Particle Physics on the Eve of LHC: Proceedings of the Thirteenth Lomonosov Conference on Elementary Particle Physics 9812837582

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Table of contents :
CONTENTS......Page 14
Thirteenth Lomonosov Conference on Elementary Particle Physics - Sponsors and Committees......Page 6
Seventh International Meeting on Problems of Intelligentsia - Presidium......Page 8
Foreword......Page 10
Fundamentals of Particle Physics......Page 22
2 The quantum number of color and colored quarks......Page 24
3 Dynamic quark models of hadrons composed of quasifree colored quarks......Page 26
3.1 Hadron form factors......Page 27
3.2 The model of quasifree quarks and the laws of scaling at high energies......Page 28
3.3 Quark counting formulae......Page 29
References......Page 30
1 Introduction......Page 32
2 Influences leading to the discovery of the hidden 3-valued color charge degree of freedom......Page 33
3 Introduction of the gauge theory of color......Page 38
4 Summary......Page 39
References......Page 40
1 Introduction......Page 41
2.2 Lomonosov, Lavoisier......Page 42
3.2 Energy and inertia......Page 44
3.6 Einstein supports Eo = mc2......Page 45
4.2 Feynman diagrams......Page 46
7.1 Natural definition of mass......Page 47
7.3 Equivalence of mass and rest energy......Page 48
7.4 Interconversion between rest energy and kinetic energy......Page 51
7.6 Mass differences of hadrons......Page 53
7.7 Some basic questions......Page 55
COROLLARY V.......Page 56
Physics at Accelerators and Studies in SM and Beyond......Page 58
2 Search for standard Higgs boson at the LHC......Page 60
3 Supersymmetry search......Page 63
5 Conclusion......Page 65
References......Page 66
1 INTRODUCTION......Page 67
2 THE FIRST PHYSICS DATA......Page 68
3.1 Low mass region (115 GeV< mH 2mz)......Page 71
4 SUPERSYMMETRIC HIGGS SEARCHES......Page 72
5 MEASUREMENTS OF THE MASS, WIDTH AND COUPLINGS OF THE HIGGS BOSONS......Page 73
ACKNOWLEDGMENTS......Page 74
1 The Standard Model and the ATLAS experiment......Page 76
2.1 Searches for new Matter......Page 77
2.2 Searches for new gauge group structure......Page 80
2.3 Searches for new Electro-weak symmetry breaking mechanisms......Page 82
References......Page 84
The Status of the International Linear Collider B. Foster......Page 86
Reference......Page 87
2 Deep-inelastic NC and CC ep scattering......Page 88
3 Structure functions F2 , XF3 and FL......Page 91
4 Charm and bottom structure functions F2cc, Fbb......Page 93
5.2 Polarisation asymmetry in NC......Page 94
6 Partonic structure of the proton......Page 95
7 Strong coupling as......Page 96
References......Page 97
2 Single top production......Page 98
3 B0 Mixing......Page 99
4 CP violation in B. decays......Page 102
Acknowledgments......Page 104
References......Page 105
Direct Observation of the Strange b Barion st L. Vertogradov......Page 106
References......Page 111
2 LHCb experiment......Page 112
3 The search for Bs —> fi+fj......Page 113
4.2 Radiative decays b --7 sry......Page 114
References......Page 115
1 Introduction......Page 116
2 Bd,s mixing phases......Page 117
3 CKM angle 7......Page 118
References......Page 119
1 Models with Extra Spatial Dimensions......Page 120
2 Collider Searches for Extra Dimensions......Page 122
3 Black Holes at the LHC......Page 125
References......Page 126
Neutrino Physics......Page 128
2 The MiniBooNE Experiment......Page 130
3 Oscillation Analysis......Page 131
References......Page 133
1 Introduction......Page 134
2.1 Analysis......Page 135
3 Prospects......Page 136
References......Page 138
The New Result of the Neutrino Magnetic Moment Measurement in the GEMMA Experiment A. G. Beda, V. B. Brudanin, E. V. Demidova, V. G. Egorov, M. G. Gavrilov, M. V. Shirchenko, A. S. Starostin, Ts. Vylov......Page 140
References......Page 141
2.1 Atmospheric neutrinos......Page 142
2.2 Search for Neutrinos from WIMP Annihilation......Page 143
2.3 A search for fast magnetic monopoles......Page 144
2.4 A search for extraterrestrial high-energy neutrinos......Page 146
3.1 Modernization of data acquisition system......Page 148
4 Conclusion......Page 149
References......Page 150
1 Introduction......Page 152
2 Pioneering developments: DUM AND......Page 154
4 The NESTOR experiment......Page 155
5 The NEMO experiment......Page 156
6 The ANTARES experiment......Page 158
References......Page 160
2.2 Neutrinoless double beta decay......Page 162
3.1 NEMO-3 experiment [26,29]......Page 163
3.2 CUORICINO [24]......Page 164
5 Conclusion......Page 165
References......Page 166
1 Introduction......Page 167
2 The original scenario......Page 168
3 Low energy beta-beams......Page 169
4 The other scenarios for CP violation searches......Page 172
References......Page 173
2.1 Intense narrow-band neutrino beam......Page 175
3.1 013 measurement,' v ve appearance......Page 176
4 Hadron production measurement......Page 177
6 Summary......Page 179
References......Page 180
1 Introduction......Page 181
2.2 Sensitivity reach to NSI......Page 182
3 Quantum Decoherence......Page 184
References......Page 186
3 Decays of sterile neutrinos......Page 187
6 Conclusion......Page 188
References......Page 189
2 The Cuoricino detector......Page 190
3 Cuoricino results......Page 191
4 From Cuoricino to CUORE......Page 192
References......Page 194
1.1 Introduction......Page 195
2.1 Precursor experiments......Page 196
2.2 MARE: phase I......Page 197
2.3 MARE: phase II......Page 198
References......Page 199
2 Angular correlation for the long range mechanism......Page 200
3 Analysis of the electron angular correlation......Page 201
Acknowledgments......Page 202
References......Page 203
Neutrino Energy Quantization in Rotating Medium A. Grigoriev, A. Studenikin......Page 204
References......Page 208
Neutrino Propagation in Dense Magnetized Matter E. V. Arbuzova, A. E. Lobanov, E. M. Murchikova......Page 209
References......Page 213
1 Introduction......Page 214
2 Cherenkov process VL VR and its crossing VL VR......Page 215
4 Does the window for the "spin light of neutrino" exist?......Page 216
References......Page 217
Astroparticle Physics and Cosmology......Page 218
1 Introduction (scientific tasks)......Page 220
2 Magnetic spectrometer PAMELA......Page 222
3 Measurements. Analysis of scientific Information,......Page 223
4 Preliminary results......Page 224
References......Page 227
Introduction......Page 228
2 AMS-02 Sensitivity for DM search......Page 229
References......Page 234
1 The DAMA/NaI experiment......Page 235
2 The DAMAjNaI model-independent result......Page 236
3 Corollary model-dependent quests......Page 237
References......Page 244
1 Search for a decay of natural Europiulll [1,2]......Page 246
2 Measurement of 2/32v decay of 100Mo to the at level of 100Ru [4]......Page 247
3 Search for 2/3 processes in 64Zn with a ZNWO4 scintillator [7].......Page 248
References......Page 249
1 Dark matter clumps......Page 250
2 Cosmological distribution function of clumps......Page 251
3 Tidal destruction of clumps......Page 252
4 Modified distribution function of clumps......Page 254
5 Annihilation of dark matter in clumps......Page 255
References......Page 257
2 Model and observable quantities......Page 258
4 Current observational constraints for inflationary models......Page 259
References......Page 261
2 The extended NJL model in curved space-time......Page 262
3 Static Einstein universe......Page 263
4 Phase transitions......Page 264
References......Page 265
1 Introduction......Page 266
3 The Method of Superpotential......Page 267
References......Page 269
3 Graviatom existence conditions......Page 270
4 Graviatom radiation......Page 271
5 Systems with neutrinos......Page 272
References......Page 273
CP Violation and Rare Decays......Page 274
2 B – / B pp puzzle......Page 276
3 Calculation of the FSI phases of B — and B — pp decay amplitudes......Page 278
4 Direct CPV in B — -decays and phases of the penguin contribution......Page 280
5 Polarizations of vector mesons in B —' VV -decays......Page 281
Acknowledgments......Page 282
References......Page 283
1 CP Violation in the Standard Model......Page 284
2 Recent Measurements of the angle /3......Page 286
4 A flavor of the CKM fits as of Summer 2007......Page 289
References......Page 291
1 -dl The phenomenology of DO-D MIXING......Page 292
2 Studies of DO-lf Mixing by the BaBar Collaboration......Page 293
3 Conclusions......Page 295
References......Page 296
1 Introduction......Page 297
2 Experimental Setup......Page 298
3 Asymmetry measurement method......Page 299
References......Page 300
2 Beam line and Detector......Page 301
3 Cusp Effect......Page 302
4 Ke4 decay......Page 304
References......Page 305
2 The radiative K......Page 306
4 Weak radiative H° decays......Page 307
References......Page 309
2 Proposal of the future experiment......Page 310
2.1 The beam line......Page 311
2.2 The experimental set-up and RfjD current status......Page 312
References......Page 313
2 The tagging technique......Page 314
3 Vus measurement......Page 315
4 Measurement of Ke2/K{L2 and SUSY Higgs constraints......Page 316
5.1 Measurement of the mass......Page 317
5.2 3 decay......Page 318
References......Page 319
Decay Constants and Masses of Heavy-Light Mesons in Field Correlator Method A. M. Badalian......Page 320
References......Page 323
Bilinear R-Parity Violation in Rare Meson Decays A. Ali, A. V. Borisov, M. V. Sidorova......Page 324
References......Page 327
Final State Interaction in K -> 21T Decay E. Shabalin......Page 328
References......Page 331
Hadron Physics......Page 332
Collective Effects in Central Heavy-Ion Collisions G. 1. Lykasov, A. N. Sissakian, A. S. Sarin, V. D. Toneev......Page 334
References......Page 338
2.1 Action vs entropy factoTs......Page 339
2.2 The Polyakov line......Page 340
3.2 Surface operators, monopoles......Page 341
3.3 Dual pictures of confinement......Page 342
4 Extra dimensions......Page 343
5.2 Magnetic component of the Yang-Mills plasma......Page 344
References......Page 345
1 Introduction......Page 347
2 Computational details......Page 348
3 Results......Page 349
References......Page 350
2 The Effective Hamiltonian in FCM......Page 351
4 Conclusions......Page 353
References......Page 354
2 Nonperturbative EoS of QGP......Page 355
3 Phase transition......Page 357
References......Page 359
Chiral Symmetry Breaking and the Lorentz Nature of Confinement A. V. Nefediev......Page 360
References......Page 363
Structure Function Moments of Proton and Neutron M. Osipenko......Page 364
References......Page 367
Higgs Decay to bb: Different Approaches to Resummation of QCD Effects A. L. Kataev, V. T. Kim......Page 368
References......Page 371
1 Introduction......Page 372
2 The Adler function......Page 373
3 Novel integral representation for Adler function......Page 374
References......Page 375
1 Introduction......Page 376
2 z-Scaling......Page 377
3 QeD test of z-scaling......Page 378
References......Page 380
Quark Mixing in the Standard Model and the Space Rotations G. Dattoli, K. Zhukovsky......Page 381
References......Page 384
1 Introduction......Page 385
3 Model-independent definitions of renormalized coupling constants......Page 386
References......Page 388
New Developments in Quantum Field Theory......Page 390
1 Approach unifying spins and charges......Page 392
2 Concluding remarks......Page 397
References......Page 398
Z2 Electric Strings and Center Vortices in SU(2) Lattice Gauge Theory M. 1. Polikarpov, P. V. Buividovich......Page 399
References......Page 406
Upper Bound on the Lightest Neutralino Mass in the Minimal Non-Minimal Supersymmetric Standard Model S. Hesselbach, G. Moortgat-Pick, D. J. Miller, R. Nevzorov, M. Trusov......Page 407
References......Page 410
Application of Higher Derivative Regularization to Calculation of Quantum Corrections in N=l Supersymmetric Theories K. Stepanyantz......Page 411
References......Page 414
1 Introduction......Page 415
2 The solution and main results......Page 416
References......Page 418
2 Khalfin's Theorem and its implications......Page 419
3 Model calculations......Page 421
4 Final remarks......Page 422
References......Page 423
1 Why we need in the Random Lattice QCD......Page 424
2 From Lattice QeD to chiral lagrangians: step by step......Page 425
References......Page 428
String-Like Electrostatic Interaction from QED with Infinite Magnetic Field A. E. Shabad, V. V. Usov......Page 429
References......Page 432
1 Introduction......Page 433
2 Scalar models......Page 434
3 Quantum Electrodynamics......Page 435
References......Page 436
2 Hydrogen-like bound state......Page 437
3 Electron in a homogeneous magnetic field. Synchrotron radiation......Page 439
References......Page 440
Particles with Low Binding Energy in a Strong Stationary Magnetic Field E. V. Arbuzova, G. A. Kravtsova, V. N. Rodionov......Page 441
References......Page 444
Triangle Anomaly and Radiatively Induced Lorentz and CPT Violation in Electrodynamics A. E. Lobanov, A. P. Venediktov......Page 445
References......Page 447
Introduction......Page 448
Starting theoretical expressions......Page 449
The study of p(J) (q) functions properties......Page 450
References......Page 452
1. Introduction......Page 453
2. Semi-classical identification of the spin light for synchrotron radiation.......Page 454
3. Spin light in the classical theory of synchrotron radiation......Page 455
4. Conclusion......Page 457
Acknowledgments......Page 458
References......Page 459
1.1 Particles in real reference frame......Page 460
1.2 Scalar potential in the phase space of coordinates and their multiple derivatives......Page 462
Reference......Page 463
1 Introduction......Page 464
2 Equations for stable leptons......Page 465
3 Extensions of the stable lepton groups......Page 466
References......Page 467
Generalized Dirac Equation Describing the Quark Structure of Nucleons A. Rabinowitch......Page 468
References......Page 471
1 Main idea......Page 472
2 Topological defect......Page 473
3 Stochastic behavior......Page 474
References......Page 475
Problems of Intelligentsia......Page 476
The Conscience of the Intelligentsia J. K. Bleimaier......Page 478
Conference Programme......Page 484
List of Participants......Page 490
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PARTICLE PHYSICS on the Eve of LHC

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Faculty of Physics of Moscow State University

INTERREGIONAL CENTRE FORADVANCEDSTUDffiS

Proceedings of the Thirteenth Lomonosov Conference on Elementary Particle Physics

PARTICLE PHYSICS on the Eve of LHC Moscow, Russia

23 - 29 August 2007

Editor

Alexander I. Studenikin Department of Theoretical Physics Moscow State University, Russia

'lit

World Scientific

NEW JERSEY· LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI

Published by

World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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

PARTICLE PHYSICS ON THE EVE OF LHC Proceedings of the 13th Lomonosov Conference on Elementary Particle Physics Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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

ISBN-13 978-981-283-758-5 ISBN-IO 981-283-758-2

Printed in Singapore by B & JO Enterprise

v

Moscow State University Faculty of Physics Centre for Advanced Studies

mll"II'II',ElIl"ll".anl,nn'!:II1

Russian Foundation for Basic Research Russian Agency for Science and Innovation Russian Academy of Sciences Russian Agency for Atomic Energy Dmitry Zimin "Dynasty" Foundation Institutions Faculty of Physics of Moscow State :::iK()be,lts,!ln institute of Nuclear Physics, Moscow State Centre for Advanced Studies Joint Institute for Nuclear Institute for Nuclear R""'"",,,,",rr-h Theoretical and i=vY,ari,rnQlnt",1 Budker Institute of Nuclear

vi

International Advisory Committee

E.Akhmedov (ICTP, Trieste & Kurchatov Inst.,Moscow), S.Selayev (Kurchatov Inst.,Moscow), VSerezinsky (LNGS, Gran Sasso), S.Silenky (JINR, Dubna), J.Sleimaier (Princeton), MDanilov (ITEP, Moscow), GDiambrini-Palazzi (Univ. of Rome), ADolgov (INFN, Ferrara & ITEP, Moscow), VKadyshevsky (JINR, Dubna), S.Kapitza (EAPS, Moscow) A.Logunov (IHEP, Protvino), V.Matveev (INR, Moscow), P.Nowosad (Univ. of Sao Paulo), L.Okun (ITEP, Moscow), M.Panasyuk (SINP MSU), VRubakov (INR, Moscow), D.Shirkov (JINR, Dubna), J.Silk (Univ. of Oxford), ASissakian (JINR,Dubna), ASkrinsky (INP, Novosibirsk), ASlavnov (MSU & Steklov Math.lnst, Moscow) ASmirnov (ICTP, Trieste & INR, Moscow), P.Spiliantini (INFN, Florence), Organizing Committee

V.Sagrov (Tomsk State Univ.), VSelokurov (MSU), VSraginsky (MSU), AEgorov (ICAS, Moscow), D.Galtsov (MSU), AGrigoriev (MSU & ICAS, Moscow), P.Kashkarov (MSU), AKataev (INR, Moscow), O.Khrustalev (MSU), VMikhailin (MSU & ICAS, Moscow) AMourao (1ST/CENTRA, Lisbon), N.Narozhny (MEPHI, Moscow), A.Nikishov (Lebedev Physical Inst., Moscow), N.Nikiforova (MSU), VRitus (Lebedev Physical Inst., Moscow), Yu.Popov (MSU) , VSavrin (MSU), D.Shirkov (JINR, Dubna), Yu.Simonov (ITEP, Moscow), AStudenikin (MSU & ICAS, Moscow), V.Trukhin (MSU)

vii

Moscow State University Interregional Centre for Advanced Studies

SEVENTH INTERNATIONAL MEETING ON PROBLEMS OF INTELLIGENTSIA "RIGHTS and RESPONSIBILITY of the NTELLIGENTSIA" Moscow, August 29, 2007

Presidium of the Meeting VASadovnichy (MSU) - Chairman VV.Belokurov (MSU) J.Bleimaier (Princeton) G.Diambrini-Palazzi (Universiry of Rome) VG.Kadyshevsky (JINR) S.P.Kapitza (Russian Academy of Sciensies) N.S.Khrustaleva (Ministry of Education and Science, Russia) A.I.Studenikin (MSU & ICAS) - Vice Chairman V.I.Trukhin (MSU)

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FOREWORD

The 13 th Lomonosov Conference on Elementary Particle Physics was held at the Moscow State University (Moscow, Russia) on August 23-29,2007 under the Patronage of the Rector of the Moscow State University Victor Sadovnichy. The conference was organized by the Faculty of Physics and SkobeJtsyn Institute of Nuclear Physics of the Moscow State University in cooperation with the Interregional Centre for Advanced Studies and supported by the Joint Institute for Nuclear Research (Dubna), the Institute for Nuclear Research (Moscow), the Budker Institute of Nuclear Physics (Novosibirsk) and the Institute of Theoretical and Experimental Physics (Moscow). The Russian Foundation for Basic Research, the Russian Agency for Science and Innovation, the Russian Academy of Sciences, the Dmitry Zimin "Dynasty" Foundation and the Russian Agency for Atomic Energy sponsored the conference. It was more than twenty years ago when the first of the series of conferences (from 1993 called the "Lomonosov Conferences"), was held at the Department of Theoretical Physics of the Moscow State University (June 1983, Moscow). The second conference was held in Kishinev, Republic of Moldavia, USSR (May 1985). After the four years break this series was resumed on a new conceptual basis for the conference programme focus. During the preparation of the third conference (that was held in Maykop, Russia, 1989) a desire to broaden the programme to include more general issues in particle physics became apparent. During the conference of the year 1992 held in Yaroslavl it was proposed by myself and approved by numerous participants that these irregularly held meetings should be transformed into regular events under the title "Lomonosov Conferences on Elementary Particle Physics". Since then at subsequent meetings of this series a wide variety of interesting things both in theory and experiment of particle physics, field theory, astrophysics, gravitation and cosmology were included into the programmes. It was also decided to enlarge the number of institutions that would take part in preparation of future conferences. Mikhail Lomonosov (1711-1765), a brilliant Russian encyclopaedias of the era of the Russian Empress Catherine the 2nd, was world renowned for his distinguished contributions in the fields of science and art. He also helped establish the high school educational system in Russia. The Moscow State University was founded in 1755 based on his plan and initiative, and the University now bears the name of Lomonosov. The 6th Lomonosov Conference on Elementary Particle Physics (1993) and all of the subsequent conferences of this series were held at the Moscow State University on each of the odd years. Publication of the volume "Particle Physics, Gauge Fields and Astrophysics" containing articles written on the basis of presentations at the 5th and 6th Lomonosov Conferences was supported by the Accademia Nazionale dei Lincei (Rome, 1994). Proceedings of the 7th and 8th Lomonosov Conference (entitled "Problems of Fundamental Physics" and "Elementary Particle Physics") were published by the Interregional Centre for ix

x Advanced Studies (Moscow, 1997 and 1999). Proceedings of the 9th , 10th , 11th and 12th Lomonosov Conferences (entitled "Particle Physics at the Start of the New Millennium", "Frontiers of Particle Physics", "Particle Phlsics in Laboratory, Space and Universe" and "Particle Physics at the Year of 250 Anniversary of Moscow University") were published by World Scientific Publishing Co. (Singapore) in 2001,2003,2005 and 2006, correspondently. The physics programme of the 13 th Lomonosov Conference on Elementary Particle Physics (August, 2007) included review and original talks on wide range of items such as neutrino and astroparticle physics, electroweak theory, fundamental symmetries, tests of standard model and beyond, heavy quark physics, nonperturbative QCD, quantum gravity effects, physics at the future accelerators. Totally there were more than 350 participants with 113 talks including 32 plenary (30 min) talks, 48 session (25-20 min) talks and 33 brief (15 min) reports. One of the goals of the conference was to bring together scientists, both theoreticians and experimentalists, working in different fields, so that no parallel sessions were organized at the conference. The Round table discussion on "Dark Matter and Dark Energy: a Clue to Foundations of Nature" was held during the last day of the 13 th Lomonosov Conference. Following the tradition that has started in 1995, each of the Lomonosov Conferences on particle physics has been accompanied by a conference on problems of intellectuals. The 7th International Meeting on Problems of Intelligentsia held during the 13 th Lomonosov Conference (August 29, 2007) was dedicated to discussions on the issue "Rights and Responsibility of the Intelligentsia". The success of the 13 th Lomonosov Conference was due in a large part to contributions of the International Advisory Committee and Organizing Committee. On behalf of these Committees I would like to warmly thank the session chairpersons, the speakers and all of the participants of the 13 th Lomonosov Conference and the 7th International Meeting on Problems of Intelligentsia. We are grateful to the Rector of the Moscow State University, Victor Sadovnichy, the Vice Rector df the Moscow State University, Vladimir Belokurov, the Dean of the Faculty of Physics, Vladimir Trukhin, the Director of the Skobeltsyn Institute of Nuclear Physics, Mikhail Panasyuk, the Directors of the Joint Institute for Nuclear Research, Alexey Sissakian, the Director of the Institute for Nuclear Research, Victor Matveev, the Director of the Budker Institute of Nuclear Physics, Alexander Skrinsky, and the Vice Dean of the Faculty of Physics of the Moscow State University, Anatoly Kozar for the support in organizing these two conferences. Special thanks are due to Alexander Suvorinov (the Russian Agency for Science and Innovations), Gennady Kozlov (JINR) and Oleg Patarakin (the Russian Agency for Atomic Energy) for their very valuable help. I would like to thank Giorgio Chiarelli, Dmitri Denisov, Francesca Di Lodovico, Hassan Jawahery, Andrey Kataev, Cristina Lazzaroni, William C. Louis, Frank Merrit, Thomas MUller, Tatsuya Nakada, Daniel Pitzl, Jacob Schneps, Claude

xi

Vallee and Horst Wahl for their help in planning of the scientific programme of the meeting and inviting speakers for the topical sessions of the conference. Furthermore, I am very pleased to mention Alexander Grigoriev, the Scientific Secretary of the conference, Andrey Egorov, Mila Polyakova, Dmitry Zhuridov, Dasha Novikova, Maxim Perfilov and Katya Salobaeva for their very efficient work in preparing and running the meeting. These Proceedings were prepared for publication at the Interregional Centre for Advanced Studies with support by the Russian Foundation for Basic Research, the Russian Agency for Science and Innovations and the Russian Agency for Atomic Energy. Alexander Studenikin

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CONTENTS Thirteenth Lomonosov Conference on Elementary Particle Physics Sponsors and Committees Seventh International Meeting on Problems of Intelligentsia Presidium ｔｨｾ＠

v vii ｾ＠

Fundamentals of Particle Physics The Quantum Number of Color, Colored Quarks and Dynamic Models of Hadrons Composed of Quasifree Quarks V. Matveev, A. Tavkhelidze

3

Discovery of the Color Degree of Freedom in Particle Physics: A Personal Perspective O. W. Greenberg

11

The Evolution of the Concepts of Energy, Momentum, and Mass from Newton and Lomonosov to Einstein and Feynman L. Okun

20

Physics at Accelerators and Studies in SM and Beyond Search for New Physics at LHC (CMS) N. K rasnikov

39

Measuring the Higgs Boson(s) at ATLAS C. Kourkoumelis

46

Beyond the Standard Model Physics Reach of the ATLAS Experiment G. Unel

55

The Status of the International Linear Collider B. Foster

65

Review of Results of the Electron-Proton Collider HERA V. Chekelian

67

Elements from Bs Recent Results from the Tevatron on CKM ｍ｡ｴｲｾ＠ Oscillations and Single Top Production, and Studies of CP Violation in Bs Decays J. P. Fernandez

77

Direct Observation of the Strange b Barion L. Vertogradov xiii

st 85

xiv

Search for New Physics in Rare B Decays at LHCb V. Egorychev

91

CKM Angle Measurements at LHCb S. Barsuk

95

Collider Searches for Extra Spatial Dimensions and Black Holes G. Landsberg

99

Neutrino Physics Results of the MiniBooNE Neutrino Oscillation Experiment Z. Djurcic

109

MINOS Results and Prospects J.P.Ochoa-Ricoux

113

The New Result of the Neutrino Magnetic Moment Measurement in the GEMMA Experiment A. G. Beda, V. B. Brudanin, E. V. Demidova, V. G. Egorov, M. G. Gavrilov, M. V. Shirchenko, A. S. Starostin, Ts. Vylov

119

The Baikal Neutrino Experiment: Status, Selected Physics Results, and Perspectives V. Aynutdinov, A. Avrorin, V. Balkanov, 1. Belolaptikov, N. Budnev, I. Danilchenko, G. Domogatsky, A. Doroshenko, A. Dyachok, Zh.-A. Dzhilkibaev, S. Fialkovsky, O. Gaponenko, K. Golubkov, O. Gress, T. Gress, O. Grishin, A. Klabukov, A. Klimov, A. Kochanov, K. Konischev, A. Koshechkin, V. Kulepov, L. Kuzmichev, E. Middell, S. Mikheyev, M. Milenin, R. Mirgazov, E. Osipova, G. Pan'kov, L. Pan'kov, A. Panfilov, D. Petukhov, E. Pliskovsky, P. Pokhil, V. Poleschuk, E. Popova, V. Prosin, M. Rozanov, V. Rubtzov, A. Sheifier, A. Shirokov, B. Shoibonov, Ch. Spiering, B. Tarashansky, R. Wischnewski, I. Yashin, V. Zhukov

121

Neutrino Telescopes in the Deep Sea V. Flaminio

131

Double Beta Decay: Present Status A. S. Barabash

141

Beta-Beams C. Volpe

146

T2K Experiment K. Sakashita

154

xv Non-Standard Neutrino Physics Probed by Tokai-to-Kamioka-Korea Two-Detector Complex

N. Cipriano Ribeiro, T. Kajita, P. Ko, H. Minakata, S. Nakayama, H. Nunokawa

160

Sterile Neutrinos: From Cosmology to the LHC

F. Vannucci

166

From Cuoricino to Cuore Towards the Inverted Hierarchy Region

C. Nones

169

The MARE Experiment: Calorimetric Approach to the Direct Measurement of the Neutrino Mass

E. Andreotti

174

Electron Angular Correlation in Neutrinoless Double Beta Decay and New Physics

A. Ali, A. Borisov, D. Zhuridov

179

Neutrino Energy Quantization in Rotating Medium

A. Grigoriev, A. Studenikin

183

Neutrino Propagation in Dense Magnetized Matter

E. V. Arbuzova, A. E. Lobanov, E. M. Murchikova

188

Plasma Induced Neutrino Spin Flip via the Neutrino Magnetic Moment

A. K uznetsov, N. Mikheev

193

Astroparticle Physics and Cosmology International Russian-Italian Mission "RIM-PAMELA"

A. M. Galper, P. Picozza, o. Adriani, M. Ambriola, G. C. Barbarino, A. Basili, G. A. Bazilevskaja, R. Bellotti, M. Boezio, E. A. Bogomolov, L. Bonechi, M. Bongi, L. Bongiorno, V. Bonvicini, A. Bruno, F. Cafagna, D. Campana, P. Carlson, M. Casolino, G. Castellini, M. P. De Pascale, G. De Rosa, V. Di Felice, D. Fedele, P. Hofverberg, L. A. Grishantseva, S. V. Koldashov, S. Y. Krutkov, A. N. Kvashnin, J. Lundquist, O. Maksumov, V. Malvezzi, L. Marcelli, W. Menn, V. V. Mikhailov, M. Minori, E. Mocchiutti, A. Morselli, S. Orsi, G. Osteria, P. Papini, M. Pearce, M. Ricci, S. B. Ricciarini, M. F. Runtso, S. Russo, M. Simon, R. Sparvoli, P. Spillantini, Y. 1. Stozhkov, E. Taddei, A. Vacchi, E. Vannuccini, G. Vasilyev, S. A. Voronov, Y. T. Yurkin, G. Zampa, N. Zampa, V. G. Zverev

199

xvi

Dark Matter Searches with AMS-02 Experiment A. Malinin

207

Investigating the Dark Halo R. Bernabei, P. Belli, F. Montecchia, F. Nozzoli, F. Cappella, A. Incicchitti, D. Prosperi, R. Cerulli, C. J. Dai, H. L. He, H. H. Kuang, J. M. Ma, X. D. Sheng, Z. P. Ye

214

Search for Rare Processes at Gran Sasso P. Belli, R. Bernabei, R. S. Boiko, F. Cappella, R. Cerulli, C. J. Dai, F. A. Danevich, A. d'Angelo, S. d'Angelo, B. V. Crinyov, A. Incicchitti, V. V. Kobychev, B. N. Kropivyansky, M. Laubenstein, P. C. Nagornyi, S. S. Nagorny, S. Nisi, F. Nozzoli, D. V. Poda, D. Prosperi, A. V. Tolmachev, V. I. Tretyak, l. M. Vyshnevskyi, R. P. Yavetskiy, S. S. Yurchenko

225

Anisotropy of Dark Matter Annihilation and Remnants of Dark Matter Clumps in the Galaxy V. Berezinsky, V. Dokuchaev, Yu. Eroshenko

229

Current Observational Constraints on Inflationary Models E. Mikheeva

237

Phase Transitions in Dense Quark Matter in a Constant Curvature Gravitational Field D. Ebert, V. Ch. Zhukovsky, A. V. Tyukov

241

Construction of Exact Solutions in Two-Fields Models S. Yu. Vernov

245

Quantum Systems Bound by Gravity M. L. Fil'chenkov, S. V. Kopylov, Y. P. Laptev

249

CP Violation and Rare Decays Some Puzzles of Rare B-Decays A. B. Kaidalov

255

Measurements of CP Violation in b Decays and CKM Parameters J. Chauveau

263

Evidence for DO_Do Mixing at BaBar M. V. Purohit

271

Search for Direct CP Violation in Charged Kaon Decays from NA48/2 Experiment S. Balev

276

xvii

Scattering Lengths from Measurements of Ke4 and K± Decays at NA48/2 D. Madigozhin

7m

->

1T± 1T 01T O

280

Rare Kaon and Hyperon Decays in NA48 Experiment N. M ala kana va

285

THE K+ -> 1T+ vD Experiment at CERN Yu. Potrebenikov

289

Recent KLOE Results B. Di Micco

293

Decay Constants and Masses of Heavy-Light Mesons in Field Correlator Method A. M. Badalian

299

Bilinear R-Parity Violation in Rare Meson Decays A. Ali, A. V. Borisov, M. V. Sidorova

303

Final State Interaction in K E. Shabalin

307

->

21T Decay

Hadron Physics

Collective Effects in Central Heavy-Ion Collisions G. 1. Lykasov, A. N. Sissakian, A. S. Sarin, V. D. Toneev

313

Stringy Phenomena in Yang-Mills Plasma V. 1. Zakharov

318

Lattice Results on Gluon and Ghost Propagators in Landau Gauge I. L. Bogolubsky, V. G. Bornyakov, G. Burgio, E.-M. Ilgenfritz, M. Miiller-Preussker, V. K. Mitrjushkin

326

ｾ＠

and 2: Excited States in Field Correlator Method I. Narodetskii, A. Veselov

330

Theory of Quark-Gluon Plasma and Phase Transition E. V. Komarov, Yu. A. Simonov

334

Chiral Symmetry Breaking and the Lorentz Nature of Confinement A. V. Nefediev

339

Structure Function Moments of Proton and Neutron M. Osipenko

343

Higgs Decay to bb: Different Approaches to Resummation of QCD Effects A. L. Kataev, V. T. Kim

347

xviii

A Novel Integral Representation for the Adler Function and Its Behavior at Low Energies A. V. Nesterenko

351

QCD Test of z-Scaling for nO-Meson Production in pp Collisions at High Energies M. Tokarev, T. Dedovich

355

Quark Mixing in the Standard Model and the Space Rotations G. Dattoli, K. Zhukovsky

360

Analytic Approach to Constructing Effective Theory of Strong Interactions and Its Application to Pion-Nucleon Scattering A. N. Safronov

364

New Developments in Quantum Field Theory On the Origin of Families and their Mass Matrices with the Approach Unifying Spin and Charges, Prediction for New Families N. S. Mankoc Borstnik

371

Z2 Electric Strings and Center Vortices in SU(2) Lattice Gauge Theory M. 1. Polikarpov, P. V. Buividovich

378

Upper Bound on the Lightest Neutralino Mass in the Minimal Non-Minimal Supersymmetric Standard Model S. Hesselbach, G. Moortgat-Pick, D. J. Miller, R. Nevzorov, M. Trusov

386

Application of Higher Derivative Regularization to Calculation of Quantum Corrections in N=l Supersymmetric Theories K. Stepanyantz

390

Nonperturbative Quantum Relativistic Effects in the Confinement Mechanism for Particles in a Deep Potential Well K. A. Sveshnikov, M. V. Ulybyshev

394

Khalfin's Theorem and Neutral Mesons Subsystem K. Urbanowski

398

Effective Lagrangians and Field Theory on a Lattice O. V. Pavlovsky

403

String-Like Electrostatic Interaction from QED with Infinite Magnetic Field A. E. Shabad, V. V. Usov

408

xix

QFT Systems with 2D Spatial Defects 1. V. Fialkovsky, V. N. Markov, Yu. M. Pismak

412

Bound State Problems and Radiative Effects in Extended Electrodynamics with Lorentz Violation 1. E. Prolov, O. G. Kharlanov, V. Ch. Zhukovsky

416

Particles with Low Binding Energy in a Strong Stationary Magnetic Field E. V. Arbuzova, G. A. Kravtsova, V. N. Rodionov

420

Triangle Anomaly and Radiatively Induced Lorentz and CPT Violation in Electrodynamics A. E. Lobanov, A. P. Venediktov

424

The Comparative Analysis of the Angular Distribution of Synchrotron Radiation for a Spinless Particle in Classic and Quantum Theories V. G. Bagrov, A. N. Burimova, A. A. Gusev

427

Problem of the Spin Light Identification V. A. Bordovitsyn, V. V. Telushkin

432

Simulation the Nuclear Interaction T. F. K amalov

439

Unstable Leptons and ({L- e - T}-Universality O. Kosmachev

443

Generalized Dirac Equation Describing the Quark Structure of Nucleons A. Rabinowitch

447

Unique Geometrization of Material and Electromagnetic Wave Fields O. Olkhov

451

Problems of Intelligentsia

The Conscience of the Intelligentsia J. K. Bleimaier

457

Conference Programme

463

List of Participants

469

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Fundamentals of Particle Physics

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THE QUANTUM NUMBER OF COLOR, COLORED QUARKS AND DYNAMIC MODELS OF HADRONS COMPOSED OF QUASIFREE QUARKS

v. Matveev a , A. Tavkhelidze b Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia Abstract. Are exposed the main stages of the early development of the hypothesis of the quantum number of color and of colored quarks

1

Introduction

At present, the dominant point of view is that all physical phenomena and processes, both terrestrial and cosmological, are governed by three fundamental forces: gravitational, electroweak and chromodynamic. The color charge serves as the source of chromo dynamic forces. In this talk we shall expose the main stages of the early development of the hypothesis of the quantum number of color and of colored quarks, put forward under the ideological influence of and in collaboration with N .Bogolubov at the JINR Laboratory of Theoretical Physics. In these works, the concept of color, colored quarks, was introduced for the first time, and a dynamical description of hadrons was given within the framework of the model of quasifree colored quarks. Introduction of the quantum number of color permitted to treat colored quarks as real physical objects, constituents of matter. Further, from the color 5U(3) symmetry, the Yang-Mills principle of local invariance and quantization of chromo dynamic fields gave rise to quantum chromo dynamics - the modern theory of strong interactions. 2

The quantum number of color and colored quarks

In 1964, when the hypothesis of quarks was put forward by Gell-Mann [lJ and Zweig [2], quarks were only considered to be mathematical objects, in terms of which it was possible, in a most simple and elegant way, to describe the properties, already revealed by that time, of the approximate unitary 5U(3) symmetry of strong interactions. At the beginning, these particles, exhibiting fractional charges and not observable in a free state, were not attributed the necessary physical interpretation. First of all, making up hadrons of quarks, possessing spin ｾＬ＠ led to a contradiction with the Pauli principle and the Fermi-Dirac statistics for systems composed of particles of semiinteger spin. The problem of the quark statistics was not, however, the sole obstacle in the path of theory. No answer existed to the following question: why were ae-mail: [email protected] be-mail: [email protected]

3

4

only systems consisting of three quarks and quark-antiquark pairs realized in Nature, and why were there no indications of the existence of other multiquark states? Especially important was the issue of the possible existence of quarks in a free state (the problem of quark confinement). In 1965, analysis of these problems led N.Bogolubov, B.Struminsky and A.Tavkhelidze [3], as well as LNambu and M.Hana [4], and LMiyamoto [5] to the cardinal idea of quarks exhibiting a new, hitherto unknown, quantum number subsequently termed color. From the very beginning creation of the relativistically invariant dynamical quark model of hadrons was based, first of all, on the assumption of quarks representing real physical objects determining the structure of hadrons. To make it possible for quarks to be considered fundamental physical particles, the hypothesis was proposed by three authors (Bogolubov, Struminsky, Tavkhelidze - 1965, January) that the quarks, should possess an additional quantum number, and that quarks of each kind may exist in three (unitary) equivalent states q

==

(ql,q2,q3)

differing in values of the new quantum number, subsequently termed color. Since at the time, when the new quantum number was introduced, only three kinds of quarks were known - (u, d, s), the quark model with an additional quantum number was termed the three-triplet model. Since the new quantum number is termed color, colored quarks may be in three equivalent states, such as, for example, red, blue and green. With introduction of the new quantum number, color, the question naturally arised of the possible appearance of hadrons possessing color, which, however, have not been observed. From the assumption that colored quarks are physical objects, while the hadron world is degenerate in the new quantum number, or it is colorless, it followed that solutions of the dynamic equations for baryons and mesons in the s-state should be neutral in the color quantum numbers [4,6]. From the requirement that baryons be colorless, the wave function of the observed baryon family in the ground state, described by the totally symmetric 56-component tensor

sss). New Physics can contribute to the latter via virtual new particles in the loop.

Table 1: CKM structure of non leptonic b decay amplitudes. The amplitude for a b --> qlq-2q3 transition is written in terms of T and/or P amplitudes with the CKM factors shown explicitly. The power of >- governing the first and second terms are given. A golden channel leads to a pure measurement of a CKM phase or UT angle 'P. Only effective phases are accessible from the non golden channels. quark process Aces"" Veb ｖ･ｾｔｳ＠ + Vub V':sPs Asss "" Veb ｖ･ｾｐｳ＠ + Vub V':sP; Aced"" Veb Vcd Teed + "lltb '-"t'dPd Auud "" Vub V,:dTuud + Vtb ｖｴｾｐ､＠

2

1st term ＾Ｍｾ＠

>-2 >-3 >-3

2nd >-4 >-4 >-3 >-3

example golden golden

Jj1/J K S,L ¢K£

D+D71"+ 71"- , pO pO

'P f3 f3 f3eff aeff

Recent Measurements of the angle /3

The most recent results are tabulated in reference [3]. Here I focus on the measurements of the b --> ccs and b --> sss channels, in particular on the golden modes. Over the last few years, much speculation was entertained by the observation that most Penguin dominated b --> sss final states were measured with sin2/3ejj lower than those from Tree dominated b --> ccs (Figure 3). A simple minded average of all the Penguin measurements fell lower than the Tree measurement with almost 3 standard deviation significance (Figure 4-b)). Figure 3 shows the latest measurements of ａＳＧｰＨｾｴＩ＠ with the golden channels B --> charmonium KO by BABAR [4] and J/1jJKo by Belle [5]: S S

= 0.714 ± 0.032 ± 0.018, C = 0.049 ± 0.022 ± 0.017 (BABAR), = 0.642 ± 0.031 ± 0.017, C = 0.018 ± 0.021 ± 0.014 (Belle),

where the first uncertainties are statistical and the second ones systematic. The average over all charmonium KO measurements is sin 2/3 = 0.678 ± 0.025 or, in the first quadrant of the (p,7}) plane: /31 = (21.3 ± 1.0)0 or /32 = (67.8 ± 1.0)0, /31 being favored by several measurements each with small statistical significance [6]. New this summer are the time dependent amplitude (Dalitz) analyses on Penguin golden channels ｋｾｨＫ＠ h- [7,8], where h refers to a 7r or a K meson.

888

Penguin channels.

direct measurements of the f3 (not a function final states as well as the non-resonant threeAfter "ll'OICl.l1lC, I focus on the

are paraman isobar model. Each term is a ",,"ull.teA COmI)lCX (isobar) coefficient whose argument incor5 a)-c) show the two-body invariant and the components from ＾ｉＢＬＮｲｮＱｾｰﾷＧ＠ for the

enriched fit distributions for the time dependent Dalitz analysis of . The full fit distributions are superimposed over the data points three invariant mass spectra of the Dalitz plot. The shaded areas correspond to the background components, and the signal. Vetoes create holes the D and J /'Ij; The pO and fo peaks are in the 7r+7rThe corresponding dependent CP asymmetries for d) and are shown at the bottom of the

which makes the minded average sin over with the measurements from the Tree processes the result from the above pre4 to the older dataset 4

'AJ.tll,Jel>L,lUJlC

and measured B meson related There is no evidence for direct OP violation from the measured time

268 dependent CP asymmetries and no compelling hint for New Physics. 3

Recent Progress on the other UT Angles

Here, I have chosen to highlight the recent progress made on the GronauLondon (GL) analysis [9] of the B -+ pp channels. This charmless b -+ uud decay is not a golden mode (Table 1). The Penguin pollution introduces a phase shift on the determination of the UT angle a and one measure ael I instead of a. The GL method exploits the SU(2) symmetry to combine all charge states in B -+ pp and determine a-aell up to trigonometric ambiguities. It has been appreciated for some time that since the branching fraction for BO -+ pO pO is measurable with fair accuracy, the GL triangle can be constructed more precisely than in the founding case B -+ 7r7L Furthermore with four charged pions in the final state, the decay vertex can be accurately reconstructed and the time dependent CP asymmetry measured, an impossible feat for B -+ 7r 0 7r 0 . With high statistics it has been possible this year to measure Acp(t) for the longitudinally polarized pOpo pairs [10]. Including these results into the GL fit yields the confidence level profile for a - ael I shown on Figure 6. It is nomore fiat as was the case when no CP asymmetry measurement was included. Some discrimination between the mirror solutions already observed with previous spin-averaged measurements of Acp(t) can be seen. There is hope that an accurate determination of a will be obtained with the full data samples of the B factories.

Figure 6: Exclusion confidence level scan for Q The red (solid) QeJ J. curve corresponds to the recent measurement of the time dependent asymmetry for longitudinally polarized pO pO pairs in neutral B decays [10]. The one and two-sigma exclusion levels are shown as horizontal intermittent lines.

4

ｾ＠

-!

1 ••••••. without ｃｾｑ＠

0.8

and ｓｾ＠

with Cro and without ｓｾ＠

0 .•

0.4

0.2

·10

10

20

30

40 «- 7r+ 7r- e± v decay is described by the five CabibboMaksymowicz variables [13]: invariant mass squared of a dip ion Sn = M;n' invariant mass squared of dilepton Sn = M;v and angles en, ee, ¢. en is the angle between 7r- and dilepton momenta in the rest frame of dipion, ee is the angle between v and dip ion momenta in the rest frame of dilepton. ¢ is the angle between the plane of dilepton and e± momenta and the plane of dipion and 7r+ momenta in the kaon rest frame. The matrix element is defined by means of axial form factors F,G and a vector form factor H. A partial wave expansion of the form factors may be restricted to sand p waves: F = Fs ei08 + FpeiOpcosen, G = Gpe iOp , H = Hpe iOp (only phase shift 5 = 5s - 5p is observable). From the 2003 data, about 670000 Ke4 decays have been selected. Reconstructed events are distributed in 10 x 5 x 5 x 5 x 12 iso-populated bins in the (M1r1r' Mev, cos(e n ), cos(ee), ¢) space. Ten independent fits (one per M1r1r bin) of five parameters (Fp, Gp, Hp, 8, and a normalization constant, that absorbs Fs) where performed in four dimensional space using the acceptance and resolution information from Monte Carlo simulation. The value of the phase difference 5 was extracted from the measured asymmetry of ¢ distribution as a function of M 1r1r . The result of 5 measurement is shown on the Fig. 3 together with results from previous experiments [2,3]. The phase shift measurements can be related to the 7r7r scattering lengths using the analytical properties and crossing symmetry of amplitudes (Roy equations [15]). One can use the Universal Band approach [16,17] to extract alone. At the center line of the Universal Band (I-parameter fit), NA48/2 = 0.256±0.006stat±0.002syst±0.018theoTl phase measurements translate as which implies a6 = -0.0312 ± O.OOl1 stat ± 0.0004 syst ± 0.013 t heor. In the case of the fit where both ag and a6 are free parameters, the result is ag = 0.233 ± 0.016 stat ± 0.007 syst, a6 = -0.0471 ± O.Ol1 stat ± 0.004 syst (the correlation is 96.7%). Finally, resent work [18] suggests that isospin symmetry breaking effects, neglected so far in the Ke4 phase shift analysis, would lead, when taken into account, to decrease of ag by ｾ＠ 0.022, and a6 - by about 0.004, leading to good compatibility between the Ke4 and cusp analyses results for pion scattering lengths.

ag

ag

284

•

,.-....0.3 ""0

t: 0.2

ｾ＠

'0

•

0

-0.1

t •• t ., • t • •

. • •

0 .1

• NA48/2 (2003 Data) • E865 [3] ... Geneva-Saclay [2]

'

ｾＧＭＬｉ＠

0.28

0.3

•

0.32

0.34

0.36

0.38

M",,( GeV/ c Figure 3: Phase shift 8 measurements from

5

Ke4

0.4

2 )

experiments

Conclusion

Two independent results of 7r7r scattering lengths measurement, obtained by NA48/2 experiment, are compatible between each other and are in agreement with current predictions of ChPT, if the isospin symmetry breaking effects are taken into account in both analyses of experimental data. References

[1] G.Colangelo, J.Gasser, H.Leutwyler, Nucl.Phys. B 603, 125 (2001). [2] L.Rosselet et al, Phys.Rev. D 15,574 (1977). [3] S.Pislak et ai, Phys.Lett. 87,221801 (2001). [4] B.Adeva et al, Phys.Lett. B 619, 50 (2005). [5] J.RBatley et al, Eur.Phys.J. C 52, 875 (2007). [6] V.Fanty et al, Nucl.lnstrum.Methods A 574, 433 (2007). [7] J .RBatley et ai, Phys.Lett. B 633, 173 (2006). [8] N.Cabibbo, Phys. Rev. Lett. 93, 121801 (2004). [9] N.Cabibbo and G.Isidori, JHEP 503, 21 (2005). [10] K.Knecht and RUrech, Nucl.Phys. B 519, 329 (1998). [11] Z.K.Silagadze, JEPT Lett 60,689 (1994). [12] S.RGevorkyan, A.V.Tarasov, O.O.Voskresenskaya. Phys.Lett. B 649, 159 (2007). [13] N.Cabibbo and A. Maksymowicz, Phys. Rev. 168, 1926 (1968). [14] G.Amoros and J.Bijnens, J.Phys G 25, 1607 (1999). [15] S.Roy, Phys. Lett. B 36, 353 (1971). [16] B.Ananthanarayan et al, Phys.Rep. 353, 207 (2001). [17] S.Descotes, N.Fuchs, L.Girlanda, J.Stern, Eur.Phys.J. C 24, 469 (2002). [18] J.Gasser, Proc. of KAON 2007 Int. Conf., Frascati, May 21-25, (2007).

RARE KAON AND HYPERON DECAYS IN N A48 EXPERIMENT N.Molokanova U

Joint Institute for Nuclear Research, 141980 Dubna, Russia Abstract. Recent results from the experiments NA48/1 and NA48/2 are reported. The first measurement of direct emission and interference terms in K± --+ 7l'±7l'0')' and the first observation of K± --+ 7l'±e+e-')' are described. Concerning NA48/1 measurements on radiative hyperon decays are presented.

1

Introduction

The series of experiments NA48 have explored many topics in the charged and neutral kaon physics. In this paper we shall discuss some of the most recent measurements produced by two stages of the experimental program: NA48/1 and NA48/2. NA48/1 (2002) has been oriented mainly to the study of rare Ks decays and has produced also results in hyperon physics. NA48/2 (2003-2004) was designed to search for direct CP-violation in K± decays, but also many other results in rare decays have been achieved.

2 The radiative K The decay channel K± ----t 7f±7f0'f' is one of the most interesting and important channels for studying the low energy structure of the QCD. Three components contribute to K± ----t 7f±7f0'f' decay amplitude: the Inner Bremsstrahlung (IB) associated with the decay K± ----t 7f±7fo in which the photon is emitted from the outgoing charged pion, Direct Emission (DE) from the vertex and the interference (INT) between these two. The K± ----t 7f±7f0'f' decays are described in terms of two kinematic variables: the kinetic energy of charged pion in kaon rest frame (T;) and invariant variable W 2 = (PK . P",!)(P7l' . P"'!)/(mKm 1r )2, where PK , P1r , P",! are the 4-momenta of the kaon, charged pion and odd gamma, respectively. About 124.103 events were selected in the range T; < 80 Me V and 0.2 < W < 0.9. In the previous measurements a lower cut T; > 55 MeV was introduced in order to suppress K± ----t 7f±7fo7fo and K± ----t 7f±7fo background. In NA48/2 measurement these backgrounds are avoided by application of a special algorithm, which detects overlapping gamma in the detector and due to the limit ±10 MeV on the deviation ofreconstructed kaon mass from its nominal value. The upper cut on T; rejects K± ----t 7f±7fo decays. The background in the selected sample is kept under 10- 4 . The probability of the photon mistagging (i.e. choice of wrong odd photon) is estimated to be less than 0.1%. ue-mail: [email protected]

285

The preliminary Yalues for the fractions of DE and INT with respect to IB are Frac(DE) = (3.35 ± 0.35 stat ± Q.25syst)% Prac(INT) = (-2.67 ± 0.81 stat ± 0.73^,*)%This is the first measurement of a non vanishing interference term in the K^ —»• 7T 7r0/y decay. 3

First observation of the decay K^ —* ir ± e + e~7

NA48/2 experiment observed for the first time the radiative decay K^ —> 7r ± e + e~7. The signal is selected between 480 MeV/e2 and 505 MeV/c2 in the invariant ir : t e + e"7 mass and requiring the invariant e + e^7 mass to be greater that 260 MeV/e2. Fig.l displays the projections of this region on the corresponding axes. The crosses represent data while the filled distribution represent different simulated background contribution. 120 candidates were selected with 7.3 ± 1.7 estimated background. The main source of BG is the K —> 7r^~ir(L'y with a lost •*¥.

Figure 1: K~^ —• it^ze^~e~~ry decay. The invariant w^e^e y (left) and e"*"e 'y (right) masses with corresponding background distributions. Black crosses represent data distribution.

By using K^ ^ w^w0 as normalization channel the branching ratio was pr€:nnnnl:try est,lmatE:C1 to be Br{K± -* W±TT°'J) = (1.19 ± 0.12stet ± 0.04,v.t) • 10^8. More details on If* —* w±e+e~'y decay analysis could be found in [1], 4

Weak radiative H° decays

Up to this day weak radiative hyperon decays as H° —> A7 and E° —* S7 are still barely understood. Several theoretical models exist, which give

287

very different predictions. An excellent experimental parameter to distinguish between models is the decay asymmetry a. It is defined by HN _ = *0(l4W), where 9 is the direction of the daughter baryon with respect to the polarization of 5° in its rest frame. For example, the decay asymmetry for H° —» A7 can be measured by looking at the angle between the incoming 5 and the outgoing proton from the subsequent A —> pir^ decay in the A rest frame. Using this method, the measurement is independent of the unknown initial H° polarization. The NA48/1 experiment has selected 48314 5° -+ A7 and 13068 H° -* £7 candidates (fig,2). The background contributions are 0.8% for H° —•> A7 and about 3% for S° —•» S7, respectively.

Figure 2: H° —•> A"/ (left) and H° —+ £7 (right) signal together with MC expectations for signals and backgrounds.

Using these data, fits to the decay asymmetries have been performed. In case of H° —* £ 7 , where we have the subsequent decay E° —+ A7, the product cos0s~-*"£>y • cosOs-tAi n a s to be used for the fit. Both fits show the, expected linear behavior on the angular parameters. After correcting for the well-known asymmetry of A —» pn~, values of a H o^A 7 = -0.684 ± 0.0203tet ± 0.061 s v s t and as^s-y = ^0.682 ± 0.031 s t a t ± 0.0653I,st are obtained. These values agree with preYious measurements by NA48 on H° ^+ A7 [2] and KTeV on H° —* £7 [3], but are much more precise. In particular the result on S° —> A7 is of high theoretical interest, as it confirms the large negative value of the decay asymmetry, which is difficult to accommodate for quark and vector meson dominance models.

288

5

First observation of 3° —- - +

la the 2002 run of NA48/1 experiment the weak radiative decay E° ~~> Ae+e~~ was detected for the first time [4]. 412 candidates were selected with 15 C:1CgroulUO events (fig. 3) The obtained brancb:ing fraction Br(E° ~* Ae + e") = (7.7 ± 0.5 stot ± 0Asyst) • HP 6 is consistent with inner bremsstrahlung-like e+e~ production mechanism.

Figure 3: The invariant mass of Ae+e-

together with the simulated background.

The decay parameter ctEAee c a n be measured from the angular distribution dN N ^^^YCl-CHAee^COS^H), (1) where cosffps is the angle between the proton from A —» JM decay relative to the H° line of flight in the A rest frame and a_ is the asymmetry parameter for the decay A —> pw^. The obtained value aSAee = -0.8 ± 0.2 is consistent with the latest published value of the decay asymmetry parameter for 5 —> A-y. References [1] [2] [3] [4]

J.R.Batley et al, CERN-PH-EP/2007-033, accepted by Phys.Lett.B. A.Lai et al, Phys.Lett. B 584, 251 (2004). A.Alavi-Haxati et al, Phys.Rev.Lett. 86 , 3239 (2001). J.R.Batley et al, Phys.Rev. B 650, 1 (2007).

THE K+ -

rr+ vi) EXPERIMENT AT CERN Yu.Potrebenikov a

Laboratory of Particle Physics, Joint Institute for Nuclear Research, 141980 Dubna Moscow region, Russia Abstmct. The P326 proposal of a new experiment NA62 aiming to perform precise measurement of the very rare kaon decay K+ -+ 7I"+vii branching ratio at CERN is described. About 80 K+ -+ 71"+ vii events with 10% of background is planned to obtain in two years of data taking. The status of the project, current status of R&D and future plans of the experiment are discussed.

1

Introduction

The K+ -) 7r+vv decay is a flavor changing neutral current process, computable with very small theoretical uncertainty of about 5% [1]. The hadronic matrix element can be parameterized in terms of the branching ratio of the well measured K+ -) 7r°ev decay [2] using isospin symmetry. The computed value is (8.0 ± 1.1) x 10- 11 , where the error is dominated by the uncertainty in the knowledge of the CKM matrix elements. Such an extreme theoretical clarity, unique in K and B physics, makes this decay (together with KL -) 7r°vv) extremely sensitive to new physics (see for example [3,4]). Only 3 K+ -) 7r+vv events have been observed by BNL-E949 experiment [5], that gives a central value of the branching ratio higher than the SM expectation. But rv 10% accuracy measurement of the branching ratio is required to provide a significative test of new physics contributions. This is the goal of the proposed NA48/3 (or NA62) experiment at CERN-SPS [6]. The aim of the experiment is to collect about 80 K+ -) 7r+vv events with the background level of 10%. 2

Proposal of the future experiment

The NA62 experiment will use kaon decays in-flight technique, based on the NA48 apparatus and the same CERN-SPS beam line which produced the kaon beam for all NA48 experiments. The R&D program for this experiment, started in 2006, is continuing in 2007. The data taking should start in 2011. The layout of the experiment is shown in fig. 1. The goal of the experiment can be reached by having 10% signal acceptance and by using a beam line able to provide the order of 10 13 kaon decays. To study K+ -) 7r+vv decay it is necessary to reconstruct one positive pion track in the downstream detector. If a beam and a pion tracking detectors provide a precise reconstruction of the decay kinematics, the missing mass allows a kinematical separation between the signal and more than 90% of the total background (fig.2); only non-gaussian tails from K+ -) 7r+7r0 and K+ -) p,+vJ.L ae-mail: [email protected]

289

290 m

\(,\'IU)

ＬＭｾＧ＠

ＭＬｹｾｭ＠

ｭｕ＼ｉｾ｟＠

VACUUM

ｾ＠

ＭＧｉｾ

---0-

1(+-75GoV

iL-_ _ _ _---,

-1

SI'IIlES.l

S.\C

-\

ｾ［Ｚｉ｡ｴｭ＠

r:,

',I: "

: :: :: : IStraw "

-2

,\,>

•.

--LKr

" .., Tube

Figure 1: Layout of the experiment

in the squared missing mass resolution will present in the defined signal region. But the kinematics only cannot provide background rejection factor of 10 13 . So, different veto (photon and muon) and particle identification (CEDAR and RICH) systems are included into experimental set-up to fulfill these needs. Moreover, the detector can provide redundancy both for kinematics reconstruction and particle identification allowing to estimate background directly from the data.

ＭｏＮｉＧｓＡＺ＼｟ｦＬｾＢｮｯｫｳ［ｆ］Ｄｃｌｩ＠

ＢＧＭＮｾｏＱＵ＠ ｾｓＡｩ＠

ＮｏｾＧＵＢＭｩＷＬ］［ｲＰｰｊｌＺｬＡ＠ GeV2/c

4

ＭｌｾｏＢＱＵ＠ ｭｾｬｳＤ＠

GeV 2/C 4

Figure 2: Squared missing mass for krum decays

2.1

The beam line

A 400 GeV Ic proton beam from the SPS, impinging a Be target, produces a secondary charged beam. 100 m long beam line selects 75 Ge V I c momentum

291

beam with 1.1 % RMS momentum bite and an average rate of about 800 MHz integrated over an area of 14 cm 2 • The beam contains 6% of K+. The average rate seen by the downstream detectors integrated on their surface is rv 11 MHz. The described beam line provides 5 x 10 12 K+ decays, assuming 60 m decay region and 100 days of run at efficiency of 60%, which is a very realistic estimation based on the decennial NA48 experience at the SPS.

2.2

The experimental set-up and RfjD current status

The experimental set-up consists of: Beam (Gigatracker) and pion (magnetic) spectrometers. The first one consists of three silicon pixel stations across the second achromat of the beam line, produced by 300 x 300 /-lm2 pixel each. The time resolution of 200 ps is provided by 0.13 microns technology of silicon detector production. The magnetic spectrometer is designed with 6 (or 4) straw chambers with 4 coordinate views each. Chambers should work in vacuum, introduce small material contribution (0.5% Xo per chamber) and have a good spatial resolution (130 microns per view). 36 /-lm mylar straw tubes with about 10 mm in diameter welded by ultrasound machine and cowered with gold inside will be used for these reasons. This spectrometer will be used as a veto as well for high energy negative pion from Ke4 decays. The R&D program has been started in 2006, a full length and reduced-size prototype has been constructed, integrated and tested in the NA62 set-up during the 2007 run at CERN. Differential Cherenkov counter CEDAR and RICH. CEDAR [7], differential Cerenkov counter existing at CERN, will be used after its upgrade for new experimental conditions for kaon tagging to keep the beam background under control. The 18 m long RICH located after magnetic spectrometer and filled with Ne at atmospheric pressure aimed for particle identification and pion momentum measurement. It will contain about 2000 PMTs in the focal plane and has to reach a time resolution of 100 ps to provide time information for downstream tracks. A full-length prototype 60 cm in diameter and 96 PMTs has been integrated in the NA62 set-up and tested during the 2007 NA62 run at CERN. Large angle (for 10-50 mrad), medium angle (for 1-10 mrad) and small angle (for

0.28

ｾ＠

0.26

1 2

3 4 5

Q)

LO

0.24

6

"

t-

0.22 0.2 0.18 0.16

+

""

Cl..

V

0.14

Ｏｾ＠

.....................................•..........

0.12 0.1

o

20

40

60 Shh 1/2,

80 [GeV]

100

120

140

Figure 2: Average square for the transverse momentum of K+-meson produced from the interaction of two hadrons one of them is in the equilibrated fireball as a function of "fShh at T = 150 MeV.

< ｐｾＫ＠

,t ＾ｾＬ＠ respectively, when IC > > Iq , whereas the curves 3 and 4 correspond to the same quantities with IC = 3 1q . The line 5 in Fig.2 corresponds to the average square for the transverse momentum of K+ produced in the free p + p collisions < p; ＾ｾｲ］＠ 0.14 GeV/c 2 . As our calculations show, the temperature dependence for < ｐｾＫＬｴ＠ ＾ｾｴ＠ is rather weak in the interval T = 100 - 150 MeV.

As is evident from Fig.2, the obtained results are sensitive to the mass value of a hadron which is locally equilibrated with the surrounding nuclear matter at ..jShh :::; 10(GeV). We found that the quark distribution in a hadron depends on the fireball temperature T. At any T the average transverse momentum squared of a quark grows and then saturates when ..jShh increases. Numerically this saturation property depends on T. It leads to a similar energy dependence for the averThe saturation age transverse momentum squared of hadron hI < pt,t ＾ｾｴＮ＠ property for < ｰｾ＠ 1, t ＾ｾｴ＠ depends also on the temperature T and it is very sensitive to the dynamics of hadronization. As an example, we studied the energy dependence of the inverse slope of transverse mass spectrum of K-mesons produced in central heavy-ion collisions and got its energy dependence qualitatively similar observed to one experimentally. We guess that our assumption on the thermodynamical equilibrium of hadrons given by eq.(l) can be applied for heavy nuclei only and not for the early interaction stage.

317

AcknowledgIllents

The authors are grateful for very useful discussions with P.Braun-Munzinger, K.A.Bugaev, W.Cassing, A.V.Efremov, M.Gazdzicki, S.B.Gerasimov, M.I.Gorenstein, Yu.B.Ivanov, A.B.Kaidalov and O.V.Teryaev. This work was supported in part by RFBR Grant N 05-02-17695 and by the special program of the Ministry of Education and Science of the Russian Federation (grant RNP.2.1.1.5409). References

[1) L.Ahle et. at., E866 and E917 Collaboration, Phys. Let. B476, 1 (2000); ibid. B490, 53 (2000). [2) S.V.Afanasiev et at. (NA49 Collab.), Phys.Rev. C66, 054902(2002); C. Alt et al., J. Phys. G30, S119 (2004); M.Gazdzicki, et at., J. Phys. G30, S701 (2004). [3) C.Adleret at., STAR Collaboration, nucl-ex/0206008; O.Barannikova et at., Nucl. Phys. A715, 458 (2003); K.Filimonov et at., hep-ex/0306056; D.Ouerdane et at, BRAHMS Collaboration, Nucl. Phys. A715,478 (2003); J.H.Lee et at., J. Phys. G30, S85 (2004); S.S.Adler et at., PHENIX Collaboration, nucl-ex/030701O; nuclex/0307022. [4) E.V.Shuryak, Phys. Rep. 61, 71 (1980). [5) E.V.Shuryak and O.Zhirov, Phys. Lett. B89, 253 (1980); Yad. Fiz. 28, 485 (1978) [Sov. J. Nucl. Phys. 28,247 (1978). [6) L. van Hove, Phys. Lett. B118, 138 (1982). [7) M.Gorenstein, M.Gazdzicki and K.Bugaev, Phys. Lett. B567, 175 (2003). [8) B.Mohanty, et at., Phys. Rev. C68, 021901 (2003). [9) M.Gazdzicki et at., Braz. J. Phys. 34, 322 (2004). [10) J.Kuti and V.F.Weiskopf, Phys. Rev. D4, 3418 (1971). [11) G.I.Lykasov, A.N.Sissakian, A.S.Sorin, D.V.Toneev, in preparation. [12) A.Capella, V.J.Tran Than Van, Z.Phys.ClO, 249 (1981). [13) O.Benhar, S.Fantoni, G.I.Lykasov, N.V.Slavin, Phys. Rev. C55, 244 (1997). [14) G.'t Hooft, Nucl. Phys., B72, 461 (1974). [15) G.Veneziano, Phys. Lett., B52, 220 (1974). [16) A.B.Kaidalov and K.A.Ter-Martirosyan, Phys. Lett. B117, 247 (1982). [17) A.B.Kaidalov and O.I.Piskunova, Z. Phys. C30, 145 (1986). [18) A.Capella, U.Sukhatme, C.L.Tan, J. Tran Thanh Van, Phys.Rep. 236, 225 (1994). [19) G.I.Lykasov and M.N.Sergeenko, Z. Phys. C70, 455 (1996).

STRINGY PHENOMENA IN YANG-MILLS PLASMA V.1. Zakharov a

INFN, Sezione di Pisa, Largo PontecoTVo 3, 56127, Pis a, Italy ITEP, B.Cheremushkinskaya 25, Moscow, 117218, Russia Abstract. We review the grounds for and consequences from the hypothesis that at the point of the confinement-deconfinement phase transition both electric and magnetic strings are released into the Yang-Mills plasma. We comment also briefly on the averaged Polyakov line as an order parameter of the deconfinement phase transition.

1

Introduction

The goal of this talk is to substantiate a phenomenological stringy picture for the confinement-deconfinement phase transition. The stringy picture for the phase transition was advocated first long time ago [1] and the topic is, in its generality, too broad for such a talk. Thus, we will concentrate on a recent proposal [2,3] that there exists a magnetic component of the Yang-Mills plasma at temperatures close and above the critical temperature Te. While the main ideas are presented in the original papers [2], there appeared most recently results of dedicated lattice measurements [4,5] which support the picture proposed although much more remains to be done before one could really claim observation of the magnetic component of the plasma. Electric strings

2 2.1

Action vs entropy factoTs

Consider quark and anti-quark separated by distance x. To make the construct gauge invariant one has to connect the quarks by a string: (1)

The path-ordered exponent is our first image of what we would call electric string. If quarks develop in time, the string sweeps an area A. Let us consider the most primitive dynamics of a closed string. The string carries color charge and, therefore has a divergent self-energy. To regularize this divergence, introduce finite thickness of the string, TO , TO « Ixl. Then the corresponding action is of order bare Sstring

=

CIg 2(TO )Aj TO2

=

O"bare'

A

.

(2)

To evaluate the renormalized, or physical string tension O"ren one has to subtract from (2) the entropy factor (see, e.g., [6]): ae-mail: [email protected]

318

319

where Nstring is the number of various surfaces with the same area constant C2 is of pure geometrical origin. As a result b:

A,

(3) the

(4) Consider first ro = a, where a is the lattice spacing. In the limit of the large coupling, g2(a) » 1, the bare action factor prevails and the renormalized tension is positive. We have the strong-coupling confinement. This string is infinitely thin but theory is not realistic because of the strong-coupling limit. In the asymptotic-freedom case, g2(a) -+ 0 the tension (4) is negative and the string is unstable in the ultraviolet. In the ultraviolet, on the other hand, free gluons is the right approximation and strings with a negative tension is not a viable alternative. Next, we can still consider the asymptotic-freedom case but choose the thickThen g2(ro) can be large enough to make the ness of the string ro rv ａｑｾｄＧ＠ renormalized tension (4) positive. Thus, we might have 'thick' strings which could be useful effective degrees of freedom in the infrared. At large temperatures g2 is limited by g2 (T), limT -+00 g2 (T) -+ 0 since the time extension of the lattice (Euclidean space-time) is liT. Thus, at an intermediate temperature the effective tension (4) vanishes and the electric strings percolate through the vacuum. 2.2

The Polyakov line

Continuing with the finite-temperature physics, another image for the electric strings is provided by the the Polyakov line which is a Wilson line winding once through the lattice in the periodic time direction:

r

llT

P == Trn

=

TrPexp}o

Ao(x,r)dr,

(5)

where the trace is taken in the fundamental representation. Imagine that we would like to use the Hamiltonian formalism and gauge Ao = O. Unlike the case of T = 0 it is not possible to fix Ao = 0 because of the periodicity in the time direction. In other words, the non-local variable (5) is gauge invariant and cannot be eliminated by gauge transformations. It is still possible to put Ao = 0 provided that the non-local degree of freedom (5) is added explicitly [1] into the partition function:

Z[n] = b Actually,

J

DAn(x,r)exp ( -

J､ＳｸｲＨ｡ｾｫＩＲＫｆｦｬ＠

we oversimplify the estimate of the entropy greatly, see, in particular, [7].

(6)

320 where Ak(x, (3) = n- 1Adx, O)n + n-10kn(X). Note, however, that by introducing a new variable we admit extra ultraviolet divergences into the theory and make the model non-renormalizable, for further references see, e.g., [8]. 3

3.1

Magnetic strings

Topology of the magnetic strings

In Yang-Mills theories, one expects that the magnetic strings are no less fundamental than electric strings. Moreover, condensation of magnetic degrees of freedom is commonly believed to be responsible for the confinement. The scenario is realize in the Abelian case, [9]. The magnetic degrees of freedom are identified in this case through violations of the Bianchi identities:

(7) where j::,on is the monopole current. In the non-Abelian case, the gauge potential can be expressed in terms of the field strength tensor [10]:

(8) where ｇｾｬ＠ is the matrix inverse to the matrix of the field strength tensor. As far as (8) holds, the Bianchi identities are valid automatically. There might exist, however, such field configurations that the inversion (8) is not possible because the matrix G- 1 does not exist. Actually, it was noted from the very beginning [10] that the inversion (8) fails in 2d case, see also below. Alternatively, in 4d case there can exist 2d defects [11] along which the matrix G- 1 is singular. These 2d defects is our image for the magnetic strings. Magnetic strings are to be added as new degrees of freedom to the standard YM theories which assume Bianchi identities valid. 3.2

Surface operators, monopoles

The action associated with the 2d defects can be readily guessed on symmetry grounds. In fact, such surfaces were considered, for other reasons, in Refs. [12,13] (in the latter reference they were labeled as surface operators). Namely, consider a surface, with area element da 1"'-' and introduce the action: S s'ur face

= const

J ｊＱＮｶｇｾ＠ da

(no summation over

j-l,

v) .

(9)

The central point is that the action (9) respects non-Abelian invariance despite of the fact that it apparently carries a color index a. The reason is that one can

321

use gauge invariance to rotate one particular component of the field strength tensor to the Cartan subgroup:

(10) where for simplicity we considered the gauge group SU(2). In other words, non-Abelian fields living on a surface are in fact Abelian. The inversion (8), on the other hand, is specific for the non-Abelian case and fails in the Abelian case. Thus, the magnetic strings replace the magnetic monopoles (7) relevant to the Abelian case. It is worth emphasizing that to carry a finite magnetic flux the surfaces (9) are to be endowed with singular fields ｇｾｶＮ＠ Note that gauge fixation (10) fails if ｇｾｶ＠ = o. Such defects are trajectories living on the surfaces and correspond to non-Abelian monopoles, for related discussion see [14].

3.3

Dual pictures of confinement

There is a deep relation between magnetic and electric strings. Namely, the expectation value of the Wilson line, < W > can be evaluated either in terms of electric strings open on the heavy quarks, or in terms of the linking number between electric and magnetic strings, [15,16]. It is useful to start with 3d and consider a tube of magnetic field which pierces a surface spanned on the Wilson line. Considering, for simplicity, the Abelian case one gets for the Wilson line:

exp(ifAJ.!dxJ.!)

=

exp(i ｾ＠

exp (- Aminacon/) ,

(12)

where Amin is the minimal area spanned on the Wilson line. One can say that (12) means that the expectation value of the Wilson line is suppressed in the strongest possible way. To get such a suppression the magnetic flux carried by the magnetic strings is to be random [17]. This, in turn, implies that the magnetic strings percolate through the vacuum. For this to happen, the magnetic strings are to have a vanishing tension, (13) amagn = O. Eq. (13) means that the heavy-monopole potential is not confining, as it should be.

322 It is remarkable that we derived (13) starting with consideration of the Wilson line, not directly of the 't Hooft line. 4

Extra dimensions

The logic outlined in the preceding sections has a weak point since we mix up two different pictures for the confining string, that is, thin and thick strings. Indeed, the string which can be open on quarks is to be infinitely thin since quarks are point-like while the string which has tension in physical units, see Eq (4) has thickness of order ａｑｾｄＮ＠ This inconsistency is in fact difficult to remove and the way out which is becoming common nowadays introduces a novel notion of extra dimensions, or running string tension, for review see [18]. Roughly speaking, one is assuming the string to be infinitely thin but with tension depending on its size. For areas A :S ａｑｾｄＧ＠ (Jeff

rv

I/A

(14)

For larger areas, the string tension is frozen, (Jeff

ｾ＠

(Jeanf

,

if A :::: ａｑｾｄ＠

,

(15)

where (Jeanf determines heavy-quark linear potential at large distances. Formulae (14), (15) are somewhat loose because area is not the only characteristic of a surface. It turns out that language of extra dimensions is much more adequate. In this framework, one postulates that there exists an extra dimension, z such that 'our' world corresponds to z = 0 while strings connecting quarks extend into z =I o. The action associated with the string is the same Nambu-Goto action which we actually discussed above but now the area is calculated with account of geometry which is a nontrivial function of z. In particular, assuming that the metric is (16) where R2 is a constant and Xi ,i = 1, .. ,4, are Euclidean 4d coordinates, one reproduces Coulomb-like heavy-quark potential. This is quite obvious from dimensional considerations and the metric (16) realizes the assumption (14). Concerning realization of (15) it is much more arbitrary since one introduces by hand a new parameter, AQCD . The following model

R2 2 ds = exp(cz 2 ) 2" (Jdt 2 +dx; +r1dz2), f(z) = 1-(nzTe)4, c ｾ＠ z

GeV 2,

(17) gives a reasonable description of broad variety of phenomena both at zero and finite temperatures [19].

323

As for the magnetic strings, they have another geometry and correspond to branes wrapped on extra compact dimensions, which are to be added to the five dimensions already introduced, for details see [18]. Magnetic monopoles, in this language, are Kaluza-Klein modes associated with the extra compact dimensions [11].

5

Stringy phenomena near the critical temperature

After all these preliminary remarks we are in position to make predictions specific for the string-based phenomenology of Yang-Mills theories.

5.1

Polyakov line as an order parameter

As argued first in Ref. [1], in pure Yang-Mills case (without quarks) the expectation value of the Polyakov line (5) serves as an order parameter:

(P) == 0 if T < Te .

(18)

In the explicit calculations [19] with the metric (15) the averaged Polyakov line exp( -constjT) contains at small temperatures exponentially small terms < P ＾ｾ＠ and Eq. (18) does not hold. Although non-observance of (18) could well be a consequence of the approximations made, it might be useful to understand the reasons for this discrepancy. The proof of (18) exploits the center symmetry. Namely, the Polyakov line is changed by a phase factor under transformations belonging to the group center of the gauge group while the lattice Yang-Mills action can be formulated as symmetrical under the center transformations. However, the lattice action might not know about the center symmetry as well, (for recent discussion and references see [20]). There is no center-group symmetry in the stringy approach, based on (15) but probably there is nothing wrong about this. Thus, violations of (18) seemingly cannot be ruled out on general grounds. There are further interesting issues to discuss in this connection. In particular, the stringy formulation (15) leads to qualitative predictions which are in accord with the lattice data [22], like fast growing entropy in the system of heavy quarks towards T = Te. On pure theoretical side, dependence of continuum physics on details of the lattice regularization (whether we have the center symmetry or not) is most challenging. Because of space considerations, we cannot go into detailed discussion of these issues here, however.

5.2

Magnetic component of the Yang-Mills plasma

We have already mentioned that at the point of the phase transition, Te one expects [1] vanishing tension of the electric string:

(Jeleetr(T) 2: 0,

T 2: Te

(19)

324 On the other hand, tension of the confining string can be evaluated in terms of the magnetic strings, see subsection 3.3. Thus, Eq. (19) implies that magnetic strings acquire non-zero tension at T > Te:

(J"magn(T) 2: 0, T 2: Te .

(20)

Thus, in the deconfining phase the magnetic strings correspond to physical degrees of freedom and are to be present in the Yang-Mills plasma [2]. The question is, how to detect this effect. On the lattice, magnetic strings are identifiable directly, for details see [11]. And, indeed, the magnetic strings do not percolate at T > Te, for references see [15]. More quantitative predictions can be made in terms of the monopoles, which are, as explained above, particles living on the strings. The word 'particles' is to be perceived with some caution, however, since we are discussing now the lattice, or Euclidean formulation and the difference between virtual and real particles is not so obvious as in the Minkowski space. Nevertheless, one can argue [2] that the density of real (in the Minkowskian sense) particles is proportional to the density of the so called wrapped trajectories [21] which are trajectories stretching in the time direction from one boundary to the other:

Preal(T) '" Pwrapped(T) , T

> Te.

(21)

This relation implies, in turn, that the density pwrapped is to be in physical and cannot depend on the lattice spacing. This is in fact a very units, '" ｩ｜ｾｃｄ＠ strong constraint on the data. Which indeed turns to be true [5].

5.3

Ghost-like matter

Measurements on the magnetic strings, reveal [4] astonishingly enough, that both energy density and pressure associated with the magnetic strings are negative: (22) tmagn(T) < 0, Pmagn(T) < 0, Te < T < 2Te . There is a proposal [23] how to accommodate this observation within the stringy picture. The basic idea is that in 2d and 4d the conformal anomaly has opposite signs and this is responsible for the ghost-like sign in case of the 2d defects (22). Acknowledgments

I am indebted to O. Andreev, M.N. Chernodub, A. Di Giacomo, M. D'Elia, A.S. Gorsky for enlightening discussions. References

[1] A. M. Polyakov, Phys. Lett. B72, 477 (1978); "Confinement and liberat'ion", [arXiv:hep-th/0407209].

325 [2] M.N. Chernodub and V.1. Zakharov, Phys. Rev.Lett. 98, 082002 (2007); "Magnetic strings as part of Yang-Mills plasma ", [arXiv:hepphj0702245]. [3] Ch. P. Korthals Altes, "Quasi-particle model in hot QCD", [arXiv:hepphj0406138]; Jinfeng Liao and E. Shuryak, Phys. Rev. C75, 054907 (2007), [arXiv:hep-phj0611131]. [4] M.N. Chernodub et al., "Topological defects and equation of state of gluon plasma", [arXiv:0710.2547]. [5] A. D'Alessandro and M. D'Elia, "Magnetic monopoles in the high temperature phase of Yang-Mills theories", [arXiv:0711.1266]. [6] A.M. Polyakov, "Gauge Fields and Strings", Harvard Academic Publishers, (1987) . [7J A.B. Zamolodchikov, Phys. Lett. B117, 87 (1982). [8] J. C. Myers and M.C. Ogilvie, "New phases of finite temperature gauge theory from an extended action", [arXiv:0710.0674J; Ph. de Forcrand, A.Kurkela, A. Vuorinen, "Center-Symmetric Effective Theory for HighTemperature SU(2) Yang-Mills Theory" [arXiv:0801.1566]. [9] A.M. Polyakov, Phys. Lett. B59, 82 (1975); M. E. Peskin, Annals Phys. 113, 122 (1978). [lOJ M.B. Halpern, Phys. Rev. D16, 1798 (1977); ibid D19, 517 (1979). [l1J V.1. Zakharov, Braz. J. Phys. 37, 165 (2007), [arXiv:hep-phj0612342J. [12J M.N. Chernodub, F.V. Gubarev, M.1. Polikarpov, V. I. Zakharov, Nucl.Phys. B600, 163 (2001), [arXiv:hep-thjOOl0265]. [13J S. Gukov and E. Witten, "Gauge Theory, Ramification, And The Geometric Langlands Program", [arXiv:hep-thj0612073J. [14] G. 't Hooft, Nucl. Phys. B190 , 455 (1981). [15J J. Greensite, Prog. Part. Nucl. Phys. 51, 1 (2003), [arXiv:heplatj0301023]. [16J V.1. Zakharov, AlP Conf. Proc. 756, 182 (2005), [arXiv:hepphj0501011]. [17] A. Di Giacomo, H. G. Dosch, V.1. Shevchenko, Yu.A. Simonov, Phys. Rept. 372, 319 (2002), [arXiv:hep-phj0007223]. [18J O. Aharony et al., Phys. Rept. 323,1832000, [arXiv:hep-thj9905111J. [19J O. Andreev, V.1. Zakharov, Phys. Rev, D74, 025023 (2006),[arXiv:hepphj0604204]; Phys. Lett. B645, 437 (2007), [arXiv:hep-phj0607026]; JHEP, 0704:100 (2007), [arXiv:hep-phj0611304]. [20J G. Burgio, PoS(LAT2007), 292 (2007), [arXiv:0710.0476J. [21] V.G. Bornyakov, V.K. Mitrjushkin, M. Muller-Preussker , Phys. Lett. B284, 99 (1992). [22J P. Petreczky, Nucl. Phys. A 785, 10 (2007), [arXiv:hep-Iatj0609040]. [23J A. Gorsky, V. Zakharov, "Magnetic strings in Lattice QCD as Nonabelian Vortices", [arXiv:0707.1284J.

LATTICE RESULTS ON GLUON AND GHOST PROPAGATORS IN LANDAU GAUGE I.L. Bogolubsky Joint Institute for Nuclear Research, 141980 Dubna, Russia

V.G. Bornyakov a Institute for High Energy Physics, 142281 Protvino, Russia and Institute of Theoretical and Experimental Physics, Moscow, Russia

G. Burgio Universitiit Tilbingen, Institut filr Theoretische Physik, 72076 Tilbingen, Germany

E.-M. Ilgenfritz, M. Miiller-Preussker Humboldt-Universitiit zu Berlin, Institut filr Physik, 12489 Berlin, Germany

V.K. Mitrjushkin Joint Institute for Nuclear Research, 141980 Dubna, Russia and Institute of Theoretical and Experimental Physics, Moscow, Russia Abstract. We present clear evidence of strong effects of Gribov copies in Landau gauge gluon and ghost propagators computed on the lattice at small momenta by employing a new approach to Landau gauge fixing and a more effective numerical algorithm. It is further shown that the new approach substantially decreases notorious finite-volume effects.

1

Introduction

The gauge-variant Green functions, in particular for the covariant Landau gauge, are important for various reasons. Their infrared asymptotics is crucial for gluon and quark confinement according to scenarios invented by Gribov [1] and Zwanziger [2] and by Kugo and Ojima [3]. They have proposed that the Landau gauge ghost propagator is infrared diverging while the gluon propagator is infrared vanishing. The interest in these propagators was stimulated in part by the progress achieved in solving Dyson-Schwinger equations (DSE) for these propagators (for a recent review see [4]). Recently it has been argued that a unique and exact power-like infrared asymptotic behavior of all Green functions can be derived without truncating the hierarchy of DSE [5]. This solution agrees completely with the scenarios of confinement mentioned above. The lattice approach is another powerful tool to compute these propagators in an ab initio fashion but not free of lattice artefacts. So far, there is no consensus between DSE and lattice results. For the gluon propagator, the ultimate decrease towards vanishing momentum has not yet been established in lattice computations. Lattice results for the ghost propagator qualitatively agree with the predicted diverging behavior but show a substantially smaller infrared exponent [6]. The lattice approach has its own limitations. The effects of the finite volume might be strong at the lowest lattice momenta. Moreover, gauge fixing is ae-mail: [email protected]

326

327 not unique resulting in the so-called Gribov problem. Previously it has been concluded that the gluon propagator does not show effects of Gribov copies beyond statistical noise, while the ghost propagator has been found to deviate by up to 10% depending on the quality of gauge fixing [7,8]. Recently anew, extended approach to Landau gauge fixing has been proposed [9]. In this contribution we present results obtained within this new method and using a more effective numerical algorithm for lattice gauge fixing, the simulated annealing (SA) algorithm. Results for the gluon propagator have been already discussed in [10], while results for the ghost propagator are presented here for the first time. 2

Computational details

Our computations have been performed for one lattice spacing corresponding to rather strong bare coupling, at f3 == 4/ g6 = 2.20, on lattices from 84 up to 324. The corresponding lattice scale a is fixed adopting ..j(ia = 0.469 [ll] with the string tension put equal to a = (440 Me V)2. Thus, our largest lattice size 32 4 corresponds to a volume (6.7 fm)4. In order to fix the Landau gauge for each lattice gauge field {U} generated by means of a Me procedure, the gauge functional

(1) is iteratively maximized with respect to a gauge transformation g(x) which is usually taken as a periodic field. In SU(N) gluodynamics the lattice action and the path integral measure are invariant under extended gauge transformations which are periodic modulo Z(N),

g(x + Lv) = z"g(x),

z"

E Z(N)

(2)

in all four directions. Any such gauge transformation is equivalent to a combination of a periodic gauge transformation and a flip Ux " ----* z" Ux " for a 3D hyperplane with fixed X". With respect to the flip transformation all gauge copies of one given field configuration can be split into N 4 flip sectors. The traditional gauge fixing procedure considers one flip sector as a separate gauge orbit. The new approach suggested in [9] combines all N 4 sectors into one gauge orbit. Note, that this approach is not applicable in a gauge theory with fundamental matter fields because the action is not invariant under transformation (2), while in the deconfinement phase of SU(N) pure gluodynamics it should be modified: only flips in space directions are left in the gauge orbit. In practice, few Gribov copies are generated for each sector and the best one over all sectors is chosen by employing an optimized simulating annealing algorithm in combination with finalizing overrelaxation.

328 3

Results

Thus, we are looking for the gauge copy with the highest value of the gauge functional among gauge copies belonging to the enlarged gauge orbit as defined above. It is immediately clear that this procedure allows to find higher local maxima of the gauge functional (1) than the traditional ('old') gauge fixing procedures employing purely periodic gauge transformations and the standard overrelaxation algorithm. Obviously the two prescriptions to fix the Landau gauge, the traditional one and the extended one, are not equivalent. Indeed, for some modest lattice volumes and for the lowest momenta it has been shown in Ref. [9] that they give rise to different results for the gluon as well as the ghost propagators. Comparing results for different lattice sizes we found that the results seem to converge to each other in the large volume limit. It is important that results obtained with the new prescription converge towards the infinite volume limit much faster. In Fig. 1 the gluon propagator D(p2) is Ｙｾ＠

••

......

8 7

N

>6 o --5 Q)

L = 1.7 fm 0 L = 2.5 fm L = 3.4 fm v L = 5.0 fm '" eL = 6.7 fm 0 " hyperon excitations. Shown are the dynamical quark masses JLi. the confinement energies Eo and the hyperon masses M (all in units of MeV).

the P-wave baryons can be obtained with a spin independent energy eigenvalues corresponding to the confinement plus Coulomb potentials. Moreover this comparative study gives a better insight into the quark model results where the constituent masses encode the QCD dynamics.

Acknowledgment This work was supported by RFBR grants 05-02-17869 and 06-02-17120. References [1] T. Burch et al., Phys, Rev. D 74, 014504 (2006) [2] H. G. Dosch, Phys. Lett. 190, 177 (1987), H. G. Dosch, Yu. A. Simonov, Phys. Lett. 202, 339 (1988) [3] 1. M. Narodetskii and M. A. Trusov, Phys. Atom. Nucl. 67, 762 (2004); Yad. Fiz. 67, 783 (2004) [4] A. Yu. Dubin, A. B. Kaidalov, and Yu. A. Simonov, Phys. Lett. B323, 41 (1994) ; B343, 310 (1995) [5] O. N. Driga, 1. M. Narodetskii, A. 1. Veselov, arXiv: hep-ph/0712.1479 [6] Yu. A. Simonov, Phys. Lett. B 515, 137 (2001) ; A. DiGiacomo and Yu. A. Simonov, Phys. Lett. B 595, 368 (2001)

THEORY OF QUARK-GLUON PLASMA AND PHASE TRANSITION E.V.Komarov a, Yu.A.Simonov b ITEP, Moscow Abstract.Nonperturbative picture of strong interacting quark-gluon plasma is given based on the systematic Field Correlator Method. Equation of state, phase transition in density-temperature plane is derived and compared to lattice data as well as subsequent thermodynamical quantities of QGP.

1

Introduction

The perturbative exploring of quark-gluon plasma (QGP) has some difficulties in describing the physics of QGP and phase transitions. However, it was realized 30 years ago that nonperturbative (np) vacuum fields are strong ( [1]) and later it was predicted ( [2]) and confirmed on the lattice ( [3]) that the magnetic part of gluon condensate does not decrease at T > Te and even grows as T4 at large T. Therefore it is natural to apply the np approach, the Field Correlator Method (FCM) ( [4]) to the problem of QGP and phase transitions, which was done in a series of papers ( [5]- [9]). As a result one obtains np equation of state (EoS) of QGP and the full picture of phase transition, including an unbiased prediction for the critical temperature Te(P,) for different number of flavors nt.

2

Nonperturbative EoS of QGP

We split the gluonic field AIL into the background field BIL and the (valence gluon) quantum field aIL: AIL = BIL + aIL both satisfying the periodic boundary conditions. The partition function averaged both in perturbative and np fields is

Z(V, T) = (Z(B

+ a))B,a

(1)

Exploring free energy F(T,p,) = -Tln(Z(B))B that contains perturbative and np interactions of quarks and gluons (which also includes creation and dissociation of bound states) we follow so-called Single Line Approximation (SLA). Namely, we assume that quark-gluon system for T > Te stays gauge invariant, as it was for T < Te , and neglect all perturbative interactions in the first approximation. Nevertheless in SLA already exist a strong interaction of gluons (and quarks) with np vacuum fields. This interaction consists of colorelectric (CE) and colormagnetic (CM) parts. The CE part in deconfinement phase creates np self-energy contribution for every quark and gluon embedded ae-mail: [email protected] be-mail: [email protected]

334

335

in corresponding Polyakov line. An important point is that Polyakov line is computed from the gauge invariant qij (gg) Wilson loop, which for np Df interaction splits into individual quark (gluon) contributions. As for CM part - its consideration is beyond the SLA, because as has been recently shown in paper ( [10]) strong CM fields are responsible for creation of bound states of white combinations of quarks and gluons. To proceed with FCM we apply the nonabelian Stokes theorem and the Gaussian approximation to compute the Polyakov line in terms of np field correlators Lfund = tr Pexp (i9 B 4 (z)dz 4 ) = c

It

J

J tr c

with

exp ( Ｍｾ＠

ISnISn daJ.L" (u)dO)'o.a (v)DJ.L"')..cr ) (2)

DJ.L"')..cr == g2 (FJ.L" (u)(u, v)F>.a(v)(V, u)) Df and DE arise from CE field strengths:

[E +DIE +U420D f] ｯｵｾ＠ +UiUk oDf oiP

1 Nc DOi,Ok = 6ik D

(3)

As a result the Polyakov loop can be expressed in terms of "potentials" VI and VD VI(T)+2VD) 9/4 (4) Lfund = exp ( 2T ,Ladj = (Lfund) , with Vl(T) == VI(oo,T), VD == VD(r*,T) ([5])

1 21

00

VI(r, T)

=

dv(l - vT)

00

VD(r, T)

=

dv(l - vT)

1T ､ｾ＠

lT ､ｾ＠

ｾｄｦＨ＠ (r -

ｊｾＲ＠ ｾＩｄｅＨ＠

+ v 2)

(5)

Je + v2 )

(6)

In what follows we use the Polyakov line fit ( [8,9]) Lfund

(x = ｾＬ＠

T)

= exp (- ＨＱＮｾＺＵＲｔ＠

(7)

)

The free energy F(T) of quarks and gluons in SLA can be expressed as a sum over all Matsubara winding numbers n with coefficients Ljund and ｌｾ､ｪ＠ for quarks and gluons respectively. For nonzero chemical potential one can keep L fund,adj independent of 1-", treating np fields as strong and unchanged by in the first approximation. The final formulas for pressure of qgp are ( [7,9])

I-"

I-"

p pq == ｾＴ＠

SLA

= nＱｦｾ＠

[(I-"-Yl.) " T + " (I-"+Yl.)] -T

(8)

336 12

l,...,

DB

ｄｾ＠

2 D.4

02

0 Ｑｾ＠ ｏｾ＠

2.5

100

3..5

°

=

300

400

500

600

Figure 2: Analytic (8), (10) and lattice ( [11]) curves for pressure of QGP with nf = 0,2+ 1,3 from ([9]).

Figure 1: Fit (7) of Polyakov line for nf = and nf = 2)(black curves) to the lattice data ( [11]).

where v

200

mq IT and

1

00

m1 + X at high energies can be described in terms of the corresponding kinematic characteristics of the constituent subprocess written in the symbolic form (x1Mt) + (X2M2) -> m1 + (x1M1 + X2M2 + m2) satisfying the condition

(1) The equation is the expression of locality of hadron interaction at a constituent level. Here Xl and X2 are fractions of the incoming momenta P1 and P2 of the colliding objects with the masses M1 and M 2. They determine the minimum energy, which is necessary for production of the secondary particle with the mass m1 and the four-momentum p. The parameter m2 is introduced to satisfy the internal additive conservation laws (for baryon number, isospin, strangeness, and so on). The quantity n is introduced to connect kinematic (X1,2) and structural (8 1,2) characteristics of the interaction. It is chosen in the form

(2) where m is the mass of nucleon and 81 and 82 are factors relating to the fractal dimensions of the colliding objects. The fractions Xl and X2 are determined to maximize the value of n(X1,X2), simultaneously fulfilling the condition (1)

(3) The fractions X1,2 cover the full phase space accessible at any energy. According to the self-similarity principle the scaling function '¢ (z) is constructed as the function depending on the single dimensionless variable z expressed via dimensionless combinations of Lorentz invariants. It is written in the form 7rS -1 d3 (T (4) '¢ (z) = - (dN/dTJ )(T,n . J E dP3 Here, Ed3 (T/dp 3 is the invariant cross section, s is the center-of-mass collision energy squared, (Tin is the inelastic cross section, J is the corresponding Jacobian. The factor J is the known function of the kinematic variables, the

357

momenta and masses of the colliding and produced particles. The function 'I/J(z) is normalized as follows

1a';)O 'I/J(z)dz =

1.

(5)

The relation allows us to interpret the function 'I/J(z) as a probability density to produce a particle with the corresponding value of the variable z. According to the fractality principle the variable z is constructed as a fractal measure z = zon- 1 for the corresponding inclusive process. It reveals the property zen) -. 00 at n- 1 - . 00. The divergent part n- 1 describes the resolution at which the collision of the constituents can be singled out of this process. The n(X1,X2) represents relative number of all initial configurations containing the constituents which carry fractions Xl and X2 of the incoming momenta.

3

QeD test of z-scaling

Here we analyze the new data obtained by the STAR and PHENIX Collaborations [3,4] on high-PT spectra of nO mesons produced in PP collisions at VB = 200 GeV. The results are compared with the NLO QCD calculations in PT and z presentations. Figure lea) shows nO meson PT-spectra obtained at ISR (see [2] and references therein) and RHIC energies [3,4]. The strong dependence of cross sections on collision energy was experimentally observed. The scaling function 'I/J(z) for the same data are presented in Figure 1 (b). The shape of the scaling function for RHIC data (*,6) is found to be in good agreement with 'I/J(z) for the ISR data shown by the dashed line. The asymptotical behavior of 'I/J(z) is described by the power law, 'I/J(z) rv z-i3. The value of the slope parameter f3 is independent of kinematical variables. ＱＰＧｾＭ

10 • 10 1

10 10

ＢＧｾ＠

'.,.

p-p

ＢｾＮ＠

ｾＬ＠

11'0

_2

...';'....

10 -.

'M'

10'" . . 10 .... ＱＰｾ＠

10 ., ＱＰｾ＠ Ｑｏｾ＠

10 -.. 10 _II 10 ＭＮｾＢｵｌｊ＠

s'IJ, GeV b.

* ...

200

" __ 90 G

"'.

--.ＮＬｾ＠

PHENIX

200 STAR 30-62 ISR

\"'"

10

a)

...

10

I

b)

Figure 1: The PT (a) and z (b) presentations of experimental data on inclusive cross sections of rro mesons produced in pp collisions at the ISR and RHIC [3,4].

358

" ....-----------, ,,' , p+p",,,,O+X

"

..

10-'

::'"

..u 10-11

'>

10-

:g;

ｾ＠

E 10'" • 10-'

). ｾｧ＠ ［Ｚｧｾ＠

ｾ＠

17)1 1t 10· ｏｾＬＺＭ＠

"';,':-', ＢＧ［ＳＰｾＺＭＵ＠ ＢＧ［Ｒｾ＠ ＢＧ［ＲｾＰ＠ p,.. GeV/e

"":"'0

b)

a) '0·...-----------, '0'

ｾ＠

10

:u

1 10 "

'>

10'"

ｾＮ＠

ｾＺＮＡ＠ ｜ｾｩﾷＮ＠ ｾＺ＠

10'''0

p+p ... .,..O+X

p+p ....nO+X

",1. aA' -(3 >.2 ) , ->. 0 _0>.3 (3 >.2 0

C

C°

0>.3 (1- e- iO )

(5)

0 o >. 3 (-1 + ei6 ) 0 0 0

°

),

where A2 is the matrix (1) with 8=0. Then the transformation following form:

V ｾ＠

PRotPCP

(1 - ｾ＠ [A2' All),

PRot

= e A2 ,

ｾＶＩ＠

V takes the

PCP =

e

Al

,

(0 1)

(7)

CThe other possibility to separate the real and the imaginary parts would be the following: A2

0 =).,

(

1

-1 _0).,2 cos

(8)

0 (3).

0).,2

cos (8) )

-(3).,

0

and Ai

=

io).,3 s in8

0 1

0

0 0

0 0

362

where the commutator is of the order 0(>,4). Thus, in this approximation the transformation, corresponding to the quark mixing with CP violation, is composed of the purely rotational part PRot, which is related to the rotation matrix (2) via the formulae (3), (4) and the CP violating part PCP

PRo'

ｾ＠

exp

ＨｾＩＬ＠

>. -a >.3

a),3

-(3 >.2 0 (3 >.2 0

)

,

PCP

ＧｩｮＲｾＩ＠ 01 O.

So, within the exact theory one can say that for real systems, the property (7) can not occur if CPT symmetry holds and CP is violated. This means that the relation (7) can only be considered as an approximation.

3

Model calculations

In this Section we discuss results of numerical calculations performed within the use of the symbolic and numeric package "Mathematica" for the model considered by Khalfin in [4,5], and by Nowakowski in [8] and then used in [11]. This model is formulated using the spectral language for the description of Ks,KL and KO, K, by introducing a hermitian Hamiltonian, H, with a continuous spectrum of decay products (for details see [1]). Assuming that CPT symmetry holds but CP symmetry is violated and using the experimentally obtained values of the parameters characterizing neutral kaon complex make it possible within this model to examine numerically the Khalfin's Theorem as well as other relations and conclusions obtained using this Theorem (for details see [8, lID. The results of numerical calculations of the modulus of the ratio ｾｭ＠ for some time interval are presented below in Fig. 1. Analyzing the results of these calculations one can find that for x E (0.01,10),

Ymax(X) - Ymin(X) ':::: 3.3 where, Ymax(X)

= Ir(t)lmax

and Ymin(X)

X

10- 16 ,

= Ir(t)lmin-

(10)

401

ｾ､ｗＭｉＧｩ

··C.2:

..

4

10

Figure 1: Numerical examination of the Khalfin's Theorem. Here y(x)

= Ir(t)1 == 1ｾｭ＠

I, x = If .t,

and x E (0.01,10).

Similarly, using "Mathematica" and starting from the amplitudes Ajk(t) and using the formulae for hll (t), h22(t) and the condition All (t) = A22(t) one can compute the difference (h ll (t) - h22 (t) for the model considered. Results of such calculations for some time interval are presented below in Fig. 2. An expansion of scale in the left panel of Fig. 2 shows that continuous fluctuations, similar to those in the right panel of Fig. 2, appear.

2.01' 10.

11

ＱＮＹＢＰＭＫｾ［ＲＧＳＺＵ＠

x

1. 9S .10. 13 1.97 '10. 13

-1.5'10. 16

1.96 '10"l) 1.95-10. 13

Figure 2: The real part (left) and the imaginary part (right) of (hll(t) - h22(t))

*' .

There is y(x) = 3{(h ll (t) - h22(t) and y(x) = u(/3,B)'

P ( /3 B) rr n"

= dW,ru = Frr (n,/3,B) dW,il

iJ>rr(/3,B)'

(2)

= dWr +dW,ru = Fu(n,/3,B) + Frr (n,/3,B) cos 2 B dWgl+dW,il

iJ>u(/3,B)+iJ>rr(/3,B)cos 2 B

Where dWqu, dWcl are the quantum and the classical angular distributions of SR respectively, B is the angle between the vector of the magnetic field and the direction of the radiation propagation. In equations (2) the following notations are used:

4+3q

4+q

iJ>u(/3, B) = 16(1 _ q)5/2' iJ>rr(/3, B) = 16(1 _ q)1/2 ' n

Fu,rr(n,/3,B) = LFu,rr(vjn,/3,B), v=l

8v 2 x[' 2

(x)

F ( /3 B) n n-v (/3) = (2n + 1); (1 + p)2' Frr v; n, , B = u V j n" 1- p

x

= v 1 + p'

Where [n,n-v(X), ｛ｾＬｮＭｶＨｸＩ＠

P=

./

2vq

V1- 2n + l'

q

= /3 2 sin2 B,

4 2[2

0 ｾ＠

v qp

()

X (in n-v + p)2

q ｾ＠

,

(3)

/3 2 < l.

are the Laguerre function and its derivative [1].

429

The remarkable thing is the following: at each fixed n the functions Pa ,7r (n, (J, 8) depend on one variable q only, where q = /3 2 sin 2 8:

Pa ,7r(n,{J,8)

= pJ':)(q).

(4)

The function P(n, /3, 8) dos not possess this property. At each fixed n these functions are strongly depend on both /3 and 8 and satisfy the inequality following from (2):

min{pJn)(q), pJn)(q)} ｾ＠ P(n,{J,8) ｾ＠ max{pJn) (q), pJn)(q)}. The study of ｰｊｾＩ＠

(5)

(q) functions properties

It is easy to see that in the nonrelativistic limit the (2, 3) from formulas (2, 3) imply

/3 « 1 (i. e.

q

< (J2

«

1)

(n)( 2n [ 37q ] (n) 2n [ 35 q ] Pa q):::::: 2n + 1 1 - 4(2n + 1) , P7r (q):::::: 2n + 1 1 - 4(2n + 1) , (6) P(n, /3, Therefore at

8) : : 2n2: 1 [1 -

q 4(2n + 1)

(35 + 1 + :OS2 8) ].

/3 « 1 we have p(n) (q) < P(n a

-

(.I

, p,

8) < p(n) (q) < ｾ＠ -

7r

-

2n + 1

< l.

(7)

The functions ｰｊｾＩＨｱＬ＠ P(n,{J,8) decrease while q increases at fixed n. While n increases at fixed q these functions increase but stay lower than unity and at n -t 00 tend to unity. Replacing the summation on v by the corresponding integration in (3) and using well-known approximations [1] of Laguerre functions by McDonald's functions K 1/ 3(X), K 2/3(x) we could easily find the following expressions in the ultrarelativistic case:

2 3

Ho

m c = --. eli

It evidently follows from (5) that the ｰｊｾＩ＠ (q) are monotonically decreasing functions on J.L (therefore at fixed q monotonically increasing functions on n

430

and at fixed n monotonically decreasing functions on q) tending to zero at J.L -+ 00 follows from (8). At J.L < < 1 we have

p(n)( ) ｾ＠ u

1_

320J.L

2l7rv'3 '

q

p(n)() ｾ＠ 1- 256J.L • ,.. q 157rv'3

(9)

It is obvious that inequality (7) changes to the opposite one:

1> pJn)(q) ｾ＠ P(n,(3,O) ｾ＠ pJn) (q).

(10)

It is obvious from (3) that the functions Fu,,..(n, (3, 0) are finite at any values of q (including q = 1). Hence, at q -+ 1 the following asympthotics always take place: pJn)(q) ｾ＠ Au(n)(l- q)5/2, pJn)(q) ｾ＠ A,..(n)(l- q)7/2. (11) Here Au,,..(n) are some numbers depending on n. This guaranties that at 1 q < < 1 the inequalities (10) hold and ｰｊｾ＠ (q), P(n, (3, 0) tend to zero at q -+ 1. The obtained results prove the validity of the following inequalities 0< min{pJn) (q), pJn)(q)} :S P(n,(3,O) :S max{pJn) (q), pJn)(q)} < 1. (12) In conclusion on the figures below we present the graphs of these functions for different n. 1.

1.

0.8

0.8

P( q- 24!SF3 q')]

= 1+

- interference of the charge radiation and radiation of intrinsic magnetic moment,

f )JL =

[1+'" 1+ＱＭＢＧＨｾＫ＠ 2

9

2

9

385.J3 432'

J];:2

- magnetic moment radiation due to the Larmor precession

fTh

)J

= (1 + ,,' 7 + 1- ,{ 1 2

9

2

9

);2

'='

434 - magnetic moment radiation due to the Thomas precession,

=[_ 1+ ｾＧ＠

jL-Th

(_! _

1 + 1- ｾＧ＠

2

fl

3

35-13

2

3

216 r;

J];:2 ':>

- interference of the Larmor and Thomas radiation,

fa

= 1+r;r;' [ 2

I'

a(,

245-13

r; 3

72

,2J_!!",2] + 1-r;r;' (49+ 175-13 J!!..e 9 2 6 r; 9

- radiation due to anomalous magnetic moment of an electron:

g-2 l1a = 11- 110 = -2-110 where Po formulae

ｱ］ＳＱＰｈｲｾｮＲ＠

=en / 2m oc is the Bohr magneton. Besides, everywhere in these

m oc

2

2 mocp

］ｾｲ＠

(2)

2H

is a quantum parameter well-known in the synchrotron radiation theory, and factors

=

r;,r;' ±1 correspond to the spin quantum numbers, magnetic field.

H*

is the Schwinger critical

3. Spin light in the classical theory of synchrotron radiation Here we will show that the purely classical theory can explain completely the origin of the spin light. The radiation of electrical charge, possessing also an intrinsic magnetic moment in the classical theory, is described by the Lienard- Wiechert potentials and Hertz tensor polarization potentials [17]

Aa

eva, QafJ ］｟ｾｮ｡ｦｊＮ＠ c RP v P

=_!

RP v P

Corresponding tensor of electromagnetic field is calculated by the formula

HUV

］ｌＮｾ｟ａ｛Ｌｉｬｮｖ｝Ａ＠

2

c

d'i

d

c2

d'i 2

Q[,IlU n n V] U'

Using then the standard technique of classical theory of radiation with equations of an electron motion and spin precession in the homogeneous magnetic field in the linear approximation by 11 one can find the spectral-angular distribution of radiation power in the form

435

2 2 3 n { cos e 2 12 dO. = 47r y4 f32 sin 2 eJ n + J n + Ｔｾ＠

2

eL

f.loOJ COS e '} eoc f3 sin e nJ n J n WSR

dwn

=

'

(3)

=

Here OJ eoH / mocy is the cyclotron frequency coincident at g 2 with the frequency of spin precession. This formula is a generalization of the Schott formula for spectral-angular distribution of synchrotron radiation with respect to radiation of intrinsic magnetic moment of an electron. It can be shown that this formula reproduces exactly all properties of spin light concerned with radiation at Larmor precession and described by the semi-classical theory (see also Ref. [8]). Here we will show this on the example of calculating of total radiation power in the most actual ultrarelativistic case when

f.loOJ _ f.lo _ 1 liOJ _ 1 H _ 1 ＠ｾ eoc - eop - 2 moc 2 - 2y H* - 3y2 . Towards that purpose one should sum up the expression in formula (3) over the spectrum and integrate it over the angles. As a result we find the same formulas for total synchrotron radiation power and its polarization components as in the semiclassical theory but without recoil effects, Thomas precession and anomalous magnetic moment of an electron

W = (1 Ｋｾ＠

)W

eL

Wa = ＸＫＶｾ＠ eL

(

W;L =

Ｈｾ＠

7

SR '

1 .;:)

+ ｾ＠

ｾ＠

WSR ,W1/" = ＸＫＶｾ＠ eL

(

1

1 ';:)

i

ｾ＠

) WSR ' W;L = ( + ｾ＠

WSR '

(4)

) WSR .

This result does not depend on sequence of foregoing operations. Thus, this radiation is non-polarized as one could expect from the origin of the spin light. The formula for spectral-angular distribution is eL

dW 27 2( 2)2{K213 2 +--2 x2 2 - - = - - 2 Y l+x K1I3 dxdy 167r 1+ X 2 f.lo y2 yx } + ＶＭｾ＠ ( )112 K1I3K2/3 WSR ' eOp 1+x2 Integration over the spectrum in this expression gives the angular distribution of synchrotron radiation power with additional term for the spin light

436

{3 [7 ---;;;- = 32 (1 +

2 5X + X2

eL

dW

X2 y12

+

Y'2

(1

35

+ 16 ｾ＠

]

X2}

(1 + X2 t 2

WSR '

If the expression (15) is integrated over the angles, one can find the spectral composition of this radiation eL

f

dW 9..[3 {co - - = - y K 5/3(X)d.x dy 81T y

ＫＭｾ＠

2

3

f

co K

I/3(X)d.x

}

W SR '

y

The terms for the spin light in the last two formulas are equal to doubled components for linear polarization of radiation. Naturally, further integration in the last formulas over the angular parameter x or over the spectrum leads us to the formulae (4). This result can be shown by another method. According to the general theory of relativistic radiation of point like magnetic moment, the part of energy corresponding to the mixed synchrotron radiation emitted per unit proper time is determined by expression ([5],see also Ref. [18], formula (6.17»

ap 2 dpaJ = 2 eof.1o (d n w ｟ｾｶ｡＠ ( d-r 3 c4 d-r 2 p c2

dwp nJ'O'w __ 1 naPw w w p ). d-r CT c2 P p

Here nap is the dimensionless classical tensor of spin, pa is the four-dimensional momentum of radiation. Its zero component gives the power of mixed radiation

W eL

=.:.. dpo r d-r

Substitution of the corresponding solution of equation of motion, and averaging over period of charge motion and over the spin precession gives

W

eL

r JwSR'

z =(l+.!.;=n 3'='

where

n z = ｲｾ＠ . As to recoil effects and Thomas precession they can be completely

described by classical methods but with use of quantum laws of conservation.

4. Conclusion

Thus, we have shown that the classical and quantum theory of spin light are in agreement with each other at the first approximation by Plank's constant. A question is arising: is the correspondence principle fulfilled in higher-order approximation with respect to the Plank constant? According to the method described earlier the answer to this question is fairly evident: all depends on the possibility of neglecting of the quantum effects and other factors like the Thomas precession.

437

An extraordinary example is radiation of a neutron in a homogeneous magnetic field which arises exclusively due to the spin flip in the quantum theory. Relativistic quantum theory of neutron radiation was developed by the group of Russian scientists (I. M. Ternov, V. G. Bagrov and A. M. Hapaev) [21]. The classical theory of neutron radiation emitted at the spin precession, which was developed by V. A. Bordovitsyn with coauthors [17-20], turned out to be in full accordance with the quantum theory but differs by a constant coefficient equal to 4, which, as it turned out, is connected with specific properties of quantum transition with spin-flip ([8]). However such radiation in the classical theory does not exist in the common interpretation. Therefore the correspondence principle in this case is inapplicable. With regard to the synchrotron radiation of an electron the correspondence principle applied to radiation of the intrinsic magnetic moment works very well in the limit case p ｾ＠ 00 and on assumption that the value of anomalous magnetic moment is large enough to neglect the Thomas precession. Note that in the mixed synchrotron radiation the terms which are proportional to Plank constant and contain the anomalous magnetic moment are in full accordance with the classical theory. Apparently, this is connected with the fact that the anomalous magnetic moment does not undergo Thomas precession (see [22]) . It is easy to show that the developed here classical theory gives the same terms 2

for radiation without spin-flip and proportional to h as are derived by the semiclassical theory for spin radiation caused by Larmor precession. Thus, we have in detail considered here the spin light identification problem when the spin radiation proceeds against the background of powerful synchrotron radiation, recoil effects, and other relativistic phenomena. In its pure form the spin light contributes to the synchrotron radiation power as a small correction 2

proportional to h • At the present time the problem of spin light radiation of the relativistic magnetic moment is particularly urgent in connection with the construction of ultrahigh energy accelerators. The procedure for experimental observation of spin dependence of synchrotron radiation power was proposed in Budker Institute of Nuclear Physics (Novosibisk), and this experiment itself was described in [23-25]]. In this experiment synchrotron radiation power proportional to h was for the first time observed to be dependent on the spin orientation of a free electron moving in a macroscopic magnetic field. Now it is possible to carry out more detailed investigation of spin light. Acknowledgments

We thank Prof. Yu. L. Pivovarov. for interesting discussion on these problems and Prof. V.Ya. Epp for his help in improving of the paper. This work was supported by RF President Grant no. SS 5103.2006.2, and by RFBR grant no. 06-02-16 719.

438 References

[1] V. Bargmann, L. Michel, V. L. Telegdi, Phys. Rev. Lett. 2 (1959) 435. [2] A A Schupp, R. V. Pidd, H. R. Crane, Phys. Rev. 121 (1961) 1. [3] V. A Bordovitsyn, I. M. Ternov, V. G. Bagrov, SOY. Phys. Usp. 165 (1995) 1083 (in Russian). [4] V. A Bordovistyn, V. S. Gushchina, I. M. Ternov, Nucl. Instr. Meth. A 359 (1995) 34. [5] VA Bordovitsyn, Izv. Vuz. Fiz. 40, N22 (1997) 40 (in Russian). [6) G. N. Kulipanov, A E. Bondar, V. A Bordovitsyn et aI., Nucl. Instr. Meth. A405 (1998)191. [7] I. M. Ternov, Introduction to Spin Physics of Relativistic Particles, MSU Press (1997) 240 (in Russian). [8] Synchrotron Radiation Theory and its Development. Ed.V.ABordovitsyn, World Scientific, Singapore, 1999. See also: Radiation Theory of Relativistic Particles, Fizmatlit, Moscow, 2002 (in Russian). [9] VA Bordovitsyn, V.Ya. Epp, Nucl. Instr. Meth. A 220 (1998) 405. V. A Bordovitsyn, [10] A Lobanov, A Studenikin, Phys. Lett. B 564 (2003) 27. [11] A E. Lobanov, Phys. Lett. B 619 (2005) 136. [12] G. J. Bhabha, G.C. Corben, Proc. Roy. Soc. 178 (1941) 273. [13] A Bialas, Acta Phys. Polon, 22 (1962) 349. [14] M. Koisrud, E. Leer, Phys. Norv. 17 (1967) 181. [15] J .Cohn, H.Wiebe, J.Math. Phys. 17 (1976) 1496. [16] J. D. Jackson, Rev. Mod. Phys. 48 (1976) 417I. [17] V. A Bordovitsyn et aI., Izv. Vuz, Fiz.21, N25 (1978) 12; N210 (1980) 33. [18] V. A Bordovitsyn, G. K. Razina, N. N. Byzov, Izv. Vuz, Fiz. 23, N210 (1980) 33. [19] V. A Bordovitsyn, R. Torres, Izv. Vuz., Fiz. 29 N25 (1986) 38. [20] V. A Bordovitsyn, V.S.Guschina, Izv. Vuz., Fiz. 37, N21 (1994) 53. [21] I. M Ternov, V.G.Bagrov, A M. Khapaev, Zh. Exp. Teor. Fiz.48 919650 921 (in Russian), SOY. Phys, JETP 21 (1965) 613. [22] VA Bordovitsyn, V.V.Telushkin, Izv. Vuz., Fiz. 49, N2 (2006). [23] V.N.Korchuganov, G.N.Kulipanov, M.N.Mezentsev, et aI., Preprint INP 7783, INP, Novosibirsk (1977) . [24] AE.Bondar, E.L.Saldin, Nucl. Instr. Meth.195 (1982) 577. [25] S.ABelomestnykh, AE.Bondar, M.N.Yegorychev, et al. Nukl. Instr. Meth., 227 (1984) 173.

SIMULATION THE NUCLEAR INTERACTION Timur F. Kamalov a Physics Department, Moscow State Open University, 107966 Moscow, Russia Abstmct. Refined are the known descriptions of particle behavior with the help of Lagrange function in non-inertial reference systems depends of coordinates and their multiple derivatives. This entails existing of circumstances when at closer distances gravitational effects can prove considerably stronger than in case of this situation being calculated with the help of Lagrange function in inertial reference systems depends of coordinates and their first derivatives. For example, this may be the case if the gravitational potential is described as a power series in sir where s is a constant correspondence for the nuclei scale.

1

Simulation in real reference frame

1.1

Particles in real reference frame

Classical physics usually considers the motion of bodies in inertial reference systems. This is a simplified and approximate description of the real pattern of the motion, as it is practically impossible to get an ideal inertial reference system. Actually in any particular reference system there always exist minor influences. Let us consider the precise description of the dynamics of the motion of bodies taking into account complex non-inertial nature of reference systems. For this end, let us consider a body in a non-inertial reference system, denoting the position of the body as r and time as t. Then, expanding into Taylor series the function r = ret), we get

_ r - ro

at 2

1.

1 ..

3

1 . (n)

4

+ vt + - 2 + ,at + ,at + ... + ,a 3. 4. n.

t

n

+ ...

(1)

Let us compare this expansion with the well-known kinematical equation for inertial reference systems of Newtonian physics relating the distance to the acceleration a,

rNewton

at 2

= ro + vt + T·

(2)

Denoting the hidden variables accounting for additional terms in non-inertial reference systems with respect to inertial ones as qr, we get 1 .

3

qr = 3! at

1 ..

4

1 . (n)

+ 4! at + ... + n! a

n

t

+ ...

(3)

Then

r

=

rNewton

ae-mail: [email protected]

439

+q

(4)

440

For inertial reference systems the Lagrangian L is the function of only the coordinates and their first derivatives, L = L(t, r, r) For non-inertial reference systems, the Lagrangian depends on the coordinates and their higher deriva. .....

·(n)

tives as well as of the first one, i.e. L = L(t, r, r, r, r, ... , r ) Applying the principle of least action, we get [1]

J ......

JL.) ｾ＠

n dn 8L (5) -1) dt n (--:(;0 ) Jrdt = O. n=O 8 r Then, the Euler - Lagrange function for complex non-inertial reference systems takes on the form JS = J

·(n)

L(r, r, r, r, ... , r )dt

=

(6) Or

(7) Denoting

p = p(2)

aL p

Or'

= ｡ｾＬ＠

ar

=

=aL

a.(4)' r

aT

p(3)

=

p(5)

=

·(4)

p(4)

aL

·(20 > s, we have the equation for the gravitational potential c.p = ｇｭｾ｡Ｌ＠ where a = l. For the short distances r < < s, we have the equation for the gravitational potential (11). For particles described by the generalized Lagrange function at small distances, i.e. when the series diverges, there shall be much stronger gravitational forces acting than it is usually considered in calculations employing the Lagrange function. This model of short-range gravitational interaction allows one to compare nuclear and gravitational interactions at small distances. Acknowledgments

I thank Professor A. Studenikin and Doctor A. Grigoriev for providing me the possibility of fruitful discussions. Reference

[lJ M. V. Ostrogradskii, M'emoires de l'Academie Imp'eriale des Sciences de Saint-P'etersbourg v. 6, 385 (1850).

UNSTABLE LEPTONS AND (p, - e - 7)- UNIVERSALITY O.Kosmachev a Joint Institute for Nuclear Research, 141980, Moscow Region, Dubna, Russia Abstract. Main advantage and virtue of proposed method is a possibility to describe and enumerate all possible types of free equations for stable and unstable leptons in the frame work of homogeneous Lorentz group by means of unique approach.

1

Introduction

Free states are necessary for description of interactions. As it is known they play the role of initial and final states. Free states equations are unique way to introduce in theory quantum numbers identifying any leptons. Such quantum numbers characterize an equation structure. They will be called structural quantum numbers. The proposed method succeed from those fundamental requirements as Dirac equation [1]: invariance of the equations relative to homogeneous Lorentz group taking into account four connected components; formulation of the equations on the base of irreducible representations of the groups, determining every lepton equation; conservation of four-vector of probability current and positively defined fourth component of the current; spin value of the leptons is proposed equal to 1/2. One can show [2] that a totality of enumerated physical requirements are necessary and sufficient conditions (together with some group-theoretical requirements) for formulation of lepton wave equation out of Lagrange formalism. As it was shown formerly [2] Dirac equation is related with three different irreducible representations of homogeneous Lorentz group.It follows from the fact that Dirac ,-matrix group contains two subgroups d" b, and dual property of d'Y. In this case standard (proper, orthochronous) representation is realized on d, group, T-conjugate representation is realized on b, group, P-conjugate representation is realized on I, group, Corresponding algebras (six-dimensional Lie algebras of homogeneous Lorentz group) are characterized completely by their commutative relations (CR). They are of the form for d, group [ai, ak]

= Cikl2a!,

rbi, bk )

=

-cikI 2a !,

[ai, bk)

= cikl 2b l,

where Cikl is Levi-Cevita tensor, i, k, l = 1,2,3; ai, bi are infinitesimal operators of three-rotations and boosts respectively and al '" ,3,2, a2 '" ,I'Y3 a3 == a}a2 '" a2ala2"1 = all, bl '" ,}, b2 '" ,2, b3 '" Here following definitions are used [1)

,2,},

[i( 'J.tPJ.t)

+ me]'ll =

,3·

0,

'J.t'v

+ 'v'J.t

ue-mail: [email protected]

443

= 28J.tv,

p" V =

1,2,3,4.

444

Commutative relations on the base of b')':

[ar, a2l

=

｛｢ｾＬ＠

=

｢ｾｬ＠

2a3, 2a3,

ｷｨ･ｲ［｢ｾ＠

[aI, ｢ｾｬ＠ = 0, [aI, ｢ｾｬ＠ = 2b3 [a2' b3l = Ｒ｢ｾＬ＠ [a3, ｢ｾｬ＠ = Ｒ｢ｾＬ＠ '" -/'1'/'4, ｢ｾ＠ the base of f')'-group:

b3l = ＭＲ｢ｾＬ＠ = Ｒ｢ｾＬ＠

[a 3,｢ｾ｝＠

｛｡ｾＬＳｬ＠

= -

｛｢ｾＬＳ｝＠

(1)

b3 '" -/'3/'4· Commutative relations on

'" -/'2/'4,

[aI, a2l = 2a3' ｛｢ｾＬ＠ ｢ｾｬ＠ = - 2a 3' [aI, ｢ｾｬ＠ = 0, [al,b 2l = 2b3, ｛｡ｾＬ＠

[a3, all = 2a2, [a2,a3] = 2al, [b3, ｾｬ＠｢ = 2a2, ｛｢ｾＬＳ｝＠ = 2al, [a3, b3l = 0, [a2' b2l = 0, [al,b 3l = ＭＲ｢ｾＬ＠ [a2' ｢ｾｬ＠ = -2b3, [a3, ｢ｾｬ＠ = ＭＲ｢ｾＬ＠

2a l,

[a3' all = Ｒ｡ｾＬ＠

[b3, ｢ｾ｝＠ = = 2al, [a3' b3l = ｛｡ｾＬ＠ ｢ｾｬ＠ = 0, [al,b 3l = -2b 2, ｛｡ｾＬ＠ ｢ｾｬ＠ = -2b3, [a3' ｢ｾ｝＠ = Ｒ｢ｾＮ＠

ＭＲ｡ｾＬ＠

0,

(2)

The last connected component c" was obtained with following commutative relations:

[aI, ｡ｾｬ＠

=

2a3'

[b"I' b"] 2 -- 2a '3 , ｛｡ｲＬ｢ｾｬ＠

[aI, ｢ｾ｝＠ ｛｡ＲＬ｢ｾ｝＠

[a'3' b"l I

= 0, = Ｒ｢ｾ＠ = ＭＲ｢ｾＬ＠

--

2b"2'

[a2, a3l = -2ar, ｛｢ｾＬ＠ ｢ｾ｝＠ = -2ar, ｛｡ｾＬ＠

｢ｾｬ＠

= 0, = ＭＲ｢ｾＬ＠

[aI, ｢ｾｬ＠ [a'2' b"l I - - 2b" 3' [a3' ｢ｾｬ＠ = ｢ｾＮ＠Ｒ

[a3' ad = ｡ｾＬ＠Ｒ [b"3' b"] I -- 2a'2'

[a 3, ｢ｾｬ＠

= 0,

(3)

The last three types of CR (1),(2),(3) ap to lately [2] were not represented in physical literature. Now we have the complete and closed set of constituents for description of lepton wave equations. 2

Equations for stable leptons

The base of every lepton equation is a corresponding /,-matrix group. Each of the /,-matrix group are produced by four generators. Three of them anticommute and ensure Lorentz invariance of different kinds. The fourth generator is a necessary condition for the formation of wave equation. The distinct nonidentical equations are became by virtue of different combinations of the four subgroups d,),' b,),' c')" fT Structural content of the groups for every type of equation has the form. 1. Dirac equation -

D')'[IIl: d,)" b,),' fl'.

445

2. Equation for doublet massive neutrino -

D'Y[I): d'Y' C'Y' IT

3. Equation for quartet massless neutrino -

D'Y[I I I): d'Y' b'Y' C'Y' IT

4. Equation for massless T-singlet -

D'Y[IV]: bT

5. Equation for massless P-singlet -

D'Y[V): cT

Every equation has its own structure allowing to distinguish one equation from other. All equations have not physical substructures, therefore leptons are stable. Obtained method allows to calculate full number of the stable leptons in the framework of starting suppositions. 3

Extensions of the stable lepton groups

Is it possible to obtain additional lepton equations on the bas of previous suppositions? This problem is attained by introducing additional (fifth) generator for new group production. As it turned out there are exist three and only three such possibilities. Each of them is equivalent to introduction of additional quantum characteristics (quantum numbers). The extension of Dirac ,-matrix group (D'Y(IJ)) by means of anticommuting generator r5 such that rg = I leads to 6.1-grouP with structural invariant [4],[3J equal to In[6.1J = -1. The extension of Dirac ,-matrix group by means of anticommuting generator ｲｾ＠ such that ｲｾＲ＠ = -I leads to 6.3-grouP with structural invariant equal to In[6.3) = o. The extension of neutrino doublet group (D'Y(I)) by means of anticommuting generator ｲｾ＠ such that ｲｾＲ＠ = -I leads to 6.2-groUP with structural invariant equal to In[6.21 = 1. al-grOUP has the following defining relations r JLr v

+ r vr JL =

28JLv,

(/-L, v

= 1,2,3,4,5)

(4)

One can show on the bas of (4) that 6. 1 contains 3 and only 3 subgroups of 32-order. As a result we have following content

6.t{D'Y(IJ),

D'Y(IIJ),

D'Y(IV)}

(5)

Relation (4) together with structural invariant In[6. 11 = -1 identify 6. 1 in physical sense. a 3 -group is obtained under extension of Dirac group by similar defining relations rsrt + rtr s = 28st , (s,t = 1,2,3,4), (6) (s = l,2,3,4),rg =-1. rsr5 + r5rs = 0, The group content was changed in this way

6.3{D'Y(IJ),

D'Y(I),

D'Y(IIJ)},

(7)

446

This corresponds to structural invariant ｉｮ｛ｾＳＱ＠ A 2 -group and it defining relations.

rS t + rtr s

rsr4 + r 4r s r ur5 + r5r u

=

2b s t,

= 0, = 0,

= O.

(s,t=1,2,3), (s = ＱＬＲＳＩｲｾ＠ =-l. (u = ＱＬＲＳＴＩｲｾ＠ =-1.

(8)

The group content differs from two previous cases ｾＲｻｄＬＨｉＩ＠

Structural invariant is equal to ｉｮ｛ｾＲＱ＠ 4

D, (III) ,

D,(V)},

(9)

= 1.

Conclusion

All examined equations have its own mathematical structure. These structures are not repeated, therefore they may be used for theoretical identification of the particles in free states. The first five equations including Dirac one have not physical substructures. Objects without structure can not disintegrate spontaneously , therefore all they are stable. The last three equations (AI, A 2 , A 3 ) have internal structures allowing of physical interpretation. If we suppose that the mass of the new particles is more than sum of masses of its constituents, they become candidates for unstable leptons. It is evidentally that equations on the base of Al and A3 may be interpreted as the equations for the massive charged leptons such as M± and T±. Their structural distinctions are the base for solving of the (M - e - T)universality problem by means of interaction descriptions. It is possible to relate A 2 -group with massive unstable neutrino. References [1] P.Dirac, Proc.Roy. Soc. A vol.117, 610 (1928). [2J O.Kosmachev, Representations of the Lorentz Group and Classification of Stable Leptons (Preprint JINR, P2-2006-6) Dubna, 2006. [3J A.Gusev, O.Kosmachev, Structural Quantum Numbers and Nonstable Leptons (preprint JINR, P4-2006-188) Dubna, 2006. [4J J.S. Lomont Applications of finite groups, (Academic Press, New York, London) 51, 1959.

GENERALIZED DIRAC EQUATION DESCRIBING THE QUARK STRUCTURE OF NUCLEONS A.Rabinowitch Abstract. We consider a generalization of the Dirac equation to describe the quark structure and anomalous magnetic moments of nucleons. The suggested generalization contains two 3 by 3 matrices consisting of the quark charges and describes a wave function of a nucleon having 12 components. It is shown that the magnetic moments of nucleons determined via the generalized Dirac equation accord with their experimental values.

As is known, the Dirac equation for the relativistic electron cannot be applied without substantial modifications to describe nucleons, since it does not give their anomalous magnetic moments. That is why a generalization of the Dirac equation was proposed in which an additional term describing non-minimal interaction of nucleons with electromagnetic fields was introduced [1,2]. However, this well-accepted generalization has two serious disadvantages. Namely, it does not describe the quark structure of nucleons and the experimental values of the anomalous magnetic moments of protons and neutrons cannot be deduced from it. Because of these reasons we seek another equation for nucleons which could be free of the two disadvantages. For this purpose let us consider the following generalization of the Dirac equation to describe nucleons:

(1) where F is the column consisting of three bispinors \{1k, k = 1,2,3, a and b are 3 x 3 matrices characterizing the quark structure of nucleons, 1 is the unit 3 x 3 matrix, are the Dirac matrices, n = 0,1,2,3, An are potentials of an external electromagnetic field, ep is the proton's charge, m is the rest mass of a nucleon when

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