Quark Confinement and the Hadron Spectrum IV

Quark Confinement and the

Hadron Spectrum IV Wolfgang Lucha and Khin Maung Maung (editors) World Scientific

Quark Confinement and the

Hadron Spectrum IV

This page is intentionally left blank

Quark Confinement and the

Hadron Spectrum IV Vienna, Austria

3 - 8 July 2000

Editors

Wolfgang Lucha Austrian Academy of Sciences, Austria

Khin Maung Maung Hampton University, USA

V f e World Scientific «•

New Jersey* London 'Singapore* * Singapore Hong * Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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

Cover photograph: Austrian Academy of Sciences Photographs by F. F. Schoberl

QUARK CONFINEMENT AND THE HADRON SPECTRUM IV Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4567-X

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V

PREFACE The series of international conferences on "Quark Confinement and the Hadron Spectrum" has been founded in 1994 by Giovanni Prosperi. The first meeting of this series took place in Italy, in the beautiful Villa Olmo at the Lake of Como. The original purpose of this series of biennial workshops was to bring together people approaching the problem of confinement of the colour degrees of freedom and the description of hadrons as bound states of quarks from different points of view. Six years later, this series of meetings already has an impressive history: 1994: Como, Italy (Chairman: Giovanni M. PROSPERI), 1996: Como, Italy (Chairman: Giovanni M. PROSPERI), 1998: CEBAF, U.S.A. (Chairman: Nathan ISGUR). The Fourth International Conference on "Quark Confinement and the Hadron Spectrum" was held from July 3 to July 8, 2000 in Vienna at the famous main building of the Austrian Academy of Sciences, the former main building of the University of Vienna, which was planned by the French architect Jean Nicolas Jadot and built in the years 1753 - 1755. Since the last meeting in this series of conferences in 1998, several exciting developments have occurred: a continuous systematic improvement of effective theories designed to imitate the effects of quantum chromodynamics for heavy quarks; the first possible hints for the production and detection of quark-gluon plasma, obtained from the totality of results of the seven dedicated experiments of the lead-beam programme at CERN; and many others. In order to take into account these developments in a proper manner, "Quark Confinement and the Hadron Spectrum IV" has been given the following organizational structure: 1. The Plenary Session, consisting of the presentations of altogether not less than 24 invited speakers. I regret very much that from the following four invited plenary talks no written contributions have been received for publication in these Proceedings: • Anthony G. WILLIAMS (Adelaide): Lattice Studies of Confinement and Chiral Symmetry Breaking in a Covariant Gauge, • FedericoANTINORI(Padova): Recent Results from the CERN-SPS Heavy-Ion Programme, • Mikhail SHIFMAN (Minnesota): Lessons for QCD from Supersymmetry, • Zoltan LIGETI (Fermilab): CKM Matrix Elements from B Decays.

VI

2. Four Parallel Sessions, covering the following topics: Session A: Vacuum Structure of QCD and Mechanism of Confinement Conveners: Michael CREUTZ (Brookhaven NL), Mikhail Igorevich POLIKARPOV (ITEP, Moscow), Giovanni M. PROSPER! (Milano). Session B : Light Quarks (and Gluons) Conveners: Martin G. OLSSON (Madison), Massimo TESTA (Roma), Nils A. TORNQVIST (Helsinki). Session C: Heavy Quarks (and Gluons) Conveners: Christine T. H. DAVTES (Glasgow), Dieter GROMES (Heidelberg). Session D :

Deconfinement

Conveners: Frithjof KARSCH (Bielefeld), Janos POLONYI (Strasbourg). 3. The Poster Session, organized by Howard GEORGI (Harvard). From the numerous posters presented there four contributions found their way into these Proceedings. I wish to express my deep gratitude to the members of the International Advisory Committee for their proposals, to all people who agreed to present an invited talk or to act as a convener of one of the Parallel Sessions, as well as to all those institutions which supported this conference in one way or the other. The complete List of Sponsors may be found on one of the following pages. Last not least, it is a great pleasure for me to thank all those members of the—more or less Local—Organizing Committee who took an active part in the organization of this meeting as well as in the preparation of these Proceedings, in particular, to thank our secretary Elisabeth Karsky and Khin Maung Maung who did most of the hard and tedious work. I am convinced that all participants of "CONFINEMENT rV" are eagerly looking forward to the next meeting in this series of conferences, which will take place in 2002 in Italy, at the beautiful Lake of Garda. Vienna, December 2001

Wolfgang Lucha Chairman of the Conference

VII

INTERNATIONAL ADVISORY C O M M I T T E E Marshall BAKER (Seattle) James S. BALL (Salt Lake City) Michael CREUTZ (Brookhaven National Lab) Hans Gunther DOSCH (Heidelberg) Bernice DURAND (Madison) Loyal DURAND (Madison) Gerhard ECKER (Vienna) Dieter GROMES (Heidelberg) Nathan ISGUR (CEBAF) Heinrich LEUTWYLER (Bern) Xue-Qian LI (Nankai Univ., Tianjin) Aneesh V. MANOHAR (San Diego) Yuval NE'EMAN (Tel Aviv) Matthias NEUBERT (Cornell) Martin G. OLSSON (Madison) Enrico PREDAZZI (Torino) Giovanni Maria PROSPERI (Milano) Mikhail SHIFMAN (Minnesota) Massimo TESTA (Roma) Hiroshi TOKI (R.C.N.P., Osaka) Nils A. TORNQVIST (Helsinki) Francisco Jose YNDURAIN (Madrid) t Fred ZACHARIASEN (Caltech)

(LOCAL) ORGANIZING C O M M I T T E E Wolfgang LUCHA (HEPHY, Vienna) [chairman] Nora BRAMBILLA (Heidelberg) Manfried FABER (Vienna Univ. of Technology) Lewis FULCHER (Bowling Green State Univ., Ohio) Howard GEORGI (Harvard) Khin MAUNG MAUNG (Hampton) Franz Ferdinand SCHOBERL (Vienna Univ.)

VIII

Supported by Austrian Airlines Aviation Group Austrian Federal Ministry of Education, Science and Culture Austrian Research Association Bank Austria Creditanstalt Institute for High Energy Physics, Austrian Academy of Sciences Institute for Nuclear Physics, Vienna University of Technology NuHEP Research Center, Hampton University (U.S.A.) The City of Vienna Verein der Freunde der Osterreichischen Akademie der Wissenschaften Vienna Convention Bureau

ix

CONTENTS

Preface Committees and Sponsors

v vii

P L E N A R Y SESSION P R E S E N T A T I O N S

1

The Dirac Operator Spectrum and Effective Field Theory Poul Henrik Damgaard

3

Applications of the Hypervirial Theorem Yi-Bing Ding, Xue-Qian Li and Peng-Nian Shen

15

Contemporary Applications of Dyson-Schwinger Equations M. B. Hecht, Craig D. Roberts and S. M. Schmidt

27

What the Infrared Behaviour of QCD Green Functions Can Tell Us about Confinement in the Covariant Gauge Lorenz von Smekal and Reinhard Alkofer

40

On the Emerging Phenomenology of ( ( ^ ) m i n ) F. V. Gubarev and Valentine I. Zakharov

52

Strangeness Production Using Electrons Oliver Keith Baker

64

Dual QCD, Effective String Theory and Regge Trajectories (Fred Zachariasen Memorial Lecture) Marshall Baker and R. Steinke Many-Body Approach to Mesons, Hybrids and Glueballs Stephen R. Cotanch and Felipe J. Llanes-Estrada

74 90

Recent Developments in Chiral Perturbation Theory Gilberto Colangelo

102

Exact Chiral Symmetry with a Non-Perturbative Cutoff Herbert Neuberger

114

X

The Role of Chiral Symmetry in Hadronic Scattering Jose Emilio Fernandes Tavares Ribeiro

126

Center Vortices and Colour Confinement in Lattice QCD Roman Bertie, Manfried Faber, Jeff Greensite and Stefan Olejnik

138

Center Vortices in Continuum Yang-Mills Theory Hugo Reinhardt and Michael Engelhardt

150

Continuum QCD and Light Mesons Pieter Maris

163

The Last Meson Thomas Devlin

176

The Bc and Other Hadrons from the Bottom Drawer of Lattice QCD . . . 185 Hugh Philip Shanahan Glueballs, Hybrid and Exotic Mesons and String Breaking Chris Michael

197

QCD Potentiology Gunnar S. Bali

209

pNRQCD: Review of Selected Results Antonio Vairo

221

Confinement IV — Summary(?) Howard Georgi

234

PARALLEL SESSION CONTRIBUTIONS PARALLEL SESSION A Vacuum Structure of QCD and Mechanism of Confinement Vortex Configurations in SO(3) x Z(2) Andrei Alexandru and Richard W. Haymaker Recent Results in the Center Vortex Model for the Infrared Sector of Yang-Mills Theory Michael Engelhardt, Hugo Reinhardt and Manfried Faber

239 241 243

246

XI

Instantons, Monopoles, and Gribov Copies in the Maximally Abelian Gauge Thomas Heinzl

249

The Gluon Condensate from Gauge Invariant Vortex Vacuum Texture .. 252 Kurt Langfeld Quark Confinement in the Analytic Approach to QCD Alexander V. Nesterenko Mass Generation, Ghost Condensation and Broken Symmetry: SU(2) in Covariant Abelian Gauges Martin Schaden Confinement Picture in Dual Formulation of Lattice Gauge Models Oleg Borisenko and Manfried Faber Renormalons on the Lattice and the OPE for the Plaquette: A Status Report Francesco di Renzo

255

258 269

272

Colour Confinement in the Lattice Landau Gauge QCD Simulation Sadataka Furui and Hideo Nakajima

275

Hamiltonian Lattice Gauge Theory Norbert E. Ligterink

279

Lattice QCD with the Overlap-Dirac Operator: Its A Parameter, and One-Loop Renormalization of Fermionic Currents Constantia Alexandrou, E. Follana, Haralambos Panagopoulos and E. Vicari

282

Comments on the Confinement from Dilaton-Gluon Coupling in QCD .. 285 Mohamed Chabab On Confinement Provided by the Spontaneous Symmetry Breaking Valentin E. Kuzmichev and V. V. Kuzmichev

289

Chiral Physics in the Quark Composites Approach to QCD Sergio Caracciolo, Fabrizio Palumbo and Roberto Scimia

292

XII

PARALLEL SESSION B Light Quarks (and Gluons)

295

Spectroscopy "Windows" of Quark-Antiquark Mesons and Glueballs with Effective Regge Trajectories Martina Brisudova, Leonid Burakovsky and Terrance Goldman

297

The Topological Susceptibility and Pion Decay Constant from Lattice QCD UKQCD Collaboration: Alistair Hart and M. Teper

300

Light-Light and Heavy-Light Mesons Spectra from Nonperturbative QCD Alexei V. Nefediev

303

Electromagnetic Properties of Light Nucleon Resonances in a Chiral Constituent Quark Model Robert Ferdinand Wagenbrunn, M. Radici and S. Boffi

306

Relationship of Pionium Lifetime with Pion Scattering Lengths in Generalized Chiral Perturbation Theory Hagop Sazdjian

309

The Infrared Behaviour of the Gluon Propagator in SU(2) and SU(3) without Lattice Gribov Copies Constantia Alexandrou, Ph. de Forcrand and E. Follana

312

The 7 —>• 37r Form Factor as a Constraint on Schwinger-Dyson Modeling of Light Quarks Dubravko Klabucar and Bojan Bistrovic

315

PARALLEL SESSION C Heavy Quarks (and Gluons) Application of the Shifted-Z Expansion Method to B and D Meson Leptonic Decay Constants in the Semi-Relativistic Wave Equation Thabit Barakat The Mass of the b Quark from Lattice NRQCD Sara Collins

319

321 325

XIII

Phenomenology of Heavy Quark Systems and NRQCD Matching Coefficients Lewis P. Fulcher The Radial Wave Functions of a Heavy-Light Meson Calculated on a Lattice UKQCD Collaboration, Anthony M. Green, J. Koponen, P. Pennanen and Chris Michael

328

331

Weak Form Factors for Heavy Meson Decays Dmitri Melikhov and B. Stech

334

Comparison of Two-Body Equations John H. Connell

337

Instantaneous Bethe-Salpeter Equation: (Semi-)Analytical Solution . . . . 340 Wolfgang Lucha, Khin Maung Maung and Franz F. Schoberl Generalized Gell-Mann-Low Theorem Applied to a Scalar Model Axel Weber PARALLEL SESSION D Deconfinement QCD Sum Rules at Finite Temperature and Density Samirnath Mallik

343

347 349

Eigenvalues of the QCD Dirac Operator at Finite Temperature and Density E. Bittner, M.-P. Lombardo, Harald Markum and R. Pullirsch

353

Evidence for Deconfinement from the J/ip Suppression Pattern in Pb-Pb Collisions by the NA50 Experiment M. C. Abreu et al. (presented by Luciano Ramello)

357

POSTER SESSION CONTRIBUTIONS Monopole Condensation and Confinement in Non-Abelian Lattice Gauge Theory P. Cea and Leonardo Cosmai

361 363

XIV

Mass of the Bottom Quark Wolfgang Lucha and Franz F. Schoberl

364

Random Matrix Theory and Chiral Condensate for QED B. A. Berg, Harald Markum, R. Pullirsch and T. Wettig

365

Strong Decays in the 't Hooft Model Yulia Kalashnikova and Alexei V. Nefediev

366

List of Participants

367

PLENARY SESSION PRESENTATIONS

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3

The Dirac Operator Spectrum and Effective Field Theory P. H. DAMGAARD The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark When chiral symmetry is spontaneously broken, the low-energy part of the Dirac operator spectrum can be computed analytically in the chiral limit. The tool is effective field theory or, equivalently in this case, Random Matrix Theory.

1

Introduction

Contrary to what one might naively have expected, it is possible to predict the analytical behavior of the Dirac operator spectrum in QCD and QCD-like theories for eigenvalues very close to zero. This is the low-energy regime of the theory, and ordinary QCD perturbation theory is completely irrelevant here. In fact, the analytical predictions turn out to be not only non-perturbative in the QCD coupling constant, but exact. These developments began eight years ago with a paper by Leutwyler and Smilga 1 , where for the first time it was shown that if QCD undergoes spontaneous chiral symmetry breaking, there are constraints on the Dirac operator eigenvalues near the origin. So the idea is not new, but there have been many new developments in just the past few months. Here I summarize some of the recent work I have been involved in myself. For a comprehensive review of the subject up to last year, see ref. 2 . What originally pushed this subject forward was the suggestion by Shuryak and Verbaarschot 3 that the lowest part of the Dirac operator spectrum could be computed analytically by means of Random Matrix Theory. Much work has gone into understanding this remarkable connection between chiral symmetry breaking in gauge theories and Random Matrix Theory. In particular, it has been shown that the chiral (flavor) symmetry breaking in gauge theories coupled to fermions can be classified precisely according to the main categories of Random Matrix Theory ensembles 4 ' 5 , 6 . In the limit relevant for applications to the Dirac operator spectrum these Random Matrix Theory ensembles actually provide universality classes7: results do not depend in detail on the particular distributions of the random matrices. Random Matrix Theory should not be required to derive results for the Dirac operator spectrum; all should follow from field theory alone. It is one of the more recent discoveries that this is indeed the case. There are now three independent means of deriving analytical results for the Dirac operator spectrum:

4

• Random Matrix Theory • The supersymmetric method • The replica method Both the so-called supersymmetric 8 and replica 9 ' 10 methods stand out, as they rely only on the field theory formulation itself. Because results are claimed to be exact, all three methods should yield identical results. This is indeed the case. One pleasing consequence is that one can now use any of the three different formulations according to what is most convenient. Generically the Random Matrix Theory formulation is the easiest for analytical calculations, but there are cases where it is actually simpler to use the effective field theory partition function n (see also refs. 1 2 ' 1 3 ). Even in those cases, the explicit derivations have usually gone through Random Matrix Theory at intermediate steps. In this short review I will cover three different topics on which there has been progress in just the last few months. These are: (i) the analytical computation of individual eigenvalue distributions 14 (and subsequent lattice measurements 16 ), (ii) the replica method as applied to the finite-volume effective partition function of QCD 9 , 1 °, and (Hi) lattice Monte Carlo measurements of the low-energy end of the Dirac operator spectrum near the chiral phase transition 1 7 ' 1 8 . 2

Distributions of individual Dirac operator eigenvalues

Analytical expressions for the distribution of the first non-zero Dirac operator eigenvalue have been known for some time in the case of all three chiral ensembles. Recently a much more general analytcial expression was found for the probability distribution of the fcth smallest Dirac operator eigenvalue (again counted above the exact zero modes) 1 4 . The formula is completely general, holds for any number of (possibly massive) fermions, and for any of the three major chiral universality classes, as labelled by their Dyson index /?. The only technical restriction is that for the universality class of /? — 1 (corresponding to gauge group SU(2) and fermions in the fundamental representation), the formula only works for a sector of odd topological charge v. While the analytical expression has been derived in the framework of Random Matrix Theory, the final formula turns out to depend only on the effective field theory partition functions. It generalizes a related formula for just the lowest eigenvalue distribution in the case of the j3 = 2 universality class, which also expressed the distribution in terms of finite volume partition functions 15 .

5

Before presenting the resulting closed analytical expression for the distribution of the kth smallest Dirac operator eigenvalue, let us first recall some basic facts about the connection to Random Matrix Theory. Because chiral symmetry is presumed spontaneously broken, the large-volume euclidean partition function is, in the chiral limit, entirely dominated by the pseudoGoldstone bosons. The dominance is very strong: contributions from all other physical excitations are exponentially suppressed in the masses, with a coefficient in the exponent that grows linearly with the size of the box. As the size of the box is sent to infinity, the Goldstone bosons dominate the euclidean partition function entirely. It is in this sense that the results for the Dirac operator eigenvalues are exact: they can be made to reach any desired accuracy by taking the volume large enough, and the quark masses small enough. By imposing on the four-volume that V ^> m~ 4 only the zero mode piece of the Goldstone field 7r(x) survives, and the effective partition function, in a sector of topological charge v becomes 1

JUiNA IU(Nf)

(1)

.2

This partition function, a zero-dimensional group integral has, surprisingly, a Random Matrix Theory representation 3 . To get it, one starts with 3 ziN'\{mi})

=

fdWY[det(iM

+ mf)

exp

'

N 2

tr V(M2)

(2)

where

The complex matrix W in eq. (2) is of size (N + v) x N, and the potential is essentially not constrained beyond yielding a non-vanishing spectral density at the origin, p(0) ^ 0 7 . The limit N -¥ co is taken in the Random Matrix Theory partition function in such a way that the combinations p,i = mi7rp(0)2N, are kept fixed. In this limit the partition functions Z\, ! and Z„ f become equal up to a /ij-independent constant 3 . For practical computations an eigenvalue representation of the above matrix integral is convenient. Taking a general Dyson index /3 = 1,2 or 4, it can, up to an overall and irrelevant normalization factor, be written as

6

where xj are the eigenvalues of M 2 . Here the potential has been chosen to be just V(x) = x. This is precisely permitted because of universality of the final results 7 . Due to symmetry under v —• — v it is convenient to restrict oneself to v > 0. The unnormalized joint probability distribution for all N eigenvalues is

p£w.. > a W {m?})=n(*f + i ) / 2 -v^ n(*«+»#) ni*« -*if • (5) Similarly, the unnormalized joint probability distribution of the k smallest eigenvalues {0 < xi < • • • < xk-\ < Xk} is given by ^^(^•••.^{m2}) = /

dxk+1---dxNp(fi\x1,...,xN;{m2})

Jxk

( +i)/2 i

=n(*r - - e-^- n^+*»?)) n i*< - ^ i=l

x/ •

j=l

dxk+1---dxN

II

/lfc

x n

t>j

(af("+1)/3-1e-^n(a:i+m?)n(*i-^)/J)

«=*+l

.7=1

i=l

I^-^I".

(6)

i>j>fc+l

Shifting Xj -> Xj + x* in the integrand gives n«« (*!,...,*,;{»»?}) = *

i>j

JV

/.oo

x f l l * * - *,f / Jo

ft

AT/

fa

e-pXi^(Xi

i=k+i

fc-1

x Y[{Xi + xk - x^f) J=l

e-^-^-n(xf+1)/2-1e-^n(xi+m|)) + xtfW*-1

^ ( x , + m2 + xk) j=i

TV

[J

\Xi - Xj f .

(7)

i>J>*=+l

To finally get the probability distributions of the Dirac operator eigenvalues one takes the microscopic limit N —> oo with, in this particular convention, £i = Trp(0)y/xi = \/8Nxi and Hj = npifyrrij = VSNTUJ kept fixed. In this large-iV limit the difference between partition functions based on N — k and N eigenvalues disappears. One immediately sees that the new terms in the

7 integrand of (7) can be interpreted as arising from new additional fermion species, with the partition function now being evaluated in a fixed topological sector of effective charge v = 1 + 2//3. The only restriction is that for /3 = 1 the topological charge v must be odd (because otherwise the number of additional fermion species will be fractional). Taking into account the definition (4), this gives:

••**j

w +

&^»

^ - i g(cf " + "-K2 - c?)«- x

j=i

^"

UM;

In the partition function in the numerator each of the Nf fermion masses have been shifted according to /x? —»• /zf + Cf • There are /?(&— 1) new masses Cj? — Co i = 1 , . . . , k — 1, each of them being /? times degenerate. Finally there are also /3(i/ + l ) / 2 — 1 new fermions, all of mass £&. The overall normalization factor C is fixed by the requirement of probability conservation. To get the individual eigenvalue distribution of the fcth eigenvalue, one simply integrates out the previous A; — 1 smaller eigenvalues, viz.,

dCk-i 40) (Ci, • • •, G-i, C; ( M ) (9)

pf (C; {JH}) = ! dCi I dC2 • • • /

Jo Jci JC-2 The general formulas (8) and (9) may look rather complicated, but they actually simplify considerably in a number of interesting situations. For instance, in the physically most interesting case of QCD (which belongs to the /3 = 2 universality class 4 ), we get in a sector of topological charge v = 0:

U*(Cl,...,a;{M) = Cc-«/4Cfc j j ( C . J J ( C 2 + / 1 2 ^ j j ( C 2 _ c 2 ) 2 x i=l

2o(M)

j=l

i>j

(10)

The finite-volume partition functions involved here are known in closed analytical form 19 , ZV({IH}) = detA({/z})/A(/x 2 ) , (11)

8

where the determinant is taken over the Nf x Nf matrix

Miri) = MT'W-ite),

(12)

and Nf

2

A(M ) = n ^ - ^ 2 ) -

(13)

i>j

With this convention the normalization factor is C = 1/2 for all values of k, Nf and v. The analytical formula (10) has very recently been tested by lattice gauge theory simulations 16 , with quite remarkable agreement even at relatively small volumes. These simulations were done with staggered fermions, which are almost totally insensitive to gauge field topology at the couplings we are concerned with here 2 0 . This means that comparisons should be done only with the v = 0 analytical predictions. Shown in figure 1 is the result of a largestatistics computation in quenched SU(3) gauge theory. The individual Dirac operator eigenvalues indeed do have distributions that fall right on top of the analytical predictions.

0

10

20

30

Fig. 1 Individual Dirac operator eigenvalue distributions from lattice Monte Carlo simulations in quenched SU(3) gauge theory , and the analytical predictions of eq. (10).

9

Simulations with dynamical fermions in SU(3) gauge theory display the same degree of accuracy (although for obvious reasons the statistics here is much lower) 16 . Recall that both masses m* and eigenvalues are rescaled with the same factor of £ V . This means that to get as good accuracy as possible one needs to probe the theory of nearly massless quarks. In figure 2 is shown an analogous plot for the theory with Nf = 1 (strong coupling) staggered fermions (at a mass value of m = 0.003 and a volume of 6 4 ).

Fig. 2 Individual Dirac operator eigenvalue distributions from lattice Monte Carlo simulations of SU(3) gauge theory with Nf = 1 , compared with the analytical predictions of eq. (10).

One sees the same kind of detailed agreement as in the quenched theory. This establishes quite clearly that not only can the microscopic spectral density of the Dirac operator be computed analytically in the relevant scaling window, also individual Dirac operator eigenvalues are falling right on the analytical predictions. 3

The replica method

In lattice gauge theory simulations one often starts out with quenched calculations: the fermion determinant is entirely ignored in the averages. This is

10

just a simple (sometimes accurate, sometimes not) approximation, introduced only in order to save computer time. A more sophisticated approach is "partial quenching". Here one does simulate the full theory with dynamical fermions, but in addition one extracts more information from the same configurations. This is done by computing quenched averages in the field configurations that already included the effects of dynamical fermions. At first sight it may sound absurd to do this, but in fact there is much genuine physics to be extracted from such partially quenched simulations. In addition, it turns out that at the analytical level partial quenching is just what is needed in order to derive properties of the Dirac operator spectrum. In this case partial quenching is not at all considered as an approximation to the full theory; rather, it is used to obtain genuine physical information about the real theory, the one with propagating dynamical quarks. To make a partially quenched average, we must measure some correlation function of new fictituous "valence" quarks that do not affect the configurations. The perhaps most simple quantity is the partially quenched chiral condensate. We can get this condensate by adding Nv valence quarks, differentiate with respect to their mass, and subsequently taking the limit Nv ->• 0:

L,

Nv-tO

f\v

OfJ,v

1

i^'

+7V

"V*,{M)

f-#-i^+JV"V»,{M)

aAr

^

(14) JV„=0

In the last line we have formally expanded the partition function as a Taylor series in Nv. It is not obvious that precise meaning can be given to such a notion, since the function involved a priori is known only at integer values of Nv. This particular way of deriving the partially quenched chiral condensate is known as the replica method. Its applicability in certain condensed matter contexts has been questioned 21 , but last year there was considerable progress towards understanding how to apply this method to derive spectral properties 22 . In the present context of QCD, the partially quenched chiral condensate in the same finite-volume scaling regime as discussed above was computed in refs. 9 10 ' . The replica method trivially works in ordinary QCD perturbation theory (it simply kills all closed fermion loops, as required). It is a more non-trivial fact that the replica method also is suited for deriving quantities that are nonperturbative in the QCD coupling constant. For instance, in the low-energy framework of effective chiral Lagrangians the replica method 23 works quite analogous to the previously known supersymmetric method 2 4 .

11

Although in principle the partially quenched chiral condensate is an unphysical quantity, it turns out that this is not quite so. There is one tiny bit of this quantity that contains important physics: this the discontinuity across the imaginary mass axis 8 , which gives the spectral density. In the microscopic scaling region:

PpiCiilH}) = 2^ D i s c E"(M«>{M)L=i Sr02