. For instance, if Of has spinparity J p = 1/2*, the partial wave of the K N state is p or s wave. The

’.

248

Figure 1 .

Decay of the pentaquark state.

different partial wave nature appears in the formula of the decay width

where denote the decay widths of the positive and negative parity O+, and g K N @ the I
-

-

-

2. Decay width The decay of the pentaquark state going into one baryon and one meson is dominated by the fall-apart process as shown in Figure 1 (left), which is associated with the annihilation of a quark and antiquark pair. The matrix element of such a process is written as a product of the so-called spectroscopic factor and an interaction matrix element, MO++KN

= S K N in @+ . hint.

(2)

The former S K N in @+ is a probability amplitude to find in the pentaquark state three-quark and quark-antiquark clusters having the quantum numbers of the nucleon and kaon, respectively. A calculation obtaining this factor was done in Ref. by writing the pentaquark wave functions explicitly. Here we briefly show how this is done using the technique of the Young tableaux. We perform a calculation for the negative parity ( O S ) ~configuration. Let us start with the fact that the color wave function of the q4 system

249

must be [211], to form the color singlet state with S of Ill],. Since all four light quarks occupy the same 0s orbit, the orbital wave function is totally symmetric [4],. Therefore, the spin-flavor wave function must have the symmetry [31]js which is combined with the color wave function [211], to form the totally antisymmetric state [ 1 1 1 1 ] , , ~ ~In . the Young tableau, this can be expressed as

The subscripts c, s, f and o in the tableaux denote color ( c ) , spin (s),flavor ( f ) and orbital (0)parts of the wave function. Furthermore, center-dot ”.” denotes the inner-product of wave functions in different functional space. The csf wave function is now decomposed into color and spin-flavor part. In the Young diagram with particle number assignment, it can be written as

The Young diagrams is convenient when projecting out the term containing the quarks 123 forming the nucleonic component and of 45 kaoninc one. The first term is the one of such, where the color wave function of 123 is totally antisymmetric [ll 1Ic and spin-flavor part is totally symmetric [3],f. Assuming that the @+ has isospin 0, the flavor wave function is expressed by [22], and so the only possible spin wave function is [31],. Therefore, in the Young diagram, the spin-flavor wave function can be expressed as

Finally the S wave function is multiplied to the above q4 wave function. The color, spin-flavor wave function of S quark is expressed by

250

This is combined with the q4 wave function to yield the O+ wave function

+... .

('

In the first term of this equation, the fourth quark and S form the desired color (singlet) and flavor (isosinglet) quantum numbers. The spin part needs one more step. For instance, the spin wave function = 1 as the coupling structure with

[[s123,

S5]St"t =

[[1/2, 1/21', 1/2]1/2

which may be recoupled for the kaon with spin

5'45

(8)

= 0:

"1/2, 1/211, 1 / 2 p 2 = XCJ[1/2[1/2, l/2]J]''2, J

4 co = -

1 C' = (9) 2 ' 2 Here the coefficients c g and c1 are the amplitude for the spin S45 = 0 and 1 components. S45 = 1 corresponds to the Ii* vector meson of spin one. Therefore, the coupling strength of K* to the O+ is l/& of that of K for the negative parity O+. Using the results of Eqs. (7) and (9), one finds the spectroscopic factor, the amplitude of finding the neutron-like u d d and kaon-like U S , to be

In other words, we have

251

Although we do not show here, but the similar manipulation can be performed for positive parity pentaquarks of ( O S ) l~p configuration. So far the decomposition of the wave function, and now for the amplitude. In the quark model, the interaction matrix element can be computed by the meson-quark interaction of Yukawa type: Lmgg

= g05AadJaq

(12)

1

where A, are SU(3) flavor matrices and dJa are the octet meson fields. The coupling constant g may be determined from the pion-nucleon coupling constant gr” = 59. Therefore, using gr” 13, we find g 2.6. In fact, the meson-quark interaction of (12) has been used for spaced like transitions, quark + quark and meson. Here we need a transition of an annihilation of a pair of quark and antiquark in the time like region. Different kinematics changes the strength of the coupling because the meson is a composite object. Pentaquark decays can provide information such quark and antiquark annihilation. Although such a kinematical dependence may affect the numerical estimation below, we assume to use the same coupling constant as given here. In Ref. 5 , the matrix element of the axial vector current was computed under the assumption of PCAC. In the quark model without meson degrees of freedom, however, the PCAC relation does not hold, and therefore, the decomposition of the current matrix element into various form factors as including the kaon pole term should be done carefully. In the present method, the kaon-quark interaction is defined in (12) which is then directly related to the K N O coupling in a rather straightforward manner. Further details of calculation can be found in Ref. 3, and here several results are summarized as follows. For the negative parity state of ( O S ) ~ , the decay width turns out to be of order of several hundreds MeV or more, typically 0.5 1 GeV, as shown in Table 1. In the calculation it has been assumed that the spatial wave function for the initial and final state hadrons are described by a common harmonic oscillator states. Also the masses of the particles are taken as experimental values, e.g., Ma+ = 1540 MeV. For the result of the negative parity state of ( O S ) ~ ,the unique prediction can be made, since there is only one quark model states. The very broad width suggests that the ( O S ) ~state couples very strongly to the K’N continuum and is hardly identified with a resonant state with a narrow width. For the positive parity state, the orbital excitation introduces more degrees of freedom for the pentaquark state. In fact, four independent configurations are available for spin-parity J p = 1/2+ ‘. Here we consider three

-

-

-

252

1 fm

1.5fm-2

I

520

74

37

12

configurations which minimize (1) a spin-flavor interaction of one meson exchange 4 , (2) a spin-color interaction of one gluon exchange, and (3) the S = I = 0 diquark correlated state as proposed by Jaffe and Wilczek '. As shown in Table 1, the resulting decay widths are about 80 MeV, 40 MeV and 10 MeV, respectively. The diquark correlation of (3) develops a spinflavor-color wave function having small overlap with the decaying channel of the nucleon and kaon. In the evaluation of these values, we did not consider spatial correlations. However, if, for instance, diquark correlation is taken into account, spatial overlap becomes less than one which further suppresses the decay width. The small values of the decay width for J p = 1/2+ as compared with the large values for J p = l/2- can be explained by the difference in the coupling structure; one is the pseudoscalar type of a' . and the other the scalar type of 1. The former of the pwave coupling includes a factor q/(2M) which suppresses the decay width significantly as compared with the latter at the present kinematics, q 250 MeV and M 1 GeV, when the same coupling constant g N K @ is employed.

-

-

3. Summary and discussions In this report we have discussed decay of the pentaquark O+. The mechanism of the fall-apart process is very sensitive to the structure, especially to the spin and parity. For J p = 1/2-, the naive ground state of ( O S ) ~can no longer survive as a narrow resonance as it couples very strongly with the ICN scattering state. In other words, the ( O S ) ~configuration is almost the scattering state with little component of a confined state. Contrary, the decay widths of 1/2+ states turn out to be of order ten MeV, but with once again strong dependence on the configuration. The present analyses can be extended straightforwardly to the O+ of spin 3/2. For the negative parity state, the spin 1 state of the four quarks in the O+ may be combined with the spin of ?. for the total spin 3/2. In

253

this case the final K N state must be in a d-wave state, and therefore, the spectroscopic factor of finding a d-wave K N state in the ( O S ) is ~ simply zero. If a tensor interaction induces an admixture of a d-wave configuration, it can decay into a d-wave K N state. However, the mixture of the d-wave state is expected to be small just as for the deuteron. There couid be a possible decay channel of the nucleon and the vector I<* of J p = 1-. This decay, however, does not occur since the total mass of the decay channel is larger than the mass of O+. Hence the J p = 3/2- state could be another candidate for the observed narrow state. This state does not have a spinorbit partner and forms a single resonance peak around its energy. For the positive parity case, the p-state orbital excitation may be combined with the spin of s for the total spin 3/2. In this case, the calculation of the decay width is precisely the same as before (See Ref. for more details). After taking the average over the angle f, however, the coupling yields the same factor as for the case J = 1/2. Hence the decay rate of spin 3/2 O+ is the same as that of O+ of spin 1/2 in the present treatment.

Acknowledgements The author would like to thank E. Hiyama, T . Hyodo, M. Kamimura, T . Nakano, S.I. Nam, M. Oka, T. Shinozaki and H. Toki, for discussions. This work supported in part by the Grant for Scientific Research ((C) No.16540252) from the Ministry of Education, Culture, Science and Technology, Japan. References 1. T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91, 012002 (2003); For the latest experimental situation, see the web site of the workshop PENTAQUARK04, www.rcnp.osaka-u.ac.jp/ penta04. 2. For a recent review, see for instance: M. Oka, hep-ph/0406211 and references therein. 3. A. Hosaka, M. Oka and T. Shinozaki, arXiv:hepph/0409102. 4. C.E. Carlson, C.D. Carone, H.J. Kwee and V. Nazaryan, Phys. Lett. B 573, 101 (2003); hepph/1012325. 5. D. Melikhov, S. Simula and B. Stech, Phys. Lett. B 594, 265 (2004) [arXiv:hep-ph/0405037]. 6. B.K. Jennings and K. Maltman, Phys. Rev. D 68, 094020 (2004). 7. R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003).

254

DYNAMICAL STUDY OF THE PENTAQUARK ANTIDECUPLET IN A CONSTITUENT QUARK MODEL

FL. STANCU University of Li&ge, Institute of Physics, B.5, Sart Tilman, B-4000 Litge 1, Belgium E-mail: fstancuOulg.ac. be Dynamical calculations are performed for all members of the flavor antidecuplet t o which the pentaquark 8+ belongs. The framework is a constituent quark model where the short-range interaction has a flavor-spin structure. From symmetry considerations the lowest state acquires a positive parity. By fitting the mass of 8+ of minimal content uud&, the mass of =--, of minimal content ddssTi, is predicted to be approximately 1960 MeV. It is shown that the octet and antidecuplet states with the same quantum numbers mix ideally due to S u ( 3 ) ~ breaking.

1. Introduction

At present there is a large variety of approaches to pentaquarks: chiral soliton or Skyrme models, constituent quark models, instanton models, QCD sum rules, lattice calculations, etc. Here I shall discuss the pentaquarks in the framework of constituent quark models. These models describe a large number of observables in ordinary hadron spectroscopy as e. g. spectra, static properties, decays, form factors, etc. Therefore it seems interesting to look for their predictions for exotics. I shall refer to two standard constituent quark models: the color-spin (CS) model where the hyperfine interaction is of one-gluon exchange type and the flavor-spin (FS) where the hyperfine interaction is due to meson exchange. There are also hybrid models where the hyperfine interaction is a superposition of CS and FS interactions. Presently the main issues of any approach to pentaquarks are: (1) The spin and parity (2) The mass of O+ (or @ for heavy pentaquarks) (3) The splitting between isomultiplets of a given s U ( 3 ) representation ~ (4) The mixing of representations due to s U ( 3 ) ~breaking

255

(5) The width ( 6 ) The production mechanism Here I shall present results for light and heavy pentaquarks obtained in the FS model. I shall cover all but the last two items. 2. P a r i t y and spin

The antidecuplet to which 8+ belongs can be obtained from the direct product of two flavor octets, one representing a baryon ( q 3 ) and the other a meson (qq) 8~ x 8 ~ = 2 7 ~ + 1 0 ~ + I O ~ + 2 ( 8 ) ~ + 1 ~

(1)

m~

The antidecuplet can mix with 8 ~for, example, because s u ( 3 ) is ~ not exact. This mixing will be considered below. To find the parity of 8+ and of its partners one looks first at the q4 subsystem with I = 0 and S = 0, i. e. with quantum numbers compatible with the content uuddS of @+. This means that the flavor and spin wave functions have symmetry [ 2 2 ] and ~ [22]s respectively. Their direct product can generate the state [ ~ ] F s .If the orbital wave function contains a unit of orbital excitation, it is described by [31]0. The color part of q4 is [211]c in order to give rise to a color singlet state after the coupling to 4. Then [31]0 x [ 2 1 1 ] ~4 [ 1 1 1 1 ] 0 ~so that the Pauli principle requires [ ~ ] F s .In the FS model the contribution of the hyperfine attraction of [ ~ ] F sis so strong that it fully overcomes the excess of kinetic energy due to the orbital excitation and the lowest state has positive parity irrespective of the flavor content of the pentaqu+rk '. The spin is either 1/2 or 3/2. In the CS model a positive parity as lowest state is also possible in principle 3. The orbital and flavor symmetries give [31]0 x [ 2 2 ] ~+ [ 2 1 1 ] 0 ~ and, due to Pauli principle, this must combine with [ 3 1 ] ~ s the , lowest allowed symmetry. Then only if the excess of kinetic energy is compensated by the attractive hyperfine interaction the parity would be positive. This does occur in realistic calculations with CS interaction, as implied by Refs. 4t5, SO that the lowest state has negative parity. More generally, parity remains a controversial issue also in QCD lattice calculations QCD sum rules lead to negative parity 8. 617.

3. The orbital wave function There are four internal Jacobi coordinates Z = ?I (FI F2 - 2?3)/&, z' = (TI F2 ?3 - 3?4)/&

+

+

+

?2,

,

y' = and t' =

256

+

+ +

/m

(71 ?2 73 F4 - 475) where 5 denotes the antiquark. The total wave function is a linear combination of three independent orbital basis vectors contributing with equal weight2l1O $1 = $0 z YIO ( E ) , $2 = $0 y Yio ( i )and $3 = $0 z YIO ( 2 ) where

contains two variational parameters: a, the same for all q4 coordinates Z, y' and 2, and 0,related to 6 the relative coordinate of q4 to ij. 4. The antidecuplet

The pentaquark masses are calculated by using the realistic Hamiltonian of Ref. ', which leads to a good description of low-energy non-strange and strange baryon spectra. It contains an internal kinetic energy term T , a linear confinement potential V, and a short-range flavor-spin hyperfine interaction V, with an explicit radial form for the pseudoscalar meson exchange. Details of these calculations are given in Ref. '. The expectation Table 1. The hyperfine interaction V, integrated in the flavor-spin space. for some q4 subsystems (for notation see text).

values of the hyperfine interaction V, integrated in the flavor-spin space, are shown in Table 1 for the three q4 subsystems necessary to construct the antidecuplet. They are expressed in terms of the two-body radial form V p Q bof Ref. ', where qaqb specifies the flavor content of the interacting qq pair and y the exchanged meson. The s U ( 3 ) ~is explicitly broken by the quark masses and by the meson masses. By taking V y = V:" = V;" and V y = V;" = 0, one recovers the simpler model of Ref. l2 where one does not distinguish between the uu, u s or ss pairs in the rpmeson exchange. Moreover, in Ref. 12, for every exchanged meson, the radial two-body matrix elements are equal, irrespective of the angular momentum of the state, C = 0 or C = 1. This is because on takes as parameters the already integrated

257

two-body matrix elements of some radial part of the hyperfine interaction, fitted to ground state baryons. Here one explicitly introduces radial excitations at the quark level. Table 2 contains the partial contributions Table 2. Partial contributions from the model Hamiltonian and total energy E = C&, mi (T) (Vc) (Vx)in MeV for various q4?j systems. The mass M is obtained from E by subtraction of 510 MeV in order to fit the mass of €I The values of the variational parameters a and 0 are indicated in the last two columns.

+

+

+

uudda

1700

1864

442

-2044

1962

1452

0.42

uuddB

1800

1848

461

-2059

2050

1540

0.42

1.01

uudsa

1800

1535

461

-1563

2233

1732

0.45

0.92

uudsB

1900

1634

440

-1663

2310

1800

0.44

0.87

ddssti

1900

1418

464

-1310

2472

1962

0.46

0.92

uussS

2000

1410

452

-1310

2552

2042

0.46

0.87

Table 3. The antidecuplet mass spectrum (MeV) for P =

+ 1.

Pentaquark

Y, I, 1s

8+

2,070

1540

Nm

1,112,112

1684

1665

"m

0,1,1

1829

1786

1962

1906

Present results Ref.

I__

a

-1,312,-312

[lo]

0.92

Carlson et al. Ref. [12]

1540

and the variational solution E of the Hamiltonian resulting from the trial wave function introduced in Sec. 3. All specified q4?j systems are needed to construct the antidecuplet and the octet. One can see that, except for the confinement contribution (VJ, all the other terms break s U ( 3 ) ~ the : 5 increases, the kinetic energy (T)decreases and the mass term Cn=lmi short range attraction (Vx)decreases with the quark masses. For reasons explained in Refs. 510 MeV are subtracted from the total energy E in order to reproduce the experimental 8+ mass. For completeness, in the last two columns of Table 2 the values of the variational parameters a and ,B of the radial wave function (Sec. 3) are indicated. The parameter a takes values around a0 = 0.44 fm. This is loyll

258

precisely the value which minimizes the ground state nucleon mass when the trial wave function is q5 o( exp[-(z2 y 2 ) / 4 a ~ ]where Z and y' are the Jacobi coordinates of Sec. 3. The quantity a0 gives a measure of the quark core size of the nucleon because it is its root-mean-square radius. The parameter is related to the coordinate
+

+

+

In comparison with Carlson et al. 12, where the mass of Q+ is also adjusted to 1540 MeV, here the masses of N a , Cm and E-- are higher. In the lowest order of s U ( 3 ) breaking, ~ one can parametrize the present result by the Gell-Mann-Okubo (GMO) mass formula, M = M m c Y . This gives M cu 1829 - 145 Y. The nearly equal spacing between isomultiplets is illustrated in Fig. 1 a).

+

5. Representation mixing The present model contains s U ( 3 ) ~breaking so that representation mixing appears naturally and it can be derived dynamically. Recall that Table 3, column 3 gives the pure antidecuplet masses. The pure octet masses are easily calculable using Table 2. These are

The octet-antidecuplet mixing matrix element V has two non-vanishing contributions, one coming from the mass term and the other from the kinetic energy hyperfine interaction. Its form is

+

V=

q ( m s- mu)+

4 [S(uudsS)- S(uuddz)] = 166 MeV

for N

q ( m s - mu)+

4[S(uussS)

for C

- S(uudsd)] = 155

MeV

(5)

259

+

where S = ( T ) (Vx).The numerical values on the right hand side of Eq. ( 5 ) result from the quark masses mu,d = 340 MeV, m, = 440 MeV and from the values of ( T ) and (Vx)exhibited in Table 2. One can see that the mass-induced breaking term is identical for N and C, as expected from simple SU(3) considerations, and it represents more than 1 / 2 of V. The masses of the physical states, the “mainly octet” N* and the “mainly

---

1962

Em

1829

Nm

1684

e+

1540

---

1962

N5

1801

E5

1719

e+

1540

I

b) mixed with the octet

a) pure antidecuplet

Figure 1. Comparison between a) the pure antidecuplet spectrum of Table 3 and b) the “mainly antidecuplet” solutions after the mixing with the octet.

antidecuplet” N5, result from diagonalizing a 2 x 2 matrix in each case. Accordingly, the nucleon solutions are

N* = N ~ C O S ~-NNmsineN, N5 = NgsinBN NmCOSeN,

+

(6)

with the mixing angle defined by

The masses obtained from this mixing are 1451 MeV and 1801 MeV respectively and the mixing angle is ON = 35.34’, which means that the “mainly antidecuplet” state N5 is 67 % Nm and 33 % Ns, and the “mainly octet” N* state is the other way round. The latter is located in the Roper resonance mass region 1430 - 1470 MeV. However this is a q46 state, i. e. it is different from the q3 radially excited state obtained in Ref. at 1493

260

MeV. A mixing of the q3 and the q4tj states could possibly be a better description of reality. The “mainly antidecuplet” solution at 1801 MeV is 70 MeV above the higher option of Ref. 13, at 1730 MeV, interpreted as the Y = 1 narrow resonance partner of 8+. In a similar way one obtains two C resonances, the “mainly decuplet” being at 1719 MeV and the “mainly octet” at 2046 MeV. The “mainly d+ cuplet” is lower than Cm because the octet-antidecuplet mixing angle has negative sign, Ox = -35.48’. Its mass is 30 MeV above the the experimental mass range 1630 - 1690 MeV of the three star C(l660) resonance and 10 MeV below the lowest seen experimental edge of the one star C(1770) resonance (see the PDG l4 full listings). As the higher mass region of C is less known experimentally, it would be difficult to make an assignment for the higher state. The pentaquark spectrum resulting from the octetantidecuplet mixing is illustrated in Fig. 1 b). One can see that the order of N and C states is reversed with respect to case a). The mixing angles ON and Ox are nearly equal in absolute value, but they have opposite signs. The reason is that M ( N m ) > M(N8) while M ( C m ) < Ad(&). Interestingly, each is close to the value of the ideal mixing angle O$ = 35.26O and O F = -35.26O. This implies that in practice the “mainly antidecuplet” N5 state carries the whole hidden strangeness and that N* has a simple content, for example uuddd when the charge is positive. 6. Heavy pentaquarks Table 4. Masses (MeV) charmed pentaquarks.

Pentaquark @ :

of the positive parity antisextet

I

Content

0

FS model Ref. [2]

Lattice Ref. [15]

u uddE

2902

2977f104 3180 f 69 3650 f 95

N :

112

uudsE

3161

E:

1

uussE

3403

Based on the same constituent quark model g , positive parity heavy charmed pentaquarks of minimal content uuddE have been proposed long before the first observation l7 of @+(uudds). Table 4 reproduces the results of Ref. where the masses represent the binding energies AE (Table 11) to

261

which threshold energies ET,(Table I) have been added. These results are compared with the only lattice calculations which predict positive parity 15. Interestingly the masses are quite similar in the two approaches. In the FS model the lightest negative parity pentaquark is a few hundreds MeV heavier l8 than 0: of Table 4. The experimental search for charmed pentaquarks is contradictory so far 19. 7. Conclusions

In the new light shed by the pentaquark studies, the usual practice of hadron spectroscopy is expected to change. There are hints that the wave functions of some excited states might contain q4ij components. These components, if obtained quantitatively, would perhaps better explain the widths and mass shifts in the baryon resonances. In particular the mass of the Roper resonance may be further shifted up or down. Also it is important to understand the role of the chiral symmetry breaking on the properties of pentaquarks 16, inasmuch as the predictions of Ref. ', which motivated this new wave of interest, are essentially based on this concept.

References 1. D. Diakonov, V. Petrov and M. Polyakov, 2. Phys. A359,305 (1997). F1. Stancu, Phys. Rev. D58, 111501 (1998). B. Jennings and K. Maltman, Phys. Rev. D69,094020 (2004).

2. 3. 4. 5.

Y. Kanada-Enyo et al., hep-ph/0404144 and these proceedings S. Takeuchi and K. Shimizu, these proceedings. 6. S. Sasaki, heplat/0310014 and these proceedings. 7. T. -W. Chiu and T. -H. Hsieh, hepph/0403020 and these proceedings. 8. J. Sugiyama, T. Doi and M. Oka, Phys. Lett. B381,167(2004);M.Oka , h e p ph/0409295 (these proceedings), S. H. Lee et al. hep-ph/O411104 and these proceedings. 9. L. Ya. Glozman, Z. Papp and W. Plessas, Phys. Lett. B381,311 (1996). 10. F1. Stancu, Phys. Lett. B595,269 (2004); ibid. Erratum, B598, 295 (2004). 11. F1. Stancu and D. 0. Riska, Phys. Lett. B575, 242 (2003). 12. C. E. Carlson et al., Phys. Lett. B579 (2004) 52. 13. R. A. Arndt et al., nucl-th/0312126. 14. S. Eidelman et al. (Particle Data Group) Phys. Lett. B592, 1 (2004). 15. T. -W. Chiu and T. -H. Hsieh, hep-ph/0404007. 16. A. Hosaka,Phys. Lett. B571,55 (2003); hep-ph/0409101. 17. T. Nakano et al., (LEPS Coll.) Phys. Rev. Lett. 91,012002 (2003). 18. M. Genovese et al., Phys. Lett. 425,171 (1998). 19. A. Aktas et al. (H1 Coll.), hep-ex/0403017; M. -J. Wang, these proceedings.

262

PENTAQUARK WITH DIQUARK CORRELATIONS IN A QUARK MODEL

SACHIKO TAKEUCHI Japan College of Social Work, Kiyose, Tokyo 204-8555, Japan E-mail: [email protected] KIYOTAKA SHIMIZU Department of Physics, Sophia University, Chiyoda-ku, Tokyo 108-8554, Japan We have investigated uud& pentaquarks by employing quark models with the meson exchange and the effective gluon exchange as qq and qq interactions. The system for five quarks is dynamically solved; two quarks are allowed to have a diquark-like qq correlation. It is found that the lowest mass of the pentaquark is about 1947 - 2144 MeV. There are parameter sets where the mass of the lowest positive-parity state become lower than that of the negativeparity states. Which parity corresponds to the observed peak is still an open question. Relative distance of two quarks with the attractive interaction is found to be by about 1.2 - 1.3 times closer than that of the repulsive one. The diquark-like quark correlation seems to play an important role in the pentaquark systems.

1. Introduction Since the experimental discovery of the baryon resonance with strangeness +1, 0(1540)+, many attempts have been performed to describe the peak theoretically.2 To describe this resonance by using a quark model, one needs at least five quarks, uudds, which is called a pentaquark. A quark model, however, seems to have difficulties to explain some of the features of this peak. Namely, (1) the observed mass is rather low, (2) the observed width is very narrow, and (3) there is only one peak is found, especially no T=l peak nearby. To reproduce the observed mass, 100 MeV above the KN threshold, it is preferable to take lowest-mass configuration, TJP=Oi-. It is reported, however, that this state will have a very large width, which contradicts to the observed narrow width.3 Other candidate, O:+, may have a rather narrow width, but it might not become as low &s the observed one. The fact that the peak is buried in the NK continuum makes the problem more difficult. As was reported in this w o r k ~ h o p , the ~ ’ ~QCD lattice

263

calculation as well as the QCD sum rule approach found this continuum problem rather serious. The quark model can deal with this problem by introducing the scattering states using the resonating group method. Also, it is reported6 that a ‘bound state’ calculated without such scattering states becomes a resonance at almost the same energy when the scattering states are introduced though the levels which couple strongly to the nucleon-kaon systems disappear. In this work, we investigate the pentquark systems without taking its breaking effect as a first step. Our main aim here is to investigate the effects of the qq correlation on the pentaquarks, which have been mainly treated only by a simple wave function. We employ several parameter sets for the hamiltonian: those with the one-boson exchange (OBE), those with the one-gluon exchange (OGE), and those with the semirelativistic or the nonrelativistic kinetic energy terms. By employing these various parameter sets, we try to estimate the mass of pentaquarks of various quantum numbers with a controlled ambiguity. 2. Model

The hamiltonian for quarks and anti-quarks is taken as: ~q

=

C ~i + + C 210

i

+ visij + vcij + Konfij)

(~0GEij

(1)

i<j

with

The two-body potential term consists of the one-gluon-exchange potential, VOGE,the one-PS-meson-exchange potential, Vps, the one-a-mesonexchange potential, V,, and the confinement potential, Vconf:

264 Table 1. Five parameter sets. Each parameter set is denoted by Ra, or Rgn, etc., according to the kinetic energy term and the qq interaction. Each meson mass is taken to be the observed one, and mrr= 675 MeV. Model ID

qq int.

mu m, as A, &/4n (90/ge)2 Ao n aconf Vo [MeV] [MeV] [h-’1 [hi’] [MeV/h] [MeV]

R U 7) Rat Rgn OGE T u 77 7’ N g d OGEn u q Ng OGE Grad n q q’

313 340 313 313 340

530 0 560 0.35 550 0.35 680 1.72 500 0

3 5 3 -

0.69 0 0.69 1 0.592 0 0 0.67 1.34

1.81 0.92 170 1.81 0.92 172.4 2.87 0.81 172.4 - - 172.4 2.87 0.81 172.4

-378.3 -381.7 -453 -345.5 -416

+Ref. 9, *Ref. 8, and §Ref. 7. Table 2. S-wave and P-wave quark pairs

T a P 7 6

0 O 1 1

S

C

(-)‘

0 1 0 1

3 6 6 3

+

+ + +

6 P 9 d

T

S

C

(-)‘

0 o 1 l

0 1 0 1

6 3 3 6

-

In V~JGE,as is the OGE strength, and A, is the gluon form factor. In Vps, g is the quark-meson coupling constant: g = 98 for T , K, and 7,and g = go far q’ meson. The value of 9 3 can be obtained from the observed nucleon-pion coupling constant, g?rNN.7’8 The term proportional to A; is originally the &function term; the form factor for the meson exchange, A,, is assumed to depend on the meson mass m, as A, = A0 K m,.7’899 As for the confinement potential for the five quark system, we replace the factor ( X i . X j ) in eq. (6) by its average value as

+

This modified confinement potential gives the same value for the orbital ( O S ) ~state as that of the original one. This replacement enables us to remove all the scattering states and to investigate only tightly bound states, which will appear as narrow peaks. After the coupling to the scattering states with an original confinement potential is introduced, some of the states we find will melt away into the continuum. The situation can be clarified by evaluating the width, which we will investigate elsewhere. We have employed four kinds of parameter sets: RT, RgT, Ngr and Ng (Table 1). R stands for the parameter sets with the semirelativistic kinetic energy term, while N stands for those with the nonrelativistic one. The parameter sets with OBE [OGE] are denoted by the name with 7r [g]. We also perform the calculation with the parameter set given by Graz group.7

265

The wave function we employ is written as:

C CGX‘A

$TSL (CAY e B , 7, R) =

4

4

i,j,n,m,w,w’,X

4q2(w,eA;ui)4q2(w’,eB;uj)$(k%vn)lL XdR;wm)

(8)

7

where d q 4 is the antisymmetrization operator over the four ud-quarks, and

eA,eB,q and R are the internal coordinates defined as:
+ r2 - 7-3 - r4)/2 ,

R = (r1+ T Z

e;

(9)

r3

r2

+ + r4)/4 - rs . r3

(10)

(11)

(w, u ) is the wave function for two quarks with quantum number w , which is one of a,p, . . , 8 listed in Table 2, with size parameter u: cPqz

-

The relative wave function between two qq pairs, $(A, q ;v) and the wave function between four quarks and 3 quark, xg(R,w), are taken as:

For the negative-parity pentaquarks (L=O), we usc‘ dl possible C=C’=J=O states. For the positive-parity pentaquarks (L=l), we use the states xt ,re as one of C C’ and X is equal to 1. The gaussian expansions are geometrical series: ui+l/ui = vn+l/vn = w7 t l / ~ m = 2. We take ti points for u (0.035-1.12 or 0.04-!.28fm), 4 points for v (0.1-O.Sfm), and 3 points for w (0.2-0.8fm). Since we use a variational method, the obtained masses are the upper-limit. They, however, converge rapidly; the mass may reduce more, but probably only by several MeV. I

3. Results and discussions

The mass of the q?j, q3, and q4p systems are shown in Table 3. Parts of these baryon masses were given in refs. 8 and 9. It is very difficult for a constituent quark model to describe the Goldstone bosom. Also, it is hard to justify the models with the kaon-exchange interaction between quarks to describe a kaon. We use the K* mass as a reference for the pentaquarks.

266 Table 3. Masses of meson, baryon, and pentaquark, P ( T J P ) , in MeV. N

A

K* NK*

P(Oi-) rn'p

mp

Rn Rgn Ngn Ng Graz

941 938 936 938 937

1261 1232 1232 1232 1239

979 931 814 814 927

1921 1869 1846 1846 1864

Exp.t 939 1232 892 1831

2109 1985 2029 1966 2231

2054 1947 1996 1971 2160

P(O:-) rn'p

P(la-) m'p

mp

mp

2141 2064 2106 2153 2240

2083 2018 2066 2144 2168

2143 2078 2106 2170 2251

2083 2021 2061 2145 2173

P(Oa+)t rn'p

mp

2165 2129 2321 2345 2248

2045 2006 2144 2197 2120

1540

tRef. 12. $Since the interaction we use is central, Oi+ and 03' are degenerated.

Contrary to the q7j system, we have more satisfactory results for the q3 baryons. The mass spectrum of the S-wave ground states is well reproduced. Moreover, since the chiral quark models have a mechanism to lower the mass of the Roper resonance than that of the negative-parity excited nucleons, the excited baryon mass spectrum can also be reproduced. On the other hand, in the OGE quark model picture, the Roper mass is considered to reduce by introducing the pion-cloud effect, which should be taken into account separately. Though it is interesting to see whether the Roper resonance has a pentaquark component,loJ1 that is out of scope of our present work. Parameter sets Rgn and Ng underestimate the negativeparity excited nucleon mass by about 70 MeV and 90 MeV, respectively. When we discuss the positive-parity pentaquarks by these models, we will have to take this underestimate into account. Now we discuss the system of the pentaquarks. There is no bound state when we use the original confinement, eq. (6). In Table 3, we show the masses of the pentaquarks with K!Lf, mp, and the mass with the correction from the confinement potential evaluated by the wave function corresponding to mp:

In Figure 1, we plot these m p (thin bars) and rn'p (thick bars) for Rgn. Among the q4 S-wave system, five spin-isospin states can couple to the orbital ( O S ) ~configuration: (TS)=(Ol), (lo), (ll), (12), and (21).13 The negative-parity pentaquarks which have a large component of these q4 states correspond to the levels under the dotted line in Figure 1. The mass difference between them and other states is about several hundred MeV. Among them, the ( T S ) = (01) and (10) states are the lowest two states, which are essentially degenerated: ie., (TS)Jp=(Ol)$-, (Ol)g-, and (lo);-. As OGE becomes stronger, (01);- goes down. For example,

267

-L-L

-L

- L - L -L

(2S)p=1- 3- 5T=O

1- 3- 5T=l

1- 3- 5T=2

1' 3' 5' T=O

1' 3' 5' T=l

1' 3' 5' T=2

Figure 1. Pentaquark mass spectrum obtained by the Rg?r parameter set.

the splitt,ing between (01)i- and $- is 71 MeV in Rgn whereas it is 32 MeV in Rn, or 9 MeV in Graz. It becomes 174 MeV in Ng, where all the hyperfine splitting comes from OGE. The remaining two levels, however, still stay close to each other. There is not such a large separation in the mass spectrum of the positive parity pentaquarks, reflecting the fact that all of the spin-isospin states can couple to the orbital (OS)~@ state. The interaction, however, makes one of the states very low: ie., (00);' (and because the interaction is central in the present work). It can actually be as low as the negative-parity states. The masses are still much higher than the observed peak, and depend on the parameters. There is a large ambiguity in the zero-point energy14 as well as in the interaction. For example, we should include the instanton induced interaction,15 which is the source of the 77-77' mass difference. It seems, however, that the relative positions of the levels do not change much. We argue that one of the above mentioned levels is observed as the peak. Because the TJP=O$- and 18- states couple to the NK state strongly and 1;- couples to AK, it is unlikely that they appear as narrow peaks. It seem that 0;- is a good candidate for the observed peak.14 Unfortunately, which of the above 0;- and 04' states is most likely seen is still an open question. The 0;- is lower than the other in Ngn and Ng, and also in the Rgn parameter set if its underestimate of the P-wave baryon mass is taking into account. On the other hand, the 04' state is lower than the other in the semirelativistic chiral models: Rn and Graz. In all the cases, however, the mass difference between these two states is not large. Except for the confinement force, all the interaction terms are shortranged in the quark model. Thus, when the two-quark correlation is intro-

4'

268 Table 4. Values of ( U F S )for each q4 state.

Parity

+

(00) (11) (01) ,(lo)

-30 (-30 ) -22 (-22) ) (8) -10 (-16 ) ) l2 (6)

2178 2342 2465

) 164 ) 123

Table 5. Size of the quark pairs of each (T2S2) spin-isospin state in the TS)Jp=(Ol)i- and (OO)%+ pentaquarks as well as that in the nucleon.

(T25‘2)pair (RT)

(T2S2)pair (Rgn)

(TS)JP

(00)

(01)

(10)

(11)

(00)

(01)

(10)

(11)

(01)1/2(00)1/2+ N

0.53 0.56 0.50

0.70 0.69 0.65

0.68 0.69 0.65

0.62 0.61 0.56

0.56 0.61 0.55

0.69 0.74 0.76

0.68 0.74 0.76

0.64 0.68 0.62

duced in bhe model, quark pairs where the interaction is attractive come closer while those with repulsion tend to stay apart from each other. Then an attractive pair may behave like a single particle, called a diquark. In the chiral quark model with a simple gaussian wave function, the matrix element of the spin-isospin operator, ( O F S )is , proportional to the hyperfine splitting (Table 4): (OFS) = -([f]TSl x ( T i * 7j)(gi ’ gj)l[f]TS) * i<j

(16)

On the other hand, suppose one takes a diquark-model picture, only the pairs between which the interaction is attractive have to be considered. The expectation values of ( O F S )in this picture are listed in the parentheses in Table 4 alongside of the original matrix elements. The ‘mass difference’ between these states is also shown in the column under ‘diff’. The ratio of the mass differences of the positive-parity states is 8/12 in the shell-model picture while it is 8/6 in the diquark-like picture. The ratio obtained from the averaged masses by our full calculation is found to support the diquarklike picture. The qq correlation plays an important role in the pentaquarks. More direct approach to see the importance of the qq correlation is to look into the size of the quark pairs, In Table 5 we show the size of quark pair for each (T2S2)state in the lowest pentaquarks. Their size is large when the interaction is repulsive while it becomes small for the attractive pairs. The ratio is about 1.2 - 1.3. The degree of the qq correlation in the pentaquarks seems similar to that in the nucleon.

m.

269

4. Summary

We have investigated uuddS pentaquarks by employing quark models with the meson exchange and the effective gluon exchange as qq and qq interactions. The system for five quarks is dynamically solved; two quarks are allowed to have a diquark-like qq correlation. The present work indicates that the TJP=O$-, 0;' pentaquark states can be almost as low as the 0;- state, which has been assigned to the observed peak, but expected to have a large width. Both of the 0;- and 0;' states are considered to have a narrow width. Which parity should correspond to the observed peak is still an open question. Relative distance of two quarks with the attractive interaction is found to be by about 1.2 - 1.3 times closer than that of the repulsive one. The diquark-like quark correlation seems to play an important role in the pentaquark systems. The 1;- state is also found to be low. Like the 0;- state, however, it couples to the NK states strongly. This may be the reason why there is no peak in the T=l channel. As for the absolute mass, our estimate is still more than 400 MeV higher than the observed one. We consider the extra attraction may come from other qq interactions as well as from the ambiguous zero-point energy. The width and the resonant energy should be investigated by including the coupling to the baryon meson asymptotic states, which is underway. This work is supported in part by a Grant-in-Aid for Scientific Research from JSPS (No. 15540289). References 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13.

14. 15.

T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91,012002 (2003). M. Oka, Prog. Theor. Phys. 112, 1 (2004), and references therein. A. Hosaka, M. Oka, and T. Shinozaki, hep-ph/0409102. S. Sasaki et al, in this issue; F. Lee et al, in this issue. M. Oka et al, in this issue. E. Hiyama et al, in this issue. L. Ya. Glozman et al, Phys. Rev. D58, 094030 (1998). M. Furuichi and K. Shimizu, Phys. Rev. C65, 025201 (2002). M. Furuichi, K. Shimizu, and S. Takeuchi, Phys. Rev. C68,034001 (2003). R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91,232003 (2003). M. Karliner and H. J. Lipkin, Phys. Lett. B 575, 249 (2003). S. Eidelman et al, Phys. Lett. B 592, 1 (2004); http://pdg.lbl.gov/ K. Shimizu and S. Takeuchi, in preparation. S. Takeuchi and K. Shimizu, hepph/0410286. T. Shinozaki et al, in this issue; hepph/0409103.

270

CONTRIBUTION OF INSTANTON INDUCED INTERACTION FOR PENTA-QUARKS IN MIT BAG MODEL

TETSUYA SHINOZAKI AND MAKOTO OKA Department of Physics, H-27, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan E-mail: [email protected] SACHIKO TAKEUCHI Japan College of Social Work, Kiyose 204-8555, Japan Roles of instanton induced interactions (111) in the masses of pentaquark baryons, O+ ( J = 1/2 and 3/2) and S--, and a dibaryon, H , are discussed using the M I T bag model. It is shown that the two-body terms in I11 give a strong attraction mainly due to the increase of the number of pairs in multi-quark systems. In contrast, the three-body u-d-s interaction is repulsive. It is found that 111 lowers the mass of negative-parity O+ as much as 100 MeV from the mass predicted by the bag model without 111.

Reports of discoveries’ of exotic baryons started intensive discussions on various possibilities of bound pentaquark states. The most essential properties are spin and parity. Predictions by the chiral soliton model2 and various quark m0dels~9~2~ claimed a 1/2+ state of Q+, while constituent quark model with all the five quarks sitting in the lowest energy level predicts a negative parity ground state, 1/2-. Furthermore, majority of QCD-based calculations, such as QCD sum rule6 and lattice QCD7, indicate that the positive parity state has a higher mass. The constituent quark model has several important dynamical ingredients, among which we here consider confinement, perturbative one-gluon exchange (OgE) interaction and nonperturbative instanton induced interaction (111). Interesting roles of I11 in the baryon spectrum were The purpose of this paper is to clarify roles of I11 in the pentaquark systems in the context of the MIT bag model. In the naive bag model, the “vacuum” inside the bag is identified as the perturbative vacuum. There-

271

fore we consider a situation that the vacuum structure is minimally modified to reproduce appropriate spectrum of the pseudoscalar mesons. On the other hand, it is expected that the confinement mechanism that successfully describes three-quark baryon states is common to the pentaquark baryons. Also there are some advantages of the bag model. The quarks inside the baryon are treated relativistically. The size of the baryons can be automatically determined from pressure balance at the bag surface. The instanton induced interaction (111), introduced by 't Hooftl0, is an interaction among quarks of N f (= 3) light flavors, The main difference from the perturbative gluon-exchange interactions is that 111is not chirally invariant and applies only on flavor singlet states of quarks. The interaction is written as a contact interaction''.

x $'L, k' (3)'$L , j (2)$'L ,i (1) f (h.c.)

1 5 where E i j k is totally asymmetric tensor, and i, j and k represent flavor. R and L are chiral indices. H ( 3 ) is the 3-body Hamiltonian and H ( 2 ) is the 2-body Hamiltonian obtained by contracting a quark pair in the 3-body 111 into a quark condensate: GfJ = 7 9 $ G ( 3 ) , GLt) = mzff/mzffGfJ. where meff is the constituent mass of the quark. We use mzff/mzff N 0.6. The (iiu)N (-225MeV)3 is the quark condensate. The 3-body interaction is repulsive, while the 2-body interaction H ( 2 ) is attractive because the quark condensate is negative. Some previous works consider I11 in the context of diquark models of the pentaquark baryons5. The mass of a hadron in the MIT Bag Model12 is given by

H ( 2 ) = G(2)EijEitjf$ ~ , i ( l ) $ ~ , j ( 2 ) ( 1-c1 . 02)$'~,jt(2)$'~,i,(1) f (h.c.),

M ( R ) = n,w(mu, R)

+ n,w(m,, R ) + 4x13 BR3 - Zo/R

+ (1- PIII) C(6.q( 2 .x J M ~ ~ ( R ) i>j

+ PIII(H(~)(R) + Id2'(R))+ Eo, where R is the bag radius, The fifth term is the color-magnetic part from OgE. PIIIis a parameter which represents the portion of the hyperfine splitting induced by 111. If PIII = 0, the mass splitting of N - A comes purely from OgE, while for PIII = 1 it comes purely from 111. The way of determining PIIIis to reproduce the 77 - q' mass difference. In the

272

case of the MIT Bag modell,, We estimated around PIII = 0.3. Eo is introduced to reproduce the mass of the nucleon. It is given roughly by Eo = 150MeV x P I I I . The Eo can be taken into account by changing 20 and B accordingly, but here we remain t o fix 20 and B. The other parameters of the bag model are taken from the original MIT bag modell,. We consider the pentaquarks O+ composed of uudds with isospin 0, spin 112 and negative parity, and E - , a partner within the flavor f0 with isospin 312. We also consider O$=,,,, which is the spin partner of O+.

1.3 1.2 1.I 1

0.9 0.8

0.7

’

0

0.2

0.4

I

1

I

0.6

0.8

1

Plll

Figure 1. The masses (left) and the bag radii (right).

In Fig. 1, we show the masses of the pentaquarks as functions of PIII. The dashed lines are the values at PIII = 0, which correspond to the masses under the influence only of OgE. The right end, PIII = 1, gives the masses when the N - A splitting is purely due t o 111. We point out that the pure OgE lowers the masses of O+ from the noninteracting 5 quark state. One sees that the O+ is affected by I11 most strongly among these states. At close to the PIII=~, the mass of O+ is lOOMeV smaller than that at PIII = 0. In contrast, O$=,,, changes significantly in all PIII. But the mass of S- is almost constant. The mass of the H dibaryon grows monotonically as PIII increases. It is found that the mass of O+ does not agree with the experimental value (1540MeV) even if the full I11 is introduced. On the other hand, the model reproduces the mass of E - . The contribution of the 3 body I11 is roughly 10% of that of the 2 body I11 for the pentaquarks. For the O$=,,,, the contribution of OgE is very small. Thus effects of I11 are most easily seen in @$=,,. In Fig. 1, the radii of the considered baryons are given. They show that O+ shrinks as PIII increases. At the realistic region , PIII = 0.3, the radii

273

of pentaquarks are about 0% 20% larger than the radius of the nucleon, 5 GeV-l. We find that the strongly attractive force of I11 makes the bag radii shrink. In fact, the radii of the pentaquarks are as small as the radius of the 3-quark baryons. We conclude that the effects of I11 have been studied using the MIT bag model in the negative parity case. We have found that I11 lowers the mass of O+ and Oi=,,,, while the mass of H increases as the strength of I11 increases. The present results can not reproduce the observed O+ mass. Possible resolutions are corrections from expected two-body (diquark type) correlations, pionic effects, which may be included in chiral bag models, and also couplings to background N K scattering states. If these effects are important, the pentaquark spectrum may be well modified. Despite these defects, the current study is worthwhile because using the simplest possible picture of the hadron, we demonstrate how large and important are the effects of instantons on the spectrum of pentaquarks. Further analysis including the above-mentioned corrections are to be performed as the next step. N

Acknowledgments

This work is supported in part by the Grant for Scientific Research (B)No.15340072, (C)No.16540236 and (C)No.15540289 from the Ministry of Education, Culture, Sports, Science and Technology, Japan. T. S. is supported by a 21st Century COE Program at Tokyo Tech "Nanometer-Scale Quantum Physics" by the Ministry of Education, Culture, Sports, Science and Technology. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

T. Nakano e t al. [LEPS Collaboration], Phys. Rev. Lett. 91,012002 (2003) D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359,305 (1997) M. Karliner and H. J. Lipkin, Phys. Lett. B 575,249 (2003) R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91,232003 (2003) N. I. Kochelev, H. J. Lee and V. Vento, Phys. Lett. B 594,87 (2004) J. Sugiyama, T . Doi and M. Oka, Phys. Lett. B 581, 167 (2004) N. Ishii et al. arXiv:hep-lat/0408030. E. V. Shuryak and J. L. Rosner, Phys. Lett. B 218,72 (1989). M. Oka and S. Takeuchi, Phys. Rev. Lett. 63,1780 (1989). G. 't Hooft, Phys. Rev. D 14,3432 (1976) M. Oka and S. Takeuchi, Nucl. Phys. A 524,649 (1991). T. DeGrand et al. Phys. Rev. D 12,2060 (1975). S. Takeuchi and M. Oka, Nucl. Phys. A 547,283C (1992).

274

FIVE-BODY CALCULATION OF RESONANCE AND CONTINUUM STATES OF PENTAQUARK BARYONS WITH QUARK-QUARK CORRELATION

E. HIYAMA Department of Physics, Nam Women’s University, Nam 630-8506, Japan M. KAMIMURA Department of Physics, Kyushu University, fikuoka, 812-8185, Japan A. HOSAKA A N D H. TOKI Research Center for Nuclear Physics (RCNP), Osaka University, Ibamki, 567-0047, Japan M. YAHIRO Department of Physics, Kyushu University, f i h o k a , 812-8185, Japan Five-body calculation for Q+ has been performed in the framework of a variational method employing Jacobi-coordinate Gaussian-basis function. The difficult calculation taking N K scattering channel explicitly was performed for the first time. We investigated whether there were resonance states or not in J = 1/2+ and J = 112- states.

+

1. Introduction

Recently, pentaquark, O+, was observed by the LEPS group at Spring8 l. This observation is giving a great impact to hadron physics. At the same time, this observation give new important issues theoretically: (1) Can we explain the observed mass of O+ at the low energy? (2)Can we explain the observed narrow decay width? (3) What is the spin parity, 1/2+(T= 0) or 1/2-(T = O)? (4) What is spatial structure of O+ Namely, what type of configuration is dominantly in O+ ? For this purpose, our strategy is as follows: [A]As for q- q and q - ij interaction, we employ potentials which can well explain important observed quantities of baryons. [B] As for model space of five-quark system, we em-

275

ploy precise five-body basis functions that are appropriate for describing the q - q and q - q correlations and for obtaining energies of 5-quark states accurately. We finally solve the five-body problem under the scattering boundary condition for the N + K channel in order to examine whether the O+ are resonance state or non-resonance continuum state. So far in the literature calculations this N + K channel was neglected due to very difficulty of this calculation. This difficult calculation is performed for the first time and is shown that by Gaussian Expansion Method five-body calculation including N K channel explicitly is very important. 2*394,

5v61778

+

2. Model and Interaction

2.1. Interaction

We use a two-body harmonic oscillator potential as the confiment force. The effective Hamiltonian is given by

H =

c +)' (mi

2mi

-TG

+ vc +vs + vi),

i

in which we have

and

vs

cmimj css

= i<j

[

exp - (Xi -

/P2]

(1 - ui

Uj)/4,

(3)

where Vc is the spin-independent confinement potential and Vs is the spindependent quark-quark correlation that acts only on the spin-0 pair. Here, mi, xi and pi are the mass, position and momentum of the i-th constituent quark. TGis the c.m. kinetic energy, and Vo is a constant parameter, which contributes to the overall shift of the resulting spectrum and is chosen to adjust the energy of the lowest state to the nucleon mass. K , Css and ,8 are the model parameters, which are taken to be the same for all the baryons considered. The quark masses are taken to be mu = md = 330 MeV and mB= 500 MeV. In this work, we do not employ tensor forces or spin-orbit forces between quarks, since they are not effective in the baryon ground states, where the s-wave component is dominant. We fix the model parameter K , P, Css/me and fi so as to reproduce the masses of nucleon and A, and the charge radius of proton. The experimentally measured proton charge radius included contributions from both

276

the valence quark core part and its meson clouds. It is reasonable to subtract the vector meson dominance contribution from the data for the proton electric charge radius (0.86f1n)~to obtain the valence quark core radius < r: >core. From this analysis, we extract < >coreN (0.6fm2). We obtain K = 0.007 GeV3(180 MeV fm-2), p = 0.55 fm, Css/mt = -1.092GeV and VO = -515 MeV, which give m ( N ) = 939 MeV, m ( A ) = 1232 MeV and = (0.60fm)2. After this determination, there remain no adjustable parameters in our calculation. The use of qq and qQinteractions reproduce the following observed quantities well: (1) the masses of baryons, (2) masses of mesons (see Fig.(l)), (3) magnetic moment and (4)non-leptonic weak decay matrix. The details are written in Ref. [9].

<

1000 K*

800 -

600 -

K

-----"---400 Figure 1. Energy levels of mesons

2.2. Method

Using this Hamiltonian, we solve the non-relativistic five-body problem. For this purpose, we adopt the Gaussian expansion method (GEM) for fewbody systems, which has been developed by the two of the present authors (E. H. and M. K.) and their collaborators Following to the GEM we consider four rearrangement Jacobian coordinates, as shown in Fig.2. 51697,8.

576979s,

277

C=l

c=3

c=2

c=4

Figure 2. Rearrangement Jacobian coordinates for Q+

We first construct five-body basis functions for the color, isospin, spin and spatial parts of the channel. (c=l) Q J M ( Q Q Q 4 4 = @JM

(c=3) + @ J(c=2) M + @JM

+

@$CC4)

(4)

Each channel amplitude is expanded in terms of basis functions for each Jacobian coordinates:

278

we take the functional form of &lm(r), $ k j i ( s ) X N L M ( and R ) &,xp(p) as

h m ( r )= r1 e-

(w~~m , (q

Juxp(p)= px e-(plpy)2Yx J 3 ) +k

9

.i ( ~= ) s k e - ( s / s i ) a yrCj(S) ,

J7

XNLM(R) = RL e - ( R l R N ) 2 Y L M ( 6 ) ,

(8) where the Gaussian range parameters are chosen to lie in geometrical progressions:

r,

= r#-1

.(

pv = p 1 a U - 1

(V'

s. 2 -

'i(

1 - nmax) , 1 - ),v .

=

1 - imax).

RN = RIAN-l ( N = 1 - ~ m a x ,) (9) These basis functions have been verified to be suitable for describing both short-range correlations and the long-range tail behavior of few-body systems The eigenenergy is determined by the Rayleigh-Ritz variational method. 576t798.

3. Results and discussion

In Fig.3, the calculated energy levels in J = 1/2+ and 1/2+ taking only C = 2 channel are shown. We see that the mass of J = 1/2+ is lower than that of the J = 1/2- state. But, the energies of the both states are much higher than the observed mass of O+. Next, we show the masses of J = 1/2+ and J = 1/2- states further including C = 3 and C = 4 channels but omitting the C = 1, namely, N + K scattering channel. We obtained many bound states. We see that the lowest state is J = 1/2-, which is close to the observed state. And, we solve fivebody system taking C = 1 and 2 channels. Then, we obtained very narrow resonance states at 558 MeV and 951 MeV in J = 1/2+ and 1/2- states. The narrow resonance appears due to the small probability of transition from the C = 2 channel configuration to the N K scattering states. Next, we solve this five-body system taking all of four channels. As shown in Fig.4, we see that there are several discerete states near experimental energy region. It is necessary to investigate that these states resonance states or non-resonance continuum states. The useful method to examine it is the real scaling method l o . In this method, we artificially scal the range parameters of Gaussian basis functions

+

279 (b) E(MeV)

t

A

1358

1300-

1R+

1/2-

1134 1OOO-

500

-

943 951 550 558

260

448

144

Figure 3. (a) Calculated energy levels taking C = 2 channel (b)Calculated energy levels taking C = 2,3 and 4 channels

E(MeV)

112-

I I

Figure 4.

Calculated energy levels in J = 1/2+ and 112- taking C = 1 N 4 channels

by multiplying a factor a. We then obtain eigenvalue with respect to a converged into the threshold in the case of non-resonance continuum states.

280

On the other hand, the energy of the real resonance is almost constant with respect to the scaling parameter a. We use this method for the five-quark system. As shown in Fig.5, in J = 1/2- state, we see that all states behave as non-resonance continuum states. Namely, there is no resonance state on J = 1/2-. The same tendency is seen in the case of J = 1/2+.

EXP.

1

a

2

1.5

Figure 5. eigenvalues in J = 1/2- state with respect to scaling parameter a

4. Conclusion

Using the interactions which explain the two-body and three-body systems so well, the five-body calculation of O+ was performed. In the case of omitting C = 1 channel, namely, K N scattering channel, we obtained several bound states close to the observed energy. However, solving the five-body problem, under the scattering boundary condition for the K N scattering channel, we understand that all states in J = 1/2+(T = 0) and J = 1/2-(T = 0) became to be non-resonance continuum states. The possibility of another spin-parity states such as J = 3/2+(T = 0),3/2-(T = 0),5/2+(T = 0),5/2-(T = 0),1/2+(T = l),1/2-(T = l),3/2-(T = 1),3/2+(T= l),5/2+(T = 1) and 5/2-(T = 1) as resonance states. If we cannot find any resonance state close to the observed data, it might give a chance to provide with any new model including the N K scattering channel explicitly. For this purpose, we expect new many systematic data about five-body quark systems in the future.

+

+

+

281

References LEPS Collaboration, T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003). '1'. Kanada-En'yo, 0. Morimatsu, T. Nishikawa, hepph/0404144 J. Carlson and R. Pandaharipande, Phys. Rev. D43, 1652 (1991). R. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003). E. Hiyama, Y. Kin0 and M. Kamimura, Prog. Part. Nucl. Phys. 51,223 (2003). M. Kamimura, Phys. Rev. A38, 721 (1988). H. Kameyama, M. Kamimura and Y . Fukushima, Phys. Rev. C40,974 (1989). E. Hiyama and M. Kamimura, Nucl. Phys. A588, 35c (1995). E. Hiyama, K. Suzuki, H. Toki, M. Kamimura, Prog. Theor. Phys. 112, 99 (2004). 10. J. Simons, J. Chem. Phys. 75, 2465 (1981).

1. 2. 3. 4. 5. 6. 7. 8. 9.

282

FLAVOR STRUCTURE OF PENTAQUARK BARYONS IN QUARK MODEL

YONGSEOK OH Department of Physics and Astronomy, University of Georgia Athens, Georgia 30602, USA E-mail: [email protected] HUNGCHONG KIM Institute of Physics and Applied Physics, Yonsei Unaversity Seoul 120-749,Korea E-mail: [email protected]

m,

The flavor SU(3) group structure of pentaquark baryons which form 1,8 , 10, 27, and 35 multiplets is investigated in quark model. The flavor wave functions of all the pentaquark baryons are constructed in SU(3) quark model and their Yukawa interactions with meson octet am obtained in general and in the special case of the octet-antidecuplet ideal mixing with the OZI rule. The ma89 sum rules of pentaquark baryons are also discussed.

1. Introduction Great interests in exotic baryons in hadron physics have been initiated by the discovery of 0+(1540) state by the LEPS Collaboration and subsequent The observation of Z--(1862) by NA49 Collaboration experiments .'*I may suggest that S(1862) forms pentaquark antidecuplet with 0+(1540) as anticipated by the soliton model study '. Later, the H1 Collaboration reported the existence of anti-charmed pentaquark state 5 , which revives the interests in the heavy pentaquark system However, the existence of pentaquark baryons are not fully confirmed by experiments yet as some high energy experiments report null results for those states 9910. A summary for the experimental situation and perspectives can be found, e.g., in Refs. 11. Theoretically, many ideas have been suggested and developed to study the exotic pentaquark states in various approaches and models "913, but more detailed studies are required to understand the properties and formation of pentaquark states. 69778.

283

As the pentaquark baryons may be produced in photon-hadron or hadron-hadron reactions, it is important to understand their production mechanisms and decay channels in order to confirm the existence of the pentaquark states and to study their properties. The present studies on the production reactions are limited by the lack of experimental and pheIn particular, those nomenological inputs on some couplings studies could not include the contributions from the intermediate pentaquark states in production mechanisms. Therefore, it is strongly desired to understand the interactions of pentaquark baryons with other hadrons. On the other hand, many theoretical speculations suggest that the physical pentaquark states would be mixtures of various multiplets 19%20,21. Thus it is necessary to construct the wavefunctions of pentaquark baryons in terms of quark and antiquark for understanding the structure of pentaquark states. In this talk, we discuss a way to construct the wavefunctions of all the pentaquark baryons in quark model and obtain their SU(3) symmetric interactions with other baryons. Then several mass relations among the pentaquark baryons are discussed. In addition, we explore the couplings in the special case of the antidecuplet-octet ideal mixing with the OZI rule. The topics presented here are discussed in more detail in Refs. 22, 23, 24. 14915*16917~18.

2. Wavefunctions and interactions of pentaquark baryons We start with the representations for quark and antiquark. We denote a quark by qi and an antiquark by qi with i = 1,2,3, so that 91, 9 2 , and 413 are u, d, and s quark, respectively. The inner products of the quark and antiquark operators are normalized as (qi,9’) = hi’, Then the diquark state is decomposed as (qi,q j ) = h i j ,

(qi,q j ) = 0.

(1)

where

and

Ti = $kAjk,

(4)

so that s j k and T irepresent 6 and 3, respectively. This shows that 3@3 = 6~3%.

284

The product of two diquarks can be written as

aa

Here, Ta, Qa, and are (1,O) type and represent 3. Taj and S i j are (0,2) type Of 8. A150 Tjk, ?;&, and sjk are (2,1) type Of 15 and Tijkl are (4,o) type of 15. Their explicit forms can be found in Ref. 24. Then it is straightforward to obtain the wavefunctions of pentaquark baryons. Since the product of two diquarks can form 3, 8, and 15, the pentaquark states can have 1 , 8 , 1 0 , m , 27, and 35, while the normal threequark baryons have 8 and 10. So the pentaquark states have much richer spectrum than the normal baryons. Their flavor wavefunctionsare obtained by direct product of (qjq&)(qlqm)and an antiquark p. For example, the 35-plet tensors TGkl and the 27-plet tensors TZ can be constructed as TGkl

= Tajkld -

s1 ( d r T j k 6 m r + d ; T i k l m r + d g T i j l m T + d f T i j k m r )

7

285

The pentaquark octet Pj and antidecuplet Tijk read

Therefore, by constructing all possible pentaquark tensors we can verify 3 €3 3 €3 3 €3 3 8 % = 35 @ (3)27 @ (2)10@ (4)lO @ (8)8 @ (3)1,

(9) where the numbers in parentheses are the number of multiplicity. The inner products of the multiplets are given in Ref. 24. With those informations at hand, one can identify the tensor representations with the baryon states of definite isospin and hypercharge. (See Ref. 24 for details.) The SU(3) symmetric Yukawa interactions of pentaquarks can be constructed by fully contracting the upper and lower indices of the three tensors representing two baryon multiplets and the meson octet. When the number of upper indices does not match that of lower indices, the LeviCivita tensors E i j k are introduced to make the interactions fully contracted. The SU(3) symmetric interactions constructed in this way give several constraints or selection rules to the pentaquaxk interactions, which should be useful to identify the pentaquark states. Since 8 @ 8 = 27@10 @ 1O@ 81 83 8 2 @ 1, 10 €3 8 = 35 @ 27 @ 10 @ 8,

l o @8 = =@

27@10@8,

2 7 8 8 = 64 @ 35 @=@

271 @ 272 @ 10@1O@8, 35 €3 8 = 81 @ 64 @ 351 @ 352 @ 28 @ 2 7 @10,

(10)

286

we find the followings. First, the pentaquaxk singlet can couple to pentaquark octet only. Second, the 27-27 and 35-35 interactions have two types (f and d types) like 8-8 interaction. Third, the interactions including 10-10,35-8, and 35-10 are not allowed as they cannot form SU(3)invariant interactions. Thus, 35-plet couplings are limited to the interactions with 27-plet and decuplet. We refer to Refs. 22, 24 for the explicit relations for pentaquark interactions. The SU(3) symmetry breaking terms can be included in a standard way 23,25926.

3. Mass sum rules Since all the particles belonging to an irreducible representation of SU(3) are degenerate in the SU(3) symmetry limit, it is required to include SU(3) symmetry breaking to obtain the mass splitting. It is well-known that the Hamiltonian which breaks SU(3) symmetry but still preserves the isospin symmetry and hypercharge is proportional to the Gell-Mann matrix As, from which we introduce the hypercharge tensor as Y = diag(l,l,-2). Then the baryon masses can be obtained by constructing all possible contractions among irreducible tensors and the hypercharge tensor. As the mass formulas contain several parameters which take different values depending on the multiplet in general, we can obtain only the mass relations. The Hamiltonian constructed in this way reads

where a, b, and c are mass parameters. Then, in addition to the well-known Gell-Mann-Okubo mass relation for the baryon octet and the decuplet equal-spacing rule, we have some interesting mass sum rules for antidecuplet, 27-plet, and 35-plet. In antidecuplet, we have the equal spacing rule *,

-

2m,3/2 -

cm = Em - Ni7;j = Nm - 0.

(12)

In the 27-plet, we find the analog of the Gell-Mann-Okubo mass relation,

2(N27

+ E27) = 3A27 + c27.

(13)

287

In addition, we find that some of the 27-plet members, i.e., 0 1 , A 2 7 , S 2 7 , 3 / 2 , and Cl27,1, satisfy two independent equal-spacing rules,

c27,2,

- s 2 7 , 3 / 2 = s 2 7 , 3 / 2 - c27,2r c27,2 - A27 = A27 - 0 1*

R27,l

(14)

Note that they are the states with maximum isospin for a given hypercharge and the equal-spacing rule holds independently for the upper half of the 27plet weight diagram and for the lower half of that weight diagram 24. For the 35-plet baryons, we observe that there are two sets of baryons which satisfy the equal-spacing rule separately 24, namely,

- E35 = s 3 5 - E 3 5 = C35 - A 3 5 = A 3 5 - 0 2 , x - %5,1 = Q35,l - %5,3/2 = %5,3/2 - C35,2 = c 3 5 . 2 - &/2035

(15)

4. Ideal mixing of antidecuplet and octet with the OZI rule

In the diquark-diquark-antiquarkmodel for pentaquarks, Jaffe and Wilczek advocated the ideal mixing of the antidecuplet with the octet 19. By referring the detailed discussion on the ideal mixing and the OZI rule to Ref. 23,here we discuss the consequence of the OZI rule in the interactions of pentaquark octet and antidecuplet. As can be seen from Eq. (lo), the pentaquark octet interaction with normal baryon octet and meson octet has two couplings, f and d. A relation between the two couplings can be found by imposing the OZI rule or the fall-apart mechanism 20. To see this, we go back to Eq. (5) and note that the pentaquark octet and antidecuplet - come together from the 8 of two diquarks and 5 of one antiquark, i.e., 6 8 3 = 10CB 8. This follows from sij

8 q k = Tijk

~[ij,k].

(16) Obviously, the last part, being an octet representation, can be replaced by a two-index field P/ such as s[ij,k] =€ljkpi lib j 1 +e Pl. (17) In this scheme, the pentaquark antidecuplet and pentaquark octet have the same universal coupling constant. It is now clear to see that the index k in Eq. (17),the index for the antiquark, should be contracted with the antiquark index of the meson field to represent the fall-apart mechanism or the OZI rule, as the usual baryon B does not contain an antiquark in the OZI limit. Hence, the interaction should follow the form as

288

Substituting Eq. (17) into Eq. (18),one has

Comparison with the standard expression for the octet baryon interactions off and d types leads to f = 1/2 and d = 3/2. Therefore, one can find that the 021 rule makes a special choice on the f / d ratio as f / d = 1/3 20y23.

5. Summary We have obtained the flavor wavefunctions of all the pentaquark baryons in quark model. Then the SU(3) symmetric interactions of the pentaquark baryons as well as their mass sum rules are derived. This will help to identify not only exotic baryons but also crypto-exotic states. At this stage, we notice that there are several recent reports about the existence of cryptoexotic pentaquark states 27 , whose existence, however, should be clarified by further experiments 28. Acknowledgments

We are grateful to Su Houng Lee for useful discussions. This work was supported in part by Forschungszentrum-Juilich, contract No. 41445282 (COSY-058) and the Brain Korea 21 project of Korean Ministry of Education. References 1. LEPS Collaboration, T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003). 2. DIANA Collaboration, V. V. Barmin et al., Phys. At. Nucl. 66,1715 (2003); CLAS Collaboration, S. Stepanyan et al., Phys. Rev. Lett. 91, 252001 (2003); SAPHIR Collaboration, J. Barth et al., Phys. Lett. B 572, 127 (2003); CLAS Collaboration, V. Kubarovsky et al., Phys. Rev. Lett. 92, 032001 (2004);

A. E. Asratyan, A. G. Dolgolenko, and M. A. Kubantsev, Phys. At. Nucl. 67, 682 (2004); HERMES Collaboration, A. Airapetian et al., Phys. Lett. B 585, 213 (2004); SVD Collaboration, A. Aleev et al., hep-ex/0401024; COSY-TOF Collaboration, M. Abdel-Bary et al., Phys. Lett. B 595, 127

(2004); P. Zh. Aslanyan, V. N. Emelyanenko, and G. G. Rikhkvitzkaya, hepex/0403044; ZEUS Collaboration, S. Chekanov et al., Phys. Lett. B 591, 7 (2004); Yu. A. Troyan et al., hep-ex/0404003; S. V. Chekanov for the ZEUS Collaboration, hep-ex/0404007. 3. NA49 Collaboration, C. Alt et al., Phys. Rev. Lett. 92, 042003 (2004). 4. D. Diakonov, V. Petrov, and M. Polyakov, Z. Phys. A 359, 305 (1997). 5. H1 Collaboration, A. Aktas et al., hep-ex/0403017.

289

6. H. J. Lipkin, Phys. Lett. B 195, 484 (1987); C. Gignoux, B. Silvestre-Brac, and J. M. Richard, Phys. Lett. B 193, 323 (1987). 7. Y. Oh, B.-Y. Park, and D.-P. Min, Phys. Lett. B 331, 362 (1994); Phys. Rev. D 50, 3350 (1994); Y. Oh and B.-Y. Park, Phys. Rev. D 51, 5016 (1995); M. Genovese, J.-M. Richard, F1. Stancu, and S. Pepin, Phys. Lett. B 425, 171 (1998); F1. Stancu, Phys. Rev. D 58, 111501 (1998). 8. H. Kim, S. H. Lee, and Y. Oh, Phys. Lett. B 595, 293 (2004). 9. BES Collaboration, M. Ablikim et al., Phys. Rev. D 70, 012004 (2004); K. T. Knopfle et al. for the HERA-B Collaboration, J. Phys. G 30, S1363 (2004); HERA-B Collaboration, I. Abt et al., hep-ex/0408048; C. Pinkenburg, nuclex/0404001; SPHINX Collaboration, Yu. M. Antipov et al., hep-ex/0407026; BABAR Collaboration, B. Aubert et al., hep-ex/0408037; hep-ex/0408064. 10. H. G. Fischer and S. Wenig, hep-ex/0401014; STAR Collaboration, S. Kabana et al., hep-ex/0406032; J. Pochodzalla, hep-ex/0406077; WA89 Collaboration, M. I. Adamwich et al., Phys. Rev. C 70, 022201 (2004); I. V. Gorelov for the CDF Collaboration, hep-ex/0408025. 11. V. D. Burkert et al., nucl-ex/0408019; V. Kubarovsky and P. Stoler, hepex/0409025; K. Hicks, hep-ph/0408001; P. Rossi, hep-ex/0409057. 12. B. K. Jennings and K. Maltman, Phys. Rev. D 69, 094020 (2004); C. E. Carlson et al., Phys. Rev. D 70, 037501 (2004); 13. R. L. JaRe, hep-ph/0409065. 14. W. Liu and C. M. KO,Phys. Rev. C 68, 045203 (2003); Nucl. Phys. A 741, 215 (2004); W. Liu et al., Phys. Rev. C 69, 025202 (2004). 15. Y. Oh, H. Kim, and S. H. Lee, Phys. Rev. D 69, 014009 (2004); Phys. Rev. D 69, 074016 (2004); hep-ph/0312229, to be published in Nucl. Phys. A. 16. A. R. Dzierba et al., Phys. Rev. D 69, 051901 (2004); W. Itoberts, nuclth/0408034; Q. Zhao and J. S. Al-Khalili, Phys. Lett. B 585, 91 (2004); 596, 317(E) (2004); B.-G. Yu, T.-K. Choi, and C.-R. Ji, nucl-th/0312075. 17. K. Nakayama and K. Tsushima, Phys. Lett. B 583,269 (2004); K. Nakayama and W. G. Love, Phys. Rev. C 70, 012201 (2004). 18. S. I. Nam, A. Hosaka, and H.-C. Kim, hep-ph/0403009; hep-ph/0405227. 19. R. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003). 20. F. E. Close and J. J. Dudek, Phys. Lett. B 586, 75 (2004). 21. J. Ellis, M. Karliner, and M. Praszalowicz, JHEP 05, 002 (2004). 22. Y. Oh, H. Kim, and S. H. Lee, Phys. Rev. D 69, 094009 (2004). 23. S. H. Lee, H. Kim, and Y. Oh, hep-ph/0402135. 24. Y. Oh and H. Kim, hep-ph/0405010, to be published in Phys. Rev. D 70. 25. S. Pakvasa and M. Suzuki, Phys. Rev. D 70, 036002 (2004). 26. S. M. Golbeck and M. A. Savrov, hep-ph/0406060. 27. L. G. Landsberg, Phys. Rep. 320, 223 (1999); SPHINX Collaboration, Yu. M. Antipov et al., Phys. At. Nucl. 65, 2070 (2002); CB-ELSA Collaboration, V. Cred6 et al., hep-ex/0311045; BES Collaboration, M. Ablikim et al., hepex/0405030; V. Kuznetsov for the GRAAL Collaboration, hep-ex/0409032. 28. P. A. Zohierczuk et al., Phys. Lett. B 597, 131 (2004).

290

PARITY OF THE PENTAQUARK BARYON FROM THE QCD SUM RULE*

SU HOUNG LEE, HUNGCHONG KIM AND YOUNGSHIN KWON Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea E-mail: [email protected]

The QCD sum rule for the pentaquark O+, first analyzed by Sugiyama, Doi and Oh,is reanalyzed with a phenomenological side that explicitly includes the contributions from the two particle reducible kaon-nucleon intermediate state. The magnitude for the overlap of the O+ interpolating current with the kaon-nucleon state is obtained by using soft kaon theorem and a separate sum rule for the ground state nucleon with the pentaquark nucleon interpolating current. It is found that the K-N intermediate state constitutes only 10%of the sum rule so that the original claim that the parity of Q+ is negative remains valid.

1. Introduction The discovery of the O+ baryon by LEPS Collaboration at Spring-8 has spurred a lot of works in the field of exotic hadrons. So far, not much is known about the properties of the Of except its mass, which is about 1540 MeV, and its small decay width, which is smaller than the experimental resolutions of around 10 MeV. The O+ baryon, being a strangeness +1 state, is exotic since its minimal quark content should be uuddi?. Other states that have positive strangeness but different charges are not observed a, which suggests that the O+ is an isosinglet. The existence of such an exotic state with narrow width with spin parity 1/2+ was first predicted by Diakonov et al. in the chiral soliton model, where the O+ is a member of the baryon anti-decuplet. Although the chiral soliton models predict a positive parity for the ground state pentaquark, it is expected in a constituent quark model *This work is supported by the Brain Korea 21 project. aFor the classification of the pentaquarks and their decay modes, see Ref.[2]

291

that the ground state of the pentaquark have a negative parity because all the quarks would be in the s-states. In such a case the expected decay width would be quite large as the quarks in the pentaquark state would easily fall into a kaon and a nucleon state. In a more sophisticated constituent quark model, where chiral dynamics are included, it is possible that the ground state pentaquark have a positive parity4. Hence, determining the parity of the pentaquark states will not only be important in establishing the basic quantum numbers of the pentaquark states, but also in understanding the QCD dynamics especially when multiquarks are involved. Initially, both Lattice QCD5 and QCD sum rule6 analysis showed the existence of a negative parity pentaquark state in the isospin zero and spin 1/2 channel. However, subsequent analysis in the lattice QCD found no stable pentaquark state in the advertised channel8. Similarly, in a different QCD sum rule analysis, the parity was found t o be positive. A major uncertainty in both approaches is associated with isolating the pentaquark contribution in the correlation functions between the pentaquark interpolating currents. Because the interpolating current can also couple to the two hadron reducible (2HR) kaon-nucleon (K-N) intermediate state, it is difficult to extract signals for the pentaquark state from the theoretical calculation of the two point correlation function. This is particularly so because the K-N threshold lies below the expected O+ state and neither the Bore1 transformation in the QCD sum rule, nor the large imaginary time behavior in the lattice gauge theory calculation can isolate the Q+ state. Hence, it is essential in both approaches to estimate the contribution coming from the K-N intermediate state in the correlation function. This point has been noted for the QCD sum rule approach by Morimatsu et.al.', who claimed that after subtracting out the 2HR part of the OPE, one finds that the parity becomes positive. However, as we will show, subtracting the 2HR contribution in the OPE level is an ill-defined approach. Instead, its contribution can be estimated in the phenomenological side. The magnitude for the overlap of the O+ interpolating current with the kaon-nucleon state is obtained by first applying a chiral rotation to the Of interpolating current and estimate the kaon overlap in the soft kaon limit. We then analyze the QCD sum rule for the nucleon with the resulting pentaquark nucleon interpolating current to estimate the nucleon overlap. It is found that the K-N intermediate states constitutes only 5% of the sum rule so that the original claim that the parity of O+ is negative remains valid.

292

2. Correlation functions

Let us begin with the correlation function between the interpolating field for 0 ,

H(q) =i

I

d4zeiq"(OIT(Je(z), &(0))10)

(1)

where

The OPE of this correlation function has been calculated by Sugiyama, Doi and Oka (SD0)6 and its extension to the anti-charmed pentaquark has been made in Ref.[7]. F'rom comparing the OPE to the phenomenological side saturated by the ground state 0 and a continuum, SDO were able to identify the parity of the 0 to be negative. However, as has been noted by Kondo, Morimatsu and Nishikawa (KMN)8, the correlation function can have two hadron reducible (2HR) contributions in addition to the two hadron irreducible (2HI) part. This means that since Je is the 5 quark current with a strangeness +1, isospin zero, it can also easily couple directly to a kaon-nucleon intermediate state or any of their excited states, namely,

+

rI(q) = nzH1r I z H R

(3)

where

Therefore, to extract information about the pentaquarks from the OPE calculation, one has to subtract the contributions from the 2HR contributions,

In the left hand side, we are interested in the ground state but the OPE part can be calculated for large -q2. The usual way of satisfying both approximations is via the Bore1 transformation. A question in this particular case is how to subtract the 2HR contribution effectively.

293

2.1. Method by K M N

KMN suggest to calculate the large -q2 limit of I12HR(q)using the OPE of the I I g p E ( p ) ,I I Z p E ( p - q ) .

However, a little inspection shows that such factorization is an ill-defined problem. The reason is the following. The OPE of the left-hand side of Eq.(6) means that it is obtained from the short distance expansion of the correlator; namely in the large -q2 limit. Also being the O P E parts, I I g p E ( p ) and I I g p E ( p - q ) are obtained in the large -p2 and - ( p - q ) 2 limit. However, as can be seen in Eq.(6), there are other important regions of p 2 , which contribute t o the OPE of the left hand side. An example of such regions are given in table 1. Another serious problem with Eq.(6) is Table 1. Typical momentum regions which contribute t o the OPE of I12HR(q). The first line represents the region which has been taken into account through Eq. (6). contribution to I12HRioPE(q)

IIK(p - q )

n ~ ( p )

comments

large -q2

large -p2

large - ( p - q ) 2

large -q2

small -p2

large - ( p

- q)2

not in Eq.(6)

large -q2

large -p2

small - ( p - q ) 2

not in Eq.(6)

in Eq.(6)

the implicit assumption of (OIJKJNIKW

=

(OlJKlIo x

(OlJN"),

(7)

which can be shown t o be not true in general.

2.2. Our method Here, we suggest t o subtract out the 2HR contribution by explicitly estimating the contribution coming from the non-interacting K-N intermediate state,

where

294

There are additional contributions coming from excited kaon or nucleon states. However, these contributions are exponentially suppressed after Bore1 transformation. Hence, to estimate the lowest 2HR contribution, we need to know the overlap strength in Eq.(9). This strength will be estimated in the following section by combining the soft-kaon limit and a sum rule for the nucleon with pentaquark interpolating field.

3. Estimating the overlap strength To calculate the overlap strength X K N , we first use the soft-kaon theorem, 1 1 (OIJolKN) - ~ ( O l [ Q ~ JoIIN) , = --(olJN,5IN) -+

f 7 r

1

= ---i75u(p)XN f7r

where

(10)

QF= Sd3ydt(y)i75s(y),and

Using Eq.(lO) in Eq.(9) we have, 1

Hence, t o know X K N , we need t o know the five-quark component of the nucleon. To do that, we first construct the sum rule for the nucleon using the following correlation function,

II(q) = 2

J'

d ~ . e ~ " ( o ~ S ( . ~ ) J ~ , ~ ( z ) J N , 5 ( 2 ) 1,0 )

(13)

where J N , is~ given in Eq.(11). We then divide the imaginary part into two parts for qo > 0, 1 -1mJqqo) = A(qo)yO B(q0) . (14) lr

+

Then the spectral density for the positive and negative parity physical states will be as follows, P*(40> = 4 q o ) F B(q0) .

(15)

Note that the signs are reversed compared to that of SDO because the nucleon current J N , as ~ given in E q . ( l l ) has an additional factor of 7 5 compared t o the usual nucleon current.

295

For the nucleon correlation function given in Eq.(13), the respective

OPE are given by

2(sg0 . Gs)- (&a . Gd)

(16)

The spectral density is assumed to have the following form, = IXN&I2S(q0 - m N d

P:hen(40)

+ q 4 0 - &)P$,t(Qo)

*

(17)

We substitute this into the following Borel transformed dispersion relation,

~ a sum rule for nucleon mass by and obtain a sum rule for I X N + ~ and taking the derivative with respect to M 2 . As can be seen from Fig.1, we

8.0X101i

........Dim 1 + Dim 3 ......+ Dim 4

1.4

......

+ Dim 5 + Dim 6

~

1.2

-

6.0~10'"

__.. ........ ...................

-- _ _ _ _ _ _ _ _ _ _ _ _ - _ _ - -

............ 0.6 -

2

.

O x l O " I 1.0 1.2 1.4 1.6 1.8 2.0

1.0 1.2 1.4 1.6 1.8 2.0

Borel mass [GeV]

Borel mass [GeV]

Figure 1. Borel curve for five quark current.

from eq.(18) and mass of the nucleon with

296

-

obtain a consistent (positive) sum rule for I X N + ~ ~ 1 x 10-l' GeV12 and a reasonable mass for the nucleon. Similarly one can obtain consistent results for the negative-parity nucleon Sll(1535) from the sum rule for I X N - ~ ~ . 4. Reanalysis of SDO sum rule

We now use I X K N ~ = ~ ~ L X N +in~Eq.(8), ~ whose imaginary part for the positive and negative parity channels are flT

Then, the sum rule of SDO with the explicit contribution from the K-N 2HR contribution subtracted out looks as,

J

0

m K f m N

where the OPE is given in Ref.[6]

5

2.0x10-10:

l.oxlO1o0.0

- -. -..- ----.-..__ .

-3.0~1 0'' 1.0 1.2 1.4 1.6 1.8 2.0

1.0 1.2 1.4 1.6 1.8 2.0

Borel mass [GeV]

Borel mass [GeV]

Figure 2. Borel curve for the left hand side of eq.(20) for the pentaquark OPE with K N 2HR contribution.

297 T h e result for (Xe*I2 is given in Fig.2. As can be seen in the figure, t h e contribution from t h e K-N 2HR state constitutes less than 10 ’% of t h e total OPE so t h a t t h e sum rule for l X 0 - 1 ~ physically makes sense while t h a t for IXo+l2 does not. Hence, t h e conclusion first given by SDO t h a t t h e OPE is consistent with t h e existence of a negative parity pentaquark state remains valid.

References 1. LEPS Collaboration, T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003). 2. Y. s. Oh and H. c. Kim, arXiv:hep-ph/0405010,to be published in P R D Y. s. Oh, H. c. Kim and S. H. Lee, Phys. Rev. D 69,094009 (2004) [arXiv:hepph/0310!17]; S. H. Lee, H. Kim and Y. s. Oh, arXiv:hep-ph/0402135. 3. D. Diakonov, V. Petrov, and M. Polyakov, Z. Phys. A 359, 305 (1997). 4. A. Hosaka, Phys. Lett. B 571, 55 (2003). 5. F. Csikor, Z. Fodor, S.D. Katz, and T.G. KovAcs, J. High Energy Phys. 11, 070 (2003); S. Sasaki, hep-lat/0310014. 6. J. Sugiyama, T. Doi and M. Oka, Phys. Lett. B 581, 167 (2004) [arXiv:hepph/0309271]. 7. H. Kim, S. H. Lee and Y. s. Oh, Phys. Lett. B 595, 293 (2004) [arXiv:hepph/0404170]. 8. Y. Kondo, 0. Morimatsu and T. Nishikawa, arXiv:hep-ph/0404285.

298

PENTAQUARK BARYONS FROM LATTICE CALCULATIONS

SHOICHI SASAKI Department of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japapn E-mail:[email protected] The present status of pentaquark spectroscopy in lattice QCD is reviewed. The current lattice simulations seem to give no indication of a pentaquark in the positive parity channel to be identified with the 0+(1540).

1. Introduction

Recently, the LEPS collaboration at Spring-8 has observed a very sharp peak resonance in the K - missing-mass spectrum of the y n 4 nK+Kreaction on 12C '. The peak position is located at 1540 MeV with a very narrow width. Remarkably, the observed resonance should have strangeness fl. Thus, 0+(1540) cannot be a three quark state and should be an exotic baryon state with the minimal quark content uudds. This discovery is subsequently confirmed by other experiments Experimentally, spin, parity and isospin are not determined yet. Non-existence of a narrow resonance in pK+ channel indicates that possibility of I = 1 has been already ruled out In addition, two other candidates for the pentaquark baryon have reported by the NA49 collaboration and the H1 collaboration '. It should be pointed out that those discoveries are not confirmed yet by other experiments 4. 213t.

173.

2. Lattice pentaquark spectroscopy

If the pentaquark baryons really exist, such states must emerge directly from first principles, QCD. Of course, what we should do is to confirm the presence of the pentaquarks by lattice QCD. Experimentally, it is rather tIt should be noted, however, that the experimental evidence for the 8+(1540) is not very solid yet since there are a similar number of negative results to be reported '.

299

difficult t o determine the parity of the 0+(1540). Thus, lattice QCD has a chance to answer the undetermined quantum numbers before experimental efforts. Lattice QCD has also a feasibility to predict the masses for undiscovered pentaquark baryons. I stress that there is substantial progress in lattice study of excited baryons recently ’. Especially, the negative parity nucleon N*(1535), which lies close to the 0+(1540), has become an established state in quenched lattice QCD Here I report that quenched lattice QCD is capable of studying the 0+(1540) as well. Indeed, it is not so easy to deal with the qqqqii state rather than usual baryons (qqq) and mesons (qq) in lattice QCD. The qqqqij state can be decomposed into a pair of color singlet states as qqq and qq, in other words, can decay into two hadron states even in the quenched approximation. For instance, one can start a study with a simple minded local operator for the 0+(1540), which is constructed from the product of a neutron operator and a K+ operator such as 0 = & , b c ( d ~ C ~ 5 u b ) d C ( B e ~ The 5 u e )two-point . correlation function composed of this operator, in general, couples not only to the 0 state (single hadron) but also t o the two hadron states such as an interacting KN system ’. Even worse, when the mass of the qqqqij state is higher than the threshold of the hadronic two-body system, the two-point function should be dominated by the two hadron states. Thus, a specific operator with as little overlap with the hadronic two-body states as possible is desired in order to identify the signal of the pentaquark state in lattice QCD. Once one can identify the pentaquark signal in lattice QCD, to determine the parity of the Of(1540) is the most challenging issue at present. Thus, it is necessary to project out the parity eigenstate from given lattice data precisely. I discuss three related issues as follows. ’i8.

2.1.

Estimation of the K N threshold

The experimentally observed 0+(1540) state is clearly a resonance state. However, its mass is near the K N threshold. We could manage t o calculate the pentaquark as a bound state if its parity were positive. Here, I recall that all momenta are quantized as p’L = 2 7 r r ( r ~Z3)on lattice in finite volume (the spacial extent L ) with the periodic boundary condition. Thus, the spectrum of energies of two hadron states such as KN states with zero total momentum should be discrete and these energies are approximately equal t o values, which are evaluated in the noninteracting case:

300 2.4

-2.1

0

1

2

3

4

5

L [fml Figure 1. The S-wave and P-wave K N threshold energies on a lattice of spatial extent L. If L 5 4.6 fm, the mass of the 0+(1540) is lower than the P-wave K N threshold.

where p , = f i ' 2 r / L and n E Z. The positive parity 0 state decay into K N in a P-wave where the K N system should have a nonzero relative momentum. The P-wave K N threshold is simply estimated at an energy level El, which is evaluated with the smallest nonzero momentum p l = 2 n / L in Eq. (1). The energy level El can be lifted by decreasing spatial extent as depicted in Fig. 1 while the lowest energy level Eo, which corresponds to the S-wave K N threshold, remains unchanged. The level crossing between El and the 0 mass takes place around 4.6 fm in this crude estimation. It implies that the positive parity 0 state m a y become a bound state in the typical size of available lattice simulations, i.e. L M 2 - 3 fm. 2.2.

Choice of operators

For the case of the negative parity 0 state, the presence of the K N scattering state must complicates the study of pentaquarks in lattice QCD. One should choose an optimal operator, which couples weakly to the K N scattering state, in order to access the pentaquark state above the (S-wave) K N threshold. For this direction, I would like to recall that the less known observation in the spectroscopy of the nucleon. There are two possible interpolating operators for the I = 112 and J p = 1/2+ state; 0: = ~ , b ~ [ ' l l ~ C ~ 5 d and O F = E ~ ~ ~ [ Z L ~ even C ~ ~if ]one T ~restricts U , , operators to contain no derivatives and to belong to the ($,O) @ (0, $) chiral multiplet under S U ( 2 ) L @ S v ( 2 ) *. ~ Of course, two operators have the same quantum number of the nucleon. The first operator O y is utilized conventionally in

301 2.0 I=l/Z, Jp =1/2+

p 4.0,V-163 X 32,DWF 0.0 0.0

0.1

0.2

0.3

0.4

0.5

(aMd2 Figure 2. Comparison of the fitted mass from correlation (0,”BF+ o,“B?) (*) 8 .

(0,”By)(o), (0:gF)

( 0 ) and

the cross

lattice QCD since the second operator O! vanishes in the non-relativistic limit. It implies that the second operator is expected t o have small overlap with the nucleon l(OlOFINuc1)I M 0. Indeed, the mass extracted from the correlator constructed by the second operator 02 exhibit the different mass from the nucleon mass as shown in Fig. 2. The operator dependence on a overlap with desired state is evident, at least, in the heavy quark regime, while the cross correlation suggests that the small overlap with the nucleon might be no longer robust in the light quark regime where is far from the non-relativistic description 8. 2.3. Parity projection

The intrinsic parity of the local baryon operator can be defined by the parity transformation of internal quark fields as

P O ( ” ) ( 2 J ) P += 77740(7))(-2,t),

(2) y50(-) for the lo-

where 77 = fl. However, due to the relation O(+) = cal baryon operator, the resulting two-point correlation functions are also related with each other as (O(+)(x)a(+)(O)) = -y5( O(-)(x)a(-)(0))y5. This means that the two-point correlation function composed of the local baryon operator can couple to both positive- and negative-parity states. However, I note that anti-particle contributions of opposite parity states is propagating forward in time. Thus, the +/- parity eigenstate in the forward propagating contributions is obtained by choosing the appropriate projection (1f77y4)/2,which is given in reference to the intrinsic parity of operators, 77. Details of the parity projection are described in Ref. 6 .

302

3. First exploratory studies

3.1. Local pentaquark operators

As I remarked previously, an optimal operator, which couples weakly to the KN scattering state, would be required to explore the pentaquark baryons in lattice QCD. For this purpose, two types of local pentaquark operator for isosinglet state are proposed in the first two studies. One is a color variant of the simple product of nucleon and kaon operators,

ojli = & a b c [ U ~ C y 5 d b ] { U e ( S e y 5 d c-) (U * d ) } ,

(3)

which is proposed by Csikor et al. l l . The other is proposed by Sasaki l2 as in a rather exotic description guided by the diquark-diquark-antiquark structure:

oj?~= E a b c E a e f E b g h [ U : C r l d f ] [ U T c r Z d h ] C S T r1,2 = 1,y5,y5yp (but rl # r 2 ) and the superscript

(4)

where “7” stands for the intrinsic parity of the operator. There are three kinds of diquarkdiquark-antiquark operator in this description, which are useful for the extended study with the 3 x 3 correlation matrix analysis. More details of the diquark-diquark-antiquark operator are described in Ref. 11. In an exploratory study, one may assume that those interpolating operators have smaller overlap with the KN-scattering state than the simple product of nucleon and kaon operators, at least, in the heavy quark regime. Because, in the non-relativistic limit, all of them give rise t o the different wave function from the KN two-hadron system. 3.2. Results

The first two lattice studies were performed with the Wilson gauge action and the Wilson fermion action at the almost same box size L 21 2.0 - 2.2 fm.The lattice spacing Csikor et al. use is rather coarse than that of Sasaki, but their calculation was employed with relatively lighter pion masses (Adn 0.4 - 0.6 GeV). The main difference between two studies is the choice of pentaquark operators. After some initial confusion about the parity assignmentt , both calculations agreed that the lowest state of the isosinglet S = +1 baryons has the negative parity as shown in Fig. 3. The main results from the first two exploratory studies can be summarized as follows. N

tSee a footnote in Ref. 10.

303

41

L O , Jp=l/2+

f

m

@

0

1-0, Jp=1/2

CFKK (p=6.0) Sasaki (p=6.2) 11' 0.0

. 0.4' . 0.8 ' M:

'

'

1.2

.

6

[GeV']

Figure 3. Masses of the isosinglet S = +1 baryons with both positive- and negativeparity as functions of pion mass squared 11,12. The experimental value for 0+(1540) is marked with a star. 0

The current lattice simulations seem to give no indication of a pentaquark in the positive parity channel to be identified with the O+ (1540). The negative parity channel can easily accommodate a pentaquark with a mass close to the experimental value.

Therefore, both authors conclude that the exploratory lattice study favors spin-parity (1/2)- and isospin 0 for the 0+(1540). In Ref. 11, the anticharmed analog of the 0 state was also explored. It is found that the Qc(uuddE) lies much higher than the D N threshold, in contrast to several model predictions 13. More detailed lattice study would be desirable to clarify the significance of those observations. 4.

Subsequent lattice studies

There are four subsequent lattice studies of pentaquark spectroscopy to be found in the literature Other two preliminary results had been also reported at some conferences I give a short review of those results as follows. Recently, Kentucky group performed their simulations near the physical pion mass region with overlap fermions 14. However, they choose the simple minded operator as the product of nucleon and kaon operators to explore the pentaquark baryons. Instead, the sequential constrained fitting method is applied in their analysis to disentangle the pentaquark signal from towers of K N scattering state. They also check carefully the volume dependence 14,15916117.

18119.

304 Table 1.

Summary of the present status of lattice pentaquark spectroscopy.

author(s) and ref. Csikor et al. Sasaki l2 Mathur et al. l4 Chiu-Hsieh l5 T I T group l6 Cyprus group l7 YITP group l8 MIT group l9

signal

parity of pentaquark

Yes Yes No YeS No Yes Yes Yes

negative negative N/A positive N/A negative negative negative

-

kind of operator color variant of K N diquark-diquark-antiquark simple K N diquark-diquark-ant iquark diquark-diquark-antiquark diquark-diquark-antiquark simple K N & color variant of K N diquark-diquark-antiquark

of the spectral weight. Then, they found that ground state in either parity channels has a characteristic volume dependence on the spectral weight, which should have l/L3 dependence for two particles. Their final conclusion is that there is no sign of pentaquark signal in either parity channels in their calculation. However, it seems that their results are consistent with the experimental fact that the 0+(1540) state has not yet been found in the K N scattering data as an elastic resonance 20. Another negative results against the first two studies are reported by TIT group 16, while they confirm that the lowest energy state appears in the negative parity channel. In their study, a new method is proposed to lift up the S-wave K N threshold by imposing a hybrid boundary condition in the spatial direction and it is also applied in their calculation. They found that the lowest state in negative parity channel seems to be the two hadron states. My critique of their results may be found in Ref. 22. There are several positive results for the first two studies. Cyprus group performed careful studies concerning the volume dependence of the spectral weight and then found that their observed state seems to be a single hadron state, i.e. the pentaquark state 17. YITP group has started the 2 x 2 correlation matrix analysis with the simple minded operator as the product of nucleon and kaon operators and its color variant 18. Their preliminary results support that there is an indication of the presence of the pentaquark state near the lowest K N scattering state in the negative parity channel. MIT group also reported that preliminary results are consistent with results of the first two studies 19. 5.

Summary and Outlook

Table 1 represents a summary of the present status for each lattice calculation. The first conclusion of the first two studies as summarized in Sec. 3.2 is confirmed by subsequent lattice studies. The currently impor-

305

tant issue is whether or not t o establish the presence of the 0+(1540) in the negative parity channel. It is necessary for this t o disentangle the pentaquark signal from the K N scattering states completely . The correlation matrix analysis is strongly required t o separate the K N scattering state and isolate the pentaquark state. We also should check the volume dependence of the spectral weight. Probably, the hybrid boundary condition is helpful t o identify the pentaquark state as a single hadron state. Needless t o say, we ought t o try non-local types of pentaquark operator in order to verify whether there is no indication of the @+(1540) in the positive parity channel. Finally I stress that all present results should be regarded as exploratory. Indeed, much detail studies are in progress in each group. The author is supported by JSPS Grant-in-Aid for Encouragement of Young Scientists (No. 15740137).

References 1. LEPS Collaboration, Phys. Rev. Lett. 91,012002 (2003). 2. DIANA Collaboration, Phys. Atom. Nucl. 66,1715 (2003); CLAS Collaboration, Phys. Rev. Lett. 91,252001 (2003). 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

SAPHIR Collaboration, hep-ex/0307083. For a recent review, see K. Hicks, hep-ph/0408001; references therein. NA49 Collaboration, Phys. Rev. Lett. 92,042003 (2004). H1 Collaboration, Phys. Lett. B 588,17 (2004). For recent reviews, see S. Sasaki, Prog. Theor. Phys. Suppl. 151,143 (2003), nucl-th/0305014; C. Morningstar, nucl-th/0308026; D. B. Leinweber et al., nucl-th/0406032 and references therein. S. Sasaki, T. Blum and S. Ohta, Phys. Rev. D 65,074503 (2002). M. Liischer, Commun. Math. Phys. 105, 153 (1986). K. C. Bowler et al., Nucl. Phys. B 240,213 (1984); D. B. Leinweber, Phys. Rev. D 51,6383 (1995). F. Csikor, Z. Fodor, S. D. Katz and T. G. Kovacs, JHEP 0311,070 (2003), hep-lat/0309090. S. Sasaki, Phys. Rev. Lett. 93,152001 (2004), hep-lat/0310014. F. Stancu, Phys. Rev. D 58,111501 (1998); R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91,232003 (2003); M. Karliner and H. J. Lipkin, hep-ph/0307343. N. Mathur et al., hep-ph/0406196. T. W. Chiu and T. H. Hsieh, hep-ph/0403020. N. Ishii et al., hep-lat/0408030. C. Alexandrou et. al., heplat/0409065. T.T. Takahashi et al., these proceedings. J. Negele, Talk presented at QNP2004. R. A. Arndt et al., Phys. Rev. C 68,042201 (2003) [Erratum-ibid. C 69, 019901 (2004)l. S. Sasaki, hep-lat/0410016.

306

EXCITED BARYONS AND PENTAQUARKS ON THE LATTICE

F.X. LEE Center for Nuclear Studies, Physics Department, The George Washington University, Washington, DC 20052, USA I report recent progress in computing the m a s spectrum of excited baryons and pentaquarks in lattice QCD.

1. QCD primer

Quantum Chromodynamics (QCD) is widely accepted as the fundamental theory of the strong interaction. The QCD Lagrangian density can be written down simply in one line (in Euclidean space) 1

LQCD= -TrF,, Fvu 2

+

+ q(ypDv + m,)q

(1)

where F,, = dA, - dA, g[A,, A,] is the gluon field strength tensor and D, = a, gA, is the covariant derivative which provides the interaction between the gluon and quark terms. The action of QCD is the integral of the Lagrangian density over space-time: SQCD= L Q C D ~ QCD ~ X . is a highly non-linear relativistic quantum field theory. It is well-known that the theory has chiral symmetry in the m, = 0 limit and the symmetry is spontaneously broken in the vacuum. At high energies, it exhibits asymptotic freedom, while at low energies it has confinement. At the present, the only tool that provides a solution to QCD with controlled systematic errors is lattice QCD which solves the theory on a discrete space-time lattice using Fig. 1 is a picture of the proton in QCD. Three valence QCD quarks propagate in time continuously from one point to another in the QCD vacuum with the quantum numbers of the proton. They can back in both space and time. Quark-antiquark bubbles pop up from the vacuum. The quark lines are dressed by any number of gluons. It‘s the interactions such as these that are responsible for most of proton’s mass. The quark themselves (5 MeV) contribute less than 1 percent to the proton mass. This is in contrast to the quark model where most of the proton mass comes from the constituent

+

307

quarks (330 MeV) with only weak pair-wise interactions via one-gluonexchange.

Figure 1.

Proton in QCD (left) vs. proton in quark model (right).

2. Baryon resonances

The rich structure of the excited baryon spectrum, as documented by the particle data group provides a fertile ground for exploring the nature of quark-quark interactions. Most of the spectrum, however, is poorly known. Traditionally quark models have led the way in making sense of the spectrum. But many puzzles remain. What is the nature of the Roper resonance, and the R(1405)? How to explain the inverted ordering of the lowest-lying states which has the order of positive and negative-parity excitations inverted between N , A and A channels? Two contrasting views have emerged about the nature of the hyperfine splittings in the baryons. One is from the constituent quark model which has the interaction dominated by one-gluon-change type, ie., color-spin A; X i & $2. The other is based on Goldstone-boson-exchange which has flavor-color A{ A$% . & as the dominant part. Lattice QCD is perhaps the most desirable tool to adjudicate the theoretical controversy surrounding these issues. Evidence from valence QCD favors the flavor-color picture. Lattice QCD has evolved to the point that the best quenched calculation, of the ground-state hadron spectrum has reproduced the observed values to within 7%, with the remaining discrepancy attributed to the quenched approximation. This bodes well for the exploration of the excited sectors of the spectrum, even in the quenched approximation. 394

e

-

308

2.1. Roper and There exist a number of lattice studies of the excited baryon spectrum using a variety of actions 7,8,9,10,11112,13,14. two independent local fields:

is the standard nucleon operator, while x2, which has a vanishing nonrelativistic limit, is sometimes referred to as the 'bad' nucleon operator. Note that baryon interpolating fields couple to both positive and negativeparity states, which can be separated by well-established parity-projection techniques. There are two problems facing these studies. First, they have not been able to probe the relevant low quark mass region while preserving chiral symmetry at finite lattice spacing (except Ref. l2 which uses the Domain Wall fermion). Since the controversy about the nature of Roper hinges on chiral symmetry, it is essential to have a fermion action which explicitly exhibits the correct spontmeously broken chiral syrmietry. Another difficulty of the calculation of the excited states in lattice QCD is that the conventional two-exponential fits are not reliable. Facing the uncertainty of the fitting procedure for the excited state, it has been suggested to use the non-standard nucleon interpolating field x 2 , in the hope that it may have negligible overlap with the nucleon so that the Roper state can be seen more readily. However, the lowest state calculated with this interpolation field (2.2GeV) is much higher than the Roper state. Employing the maximum entropy method allows one to study the nucleon and its radial excitation with the standard nucleon interpolation field 15. However, with the pion mass at N 600 MeV, the nucleon radial excitation is still too high ( w 2GeV). So the ordering of the nucleon, Sll(1535) and the Roper in these studies remains the same &s that from quark models. Our final result The details of the calculation can be found in Ref. is shown in Fig. 2. We see that for heavy quarks (m, 2 800 MeV), the Roper, Slit and nucleon splittings are like those of the heavy quarkonium. When the quark mass becomes lighter, the Roper and S11 have a tendency to coincide and cross over around m, = 220MeV. The masses extrapolated to the physical limit are consistent with the experimental values. Our result confirms the notion that the order reversal between the Roper and S11 (l535) compared to the heavy quark system is caused by the flavor-spin interaction between the quarks due to Goldstone boson exchanges '. It x1

309 I

I

I

I

I

I

I

2 0

0

1.5

0

0

W

Nucleon

c/)

: $

1

1.7

1.5

0.5

1.3 0

0

I

0

I

I

0.1

0.2 I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 rn;(GeV2)

Figure 2. Nucleon, Roper, and S11 massea as a function of m?, using the standard nucleon interpolating field XI. The insert is the ratio of Roper to nucleon mass. The experimental values are indicated by the corresponding open symbols.

serves to verify that the Roper (1440) is a radial excitation of the nucleon with three valence quarks. It also cast doubts on the viability of using the non-standard interpolation field for the Roper. We further support the notion that there is a transition from heavy quarks (where the SU(6) symmetry supplemented with color-spin interaction for the valence quarks gives a reasonable description) to light quarks (where the dynamics is dictated by chiral symmetry). It is suggested that this transition occurs at m, 400MeV for the nucleon. N

2.2. The q' ghost

The above result is obtained only after the special effects of the so-called r]' ghost are removed. In full QCD, the r]' meson contributes to the proton via vacuum polarizations, as shown in Fig. 3. Being a relatively heavy meson, its contribution is much smaller than that of the pion. However, in quenched QCD, the vacuum loops are suppressed, as shown in Fig. 4 (hairpin diagram), resulting in the following peculiar properties. First, it becomes a light degree of freedom! with a mass degenerate with that of

310

the pion. Second, it is present in all hadron correlators. Third, it gives a negative-metric contribution to the correlation function. For these reasons, it is termed the q’ ghost: it is an unphysical state, and a pathology of the quenched approximation. The effects of q’ ghost were first observed in the a0 meson channel 20, where the ghost S-wave q’n state lies lower than a0 for small quark mass. The situation here is similar with the excited state of the nucleon where the P-wave q’N appears in the vicinity of the Roper. Since this is not clearly exhibited in the nucleon correlator where the nucleon is the lowest state in the channel and dominates the long-time behavior of the correlator, we can look at the parity partner of the nucleon ( N a - or ,911) with I = 1/2. There, the lowest S-wave $ N state with a mass close to the sum of the pion and nucleon masses can be lower than ,911 for sufficiently low quark mass. Due to the negative-metric contribution of the hairpin diagram, one expects that the &1 correlator will turn negative at larger time separations as is in the case of the a0 20. This was indeed observed 18, the first evidence of q’ ghost in a baryon channel.

P

U

Figure 3. Quark-line diagram for the ?I’ contribution to the proton in full QCD (left) and its hadronic representation (right). Any number of gluon lines can be present in the quark-line diagram.

d

D

p

$ ghost .._...._ ..

/‘

. 3

P

P

U

Figure 4.

The

4 contribution

to the proton in quenched QCD.

Using our constrained curve fitting algorithm 19, we are able to distinguish the physical %per and ,911 from the ghost two-particle intermediate

311

states (7”) by checking their volume dependence and their weights as a function of the pion mass. Our results demonstrate that the effects of 7’ ghost must be reckoned with in the chiral region (below mrr MeV ) in all hadron channels in quenched QCD. N

3. Pentaquarks Since the report on the discovery two years ago of an exotic pentaquark, named as B+(uuddS)?with a mass of about 1540 MeV and a narrow width of less than 20 MeV 21, there has been an explosive growth of interest in the subject 22. Here I focus on an overview of developments on the lattice. So far, there are four lattice calculations Here we use 23,27 to recap the essential elements of a lattice calculation of the pentaquark. Pentaquarks do not have a unique color structure aside from being a color singlet. For a pentaquark of the type uudds, we consider both isospin 1=0 and 1=1 states with the following interpolating field 23924i25726327.

x = €aac(uTaCy5db)[uc(~ygde) F { u 41

(2)

where the minus sign is for 1=0 and the plus sign for I=l. The explicit spinparity of this interpolating field is $-, but it couples to both parities. Under the anti-periodic boundary condition used in this work, the positive-parity state propagates in the forward time direction in the lower component of the correlation function, while the negative-parity state propagates backward in the same lower component. The left half of Fig. 5 shows the results in the I ( J p ) = 1 (1/2-) channel as a function of mz. Also plotted is the KN threshold energy in the S-wave E K N ( =~ 0) = m K mN which is the same on both lattices. There is no need to consider ghost states in this channel, which is supported by the fact that the correlation function is positive throughout. The calculated ground state is plotted in the figure for the two lattices. The energy on the smaller lattice (L=2.4 fm) is consistently higher than that on the larger one (L=3.2 fm). This is the expected volume dependence for two particles in a box with a negative scattering length for medium and heavier quark masses. At the lowest mass, the energy coincides with the S-wave threshold, meaning that there is little interaction, consistent with the experimental fact of zero scattering length. The first excited state in this channel is much higher than 1.54 GeV near the physical pion mass: and we identify it as the p=l KN scattering state. There is no candidate for a pentaquark. As far as the ground state is concerned, our results more or less agree with

+

312

those of Ref. 24 and 25, but disagree with those of Ref. 26. It is noted in Ref. 24 and 25 that they have seen an low-lying excited state above the K N mass thrcshold and thcy intcrprct it as thc pcntquark statc. Wc tricd but could not accommodate an extra low-lying pentaquark state within 100 MeV above the K N threshold in our one-channel calculation. In the I ( J p ) = 1 (1/2+) channel shown in the the right half of Fig. 5 the NKq’ ghost state, pentaquark, and KN pwave scattering state are the lowest states. We found a ghost state and KN scattering state, but not a pentaquaxk state near 1.54 GeV. We have tried to see if our data could accommodate three states, but the x2/dof would simply reject it. The energy of the K N scattering state lies higher on the smaller lattice (L=2.4 fm) than that on the larger lattice (L = 3.2 fm). This mainly reflects the fact that p l is larger on the L = 2.4 fm lattice than the one on the L = 3.2 fm lattice. At the lowest mass, the energies almost coincide with the P-wave thresholds, meaning that the K N interaction is weak, consistent with experiment. N

3.5 3

p

3

Y

2

2.5

2.5

5

2

g 2

g 2

1.5

1.5

’I

3

0

0.2

0.4

0.6

0.8

1

1.2

0

1.4

0.2

0.4

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0.8

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1.2

4

m,2 (GeV*)

%’ CGeV’,

Figure 5. Left: the computed mass in the Z ( J p ) = 1 (1/2-) channel as a function of m$ for the two lattices L=2.4 fm and L=3.2 fm.The curve is the KN threshold energy in the S-wave E K N @ = 0) = m K + m N . Right: same, but for I ( J p ) = 1 (1/2+) channel. The two lower curves are the KN threshold energies in the P-wave E K N ( = ~ 1). The two higher curves are for the non-interacting ghost states.

There is one relatively easy test that can check whether a state is a genuine pentaquark or a KN scattering state. It is the volume dependence of spectral weight w as in G(t) = we-mt. For a one-particle state, there is almost not volume dependence for w. But for a two-particle state, there is an inverse volume dependence w 1/V. Fig. 6 shows our results in the 1(1/2*) channels. The results in the 0 (1/2*) are similar. N

313

I

0 0

0.2

0.4

0.6

0.8

1

01 0

0.2

as

0.4

I 0.8

1

m,

mn

Figure 6. Volume dependence of the spectral weight in the 1 (l/2-) channel (left) and 1 (1/2+) channel (right). The line at 2.37 is the expected volume dependence of the spectral weight.

4. Conclusion

It appears that the ordering of low-lying baryons can be reproduced on the lattice with standard interpolating fields built from three quarks. We observed the cross-over of the Roper and &I in the region of pion rntlss 300 MeV. This shows the importance of pushing into the light quark region mass where chiral dynamics dominates. Our results support the notion that there is a transition from color-spin to flavor-spin in the hyperfine interaction from heavy to light quark masses. However, additional complications arise due to the r)' ghost states in the light mass region in the quenched approximation. This was clearly exposed in the 5'11 channel. More advanced fitting algorithm that incorporates these ghost states has to be used. As long as the ghost states are properly dealt with, our results show that the quenched approximation can be used to explore the baryon spectrum deep in the chiral region. As for the pentaquarks, there is no consistent picture emerging on the lattice. Our results based on the overlap fermion and pion mass as low as 180 MeV seem to reveal no evidence for a pentaquark state of the type uuddS with the quantum numbers I ( J p ) = O( near a mass of 1540 MeV. Instead, the correlation functions are dominated by KN scattering states and the ghost KNr)' states in the 1/2+ channel at low quark mass (pion mass less than 300 MeV). Our results are consistent with the known features of the KN scattering phase-shifts analysis 2s). We have checked that the K N states exhibit the expected volume dependence in the spectral weight for two-particle scattering. Our conclusion is in contradiction with the other lattice calculations N

i")

314

which has claimed a pentaquark signal of either negative parity 24,25, or positive parity 26, in the vicinity of 1.54 GeV. These claims should be takcn with caution. Thc ccntral issuc is how to rcliably scparatc a gcnuine pentaquark from the KN scattering states. We propose a simple test, namely volume dependence in the spectral weight, that can distinguish one kom the other. We advocate this volume dependence to test the character of extracted states. This work is supported in part by U.S. Department of Energy under grants DEFG02-95ER40907 and DEFG05-84ER40154. The computing resources at NERSC (operated by DOE under DEAC03-76SF00098) are also acknowledged. Collaboration with N. Mathur, K.F. Liu, C. Bennhold, Y. Chen, S.J. Dong, T. Draper, Horvath and J.B. Zhang is gratefully acknowledged.

References Particle Data Group, Eur. Phys. J. C 15,1 (2000). Phys. Rev. Lett. 84, 238 (2000) N. Isgur and G. Karl, Phys. Rev. D 18,4187 (1978). S. Capstick and N. Isgur, Phys. Rev. D 34,2809 (1986). 5. L. Ya. Glozman and D.O. &ka, Phys. Rep. 268,263 (1996); L. Ya. Glozman et al., Phys. Rev. D 58,0903 (1998). 6. K.F. Liu et ul.,Phys. Rev. D 59, 112001 (1999). 7. D.B. Leinweber, Phys. Rev. D 51,6383 (1995). 8. F.X. Lee, D.B. Leinweber, Nucl. Phys. B (Proc. Suppl.) 73,258 (1999); F.X. Lee, Nucl. Phys. B (Proc. Suppl.) 94,251 (2001); F.X. Lee et al., Nucl. Phys. B (Proc. Suppl.) 106,248 (2002) 9. S. Sasaki, Nucl. Phys. B (Proc. Suppl.) 83, 206 (2000); hepph/0004252; T. Blum, S. Sasaki, hep-lat/0002019; S. Sasaki, T. Blum, S. Ohta, heplat/0102010. 10. D. Richards, Nucl. Phys. B (Proc. Suppl.) 94,269 (2001); M. Grokeler et al., heplat/0106022. 11. W. Melnitcho& et al., heplat/0202022. 12. S. Sasaki,T. Blum, and S. Ohta, Phys. Rev. D65, 074503 (2002). 13. D.G. Richards et al., Nucl. Phys. (Proc. Suppl.) B109, 89 (2002). 14. R.Edwards, U. Heller, D. Richards, hep-lat/0304. 15. S. Sasaki, K. Sasaki, T. Hatsuda, and M. Asakawa, heplat/O209059; S. Sasaki, nucl-th/0305014. 16. S.J. Dong, F.X. Lee, K.F. Liu, and J.B. Zhang, Phys. Rev. Lett. 85, 5051 (2000). 17. S.J. Dong, T. Draper, I. HorvAth, F.X. Lee, K.F. Liu, N. Mathur, J.B. Zhang, Phys. Rev. D (in print); hep-lat/0304005. 18. S.J. Dong, T. Draper, I. HorvAth, F.X. Lee, K.F. Liu, N. Mathur and J.B. Zhang, hepph/0306199.

1. 2. 3. 4.

315 19. Y. Chen, S.J. Dong, T. Draper, I. Horvhth, F.X. Lee, K.F. Liu, N. Mathur, C. Srinivasan, S. Tamhankar, J.B. Zhang, heplat/0405001. 20. W. Bardeen, A. Duncan, E. Eichten, N. Isgur, H. Thacker, Phys. Rev. D65, 014509 (2002). 21. T. Nakano et &. (LEPS Collaboration), Phys. Rev. Lett. 91,012002(2003). 22. See these proceedings on the subject of pentaquarks. A search at SPIRES (http://www.slac.stanford.edu/spires/find) or the e-print archive (http://arxiv.org) would reveal more than 300 papers so far. Or more than 6000 entries on google. 23. F.X. Lee eta al, "A search for pentaquarks on the lattice", (unpublished), presented at Lattice03 and Cairns Workshop in summer 2003. 24. F. Csikor, Z. Fodor, S.D. Katz and T.G. Kovks, JHEP 0311,070 (2003), hep-lat /030909O. 25. S. Sasaki, heplat/0310014. 26. T.W. Chiu and T.H. Hsieh, hep-ph/0403020, hepph/0404007. 27. N. Mathur, F.X. Lee, C. Bennhold, Y. Chen, S.J. Dong, T. Draper, I. Horvath, K.F. Liu, J.B. Zhang, "A lattice study of pentquarks with overlap fermions" , (to be published). 28. J.S. Hyslop, R.A. Arndt, L.D. Roper, and R.L. Workman, Phys. Rev. D 46, 961 (1992).

316

ANISOTROPIC LATTICE QCD STUDIES OF PENTAQUARK ANTI-DECUPLET *

N. ISHII', T. DOI', H. IIDA', M. OKA', F. OKIHARU3, H. SUGANUMA'

' Dept. of Phys., Tokyo Institute of Technology, Meguro, Tokyo 152-8551,Japan RIKEN B N L Research Center, BNL, Upton, New York 11973, USA Faculty of Science and Tech., Nihon Uniu., Chiyoda, Tokyo 101-8308,Japan Anti-decuplet penta-quark baryon is studied with the quenched anisotropic lattice QCD for accurate measurement of the correlator. Both the positive and negative parity states are studied using a non-NK type interpolating field with I = 0 and J = 1/2. After the chiral extrapolation, the lowest positive parity state is found a t m e N 2.25 GeV, which is too massive to be identified with the experimentally observed S+(1540). The lowest negative parity state is found at me N 1.75 GeV, which is rather close to the empirical value. To distinguish a compact 5 Q resonance state from an NK scattering state, a new method with "hybnd boundary wnditaon (HBC)" is proposed. The HBC analysis shows that the observed state in the negative parity channel is an NK scattering state.

1. Introduction

LEPS group at Spring-8 has discovered a narrow resonance O+(1540), which is centered at 1.54 f 0.01 GeV with a width smaller than 25 MeV.' This resonance is confirmed to have baryon number B = 1, charge Q = +1 and strangeness S = +1 implying that it is a baryon containing at least one 3. Hence, its simplest configuration is uudd3, i.e., a manifestly exotic penta-quark (5Q) state. The experimental discovery of O+ was motivated by a theoretical prediction.' Tremendous theoretical efforts have been and are still being devoted among which its parity is one of the most to the investigation of 0+,3,4 important topics. Experimentally, the parity determination of Of is quite ~ h a l l e n g i n g ,while ~ ? ~ opinions are divided in the theoretical side.3 There are several quenched lattice QCD studies of the 5Q ~ t a t e , However, the results have not yet reached a consensus. One group claims the existence of a low-lying positive parity 5Q resonance. Negative parity 5Q resonance is claimed by two g r o ~ p s , 'among ~~ which Ref. 8 has omitted a quark-exchange diagram between diquark pairs assuming the highly *Lattice QCD numerical calculation has been done with NEC SX-5 at Osaka University.

317

correlated diquark picture. Note that these three groups employed non-NK type interpolating fields. In contrast, Fkf. 10 has employed the NK-type interpolating field, and performed solid analysis concluding that no signal for a 5Q resonance state is observed. There is another type of lattice QCD studies of the static 5Q potential aiming at providing physical insights into the structure of 5Q baryons. In this paper, we study the 5Q baryon O+ for both parities by using high-precision data generated with the quenched anisotropic lattice QCD. We employ the standard Wilson gauge action at p = 5.75 on the 123 x 96 lattice with the renormalized anisotropy a,/at = 4. The anisotropic lattice method is a powerful technique, which can provide us with high-precision data quite e f f i ~ i e n t l y The . ~ ~lattice ~ ~ ~spacing ~ ~ ~ is ~~ determined ~ from the static quark potential adopting the Sommer parameter r i l = 395 MeV leading to a;l = 1.100(6) GeV (a, N 0.18 fm).13 The lattice size 123 x 96 amounts to (2.15fm)3 x 4.30fm in the physical unit. The O(a)-improved Wilson quark (clover) action is employed l3 with four values of hopping parameters as K = 0.1210(0.0010)0.1240, which roughly covers m, 5 rn, 5 2m, corresponding to mr/rnp = 0.81,0.77,0.72 and 0.65. By keeping K , = 0.1240 fixed for s quark, we change 6 = 0.1210 - 0.1240 for u and d quarks for chiral extrapolation. Unless otherwise indicated, we use (K,, K )

= (0.1240,0.1220),

(1)

as a typical set of hopping parameters. Anti-periodic boundary condition (BC) is imposed on the quark fields along the temporal direction. To enhance the low-lying spectra, we adopt a smeared source with the gaussian size p N 0.4 fm. We use 504 gauge configurations to construct correlators of O+. For detail, see Ref. 16. In the former part of this paper, we present the standard analysis of 5Q correlators in both the positive and the negative parity channels adopting the standard periodic boundary condition along spatial directions. Latter half of this paper is devoted to a further investigation of the negative parity state. Proposing a new general method with “hybrid boundary condition (HBC)” , we attempt to determine whether it is a compact resonance state or a NK scattering state. 2. Parity projection

We consider a non-NK type interpolating field for O+ as

318

where a-g denote color indices, and C = 7 4 7 2 denotes the charge conjugation matrix. The quantum number of 0 is spin J = 1/2 and isospin I = 0. Under the spatial reflection of the quark fields, i.e., q(t,5) -+ 74q(t,-5), 0 transforms exactly in the same way, i.e., O(t,5) -+ +74O(t, -5), which means that the intrinsic parity of 0 is positive. Although its intrinsic parity is positive, it couples to negative parity states as weii.17 We consider the asymptotic behavior of the correlator in the 5Q CM frame as

where V denotes the spatial volume. In the region of 0 << t << Nt (Nt : temporal lattice size), the correlator is decomposed into two parts as

~ ( t3 )p+

(,,-,ti

+ p- c-e-m-t

(

- C-e-m-(Nt-t) - C+e-mt(Nt-t)

) ),

(4)

where rn* refer to the energies of lowest-lying states in positive and negative parity channels, respectively. P6 = (1 fy4)/2 serve as projection matrices onto the “upper” and “lower” Dirac subspaces, respectively, in the standard Dirac representation. Eq. (4)suggests that, in the region of 0 << t << Nt/2 the backwardly propagating states can be neglected. Hence, “upper” Dirac subspace is dominated by the lowest-lying positive parity state, whereas “lower” Dirac subspace is dominated by the lowest-lying negative parity state. We utilize this property in parity projection. 3. Numerical Result with standard BC

In Fig. 1, we show the effective mass plots for both the parity channels adopting Eq. (1). The effective mass is defined as

meffft) log(g(t)/g(t + I)),

(5)

where g(t) denotes correlator in Dirac “upper” or “lower” subspaces. Formally, g(t) can be expressed as a sum of exponentials. In the asymptotic region 0 << t << N t / 2 in Euclidean time, contaminations of excited states are expected to be reduced. If g(t) is dominated by single exponential corresponding to the lowest-lying mass m, then the effective mass behaves as constant in this region, i.e., men(t) m. Owing to this property, the effective mass plot is often used to determine the fit range.17 N

319

For both the parity channels, we find plateaus in the region 25 5 t 5 35, where single-exponential dominance is expected to be achieved. We simply neglect the data for t > 35, where contributions from the backward propagations are seen to become less negligible. The single-exponential fit is performed in the plateau region. The results are denoted by solid lines. The dotted lines indicate the p-wave (s-wave) NK thresholds for positive (negative) parity channels on the spatial lattice size L N 2.15 fm. Note that due to the quantized spatial momentum in the finite box, with the p-wave threshold is raised as E t h N 15minI = 2r/L*

d m +d

K

4.0

s

f

s9

3.5

3.5

3.0 2.5

2.5 10

20 t [&I

30

40

20

10 t

30

40

[&I

Figure 1. The effective mass plots of positive and negative parity Q+ adopting Eq. (1). The solid lines denote the result of the single-exponential fit performed in the region, 25 5 t 5 35. The dotted lines denote the pwave (s-wave) NK threshold energy for positive (cegative) parity channels on the spatial lattice size L E 2.15 fm.

In Fig. 2, the masses of positive (triangle) and negative (circle) parity O+ are plotted against m l . The open symbols denote direct lattice data. We find that the data behaves linearly in m:. Such a linear behavior against rn: is also observed for ordinary non-PS mesons and baryons.13J4 We extrapolate the lattice data linearly to the physical quark mass region. The results are denoted by closed symbols. For convenience, we show p wave (upper) and s-wave (lower) NK threshold with dotted lines. In the positive parity channel, the chiral extrapolation leads to me = 2.25 GeV. Since it is too massive, it cannot be identified with the experimentally observed Q+(1540). In contrast, in the negative parity channel, the chiral extrapolation leads to me = 1.75 GeV, which is rather close to the empirical value. In order for this to be identified with O+(1540), it should be confirmed that the observed state is not a NK scattering state but a compact 5Q resonance state. We will pursue this direction in the next section.

320

....." - / a _.' _,,,,_...... ./." __..' ,... ._.... ._...-.

2.5

d

+

,......'....2.0 .._... -.''z

__..

_,....-.. _...-.

...

__/

J .,._d"' ....

1.5

0.0

0.5

1 .O

n4rZ [GeV*I

Figure 2. me for both parity channels against m:. The triangles correspond t o the positive parity, while the circles correspond to the negative parity. The open symbols denote direct lattice data, whereas the closed ones the results after the chiral extrapolation. The dotted lines indicate the NK threshold energies for pwave (upper) and s-wave (lower) cases.

4. Further investigation with hybrid BC

In the positive parity channel, NK scattering states are in p-wave. Hence, in the 5Q CM frame, the minimum momenta of N and K are non-zero, i.e., \&in\ = 2n/L ( L : the spatial size of the lattice), through which the NK threshold energy acquires the explicit volume dependence as &h N d K + This can be utilized to determine whether the state of concern is a compact resonance or a NK scattering state. On the other hand, in the negative parity channel, NK scattering states are in s-wave. Hence, in the 5Q CM frame, N and K can have zero spatial momentum, and the NK threshold energy does not have an explicit volume dependence, i.e., &h N mN mK. This is not convenient for our purpose. We may ask ourselves whether there could be some prescription which can provide s-wave NK threshold energy with an explicit volume dependence as the p-wave one. This is achieved by twisting the spatial boundary

d-.

+

Table 1. The hybrid boundary wndataon (HBC)imposed on the quark fields. The second line shows the standard (periodic) BC for comparison.

HBC standard BC

u quark anti-periodic periodic

dquark anti-periodic periodic

squark periodic periodic

condition (BC) of quark fields in a flavor dependent manner as follows. We impose the anti-periodic BC on u and d quark fields, while periodic BC on s quark field. We will refer to this boundary condition as "hybrid b o u n d a y condition (HBC)" . (See Table 1.) Under HBC, hadrons are subject to their own spatial BC. For instance,

321

since N(uud, udd)and K(u3, di?) contain odd numbers of u and d quarks, they are subject to the anti-periodic BC. In contrast, since @+(uuddi?) contains even numbers of u and d quarks, it is subject to the periodic BC. (See Table 2.) Recall that, in the box of the size L , the spatial momenta Table 2. The consequence of the HBC on hadrons N K

8+

spatial BC anti-periodic anti-periodic periodic

quark content uud,udd u8,dS UUddB

minimum momentum ( f x / L ,f n / L , f a / L ) ( f x / L ,f a / L , f a / L ) (0,070)

= J3a/L = &/L IAninl = 0 l&,inl

Igrninl

+

are quantized as pi = 2nix/L for periodic BC and (2ni l ) x / L for antiperiodic BC with ni E Z. Therefore, @+ can have zero spatial momentum in 0.[ In contrast, N and K have non-zero minimum spatial moas Ism= menta as IgminI = &/L. Thus, under HBC, s-wave NK threshold energy is raised as Eth N

-4

+ J-,

Ismin1 = &T/L,

(6)

whereas a compact 5Q resonance state is expected to be unaffected. In order to see that HBC does not affect the spatially localized resonance states, we show an example of an established resonance C(uds) baryon. We select C, because it is subject to the periodic BC unlike N. In Fig. 3, we show Standard BC

Hybrid BC

2.0

2.0

s

1.9

1.8

2

1.8

1.7

'

1.7

1.9

6F

'

1.6

1.5

1.6 1.5

10

20 t

[ql

30

40

20

10 t

30

40

[ql

Figure 3. The effective mass plots of C ( u d s ) under the standard BC and HBC adopting Eq. (1). Solid lines denote the best-fit results performed in the region 20 5 t 5 30.

the effective mass plots of C for the standard BC and HBC. The solid lines denote the best-fit results performed in the plateau region as 20 5 t 5 30. There is no significant difference between the two best-fit masses, which shows that HBC does not affect the localized resonance states. Now we present the HBC result of the 5Q effective mass in the negative parity channel in Fig. 4. The dotted line denotes the modified NK threshold

322

20

10

30

40

1141 Figure 4. The effective mass plot for the negative parity 8+ under HBC adopting Eq. (1). The solid line denotes the result of the best-fit performed in the plateau region as 25 5 t 5 35. The dotted line denotes the s-wave NK threshold Eq. (6). This figure should be compared with the r.h.s. in Fig. 1. t

energy due to HBC. Note that the shift of the NK threshold amounts to about 200 MeV in this case, i.e., L N 2.15 fm with Eq. (1). We observe that the plateau is raised by consistent amount as the shift of the threshold, which shows that there is no compact 5Q resonance in the region as

In particular, the plateau observed in the negative parity channel in the previous section turns out to be an NK scattering state. In Fig. 5, we show the comparison of the results of standard BC (1.h.s.) and HBC (r.h.s.) for each K . Dots denote the best-fit mass obtained in their plateau region. the solid lines denote the NK threshold energies. We see that, for all K , the plateaus are raised in a consistent amount as the shift of the NK threshold energies.

3.0 -

d.1210

:

I 2.0

Standard BC

Hybrid BC

Figure 5 . Comparison of the results of standard BC (1.h.s) and HBC (r.h.s.) for each n. Dots denote the best-fit mas obtained in their plateau regions. The solid lines denote the corresponding NK threshold energies.

323

5. Summary and Discussion We have studied the penta-quark O+ state with the quenched anisotropic lattice QCD to provide high-precision data. After the chiral extrapolation, we have obtained me = 2.25 GeV in the positive parity channel. Since it is too massive, we have concluded that it cannot be identified with the experimentally observed O+( 1540). In contrast, in the negative parity channel, we have obtained me = 1.75 GeV, which is rather close to the empirical value. In order to confirm that this state is a compact 5Q resonance, we have proposed a new general method with “hybrid boundary condition (HBC)”. The H B C analysis has showed that the observed states in the negative parity channel are NK scattering states for all values of K . We have thus observed no relevant signals on the compact 5Q resonance in both the parity channels. To reveal the mysterious nature of O+(1540), more systematic investigations seem to be necessary. In the future study, it is desirable to examine the large volume effect, the dynamical quark effect, different interpolating fields including highly non-local ones, and different quantum numbers other than J = 1/2,I = 0.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14.

15. 16. 17.

LEPS Collaboration, T. Nakano et al., Phys. Rev. Lett. 91,012002 (2003). D. Diakonov, V. Petrov and M.V. Polyakov, 2. Phys. A359, 305 (1997). For a review, M. Oka, Prog. Theor. Phys. 111,1 (2004) and its references. S.L. Zhu, Int. J. Mod. Phys. A19, 3439 (2004) and its references. T. Nakano and K. Hicks, Mod. Phys. Lett. A19, 645 (2004). A.W. Thomas, K. Hicks and A. Hosaka, Prog. Theor. Phys. 111, 291 (2004). F. Csikor, Z. Fodor, S.D. Katz and T.G. KOVXS,JHEP 0311,070 (2003). S. Sasaki, Phys. Rev. Lett. 93, 152001 (2004). T.W. Chiu and T.H. Hsieh, hegph/0403020. N.Mathur, F.X.Lee, A.Alexandru, C.Bennhold, Y.Chen, S.J.Dong, T.Draper, I.HorvBth, K.F.Liu, S.Tamhankar and J.B.Zang, hep-ph/0406196. H. Suganuma, T.T. Takahashi, F. Okiharu and H. Ichie, Pmc. of QCD Down Under, Adelaide, March 2004, Nucl. Phys. B (Proc. Suppl.) in press; F. Okiharu, H. Suganuma and T.T. Takahashi, heplat/0407001. T.R. Klassen, NucZ. Phys. B533, 557 (1998). H. Matsufuru, T. Onogi and T. Umeda, Phys. Rev. D64, 114503 (2001). Y. Nemoto, N. Nakajima, H. Matsufuru and H. Suganuma, Phys. Rev. D68, 094505 (2003). N. Ishii, H. Suganuma and H. Matsufuru, Phys. Rev. D66, 094506 (2002); Phys. Rev. D66, 014507 (2002). N. Ishii, T. Doi, H. Iida, M. Oka, F. Okiharu, H. Suganuma, heplat/0408030. I. Montvay and G. Munster, “Quantum Fields on a Lattice”, (Cambridge Univ. Press, Cambridge, England, 1994), p. 1.

324

LATTICE QCD STUDY OF THE PENTAQUARK BARYONS

T. T. TAKAHASHI, T. UMEDA, T. ONOGI AND T. KUNIHIRO Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa-Oiwakecho,Sakyo, Kyoto 606-8502, Japan

We study the spin hadronic state in quenched lattice QCD t o search for a possible S = +1 pentaquark resonance. Simulations are carried out on 83 x 24, lo3 x 24, 123 x 24 and 163 x 24 lattices at /?=5.7 at the quenched level with the standard plaquette gauge action and Wilson quark action. We adopt two independent operators with I = 0 and J p = f to construct a 2 x 2 correlation matrix. After the diagonalization of the correlation matrix, we successfully obtain the energies of the ground-state and the 1st excited-state in this channel. The volume dependence of the energies suggests the existence of a possible resonance state slightly above the NK threshold in I = 0 and J p = channel.

4-

1. Introduction

After the discovery of Of(1540), the property of the particle is one of the central issues in hadron physics. While it is very likely that the isospin of Of is zero, the spin and the parity of Of and the origin of its tiny width of a few MeV still remain open questions. In spite of many theoretical studies on O+, the nature of the exotic particle is still controversial. As for the theoretical approaches, the lattice QCD is considered as one of the most reliable tools for studying the properties of hadron states from first principle, which has been very successful in reproducing the non-exotic hadron mass spectra. Up to now, several lattice QCD studies have been reported, which mainly aim to look for pentaquarks. However, the results are unfortunately contradictory with each other. On one hand, in refs. the authors conclude the parity of O+ is negative. On the other hand, in ref. 6, O+ in the positive parity channel is reported. In refs. even the absence of Of is suggested. A difficulty in the study of Of using lattice QCD comes from the contaminations of the scattering states of nucleon and Kaon, since the mass of O+ is above the NK threshold. In order to verify the existence of a resonance state, we need to properly discriminate the first few lowest states, 415,

718,

325

pick up one state as a candidate, and determine whether it is a resonance state or not. In this paper, we study I = 0 and J = channel in quenched lattice QCD. With the aim to extract the possible resonance state slightly above the NK threshold in this channel, we adopt two independent operators with I = 0 and J = and diagonalize the 2 x 2 correlation matrix of the operators. After the successful separation of the states, we investigate the volume dependence of the energy so that we can distinguish the resonance state from simple scattering states.

2. Formalism to separate the states

In lattice QCD calculations, the energies of states can be measured using the operator correlations. Let us consider the correlation (O(T)O(O)t)with O ( T ) an interpolating operator on the time plane t = T with a certain quantum number. The correlation (O(T)O(O)t ) physically represents the situation in which the states with the same quantum number as O ( t ) are created at t = 0 and propagate into Euclidean time direction and finally are annihilated at t = T . Taking into account the fact that the created state Otlwuc) can be expressed by the complete set of QCD eigenstates as Otlwac) = ~ 1 0 ) c l l l ) .. = C c i l i ) , we can write down the correlation (o(T)o(o)~) as (o(T)o(o)~) = Cicj(ile-fiTlj) = lci12e-EaT in

+

+

xi

xicj

terms of the QCD transfer matrix e-HT, with QCD Hamiltonian I? and nt h excited-state In) normalized as (mln)= dmn. We can extract the groundstate energy Eo by taking large T so that the correlation (O(T)O(O)t) behaves as a single-exponential function (O(T)O(O)t) IQ12e--EoT. It is however rather difficult to obtain the excited-state energy Ei(i > 0) from the multi-exponential function of T , which is a problem for extracting the possible pentaquark signal above the threshold. To overcome this difficulty, we adopt the variational method using correlation matrices of independent operators. In this method, we need a set of independent operators {O'}. We define the correlation matrix C I J as C I J ( T )3 ( O ' ( T ) O J t ( 0 ) ) .Then, from the product C-l(T l ) C ( T ) ,we can extract the energies { E i } as the logarithm of eigenvalues {eEz} of the matrix C 1 ( T l ) C ( T ) .In the case when we prepare N independent operators, the correlation matrix is an N x N matrix and we can obtain N eigenenergies { E i } (0 5 i 5 N - 1). N

+

+

326

3. Lattice set up

As interpolating operators {0'}, we take two independent operators; Ol(2) = - € " b c [ u ~ ( ~ ) ~ ~ 5 d b ( ~ ) ] u , ( ~ )[ ~(u ( ~H ) ~ ~dd) , which ( ~ ) ] is expected to have a larger overlap with O+ state, and 02(z) = & " b c [ ~ ~ ( ~ ) C ~ 5 d b ( ~ ) ] u , ( ~ ) [ S, ( (u ~ )H ~ 5dd) ,which ( z ) ] we expect to have larger overlaps with NK scattering states. Both of them have the quantum number of ( I ,J ) = (0, Here, Dirac fields u(x),d(z) and s(z) are up, down and strange quark field, respectively and the Roman alphabets {a,b,c,e} denote color indices. In this case, the correlation matrix is a 2 x 2 matrix. Furthermore, for a source, we construct and use "wall" operators Okall(t)and Oiall(t)defined using spatially spread quark fields q(z) as

i).

xz

5

-

(u ++ d ) . We fix the source operator e,,ll(t) on t=6 plane to reduce the effect of the Dirichlet boundary on t = O plane, and calculate C r J ( T= ) CZ(O'(2,T 6)Gia11(6)). We adopt the standard plaquette (Wilson) gauge action at p = 5.7 and Wilson quark action with the hopping parameters as (rcu,d, rc,)=(0.1600,0.1650), (0.1625,0.1650), (0.1650,0.1650), (0.1600,0.1600) and (0.1650,0.1600), which correspond to the current quark masses ( m u , d , m,) (240, loo), (170, loo), (100, loo), (240,240) and (100,240), respectively in the unit of MeV. We perform lattice QCD calculations at p = 5.7 employing four different sizes of lattices, 83x 24, lo3 x 24, 123 x 24 and 163 x 24 with 2900, 2900, 1950 and 950 gauge configurations. At p = 5.7, the lattice spacing a is set to be 0.17 fm so that the p meson mass is properly reproduced. Then, in the physical unit, the lattice sizes are 1.43 x 4.0 fm4, 1.73 x 4.0 fm4, 2.03 x 4.0 fm4 and 2.73 x 4.0 fm4. We take periodic boundary conditions in the directions for the gauge field, whereas we impose periodic boundary conditions on the spatial directions and the Dirichlet boundary condition on the temporal direction for the quark field. Many lattice studies about Of adopted a periodic or antiperiodic boundary condition on the temporal direction for quarks. In the case when the temporal length Nt is not long enough, we have to be careful about the contaminations by particles which go beyond the temporal boundary. For example, with the periodic boundary condition, the correlation (O(T)G(O)) contains the unwanted contributions such as (KIO(T)IN)(NIG(O)IK) e--ENTfEK(T-Nt). While no quark can go over the boundary on t = O in the temporal direction, the boundary is transparent for the particles composed only by gluons; ie. glueballs, due to the

+

N

N

''

327

periodicity of the gauge action. Taking into account that these particles are rather heavy, it would be however safe to neglect these gluonic particles going beyond the boundary. Then, the correlation (O(T)G(O)) mainly contains such terms as (w~c~~(T)~NK)(NK~~(O)~wac) and we can use the exactly same prescription mentioned in the previous section. 4. Ground-state and 1st Excited-state in (I,J p ) = ( 0 , $-)

channel In this section, we investigate the ground-state and the 1st excited-state in ( I ,J p ) = (0, f-) channel. We focus on the volume dependence of the energy of each state, to distinguish a possible resonance state from NK scattering states. We expect a resonance state to have almost no volume dependence, while the energies of NK scattering states are expected to scale

+d

as

m -

according to the relative momentum

9ii between N and K on finite periodic lattices on the assumption that the NK interaction is negligible. Note here that there are other possible finite volume effects due to the finite size of N and K or possible pentaquark

Fii.

states, than the volume dependence arising from We then need to take account of this fact in the following analysis. Now we show the lattice QCD results of the ground-state and the 1st excited-state in I = 0 and J p = f- channel. In Fig. 1, we plot the ground(K,,~,K,)=(O.I 650,0.1650) (~,,~,~,)=(0.1600,0.1600)

$

3

3

2.5

2.5

2

2

x

Y

1.5

K

1

1

0.5

0.5

w

0

-

1.5

x

F

-

1

2 Extent L [fm]

3

0

1

2

3

Extent L [fm]

i-)

Figure 1. The lattice QCD data of the ground-state in ( I , J p ) = (0, channel are plotted against the lattice extent L. The left figure is for ( ) E , J , ns) = (0.1650,0.1650) and the right figure is for (nu,d, ns) = (0.1600,0.1600). The solid line denotes the simple sum M N M K of the masses of nucleon M N and Kaon M K . We use the central values of M N and M K obtained on the lattice with L = 16

+

328

:-

state energies in I = 0 and J p = channel on four different volumes. Here the horizontal axis denotes the lattice extent L in the physical unit and the vertical axis is the energy of this state. The symbols with error-bars are lattice data and the solid line denotes the sum M N M K of the nucleon mass M N and Kaon mass M K . At a glance, we find that the energy of this state takes almost constant value against the volume variation and coincides with the simple sum M N M K . We can then conclude the ground-state in I = 0 and J p = $- channel is the NK scattering state with the relative momentum IpI = 0.

+

+

(K,,d,K,)=(O.I

(K,,,,,Ks)=(O.16OO,O.I

6 5 0 , O . i 650) 3.5

3.5

z9 >r

F Ill

600)

3 2.5

c

w

2 1.5

Figure 2.

2 3 Extent L [fm]

1

2 3 Extent L [fm]

a-) channel are d m + d m

The lattice QCD data of the 1st excited-state in ( I ,J p ) = (0,

plotted against the lattice extent L. The solid line represents with IpI = 2 x / L the smallest relative momentum on the lattice. The dashed line denotes

MN + M K .

Fig. 2 shows the 1st excited-state energies in this channel again in terms of L. The solid line shows J M & ( ~ T / L ) ~J M g ( ~ T / L as ) ~ the function of L. In this case, the lattice data exhibit clearly some volume dependence. One would naively consider this dependence to be that of the 2nd lowest NK scattering state, which scales as J M & (~T/L)~ JM$ ( ~ T / L ) and ~ , would take the deviations as from other possible finite size effects. However, if it is the case, the significant deviations in 1.5 5 L 5 3 fm in the right figure may not be justified. In the range of L where the lattice data in the left figure scale just as the solid line, we should observe the same behavior in the right figure. Because, in the case when quarks are heavy, composite particles will be rather compact and we expect smaller finite size effects. We can understand this behavior on the assumption that this state is

+

+

+

+

+

+

329

a resonance state rather than a scattering state. In fact, while the data in the left figure rapidly go above the solid line as L is decreased, which can be considered to arise due to the finite size of a resonance state, the lattice data exhibit almost no volume dependence in the right figure especially in 1.5 5 L 2 3 fm,which can be regarded as the characteristic of resonance states. 5 . ( I ,J p )

= ( 0 , f')

channel

3.5

F 2 x

9 a

3 2.5

c

w

2

1.5

1

2

1.5 I

3

2

1

Extent [frn]

3

Extent [frn]

3')

Figure 3. The lattice QCD data in the (I,J p ) = (0, channel are plotted against the lattice extent L. The solid line denotes the simple sum MN* M K of the masses of the ground-state negative-parity nucleon MN*and Kaon M K .

+

In the same way as ( I ,J p ) = (0, $-) channel, we have attempted to diagonalize the correlation matrix on ( I ,J p ) = (0, channel using the wall-sources gwall(t). In this channel, the diagonalization is rather unstable and we find only one state except for tiny contributions of possible other states. We plot the lattice data in Fig. 3. One finds that they have almost no volume dependence and that they coincide with the solid line which represents the simple sum M N . M K of M N . and M K , with MN* the mass of the ground-state of the negative-parity nucleon. From this fact, the state we observe is concluded to be the N * K scattering state with the relative momentum IpI = 0. It is rather strange because the p-wave state of N and K with the relative momentum IpI = 2 7 ~ / Lwill be lighter than the N * K scattering state with the relative momentum IpI = 0. We miss this lighter state in our analysis. This failure would be due to the walllike operator Owall(t): The operator @,,,ll(t) is constructed by the spatially spread quark fields C2q(x)with zero momentum. This may lead to the

b')

+

330

large overlaps with the scattering state with zero relative momentum. It is also desired to try another operator which couples to p-wave NK scattering state. 6. Summary We have performed the lattice QCD study of the S=+l pentaquark baryons on g3 x 24, lo3 x 24, 123 x 24 and 163x 24 lattices at p=5.7 at the quenched level with the standard plaquette gauge action and Wilson quark action. With the aim to separate states, we have adopted two independent operators with I = 0 and J p = $ so that we can construct a 2 x 2 correlation matrix. From the correlation matrix of the operators, we have successfully obtained the energies of the ground-state and the 1st excited-state in ( I ,J p ) = (0, +-) channel. The volume dependence of the energies shows that the 1st excited-state in this channel is rather different from a simple NK scattering state and is likely to be the resonance state located slightly above the NK threshold, and also indicates that the ground-state is the NK scattering state with the relative momentum IpI = 0. As for the ( I ,J p ) = (0,i') channel, we have observed only one state in the present analysis, which is likely to be a N*K scattering state of the negative-parity nucleon N* and Kaon with the relative momentum Ip( = 0.

References 1. T. Nakano et al., Phys. Rev. Lett. 91,012002 (2003). 2. M. Oka, Prog. Theor. Phys. 112, 1-19 (2004). 3. For instance, CP-PACS Collaboration (A. Ali Khan et al.), Phys. Rev. D65, 054505 (2002); Erratum-ibid. D67,059901 (2003). 4. F. Csikor, Z. Fodor, S. D. Katz, T. G. Kovacs, JHEP 0311, 070 (2003). 5. S. Sasaki, hep-lat/0310014. 6. T-W. Chiu, T-H. Hsieh, hep-ph/0403020. 7. N. Mathur et al., hep-lat/0406196. 8. N. Ishii, T. Doi, H. Iida, M. Oka, F. Okiharu, H. Suganuma, hep-lat/0408030. 9. T. T . Takahashi and H. Suganuma, Phys. Rev. D70,074506 (2004) and references therein. 10. CP-PACS collaboration (T. Yamazaki et al.), hep-lat/0402025.

331

SIGNAL OF O+ IN QUENCHED LATTICE QCD WITH EXACT CHIRAL SYMMETRY*

TING-WAI CHIU, TUNG-HAN HSIEH Physics Department and National Center for Theoretical Sciences at Taipei National Taiwan University, Taipei, Taiwan 106, Taiwan

We investigate the mass spectrum of the pentaquark baryon ( d u d s ) in quenched lattice QCD with exact chiral symmetry. Using 3 different interpolating operators, we measure their 3 x 3 correlation matrix and obtain the eigenvalues A* (t) with f parity, for 100 gauge configurations generated with Wilson gauge action a t 0 = 6.1 on the 203 x 40 lattice. For the lowest-lying J p = 1/2- state, its effective mass is almost identical to that of the KN s-wave, while for the lowest-lying J p = 1/2+ state, its effective mass is smaller than that of the KN p-wave, especially for the regime mu < m,. By chiral extrapolation (linear in m:) to mT = 135 MeV, we obatin the masses of the lowest-lying states: m(1/2-) = 1424(57) MeV and m(l/2+) = 1562(121) MeV, in agreement with the masses of m~ m~ N 1430 MeV and 0+(1540) respectively.

+

1. Introduction

The recent experimental observation of the exotic baryon W(1540) (with the quantum numbers of K+n) by LEPS collaboration1 at Spring-8 and the subsequent confirmation from some experimental groups has become one of the most interesting topics in hadron physics. The remarkable features of 8+(1540) are its strangeness S = +1, and its exceptionally narrow decay width (< 15 MeV) even though it is 100 MeV above the KN threshold. Its strangeness S = +1 immediately implies that it cannot be an ordinary baryon composed of three quarks. Its minimal quark content is ududs. Nevertheless, there are quite a number of experiments2 which so far have not observed 0+(1540) or any pentaquarks. This casts some doubts about the existence of W(1540). Historically, the experimental search for 0+( 1540) was motivated by the predictions of the chiral-soliton model3, an outgrowth of the Skyrme N

*This work was supported in part by the National Science Council, ROC.

332

model. Even though the chiral solition model seems to provide very close predictions for the mats and the width of 8+(1540), obviously, it cannot reproduce all aspects of QCD, the fundamental theory of strong interactions. Now the central theoretical question is whether the spectrum of QCD possesses 6+ with the correct quantum numbers, mass, and decay width. At present, the most viable approach to solve QCD nonperturbatively from the first principles is lattice QCD. Explicitly, one needs to construct an interpolating operator which has a significant overlap with the pentaquark baryon states. Then one computes the time-correlation function of this interpolating operator, and from which to extract the masses of its even and odd parity states respectively. However, since any (ududa) operator must couple to hadronic states with the same quantum numbers, it is necessary to disentangle the lowest lying pentaquark states from the K N scattering states, as well as the excited pentaquark states. To this end, we adopt the variational methodls~lg,and use three different interpolating operators of (udud.5) to compute their 3 x 3 correlation matrix, and from its eigenvalues to extract the masses of the even and odd parity states. These three interpolating operators (with I = 0) are 4,5,6,7,8,9,10:

(01

Iza

= [ u T c y 5 d I z c {Szpe ( ~ 5 ) ~ 7 7 u z ,(e~ 5 d ) z a c --Szpe(y5)p&z,e

(021,~

(75u)zac

1

(1)

= [uTcy5d]zc { ~ z p e ( y 5 ) / 3 9 ~ z 9 c ( y 5 d ) z a e

1

--Bzpe(y5)pqdzqc ( ~ 5 u ) z a e

( 0 3 ) ~=~ ~ c c i e [ ~ ~ ~ y 5 [du ]Tzc cd ] z d (CST)zae

(2) (3)

where u, d and s denote the quark fields; €abc is the completely antisymmetric tensor; 2, { a , b , c } and { a , p , q } denote the lattice site, color, and Dirac indices respectively; and C is the charge conjugation operator. Here the diquark operator is defined aa

where r a p = -J?pa. Thus the diquark transforms like a spin singlet (Is), color anti-triplet (gC),and flavor anti-triplet (35). For r = C y 5 , it transforms as a scalar, while for r = C , it transforms like a pseudoscalar. Note that 01, 0 2 , and O3 all transform like an even operator under parity. In the Jaffe-Wilzcek model7, each pair of [ud]form a diquark. Then the pentaquark baryon e([ud][ud]S)emerges &s the color singlet in ( 3 , x 3,) x 3, = 1, -t 8, 8, and a member (with S = +1 and I = 0)

+ + m,

333

+ + + mf.

of the flavor anti-decuplet in 5 f x 3 f x 35 = 15 85 85 NOW, if one attempts t o construct a local interpolating operator for [udJ[ud]B, then these two identical diquarks must be chosen to transform differently (i.e., one scalar and one pseudoscalar), otherwise €abc[Ud]zb[Ud]zc%aa is identically zero since diquarks are bosons. Thus, when the orbital angular momentum of this scalar-pseudoscalar-antifermionsystem is zero (i.e., the lowest lying state), its parity is even rather than odd. Alternatively, if these two diquarks are located at two different sites, then both diquark operators can be chosen to be scalar, however, they must be antisymmetric in space, i.e., with odd integer orbital angular momentum. Thus the parity of lowest lying state of this scalar-scalar-antifermion system is even, as suggested in the original Jaffe-Wilzcek model. (Note that all correlated quark models e.g., Karliner-Lipkin modelll and flavor-spin model12, advocate that the parity of 9+(1540) is positive.) 2. Computation of quark propagators

Now it is straightforward to work out the pentaquark propagator (9,,e,a) in terms of quark propagators. In lattice QCD with exact chiral symmetry, quark propagator13 with bare mass m, is of the form (D, mq)-'where D, is exactly chirally symmetric at finite lattice spacing. In the continuum limit, (D, m,)-' reproduces [r,(a, iA,) m,]-'. For the optimal domain-wall fermion14 with N , + 2 sites in the fifth dimension,

+

+

+

+

where Owis the standard Wilson Dirac operator plus a negative parameter -mo (0 < rno < 2), and {us}are a set of weights specified by an exact formula such that D, possesses the optimal chiral symmetry14. Since

+

and D(m,) = m, (mo - m,/2)[1+ ry5S(Hw)],thus the quark propagator can be obtained by solving the system D(m,)Y = I with nested conjugate gradient15, which turns out to be highly efficient (in terms of the precision of chirality versus CPU time and memory storage) if the inner conjugate gradient loop is iterated with Neuberger's double pass algorithm16.

334

We generate 100 gauge configurations with Wilson gauge action at 0 = 6.1 on the 203 x 40 lattice. Then we compute two sets of (point-to-point) quark propagators, for periodic and antiperiodic boundary conditions in the time direction respectively. Here the boundary condition in any spatial direction is always periodic. Now we use the averaged quark propagator to compute the time correlation function of any hadronic observable such that the effects due to finite T can be largely reduced17. Fixing mo = 1.3, we project out 16 low-lying eigenmodes of IH,I and perform the nested conjugate gradient in the complement of the vector space spanned by these eigenmodes. For N , = 128, the weights {us}are = 6.3, where Amin 5 A(IH,I) 5 ,A ,, fixed with Amin = 0.18 and ,A,, for all gauge configurations. For each configuration, (point to point) quark propagators are computed for 30 bare quark masses in the range 0.03 5 mqa 5 0.8, with stopping criteria and 2 x for the outer and inner conjugate gradient loops respectively. Then the norm of the residual vector of each column of the quark propagator is less than 2 x and II(D, m,)Y - 111 < 2 x the chiral symmetry breaking due to finite Ns(=128) is less than (T = (YtS2Y/YtY- 11 < for every iteration of the nested conjugate gradient.

+

3. Determination of a-1 and m,

After the quark propagators have been computed, we first measure the pion propagator and its time correlation function, and extract the pion mass (m,a) and the pion decay constant (f,a). With the experimental input fr = 132 MeV, we determine = 2.237(76) GeV. The bare mass of strange quark is determined by extracting the mass of vector meson from the time correlation function

At mqa = 0.08, M v a = 0.4601(44), which gives Mv = 1029(10) MeV, in good agreement with the mass of 4(1020). Thus we take the strange quark bare mass to be m,a = 0.08. Then we have 10 quark masses smaller than m,, i.e., m,a = 0.03,0.035,0.04,0.045,0.05,0.055,0.06,0.065,0.07,0.075. In this paper, we work in the isospin limit mu = md.

335

4. The 3

x 3 correlation matrix for 0

Next we compute the propagators ((Oi)za(Oj)ya) with fixed 9 = (6,0),and their time correlation functions C$(t) with f parity

where the trace sums over the Dirac space, and the subscripts f and U denote fermionic average and gauge field ensemble average respectively. Then the 3 x 3 correlation matrix C*(t) = (C;(t)} can be constructed. Now with a judiciously chosen t o , we diagonalize the normalized correlation matrix C * ( t 0 ) - l / ~ C * ( t ) C * ( t 0 ) -and ~ / ~obtain eigenvalues {A:(t)}, and from which t o extract the masses { m f }of the lowest lying and two excited ~ states for f parity respectively. This is the variational m e t h ~ d to disentangle the lowest and the excited states. Then the mass m: can be extracted by single exponential fit to A'(t), for the range o f t in which the effective mass M e ~ ( t = ) ln[A(t)/A(t + l)]attains a plateau. 6 ,

I

EWecWmassof + lowest wng state J ~ 5-

-6 -7

a

mua = 0.1

-9 -10 -11

rn p

4 .

-

0.10

3 1;

-14 -15 -16 -17 -18 -19

-20 -21 -22 -23

0

5

10

t

15

20

04

0

5

10

15

I

20

t

Figure 1. (a) The eigenvalue A+(t) of the lowest positive parity state, for m,a = 0.1. The solid line is the single exponential fit for 9 5 t 5 14. (b) The effective mass Meff(t) = ln[A(t)/A(t + l)]of A+(t) in Fig. la.

In Fig. la, the eigenvalue A+(t) corresponding to the lowest J p = 1/2+ state is plotted versus the time slices, for m,a = 0.1, while the corresponding effective mass is shown in Fig. lb. Here we have suppressed any data

336

point which has error (jackknife with single elimination) larger than its mean value. Similarly, the eigenvalue A - ( t ) corresponding to the lowest J p = 1/2- state and its effective mass are plotted in Fig. 2a and Fig. 2b. First, we observe that the effective mass of the lowest J p = 1/2+ state in Fig. l b attains a plateau for t E [9,14],and its mass can be extracted by single exponential fit (Fig. la). Similarly, the effective mass of the lowest J p = 1/2- state in Fig. 2b attains a plateau for t E [ll,171, and its mass can be extracted by single exponential fit (Fig. 2a). -2 -3 -4

Lowest lying state JP=$

-5

0

5

15

10

20

25

0

5

1

0

t

(4

1

5

2

0

2

5

t

(b)

Figure 2. (a) The eigenvalue A - ( t ) of the lowest negative parity state, for mua = 0.1. The solid line is the single exponential fit for 11 5 t 5 17. (b) The effective mass of A - ( t ) in Fig. 2a.

In Fig. 3a, the masses of the lowest lying J = (1/2)* states are plotted versus m:, where all mass fits have x 2 / d . o . f .< 1. Using the four smallest masses (i.e., with mua = 0.03,0.035,0.04,0.045) for chiral extrapolation (linear in rn; ) t o physical pion mass m, = 135 MeV, we obtain the mass of the lowest lying states: m(l/2-) = 1424(57) MeV, and m(l/2+) = 1562(121) MeV, in agreement with the masses of mK mN N 1430 MeV, and e(1540) respectively.

+

5. Distinguishing the K N scattering states

Now the question is whether they are scattering states or bound states. In order to obtain the mass spectrum of K N scattering states, we consider the

337

time correlation function of K N operator without any exchange of quarks between K and N in its propagator, i.e., the interaction between K and N is only through the exchange of gluons. Explicitly,

where N = [uTCy5d]d,and K = gy5u. The masses of lowest lying K N scattering states are plotted in Fig. 3b. For the J p = 1/2- state, using the four smallest masses for chiral extrapolation to m, = 135 MeV, we obtain m K N ( 1 / 2 - ) = 1433(72) MeV, in agreement with the mass of mK mN N 1430 MeV, the KN s-wave. Further, its mass spectrum is almost identical to that of the lowest J p = 1 / 2 - state of 8 in Fig. 3a, for the entire range of mu. Thus we identify the lowest J p = 1 / 2 - state of 63(ududS) with the KN s-wave scattering state. On the other hand, for the J p = 1/2+ state, its mass is much higher than the naive estimate Jm& ( ~ T / L )Jm& ~ ( ~ T / Lwhere ) ~ , L is the lattice size in spatial directions. This suggests that the KN p-wave (in the quenched approximation) in a finite torus is much more complicated than two free particles with momenta J?K = -J?N = 27r&/L. Further, the mass of KN p-wave scattering state in Fig. 3b is always larger than the mass of the J p = 1/2+ state in Fig. 3a. In particular, for mu < m,, the former is significantly larger than the latter. This seems to suggest that the lowest J p = 1/2+ state of 8(ududs) is difleerent from the KN p-wave scattering state. In other words, it is likely to be a bound state with mass 1562(121) MeV. If it is identified with 8+(1540), then it predicts that the parity of 8+(1540) is positive.

+

+

+

+

6. Concluding remarks

It is vital t o re-confirm the above picture with a larger lattice, smaller lattice spacing, and higher statistics, especially for the regime mu < m,, which is crucial for the chiral extrapolation. Further, to ensure that the lowest lying J p = 1/2+ is not a scattering state, one can also compute its spectral weight for two different volumes4, since for a scattering state, its spectral weight is inversely proportional to the volume. Even if it is confirmed to be a bound state, one still has to find out whether its decay width could be as small as 20 MeV, compatible to that of 8+(1540). Our present data only shows an unambiguous signal of pentaquark (ududs) resonance around 1540 MeV, with S = +1 and I ( J p ) = 0(1/2+).

338 4.47

KN scattering states

Lowest lying states of 6mdudS) 3.38

z =I

2.24

1.12

. 0.10 0.20 0.30 0.40 0.50 0.80 0.70

m (: GeV')

.

.

.

.

.

.

0.10 0.20 0.30 0.40 0.50 0.80 0.70

mn2(GeV2)

Figure 3. (a) The masses of the lowest lying states of 0(udu&). The solid lines are chiral extrapolation (linear in m:) using four smallest masses. (b) The masses of the lowest lying K N scattering states. The solid line (for J p = 1/2-) is the chiral extrapolation using four smallest masses.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16. 17. 18. 19.

T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91,012002 (2003) For a recent review, see K. Hicks, hep-ex/0412048, and this proceedings. D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359,305 (1997). N. Mathur et al., hep-ph/0406196. S. L. Zhu, Phys. Rev. Lett. 91,232002 (2003) F. Csikor, Z. Fodor, S. D. Katz and T. G.Kovacs, JHEP 0311,070 (2003) R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91,232003 (2003) J. Sugiyama, T. Doi and M. Oka, Phys. Lett. B 581,167 (2004) S. Sasaki, hep-lat/0310014. T. W. Chiu and T. H. Hsieh, hep-ph/0403020. M. Karliner and H. J. Lipkin, Phys. Lett. B 575,249 (2003) F. Stancu and D. 0. Riska, Phys. Lett. B 575,242 (2003) T. W. Chiu, Phys. Rev. D 60, 034503 (1999) T. W. Chiu, Phys. Rev. Lett. 90, 071601 (2003); Phys. Lett. B 552, 97 (2003); hep-lat/0303008, Nucl. Phys. Proc. Suppl. 129,135 (2004). H. Neuberger, Phys. Rev. Lett. 81, 4060 (1998); Phys. Lett. B 417, 141 (1998) H. Neuberger, Int. J. Mod. Phys. C 10, 1051 (1999); T. W. Chiu and T. H. Hsieh, Phys. Rev. E 68, 066704 (2003) T. W. Chiu and T. H. Hsieh, in preparation. C. Michael, Nucl. Phys. B 259, 58 (1985) M. Luscher and U. Wolff, Nucl. Phys. B 339,222 (1990)

339

THE STATIC PENTAQUARK POTENTIAL IN LATTICE QCD* FUMIKO OKIHARU Nihon University, 1-8-14 Kanda Surugadai, Chiyoda, Tokyo 101-8308, Japan

HIDE0 SUGANUMA Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo 152-8551, Japan

TORU T. TAKAHASHI Y I T P , Kyoto University, Kitashirakawa, Sakyo, Kyoto 606-8502, Japan We perform the first study for the static penta-quark (5Q) potential in lattice QCD with P=S.O and 163 x 32 at the quenched level. Accurate results of the 5Q potential are extracted from the 5Q Wilson loop using the smearing method, which enhances the ground-state component. The tetra-quark potential for the QQ-QQ system is also studied in lattice QCD. The multi-quark potentials are found to be well described as a sum of the one-gluon-exchange Coulomb term and the multi-Y linear confinement term based on the flux-tube picture.

1. Introduction

The recent experimental discoveries of O+(1540),ly2 E--(1862)3 and 0,(3099)4 as the candidates of penta-quark (5Q) baryons are expected to reveal new aspects of QCD and hadrons. Such experiments were motivated by the theoretical prediction by Diakonov et aL5, and many theoretical studies have been done to clarify the 5Q baryon^,^*^-^^ However, there are so many open problems for several remarkable features of the 5Q baryons. Experimental data indicate extremely narrow decay widths and small masses of the 5Q baryons, and the parity determination is also an open problem. For the physical understanding of these features, theoretical analyses are necessary as well as the experimental studies. In particular, to clarify the inter-quark force in the multi-quark system based on QCD is required for the realistic modeling of the multi-quark system. For this purpose, we perform the static penta-quark potential V& in lattice QCD. We here investigate QQ-Q-QQ type configurations for the 5Q *The lattice QCD simulation has been done on NEC-SX5 a t Osaka University.

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system8y9where the two QQ clusters belong to 3*representation in SU(3),, since the 3* diquark has a smaller energy than the 6 diquark.6 Similarly, we investigate QQ-QQ type configurations for the 4Q system where the QQ and the QQ clusters belong to 3* and 3 representation, respectively. 2. Theoretical Ansatz: OGE Coulomb plus multi-Y Ansatz

To begin with, we give a theoretical consideration for the multi-quark potential. From recent lattice QCD studies, the static three-quark (3Q) potential for baryons is found to obey the Coulomb plus Y-type linear potential, i e . , the Y-Ansatz.18 In fact, the lattice data for the QQ and the 3Q potentials can be well described as

where A denotes the Coulomb coefficient, D the string tension, C a constant and Lmin the minimal length of the flux tube linking the valence quarks. These lattice QCD results indicate the flux-tube picture for hadrons. We generalize the Y-Ansatz to the static 5Q potential v5Q, and conjecture the one-gluon-exchange (OGE)Coulomb plus multi-Y linear potentia18J’ for V;Q, i.e., OGE Coulomb plus multi-Y Ansatz,

= -A~Q{(-

1

1 1 1 1 1 + -)7-34 +1-(+ - + - + -) 2 T15 T25 7-45

~ 1 2

735

where the two quarks at (rl, 1 2 ) and those at ( 1 3 , r4) belong to 3*representation, respectively, and the antiquark locates at r5. Here, Lmin is defined as the minimal length of the flux tube linking all the valence (anti-)quarks. For the 4Q system, OGE Coulomb plus multi-Y Ansatz is expressed as

where two quarks locate at (rl, 1-2) and two antiquarks at (r3, r4). We theoretically expect +AQO = A ~ Q =A ~ Q =A ~ Q as the OGE result, and O ~ Q = D3Q = D4Q = O5Q as the universality of the string tension.

341

3. Multi-quark Wilson loops and multi-quark potentials In QCD,the static potential V is derived from the Wilson loop W as 1 V = - lim -In(W). T+m

T

(5)

The static multi-quark potentials can be also obtained from the corresponding multi-quark Wilson loops. As shown in Figs.1 and 2, we define the 5Q Wilson loop W S Q and ~ the 4Q Wilson loop W4Q as 1 W5Q E -eabcea ll;raa1(x3x12x4)bb’(ii3ii12ii4)CC1, (6) 3! 1 W4Q 3 ~tr(&flxIZM2RlZ), (7) I

where xi, R i , M , M

j

l

l

(i = 1 , 2 , 3 , 4 , j = 1 , 2 ) are given by

xi,&,M ,fij E Pexp{ig J ~ ’ ” A ’ ” ( 4E) SU(3),, (8) i e . , ii,a,I@, (i = 3 , 4 , j = 1,2) are line-like variables and ti, & (i = Li ,Ri , M , M j

ll;rj

1,2) are staple-like variables, and

x12,

R l z are defined by

Note that both the 4Q Wilson loop W4Q and the 5Q Wilson loop W5Q are gauge invariant.

!IL

R2

Figure 1. The penta-quark (5Q) Wilson loop w6Q. A gauge-invariant 5Q state is generated at t = 0 and annihilated at t = T . Four quarks and an antiquark are spatially fixed for 0 < t < T .

Figure 2. The tetra-quark (4Q) Wilson loop w4Q. A gauge invariant 4Q state is generated at t = 0 and annihilated at t = T . Two quarks and two antiquarks are spatially fixed for 0 < t < T .

In general, the multi-quark operator in the multi-quark Wilson loop contains excited-state components. In order to extract the ground-state potential, we use the smearing method.lE By this procedure, we obtain the quasi-ground-state operators for the static multi-quark systems, and thus perform the accurate calculations for the multi-quark potentials with them.

342

4. Lattice QCD results and Concluding Remarks

The lattice QCD simulation is performed at O, = 6.0, i.e., a N 0.1 fm, on the 163 x 32 lattice at the quenched l e ~ e l In . ~this ~ ~paper, we investigate the planar and twisted configurations for the 5Q system as shown in Figs.3 and 4, and demonstrate the case of dl = d2 = d3 = d4 d and hl = h2 h/2.

Figure 3. A planar configuration of the penta-quark system.

Figure 4. A twisted configuration of the penta-quark system with Q1Q2 IQ3Q4.

In Fig.5, we show the lattice QCD results of the 5Q potential v5Q.The lattice data denoted by the symbols are found to be well reproduced by the theoretical curves of the OGE plus multi-Y an sat^^*^^ with ( A ~ Q05Q) , fixed to be ( A 3 ~O3Q) , N (0.1366, 0.046aW2)in the 3Q potential.lEb 2.6

2.6

2.4

2.4

2.2

2.2

2

2

1 .8

1.8

1.6

1.6

1.4 1.2’ 1

1.4 ’

’

2

3

.

4

’

5

’

’

6 h

7

’

8

I

’

9

1

0

1

2

3

4

5

6

7

8

9

10

h

Figure 5. Lattice QCD results of the 5Q potential VSQin the lattice unit for the planar configurations (left) and the twisted configurations (right). The symbols denote the lattice data. The theoretical curves of the OGE plus multi-Y Ansatz are added.

Figure 6 shows the lattice results of the 4Q potential V4Q. For large h, V4Q coincides with the energy h 4 Q of the connected 4Q system. For small h, v4Qcoincides with the energy 2vQQ of the “two-meson” system composed of two flux-tubes. Thus, we get the relation of V4Q = min(&Q,2VQQ), and find the “flip-flop” between the connected 4Q system and the “twoaFor the extreme case, e.g., d > &hi, we here assume that the flux-tube is formed on the straight lines of QlQtj and Q2Qs. bDue to this fixing, there is no adjustable parameter except for an irrelevant constant.

343 2 1.8 1.6 1.4 1.2 1 0.8

. . .

,

,

,

,

,

.

1

4

5 h

6

7

8

9

2

3

0.8 1

2

3

4

5 h

6

7

8

9

Figure 6. The 4Q potential V ~ Q for d = 1 (left) and d = 2 (right) for the planar 4Q configuration similar to Fig.3. The horizontal axis h corresponds to hl+h2. The symbols denote the lattice QCD results. The theoretical curves are added for the connected 4Q system (the solid curve) and for the “two-meson”system (the dashed curve).

meson” system around the level-crossing point where these two systems are degenerate as V c 4 ~ = 2vQQ. This result also indicates the flux-tube picture. To summarize, we have performed the static 5Q and 4Q potentials in lattice QCD, and have found that the multi-quark potentials are well described with the OGE Coulomb plus multi-Y linear potential except for extreme cases. For the static 4Q potential, we have found the “flip-flop” between the connected 4Q system and the “two-meson” system. The present lattice QCD results for the multi-quark potentials provide a guiding principle in modeling the multi-quark system. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

LEPS Collaboration (T. Nakano et d.), Phys. Rev. Lett. 91, 012002 (2003). S.L. Zhu, Int. J. Mod. Phys. A19, 3439 (2004) and references therein. NA49 Collaboration (C. Alt et al.), Phys. Rev. Lett. 92, 042003 (2004). H1 Collaboration (A. Aktas et al.), Phys. Lett. B588, 17 (2004). D. Diakonav, V. Petrov and M. Polyakov, 2. Phys. A359, 305 (1997). R.L. JafTe and F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003). F1. Stancu and D.O. Riska, Phys. Lett. B575, 242 (2003). F. Okiharu, H. Suganuma and T.T. Takahashi,hep-lat/0407001 (2004). H. Suganuma, T.T. Takahashi, F. Okiharu and H. Ichie, Proc. of QCD D o m Under, March 2004, Adelaide, Nucl. Phys. B (Proc. Suppl.) (2004) in press. M. Karliner and H.J. Lipkin, hep-ph/0307243 (2003). N. Mathur et al., hep-ph/0406196 (2004), and references therein. I.M. Narodetskii et al., Phys. Lett. B578, 318 (2004). A. Hosaka, Phys. Lett. B571, 55 (2003). P. Bicudo and G.M. Marques, Phys. Rev. D69, 011503 (2004). X.-C. Song and S.-L. Zhu, hep-ph/0403093 (2004). Y. Kanada-Enyo, 0. Morimatsu and T. Nishikawa, hep-ph/0404144 (2004). For a review, M. Oka, Pmg. Theor. Phys. 111,1 (2004) and its references. T.T. Takahashi et al., Phys. Rev. Lett. 86,18 (2001); Phys. Rev. D65,114509 (2002); Phys. Rev. Lett. 90, 182001 (2003); Phys. Rev. D70, 074506 (2004).

344

QCD SUM RULES OF PENTAQUARKS

MAKOTO OKA Department of Physics, H-27,Tokyo Institute of Technology Meguro, 152-8551, Japan E-mail: [email protected] QCD sum rule is applied t o the pentaquark spectroscopy. It is concluded that no pmitive parity state is seen in low energy region, while there may exist negative parity states at around 1.5 GeV. Choice of interpolating local operators and relation to the lattice calculations are discussed.

1. Introduction

The newly discovered pentaquark Q+1*2 and its siblings are quite mysterious from the QCD viewpoint. Many questions are raised including 0

0 0

Why are they so light? The masses Ad(@+) = 1540 MeV and M ( E - - ) = 1862 MeV are unexpectedly small as states with five constituent quarks. Why are they so narrow? The width I?(@+) < 10 MeV is unusually narrow. This may be the biggest problem in understanding the structure of @+. What are the spin, parity, SU(3) irrep., .. .? How are they produced? This is a question raised from the beginning, but recent reports of “non-observation” of pentaquarks make this question more important. In which process do we have to see the state?

The spin and parity seem to be crucial to determine the structure of the pentaquarks. In the naive quark models, the ground state in which all the quarks occupy the lowest energy mode should have negative parity. We expect both the 1/2 or 3/2 states, whose splitting is given by the spin-spin interaction between quark^.^*^ On the other hand, various model calculations predict a positive parity state. In the chiral quark model,5 whose prediction motivated the experimental search at the Spring-8, the expected state has 1/2+. Another popu-

345

lar type of models are diquark models, in which a ud (S = 0, I = 0) diquark plays a crucial r0le.~7’ The ground states in those models are predicted to be a positive parity state with spin 1/2 or 3/2. It is thus very important to determine the parity, but experimentally its determination turns out to be non-triviaL8 So while our experimental colleagues are working hard to determine the spin and parity of the pentaquarks, theorists should make predictions directly from QCD. This is the subject of this talk. The main part of this talk is based on the research carried out by the Tokyo Tech. group including J. Sugiyama, T. DO^,^ N. Ishii, H. Iida, F. Okiharu, and H. Suganuma.lo 2. Two-point Correlator in QCD

A tool to attack the pentaquark question in QCD is two-point correlator (TPC). TPC defined by

n(p) =

J

d4zeip.’(01T(J(s)~(0))(O)

(1)

contains all information of the spectrum of the channel specified by the employed interpolating field local operator, J ( z ) . In the lattice QCD, TPC with the imaginary time variable T is considered, By integrating over 3-D coordinate 2 and taking the large T limit, it will give the maas of the ground state.

rII(p’= 0;). =

J d3z (OpyJ(Z,7-)9(0,0))(0) i

The QCD sum rule is a relation obtained by calculating TPC in two ways. In one hand, it is calculated in the deep Euclidean region p i + 00. This is the region where the perturbation theory is -p2 valid because the QCD coupling constant ctd(pg)is suppressed according to the asymptotic freedom of QCD. In fact, QCD allows a systematic expansion called the operator product expansion (OPE) in this regime and nonperturbative effects are taken into account as vacuum condensates of local operators.

346

where On represents a local operator of dimension n. For higher n, Cn is suppressed as the n-th power of l / p i at large &, which leads to convergence of OPE. The other side of the sum rule is TPC calculated at the physical region, where it is represented by a spectral function p(s) at s = p2 = m2: 1 p(s) = -ImrI(s) = JX$6(S - m!) (4)

c

7r

i

where i labels hadrons (poles) and scattering states (cut). Incidentally a common assumption in the QCD sum rule is to parametrize the spectral function by a sharp resonance and continuum at s > SO, p(s)

N

+

[XI26(s - m2) B(s - so) p ( s ) o p ~

90:

threshold

(5)

Then the two expressions of TPC at different kinematical regimes are related by using analyticity of ll(p2) in complex s = p2 variable. The analytic continuation is achieved by a dispersion relation,

In order to improve sum rule performance, it is common to apply the Borel transform defined by

where M 2 is a mass scale introduced in the transform and is called Borel mass. An example of the Borel transform is ds =

1"

e-8/M21rnII(s)ds,

which shows its role clearly, that is, it enhances the pole region by the weight function e-s/M2. It also accelerates the convergence of OPE by an extra numerical factor proportional to l / n ! .

2.1. Parity Projection Our main aim is to determine the parity of pentaquarks, for which we need parity projection. Parity projection for J = 1/2 baryons was formulated by Jido et al.ll There the authors employ the forward-time TPC,

II( p ) =

J

d4zeipZB(zo)(01 JB (z).JB (0) lo),

(9)

347

then its imaginary part at p" 0 can be projected into positive parity and negative parity parts as

+ (continuum) = To A(m)+ B(po).

OPE of the correlator shows that A(p0) contains chiral even parts, such as the gluon condensate, while B(po) corresponds to the chiral odd parts, such as the quark condensate, (qq). Now one sees that the combinations, A + B and A - B , represent spectral functions of the positive and negative panty baryons, respectively. By constructing the QCD sum rule for A f B , one can extract masses and coupling strengths of the positive and negative parity baryons. It is concluded that splitting of positive and negative parity baryons is determined by chiral odd terms in OPE, and thus chiral symmetry breaking of the QCD vacuum. The h a 1 sum rules for the positive and negative parity baryons are given by

2.2. Choice of Interpolating Local Operator

In principle, any local operator with appropriate quantum numbers can be employed for sum rules, or lattice QCD calculations. It is, however, not the case in practice. Because one has continuum contribution from hadronic background, it is important to enhance the pole term as much as possible by choosing the most appropriate operator. This situation is well demonstrated in the case of the sum rule and the lattice calculation for the nucleon. It is known that there are two independent local operators (without derivatives) for the nucleon, ~1 (z)

= eabc[uT(z)~~5db(z)] uc(z)

~ ~ ( =2e a1b c [ u T ( z ) ~ d b ( z >75uc(z) I ,

(13) where C denotes the gamma matrix for charge conjugation and a, b, .. . are color indices. Then a general $quark operator can be written as

+

J ( z ) = B2(z) tBl(Z)

(14)

348

B1 and B2 have different chiral properties, which can be seen in some particular limits: For t = -1, J belongs to the (3,3) (3,3) representation of S U ( ~ ) L X S U ( ~which ) R , is called the Ioffe current. For t = +1, it belongs to ( 8 , l ) (1,8), and for t = 0, it is purely B2, which vanishes in the NR limit. It is practically important in lattice QCD and sum rules to determine which is the best operator for the positive parity and negative parity nucleons. Such a study was performed for lattice QCD by Sasaki et al.I2 by using the domain wall fermion formalism. They demonstrated that the B1 operator does not couple to the nucleon N(940)and gives its excited state N* (1440) fairly well. An analysis for the sum rule was performed by Jido.13 He showed that the positive parity baryon has strong coupling t o the Ioffe current, while the negative parity strength is most strong for a positive t , i.e., t N 0.8,1.2.

+

+

3. S u m Rule for Pentaquark

A local operator for Q+, employed by

is

This operator is chosen because its overlap with N K states is suppressed, which can be seen from the Fiertz rearrangement,"

of(.) x

;{

= eade ( u Z C 7 5 d e )

- Y 5 d a ( 3 c 7 5 u c ) + 7 5 u a (SC75dC) - d a ( S c u c ) + uc ( S C d C )

The residue of the pole, /XI2, in Eq. (5) represents the strength with which the interpolating operator couples to the physical state, and it should be positive if the pole is real. We use this condition to determine the panty of the pentaquark. In Fig. 1, we plot the OPE side (as a function of M ) corresponding to exp(-rn2/M2). We fmd that the dimension-:, which condensate, (3g8a- Gs),gives a large negative contribution to makes /A+ l2 to be nearly zero or even slightly negative. This suggests that the pole in the positive-parity spectral function is spurious. In contrast, the large ( S g 8 a . Gs) contribution makes ILI2positive. We thus conclude that the obtained negative-parity state is a red state.

349

............ ----J.. -1e-10

-

-2e-lo

-

.......... -___ ---_ ............. --._ ........................................................ .......

-..__ -..---..-.

--..

-3e-10

-1e-10 -2e10 -3e-10

1

1.2

1.4

1.6

1.B

2

M l&Y

Figure 1. Contributions from the terms of each dimension added up successively for the negative-parity and positive-parity sum rules with 6= 1.8 GeV.

The mass of the negative parity O+ is estimated using the sum rule, and we find that the M dependence is rather weak. Therefore the reliability of the sum rule seems good. The mass is, however, sensitive to choices of the continuum threshold S t h . Nevertheless, we confirm that the result is consistent with the observed value. Sum rules for other pentaquark baryons, such as F - ( I = 3/2), have been constructed. The results are similar to O+,that is, only negative parity correlator has significant strength below 2 GeV. We also find that SU(3) breaking effects seem small in the negative-parity pentaquark states. See Ftef.l4 for details.

350

4. Discussion

How do we interpret the results? Are they consistent with what our experimental colleagues have observed? Are they really pentaquark states or something else? What we see in the QCD sum rule may or may not be a sharp resonance state, because the sum rule cannot tell the width of the state. (Coupling strength to decay channels may be calculated, which is related to the width indirectly.) It is important that the interpolating local operator has small overlap with the decay channel so that the signal to background ratio is enhanced. It has been shown by Lee15 in this workshop that contribution of the N K states in the non NK-type correlator is sufficiently small. He indeed demonstrated that using the soft-meson technique, the 8-vacuum matrix element of the local operator can be rewritten as a N-vacuum matrix element of a 5-quark nucleon operator.

Constructing a (new) nucleon sum rule using the 5-quark operator in the right hand side, Lee found that it is consistent with positive parity nucleon and that the NK continuum contribution is less than 5% in the Of sum rule. Similarly, choices of the operator may be critically important in lattice QCD calculations. In the workshop, several lattice calculations of the mass of Of with various local operators have been presented.16~17*18~1g~10 All but one agree that signal of 1/2+ state is found only at very high mass, i.e., above 2 GeV. They disagree, however, on the existence of a 1/2- state. Lee1* presented their results of an extensive study by using the operator that is a product of N and K. They claim that no signal for a compact 5-quark state is found. It is naturally expected that the NK operator does couple to the continuum state strongly and therefore is not appropriate to look for a signal of a compact resonance state. In contrast, Sasaki17 carried out a pioneering study employing the non NK operator and claimed that TPC shows a double-plateau structure, one of which corresponds to the resonance above the other one, the N K threshold. Ishii et al.l0 studied 8+using an anisotropic lattice in order to enhance precision of the mass determination. They employ quenched approximation with the non NK interpolating operator. It is found a smeared source is useful in extracting the lowest energy state effectively. They have also introduced a new technique called hybrid boundary method and demon-

351

strated that existence of a 1/2- state is unlikely near the N K threshold. Details of their results are given in ref.”. 4.1.

PP Strikes Back?

Several authors have suggested existence of low-lying positive parity state in QCD. In lattice QCD, Chiu and Hsiehlg carried out quenched calculation with the domain-wall quark and concluded that the lowest mass state is an unstable negative parity state, and a positive-parity state appears around the right mass range. QCD sum rule calculations also predicted the change of the order of the parity. Kim et aL21 constructed a sum rule for E pentaquark baryon and found that a positive parity state comes lower than negative parity states. They found also that parity inversion has occurred due to new terms with the charm quark mass and therefore their result is consistent with negativeparity Of. Another study by Kondo et d.22 claimed that the conventional sum rule is contaminated by “baryon-meson reducible” diagrams and removal of such diagrams may reverse the conclusion on the parity assignment. They define a so-called two-hadron irreducible TPC as

( T ( J ~ ( ~ > Z ~ (=O(T(Je(s)&(O)) )>~~’

(.,Ji.f

- C(T(S,(s)JL(O))WJif

(ON,

(18)

ij

where the second term represents contribution of two hadrons propagating without interacting each other. Then as two-hadron reducible terms they assign the diagrams in OPE that can split into two color-singlets with no interaction between them. In principle, it is important to suppress contribution from N K (and other hadronic) scattering states as much as possible to isolate a sharp resonance state on top of it. It is, however, shown that the subtraction of N K reducible part is not simply accomplished by throwing away the diagrams which have no connection between two color singlet parts. The following problems are pointed out. (1) The 3-quark and qq operators are connected at the vertex, where the quark operators are normal-ordered so that divergence from vertex corrections is subtracted. Namely, a renormalization is required to isolate noninteracting hadrons, J S ( 2 ) = JN(Z)JK(Z)x

z,,,

+ ...

(19)

352

Therefore the subtraction of the two-hadron reducible part should also take care of the renormalization factor, which cannot be done simply by eliminating some of the perturbative QCD diagrams. (2) It is also pointed out that analytic continuation should bring full interactions among quarks (and gluons) in QCD sum rule and their interactions are determined by the local vertices. (3) Another problem is that the %quark and qg TPCs are not independent in the sum rule, because quark (gluon) condensates of one correlator and the other should be correlated with each other. (4) The validity of the OPE for the two-hadron reducible part was pointed out by Lee.15 Thus defining “non-interacting” part is not trivial. It is therefore concluded that subtraction of non interacting part should be done more carefully, even if it is possible, and the results of ref.22 seem not correct. The same conclusion was reached by Lee.15

5. Conclusion

The conclusion from QCD as of today is simple. (1) QCD predicts no JT = 1/2f ( F = 10’) pentaquark. Most results indicate its mass to be 2 GeV or higher. (2) Some calculations predict negative parity pentaquark state, but it may well be buried in the NK continuum. It certainly requires confirmation.

What are possible remedies for the discrepancy between the QCD predictions and most other model calculations. Are there strong pionic effects, which may not be taken into account properly in the sum rule nor the quenched QCD calculations? It is, however, noted that the Skyrmion model predicts no less “pionic” effects in the nucleon and A. Why, then, the QCD calculations, even the quenched approximation, do so well for the ordinary baryons? Another possibility is that the interpolating local operator is completely wrong. Such a possibility may include that this is a state with 7 quarks or more. If the state is a N K T bound state, for instance, a 5-quark lattice QCD calculation hardly reproduces it. It should be interesting to look for some non QCD possibility for the “pentaquark” state.

353

References 1. T. Nakano et al., Phys. Rev. Lett. 91,012002 (2003). 2. K. Hicks, Summary talk of the Workshop in this Proceedings. 3. C.E. Carlson, C.D. Carone, H.J. Kwee and V. Nazaryan, Phys. Lett. B573, 101 (2003); F1. Stancu and D.O. Risk, Phys. Lett. B575, 242 (2003); F1. Stancu, Phys. Lett. B595, 269 (2004); B.K. Jennings and K. Maltman, Phys. Rev. D68, 094020 (2004); R.Bijker et al., hep-ph/0310281. 4. T. Shinozaki, S. Takeuchi and M. Oh,hep-ph/0409103. 5. D. Diakonov, V. Petrov and M. Polyakov, 2.Phys. A 359,305 (1997). 6. R.L. JaEe and R. Wilczek, Phys. Rew. Lett. 91,55 (2003); M. Karliner and H.J. Lipkin, Phys. Lett. B575, 249 (2003). 7. Y. Kanada-En’yo, 0. Morimatsu and T. Nishikawa, hep-ph/0404144. 8. A.W. Thomas, K. Hicks and A. Hosaka, Pmg. Theor. Phys. 111, 291 (2004); C. Hanhart et al., Phys. Lett. B590, 39 (2004). 9. J. Sugiyama, T. Doi and M. Oh,Phys. Lett. B581, 167 (2004). 10. N. Ishii et al., hep-lat/0408030. 11. D. Jido, N. Kodama and M. Oh,Phys. Rev. D54, 4532 (1996). 12. S. Sasaki et al., Phys. Rev. D65, 074503 (2002). 13. D. Jido, to be published; D. Jido and M. Oka, hep-ph/9611322; M. Oka, D. Jido and A. Hosaka, Nucl. Phys. A629, 156c (1998) (hep-ph/9702351). 14. J. Sugiyama, T. Doi and M. Oh,in this Proceedings. 15. S.H. Lee, in this Proceedings and to be published. 16. F. Csikor, Z. Fodor, S. D. Katz and T. G. Kovacs, JHEP 0311,70 (2003). 17. S. Sasaki, hep-lat/0310014. 18. N. Mathur et al., hep-ph/0406196. 19. T.-W. Chiu and T.-H. Hsieh, hep-ph/0403020. 20. T. Takahashi, in this Proceedings. 21. H. Kim, S.H. Lee, Y. Oh, Phys. Lett. B595, 293 (2004). 22. Y. Kondo, 0. Morimatsu and T. Nishikawa, hep-ph/0404285.

354

PENTAQUARK BARYON FROM THE QCD SUM RULE WITH THE IDEAL MIXING

'J. SUGIYAMA, 2T. DO1 AND

'M.OKA

' D e p t . of Physics, Tokyo Institute of Technology, H27 Meguro, Tokyo, 152-8551, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, N Y 11973-5000, USA

roof

QCD sum rules for the and 8f pentaquark baryons are presented. Parity projection is carried out by combining chiral even and odd terms of OPE. It is found that the negative parity states appear in lowest mass region for almost all the pentaquarks. Effects of SU(3) breaking is discussed.

1. Introduction

Pentaquark baryon Of was discovered by the LEPS Group a t SPring-8l in 2002 and it has been confirmed by several other experiments since then. From its quantum numbers, B = 1 and S = 1, one sees that O+ must consist of at least five quarks. Its width, which seems t o be less than 10MeV, is very narrow. One of the key issue is its spin and parity. The naive quark model suggests a negative parity state, in which all the quarks are in the lowest orbit with L = 0. But this picture can hardly explain its narrow width. On the other hand, Diakonov et aL2 predicted a positive parity state, which was supported by other models, such as a diquark model3. The shortcoming of this case is that the mass may become too large because of larger kinetic energy expected typically from the mass difference between N(940) and N(1535). It is therefore very important t o determine the parity (and spin) of Of directly from QCD. Last year, another pentaquark candidate, K-, was reported by NA49 group4. E-- is a baryon with S = -2 and I3 = -312 and its width is also very narrow despite its larger gap from the threshold. From the flavor SU(3) symmetry it is natural t o expect that O+ and E-- belong t o the same SU(S)multiplet, f0f. Jaffe and Wilczek3 pointed out that while O+ and are purely in fOf, the other members of the multiplets, which are

355

also pentaquarks, are likely to be ideally mixed with a 8f multiplet. We then have to observe altogether 18 pentaquarks a t the mass range from 1.5 GeV t o 2 GeV. We here study these pentaquarks in the framework of the QCD sum rule. 2. 0’ from the QCD Sum Rule The QCD sum rule connects the phenomenological hadron spectrum and the OPE spectral function calculated in the deep-Euclidean region with the analytic continuation and provides the hadron properties. We choose the interpolating field which consists of two diquarks and an anti-quark. ‘&bcEdefEcfg

{u: ( I C ) C C E b ( x ) } { u ~ ( x ) C Y 5 d (x)}csT e (x)

(1)

The parity projection5 is needed because our purpose is determining the parity. We can describe the spectral function of the positive(negative) parity state p*(qo) as the sum(difference) of the chiral even terms A(q0) and the chiral odd terms B(q0) by taking the rest frame and using the retarded Green function. P * k O ) = 4 q o ) fW q o )

(2)

We assume that the phenomenological side consists of the sharp peak which is regarded as the delta function and the continuum which is approximated as the same form of the OPE. f Pphen(qO) = IA$ls(qO -m*)

+ e ( q 0 - &)POPE(qO)

(3)

If a sharp peak resonance exists in the low energy region, lA$l must have a positive value. This is our criterion to determine the parity of the pentaquark8. From the (A:( calculated for each parity, one finds that (A:( for positive parity pentaquark is almost zero or slightly negative. On the other hand, IA?( for negative parity state is significantly positive. We notice that the dimension-3 term ( ( 4 4 ) term) and the dimension-5 term ((tjaGq) term) in OPE are dominafit. They play dominant roles in determining the parity because their signs are reversed when we change the parity. We calculate the mass for the negative parity me from the QCD sum rule. As it is insensitive to the Bore1 mass. the sum rule works fairly well. The predicted mass is around 1.5 GeV and is consistent with experiment. However, the result depend on the choice of the threshold parameter, 6 , and therefore the mass prediction has significant ambiguity.

356

3. 8f and 1&

Pentaquark Baryons

We consider all the 8f and fOf pentaquarks assuming the ideal mixing scheme. For instance, we have two “pentaquurk” N states, whose quark contents are u d u d f i , and u d d s s . Similarly two C states appear in this scheme. The interpolating fields for these states are given symbolically in the Table. In this study, we ignore annihilation diagrams, which appear for N and C sum rules. We determine the parity of these pentaquark baryons using the same method as the O+ case. We find that all the 8f and fOfpentaquark baryons have negative parity. Only possible exception is C,, where ]A2] is nearly zero. Again the predicted masses from the sum rule is insensitive to the Bore1 mass, but depends strongly on +. The we plot the pentaquark masses versus 6in Fig. 1. It is found that the pentaquarks with more s-quarks look lighter for a fixed value of 6. For example, Z--, which has two s-quarks, is lighter than Of with one s-quark. This is obviously against our intuition. In the sum rule, effects of SU(3)f breaking come from three origins, (1) finite s-quark mass m,, (2) the ratio of the quark condensate (ss)/(@) and (3) difference of from a channel to another. In Fig. 2, dependences on the former two values are illustrated for @+ with a fixed = 1.8GeV. We compare this with the same quantity for the ordinary three-quark octet baryons. One sees that the direction of m, dependence is reversed for the pentaquarks, although its slope is milder than that for the three-quark baryons. Effects of the differences in 6, then, may be dominant origin of the SU(3) breaking.

Interpolating fields

f a b c UPUSg a b c

1 0 2 2 3

f a b c u f u : fic

2

SPSSs b c

O

fabc a

N

fabcS:Stfic

Ns

z 1f a b c UaP S S bsc

C C,

&fabcufstfic

Y

-

# of s plus

-

+ $. +

SPUSs b c

Zcabc a

&fabcSrU:fic

quark contents ududs ududfi uddss uddsii dsdss dsdsii

Note: The capital letter represents a diquark operator with corresponding conjugate flavor. Its superscript denotes its Lorentz structure, S: scalar or P: pseudoscalar.

357

0

Figure 1. Masses of the pentaquarks vs. &. The pentaquark masses are determined from the condition that it is equal to the Bore1 mass.

Figure 2. rn, and ( S s ) / ( q q ) dependences of the mass of @+, me, for fixed & = 1.8GeV.

4. Conclusion

We have studied the pentaquark states in the ideally mixed SU(3) multiplets, 8f and COf. Parity projection predicts that all the pentaquarks are likely to have negative parity if they exist in low energy region, around 1.5 to 2 GeV. One possible exception is C,, which contains three s quarks. We find that the strength for C, is weak. The results are consistent with most of the lattice QCD calculations carried out so far.7 Effects of SU(3) breaking are studied in detail. The strange quark mass and the difference in quark condensates are both making the “stranger” baryons lighter. It is counter-intuitive, but the masses of the pentaquarks may be controlled by the threshold difference. We acknowledge the support from the MEXT of Japan through the Grant for Scientific Research (B)No.15340072. References 1. 2. 3. 4. 5. 6. 7.

T. Nakano, et al., Phys. Rev. Lett. 91,012002 (2003). D. Diakonov, V. Petrov and M. Polyakov, 2.Phys. A359,305 (1997). R.L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91,232003 (2003). C. Alt, et al., Phys. Rev. Lett. 92,042003 (2004). D. Jido, N.Kodama and M. Oka, Phys. Rev. D54,4532 (1996). J . Sugiyama, T. Doi and M. Oka, Phys. Lett. B581,167 (2004). See these proceedings and also N.Ishii et al., heplat/0408030.

358

MASS AND PARITY OF PENTAQUARK FROM TWO-HADRON-IRREDUCIBLE QCD SUM RULE

T. NISHIKAWA* Institute of Particle and Nuclear Study, KEK, 1-1 Ooho, Tsukuba, Ibaraki 305-0801, Japan E-mail: [email protected]

We point out that naive pentaquark correlations function include two-hadronreducible contributions, which are given by convolution of baryon and meson correlation functions and have nothing t o do with pentaquark. We show that t h e two-hadron-reducible contributions can be large in the operator product expansion of the correlation functions. We propose t o use t h e two-hadron-irreducible correlation function, which is obtained by subtracting the two-hadron-reducible contribution from t h e naive correlation function.

Possible existence of an S = +1 exotic baryon has recently been reported by LEPS collaboration in Spring-gl. In this experiment, the mass is 1540MeV and the width is bounded by 25MeV. This state cannot be a three-quark state since it has S = +1, and the minimal quark content is (uuddg). It has come to be called "pentaquark O+". The discovery of O+ has triggered an intense experimental and theoretical activity to clarify the quantum numbers and to understand the structure of the Of. In this talk, we focus on the application of the QCD sum rule" to the pentaquark and discuss an issue which is characteristic for exotic hadrons. Up to now, several groups have reported results of QCD sum rules for the pentaquark7i8ig. We point out that their naive pentaquark correlation functions include two-hadron-reducible contributions, which are due to noninteracting propagation of the three-quark baryon and the meson and therefore have nothing to do with the pentaquark". These contributions exist in the correlation function only for exotic hadrons and are potentially large. Instead, we propose to use the two-hadron-irreducible correlation N

23374,5,6,79839

'This work is done in collaboration with Y. Kondo (Kokugakuin univ.) and 0. Morimatsu (KEK)

359

function, which is obtained by subtracting the two-hadron-reducible contribution from the naive correlation function. The basic object of the study is the correlation function of baryon interpolating field 7 : n ( ~=) -2 d4xei~””(OlT(~(x)~(0))10). The spectral function, p ( p 0 ) = -Imn(po k ) / 7 r , in the rest frame p = 0 can be written as PO) = P+P+(Po) P-P-(Po),where P* = (TOf 1)/2. In P&(Po), there exists not only positive but negative parity states contribution. For example, positive parity states contribute to p+(po > 0) and negative party states to p + ( p o < 0). This is because the interpolating field couples to positive and negative parity states. On the other hand, in the deep Euclid region, pg -00, p&o) can be evaluated by an OPE. Using the analyticity we obtain the QCD sum rule as J-“, ~ P O P ~ ~ ~ ( P= OJ-”, ) W~( PPOOP)& ~ P O ) where W ( p 0 ) is an analytic function of PO. p + ( p o ) are parameterized by a pole plus continuum contribution, p*(po) = IX*126(po - m*) IXT,12S(po m?) [(?(Po- w + ) (?(-Po - wr)]poPE(po), where mh and w s are the masses and the continuum threshold parameters of positive (negative) parity states, resepctively. Substituting this equation to the right-hand side of the sum rule and using the Borel weight, W ( p 0 ) = pg exp(-p$/M2), we obtain the Borel sum rules for positive and negative parity baryons. From the sum rules for n = 0 4, we can eliminate the pole residues 1X*I2 and obtain the sum rules for mka. First we apply the above formalism to the nucleon to test its predictability. We use the general nucleon interpolating field13 given by VN = ~ , ~ [ ( u ~ C d b t(uaC~5db)uc], ) ~ ~ u ~ where u and d are field operators of up and down quarks, C denotes the charge conjugation operator and a, b and c are color indices. We choose the effective continuum threshold as w+ = 1.44 GeV and w- = 1.65 GeV which correspond to the masses of N(1440) and N(1650), respectively. The Borel curves of mN+ and m N - with t = -0.7 have the stable plateau as a function of the Borel mass. We obtain the masses of the positive and negative-parity nucleons as m N + = 1.0 GeV, m N - = 1.6 GeV, which well agree with experimental values. Now we consider the pentaquark sum rule. A remarkable feature of the pentaquark is that it can be decomposed into a color-singlet three-quark state, baryon, and a color-singlet quark-antiquark state, meson. (Hereafter,

+

+

s

--f

+

+

+

+

N

+

aOur method for deriving QCD sum rules for positive and negative parity baryons are different from that of the previous work12. For more details, we will report in a future p~blication’~.

360

we use the term baryon when its minimal quark-content is qqq.) Therefore, the interpolating field for the pentaquark can be expressed as a sum of the ~&(z)&(x product of baryon and meson interpolating fields: v p ( z )= where &(z) and &(x) are color-singlet baryon and meson interpolating fields, respectively. Due to this separability, the pentaquark correlation function has a part in which the baryon and the meson propagate independently without interacting each other. We define this part as the twohadron-reducible (2HR) part and the rest of the correlation function as the two-hadron-irreducible (2HI) part. Diagrammatically, the 2HR and 2HI parts are represented as Figures (a) and (b), respectively. Clearly, the 2HR part is completely determined by the baryon and meson correlation functions and has nothing to do with the pentaquark.

xi

Figure 1. (a): Two-Hadron-Reducible (2HR) diagram. (2HI) diagram.

(b):Two-Hadron-Irreducible

Let us next look at the separation of the 2HR and 2HI parts in the spectral function. We suppose that the lowest states generated by 7~ and 7~ are spin-1/2 baryon B and spin-0 meson M , respectively. Consider only the contribution of the B M scattering states in the spectral function, p F M ( p ) ,just for simplicity. One can divide p F M ( p ) into two parts by means of the reduction formula. One of them is the 2HR contribution due to the trivial noninteracting contribution of the B M intermediate states. The other is related with the T-matrix for the B M scattering and corresponds to the 2HI contribution. If the pentaquark is a resonance in the BA4 channel the pentaquark state lies in the B M T-matrix as a pole at a complex energy. The 2HR contribution is therefore not related to the pentaquark. Some comments are in order here. 2HR contributions discussed here exist commonly in the correlation functions for exotic hadrons but not for ordinary hadrons. Crucial assumption here is confinement. Namely, we assume that only color-singlet states contribute to the spectral function. Therefore, the separability of the pentaquark into color-singlet baryon and meson is the origin of the existence of the 2HR contribution. Let us turn to the separation of the 2HR and 2HI parts in the OPE. We calculated the 2HR parts of the correlation functions for interpolating fields used in Refs.[7-9]. We found that the 2HR part is large at least of

361

the same order as the 2HI part. In particular, for the interpolating field used in Ref.[9] , the Wilson coeffcients of the operators in the 2HR part are -15/7 of the 2HI part up to dimension 6 except for the operator, %G2. Now, taking the interpolating field employed in Ref.[9], we will demonstrate how the results of the sum rule can change if we remove the 2HR part. When we use the naive correlation function, we obtain positive IX- l2 and negative )A+/'. It was concluded that the obtained negative parity state is a real one but the pole in the positive parity spectral function is spurious. When we replace the total spectral function by the 2HI part, /XI2 is positive for the positive-parity state but negative for the negative-parity state. This result was expected because the Wilson coeffcients of the operators in the 2HR part are -15/7 of those in the 2HI part up to dimension 6 except for the operator, %G'. Therefore, the sum rule for the 2HI part of the spectral function leads us to the opposite conclusion that the obtained positive parity state is a real one but the pole in the negative parity spectral function is spurious. The mass of the O+ is estimated to be 1.6GeV b. Some final comments are in order here. Logically, there is nothing wrong to use the total correlation function. It is much better if the background can be exactly separated, which is what we proposed in this talk. N

References 1. 2. 3. 4. 5. 6. 7. 8.

LEPS collaboration, T. Nakano et al. , Phys. Rev. Lett. 91 (2003) 012002. R. Jaffe and F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003.

S. Capstick, P. R. Page and W. Roberts, Phys. Lett. B570 (2003) 185. M. Karliner and H. J. Lipkin, Phys. Lett. B575 (2003) 249. S. Sasaki, hep-lat/0310014. F. Csikor, Z. Fodor, S.D. Katz and T.D. Kovacs, JHEP11(2003)070 Shi-Lin Zhu, Phys. Rev. Lett. 91 (2003) 232002. R. D. Matheus, F. S. Navarra, M. Nielsen, R. Rodrigues d a Silva and S. H. Lee, Phys. Lett. B578 (2004) 323 9. J. Sugiyama, T. Doi and M. Oka, Phys. Lett. B581(2004) 167. 10. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147 (1979) 385; Nucl. Phys. B147 (1979) 448. 11. Y. Kondo, 0. Morimatsu and T. Nishikawa, hep-ph/0404285. 12. D. Jido, N. Kodama and M. Oka, Phys. Rev. D54 (1996) 4532. 13. D. Espriu, P. Pascual, and R. Tarrach, Nucl. Phys. B214 (1983) 285. 14. Y. Kondo, 0. Morimatsu and T. Nishikawa, in preparation.

bThe way of evaluating the 2HR part employed in this paper may be incorrect. If so, we will report the results obtained by correctly removing the 2HR part in a future publication.

362

THREE-QUARK FLAVOUR-DEPENDENT FORCE IN PENTAQUARKS

v. DMITRASINOVIC VinEa Institute of Nuclear Sciences P . 0.Box 522, 11001 Belgmd, Serbia E-mail: dmitraOuin.bg.ac.yu We review critically the predictions of pentaquark masses in the quark model, in particular those based on the Glozman-Risks hyperfine interaction. This leads us to the Va(1) symmetry breaking Kobayashi-KondeMaskaw&’tHooft interaction. We discuss its phenomenological consequences in pentaquarks.

1. Introduction

Experimental facts relating to pentaquarks have been reviewed at great length at this Workshop, so I shall not touch upon that subject. I take the existence of 0+(1540) for granted, but make no assumptions about the mass of the Ed- resonance. We shall quickly review the predictions for the E-- mass in the quark model with various hyperfine interactions (HFI). As for any new particle there are three main issues: (i) Absolute mass and mass splitting within multiplets? W e cannot predact absolute mass (as yet), only mass splittings! (ii) Spin & parity, spin or parity partners? We assume span 1/2, of either parity. (iii) Decay half-life? We shall not discuss the decay width here. To these questions there are three classes of model answers: 1) Chiral soliton models (reviewed elsewhere in this Workshop); 2) Constituent quark models (subject of this, and many other talks in these Proceedings); 3) Hadronic molecule models ( K r N or “Heptaquark” models), see Ref. . 2. Basic pentaquark theory 2.1. Pentaquarka in the constituent quark model Flavour SU(3) pentaquark content:

363

The observed 0+(1540) state is an isoscalar Flavour-spin SU(6) pentaquark content

+

O+((uvffi) C

10~.

686636868.8 = 4(20)@8(70)@m@4( 56)@3(540)@2(560)@700@3( 1134)

Three colour singlets + Pauli principle allows more than one SU(6) multiplet with orbital angular momentum 1 = 0 waz. [56,0-], [70,0-] and [1134,0-1. Excitation energy tiw may depend on the colour state the pentaquark is in.

Figure 1. Pauli allowed pentaquark states in the harmonic oscillator potential.

Many SU(6) multiplets contain the mp-plet: [1134,0-] with odd parity, and [70,1+], [540,1+], [560,1+] and [700,1+] with even parity + u prPori one cannot predict the parity of the observed Q+ state. Whichever SU(6) multiplet we choose, the mF-plet will have many “spin partners” regardless of parity.

+ Cannot u priori predict the spin of the observed 0+ state, except to note that the lowest possible spin S = 1/2 is preferred by two “standard” hyperhe interactions (HFI), see below. Spin-orbit forces are weak in hadrons, so one may expect “spin-orbit partners” of the mp-plet with different Js within 100 MeV from the ground state, if parity is even. A rich spectrum is expected.

364

There are two other exotic pentaquarks that are (almost) degenerate in the and the E+.The NA49 observation of the 5-at 1860 10F-plet: the

=--

MeV has been challenged by WA89. We wish to predict the E-- mass. 2.2. Mass splitting i n the i & - p l e t

Quark mass difference m,-m = 150f30 MeV (standard chiral perturbation theory value) induces E - 0 pentaquark mass splitting in the mF-plet:

M ( 3 ) - M ( Q ) = (m, - m) = 150 f 30 MeV. Confining interaction is mass independent + the only other source of pentaquark mass splitting is the hyperfine interaction (HFI). As the free quark model predicts E-- mass at 1700 MeV, if NA49 were right, then the HFI contribution to mass splitting would have to be very large (100 %). There are two widely accepted HF interactions: the colour-spin and the flavour-spin dependent type. We shall treat both in the “schematic” approximation where the spatial matrix elements of the HFI for all hadrons are taken to be the same. 1) Colour-Spin HF interaction (one gluon exchange Breit interaction in QCD)

A-N mass splitting determines the coupling constant:

The reasults for either parity are shown in the Table below Parity

I

Z-Q

P = + (m,-m)[1+4%] p=-

(m, - m)

(MeV)

=(MeV)

=160&321720&32 = 150 f 30 1690 f 30

If N a49 experiments value M- 1860 Mev were right then the CS HFI

365

2) Flavour-Spin interaction

^--*<£H-'isf*-'' Note the phase (—)a'3' = — for qq pairs, and + otherwise. A-N mass splitting determines the coupling constant: CY — TZ(

30 MeV.

In all previuos pentaquark calculations this interaction has been used only among the four quarks, whereas the qq interaction has been ignored. We do not make that "approximation" here. In that sense our results shown in the Table below are new. Parity

E-Q

(MeV)

B(MeV)

P = + (m. - m) [l + 28-=•*] = 550 ± 100 2090 ± 100 P = - (m, - m) [l + 8^*] =265 ±50 1805 ± 50 If one accepts the NA49 value of Ms=1860 MeV, then one is led to odd parity of 0+(1540) as a result of this model. Note the difference in the predictions of the two HF interactions for the E mass. If the experimental value should fall into a gap between the predictions of two models it only means that the "true" HFI is a mixture of the CS and FS type. There must be other predictions characteristic of the GR model that can be tested before E is definitively detected. 3. Pentaquark spectrum in the Glozman-Riska model GR interaction can be evaluated using

-20 , (1) « < 3

a function of the total pentaquark spin 5, SU(6) and SU(3) quadratic Casimir operators C^tot], c£2)[tot]. =^ all states with identical spin S and SU(3) multiplet that belong to the same SU(6) multiplet have the same GR HFI energy, i.e. median mass.

366

This result is contrary to the “approximation” in which the antiquark interaction is neglected 2; there identical spin and flavour SU(3) states are not degenerate, whereas different SU(3) states belonging to Merent SU(6) irreps are degenerate if only their 1q4) flavour SU(3) and SU(6) contents are the same. Addition of the antiquark interaction leaves the exotic multiplets’ OF, 2 7 ~3, 5 ~ masses ) essentially unchanged, whereas some of the non-exotic multiplets change even the signs of their HFI energies. There is an additional degeneracy between flavour SU(3) multiplets and - the ms-plet are degenerate. Such a their conjugates: thus the 1 0 ~and 10F-plet has not been observed thus far 2. The exotic states m ~2,7 ~ 351;. , C [700,1f] are pushed down, and d e pending on the size of tiw may even fall below the [1134,0-1.

0.75

F

% 0.5

2

0.25

1 ~(135)

c, = 0 Figure 2.

c,>O

SU(3) S.break.

Expt

Pseudoscalar meson spectrum in the Glozman-Riska model.

367

The GR model makes reasonable postdictions of the q3 baryon spectra, but the pattern of the q?j meson spectrum turns out wrong with this simple flavour-spin interaction, see Fig. 2. As 0 pentaquarks decay into a baryon and a K meson, one must have a Must look for another flavourmodel that adequately describes both. dependent HFI that does not spoil the meson spectra! The answer is given by QCD instanton-induced UA(1) symmetry breaking Kobayashi-KondoMaskawa-’t Hooft interaction. 4. KMT flavour dependent interaction

Kobayashi-Kondo-Mask’t Hooft instanton-induced UA(1) symmetry breaking flavour-dependent interaction 394

-@ = - K

[detf ($(I + ~ 5 ) $ ) + detf ($(I - ~ 5 ) $ ) ]

where d e t r ( q ( l + 75)$) is a determinant only in the flavour space. This interaction changes the q , ~ masses, ’ but it also affects the spinless meson, and the baryon spectra 5 . Two- and threequark KMT potentials

Pentaquark HF energy due to the KMT interaction EF-!zdy

KMT E3-body

+28S(1

= -A( =

B

- g1 - s(1+ s)4- i(-Cp’[Q4] 1 + ci2)[tOt]+ &)[tot])) (13801 - 744S(1+

s) - 36Cf)[t0t] + 72Cf)[Q4]+ 18(-111

+ S))Cf)[Q4]- 9(-125 + 28S(1+ S))Cp’[tot] - 702Cp)[tot]) (2)

where

A = -4K(?jq)o(b(ri - rj)) 11 95 f 20 MeV B = 4K(b(ri - rj)6(rj - rk)) 11 50 f 10 MeV.

368

2

0 I-

3

B

w

--

-2

8

-4

8

8

1

-6

3

K=O

5

15

10

15’

KXI

Figure 3 . Pentaquark spectrum in general with the KMT interaction. HFI energy E K M T in units of A = 95 f20 MeV. Numbers at the bottom of columns indicate the q*HF states’ s u ( 3 ) F multiplets.

The cubic Casimirs of SU(3) and SU(6) Cp),Cf) in the three-body interaction remove “accidental” degeneracies present in the GR model: the 10pplet and the mp-plet are not degenerate, see Fig. 3. Moreover, there is splitting that depends on the flavour SU(3) content of the q4 subsytem due to the quadratic Casimir Cf)[Q4] in the two-body interaction. 2 7 ~3 ,5 ~masses ) are significantly modified by the Exotic multiplets’ (m~, KMT interaction as compared to GR model, see Fig. 3. Most of the , ~ are ) also modified. They split according to both nonexotics ( l ~8, ~1 0 their q4 and total (&) flavour contents, indicated at the bottom of columns in Fig. 3.

369

5. Summary 0

0

0

0

0

Constituent quark model predicts many exotic pentaquark states close to the O’(1540) Pentaquark spectrum is very sensitive to the type of hyperfine interaction: small 3 - 0 mass splitting (150MeV) for CS, and a large one (> 270MeV) for FS HFI. Established baryons N(1440) (“Roper”) and N(1710) can NOT be the result of 1 0 -~8~ mixing in either HFI model. Z mass < 1750MeV would indicate dominance of colour-spin HF forces, E mass > 1850MeV would indicate dominance of flavourspin HF forces.

Both CS and FS model predict many cryptoexotic pentaquarks lying below O(1540). None have been observed thus far. Meson spectra predicted by the GR model are wrong. =+ Must find a better HF interaction that describes well both baryons and mesons. We use the Kobayashi-Kondo-Maskawa-’t Hooft (KMT) u A ( 1 ) symmetry breaking interaction in QCD contains two- and threebody interactions. We show new results for pentaquark spectra with the KMT interaction.

Acknowledgments

The author wishes to acknowledge financial support from the organizers of PentaO4 and the kind hospitality of RCNP theory group. References 1. T. Kishimoto and T. Sato, hep-ex/0312003. 2. V. DrnitraSinoviC and F. Stancu, hep-ph/0402190v2. 3. M. Kobayashi, H. Kondo and T. Maskawa, Prog. Theor. Phys. 45, 1955 (1971). 4. G. ‘t Hooft, Phys. Rev. D 14,3432 (1976), (E)ibid. 18,2199 (1978). 5. T. Kunihiro and T. Hatsuda, Phys. Lett. B 206, 385 (1988); V. Bernard, R.L. Jaffe and U.-G. Meissner, Nucl. Phys. B308, 753 (1988); V. DmitrSinoviC, Phys. Rev. C 53, 1383 (1996).

370

INTERACTION OF THE O+ WITH THE NUCLEAR MEDIUM

M. J. VICENTE VACAS, D. CABRERA, Q. B. LI, V. K. MAGAS AND E. OSET Departamento de Fisica Tedrica and IFIC, Centro Mixto Universidad de Valencia - CSIC, Institutos de Investigacidn de Paterna, Apdo. correos 82085, 46071, Valencia, Spain We study the selfenergy of the Q+ pentaquark in nuclei associated with two types of interaction: the K N decay of the Q+ and two meson baryon decay channels of the Q+. Whereas the potential related to the first source is quite weak, the second kind of interaction produces a large and attractive potential that could lead to the existence of Q+ nuclear bound states.

1. Introduction

The discovery of the @+ resonance' opens the possibility of forming exotic @+ hypernuclei, which, like in the case of negative strangeness, could provide complementary information on the structure and properties of the pentaquark to that obtained from elementary reactions. In Ref. 2, a schematic model for quark-pair interaction with nucleons was used to describe the O+, which suggested that @+ hypernuclei, stable against strong decay, might exist. Later, in Ref. 3, the @+ selfenergy in the nuclei is calculated, based on the @ + K N interaction. The results show a selfenergy too weak to bind the @+ in nuclei. In this talk, we present some selected results from Ref. 4. There, we redo the calculations of Ref. 3 modifying some of the assumptions. The results are similar to those of Ref. 3 and a quite small potential is obtained from this source. Additionally, we consider other new selfenergy pieces related to the coupling of the O+ resonance to a baryon and two mesons under certain assumptions. We find that the in-medium renormalization of the two meson cloud of the @+ could lead to a sizable attraction, enough to produce bound and narrow @+ states in nuclei. The coupling of the @+ to two mesons and a nucleon is studied using

371

a S U ( 3 ) symmetric Lagrangian5, constructed to account for the coupling of the antidecuplet to a baryon and two pseudoscalar mesons. With this Lagrangian an attractive selfenergy is obtained for all the members of the antidecuplet coming from the two meson cloud.

2. The O+ selfenergy in nuclear matter 2.1. Selfeneryy from the K N decay channel

The @+ selfenergy diagram in vacuum is shown in Fig. 1. For the L = 0

Figure 1. Q+ selfenergy diagram related to the KN decay channel.

case, the free O+ selfenergy from this diagram is given by

+ &+,a2.

where M is the nucleon mass, E N ( k ) = d k2. r The results for L = 1 are obtained by the substitution g L + , + Next, we evaluate the O+ selfenergy in infinite nuclear matter with density p. The nucleon propagator is changed in the following way, 1 p o - qo - EN

+ ic

---f

1-nF-g po - qo - EN

+ zc

nw+

4

p o - qo - EN - ic '

(2)

where n(z) is the occupation number. The vacuum kaon propagator is replaced by the in-medium one, 1 q2 - m&

+ ic

1

+ q2

- m: - n K ( q , p ) '

(3)

where n K ( q o , 14,~) is the kaon selfenergy. The s-wave part of this selfenergy is well approximated by6l7 I I $ ' ( p ) = 0.13mkplpo , where po is

372

the normal nuclear density. The p-wave part is taken from the model of Ref. 8, which accounts for Ah, C h and C*(1385)h excitations. After some approximations, the go integration leads to

k2

+ -2

with go = po - EN@- 43 - V N ,VN = -&, FO = 1, Fl = and qon the Qon on shell kaon momentum. In Eq. 4 we have taken into account the nucleon For the binding by using the Thomas-Fermi potential, VN = -ICF(T)~/~M. calculations, we have taken an average value of the momentum of the O+ in eventual bound states of p = 200 MeV, similar to that of bound nucleons in nuclei. We show the results in Figs. 2, 3, where we assume that I? = 15 MeV. As it is obvious from Eq. 4, the in medium selfenergy is proportional to the vacuum width, thus the results should be scaled when the width is better determined. In any case, even if I' = 15 MeV in vacuum, inside the nucleus the width is small, basically because of the Pauli blocking. For 20 MeV of Of binding, the width would go down from 15 MeV to less than 6 MeV at p = po. This width could be smaller than the separation between different bound levels. As for the real part of the O+ selfenergy, we find, in qualitative agreement with Ref. 3, that the O+ potential in the medium is small, of the order of 1 MeV or less and not enough to bind O+ in nuclei.

2.2. The O+ selfenergy tied to the two-meson cloud In this section we consider contributions to the O+ selfenergy from diagrams in which the O+ couples to a nucleon and two mesons, like the one in Fig. 4. There is no direct information on these couplings since the O+ mass is below the two-meson decay threshold. To proceed, we do several assumptions. First, the O+ is assumed to have J p = 1/2+ associated to an SU(3) antidecupletg. Also the N*(1710) is supposed to couple strongly to the same antidecuplet. From the PDG data on N*(1710) decays we determine the couplings to the two-meson channels, and using SU(3) symmetry obtain the corresponding couplings for the O+ pentaquark.

373

0

.I

1

I

I

L=O

I

-p q 0 , p = O MeV ---_p q O , p = 200 MeV .........p=0.75*p0, p=O MeV .-. -. . p=0.75*p0. p=200 MeV

-

-10 -

-

1500

I

I

I

I

1520

1540

1560

1580

1600

E (MeV) Figure 2. L = 0.

Imaginary part of the Q+ selfenergy associated to the K N decay channel for

0

:

I

I

I

I

......... p=0.75*p0,p=0MeV.-.-.. p=0.75*p0, p = 200Mef-

-15 -

1500

I

1

I

I

1520

1540

1560

1580

1600

E (MeV) Figure 3. L = 1.

Imaginary part of the Q+ selfenergy associated to the K N decay channel for

374

,

.

#'

'

o+(p)

,#*

;;

-n- (9)- -

-b-

-*.\,

'\

'; \\ II

I , #

o+

N Figure 4.

Two-meson Q+ selfenergy diagram.

In order to account for the N*(1710) decay into N ( m , p - wave, I = 1) and N(m,s - wave, I = 0) we use the following lagrangians

L1 = igf@ilmTijkyPBj 1 ( vPIm.7

(5)

with

where f = 93 MeV is the pion decay constant and Tijl, Bi, q5k SU(3) tensors which account for the antidecuplet states, the octet of bafyons and the octet of 0- mesons, respectively. The second term is given by

4'

which couples two mesons in L = 0 to the antidecuplet and the baryon and they are in I = 0 for the case of two pions. From the Lagrangian terms of Eqs. (5, 7) we obtain the transition amplitudes N* + ~ T NTaking . the I = 1) central values from the PDGlO for the N*(1710) -+ N ( n ~ , p wave, and for the N*(1710) -+ N ( m , s - wave,I = 0), the resulting coupling constants are g13 = 0.72 and i i o = 1.9. The implementation of the medium effects is done by including the medium selfenergy of the kaon and modifying the nucleon propagator, as before. On the other hand, for the pion we modify the propagator using the p-wave selfenergy from p h and Ah excitation^^^^^^^^^. Once the O+ selfenergy at a density p is evaluated, the optical potential felt by the O+ in the medium is obtained by subtracting the free O+ selfenergy. We should also note that while the O+ ---t KTN decay is forbidden, in the medium the T can lead to a p h excitation and this opens a new decay channel O + N -+ NNK, which is open down to 1432 MeV, quite below the free O+ mass. We have shown4 that the width from this channel is also very small, but should the O+ free width be of the order of 1 MeV as

375

suggested in Refs. 14, 15, the new decay mode would make the width in the medium larger than the free width. We present the results in Figs. 5 and 6. We can see there that the potential for p = po is sizable and attractive and goes from -70 to -120 MeV depending on the cut-off used in the selfenergy evaluation.

Figure 5.

Real part of the two-meson contribution to the 8+ selfenergy at p = po.

Even with the quoted large uncertainties we conclude that there could be a sizable attraction of the order of magnitude of 50-100 MeV at normal nuclear density, which is more than enough to bind the O+ in any nucleus. In Fig. 6 we show the imaginary part of the O+ selfenergy related to the two-meson decay mechanism. We find that r would be smaller than 5 MeV for bound states with a binding of -20 MeV and negligible for binding energies of -40 MeV or higher. This, together with the small widths associated to the K N decay channel, would lead to Of widths below 8 MeV, assuming a free width of 15 MeV, and much lower if the O+ free width is of the order of 1 MeV. In any case, for most nuclei, this width would be smaller than the separation of the deep levels16.

376

1450 Figure 6.

1500

1550

E (MeV)

Imaginary part of the two-meson contribution to the Q+ selfenergy a t p = po.

3. Conclusions The selfenergy of the O+ in the nuclear medium associated to the K N decay channels is quite weak, even assuming a large free width of around 15 MeV for the O+. However, there is a large attractive O+ potential in the nucleus associated to the two meson cloud of the antidecuplet. New N N K , but the decay channels open for the O+ in the medium, O+N width from this new channel, together with the one from K N decay, is still small compared to the separation of the bound levels of the O+ in light and intermediate nuclei. This conclusions depend on several assumptions, namely: the O+ is assumed to be 1/2+ associated to an S U ( 3 ) antidecuplet, the N*(1710) is supposed to couple largely to the same antidecuplet, two specific Lagrangians have been chosen, the average value of the N*(1710) width and the partial decay ratios, which experimentally have large uncertainties, have been taken to fix the couplings. It is clear that with all these assumptions one must accept a large uncertainty in the results. So we can not be precise on the binding energies of the O+. However, the order of magnitude obtained for the potential is such that even with a wide margin of uncertainty, --$

377

the conclusion that there would be bound states is quite safe. In fact, with potentials with a strength of 20 MeV or less one would already get bound st.ates. Furthermore, since the strength of the real part and the imaginary part from the N K p h decay are driven by the same coupling, a reduction on the strength of the potential would also lead to reduced widths such t.hat the principle that the widths are reasonably smaller than the separation between levels still holds.

Acknowledgments This work is partly supported by DGICYT contract number BFM200300856, the E.U. EURIDICE, network contract no. HPRN-CT-2002-00311 and by the Research Cooperation Program of the Japan Society for t.he Promotion of Science (JSPS) and the Spanish Consejo Superior de Investigaciones Cientificas (CSIC). D. C. acknowledges financial support from MCYT and Q. B. Li acknowledges support. from the Ministerio de Educaci6n y Ciencia in the program of Doctores y Tecn6logos Extranjeros.

References 1. T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91 (2003) 012002. 2. G. A. Miller, Phys. Rev. C70 (2004) 022202. 3. H. C. Kim, C. H. Lee and H. J. Lee, arXiv:hepph/0402141. 4. D. Cabrera, Q. B. Li, V. K. Magas, E. Oset and M. J. Vicente Vacas, arXiv:nucl-th/0407007. 5. T. Hyodo, F. Llanes-Estrada, E. Oset, J.R. Pelaez, A. Hosaka and M. J. Vicente Vacas. in preparation. 6. N. Kaiser, T. Waas and W. U'eise, Nucl. Phys. A 612 (1997) 297. 7. E. Oset and A. Ramos, Nucl. Phys. A 679 (2001) 616. 8. D. Cabrera and M. J. Vicente Vacas, Phys. Rev. C 67 (2003) 045203. 9. D. Diakonov, V. Petrov and M. V. Polyakov, Z.Phys. A 359 (1997) 305. 10. S. E,idelman et. al., Phys. Lett. B 592 (2004) 1. 11. E. Oset, H. Toki and Mi. Weise; Phys. Rept. 83 (1982) 281. 12. H. C. Chiang, E. Oset and M. J. Vicente Vacas, Nucl. Phys. A 644 (1998) 77. 13. D. Cabrera and M. J. Vicente Vacas, Phys. Rev. C 69 (2004) 065204. 14. W. R. Gibbs, arXiv:nucl-th/0405024. 15. A. Sibirtsev, J. Haidenbauer, S. Krewald and U. G. Meissner, arXiv:hep ph/0405099. 16. H. Nagahiro, S. Hirenzaki, E. Oset and M. J. Vicente Vacas, arXiv:nuclt h/0408002.

378

PRODUCTION OF O+ HYPERNUCLEI WITH THE (K+, ?r+) REACTION

H. NAGAHIRO AND S. HIRENZAKI Department of Physics, Nara Women’s University, Nara 630-8506, Japan E. OSET AND M.J. VICENTE VACAS Departamento de Fisica Tedrica and IFIC, Centro M b t o Universidad de Valencia- CSIC, Institutos de Inveatigacidn de Paterna, Aptd. 22085, 46071 Valencia, Spain

We investigate the formation cross sections of the @+-hypernucleitheoretically. In the (K+,T+) reaction, we obtain the excitation spectra with clearly differentiated peaks corresponding to the O+ bound states.

The discovery of the O+ at SPring-8/Osaka1, followed by its confirmation in different other experiments, has made a substantial impact in hadronic physics (see Ref. 2 for a compilation of experimental and t h e e retical works on the issue). The possibility that there would be O+ bound states in nuclei has not passed unnoticed and the O+ selfenergy in the nucleus was evaluated3, however with only the part tied to the K N decay, which is known experimentally to be very small. As a consequence, the O+ potential obtained was too weak to bind @+ in nuclei. Suggestions of possible bound states, with qualitative arguments, were also made.4 In a recent paper5 the possibility of having O+ bound states in nuclei was investigated and it was concluded that there is an attractive O+ potential, which, within uncertainties, is strong enough to bind the O+ in nuclei. Restrictions from Pauli blocking and binding reduce the O+ width in nuclei to about one third or less of the free width, and with attractive O+ nucleus potentials ranging from 60 to 120 MeV at normal nuclear matter density, the separation between the deeper O+ levels in light and medium nuclei is larger than the width, even in the case that the free Q+ width were as big as 15 MeV. This is a desirable experimental situation in which clear peaks could be observed provided an appropriate reaction is used.

379

In Ref. 5 the O+ selfenergy tied to the K N decay was also studied and found to be very small like in Ref. 3. The large attraction found in Ref. 5 is tied to the coupling of the O+ to two mesons and a baryon which was related to the strong decay of the N'( 1710) resonance to a nucleon and two pions. In the present paper we present a suitable reaction, ( K + ,A + ) in nuclei, which leads to clear peaks and a fair strength in the spectra of O+ nuclear states.' The reaction is of the substitutional type, in which one of the nucleons in the nucleus will be substituted by the O + . Hence, on top of the different O+ bound states, we shall also have to take into account the binding energies of the nucleon levels in the nucleus when we look for the spectra of this reaction. The analogous ( K - , A - ) reaction has been a standard tool to produce A hypernuclei7 and it was also used to see the only E hypernucleus known so There is an added difficulty here since one would wish to have a kinematics as close as possible to a recoilless condition, which makes the production cross sections larger. This is the case of the (d13He)reaction, which was used with success to find the deeply bound pionic atorns.l0i1l In the present case, given the fact that one has to create the excess mass of the O+ over the nucleon from the kinetic energy of the kaon, this induces inevitably a momentum transfer, which reach a minimal value of the order of 450 MeV/c for an initial K + momentum of around 620 MeV/c. Although this quantity is larger than the Fermi momentum, it can still be accommodated in the O+ wave function without much penalty, such that the cross sections obtained are still &able. We show the momentum transfer of the forward ( K + , A + )reactions for the O+ nuclei formation as a function of the incident K energy in Fig.1. We can also expect to produce O+ nuclear states by the ( y , K - ) and ( A - , K - ) reactions. However, the typical momentum transfer of these reactions are much larger than that of the ( K + , A + )reaction, and the formation rate of these reactions will be suppressed significantly. Also the distortion of the K + is substantially weaker than that of the K - . The cross section for the ( K + , A + )reaction for the formation of the O+ bound states is given by,

380

where $0 (3and $ p ( f l are the 0 and proton wavefunctions in bound states, 6 and @arethe momenta of the incident kaon and the emitted pion, respectively, and J? is the decay width of the final 0+ nuclear states. In Eq. (1) A E is defined as,

+

A E = w ( K + ) ( M p - Sp)- ( M e - B e ) - w ( T + ) ,

(2)

where w ( K + ) and @ ( A + ) are the relativistic energies of K + and w + , Sp is the proton separation energy from the target nucleus, Be is the 0 binding energy in the final states and M,,, M e are the proton and 0 masses, respectively. We sum up all contributions of proton single particle states in the target nucleus and 0 bound states in the final states in Eq. (1). In order to take into account the distortion affecting the pion and kaon, we use the eikonal approximation, in which the distorted waves are a p proximated by plane waves with a distortion factor. The distortion factor D(&z) appearing in Eq. (1) is defined as,

where PK and P, are the probabilities per unit length of kaon and pion flux loss in the nuclear medium. We evaluate PK using the experimental K p and K n total cross sections and the target nuclear density distributions as PK = U K ~ ~ UP K ~ ~ , ,On . the other hand, P, is obtained from the theoretical pion selfenergy, lI,in Refs. 12, 13 as P, = -ImII/p with p the pion momentum. The matrix element t in Eq. (1) is the K + p + w + 0 + transition t - matrix. We get this magnitude from the same Lagrangian used in Ref. 5 to obtain the O+ nuclear potential. The O+ selfenergy in Ref. 5 was obtained studying the excitation of intermediate states K N and K w N , and incorporating the medium effects in the mesons and the nucleon. The part of the selfenergy tied to the K N channel was found very small, but the one tied to the KwN channel led to a sizable attraction. In Ref. 5 the following two Lagrangians were used coupling to the Of to KwN

+

L = igioPlrnT~jky’B:’(V,)~,

L = -#13ci1mF.. 1 v k (4 . d)jBk I mi 2f

(4)

(5)

with V, the two mesons vector current, f the pion decay constant and z j k , B; SU(3) tensors which account for the antidecuplet states, the octet of

,tpi

381

'f

baryons and the octet of 0- mesons, respectively. The amplitude provided by the first Lagrangian, in the nonrelativistic limit, is proportional to the difference of the meson energies in the @+ + NlrK process. Here, since we have an incoming and an outgoing meson, the difference of energies is substituted by their sum and this makes the vertex larger than in the two meson production. Given the fact that the two terms gave equivalent contributions to the @+ binding energy, now the vector term will be dominant, and for the estimations that we are doing here it is sufficient to take this term. Hence, the K + p + @+lrf amplitude is written as,

1 lo(@+) ++I), 4f a with gla = 0.723 and f = 93 MeV. The wave functions of the @+ are obtained by solving the SchrGdinger equation with two potentials, one with a strength

+

t = --(-&)g-

V ( T )= -6O-[MeV], P(T) Po

(7)

and the other one

V ( T )= -120-[MeV]. P(4 Po

The calculated binding energies with these potentials are reported in Ref. 5. In this exploratory level, we only take into account a volume type potential and we have no LS splitting in the 0 energy spectrum. As to the width, it was found in Ref. 5 that assuming the free width to be 15 MeV the width in the medium was smaller than 6-7 MeV, due to Pauli blocking effects mostly. We evaluate the level widths using the imaginary part of the 0 selfenergies calculated in Ref. 5 at the appropriate 0 energies. We show the calculated spectra for the formation of the 0 bound states in Fig. 2 and Fig. 3. We selected carbon as target since the level spacing of each subcomponent is expected to be comfortably large to observe the isolated peaks structure. The incident kinetic energy of the kaon beam is fixed at 300 MeV to miniiise the momentum transfer of the reaction. The results with the shallow 0 potential V ( T )= -6Op/p0 (MeV) are shown in Fig. 2. We find three isolated peaks in the spectrum. We also find that the magnitude of the formation cross section is around a few [pb/sr MeV] which is expected to be reachable in experiments. The results with the deep 0 potential V ( T )= -120p(~)/p,-, (MeV) are shown in Fig. 3, where we find the separated peaks in the cross section

382

again. In this case, we have six clear peaks in the spectrum according to the existence of more 0 bound states due to the deeper potential. The magnitude of the cross section is around 10 times larger than with the shallow potential case because the width of the 0 state is smaller for the deeper bound states which makes the peaks higher. The number of subcomponents of the spectrum is increased for the case of the deeper potential and we find again reasonably large cross section for the reaction. In these calculated results, we have not included the quasi-free 0 production, which will have certain contribution to the spectrum above the 0 production threshold. The threshold is shown by the vertical lines in Fig. 2 and 3. The spectrum for w, below the threshold would be modified by the inclusion of the quasi-free 0 processes. However, the spectrum in the bound 0 region will not be affected by them. We also calculate the spectrum at another incident kinetic energy of the kaon beam TK = 800 MeV, which corresponds to p~ 1.2 GeV/c. The results are shown in Fig. 4 with the shallow potential V ( r )= -6Op/po (MeV). We find that the magnitude of the cross section with TK = 800 MeV is about four times larger than that with TK = 300 MeV which is shown in Fig. 2. This energy dependence of the cross section is mainly caused by the energy dependence of the t-matix in Eq. (6). In summary, the results of Ref. 5 indicate that there should be bound states of 0+ in nuclei, with separation energies reasonably larger than the width of the states. In view of that, we investigated the (K+,n+) reaction to produce these states and obtained excitation spectra of O+ states for a lactarget with two different potentials which cover the likely range of the O+ nucleus optical potential according to the calculations of Ref. 5 . We obtain reasonable production rates in spite of the fact that the momentum transfer is not too small. We would like to mention here that the strength of the cross sections presented here could change as a result of recent advances in Ref. 14, but the shape of the curves and the separability of the peaks remains unchanged. Measurements of binding energies and partial decay widths in nuclei would provide precise information on the coupling of the 0+to two meson channels and about the K n N component in the 0+ wave function. The results obtained here should strongly encourage to do this experiment which could open the doors to the new field of 0+hypernuclei.

-

383

A dcnowledgments This work is partly supported by DGICYT contract number BFM200300856, and the E.U. EURIDICE network contract no. HPRN-CT-200200311. 600

550

,

[

1 (K+,%+)reaction

300 150

200

250 300 350 400 K+ kinetic energy pdev]

450

500

Figure 1. Momentum transfer of the (K+,?r+)reaction for the formation of the O+ nuclear states plotted as a function of the incident kaon kinetic enevy. The solid line shows the result with Sp- Bo e 0 and the dashed line with Sp Bo = -50 MeV, where S, and Bo are the proton separation energy and 0 binding energy, respectively.

-

References 1.

T.Nalcano e t al. [LEPS Collaboration], neutron,” Phys. Rev. Lett. 91 (2003)

012002 [arXiv:hepex/0301020]. 2. T.Hyodo, http://www.rcnp.osaka-u.ac.jp/ hyodo/research/Thetapub.html 3. H. C. Kim,C. H. Lee and H. J. Lee, arXiv:hepph/0402141. 4. G. A. Miller, arXiv:nucl-th/0402099. 5. D. Cabrera, Q.B. Li, W. Magas, E. Oset and M.J. Vicente Vacas, nuclth/0407007. 6. H. Nagahiro, S. Hirenmki, E. Oset and M.J. Vicente Vacas, nucl-th/0408002. 7. A. Sakaguchi e t al., Nucl. Phys. A 721 (2003) 979. 8. T.Nagae et al., Reaction At Phys. Rev. Lett. 80 (1998) 1605. 9. S. Bart et al., Phys. Rev. Lett. 83 (1999) 5238. 10. H. TOE,S. Hirenzaki and T.Yammaki, Pionic Nucl. Phys. A 530 (1991) 679. 11. S. Hiremaki, H. TOE,T.Yamazaki, Phys. Rev. C 44 (1991) 2472. 12. E. Oset and M.J. Vicente-Vacas, Nucl. Phys. A454 (1986) 637-652. 13. J. Nieves, E. Oset and C. Garcia-Recio, Nucl. Phys. A654 (1993) 554-579. 14. A.Hosaka, T.Hyodo, F.J.Llanes-Estrada, E. Oset, J.R. Pelaez, M.J. Vicente Vacas, to be published.

384

F 0.004 2 0.003

E

Y

0.001 0

140

180

160

200

220

240

50

QI [MeV]

Figure 2. Calculated 0 bound states formation cross section shown as a function of the emitted pion energy w* a t forward angles for a lac target. The incident kaon kinetic energy, TK,is 300 MeV and the shallow 0 nuclear potential V ( r )= --BOp(r)/po MeV is used. The total spectrum is shown by the thick-solid line and the dominant subcomponents are also shown by the thin lines as indicated in the figure. The 0 production threshold leaving the residual nucleus in its ground state is shown by the vertical line.

0.06

sF

0.05

s

0.04

Y

0.03

B E

$12

0.02 0.01

0 140

160

180

200

220

240

260

280

300

QI [MeV1

Figure 3. Calculated 0 bound states formation cross section shown as a function of the emitted pion energy W~ a t forward angles for a lac target. The incident kaon kinetic energy, TK,is 300 MeV and the deep 0 nuclear potential V ( r )= - 1 2 0 p ( r ) / m MeV is used. The total spectrum is shown by the thick-solid line and the dominant subcomponents are also shown by the thin lines as indicated in the figure. The 0 production threshold leaving the residual nucleus in its ground state is shown by the vertical line.

385 0.02

0.015

0.01

0.005

0

Figure 4. Calculated 0 bound states formation cross section shown as a function of the emitted pion energy Y= a t forward angles for a lac target. The incident kaon kinetic energy TK is 800 MeV, and the shallow 0 nuclear potential V(r) = - 6 O p ( r ) / p g MeV is used. The total spectrum is shown by the thick-solid line and the dominant subcomponents are also shown by the thin lines BB indicated in the figure. The 0 production threshold leaving the residual nucleus in its ground state is shown by the vertical line.

386

DYNAMICS OF PENTAQUARK IN COLOR MOLECULAR DYNAMICS SIMULATION

w u MAEZAWA~, TOSHIKI MARUYAMA~, NAOYUKI ITAGAKI~,AND TETSUO HATSUDA~ Department of Physics, University of Tokyo, Tokyo 113-0033, Japan Advanced Science Research Center, Japan Atomic Research Institute, Tokai, Ibaraki, 31 9-11 95, Japan Life time of the pentaquark Q+ is investigated on the basis of the color molecular dynamics simulation. We find that it takes a long time (typically of 50 - 100 f m / c ) for the initial pentaquark-state to rearrange its color and spatial configurations to decay into the nucleon+kaon final state. The potential surface in the color and position spaces also supports this picture: Pentaquark wanders on the potential surface to find a narrow channel to decay.

1. Introduction

The pentaquark O+ composed of uuddz with the mass N 1540MeV and the width (< 10 MeV) has been first reported by LEPS collaboration at Spring-8 l . It is exceptionally narrow state despite the fact that the mass is 110 MeV above the nK+ threshold. In this article, we report our recent study on a possible mechanism of the long life time of the pentaquark on the basis of the constituent quark model and color molecular dynamics (CMD) '. 2. Basic formulations of CMD

The CMD originally developed in ref.3 is a quantum molecular dynamics for constituent quarks, in which single quark wave function is parameterized by a Gaussian wave pocket in coordinate space and by a color coherent state in the SU(3) color space with time-dependent variables. We express the total wave function of the pentaquark as a direct product of single-particle quark wave-functions. Time evolution of the system is given by solving the equations of motion obtained from the time-dependent variational principle. Then the position and color of each quark change as a function of time. Hamiltonian consists

387

of the kinetic term and the color-dependent potential. In order to reproduce the mass difference between O+ and nK+ states, we introduce also a mesonexchange type potential. 3. Key-parameters to characterize the decay

+

Since the decay of O+ to the nucleon kaon ( N K ) state is a dynamical process in which 50 variables change in time, it is convenient to define some particular combinations of variables which can qualitatively characterize the decay process. a and D defined below are such key-parameters. a is a measure how well the colors are mixed among five quarks. For a = 0, not only the five quarks are color-white as a whole, but also there are color-white qQ and qqq subclusters. As a increases, the mixture of color in the five quarks increases. The system cannot be separated into two white clusters any more for large a. Note that this parameter is defined only in the color space. Therefore, even if subclusters formed at small a are color-while, they do not have to be clusters in coordinate space. The effective distance D is a measure for the rate of clustering of the five quarks in the coordinate space. For small D, the five quarks are in one unit, while for large D, they split into two spatially separated subclusters. Since D is defined only in the coordinate space, each subcluster is not necessarily color-white. By using these parameters, the pentaquark Of is characterized as a state with large (Y and small D, while the N K scattering state is characterized by small a and large D. 4. Correlation between color and distance, and the life time

We simulate the time development of the five quarks from initial conditions with various color and spatial configurations (a;,;t and Dinit). Then we estimate the lifetime of the system until its decay into the N K state. Figure l(a) shows the relation between the initial color mixing rate a i n i t and the lifetime T. The solid point for a given T is obtained by averaging over randomly distributed values of ainit. The error-bars show the 1u variance of a;,+ This figure clearly shows a positive correlation between T and ai,it. The result can be easily understood: it takes more time for the state with large color mixing to rearrange their colors to two white systems, which decays into the N K final state. Figure l(b) shows the relation between the initial spatial distance Dinit and the lifetime T. The meaning of the solid points and error-bars is the

388

same as Fig. l(a). The figure shows a negative correlation between T and Dinit. The result can be easily understood: it takes more time for the state with small spatial distance between subclusters to rearrange their coordinates to the N K final state. If we choose a i n i t = 0.5 and Dinit = 0 as typical initial values for the pentaquark, we have the life time (decay width) of about 100 fm/c (2 MeV). It is too hasty at the moment to compare this number with the experimental upper bound of 10 MeV, but the result is suggestive.

Or

0.1

03

01

0.4

I

0.5

0

I

0.1

02

%it

0.3 %it

0.4

I 0.5

[k]

Figure 1. (a) Relation between the initial color mixing rate ainit and the lifetime T. (b) Relation between the initial spatial distance D;,;t and the lifetime T.

5. Potential surface as a function of a and D To study the effect of a and D on the dynamical decay process in more details, we calculate an effective potential V between the qQ and qqq subclusters as function of a and D. Shown in Fig. 2(a) is the result of such calculation using the potential part of the Hamiltonian. In general, V increases as D increases due to the effect of color confinement potential. The only exception is a = 0 where the potential energy decreases as D increases. This is because there is no resistance from the confinement force. Note also that the potential surface is flat along the a direction for small D. Now, suppose that O+ is located at a point with small D and large a as indicated by the solid circle in Fig. 2(a). For this state to decay into the N K state indicated by the open circle in Fig. 2(a), it has to rearrange its color to find a narrow channel near a = 0. This takes a long time since the potential surface is flat in a direction and thus the system goes back and forth before reaching the channel. Once it reaches the region around a = 0,

389

it quickly decays into the N K state along the narrow channel. Namely, the flat potential along the a-axis near D = 0 and the narrow channel along the D-axis near a = 0 are two essential sources of the long life time of the pentaquark. Figure 2(b) shows the actual path in the simulation of the pentaquark state decaying into the N K state. The initial condition is taken to be (Yinit = 0.4 and Dinit = 0.25fm, and it takes 25fm/c to decay. Wandering in color space of the pentaquark before the decay can be seen explicitly from this figure.

@>

0

0

0.2

0.4

0.6

0.8

a!

Figure 2. (a) An effective potential V on the a - D plain. (b) Actual motion of the pentaquark decaying into the N K state is superimposed on the top view of the effective potential surface.

6. Summary

In order to study the narrow decay width of O+, we have carried out color molecular dynamics (CMD) simulation for five quarks. The results of the simulation show that there is a positive (negative) correlation between ainit (Dinit) and the life time T of the five-quark state. T can reach to even 100 fm/c if a i n i t is large and Dinit is small. Narrow channel in the effective potential surface V ( a ,D ) is found to be the physical origin to cause the long life time. The pentaquark wanders around the potential surface. References 1. T. Nakano et al., Phys. Rev. Lett. 91, 012002 (2003). 2. Y. Maezawa, T. Maruyama, N. Itagaki and T. Hatsuda, hep-ph/0408056. 3. T.Maruyama and T.Hatsuda, Phys. Rev. C61,062201 (2000).

390

EXOTIC PENTAQUARKS, CRYPTO-HEPTAQUARKS AND LINEAR THREE-HADRONIC MOLECULES

P.BICUDO Dep. FiSica and CFIF? Instituto Superior Tdcnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal, email : [email protected] In this talk, multiquarks are studied microscopically in a standard quark model. In pure ground-state pentaquarks the short-range interaction is computed and it is shown t o be repulsive, a narrow pentaquark cannot be in the groundstate. As a possible excitation, an additional quark-antiquark,pair is then considered, and this is suggested t o produce linear molecular system, with a narrow decay width. This excitation may be energetically favourable to the pwave excitation suggested by the other pentaquark models. Here, the quarks assemble in three hadronic clusters, and the central hadron providm stability. The possible crypto-heptaquark hadrons with exotic pentaquark flavours are studied.

Exotic multiquarks are expected since the early works of Jaffe and the masses and decays in the SU(3) exotic anti-decuplet The penwere fist predicted within the chiral soliton model 2. taquarks have been revived recently by several searches of the 8+(1540) 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, first discovered at LEPS 3, and by searches of the E:--(1860) and of the D*-p(3100) 24: observed respectively at NA49 21 and at H1 24. Pentaquark structures have also been studied on the lattice Moreover multiquarks are favoured by the presence of several different flavours The observation of the D*-p(3100) at H1, the observation of doubl,le-char~lied baryons at SELEX 36, and the future search of double-charmed baryons at COMPASS 37 suggest that new pentaquarks with heavy quarks may be discovered. In this talk it is shown that the pentaquarks cannot be in the groundstate. The lowest excitation consists in including a light quark-antiquark pair in the system. This results ifi a heptaquark and in a linear molecular system. The possible crypto-heptaquark hadrons with exotic pentaquark 21322923

25926927928129130331,32.

33,34135.

391

flavours, with strange, charmed and bottomed quarks, are studied. Recently this principle was used to suggest that the 8+(1540) is a K 0 7c * N molecule with binding energy of 30 MeV 38~39~40, and the ?-(1862) is a K 0 N 0 K molecule with a binding energy of 60 MeV 38941.I also suggest that the new positive parity scalar D,(2320) and axial D,+(2460) are I?oD and K 0 D* multiquarks 42, and that the D*-p(3100) is consistent with a D" 0 7c 0 N linear molecule with an energy of 15 MeV above threshold A systematic search of similar structures has also been performed 44. These recent results are now reviewed. Here I study multiquarks microscopically with a standard quark-model (QM) Hamiltonian. The energy of the multiquark state, and the short range interaction of the mesonic or baryonic subclusters of the multiquark are computed with the multiquark matrix element of the QM Hamiltonian, 38j43,

Each quark or antiquark has a kinetic energy Ti. The colour-dependent twobody interaction V& includes the standard confining and hyperfine terms,

The potential of eq. (2) reproduces the meson and baryon spectrum with quark and antiquark bound states (from heavy quarkonium to the light pion mass). Moreover, the Resonating Group Method (RGM) 45 was applied by Ribeiro, 46 Toki 47 and O h 48 to show that in exotic N N scattering the quark two-body-potential, together with the Pauli repulsion of quarks, explains the N N hard core repulsion. Recently, a breakthrough was achieved in chiral symmetric quark models. These models are inspired in the original work of Nambu and Jona-Lasinio 49. Addressing a tetraquark system with 7c 7c quantum numbers; it was shown that the QM with the quark-antiquark annihilation Aij also fully complies with chiral symmetry, including the Adler zero and the Weinberg theorem For the purpose of this talk, only the matrix elements of the potentials in eq. (1) matter. The hadron spectrum constrains the hyperfine potential,

+

+

+

50,51352.

3

The pion mass

53, constrains

the annihilation potential,

392

and this is correct for the annihilation of 'u, or d quarks. The annihilation potential only shows up in non-exotic channels, and it is clear from eq. (4) that the annihilation potential provides an attractive (negative) interaction. The quark-quark(antiquark) potential is dominated by the interplay of the hyperfine interaction of eq. (3) and the Pauli quark exchange. In s-wave systems with low spin this results in a repulsive interaction. Therefore, I arrive at the attraction/repulsion criterion for groundstate hadrons: - whenever the two interacting hadrons have quarks (or antiquarks) with a common flavour, the repulsion is increased by the Paula principle; - when the two interacting hadrons have a quark and an antiquark with the same flavour, the attraction is enhanced by the quark-antiquark annihilation. For instance, uud - s.li is attractive, and uud - US is repulsive. This qualitative rule is confirmed by quantitative computations of the short-range interactions of the K ; N , K , D,D*, B , B* The attraction/repulsion criterion shows clearly that the exotic groundstate pentaquarks, containing five quarks only, are repelled. If the pentaquark could be forced to remain in the groundstate, this repulsion would provide a mass of 1535 MeV, close to the 8+ mass. There is an evidence of such a negative parity state both in quark model calculations and in lattice computations. However the existence of this groundstate can only appear as an artifact in simulations that deny the decay into the K - N channel. Actually the groundstate is completely open to a strong decay into the K - N channel, and this decay is further enhanced by the repulsion. It is indeed well known that any narrow pentaquark must contain an excitation, to prevent a decay width of hundreds of MeV to a meson-baryon channel. This is understood in the diquark and string model of J&e and Wiczek 54 and Karliner and Lipkin 5 5 , and in the Skyrme model of Diakonov, Petrov and Polyakov '. These models suggest that the pentaquarks include a pwave, or rotational excitation. However this excitation usually leads to a higher energy shift than the one observed, and a novel energy cancellation remains to be consistently provided. A candidate for the energy cancellation is the flavour-hyperfine interaction of Stancu and Riska 56. Although these models are quite appealing, and they have been advocating pentaquarks for a long time, here I propose a different mechanism, which is more plausible in a standard quark model approach. Moreover this mechanism is in a sense confirmed in recent lattice computations, where pentaquarks with pwave excitations indeed have a higher mass than the observed pentaquarks. 38142,43741950751~52.

393 Table 1. Exotic-flavour pentaquarks with no heavy quark. mass [GeV] decay channels

flavour

linear molecule

Z = 112, ssssi(+3 it) :

five-hadron molecule

Z = 1, sss1~(+21i):

four-hadron molecule

z = 312, ssiii(+ii) =

si. iii si: K. N.K = z--

1.86

z = 2, siiii(+ii) =

d. iii ii:

pion unbound

z = 512, iiiii(+ii) =

ii. iii ii:

pion unbound

I = 0, llll%(+li) =

1% i f . iii : K r N = C3+

1.54

.

K + + ,r f 2

K+N

In this talk I consider that a s-wave flavour-singlet light quark-antiquark pair ldis added to the pentaquark M . The resulting heptaquark M’ is a state with parity opposite to the original M 5 7 , due to the intrinsic panty of ferIrLioIis and anti-fermions. The gTouIid-state of M’ is also Iittturally rearranged in a s-wave baryon and in two s-wave mesons, where the two outer hadrons are repelled, while the central hadron provides stability. Because the s-wave pion is the lightest hadron, the minimum energy needed to create a quark-antiquark pair can be as small as 100 MeV. This energy shift is lower than the typical energy of 300-600 MeV of spin-isospin or angular excitations in hadrons. Moreover, the low-energy p-wave decay of the heptaquarks M’ (after the extra quark-antiquark pair is annihilated) results in a very narrow decay width, consistent with the observed exotic flavour pentaquarks. I now detail the strategy to find the possible linear heptaquark molecules, neglecting higher Fock space excitations. a) The top quark is excluded because it is too unstable. To minimise the short-range repulsion and to increase the attraction of the three-hadron system, I only consider pentaquarks with a minimally exotic isospin, and with low spin. b) Here the flavour is decomposed in an s-wave system of a spin 1/2 baryon and two pseudoscalar mesons, except for the vectors D* and B* which are also considered. c) I consider as candidate3 for narrow pentaquarks systems where one hadron is attracted by both other ones. The criterion is used to discriminate which hadrons are bound and which are repelled.

394 Table 2. Exotic flavour pentaquarks with one heavy quark. flavour

linear molecule

Z = 1/2, Hsssi(+lli) :

four-hadron molecule

I = 2, Hiiii(+ii) =

l i e 111 H i :

I = 112, Hlll#(+li) =

I = 112, Hl111(+li) =

l i e 11H : KeneCc K*Ir*&, 11 Him 111 : KeDeN KeD*eN KeBeN K e P e N

mass IGeVl

decay channels

pion unbound

IS

3.08 f0.03 6.41 & 0.1 3.25 f0.03 3.39 f0.03 6.66 f 0 . 0 3 6.71 k 0.03

+

+ Cc, Da + N + Cb, Ds N K + A,, K + Cc, Da + N K + h c , K + Cc, Of + N K + h b , K + Cb: Ba + N K + h b , K + Cb, B f + N K hc, K K 4- h b , K

d) In the case of some exotic flavour pentaquarks, only a four-hadronmolecule or a five-hadron-molecule would bind. These cases are not detailed, because they are difficult to create in the laboratory. e ) hiloreover, in the particular case where one of the three hadrons is a ?r: binding is only assumed if the ?r is the central hadron, attracted both by the other two ones. The 7r is too light to be bound by just one hadron 38. f) The masses of the bound states with a pion are computed assuming a total binding energy of the order of 10 MeV, averaging the binding energy of the @+ and of the D*-p system in the molecular perspective. The masses of the other bound states are computed assuming a total binding energy of the order of 50 MeV, averaging the binding energies of the "t-- and of the new positiveparity Ds mewus. g ) This results in an error bar of f 30 MeV for the mass. When one of the three hadrons is not listed by the Particle Data Group 58, its mass is extracted from a lattice computation 59, and the error bar is f 100 MeV. e ) Although three-body decay channels are possible through quark rearrangement, their observation requires high experimental statistics. Only some of the different possible two-body decay processes are detailed here.

395 Table 3. Exotic flavour pentaquarks with one heavy anti-quark. flavour

linear molecule

I = 0. ssssHl+3li~:

five-hadron molecule

I = 112. ssslH(+2lfi :

four-hadron molecule

I = 0,ssiiH(+ii) =

iH if. iss D.7r.E D'o7r.8

-

Berm2 B*OTO;

I = 112, slllH(+lt) =

iH if. iis D.7r.C D'07r.C BonoE B*o7roC i H si. iii DoKoN 8'oK.N BORON B*oKoN

I = 112, siiiH(+ii) =

.

I = O,ilZlH(+1ij =

b*

iH ii. iii D.7r.N . T O N = D*-p BoroN B*o?roN

decay channels

mass lGeVl

3.31 f0.03 3.45 f0.03 6.73 f 0.03 6.77 f 0.03 3.19 & 0.03 3.33 f0.03 6.60 f0.03 6.64 f0.03 3.25 f0.03 3.39 f0.03 6.66f0.03 6.71 f0.03 2.93 f0.03 3.10 6.35 f0.03 6.39 f0.03

D

+ 8, D. + A

+ A, D 8 B + 8 , B.+A B* 2, B: + A, B ,

D* f E , D:

+

+A +A

b + A; b + C, b.

+N + A, D* + C, Df + N B + A , B + C, B. + N B* + A, B* + C, B: + N D + A, D + C: D, + N D* + A, D* + C, Of + N D*

B+h, B+C, B,+N + C, B: + N

B* + A, B'

D+N P+N,D+N B+N B* N, B N

+

+

To conclude: this work has performed a systematic search of exoticflavour pentaquarks, using the heptaquark, or linear three-body hadronicmolecule perspective. This perspective is the result of standard QM computations of pentaquarks and hepatquark masses and of hadron-hadron shortrange interactions. A large number of new exotic flavour-pentaquarks are predicted in Tables 1 , 2 and 3 together with their two-body decay channels. The systems with more than one heavy antiquark are very numerous and they are omitted here. Moreover, some new multiquarks may be easier to bind than the presently observed exotic pentaquarks.

Acknowledgments

I thank the organisers of Pentaquark04, and I am grateful to Chris Hanhart, Eulogio Oset; Dimitri Diakonov, Frank Lee, Fumiko Okiharu, Hiroshi Toki, Makoto Oka, Silvia Nicolai, Takashi Nakano and Ting-Wai Chiu for lively

396 discussons during t h e P e n t q u a r k 0 4 conference.

References 1. R.L. J&e, SLAC-PUB-1774, talk presented at the Topical Conf. on Baryon

2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

Resonances, Oxford, England, July 5-9, 1976; R. L. J&e, Phys. Rev. D 15 281 (1977). D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A 359 305 (1997) [arXiv:hep-ph/9703373]. T. Nakano et d.[LEPS Collaboration], Phys. Rev. Lett. 91,012002 (2003) [arXiv:hep-ex/0301020]. V. V. Bannin et d.[DIANA Collaboration], Phys. Atom. Nucl. 66, 1715 (2003) [Yad. Fiz. 66,1763 (2003)] [arXiv:hepex/0304040]. S. Stepanyan et al. [CLAS Collaboration], Phys. Rev. Lett. 91,252001 (2003) [arXiv:hep-ex/0307018]. J. Barth et al. [SAPHIR Collaboration], arXiv:hep-ex/0307083. A. E. Asratyan, A. G. Dolgolenko and M. A. Kubantsev, arXiv:hepex/0309042. V. Kubarovsky et al. [CLAS Collaboration], Phys. Rev. Lett. 92, 032001 (2004) [Erratum-ibid. 92, 049902 (2004)l [arXiv:hep-ex/0311046]. A. Airapetian et al. [HERMES Collaboration], Phys. Lett. B 585,213 (2004) [arXiv:hep-ex/0312044]. H. G. Juengst [CLAS Collaboration], arXiv:nucl-ex/0312019. A. Aleev et al. [SVD Collaboration], arXiv:hep-ex/0401024. J. Z. Bai et al. [BES Collaboration], arXiv:hep-ex/0402012. M. Abdel-Bary et al. [COSY-TOF Collaboration], arXiv:hepex/0403011. K. T. Knopfle, M. Zavertyaev and T. Zivko [HEM-B Collaboration], arXiv:hepex/0403020. P. Z. Aslanyan, V. N. Emelyanenko and G. G. Rikhkvitzkaya, arXiv:hep ex/0403044. S. Chekanov et d.[ZEUS Collaboration], arXiv:hep-e~/0403051. C. Pinkenburg, arXiv:nucl-ex/0404001. Y . A. Troyan, A. V. Beljaev, A. Y. Troyan, E. B. Plekhanov, A. P. Jerusalimov, G. B. Piskaleva and S. G. Arakelian, arXiv:hepex/0404003. S. Raducci, P. Abreu, A. De Angelis, DELPHI note 2004-002 CONF 683, March 2004. I. Abt et al. [HEM-B Collaboration], arXiv:hepex/0408048. C. Alt e t al. “A49 Collaboration], Phys. Rev. Lett. 92, 042003 (2004) [arXiv:hep-ex/0310014]. H. G. Fischer and S. Wenig, arXiv:hepex/0401014. J. W. Price, J. Ducote, J. Goetz and B. M. K. Nefkens [CLAS Collaboration], arXiv:nuc1-ex/0402006. [Hl Collaboration], arXiv:hep-ex/0403017. F. Csikor, Z. Fodor, S. D. Katz and T. G. Kovacs, JHEP 0311 (2003) 070 [arXiv:heplat/0309090]. S. Sasaki,arXiv:hep-lat/0310014.

397 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

T. W. Chiu and T. H. Hsieh, arXiv:hep-ph/0403020. T. W. Chiu and T. H. Hsieh, arXiv:hep-ph/0404007. N. Mathur et al., arXiv:hep-ph/0406196. F. Okiharu, H. Suganuma and T. T. Takahashi, arXiv:heplat/0407001. N. Ishii, T. Doi, H, Iida, M. Oka, F. Okiharu and H. Suganuma, arXiv:heplat/0408030. C. Alexandrou, G. Koutsou and A. Tsapalis, arXiv:heplat/0409065. J. M. Richard, arXiv:hep-ph/0212224. K. Cheung, arXiv:hep-ph/0308176. M. F. M. Lutz and E. E. Kolomeitsev, arXiv:nucl-th/0402084. M. Mattson et d.[SELEX Collaboration], Phys. Rev. Lett. 89, 112001 (2002) [arXiv:hep-ex/0208014]. L. Schmitt, S. Paul, R. Kuhn and M. A. Moinester, arXiv:hep-ex/0310049. P. Bicudo and G. M. Marques, Phys. Rev. D 69,011503 (2004) [arXiv:hepph/0308073]. F. J. Llanes-Estrada, E. Oset and V. Mateu, arXiv:nucl-th/0311020. T. Kishimoto and T. Sato, arXiv:hepex/0312003. P. Bicudo, arXiv:hepph/0403146. P. Bicudo, arXiv:hepph/0401106. P. Bicudo, arXiv:hepph/0403295. P. Bicudo, arXiv:hepph/0405086 J. Wheeler, Phys. Rev. 52, 1083 (1937); ibidem 1107. J. E. Ribeiro, Z. Phys. C 5,27 (1980). H. Toki, 2. Phys. A 294, 173 (1980). M. Oka and K. Yazaki, Prog. Theor. Phys. 66, 556 (1981); M. Oka and K. Yazaki, Prog. Theor. Phys. 66,572 (1981). Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 (1961). P. Bicudo, Phys. Rev. C 67,035201 (2003). P. Bicudo, S. Cotanch, F. Llanes-Estrada, P. Maris, J. E. Ribeiro and A. Szczepaniak, Phys. Rev. D 65,076008 (2002) [arXiv:hep-ph/0112015]. P. Bicudo, M. Faria, G. M. Marques and J. E. Ribeiro, Nucl. Phys. A 735, 138 (2004) [arXiv:nucl-th/0106071]. P. Bicudo and J. E. Ribeiro, Phys. Rev. D 42, 1611 (1990); ibidem D 42, 1625; ibidem D 42, 1635. R. L. Jaf€eand F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003) [arXiv:hepphj03073411. M. Karliner and H. J. Lipkin, arXiv:hep-ph/0307243. F. Stancu and D. 0. Riska, Phys. Lett. 13 575, 242 (2003) [arXiv:hepph/0307010]. M. A. Nowak, M. Rho and I. Zahed, Phys. Rev. D 48,4370 (1993) [arXiv:hepph/9209272]. K. Hagiwara et al. [Particle Data Group Collaboration], Phys. Rev. D 66, 010001 (2002). N. Mathur, R. Lewis and R. M. Woloshyn, Phys. Rev. D 66,014502 (2002) [arxiv:hep-ph/0203253].

398

HADRONIC ASPECTS OF EXOTIC BARYONS

E. OSET, S. SARKAR, M.J. VICENTE VACAS, V. MATEU

Departamento de Fisica Tedrica and IFIC, Centm Mixto Uniuersidad de Valencia-CSIC, Institutos de Inuestigacidn de Paterna, Aptd. 22085, 46071 Valencia, Spain T. HYODO, A. HOSAKA

Research Center for Nuclear Physics (RCNP), Ibamlci, Osaka 567-0047, Japan F. J. LLANES-ESTRADA Departamento de Fisica Tedrica I, Uniuersidad Complutense, Madrid, Spain In this talk I look into three different topics, addressing first the possibility that the 8+ is a bound state of KnN, exploiting the results of this study to find out the contribution of two meson and one baryon components in the baryon antidecuplet and in the t h u d place I present results on a new resonant exotic baryonic state which appears as dynamically generated by the Weinberg Tomozawa

OK interaction.

1. Is the O+ a K T N bound state?

The experiment by LEPS collaboration at SPring-8/0saka

has found a

clear signal for an S = +1 positive charge resonance around 1540 MeV. The signal is also found in many other experiments and not found in some experiments at high energy, and is subject of intense study in different labs to obtain higher statistics. A list of papers on the issue can be found in 2.

At a time when many low energy baryonic resonances are well described

as being dynamically generated as meson baryon quasibound states within

chiral unitary approaches

1073949576

it looks tempting to investigate the pos-

399 /

\

/

\

\

/

\

/

Figure 1. Diagrams considered in the n N interaction.

sibility of this state being a quasibound state of a meson and a baryon or two mesons and a baryon. Its nature as a K N s-wave state is easily ruled out since the interaction is repulsive. K N in a p-wave, which is attractive, is too weak to bind. The next logical possibility is to consider a quasibound state of K T N , which in s-wave would naturally correspond to spin-parity 1/2+, the quantum numbers suggested in 7 . Such an idea has already been put forward in

l1

where a study of the interaction of the three body system

is conducted in the context of chiral quark models. A more detailed work is done in

12,

which we summarize here.

Upon considering the possible structure of O+ we are guided by the experimental observation that the state is not produced in the K + p final state. This would rule out the possibility of the 0 state having isospin I=l. Then we accept the O+ to be an 1=0 state. As we couple a pion and a kaon to the nucleon to form such state, a consequence is that the K n substate must combine to I=1/2 and not I=3/2. This is also welcome dynamically since the s-wave K T interaction in I=1/2 is attractive (in I=3/2 repulsive) 13.

The attractive interaction in I=1/2 is very strong and gives rise to the

dynamical generation of the scalar a large width

-

IE

resonance around 850 MeV and with

13.

In order to determine the possible O+ state we search for poles of the

KnN

K n N scattering matrix. To such point we construct the series

of diagrams of fig. 1, where we account explicitly for the K T interaction by constructing correlated KT pairs and letting the intermediate Kn and nucleon propagate. This requires a kernel for the two meson-nucleon interaction which we now address. We formulate the meson-baryon lagrangian in terms of the SU(3) matrices, B , F p , up and the implicit meson matrix

400

standard in ChPT 9 ,

L=

(Bi’fVpB)-MB%

1

(BB)+ s D n Py’y.5

1

{ u p , B})+2FTr

(BY’lyS

[ u p ,B ] )

(1)

with the definitions in

g.

First there is a contact three body force simultaneously involving the pion, kaon and nucleon, which can be derived from the meson- baryon Lagrangian term containing the covariant derivative V, . By taking the isospin I=1/2

K

states and combining them with the

nucleon we generate I=O,1 states which diagonalize the scattering matrix associated to

t,B

and we find that the interaction in the 1=0 channel is

attractive, while in the 1=1 channel is repulsive. This would give chances to the KNt-matrix to develop a pole in the bound region, but rules out the 1=1 state. The series of terms of Fig. 1 might lead to a bound state of K N which would not decay since the only intermediate channel is made out of KTN with mass above the available energy. The decay into K N observed experimentally can be taken into account explicitly and this and other diagrams accounting for the interaction of the mesons with the other meson or the nucleon are taken into account in the calculations

12.

What we find at the end is that, in spite of the attraction found, this interaction is not enough to bind the system, since we do not find a pole below the K T N threshold. In order to quantify this second statement we increase artificially the potential t mby~adding to it a quantity which leads

/, = 1540 MeV with a width of around I? to a pole around S

=

40 M e V .

This is accomplished by adding an attractive potential around five or six times bigger than the existing one. We should however note that we have not exhausted all possible sources of three body interaction since only those tied to the Weinberg Tomozawa term have been considered. We think that some more work in this direction should be still encouraged and there are already some steps given in

14.

40 1

2. Coupling of the O+ t o K?rN

Although not enough to bind the K T N system, the interaction has proved attractive in L=O and I=O. This, together with the proximity of the O+ mass to the K T N threshold ( 30 MeV) suggests that the O+ should have

a non negligible K T N component in its wave function. The procedure followed in l5 to find out the contribution to the binding is the following: 1) one assumes that the O+ belongs to the standard antidecuplet of baryons suggested in

7.

2) The N*(1710) is assumed to have a large component

corresponding to this antidecuplet. 3) From the large decay of the N*(1710) into TTN,both in s-wave and pwave, we extract the strength for two

SU(3) invariant phenomenological potentials which allow us to extend the coupling to different meson meson baryon components of all baryons of the antidecuplet. 4) A selfenergy diagram is constructed with two vertices from these Lagrangians and two meson and a baryon intermediate states. 5 ) Regularization of the loops is done with a cut off similar to the one needed in the study of the l?N interaction and this leads to attractive selfenergies of the order of 100-150 MeV. At the same time one finds an energy splitting between the different members of the antidecuplet of the order of 20 MeV,

or 20 percent of the empirical values, with the right ordering demanded by the Gell-Mann-Okubo rule, and hence a maximum binding for the O+. This finding means that detailed studies of the O+ should take into consideration this important component of K T N which helps produce extra binding for the O+,one of the problems faced by ordinary quark models. The finding of this work has repercussions in the selfenergy of the O+ in nuclei. Indeed, as found in

16, when

one takes into account the pionic

medium polarization, exciting ph and Ah components with the pion, the mechanism leads to an extra attraction in the medium which is of the order of 50-100 MeV at normal nuclear matter density. This, together with the other finding of a very small imaginary part of the selfenergy, leads to levels of the O+ which are separated by energies far larger than the width

402

of the states. This makes it a clear case for experimental observation and suggestions of reactions have already been made

17.

3. A resonant AK state as a dynamically generated exotic

baryon Given the success of the chiral unitary approach in generating dynamically low energy resonances from the interaction of the octets of stable baryons and the pseudoscalar mesons, in

l8 the

interaction of the decuplet of 3/2+

with the octet of pseudoscalar mesons was studied and shown to lead to many states which were associated to experimentally well established resonances. The purpose of the present work is to show that this interaction leads also to a new state of positive strangeness, with I = 1 and J p = 3/2-, hence, an exotic baryon described in terms of a resonant state of a A and

a K. The lowest order chiral Lagrangian for the interaction of the baryon decuplet with the octet of pseudoscalar mesons is given by

l9

where TCbcis the spin decuplet field and D” the covariant derivative given in

19.

The identification of the physical decuplet states with the T:bc can

be seen in 20. For strangeness S = 1 and charge Q = 3 there is only one channel

A++K+ which has I = 2. For S = 1 and Q = 2 there are two channels A++Ko and A + K + . From these one can extract the transition amplitudes for the I = 2 and I = 1 combinations and we find 2o 3

V ( S = 1,I = 2) = -(kO+k’O); 4f2

1 V ( S = 1,I = 1) = --(kO+k’O),

4f

(3)

where k(k’) indicate the incoming (outgoing) meson momenta. These results indicate that the interaction in the I is attractive in I = 1.

=2

channel is repulsive while it

403

The use of V as the kernel of the Bethe Salpeter equation 3, or the N/D unitary approach of

both lead to the scattering amplitude

t

= (1 - VG)-lV

In eq. (4), V factorizes on shell

314

(4)

and G stands for the loop function of

the meson and baryon propagators, the expressions for those being given in

for a cut off regularization and in

for dimensional regularization.

Next we fix the scale of regularization by determining the cut off, qmax, in the loop function of the meson and baryon propagators in order to reproduce the resonances for other strangeness and isospin channels. They are one resonance in ( I ,S ) = (0, -3), another one in ( I ,S) = (1/2, -2) and another one in

(1,s)= (1,-1).

The last two appear in

l8

around 1800

MeV and 1600 MeV and they are identified with the E(1820) and C(1670). We obtain the same results as in

l8 using

a cut off qmax = 700 MeV.

With this cut off we explore the analytical properties of the amplitude for S = 1, I = 1 in the first and second Riemann sheets. First we see that there is no pole in the first Riemann sheet. Next we explore the second Riemann sheet which we obtain by changing the sign of the momentum in the expresssion for the meson baryon loop function. We find a pole at , b = 1635 MeV in the second Riemann sheet. The situation in the scattering matrix is revealed in fig. 2 which shows the real and imaginary part of the K A amplitudes for the case of I = 1. For the case of I = 2 the imaginary part follows the ordinary behaviour of the opening of a threshold, growing smoothly from threshold. The real part is also smooth. For the case of I = 1, instead, the strength of the imaginary part is stuck to threshold as a reminder of the existing pole in the complex plane, growing very fast with energy close to threshold. The real part has also a pronounced cusp at threshold, which is also tied to the same singularity.

404

Re t (with width)

-0.05

-0.07

1400

1500

1600 1700 1800 1900 C. M. Energy (MeV)

Figure 2. Amplitudes for AK

+ AK

2000

2100

for I = I

In figure 2 and we see that the peak around threshold becomes smoother and some strength is moved to higher energies when we consider the width of the A in the intermediate states. Even then, the strength of the real and imaginary parts in the I = 1 are much larger than for I = 2. The modulus squared of the amplitudes shows some peak behavior around 1800 MeV in the case of I = 1, while it is small and has no structure in the case of I = 2. We propose the study of the following reactions: 1) pp

pp

+

C-A++K+, 3) pp

--t

-+

AA+K+, 2)

CoA++Ko. In the first case the A+K+ state

produced has necessarily I = 1. In the second case the A++K+ state has

I = 2. In the third case the A++Ko state has mostly an I = 1 component. The experimental confirmation of the results found here through the study of the AK invariant mass distribution in these reactions would give support to this new exotic baryonic state which stands as a resonant AK state.

405

Acknowledgments This work is partly supported by DGICYT contract number BFM2003-

00856, t h e E.U. EURIDICE network contract no. HPRN-CT-2002-00311 and the Research Cooperation program of t h e JSPS and the CSIC.

References 1. T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91 (2003) 012002.

2. T. Hyodo, http://www.rcnp.osaka-u.ac.jp/ hyodo/research/Thetapub.html 3. E. Oset and A. Ramos, Nucl. Phys. A 635 (1998) 99. 4. J. A. Oller and U. G. Meissner, Phys. Lett. B 500 (2001) 263. 5. D. Jido, J. A. Oller, E. Oset, A. Ramos and U. G. Meissner, Nucl. Phys. A 725 (2003) 181. 6. J. Nieves and E. Ruiz Arriola, Phys. Rev. D 64 (2001) 116008. 7. D. Diakonov, V. Petrov and M.V. Polyakov, Z. Phys. A359 (1997) 305. 8. J. Barth et al. [SAPHIR Collaboration], Phys. Lett. B 572 (2003) 127.

9. V. Bernard, N. Kaiser and U.G. Meissner, Int. J. Mod. Phys. E4 (1995) 193. 10. N. Kaiser, P. B. Siege1 and W. Weise, Nucl. Phys. A 594 (1995) 325. 11. P. Bicudo and G. M. Marques, Phys. Rev. D 69 (2004) 011503.

12. F. J. Llanes-Estrada, E. Oset and V. Mateu, Phys. Rev. C 69 (2004) 055203.

13. J. A. Oller, E. Oset and J. R. Pelaez, Phys. Rev. D 59 (1999) 074001 [Erratum-ibid. D 60 (1999) 0999061. 14. T. Kishimoto and T. Sato, arXiv:hepex/0312003. 15. T. Hyodo, F. Llanes, E. Oset, A. Hosaka, J.R. Pelbz, M.J. Vicente Vacas, to be submitted. 16. D. Cabrera, Q. B. Li, V. K. Magas, E. Oset and M. J. Vicente Vacas, arXiv:nucl-t h/0407007. 17. H. Nagahiro, S. Hirenzaki, E. Oset and M. J. Vicente Vacas, arXiv:nuclth/0408002. 18. E. E. Kolomeitsev and M. F. M. Lutz, Phys. Lett. B 585, 243 (2004).

19. E. Jenkins and A. V. Manohar, Phys. Lett. B 259 (1991) 353. 20. S. Sarkar, E. Oset and M. J. Vicente Vacas, arXiv:nucl-th/0404023.

406

PENTAQUARK BARYONS IN STRING THEORY

M. BAND0 Physics Division, Aichi University, Aichi 4 70-0296, Japan

T. KUGO Yukawa Institute, Kyoto University, Kyoto 606-8502, Japan A. SUGAMOTO AND S. TERUNUMA Department of Physics and Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo 112-8610, Japan

Pentaquark baryons 8+ and E-- are studied in the dual gravity theory of QCD in which simple mass formulae of pentaquarks are derived in the Maldacena prototype model for supersymmetric QCD and a more realistic model for ordinary QCD. In this approach it is possible to explain the extremely narrow decay widths of pentaquarks. With the aim of constructing more reliable mass formulae, a sketch is given on spin and the hyperfine interaction in the string picture.

1. Introduction We are very happy to give a talk at this pentaquark 04 conference, on the bassis of our recent papers l, It made a great impact on us when the pentaquark baryon O+ was found here at Spring-8 by T. Nakano et al. last year. O+ is an exotic baryon consisting of five quarks, (ud)(ud)S, having the mass, &I(@ = 1,540 f lOMeV, and the width, I?(@+) 5 25MeV. Subsequently, other pentaquarks, Z:--((ds)(ds)C) and @((ud)(ud)E),were reported to be observed at CERN NA49 and H E M H1, respectively. At this conference, we have also learned that there are positive and negative indications on the observation of these pentaquarks, depending on the experimental apparatuses. Pentaquarks were predicted by Diakonov et al. in 1997 as chiral solitons. As is well known, in the naive quark model, hadron masses are estimated as the sum of masses of constituent quarks and energy of the hyperfine or the color magnetic interactions. Masses of triquarks calculated in this way

407

are in good agreement with the observed values, but for pentaquarks, the observed masses are about 200 MeV lower than the predicted values, and the observed widths are very narrow, being about 1/100 of their Q-values 2 . Therefore, this is a very interesting problem to inquire. Jaf€e and Wilczek treated the penatquarks as being composed of two diquark pairs (ud)(ud) and one anti-quark S , while Karliner and Lipkin considered they are made of two clusters, diquark (ud) and triquark (uda). The pentaquarks were also studied in lattice QCD and QCD sum rules. In this conference, we had a number of good talks on the QCD flux tube models of pentaquarks, which were presented by Y . Enyo, E. Hiyama, S. Takeuchi, F. Okiharu, and H. Suganuma *. The purpose of this talk is to study the pentaquark baryons in colored string theory, using the recent development in string theories started by J. Maldacena in 1998. The best way to understand this picture is to draw a picture of O+ as quarks connected by colored strings of red, green and blue, which gives a very beautiful shape displayed in Fig.1.

Figure 1. Three-dimensional view of the pentaquark 8+ in the string picture.

408

In this picture, the mass of O+ is estimated as the total length of the colored strings located in the curved space with extra dimensions. Furthermore, there may be an unexpected merit of this picture; that is, this branched web is quite stable. This is because, for a pentaquark to decay into a meson and a triquark baryon, string configurations with a loop must appear in the intermediate stage of decay, but this may be a rare occurrence. Intuitively speaking, in order for the pentaquark to decay, the string con+ necting two junctions JO and J1(l#), say with red color, should be replaced by two strings with anti-blue and anti-green colors. This replacement is energetically and stochastically very dificult to ocuur.

2. Dual gravity model of QCD

The correspondence betwen dual gravity model and QCD, usually called AdS/CFT correspondence, is a very difficult concept for us, but it may be understood by using factorization and vacuum insertion. QCD consists of quarks and gluons; quarks possess both color ( r , g , b ) and flavor (u, d, s,etc.), while gluons possess color ( T , g, b) and anti-color ( F , g , 6 ) but not flavor. An open string (a string with two endpoints) is ideally suited to account for such quantum numbers at its two ends. For quarks, one end represents color and the other end flavor. For gluons, one end represents color and the other anti-color. In recently developed string theory, we prepare “branes” (higher dimensional extended objects that are generalized membranes) on which the endpoints of these open strings are confined to move. Applying this idea to QCD, we introduce N,(= 3) “colored branes” and N f “flavored branes” at which open strings corresponding to quarks and gluons terminate. Because the classical energy of a string is proportional to its length and because gluons are massless, Nc colored branes should lie on top of one another. On the other hand, quarks possess intrinsic masses, and therefore the endpoints of a quark string, namely, a flavored brane and a colored brane should be separated from each other by a nonvanishing distance U . If the direction of the separation is chosen along the fifth (extra) dimension u, the energy stored by this separation is the internal one. Then, the intrinsic quark mass m, can be represented as m, = U x (string tension), where the string tension is the energy stored inside a unit length of string. To evaluate the amplitude for a certain process to occur in the above picture, we have to sum up all the possible two-dimensional world sheets swept by the string with the weight exp(iS), where the action S is given by

409

S=(energy) x (time)=(area of the string’s world sheet) x (tension). As stated above, the endpoints of the strings are confined to the colored branes or flavored branes, so that the world sheet has boundary trajectories {Ci}(i= 1,2, ...) on colored branes and { F j } ( j = 1,2, ...) on flavored branes. This amplitude is denoted as A({Ci},{F’}). Let us make a simple approximation using factorization and a vacuum insertion, which is frequently used in ordinary QCD. For example, in the decay of Bo --t K-e+v, we factorize the current-current interaction and use the vacuum insertion ~vac)(wac~:

( B ’ ~ J + P JJK-, ~ e+, v) M (BO, K+ ( J + P lvuc)(wml~; le+, v).

(1)

In the same way, the string amplitude can be approximated by the factorized amplitude with a vacuum insertion:

A( { ci1, {Fj 1) = ({ Ci1I v 4 (vaclP

j

1).

(2)

Summing up all the possible configuration of {Ci}(i= 1,2, ...) gives

Now, the remaining problem is to determine what the vacuum state is. As seen from the first factor, ~ ~ , ~ ( { C i } ~ v the a c )existence , of N, colored branes deform the flat vacuum to the curved space with compactification. The various curved spaces with compactification (vacua) are known after Maldacena’s work. If we prepare N,(= 3) four dimensional Minkowski spaces (world sheets of D3-branes) for the colored branes, the vacuum becomes the five-dimensional Anti de Sitter space Ads5 x S5. We call this Maldacena’s prototype model which corresponds to N = 4 supersymmetric SU(N,) Yang-Mills theory, but it is not the ordinary QCD. To describe the five-dimensionalspace, we introduce an extra coordinate u which measures the intrinsic quark masses in addition to the Minkowski space, ( t , z , x ~ )along , which the world volume of the stuck N, colored branes extend. In more realistic models, we need to break the supersymmetries. For this purpose, an effective method is to compactify one space-like dimension to a circle, a variant of the method of Witten. Then, we obtain AdS Schwartzshild spaces. In this way, the difference in boundary conditions between fermion and boson in the compactified dimension breaks the supersymmetries completely. Therefore, if the radius of the compactified circle is RKK, then the mass scale MKK = ~ I T / R K is K introduced.

410

The metric of the vacuum deformed by the existence of N, colored branes whose world sheets are Minkowski space times the circle, is known. We have used this curved space to describe a more realistic model of QCD. 3. General formulation of Pentaquarks

As discussed in the previous section, to evaluate the amplitude or the energy of pentaquarks, we have to evaluate (vacl{Fj}).We are interested in evaluating the static energy of pentaquarks. So, we first fix the position of flavors, or fix the static trajectories of five quarks, Fj=u,d,ut,dt,B,on the flavored branes. The u- and d-quarks are placed on the same flavored brane, since u and d have an almost equal mass. On the other hand, s-quark is heavior than u and d, so that s-quark is placed on another brane located farther from the colored branes than the brane of u and d. These five quarks are connected by colored strings as in Fig.1. This picture shows the three dimensional view. In our treatment, however, the pentaquark is located in the five dimensional curved space determined by the dual gravity theory of QCD. Therefore, the strings can extend also in the fifth dimension (u-direction). This is the same problem of finding the shape and length of a string placed under the gravity, where both ends of string are picked up by hands. In our problem the virtical coordinate corresponds to u, while the horizontal coordinates on the earth correspond to x’s and z‘s in Fig. 1. Therefore, we can solve this problem easily and obtain the energy stored inside the strings of pentaquarks as the function of coordinates z’s and z’s. Subtracting the rest masses of quarks we obtain the potential of the pent aquark . 4. Maldacena prototype model

By solving the non-relativistic Schodinger equations in the Maldacena prototype model, following the method just mentioned, the mass formula of the pentaquark family of Q+ is obtained as

M((qlqz)(q;q;)G3)= 2(ml + m z ) ( A+ B ) + m3A,

(4)

while that of the triquark family of nucleons reads

M(qlQ2q3)= (ml+ m2 where A = 1 - acNca2/.rrand 8 0.236.

=

+ m3)A,

(5)

-acNcb2/.rr, with a x 0.359 and b

NN

411

5. Pentaquarks in a QCD like model In this model, we obtain the following mass formulae: &f(pentaquark) = m l

@(triquark) =

+ + 7h3 + 63{(m1)3 + (m2)4} 7%~

+ a2+ a3 + -32( c ~ , N , ) ~ (mi)-;.(7) i=1-3

Here, the pentaquark and triquark are considered to be ((qlq2)2i&) (414243), respectively, and the dimensionaless masses with bars are normalized by MKK/(~~N,). We choose the input parameters, M ( N ) =939 MeV and M ( C ) =1,193 MeV, and M(C,) =2,452MeV. Then, fixing a, to 0.33 (or Nca, = l ) , the quark masses and the pentaquark masses are estimated respectively as mu = m d =313-312 MeV, m, =567-566 MeV, m, =1,826 MeV, and M ( O ) =1,577-1,715 MeV, M(E) =1,670-1,841 MeV, M(O,) =2,836-2,974 MeV, M(Ec) =3,266-3,556 MeV, corresponding to the KK mass scale of M = M K K =2-5 MeV. Here, the pentaquark masses with c-quark are the new predictions, not included previously l . 6. Decay process of pentaquarks

+

In the string picture, the decay processes of O+ ---* n K+ are displayed in Fig.2. The main step is the recombination of two strings. In each channel, neutron n or a K meson accompanied by a “string loop” is created. This is the key point of having narrow widths for pentaquarks. The recombination of strings can be replaced by the other process in which a string segment is firstly splitted by the pair production of quarks, producing a baryon with five quarks or a meson with four quarks. The importance of these states are also pointed out by D. Diakonov in this conference. Subsequently, these exotic baryon or meson becomes neutron or K meson with the string loop by the pair annihilation of quarks. If the state with a “sring loop” is denoted with tilde, two decay channels can be written as follows:

e+-{

f i 0 + K+ N o + I?+

--

N o + K+ (channel l ) , N o + K+ (channel 2).

(8)

The narrow width, O( 1) MeV, of the pentaquarks, comes from the difficulty of forming the “string loop” states in the decay process. Using PCAC

412

U

2

L!

a*'

L..'

/

Figure 2.

d

\ d

s

O+OU'

K+

Decay processes of 8+ -t K +

+ N o (neutron).

we can show that the mass mixing between states with and whithout the string loop should be small, being roughly 1/10 as large as their masses. In this conference H. Suganuma 4 , starting from our decay mechanism, identified the states with a string loop to be the f i s t gluonic excitation of hadrons. He claimed that the excited state is about 1 GeV heavier than the ordinary hadrons by the lattice calculation, and that the decay amplitude has the suppression factor of about 1/150. This Suganuma's talk reinforced our suppression mechanism of pentaquark's decay. 7. Preliminary sketch of spin and the hyperfine interaction

In string theory we have the fermionic variables @'(T, a) in addition to the . former is the distribution function of y bosonic variables X ~ ( T , O )The matrices, so that the "spin" is distributed on the whole string in the string picture. This is probably useful to explain the spin crisis of hadrons. In this string picture we have obtained a formula of the hyperfine interactions. Detailed anaysis of the stringy hypefine interaction will make the study of pentaquarks more realistic.

8. Conclusion 1) Pentaquark baryons are studied in the dual gravity model of QCD. 2) This model may be understood by using factorization and vacuum insertion. 3) Simple connection conditions are derived at junctions of string webs. 4)

413

In the extremely naive approximation, mass formula is obtained, and the decay rate is roughly estimated. 5 ) Nevertheless, the predictions do not differ significantly from the experiment values. 6) Spins and the hyperfine interaction are sketched in string theory in order to approach more realistic study. 7) The concept that colors and flavors are located on the ends points of strings while spins are distributed on the whole string may give new insights on hadron physics. Now, the string theory comes down to the real world ?! Acknowledgments

The authors give their sincere thanks to Prof. H. Toki, Prof. A. Hosaka, and all the staff of Pentaquark 04 for their excellent organization and for giving the opportunity of presenting this talk. References 1. M. Bando, T. Kugo, A. Sugamoto, S. Terunuma, Prog. Theor. Phys. 112,325 (2004):hepph/0405259; A. Sugamoto, talk given at 2nd international symposium on “New Developments ofhtegruted Sciences” held at Ochanomizu U. on March 16 (2004):hep ph/0404019. 2. M. O h , Pmg. Theor. 112, 1 (2004):hepph/0406211. 3. Y.Kanada-Enyo, 0. Morimatsu, T. Nishikawa, hep-ph/0404144; E. Hiyama, talk at Pentaquark 04. T. Shinozaki, M. Oka, S. Takeuchi: hepph/0409103. F. Okiharu, H. Suganuma, T. T. Takahashi: hep-lat/0407001. 4. H. Suganuma, talk at Pentaquark 04.

414

NARROW WIDTH OF PENTAQUARK BARYONS IN QCD STRING THEORY

HIDE0 SUGANUMA AND HIROKO ICHIE Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo 152-8551,Japan [email protected]. ac.jp FUMIKO OKIHARU Department of Physics, Nihon University, Chiyoda, Tokyo 101-8308,Japan TORU T. TAKAHASHI Y I T P , Kyoto University, Kitashirakawa, Sakyo, Kyoto 606-8502,Japan Using the QCD string theory, we investigate the physical reason of the narrow width of penta-quark baryons in terms of the large gluonic-excitation energy. In the QCD string theory, the penta-quark baryon decays via a gluonic-excited state of a baryon and meson system, where a pair of Y-shaped junction and anti-junction is created. On the other hand, we find in lattice QCD that the lowest gluonicexcitation energy takes a large value of about 1 GeV. Therefore, in the QCD string theory, the decay of the penta-quark baryon near the threshold is considered as a quantum tunneling process via a highly-excited state (a gluonic-excited state), which leads to an extremely narrow decay width of the penta-quark system.

1. Introduction

In 1969, Y . Nambu first presented the string picture for hadrons' to explain the Veneziano amplitude2 on the reactions and the resonances of hadrons. Since then, the string picture has been one of the important pictures for hadrons and has provided many interesting ideas in the wide region of the particle physics. Recently, various candidates of multi-quark hadrons have been experimentally o b s e r ~ e de+ . ~(1540),3 ~ ~ ~2--(1862)4 ~ ~ ~ ~ ~ and 8,(3099)5 are considered to be penta-quark (4Q-Q) states and have been investigated with various theoretical frameworks.8397 10,11912,13914,15716717718,w O , 2 1 , 2 2 X (3872)6 and 0,(2317)7 are expected to be tetra-quark (QQ-QQ) states23>24725 from the consideration of their mass, narrow decay width and decay mode.

415

As a remarkable feature of multi-quark hadrons, their decay widths are extremely narrow, and it gives an interesting puzzle in the hadron physics. In this paper, we investigate the physical reason of the narrow decay width of penta-quark baryons in the QCD string theory, with referring the recent With lattice QCD, we discuss the lattice QCD results.15’16,17,26’27’28,29,30,31 flux-tube picture and the gluonic excitation in Sects.:! and 3, respectively. In Sect.4, we apply the QCD string theory to penta-quark dynamics, and try to estimate the decay width of the penta-quark baryon near the threshold. 2. The Color-Flux-Tube Picture from Lattice QCD

To begin with, we show the recent lattice QCD studies of the inter-quark potentials in 3Q, 4Q and 5Q systems,15~16~17~26~27 and revisit the color-fluxtube p i c t ~ r e ~for’ ~hadrons, ~~ which is idealized as the QCD string theory.

2.1. The Three-Quark Potential in Lattice QCD For more than 300 different patterns of spatially-fixed 3Q systems, we calculate the 3Q potential from the 3Q Wilson loop in SU(3) lattice QCD with (p=5.7, 123 x 24), (p=5.8, 163 x 32), (p=6.0, 163 x 32) and (p = 6.2, 244). For the accurate measurement, we construct the ground-state-dominant 3Q operator using the smearing method. To conclude, we find that the static ground-state 3Q potential V!;’ is well described by the Coulomb plus Y-type linear potential, i.e., Y-Ansatz,

within 1%-level d e ~ i a t i o n He . ~re,~ Lmin ~ ~ ~is ~the ~ ~minimal value of the total length of the flux-tube, which is Y-shaped for the 3Q system. To demonstrate this, we show in Fig.l(a) the 3Q confinement potential i.e., the 3Q potential subtracted by the Coulomb part, plotted against the Y-shaped flux-tube length Lmin. For each p, clear linear correspondence is found between the 3Q confinement potential V:rf and Lmin, which indicates Y-Ansatz for the 3Q potential. Recent-ly,as a clear evidence for Y-Ansatz, Y-type flux-tube formation is actually observed in maximally- Abelian (MA) projected lattice QCD from the measurement of the action density in the spatially-fixed 3Q ~ y s t e m Thus, together with recent several other analytical and numerical s t ~ d i e s Y-Ansatz , ~ ~ ~ for ~ ~the~ static ~ 3Q potential seems to be almost settled. This result indicates the color-flux-tube picture for baryons.

Qrf,

416

Figure 1. (a) The 3Q confinement potential V;gf, i.e., the 3Q potential subtracted by the Coulomb part, plotted against the Y-shaped flux-tube length Lmin in the lattice unit. (b) The lattice QCD result for Y-type flux-tube formation in the spatially-fixed 3Q system. The distance between the junction and each quark is about 0.5 fm.

2.2. Tetra-quark and Penta-quark Potentials

Motivated by recent experimental discoveries of multi-quark hadrons, we perform the first study of the multi-quark potentials in SU(3) lattice QCD. We calculate the multi-quark potentials from the multi-quark Wilson loops, and find that they can be expressed as the sum of OGE Coulomb potentials and the linear potential based on the flux-tube p i ~ t ~ r e , ~ ~ i

where Lmin is the minimal value of the total length of the flux-tube linking the static quarks. Thus, the lattice QCD study indicates the color-flux-tube picture even for the multi-quark systems. Also, this lattice result presents the proper Hamiltonian for the quark-model calculation of the multi-quark systems.

3. The Gluonic Excitation in the 3Q System Next, we study the gluonic excitation in lattice QCD.28*29i30 In the hadron physics, the gluonic excitation is one of the interesting phenomena beyond the quark model, and relates to the hybrid h a d r o n ~such ~ ~ ?as~qqG ~ and qqqG in the valence picture. In QCD, the gluonic-excitation energy is given by the energy difference A E ~ Q = V$$ - Vt;' between the ground-state poand the excited-state potential V;;., and physically means the tential excitation energy of the gluon-field configuration in the static 3Q system. For about 100 different patterns of 3Q systems, we perform the first study of the excited-state potential in SU(3) lattice QCD with 163 x 32 at p=5.8 and 6.0 by diagonalizing the QCD Hamiltonian in the presence of

sf$'

417

three quarks. In Fig.2, we show the 1st excited-state 3Q potential V;$' and the ground-state potential V:;'. The gluonic-excitation energy O&Q G V,;. - V:;' in the 3Q system is found to be about lGeV in the hadronic scale as 0.5fm 5 Lmin 5 1.5fm. This result predicts that the lowest hybrid baryon qqqG has a large mass of about 2 GeV.

0

5 10 L,, [lattice unit]

15

Figure 2. The 1st excited-state 3 Q potential V
Note that the gluonic-excitation energy of about lGeV is rather large compared with the excitation energies of the quark origin. Also for the QQ system, the gluonic-excitation energy is found to take a large value of about 1GeV37. Therefore, for low-lying hadrons, the contribution of gluonic excitations is considered to be negligible, and the dominant contribution is brought by quark dynamics such as the spin-orbit interaction, which results in the quark model without gluonic m ~ d eIn Fig.3, ~ .we present ~ ~a ~ possible scenario from QCD to the massive quark model in terms of color confinement and dynamical chiral-symmetry breaking (DCSB). 4. The QCD String Theory for the Penta-Quark Decay

Our lattice QCD studies on the various inter-quark potentials indicate the flux-tube picture for hadrons, which is idealized as the QCD string model. In this section, we consider penta-quark dynamics, especially for its extremely narrow width, in terms of the QCD string theory. The ordinary string theory mainly describes open and closed strings corresponding t o QQ mesons and glueballs, and has only two types of the reaction process as shown in Fig.4:

~

418

Quantum Chromodynamics

I

(Strong) Color Confinement

I

Dynamical Chiral Symmetry BRaking

Color Flux-Tube Formation with a Large String Tension u 2 1 GeVffm

0

Large Excitation Energy of the Flux-Tube Vibration

Large Constituent Quark Mass Mp N 300 MeV

0

Large Gluonic Excitation Energy A E N 1 GeV

0

Absence of Gluonic Excitation Modes in Low-lying Hadrons

Non-Relativistic Tleatment on Quark Dynamics in Low-lying Hadrons

Massive Quark Model for Low-lying Hadrons Figure 3. A possible scenario from QCD to the quark model in terms of color confinement and DCSB. DCSB leads to a large constituent quark maSS of about 300 MeV, which enables the non-relativistic treatment for quark dynamics approximately. Color confinement results in the color flux-tube formation among quarks with a large string tension of u N 1 GeV/fm. In the flux-tube picture, the gluonic excitation is described as the flux-tube vibration, and its energy is expected to be large in the hadronic scale. The large gluonic-excitation energy of about 1 GeV leads to the absence of the gluonic mode in low-lying hadrons, which plays the key role to the success of the quark model without gluonic-excitation modes.

1. The string breaking (or fusion) process. 2. The string recombination process. On the other hand, the QCD string theory describes also baryons and anti-baryons as the Y-shaped flux-tube, which is different from the ordinary string theory. Note that the appearance of the Y-type junction is peculiar to the QCD string theory with the SU(3) color structure. Accordingly, the QCD string theory includes the third reaction process as shown in Fig.5: 3. The junction (J) and anti-junction process.

(5) par creation (or annihilation)

Through this J-5 pair creation process, the baryon and anti-baryon pair creation can be described.

419

Figure 4. The reaction process in the ordinary string theory: the string breaking (or fusion) process (left) and the string recombination process (right).

Figure 5. The junction (J) and anti-junction peculiar to the QCD string theory.

( 5 ) par

creation (or annihilation) process

As a remarkable fact in the QCD string theory, the decay/creation process of penta-quark baryons inevitably accompanies the J-5 creation20 as shown in Fig.6. Here, the intermediate state is considered as a gluonicexcited state, since it clearly corresponds to a non-quark-origin excitation. As shown in the previous section, the lattice QCD study indicates that such a gluonic-excited state is a highly-excited state with the excitation energy above 1GeV. Then, in the QCD string theory, the decay process of the penta-quark baryon near the threshold can be regarded as a quantum tunneling, and therefore the penta-quark decay is expected to be strongly suppressed. This leads to a very small decay width of penta-quark baryons.

Figure 6. A decay process of the penta-quark baryon in the QCD string theory. The penta-quark decay process inevitably accompanies the 5-5 creation, which is a kind of the gluonic excitation. There is also a decay process via the gluonic-excited meson.

420

Now, we try to estimate the decay width of penta-quark baryons near the threshold in the QCD string theory. In the quantum tunneling as shown in Fig.6, the barrier height corresponds to the gluonic-excitation energy A E of the intermediate state, and can be estimated as A E N 1GeV. The time scale T for the tunneling process is expected to be the hadronic scale as T = 0.5 lfm, since T cannot be smaller than the spatial size of the reaction area due to the causality. Then, the suppression factor for the penta-quark decay can be roughly estimated as I exp(-AET)I2 N (e- lGeVx(0.5-l)fm12 10-2 10-4. (3)

-

Note that this suppression factor I exp(-AET)I2 appears in the decay process of low-lying penta-quarks for both positive- and negative-parity states. For the decay of 8+(1540) into N and K , the Q-value Q is Q = M ( 0 + ) M(N) - M(K) N (1540 - 940 - 500)MeV N 100MeV. In ordinary sense, the decay width is expected to be controlled by rhadron N Q N 100MeV. Considering the extra suppression factor of I exp(-AET)I2, we get a rough order estimate for the decay width of 0+(1540) as r[8+(1540)]II rhadron

X

I exp(-AET)I2

N

1

N

10-2MeV.

(4)

Acknowledgements

H.S. thanks Profs. T. Kugo and A. Sugamoto for useful discussions on the QCD string theory. H.S. is also grateful to Pro&. K. Hicks and C. Hanhart for useful discussions on the penta-quark decay. The lattice QCD Monte Carlo simulations have been performed on NEC-SX5 at Osaka University. References 1. Y . Nambu, in Symmetries and Quark Models (Wayne State University, 1969); Lecture Notes at the Copenhagen Symposium (1970). 2. G. Veneziano, Nuovo Cam. A 5 7 , 190 (1968). 3. LEPS Collaboration (T. Nakano et al.), Phys. Rev. Lett. 91, 012002 (2003); DIANA Collaboration (V. V. Narmin et al.), Phys. Atom. Nucl. 66, 1715 (2003); CLAS Collaboration (S. Stephanian et al.), Phys. Rev. Lett. 91, 252001 (2003); SAPHIR Collaboration (J. Barth et al.), Phys. Lett. B572, 127 (2003). 4. H1 Collaboration (A. Aktas et al.), Phys. Lett. B588, 17 (2004). 5. NA49 Collaboration (C. Alt et al.), Phys. Rev. Lett. 92, 042003 (2004). 6. Belle Collaboration (S. K. Choi et al.), Phys. Rev. Lett. 91, 262001 (2003); CDF I1 Collaboration (D. Acosta et aZ.), Phys. Rev. Lett. 93, 072001 (2004); DO Collaboration (V. M. Abazov et al.), Phys. Rev. Lett. 93, 162002 (2004); BABAR Collaboration (B. Aubert et aZ.), Phys. Rev. Lett. 93, 041801 (2004).

421 7. BABAR Collaboration (B. Aubert et aZ.), Phys. Rev. Lett. 90, 242001 (2003); Belle Collaboration (P. Krokovny et al.), Phys. Rev. Lett. 91, 262002 (2003). 8. D. Diakonov, V. Petrov and M. Polyakov, 2. Phys. A359, 305 (1997). 9. For recent reviews, M. Oka, Prog. Theor. Phys. 111, 1 (2004); S. L. Zhu, Int. J. Mod. Phys. A19, 3439 (2004) and their references. 10. S.-L. Zhu, Phys. Rev. Lett. 91, 232002 (2003). 11. R.L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003). 12. A. Hosaka, Phys. Lett. B571, 55 (2003). 13. F1. Stancu and D.O. Riska, Phys. Lett. B575, 242 (2003). 14. M. Karliner and H.J. Lipkin, hep-ph/0307243 (2003). 15. H. Suganuma, T. T. Takahashi, F. Okiharu and H. Ichie, in QCD Down Under, March 2004, Adelaide, NucZ. Phys. B (Proc. Suppl.) (2004) in press. 16. F. Okiharu, H. Suganuma and T. T. Takahashi, “First study for the pentaquark potential in SU(3) lattice QCD” , hep-lat/0407001. 17. F. Okiharu, H. Suganuma and T. T. Takahashi, this proceedings. 18. P. Bicudo and G.M. Marques, Phys. Rev. D69, 011503 (2004); P. Bicudo, this proceedings. 19. I.M. Narodetskii, Yu. A. Simonov, M.A. Tkusov and A.I. Veselov, Phys. Lett. B578, 318 (2004). 20. M. Bando, T. Kugo, A. Sugamoto and S. Terunuma, this proceedings; Prog. Theor. Phys. 112, 325 (2004). 21. X.-C. Song and S.-L. Zhu, Mod. Phys. Lett. A19, 2791 (2004). 22. Y . Kanada-Enyo, 0. Morimatsu and T. Nishikawa, this proceedings. 23. F. E. Close and S. Godfrey, Phys. Lett. B574, 210 (2003). 24. F. E. Close and P. R. Page, Phys. Lett. B578, 119 (2004). 25. E. S. Swanson, Phys. Lett. B588, 189 (2004); ibid. B598, 197 (2004). 26. T.T. Takahashi, H. Matsufuru, Y . Nemoto and H. Suganuma, Phys. Rev. Lett. 86, 18 (2001); in Dynamics of Gauge Fields, p.179 (2000). 27. T.T. Takahashi, H. Suganuma, Y . Nemoto and H. Matsufuru, Phys. Rev. D65, 114509 (2002); AIP Conf. Proc. 594, 341 (2001). 28. T.T. Takahashi and H. Suganuma, Phys. Rev. Lett. 90, 182001 (2003). 29. T.T. Takahashi and H. Suganuma, Phys. Rev. D70, 074506 (2004). 30. H. Suganuma, T.T. Takahashi and H. Ichie, in Color Confinement and Hadrons an Quantum Chromodynamics, p.249 (World Scientific, 2004). 31. H. Ichie, V. Bornyakov, T. Streuer and G. Schierholz, Nucl. Phys. A721, 899 (2003); Nucl. Phys. B (Proc. Suppl.) 119, 751 (2003). 32. Y . Nambu, Phys. Rev. D10, 4262 (1974). 33. J. Kogut and L. Susskind, Phys. Rev. D11, 395 (1975); J. Carlson, J. Kogut and V. Pandharipande, Phys. Rev. D27, 233 (1983); ibid. D28, 2807 (1983). 34. D.S. Kuzmenko and Yu. A. Simonov, Phys. Atom. Nucl. 66, 950 (2003). 35. J.M. Cornwall, Phys. Rev. D69, 065013 (2004). 36. P.O. Bowman and A.P. Szczepaniak, Phys. Rev. D70, 016002 (2004). 37. K.J. Juge, J. Kuti and C. Morningstar, Phys. Rev. Lett. 90, 161601 (2003); AIP Conf. Proc. 688, 193 (2004). 38. S. Capstick and P.R. Page, Phys. Rev. (266, 065204 (2002).

422

WORKSHOP SUMMARY: EXPERIMENT

KENNETH HICKS * Department of Physics and Astonomy Athens, OH 45701, USA E-mail: [email protected]

A summary of experimental results from the Pentaquark 2004 Workshop held at the Spring-8 facility in Japan is given. New results from the LEPS collaboration are highlighted, and older results are reviewed. Non-observations are also discussed in light of theoretical estimates of possible 8+production mechanisms. The problem of the narrow width and the parity of the Q+ are explored and point to future experimental work that is needed.

1. Introduction

This was an exciting workshop with many new results in the rapidly changing field of exotic baryons. Both theoretical and experimental advances have been made, with new ideas by the theorists to explain how a very narrow resonance can be constructed. On the other hand, the experimental results are mixed, with some new positive evidence and some new null measurements, and little hope to clarify the questions of width and parity of the @+ within the next few years. Clearly, the existence of the pentaquark is an experimental issue, and it must be resolved before the physics community can take seriously the theoretical explanations. Here, I will focus on the experimental results and leave the theoretical summary to Carlson With both positive and null measurements of several possible pentaquark resonances, there are strong statements being made on both sides of the argument. In fact, some might even say that it has become an emotionally-charged issue in hadronic physics. It is important to realize that it takes time to do good experiments, and that nature sometimes has surprises in store for us. For this reason, caution and patience are advised while we wait for progress. If we let science take its course, then in the end the truth will emerge. *Supported in part by RCNP (Japan) and the NSF/DOE (USA).

423

Since there is uncertainty in the existence of the pentaquark, the best we can do is ask whether there is good reason to be optimistic or pessimistic for the future of the pentaquark. I will return to this question periodically.

2. Reasons to be Optimistic

Over ten experiments have published evidence for the O+ pentaquark While the number of measurements may be impressive, one must keep in mind that most of these are in the range of 4-6 standard deviations above a background that is difficult to quantify. In fact, these statistical estimates assume a smooth background shape, and hence the statistical significance may be overestimated. Still, it seems unlikely that all of the results can be explained as a statistical fluctuation of the background, since so many different reactions (with different backgrounds) have been used in the analysis. Of course, such hand-waving arguments are not proof that the O+ exists, but the variety of reactions and the quality of the experiments does provide some encouragement that the O+ exists. The HERMES and ZEUS experiments are well-known and respected (as are other groups that have published positive evidence). However, these experiments have been criticized l3 because they cannot determine the strangeness in the Kop final state. In this workshop, the HERMES collaboration l4 showed that the Kop peak in their results is not consistent with an interpretation as a C*+ resonance. They also showed that the peak to background ratio in their data can be enhanced by applying cuts that remove the known K* and A* resonances. This strengthens their case that the peak is real, although more statistics are needed, which will occur in the next year. This is some reason for optimism. An intriguing result was presented by the GRAAL collaboration l5 which did not search for the O+ but instead have evidence for a narrow N* resonance near 1680 MeV. In their measurement, q-photoproduction on deuterium, they separate events that occurred on the neutron or the proton. In a theoretical prediction by Polyakov and Rathke 16, transitions from octet baryons to antidecuplet baryons are suppressed on the proton (due to an isospin factor) but allowed on the neutron. In fact, this is what the GRAAL collaboration see, albeit with limited statistics. If this claim can be confirmed by other experiments, then this narrow N* resonance fits better into the symmetry group that includes the O+ as well. But until the GRAAL result is confirmed, it is wise to resist the urge for optimism. Perhaps the best evidence so far for the O+ was shown at this work213,4,51617,8,9,10111,12.

424

shop by the LEPS collaboration 17. This analysis of the yd 3 K + K - X , where X is restricted to have the mass of deuterium, is an improvement over analysis shown at the MESON04 conference 13. Additional cuts to remove coherent product ion and an energy-dependent &meson exclusion were shown to enhance the A(1520) peak in the K+ missing mass spectrum (corrected for Fermi motion). These same cuts also enhance the peak in the K - missing mass, where the O+ might be expected. A new develop ment is the use of event mixing to get the shape of the background. Event mixing uses a K+ from one event and a K - from a different event. The missing mass of the mixed events is still required to be at the deuterium mass, which ensures energy conservation. The advantage of using event mixing is: (1) the statistics can be increased because of the number of combinations, (2) the real angle and momentum distribution is used for the kaons, which is better than a phase-space Monte Car10 that is sometimes used to determine the detector acceptance, (3) correlations between the K+ and K - are removed, so that the generated background goes smoothly under real peaks (which by definition have correlated K t K - pairs). The event mixing was shown l7 to work for the background under the A( 1520) and using the same procedure, the O+ peak also comes up clearly above the mixed-event background. This quantitative analysis of the background shape gives more credence to the O+ peak seen in the new LEPS deuterium data. Finally, the best reason for optimism is that there are several new results on the horizon that have the potential to convincingly settle the question of whether the O+ exists or not. The CLAS collaboration has taken new data on both a deuterium target l8 and a proton target l9 with about 10 times the statistics of earlier data, and expect to have results on several different O+ search topologies by early 2005. The COSY-TOF collaboration 2o will upgrade their detector and will take more data in 2005, thus increasing their statistics by (perhaps) a factor of five. As already mentioned, the HERMES collaboration will double their statistics on a deuterium target soon, which can substantially help their 0+ search. With these new results on the horizon, 2005 will be an exciting year.

3. Reasons to be Pessimistic

There are a number of experiments that give null results in a search for the O+ pentaquark. At the time of this conference, only 2 were published and a number had been presented at an earlier conference 23. At this 21i22

425

meeting, there were presentations of pentaquark searches by the BaBar 24, Belle 25 and Fermilab E690 26 collaborations. Naively, one would expect to see the O+ (and perhaps the =--) in these experiments, and the null results imply that either: (1) the pentaquark does not exist, or (2) the production mechanism of pentaquarks differs from that of 3-quark baryon resonances. In any case, these results are good reason for one to be pessimistic (or at the very least very cautious) about the existence of pentaquarks. The BaBar results 24 have high statistics and reasonable signals for the established N* and Y*resonances. Here, a baryon-antibaryon pair is created from e+e- collisions at f i = 10.58 GeV. The A(1520) resonance is seen clearly in the p K - invariant mass, but no structure is seen in the pKo mass spectrum in the region of the O+ mass. Based on systematics of baryon production rates as a function of baryon mass, one can estimate the number of O+ baryons that should have been produced. However, this assumes that pentaquark production (involving 5 quarks and 5 antiquarks) follows the same systematics as 3-quark baryons. Due to the uncertainty in the production mechanism, theoretical calculations are needed to understand the true significance of these null results. The Belle experiment 25 took a different approach. They used secondary scattering of mesons (from e+e- collisions) in their silicon vertex detector to produce known Y* resonances. If the O+ exists, it could be produced with a K+ beam of the right energy. Unfortunately, the hadrons incident on the silicon target have unknown identity and unknown energy. Only a small fraction of these data could result in production of the O+ and detected by its decay into the pKo channel. With the high resolution of Belle, even a small signal (with a narrow width) might be visible, but none was seen. Again, we need better calculations of the expected number of counts (based on Belle's spectrum of hadrons incident on silicon) before we can interpret their null result. The E690 experiment 26 uses protons of about 800 GeV in peripheral collisions with target protons. By putting cuts on the missing transverse momentum and the longitudinal energy, exclusive reactions can be measured. In the p K - mass spectrum, about 5000 A(1520) events are seen, but no structure is seen in the pKo mass spectrum. Because of the exclusive reaction, the strangeness of the pKo system is known and so this spectrum is not contaminated with C* resonances, such as the C(1660). Hence, this is a significant null result, and suggests one should be pessimistic about the O+ existence. One interesting development of this workshop was a calculation pre-

426

sented by Titov 27, using quark constituent counting rules to estimate the ratio of O+ to A(1520) production in fragmentation reactions. F’ragmentation functions are well established 28 and have been used for years to describe the distribution of hadrons from high-energy collisions, based on the number of constituent partons in the projectile and target. Using this model, Titov shows that production of the 8+ is suppressed relative to the A(1520) resonance by about 3 orders of magnitude for experiments such as E690 and BaBar. Of course, the simple model used for this estimate may not be a good approximation for all kinematics, but it is consistent with the null experimental results at high energies. The optimist would argue that we could have expected null results from fragmentation reactions in high-energy experiments. It is easy to see that there is reason for pessimism, but the evidence is not entirely convincing. It is difficult to know how many O+ events should have been seen in the high-energy experiments with null results. In fact, there is even a reasonable explanation from Titov, using fragmentation functions, for these null results. The case for “killing” the O+ has not been made.

4.

The Problem of the Width

Perhaps the most disturbing fact of the O+ evidence is that its width appears to be very narrow. Direct evidence limits the width to be less than about 10 MeV. Indirect evidence, based on analysis of KN scattering data estimates the width at a few MeV or less. Such a narrow width for a resonance 100 MeV above its strong decay threshold would be unprecedented. Coupled with the narrow width problem is the question of parity. The spin of the lowest-lying O+ is expected to be J = 1/2 with either negative (S-wave) or positive (P-wave) parity. A narrow width from an S-wave resonance makes no sense 34 whereas a P-wave would allow a centrifugal barrier making a narrow width at least possible 34,35. It seems logical that if the O+ width is narrow, its parity must be positive. This idea was beautifully presented by Hosaka 36. What do lattice QCD calculations say about the parity? Several lattice results were presented at this conference, and except for one result 37, only the negative parity projection gives a result consistent with the O+ So we have an apparent contradiction between the parity deduced from quark 3y899

29!30931732733,

38p39.

models (above) and the parity deduced from (most) lattice calculations.

427

One obvious resolution to this dilemma is to conclude that the O+ does not exist. However, we must realize that the lattice calculations for exotic baryon resonances should be regarded as exploratory 38. Extrapolating to the chiral limit from the heavy quarks used in lattice calculations must be done properly and furthermore, all lattice calculations are done in the quenched approximation. Hence we should be cautious about parity statements based on current lattice results. Even if the O+ exists with positive parity, a width as narrow as 1 MeV is theoretically difficult to understand 40. However, several new theoretical ideas were presented showing that such a narrow width is consistent with theory. Using a two-state model, Lipkin showed 41 that the mass eigenstates of two pentaquarks (e.g., mixtures of the Jaffe-Wilczek model and the diquark-triquark model) can mix, resulting in one coupling strongly to KN decay (with a wide width) and one decoupling (with a narrow width). Another approach, this time with mixing between the octet and the shown by Praszalowicz 40, can suppress the width by a correction factor that depends on the value of the pion-nucleon C term. From a completely different angle, using the QCD string model, Suganuma et al. showed 42 that the pentaquark does not just “fall apart” as predicted by the quark model, but must overcome a sizeable potential barrier to decay into a KN final state. This results in a very narrow width for the O+ in their model. In all, it is interesting that a narrow width of 1 MeV can be accomodated in the quark model, the chiral soliton model and the QCD string model. Clearly, experimental information is needed before one can test the various ideas about the O+ width. Proposals at KEK and Jefferson Lab for high resolution spectrometer experiments are being considered. Other facilities alredy mentioned (COSY-TOF, HERMES, ZEUS) will gather more statistics, which should enable a better determination of the O+ width. In addition to width measurements, we need to know the O+ parity. This will likely be done at COSY-TOF 2o using polarized target and polarized beam, which has a clear theoretical interpretation as shown by Hanhart 45. If the O+ exists, then we have the experimental tools to learn about its width and parity. 37p39

n,

5.

Summary

This workshop was filled with exciting new developments, both experimental and theoretical. The LEPS collaboration showed new data, this time using a deuterium target, which appears to confirm the existence of the O+

428

although the results are still preliminary. A possible narrow N* state was seen in 7 photoproduction from the neutron at GRAAL, but not from the proton, in agreement with theoretical predictions. While no new data were shown for the O+ width or parity, several theoretical models showed that a narrow width for the O+ is not unreasonable. The null results from high-energy experiments are worrisome, but theoretical estimates from fragmentation functions suggest that O+ production is very suppressed relative to baryon resonances like the h(1520).If so, then this is not negative evidence for the O+ but just non-observation. Of course, the proof of O+ existence must be convincing in the medium energy experiments, with high-statistics data, before one can believe it is suppressed in high-energy data. So should we be optimistic or pessimistic about the existence of pentaquarks? At the present time, there is no clear choice. However, new data will be available soon that will clarify the situation. If the O+ exists, then we have a rich new spectroscopy to explore.

Acknowledgments I am grateful for the support I received from RCNP (Osaka University) and Kyoto University during my sabbatical in Japan. It has been a great pleasure to work closely with Takashi Nakano and Atsushi Hosaka, along with the members of their groups, during my stay. I congratulate Hosakasan for the excellent organization of the Pentaquark 2004 workshop.

References C. Carlson, these proceedings. T. Nakano et al., (LEPS), Phys. Rev. Lett. 91, 012002 (2003). V.V. Barmin et al., (DIANA), Phys. Atom. Nuclei 66, 1715 (2003). S. Stepanyan et al., (CLAS), Phys. Rev. Lett. 91, 25001 (2003). V. Kubarovsky et al., (CLAS), Phys. Rev. Lett. 92, 032001 (2004). J. Barth et al., (SAPHIR), Phys. Lett. B572, 127 (2003). A.E. Asratyan, A.G. Dolgolkenko and M.A. Kubantsev, Phys. Atom. Nucl. (2004); hepex/0309042. 8. A. Airapetian et al., (HERMES), Phys. Lett. B585 (2004) 213; h e p ex/03 12044. 9. The ZEUS collaboration, Phys. Lett. B591 (2004) 7-22; hepex/0403051. 10. C. Alt et al., (NA49), Phys. Rev. Lett. 92, 042003 (2004); hepex/0310014. 11. M. Abdel-Barv, et al., (COSY-TOF), Phys. Lett. B595, 127 (2004); nuclex/0403011. 12. A. Aleev et al., (SVD), submitted to Yad. Fiz.; hepex/0401024. 1. 2. 3. 4. 5. 6. 7.

429 13. K. Hicks, hepph/0408001. 14. W. Lorenzon (HERMES), these proceedings. 15. C. Schaerf (GRAAL), these proceedings. 16. M.V. Polyakov and A. Rathke, hepph/0303138. 17. T. Nakano et al., these proceedings. 18. D. Tedeschi (CLAS), talk in this workshop. 19. R. DeVita (CLAS), these proceedings. 20. W. Eyrich (COSY-TOF), these proceedings. 21. J.Z. Bai et al., (BES), hepex/0402012. 22. K.T. Knopffe et al., (HEM-B), hepex/0403020. 23. Quarks and Nuclear Physics, Indiana University, 2004; website www.qnp2004.org. 24. V. Halyo (BaBar), these proceedings. 25. R. Mizuk (Belle), these proceedings. 26. E. Gottschalk (E690), these proceedings. 27. A.I. Titov, A. Hosaka, S. Date, Y. Ohashi, these proceedings and Phys. Rev. C 71 (2004). 28. F.E. Close, A n Introduction to Quarks and Partons, Academic Press, London, 1979. 29. S. Nussinov, hepph/0307357. 30. R.N. Cahn and G.H. Trilling, Phys. Rev. D69, 011501 (2004). 31. R.A. Arndt, 1.1. Strakovsky and R.L. Workman, Phys. Rev. C68, 042201(R) (2003). 32. A. Sibirtsev, J. Haidenbauer, S. Krewald and Ulf-G. Meissner, hepph/0405099. 33. W.R. Gibbs, nucl-th/0405024. 34. R. Jaffe and F. Wilczek, Phys. Rev. Lett. 91:232003 (2003). 35. M. Karliner and H.J. Lipkin, Phys. Lett. B575 (2003) 249; hepph/0307243. 36. S.-I. Nam, A. Hosaka, and H.-C. Kim, these proceedings. 37. T.-W. Chiu and T.-H. Hsieh, these proceedings. 38. S. Sasaki, these proceedings. 39. F.X. Lee, these proceedings. 40. M. Praszalowicz, these proceedings. 41. H. Lipkin, these proceedings; see also hepph/0401072. 42. H. Suganuma, H. Ichie, F. Okiharu, T. Takahashi, these proceedings; h e p ph/0412271. 43. K. Imai et al., KEK experiment E559. 44. B. Wojtsekhowski and G. Cates, proposal P05-009. 45. C. Hanhart, J. Haidenbauer, K. Nakayama, U.-G. Meissner, these proceedings; hepph/0407107.

430

PENTAQUARKS: THEORY OVERVIEW, AND SOME MORE ABOUT QUARK MODELS

CARL E. CARLSON* Particle The0y Group, Physics Department College of William and M a y Williamsburg, VA 23187-8975,USA E-mail: [email protected]

After reviewing the basics, topics in this talk include an attempted survey of theoretical contributions to this workshop, some extra specific comments on quark models, and a summary.

1. Basics

At the PANIC conference in Osaka in October 2002, a collaboration working at Spring-8 announced a baryon state’, the 0+(1540) with a decay mode 0+ -, nK+. Assuming the decay is not Weak, that meant it had strangeness +1, and therefore was an exotic state (“exotic” in the sense that it is a baryon that cannot be a 8 state). It can be a ududa state: a pentaquark. It has since been found at other labs by other collaborations, and also has failed to be seen in some experiments: an experimental review follows this talk2. In terms of flavor, considering u, d, and s quarks, a quark is in a 3 representation and an antiquark is in a 3, and the relevant product is 3 @ 3 8 3 @ 3 @ 3 = 1@ 8@ l O @ I V @ 27@35

(1)

(with multiplicities omitted on the right hand side). An exotic must be in a a 27, or a 35. We have room to show two of these in Fig. 1; the 35, not shown, has an isotensor multiplet (quintet) of 0’s. Since searches for a C W don’t find it, we believe the 0+(1540) is isosinglet, and in a flavor The spin is not known experimentally; we

m,

m.

*Work partially supported by the National Science Foundation (USA) under grant PHY0245056.

431

will assume it is 1/2. Neither is the parity known, and we shall consider both possibilities.

Figure 1. Two flavor multiplets with strangeness +1 states, the 10 on the left and the 27 on the right. The shaded states are guaranteed to be exotic. (Note: the Particle Data Group has proposed that the Z = 3/2, S = -2 be called Q rather than 2.)

2. Talks at this conference

I think it worthwhile to catalog the talks we heard at this workshop. It gives some idea of the effort that has gone into understanding the pentaquark, and of the variety of starting points for that understanding. (In some places I have also listed work not directly reported here.) Let me start with lattice gauge theory and with QCD sum rules, with a short comment following. 0 Lattice gauge theory. There were talks at this conference by Chiu, Ishii, Takahashi, Sasaki, and F. Lee. The table notes what parity the pentaquark was found to have by each collaboration-if they saw a pentaquark signal at all. Mather (Kentucky) et al. no pentaquark parity Chiu and Hsieh (Taiwan) Sasaki (Tokyo) - parity - parity Csikor et al. (Hungary/Germany) MIT - parity Ishii et al. no pentaquark - parity Takahashi et a1

+

0

-

QCD sum rules Sugiyama, Doi, Oka Nis hikawa S. H. Lee

negative parity positive parity negative parity again

432

Lattice gauge theory and QCD sum rules share a problem (of which the practitioners are well aware): The O+ state we want lies within an N K continuum. The threshold for the continuum is below the O+. That means the desired signal is exponentially suppressed relative to continuum. Part of the practitioners skill is in choosing test operators with little overlap with the continuum. Hence the discussions-and also hope of progress. 0 Chiral soliton or Skyrme models. These were the original motivational springboard of the modern pentaquark era. We should mention the work of Diakonov, Petrov, Polyakov; of Manohar; of Weigel; of Praszalowicz; of Hirada; of Chemtob; of Oh, Park, & Min; a good fraction of these workers were represented here. 0 Quark model. The quark model allows one to explicit calculations of many pentaquark properties. There were contributions regarding wave functions, masses, production rates, magnetic moments, etc., and I also list a f ew authors who were not represented here. Quark cluster states H~gaasenand Sorba Lipkin and Karliner Jaffe and Wilczek Consequences of Flavor-Spin Riska and Stancu hyperfine interactions Jennings and Maltman; C., Carone, Kwee, Nazaryan Real 5q calculations Takeuchi; Enyo; Hiyama Maezawa (w/movies!); Okiharu (given a choice of 7-1) Shinozaki Instanton motivated potentials Higher representations Dmitrasinovic Y . Oh Manohar & Jenkins H. C. Kim Magnetic moments C. M. KO Production S. I. Nam;

+

0 Consequences of strong coupling of 0 t o nucleon two mesons. Under this heading comes the possibility that the 0 itself is a “molecular” state, i.e., an N K T w/40 MeV binding, as well as binding of the 0 within a nuclear medium. Workers here include Vacas, Nagahiro, Bicudo, Oset, Kishimoto, and Sato. And to close this catalog, we have 0 String views of pentaquarks, reported by Suganuma and by Sugamoto. 0 Methods of parity measurement-theory. Discussed here by Hanhardt. Why pentaquarks are unseen at high energy. Discussed by Titov.

433

We continue with some additional remarks in the context of quark models of pentaquarks. An important question is the parity of the state. 3. Negative parity

The easiest q4Q,Fig. 3, state to consider in a quark model is one where all the quarks are in lowest spatial state. Since the Q has negative intrinsic parity, the state overall will have negative parity. This can be good in that it may be supported by lattice calculations and QCD sum rules, or bad in that it disagrees with results of chiral soliton models, or really bad in that a straightforward version will lead to very broad decay width.

Figure 2.

Pentaquark state.

It is, of course, crucial that the q4 part of the pentaquark wave function be totally antisymmetric, as required by Fermi statistics. It is useful to write out the full state. One may choose different ways to express it, and we will display the result of breaking the state into a sum of terms where each term is a product of a q3 state and a qQ state. For the q4Qstate that has the quantum numbers of the O+, the result is3

The coefficients are fixed uniquely (up to a phase) by the requirement of antisymmetry when a quark in the q3 is exchanged with the quark in the

m. The notation “N” in the above expansion means a state with the color, flavor, and spin structure of a nucleon. The spatial wave function may be different, for example, it may be more spread out than for an isolated nucleon state: it is, after all, embedded in a O+. Similar comments hold for the K and K*. However, the N K quantum number part of the state is the one that, at least naively, can easily fall apart into a real nucleon plus kaon final state. The other pieces of the state, when taken apart, either weigh more than the 0’- or are colored. Hence only 25% of state can decay

434

into kinematically allowed final states (123% each for nK+ and p K o ) . We shall see that 25% is a big number, after we discuss the positive parity case. 4. Positive parity

Positive parity requires that one of the quarks be in a P-state excitation. One then expects the state to get heavier and one has to discuss counter mechanisms that could keep the state light. One countermechanism is to cluster the quarks, and ignore interactions among quarks in different clusters. Quarks within a given cluster do have color-spin interactions. An illustration with two clusters is shown in Fig. 4. /

/

I

/

I I

‘

I

\

I

\

/

‘

/

/ “triquark”[color 31

diquark [color?I

Figure 3. Clustered quark possibility

One seeks cluster possibilities that have maximal binding. Ignoring intercluster interactions, it can be made to work. For example, looking at the figure, choose the diquark as C = 3, F = 3, S = 0, and the qqq ‘‘triquark” as C = 3, F = G , S = 1/2, and put the clusters in a relative spatial P-wave. This is a suggestion of Karliner and Lipkin; one should see6 and a version by Jaf€e and Wilczek with clustering into diquark-diquark-8. However the requirement to keep the clusters separated so that intercluster qq and qQ interactions small-and to dare neglect Fermi statistics requirements of quarks in separate clusters-seems dangerous. We wish also to consider uncorrelated states, with Fermi statistics fully satisfied. One the needs different mechanism to make positive parity lighter than negative parity, since the color-spin attractions among quarks prove unsuccessful. However, another possibility is at hand, in the form of flavorspin interactions for quark pairs. Consider the mass shift operator, = -Cx

C(XF~), (XFO)~ *

(3)

This flavor-dependent mass operator was originally put forward because

435

it had good effects on 43 baryon spectroscopy4. Later comparative studies of mass operators for excited states indicated flavor-spin operators were necessary to describe the observed mass spectrum5. Be all that as it may, the question remains: For the q4 part of state, how can an S3P configuration be lighter than an S4? Recalling the q3 baryon case is useful. It shows the mass advantage of a totally symmetric flavor-spin state (the relevance of this to the pentaquark will become clear) and sets the value of the parameter C,. The problem in the q3 sector is that the Roper or N*(1440) [an excited S-state] is lighter than the P-states, e.g., the &1(1535). With spin-dependent interactions, the situation is like the intermediate part of Fig. 4, drawn for a potential like the harmonic oscillator. The Roper is heavier than the nucleon by two oscillator spacings, the P-state by only one. However, the Nucleon, Roper, A all have symmetric flavor-spin wave functions, and the spin-dependent interactions can pull them down much more that the 5'11. Explicitly,

AMx=

{

-14Cx -4Cx -2Cx

N(939), N*(1440) A(1232) N* (1535)

(4)

and the masses become like the final part of Fig 4. One picks Cx = 30 MeV from N-A mass splitting, and chooses an oscillator spacing fiw M 250 MeV. R

F I

;,'w

I I

-14C,

1

N,A

R

A-

\

-14CI\\ \

N

Figure 4.

Some q3 baryons.

For the pentaquark, the S3P state has, or can have, a totally symmetric flavor-spin wave function, the S4 cannot. The effect is large: 56 M ( S 3 P )- M(S4) = fuJ - -C, M -310 MeV . (5) 3

436

The flavor-spin mass interaction makes positive parity pentaquark states lighter than the negative parity ones. As for the negative parity case, we can write the q4q positive parity state as a unique sum of products of 8 and qq states, where the q3 or qq may have the quantum numbers of observed hadrons, or may (more often) be color octet states. The details are in7l8. The only q3 and qq combination with quantum numbers allowing a color and kinematically possible decay is the N K combination, and the overlap is small. We quote 1 ( N K ( Q + )( 2 = 5 . 5.

Narrowness

Given the overlap just calculated, one could simply estimate that for the positive parity case, I'(Q+)is 5% of typical strong interaction decay. One can do better by invoking effective field theory ideas, using a Lagrangian (for the positive parity case)

Lint = ig+

+ hermitian conjugate.

4 ~ ~ 5 $ q5 1 p

(7)

One gets the naive size of the coupling parameter from naive dimensional analysis, and modifies it with using the just calculated 0 - N K overlap, as gt = Poqjerlap ( g y v e ) ) 2 . The consequence of naive dimensional analysis is g y e ) = 47r, incidentally the s a m e q u i t e accurateresult one obtains for & N N , Working out the phase space for the positive parity O+, one gets7?'

r + --9 6

x85MeV=4.4MeV.

(8)

The overlaps taken into account are only for the color-flavor-spin part of the wave function. The physical size of the spatial wave functions of the Q+, nucleon, and K are surely not the same, and taking overlaps of spatial wave functions also into account will reduce the above estimate, possibly significantly. For the negative parity state, the bigger Poverlapand bigger phase space (S-wave rather than P-wave) leads to the astonishing result

r- M

1.0 GeV.

(9)

6. Closing comments

There has been lots of theoretical work on pentaquarks, from assorted starting points. It was exciting to hear the various talks presented here.

437

Lattice gauge theory and QCD sum rule workers reported a number of independent calculations. The majority found a signal for the pentaquark, and found it with negative parity. But right now it would be safer to say that the parity and existence of pentaquarks not settled within the context of these two techniques. Chiral quark soliton models (still) predict the existence of pentaquarks, and put the lightest one in a flavor with mass in 1500 Mev range and with positive parity. The quark model can accommodate either parity. It seems easier to explain the narrowness of the state if the parity is positive. Numerical estimates of the widths of positive and negative parity pentaquark are dramatically different, and well justified estimates of the positive parity widths are as narrow as the experiments are requiring. Then there is the possibility that O+ is N K T bound molecular bound state. Let me describe this idea as intriguing, and note that it could render much of the other work moot. There are still things to do. In particular, one thing somewhat but not heavily represented here were studies of excited pentaquarks. Greatest thanks to all the organizers of this very excellent workshop. References 1. T. Nakano e t al. [LEPS Collaboration], Phys. Rev. Lett. 91, 012002 (2003)

[arXiv:hep-ex/0301020]. 2. K. Hicks, this volume. 3. C. E. Carlson, C. D. Carone, H. J. Kwee and V. Nazaryan, Phys. Lett. B 573, 101 (2003) (arXiv:hepph/0307396]. 4. L. Y. Glozman and D. 0. Riska, Phys. Rept. 268, 263 (1996) [arXiv:hep ph/9505422]; F. Stancu and D. 0. Riska, Phys. Lett. B 575, 242 (2003) [arXiv:hepph/0307010]. 5. C. E. Carlson, C. D. Carone, J. L. Goity and R. F. Lebed, Phys. Lett. B 438,327 (1998) [arXiv:hepph/9807334] and Phys. Rev. D 59, 114008 (1999) [arXiv:hepph/9812440]; H. Collins and H. Georgi, Phys. Rev. D 59, 094010 (1999) (arXiv:hepph/9810392]. 6. H. Hogaasen and P. Sorba, arXiv:hepph/0410224; T. Burns, F. E. Close and J. J . Dudek, arXiv:hep-ph/0411160; B. K. Jennings and K. Maltman, Phys. Rev. D 69, 094020 (2004) [arXiv:hepph/0308286]; and references therein. 7. C. E. Carlson, C. D. Carone, H. J. Kwee and V. Nazaryan, Phys. Rev. D 70, 037501 (2004) [arXiv:hepph/0312325]. 8. C. E. Carlson, C. D. Carone, H. J. Kwee and V. Nazaryan, Phys. Lett. B 579, 52 (2004) [arXiv:hep-ph/0310038]. 9. A. Hosaka, M. Oka and T. Shinozaki, arXiv:hep-ph/0409102.

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List of Participants

Abe, Kazuo Akimura, Yuka Ahn, Deuk Soon Arimoto, Natsuko Babuchenko, Alex Battaglieri, Marco A. Bicudo, Pedro Carlson, Carl Chang, Wen-Chen Chiu, Ting-Wai Date, Shin De Vita, Raffaella Diakonov, Dmitri Dmitrasinovic, Veljko Doering, Michael

KEK, Japan [email protected] p Saitama University, Japan [email protected] p Osaka University, Japan [email protected] RCNP, Osaka University, Japan [email protected] University of Liege, Bergium Istituto Nazionale di Fisica Nucleare, Italy [email protected]. it Instituto Superior Tecnico, Portugal [email protected]. pt William-Mary Colledge, USA [email protected] Academia Sinica, Taiwan [email protected] National Taiwan University, Taiwan [email protected] JASFU, Spring-8, Japan [email protected] Istituto Nazionale di Fisica Nucleare, Italy [email protected] NORDITA, Denmark [email protected] Belgrade, Serbia [email protected] University of Valencia, IFIC, Spain [email protected]

440

Ejiri, Hiro Emori, Takashi Enyo, Yoshiko Eyrich, Wolfgang Fujimura, Hisakao Fujiwara, Mamoru Gilad, Shalev Gottschalk, Erik Halyo, Valerie Hanhart, Christoph Harada, Koji Hayata, Masashi Hicks, Ken Hiramoto, Makoto Hirenzaki, Satoru Hiyama, Emiko Hori, Ryuma Horie, Keito Hosaka, Atsushi Hotta, Tomoaki

NS/ICU and RCNP/SPring-8, Japan [email protected] Kyoto University, Japan [email protected] KEK, Japan [email protected] University of Erlangen, Germany [email protected] Kyoto University, Japan [email protected] RCNP, Osaka University, Japan [email protected] MIT, USA [email protected] Fermilab, USA [email protected] SLAC/BaBar, USA [email protected] Juelich, Germany [email protected] Kyushu University, Japan [email protected] Kyoto University, Japan [email protected] Ohio State University, USA [email protected] Tokyo Institute of Technology, Japan [email protected] .titech. ac .j p Nara Woman's University, Japan zaki@cc. nara-wu .ac.jp Nara Woman's University, Japan [email protected] University of Tokyo, Japan [email protected] RCNP, Osaka University, Japan [email protected] RCNP, Osaka University, Japan [email protected] RCNP, Osaka University, Japan [email protected]

441

Hyodo, Tetsuo Igi, Keiji Imai, Ken Ishii, Noriyoshi Jiang, Xiaodong Kadija, Kreso Kamimura, Masayasu Kang, Ji-Young Karshon, Uri Kelkar, Neelima Kim, Hyun-Chul Kino, Koichi KO, Che-Ming Kohri, Hideki Kojo, Toru Kotani, Tsuneyuki Kumagai, Noritake Kunihiro, Teiji Kwon, Young-Shin Lee, Su-Houng

RCNP, Osaka University, Japan [email protected] Riken , Japan [email protected] Kyoto University, Japan [email protected] Tokyo Institute of Technology, Japan [email protected] Rutgers University and Jefferson Lab, USA [email protected] Rudjer Boskovic Institute, Croatia [email protected] Kyushu University, Japan [email protected] Tsukuba University, Japan [email protected] Weizmann Institute of Science, Israel [email protected] .il Universidad de Los Andes, Colombia [email protected] Pusan National University, Korea [email protected] RCNP, Osaka University, Japan [email protected] Texas A& M, USA [email protected] .edu RCNP, Osaka University, Japan [email protected] Kyoto University, Japan [email protected] Osaka University, Japan [email protected] JASRI, Spring-8, Japan YITP, Kyoto University, Japan [email protected]. ac.j p Yonsei University, Korea [email protected] p Yonsei University, Korea [email protected]

442

The George Washington University, USA [email protected] Weizmann Institute of Science, Israel Lipkin, Harry J. [email protected] University of Michigan, USA Lorenzon, Wolfgang [email protected] The University of Tokyo, Japan Maezawa, Yuu [email protected] Wakayama Medical University, Japan Makino, Seiji [email protected] p FMIPA, Universitas Indonesia, Indonesia Mart, Terry [email protected] Tokyo Kasei University, Japan Matsuki, Takayuki [email protected] p National Defense Academy in Japan, Japan Matsumura, Toru [email protected] Osaka University, Japan Mibe, Tsutomu [email protected] Kyoto University, Japan Miwa, Koji [email protected] ITEP, Russia Mizuk, Roman [email protected] Kobe University, Japan Morii, Toshiyuki [email protected] KEK , 3 apan Morimatsu, Osamu [email protected] Muramatsu, Norihito RCNP, Osaka University, Japan [email protected] Nara Women’s University, Japan Nagahiro, Hideko [email protected] RCNP, Osaka University, Japan Nagata, Keitaro [email protected] Nakagawa, Yoshiyuki RCNP, Osaka University, Japan [email protected] Suzuka College of Technology, Japan Nakamoto, Choki [email protected] RCNP, Osaka University, Japan Nakano, Takashi [email protected] RCNP and NuRi, Japan and Korea Nam, Seung-I1 [email protected] .j p Lee, Frank X.

443

Kyoto University, Japan [email protected] KEK, Japan Nemura, Hidekatsu hidekat [email protected] p IPN Orsay, France Niccolai, Silvia [email protected] KEK, Japan Nishikawa, Tetsuo [email protected] Universidad de Los Andes, Colombia Nowakowski, Marek [email protected] University of Georgia, USA Oh, Yongseok [email protected] JASFU, Spring-8, Japan Ohashi, Yuji [email protected] Tokyo Institute of Technology, Japan Oka, Makoto oka@th. phys.t itech.ac.j p Nihon University, Japan Okiharu, Fumiko [email protected] RCNP, Osaka University, Japan Onuma, Takenobu [email protected] University of Valencia, IFIC, Spain Oset, Eulogio [email protected] Princeton University, USA Ouyang, Peter [email protected] Jagellonian University, Poland Praszalowicz, Michal [email protected] Rangacharyulu, Chary University of Saskatchewan, Canada [email protected] Argonne National Laboratory, USA Reimer, Paul E. [email protected] University of Valencia, IFIC, Spain Roca, Luis [email protected] Osaka University, Japan Sakaguchi, Atsushi [email protected] p University of Tokyo, Japan Sasaki, Shoichi ssasaki62phys.s.u-tokyo.ac.jp KEK, Japan Sawada, Shinya [email protected] Osaka University, Japan Sawada, Takahiro [email protected]

Nawa, Kanabu

444

Schaerf, Carlo Seki, Yoshichika Sekimoto, Michiko Shimizu, Kiyotaka Shimizu, Hajime Shinozaki, Tetsuya Stancu, Florica Sugamoto, Akio Suganuma, Hideo Sugiyama, Jun Takahashi, Toru Takeuchi, Sachiko Takizawa, Makoto Tedeschi, David John Titov, Alexander I. Toki, Hiroshi Toyokawa, Hidenori Tsumura, Kyosuke Uchida, Makoto Umeda, Takashi

University Roma "Tor Vergata" , Italy schaerfQroma2 .infn.it Kyoto University, Japan [email protected] KEK, Japan [email protected] Sophia University, Japan [email protected] p LNS, Tohoku University, Japan

[email protected] Tokyo Institule of Technology, Japan shinozk@th. p hys.t itech. ac .j p University of Liege, Belgium [email protected] Ochanomizu University, Japan [email protected] Tokyo Institute of Technology, Japan [email protected] Tokyo Instisute of Technology, Japan sugiyama@th. p hys.t itech .ac.j p YITP, Kyoto University, Japan [email protected] Japan College of Social Work, Japan [email protected] Showa Pharmaceutical University, Japan [email protected] University of South Carolina, USA [email protected] JAERI/JINR, Russia [email protected] RCNP, Osaka University, Japan [email protected] JASRI, Japan [email protected] p Kyoto University, Japan [email protected] RCNP, Osaka University, Japan [email protected] YITP, Kyoto University, Japan [email protected]. ac.j p

445

Vacas, Manuel Vicente Wang, Ming-Jer Weigel, Herbert Yamagat a, J unko Yang, Ghil-Seok Yosoi, Masaru You, Hyeg-Min

University of Valencia, IFIC, Spain Manuel.J [email protected] Taiwan, Inst. Phys. and Fermilab, Taiwan and USA [email protected] Siegen University, Denmark [email protected] Nara Women’s University, Japan [email protected]. ac.j p Pusan National University, Korea [email protected] Kyoto University, Japan [email protected] 0-u .ac.j p Seuol National University, Korea [email protected]

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