Nucleus-Nucleus Collisions: Proceedings of the Conference Bologna 2000

THE SCIENCE AND CULTURE SERIES — ADVANCED SCIENTIFIC CULTURE Series Editor: A. Zichichi

NUCLEUS-NUCLEUS COLLISIONS PROCEEDINGS OF THE CONFERENCE: BOLOGNA 2000 STRUCTURE OF^HE NUCLEUS AT THE DAWN OF THE^NTURY

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I0RI, M. BRUNO, A. VENTURA, D. VRETENAR World Scientific

NUCLEUS-NUCLEUS COLLISIONS PROCEEDINGS OF THE CONFERENCE: BOLOGNA 2000 STRUCTURE OF THE NUCLEUS AT THE DAWN OF THE CENTURY

THE SCIENCE AND CULTURE SERIES — ADVANCED SCIENTIFIC CULTURE Series Editor: A. Zichichi, European Physical Society, Geneva, Switzerland Series Editorial Board: P. G. Bergmann, J. Collinge, V. Hughes, N. Kurti, T. D. Lee, K. M. B. Siegbahn, G. 't Hooft, P. Toubert, E. Velikhov, G. Veneziano, G. Zhou

1.

Nucleus-Nucleus Collisions Bologna 2000. Structure of the Nucleus at the Dawn of the Century

2.

Nuclear Structure Bologna 2000. Structure of the Nucleus at the Dawn of the Century

3.

Hadrons, Nuclei and Applications Bologna 2000. Structure of the Nucleus at the Dawn of the Century

NUCLEUS-NUCLEUS COLLISIONS PROCEEDINGS OF THE CONFERENCE: BOLOGNA 2000 STRUCTURE OF THE NUCLEUS AT THE DAWN OF THE CENTURY

Bologna, Italy

29 May - 3 June 2000

Editors

Giovanni C. Bonsignori Mauro Bruno Dipartimento di Fisica dell'Universita di Bologna and INFN-Sezione di Bologna, Italy

Alberto Ventura Ente Nuove Tecnologie, Energia e Ambiente and INFN Bologna, Italy

Dario Vretenar Physics Department, University of Zagreb, Croatia

Series Editor A. Zichichi

V f e World Scientific M

New Jersey • London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pre. 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.

NUCLEUS-NUCLEUS COLLISIONS Proceedings of the Conference: Bologna 2000 Structure of the Nucleus at the Dawn of the Century Copyright © 2001 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.

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ISBN 981-02-4731-1

Printed in Singapore.

CONTENTS

Foreword The Editors

xv

Welcome Address

xix

F. Rover si-Monaco Committees and Sponsors Photographs

xxi xxiii

Opening Lecture Antimatter — Past, Present and Future A. Zichichi

3

Introductory Talk What Can Nuclear Collisions Teach Us about the Boiling of Water or the Formation of Multi-Star Systems? D. H. E. Gross Section I.

53

Quark and Gluon—Plasma Phase Transition and Relativistic Heavy-Ion Reactions

Results from Pb-Pb Collisions at CERN SPS L. Riccati and E. Scomparin

63

The Search for the QGP: A Critical Appraisal H. Satz

71

Strange Baryon Signals of a New State of Matter in Lead-Lead Collisions at the CERN SPS F. Antinori for the WA97 Collaboration

73

vi

Cooper-Mesons in the Color-Flavor-Locked Superconducting Phase of Dense QCD A. Wirzba

83

Charmonium Suppression in P b - P b Collisions and Quark-Gluon Deconfinement A. B. Kurepin for the NA50 Collaboration

91

Particle Production in P b + P b Collisions at 158 GeV/Nucleon in the NA49 Detector H. Strobele for the NA4-9 Collaboration

99

Exploring the Chiral Phase Transition in High-Energy Collisions J. Randrup

105

Nuclear Collective Flow in Heavy Ion Collisions at SIS Energies N. Bastid for the FOPI Collaboration

111

Hadron Observables from Hadronic Transport Model with Jet Production at RHIC Y. Nara

115

Simultaneous Heavy Ion Dissociation at Ultrarelativistic Energies /. A. Pshenichnov, J. P. Bondorf, S. Masetti, I. N. Mishustin and A. Ventura

119

Direct Photon Production in 158A GeV 2 0 8 P b + 2 0 8 P b Collisions T. Peitzmann for the WA98 Collaboration

123

Deformation and Orientation Effects in Uranium-on-Uranium Collisions at Relativistic Energies Bao-An Li

129

Subthreshold Heavy-Meson and Antiproton Production in the Nucleus-Nucleus Collisions A. T. D'Yachenko

133

Pion Imaging at the AGS S. Y. Panitkin for the E895 Collaboration

137

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Directed Flow in 4.2A GeV/C C+C and C+Ta Collisions J. Milosevic and L. J. Simic

145

Fragmentation of Very High Energy Heavy Ions M. Giorgini and S. Manzoor

149

Section II.

Liquid—Gas Phase Transitions in Nuclear Matter

Phase Transition in Finite Systems Ph. Chomaz, V. Duflot and F. Gulminelli

155

Present Status and Future Prospects of Investigations of the Liquid-Gas Phase Transition C.-K. Gelbke

167

Critical Phenomena in Finite Systems A. Bonasera, T. Maruyama and S. Chiba

179

Experimental Signals of the First Phase Transition of Nuclear Matter B. Borderie

187

Nuclear Fragmentation, Phase Transitions and their Characterization in Finite Systems of Interacting Particles J. M. Carmona, N. Michel, J. Richert and P. Wagner

197

Topology and Phase Transitions: Towards a Proper Mathematical Definition of Finite N Transitions M. Pettini, R. Franzosi and L. Spinelli

203

Introduction to the Phase Transition Discussion Ph. Chomaz

209

Finite Nuclear Fragmenting Systems: An Experimental Evidence of a First Order Liquid-Gas Phase Transition M. D'Agostino, F. Gulminelli, Ph. Chomaz, M. Bruno, F. Cannata, N. Le Neindre, R. Bougault, M. L. Fiandri, E. Fuschini, F. Gramegna, I. Iori, G. V. Margagliotti, A. Moroni, G. Vannini and E. Verondini

215

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Phase Transition in XE+SN Central Events between 32 and 50A MeV N. Le Neindre, R. Bougault, Ph. Chomaz, F. Gulminelli and the INDRA Collaboration

221

Nuclear Caloric Curve: Influence of the Secondary Decays on the Isotopic Thermometers A. H. Raduta and A. R. Raduta

227

Phase Transition Signals in Thermally Excited Nuclei V. E. Viola for the E900 Collaboration

231

Hydrogen Cluster Multifragmentation and Percolation Models F. Gobet, B. Farizon, M. Farizon, M. J. Gaillard, J. P. Bucket, M. Carre, P. Scheier and T. D. Mark

237

Single Quasiparticle Entropy in Excited Nuclei with T < 1 MeV M. Guttormsen, M. Hjorth-Jensen, E. Melby, J. Rekstad, A. Schiller and S. Siem

243

Section III.

Nuclear Caloric Curve and Thermodynamics of Heavy Ion Collisions

Surveying Temperature and Density Measurements in Nuclear Calorimetry G. Raciti for the ALADIN Collaboration

249

Caloric Curve of Fragmenting Systems C O. Dorso and A. Chernomoretz

257

Thermodynamics of Explosions G. Neergaard, J. P. Bondorf and I. N. Mishustin

263

Microcanonical Investigation of the Recent Nuclear Caloric Curve Experimental Evaluations A. H. Raduta and A. R. Raduta

269

Can We Determine the Nuclear Equation of State from Heavy Ion Collisions? T. Gaitanos, H. H. Wolter, C. Fuchs and A. Faessler

273

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Thermodynamical Description of Heavy Ion Collisions T. Gaitanos, H. H. Wolter and C. Fuchs Section IV.

279

Statistical and Dynamical Aspects of Fragmentation

Exact Solution of the Statistical Multifragmentation Model and the Liquid-Gas Mixed Phase K. A. Bugaev, M. I Gorenstein, I. N. Mishustin and W. Greiner

285

What Can We Learn from Nuclear Matter Instabilities? V. Baran, M. Colonna, M. Di Toro, M. Zielinska-Pfabe and H. H. Wolter

293

Energetic Proton Emission and Reaction Dynamics in Heavy Ion Reactions Close to the Fermi Energy R. Coniglione, P. Sapienza, E. Migneco, C. Agodi, R. Alba, G. Bellia, M. Colonna, A. Del Zoppo, P. Finocchiaro, V. Greco, K. Loukachine, C. Maiolino, P. Piattelli, D. Santonocito, P. G. Ventura, N. Colonna, M. Bruno, M. D'Agostino, M. L. Fiandri, G. Vannini, P. F. Mastinu, F. Gramegna, I. Iori, L. Fabbietti, A. Moroni, G. V. Margagliotti, P. M. Milazzo, R. Rui, F. Tonetto, Y. Blumenfeld and J. A. Scarpaci

299

New Results on Preequilibrium 7-Ray Emission and GDR Saturation on Reactions at 25A MeV G. Cardella, F. Amorini, A. Di Pietro, P. Figuera, G. Lanzalone, J. Lu, A. Musumarra, M. Papa, G. Pappalardo, S. Pirrone, F. Rizzo and S. Tudisco

305

The Onset of Mid-Velocity Emissions in Symmetric Heavy Ion Reactions E. Plagnol, J. Lukasik and the INDRA Collaboration

309

Contribution of Prompt Emissions to the Production of Intermediate Velocity Light Particles in the 3 6 Ar+ 5 8 Ni Reaction at 95 MeV/Nucleon P. Pawlowski, B. Borderie and the INDRA Collaboration

313

Proton Emission Times in Spectator Fragmentation C. Schwarz for the ALADIN Collaboration

317

Experimental Evidence for Spinodal Decomposition in Multifragmentation of Heavy Systems M. F. Rivet for the INDRA Collaboration

321

Non-Equilibrium Effects on a Second-Order Phase Transition V. Latora and A. Rapisarda

327

Multifragmentation of Expanding Nuclear Matter S. Chikazumi, T. Maruyama, K. Niita, A. Iwamoto and S. Chiba

331

Contemporary Presence of Dynamical and Statistical Intermediate Mass Fragment Production Mechanisms in Midperipheral Ni+Ni Collisions at 30 MeV/Nucleon P. M. Milazzo, M. Sisto, G. V. Margagliotti, R. Rui, G. Vannini, M. Bruno, M. D'Agostino, N. Colonna, C. Agodi, R. Alba, G. Bellia, M. Colonna, R. Coniglione, A. Del Zoppo, P. Finocchiaro, C. Maiolino, E. Migneco, P. Piattelli, D. Santonocito, P. Sapienza, F. Gramegna, P. F. Mastinu, L. Fabbietti, I. Iori, A. Moroni and M. Belkacem

335

External Coulomb and Angular Momentum Influence on Isotope Composition of Nuclear Fragments A. S. Botvina

341

Section V.

Intermediate Energy Heavy-Ion Reactions

Isospin Fractionation in Excited Nuclear Systems S. J. Yennello, M. Veslesky, E. Martin, R. Laforest, D. Rowland, E. Ramakrishnan, A. Ruangma and E. M. Winchester

347

The Disappearance of Flow and the Nuclear Equation of State G. D. Westfall

353

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The Backtracing Procedure in Nuclear Physics P. Desesquelles

359

Ni+Ni Collisions at 32 MeV/U: Experimental Insight with the INDRA Multidetector A. M. Maskay, P. Lautesse, P. Desesquelles, E. Gerlic, J. L. Laville and the INDRA Collaboration

363

Two-Fragment Correlation Function for Quasi-Projectile and Mid-Rapidity Emission Z. He, R. Roy, L. Gingras, Y. Larochelle, D. Ouerdane, L. Beaulieu, P. Gagne, X. Qian, C. St-Pierre, G. C. Ball and D. Horn

367

Fermion Interferometry in Ni-Induced Reactions at E/A=45 MeV R. Ghetti

371

Investigation of an Angular Distribution of in Peripheral and Central Nucleus-Nucleus at the Momentum of 4.2A GeV/c M. K. Suleymanov, O. B. Abdinov, Z. N. S. Angelov, A. C. Vodopyanov and

375

Protons Collisions Y. Sadigov, A. A. Kuznetsov

REVERSE Experiment at Laboratori Nazionali del Sud E. Geraci for the REVERSE Collaboration

379

REVERSE: The First Experiment with the Chimera Detector G. Politi for the REVERSE Collaboration

383

Non Equilibrated IMF Emission in Heavy Ion Collisions around the Fermi Energy S. Piantelli, L. Bidini, M. Bini, G. Casini, P. R. Maurenzig, A. Olmi, G. Pasquali, G. Poggi, S. Poggi, A. A. Stefanini and N. Taccetti

387

The Complete Fusion and the Competitive Processes in the 3 2 S+ 1 2 C Reaction at E( 3 2 S)=20 MeV/A S. Pirrgjne, G. Politi, G. Lanzalone, S. Aiello, N. Arena, Seb. Cavallaro, E. Geraci, F. Porto and S. Sambataro

391

A Simple Pulse Shape Discrimination Method Applied to Silicon Strip Detector J. Lu, F. Amorini, G. Cordelia, A. Di Pietro, P. Figuera, A. Musumarra, M. Papa, G. Pappalardo, F. Rizzo and S. Tudisco Section VI.

395

Reaction Mechanisms around the Barrier. Fusion and Fission in Heavy-Ion Reactions

Cross-Sections for Coulomb and Nuclear Breakup of Three-Body Halo Nuclei E. Garrido, D. V. Fedorov and A. S. Jensen

401

Multinucleon Transfer Reactions Studied with the PISOLO Spectrometer L. Corradi, A. M. Stefanini, A. M. Vinodkumar, S. Beghini, G. Montagnoli, F. Scarlassara and G. Pollarolo

405

Study of the Cluster Emission Barrier in 1 2 C+ 2 0 8 Pb Elastic Scattering and Possible Observation of Quasimolecular Configuration A. A. Ogloblin, K. P. Artemov, Yu. A. Glukhov, A. S. Dem'yanova, V. V. Paramonov, M. V.Rozhkov, V. P. Rudakov and S. A. Goncharov

409

Very Strong Reaction Channels at Barrier Energies in the System 9Be+ 209 Bi C. Signorini, A. Andrighetto, J. Y. Guo, L. Stroe, A. Vitturi, M. Ruan, M. Trotta, F. Soramel, K. E. G. Lobner, K. Rudolph, I. Thompson, D. Pierroutsakou and M. Romoli

413

Nuclear Rainbows, Nuclear Matter and the 1 6 0 + 1 6 0 System W. von Oertzen, A. Blazevic, H. G. Bohlen, D. T. Khoa, F. Nouffer, P. Roussel-Chomaz, W. Mittig and J. M. Cassandjian

419

Bremsstrahlung by Nonrelativistic Particles in Matter A. V. Koshelkin

427

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Pronounced Airy Structure in Elastic 1 6 0 + 1 2 C Scattering at Eiab=200 MeV Y. A. Gloukhov, A. S. Dem'yanova, A. A. Ogloblin, M. V. Rozhkov, S. A. Goncharov, R. Julin and W. Trzaska

431

Fusion Energy Thresholds Predicted with an Adiabatic Nucleus-Nucleus Potential J. Wilczynski and K. Siwek- Wilczynska

435

Near-Barrier Fusion of 36 S+ 90 - 96 Zr: What is the Effect of the Strong Octupole Vibration of 96 Zr? L. Corradi, A. M. Stefanini, A. M. Vinodkumar, S. Beghini, G. Montagnoli, F. Scarlassara and M. Bisogno

441

Dynamical Model of Fission Fragment Angular Distribution V. A. Drozdov, D. 0. Eremenko, 0. V. Fotina, S. Yu. Platonov and O. A. Yuminov

445

Fusion-Fission Reaction 1 2 C, 1 6 O+ 2 0 8 Pb at Subbarrier Energies S. P. Tretyakova, M. G. Itkis, E. M. Kozulin, N. A. Kondratiev, I. V. Pokrovskii, E. V. Prokhorova, L. Calabretta, C. Maiolini, A. Ya. Rusanov and T. Yu. Tretyakova

449

Sub-Barrier Fusion with a Halo Nucleus: The 6 He Case M. Trotta, J. I. Sida, N. Alamanos, F. Auger, A. Drouart, D. J. C. Durand, A. Gillibert, C. Jouanne, V. Lapoux, A. Lumbroso, F. Marie, S. Ottini, C. Volant, A. Andreyev, D. L. Balabanski, N. Coulier, G. Georgiev, M. Huyse, G. Neyens, R. Raabe, S. Ternier, P. van Duppen, K. Vivey, C. Borcea, J. D. Hinnefeld, A. Musumarra, A. Lepine and R. Wolski

453

Concluding Remarks Impact of Nuclear Science on Modern Society R. A. Ricci

459

Author Index

465

FOREWORD

The International Nuclear Physics Conference " Bologna 2000. Structure of the Nucleus at the Dawn of the Century", held from May 29 to June 3, 2000 in Bologna, Italy, in the Aula Magna of the University and in the medieval Palazzo dei Notai, was attended by more than 340 scientists from 43 countries and five continents. It was the first time that an International Nuclear Physics Conference of this size had taken place in Bologna. The Conference was organized by the Physics Department of the University and by the Bologna Section of the Italian National Institute for Nuclear Physics (INFN). It was sponsored, among others, by the Research Directorate General of the European Commission, the University of Bologna, the Italian Physical Society (SIF), the INFN National Laboratories in Legnaro (LNL) and Catania (LNS), the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) in Trento, the Italian National Agency for New Technologies, Energy and the Environment (ENEA). The idea for this Conference emerged from the desire to discuss perspectives in fundamental Nuclear Physics. Some of the most important European research facilities are located in Italy, a country with a long and rich tradition in this field. The Conference took place at a particularly symbolic moment when, celebrating the 900"* anniversary of its famous University, Bologna was designated as European City of Culture for the Year 2000. The aim of the Conference was also a cultural contribution to these celebrations and to the commemoration of the 200"1 anniversary of the death of Luigi Galvani, one of Bologna's most renowned scholars. In 1997, when Thessaloniki was European City of Culture, the Aristotle University organized a very successful Conference: "Advances in Nuclear Physics and Related Areas". The Bologna event was organized as the second in a hopefully long series of Nuclear Physics Meetings in the European Cities of Culture. This International Conference Bologna 2000 was devoted to a discipline which has seen a strong revival of research activities in the last decade. New experimental results and theoretical developments in the field of Nuclear Physics will certainly produce important contributions to our knowledge and understanding of Nature's fundamental building blocks. The interest aroused by the Conference in the scientific community is clearly reflected in the large number of participants. They represented the most important nuclear physics laboratories in the world. The generous grants from our sponsors allowed the financial support for a considerable number of

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participants who, otherwise, would not have been able to attend. The Conference covered five major topics of modern nuclear physics: nuclear structure, nucleus-nucleus collisions, hadron dynamics, nuclear astrophysics and trans-disciplinary and peaceful applications of nuclear science. The scope of the Conference was to present a review of recent progress in the field and to provide a forum for the discussion of current and future research projects. The Conference consisted of plenary and parallel sessions. The plenary sessions focused on comprehensive reviews, concepts and perspectives. Specialized talks on recent developments were presented in the parallel and poster sessions. More than 200 participants, among them many young researchers, had the opportunity to present their work in oral contributions. Two guided tours helped participants and accompanying persons to discover Bologna and the historical roots of its culture and traditions. On the last day most participants joined the excursion to Ravenna and its magnificent Byzantine monuments. We sincerely hope that all the participants enjoyed the Conference and their stay in Bologna. Acknowledgements We are very grateful to the University of Bologna and to its Rector Fabio Roversi Monaco for the generous financial support and the hospitality extended to the Conference in the beautiful Aula Magna of Santa Lucia. We would like to thank the Rector for his warm welcome address. The Pro-rector of the University Ettore Verondini gave an important and encouraging contribution to the organisation of the Conference. We thank professor Enzo Iarocci, President of the National Institute for Nuclear Physics, for his welcome address. Our special thanks go to Professor Antonino Zichichi, distinguished senior member of the Department of Physics of the University of Bologna. His promotional activities, his help in the publishing of the Proceedings, and finally his extremely interesting opening lecture, made an essential and animating contribution to the success of the Conference. The Conference was closed with the most interesting review of the past and the future of Nuclear physics by Professor Renato Angelo Ricci, former President of the Italian Physical Society. We acknowledge the help of the Department of Physics of the University. We are very grateful to its Director, Professor Attilio Forino, for his numerous suggestions, encouragement and also generous financial help which was essential in the initial stages of the organisation of the Conference. Our spe-

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cial thanks go to the Bologna section of the INFN and its Director Professor Paolo Giusti for the financial and logistic support. The Research Directorate General of the European Commission, the framework of Human Potential Programme, High-Level Scientific Conferences (under contract no. HPCF-CT-1999-00131), generously supported the participation of young researchers from the European Community and Associate States. It is a pleasure to thank the President of ENEA, Nobelist Carlo Rubbia, and the head of the ENEA Accelerator-Driven-System Project, Giuseppe Gherardi, for the financial and scientific support offered by ENEA to the Conference. Essential for the success of the Conference was the dedicated work of the Conference Secretariat: Barbara Simoni and Luisa De Angelis, the Conference Agency: Angela Rizzi, and the kind assistance of Erika Giorgianni. Special thanks go to Angela Belluzzi from the Office of the Rector of the University for her help in organising the Conference venue. The commitment of ECT* Trento and its Scientific Secretary Renzo Leonardi was invaluable in the organisation and during the Conference. The very efficient work of Ines Campo and Rachel Weatherhead during the five days of the Conference, and the kind assistance of Mauro Meneghini should be particularly mentioned. We express our warmest thanks to the Directress of the Centro Documentazione delle Donne, Mrs Grazia Negrini, for the kind hospitality extended to the Conference in the magnificent Palazzo dei Notai. We are very grateful to the municipality of Bologna for hosting the welcome party in the splendid Sala d' Ercole. The visit to the Museo Morandi and to the Town Hall art collections was particularly appreciated. The financial contribution of the "Fondazione Carisbo" is gratefully acknowledged. Further financial support was provided by the listed in the following pages. Informative brochures were kindly provided by the University of Bologna, by Professor Giorgio Giacomelli, the LNS and LNL and the " Assessorato alia Cultura" of the Bologna Province. The plenary sessions were recorded and broadcasted in real time via Internet by the INFN informatics and multimedia! groups. We express our warmest thanks to the Director Paolo Mazzanti and to Gianfranco Artusi, Daniela Bortolotti, Roberto Giacomelli, Alessandro Italiano, Franco Martelli, Gianluca Peco, Ombretta Pinazza, Giuliano Scandellari, Stefano Zani. The facilities and services provided by the Physics Department are gratefully acknowledged. We thank Attilio Capponi, Nino Muzzupappa, Romano Serra, Simonetta Salsini and Giuseppe Zaccarelli. The work of Paolo Finelli was invaluable in preparing the proceedings of

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the Conference. Our warmest thanks go to the colleagues and friends, members of the Organizing Committee, and to the International Advisory Committee, for the excellent job done in preparing the scientific programme and selecting the speakers. The plenary and parallel sessions were run very smoothly and efficiently by our Session chairpersons. Finally, we would like to thank all the participants whose enthusiasm, interest and scientific ideas represent the main contribution to the success of the Conference. They presented an exciting variety of new results, demonstrating the vitality of Nuclear Physics and shedding light on future perspectives at the turn of the century.

The Editors

WELCOME ADDRESS FABIO ROVERSI MONACO Rector of the University of Bologna Ladies and Gentlemen, distinguished Colleagues, It is my pleasure to warmly welcome all the participants in this Conference. The Bologna studium, as you know, is the oldest of the western world: here, the very concept of University as an institution based on independent scientific research and teaching first came into being. In 1988 the University of Bologna celebrated its nine-hundreth anniversary and, on that occasion, the Magna Charta Universitatum, i.e. the Universal Declaration of Academic rights, was signed by more than 400 Rectors of Universities all over the world. A few months ago, following the same inspiration, an Observatory on Academic Freedom was founded, with the purpose of controlling the respect of the fundamental principles on which scientific research and teaching are based. This Conference opens at the turn of a millennium in a particularly symbolic moment: the year 2000, when Bologna takes its turn as Cultural Capital of Europe. A century is passing that saw a tremendous progress in understanding the ultimate structure of matter: technological and social consequences have deeply affected the whole mankind in good and evil. Only one hundred years ago Henry Becquerel and Marie Curie discovered radioactivity. Some ten years later, Ernest Rutherford proved the existence, at the core of the atom, of a very small nucleus, where the same kind of energy is stored as the one that feeds our life from the sun. Even the atoms that make up our bodies were synthesized by nuclear reactions in the core of stars and supernova explosions: there is thus an impressive link between macro- and microcosm. Nevertheless, we assist today to a growing disaffection for science, mainly due to nuclear weapons and improper use of nuclear energy. Let me remind you that the term nuclear was recently dropped from such medical applications as Nuclear Magnetic Resonance. Does this mean that we must give up studying Nature? The Magna Charta states that the future of mankind depends to a large extent on cultural, scientific and technical development. This means that we must go on investigating the deepest problems Nature faces us with, but, at the same time, we must recover the sense of Nature as a common mother. What we need is a new sense of ethic with a direct commitment of scientists for peace, justice, development and mutual understanding.

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It is the first time a Nuclear Physics Conference of large size takes place in Bologna. In 1977, when Thessaloniki was the Cultural Capital, the Aristotle University organized a very successful Conference " Advances in Nuclear Physics and Related Areas". I hope we are participating in the beginning of a long series of Nuclear Physics meetings to be held in the Cultural Capitals of Europe. Starting from Enrico Fermi, Italy has a long and rich tradition in this field, and the contribution of Bologna to the advances of nuclear physics covers the main trends in experimental and theoretical fundamental research, carried on mainly by University and INFN, in collaboration with major institutions and laboratories all over the world. It ranges from light nuclei to heavy-ion physics, from nuclear models to hadron structure with a prominent emphasis on high-energy physics and fundamental interactions. Let me mention now a few words about some troubles that nuclear physics is actually suffering in Bologna. While experimental nuclear physics is represented by many active groups, theoretical nuclear physics runs the risk of disappearing, since young people are moving towards other fields of research where more positions are available. This is a pity in many respects: firstly, a long tradition is disappearing and the richness of knowledge and experience in this field will be definitely lost in a short time; secondly, even if they are considered less fundamental than subnuclear physics, nuclear problems show many interesting and puzzling facets, which may stimulate imagination and development of new models and techniques. For these reasons, I think that the Physics Department, the Faculty of Sciences and INFN should make a serious effort in term of positions - professors and researchers - to avoid the loss of such a relevant cultural legacy. The aim of this Conference is to promote the exchange of ideas and future collaboration among nuclear physicists coming from developed countries and developing regions. The high quality of the programme, with special emphasis on new directions and opportunities, the quality of contributions and the level of speakers, who are among the most prominent scientists in this field, the large number of participants, illustrate the vitality of nuclear science and are favourable auspices for the success of this International Conference. Let me conclude by warmly thanking Giovanni Carlo Bonsignori, Mauro Bruno and the whole organizing committee for their engagement in arranging this meeting. Finally, I wish the Bologna-2000 International Conference on Structure of the Nucleus at the Dawn of the Century the best success, and I hope you will enjoy not only the conference, but also your stay in Bologna and its University.

Conference Organization CONFERENCE CHAIRMEN:

G. Bonsignori (Bologna), M. Bruno (Bologna)

SCIENTIFIC SECRETARIES:

A. Ventura (ENEA/Bologna), D. Vretenar (Zagreb)

INTERNATIONAL ADVISORY COMMITTEE M. Arnould (Bruxelles) J. Aysto (Jyvaskyla) C. Baktash (Oak Ridge) D. Bazzacco (Padova) J.-F. Berger (CEA/Bruyeres-le-Chatel) J.-P. Blaizot (CEA/Saclay) S. Born (Pavia) J. Bondorf (Copenhagen) R. Bougault (Caen) T. Bressani (Torino) R. Broglia (Milano) P. Chomaz (Caen) C. Ciofi degli Atti (Perugia) M. Di Toro (Catania) C. Fahlander (Lund) G. Fiorentini (Ferrara) B. Frois (CEN Saclay) K. Gelbke (NSCL/East Lansing) W. Gelletly (Surrey) G. Gherardi (ENEA/Bologna)

D.H.E. Gross (Berlin) M. Grypeos (Thessaloniki) W. Haxton (Seattle) F. Iachello (New Haven) A.V. Ignatyuk(Obninsk) M. Ishihara (Tokyo) F. Kaeppeler (Karlsruhe) P. Kienle (Munchen) R. Malfliet (Trento) I. Massa (Bologna) V.Metag (Giessen) E. Migneco (Catania) Y.Ts. Oganessian (Dubna) T. Otsuka (Tokyo) R.A. Ricci (Padova) P. Ring (Munchen) F.-K. Thielemann (Basel) A.W. Thomas (Adelaide) A. Zichichi (Bologna)

LOCAL ORGANIZING COMMITTEE S. Lunar di (Padova) A. Mengoni (ENEA/Bologna) A. Pagano (Catania) M. Savoia (Bologna) A. Vitturi (Padova)

F. Cannata (Bologna) A. Covello (Napoli) G. De Angelis (Legnaro) P. Finocchiaro (Catania) R. Leonardi (Trento) G. Lo Bianco (Camerino)

XXI

XX ii

SPONSORS • Dipartimento di Fisica dell'Universita' di Bologna • Istituto Nazionale di Fisica Nucleare - Sezione di Bologna • Universita' degli Studi di Bologna • Provincia di Bologna • INFN - Laboratori Nazionali del Sud - Catania • INFN - Laboratori Nazionali di Legnaro - Legnaro PD • ENEA • Research Directorate General of the European Commission (Under contract no. HPCF-CT-1999-00131) • ECT* - European Centre for Theoretical Studies in Nuclear Physics and Related Areas • Societa' Italiana di Fisica • Fondazione Carisbo, Bologna • Banca di Roma, Roma • Bologna 2000 Committee • EG&G ORTEC Italia • CAEN - Nuclear Physics - Viareggio LU • 3M Italia • Consorzio Marchio Storico dei Lambruschi Modenesi • Consorzio tutela Parmigiano Reggiano • Consorzio produttori di aceto balsamico tradizionale di Modena • Alcisa Bologna • Consorzio tutela aceto balsamico • Monari e Federzoni - Bomporto (Modena)

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BOLOGNA 2000 Structure of the Nucleus at the Dawn of the Century

Opening Lecture

ANTIMATTER PAST, PRESENT AND FUTURE

Antonino Zichichi

Monday, 29 May 2000, h. 11.00 a.m.

Aula Magna di Santa Lucia, Universitd di Bologna

ANTIMATTER PAST, PRESENT AND FUTURE Antonino Zichichi INFN and University of Bologna, Italy CERNk Geneva, Switzerland World Federation of Scientists - Beijing, Geneva, Moscow, New York

ABSTRACT In order to have matter we need fundamental fermions (quarks and leptons), particles (mesons and baryons) and nuclei. For antimatter to exist, the antifundamental fermions, as well as the antiparticles and the antinuclei, are needed. The masses associated with these components of matter are the "intrinsic" (quarks and leptons), the "confinement" (mesons and baryons) and the "binding" [either nuclear (nuclei), or electromagnetic (atoms)]. The first two are positive, the two "binding" ones are negative. These masses have different origins. No one has been able to establish the origin of the "intrinsic" masses (it could be the Higgs mechanism, but this lacks experimental confirmation so far). The "confinement" masses are QCD non-perturbative effects. The nuclear "binding" masses are QCD-induced colour neutral effects; the electromagnetic "binding" is due to QED and, since QED is the best experimentally checked RQFT, its validity in terms of the CPT symmetry cannot easily be questioned and this is why the electromagnetic "binding" is not included in this review. If CPT were theoretically well established as it was when discovered, all mass differences, between any matter and its antimatter partner, should be zero. The best limits for the validity of CPT invariance in the field of masses are two: i) the determination of a very small upper limit on A r n - (the mass difference between a meson and an antimeson) derived from the mass difference between the long- and the short-lived K-mesons, Ani|, K ; ii) the measurements of the charge over mass ratio ( —) for protons and antiprotons. These experimental results are discussed in terms of their relevance to check CPT invariance for the "intrinsic" and for the "confinement" masses. It is pointed out that the value of Am - could be totally insensitive to a large CPT breaking in the intrinsic quark masses. Furthermore, the physics of "intrinsic" and "confinement" masses has nothing to do with the nuclear "binding" masses, where the limits are very poor. A deuteronantideuteron mass difference Anv - , many orders of magnitude larger than the present limits in the "confinement" and in the "intrinsic" masses, could exist and have no effects on the CPT limits known so far, including the very high precision measurements on Am - and on ( r j ) • New experiments need to be performed in nuclear matter-antimatter in order to establish the validity of CPT invariance in non-perturbative — colour neutral — QCD effects, such as the nuclear binding forces which no one is able to describe in terms of a fundamental RQFT. 3

ANTIMATTER PAST, PRESENT AND FUTURE

Antonino Zichichi

CONTENTS 1

-

Introduction

5

2

-

The origin and the seven decades of developments

8

3

-

Definitions. Intrinsic, confinement and binding masses. Matter and antimatter

14

4

-

What we know about the sources of the three types of masses

20

5

-

Meson-antimeson and baryon-antibaryon mass differences

22

6

-

The value of AmK

K

and CPT invariance in the "intrinsic" quark

masses 7

-

26

The experiment of Gabrielse et al. and its relevance for the validity of CPT invariance in the "intrinsic" quark masses

29

8

-

The validity of CPT invariance in the nuclear "binding" mass

31

9

-

Antimatter in cosmology and in space

33

Conclusions

35

10 -

APPENDIX 1: Matter, mass, energy and charges: Origin and status

37

APPENDIX 2: Antinuclear matter, isotopic spin, gauge and planet

39

APPENDIX 3: From (9° 9"°) and (9° 9°), to (K°K°), (K+K ) and (KLKS) ...

40

4

ANTIMATTER PAST, PRESENT AND FUTURE Antonino Zichichi INFN and University of Bologna, Italy CERN, Geneva, Switzerland World Federation of Scientists - Beijing, Geneva, Moscow, New York

1 —

Introduction In K-meson physics, CP invariance is violated. This is probably the reason why it is

often stated that the non zero value of the mass difference between the long- and the shortlived K -mesons, AmK value of AmK

K

K

* 0 , is evidence for "matter-antimatter" asymmetry. Since the

is extremely small, it is argued that there is no point in performing

experiments able to detect mass differences, Am, if their sensitivity cannot compete with Anw

K

. None of these points is correct. In fact AmK

K

is a mass difference between two

states of particles, K L and K s , each one being a meson-antimeson mixture, and provides only an upper limit on A m ^ , the mass difference between a meson and its antipartner. This KK

upper limit is not between matter and antimatter. We will see that matter consists of masses and charges. There are in fact three types of masses: "intrinsic", "confinement" and "binding". We will see the reasons why Am - cannot be considered an example of matterKK

antimatter mass difference. Moreover, only the violation of CPT invariance would give rise to A m ^ * 0, not the violation of CP symmetry. On the other hand, if CPT invariance KK

would break down, it could happen that Am - be very small and even zero, while large KK,

effects be present in other types of mass differences. For instance in the nuclear binding, the first example of which is the mass difference between the deuteron and the antideuteron Am =• . The effects of CPT breaking on the value of Am^r could be very large and have no consequences on the present upper limit on A m ^ and on the value of AmK

K

. The need

for new experiments is emphasized and the distinction between the various sources of masses, ["intrinsic" (i.e. the fundamental fermion masses), QCD "confinement" and nuclear "bindipg"] is illustrated, together with the development of the concept of matter and antimatter during the seven decades following the discovery by Dirac of his equation. In the context of new experiments on the validity of CPT invariance, it is pointed out that, in order to have symmetry between matter and antimatter, all processes responsible for the existence of the constituents of matter and of their interactions have to obey CPT invariance, but the validity of CPT is nowadays "effective", no longer "fundamental" as it was when discovered. 5

6 Purpose of this review is to put in clear terms the reasons why it is important not to abandon the experimental checks of CPT invariance in the field of mass measurements. There are in fact four types of masses: two are positive, two are negative. The validity of CPT should be checked for each of these four sources of masses. The two positive sources are: the intrinsic one, which refers to fundamental fermion masses (quarks and leptons) and the QCD confining effects which contribute to the masses of mesons and baryons. In the case of baryons, the mass originated by the QCD confining forces exceeds the mass possessed by the fundamental fermions (quarks of the first family) by at least two orders of magnitude. The negative masses are in the nuclear "binding" energy of nuclei and in the "electromagnetic" binding energy of the atoms. Since CPT loses its foundation at the Planck scale, and no one knows either the origin of the fundamental fermion masses (quarks and leptons), or how to derive, from first principles, the QCD confinement masses and the nuclear binding, possible CPT breaking effects could be present in the mass difference between the fundamental fermions Amff as well as between baryons and antibaryons, meson-antimesons, and between nuclei-antinuclei. Mass differences like Amff can be washed out for mesonic states (made of quark-antiquark pairs) and have to be investigated via the study of the mass differences between baryons and antibaryons (made of three quarks and antiquarks, respectively). In chapter two the development of the concept of antiparticle and the birth of the concept of antimatter with its properties and problems are reviewed, together with the basic conceptual steps which occurred during the seven decades since the discovery of the Dirac equation. The third chapter illustrates the reasons why the terminology should be clearly defined in order to disentangle three distinct problems in the symmetry or asymmetry between fundamental fermion-antifermion, particle-antiparticle and matter-antimatter states. The meson-antimeson, proton-antiproton and deuteron-antideuteron mass differences are discussed. The fourth chapter briefly reviews the different sources of masses: intrinsic, confinement and binding. The fifth chapter is focused on the basic difference which exists between a mesonic and a baryonic state, since a meson is made of a quark-antiquark pair, while a baryon consists of a quark triplet. This is of great relevance when mass differences between mesonsantimesons and baryons-antibaryons are used to check CPT invariance. Chapter six is devoted to a discussion of the real meaning of the mass difference between the long and the short lived neutral K-meson: Am,,. K . It is pointed out that the non

7 zero value of this mass difference can be used to establish the validity of CPT symmetry on the "confinement" masses, but neither on the "intrinsic" nor on the nuclear "binding" ones. In other words Anw

K

* 0 has no consequences on the matter-antimatter asymmetry. In

fact, the value of Am „ K

would remain as it is, even if there were a perfect symmetry

between the mass of the fundamental fermions and antifermions, i.e. Am = (m - m^ ) = 0 (where i stands for the "flavour" of the given quark, q). On the other hand the value of Anw „

would remain as it is if there were an asymmetry between the mass of the

fundamental fermions-antifermions, Amj * 0, many orders of magnitude higher than the present known upper limit. And even if there were a large mass difference between nuclear matter and antimatter, AmK

K

would remain as it is.

Chapter seven deals with the high precision measurements on the charge over mass ratio ( T T ) for protons and antiprotons. These data can be used to establish, at the level of three parts in a hundred billion parts, the validity of CPT invariance in the intrinsic quark masses. In chapter eight the problem of the validity of CPT invariance in the nuclear binding which produces negative masses is discussed. Here the CPT breaking could be very large, compared with the other known limits, since these limits are not in the physics of the colour neutral QCD non-perturbative effects. Thus the need for new experiments to effectively measure if matter-antimatter symmetry really exists, is emphasized. The problem of antimatter in cosmology and the search for antimatter in space are in chapter nine. Conclusions in chapter ten.

8 2 —

The origin and the seven decades of developments The development of our understanding in the field of what is generally called "matter"

and "antimatter" spans nearly 70 years of physics research. It all started in 1927 with Paul Adrien Maurice Dirac, his equation [1] and the discovery by Herman Weyl of the Electric Charge Symmetry Operator C [2]. It went on with the discovery by Wigner [3], Wick, Wigner and Wightman [4] of the Parity Symmetry Operator P; and also with the discovery by Wigner [5, 6], Schwinger [7] and Bell [8] of the Time Reversal Symmetry Operator T. The birth of the Symmetry Operators C, P, T, and the construction of the mathematical formalism able to describe a fundamental force of Nature using the concept of a quantized local field obeying the laws of relativity, i.e. the Relativistic Quantum Field Theory (RQFT), culminated with the discovery by Pauli, Schwinger, Liiders, Zumino and Jost [9], of the CPT Symmetry, a fundamental property of all RQFTs. The final step is the unification of all gauge forces [10] and the proof by T.D. Lee (1995) [11] that at the Planck scale the CPT theorem loses its foundations. Let us recall: CPT invariance means that if we reverse all charges using the Symmetry Operator C, invert the three axes of space using P and the time arrow using T, there should be no change in the physics results obtained before and after all these inversions. A consequence of the CPT theorem is the following. If we define "matter" as a stable state made with masses coupled to additively conserved quantum numbers (e.g. electric charges, baryonic numbers, leptonic numbers, etc. ...), a CPT reflected state must exist with opposite values of all additively conserved quantities (electric charges, baryonic numbers, etc. ...). This reflected state is antimatter. In other words, the existence of matter implies, if CPT invariance is valid, that antimatter must exist and that the masses of matter and antimatter be the same. Suppose that only C invariance holds true. The existence of matter implies the existence of antimatter. But the reverse is not true. For example if C invariance breaks down, then the existence of antimatter is granted by the validity of CPT. Suppose that CP invariance holds true. The existence of matter implies again the existence of antimatter, but the reverse is not true. If CP invariance breaks down, here again the existence of antimatter is granted by the validity of CPT. When in 1930 the C operator was discovered, its validity was thought to be proved by the existence of the antielectron [12]. Later on, the equality of the lifetimes, T + =

x _ H n > of the positive and of the negative muons was also considered a further proof for the validity

9 of C invariance. This belief went on with the discoveries of the antiproton (p) by Segre et al. [13], of the antineutron (n) by Piccioni et al. [14] and even with the discovery (see later) of the second neutral "strange" meson (called at the time 9° ) by Lederman et al. [15]. While the validity of C invariance appeared to be experimentally established on firm grounds, and the CPT theorem was discovered, a "new era" started. Let us recall what the "old era" was. The edifice of antimatter was built on the validity of C invariance. In fact, as mentioned above, if C invariance holds in all physics processes, then the existence of the electron is sufficient to guarantee the existence of the antielectron, and the existence of a particle that of its antiparticle. Dirac in his Nobel Lecture [16] proposed, in addition to the existence of all antiparticles, that of antimatter, antistars and antigalaxies. The "new era" had an important (later proved to be false) component in the domain of the crisis for all known RQFTs. Before the "new era" started, there was a period of great enthusiasm and success in the description of: i) electromagnetic interactions (QED), thanks to the discovery of the renormalization; ii) weak and iii) nuclear interactions, respectively, thanks to the Fermi theory of the weak processes and to the Yukawa prediction of the ii-meson. After these original successes, serious difficulties came in. In QED with the Landau poles, in weak interaction with the unitarity at 300 GeV being violated, and in the field of strong interactions with the proliferation of mesons and baryons. This proliferation was totally out of any theoretical understanding in terms of a RQFT. Moreover a new mathematical formalism came into the limelight: the Scattering matrix. The S-matrix was the negation of the "field" concept and needed only analyticity, unitarity and crossing. The "new era" had its real fundamental contribution in the sector of the breakdown of all Symmetry Operators. In 1953 the ( 9 - x ) puzzle found by R.H. Dalitz [17] prompted T.D. Lee and C.N. Yang [18] to discover that there were no experimental checks for the validity of the Symmetry Operators C and P in the field of weak interactions. Within a year, C.S. Wu et al. discovered [19] that weak interactions do not obey C and P invariance, but CP seemed to be conserved [20]. By the end of 1957 the world appeared to have, as counterpart, the antiworld made like our own world with all charges and parity reversed. So, antimatter, antistars and antigalaxies could still exist with just one little additional detail: all intrinsic charges had to be reversed, together with the parity of each state. But in 1964, J. Christenson, J.W. Cronin, V.L. Fitch and R. Turlay [21] discovered that the combined Symmetry Operator CP was not valid in the decay of the neutral K-meson. T.D. Lee, R. Oehme and C.N. Yang (LOY) [22], before the experimental discovery by

10 C.S. Wu et al. [19] that, in weak interactions, the two Symmetry laws C and P were fully violated, pointed out that the existence of two "strange" mesons, 9j with decay mode 2 it and lifetime x =. 10

sec, 0 2 with decay mode 3 jt and much longer lifetime, could not be

considered a proof for the validity of either C invariance or P invariance or CP invariance. The (1957) LOY paper [22] anticipated the 1964 discovery [21], i.e. that the CP Symmetry is violated. In fact the LOY paper established, on firm grounds, the clear distinction between meson-antimeson mixing and CP invariance [see Appendix 3]. The breakdown of CP implied that also T had to be broken if CPT had to remain a fundamental invariance. For some founding fathers of our physics, the time reversal invariance was sacred. Thus CPT had to be broken. After all, why should we believe in CPT if RQFTs were encountering big difficulties? The "new era" had no need for the validity of the Symmetry Operators nor, apparently, for the description of the fundamental forces in terms of a RQFT. What about matter-antimatter symmetry? The discovery of CP violation was announced at the first High Energy Conference, held in the USSR, where I was reporting on the results obtained at CERN with the most intense beam of negative particles implemented in order to study the time-like structure of the proton [23, 24], to search for a new Heavy Lepton [25] in addition to the muon, and to study the existence of nuclear antimatter at a level ten times better than previous searches. The purpose of this negative beam was to compete with the physics of "Bubble-chamber" which dominated nearly all the physics research activities of the time. The breakdown of the Symmetry Operators (C, P, CP) and the apparent triumph of the S-matrix were coupled with the lack of evidence for the first example of nuclear antimatter (the antideuteron) found not to be there at the 10

level: i.e. no antideuteron

observed despite the 10 pions, produced in the same interactions. Question. If C invariance is broken, what is expected to happen to the existence of antimatter? The answer — as said before — is not that antimatter should not be there, but that its existence is granted by the validity of CPT. And what about the well established equality between x^+ and x^- ? The answer is the same. The equality X^i+

— T^-

is granted by the validity of CPT invariance. Another question. Since CP invariance does not hold in K-physics, what happens to matter-antimatter symmetry? Is it broken? The answer is no, since — as mentioned earlier — matter-antimatter symmetry, despite the breaking of CP invariance, is granted by the validity of CPT. In other words it does not matter if the Symmetry Operators, C, P, and CP

11

are broken. The existence of antimatter depends on the validity of CPT. But the validity of CPT was (and is) deeply rooted in RQFT. Here comes the key point: the theoretical description of the strong interactions. No one knew how to describe the nuclear forces in terms of a mathematical formalism of the RQFT type. We have said that those were years of triumph for the negation of RQFT: i.e. the S-matrix theory. Since this was the status of the strong interactions, the existence of the antideuteron could not be "theoretically" predicted. In fact, the strong interactions seemed to be dominated by the S-matrix theory and the Symmetry Operators C, P and CP were all experimentally found to be broken with the T Symmetry Operator imagined to have a privileged role. All this conspired towards a breakdown of CPT invariance. Table 1 illustrates the basic steps, during seven decades, from the antielectron and the properties of an antiparticle to the properties of antimatter and the unification of all gauge forces (QED, QFD, QCD). It is the unification of all forces at E G U = 10 16 GeV, and eventually at the Planck scale, which creates problems to the validity of CPT [11]. Let me elaborate a bit on these developments, starting with the discovery of the first example of nuclear antimatter (or, better, "antinuclear matter": see later). For this to happen, two basic technical developments were needed. One was the most intensive beam of negative particles we had built at CERN. The other was the most precise Time-Of-Flight (TOF) device we had implemented [26] to reach a time accuracy of ± 100 psec (psec s 10" sec). These technical developments allowed us to improve, by an order of magnitude, the search for nuclear antimatter. And this is how — contrary to the general theoretical trend of the time and to the experimental failure to observe the first example of nuclear antimatter — we were able to prove that the antideuteron was produced at the level of 10" , thus providing a firm basis for the validity of the CPT symmetry in the theory of the nuclear forces, despite the fact that no one knew how to theoretically describe these forces. This was in 1965. The following years were even more dramatic for the validity of a mathematical description of the strong interactions in terms of a RQFT. Not only the Smatrix theory was dominating but an experimental discovery at SLAC came in. It was called "scaling" [27]: its physical implication was (and is) that the proton had to have elementary constituents (called partons: quarks and gluons) and that these, at high energy, had to be free inside a proton. On the other hand, it was experimentally found at CERN [28] that, in violent (pp) collisions, there were no free quarks produced. The general trend in the theoretical interpretation of these experimental results was that RQFT was not able to describe the physics of the strong interactions. The proof that the first example of nuclear antimatter exists was — during those years — the only source for confidence in a RQFT

12 description of the strong interactions. There were exceptions in the theoretical front. In the volume "The Discovery of Nuclear Antimatter" [26] Luciano Maiani says: «The discovery of the antideuteron gave more confidence to the search of a field theoretical basis for the strong interactions, that today seems so obvious to us». The most interesting of these searches was the one by G. 't Hooft who discovered in 1971 that the (3-function has negative sign for nonAbelian RQFT. By now — after many decades — our understanding of the nuclear forces is not in terms of a fundamental RQFT but in terms of an "effective" theory which should be derived from the fundamental one - QCD: i.e. the non-Abelian force acting between quarks and gluons. This is a RQFT; however, the discovery of the renormalization group equations (RGEs) generates the running with energy of the three couplings (ctj a 2 a 3) of the gauge forces (QCD, QFD, QED) towards a unique point at E G U = 1016 GeV where a

GU ~ 74 [10]

anl

^ eventually at the Planck scale. Here the CPT theorem loses its

foundation [11], therefore all gauge forces, despite being examples of Relativistic Quantum Field Theories (RQFTs), have no fundamental reasons to obey CPT invariance, if their common origin is at the Planck scale. How the loss of such a theoretical fundamental reason affects the low energy domain, where the colour-neutral-QCD-binding nuclear forces act, is not known. Only high precision experiments on the mass difference Am, between nuclei and antinuclei, as for example, deuteron-antideuteron, Am - , can determine these consequences. At present all we know is very simple: antinuclear matter exists despite that the theoretical foundations whose existence it was based upon have vanished.

13

SEVEN

DECADES FROM THE ANTIELECTRON TO ANTIMATTER AND THE UNIFICATION OF ALL GAUGE FORCES

• The validity of C invariance from 1927 to 1957 After the discovery by Thomson in 1897 of the first example of an elementary particle, the Electron, it took the genius of Dirac to theoretically discover the Antielectron thirty years after Thomson. 1927 -> Dirac equation [1]; the existence of the antielectron is, soon after, theoretically predicted. Only a few years were needed, after Dirac's theoretical discovery, to experimentally confirm (Anderson, Blackett and Occhialini [12]) the existence of the Dirac antielectron. 1930-1957 -* Discovery of the C operator (charge conjugation) [H. Weyl and P.A.M. Dirac [2]]; discovery of the P Symmetry Operator (Parity) [E.P. Wigner, G.C. Wick and A.S. Wightman [3, 4]]; discovery of the T operator (time reversal) [E.P. Wigner, J. Schwinger and J.S. Bell [5, 6, 7, 8]]; discovery of the CPT Symmetry Operator from RQFT (1955-57) [9]. 1927-1957 -» Validity of C invariance: e+ [12]; p [13]; n [14]; K° — 3JI [15] but see LOY [22]. • The new era starts: C * ; P * ; CP * ( , ) 1956 -» Lee & Yang P * ; C * [18]. 1957 -* Before the experimental discovery of P * & C * , Lee, Oehme, Yang (LOY) [22] point out that the existence of the second neutral K-meson, K 2 -» 3JI , is proof neither of C invariance nor of CP invariance. Flavour antiflavour mixing does not imply CP invariance. 1957 -» C . S . W u e t a l . P x ; C * [19]; CP ok [20]. 1964 — K° - . 2it M K L : CP * [21]. 1947-1967 -» QED divergences & Landau poles. 1950-1970 -» The crisis of RQFT & the triumph of S-matrix theory (i.e. the negation of RQFT). 1965 -» Nuclear antimatter is (experimentally) discovered [29]. See also [26]. 1968 -» The discovery [27] at SLAC of Scaling (free quarks inside a nucleon at very high q 2 ) but in violent (pp) collisions no free quarks at the ISR are experimentally found [28]. Theorists consider Scaling as being evidence for RQFT not to be able to describe the Physics of Strong Interactions. The only exception is G. 't Hooft who discovers in 1971 that the p-function has negative sign for non-Abelian theories [30]. 1971-1973 -» p = - ; 't Hooft, Gross & Wilczek. The discovery of non-Abelian gauge theories. Asymptotic freedom in the interaction between quarks and gluons [30]. 1974 -» All gauge couplings a, o^ c^ run with q 2 but they do not converge towards a unique point. 1979 -» A.P. & A.Z. point out that the new degree of freedom due to SUSY allows the three couplings a a . a , , to converge towards a unique point [31]. 1980 -» QCD has a "hidden" side: the multitude of final states for each pair of interacting particles: ( e + e _ ; pp ; up; Kp; vp; pp; etc.). The introduction of the Effective Energy allows to discover the Universality properties [32] in the multihadronic final states. 1992 -» All gauge couplings converge towards a unique point at the gauge unification energy: E G U = 10 16 GeV with aGV = ^ [33, 3 4 ] . 1994 —• The Gap [35] between E G U & the String Unification Energy: E s u ~ E P l a n c k . 1995 -» CPT loses its foundations at the Planck scale (T.D. Lee) [11]. 1995-1999 -» No CPT theorem from M-theory (B. Greene) [36]. 1995-2000 -* A.Z. points out the need for new experiments to establish if matter-antimatter symmetry or asymmetry are at work. (*)

The symbol ^ stands for "Symmetry Breakdown". Table 1

14 3—

Definitions. Intrinsic, confinement and binding masses. Matter and antimatter Richard Feynman would have said: «let us first agree on what we are talking about*.

In the field of "matter-antimatter" this is badly needed. We have learned during these seven decades since the discovery of the Dirac equation that matter consists of fundamental fermions (leptons and quarks) which show up as free objects if they are leptons or manifest in a confined state which can be made of a quark antiquark pair (mesons) or a three quark state (baryons). These states are confined by QCD colour forces. During many decades mesons and baryons have been called "elementary particles". We know they are not elementary but composite of quarks and gluons, nevertheless we can go on calling them "particles". Baryons and mesons interact in such a way as to build up the nuclei of the atoms. Matter and antimatter are made of some constituents and their antis, respectively. They are listed in Table 2. If a basic symmetry has to exist between matter and antimatter, from this symmetry necessarily follows the symmetry between the constituents and their antis. The reverse is not true. If a basic symmetry principle exists between the constituents and their antis, this does not imply that the symmetry can be extended to matter and antimatter. In fact, in order to have matter, we need to put together the constituents of which matter consists. In order to have antimatter, we need to put together the anticonstituents of which antimatter consists. The way we put constituents together depends on the interactions which are responsible for their existence and for their properties, including the mutual interactions between them. The same is true for the anticonstituents. charges;

antimatter consists of mass plus anticharges.

Matter consists of mass and The properties of matter and

antimatter include all these fundamental problems. If we apply the CPT operator to matter we will get mass plus anticharges. If the charges are of nuclear origin we will have antinuclear charges with the same mass, if CPT invariance holds. If not, the mass associated with the antinuclear charges will be different from the mass associated with the nuclear charges. We have also the case where the mass is associated with the electric charges.

All atoms are made of electrons glued

electromagnetically to nuclei. The antielectrons will be electromagnetically glued to the antinuclei. There is therefore an electromagnetic binding which ensures the existence of all atoms. The masses produced by the electromagnetic binding are negative, like the nuclear binding. The negative masses of nuclear and electromagnetic origin should be the same for

15

CONSTITUENTS AND ANTICONSTITUENTS FUNDAMENTAL FERMIONS AND ANTIFERMIONS Leptons (£) vP \ Ve

(

V(X

V

)

l

/

Quarks

and

HL

HL

;

c

(q)

/u \

I )

(t

Id ,

,s ,

UJ

t

\

© Antileptons (£)

Antiquarks

and

PARTICLES esons and Baryo ns (q q q)

(qq) and

ANTIPARTICLES Ai itimesons and Antibarycjns (qqq )

( qq)

NUCLEI

AND ANTINUCLEI

=

D

;

(p5)

( ann) =

H

;

(piin)

; P n)

( ppn) a

He

;

(ppn)

^ D 3 B

-

3

H

He

(q)

16 atoms and antiatoms only if CPT invariance holds for the electromagnetic and the nuclear forces. There is a difference between electromagnetic and nuclear forces. The electromagnetic forces are (fundamental forces)'*' described by a Relativistic Quantum Field Theory (RQFT), (QED); the nuclear forces are secondary effects originated by the fundamental colour force QCD acting between quarks and gluons inside the "particles". The nuclear forces mimic the existence of nuclear charges and of a nuclear field whose source was believed to be "elementary", the nucleon. We know that the nucleon is not elementary but composed of a ( q q q ) triplet. The quanta of the nuclear field were also thought to be elementary, the JI -mesons. We now know that not only the Jt-meson, but all mesons, are not elementary entities but composed of a (q q) pair. Nuclei exist if protons and neutrons can be bound together by nuclear forces. Proton and neutrons (nucleons) are QCD colour neutral, nevertheless these residual QCD effects generate attraction between nucleons. These attractive forces — before QCD was discovered — were called "Nuclear Forces" and, as mentioned above, were supposed to be originated by "Nuclear" charges. We know now that nuclear charges do not exist as fundamental charges. Their effects are the residues of QCD, once we go from the QCD colour-full world of quarks and gluons to the QCD colour neutral world of baryons and mesons. Nevertheless if we suppose that the QCD colour neutral residual effects behave as if nuclear charges exist, and apply the C operator to a "nuclear" charge, it reverses the sign of this "charge". If C invariance holds, the mass associated with the opposite nuclear charge remains the same as before.

Thus

an

antinucleus, composed of a number of antinucleons (larger or equal to two), has all nuclear charges reversed. The antinuclear charges are associated with the mass of the antinuclei. Thus the antideuteron is an example of antinuclear matter. When the anticharges are of electromagnetic nature, the appropriate terminology would be: antielectromagnetic matter. Since matter is made of mass plus charges, the terms antinuclear

and

antielectromagnetic specify the reversal of the charges and there is no reason to specify further this reversal with the term "anti" repeated for matter. Antinuclear antimatter is redundant, but to eliminate the redundancy the most appropriate choice is "antinuclear matter".

Certainly not "nuclear antimatter", as it is in the present-day terminology

(see

Appendix 2). The word "matter" should be used to specify the elements of the Mendeleev table. If

(*)

They originate from the fundamental gauge forces SU(2)xU(l) broken at the Fermi scale (see for example [30]).

17

matter was made only of Hydrogen, we would not need the nuclei. Hydrogen is the only element whose nucleus is a single particle, the proton, and the atom is obtained by adding to it a fundamental fermion, the electron. Matter^ consists of nuclei surrounded by electron clouds. In order to have an atom of iron we need the leptonic cloud (made of electrons) and the nucleus (made of baryons glued by mesons). To have anti-iron we need the fundamental antifermions (positrons), the antiparticles (antimesons and antibaryons) and the property that these antiparticles must interact like the particles in order to have the antinucleus of the iron. We cannot define antimatter as the fundamental antifermions and the antiparticles. We need to specify the property which allows all antielements of the Mendeleev table to exist. This property (in addition to the electromagnetic binding) is the nuclear binding which allows antiprotons and antineutrons (antinucleons) to form the antinuclei. Thus it seems appropriate to use the word antimatter when we refer to the specific need to build the antielements of the Mendeleev table. The nucleus of the anti-Hydrogen atom is an antiparticle (the antiproton). The simplest element where an antinucleus is needed is the antideuterium. This is the reason why the antideuteron ( p n ) is referred to as being the first example of "nuclear antimatter"; (as mentioned earlier, "antinuclear matter" would be more appropriate). To be more precise, in order to build matter it is necessary to have quarks and leptons (fundamental fermions), mesons and baryons (particles) and nuclei (protons and neutrons glued together). As shown in Table 2 there are three classes of constituents to which masses are associated: i) the fundamental fermions (quarks and leptons): f; i.e. "intrinsic" masses; ii) the particles (mesons and baryons): p A ; i.e. "intrinsic" plus "confinement" masses; iii) the nuclei (Deuteron, Tritium, Helium etc.): N ; i.e. "intrinsic", "confinement" plus the nuclear "binding" masses. To these correspond three types of anticonstituents. The three types of mass differences Am follow from these three classes and are: i) Amf7 (example: the mass-difference quark-antiquark, Am - ). ii) Am„P P= (example: the mass-difference meson-antimeson, A mK^K; or the >• r A

A

mass-difference proton-antiproton Am _). iii) Am^-: (example: the mass-difference deuteron-antideuteron, AmDg). An antiatom of iron would be made of an antinucleus surrounded by a cloud of (*)

The word "matter" should not be used to indicate "particles" (mesons and baryons) or fundamental fermions: quarks and leptons.

18

antielectrons. To its mass, all types of masses will contribute: "intrinsic", "confinement" and "binding" (nuclear plus electromagnetic). In fact the mass of the antiatom of iron would have: i) a component which is in the antinuclear mass of the anti-iron nucleus; and ii) a component which is in the antielectromagnetic mass of the antielectrons surrounding the antielectric charges of the antiprotons in the nucleus (in addition to other possible effects originating in the antielectromagnetic properties of the antineutrons in the same nucleus). If all forces, including both the residual colour-neutral QCD forces, called nuclear forces, and the QED forces, obey CPT symmetry, the mass of the anti-iron element of the Mendeleev table will be exactly equal to the mass of our iron. Thus matter and antimatter would be identical. But if CPT breaking mechanisms are active in any of the processes needed to build up the antiatom of iron, the matter-antimatter identity will be lost. In Table 3 there is a synthesis of all quantities needed to build Matter and Antimatter. In the column "masses", it is shown where the various masses come in.

19

Table 3: If the Hydrogen atom were the only element in the Mendeleev table, then the only form of antimatter would be the anti-Hydrogen atom. There are, by now, 118 elements in the Mendeleev table and the first element, Hydrogen, is the only example of atom made of two elementary particles: the proton and the electron. In order for (what we call) matter to exist, it is necessary that protons and neutrons can attract each other to form nuclei.

20 4—

What we know about the sources of the three types of masses To sum up, our understanding of the physical quantity called mass can be described in

terms of three sources: i)

the "intrinsic" mass of the fundamental fermions (quarks and leptons); the origin of these "intrinsic" masses is not known but is certainly not due to Quantum ChromoDynamics (QCD).

ii)

the "confinement" masses : m B a g , due to QCD confining colour forces; these effects produce positive masses.

iii) the "binding" masses which can be either of electromagnetic or of nuclear origin. These effects produce negative masses. All these sources of masses have different origins. Two of them, the QCD "confinement" masses and the electromagnetic "binding" masses, are associated with well known forces, QCD and QED. The other two ("intrinsic" and "nuclear binding") have some peculiar features. We briefly review all of them. "Intrinsic mass". It has still not been established how the masses of the fundamental fermions (quarks and leptons) are generated and what is the energy scale where this mechanism becomes operative. It could be the so-called "Higgs" mechanism and it could also be a new force of nature operating at the Planck scale. If it is at the Planck scale, here the CPT theorem loses its foundations [11] and it cannot be excluded that a CPT breaking occurs. Since no one knows either the energy scale where the masses of the fundamental fermions are generated, or if quark and lepton masses have the same origin, it cannot be excluded that at the energy scale where these masses are generated — for example the SUSY breaking scale — the CPT invariance breaks down. It is therefore worthwhile to investigate if the "intrinsic" masses obey CPT invariance. "Confinement mass". We will assume for simplicity that the "confinement" masses respect CPT invariance. Since QCD is a Relativistic-Local-Quantum-Field-Theory (RQFT), it could appear reasonable to expect that QCD respects CPT invariance [37]. But the convergence of the gauge forces at the Planck scale creates problems with the foundations of CPT [11]. On the other hand, the "confinement" mass is a QCD non-perturbative phenomenon and no one knows how to deal with these divergent effects. The "binding" masses have two origins. Electromagnetic "binding" masses. All atoms exist because of the electromagnetic binding effects, which produce negative masses.

Their origin is in Quantum

ElectroDynamics (QED). If QED were a RQFT without the problems created by the

21 convergence of the gauge forces at the Planck scale, here again it would be reasonable to expect no CPT breaking effects. Nuclear "binding" masses. All nuclei exist because of the nuclear "binding" forces. These are residual QCD effects, acting between neutral colour quark-triplets (qqq), the nucleons, and neutral colour quark-antiquark pairs (q q), the mesons. Nuclear physics is based on the interaction between nucleons mediated by mesons. Before the discovery of QCD [32], nuclear physics was thought to be an example of a RQFT; by now we know that the origin of nuclear forces is the non-Abelian theory describing the interaction between quarks and gluons, Quantum Chromodynamics, QCD; nevertheless no one has been able so far to derive the nuclear forces from QCD. What is clear is that what we call nuclear processes represent the transition from the QCD world (coloured quark and gluons) to the QCD colour neutral world of baryons and mesons. Since no one is able to describe this transition, it is probably wise to perform some experimental checks on the validity of CPT invariance in the nuclear "binding" masses, which are negative. The conclusion is that, out of the four types of masses, two of them appear to be associated with problems still to be understood. These two are: i) the exact origin of the intrinsic quark masses and ii) the nuclear binding (negative) masses which are associated with the transition from the QCD colour-full world of quarks and gluons to the colour neutral world of mesons and baryons. This is why it is important to check CPT invariance in the field of "intrinsic" and of nuclear "binding" masses. Let us start focusing our attention on the "intrinsic" quark masses and on their CPT invariance. The first point to clarify is the basic difference between a meson and a baryon.

22 5 —

Meson-antimeson and baryon-antibaryon mass differences A meson is made of a quark-antiquark pair, while a baryon consists of a quark triplet.

This difference is particularly relevant when mesons-antimesons and baryons-antibaryons are used to check CPT invariance via the measurement of the masses which make up the mesons, the antimesons, the baryons and the antibaryons. The first input we need is how to construct the mass of a meson and the mass of a baryon. The simplest proposal is to assume that the mass of these bound states is the sum of the masses of the various components, as it is the case for the "binding" masses' '. For the "binding" masses, "electromagnetic" and "nuclear", it is taken for granted that the "binding" masses (always negative) are the result of the sum of all masses of the free particle states minus the mass of the bound state, be it a molecule, an atom or a nucleus. When dealing with molecules and atoms, the intrinsic "leptonic" masses come in. These are the only type of "intrinsic" masses accessible to direct measurements since leptons exist as free states. Quarks exist only as bound states of mesons and baryons and their masses can only be indirectly determined. When dealing with mesons and baryons, life is not easy because we have no way to directly measure the masses of their constituents, quarks and antiquarks, since they do not exist as free states. What about the linear versus the quadratic expressions for the fermionic and bosonic masses?

In our Lagrangians the fermionic masses appear linearly, but the bosonic ones

quadratically: the origin being in their respective equations of motion. Since there is no theory describing CPT breaking effects in the masses of either fermions or bosons, we will consider the linear case for both fermions and bosons and, for bosons, also the quadratic expression. The linear case for bosons will be in § 6. The reason is that we want to have at least a preliminary idea on what could be the consequences of CPT breaking for the masses of mesons and antimesons, baryons and antibaryons. In fact, as pointed out earlier, a meson is a boson and it is composed of a quark and an antiquark, while its antipartner is again an antiquark-quark pair. So in the bosonic state we have the presence of both a quark and its anti. We will see that, if the CPT breaking has some simple property, its effects on the masses of quarks and antiquarks may be washed out for mesonic states and be present in baryonic states, since a baryon has no antiquarks in its

(*)

We are aware of the fact that this is a severe simplification. Masses originate from the Hamiltonian which is an Operator, not a C number.

23 composition. This means that we cannot neglect the basic structure of a meson and of a baryon when searching for CPT breaking effects in their masses. It is generally stated that the CPT invariance in the field of masses has its best check on the very high precision determination of the mass difference between the neutral K-meson, K , and its antipartner, K . There is in fact an upper limit oni this this mass difference | mKo - mR0 |

10 = Am KR * 4 x l(T-10 eV/c 2

(1)

which is derived from the mass difference [38] between the long-lived and the short-lived neutral K-mesons: (m K0 ~ n v )

-

AmKLKs = (3.491 ± 0.009) x 10"6 eV/c2 .

(2)

When the result (1) is compared with the value of the K -meson mass, the result is -

^

*

ID"18 •

(3)

K

If the result (3) were not linked to the mesonic mass, this would allow to conclude that CPT invariance is checked at the level of a few parts in 10 18 parts in the field of massmeasurements. This is not the case; it could very well happen that the result (3) be exactly zero, despite CPT invariance not being valid in the field of "intrinsic" and of nuclear "binding" masses. The reason being that a meson is composed of a quark-antiquark pair, and its antipartner is again an antiquark-quark pair. To avoid this cancellation, the best way to check possible CPT breaking effects is by comparing states composed of quarks, such as a baryon (qqq), with states composed of antiquarks, such as an antibaryon (qqq). If we compare baryon (qqq) and antibaryon (qqq) masses, the possibility of cancellation effects, which are present when comparing a meson (qq) with its antistate (qq), does not exist. According to a QCD model proposed by G. 't Hooft [39], the mass squared of a meson, (qq) pair, could be expressed by the sum of the intrinsic quark masses times a QCD factor, K,?™ n e m e n , coming from QCD confining forces acting on the quarks. For a baryon, quark triplet (qqq), the 't Hooft model [39] would suggest again the sum of the various quark masses plus a mass term F 0 ™ i n e m e n due to the QCD confining effects. It can be argued that mesonic as well as baryonic masses are linked to eigenstates and that the eigenvalue of the given mass-matrix could have strong contributions from offdiagonal elements of the same mass-matrix. On the other hand, it cannot be excluded that CPT breaking effects have consequences that are fundamentally different for a mesonic and a baryonic state. As emphasized above, a meson is made with a quark-antiquark pair, (qq),

24 while a baryon consists of a quark triplet (qqq). If our simple hypothesis is followed by nature, then care is needed when dealing with the experimental results on CPT invariance coming from the mass difference measured by comparing the mass of a meson and its antipartner. Following G. 't Hooft [39], the masses of a meson (qfL) (and of its antistate (c^q,)) are given by c™

\ 2 _ ,

(mq.q.) where m mand

• „

\ ^Confinement

= (m q . + m - )

=

the mass of the quark flavour "i"

=

the mass of the antiquark flavour " j "

KOCD

=

K ^

a factor — with dimension of mass — originated by the QCD

confinement forces needed to keep the (qjCh) pair into the bound state called meson. We assume that K ^ i n e m e n t is CPT invariant^. If this is the case, when we work out the meson-antimeson mass difference we have: \2 m qi

qj)

/„

\2T

r/

\

/

M ^Confinement

" ( m q i q j ) ] = [( m qi " mq;) + ( ^ .

=

" « , . )] K Q C D

[(Amqi-)-(Am-jq.)]^inemeni.

If CPT invariance holds for the "intrinsic" mass differences, we expect Am q = zero and Am q j - = zero . If CPT breaking effects are present in the intrinsic quark masses, we expect Am

qiqi

"

°

and Am q .-. *

0 .

Suppose that CPT breaking lowers the antiquark masses (assuming m

<m-

would

not change the conclusion) m qi

>

m-.

and suppose that the CPT breaking is "flavour" independent. In this case Am

(*)

qiqi

= ^qjqj =

Am

"

We make this assumption in spite of the fact that KJ°" ",emenl does contain higher order contributions from the quark masses. Nevertheless we are assuming for simplicity that these contributions in K are CPT invariant.

25 If this is true, we expect the difference between the mesonic and the antimesonic masses to be given by r

m

2

[ m(q;qj) "

m

2

i

r

.

(q iqj )] = [ ^

A

"

Am

i ^Confinement

1 KQCD

=

ZeTO

'

This cancellation effect does not exist when we compare baryonic and antibaryonic states. In fact, the masses of a baryon and of its antibaryonic partner are expected to be, following 't Hooft [39] : m m

_

/ -,

-.

,

q iqiqj

=

(2mq. + m q .)

,

+

FQCD

qiqiqj

_ =

/~ <^

+

^Confinement F QCD

, +

s *% >

~ Confinement

where K™ D ' nemen is a mass term factor due to the QCD confinement forces which act on the quark triplet (qqq). As before we assume the same flavour invariant CPT breaking and no CPT breaking for F Confinement The mass difference is given by: (<\qiqj "

m

qiqiqj > = P K

" "Mi > + K j " "Mj ^ =

3

^

'

The conclusion of this section is that, if we want to search for CPT breaking effects in the "intrinsic" quark masses whose origin is unknown, it is dangerous to study mesonantimeson mass differences while it is safe to study baryon-antibaryon mass differences. We now turn to the specific cases where upper limits on mass differences have been experimentally measured: KK, pp .

26

6 —

The value of AmK .j and CPT invariance in the "intrinsic" quark masses The starting point is the experimental determination of the upper limit Am K - = | m ^ -m R o

| < 4 x 10-1° eV/c2 ,

(1)

and the fact that a K-meson depends on the "intrinsic" quark masses and on the QCD-Bag effects. An assumption different from the one in § 5 is that the mass of the K-meson and that of its antiparticle state K° are given by the sum of the various components J

mRo = md

d?

T

+ ms + m ^ +

m

di

Rad mjs

(4)

j

where "d" and "s" indicate the quark flavours and where, md

= the mass of the fundamental fermion "d".

nig

= the mass of the fundamental fermion "s".

mj

a the mass of the fundamental antifermion "d".

m-

m the mass of the fundamental antifermion " s". a

md |

s

the QCD-Bag energy needed to confine the pair "d s ".

mj^

s

the QCD-Bag energy needed to confine the pair "ds".

mRad

m

rat ji a tiye

corrections.

Formula (4) makes sense if quarks are non-relativistic. If the interaction responsible for the intrinsic mass of a quark is CPT invariant, if the QCD confining effects and the radiative effects are all CPT invariant, the result is expected to be (mKo - mRo) = ArnKR = z ero If there are CPT breaking effects on the fundamental fermion masses and no effects on QCD confining forces and on radiative effects, then, from (4) we see that the only quantities left for CPT breaking are the "intrinsic" quark masses. The simplest CPT breaking possibility is that all Artij are equal: i.e. that the CPT breaking mechanism is "flavour" invariant: all flavours differ from their antiquark states by the same amount A m: mq.

-

m-

±

Am; .

27 Out of the two possibilities: either m

> m ;

m

=

m

qi

or +

qi

m

Am

< m - , we choose the first one: (5)

i •

In both cases, the final result would be the same; choosing m Am

K°K°

=

m

K° ~

despite the fact that m

m

R°

*

=

(md~

m

+

d )

( m ?~

m

s)

=

> m- , we have:

Am

d ~

Am

s

=

0 (6)

m- as in (5). This is because a meson is a mixture of a quark

plus an antiquark and we have adopted the simplest CPT breaking mechanism. Since the CPT breaking effects in the "intrinsic" quark masses can be washed out by the simplest CPT breaking mechanism as shown in (6), the limit (1) cannot be used for the validity of CPT invariance. Therefore the ratio i^KL m K

~

1 0 -18

(7)

may not necessarily represent the upper limit for the validity of CPT invariance in the field of "intrinsic" quark masses. In § 5 we have considered a QCD model due to G. 't Hooft [39]. According to this model the mass of a K-meson is given by the formula m

(md

K° =

+

m

( M QCD +

l)

0(mu,md,m-))

where O is a complicated function which starts linear in m u , m d , m s . A simplified version of this formula is: ™2„ mKo

where K 0 p n n i n e m e n

=

(„

. „,- \

^Confinement

/a^

(md

+

KQCD

(8)

ms)

is a factor with dimension of a mass which represents the effects on the

K-meson mass of the QCD confining forces. Assuming, as before, the simplest CPT, flavour invariant, breaking mechanism the result is 2 (mK° -

m

2 K°)

= [ K "

1

^ )

+

(m7-ms)]

K

QCD = ( A m _

Am

)

K

QCD =

zer0

-

(9)

Thus, also in this case (8), the result is: Am

KK

=

zero

'

despite CPT breaking effects in the intrinsic quark masses. Of course, if other mechanisms are at work, our conclusions (6) and (9) are to be revisited. However, since the crucial issue is CPT invariance in the "intrinsic" quark masses,

28 it is obvious that the result (7) cannot be taken as a proof for the CPT invariance in the "intrinsic" quark masses. The reason being, as repeatedly emphasized, that a meson is a mixture of a quark plus an antiquark. To search for a CPT violation in the "intrinsic" quark masses, it is necessary to study those particle states which consist of quarks only (such as the baryons), and not of quark antiquark mixtures (such as the mesons), i.e. in the physics of protons and antiprotons.

29 7 —

The experiment of Gabrielse et al. and its relevance for the validity of CPT invariance in the "intrinsic" quark masses Following what has been said in § 5, when the lightest nucleon (the proton) is

considered, its mass can be given by the very simple expression: 2mu

+

md + m ^

+ mu™d

(10)

where: i)

m , m . are the intrinsic masses of the elementary fermions, the quarks of the first family;

ii)

m

Bag

is the mass produced by the QCD colour forces acting between

quarks and gluons and confining them within the proton radius; iii) '

m ^ , is the effect of the radiative corrections on the mass. uud

The same parts appear in the mass of an antiproton: m_ = 2 m _ + m - + m -?ai p

u

m Rad .*.,

t

u u d

d

u u d

If CPT invariance holds for all effects, we expect: Am A _ = m PP

P

- m_

P

the experimental limit is [ 3 0 ] : Am _ = PP

(22 ± 40) eV/c 2 . '

As for the meson case, we assume the simplest CPT breaking effect: i.e. all quark flavours differ from their antiquark states by the same amount Am : m. Hi

=

m^

±

Am

Mi

i

and no effects of CPT breaking are present in the QCD-Bag which produces "confinement" masses and in the Radiative parts. Out of the two possibilities, m

> nfc or m Hi

we choose m

4i

< m-j. , Hi

Hi

= m^. + Am: . In both cases the final result would be the same. Hi

Hi

If this type of CPT violation occurs, we have: A m p - = (m p - m ? ) = ( 2 2 ± 4 0 ) e V / c 2 = 3 A m ^ .

(11)

Assuming the simplest CPT breaking mechanism quoted above, the result (11) is a direct experimental determination of a mass difference between the fundamental fermions of the first family and their antistates. For the sake of the argument, let us suppose that its value is Am p p = 60 eV/c 2 .

(12)

30

In this case, Am q - = 20 eV/c2 .

(13)

Since free quarks do not exist, we can only measure indirectly the quark masses; they are called "current" masses and the present estimates for mu and m^ are in the (5 + 10) MeV range; the result (13) gives f ^ ^ r I mq

i n S i C

2 eV/c2

<~

° , fi (5 + 10) x 106 eV/c2

)

= (4 + 2) x 10-' .

(14)

Recently, a high precision experiment [40] on the ratio (^) (electric charge over mass) for protons and antiprotons has established the equality of these ratios to within nine parts in a hundred billion parts

(M)„ -Qf-

=

1 ±

9 x 10-11 .

(15)

KM)V

This experimental result is the most precise CPT test using baryons. The result of Gabrielse et al. [40] can be used to establish a better limit than (14) on (Am) experimental accuracy is ± 9 x 10

since here the

. We thus have, if we assume that CPT invariance

holds for the equality of the electric charges associated with the proton and the antiproton, the final result: Am

.Intrinsic

**&-) m

< 3x10-",

(16)

q

which is the best limit on the validity of CPT invariance in the processes responsible for the production of the "intrinsic" quark masses. New experiments to continue improving the present knowledge on the validity of CPT invariance in the field of "intrinsic" masses should be encouraged. We now turn to the case of nuclear "binding" masses.

31

8 —

The validity of CPT invariance in the nuclear "binding" mass The first example of an interaction where the nuclear "binding" negative mass is

active is in the case of the deuteron, a nucleus made of one proton and one neutron. Suppose that the source of the nuclear field was the "nuclear charge" of the nucleon (proton or neutron) and the field quantum was the pion [41]. If this was true, the next step would be to theoretically formulate in terms of a RQFT the interaction between an antiproton and an antineutron like the interaction which allows the deuteron to exist. This is not the case: we do not have a fundamental RQFT formalism describing the nuclear forces. No one has so far succeeded in deriving this force from the fundamental one: QCD. Since CPT is deeply rooted in the mathematical structure of a RQFT, we have no way to know if the transition from the QCD world (quark and gluons) to the world of baryons and mesons will or will not be described by a RQFT. To check if CPT invariance holds in the nuclear "binding" masses is therefore fully justified. To this end we need to study the mass difference between nuclear matter and antimatter. As mentioned earlier, the most elementary example of nuclear matter is the deuteron (pn) whose counterpart is the antideuteron (pd). A very simple approximation to their masses is: •~/r\\ — «,

-L ™

_

™ Binding

.

__ Rad

6 m(D) = m + m m + m v / p n p n pn in ding m(D) = m_p + m_n - m ?p n+ mM v / pn

.

In addition to the particle masses (m , m ) and the antiparticle masses (m - , m - ), we now have the nuclear binding effects, m BlndmS and m - n - n g , which, contrary to the b

pn

p n

J

QCD "Bag" effects (that produce positive masses), subtract mass to the (p n) and (p n) systems, respectively. If all these processes are CPT invariant, we expect the mass difference between the deuteron and the antideuteron to be zero ^ D D

=

m

D "

m

D

=

Z6r0

'

The experimental limit is [30] : AmDg = zero ± 80 MeV/c2.

(17)

The result (17) implies AmmL < mD5 ~

80MeV/C2 2000 MeV/c

= 4 x 10 -2

(18)

32 This is the best determination of CPT invariance when nuclear matter and antimatter are concerned. Since CPT, within the limits given by (15), is valid in baryon antibaryon physics, the result (17) can be used to estimate the validity of CPT invariance in the nuclear binding mass, which is in the MeV range. Thus: I

Ampp

\

DD nuclear binding mass

1

1(T

(19)

The conclusion is that there is practically no limit on CPT invariance in the field of nuclear binding masses, since the experimental uncertainty is bigger than the binding mass itself. It should be emphasized that the nuclear "binding" mass is the result of a transition from the QCD colour-full world of quarks and gluons to the colour neutral QCD world of baryons and mesons: a field where a series of new experiments is being implemented [42] and where, despite all efforts, no one is able to formulate in terms of a RQFT this new state of subnuclear matter, called "quark-gluon plasma".

An exact determination of the mass

differences between nuclear and antinuclear matter states would allow to check CPT invariance having as reference a physics quantity — the mass — which is strongly correlated with this field of physics still to be understood.

33 9 — Antimatter in Cosmology and in Space Modern cosmological theories claim to be able to account for the nine orders of magnitude which exist between the predictions of the simplest baryon symmetric big-bang model and real life. We exist because the ratio of baryons to photons is of the order of 10"9. On the other hand, according to the simplest baryon symmetric big-bang model, when the universe is expanded and matter is "frozen-out", only one residual baryon or antibaryon survives out of 10 18 and therefore the ratio of baryons to photons should be 10" 18 . Before the discovery of CP violation [21], the universe was "theoretically" predicted to be with an equal number of galaxies and antigalaxies, well separated in distinct volumes, to avoid annihilation. It is the discovery of CP violation [21] which prompted A. Sakharov [43] to imagine a mechanism in order to have a matter-asymmetric universe. For this to happen, it is necessary that a violation of both C and CP takes place, in addition to baryon non-conserving interactions and a departure from thermal equilibrium. These three Sakharov conditions do not necessarily end up with a matter-dominated universe. This is one option. The other option is that the CP symmetry violation occurs with opposite signs, thus producing well-separated microscopic domains, fully dominated by baryons or antibaryons. The inflation mechanism would later expand these microdomains into astronomical dimensions. The."theoretical" predictions for the existence of galaxies and antigalaxies cannot be compared in rigour with the "theoretical" predictions based on the validity of the CPT theorem as it was the case for the problem of the existence of matter and antimatter. The crisis of CPT was a fundamental one, during the triumph of the S-matrix theory. The new CPT crisis is again of fundamental origin and it arises from the convergence of the gauge couplings at the Planck scale. The crisis of the cosmological theories predicting a matter-dominated universe is by far less rigorous, as we have seen with the Sakharov model. Only experiments can establish on firm bases if our universe is matter-antimatter symmetric or not. This is the reason why an experiment, called AMS (Alpha Magnetic Spectrometer) has been designed by S.C.C. Ting [44] in order to search for antimatter in space. If the CERN group [26] had proved in 1965 that antimatter did not exist, it would not have been possible to think of the AMS Project [44]. Thirty-five years ago, to establish the existence of antimatter had to do with the CPT theorem. Now, the existence of antimatter in Space has to do with the beginning of the universe and its evolution. Despite these nine orders of magnitude quoted above, the fashionable theoretical predictions are for the non-

34 existence of antimatter in Space. Having in mind the nine orders of magnitude, it is obvious that these predictions, not based on a theorem but on a model, should be experimentally tested as precisely as possible.

35

10 — Conclusions The extremely small value of AmKj^ s 4 x 10 - 1 0 eV/c2 gives the ratio

^ - s 10~18

as the best check of CPT invariance in the field of mass-measurements. This conclusion is not as straight-forward as generally believed and could even be wrong. In fact the K° -meson consists of a quark-antiquark pair (d s) confined by QCD colour forces which act on the (d s) pair. Thus the mass of a K-meson receives contributions from two different sources. One is the QCD-Bag, which is a QCD non-perturbative confinement effect and, since QCD is a RQFT, it could be CPT invariant despite the problems created by the convergence of the gauge couplings at the Planck scale. The other contribution is coming from the intrinsic quark masses, whose origin is totally unknown and certainly not due to QCD or to any other well established CPT invariant mechanism. It is therefore crucial to check if CPT invariance holds true for the source of the intrinsic quark masses. Here A m ^ fails to be the best check. In fact, if CPT invariance is broken in the process giving rise to the intrinsic quark masses (more precisely m d and m s ), and if QCD respects CPT invariance, AmKjj could be exactly zero, despite a strong CPT breaking in the intrinsic quark masses. There is no theory describing CPT breaking effects but Am^ = 0 is the result of the simplest possible flavour invariant model of CPT breaking. The reason for Am jg = 0, despite a strong CPT breaking in the intrinsic quark masses, is due to the basic property of a meson: its structure in terms of a (quark-antiquark) pair. Experiments in baryon-antibaryon physics are free from this danger. In fact, it is thanks to a recent experiment by Gabrielse et al. on the measurement of the charge over mass ratio (^j) for protons and antiprotons that a CPT check on the intrinsic quark masses can be well established. This limit allows to conclude that, if QCD confinement effects are CPT invariant, the CPT breaking in the intrinsic quark masses (more precisely m u , md) cannot be larger than - ^ 3 5 - < 3 x 10- 11 . m

(20)

q

This limit is seven orders of magnitude larger than the one existing in K-meson physics but it is free from any washing effect, since a baryon has no quark-antiquark structure but only quarks. Thus, if the final result of a Gedanken experiment would be AmK£ = zero , the CPT Symmetry in the intrinsic quark masses could be broken at a level as large as (20) and have no effect on the "Gedanken" result.

36 Experiments to push the limit (20) as much as possible should not be discouraged on the basis of Am K g being nearly zero. Other remarks on the transition from the QCD world (quark and gluons) to the QCD vacuum (mesons and baryons) should encourage new experiments to check the validity of CPT invariance in comparing the masses of nuclear matter and antimatter, as for example the deuteron-antideuteron mass difference. The "intrinsic", "confinement" and "binding" masses have different origins. The "intrinsic" is not known. The nuclear "binding" exists because of the transition from the QCD world (quarks and gluons) to the QCD colour neutral world of baryons and mesons. Contrary to the general belief, it is pointed out that the high precision mass difference Am K

K

, which yields the upper limit A m K g , cannot be taken as a proof of CPT

invariance at the 10

level because a simple flavour invariant, CPT breaking, mechanism

would allow a large CPT breaking in the intrinsic quark masses but A m K g = 0. The CPT invariance for the "intrinsic" masses is at the 10

level and comes from proton-antiproton

physics experiments. The accuracy in CPT invariance for the nuclear "binding" is very poor and the need for new experiments is emphasized because the study of CPT invariance in the nuclear "binding" masses allows to investigate the transition from the QCD world (quarks and gluons) to the QCD colour neutral world of baryons and mesons.

37 APPENDIX 1 Matter, mass, energy and charges: origin and status It was in 1905, with the discovery of equation (21), that the need to distinguish "mass" and "matter" appeared for the first time. In fact, when Einstein discovered his equation E = mc2

(21)

he could not sleep. If charges (electric and nuclear) were not there, we could not exist; why does a glass of water not transform itself into energy? Why do all rest-masses not transform themselves into energy? Fortunately in 1897, J.J. Thomson had discovered the first example of an elementary particle carrying the electric charge: the electron. The reason why the restmass of the electron does not transform itself into energy is because the electric charge cannot disappear. It is this property that guarantees the stability of matter. In fact, in (21) "m" stands for "mass", not for matter. In the previous decades and centuries, the stability of a piece of stone or a piece of iron was thought to be associated with its weight. The Greek temples were constructed using very robust heavy columns. Why robust? Because they were heavy (ancient wisdom). The advent of equation (21) prompted the need to separate the concept of "mass" from that of "matter": the stability of matter stopped to be associated with its mass and became indeed a property linked with the existence of a completely new physical entity: the electric charge. The J.J. Thomson discovery was enough to understand why a glass of water does not transform itself into energy and Einstein could relax, sleep at night and publish his paper. Question: would this paper have been published by Einstein in 1905 if J.J. Thomson had not discovered the electron in 1897? Or would Einstein have followed his "wisdom" of inventing something like the "cosmological constant" when, in 1917, he realized that his general-relativity-gravitational equation implied that our universe could not be static? At that time the "fixed stars" and our galaxy were taken to be the whole universe. Twelve years were needed before Hubble (1929) could discover that some of the "fixed" stars were galaxies moving away from us. Our galaxy was only a small part of the whole universe which became by far larger than previously thought. If the expansion of the universe had been known in 1917, Einstein would not have introduced the cosmological constant in his equation. Fortunately, in 1905, the existence of the electric charge was known and Einstein did not need to introduce anything in his equation (21) but the distinction between "mass" and "matter". The rest-mass is the relativistic invariant associated with the quadrimomentum of a particle ( q s { i p ; E }). The energy is the 4th component of the same quadrimomentum. When the particle is at rest, the square root of the Relativistic invariant, associated with its quadrimomentum V q 2 , corresponds to the rest energy: q 2 = E 2 - | p |

= m 2 . The "rest

38 energy" is the "mass". When the particle is moving, the relativistic invariant associated with its quadrimomentum is obviously the same as before (q 2 = m 2 ) , but the total energy of the particle (4th component of q) does not correspond to the rest-mass of the particle. In a collision, the 4th component of q can change while the rest-mass must remain the same (it is a relativistic invariant). The term "rest-mass" is redundant and contradictory since the mass is a relativistic invariant and must therefore be the same if the particle is at rest or moving. From now on we will use the correct terminology: mass. If the particle is associated with an electric charge, the charge remains attached to the mass of the particle, not to the 4th component of its quadrimomentum q. The electric charge is a conserved quantity and this is why "matter" = "mass + charge" is stable. In the decades following 1905 other physical properties were discovered which behaved as if they were new types of "charges". The "baryonic" and the "leptonic" quantum numbers were discovered to be additively conserved like the electric charge. Other quantum numbers come in, each one being additively conserved. They are now called "flavours". They can be associated either with quarks or with leptons. The total number of quark flavours is six ( u, d, c, s, t, b ), as well as the total number of leptonic flavours ( v e , e; v , \JL ; v H I j T ). These "flavours" are associated with the fundamental fermions (quarks and leptons) which enter in the structure of matter together with particles and nuclei (see Table 2 in § 3).

39 APPENDIX 2 Antinuclear matter, Isotopic Spin, Gauge and Planet The reader should not worry too much. "Nuclear antimatter" will not be changed into "antinuclear matter" like the words "isotopic spin" and "planet". "Isotopic spin" should be replaced by "isobaric spin" since "isotopic" means the "same topic" and specifies the same "place" of an element in the Mendeleev table. The "same-place" means the same electric charge. To say that the isotopic spin of the pion is 1 corresponds to the fact that the three pions, instead of being JI° , should have the same electric charge. A few words on "gauge", which is the English translation of "Eich", used by H. Weyl for his proposal to Einstein: physics cannot depend on the choice of the "gauge" used to measure space and time. If physics remains invariant for transformations as complicated as the Lorentz transformation, how can it be that it must depend on the "Eich", such as the distance between two railway tracks? Einstein reacted very positively: "your idea is very correct. Unfortunately it would produce atomic spectra which depend on the history of each atom. It is an excellent proposal but God did not like it". With the advent of quantum mechanics, Weyl realized that a factor "i" was missing in his proposal. So he wrote another paper where the word "Eich" remained unchanged, but what was thought to be a "gauge" invariance became a "phase" invariance. Weyl kept the word "Gauge" to specify "phase" invariance. When we say "gauge principle" we mean: "invariance for local transformations in a fictitious space which can have either 1 or 2 or 3 complex dimensions. Physics remains invariant if we operate in these fictitious spaces with 1, 2, 3 dimensions obeying the orders of the symmetry groups U(l) (the original phase of Weyl), SU(2) and SU(3). There is no "Eich" but the word "gauge", yes. "Planet" in Greek means "errant". When the term "planet" was coined, the "errant" stars were the Sun, the Moon, Mercury, Venus, Mars, Jupiter and Saturn. The Earth was at the centre of the Universe, immobile. Thanks to Galilei we know that the Earth is a satellite of the "planet" Sun. The word "our planet" now refers to our Earth; like isotopic spin it is used to specify properties very different from those which the word itself originates from.

40 APPENDIX 3 From (9° 6 ° ) and (ej 9°), to (KJ K!J), (K+K_) and ( K ^ ) . For the convenience of the reader, we recapitulate the sequence of the various terminologies originating from the neutral strange meson antimeson physics. As shown in Table 1, during nearly three decades the validity of the Symmetry Operator C was thought to be well established and was generalized to act on all the additively conserved quantum numbers, including "strangeness". But important developments did occur in the physics of the neutral mesons carrying the strangeness quantum number. The V -particles discovered by the Blackett group [45] were identified as being of baryonic and mesonic origin; the mesonic one was named 9 and was known to decay into 2n , 9°^

2K

with a lifetime of the order of 10" sec. The first attempt to understand what happens when a system made of a particle, 9 , and its antiparticle, 9 , can be mixed, is due to Gell-Mann and Pais [46]. Strong interactions distinguish the meson 9 from its antistate: | 9 ) * | 6 ) . In fact the strangeness quantum number, S , is - 1 for 9 and + 1 for 9 . These two states are not eigenstates of C, C | 9°) = | 9°) . Since strangeness is not conserved in weak interactions, the transitions 9° ~

9°

are allowed and will occur. Thus, it is physically meaningful to construct two states which are eigenstates of the C operator:

|9°)

+

|9°>

V2

and

(22)

l# =

1 e°> - 19°) n

with opposite C eigenvalues: C |e?)

= + |e?>

and

c| $>

= -

e£>

Following Gell-Mann and Pais, the 9 -meson was identified as being the 9 j and the second meson, 9 2 , was expected to decay into 3 JI with a longer lifetime, due to phase-space.

41

A dedicated experiment was designed by Lederman in order to detect, with a Cloud-chamber, the long-lived 9 2 meson. This was the last discovery where the glorious instrument, the "Cloud-chamber" (also called the Wilson-chamber), was used in particle physics. The discovery was interpreted as a further proof of the validity of C invariance and for a short period of time 6 1 and 9 2 were considered as being the two eigenstates of C, as suggested by Gell-Mann and Pais. The (9 - x) mesons [17], first became 9j 9 2 , soon later K+ K_ and K j K2, then K s K L (i.e. K+ K_ again). The choice (9j 92) was based on C invariance;

&+ K _ was

based on the fact that the Symmetry Operators C and P were not valid in the weak interactions and therefore CP invariance had to be experimentally established; (K j K 2 ) was based on the assumption of CP invariance and (K s KjJ on the violation of CP invariance; and this means back to the original proposal where the validity of CP invariance was not taken for granted [22]; in fact | K^> - K+ and | K°) = K_ . As discussed in § 2, thanks to LOY [22], it was immediately realized that meson and antimeson mixing, K ** K , does not imply that the CP Symmetry law has to be valid in the decays of the K-mesons. Many decades later, with the discovery of new flavours in addition to "strangeness", this problem is still of great interest. It is probably worth recalling the basic points, since not many people know the origin of the fundamental difference between flavour-antiflavour mixing and CP invariance [22]. Thanks to the work of Lee and Yang [18], both Symmetry Operators, C and P, were discovered not to be conserved in weak processes [19] while their product CP was proposed to be conserved in weak interactions [20]. Thus two new eigenstates of CP come in:

K?> =

| K ° > - | K°>

V2 (26) |K°> + | K ° )

K,

V2

with

14 = CP 14 = CP

and and

K°

+ -

4 4

2it 3 it

Before the discovery of C and P violation in weak processes, Lee, Oehme and Yang [22]

42

pointed out that the problem of (meson-antimeson) mixing had not to be confused with the problem of CP conservation. In other words the transitions

K ° « K° could give rise to mixed states p|K°)-q|K°) v+

-

V |P|2 + |q|2 (27)

K

=

p|K^+q|K°> V IPI2 + |q|2

with neither K+ nor K_ being eigenstates of the CP operator. Only if -E--1 are the two states | K+ ) and | K_ ) eigenstates of CP, with eigenvalues CP | K+ ) = + | K+ ) and CP |K_) = -

|K_). If CP is broken, * 1. |q|

but the two mixed states

|K + ) and |K_) exist as well. This is not a trivial remark since

for a long time the existence of two neutral strange mesons was considered as being, on a first attempt as the result of C invariance, (9 9,), and on a second attempt as the result of CP ,„0

<\

v invariance, (Kj K^) •

The discovery by Cronin, Fitch and collaborators [21] that the long-lived K -meson, ,0

K L , decays into 2 it 2jt

with a branching ratio of 2K

K,

-3 = 10

(28)

•all

confirms the validity of the LOY description of the strangeness mixture: i.e. flavour antiflavour mixing does not imply CP conservation. Thus, the discovery by Lederman et al. of the long-lived 9,

-> K^ -> K_

K,

3n

is no proof of CP invariance. In fact what was discovered by Lederman is the 3 jt decay mode of K_, now called K L.

43

Still nowadays no one knows why flavour mixing is as it is [i.e. described by the Cabibbo [47] —> GIM [48] —» KM [49]) mechanisms] and how the flavour mixing is related to CP violation. The latest results on (E'/E) is the most recent example of the link between flavour mixing and CP violation. We have known since 1964 [21] that CP invariance does not hold and therefore the (K°K°) mixture needs to be described by (27), as suggested by LOY [22]. The present terminology is using KL for the long-lived K-mixture and K s for the short-lived Kmixture. This means that we go back to the physics suggested by LOY [22], in fact: |K + ) -

|Ks>

|K_>-

|K L ) .

The recent discovery that E'/E^O [50] rules out the milliweak proposal [51], i.e. that K L is a mixture of K2 with a per mill fraction of Kj as the sole source of CP violation.

44 References and Notes [1]

P.A.M. Dirac, "The Quantum Theory of the Electron", Proc. Roy. Soc. (London) A117. 610 (1928); "The Quantum Theory of the Electron, Part IP', Proc. Roy. Soc. (London) A118. 351 (1928).

[2]

H. Weyl, " Gruppentheorie und Quantenmechanik", 2nd ed., 234 (1931).

[3]

E.P. Wigner, " Unitary Representations Math., 40, 149(1939).

[4]

G.C. Wick, E.P. Wigner, and A.S. Wightman, "Intrinsic Particles", Phys. Rev. 88,101 (1952).

[5]

E.P. Wigner, "Uber die Operation der Zeitumkehr in der Quanten-mechanik", Gott. Nach. 546-559 (1931). Here for the first time an anti-unitary symmetry appears.

[6]

E.P. Wigner, Ann. Math. 40,149 (1939).

[7]

J. Schwinger, Phys. Rev. 82, 914 (1951).

[8]

J.S. Bell, "Time Reversal in Field Theory", Proc. Roy. Soc. (London) A231, 479-495 (1955).

[9]

To the best of my knowledge, the CPT theorem was first proved by W. Pauli in his article "Exclusion Principle, Lorentz Group and Reflection of Space-Time and Charge", in "Niels Bohr and the Development of Physics" [Pergamon Press, London, page 30 (1955)], which in turn is an extension of the work of J. Schwinger [Phys. Rev. 82, 914 (1951); "The Theory of Quantized Fields. II", Phys. Rev. 9 1 , 713 (1953); "The Theory of Quantized Fields. Ill", Phys. Rev. 9 1 , 728 (1953); "The Theory of Quantized Fields. VI.", Phys. Rev. 94, 1362 (1954)] and G. Luders, "On the Equivalence of Invariance under Time Reversal and under Particle-Antiparticle Conjugation for Relativistic Field Theories" [Dansk. Mat. Fys. Medd. 28, 5(1954)], which referred to an unpublished remark by B. Zumino. The final contribution to the CPT theorem was given by R. Jost, in "Eine Bemerkung zum CPT Theorem" [Helv. Phys. Acta 30, 409 (1957)], who showed that a weaker condition, called "weak local commutativity" was sufficient for the validity of the CPT theorem.

[10]

The Simultaneous Evolution of Masses and Couplings: Consequences on Supersymmetry Spectra and Thresholds F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, Nuovo Cimento 105A. 1179 (1992); see also A. Zichichi in "Subnuclear Physics - The first fifty years", O. Barnabei, P. Pupillo and F. Roversi Monaco (eds), a joint publication by University and Academy of Sciences of Bologna, Italy (1998); World Scientific Series in 20th Century Physics, Vol. 24 (2000). See also References [33], [34], [35].

of the Inhomogeneous Lorentz Group", Ann.

Parity

of

Elementary

45 [ll]

Are Matter and Antimatter Symmetric? T.D. Lee, in Proceedings of the "Symposium to celebrate the 30th anniversary of the Discovery of Nuclear Antimatter", L. Maiani and R.A. Ricci (eds), Conference Proceedings 53, page 1, Italian Physical Society, Bologna, Italy (1995).

[12]

The Positive Electron C D . Anderson, Phys. Rev. 43, 491 (1933); Some Photographs of the Tracks of Penetrating Radiation P.M.S. Blackett and G.P.S. Occhialini, Proc. Roy. Soc. A139. 699 (1933).

[13]

Observation of Antiprotons O. Chamberlain, E. Segre, C. Wiegand, and T. Ypsilantis, Physical Review 100. 947 (1955).

[14]

Anti-Neutrons Produced from Anti-Protons in Charge Exchange Collisions B.Cork, G.R. Lambertson, O. Piccioni, W. A. Wenzel, Physical Review 104, 1193 (1957).

[15]

Observation of Long-Lived Neutral V Particles K. Lande, E.T. Booth, J. Impeduglia, L.M. Lederman, and W. Chinowski, Physical Review 103,1901 (1956).

[16]

Theory of Electrons and Positrons P.A.M. Dirac, Nobel Lecture, December 12,1933.

[17]

On the Analysis of r-Meson data and the Nature of the x-Meson R.H. Dalitz, Phil. Mag. 44,1068 (1953); Isotopic Spin Changes in r and 6 Decay R.H. Dalitz, Proceedings of the Physical Society A69, 527 (1956); Present Status ofx Spin-Parity R.H. Dalitz, Proceedings of the "Sixth Annual Rochester Conference on High Energy Nuclear Physics", Interscience Publishers, Inc., New York, 19 (1956); for a detailed record of the events which led to the (6-x) puzzle see R.H. Dalitz "Kaon Decays to Pions: the t-d Problem" in "History of Original Ideas and Basic Discoveries in Particle Physics", H.B. Newman and T. Ypsilantis (eds), Plenum Press, New York and London, 163 (1994).

[18]

Question of Parity Conservation in Weak Interactions T.D. Lee and C.N. Yang, Phys. Rev. 104, 254 (1956).

[19]

Experimental Test of Parity Conservation in Beta Decay C.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes, Phys. Rev. 105,1413 (1957); Observation of the Failure of Conservation of Parity and Charge Conjugation Meson Decays: The Magnetic Moment of the Free Muon R. Garwin, L. Lederman, and M. Weinrich, Phys. Rev. 105, 1415 (1957);

in

46 Nuclear Emulsion Evidence for Parity Non-Conservation in the Decay Chain J.J. Friedman and V.L. Telegdi, Phys. Rev. 105, 1681 (1957).

jti~n+e+

[20]

On the Conservation Laws for Weak Interactions L.D. Landau, Zh. Eksp. Teor. Fiz. 32, 405 (1957).

[21]

Evidence for the 2 n Decay of the K°2 Meson J. Christenson, J.W. Cronin, V.L. Fitch, and R. Turlay, Physical Review Letters 113. 138 (1964).

[22]

Remarks on Possible Noninvariance under Time Reversal and Charge Conjugation T.D. Lee, R. Oehme, and C.N. Yang, Physical Review 106, 340 (1957).

[23]

Search for the Time-Like Structure of the Proton M. Conversi, T. Massam, Th. Muller and A. Zichichi, Phys. Lett. 5, 195 (1963).

[24]

The Leptonic Annihilation Modes of the Proton-Antiproton System at 6.8 (GeV/c)2 Timelike Four-Momentum Transfer M. Conversi, T. Massam, Th. Muller and A. Zichichi, Nuovo Cimento 40, 690 (1965).

[25]

C.S. Wu, T.D. Lee, N. Cabibbo, V.F. Weisskopf, S.C.C. Ting, C. Villi, M. Conversi, A. Petermann, B.H. Wiik and G. Wolf The Origin of the Third Family, O. Barnabei, L. Maiani, R.A. Ricci and F. Roversi Monaco (eds), Academy of Sciences, Bologna University, INFN, SIF, Rome (1997); and World Scientific Series in 20th Century Physics, Vol. 20 (1998).

[26]

The Discovery of Nuclear Antimatter L. Maiani and R.A. Ricci (eds), Conference Proceedings 53, Italian Physical Society, Bologna, Italy (1995); see also A. Zichichi in "Subnuclear Physics - The first fifty years", O. Barnabei, P. Pupillo and F. Roversi Monaco (eds), a joint publication by University and Academy of Sciences of Bologna, Italy (1998); World Scientific Series in 20th Century Physics, Vol. 24 (2000).

[27J

The first report on "scaling" was presented by J.I. Friedman at the 14th International Conference on High Energy Physics in Vienna, 28 August-5 September 1968. The report was presented as paper n. 563 but not published in the Conference Proceedings. It was published as a SLAC preprint. The SLAC data on scaling were included in the Panofsky general report to the Conference where he says «... the apparent success of the parametrization of the cross-sections in the variable v / q 2 in addition to the large cross-section itself is at least indicative that point-like interactions are becoming involved». "Low q2 Electrodynamics, Elastic and Inelastic Electron (and Muon) Scattering", W.K.H. Panofsky in Proceedings of 14th International Conference on High Energy Physics in Vienna 1968, J. Prentki and J. Steinberger (eds), page 23, published by CERN (1968). The following physicists participated in the inelastic electron scattering experiments: W.B. Atwood, E. Bloom, A. Bodek, M. Breidenbach, G. Buschhorn, R. Cottrell, D. Coward, H. DeStaebler, R. Ditzler, J. Drees, J. Elias, G. Hartmann, C. Jordan, M. Mestayer, G. Miller, L. Mo, H. Piel, J. Poucher,

47 C. Prescott, M. Riordan, L. Rochester, D. Sherden, M. Sogard, S. Stein, D. Trines, and R. Verdier. For additional acknowledgements see J.I. Friedman, H.W. Kendall and R.E. Taylor, "Deep Inelastic Scattering: Acknowledgements", Les Prix Nobel 1990, (Almqvist and Wiksell, Stockholm/Uppsala 1991), also Rev. Mod. Phys. 63, 629 (1991). For a detailed reconstruction of the events see J.I. Friedman "Deep Inelastic Scattering Evidence for the Reality of Quarks" in "History of Original Ideas and Basic Discoveries in Particle Physics", H.B. Newman and T. Ypsilantis (eds), Plenum Press, New York and London, 725 (1994). [28]

Quark Search at the ISR T. Massam and A. Zichichi, CERNpreprint,

June 1968;

Search for Fractionally Charged Particles Produced in Proton-Proton Collisions at the Highest ISR Energy M. Basile, G. Cara Romeo, L. Cifarelli, P. Giusti, T. Massam, F. Palmonari, G. Valenti and A. Zichichi, Nuovo Cimento 40A, 41 (1977); and A Search for quarks in the CERN SPS Neutrino Beam M. Basile, G. Cara Romeo, L. Cifarelli, A. Contin, G. D'Ali, P. Giusti, T. Massam, F. Palmonari, G. Sartorelli, G. Valenti and A. Zichichi, Nuovo Cimento 45A, 281 (1978). [29]

Experimental Observation of Antideuteron Production T. Massam, Th. Muller, B. Righini, M. Schneegans, and A. Zichichi, Cimento 39,10 (1965).

Nuovo

[30]

A. Zichichi in "Subnuclear Physics - The first fifty years", O. Barnabei, P. Pupillo and F. Roversi Monaco (eds), a joint publication by University and Academy of Sciences of Bologna, Italy (1998); World Scientific Series in 20th Century Physics, Vol. 24 (2000).

[3l]

New Developments in Elementary Particle Physics A. Zichichi, Rivista del Nuovo Cimento 2, n. 14,1 (1979). The statement on page 2 of this paper, ^.Unification of all forces needs first a Supersymmetry. This can be broken later, thus generating the sequence of the various forces of nature as we observe them», was based on a work by A. Petermann and A. Zichichi in which the renormalization group running of the couplings using supersymmetry was studied with the result that the convergence of the three couplings improved. This work was not published, but perhaps known to a few. The statement quoted is the first instance in which it was pointed out that supersymmetry might play an important role in the convergence of the gauge couplings. In fact, the convergence of three straight lines (a~J a~2 a ~ 3 ) w i m a change in slope is guaranteed by the Euclidean geometry, as long as the point where the slope changes is tuned appropriately. What is incorrect about the convergence of the couplings is that, with the initial conditions given by the LEP results, the change in slope needs to be at M S U S Y ~ 1 TeV as claimed by some authors not aware in 1991 of what was known in 1979 to A. Petermann and A. Zichichi.

48 [32]

V.N. Gribov, G. 't Hooft, G. Veneziano and V.F. Weisskopf "The Creation of Quantum ChromoDynamics and the Effective Energy", L.N. Lipatov (ed), a joint publication by the University and the Academy of Sciences of Bologna, Italy (1998); World Scientific Series in 20th Century Physics, Vol. 25 (2000).

[33]

The Effective Experimental Constraints on MsuSY ond MGUT F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, Nuovo Cimento 1817(1991).

104A,

[34]

The Simultaneous Evolution of Masses and Couplings: Consequences on Supersymmetry Spectra and Thresholds F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, Nuovo Cimento 105A. 1179(1992).

[35 J

A Study of the Various Approaches to MGUT and aour F. Anselmo, L. Cifarelli and A. Zichichi, Nuovo Cimento 105A, 1335 (1992).

[36]

String Theory: the Basic Ideas B. Greene, Erice Lectures - Discussion 1999 in "Basics and Highlights Fundamental Physics", A. Zichichi (ed), World Scientific (to be published).

in

[37]

No matter what the physics processes are, if their mathematical formulation can be expressed in terms of a relativistic, local, quantized, field theory (RQFT), these processes have to obey CPT invariance. Thus to violate this fundamental invariance of nature corresponds to break the basic conceptual structure of a RQFT. It took many decades to discovery CPT invariance, as discussed in Reference 9. And many other decades were needed to discover the convergence of the gauge forces at the Planck scale.

[38]

C. Caso et al., Particle Data Group, Eur. Phys. J. C3,1 (1998).

[39]

G. 't Hooft, The Physics of Instantons in the pseudoscalar and vector meson mixing, Hep-th/9903183, to be published in the volume "Quark flavour mixing in meson physics"; and private communication.

[40]

Precision Mass Spectroscopy of the Antiproton and Proton Using Simultaneously Trapped Particles G. Gabrielse, A. Khabbaz, D.S. Hall, C. Heimann, H. Kalinowsky and W. Jhe, Phys. Rev. Lett. 82, 3198 (1999).

[41]

As discussed in § 3, the nuclear forces mimic the existence of nuclear charges. If despite the non-existence of "nuclear charges", CPT invariance holds, the mass associated with the opposite nuclear charges is the same as the one associated with the original nuclear charges. Thus an antinucleus must have the same mass as the nucleus. And this means that the nuclear "binding" masses obey CPT invariance. The mass of the deuteron has in its value all transition processes needed for the QCD colour-full world of quarks and gluons to become the world made of mesons and

49 baryons. The equality of the "binding" masses in nuclei and antinuclei is the proof that CPT invariance holds in all these processes. [42]

G. 't Hooft, The Creation of Quantum ChromoDynamics, in "The Creation of Quantum ChromoDynamics and the Effective Energy", page 25, V.N. Gribov, G. 't Hooft, G. Veneziano and V.F. Weisskopf; N.L. Lipatov (ed), Academy of Sciences and University of Bologna, INFN, SIF, joint publication (1998); World Scientific Series in 20th Century Physics, Vol. 25 (2000); T.D. Lee, Rev. Mod. Phys., Vol. 47, 267 (1975); T.D. Lee, Nucl. Phvs..A553. 3c (1993); and A. Zichichi, INFN - Report on the ALICE project for LHC to the INFN-Comm. Ill, Rome - Italy, May 1999.

[43]

A. Sakharov, JEPTLett.,5,

24 (1967).

[44]

The Discovery of Nuclear Antimatter and the Origin of the AMS Experiment S.C.C. Ting, in Proceedings of the Symposium to celebrate the 30th anniversary of the Discovery of Nuclear Antimatter, L. Maiani and R.A. Ricci (eds), Conference Proceedings 53, 21, Italian Physical Society, Bologna, Italy (1995).

[45]

Evidence for the Existence of New Unstable Elementary Particles G.D. Rochester and C.C. Butler, Nature 160, 855 (1947).

[46]

Behavior of Neutral Particles under Charge Conjugation M. Gell-Mann and A. Pais, Phys. Rev. 97,1387 (1955).

[47]

Unitary Symmetry and Leptonic Decays N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).

[48]

Weak Interactions with Lepton-Hadron Symmetry S.L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2, 1285 (1970).

[49]

CP-Violation in the Renormalizable Theory of Weak Interaction M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).

[50]

Observation of direct CP violation in KL s -* mi decays A. Alavi-Harati et al., Phys. Rev. Lett. 83.', 22 (1999).

[51]

Violation of CP invariance and the possibility of very weak interactions L. Wolfenstein, Phys. Rev. Letters 13, 562 (1964); T.D. Lee and L. Wolfenstein, Phys. Rev. 138, B1490 (1965); see also L. Wolfenstein in "Theory and Phenomenology in Particle Physics", Proceedings of the 1968 Erice School, 219, Academic Press, New York and London (1969), A. Zichichi (ed).

Introductory Talk

W h a t can nuclear collisions teach us about the boiling of water or the formation of multi-star systems D.H.E. Gross Hahn-Meitner-Institut Berlin, Bereich Theoretische Physik,Glienickerstr. 100 14109 Berlin, Germany and Freie Universitat Berlin, Fachbereich Physik Phase transitions in nuclei, small atomic clusters and self-gravitating systems demand the extension of thermo-statistics to "Small" systems. The main obstacle is the thermodynamic limit. It is shown how the original definition of the entropy by Boltzmann as the volume of the energy-manifold of the N-body phase space allows a geometrical definition of the entropy as function of t h e conserved quantities. Without invoking the thermodynamic limit the whole "zoo" of phase transitions and critical points/lines can be unambiguously defined. The relation to the Yang-Lee singularities of the grand-canonical partition sum is pointed out. It is shown that just phase transitions in non-extensive systems give the complete set of characteristic parameters of the transition including the surface tension. Nuclear heavy-ion collisions are an experimental playground to explore this extension of thermo-statistics

Thermodynamicists will certainly answer our question by: "Nothing" and yet: 1

There is a lot t o add t o classical equilibrium statistics from our experience with "Small" systems:

Following Lieb 1 extensivity a and the existence of the thermodynamic limit N —• oc\N/v=const are essential conditions for conventional (canonical) thermodynamics to apply. Certainly, this implies also the homogeneity of the system. Phase transitions are somehow foreign to this: The essence of first order transitions is that the systems become inhomogeneous and split into different phases separated by interfaces. In general are phase transitions consequently represented by singularities in the grand-canonical partition sum (Yang-Lee singularities). In the following we show that the micro-canonical ensemble gives much more detailed insight. There is a whole group of physical many-body systems called "Small" in the following which cannot be addressed by conventional thermo-statistics: • nuclei,

"Dividing extensive systems into larger pieces, the total energy and entropy are equal to the sum of those of the pieces. 53

54

• atomic cluster • polymers • soft matter (biological) systems • astrophysical systems • first order transitions are distinguished from continuous transitions by the appearance of phase-separations and interfaces with surface tension. If the range of the force or the thickness of the surface layers is such that the number of surface particles is not negligible compared to the total number of particles, these systems are non-extensive. It is common to all these examples, that the systems are non-extensive. For such systems the thermodynamic limit does not exist or makes no sense. Either the range of the forces (Coulomb, gravitation) are of the order of the linear dimensions of these systems, and/or they are strongly inhomogeneous e.g. at phase-separation. Before inventing any new type of non-extensive thermodynamics or entropy like the one introduced by Tsallis 2 we should realize that Boltzmann's definition:

1 S - k * InW 1 with

W(E,N,V)=e0tr6(E-HN)

^E-H^ =

(1)

lm06iE'HN)-

(eo a suitable small energy constant) does not invoke the thermodynamic limit, nor additivity, nor extensivity, nor concavity of the entropy S(E,N) (downwards bending). This was largely forgotten since hundred years. We have to go back to pre Gibbsian times. It is a purely geometrical definition of the entropy and applies as well to "Small" systems. Moreover, the entropy S(E, N) as defined above is everywhere single-valued and multiple differentiate. There are no singularities in it. This is the most simple access to equilibrium statistics. We will explore its consequences in this contribution. Moreover, we will see that this way we get simultaneously the complete information about the three crucial parameters characterizing a phase transition of first order: transition temperature Ttr, latent heat per atom qiat and surface tension aSUrf- Boltzmann's famous epitaph above contains everything what can be said about equilibrium thermodynamics in its most condensed form. W is the volume of the sub-manifold at sharp energy in the 6iV-dim. phase space.

55

2

Relation of the topology of S(E, N) to the Yang-Lee singularities

In conventional thermo-statistics phase transitions are indicated by singularities of the grand-canonical partition function Z(T, n,V), V is the volume. See more details in 3>4>5'6 Z(T,(i,V)=

— diV -[E-fiN-TS{E)]/T JJo £ o i7 0 ° d e d n e -V[ e -Mn-T.(«,„)]/T = Y1 eo JJo ^ const.+lin.+quadr.

(2)

in the thermodynamic limit V —> oo\N/v=constThe double Laplace integral (2) can be evaluated asymptotically for large V by expanding the exponent as indicated in the last line to second order in Ae, An around the "stationary point" es,ns where the linear term vanish: 1

as

dE a OS A* _ T dN P dS T ~ dv

(3)

the only term remaining to be integrated is the quadratic one. If the two eigen-curvatures Ai < 0, A2 < 0 this is then a Gaussian integral and yields: T/2 Z{T,H,V)

=

rr°°

}-.e-V[e,-»n,-Ts(e.,n.)}/T

eo Z{T,n,V) F(T,n,V) V

//

^

^

^[X^+X^/2

(5)

e~F^^

=

(4)

JJo

, Tln(Vdet(es,ns)) < \nV es - ims - Tss + *— + o(-^-).

(6)

Here det(e s , ns) is the determinant of the curvatures of s(e, n). det(e,n) =

d2s

d2(

df

dnfie

dedn

dri2

— A1A2,

Ai > A2

(7)

In the cases studied here A2 < 0 but Ai can be positive or negative. If det(e s ,n s ) is positive (Ai < 0) the last two terms in eq.(6) go to 0, and we

56

Figure 1: MMMC simulation of the entropy s(e) (e in eV per atom) of a system of 1000 sodium atoms with realistic interaction at an external pressure of 1 atm. At the energy per atom e\ the system is in the pure liquid phase and at e$ in the pure gas phase, of course with fluctuations. The latent heat per atom is qiat — ez — e,\- Attention: the curve s(e) is artifically sheared by subtracting a linear function 25 + e * 11.5 in order to make the convex intruder visible. s(e) is always a steeply monotonic rising function. We clearly see the global concave (downwards bending) nature of s(e) and its convex intruder. Its depth is the entropy loss due to the additional correlations by the interfaces. From this one can calculate the surface tension aSUTj /Ttr = Assurf * N/Nsurf. The double tangent is the concave hull of s(e). Its derivative gives the Maxwell line in the caloric curve T{e) at Ttr

obtain the familiar result f(T,/j,,V) = es — /m s — Tss. I.e. the curvature Ai of the entropy surface s(e, n, V) decides whether the grand-canonical ensemble agrees with the fundamental micro ensemble in the thermodynamic limit. If this is the case, Z(T, /z, V) is analytical and due to Yang and Lee we have a single, stable phase. Or otherwise, the Yang-Lee singularities reflect anomalous points/regions of \\ > 0 (det(e,n) < 0). This is crucial. As det(e s ,n s ) can be studied for finite or even small systems as well, this is the only proper extension of phase transitions to "Small" systems. 3

The physical origin of the wrong (positive) curvature Ai of s(es, ns)

Before we proceed with the general micro-canonical classification of the various types of phase transitions we will now discuss the physical origin of convex (upwards bending) intruders in the entropy surface. As nuclear matter is not accessible to us we should investigate atomic clusters and compare their thermodynamic behavior with that of the known bulk. In the figure (1) we compare the "liquid-gas" phase transition in sodium clusters of a few hundred atoms with that of the bulk at 1 atm..

57

^0

Ttr [K] qiat [eV] Na

Sboil

&ssurf ly

e.ff V/Ttr

200 940 0.82 10.1 0.55 39.94 2.75

1000 990 0.91 10.7 0.56 98.53 5.68

3000 1095 0.94 9.9 0.45 186.6 7.07

bulk 1156 0.923 9.267 CO

7.41

Table 1: Parameters of the liquid-gas transition of small sodium clusters (MMMCcalculation) in comparison with the bulk.

In the table (1) we show in comparison with the known bulk values the four important parameters, transition temperature Ttr, latent heat per atom qiat, the entropy gain for the evaporation of one atom Sb0u as proposed by the empirical Trouton's rule (~ 10) and A s s u r / , the surface entropy per atom as defined above. Nei, is the average number of surface atoms of the clusters. cr/Ttr is the surface tension over the transition temperature.

4

Phase transitions in the micro-ensemble: The topology of the determinant of curvatures of s ( e , n ) :

Now we can give a systematic and generic classification of phase transitions which applies also to "Small" systems and their relation to the Yang-Lee singularities: • A single stable phase by d(e, n) > 0 (Ai < 0). Here s(e,n) is concave (downwards bended) in both gdirections. Then there is a one to c one mapping of the grand-canonical «-»the micro-ensemble. In the two examples on the right the order parameter, the direction vi of the eigenvector of largest curvature Ai was simply assumed to be the energy.

enerffv

58

• A transition of first order with phase separation and surface tension is indicated by d(e,n) < 0 (Ai > 0). s(e,n) has a convex intruder (upwards bended) in the direction vi of the largest curvature. The whole convex area of {e,n} is mapped into a single point in the canonical ensemble.. I.e. if the curvature of S(E, N) is Xi > 0 both ensembles are not equivalent. • A continuous ("second order") transition with vanishing surface tension, where two neighboring phases become indistinguishable. This is at points where the two stationary points move into one another. I.e. where d(e, n) = 0 and v^=o • Ve! = 0. These are the catastrophes of the Laplace transform E -> T. Here v\=0 is the eigenvector of d belonging to the largest curvature eigenvalue A = 0 (—> order parameter). • Finally, a multi-critical point where more than two phases become indistinguishable is at the branching of several lines in the {e, n}-phasediagram with d = 0, V d = 0. An example showing all these possible types of phase transitions in a small system is discussed in 3 ' 4 , 5 5

On the statistical formation of multi-star systems under rotation

Having discussed the micro-canonical description of phase transitions in "Small" systems we sketch here the relevance of our new formulation of thermo-statistics for astrophysical systems. A finite self-gravitating system is controlled by its accessible phase space and its topology: • Due to the long-range gravity the system is non-extensive. • Because there is no heat-, no angular momentum bath a microcanonical treatment under the observation of the conservation laws is neccessary. Attractive gravity leads to a collaps (phase transition of first order). • Under rotation the collapsed system may multi-fragment and multi-star systems will be formed.

59

• Also the fragmentation of Shoemaker-Levy 9 may be viewed as statistical multifragmentation under the linear stress of Jupiter's gravitation field. With the analogy to the long-range Coulomb force one may study many aspects by the study of collision of heavy nuclei. E.g.: • At higher rotation the system prefers larger moment of inertia. This leads to more symmetric heaviest fragments. For the astro-problem higher angular momentum leads to double or multiple stars instead of a mono star. • For heavy-ion collisions, rotation leads to higher radial energy of fragments, larger variance in the statistical fragmentation which must be distinguished from from radial flow ! 6

Conclusion

Instead of using the boiling of water we used the boiling of sodium to demonstrate the physics of first order phase transitions and especially the surface tension. "Small" systems show clearly how to determine all important characteristica of first order transitions: transition temperature, latent heat, and surface tension. This is possible just because these systems are not extensive and their entropy s(e,n) has convex intruders from which the surface entropy can be determined. In the thermodynamic limit this fact becomes more obscured than illuminated. We also mentioned collapsing of self-gravitating and rotating hydrogen clouds towards multi-star systems. As far as the structure of the phase space is concerned there is a striking analogy to the phase space of fragmenting hot nuclei under rotation. The experimental study of the latter is also interesting in view of this analogy. One of the main differences is of course that the latter are experimentally accessible. Another important question is whether there is a kind of statistical equilibrium established in the astrophysical systems and if what the equilibration mechanism might be. Here our learning process has just started. 1. Elliott H. Lieb and J. Yngvason. The physics and mathematics of the second law of thermodynamics. Physics Report,cond-mat/9708200, 310:196, 1999. 2. C. Tsallis. Possible generalization of Boltzmann-Gibbs statistics. J.Stat.Phys, 52:479, 1988. 3. D.H.E. Gross. Microcanonical thermodynamics: Phase transitions in "Small" systems. To be published in Lecture Notes in Physics. World Scientific, Singapore, 2000.

60

4. D.H.E. Gross and E. Votyakov. Phase transitions in "small" systems. Eur.Phys.J.B, 15:115-126, (2000), http://arXiv.org/abs/condmat/?9911257. 5. D.H.E. Gross. Micro-canonical statistical mechanics of some nonextensive systems. Invited talk to the International Workshop on Classical and Quantum Complexity and Nonextensive Thermodynamics (Denton-Texas, April 3-6, 2000) http://arXiv.org/abs/astro-ph/9condmat/0004268. 6. D.H.E. Gross. Phase transitions in "Small" systems - a challenge for thermodynamics. Invited talk to CRIS 2000, 3rd Catania Relativistic Ion Studies, Phase Transitions in Strong Interactions: Status and Perspectives, Acicastello, Italy, May 22-26, 2000 http://arXiv.org/abs/condmat/?0006087.

Section I Quark and Gluon-Plasma Phase Transition and Relativistic Heavy-Ion Reactions

RESULTS FROM P b - P b COLLISIONS AT C E R N SPS

L. R I C C A T I a n d E. S C O M P A R I N INFN

- Sezione

di Torino,

Via P. Giuria 1, 1-10125

Torino

(Italy)

The CERN heavy ion program, started in 1986, has accumulated an impressive amount of experimental results on the study of high energy density nuclear matter. One experiment (NA50) has reported evidence for deconfinement of quarks and gluons from the study of the J/ip suppression pattern measured in Pb-Pb collisions. T h e enhancement of multistrange hyperon production observed by WA97 proves that strangeness is equilibrated on a very short timescale, a behaviour expected in case of transition to a deconfined state. Finally, the study of hadron yields has allowed to establish the velocity of the fireball expansion, and to calculate the freeze-out temperature of the created system. All the experiments agree in concluding that at least in central Pb-Pb interactions a state with many of the characteristics foreseen for a Quark-Gluon Plasma(QGP) has been created.

1

INTRODUCTION

The aim of heavy ion experiments in the CERN SPS energy range (A/S ~ 20 GeV) is the study of the phase transition from ordinary nuclear matter towards a state where quarks and gluons behave as free particles, and possibly reach thermal kinematic distributions (QGP). Lattice QCD calculations estimate the occurrence of this transition at T~170 MeV 1 . As will be discussed later, such a temperature is actually reached in the collisions studied at the SPS. Therefore this is the ideal environment for the study of this transition, which can be experimentally detected looking at specific signatures that at the end of the system evolution still keep memory of the early stage of the reaction. The study of ultrarelativistic ion collisions at CERN started in 1986; at that time 1 6 0 ions were first accelerated to 60 GeV/nucleon, and shortly afterwards 32 S ions at 200 GeV/nucleon became available. With the delivery of Pb ions in 1994 the program entered its crucial phase. In this short review, I will try to focus on the aspects of the experimental results which have proven to be more relevant. In particular, after short considerations on the description of the event geometry, I will discuss the state of our knowledge on the J/ip suppression and on the strangeness enhancement, proposed long time ago as signatures of deconfinement, and on the evolution of the produced hadronic final state. 63

64

2

THE SPS EXPERIMENTAL PROGRAM

At present, still four experiments are taking data at the SPS Pb beam. One of them, NA49, is a large acceptance spectrometer for the study of hadron production, which can access a large variety of signals, from strangeness to HBT interferometry. The other experiments focus on specific signals relevant to the search for QGP: NA50 studies dimuon physics, with emphasis on charmonium suppression, NA57 explores the strange hyperon production, and NA45 (CERES) measures the dielectron yield in the invariant mass region below 1 GeV/c 2 . In this short review many important topics will be unfortunately omitted. 2.1

Event geometry

There is a rather general consensus on the determination of the centrality variables (impact parameter b, number of participant nucleons Npart) as a function of the related measured quantities, namely the charged hadron multiplicity Nch, the transverse energy ET around midrapidity, and the zero degree energy EZDC- Assuming that Nch and ET are directly proportional to Npart or, similarly, that EZDC scales linearly with the number of spectator nucleons (wounded-nucleon model, WNM) it is possible to describe with great accuracy the measured distributions in the frame of the Glauber model of the nucleusnucleus collision. A typical example can be seen in Fig. 1.

55 •a

10

4

-5

10

0

200

400

600

800

1000 1200

Figure 1: Fit of the NA57 charged hadron multiplicity distribution with the wounded-nucleon model.

In this way it is possible to calculate the relationship between the measured quantities and Npart, and therefore to compare the results of various experiments as a function of a common centrality variable. The study of the

65

ET distributions is also important since it allows, in the frame of the Bjorken model, to estimate the energy density e reached in the collision. The estimates give e ~ 3 GeV/fm 3 for central Pb-Pb interactions, a value significantly higher than the ones predicted for the occurrence of the phase transition towards a QGP (of the order of 1 GeV/fm 3 ). 2.2

Charmonium suppression

Up to now, the most evident indication that SPS experiments have reached the threshold energy density needed for deconfinement comes from the measured charmonia suppression pattern. In case of phase transition to QGP, color screening prevents charmonium binding. Furthermore, one expects a hierarchy of suppression, due to the different binding energies of the cc states; the loosely bound ip' and Xc should start to be suppressed at a lower temperature with respect to the strongly bound J/ip state. This signature has been studied in detail for Pb-Pb collisions by the NA50 experiment. Their set-up consists essentially of a dimuon spectrometer complemented by various centrality detectors. The results are summarized in Fig. 2 and 3. The quantity plotted on the vertical axis is the ratio between the measured J/tp and Drell-Yan cross section. Since Drell-Yan is a hard process whose cross section scales with the number of nucleon-nucleon collisions, the ratio (JJ/^/ODY represents the J/ip cross section per nucleon-nucleon collision. Being the ratio of two measured yields, this quantity is free from most systematical errors, such as detector inefficiencies and flux uncertainties. In Fig. 2 (TJ/^/CTDY for p-A, S-U and Pb-Pb collisions is shown as a function of the so-called L variable, the mean length of nuclear matter seen by the cc pair in its way through the colliding nuclei. While p-A, S-U, and peripheral (L <8 fm, corresponding to b >8 fm) Pb-Pb collisions follow an exponential scaling law, which has been shown to be compatible with the nuclear absorption of a pre-resonant cc state, the central Pb-Pb points suddenly deviate from this behaviour, indicating the onset of an anomalous J/ip suppression. In Fig. 3 the Pb points alone are shown as a function of the measured transverse energy ET- Using an upgraded analysis method, which allows to calculate the Drell-Yan ET distribution starting from the huge sample of minimum bias events (the so-called "minimum bias analysis"), it is possible to observe a two-step suppression pattern. In a deconfinement scenario this behaviour can be explained assuming that the Xc state (not directly identified by NA50, which only sees its decay J/ip from the Xc —> J/ip + 7 process) melts for Pb-Pb collisions producing ET >40 GeV and that the J/ip state melts when ET >90 GeV. On the contrary, the conventional hadronic models actually available cannot reproduce this pattern (see 2

66

and references therein). Finally, the study of the V' suppression pattern has revealed that this resonance is already suppressed in peripheral S-U collisions. However it is so loosely bound that it can be easily broken by soft interaction with comoving hadrons and therefore its study is not directly relevant for the investigation of deconfinement. •f 3

O

90 80

•

?™ £!

60

T3

50

f« ca

p(450"GeV/c)-p,d(NA51)

* {-.{SW-'CeVAiS-A',, ^W,U, (NA*«>

•

tt

S{32x200"GeV/c)-U(NA38)

* e inn2(fSxl5UH.VfcM'L- '.NA50I

t

: ^k t

20

0^ = 5.8 ± 0.6 mb

% _

4 ,

2.5

5

7.5

10

Lffm) Figure 2: The ratio J/tp/DY as a function of L, for p-A, S-U and Pb-Pb collisions.

0

20

40

60

80

100

120

140

ET(GeV)

Figure 3: The ratio J/tp/DY as a function of ET, for Pb-Pb collisions. Circles represent the results obtained with measured Drell-Yan events, squares and triangles the results of the "minimum bias" analysis.

As a general remark to NA50 results, it can be observed that the crucial feature of this experiment is the capability of studying physics observables {J/4> production in this case) in small steps of centrality. Thanks to this fact it has been possible to observe the two-step pattern of Fig. 3, which is in a sense a model-independent indication of the occurrence of the phase transition. However, the peripheral Pb-Pb collisions should be explored in more detail in order to give a really convincing evidence for the occurrence of the drop at ET ~ 40 GeV. The set-up for the year 2000 run will in fact be optimized in view of such measurement. Finally, the observation of the onset of the anomalous J/V> suppression in collisions induced by intermediate mass projectiles (Ag, In) would also be of great interest. 2.3

Strangeness

enhancement

Strangeness enhancement has been proposed long ago as a signature of the transition to QGP. Gluon abundances in the plasma phase, (partial) restora-

67

tion of chiral symmetry and Pauli blocking of u and d quark states rise the strangeness content of the measured final state. The most spectacular manifestation of this signature at the SPS has been seen in the study of the production of multistrange hyperons close to midrapidity. Very crudely, if what happens at SPS is the statistical hadronization of a strangeness-enhanced deconfined phase, with global enhancement factor Ea, one expects hadrons containing N strange quarks to be produced with a rate E% times higher than in an environment with no strangeness enhancement, such as p-A collisions. This implies EQ > Ez > E\, where Ei is the enhancement of the particle i with respect to p-A interactions. On the contrary, models based on hadron rescattering give an opposite kind of behaviour, due to high production thresholds (~3 GeV for the process TCTT —> ClCl) and low cross sections. Hyperon production has been experimentally studied in Pb-Pb collisions by WA97. The detection of multistrange particles in nucleus-nucleus interactions poses non-negligible experimental problems, since their decay products have to be identified in a very high multiplicity hadron environment. In Fig.4 the measured multiplicity of strange particles per participant is shown, using p-Be results as a reference. The hyerarchy of enhancements foreseen in case of QGP formation is clearly observed, with an D, enhancement factor of about 17 3 .

1

ff+n*

'.

_+_-*-*•• H"

-*--*-•++

-++*" ;

A

- K h f T I"'

1

1 pBe 1

pPb 10

PbPb 102

pBe 10?

-+-++t A

t

:

pPb 10

PbPb 10 2

10°

Figure 4: Negative hadrons and hyperon yields per participant, normalized to p-Be results, measured by WA97.

The other remarkable result is that the for Npart >100 a saturation of the enhancement is observed. It would of course be very interesting to determine the onset point of such enhancement, to see if it occurs in a limited Npart range, and, if this would be the case, if its position on the Npart axis agrees with the onset of the anomalous J/ip suppression observed by NA50. Fortunately NA57

68

covers the region of peripheral Pb-Pb interactions down to Npart ~50, and its first results are expected to appear very soon. The other possible approach to the study of strangeness is the measurement in a large kinematical region. In this way a "global" strangeness enhancement factor can be extracted. The NA49 experiment, using its large acceptance spectrometer, has studied in detail if-meson production (kaons carry about 75% of the global strangeness content of the final state). A global enhancement factor of about 2 has been found for central Pb-Pb collisions relatively to p-A (see Fig. 5) 4 . An enhancement of the same size is also visible in central S-S and S-Ag collisions. The same study has also been performed as a function of the collision centrality. As a function of Npart (Fig. 6), it is possible to see that in Pb-Pb collisions the kaon production smoothly increases from peripheral to central reactions, with a possible flattening only for Npart >300. P b + P b , NA4-9 P r e l i m i n a r y

NA4(i preliminary

A K

0.16

S+S

V_ 0 . 2

0.14

* •

s+s

A v: V

Pb+Pb

*

'+=0.15

S °-1 s. „ 0.08

•

1

S+AQ

0.1

•a 0.12
s

ft ,

0.05

pA J

0.04

;

*

10 <

Np >

Figure 5: Phase space integrated kaon to pion multiplicities for central nucleusnucleus, p-p and p-A collisions.

++ K°sl

4- 1 ^

•4-

-A-

- A - P _A_

T

*

0.02

10

;

-*-

CD

> 0.06

K

1 + +

50

-&-

,&.

100 150 200 250 300 350 400 Number of participants

Figure 6: Phase space integrated K/ir (and p/tr) multiplicities as a function of Npart, for p-p, S-S and Pb-Pb collisions.

It must also be noted that the S-S point is not compatible with the Pb-Pb point corresponding to the same Npart. The results on kaons pose a certain number of questions from the point of view of the comparison with the results of the other experiments. In fact, if kaon and hyperon production can be explained in the frame of a common statistical hadronization mechanism, one would expect in Pb-Pb the same kind of centrality dependence for the yields. Experimentally it is found that the mulitplicity per participant is flat for multistrange hyperons and smoothly increasing for kaons. Finally, it must also be stressed that the " pattern" of global strangeness enhancement is rather different from the one seen by NA38/NA50 for the charmonium suppression; the last one, in fact, does not exhibit any "anomaly" for S-induced collisions,

69

while global strangeness is already enhanced there. 2.4

The evolution of the final state

Thanks to the combined progress of theory and experiment, our knowledge of the late stages of the collision is today much deeper than a few years ago. The essential parameters needed to characterize the evolution towards the final state have been fixed with good precision. Very briefly, the chemical freeze-out temperature and baryon chemical potential have been determined in the frame of statistical models of the hadronization process. The availability of hadron data in large phase space region, as the ones produced by NA49, has allowed to have accurate fits for the free parameters. Several different version of these models exist, but they agree in indicating T ~170 MeV for the chemical freezeout temperature, and /XB ~270 MeV. Since such a temperature is very close to the one predicted for the phase transition to QGP, it is quite probable that the system crosses the phase boundary shortly before the chemical freeze-out point. Furthermore, the almost complete strangeness saturation indicated by these models, shows that the observed strangeness enhancement comes essentially from the partonic phase. The following step in the history of the collision, namely the expansion of the chemically equilibrated hadronic system, has been investigated using as tools the study of the transverse mass spectra of identified particles and the results from HBT analysis. The study of the transverse mass spectra has been carried out by several experiments (NA49, NA44, WA97). It was soon realized that the inverse slope T of the transverse mass distribution, which should be connected with the temperature of the system at decoupling time, increases steadily with the mass of the considered particle. This can be seen in Fig. 7 5 . This fact can be explained as arising from the relativistic superposition of the thermal motion of the expanding system plus a radial ordered collective flow, caused by the explosion of the fireball. Since we observe this "Little Bang" from the outside, we see a blue-shift of the temperature, conceptually similar to the red-shift we observe for the background radiation from the Big Bang. The departure of the fi from the systematics can be due to an early freeze-out of such particle. In fact the fi, having zero isospin, cannot form resonances with the pions and therefore decouples very soon from the expanding hadron gas. By the way this is a further indication that the enhancement of the O, does not originate in the late stages of the reaction but it is rather due to the pre-hadronic phase. Different ways are in principle available to disentangle thermal motion and radial flow. The most accurate of them combines the information from

70

Figure 7: Dependence of the m y spectra inverse slope T on the particle mass m for Pb-Pb collisions.

Figure 8: Allowed region of freeze-out temperature vs. radial flow velocity for central Pb-Pb collisions.

rriT spectra with the ones from Bose-Einstein pion correlations. Very essentially, by fitting the dependence of the HBT transverse radius on the transverse momentum of the pion pair, it is possible to extract, in a model dependent way, the ratio 0^/T between the radial flow velocity and the freeze-out temperature of the system. The remaining ambiguity, as shown in Fig. 8, can be eliminated using the information from h~ and d transverse spectra. This finally gives T = (120 ± 12) GeV and pT = 0.55 ± 0.12 (WA98 has obtained with a similar analysis slightly lower T values). This means that the fireball explodes at half the speed of light, justifying therefore the term "Little Bang" used to describe the expansion of the hadronic phase. 3

CONCLUSIONS

In summary at the CERN SPS in the lead-lead collisions a new state of matter has been created and a "little bang" reproduced in laboratory. Looking at the results on charmonium suppression and multistrange hyperons production this state cannot be explained by conventional models of hadronic interactions. 1. 2. 3. 4. 5.

F. Karsch, hep-lat/9909006. M.C. Abreu et al. (NA50 collaboration), Phys. Lett. B477 (2000) 28. E. Andersen et al. (WA97 collaboration), Phys. Lett. B449 (1999) 401. S. Margetis et al. (NA49 collaboration), J. Phys. G25 (1999) 189. F. Antinori et al. (WA97 collaboration), Eur. Phys. Journ. C14 (2000) 633.

T H E SEARCH FOR T H E QGP: A CRITICAL A P P R A I S A L HELMUT SATZ University of Bielefeld, Faculty for Physics, P.O.Box 10 01 31, D-33501 Bielefeld, Germany E-mail: [email protected]

1

Abstract

Over the past 15 years, an extensive program of high energy nuclear collisions at BNL and CERN has been devoted to the experimental search for the quarkgluon plasma predicted by QCD. The start of RHIC this year will increase the highest available collision energy by a factor 10. This seems a good time for a critical assessment: what have we learned so far and what can we hope to learn in the coming years?

Contents: 1. Expecting the Unexpected 2. Charmonium Suppression 3. In-Medium Hadron Modifications 4. Strangeness and Thermalization 5. Catching the Elusive

An extended version of this was presented in a plenary talk at Lattice 2000: XVIII International Symposium on Lattice Field Theory, August 17-22, 2000, Bangalore/India, proceedings to be published in Nucl. Phys. B. It is also accessible on the Server under hep-ph/0009099. 71

STRANGE BARYON SIGNALS OF A NEW STATE OF MATTER IN LEAD-LEAD COLLISIONS AT THE CERN SPS Presented by: FEDERICO ANTINORI INFN, Sezione di Padova, via Marzolo 8, 1-35131 Padova, Italy E-mail: federico. antinori @padova. infn. it on behalf of the WA97 Collaboration:

F.ANTINORIj, W.BEUSCH f, I.J.BLOODWORTH d, R.CALJANDROa, N.CARRERf, D.DI BARI a , S.DI LJBERTO', D.ELIAa, D.EVANS d, K.FANEBUST b, F.FAYAZZADEH \ R.A.FINIa, J.FTA* NIK g, B.GHIDINIa, G.GRELLA m, M.GUIJNO e , H.HELSTRUPc, M.HENRIQUEZ \ A.K.HOLME \ D.HUSS h, A. JACHOLKOWSKIa, G.T.JONES d, J.B.KINSON d, K.KNUDSON f, I.KRALIK g, V.LENTIa, R.LIETAVAf, R.A.LOCONSOLE a , G.L0VH0IDEN\ V.MANZARI a, M.A.MAZZONI', F.MEDDIf4, A.MICHALON n, M.E.MICHALON-MENTZER n, M.MORANDO', P.I.NORMAN d, B.PASTIR' AK 8 , E.QUERCIGH fj, D.ROHRICH b, G.ROMANO m, K.SAFA* IK f , L.§ANDOR fg , G.SEGATOj, P.STAROBAk, M.THOMPSON d, J.URBAN g, T.VK 4 , O.VILLALOBOS BAILLIE d, T.VIRGILIm, M.F.VOTRUBA d and P.ZAVADA k. Dipartimento La. di Fisica dell'Universita e del Politecnico di Bari and Sezione INFN, Ban, Italy Fysisk Institutt, Universitetet i Bergen, Bergen, Norway H0gskolen i Bergen, Bergen, Norway School of Physics and Astronomy, University of Birmingham, Birmingham U.K. Unversitil di Catania and INFN, Catania, Italy CERN, European Laboratory for Particle Physics, Geneva, Switzerland g Institute of Experimental Physics, Slovak Academy of Sciences, KoSice, Slovakia GRPHE, Universite de Haute Alsace, Mulhouse, France Fysisk Institutt, Universitetet i Oslo, Oslo, Norway J Dipartimento di Fisica "G.Galilei" dell'Universita and Sezione INFN, Padua, Italy Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic Dipartimento di Fisica dell'Universita "La Sapienza" and Sezione INFN, Rome, Italy Dipartimento di Scienze Fisiche "E.R.Caianiello " dell 'Universita and INFN, Salerno, Italy Institut de Recherches Subatomiques, IN2P3/ULP, Strasbourg, France The evidence for the formation of a deconfined state of matter in Pb-Pb collisions at the CERN SPS coming from the study of the production of strange and multi-strange particles in p-Be, p-Pb and Pb-Pb collisions in Experiment WA97 is reviewed and discussed.

73

74

1

Introduction

It was suggested almost 20 years ago [1,2] that strange particles would be a sensitive probe in the search for the colour deconfinement phase transition in ultrarelativistic nucleus-nucleus collisions. Deconfinement is expected to be accompanied by a partial restoration of the chiral symmetry: as partons are liberated from the confining effects of strong interaction, the masses of the quarks would decrease from the effective value they take on inside hadrons to their current values. While the effective mass of a strange quark inside a hadron is of the order of 500 MeV, the current value of the strange quark mass, about 150 MeV, is of the same order of the critical deconfinement temperature. Therefore, one expects, in the deconfined phase, to have a rapid and copious production of strange quarkantiquark pairs, mostly by gluon-gluon fusion. If such an enhancement of strangeness production survives hadronization, it will result in enhanced yields of strange particles, and especially of particles containing multiple strange quarks: the increase in the relative abundance of the various species of hyperons should be larger and larger as one goes from lsl=l (A) to lsl=2 (2) to lsl=3 (Q). Some enhancement in the abundance of strange particles when going from protonnucleus to nucleus-nucleus collisions could be present, due to inelastic rescattering among the reaction products, even if the reaction proceeds through purely hadronic stages, with no deconfinement. However, if the production of A and K via inelastic rescattering among the hundreds of hadrons produced in the nucleus-nucleus collision, may be relatively easy (throughror—»KK,TIN-»KA) , it is more and more difficult (i.e. slow) to produce in this way baryons with higher and higher strangeness content, as the thresholds and the number of required multiple interactions increase. This crucial difference between the hadronic and deconfined scenarios makes the study of the production of particles with lsl>l of particular interest in this context. 2

The WA97 Experiment

Experiment WA97 at the CERN-SPS has been designed to study the production of lsl=l,2,3 particles at central rapidity in Pb-Pb collisions and, as a reference, in p-A collisions at a beam momentum of 158 GeV/c per nucleon. The layout of the experiment is sketched in figure 1 and is described in [3,4,5]. In order to cope with the large density of tracks emerging from the nucleus-nucleus collisions, tracking is performed in a high granularity Pixel Tracking Chamber (PTC). The PTC, with a cross section of 5x5cm2 and a total length of 90cm, is placed above the beam line and is inclined with its lower edge pointing back to the

75

target. Strange particles are detected by reconstructing in the PTC the tracks emerging from their weak decays. At the chosen inclination setting (40mrad), the telescope is sensitive to K°s, A, 5" and UT produced over about one unit of rapidity around mid-rapidity, with a transverse momentum larger than a few hundreds MeV. The target, the multiplicity detectors (see below) and the PTC are placed inside the 1.8T magnetic field of the OMEGA magnet, which is followed by a system of pad-cathode multiwire proportional chambers used as lever-arm detectors pad chambers

, beam PTC silicon 6 '• * telescope; somip^/ 0.5111 channels 5A

scintillatorA _ 7 , /t petals / V j Pb t a r g e t > £ V

mufti

ta

P|icity

det6Ct

°

rS

Figure 1. Sketch of the layout of the WA97 experiment «* 600

1.3 +

1.35

Ajt"+Ajr mass, GeV

1.65

1.7

1.75

AK*+AK* mass, GeV

Figure 2. 5" and QT signalsfromthe WA97 Pb-Pb sample

76

Scintillator petals placed behind the target provide an interaction trigger selecting approximately the most central 40% of the inelastic Pb-Pb cross section. The petals are followed by two stations of MicroStrip Detectors (MSD) which sample the charged multiplicity at central rapidity providing information on the event centrality (i.e. number of wounded nucleons) which is used for the off-line study of the centrality dependence of the particle yields. The Pb-Pb data sample is divided into four MSD multiplicity classes and the average number of wounded nucleons for each class is determined from a Wounded Nucleon Model [6] fit to the charged multiplicity distribution da/dNch. This analysis is discussed in detail in [7,8]. In the proton beam reference runs, Be and Pb targets were used. In these runs, a trigger was applied demanding at least two tracks in the telescope, as required to find V°s (two-track trigger). Data were also collected requiring at least one track in the telescope (one-track trigger). These data were used for the study of the production of negative particles. The numbers of wounded nucleons corresponding to p-Be and p-Pb collisions have been determined as an average value for inelastic collisions in the framework of the Glauber model [9]. The details of the analysis, i.e. the reconstruction procedure, the extraction of the signals and the event weighting procedures are described in [4,10,11]. As an example of the quality of the data, the 5 " and Q invariant mass spectra are shown in figure 2. 3

Hyperons' transverse mass spectra

The analysis of the transverse mass distributions is described in detail in [11]. The ntj- = -y/m2 + pT the form: dN dm?

, oc nij- exp(

distributions of negatives and strange particles are fitted to

m,. ) T

The results of the fits for hyperons are shown in figure 3. The values of the fitted transverse mass slope parameter T are plotted in figure 4 as a function of the rest mass of the particles, along with results from other CERN heavy-ion experiments (see e.g. [11] and references therein). The presence of strong radial flow in Pb-Pb collisions at the SPS was deduced from the systematics of experimental data on non-strange and singly-strange hadron, which suggests a linear increase of the

77

inverse slope with the particle mass [12,13], as indicated by the line in figure 4. From figures 3 and 4 we note that the slopes for all singly- and multiply-strange baryons transverse mass spectra are instead remarkably similar. Such a behaviour can be expected both in a collective expansion scenario [14] as due to an early freeze-out of fi and E [15,16] or in a scenario where a fireball of deconfined partons suddenly hadronizes [17,18]. In both cases, the £2 and 5 abundances would be frozen in at the point of hadron formation.

mT (GeV/c2) Figure 3. Transverse mass distributions for hyperons in Pb-Pb collisions.

78

0.5

.

0

O

o NA49 • NA44 • WA97

0.4

0.3

/ \ *

-

f a'+a*

ir+WK0 0.2

K

'• fX

J% 0.1

0.5

2 m(GeV)

1.5

Figure 4. Dependence of the transverse mass inverse slope on the particle rest mass (see text).

W3

-

2

i

2

.810 i 10 i

»-»•

_- "

t

1 i i -2

—°-

™

-1:

10

-= :

s

• » 0 •

10 -1

a

»f

•4

10 10

-

pBe pPb '~1,HIIII|

PbPb

pBe pPb

10

3

101

10

:

PbPb 1

I

2

10

1

; 1 ; 1 ; 1

O

10 i -3:

—t

^ \ -*- n+nS

" " " I

10

' '

'1

10

(N rari > Figure S. Yields per unit rapidity at central rapidity as a function of the number of wounded nucleons (Nput). See text for details.

79

4

Particle yields

The yields per unit of rapidity at central rapidity for negative particles and for A, 5", QT and their antiparticles are plotted in figure 5 as a function of the average number of wounded nucleons for p-Be and p-Pb collisions and for the four centrality classes in Pb-Pb collisions. All yields are extrapolated to a common phase space window covering full pT and one unit of rapidity centered at midrapidity. The horizontal error bars of the Pb-Pb points represent the FWHM of the Npart distributions in the four multiplicity classes [7]. In figure 5, as well as in the following figures, particles are divided in two classes: those with no valence quark in common with the nucleon are plotted on the right, the others are shown on the left. The two groups have been kept separate since it is known that they may exhibit different production features (for instance, A and A have different rapidity spectra both in p-S and S-S collisions [19]). Figure 6 shows the yields relative to those measured in p-Be collisions. The straight line indicates the expectation in case the yields per participant (wounded) nucleon did not change from p-Be collisions all the way to central Pb-Pb collisions. As can be seen, while the p-Pb points are compatible with such an expectation, there is a clear excess for all particle species in Pb-Pb collisions. The effect is more and more pronounced as the strangeness content of the particles increases. Figure 7 displays the yields per participant relative to the p-Be ones. The constant-yieldper-participant expectation now lays parallel to the horizontal axis. The yields per participant appear to be rather constant within the centrality range covered by WA97. The enhancement factors can be read off the vertical axis. The largest effect is observed for the triply-strange ft", which are enhanced by a factor between 15 and 20. This enhancement a pattern, as discussed above, is naturally expected in a deconfined scenario, while it is not explained by any known hadronic microscopic model [5,20,21].

80 0>

jj-n+n + '

+" + - t i

rfE" "*>A

•8

-t- -«»A

•aio 2 VI

'C

"3 P

'

U > :

-I :

A

1 -

~s

PbPb

pBepPb

li.l.n

pBe pPb

I l .mill

2

10

10

J 1

i

i

|

i

' 3

10

1

10

1

PbPb 1 J " " " ' 1 '_'•' 2 3

10

10

Figure 6. Yields per unit rapidity (same as in figure 5) relative to the p-Be yields.

3 0.

* £1+n*

10 -

1

• + * *

-H#^ ;

-+-"%A

I

-H+rTh"

J 1

.«t

f pBe pPb

PbPb

10

1

:

pBe pPb • ' -

10

-f-f-^x ' 1

10f

1

'

PbPb 1

10

'

2

10

3

10
Figure 7. Yields per unit rapidity (same as in figure 5) per participant relative to p-Be.

5

Summary and conclusions

The enhancement of the production of strange particles per participant (wounded) nucleon in Pb-Pb collisions with respect to p-A collisions increases with the strangeness content of the produced particle, as naturally expected in a deconfinement scenario, while no known hadronic model reproduces the WA97 data.

81

The analysis of the transverse mass spectra indicates that the multistrange hyperons, are emitted rather early in the collision, either as a result of an early freeze-out or of a sudden hadronization. The transverse mass distributions of the S and especially Q baryons put therefore stringent constraints on any future attempt to explain the enhancement pattern as a consequence of secondary interactions in a purely hadronic system. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Rafelski J., Phys. Rep. 88 (1982) 331. Rafelski J. and Miiller B., Phys. Rev. Lett. 48 (1982) 1066. Antinori F. et al., Nucl. Phys. A590 (1995) 139c. Andersen E. et al., Phys. Lett. B433 (1998) 209. Antinori F. et al., Nucl. Phys. A661 (1999) 130c. Bia« as A., Bleszy ski M. and Czy W., Nucl. Phys. B i l l (1976) 461. Antinori F. et al., Nucl. Phys. A661 (1999) 357c. Antinori F. et al., CERN-EP 2000-002, to appear in Eur. Phys. J. C. Wong C.Y., Introduction to High-Energy Heavy-Ion Collisions (World Scientific Publishing, Singapore, 1994) 251-264. Andersen E. et al., Phys. Lett. B449 (1999) 401. Antinori F. et al., Eur. Phys. J. C14 (2000) 633. Bearden I.G. et al., Phys. Rev. Lett. 78 (1977) 2080. Alber T. et al., J. Phys. G23 (1997) 1817. Tomasik B., Wiedemann U. and Heinz U., Nucl. Phys. A663 (2000) 753. Van Hecke H., Sorge H. and Xu N., Phys. Rev. Lett. 81 (1998) 5764. Dumitru A. et al., nucl-th/9901046. Letessier J. and Rafelski J., nucl-th/0003014, to appear in Int. J. Mod. Phys. E. Hamieh S., Letessier J. and Rafelski J., hep-ph/0006085. Alber T. et al., Z. Phys. C46 (1994) 195. Antinori F. et al., Eur. Phys. J. C l l (1999) 79. Soff S. et al., nucl-th/9907026.

C O O P E R - M E S O N S I N T H E COLOR-FLAVOR-LOCKED S U P E R C O N D U C T I N G P H A S E OF D E N S E Q C D ° A. Wirzba Institut fur Kernphysik (Theorie) Forschungszentrum Jiilich D-52425 Jiilich, Germany E-mail: [email protected] QCD superconductors in the color-flavor-locked (CFL) phase sustain excitations ("Cooper" mesons) that can be described as pairs of particles or holes around a gapped Fermi surface. In weak coupling and to leading logarithm accuracy the masses, decay constants and form factors of the scalar, pseudoscalar, vector and axial-vector excitations, which explicitly are of finite size, can be calculated exactly. Furthermore, the constraints of this microscopic calculation on the effectivelagrangian description and the computation of the generalized triangle anomaly are discussed.

1

Introduction

Quantum chromodynamics (QCD) at high density, relevant to the physics of the early universe, compact stars and relativistic heavy ion collisions, is presently attracting a renewed attention from both nuclear and particle physicists. For large chemical potential, /x » AQCD, quarks at the edge of the Fermi surface interact weakly, although the high degeneracy of the Fermi surface causes perturbation theory to fail. Thus particles can pair as diquarks and condense at the boundary of the Fermi surface leading to energy gaps and therefore to a superconducting ground state 1 ' 2 ' 3 ' 4 ' 5 ' 6 , 7 . The resulting QCD superconductor breaks color and flavor symmetry spontaneously. Therefore, the ground state exhibits Goldstone modes that are either particle-hole excitations (ordinary pions) or particle-particle and hole-hole excitations (BCS pions TT) with a mass that vanishes in the chiral limit. Effective-lagrangian approaches to QCD in the color-flavor-locked (CFL) superconducting phase 4 provide a convenient description of the long-wavelength physics structured by global flavor-color symmetries (incl. anomalies) 8,9,10 ' 11 : C = ^ T r (doUdofr)

- ^ T r (diUdiU^

+ £ m a s s + £ w z w + £ h . 0 . , (1)

where U = exp(ira • na/FT) is the pertinent unitary chiral matrix of the pseudoscalar Goldstone modes 7f° which are of diquark nature here. The nonlinear °FZJ-IKP(TH)-2000-22 83

84

sigma-model part of this effective lagrangian is of the nonrelativistic antiferromagnetic type introduced by Leutwyler 12 . Note that the space-like pion decay constant Fs is smaller than the time-like one FT- This is a common property for the propagation of Goldstone modes in a matter background 13<14>15. £ m a s s is the explicitly symmetry-breaking part, £wzw is the (generalized) WessZumino-Witten term 8 , and £h.o. stands for all higher order derivative or mass contributions. However, such an effective-lagrangian approach is intrinsically based on a point-like description and does not allow a direct calculation of the underlying parameters that are important for a quantitative description of the bulk (thermodynamic and transport) properties of the QCD superconductor. This requires a more microscopic description. In this talk, I report about such a work 16 ' 17 ' 18 where the excitations of QCD superconductors in the CFL phase are described microscopically as composite (finite size) pairs of quasi-particles or quasi-holes around the gaped Fermi surface. The results of microscopic calculations in the leading logarithm approximation of the masses and decay constants of these generalized (alias "Cooper") mesons are reviewed. 2

Q C D Superconductor in the CFL Phase

In the CFL phase of the QCD superconductor 4 , the quarks are gapped. Their propagation in the chiral limit is conveniently described in the Nambu-Gorkov formalism where the fermi field i\> is doubled to (tjj, ipc) with ipc = CT/>T. 6 For large chemical potential fi, the antiparticles decouple: the particle/hole energies are eq « =F(<7M + IGfa)! 2 ) 1 / 2 , whereas the antiparticle/antiholes ones are eq » T 2 ^ J with
„ f7°fa>+«||)A-(q) ~ ^ MG(«)A-(q)

-MtG*(?)A+(q) \ 1 7°(
, . ^

in terms of the projectors A ± (q) = j ( l ± a • q) onto positive and negative energies and the color-fiavor-locking matrix M ? ^ = e*/j/«e*c,c* 7s of the CFL phase. G(q) is the gap function (not a constant) that satisfies the gap equation of Fig. 1:

sw4P% ,G(<||) h ( K 6 Jo

^/«f| + |G(<7||)|2

X

1A

A

L Q3

CPl|-«ll)

( 1+ |P|rfl||l 8 + }mi,|p,|-fl|,|) } '

+m

*/ (3)

85

2

_

1

Figure 1: BCS gap equation. The thin and thick lines are the free and dressed quark propagator, respectively, whereas the wiggly line is the gluon propagator. The indices refer to the off-diagonal components in the Nambu-Gorkov space

where /i» = § £p in terms of the strong coupling constant g}9 The gap equation is not of a conventional BCS-type because of the long-range structure of the gluon propagator, i.e.

in Euclidean space. The gluon propagator is Debye screened ( m% = mf> « Nf(g/j,)2/2w2) and Landau damped (m^, « 7rm|>|g4|/|4q|), where mo is the Debye mass, TUM is the magnetic screening generated by Landau damping and Nf the number of flavors. In weak coupling, the upper limit for the transverse momenta is Ax = 2/4 > rriB,mM and the logarithms in Eq. (3) cannot be expanded. Instead logarithmic scales x = ln(A*/g||) and xn = ln(A*/Gn) (with A* = (4Aj_/7rm^)) have to be introduced. To leading logarithm accuracy, the gap function G(q) is then solely a real-valued function of qf|| given by 5 ' 1 9 G{x) = Go sin (~-\ 3

= G0 sin (h.x/V?}

with G 0 « ( ^ )

e ~ ^ . (5)

Generalized Scalar and Pseudoscalar Mesons

The generalized mesons of the CFL phase are excitations in qq, whereas the conventional mesons are excitations in qq. Their wave-functions follow form the Bethe-Salpeter equation pictorially described in Fig. 2 1 6 ' 1 7 . In particular, in leading logarithm approximation, the scalar vertex operator of 'isospin' A for a pair of particles or holes with momenta P/2 ± p is given in the NambuGorkov space by TA(p,P)

= ±(

°

iT*s{p,P)

(M^\

(6)

with MA = Mia (TA)ia and Mia = efe%y5, where (e a ) 6c = eabc is the totally antisymmetric tensor in flavor (/) and color (c) space, respectively. The

86

corresponding composite pseudoscalar vertex reads TA(VP) = —

(

0

(MA)f)

-iry5r*PS(p,P)

(7)

In the chiral limit, the scalar and pseudoscalar vertex reduces to the gap, i.e. T(p)s(p, 0) = G{p). Therefore the scalar and pseudoscalar modes are massless Goldstone modes. In Ref.16 the generalized PCAC relation (BCS|A»(0)|*£(P)> = iFP» F*

(8)

was studied. It was found there (see also Ref. 10 ) that the temporal decay constant of the generalized pion is given by FT = fi/ir whereas the spatial one, Fs, is smaller by a factor 1/V5. Thus the velocity of the Goldstone modes, v = FS/FT — 1/V5, is less than the speed of light (c = 1). The latter property mirrors the propagation of ordinary pions in nuclear matter 13 > 14 ' 15 . The decay constants of the generalized pions are very large compared with the gap function which is bounded by Go. Thus the Cooper-pions have a very small (spatial) size (of the order of the inverse momentum exchanged between quark pairs at the Fermi surface), irrespectively of any screening. The dependence of the mass of the generalized pion on the current quark mass can be estimated with similar methods and was studied in Refs. 10 ' 16 . 4

Higgs M e c h a n i s m

The generalized scalar mesons mix with the longitudinal gluons in the CFL phase 17 . In fact, all the 8 generalized scalars are eaten up by the longitudinal gluons, such that the 8 originally massless gluons (with two transverse polarizations) acquire masses (Meissner effect). This Meissner mass refers to the inverse penetration length of static colored magnetic fields in the QCD superconductor which is unexpectedly small, i.e. \jg\i.17 In weak coupling, the Meissner mass is of the order of the electric screening mass TUE »
»i = ~T

-i

^^.

"

qp

ii

-

^

^

^

-T

Figure 2: Bethe-Salpeter equation for the generalized mesons in the QCD superconductor.

87

is not of the order of gGo as in a conventional superconductor with a constant (energy independent) gap. Note, however, that the nonstatic gluonic modes with energy Qo > Go sense "free quarks" for which there is electric screening, but no magnetic screening. 5

Generalized Vector and Axial-Vector Mesons

The pairs of (quasi-)quarks or (quasi-)holes at the Fermi surface can also form generalized vector and axial-vector meson-states in the CFL phase 1 7 . These composites have a similar form factor as the Cooper-pions and therefore finite size. Because Lorentz invariance is absent, there are electric and magnetic vector-composites. The difference to the (pseudo-)scalar case is the reduced strength of the coupling in the Bethe-Salpeter (BS) equation. Thus the (pseudo-)scalar BS equation admits massless modes, while the (axial-)vector one does not. The mass of the composite vector excitations is close to (but smaller than) twice the gap in weak coupling, but goes asymptotically to zero with increasing coupling thereby realizing Georgi's vector limit in cold and dense matter 1 7 . Both the vector and axial-vector Cooper-octets are degenerate in leading-log approximation, in spite of the chiral symmetry breaking of the CFL phase. Furthermore, it is shown in Ref. 17 that the composite vector mesons decouple from the Noether currents and that they do not decay to pions in leading logarithm accuracy, contrary to their analogues in the QCD vacuum. Moreover, they decouple from the gluons and scalars in the CFL phase as well. 6

Finite Size and Hidden Gauge

The quark pairs of the mesons in the QCD superconductor have finite size. Though the space like separation is of the order 1/fi and therefore small, the time-like separation is governed by the scale of the "magnetic mass" l / m M « l / ( m | G 0 ) 1 / 3 » 1/M-

(9)

Hidden gauge symmetry and vector meson dominance (VMD) can therefore be only approximate concepts and do not hold in weak coupling and to leading-log approximation 17 . 7

Modified Triangle Anomaly

In the CFL phase the ordinary photon is screened, and the gluons are either screened or higgsed. However, it was pointed out in Ref. 4 that the CFL phase

88

(a)

(b) 7 7 in the CFL phase. The crosses refer to pair-

Figure 3: Leading contributions to n° condensate insertions.

is transparent to a modified or tilde photon, A,, = An cos 9 + Hp sin 6 ,

(10)

where A^ is the ordinary photon field, which couples to the charge matrix eQero = e d i a g ( 2 / 3 , - 1 / 3 , - 1 / 3 ) of the quarks, and Hp is the gluon field for U(1)Y where gY = (?diag(—2/3,1/3,1/3) is the color-hypercharge matrix. Furthermore, cos# = g/\/e2 + g2, and sin0 = e/^/e 2 + g2- A^ carries colorflavor and tags to the charges of the Goldstone modes. The quark coupling to the tilde photon is in units of e = ecos#. As a result, the CFL phase is characterized by generalized flavor-color anomalies 8 . In Ref.18 it was shown how the triangle anomaly emerges from a direct calculation of the decay of the generalized pion into two generalized photons, w° —• 77, in the leading logarithm approximation, i.e. ~2

1

»'Jr ^ j T r (TA Q 2 ) e M „ o / } r f ^

T^(jn,pa)=

(11)

(with Q = Qem)- This result is consistent with the modified triangle anomaly fV* A 3 — _ _

"

_

967r2

e

t , r

" '"

pii"

ppc

(12)

suggested by the generalized WZW term 8 . Here FM" is the field strength associated to Eq. (10). Note that the conventional triangle anomaly is larger by a factor Nc. This factor is absent here, because of the color-flavor locking of the quarks. Furthermore, the internal line between the two generalized-photon vertices in Fig. 3 has to correspond to an antiparticle. Therefore, the amplitude (11) does not have an explicit [i2 dependence which naively would be expected from the momentum integration around the Fermi surface. In fact, because of the dependence on FT = ju/7r, the radiative decay of the "Cooper" pion (11) vanishes as 1/fi'm dense matter.

89

8

Conclusion

In this talk, I have reported about the so-called Cooper-mesons: generalized scalar, pseudoscalar, vector and axial-vector particle-particle or hole-hole excitations in the CFL superconductor of QCD in the weak coupling limit. The octet scalar and pseudoscalar excitations are both massless, but only the pseudoscalars survive as Goldstone modes, while the scalars ones are higgsed by the gluons leading to the Meissner effect. The vectors and axial vectors are bound and degenerate irrespectively of their polarization. Their mass is less than twice the gap. Chiral symmetry is explicitly realized in the vector spectrum in the CFL phase in leading logarithm approximation, in spite of its breaking in general. In the CFL superconductor the vector mesons are characterized by form factors that are similar but not identical to those of the generalized pions. These self-generated form factors provide a natural cutoff to regulate the effective calculations at the Fermi surface. The composite vector mesons can only be viewed as hidden-gauge excitations if their size is ignored (their form factor set to one). Only in this approximate limit the effective lagrangian description, vector dominance and universality are valid. The zero-size limit is not compatible with the weakcoupling limit, because of the long-range pairing mechanism at work at large quark chemical potential. It is an open question whether going beyond the weak-coupling and leading-log approximations would render the hidden-gauge or vector-meson-dominance concepts of effective field theories more appropriate. The finite-size of the generalized pion, however, does not upset the geometrical normalization of the triangle anomaly in the generalized WZW form of Ref.8. Much like in the vacuum, the radiative decay of the generalized pions is dictated by geometry in leading order. Furthermore, it vanishes at asymptotic densities. The existence of bound light vector and axial-vector mesons in QCD at high density may have interesting consequences on dilepton and neutrino emissivities in dense environments: e.g., in young and hot neutron stars neutrino production via quarks in the superconducting phase can be substantially modified if the vector excitations are deeply bound with a non-vanishing coupling. These excitations may be directly seen by scattering electrons off compressed nuclear matter (with densities that allow for a superconducting phase to form) and may cause substantial soft dilepton emission in the same energy range in "cold" heavy-ion collisions.

90

Acknowledgments It is a pleasure for me to thank my collaborators on this project: Ismail Zahed, Mannque Rho, Maciej Nowak, Byung-Yoon Park, and Edward Shuryak. References 1. B.C. Barrois, Nucl. Phys. B 129, 390 (1977); S. Frautschi, Proceedings of workshop on hadronic matter at extreme density, Erice 1978. 2. D. Bailin and A. Love, Phys. Rept. 107, 325 (1984). 3. M. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B 422, 247 (1998); R. Rapp, T. Schafer, E.V. Shuryak and M. Velkovsky, Phys. Rev. Lett. 81, 53 (1998). 4. M. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B 537, 443 (1999); T. Schafer and F. Wilczek, Phys. Rev. Lett. 82, 3956 (1999). 5. D.T. Son, Phys. Rev. D 59, 094019 (1999). 6. R.D. Pisarski and D.H. Rischke, Phys. Rev. D 60, 094013 (1999). 7. For recent reviews, see K. Rajagopal, to appear in Proceedings of Quark Matter '99, hep-ph/9908360; F. Wilczek, to appear in Proceedings of PANIC '99, hep-ph/9908480; T. Schafer, nucl-th/9911017; M. Alford, to appear in Proceedings of TMU-Yale, Dec 1999 hep-ph/0003185. 8. D.K. Hong, M. Rho and I. Zahed, Phys. Lett. B 468, 261 (1999). 9. R. Casalbuoni and R. Gatto, Phys. Lett. B 464, 111 (1999). 10. D.T. Son and M.A. Stephanov, Phys. Rev. D 61, 074012 (2000). 11. K. Zarembo, Phys. Rev. D 62, 054003 (2000). 12. H. Leutwyler, hys. Rev. D 49, 3033 (1994). 13. V. Thorsson and A. Wirzba, Nucl. Phys. A 589, 633 (1995) [nucl-th/9502003]; A. Wirzba and V. Thorsson, hep-ph/9502314. 14. M. Kirchbach and A. Wirzba, Nucl. Phys. A 604, 395 (1996) [nucl-th/9603017]; hep-ph/9609291; Nucl. Phys. A 616, 648 (1997) [hep-ph/9701237]. 15. R.D. Pisarski and M. Tytgat, Phys. Rev. D 54, 2989 (1996). 16. M. Rho, A. Wirzba and I. Zahed, Phys. Lett. B 473, 126 (2000) [hep-ph/9910550]. 17. M. Rho, E. Shuryak, A. Wirzba and I. Zahed, Nucl. Phys. A 676, 273 (2000) [hep-ph/0001104]. 18. M. A. Nowak, M. Rho, A. Wirzba and I. Zahed, hep-ph/0007034. 19. B.-Y. Park, M. Rho, A. Wirzba and I. Zahed, Phys. Rev. D 62, 034015 (2000) [hep-ph/9910347].

C H A R M O N I U M S U P P R E S S I O N I N P B - P B COLLISIONS A N D QUARK-GLUON DECONFINEMENT A.B. KUREPIN for the NA50 Collaboration M.C. ABREU 6 " 4 , B. ALESSANDRO 10 , C. ALEXA 3 , R. ARNALDI 1 0 , M. ATAYAN 12 , C. BAGLIN 1 , A. BALDIT 2 , M. BEDJIDIAN 1 1 , S. BEOLE 1 0 , V. BOLDEA 3 , P. BORDALO 6 ' 3 , A. BUSSIERE 1 , L. CAPELLI 1 1 , L. CASAGRANDE 6 ' 0 , J. CASTOR 2 , T. CHAMBON 2 , B. CHAURAND 9 , I. CHEVROT 2 , B. CHEYNIS 11 , E. CHIAVASSA 10 , C. CICALO 4 , T. CLAUDINO 6 , M.P. COMETS 8 , N. CONSTANS 9 , S. CONSTANTINESCU 3 , N. DE MARCO 1 0 , A. DE FALCO 4 , G. DELLACASA 10 ' 15 , A. DEVAUX 2 , S. DITA 3 , O. DRAPIER 1 1 , L. DUCROUX 11 , B. ESPAGNON 2 , J. FARGEIX 2 , P. FORCE 2 , M. GALLIO 10 , Y.K. GAVRILOV 7 , C. GERSCHEL 8 , P. GIUBELLINO 1 0 , M.B. GOLUBEVA 7 , M. GONIN 9 , A.A. GRIGORIAN 1 2 , J.Y. GROSSIORD 1 1 , F.F. GUBER 7 , A. GUICHARD 1 1 , H. GULKANYAN 12 , R. HAKOBYAN 1 2 , R. HAROUTUNIAN 11 , M. IDZIK 1 0 ' E , D. JOUAN 8 , T.L. KARAVITCHEVA 7 , L. KLUBERG 9 , A.B. KUREPIN 7 , Y. LE BORNEO 8 , C. LOURENQO 5 , P. MACCIOTTA 4 , M. MAC CORMICK 8 , A. MARZARI-CHIESA 1 0 , M. MASERA 10 , A. MASONI 4 , S. MEHRABYAN 1 2 , M. MONTENO 1 0 , A. MUSSO 10 , P. PETIAU 9 , A. PICCOTTI 1 0 , J.R. PIZZI 1 1 , F. PRINO 1 0 , G. PUDDU 4 , C. QUINTANS 6 , S. RAMOS 6 ' B , L. RAMELLO 1 0 ' D , P. RATO MENDES 6 , L. RICCATI 1 0 , A . R O M A N A 9 , 1 . ROPOTAR 5 , P. SATURNINI 2 , E. SCOMPARIN 10 S. SERCI 4 , R. SHAHOYAN 6 "", S. SILVA6, M. SITTA 1 0 " 0 , C. SOAVE 10 , P. SONDEREGGER 5 ' 3 , X. TARRAGO 8 , N.S. TOPILSKAYA 7 , G.L. USAI 4 , E. VERCELLIN 1 0 , L. VILLATTE 8 , N. WILLIS 8 . 1 LAPP, CNRS-IN2P3, Annecy-le-Vieux, France. 2 LPC, Univ. Blaise Pascal and CNRS-IN2P3, Aubiere, France. 3 IFA, Bucharest, Romania. 4 Universitd di Cagliari/INFN, Cagliari, Italy. 5 CERN, Geneva, Switzerland. 6 LIP, Lisbon, Portugal. 7 INR, Moscow, Russia. 8 IPN, Univ. de Paris-Sud and CNRS-IN2P3, Orsay, France. 9 LPNHE, Ecole Polytechnique and CNRS-IN2P3, Palaiseau, France. 10 Universitd, di Torino/INFN, Torino, Italy. n IPN, Univ. Claude Bernard Lyon-I and CNRS-IN2P3, Villeurbanne, France. 12 YerPhI, Yerevan, Armenia. a) also at UCEH, Universidade de Algarve, Faro, Portugal b) also at 1ST, Universidade Tecnica de Lisboa, Lisbon, Portugal c) now at CERN d) Universitd del Piemonte Orientale, Alessandria and INFN-Torino, Italy e) now at Faculty of Physics and Nuclear Techniques, University of Mining and Metallurgy, Cracow, Poland f) on leave of absence of YerPhI, Yerevan, Armenia

91

92 An anomalous charmonium suppression in central P b - P b collisions at 158 GeV/c per nucleon was observed in the NA50 experiment at CERN. The ratio of the J/tp production cross section to the Drell-Yan cross section measured for several lighter colliding systems extending from p-p to S-U exhibits the exponential decrease due to ordinary absorption of the J/tp in nuclear matter. For Pb-Pb interactions the experimental d a t a agree with this ordinary absorption curve only for the most peripheral collisions. By increasing the centrality of the interaction, a threshold effect is observed for the onset of the anomalous suppression. T h e experimental d a t a agree well with t h e predictions of a quark-gluon phase transition formed during the collision and with the interpretation of charmonium suppression due to Debye color screening in a deconfined plasma.

1

Introduction

Although the existence of a deconfinement state of quarks and gluons or quarkgluon plasma is predicted by non-perturbative lattice calculations 1, only the experimental observations of this new state of matter could provide the real values of critical temperature and energy density of phase transition. An extensive search of the new physical phenomena by collisions of high energy lead ions with nuclear targets was performed in CERN experiments during last years. Several experimental observables sensitive to quark-gluon plasma formations were tested. In particular the NA50 experiment extends to lead-lead collisions the early study of charmonium states production in the NA38 experiment made with protons and light ions like sulphur and oxygen. The physical justification of the NA50 experiment was based on the prediction of Matsui and Satz 2 on a suppression of charmonium production in case of quark-gluon plasma formation during heavy ion collisions due to Debye color screening of the binding potential between c and c quarks preventing to form charmonia states. As a reference process for comparison of charmonium production cross section with different projectiles and target nuclei the yield of Drell-Yan dimuons was chosen. It was found that the Drell-Yan cross section is proportional to the product of numbers of nucleons in colliding nuclei cr{DY) ~ A-B. Therefore one can assume that the Drell-Yan process in nucleus-nucleus collisions proceeds through nucleon-nucleon interactions. On the other hand in the NA38 experiment with light nuclei projectiles 3 the deviation of the J/ip production cross section from a A • B dependence was observed. A decrease of the charmonium production cross section compared to the Drell-Yan cross section for heavier colliding nuclei was accounted for by nuclear absorption of the cc pair before it forms the J/ip state 4 . This ordinary suppression could be described by one parameter - the absorption cross section aabs = 6.1 ± 0.7 mb 5 .

93

The first experimental data on Pb-Pb obtained by NA50 in 1995 5 exibit a strong departure from this smooth behaviour. The value of the J/tp production cross section in Pb-Pb collisions was found to be by a factor 0.77 ± 0.04 below the one expected from the extrapolation of the ordinary absorption dependence established from the p-A and S-U data. This effect of anomalous charmonium suppression increased for more central collisions. The centrality of the collision was determined by the measurement of the neutral transverse energy ET with an electromagnetic calorimeter. The most central coUisions correspond to ET > 100 GeV. In 1996 new high statistical measurements were made by increasing the target thickness 6 . A detailed study of the centrality dependence was possible, and it was shown that the charmonium production cross section for low ET, i-d. for the most peripheral collisions could be described by the ordinary absorption curve. The statistical accuracy of the experimental results was considerably increased by using as a reference process the minimum bias events. The procedure of minimum bias analysis is described in detail in 6 . The data obtained in 1996 have shown unambiguously that a threshold suppression effect was observed at ET ~ 40 GeV corresponding to an impact parameter of about 8 fm. However a comparision made for the high ET value of the (thin target) 1995 data and the (thick target) 1996 data has revealed some excess for the latter sample of events. This distortion is due to reinteractions of spectator fragments in the thick target, about 30% of an interaction length. Some secondary peripheral collisions with larger cross sections ratio could be erroneously taken as more central collisions without reinteraction. Therefore the experiment in 1998 was performed with one thin target 7 . The result is in perfect agreement with the 1995 data. The review of charmonium suppression in Pb-Pb collisions obtained in the three running periods is presented in this report. 2

Experimental installation

The main detector of the N A50 installation is a dimuon spectrometer equipped with tracking wire chambers, trigger scintillator detectors, a hadron absorber with a preabsorber made of BeO 5 . Additional detectors were used specifically designed to cope with the high radiation level, the huge multiplicities and the background induced by incident Pb ions: a beam hodoscope and several antihalo detectors. A set of quartz counters located near each piece of the segmented target was used to determine the vertex of the interaction and reject events where spectator fragments produce a second interaction in a

94

downstream subtarget. The centrality of the collisions is determined by three detectors: an electromagnetic calorimeter, a multiplicity detector and a zero degree calorimeter. For the extraction of the J/tp and ip' yield from the muon pair invariant mass spectra, the contributions from different processes were taken into account: combinatorial background originating from uncorrelated pion and kaon decays, open charm and Drell-Yan production.

O Ptt.PMWS D P6 P»1988«0iMnrmjma
20

40

60

80

100

120 140 E^GeV)

Figure 1. cr^/a-py ratio as a function of ET, obtained with the standard and minimum bias analyses of the 1996 and 1998 data samples. The curve represents the J/if) suppression due to the ordinary nuclear absorption.

3

Results from the N A 5 0 experiment

A detailed investigation of the anomalous charmonium suppression in PbPb collision is the main goal of the NA50 experiment. The results obtained up to now are presented in Figs. 1, 2, 3. The ratio of the measured cross section for J/ip production to the Drell-Yan cross section as a function of the neutral transverse energy ET is shown in Fig. 1. Results with better statistics and more ET bins have been obtained using the minimum bias events, and calculating with the Glauber model the ratio of Drell-Yan to

95

minimum bias yield. They are normalized to the standard analysis results in the ET interval between 50 and 75 GeV. The most central data of the 1996 sample for Ej- > 100 GeV are not shown due to systematics effects due to reinteractions within the thick target. . 40-r

120

140

Figure 2. Comparison between our data and several conventional calculations of J/il> suppression.

The solid line, shown in Fig.l, corresponds to the extrapolation of the ordinary nuclear absorption curve obtained from proton-nuclei and light nuclei collision systems studied by the NA38 and NA51 collaborations 3 . A clear onset of the sudden drop near ET = 40 GeV is seen. This anomalous threshold effect first observed by the NA50 collaboration in 1995 is confirmed by results obtained with the high statistics data in 1996. No conventional models could represent this sharp threshold effect with increasing centrality of collision. The data of the 1998 experimental run are shown in Fig. 1 for ET > 40 GeV since for lower ET the contribution from Pb interactions on air is too big as compared to the thin target used. It was found with the special empty target run that this background is negligible for the events with ET above 40 GeV. The 1998 data show a second drop in the J/tjj suppression pattern at ET around 90 GeV. By increasing the centrality of the collision a steady decrease of the ratio

96

of the J/tp to Drell-Yan cross section is observed. This centrality dependence is in clear disagreement with several hadronic models which assume that the charmonia states are absorbed by interactions with comoving hadrons [8-11]. All these models predict a smooth decrease of the J/ip production cross section from p-p to Pb-Pb collisions, tending to a saturation of the J/ip suppression in the most central collisions. The comparision is presented in Fig. 2. On the contrary, the centrality dependence of the anomalous J/ip suppression measured in the NA50 experiment agrees with the pattern expected in the framework of production of a deconfined state of quarks and gluons 12 . $1.4 "1-2

•fFHi'f#H

1-

•0.8 0.6 0.4 02

•I • a r • 4 o

F* - Pt> 1 MS vrtlh Minimum Bias Pb-Pt>1 MS with Mlrtmiffi Bias Pb-Pb 1MB 3-U NAM P-A NWS p.p(d> NASI " ' • • ' ' l

as

1

'•'""

, „ , . , . , , . , , r „ | ,„,,„|

1.5

2.S

3

3,5

Figure 3. Measured J/ip production yields, normalized to the yields expected assuming that the only source of suppression is the ordinary absorption by the nuclear medium. Results are shown as a function of the energy density reached in the collision.

The threshold effect and the anomalous charmonium suppression pattern are illustrated in Fig. 3, where the ratio of the J/tp production yield to the expected one due to only ordinary nuclear absorption is plotted as a function of the energy density reached in the nucleus- nucleus collision. In this plot are compared the data for different colliding systems from the NA38 and NA50 experiments. The energy density was computed according to Bjoken's model 13 . It reaches the value 3.5 GeV/fm3 for the most central Pb-Pb collisions and is in accordance with other experimental estimates and with cascade model

97

calculations. The two drops observed in the charmonium production cross section seen in Fig.3 at energy densities around 2.3 and 3.1 GeV/fm3 could be understood as due to the successive melting of first the more loosely bound state Xc a n ( l then the J/tp charmonium states. The fraction of J/tp produced by Xc radiative decay is not known. For p-A collision this fraction is about 30-40 %. 4

Conclusions

The combined results of the NA38 and NA50 experiments clearly show a two step-wise charmonium suppression pattern at high energy densities with no visible saturation for the most central collisions. This observation exclude the presently available models of J/ip suppression based on the interaction of charmonium with surrounding hadronic confined matter. On the contrary the step-wise behaviour of charmonium suppression is easily understood as a result of Debye color screening if quark-gluon plasma is produced during the collisions. Therefore, the experimental study of charmonium suppression performed by the NA50 collaboration provides evidence for the deconfinement of quarks and gluons in central Pb-Pb collisions at SPS energy. This work was partially supported by Russian Fondation for Fundamental Research, grant 99-02-16003. References 1. G. Boyd et al., Nucl. Phys, B 469 (1996) 419. 2. T. Matsui and H. Satz, Phys. Lett., B 178 (1986) 416. 3. M.C. Abreu et al., NA38 Collaboration, Phys. Lett. B 444 Phys. Lett. B 449 (1999) 128. 4. C. Gerschel, J.Hufner, Z. Phys., C 56 (1992) 171. 5. M.C. Abreu et al., NA50 Collaboration, Phys. Lett., B 410 6. M.C. Abreu et al., NA50 Collaboration, Phys. Lett., B 450 7. M.C. Abreu et al., NA50 Collaboration, Phys. Lett., B 477 8. J. Geiss et al., Phys. Lett., B 447 (1999) 31. 9. C. Spieles et al., Phys. Rev. C 60 (1999) 054901. 10. D.E. Kahana and S.H. Kahana, nucl-th/9908063. 11. N.Armesto, A. Capella and E.G.Ferreiro, Phys. Rev., C 59 12. H.Satz, Nucl. Phys, A 661 (1999) 104c. 13. J.D. Bjorken, Phys. Rev. D 27 (1983) 1409.

(1998) 516;

(1997) 337. (1999) 456. (2000) 28.

(1999) 395.

PARTICLE P R O D U C T I O N I N P B + P B COLLISIONS AT 158 G E V / N U C L E O N IN T H E N A 4 9 D E T E C T O R H. S T R O B E L E (FOR T H E NA49 COLLABORATION) Universitat

Frankfurt/Main,

Germany

The NA49 collaboration studies hadron production with a large acceptance T P C detector in P b and proton induced collisions at the CERN SPS. Results from P b + P b collisions are summarized in terms baryon stopping, flow, pion rapidity distributions, kaon and cj> production, as well as event-by event fluctuations of the k/w ratio and the mean transverse momentum.

1

Introduction

Experiment NA49 at the CERN SPS was designed to search for evidence of a transient phase of deconfmed matter, the quark-gluon-plasma (QGP), in ultrarelativistic nucleus-nucleus collisions via the measurement of hadronic observables. The detector 1 consists of 2 large dipole magnets, 4 time projection chambers (TPC) inside and downstream of the magnets, and scintillation counter time-of-flight (TOF) arrays. Precision tracking and identification via energy loss in the TPC gas (4% resolution) and TOF (60 ps resolution) is accomplished for about 60% of the produced charged particles and the decay products of strange particle decays. NA49 measures inclusive particle spectra, correlations, and hadronic resonance production with high statistics and wide coverage of phase space. The high produced particle multiplicity in P b + P b collisions combined with the large acceptance allow for the first time a meaningful measurement of global properties and their fluctuations in individual events.

2

Results

This report summarizes results obtained from P b + P b collisions at varying impact parameters. Central collisions represent 5% of the total cross section or impact parameters b < 3.5 fm with 360 participant nucleons whereas minimum bias collisions are divided up in 6 centrality windows. Emphasis is put on baryon stopping and particle production in terms of a statistical/thermodynamical interpretation. 99

100

i*

NA49 Preliminary

•

Pb+Pb (1)-4(p-p)+X

• pwbm •

Pb+Pb (3)

* Pb+Pb (
10 --•—#--a—»—

Rapidity

Figure 1. Normalized rapidity distributions of p — p for Pb+Pb collisions for different centralizes.

2.1

Net baryon and negatively charged hadron distributions

The rapidity (y) distribution of the net baryons (protons) is one way to illustrate the degree of stopping of the incoming nucleons. It is well established that in p + p interactions the net protons exhibit peaks at low and high y. Such a behavior is also seen for peripheral P b + P b collisions (lowest curve in Fig. 1). For more central collisions the peak of the distribution is moved towards midrapidity by about 1 unit. This is clear evidence for multiple scattering of the incoming nucleons 2 . The yield of negatively charged hadrons (predominantly Tt~) is plotted versus the rapidity y in Fig. 2 3 . It is compared to results from central S+S and nucleon-nucleon collisions scaled by the ratio of the number of participating nucleons. One observes similar y-distributions; however an overall excess of about 20% is seen in central nucleus-nucleus collisions. Since pions carry most of the entropy in a hadron gas, this finding has been attributed to increased specific entropy production 4 which could indicate a QGP phase in the early stage of nucleus-nucleus reactions.

101 250 negative hadrons 200 150

;*??

if $

¥

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• Pb+Pb, central 5%

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• scaled N+N

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i

I

i

I

i

^ I

i

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i

-3

Figure 2. Normalized rapidity distributions of h~ for central P b + P b collisions (NA49) compared to results from central S+S (NA35) and nucleon-nucleon reactions scaled by the ratio of participant nucleons.

A

1 5 8 - 2 0 0 A GeV/c

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fc 0.014 PbPWNMg;

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0.O02 l

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10

10 < NP>

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1 800

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Figure 3. (a) ratio of the numbers of produced Kaons and pions in various collision systems at 158-200 GeV/nucleon beam energy, (b) ratio of the number of produced ^-mesons and •K~ in central P b + P b and p + p collisions versus the square s of the nucleon-nucleon cms energy.

2.2

Strangeness production and hadro-chemical freezeout

About 75% of s and s quarks in the produced hadrons are contained in kaons. Due to the large acceptance, NA49 can determine the ratio (K + K)/(ir) in

102

Figure 4. (a) allowed regions of freezeout temperature T and radial expansion velocity /3± from mt spectra and BE correlations in central P b + P b collisions, (b) elliptic flow parameter V2 versus impact parameter b compared to model calculations.

full phase space. The result is plotted in Fig. 3a for central P b + P b together with measurements in S+S, S+Ag as well as p + p and p+A reactions. Kaon production is seen to be enhanced by a factor of two in nucleus-nucleus collisions. As shown in Fig. 3b this effect is even more pronounced for the hidden strangeness (j>{ss) meson. Again the QGP hypothesis provides a natural explanation of these observations 4 , while it appears difficult to achieve the observed kaon yield in a hadron gas. The particle composition in fact suggests hadron production according to phase space probabilities at a universal temperature of about 180 MeV, with strangeness suppression being reduced by a factor two in nucleus-nucleus collisions5. 2.3

Kinetic freezeout and anisotropic flow

Bose-Einstein (BE) correlations of pion pairs together with inclusive momentum spectra have been used to derive the size and internal dynamics of the produced particle system at freezeout6, i.e. when the interactions among particles cease. The transverse momentum dependence of the correlations and the mass dependence of the inverse slope parameters of the mt distributions are signatures of strong longitudinal and transverse expansion velocities. Fig. 4a illustrates that the measurements are described by a freezeout temperature of about 120 MeV and a transverse velocity of P±_ = 0.55. Moreover the analysis obtained a lifetime of the fireball of 8 fm/c during which its transverse

103

. 10

0.3

0.4

0.5

0.1 0.2 0.3 0.4 0.5

Single Event [GeV/c]

Single Event K/rc ratio

Figure 5. Distibutions of (pt) and the ratio K/ir measured in individual events (data points) compared to expectations from finite number fluctuations (histograms).

size increases more than twofold. For this method of analysis a scheme was used which treats BE-correlations and inclusive particle spectra consistently. However, it overestimates the temperature at large /3j_. Further information on the effective equation of state of the produced matter can be deduced from the study of non-central collisions, where pressure driven flow in the initial almond-shaped interaction region may lead to an anisotropic azimuthal distribution at freezeout. This anisotropy, quantified by the Fourier coefficient Vi of the azimuthal distribution of particles 7 , is plotted in Fig. 4b as a function of the impact parameter b of the collision. Significant in-plane elliptic flow is observed; its order of magnitude is reproduced by the rescattering model RQMD 8 . Hydrodynamical calculations based on a pion gas 9 overpredict the measurements by a factor 5. 2.4

Fluctuations of temperature and flavor content

Fluctuations of average event properties reflect both the equilibration and thermodynamical properties of the produced matter as well as possible longrange correlations which may occur in a medium close to the phase transition. The distributions of the average transverse momentum (p t ) and the ratio of the numbers of charged kaons and pions K/iv per event are displayed in Fig. 5.

104

They closely agree with expectations from finite number statistics as shown by the histograms derived by mixing particles from different events. It appears that central P b + P b collisions produce rather uniform particle ensembles. The lack of dynamical fluctuations in temperature (upper limit on fluctuations in (pt) < 1%) and flavor composition (fluctuations in K/TT < 5%) indicates a high degree of equilibration in the produced matter. This naturally occurs in a QGP phase where equilibration processes are expected to proceed much faster than in a hadron gas. 3

Summary and outlook

Results of NA49 and other SPS experiments found predicted signatures of the QGP, unfortunately none of them are unequivocal. When the produced hadron density is extrapolated back to the initial stage of the reaction, these hadrons are found to be so densely packed that it is hard to imagine how they can exist as separate entities. Whether this dense state is a truly equilibrated QGP phase remains to be elucidated. In the remaining SPS heavy-ion program NA49 will attempt to locate the threshold for QGP production at lower beam energies and with lighter nuclei in order to more clearly confirm its fleeting existence. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

S. Afanasiev et al, CERN-EP-99-001, NIM A430(1999)210 G.E. Cooper et al, Nucl. Phys. A661(1999)362c-365c H. Appelshaeuser et al, Phys. Rev. Lett. 82 (1999) 2471 M. Gazdzicki and M. Gorenstein, Acta Phys. Pol. B30(1999)2705 F. Becattini et al, Z. Phys.C76(1997)269; Eur. Phys. J. C5(1998)143 H. Appelshaeuser et al, Eur. Phys. J. C2 (1998) 661 A. Poskanzer and S. Voloshin, Phys. Rev. C58 (1998) 1671 H. Sorge, Phys. Rev. Lett. 82 (1999) 2048 J.-Y. Ollitrault, Phys. Rev. D46 (1992) 229

E X P L O R I N G T H E CHIRAL P H A S E T R A N S I T I O N IN H I G H - E N E R G Y COLLISIONS J 0 R G E N RANDRUP Nuclear Science Division, Lawrence Berkeley Laboratory, Berkeley, California 94720 E-mail: [email protected] Chiral symmetry plays an important role in nuclear physics but its manifestations have so far been testable only in the normal phase where the symmetry is spontaneously broken. More global explorations can be made by means of high-energy nuclear collisions, since a partial restoration of chiral symmetry is expected to occur within the highly agitated collision zone. Our current understanding of the chiral phase diagram will be reviewed, the special character of the induced nonequilibrium relaxation dynamics will be elucidated, and two areas of intensified future research will be highlighted.

1

Introduction

As the new century dawns, the field of nuclear physics encompasses a broad spectrum of research frontiers at which many promising research issues present themselves. Chiral symmetry constitutes a central concept in modern nuclear physics1 and the present discussion focusses on the prospects for exploring its manifestations over a wide range of environments by means of high-energy nuclear collisions. The strong interaction possesses chiral symmetry only to the degree that the basic quark masses can be neglected. Moreover, the self-interaction of the gluon fields encourages the spontaneous breaking of the symmetry in systems at low excitations, causing the familar vacuum to be a chiral condensate with a finite expectation value. Although chiral symmetry has proven to provide a successful framework for understanding many features in hadronic physics, the confrontation with experimental data has been primarily limited to the strongly broken phase. In the highly excited environments produced in high-energy collisions, the chiral condensate is significantly weakened and the system may (briefly) achieve approximate chiral symmetry. The subsequent non-equilibrium relaxation process may then produce observable signals that could be exploited to test our global understanding of chiral symmetry in strong interactions 2 ' 3,4 . This presentation reviews our current expectations regarding this novel phenomenon and identifies two important topics for intensified research, namely the development of a practical treatment of real-time non-equilibrium quantumfield dynamics in a non-uniform environment and the extension of the models from the ud SU(2) sector to the uds SU(3) sector of the strong interactions. 105

106

2

Equilibrium Properties

A simple theoretical tool for the global exploration of chiral symmetry in baryon-free systems is provided by the linear a model. It describes the 0(4) chiral field 0(r, i) = (a, n) by means of a simple effective quartic interaction, c = \d»<\>°d»4>-

2 \{4>°- v f

+ Ha

, 0 o 0 = (r(r)2+7r(r)-7r(r) . (1)

The parameters, A, v, H, may be fixed to reproduce the pion decay constant fn, the free pion mass m w , and the mass of the schematic a meson, mCT. Useful insight can be gained by considering macroscopically uniform matter. The chiral field can then be decomposed, 0(r, t) — 0(t) + 64>(r,t), where 5(f) represents quasi-particle agitations relative to the (false) vacuum characterized by the 0(4) chiral order parameter . Figure 1 {left) shows the associated free energy density FT = Vr(^o,Xo) _ TST(<{>O) along the a axis, with (To = ocosxo- At each temperature T, the minimum in FT represents the equilibrium value of the order parameter and it shrinks steadily as T is increased, thus producing a crossover from the strongly broken to the approximately symmetric chiral phase: Cooling of uniform matter 7"o=400 MeV

&100

-120 -80 -40 0 40 80 Order parameter a0 (MeV)

120

0

20 40 60 80 100 Order parameter $0 = l«l»l (MeV)

Figure 1: Left: The free energy density FT calculated with the semi-classical mean-field approximation to the linear a model, as a function of the order parameter tro for various values T of the quasi-particle gas. At each temperature, the solid dots indicate the equilibria. For T < To = V%v, the order parameter must exceed a certain minimum value before all quasiparticle modes are stable; the corresponding end points are connected by the dotted curve. The top dashed curve is the bare potential obtained for T = 0 . Right: The combined dynamical evolution of the order parameter ao = (c) and the field fluctuations {(50 2 } 1 / 2 . The dashed curve connects the equilibria from T=0 to above 500 MeV and the unstable region within which fi% < 0 is shown by the shaded region. Each system has been prepared in thermal equilibrium at To — 400 MeV and then subjected to a Rayeligh cooling, emulating a uniform expansion in D dimensions.

107

3

D C C Dynamics

If the field fluctuation is employed as a measure of the agitation, as in fig. 1 (right), the resulting diagram applies to any state of the system and is thus suitable for displaying non-equilibrium dynamics. In particular, it is possible to display the trajectories that result from the application of a Rayleigh cooling to initially hot chiral matter in equilibrium,

[D + A(0o0-« J )]0(r,t) =

He„---,

(2)

which emulates a Bjorken-type scaling expansion in D dimensions.6 Two general features are apparent: 1) only very rapid cooling rates lead into the supercritical region where spontaneous pion creation occurs, thus suggesting that this interesting quench scenario may be dynamically unreachable; 2) for any cooling rate, the order parameter ultimately settles into rather regular damped oscillations around the true vacuum which may be expected to produce parametric amplification of pionic modes with frequencies u^ « \ma. More refined scenarios considering rods endowed with a longitudinal Bjorken scaling expansion confirm the above expectations, as is illustrated in fig. 2 for either the dynamical trajectory (left) or the effective pion mass (right)?

T„= 240 MeV

T, = 240 MeV

— Bjorken matter

*

- Bjorken rod {R0 =6 fm) — Equilibrium path — Critical boundary "\ — Bjorken matter ot=1,2,3,4,5,10,20,30 fm/c -- Bjorken rod (R0=6 fm) •t=1,2,3,4,5,10,20,30 tm/c

Order parameter rji0 (MeV)

11A

11

Proper time t (fm/c)

Figure 2: Left: Phase evolution of the interior of a Bjorken rod prepared with a bulk temperature of Tn=240 MeV and a radius of i?o=6 fm and the corresponding evolution of Bjorken matter (obtained in the limit of a very thick Bjorken rod, Ro —> oo) In order to be able to display an arbitrary non-equilibrium scenario, the average field fluctuation ( o ^ 2 ) 1 / 2 = (oV2 4- (57T2)1/2 has been used in place of the temperature on the vertical axis. Also shown are the equilibrium path and the boundary of the supercritical region. The rod has been probed over a hollow cylindrical volume, K p < 3 (fm). The paths are marked at successive proper times T=1,2,3,4,5,10,20,30 fm/c. The two evolutions are similar early on, but the rod then relaxes significantly faster due to the generated transverse expansion. Right: The corresponding evolutions of the squared effective pion mass ^ 2 .

108

4

Vacuum Fluctuations

A full quantum-field treatment of the chiral dynamics is beyond current reach and the dynamical simulation studies have employed classical fields. Although much valuable insight can been gained in this manner, it is important to recognize that such treatments are not quantitatively reliable. This is perhaps best illustrated by considering a free pionic mode with a time-dependent mass, [D + ojl{t)}
n£ nal = Xk [nt% + \] - | > Xkn™«

(3)

which exceeds the classical result X^nj^11', since the amplification coefficient Xk is generally larger than unity. The expression brings out the fact that the quantum fluctuations and the statistical fluctuations combine in a democratic fashion as seeds for the parametric amplification. A quantitative impression of their relative importance can be gained from fig. 3 (left). It is evident that the thermal occupancies are never large compared to one half. Consequently, the vacuum fluctuations are never negligible. Efforts to develop a suitable quantum-field treatment are underway10 and an illustration is given in fig. 3 (right). It is seen that pions with momenta around 200 MeV/c are produced, even if the initial state contained no quasiparticles at all. The initial presence of thermal excitations will then typically increase the yield several fold. Thermal Equilibrium

100 200 300 Temperature T(MeV)

400

-200

0

200

Momentum p (MeV/c)

Figure 3: Left: The temperature dependence of the thermal occupanices of the quasipions as obtained with either a semi-classicial treatment 5 or an optimized perturbation method. 9 Right: The final pion spectrum arising solely from amplification of the vacuum fluctuations in one-dimensional systems with given effective mass functions n\(x,t) of forms similar to those obtained for Bjorken rods (see fig. 2 (right)).10

109

5

Strangeness

Since the temperatures of interest easily exceed the mass of the s quark, it is expected that strangeness must be incorporated. Within the linear a model, this involves an extension from its familiar SU(2) form to SU(3). The number of meson fields then increases from 4 to 18,

ag,C; a,0

,n+

a; 7r°,7r

Vo;

•*'

> a0 ' K* ", K*+, K*°, K*°; -,n+,K~ , K+, K°, K°

(4)

and the equilibrium value of the order parameter acquires a strange component, a -> (a, £), where ( = (ss). As is evident from Fig. 4 (left), the inclusion of strangeness severely impedes the restoration of chiral symmetry as the temperature is raised, thus casting doubt on the standard scenario in which approximate restoration is assumed to occur once T exceeds a few hundred MeV. On the other hand, as illustrated in Fig. 4 (right), the dynamics of the order parameter becomes more intricate. As a result, the kaon modes may experience parametric amplification in analogy to what happens for the pions. The neutral kaon fraction fK = (K° + K°)/(K+ + K~ + K° + K°) then has a uniform distribution, PK(/K) = 1, for an idealized source, while the neutral pion fraction /„ = Tr°/(n+ + -K~ + TT°) retains its SU(2) distribution. 11

•.s 60

S * ' I

D=3

% 100

1 7

2

ft i

60

*

40

AT=100MeV

80

— • SU(2) — • SU(3). m„=eoo MeV ~ • SU(3), mo=800 MeV — • S U ( 3 ) , m„-1000MeV — • SU(3), no kaon fluct.

•••-

20

0

20

40

60

m =1 GeV 80

Order parameter a [MeV]

100

20

40

60

80

100

Order parameter a [MeV]

Figure 4: Left: The thermal path of the two order parameter (
110

6

Outlook

Chiral symemtry is a fundamental concept in strong-interaction physics and hence it is of general interest to explore its properties over a wide range of environments. High-energy collisions between heavy nuclei provide a promising means for this undertaking, as dynamical explorations with the linear a model suggest that specific signatures may be practically observable. It is therefore worthwhile to intensify the research efforts in this area. On the theoretical side, many challenges remain and the present discussion has merely highlighted two of them: 1) the need for taking account of the vacuum fluctuations by developing a proper real-time non-equilibrium quantum-field treatment of the chiral dynamics and 2) the need for including the strangeness degrees of freedom into the description. Although both problems pose considerable difficulties, formal as well as practical, the available results do provide specific guidance with respect to what may constitute particularly informative observables. These include enhancements in the soft part of the transverse pion spectra, anomalous fluctuations in their multiplicities, azimuthal correlation patterns, and a widening of the distribution of the neutral pion fraction.7 Thus (apart from this latter observable which may be beyond early reach), there is reason to hope for the appearance of interesting data in the upcoming first round of RHIC experiments. Acknowledgments This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Nuclear Physics Division of the U.S. Department of Energy, under Contract No. DE-AC03-76SF00098. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

V. Koch, Int. J. Mod. Phys. E6, 203 (1997). K. Rajagopal, Quark-Gluon Plasma 2 (R. Hwa), World Scientific (1995). J.-P. Blaizot and A. Krzywicki, Acta Phys. Polon. B27 (1996) 1687. J.D. Bjorken, Acta Phys. Polon. B28 (1997) 2773. J. Randrup, Phys. Rev. D55, 1188 (1997); Nucl. Phys. A616, 531 (1997). J. Randrup, Phys. Rev. Lett. 77, 1226 (1996). T.C. Petersen and J. Randrup, Phys. Rev. C61, 024906 (2000). J. Randrup, Heavy Ion Phys. 9, 289 (1999). S. Chiku and T. Hatsuda, Phys. Rev. D58, 076001 (1998). J. Randrup, LBNL-00000 (2000) in preparation. J. Schaffner-Bielich and J. Randrup, Phys. Rev. C59, 3329 (1999).

N U C L E A R COLLECTIVE FLOW IN HEAVY ION COLLISIONS AT SIS ENERGIES

LPC

N . B A S T I D for t h e F O P I Collaboration a Clermont-Ferrand, IN2P3-CNRS and Universite Blaise 63177 Aubiere Cedex, FRANCE E-mail: [email protected]

Pascal,

Directed flow and elliptic flow have been studied in heavy ion collisions at SIS energies with the FOPI detector. A particular interest is devoted to the results obtained from the Fourier expansion method which is used to describe the anisotropies of the particle azimuthal distributions. Comparisons with the predictions of the IQMD transport model are performed.

1

Introduction

Collective flow effects in heavy ion collisions are of great interest since they allow to extract information on the complex dynamics with the goal of drawing conclusions about the hot and dense nuclear matter and its underlying Equation of State (EoS). The FOPI Collaboration has realized a systematic study of in-plane flow component and out-of-plane flow component providing a complete and accurate set of experimental data. In this contribution we focus on flow results of charged baryons in Ru + Ru collisions at 400 AMeV and 1.69 AGeV. 2

Experimental setup and analysis method

The data have been measured with the FOPI detector 1 at GSI-Darmstadt. The FOPI detector is an azimuthally symmetric apparatus made of several sub-detectors which provide charge and mass determination over nearly the full 47r solid angle. The central part (33° < 9iab < 150°) is placed in a superconducting solenoid and consists of a drift chamber surrounded by a barrel of plastic scintillators. The forward part (1.2° < 6iab < 30°) is composed of a wall of plastic scintillators and an other drift chamber. The events are sorted out according to their degree of centrality by imposing cuts on the multiplicity of charged particles 2 . The global variable Erat 3 a

F O P I Collaboration : NIPNE Bucharest, Romania ; KFKI Budapest, Hungary ; LPC Clermont-Ferrand, France ; GSI Darmstadt, Germany ; FZR Dresden, Germany ; Seoul Korea University, South Korea ; Heidelberg University, Germany ; I T E P Moscow, Russia ; KI Moscow, Russia ; IReS Strasbourg, France ; Warsaw University, Poland ; RBI Zagreb, Croatia

in

112

is used for the selection of the most central collisions. The reaction plane is reconstructed using the standard transverse momentum analysis method 4 . The flow observables are corrected for the reaction plane dispersion and for autocorrelation and momentum conservation effects according to the methods described in 5 . 3

Results

Flow results obtained from light and intermediate mass fragments measured in semi-central Au + Au collisions between 100 AMeV and 800 AMeV can be found i n 6 . We report here on recent flow results obtained with charged baryons measured in semi-central Ru + Ru collisions with the central part of the FOPI detector. - u

-0.1

< b > = 2.9 fm - 0.7< y0 < - 0.5

L

CM0.2
geo>

=

4

-7fm

- 0.3 < y0 < - 0.1

-0.2 -0.3 -0.4

f

-0.5 -0.6 '-

V

A

% -m * A ^m

>J

ir

****, T

-0.7 -0.8 C)

•

0.4

0.8

*VHe A->A = 3 • ->d •~*P 1.2 1.6 2

Pt°

0.4

0.8

1.2

1.6

2

Pt°

Figure 1: vi (left) and V2 (right) Fourier coefficients as a function of the scaled transverse momentum p° for the normalized rapidity bins j/o mentioned on the figures. p° is the particle transverse momentum normalized t o the projectile momentum in the center-of-mass system and j/o denotes the particle rapidity divided by the beam rapidity in the center-of-mass system. T h e results concern charged baryons in semi-central Ru + Ru reactions a t 400 AMeV.

The in-plane flow component and the out-of-plane flow component are investigated by means of a Fourier expansion of azimuthal distributions, (p being the azimuthal angle of a particle with respect to the reconstructed reaction plane,

113

the dN/dip distributions can be parametrized by VQ (1 + 2J2li=i vn cos{rup)) where vn = < cos(rup) > are the Fourier coefficients. The first and the second coefficient, ^i and V2, quantify the strength of directed and elliptic flow, respectively. This procedure allows to simultaneously study the two flow components as a function of the particle transverse momentum pt and rapidity. The comparisons of the results with the predictions of transport models should provide deeper insight into the properties of the hot and dense nuclear matter. Figure 1 displays the pt dependence of v\ and v% of charged baryons measured in semi-central Ru + Ru collisions at 400 AMeV. Charged baryons have a negative «i for all pt. Since the rapidity window is located in the backward hemisphere, that means that charged baryons are positively flowing. The magnitude of directed flow increases with pt and tends to saturate in the highest pt range. The V2 coefficient is plotted near mid-rapidity where the magnitude of elliptic flow is the largest. This second order Fourier coefficient is negative (i.e. out-of-plane preferred emission) and strongly correlated with pt. Both directed flow and elliptic flow increase with the particle mass hence confirming that composite particles provide a good probe for collective flow effects.

^ o,

-1.2
1

-0.9<-0.6

,

-0.6 < y(0) < -0.3

1

0 0.20.40.60.8 0 0.20.40.60.8 0 0.20.40.60.8 1 pt (GeV/c) Figure 2: v\ versus pt for protons in semi-central Ru -I- Ru reactions at 1.69 AGeV. The points refer to the data and the curves correspond t o t h e IQMD predictions. The results are shown for three rapidity windows mentioned on the figures.

The pt dependence of the proton directed flow data is compared with the predictions of the IQMD (Isospin Quantum Molecular Dynamics) model 7 in figure 2 for semi-central Ru + Ru collisions at 1.69 AGeV and three rapidity windows in the backward hemisphere. Two parametrizations of the EoS have

114

been studied, soft and hard, both including momentum dependent interactions, labelled SM and HM, respectively. The magnitude of directed flow is maximum in the target rapidity region and decreases when going towards mid-rapidity both in the model and in the data. The model fails in consistently describing the pt dependence of v\, although a reasonable agreement is found when ptintegrated flow data are used 8 . This seems to be a general feature of transport model calculations since a similar discrepancy has been observed from the comparison of experimental data with the RQMD model predictions for Au + Au reactions at 11 AGeV 9 and with the RBUU model predictions for Ni + Ni reactions at 1.93 AGeV and Ru + Ru reactions at 1.69 AGeV 1 0 . 4

Conclusion

The present experimental results reveal detailed information on the various dependences of directed and elliptic flow at SIS energies. The study of the Pt-differential flow should allow to constrain the main parameters used in the microscopic models. References 1. A. Gobbi, FOPI Collaboration, Nucl. Instrum. Methods A 324, 156 (1993) ; J. Ritman, FOPI Collaboration, Nucl. Phys. B (Proc. Suppl.) 44, 708 (1995). 2. J.P. Alard, FOPI Collaboration, Phys. Rev. Lett. 69, 889 (1992). 3. W. Reisdorf, FOPI Collaboration, Nucl. Phys. A 612, 493 (1997). 4. P. Danielewicz and G. Odyniec, Phys. Lett. B 157, 146 (1985). 5. C.A. Ogilvie et al., Phys. Rev. C 40, 2592 (1989) ; J.Y. Ollitrault, Nucl-ex/9711003 (1997). 6. V. Ramillien, FOPI Collaboration, Nucl. Phys. A 587, 802 (1995) ; N. Bastid, FOPI Collaboration, Nucl. Phys. A 622, 573 (1997) ; P. Crochet, FOPI Collaboration, Nucl. Phys. A 624, 755 (1997) ; P. Crochet, FOPI Collaboration, Nucl. Phys. A 627, 522 (1997) ; F. Rami, FOPI Collaboration, Nucl. Phys. A 646, 367 (1999) ; A. Andronic, FOPI Collaboration, Nucl. Phys. A, in press. 7. J. Aichelin, Phys. Rep. 202, 233 (1991) ; C. Hartnack et al., Eur. Phys. Journ. A 1, 151 (1998). 8. G.Q. Li and G.E. Brown, Nucl. Phys. A 636, 487 (1998) ; P.K. Sahu et al., Nucl. Phys. A 640, 693 (1998). 9. J. Barrette et al., Phys. Rev. C 56, 3454 (1997). 10. P. Crochet, FOPI Collaboration, Phys. Lett. B 486, 6 (2000).

H A D R O N OBSERVABLES F R O M H A D R O N I C T R A N S P O R T MODEL W I T H JET P R O D U C T I O N AT RHIC YASUSHI N A R A RIKEN

BNL

Research

Center,

Brookhaven National Laboratory, 11974, USA E-mail: [email protected]

Upton, New

York

The hadronic transport model with jet production is applied to explore heavy ion collision at RHIC energies. Higher hadronic resonance states up to 2GeV are also included in this model to handle hadronic resonance gas system. After showing some results on SPS data, we will show the predictions at RHIC energies. It is found that pQCD like transverse momentum shape becomes exponential at pj_ < 4GeV by hadronic rescattering.

1

Introduction

Relativistic heavy ion collider (RHIC) experiments at Brookhaven National Laboratory have started to explore a new form of matter. Microscopic transport models, for example, RQMD 3 or UrQMD 4 have been well tested at SPS or AGS energies. They have been recognized as a powerful tool to study relativistic heavy collisions. Those models are also used to give some predictions at RHIC 5 ' 6 , especially, observables which is considered to be related to the soft physics. Event generators based on perturbative QCD (pQCD) are proposed such as HIJING 1 , VNI 2 , emphasizing the importance of mini-jet productions. In this work, the shape of the high momentum part in the transverse momentum distributions will be investigated by the transport model JAM 7 . Particle production in JAM is based on hadronic resonances or strings formation and decay and pQCD hard scattering to simulate large energy range of particle interactions (lGeV < v ^ < 200GeV). We are interested in the particle spectra affected by the interactions in the hadronic resonance gas stage of the heavy ion collisions. 2

Model description

The main elastic hh idea from excitation

features included in JAM are as follows. (1) At low energies, incollisions are modeled by the resonance productions based on the RQMD and UrQMD. (2) Above the resonance region, soft string is implemented along the lines of the HIJING model 1 . (3) Multiple 115

116

. Pb(158AGeV)+Pb h' b<3.2fm (a)

10 ^

> 3.7
O

10

10 Z -a 10

T3

1

2 p ± (GeV/c)

3

JAM, Au+Au at RHIC b<3.0fm pion (b)

10

2 4 Px (GeV/c)

Figure 1. Fig. (a) shows transverse momentum distributions of negative particles (left) in Pb(158AGeV) + P b , 6 < 3.2fm together with the NA49 experimental data from 1 0 . (b) shows transverse momentum distributions of pions in Au + Au, y/s = 200AGeV, b < 3.0fm. Solid lines correspond to the JAM results with rescattering among hadrons, dotted lines are for without meson-baryon and meson-meson collisions.

minijet production is also included in the same way as the HIJING model in which jet cross section and the number of jet are calculated using an eikonal formalism for perturbative QCD (pQCD) and hard parton-parton scatterings with initial and final state radiation are simulated using PYTHIA 9 program. (4) Rescattering of hadrons which have original constituent quarks can occur with other hadrons assuming the additive quark cross section within a formation time. Note that in this calculations, neither nuclear shadowing effects, nor the effect of jet quenching is not included. 3

Results

In Fig. 1(a), we compare JAM results to NA49 experimental data 10 for the central collision of P b + P b at 158AGeV. The calculations without mesonbaryon and meson-meson interactions (dotted lines) give totally different shape, however it can be seen that the calculations with full rescattering among hadrons well reproduce experimental data. In Fig. 1(b), we give a predictions from JAM calculations on pion distributions with and without rescattering. The shape of the pion transverse momentum below pi. < 3 - 4GeV/c becomes exponential from the pQCD like spectrum after hadronic rescatter-

117

60

Net proton (p-p)

(a) full no.reseat.

0 rapidity

Figure 2. Solid lines denote calculations with full rescattering. Dotted lines denote calculations without any meson-baryon and meson-meson collisions, (a) Rapidity distribution of net protons, (b) Rapidity distribution of net baryons. (c) Rapidity distribution of negative pions. (d) Rapidity distribution of antiprotons. (e) Rapidity distribution of negative kaons. (d) Rapidity distribution of positive kaons.

ing.

Figs. 2 show rapidity distributions of (a) net protons ,(b) net baryons, (c) negative pions, (d) antiprotons, (e) negative kaons, (d) positive kaons in central Au + Au collisions (b < 3fm) at ^/s = 200AGeV. Solid lines correspond to calculations with full hadronic rescattering, while dotted lines are for without

118

meson-baryon and meson-meson interactions. The yield of kaons are largely affected by rescattering. This is consistent with AMPT 8 and UrQMD 6 calculations. We have to care about this increase of the strangeness by hadronic interactions when we discuss about the enhancement of strangeness by QGP formation. We also see that antiproton yield is almost the same after finalstate hadronic interactions consisting with AMPT ampt and UrQMD 6 . 4

Conclusions

In conclusion, the JAM model has been used to study Au + Au reactions at RHIC energies to give predictions on final hadronic spectra. We saw the effect of rescattering on the transverse momentum spectra at both SPS and RHIC energies. Hadronic rescattering generally increase the slope of the transverse momentum spectra at both SPS and RHIC energies. References 1. X. N. Wang, Phys. Rep. 280, 287 (1997); X. N. Wang and M. Gyulassy, Comp. Phys. Comm. 83, 307 (1994); 2. K. Geiger, Phys. Rep. 258, 238 (1995); Comp. Phys. Comm. 104, 70 (1997); 3. H. Sorge, Phys. Rev. C 52, 3291 (1995). 4. S.A. Bass, M. Belkacem, M. Bleicher, M. Brandstetter, L. Bravina, C. Ernst, L. Gerland, M. Hofmann, S. Hofmann, J. Konopka, G. Mao, L. Neise, S. Soff, C. Spieles, H. Weber, L.A. Winckelmann, H. Stocker, W. Greiner, C. Hartnack, J. Aichelin and N. Amelin, Prog. Part. Nucl. Phys. 41, 225 (1998); 5. T. Schonfeld, H. Stocker, W. Greiner, and H. Sorge, Mod. Phys. Lett. A 8, 2631 (1993). 6. M. J. Bleicher, S. A. Bass, L. V. Bravina, W. Greiner, S. Soff, H. Stocker, N. Xu, and E. E. Zabrodin, Phys. Rev. C 62, 024904-1 (2000); 7. Y. Nara, N. Otuka, A. Ohnishi, K. Niita, S. Chiba, Phys. Rev. C 61, 024901 (2000); Y. Nara, Nucl. Phys. A 638, 555c (1998), http://quark.phy. bnl.gov/~ ynara/jam/. 8. B. Zhang, C. M. Ko, B. A. Li, and Z. Lin, Phys. Rev. C 61, 067901 (2000). 9. T. Sjostrand, Comp. Phys. Comm. 82, 74 (1994); PYTHIA 5.7 and JETSET 7.4 Physics and Manual, http://thep.lu.se/tf2/staff/torbjorn/Welcome.html. 10. NA49, H. Appelshaueser et al, Phys. Rev. Lett. 82, 2471 (1999).

SIMULTANEOUS HEAVY ION DISSOCIATION AT ULTRARELATIVISTIC ENERGIES I.A. P S H E N I C H N O V 1 ' 2 " , J . P . B O N D O R F 3 , S. M A S E T T I 2 , I.N. M I S H U S T I N 3 ' 4 , A. V E N T U R A 2 1 Institute for Nuclear Research, Russian Academy of Science, 117312 Moscow, Russia 2 Italian National Agency for New Technologies, Energy and Environment, 40129 Bologna, Italy 3 JViels Bohr Institute, DK-2100 Copenhagen, Denmark 4 Kurchatov Institute, Russian Research Center, 123182 Moscow, Russia We study the simultaneous dissociation of heavy ultrarelativistic nuclei followed by the forward-backward neutron emission in peripheral collisions at colliders. The main contribution to this particular heavy-ion dissociation process, which can be used as a beam luminosity monitor, is expected to be due to the electromagnetic interaction. The Weizsacker-Williams method is extended to the case of simultaneous excitation of collision partners which is simulated by the RELDIS code. A contribution to the dissociation cross section due to grazing nuclear interactions is estimated within the abrasion model and found to be relatively small.

1

Single and mutual electromagnetic dissociation

Let us consider a collision of heavy ultrarelativistic nuclei with the masses and charges (Ai,Z\) and {A^iZ-i) at the impact parameter b exceeding the sum of nuclear radii, b > Ri + R2. According to the Weizsacker-Williams (WW) method, the impact of the Lorentz-boosted Coulomb field of the nucleus At on the collision partner A^ is treated as the absorption of equivalent photons 1 . The mean number of photons absorbed by the nucleus Ai in such collision is: mA2(b) = JNz^Eutyo-A^E^dEx,

(1)

where the spectrum of virtual photons, Nz^iEx^b)l, and the total photoabsorption cross section, GA2{EX), are used. Assuming that the probability of multiphoton absorption is given by the Poisson distribution with the mean multiplicity m,i 2 (b), one has the cross section for the single electromagnetic dissociation to a given channel il: oo

a 1 (i) = 27r fbdbPAAb),

bc = 1.34{A\/3 + A\/3 - 0.75(A~1/3

°e-mail: [email protected] 119

+ A~1/3)),

(2)

120

At —* Ei y

At •

»

Ei

A*

A*

•"•2

" 2

Figure 1: Mutual electromagnetic dissociation of relativistic nuclei. The open and closed circles denote the elastic and inelastic vertices, respectively.

where bc is in fm and the probability of dissociation at impact parameter b is given by: PAl(b) = c -""i(») JdE1NZl(E1,b)aA2(E1)fA2(E1,i).

(3)

Here fAs (EI , i) is the branching ratio for the considered channel i in the absorption of a photon with the energy E\ on the nucleus A2. These values are calculated by photonuclear reaction models 2 ' 3 or taken from experiments. In the WW method, the graph for the mutual electromagnetic dissociation, Fig. 1, may be constructed from two graphs of the single dissociation by interchanging the roles of "emitter" and "absorber" at the secondary photon exchange. Such procedure is possible since the first emitted photon with Ei < Emax ~ 7/i? does not change essentially the total energy, EA — ~fMA, of the emitting nucleus, Emax/EA sa 1/RMA ~ 10~ 4 , and there are no correlations between the energies Ex and E2. In other words, both photon exchanges may be considered as independent processes. Moreover, at ultrarelativistic energies the collision time is much shorter then a typical deexcitation period when a nucleus loses its charge via the proton emission or fission. It means that the equivalent photon spectrum from the excited nucleus, A%, is equal to the spectrum from the nucleus in its ground state, A2, see Fig. 1. Therefore the cross section for the mutual electromagnetic dissociation of the nuclei Ai and A2 to given channels i and j is given by: oo


(4)

121

Substituting PA (b) for each of the nuclei one has:

<>m(i,j) = J J

dE1dE2Mm(EuE2)aA2(E1)aAl(E2)fA2(E1,i)fAl(E2,j), (5)

with the spectral function Mm{Ei,E2)

denned for the mutual dissociation:

oo

Nm (Ex, E2) = 2TT J bdbe-2m^NZl {E1, b)NZ2 (E2, b).

2

(6)

Nucleon removal in grazing nuclear collisions

The cross section for the abrasion of n neutrons and z protons from the projectile (Ax, Z\) in a peripheral collision with the target (A2, Z2) may be derived from the Glauber multiple scattering theory 4 ' 5 . A simple parameterization exists for the single neutron removal cross section 5 : —-r—
aG = 2TT\OC — — y

where A6 « 0.5 fm and Pesc w 0.75 is the probability for a neutron to escape without suffering the interaction with a spectator fragment. This can be extended to the case of mutual (In,In) emission:

<w(ln,ln) = ( ^ l - ) ( ^ i y 3

G

P l

c

.

(7)

Results and discussion

The results of the abrasion model for the charge changing cross sections of the single dissociation of 158A GeV 2 0 8 Pb ions are shown in Fig. 2 in addition to the electromagnetic contribution calculated by the RELDIS code. As one can see, the electromagnetic contribution dominates for few nucleon removal process. The interaction of knocked out nucleons with spectators and spectator de-excitation process itself were neglected in this version of the abrasion model. However, good agreement with the experimental data 6 is found. After this verification the model can be extrapolated to the energies of RHIC and LHC heavy-ion colliders. There is a proposal to use the simultaneous neutron emission for beam luminosity monitoring via the correlated registration of forward neutrons in zero degree calorimeters at RHIC 7 . The model predicts er m (ln, In) « 750 and 970 mb for AuAu and PbPb collisions at RHIC and LHC, respectively, for the correlated single neutron emission.

122

158AGeVPbPb

- .!•-•"'"I

70

72

74

76

78

80

82 Z

Figure 2: Charge changing cross sections of 2 0 8 P b ions at CERN SPS energies. The solid and dashed-line histograms are the RELDIS code results with and without contribution of the abrasion process, respectively. The points are experimental d a t a 6 .

We found that an important part of the dissociation events leading to the single neutron emission is accompanied by the emission of charged particles and nuclear fragments. Such events are due to the equivalent photon absorption above the Giant Resonance region. For grazing nuclear collisions Eq. (7) gives anuc(ln, In) as 100 mb. Our detailed consideration of the mutual nuclear dissociation is in progress. We are grateful to A.S. Botvina, G. Dellacasa, J.J. Gaardh0je, G. Giacomelli, A.B. Kurepin and S. White for useful discussions. I.A.P. and I.N.M. are indebted to the Organizing Committee of Bologna 2000 Conference for the kind hospitality and financial support. References 1. LA. Pshenichnov et al., Phys. Rev. C60, 044901 (1999) and references therein. 2. M.B. Chadwick et al., Acta Phys. Slov. 45, 633 (1995). 3. A.S. Iljinov et al., Nucl. Phys. A616, 575 (1997). 4. J. Hiifner et al., Phys. Rev. C12, 1888 (1975). 5. C.J. Benesh et al., Phys. Rev. C40, 1198 (1989). 6. H. Dekhissi et al., Nucl. Phys. A662, 207 (2000). 7. A.J. Baltz et al., Nucl. Instrum. Methods A417, 1 (1998).

D I R E C T P H O T O N P R O D U C T I O N I N 158 A G E V 2 0 8 p B + 2 0 8 p B COLLISIONS

T. PEITZMANN University

of Minster,

D-48149

Miinster,

Germany

F O R T H E WA98 C O L L A B O R A T I O N A measurement of direct photon production in Pb+ P b collisions at 158 A GeV has been carried out in the CERN WA98 experiment. The invariant yield of direct photons in central collisions is extracted as a function of transverse momentum in the interval 0.5 < px < 4 GeV/c. A significant direct photon signal, compared to statistical and systematical errors, is seen at py > 1.5 GeV/c. The results constitute the first observation of direct photons in ultrarelativistic heavy-ion collisions which could be significant for diagnosis of quark gluon plasma formation.

1

Introduction

The observation of a new phase of strongly interacting matter, the quark gluon plasma (QGP), is one of the most important goals of current nuclear physics research. To study QGP formation photons (both real and virtual) were one of the earliest proposed signatures 3 ' 4 . They are likely to escape from the system directly after production without further interaction, unlike hadrons. Thus, photons carry information on their emitting sources from throughout the entire collision history, including the initial hot and dense phase. Recently, it was shown by Aurenche et al. 5 that photon production rates in the QGP when calculated up to two loop diagrams, are considerably greater than the earlier lowest order estimates 6 . Following this result, Srivastava 7 has shown that at sufficiently high initial temperatures the photon yield from quark matter may significantly exceed the contribution from the hadronic matter to provide a direct probe of the quark matter phase. A large number of measurements of prompt photon production at high transverse momentum (p? > 3GeV/c) exist for proton-proton, protonantiproton, and proton-nucleus collisions (see e.g. 8 ) . First attempts to observe direct photon production in ultrarelativistic heavy-ion collisions with oxygen and sulphur beams found no significant excess 9>10>u>12. The WA80 collaboration 12 provided the most interesting result with a p y dependent upper limit on the direct photon production in S+Au collisions at 200AGeV. In this paper we report on the first observation of direct photon production in ultrarelativistic heavy-ion collisions.

123

124

2

Data Analysis

The results are from the CERN experiment WA98 13 which consists of large acceptance photon and hadron spectrometers. Photons are measured with the WA98 lead-glass photon detector, LEDA, which consisted of 10,080 individual modules with photomultiplier readout. The detector was located at a distance of 21.5 m from the target and covered the pseudorapidity interval 2.35 < rj < 2.95 (ycm = 2.9). The particle identification was supplemented by a charged particle veto detector in front of LEDA. The results presented here were obtained from an analysis of the data taken with Pb beams in 1995 and 1996. The 20% most peripheral and the 10% most central reactions have been selected from the minimum bias cross section (a-minbia, » 6300mb) using the measured transverse energy ET- In total, sb 6.710 6 central and » 4.3-106 peripheral reactions have been analyzed. The extraction of direct photons in the high multiplicity environment of heavy-ion collisions must be performed on a statistical basis by comparison of the measured inclusive photon spectra to the background expected from hadronic decays. Neutral pions and 77 mesons are reconstructed via their 77 decay branch. For a detailed description of the detectors and the analysis procedure see 14 . The final measured inclusive photon spectra are then compared to the calculated background photon spectra to check for a possible photon excess beyond that from long-lived radiative decays. The background calculation is based on the measured 7T° spectra and the measured 7//7r°-ratio. The spectral shapes of other hadrons having radiative decays are calculated assuming rayscaling 15 with yields relative to 7r°'s taken from the literature. It should be noted that the measured contribution (from 7r° and 77) amounts to « 97% of the total photon background. 3

Results

Fig. 1 shows the fully corrected inclusive photon spectra for peripheral and central collisions. The spectra cover the pr range of 0.3 — 4.0 GeV/c (slightly less for peripheral collisions) and extend over six orders of magnitude. Fig. 1 also shows the distributions of neutral pions which extend over a similar momentum range with slightly larger statistical errors. The ratio of measured photons to calculated background photons is displayed in Fig. 2 as a function of transverse momentum. The upper plot shows the ratio for peripheral collisions which is seen to be compatible with one, i.e. no indication of a direct photon excess is observed. The lower plot shows the

125

158AGeV ^ P b + ^ P b 10I. Central

cir 10

5

o o o a o a

10' 10

o

a

•

••

'• o

10

Peripheral

o

o

o

a

*

"•••ttt

••

10

'% •

tit til

10 10

0

0.5

2

7.5

2

2.5

J

i.5

4

Transverse Momentum (GeV/c)

Figure 1. The inclusive photon (circles) and w° (squares) transverse momentum distributions for peripheral (open points) and central (solid points) 158 A GeV 2 0 8 P b + 2 0 8 P b collisions. The d a t a have been corrected for efficiency and acceptance. Only statistical errors are shown.

same ratio for central collisions. It rises from a value of » 1 at low pr to exhibit an excess of about 20% at high PT • A careful study of possible systematical errors is crucial for the direct photon analysis. The largest contributions are from the 7 and 7T° identification efficiencies and the uncertainties related to the 77 measurement. It should be emphasized that the inclusive photon and neutral meson (the basis for the background calculation) yields have been extracted from the same detector for exactly the same data sample. This decreases the sensitivity to many detector related errors and eliminates all errors associated with trigger bias or absolute yield normalization. Full details on the systematical error estimates are given in 14 . The total py-dependent systematical errors are shown by the shaded regions in Fig. 2. A significant photon excess is clearly observed in

126

l.S

-

1.6

-

1.4

r

1.2

-

158AGeV ^ P b + ^ P b Peripheral Collisions

^aa»y»T*«^fr"|J£

1

l°*

'-

Z . 0.6

-

i:

1.4

Central Collisions

:

1.2

ii

S

I 7

•

-

•

••

-

^

IT ;

0.8 0.6

i • • .i,in.i,i, , i

'

0.5

I

1.5 2 2.5 3 3.5 Transverse Momentum (GeV/c)

4

Figure 2. The TMeas/^Bkgd ratio as a function of transverse momentum for peripheral (part a)) and central (part b)) 158 A GeV 2 0 8 P b + 2 0 8 P b collisions. The errors on the d a t a points indicate the statistical errors only. The p-r-dependent systematical errors are indicated by the shaded bands.

central collisions for pr > 1.5GeV/e. The final invariant direct photon yield per central collision is presented in Fig. 3. The statistical and asymmetric systematical errors of Fig. 2 are added in quadrature to obtain the total upper and lower errors shown in Fig. 3. An additional px-dependent error is included to account for that portion of the uncertainty in the energy scale which cancels in the ratios. In the case that the lower error is less than zero a downward arrow is shown with the tail of the arrow indicating the 90% confidence level upper limit ("/Excess + 1-28 C[/ pper ). No published prompt photon results exist for proton-induced reactions at the y/s of the present measurement. Instead, prompt photon yields for proton-induced reactions on fixed targets at 200 GeV are shown in Fig. 3 for comparison 16 . 17 . 18 . These results have been scaled for comparison with the present measurements according to the calculated average number of nucleon-

127

1.5

2

2.5

3

3.5

4

4.5

Transverse Momentum (GeV/c)

Figure 3. The invariant direct photon multiplicity for central 158 A GeV 2 0 8 P b + 2 0 8 P b collisions. The error bars indicate the combined statistical and systematical errors. Data points with downward arrows indicate unbounded 90% CL upper limits. Results of several direct photon measurements for proton-induced reactions have been scaled to central 208 P b - | - 2 0 8 P b collisions for comparison.

nucleon collisions (660) for the central P b + P b event selection and according to the beam energy under the assumption that Ed^a-^/dp3 = f(xT)/s2, where XT = 2pr/y/s 19 . This comparison indicates that the observed direct photon production in central 2 0 8 Pb+ 2 0 8 Pb collisions has a shape similar to that expected for proton-induced reactions at the same -y/s but a yield which is enhanced. 4

Summary

The first observation of direct photons in ultrarelativistic heavy-ion collisions has been presented. While peripheral P b + P b collisions exhibit no significant

128

photon excess, the 10% most central reactions show a clear excess of direct photons in the range of pr greater than about 1.5GeV/c. The invariant direct photon multiplicity as a function of transverse momentum was presented for central 208 pb-(- 208 Pb collisions and compared to proton-induced results at similar incident energy. The comparison indicates excess direct photon production in central 2 0 8 Pb+ 2 0 8 Pb collisions beyond that expected from protoninduced reactions. The result suggests modification of the prompt photon production in nucleus-nucleus collisions, or additional contributions from preequilibrium or thermal photon emission. The result should provide a stringent test for different reaction scenarios, including those with quark gluon plasma formation, and may provide information on the initial temperature attained in these collisions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

M.C. Abreu et al., Phys. Lett. B 410, 337 (1997). E. Andersen et al., Phys. Lett. B 449, 401 (1999). E.L. Feinberg, Nuovo Cimento 34 A, 391 (1976). E. Shuryak, Phys. Lett. B 78, 150 (1978). P. Aurenche, F. Gelis, H. Zaraket, and R. Kobes, Phys. Rev. D 58, 085003 (1998). J. Kapusta, P. Lichard, and D. Seibert, Phys. Rev. D 44, 2774 (1991). D.K. Srivastava, Eur. Phys. J. C 10, 487 (1999). W. Vogelsang and M.R. Whalley, J. Phys. G: Nucl. Part. Phys. 23, Al (1997). HELIOS Collaboration, T. Akesson et al., Z. Phys. C 46, 369 (1990). WA80 Collaboration, R. Albrecht et al., Z. Phys. C 5 1 , 1 (1991). CERES Collaboration, R. Baur et al, Z. Phys. C 71, 571 (1996). WA80 Collaboration, R. Albrecht et al., Phys. Rev. Lett. 76, 3506 (1996). WA98 Collaboration, Proposal for a large acceptance hadron and photon spectrometer, 1991, Preprint CERN/SPSLC 91-17, SPSLC/P260 WA98 Collaboration, M.M. Aggarwal et al, nucl-ex/0006007, submitted to Phys. Rev. C. M. Bourquin and J.-M. Gaillard, Nucl. Phys. B 114, 334 (1976). E704 Collaboration, D.L. Adams et al., Phys. Lett. B 345, 569 (1995). E629 Collaboration, M. McLaughlin et al., Phys. Rev. Lett. 5 1 , 971 (1983). NA3 Collaboration, J. Badier et al., Z. Phys. C 3 1 , 341 (1986). J.F. Owens, Rev. Mod. Phys. 59, 465 (1987).

D e f o r m a t i o n a n d O r i e n t a t i o n Effects i n U r a n i u m - o n - U r a n i u m C o l l i s i o n s at R e l a t i v i s t i c E n e r g i e s

Bao-An Li Department of Chemistry and Physics, P.O. Box 419> Arkansas State University, State University, AR 72467-0419, USA Email: [email protected] Deformation and orientation effects on compression, elliptic flow and particle production in uranium on uranium collisions (UU) at relativistic energies are studied within the transport model ART. The density compression in tip-tip UU collisions is found to be about 30% higher and lasts approximately 50% longer t h a n in bodybody or spherical UU reactions. The tip-tip UU collisions are thus more probable to create the QGP at AGS and SPS energies while the body-body UU collisions are more useful for studying properties of the QGP at higher energies.

1

Introduction

Prospects for new physics in uranium-on-uranium (UU) collisions due to deformation and orientation effects have recently generated much interests in the relativistic heavy-ion community 1 ' 2 ' 3 ' 4 . In particular, UU collisions have been proposed to extend beyond P b + P b collisions to better understand the J/ip suppression mechanism at C E R N ' s SPS. Many other outstanding issues m a y also be resolved by studying deformation and orientation effects in UU collisions at relativistic energies. One of the most critical factors to all of these issues is the m a x i m u m achievable energy density in UU collisions. Because of the deformation, UU collisions at the same b e a m energy and impact parameter b u t different orientations are expected to form dense m a t t e r with different compressions and lifetimes. In particular, the large deformation of u r a n i u m nuclei lets one gain particle multiplicity, energy density a n d longer reaction time by aligning the two u r a n i u m nuclei with their long axes head-on (tip-tip). We report here some results of a study on UU collisions within the relativistic transport model ART 6 . 2

D e f o r m a t i o n a n d O r i e n t a t i o n Effects o n R e a c t i o n D y n a m i c s

Uranium is approximately an ellipsoid with a long and short semi-axis Ri — R • (1 + | 6 ) and R, = R • (1 — | 6 ) , where R is the equivalent spherical radius and 6 is the deformation parameter. For 238U, one has 6 = 0.27 and thus a long/short axis ratio of about 1.3. Among all possible orientations between two colliding u r a n i u m nuclei, the tip-tip (with long axes head-on)and body-body 129

130

Figure 1: The evolution of central baryon density in Au-Au (filled circles), tip-tip (solid line), body-body (dotted line) and sphere-sphere (dashed line) UU collisions at a beam energy of 20 GeV/nucleon and an impact parameter of 0 (upper panel) and 6 fm (lower panel), respectively.

(with short axes head-on and long axes parallel in the reaction plane) collisions are the most interesting ones. Shown in Fig. 1 are the evolution of central baryon densities in the UU collisions at a beam energy of 20 GeV/nucleon and an impact parameter of 0 and 6 fm, respectively. While the body-body UU collisions lead to density compressions comparable to those reached in the Au-Au and spherical UU collisions, a 30% more compression is obtained in the tip-tip UU collisions at both impact parameters. The high density phase (i.e., with p/po > 5) in the tip-tip collisions lasts about 3-5 fm/c longer than the body-body collisions. The higher compression and longer passage time render the tip-tip UU collisions the most probable candidates to form the QuarkGluon-Plasma (QGP). 3

Deformation and Orientation Effects on Elliptic Flow

The elliptic flow reflects the anisotropy in the particle transverse momentum (pt) distribution at midrapidity, V2=<(pl-P2y)/P2t>,

(1)

where px (py) is the transverse momentum in (perpendicular to ) the reaction plane and the average is taken over all particles in all events. Shown in Fig. 2 are the nucleon elliptic flow as a function of impact parameter in UU collisions with different orientations at a beam energy of 10 GeV/nucleon and an impact parameter of 6 fm. We initialized the two uranium nuclei such that their long axes are in the reaction plane in both tip-tip and body-body collisions. It is seen that both the tip-tip and sphere-sphere collisions lead to a strong "in-plane

131

a

XJ-*-Ba**TJ. t i p — t i p

E / A = 1 0

GOV

a p h o r o - ^ s g l i o i

a^8*,

Figure 2: The impact parameter dependence of nucleon elliptic flow in the tip-tip (solid line), body-body (dotted line) and sphere-sphere (dashed line) UU collisions at a beam energy of 10 GeV/nucleon.

flow" (positive t^) while the body-body reactions result in a large "squeezeout" (negative i^)- Unique to the body-body UU collisions, the strength of elliptic flow is the highest in the most central collisions where the shadowing effect in the reaction plane is the strongest. While in tip-tip and sphere-sphere UU collisions the elliptic flow vanishes in the most central collisions due to symmetry. Similar results are also found for pions.

4

Deformation and Orientation Effects on Particle Production

Shown in Fig. 3 are the multiplicities of pions and positive kaons as a function of impact parameter. The maximum impact parameter for the tip-tip and body-body UU collisions are 2RB and 2Ri, respectively. As one expects the central (with 6 < 5 fm) tip-tip UU collisions produce more particles due to the higher compression and the longer passage time of the reaction. While at larger impact parameters, the smaller overlap volume in the tip-tip collisions leads to less particle production than the body-body and sphere-sphere reactions. Also as one expects from the reaction geometry, the multiplicities in the bodybody collisions approach those in the sphere-sphere collisions as the impact parameter reaches zero. In the most central collisions, the tip-tip UU collisions produce about 15% (40%) more pions (positive kaons) than the body-body and sphere-sphere UU collisions. Compared to pions, kaons are more sensitive to the density compression since most of them are produced from second chance particle (resonance)-particle (resonance) scatterings at the energies studied here.

132 a o o i • . . - — i — • • • • ! • • — • • ! • • • • — i

• • • •—r

*> CTm.) Figure 3: The impact parameter dependence of pion (upper panel) and positive kaon (lower panel) production in UU collisions.

5

Summary

In summary, using A Relativistic Transport model we have studied the deformation and orientation effects on the compression, elliptic flow and particle production in uranium on uranium (UU) collisions at relativistic energies. The compression in the tip-tip UU collisions is about 30% higher and lasts approximately 50% longer than in the body-body or spherical UU collisions. Moreover, we found that the nucleon elliptic flow in the body-body UU collisions have some unique features. We would like to thank P. Braun-Munzinger, W.F. Henning, C M . Ko, A.T. Sustich, Matt Tilley and B. Zhang for helpful discussions. This work was supported in part by the National Science Foundation under Grant No. PHY0088934 and Arkansas Science and Technology Authority Grant No. OO-B-14. References 1. W.F. Henning, private communications. 2. P. Braun-Munzinger, Memo, to RHIC management, Sept. 18, 1992. 3. B.A. Li, Phys. Rev. C61, 021903 (2000); E.V. Shuryak, Phys. Rev. C61, 034905 (2000). 4. Ben-Hao Sa and A. Tai, nucl-th/9912028; S. Das Gupta and C. Gale, nucl-th/0003005; P.F. Kolb, J. Solfrank abd U. Heinz, hep-ph/0006129; C. Nonaka, E. Honda and S. Muroya, hep-ph/0007187. 5. B. A. Li and C. M. Ko, Phys. Rev. C52, 2037 (1995).

S U B T H R E S H O L D HEAVY - M E S O N A N D A N T I P R O T O N P R O D U C T I O N IN THE N U C L E U S - N U C L E U S COLLISIONS

V.G. Khlopin

Radium

A. T . D ' Y A C H E N K O Institute, 197022 St. Petersburg,

Russia

Production of K- mesons and antiprotons in heavy ion collisions is considered at energies below the production threshold in the free nucleon - nucleon collisions. The fluid dynamics model was used to describe the experimental data.For strange particles and antiparticles the associative character of the production was taken into account, wich reduced considerably the absolute value of the production crosssection.

1

Introduction

The collisions of heavy nuclei at intermediate energies (800-2000 MeV/nucleon) are of great interest when studying nuclear matter at high temperature and density. The determination of characteristics of secondaries allows to determine the thermal and the collective energy of the fireball, which are connected with an equation of state (EOS) of nuclear matter and the module of compression of nuclear matter [1-4]. The thresholds of the K+ and K~ - meson production in nucleon - nucleon collisions are equal to 1.58 GeV and 2.5 GeV respectively, the threshold of the p - production amounts to 5.6 GeV. The production of K - mesons and antiprotons at subthreshold energies is a result of the collective dynamics in the nucleus - nucleus collisions. The associative character of the strange particle production and antiparticle production processes should be included in the thermodynamic description of the system [5]. 2

T h e M o d e l for Calculations

The fluid dynamics model is most suitable for studying the nuclear matter EOS ( the dependence of pressure and density of energy on temperature and nucleon density ). The equation of state determining the dependence of the pressure P and the energy density e on the density p is a sum of kinetic terms and interaction terms (P = Pkin + Pint and e = efc,„ + eint). The invariant differential crosssection of the reaction A + B —>• K (p) + X with the emission of kaons and antiprotons can be written as

E

$=jS^IidiI

d d

* ^ -pvcose)f-

133

(1)

134

Here / is a distribution function for K - mesons (antiprotons). It reads as f(E,-p,t)=g(exp(-

,l(E — pv cos6 — u). -^ —)±l)-\

. i (2)

where g = 1 for kaons, and 2 for antiprotons, p is the momentum of the produced particle, E = y/p2 + m2, v(r, t), T(r, t) are the fields of velocites and temperatures respectively, which are the solutions of relativistic fluid dynamics equations, 7 = (1 — v 2 ) - ? is the Lorentz factor, I is the impact parameter, <j> is the azimuthal angle, \i is the chemical potential. The sign ± concerns fermions or bosons respectively. The absorption of particles, interacting with the environment can be taken into account by introducing the factor

Q = exp(-^p) )

(3)

where < R > is the mean radius of the area, from which the secondary particles are emitted, A being the particle free-path-length. 3

Results

The calculation of the K- meson and antiproton production cross-sections was carried out according to expressions (1) - (3) with Skyrme's interaction parameters equal to 60 = -1033 MeV fm 3 , 63 = 14000 MeV fm 6 [6,7]. The chosen parametrization of the effective forces corresponds to realistic values of compression module K = 398 MeV (hard EOS) and the normal density po = 0.17 fm"3. The associative character of the production process should be included in the thermodynamic description by the replacement of the local equilibrium distribution function (2) with the conditional distribution function [5] The comparison of the calculated differential K+ - meson production cross-section with experimental data [9] is shown in fig. 1 for the reaction 58 Ni + 5 8 Ni at angles 40° < 9 < 48° and energies of the Ni ions equal to 1.8 GeV/nucleon(l), 1 GeV/nucleon(2) and 0.8 GeV/nucleon(3), respectively. The effective free-path-length of the K+ - mesons in the nuclei was taken equal to 6 fm. As seen from fig. 1, not only the calculated and the experimental spectra slopes are in agreement, but also the absolute values of the experimental and the calculated differential cross-sections of the K+ - meson production. In fig. 2 the calculated and the experimental [10] invariant differential production cross-sections in 28Si + 2 8 Si collisions (2 GeV/nucleon) are shown for 7T_ - mesons (1), K+ - mesons (2), K~~ - mesons (3) and antiprotons(4) as

135

0.0

0.1

0.2

0.3

Fie. 1.

0.4 Ecm

'

0.5 GeV

0

200

400

600

Fig. 2.

800 Ecm

'

1000 MeV

Figure 1: The calculated (solid lines) and experimental (points) invariant differential A""*" - meson production cross-sections [9], in the reaction 58 iVi + 5 8 JVt at the angles 40 — 48° and energies of the Ni ions, equal to 1.8 GeV/nucleon (1), 1.0 GeV/nucleon (2) and 0.8 GeV/nucleon (3). Figure 2: The calculated (solid lines) and the experimental (points) [12] invariant differential production cross-sections in the reaction 2 8 5i + 2 8 Si at the energy of the Si ions 2 GeV/nucleone for 7r— - mesons (diamonds) (1), K+ - mesons (squares) (2), K - mesons (triangles) (3), antiprotons (circles) (4) as functions of the energy of the particles in the centre of mass system.

functions of energy of the particles in the center of mass system of the colliding nuclei. The calculation reproduces the effective experimental spectra slopes. The solid curve (1) obtained at A = 5 fm is closer to the experimental data, than the dashed curve (2) obtained at A = 3 fm. The pion production from Adecay as well as the thermal pion production should be taken into account. The calculated production cross-sections for K~ -mesons are overestimated compared with experimental data (dashed curve), which can be connected with a smaller antikaon free-path-length in the nucleus equal to 2 fm (see, e.g.[8]) (solid curve) compared with that for K+ - meson (Aj^ + ~ 6 fm). The calculation reproduces the absolute value of the experimental antiproton production cross-section. The free-path-length of the antiproton in the nucleus was taken to be equal to 1.6 fm. It is more difficult for antiprotons compared with mesons to leave the interaction zone. The calculation reproduces the energy spectrum of the produced particles as well as the absolute value of the K~ - meson production cross-section at the energies 1.85 GeV/nucleon and 1.66 GeV/nucleon

136

for the Ni + Ni reactions if the antikaon free-path-lenght is equal to A = 2 fm [11]. A m o m e n t u m dependence of the antiproton production cross-section at various energies and combinations of the colliding nuclei [12] is reproduced in the calculation. T h e relative value of the cross-section for various reactions is reproduced also [11] within the experimental errors, though experimental d a t a are rather scarce. Thus, the calculation within the framework of relativistic fluid dynamics model of heavy ion interactions can reproduce the experimental energy spect r a of the subthreshold antiprotons and K- mesons at various energies and various combinations of the colliding nuclei. For strange particle production and antiparticle production the associative character of the reaction was taken into account, which reduced considerably the absolute value of the A'-meson production cross-section. Though the experimental points are not too numerous, the calculation reproduces the experimental antiproton energy spectrum as well as the relative value of the production cross-section for various energies and various combinations of the colliding nuclei. A cknowledgment s T h e author expresses his gratitude to Professors O.V. Lozhkin, V.P. Eismont and A.A. Rimski-Korsakov for their interest in the work and support. References 1. H. Stocker and W . Greiner, Phys. Rep. 137, 277 (1986). 2. W . Cassing, V. Metag, U. Mosel and K. Niita, Phys. Rep. 1 8 8 , 363 (1990). 3. I.N. Mishustin, V.N. Russkikh and L.M. Satarov, Yad. Fiz. 5 4 , 429 (1991). 4. C. Hartnack, J. Aichelin, H. Stocker and W . Greiner, Phys. Rev. Lett. 72, 3767 (1994). 5. K.K. G u d i m a a n d V.D. Toneev, Yad. Fiz, 4 2 , 645 (1985). 6. A . T . D'yachenko, Bull. Russ. Acad. Set. (Phys.Ser.) 6 0 , No. 2, 195 (1996). 7. A . T . D'yachenko, Nucl. Phys. A 6 2 6 , 273 (1997). 8. W . Zwermann and B. Schiirmann Nucl. Phys. A 4 2 3 , 525 (1984). 9. R. B a r t h et al. (KaoS collaboration) Phys. Rev. Lett. 78, 4007 (1997). 10. A. Shor A et al. Phys. Rev. Lett. 6 3 , 2192 (1989). 11. A . T . D'yachenko, J. Phys. G: Nucl. Part. Phys. 26, 861 (2000). 12. A. Schrotter et al. Z. Phys. A 3 5 0 , 101 (1994).

P I O N I M A G I N G AT T H E AGS S.Y. PANITKIN, 7 N.N. AJITANAND, 12 J. ALEXANDER, 12 M. ANDERSON, 5 D. BEST, 1 F.P. BRADY, 6 T. CASE, 1 W. CASKEY, 6 D. CEBRA, 5 J. CHANCE, 5 P. CHUNG, 12 B. COLE, 4 K. CROWE, 1 A. DAS, 10 J. DRAPER, 5 M. GILKES, 11 S. GUSHUE, 2 M. HEFFNER, 5 A. HIRSCH, 11 E. HJORT, 1 1 L. HUO, 6 M. JUSTICE, 7 M. KAPLAN, 3 D. KEANE, 7 J. KINTNER, 8 J. KLAY, 5 D. KROFCHECK, 9 R. LACEY, 12 M.A. LISA, 10 H. LIU, 7 Y.M. LIU, 6 R. MCGRATH, 12 Z. MILOSEVICH, 3 , G. ODYNIEC, 1 D. OLSON, 1 C. PINKENBURG, 1 2 N. PORILE, 1 1 G. RAI, 1 H.G. RITTER, 1 J. ROMERO, 5 R. SCHARENBERG, 11 L.S. SCHROEDER, 1 B. SRIVASTAVA,11 N.T.B. STONE, 2 T.J.M. SYMONS, 1 S. WANG, 7 R. WELLS, 1 0 J. WHITFIELD, 3 T. WIENOLD, 1 R. WITT, 7 L. WOOD, 5 X. YANG, 4 W.N. ZHANG, 6 Y. ZHANG 4 (E895 COLLABORATION) 1 Lawrence Berkeley National Laboratory, Berkeley, California 94720 2 Brookhaven National Laboratoryj Upton, New York 11973 3 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 4 Columbia University, New York, New York 10027 5 University of California, Davis, California 95616 6 Harbin Institute of Technology, Harbin 150001, P. R. China 7 Kent State University, Kent, Ohio 44242 6 St. Mary's College of California, Moraga, California 94575 9 University of Auckland, Auckland, New Zealand 10 The Ohio State University, Columbus, Ohio 43210 11 Purdue University, West Lafayette, Indiana 47907 12 State University of New York, Stony Brook, New York 11794 Source imaging analysis was performed on two-pion correlations in central Au + Au collisions at beam energies between 2 and 8A GeV. We apply the imaging technique by Brown and Danielewicz, which allows a model-independent extraction of source functions with useful accuracy out to relative pion separations of about 20 fm. We found that extracted source functions have Gaussian shapes. Values of source functions at zero separation are almost constant across the energy range under study. Imaging results are found to be consistent with conventional source parameters obtained from a multidimensional HBT analysis.

1

Introduction

Measurements of two-particle correlations offer a powerful tool for studies of spatial and temporal behaviour of the heavy-ion collisions. The complex nature of the heavy-ion reactions requires utilization of different probes and different analysis techniques in order to obtaine reliable and complete picture

137

138

of the system created in the collision. We will discuss application of imaging technique of Brown-Danielewicz which allows one to reconstruct the entire source function, phase space density and entropy from any like-pair correlations in a model independent way. We will present recent results of the two-pion imaging analysis performed by the E895 Collaboration and discuss natural connection between imaging and traditional HBT. 1.1

E895 Experiment and Analysis Details

Detailed description of the experiment E895 can be found elsewhere 1 . The Au beams of kinetic energy 1.85, 3.9, 5.9 and 7.9A GeV from the AGS accelerator at Brookhaven were incident on a fixed Au target. The analyzed ir~ samples come from the main E895 subsystem — the EOS Time Projection Chamber (TPC) 2 , located in a dipole magnetic field of 0.75 or 1.0 Tesla. Results of the E895 w~ HBT analysis were reported earlier 3 , and in this paper, we focus on an application of imaging to the same datasets. 1.2

Source Imaging

In order to obtain the two-pion correlation function Ci experimentally, the standard event-mixing technique was used 4 . Negative pions were detected and reconstructed over a substantial fraction of 47r solid angle in the centerof-mass frame, and simultaneous measurement of particle momentum and specific ionization in the TPC gas helped to separate ir~ from other negatively charge particles, such as electrons, K~ and antiprotons. Contamination from these species is estimated to be under 5%, and is even less at the lower beam energies and higher transverse momenta. Momentum resolution in the region of correlation measurements is better than 3%. Event centrality selection was based on the multiplicity of reconstructed charged particles. In the present analysis, events were selected with a multiplicity corresponding to the upper 11% of the inelastic cross section for Au -I- Au collisions. Only pions with pr between 100 MeV/c and 300 MeV/c and within ±0.35 units from midrapidity were used. Another requirement was for each used 7r - track to point back to the primary event vertex with a distance of closest approach (DCA) less than 2.5 cm. This cut removes most pions originating from weak decays of long-lived particles, e.g. A and K°. Monte Carlo simulations based on the RQMD model 5 indicate that decay daughters are present at a level that varies from 5% at 4A GeV to 10% at 8A GeV, and they he preferentially at pr < 100 MeV/c. Finally, in order to overcome effects of track merging, a cut on spatial separation of two tracks was imposed. For pairs from both

139

2.5 it Ih

2

IS IS

O

-*-•-•-

_»

• * * * - •

1

• • • • • • •

• 2GeV

OS

• 4GeV

,

0

j

2 IS

" -W-

1

-^M^MM>

03

* 6GeV

0

i

T U T

'

0.05 0.1 Q lnv ,(GeWc)

-ChCbO^0-0-0-©

o 8GeV

, 0.05 0.1 Q^GeV/c)

0.15

Figure 1. Measured two-pion correlation functions for Au + Au central collisions at four beam energies.

"true" and "background" distributions, the separation between two tracks was required to be greater than 4.5 cm over a distance of 18 cm in the beam direction. This cut also suppresses effects of track splitting 3 . Correlation functions were corrected for pion Coulomb interaction 3 . Imaging 6 ' 7 was used to extract quantitative information from the measured correlation functions. It has been shown 8 that imaging allows robust reconstruction of the source function for systems with non-zero lifetime, even in the presence of strong space-momentum correlations. The main features of the method are outlined below; see Refs 6 ' 7 ' 9 for more details. The two-particle correlation function may be expressed as follows:

CP(Q)-1 =

JdrK(Q,r)SP(T),

(1)

140

•3

10

• • • O

2GeV 4GeV 6GeV SGeV

10

"'«|

"MM «!

GO

10

i

1A

0

2

4

. .

i i i i i i

6

8

10

12

14

16

18

20

r,(fm) Figure 2. Relative source functions extracted from the pion correlation data at 2, 4, 6 and 8A GeV.

where AT = |$Q^(r)| 2 — 1 andQ^ is the relative wavefunction of the pair. The source function, Sp(r), is the distribution of relative separations of emission points for the two particles in their center-of-mass frame. The total momentum of the pair is denoted by P , the relative momentum by Q = p i — P2, and the relative separation by r. Since the correlation data is already Coulomb corrected, we may approximate the relative wavefunction of the pions with a noninteracting spin-0 boson wavefunction:

*¥(r) = ±(eW+e-«*'*').

(2)

The goal of imaging is the determination of the relative source function (Sp(r) in Eq. (1)), given Cp(Q). The problem of imaging then becomes the problem of inverting K(q, r) 1 0 . For pions, one might think that because Eq. 1 is a Fourier cosine transform it may be inverted analytically 6 to give the source

141 Table 1. Radius parameters of Gaussian fits to the extracted source (Rs) measured correlation functions(Rc , 2) for different energies.

Eb (AGeV) i?s(fm) Rc2 (fm)

2 6.70±0.04 6.39±0.2

4 6.35±0.03 6.05±0.1

6 5.56±0.03 5.51±0.15

functions and

8 5.53±0.05 5.61±0.28

function directly in terms of the correlation function: 5 P (r) = J^ysj
cos (Q • r) (C P (Q) - 1).

(3)

While this might work for vanishing experimental uncertainty, for realistic situations, this is a poor way to perform the inversion. When Fourier transforming, it is impossible to distinguish between statistical noise and real structure in the data n . Instead, we proceed as in 13 and expand the source in a Basis Spline basis: S(r) = E • SjBj(r). With this, Eq. (1) becomes a matrix equation d = J2j KijSj with a new kernel: Kij = jdrKiQi^Bjir).

(4)

Imaging reduces to finding the set of source coefficients, Sj, that minimize the X2. Here, X 2 = E<(<* " E ^ ^ / A 2 ^ . This set of source coefficients is Sj = Yli[(KT(A2C)-1K)-1KTB]ji(Ci - 1) where KT is the transpose of the kernel matrix. The uncertainty of the source is the square-root of the diagonal elements of the covariance matrix of the source, A 2 S = ( J ftT T (A 2 C)- 1 i£:)- 1 . It has been shown 9 that for the case of noninteracting spin-zero bosons with Gaussian correlations, there is a natural connection between the imaged sources and "standard" HBT parameters. Indeed, the source is a Gaussian with the "standard" HBT parameters as radii: S(r) =

_

X

exp (- \nri[R2]^11

,

,

(5)

where A is a fit parameter traditionally called the chaoticity or coherence factor in HBT analyses. Here, [-R2] is the real symmetric matrix of radius parameters [R2] =

Rl

Rl R\t

.

(6)

142

• S"V->0) o

S^fromHBT

£ 10'

6 , (GeV)

Figure 3. Values of inverse relative source functions at zero separation as a function of beam energy, obtained using imaging (solid circles) and HBT (open circles).

Eq. (5) is the most general Gaussian one may use, but usually one assumes a particular symmetry of the single particle source so that there is only one non-vanishing non-diagonal element R^. The asymptotic value of the relative source function at zero separation S(T -»• 0) is related to the inverse effective volume of particle emission, and has units of fm - 3 : S(r - • 0) =

A

(2V5F) V d e t [R>]

(7)

As it was shown in 9 5(r -+ 0) is an important parameter needed to extract the space-averaged phase-space density. Figure 1 shows measured angle-averaged two-pion correlation functions for central Au + Au collisions at 2, 4, 6 and &A GeV. Figure 2 shows relative source functions S(r) obtained by applying the imaging technique to the measured two-pion correlation functions. Note that the plotted points are for representation of the continuous source function and

143

Table 2. Fit parameters for the Bertsch-Pratt HBT parameterization of the pion correlation functions for E895 beam energies used for evaluation of S(r —• 0).

Eb (AGeV) A R0 (fm) Ra (fm) Ri (fm)

Rl (fm)

2 0.99±0.06 6.22±0.26 6.28±0.20 5.15±0.19 -2.43±1.71

4 0.74±0.03 5.79±0.16 5.37±0.11 5.15±0.14 0.43±1.03

6 0.65±0.03 5.76±0.23 5.05±0.12 4.72±0.18 2.17±1.20

8 0.65±0.05 5.49±0.31 4.83±0.21 4.64±0.24 -0.65±1.85

hence are not statistically independent of each other as the source functions are expanded in Basis Splines 13 . Since the source covariance matrix is not diagonal, the coefficients of the Basis spline expansion are also not independent which is taken into account during x2 calculations. With this technique, we may reconstruct the distribution of relative pion separations with useful accuracy out to r ~ 20 fm. The images obtained at each of the four E895 beam energies are rather similar in shape, and upon fitting with a Gaussian function, values of x 2 per degree of freedom between 0.9 and 1.2 are obtained. Results of fits to the source function and correlation functions are shown in Table 1. One can see that source radii extracted via both techniques are similar, further confirming the validity of a Gaussian source hypothesis. Figure 3 compares values related to effective volumes of pion emission inferred from the standard Bertsch-Pratt pion HBT parametrization (open circles) with the effective volumes derived from the image source functions shown in Fig. 2 (solid circles). Essentially, this figure plots the inverse of the left- and right-hand sides of Eq. 7. Results of the multidimensional Bertsch-Pratt fit to the pion correlation data have already been published in Ref. 3 and are reproduced in Table 2. It can be seen from Fig. 3 that the agreement between imaging and the HBT parametrization is fairly good. Values of source functions at zero separations estimated via either technique are approximately constant within errors across the 2 to 8A GeV beam energy range. In summary, we present measurements of one-dimensional correlation functions for negative pions emitted at mid-rapidity from central Au + Au collisions at 2, 4, 6 and 8A GeV. These correlation functions are analyzed using the imaging technique of Brown and Danielewicz. It is found that relative source functions S(r) have rather similar shapes and zero-separation intercepts S(r —> 0). Distributions of relative separation have been measured out to 20 fm, and the extracted source functions are approximately Gaussian. We have performed the first experimental check of the predicted connection

144

between imaging and traditional meson interferometry techniques and found that the two methods are in good agreement. This agreement paves the way for applications of the imaging method to the interpretation of pair correlations among strongly interacting particle such as protons, antiprotons, etc. Values of source functions at zero separation which are related to the pion effective volumes of emission are almost constant across the range of bombarding energies under study. Acknowledgments Parts of this paper are based on work done in collaboration with D. Brown and G.F. Bertsch 9 . This research is supported by US DOE, NSF and other funding, as detailed in Ref. 3 ' 9 . References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

G. Rai et al., proposal LBL-PUB-5399 (1993). G. Rai et al., IEEE Trans. Nucl. Sci. 37, 56 (1990). E895 Collaboration, M.A. Lisa et al., Phys. Rev. Lett. 84, 2798 (2000). G. Kopylov, Phys. Lett. 50B, 472 (1974). H. Sorge, Phys. Rev. C 52, 3291 (1995). D.A. Brown and P. Danielewicz, Phys. Lett. B398, 252 (1997). D.A. Brown and P. Danielewicz, Phys. Rev. C57, 2474 (1998). S.Y. Panitkin and D.A. Brown, Phys. Rev. C61, 021901 (2000). D.A. Brown, S.Y. Panitkin and G.F. Bertsch, Phys. Rev. C62, 014904 (2000). A. Tarantola, Inverse Problem Theory, Elsevier, (1987). D. Brown, nucl-th/9904063. D.A. Brown, nucl-th/0003021. D.A. Brown and P. Danielewicz, in preparation; C. de Boor, A Practical Guide to Splines, Springer-Verlag, (1978).

D I R E C T E D F L O W IN 4.2A G E V / C C + C A N D COLLISIONS

C+TA

J . M I L O S E V I C A N D L J . SIMIC Institute

of Physics, E-mail:

P.O. Box 68, 11080 Belgrade, Yugoslavia [email protected]

The directed flow of protons and ir— in 4.2A GeV/c C + C and C + T a collisions is studied using the Fourier analysis of the azimuthal distributions. It is found that protons exhibit strong directed flow. In C + C collisions, the directed flow of pions is too weak to be detected experimentally, while in C + T a collisions pions show significant directedflow. Quark-gluon-string model calculations confirm these results. This model suggests that the decay of resonances and rescattering of secondaries determine the proton and 7r~ directed flow.

T h e directed flow of protons a n d x - in 4.2A G e V / c C + C and C + T a collisions is studied using the Fourier analysis of the azimuthal distributions *. We analysed 9500 C + C and 1000 C + T a semicentral and central collisions obtained with the 2m propane bubble chamber, exposed a t JINR, D u b n a synchrophasotron 2 . T h e same type of analysis is performed using the events generated by the Quark-Gluon-String Model (QGSM) 3 . Additionally, we used QGSM to clarify which physical processes are responsible for flow effects. For analysing directed flow, a Fourier expansion — - « — [l + 2t>! cos(<£) + 2v2 cos{2)}, a
(1)

Zit

of the azimuthal distribution of particles is performed. T h e coefficients v 1 =(cos(<^)) and t>2=(cos(2<£)) quantify the directed and elliptic flow respectively. In Eq. (1), 4> is the particle azimuthal angle measured with respect to the (true) reaction plane. Since b o t h Pbeam and b are available in Q G S M , they are used to determine the true reaction plane. In the experiment, the reaction plane is estimated, for the i-th proton in each event, using the vector 4 ' 5

where rrij and p ^ j represent mass and transverse m o m e n t u m of the j'-th proton emitted in the forward (y > ycm +6, Wj = 1), or backward (y < ycm — 6, ujj = — 1) hemisphere. Here, ycm denotes the center of mass rapidity while the quantity 8 (=0.2) removes the protons emitted around the ycm which are not contributing to the determination of the reaction plane. In Eq. (2),

145

146

mays is the sum of the projectile and target masses. Determining the reaction plane in this way, we avoid an autocorrelation as well as a correlation due to momentum conservation. Because of the finite resolution of the experimentally determined reaction plane, we need to correct the Fourier coefficients v[ obtained with respect to this reaction plane. The relation between the 'true' Fourier coefficients v\ and the coefficients v[ is t>i = v[/ (cos(A#)). For the determination of the correction factor (cos(A<&)) we used the subevent method 1>6. We found (cos(A*))=0.56 (0.59) for C+C (C+Ta) collisions.

C+Ta protons 0.2

:

/ $ &

f+ y* -0.2 -

1

0

i

.

.

.

1 JC

.

i

.

.

.

.

i

,

,

Figure 1. Rapidity dependence of vi for protons and n— for 4.2A GeV/c C+C and C+Ta collisions. Filled circles represent the experimental v\ values, while the solid (dashed) line represents QGSM calculation for «i relative to the true (estimated) reaction plane.

0.1 . • data

=r QGSM

-1 o , , ,

i ,

,

, .

i

,

,

,

,

i

.

.

.

1

-1

rapidity Fig. 1 shows the experimentally determined v\ coefficient vs. y for protons and ic~ together with the v\ calculated by QGSM relative to the true and to the estimated reaction plane for both collision systems. The values of the two QGSM results for v\ are quite close for protons. The dependence of t;i on rapidity is characterized by a curve with a positive slope and with a zero-crossing at y = 0. This indicates positive directed flow with magnitude v\ fa 0.15 at beam rapidity. QGSM reproduces satisfactorily the shape and the magnitude of the v\ (y) data. For protons we also determined the slope F = d(px)/dy at midrapidity, using the relation vi = {px)/(pr)- After the normalisation to the mass number of the colliding systems we obtained the so called scaled flow, Fs = F/{Ayz + A\lz) = 22 ± 2 MeV/c (26 ± 2) MeV/c for C+C (C+Ta) collisions. These are in agreement with the observed trend 7 of a slow decrease of the directed flow values with increasing beam energy. For ir~ the experimental values of Vi indicate that the directed flow is non-existent

147

in C + C collision. This result is confirmed by Q G S M calculations of vi with respect t o t h e estimated reaction plane. However, t h e model calculations of v\ with respect t o t h e true reaction plane show t h e existence of a very weak negative directed flow of pions with value of 0.02 a t t h e target rapidity. This further suggests t h a t in t h e collisions of light nuclei, like C + C , t h e very weak flow can not b e observed experimentally because of the limited accuracy in the determination of the reaction plane. For C + T a collisions, v-± values are positive a t all rapidity bins. It means t h a t pions are preferentially emitted in t h e reaction plane from t h e target t o projectile. This behaviour could be explained by t h e shadowing effect of the T a target. Q G S M calculations show the same behaviour, b u t they overestimate t h e flow values.

C+C protons 0.2

J? * * * * * * ^ « ••*

S^* 0 * -0.2

C+Ta

_t3r

•PTi

. . , , i , , , , i , ,

-1

0

-••

0.1

1

1

0

0.4 -

fefifcogU±>To '^^FfcjfcjK* Vo^, -0.1 -•.

.

.

i

.

.

1

,

.

,

,

i

1

,

,

• •

0.2

.

- 1 0

i

JC"-*-

Jl"

Figure 2. R a p i d i t y d e p e n d e n c e of v\ for p r o t o n s a n d ir~ [solid line); for p r o t o n s a n d ir — originati n g from r e s o n a n c e s d e c a y (stars), p r i m a r y (full circles), a n d s e c o n d a r y (open circles) n o n - r e s o n a n t interact i o n s for 4 . 2 A G e V / c C + C and C + T a events generated with QGSM.

,

"

'£. -1

0

1

rapidity According t o Q G S M , 39% (43%) of the protons a n d 7 1 % (83%) of t h e TT~ in C + C ( C + T a ) interactions originate from decays of A ' s , p, w, t) a n d rf resonances. T h e rest originates from t h e 'non-resonant' primary a n d secondary interactions. Some of the protons a n d ir~ from primary interactions (5% in C + C a n d 1% in C + T a collisions) escape t h e collision zone without further rescattering. Therefore, in Q G S M , we separately evaluated t h e flow of protons a n d v~ originating from: (i) decay of resonances, (ii) primary and ( m ) secondary non-resonant interactions. Fig. 2 shows distributions of vi vs. y for protons a n d ir~ from these sources. In C + C collisions, protons (%~) from resonances decay a n d rescattered protons (ic~) show directed flow

148

(antiflow). For nonrescattered protons the vi(y) distribution is relatively flat, while nonrescattered pions show strong negative directed flow at the level of 0.12 at beam rapidity. These particles are produced a t the early stage of the collision. Antiflow of pions can be explained by the shadowing efFect. However, the shadowing m a t t e r is different for pions at different collision stages. At early times (less t h a n passing time) pions are shadowed by the cold spect a t o r s . Later, after the spectator m a t t e r leaves the collision zone, pions are shadowed by the participant nucleons. This m a y be the underlying mechanism t h a t leads to a different behavior of Vi for ir~ and protons. As in the case of C + C collisions, the decay of resonances a n d rescattering of secondaries determine the proton and ir~ flow. For nonrescattered particles, shadowing of the cold spectators is much stronger because of the much bigger size of the Ta target. This leads to the large positive vi values for nonrescattered particles. In summary, the directed flow of protons and ic~ in 4.2A G e V / c C + C a n d C + T a collisions was examined using the Fourier analysis of the azimuthal distributions. Additionally, we used Q G S M events t o clarify which physical processes are responsible for flow effects. It was found t h a t the protons exhibit strong positive directed flow in b o t h collisional systems. Q G S M reproduces satisfactorily the magnitude of this flow. For ir~ in C + C collisions, a directed flow was not found experimentally due to the poor reaction plane resolution. But, Q G S M calculations show a very small negative directed flow. In C + T a collisions, because of the target shadowing, v\ values of -K~ are positive at all rapidities. According to Q G S M two factors t h a t dominantly determine the proton and ir~ flow are the decay of resonances and rescattering of secondaries. Shadowing of the cold nuclear m a t t e r affects only the flow of nonrescattered particles produced at the early stage of the reaction, except fot protons in C + C collisions because of the smallness of the colliding nuclei. References 1. A. M. Poskanzer a n d S. A. Voloshin, Phys. Rev. C 5 8 , 1671 (1998). 2. E. A b d r a h m a n o v et al, Z. Phys. C 5, 1 (1980); H. Agakishiev et al, Z. Phys. C 12, 283 (1982); H. Agakishiev et al, Z. Phys. C 16, 307 (1983). 3. N. S. Amelin, K. K. G u d i m a and V. D. Toneev, Yad. Fiz. 5 1 , 512 (1990); Sov. J. Nucl. Phys. 5 1 , 327 (1990). 4. P. Danielewicz and G. Odyniec, Phys. Lett. B 157, 146 (1985). 5. C. A. Ogilvie et al, Phys. Rev. C 4 0 , 2592 (1989). 6. J.-Y. Ollitrault, nucl-ex/9711003. 7. J. Chance et al, EOS Collaboration, Phys. Rev. Lett. 7 8 , 2535 (1997).

F R A G M E N T A T I O N OF VERY HIGH E N E R G Y HEAVY IONS

Dipartimento

M. G I O R G I N P di Fisica and INFN, Sezione di Bologna, V. It C. Berti Pichat 1-40127 Bologna, Italy E-mail: [email protected]

Radiation E-mail:

6/2,

S. M A N Z O O R 0 Division, PINSTECH, P. 0. Nilore, Islamabad, Pakistan, [email protected] , [email protected] Physics

A stack of CR39 ( C i 2 H i s 0 7 ) n nuclear track detectors with a Cu target was exposed to a 158 A GeV lead ion beam at the CERN-SPS, in order to study the fragmentation properties of lead nuclei. Measurements of the total, break-up and pick-up charge-changing cross sections of ultrarelativistic P b ions on Cu and CR39 targets are presented and discussed.

1

Introduction

We present experimental results on fragmentation charge-changing cross sections of 158 A GeV lead ions (charge Z = 82e) incident on Cu and CR39 targets. To detect and identify the relativistic ions, the nuclear track detector CR39 was used. When an ion crosses a nuclear track detector foil, it produces damages at the level of molecular bonds, forming the so called "latent track". During the chemical etching of the detector in a basic water solution, etch-pit cones are formed on both sides of the foil. The base area and the height of each cone are functions of the Restricted Energy Loss (REL) of the incident ion and thus of its charge Z1'2. 2

Experimental procedure

A stack made of CR39 nuclear track detectors with a Cu target was exposed in November 1996 at the CERN-SPS to a beam of 158 A GeV Pb ions. The exposure was performed at normal incidence. The total number of lead ions incident on the stack was about 7.8 x 104, distributed in 8 spots. The central density in each spot was around 1500 ions/cm 2 . The stack had the following composition: 12 CR39 sheets ~ 0.6 mm thick, a Cu target ~ 10 mm thick; 38 CR39 sheets ~ 0.6 mm thick. In the present °for the EMU18 Coll.: G. Giacomelli, M. Giorgini, H.A. Khan, G. Mandrioli, S. Manzoor, L. Patrizii, V. Popa, I.E. Qureshi, M.A. Rana, M. Sajid, P. Serra, M.I. Shahzad, G. Sher and V. Togo. 149

150

350 Z = 82

300 3

250

£

200

£

150

C O

Z = 80

6 Z

Z = 65 Z = 70

100

83

Z = 75

50

20

40

60

80

100

120

140

160

Cone height (arbitrary units) Figure 1: Cone height distribution for P b ions and heavy fragments measured on one face of the CR39 sheet immediately after the Cu target.

analysis, the CR39 sheets immediately before and after the Cu target and the last sheet of the stack were used. After exposure, the sheets were etched for 72 h in a 4N KOH water solution at a temperature of 45 °C. Previous calibrations of the detectors have shown that for high Z nuclei, the height of the etched cone is more sensitive to Z than its base area or diameter 3 . In order to separate the lead ions from the nuclear fragments with charge Z > 63e, we performed manual measurements of about 6300 cone heights using an optical Zeiss microscope with a magnification of 40 x. Fig. 1 shows the cone height distribution of Pb ions and heavy fragments measured on a single face of the CR39 sheet located after the Cu target. The charge resolution obtained is about 0.2e. 3

Total charge-changing cross sections

Using the survival fraction of lead ions for the Cu and CR39 targets, we measured the total charge-changing cross sections of lead ions using the formula: Nin AT (rtot = — Z T T l n TT~ (!) PTtrNA Nout where Ni„ and Nout are the numbers of lead ions before and after the target, respectively; NA is Avogadro's number; pr, AT,
151

mass and the thickness of the target. The data are indicated by the black points in Fig. 2, the uncertainties are statistical only.

Figure 2: Measured total fragmentation charge-changing cross sections otot of 158 A GeV P b projectiles versus the target mass number A?: the black points are our d a t a on Cu and CR39, the open points refer to d a t a obtained by a similar experiment 4 using the same beam. The solid line represents the fit of all the d a t a to formula (2) of ref. [4]. Results from a 10 A GeV Au beam incident on various targets I s - 7 ! are also shown.

As shown in Fig. 2, the data are in agreement with previous data obtained by a similar experiment using the same beam and different targets with atomic masses ranging from 4.7 a.m.u. (CH 2 ) to 207 a.m.u. (Pb) 4 . The solid line in Fig. 2 is the fit of all the data to formula (2) of ref. [4] which yields X2/D.o.F. — 0.7. Results from other experiments using a 10 A GeV Au beam incident on various targets t5_7J are also shown in Fig. 2. 4

Partial fragmentation charge-changing cross sections

The partial fragmentation charge-changing cross sections of Pb ions yielding fragments with charge 64e < Z < 82e were calculated for the Cu and CR39 targets using the formula 8 : aZ

—

AT

(9)

~ prtTNA N82 where Z = 64e -i- 81e, Nz is the number of fragment nuclei with charge Z produced in the target, 7^82 is the number of unfragmented beam nuclei and PT, AT, *T> NA have the same meaning as in Eq. (1). In this procedure, the successive fragmentation processes are neglected. The results for the partial fragmentation cross sections of incident lead ions on Cu and CR39 targets are shown versus AZ in Fig. 3.

152

i i i i i i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i

: • • o-Cu

.

- O D

CTCR39

^

0

D

'n*

E

102

*5 « S . .

r-*

b

1

-20

5

• • • I • • • • I ' • • • I • • • • I • ' ' ' I • ' ' • I ' ' • ' I • • ' ' I '

-17.5

-15

-12.5

-10

-7.5

-5

-2.5

' '

0

AZ Figure 3: Partial fragmentation charge-changing cross sections for incident lead ions, uz versus AZ for the Cu and CR39 targets computed with Eq. (2). The black points refer to &z for Cu, the open points refer to oz for CR39. The squares refer to the pick-up cross sections. The errors are only statistical.

The square points in Fig. 3 refer to the charge pick-up cross sections, determined using Eq. (2) where Nz is the number of nuclei with Z = 83e produced in the target. Acknowledgments We thank the CERN SPS staff for the good performance of the exposure, the technical staff of the Bologna INFN and of the PINSTECH Laboratory. References 1. R. Fleischer, P.B. Price and R.M. Walker in Nuclear Tracks in Solids, (University of California Press, 1975). 2. S. Cecchini et al, Nuovo Cimento A 109, 1119 (1996). 3. G. Giacomelli et al., Nucl. lustrum. Methods A 411, 41 (1998). 4. H. Dekhissi et al, Nucl. Phys. A 662, 207 (2000). 5. S.E. Hirzebruch et al., Phys. Rev. C 5 1 , 2085 (1995). 6. L.Y. Geer et al., Phys. Rev. C 52, 334 (1995). 7. Y. He and P.B. Price, Z. Phys. A 348, 105 (1994). 8. D.P. Bhattacharyya et al., Nuovo Cimento C 16, 249 (1993).

Section II Liquid-Gas Phase Transitions in Nuclear Matter

Phase Transition in Finite Systems Philippe CHOMAZW, Veronique DUFLOT^ 1 - 2 ) and Francesca GULMINELLI^ 2 ) f1' G.A.N.I.L.(CEA-DSM/IN2P3-CNRS), BP 5027, U076 Caen cedex 5, France LPC Caen, (IN2P3-CNRS/ISMRA et Universite), F-14050 Caen cedex, France

(2)

In this paper we present a review of selected aspects of Phase transitions in finite systems applied in particular to the liquid-gas phase transition in nuclei.

1

INTRODUCTION

Phase transitions are universal properties of matter in interaction. They have been widely studied in the thermodynamical limit of infinite systems. However, in many physical situations this limit cannot be accessed and so phase transitions should be reconsidered from a more general point of view. This is for example the case of matter under long range forces like gravitation. Even if these self gravitating systems are very large they cannot be considered as infinite because of the non saturating nature of the force. Other cases are provided by microscopic or mesoscopic systems Metallic clusters can melt before being vaporized. Quantum fluid may undergo Bose condensation or superfluid phase transition. Dense hadronic matter should merge in a quark and gluon plasma phase while nuclei are expected to exhibit a liquid -gas phase transition. For all these systems the theoretical and experimental issue is how to sign a possible phase transition in a finite system. In this contribution we show that the problem of the non existence of boundary conditions can be solved by introducing a statistical ensemble with an averaged constrained volume. In such an ensemble the microcanonical heat capacity becomes negative in the transition region. We show that the caloric curve explicitly depends on the considered transformation of the volume with the excitation energy and so does not bear direct informations on the characteristics of the phase transition. Conversely, partial energy fluctuations are demonstrated to be a direct measure of the equation of state. Since the heat capacity has a negative branch in the phase transition region, the presence of abnormally large kinetic energy fluctuations is a signal of the liquid gas phase transition. 155

156

2 2.1

MODELS FOR P H A S E T R A N S I T I O N IN FINITE SYSTEMS Lattice-gas model

Let us first consider an exactly solvable model for second and first order phase transitions, the Lattice Gas Model of Lee and Yang 1 . This is a simplified model which can be interpreted as a schematic representation of a classical fluid with a Van der Waals type of equation of state. In our numerical implementation the TV sites of a lattice are characterized by an occupation number r = 0 or 1 (see Fig. 1). Particles occupying nearest neighboring sites interact with an energy e. A kinetic energy term is also included so that the Hamiltonian is given by:

i=i

ij

where the second sum runs only over neighboring sites. In the liquid-gas phase transition, since the order parameter is the density difference between the two phases, the volume is essential in determining thermodynamical properties. Many studies have been performed considering periodic boundary conditions in order to avoid the effects of the surface 2>3>4. Systems in a fixed cubic volume have also been investigated 5,6 . In the experimental case however the volume is not defined through boundary conditions because we are dealing with an open system. However, a typical average size of the fragmenting system might be deduced from experimental observables. For example the (average) radius of a hot source can be defined through interferometry or through comparisons with statistical models. From a theoretical point of view this implies that at equilibrium the entropy of the system should be maximized under the constraint of a specific value for the average volume. In the absence of a preferred direction, an average volume can be defined through the one-body observable 7,8 :

? = &£*'

<2>

i=i

where r; is the distance to the center of the lattice. Introducing first a canonical description in which the energy observable H as well as the volume V are known in average we have to introduce the associated Lagrange multipliers /?, A so that the partition function reads:

ZPiX = J2 ex P ( - / ^ ( n ) - A^ (n) ) (»)

(3)

157

Here, U^ and lA") are the expectation values of the operators H and V in the nth event, (3 is the inverse of the canonical t e m p e r a t u r e (3 = l/Tcan, and the quantity P — —A//? has the dimension of a pressure. In other words, the experimental fact t h a t the volume is known only in average means t h a t the pressure, interpreted as the Lagrange multiplier associated with the volume observable, can be considered as the relevant state variable. A statistical ensemble of events associated with an average volume can easily be generated through a canonical sampling using a constrained Hamiltonian E— H—P V where P V can be considered as a constraining one-body external field. T h e averaged constrained energy E — U — PV can be interpreted as an enthalpy. The actual value of the pressure parameter must be defined to get the desired average volume. For finite systems the various ensembles are not equivalent because fluctuations cannot be neglected. Since energy is a directly accessible observable in each event, t h e correct statistical framework is the microcanonical ensemble. An easy way to access microcanonical quantities is to sort the canonical partitions according to their total energy. It is i m p o r t a n t to notice t h a t this procedure m a y be in some cases numerically time consuming but it is an exact method to generate constant energy events with the correct microcanonical weight (see eq.(4) below). W h e n b o t h energy and volume are considered as thermodynamical observables one should in principle sort the canonical events sampled for a given t e m p e r a t u r e and pressure as a function of the energy and the volume. However, if the pressure is fixed it is sufficient to sort the events as a function of the constrained energy E. At a given t e m p e r a t u r e (3 the canonical distribution reads: Pp(E)

= ^±exp(-/3E)

(4)

where W is the degeneracy of the state. In a sampling of NQ events the probability Pp can be estimated from the number Np of events falling in the enthalpy bin of size AE around E, Pp [E) AE « Np {E) /N0. Equation (4) can be inverted leading to the entropy S (E) = log(W (E)). This allows a direct estimation of the microcanonical caloric curve

r^tm-t+'JsgiS.

(5)

which is valid for every /3. W i t h a single microcanonical sampling at an arbitrary (3 it is in principle possible to directly compute the whole microcanonical caloric curve (5) without any approximation. In the calculations shown below a number A — 216 of particles is fixed; to illustrate a first order phase transition the pressure P is chosen in such a

158 way t h a t the isobar crosses the canonical coexistence line at about the half of the critical t e m p e r a t u r e . Canonical statistical averages are taken over events obtained with a s t a n d a r d Metropolis sampling of the lattice occupations according to the partition function (3). It is interesting to compare the microcanonical and canonical caloric curves as shown in Fig. 1. In finite systems two canonical caloric curves can be defined, corresponding to the average and most probable energy associated to a given j3. In infinite systems these two energies are equal because fluctuations can be neglected. Far from the phase transition the canonical and microcanonical curves agree. Indeed, from equation (5) we can see t h a t the most probable canonical energy is characterized by the equality of the microcanonical and canonical temperatures. In the coexistence region however the predictions of the two ensembles differ in an noticeable manner. T h e canonical caloric curves are by definition monovaluated while this restriction does not apply to the microcanonical case. T h e microcanonical caloric curve presents a back bending while in the back-bending region the canonical caloric curve associated with the most probable energy presents a discontinuity equivalent to the Maxwell construction. T h e observed energy j u m p is directly related to the latent heat of the first order phase transition. Allowing a fluctuating volume is essential

Figure 1: Right: schematic drawing of the lattice-gas model. Left: Microcanonical caloric curve (full line) compared with the most probable (dots) and average (line) canonical energy for each /3. to obtain the caloric curves of Fig. 1: constant volume lattice gas calculations produce smoothly increasing caloric curves 9 even within the microcanonical ensemble as we will discuss later. It should be noticed t h a t the partitions which fall in the energy region corresponding to the canonical t e m p e r a t u r e j u m p are hardly sampled by the canonical ensemble, b u t are accessible in the micro-

159

canonical ensemble. Therefore, in a finite system the microcanonical sorting of events allows to study regions of the phase diagram which are forbidden in the canonical formalism. These regions are characterized by specific properties such as negative heat capacities which we will later on study in more detail. In particular it is i m p o r t a n t to identify experimental observables which can directly inform us about these peculiar properties. 2.2

Analytical

quantum

models

In the lattice-gas model the connection between a phase transition and a back bending in the caloric curve appears evident. However, one may worry about the generality of such a statement. Is it a general property found in m a n y systems? Do such anomalies also exist in q u a n t u m systems?

40 20

a^fe

10

•K.

V J\ £.AA.
•Iff

» f i 6&&4AA&.&

20

30

40

H=e,(iS|-A12+A2)+e,N2

SO

Energy Figure 2: Right: schematic drawing of the double quantum oscillator model with the associated Hamiltonian. Left: entropy, temperature and heat capacity as a function of the excitation energy. In order t o address this question we have investigated a model of A particles which can j u m p from one harmonic oscillator to an other. In the first one all particles strongly interact while in the second one they are free. T h e curvature

160

of the second well plays the role of a confining potential i. e. of a pressure. The corresponding Hamiltonian reads: H = s1 (jVi -A\+

A2) + e2Ni

(6)

with the operators A

A

s a a

Ni = J2 L i n

Ai = Y,5l

n=l

(7)

n=l

where in is the harmonic well occupied by the particle i. Using this Hamiltonian we can compute the level density and so the entropy. To simplify the calculation we have chosen S\ and 62 to be commensurable. Then, we can compute the temperature and the associated heat capacity (see Fig. 2). We observe that the system indeed presents an anomaly in the curvature of the entropy. Back-bending and negative heat capacities automatically follow. 3

ROLE OF T H E V O L U M E

As far as the liquid-gas phase transition is concerned it is essential to discussed the role of the volume since it is the order parameter and since we know that the divergency of the heat capacity depends upon the isobar or isochore character of the considered transformation. 3.1

Caloric curves and abnormal fluctuations

The normalized fluctuations < T | / 7 \ 2 obtained in the microcanonical ensemble with a constrained average volume, \, are shown in the energy-A plane in Fig. 3 together with the isotherms 10 . One can clearly see that up to the critical temperature the fluctuations are abnormally large in the coexistence region. From Fig. 3 it is apparent that the phase transition signal is visible in the temperature as well as in the fluctuation observable. However the experimentally measured caloric curves are not bidimensional. Indeed, even if different sources with different excitation energies can be prepared, the other thermodynamical parameters are not controlled even if they can be measured. In particular an average value for the freeze-out volume of a selected ensemble of events cannot be varied independently of the deposited energy. This means that experiments are sampling a monodimensional curve on the equation of state surface. The resulting caloric curve therefore depends on the actual transformation in the thermodynamical parameters plane. As an example the behavior of the

161 t e m p e r a t u r e as a function of energy at a constant pressure or a constant average volume in the subcritical region is displayed in the upper part of Fig. 3. At

= cst

*-WSM

l(A,HeV)

°

^iA-MeV).

Figure 3: Left: Isotherms and contour plot of the kinetic energy fluctuations in the X — E plane. The level corresponding to the canonical expectation