physics at the japan hadron facility: proceedings of the workshop adelaide, australia, 14-21 march...
TRANSCRIPT
.+*
Proceedings of the Workshop on
Physics at the Japan Hadron Facility
CJHF)
Editors
V. Guzey, A. Kizilersii, T. Nagae & A. W. Thomas
NATIONAL INSTITUTE FOR
Theoretical Physics
CIMTHI f OR T i l
SUBATQMK
s * ^ i
Physics at the Japan Hadron Facility
UHFl
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SPECIAL RESEARCH CENTRE FOR THE
>° * • SUBATOMIC NATIONAL INSTITUTE FOR •vmM.""^
^Theoretical Physics
Proceedings of the Workshop on
Physics at the Japan Hadron Facility
(JHFl
Adelaide, Australia 14-21 March 2002
Editors
V. Guzey, A. Kizilersu & A. W. Thomas CSSM, University of Adelaide, Australia
T. Nagae KEK, Japan
J*) World Scientific New Jersey • London • Singapore • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
PHYSICS AT THE JAPAN HADRON FACILITY Proceedings of the Workshop
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-238-147-3
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Printed in Singapore by World Scientific Printers (S) Pte Ltd
Foreword
The decision to proceed with construction of the Japan Hadron Facility (JHF) is of momentous importance for the international hadronic physics community. With its intense beams of protons, nuclei, kaons and neutrinos it will complement other major facilities in this field, such as Jefferson Laboratory and GSI. For scientists in the Asia-Pacific region this is an especially exciting opportunity. It was therefore a considerable honour that the Director of JHF, Professor Shoji Nagamiya, agreed to hold the first workshop outside of Japan dealing with the physics opportunities which it might provide here in Adelaide at CSSM.
This meeting was attended by almost 50 delegates from eight countries, with the majority (of course) from Australia and Japan. After an introduction to the various facilities that will be available consideration turned to particular physics issues that can be addressed. These have been arranged in this volume under the general headings of strangeness in nuclear matter, neutrino oscillations and interactions, hadron structure and properties, nuclear and nucleon structure functions and hadronic properties of nuclei. These articles provide a tantalising glimpse of just some of the issues which will be addressed at JHF.
During the meeting all delegates had access to library, internet and computing facilities and as usual at CSSM the early afternoons were kept clean for informal discussions and possible collaborative work. The organisation of the meeting was handled in an extremely friendly and efficient way by Sara Boffa and Sharon Johnson, while Ramona Adorjan provided assistance in computational matters.
All contributions in this volume have been independently refereed.
V. Guzey A. Kizilersii
T. Nagae A.W. Thomas
Adelaide, June 2002
v
1. F.-G. Cao 2. A. Chian
R. Adorjan M.Oka S. BofFa
6. H. Toki 7. T. Thomas 8. A. Kizilersii 9. S. Johnson 10. C. Allton 11. H. Mineo
12. E. Rempel 13. J. Haidenbauer 14. M. Lohe 15. S. Tovey 16. A. Hosaka 17. T. Nagae 18. A. Kalloniatis 19. V. Guzey 20. W. Kamleh 21. G. Krein 22. S. Sawada
23. T. Hatsuda 24. F. Khanna 25. S. Kvinikhidze 26. J. Ashley 27.1. Bojak 28. J. Hedditch 29. R. Young 30. W. Bentz 31. X.-H. Guo 32. J. Zhang 33. D. Leinweber
34. O. Leitner 35. S. B.-Thompson 36. P. Coddington 37. K. Tsushima 38. M. Stanford 39. J. Zanotti 40. M. Burkardt 41. A. Schreiber 42. M. Oettel 43. W. Detmold
Contents
Foreword v
1. OVERVIEW OF JAPAN HADRON FACILITY
KEK/JAERI Joint Project on High-Intensity Proton Accelerators S. Sawada 3
JHF Overview — Strangeness Nuclear Physics and Particle Physics Programs
T. Nagae 25
2. STRANGENESS IN NUCLEAR MATTER
Weak Decays of Hyperon and Hypernuclei M. Oka, K. Takayama, K. Sasaki and T. Inoue 41
The A — A Interaction and Strangeness -2 Hypernuclei I. R. Afnan 51
Hyperon-Nucleon Interaction and Strangeness Production in pp Collisions
J. Haidenbauer 60
3. NEUTRINO OSCILLATIONS A N D INTERACTIONS
Implications of the JHF-Kamioka Neutrino Oscillation Experiment R. R. Volkas 73
Some Implications of the NuTeV Anomaly B. H. J. McKellar, M. Garbutt, G. J. Stephenson, Jr. and T. Goldman 83
Non-Standard Interactions and Neutrino Oscillations M. A. Garbutt and B. H. J. McKellar 93
Neutrino Oscillation Searches at Accelerators and Reactors S. N. Tovey 103
vii
VIII
4. H A D R O N STRUCTURE A N D PROPERTIES
Lattice QCD and Hadron Structure A. W. Thomas 115
Baryon Resonance Phenomenology I. C. Cloet, D. B. Leinweber and A. W. Thomas 125
Lattice QCD, Gauge Fixing and the Transition to the Perturbative Regime
A. G. Williams and M. Stanford 136
Quark Model and Chiral Symmetry Aspects of Excited Baryons A. Hosaka 145
Quenched Chiral Physics in Baryon Masses R. D. Young, D. B. Leinweber, A. W. Thomas and S. V. Wright 155
QCD at Non-Zero Chemical Potential and Temperature from the Lattice
C. R. Allton. S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, Ch. Schmidt and L. Scorzato 164
Hadron Masses from a Novel Fat-Link Fermion Action J. M. Zanotti. S. Bilson-Thompson, F. D. R. Bonnet, D. B. Leinweber, A. G. Williams, J. B. Zhang, W. Melnitchouk and F. X. Lee 174
5. NUCLEAR A N D NUCLEON STRUCTURE FUNCTIONS
Small-a; Nuclear Effects in Parton Distributions V. Guzey 187
The NuTeV Anomaly and Symmetry Breaking in the Parton Distribution Functions
F. G. Cao and A. I. Signal 195
Nucleons as Relativistic Three-Quark States M. Oettel 203
ix
(Polarized) Hadroproduction of Open Charm at the JHF in NLO QCD
I. Bojak 212
Covariant Light-Front Dynamics and Its Application to the Meson Wave Functions
O. M. A. Leitner, A. W. Thomas and J.-F. Mathiot 222
Nucleon Structure Functions at Finite Density in the NJL Model H. Mineo, W. Bentz, A. W. Thomas, N. Ishii and K. Yazaki 232
Violation of Sum Rules for Twist-3 Parton Distributions in QCD
M. Burkardt and Y. Koike 241
6. HADRONIC PROPERTIES OF NUCLEI
Properties of Hadrons in Nuclear Matter F. C. Khanna and D. U. Matrasulov 255
Relativistic Mean-Field Theory with Pion in Finite Nuclei H. Toki, K. Ikeda and S. Sugimoto 263
Equation of State of Quark-Nuclear Matter G. Krein and V. E. Vizcarra 274
Equations of State for Nuclear Matter and Quark Matter in the NJL Model
W. Bentz, T. Horikawa, N. Ishii and A. W. Thomas 285
Color Superconductivity in Dense QCD and Structure of Cooper Pairs
H. Abuki, T. Hatsuda and K. Itakura 294
Hadron Properties in Nuclear Medium and Their Impacts on Observables
K. Tsushima 303
List of Participants 313
1. Overview of Japan Hadron Facility
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KEK/JAERI JOINT PROJECT ON HIGH-INTENSITY PROTON ACCELERATORS
S. SAWADA
KEK - High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba-shi, Ibaraki-ken 305-0801, Japan
E-mail: [email protected]
The KEK/JAERI Joint Project for High-Intensity Proton Accelerators aims to deliver 3 GeV (1 MW) and 50 GeV (0.75 MW) proton beams. An overview of the planned facility and the scientific possibilities are presented. Especially, nu-clear/hadron physics possibilities are described in some detail.
1. Introduction
The KEK/JAERI Joint Project1 is a new accelerator project to produce 1 MW power for proton beams, where various sciences will be conducted, from nuclear and particle physics to material and life sciences. The project has proceeded into a construction phase in JFY2001, which started on April 1, 2001, and it is expected to complete the phase 1 construction in JFY2006. In this paper, a general introduction to the project, including the accelerator configuration and the schedule, is given in Sec. 2. An overview of the sciences with the Joint Project is described in Sec. 3. In Sec. 4, possibilities of nuclear/hadron physics experiments are discussed in some detail, while high-energy physics experiments and strangeness nuclear physics experiments are discussed in the contribution by T. Nagae2.
2. Facility
Originally, KEK (High Energy Accelerator Research Organization) had a hadron accelerator project called the Japan Hadron Facility (JHF) which consisted of a 50 GeV Proton Synchrotron (PS) and a 3 GeV booster synchrotron, where the projected power of the latter was 0.6 MW. On the other hand, JAERI (Japan Atomic Energy Research Institute) had a high-power spallation neutron source project with a proton linac, in which 3 MW pulsed beams were planned for neutron scattering and 5 MW continuous beams were planned for nuclear transmutation. Since both projects have a common goal to accelerate
3
4
high-power proton beams, these two projects were combined into one Joint Project.
The accelerator complex of the project consists of the following components :
• A 400 MeV proton linac (normal conducting) to inject beams into the 3 GeV PS.
• A superconducting linac to accelerate protons from 400 MeV to 600 MeV. The 600 MeV proton beams will be used for R&D toward nuclear transmutation.
• A 25 Hz 3 GeV proton synchrotron with 1 MW power. This will be used primarily for material and life sciences with neutrons and muons.
• A 50 GeV proton synchrotron with slow extraction for kaon beams etc., and fast extraction for neutrino beams to Super Kamiokande. The beam current will be 15 \xA, which corresponds to a beam power of 0.75 MW.
The accelerators have the highest beam power in the world in their energy regions, as illustrated in Fig. 1. With these intense proton beams, various secondary particle beams (neutrons, mesons, antiprotons, etc.) will be produced in proton-nucleus reactions (Fig. 2).
In order to utilize these beams, an experimental hall and a neutrino beam line will be constructed for the 50 GeV facility. At the experimental hall, proton beams with slow extraction will be used for various fixed-target experiments, while the neutrino beam line will send neutrino beams to Super Kamiokande, 290 km away. The 3 GeV experimental hall will be used by neutron and muon users, and the experimental facilities for ADS (Accelerator Driven transmutation System) will utilize 600 MeV proton beams.
The facility will be located at the Tokai campus of JAERI, about 70 km northeast from KEK (about 140 km northeast from Tokyo).
The total cost of the project is estimated to be 189 billion Yen. In March, 2001, the budget for the phase 1 construction of the project was officially approved by the Diet. Phase 1 costs 134 billion Yen, and consists of accelerators and a part of the experimental facilities, as shown in Fig. 3. As for the facilities for nuclear and particle physics experiments at the 50 GeV PS, a slow-extracted primary beam line and a half-sized experimental hall will be constructed in phase 1. The construction period of phase 1 is 6 years (Fig. 4), and the first beam from the accelerators is expected in the spring of 2007. The project team is making great effort to have the remaining part of the project (phase 2) approved soon to realize the entire project. We hope that phase 2 construction will start immediately after the phase 1 construction, or even
iod
Proposed • " H * * Om Existing construction
10000
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I o m ^KEK-500MeV/7" -^poster
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0.1 10 100 1000 10000
Energy (GeV)
Figure 1. World's fixed-target proton accelerators. The product of the horizontal axis (beam energy) and the vertical axis (beam intensity) makes beam power.
before the completion of phase 1. There are various possibilities for future upgrades of the 50 GeV PS fa
cility. Heavy-ion beams at about 20 GeV per nucleon, which were planned in the original JHF project at KEK, will be realized if a heavy-ion injector is added. Although the acceleration of polarized proton beams needs further technical considerations by accelerator experts, strong demand from the user community will promote its realization. Those who now work at CERN An-tiproton Decelerator facility have expressed their interest to have a low-energy antiproton facility for atomic and fundamental physics. A muon storage ring with a FFAG (Fixed Field Alternating Gradient) ring, which can store intense muons, is also one of the future possibilities. There are many other ideas for future plans, such as a muon factory, a neutrino factory, a facility for ultra-cold neutrons, etc.
The project team will call for Letters of Intents (LOIs) from potential
6
Target Nucleus
Proton (p)
Neutron (n)
\
Proton (p) 3 GeV, 50 GeV
Radioactive Nuclei Separation and acceleration of various radioactive nuclei produced with 3-GeVproton beam
^
o
; ««tete ;; ffics, Super-heavy
Nuclei
Muon Science uSR, hlah-Tc superconductor.
Muonlum, pCF
M u o n ( u ) Production of high iritonsi'} pulsed muon beams from pion decay
"**^0 Neutrino (v)
K Nuclear/Particle Physics • * Hypsmuctei, Manna In-Nuclear Matter,
Noutrino Oscfltarton. K Rare Decays Antimatter
Nuclear Transmutation
Neutron (n) High-mtonsil> pulsed spallation neutron source produred with 3-Gev 3 3 3 - M A proton beam
Neutron Science Magnetism. Fractals, Porymwu, StruetudrBiotegy
Figure 2. Examples of the secondary particles produced by proton beams from the planned accelerators and scientific possibilities.
Phasel
Phase2
Nuclear and Particle
Physics Facility
3 GeV Synchrotron
(1MW, 333uA, 25Hz)
R&D for Nuclear j - ^ i
Transmutation
400-600 MeV Linac
(superconducting)
\
r 400 MeV Linac (normal conducting)
Neutrinos to f"* l
Super Kamiokande
50 GeV Synchrotron
(15uA)
Figure 3. Schematic view of the planned accelerator complex.
users around the world in the near future. I think the LOIs should not only be on physics with the phase 1 facilities but also on physics with the phase 2 or further upgraded facilities. With these LOIs, a kind of PAC (Program Advisory Committee), which will be formed soon, will be able to discuss a necessary scheme in the phase 1 construction and future upgrades.
7
Linac Superconducting Linac 3 GeV Synchrotron 50 GeV Synchrotron
Neutron Scattering Facility Muon Facility Transmutaion R&D
Nuclear/Particle Phys. Facility Neutrino Facility
FYOO FY01 FY02
•>
FY03
?
?
FY04 FY05 FY06 FY07 FY08
•M « B
• • » • • -
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Construction budget i
R&D budget
? indicates Phase 2
1 MW
R&D Budget for FYOO = 38 Oku Yen
Phase 1 Project Cost = 1,335 Oku Yen (Total Project Cost a 1 -890 Oku Yen)
Figure 4. Construction schedule.
3. Overview of the Scientific Possibilities
Sciences to be conducted at the project are well summarized by Fig. 2. The spallation neutrons produced by the 3 GeV PS will be used mainly for material and life sciences. Muons produced by the 3 GeV protons will also be used for various kinds of science, such as muonium science with the //SR technique. Proton-nucleus collisions at 50 GeV will produce kaons, anti-protons, neutrinos and other secondary particles. The use of these particle beams will open frontiers in nuclear/particle physics.
3.1. Material and Life Sciences
In material and life sciences, neutron beams are very important and useful. The neutron carries two unique features. One is that the neutron does not have any electric charge, and has a mass which is close to the proton mass. Thus, neutrons are scattered by atomic nuclei, in particular, by light-mass nuclei. This feature is unique if one compares neutrons with synchrotron X-rays. Because X-rays are scattered by electrons, they can observe atoms with large atomic numbers. As one can see from the example of a lithium battery3 in Fig. 5, neutrons can probe lighter atoms, such as oxygen and lithium, while X-rays can probe heavier atoms (Manganese). The other unique feature of the neutron is that it carries a magnetic moment. The neutron is a microscopic magnet. Thus, magnetic scattering of neutrons will reveal the microscopic magnetic
8
structure of a material. For example, a typical high-Tc superconducting material, YBa2Cu30e, has an antiferromagnetic structure; this magnetic structure was determined by a neutron scattering experiment. Recently, fluctuations of macroscopic magnetic layers have also been discovered by neutron scattering. Magnetic structures and their fluctuations might provide us with a deep insight into the superconducting mechanisms. Basic studies of superconducting materials are also useful for the development of applications of high-Tc superconducting materials for industries. This project will accommodate up to 25 beam channels for neutron sciences.
Fourier map of a Li-ion battery cathode material
NEUTRON IS X-ray
&
Figure 5. Structure for a Li-battery seen by neutrons (left) and X-rays (right).
In addition, we plan to install powerful muon beam lines for material and life sciences. Muons also have two aspects. Positive muons can be considered as light protons, while negative muons can be considered as heavy electrons. A muonium (a bound state of a muon and an electron) is an example of the aspect as a "light proton" of a muon, while a muonic atom (a bound state of a proton and a negative muon) is one as a "heavy electron". These unique features as well as the magnetic property of muons are important for investigating
9
materials. In addition, muon-catalyzed fusion is an example of the application of muons.
3.2. R&D Toward Nuclear Transmutation
One goal of this project is to conduct R&D for nuclear transmutation, which aims to reduce long-lived radioactivities produced in nuclear fuel plants. It is believed that the proton power required for the real industrial treatment of nuclear waste transmutation is of the order of 20-50 MW. Our project has much smaller proton power compared to these numbers. Thus, we plan to perform R&D experiments to establish the concept of accelerator-driven nuclear transmutation. A variety of mechanical tests as well as determinations of nuclear reaction cross sections under various critical conditions will be performed.
3.3. Nuclear and Particle Physics
As listed in the proposal of our project1 (Tables 1 and 2), various ideas for experimental programs have been proposed for nuclear and particle physics. They utilize a variety of intense particle beams, such as pions, kaons, protons, anti-protons and neutrinos. These beams are unique to this project. Among these subjects, some ideas have already been published in the form of Letters of Intent or Expression of Interest, while an official call for Letters of Intents is scheduled in the near future. They are :
• Letter of Intent for the Experiments on Strangeness Nuclear Physics at the 50 GeV Proton Synchrotron4,
• Expression of Interest for Nuclear/Hadron Physics Experiments at the 50 GeV Proton Synchrotron5,
• The JHF-Kamioka neutrino project6.
The subjects of strangeness nuclear physics and neutrino physics as well as other particle-physics related topics are discussed by T. Nagae2. Nu-clear/hadron physics is discussed in Sec. 4 in some detail.
Most of these experiments will use slow-extracted beams from the 50 GeV PS. In order to accommodate these experiments, an experimental hall will be constructed. Figure 6 shows a schematic layout of the experimental hall. In phase 1, about half (the left part of the hall) will be constructed as well as the full-sized (150m long) switching yard, which will deliver primary proton beams from the 50 GeV PS to the experimental hall. The 0.75 MW proton beam will produce a variety of intense secondary beams. For example, the K-1.8 beam line, which is designed mainly for experiments with 1.0-2.0 GeV/c kaon beams, will provide 1.5 xlO7 kaons (1.8 GeV/c). They will be the most intense kaon
10
Table 1. List of nuclear physics programs along with their goals considered in the project proposal.
Strangeness nuclear physics
Chiral Symmetry
Structure Function
Hadron Spectroscopy
Heavy Ion Physics
Atomic Physics
Topics
A-hypernuclear spectroscopy
S=-2 hypernu-clei Hyperon-nucleon scattering KN interaction Charmed-hypernuclei Vector mesons in nuclear medium Nucleon and nuclear structure function Exotic searches
Antiproton beam Normal baryon /meson spectroscopy High density matter Multi-strangeness fragments Anti-hydrogen physics
Motivation
Precise studies of A-hypernuclei
Goal
AE ~ 2 KeV resolution by ~y-ray spectroscopy
E-hypernuclei, AA-hypernuclei, search for H particle
Study of YN interaction, Flavour SU(3)
Collect Ap, Ep, Hp data as in NN data
Measurement of SA'JV
Production of Ac hypernuclei
Restoration of chiral symmetry breaking
Quark-gluon (parton) distribution
<j>, p, J/ip in nuclear medium (dilepton)
Low Q 2 , high x region
Search for glueball and hybrid
Energy region above LEAR
QCD confinement (DGL theory etc.)
Complete SU(3) baryon/meson spectra
Search for QGP
Search for S<-2 strangeness mat ter
Precision spectroscopy, CPT test
beams in the world and about an order of magnitude higher intensity than the beams available in the existing facilities.
In phase 2, the neutrino beam line and the rest of the experimental hall will be constructed. At the end of phase 2 construction, 3 primary beam lines for slow-extracted proton beams will be available, which will be used for experiments, such as kaon rare decays, hadron spectroscopy experiments, etc. In addition to the phase 2 construction, there are upgrade plans for facilities, since a fixed-target proton machine can be used for various purposes. One example is the PRISM project, which intends to store high intensity muons for a rare decay experiment; another is a low-energy anti-proton storage ring for fundamental and atomic physics experiments. Because they use fast-extracted
11
Table 2. List of particle physics programs along with their goals considered in the project proposal.
Rare K Decays
Fundamental Symmetry
Muon Lep-ton Flavour Violation
Neutrino Physics
Neutron Physics
Topics
Study of K+ ->• 77+1/(3 decay
Search for K"L -* •K°UU
K —s- 7T77, K -+ 7T7T7
Search for P y in K"+ -+ TTV+I / de
cay Am;{ in K —y -KIT
Search for yT + N -s- e~ + AT
Search for / i + —>• e"*"7 and ^+ —>• e+e-e+ Neutrino oscillation
Neutrino scattering Search for EDM of neutron
Motivation
Determination of CKM matrix element (\Vtd\)
Determination of CP-violating phase (r)) in CKM matrix
Goal
About 100 events, B(A'+ -»• 7T+I/P) ~ 9 x 1 0 - 1 1 is predicted. About 1000 events, B(K°L - • 7r°j/P) ~ 3 X 1 0 _ u is predicted.
Test of chiral perturbation theory
Search for T violation
Search for CPT violation SUSY-GUT, SUSY and heavy VR
SUSY-GUT, SUSY and heavy I/R
Determination of neutrino mass and mixing
PT < 10~4
Amfc/rriK ~ 1 0 -18
B(^+ - • e+7) ~ io - l s
B ( M + -y e + 7 ) ~ 1 0 - 1 5 , B(ti+ -»• e + e - e + ) ~ l O " 1 5
Am 2 < 10~ 3 (for long baseline)
Determination of electroweak form factors New physics with CP violation
d < 10 - ' 2 8 ecm
proton beams, a new beam line is under consideration and discussion at the east side of the 50 GeV PS (between the ring and the sea). Their design work is now being undertaken. Also, experiments with heavy-ion beams or polarized beams, which were included in the original JHF project at KEK, are considered for the future. Some of them are illustrated in Sec. 4.
4. Nuclear/Hadron Physics Experiments
Nuclear/hadron physics conducted here includes a wide range of physics, which intends to understand the properties of hadrons, nucleons, and nuclei. Examples are listed in Table 1, and some of them are discussed in "Expression of Interest"5, which include:
• Study of the chiral property of dense nuclear matter through measurements of the meson-spectral-change in medium,
• Physics of high-mass dimuon production at the 50 GeV Proton Syn-
12
[ Experimental HaB j
Apron Test Exp. Stage
Apron Stage (E.)
Future Extension
L _L
Primary
;::::::::::::::3iuL, i— Control/User's BLDG i| | [ | Apron
' (W.)
Figure 6. Experimental hall for slowly extracted beams. Primary proton beams come from the left and the target stations (T l , T2 and TX) produce various secondary beams. The "primary" beam line will also be used as a high-momentum beam line for pions, kaons and anti-protons. In the future, when heavy ion beams are available, this "primary" beam line will also be used for heavy ion experiments.
chrotron, • Multifragmentation, • Strangeness nuclear physics with high-energy heavy-ion beams, • Systematic study of the collective behaviour in hadron production, • Polarized beam/target experiments.
In Sec. 4.1, possibilities of nuclear matter physics are discussed and in Sec. 4.2, a study of sea-quark distributions is introduced.
4 .1 . Nuclear Matter Physics
" Desire to explore the ultimate form of matter" - this has motivated human beings to study philosophy, sciences and technologies throughout our history. The ultimate dense and heavy matter in the universe is nuclear matter. In nuclear matter, because the interactions are so strong, hadrons cannot be the same as they are in free space, and thus new particle degrees of freedom might emerge. For example, inside a neutron star, the heaviest nuclei in the universe, where cold and dense nuclear matter is formed, various states of matter have been theoretically speculated, such as pion condensation, kaon condensation, hyperon matter, and quark matter. Even in normal nuclei, which are already
13
very dense, it is strongly suggested that the properties of strongly interacting particles, such as vector mesons, would be different from those in free space. In heavy-ion collisions, extremely hot and/or dense nuclear matter is expected to be formed. In order to understand nuclear matter, including these states and phenomena, we have to investigate various properties of nuclear matter in a variety of states as much as possible as well as the fundamental principle of the strong interaction, quantum chromodynamics (QCD).
At the 50 GeV PS, nuclear matter will be studied by different approaches. The first approach is a study of chiral properties of matter through measurements of the meson-spectral change in a medium. In modern solid state physics, or " quantum electro many-body physics", the first things to be studied are elementary excitations in matter, their change as a function of temperature, pressure, etc., and the interactions with each other, and so on. Then, a phase transition is often understood in terms of elementary excitations or effective degrees of freedom. In nuclear physics, however, a consensus has not yet been reached about the effective degrees of freedom in matter.
Nuclei are the only laboratory which is practically available with sufficiently high and known density and with a uniformity sufficient to measure excitations in strongly interacting matter, such as the w meson, <j> meson, and so on. Obtaining information on elementary excitations in nuclear matter will deepen our understanding of QCD as a quantum field theory in the non-perturbative region, and should take it far beyond understanding by such somewhat classical concepts as the mean potential, effective mass, etc. To measure the vector spectral function in nuclei through dilepton decays of vector mesons will provide invaluable clues to such information. The dynamical breaking of chiral symmetry in the QCD vacuum induces an effective mass of quarks, which has been known phenomenologically as "constituent quark mass". The quark condensate (qq), which takes —(225 ± 25) MeV3 in the vacuum, is an order parameter of this dynamical breaking of chiral symmetry. Numerical simulations in lattice QCD as well as model calculations show that (qq) is subject to change in hot (T ^ 0) and/or dense (p ^ 0) matter where chiral symmetry is restored or partially restored (Fig. 7). Although {qq} is not a direct experimental observable, the spectral changes of mesons and baryons may provide a good measure of such partial restoration of chiral symmetry and the associated change7,8 of (qq). Neutral vector mesons («, </>, etc.) are suitable for studying the spectral changes because they can decay into dileptons which do not suffer from strong interactions, and thus carry information of the meson properties in matter. In fact, experimental hints for the spectral changes have been reported by measuring dileptons. For example, an excess of low-mass dileptons was observed by CERN heavy-ion experiments9.
14
* kqq>p/F]
Figure 7. (qq) as a function of the nuclear density and temperature.
Recently, more direct measurements of the mass of vector mesons have been tried by a group at the KEK 12-GeV PS. They used proton-nucleus reactions to produce mesons inside a nucleus, and measured the invariant mass of di-electrons. Their result (Fig. 8) shows an excess in the mass spectrum just below the p and u peak, which may be an indication of a mass modification of the vector meson inside a nucleus10. At the 50 GeV PS, an extended experiment is planned, which will utilize the merit of the new accelerator and a new beam line, in order to obtain firm results on the mass modification and chiral properties of nuclear matter. The new beam line should deliver a proton beam of 109/sec with a beam spot size of less than 1 mm. We can therefore expect a smaller halo around the beam, and thus the counter acceptance can be extended, and the momentum resolutions can be significantly improved. Also, the new accelerator can deliver not only proton beams, but also slow separated beams, such as ?r, K, and p with a momentum below 2 GeV/c. With these beams, one can study, for example, slow <f> meson production with a K -
beam, slow u meson production with a tr~ beam, and conduct an exotic search with a p beam. K* -» K+ + 7 can also be studied. After establishing mass modification with hadron beams, meson production in heavy-ion reactions will be studied with high baryon density and, thus, many more modifications are expected.
The 50 GeV PS will be able to accelerate (light) heavy ions in the future with a minimum cost for an injector accelerator. The maximum energy per nucleon of the heavy ion beams will be around 25 GeV/A, where one expects
15
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- fit result X ,X,Q~ e+e' x - N&ef
combinatorial
Hen—n..... 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
[ G e W ]
e+e- invariant mass
[events / 50MeV/c2]
30
25
20
15
10
5
0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
[GeV/c']
e+e- invariant mass
:b)
n l *N.
"•- ' • • - :
from heavy target
- fit result • x . ^ e - e V - x-Ne-e1
- combinatorial
Figure 8. Di-electron spectra (a) from a carbon target and (b) from a copper target. One can see an excess just below the p/u peaks.
that the highest baryon density (about 10 times the normal nuclear density) can be reached (Fig. 9). In other words, measurements at the highest baryon density which we can reach experimentally can be realized. There, experiments with various probes can be made:
• Hadronic probe physics
- Origin of collective force — Flow
- Properties of high density nuclear matter
• Leptonic probe physics
- Low mass spectroscopy — Onset of mass modification
- Vector meson mass — Chiral symmetry
• Production of multi-strangeness baryons
- Onset of strange quark enhancement - Short-lived strange matter search
• Anti-nucleus production
- Anti-helium production
- Long-lived strange matter search
• Some exotics
- HBT of direct 7's - Mass of unflavoured meson »7'(958)
16
10
Q. ""3-
5 -
p 2 Tom
1 t Bevalac
I
i.,,« i | H Dense (and Hot) W Region i
10 f f 100 f A»Ge
AGS
50-Gc sVPS
V/c
SPS
Figure 9. Nuclear density in heavy-ion reactions as a function of incident kinetic energy per nucleon.
4.2. Dimuon Measurement with 50 GeV Protons
Parton structure of nuclear matter is a basic concept in QCD and hadron-nucleus interactions are also an appropriate playground to study QCD through the parton structure3.
The detection of high-mass dileptons produced in high-energy hadronic interactions has a long and glorious history. The charm and beauty quarks were discovered in the 1970's via the dilepton decay modes of the J/^ and T resonances. The data on the Drell-Yan process has been a source of information concerning the antiquark structure of the nucleon11. Furthermore, Drell-Yan production with pion and kaon beams has yielded the parton distributions of these unstable particles for the first time. A generalized Drell-Yan process was also responsible for the discovery of the W and Z gauge bosons in the 1980's.
To lowest order, the Drell-Yan process depends on the product of the quark and antiquark distributions in the beam and target as
, T = „ y2e2a[qa(x1)qa(x2) + qa{xi)qa(x2)]. (1)
Here, qa(x) are the quark or antiquark structure functions of the two colliding hadrons evaluated at momentum fractions x\ and x2- The sum is over the quark flavours, and s is the centre-of-mass energy squared.
a This part of our Expression of Interest5 can be downloaded from the e-print archive as hep-ph/0007341.
17
The kinematics of the virtual photon - longitudinal centre-of-mass momentum (pjj), transverse momentum (pip) and mass (M7) - are determined by measuring the two-muon decay of the virtual photon. These quantities determine the momentum fractions of the two quarks:
*F=P | | /P | | = XX-X2, (2)
M,2 = Xlx2s, (3)
where pjj is the virtual photon centre-of-mass longitudinal momentum and p?,'max is the maximum value it can have.
To gain sensitivity to the antiquark distribution of the target, one chooses a proton beam and selects the kinematic region of positive xp and large x\. In this limit, the contribution from the second term in Eq. (1) is small and the first term is dominated by the u(x\) distribution of the proton. Under these circumstances, the ratio of the cross sections for two different targets, X and Y, which have Ax and Ay nucleons is approximately the ratio of the u(x2) distributions:
1 / dax \ Ax \dx1dx2 )
1 ( doY \ AY ydxidx? J
"X(*» ( 4 ) uY (x2) Xi^>X2
In this relation the cross sections are defined per nucleus, but the parton distributions are conventionally defined per nucleon.
Equation (4) demonstrates the power of Drell-Yan experiments in determining relative antiquark distributions. This feature was explored by recent Fermilab experiments using 800 GeV proton beams12. The 50 GeV PS provides a unique opportunity for extending the Fermilab measurements to larger x2 (x2 > 0.25). For a given value of x\ and x2, the Drell-Yan cross section is proportional to 1/s (see Eq. (1)). Hence, the cross section at 50 GeV is roughly 16-times greater than that at 800 GeV (The price one pays at lower beam energies is that one has limited reach for small x2, which could be best studied at higher energies). Furthermore, to the extent that the radiation dose scales as beam power, one can take an m 16 times higher beam flux at 50 GeV relative to 800 GeV. The combination of these two effects could lead to two orders of magnitude improvement in the statistics at high x2 over the previous Fermilab experiments.
Among the physics issues which can be accessed by dilepton (dimuon) measurements, two subjects are described here as examples : d/u asymmetry and partonic energy loss.
18
d/u of the proton : The 50 GeV PS offers a unique opportunity to extend the existing measurements of antiquark distributions to much larger values of Bjorken-x. Such information is crucial for understanding the origins of flavour asymmetry in the nucleon sea, and for illuminating the nuclear environment effects on parton distributions. Until recently, it had been assumed that the distributions of u and d quarks were identical. Although the equality of u and d in the proton is not required by any known symmetry, this is a plausible assumption for sea quarks generated by gluon splitting. Because the masses of the up and down quarks are small compared to the confinement scale, nearly equal numbers of up and down sea quarks should result. The assumption of u{x) = d(x) can be tested by measurements of the Gottfried integral13, defined
by
IG = J [Fr(x,Q2)-F?(X,Q2)]/xdx=± + j j [up(x) - dp(x)]dx, (5)
where F$ and i*^ are the proton and neutron structure functions measured in DIS experiments. Under the assumption of a symmetric sea, u — d, the Gottfried Sum Rule (GSR) IG — 1/3, is obtained.
The most accurate test of the GSR was reported in 1991 by the New Muon Collaboration (NMC)14, which measured F% and F£ over the region 0.004 < x < 0.8. They determined the Gottfried integral to be 0.235 ± 0.026, significantly below 1/3. This surprising result has generated much interest, and it strongly suggests that the assumption u = d should be abandoned. Specifically, the NMC result implies
f [d{x)-u(x)]dx = 0.148 ±0.039. (6) Jo
Equation 6 shows that only the integral of d — u was deduced from the DIS measurements. The x dependence of d — u remained unspecified.
The proton-induced Drell-Yan process provides an independent means to probe the flavour asymmetry of the nucleon sea. An important advantage of the Drell-Yan process is that the x dependence of d/u can be determined. After measurements by the Fermilab E772 and CERN NA51 experiments, a Drell-Yan experiment (E866), aiming for higher statistical accuracy and wider kinematic coverage than NA51, was recently completed15,16 at Fermilab. This experiment also measured the Drell-Yan muon pairs from 800-GeV/c protons interacting with liquid deuterium and hydrogen targets. Equation (4) shows that the Drell-Yan cross section ratio at large XF is approximately given as
*py(p +4_ ~ I (i + ^ \ (7) 2(TDY(P + P) 2 V Ufa)
19
Values for d/u were extracted by the E866 collaboration at Q2 = 54 GeV2/c2
over the region 0.02 < x < 0.345. These results show that the d/u ratio increases linearly for x < 0.15, has a peak around x = 0.15 and rapidly decreases beyond x = 0.2. This feature was not in agreement with parameterizations of parton distributions before the E866 experiment. Various theoretical models have been proposed to illustrate the d/u ratio, such as the pion cloud model and the chiral model. The interplay between the perturbative and non pertur-bative components of the nucleon sea remains to be determined better. Since the perturbative process gives a symmetric d/u while a non-perturbative process is needed to generate an asymmetric d/u sea, the relative importance of these two components is directly reflected in the d/u ratios. Thus, it would be very important to extend the Drell-Yan measurements to kinematic regimes beyond the current limits.
The 50 GeV PS presents an excellent opportunity for extending the d/u measurement to larger x (x > 0.25). As mentioned earlier, for given values of X\ and x2 the Drell-Yan cross section is proportional to 1/s. Hence, the Drell-Yan cross section at 50 GeV is roughly 16-times greater than at 800 GeV. Figure 10 shows the expected statistical accuracy for a(p -f d)/2a{p-\- p) at the 50 GeV PS compared with the data from E866 and a proposed measurement17 using the 120 GeV proton beam at the Fermilab Main-Injector. The experimental apparatus and assumptions are described later. A definitive measurement of the d/u over the region 0.25 < x < 0.7 could indeed be obtained at the 50 GeV PS.
Partonic energy loss in nuclei : The subject of energy loss of fast par-tons propagating through hadronic matter has attracted considerable interest recently. For example, there are many discussions on recent results from the Relativistic Heavy Ion Collider (RHIC) about partonic energy loss inside the Quark Gluon Plasma (QGP). The nuclear dependence of the Drell-Yan process provides a particularly clean way to measure the energy loss of incident quarks in a cold nuclear medium, which is also important to understand the energy loss in a hot nuclear medium, QGP. Partonic energy loss would lead to a degradation of the quark momentum prior to annihilation, resulting in a less energetic muon pair. Therefore, one expects the Drell-Yan cross sections for heavier targets to drop more rapidly at large x\ (or XF).
Analyses on nuclear-dependence data from Fermilab E772 and E866 experiments18,19 showed different results, and there are several models for the expression of the partonic energy loss :
A « i — — K\x\Ax (8)
20
Figure 10. (p + d)/(p + p) Drell-Yan ratios from E866 (open circles) are compared with the expected sensitivities at the 120 GeV Main Injector (solid circles) and the 50 GeV PS (solid squares).
A i i « 2
A1'3, (9)
or
Azi S
(10)
With these backgrounds, the partonic energy loss should be determined more clearly by an experiment. At the 50 GeV PS, a much more sensitive study of the partonic energy loss could be carried out. We have simulated the effect of the initial-state energy loss on the p + W Drell-Yan cross sections; the results are shown in Fig. 11. Assuming a 60-day run with the nominal spectrometer configuration (see our experimental apparatus below), the expected x\ distribution for p + d is shown as the solid curve. The dashed, dotted, and dash-dotted curves in Fig. 11 correspond to the p+Wxi spectra assuming a partonic energy loss form of Eq. (9) with dE/dz of-0.1, -0.25, -0.5 GeV/fm, respectively. The ratios of p + W over p + d, shown in Fig. 11, are very sensitive to the quark energy loss rate, and the expected statistical accuracy can easily identify an energy loss as small as 0.1 GeV/fm. The greater sensitivity
21
at 50 GeV is due to the 1/s factor in Eq. (9) and Eq. (10). Another important advantage at 50 GeV is the absence of a shadowing eifect, and no shadowing correction is required.
10 4
10 3
(/I
l i o o O
10
'o
0.9
0.8
„ 0 . 7
^ 0 . 5
+ 0.4 Q.
" 0 . 3
0.2
0.!
°0
Figure 11. a): Solid curve is the expected p + d spectrum for a 60-day run at 50 GeV. The dashed, dotted, and dash-dotted curves correspond to p + W spectra assuming a partonic energy loss rate of 0.1, 0.25, 0.5 GeV/fm, respectively, b): Solid circles show the expected statistical errors for (p+W)/(p+d) ratios in a 60-day run each for p+W and p+p. The solid, dashed, and dotted curves correspond to a partonic energy loss rate of 0.1, 0.25, 0.5 GeV/fm, respectively.
The Drell-Yan A-dependence data could further be used to determine whether the energy loss follows an L (as in Eq. (9)) or an L2 (as in Eq. (10)) dependence. This is illustrated in Fig. 12, where the solid circles correspond to (p + A)/(p + d), assuming an energy-loss rate of 0.25 GeV/fm using Eq. (9). The open squares correspond to the situation when energy loss is described by
22
Eq. (10) (the value of K3 is selected by matching the (p + W)/(p+ d) values for both cases). Figure 12 shows that one can easily distinguish an L- from an independence even when the energy loss rate is as small as 0.25 GeV/fm.
Figure 12. a): Solid circles correspond to the expected (p + J 4 ) / ( P + d) ratios assuming a partonic energy loss rate of 0.25 GeV/fm with a nuclear dependence given by Eq. 9. The open squares correspond to partonic energy loss given by Eq. 10. The statistical errors were calculated assuming a 60-day run for each target, b) Same as the top figure, but for a different xx bin (0.7 < x < 0.8).
Experimental apparatus : The spectrometer considered here is designed to measure muon pairs at MM+/J- > 1 GeV with 50 GeV proton beam. The E866 spectrometer and its daughter, a proposed P906 spectrometer17, are taken as a starting point. The E866 spectrometer is shown in Fig. 13. H+p~ pairs produced at the target were analysed by a vertical-bending spectrometer. Taking into account the differences in the kinematic conditions, a spectrometer with
23
larger width and shorter length has been designed. The total length of the spectrometer system from the entrance of the first magnet to the end of the detector system is about 15 m. The assumptions for Monte-Carlo simulations shown here are as follows :
• The beam intensity is 1 x 1012 protons/(3 sec), • The net efficiency of data acquisition is 0.5, • Data are taken for 60 days for 50-cm long liquid proton and deuteron
targets.
Ring-Imaging Cherenkov Counter
SM3 Analyzing Magnet
Muon Detectors
Electromagnetic Calorimeter
-SMO
Figure 13. Schematic layout of the Meson-East focusing spectrometer at Fermilab.
5. S u m m a r y
In this talk, an overview of the High Intensity Proton Accelerator project, under construction at Tokai, Ibaraki, Japan, has been presented. The accelerator complex, which will deliver the most intense proton beams, provides very unique opportunities for studies of broad sciences from nuclear and particle physics to material and life sciences. In addition, R&D studies on accelerator-driven nuclear transmutations will be pursued. Turning our view to nuclear/particle physics, high intensity secondary particles, such as kaons, pions and anti-protons, are key tools. In addition, primary proton beams and heavy-ion beams will provide very interesting possibilities in these fields. The subjects presented here are just examples of the possibilities. Ideas from the world-wide community are very much welcome.
24
R e f e r e n c e s
1. The Joint Project for High-Intensity Proton Accelerators, M. Furusaka et al., KEK Report 99-4, JAERI-Tech 99-056 (1999).
2. T. Nagae, these proceedings. 3. T. Kamiyama in Materials Research Using Cold Neutrons at Pulsed Neutron
Sources, ed. P. Thiyagarajan, F. R. Trouw, B. Marzec and C. Loong (World Scientific, 1999).
4. Letter of Intent for the Experiments on Strangeness Nuclear Physics at the 50 GeV Proton Synchrotron, K. Imai et al., http://www-jhf.kek.jp/JHF.WWW/LOI/50GeVNP-LOI-vl.0-pdf http://www-jhf.kek.Jp/JHF.WWW/LOI/50GeVNP-LOI-vl.0.ps.
5. Expression of Interest for Nuclear/Hadron Physics Experiments at the 50 GeV Proton Synchrotron, M. Asakawa et al., KEK Report 2000-11 (2000), http://psuxl.kek.jp/50GeV-PS/EOI-nucl-hadron.pdf, http://psuxl.kek.jp/50GeV-PS/EOI-nucl-hadron.ps.gz.
6. The JHF-Kamioka neutrino project, Y. Itow et al., KEK Report 2001-4, ICRR-report-477-2001-7, TRI-PP-01-05, hep-ex/0106019 (2001).
7. T. Hatsuda and T. Kunihiro, Phys. Rep. 247, 221 (1994). 8. G. Brown and M. Rho, Phys. Rep. 269, 333 (1996). 9. G. Agakchiev et al. (CERES collaboration), Phys. Lett. B422, 405 (1998).
10. K. Ozawa et al., Phys. Rev. Lett. 86, 5019 (2001). 11. I. R. Kenyon, Rep. Prog. Phys. 45, 1261 (1982). 12. P. L. McGaughey, J. M. Moss and J. C. Peng, Ann. Rev. Nucl. Part. Sci. 49,
217 (1999). 13. K. Gottfried, Phys. Rev. Lett. 18, 1174 (1967). 14. P. Amaudruz et al., Phys. Rev. Lett. 66, 2712 (1991); M. Arneodo et al., Phys.
Rev. D55, R l (1994). 15. E. A. Hawker et al., Phys. Rev. Lett. 80, 3715 (1998). 16. J. C. Peng et al., Phys. Rev. D58, 092004 (1998). 17. Proposal for Drell-Yan Measurements of Nucleon and Nuclear Structure with
FNAL Main Injector, P906 Collaboration, http://p25ext.lanl.gov/e866/papers/p906/proposal_final.ps.
18. S. Gavin and J. Milana, Phys. Rev. Lett. 68, 1834 (1992). 19. M. A. Vasiliev et al., Phys. Rev. Lett. 83, 2304 (1999).
JHF OVERVIEW - STRANGENESS NUCLEAR PHYSICS A N D PARTICLE
PHYSICS PROGRAMS -
T . N A G A E
High Energy Accelerator Research Organization (KEK), Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan
E-mail: [email protected]
Strangeness nuclear physics is one of the major subjects in nuclear physics to be explored at the 50 GeV PS. Spectroscopic studies of S——2 systems are quite unique to the new facility. Also, high-resolution 7-ray spectroscopy will be carried out for a wide range of hypernuclei at high precision. In particle pysics, there are three interesting subjects: neutrino long-baseline oscillation experiments, kaon rare decays, and muon rare decays. An overview of these topics is presented in this paper.
1. Strangeness Nuclear Physics programs at the 50 GeV PS
Many types of nuclear physics programs have been proposed using high-intensity secondary beams of kaons, pions, anti-protons, and primary proton beams at the 50 GeV PS. A letter of intent for experiments on strangeness nuclear physics1 well summarizes the initial experimental programs in this field. Among the various topics discussed in the letter of intent, I introduce two interesting programs: the investigation of new hadronic many-body systems with strangeness 5=—2 and high-resolution 7-ray spectroscopy.
1.1. Spectroscopic Study of S=—2 Systems
The high-intensity K~ beam at ~1.8 GeV/c available at the 50 GeV PS is quite unique to open a new frontier of strangeness nuclear physics in the spectroscopic studies of strangeness S——2 systems. Here, the S=—2 systems include S-hypernuclei, double-A hypernuclei, and possibly H-hypernuclei. This is not only a step forward from the S=—l systems as a natural extension, but also a significant step to explore multi-strangeness hadronic systems. Along the way we can get some idea of the properties of strange hadronic matter (S — —00) in the core of a neutron star. Also, it is important to extract some information on EN and A-A interactions from the spectroscopic data.
25
26
The (K~ ,K+) reaction is one of the best tools to implant the 5=—2 through an elementary process K~+p —>• K++E~, the cross section in the forward direction of which has a broad maximum around this energy.
At present, the experimental information on S=—2 systems mainly comes from several emulsion data in limited statistics. As for the S-hypernuclei, there are some hints for there existence from emulsion events. However, it is still not conclusive. Some upper limits on the S-nucleus potential have been obtained from the production rate in the bound region of a H-hypernucleus via the (K~ ,K+) reaction.
There have been three emulsion events which show the existence of double-A hypernuclei. A recent counter experiment reported the production of the
AAH hypernucleus2. The binding energy of a double-A hypernucleus, AAHe, has been measured for the first time in a recent emulsion experiment at KEK3. In this measurement, the hypernuclear species was uniquely identified, and the binding energy was measured without ambiguities from possible population of excited states.
A lot of searches for .ff-dibaryon have been carried out since the late 1980s to 1990s. No evidence has been observed so far. The observation of weak decays from the double-A hypernuclei limits the allowed mass range of the H-dibaryon being very close to 2xm\. There are suggestions that the .ff-particle may exist as a resonance and/or the "/T'-type configuration might be mixed in the S=—2 systems.
In Fig. 1, a typical energy spectrum and decay thresholds for S- and double-A hypernuclear configurations are shown. The energy difference between the (S~p) system and the (AA) system is only 28.3 MeV in free space. Therefore, a relatively large configuration mixing between E~+A and AA+(yl — 1) states is suggested. It should be noted that a mixing of the S component in A-hypernuclei is suggested to be several %, in which the energy difference between the two states is ~75 MeV. This mixing would be quite significant in heavy targets, because the E~-hypernuclear levels are deeply bound with the aid of the large Coulomb potential. It is very interesting to investigate whether the single-particle picture of S~ is valid or not in such a system.
Several ways to explore the S=—2 systems are proposed. One promising way is the spectroscopy of S-hypernuclei with the (K~ ,K+) reaction. This method investigates the entrance channel of the S=—2 world.
1.1.1. Spectroscopy ofE Hypernuclei
The S-hypernuclei will play an essential role in our investigation of the S=—2 baryon-baryon interaction.
27
Energy Spectrum of S=-2 systems
A-1(Z-1)+H- JI B
E-hypernucleus
(A-1)g.s.®Ps H _ N interaction
??T
2xB/
ABAA""1 r
A"2(Z-2)+A+A
*-\(Z-2)+A
(A-1)g.s®SE
Double-A hypernucleus BAA (A-2) g . s ®S A P A
- " A A ( Z - 2 ) + n
(A-2)g.s.®SA2
j H particle mass H + (A-2)
Weak Decays
Figure 1. Typical energy spectrum and decay threshold for H- and double-A hypernuclear configurations.
Unlike A and double-A hypernuclear ground states which are long lived and
decay via the weak interaction, S-hypernuclei decay via the strong interaction
through the S~p—> AA (Q = 28.3 MeV) conversion. In this sense, the situation
is very similar to the S-hypernuclei in which the strong conversion process
EN—>• AN (Q ~ 75 MeV) exists and broadens the state. A naive semi-classical
est imate of the spreading width for a S~ single-particle state for an infinite
nuclear mat te r gives Tj. ~ 1 3 MeV. However, for finite nuclei, the width would
be reduced to be < 1 MeV due to the reduction of phase space and overlap of
wave functions, etc.
Therefore, it is expected that the spectroscopy of S-hypernuclei is promis
ing. Here we use the (K~ ,A'+) reaction in which we can use the same method
as in the (n+,K+) reaction in the A-hypernuclear spectroscopy. In fact, two
reactions have a very similar characteristics of the large recoil momentum of
28
a produced hyperon: p=- ~500 MeV/c and p\ ~350 MeV/c . Therefore,
even for heavy targets well-separated peak structures are expected in spite
of many possible excitations, because the spin-stretched configurations with
(•p + fe + J = even are strongly populated as in the case of the (TT+ ,K+)
reaction, or even more strongly.
Convincing evidence for S single-particle states would yield information on
the H single-particle potential and the effective SN interaction. Knowledge
of the depth of the S-nucleus potential is important also for estimating the
existence of strange hadronic mat ter with S's.
For the spectroscopy of the (K~ ,A'+) reaction, we need two spectrometers
as in the (ir+ ,K+) reaction: a beam line spectrometer for the incident K~ and
a K+ spectrometer.
At this moment , it is proposed to construct a new beam line with a good
K~/ir~ ratio of > 1 and a good momentum resolution of ~ 5 x l 0 - 4 (FWHM).
As for the K~ /ir~ ratio, the 2-GeV/c beam line at BNL-AGS is an excellent
example. This is essential to handle such a high intensity beam of 1 x 10 7 / s .
The momentum resolution in the beam line, however, is too poor to enable us
to perform spectroscopic studies. Good momentum resolution, beteter than
l x l O - 3 , has been already achieved at the K6 beam line of the KEK 12-GeV
PS. The last part of the beam line, after the mass separation, consists of a
QQDQQ system (system with four quadrapole magnets and one dipole magnet)
to reconstruct the incident momentum. Thus, we need a combination of the
two beam lines for the new K~ beam line.
A 2-GeV/c kaon beam line for the J H F was designed by J. Doornbos4 . A
schematic layout of the beam line is shown in Fig. 2. A beam line spectrom
eter is installed in the last par t of the beam line. It consists of a Q D Q D Q
system (system with three quadrapole magnets and two dipole magnets) . It is
estimated tha t the momentum resolution of 2 x l 0 - 4 could be achievable.
For the K+ spectrometer, we will use the existing SKS spectrometer with
some modifications. In the (K~ ,K+) reaction, the K+ momentum correspond
ing to the production of S-hypernuclei is around 1.2 GeV/c . In the (TT+ ,K+)
reaction, the SKS magnetic field is 2.2 T for 0.72 GeV/c . Therefore, the SKS
maximum magnetic field of ~2 .7 T does not allow us to put the central beam
at 1.2 GeV/c . In Fig. 3, the setup of the SKS spectrometer for the (K~ ,K+)
reaction is shown schematically.
Since the radius for the central momentum is larger than tha t for the
(TT+,K+) reaction, the target point is moved away from the magnet so tha t
the acceptance of the spectrometer is reduced to be ~50 msr.
The optical property and the acceptance could be improved by installing a
small dipole magnet and/or a quadrupole magnet at the entrance of the SKS
29
Experimental Target
Figure 2. Schematic layout of the 2-GeV/c kaon beam line designed by J. Doornbos.
KEK-SKS '' x
Figure 3. Schematic layout of the SKS spectrometer for the (K~,K+) reaction a t 1.65 GeV/c.
magnet. The design study is still underway. The overall energy resolution is estimated to be 2 MeV(FWHM) for a 2-g/cm2 target thickness.
The production cross section of H-hypernuclei in the (K~ ,K+) reaction is calculated by Akaishi et al.5 within the framework of the distorted-wave impulse approximation (DWIA) using the Green's function method. The cal-
30
culation suggests the production cross section to be ~0,1 /A/sr/MeV, around the middle of the bound region for various types of potentials.
The yield for the 208Pb target with 2-g/cm2 thickness is estimated to be ~ 6 events/day. So, even for the heaviest case, we could get enough statistics within ~20 days to obtain spectroscopic information. Several peak positions for S-orbitals with high angular momenta would be measured within the precision of <1 MeV, so that we can accurately determine the potential depth of the E~~nucleus potential. For lighter targets such as 28Si and 58Ni, the yields are several times higher with the normalized target thickness of 2 g/cm2.
1.2. High Resolution Hypernuclear f-ray Spectroscopy
High-resolution 7-spectroscopy using Germanium (Ge) detectors, which is one of the most powerful means to study nuclear structure, has recently been introduced in hypernuclear physics in order to drastically improve the energy resolution of hypernuclear levels from 1-2 MeV (FWHM) to a few keV (FWHM). A large-acceptance Ge detector array, called Hyperball, has been constructed for hypernuclear 7-ray spectroscopy6, and hypernuclear 7 transitions were successfully observed for the first time with Ge detectors7'8.
Figure 4. Ge detector system for hypernuclear 7 spectroscopy.
31
These studies will be extended further and exciting new physics fields will be opened up at the 50-GeV PS. The following three subjects will be pursued.
(1) Baryon-Baryon Interactions
The AN interactions will be further studied from detailed hypernuclear level structures. The experimental information will be compared with theoretical models in order to understand the baryon-baryon interactions. In the JHF, the ANN three-body force and charge symmetry breaking A7V interaction will also be clarified with plenty of data for hypernuclei. In addition, the A A interaction will be studied using 7 spectroscopy of double A hypernuclei, and the E.N interaction can be investigated by measuring X-rays from hyperon atoms.
(2) Impurity Nuclear Physics
Since hyperons are free from Pauli blocking and feel nuclear forces different from those by nucleons in a nucleus, only one (or two) hyperon(s) introduced in a nucleus may give rise to drastic changes in the nuclear structure, such as changes of the size and the shape, change of the cluster structure, emergence of new symmetries, change of collective motions, etc. The level scheme and B(E2) (the reduced £'2-transition probability) of A hypernuclei studied by 7 spectroscopy will reveal such interesting phenomena, and a new field to be called "impurity nuclear physics" will be exploited.
(3) Medium Effect of Baryons
Using hyperons free from Pauli blocking, we can investigate possible modification of baryons in nuclear matter, by deriving the magnetic moment of a A from measurement of B(M1) (the reduced Ml— transition probability) in hypernuclei, for example.
1.2.1. Method and Setup
The (K~,ir~) reaction at 1.1 GeV/c is used in order to produce A hypernuclei in most cases. This reaction has a large spin-flip amplitude and allows population of various hypernuclear states including spin-flip states with unnatural parities.
We require a secondary beam line of which intensity is optimized to 1.1 GeV/c. It should have a double-stage mass separator to obtain pure K~ beams in order to minimize the counting rates of Ge detectors and tracking
32
devices in the spectrometer. The beam intensity of 1.1 GeV/c K is expected to be 1.9xl07 K~ per spill (2xl01 4 protons) in 3.4 sec cycle.
The momentum of the K~ beam is measured event-by-event with a beam-line spectrometer having < 0.2% FWHM resolution. The outgoing n~ is measured with a spectrometer similar to SKS, which is required to have acceptance of more than 50 msr and momentum resolution of < 0.2% FWHM. The overall mass resolution better than 3 MeV is necessary.
Around the target, we install a new Ge detector system, which is similar to the present Hyperball but has a much larger efficiency. In the present design, we expect to use 14 sets of "Segmented Super Clover Ge detectors", which have recently become commercially available. One detector set consists of four Ge crystals of 7cm<̂ > x 14 cm, and the electronode of each crystal is segmented into 4 readout channels. Such a fine segmentation is necessary for Doppler shift correction. The detectors are installed so that the crystal surface is located at least 20 cm from the target. This distance is necessary for counting rate of the detectors as well as for the Doppler shift correction. The Ge crystals cover about 40% of the total solid angle. The Ge detector system has a photo-peak efficiency of 12% at 1 MeV in total. Each of the Ge detector is surrounded by a set of BGO or GSO counters, which are used to veto Compton scattering and high-energy 7 rays from 7r°. These counters should be finely segmented to reduce the counting rate.
1.2.2. ^2C - Yields and Expected Results
Figure 5 shows level energies, cross sections, and 7-ray branching ratios assumed in a yield estimate. The production cross sections of the ^2C states (except for the first 2 + state) were calculated by Itonaga et al.9 for 1.1 GeV/c (K~ ,K~) reaction. The energy levels are taken from the experimental values from KEK-E369, but the doublet spacing energies are taken from the new parameter set of the spin-dependent interactions by Millener10.
In the yield estimation, we assumed the following values :
• K~ beam intensity: 1.9xl07 /spill(2xl014ppp) • PS cycle: 3.4 sec • Target: 10 g/cm2 / A x6x l0 2 3
• Effective spectrometer solid angle: Qeff = 003 sr • Spectrometer tracking efficiency: e$p = 0.4 • Energy dependent Ge detector efficiency • Ge detector live time: tQe nve = 0.6
The estimated 7 ray yields for 5 days run are shown in Fig. 5.
33
o1-
MeV 5g J O
£; p"- o* d- eo en -co S i c i n . i . , • i i " -CO CN CO
2+ 8.3
0+
-22
ii;:ooo
1fc
do/d£J(e=10°) ;x 1.1 GeWc(K,7t) (MeV) (nb/sr)
8
) 0.201
$0,663
•1 Q (0.07
4.5 9*
9 18*
25* 50*
Exp Calc *Calc (E369)(Millener) (Itonaga)
Expected Yield (5 days)
a
b c d e f
g
sing
112000
2600 12000
7500 1100
240 1300
le
h i i k I m
1700 64
580 1400 2000
230
c f 9 i k m
with a 1900
50 150 94
170 31
coi ncidnece withe
a 1900 g 39 k 50
withd h 55
Figure 5. Top: Expected level energies and 7 transitions of j^2C used in the simulation. Bottom: Expected yields of 7 transitions of ^2C for 5 days run. Yields of 7-7 coincidence events are also shown.
Assuming the same continuous background level as in the KEK experiment, we can simulate the spectrum for the jy2C run. We will have enough yields for almost all the 7 transitions. Figure 6 shows some examples of 7-7 coincidence spectra. Those spectra enable us to completely reconstruct the level scheme.
2. Particle Physics at the 50 GeV PS
Particle physics in the 20th century established the Standard Model of high energy physics. However, many mysteries remain to be explored. One of
34
coincidence with a
IT) \ 6 0 0 in
S 400 o o
200
0
1-2-2-1
h 1-3-0-
k 2Y--1- :
JJu^ «LoU • •l-ft.*-. . A . , 2.25 2.75 3.25 3.5
E (MeV)
6.75 7 E (MeV)
coincidence with c
^ 15 LO \ {n
•H 10 D O
0 5
0
1-3-0-
ti, infS L at 1 III L > l.ilhn im mm nl Wimnl hmJi,«r,U.
k 2 - 2 - 1-2
2.25 2.5 2.75 3.25 3.5 3.75 4 E (MeV)
Figure 6. Simulated 7-7 coincidence spectrum of ^ 2 C. Top two: coincidence with 7-ray "a" (1J* —ylj"). Bottom: coincidence with 7-ray "c" (1J" —*-2j~).
the most profound one is the asymmetry of particles and anti-particles in the universe. Another one is the generation problem. In these problems, mass hierarchy and mixing in three generations of quarks and leptons are to be investigated. By using the high-intensity neutrino, kaon, and muon beams at the 50 GeV PS, several important experiments to explore physics beyond the Standard Model will be carried out.
2 .1 . Neutrino Long-Baseline Oscillation Experiment
The discovery of the existence of neutrino oscillation in the atmospheric neutrinos by Super-Kamiokande has opened the possibility of studying the masses and mixing in the lepton sector in detail. In order to confirm the result, the first accelerator-based long baseline neutrino oscillation experiment, called K2K, is taking its data, from which first results already indicate oscillations.
A second generation neutrino oscillation experiment11 is proposed by using
35
the much higher intensity neutrino beam available at the 50 GeV PS. It aims at high precision measurements of the lepton mixing parameters described by a 3x3 unitary matrix (Maki-Nakagawa-Sakata matrix12), in which three mixing angles (912, #23, and 913) and a complex phase (S) are the parameters to be measured. A unique feature of the proposed experiment is that the neutrino energy is tuned to the oscillation maximum at ~1 GeV for a baseline length of 295 km between the JAERFs Tokai site and the Super-Karniokande detector. An off-axis beam is used to have a narrow energy band. The world's largest water Cerenkov detector has advantages not only in event rate but also in energy resolution and particle identification. The charged-current quasi-elastic events with low energy neutrinos enable us to have a good energy resolution of ~80 MeV.
Super-Kamlokande Detector
Figure 7. Baseline of the JHF-Kamioka neutrino oscillation experiment.
In the first phase of the experiment, the physics goal is an order of magnitude improvement in precision in the v^ -¥ vT oscillation measurement (<J(Ami3)-10~4 eV2 and <J(sin22023)=O.Ol), a factor of 20 more sensitive search
36
in the v^ —> ve appearance (sin22^ e ~0.5 sin22#i3 >0.003), and confirmation of the v^ —> vr oscillation or discovery of sterile neutrinos by detecting the neutral current events. A comparison between the currently planned long-baseline experiments is summarized in Table 1.
Table 1. A comparison between the currently planned long-baseline experiments. 5(s in 2 20 2 3 )and5(A
m 23) a r e e x P e c t e d precisions of each parameter. The last line, sin22#i3 indicates the sensitive region for the ve appearance.
Baseline
5(sin22023) « (Am| 3 ) sin22013
MINOS(FNAL) FNAL-Soudan mine 730 km 0.05 10% 0.08
ICARUS(CERN) CERN-Gran Sasso 732 km 0.05 10% 0.015
50-GeV PS(JHF) JAERI-Kamioka 295 km 0.01 3% 0.006
In the second phase of the experiment, it is proposed to increase the proton intensity up to 4 MW and to have a 1 Mt water Cerenkov detector, called Hyper-Kamiokande, aiming to search for CP violation in neutrino oscillations. The low energy neutrino beam has the advantage of large CP asymmetry and small matter effects.
2.2. Kaon Decay Physics
One of the most important subjects in particle physics today is the confirmation of the Standard Model in conjunction with the Cabibbo-Kobayashi-Maskawa (CKM) matrix and the search for new physics in deviations. The CKM scheme can be tested in a most rational way in terms of the unitary triangle. Here, kaon decays play an important role together with the B meson decays. In particular, flavor-changing neutral-current kaon decays of A'£ —)• ifivv and A'+ —)• n+vD provide us with important constraints on the unitarity condition. The most important feature here is the fact that the appearance of new physics is in different ways between K and B mesons in the unitary triangles. For example, in the MSSM the Standard Model prediction is shifted by 10-20% for the K meson, but no change appears for the B meson. Therefore, it becomes very urgent to compare both, once the triangle from B decays will be established. As typical and urgent kaon-decay experiments, the following two experiments are discussed.
(1) K°L - > n°vf>
The decay amplitude of this rare decay is proportional to fj, one of the Wolfen-stein parameters corresponding to the height of the unitary triangle. It is
37
known tha t this channel is the purest channel with no theoretical ambiguity.
At the moment a pilot experiment E391a is being prepared at the KEK-PS.
Basically the same detector can be used at the J H F . The experimental group
expects a few hundred events/ 107s (one year) for the present Standard Model
prediction. One may determine the fj parameter with an accuracy of several
% within one year of da ta taking.
The KOPIO experiment at BNL-AGS will run from 2007 almost a t the
same t ime as J H F . They expect ~ 5 0 events by three year run. The J H F with
higher beam flux can exceed KOPIO by a factor of 20.
(2) K+ -> K+VV
This charged-K + rare decay mode provides the length of a triangle in the (p, fj)
plane and is rather effective to determine p giving the mean to determine (p, fj)
together with A'£ —> 7r°z/i>. Currently, the BNL-AGS E949 is running aiming
for 5-10 events in two years (6000 hrs) of da t a taking after the discovery of two
events in the preceding E787. At the J H F the experiment group expects ~100
events using the same technique of stopped K+ decay after a similar length of
running t ime. A combination of the CKM parameters will be determined with
an accuracy better than 5%.
At the Main Injector of FNAL, the CKM experiment has been approved
and started R&D. They will employ the in-flight decay method. It is now
planned to run it around 2007 almost at the same t ime as the J H F . They will
be able to accumulate ~100 events.
2.3. Muon Decay Physics
The physics of muon rare decays is that of the lepton flavor violation. Rare
decays such as p —> ef, p. —>3e, p —>-e conversion, p —>• ~p conversion have
been intensively studied for some t ime and their branching ratio upper bounds
have steadily pushed down. According to recent theoretical SUSY-GUT cal
culations, the current limits on muon rare decays are not very far from the
possible model parameter region and muon decays possess high potentiality to
find new physics. In this regards, the p~ (A,Z)—> e~(A,Z) conversion is the
most interesting channel.
The BNL-AGS MECO experiment which will be prepared from now is
aiming for sensitivity of 1 0 - 1 6 in the branching ratio.
The PRISM collaboration in the J H F is proposing to build a high intensity
and monochromatic muon source by means of a phase rotation technique using
FFAG synchrotrons. A muon yield of 10 1 2 / s may be possible. This beam
enables us to go down to the 1 0 - 1 8 level, improving the MECO limit further
38
by a factor of 100.
3. S u m m a r y
Here, I briefly described several interesting experiments in strangeness nuclear physics and particle physics. Most of them are very important and urgent. Unfortunately, the facilities now approved for construction in Phase-1 of the project are not ready for carrying out all the experiments, while they have such potential. I eagerly hope that the construction of Phase-2, which includes a neutrino beam line and the extension of the slow-extraction experimental area, will be approved in time.
Acknowledgments
I would like to thank all members of the KEK/JAERI Joint Project Team; Project Director Shoji Nagamiya, in particular. The author also acknowledges Prof. H. Tamura and Prof. J. Imazato.
References
1. JHF Strangeness Nuclear Physics Group, Letter of Intent for Experiments on Strangeness Nuclear Physics at the 50-GeVProton Synchrotron, July 2000 (tin-published, http://www-jhf.kek.Jp/JHF.WWW/LOI/50GeVNP-LOI-vl.0.pdf).
2. J.K. Ahn et al., Phys. Rev. Lett. 87, 132504 (2001). 3. H. Takahashi et al., Phys. Rev. Lett. 87, 212502 (2001). 4. J. Doornbos, KEK Report 97-5 (1997). 5. S. Tadokoro, H. Kobayashi, and Y. Akaishi, INS-Rep.-1058, INS, Univ. of Tokyo,
(1994). 6. H. Tamura, Nucl. Phys. A639 (1998) 83c. 7. H. Tamura et al., Phys. Rev. Lett. 84, 5963 (2000). 8. K. Tanida et al., Phys. Rev. Lett. 86, 1982 (2001). 9. K. Itonaga et al., Prog. Theor. Phys. Suppl. 117, 17 (1994).
10. D.J. Millener, Proc. Int. Conf. on "Hypernuclear Physics with Electromagnetic Probes" (HYPJLAB99), Hampton (1999).
11. Y. Itow et al., http://neutrino.kek.jp/jhfnu, KEK Report 2001-4, hep-ex/0106019. 12. Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys. 49, 652 (1973).
2. Strangeness in Nuclear Matter
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WEAK DECAYS OF HYPERON A N D HYPERNUCLEI
M. OKA AND K. TAKAYAMA Department of Physics, Tokyo Institute of Technology
Meguro, Tokyo 152-8551, Japan E-mail: [email protected] ; [email protected]
K. SASAKI Institute of Particle and Nuclear Studies,
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan E-mail: [email protected]
T. INOUE Departamento de Fisica Teorica and IFIC,
Centro Mixto Universidad de Valencia-CSIC Institutes de Investigacion de Paterna,
Apdo. correos 22085, 46071, Valencia, Spain E-mail: [email protected]
Weak decays of hyperons and hypernuclei are studied from the viewpoints of chiral symmetry of QCD and roles of the quark structure of hadrons. First we point out that the soft pion relations are useful in understanding the isospin properties of the hyperon decays. The chiral effective theory approach to the hyperon decays is then introduced. Secondly, we discuss the short-range part of the AN —• NN weak transitions using the quark model of baryons. Recent studies show a satisfactory account of the non-mesonic weak decays of light hypernuclei. We also discuss 7T+ decays of light hypernuclei and show that the 7r+ decay amplitudes are related to the A / = 3/2 amplitudes of the non-mesonic decay in the soft pion limit.
1. In t roduc t ion
The high intensity 50 GeV proton synchrotron facility (JHF) is expected to produce a large amount of strange particles as well as hypernuclei. The strangeness hadron physics gives qualitatively new aspects of QCD as the mass of the strange quark is of the same order of the scale of QCD, AQCD- Thus the dynamics of QCD is most strongly reflected in the strangeness sector of hadrons.
An interesting feature of strange hadrons lies in their weak decays. Main decay modes of strange hadrons are nonleptonic with lifetimes ~ 10 - 1 0 sec.
41
42
Experimental studies of the weak decays of the kaons, hyperons and hyper-nuclei have revealed that the weak interactions of the standard theory are significantly modified by the strong interaction. Therefore they provide us with an important opportunity for understanding nonperturbative aspects of QCD. In this article, we consider how the chiral structure of QCD is relevant for the weak decays of hyperons and also how the structures of baryons can be detected in the hypernuclear weak decays.
2. Chira l Symmet ry
Chiral symmetry is a powerful tool in understanding properties of low lying hadrons and their interactions. A useful tool to take into account the chiral symmetric dynamics of hadrons is the soft pion theorem1,
(awa(q)\0\/3) ^ 4 ° - f {a\[Ql6m + (Pole terms). (1) JIT
This can be applied to the weak decays of hyperons, such as,
{n«\q)\Hpv \A) g^° - Un\[Ql HPV]\A) = - ^ < n | t f ^ | A > . (2)
in tjn
Here Hpv is the parity violating part of the weak Hamiltonian, which contains only the left-handed currents qll^ll an<3 the flavour singlet right-handed current induced by the penguin type QCD corrections. This allows us to relate the commutator [Q^, Hpv] in the second expression of Eq. (2) to Hpc in the final expression as
[QR,HW] = 0
[Ql, Hw] = -[Ql,Hw] = -[Ia,Hw]
-(n\[Qa5,H
PV]\A) = (n\[Ia,Hpc}\\) = -l-(n\Hpc\A). (3)
Thus, the parity violating amplitudes, or the S-wave decay amplitudes, of pionic decays of various hyperons can be expressed in terms of the baryonic matrix elements of the parity conserving weak Hamiltonian. As a result, for instance, the AI — 1/2 dominance of A -> Nn decays follows immediately since "A —>• n transition" is purely AI = 1/2. Furthermore, if we conjecture that Hw is purely flavour octet, then various matrix elements, (n\Hpc\Yi), (T,\HPC\E), are all related to (n\Hpc\A) and thus several relations among the pionic decay amplitudes of hyperons are obtained. Such relations are known to be satisfied fairly well for the 5-wave decay amplitudes of the hyperon decays2.
The parity conserving decays belong to exceptions of the soft pion theorem, in which the pole terms cannot be neglected. The pole terms are such that
43
A -> n -> WK0 or A -> E°7r° -> n^r0, and their amplitudes are
(nir°{q -> 0)|tfpc |A> ~ (n7r°|n) (n\Hpc\A) mA - mn
+ {n\Hpc\E°) i ( S V | A ) . (4)
An interesting result is given for the £+ —• mr+ decay, that is
<n7r + | f f w |E + > 8 o f t . p i o n = 0, (5)
since [I~, H ] = 0. On the other hand, there is no such a constraint for the pole terms of the parity conserving amplitudes, where the p, S° and A intermediate states with different energy denominators contribute. Therefore the soft-pion theorem suggests that the E + —> nir+ decay goes only through the parity conserving P-wave channel. This is indeed what is observed experimentally, the PV amplitude of 0.13 is to be compared to the PV one of 42.2. This example shows that the AI = 1/2 dominance and the soft-pion relation are very well satisfied in this decay.
We later consider 7r+ decays of hypernuclei in this context and see that the soft-pion theorem suggests the TT+ decays are induced only by the AI = 3/2 part of the weak Hamiltonian.
3. A I = 1/2 Rule
In the above discussion, we have assumed the AI = 1/2 dominance of the weak matrix element, (N\HPC\T,). An explanation of the "A/ = 1/2 rule" has been a long standing problem of the weak decay of kaons and the hyperons. It was shown long time ago that the perturbative QCD corrections to the standard model weak vertex enhance the AI = 1 / 2 component, while suppressesing the counterpart, the AI = 3/2 component3. This mechanism can be understood easily by decomposing the weak s + d —> u + d transition into the isospin-spin-colour eigenstates. As the strangeness changing transition is induced only by the charged current, or the py-boson exchange, the vertex at low energy is given, without the QCD corrections, by
(^7"«£) (^7M«i ) = (^7"«Z)(«27^«i) - (6)
where a and (5 are colour indices and the equality comes from the Fierz transformation. From this we observe that the colour+isospin combination of the final u + d quarks is always symmetric, namely, (// = 0, Colour 3) or (// = 1, Colour 6). In both cases, the total spin of the final quarks must be zero. When we consider gluon corrections to this vertex, we notice that the gluon exchange,
44
or its colour-magnetic component, —(Ai • A2)(o"i • cr2) term, between the final u and d is attractive for (1/ = 0, Colour 3),
(S = 0, C = 3|(A! • \2)(<n • <r2)\S = 0, C = 3) = - 8 , (7)
while it is repulsive in the other,
(S = 0,C = 6|(Ai • A2)(<r1 • a2)\S = 0,C = 6) = +4. (8)
Therefore the QCD correction tends to enhance the final If = 0 amplitude. The above heuristic explanation of the AI = 1/2 enhancement can be confirmed in the renormalization group improved effective action of the strangeness-changing weak interaction3. It was also shown that a further enhancement of AI — \j1 is resulted due to the "Penguin" diagrams3, which is regarded as a QCD-corrected s —*• d transition, and is purely AI = 1/2.
The perturbative QCD corrections are not the only source of the enhancement, but we expect further effects of the nonperturbative origin. In fact, it is known that the enhancement in the effective interaction is not large enough to explain the observed dominance of AI = 1 / 2 . Here we here concentrate on the baryonic weak interaction. Miura-Minamikawa and Pati-Woo4 (MMPW) pointed out that the AI = 3/2 part of the nonleptonic hyperon decays is suppressed due to the colour symmetry of the valence quarks in the baryon. It is understood easily by considering the colour structure of the 4-quark operator which belongs to the 27-dimensional irrep. of the flavour SU(3) :
0(27) = (uirsD&lA) + (JL1">LMJ^L) - (9) which is the part responsible for the AI = 3/2 transition. From the symmetry of the final u and d quarks, this operator creates two quarks with their colour part being symmetric, i.e. the colour 6 state. As the colour wave function for the three valence quarks of the baryons, the hyperon in the initial state or the nucleon in the final state, this operator cannot be connected to two quarks inside the baryon. Thus, the only possibility is the external diagram in which both the quark and the antiquark of the meson (pion) are connected to the weak vertex directly. Such a diagram is not allowed either because the external qq created by the operator of Eq. (9) should be both left-handed and therefore not form a pseudoscalar meson in the chiral limit. Gluon exchanges among the initial or final quarks do not help, while an "exotic" component such as a valence gluon will change the situation. Thus we observe that the AI = 3/2 part of the Y —> N + PS meson is strongly suppressed as far as we consider the valence quark picture of the baryons. It should be noted, however, that the vector mesons may couple to AI = 3/2 vertices directly5.
Another possible explanation often given for the AI = 1 / 2 enhancement is the dominance of the scalar diquark, 0+ ud(I — 0), which is dynamically
45
favoured under the gluon exchange interaction in the baryon. This is in fact a nonperturbative version of the AI = 1/2 enhancement in the perturbative correction discussed above. It is, however, noted that such a strong correlation of the diquark breaks the SU(6) symmetry and therefore has a large influence on the static properties of baryons, such as charge distributions, magnetic moments, etc.
4. Chiral Perturbation Theory for the Nonleptonic Weak Decays
So far, we have seen that the chiral symmetry plays an important role in the weak decays of strangeness. However, in order to describe the hyperon decays quantitatively, it is necessary to go beyond the chiral limit. There are corrections due to finite quark mass, flavour SU(3) breaking, which can be treated by the chiral perturbation theory approach. In view of the qualitative success in the soft-pion approach, it is promising to apply the chiral effective theories to the hyperon and hypernuclear decay processes. Several recent studies have employed such approaches for the hyperon decays6,7.
First we consider "hadronization" of the effective weak Hamiltonian,
Hweak = X>(f*if VH^rpV), (io) i
where I\ 's are relevant 7 matrices and c,-'s are the coefficients given as solutions of the renormalization group equation. The B —• B'M transition is induced by the matrix element,
(MB'\Hweak\B) = Y^Ci(M\qaT<.1)qa\0)(B'\fT<l'i)q^\B) i
+ (lf+ 2) + (Fierz term 0(1/NC))
+ (internal diagrams; AI ^ 3/2). (11)
The first two terms are the factorization terms where the meson is produced directly from the weak vertex. The third term is the Fierz transform term, which is suppressed by the l/Nc factor. The fourth term is the non-factorized term, where three or four quark operators are contracted with the quarks inside the baryons B or B'. This term is called the "internal diagrams".
Now we assume that the main contribution comes from the valence quarks of the baryons, which is reasonable for the weak transition as it changes the flavour quantum numbers of the baryons. Under this assumption, we conclude that the AI = 3/2 operator, Eq. (9), does not contribute to the internal diagrams according to the MMPW theorem. Therefore the AI = 3/2 transition can be induced only by the factorized terms. The factorized terms are written
46
in terms of the matrix elements of the quark bilinear operators and therefore can be expressed by mesonic and baryonic currents.
The SU(3) chiral perturbation theory for baryons in the heavy baryon formulation is employed and is combined with the above construction of the effective weak Hamiltonian. The replacement is, for instance,
•/,§" = <iiLlMqjL
-»• Trlv"B[^hijt B]] - 2£>Tr[5S" {£%•£, B}]
-2FTr[BS'l[?hijt,B]], (12)
where B is the octet baryon field, £ the meson fields, v^ (S^) is the velocity (spin) operator and hij is the flavour matrix that changes s —¥ d. Similar current operators for the mesons can be easily constructed. For the details, the readers are encouraged to refer to the paper7.
Using the current operators such as J^, one can construct an effective weak Hamiltonian given as a sum of the current-current operators. Such construction is possible for the factorized and Fierz terms in Eq. (11), while the internal terms cannot be written in the current-current form. As the AI = 3/2 transitions are free from the internal diagrams, we mainly consider the AI = 3/2 transitions.
The hyperon decay amplitudes have been calculated to the tree level and also to the one-loop level in the chiral perturbation theory7. The results are summarized as follows :
(1) We compare the calculated AI = 3/2 decay amplitudes with experimental data. It was shown that at the tree level the S-wave (PV) amplitudes are reproduced reasonably well, while the P-wave (PC) ones have some difficulty.
(2) For both the S- and P- waves, the chiral-log one-loop terms are disturbingly large.
(3) Nevertheless, in overall, the smallness of the AI = 3/2 amplitudes is confirmed. On the other hand, the resulting AI = 1/2 amplitudes are also small if the internal terms are omitted. It leads to the conclusion that the internal diagrams should be dominant in the nonleptonic weak decays.
The difficulty in reproducing the P-wave amplitudes simultaneously with the S-wave ones is a common problem known in phenomenological approaches. In the SU(3) symmetry analysis, the f/d ratios for the S-wave and P-wave amplitudes do not agree with each other. However, they come from the same source in the chiral Lagrangian. Often, the P-wave amplitudes require higher order and/or non-tree contributions. The problem of convergence is common
47
too. Jenkins has shown tha t contribution from the decuplet baryons suppresses
large one-loop corrections. Even then the fitting to the experimental da t a is
not satisfactory yet6 .
5. D i r e c t Quark M e c h a n i s m a n d W e a k D e c a y o f H y p e r n u c l e i
The pionic decay of A is known to be suppressed in nuclear medium as the final
nucleon does not have enough momentum to go above the Fermi energy. Thus
the main decay mode is non-mesonic, which is induced by another nucleon,
tha t is, \p —> pn and An —>• nn. These processes are viewed as weak baryonic
interactions, which are unique and interesting themselves as a new type of the
nonleptonic weak interactions of baryons. Furthermore, this is a reaction in
which the momentum transfer is so large tha t the quark substructures of the
baryons may be significant.
Recently, we proposed the direct quark (DQ) transition mechanism to ac
count for the short-range part of the YN —> NN weak interaction8 , 9 . The
DQ transition potential is obtained by evaluating su —> ud and sd —> dd t ran
sitions among the valence quarks in two baryons. While the DQ is expected
to describe the short-range part of the transition, the long-range part of the
transition is to be described by the light meson exchanges. We consider the 7r
and K exchanges on the top of DQ. The role of K is found to be essential in
describing the experimental da ta .
The decay rates of the A in nuclear mat ter and in light hypernuclei are
calculated. The results are compared with those without DQ, and also with
experiment. We leave the details to the l i terature1 0 '1 1 '1 2 , while the conclusions
of our study are summarized here : (1) The DQ transition is significantly large,
and shows qualitative differences from the meson exchanges. (2) The one pion
exchange (OPE) mechanism yields a large tensor amplitude, i.e. the transit ion
from Ap :3 S\ to np :3 D\. This large tensor ampli tude causes difficulty tha t
the nn/np ratio is too small compared to the value suggested by experiment,
so-called the n/p ratio problem. We found tha t this problem is solved by the
introduction of the short range part of the transition potential as well as the
soft form factor for OPE . Both the DQ transition and the kaon exchange are
significant in enhancing the ratio. The kaon exchange mainly reduces the ten
sor part of the pion exchange and also enhances the nn decay rate, while the
DQ is strong in the neutron induced decay. (3) The AI = 3/2 contribution
is significant for the J = 0 transition amplitudes in the DQ mechanism, while
the meson exchanges are assumed purely in the AI = 1/2 transit ion. Un
fortunately, it is not possible at present to determine the importance of the
AI — 3/2 contribution from the experimental da ta 1 3 . We suggest tha t the ^ H
decay is the key quanti ty for clarifying the role of the AI — 3 /2 contribution.
48
(4) Virtual excitation of E as an intermediate state in the AN —>• NN process has been discussed in Ref. [12]. We have found that coherent S mixing of order 1% in the A = 4 hypernuclei will give significant effects on their weak decays. An advantage of the decay is that the sign of the mixing can be determined by comparing the calculation and the experimental data. In fact, we found that the agreement is better when we include the E effect with the sign consistent with the one-pion exchange transition.
The high intensity kaon beam at the JHF will give us a new opportunity to study weak decays of double hypernuclei. There are new non-mesonic modes, AA —J- An and —>• En, which are complementary to the AN —> NN modes. In the new modes, only the decays in the spin singlet channels are allowed, while in the AN —• NN, the spin triplet channels are much stronger so that the spin singlet channels are often hidden behind. Calculations of the decays of double hypernuclei in the DQ and meson exchange mechanisms is underway.
6. 7r+ Decay Mode and AI = 3 / 2 Amplitudes
Light hypernuclei may decay weakly by emitting a pion. While the free A decays into pir~ or nn°, the 7r+ decay requires an assistance of a proton, i. e.A + p —> n + n + n+. Some old experimental data14 suggest that the ratio of 7r+ and TT" emission from ^He is about 5%. The most natural explanation of this process is A —> nir° decay followed by the n°p —»• 7r+n charge exchange reaction. It was evaluated for realistic hypernuclear wave functions and found to explain only 1.2% for the TT+ /n~ ratio15. Another possibility is to consider the E + —• 7r+n decay after the conversion Ap —• E + n by the strong interaction such as pion or kaon exchanges. It was found, however, that the free E + decay, which is dominated by P-wave amplitude, gives at most 0.2% for the n+ /ir~ ratio.
In order to solve this problem, we have applied the soft-pion technique to the 7T+ decay of light hypernuclei16. The soft-pion theorem for the process Ap —>• nn;r+ (q —> 0) reads
lim(nmr+(q)\Hw\Ap) = —^-(nn\[Q^,Hw]\Ap). (13)
Again, because of
[Qs,Hw] = -[I-,Hw], (14)
it discriminates the isospin properties of Hw • Similarly to the case of the E +
decay, we see that the AI = 1/2 part vanishes as
[I.,HW{AI = 1/2 , AIZ = -1/2)] = 0
49
[I„,HW(AI = 3/2 , AIZ = - 1 / 2 ) ] =
y/3Hw(M = 3 /2 , A / 3 = - 3 / 2 ) . (15)
We then obtain
lim(nnK+(q)\Hw\kp) = - t f - ( n n \ H w { A I = 3 /2, A / 3 = - 3 / 2 ) | A p ) . (16) 9-*-° vlf-n
Thus we conclude tha t the soft TT+ emission in the A decay in hypernuclei
is caused only by the AI — 3/2 component of the strangeness changing weak
Hamiltonian. In other words, the ir+ emission from hypernuclei probes the
AI = 3/2 transition of AN —» NN. Available experimental da ta are old and
not enough to convince ourselves to see the AI — 3 /2 effect. We hope the
future experiments at J H F will reveal the isospin structure of the non-mesonic
weak decays.
7. C o n c l u s i o n
In this article, we first discussed the roles of chiral symmetry in the weak
hyperon transitions. We have pointed out tha t the soft pion relations are
useful in understanding the isospin properties of the hyperon decays. The
chiral effective theory approach to the hyperon decays is introduced for further
quanti tat ive study, but it is found to have some difficulties. Secondly, we have
shown tha t the DQ mechanism for the short-range par t of the hypernuclear
non-mesonic weak decays is important . Our DQ + IT + K model gives a
satisfactory account of the weak decays of hypernuclei. The n+ decays of light
hypernuclei are found to come from the AI — 3/2 amplitudes given by the
DQ.
Our goal is to quantitatively understand the QCD effects on the weak
interaction. The JHF will be the most effective tool to study the strangeness
changing weak interactions. Further cooperative effort of the theorists and the
experimentalists is essential for the success.
R e f e r e n c e s
1. S.B. Treiman, "Current Algebra and Its Applications", (Princeton Univ. Press, 1972); J.J. Sakurai, "Currents and Mesons", (Univ. Chicago Press, 1969).
2. J.F. Donoghue, E. Golowich and B. Holstein, Phys. Rept. 131, 319 (1986). 3. M.K. Gaillard and B.W. Lee, Phys. Rev. Lett. 33,108 (1974); A.I. Vainshtein,
V.I. Zakharov and M.A. Shifman, Sov. Phys. JETP 45, 670 (1977); F.J. Gillman, M.B.Wise, Phys. Rev. D20, 2382 (1979); E.A. Paschos, T. Schneider and Y.L. Wu, Nucl. Phys. B332, 285 (1990).
4. J.C. Pati and C.H. Woo, Phys. Rev. D 3 , 2920 (1971); K. Miura and T. Mi-nanikawa, Prog. Theor. Phys. 3 8 , 954 (1967).
50
5. K. Maltman and M. Shmatikov, Phys. Rev. C51, 1576 (1995). 6. E. Jenkins, Nucl. Phys. B375, 561 (1992); B. Borasoy and B.R. Holstein, Euro.
Phys. Jour. C6, 85 (1999); Phys. Rev. D59, 094025(1999). 7. K. Takayama and M. Oka, hep-ph/9809388, hep-ph/9811435. 8. T. Inoue, S. Takeuchi and M. Oka, Nucl. Phys. A577, 281c (1994); Nucl. Phys.
97, 563 (1996). 9. K. Maltman and M. Shmatikov, Phys. Lett. B331, 1 (1994). 10. T. Inoue, M. Oka, T. Motoba and K. Itonaga, Nucl. Phys. A633, 312 (1998). 11. K. Sasaki, T. Inoue and M. Oka, Nucl. Phys. A669, 331 (2000). 12. K. Sasaki, T. Inoue and M. Oka, to be published in Nucl. Phys. A (nucl-
th/0204057). 13. R.A. Schumacher, Nucl. Phys. A547, 143c (1992); R.A. Schumacher for the
E788 Collaboration "Properties & Interactions of Hyperons", ed. by B. F. Gibson, P. D. Barnes and K. Nakai (World Scientific, 1994), p. 85.
14. C. Mayeur, et al., Nuovo Cim. 44, 698 (1966); G. Keyes, J. Sacton, J.H. Wickens and M.M. Block, Nuovo Cim. 31A, 401 (1976).
15. R.H. Dalitz and F. von Hippel, Nuovo Cim. 34, 779 (1964); F. von Hippel, Phys. Rev. 136, B455 (1964); A. Cieply and A. Gal, Phys. Rev. C55, 2715 (1997).
16. M. Oka, Nucl. Phys. A647, 97 (1999).
THE A - A INTERACTION A N D STRANGENESS -2 HYPERNUCLEI
I. R. AFNAN School oj Chemistry Physics and Earth Sciences
Flinders University, GPO Box 2100, Adelaide 5001, Australia E-mail: iraj. afnan Oflinders. edu. au
The analysis of the one-boson exchange potential for the 'So S = — 2 channel is presented with emphasis on the role of coupling between the channels. The contribution of the different mesons is examined with the surprising result that the pion exchange component is almost negligible. The singlet scalar exchange, an approximation to the two pion exchange, plays an important role in all channels. The variation of the coupling between channels as function of the AA scattering length is examined. The implication of these observations for , ?He is considered, with the possibility that the new experimental data will give a consistency between the experimental AA matrix elements and the SU(3) analysis of the baryon-baryon interaction within the one-boson exchange potential.
1. In t roduc t ion
The recent measurement of the binding energy of AAHe suggests that the effective AA interaction1
AS A A = 5A A(A lHe) - 2BA(lHe)
= 1 .01±0 .20J ;^MeV, (1)
in light nuclei is considerably smaller than the old measurement m 4.7 MeV reported by Prowse2. In the present analysis we examine the implication of this new result within the framework of one-boson exchange (OBE) models that employ SU(3) symmetry to determine the baryon-baryon interaction in channels with strangeness S — — 2.
If we assume flavour SU(3) is a good symmetry, we can write the matrix elements of the potential in terms of the irreducible representation of 8 ® 8 as
(nn\V\nn) = V27 Q « 4
(AN\V\AN)=-V27+-VS,
{\\\V\AA) = ?j5V„+±VB. + ^V1. (2)
51
52
Considering the fact that V8, and V\ are repulsive while V27 is attractive3, we may conclude that
(Vnn)>(VAN)>(VAA) . (3)
With the three old measurements of AA hypernuclei (A6AHe2, A°ABe4'5,
and AAB 6 ' 7 ) which predicted that the AA matrix element (AA|V|AA) was sa 4 — 5 MeV, there were suggestions that the breaking of SU(3) symmetry and the coupling between the AA, the EN and the EE channels in the 1So partial wave could bridge the gap between experiment (VAA) ^ {Vnn) > (V\N) and the predictions of SU(3).
To examine this problem and its implication for the new experimental result, we consider the one-boson exchange (OBE) potential of Nijmegen model8
D. Here, if we require all coupling constants to be determined by the SU(3) rotation of those parameters as given in the nucleon-nucleon (NN) and hyperon-nucleon (YN) channels, then the only free parameters are those of the short range interaction. These we vary within the limitation that the long range part of the potential is predominantly OBE in origin. This allows us to examine the consequence of the new measurement of the AA matrix element on the strength of the A A interaction, and the importance of the coupling of the A A channel to the 3N and EE channels.
2. The OBE Potential in the S = - 2 Channel
To perform an SU(3) rotation on an OBE potential defined in the 5 = 0, —1 channels, we need to write the Lagrangian in terms of the baryon octet with the mesons as either a singlet or a member of an octet. If the interaction is taken to be of the Yukawa type, the interaction Lagrangian takes the form9
{gs[B^B]sMs + g8l[B^B]8lM8 + 5s2[S tB]s2M8} , (4)
where B and M are the field operators for the baryons and mesons. In writing the above Lagrangian, which is a scalar, we have coupled the initial and final baryons to a flavour singlet or an octet. Since there are two irreducible octet representations, we need a different coupling constant for each of the representations. This Lagrangian has one coupling constant for each singlet meson ga, and two coupling constants <7{8!} a n d 9{82} f° r each meson octet. These coupling constants can then be determined by fitting to the experimental data.
The Nijmegen potential8 model D takes for the exchange mesons the pseu-doscalar octet {TT,J],T]',K}, the vector octet {p, <j>,w, K*} and a scalar meson {e}. The masses of the mesons and baryons are taken from experiment, while the coupling constants are adjusted to fit the data in the S — 0 , - 1 with a hard core for the short range interaction. This, in principle, determines
53
the long range part of the potential which should be described in terms of meson-baryon degrees of freedom. The same coupling constants can be used to construct the OBE potential for S < - 2 . The breaking of flavour SU(3) is a result of using physical masses for the baryons and mesons and the difference in the short range behaviour of the potential as we proceed from the S = 0 to the S = — 1 and S = — 2 channels.
This procedure was followed by Carr et a/10. They only considered the 5-wave interaction and ignored the tensor interaction. Their potential for the exchange of the ith meson was of the form
VPtf + ffi-fftVPir), Vi(r) (5)
where the radial potential V« , a (0
Vj'">(r) = V0 (*) -m^r
mir -C
c, a for a meson of mass rrii is taken to be Mr'
a = c,a . (6) M\ mi) Mr
> 2
r(fm)
Figure 1. The AA potential in 'So channel. The solid and dashed line are the contributions of the ui- and e-exchange, while the dotted line is the total potential. Here C is adjusted so that the AA scattering length a^A = —1-91 frn.
To guarantee one-parameter short-range repulsion, the mass M = 2500 MeV is chosen in all partial waves. Then the parameter C determines the strength of the short range interaction. This new parameter C is constrained to ensure that the potential for r > 1.0 fm is unchanged and the short range interaction is always repulsive. In Fig. 1 we present the AA potential in the 1SQ channel.
54
Included in the figure are the contributions from the e (dashed line) and w (solid line/) exchange as well as the full potential which includes the sum of all the contributions from all the meson exchanges allowed. In this case the parameter C is adjusted for the potential to give a A A scattering length <ZAA = —1.91 fm. Here we observe that the dominant contribution to the potential is from the e-exchange which is not part of any meson octet, and was introduced to give medium range attraction and emulate two pion exchange.
>
in in
>
r(fm)
Figure 2. The 'So EiV — HAT potential. The contributions of the IT-, UJ- and e-exchange are represented by solid, dashed and dotted lines respectively. The total potential is represented by a dense dotted line. Here C is adjusted to give a^A = —1-91 fm.
We now turn to the EN — EN potential where 7r-exchange is allowed. In Fig. 2 we present the most important contributions to the potential as well as the contribution from rr-exchange. Here again the dominant contribution is from the e-exchange. But surprisingly, the 7r-exchange is negligible. One can make the same observation for the EE — EE potential where the 7r-exchange is an order of magnitude smaller than the e-exchange. This result is a reflection of the fact that in the 1So channel the strength of the 7r-exchange had a factor of 5̂f", where m is a hadron mass. Thus we may conclude that the diagonal elements of the potential have little contribution from the 7r-exchange and are dominated by the e-exchange. If one examines the coupling between the three channels AA, EN, and EE, then one observes that the 7r-exchange is possible in the transition between the AA and EE channels. However, in this case the
55
other isovector exchange, the p, is dominant. We now turn to the importance of the coupling between the three channels.
Here, we should point out that if the coupling is important, then the extraction of the AA interaction from light S = - 2 hypernuclei will require that we include this coupling in the analysis of the data. To illustrate this point let us consider the effective matrix element of the AA interaction to second order in perturbation theory, i.e.
<«<AA|l / |AA>-K^M, (7)
0.8 1 1.2 1.4 1.6 1.8 2
r(fm)
Figure 3. The AA potential in the 1SQ channel. The solid line labelled V0 is the OBE with no cutoff. The curves V\, V2 and V3 correspond to the potential with no channel coupling, with coupling to the EAT channel only, and with the full coupling to the 3.N and SE channels. The parameter C was adjusted to get a scattering length »AA = —1.91 frn-
where AE ss 25 MeV. In free space due to the small difference between the AA and EN threshold, this coupling is more important than that between the NN and NA in the S — 0 channel. On the other hand, in the nuclear medium, the transition from A A to EN is Pauli blocked. As a result, the additional attraction from the second order term is suppressed in nuclei. This indicates that the effective AA matrix element is less attractive in the nuclear medium than in free space. This is all true provided the coupling is in general large in free space. To examine this question, we consider the importance of the coupling as we change the A A scattering length OAA-
56
In Fig. 3 we present the AA potential with no coupling (Vi), with coupling to the EN channel (V2) and with the full coupling to both EN and EE channel (V3). The short range parameter C was adjusted so that the potentials have a scattering length aAA = —1.91 fm. The potential V2 gives a binding energy for A^He10 that is in agreement with the old experimental result of Prowse2. Here, we observe that as one includes first the EN and then EE channels, the AA potential becomes shallower. This suggests that the coupling will reduce the binding energy of AA hypernuclei as first observed by Carr et al.w The surprise here is the importance of the coupling to the EE channel even though the threshold for the EE channel is some 160 MeV above the AA threshold.
0.8 1 1.2 1.4 1.6 1.8 2
r(fm)
Figure 4. The 'So AA potential for the case when the potential, including coupling to all the channels gives a scattering length OAA = —0.5 fm. The curves have the same labelling as in Fig. 3.
The new measurement1 of the matrix element of the AA interaction in
AAHe suggests that the AA potential is in fact much weaker than assumed in the past. We therefore have considered a potential that gives a scattering length <2AA = —0.5 fm. This is consistent with the results using the recent Nijmegen potential11. In Fig. 4 we present the AA potential with no coupling (Vi), with coupling to the EN channel (V2) and with coupling to both the EN and EE channels for a\\ = —0.5 fm. There are two distinct differences between the results for <2AA = —0.5 fm and those for OAA = —1-91 fm. These are : (i) In general the smaller scattering length gives a potential that is 30%
57
shallower, (ii) More important is the fact that the coupling is not as important. However even in this case, the coupling to the EE channel is more important than just including the coupling to the EN. This suggests that as we reduce the strength of the AA potential, the role of the coupling is reduced. Maybe more important is the fact that we may need to include the coupling to the EE channel even though the EE threshold is some 160 MeV above the AA threshold, while the EN threshold is only 25 MeV above the AA threshold.
To give some quantitative measure to the variation in the AA matrix element with changes in the scattering length, we recall the results of Carr et al.10
in Table 1 for AAHe. Here we tabulate the AA scattering length OAA, and the binding energy of AAHe with and without coupling between the AA and EN channels. Also included are the AA matrix elements (AB = [BE (AA — EN) — 6.14]) in this hypernucleus. These results confirm our expectation that the coupling between the channels becomes negligible as the scattering length «AA becomes small and negative, i.e. as the AA interaction becomes weak.
Table 1. The variation in the AA interaction with changes in the strength of the AA potential as measured by a^A-
aAA fm -1.91 -21.1 7.82 3.37
BE(AA<* - S.Na) MeV 9.738 12.268 15.912 19.836
BE(AAa) MeV 10.007 14.138 17.842 23.342
A B MeV 3.6
6.13 9.76 13.7
This change in the binding energy with and without the coupling to the EN channel as one varies the AA scattering length, is illustrated in Fig. 5. Here we have a plot of the binding energy as a function of aAA. In particular, the (+) and (x) are the results of Carr et al.10 with and without coupling between the AA and EN channels. Also included are the recent results of Filikhin and Gal12 with just the AA channel. From the results of Carr et al.10
we can clearly see that the role of coupling for small and negative scattering length is negligible, while the results of Filikhin and Gal12 suggest that the new experimental result1 for the binding energy of AAHe of 7.25 ± 0.2j;°-\\ MeV will give us a scattering length for the A A of « —0.5 fm.
3. Conclusion
We have demonstrated that within the framework of the one-boson exchange model (OBE) and flavour SU(3) symmetry one can generate a one-parameter set of potentials that preserve the OBE tail. The short range repulsion can then be adjusted for the potential to give the A A scattering length. The only
58
(MeV
)
w CD
z , t
22
20
18
16
14
12
10
8
c
--
-
+
X
*
#
1 1 1
Carr with coupling
Carr without coupling
Filikhin & Gal
X
i
*
1 x
-
+ -
X
+
X
+
* ¥ *
i i
-3 -2.5 -1.5 -1
A A
-0.5 0 0.5
Figure 5. Plot of the binding energy (B.E.) of A ^He as a function of a^A . Here ( + ) and (x) are the results of Carr et ah with and without coupling to the zLN channel. Also included are the results of Filikhin and Gal (*).
problem with this procedure is the fact that the potential is dominated by the exchange of the scalar e meson. This meson was introduced in the strangeness S = 0 to give medium range attraction and to model the two pion exchange. Its dominance in the S = —2 channel suggests that we need to go back and include the explicit two pion exchange within a framework that will still allow us to perform a flavour SU(3) rotation on the potential to generate the AA interaction.
From the above analysis of the importance of coupling between the channels (AA, HAT and EE) in the 1So strangeness S = —2 partial wave, we found that for small negative AA scattering length, the coupling between the channels is relatively weak. If we now use this observation along with the recent measurement of the binding energy of A^He, we may conclude that confirmation of this measurement could constrain the A A scattering length a\\ ?« — 0.5 fm to good accuracy. This could be achieved without the inclusion of the coupling to the EN and ES channels, which is a complication in the calculation of energies of light hypernuclei.
Finally, the new measurement of the AA matrix element1 restores the va-
59
lidity of SU(3) for predicting the relative strength of the interactions in the
S = 0, — 1 and —2, as stated in Eq. (3).
A c k n o w l e d g e m e n t s
The author would like to thank the Australian Research Council for their
financial support during the course of this work. He is indebted to his collab
orator B.F. Gibson for the many stimulating and enlightening discussions and
correspondence on the subject.
R e f e r e n c e s
1. H. Takahashi et al., Phys. Rev. Lett. 87, 212502 (2001). 2. D. J. Prowse, Phys. Rev. Lett. 17, 782 (1966). 3. Th. A. Rijken, P. M. M. Maessen and J. J. de Swart, Nucl. Phys. A547, 245c
(1992). 4. M. Danysz et al., Nucl. Phys. 49, 121 (1963). 5. R. H. Dalitz et al., Proc. R. Soc. London A426, 1 (1989). 6. S. Aoki et al., Phys. Rev. Lett. 65, 1729 (1990); Prog. Theor. Phys. 85, 1287
(1991). 7. C. B. Dover, D. J. Millener, A. Gal and D. H. Davis, Phys. Rev. C44, 1905 (1991). 8. M. M. Nagel, Th. A. Rijken and J. J. de Swart, Phys. Rev. D15, 2547 (1977). 9. J. J. de Swart, Rev. Mod. Phys. 35, 916 (1963). 10. S. B. Carr, I. R. Afnan and B. F. Gibson, Nucl. Phys. A625, 143 (1997). 11. V. G. J. Stoks and Th. A. Rijken, Phys. Rev. C59, 3009 (1999). 12. I. N. Filikhin and A. Gal, Phys. Rev. C65, 041001 (2002), nucl-th/0110008,
nucl-th/0203036.
H Y P E R O N - N U C L E O N I N T E R A C T I O N A N D S T R A N G E N E S S
P R O D U C T I O N IN PP C O L L I S I O N S
J. H A I D E N B A U E R
Institut Jiir Kernphysik, Forschungszentrum Julich, D-524S5 Julich, Germany
E-mail: [email protected]
A new model for the hyperon-nucleon (AN, EiV) interaction is presented. The model incorporates the standard one-boson exchange contributions of the lowest pseudoscalar and vector meson multiplets with coupling constants fixed by SU(6) symmetry relations. As the main feature of the new model, the exchange of two correlated pions or kaons, both in the scalar-isoscalar (<r) and vector-isovector (p) channels, is included. Furthermore, results of a model calculation for the reactions pp —y NAK and pp —y N'EK near their thresholds are reported. Special attention is paid to the cross section ratio <Tpp-i.pAK+ /app^pi;0K+ which was found to be unexpectedly large in recent experiments.
1. I n t r o d u c t i o n
There is solid experimental evidence that the strong interaction obeys, besides
fundamental symmetries like Lorentz invariance or space and t ime reversal, an
additional symmetry principle namely the isospin invariance. The correspond
ing symmetry group is SU(2). However, the situation is much less clear when
it comes to strangeness and the validity of SU(3) symmetry. On the hadronic
level the breaking of the SU(3) symmetry is clearly visible in the mass splittings
of the particles within SU(3) multiplets. For instance the "strange" members
of the octet to which the nucleons belong are between 200 MeV (the A and
S) and 400 MeV (the cascade particle S) heavier than the proton and /or neu
tron. The interesting question is, however, whether there is also a dynamical
breaking of the SU(3) symmetry - besides these obvious "kinematic" effects.
Such a dynamical breaking would manifest itself, among other things, in the
coupling strengths at the hadronic vertices.
An excellent possibility to investigate the validity of SU(3) symmetry is
offered by the study of the AN and Y,N systems. Assuming SU(3) symmetry
one can relate the coupling constants of the hyperon-nucleon (AN and Y,N)
interaction to those of the nucleon-nucleon (NN) interaction which are much
better constrained by experimental information and, therefore, much better
60
61
known. The predictions from the resulting hyperon-nucleon model can then
be compared with corresponding experiments and, thereby, could provide in
dications for a possible failure of SU(3) symmetry. Such a strategy has been
followed in theoretical investigations of the Jiilich1 '2 and Nijmegen3 '4 groups.
However, due to the rather limited AN and EN scattering da t a so far, no
clear conclusion could be reached. Among the recent experiments involving
hyperons the associated strangeness production in the reactions NN —)• NAK
and NN —> NEK5'6'7,8 is certainly the most promising one to yield new and
stringent constraints for the A AT and EN interactions and, therefore, for the
validity of the SU(3) symmetry. Since the measurements can be done practi
cally at the reaction thresholds, the relative momentum in the outgoing AN
or EN systems will be minimal. Thus one is able to gain information about
the interaction in those systems at much smaller energies than ever before.
Besides the interaction in the (final) AN and EN states SU(3) symmetry
plays also a role in the production amplitudes of the reactions NN —> NAK
and AW —>• KEK. In microscopic models of the production ampli tude the
NAK and NEK coupling constants enter various contributions. Accordingly,
in such models the magnitude of the predicted NAK and NEK cross sections
depends crucially on their values. Thus, a comparison of those predictions
with experiments is expected to allow also conclusions on the validity of SU(3)
symmetry.
2 . T h e H y p e r o n - N u c l e o n I n t e r a c t i o n
The hyperon-nucleon (YN) interaction is an ideal testing ground for studying
the importance of SU(3) flavour symmetry breaking in hadronic systems. Ex
isting meson exchange models of the YN force usually assume SU(3) flavour
symmetry for the hadronic coupling constants, and in some cases1 ,2 even the
SU(6) symmetry of the quark model. The symmetry requirements provide re
lations between couplings of mesons of a given multiplet to the baryon current,
which greatly reduce the number of free model parameters. Specifically, cou
pling constants at the strange vertices are connected to nucleon-nucleon-meson
coupling constants, which in turn are constrained by the wealth of empirical
information on NN scattering. Essentially all YN interaction models can re
produce the existing YN scattering data , so that at present the assumption of
SU(3) symmetry for the coupling constants cannot be ruled out by experiment.
One should note, however, tha t the various models differ dramatically
in their t rea tment of the scalar-isoscalar meson sector, which describes the
baryon-baryon interaction at intermediate ranges. For example, in the Ni
jmegen models3 '4 this interaction is generated by the exchange of a genuine
scalar meson SU(3) nonet. The Tubingen model9 , on the other hand, which
62
is essentially a constituent quark model supplemented by n and a exchange at intermediate and short ranges, treats the <r meson as an SU(3) singlet.
In the YN models of the Jiilich group1'2 the a (with a mass of fa 550 MeV) is viewed as arising from correlated 7T7T exchange. A rough estimate for the ratios of the cr-coupling strengths in the various channels can then be obtained from the relevant pion couplings. In practice, however, in the Jiilich YN models, which start from the Bonn TV TV potential, the coupling constants of the fictitious a meson at the strange vertices (AAa, EE<r) are free parameters — a rather unsatisfactory feature of the models.
These problems can be overcome by an explicit evaluation of a correlated 7T7T exchange in the various baryon-baryon channels. A corresponding calculation was already performed for the NN case in Ref. [10]. The starting point was a field theoretic model for both the TV TV —>• 7T7T Born amplitudes and 7T7T and KK elastic scattering11. With the help of unitarity and dispersion relations, the amplitude for the correlated XTT exchange in the NN interaction was computed, showing characteristic differences compared with the a and p exchange in the (full) Bonn potential.
In a recent study12 the Jiilich group presented a microscopic derivation of a correlated 7T7T exchange in various baryon-baryon (BB1) channels with strangeness S — 0 , - 1 and —2. The KK channel was treated on an equal footing with the TTTT channel in order to reliably determine the influence of KK correlations in the relevant t-channels. In this approach one can replace the phenomenological a and p exchanges in the Bonn NN13 and Jiilich YN1
models by correlated processes, and eliminate undetermined parameters such as the BB'a coupling constants. As a first application of the full model14 for the correlated TTTT and KK exchange, we present here new results for Y7V cross sections for various YN channels, and compare them with the available data.
Alternative approaches to describe baryon-baryon interactions using an effective field theory, based on chiral power counting schemes, have recently been applied to the TV TV interaction. However, at present a quantitative description of TV TV scattering within a consistent power counting scheme is still problematic15. Furthermore, it is not clear that such a description can be applied in the strangeness sector, where the expansion parameter, mx/m^, may no longer be small enough to allow an accurate low order truncation. In addition, contact terms, which parameterize the intermediate and short range interaction, do not fulfil any SU(3) relations, and cannot be fixed by currently available YTV data. At present, therefore, to obtain a quantitative description of Y TV scattering data over a large energy range one is forced towards a more traditional approach, such as that adopted here.
63
2.1. Potential from Correlated ivir + KK Exchange
Let us briefly describe the dynamical model10,12 for the correlated two-pion and two-kaon exchange in the baryon-baryon interaction, both in the scalar-isoscalar (er) and vector-isovector (/?) channels. The contribution of the correlated 7T7T and KK exchange is derived from the amplitudes for the transition of a baryon-antibaryon state {BB') to a -K-K or KK state in the pseudophysical region by applying dispersion theory and unitarity. For the BB' -»• TTX, KK amplitudes, a microscopic model is constructed, which is based on the hadron exchange picture.
The Born terms include contributions from baryon exchange as well as p-pole diagrams (see. Ref. [16]). The correlations between the two pseudoscalar mesons are taken into account by means of a coupled channel (TTTT, KK) model11 '16 generated from s- and ^-channel meson exchange Born terms. This model describes the empirical Kir phase shifts over a large energy range from threshold up to 1.3 GeV. The parameters of the BB' —> mr, KK model, which are interrelated through SU(3) symmetry, are determined by fitting to the quasiempirical NN' —> mr amplitudes in the pseudophysical region, t < 4ra2, obtained by analytic continuation of the empirical ITN and 7T7T data12.
From the BB' —>• TTTT helicity amplitudes one can calculate the corresponding spectral functions (see Ref. [12] for details), which are then inserted into dispersion integrals to obtain the (on-shell) baryon-baryon interaction in the <7 (0+) and p (1~) channels :
Note that the spectral functions characterize both the strength and range of the interaction. For the exchange of an infinitely narrow meson the spectral function becomes a ^-function at the appropriate mass.
2.2. Results and Discussion
As shown by Reuber et al.12, the strength of the correlated niv and KK in the a channel exchange decreases as the strangeness of the baryon-baryon channels becomes more negative. For example, in the hyperon-nucleon systems (AN, SAT) the scalar-isoscalar part of the correlated exchanges is about a factor of 2 weaker than in the NN channel, and, in particular, is also weaker than the phenomenological <r meson exchange used in the original Jiilich YN model1. Accordingly, we expect that the microscopic model with correlated 7T7T exchange will lead toaYiV interaction which is less attractive.
64
2Tp - > E°n
°Sactii-Zon)Mal-°Atox«nd*r«al. •Kadykeiai. 250
ZOO
3 " _£_ iso
100
so
\
-- ixrX.
o Engatmam et al. - - JuMch-BB — Juefch'01
T* 1 ^- -
100 2 0 0 3 0 0 4 0 0 5 0 0 0 0 0 7 0 0 9 0 0 P« (MeV/c)
100 120 140 100 ISO JUtMeV/c)
140 180 180 P»(MeV/c)
I"p - > E~p £*p -> E'p
~ ^ \ .
• ^
"ElHttll lL — Jurtrfi'80 — JuafctiTM
(UlMoV/c) PUfMeV/c)
Figure 1. Total KjV scattering cross sections as a function of laboratory momentum, Piab-The solid lines are results of the new YN model, based on correlated irir and KK exchange, while the dashed lines are results of the Jiilich YN model A 1 . The da ta are from Ref. [17].
Besides replacing the conventional a and p exchanges by correlated nn and KK exchange, there are in addition several new ingredients in the present YN model. First of all, we now take into account contributions from an an(980) exchange. The ao meson is present in the original Bonn NN potential13, and for consistency should also be included in the YN model. Secondly, we consider the exchange of a strange scalar meson, the K, with mass ~ 1000 MeV. Let us emphasize, however, that these particles are not viewed as being members of a scalar meson SU(3) multiplet, but rather as representations of strong meson-meson correlations in the scalar-isovector (nrj-KK)16 and scalar-isospin-1/2 {KK) channels, respectively. In principle, their contributions can also be evaluated along the lines of Ref. [12], however, for simplicity in the present model
65
they are effectively parameterized by one boson exchange diagrams with the appropriate quantum numbers. In any case, these phenomenological pieces are of rather short range, and do not modify the long range part of the YN interaction, which is determined solely by SU(6) constraints (for the pseudoscalar and vector mesons) and by correlated 7T7T and KK exchange.
In Fig. 1 we compare the integrated cross sections for the new YN potential (solid curves) with the YN —> Y'N scattering data as a function of the laboratory momentum, piab- The agreement between the predictions and the data17 is clearly excellent in all channels. Also shown are the predictions from the original Jiilich YN model A1 (dashed curves). The main qualitative differences between the two models appear in the Ap —>• Ap channel, for which the Jiilich model1 (with standard a and p exchange) predicts a broad shoulder at piab PH 350 MeV/c. This structure, which is not supported by the available experimental evidence, is due to a bound state in the x5o partial wave of the EJV channel. It is not present in the new model. The agreement in the other channels is equally good, if not better, for the new model. Further results and more details about the model will be presented in a forthcoming paper18.
3. Strangeness Production in NN Collisions
3.1. The Model
In a recent measurement of the reactions pp —i pAK+ and pp —• pE°A'+ near their thresholds it was found that the cross section for E° production is about a factor of 30 smaller than the one for A production7. We want to report on an exploratory investigation of the origin of this strong suppression of the near-threshold E° production19. In particular we want to examine a possible explanation that was suggested in Ref. [7], namely effects of the strong EJV final state interaction (FSI) leading to the EJV —>• AJV conversion. We treat the associated strangeness production in the standard distorted wave Born approximation. We assume that the strangeness production process is governed by the 7r- and K exchange mechanisms as depicted in Fig. 2. In order to have a solid basis for our study of possible conversion effects we employ a microscopic YN interaction model developed by the Jiilich group (specifically model A of Ref. [1]). This model is derived in the meson-exchange picture and takes into account the coupling between the AJV and EJV channels.
The vertex parameters (coupling constants, form factors) appearing at the JVJV7T and NYK vertices in the production diagrams are taken over from the Jiilich YN interaction. The elementary amplitudes TKN and T^jv-j-ify are taken from the corresponding microscopic models20,21 that were developed by our group. However, for simplicity reasons we use the scattering length and on-
66
K
Figure 2. Mechanisms for the reaction pp —> pYK (Y investigation: (a) kaon exchange; (b) pion exchange.
A, E) considered in the present
shell threshold amplitudes, respectively, instead of the full (off-shell) KN and TTN —• KY tf-matrices. The off-shell extrapolation of the amplitudes is done by multiplying those quantities with the same form factor that is used at the vertex where the exchanged meson is emitted. Only s-waves are considered.
We do not take into account the initial state interaction (ISI) between the protons. Therefore we expect an overestimation of the cross sections by a factor of around 3 in our calculation22. But since the thresholds for the A and E° production are relatively close together and the energy dependence of the NN interaction is relatively weak in this energy region, the ISI effects should be very similar for the two strangeness production channels and therefore should roughly drop out when ratios of the cross sections are taken.
3.2. Results and Discussion
The cross section ratio for the K exchange alone and based on the Born diagram is 16, (see Table 1). Including the YN FSI, i.e. possible conversion effects UN —• AN, leads to a strong enhancement of the cross section in the A channel but only to a moderate enhancement in the S° channel. As a consequence, the resulting cross section ratio becomes significantly larger than the value obtained from the Born term and, in fact, exceeds the experimental value. In case of the pion exchange, the Born diagram yields a cross section ratio of 0.9. Adding the FSI increases the cross section ratio somewhat, but it remains far below the experiment.
Thus, it is clear that, in principle, the K exchange alone could explain the cross section ratio - especially after inclusion of FSI effects. However, we also see from Table 1 that the n exchange is possibly the dominant production mechanism for the E° channel and therefore it cannot be neglected. Indeed, the two production mechanisms play quite different roles in the two reactions under consideration. The K exchange yields by far the dominant contribution for PP ~• pAK+. The influence of ir exchange is very small. In case of the reaction
67
pp ->• pT,°K+, however, the TT- and K exchange give rise to contributions of comparable magnitude. This feature becomes very important when we now add the two contributions coherently and consider different choices for the relative sign between the TT and K exchange amplitudes. In one case (indicated by "K + TT" in Table 1) the TT and K exchange contributions add up constructively for pp —> pTPK+ and the resulting total cross section is significantly larger than the individual results. For the other choice (indicated by "K — TT") we get a destructive interference between the amplitudes yielding a total cross section that is much smaller. Consequently, in the latter case the cross section ratio is much larger and, as a matter of fact, in rough agreement with the experiment (see Table 1) - suggesting a destructive interference between the TT and K exchange contributions as a possible explanation for the observed suppression of near-threshold S° production.
Table 1. Total cross section' of the reactions pp —¥ pAK+ (<rA) and pp —*• pE°A"+ (ego) at the excess energies Q = 13.2 MeV (<TA) and Q — 13.0 MeV (<TJ;O).
diagrams K (Born) K (FSI)
•K (Born) 7T (FSI)
"K ' + TT" (FSI) "K - TT" (FSI)
exp.7
<rA [nb] 739
2426 71 113
2471 2607
505± 33
o-j-o [nb] 46 57 77 105 251 73
20.1±3.0
^ A / < 7 £ 0
16 43 0.9 1.1 9.9 36
25±6
In Fig. 3 we show the total cross sections as a function of the excess energy. It is clear from Table 1 that our model calculations overestimate the absolute values of the empirical cross sections by roughly a factor 4. This is not too surprising because, as already mentioned earlier, effects from the ISI are not taken into account. Therefore, we re-normalized all results by a factor19 of 0.3 to compensate for the ISI effects22. Then it can be easily seen that the model calculations yield an energy dependence that is in rather nice agreement with the experiment. This suggests that, like in the case of pion production, the energy dependence of the cross section is primarily influenced by the FSI between the baryons and that effects from other possible FSI (in the KN and/or KY systems) might play a minor role.
It is now interesting to look also at the corresponding results for other S production channels. For instance, for the reaction pp -)• n S + A ' + the predicted cross sections at the excess energy of 13 MeV are 86 ("K + TT") and
68
5 10 excess energy [MeV]
5 10 excess energy [MeV]
Figure 3. Total cross sections for the reactions pp -4- pAA'+ and pp —• pE°A'+. The solid curve corresponds to the choice "K — 7r" and the dashed curve to "K + 7r", cf. text. All curves are re-normalized by a factor of 0.3. The experimental data are from Ref. [7].
229 nb ("K — 7r"), respectively. Thus, the interference pattern is just the opposite as for pp -> pT,°K+, (see Table 1). For the "K — n" case favoured by the experimental O"A/<TSO ratio, our calculation yields a cross section for pp —> n S + A ' + that is about 3 times larger than the one for pp —» pYpK^. Such a ratio is in fair agreement with the data and model calculations at higher energies, see, e.g. Ref. [23]. The other choice, "K + ir", leads to a <Tpp^n'£+K+ that is about a factor of about 3 smaller than (7pp-ypT;0K+ ~ a
result which is rather difficult to reconcile with the present knowledge about these reactions at higher energies. These features are displayed graphically in Fig. 4 as a function of the excess energy.
%
TOO
10
1
0
pp-»nZ*K*
^•"""T---"— /^.--'''
/''' K+it // K-it
•E %
100
10
1
pp->pxV „---
/'\^^^ //^
/ / — *-* 1/ K+Jt
I 1
0 5 10 15 excess energy [MeV]
0 5 10 15 excess energy [MeV]
Figure 4. Total cross sections for the reactions pp -4 n£+ A"+ and pp —• p E + K°. The solid curve corresponds to the choice UK — n" and the dashed curve to "K + 7r". All curves are renormalized by a factor of 0.3.
For the reaction pp —¥ pE+A'° the predicted cross sections at the excess
69
energy of 13 MeV are 725 ("A+TT") and 423 nb ( "A ' -TT") , respectively. Thus, in
this case the interference pattern is the same as for pp —>• pS°K + . Furthermore,
for either choice ("K±ir"), the cross section for pp ->• pS+ A 0 is about 3.3 times
larger than the one for pp -> pS°A'+ . Note that the experimental evidence at
higher energies suggests a ratio of around unity for those channels.
Acknowledgments
I want to thank A. Gasparian, C. Hanhart, W. Melnitchouk, and J. Speth
for collaboration on the topics presented here. Furthermore, I would like to
acknowledge the financial support by the Grant No. 477 AUS-113/14/0 from
the Deutsche Forschungsgemeinschaft.
References
1. B. Holzenkamp, K. Holinde and J. Speth, Nucl. Phys. A500, 485 (1989). 2. A. Reuber, K. Holinde and J. Speth, Nucl. Phys. A570, 543 (1994). 3. P.M.M. Maessen, T.A. Rijken and J.J. de Swart, Phys. Rev. C40, 2226 (1989). 4. Th.A. Rijken V.G.J. Stoks and Y. Yamamoto, Phys. Rev. C59, 21 (1999). 5. J.T. Balewski et al., Phys. Lett. B420, 211 (1998). 6. R. Bilger et al., Phys. Lett. B420, 217 (1998). 7. S. Sewerin et al., Phys. Rev. Lett. 83, 682 (1999). 8. S. Marcello et al., Nucl. Phys. A691, 344c (2001). 9. U. Straub et al., Nucl. Phys. A483, 686 (1988); Nucl. Phys. A508, 385c (1990). 10. H.-C. Kim, J.W. Durso and K. Holinde, Phys. Rev. C49, 2355 (1994). 11. D. Lohse, J.W. Durso, K. Holinde and J. Speth, Nucl. Phys. A516, 513 (1990). 12. A. Reuber, K. Holinde, H.-C. Kim and J. Speth, Nucl. Phys. A608, 243 (1996). 13. R. Machleidt, K. Holinde and Ch. Elster, Phys. Rep. 149, 1 (1987). 14. J. Haidenbauer, W. Melnitchouk and J. Speth, Nucl. Phys. A663, 549 (2000). 15. D.R. Entem and R. Machleidt, Phys. Lett. B524, 93 (2002). 16. G. Janssen, B.C. Pearce, K. Holinde, and J. Speth, Phys. Rev. D52, 2690 (1995). 17. G. Alexander et al., Phys. Rev. 173, 1452 (1968); B. Sechi-Zorn et al., Phys. Rev.
175, 1735 (1968); F. Eisele et al., Phys. Lett. 37B, 204 (1971); R. Engelmann et al., Phys. Lett. 21, 587 (1966).
18. J. Haidenbauer, W. Melnitchouk and J. Speth, in preparation. 19. A.M. Gasparian et al., Phys. Lett. B480, 273 (2000). 20. M. Hoffmann et al., Nucl. Phys. A593, 341 (1995). 21. M. Hoffmann, Jiilich report, No. 3238 (1996). 22. M. Batinic et al., Phys. Scripta 56, 321 (1997). 23. J.M. Laget, Phys. Lett. B259, 24 (1991).
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3. Neutrino Oscillations and Interactions
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IMPLICATIONS OF THE JHF-KAMIOKA NEUTRINO OSCILLATION EXPERIMENT
R. R. VOLKAS School of Physics, Research Centre for High Energy Physics
The University of Melbourne, VIC. 3010, Australia E-mail: [email protected]
After briefly reviewing the existing evidence for neutrino oscillations, I summarise the goals and capabilities of the JHF-Kamioka long baseline superbeam experiment. Theoretical implications of what this experiment could potentially discover are then discussed.
1. Introduction
Neutrino oscillations arise when there is a mismatch between the neutrino states produced in weak interaction processes ("weak eigenstates") and the Hamiltonian eigenstates ("mass eigenstates"). The three known weak eigenstates are the familiar ve,u,T flavours. If neutrinos have non-degenerate masses, then the neutrino mixing matrix U defined through
VOL = ] P UaiVi, (1) i = l,2,3
gives rise to non-trivial effects including oscillations. In this equation, a = e,n,T and V{ is the state of definite mass m*. The complex numbers Uai constitute the mixing matrix. Additional light neutral fermions usually known as "sterile neutrinos i/s" may also exist. If so, then Eq. (1) must be generalised in the obvious way. For the three flavour case, the mixing matrix can be parameterised in terms of three physical mixing angles and some CP violating phases (the precise number of which depends on whether the neutrino masses are of Dirac or Majorana form). If light sterile neutrinos exist, then there are additional mass and mixing parameters.
Oscillations arise due to relative phases between the i/,- induced by time evolution. Considering two flavours only for simplicity, the mixing pattern
va = cos Oi/i + sin 9v2
up = — sin Bv\ + cos 0i/2 , (2)
73
74
implies the transition probability
P(va-H'p)= sin2 20 sin2 - ^ - . (3)
after the state which begins life as a va propagates through a distance L. The mixing angle sets the magnitude of the oscillations, and Am? /E determines the oscillation length, where E is energy and Am2 = m\ — m\. It is straightforward to generalise this formula to multiflavour cases. Nature chooses the Am2 and 6 parameters, while experimentalists have some control over E and L. This partial freedom is utilised in the design of the JHF-Kamioka experiment that is the focus of this talk1.
Extremely convincing evidence for the disappearance of muon-neutrinos has been provided by SuperKamiokande and other experiments through observations of atmospheric neutrinos2. The upper atmosphere acts as a beam dump for cosmic rays, with ve^ and their antiparticles produced as byproducts. The zenith angle pattern of the contained //-like events reveals a clear deficit of up-going relative to down-going progenitor v^s, while the e-like events show no anomalous angular dependence. These data are consistent with v^ —*• vx oscillation, where x ^ e. Doing a detailed fit assuming either v^ -4 vr or v^ —¥ vs
produces allowed regions which can be roughly described as sin2 26 > 0.85 and 1 0 - 3 < Am2 /eV2 < 8 x 10 - 3 . Other aspects of the atmospheric neutrino data show a preference for v^ —> vT over v^ -4- vs, the statistical significance of which has been under dispute4. JHF-Kamioka and the other long baseline experiments have been designed to reproduce the atmospheric neutrino effect in a terrestrial context where the neutrino source as well as the detector are under experimental control. Indeed, the pioneering K2K long baseline experiment has already reported a v^ deficit roughly consistent with the atmospheric effect3.
All solar neutrino experiments have revealed a deficit by a factor of 2 — 3 in the vt flux relative to standard solar model expectations5'6. The new data from SNO provide strong evidence that solar ue's oscillate into other active flavours en route to the Earth6. The spectrally undistorted nature of the 8B neutrino flux, when combined with the strong deficit factor, limits the oscillation parameter space to sin2 16 ~ 0.7. It is interesting that both the solar and atmospheric mixing angles are large, quite unlike their quark analogues. The solar Ara2 is constrained to be at least an order of magnitude smaller than its atmospheric counterpart.
The LSND experiment7 has provided fully terrestrial evidence for V^ -+ Ve
oscillations, with a small mixing angle and a relatively large Am2 of about 1 eV2. This as yet uncorroborated but fascinating result will soon be checked by MiniBooNE. Following a common practice that I do not condone, I will
75
sometimes "bury my head in the sand" throughout this paper by assuming that the LSND anomaly8 is not due to oscillations. If all three anomalies are due to oscillations, then the incommensurate Am2 values imply that at least one additional flavour, necessarily sterile, must exist.
So, in summary, with head in the sand: the atmospheric and solar anomalies imply that two out of the three mixing angles in U are large (and at least the atmospheric one can even be maximal). These two angles are usually denoted #12 and 623. The third mixing angle, #13, is constrained to be small through neutrino disappearance bounds, and we have no constraints on the CP violating phase <P. Coming clean with LSND forces us to also confront the possible existence of vs flavours and additional parameters.
In light of the above, the scientific goals of JHF-Kamioka are well motivated. The main ones are :
• precision measurement of the atmospheric neutrino oscillation parameters;
• discrimination between v^ —»• vT and v^ —• vs; • search for 1/̂ —>• ve; • search for CP violation in the lepton sector.
The first three of these goals can happen during phase 1 of the project, while the fourth will have to wait for phase 2.
2. The Capabilities of JHF-Kamioka
The JHF-Kamioka project envisages a high flux, narrow band v^ or v^ beam with peak energy in the few-GeV regime being directed from the Japan Hadron Facility to the Kamioka laboratory located about 295 km away. Contamination due to ue will be reduced by having a relatively short decay volume for the muons produced by pion decay. Various types of beams will be possible, depending on momentum selection of the parent pions and the choice of beam direction (off axis or well-directed). The peak energy will be tunable and the beam spectrum well known, enhancing sensitivity to oscillation effects. The far detector in phase 1 will be the existing Super-Kamiokande 50 kt water Cerenkov detector. Phase 2 envisages an increase in beam power and the construction of a 1 Mt water Cerenkov "Hyper-Kamiokande" as a second far detector. Plots of typical beam spectra including flavour composition are available from the collaboration's Letter of Intent1.
aNote that the additional phases of the Majorana case do not affect oscillation probabilities.
76
2.1 . Precision Measurement of "Atmospheric" Oscillation Parameters
Figure 1 depicts the expected precision for measurements of the oscillation parameters relevant for solving the atmospheric neutrino problem15. The sensitivity depends on the type of beam used and on the actual values of the parameters. Precision measurements down to about 0.01 in sin216 and few x 10 - 5
in Ara2/eV2 are envisaged.
0.002 0.004 0.006 , Am2 (eV2)
<5sin!(2i9)
0.002 0.004 0.0,06 , Am2 (eV2)
6Am2
Figure 1. Sensitivity of the atmospheric oscillation parameters for the case sin2 26 — 0.9 (dashed line) compared to the case sin2 26 = 1 (solid line), as a function of the true A m 2 . The beam choice has been optimised for a true Am 2 of 3 X 10—3 eV2 in this illustration. See the LOI for further details.
' M 2.2. Discrimination of u^ —*• vT and u^
This relies on the observation of neutral current (NC) induced single pion production in the far detector. The 7r° events will be the most useful, because of the relatively clean nature of the 77 decay mode. Figure 2 compares the expected rates for the vT and us cases, with a clear suppression evident in the latter for Am2 > 1 - 2 x 1 0 - 3 eV2.
2.3. Search for v^ —¥ ve and CP Violation
MiniBooNE will confirm or disconfirm the v^ —»• Fe interpretation of the LSND anomaly. Assuming disconfirmation, the existence of such an oscillation mode
bThese and all subsequent figures come from the LOI. '
77
again becomes an open question. In the three neutrino picture, the v^ —> ve
transition probability is proportional to the small parameter sin2 2#i3. Figure 3 displays the expected sensitivity for ve appearance in JHF-Kamioka, with the region already excluded by CHOOZ and Palo Verde superimposed9. The angle #13 must be sufficiently large for CP violation effects to be observable in oscillation experiments.
Figure 2. Comparison of single 7r° production for the i/^ -> vT and u^ —t u3 atmospheric neutrino channels. The plots assume maximal mixing, with the three panels corresponding to different possible beams. Discrimination can be achieved in the Am 2 range of interest for resolving the atmospheric anomaly.
3. Theoretical Implications
I will now briefly discuss possible theoretical ramifications of the type of information JHF-Kamioka could provide.
3 .1 . ug or no ug
Particles, arranged into multiplets, form the raw ingredients for spontaneously broken gauge theories such as the standard model. A very basic activity in theoretical particle physics is to understand how the standard model Lagrangian might emerge in a effective sense out of a more fundamental theory. To pursue these studies, we really need to know what the fundamental low-mass degrees of freedom are. The possible existence of light sterile neutrinos is therefore a very interesting loose end from the theoretical perspective as well as the phenomenological.
One of the famous issues arising from the standard model is the flavour problem: can the values of the quark and lepton mass and mixing angle parameters, and the family structure, be understood through a standard model extension? Neutrinos could well provide very important clues, because of the
78
-1 10
i -2
10
-3 10
-4 1 0 * -a -1
10 10 * * | . 1
Figure 3. Expected reach of the fM -*• j / e oscillation search via pe appearance after 5 years of running. The three contours correspond to different beam choices.
contrast they provide to the other fermions. Neutrinos are unusually light, and the large vacuum mixing angles required look qualitatively very different to the small Kobayashi-Maskawa mixing angles of the quark sector. But before we can properly reflect on how they might help resolve (or deepen!) the flavour puzzle, we need to know exactly how many neutrino-like degrees of freedom exist. The discovery of sterile neutrinos would be roughly as important as the discoveries of c, r and 6 in the 1970's.
In the near future, we await results from MiniBooNE. While this experiment is very important for sterile neutrino research, it can only provide indirect evidence for their existence. Irrespective of what is found by MiniBooNE, the ability of experiments such as JHF-Kamioka to perform neutral current measurements and thus potentially discover sterile neutrinos directly is very welcome. SNO has of course recently provided strong constraints on the sterile neutrino component of the solar neutrino flux6.
A famous theoretical problem posed by light sterile neutrinos is: Why are they light? The most sterile of possible sterile neutrino candidates are fermions with the gauge quantum numbers of the vacuum10. Such states obviously have gauge invariant Majorana mass terms, and there is no a priori reason to expect them to be of similar magnitude to the active neutrino masses. In fact they can
90% C.L. sensitivities
79
be arbitrarily large. Mirror symmetry has been proposed to explain both why
apparently sterile states exist and why they are light1 1 '1 2 . If the mirror mat te r
idea is correct, then sterile neutrinos would be just the t ip of iceberg, because
mirror partners would be expected for all known particles. The ramifications
of this would obviously be enormous.
3.2. Precision Measurements of the Atmospheric Parameters
As well as performing a degree of freedom audit, we of course also need as
much information as possible on the precise values of mass and mixing angle
parameters . We can dream tha t one day a predictive theory for flavour will
emerge, and an important test will be a direct comparison of those predictions
with measured neutrino parameters. In the meantime, we should try to at
least correlate aspects of the neutrino flavour problem with new theoretical
principles.
The existence of large neutrino mixing angles is thought provoking. The
mirror symmetry idea allows two-flavour active-sterile maximal mixing to be
understood on the basis of a simple theoretical principle. We had hoped tha t
the solar and atmospheric neutrino problems could be solved in a unified way
through the maximal oscillations of ve 's and v^ 's into their respective mirror
(sterile) par tners 1 1 . Alas, the SNO results appear to have ruled out the solar
neutrino part of this hypothesis (they imply an upper bound on the j/e~mirror-
ve A m 2 parameter) . As discussed above, it is important for experiments such
as JHF-Kamioka to check the claim from Super-Kamiokande that the a tmo
spheric mode is predominantly into vT13.
But there is also the question of the atmospheric mixing angle: is it maxi
mal or merely large? A precision measurement of sin 29 at the 0.01 level has
the potential to rule out exact maximal mixing, or point more strongly towards
it. This is important theoretically, because exact maximal mixing is a special
point in parameter space. The atmospheric neutrino da ta have always pre
ferred true maximal mixing, though it is possible tha t the actual value is, say,
sin2 29 = 0.93. If so, then JHF-Kamioka should be able to rule out maximal
mixing to a high level of statistical significance.
Maximal mixing would point to an underlying new symmetry of nature. If
the mode is u^ —> vs, then mirror symmetry would be the prime candidate. But
we should in general endeavour to discover new symmetry principles through
neutrino oscillation physics. It is perhaps useful to categorise such a t t empts
according to whether the symmetry is exact (e.g. the Melbourne version of mir
ror symmetry) , spontaneously broken (e.g. broken mirror symmetry, horizontal
symmetry) or approximate (e.g. Le±Lll—LT). There is historical precedent for
the first and third possibilities (e.g. colour and electromagnetic gauge invari-
80
ance, and Gell-Mann-Neeman SU(3), respectively), while the second awaits discovery of the Higgs boson.
3.3. Ufj, —> ue Search and CP Violation
Let us assume that LSND has not already discovered the anti-particle version of this oscillation mode. Then the connection between #13 and the existence of CP violation indicates that the former in a sense quantifies the extent to which the neutrino mixing is "truly three-flavour". This is an important part of the flavour puzzle.
Observing CP violation in the lepton sector would allow comparison with similar effects in the quark sector, with information about the latter on the rise because of the 5-factory experiments. The neutrino sector already displays a difference from the quark sector through its large mixing angles. How will CP violation compare, and what implications will that have on theories of quark-lepton symmetry?
CP violation is of course important in theories of baryogenesis. While its establishment in neutrino oscillations would have no direct consequence for baryogenesis, it would show that matter-antimatter asymmetry is not confined to strongly interacting particles. Baryogenesis can proceed through the sphaleron reprocessing of a lepton asymmetry created, for example, from out-of-equilibrium and CP violating decays of "heavy neutral leptons"14. The latter (hypothetical) species are neutrino-like, but very massive (perhaps they are the heavy gauge singlets needed for the see-saw mechanism). Unfortunately, the CP violating parameters in the heavy neutral fermion sector need not be related to those in the light neutral fermion (i.e. neutrino) sector. While these interconnections are not mandatory, one can hope for relations within specific and predictive standard model extensions. We are a long way from having such a theory, but all the experimental information we can get will help. Switching perspective, plausible theoretical proposals for connecting the neutrino sector parameters to baryogenesis would be welcomed by experimentalists as a spur to their leptonic ambitions. The possible existence of Majorana phases in addition to Dirac phase(s) is an important consideration.
4. Conclusion
Long baseline superbeam experiments such as JHF-Kamioka promise to supply very important new information about the neutrino sector, from precision measurements of parameters through to possible discovery of sterile neutrinos and/or CP violation. These are of great importance in the quest to understand the flavour problem.
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While not discussed fully in this talk, these results will be of great relevance
for astrophysics and cosmology as well as for particle physics, especially if light
sterile neutrinos are discovered15. The discovery of leptonic CP violation would
also be (indirectly) important for the baryogenesis puzzle.
Acknowledgments
I would like to thank Tony Thomas for inviting me to this workshop and to
the Special Research Centre for the Subatomic Structure of Matter at the Uni
versity of Adelaide for partial financial support. This work was also partially
supported by the University of Melbourne. I would also like to thank the par
ticipants in the neutrino stream at the recent WIN meeting in Christchurch
for stimulating some of the thoughts expressed during this talk.
References
1. Y. Itow et al., hep-ex/0106019. 2. Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 82, 1562
(1998); Phys. Lett. B436, 33 (1998); Phys. Lett. B433, 9 (1998); Soudan 2 Collaboration, W. W. Allison et al., Phys. Lett. B449, 137 (1999).
3. K2K Collaboration, S. H. Ahn et al., Phys. Lett. B511, 178 (2001). 4. Super-Kamiokande Collaboration, S. Fukuda et al., Phys. Rev. Lett. 85, 3999
(2000); R. Foot, Phys. Lett. B496, 169 (2000). 5. Homestake Collaboration, B. T. Cleveland et al., Astrophys. J. 496, 505 (1998);
Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 77, 1683 (1996); Super-Kamiokande Collaboration, Phys. Rev. Lett. 86, 5651 (2001); Sage Collaboration, J. N. Abdurashitov, et al., Phys. Rev. Lett. 83, 4686 (1999); Gallex Collaboration, W. Hampel et al., Phys. Lett. B447, 127 (1999); GNO Collaboration, M. Altann et al., Phys. Lett. B490, 16 (2000).
6. SNO Collaboration, Q. R. Ahmad et al., nucl-ex/0204008; nucl-ex/0204009; Phys. Rev. Lett. 87, 071301 (2001).
7. LSND Collaboration, C. Athanassapoulos et al., Phys. Rev. Lett. 81, 1774 (1998); Phys. Rev. C58, 2489 (1998).
8. As far as I know, this terminology was proposed by A. de Rujula. 9. CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B466, 415 (1999); Palo
Verde Collaboration, F. Boehm et al., Nucl. Phys. Proc. Suppl. 91, 91 (2001). 10. For a pedagogical introduction to sterile neutrino theories see R. R. Volkas, hep-
ph/011326, Prog. Part. Nucl. Phys. (in press). 11. R. Foot, H. Lew and R. R. Volkas, Mod. Phys. Lett. A7, 2567 (1992); R. Foot,
Mod. Phys. Lett. A9, 169 (1994); R. Foot and R. R. Volkas, Phys. Rev. D52, 6595 (1995).
12. Z. G. Berezhiani and R. N. Mohapatra, Phys. Rev. D52, 6607 (1995). 13. R. Foot and R. R. Volkas, hep-ph/0204265. 14. M. Fukugita and T. Yanagida, Phys. Lett. B174, 45 (1986). 15. For reviews on neutrino cosmology see, M. Prakash, J. M. Lattimer, R. F. Sawyer
and R. R. Volkas, Ann. Rev. Nucl. Part. Set. 51, 295 (2001); A. D. Dolgov, hep-
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ph/0202122; for recent interesting work on neutrino oscillations and cosmology see A. D. Dolgov et a l , hep-ph/0201287; Y. Y. Y. Wong, hep-ph/0203180; K. N. Abazajian, J. F. Beacom and N. F. Bell, astro-ph/0203442.
SOME IMPLICATIONS OF THE NUTEV ANOMALY
B. H. J. M C K E L L A R AND M. G A R B U T T
School of Physics, University of Melbourne, Victoria, Australia. 3010 E-mail:[email protected], [email protected]
G. J. STEPHENSON, JR. Dept. of Physics & Astronomy, University of New Mexico,
Albuquerque, NM 87131 USA E-mail: [email protected]
T. G O L D M A N
MS B283, Theoretical Division, Los Alamos National Laboratory Los Alamos, New Mexico 87545 USA E-mail: [email protected]
In this paper we disscussed the NUTeV anomaly and its implications for neutrino oscillations and new physics.
1. Introduction
In October 2001, the NUTeV collaboration1 published results on an analysis of the scattering of neutrinos and antineutrinos leading to the claim that the value of the Weinberg angle extracted from the data is about 3 a above the world average value. This raises the interesting question that there may be a hint of new physics here, as the neutrino energy, averaging about 60 GeV, but extending above 100 GeV is larger than analysed previously.
In trying to decide on the presence or absence of new physics in the result, it is of course important to question the experimental validity of the result — a task we are not competent to perform, and to question whether some Standard Model Physics, ignored in the analysis, could produce the effects, and only then to speculated about possible new physics effects. In this paper we do the last two of these.
One of the reasons for choosing this subject is of course its topicality. But, as NuTeV is not taking any more data, it is worth considering whether or not there are opportunities for experiments to be done at the JHF neutrino beam to check the NuTeV result, albeit at lower energies. We leave this as a
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challenge to our experimental colleagues. Some papers which became available after the workshop have also been
used as input for this written version of the talk.
2. The NuTeV Experiment
The key ingredient in the experiment is the determination of the ratio of neutral current to charged current events induced by the neutrino beam, R^xp, and by the antineutrino beam, R^xp- The subscript "exp" is a reminder that these ratios are derived directly from the data, and no correction is applied to it to extract values of Rv = ^ N C / ^ C C
an<^ ^ = ° N C / ° C C which can be
compared with one's calculations. Instead of the Standard Model predictions, QCD models are used as input to a Monte Carlo calculation of the R^p. The results are given in Table 1.
Table 1. The observed and expected ratios of charged current to neutral current events.
exp ratios observed SM prediction
v 0.3918 0.3950
v 0.4050 0.4066
The value of sin 9w determined from a fit to the data is 3<r above the value determined from other experiments. Explicitly the NuTeV result is
sin2 6W (on_she l l ) = 0.2277 ± 0.0013(stat) ± 0.0009(syst)
Leading terms in the one-loop electroweak radiative corrections produce the small residual dependence of the result2 on M t o p and Mniggs- The value of sin2 6w determined by a fit to other electroweak measurements3 is 0.2227 ± 0.0004.
Equivalently, the results may be quoted in terms of the value of rriyy, determined from the "on-shell renormalisation condition" sin2 0w = 1 — ™JV / m z • The NuTeV value is 80.136 ± 0.084, about 2.8<r below the value 80.376 ± 0.023 determined by other experiments.
The neutrino beams are well defined. There is about a 3 x 10 - 4 component of v in the v beam and a 4 x 10 - 3 component of v in the v beam. The electron neutrino component of the beam is at the 1.6% level.
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The target and detector are combined, and consist of 168 3m x 3m x 5.1cm iron plates, interleaved with 84 liquid scintillation counters, and 42 drift chambers.
A feature of the analysis which is worth noting is the technique used for distinguishing charged current and neutral current events. Almost all of the charged current events will involve fast muons, which will traverse many of the steel plates, while neutral current events will not involve any penetrating charged leptons, and will not travel through many of the plates. Neutral current events are thus identified as "short events", and charged current events as "long events". Even without oscillations, there are some electron neutrinos in the beam, and these generate charged current events in which electrons are produced. These ve CC events will be short events and need to be handled in the Monte Carlo extraction of ratios of cross sections.
Many corrections need to be applied to these ratios to obtain the ratios to which one should compare calculations. While this is not the procedure adopted by the NuTeV group, it is useful to do it here to get a feeling for the effects of modifications of the analysis of the results, without having to carry out a full Monte Carlo analysis. The major corrections, together with the total correction are in Table 2, leading to the corrected values
/T = 0.3086 HF = 0.3453, (2)
which may be compared to theoretical calculations.
Table 2. Corrections to the experimental ratios. R = Rexp + SR.
Correction
CR background
Short CC events
Total correction
SR"
-0.0036
-0.068
-0.021
-0.0830
SR"
-0.019
-0.086
-0.024
-0.0597
3. Under s t and ing R
In the tree graph approximation, the effective Lagrangian is
£eff = -2V2GF {[0^avL] [dLlauL] + h.c.) (3)
- 2 V 2 G F ^ Yl 3Ag [vulav^] [qAlaiA • (4) A=L,R q=u,d
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Assuming isospin symmetry we introduce
9L = 9LU + 9ld a n d 9R = 9RU + 9Rd > (5)
and find, in lowest order, following the standard analysis4
Ru = (39l+9R)q+(ZgR+9l)q ( )
3? + q [ ' & _ (gi + 3g^)g + (ff| + 3gi)g R ~ 7^9 — (7)
Here q and q are the fraction of the momentum carried by quarks and anti-quarks, respectively, and isospin symmetry is assumed, so that q — (u + d)/2. Introducing the parameter
r = ^ c c / ^ c c = (q + 3?)/(3<z + q), (8)
the ratios may be rewritten as
IT=gL + rg% and R° = g\ + r " 1 ^ . (9)
The data then give
^L.NuTeV = 0-3005 ± 0.0014 cf g2LWA = 0.3042
5fl,NuTeV = 0.0310 ±0.0011 cf 5i,wA = 0.0301. (10)
To explain the data we must decrease g\, and possibly increase gR, although the latter is consistent with the world average (WA) value. The key question whether it is old physics or new physics which is giving the discrepancy. A comprehensive analysis has been given by Davidson et a/.4
To see whether this simplified analysis is reasonable we can determine the Weinberg angle from the coupling constants in Eq. (10), which gives s i n 2 ^ = 0.2277±0.0016, and is identical in the central value to Eq. (1), showing that this "theorists" way of thinking about the data is consistent.
4. Old Physics
4 .1 . Standard Model Radiative Corrections
Standard model radiative corrections are included in the NuTeV analysis, where they give rise to the dependence of the extracted value of the Weinberg angle on the masses to the top quark and the Higgs. These corrections enter the coupling constants by
9L = p2 ($ - sin2 0W + S. sm4ewk2) and gR = %p2 sin4 6wk2, (11)
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with
, = 1.0087+0.000. ( i ^ j - - , ) - 0.0006b ( j ^ )
* = L0035+M004 ( irikv " 0 - M0291" (iSKv) • <12> as long as the values of the top and Higgs masses are not too far from 175 GeV, and 100 GeV, respectively.
These corrections are well tested by the mass of LEP data, and by the fact that the observed top quark mass is in good agreement with that implied by the radiative corrections to the LEP data. The Higgs mass is not so well defined, and indeed the LEP data on lepton asymmetries show almost 3<r discrepancies in the couplings from the overall best fit. The lepton asymmetries prefer a light Higgs, and the hadron asymmetries prefer a heavy Higgs4. As increasing the Higgs mass moves g\ in the right direction, it is tempting to seize on this. But even using the Higgs mass favoured by the hadron asymmetry data does not remove the discrepancy.
4.2. QCD Corrections
The parton distribution functions enter the cross section ratios via the parameter r of Eq. (8). Paschos and Wolfenstein5 noted that the ratio
P _ °NC ~ ^NC _ R" ~ r^'/ _ .2 „2 _ j - 2 a c\ i\ -"PW
= ~ ~ - —i = 3L ~ 9R - 3 ~ sin 0W , (M)
""cc ''"cc i — r
is independent of the parton distribution functions. While the Paschos-Wolfenstein ratio is not directly measured in the NuTeV experiment, it is useful as a framework to discuss the possible QCD corrections.
Davidson et a/.4 give the relation
RPW =g\-sk+ (M~~<nQ
+-(C~~S~) {[lArf + iArf]
+ %(d- al) (̂ C-x - SCs)} + o (j^y) - (14)
which incorporates corrections due to isospin breaking (u~ — d~), sea quarks (q~ = q — q, and c~ —s~), and the next-to-leading order (SCi). The parameter is defined by Q~ = (w + d~)/2. The next-to-leading order corrections appear as corrections to the already small isospin breaking and sea quark effects, and will be neglected in the present analysis.
A conservative overestimate of the isospin breaking effects, remembering that the fact that iron has more neutrons than protons has already been in-
88
eluded in the analysis, is that the isospin correction is uncertain by an amount
s (^F~) - 00L (15)
The maximum uncertainty would lead to a theoretical uncertainty in sin 6\y of 0.002, reducing the significance of the discrepancy to 1.9<r — perhaps to the level where it should no longer concern us.
Sea quark effects have been the subject of some controversy. The estimate that c~ is negligible is well established, but estimates of s~ vary4. The large value of s~ fts 0.002 would lower sin2 8w by 0.0026, reducing the discrepancy to about 1.5<7, and making it insignificant. Zeller et al.e, however, assert that the violation of (s(x)) — (s(:c)} increases the discrepancy with the standard model.
Recently, Miller and Thomas7, have pointed out that, since a significant fraction of the NuTeV data are at low x (x < 0.1), shadowing in the nuclear target must be taken into account in the analysis. An analysis based on vector meson dominance suggests that the effect of shadowing will be to increase the neutral current cross sections relative to the charged current cross sections, and that the effect will be larger for antineutrinos than for neutrinos. If Rv
is increased by a factor (1 + e), and R" by a factor (1 + e), then the effect of shadowing is to change the Paschos-Wolfenstein ratio from the nucleon value iipw to the nuclear value RA,PW-
RA,PW = RPW ( l + e - ^ — ( e - e ) V (16)
To remove the discrepancy requires e — 2e & 0.03, a value not inconsistent with the estimates in other circumstances.
Thus there are three old physics effects which could, alone or in combination, reduce the NuTeV discrepancy to values where it is no longer significant :
(1) isospin breaking effects, (2) effects of strange quarks in the sea, (3) nuclear shadowing effects.
They are all deserving of future study, to the extent that they should be incorporated in the Monte Carlo analysis of the experiment.
5. New Physics
5.1. Oblique Physics
Heavy particles which do not couple directly to the ordinary fermions give rise to effects which are called oblique physics. These couplings can be parame-
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terised in various ways, for example by the S,T and U parameters8. They are so tightly constrained by the LEP data that they cannot contribute significantly to the NuTeV discrepancy4.
5.2. Anomalous Z Couplings
The NuTeV paper noted that a shift in the v^ — Z coupling could resolve the discrepancy. The fractional shift required is Sgl z/gl z — —(1.16 ± 0.42)%. As the Z width to vv is (0.53 ± 0.08)% less than the best fit, there may appear to be room for this explanation. However, it is very difficult to create models in which a variation of gv^z does not also vary the coupling gV\iW, which is severely constrained by fi, 7r and r decays. Within the SU(2)x U(l) electroweak symmetry group, these data require that Sgl z/g1 z\ < 0.04%. Either new interactions are implied or this is not an explanation of the discrepancy. However, the other constraints on new interactions make many of the candidates unlikely possibilities.
5.3. Mixing with a Z'
One way to vary the Z — v — v coupling is to introduce an additional neutral vector boson, the Z', which mixes with the Z, with the mixing angle 8, which has a mass mz>, and couples to a current Jz>. To lowest order in the mixing angle and the mass ratio mz/mz', this is equivalent to a shift in the normal Z mass and current;
m z ^mz,SM-e mZ' a n d JZ^JZ,SM-0JZ', (17)
and a new interaction —(1/2) \jfr) , where coupling constants have been
absorbed into the currents. The constraints discussed in the previous section show that we can hope
to resolve the discrepancy only if the new couplings do not respect SU{2)L
symmetry. In this case an improvement on the \2 is possible, but it is not easy to construct the necessary models for these fits.
5.4. Minimal Supersymmetric Standard Model
Davidson et al. present the results of an analysis of the loop contributions from the minimal supersymmetric Standard Model, and show that, although the couplings of the leptons to the W and Z are shifted, the shifts are all small and generally go in the direction opposite to that required to explain the discrepancy.
90
5.5. New Four Fermion Interactions
The previous discussion suggests that it is very difficult to explain the discrep
ancy by introducing modifications to the parameters or the structure of the
s tandard model. Tha t leaves open the possibility tha t there are new interac
tions which are low energy relics of new interactions, which have a characteristic
high mass scale. At low energies these generically appear as non-renormalisable
four fermion interactions.
In the context of charged currents, we have earlier discussed a number of
possibilities in terms of the anomalous results in the a t t empts to measure the
electron neutrino mass in t r i t ium beta decay9 , and we have investigated the
possibility that these additional interactions could be the explanation of the
NuTeV discrepancy. However, we find that , while the inclusion of a scalar
interaction goes in the right direction to explain the effect, as one should
expect because the scalar interference has the opposite sign for neutrinos and
antineutrinos, the constraints from other physics are such tha t the effects are
about two orders of magnitude too small to explain the NuTeV result1 0 .
Varying the vector L — L interactions by about 1% can explain the NuTeV
anomaly, and is allowed by the Tevatron da ta on second generation leptons.
So this represents a possible new physics explanation.
Tensor interactions can contribute in a way similar to the scalar interac
tions, and so can provide a new physics explanation. But it is difficult to
construct models in which there are tensor interactions, without introducing
scalar interactions, and the strengths of the two interactions are related. The
constraints tha t we have noted against scalar interactions then preclude the
possibility tha t tensor interactions are sufficiently strong to account for the
NuTeV data .
The required interactions can be generated by either leptoquark exchange,
or Z' exchange (where the Z' is now an unmixed new neutral vector boson4 .
Perhaps the most interesting case is the Z' case, because the LEP and Tevatron
d a t a 1 1 allow a Z' with a mass below 10 GeV, as well as above 600 GeV.
A particular example is a new U(\)' symmetry, with the Z' coupled to the
current conserving B—3LM, which is anomaly free and avoids constraints on the
production of Z' from e+e~ colliders as the Z' does not couple to electrons.
However, such a coupling is at variance with the suggestions from neutrino
oscillation da ta , which seem to prefer conservation12 of Le — Lf, — LT, but by
suitable adjustment of the coupling constants we can avoid the conflict.
It is possible to fit the NuTeV da ta for \fm2z, — t/gz> & 3 TeV where the
momen tum transferred is such that , on average, t w —20 GeV . Since new neu
tral vector bosons were proposed as an explanation of the erstwhile discrepancy
91
in the muon g - 2, the present limits on the deviation of aM from the calculated
value in the Standard Model, JaM = (20 ± 18) x 10" 1 0 , provide limits on the
coupling and mass of the new Z': 8a^ = 8.4 x 1 0 - 1 0 (3TeV/[mZ'/gz>]) • Con
sequently, this Z' model can provide an explanation of the NuTeV anomaly,
without being in conflict with the muon g — 2 data .
6. N e u t r i n o Osc i l la t ion E x p l a n a t i o n s
Neutrino mixing and oscillations are now an established phenomenon, and it
is na tura l to ask whether the NuTeV discrepancy can be further evidence for
oscillations, particularly as the L/E scale of the experiment, of order l K m / 1 0 0
GeV, is different to the scales probed by the solar and atmospheric neutrinos.
Giunti and Laveder13 have suggested tha t ve —»• v, oscillations may be re
sponsible. They will reduce the electron-neutrino flux at the detector, and thus
the subtraction of the expected ve CC induced short events overcompensates
for the actual events. This increases the s tandard model prediction for i?£xp in
Table 1, by increasing the denominator, and thus can resolve the discrepancy.
Giunti and Laveder obtain the values
A m 2 ~ 1 0 - 1 0 0 e V 2 and sin2 0 ~ 0.4, (18)
which are barely allowed by the reactor neutrino data 8 , and could be in conflict
with nucleosynthesis.
Other oscillation possibilities have been analysed in detail elsewhere14 . The
summary result is tha t most oscillation possibilities enhance, rather than di
minish, the discrepancy. While oscillation to "sterile" neutrinos with only
V + A interactions move the ratios in the right direction, other limits require
tha t the effect be too small to explain the NuTeV result.
7. C o n c l u s i o n
These days there is great fervour associated with the search for physics beyond
the Standard Model. Nevertheless, calm reflection on the NuTeV result and
analysis suggests that the most likely explanations are "old physics" explana
tions.
For experimentalists, it is important to note that an experimental check
of the results given by NuTeV is desirable. Although the J H F will not have
the neutrino energy of the NuTeV beam, it will have a high intensity neutrino
beam, and I suggest that it is important that consideration be given to a
determination of the Weinberg angle by comparing charge and neutral current
events in that beam.
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Acknowledgements
This research is partially supported by the Department of Energy under contract W-7405-ENG-36, by the National Science Foundation and by the Australian Research Council.
References
1. G. P. Zeller et al (The NuTeV collaboration), Phys. Rev. Lett. 88, 091802 (2002) [hep-ex/0110059]. See also "A Departure From Prediction: Electroweak Physics at NuTeV", Kevin McFarland, FNAL Wine and Cheese, October 26, 2001. Available through the University of Rochester website.
2. D. Bardin and V. A. Dokuchaeva, Report No. JINR-E2-86 260, (1986). 3. D. Bardin et al., Report No. CERN-EP/2001-98, hep-ex/0112021. 4. S. Davidson et al., hep-ph/0112302 5. E. A. Paschos and L. Wolfenstein, Phys. Rev. D7, 91 (1973). 6. G. P. Zeller et al, hep-ex/0203004, hep-ex/0205080. 7. G. A. Miller and A. W. Thomas, hep-ex/0204007. 8. D. E. Groom et al, Eur. Phys. J., C15, 1 (2000). 9. G. J. Stephenson, Jr., T. Goldman and B. H. J. McKellar, Phys. Rev. D62, 093013
(2000), B. H. J. McKellar, M. Garbutt, G. J. Stephenson, Jr., and T. Goldman, Proceedings of the EPS International Conference on High Energy Physics, Budapest, 2001 (D. Horvath, P. Levai, A. Patkos, eds.), JHEP (http://jhep.sissa.it/) Proceedings Section, PrHEP-hep2001/193; [hep-ph/0106122].
10. G. J. Stephenson, Jr., T. Goldman, B. H. J. McKellar, and M. Garbutt, APS April meeting: "NuTeV Constraints on Effects in TVitium Beta Decay", abstract [115.005].
11. The CDF Collaboration, Phys. Rev. Lett. 79, 2192 (1997). 12. R. Barbieri et al, Phys. Lett. B445, 407 (1999). 13. C Giunti and M. Laveder, hep-ph/0202152. 14. T. Goldman, G. J. Stephenson, Jr., B. H. J. McKellar, and M. Garbutt, APS
April meeting: "Neutrino Oscillations and the Paschos-Wolfenstein Ratio", abstract [115.002].
NON-STANDARD INTERACTIONS AND NEUTRINO OSCILLATIONS
M.A. G A R B U T T AND B.H.J . M C K E L L A R
School of Physics, The University of Melbourne 3010, Australia
E-mail: rn.garbuttQphysics.unimelb.edu.au; [email protected]
The impact of new physics in the form of general four-fermion interactions on neutrino oscillation signals at future neutrino factories is investigated. We develop a field theoretic description of the overall oscillation process including a source and detector. This formalism is found to be ideal for incorporating interactions at the source and detector with different chiral structures. A study of the v^ —¥ vT
oscillation channel is undertaken for a medium baseline. It is found that neutrino factories are ideal for establishing whether a non-standard interaction is diagonal in the standard weak basis.
1. Introduction
The observation of neutrino oscillations are of prime importance to our understanding of particle physics. The observation of atmospheric neutrino oscillations at the Super-Kamiokande experiment and subsequent confirmation at SNO indicate1 that i/u oscillates to a vT with a mass squared difference of sa 3 x 10 -3eV2 . The simplest interpretation of this result is that at least one of the neutrino mass eigenstates has a non-zero mass of m > 5.5 x 10 -2eV, and as such provides the first glimpse of physics beyond the Standard Model (SM).
The standard treatment of neutrino mixing has three active flavour eigenstates mixed with three mass eigenstates. This mixing is described by the rotation of one basis into the other by the angles #13, #12 and #23 and a complex phase. The atmospheric data constrains sin #23 to be almost maximal with a mass squared difference of \8m\z\ « 3 x 10_3eV2, while the solar neutrino data2 also constrains #i2 to be large and |<5mi3| < 3 x 10 -4eV2 . Furthermore 013 is bound by reactor data3 such that sin #13 < 0.1.
The signs of the mass differences are still unknown, while information on the complex CP violating phase is extremely scant. In order to overcome this there have been a number new experimental facilities proposed, the JHF neutrino beam is one, while others are based on a muon storage ring. All provide a high
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intensity neutrino beam over a wide range of base lengths and energies4. Alongside the neutrino oscillation industry there has been an ongoing and
long established program of precision experiments aimed at measuring the properties of the weak interaction. Measurements of the tritium beta decay spectrum end-point is one such program while the determination of the muon and tau Michel parameters is another5'6. Originally the focus of these experiments was to establish the Lorentz structure of the weak interaction, that is vector minus axial-vector (V-A) or some other combination of scalar and tensor operators7. More recently, in the case of the tritium experiments the focus has shifted to measuring the electron neutrino mass8. Not until very recently have the possible effects of neutrino mixing on these experiments been considered9.
In this paper a different scenario is examined, that is the impact of nonstandard interactions such as additional Lorentz structures in the weak interaction on future neutrino oscillation experiments. In particular, we derive a formalism that will allow the description of the oscillation of a neutrino produced by one type of general interaction and then its subsequent detection via another interaction.
2. Non-Standard Interactions
What is meant by non-standard interactions (NSI) is now defined. In the SM, weak decay processes are mediated via the exchange of a charged vector boson, which is successfully described at low energies by a V — A current-current interaction with an effective coupling strength Gf. If new physics is present at higher masses the low energy result will be to induce new effective interactions with a coupling strength weaker than the coupling of the dominant V — A interaction. Additionally, there is no reason to expect that the new physics has the same Lorentz structure as the weak interaction, hence new currents may be a scalar (S) or pseudo-scalar (P) in nature or comprise of different combinations of vector and axial vector operators. Given the dominance of the left-handed V — A interaction, it is convenient to cast these new interactions into forms with definite handedness, L = V — A, R= V + A, SL = S — P and SR = S + P.
Assuming lepton universality the most general interaction Hamiltonians for low energy, semi-leptonic and leptonic processes are respectively given by
HSL =Y,GaPWe*»?> (V^Vy)1 + h.c., (1) a/3
and
HL=YJ9a^i^^^(^m^^u^) +h.c, (2) a/3
95
where a/ji — L, R, SL, SR, while ipq and tf>q> are quark fields, in and 4>m
are charged lepton fields. The neutrino field is V v > representing a neutrino produced by an (a/3) type interaction and associated with a charged lepton of flavour /. The operators T\ are a combination of one of the five bilinear covariants
TL = 7^(1-75)
Tfl = 7 4 1 + 75)
r s t = ( i - 7 8 )
r 5 „ = (i + 75) . (3)
In the SM only a = (L, R) and f3 = L are present in the semi-leptonic case with ip^ae = iv*e • And in the leptonic interaction only a = L and ji = L are allowed.
The fact that neutrinos oscillate indicates that the interaction eigenstate is not the same as the mass eigenstate. Typically the two are related via a unitary transformation
K"> = £M"'->- (4) i
Phenomenologically, there is no reason why a neutrino produced by a scalar interaction needs to be coupled to the same combination of mass eigenstate as a neutrino produced by another interaction. So in this work neutrino mixing is described by
K0"1) = E <!«">• (5) i
Spectral measurements such as the tritium and Michel parameter experiments yield little information about the mixing elements Uff, hence it is prudent to investigate the impact of NSI on oscillation experiments. To do this a formalism that describes the oscillation of a neutrino with information about both the production and detection processes is needed.
This may be done by treating the whole production, propagation and detection process as a single scattering event. In this scenario the neutrino and charged vector bosons at the production and detection sites are treated as unobserved intermediate states. The oscillation phenomena arise through the interference between these diagrams. Now, if NSI are present, additional diagrams with the non-standard boson at the production and/or detection site need to be included.
Some calculations along these lines have been performed previously10. In the work by Johnson and McKay10 (JM) the NSI lead to direct flavour violation. JM used the framework of weak process (WP) states to perform their
96
calculation. The WP state of flavour a is defined as
\va)wp w Y, \vi)("iF\Hint\I) , (6) i
where {viF\Hint\I) describes the transition from an initial state \I) to a final state of \v»F) with a neutrino of mass m,- present. In this formalism the oscillation probability is interpreted as
*>„_>, ~ | ^ ( F ' | ^ n t | / V l ) e - ' m ^ / 2 £ ( ^ F | / f i n t | / ) | 2 , (7) i
where the primed states represent the interaction at the detector. This formalism works very well in some circumstances, however it is not
immediately clear as to how one would predict a normalised experimental event rate for a given situation. A scattering theory approach where spin and chiral degrees of freedom and neutrino masses are explicitly taken into account will allow a transparent derivation of a normalised event rate.
3. Field Theory Analysis
Field theoretic (scattering theory) calculations of neutrino oscillations have been performed by a number of authors11. For a more complete list of referenced see the work by Beuthe. In this note the formalism developed by Cardall and Chung12 is adapted to cater for NSI.
To keep the notation transparent a specific example will be studied, that of muon decay at the source, muon neutrino3 oscillation and subsequent detection of a negatively charged tau at the detector. The scattering amplitude is given by the time ordered product of interaction Hamiltonians
A = (i>ee-X\T[jd\ JdAyHL{x)HSL{y)]\n-N)
dx° / dy° / d3x / d3y ,
-co 7 -00 Jvs JvD WJ^WWE
exp H(p , j - pe - Pv) • x] exp [-i(pN - Pr - Px) • y] ,av
* JVtBMUDVR,) """•»'»•'*•'> ^WrW*) ' " where E^, Ee and Ev, are the energies of the muon, electron and electron neutrino, respectively, with corresponding four momenta p^, pe and p„. The reaction at the source is evaluated at a space-time i in a volume Vs. The
a T h e term muon neutrino is used loosely here since neutrinos produced via different interactions and associated with the decay of muons are not necessarily the same. This is due to the various mixing matrices associated with the different interactions. When it matters, a more specific language will be used.
97
energies of the particles at the detector are EN , ET, Ex with associated four-momenta PN,PT,PX- The reaction at the detector is evaluated at a space-time y in a volume VD- Furthermore, the quantity Mfi(x, y\pi,Pf,q) is given by
Mfi(x,y\Pi,pj,q) = Y<Y<Yl9af>GX°K?Ur?JX{PN>Px) i a/3 ACT
x (u{pT)?x&{x,y)Tlufa))-{u{pe)Ypv{pv)) . (9)
Here p; and pj are the momenta of the particles in the initial and final state respectively while JX{PN,PX) is the nuclear matrix element for the transition operator T\. The sums are over neutrino masses, source and detector interactions, respectively. The function G'(x, y) is the neutrino propagator or Green's function. Due to the chiral nature of the operators T\ and r £ , it is convenient to write the Green's function in chiral blocks
where GXY is the element projected out by PxG(x, y)Py where Px = PR/L — (1 ± 75). It can be shown that in the ultra-relativistic limit, the diagonal components of the Green's function vanish and that
2
GLR = / JTT-J-^ ~x M i - S - L ) V i r • (n)
where u is the now on-shell energy of the neutrino. To perform this calculation an expansion about the points xs and yo has been performed with
L = (yD-Ss)/\yD-xs\. (12)
A similar expression can be found for GRL with the sign of the Pauli spinor changed. For details of the calculation see Cardall and Chung12. In this limit we can write for, say, G^R
PLG(x,y)PR= f 7 ^ e ^ t o B - - ° « ( t « ( a , ) M ^ ) e t " f ) A t T ' ^ J-00 ( 2 7 r ) 4n\yD - x s \
where u(u)^ is a left-handed neutrino spinor. Note that in this limit the right-handed state is completely disconnected. The other surviving component of the Green's function is
PRG(x,y)PL=r ^ e [ - ^ t o ' - - - ) ] ( t t ( w ) t f i ( u ) t ) ! ! ^ ^ . ( 1 4 ) J-00 ( 2 7 r ) 47T|yD -XS\
The fact that only one spin state contributes allows M/,- to split into two different matrix elements, one corresponding to the production process and the
other to the detection process. The matrix element now reads
/
OO J eiwL(y-x)
4n\L\
J2 MfiMjie-'Zi-1-®-*) (15)
where
and
K = T,^U:fu(oJ)TU(p)i) • (u(pe)I>(p„)) a0
Mfi ^J2GXaU^jX(PN'Px)u(Pr)rxu(uj) . ACT
(16)
(17)
A per particle event rate is found by squaring the amplitude, dividing by the characteristic time of the process and integrating over the phase space of the external particles. The event rate that one may anticipate in an experimental situation is found by integrating the per particle rate over the initial state distribution functions. The experimental rate can eventually be expressed as
dNTdu - J d3xs / d3pfifli(xs,pli) / d3yD / d3pNfN(yD,pN)—du), (18)
where
dT duj = £ d?Nu
dwdQ,v
1 / <Jra?- -. \ (19)
The neutrino energy distribution is given by
d?Nv
dwdSlv (20)
where the mass eigenstate indices are carried by the muon neutrino mixing elements. In general it is not possible to factor these quantities out since neutrinos produced by different interactions are not necessarily mixed in the same way. The tau neutrino-nucleon cross section is given by aT((^)\ij, where again it is not possible to factor out the mixing elements. Explicit expressions for these quantities will be given in the next section.
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4. Resu l t s for a Neu t r ino Factory
Here the model of neutrino oscillations with NSI developed in the previous sections is applied to a generic neutrino factory scenario. No attempt has been made at a full simulation of this type of experiment, rather regions of interesting parameter space are determined and a quantitative understanding of the effects of NSI is gained. The v^ —> vr oscillation channel is explored.
For the sake of exposition only a left-chiral scalar current at the detector is considered in addition to the standard V — A current. The new physics is not considered at the source since the muon beam was most likely produced in a left-chiral eigenstate, while the scalar current requires right-handed muons.
The standard and non-standard mixing matrices are
uLL = / cos* s i n 0 \ ^ USLSL = / COB* s i n 0 \ y — sin 0 cos 0 J \ — sin <p cos <p J
respectively. As an upper bound on the NSI, we assume for the coupling strengths that GSLSL < O.QlGLL. The neutrino energy distribution, assuming that the electron mass can be neglected, is
d2N„
dzd£lv
2 z 'TLLTTLL, Tll—UpUtf{3-2z), (22)
where T^ is the muon decay rate, z = LJ/E^ and E^ is the muon beam energy. The tau production cross section is
•Mb = f"J«£& (23)
The differential cross section expressed in terms of the Bjorken scaling variables x' and y1 has the general form
d2aT
dx'dy' _ \rLL\lTjLLTjLL^ aLL \(~iSLSLi2TTSLSLTrSLSL ^ aSLSL - H* I "ri urj dx,dyl I" 1^ I "ri "rj fa,dyl
+ GLLGS^ {UtfU%s<- + UTfU?^) d^^r . (24)
The dependence on i and j has been made explicit, the standard term dxf j;h ,
the interference term dJ;ffi , and the scalar term ^dy'L m a y 'De calculated using a parton model. Note that the interference term vanishes in the limit where the parton and tau masses are zero.
The sensitivity of a future neutrino factory to NSI is investigated by defining a x2 function which determines the required detector mass and number of
100
useful muon decays in order to claim new physics. The x 2 function for a detector mass and number of muons (N^ • Mdet = 102lkt) is defined as
|7VSM — NNSII2
XNM21 = 2 ^ jfsW ' (25) k *
where N^M is the expected number of tau producing charged current events in energy bin k in the absence of NSI for this detector mass and number of muons, while A^* 7 is the number expected with the NSI present. The required number of detector mass-number of muon units, NMrec, is
NMrec > 4 ^ , (26) X/VM21
where xlo¥ ls ^ e c m s c lu a r e c l value at 90% confidence level, and one detector mass-muon number unit corresponds to 1021kt. The proposed muon storage ring at CERN for example would provide ~ 1021 useful muon decays per year13.
The number of detector mass-muon number units required to detect NSI at 90%CL versus the non-standard mixing angle is shown in Fig. 1. The values of the parameters used in producing this plot are explained in the caption.
The main result is the variation of NMree over the full range of <f>. This simplified analysis indicates that for some values of <f> a detector mass of ~ 30kt would be in a good position to either detect NSI or increase the upper bound on the non-standard coupling strength. Points of particular interest are <j> = 7r/4, 57r/4 and (j> = 3^/4, 77r/4. The first case corresponds to the situation V!*LSL = v^h or XJ^SL = U^L, up to a phase. For this case NMrec
is at a maximum. In the second case, V^LS^ = u^L or U^LSL — U^L, up to a phase, these points corresponding to direct flavour violation. In this case NMrec is at a minimum. This result is to be expected, since at this energy and over a medium base line the neutrino beam will comprise mostly of non-oscillated i^'s. Any flavour violating non-standard coupling will in essence be picked out over the standard term. This effect was noted in previous work by Datta et al.14
5. Conclusion
In this study we have shown that in order to arrive at a fully normalised experimental event rate resulting from neutrino oscillations with non-standard interactions one needs to utilise a field theoretic description of the process. This formalism has the added advantage of providing an explicit treatment of oscillations of non-chiral neutrinos.
The numerical study of a generic neutrino factory has shown the importance of allowing for an arbitrary mixing when looking for NSI. This is exemplified
101
10000 F
1000 r-
Figure 1. <f> is the non-standard mixing angle and NMrec is the required detector mass-muon number units defined in the text. The beam energy is taken to be 50 GeV, the mass difference is Sm2 — 2.5 x 1 0 - 3 eV2 , the base line is \L\ — 732 Km and the standard mixing angle is taken to be maximal, sin(20) = 1.
by the results showing tha t the neutrino flux and detector mass required to
detect new physics is greatest when the non-standard interaction eigenstate is
also an eigenstate of the corresponding weak interaction. The converse is t rue
for NSI tha t are not eigenstates of the weak interaction.
Any results obtained from precision experiments directly measuring NSI or
from oscillation experiments must include the possibility tha t a weaker force
may be present and may not be diagonal in the weak basis.
More dramat ic results may be expected from the non-standard interaction
of the electron neutrino with a fermion background (a non-standard MSW
effect), however we leave this to future work.
R e f e r e n c e s
1. Y. Fukuda et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 81 , 1562 (1998).
2. M. C. Gonzalez-Garcia, P. C. de Holanda, C. Pena-Garay and J. W. Valle, Nucl. Phys. B573, 3 (2000).
3. M. Apollonio et al., CHOOZ Collaboration, Phys. Lett. B466, 415 (1999). 4. Y. Itow et al., hep-ex/0106019. 5. M. Fritschi, E. Holzschuh, W. Kundig, J. W. Petersen, R. E. Pixley and H. Stussi,
Phys. Lett. B173, 485 (1986); R.G. Robertson, T.J. Bowles, G.J. Stephenson, D.L. Wark, J. F. Wilkerson and D.A. Knapp, Phys. Rev. Lett. 67, 957 (1991);
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A.I. Belesev et al, Phys. Lett. B350, 263 (1995); V.M. Lobashev et al., Phys. Lett. B460, 227 (1999); C. Weinheimer et al., Phys. Lett. B300, 210 (1993).
6. A. Rouge, Eur. Phys. J. C18, 491 (2001). W. Fetscher and H. J. Gerber, Eur. Phys. J. C15, 316 (2000).
7. J.D. Jackson, S.B. Treiman and H.W. Wyld, Jr., Phys. Rev. 106, 517 (1957). 8. C. Weinheimer et al., Phys. Lett. B460, 219 (1999); Talk by V. Aseev et. al.,
at the International Workshop on Neutrino M asses in the sub-eV Range, Bad Liebenzell, Germany, January 18th-21st, 2001. See also homepage: http://www-ikl.fzk.de/tritium/.
9. B. H. McKellar, M. Garbutt, G. J. Stephenson and T. Goldman, hep-ph/0106122; G. J. Stephenson, T. Goldman and B. H. McKellar, Phys. Rev. D62, 093013 (2000); G. J. Stephenson and T. Goldman, Phys. Lett. B440, 89 (1998); Y. Farzan, O. L. Peres and A. Y. Smirnov, Nucl. Phys. B612, 59 (2001); F. Vis-sani, Nucl. Phys. Proc. Suppl. 100, 273 (2001).
10. L. M. Johnson and D. W. McKay, Phys. Rev. D61, 113007 (2000); L. M. Johnson and D. W. McKay, Phys. Lett. B433, 355 (1998); T. Ota, J. Sato and N. a. Ya-mashita, hep-ph/0112329.
11. M. Beuthe, hep-ph/0109119; C. Y. Cardall, Phys. Rev. D61, 073006 (2000); W. Grimus, S. Mohanty and P. Stockinger, hep-ph/9909341; W. Grimus and P. Stockinger, Phys. Rev. D54, 3414 (1996); C. Giunti, C. W. Kim, J. A. Lee and U. W. Lee, Phys. Rev. D48, 4310 (1993).
12. C. Y. Cardall and D. J. Chung, Phys. Rev. D60, 073012 (1999). 13. B. . Autin, A. . Blondel and J. R. Ellis, "Prospective study of muon storage rings
at CERN," CERN-99-02. 14. A. Datta, R. Gandhi, B. Mukhopadhyaya and P. Mehta, hep-ph/0105137.
NEUTRINO OSCILLATION SEARCHES AT ACCELERATORS A N D REACTORS
S. N. T O V E Y
Research Centre for High Energy Physics, University of Melbourne, Parkville, VIC. 3010, Australia
E-mail: S. [email protected]
This talk will review recent searches for neutrino oscillations using neutrinos produced at accelerators and reactors.
1. Introduction
The theory of neutrino oscillations will be briefly reviewed.
Results will be presented on the following experiments :
(1) NOMAD and CHORUS at CERN, (2) KARMEN at Rutherford and LSND at Los Alamos, (3) K2K in Japan, (4) CHOOZ as a typical reactor experiment.
2. The Theory of Neutrino Oscillations
These results are well known and only a few equations will be quoted without derivation. Oscillations can only occur if the neutrino flavour or weak eigenstates differ from the neutrino mass eigenstates. The two representations (ue,i/^,i>T) and (v\,U2,vz) a r e related via a unitary mixing matrix. For simplicity consider a two-neutrino scenario in which we ignore (say) the electron neutrino.
Then the mass and flavour eigenstates are related via a simpler (unitary) matrix :
103
104
cosO sinO —sinO cosO Z) • (I»
If one produces a beam of one flavour (say v^) then after a distance L the probability that it has become a vT is :
P{yT,L) = sin2 (26) x sin2 (nL/Losc), (2)
where the characteristic oscillation length is given by :
Losc = (47r^) /Am 2 , (3)
and
Am2 = m\—m\. (4)
This equation is in natural units (h = c = 1 ) . If we measure L in km, E in GeV and Am2 in eV2, then :
Losc » (2 .5^) /Ara 2 . (5)
By conservation of particles, the probability that the v^ remains a v^ is :
P ( I / „ , L ) = 1 . 0 - P ( I / T , I ) . (6)
3. The N O M A D and CHORUS Experiments at CERN
These experiments were situated in the West Area Neutrino Facility (WANF) at CERN. Neutrinos were produced by allowing a beam of 450 GeV/c protons to strike a Berylium target. The resulting spectrum of neutrinos at NOMAD is shown in Fig. 1. The spectrum at CHORUS is slightly harder as the cross-sectional area of that experiment is smaller than NOMAD and neutrinos near the beam axis have slightly higher energies.
The methods used by NOMAD and CHORUS are completely complementary, and they are described below. Both seek to observe the appearance of vT
via the reaction :
vT + N->T~ + X. (7)
3.1. The NOMAD Experiment
A full description of the NOMAD detector can be found in Altegoer et al.1. NOMAD does not have the spatial resolution to see the track of a T~
which typically travels less that 1 mm. Instead it relies on Kinematic Cuts, Topological Constraints and Likelihood Methods. And it uses the novel (at
105
Neutrino Flux
80 100 120 Neutrino Energy (GeV)
Figure 1. The energy spectra of the various neutrino flavour states at NOMAD.
least to HEP) techniques of Blind Analyses and Frequentist Statistics. A full description and final results can be found in Astier et al.2.
In summary NOMAD sees no evidence for oscillations. The number and shape of the observed events are consistent with background. The number of candidates is 52 events, to be compared to a total predicted background of 51.1 ± 5.4 events. The signal and background are compatible in each decay channel.
In the most sensitive region (i.e. that with the smallest background) NOMAD observes 1 event, as compared to a calculated background of 1.62 (+1.89-0.38).
3.2. The CHORUS Experiment
CHORUS had a target of photographic emulsion with a total mass of 770 kg. The spatial resolution is very good and would allow the tracks produced by
106
a T~ to be seen. Potential r vertices are located via a Fiber target tracker, with an angle resolution of about 2 mrad and a position resolution of 150 mm. A calorimeter and spectrometer are situated downstream of the target and tracker. More details can be found in Eskut et al.3
CHORUS has completed what they term their "Phase I" analysis, and no candidates were observed with negligible backgrounds, see Eskut et al.4 A complete rescanning of the emulsion (Phase II) is underway. This will allow a signicant improvement of their current published limits.
3.3. The CERN Results
The final NOMAD limit and the Phase I CHORUS limits are summarised in the plot below in Fig. 2.
^Tio-> u
10 ' 2 _ NOMA©
10
1 T
T 1—i i i l l "
10
V —» V H x
90% C.L.
.1 j i i i i i i i l
10 10
CDIIS -- J • • i t • i
10 -1
sin2 28
Figure 2. The regions of parameter space excluded by the two CERN experiments.
As can be seen from the exclusion plot, at high values of Am2 the NOMAD
107
experiment produced a limit about an order of magnitude better than previous results. CHORUS (Phase I) is not quite as good but in Phase II it is expected to better the NOMAD limit. It is also clear that the CERN experiments are only sensitive for values of Am2 > 1 eV2. For smaller mass differences significant oscillations would not have had time to develop.
4. Experiments with Stopped Pion Beams
Two experiments have searched for oscillations using neutrinos produced by stopping a beam of n mesons in a massive target. The results are controversial.
The LSND experiment5 at Los Alamos in the U.S.A. claims to have seen a signal. The KARMEN experiment6 at the Rutherford Appleton Laboratory in the U.K. does not see any sign of oscillations A careful comparison (see Fig. 3) shows that the two groups are not quite in conflict. There is a very small region of parameter space where a signal from LSND cannot be excluded by KARMEN.
Figure 3. Results from the KARMEN and LSND experiments.
However the jury is still out. A new experiment MiniBoone, which will use the 8 GeV booster at Fermilab in the U.S.A., should resolve this issue soon. It has a custom-designed beam stopper, as does KARMEN, whereas the beam stopper at Los Alamos was not optimised for neutrino oscillation searches.
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5. Long Baseline Experiments
The low values of Am2 suggested by positive results for oscillations using neutrinos produced in the Sun and in the atmosphere have led to a number of proposals to send accelerator-produced neutrinos to distant detectors.
Two of these, from Fermilab to the Soudan mine and from CERN to Gran Sasso in Italy, have yet to collect data. Coincidently both have baselines of about 750 km. One experiment in Japan is already producing results, and will be discussed here.
5.1. K2K
The K2K experiment7 sends neutrinos produced in a 12 GeV proton accelerator at the KEK laboratory to the giant SuperKamiokande (SuperK) detector in western Japan. The baseline is about 250 km.
Unfortunately K2K has ceased operations due to an accident at SuperK in which many photomultipliers were destroyed. The prior results were tantalising.
The mean energy of the neutrinos (mostly u^) produced at KEK means that, should they oscillate into i/T, then those vT would be below threshold for producing charged tauons via a charged current reaction. K2K is thus essentially a disappearance experiment in which the "signal" would be too few neutrinos arriving at SuperK.
The preferred values of the oscillation parameters to explain the SuperK atmospheric neutrino data are Am2 = 0.3 x 10 - 3 eV2 with maximal mixing. If those values are assumed then the K2K experiment expects to observe a deficit of events at a neutrino energy of about 600 MeV. Preliminary data, see Fig. 4, show just such a deficit.
As noted K2K has been halted by the accident at SuperK. Reconstruction of that detector is proceeding at pace, and K2K hopes to resume operations by the end of 2002.
6. Reactor Experiments
Nuclear reactors produce copious numbers of neutrinos, in particular the anti-iv In the absence of oscillations the flux should fall off like an inverse square law. A departure from such behaviour would indicate oscillations into another type of neutrino which, at these low energies, would not interact.
The two most powerful reactors being used are at CHOOZ in France and at Palo Verde in the U.S.A. Figure 5 shows the CHOOZ detector. Neither experiment sees a departure from the fluxes predicted in a no-oscillation scenario.
109
m
15
10
Note: Am2=3xMF aV2
corresp onds to 600 MeV Ev
1
4
Figure 4. The spectrum of i/MCharged Current neutrino interactions observed in the Su-perKamiokande detector using a ^ b e a m from KEK. See the text for a discussion.
Results from CHOOZ8 are shown below in Fig. 6. The Palo Verde experiment has very similar limits.
7. Conclusions
Positive results (discussed by other speakers at this workshop)from experiments studying neutrinos produced in the Sun and in the Earth's atmosphere do disappear before arriving at a detector. By far the most likely scenario is that they have oscillated into another neutrino species which cannot be detected. These experiments indicate that the differences in the neutrino masses (or more precisely the quantities Am2) are small.
Short baseline experiments at high energy accelerators have all yielded negative results, which is now thought to be because they cannot explore the region of small Am2 . Planned long baseline experiments at CERN and Fermilab should (just) be able to explore that region.
110
CZZZZZTZ. .. j ~ n x r z r r r i . . ._..; „u „;.:;:•••. :i .'*>' 1 2 3 4 5 6fn
Figure 5. The CHOOZ detector in France
The one positive hint of an oscillation signal has come from a low-energy accelerator produced beam in a medium baseline experiment from KEK to SuperK. The resumption of that experiment is eagerly awaited.
Finally it should be noted that no experiment has seen the appearance of a new species of neutrino in a beam that was originally of a different species. When, or if, observed that will be the decisive evidence that oscillations do occur.
Acknowledgements
I would like to thank Dr. Jaap Panman, the spokesman of the CHORUS experiment for providing me with the powerpoint version of a recent CHORUS presentation.
111
— analysis C
YA 90% CL Kamiokande (multi-GeV)
H 90% CL Kamiokande (sub+multi-GeV) I . . . . i i . . , . i . . • .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sin2(28)
Figure 6. Oscillation limits from CHOOZ
R e f e r e n c e s
1. J . Altegoer et al., NIM A 4 0 4 , 96 (1998). 2. P. Astier et al., Nucl. Phys. B 6 1 1 , 3 (2001). 3. E. Eskut et a.l, NIM A 4 0 1 , 7 (1997). 4. E . Eskut et al., Phys. Lett. B 4 9 7 , 8 (2001). 5. C.Athanassopolus et al., Phys. Rev. C 5 4 , 2658 (1996). 6. B . Armbrus t e r et al., Phys. Rev. C 5 7 , 3414 (1998).
7. Y. Oyama, K E K Prepr int 2001-7, copy of a talk a t the "Cairo Int. Conference",
J anuary 2001. 8. M. Apollonio et al., Phys. Lett. B 4 6 6 , 415 (1999).
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4. Hadron Structure and Properties
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LATTICE QCD A N D HADRON STRUCTURE
A. W. THOMAS
Special Research Centre for the Subatomic Structure of Matter and Department of Physics and Mathematical Physics,
Adelaide, SA 5005, Australia E-mail: [email protected]
One of the great challenges of lattice QCD is to produce unambiguous predictions for the properties of physical hadrons. We review recent progress with respect to a major barrier to achieving this goal, namely the fact that computation time currently limits us to large quark mass. Using insights from the study of the lattice data itself and the general constraints of chiral symmetry we demonstrate that it is possible to extrapolate reliably from the mass region where calculations can be performed to the chiral limit.
1. Introduction
At the present time we have a wonderful conjunction of opportunities. Modern accelerator facilities such as Jlab, Mainz, DESY and CERN are providing data of unprecedented precision over a tremendous kinematic range at the same time as numerical simulations of lattice QCD are delivering results of impressive accuracy. It is therefore timely to ask how to use these advances to develop a new and deeper understanding of hadron structure and dynamics1.
Let us begin with lattice QCD. Involving as it does a finite grid of space-time points, lattice QCD requires numerous extrapolations before one can compare with any measured hadron property. The continuum limit, a —• 0 (with a the lattice spacing), is typically under good control2. With improved quark and gluon actions the 0(a) errors can be eliminated so that the finite-a errors are quite small, even at a modest lattice spacing3 - say 0.1 fm. In contrast, the infinite volume limit is much more difficult to implement as the volume, and hence the calculation time, scales like TV4. Furthermore, this limit is inextricably linked to the third critical extrapolation, namely the continuation to small quark masses (the "chiral extrapolation"). The reason is, of course, that chiral symmetry is spontaneously broken in QCD, with the pion being a massless Goldstone boson in the chiral limit. As the lattice volume must contain the pion cloud of whatever hadron is under study one expects that the box size, L, should be at least 4m"1 . At the physical pion mass this is a box
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116
5.6 fm on a side, or a 564 lattice, with a — 0.1 fm. This is roughly twice as big as the lattices currently in use.
Since the time for calculations with dynamical fermions (i.e. including quark-antiquark creation and annihilation in the vacuum) scale4 as m~36, current calculations have been limited to light quark masses 6-10 times larger than the physical ones. With the next generation of supercomputers planned to be around 10 Teraflops, it should be possible to get as low as 4 times the physical quark mass, but to actually reach the physical mass on an acceptable volume will need at least 500 Teraflops. This is 10-20 years away.
Since a major motivation for lattice QCD must be to unambiguously compare the calculations of hadron properties with experiment, this is somewhat disappointing. The only remedy for the next decade at least is to find a way to extrapolate masses, form-factors, and so on, calculated at a range of masses considerably larger than the physical ones, to the chiral limit. In an effort to avoid theoretical bias this has usually been done through low-order polynomial fits as a function of quark mass. Unfortunately, this is incorrect and can yield quite misleading results because of the Goldstone nature of the pion.
Once chiral symmetry is spontaneously broken, as we have known for decades that it must be in QCD and as it has been confirmed in lattice calculations, all hadron properties receive contributions involving Goldstone boson loops. These loops inevitably lead to results that depend on either logarithms or odd powers of the pion mass. The Gell-Mann-Oakes-Renner relation, however, implies that mn is proportional to the square root of mq, so logarithms and odd powers of m , are non-analytic5 in the quark mass, with a branch point at mq = 0. One simply cannot make a power series expansion about a branch point.
On totally general grounds, one is therefore compelled to incorporate the non-analyticity into any extrapolation procedure. The classical approach to this problem is chiral perturbation theory, an effective field theory built upon the symmetries of QCD6. There is considerable evidence that the scale naturally associated with chiral symmetry breaking in QCD, AXSB, is of order 4irfn, or about 1 GeV. Chiral perturbation theory then leads to an expansion in powers of mff/AxsB and P / A X S B , with p a typical momentum scale for the process under consideration. At 0(p4), the corresponding effective Lagrangian has only a small number of unknown coefficients which can be determined from experiment. On the other hand, at 0(p6) there are more than 100 unknown parameters, far too many to determine phenomenologically.
Another complication, not often discussed, is that there is yet another mass scale entering the study of nucleon (and other baryon) structure7,8. This scale is the inverse of the size of the nucleon, A ~ R"1. Since A is naturally more like
117
a few hundred MeV, rather than a 1 GeV, the natural expansion parameter ,
mn/A, is of order unity for mn ~ 2 - 3mPh y s - the lowest mass scale at which
lattice da ta exists. This is much larger than mK/AxsB ~ 0.3 — 0.4, which
might have given one some hope for convergence. As it is, the large values
of m,r /A at which lattice da ta exist make any chance of a reliable expansion
in chiral perturbat ion theory (xPT) fairly minimal9 . Even though one has
reason to doubt the practical utility of xPT, the lattice da t a itself does give
us some valuable hints as to how the di lemma might be resolved. The key is
to realize tha t , even though the masses may be large, one is actually studying
the properties of QCD, not a model. In particular, one can use the behaviour
of hadron properties as a function of mass to obtain valuable new insights into
hadron structure.
The first thing that stands out, once one views the da ta as a whole, is
just how smoothly every hadron property behaves in the region of large quark
mass. In fact, baryon masses behave like a + bmq, magnetic moments like
(c + t / m q ) - 1 , charge radii squared like (e + / m q ) _ 1 and so on. Thus, if one
defined a light "constituent quark mass" as M = Mo + cmq (with c ~ 1), one
would find baryon masses proportional to M (times the number of u and d
quarks), magnetic moments proportional to M _ 1 and so on - just as in the
constituent quark picture. There is simply no evidence at all for the rapid, non-
linearity associated with the branch cuts created by Goldstone boson loops.
Indeed, there is not even any evidence for a statistically significant difference
between properties calculated in quenched versus full QCD! How can this be?
The natural answer is readily found in the additional scale, A ~ Br1, men
tioned earlier. In QCD (and quenched QCD), Goldstone bosons are emit ted
and absorbed by large, composite objects built of quarks and gluons. Whenever
a composite object emits or absorbs a probe with finite momentum one must
have a form-factor which will suppress such processes for momenta greater
than A ~ Br1. Indeed, for m* > A we expect Goldstone boson loops to be
suppressed as powers of A / m r , not mn/A (or mn/Axss). Of course, this does
not mean tha t one cannot in principle carry through the program of xPT.
However, it does mean that there may be considerable correlations between
higher order coefficients and tha t it may be much more efficient to adopt an
approach which exploits the physical insight we jus t explained.
Over the past three years or so, our group in Adelaide has worked with
a number of colleagues around the world to do just this. Tha t is, we have
developed an efficient technique, using very few free parameters, to extrapolate
every hadron property which can be calculated on the lattice from the large
mass region to the physical quark mass - while preserving the most impor tant
non-analytic behaviour of each of those observables. This task is not trivial,
118
in that various observables need different phenomenological treatments. On the other hand, there is a unifying theme. That is, pion loops are rapidly suppressed for pion masses larger than A (mff > 0.4 — 0.5 GeV). In this region the constituent quark model seems to represent the lattice data extremely well. However, for mw below 400-500 MeV the Goldstone loops lead to rapid, non-analytic variation with mq and it is crucial to preserve the correct leading non-analytic (LNA) and sometimes the next-to-leading non-analytic (NLNA) behaviour of xPT.
In order to guide the construction of an effective, phenomenological extrapolation formula for each hadron property, we have found it extremely valuable to study the behaviour in a particular chiral quark model - the cloudy bag model (CBM)8 '10,11. Built in the early 80's it combined a simple model for quark confinement (the MIT bag) with a perturbative treatment of the pion cloud necessary to ensure chiral symmetry. The consistency of the perturbative treatment was, not surprisingly in view of our earlier discussion, a consequence of the suppression of high momenta by the finite size of the pion source (in this case the bag). Certainly the MIT bag, with its sharp, static surface, has its quantitative defects. Yet the model can be solved in closed form and all hadron properties studied carefully over the full range of masses needed in lattice QCD. Provided one works to the appropriate order the model preserves the exact LNA and NLNA behaviour of QCD in the low mass region while naturally suppressing the Goldstone boson loops for m^ > A ~ R~1. Finally, it actually yields quite a good description of lattice data in the large mass region for all observables in terms of just a couple of parameters.
With this lengthy explanation of the physics which underlies the superficially different extrapolation procedures, we now summarise the results for some phenomenologically significant baryon properties.
2. Electromagnetic Properties of Hadrons
While there is only limited (and indeed quite old) lattice data for hadron charge radii, recent experimental progress in the determination of hyperon charge radii has led us to examine the extrapolation procedure for extracting charge radii from the lattice simulations. Figure 1 shows the extrapolation of the lattice data for the charge radius of the proton, including the In mff (LNA) term in a generalised Pade approximant12'13:
{r}*~ l + c2ml • {i}
Here ci and c2 are parameters determined by fitting the lattice data in the large mass region (m2 > 0.4 GeV2), while \i, the scale at which the effects of
119
pion loops are suppressed, is not yet determined by the data but is simply set to 0.5 GeV. The coefficient \ is model independent and determined by chiral perturbation theory. Clearly the agreement with experiment is much better if, as shown, the logarithm required by chiral symmetry is correctly included -rather than simply making a linear extrapolation in the quark mass (or ml). Full details of the results for all the octet baryons may be found in Ref. [12].
The situation for baryon magnetic moments is also very interesting. The LNA contribution in this case arises from the diagram where the photon couples to the pion loop. As this involves two pion propagators the expansion of the proton and neutron moments is:
MP(") = A(J (n )Tam, + O K ) (2)
Here //Q is the value in the chiral limit and the linear term in mn is propor-tional to m | , a branch point at mg = 0. The coefficient of the LNA term is a = 4.4//jvGeV_1. At the physical pion mass this LNA contribution is 0.6pjv, which is almost a third of the neutron magnetic moment.
1.2
1.0
0.8
*A 0.6 -
0.4 v
C o "o 0.2 u
0.0 }r
- 0 . 2
• - - + - — • - - 1 - * - . -
A J _
0.0 0.2 0.4 0.6 2 (GeV2)
0.8 1.0 m
Figure 1. Fits to lattice results for the squared electric charge radius of the proton - from Ref. [12]. Fits to the contributions from individual quark flavours are also shown (the u-quark results are indicated by open triangles and the d-quark results by open squares). Physical values predicted by the fits are indicated at the physical pion mass. The experimental value is denoted by an asterisk.
Just as for the charge radii, the chiral behaviour of fj.p^ is vital for a correct extrapolation of lattice data. As shown in Fig. 2, one can obtain a very
120
satisfactory fit to some rather old data, which happens to be the best available, using the simple Pade approximant14:
^ = pin)
li-jfom^+^m? (3)
Existing lattice data can only determine two parameters and Eq. (3) has just two free parameters while guaranteeing the correct LNA behaviour as m^ —¥ 0 and the correct behaviour of HQET at large ml. The extrapolated values of fj,p and n" at the physical pion mass, 2.85 ± 0.22/JJV and —1.90 ± 0.15pjv are currently the best estimates from non-perturbative QCD14. (Similar results, including NLNA terms in chiral perturbation theory, have been reported recently by Hemmert and Weise15.) For the application of similar ideas to other members of the nucleon octet we refer to Ref. [16], while for the strangeness magnetic moment of the nucleon we refer to Ref. [17]. The last example is another case where tremendous improvements in the experimental capabilities, specifically the accurate measurement of parity violation in ep scattering18, is giving us vital information on hadron structure.
Hs-
3 +->
c
^om
e
-s o
-1-5
CI)
a M cO S C o CD U 3
55
3.5 3.0 2 5
2.0 1.5 1.0
0.5 ( ) . ( )
- 0 5 - 1 0 - 1 . 5 - 2 0
- 2 . 5
Figure 2. Extrapolation of lattice QCD magnetic moments for the proton (upper) and neutron (lower) to the chiral limit. The curves illustrate a two parameter fit to the simulation data, using a Pade approximant, in which the one-loop corrected chiral coefficient of mn is taken from xPT. The experimentally measured moments are indicated by asterisks. The figure is taken from Ref. [14].
121
In concluding this section, we note tha t the observation tha t chiral correc
tions are totally suppressed for m^ above about 0.5 GeV and tha t the lattice
da ta looks very like a constituent quark picture there suggests a novel approach
to modelling hadron structure. It seems tha t one might avoid many of the com
plications of the chiral quark models, as well as many of the obvious failures
of constituent quark models by building a new constituent quark model with
u and d masses in the region of the strange quark - where SU(3) symmetry
should be exact. Comparison with da ta could then be made after the same
sort of chiral extrapolation procedure that has been applied to the lattice da ta .
Initial results obtained by Cloet et al. for the octet baryon magnetic moments
using this approach are very promising indeed1 9 . We note also the extension
to A-baryons, including the NLNA behaviour, reported for the first t ime in
these proceedings2 0 .
3 . M o m e n t s of S t r u c t u r e F u n c t i o n s
The moments of the parton distributions measured in lepton-nucleon deep
inelastic scattering are related, through the operator product expansion, to
the forward nucleon mat r ix elements of certain local twist-2 operators which
can be accessed in lattice simulations1 . The more recent da ta , used in the
present analysis, are taken from the Q C D S F 2 1 and MIT 2 2 groups and shown
in Fig. 3 for the n = 1, 2 and 3 moments of the u — d difference at NLO in the
MS scheme.
To compare the lattice results with the experimentally measured moments ,
one must extrapolate in quark mass from about 50 MeV to the physical value.
Naively this is done by assuming tha t the moments depend linearly on the
quark mass. However, as shown in Fig. 3 (long dashed lines), a linear extrap
olation of the world lattice da ta for the u — d moments typically overestimates
the experimental values by 50%. This suggests that important physics is still
being omit ted from the lattice calculations and their extrapolations.
Here, as for all other hadron properties, a linear extrapolation in m ~ m 2
must fail as it omits crucial nonanalytic structure associated with chiral sym
metry breaking. The leading nonanalytic (LNA) te rm for the u and d dis
tributions in the physical nucleon arises from the single pion loop dressing of
the bare nucleon and has been shown2 3 , 2 4 to behave as m 2 log mn. Experience
with the chiral behaviour of masses and magnetic moments shows tha t the
LNA terms alone are not sufficient to describe lattice da ta for m^ > 200 MeV.
Thus, in order to fit the lattice da ta at larger m^, while preserving the correct
chiral behaviour of moments as mn —>• 0, a low order, analytic expansion in
m 2 is also included in the extrapolation and the moments of u — d are fitted
122
0.4
0.3
0 .1
0.12-
0.04
0.06-
0.02-
1 1 1
QCDSF Gockeler e t al.. 1996 QCDSF Gockeler et al., 1997 QCDSF Best e t al., 1997 MIT-DD60Q (Quenched) MIT-SESAM (Full)
Experiment Meson Cloud Model
•+ - h -i h
-l h H 1- •H h
-jbn-iL-
o.o 0.2 0.4 0.6 [GeV2]
0.8 1.0
Figure 3. Moments of the u — d quark distribution from various lattice simulations. The straight (long-dashed) lines are linear fits to this data, while the curves have the correct LNA behaviour in the chiral limit — see the text for details. The small squares are the results of the meson cloud model and the dashed curve through them best fits using Eq. (4). The stars represent the phenomenological values taken from NLO fits in the MS scheme. The figure is taken from Ref. [25].
with the form25:
(xn)u-d = an + K ml + an cLNA ml In ml ml + (i<
(4)
123
where the coefficient24, cLNA = - ( 3 ^ + l ) / (47r/ f f)2 . The parameters a„ , bn
are determined by fitting the lattice data . The mass /j determines the scale
above which pion loops no longer yield rapid variation and corresponds to
the upper limit of the momentum integration if one applies a sharp cut-off in
the pion loop integral. Consistent with our earlier discussion of this scale it
is taken to be 550 MeV. Multi-meson loops and other contributions cannot
give rise to LNA behaviour and thus, near the chiral limit, Eq. (4) is the most
general form for moments of the PDFs at (^(m2) which is consistent with chiral
symmetry. We stress tha t \i is not yet determined by the lattice da t a and it is
indeed possible to consistently fit both the lattice da ta and the experimental
values with \i ranging from 400 MeV to 700 MeV. This dependence on \i is
illustrated in Fig. 3 by the difference between the inner and outer envelopes
on the fits. Data at smaller quark masses, ideally m 2 ~ 0.05-0.10 GeV 2 , are
therefore crucial to constrain this parameter in order to perform an accurate
extrapolation based solely on lattice da ta .
4 . C o n c l u s i o n
The next few years will see tremendous progress in our understanding of hadron
structure. In combination with the very successful techniques for chiral ex
trapolat ion, which we have illustrated by jus t a few examples, lattice Q C D
will finally yield accurate da ta on the consequences of non-perturbative Q C D .
Furthermore, the physical insights obtained will guide the development of new
quark models which provide a much more realistic representation of QCD.
A c k n o w l e d g m e n t s
This work was supported by the Australian Research Council and the Univer
sity of Adelaide.
R e f e r e n c e s
1. A. W. Thomas and W. Weise, "The Structure of the Nucleon," ISBN 3-527-40297-7 Wiley-VCH, Berlin 2001.
2. D. G. Richards et al. [LHPC Collaboration], Nucl. Phys. Proc. Suppl. 109, 89 (2002) [arXiv:hep-lat/0112031].
3. J. M. Zanotti et al., Nucl. Phys. Proc. Suppl. 109, 101 (2002) [arXiv:hep-lat/0201004].
4. T. Lippert, S. Gusken and K. Schilling, Nucl. Phys. Proc. Suppl. 83, 182 (2000). 5. L. F. Li and H. Pagels, Phys. Rev. Lett. 26, 1204 (1971). 6. J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984). 7. W. Detmold et al., Pramana 57, 251 (2001) [arXiv:nucl-th/0104043]. 8. A. W. Thomas, Adv. Nucl. Phys. 13, 1 (1984).
124
9. T. Hatsuda, Phys. Rev. Lett. 65, 543 (1990). 10. G. A. Miller, A. W. Thomas and S. Theberge, Phys. Lett. B91, 192 (1980). 11. S. Theberge, G. A. Miller and A. W. Thomas, Can. J. Phys. 60, 59 (1982). 12. E. J. Hackett-Jones et at., Phys. Lett. B494, 89 (2000) [hep-lat/0008018]. 13. G. V. Dunne et al., Phys. Lett. B531, 77 (2002) [hep-th/0110155]. 14. D. B. Leinweber et al., Phys. Rev. D60, 034014 (1999). 15. T. R. Hemmert and W. Weise, arXiv:hep-lat/0204005. 16. E. J. Hackett-Jones et al., Phys. Lett. B489, 143 (2000). 17. D. B. Leinweber and A. W. Thomas, Phys. Rev. D62, 074505 (2000). 18. K. S. Kumar and P. A. Souder, Prog. Part. Nucl. Phys. 45, S333 (2000). 19. I. C. Cloet, D. B. Leinweber and A. W. Thomas, arXiv:hep-ph/0203023. 20. I. C. Cloet, D. B. Leinweber and A. W. Thomas, "Baryon Resonance Phe
nomenology" , in these proceedings. 21. M. Gockeler et al, Nucl. Phys. Proc. Suppl. 53, 81 (1997). 22. D. Dolgov et al., Nucl. Phys. Proc. Suppl. 94, 303 (2001). 23. A. W. Thomas et al., Phys. Rev. Lett. 85, 2892 (2000). 24. D. Arndt and M. J. Savage, nucl-th/0105045;
J. Chen and X. Ji, hep-ph/0105197. 25. W. Detmold et al., Phys. Rev. Lett. 87, 172001 (2001). 26. P. A. Guichon, Phys. Lett. B163, 221 (1985).
BARYON RESONANCE PHENOMENOLOGY
I.C. CLOET, D.B. LEINWEBER AND A.W. THOMAS Special Research Centre for the Subatomic Structure of Matter and
Department of Physics and Mathematical Physics, University of Adelaide, SA 5005, Australia
E-mail: dleinweb6physics.adelaide.edu.au; icloetQphysics.adelaide.edu.au; [email protected]
The Japan Hadron Facility will provide an unprecedented opportunity for the study of baryon resonance properties. This paper will focus on the chiral nonanalytic behaviour of magnetic moments exclusive to baryons with open decay channels. To illustrate the novel features associated with an open decay channel, we consider the "Access" quark model, where an analytic continuation of chiral perturbation theory is employed to connect results obtained using the constituent quark model in the limit of SU(3)-flavour symmetry to empirical determinations.
1. Introduction
The Japan Hadron Facility will present new opportunities for the investigation of baryon resonance properties. In particular, access to the hyperons of the baryon decuplet will be unprecedented. This contribution serves to highlight the novel and important aspects of QCD that can be explored through an experimental program focusing on decuplet-baryon resonance phenomenology.
To highlight the new opportunities, it is sufficient to address the magnetic moments of the charged A baryons of the decuplet. The magnetic moments of these baryons have already caught the attention of experimentalists and hold the promise of being accurately measured in the foreseeable future. Experimental estimates exist for the A + + magnetic moment, based on the reaction 7T+ p ->• 7T+ 7' p. The Particle Data Group1 provides the range 3.7-7.5 /z;v for the A + + magnetic moment with the most recent experimental result2 of 4.52 ± 0.50 ± 0.45 p.^. In principle, the A + magnetic moment can be obtained from the reaction y p —̂ 7r° 7' p, as demonstrated at the Mainz microtron3. An experimental value for the A + magnetic moment appears imminent.
Recent extrapolations of octet baryon magnetic moments4,5,6 have utilized an analytic continuation of the leading nonanalytic (LNA) structure of Chiral Perturbation Theory (xPT), as the extrapolation function. The unique feature of this extrapolation function is that it contains the correct chiral behaviour
125
126
as mq —y 0 while also possessing the Dirac moment mass dependence in the heavy quark mass regime.
The extrapolation function utilized here has these same features, however we move beyond the previous approach by incorporating not only the LNA but also the next to leading nonanalytic (NLNA) structure of xPT in the extrapolation function. Incorporating the NLNA terms contributes little to the octet baryon magnetic moments, however it proves vital for decuplet baryons. The NLNA terms contain information regarding the branch point at m^ = M A — MN, associated with the A —> Nn decay channel and play a significant role in decuplet-baryon magnetic moments.
2. Leading and Next-to-Leading Nonanalytic Behaviour
We begin with the chiral expansion for decuplet baryon magnetic moments7. The LNA and NLNA behaviour is given by
<HGK + \ra{G* - QK) , (i)
where qt and 1^ are charge and isospin respectively, and Gj (j = T , K) is given
by
Sj = 3 ^ \^H2T%milN) +Cin-6Ntmj,N)} . (2)
% describes the meson coupling to decuplet baryons and C describes octet-decuplet transitions. We take 7i = —2.2 and C = —1.2. Here we omit the Roper7 as this transition requires significant excitation energy and is strongly suppressed by the finite size of the meson source. The octet-decuplet mass splitting <5jv is assigned its average value
SN = Mio - M8 = 1377 - 1151 = +226 MeV.
We take fn - 93 MeV and fK = 112 MeV. The function F(5,m,fi) has the forma
m > | 6 | ,
H5,m,„) = - £ l n (%) + V ^ ^ f j - t a n " 1 - 7 4 = ^ \ >
m<\&\,
HS,rn,,) = J - l n ( 4 ) +1 - V s ^ ^ l n ( ' - V ^ Z ^ ) . (3)
7T \fi2 J n ys + y/S2 - m 2 J
"This definition for F(8, m, n) corrects a sign error in Ref. [7]. It differs by an overall minus sign and suppresses additive constants which are irrelevant in our analysis. As S —y 0, f ( J , m i , / i ) ->• rn.Tr.
127
Hence the LNA behaviour of decuplet magnetic moments is given by
XTT mn + XK mK + X'„ ^(-^N,mn,n7r) + X'K F{-6N,mK,nK),
where
(4)
Xvr =
XK
108 MNH2
MNn2
*(fi<)2 V108 qi_
72 XK =
MNC2
* CM2
MNC2
48
T (/x)2 V48 32 (5)
The values for the above chiral coefficients, Eq. (5), describing the strength of various meson dressings of the A baryons, are summarized in Table 1 for the four A baryons of the decuplet.
Table 1. The baryon chiral coefficients for the four A baryons of the decuplet. Coefficients are calculated with H — -2 .2 and C = - 1 . 2 , where H is the decu-plet-decuplet coupling constant and C is the decuplet-octet coupling constant. We have suppressed the kaon loop contribution by using SK = 1-2 /jr and jn — 93 MeV.
X*r XK X'* XK
A++
-2 .33 -1 .61 -1 .56 -1 .08
A+
-0 .777 -1.070 -0 .518 -0 .719
A0
+0.777 -0.535 +0.518 -0 .361
A ~
+2.33 0
+ 1.56 0
3. Analytic Continuation of x P T
It is now recognized that in any extrapolation from the heavy quark regime (where constituent quark properties are manifest) to the physical world, it is imperative to incorporate the quark-mass dependence of observables predicted by xPT in the chiral limit. However, as results are often obtained using methods ideally suited for heavy quark masses, it is imperative for the extrapolation function to correctly reflect the behaviour of the physical observable in the heavy quark mass regime as well. An extrapolation function for the A-baryon magnetic moments satisfying these criteria is
^~ l-T(mn)/^lQ + |3m^, W
where JIQ and /3 are parameters optimized to fit results obtained near the strange quark mass and r(m7r) is taken from the chiral expansion for decuplet magnetic moments in Sec. 2
128
IXm*) = X-K m* +XK (mK - m^) +x'n {^r{SN,mir,/ir) - Tv) ^v^ - v , N v ,
A B C
+XK {.H-&N,mK,pK)-TK\, (7)
where m^- , Tw and ^ x are constants defined to ensure that each term A, B, C and D, vanishes in the chiral limit. Utilizing the following relations provided b y X P T
mK2 = m{°) +-mv
2, (8)
mf = yjim^y-lim^y , (9) the four terms of Eq. (7) vanish in the chiral limit provided
' v " / * v2 M 0 ) ) 8 - ^ v (10)
Figure 1 presents a plot of each of the four terms of Eq. (7) (without the chiral coefficient pre-factors) as a function of m j . Figure 2 presents a plot of the four terms summed with the appropriate weightings of Table 1 for each of the four charge states of the A.
The extrapolation function of Eq. (6) is designed to reproduce the leading and next-to-leading nonanalytic structure expressed in Eq. (7) for expansions about mn = 0. Equation (6) may be regarded as an analytic continuation of Eq. (7), preserving the constraints imposed by chiral symmetry and introducing the heavy quark mass regime behaviour to the extrapolation function. The LNA behaviour of Eq. (7) is complemented by terms analytic in the quark mass with fit parameters po and 0 adjusted to fit additional constraints on the observable under investigation.
Hence the extrapolation function guarantees the correct nonanalytic behaviour in the chiral limit. Further, as mn becomes large, Eq. (6) is proportional to l /m£. As mw
2 oc mq over the applicable mass range, the magnetic moment extrapolation function decreases as \/mq for increasing quark mass, precisely as the Dirac moment requires. This extrapolation function therefore provides a functional form bridging the heavy quark mass regime and the chiral limit.
129
3
in
a 0 -t->
o 3
"ffl I H
£1 u
1.6
1.4
1.2
1.0
( I H
0.6
0.4
0.2
0.0
-0 .2
-0 .4
i
-
^
7 / ^ - r ^ ^ ^
i
1 1
C ^
^^\^^^~~k
!)___
r^^-^ ET^
i i
_
——=
-
-
-
0.0 0.2 0.4 0.6 m 2 (GeV2)
0.8 1.0
Figure 1. Plots of the four chiral expansion functions (without the chiral coefficient pre-factors) of Eq. (7), labelled A, B, C, D in Eq. (7).
0.4 0.6 m / (GeV 2 )
Figure 2. Plots of the sum of all four chiral expansion terms of Eq. (7), for each A baryon.
4. Resul t s
The method employed to obtain our theoretical predictions is analogous to that presented in our previous analysis of octet baryon magnetic moments4. We take the established input parameters, the strange-constituent and strange-current
130
quark masses (Ms and c m^hys respectively6), obtained by optimizing agreement between the AccessQMc and octet baryon magnetic moments. There, Ms = 565 MeV and c ra.Phys = 144 MeV provides optimal agreement.
The constituent quark model (CQM) provides the following formulas that relate the constituent quark masses to the delta magnetic moments:
HA++ = 3/iu , fiA+ = IVu + Hd , f^^o = flu + 2/J.d , ^ A - = 3 ^ d , (11)
with
2M W IMN \MN ,1 0 . ^ = 3 1 ^ ^ ' ^ = " 3 M 7 m ' ^ - S M T ^ ' (12)
These formulas are used to obtain two magnetic moment data points near the SU(3)-flavour limit where u and d quarks take values near the s-quark mass.
To fit Eq. (6), which is a function of mn, to the magnetic moments given by the CQM in Eq. (11) with constituent-quark masses Mu = Md = Mi (i = 1,2), we relate the pion mass to the constituent quark mass via the current quark mass4. Chiral symmetry provides
- ^ - = m ' (13) m? h y s (m£h y s)2 ' l ;
where mj?hys is the quark mass associated with the physical pion mass, mfy s . From lattice studies, we know that this relation holds well over a remarkably large regime of pion masses, up to m r ~ 1 GeV. The link between constituent and current quark masses is provided by
M = Mx + cmq, (14)
where Mx is the constituent quark mass in the chiral limit and c is of order 1. Using Eq. (13) this leads to
c m\ phys
M = Mx + ̂ t^2ml- (15) (m? r
The link between the constituent quark masses M; and mn is thus provided by
The parameter c is expected to be the order of 1. c The name indicates the mathematical origins of the model: Analytic Continuation of the Chiral Expansion for the SU(6) Simple Quark Model.
131
Table 2. Theoretical predictions for the charged A baryon magnetic moments. The fit parameters po and /3 are given for each scenario. The only currently known experimental value for the A baryon magnetic moments is the A++ moment, where recent measurements provide2 / iA++ = 4.52 ± 0.50 ± 0.45 /ijy.
Baryon
A++ A+ A"
Mo
+5.67 +2.69 -3 .22
0 0.16 0.20 0.06
AccessQM (AIJV)
+5.39 +2.58 -2 .99
where Ms — c mphys = Mx encapsulates information on the constituent quark mass in the chiral limit, and c rafys provides information on the strange current quark mass. We use the ratio
rnPhys
-phys = 24.4 ± 1 . 5 . (17)
provided by x ? T 8 to express the light current quark mass, mPhys, in terms of the strange current quark mass, mfy s , in Eq. (16).
The analytic continuation of xPT, Eq. (6), is fit to the constituent quark model (CQM) as a function of mn
2. Results are presented in Figs. 3 and 4. The magnetic moments given by the CQM either side of the SU(3)-flavour limit are indicated by a dot (•) and the theoretical prediction is indicated at the physical pion mass by a star (*). These results along with the parameters Ho and /3, are summarized in Table 2.
The interesting feature of these plots is the cusp at mw2 = &M2 which
indicates the opening of the octet decay channel, A -> Nir. The physics behind the cusp is intuitively revealed by the relation between the derivative of the magnetic moment with respect to ml and the derivative with respect to the momentum transfer q2, provided by the pion propagator l/(q2 + ml) in the heavy baryon limit. Derivatives with respect to q2 are proportional to the magnetic charge radius in the limit q2 -> 0,
(r2M) = -^GM(q2)\q,=0. (18)
If we consider for example A++ ->• pn+ with | j , rrij) = |3/2, 3/2), the lowest-lying state conserving parity and angular momentum will have a relative P-wave orbital angular momentum with \l,mj) — |1,1). Thus the positively-charged pion makes a positive contribution to the magnetic moment. As the opening of the p n+ decay channel is approached from the heavy quark-mass regime, the range of the pion cloud increases in accord with the Heisenberg uncertainty principle, AE At ~ h. Just above threshold the pion cloud extends
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6 i —i 1 1 1 1 r
M -
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 mj1 (GeV2)
Figure 3. The extrapolation function fit for A++ and A+ magnetic moments. The magnetic moments given by the CQM either side of the SU(3)-flavour limit are indicated by dots (•) and the theoretical prediction for each baryon is indicated at the physical pion mass by a star (*). The only available experimental data is for the A + + and is indicated by an asterisk (*). The proton extrapolation4 (dashed line) is included to illustrate the effect of the open decay channel, A —¥ N TT, in the A"*" extrapolation. The presence of this decay channel gives rise to a A+ moment smaller than the proton moment.
towards infinity as AE -» 0 and the magnetic moment charge radius diverges. Similarly, (d/drn^)GM —*• —oo. Below threshold, GM becomes complex and the magnetic moment of the A is identified with the real part. The imaginary part describes the physics associated with photon-pion coupling in which the pion is subsequently observed as a decay product.
It is the NLNA terms of the chiral expansion for decuplet baryons that contain the information regarding the decuplet to octet transitions. These transitions are energetically favourable making them of paramount importance in determining the physical properties of A baryons. The NLNA terms serve to enhance the magnitude of the magnetic moment above the opening of the decay channel. However, as the decay channel opens and an imaginary part develops, the magnitude of the real part of the magnetic moment is suppressed. The strength of the LNA terms, which enhance the magnetic moment magnitude as the chiral limit is approach, overwhelms the NLNA contributions such that the magnitude of the moments continues to rise towards the chiral limit.
The inclusion of the NLNA structure into octet baryon magnetic moment
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-1 .0
-1 .5
r̂ -2.o c
6
o - 2 . 5
- 3 . 0
- 3 . 5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 m 2 (GeV2)
Figure 4. The extrapolation function fit for the A - magnetic moment. The magnetic moments given by the CQM either side of the SU(3)-flavour limit are indicated by dots (•) and the theoretical prediction is indicated at the physical pion mass by a star (*). There is currently no experimental value for the A - magnetic moment.
extrapolations is less important for two reasons. The curvature associated with the NLNA terms is negligible for the N and E baryons and small for the A and S baryons. More importantly one can infer the effects of the higher order terms of xPT, usually dropped in truncating the chiral expansion, through the consideration of phenomenological models. If one incorporates form factors at the meson-baryon vertices, reflecting the finite size of the meson source, one finds that transitions from ground state octet baryons to excited state baryons are suppressed relative to that of xPT to finite order, where point-like couplings are taken. In xPT it is argued that the suppression of excited state transitions comes about through higher order terms in the chiral expansion. As such, the inclusion of NLNA terms alone will result in an overestimate of the transition contributions, unless one works very near the chiral limit where higher order terms are indeed small. For this reason octet to decuplet or higher excited state transitions have been omitted in previous studies4'5'6.
In the simplest CQM with mu = md, the A + and proton moments are degenerate. However, spin-dependent interactions between constituent quarks will enhance the A + relative to the proton at large quark masses, and this is supported by lattice QCD simulation results9. As a result, early lattice QCD predictions based on linear extrapolations9 report the A + moment to be
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greater than the proton moment . However with the extrapolations presented
here which preserve the LNA behaviour of x P T , the opposite conclusion is
reached. We predict the A + and proton magnetic moments of 2.58 //JV a n d
2.77 /ijy respectively. The proton magnetic moment extrapolation4 is included
in Fig. 3 as an illustration of the importance of incorporating the correct nonan
alytic behaviour predicted by %PT in any extrapolation to the physical world.
An experimentally measured value for the A + magnetic moment would offer
important insights into the role of spin-dependent forces and chiral nonanalytic
behaviour in the quark structure of baryon resonances.
5. C o n c l u s i o n
An extrapolation function for the decuplet baryon magnetic moments has been
presented. This function preserves the leading and next-to-leading nonanalytic
behaviour of chiral perturbation theory while incorporating the Dirac-moment
dependence for moderately heavy quarks. Interesting nonanalytic behaviour of
the magnetic moments associated with the opening of the n N decay channel
has been highlighted. It will be interesting to apply these techniques to existing
and forthcoming lattice QCD results, and research in this direction is currently
in progress.
An experimental value exists only for the A + + magnetic moment where
P A + + = 4.52 ± 0.50 ± 0.45 (IN- This value is in good agreement with the
prediction of 5.39 HN given by our AccessQM as described above. Arrival of
experimental values for the A + and A - magnetic moments are eagerly antic
ipated and should be forthcoming in the next few years. More importantly,
these techniques may be applied to the decuplet hyperon resonances where the
role of the kaon cloud becomes important . We look forward to new J H F results
in this area in the future.
A c k n o w l e d g e m e n t
This work was supported by the Australian Research Council.
R e f e r e n c e s
1. Particle Data Group, Eur. Phys. J., C15, (2000). 2. A. Bosshard et al., Phys. Rev. D44, 1962 (1991). 3. M. Kotulla [TAPS and A2 Collaborations], Prepared for Hirschegg '01: Structure
of Hadrons: 29th International Workshop on Gross Properties of Nuclei and Nuclear Excitations, Hirschegg, Austria, 14-20 Jan 2001.
4. I. C. Cloet, D. B. Leinweber and A. W. Thomas, Phys. Rev. C65, 062201 (2002). 5. D. B. Leinweber, D. H. Lu and A. W. Thomas, Phys. Rev. D60, 034014 (1999).
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6. E. J. Hackett-Jones, D. B. Leinweber and A. W. Thomas, Phys. Lett. B489, 143 (2000).
7. M. K. Banerjee and J. Milana, Phys. Lett. D54, 5804 (1996). 8. H. Leutwyler, Phys. Lett. B378, 313 (1996). 9. D.B. Leinweber, R.M. Woloshyn, T. Draper, Phys. Rev. D46, 3067 (1992).
LATTICE QCD, GAUGE FIXING A N D THE TRANSITION TO THE PERTURBATIVE REGIME
A. G. WILLIAMS AND M. S T A N F O R D
Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, SA 5005, Australia
E-mail: Anthony. [email protected]; [email protected]
The standard definition of perturbative QCD uses the Faddeev-Popov gauge-fixing procedure, which leads to ghosts and the local BRST invariance of the gauge-fixed perturbative QCD action. In the nonperturbative regime, there appears to be a choice of using nonlocal Gribov-copy free gauges (e.g. Laplacian gauge) or of at tempting to maintain local BRST invariance at the expense of admitting Gribov copies and somehow summing or averaging over them. It should be recognized that the standard implementation of lattice QCD corresponds to the former choice even when only physical (i.e. colour singlet) observables are being calculated. These issues are introduced and briefly explained.
1. Introduction
Perturbative quantum chromodynamics (QCD) is formulated using the Faddeev-Popov gauge-fixing procedure, which introduces ghost fields and leads to the local BRST invariance of the gauge-fixed perturbative QCD action. These perturbative gauge fixing schemes include, e.g. the standard choices of covariant, Coulomb and axial gauge fixing. These are entirely adequate for the purpose of studying perturbative QCD, however, they fail in the nonperturbative regime due to the presence of Gribov copies, i.e. gauge-equivalent gauge-field configurations survive in the gauge functional integration after gauge fixing. One could define nonperturbative QCD by imposing a non-local Gribov-copy free gauge fixing (such as Laplacian gauge) or, alternatively, one could attempt to maintain local BRST invariance at the cost of admitting Gribov copies. We will see that by definition the standard ensemble-averaging technique of lattice QCD corresponds to the former definition.
We begin by reviewing the standard arguments for constructing QCD perturbation theory, which use the Faddeev-Popov gauge fixing procedure to construct the perturbative QCD gauge-fixed Lagrangian density. The naive La-
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137
grangian density of QCD is
£QCD = —F^F^ + -£qj(iP- msMh (1)
where the index / corresponds to the quark flavours. The naive Lagrangian is neither gauge-fixed nor renormalized, however it is invariant under local SU(3)C gauge transformations g(x). For arbitrary small uia{x) we have
g(x) = expi-igs (j) ua(x)\ £ SU(Z), (2)
where the \a/2 = ta are the generators of the gauge group SU(3) and the index a runs over the eight generator labels a = 1, 2,..., 8.
Consider some gauge-invariant Green's function (for the time being we shall concern ourselves only with gluons)
wnmm = sv;™£*. ... where 0[A] is some gauge-independent quantity depending on the gauge field, A^(x). We see that the gauge-independence of 0[A] and S[A] gives rise to an infinite quantity in both the numerator and denominator, which must be eliminated by gauge-fixing. The Minkowski-space Green's functions are defined as the Wick-rotated versions of the Euclidean ones.
The gauge orbit for some configuration A^ is defined to be the set of all gauge-equivalent configurations. Each point A9 on the gauge orbit is obtained by acting upon A^ with the gauge transformation g. By definition the action, S[A], is gauge invariant and so all configurations on the gauge orbit have the same action, e.g. see the illustration in Fig. 1.
.gauge orbit
Figure 1. Illustration of the gauge orbit containing Ap and indicating the effect of acting on A^ with the gauge transformation p. The action S[A] is constant around the orbit.
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The integral over the gauge fields can be written as the integral over a full set of gauge-inequivalent (i.e. gauge-fixed) configurations, fVAs{, and an integral over the gauge group fVg. In other words, fVAst is an integral over the set of all possible gauge orbits and fDg is an integral around the gauge orbits. Thus we can write
I VA= fvA6t fvg. (4)
To make integrals such as those in the numerator and denominator of Eq. (3) finite and also to study gauge-dependent quantities in a meaningful way, we need to eliminate this integral around the gauge orbit, fT>g.
2. Gribov Copies and the Faddeev-Popov Determinant
Any gauge-fixing procedure defines a surface in gauge-field configuration space. Figure 2 is a depiction of these surfaces represented as dashed lines intersecting the gauge orbits within this configuration space. Of course, in general, the gauge orbits are hypersurfaces and so are the gauge-fixing surfaces. Any gauge-fixing surface must, by definition, only intersect the gauge orbits at distinct isolated points in configuration space. For this reason, it is sufficient to use lines for the simple illustration of the concepts here. An ideal (or complete) gauge-fixing condition, F[A] — 0, defines a surface that intersects each gauge orbit once and only once and by convention contains the trivial configuration A,, = 0. A non-ideal gauge-fixing condition, F'[A] = 0, defines a surface or surfaces which intersect the gauge orbit more than once. These multiple intersections of the non-ideal gauge fixing surface(s) with the gauge orbit are referred to as Gribov copies1'2,3,4'5'6. Lorentz gauge (dflA^(x) = 0) for example, has many Gribov copies per gauge orbit. By definition an ideal gauge fixing is free from Gribov copies. We refer to the ideal gauge-fixing surface ^[^4] = 0 as the Fundamental Modular Region (FMR) for that gauge choice. Typically the gauge fixing condition depends on a space-time coordinate, (e.g. Lorentz gauge, axial gauge, etc.), and so we write the gauge fixing condition more generally as F([Aj;a:) = 0.
Let us denote one arbitrary gauge configuration per gauge orbit as A° and let this correspond to the "origin" of gauge configurations on that gauge orbit, i.e. to g — 0 on that orbit. Then each gauge orbit can be labelled by A° and the set of all such A°„ is equivalent to one particular, complete specification of the gauge. Under a gauge transformation, g, we move from the origin of the gauge orbit to the configuration, A3^, where by definition
A° - ^ A9^ — gA°^g^ - (i/gs)(d^g)g^. Let us denote for each gauge orbit the gauge transformation, g = #[J4°], as the transformation which takes us
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F'[A]=0
Figure 2. Ideal, F[A], and non-ideal, F'[A], gauge-fixing.
from the origin of that orbit, A° , to the configuration, Aff-, which lies on the ideal gauge-fixed surface specified by F([A];x) = 0. In other words, we have F{[A];x)\Ai = 0 for A« = A**- G FMR.
The inverse Faddeev-Popov determinant is defined as the integral over the gauge group of the gauge-fixing condition, i.e.
fSF([A];x" - 1
-,l[A^f} = jvg8[F[A}]jv98{g-g) det V Sg(y) (5)
Let us define the matrix Mf [A] as
MF([A];x,y)ab (6)
SFa([A];x)
Sgb(y)
Then the Faddeev-Popov determinant for an arbitrary configuration A^ can be defined as Ap[A] = |det Mf[^4]|. (The reason for the name is now clear). Note that we have consistency, since A ^ 1 ^ 6 ^ ] = A ^ 1 ^ ] = fVg S(g — g)AF
l\A\. We have 1 = fVg AF[A] £[F[y4]] by definition and hence
I VAef- = f VA^- f Vg AF[A] S[F[A]] = f VA AF[A] S[F[A]]. (7)
Since for an ideal gauge-fixing there is one and only one g per gauge orbit, such that -F([A|; x)\§ = 0, then |detMf[^4]| is non-zero on the FMR. It follows that if there is at least one smooth path between any two configurations in the FMR and since the determinant cannot be zero on the FMR, then it cannot change sign on the FMR. The Gribov horizon is defined by those configurations with det M F [ A ] = 0 which lie closest to the FMR. By definition the determinant can change sign on or outside this horizon. Clearly, the FMR is contained within the Gribov horizon and for an ideal gauge fixing, since the sign of the determinant cannot change, we can replace IdetM^I with det M^, [i.e. the overall sign of the functional integral is normalized away in Eq. (3)].
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These results are generalizations of results from ordinary calculus, where
d e t ( | ^ ) _ = jdx1---dxnS^(f(x)), (8)
and if there is one and only one x which is a solution of f(x) — 0 then the matrix Mjj = dfi/dxj is invertible (i.e. non-singular) on the hypersurface f(x) = 0 and hence det M ^ 0.
3. Generalized Faddeev-Popov Technique
Let us now assume that we have a family of ideal gauge fixings F([i4];i) = f([A];x) — c(x) for any Lorentz scalar c(x) and for /([A]; x) being some Lorentz scalar function, (e.g. d^A^x) or n^A^x) or similar or any nonlocal generalizations of these). Therefore, using the fact that we remain in the FMR and can drop the modulus of the determinant, we have
fvA«f = JVA det MF[A] S[f[A] - c]. (9)
Since c(x) is an arbitrary function, we can define a new "gauge" as the Gaussian weighted average over c(x), i.e.,
fvA**- oc fvcexpl-^- fd4xc(x)2\ fVA detMF[A] S[f[A] - c]
oc / x > y l d e t M F [ ^ ] e x p | - ^ f d4xf{[A];x)2\
ex jvAVXVx expl.-ijd4xd4y x x(x)MF([A];x,y)X{y)\
xexp{~Jd4xf([A];x)2}, (10)
where we have introduced the anti-commuting ghost fields \ and \. Note that this kind of ideal gauge fixing does not choose just one gauge configuration on the gauge orbit, but rather is some Gaussian weighted average over gauge fields on the gauge orbit. We then obtain
where
\Fa,u,F% - 1 (f([A];x))2 + Y,9f(ip- "»/)«/
+ J d4xdAy x(x)MF([A]; x, y)X(y). (12)
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4. S t anda rd Gauge Fixing
We can now recover standard gauge fixing schemes as special cases of this generalized form. First consider standard covariant gauge, which we obtain by taking f([A];x) = dliA>1(x) and by neglecting the fact that this leads to Gribov copies. We need to evaluate Mf[A] in the vicinity of the gauge-fixing surface for this choice:
MJIWx vYb 8 F a ^ x ) - *[^A«>(x)-c{x)\ _ SA°»{*) m ) MF([A],x,y) - Sgb{y) - Sgb{y) - 0 „ ^ ( t f ) ] • (13)
Under an infinitesimal gauge transformation g we have
A$(x)Sm-^9 (A%(x) = Al(x) + 9sfab^b(x)Al(x) - d^a{x) + 0(J){\A)
and hence in the neighbourhood of the gauge fixing surface (i.e. for small fluctuations along the gauge orbit around A& f•), we have
MF{[A];x,yyb=d*JAail{x) (15) u-0 6u»(y)}
= d; ( [ - c W +g,fabeAcHx)] x S^(x - y)) .
We then recover the standard covariant gauge-fixed form of the QCD action
+(dl>Xa)(d"6ab-gfabcA2)Xb. (16)
However, this gauge fixing has not removed the Gribov copies and so the formal manipulations which lead to this action are not valid. This Lorentz covariant set of naive gauges corresponds to a Gaussian weighted average over generalized Lorentz gauges, where the gauge parameter £ is the width of the Gaussian distribution over the configurations on the gauge orbit. Setting £ — 0 we see that the width vanishes and we obtain Landau gauge (equivalent to Lorentz gauge, dl'All(x) = 0). Choosing £ = 1 is referred to as "Feynman gauge" and so on.
We can similarly recover the standard QCD action for axial gauge, where n^A^(x) = 0. Proceeding as for the generalized covariant gauge, we first identify /([.A]; a;) = ntiA
>i(x) and obtain the gauge-fixed action
SdQ,q,A] = j <Px \Fa^F;u - ^ (n.A^x))2 +Y,q,(iP - mj)qf (17)
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Taking the "Landau-like" zero-width limit £ —> 0 we select nltA,t(x) = 0
exactly and recover the usual axial-gauge fixed QCD action. Axial gauge does
not involve ghost fields, since in this case
MF{[A^);x,yYb=n,8-£^ _ = „„ {[-d^Sab]S^(x - yj) , (18)
which is independent of A^ since n^Aff(x) = 0. In other words, the gauge
field does not appear in Mp[A] on the gauge-fixed surface. Unfortunately axial
gauge suffers from singularities which lead to significant difficulties when trying
to define perturbat ion theory beyond one loop. A related feature is tha t axial
gauge is not a complete gauge fixing prescription. While there are complete
versions of axial gauge on the lattice, these always involve a nonlocal element,
or reintroduce Gribov copies at the boundary so as not to destroy the Polyakov
loop.
5. D i s c u s s i o n a n d Conc lus ions
There is no known Gribov-copy-free gauge fixing which is a local function of
A^x). In other words, such a gauge fixing cannot be expressed as a function
of A^(x) and a finite number of its derivatives, i.e. F([yt];;c) ^ F{d,j,,A^{x))
for all x. Hence, the gauge-fixed action, 5 j [• • •], in Eq. (12) becomes non-local
and gives rise to a nonlocal quantum field theory. Since this action serves as
the basis for the proof of the renormalizability of QCD, the proof of asymptotic
freedom, local BRS symmetry, and the Schwinger-Dyson equations (to name
but a few) the nonlocality of the action leaves us without a reliable basis from
which to prove these features of QCD in the nonperturbative context.
It is well-established tha t Q C D is asymptotically free, i.e. it has weak-
coupling at large momenta . In the weak coupling limit the functional integral is
dominated by small action configurations. As a consequence, momentum-space
Green's functions at large momenta will receive their dominant contributions
in the pa th integral from configurations near the trivial gauge orbit, i.e. the
orbit containing A^ = 0, since this orbit minimizes the action. If we use
s tandard gauge fixing, which neglects the fact that Gribov copies are present,
then at large momenta J VA will be dominated by configurations lying on the
gauge-fixed surfaces in the neighbourhood of each of the Gribov copies on the
trivial orbit. Since for small field fluctuations the Gribov copies cannot be
aware of each other, we merely overcount the contribution by a factor equal
to the number of copies on the trivial orbit. This overcounting is normalized
away by the ratio in Eq. (3) and becomes irrelevant. Thus it is possible to
understand why Gribov copies can be neglected at large momenta and why it
is sufficient to use s tandard gauge fixing schemes as the basis for calculations
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in perturbative QCD. Since renormalizability is an ultraviolet issue, there is no question about the renormalizability of QCD.
Lattice QCD has provided numerical confirmation of asymptotic freedom, so let us now turn our attention to the matter of Gribov copies in lattice QCD. Since the observable 0[A] and the action are both gauge-invariant it does not matter whether we sample from the FMR of an ideal gauge-fixing or elsewhere on the gauge orbit. The trick is simply to sample at most once from each orbit. Since there is an infinite number of gauge orbits (even on the lattice), no finite ensemble will ever sample the same orbit twice. This makes Gribov copies and gauge-fixing irrelevant in the calculation of colour-singlet quantities on the lattice.
The calculation of gauge-dependent Green's functions on the lattice does require that the gauge be fixed. The standard choice is naive lattice Landau gauge, which selects essentially at random between the Landau gauge Gribov copies for the gauge orbits represented in the ensemble. This means that, while the gauge fixing is well-defined in that there are no Gribov copies, the Landau gauge-fixed configurations are not from a single connected FMR. For this reason lattice studies of gluon and quark propagators are now being extended to Laplacian gauge for comparison. Laplacian gauge is interesting because it is Gribov-copy-free (except on a set of configurations of measure zero) and it reduces to Landau gauge at large momenta. Lattice calculations of the Laplacian gauge and Landau gauge quark and gluon propagators converge at large momenta and hence are consistent with this expectation.
In conclusion, it should be noted that throughout this discussion there has been the implicit assumption that nonperturbative QCD should be defined in such a way that each gauge orbit is represented only once in the functional integral, i.e. that it should be defined to have no Gribov copies. This is the definition of nonperturbative QCD implicitly assumed in lattice QCD studies. We have seen that this assumption destroys locality and the BRS invariance of the theory. An equally valid point of view is that locality and BRS symmetry are central to the definition of QCD and must not be sacrificed in the nonperturbative regime, (see, e.g. Ref. [3,4,5,6]). This viewpoint implies that Gribov copies are necessarily present, that gauge orbits are multiply represented, and that the definition of nonperturbative QCD must be considered with some care. Since these nonperturbative definitions of QCD appear to be different, establishing which is the one appropriate for the description of the physical world is of considerable importance.
144
References
1. L. Giusti, M. L. Paciello, C. Parrinello, S. Petrarca and B. Taglienti, Int. J. Mod. Phys. A16, 3487 (2001) [arXiv:hep-lat/0104012] and references therein.
2. P. van Baal, arXiv:hep-th/9711070. 3. H. Neuberger, Phys. Lett. B183, 337 (1987). 4. M. Testa, Phys. Lett. B429, 349 (1998) [arXiv:hep-lat/9803025]. 5. M. Testa, arXiv:hep-lat/9912029. 6. R. Alkofer and L. von Smekal, Phys. Rep. 353, 281 (2001) [arXiv:hep-ph/0007355]
and references therein.
QUARK MODEL A N D CHIRAL SYMMETRY ASPECTS OF EXCITED BARYONS
A. H O S A K A
Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0047 Japan
E-mail: [email protected]
We investigate properties of excited baryons from two different points of view. In one of them, the argument is based on a quark model with an emphasis on flavour independent nature of the baryon spectra. This easily indicates positions of missing resonances. In the other, we discuss how baryon resonances are classified by the chiral symmetry group. It is shown that there are two baryon representations. We present nir production experiments to distinguish the two baryon representations.
1. Introduction
It is important that we are able to control meson-baryon dynamics quantitatively as much as possible. This will provide not only a clue to understand fundamentals of hadron physics but also a basis for the description of more complicated system such as hadronic and/or quark matter in extreme conditions. The latter is one of the important subjects of the JHF project1.
Quark models provide now standard methods to describe hadrons, as we know empirically that they work well over a wide range of phenomena2. Flavour symmetry is implemented by regarding quarks as fundamental representations of SU(6) spin-flavour symmetry. By choosing constituent quark masses appropriately, baryon magnetic moments are well reproduced. Introducing another scale parameter for quark distributions (oscillator parameter for a quark confining force), numbers of excited states are predicted which can be compared with data, although fine tuning is necessary to get reasonable results.
We also know that chiral symmetry dictates much of hadron dynamics3. Chiral symmetry with its spontaneous breaking is indeed one of important aspects of low energy QCD. Interactions of pions and kaons, which are the Nambu-Goldstone bosons, are described well by the current algebra and chiral perturbation theory4.
To the present date, it is not obvious how chiral symmetry and the quark
145
146
model aspects can merge into a single framework. Nevertheless, here we attempt to demonstrate examples based on the above two aspects for excited baryons. In Sec. 2, we study masses and transition amplitudes of excited baryons in a quark model. We point out that baryon spectra resemble very much rotational bands of a deformed harmonic oscillator model. As an illustration, we compute not only masses but also pion transition amplitudes between the same rotational band. In Sec. 3, we propose another point of view where positive and negative parity baryons are considered as members of a multiplet of chiral symmetry. Two distinguished assignments of chiral symmetry of baryons are discussed from a group theoretical point of view. Experiments which can observe the two assignments are proposed. The final section is devoted to a brief summary of this report.
2. Quark Model Description and Deformed Excited States
2.1 . Masses
Let us first look at experimental data as shown in Fig. 1. We take 49 states out of 50 states of three and four stars, and several states with one and two stars5. In showing experimental data we follow the prescriptions: (1) Masses are measured from the ground states in each spin-flavour multiplet, in order to subtract the spin-flavour dependence which is well established by the Gell-Mann-Okubo mass formula. (2) Masses of 2 8 M S , 4&MS states for positive parity, and of 48JWS states for negative parity are reduced by 200 MeV. Then the resulting mass spectra show a very simple systematics, which remarkably resembles rotational bands. It is also important that the regularity is common to all channels independent of flavour.
Hence, we consider a quark model with a deformed harmonic oscillator potential6:
HDOQ - 22 7^ + 2 m^lx2i + "&? + w^.2) (1) » = l L J
Here we ignore interactions due to gluons and mesons. The only dynamics here is the shape change which is described by the deformed oscillator potential, u)x ^ uiy 7̂ <JJZ . The Hamiltonian looks very simple, but it can indeed reproduce the structure of the mass spectra in Fig. 1.
After removing the centre of mass motion, we find an intrinsic energy Eint(Nx,Ny,Nz) = (Nx + l)cux + [Ny + l)w„ + {Nz + l )w z , where Nx,Ny, Nz
are the sum of principal quantum numbers for the internal degrees of freedom for the p and A coordinates. Imposing the volume conservation condition LOxwyu)z = w3, the minimum energy occurs when the system is deformed. When
147
Nx = Ny = 0,NZ = N, one obtains a prolate deformation. Our discussions in what follows are based on this prolate deformation, since it is energetically most favourable.
Physical states are obtained by rotating the deformed states, whose energies are computed by the standard cranking method6
^rot — f^int <L2) + 2 , ( 1 + 1)
(2) 2/ 21
Here the moment of inertia / and the expectation value of the angular momentum fluctuation (L2) are given in the literature6. We note the presence of the second term which reduces the energy. This is crucially important to bring the mass of the l / 2 + excited states down closer to the observed values (the aspect which the naive spherical quark model can not explain). Hence the Roper resonance of l / 2 + , A^*(1440), appears as the band head of the N = 2 rotational band.
2000
1500
500
o -
Fl9<2000)
P„(1720) , J — " P„ ( 'M°) F » ' 2 " ° » F1S(1S15)
-> , , ,w-p^r ;—;• "=" "—--P„(14«) " P0,(1M0> P01(1810) P„(1W0)
P„(1880) F17<2030)
' -H,„(2420) F^(2390)
P,a(2M0) F„(1950)
F„(20701 .PM(1«0) . .
" " ~ ~ P},(1910)
13 P^deoo)
P,3(1385) P33I1232)
N ' 1 0 ,
L-2
L=0
10 0
JJ1500
£1000 -
B 500
Dlg(2200)
G^piOO)
S„ (153a D„(1700) DtelieSOl „ ( 1 B 3 ( ) 1 S„(1750)
S„(1650)
• 1^(2350) " •
I)
(1800) -D13(1670) • D1S(1775) Ss,(1620)
q,(210O)
0^(1520)
s„,<uos)
8MS % 10„
N
Figure 1. SU(3) baryon masses.
148
The only parameter in the DOQ model, w, is determined by the average of the excitation energies of the first l / 2 + excited states in the 28 multiplet. We find u — 644 MeV. The resulting mass spectra are compared with the observed spectrum in Fig. 1. The agreement between the theoretical and observed states is remarkable. In Fig. 1 it is easily recognized that there are many missing states in the region of L — 4 and L = 3. It would be interesting if those states are observed in experiments.
2.2. Pion Transition
If the deformation of the system is large, it has significant influence on transition amplitudes. A familiar example is the E2 transition between the same rotational band of deformed nuclear states, which is called intra-band transitions (see Fig. 2). Here a characteristic feature is that the decay rates are written as products of common factors of intrinsic moments and geometric factors of rotation (Clebsch-Gordan coefficients). Consequently, ratios between decay rates are given solely by the geometric factors. This is called the Alaga rule in nuclear physics7.
In the case of baryon decays, the strong interaction is dominant and, therefore, we investigate decays accompanied by a pion emission. We are interested in the intra-band transitions where the orbital angular momentum changes by AL = 2. Due to the pseudoscalar nature, the angular momentum and parity of the pion are l^ = 1 + or 3 + . The selection rule of the transitions for positive parity excited states is shown in Fig. 2.
Selection rules (even parity)
Figure 2. Inter- and intra-band transitions. Selection rules for even parity transitions are also shown.
Assuming the semiclassical nature of large deformation, we compute the amplitudes in the collective coordinate cranking method. Let us briefly outline how the decay amplitudes are computed8. Write a deformed excited state as V'intC*)- Here, for simplicity, x denotes the internal coordinates of three quarks. In the semiclassical picture, the deformed intrinsic state V'int(^) rotates
149
adiabatically: rj;TOt(A,x) — R(A)ipint(x), where R(A) is the rotation matrix of angle A. By taking a matrix element of the nqq interaction Hamiltonian, hwqq(x) = —#/(2m)V • ara (g is the nqq coupling constant, m the constituent quark mass and a the isospin index of the pion.), in the rotating state i>TOt, we obtain the collective Hamiltonian for the decay :
Hco\\{A) = J d3xi>lot{A,x)Kqq{x)j;rot{A,x). (3)
Now a collective rotational wave function of orbital angular momentum L is given by the standard D function of rank L, which is combined with a spin wave function Xm to form the state of total spin (J, M): ^JM(A) = [D%l0(A),xli]
JM • The decay amplitude of N*(JM) -»• •KN*(J'MI) is then given by
= jd[A] &J,M,(A)HCO]](A)*JM(A) . (4)
Computation of this matrix element is tedious but straightforward. The final expression is rather lengthy, but as anticipated, it is expressed as products of geometric factors and intrinsic moments of the type :
Q'-(A) = (mt\jlw(kx)Y,w0(x)\int). (5)
In Table 1 the decay widths for even parity intra-band transitions (L = 4 —> V — 2) are summarized. There is one large number, T(7/2+ —> 5/2+), but others are small. In particular, those of L — 2 —> V = 0 are negligible due to small phase space. The large decay width of T(7/2+ -» 5/2+) may explain the missing nucleon resonance around the mass 2200 MeV in the G71 pion channel. Experimentally, observation of intra-band transitions requires measurements of at least two pions, since the daughter resonance of the intra-band transition still decays into the ground state.
Table 1. Decay widths of L = 4 nucleon resonances in MeV.
9/2 -4 5/2 9/2 -> 3/2 7/2 -4 5/2 7/2 ->• 3/2 18.9 35.3 115 0.5
Apart from the absolute values of the various decay widths, it is of interest to see, in the long wave length limit, the ratios of transitions which are dominated by the geometric factors. Consider, for instance, 1% = 1 + transitions of L + 2 (Ji = L + 3/2) -J- L(Jf = L+ 1/2) and of L (J; = L - 1/2) -> L - 2 {Jf = L - 3/2). When L = 2 these transitions are L = 4 ( J f = 7/2+) ->• L = 2 (Jf = 5/2+) and L = 2 (Jf = 3/2+) -> L = 0 (jf = 1/2+) (see the
150
GeV
1.5
1 0
0.5
Mesons
[ . ° l (1260)
p(770)
<T('606) '
„- *(139)
/ , (1285) _ 1(1300)
' ' ° 1(1295)
,a 0(980)
OX 780) ° /-0(975)
AT, (1400) I*
K,<1270)
1 , K (890)
Baryons
,_ AT<1535) 5"
j - , _
£(1190) , . " ( ' 2 3 2 ) /1(1120) 2
. W(939) j . « — ^
Figure 3. Mass splittings of positive and negative parity hadrons in various channels. The uncertain mass of sigma (<r) is hatched.
right part of Fig. 2. Apart from the k7 (pion momentum) dependence due to the phase space, the ratio of the decay widths becomes
ri+2(J,=L+3/2)-)-L(J /=L+l/2) _ (L + 1)(2L - 1)
^L(Ji=L-l/2)-yL-2(Jf=L-3/2) (L ~ 1)(2L + 3)
This is analogous to the Alaga rule in nuclear physics7.
-> 9/7 (for L = 2). (6)
3. Chiral Symmetry of Baryons
3.1. Chiral Partner Baryons
Let us now turn to another topic : the role of chiral symmetry. Naively, if chiral symmetry is a good symmetry, one would expect that hadrons are classified by linear representations of chiral symmetry, where positive and negative parity states form a representation. In reality, however, due to spontaneous breaking, such a symmetry pattern may not necessarily be realized. Still, one can imagine that observed hadrons would belong to linear representations when chiral symmetry is restored. This is suggested by a common pattern of the mass splitting between the positive and negative parity states as shown in Fig. 3. In particular, all the mass splittings are of the same order of about half GeV, which could be generated by the spontaneous breaking.
In this way, one could imagine, for instance, that the ground state nucleoli belongs to the representation, ( | , 0) © (0, ^), of the isospin chiral group SU(2)/{X SU(2)i. The pions are assigned to ( | , i ) which is supplemented by the still controversial particle, sigma. Another example is a representation for p and ai , which can be identified with (1,0) ® (0,1).
In this report, we would like to discuss particle classifications for baryons9,10,11,12. So far, such an investigation has not been done very often, although it is a very fundamental question in hadron physics.
151
3.2. Naive and Mirror Assignments of Baryons
Chiral symmetry is a flavour symmetry for the right and left handed fermions. In QCD, the current quarks possess this symmetry. The right and left handed components (of the Lorentz group) are defined in terms of a four component nucleon field TV by
N = Nr + Nr, Nr = ^ ^ N , Nl=]—^N. (7)
For SU(2).RX S U ( 2 ) L , chiral (symmetry) transformations are defined as flavour transformations for the right and left handed components: Nr -> giiNr , Wj —>• 9LNI , where gR and gL are elements of SU(2)i{ and SU(2)^, respectively.
Now consider two fermions and their transformation rules. Naively, one would expect the same transformations for both N\ and N2 >
Nir - • gRNlr, Nu -> gLNu ; N2r -»• gnN2r, N2l -> gLN2i. (8)
As first noticed by Lee13, it is also possible to assign the opposite transformation for the second nucleon,
Nir - > gRNlr , Nu - > 9LNu \ A ^ ^ O L # 2 r , N2, -> gRN2l • (9)
The point is that it is sufficient to assign different internal symmetries for the right and left handed components. It does not matter whether which of SU(2)ft or SU(2)x is assigned. Since the transformations of the right and left handed components are interchanged, the second assignment of fiq. (9) is called mirror, while the first assignment of Eq. (8) is called naive. From Eqs. (8) and (9), it is obvious that the sign of the axial charge of N2 is opposite to that of Ni in the mirror assignment. The sign of g& was discussed by Weinberg for the case of single fermion15. It becomes significant when there are two fermions (nucleons here).
Another feature of the mirror assignment is that it is possible to introduce the chiral invariant mass term in the form of interaction between Ni and N2: mo(NiN2 + N2Ni), where mo is a parameter which can not be determined within the present theoretical arguments. Such a mass term was used by DeTar and Kunihiro14 in order to explain a possibility of finite mass of baryons when chiral symmetry is restored.
3.3. iv and rj Productions at Threshold Region
In order to distinguish the two chiral assignments, we propose an experimental method to study the two chiral assignments10. As discussed in the preceding sections, one of the differences between the naive and mirror assignments is the relative sign of the axial coupling constants of the positive and negative
152
(1) (2) (3)
VJV_/ / \ / / % / / N— 1 i 4 — w a -it — 3 >• 1
SnNN &AI*W gxNN
Figure 4. Dominant three diagrams for 7r and 77 productions. The incident wavy line is either a pion of photon. In fact, there are in total 12 resonance dominant diagrams.
parity nucleons. In the following discussions, we identify N+ ~ JV(939) and N_ ~ 7V(1535) = N*. Strictly, the identification of the negative parity nucleon with the first excited state Af(1535) is no more than an assumption. From the experimental point of view, however, ./V(1535) has a distinguished feature that it has a strong coupling with an rj meson, which can be used as a filter for the resonance. In practice, we observe the pion couplings which are related to the axial couplings through the Goldberger-Treiman relation gnN±N±fw — SuM± .
Let us consider n and 77 productions induced by a pion or photon. Suppose that the two diagrams of (1) and (2) as shown in Fig. 4 are dominant in these processes. Modulo energy denominator, the only difference of these processes is due to the coupling constants g^NN and gnN'N'- Therefore, depending on their relative sign, cross sections are either enhanced or suppressed. In the pion induced process, due to the p-wave coupling nature, another diagram (3) also contributes substantially.
In actual computation, we take the interaction Lagrangians :
L„NN = g-wNNNi-frT • TTN , LvNN. = gnNN' {Nr)N* + N*T)N),
L*NN> = g*NN> (NT • TTN* + N*T • nN), (10)
LnN'N' - gnN'N'{N*if5T • nN*) .
We use these interactions both for the naive and mirror cases with empirical coupling constants for gVNN ~ 13, g^NN* ~ 0.7 and gVNN* ~ 2. The coupling constants g-xNN' ~ 0.7 and gVNN' ~ 2 are determined from the partial decay widths5, rjv(i535)->irjv ^ rw(i535)-n?N ~ 70 MeV. The unknown parameter is the gwN*N' coupling. One can estimate it by using the theoretical value of the axial charge g*A and the Goldberger-Treiman relation for ./V*. When g*A = ±1 for the naive and mirror assignments, we find gnN'N' = ^ " ^ w / / ^ ~ ±17. Here, just for simplicity, we use the same absolute value as gnNN • The coupling values used in our computations are summarized in Table 2.
Several remarks are in order here:
• We assume resonance (N*) pole dominance. This is considered to be good, particularly for the t] production at the threshold region, since •q is dominantly produced by N*.
153
Table 2. Parameters used in our calculation.
mN
938 (MeV)
m^i'
1535 (MeV)
TJV*
140 (MeV)
9TTNN
13 9nNN*
0.7 9r)NN'
2.0 9nN'N'
13 (naive) -13 (mirror)
I Tip -> JtriN ]
~ 60
- 40
<j | 20 H
0
Total Cross Section
540 560 580 600
Pen, ( M e V / <=)
14-
12-
8 -
6 -
4 -
Pion momentum distribution
IP 1 = 575 MeV/c 1 am.1
Mirror
Naive
-0.5 0.0 0.5 10 „„_D c m .
>JtT)N 1
560 580 600 620 640 660 680
Photon energy [MeV]
[mbf Angular distribution
^ s . Naiv
Mirr
E = 590 MeV
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cog
Figure 5. Various cross sections for w and r\ productions.
• There are in total twelve resonance dominant diagrams. Due to the
energy denominator, the three diagrams in Fig. 4 are dominant .
• Background contributions, in which two meson (seagull) or three me
son vertices appear, are suppressed due to G-parity conservation.
We show various cross sections for the pion and photon induced processes
in Fig. 5. We observe tha t :
(1) Total cross sections are of the order of micro barn tha t is well accessible
by experiments.
(2) In the photon induced process, the diagrams (1) and (2) interfere de
structively. In the pion induced case, due to the momentum dependence
of the initial vertex, the third term (3) becomes dominant .
(3) In the pion induced reaction, the angular distribution of the final s ta te
pion differs crucially depending on the sign of the wNN and nN*N*
couplings.
154
4. S u m m a r y
In this report, we have discussed two important aspects of baryon dynamics:
one is a quark model aspect and the other is tha t of chiral symmetry. In
baryon mass spectra, it is emphasized once again tha t a simple non-relativistic
quark model of spatial defamation works well for almost all SU(3) baryon
masses independent of flavour. This suggests that this important dynamics
has gluonic nature. At the same t ime, chiral symmetry and relevant dynamics,
especially due to the chiral mesons, have also significant roles. Evidently the
coupling of the chiral mesons renormalizes the simple quark model calculations.
A more fundamental question is how multiplets of chiral symmetry are related
to those of the quark model states. In particular, from the chiral symmetry
argument, we have shown tha t two different types of baryons are possible which
are distinguished by the sign of the axial charge. It is interesting to investigate
a microscopic origin of such states, perhaps from meson effects, relativistic
quark effects and etc.
A c k n o w l e d g m e n t s
The author thanks his collaborators, Miho Koma, Hiroshi Toki, Daisuke Jido
and Makoto Oka on the subjects presented here.
R e f e r e n c e s
1. http://jkj.tokai.jaeri.go.jp/ 2. N. Isgur and G. Karl, Phys. Rev. D18, 4187 (1978); ibid. D19, 2653 (1979). 3. A. Hosaka and H. Toki, "Quarks, baryons and chiral symmetry", World Scientific
(2001), Singapore. 4. J. Gasser and H. Leutwyler, Ann. Phys. 158, 142 (1984). 5. Particle Data Group, Review of particle physics, Eur. Phys. J. C15 (2000). 6. M. Takayama and H. Toki and A. Hosaka, Prog. Theor. Phys. 101, 1271 (1999). 7. G. Alaga and V. Paar, Phys. Lett. B61, 129 (1976). 8. M. Koma, PhD. Thesis, Osaka University (2002). 9. D. Jido M. Oka and A. Hosaka, Prog. Theor. Phys. 106, 873 (2001). 10. D. Jido M. Oka and A. Hosaka, Prog. Theor. Phys. 106, 823 (2001). 11. D. Jido, Y. Nemoto, M. Oka and A. Hosaka, Nucl. Phys. A671, 471 (2000). 12. D. Jido, T. Hatsuda and T. Kunihiro, Phys. Rev. Lett. 84, 3252 (2000). 13. B. W. Lee, Chiral Dynamics, Gordon and Breach, New York, (1972). 14. C. DeTar and T. Kunihiro, Phys. Rev. D39, 2805 (1989). 15. S. Weinberg, Phys. Rev. Lett. 65, 1177 and 1181 (1990).
QUENCHED CHIRAL PHYSICS IN BARYON MASSES
R. D. Y O U N G , D. B. L E I N W E B E R A N D A. W. T H O M A S
Special Research Centre for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, University of Adelaide, Adelaide SA 5005, Australia
E-mail: [email protected]; [email protected]; athomas ©physics, adelaide.edu.au
S. V. W R I G H T
Division of Theoretical Physics, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, U.K.
E-mail: [email protected]
Recent work has identified that the primary differences between quenched and dynamical spectroscopy can be described by chiral loop effects. Here we highlight the features of this study.
1. Introduction
Chiral symmetry has long been known to play an important role in the low energy properties of QCD. Theoretical studies of the non-perturbative features of QCD, expressed in terms of the fundamental theory, have proven to be most successful in the field of lattice gauge theory. Lattice studies are typically restricted to relatively large values of the u and d masses, where chiral effects are highly suppressed. As a consequence, direct observation of chiral properties is a challenge in lattice simulations. The study of low-lying baryon spectroscopy in both quenched and full QCD has provided a direct connection between lattice results and chiral physics1,2.
The fact that one is restricted to quark masses much larger than the physical values means that, in addition to all the usual extrapolations (e.g. to the infinite volume and continuum limits), if one wants to compare with empirical hadron observables, one must also have a reliable method of extrapolation to the chiral limit. Any such extrapolation must incorporate the appropriate chiral corrections, arising from Goldstone boson loops, which give rise to rapid, non-linear variations as the chiral limit is approached. The importance of incorporating such behaviour has been successfully demonstrated for a number of hadronic observables, see Ref. [3] for a review and the references contained
155
156
therein. The quenched approximation is a widely used tool for studying non-
perturbative QCD within numerical simulations of lattice gauge theory. With an appropriate choice of the lattice scale and at moderate to heavy quark masses, this approximation has been shown to give only small, systematic deviations from the results of full QCD with dynamical fermions. Although no formal connection has been established between full and quenched QCD, the similarity of the results has led to the belief that the effects of quenching are small and hence that quenched QCD provides a reasonable approximation to the full theory4. Under a more reliable choice of lattice scale, where chiral effects are negligible, clear differences are observed between quenched and dynamical results5.
2. Chiral Physics and the Quenched Approximation
Chiral symmetry is spontaneously broken in the ground state of QCD. As a consequence, the pion is a Goldstone boson characteristic of this broken symmetry. The pion mass then behaves as m^ oc mq, the well known Gell-Mann-Oakes-Renner (GOR) relation. In principle, this relation is only guaranteed for quark masses near zero. Explicit lattice calculations show that it holds over an enormous range, as high as m , ~ 1 GeV. These almost massless Goldstone bosons couple strongly to low-lying baryon states — particularly the nucleon (N) and delta (A).
Meson-loop diagrams involving Goldstone bosons coupling to baryons give rise to non-analytic behaviour of baryon properties as a function of quark mass. At light quark masses these corrections are large and rapidly varying. At heavy quark masses these contributions are suppressed and hadron properties are smooth, slowly varying functions of quark mass. The scale of this transition is characterised by the inverse size of the pion-cloud source. Below pion masses of about 400-500 MeV these non-analytic contributions become increasingly important, while they rapidly become negligible above this point. Since lattice simulations are typically restricted to the domain where mn > 500 MeV, these rapid-varying chiral effects must be incorporated phenomenologically.
Within the quenched approximation dynamical sea quarks are absent from the simulation. As a consequence the structure of meson-loop contributions is modified. In the physical theory of QCD, meson-loop diagrams can be described by two topologically differing types. A typical meson-loop diagram may be decomposed into those where the loop meson contains a sea quark, such as Fig. 1(a), and those where the loop meson is comprised of pure valence quarks, see Fig. 1(b). These diagrams involving the sea quark, type (a), are obviously absent in the quenched approximation and consequently only
157
(a) (b)
Figure 1. Quark flow diagrams of pion loop contributions appearing in QCD.
(a) (b)
Figure 2. Quark flow diagrams of chiral vf loop contributions appearing in QQCD: (a) axial hairpin, (b) double hairpin.
a subset of the contributions to the physical theory are included. This type of argument, together with SU(6) symmetry, is precisely that described for the evaluation of non-analytic contributions to baryon magnetic moments by Leinweber6. This quark flow approach is analogous to the original approach to chiral perturbation theory for mesons performed by Sharpe7'8.
In addition to the usual pion loop contributions, quenched QCD (QQCD) contains loop diagrams involving the flavour singlet rf which also give rise to important non-analytic structure. Within full QCD such loops do not play a role in the chiral expansion because the rf remains massive in the chiral limit. On the other hand, in the quenched approximation the rf is also a Goldstone boson8 and the rf propagator has the same kinematic structure as that of the pion.
As a consequence there are two new chiral loop contributions unique to the quenched theory. The first of these corresponds to an axial hairpin diagram such as that indicated in Fig. 2(a). This diagram gives a contribution to the chiral expansion of baryon masses which is non-analytic at order m^. The second of these new rf loop diagrams arises from the double hairpin vertex as pictured in Fig. 2(b). This contribution is particularly interesting because it involves two Goldstone boson propagators and is therefore the source of more singular non-analytic behaviour - linear in m^. In studying the extrapolation of quenched lattice results it is essential to treat these contributions on an equal footing to the pion-loop diagrams discussed earlier.
158
3. Chiral Extrapolations
In general, the coefficients of the leading (LNA) and next-to leading non-analytic (NLNA) terms in a chiral expansion of baryon masses are very large. For instance, the LNA term for the nucleon mass is 8rrvN ' = —5.6 m% (with m , and 5mN in GeV). With mw = 0.5GeV, quite a low mass for current simulations, this yields Sm^N ' — 0.7 GeV — a huge contribution. Furthermore, in this region hadron masses in both full and quenched lattice QCD are found to be essentially linear in rn\ or equivalently quark mass, whereas 8rrvN ' is highly non-linear. The challenge is therefore to ensure the appropriate LNA and NLNA behaviour, with the correct coefficients, as mn —• 0, while making a sufficiently rapid transition to the linear behaviour of actual lattice data, where ra, becomes large.
A reliable method for achieving all this was proposed by Leinweber et al.9
They fit the full (unquenched) lattice data with the form:
MB =aB+pBml +J2B(mn,A), (1)
where Y,B is the total contribution from those pion loops which give rise to the LNA and NLNA terms in the self-energy of the baryon. The extension to the case of quenched QCD is achieved by replacing the self-energies, S s , of the physical theory by the corresponding contributions of the quenched theory2, tB.
The linear term of Eq. (1), which dominates for mn S> A, models the quark mass dependence of the pion-cloud source — the baryon without its pion dressing. This term also serves to account for loop diagrams involving heavier mesons, which have much slower variation with quark mass.
The diagrams for the various meson-loop contributions are evaluated using a phenomenological regulator. This regulator has the effect of suppressing the contributions as soon as the pion mass becomes large. At light quark masses the self-energies, T,B, provide the same non-analytic behaviour as xPT, independent of the choice of regulator. Therefore the functional form, Eq. (1), naturally encapsulates both the light quark limit of %PT and the heavy quark behaviour observed on the lattice.
We consider the leading order diagrams containing only the lightest Gold-stone degrees of freedom. These are responsible for the most rapid non-linear variation as the quark mass is pushed down toward the chiral limit. In the physical theory we consider only those diagrams containing pions, as depicted in Fig. 3. In QQCD there exist modified pion loop contributions and the additional structures arising from the rf behaving as a Goldstone boson. The diagrams contributing to the nucleon self-energy in the quenched approximation are shown in Fig. 4, the A can be described by analogous diagrams.
159
+ E J V = L : '
+
Figure3. Illustrative view of the meson-loop self-energies, S s , infullQCD. These diagrams give rise to the LNA and NLNA contributions in the chiral expansion. Single (double) lines denote propagation ofaJV (A).
+ -'N
+ — - * — — < • i—
Figure 4. Illustrative view of the meson-loop self-energies, Eg, in quenched QCD. These diagrams give rise to the LNA and NLNA contributions in the chiral expansion. A cross represents a hairpin vertex in the 77' propagator. Single (double) lines denote propagation of a AT (A).
For the evaluation of the quenched quantities we assume that the parameters of the chiral Lagrangian exhibit negligible differences between quenched and 3-flavour dynamical simulations. This is a working hypothesis with no better guidance yet available, but the successful results of this work demonstrate the self-consistency of such an assumption1. Only with further accurate lattice simulations at light masses will one be able to determine the extent to which our hypothesis holds.
To highlight the differences in the self-energy contributions we show the net contributions to the A in full ( S A ) and quenched ( E A ) QCD in Fig. 5. For details of the breakdown of the individual contributions we refer the reader to our longer article2. The significant point to note is that whereas the meson cloud of the A is attractive in full QCD, it exhibits repulsive behaviour within the quenched approximation.
4. Fitting Lattice Data
The lattice data considered in this analysis comes from the recent paper of Bernard et al.h These simulations were performed using an improved Kogut-Susskind quark action, which shows evidence of good scaling10. Unlike the standard Wilson fermion action, masses determined at finite lattice spacing
160
0.2
0.1
P 0.0 O,
-0 .2
- 0 . 3 0.0 0.1 0.2 0.3 0.4 0.5 0.6
mj* (GeV2)
Figure 5. Net contributions to the A self-energy evaluated with dipole regulator at A = 0.8 GeV.
are good estimates of the continuum limit results. We are particularly concerned with the chiral extrapolation of baryon
masses and how their behaviour is affected by the quenched approximation. In such a study, it is essential that the method of scale determination is free from chiral contamination. One such method involves the static-quark potential. As low-lying pseudoscalar mesons made of light quarks exhibit negligible coupling to hadrons containing only heavy valence quarks, the low energy effective field theory plays no role in the determination of the scale for these systems. In fixing the scale through such a procedure one constrains all simulations, quenched, 2-flavour, 3-flavour etc., to match phenomenological static-quark forces. Effectively, the short range (0.35 ~ 0.5 fm) interactions are matched across all simulations.
A commonly adopted method involving the static-quark potential is the Sommer scale11,12. This procedure defines the force, F(r), between heavy quarks at a particular length scale, namely TQ ~ 0.5 fm. Choosing a narrow window to study the potential avoids complications arising in dynamical simulations where screening and ultimately string breaking is encountered at large separations. The lattice data analysed in this report uses a variant of this definition, choosing to define the force5 at r\ = 0.35 fm via rlF(r\) = 1.00.
The non-analytic chiral behaviour is governed by the infrared regions of the self-energy integrals. Due to the finite volume of lattice simulations much of
161
Table 1. Best fit parameters for both full and quenched data sets with dipole regulator, A = 0.8 GeV. All masses are in GeV.
Simulation apf 0ff » A /?& Physical 1.27(2) 0.90(5) 1.45(3) 0.74(8) Quenched 1.24(2) 0.85(6) 1.45(4) 0.72(11)
this structure will not be captured. For this reason we evaluate the self-energy corrections with pion momenta restricted to those available on the particular lattice2,13. In this way we get a first estimate of the discretisation errors in the meson-loop corrections. In no way does this account for any artefacts associated with the pion-cloud source.
Current lattice data is insufficient to reliably extract the dipole regulator parameter, A. We fit all data choosing a common value to describe all vertices, A = 0.8 GeV. This choice has been optimised2 to highlight the main result of this analysis. We note that the value of A which we find is indeed consistent with phenomenological estimates which suggest that this should be somewhat less than 1 GeV.
We fit both quenched and dynamical simulation results to the form of Eq. (1) with appropriate discretised self-energies. These fits are shown in Fig. 6. It is the open squares which should be compared with the lattice data. These points correspond to evaluation of the self-energies on the discretised momentum grid. The lines represent a restoration of the continuum limit in the self-energy evaluation. Discrepancies between the continuum and discrete version only become apparent at light quark masses, this corresponds to the Compton wavelength of the pion becoming comparable to the finite spatial extent of the lattice.
The important result of this study is that the behaviour of the pion-cloud source is found to be quite similar in both quenched and dynamical simulations. Once the self-energies corresponding to the given theory are incorporated into the fit, the linear terms are found to be in excellent agreement. Our best fit parameters, for the selected dipole mass A = 0.8 GeV, are shown in Table 1. Here we observe the remarkable agreement between quenched and dynamical data sets for N and A masses over a wide range of pion mass. This leads to the interpretation that the primary effects of quenching can be described by the modified chiral structures which give rise to the LNA and NLNA behaviour of the respective theories.
The success of fitting the iV and A data sets with a common regulator lends confidence to an interpretation of the mass splitting between these states. Examination of the self-energy contributions in full QCD suggests that only about 50 MeV of the observed 300 MeV N-A splitting arises from pion loops2.
162
0.8 ' ' ' ' ' ' ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6
mwB (GeV3)
Figure 6. Fit (open squares) to lattice data 6 (Quenched A, Dynamical A) with adjusted self-energy expressions accounting for finite volume and lattice spacing artifacts. The infinite-volume, continuum limit of quenched (dashed lines) and dynamical (solid lines) are shown. The lower curves and data points are for the nucleon and the upper ones for the A.
The dominant contribution to the hyperfine splitting would then naturally be described by some short-range quark-gluon interactions.
5. Conclusions
We have demonstrated the strength of fitting lattice data with a functional form which naturally interpolates between the domains of heavy and light quarks. The extrapolation formula gives a reliable method for the extraction of baryon masses at realistic quark masses. Although the quenched approximation gives rise to more singular behaviour in the chiral limit, these contributions are quickly suppressed with increasing pion mass. Within the quenched approximation only limited curvature is observed for the N down to low quark masses. In contrast, we find some upward curvature of the A mass in the light quark domain.
The observation that the source of the meson cloud has remarkably similar behaviour within both quenched and physical simulations is of considerable importance. One can describe the primary effects of quenching by the meson-loop contributions which give rise to the most rapid, non-linear variation at light quark masses. This leads one to the possibility of applying this result to obtain more physical results from quenched simulations. The structure of
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the meson-cloud source can be determined from quenched simulations and then
the chiral structures of the physical theory can be incorporated phenomenolog-
ically. Natural extension of this work leads to the analysis of further hyperons
to investigate the applicability over a range of particles.
R e f e r e n c e s
1. R. D. Young, D. B. Leinweber, A. W. Thomas and S. V. Wright, hep-lat/0111041. 2. R. D. Young, D. B. Leinweber, A. W. Thomas and S. V. Wright, hep-lat/0205017. 3. W. Detmold, D. B. Leinweber, W. Melnitchouk, A. W. Thomas and S. V. Wright,
Pramana 57, 251 (2001); nucl-th/0104043. 4. S. Aoki et al., CP-PACS Collaboration, Phys. Rev. Lett. 84, 238 (2000). 5. C. W. Bernard et al., Phys. Rev. D64, 054506 (2001). 6. D. B. Leinweber, Nucl. Phys. Proc. Suppl. 109, 45 (2002). 7. S. R. Sharpe, Phys. Rev. D41, 3233 (1990). 8. S. R. Sharpe, Phys. Rev. D46, 3146 (1992). 9. D. B. Leinweber, A. W. Thomas, K. Tsushima and S. V. Wright, Phys. Rev.
D61, 074502 (2000). 10. C. W. Bernard et al., MILC Collaboration, Phys. Rev. D61, 111502 (2000). 11. R. Sommer, Nucl. Phys. B411, 839 (1994). 12. R. G. Edwards, U. M. Heller and T. R. Klassen, Nucl. Phys. B517, 377 (1998). 13. D. B. Leinweber, A. W. Thomas, K. Tsushima and S. V. Wright, Phys. Rev.
D64, 094502 (2001).
QCD AT NON-ZERO CHEMICAL POTENTIAL AND TEMPERATURE FROM THE LATTICE
C.R. ALLTON* Department of Physics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP,
U.KJ and
Department of Mathematics, University of Queensland, Brisbane 4072, Australia E-mail: [email protected]
S. E J I R I , S.J. H A N D S
Department of Physics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, U.K.
E-mail: [email protected] ; [email protected]
O. K A C Z M A R E K , F . K A R S C H , E. L A E R M A N N , CH. S C H M I D T
Fakultat fur Physik, Universitat Bielefeld, D-33615 Bielefeld, Germany E-mail: [email protected] ; [email protected] ;
[email protected] ; [email protected]
L. S C O R Z A T O
Department of Physics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, U.K. and
DESY Theory Division, Notkestrasse 85, D-2260S Hamburg, Germany. E-mail: [email protected]
A study of QCD at non-zero chemical potential, 11, and temperature, T, is performed using the lattice technique. The transition temperature (between the confined and deconfined phases) is determined as a function of /i and is found to be in agreement with other work. In addition the variation of the pressure and energy density with [i is obtained for n > 0. These results are of particular relevance for heavy-ion collision experiments.
'speaker at the workshop 'permanent address
164
165
1. Introduction
The QCD phase diagram has come under increasing experimental and theoretical scrutiny over the last few years. On the experimental side, very recent studies of compact astronomical objects have suggested that their cores contain "quark matter", i.e. QCD in a new, unconfined phase where the basic units of matter are quarks, rather than nuclei or nucleons1. More terrestrially, heavy ion collision experiments, such as those performed at RHIC and CERN, are also believed to be probing unconfined QCD2. On the theoretical side, the study of QCD under these extreme densities and temperatures has proceeded along several fronts. One of the most promising areas of research is the use of lattice techniques to study either QCD itself, or model theories which mimic the strong interaction3. Clearly the most satisfying approach would be the former, i.e. a direct lattice study of QCD at various coordinates (T, p) in its phase space (fi is the chemical potential for the quark number). However, until recently, this has proved intractable at a practical level for very fundamental reasons. This is because the Monte Carlo integration technique, which is at the heart of the (Euclidean) lattice approach, breaks down when \i ^ 0. This work summarises one new approach which overcomes this problem and has made progress for /i ^ 0 and T =£ 0.
In the next section a summary is given of the lattice technique and the problem incurred when p ^ 0. Section 3 describes the method used to overcome these difficulties, and Sec. 4 outlines the simulation details. The next two sections apply the method to variations in m and fi, and section Sec. 7 describes calculations of the pressure and energy density as functions of fi.
A full account of this work is published elsewhere4.
2. Lattice Technique
On the lattice, the quark fields, 4>(x), are defined on the sites, x, and the gluonic fields, U^(x), on the links x —• x + fi. Observables are then calculated via a Monte Carlo integration approach :
<"> = A T ^ — £ ' "(v^.to. (i)
where ^ represents a sum over configurations {U, ^,r(>} which are selected with probability proportional to the Boltzmann weight P{{U,il>,xl>}) oc e-s({u,i>,i>}) w i th S a suitably defined (Euclidean) gauge-invariant action. The fermionic part of this action is
166
SF = YJ^)iP+rn)^{x). (2)
= M
For /i = 0 it can be shown that this action produces a (real-valued) positive Boltzmann weight.
Calculations at non-zero temperature, T ^ 0, can be performed by using a lattice with a finite temporal extent of Nta — 1/T, where Nt is the number of lattice sites in the time dimension. In practice, T is varied by changing the gauge coupling, go, and hence (through dimensional transmutation) the lattice spacing, a, rather than by changing Nt (which can only be changed in discrete steps!).
The chemical potential is introduced into the system via an additional term in the quark matrix M, proportional to the Dirac gamma matrix, 70,
M^M + mo- (3) For fi ^ 0, this leads to a complex-valued Boltzmann "weight" which can therefore no longer be used as a probability distribution, and, hence, the Monte Carlo integration procedure is no longer applicable. This is known as the Sign Problem and has plagued more than a decade of lattice calculations of QCD at fj, 7̂ 0.
3. Reweighting
This section outlines the Ferrenberg-Swendsen reweighting approach5 which is used to overcome the sign problem detailed in the previous section. Observ-ables at one set of parameter values {fi,m,pi) (where /? = Q/gl, and m is the quark mass) can be calculated using an ensemble generated at another set of parameters (/3o,mo,fio) as follows,
<fi)(/3,m,M) = z{/3 l
m v JDUQdetM(m,^e-s^
/ Q e(lndetM(m,/i)-lndetM(mo,/Jo))e-S,,(/?)+S,,(/?o)\
~ /e(lndetM(m,M)-lndetM(mo,/Jo))e-S9(f8)+S<j(/8<>)\ ' ^ '
Here Sg is the gauge action. In principle, Eq. (4) can be used to map out the entire phase diagram of
QCD. However, it has been found that its naive application fails since the relative size of fluctuations in both numerator and denominator tend to grow exponentially with the volume of the system studied. This is a signal that the
167
overlap between the ensemble at (/3o,m0,//o) is small compared with that at (P,m,n).
One study which has had success in using Eq. (4) is that of Fodor and Katz6. They apply the reweighting approach not at an arbitrary point, (T,//), in the phase diagram, but rather they trace out the phase transition line Tc (/•*)• This method is successful presumably because the overlap between the ensembles remains high along the "coexistence" line which defines the transition.
This paper utilises an alternative approach which Taylor expands Eq. (4) as a function of fj, (or m) and hence estimates the derivatives of various quantities w.r.t. fi (or m). Derivatives up to the second order are considered, thus for (ft) (in the case of a Taylor expansion in /J.) we have
. _ ((ftp + Qifi + ft2p2) expjRin + R2H2 ~ Agg))(/?0i<<o)
{ >iM (exp(R^ + R2^-ASg)){0Otlio) ' W
where [io = 0. In Eq. (5), Rn is the n—th derivative of the fermionic reweighting factor in Eq. (4), i.e.
/ d e t M ( / i ) \ _ ^ ^ a " l n d e t M ( 0 ) _ ^
v v ' ' n—\ r n= l
The Q„ are similarly the nth derivatives of the observable fi. Two observables are studied: the chiral condensate (tp1^) and the Polyakov
Loop, L. Since L is a pure gluonic quantity, defined as
1 Nt
(L) = (VJ2Trl[Ut(x,t)), (7) X t = U
all of its derivatives are zero. (Here, Vs is the spatial volume.) However, the expansion of (iprp) is more challenging since it is defined as
(W) ~ ( t rM - 1 ) , (8)
and hence the application of Eq. (5) requires determinations of dnM~1/dfin. The susceptibilities, x, of both (iptp) and L are defined as usual by their
fluctuations, e.g.
XL = (volume factor) x {{L2) - (L)2). (9)
These susceptibilities have a maximum at the transition point, j3c, and hence can be used to determine the transition point /Jc(m,//).
168
4. Simulation Details
The lattice calculations were performed using a "p4-improved" discretisation of the continuum action which is a sophisticated lattice action7,8 maintaining rotational invariance of the free fermion propagator up to 0(p4) (p is the momentum here). Two dynamical flavours of quarks were used with a 163 x 4 lattice. Simulations were performed at quark mass, m = 0.1 and 0.2 which correspond to (unphysically heavy) pseudoscalar-vector meson mass ratios of MPS/MV « 0.70 and 0.85. Approximately 400,000 configurations were generated in total using around 6 months of a 128-node (64Gflop peak) APEmille in the University of Wales, Swansea.
5. Results for Mass Reweighting
As a check of our method, we first use mass reweighting, since, unlike the \i ^ 0 case, there is no theoretical difficulty in simulating at virtually any value of m, and hence there are published data at a number of different ra values readily available for comparison. Reweighting in quark mass is simply a matter of setting \i = ^o = 0 in Eq. (4) and Taylor expanding in m rather than fi in Eqs. (5,6). (Note that dM/dm = 1 and dnM/dmn = 0 for n > 1.)
We use the peak position in both the chiral condensate and Polyakov Loop susceptibilities, x ^ L, to determine the phase transition point /?c("i). Figure 1 illustrates this by plotting XM as a function of ft (for m = 0.2). This shows the variation in the peak position of xsw, as ro is changed in steps of 0.01 around mo = 0.2.
Figure 1. The quark mass dependency of xr. as a function of P in the neighbourhood of m = 0.2 for the Chiral Susceptibility.
169
Once these peak positions have been determined (for both XM a n d XL), 0C can be plotted as a function of m and a comparison can be made with other determinations. This is done in Fig. 2 where the f3c values from earlier work7 are also shown. The line segments around our data points at m = 0.1 and 0.2 indicate the gradient of (5C as a function of m using our Taylor-expanded reweighting technique. Shown in Fig. 2 are results from both the chiral condensate and Polyakov loop. These are both in perfect agreement (as expected). Furthermore, our method agrees with previously published data confirming the validity of the approach.
J.O
3.75
3.7
CO." 3.65
3.6
3.55 " 8
1
i
i
/
, i ,
V
1
-
-
O 8x4 (previous)
A 16 x4 (previous)
• 163x4 (chiral)
• 163x4 (Polyakov)
l . l . l .
0 0.05 0.1 0.15 0.2 0.25 0.3
m
Figure 2. The transition "temperature", f)c, as a function of m in comparison with previous work .
6. Results for ft Reweighting
We now turn to reweighting in chemical potential, fi. As in6, rather than applying the method to arbitrary parameter values, we trace out the transition point /3c(fi). Using Eq. (5), (ipij)) and the Polyakov Loop, L, are calculated as a function of y, together with their susceptibilities, x ^ , t ( s e e Eq. (9)). In Fig. 3, we plot XZ& against ft for various fi. Note that the peak position moves as fi changes. The determination of the transition point ftc from both the chiral condensate and the Polyakov loop (not shown here) are found to be in agreement.
Because we have calculated all quantities to 0(/i2) we can extract /3C and fit it to a quadratic in fi. (In fact, it can be shown4 that the first derivative
170
0.8
, , ___,_ — |lH).05 .U&£&gi
Figure 3. The n dependence of x x ^ as a function of /? in the neighbourhood of m = 0.2 for the chiral susceptibility.
d/Hc/dfi = 0.) We find d2/3c/d^2 = 1.1 ± 50% for both quark masses m and 0.2.
We now use
d2re
d/j,2
1 d2/?c
N?TC d^ / ( •
d£ da
0.1
(10)
to convert d2/?c/d/i2 into physical units, with the beta-function dp /do coming from string tension data7. We find ?c^npf « —0.14 at ma = 0.1.
Figure 4 shows the phase transition curve Tc(n) obtained from this method. On this graph we have plotted the Fodor-Katz point6 which is within our errors, confirming our method. Also shown is the n value corresponding to RHIC. It is interesting to extrapolate the curve Tc(fj) to the horizontal axis (as shown). It is known that the transition (at T fn 0) between ordinary hadronic matter and quark matter occurs at around ft ss 400 MeV. This is at a smaller value of fj. than the horizontal intercept of our data indicating (not surprisingly) the presence of higher order terms in the Taylor expansion and/or a breakdown in our method at these large values of fi.
This motivates the question: for what range in fi do we expect our method to be accurate (and converge to the correct answer)? We have studied this issue by calculating the complex phase, 9, of the fermionic determinant (which enters in the reweighting factor in Eq. (4)), i.e.
d e t M = | d e t M | e " . (11)
171
200
150
u £100
50
"0 200 400 600
(i, (Mev)
Figure 4. The transition temperature, Tc, as a function of fj.. The diamond symbol is the endpoint of the first order transition obtained by Fodor and Katz 6 .
The reweighting method will fail when the fluctuations (standard deviation) in 9 are larger than 0(7r/2). Taylor expanding Eq. (11) and noting that only odd derivatives contribute to the complex phase 6, we find that the standard deviation A0 ~ 0(TT/2) at around p/Tc ~ 0.5. Since this is around five times the fi value of RHIC, we can confirm that our method is applicable for RHIC physics.
An interesting dynamical quark effect can be uncovered when studying the Polyakov Loop susceptibility, \L- For n < 0, anti-quarks are dynamically generated which screened colour charge. This leads to a reduction in the free energy of a single quark, and a corresponding reduction in the strength of the singularity at the transition. We observe this effect by noting that the peak height of XL is smaller for p, < 0 compared with // > 0 in Ref. [4].
7. Pressure and Energy Density
Of great interest for heavy ion collision experiments is the study of the pressure, p, and energy density e and their p dependence. We can obtain estimates of p by employing the integral method9:
P=^\nZ. (12)
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The first derivative of p w.r.t. p. is related to the quark number density,
TdlnZ , x
^=v-aT' (13)
and the second derivative to the singlet quark number susceptibility, xs • Both nq and xs can be calculated in terms of the quark matrix, M.
Using the above to estimate p at the RHIC point we find that p increases by around 1% from its p. = 0 value.
The energy density, e, can be obtained from
e — 3p _ 1 dlnZ (I da\~
T4 VT3 dp yadp The derivatives of Eq. (14) can be expressed in terms of nq and xs- Combining this with the above calculation of p, we obtain a value for e alone. We find that, at the RHIC point, there is again only a 1% deviation from e(p = 0).
Finally we study the variation of p and e along the transition line Tc (fi). (The above calculations were performed at fixed T.) Our aim is to determine whether p and e are constant along Tc(p). The constant p line is defined as
Ap=^AT+-^-)A(^) = 0, (15)
with a similar expression for the constant e line. Using the above and the value determined earlier for the rate of change of Tc with fi, we find that the value of both p and e along the transition line Tc(p) is consistent with zero with our current precision4.
8. Conclusions
This work (which is published in full elsewhere4) has outlined a new method of determining thermal properties of QCD at non-zero chemical potential from the lattice. This approach is based on Taylor expanding the Ferrenberg-Swendsen re weighting scheme.
Using this method, the susceptibilities in the chiral condensate and Polyakov loop were determined and their peak positions used to define the transition point, Tc. As a warmup exercise, the re weighting technique was used to determine the transition point as a function of the quark mass, m confirming earlier work. The method was then applied to obtaining Tc as a function of chemical potential, fi, confirming the work of Fodor and Katz6. Very recent work of de Forcrand and Philipsen, who studied the transition temperature for imaginary \i and then analytically continued these results to real-valued p,, also confirm our results10.
173
The region of applicability of our method was studied by calculating the fluctuations in the phase of the reweighting factor. This region was found to be substantial and easily covers the physically interesting values of fj, appropriate for RHIC physics.
We also extracted information about the pressure, p, and energy density, e, as a function of chemical potential. We found that the variation in these quantities from their values at p. = 0 is tiny. This leads us to conclude that RHIC physics well approximated by p — 0 physics. Furthermore we find that p and e are approximately constant along the transition line Tc(p).
The success of this work motivates the use of lighter, more physical quark masses, and the study of (2+1) dynamical flavours to correctly model real world physics.
A cknowledgment s
The authors would like to acknowledge the Particle Physics and Astronomy Research Council for the award of the grant PPA/G/S/1999/00026 and the European Union for the grant ERBFMRX-CT97-0122. CRA would like to thank the Department of Mathematics, University of Queensland, Australia for their kind hospitality while part of this work was performed.
References
1. See http://wwwl.msfc.nasa.gov/NEWSROOM/news/releases/2002/02-082.html (NASA Press release); and J. Drake et al., Ast. J., June 20 2002 to appear.
2. For a recent announcement from CERN, see http://cern.web.cern.ch/CERN/Announcements/2000/NewStateMatter.
3. For a review, see S.J. Hands, Nucl. Phys. Proc. Suppl. 106, 142 (2002), hep-lat/0109034.
4. C.R. Allton, S. Ejiri, S.J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, Ch. Schmidt, L. Scorzato, hep-lat/0204010 .
5. A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988); Phys. Rev. Lett. 63, 1195 (1989).
6. Z. Fodor and S.D. Katz, hep-lat/0104001; JHEP 0203, 014 (2002), hep-lat/0106002.
7. F. Karsch, E. Laermann, and A. Peikert, Phys. Lett. B478, 447 (2000); Nucl. Phys. B605, 579 (2001).
8. U.M. Heller, F. Karsch, and B. Sturm, Phys. Rev. D60, 114502 (1999). 9. J. Engels, J. Fingberg, F. Karsch, D. MiUer and M. Weber, Phys. Lett. B252,
625 (1990). 10. Ph. de Forcrand and O. Philipsen, hep-lat/0205016.
H A D R O N MASSES FROM A NOVEL FAT-LINK FERMION ACTION
J. M. ZANOTTI, S. BILSON-THOMPSON, F. D. R. BONNET, D. B. LEINWEBER, A. G. WILLIAMS AND J. B. ZHANG
Special Research Center for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, Adelaide University, 5005, Australia
E-mail: [email protected]; [email protected] [email protected]; [email protected] [email protected]; [email protected]
W. M E L N I T C H O U K
Special Research Center for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, Adelaide University, 5005, Australia,
and Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, U.S.A.
E-mail: [email protected]
F. X. LEE Center for Nuclear Studies, Department of Physics,
The George Washington University, Washington, D.C. 20052, U.S.A., and
Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, U.S.A E-mail: [email protected]
Hadron masses are calculated in quenched lattice QCD in order to probe the scaling behaviour of a novel fat-link clover fermion action in which only the irrelevant operators of the fermion action are constructed using APE-smeared links. Light quark masses providing a m^/mp mass ratio of 0.36 are considered to assess the exceptional configuration problem of clover-fermion actions. The simulations are performed with an order a2 mean-field improved plaquette plus rectangle gluon action on a 163 X 32 lattice with a lattice spacing of a = 0.125 fm. We compare actions with n = 4 and 12 smearing sweeps with a smearing fraction of 0.7. The n = 4 Fat Link Irrelevant Clover action provides scaling which is superior to mean-field improvement, and offers advantages over nonperturbative 0 ( a ) improvement including a reduced exceptional configuration problem.
1. Introduction
Understanding the generation of hadron mass from first principles has proved to be challenging. The only method for deriving hadron masses directly from
174
175
QCD is a numerical calculation on the lattice. The high computational cost required to perform accurate lattice calculations at small lattice spacings, however, has led to an increased interest in quark action improvement. To avoid the famous doubling problem, Wilson fermions1 introduce additional terms which explicitly break chiral symmetry at 0(a). To extrapolate reliably to the continuum, simulations must be performed on fine lattices, which are computationally very expensive. The scaling properties of the Wilson action at finite a can be improved by introducing any number of irrelevant operators of increasing dimension which vanish in the continuum limit.
The Sheikholeslami-Wohlert (clover) action2 introduces an additional irrelevant dimension-five operator to the standard Wilson1 quark action,
Ssw = Sw j — i>(x)(r^Fllvip(x) , (1)
where S\y is the standard Wilson action and Csw is the clover coefficient which can be tuned to remove 0(a) artifacts. Nonperturbative (NP) 0(a) improvement3 tunes Csw to all powers in g2 and displays excellent scaling, as shown by Edwards et al.4 who studied the scaling properties of the nucleon and vector meson masses for various lattice spacings (see also Sec. 4 below). In particular, the linear behaviour of the NP-improved clover actions, when plotted against a2, demonstrates that 0(a) errors are removed. It was also found in Ref. [4] that a linear extrapolation of the mean-field improved data fails, indicating that 0(a) errors are still present.
A drawback to the clover action, however, is the associated problem of exceptional configurations, where the quark propagator encounters singular behaviour as the quark mass becomes small. In practice, this prevents the use of coarse lattices (/3 < 5.7 ~ a > 0.18 fm)5'6. Furthermore, the plaquette version of F^, which is commonly used in Eq. (1), has large 0(a2) errors, which can lead to errors of the order of 10% in the topological charge even on very smooth configurations7.
The idea of using fat links in fermion actions was first explored by the MIT group8 and more recently has been studied by DeGrand et al.6'9, who showed that the exceptional configuration problem can be overcome by using a fat-link (FL) clover action. Moreover, the renormalization of the coefficients of action improvement terms is small. A drawback to conventional fat-link techniques, however, is that in smearing the links gluon interactions are removed at the scale of the cutoff. While this has some tremendous benefits, the short-distance quark interactions are lost. As a result decay constants, which are sensitive to the wave function at the origin, are suppressed.
A solution to these problems is to work with two sets of links in the fermion action. In the relevant dimension-four operators, one works with the untouched
176
links generated via Monte Carlo methods, while the smeared fat links are introduced only in the higher dimension irrelevant operators. The effect this has on decay constants is under investigation and will be reported elsewhere.
In this paper we present the first results of simulations of the spectrum of light mesons and baryons at light quark masses using this variation of the clover action. In particular, we will start with the standard clover action and replace the links in the irrelevant operators with APE smeared10, or fat links. We shall refer to this action as the Fat-Link Irrelevant Clover (FLIC) action11.
2. Gauge Action
The simulations are performed using a tree-level (9(a2)-Symanzik-improved12
gauge action on a 163 x 32 lattice at 0 = 4.60, providing a lattice spacing a = 0.125(2) fm determined from the string tension with yfa = 440 MeV. A total of 50 configurations are used in this analysis, and the error analysis is performed by a third-order, single-eliminationjackknife, with the \ 2 P e r degree of freedom (Nj^p) obtained via covariance matrix fits. Further details of this simulations may be found in Ref. [11].
3. Fat-Link Irrelevant Fermion Action
Fat links6'9 are created by averaging or smearing links on the lattice with their nearest neighbours in a gauge covariant manner (APE smearing). The smearing procedure10 replaces a link, U^x), with a sum of the link and a times its staples
4
+Ul(x - ua)Ufi(x - va)Uu{x - ua + fia) , (2)
followed by projection back to SU(3). We select the unitary matrix J/JL which maximizes
Heti(U^U',!), (3)
by iterating over the three diagonal SU(2) subgroups of SU(3). We repeat this procedure of smearing followed immediately by projection n times. We create our fat links by setting a = 0.7 and comparing n — 4 and 12 smearing sweeps. The mean-field improved FLIC action now becomes
S & = Sw - ^ J ^x)a^F^(x) , (4)
where F^ is constructed using fat links, and where the mean-field improved Fat-Link Irrelevant Wilson action is
177
Table 1. The value of the mean link for different numbers of smearing sweeps, n.
n
0
4
12
"o
0.88894473
0.99658530
0.99927343
K L ) 4
0.62445197
0.98641100
0.99709689
X
(5)
x,p
In ip(x+n)-wo «o
ip(x - p.)
— V -* 4>{X +fl)+ „ F L nX
U FL 0 u;
FL 0
•/*) (6)
with K = l /(2m + 8r). We take the standard value r = 1. The 7-matrices are hermitian and a^ = [7^, 7„]/(2i).
As reported in Table 1, the mean-field improvement parameter for the fat links is very close to 1. Hence, the mean-field improved coefficient for Csw is expected to be adequatea. In addition, actions with many irrelevant operators (e.g. the D234 action) can now be handled with confidence as tree-level knowledge of the improvement coefficients should be sufficient. Another advantage is that one can now use highly improved definitions of F^ (involving terms up to UQ2), which give impressive near-integer results for the topological charge13.
In particular, we employ an 0(a4) improved definition13 of F^ in which the standard clover-sum of four l x l Wilson loops lying in the fi, v plane is combined with 2 x 2 and 3 x 3 Wilson loop clovers.
Work by DeForcrand et a/.14 suggests that 7 cooling sweeps are required to approach topological charge within 1% of integer value. This is approximately15 16 APE smearing sweeps at a = 0.7. However, achieving integer topological charge is not necessary for the purposes of studying hadron masses, as has been well established. To reach integer topological charge, even with improved definitions of the topological charge operator, requires significant smoothing and associated loss of short-distance information. Instead, we regard this as an upper limit on the number of smearing sweeps.
"Our experience with topological charge operators suggests that it is advantageous to include «o factors, even as they approach 1.
178
Using unimproved gauge fields and an unimproved topological charge operator, Bonnet et al.7 found that the topological charge settles down after about 10 sweeps of APE smearing at a — 0.7. Consequently, we create fat links with APE smearing parameters n = 12 and a = 0.7. This corresponds to ~ 2.5 times the smearing used in Refs. [6,9]. Further investigation reveals that improved gauge fields with a small lattice spacing (a = 0.125 fm) are smooth after only 4 sweeps. Hence, we perform calculations with 4 sweeps of smearing at a = 0.7 and consider n = 12 as a second reference. Table 1 lists the values of UQL for n = 0, 4 and 12 smearing sweeps.
We also compare our results with the standard Mean-Field Improved Clover (MFIC) action. We mean-field improve as defined in Eqs. (4) and (6) but with thin links throughout. The standard Wilson-loop definition of F^ is used.
A fixed boundary condition is used for the fermions by setting
Ut{x, nt) = 0 and UfL(x, nt) = 0 V x , (7)
in the hopping terms of the fermion action. The fermion source is centred at the space-time location (x,y,z,t) — (1,1,1,3), which allows for two steps backward in time without loss of signal. Gauge-invariant gaussian smearing16
in the spatial dimensions is applied at the source to increase the overlap of the interpolating operators with the ground states.
4. Resul ts
Hadron masses are extracted from the Euclidean time dependence of the calculated two-point correlation functions. The effective masses are given by
M(t + 1/2) = log[G(<)] - Iog[G(t + 1)] • (8)
The critical value of f\l y Kg ) IS determined by linearly extrapolating m\ as a function of mq to zero. We used five values of quark mass and the strange quark mass was taken to be the second heaviest quark mass.
Effective masses in Eq. (8) are calculated as a function of time and various time-fitting intervals are tested with a covariance matrix to obtain XV-^DF-
Good values of X2/NUF are obtained for many different time-fitting intervals as long as one fits after time slice 8. All fits for this action ("FLIC4") are therefore performed on time slices 9 through 14. For the Wilson action and the FLIC action with n - 12 ("FLIC12") the fitting regimes used are 9-13 and 9-14, respectively.
The behaviour of the p, nucleon and A masses as a function of squared pion mass is shown in Fig. 1 for the various actions. The first feature to note is the excellent agreement between the FLIC4 and FLIC12 actions. On the other hand, the Wilson action appears to lie somewhat low in comparison. It is also
179
A
N
P
i
*
* *
*
* *>
i
*
*
&
i
** *
*|
**
**
* *
_ j i
1
* «
IO.
1
* * *
-
~ • FLIC12 • FLIC4 0 Wilson
i
0.0 0.2 0.4 0.6 0.8 1.0 m/ (GeV8)
Figure 1. Masses of the nucleon, A and p meson versus m\ for the FLIC4, FLIC12 and Wilson actions.
reassuring that all actions give the correct mass ordering in the spectrum. The value of the squared pion mass at m^/mp = 0.7 is plotted on the abscissa for the three actions as a reference point. This point is chosen in order to allow comparison of different results by interpolating them to a common value of m^/mp = 0.7, rather than extrapolating them to smaller quark masses, which is subject to larger systematic and statistical uncertainties.
The scaling behaviour of the different actions is illustrated in Fig. 2. The present results for the Wilson action agree with those of Ref. [4]. The first feature to observe in Fig. 2 is that actions with fat-link irrelevant operators perform extremely well. For both the vector meson and the nucleon, the FLIC actions perform significantly better than the mean-field improved clover action. It is also clear that the FLIC4 action performs systematically better than the FLIC12. This suggests that 12 smearing sweeps removes too much short-distance information from the gauge-field configurations. On the other hand, 4 sweeps of smearing combined with our 0(a4) improved F^„ provides excellent results, without the fine tuning of Csw m the NP improvement program.
Notice that for the p meson, a linear extrapolation of previous mean-field improved clover results in Fig. 2 passes through our mean-field improved clover result at a2a ~ 0.08 which lies systematically low relative to the FLIC actions. However, a linear extrapolation does not pass through the continuum limit result, thus confirming the presence of significant 0(a) errors in the mean-field improved clover fermion action. While there are no NP-improved clover plus improved gluon simulation results at a2<r ~ 0.08, the simulation results that are available indicate that the fat-link results also compete well with those
180
a NP Clover
•» FLIC4, 200conf * N p Clover+imp glue |» i Opr jTilgon I , , , , I , , , , I
0.00 0.05 0.10 0.15 a2 a
Figure 2. Nucleon and vector meson masses for the Wilson, NP-improved and FLIC actions obtained by interpolating our results of Fig. 1 to mTjmp = 0.7. Results from the present simulations are indicated by the solid points. The fat links are constructed with n = 4 (solid squares) and n = 12 (stars) smearing sweeps at a = 0.7.
obtained with a NP-improved clover fermion action. Having determined FLIC4 is the preferred action, we have increased the
number of configurations to 200 for this action. As expected, the error bars are halved and the central values for the FLIC4 points move to the upper end of the error bars on the 50 configuration result, further supporting the promise of excellent scaling.
Finally, in order to search for exceptional configurations by pushing the bare quark mass down, we would like our preferred action to be efficient when inverting the fermion matrix. In Fig. 3 we compare the convergence rates of the different actions by plotting the number of stabilized biconjugate gradient17
iterations required to invert the fermion matrix as a function of mn/mp. For any particular value of mv/mp, the FLIC actions converge faster than both the Wilson and mean-field improved clover fermion actions. Also, the slopes of the FLIC lines are smaller in magnitude than those for Wilson and mean-field improved clover actions, which provides great promise for performing cost effective simulations at quark masses closer to the physical values. Problems with exceptional configurations have prevented such simulations in the past.
The ease with which one can invert the fermion matrix using FLIC fermions leads us to attempt simulations of three lighter quark masses corresponding
b
>
181
450
400
350
M 300
.2 250
£ 2 0 0 <u
" 150
100
50
0
-
-
-
--
1 1 1
N - - - FLIC12 \ - - - F L I C 4
\ - - - W i l s o n x \ - - - MFIC
x x x \ x x \ \ «.« . N
N \ xx X \ X N X \
x x ^ \ X X X \ x x S N
xx s ^
1 1 1
-
-
-
-
"
0.5 0.6 0.7 m / m
0.8 0.9
Figure 3. Average number of stabilized biconjugate gradient iterations for the Wilson, FLIC and mean-field improved clover (MFIC) actions plotted against mn /mp. The fat links are constructed with n = 4 (solid squares) and n = 12 (stars) smearing sweeps at a = 0.7.
to mn/mp = 0.53, 0.45, 0.36. Previous attempts with Wilson-style fermion actions have only succeeded in getting down18 to m„/mp = 0.47. Figure 4 shows the behaviour of the TV, A, E, S masses as a function of m£ for all eight masses considered. The first two of these lighter quark masses show no sign of exceptional configurations, however at the lightest mass tested, the pion correlator exhibited divergences on 1% of the configurations. These two exceptional configurations were omitted in producing Fig. 4.
At the lightest quark mass, the Compton wavelength of the pion is A^ = l/mn ~ 0.66 fm. The general rule for studying hadrons at light quark masses is to ensure Xw < ( l /4x length of spatial dimensions). The physical length of the spatial dimensions of our lattice is ~ 2 fm which is < 4 x A -̂. Hence it is important to investigate the impact of finite volume effects on the exceptional configuration problem. Simulations on a 203 x 40 lattice with a physical length ~ 2.7 fm are now under way using the precision 5-loop improved from of FM„ complemented by two additional APE smearing sweeps.
5. Conclusions
We have examined the hadron mass spectrum using a novel Fat Link Irrelevant Clover (FLIC) fermion action, in which only the irrelevant, higher-dimension operators involve smeared links. One of the main conclusions of this work is that the use of fat links in the irrelevant operators provides excellent results. Fat links promise improved scaling behaviour over mean-field improvement. This technique also solves a significant problem with 0(a) nonperturbative
182
a> O
1.8
1.7
1.6
1.5
3 I-*
g 1.3
a 1.2
1.1 -
1.0
~
9 H *
-
A *
$ -•/v}1
i
i
i 5
i
s
§ 5
$
1
s
i 5
i
1
z i 3
i
1
s 1 s s
--
-.
' 0.0 0.2 0.4
TO ' 0.6
(GeV2) 0.8 1.0
Figure 4. Masses of the nucleon, A, E and H versus m\ for the FLIC4 fermion action.
improvement on mean field-improved gluon configurations. Simulations are possible and the results are competitive with nonperturbative-improved clover results on plaquette-action gluon configurations. We have found that minimal smearing holds the promise of better scaling behaviour. Our results suggest that too much smearing removes relevant information from the gauge fields, leading to poorer performance. Fermion matrix inversion for FLIC actions is more efficient and results show no sign of exceptional configuration problems down to mn/mp = 0.45. However we encounter divergences in the pion correlator at m^/rrip = 0.36 on 1% of the configurations analysed on this particular lattice.
This work paves the way for promising future studies. It will be of great interest to consider different lattice spacings to further test the scaling of the fat-link actions. Current work is under way to further explore the exceptional configuration problem where a precision field-strength tensor and additional smearing hold promise. A study of the spectrum of excited hadrons using the fat-link clover actions is currently in progress19.
Acknowledgements
This work was supported by the Australian Research Council. We would also like to thank the National Computing Facility for Lattice Gauge Theories for the use of the Orion Supercomputer. W.M. and F.X.L. were partially supported by the U.S. Department of Energy contract DE-AC05-84ER40150, under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility (Jefferson Lab).
183
References
1. K.G. Wilson, in New Phenomena in Subnuclear Physics, Part A, A. Zichichi (ed.), Plenum Press, New York, p. 69, 1975.
2. B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259, 572 (1985). 3. M. Luscher et al., Nucl. Phys. B478, 365 (1996) [hep-lat/9605038]; M. Luscher
et ai, Nucl. Phys. B491, 323 (1997) 323 [hep-lat/9609035]. 4. R.G. Edwards, U.M. Heller and T.R. Klassen, Phys. Rev. Lett. 80, 3448 (1998).
[hep-lat/9711052]; see also R.D. Kenway, Nucl. Phys. Proc. Suppl. 73, 16 (1999) [hep-lat/9810054] for a review.
5. W. Bardeen et al., Phys. Rev. D57, 1633 (1998) [hep-lat/9705008]; W. Bardeen et al., Phys. Rev. D57, 3890 (1998).
6. T. DeGrand et al. (MILC Collaboration), [hep-lat/9807002]. 7. F.D. Bonnet et al., Phys. Rev. D62, 094509 (2000) [hep-lat/0001018]. 8. M.C. Chu et al., Phys. Rev. D49, 6039 (1994) [hep-lat/9312071]. 9. T. DeGrand (MILC collaboration), Phys. Rev. D60, 094501 (1999) [hep-
lat/9903006]. 10. M. Falcioni et al., Nucl. Phys. B251, 624 (1985); M. Albanese et al., Phys. Lett.
B192, 163 (1987). 11. J.M. Zanotti et al. Phys. Rev. D60, 074507 (2002) [hep-lat/0110216];
Nucl.Phys.Proc.Suppl. 109, 101 (2002) [hep-lat/0201004]. 12. K. Symanzik, Nucl. Phys. B226, 187 (1983). 13. S. Bilson-Thompson et al., Nucl.Phys.Proc.Suppl 109 116 (2002) [hep-
lat/0112034]; hep-lat/0203008. 14. P. de Forcrand et al., Nucl. Phys. B499, 409 (1997) [hep-lat/9701012]; P. de
Forcrand et ai, [hep-lat/9802017]. 15. F.D. Bonnet et al., [hep-lat/0106023]. 16. S. Gusken, Nucl. Phys. Proc. Suppl. 17, 361 (1990). 17. A. Frommer et al., Int. J. Mod. Phys. C5 , 1073 (1994) [hep-lat/9404013]. 18. M.D. Morte et al., hep-lat/0111048. 19. W. Melnitchouk et al., Nucl. Phys. Proc. Suppl 109, 116 (2002),
hep-lat/0201005; hep-lat/0202022.
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5. Nuclear and Nucleon Structure Functions
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SMALL-X NUCLEAR EFFECTS IN PARTON DISTRIBUTIONS
V. G U Z E Y
Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, Adelaide, 5005, Australia
E-mail: [email protected]
We summarize the status of nuclear parton distribution functions in nuclei at low x and show that they are not constrained well enough by the available data . Measurements of Drell-Yan dimuon pair production on nuclei at the JHF is singled out as a good candidate to significantly increase information about antishadowing and shadowing of valence and antiquark parton distributions in nuclei. A brief description of the leading twist approach to nuclear shadowing is given and outstanding theoretical problems are reviewed.
1. Parton Distributions in Nuclei
It is now very well established that the distributions of partons (quarks and gluons) in nuclei differ from those in free nucleons. Experimental information on nuclear parton distribution functions (nPDFs) is obtained from the following fixed nuclear target experiments: inclusive deep inelastic scattering (DIS) of leptons, Drell-Yan dimuon production by hadron beams, and lepto- and hadroproduction of vector mesons. Each of the afore mentioned processes is sensitive to a particular type or combination of nPDFs.
Inclusive DIS, l + A—tl' + X, measures the nuclear structure function F£, which, to leading order, takes the form
F2A(x,Q*)=Yje](f?(x,Q2) + fi(x,Q*)). (1)
i
Here f* is the quark nPDF; f£ is that of the antiquark. Inclusive DIS constrains best the combination f£ + f£. It is important to note that the fixed-target kinematics correlates the values of x and Q2 entering Eq. (1). For instance, for the NMC experiment1, the condition Q2 > 1 GeV2 means that x > 5 x 10 - 3 . Hence the data set eligible for the perturbative QCD analysis becomes reduced, and the reliable extraction of nPDF, especially at low x, becomes more difficult.
Once F£ is measured, the gluon distribution in nuclei, gA(x,Q2), can be
187
188
determined indirectly and approximately through scaling violations of FA :
^ ^ ? f 2 l s W > . (2) However, the accuracy and extent in x and Q2 variables of the available data do not offer enough statistics to pin down gA{x, Q2) reliably enough.
Alternatively, gA{x,Q2) can be studied directly via the "proton-gluon fusion" mechanism :
7*-|V ">•//* + A (3) which dominates lepto- and hadroproduction of vector mesons (J/^f, W, T) at high energies.
Another important source of information on nPDFs is Drell-Yan high-mass dimuon production, p + A —> fi+^~ + X. To leading order, the differential cross section of this process reads
£k - C £ E - J ( ' . < ' ' > * <»>>+tt-itfc))' (4)
i
where y/s is the centre-of-mass energy; x\ and x-i are the Bjorken variables of the projectile and target, respectively. In the kinematics, where X\ ^> x%, the first term dominates, and Eq. (4) gives a measure of the distribution of antiquarks in nuclei.
How well do we know nPDFs after almost 20 years of experiments? The dispersion of predictions of modern models for nPDFs indicates that nPDFs are not constrained sufficiently. Figures 1 and 2 demonstrate predictions for the ratios uA/(AuN) and gA/(AgN) at Q = 1.5 GeV as a function of x for the heavy nucleus of 208Pb. The curves are results of Frankfurt et al.2 (solid curve labelled FGMS), Eskola et al.3 (dotted curve labelled ESK98), Li and Wang4 (dotted curve labelled HIJING), and Kumano et al.5 (dash-dotted curve labelled HKM). While the solid curve is a theoretical prediction made within the framework of a leading twist approach (see Sec. 3), the other three curves are fits to the available data with several simplifying assumptions. There is no region in x, where the four results would be consistent with each other.
2. Drell-Yan Dimuon Production on Nuclei at JHF
The Japan Hadron Facility (JHF) offers a unique possibility to perform very precise measurements of Drell-Yan high-mass dimuon production on nuclei, and thus to determine the nuclear sea and valence quark distributions6.
Let us briefly discuss relevant kinematics of the reaction (see Eq. (4)). To leading order, Drell-Yan dimuon production can be envisioned as annihilation
189
0.9
0.8
0.7
0.6
0.5
-
/
J ^
" y /
'/
£ • • • • •
* * f
. . /
Q-1.5 GeV 208Pb
i i
I
FGMS
ESK98
HIJING HKM
Figure 1. The ratio of the antiquark nPDF, uA/(AuN), within four different models.
of a quark (antiquark) from the beam by an antiquark (quark) from the target into a virtual photon with subsequent production of a pair of oppositely charged muons. The Bjorken variable x\ refers to the beam (proton), while xi refers to the nuclear target. These variables are related by x\X2 = Q2/s, where Q2 is the virtuality of the photon. Q2 is equal to the invariant mass squared of the detected dimuon pair, and, since one wants to stay away from the n+(i~ resonances, Q2 is large, Q2 > 16 GeV2.
Drell-Yan dimuon production on nuclei was studied by the Fermilab E772 experiment with 800 GeV/c protons7. In this case, 0.05 < x2 < 0.3 and the second term in Eq. (4) can be safely neglected. Thus, the experiment gave a direct measure of the antiquark nPDFs in several nuclei with the conclusion that antiquarks are not modified in nuclei for 0.1 < x < 0.3. This finding imposes powerful constraints on models of nuclear shadowing and antishadowing for antiquarks in nuclei: the scenario of Kumano et al. is ruled out. Also, the E772 result rules out models of the EMC effect with significant pion excess.
JHF will have two advantages over the E772 experiment. First, the beam energy is lower, Ep = 50 GeV, which results in an increase by factor 16 of the
190
FGMS ESK98
Q=1.5GeV HIJING HKM
208 p b
Figure 2. The ratio of the gluon nPDF, gA/(Ag ), within four different models.
differential cross section in Eq. (4) because of the factor 1/s. Second, a much higher flux will give another factor 16 gain. The values of xi will be higher, #2 > 015 — 0.2, which means that both terms in Eq. (4) should be retained. This will allow one to study both antiquark and valence quark nPDFs. One should note that the depletion of nPDFs at low x (nuclear shadowing), is followed by some enhancement (antishadowing) at 0.05 < x < 0.2 (see Fig. 2). Thus, in the available domain of X2, JHF will be able to determine the pattern of antishadowing with much better accuracy than any previous experiment.
Among other exciting possibilities one should also mention measurements of hotly debated parton energy loss, using Drell-Yan dimuon production6. Besides very high statistics, the values of x-i values are higher at JHF than at Fermilab (Fermilab experiment8 E866), which would enable a much cleaner analysis of the data.
191
3. Leading Twist Nuclear Shadowing and Antishadowing of Singlet PDFs
In Figs. 1 and 2 we presented our predictions for the singlet (antiquark and gluon) nPDFs. Inspecting Fig. 2 one notes that the ratio gA/(AgN) < 1 for 10~4 < x < 0.03, which means that the gluon distribution is depleted, or shadowed, in nuclei. For 0.03 < x < 0.2, gA/(AgN) > 1, which represents an enhancement, which is termed antishadowing. Both effects are characteristic to small x and are believed to arise from the coherent (simultaneous) interaction of the projectile with several nucleons in the target. In what follows we shall give a brief account of the derivation of our leading twist result. For more details, we refer the reader to the original publication2.
Our leading twist approach to nuclear shadowing in DIS on nuclei is based on the space-time picture of hadron-deuteron scattering developed by Gribov9. It was observed that at high energies the nuclear shadowing correction to the total cross section, which arises from the simultaneous interaction of the projectile with both nucleons, is dominated by the excitation of diffractive intermediate states. Ultimately this enables one to relate nuclear shadowing in hadron-deuteron scattering to diffraction in hadron-nucleon scattering.
These ideas can be generalized to lepton DIS on any nucleus10. Moreover, using the QCD factorization theorems for inclusive and hard diffractive processes, nuclear shadowing can be formulated for each nuclear parton distribution, fj/A, separately. Introducing the shadowing correction Sfj/A as Sfj/A = fj/A - Afj/N, we obtain
Sfj/A = A i \ ^lQxRe (1-itf /
/•OO /»00
P dxrfflrl(P,Q'>,xr,0) pA{b,z^) pA{b,*,)<,<•'•""<•' J X
x e -{A/2)(l-iV)aift / / * dzpA(z)
(5)
Here j denotes the parton species (flavour of the quark or the gluon); 77 is the ratio of the imaginary to real part of the diffractive scattering amplitude; 6 and z\ and z-i are the transverse and longitudinal coordinates of any two nucleons of the target nucleus; xo is a cut-off parameter (xo = 0 . 1 for antiquarks and #0 = 0.03 for gluons); PA is the nuclear density normalized to unity. The exponential factor takes into account the suppression due to the interaction with three and more nucleons. The essence of the leading twist approach is that Sfj/A is proportional to the diffractive parton distribution of the nucleon, ff/N, which is a function of the usual diffractive variables (3 = x/xp, xp and t (note
192
that t — 0). Since fP,N obeys the leading twist DGLAP evolution equations, so does Sfj/A- So a slow, logarithmic, or leading-twist Q2-dependence of nuclear shadowing is expected.
One of the characteristic features of Eq. (5) is that nuclear shadowing for gluons is more sizable than for the sea quarks. Indeed, as observed at HERA, DIS diffraction is dominated by the gluon diffractive distribution, i.e. fq/N -^ fq'/N- Hence, &9A > SuA. This trend can be seen in Figs. 1 and 2.
The cut-off parameter xo cannot be determined in our approach from the first principles and should be inferred from the comparison to the data. This parameter defines the transition point from shadowing to antishadowing. As discussed above, antishadowing for the sea quarks was not observed by the E772 experiment so that xo can be taken arbitrary large (xo = 0.1 in our case). The analysis of scaling violations of FA indicated that gluons in nuclei might be enhanced11 for 0.03 < x < 0.2. Thus, in our analysis we choose XQ = 0.03 for gluons and complement Eq. (5) by modelling the enhancement of the gluon nPDF in the interval 0.03 < x < 0.2 by requiring that the fraction of the total momentum carried by gluons is the same in nuclei and the proton:
/•0.2 /-0.2
/ dxxgA(x,Q2)/A= dxxgN{x,Q2). (6)
Finally, we note that the leading twist approach is applicable for the evaluation of nuclear shadowing of the antiquark and gluon nPDFs. Indeed, high energy diffraction is believed to be dominated by the 2-channel exchange with vacuum quantum numbers, the Pomeron exchange. Exactly this type of exchange drives singlet parton distributions (antiquarks and gluons) at low x.
4. Shadowing and Antishadowing of Valence nPDFs
The situation with nuclear shadowing of the valence nPDFs is less transparent. The coherent theoretical picture of nuclear shadowing is absent, and the available fixed-target data on FA are not sensitive to the valence distributions in nuclei at low x. From the theoretical point of view, nuclear shadowing for valence nPDFs could be formulated in a form similar to Eq. (5) with the important modification that the Pomeron exchange is replaced by the interference between the Pomeron and Reggeon (non-vacuum quantum numbers) exchanges. However, the experimental information on such type of interference is absent.
It is interesting to note that high precision measurements of Drell-Yan dimuon productions at JHF for x2 > 0.15 — 0.2 should significantly clarify the situation with the valence quarks. Indeed, the baryon charge sum rule, for
193
instance,
A
dxut,(x,Q2)/A = 2, (7)
combined with the measured expected enhancement for x > 0.05, would give
a tight constraint on the net contribution of the low-a: shadowing region to the
sum rule in Eq. (7).
Finally, we cannot help but reflect on outstanding theoretical challenges in
the theory of nuclear shadowing and antishadowing. First, while the origin of
nuclear shadowing is understood, the dynamics of antishadowing is unknown:
Eq. (5) does not include antishadowing in nPDFs . Ideally one would like to
have a theory, which would include both effects on equal footing, and would
also automatically satisfy the baryon charge and momentum sum rules. Sec
ond, a theory of small-a; nuclear effects for the valence and non-singlet n P D F s
is lacking. Once available, one can immediately analyse low-a; behaviour of
such quantities as non-singlet combinations of nuclear structure functions,
^ 2 H e - -F2H. o r polarized nPDFs .
5 . C o n c l u s i o n s a n d D i s c u s s i o n
The Japan Hadron facility (JHF) can significantly contribute to studies of low-
x nuclear phenomena, shadowing and antishadowing, using Drell-Yan dimuon
production on nuclear targets. Such measurements should be precise enough
to study the upper-a; end of nuclear shadowing and the whole range of an
tishadowing of antiquark and valence nPDFs . This will help understand the
presently unknown dynamics of antishadowing and shadowing for the valence
n P D F s and will have implications for small-a; DIS on polarized nuclear targets .
As experiments complimentary to the main physics program of J H F , we
suggest to study Drell-Yan dimuon production on nuclei induced by pion and
kaon beams, which should allow one to eliminate the resonance background and
decrease the mass of the measured dimuon pair and, thus, Q2 of the probed
partons. This will enable to take da t a in previously unat tainable kinematical
region, which will facilitate a comparison to the inclusive da ta generally taken
at lower Q2.
One can also suggest to measure inelastic J/\I? production on nuclei. Wi th
the luminosity of J H F , one should be able to extract the gluon n P D F in the
antishadowing region with unprecedented accuracy.
A c k n o w l e d g m e n t s
This work was supported in part the Australian Research Council and the
University of Adelaide.
L
194
References
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2. L. Frankfurt, V. Guzey, M. McDermott, and M. Strikman, JHEP 202, 27 (2002). 3. K. J. Eskola, V. J. Kolhinen, and P. V. Ruuskanen, Nucl. Phys. B535, 351 (1998). 4. Shi-yuan Li and X. N. Wang, Phys. Lett. B527, 85 (2002). 5. M. Hirai, S. Kumano, and M. Miyama, Phys. Rev. D64, 034003 (2001). 6. M. Asakawa et al., Expression of interest for Nuclear/Hadron Physics Experiments
at the 50-GeVProton Synchrotron, KEK Report 2000-11, October 2000. 7. D. M. Aide et al., Phys. Rev. Lett. 64, 2479 (1990). 8. M. A. Vasiliev et al., Phys. Rev. Lett. 83, 2304 (1999). 9. V. N. Gribov, Sov. Phys. JETP 29, 483 (1969). 10. L. Frankfurt and M. Strikman, Eur. Phys. J. A5, 293 (1999). 11. T. Gousset and H. J. Pirner, Phys. Lett. B375, 349 (1996).
THE N U T E V ANOMALY A N D SYMMETRY BREAKING IN THE PARTON DISTRIBUTION FUNCTIONS
F. G. CAO AND A. I. SIGNAL Institute of Fundamental Sciences, Massey University
Palmerston North, 5301, New Zealand E-mail: [email protected]; [email protected]
The NuTeV collaboration recently reported a measurement of sin 8^ which is about 3 standard deviations above the standard model prediction. Quark-antiquark symmetry breaking in the strange sea and charge symmetry breaking in the parton distribution functions may affect the NuTeV result. We present theoretical calculations of these symmetry breakings a using meson cloud model. It was found that the corrections from these symmetry breakings would not be significant enough to explain the NuTeV anomaly.
1. The NuTeV Result
The NuTeV collaboration recently measured1 the electroweak parameter sin2 6w using neutrino (anti-neutrino) -nucleon scattering. In order to reduce the uncertainties associated with charm production, the Paschos-Wolfenstein ratio2 was measured,
P - _ aNC ~ aNC _ 1 0 ; „ 2 f l M \
"cc acc z
where cr^c and <TQQ a r e ^n e neutral and charged current cross sections for neutrino (anti-neutrino) respectively. By measuring the cross sections in Eq. (1) the NuTeV extracted the value sin2 9W = 0.2277 ± 0.0013(sicrf.) ± 0.0009(sj/si.), which is about 3 standard deviations above the standard model prediction of 0.2227 ± 0.0004. The possible explanations for the NuTeV anomaly could be :
(1) The experimental uncertainties (systematic error and statistical error) may be larger than that have been estimated in the NuTeV analysis3;
(2) Theoretical corrections to the Paschos-Wolfenstein relation (Eq. (1)) may be important4,5;
(3) The anomaly comes from anomalous neutrino interactions6; (4) The anomaly may be an indication of new physics4.
195
196
In this paper we will study explanation (2), particularly the corrections from possible symmetry breaking in the parton distribution functions (PDFs). The Paschos-Wolfenstein relation (Eq. (1)) is obtained under the assumption of parton distribution functions of the nucleon being strange quark-antiquark symmetric (s — s) and charge symmetric (dp — un and up = dn). There have been experimental analyses7,8'9 and several theoretical investigations10,11,12'13
on the possible breaking of s-s symmetry and charge symmetry in the PDFs since the discovery and establishment of flavour symmetry breaking in the nucleon sea14. Taking into account these symmetry breakings, the Paschos-Wolfenstein relation, Eq. (1), becomes4
R~ — — — sin2 0w 2
(Suv) - (Sdv) - (Ss) 2 (9LU ~9RJ+ ^Id ~ 92Rd) (2) ((u) + (d))/2
where (q) — f dxxq(x) and (Sq) — f dxxSq(x) are the second moments of the corresponding distributions, and giq and gRq (q — u, d) are the Z couplings. 8dv — dv
v — u" and Suv = up — d™ 'measure' charge symmetry breaking (CSB) in the PDFs, and Ss — sp -sp — sn - sn 'measures' quark-antiquark symmetry breaking in the strange sea. Eq. (2) clearly shows that s-s symmetry breaking and CSB in the PDFs will affect the extraction of sin2 Ow in the NuTeV measurement. It is the second moments of the symmetry breakings that enter into the Paschos-Wolfenstein relation, and a value {Suv) — {Sdv) — (6s) m —0.0038 is needed4 to make the NuTeV result to be consistent with the standard model prediction. We investigated this breaking11,13 using a meson cloud model which can provide a natural explanation of the flavour symmetry breaking in the nucleon (d > u for the proton)14.
2. s-s Symmetry Breaking
In the meson cloud model (MCM)15 the nucleon can be viewed as a bare nucleon surrounded by a mesonic cloud. The physical nucleon state can be expressed in terms of a series of baryon and meson components,
|P>phys. = | p > t o r e + | J V 7 r ) + | A j r } + | A ( E ) / i 0 + . . . . ( 3 )
The pion is needed for chiral symmetry in quark models and plays an important role in understanding of the nucleon structure at lower energy. A crucial observation was made by Sullivan16 in the early 70's about the importance of these baryon-meson components in high energy processes, for example, about the contributions to the nucleon structure functions measured in deep inelastic scattering. If the lifetime of the baryon-meson Fock states is much longer
197
than the strong interaction time between the hard photon and nucleon, then the interaction between the hard photon and the baryons and mesons in the expansion of Eq. (3) may become important. The contribution to the PDFs of the nucleon from such kind of processes can be written as convolution of the fluctuation function, which describes the fluctuation N —¥ BM, with parton distributions of the baryon and meson. Then the flavour asymmetry d > u emerges naturally since the probability of finding the Fock state \TIK+) is much larger than that of | A + + T T _ ) .
For the strange content of the nucleon sea, the important baryon-meson components are AA' and EA'. The non-perturbative contributions to the strange and anti-strange distributions in the proton can be written as11
dyfBK{y)sBC-) (4) y y Jx
*"(*) = fdJ-fKB{y)sK{X-), (5) Jx y y
where /sjf(y) = / K B ( 1 — y) (y and 1 — y being the longitudinal momentum fractions of the baryon and meson) are the fluctuation functions, and s (B = A, E) and sK are the s and s distributions in the A (E) baryons and K meson, respectively. The fluctuation functions are calculated11 from an effective meson-baryon-nucleon interaction Lagrangian using time-ordered perturbation theory in the infinite momentum frame,
*MCM,.A _ 9NBK P dkl G%K(y,kl) {ymN-mB)2 + kl fBK ( » ) - l 6 ^ - y 0 , (1 _ y) {m% - m%KY y ' ( 6 )
where gNBK is the effective coupling constant; mBK — (m2B + k]_)/y+ (m2
K + &j_)/(l—y) is the invariant mass squared of the BK Fock state; and GBK{V^ k\) is a phenomenological vertex form factor for which we adopt an exponential form
GBK{y, fc±) = exp N-mBK(y,kl) 2A?
(7)
with Ac being a cut-off parameter. From Eqs. (4) and (5) we know that the non-perturbative contributions
to s and s distributions in the nucleon are different, and the difference s — s depends on both the fluctuation functions (/BK ano> IKB) and the parton distributions in the baryon and meson (sB and sK). Due to the A (S) baryons being heavier than the K meson, fBx(y) is harder than fxB(y), which suggests
giv y gN -m £ne j a r g e x r e gi o n . On the other hand, the s distribution of the K meson (sK(x)) is generally believed to be harder than the s distribution of
198
the baryon (sB(x)) as the baryon contains one more valence quark than the meson, which implies sN < sN in the large x region. The final prediction of the s-s asymmetry will depend on these two competing effects.
We employed the following two prescriptions for the strange and anti-strange distributions in the A (£) baryons and K meson :
(1) Use SU(3) symmetry for the parton distribution functions of the baryons, i.e. sA = s s = \uN where the MRST9817 parameterization for uN is adopted, and GRS9818 parameterization for s , which is obtained by connecting sK to the valence quark distribution in the pionic meson based on the constituent quark model;
(2) Calculate the strange distributions in the bag model in order to take into account the SU(3) symmetry breaking effect in the PDFs of the baryons, sA ^ s s ^ \uN, which could be important19.
The numerical results are given in Fig. 1. It can be seen that the two prescrip-
0.0015 - -
0.001
0.0005
0
-0 .0005 0 0 .1 0.2 0.3 0.4 0.5 0.6
Figure 1. x(s — s) versus x. The solid and dotted curves are the results using prescription (1) and (2) for sB and SK, respectively.
tions give very different predictions for the s-s asymmetry. The calculations also depend on the cut-off parameter Ac introduced in
Eq. (7). The baryon and meson production data can impose some constrains on this parameter and a value of about 1.1 GeV is preferred20. We studied the dependence on this parameter by allowing Ac to change from 0.8 GeV to 1.5 GeV. The probabilities of finding the AA' (£A') Fock states in the proton wavefunction change from 0.18% (0.09%) to 5.4% (3.5%). Consequently the independence and the magnitude of the s-s asymmetry change dramatically (see Fig. 2). The Ac-dependence of (Ss) is plotted in Fig. 3. (6s) depends on Ac
much less dramatically than the difference x(s — s) because of the cancellation
199
SU3 Bag
0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6
Figure 2. x(s — s) versus x. The solid, dashed and dotted curves are the results with Ac = 1.5, 1.08 and 0.8 GeV, respectively.
0.8 0.9 1.1 1.3 1.5
Figure 3. (Ss) versus A c . The dotted and solid curves are the results using prescription (1) and (2) for s and s , respectively.
between the contributions in the small x and large x regions (see Fig. 2). (Ss) changes from being positive to negative with the increase of the cut-off Ac and has a maximum at about Ac = 1.1 GeV. So the theoretical estimates for (Ss) lie in the range from -0.0005 to 0.0001, which is about an order of magnitude smaller than that is needed to explain the NuTeV anomaly.
3. Charge Symmet ry Breaking
It has been generally believed that charge symmetry is highly respected in the nucleon system. However some theoretical calculations12 have suggested that the charge symmetry breaking (CSB) in the valence quark distributions may be as large as 2% - 10%, which is rather large compared to the low-energy
200
results. Any unexpected large CSB will greatly affect our understanding of non-perturbative dynamics and hadronic structure, and also the extraction of sin2 6w from neutrino scattering.
CSB in both the valence and sea quark distributions comes from non-perturbative dynamics. Thereby the meson cloud model (MCM), describing the non-perturbative structure of the nucleon, can provide a natural explanation of CSB in the valence and sea quark distributions of the nucleon13. The fluctuations we consider include TV —>• Nir and N —»• An. The baryons (mesons) in the respective virtual Fock states of the proton and neutron may carry different charges. If we neglect the mass differences between these baryons (mesons), the fluctuation functions for the proton and neutron will be the same, and the contributions to the parton distributions of the nucleon from these fluctuations will be charge symmetric. The electromagnetic interaction induces mass differences3, among these baryons (mesons)21,
trip — f^n = —1-3 MeV, mn± — m„o = 4.6 MeV,
mAo - mA+ = 1.3 MeV, mA- - mAo = 3.9 MeV. (8)
Due to these mass difference the probabilities of corresponding fluctuations for the proton and neutron may be different, thereby the contributions to the PDFs of the proton and neutron from these fluctuations may be different.
The numerical result for CSB in the valence quark sector is given in Fig. 4. We find that xSdv and xSuv have similar shapes and both are negative, which
° T
•S § -0.0002-a* u
f -0.0004-
> -0.0006-
u -0.0008 -•
0 0.1 0.2 0.3 0.4 0.5
X
Figure 4. Charge symmetry breaking in the valence sector.
is quite different from the quark model prediction12 for x6dv being positive for
aUnlike the other mass splittings, the splitting between charged and uncharged pion masses is not a violation of charge symmetry - it is a violation of isospin symmetry.
xSdv
x5uv
201
most values of x. Our numerical results are about 10% of the quark model estimates12. We did not find any significant large-x enhancement of the ratio Rmin = 8dv/d%, which is predicted in the quark model calculations12. The smallness of any CSB effect as x -^ 1 is natural in the MCM, as all the fluctuation functions go to zero as y —>• 1, and hence there is no non-perturbative contribution to the parton distributions at large x. We estimate (Su)—{Sd) is about —0.0003, which is much smaller than required to explain the NuTeV anomaly. Considering both s-s symmetry breaking and charge symmetry breaking we have (Su) — (Sd) — (5s) ca 0.0002 0.0004, which is an of magnitude order smaller than needed to explain the NuTeV discrepancy.
4. Summary
The NuTeV measurement of sin2 6W being 0.2277 ±0.0013(stat.)±0.0009{sy si.) is about 3 standard deviations above the standard model prediction. Possible symmetry breaking in the parton distribution functions of the nucleon, strange-antistrange symmetry breaking and charge symmetry breaking may affect the interpretation of NuTeV result. We reported theoretical calculations for these symmetry breakings using a meson cloud model, which could naturally explain flavour symmetry breaking in the nucleon sea (d > u). It was found that the corrections from the two symmetry breakings would not be significant enough to alter the NuTeV result. Moreover there is no established experimental evidences for or against quark-antiqu ark symmetry and charge symmetry in the parton distribution functions of the nucleon. More studies are needed to explain the NuTeV anomaly.
Acknowledgments
This work was partially supported by the Science and Technology Postdoctoral Fellowship of the Foundation for Research Science and Technology, and the Marsden Fund of the Royal Society of New Zealand.
References
1. G. P. Zeller et al, NuTeV Collaboration, Phys. Rev. Lett. 88, 091802 (2002). 2. E. A. Paschos and L. Wolfenstein, Phys. Rev. D7, 91 (1973). 3. G. A. Miller and A. W. Thomas, hep-ex/0204007. 4. S. Davidson et al., hep-ph/0112302. 5. G. P. Zeller et al., NuTeV Collaboration, hep-ex/0203004. 6. E. Ma and D. P. Roy, Phys. Rev. D65, 075021 (2002); S. Barshay and G.
KreyerhofT, Phys. Lett. B535, 201 (2002); C. Giunti and M. Laveder, hep-ph/0202152.
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7. A. O. Bazarko et al., CCFR Collaboration, Z. Phys. C65, 189 (1995). 8. M. Goncharov et al., NuTeV Collaboration, Phys. Rev. D64, 112006 (2001). 9. V. Barone, C. Pascaud and F. Zomer, Eur. Phys. J. C12, 243 (2000).
10. A. I. Signal and A. W. Thomas, Phys. Lett. B191, 205 (1987); S. J. Brodsky and B. Q. Ma, Phys. Lett. B381, 317 (1996).
11. F.-G. Cao and A. I. Signal, Phys. Rev. D60, 074021 (1999). 12. For a recent review see J. T. Londergan and A. W. Thomas, in Progress in
Particle and Nuclear Physics, Volume 41, P. 49, ed. A. Faessler (Elsevier Science, Amsterdam, 1998); E. Sather, Phys. Lett. B274, 433 (1992); E. Rodionov, A. W. Thomas and J. T. Londergan, Mod. Phys. Lett. A9, 1799 (1994).
13. F.-G. Cao and A. I. Signal, Phys. Rev. C62, 015203 (2000). 14. E. A. Hawker et al., E866/NuSea Collaboration, Phys. Rev. Lett. 80, 3715
(1998); For a recent review see J.-C. Peng and G. T. Garvey, hep-ph/9912370. 15. A. W. Thomas, Phys. Lett. B126, 97 (1983). 16. J. D. Sullivan, Phys. Rev. D5, 1732 (1972). 17. A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thome, Eur. Phys. J.
C5, 463 (1998). 18. M. Gliick, E. Reya and M. Stratmann, Eur. Phys. J. C2, 159 (1998). 19. C. Boros and A. W. Thomas, Phys. Rev. D60, 074017 (1999). 20. H. Holtmann, A. Szczurek and J. Speth, Nucl. Phys. A569, 631 (1996). 21. D. E. Groom et al., Particle Data Group, Eur. Phys. J. C15, 1 (2000).
NUCLEONS AS RELATIVISTIC THREE-QUARK STATES
M. OETTEL Max-Planck~Institut fur Metallforschung
Heisenbergstr. 3, 70569 Stuttgart, Germany E-mail: [email protected]
A covariant nucleon model is formulated which uses dressed quarks compatible with recent lattice data and Dyson-Schwinger results. Two—quark correlations are modelled as a series of quark loops in the scalar and axial vector channel. Faddeev equations for nucleon and delta are solved and nucleon form factors are calculated in a fully covariant and gauge-invariant scheme. The insufficience of a pure valence quark description becomes apparent in results for the electric form factor of the neutron and the ratio HGE/GM for the proton.
1. Introduction to the Model
Virtually all intricacies and complications of QCD are present in the analysis of the simplest baryonic bound state, the nucleon. Spectroscopy suggests that it is mainly composed of three confined constituent quarks of mass ~ 0.33 GeV. Probing its structure in deep inelastic scattering (DIS) reveals a completely different picture: Depending on the energy of the incoming photon, the proton appears to be a complicated mixture of a three-quark valence core supplemented with higher qq and gluonic components. To be precise, such a Fock-state picture is rigorously valid only in the light-cone gauge, and the quark states should correspond to the nearly massless current quarks from the QCD Lagrangian. From both points of view, the nucleon is a complicated relativistic bound state which calls for a covariant description since already the simplest processes which are described by the various nucleon form factors involve considerable momentum transfers.
It has become common wisdom that massive constituent quarks are generated by the spontaneous breaking of the approximate chiral symmetry of QCD. Assuming that the gluon propagator and the quark-gluon vertex possess enough integrated strength, Dyson-Schwinger (DS) equation studies1'2
have predicted a running quark mass which is about the constituent quark mass in the infrared and drops to the current mass in the ultraviolet. This has been confirmed on the lattice3 '4. Although all these results have been obtained
203
204
0 02 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
rfm
Figure 1. Lattice results for the static three-quark potential. The data points for the potential (little dashes with error bars) are compared to static, string-like potentials of Y and A type between the three quarks. (Adapted from Ref. [8].)
in Landau gauge and not in the light-cone gauge, we may assume that a good part (though by no means all) of the complicated Fock state structure of the proton is buried in the structure of the constituent quark.
Covariant bound states wave functions may be obtained by use of the co-variant Bethe-Salpeter (BS) equation. The properties of pseudoscalar mesons are described successfully in combined DS/BS-studies2 which emphasize their dual role as both q — q bound states and Goldstone bosons. Along these lines the nucleon's bound state amplitude can be obtained by solving a relativistic Faddeev equation if irreducible three-quark correlations are neglected. This is a strong assumption which has been tested on the lattice only for static quarks, see Fig. 1. The Faddeev equation needs as input the full solution for the q — q scattering kernel (whose determination is itself almost an unsurmount-able task). Truncating the interaction between the two quarks to lowest order (one effective gluon which generates the constituent quark mass) one finds di-quark poles with scalar (0+) diquarks (RS 0.7 — 0.8 GeV) and axial vector (1+) diquarks (m 0.9 — 1.0 GeV) having the lowest masses2,5. Although the poles may not survive in higher orders6 the dominance of scalar and axial vector correlations in the q — q matrix has also be seen on the lattice (in Landau gauge), see Fig. 2. Despite these signals for pronounced q — q correlations, no evidence for compact (i.e. pointlike) diquarks has been seen in DIS or other processes.
205
40
35
30
25
20
15
10
5
p(o>)
.
'
.
•
(a)
• • i - i 1 1 1
I scalar (303) diquark 1 axial vector (613) diquark
! "mass" peak K = 0.147
continuum K / ••-.,. 0)[GeV]
0 0.5 1 1.5 2.5 3.5
Figure 2. Lattice results for the spectral function in the colour antitriplet (3) q — q channel. There is an almost perfect mass peak in the scalar diquark channel while for the axial vector diquark the mass peak and the continuum are of equal importance. The spectral functions of other diquarks is of no significance. (Adapted from Ref. [7].)
Putting all these motivations together, we formulate a covariant model which incorporates9
• quark propagators with a running mass function in accordance with DS and lattice results. A parameterization in terms of entire functions is used which has been fitted to a number of low-energy meson observables10. Although the general shape agrees with recent lattice dataa (see Fig. 3) the latter suggest a somewhat broader mass function. The relevance of this for the electromagnetic form factors will be discussed.
• separable q — q correlations truncated to the 0+ and 1 + channels, see Fig. 3. Their parameterization involves diquark vertex functions x> which are described by their dominant Dirac structure, and a scalar function F, which assigns a width wo+[i+] to the diquarks. The diquark propagator is obtained as a sum of quark polarization loops. An external probe (e.g. a photon) will resolve the quark substructure of the diquarks by coupling to the quarks in the loop.
• strict covariance in solving the resulting Faddeev equations.
Thanks to the separability assumption the Faddeev equations reduce to an
aNote that the fit has been obtained some years earlier than lattice data became available.
206
A Lattice, m ^ = 91 MeV
— u/d quark parametrizatkm
m K . . = 5MeV
" Z 2€>=CC
*/"'"
a=/0+,l+}
= A,,
/ = P o + ( 7 5 C ) F ( » o + )
Figure 3. Left panel: Parametrized quark mass function compared to lattice da ta from Ref. [3]. Right panel: The separability assumption for the q — q t matrix.
effective diquark-quark Bethe-Salpeter equation where the attractive interaction is effected through the quark exchange between the quark and diquark, see Fig. 4. With the masses of the nucleon and delta as input parameters, we can vary only the width of the scalar diquark which in turn determines the diquark masses through the zeros of the inverse diquark propagators. The rich structure of the BS wave functions and the numerical procedure to solve the BS equation can be found in Refs. [11,12] respectively. Exemplary width parameters for which a solution can be found together with the resulting diquark masses are given in Table 1. We see that for a scalar diquark width of ~ 0.3 fm the diquark masses are in agreement with the position of the lattice peaks. Smaller widths give approximately equal masses for the scalar and axial vector diquark while larger widths tend to increase the mass difference. We remark that the width of the nucleon BS wave function (which can be interpreted as a quark-diquark separation) is of about 0.4 fm. On the one hand, this indicates no strong diquark clustering (for which there is no evidence anyway). On the other hand the diquark width should not exceed the quark-diquark width for
a,i={0, f }
Figure 4. The Bethe-Salpeter equation for the effective baryon-quark-diquark vertex function »o+[i+l.
207
our diquark-quark picture to make sense.
Table 1. Parameter sets which give a solution to the BS equation with the physical mass of nucleon and delta.
Set w0+ wl+ m0+ TOj+ Mff M& - f m - - G e V -
I 0.34 0.25 0.75 0.92 II 0.28 0.17 0.80 0.89 0.94 1.32
III 0.25 0.14 0.86 0.87
2. Electromagnetic Form Factors
The so-called gauging method13 gives the correct, gauge invariant prescription how to treat processes where an external photon couples to a bound state whose wave function in turn is described by the solution of an integral equation. It basically consists of coupling the photon to all momentum dependent terms (propagators, vertices) in the kernel of the integral equation. Consequently we have impulse approximation diagrams where the external photon couples to the spectator quark and the quarks within the diquark. Inspecting the kernel of the BS equation, we find furthermore diagrams where the photon couples to the exchange quark and directly to the quark-diquark vertices (seagull graphs). The most important diagram, though, is the impulse approximation quark diagram, and its central element is the quark-photon vertex. It consists of a longitudinal and a transverse part,
r£ = r£ L + r £ T ) (l)
where the longitudinal part T^L (the Ball-Chiu vertex), which is fixed by the Ward-Takahashi identity, is entirely determined by the form of the quark propagator13. The remaining transverse part might receive dynamical contributions of which the p — w meson poles in the q — q vector channel are presumably the most important ones. The most thorough study of these contributions has been made in Ref. [14] where the quark-photon vertex is analysed in the DS/BS framework. It was shown/that the transverse vertex contributes about one half to the pion charge radius and that it is essential for reproducing the
bTo be precise, the Ball-Chiu vertex contains a fixed transverse part since it should be free of kinematical singularities.
208
Figure 5. Electric form factor of the proton (left panel) and neutron (right panel). The small dotted curve in the right panel refers to a calculation from Ref. [15], assuming an electromagnetically compact (pointlike) diquark.
experimental results for the pion form factor. Following the analysis of Ref. [14] we write for the transverse vertex
r I * r f = ^ Q ^ ^ ^ W = *-P). (2)
Here the BS wave function of the vector meson ^ is modelled by its two dominant structures. It is properly normalized and reproduces the experimental decay constant fp. The exponential describes an off-shell damping of the vector meson propagator, and we fit the constant a to the pion form factor, for details see Ref. [15].
The resulting electric form factors for proton and neutron are plotted in Fig. 5. We find for all sets that the transverse part of the quark-photon vertex contributes about 25% to the proton electric charge radius. Not surprisingly the form factor becomes softer with increasing diquark size which needs to be above 0.3 fm to bring the theoretical curve close to the data. Turning to the neutron, we see that all sets give a positive GE (a relativistic effect due to lower components in the BS wave function) but miss the experimental low-Q2 behaviour completely. For comparison we have given results from a calculation with pointlike diquarks which are roughly compatible with the data. It appears that the latter assumption mimics non-valence effects that may arise if e.g. a pion-cloud is added to the nucleon. A pion-cloud would not only influence neutron's GE but also induce a mass shift for the nucleon between16 200 and 300 MeV and somewhat less for the delta. We therefore solved the BS equations for higher core masses of the nucleon and delta and found quantitatively little difference to the old form factor results. Especially
209
• Jones etal.PRL 84(2000) 1398 0.1 -
0I , 1 , 1 . 1 , 1 0 0.5 1 1.5 2
Figure 6. Proton's ratio HGE/GM in comparison with the experimental data .
neutron's GE remains quenched. Thus, a covariant calculation of pionic effects
would clearly be desirable.
For realistic diquark sizes, proton's ratio /J,GE/GM is underest imated
in our calculations. This discrepancy can be traced back to the structure
of the Bal l-Chiu vertex. After some reshuffling of its tensor s t ructure 9
one can isolate a transverse term which is proportional to the difference
in the quark mass function for the outgoing and the incoming quark, i.e.
~ [M(k2) — M(p2)]/[k2 — p 2 ] . This term (which is absent in cons tant -mass
constituent models) produces quite large negative contributions to GE for in
termediate Q2 and therefore quenches the ratio. To investigate this in more
detail, we replaced the fit of the running quark mass in Fig. 3 by a fit to the
chiral extrapolation of latest lattice da ta 4 , see the left panel of Fig. 7.
Replacing the quark mass in the Ball-Chiu vertex by (a) the new fit and
(b) by a constant we find results for (IGE/G/M which are depicted in the
right panel of Fig. 7. Surprisingly enough the constant constituent mass does
the best job but also the new fit to the lattice da ta causes some noticeable
change compared to the results with the original meson fit. Since the running
quark mass is now well established and should be an element of any serious
nucleon model we therefore conclude tha t the observed ratio is most likely a
consequence of a subtle interplay of various contributions such as a realistic
quark propagator, vector mesons in the quark-photon vertex and possibly also
pions from the cloud. Further work which analyses these contributions in much
more detail than could be done here is clearly required.
To summarize, we have investigated a covariant nucleon model involving
three constituent quarks whose propagators capture the essential features of
lattice simulations and DS calculations. Three-quark irreducible interactions
210
p[GeV] o*[GeV*]
Figure 7. Left panel: lattice data and the two fits to the quark mass function. Right panel: the ratio I*GE/GM for calculations with different quark mass functions in the Ball-Chiu vertex. The parameter for the scalar diquark width is here w0+ = 0.30 fm which results in the diquark masses rn0+ = 0.77 GeV and mj+ = 0.91 GeV.
were discarded and the problem was reduced to an effective quark-diquark
problem. The form factor results, especially for neutron's GE, point to the
necessity of incorporating meson cloud contributions. The observed rat io
[IGE/GM eludes a simple interpretation and is possibly a consequence of a
number of mechanisms.
A c k n o w l e d g m e n t s
The author wants to thank the Special Research Centre for the Subatomic
Structure of Matter (CSSM) in Adelaide where most of this study was con
ducted. He is also grateful to the Alexander-von-Humboldt foundation which
supported this research by a Feodor-Lynen grant. Special thanks go to Rein-
hard Alkofer with whom the work was done and to Tony Thomas for many
insightful remarks.
R e f e r e n c e s
1. R. Alkofer and L. von Smekal, Phys. Rep. 353, 281 (2001). 2. C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. 45, SI (2000). 3. J. Skullerud, D. B. Leinweber and A. G. Williams, Phys. Rev. D64, 074508
(2001). 4. P. O. Bowman, U. M. Heller and A. G. Williams, hep-lat/0203001. 5. P. Maris, nucl-th/0204020. 6. A. Bender, W. Detmold, C. D. Roberts and A. W. Thomas, nucl-th/0202082. 7. C. Alexandrou, P. De Forcrand and A. Tsapalis, Phys. Rev. D65, 054503 (2002). 8. I. Wetzorke and F. Karsch, hep-lat/0008008. 9. M. Oettel and R. Alkofer, hep-ph/0204178.
10. C. J. Burden, C. D. Roberts and M. J. Thomson, Phys. Lett. B371, 163 (1996).
211
11. M. Oettel, G. Hellstern, R. Alkofer and H. Reinhardt, Phys. Rev. C58, 2459 (1998).
12. M. Oettel, L. von Smekal and R. Alkofer, Comp. Phys. Comm. 144, 63 (2002). 13. A. N. Kvinikhidze and B. Blankleider, Phys. Rev. C60, 044003 (1999). 14. P. Maris and P. C. Tandy, Phys. Rev. C61 , 045202 (2000). 15. M. Oettel, R. Alkofer and L. von Smekal, Eur. Phys. J. A8 , 553 (2000). 16. M. Oettel and A. W. Thomas, nucl-th/0203073.
(POLARIZED) HADROPRODUCTION OF OPEN CHARM AT THE JHF IN NLO QCD
I. B O J A K
CSSM, University of Adelaide Adelaide, SA 5005, Australia
E-mail: [email protected]
We present the complete next-to-leading order QCD corrections to (polarized) hadroproduction of heavy flavors and investigate how they can be studied experimentally in (polarized) pp collisions at the JHF in order to constrain the (polarized) gluon density. It is demonstrated that the dependence on the unphysical renor-malization and factorization scales is strongly reduced beyond the leading order. We also briefly discuss how the high luminosity of the JHF can be used to control the remaining theoretical uncertainties.
1. Introduction
Although we have gained precise information concerning the total quark spin contribution to the nucleon spin in the last decade, the spin-dependent gluon density Ag remains elusive, see Fig. 1. Hence current and future experiments focus strongly on the issue of constraining Ag. The JHF could play a prominent role in these efforts using production of open charm. The gluon participates are dominantly in heavy flavor pair creation in longitudinally polarized pp collisions. Figure 1 also shows that even the unpolarized gluon distribution has considerable uncertainties at large x, and hence could be pinned down by a corresponding unpolarized measurement.
In leading order (LO), heavy flavor pair production in hadron-hadron collisions proceeds through two parton-parton subprocesses,
gg^QQ and qq^QQ . (1)
Gluon-gluon fusion is by far the most dominant mechanism for charm and bottom production in the unpolarized case in all experimentally relevant regions of the phase space1,2 '3,4. This should hold true in the polarized case unless Ag is very small. However, it is necessary to include next-to-leading order (NLO) QCD corrections for a reliable description. The LO results depend strongly on the arbitrary factorization and renormalization scales. Furthermore, in the unpolarized case the NLO corrections are known to be large1,2. Finally, new
212
213
1.2
1
0 . 8
0 . 6
0 . 4
0 . 2
0
xAG(x) NLO
5 G e V 2
xG(x) (GRV)
1 0 1 0 —x- ^T 1 0 "* 1
x Figure 1. Uncertainty in Ag (shaded la error bands) and in g at large a; (hatched). This is a copy of Fig. 5 of the analysis of Bliimlein and Bottcher 7 (BB).
processes with a single light quark in the initial state contribute for the first time in NLO. Unpolarized NLO results1 '2,3,4 and polarized LO expressions5'6
have been available before, but the complete NLO results are presented here for the first time.
2 . Technical Framework
The 0{oP$) NLO QCD corrections to heavy flavor production consist of the one-loop virtual and the real "2 —• 3" corrections, the latter include the new production mechanism <?g[f] —̂ QQg[?]. We use n-dimensiona! regularization, with n = 4 + e , to deal with the spurious singularities. Ultraviolet singularities are removed by mass and coupling constant renormalization at the renormal-ization scale (JLR. The employed scheme is a variant of MS typical for heavy flavor production3'4'2, which yields riij = nj — 1 light and one heavy flavor. Infrared divergencies and poles from soft gluon brernsstrahlung cancel. The remaining collinear singularities are absorbed into the bare parton densities at the factorization scale fj,p using the MS scheme.
The unpolarized and polarized squared matrix elements can be obtained simultaneously using arbitrary helicities A ^ ="±" for the incoming quarks or
214
gluons:
|M|2(A1)A2) = |M | 2 + A1A2A|M|2 . (2)
The unpolarized \M\ in Eq. (2) can be compared to the literature1,2,3,4. This is an important consistency check for the correctness of our new A|M| 2 . We treat the Levi-Civita e-tensor and 75 in n-dimensions by applying the HVBM prescription8. However, this leads to e-dimensional scalar products appearing alongside the usual n-dimensional ones. These contributions can be accounted for by an appropriately modified phase space formula9,10.
The spin-dependent parton densities and cross sections are denned by
Af(x^F) = f+(x,fiF)-ft(x,fiF) (3)
Acr = ![*(+,+)-*(+,-)] • (4)
Here /+ (ft) gives the probability to find a parton / = q,q,g at a scale HF with momentum fraction x and helicity + (—) in a proton with helicity +. Similarly, cr(+, +) is the hadronic (<r partonic) cross section for the scattering of two hadrons (partons) with helicities +. The familiar unpolarized quantities are obtained by taking the sum in Eqs. (3) and (4), respectively . In the following <f> denotes both the unpolarized quantity <f> and the polarized analogue A^.
The virtual cross sections are obtained from the interference between the virtual and Born amplitudes and contain tensor loop integrals, which are Passarino-Veltman-reduced to scalar integrals11. The required scalar integrals can be found in1,10. Phase space integrations for the real 2 -» 3 gluon bremsstrahlung corrections require extensive partial fractioning to transform them to a standard form1,9,10. A sufficient set of four- and n-dimensional integrals can again be found in1 '10. The final color-averaged results for gluon-gluon fusion can be decomposed according to color structure as
\Mgg |2 ~ [(2CF)2MqED + C2AM0q + MKQ + 2CFMRF + CAMQL\ , (5)
with Casimir operators CF = (N^ — l)/2Nc and CA = Nc, Nc denoting the number of colors. The choice in Eq. (5) ensures that the "Abelian" MQED is identical to the QED part9 of 75 -» QQ with the usual factor l/(2Nc) for replacing a photon by a gluon. Similar color decompositions are made for the other subprocesses.
3. Numerical and Phenomenological Results
The total partonic subprocess cross sections &ij, i,j — q,q,g can be expressed
in terms of LO and NLO functions /,-•' and f>-', /> •', respectively, which
215
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
Alff+3.7<
f (0) + 3 7 . f (l)
f(0)
Tt| |
le-05 0.0001 0.001 0.01 0.1 1 r|=s/(4 m2) -1
100 1000 10000
Figure 2. (m2 j'a\)agg in NLO (MS) and LO as a function of rj according to Eq. (6), with fip = HR = "i for simplicity and Airas = 3.7 as appropriate for charm production.
depend only on a single scaling variable n = s/(4m2) — 1:
m W + *™* ?(i) W+f. I)
)ln4 + A^»ln41}! (6) 3 m2 87r2 3 HF\\
where s is the available partonic center of mass (c.m.s.) energy squared, m is the heavy quark mass, as = as(pfj), and /30 = (HCU — 2nj/)/3. The last term in Eq. (6) vanishes for the standard choice (ip = MR- In Fig. 2 we show {tn21a2
s)agg in LO and NLO for fiF — fj,R — m as a, function of r] in the MS scheme. The NLO corrections are significant in the entire n range. In LO
0 at threshold, but in NLO & ^ [(2CF)2 - C\ + §] ^ 'gg ~r " " " " " i ^ " " * " ) " " " - " ^ ~ ~ ~ , 8 m 2 2 ( A T £ - I )
due to a "Coulomb singularity". Logarithms from soft gluon emissions also contribute significantly even at the lowest 7] shown. At high energies gluon exchanges in the t-channel drive the unpolarized NLO result to a plateau value1, whereas the polarized NLO cross section vanishes. The large deviations between NLO and LO arise from Feynman diagram topologies that occur for the first time at NLO.
Figure 3 shows the decomposition of the polarized NLO (MS) coefficient function A / ^ into contributions with different color structure as defined in
216
0.04
0.02
-0.02
-0.04
A f ^ Q E D )
Af^COQ)
Af^CKQ)
50-Afg^CQL)
le-06 le-05 0.0001 0.001 0.01 0.1 1 10 T|=s/(4 nT) - 1
0)
100 1000
Figure 3. NLO (MS) coefficient function Afgg color decomposed according to Eq. (5). The
quark loop (QL) contribution is enhanced by a factor 50. Also included are A / J g and fgg'.
Eq. (5). Notable are the large cancellations between the QED and the first non-Abelian (OQ) contributions in the threshold region t] —> 0. For completeness we also present here the coefficient functions Afgg' and fgg in Eq. (6), which arise from the mass factorization procedure. The RF contribution just moved into the last term in Eq. (6) and is not shown. The physical total cross section is obtained by convoluting the partonic cross sections in Eq. (6) with the parton densities
a(S,m2) = 5Z / dxi dx2fi(xi,iiF)fj{x2,VF)vij{s,m2) (7)
where 5 is the hadronic cm. energy squared, s = X1X2S, and xmin = Am2 IS. The differential heavy (anti-)quark inclusive distributions like d2a/dpT(lpL are obtained similarly by convoluting with our double differential partonic cross sections. In Fig. 4 we plot xix2Ag(xi,fij)Ag(x2,Hf)Aagg(xiX2S)/(a2/m2) with fij = \iT — m, 4iras = 3.7, and GRSV'OO val. helicity densities12. This shows the gluon-gluon fusion contribution in Eq. (7), with £1X2 included to display the right volume with logarithmic axes. We see that the z-region xia;2 — Zmin = 4m2/S is dominant. Furthermore, a peak on that line is situ-
217
L o g x i
^ 0 . 7 5 . 0.25
- 0 . 2 5
Logx2
0 o 0
0
0.
0
.008
.006
004
002
Figure 4. The gluon-gluon fusion contribution xiX2&g(xi,iif)Ag(x2,Hf)&<7gg(xiZ2S)/ (a2Jm2) for Eq. (7) with nf = tir = m, 4wa, = 3.7, and GRSVOO val. helicity densities 1 2 .
ated at xx ~ x%. Thus at VS ~ 10 GeV the JHF will probe x ~ ^/xmin ~ 0.3. The situation is similar for the unpolarized case. Note that in photoproduc-tion, the region x ~ xmin is dominant. Thus the COMPASS13 measurement will probe smaller x ~ 0.08 even if using the same cm. energy as the JHF.
Experiments will usually study the longitudinal spin asymmetry defined by
A(S,m2) = Aa(S, ro2) <r(S, m2)
(8)
The advantage is that one does not need to determine the absolute normalization of the cross section a(S, m2). In Fig. 5 we demonstrate that the NLO results for the total charm spin asymmetry are indeed more robust under scale variations than LO estimates. We vary HF and mc independently of each other. The range for / i | = / r o 2 is given by 1 < / < 4.5, with the conventional choice pit = HF. The range for the charm mass mc = cGeV is 1.2 < c < 1.6. The JHF beam energy of Eb = 50 GeV and the GRSVOO val. helicity densities12
have been used. We show A(f,c)/A(f = 2.5,c = 1.4) - 1, i.e. the deviation in percent of the asymmetry according to Eq. (8) with respect to a reference asymmetry taken at fixed nF = fi2
R = 2.5m2 with rac = 1.4 GeV. To better guide the eye, contour lines in steps of 30% are plotted at the base of each plot. Here we also indicate mc = 1.4 GeV by a thin solid line. The NLO result in the right part of Fig. 5 is considerably "flatter" than the LO result in the
218
Ajp(f,cyAjp(2.5,1.4)-l
Eb=50 GeV
f=|if/nig, mc=c GeV, Mr=Mf
Figure 5. Deviation [in %] of the total charm spin asymmetry in LO (left) and NLO (right) from a reference choice ("0-pin" marker, see text) as a function of fip and mc with (J.R = fip. The LO result is multiplied by a factor (-1). Corresponding contour lines in steps of 30% are given at the base of each plot.
left part with respect to the variations. Variations of the charm mass cause the major uncertainty of about ±30% in the NLO predictions. In LO we find unacceptably large theoretical uncertainties.
We predict the charm production asymmetry at the JHF in Fig. 6 using a range of recent12'14,15 and older16'17 helicity densities. These sets mainly differ in their assumptions about Ag. For calculating the required unpolarized a, we have used the associated unpolarized parton distributions specified in the references. For consistency, mc is also taken as in these fits. All results are obtained for the choice pip — p?R = 2.5(m2 +Py) . Figure 6 shows that open charm production at the JHF can be very useful in pinning down Ag. The estimated statistical error of such a measurement, 8A = p i ,—, 1± with luminosity C = 7.66 fb_ 1 in 120 days, charm detection efficiency ec = 10 - 3 , polarizations for beam and target P^Pt = 1/2, and dilution factor fu = 0.15, is significantly smaller than the total spread of the predictions. Gluons of small size, e.g. the oscillating A<7 of GS C (see Ref. [17]), cannot be detected.
219
0.8 pTM
Figure 6. The NLO charm asymmetry A for a JHF beam energy Eb — 50 GeV as a function of p™'n with px > p™ln. Also shown is an estimate for the statistical error (see text) .
In Fig. 7 we show the dependence of the asymmetry prediction on the beam energy for the GRSV'OO val. helicity densities, including the theoretical error from HF and [MR variations. At small p™in the sensitivity to the differences in A<7 is largest, see Fig. 6, but here also any uncertainties in the beam energy can have large effects. Hence it is crucial that the beam energy will be known with good precision.
The theoretical uncertainties of polarized Aa and unpolarized cross sections a are much larger than those of the asymmetries A, since they cancel to a large extent in the ratio. Figure 8 shows the huge uncertainty in <r upon varying mc, fir, and fij. However, if the JHF can measure a directly, then one can try to pin down the appropriate scale choice experimentally. For small p™"1
the uncertainty in er stemming from uncertainties in the parton distribution functions is currently about ±40%, as estimated using the complete uncertainty information of the CTEQ6 set19. However, the statistical error of the JHF measurement will be negligible after only one day of data taking, due to very high luminosity and the large cross section. Hence we can envision a data point with an error of about ±40% in the low p^m region of Fig. 8. Obviously this will severely restrict the possible scale choices.
220
&S
10
5
0 E i-
Al - 5 i -
< -10
-15
-20
~ I T" T I I I [ I 1 1 T"
r * r
2<u,2<4m
m 2 % ^ 4 m 2
c * ? c
GRSV'OO va
E =40 GeV b
E =50 GeV D
E =60 GeV b
- j i i i_
0.0 0.5 1.0 1.5 p™n [GeV]
2.0
Figure 7. The NLO charm asymmetry A in percent at JHF beam energies Eb = 40,50,60 GeV with the GRSV'OO val. densities 12 as a function of p^in. The bands show the uncertainty due to varying mc < (fj-r,^f) < 2m c .
4. Summary
Determining the gluon helicity density Ag is one of the most pressing problems in spin physics today. The JHF could contribute significantly to pinning down Ag in the large x region using polarized production of open charm. The NLO QCD corrections20 presented here will make this feasible by reducing the theoretical uncertainties to a viable level. However, it would be highly advantageous if the JHF limited those uncertainties further by first determining the unpolarized cross section. Since polarized beam and target are planned for a later stage of the JHF, this is in accordance with the current experimental plans at the JHF anyway.
Acknowledgments
I thank M. Stratmann for fruitful collaboration. This work has been supported by the Australian Research Council.
221
1000
,—. 800 _o c i i
T~ 600 E t-Q.
.cT 400 b
200
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
p™n [GeV]
Figure 8. The unpolarized NLO charm cross section a at the JHF for different mc = 1.3,1.4,1.5 GeV with the GRV'98 densities 18. The bands show the uncertainty due to varying mc < (^ r ,/i/) < 2mc.
References
1. W. Beenakker et al., Phys. Rev. D40, 54 (1989). 2. W. Beenakker et al, Nucl. Phys. B351, 507 (1991). 3. S. Dawson, R.K. Ellis, and P. Nason, Nucl. Phys. B327, 49 (1988). 4. S. Dawson, R.K. Ellis, and P. Nason, Nucl. Phys. B303, 607 (1988). 5. A.P. Contogouris, S. Papadopoulos, and B. Kamal, Phys. Lett. B246, 523 (1990). 6. M. Karliner and R.W. Robinett, Phys. Lett. B324, 209 (1994). 7. J. Bliimlein and H. Bottcher, arXiv:hep-ph/0203155. 8. G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972);
P. Breitenlohner and D. Maison, Comm. Math. Phys. 52, 11 (1977). 9. I. Bojak and M. Stratmann, Phys. Lett. B433, 411 (1998); Nucl. Phys. B540,
345 (1999); Erratum, ibid. B569, 694 (2000). 10. I. Bojak, Ph.D. thesis, Univ. of Dortmund, April 2000, arXiv:hep-ph/0005120. 11. G. Passarino and M. Veltman, Nucl. Phys. B160, 151 (1979);
W. Beenakker, Ph.D. thesis, Univ. of Leiden, 1989. 12. M. Gliick et al., Phys. Rev. D63, 094005 (2001). 13. COMPASS Collaboration, G. Baum et al., report CERN/SPSLC 96-14,30. 14. AA Collaboration, Y. Goto et al., Phys. Rev. D62, 034017 (2000). 15. D. de Florian and R. Sassot, Phys. Rev. D62, 094025 (2000). 16. M. Gliick et al., Phys. Rev. D53, 4775 (1996). 17. T. Gehrmann and W.J. Stirling, Phys. Rev. D53, 6100 (1996). 18. M. Gliick, E. Reya and A. Vogt, Eur. Phys. J. C5 , 461 (1998). 19. J. Pumplin et al., arXiv:hep-ph/0201195. 20. I. Bojak and M. Stratmann, arXiv:hep-ph/0112276 and in preparation.
r l _ J ^ I 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l T 1 I l - I - p - l I I I '
\ m2^u2^4m2
\ c < r c m 2 % 2 ^4m 2 '-
- \ C ™ f C -\ '• \ GRV'98 :
hr-^ \ m c = 1 - 3 GeV ";
COVARIANT LIGHT-FRONT DYNAMICS A N D ITS APPLICATION TO THE MESON WAVE FUNCTIONS
O. M. A. L E I T N E R 1 ' 2 A N D A.W. T H O M A S 1
Department of Physics and Mathematical Physics, and Special Research Centre for the Subatomic Structure of Matter,
University of Adelaide, Adelaide 5005, Australia E-mail: oleitnerQphysics.adelaide.edu.au ; [email protected]
J . -F . M A T H I O T 2
2 Laboratoire de Physique Corpusculaire, Universite Blaise Pascal, CNRS/IN2PS, 24 avenue des Landais, 63177 Aubiere Cedex, France
E-mail: [email protected]
In the framework of covariant formulation of light-front dynamics, we determine phenomenologically various meson wave functions. We focus on the pseudo-scalar particles B, D, w and K, as well as on the vector mesons J/ip, p and w. All these relativistic wave functions are defined on the light-front plane w-x = 0, in four dimensional space. Their relativistic structure is fully determined. The parameters are well constrained by several experimental data sets: decay constants, transition form factors, electromagnetic form factors and charge radii.
1. Covariant Light-Front Dynamics
In CLFD1, the state vector which describes the physical bound state is defined on the light-front plane given by the equation ui-x = 0, where u> is an unspecified light-like four vector (w2 = 0). It defines the position of the light-front plane. This approach is a generalization of the standard Light-Front Dynamics (LFD). The latter can easily be recovered with a special choice of the light-front orientation, u> = (1 ,0 ,0 , -1) . The description of relativistic systems in LFD provides several advantages compared with the standard formulation. The main properties are the following1:
• The formalism does not involve vacuum fluctuation contributions. Therefore, the state vector describing the physical bound state contains a definite number of particles, described by Fock state components.
• The Fock components of the state vector satisfy a three dimensional equation.
222
223
• The relativistic wave function and off-shell amplitude depend on the orientation of the light-front plane which is fully parametrized in terms of the four vector u>.
The state vector describing a meson of momentum p, denned on a light-front plane characterized by w, is given by (retaining the two-body component only):
M L = <f>J\v) = (^)3/2j */xA
fflia<Ta(*i,*2,P,«r)ati(k1)4a(k2)|0)
d3k\ d3k-> x f W f r + fc2 - p - UT) e x p f t r ^ f r • p)dr ( 2 ) r ) , / 8 ^ ^ ( 2 i r ) , / a ^ • W
where £/. = y/\s? + m2. We emphasize that the bound state wave function (see Fig. 1) is always an off-energy-shell object (r ^ 0 because of its binding energy). The parameter r is entirely determined by the on-mass-shell condition for the individual constituents.
>
q,<b (p)
Figure 1. Two-body relativistic wave function.
The two-body wave function $(/:i, k^,p,UT) can be parametrized in terms of several sets of variables. In order to make a close connection to the non-relativistic case, it will be convenient to introduce the pair of variables1 (k, n). The relativistic momentum k corresponds, in the frame where ki + k2 = 0, to the usual relative momentum between the two particles. The unit vector, n, corresponds in this frame to the spatial direction of w. The second set of variables which we shall use in the following is the usual light front coordinates ( I , R J _ ) . All details can be found in Ref. [1].
The two-body component $ is the solution of a three dimensional equation, Eq. (2). For simplicity, let us consider the case of a two-body scalar
224
system composed of two scalar particles. The wave function is scalar, and can be parametrized in terms of $(x, R-jJ- The homogeneous equation, which <3> should satisfy, is given by :
R]_ + n x(l — x)
M2)^(X,H2L) =
^ f » ( « ' , B f l A C ( « ' , R ' ± ; x , R x , M 2 ) £ * ^ , (2) (2
where m is the constituent quark mass (for two identical particles) and M is the bound state mass. In this form, this equation is similar to the Weinberg equation. The kernel, K, depends on the dynamics of the system. In our case, we shall use a kernel with one-gluon exchange.
2. Pseudo-Scalar Mesons
The explicit covariance of our approach allows us to write down the general structure of the two-body bound state. For a pseudo-scalar particle composed of an antiquark and a quark of mass mi and m,2, respectively, it has the form :
*PS = ^ " ( * 2 )
1 ih 75"(*i) , (3)
where u(k^) and v(k\) are the usual Dirac spinors, and A\, Ai are the two scalar components of the wave function. The mass mr is defined by mr = 2 m ' m a and
^ ' J ' mi+m2
is chosen here just for convenience. A\ and Ai will be expressed in terms of ^i and 52 which are Gaussian wave functions written as gt = 4.7T2a,-/J,- exp(—/3,k2), with oti and /?,- two parameters to be determined from experimental data.
2 .1 . Normalization
In the spirit of the constituent quark model, the state vector is decomposed in Fock components, and only the two-body component is retained. Thus one gets, for a state of zero total angular momentum, the following normalization condition1 :
N* = l = l E ^ A i A ^ A ^ , (4) A1A2
where D is an invariant phase space element given by,
1 d3ki _ 1 d3k _ 1 d2Rxda;
~ (2?r)3 (1 - x)2ekl ~ (2?r)3 ek ~ (2ir)3 2x(l - x) ' ( '
225
With the pseudo-scalar wave function written in Eq. (3), the expression for the normalization is therefore :
1 • / .
R j + (m2x + mi (1 - z))2
m2x(l — x) '
+4AlA2 | " i ( ! - « ) + " * » _ D.(Q)
2.2. Decay Constant
According to the usual definition, the semi-leptonic decay amplitude is TM = (0\Jfi\PS) where JM is the 7^75 current. Since our formulation is explicitly covariant, we can decompose T^ in terms of all momenta available in our system, i.e. the incoming meson momentum p^ and u^. We have therefore :
T^ = fps Pp + Bun , (7)
where fps is the physical decay constant. Using the diagrammatical rules of CLFD, we can calculate T^, and then the decay constant is given by :
/PS = 2N/6/ n<'-"-'> + "1">U + :te(i-,M, -W.K! D. (8)
2.3. Electromagnetic Form Factor
The electromagnetic form factor is one of the most useful tools to understand the internal structure of a bound state. Moreover, from the electromagnetic form factor at low Q2, it is possible to determine its charge radius. This physical observable is therefore very powerful in order to constrain the phenomeno-logical structure of the wave function. By definition, the general physical electromagnetic amplitude of a spinless system1 is :
JP = (P + P')P FPS (Q2) + ^B(Q2) , (9)
where Fps(Q2) is the physical form factor. In any exact calculation, B(Q2) should be zero, while it is non-zero for any approximate calculation. It should therefore be removed since it is a non-physical contribution. We choose for convenience u> • q = 0. This implies automatically that the form factors Fps (Q2) and B(Q2) depend on Q2 = — q2 only since, for homogeneity arguments, they can only depend on u> • p/ui • p' = 1. The physical electromagnetic form factor, Fps{Q2), can be obtained by contracting both sides in Eq. (9) with u>p. One then has :
F?^=Bp- ^
226
After applying the diagrammatical rules of CLFD, one gets for the electromagnetic form factor :
2 f r (TOl(l-,) + m 2 ^ + R i - , R x . A
JD L A1 ~ x)mr
+ 2(A1A'2 + A[A2)imii]-~x) + m2X) + 4z(l - x)A2A'2] D . (11)
mr J
From this form factor, we can extract two major pieces of information : the first one is the charge radius of the bound state is defined by :
(rls) = -^FPS{Q2) _ „ , (12) dQ2
Q2=0
and the second one is the behaviour of the electromagnetic form factor at high Q2 (asymptotic form). It is now well accepted that the asymptotic behaviour of the pion form factor, Fn(Q
2) ~ l/Q2, is fully determined by the one-gluon exchange mechanism. This mechanism can either be considered explicitly in the hard scattering amplitude, or incorporated in the relativistic wave function of the meson. Here we adopt the second strategy. At asymptotically large Q2, the form factor is dominated by the contribution from the relativistic component A% in Eq. (3). The high momentum tail of the wave function is generated by the one-gluon exchange kernel.
2.4. Pion Transition Form Factor
The quantum numbers of the TT transition amplitude, -K -> 7*7, are similar to the deuteron electro-disintegration amplitude near threshold, as detailed in Ref. [1]. Thus, the exact physical amplitude has the form :
F„ = ^epllvXq»PxF^ , (13)
where P — p+p' and q — p'—p. In any approximate calculation, the amplitude Fpp has to depend on u. It should thus be decomposed in terms of all possible tensor structures compatible with the quantum numbers of the transition, as we did above for the semi-leptonic decay constant and electromagnetic form factor. Therefore, we can extract the physical form factor F*"7 by the following contraction :
e^p^xq^xFup . (14) 2Q2(Wp)
At leading order, the transition form factor is given by two relevant diagrams (other diagrams which should be taken into account in leading order either
227
correspond to vacuum diagrams or are equal to zero for u • q = 0) and the total amplitude reads :
AD
R , .A 1 , ( 1 5 )
s ) - 2 R j . - A + xQ2
Rj^A
~Q2 x\Ax+ 2A2x(l -x) + A2 "- (1 - x)
where eq is the quark electric charge. Two physical constraints can be extracted from the transition form factor. The first is the axial anomaly which is defined at Q2 = 0 GeV . It gives the transition form factor when both photons are on their mass-shell. One should have :
FW1(Q2 = 0) = -\- . (16)
The second one is the asymptotic behaviour : at high Q2, the transition form factor behaves like l/Q2.
3. Vector Mesons
3.1. Formalism
We shall now concentrate on the structure of vector mesons, Jp = 1~. The wave function is decomposed in terms of all possible, independent spin structures. One gets1 :
d ( * i - * J . P . W T ) = V ^ e £ ( p ) ^{k2) ^ v^ih) , (17)
with (f)^ = Q(<pi-i 6). e£ is the polarization vector of the vector meson, and m is the mass of the quark and antiquark. We consider here only the case of equal masses for the quark and antiquark. The wave function is determined by six invariant functions y>i_6, which depend on two invariant scalar variables. In terms of the variables k and n, the wave function takes the form1 :
^20i(k,n) = V^42V>A(k,n)w0l , (18)
with V(k,n) = T{fi-\fi). The relation between ipx and if) is the same as the relation between the spherical function Y]*(n) and n.
The coefficients of the spin structures in ^(k, n) are scalar functions of two independent invariants, which we can choose to be k2 and k-n , since these variables are only rotated by a Lorentz boost1. In the non-relativistic limit, only two components remain, f\ and f2, and they only depend on k2. We shall neglect in first approximation the tensor component, f2, so one is left with the
228
non-relativistic wave function / i = ^ ^ ^ ( k 2 ) . The wave function ^ ^ ( k 2 ) should be normalized2 and one gets,
N2=l = m f \<j>NR(k2)\2D . (19) JD
3.2. Decay Constant
The leptonic vector decay diagram is similar to the semi-leptonic one. The decay amplitude M^ to produce a photon with polarisation e^ from a vector state of polarisation ep can be decomposed in terms of all possible tensor structures2. Therefore we can write M.>ip as :
M"> = Fa\> + ^ - < + P-<P + T^C< + Dal" , (20)
where the tensors2 a f describe the kinematics. By simple algebraic manipulation2, we can isolate the physical amplitude F as a function of Mfip, and one obtains :
F=^{I1- 2/2 + h + h) , where U = M^af . (21)
On the other hand, we can calculate M^p from the CLFD diagrammatical rules and thus extract the physical amplitude, F, which can be written as :
' -2V6m 3 / 2
i - "̂ I L 1 - 2 ( ^ + 5*) Vet m)
^ " ( k 2 ) D . (22)
In this approach, the non-relativistic wave function ^ ^ ( k 2 ) is parametrized as a Gaussian which includes a and 8, the two parameters determined from experimental data sets. After averaging over the polarization states of the vector meson, one gets for the decay width :
r ( ^ e + e - ) = ^ a 2 £ 2 | F | \ (23)
where Eq is a factor arising from the electric charges of the constituent quarks, M is the mass of the vector meson and ae is the electromagnetic fine-structure constant.
3.3. Radiative Corrections to the Wave Function
Since we determine the parameters of the wave function from the leptonic decay width only, it is more appropriate here to correct the decay amplitude directly for radiative corrections. In the non-relativistic limit, this gives rise to the well known correction :
^NR rr (i - ^ ) , (24)
229
which implies a huge correction, even for the J/^ (50% correction). However it has been shown in Ref. [3] that the relativistic calculation of these radiative corrections leads to a result that is better founded physically. The kinematical relativistic corrections induced by the finite relative momentum between the quark-antiquark pair, as detailed above, gives rise to a first reduction. The dynamical correction induced by the one-gluon exchange contribution to the amplitude leads to a reduction of the correction 1 | " s by 70%. The latter reduction is indeed rather important since the radiative correction is known to be of the order of 15% for the J/ty. This gives us somewhat more confidence in the expansion in as.
4. Results
We used all available physical constraints to determine the meson wave functions. In regard to pseudo-scalar particles, we studied heavy (B and D) and light (TT and K) particles. For vector particles, we also studied both heavy {J/ip) and light (p and w) particles . The parameters included for the phe-nomenological description of these mesons have been extracted from physical constraints such as decay constants, electromagnetic form factors, charge radii, transition form factors and the normalization.
For B and D particles, two constraints were used : the decay constant and the normalization condition. Their distribution amplitude is shown Fig. 2 and emphasizes their internal structure (heavy quark b or d). As output, the average transverse momentum is A / ( R ^ ) — 0.556 GeV for B and >/(S^y = 0.494 GeV for D.
For the pion, the electromagnetic form factor at low Q2 is in total agreement with all experimental data sets. From this, we found the pion charge radius squared equal to 0.417 fm2, which is within experimental uncertainties. At high Q2, the electromagnetic form factor is in good agreement with the latest experimental data sets, because of the inclusion of the one-gluon exchange. For the transition form factor, we found an asymptotic limit in agreement with experimental data.
For the kaon, the lack of data does not allow us to extract accurate results, but we obtained a nice agreement with a previous study4. At low Q2, the electromagnetic form factor is compatible with different studies and with experimental data sets even though they are not sufficiently numerous and precise. Moreover, we found a good approximation for the kaon charge radius squared since one gets 0.388 fm . This result is still in agreement with the experimental value. At high Q2, we found the same behaviour for this one as for the pion.
230
l
0.8
0.6 *(x)
0.4
0.2
"0 0.2 0.4 0.6 0.8 1
x
Figure 2. Pseudo-scalar distribution amplitude. Full line, dotted line, dashed line and dot-dashed line represent the results for B, D, K and TT, respectively.
The distribution amplitudes for the pion and kaon are also shown in Fig. 2. As numerical outputs, the value of the transverse momentum is V^Rj.} = 0.320 GeV for the pion and y/(R?±) = 0.340 GeV for the kaon.
Considering vector particles, we studied three particles : J/tp,p and w. Since our approach has been phenomenological, we only concentrated on the first two components of the wave function. That is the reason why we just used the normalization and decay constant. We found results in total agreement with the quark model. The distribution amplitudes for the vector mesons are shown in Fig. 3. As usually observed, the distribution amplitudes for p and w are very close because of the similar internal quark structure.
In our approach, the radiative corrections have been taken into account only for J/ip and explain the main features of the distribution amplitude. The p and u> structure (light quarks) does not require the same treatment, but they should be analysed in the same way in the future. As output, the average transverse momentum has been calculated, and one gets \ / ( R l ) = 0.517,0.459,0.434 GeV, for the J/ip, p and u, respectively.
This study has shown that CLFD is a very powerful framework which takes into account the dynamical spin structure of the wave functions of pseudo-scalar and vector particles. We stress that both wave functions and transition form factors are described in a relativistic formalism. Through this work, we found similar results, in total consistency with soft QCD4 (for n and K). Moreover, the results obtained also reproduce the heavy quark effective theory5
(for B and D). All particles can be derived by following this formalism and by using experimental data sets as constraints.
— B
.... D — • K —- Jt
\--' I • • . - • • • I
231
*(x)
1
0.8
0.6
0.4
0.2
n
----~
-
.
1 1 '
— JA|/ CO
._ . p
SL'
//
// '•' I
/ /
y i
i /
/ L *:"•'•"••
i
\ i
\ -•.T-.-.O
1
' 1 '
-i
\ A
\ ')
, \ \ " 0.2 0.4 0.6 0.8
Figure 3. Vector meson distribution amplitude. Full line, dotted line and dot-dashed line represent the results for J/ip, w and p, respectively.
A c k n o w l e d g e m e n t s
This work was supported in part by the Australian Research Council and the
University of Adelaide.
R e f e r e n c e s
1. J. Carbonell, B. Desplanques, V.A. Karmanov and J.-F. Mathiot, Phys. Rep. 300, 215 (1998).
2. S. Louise, J.-J. Dugne and J.-F. Mathiot, Phys. Rev. Lett. 472, 357 (2000). 3. F. Bissey, J.-J. Dugne and J.-F. Mathiot, hep-ph/0112079. 4. P. Maris and P.C. Tandy, Phys. Rev. C62, 055204 (2000). 5. H.-Y. Cheng, C.-Y. Cheung, C.-W. Hwang and W.-M. Zhang, Phys. Rev. D57,
5598 (1998).
NUCLEON STRUCTURE FUNCTIONS AT FINITE DENSITY IN THE NJL MODEL
H. M I N E O
Department of Physics, National Taiwan University 1 Roosevelt, Section 4t Taipei, Taiwan
E-mail: [email protected]
W . B E N T Z
Department of Physics, Tokai University Hiratsuka-shi, Kanagawa 259-1207, Japan
E-mail: [email protected]
A.W. T H O M A S
Special Research Centre for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, Adelaide University
Adelaide, SA 5005, Australia E-mail: athomas©physics.adelaide.edu.au
N. ISHII
The Institute of Physical and Chemical Research (RIKEN) 2-1, Hirosawa, Wako-shi, Saitama 351-0198, Japan
E-mail: [email protected]
K. YAZAKI
Department of Physics, Tokyo Woman's Christian University Suginami-ku, Tokyo 167-8585, Japan E-mail: [email protected]
In this work we discuss the EMC effect of the nucleon structure functions in nuclear matter , using a simple approximation to the relativistic Faddeev description of the nucleon in the framework of the Nambu-Jona-Lasinio model. We adopt a stable nuclear matter equation of state, calculated in the Nambu-Jona-Lasinio model, which incorporates confinement effects phenomenologically so as to avoid unphysical thresholds for the decay into quarks. We will compare our results for the BMC ratio in nuclear matter within the Nambu-Jona-Lasinio model with the parametrized fits to the experimental data.
232
233
1. Introduction
The investigation of the nucleon structure functions is a very active field of current research1,2,3. On the theoretical side, it is not yet possible to directly use QCD for their description, although it is now possible to calculate the first few moments in lattice QCD and discuss the extrapolation to the chiral limit4. For a direct description of structure functions, however, effective quark theories, like bag models5, soliton models6 and Faddeev descriptions7, play important roles.
In this work we calculate the quark light-cone (LC) momentum distributions in the nucleon7 and discuss the medium effects on the nucleon structure functions, that is, the EMC effect in nuclear matter8. Our approach is based on the relativistic Faddeev equation9 in the NJL model10, assuming a simple "static approximation"11 for the Faddeev kernel. In our approach, the nucleon is described as a bound state of a quark and a scalar diquark (Jp = 0 + , T — 0). Axial vector diquarks (Jp = 1+ ,T = 1) have been investigated in Ref. [12], and it was shown that their effects should be subordinate to the scalar diquark ones. Pion cloud effects can be taken into account perturbatively, as has been done in Ref. [7,12] for the free nucleon structure functions. In this work, however, axial vector diquark and pion cloud effects will not yet be included.
In a recent work13, a stable nuclear matter equation of state has been obtained in the NJL model by combining the quark-scalar diquark description of the nucleon with the mean field (Hartree) description of the nuclear matter ground state, using a 'hybrid model'. For the stability of nuclear matter it was essential to incorporate confinement effects phenomenologically so as to avoid unphysical decay thresholds. This has been done by introducing an infrared (IR) cut-off, in addition to the usual ultraviolet (UV) one, in the proper time regularization scheme14. We will present our results for the EMC ratio in nuclear matter based on this equation of state.
2. Calculations
2.1. Model
The Lagrangian of the flavour SU(2) NJL model has the form C = V> (i$ — m) ifr + Ci, where m is the current quark mass and Li is a chiral invariant 4-Fermi interaction. By using Fierz transformations, any given £ / can be decomposed into various qq and qq channel interactions. The ratio r, = G3/Gn
of the effective coupling constant in the scalar diquark channel (Gs) to the one in the pionic qq channel (Gn) depends on the assumed form of £ / , but in this work we will use rs as a parameter reflecting different forms of the interaction Lagrangian. We fix the constituent quark mass in the vacuum, which is related
234
to the current quark mass via the gap equation, as M = 400 MeV. The UV cut-off A is fixed by the pion decay constant (/* = 93 MeV), and Gn is fixed by reproducing the pion mass at zero density (mn = 140 MeV) as the pole of the qq t-matrix in the ladder approximation to the Bethe-Salpeter equation. Then the ratio rs is determined so as to reproduce the nucleon mass at zero density {MM = 940 MeV) as the pole of the quark-scalar diquark t-matrix in the ladder approximation to the relativistic Faddeev equation7. This automatically determines also the value of the scalar diquark mass Ms as the pole of the t-matrix in the scalar qq channel. In this work we neglect the momentum dependence of the propagator of the exchanged quark in the Faddeev kernel, that is, we work in the "static approximation". In this approximation the Faddeev equation can be solved almost analytically.
Based on this quark-scalar diquark picture of the single nucleon, a nuclear matter equation of state has been constructed in Ref. [13] using the mean field approximation for the nuclear matter ground state. The essential difference to chiral point-like nucleon models, like the linear <r-model, is that the nucleon mass in the self consistent scalar field is calculated from the underlying quark-diquark equation, that is, the scalar field couples to the quarks and not to the nucleon as a whole. For the stability of nuclear matter it is necessary to avoid the unphysical threshold for the decay into quarks, and this can be achieved by introducing an IR cutoff p, in addition to the UV cutoff A in the proper time cutoff scheme. We will fix the IR cutoff at p — 200 MeV, since, as long as p > 100 MeV, the results are very similar13.
The resulting equation of state gives the density dependence of the effective masses in the medium, M*, M* and MN, which are less than those at zero density, while the pion mass in the medium, m*, is somewhat larger than mn, since in this mean field approach only the (slightly) repulsive S-wave interaction between the pion and the nucleon is taken into account. The values of these masses are listed in Table 1 for the case of zero density, p = 0.16 f m - 3 (normal nuclear matter density) and p = 0.21 fm - 3 (saturation point of the nuclear matter equation of state in this model a ) .
a Since in this model there is only one additional parameter (the strength of the 4-Fermi vector interaction Gu) as we go from the zero density to the finite density case, one cannot fit the empirical saturation point. In Ref. [13] the value of Gw has been adjusted so that the binding energy curve passes through the empirical saturation point.
235
Table 1. The effective masses in the medium with density p
p, m i " 3
Af*, MeV M*, MeV M*N, MeV ml, MeV
0 400 576 940 140
0.16 308 413 707 159
0.21 282 368 649 167
2.2. Quark LC Momentum Distributions
The quark and antiquark light-cone (LC) momentum distributions in the proton can be obtained from the function15
? » = \ j ^ • > - « - ( p | T ( ? , ( 0 ) 7+ l M * - ) ) \P), (1)
with q(x) — q(x) and q(x) = — q~(—x) for 0 < x < 1. Here x is the fraction of the proton's LC momentum component p_ = (po - P3)/V% carried by the quark (or antiquark) with flavour q — u,d, and \p) denotes the proton state with momentum p. The valence and sea quark distributions are then given by qv(x) — q(x) — q(x) and qs(x) = q(x). As explained in15, the evaluation of the distribution in Eq. (1) can be reduced to a straightforward Feynman diagram calculation by noting that it is nothing but the Fourier transform of the quark two-point function in the proton traced with 7 + , where the component k- of the quark LC momentum is fixed as k- — xp_ and the other quark momentum components (fc+,kj_) are integrated out.
Since the proper time regularization scheme is defined in Euclidean space, it is very difficult to calculate the quark LC distributions directly from Eq. (1). Instead of direct calculation, we first calculate the moments and then construct the distribution functions in terms of its moments15, which can be done analytically within our model. The resulting expression becomes
q(x) = ^[Tq(x - ie) -Tq(x + ie)] (2)
oo
r,(*)=x;j?/*n.
where J£ is the nth moment of q(x) defined as J£ = £1dxxn-1q(x). Since the moments are given by the expectation values of local quark operators, they can be calculated also in Euclidean regularization schemes.
The equation of state of Ref. [13] is also used to calculate the nucleon LC momentum distribution in nuclear matter. This function, /N/A^A), where yA
is A times the fraction of the LC momentum of the whole system carried by
236
the nucleon (0 < yA < A), was discussed in detail in Ref. [16]. The quark LC momentum distributions in the medium are then obtained by applying the standard convolution formalism3:
%/A(XA) = I dyA J dzS(xA - yAz)q*(z)f^/A(yA), (3)
where the asterisk indicates that the in-medium masses are used, and the variables are defined as
O2 k+
xA=A - 2 — =A-T (0 < xA < A) %PA • g p\
yA=A-?^- = A^r (0<yA<A) PA-q P\ ~
kq k+
z = — - = —- ( 0 < z < 1). pq P+ ~ ~
Here PA , p and k are the 4-momenta of the nucleus, the nucleon and the quark inside the nucleon, respectively, and Q is the 4-momentum transfer to the nucleon. The usual Bjorken variable x^ is related to XA by
MA ~MN ... -rj MA XN = xA-rj-j— = aiATT- with MN = ——.
A M/v MN A By using the Q2 evolution up to next-to-leading order17, the results for
the quark LC momentum distributions are evolved from the low energy scale Q2 = 0.16 GeV2 to the scale Q2 where parametrized fits to the experimental data18 and empirical parametrizations2 are available. The value of Q\ has been determined so as to reproduce the qualitative features of the valence quark distributions in the free nucleon.
3. Results
We first show the results for the valence quark distributions in the proton at zero density in Figs. 1 and 2. The results for uv(x) and dv(x) after the Q2 evolution (dashed lines) are compared to the empirical distributions2 at Q2 = 4.0 GeV2 (dotted lines). The overall properties of the valence quark distributions, in particular the fact that dv(x) is softer than uv(x), are qualitatively reproduced. Figure 1 shows that uv(x) at the low energy scale QQ — 0.16 GeV2 has a non-zero support at x — 1, although the support vanishes for x > 1. Since in general the distributions near x = 1 are very sensitive to the regularization procedure, the discontinuity at x = 1 must be considered as an artifact of the proper time cut-off scheme. (It does not appear, for example, in the regularization scheme used in Ref. [7].) This behaviour will affect the result for the EMC ratio at large x, as will shortly be seen.
237
xu. (x)
•
•
/ / / /
/ \ ^ O l = 0.16 GeV2
/ Q2 = 4.0 GeV2 \ -/--< \
f"-'\ \ / x \ emp i r i oa I %N \ .
Figure 1. The valence u-quark distribution at zero density. The result after the Q2 evolution is compared with the empirical distribution of Ref. [2] at Q2 = 4.0 GeV2 . Here the variable x is the same as the Bjorken variable Xff.
0.7
0.6
0.5
0.4
0 J
0.2
0.1
xd,(x) L
/—s^
/ \ ^ Do = 0.16 GeV2
/ \ / Q' = 4.0 Gev\
• / • • / " - . . \ \
• / / t %, \ j / empirical " \ \
/ *,N*-* ^^
Figure 2. The valence d-quark distribution at zero density. The result after the Q2 evolution is compared with the empirical distribution of Ref. [2] at Q2 = 4.0 GeV2 . Here the variable x is the same as the Bjorken variable x^j.
Since our aim is to calculate the EMC ratio for isospin symmetric nuclear matter, we show in Fig. 3 the isoscalar quark distributions at zero and finite density, and also the result after performing the convolution with the nucleon momentum distribution in nuclear matter according to Eq. (3). We see that
238
the isoscalar quark distribution in the nucleon at finite density is shifted towards smaller x, i.e. it is softer for a bound nucleon in comparison to a free nucleon. (This can be related to the increase of the radius of the distribution of baryon charge, which in our calculation amounts to 3 ~ 4%.) After the convolution with the nucleon momentum distribution, the isoscalar quark distribution becomes a smooth function with a small non-zero support also for x > 1. Nevertheless, for x < 1 it always stays below the zero density distribution at large x.
0 0.2 0.4 0.6 OS 1 1.2
X
Figure 3. Isoscalar quark distributions at zero density (solid line) and finite density (dashed line). The isoscalar quark distribution after convolution with the nucleon momentum distribution in nuclear matter is shown by the dotted line. Here x corresponds to the Bjorken variable Xff for the zero density quark distributions, the variable z (see Eq. (3)) for the finite density quark distributions, and XA for the quark distributions after convolution with the nucleon momentum distribution.
In Fig.4 we plot the EMC ratio F^(x)/F2>(x) in comparison to the parametrized fits to the experimental data18. Here F*(x) is the nuclear matter structure function per nucleon and F ^ x ) is the isoscalar nucleon structure function at zero density. In the intermediate x region, where the quark distributions are softened by medium effects, our calculated ratio can reproduce the trend of the data. In the large x region, however, our results continue to decrease, which is due to the artificial end-point behaviour of the quark distributions at zero density due to the regularization procedure, see Fig. 3. In the small x region our calculation gives an enhancement of the EMC ratio, which can be understood from the baryon number sum rule, but for very small x the calculated ratio does not show the empirically observed suppression, which is
239
related to shadowing.
F£(X) /F 2 ° (X )
0 0.2 0.4 0.6 0£ 1
X
Figure 4. The ratio of nuclear to deuteron structure functions. The parametrized fits to the experimental data are taken from Ref. [18]. The variable x here corresponds to the Bjorken variable x^.
4. Conclusion
In this work we calculated the spin independent nucleon structure functions at zero and finite density in the NJL model based on the relativistic Faddeev approach. One advantage of our calculation is that we treated the free nucleon, nuclear matter, and the structure functions at zero and finite density in one consistent framework. In particular, we wish to emphasize that the vector potential has been consistently included not only for the nuclear matter equation of state, but also for the description of the nucleon and quark momentum distributions. This has the important consequence that the baryon number and momentum sum rules for the nucleon and quark momentum distributions are satisfied automatically.
Our model can explain the softening of the valence quark distributions within the nucleon bound in nuclear matter, corresponding to an increase of the radius associated with the distribution of the baryon number. The observed suppression of the EMC ratio for intermediate values of x can therefore be reproduced qualitatively. However, for large x our quark distributions show a discontinuity which is an artifact of the cut-off procedure (proper time scheme), and therefore the EMC ratio does not increase as x —> 1, although the Fermi
240
motion of nucleons has been properly taken into account. It is possible to
remedy this situation by improving the higher moments . Also, the overall
shape of the quark distributions in the free nucleon is expected to improve if one
includes the pion cloud effects7,12. These points are now under investigation
and will be discussed in detail elsewhere19 .
A c k n o w l e d g e m e n t s
The authors thank M. Miyama and S. Kumano for the Q2 evolution code of
Ref. [17]. This work was supported by the Australian Research Council, Ade
laide University, and the Grant in Aid for Scientific Research of the Japanese
Ministry of Education, Culture, Sports, Science and Technology, Project No.
C2-13640298.
R e f e r e n c e s
1. J. Gomez et al., Phys. Rev. D49, 4348 (1994); M. Arneodo et al., Nucl. Phys. B483, 3 (1997); D. F. Geesaman, K. Saito and A. W. Thomas, Ann. Rev. Nucl. Part. Sci. 45, 337 (1995).
2. A.D. Martin, R.G. Roberts, W.J. Stirling and R.S. Thone, Eur. Phys. J. C14, 133 (2000).
3. A.W. Thomas and W. Weise, The Structure of the Nucleon, Wiley-VCH, 2001. 4. W. Detmold, W. Melnitchouk, J. W. Negele, D. B. Renner and A. W. Thomas,
Phys. Rev. Lett. 87, 172001 (2001). W. Detmold, W. Melnitchouk and A. W. Thomas, Eur. Phys. J. C13, 1 (2001).
5. A. W. Schreiber, A. I. Signal and A. W. Thomas, Phys. Rev. D44, 2653 (1991). 6. M. Wakamatsu and T. Kubota, Phys. Rev. D57, 5755 (1998). 7. H. Mineo, W. Bentz and K. Yazaki, Phys. Rev. C60, 065201 (1999). 8. European Muon Collab., Phys. Lett. B123, 275 (1983); Nucl. Phys. B293, 740
(1983); A. Bodek et al. (SLAC), Phys. Rev. Lett. 50, 1431 (1983); R. G. Arnold et al. (SLAC), Phys. Rev. Lett. 52, 727 (1984).
9. N. Ishii, W. Bentz and K. Yazaki, Nucl. Phys. A578, 617 (1995). 10. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1960); 124, 246 (1961). 11. A. Buck, R. Alkofer and H. Reinhardt, Phys. Lett. B286, 29 (1992). 12. H. Mineo, W. Bentz, N. Ishii, and K. Yazaki, Nucl. Phys. A703, 785 (2002). 13. W. Bentz and A. W. Thomas, Nucl. Phys. A696, 138 (2001). 14. D. Ebert, T. Feldmann and H. Reinhardt, Phys. Lett. B388, 154 (1996).
G. Hellstern, R. Alkofer and H. Reinhardt, Nucl. Phys. A625, 697 (1997). 15. R. L. Jaffe, 1985 Los Alamos School on Relativistic Dynamics and Quark Nuclear
Physics, eds. M.B. Johnson and A. Pickleseimer, Wiley, New York, 1985; R. L. Jaffe, Nucl. Phys. B229, 205 (1983).
16. G.A. Miller and J.R. Smith, Phys. Rev. C65, 015211 (2002). 17. M. Miyama and S. Kumano, Comp. Phys. Commun. 94, 185 (1996). 18. I. Sick and D. Day, Phys. Lett. B274, 16 (1992). 19. H. Mineo, W. Bentz, N. Ishii, A.W. Thomas and K. Yazaki, to be published.
VIOLATION OF SUM RULES FOR TWIST-3 PARTON DISTRIBUTIONS IN QCD
M. B U R K A R D T
Dept. of Physics, New Mexico State University, Las Cruces, NM 88003, USA E-mail: burkardtQnmsu.edu
Y. K O I K E
Dept. of Physics, Niigata University, Niigata 950-2181, Japan
E-mail: [email protected]
Sum rules for twist-3 distributions are re-examined. Integral relations between twist-3 and twist-2 parton distributions suggest the possibility for a <5-function at x = 0. We confirm and clarify this result by constructing hi and hL (the quark-gluon interaction dependent part of hi) explicitly from their moments for a one-loop dressed massive quark. The physics of these results is illustrated by calculating hi(x,Q2) using light-front time-ordered pQCD to O(cts) on a quark target.
1. Introduction
Ongoing experiments with polarized beams and/or targets conducted at RHIC, HERMES and COMPASS etc. are providing us with important information on the spin distribution carried by quarks and gluons in the nucleon. They are also enabling us to extract information on the higher twist distributions which represent the effect of quark-gluon correlations. In particular, the twist-3 distributions gr{x,Q2) and IIL(X,Q2) are unique in that they appear as a leading contribution in some spin asymmetries. For example, gr can be measured in the transversely polarized lepton-nucleon deep inelastic scattering and hi appears in the longitudinal-transverse spin asymmetry in the polarized nucleon-nucleon Drell-Yan process1. The purpose of this paper is to reexamine the validity of the sum rules for these twist-3 distributions.
A complete list of twist-3 quark distributions is given by the light-cone correlation functions in a hadron with momentum P, spin S and mass M:
J ^eiX'(PSmOh"^(Xn)\Q,\PS)
= 2 [gi(x, Q2)pf(S • n) + gT(x, Q2)S"± + M2g3(x, Q2)n»(S • n)] , (1)
241
242
/
/
^eiXx(PS\i>{Q)a^il5xl;{\n)\Q,\PS) = 2 [^(x,Q2)(Stf - S1_p»)/M
+hL(x, Q2)M{ptinv - p"n".)(5 • n) + h3(x, Q2)M(S^nv - Sin'1)] (2)
^eiX*(PSMO)^n)\Qi \PS) = 2Me(x, Q2). (3)
The light-like vectors p and n are introduced by the relation p2 — n2 = 0, n+ = p~ = 0, P" = p" + ^-n" and S" = (5-n)p" + (5-p)n'1+S^. The variable a; G [—1,1] represents the parton's light-cone momentum fraction. Anti-quark distributions <h,T,3(a:,<32), ^I . -MC^, <32) a r e obtained by replacing V by its charge conjugation field CrpT in Eqs. (l)-(3) and are related to the quark distributions as </I,T,3(Z,Q2) = fli,T,3(-z, Q2) and hij^x,Q2) - -hiiT}3{-x,Q2). The sum rules of interest to us are obtained by taking the first moment of the above relations. For example, from Eq. (1), one obtains
(ps\mrl5m\Q>\ps) (4) = 2 J dx [9l(x,Q2)p^S-n) + gT(x,Q2)Sl + M2g3(x,Q2)n"(S-n)].
From rotational invariance, it follows that the left hand side of Eq. (4) is proportional to the spin vector 5^ and thus gi,T,3(x<Q2) m u s t satisfy
J dx9l(x,Q2) = J dxgT(x,Q2), (5)
J dxgx(x,Q2) = 2 j dxg3(x,Q2). (6)
The same argument for Eq. (2) leads to the sum rule relations for h\tLi3(x, Q2):
j dxh1(x,Q2)=f dxhL(x,Q2), (7)
f dxh1(x,Q2) = 2 f dxh3{x,Q2). (8)
The sum rule in Eq. (5) is known as Burkhardt-Cottingham sum rule2 and Eq. (7) was first derived in Refs. [3,4]. Since the twist-4 distributions g3, h3 are unlikely to be measured experimentally, the sum rules involving those functions, Eqs. (6) and (8), are practically useless and will not be addressed in the subsequent discussions. As clear from the above derivation, these sum rules are mere consequences of rotational invariance and there is no doubt in their validity in a mathematical sense. However, if one tries to confirm those sum rules by experiment, great care is required to perform the integral including x — 0. In DIS, x is identified as the Bjorken variable XB — Q2/2P • q and
243
x = 0 corresponds to P • q -> oo, which can never be achieved in a rigorous sense. Accordingly, if fiL(x,Q2) has a contribution proportional to S(x) and h\(x, Q2) does not, experimental measurement would claim a violation of the sum rule.
In this paper we re-examine the sum rule involving the first moment of the twist-3 distribution h^{x, Q2). In particular we argue that hi(x, Q2) has a rf(x)-singularity at x — 0. Starting from the general QCD-based decomposition of fiL, we show that it contains a function h™ which has a <$(x)-singularity. In Sec. 3, we construct hi for a massive quark from the moments of h\ at the one-loop level and show that h\ also has a <J(a;)-singularity, which together with the singularity in /i™ gives rise to a <J(a;)-singularity in h^ itself. In Sec. 4, we perform an explicit light-cone calculation of hi in the one-loop level to confirm the result of the previous sections. Details can be found in Ref. [5].
2. <S(a;)-Functions in HL(X, Q2)
The OPE analysis of the correlation function in Eq. (2) allows us to decompose hi(x, Q2) into a contribution expressed in terms of twist-2 distributions and the rest which we call h\(x,Q2). Since the scale dependence of each distribution is inessential in the following discussion, we shall omit it in this section for simplicity. Introducing the notation for the moments on [—1,1], Mn[hi] = f_1 dx xnhi{x), this decomposition is given in terms of the moment relation1:
Mn[hi] = -^-Mn[h1] + ^-1^-Mn-1[g1} + Mn[hl], ( n > l ) (9)
M0[hL} = M0[/ii], (10)
with the conditions
M„[fti] = Mi[Ai] = 0. (11)
By inverting the moment relation, one finds
hL(x) = hYw(x) + h>E(x) + hl(x) (12)
kiM .o. f\jjM\+hi{x) {x>0)
(13) 2x[ dy±M + ^\!!M-2xr dy
Jx y2 M [ x Jx
J_y y2 M [ x 7-i T + h3
L(x), (x<0)
where the first and second terms in Eq. (12) denote the corresponding terms in (13). In this notation the sum rule in Eq. (10) and the condition of Eq. (11)
244
imply3,
M0[h]>] = 0. (14)
Naively integrating Eq. (13) over a; for a; > 0, while dropping all surface terms6 one arrives at fQ dxhi(x) — JQ dxhi(x) + JQ dxh3
L{x) and likewise for f_xdxhi(x). Together with Eq. (11), this yields f_1dxhi = f_1dxhi. However, this procedure may be wrong due to the very singular behaviour of the functions involved near x — 0. Investigating this issue will be the main purpose of this paper.
We first address the potential singularity at x = 0 in the integral expression for h™(x) in Eq. (13). In order to regulate the region near x — 0, we first multiply h™(x) by x&, integrate from 0 to 1 and let /3 —• 0. This yields
lldxhtix) = % }zpj\xx^9l{y) = ^ l ( 0 + ) , (15)
while multiplying Eq. (13) by \x\& and integration from —1 to 0 yields
J°~ dxhT(x) = -£L\ijnpJ0~ dx\xf~lgi(y) = - ^ i ( O - ) , (16)
where we have assumed that gx (0±) is finite. Adding these results we have
j°^dxh'E(x) + lo dxhZ(x)=^(gi(0+)-gi(0-)). (17)
Equation (14) and the fact that, in general, limx^o gi(%) — gi{—x) ^ 0, implyb
TYI
h?(x) = h?(x)reg - ^L ( 5 l ( 0 + ) - 5 1 (0 - ) ) 6(x), (18)
where h™(x)reg is defined by the integral in Eq. (13) at x > 0 and x < 0 and is regular at x = 0. Equation (18) indicates that hi has a S(x) term unless h\{x) has a S(x) term and it cancels the above singularity in h'£{x).
Equation (18) demonstrates that the functions constituting h^x) are more singular near x — 0 than previously assumed and great care needs to be taken when replacing integrals over nonzero values of x by integrals that involve the origin. In particular, if HL{X) itself contains a S(x) term, then Eq. (10) implies
f ( M * ) ~ M * ) ) + / (hL(x)-h1(x))^0, (19) JQ+ J-i
aMore precisely, the original OPE tells us M0[h\ + h^] = 0. But as long as 31 (0±) is finite, which we will assume, this is equivalent to the stronger relations, Eqs. (11) and (14). bFor example, dressing a quark at O(as) yields fli(0+) 5̂ 0 and gi(0—) = 31 (0+) = 0.
245
and, since h\{x) is singularity free at x — 0:
/
o - ,.1 ,.1
dxhi(x) + I dxhi(x) = I dxhi(i • l Jo+ J-i
(x). (20) 'o+
Accordingly, an attempt to verify the "/it-sum rule"3 would obviously fail. However, in order to see whether the S(x) identified in Eq. (18) eventually
survives in /IL(X), we have to investigate the behaviour of hL(x) at x = 0. To this end we will explicitly construct hi{x) for a massive quark to 0(as).
3. hj,(x, Q2) from the Moment Relations
In this section we will construct IIL(X,Q2) for a massive quark (mass mq) to O(as) from the one-loop calculation of Mn[h\\:
hL(x,Q2)^hL°\x) + ^CFln^hL1)(x), (21)
where the scale Q2 is introduced as an ultraviolet cutoff and the Cp — 4/3 is the colour factor. hL ' 'm ( ' \x) are denned similarly. g[°'(x) — h\'(x) = S(l — x) gives hL '(x) = S(l - x), as it should. One-loop calculations for g\{x) and h\(x) for a quark yield the well known splitting functions7,8:
'"'""" = p ^ + 1 4 " - > <22> *S"w = i r ^ i : +14*1 - *>• <23'
Inserting these equations into the defining equation in Eq. (13), one obtains
h^1\x)=3x + Ax\n^- (24)
h™{1){x) = (i^x]7 - 4 * l n i i r _ 3 + 3 * _ \S{x)- (25)
In the first line of Eq. (25), the term (3a:-§<5(l — x)) comes from the self-energy
correction, i.e. from expanding M = mq 1 + f f C F f l n ^ in Eq. (12), and
-\&{x) — -^g\(G+)S(x) in h™^\x) accounts for the second term on the right
hand side of Eq. (18). We also note that hLVW does not have any singularity
at x = 0 and satisfies f„ dx hL ( ' (x) = /„ dx h\ (x) as it should.
The structure function hL' (x) can be constructed if we know the purely twist-3 part /iL
v (x) at the one-loop level. One-loop renormalization of hL was completed in9 and the mixing matrix for the local operators contributing to
246
the moments of hi,(x,Q2) was presented. We obtain for the moment of the quark distribution5
f1 dx xnhf\x) = -J_ - J L + I, (26) y_! L n+l n + 2 2 K '
for n > 2. From this result, together with Eq. (11), we can construct h^ '(x) as
hf\x) = 3 - 6x + l-8{\ -x)~ \s(x). (27)
We emphasize that the — l/2S(x) in Eq. (27) is necessary to reproduce the n = 0 moment of h3
LW(x). From Eqs. (24), (25) and (27), one obtains
M*. Q2) = <*(! - *) + ^n%°F 2n m2
2 . + ls(l-z)-S(x) • (28) .[1-*]+ 2
We remark that the above calculation indicates that the S(x) term appears not only in h™ but also in h\. Furthermore, they do not cancel but add up to give rise to — S(x) in hi{x,Q2) itself. In the next section we will confirm Eq. (28) through a direct calculation of hi(x, Q2) for a quark.
4. Light-Cone Calculation of hz,(x, Q2)
In order to illustrate the physical origin of the S(x) terms in fiL(x) and to develop a more convenient procedure for calculating such terms, we now evaluate /IL(X) using the time-ordered light-front (LF) perturbation theory. The method has been outlined in Ref. [10] and we will restrict ourselves here to the essential steps only. There are two equivalent ways to perform time-ordered LF perturbation theory: one can either work with the LF Hamiltonian of QCD and perform old-fashioned perturbation theory10, or one can start from Feyn-man perturbation theory and integrate over the LF-energy k~ first. In the following, we will use the latter approach for the one-loop calculation of hi(x).
In LF gauge, A+ — 0, parton distributions can be expressed in terms of LF momentum densities (&+-densities). Therefore, one finds for a parton distribution, characterized by the Dirac matrix T at O(as) and for 0 < x < 1
/
H4k 1 1 79^4 7" 7 ^ 7 r7"«(p) (zir) k — m a + ie k — m„ + is
xS (x- — j D^lp-k), (29)
where Dfll/(q) = ~rfe g^ - fy""*"*19" is the gauge field propagator in LF
gauge, and n^ is a light-like vector such that n • A = A+ ~ (̂ 1° + A3) /y/2 for
247
any four vector A*. The k integrals in expressions like Eq. (29) are performed using Cauchy's theorem, yielding for 0 < k+ < p+
I dk-
2 T ( p _ m 2 + i£y (p - k)2 + ie
1 1 1 kx-oo 1 1
(30)
(2fc+)2 2(p+ - k+)
where we used k+ = xp+. In order to integrate all terms in Eq. (29) over k~, Cauchy's theorem is used to replace any factors of k~ in the numerator of Eq. (29) containing k~ by their on-shell value at the pole of the gluon propagator
k ^ k ~P 2 ( p + - * + ) ' ( 3 1 )
In the following we will focus on the UV divergent contributions to the parton distribution only. This helps to keep the necessary algebra at a reasonable level. We find for 0 < x < 1 to 0(as)
M-,« a ) = ^ l n ^ i r J ^ - , (32)
where the usual +-prescription for >1_1^ applies at x = 1, i.e. n 1 ^ = j ^ for
x < 1 and f0 t fangi = 0. Furthermore, /IL(X) = 0 for x < 0, since antiquarks do not occur in the O(as) dressing of a quark. In addition to Eq. (32), there is also an explicit S(x — 1) contribution at x = 1. These are familiar from twist-2 distributions, where they reflect the fact that the probability to find the quark as a bare quark is less than one due to the dressing with gluons. For higher twist distributions, the wave function renormalization contribution is T^-CF In Qj^5(x — 1). The same wave function renormalization also contributes at twist-3. However, for all higher twist distributions there is an additional source for S(x — 1) terms which has, in parton language, more the appearance of a vertex correction, but which arises in fact from the gauge-piece of self-energies connected to the vertex by an 'instantaneous fermion propagator' 3-f. For gr{x, Q2) these have been calculated in Ref. [10] where they give an
additional contribution — ff-Cf In §pS(x - 1), i.e. the total contribution at
x = 1 for gr(x, Q2) was found to be ff-Cf \n^^S(x - 1). We found the same
S(x - 1) terms also for /iz,(x, Q2)c. Combining the S(x - 1) piece with Eq. (32)
c In LF gauge, different components of the fermion field acquire different wave function
248
we thus find for 0 < x < 1
hL(x,Q2) = 8(x-l) + ^CFln^ f r ^ - ^ y - + U(x - 1)1 • (33)
Comparing this result with the well known result8 for hi
h1(x,Q2) = S(x-l) + ^\n^CF\j-^r + h(x-l)}, (34)
one realizes that
lim / dx [hL(x,Q2) - hi{x,Q2] = ~\n %CF ± 0, (35)
i.e. if one excludes the possibly problematic region x = 0, then the /i^-sum rule3 is violated already for a quark dressed with gluons at O(as).
In the above calculation, we carefully avoided the point x — 0. For most values of k+, the denominator in Eq. (29) contains three powers of k~ when k~ —» oo. However, when k+ = 0, k2 — m2 becomes independent of k~ and the denominator in Eq. (29) contains only one power of k~. Therefore, for those terms in the numerator which are linear in k~, the A;~-integral diverges linearly. Although this happens only for a point of measure zero (namely at k+ = 0), a linear divergence is indicative of a singularity of HL(X, Q2) at that pointd. To investigate the k+ & 0 singularity in these terms further, we consider
/(*+, kj.) = Jdk-n9 k
9 , ^ ,_ ,*, , ^ (36)
/ ' = dk
(k2 — m2 + is)2 (p — k)2 + is
(k2 — m2 + is)2 \{p — k)2 + is] — Jcan.yk , k x ) + / s j n . ( « j ^ x ) ,
where the 'canonical' piece fcan. is obtained by substituting for k~ its on energy-shell value k~ — p~ — 2tp+-k+\ [^e v a- l u e a* the pole at (p — k)2 = 0 , Eq. (31)]. For k+ = xp+ ^ 0, it is only this canonical piece which contributes. To see this, we note that k~ — k~ = — 2r -T-1-H, and therefore
Jsinyk i KJ_ • /
dk~ k~ 1
(k2 — m2 + ie)2 (p — k)2 + is
2(p+ - *+) / dk'
(k2 m2 + ie)2 (37)
renormalization. However, since all twist-3 parton distributions involve one LC-good and one LC-bad component, one finds the same wave function renormalization for all three twist-3 distributions. d Note that the divergence at fc+ = p~^ is only logarithmic.
249
Obviously1 2 , / dk (2fc+fc-_k2~ _m2+ < e )2 = 0 for Ar+ ^ 0 because one can always
m 2 +k 2 — ie
avoid enclosing the pole a t k~ = q 2k+— by closing the contour in the
appropriate half-plane of the complex A;~-plane. However, on the other hand / d 2 * * ( * £ - k i i m ; + , - 0 ' = k i + ^ f a n d t h e r e f o r e
fsin{k+,^) = -Li^ll. (38)
Upon collecting all terms oc k~ in the numerator of Eq. (29), and applying
Eq. (38) to those terms we find for the terms in hi(x, Q2) tha t are singular at
x~0
hL,sin{x, Q2) = - ^ In Q?CFS(x). (39)
Together with Eq. (33), this gives our final result for / IL , up to 0(as), valid
also for x = 0
a s ^ , , Q hL(x,Q') = 6(x-l) + -lCF\n
Z7T ?7l~ L - iw + H^ + ^- 1 » .(40)
As expected, hi from Eq. (40) does now satisfy the hi-sum rule, provided of
course the origin is included in the integration.
This result is important for several reasons. First of all, it confirms our
result for hi(x,Q2) as determined from the moment relations. Secondly, it
provides us with a method for calculating these S(x) terms and thus enabling
us to address the issue of validity of the naive sum rules more systematically.
And finally, it shows tha t there is a close relationship between these S(x) terms
and the infamous zero-modes in LF field theory1 3 .
The result of Ref. [11], where canonical Hamiltonian light-cone perturbat ion
theory is used to calculate hi(x) for x ^ 0, agrees with ours which provides an
independent check of the formalism and the algebra. However, the canonical
light-cone perturbation theory used in Ref. [11] is not adequate for studying
the point x — 0. From the smooth behaviour of hi{x) near x = 0 the authors
of Ref. [11] conclude tha t the sum rule for the parton distribution hi{x) is
violated to O(as)- Our explicit calculation for hi[x) not only proves tha t the
sum rule for hi(x) is not violated to this order if the point x — 0 is properly
included, but also shows that it is incorrect to draw conclusions from smooth
behaviour near x — 0 about the behaviour at x = 0.
5. S u m m a r y
We have investigated the twist-3 distribution h^x) and found tha t the sum
rule for its lowest moment is violated if the point x = 0 is not properly included.
250
For a massive quark, to O(as) we found
9 1 1 {41) 2n mf [1 - ar]+ 2
AL (*, QJ) = 8(x - 1) + T ^ - C F In
At 0(0:5), fti(a;, Q2) does not satisfy its sum rule if one excludes the origin from the region of integration (which normally happens in experimental attempts to verify a sum rule). Of course, QCD is a strongly interacting theory and parton distribution functions in QCD are nonperturbative observables. Nevertheless, if one can show that a sum rule fails already in perturbation theory, then this is usually a very strong indication that the sum rule also fails nonperturbatively (while the converse is often not the case!).
From the QCD equations of motion, we were able to show nonperturbatively that the difference between fiL(x,Q2) and h\(x,Q2) contains a S(x) term
[hL{*,Q2) - hl(x,Q2)]singular = - ^ M 0 + , Q 2 ) - <h(0-,Q2)) S(x)(42)
Since gi(0+,Q2) - gi{0-,Q2) = \imx-+0gi(x,Q2) - g\(x.Q2) seems to be nonzero (it may even diverge6), one can thus conclude that either hi(x,Q2) or h3
L(x, Q2) or both do contain such a singular term. We checked the validity of this relation to O(as) and found that, to this
order, both h\ and /i£ contain a term oc 6(x). We also verified that even though the sum rule for 1IL(X) is violated if x = 0 is not included, it is still satisfied to O(as) if the contribution from x = 0 (the S(x) term) is included.
Acknowledgments
M.B. was supported by a grant from the DOE (FG03-95ER40965). Y.K. is supported by the Grant-in-Aid for Scientific Research (No. 12640260) of the Ministry of Education, Culture, Sports, Science and Technology (Japan). We are also grateful to JSPS for the Invitation Fellowship for Research in Japan (S-00209) which made it possible to materialize this work.
References
1. R.L. Jaffe and X. Ji, Nucl. Phys. B375, 527 (1992). 2. H. Burkhardt and W.N. Cottingham, Ann. of Phys. 56, 453 (1970). 3. M. Burkardt, Proc. to Spin 92, Eds. Hasegawa et al., (Universal Academy Press,
Tokyo, 1993); R.D. Tangerman and P.J. Muilders, hep-ph/9408305. 4. M. Burkardt, Phys. Rev. D52, 3841 (1995). 5. M. Burkardt and Y. Koike, to appear in Nucl. Phys. B, hep-ph/0111343.
e In the next-to-leading order QCD for a quark, \imx-,.o 9l(x) — 9l(x) >s logarithmically divergent14.
251
6. D. Boer, Ph.D. Thesis, Vrije Univ., Amsterdam 1998. 7. V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972); Yu. L. Dok-
shitser, Sov. Phys. JETP, 46, 641 (1977); G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
8. X. Artru and M. Mekhfi, Z. Phys. C45, 669 (1990). 9. Y. Koike and K. Tanaka, Phys. Rev. D51, 6125 (1995). 10. A. Harindranath and W.-M. Zhang, Phys. Lett. B408, 347 (1997). 11. R. Kundu and A. Metz, Phys. Rev. D65, 014009 (2002). 12. S.-J. Chang and T.-M. Yan, Phys. Rev. D7, 1147 (1972). 13. M. Burkardt, Advances Nucl. Phys. 23, 1 (1996). 14. W. Vogelsang, Phys. Rev. D54, 2023 (1996).
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6. Hadronic Properties of Nuclei
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PROPERTIES OF HADRONS IN NUCLEAR MATTER
F.C.KHANNA Physics Department University of Alberta Edmonton Alberta, T6G 2J1 Canada *
and Center for the Subatomic Structure of Matter, University of Adelaide,
Adelaide, 5000, Australia E-mail: khannaQphys.ualberta.ca
D . U . M A T R A S U L O V
Physics Department University of Alberta -Edmonton Alberta, T6G 2J1 Canada
Nucleon properties change in nuclear matter at normal and high densities as well as with increasing momentum transfer (q2). It is known that with nuclear matter at high temperature, the mass and coupling constant of mesons are changed. The Japan Hadron Facility will provide an opportunity to increase the nuclear density, temperature and q2 of nuclear matter simultaneously. This provides a unique opportunity to study all these variations at once. Furthermore, with a proton beam of 50 GeV incident on a target, there will be production of mesons and hadrons with one and two charm and beauty quarks. This will provide an opportunity to change the flavour of quarks in hadrons and then study the interaction and decay of these hadrons in nuclear matter .
1. Introduction
The Japan Hadron Facility1 (JHF) will produce a proton beam of 50 GeV. Such a beam of protons incident on finite nuclei will produce local regions of high density, 2-3 times the nuclear matter density, and of high temperature, a few tens of MeV. Furthermore, in interaction with nucleons it will produce a very large variety of mesons, hyperons and hyperons with heavy flavor like charm and beauty. The phase space alone suggests that there will be copious production of charm mesons and nucleons. As in the case of hyperons2 with one or two strange quarks, the charm hyperons (Ac) and beauty hyperons (At) will be absorbed in nuclear fragments or in target nuclei. This will initiate the first study of nuclei with charm and beauty baryons in them and will
* permanent address
255
256
provide an opportunity for the study of the behaviour and motion of heavy quarks in nuclear matter. There is also the possibility that the baryons with two heavy quarks are produced. Certainly baryons with two charm quarks are more likely than with two beauty quarks. Spectroscopy of such baryons would be most interesting and the motion of such baryon in nuclear matter would be very exciting. JHF is thus expected to open a window to new phenomena in nuclear systems. Measurements to isolate these phenomena and attribute them to heavy quarks will require ingenuity and clever experimentation. But it will be well worth it to understand these new phenomena with new unusual probes.
2. Density Dependence of Nucleon and Meson Properties
It is well-known that nucleon and meson properties like masses, coupling constants and vertex form factors depend on density and momentum transfered (q2). We have proposed a medium modified Skyrme-like Lagrangian which includes the distortion of the nonlinear meson field by the medium. A scalar-isoscalar sigma meson is added that is identified with the quantum dilaton. The medium effect on the pion is introduced through the self-energy operator while its effect on the dilaton field is limited to renormalization of the gluon condensate.
Table 1. Change of masses of 7r, a, N and F with density, (all quantities are given in MeV).
P/PO 0
0.5
1.0
TOir
139.0 144.90 144.06
ma
550.1 513.8 493.8
mN
1413 1217 11.57
1 a—t-7T7r
251.2 88.7 34.6
In Table 1 we show the change of masses of 7r, a and N in the medium with density p, po is the nuclear matter density. Both a and N masses decrease. The IV-^Tr is also shown. It indicates that a becomes a sharp resonance at nuclear matter density while it is a very broad resonance in the free state.
Table 2. Change of various coupling constants with density.
P/PO 0.5
1.0
3"A/9 A 0.96 0.92
9*NN/9*NN
0.91 0.80
9^NN/9<rNN 0.88 0.78
AJ/A, 0.70 0.56
A;/AC T
0.90 0.84
In Table 2 we show various quantities normalised to the free space value.
257
The axial vector coupling constant decreases towards the value of nuclear ma t
ter (0.8). This value is realised without the dilaton field. The coupling con
stants of 7T and a decrease by 20% while the decrease of A,r in medium is quite
drastic, approaching a factor of 2.
Finally, the 7TJVJV and <TNN vertex form factors change significantly at finite
density and as a function of momentum transfer, q2. Since the high energy
protons at J H F are capable of large momentum transfer, the dependence on
q2 becomes an important factor in any analysis of the data .
Overall this model study indicates tha t the various nucleonic properties
change significantly with density and q2 and, therefore, must be included in
a systematic and consistent manner in any detailed study of the interaction
of high energy protons with mat ter . In fact it is anticipated tha t locally the
density may approach 3-4 times the nuclear density. In a such case new and
improved models have to be constructed such that proper consideration is given
to the change in nucleonic properties. In practice it is a nontrivial task since
there is no da ta or information that guides us in such a study.
3. T e m p e r a t u r e D e p e n d e n c e of N u c l e o n i c P r o p e r t i e s
The temperature dependence of coupling constants and vertex form factors
is of crucial importance for investigations in nuclear physics, astrophysics,
cosmology and particle physics. In fact for studying phase transitions and
thermodynamic properties of nuclear systems under extremal conditions it is
essential to determine temperature (T) and density (p) dependence of prop
erties of mesons and nucleons in interaction. This is certainly t rue below the
critical temperature (Tc) where the quark and gluon degrees of freedom are
masked, while above Tc the deconfinement of quarks leads to a quark-gluon
plasma (QGP) tha t requires a study in a different regime. The studies a t Rel-
ativistic Heavy Ion Collider (RHIC) are primarily directed towards achieving
a QGP, while the research program at J H F would lead to a nucleonic mat te r
t ha t achieves temperatures of a few tens of MeV. In the lat ter case it is im
perative tha t reliable studies are undertaken to understand the role of density
and temperature variations of parameters . A few a t tempts have been made
to gain some understanding of nucleon mat te r at finite temperature , while the
density dependence at T = 0 has been studied extensively.
We have studied temperature dependence of coupling constants and vertex
form factors using Thermofield Dynamics (TFD) . This formalism, proposed by
Takahashi and Umezawa, is a real-time finite temperature formulation of the
quan tum filed theory. There are two essential ingredients :
i) the Hilbert space has to be doubled by introducing a second Hilbert space
(called tilde space) and so are the corresponding operators such as Lagrangian,
258
Hamiltonian etc. and ii) a pure finite temperature vacuum state of this new Hilbert spaces defined by using Bogoliubov transformations.
The propagators are 2 x 2 matrices. In TFD it is possible to use Wick's theorem and the Feynman diagrammatic approach as in the case of zero temperature field theory. The free meson-nucleon vertex function is independent of temperature. The temperature dependence is calculated at one-loop level and the results are limited to terms linear in the fermi- and bose distribution function. The result may be written as
TA(t,T) = TA(t) + i^AAB(t,T), (1) B
with
/
dAk —^TB(k2)(a + M)TA{t){b + M)TB{k2)
x [Go(a)Go(b)D$(k2) + 2Go(a)GT(b)DB(k2)}, (2)
where
t = q2 = q2o-q2 = (p-p')2, a = p'-k, b=p-k, (3) 1
M2 + is _1
k2 - M2 + is '
and
Go^ = a2-M2 + is ' GHa) = 2wiS{a2 ~ M 2 ) ^ ( f l ) (4)
D°(k2) = 15 iTTTTT - DT(*2) = ~^iS(k2 - m2)NB(k) (5)
with
NF(a) - 0(ao)nf{a) + 0(-a0)nf{a), (6)
nF(a) = [^ (M-^) + i ] - i f nB(fc) = [ e ^ 0 l - l ] - 1 (7)
n / (a) = [e/ ,(l°o|+'l) + l ] - 1 . (8)
The coupling constants (monopole) at zero temperature and the vertex form factors are taken from the BONN-A potential. The results indicate some very interesting features when variation of coupling constants with temperature and density and the corresponding change of vertex form factors with q2 are considered. They may be summarised as follows :
i) There is a critical temperature (Tc) where the coupling constants for different mesons show a very sharp change. Tc is different for different mesons: 77 = 360 MeV, T° = 95 MeV, T/ = 200 MeV and T? = 175 MeV. The most notable point is that the w—meson coupling constant increases while the
259
other coupling constants decrease to zero at some temperature. It is worth pointing out that g„NN, goes to zero at the lower temperature suggesting that the strong attraction provided by the cr-meson disappears. What impact this has on the overall stability of the nucleus needs to be carefully considered. The sharp increase of gWNN arises due to the isospin factors for p- and w-meson. This implies that the overall repulsion increases making it much more difficult to keep a stable nucleus. It is possible that these variations depend on the choice of the vertex form factors.
ii) The vertex form factors as a function of q2 decrease more rapidly as the temperature is incresed, thus approaching zero at much smaller values of q2.
iii) The change of the coupling constant and the vertex form factors is not as drastic with the changing density as it is with a change of temperature. For example, the density dependence of Tc for different mesons is shown in Table 1 and the change is small.
iv) The usual procedure of finding the critical temperature of transition from hadronic state to the quark-gluon plasma is based on lattice QCD. In such calculations it is usual to look for critical temperature for hadronization of the quark-gluon system and it is conceivable that such a transition temperature is unique. In the present study, starting with a hadronic system as the temperature is raised the role of different mesons in maintaining an equiliblium state of nuclear matter is disturbed by a different response of various mesons. It appears that the attractive parts of the interaction are depleted while the repulsive parts are enhanced. This would suggest that the nucleon system will approach hard-sphere gas and thus would be much harder to compress it to high densities. What may be the magnitude of such a density is not established in these calculations.
v) Since the experiments at RHIC and other heavy ion facilities start with heavy ions, colliding beam or fixed target, it is imperative to understand more carefully the approach to the critical temperature as the nuclei are heated and compressed at the same time. Will the transition temperature for de-confinement of hadrons and the transition temperature for hadronization of quark-gluon plasma be the same? A resolution will be possible if we had a clear and an unambigous signal for the formation or presence of a quark-gluon plasma.
4. Production of New Particles
The Japan Hadron Facility with a proton beam of 50 GeV incident on a nuclear target will produce new exotic particles profusely, in particular the charm and beauty hadrons. Certainly there will be many more charm hadrons than beauty hadrons. Similar to the case of hypernuclei with one hyperon (A in particular)
260
there will be hypernuclei with Ac and Af,. These particles will propagate in
nuclei. Interactions of Ac or At, with nucleons, their propagation and finally
their decay will give valuable information on the c- and b - quarks in a sea of
u- and d-quarks.
No doubt the production of unusual hyperons with an s-quark will be higher
and thus will open a new era in our understanding of A in nuclear mat ter , its
decay and propagation. The variety of new physics may be put in several
different categories :
i) Large scale production of n, K and other mesons at such energies. The
cross sections are not measured very well so far.
ii) Production of oniums, charmoniums and beautyoniums, will be large
and will provide an opportunity to study their decay in detail. The system has
been studied extensively within Schrodinger and Bethe-Salpeter approaches7 .
The spectrum is understood reasonably well but the decays are not known or
understood very well. Though it is possible that separating such events among
the massive debris may be a difficult task.
iii) Mesons with one heavy and one light quarks, mesons like cq, bq, cq
and bq where q is a light quark («, d or s) will be produced in large numbers
and will then decay. The spectrum of such mesons has been studied within
Schrodinger, Dirac and Bete-Salpeter equations7 , 8 , 9 . The decay of such mesons
will again give us valuable information about heavy quarks.
iv) Baryons with two heavy quarks. It is possible to produce hyperons
with strangeness and these particles are absorbed by nuclei before decaying.
These are accompanied by production of two /^-mesons. At J H F there will be
large flux of such particles in the outgoing particles of various kind. However
in addition there are baryons, i.e. hyperons with two c- or 6-quarks. Such
particles, as the strangeness baryons will have interactions with the nuclei, they
will propagate and eventually decay. Observation and study of such particles
would give first evidence for c — c interaction and interaction of such baryon
with nuclear mat ter . Study of such individual baryons with two heavy quarks
(s, c or b) and one light quark have been done with various approximations.
Studies with Schrodinger and Dirac approach8 '9 using a two-center Coulomb
plus linear confinement potential have been carried out. These give spectrum
of the hadrons. Bethe-Salpeter approach1 0 using the two heavy quarks as a
diquark and light quark motion in this potential have been carried out . In
this approach, since the wave functions of the effective two particle system are
obtained, it is possible to study transition rates also. No studies have been
done where these heavy hadrons are inserted in nuclear mat ter . How does
the presence of finite density of ordinary hadrons inhibit the weak decay of
these hadrons? Wha t is the interaction of nucleons with these hadrons? Is
261
it attractive? The JHF machine with its capability to produce many of these exotic particles will eventually open the door to a vast array of new physics that would eventually produce a clear and better understanding of interaction among quarks and the corresponding interaction among hadrons with one or two heavy quarks in them. Such studies at JHF will truly open the door to "charm"ing and " beauty"ful nuclei and to a wide array of new physics.
5. Conclusions
To summarise, we have indicated that increasing the density and increasing the temperature have a complementary though somewhat different effect on the production of mesons, nucleons and their interactions. But the changes are sufficiently profound that they cannot be ignored in any realistic analysis of the results from JHF which is expected to produce densities up to few times of nuclear matter density and create temperatures at high as tens of MeV. The production of a wide variety of hadrons with heavy quarks (c and b) and strange quarks opens a new era in the study of nuclear physics. The physics of hypernuclei will be extended to include Ac and Aj, in addition to Ec, Sc , fic, £{,, Eb and fi{, as possible baryons in nuclear matter. The study of their interaction with nucleons would open a new chapter in the study of nuclear matter. JHF will be a unique facility that will allow us to change temperature, density, momentum transfer and the type of constituent quarks that make up a large nucleus. That is a great deal more than it has been possible to do so far.
Acknowledgements
The authors wishes to thank their collaborators, the Tashkent group, M. M. Musakhanov, A. Rakhimov and U. Yakhshiev for valuable contributions towards several of the conclusions reached in this talk. The research is supported in part by Natural Science and Engineering Research Council of Canada. Finally, the author (FCK) wishes to thank Prof. Tony Thomas for hospitality at CSSM where this talk was written up.
References
1. S. Sawada, in this proceedings; T. Nagae, in this proceedings. 2. A. Gal, Adavances in nuclear physics (Eds. M. Baranger and E.Vogt) Vol. 8
(Plenum Press, N. Y. 1975); B. Povh, Rep. Prog. Phys. 39, 824 (1976). 3. A. M. Rakhimov, M. M. Musakhanov, F. C. Khanna and U. T. Yakhshiev, Phys.
Rev. C58, 1738 (1998); A. M. Rakhimov, F. C. Khanna, U. T. Yakhshiev and M. M. Musakhanov, Nucl. Phys. A643, 383 (1998); M. Rho, Ann. Rev. Nucl. Sci. 34,
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531 (1984); G. E. Brown and M. Rho, Phys. Rep. 269, 333 (1996); K. Saito, K. Tsushima and A. W. Thomas, Phys. Rev. C55, 2637 (1997); M. K. Bannerjee and J. A. Tjon, Phys. Rev. C56, 497 (1997) The literature on this topic is extensive.
4. A. M. Rakhimov, U. T. Yakhshiev and F. C. Khanna, Phys.Rev. C61, 024907 (2000); C. A. Dominguez, C. van Gend and M. Loewe, Phys. Lett. B429, 64 (1998); Y.-J. Zhang, S. Gao and R.-K. Su, Phys. Rev. C56, 3336 (1997); A. M. Rakhimov and F. C. Khanna, Phys. Rev. C64, 064907 (2001).
5. Y. Takahashi and H. Umezawa, Collect. Phenom. 2 55 (1975); This article is reproduced in Int. J. Mod. Phys. A10 1755 (1996); A. Das, Finite temperture field theory (World Scientific, 1977); H. Umezawa, Advanced field theory : Micro, Macro and Thermal physics (AIP, New York, 1994).
6. R. Machleidt, Adv. Nucl.Phys. 19, 189 (1989). 7. X. Q. Zhu, F. C. Khanna, R. Gourishankar and R. Teshima Phys. Rev. D47, 1155
(1993); S. Godfrey and N. Isgur, Phys. Rev.T>23, 189 (1985). 8. D. U. Matrasulov, M. M. Musakhanov and T. Morii, Phys. Rev. C61, 045204
(2000); V. V. Kiselev, A. K. Likhoded and A. I. Onishchenko, Phys. Rev. D60, 014007 (1999); S. S. Gershtein, V. V. Kiselev, A. K. Likhoded and A. I. Onishchenko, Phys. Rev. D62, 054021 (2000).
9. D. U. Matrasulov, F. C. Khanna, K. Y. Rakhimov and K. T. Butanov to appear in Eur. J. Phys. A.
10. S.-P. Tong et al Phys. Rev. D62, 054024 (2000).
RELATIVISTIC MEAN-FIELD THEORY WITH PION IN FINITE NUCLEI
H. TOKI Research Center for Nuclear Physics (RCNP),
Osaka University, Ibaraki, Osaka, 567-0047, Japan, and RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0105, Japan
E-mail: [email protected]
K. IKEDA AND S. SUGIMOTO RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0105, Japan E-mail: [email protected]; [email protected]
First we review the present relativistic mean-field theory from the view point of the missing pion contribution. Then we discuss the role played by pions in hadrons and light nuclei. We introduce the interesting experimental data on pionic states taken at RCNP. These data seem to suggest the occurrence of surface pion condensation in finite nuclei. We show the results of the relativistic mean-field theory with the pion. We find that the pion mean-field becomes finite with a reasonable parameter choice. A qualitative discussion is presented on the consequence of a finite pion mean-field on Gamow-Teller transitions and spin response functions as well as other quantities.
1. Relativistic Mean-Field Theory
We start with the relativistic mean-field theory. The relativistic mean-field (RMF) theory is quantitatively very successful. This is mainly because RMF naturally includes the strong three-body repulsion, which is responsible for saturation, and a strong spin-orbit force, which is responsible for magic numbers, if expressed in the non-relativistic framework. However, this theory does not include pions, which should be the most important ingredient in hadron physics.
In fact, the Lagrangian of the relativistic mean-field (RMF) theory is
L = i^da -M-9o<r- ^ 7 " ^ - g^pP" - e7/J ( 1 ~ Ts) A»ty
+ i duad»a - \mW - ±g2a3 - \g3<r4 - \H,„H>"' + \ m l ^ ^
+ iC 3(W / i W«)2 - \G%G^ + \rn\W - \F,VF^ , (1)
263
264
where the field tensors H, G and F for the vector fields are defined through
H^v = d„w„ - d„u,i
F^ = d»Av - dvA„ , (2)
and other symbols have their usual meanings. Here, a denotes the scalar
meson, u> the vector meson and p the isovector-vector meson. A denotes the
photon.
This Lagrangian apparently does not include the pion. The pion term
should be there, but this term does not contribute in the R M F approximation
due to the conservation of parity and isospin of single particle states. There are
7 parameters in the R M F Lagrangian. Under the mean-field approximation we
can construct coupled differential equations for nucleons, mesons and photon
field, which could be easily solved numerically. By adjusting the parameters
to the existing da t a on binding energies and radii of proton magic nuclei, these
parameters are fixed by Sugahara and Toki1 . These parameter sets were named
T M l , TM2 and TMA, which have non-linear sigma and omega self-coupling
terms. The T M l parameter set is used for the nuclei with A > 40 and TM2
for A < 40. The T M A parameter set has a smooth mass dependence in order
to describe nuclei in the entire mass region. We found good descriptions of
binding energies and radii (the binding energies are shown in Fig. 1). Good
quality of the parameter sets has been demonstrated by calculating all the
even-even mass nuclei in the entire mass region2 .
In addition, we have calculated giant resonances, equation of s tate of nu
clear mat te r and superheavy elements. We are very much satisfied with the
performance of the R M F theory with the TM parameter sets. However, if we
look at hadron physics, chiral symmetry and its spontaneous breaking are the
essential ingredients of the successful description of the experimental da ta .
2. Tensor Force in Light N u c l e i
The pion exchange interaction is discussed in nuclear physics in terms of the
tensor force. Let us see this connection first. The pion exchange force in a
non-relativistic form can be written as
5=i • qa2 • q = #i • a2q2 + -S12 , (3)
where Su is called the tensor force. We can then see by direct calculations
tha t the tensor force is much larger than the central spin-spin force. Therefore,
we can say tha t the pion exchange interaction is the tensor force. Knowing
265
this, let us see what we know about the contribution of the tensor force in light nuclei.
There is a variational calculation on the a particle3. According to this calculation, about half of the potential energy is caused by the tensor force. In addition, the wave function of the a particle contains about 10% of the 2p-2h components, which consist of 2 particles in the p orbit and 2 holes in the s orbit, even though the energy difference between the two orbits is at least 20 MeV. This large admixture is caused by the strong tensor force coming from the pion exchange. We further remind ourselves of the findings of few-body systems, which are treated rigorously without the restriction of model space with a realistic nucleon-nucleon interaction, which has explicitly a short-range repulsive core and a tensor force4,5. The recent variational calculations of the Argonne group for the up to A=8 systems, with the use of a realistic
9.5
8.5
% "
< U l 6 . 5
5 . 5 -
4 5 i i i i i 1 i i i i I i i i i I i i i i I i i i i I i i i i
0 50 100 150 200 250 300
Figure 1. Binding energy per particle as a function of the mass number for proton magic number nuclei. The experimental data are shown by the dots, while the TMA results are shown by the solid curves and the NL1 results by the dashed curves.
266
two-body interaction, provide very good description of the light nuclei5. In addition, these calculations show the dominant role of the pion, which amounts to 70-80% in two-body interaction part in the 3<A<8 systems as well as the importance of the three-body interaction. Why does the RMF theory without the pion work so well? We can say that the use of such parameters as ga and gu mocks up the pion attraction effect. Is there any place for the necessity of the pion? The interesting place to look for an answer to this question is the pionic excitation of nuclei.
3. R C N P (p,n) Spin Experiments
There are two important experiments on pionic excitations performed at RCNP using the (p,n) reactions by Sakai group7'8. They are the zero-degree spectra in the (p,n) reactions and the large momentum transfer (p,n) reactions at E ~ 300 MeV.
We start with the zero-degree (p,n) reactions. We take 90Zr as an example7. Taking a simple shell model, we expect two states being populated by the Gamow-Teller operator : #7/2<7 /̂2 particle-hole state and <79/2<7̂ /2 state. Due to a repulsive interaction between these two states, we expect these two states mix and the higher state carries most of the strength, which is called the giant Gamow-Teller state. The experimental data are obtained at forward angles with the incident energy of 300 MeV and are analysed in terms of multipole components. The large fraction of the GT strength is then found to be shifted further up to the higher states than the giant GT state by 30 to 40 percent. This shift of the GT strength to higher states was studied theoretically as the coupling of the GT state to 2p-2h states due to the strong tensor force. With the coupling of highly excited states, the perturbative calculations are able to provide the large strengths in the continuum. However, the change of the GT strengths seems too large considering the calculations are done perturbatively.
The GT strength seems to be exhausted by nucleon degrees freedom. This fact suggests that the coupling of the GT states to delta isobars is very small. In terms of g' for delta-hole coupling in the spin-isospin channel, g'A < 0.2. This fact indicates that the delta-hole states should contribute largely to pion condensation and its precritical phenomenon. This possibility is rejected by the experimentally missing precritical phenomenon9,10. Relativity arguments are suggested to reduce the pionic collectivity, but it seems not enough11. The precritical phenomenon is expected also in the spin response functions12. Hence, we discuss now the second experiment of the Sakai group.
It is the (p,n) reaction with the measurement of polarizations in the initial and final channels at large momentum transfer. In addition, the isovector
267
transfer is identified as unity by the use of the (p,n) reaction. This spin measurement is able to provide the response functions in the pion and the rho meson channels8.
In the standard model of spin correlations, we use the interactions in the pion (longitudinal) and rho meson (transverse) channels as
P F m2 q2 + m2.
P f Vp = —^(g' - Cp-^r- ~)#i xq-a2x qn • f2 . (5)
m J q2 + my
These interactions provide strong attraction in the pion channel and strong repulsion in the rho meson channel at high momentum region due to small pion mass, m^, and large rho meson mass, mp. Naturally then we expect that the pion response is softened from the non-interacting case and the rho meson response is hardened. This tells us that the ratios of the spin longitudinal (TT) and the spin transverse (p) responses are larger than unity at smaller excitation energies.
The experimental data show surprisingly enough that the ratios are not larger than unity8. This fact is difficult to understand in the present framework. The relativistic description of the response functions is not enough to turn around the ratios unless we use different g' between the pion and the rho meson channels13. This experimental data indicate that there is a serious problem in the present theoretical framework in the pion channel (spin-isospin channel).
4. Surface Pion Condensation
We propose to take the pion terms seriously now, which should be present in the Lagrangian14. For simplicity, we write explicitly only the sigma and pion terms in the Lagrangian density
L = ^[ii^dn -M-g„(T-g«i<>-i,idliTa'Ka}i> + Lmeson • (6)
We then assume that the expectation value of the pion field is finite. We write the equation of motion for nucleons and pions as
[i^d^ -M-gaa- g*VnaWTa]ijj = 0 , (7)
and
[V2 - m2]7ra = -gMHlTat/j). (8)
The sigma meson and others follow the same equations of motion as in the standard case. These equations tell us the reason why we have not included
268
the pion mean-field until now. The source term of the pion Klein-Gordon
equation is non-vanishing only when the parity and the isospin are mixed in
the single particle state. This violation of the parity and isospin is caused
by the pion term in the above Dirac equation for nucleons. Hence, the single
particle s tate can be expressed as
1>njm(x) = Y, W"t<t>Kjm,t • (9) K,t
Here 4>K,jmit denotes the eigenfunctions of nucleons without the pion mean-field
term. The sum over K implies parity mixing and the sum over t isospin mixing.
We call a non-vanishing pion mean-field surface pion condensation, since the
pion source term involves derivatives of the mean-field.
5. N u m e r i c a l R e s u l t s w i t h F i n i t e P i o n F i e l d
In this section, we present our numerical results. We take the T M 1 parameter
set of Ref. [1] for all the parameters except for the pion-nucleon coupling. For
the latter we take the value for the Bonn A potential1 6 , which corresponds to
<7?r = fn/irin and fn ~ 1. We stress again tha t we use all the terms given in the
Lagrangian, including the non-linear terms. This means tha t the saturat ion
property is guaranteed and the bulk part of the nucleus tends to have the
saturat ion density. Since we are especially interested in the occurrence of
finite pion mean-field and want to see its effect under the simplest condition,
we neglect the Coulomb term. We calculate such N=Z closed shell nuclei as 1 2 C , 1 6 0 , 4 0 Ca, 5 6Ni, 8 0Zr, 1 0 0Sn and 1 6 4 P b .
Next we discuss the results with and without the pion mean-field. Wi th
the present choice of the parameters, we find the actual occurrence of the finite
pion mean-field for all the nuclei studied here. This is not a trivial result tha t
the pion mean-field becomes finite. As we will show later, the pion mean-field
become finite only when the pion-nucleon coupling constant is larger than the
critical value. There we will see the dependence of the pion mean-field on
the pion-nucleon coupling strength together with the critical strength for the
onset of the pion mean-field. The rho meson mean-field vanishes for N = Z
nuclei. The case with the finite pion mean-field provides total binding energies
larger than those obtained without pion mean-field. The pion term reduces
the total energy, while all the other terms such as kinetic energy, the combined
sigma and omega energy do not favour a finite pion mean-field. This means
tha t a single particle state is brought up to the next major shell by the pionic
correlation and, therefore, the kinetic energy becomes larger. However, the net
energy is smaller for the case with finite pion mean-field.
269
We show the mass number dependence of the pion energy per nucleon in Fig. 2. We note that the kinetic energy and the sigma and omega energies per particle are almost constant (independent of the mass number), and hence they are volume-like. We see, on the other hand, a peculiar behaviour in the pion energy. The magnitudes of the pion energy are clearly separated into two groups. One group is large and the common feature is jj-closed shell nuclei; the magic number nuclei due to a larger spin-orbit partner (j-upper) being filled. The other group is smaller and consists of LS-closed shell nuclei. The pion energy per nucleon for jj-closed shell nuclei decreases monotonically with the mass number. The rate of the decrease is stronger than A-1/3. This means that the pion mean-field energy behaves in proportion to the nuclear surface or even stronger than that. Hence, we name this phenomenon surface pion condensation.
We also mention that the separation of the LS-closed and jj-closed shell
> 0
J
10 100
A
10
1
I
s
I.
I!
1
^ ^
> • ' - ' '
^ - ^ ^ * N i
40Ca
\
J-J
L-S •
20A"1/3 •
o 100Sn
"f>Zr . * . .
O : • • • •
O 164Pb
-
Figure 2. The pion energy per nucleon as a function of the mass number on the log-log plot. There are two groups: first is the jj-closed shell nuclei denoted by open circle and the second is the LS-closed shell nuclei denoted by filled circles. The pion energy per nucleon for the jj-closed shell nuclei decreases monotonically and follows more steeply than A~ '3, which is shown by solid line.
270
nuclei into two groups of the pion energy is related to the spin-orbit splitting. The spin-orbit partner having larger total spin near the Fermi-surface does not contribute to the total energy in the case of surface pion condensation in the LS-closed shell nuclei14.
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
f2/f02
Figure 3. The pion energy per nucleon as a function of the pion-nucleon coupling constant squared. In this systematic study, we have fixed the set of the filled single particle states with the same spins as in the closed shell configurations.
In Fig. 3 we present our results of the finite pion mean-field by plotting the pion energy per nucleon for various nuclei as a function of the pion nucleon coupling constant (f„) in Fig. 3. If the pion nucleon coupling constant is decreased by 5 %, the onset of the finite pion mean-field disappears except for 12C. This fact indicates that the symmetry breaking mean-field is fragile and various effects as delta isobar contributions, the rho meson tensor coupling effects, the effects of the Fock (exchange) term and several others would influence the occurrence of the finite pion mean-field in finite nuclei. Quantitatively, this aspect will be worked out in the subsequent paper15.
Next we discuss examples of the qualitative consequence of finite pion mean-field. First, we concentrate on the Gamow-Teller (GT) transitions. Without pion condensation, there exist no transitions for LS-closed shell nuclei such as 1 6 0 , 40Ca and only two transitions, for example, for 90Zr. However, the
271
mixing of parity in the intrinsic state allows transitions of 2p-2h states. This makes the spectrum of the GT transitions with some GT strengths to lie above the two dominant peaks7 in 90Zr. Hence, naturally we have strengths in the region of the higher excitation energy in the mean-field theory with the pion, as the experiment demands. Second, the longitudinal spin response functions, which are caused by the pionic correlations, should be largely modified due to pion condensation. Since the large pionic strength is used up to construct the nuclear ground state, the pionic fluctuation should be reduced largely. This should make the spin response in the pion channel weak. This remains to be demonstrated in the future work.
Until now the phenomena involving high momentum components have been separated into two groups: nuclear correlations induced by the repulsive core and the tensor force. Surface pion condensation provides us with the possibility to describe the part of the correlation effect induced by the tensor force. This fact indicates that we are now able to separate the short range correlation phenomena into two classes: those associated with the pion and those associated with as the repulsive core. In fact, the surface pion condensation automatically provides a large amount of the 2p-2h excitations in the nuclear ground state. There should be many other consequences of surface pion condensation in nuclear phenomena. The pairing correlations and the spin-orbit couplings are all surface phenomena, and surface pion condensation would couple with these correlations and provide rich phenomena.
6. Conclusion
We have discussed the possible occurrence of the finite pion mean field in finite nuclei by introducing the pion field in the relativistic mean-field (RMF) theory. We have extended the RMF theory by introducing the parity-mixed single particle basis to accommodate the finite pion mean-field. We have taken the TM1 parameter set in the RMF theory and introduce the pion field in the pseudovector coupling with the nucleon. Making use of the pion-nucleon coupling constant in free space, we have made calculations for N=Z closed shell nuclei and demonstrated the actual occurrence of finite pion mean-field. We have demonstrated that the potential energy associated with the pion behaves as proportional to or even stronger than the nuclear surface. Hence, we name the onset of the finite pion mean-field as surface pion condensation. We qualitatively discussed the consequence of surface pion condensation on the Gamow-Teller strengths, the spin response functions and the short range correlations. The large difference of the pion energies in jj-closed shell and LS-closed shell nuclei has been found. This may be connected to the mechanism of spin-orbit splitting due to the tensor force discussed long time ago by Takagi
272
et al. and Terasawa .
We would like to stress tha t we are at the initial stage of our investigation
of the role of the pion in finite nuclei. We have to perform various studies
in order to establish the mean-field theory with the inclusion of the pion,
which was pursued in this paper for medium and heavy nuclei. We have to
definitely introduce the rho meson tensor coupling term, which acts against the
pion finite mean-field. We should include also the delta isobar pion coupling
terms, which favour a finite pion mean-field. We have to study carefully the
effect of the short range repulsions, the so-called g' term in the non-relativistic
framework. We would also like to work out the exchange terms as well as the
Fock terms in the relativistic many-body theory. We then have to work out
the parameter search for the coupling constants in the Lagrangian. We should
further perform the parity projection. Further studies are necessary in order
to establish the occurrence of surface pion condensation. We are at the gate
of exploring both theoretically and experimentally the phenomena caused by
the most essential boson, the pion, in nuclear physics.
At the Japan Hadron Facility (JHF), we expect copious da ta on physics
with strangeness. We expect flavour SU(3) symmetry in various observables.
We shall extend the concept of surface pion condensation to hypernuclei, in
which the role of the pion is expected to be extremely important .
A c k n o w l e d g m e n t s
This is an invited talk presented by H. Toki in International Workshop on
Physics at J H F held through March 14-21, 2002 in Adelaide. H.T. is grateful
to Prof. Thomas for the organization of this workshop and providing exciting
discussions on the physics to be studied at JHF.
R e f e r e n c e s
1. Y. Sugahara and H. Toki, Nucl. Phys. A579, 557 (1994). 2. D. Hirata, K. Sumiyoshi, I. Tanihata, Y. Sugahara, T. Tachibana and H. Toki,
Nucl. Phys. A616, 438c (1997). 3. Y. Akaishi, Cluster Models and Other Topics (World Scientific, Singapore, 1986),
p. 259. 4. J. Carlson and R. Schiavilla, Rev. Mod. Phys. 70, 743 (1998). 5. R.B. Wiringa, S.C. Pieper, J. Carlson, and V.R. Pandharipande, Phys. Rev.
C62, 014001 (2000); S.C. Pieper and R.B. Wiringa, Annu. Rev. Nucl. Part. Sci. 51, 53 (2001).
6. Y. Suzuki and K. Varga, Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems (Springer-Verlag, Heidelberg, 1998), Chap. 11.
7. T. Wakasa et al., Phys. Rev. C55, 2909 (1997). 8. T. Wakasa et al., Phys. Rev. C59, 3177 (1999).
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9. H. Toki and W. Weise, Phys. Rev. Lett. 42, 1034 (1979). 10. E. Oset, H. Toki and W. Weise, Phys. Rep. 83, 281 (1982). 11. H. Toki and I. Tanihata, Phys. Rev. C59, 1196 (1999). 12. E. Shiino, Y. Saito, M. Ichimura and H. Toki, Phys. Rev. C34, 1004 (1986). 13. K. Yoshida and H. Toki, Nucl. Phys. A648, 75 (1999). 14. H. Toki, S. Sugimoto and K. Ikeda, RCNP preprint (2001), nucl-th/0110017. 15. S. Sugimoto, H. Toki, and K. Ikeda, in preparation. 16. R. Brockmann and R. Machleidt, Phys. Rev. C42, 1965 (1990). 17. S. Takagi, W. Watari, and M. Yasuno, Prog. Theor. Phys. 22, 549 (1959);
T. Terasawa, Prog. Theor. Phys. 23, 87 (1960); A. Arima and T. Terasawa, Prog. Theor. Phys. 23, 115 (1960).
EQUATION OF STATE OF QUARK-NUCLEAR MATTER
G. K R E I N A N D V. E. V I Z C A R R A
Instituto de Fisica Teorica, Universidade Est a dual Paulista Rua Pamplona, 145, 01405-900 Sao Paulo, SP, Brazil
E-mail: gkreinQijt.unesp.br; [email protected]
Quark-nuclear matter is a many-body system containing hadrons and deconfined quarks. Starting from a microscopic quark-meson coupling Hamiltonian with a density dependent quark-quark interaction, an effective quark-hadron Hamiltonian is constructed via a mapping procedure. The mapping is implemented with a unitary operator such that composites are redescribed by elementary particle field operators that satisfy canonical commutation relations in an extended Fock space. Application of the unitary operator to the microscopic Hamiltonian leads to effective, hermitian operators that have a clear physical interpretation. At sufficiently high densities, the effective Hamiltonian contains interactions that lead to quark deconfinement. The equation of state of quark-nuclear matter is obtained using standard many-body techniques with the effective quark-hadron Hamiltonian. At low densities, the model is equivalent to quark-meson coupling model with confined quarks. Beyond a critical density, when quarks start to deconfine, the equation of state predicted for quark-nuclear matter is softer than the quark-meson coupling equation of state with confined quarks.
1. Introduction
One of the most exciting open questions in the study of high density hadronic matter is the identification of the appropriate degrees of freedom to describe the different matter phases. For systems with matter densities several orders of magnitude larger than the nuclear saturation density, one expects a phase of deconfined matter composed of quarks and gluons whose properties very likely can be described by perturbative QCD. For ground state nuclei, there is a large body of experimental evidence that their gross properties can be described more economically employing hadronic degrees of freedom, rather than quarks and gluons. On the other hand, for matter at densities not asymptotically higher than the saturation density, like the ones in dense stars and produced in high-energy nuclear collisions, the situation seems to be very complicated, since hadrons and deconfined quarks and gluons can be simultaneously present in the system. Presently, it is not possible to employ QCD directly to study such systems and the use of effective, tractable models are essential for making
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progress in the field. One attractive model to study the different phases of hadronic matter in
terms of explicit quark-gluon degrees of freedom is the quark-meson coupling (QMC) model, originally proposed by Guichon and subsequently improved by Saito and Thomas1. For a list of references on further improvements of the model and recent work, see Ref. [2]. In the QMC model, matter at low density is described as a system of nonoverlapping MIT bags interacting through effective scalar and vector meson degrees of freedom. The effective mesonic degrees of freedom couple directly to the quarks in the interior of the baryons. At very high density and/or temperature, when one expects that baryons and mesons dissolve, the entire system of quarks and gluons becomes confined within a single, big MIT bag.
In a regime of very high density, the description of hadronic matter in terms of nonoverlapping bags should of course break down, since once the relative distance between two bags becomes much smaller than the diameter of a bag, the individual bags loose their identity. The density at which this starts to happen is presently unknown within QCD.
In the present communication we introduce a generalization of the QMC model that allows to include quark deconfinement at high density. Our starting point is a relativistic quark potential model3. From the model quark Hamiltonian, we construct a unitarily equivalent Hamiltonian that contains quark and hadron degrees of freedom. Starting from the Fock space representation of single-hadron states, a unitary transformation is constructed such that the composite-hadron field operators are redescribed in terms of elementary particle field operators in an extended Fock space. When the unitary transformation is applied to the quark Hamiltonian, effective, hermitian Hamiltonians with a clear physical interpretation are obtained4. In particular, one of such effective Hamiltonians describes the deconfinement of quarks from the interior of hadrons. The equation of state of quark-nuclear matter (QNM) can be calculated using standard many-body techniques with the quark-hadron Hamiltonian. We will show that at low densities, the model is equivalent to the QMC model and, beyond a critical density, when quarks start to deconfine, the equation of state predicted for QNM is softer than the QMC equation of state.
2. QMC Model with Confined Quarks
The nucleons are bound states of three constituent quarks. Constant scalar (<T0) and vector (u>o) meson fields couple to the constituent quarks in the interior of the nucleons. Each constituent quark satisfies a Dirac equation of the
276
form
where
-iS • V + 0°m* + 1/2(1 + p)V(rj\ ip(r) = Elr/>(r), (1)
m„ .q-,nq-gla0, E*q = s* - g * w 0 , V(r)=<rr. (2)
The only difference from the model of Toki et al.3 is the form of the potential, while theirs is a harmonic oscillator, ours is a linearly rising one. For a linearly rising potential, the Dirac equation cannot be solved analytically. We use the saddle point variational principle (SPVP)5 to obtain an approximated solution. Since the Dirac Hamiltonian does not have a lower bound for the energy, the traditional nonrelativistic variational method cannot be employed. The SPVP amounts to minimizing (maximizing) the energy expectation value with respect to the variational parameters corresponding to the upper (lower) component of the Dirac wave function. We use as an ansatz for the Dirac wave function5
* ( r ) = ( . - U ( . r )M ) * , (3)
\i<T • rv(r) J
with
u(r) = Ne-x2r2'2, v{r) = if/\ff-Vu(r), (4)
where N is a normalization constant, and A and 7 are the variational parameters. The parameters are found by minimizing the energy eigenvalue e with respect to A and maximizing it with respect to 7. We note here that for a harmonic oscillator potential, the SPVP with this ansatz leads to the exact solution.
Following the traditional path in the QMC model, we initially fix the parameters of the model in vacuum. The nucleon mass in vacuum (<TO = 0 = wo) is given by
Mjv = 3 e - £ 0 A , (5)
where the last term above is used to take into account the era. energy of the three-quark state and other short-distance effects are not taken into account by the confining potential, such as gluon exchanges. £0 is fitted to obtain MJV = 939 MeV. The value of the string tension is taken to be a = 0.203 GeV2
and mq = 313 MeV. With these parameters, the SPVP leads to A = 2.38 frn -1
and 7 = 0.346. The value required for £0 to fit the nucleon mass is 4.67 MeV fm. Next, we proceed to obtain the energy of nuclear matter. Nuclear matter
in the QMC model is modelled as a system of nucleons treated in the mean field approximation, in which the quarks remain confined within the nucleons. The nucleon mass is now obtained as above, but now including the mean fields
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coupled to the quarks. The energy density of symmetrical nuclear matter is given by the traditional expression in the QMC model
E , fkF d3P r,. / N n a 1 2 2 1 2 2 V= J 7MEEN(P)
+^C^oPB + ^maa0--rnuw0, (6)
where E'fc = \ / p 2 + Mjy2; />s is the baryon density; ma and raw are the masses of the mesonic excitations. The next step consists in determining the mean fields. The vector mean field, from its equation of motion, is simply given in terms of pg, and the scalar field is obtained by minimizing E with respect to <TQ, as usual. The coupling constants are then obtained by fitting E to the binding energy of nuclear matter at the saturation density, i.e. E/B - MN = -15.7 MeV at pB = p0 = 0.17 fm"3 (or kF = 1.36 fm"1), where B is the baryon number (in this case B is equal to the number of nu-cleons). Note that for each value of pg, one has to use the SPVP to obtain the in-medium values of A and 7. The values obtained for the coupling constants are g% — 6.1355 and 3</£ = 6.285. The incompressibility is found to be K = 248 MeV.
In the next section we generalize the model to allow for the deconfinement of quarks.
3. QMC Model with Quark Deconfinement
Here we construct an effective Hamiltonian that contains hadron and quark degrees of freedom. The starting point is the quark model discussed in the previous section. In this model, the one-nucleon state can be written in the second-quantized notation as
|a> = Bt |0>, Bi = ^ v l W ^ t ^ t ^ ( 7 )
where the q^'s are constituent-quark creation operators and ip£1'i2'i3 is the Fock space nucleon amplitude. For independent quarks, this is simply a product of three single-quark wave functions. The convention of summing over repeated indices is used throughly. The quark creation and annihilation operators satisfy usual canonical anticommutation relations
{<!V,^} = <W' {«V,^} = {«,J,,«,!.} = 0- (8)
The index a denotes the spatial and internal quantum numbers, such as internal and cm. energies and the spin-isospin quantum numbers of the nucleon. Similarly, the quark indices p. identify the spatial and internal quantum numbers as momentum, spin, flavour and colour. The amplitude ty^^^3 is taken to be orthonormalized:
(a\0) = y*aw'i>*ypi»i» = SaP . (9)
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In the abbreviated notation we are using, the Hamiltonian corresponding to Eq. (1) can be written as
Hq = Tq + Vqq = T (fi) q\qii + -Vqq <jxv; ap) q^Upq^ (10)
where Vqq is a confining potential. Using the quark anticommutation relations of Eq. (8) and the normalization
condition of Eq. (9), one can shown that the nucleon operators, Ba and Ba, satisfy the following anticommutation relations
{Ba,Bl} = Safi-Aa0, {Ba,B0}=O, (11)
where 9
Hap — o * a v ^ 9j/39^3 2 a p iV3qV2q^iii3- \1*)
In addition, one has
{«/..*£} = y/ln^'qlql, ii^Ba] = o. (13)
The term Aap is responsible for the noncanonical nature of the baryon an-ticommutator. This term and the nonzero value of the first commutator in Eq. (13) are manifestations of the composite nature of the baryons and the kinematical dependence of the quark and nucleon operators. This fact complicates enormously the mathematical treatment of many-body systems in which deconfined qur.rks and nucleons are simultaneously present. The mapping formalism of Ref. [4], known as the Fock-Tani (FT) representation6, is a way to circumvent such complications. We will shortly review this formalism in the context of the present model. For further details, and applications for more general models, the reader is referred to Ref. [4].
The basic idea is to extend the original Fock space by introducing fictitious, or ideal nucleons that satisfy canonical anticommutation relations. The unitary operator is constructed in the extended Fock space such that
| a ) = 5 t | 0 ) - ^ t / - 1 | a ) = |a) = 6t,|0), (14)
where the ideal baryon operators 6^ and ba satisfy, by definition, the canonical anticommutation relations
{6a, *£} = *«/», {ba, W = 0. (15)
The state |0) is the vacuum of both q and 6 degrees of freedom in the new representation. In addition, in the new representation, the quark operators gt and q are kinematically independent of the ba and ba
to..*-} = {«,..*!,} = o. (16)
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The unitary operator U can be constructed as a power series in the bound state amplitude \?. The rational for this is clear : in situations the quarks remain confined in the interior of the nucleons, the term Aa/3 plays no role and it can be taken to be zero and the unitary operator becomes trivial6'4. This is the situation for low densities, when the internal structures of the nucleons do not overlap significantly in the system. As the density of the system increases, the quark structures of different nucleons start to overlap. An expansion in powers of VP's offers a power counting procedure to construct the unitary operator.
The effective Hamiltonian is constructed by applying the unitary operator to the microscopic quark Hamiltonian of Eq. (10), -ffe// = U~1HqU. The zeroth-order U is trivial and not interesting. The first-order U brings interesting effects. At this order, we denote the effective Hamiltonian by H^J,, where the superscript (1) means that U has been evaluated up to the first order in $ . H^J. can be written as
H{e)) = Tq + Hb + Vqq + Vqb + ---. (17)
The • • • refer to terms not relevant for our discussion here; Hb = Tb + Vbb, where Tb is a single-nucleon energy and Vbb is an effective nucleon-nucleon interaction without a quark exchange. This term leads to the normal QMC model, in which the many-body system is described by nonoverlapping nucleons - no quark-exchange. In particular, it can describe Fock terms in the QMC model7. The term Vqq contains two-quark and three-quark interactions. It can be shown that if ip is a bound-state eigenstate of the original quark Hamiltonian, Vqq
collapses to
Vqq = 2 V11^V'^ °P) <l\lllp1o ~ EaB\Ba. (18)
It is not difficult to show that this interaction leads to a quark Hamiltonian that has a positive semidefinite spectrum. That is, after the transformation, the resulting Hamiltonian involving only quark operators is unable to bind three quarks to form a nucleon, it describes only states in the continuum. The term Vqb is given by
Vbq = ^=[H(nm;ap)^y^
- H(wap)^pT3A(w2Mpvri)]ql1qlaqlabf)+h.c., (19)
where h.c. denotes hermitian conjugation and A((XVT; crp\) = J^a ^aVT^*apX
is known as the bound-state kernel. If $ is a stationary state of the microscopic quark Hamiltonian, one obtains
Vbq = 0, (20)
280
since A(pvT;<Tp\)V°pX = tfg"7". This result reflects the stability of the baryon bound state to spontaneous decay in the absence of external perturbations. This is clearly the case for a nucleon in vacuum. Also, in the QMC model, when one constructs the unitary transformation U with W that is an eigenstate of the microscopic quark Hamiltonian with the mean fields <r0 and wo in it, the term Vbq continues to be zero, and explicit quark degrees of freedom will not be present in the system at this order of iS.
Now, in a many-body system, the confining quark-quark interaction will become modified due to a variety of effects. Some of such effects, as self-energy corrections from quark loops, can be calculated within the model using standard many-body techniques. However, in a high density system there are other QCD effects that are not captured by the model, such as pair creation and gluonic interactions, that will eventually lead to quark deconfinement.
The formalism we just described allows to include, in an effective way, deconfinement in the QMC model. One generates an effective quark-hadron Hamiltonian as above using \t's that are eigenstates of the QMC Hamiltonian, see Eq. (1). Now, if V(r) is modified such as that it does not lead to absolute confinement, the term Vbq, given by Eq. (19), is not zero. In a mean field perspective, the Hamiltonian of Eq. (17) leads to two Fermi seas, one for baryons and one for quarks. The crucial, and difficult point here, is to obtain the relative abundances of baryons and quarks in the system. This can be evaluated in an approximated way as follows.
Let Z be the fraction of baryons in the system,
Y,Q>lK) = ZB, $>],?„> = (1 - Z)B, (21) a \i
where B is, as in the previous section, the total baryon number. In the mean field approximation - or independent-particle approximation - and for sufficiently small Vbq, Z can be estimated by the perturbative formula
1 + 6 b , (22) V^H0-E,Vbq
where Ho is Tq + Tb, and P = 1 — |6)(6| is a projection operator. In order to evaluate Eq. (22), we postulate a density dependence of the
confining interaction in the form8
V{r)=are-"2r\ (23)
where p is a prescribed function of ps- We use a simple formula for p, such that it is zero for baryon densities below three times the normal nuclear matter density po, and for higher densities it increases linearly with the density as
281
p = PB/%PO — 1- In Fig- (1) we show the potential of Eq. (23) for zero density, and 5 and 10 times the saturation density of normal nuclear matter.
1.0
>
0.75 -
0.5 -
0.25 -
0.0
-
-
-
— //'
1 '
//
- • x PB/PO =
1
yi
.-•—
10
-,.!.,
1 1
PBIPQ =
~ ^ \
i 1
- 1 —
= 5
\
•
-
-
N —
0.0 0.5 1.0
r(fm)
1.5 2.0
Figure 1. The confining potential in vacuum (solid line) and in matter for two different baryon densities.
The energy density of the system can be written as
V J0 —^g E*N{p) + 3g%u0 PB + ^mlal - -m2
uuj2
fk? d3k + 1 2 l <55^<"- (24)
where E*(k) = \/k2 + m*2, and the Fermi momenta kF and kqF are related to
the nucleon density and quark density as
Pb ZPB = 2 ( ^ ) 2 / 3 T T 2 , Pq = (1 - Z)pB = 2(kqF)2/7T2 . (25)
At this point, it is important to notice that since p only starts to operate for densities larger than three times the normal density, the coupling constants g% and g% are the same as before. Of course, for higher densities, there is a somewhat complicated self-consistency problem to be solved, since Z is density dependent. Therefore, in the process of obtaining a, the iterative problem becomes more complicated.
282
1.2
1.1 -
^ 1.0
0.9
1 1
_
1
1 1 ' 1 ' \ / '
/ s / s / s / s / s / s / ^' — / s / s / s / s / / / s / s /y // // /s
/ ""
1 , 1 . 1 ,
4 6
PB/PO
10
Figure 2. Equation of state of quark nuclear matter . The solid line is for matter composed of nucleons only and the dashed line is for matter composed by nucleons and quarks.
In order to proceed, we need Z as a function of pg. It can be calculated numerically with our ansatz wave function given above. The calculation, however, involves multidimensional integrals that must be done using a Monte Carlo integrator. For our purposes here, in this initial investigation we make some approximations. Initially we neglect the lower component of the Dirac spinor. This does not seem to be too drastic an approximation, since 7 in Eq. (4) is a small quantity. In this approximation, one obtains
/ 1 + / d3M3M3fc3|$P(A;i,&2,&3) \F(h
AE(p,kuk2,k3)
with F(q) given by
F{q) = fd3kAV(k)e-k2'x2[e^x2 - l ) ,
(26)
(27)
where $p(&i, Ar2, ^3) is the three-quark wave function of the nucleon with cm. momentum p (see Ref. [4]), AE(p, k\, fo, ^3) is the difference between of the energies of the three unbound quarks and of the three quarks bound in the potential, and AV(k) is the Fourier transform of AV(r), where
AV(r) = <rr(e-f'2r2 -l). (28)
This clearly shows that once p, = 0, i.e. the potential is density independent, one regains the original QMC model.
283
Now, Eqs. (26) and (27) still require a lot of numerical work. We simplify
them further by making two additional approximations. The first one consists
in neglecting the momentum dependence of the energy denominator, and the
second one is to use an average value for q2 in F(q2). Both approximations
taken together seem not to be a bad approximation, since the energy denom
inator under the integral is dominated by low momenta . Now the problem
consists in a single one dimensional integral tha t can easily be performed with
a Gaussian integration method.
In Fig. 2 we present the results for the energy per baryon number, E/B,
as a function of the ratio PB/PO- The solid line in this figure is the result
for the QMC model of the previous section. The dashed line shows tha t the
deconfming of quarks leads to softening of the equation of state. It would be
interesting to investigate the consequences of this softening for neutron star
phenomenology. Soft equations of state seem to be required to explain recent
observational da ta on compact stellar objects.
4. C o n c l u s i o n s
We have generalized the QMC model to include quark deconfinement in mat
ter. The model is based on an effective quark-hadron Hamiltonian obtained via
a mapping procedure from a relativistic microscopic quark Hamiltonian with
a density dependent quark-quark interaction. The equation of s tate of QNM
was obtained using the effective quark-hadron Hamiltonian. It was found tha t
beyond a critical density, when quarks start to deconfine, the equation of state
predicted for QNM is softer than the usual QMC equation of state. Implica
tions of this equation of state for the phenomenology of compact stellar objects
were pointed out.
A c k n o w l e d g e m e n t s
This work partially supported by the Brazilian agencies CNPq and FAPESP.
R e f e r e n c e s
1. P. A. M. Guichon, Phys. Lett. B200, 235 (1988); K. Saito and A.W. Thomas, Phys. Lett. B327, 9 (1994).
2. P.K. Panda, M.E. Bracco, M. Chiapparini, E. Conte and G.Krein, Phys. Rev. C (in press), nucl-th/0205051.
3. H. Toki, U. Meyer, A. Faessler, and R. Brokmann, Phys. Rev. C58, 3749 (1998). 4. D. Hadjimichef, G. Krein, S. Szpigel and J.S. da Veiga, Phys. Lett. B367, 317
(1996); Ann. Phys. (NY), 268, 105 (1998). 5. J. Franklin and R.L. Intemann, Phys. Rev. Lett. 54, 2068 (1985); J.D. Talman,
Phys. Rev. Lett. 57, 1091 (1986).
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6. M.D. Girardeau, Phys. Rev. Lett. 27, 1416 (1971); J. Math. Phys. 16, 1901 (1975). 7. G. Krein, A.W. Thomas and K. Tsushima, Nucl. Phys. A650, 313 (1999). 8. W. Alberico, P. Czerski and M. Nardi, Eur. Phys. J. A4, 195 (1999).
EQUATIONS OF STATE FOR NUCLEAR MATTER A N D QUARK MATTER IN THE NJL MODEL
W. B E N T Z AND T . HORIKAWA
Department of Physics, Tokai University Hiratsuka-shi, Kanagawa 259-1207, Japan
E-mail: [email protected]; [email protected]
N. ISHII
The Institute of Physical and Chemical Research (RIKEN) 2-1, Hirosawa, Wako-shi, Saitama 351-0198, Japan
E-mail: [email protected]
A.W. T H O M A S
Special Research Centre for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, Adelaide University
Adelaide, SA 5005, Australia E-mail: [email protected]
Using the Nambu-Jona-Lasinio model as an effective quark theory, we construct the nuclear matter and quark matter equations of state at zero temperature. The nuclear matter equation of state is based on the quark- diquark description of the single nucleon, and the quark matter equation of state includes the effects of scalar diquark condensation. The phase transition from nuclear matter to quark matter is discussed in this framework.
1. Introduction
In recent works1 the successful quark-diquark description of the single nucleon, which is based on the relativistic Faddeev method2,3, has been extended to describe the equations of state (EOS) of nuclear matter (NM) in the Nambu-Jona-Lasinio (NJL) model4. The essential ingredient to obtain a saturated EOS was the introduction of an infrared cut-off (in addition to the usual ultraviolet one)5 in order to avoid unphysical thresholds for the decay of hadrons into free quarks. The resulting NM EOS has many interesting features, for example a natural suppression of the famous "Z-graph contributions" and a very mild density dependence of the sigma meson mass which is essential for saturation.
285
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In this work we extend this successful description of matter at normal densities to the high density region. Since one expects a phase transition from NM to quark matter (QM) at densities of several times the normal nuclear matter density, we first construct the EOS of QM in the NJL model. Since our description of the single nucleon is based on the interactions in the scalar diquark channel, it is natural to consider the possibility of diquark condensation in QM6. We will discuss the static phase transition from NM to QM, in particular the role played by the pairing interaction. Throughout this work we will limit ourselves to flavour SU(2) and isospin symmetric matter.
2. Model and Nuclear Matter EOS
The Lagrangian of the flavour SU(2) NJL model has the form C — ip (i$ — m) ip + Cj, where m is the current quark mass and JCI a chirally symmetric 4-Fermi interaction. Any 4-Fermi interaction can be decomposed into various qq (meson) and qq (diquark) channels2, and for our purposes we need the interactions in the 0+ , 0~ and l~ meson (m) channels, as well as the 0+ diquark (d) channel :
Ci,m = GV ((^V>)2 - (^(75r)V)2) - Gu ( V W ) 2 (1)
Ci,d = Gs (?(75C) r 2 /?V) (V>T ( C - S ) r2PA^) - (2)
Here flA — \J"&/1 XA (A — 2,5,7) are the colour 3 matrices, and C = 27270-We consider the ratio rs = Gs/Gw as a free parameter, reflecting the form of the original interaction Lagrangian.
Separating the scalar and vector mean field parts from the Lagrangian in Eq. (1) and adding them to the free part, we obtain the mean field Lagrangian :
CMF =^{i$-M- 2GW7"",.) V> - ( M ~ m ) + Guu^, (3)
where M = m — 2Gn(p\ipip\p) is the constituent quark mass and u11 — (/9|V'7MV,|p)- (The "physical value" of M, i.e. the solution of the gap equation, will be denoted by M* for finite density, and by Mo for zero density.) Here \p) refers to the ground state under consideration characterized by the baryon density p, i.e. the vacuum (p = 0), NM or QM.
To construct the nucleon as quark-scalar diquark state, we use Eq. (3) as the one-body part together with the residual interaction in Eq. (2) to calculate the quark-diquark t-matrix (T). If the total system is at rest it has a pole at €N{P) = 6Ga,w° ± \fp1 + Mjy, where the nucleon mass Mjv is a function of the constituent quark mass MN(M). For the present finite density calculation, we will restrict ourselves to the "static approximation"7 to the relativistic Faddeev
287
equation for T, where the momentum dependence of the quark exchange kernel is neglected.
Based on the quark-diquark description of the single nucleon, the EOS of NM in the mean field approximation has been derived in Ref. [1] by using a kind of "hybrid assumption" for the NM ground state, similar in spirit to the model of Guichon and collaborators8. The resulting effective potential V{n), where fj. is the chemical potential for the baryon number, differs from the familiar expression in chiral point-nucleon models9 essentially only by the fact that MN{M) is a non-linear function of the scalar potential $ = Mo — M. (The expression for the energy density can be found in Ref. [1].) Like most other models based on a linear realization of chiral symmetry, the vacuum part of V(fi) is characterized by the "Mexican hat" shape as a function of M. Its curvature decreases rapidly as one moves away from the vacuum state (M = Mo) towards smaller M, which implies a decreasing effective a meson mass and therefore an increasing attraction between nucleons with increasing $ (or p). This leads to an instability of the nuclear matter ground state in point-like nucleon models like the linear cr-model9. In our composite nucleon model, however, the function M^{M) has a finite curvature ("scalar polarizability"10), which can lead to a repulsive contribution to the sigma meson mass and stabilization of the system. However, a sufficiently strong repulsive contribution arises only if confinement effects are incorporated phenomenologically so as to avoid the unphysical thresholds. The simplest method is to introduce an infrared cut-off (A//?) in the framework of the proper time regularization scheme5. By choosing A/.R > 0.1 GeV one obtains a saturated NM EOS characterized by an effective cr-meson mass which is almost independent of the density1.
The stabilization of the system due to the scalar polarizability is demonstrated in Fig. 1, which shows the nucleon mass as a function of $ = Mo — M and the resulting binding energy per nucleon as a function of the density a .
3. Quark Matter EOS and Phase Transition
To construct the EOS of QM, we allow for the possibility that a nonzero scalar diquark condensate develops in one particular direction in colour space, say A — 3, due to the interaction in Eq. (2). The mean field Lagrangian is then
a T h e parameters m, Auv, G„ and r3 are determined by reproducing m* — 0.14 GeV, f„ = 0.093 GeV, Mo = 0.4 GeV, and the nucleon mass in the vacuum Mjvo = 0.94 GeV. ru. = Gu/G„ is fixed so that the binding energy curve passes through the empirical saturation point. In the proper time scheme with \ I R = 0.2 GeV, this leads to m = 0.0169 GeV, Auv = 0.6385 GeV, G„ = 19.60 G e V - 2 , rs = 0.508 and rw = 0.369.
288
Figure 1. Nucleon mass as a function of scalar potential (left) and binding energy per nucleon as function of density (right) for the cases hm = 0.2 GeV and AJR = 0.
given by Eq. (3) supplemented by
SCMF = - \ tyiskfaCvif - ^C^^i^A^) , (4)
where the real field A is defined by
A = -G.iptfinfoCvif - i>TC-lT2i^M\p). (5)
We then introduce a chemical potential for the quark number (/iq — p/3) into CMF and apply the Hartree approximation. For QM at rest the energy spectrum of the quarks is given by eq(p) — \iq — ±J{EV ± (i*)2 + A2, where
Ep = s/p2 + M2, A = sfzj2&, and //* = fiq - 2Gwu°. The effective potential naturally separates into the vacuum quark loop term, mean field terms due to M, w° and A, the Fermi motion part of the quarks, and a contribution which arises from the change of the quark energy spectrum and occupation numbers due to the finite gap.
In the numerical calculations we use the proper time regularization scheme with the same values of m, G^, Auv and AIR = 0.2 GeV as in the NM caseb. However, we will allow for variations of rs and ru in order to show the dependencies.
Before discussing the phase transition, we compare in Fig. 2 the effective quark mass M* in QM for the case without pairing (r, = 0 in QM) to the one in NM. From this figure one can expect that, if there is a transition from NM to QM at several times the normal NM density, QM will already be in the
b We note that the choice AIR = 0, which might be favourable since in QM the quark decay thresholds would be welcome in contrast to the NM case, leads to results for the EOS which are qualitatively similar to AIR = 0.2 GeV, except for the behaviour of A at small fj, see Ref. [11].
289
0.4
0.3
0.1
" 0.0 0.2 0.4 _ 0.6 0.8 1.0
p[fm-3]
Figure 2. Effective quark mass in nuclear matter (NM) and quark matter (QM) as functions of the density.
"chirally restored phase" (M* ~ 0).
4. Phase Transition
The solid line in Fig. 3 shows the P-fi plot for NM, where the density increases along the line starting from the point n = MN0 = 940 MeV. (The vacuum solution P = 0 exists in the range 0 < fi < MN0, but is not drawn in the figure.) In the low density region there is the familiar gas-liquid phase transition, see the insert in Fig. 3. The EOS for QM without the effects of diquark condensation is shown by the dashed lines for the cases rw = Gu/Gn = 0.369 (same as in NM) and rw = 0. They start at // = 3M0 = 1.2 GeV. In the former case the "chiral phase transition" is of second order and therefore not directly visible on this plot, while for the latter case it is of first order and similar to the gas-liquid phase transition in nuclear matter12. The lower dotted line shows the case when M* is set equal to zero by hand for rw = 0. It corresponds to the massless quark gas EOS with the choice BllA = BlJjL = 0.181 GeV for the bag constant. For comparison we also show the EOS for a massless quark gas with B1/4 = B^4
IT = 0.145 GeV by the upper dotted line.
Figure 3 shows that there is no phase transition from NM to QM if one uses the same strength of the vector interaction in QM as determined from the saturation properties of NM. It turns out that this situation does not change
290
>
200
100
0
4 - 3
2 1 0
— - 1 . - 2
9
B=
0 920
'Bm
930
NU
QM
quai
' . . | . . , . | y . . ,
/- * 940 950 /y /
A- •*
y/ / y / -•
/• / r„=0.369 ,-' •k gas y * ,' -y t y / y :
y s r„=o y'
,-•. . . 1 . . . . 1 . . . . -1000 1200
/4MeV] 1600
Figure 3. P — \i plots for nuclear matter (NM) and quark matter (QM). The insert shows the behaviour of the solid line in the low density region. For the explanation of the lines see text.
even for a finite gap in QM, i.e. NM is the ground state for all densities. Since this situation seems unrealistic, one might conclude that the use of a vector interaction in the high density QM phase is physically unreasonable. From Fig. 3 we also see that, for the case rw = 0 in QM, there is still no phase transition, but this situation changes if the effect of the gap is included, as will be discussed below. In the following we will set rw = 0 in QM and investigate the effect of the diquark condensation.
Figure 4 shows the P—fi plots for several values of the pairing strength r, in QM in comparison to the NM case. We see that diquark condensation favours the NM —> QM phase transition, but r„ in QM should not be larger than ~ 0.3: For rs = 0.3 the NM phase would be stable only up to p = 0.45 fm~3, and for rs = 0.4 the QM phase would be the ground state for all densities. Also, the gap becomes unreasonably large for the case rs > 0.3, see Fig. 5. This figure shows that the trend towards a chiral phase transition in QM persists also for finite gap, although the corresponding density region is shifted upwards, and that the gaps are typically of the order of 100-300 MeV, which is large
291
300
200
I
6 *t-i
>
S 10° PL.
0
800 1000 1200 1400 1600
fj. [MeV]
Figure 4. P — /x plots for nuclear matter (NM) and quark matter (QM) for various pairing strengths rs.
compared to the results obtained in the 3-momentum sharp cut-off scheme0. From this we conclude that the strength rs — 0.508 determined by reproducing the experimental nucleon mass in the pure quark-scalar diquark model is too strong to be used as the pairing strength in QM. We have to note, however, that rs = 0.508 is probably an overestimate, since pion cloud13 and/or axial vector diquark14 contributions will reduce the value of rs needed to reproduce the nucleon mass.
In order to discuss the NM -» QM phase transition including diquark condensation, we use rs = 0.25 in QM as an example. The path of the system in the P-\i plane (as shown in Fig. 4) is the following: (i) Vacuum (P = 0) up to the saturation point of NM, (ii) NM up to the transition point, and (iii) QM. The corresponding p — p. and M* — p (or A* — p) plots are shown in Fig. 6, where for any fixed p only the quantities in the stable phase (VAC, NM or QM) are indicated. The density jump during the NM - • QM phase transition is quite large. The P-p and S-p plots are shown in Fig. 7, where the dashed lines indicate the mixed phases.
c The momenta which give the most important contributions to the gap equation are p ~ fi*, and in the density region of interest this is very similar to the UV cut-off. A sharp 3-momentum cut-off therefore eliminates a part of the peak around p ~ fi*, while the proper time scheme corresponds to a smooth cut-off function.
292
f i i a
0.3
0.2
0.1
. . . .
\̂ \\ v N J \ ••-; \ S i
-
V\
. . . . ! . . , .
' ' ' ' 1 ' ' ' ' 1 ' ' ' ' .
ini
- - - QM (r,-O.Z) -
on (r>»o.a) ' QM (r,-0.4)
v-\^^ -
V-, -
. r : : n S f f i " 5 -^»»?»^
>
"<
0.3
0.2
0.1
i • • • • i • • • • r •
OH ( r , - O . Z )
OH ( r . - 0 . 3 )
QM (r.-Q.4) _ . . - •• ' "
. / . . 1 . . . . 1 . . . . 1 .
. , . , . . . . ...'
. . - • - " " "
--""" :
0.0 0.2 0.4
[ t a-r 0.8 1.0 0.0 0.2 0.4
P I'm 3J 0.0 1.0
Figure 5. Effective quark mass (left) and gap (right) in quark matter as functions of the density for various rs.
G 0.7i
-
-
VAC
1 ' ' ' ' 1 '
r.(0JI)-0.2S
1 • . . . 1 .
, , , , . , , , ,
NU
. . . 1 . . . . 1
/ o n
,
-
"
!
a
400
300
200
100
M*(VAC)
. . . , . . . . , . . . . ,
V
M"(M(P" "
r,(qu)-0.20 '
"VftM)
-
Ktw 1300 1400 BOO 000 1000 1100 1200 1300 1400 1100
/* [MeV]
Figure 6. Density (left) and M*, A* (right) of the stable phases (vacuum, nuclear mat ter or quark matter) as functions of the chemical potential; r s =0.25 is used in QM.
- r.(Qll)-0.2S
-
• • VAC->IOI y
AM
. 1 . .
' 1 ' ' ' ' 1 '
NM->QM y
. . 1 . . . . 1 .
' ' ' 1 ' '
/<W
, . . ! . .
/ -
-
1500
1000
600
r.(QK)-0.26
-
, .-'VAC->mi
1 " " 1 ' '
, . ' ' 'H«->QII
, . | . -
-
-
0.00 0.26 0-60 0.75 _ ,
P ['Hi"3] 1.00 1.25 1.60 0.00 0.21 0.60 r0«__n
p [fm 3] 1.00 1.S0 1.S0
Figure 7. Pressure (left) and energy density (right) of the various phases as functions of the density; r s =0 .25 is used in QM.
5. Summary
In this work we used the NJL model as an effective quark theory to describe the single nucleon, NM and QM including the effects of diquark condensation.
293
We focused on the possibility of a phase transition NM -> QM. Our results can
be summarized as follows: (i) The strength of the vector interaction in QM has
to be reduced drastically as compared to the NM case, where it can be related
to the saturat ion properties, in order to make the phase transition possible.
(ii) Diquark condensation favours the phase transition, but reasonable values
for the strength of the pairing interaction are only about half of those derived
from the pure quark-scalar diquark model of the single nucleon. (iii) For any
reasonable scenario, the quark phase is already in the chiral symmetric, colour
broken phase (M* ~ 0, A* > 0) at the transition density.
A c k n o w l e d g e m e n t s
This work was supported by the Australian Research Council, Adelaide Uni
versity, and the Grant in Aid for Scientific Research of the Japanese Ministry of
Education, Culture, Sports, Science and Technology, Project No. C2-13640298.
R e f e r e n c e s
1. W. Bentz and A.W. Thomas, Nucl. Phys. A696, 138 (2001). 2. N. Ishii, W. Bentz and K. Yazaki, Nucl. Phys. A578, 617 (1995). 3. U. Zuckert, R. Alkofer, H. Weigel and H. Reinhardt, Phys. Rev. 55, 2030 (1997). 4. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1960); 124, 246 (1961). 5. G. Hellstern, R. Alkofer and H. Reinhardt, Nucl. Phys. A625, 697 (1997);
D. Ebert, T. Feldmann and H. Reinhardt, Phys. Lett. B388, 154 (1996). 6. M. Afford, K. Rajagopal, and F. Wilczek, Phys. Lett. B422, 247 (1998);
O. Kiriyama, S. Yasui and H. Toki, Int. J. Mod. Phys. E10, 501 (2001). 7. A. Buck, R. Alkofer and H. Reinhardt, Phys. Lett. B286, 29 (1992). 8. RA.M. Guichon, Phys. Lett. B200, 235 (1988);
P.A.M. Guichon, K. Saito, E. Rodionov and A.W. Thomas, Nucl. Phys. A601, 349 (1996).
9. W. Bentz, L.G. Liu and A. Arima, Ann. Phys. 188, 61 (1988). 10. M.C. Birse, Phys. Rev. C51, R1083 1995);
S.J. Wallace, F. Gross and J.A. Tjon, Phys. Rev. Lett. 74, 228 (1995). 11. W. Bentz, T. Horikawa, N. Ishii and A.W. Thomas, to be published. 12. M. Buballa, Nucl. Phys. A611, 393 (1996). 13. N. Ishii, Phys. Lett. B 431, 1 (1998). 14. H. Mineo, W. Bentz, N. Ishii and K. Yazaki, Nucl. Phys. A703, 785 (2002).
COLOUR SUPERCONDUCTIVITY IN DENSE QCD A N D STRUCTURE OF COOPER PAIRS
H. ABUKI AND T. HATSUDA Department of Physics, University of Tokyo, Tokyo 113-0033, Japan E-mail: [email protected]. ac.jp; hatsuda&phys.s.u-tokyo. ac.jp
K. I T A K U R A
RIKEN BNL Research Center, BNL, Upton, NY 11973, USA E-mail: [email protected]
Two-flavour colour superconductivity is examined over a wide range of baryon densities within a single model. To study the structural change of Cooper pairs, quark correlations in colour superconductor is calculated both in momentum space and in coordinate space. At extremely high baryon density (~ O(1010po))i our model becomes equivalent to the usual perturbative QCD treatment and the gap is shown to have a sharp peak near the Fermi surface due to the weak-coupling nature of QCD. On the other hand, the gap is a smooth function of the momentum at lower densities (~ O(10po)) due to strong colour magnetic and electric interactions. The size of the Cooper pair is shown to become comparable to the averaged inter-quark distance at low densities, which indicates that a crossover from a BCS state to Bose-Einstein condensate of tightly bound Cooper pairs may take place at low density.
1. Introduction
Because of asymptotic freedom and Debye screening in QCD, deconfined quark matter is expected to be realized for baryon densities much larger than the normal nuclear matter density1. This weak-coupling fermionic matter is, however, qualitatively different from a perturbed free Fermi gas system, if the temperature is low enough. This is because any attractive quark-quark interaction in the cold quark matter causes an instability of the Fermi surface due to the formation of Cooper pairs and leads to the colour superconducting phase2 '3,4.
Current understanding of colour superconductivity has been based on two different theoretical approaches. One is an analysis of Schwinger-Dyson equations with a perturbative one-gluon exchange, which is valid in the weak coupling limit at asymptotically high densities5. The dominant contribution to the formation of Cooper pairs comes from collinear scattering through the long range magnetic gluon exchange6. In such a weak-coupling regime, formation
294
295
of Cooper pairs takes place only in a small region near the Fermi surface. The second approach is the mean-field approximation with the QCD-inspired 4-fermion model introduced to study lower density regions3'7. In this approach, the magnitude of the gap becomes as large as 100 MeV and is almost constant in the vicinity of the Fermi surface.
The main purpose of this contribution is to discuss the superconducting gap over a wide range of baryon densities within a single model and to make a bridge between high and low density regimes8. To this aim, we make an extensive analysis on the structural change in spatial-momentum dependence of a Cooper pair in 2-flavour colour superconductivity at zero temperature. The momentum dependence of the gap, diffuseness of the Fermi surface, the quark-quark correlations in the superconductor are the characteristic quantities reflecting the departure from the weak-coupling picture. In particular, we show that the spatial size of Cooper pairs which is calculated from the quark-quark correlations, indicates a clear deviation from the weak-coupling BCS theory. A more complete and detailed analysis is given in Ref. [8].
2. Gap Equation with Spatial Momentum Dependence
In the standard Nambu-Gor'kov formalism for a two component Dirac spinor \p = (^ , ^ ' ) , the quark self-energy E(fcp) with Minkowski 4-momentum A; p
satisfies J4
where Da£v is the gluon propagator in medium, S(qp) the full quark propagator, and T£ the quark-gluon vertex. We study the gap function in the flavour antisymmetric, colour anti-symmetric and Jp — 0 + channel (the most attractive channel within the one-gluon exchange model)5:
A(*„) = (A2r2C75) (A+(*p)A+(fc) + A_(fcp)A_(fe)) , (2)
where r2 is the Pauli matrix acting on the flavour space, A2 a colour anti-symmetric Gell-Mann matrix, and C the charge conjugation operator. A±(fc) = ( l i f e - a ) / 2 is the projector on positive (+) and negative ( - ) energy quarks.
For the vertex in Eq. (1), we use T£ = diag(7^Aa/2, -(7 MA a /2) f) . For g2
in Eq. (1), we use a momentum dependent coupling g2{q,k) in the "improved ladder approximation"9 (ft, = (11JVC - 2iV»/3 = 29/3):
296
where p2 plays a role of a phenomenological infrared regulator which prevents the coupling constant from being too large at low momentum q, k ~ AQCD-
At high momentum, g2 shows the same logarithmic behaviour as the usual running coupling with A identified with AQCD- The quark propagator in the improved ladder approximation in vacuum is known to have a high momentum behaviour consistent with that expected from the renormalization group and the operator product expansion. We adopt A=400 MeV and p%=1.5 A2 which are determined to reproduce the low energy meson properties for the Nj = 2 vacuum10.
The gluon propagator in Eq. (1) in the Landau gauge reads,
/?„„(*/>; *o < |*|) = - .2^. ,ff in/ i i , i - wf~r> (4)
k +iM2\k0\/\k\ fc +mJ3 where Pj^ are the transverse and longitudinal projectors. The longitudinal part of the propagator (the electric part) has static screening by the Debye mass rrijy = (Nj /2-K2)g2fj,2, while the transverse part (the magnetic part) has dynamical screening by Landau damping M2 = (7r/4)m2
3. This form is a quasi-static approximation of the full gluon propagator in the sense that only the leading frequency dependence is considered u .
2.1. Full Gap Equation
The gap equation with the approximations shown above is obtained from the 2-1 element of the Schwinger-Dyson (matrix) equation, Eq. (1), which reads
A±(*) = [°° dqV±{q,k;e+e±)-=^±M= Jo q' feV^+(?)2 + |A+(?)l2
+ [ dqV^Sq^-^-^t) , A ~ ^ (5) Jo ^ q * V^-(ff)2 + |A_(g)|2 K)
Here we used a simplified notation: A±(efc ,k) -* A±(fc) with E±(q) = q =F fi and ejr being the quasi-particle energy as a solution of (e*)2 = E"±{q) + A\(e^, q). The explicit form of V± is given in Ref. [8].
At extremely high density, the Cooper pairing is expected to take place only near the Fermi surface due to the weak coupling property. In this case, we can safely neglect the antiquark-pole contribution for calculating A + . Furthermore, one may replace the momentum dependent coupling constant in Eq. (3) by that on the Fermi surface g2(fi,fi). On the other hand, at low densities, sizable diffusion of the Fermi surface occurs and the weak-coupling approximation is not justified, and we need to solve the coupled gap equations, Eq. (5), numerically.
297
2.2. Occupation Number, Correlation Function and Coherence Length
To clarify the structural change of the colour superconductor from high to low densities, it is useful to examine the following physical quantities : (I) The quark and antiquark occupation number which is related to the diagonal (1-1) element of the quark propagator, (V>j (t,y)ii>i{t,x))supeT, in the Nambu-Gor'kov formalism;
ni'2(" " i ('" T S P T E w ) • 4<,) = 9("""• "S-(,) = ° ' ( 6 )
where the superscripts (1 ,2 and 3) stand for colour indices. Since the third axis in the colour space is chosen to break colour symmetry, quarks with the third colour remains ungapped. (II) The q-q and q-q correlation functions in momentum space <f>±{q) and in coordinate space <p± (r): They reflect the internal structure of the Cooper pairs in the colour superconductor. These correlations are defined through the off-diagonal (1-2) element of the quark propagator, {tpf(t,x)^j(t,y))SUper',
<P±(q) = — A±^ . <P±(r) = N f T ^ 4 X <p±(q) eiqr, (7) 2 ^ ± ( ? ) 2 + | A ± ( 9 ) P ' ^ ± l > J (2TT)3 ^ W ' ^ '
where N is a normalization constant determined by f d3r\ip+(r)\2 = 1.
(III) The coherence length £c characterizing the typical size of a Cooper pair: It is defined simply as the root mean square radius of ip+ (r):
Jd\ r 2 | y + ( r ) | ' f0°° dk fc2 \dy+(k)/dk\2
^ f<Pr\r+{r)\* tfdkVtf+ik)? ' U
It can be shown8 that the quark correlation <p+ (r) in the weak-coupling limit behaves as
where the Pippard length is given by £p = (irA+(fi))~1. In a typical type-I superconductor in metals, the Pippard length is of the
semi-macroscopic order £p ~ 10~4 cm, whereas the inverse Fermi momentum is of microscopic order kp1 ~ 10~8 cm. Besides, there is another scale uijy, the Debye cutoff, which limits the range of attractive interactions to be just around the Fermi surface \k — fcp| < WQ. The inverse of the Debye cutoff is in between the two scales above: w^1 ~ 10 - 6 cm. Therefore there is a clear scale hierarchy, A < C U D C kp. On the other hand, since there is no intrinsic scale WD in QCD, scale hierarchy at extremely high density simply reads A ~ pe~c'3 <C kp ~ p.
298
At lower densities, however, such scale separation becomes questionable for coupling constants which are not small.
3. Momentum Dependent Gap from High to Low Densities
In Fig. 1(a), we show the gap A+(k) as a solution of the full gap equation for a wide range of densities. Since the actual position of the Fermi surface moves as we vary the density, we use k/fi as a horizontal axis in the figure which helps us to understand the change of global behaviour. The figure shows that the sharp peak at high density gradually gets broadened and simultaneously the magnitude of the gap increases as we decrease the density.
The characteristic features at high density are: (i) There is a sharp peak at the Fermi surface, and (ii) The gap decays rapidly but is nonzero for momentum far away from the Fermi surface. The property (i) is similar to the standard BCS superconductivity but (ii) is not, due to the absence of an intrinsic ultraviolet Debye-cutoff of the gluonic interaction in QCD. As for the magnitude of the gap at high density, if one estimates the several contributions to the kernel V± in the gap equation, Eq. (5), separately one finds that the colour-electric interaction enhances the gap considerably. This may be understood as follows. In coordinate space, the Debye-screened electric interaction is a Yukawa potential. Such a short-range interaction alone can form only loosely bound Cooper pairs and a very small gap. However, if the magnetic and electric interactions coexist, small size Cooper pairs are formed primarily by the long-range magnetic interaction. Then, even the short-range electric interaction becomes effective to generate further attraction between the quarks.
The characteristic features at low density are : (iii) The sharp peak at the Fermi surface disappears, and (iv) All the contributions neglected in the weak-coupling limit are actually not negligible. A close look at the effects of each contribution to the gap leads us to the conclusion that the colour superconductivity at low density is not a phenomenon just around the Fermi surface. This is confirmed by computing the occupation number following Eq. (6). The result shows that the Fermi surface is substantially diffused at low density.
In Fig. 1(b), the gap at the Fermi surface A+(/t) is shown as a function of the chemical potential. It decreases monotonically as fi increases as is shown on the figure, but increases at much higher density ft > 106 MeV. An analytic solution based on the weak coupling approximation (includes only the magnetic gluons, fixed coupling, and ignores the antiquark pole contribution) is also shown in Fig. 1(b). The magnitude of the analytic solution is normalized to the numerical solution at the highest density (i = 212A ~ 1.6 x 106 MeV. At high density, the /x-dependence of the numerical result is in good agreement with an
299
0 1 2 3 , „ » 1 0 « 1 0 » 1 0 «
Figure 1. (a) A+ (A;) as a function of &/M for various densities n = 2™ A with n = 1,2,3,12. (A =400MeV). All the calculations are done with the momentum dependent vertex g2(q, k) and with the antiquark-pole contribution, (b) Chemical potential dependence of the gap A+ (k = //) in the full calculation compared with the analytic result which is normalized at the highest density n = 2 A.
analytic form with a parametric dependence A+(//) oc g~5(iexp(—Sir2/\/2g) with <72 = g2(fi,fi). On the other hand, the difference between the two curves at low density implies the failure of the weak-coupling approximation.
4. Correlation Function and Coherence Length
One of the advantages of treating the momentum dependent gap is that we are able to calculate the correlation function which physically corresponds to the "wavefunction" of the Cooper pair. Such correlations have first been studied in Ref. [12] in the context of colour superconductivity (but with a much simpler model than Eq. (5), and only at lower density).
Figure 2(a) shows the coherence length £c of a quark Cooper pair defined as the root mean square radius of the correlation function [see Eq. (8)]. The size of the Cooper pair becomes smaller as we go to lower densities. This tendency is understood by the behaviour of the Pippard length £p = l/jrA+(/i) (which gives a rough estimate of the coherence length) together with the behaviour of A+(/i) shown in Fig. 1(b).
Also, the Cooper pair becomes smaller as density increases beyond fi = 212A. However, it does not necessarily imply the existence of tightly bound Cooper pairs. In fact, the size of a Cooper pair makes sense only in comparison to the typical length scale of the system, namely the averaged inter-quark
300
10 10 /j[MeV]
10 10 fi[MeV]
Figure 2. (a): Density dependence of the coherence length, (b): Ratio of the coherence length £c and the average inter quark distance dq as a function of the chemical potential.
distance dq for free quarks defined as
dq = 7 T 2 \ 1 / 3 1
(10)
As we go to higher densities, the ratio £c/dq increases monotonically as shown in Fig. 2(b). Namely, loosely bound Cooper pairs similar to the BCS superconductivity in metals are formed at extremely high densities. (Recall that the typical ratio in superconductivity is kp/A ~ 104.)
At the lowest density in Fig. 2, the size of the Cooper pair is less than 4 fm and the ratio £c/dq is less than 10. This is not similar to the usual BCS system. The transition from £c/dq S> 1 to £c/dq ~ 1 as \i decreases is analogous to the crossover from the BCS-type superconductor to the Bose-Einstein condensation (BEC) of tightly bound Cooper pairs13. Our result here suggests that quark matter possibly realized in the core of neutron stars may be rather like the BEC of tightly bound Cooper pairs.
For better understanding of the internal structure of the quark Cooper pair, let us consider the correlation function in coordinate space. Figure 3 shows the spatial correlation of a Cooper pair at various chemical potentials normalized a s / d ^ + O O I ^ l .
At high density, most of the quarks participating in forming a Cooper pair have the Fermi momentum fcp = \i giving a sharp peak in the momentum space correlation. In the coordinate space, this corresponds to an oscillatory distribution with a wavelength A = l//i without much structure near the origin. (The oscillation is also evident from the factor sin(pr) in the approximate correlation function in Eq. (9).) At lower densities, accumulation of the correlation
301
o X
Figure 3. The quark-quark correlation function <p+(r) in the coordinate space y + ( r ) for several different chemical potentials. <p+(r) is normalized to be unity.
near the origin in coordinate space is much more prominent in Fig. 3. This implies a localized Cooper pair composed of quarks with various momentum.
5. Summary and Discussion
We have studied the spatial-momentum dependence of a superconducting gap and the structure of Cooper pairs in two-flavour colour superconductivity, using a single model in a very wide region of density. A nontrivial momentum dependence of the gap manifests itself at low densities, where relatively large QCD coupling allows the Cooper pairing to take place in a wide region around the Fermi surface. Note that our results can be easily extended to the Nj = 3 case. Our results imply that quark matter which might exist in the core of neutron stars or in the quark stars could be rather different from that expected from the weak-coupling BCS picture and could be more like a BEC of tightly bound Cooper pairs. A study of the finite T phase transition of this strong coupling system is currently under investigation.
Acknowledgments
T.H. would like to thank A. W. Thomas, A. G. Williams, D. Leinweber and the members of the local organizing committee for giving him the opportunity to give a talk at the Joint CSSM/JHF Workshop on Physics at Japan Hadron Facility (March 14-21, Adelaide, 2002). This research was supported in part by the National Science Foundation under Grant No. PHY99-07949.
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G. W. Carter and D. Diakonov, Phys. Rev. D60, 016004 (1999). 8. H. Abuki, T. Hatsuda and K. Itakura, Phys. Rev. D65, 074014 (2002). 9. K. Higashijima, Phys. Rev. D29, 1228 (1984); Prog. Theor. Phys. Suppl. 104, 1
(1991). V. A. Miransky, Sov. J. Nucl. Phys. 38, 280 (1983). 10. See e.g., T. Ikeda, Prog. Theor. Phys. 107, 403 (2002); hep-ph/0107105 and
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HADRON PROPERTIES IN NUCLEAR MEDIUM A N D THEIR IMPACTS ON OBSERVABLES
K. TSUSHIMA Department of Physics and Astronomy, University of Georgia
Athens, GA 30602, USA E-mail: [email protected]
The effect of changes in hadron properties in a nuclear medium on physical ob-servables is discussed. Highlighted results are: (1) hypernuclei, (2) meson-nuclear bound states, (3) K-meson production in heavy ion collisions, and (4) J/<i! dissociation in a nuclear medium. In addition, results for the near-threshold w- and ^-meson production in proton-proton collisions are reported.
1. Hadrons in Nuclear Medium: Treatment in QMC
Here we discuss hadrons in a nuclear medium in the quark-meson coupling (QMC) model1. The model has been extended and successfully applied to many problems2 '3 '4,5 '6 '7 '8. A detailed description of the Lagrangian density, the mean-field equations of motion, and the treatment of finite nuclei are given in Refs. [2,3]. As examples, we show in Fig. 1 results for the 40Ca nucleus in QMC3, which is based on the quark structure of nucleon or nucleus.
*°CCL single particle energies prattma nmrttrona
Exp. QHD QMC Exp. QHO QUO J
0 1 2 3 4 5 6 r(fm)
Figure 1. Charge density distribution (the left panel) and energy spectrum (the right panel) for the 4 0 Ca nucleus, compared with the experimental data and predictions using Quantum Hadrodynamics (QHD, also labelled by "Walecka-Serot")10.
The Dirac equations for quarks and antiquarks in hadron bags (q = u,u, d or d, hereafter) neglecting the Coulomb force, are given by (|as| < bag radius)2 '3 '4 '5 '6 '7 '8 :
il • dx - (m, - V§) =F 7° (v* + \v^j
c-.«
f £
-so
-30
—*o
-SO
-60
\M*)J (i)
303
304
[*7 • dx - rnS:C] ipSiC{x) (or ipj,e(x)) = 0. (3)
The mean-field potentials for a bag in nuclear matter are defined by V§=g%a, v2= 9%u and V^=gjb, with g%, g* and gj the corresponding quark-meson coupling constants. The bag radius in medium for a hadron ft, R*h, will be determined through the stability condition for the mass of the hadron against the variation of the bag radius1,2 (see Eq. (4)). The hadron masses in a nuclear medium m*h (free mass will be denoted by m/j), are calculated by
v njWj-ZH 4 3 dml
j=q,q,Q,Q
= 0, (4) Rh—R^
where Q* = SI* = [x2g + ( ^ m ^ ) 2 ] 1 / 2 (q = u,d), with m* = mq-g$a,
fig = ft^- = [x^ + (^mQ) 2 ] 1 / 2 (Q = s, c), and X9IQ being the bag eigen-frequencies. B is the bag constant, nq(riq) and TIQ^-Q) are the lowest mode quark (antiquark) numbers for the quark flavours q and Q in the hadron ft, respectively, and the ZH parametrize the sum of the centre-of-mass and gluon fluctuation effects (assumed to be independent of density). The parameters are determined in free space to reproduce the corresponding masses. We chose the values (mg,ms,mc) = (5,250,1300) MeV for the current quark masses, and -R/v = 0.8 fm for the bag radius of the nucleon in free space. The quark-meson coupling constants, g%, g% and g^, are adjusted to fit the nuclear saturation energy and density of symmetric nuclear matter as well as the bulk symmetry energy2. However, in the studies of the kaon system, we found that it was phenomenologically necessary to increase the strength of the vector coupling to the non-strange quarks in the K+ (by a factor of 1.42, i.e. gq
Ku = 1.42#2,) in order to reproduce the empirically extracted A'+-nucleus interaction5. We assume this also for the D and D mesons7,8,9. The scalar (Vj1) and vector (Vv
h) potentials felt by the hadrons ft, in nuclear matter, are given by
Vsh = ml - mh, Vv
h = (n, - nq)V* - hV« (V« -> 1.42Vtf for K,~K, D,D),(S)
where I3 is the third component of isospin projection of the hadron ft. In Fig. 2 we show effective masses and mean field potentials for various
hadrons in symmetric nuclear matter. Several comments on the results shown in Fig. 2 are in order :
(1) Physical 7/ and 7/ mesons are treated including a mixing angle, 8p — — 10°, as (77,77') = (r]scos9p — rjisindp, rig sin 6p +rji cosOp), where 771 = (l/\/3)(«w + dd+ ss) and 778 = (1/V /6)(MU + dd — 2ss).
305
Figure 2. Effective mass ratios and mean field potentials for hadrons in nuclear mat ter (po = 0.15 f m - 3 ) . "H" stands for the H particle with an input mjj — 2m\.
(2) The H particle is treated as a 6-quark bag, uuddss, with ra# = 2mA-(3) The scalar potential for the hadron h, Vs
h, shows a universal quark number scaling rule, Vs
h/VSN ~ (nq + n ? ) /3 , where V^ is the scalar
potential for the nucleon. (See Eq. (5).) (4) The scalar potential for the <j> meson arises entirely from the (j> — u
mixing in QMC, and is tiny. (5) Reduction of the effective mass of A+ implies the role of the light
quarks in partial restoration of chiral symmetry in a heavy-light quark system, and merits further studies.
In following sections we will discuss how these changes in hadron properties in a nuclear medium impact physical observables.
306
2. Hypernuclei
When a hyperon, Y, is embedded in a nucleus, the potential felt by the hyperon at a given position of the nucleus can be calculated self-consistently4. Simultaneously one can also get energy levels for the hyperon Y in a shell-model calculation4. Because the treatment is based on the quark degrees of freedom, we need to consider the possibility of a Pauli blocking effect at the quark level, and also the effect of E-A channel coupling for the S- and A-hypernuclei. We included these effects in a phenomenological way4. Nevertheless, we can study all A-, £-, S-hypernuclei in a systematic way, due to the quark degrees of freedom, because the hyperon-meson coupling constants in nuclear medium are automatically determined4. In Table 1 we present QMC predictions for the energy levels of hyperons in various hypernuclei. Results for other heavier mass hypernuclei are given in Ref. [4].
Table 1. Energy levels for hyperons Y (in MeV), for ^O, ^ C a and ^?Ca hypernuclei, calculated in QMC. Experimental data are taken from Ref. [11].
l S l / 2
1P3/2
l P l / 2
l S l / 2
1P3/2
l P l / 2
14/2 2 s l / 2
1^3/2
1*1/2
1P3/2
l P l / 2
14/2 2 « l / 2
14/2 1/7/2
™0 (Expt.)
-12.5
-2.5 (lp) ^°Ca (Expt.)
-20.0
-12.0 (lp)
—
X'o -14.1 -5.1 -5.0 " C a -19.5 -12.3 -12.3 -4.7 -3.5 -4.6
4 9 Ca
-21.0 -13.9 -13.8 -6.5 -5.4 -6.4
—
J?-° -17.2 -8.7 -8.0
*LCa -23.5 -17.1 -16.5 -10.6 -9.3 -9.7
*?_Ca -19.3 -11.4 -10.9 -5.8 -6.7 -5.2 -1.2
17 O
-9.6 -3.2 -2.6
4 1 Cn
-13.4 -8.3 -7.7 -2.6 -1.2 -1.9
4 9 C a
-14.6 -9.4 -8.9 -3.8 -2.6 -3.1
—
17 O
-3.3
— —
4 1 f a
-4.1
— — — — —
4 9 P a
-11.5 -7.5 -7.0 -2.0
— -1.2
—
i7_o -9.9 -3.4 -3.4
tlCa -17.0 -11.2 -11.3 -5.5 -5.4 -5.6
i9_Ca -14.7 -8.7 -8.8 -3.8 -4.6 -3.8
—
17 O
-4.5
— —
i^Ca -8.1 -3.3 -3.4
— — —
itt.Ca -12.0 -7.4 -7.4 -2.1 -1.1 -2.2
—
3. Meson-Nucleus Bound States
Next, we discuss the meson-nuclear bound states. We have solved the Klein-Gordon equation for mesons j (j = w, 77,rf, D, D) with almost zero momenta, using the potentials calculated in QMC6,7 :
[V2 + Ef - fhf(r)] <f,j(r) = 0, E* = Ej + mj - iTj/2 (6)
307
m*{r) = m){r) - %- [(mj - m*(r))7 j + Tj) = m*(r) - '-Tfr), (7)
where Ej is the complex valued, total energy of the meson, and we included the widths of the mesons in a nucleus assuming a specific form using */j, which are treated as phenomenological parameters. We calculate the single-particle energies for the values j u = 0.2, and 7, = 0.5, which are expected to correspond best to experiments6, while for the 77', D and U, the widths H = 0 are assumed7. For a comparison we present also results for the w calculated using the potentials obtained in QHD12. Results are given in Tables 2 and 3.
Table 2. Calculated u-, T)- and ^'-nuclear bound state energies (in MeV), E-j = Re(E* - m}) (j = w, r;, ri'), in QMC 6 and those for the w in QHD with cr-w mixing effect 1 2 . The complexeigenenergiesare given by, E* = Ej+m,j —iVj/2. (* not calculated)
?He ! ' B
fMg
l e O
fCa
?°Zr
208 p b
I s
I s
I s
I P 2s
I s
l p I s
l p 2s
I s
I P 2s
I s
l p 2s
IT, = 0.5
Er, -10 .7
-24.5
-38 .8
-17 .8
— -32.6
-7 .72
-46 .0
-26 .8 -4 .61
-52.9
-40 .0 -21 .7
-56 .3
-48 .3
-35.9
(QMC) r„
14.5
22.8
28.5
23.1
— 26.7
18.3
31 .7
26.8 17.7
33.2
30 .5 26.1
33.2
31 .8
29.6
( Q M C )
* V * * * * *
-41 .3
-22 .8
-51 .8
-38.5
-21.9
-56.0
-47 .7
-35.4
-57.5
-52.6
-44.9
lw = 0.2
fcjtjj
-55 .6
-80 .8
-99 .7
-78.5
-42 .8
-93.4
-64 .7
-111
-90 .8
-65.5
-117
-105
-86.4
-118
-111
-100
( Q M C )
rw 24.7
28.8
31.1
29.4
24 .8
30 .6
27 .8
33 .1
31 .0 28.9
33.4
3 2 . 3 30 .7
33 .1
32.5
31.7
•yu = 0.2
hiuj
-97.4
-129
-144
-121
-80 .7
-134
- 1 0 3
-148
-129 -99 .8
-154
- 1 4 3 - 1 2 3
-157
-151
-139
( Q H D )
r „ 33 .5
38.5
39 .8
37 .8
33 .2
38 .7
35 .5
40 .1
38 .3
35 .6
40 .6
3 9 . 8
38 .0
40 .8
40.5
39.5
Table 3. D~, D° and D° bound state energies (in MeV). The widths are all set to zero,
s t a t e
I s
l p 2s
D - ( 1 . 4 2 V J )
-10.6
-10.2
-7 .7
D~(V2) -35.2
-32.1
-30.0
£ > - ( V j , n o C o u l o m b )
-11.2
-10.0
-6 .6
£>° (1 .4 2 VJ)
u n b o u n d
u n b o u n d
u n b o u n d
£>°(VJ)
-25.4
-23 .1
-19 .7
D°{V2) -96.2
-93 .0
-88 .5
Our results suggest that w, r) and rj mesons should be bound in all the nuclei considered. Furthermore, the D~ meson should be bound in 2 0 8Pb in any case, assisted by the Coulomb force .
308
4. i£-Meson Production in Heavy Ion Collisions
Here we focus on the kaon production reactions, -KN —¥ AA", in nuclear matter. We calculate the in-medium reaction amplitudes, taking into account the scalar and vector potentials for incident, final and intermediate mesons and baryons. The processes considered in the calculation, which were already established in studies for free space13, are shown in Fig. 3.
Figure 3. Processes included for the TTN —> AA' reactions.
(GeV) (GeV)
Figure 4. Energy dependence of the total cross sections. The solid, dashed, dotted lines in the left panel correspond to nuclear densities pg = (0,po,3po)- The solind and dotted lines in the right panel correspond to pg = 0, and the result averaged over the nuclear density distribution in Au+Au collisions at 2A GeV1 4 . The dots are data in free space15 , pg = 0.
Results are shown in Fig. 4. (See caption for explanations.) From the results we conclude that if one accounts for the in-medium modification of the production amplitude correctly, it is possible to understand A"+ production data in heavy ion collisions at SIS (Darmstadt, Germany) energies, even if the K+-meson feels the theoretically expected, repulsive mean field potential. The apparent failure to explain the K+ production data, if one includes the purely kinematic effects of the in-medium modification of the A'+-meson and hadrons, appears to be a consequence of the omission of these effects in the reaction amplitudes.
309
5. J/$? Dissociation in Nuclear Matter
There is a great deal of interest in possible signals of Quark-Gluon Plasma (QGP) formation, and J / $ suppression is a promising candidate as suggested by Matsui and Satz16. Our interest here is how much the J/ty absorption cross sections in hadronic dissociation processes will be modified, if the in-medium hadron potentials are included, which has never been addressed in QGP analyses.
We consider the reactions involving the J/$!, shown in Fig. 5. Recent cal-
: D * ID ID* ID D D ID ID
ID ID ID ID
Figure 5. Processes included for J/Vt dissociation.
culations for the processes in free space17 indicate a much lower cross sections than necessary to explain the data for the J / \ t suppression.
However, this situation changes when the in-medium potentials of the charmed (and also p) mesons are taken into account, as shown in the left panel of Fig. 6. Clearly, the J/\P absorption cross sections are substantially
o> r + J / .
2 0
10
\ |, Pb + Pb
V-. * 1996 \ \ • 1 996-Min. Bios
j ^ • 1998- Min. Bias
% * ^ .
E, (GeV)
Figure 6. Energy dependence of the total cross section.
enhanced not only because of the downward shift of the reaction threshold, but also because of the in-medium effect on the reaction amplitude.
In order to compare our results with the NA38/NA50 data18 on J / $ suppression in Pb+Pb collisions, we have adopted the heavy ion model proposed in Ref. [19] with the ET model from Ref. [20]. Our result is shown in the right panel of Fig. 6 by the solid line, using the density dependent, thermally
310
averaged cross section8. The dashed line shows the result reported in Ref. [20] using the phenomenological constant cross section, 4.5 mb. Both lines clearly reproduce the data18 quite well, including most recent results from NA50 on the ratio of J/vp over Drell-Yan cross sections, as a function of the transverse energy ET up to about 100 GeV.
It is important to note that if one neglects the in-medium modification of the J/9 absorption cross section, the large value of the J/9 dissociation cross section of 4.5 mb used in Ref. [20], could not be justified by microscopic theoretical calculations, and thus the NA50 data18 could not be described.
6. pp —• ppu and pp —¥ pp<f> Reactions at Near Thresholds
Here we report results for the vector meson production reactions near threshold, pp —> ppv (v = w, 4>)21'22. Description of the model and parameters determined by other reactions are given in Ref. [23]. The vector meson production amplitude, M, is depicted in Fig. 7.
Our main results for vector meson production are :
(1) Tensor to vector coupling ratio, KV = fVNN/gvNN, for the w (KW), where a typical NN interaction model 24 sets KU = 0;
(2) <j)NN coupling constant, g<j>NN, in connection with ss component in the nucleon wave function and the OZI rule violation.
- t -
( i - = ~ 2 > —[—- o> o o>
Figure 7. Amplitudes, M , for the pp -»• ppv (v = u, <f>) reactions considered in the study. TMNI ISI and FSI stand for, meson-nucleon (MN) T-matrix, initial state interaction and final state interaction, respectively.
Results for angular distributions are shown in Fig. 8, together with the experimental data for the u in Ref. [25] and <£in Ref. [26]. The total contribution comes from following two sources. First, the nucleonic source : the vector meson is produced from the vNN (meson-nucleon-nucleon) vertex. Second, the mesonic one : the vector meson is produced from the vpir (mesonic vertex),
311
where the n and p are attached to two different protons. The u meson production is dominated equally by the nucleonic and mesonic mechanisms, while for the <j>, the mesonic one is strongly dominant. Thus, production mechanisms for these two vector mesons are different. For the u meson, a large value
pp ->pp$: TJab = 2.85 GeV(K=-2.0, KH=ll90MeV) PP —» ppta: Q—173 MeV(K^~1000 MeV)
s A.o2240MeV
A.^I20M«V
IJTHH-^fTiViai-
piaiE,
-1.2 V2200MeV
A.=2O40M«V
r]~ih*+h4T*M
j * t f ^
l„=-2.0 A.*2220MeV
-^a±mi^t i .
c o s ( e ) oo«(0) ° » ( 9 ) C0S(6)
Figure 8. Angular distributions for pp -¥ ppui at excess energy Q = 173 MeV (the left panel), and for pp -»• ppcj> at excess energy Q = 83 MeV (the right panel). They are normalized to the total cross sections, <r(pp —¥ ppui) = 30.8 /j.b in Ref. [25] and o(pp —} pp<f>) = 190 nb in Ref. [26], respectively. The (solid, dash-dotted, dotted) lines correspond to the (total, nucleonic, mesonic) contributions, respectively.
KU = —2.0 with g^NN = (—9.0)2 = 81.0 fits best the data25, and turned out to be insensitive to the value of gWNN21'22, where in the Bonn NN potential model24
KW = 0 with g%NN ~ 24(4TT) ~ 301.6 is used. As for the <j> meson, several parameter sets can possibly reproduce the
data26. Surprisingly, for K4, = —2.0, a large absolute value |<70AW| = I — 2.0| can also reproduce the data. This implies a large violation of the OZI rule. However, this turned out to give a distinguish ably different energy dependence for the total cross section at near threshold22. Thus, we can distinguish if the energy dependence of the total cross section is measured at near threshold.
7. Conclusion
In conclusion, there is strong motivation to perform experiments to study the changes in hadron properties in nuclear medium or the partial restoration of chiral symmetry in nuclear medium.
Acknowledgments The author would like to thank, D.H. Lu, K. Nakayama, K. Saito, A. Sibirtsev and A.W. Thomas for exciting collaborations, and for warm hospitality at CSSM during the workshop. Special thanks go to Prof. A.W. Thomas for the support enabling me to attend the workshop. This work was partially
312
supported by the Forschungszentrum-Jiilich, contract No. 41445282 (COSY-
058).
R e f e r e n c e s
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A.W. Thomas, Phys. Rev. C55, 2637 (1997); Phys. Rev. C56, 566 (1997); D.H. Lu et al., Nucl. Phys. A634, 443 (1998); G. Krein, A.W. Thomas, K. Tsushima, Nucl. Phys. A650, 313 (1999).
3. K. Saito, K. Tsushima, A.W. Thomas, Nucl. Phys. A609, 339 (1996). 4. K. Tsushima, K. Saito, A.W. Thomas, Phys. Lett. B411, 9 (1997); (E) ibid
B421, 413 (1998); K. Tsushima et al., Nucl. Phys. A630, 691 (1998). 5. K. Tsushima et al., Phys. Lett. B429, 239 (1998); (E) ibid B436, 453 (1998). 6. K. Tsushima et al., Phys. Lett. B443, 26 (1998); K. Tsushima, in the proceeding,
ISHEP 98, Dubna, Russia, 17-22 Aug 1998, nucl-th/9811063; Nucl. Phys. A670, 198c (2000); K. Tsushima et al., Nucl. Phys. A680, 280c (2001).
7. K. Tsushima et al., Phys. Rev. C59, 2824 (1999). 8. A. Sibirtsev et al., Phys. Lett. B484, 23 (2000). 9. A. Sibirtsev, K. Tsushima, A.W. Thomas, Eur. Phys. J. A6, 351 (1999).
10. J.D. Walecka, Ann. Phys. (NY) 83, 491 (1974); B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16, 1 (1986).
11. R.E. Chrien, Nucl. Phys. A478, 705c (1988). 12. K. Saito et al., Phys. Rev. C59, 1203 (1999). 13. K. Tsushima, S.W. Huang, A. Faessler, Phys. Lett. B337, 245 (1994); J. Phys.
G21, 33 (1995); Aust. J. Phys. 50, 35 (1997); K. Tsushima et al., Phys. Rev. C59, 369 (1999); ibid, (E) C61, 029903 (2000).
14. K. Tsushima, A. Sibirtsev, A.W. Thomas, Phys. Rev. C62, 064904 (2000); J. Phys. G27, 349 (2001).
15. Landolt-Bornstein, New Series, ed. H. Schopper, 8 (1973). 16. T. Matsui and H. Satz, Phys. Lett. B178, 416 (1986). 17. S.G. Matinyan and B. Miiller, Phys. Rev. C58, 2994 (1998); B. Miiller, Nucl.
Phys. A661, 272 (1999). 18. Quark Matter '97, Nucl. Phys. A638 (1998); M. C. Abreu et al. (NA50 Collab
oration), Phys. Lett. B410, 337 (1997); Phys. Lett. B450, 456 (1999); M. C. Abreu et al. (NA50 Collaboration), Phys. Lett. B477, 28 (2000).
19. N. Armesto and A. Capella, J. Phys. G23 1969, (1997); Phys. Lett. B430, 23 (1998); N. Armesto, A. Capella and E.G. Ferreiro, Phys. Rev. C59, 395 (1999).
20. A. Capella, E.G. Ferreiro and A.B. Kaidalov, Phys. Rev. Lett. 85, 2080 (2000). 21. K. Nakayama and K. Tsushima, in preparation. 22. K. Tsushima and K. Nakayama, in preparation. 23. K. Nakayama et al., Phys. Rev. C57, 1580 (1988); Phys. Rev. C60, 055209
(1999); Phys. Rev. C61, 024001 (1999); K. Nakayama, J. Speth, T.-S.H. Lee, Phys. Rev. C65, 045210 (2002).
24. R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989). 25. S. Abd El-Samad et al. (COSY-TOF collaboration), Phys. Lett. B522, 16 (2001). 26. F. Balestra et al. (DISTO collaboration), Phys. Rev. C63, 024004 (2001).
List of Participants
Prof. Iraj A F N A N SoCPES Flinders University GPO box 2100 Adelaide, SA, 5001 AUSTRALIA [email protected]
Wesley A R M O U R 9 Beaufort Court Cadle Swansea Wales, SA5 4PH U.K. [email protected]
Dr. Wolfgang B E N T Z Dept. of Physics, School of Science Tokai University, Japan 1117 Kitakaname, Hiratsuka-shi Kanagawa 259-1292 JAPAN [email protected]
Dr. Ingo BO JAK CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]
Dr. Chris ALLTON University of Wales Swansea-UK University of Queensland Department of Mathematics St. Lucia, Brisbane, 4072 AUSTRALIA c.allton@swan. ac. uk
Jonathan ASHLEY CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Sundance B I L S O N - T H O M P S O N CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Prof. Matthias B U R K A R D T Department of Physics New Mexico State Univ Las Cruces NM 88003 U.S.A. [email protected]
313
314
Dr. Fu-Guang CAO Institute of Fundamental Sciences Massey University Private Bag 11-222 Palmerston North NEW ZEALAND [email protected]
Ian CLOET CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Dr. Will D E T M O L D CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Dr. Xin-Heng GUO CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Dr. Johann H A I D E N B A U E R Forschungszentrum Juelich IKP D-52425 Juelich GERMANY [email protected]
Prof. Abraham C H I A N WISER/NITP University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]
Benjamin C R O U C H CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Matthew G A R B U T T Theory Group School of Physics The University of Melbourne VIC,3010 AUSTRALIA [email protected]
Dr. Vadim GUZEY CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]
Prof. Tetsuo H A T S U D A University of Tokyo Physics Department Tokyo 113-0033 JAPAN [email protected]
315
John H E D D I T C H CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Prof. Atsushi H O S A K A RCNP Osaka University Ibaraki 567-0047 JAPAN [email protected]
Dr. Alexander KALLONIATIS CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Waseem KAMLEH CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Prof. Faqir K H A N N A Physics Department Theoretical Physics Institute University of Alberta Edmonton,Alberta,T6G 2J1 CANADA khanna@phys .ua lbe r t a . ca
Dr. Ayse KIZILERSU CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA akiziler ©physics . adelaide.edu.au
Prof. Gastao KREIN Instituto de Fisica Teorica Rua Pamplona, 145 01405-900 Sao Paulo, SP BRAZIL [email protected]
Ben LASSCOCK CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA 5000 AUSTRALIA [email protected]
Dr. Derek L E I N W E B E R CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA dle inweb@physics .adela ide .edu.au
Dr. Olivier LEITNER CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
316
Dr. Max LOHE University of Adelaide Department of Phys. and Math. Phys. Adelaide, SA, 5005 AUSTRALIA [email protected]
Prof. Bruce MCKELLAR School of Physics University of Melbourne VIC,3010 AUSTRALIA [email protected]
Hirobumi MINEO Department of Physics University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033 JAPAN [email protected]
Prof. Tomofumi N A G A E KEK 1-1 Oho Tsukuba 305-0801 JAPAN [email protected]
Dr. Martin OETTEL Max-Planck-Institut fiir Metallforschung Heisenbergstr. 3, 70569 Stuttgart GERMANY oettel@mf . mpg.de
Prof. Makoto OKA Department of Physics Tokyo Institute of Technology Oh-Okayama, Meguro, Tokyo 152-8551 JAPAN [email protected]
Dr. Shin'ya SAWADA KEK 1-1 Oho Tsukuba Ibaraki 305-0801 JAPAN [email protected]
Kai S C H M I D T - H O B E R G CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Dr. Andreas SCHREIBER CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
Mark S T A N F O R D CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
317
Prof. Anthony T H O M A S CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]
Dr. Stuart T O V E Y Research Centre for HEP School of Physics University of Melbourne V I C , 3010 AUSTRALIA [email protected]
Dr. Raymond VOLKAS School of Physics The University of Melbourne Melbourne VIC,3010 AUSTRALIA r .volkas@physics .unimelb.edu.au
Ross Y O U N G CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]. au
Dr. Jioanbo ZHANG CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]
Prof. Hiroshi TOKI RCNP, Osaka University MihogaokalO-l Ibaraki Osaka 567-0047 JAPAN [email protected]
Dr. Kazuo T S U S H I M A Department of Physics and Astronomy University of Georgia Athens GA 30602 USA tsushima@physast. uga. edu
A/Prof. Anthony WILLIAMS CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]
James ZANOTTI CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]
