Lattice Hadron Physics

Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglb¨ock, Heidelberg, Germany W. Domcke, Garching, ...

Author: Alex Kalloniatis | Derek Leinweber | Anthomy Williams

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Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglb¨ock, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France P. H¨anggi, Augsburg, Germany G. Hasinger, Garching, Germany K. Hepp, Z¨urich, Switzerland W. Hillebrandt, Garching, Germany D. Imboden, Z¨urich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Lipowsky, Golm, Germany H. v. L¨ohneysen, Karlsruhe, Germany I. Ojima, Kyoto, Japan D. Sornette, Nice, France, and Los Angeles, CA, USA S. Theisen, Golm, Germany W. Weise, Garching, Germany J. Wess, M¨unchen, Germany J. Zittartz, K¨oln, Germany

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A.C. Kalloniatis D.B. Leinweber A.G. Williams (Eds.)

Lattice Hadron Physics

123

Editors Alex C. Kalloniatis Derek B. Leinweber Anthony G. Williams University of Adelaide School of Chemistry & Physics Adelaide SA 5005 Australia

A. Kalloniatis D. Leinweber A. Williams (Eds.), Lattice Hadron Physics, Lect. Notes Phys. 663 (Springer, Berlin Heidelberg 2005), DOI 10.1007/b103529

Library of Congress Control Number: 2004115525 ISSN 0075-8450 ISBN-10 3-540-23911-1 Springer Berlin Heidelberg New York ISBN-13 978-3-540-23911-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by the authors/editor Data conversion by TechBooks Cover design: design & production, Heidelberg Printed on acid-free paper 54/3141/jl - 5 4 3 2 1 0

Preface

This year marks 30 years since Ken Wilson’s seminal paper of October 1974 formulating non-Abelian gauge theory on a space-time lattice, and proposing its application to the solution of the theory of quark-gluon dynamics, QCD. Since then the ﬁeld has become vast, with several major symposia and workshops per year on the ﬁeld of lattice gauge theory. In particular, the 1990s brought overlap fermions and improved operators which overcame two of the major hurdles of lattice ﬁeld theory – chirality on the lattice and suppression of discretisation errors. It is now a daunting prospect for someone outside the ﬁeld to quickly become literate and active in the area. The present volume in Lecture Notes in Physics attempts to overcome this state of aﬀairs. In their earliest form, the various contributions in this volume were ﬁrst presented by their authors in July 2001 at the international workshop on lattice hadron physics, LHP2001, in Cairns, Australia, under the auspices of the Centre for the Subatomic Structure of Matter based at the University of Adelaide in South Australia. Those conference contributions, and those of the other participants at LHP2001 have been presented elsewhere. Nevertheless those participants selected to extend their papers for this volume have put considerable eﬀort into shaping the contributions to be accessible both pedagogically, as well as up to date in light of developments since 2001. They represent a broad spectrum of workers in hadronic physics, from graduate students to senior researchers of many years of experience and a solid body of signiﬁcant work in hadronic physics. The hope is that precisely this mix of newcomers and established experts in the ﬁeld has enhanced the pedagogical value of the volume. Thematically, this series of lectures draws upon the developments made in recent years in implementing chirality on the lattice via the overlap formalism, exploiting chiral eﬀective ﬁeld theory in order to extrapolate lattice results to physical quark masses, new forms of improving operators to remove lattice artefacts, analytical studies of ﬁnite volume eﬀects in hadronic observables, studies of quark propagators on the lattice, and state of the art lattice calculations of excited resonances, including the heavy pseudoscalar eta and eta-prime mesons. We trust that these contributions will assist graduate students and experienced researchers in other areas of hadronic physics to appreciate, if not become active in, contemporary lattice gauge theory and it’s application to hadronic phenomena. We thank the contributing authors for the eﬀorts they put in and

VI

Preface

their patience as this volume was put together, and warmly thank John Hedditch for his assistance in the ﬁnal stages of compilation. Adelaide, January 2005

Alex Kalloniatis Derek Leinweber Tony Williams

Contents

Quenching Eﬀects in the Hadron Spectrum C. Allton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 The Quenched Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Results from the Quenched Approximation . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Results from Full (Unquenched) Simulations . . . . . . . . . . . . . . . . . . . . . . . 7 4 Quantifying Quenching Eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Quark Propagator from LQCD and Its Physical Implications P.O. Bowman, U.M. Heller, D.B. Leinweber, A.G. Williams, and J.B. Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Euclidean Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Staggered Quark Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Lattice Quark Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Analysis of Lattice Artefacts for Staggered Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Overlap Quark Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Analysis of Lattice Artefacts for Overlap Quarks . . . . . . . . . . . . . . . . . . . 9 The Quark Propagator in Landau Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Laplacian Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Applications: The Condensate and Running Mass . . . . . . . . . . . . . . . . . . 12 Modelling the Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalised Spin Projection for Fermion Actions W. Kamleh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Standard Spin-Projection Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Generalised Spin-Projection Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The FLIC Fermion Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 23 26 30 32 38 42 47 49 51 56 59 61 65 65 65 66 67 68 69

VIII

Contents

Baryon Spectroscopy in Lattice QCD D.B. Leinweber, W. Melnitchouk, D.G. Richards, A.G. Williams, and J.M. Zanotti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 History of Lattice N ∗ Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Lattice Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Interpolating Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Operators for Spin- 12 and Spin- 32 Baryons . . . . . . . . . . . . . . . . . . . . . . . . . 6 Survey of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Appendix − Correlation Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 74 75 78 82 92 104 105 110

Hadron Structure and QCD: Eﬀective Field Theory for Lattice Simulations D.B. Leinweber, A.W. Thomas, and R.D. Young . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Eﬀective Field Theory for QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chiral Expansion of the Nucleon Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 114 118 123 127 128

Lattice Chiral Fermions from Continuum Defects H. Neuberger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Inﬁnite Flavor Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Two Dimensional Flavor Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Compactiﬁcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Generalization of the Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 What Next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 131 133 141 143 144 144

The Computation of the η and η Mesons in Lattice QCD K. Schilling, H. Neﬀ, and T. Lippert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Prolegomena: Pseudoscalars in LQCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Real World with n¯ n s¯ s Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Three Computational Bottlenecks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Towards Realistic Physics Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 147 149 154 157 165 173 174

Contents

IX

Strong and Weak Interactions in a Finite Volume M. Testa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Summation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The L¨ uscher Quantization Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Matrix Elements of Scalar Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Nature of the LL Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summation Theorems, Locality and the LL Formula . . . . . . . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 179 183 190 191 192 194 197

Hadron Properties with FLIC Fermions J.M. Zanotti, D.B. Leinweber, W. Melnitchouk, A.G. Williams, and J.B. Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Lattice Quark Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fat-Link Irrelevant Fermion Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Lattice Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Scaling of FLIC Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Search for Exceptional Conﬁgurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Octet-Decuplet Mass Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 199 200 205 208 210 215 218 223 223

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

List of Contributors

Chris Allton Department of Physics University of Wales Swansea Swansea SA28PP, U.K [email protected] Patrick Bowman Indiana University Nuclear Theory Center 2401 Milo B. Sampson Lane Bloomington, Indiana 47408, USA [email protected] Urs M. Heller American Physical Society One Research Road Box 9000 Ridge, NY 11961-9000, USA [email protected] Alex Kalloniatis Special Research Center for the Subatomic Structure of Matter Department of Physics University of Adelaide Adelaide 5005, Australia [email protected] Waseem Kamleh Special Research Center for the Subatomic Structure of Matter Department of Physics University of Adelaide Adelaide 5005, Australia [email protected]

Derek B. Leinweber Special Research Center for the Subatomic Structure of Matter Department of Physics University of Adelaide Adelaide 5005, Australia [email protected] Thomas Lippert Wuppertal University D42097 Wuppertal Germany [email protected]. uni-wuppertal.de Wally Melnitchouk Jeﬀerson Lab 12000 Jeﬀerson Avenue Newport News, VA 23606 and Centre for the Subatomic Structure of Matter University of Adelaide Adelaide 5005, Australia [email protected] Hartmut Neﬀ Department of Physics Boston University Boston MA02215, USA [email protected] Herbert Neuberger School of Natural Sciences Institute for Advanced Study Princeton, NJ 08540

XII

List of Contributors

and Rutgers University Department of Physics and Astronomy Piscataway, NJ 08855, USA [email protected] David G. Richards Jeﬀerson Lab 12000 Jeﬀerson Avenue Newport News, VA 23606 [email protected] Klaus Schilling Wuppertal University D42097 Wuppertal, Germany and Department of Physics Boston University Boston MA02215, USA [email protected]. uni-wuppertal.de Massimo Testa Dip. di Fisica, Univ. di Roma “La Sapienza” and INFN Sezione di Roma Piazzale Aldo Moro 2, I-00185 Rome Italy [email protected]

Anthony G. Williams Special Research Center for the Subatomic Structure of Matter Department of Physics University of Adelaide Adelaide 5005, Australia [email protected] Ross Young Special Research Center for the Subatomic Structure of Matter Department of Physics University of Adelaide Adelaide 5005, Australia [email protected] James Zanotti John von Neumann-Institut f¨ ur Computing NIC Deutsches Elektronen-Synchrotron DESY D-15738 Zeuthen, Germany [email protected] Jianbo Zhang Special Research Center for the Subatomic Structure of Matter Department of Physics University of Adelaide Adelaide 5005, Australia [email protected]

Quenching Eﬀects in the Hadron Spectrum C. Allton Department of Physics, University of Wales Swansea, Swansea SA2 8PP, U.K.

Abstract. Lattice QCD has generated a wealth of data in hadronic physics over the last two decades. Until relatively recently, most of this information has been within the “quenched approximation” where virtual quark–anti-quark pairs are neglected. This review presents a descriptive discussion of the eﬀects of removing this approximation in the calculation of hadronic masses.

1 The Quenched Approximation In a quantum ﬁeld theory involving gauge and fermion degrees of freedom, such as QCD, we have the following path integral formalism for the expectation value of a quantity Ω: 1 ¯ ¯ ¯ A) e−SE (ψ,ψ,A) (1) Ω = DψDψDA Ω(ψ, ψ, Z where Z is the usual path integral and the Euclidean action SE is deﬁned in terms of the usual ﬁeld strength tensor Fµν , 1 ¯ / + m)ψ(x) + Fµν Fµν . SE = d4 x ψ(x)(D (2) 4 The gauge degrees of freedom, A, are bosonic, but the fermionic degrees of freedom, ψ are fermionic and hence anti-commute. These are diﬃcult to deal with in a computer simulation, but fortunately, since they occur as they can be integrated ψ analytically resulting in the usual determinant factor: 1 ¯ ¯ A)det(D DA Ω(ψ, ψ, / + m) e−SE (ψ,ψ,A) . (3) Ω = Z Simulations of this quantum ﬁeld theory are performed on a space-time lattice by simply replacing all the continuous derivatives and integrals with ﬁnite diﬀerences and sums over gauge conﬁgurations. Hence, we have on the lattice: Ω =

1 ¯ A) e−Sg det(∆ Ω(ψ, ψ, / + m) Z

(4)

{U }

The (naive) lattice Euclidean action is C. Allton: Quenching Eﬀects in the Hadron Spectrum, Lect. Notes Phys. 663, 1–16 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

2

C. Allton

S = SF + S g =

x

¯ / + m)ψ(x) − 1 ψ(∆ T r(Up + Up† ) 2 g0 p

(5)

where a is the lattice spacing, the link variable Uµ (x) now carries the gauge degrees of freedom, and Up is the trace of the product of link variables around a plaquette, ˆa)Uµ† (x + νˆa)Uν† (x) . (6) Up = Uµ (x)Uν (x + µ This formalism maintains gauge invariance even on a lattice [1]. Variations of the naive lattice action can be made to improve its convergence to the continuum action in two areas: • the naive action suﬀers from fermion doubling – each lattice quark ﬂavour corresponds to 2d continuum ﬂavours, where d is the space-time dimension; • lattice actions in general suﬀer from discretisation errors which enter when the continuum derivatives in (2) is replaced by the ﬁnite diﬀerence in (5). Two methods are generally used to overcome the ﬁrst diﬃculty – the Wilson/ clover family of actions, and the staggered action. Both of these actions can be tweaked so that their lattice systematic error (the second diﬃculty above) are reduced, and then they are termed “improved”. Simulations using the lattice formalism can be performed by replacing the naive sum in (4) with a Monte Carlo estimate. This introduces a statistical error O(1/ Ncf g ) in the estimate of Ω where Ncf g is the number of conﬁgurations in the Monte Carlo sum. The lattice prescription of formulating a Quantum Field Theory has a´ priori no model assumptions – it is derived exactly from the full continuum formalism with no approximations. However the parameter values in real computer simulations of lattice QCD are far from their experimental values. This is due to limitations in current computer power! Table 1 lists the values of the parameters in typical lattice simulations along with their experimental values. Thus typical lattice simulations must inevitably rely on some extrapolation of lattice data. Note that the bare lattice gauge coupling, g0 , is not listed in Table 1. This because the information about g0 is contained within the a value, through dimensional transmutation. Our usual intuition about high momentum transfers Table 1. Typical parameter values in current lattice simulations Parameter

Typical Value

Experimental Value

mq

mu , md ≈ 5 MeV

a

> ms /2 ≈ 50 MeV ∼ 0.05−0.20 fm

0

L

2−4 fm

∞

Ncf g

O(100)

∞

Nf

0, 2 or 2 + 1

“2 + 1”

Quenching Eﬀects in the Hadron Spectrum

3

(short-distance physics) corresponding to the weakly coupled regime (small values of g0 ) in asymptotically free theories such as QCD, is directly applicable to lattice simulations. So we have g0 → 0 as a → 0. Equations (4 & 5) correctly deﬁne the full continuum theory in the limit as the lattice spacing a → 0. However, it is extremely expensive to simulate with (4 & 5). Figure 1 shows the estimated cost of lattice calculations as a function of quark mass using the formula in [2] for the “clover” action. (Actually, the horizontal axis of this plot is MP S , but, from the PCAC relation, we have MP S ∝ √ mq .) 10

Tflop years

8

6

4

2

0

MK

Mπ

0

200

400

600

800

1000

MPS [MeV] Fig. 1. The computer time in Teraﬂop – years required for a full lattice QCD simulation as a function of pseudoscalar meson mass using the formula for clover actions in [2]. We have assumed (i) a lattice spacing of a = 0.1 f m; (ii) a lattice volume of (3 f m)4 ; and (iii) that 200 conﬁgurations in the ensemble sum in (4) are required. The physical points corresponding to the π− and K-mesons are shown by vertical lines

In Fig. 1 we have assumed: (i) a lattice spacing of a = 0.1 f m; (ii) a lattice volume of L4 = (3 f m)4 ; and (iii) that there are Ncf g = 200 conﬁgurations in the ensemble sum in (4). (These are very conservative assumptions!) As can be seen, for even modest values of MP S ≈ 12 MK ∼ 250 M eV , full simulations require Tera-scale computing.1 For this reason, [4] introduced the “quenched” approximation where the true QCD vacuum is replaced with one with no quarks present (i.e. Nf = 0 in Table 1). Speciﬁcally the quenched approximation is deﬁned as follows: • det(∆ / + m) is replaced by unity, thereby removing the quark-anti-quark loops from the vacuum conﬁgurations; • the coupling β = 6/g02 is shifted to try to counteract (as much as is possible) the removal of these q − q¯ pairs. Typically this shift is of the following order 1

Recent advances in lattice actions, e.g. using an improved staggered action, have meant that CPU requirements are not quite so pessimistic [3].

4

C. Allton

β Q ≈ β f ull + 0.6 where β Q,f ull refer to the coupling in the quenched and full theories. In this way the quenched approximation can be viewed as an eﬀective ﬁeld theory, i.e. it contains a subset of all the interactions, and the couplings of the quenched theory have to be tuned to take care of these missing diagrams. Figure 2 shows the diagrams which are present in both the quenched and full theory, and those which are present only in the full theory.

(a)

(b)

Fig. 2. Diagrams which are present in (a) both quenched QCD and full QCD, and (b) present only in full QCD. The full lines are quarks and the spirals are gluons

Quenched simulations are several orders of magnitude faster than full (unquenched) simulations, and full simulations have only been performed in earnest in the last 5 years or so. Typical statistical and systematic errors of state-of-theart full simulations are of the same order now as quenched simulations’ errors were a decade ago. Particle physics is primarily concerned with the comparison of theory with experiment, and when theoretical calculations have inherent errors, it is crucial to understand and quantify their scale. The main aim of this chapter is to determine the systematic eﬀect introduced in the hadron spectrum by the quenched approximation. We will ﬁnd that uncovering quenching eﬀects is more diﬃcult than one would ﬁrst imagine for two reasons: • the quenched approximation proves to be surprisingly successful for many hadronic quantities, i.e. it reproduces much of the hadron spectrum at the 5– 10% level. Assuming that QCD is the theory of the strong interaction implies that removing this approximation makes a relatively small eﬀect!

Quenching Eﬀects in the Hadron Spectrum

5

• current full simulations have statistical errors of a few percent (since they are highly cpu-intensive) and so discerning the quenching eﬀects with this relatively noisy data can prove diﬃcult; While the quenched approximation proves to be unintuitively accurate for many hadronic quantities, there are some quantities where it either fails drastically, or has pathologies when the valence quark masses in the hadrons becomes vanishing. Examples of this include: • the η and η mesons are degenerate in the quenched theory, whereas they are not degenerate in full QCD. This is because the quenched theory excludes diagrams involving disconnected q − q¯ loops. (See [5] for a description of lattice simulations of η and η .) • the chiral limit of quenched QCD suﬀers from “chiral logs” ∼log(mq /Λχ ) where mq is the quark mass and Λχ is a mass parameter proportional to pion decay constant. These logarithms enter the chiral perturbation theory expansion of various hadronic masses in the quenched theory, spoiling their chiral limit [6]. • the hyperﬁne mass splittings in heavy-mesons in the quenched theory is wrong by up to 10% or more, with the sign of the discrepancy depending on the states considered. This systematic error is greatly reduced when the full theory is considered. We will discuss some of these issues in later sections. The next section brieﬂy reviews the best current results obtained from the quenched approximation. It outlines the accuracy of this approximation for the hadronic spectrum. Section 3 presents recent results from full (i.e. unquenched) simulations and we attempt to uncover estimates of quenching eﬀects in Sect. 4.

2 Results from the Quenched Approximation While there have been many papers published using lattice simulations in the quenched approximation2 we will concentrate on the work of the CP-PACS collaboration who have produced one of the most accurate quenched study of the light hadron spectrum in [7]3 . Their calculations used an improved clover action [9] simulated at volumes of around 2.53 fm3 with several quark masses and lattice spacings. They are thus able to perform continuum (a → 0) and chiral (mq → mu,d ) extrapolations (see Table 1). An impressive summary of their quenched spectrum for the light hadrons is shown in Fig. 3 as the open symbols. Their lattice data are shown after the 2

3

A search on the SPIRES database for “quenched” returns more than 500 papers, and this does not include papers which use quenched simulations but where the authors have not included this word in the paper’s title! Another paper from the CP-PACS Collaboration studies even larger lattice of up to 643 × 112 but the lattice action employed in this work is the pure Wilson action which has O(a) errors [8].

6

C. Allton 1.8 1.6

non-strange sector

strange sector

M [GeV]

1.4

Σ∗

1.2

∆

1.0 0.8

K*

N K

N ∆ rel.dev.

rel. deviation

0.6

1.1 1.0 0.9 0.8

φ

K K* φ

Λ

Σ

{ {

Quenched QCD

Σ

Ω

Ξ

full QCD

Λ

Ξ∗

Ξ

experiment non-strange baryons ms from K ms from φ non-strange baryons ms from K ms from φ

Σ∗ Ξ∗ Ω

1.05 1.00 0.95

Fig. 3. The light hadron spectrum from CP-PACS [7], Tables XV and XII. Both the quenched and full QCD results are shown, together with the experimental value. The two hadrons on the left do not contain strange quarks, whereas the other hadrons do. The lattice spacing was set from the ρ− mass. Two methods were used to set the strange quark mass: from (i) the K and (ii) the φ− mass. The results of both deﬁnitions are shown. In the middle and lower plots, the relative deviation (= Mlattice /Mexpt ) is shown. The lower plot is a close-up of the middle plot showing the relative deviation for the strange mesons

appropriate continuum and chiral extrapolations and is taken from table XV of [7]. As can be seen from the middle panel of Fig. 3, discrepancies between the quenched lattice results and the experimental values are O(5−10%). It is important to note that this relatively small diﬀerence is only discernible due to the tiny errors in the lattice data of O(1−2%). Quenched calculations of an earlier generation [10], with correspondingly larger errors, were not able to uncover deviations from experimental values. Figure 3 contains two sets of lattice data points: those obtained from the K and φ inputs. These refer to the hadron whose mass was used to set the strange quark mass in the lattice calculation. (The ρ mass was used to deﬁne the lattice spacing, a.) The fact that there are diﬀerences between these two sets of data is itself an indication of the failure of the quenched approximation, i.e. an exact calculation’s result wouldn’t depend on how the scale was set. The CP-PACS collaboration also ﬁnd the mass splittings, such as the hyperﬁne splitting in the meson sector and the splittings in the decuplet (baryon) sector are smaller than experimental values.

Quenching Eﬀects in the Hadron Spectrum

7

A further indication of the failure of the quenched approximation is in the determination of the strange quark mass ms . As mentioned above, this quantity can again be calculated using either the K- or φ-meson as input, but the deviation between the two results is at the 3−4σ level. Moving to the heavy-hadron mass spectrum, a recent publication, using an improved staggered action, studied the splittings in the heavy-meson sector [3]. The left-hand plot in Fig. 7 (taken from [3]) shows the quenched predictions of various splittings from [3]. This clearly shows a discrepancy between the quenched results and experimental values. As we will see in Sect. 3.4, this discrepancy disappears when we remove the quenched approximation.

3 Results from Full (Unquenched) Simulations This section will give a ﬂavour of current full QCD lattice simulations by concentrating on the CP-PACS [7] and UKQCD [11] collaborations’ results for the light hadron spectrum, and the work of [3] for the heavy-meson spectrum. Both the CP-PACS and UKQCD collaborations used two ﬂavours of improved clover fermions whereas the collaboration in [3] used an improved staggered action (which has a cpu advantage over the Wilson action, see Sect. 1 and [12]). Table 2 displays the parameters used in these collaborations’ work. Note that we have diﬀerentiated the sea and valence quark masses in this table (cf. Table 1). The sea quarks are those which always appears in quark loops and are not connected to the source/sink operators (e.g. the quark loop in Fig. 2), and the valence quarks are those which enter the source/sink interpolating operators. Table 2. Indicative parameter values used in full QCD simulations by the CP-PACS [7] and UKQCD Collaborations [11] in their study of light hadrons, and in the study of the heavy-meson spectrum in [3] Parameter

CP-PACS [7]

UKQCD [11]

Davies et al. [3]

msea q

0.5ms −1.8ms

0.6ms −2.0ms

0.17ms −0.5ms

i.e. O(50−180) MeV

i.e. O(60−200) MeV

i.e. O(17−50) MeV

0.25ms −2.1ms

0.6ms −2.4ms

0.12ms −ms

i.e. O(25−210) MeV

i.e. O(60−240) MeV

i.e. O(12−100) MeV

a

0.09−0.25 fm

∼0.11 fm

0.09 fm & 0.12 fm

L

∼2.5 f m

∼1.7 f m

∼2.5 fm

Ncf g

O(1000)

O(200)

Nf

2

2

mval q

2+1

8

C. Allton

As can be seen from Table 2, the CP-PACS collaboration have performed QCD simulations at parameter values closer to the experimental values and has larger statistics than the UKQCD collaboration (see also Table 1). However the UKQCD collaboration chose a subset of its parameter values so that the lattice spacing remained ﬁxed as the sea quark mass, msea q , varied. This meant that O(a) eﬀects could more readily be disentangled from dynamical quark eﬀects. Furthermore, the UKQCD lattice action has the technical advantage that it has no O(a) lattice systematic errors. In the simulations of [3] extremely light quarks were able to be studied due to the use of the improved staggered action. (This seems to have become the action of choice for most dynamical simulations.) Also [3] simulates with the more physical value of 2 + 1 quark ﬂavours (see Table 1). Rather than give the full details of the results from these collaboration’s work, a summary is presented in the following. In the next section we attempt to understand the discrepancies between this section’s full QCD results and quenched simulations from Sect. 2. 3.1 Meson Spectrum In Fig. 4 we plot the vector and pseudoscalar meson mass taken from [7] together with the experimental points. In this ﬁgure, the lattice spacing, a, was determined from the K and K ∗ meson masses using the method described in [13]. The huge number of data points corresponds to the fact that there are 16 diﬀerent (β, msea q ) combinations in [7] and that there are 9 diﬀerent valence quark combinations for each of these (β, msea q ) ensembles. Also plotted are the experimental data points corresponding to the (π, ρ), (K, K ∗ ), and (ηs , φ).4 One of the main points to be taken from this plot is that the systematics involved in lattice simulations are clearly under control. Variations amongst this data in Fig. 4 is < ∼1%, while the lattice spacing and sea quark mass vary by ∼ 0.5−1.8ms . A close around a factor of three: a ∼ 0.09−0.25 fm, and msea q analysis of the data has been used to extrapolate away these residual systematic and a [7] (see also [14]). eﬀects in msea q Figure 3 also shows the hadronic spectrum, including the three mesons, K, K ∗ and φ taken from [7]. These have been obtained by chirally extrapolating data analogous to that in Fig. 4 to the physical quark masses. As can be seen (particularly in the lower panel of Fig. 3) the full QCD simulated results are in very good agreement with experiment. As mentioned above, the K, K ∗ mass is used to set the lattice spacing, a, in Fig. 4 [13]. This means that the lattice data and the experimental K, K ∗ point agree by construction. However, the slope of the lattice data is a real lattice prediction. In the next subsection we study this gradient. 4

Note that ηs is not a physical particle, since there is no pure s − s¯ pseudoscalar meson due to mixing with the u, d quarks. The ηs mass shown here is deﬁned as 2 − Mπ2 which follows from the PCAC relationship MP2 S ∝ mq . Mη2s = 2MK

Quenching Eﬀects in the Hadron Spectrum 1.2

9

Mηs , Mφ

MV [GeV]

MK , MK* M π , Mρ 1

0.8 0

0.2

0.4

0.6

0.8

1

MPS2 [GeV2] Fig. 4. The light meson masses from CP-PACS [7] (see also [14]). In this ﬁgure, we have set the lattice spacing from the method described in [13]. Also shown are the experimental points

3.2 J -Parameter In this section we analyse the gradient dMV /dMP2 S of the lattice data. The dimensionless quantity used to study this is deﬁned [15] dMV . (7) J = MV dMP2 S K,K ∗ Note that we deﬁne the experimental value of the J-parameter by approximating the derivative in (7) by a ﬁnite diﬀerence: J discrete = MK ∗

MK ∗ − Mρ 2 − M2 . MK π

(8)

Therefore the lattice estimate of J is obtained by taking the derivative in (7) val sea but varying valence quark w.r.t. variations in MP2 S (msea q , mq ) at ﬁxed mq val mass, mq (i.e. the experimental/physical sea quark mass is clearly ﬁxed!). The J-parameter has been studied in both [7] and [11], but we concentrate here on the analysis in [11]. Figure 5 plots the J-parameter from [11]. The Jvalues separately. These J(msea parameter is calculated at each of three msea q q ) sea 2 ≈ 0. values are then extrapolated in mq ∝ MP S to the physical point msea q This extrapolated J value is shown as a banded region in Fig. 5. As can be seen from Fig. 5 the individual J(msea q ) values are signiﬁcantly smaller than the experimental value. However there is a clear trend in msea q which tends towards the experimental point.

10

C. Allton 0.55

0.5

Experiment

J

0.45

Extrapolated

0.4

0.35

(Quenched)

0.3

0.25

0

0.1

0.2

0.3

0.4

0.5

0.6

(MPS

0.7

0.8

unitary 2

)

0.9

1

1.1

1.2

1.3

1.4

1.5

2

[GeV ]

Fig. 5. J versus (MPunitary )2 using the approaches described in the text from S unitary is the pseudoscalar meson mass comprising of degenerate valence [11]. MP S quarks which are themselves degenerate with the sea quarks. (Note, from PCAC, unitary 2 )2 ∝ msea ) = (MPunitary q .) The quenched data points have been plotted at (MP S S 2 1.3 GeV for convenience. The banded region at the left of the graph is the result of the → 0 for the full QCD data. The experimental value of J = 0.48(2) extrapolation msea q is also shown

3.3 Baryons The vector and pseudoscalar light-meson sector at various quark masses is deﬁned by the plot in Fig. 4. Traditionally, the corresponding plot containing information about the light – baryon sector (speciﬁcally the nucleon) is the “Edinburgh” plot where the nucleon mass is plotted against the pion mass (with both masses normalised by the vector meson mass). Figure 6 shows this plot for the the CP-PACS [7] and UKQCD collaborations for their unitary data, i.e. where ≡ msea mvalence q q . As can be seen, there is a relatively large spread in the data, but there is a tendency for the data to approach the experimental point as the quark masses decrease. After this chiral extrapolation is performed, the CP-PACS collaboration [7] obtained the baryonic spectrum seen in Fig. 3. This is an impressive array of data spanning octet and decuplet sectors. As can be seen Fig. 3 the nucleon and ∆ diﬀer from experiment by around 10%, whereas the Ξ, Ξ ∗ (with quark content lss) and particularly the Ω (with quark content sss) are in perfect agreement with experiment. This implies that lattice simulations become more accurate as the strange quark content increases [7]. Since lattice simulations normally have valence quarks which span the mass of the strange (see Table 2), the spectrum calculation of baryons containing purely strange valence quarks requires no valence quark chiral extrapolation. However, the level of chiral extrapolation required obviously becomes more and

Quenching Eﬀects in the Hadron Spectrum

11

CP-PACS UKQCD 1.5

MN / MV

Static

1.4

1.3

Experimental 1.2

0.2

0.4

0.6

0.8

1

MPS / MV Fig. 6. The “Edinburgh plot” for selected full QCD data from [7, 11]. The lattice data ≡ msea points shown are the unitary points (i.e. mvalence q q ). The experimental point is shown, along with the static limit (mq → ∞)

more signiﬁcant as the light quark (i.e. u and d) content of the baryon increases. This suggests that chiral extrapolation procedures need more careful consideration in order to resolve the discrepancy above (see [16]). Note that the authors of [7] themselves argue that this discrepancy could be due to ﬁnite volume eﬀects which impinge upon baryons composed of light quarks more than those composed of strange quarks. This could presumably also be a factor, particularly since ﬁnite volume eﬀects are likely to increase the mass (which is in the direction of the observed discrepancy in Fig. 3) and be most relevant for the lightest baryons. 3.4 Heavy-Meson Mass Splittings There has been a recent study of the heavy-meson spectrum in [3] which uses 2 + 1 ﬂavours of quarks, i.e. 2 light degenerate ﬂavours which play the role of the u and d quarks, and one heavier (but still dynamical) quark which plays the role of the s quark. This is obviously closer to the real world than the simulations of the CP-PACS and UKQCD collaborations (see Tables 1 & 2). We reproduce, in Fig. 7 (taken from [3]) a graph showing the ratio of lattice prediction to experiment for some heavy-meson mass splittings. As can be seen, the lattice results in the full theory (right-hand plot) are within 1σ of their experimental values.

12

C. Allton fψ fK 3MΞ − MN 2MBs − MΥ ψ(1P − 1S) Υ(1D − 1S) Υ(2P − 1S) Υ(3S − 1S) Υ(1P − 1S) 0.9 1 1.1 LQCD/Exp’t (nf = 0)

0.9 1 1.1 LQCD/Exp’t (nf = 3)

Fig. 7. Heavy-meson mass splitting (together with some light hadron quantities) taken from [3]

4 Quantifying Quenching Eﬀects 4.1 Hadron Spectrum In Sects. 2 and 3, we have outlined some results for the hadronic spectrum for both the quenched approximation and full QCD. Comparing these results we note ﬁrstly that quenched results are generally within 10% of their experimental value for a wide variety of quantities. This is an unexpectedly good level of agreement which will be discussed later in this section. Studying the light-meson sector, we note that the full theory is able to accurately reproduce the K, K ∗ and φ masses to within around 1%, far better than the quenched theory (see lower panel of Fig. 3). We note, however, that the “chiral” slope, deﬁned via the J-parameter, (7), is still several σ away from its experimental value at the simulated values of msea (see Fig. 5), and that the chiral extrapolation, msea → mu,d is required to make contact with experiment. This situation is mirrored in the baryonic spectrum. Figure 3 shows the remarkable prediction from the CP-PACS collaboration of eight baryonic masses. In general terms, the agreement between theory and experiment is enhanced when the quenched approximation is removed. Note also that there is no discrepancy in the full theory between predictions using the K and φ mesons to set the strange quark mass. The same is not true in the quenched data (see Fig. 3). The level of agreement between the full theory prediction and experiment is most profound for baryons containing the largest strange quark content. We argued in Sect. 3.3 that this could imply that the lattice data at the simulated values of mq (roughly around ms ) are correct, but that the chiral extrapolation procedure mq → mu,d is going astray. As can be seen in Fig. 6, which is roughly the baryonic equivalent of Fig. 4, the chiral extrapolation required to reach the u, d quarks is substantial.

Quenching Eﬀects in the Hadron Spectrum

13

Moving to the heavy-meson sector, we summarised in Sect. 3.4 results from [3]. These show excellent agreement between full simulation results and experiment for a variety of quantities, especially splittings in the υ spectrum. A corresponding quenched analysis shows discrepancies of ∼10%. 4.2 Why is the Quenched Approximation So Good? While there are obvious failures in the quenched approximation’s ability to reproduce the real world, it does much better than naive expectations: one would imagine that removing all q − q¯ diagrams from the theory would have a drastic eﬀect on the hadron spectrum. Figure 3 shows that this is not the case. Why then does the quenched approximation perform so well? One can obtain a handle on this issue by studying the static quark potential (which is the quantum mechanical potential between two inﬁnitely heavy quarks). Figure 8 shows UKQCD results for this quantity for both the quenched and full theory [11]. The curve shown in the graph is the “string model”, V (r) = e/r + σ/r + const [17]. Note that the data is deﬁned to agree in value and slope exactly at r = r0 (the hadronic scale deﬁned in [18]) [11]. A close up of the diﬀerence between the lattice potential and the string model at short distances is shown in Fig. 9. As can be seen from Figs. 8 & 9, the discrepancy between the quenched and full theories is negligible across the whole range of r except at very small distances where the deviation is discernible, but small. This implies that only physical quantities particularly sensitive to this short-distance scale will be aﬀected by the quenched approximation. Hadronic states are most

2.5

[V(r)V(r0)]*r0

1.5

0.5 5.93 Quenched, 623 5.29, c=1.92, k=.13400 5.26, c=1.95, k=.13450 5.20, c=2.02, k=.13500 5.20, c=2.02, k=.13550 5.20, c=2.02, k=.13565 Model

−0.5

−1.5

−2.5

0

1

2 r/r0

Fig. 8. The static quark potential from the UKQCD Collaboration [11]. The parameters c and k refer to a coeﬃcient of an improvment term in the action and the sea quark mass parameter respectively

14

C. Allton 0.5

[V(r)V(r0)]*r0 string model

5.93 Quenched 5.29, c=1.92, k=.13400 5.26, c=1.95, k=.13450 5.20, c=2.02, k=.13500

−0.5

0

1 r/r0

Fig. 9. The deviation of the static quark potential in Fig. 8 from the string model [11]

sensitive to “medium” distance scales r ∼ r0 ≈ 0.5 f m where (from Fig. 8) the two theories’ data overlay each other. Thus the quenched and full theories should agree at the same level as quarks in QCD can be approximated as moving in a static quark potential. This observation presumably has relevance to the age-old question: Why does the (nonrelativistic) quark model perform so well? It is worth noting that, from Sect. 1, the quenched approximation is deﬁned not just by replacing the quark determinant by unity, but also by renormalising the coupling g0 . In fact, if you attempt to perform quenched and full simulations at the same value of g0 , then the lattice spacing, a (or equivalently the cut-oﬀ ∼1/a) will diﬀer by a factor of around four. This is telling us that the virtual quark loops really are aﬀecting the dynamics of the simulation. The apparent contradiction between this fact, and what we have seen above, i.e. that the quenched approximation reproduces the full theory (at the 5−10% level) is resolved as follows. The lattice only actually predicts dimensionless quantities, normally expressed as, e.g. M × a, where M is some mass. In this way the lattice is able to predict dimensionless ratios of physical quantities only, e.g. M1 /M2 . Although switching the quark determinant on and oﬀ does directly aﬀect the lattice spacing, and therefore M a, it seems to have little eﬀect on the ratio M1 /M2 . In other words the physical prediction from the lattice of M1 , which can be obtained from M1 /M2 × M2expt , doesn’t seem to be greatly aﬀected by quenching. This is telling us something remarkable: for a wide variety of hadronic masses (and the static quark potential), the removal of virtual quark loops from the theory can be counter-balanced simply by an adjustment in the coupling, g0 .

Quenching Eﬀects in the Hadron Spectrum

15

There is one ﬁnal reason why quenching has only a modest aﬀect on the hadronic spectrum, compared to other physical quantities. In order to extract a hadron mass from a lattice simulation, the quantity Ω = C(t) is calculated (see (4)) where C(t) is a two-point correlation function between hadronic currents. In Euclidean space-time, we have C(t) → M2 e−M t

as

t→∞

(9)

where M is a matrix element between the vacuum and the hadronic ground state. Lets assume that we are performing a quenched calculation of C(t) and that it has a relative error of ε due to this quenched approximation, i.e. C(t)Q = C(t)f ull (1 + ε) ,

(10)

where C(t)Q,f ull are the quenched and full correlation functions respectively. Because the mass, M , appears in the argument of the exponential, a relatively small adjustment in M can mop up the quenching error ε, whereas a larger relative change would be required of the matrix element, M. Obviously this analysis is a little simplistic but it does illustrate that we can expect quenching errors in matrix elements (such as decay constants) to be larger than in masses.

5 Conclusions Lattice QCD is an approach to solving ﬁeld theories, such as QCD, which involves no model assumptions. Given a fast enough computer, the lattice can be used to solve QCD on any ﬁnite volume and with any non-zero quark mass resulting in an absolute theoretical prediction of QCD. However, in order to make the problem tractable on current computers, certain parameters of QCD need to take non-physical values (see Table 1). The parameter under study in this chapter is the number of quark ﬂavours in the vacuum, Nf . Setting Nf = 0 is called the quenched approximation and corresponds to ignoring virtual q − q¯ pairs in the vacuum. The approximation Nf = 0 is seemingly a particularly brutal approximation and, furthermore, there is little theoretical guidance as to its eﬀect. Thus we are usually forced to “measure” its eﬀect a posteriori by analysing data from lattice simulations. In this chapter we have studied the hadronic spectrum with and without the quenched approximation, in particular light mesons and baryons, and heavymesons. By comparing quenched data with experimental masses, we have shown that quenching eﬀects in the light-hadron spectrum are relatively small (5−10%), with a slightly larger discrepancy in the heavy-meson spectrum. It is only recently that full QCD simulations have been able to produce data with statistical and systematic errors beneath this level. With this new generation of data, we are now able to state that full QCD lattice results have better agreement with experiment than quenched results.

16

C. Allton

We have outlined some reasons why the quenched approximation is so relatively successful, and we have found evidence that the chiral extrapolation techniques currently being used in full QCD simulations require further consideration. In the future, more precise calculations with, and without the quenched approximation will surely enhance our understanding of the underlying physics of QCD.

Acknowledgements The author would like to thank the CSSM in Adelaide for their kind hospitality.

References 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

K. G. Wilson, Phys. D 10 (1974) 2445. 2 H. Wittig, Nucl. Phys. Proc. Suppl. 106 (2002) 197–198, hep-lat/0203021. 3 C. T. H. Davies et al., hep-lat/0304004. 3, 7, 8, 11, 12, 13 H. Hamber and G. Parisi, Phys. Rev. Lett 47 (1981) 1792, E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett 47 (1981) 1795, D. Weingarten Phys. Lett 109 B (1982) 57. 3 C. Michael, Phys. Scripta T 99 (2002) 7, hep-lat/0111056. 5 S. Sharpe, Phys. Rev. D 46 (1992) 3146, C. Bernard and M. Golterman, Phys. Rev. D 46 (1992) 853. 5 CP-PACS Collaboration, A.A. Khan et al., Phys.Rev. D 65 (2002) 054505, heplat/0105015. 5, 6, 7, 8, 9, 10, 11 CP-PACS Collaboration, S. Aoki et al., Phys. Rev. D 67 (2003) 034503, heplat/0206009. 5 B. Sheikholeslami and R. Wohlert, Nucl. Phys. B 259 (1985) 572. 5 F. Butler, H. Chen, J. Sexton, A. Vaccarino, D. Weingarten Phys. Rev. Lett. 70 (1993) 2849, hep-lat/9212031. 6 UKQCD Collaboration, C.R. Allton et al., Phys. Rev. D 65 (2002) 054502, hep-lat/0107021. 7, 9, 10, 11, 13, 14 C. Bernard et al., Nucl. Phys. B (Proc. Suppl.) 106 (2002) 1999. 7 C.R.Allton, V. Gimenez, L. Giusti, F. Rapuano, Nucl. Phys. B 489 (1997) 427– 452, hep-lat/9611021. 8, 9 W. Armour, C.R. Allton, D. Leinweber, A.W. Thomas, R. Young in preparation. 8, 9 UKQCD Collaboration, P. Lacock and C. Michael, Phys. Rev. D 52 (1995) 5213, hep-lat/9506009. 9 A. Thomas et al. Hadron Structure and QCD: Eﬀective Field Theory for Lattice Simulations, Lect. Notes Phys. 663, 113–129(2005). 11 M. L uscher, Nucl. Phys. B 180 (1981) 317. 13 R. Sommer, Nucl. Phys. B 411 (1994) 839, hep-lat/9310022. 13

Quark Propagator from LQCD and Its Physical Implications P.O. Bowman1 , U.M. Heller2 , D.B. Leinweber1 , A.G. Williams, and J.B. Zhang1 1

2

Special Research Centre for the Subatomic Structure of Matter and The Department of Physics, University of Adelaide, Australia 5005 American Physical Society, One Research Road, Box 9000, Ridge, NY 11961–9000

Abstract. The quark propagator lies at the core of lattice hadron spectrum calculations as well as studies in other nonperturbative schemes. It manifestly displays dynamical chiral symmetry breaking and can be used to connect nonperturbative methods with perturbation theory. The quark propagator can also provide direct information on lattice simulations and act as a test-bed for improvement. We investigate the lattice quark propagator with standard (Kogut–Susskind) and improved (Asqtad) Staggered actions and with the Overlap action. These actions are seen to have excellent ultraviolet behaviour as well as being useful for studying infrared properties. By using a variety of actions as well as varying the lattice parameters we are able to obtain good control over systematic errors in the quenched case. Gauge dependent quantities from lattice simulations may be aﬀected by Gribov copies. We explore this by studying the quark propagator in both Landau and Laplacian gauges. Laplacian gauges are unambiguous, i.e. free of Gribov copies. Landau and Laplacian gauges are found to produce very similar results for the quark propagator. From the quark propagator we are able to compute the chiral condensate and running quark mass. Results from Staggered and Overlap actions are in excellent agreement. The data is also used to construct ans¨ atze suitable for model hadron calculations as well as adding to our intuitive understanding of QCD.

1 Introduction Quantum Chromodynamics is widely accepted as the correct theory of the strong interactions and the quark propagator is its most basic quantity. In the low momentum region it exhibits dynamical chiral symmetry breaking (which cannot be seen from perturbation theory) and at high momentum it can be used to extract the running quark mass (which cannot be directly measured from experiment). In lattice QCD, quark propagators are tied together to calculate hadron masses and other properties; and the quark propagator is a required input for any number of hadronic models. Lattice gauge theory provides a way to calculate the quark propagator nonperturbatively, providing quantities such the chiral condensate and ΛQCD , and in turn, such a calculation can provide technical insight into lattice gauge theory simulations. Quark propagator studies are now reaching a state of maturity. The quark propagator has now been studied using Clover [1, 2], Staggered [3, 4] and Overlap [5, 6] actions. All these actions have diﬀerent systematic errors and the P.O. Bowman, U.M. Heller, D.B. Leinweber, A.G. Williams, and J.B. Zhang: Quark Propagator from LQCD and Its Physical Implications, Lect. Notes Phys. 663, 17–63 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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combination of these studies gives us an excellent handle on the possible lattice artefacts. As well as a summary of the state of this reseach, in this report we will provide the latest results from improved Staggered and Overlap lattice actions. The quark propagator is gauge dependent and we choose the ever popular Landau gauge and the interesting Laplacian gauge [7, 8]. Laplacian gauge ﬁxing is an unambiguous gauge ﬁxing and, although it is diﬃcult to understand perturbatively, it is equivalent to Landau gauge in the asymptotic region. It has been used to study the gluon propagator [9, 10, 11]. In SU (N) there are various ways to implement a Laplacian gauge ﬁxing. Three varieties of Laplacian gauge ﬁxing are used, and these form three diﬀerent, but related gauges. The systematic study of the quark propagator on the lattice has also provided fruitful interaction with other approaches to hadron physics, such as instanton phenomenology [12], chiral quark models [13] and Dyson–Schwinger equation studies [14]. The lattice is a ﬁrst principles approach and has provided valuable constraints for model builders. In turn, such alternative methods, by preserving features of QCD that the lattice lacks, can provide feedback on regions that are diﬃcult to access directly, such as the deep infrared and the chiral limit. This report is organised as follows. In an attempt to make it self-contained we start with a brief discussion of Green’s functions in Euclidean ﬁeld theory with particular reference to lattice gauge theory. This is followed by an explanation of the gauge ﬁxing we will use, especially Laplacian gauge. As the Staggered formalism is less widely understood than Wilson’s discretisation, it is described before showing some results for the quark propagator in Landau gauge. The Overlap quark formalism is then introduced, with a discussion of the particular issues for the quark propagator with that discretisation followed by a summary of results. Then, results for the quark propagator in the Laplacian gauge are shown. We will then focus on the asymptotic behaviour of the quark propagator to calculate the running mass and chiral condensate. Finally, we suggest a model form for the quark mass function and compare the data with various published models.

2 Euclidean Green’s Functions Quantum ﬁeld theories may be deﬁned by a path integral over a function space. If the coupling of a given theory is weak, α 1, then it can be expanded in terms of its bare n-point functions in a perturbative series. Perturbation theory is extremely successful in Quantum Electrodynamics and is useful in the high energy regime of Quantum Chromodynamics, but no matter what the coupling, some features of a theory will not appear at any ﬁnite order in perturbation theory. If the coupling that characterises a theory is strong, which is the case in QCD at intermediate to low energies, then the perturbative series will converge poorly, or not at all. So to make the most out of QFT we need nonperturbative methods. Lattice gauge theory provides a direct method for summing the functional path integral. In fact, it is the only way to sum the functional integral that

Quark Propagator from LQCD and Its Physical Implications

19

does not require us to mutilate the theory in a way that will have unpredictable results. It involves approximations, like any other practical method in QFT, but they are systematically removable. It is this possibility of controlling the systematic errors that makes lattice gauge theory invaluable. Although the real, four-dimensional world is, at least locally, a Minkowski space, much work in quantum ﬁeld theory is done in Euclidean space. This is not an approximation, but a complete transformation of a theory from one geometry to the other [15]. This may be a temporary measure, with a further Wick rotation back to Minkowski space to obtain physical answers, or it may be possible to extract observable quantities directly from the Euclidean form of the theory. Quantum Chromodynamics is perfectly well deﬁned in Euclidean space and as we shall at no stage ﬁnd it necessary to explicitly rotate to Minkowski space, we shall restrict ourselves to Euclidean space from the start. Suppressing spin and colour indices, continuum QCD is described by a generating functional 4 1 (1) DψDψDAe−SG [A]−SF [ψ,ψ,A]+ d xηψ+ηψ+Jµ Aµ Z[η, η, J] = Z[0] built from a gluonic action SG [A] =

1 2

2 d4 xTr Fµν (x)

(2)

where the ﬁeld strength tensor is

Fµν (x) = ∂µ Aν (x) − ∂ν Aµ (x) + ig Aµ (x), Aν (x) , a fermionic action

(3)

SF [ψ, ψ, A] =

d4 xψ(x)(γµ Dµ + m)ψ(x)

(4)

where the covariant derivative is Dµ = ∂µ + igAµ ,

(5)

and source terms η, η and Jµ . The Euclidean γ matrices are hermitian and satisfy {γµ , γν } = 2δµν . With these conventions, γ5 ≡ γ1 γ2 γ3 γ4 . One representation is

0 iσi 1 0 0 1 γi = γ4 = γ5 = (6) −iσi 0 0 −1 1 0 where the σi are the Pauli matrices. The generating functional has the normalisation (7) Z[0] = DψDψDAe−SG [A]−SF [ψ,ψ,A] , i.e., Z with all sources set to zero. For the moment, we have ignored all issues of gauge-ﬁxing. From this generating functional all physical quantities can be obtained.

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The elements of the ﬁeld theory – the n-point Green’s functions – are constructed from (1) by diﬀerentiating the generating functional with respect to the sources, then setting those sources to zero. For example, the two-point fermion Green’s function – the quark propagator – is δ δ Z S(x, y) = δη(x) δη(y) η=0,η=0 1 = DψDψDAµ ψ(x)ψ(y)e−SQCD Z[0] = ψ(x)ψ(y) .

(8)

Although there is no known method to directly solve these equations, they can be cast into a form suitable for numerical methods by replacing continuous spacetime with a (four dimensional) grid or lattice. This reduces the number of degrees of freedom of the system from inﬁnity to merely a very large number. On the lattice the gauge ﬁelds, Aµ , are replaced by parallel transport operators or links 1 Aµ (x + atˆ µ)dt ∈ SU(N) (9) Uµ (x) = P exp iag 0

eiagAµ (x)

(10)

where P denotes path ordering and µ ˆ is a unit vector in the xµ direction. The lattice spacing, a, is the distance between sites on the lattice. It is the smallest unit of length in a lattice regularised theory. So Uµ (x) is the link connecting lattice sites x and x + aˆ µ. Taking the inverse of a link reverses its direction, and ˆ), where we have dropped since they are unitary, we see that: Uµ† (x) = U−µ (x + µ the lattice spacing, a, as implied. The gauge ﬁelds are replaced by link variables for the purpose of maintaining explicit gauge invariance on the ﬁnite lattice. It can be shown that the trace of any closed loop of links is gauge invariant, and it is from such elements, called Wilson loops, that a lattice gauge action can be built. The most basic Wilson loop (1×1) is called the plaquette, and it is deﬁned to be ˆ)Uµ† (x + νˆ)Uν† (x) . Pµν (x) = Uµ (x)Uν (x + µ

(11)

The plaquette also provides us with one lattice version of the ﬁeld strength tensor and the simplest lattice gauge action – called the Wilson gauge action – can be written: β 1 W † SG [U ] = Tr 1 − Pµν + Pµν (12) N 2 plaquettes

= SG + O(a2 ) .

(13)

The coupling constant of the theory has been absorbed into the parameter,

Quark Propagator from LQCD and Its Physical Implications

β=

2N g2

21

(14)

1 ). giving (13) a form that emphasises the analogy with statistical physics (cf. kT All lattice quantities are related to their continuum counterparts by terms proportional to some power of the lattice spacing, a, thus recovering continuum QCD in the limit a → 0. This means that there are many formally equivalent ways to deﬁne any one lattice quantity. Lattice fermion actions are constructed by a suitable discretisation of the covariant Dirac operator. The simplest is

Dµ (x, y) =

1 (Uµ (x)δy,x+ˆµ − Uµ† (x − µ)δy,x−ˆµ ) , 2a

(15)

which is equivalent to (5) up to terms of O(a2 ). This “na¨ıve” Dirac operator is, however, problematic on a ﬁnite lattice. We will return to this point later. However the discretisation is accomplished, the fermion action may be written in terms of a fermion matrix ψ α (x)Kαβ (x, y, U )ψβ (y) (16) SF [ψ, ψ, U ] = x,y

where the fermion and antifermion ﬁelds, ψ and ψ respectively, dwell on the lattice sites x and y. Our lattice gauge theory is now deﬁned by the generating functional 1 Z[η, η, J] = (17) DψDψDU e−SG [U ]−SF [ψ,ψ,U ]+ x ηψ+ηψ+Jµ Uµ Z[0] in terms of the link variables, Uµ . The measure, DU = dUµ (x)

(18)

x,µ

which properly deﬁnes integration over a compact group element is called the Haar measure [16]. The discretisation acts as an ultraviolet cutoﬀ, pmax = πa , so loops will be ﬁnite if perturbation theory is applied to the lattice generating functional. Furthermore, the lattice formulation maintains exact gauge invariance, unlike ﬁeld theory with a na¨ıve UV cutoﬀ [17]. The elements of the gauge ﬁeld, the links, are elements of the compact group SU (N) and are bounded above and below. If we limit ourselves to nonperturbative methods, gauge ﬁxing is not required for the calculation of physical observables in this formalism. Calculations may be performed using lattice perturbation theory or strong coupling expansions, but the most important method is Monte Carlo integration. Equation (17) resembles a statistical system and once the Grassman variables have been integrated out (see below) we may apply standard statistical methods. The functional integral is estimated by an average over a statistical ensemble of gauge conﬁgurations, Uµ (x). These conﬁgurations are generated with

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the Boltzmann distribution given by the exponential of the action. This is a form of importance sampling and is essential for the Monte Carlo integration to converge on a practical number of conﬁgurations. As the action is positive semi-deﬁnite, statistical uncertainty in the resulting expectation values vanishes like one over the square-root of the number of conﬁgurations. This is a generic feature of Monte Carlo methods. We will now brieﬂy discuss the calculation of the quark propagator (8). Making a the change of variables

ψ → ψ = ψ − ηK −1

ψ →ψ =ψ−K

−1

η

we can rewrite the fermionic part of the generating functional as −1 1 Z[η, η] = Dψ Dψ e x −ψ Kψ e x ηK η . Z[0]

(19) (20)

(21)

Under the rules of Grassman integration the measure is invariant under the above transformation, which enables us to write Z[η, η] =

−1 1 det Ke x ηK η . Z[0]

(22)

In the interacting theory, the fermion matrix K is a function of the gauge links. Returning to the full theory, we can write −1 1 Z[η, η, J] = (23) DU det K[U ]e−SG [U ]− x ηK η+Jµ Uµ Z[0] and

δ2 Z δη(x)δη(y)

= η=0,η=0

1 Z[0]

DU K[U ]−1 det K[U ]e−SG [U ] .

(24)

Thus we see that the calculation of the quark propagator amounts to the inversion of the fermion matrix T ψ(x)ψ(y) =

N 1 K(x, y, U (i) )−1 , N i=1

(25)

where the functional integral is estimated by a ﬁnite sum over an ensemble of (i) gauge conﬁgurations, {Uµ } which have themselves been generated with the distribution det K[U ] exp −SG [U ]. Computing the determinant of the fermion matrix, det K, in (24) is incredibly expensive. For this reason, most lattice simulations use the ansatz det K = 1 (26) which is equivalent to ignoring all fermion loops in the theory. This is called the quenched approximation.

Quark Propagator from LQCD and Its Physical Implications

23

3 Gauge Fixing We will brieﬂy discuss gauge ﬁxing on the lattice, its necessity and application. More thorough discussion can be found in [18, 19] and references contained therein. Continuum gauge ﬁelds, Aµ , are acted upon by gauge transformations, G(x) ∈ SU(N) such that † Aµ (x) → AG µ (x) = G(x)(Aµ (x) + i∂µ )G (x) ,

(27)

under which the action (2), is invariant. The link variables obey the simple gauge transformation law UµG (x) = G(x)Uµ (x)G†µ (x + µ ˆ) . (28) which leaves the lattice QCD action unchanged. This is equivalent to (27) up to terms of O(a2 ). In continuum QCD, gauge symmetry leaves the generating function illdeﬁned and gauge ﬁxing is required. This is usually accomplished through the Fadde’ev–Popov procedure, which ensures that only gauge ﬁelds satisfying some gauge condition are included in the functional integral, but also introduces new particles called ghosts. These ghosts are unphysical in that they do not appear in the spectrum. One of the celebrated facts of lattice gauge theory, at least in this formulation, is that gauge ﬁxing is unnecessary for calculating observable quantities; (17) is ﬁnite. Gauge ﬁxing is, however, necessary for various tasks including the study of gauge dependent quantities, such as the quark propagator. We consider the quark propagator in Landau and Laplacian gauges. Landau gauge ﬁxing is performed by enforcing the Lorentz gauge condition, µ ∂µ Aµ (x) = 0 on a conﬁguration by conﬁguration basis. This can be achieved in the continuum through the minimisation of the L2 norm 2 AG F G [A] = d4 xTr . (29) µ (x) µ

On the lattice this may be formulated as the maximisation of a functional such as, 1 F= Tr Uµ (x) + Uµ† (x) , (30) 2 x,µ which is equivalent to the continuum condition up to O(a2 ), or FImp =

1 4 1 G Uµ (x)UµG (x + µ Tr Uµ (x) + Uµ† (x) − ˆ) + h.c. . 2 x,µ 3 12u0

(31)

(where “h.c.” means hermitian conjugate) which has O(a4 ) errors. The meanﬁeld or tadpole factor

14 1 ReT rPµν u0 = , (32) 3

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compensates for leading errors in lattice perturbation theory [20]. In this work, conﬁgurations generated using the Wilson gauge action will be gauge-ﬁxed using the standard functional, (30) and conﬁgurations generated with the improved action will be gauge-ﬁxed with the improved functional, (31). For any given conﬁguration there are, in general, many local extrema of the gauge ﬁxing functional. This was ﬁrst noticed by Gribov [21] and is true in the continuum and on the lattice for any local gauge ﬁxing functional. This means that the gauge ﬁxed conﬁguration is not uniquely deﬁned. The literature on these “Gribov copies” is huge; see [22, 23, 24] for some recent additions. This ambiguity remains a largely “in principle” problem as it has never been demonstrated to aﬀect QCD. The ghost propagator has been shown to be sensitive to Gribov copies in SU (2) lattice gauge theory [25] and various eﬀects have been attributed to Gribov copies in various models [26]. Despite these problems we persist in studying gauge dependent quantities in Landau gauge, primarily because it is easy to compare results with those derived in the same gauge from other methods. An alternative, which has recently received considerable attention is Laplacian gauge [7]. Laplacian gauge is a nonlinear gauge ﬁxing that respects rotational invariance, has been seen to be smooth, yet is free of Gribov ambiguity. It is also computationally cheaper then Landau gauge. There is, however, more than one way of obtaining such a gauge ﬁxing in SU (N). The three implementations of Laplacian gauge ﬁxing employed here are (in our notation): 1 ∂ 2 (I) gauge (QR decomposition), used by Alexandrou et al. [9]. 2 ∂ 2 (II) gauge, where the Laplacian gauge transformation is projected onto SU(3) by maximising its trace [11]. 3 ∂ 2 (III) gauge (Polar decomposition), the original prescription described in [7] and tested in [8]. All three versions reduce to the same gauge in SU (2). The gauge transformations employed in Laplacian gauge ﬁxing are constructed from the lowest eigenvectors of the covariant lattice Laplacian ∆(U )(x, y)ij v(y)sj = λs v(x)si , (33) y

where ∆(U )(x, y)ij ≡

j

2δ(x − y)δ ij − Uµ (x)ij δ(x + µ ˆ − y) − Uµ (y)†ij δ(y + µ ˆ − x) , µ

(34) i, j = 1, . . . , N for SU (N) and s labels the eigenvalues and eigenvectors. Under gauge transformations of the gauge ﬁeld, Uµ (x) → UµG (x) = G(x)Uµ (x)G† (x + µ) ,

(35)

the eigenvectors of the covariant Laplacian transform as v(x)s → G(x)v(x)s

(36)

Quark Propagator from LQCD and Its Physical Implications

25

and this property enables us to construct a gauge ﬁxing that is independent of our starting place in the orbit of gauge equivalent conﬁgurations. The three implementations discussed diﬀer in the way that the gauge transformation is constructed from the above eigenvectors. In all cases the resulting gauge should be unambiguous so long as the Nth and (N+1)th eigenvectors are not degenerate and the eigenvectors can be satisfactorily projected onto SU (N). For SU (2) the eigenvectors of the covariant Laplacian are related such that they form the columns of a matrix proportional to an element of SU (2); the projection is thus trivial. This is not the case for SU (N), N > 2. In that case we can think of the projection method as deﬁning its own, unambiguous, Laplacian gauge ﬁxing. In the original formulation [7], ∂ 2 (III) in our notation, the lowest N eigenvectors are required to gauge ﬁx an SU (N) gauge theory. These form the columns of a complex N × N matrix, M (x)ij ≡ v(x)ji

(37)

which is then projected onto SU (N) by polar decomposition. Speciﬁcally, it is possible to express M in terms of a unitary matrix and a positive hermitian matrix: M = U P . This decomposition is unique if P = (M † M )1/2 is invertible, which will be true if M is non-singular, i.e., if the eigenvectors used to construct M are linearly independent. The gauge transformation G(x) is then obtained by factoring out the determinant of the unitary matrix G† (x) = U (x)/ det[U (x)] .

(38)

The gauge transformation G(x) obtained in this way is used to transform the gauge ﬁeld (i.e., the links) to give the Laplacian gauge-ﬁxed gauge ﬁeld. G(x) can be uniquely deﬁned by this presciption except on a set of gauge orbits with measure zero (with respect to the the gauge-ﬁeld functional intregral). Note that if we perform a random gauge transformation Gr (x) on the initial gauge ﬁeld used to deﬁne our Laplacian operator, then we will have v(x)s → v (x)s = Gr (x)v(x)s and M (x) → M (x) = Gr (x)M (x). We see that P ≡ (M † M )1/2 → P = P and hence G(x) → G (x) = G(x)G†r (x). When this is applied to the transformed gauge ﬁeld it will be taken to exactly the same point on the gauge orbit as the original gauge ﬁeld went to when gauge ﬁxed. Thus all points on the gauge orbit will be mapped to the same single point on the gauge orbit after Laplacian gauge ﬁxing and so it is a complete (i.e., Gribov copy free) gauge ﬁxing. This method was investigated for U (1) and SU (2) [8]. It is clear that any prescription for projecting M onto some G† (x), which preserves the property G(x) → G (x) = G(x)G†r (x) under an arbitrary gauge transformation Gr (x), will also be a Gribov copy free Laplacian gauge ﬁxing. Every diﬀerent projection method with this property is an equally valid but distinct form of Laplacian gauge ﬁxing. The next approach was used in [9], and we shall refer to it as ∂ 2 (I) gauge. There it was noted that only N − 1 eigenvectors are actually required. To be

26

P.O. Bowman et al.

concrete, we discuss N = 3. First, apply a gauge transformation, G(x)1 , to the ﬁrst eigenvector such that

G(x)1 v(x)1 1 = ||v(x)1 || (39) and

G(x)1 v(x)1 2 = G(x)1 v(x)1 3 = 0 ,

(40)

where subscripts label the vector elements, i.e., the eigenvector – with dimension 3 – is rotated so that its magnitude is entirely in its ﬁrst element. Another gauge transformation, G(x)2 , rotates the second eigenvector, v(x)2 , such that

G(x)2 v(x)2 2 = (v22 )2 + (v32 )2 , (41) and

G(x)2 v(x)2 3 = 0 .

(42)

This second rotation is an SU (2) subgroup, which does not act on v12 (x). The gauge ﬁxing transformation is then G(x) = G(x)2 G(x)1 . Compare this to QR decomposition, where a matrix, A, is rewritten as the product of an orthogonal matrix and an upper triangular matrix. The gauge transformations are thus like Householder transformations. In addition, we explore a third version, ∂ 2 (II) gauge, where G(x) is obtained by projecting M (x) onto SU (N) by trace maximisation. M (x) is again composed of the N lowest eigenvectors and the trace of G(x)M (x)† is maximised by iteration over Cabbibo–Marinari subgroups.

4 Staggered Quark Actions Consider the na¨ıve quark action (15) at tree-level, i.e., with Uµ (x) = 1 everywhere. In momentum space, the propagator is 1 γ sin(p µ µ ) + m0 µ

S0 (p) = where the lattice momentum 2πnµ pµ = Lµ

nµ ∈

−Lµ Lµ , 2 2

(43)

(44)

covers the ﬁrst Brillouin zone. In the massless case, this propagator has a pole at pµ = 0, π for each µ. So a mode with momentum near zero will describe a low momentum continuum quark, but so will a mode with momentum near π. That means the na¨ıve quark action describes 2d = 16 species of degenerate fermions. This is the (in)famous “doubling problem.” Note that QCD only has asymptotic freedom for Nf < 33 2 , and 16 ﬂavours is rather close to this transition. Another consequence of doubling is the loss of the axial anomoly. In continuum QCD, the

Quark Propagator from LQCD and Its Physical Implications

27

axial symmetry is broken by quantum eﬀects, but on the lattice the doublers of the na¨ıve quark action conspire such that the symmetry breaking terms exactly cancel. As with gauge actions, there are many possible lattice fermion actions that reduce to the desired continuum action. The best known solution to the doubling problem is the Wilson fermion action, which uses a chiral symmetry breaking term to give the doublers mass of O(a−1 ) (the cutoﬀ). This is described in many places [16]. Here we will brieﬂy discuss the other main type of lattice fermion action, the Kogut–Susskind or Staggered action. This presentation follows that of [16, 27]. The principle observation that leads to the staggered formalism is that the doubling problem occurs at the edge of the Brillouin zone. If the available momenta only spanned half the Brillouin zone, there would be no problem. The diﬃculties lie in recovering the correct theory in the continuum limit. Halving the Brillouin zone is equivalent to doubling the “eﬀective” lattice spacing, so imagine a situation where the ﬁeld of a particular fermion species is spaced out by two links in every direction. We shall see how it is possible to reduce the 2d Dirac spinors – each with 2d/2 components – to 2d/2 by distributing the degrees of freedom across the hybercube. This process is called “spin diagonalisation.” For concreteness, we shall restrict ourselves to d = 4 (Euclidean) space-time dimensions. Consider the local change of variables ψ(x) = Γ (x)χ(x) ψ(x) = χ(x)Γ † (x)

(45)

where Γ (x) is a unitary 4 × 4 matrix that satisﬁes Γ † (x)γµ Γ (x + µ) = ηµ (x) .

(46)

Γ (x) = γ1x1 γ2x2 γ3x3 γ4x4

(47)

This is accomplished by

where the staggered phases are: ηµ (x) = (−1)

ν<µ xν

. Explicitly

η1 (x) = 1 η2 (x) = (−1)x1 η3 (x) = (−1)x1 +x2 η4 (x) = (−1)x1 +x2 +x3 Note that (45) is satisﬁed if χ is a Grassman variable with just one component (although it could have more). We can now write the Kogut–Susskind action S=

1 ηµ (x)χ(x)Dµ χ(x) + m χ(x)χ(x) 2 x,µ x

(48)

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where D is the lattice covariant derivative given in (15). The Dirac structure has been completely absorbed into the staggered phases, η. To see more clearly what has been done, consider the Staggered action in terms of the hypercubes. We re-label the position vector xµ = 2hµ + ρµ where hµ = 0, . . . ,

Lµ 2

(49)

− 1 and ρµ = 0, 1. Then we deﬁne χρ (h) = χ(x)

(50)

giving us a relationship between the single component Grassman ﬁelds on a lattice with spacing a and 16 component Grassman ﬁelds on a lattice with spacing 2a. From these components one then constructs the four ﬂavoured3 , four component Dirac ﬁelds by taking the appropriate linear combinations: (Γρ )αf χρ (h) , (51) ψαf (h) = N ρ

where the Γρ are now

Γρ = γ1ρ1 γ2ρ2 γ3ρ3 γ4ρ4

(52)

and N is a normalisation. So spin-diagonalisation has reduced our 16 Dirac ﬁelds to four distributed around a hypercube and the continuum spin-ﬂavours are reconstructed from linear combinations of the staggered. One major advantage of the staggered formulation over the Wilson is that it retains a U(1) × U(1) chiral symmetry. It is not the same symmetry as continuum QCD, but it protects the theory from additive mass renormalisation and ensures that the pion is a genuine Goldstone boson. We will work mostly in momentum space, so it will be necessary to construct the Fourier transform of the Staggered action. In analogy to the “blocking” we performed in coordinate space, we write kµ =

2πnµ Lµ

| nµ = 0, . . . , Lµ − 1

(53)

as kµ = pµ + παµ , where pµ =

2πmµ Lµ

|

mµ = 0, . . . , αµ = 0, 1,

(α is like ρ above) and deﬁne 3

k

≡

1 V

k.

Lµ −1 2

(54) (55)

Then

In modern parlance the term “taste” is used to label the would-be degenerate fermion ﬁelds, “ﬂavour” referring to the staggered ﬁelds themselves [30].

Quark Propagator from LQCD and Its Physical Implications

= k

(56)

p α =0 µ

eik·x χ(k) =

χ(x) =

1

29

k

ei(p+πα)·x χα (p) .

(57)

p α

Using δ αβ = Πµ δαµ βµ | mod 2 1 if ν < µ θν (µ) = , 0 otherwise

(58) (59)

we can deﬁne (γ µ )αβ = (−1)αµ δ α+θ(µ),β

(60)

{γ µ , γ ν }αβ = 2δµν δ αβ

(61)

where the γ µ satisfy

γ †µ

=

γ Tµ

=

γ ∗µ

= γµ ,

(62)

forming a “staggered” Dirac algebra. Putting all this together, we can derive a momentum space expression for the KS action, χα (p) i (γ µ )αβ sin(pµ ) + mδ αβ χβ (p) . (63) S= p αβ

µ

This will be very convenient as it casts the action into a familiar form. The quark action that will be of particular interest to us is the Asqtad quark action [28]. It is a type of fat-link Staggered action using three-link, ﬁve-link and seven-link terms to cancel O(a2 ) errors and improve ﬂavour symmetry. There are a number of ingredients to this improvement. In the standard KS action, the derivative term is estimated by a simple nearest-neighbours ﬁnite diﬀerence. This can be improved by the addition of a three-link, next-to-nearest-neighbours piece, called the Naik term [29]. The coeﬃcients are chosen to eliminate O(a2 ) errors, improving the rotational symmetry of the action. This resulting action is 9 1 Uµ (x)δy,x+ˆµ − Uµ† (x − µ)δy,x−ˆµ ηµ (x)χ(x) 2 x,y,µ 8 1 † U3µ (x)δy,x+3ˆµ − U3µ − (x − 3µ)δy,x−3ˆµ χ(y) + m χ(x)χ(x) 24 x

S=

(64)

Such an action was tested in [27]. Flavour symmetry breaking in this action is still large, but this can be reduced by “fattening” with three, ﬁve and seven link terms [28, 31]. The result of this fattening is to reduce the coupling of the quarks to hard gluons with momentum of the order of the cutoﬀ, as this type of gluon

30

P.O. Bowman et al.

exchange results in the ﬂavour-changing interactions that give rise to ﬂavour symmetry breaking. These terms can, however, introduce errors into the lowmomentum behaviour of the quark propagator, but these are compensated for by the addition of a planar ﬁve-link Lepage term [32]. All coeﬃcients are tadpole improved, producing the rather sophisticated Asqtad (a2 , tadpole improved) action. At tree-level (i.e. no interations, links set to the identity), the staples in this action make no contribution, so the action reduces to the Naik action. We can perform the same analysis as above to derive the momentum space action which has the tree-level form

9 1 sin(pµ ) − sin(3pµ ) + mδ αβ χβ (p) . χα (p) i (γ µ )αβ sin(pµ ) S= 8 24 p µ αβ

(65)

5 The Lattice Quark Propagator Equation (25) says that the quark propagator is calculated by inverting the fermion matrix. The calculation proceeds thus: 1 A statistical ensemble of lattice gauge conﬁgurations is generated by some method, in this case the pseudo heat-bath algorithm. All conﬁgurations used here will be quenched (see above). 2 Each conﬁguration is rotated to the desired gauge: Landau or Laplacian. 3 On each conﬁguration, the quark propagator is calculated by S(x, y)K(y, z) = δ(x, z) (66) y

where K is the fermion matrix for either the regular Kogut–Susskind quark action or the Asqtad quark action. The inversion is computed by a suitable method, in this case a multi-mass conjugate gradient algorithm [33]. 4 The propagator is Fourier transformed to momentum space. 5 The result is averaged over all gauge conﬁgurations in the ensemble. All quantities calculated on the lattice are dimensionless. While it is possible (in some ways, even preferable) to calculate dimensionless ratios, e.g. ratio of the pion mass to that of the rho: mπ /mρ , we will be mostly interested in dimensionful quantities. To set the scale, the relevant quantity is the cutoﬀ, that is, the lattice spacing, a. Note that it is the coupling, β, that is set in the lattice simulation, not the lattice spacing itself. The lattice spacing must be deduced from some physical quantity such as mρ . We will use the static quark potential [34]. All the simulations discussed will be quenched, as described above. Since the quenched approximation will have a diﬀerent eﬀect on diﬀerent quantities, it will also introduce some ambiguity into any determination of that lattice spacing. For this reason, it must be remembered that there is a ∼10% systematic uncertainty

Quark Propagator from LQCD and Its Physical Implications

31

on all quantities mentioned here in physical units. This is in addition to the quoted statistical uncertainty. In the continuum, Lorentz invariance allows us to decompose the full propagator into Dirac vector and scalar pieces, so in momentum space

or, alternatively,

S −1 (p2 ) = iA(p2 )γ · p + B(p2 )

(67)

S −1 (p2 ) = Z −1 (p2 )[iγ · p + M (p2 )] .

(68)

This is the bare propagator which, once regularised, is related to the renormalised propagator through the renormalisation constant S(a; p2 ) = Z2 (a; µ)S ren (µ; p2 ) ,

(69)

where a is some regularisation parameter, e.g. lattice spacing. Asymptotic freedom implies that, as p2 → ∞, S(p2 ) reduces to the free propagator S −1 (p2 ) → iγ · p + m0 ,

(70)

where m0 is the bare quark mass. From (63) we can see that, in momentum space, the tadpole improved, treelevel form of the KS quark propagator is (including spinor indices, but omitting colour) −1 Sαβ (p; m) = u0 i (γ µ )αβ sin(pµ ) + mδ αβ (71) µ

where pµ is the discrete lattice momentum given by −Lµ Lµ 2πnµ pµ = , nµ ∈ . Lµ 4 4

(72)

Assuming that the full propagator retains this form (in analogy to the continuum case) we write −1 (p) = i (γ µ )αβ sin(pµ )A(p) + B(p)δ αβ Sαβ µ

=Z

−1

(p) i

(γ µ )αβ sin(pµ ) + M (p)δ αβ

.

(73)

µ

For the KS action, it is thus convenient to deﬁne qµ ≡ sin(pµ ) .

(74)

We can then decompose the inverse propagator 1 Tr{γ · qS −1 } 16Nc iq 2 Z(q) M (q) = Tr{S −1 } , 16Nc

Z −1 (q) =

(75) (76)

32

P.O. Bowman et al.

where the factor of 16 comes from the trace over the spin-ﬂavour indices of the staggered quarks and Nc from the trace over colour. In actual fact we are able to construct these quantities without inverting the propagator; the above is intended as an instructive example. See the appendix in [3] for details. Note that the decomposition means that we cannot directly measure either of the invariants at zero four-momentum. Comparing (68) and (73) we see that dividing out q 2 in (75) is analagous to dividing out p2 in the continuum and ensures that that Z has the correct asymptotic behaviour. So by considering the propagator as a function of qµ , we ensure that the lattice quark propagator has the correct tree-level form, i.e., 1 (77) S tree (qµ ) = iγ · q + m and hopefully better approximates its continuum behaviour. This is the same philosophy that has been used in studies of the gluon propagator [35, 36] and Clover and Overlap quark propagators [1, 2, 5, 6]. Turning now to the Asqtad action, from (65) we see that the quark propagator with this action has the tree-level form 1 −1 Sαβ (p; m) = u0 i (γ µ )αβ sin(pµ ) 1 + sin2 (pµ ) + mδ αβ , (78) 6 µ so we repeat the above analysis, this time deﬁning 1 2 qµ ≡ sin(pµ ) 1 + sin (pµ ) . 6

(79)

The asymptotic behaviour of the quark propagator, on the lattice and in the continuum, will be discussed in more detail later.

6 Analysis of Lattice Artefacts for Staggered Quarks The quark propagator has been calculated using the Kogut–Susskind and Asqtad actions on Wilson gauge conﬁgurations at β = 5.85 and using the Asqtad action on O(a2 ) improved gauge conﬁgurations at β = 4.38 and 4.60. Details are given in Table 1. The bare quark masses were chosen to be the same in physical units on each lattice, but diﬀerences between the actions will mean that the running masses will not quite match at ﬁnite lattice spacing. The √lattice spacings quoted were determined from the static quark potential using σ = 440 MeV [34]. 6.1 Tree-Level Correction As mentioned above, the idea of “kinematic” or “tree-level” correction has been used widely in studies of the gluon propagator [35, 36] and the quark propagator [1, 2, 5, 6] and we investigate its application to our quark propagators.

Quark Propagator from LQCD and Its Physical Implications

33

Table 1. Details of the lattice simulations Gauge Action

β

Dimensions

a (fm)

Masses (ma)

Wilson

5.85

163 × 32

0.130

0.0125, 0.025, 0.0375, 0.05,

4.60

12 × 24

0.0625, 0.075 Improved

3

0.124

0.012, 0.018, 0.024, 0.036, 0.048, 0.072

Improved

4.60

163 × 32

4.38

16 × 32

0.124

0.012, 0.018, 0.024, 0.036, 0.048, 0.072, 0.108, 0.144

Improved

3

0.166

0.016, 0.024, 0.032, 0.048, 0.064, 0.096, 0.143, 0.191, 0.278

To help us understand the lattice artefacts, we separate the data into momenta lying entirely on a spatial cartesian direction (squares), along the temporal direction (triangles), the four-diagonal (diamonds) or some other combination of directions (circles). All the ﬁgures are shown in physical units, as this makes it easiest to compare results from the diﬀerent lattices. For the moment we shall restrict ourselves to Landau gauge. The Z function is plotted for the KS action in Fig. 1 for a representative quark mass, comparing the results using p and q. Firstly, we comment on the general features of this function. Z is reasonably ﬂat in the ultraviolet, reaching a value around one. This agrees with our expectation from continuum perturbation theory. The function has a dip in the infrared, which has also been seen in Dyson–Schwinger equation studies [37]. In the top of Fig. 1 we see substantial hypercubic artefacts (in particular look at the diﬀerence between the diamond and the triangle at around 2.5 GeV). We can suggest that this is caused by violation of rotational symmetry because the agreement between triangles and squares suggests that ﬁnite volume eﬀects are small in the region of interest. In the plot below, where q has been used, we see some restoration of rotational symmetry. The same study is made for the Asqtad action in Fig. 2. The general properties are the same as for the KS action except that the infrared dip is more pronounced. The data is signiﬁcantly smoother and rotational symmetry is much better. Nevertheless there are some signs of rotational symmetry breaking when p is used, and this is almost entirely eliminated when q is the abscissa. The analysis of Sect. 5 does not indicate any preferred momentum variable for the mass function. The question is, “What choice of momentum variable results in the most rapid convergence to the continuum limit?” To explore this we show the quark mass function at two lattice spacings in Fig. 3. The eﬀect is not strong, and for the ﬁner lattice there is negligible asymmetry in any case, but the “action momentum”, q, does improve rotational symmetry on the coarser lattice compared with the standard momentum, p. As remarked above,

34

P.O. Bowman et al.

Fig. 1. Z function for quark mass ma = 0.05 (m 76 MeV) for the KS action in Landau gauge. Top ﬁgure is plotted using the standard lattice momentum, p and the bottom uses the “action” momentum, q. Note that this choice aﬀects only the horizontal scale

one choice of bare mass will produce somewhat diﬀerent running masses in the interacting theory, and this accounts for the slight mismatch between these two mass functions. 6.2 Comparison of the Actions In Fig. 4 the mass function is plotted, in Landau gauge, for both actions with quark mass ma = 0.05. We see that the KS action gives a much larger value for M (0) than the Asqtad action, which is consistent with the mass diﬀerence seen in spectroscopic calculations [27]. The KS action also approaches asymptotic behaviour more slowly. The mass function from the Asqtad action shows

Quark Propagator from LQCD and Its Physical Implications

35

Fig. 2. Z function for quark mass ma = 0.05 (m 76 MeV) for the Asqtad action in Landau gauge. Top ﬁgure is plotted using the standard lattice momentum, p and the bottom uses the “action” momentum, q

better rotational symmetry. The relative improvement between the two actions increases as quark mass decreases. Next, we look at the eﬀect of improving the gauge action. In Fig. 5 we show the mass function from the Asqtad quark action and the Wilson gauge action with standard gauge ﬁxing (top) and the Symanzik improved gauge action with O(a2 ) improved gauge ﬁxing (bottom). We see that improving the gauge action does indeed improve the quark propagator, both by providing smoother conﬁgurations and because the improved quark action is able to take advantage of the improved behaviour of the gluon action. This suggests that the improvement terms in the Asqtad action are eﬀective at removing O(a2 ) errors.

36

P.O. Bowman et al.

Fig. 3. Choice of momentum: Asqtad quark mass function at β = 4.60 (a 0.124 fm) and β = 4.38 (a 0.167 fm) for bare quark mass m 114 MeV. Top ﬁgure uses the na¨ıve lattice momentum, while the bottom ﬁgure uses the momentum determined from the tree-level behaviour of the action

6.3 Finite Volume The calculation of any lattice quantity may be aﬀected by the necessarily ﬁnite volume. All of the lattices used here are asymmetric in that they are longer in one direction, which we arbitrarily call the time direction. As we have seen above, this asymmetry can be useful in detecting ﬁnite volume artefacts. Another way is to perform the calculation on two or more lattices of diﬀerent size but with ﬁxed lattice spacing. We have done this for the quark propagator and the results are shown in Fig. 6. The change in volume has had a negligible eﬀect on the mass function, from which we deduce that, on the lattices used in this study, ﬁnite volume eﬀects are small. There is, however, a signiﬁcant change in the infrared behaviour of the Z function. It would appear that some of the suppression could be an artefact. The quark mass may have some aﬀect on both of these statements.

Quark Propagator from LQCD and Its Physical Implications

37

Fig. 4. Mass function for quark mass ma = 0.05 (m 76 MeV), KS action (top) and Asqtad action (bottom) in Landau gauge

Fig. 5. O(a2 ) errors: Asqtad quark mass function for mass m 57 MeV, with the Wilson gauge action at β = 5.85 (a 0.13 fm) (top) and O(a2 ) improved gauge action at β = 4.60 (a 0.124 fm) (bottom) in Landau gauge

38

P.O. Bowman et al.

Fig. 6. Finite volume eﬀects: Asqtad quark mass (top) and Z (bottom) functions for mass m 57 MeV, with the O(a2 ) improved gauge action at β = 4.60 (a 0.124 fm) in Landau gauge. Comparison on 123 × 24 lattice (open circles) and 16 × 32 lattice (ﬁlled squares)

7 Overlap Quark Actions Another way of constructing quarks on the lattice is by the overlap fermion formalism [38, 39, 40, 41]. Overlap fermions are free from doublers, that is, there is only one continuum fermion for each lattice fermion, and they realise an exact chiral symmetry even at non-zero lattice spacing. A consequence of this is that they must be O(a) improved, since any O(a) error would necessarily violate chiral symmetry [42]. This lattice realisation of chiral symmetry is called the Ginsparg–Wilson relation [43]. An overlap-Dirac operator is constructed in the following fashion: the massless coordinate-space overlap-Dirac operator can be written in dimensionless lattice units as [41]

Quark Propagator from LQCD and Its Physical Implications

39

1 [1 + γ5 Ha ] , (80) 2 where Ha is an Hermitian operator that depends on the background gauge ﬁeld and has eigenvalues ±1. Any such D(0) is easily seen to satisfy the Ginsparg– Wilson relation (81) {γ5 , D(0)} = 2D(0)γ5 D(0) D(0) =

and, provided that its Fourier transform at low momenta is proportional to the momentum space covariant derivative, will satisfy a lattice form of chiral symmetry. It immediately follows from (80) that D† (0)D(0) = D(0)D† (0) = and that

1 † [D (0) + D(0)] 2

(82)

D† (0) = γ5 D(0)γ5 .

(83) −1 It also follows easily that {γ5 , D (0)} = 2γ5 and by deﬁning D (0) ≡ [D−1 (0) − 1] we see that the Ginsparg–Wilson relation can also be expressed in the form −1 (0)} = 0 . (84) {γ5 , D −1

With a suitable choice of Ha (x, y), or kernel, (80) represents a single, massless Dirac fermion. The standard choice of kernel is the sign function of the Hermitian Wilson Dirac operator, Hw , Ha = (Hw ) ≡ Hw /|Hw | = Hw /(Hw† Hw )1/2 ,

(85)

where Hw (x, y) = γ5 Dw (x, y), Dw being the usual (non-Hermitian) Wilson-Dirac operator. However, in the overlap formalism the Wilson mass parameter mw is large and enters the Wilson-Dirac operator with a negative sign. The Wilson mass acts here as a regulator and is chosen such that mw a ∈ (m1 a, m2 a)

(86)

The condition mw a > m1 a ensures that the Wilson operator has zero-crossings and, in turn, that D(0) has nontrivial topological charge. At tree-level, m1 a = 0 and numerical studies have found that in the interacting case, m1 mc , where mc is the usual critical mass for Wilson fermions [44]. The constraint mw a < m2 a arises from the fact that Wilson doublers reappear above this point. At tree-level m2 a = 2. In summary, we use here Hw (−mw ) = γ5 Dw (−mw ). The mean-ﬁeld improved Wilson-Dirac operator can be written as

Dw (x, y) = (−mw a) + 4r δx,y 1 (r − γµ )Uµ (x)δy,x+ˆµ + (r + γµ )Uµ† (x − aˆ − µ)δy,x−ˆµ 2 µ u0 Uµ (x) = δx,y − κ δy,x+ˆµ (r − γµ ) 2κ u0 µ µ) Uµ† (x − aˆ . (87) + (r + γµ ) δy,x−ˆµ u0

40

P.O. Bowman et al.

The negative Wilson mass (−mw a) is then related to κ by κ≡

u0 , 2(−mw a) + (1/κc )

(88)

and mean-ﬁeld improvement means that we can use the tree-level value κc = 18 . The Wilson parameter is typically chosen to be r = 1 and we will also use r = 1 here in our numerical simulations. With this choice for the sign function, we have the overlap operator † −1/2 1 , 1 + Dw Dw Dw 2

D(0) =

(89)

and it’s not immediately obvious what connection this has with a Dirac particle. In coordinate space the Wilson-Dirac operator has the form Dw = ∇ / + (r/2)∆ + (−mw a), where ∇µ is the symmetric dimensionless lattice ﬁnite diﬀerence operator, and ∆ is the dimensionless lattice Laplacian operator. Setting the links to the identity, i.e. the free ﬁeld, gives ←

→

← →

Dw = (1/2)( ∂ µ + ∂ µ )γµ + (r/2)(− ∂ µ ∂ µ ) + (−mw a) ,

(90)

where the partial derivatives are the forward and backward lattice ﬁnite diﬀerence operators, f (x) − f (x + a) a → f (x − a) − f (x) . ∂ µ f (x) = a ← ∂µ

f (x) =

Hence we have from (89) that ∇ / + (r/2)∆ − mw a 1 1+ D(0) = 2 (mw a)2 + O(∂ 2 ) →

∇ / , 2mw a

(91)

where the last line is a limit approached when operating on very smooth functions such that only ﬁrst powers of derivatives are kept. The reason for needing a negative Wilson mass (−mw a) is now apparent, i.e., it is needed to cancel the 1 in D(0). We see that at suﬃciently ﬁne lattice spacings and for pa 1 Dc (0) ≡ 2mw D(0) ,

(92)

where Dc (0) in the continuum limit becomes the usual fermion covariant deriv/ as a → 0. To extract ative contracted with the γ-matrices, i.e., Dc (0) → ∇ the propagator for a massless external quark it is necessary to make a simple subtraction

Quark Propagator from LQCD and Its Physical Implications

41

1 1 −1 1 −1 D (0) . D (0) − 1 = = 2mw 2mw 2mw (93) This deﬁnition of the massless overlap quark propagator follows from the overlap formalism and ensures that the massless quark propagator anticommutes with γ5 , i.e., {γ5 , S bare (0)} = 0 just as it does in the continuum [41]. In order to complete the discussion we now generalise the above to the massive case. This is done by simply adding a bare mass to the inverse of the massless propagator, (94) (S bare )−1 (m) ≡ (S bare )−1 (0) + m . c−1 (0) ≡ Dc−1 (0) − S bare (0) ≡ D

It is conventional to deﬁne the dimensionless overlap mass parameter µ≡

m 2mw

(95)

c−1 (µ) in analogy with (93) and then deﬁne D c−1 (µ) , S bare (m) ≡ D

(96)

c−1 (µ) is a generalisation of D c−1 (0) to the case of nonzero mass. Extending i.e., D the analogy with the massless case we introduce D(µ) and D(µ), which are generalised versions of D(0) and D(0), through the deﬁnitions c−1 (µ) ≡ D

1 −1 D (µ) 2mw

−1 (µ) ≡ and D

1 −1 D (µ) − 1 . 1−µ

(97)

We can now use (94) and the above deﬁnitions to obtain an expression for D(µ). From (94), (95) and (97) we see that we must have D(µ) = D(0) + µ. Inverting this gives

−1 1 −1 D (µ) − 1 = D(0) +µ (98) 1−µ and so

−1

−1 D−1 (µ) = (1 − µ) D(0) +µ + 1 = D(0) + 1 D(0) +µ . Inverting gives

−1 D(µ) = D(0) + µ D(0) +1

(99)

−1 and also D(0) = D(0) D(0) +1 (100)

and so ﬁnally

−1 D(µ) = (1 − µ)D(0) + µ D(0) + 1 D(0) +1 1 = (1 − µ)D(0) + µ = [1 + µ + (1 − µ)γ5 Ha ] . 2

(101)

We have recovered the standard expression for D(µ) found, for example, in [41] and elsewhere. We see that the massless limit, m → 0, implies that µ → 0 and −1 (0) and D c−1 (µ) → D c−1 (0). For non-negative bare −1 (µ) → D D(µ) → D(0), D

42

P.O. Bowman et al.

mass m we require µ ≥ 0. In order that the above expressions and manipulations be well-deﬁned we must have µ < 1. Hence, 0 ≤ µ < 1 deﬁnes the allowable range of bare masses. Due to the square-root in (89), the Overlap operator at any one point on the lattice depends on every point on the lattice. This makes it particularly expensive to calculate. Instead we approximate the sign function with the optimal rational polynomial approximation of Zolotarev [45, 46]. A 14th order polynomial is seen to be suﬃcient. As the approximation can be poor when the argument is near zero, the lowest 10–20 eigenvectors are ﬁrst projected out and handled exactly. The speed with which the Overlap operator can be evaluated is strongly dependent on the density of low-lying eigenmodes and some improvement can be made through diﬀerent gauge actions [47] and diﬀerent Overlap kernels [48, 49].

8 Analysis of Lattice Artefacts for Overlap Quarks The quark propagator has been calculated using the Overlap-Dirac operator with the Wilson kernel on O(a2 ) improved gauge conﬁgurations at β = 4.286 and 4.60. There are also preliminary results from an ensemble with β = 4.80. These lattices were selected to have the same physical volume of approximately 1.53 × 3.0 fm4 . Details are given in Table 2. Again, the bare quark masses were chosen to be the same in physical units on each lattice, but at these couplings the resulting running masses will be slightly diﬀerent. According to (95), the chosen mass parameters, µ, approximately correspond to bare masses in physical units of m = 106, 124, 142, 177, 212, 266, 354, 442, 531 and 620 MeV. For the regulator mass in the Wilson-Dirac operator κ = 0.19163 was used for all three lattices, corresponding to mw a = 1.391. Table 2. Details of the lattice simulations with the Overlap quark action β

Dimensions

a (fm)

Mass Parameter (µ)

4.286

83 × 16

0.190

0.0368, 0.0429, 0.0490, 0.0613, 0.0735,

4.60

12 × 24

0.0919, 0.1226, 0.1532, 0.1839, 0.2145 3

0.124

0.024, 0.028, 0.032, 0.040, 0.048, 0.060, 0.080, 0.100, 0.120, 0.140

4.80

163 × 32

0.093

0.018, 0.021, 0.024, 0.030, 0.036, 0.045, 0.060, 0.075, 0.090, 0.105

8.1 Tree-level Correction The tree-level form of the overlap quark propagator was derived in the appendix of [5]. It can also be deduced by calculating the quark propagator on a trivial conﬁguration (i.e., by setting all links to the identity). The result is shown in Fig. 7.

Quark Propagator from LQCD and Its Physical Implications

43

Fig. 7. Tree-level propagator: This ﬁgure shows the Dirac vector part of the Overlap propagator, qµ (open circles) as a function of the momentum, pµ calculated on the trivial conﬁguration. The line shows the ideal, continuum case. For comparison, the same part of the Asqtad (Naik) propagator is also shown (open triangles)

It can be seen from Fig. 7 that q approaches p for small momenta, which is necessary for the Overlap operator to describe a Dirac fermion, and q takes a single value for each p, which means that there are no doublers. In the large momentum region, p and q diverge rapidly, which means that – unlike the staggered case – the choice of kinematic momentum can strongly aﬀect the propagator. Unfortunately, for the Overlap propagator we do not see the clear restoration of rotational symmetry when the kinematic momentum is used. For this reason, we need to be a little more sophisticated in our approach to tree-level correction. The question is: for what choice of momentum does the Overlap quark propagator most rapidly approach its continuum form? To answer this we show the quark Z function for each of the lattice spacings available in Fig. 8. The data has been cylinder cut which trims some of the most obvious lattice artefacts. This is useful for when comparing two data sets and is explained in [36]. The three data sets do indeed appear to be converging to a common result when considered as a function of q, while there is a lack of consensus as a function of p. We also show a direct comparison with the Asqtad results in Fig. 10. For closest agreement, the kinematic momenta for the respective actions should be used. As with the Asqtad action, the tree-level analysis does not suggest an optimal choice of momentum variable for the quark mass function. Once again, we shall take an empirical approach, testing the p and q. Figure 9 shows M as a function of p and of q for each lattice spacing. We see better agreement between the results for diﬀerent lattice spacings when the standard lattice momentum, p is used. It goes without saying that the choices converge in the continuum limit. In fact, at β = 4.80 the diﬀerence is quite small.

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Fig. 8. Z function for the Overlap action at each lattice spacing for one quark mass (m 106 MeV). Data has been cylinder cut. In the top ﬁgure the kinematic momentum derived from the tree-level form of the action, q, is used, while in the bottom ﬁgure the standard lattice momentum, p, is used. Greatest consistency between the results for the diﬀerent lattice spacings is seen when q is used

8.2 Comparison with the Asqtad Action We have calculated the quark propagator using Asqtad and Overlap quark actions on one common set of gauge conﬁgurations, 123 × 24 at β = 4.60. This allows us to closely study diﬀerences induced only by choice of quark action. In all cases, we show results using 50 conﬁgurations for the Overlap action and 100 conﬁgurations for the Asqtad action. Figure 10 shows Z for the two actions at comparable masses (approximately 110 MeV). As these are bare quantites, they have been normalised at 3 GeV for the purpose of close comparison. While the Asqtad action produces a Z with near perfect rotational symmetry, the Overlap action shows some clear rotation

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Fig. 9. Mass functions for the Overlap quark propagator for ﬁxed bare mass at each lattice spacing. The top ﬁgure uses the standard lattice momentum and the bottom ﬁgure uses the action-derived momentum. Convergence appears to be most rapid using the standard lattice momentum, p. Data has been cylinder cut

symmetry breaking. This is consistent with the O(a4 ) improvement of the Asqtad action. Nonetheless, the two are in good agreement. Tree-level correction is an important ingredient in producing this agreement. Figure 11 shows the mass function for the two actions at comparable quark masses while Fig. 12 shows a comparison in the chiral limit. Both of the calculations were performed on the same gauge conﬁgurations. At ﬁnite quark mass the two actions are in excellent quantitative agreement and both show good rotational symmetry and ultraviolet behaviour. In the chiral limit there is good qualitative agreement, the chiral extrapolation probably accounting for the small diﬀerence.

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Fig. 10. Comparison between Asqtad and Overlap actions for comparable quark masses (∼110 MeV). The Z function is shown for β = 4.60 on the 123 × 24 lattice using the kinematic momenta of the respective actions. The data sets have been cylinder cut and normalised at 3 GeV

Fig. 11. Mass functions for the Asqtad and Overlap quark actions for comparable quark masses (ma = 0.072 and µ = 0.024). Data is from the 123 × 24 lattice at β = 4.60. For the Asqtad action, the mass function is shown as a function of q, while for the Overlap action we have used p. The Asqtad result represents 100 conﬁgurations while the Overlap results comes from 50 conﬁgurations. They are in good agreement and both actions show good ultraviolet behaviour

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Fig. 12. Mass functions in the chiral limit for the Overlap (ﬁlled squares) and Asqtad (open circles) quark actions. Data is from the 123 × 24 lattice at β = 4.60 and has been extrapolated to the chiral limit with a straight line ﬁt. The Overlap result was extrapolated from rather heavier quarks than the Asqtad

9 The Quark Propagator in Landau Gauge Now informed by the previous sections as to the strengths and weaknesses of our simulations, we shall draw some conclusions as to the overall behaviour of the quark propagator in Landau gauge. In all cases the quark Z function is supressed in the infrared. For the lightest masses we see little mass dependence – see Fig. 13 – but for heavier masses Z does start to ﬂatten out. Unfortunately, data for the widest range of masses is only available for the small lattices and as we saw in Fig. 6 the quark Z function suﬀers from some ﬁnite size eﬀects at this volume. In any case, at small quark masses the infrared behaviour of Z deviates signiﬁcantly from it’s perturbative form. By aﬀecting the propagator’s analytic properties, non-trivial behaviour in Z could indicate conﬁnement. One possible quark conﬁnement scenario is that Z vanishes at the mass shell, preventing a pole [50]. Calculations on a larger volume may help to shed light on this matter. The quark mass function manifestly shows dynamical mass generation, with a dramatic infrared enhancement starting around 1.5 GeV. A sample of mass functions is shown in Fig. 14. One question that is often asked about quarks is, “How light is ‘light’ ?”. For our lightest quark masses we see that while the asymptotic value of M (p) is proportional to the bare quark mass, the infrared value is independent of the bare quark mass. One measure of this can be seen in Table 4. In this regime dynamical chiral symmetry breaking clearly dominates. As the bare quark mass increases, explicit chiral symmetry breaking takes over. The infrared behaviour becomes modiﬁed and the mass function eventually becomes ﬂat. This transition occurs around the strange quark mass. This transition deserves further study. For the mass function there are no noticeable ﬁnite volume eﬀects on the lattices studied.

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Fig. 13. Mass dependence: Comparison of the Landau gauge quark Z functions for three bare quark masses. The data is for the Asqtad action with quark masses ma = 0.036, 0.075 and 0.144. Data has been cylinder cut. We see that over this range of values the mass dependence of the Z function is very weak

Fig. 14. Mass dependence: Comparison of the Landau gauge quark mass functions for a selection of bare quark masses. The data is for the Overlap action with quark masses µ = 0.024, 0.032, 0.048, 0.080 and 0.120. Data has been cylinder cut. We see that the mass function clear displays dynamical mass generation, but ﬂattens out as the quarks become heavy

To study the chiral limit we extrapolate the mass function to zero bare quark mass using a quadratic ﬁt. The result is shown in Fig. 15. At large momenta the mass function should go like the running mass, so we expect that the chiral extrapolation should follow a straight line in this region. In the infrared however, some nonlinear behaviour is expected, but we have no detailed knowledge of what form it should take. The quadratic ﬁt does a good job of describing the

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Fig. 15. Chiral limit: Landau gauge quark mass function extrapolated to the chiral limit by quadratic ﬁt to the Asqtad mass function at each quark mass. At large momentum, the mass function is zero, consistent with a massless quark, but is steeply enhanced in the infrared

data over a wide range of momenta, however there is some indication of more complicated behaviour for the lowest masses at the lowest momenta. A recent Dyson–Schwinger equation study suggests that the infrared enhancement of the mass function should be occur more rapidly (as a function of momentum) in the chiral limit [37]. In other words, the peak should be narrower. The small diﬀerence between the Asqtad and Overlap results in this region may support that conclusion. This will be studied in more detail in a future publication.

10 Laplacian Gauge Now we will consider the quark propagator in the Laplacian gauges as well as Landau gauge. This was ﬁrst done in [3]. The results shown here are from the Asqtad action. 10.1 Comparitive Performance of Landau and Laplacian Gauges Figure 16 shows the mass function for the Asqtad action in ∂ 2 (I) and ∂ 2 (II) gauges and it should be compared with the equivalent Landau gauge result at the bottom of Fig. 5. We see ﬁrstly that these three gauges give very similar results (we shall investigate this in more detail later) and secondly that they give similar performance in terms of rotational symmetry and statistical noise. Looking more closely, we can see that the Landau gauge gives a slightly cleaner signal at this lattice spacing. ∂ 2 (III) was seen in [3] to be a numerically poor algorithm. The gauge ﬁxing procedure failed for four of the conﬁgurations and eight of the remaining

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Fig. 16. Mass function for quark mass ma = 0.036 (m 57 MeV), Asqtad action in ∂ 2 (I) and ∂ 2 (II) gauges. Figure 19 shows a direct comparison with Landau gauge

conﬁgurations produced Z and M functions with pathological negative values. We have seen that this type of gauge ﬁxing fails to produce a gluon propagator that has the correct asymptotic behaviour [11]. We are dealing with matrices with vanishing determinants, which are destroying the projection onto SU (3). We expect the degree to which this problem occurs to be dependent on the simulation parameters and the numerical precision used (in this work the gauge transformations were calculated in single precision). 10.2 Gauge Dependence Now we investigate the quark mass and Z functions in Landau, ∂ 2 (I) and ∂ 2 (II) gauges. Figure 17 shows the Z function for the Asqtad action in Landau, ∂ 2 (I) and ∂ 2 (II) gauges. They are in excellent agreement in the ultraviolet, but diﬀer

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Fig. 17. Comparison of the quark Z functions for the Asqtad action for quark mass ma = 0.036. Data has been cylinder cut

signiﬁcantly in the infrared. The Z function in the Laplacian gauges is more strongly infrared suppressed than in the Landau gauge. There is a small diﬀerence in Z(q 2 ) between ∂ 2 (I) and ∂ 2 (II) gauges. In all cases the quark Z function demonstrates little mass dependence. Deviation of Z from its asymptotic value of 1 is a sign of dynamical symmetry breaking, so we expect the infrared suppression to go away in the limit of an inﬁnitely heavy quark. The two are the same, to within errors, although if we look at the lowest momentum data we see that the point for the low mass lies below the high mass one. Figure 18 shows Z in ∂ 2 (II) gauge for three quark masses. Again, the data are consistent, to within errors, but there is a systematic ordering of lightest to heaviest. Compare with Fig. 13. We conclude from this that the behaviour is consistent with expectations, dependence on the quark mass – if any – is very weak. One possible explanation is that all the masses studied are light – less than or approximately equal to the stange quark mass – and that heavier masses will aﬀect the Z function more clearly. The mass functions in Landau, ∂ 2 (I) and ∂ 2 (II) gauges – shown in Fig. 19 – agree to within errors. The level of agreement is greater than that reported in [3], which further suggests that they are the same in the continuum limit.

11 Applications: The Condensate and Running Mass As we have already remarked, although the quark propagator is gauge dependent, and hence not observable, certain gauge invariant quantities can be extracted from it directly. In the continuum, the quark mass function has the asymptotic form, dM −1 c

m ln(q 2 /Λ2QCD ) +

(102) M (q 2 ) 2 = dM 2 q →∞ q 2 ln(q /Λ2 ) QCD

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Fig. 18. Comparison of the quark Z functions for three quark masses ma = 0.012, 0.075 and 0.144, with the Asqtad action in ∂ 2 (II) gauge. Data has been cylinder cut. As in Landau gauge, Z shows little sensitivity to changing quark mass over this range

Fig. 19. Comparison of the quark mass functions for the Asqtad action, ma = 0.036. Data has been cylinder cut

(see equation (6.15) in [50]) where m is the RGI (renormalisation group invariant) mass and the anomalous dimension of the quark mass is dM = 12/(33 − 2Nf ) for Nf quark ﬂavours (zero here). In this form the fact that the mass function is independent of the choice of renormalisation point is manifest. The ﬁrst term in (102) is of nonperturbative origin and, in the chiral limit, a non-zero value for c indicates the spontaneous breaking of chiral symmetry. In this case it is related to the usual chiral condensate by c −

ψψ 4π 2 dM , 2 3 [ln(µ /Λ2QCD )]dM

(103)

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Fig. 20. Fit of the asymptotic form of the mass function (102) to the Asqtad lattice data in the chiral limit. In the infrared, this form diverges rapidly

where µ2 is a choice of renormalisation point. The second term in (102) comes from perturbation theory (one-loop) and breaks chiral symmetry explicitly. The RGI quark mass in (102) can be replaced by the running quark mass, m = m(µ2 )[ln(µ2 /Λ2QCD )]dM

(104)

which can, in turn, be related the bare quark mass through some renormalisation condition. Equation (102) has been ﬁt to the data from the chiral extrapolation of the Asqtad action, shown in Fig. 20. For that action the ﬁt region corresponds to 1.9–2.9 GeV, which includes 51 data points, a ﬁt region chosen to minimise χ2 . We use ΛMOM QCD at 691 MeV. This was computed by taking the ALPHA result of ΛMS QCD = 239 MeV [51] and converting it to the MOM scheme as described in [52]. With good enough data, it would be possible to simultaneously ﬁt for the quark mass (or chiral condensate) and ΛQCD , but it is diﬃcult to extract a quantity on which there is only a logarithmic dependence. A ﬁner lattice would be necessary, so that more ultraviolet physics could be sampled, with more conﬁgurations to improve the statistics. From (103), we ﬁnd for the condensate, (−ψψ)1/3 = 274(24) MeV , at the renomalisation point µ = 2 GeV; the quoted error is purely statistical. It is remarkable that the asymptotic form provides such a good description of the lattice data from ∼1 GeV. Having ﬁt the asymptotic form to the quark mass function we should also be able to calculate the quark mass, but there is one piece of information missing from Table 3. The eﬀect of the bare quark mass on physical quantities is theory dependent; we have no a priori knowledge of where the physical point lies. We can

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Table 3. Results for the ﬁt to (102). The ﬁt region corresponds to 1.9–2.9 GeV, which includes 51 data points, and was chosen to minimise χ2 /d.o.f. In the chiral limit this gives a value for the condensate of (−ψψ )1/3 = 274(24) MeV at the renomalisation point µ = 2 GeV am

a3 c

m (MeV)

χ2 / d.o.f.

0

0.018(6)

0

0.52

0.012

0.019(3)

28(2)

0.57

0.018

0.014(12)

44(5)

0.54

0.024

0.0096(47)

59(3)

0.52

0.036

0.0026(104)

90(5)

0.49

0.048

0.086(3)

137(4)

0.46

0.072

0.130(3)

204(5)

0.40

0.108

0.190(3)

300(5)

0.31

0.144

0.251(5)

397(8)

0.25

match a physical quantity to its real world value and thus extract the relevant bare mass for that particular action, with those particular lattice parameters. In this case we choose the Asqtad quark action on the 163 × 32 lattice at β = 4.60 and set the “physical” bare quark mass by extrapolation to the physical pion mass. We ﬁt the ﬁve lowest π masses to the form √ amπ = a2 Bm , (105)

Fig. 21. m2π versus m for the Asqtad quark action and a ﬁt to (105)

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the lowest order result from chiral perturbation theory. For the higher masses, the data deviates from this form due to the contribution from higher order terms; in the heavy quark limit mπ ∝ m. The U(1) chiral symmetry of staggered quarks ensures that there is no additive mass renormalisation. To ﬁnd the physical point we ﬁnd the bare mass corresponding to mπ = 140 MeV for each jackknife bin and get am = 0.00215(1). Then we ﬁt the lowest four RGI masses in Table 3 to the form am = Aam (106) and ﬁnd A = 1.57(16). Putting these together we get m = 5.3(6) MeV . The RGI quark mass in (102) is related to the running quark mass through (104). Thus by choosing the renormalisation point µ = 2 GeV and inserting ΛMS QCD = 239 MeV, we get m(µ = 2 GeV)MS = 3.1(4) MeV ,

or with ΛMOM QCD = 691 MeV,

m(µ = 2 GeV)MOM = 4.0(5) MeV . This result compares favourably with the Particle Data Group’s number for the average of the up and down quark masses [53], m(µ = 2 GeV)MS = 2.5 − 5.5 MeV . The curious behaviour of the condensate term at small masses is easily explained. The asymptotic behaviour of the quark mass function quickly becomes dominated by the explicit chiral symmetry breaking mass; at the same time, large ﬂuctuations make the mass function relatively noisy at these masses. Together, that means that sensitivity of the ﬁt to the ﬁrst term in (102) is lost. At larger masses, the signal becomes cleaner and statistical errors shrink, restoring stability to the ﬁts. One way of countering this eﬀect is to ﬁx ac at each light mass to some reasonable value given the chiral and heavy quark values. The resulting RGI masses are consistent with the ones in Table 3, so we expect the systematic error from this eﬀect to be small. This calculation is, of course, quenched. There are also systematic errors associated with the extrapolations. A more sophisticated approach, such as an extrapolation motivated by ﬁnite-range regulated [54] staggered chiral perturabtion theory [55], is certainly to be desired. Such an approach could also provide a handle on the uncertainty associated with quenching. Finally, we should realise that this is a rather audacious approach: we are exploiting ultraviolet physics on a relatively coarse lattice. It is very encouraging, however, that we are able to get such a promising result with such modest resources. The choices of improved quark and gauge actions are essential to this.

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12 Modelling the Mass Function The QCD Green’s functions continue to provide a fruitful area of interaction between various hadronic methods. As stated above, the quark propagator is used in a number of nonperturbative schemes; see [12, 13, 14, 56, 57, 58, 59] for a few examples. As a ﬁrst principles approach, lattice QCD can provide valuable input, and by exploiting the strengths of other approaches we can gain insight to possible diﬃculties of current lattice simulations. With these things in mind we attempt to model the lattice data and to compare the data directly to other models. We look once again at the Asqtad data for the 163 × 32, β = 4.60 ensemble. A ﬁt to the mass function for each bare mass was performed using the ansatz M (q 2 ) =

cλ2(α−1) +m . q 2α + λ2α

(107)

This is the same as the ansatz used in [3], but written slightly diﬀerently: in this case c has dimension [mass3 ]. This model was used to approximate derivatives of the quark propagator in a study of the quark-gluon vertex [60]. Fit results are shown in Table 4 and sample ﬁts are shown in Fig. 22. From these we can see that this three-parameter form does indeed describe the data well, across the whole range of quark masses studies (m 0 → 230 MeV). Also, to make some assessment of the dependence of the chiral limit result to our extrapolation, we use both a linear and a quadratic extrapolation and compare the ﬁt results. The two results agree, but statistical errors are large. At the smallest masses there is a lot of noise in the lowest momentum points, resulting in some uncertainty in this important region. The existence of a “light quark regime,” where the low momentum physics is essentially independent of the quark mass, has Table 4. Best-ﬁt parameters for the ansatz (107), in Landau gauge. The ﬁrst set of data at zero mass is for a linear extrapolation, the second for a quadratic extrapolation am

a3 c

λ (MeV) m (MeV) α

M (0) (MeV) χ2 / d.o.f.

linear quadratic 0.012 0.018 0.024 0.036 0.048 0.072 0.108 0.144

0.042(3) 0.040(3) 0.043(2) 0.044(2) 0.045(2) 0.046(2) 0.050(3) 0.059(2) 0.071(5) 0.090(9)

757(48) 719(50) 777(35) 795(28) 820(29) 802(27) 838(27) 946(29) 997(27) 1100(32)

289(22) 302(33) 308(19) 311(13) 313(11) 353(10) 370(9) 397(12) 478(4) 547(4)

0 0 25(2) 37(2) 49(2) 70(3) 92(3) 138(5) 199(6) 255(10)

1.43(9) 1.47(11) 1.51(2) 1.50(2) 1.48(2) 1.31(1) 1.28(1) 1.30(8) 1.12(3) 1.01(5)

0.21 0.72 0.70 0.65 0.60 0.53 0.48 0.44 0.37 0.35

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Fig. 22. Modelling the mass function: sample ﬁts of the ansatz (107). The ansatz is ﬁt to data in chiral limit (top) and for am = 0.144 (bottom). In both cases this ansatz provides an excellent description of the data. The ﬁt parameters are given in Table 4

already been remarked upon. Table 4 would suggest that this regime ends where the asymptotic mass is around 70 MeV. In this model, α is acting as a function of the bare mass, controlling the dynamical symmetry breaking. The model gives a value for the quark mass at zero four-momentum of around 300 MeV, consistent with the idea that this should be the constituent quark mass. Although this model does not have the correct asymptotic behaviour, this is simple to correct. As the model ansatz falls away faster than q −2 , the correct asymptotic behaviour, (102), can be added on. Such an ansatz does not, however, improve the overall ﬁt to this data. More points in the ultraviolet are required. In [57] a model quark propagator was used as input for the Fadde’ev equation, which was used for the calculation of electromagnetic form factors. One of the ans¨ atze considered was

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q 2 + c22 c23

2 1 − e−y M (q 2 ) = c1 y y=

(108) (109)

which was ﬁt to some earlier lattice data [3] with c3 ﬁxed at 1 GeV. This model has the advantage of being an entire function. We have performed a ﬁt of this generalised form – where c3 is determined by the ﬁt – to this newer data, the results of which are shown in Fig. 23. It is clear that this describes the data well, and so may ﬁnd more general application.

Fig. 23. Fit to the mass function in the chiral limit using (109). This ansatz has the virtue of being analytic everywhere. Parameters were determined by least χ2 ﬁt. c1 = 440(32) MeV, c2 = 539(18) MeV, c3 = 831(40) MeV, χ2 /d.o.f = 0.89

In [56] the authors suggested a novel form of the quark mass function which actually vanishes at zero four-momentum. The ansatz 1 − 1 − q 2 Q2 (q 2 ) 2 , (110) M (q ) = Q(q 2 ) where

−ψψ 16π 2 −q2 /λ2 e , (111) 2Nc λ4 is based on the Constrained Instanton Model and QCD sum rules with nonlocal condensates. In Fig. 24 we show two versions, one using the values for the parameters published in [56] and one where the parameters have been determined by best ﬁt to the lattice data. This form is not favoured by the lattice data. It is fascinating that such a form should be consistent with the phenomenological constraints discussed in [56] and demonstrates the utility of the ﬁrst-principles calculations presented here. Q(q 2 ) =

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Fig. 24. Fit to the mass function in the chiral limit using (110). The long dashes are for the parameters of [56]: λ = 554 MeV, ψψ 1/3 = −225 MeV, χ2 /dof = 6.2; short dashes are for a best ﬁt to the data: Λ = 719(31) MeV, ψψ 1/3 = −246(5) MeV, χ2 /d.o.f = 4.1

Finally, we consider the instanton model of Diakonov. There the quark mass function takes the form [12] M (q) = M (0)F 2 (z)

K1 (z) F (z) = 2z I0 (z)K1 (z) − I1 (z)K0 (z) − I1 (z) z ρ z =q 2

(112) (113) (114)

where the I’s and K’s are the modiﬁed Bessel functions. The parameters M (0) = 350 MeV and ρ−1 = 600 MeV are derived from the standard values of the instanton ensemble. This two parameter form has a good correspondance with the lattice data, as shown in Fig. 25, without any ﬁt parameters. Asymptotically, for this model M (q) ∼ q −6 . There is also good agreement between these results and recent Dyson– Schwinger equation studies [14, 37, 59]. In return, some studies suggest that the mass function should be somewhat narrower in the chiral limit than the results presented here [37, 57]. Better control of the chiral extrapolation is clearly necessary, and studies in this area are underway. Other discrepancies may be due to the quenched approximation and calculations with dynamical conﬁgurations are also now underway.

13 Conclusions We have seen that the Overlap and Asqtad actions provide a quark propagator with excellent rotational symmetry, scaling and ultraviolet behaviour. Both are

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Fig. 25. The quark mass function in the chiral limit from the instanton model of [12] compared with our data. There has been no ﬁt, instead we have used the published parameters M (0) = 350 MeV and ρ−1 = 600 MeV

therefore suitable for quark propagator studies. Results from the two actions are in good agreement. Overall the Landau gauge results shown in this work are consistent with the results of earlier studies [1, 2]. For the purpose of a detailed study of the quark propagator, the actions employed in the present work are, however, clearly superior to the Clover action. Quark-gluon vertex studies should derive great beneﬁt from the use of Overlap and/or Asqtad actions. Our results show that the quark mass function in Landau gauge has a large enhancement near zero four-momentum and drops rapidly to something like its expected asymptotic behaviour by around 2 GeV. The mass function is – to within statistics – the same in ∂ 2 (I) and ∂ 2 (II) gauges as it is in Landau gauge. We observe somewhat greater noise in the Laplacian gauges at these lattice spacings. The wavefunction renormalisation, Z appears to be infrared suppressed. It exhibits a weak mass dependence, but ﬁnite volume eﬀects may be signiﬁcant with the present simulation parameters. The infrared dip is even stronger in the Laplacian gauges. ∂ 2 (I) and ∂ 2 (II) diﬀer by a small amount. Dynamical chiral symmetry breaking is manifest in the quark propagator, but the other striking feature of QCD, conﬁnement, is not so obvious4 . By matching the tree-level corrected quark propagator to its continuum asymptotic form we have been able to extract quenched estimates of the isospinaveraged chiral condensate and running quark mass. Both results are consistent with those from other studies and can be greatly improved upon by using ﬁner lattices. With good enough data we could also calculate ΛQCD . Most importantly, the techniques used are immediately applicable to unquenched data, when it is available. We have also proposed a simple model for the mass function, and given 4

As this article was being completed a discussion of this issue appeared [61].

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a brief example of the light that this line of research can shed on other hadronic calculations. In this work the chiral limit was obtained by simple linear and quadratic extrapolations, which may provide a source of systematic error. Both the Asqtad and Overlap actions provide the possibility of studying the propagator at very light masses, but larger lattices are required to properly pin-down the correct behaviour. It will be very interesting to compare this behaviour with recent DSE results [37]. All the data shown here in this work was quenched, but calcuations with dynamical conﬁgurations are underway and a detailed study of the eﬀects of unquenching will be forthcoming.

Acknowledgements The authors wish to thank Jonivar Skullerud, Constantia Alexandrou and Waseem Kamleh for useful discussions, as well as the organisers of the Lattice Hadron Physics workshop (Cairns, 2001). We also thank Ryan Coad for careful reading of the manuscript and constructive feedback. Generous grants of supercomputer time from the Australian Partnership for Advanced Computing (APAC) and the Australian National Computing Facility for Lattice Gauge Theory are gratefully acknowledged. This work was supported in part by the Australian Research Council.

References 1. J-I. Skullerud, A.G. Williams: Quark Propagator in Landau Gauge, Phys. Rev. D 63, 054508 (2001) 17, 32, 60 2. J-I. Skullerud, D.B. Leinweber, A.G. Williams: Nonperturbative Improvement and Tree-level Correction of the Quark Propagator, Phys. Rev. D 64, 074508 (2001) 17, 32, 60 3. P. O. Bowman, U.M. Heller, A.G. Williams: Lattice quark propagator with staggered quarks in Landau and Laplacian gauges, Phys. Rev. D 66, 014505 (2002) 17, 32, 49, 51, 56, 58 4. P. O. Bowman, U. M. Heller, D. B. Leinweber, A.G. Williams: Modelling the quark propagator, hep-lat/0209129 17 5. F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. G. Williams, J. Zhang: Overlap Quark Propagator in Landau Gauge, Phys. Rev. D 65, 114503 (2002) 17, 32, 42 6. J. B. Zhang, F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. G. Williams: Scaling behavior of the overlap quark propagator in Landau gauge, hep-lat/0301018 17, 32 7. J. C. Vink, U-J. Wiese: Gauge Fixing on the Lattice without Ambiguity, Phys. Lett. B 289, 122 (1992) 18, 24, 25 8. J. C. Vink: Investigation of Laplacian Gauge Fixing for U(1) and SU (2) Gauge ﬁelds, Phys. Rev. D 51, 1292 (1995) 18, 24, 25 9. C. Alexandrou, Ph. de Forcrand, E. Follana: The Gluon Propagator without lattice Gribov copies, Phys. Rev. D 63, 094504 (2001) 18, 24, 25 10. C. Alexandrou, Ph. de Forcrand, E. Follana: The Gluon Propagator without lattice Gribov copies on a ﬁner lattice, Phys. Rev. D 65, 114508 (2002) 18 11. P. O. Bowman, U. M. Heller, D. B. Leinweber, A.G. Williams: Gluon propagator on coarse lattices in Laplacian gauges, Phys. Rev. D 66, 074505 (2002) 18, 24, 50

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Generalised Spin Projection for Fermion Actions W. Kamleh CSSM Lattice Collaboration, Special Research Centre for the Subatomic Structure of Matter and Department of Physics and Mathematical Physics, University of Adelaide 5005, Australia. Abstract. The majority of compute time doing lattice QCD is spent inverting the fermion matrix. The time that this takes increases with the condition number of the matrix. The FLIC (Fat Link Irrelevant Clover) action displays, among other properties, an improved condition number compared to standard actions and hence is of interest due to potential compute time savings. However, due to its two diﬀerent link sets there is a factor of two cost in ﬂoating point multiplications compared to the Wilson action. An additional factor of two has been attributed due to the loss of the so-called spin projection trick. We show that any split-link action may be written in terms of spin projectors, reducing the additional cost to at most a factor of two. Also, we review an eﬃcient means of evaluating the clover term, which is additional expense not present in the Wilson action.

1 Introduction The FLIC(Fat Link Irrelevant Clover) action [1] has become of interest recently as an alternative to standard actions (such as Wilson or Clover) due to its superior condition number [2]. This allows for more eﬃcient fermion matrix inversion [1], which is used in the calculation of propagators and dynamical conﬁgurations, and in evaluating the matrix sign function in the overlap fermion formalism [3, 4]. Hence, actions with an improved condition number have the potential to save signiﬁcant compute time. To begin, we review the spin-projection trick [5] for the Wilson action, which utilises projection operators in spinor space to reduce the computation required to evaluate the Wilson action. We then generalise this trick to the broader class of split-link actions. Finally, we examine the FLIC action speciﬁcally, and discuss a similar trick for reducing the cost of evaluating the clover term.

2 Standard Spin-Projection Trick The Wilson operator 1 Dw = ∇ /+ ∆+m 2

(1)

can be written as

W. Kamleh: Generalised Spin Projection for Fermion Actions, Lect. Notes Phys. 663, 65–69 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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1 (1 − γµ )Uµ (x)ψx+µ + (1 + γµ )Uµ† (x) ψx−µ 2 µ = (4 + m)ψx − Uµ (x)Γµ− ψx+µ + Uµ† (x)Γµ+ ψx−µ , (2)

(Dw ψ)x = (4 + m)ψx −

µ

where we have deﬁned the spin projectors Γµ± =

1 1 ± γµ . 2

If we now examine, for example, Γ2± we see that  1  1  ψ ∓ ψ4 ψ ψ 2   ψ 2 ± ψ 3     Γ2±  ψ 3  = ±ψ 2 + ψ 3  . ψ4 ∓ψ 1 + ψ 4

(3)

(4)

Similar expressions for µ = 1, 3, 4 allow us to deduce that we only need to evaluate the action of the links on the upper half (in spinor space) of Γµ± ψx∓µ , as the lower components are equal to the upper components multiplied by ±1 or ±i. In doing so we can halve the number of ﬂoating point multiplications needed in the evaluation of Dw , and also reduce intermediate memory usage. This trick can be applied in any of the standard representations for the Euclidean-space γ matrices.

3 Generalised Spin-Projection Trick We now consider the case where there are two sets of links, Uµ (x) for the naive Dirac operator ∇ / and Uµ (x) for the irrelevant Wilson term (denoted by ∆ to indicate that it contains only the links U ). In the case of a FLIC action the irrelevant links are APE-smeared, but what follows is perfectly general and does not depend upon any particular relationship between U and U . Now our “splitlink” operator is # $ 1 (Dsplit ψ)x = ∇ / + ∆ + m ψ 2 x 1 Uµ (x) − γµ Uµ (x) ψx+µ + U †µ (x) + γµ Uµ† (x) ψx−µ =− 2 µ +(4 + m)ψx .

(5)

We can observe that our projectors do not present themselves immediately as they did before. At this point, compared to the standard Wilson action, we must perform four times as many ﬂoating point multiplications, two for the split links, and two for the loss of the spin projectors. However, we have 1 = Γµ+ + Γµ− and γµ = Γµ+ − Γµ− ,

(6)

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which implies 1 Uµ (x) − Uµ (x) Γµ+ ψx+µ + Uµ (x) + Uµ (x) Γµ− ψx+µ 2 µ † + U µ (x) + Uµ† (x) Γµ+ ψx−µ + U †µ (x) − Uµ† (x) Γµ− ψx−µ +(4 + m)ψx . (7)

(Dsplit ψ)x = −

It is now clear that by deﬁning symmetrised and anti-symmetrised links, Uµ+ (x) =

1 1 Uµ (x) + Uµ (x) and Uµ− (x) = Uµ (x) − Uµ (x) 2 2

(8)

we can write (Dsplit ψ)x = −

Uµ− (x)Γµ+ ψx+µ + Uµ+ (x)Γµ− ψx+µ

µ

+Uµ+† (x)Γµ+ ψx−µ + Uµ−† (x)Γµ− ψx−µ + (4 + m)ψx .

(9)

Immediately we see that the Wilson spin projection trick is simply a special case of the split link trick where U = U . The same saving in multiplications that we received in the Wilson case applies here, so we have in principle a factor of two compared to the Wilson action because U − is not zero. In actuality, eﬃcient cache usage will reduce this to less than a factor of two.

4 The FLIC Fermion Action The FLIC action is a split-link action with clover term [6], /+ Dﬂic = ∇

$ 1 # csw ∆ − σ·F +m, 2 2

(10)

where 1 1 † [γµ , γν ], Fµν (x) = Cµν (x) − Cµν (x) , 2 2 1 Cµν (x) = Uµν (x) + U−νµ (x) + Uν−µ (x) + U−µ−ν (x) . 4 σµν =

(11) (12)

APE-smearing [7, 8, 9, 10] is carried out on the individual links in the irrelevant operators by making the replacement   α Uµ (x) → Uµ(α) (x) = P (α − 1)Uµ (x) + Uν (x)Uµ (x + aeν )Uν† (x + aeµ ) . 6 ±ν=µ

(13) Here P denotes projection of the RHS of (13) back to the SU (3) gauge group. That is, each link is modiﬁed by replacing it with a combination of itself and the

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surrounding staples to give a set of “fat links”. The means by which one projects back to SU (3) is not unique. If Xµ (x) is the smeared link before projection, that (α) is Uµ (x) ≡ PXµ (x), then we choose [11, 12] Uµ(α) (x) =

1 1

det Wµ (x) 3

Wµ (x), where

Wµ (x) = Xµ (x)[Xµ† (x)Xµ (x)]− 2 . 1

(14) (15)

As the process of APE-smearing removes short-distance physics, it is preferable to only smear the irrelevant operators. Here α is the smearing fraction and nape is the number of smearing sweeps (13) we perform. Finally, as in [1], we can perform tadpole or mean-ﬁeld improvement (MFI) [13] to bring our links closer to unity. This consists of updating each link with a division by the mean link, which is the fourth root of the average plaquette, % u0 =

& 14 1 ReTrUµν (x) . 3 x,µ<ν

(16)

For completeness, we review a (well-known) similar trick for the clover term that exploits the structure of σµν . In the evaluation of the clover term, we note that in the chiral representation of γ matrices,

0 1 1 0 γ5 = , (17) γ4 = 1 0 0 −1 the matrix σµν satisﬁes the following (in 2 × 2 block notation), ψ −ψ ψ ψ ψ −ψ σ12 = σ34 , σ13 = σ24 and σ14 = σ23 . χ χ χ −χ χ χ (18) So we have, for example,

ψ ψ (F12 − F34 )σ12 ψ . (19) + F34 σ34 = F12 σ12 (F12 + F34 )σ12 χ χ χ This means that if we store the combinations F12 ± F34 , F13 ± F24 , F14 ± F23 we can halve the number of ﬂoating point multiplications needed in the evaluation of the clover term, further improving the computational eﬃciency of the FLIC action.

5 Conclusion We have presented a generalised version of the spin-projection trick which is applicable to any split-link action. This allows us to halve the number of ﬂoatingpoint multiplications the evaluation of the action of the links upon the fermion

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ﬁeld. We have also recalled some symmetries of σµν in the chiral γ matrix representation which allow us to perform a similar cost reduction in the evaluation of the clover term. The results presented here reduce the cost of evaluating the FLIC action to about twice that of the standard Wilson action. The exact diﬀerence will vary depending upon the base architecture, but on our architecture we have veriﬁed that the cost of FLIC is almost exactly twice that of the Wilson, including the cost of the clover term. Additionally, the formulation of the split link action in (9) allows groups who have eﬃcient code for the Wilson action to simply implement eﬃcient code for the FLIC action. Given the beneﬁts of the FLIC action [1, 2] we hope that this work encourages groups to consider using the FLIC action for their calculations.

Acknowledgements The author wishes to thank Urs Heller, Robert Edwards and Herbert Neuberger for discussions at Cairns in 2001 that led to this investigation. Additional thanks go to Patrick Bowman for helpful discussions immediately after Cairns, and to Tony Williams and Derek Leinweber for valuable discussions and their contribution to the manuscript. Thanks also go to David Richards for correspondence. This work was supported by the Australian Research Council.

References 1. CSSM Lattice, J. M. Zanotti et al.: Phys. Rev. D 65, 074507 (2002), hep-lat/ 0110216. 65, 68, 69 2. W. Kamleh, D. H. Adams, D. B. Leinweber, and A. G. Williams: Phys. Rev. D 66, 014501 (2002), hep-lat/0112041. 65, 69 3. H. Neuberger: Phys. Rev. Lett. 81, 4060 (1998), hep-lat/9806025. 65 4. R. G. Edwards, U. M. Heller, and R. Narayanan: Nucl. Phys. B 540, 457 (1999), hep-lat/9807017. 65 5. M. G. Alford, T. R. Klassen, and G. P. Lepage: Nucl. Phys. B 496, 377 (1997), hep-lat/9611010. 65 6. B. Sheikholeslami and R. Wohlert: Nucl. Phys. B 259, 572 (1985). 67 7. M. Falcioni, M. L. Paciello, G. Parisi, and B. Taglienti: Nucl. Phys. B 251, 624 (1985). 67 8. APE, M. Albanese et al.: Phys. Lett. B 192, 163 (1987). 67 9. F. D. R. Bonnet, D. B. Leinweber, A. G. Williams, and J. M. Zanotti: (2001), hep-lat/0106023. 67 10. M. C. Chu, J. M. Grandy, S. Huang, and J. W. Negele: Phys. Rev. D 49, 6039 (1994), hep-lat/9312071. 67 11. Y. Liang, K. F. Liu, B. A. Li, S. J. Dong, and K. Ishikawa: Phys. Lett. B 307 (1993) 375, hep-lat/9304011. 68 12. W. Kamleh, D. B. Leinweber, and A. G. Williams: In preparation. 68 13. G. P. Lepage and P. B. Mackenzie: Phys. Rev. D 48, 2250 (1993), hep-lat/9209022. 68

Baryon Spectroscopy in Lattice QCD D.B. Leinweber1 , W. Melnitchouk2 , D.G. Richards2 , A.G. Williams1 , and J.M. Zanotti3 1

2 3

Department of Physics and Mathematical Physics and Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, 5005, Australia Jeﬀerson Lab, 12000 Jeﬀerson Avenue, Newport News, VA 23606, USA John von Neumann-Institut f¨ ur Computing NIC, Deutsches Elektronen-Synchrotron DESY, D-15738 Zeuthen, Germany

Abstract. We review recent developments in the study of excited baryon spectroscopy in lattice QCD. After introducing the basic methods used to extract masses from correlation functions, we discuss various interpolating ﬁelds and lattice actions commonly used in the literature. We present a survey of results of recent calculations of excited baryons in quenched QCD, and outline possible future directions in the study of baryon spectra.

1 Introduction and Motivation One of the primary tools used for studying the forces which conﬁne quarks inside hadrons, and determining the relevant eﬀective degrees of freedom in strongly coupled QCD, has been baryon and meson spectroscopy. This is a driving force behind the current experimental N ∗ programme at Jeﬀerson Lab, which is accumulating data of unprecedented quality and quantity on various N → N ∗ transitions [1]. The prospects of studying mesonic spectra, and in particular the role played by gluonic excitations, has been a major motivation for future facilities such as CLEO-c [2], the anti-proton facility at GSI (PANDA) [3], and the Hall D programme at a 12 GeV energy upgraded CEBAF [4]. With the increased precision of the new N ∗ data comes a growing need to understand the observed spectrum within QCD. QCD-inspired phenomenological models, whilst successful in describing many features of the N ∗ spectrum [5], leave many questions unanswered. One of the long-standing puzzles in baryon spectroscopy has been the low mass of the ﬁrst positive parity excitation of the nucleon, the N ∗ (1440) Roper resonance, compared with the lowest lying odd parity excitation. In valence − quark models with harmonic oscillator potentials, the J P = 12 state naturally + occurs below the N = 2, 12 excitation [6]. Without ﬁne tuning of parameters, valence quark models tend to leave the Roper mass too high. Similar diﬃcul+ + ties in the level orderings appear for the J P = 32 ∆∗ (1600) and 12 Σ ∗ (1690) resonances, which have led to speculations that the Roper resonances may be more appropriately viewed as hybrid baryon states with explicitly excited glue ﬁeld conﬁgurations [7], rather than as radial excitations of a three-quark conﬁguration. Other scenarios portray the Roper resonances as “breathing modes” of D.B. Leinweber, W. Melnitchouk, D.G. Richards, A.G. Williams, and J.M. Zanotti: Baryon Spectroscopy in Lattice QCD, Lect. Notes Phys. 663, 71–112 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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the ground states [8], or states which can be described in terms of meson-baryon dynamics alone [9]. Another challenge for spectroscopy is presented by the anomalously small mass of the odd parity Λ(1405) hyperon. This is alternatively interpreted as a true three-quark state, or as a hadronic molecule arising from strong coupled ¯ ,. . . states [10]. In fact, the role played by Goldchannel eﬀects between Σπ, KN stone bosons in baryon spectroscopy has received considerable attention recently [11, 12, 13]. Mass splittings between states within SU (3) quark-model multiplets provide another important motivation for studying the baryon spectrum. The dynamical origin of hyperﬁne splittings in quark models has traditionally been attributed to the colour-magnetic one gluon exchange mechanism [14]. On the other hand, there have been attempts recently to explain the hyperﬁne splittings and level orderings in terms of a spin-ﬂavour interaction associated with the exchange of a pseudoscalar nonet of Goldstone bosons between quarks [15]. Understand− − ing the mass splitting between the J P = 12 N ∗ (1535) and 32 N ∗ (1520), or − − between the 12 ∆∗ (1620) and 32 ∆∗ (1700), can help identify the important mechanisms associated with the hyperﬁne interactions, and shed light on the spin-orbit force, which has been a central mystery in spectroscopy [16]. In va− − lence quark models, the degeneracy between the N 12 and N 32 can be broken by a tensor force associated with mixing between the N 2 and N 4 representations − of SU (3) [5], although this generally leaves the N 32 at a higher energy than − − − the N 12 . In contrast, a spin-orbit force is necessary to split the ∆ 32 and ∆ 12 states. In the Goldstone boson exchange model [15], both of these pairs of states are degenerate. Similarly, neither spin-ﬂavour nor colour-magnetic interactions is able to ac− count for the mass splitting between the Λ(1405) and the J P = 32 Λ∗ (1520). A splitting between these can arise in constituent quark models with a spin-orbit interaction, but the required strength of the interaction leads to spurious mass splittings elsewhere [5, 17]. In fact, model-independent analyses of baryons in the large Nc limit [18] have found that these mass splittings receive important contributions from operators that do not have a simple quark model interpretation [18], such as those simultaneously coupling spin, isospin and orbital angular momentum, as anticipated by early QCD sum-rule analyses [16]. Of course, the coeﬃcients of the various operators in such an analysis must be determined phenomenologically and guidance from lattice QCD is essential. The large number of “missing” resonances, predicted by the constituent quark model and its generalisations but not observed experimentally, presents another problem for spectroscopy. If these states do not exist, this may suggest that perhaps a quark–diquark picture (with fewer degrees of freedom) could aﬀord a more eﬃcient description. Alternatively, the missing states could simply have weak couplings to the πN system [5]. Apart from these long-standing puzzles, interest in baryon spectroscopy has been further stimulated with the recent discovery of the exotic strangeness

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S = +1 Θ+ pentaquark [19], which has a minimum 5 valence quark (uudd¯ s) content. Postulated long ago, the Θ+ appears to have eluded searches until now, and understanding its properties and internal structure has become a major challenge for spectroscopy. A consistent description of the baryon spectrum within low-energy models requires a satisfactory resolution of the old questions of spectroscopy, as well as insight into the new puzzles. On both of these fronts real progress may only come with the help of ﬁrst-principle calculations of the spectrum in lattice QCD, which at present is the only method capable of determining hadron properties directly from the fundamental quark and gluon theory. Considerable progress has been made in the past few years in calculating hadronic properties in both quenched and full QCD, with the ground state masses now understood at the few percent level. Compared with simulations of hadron ground state properties, however, the calculation of the excited nucleon spectrum places particularly heavy demands on lattice spectroscopy. The excited nucleon states are expected to be large, with the size of a state expected to double with each increase in orbital angular momentum. Lattice studies of the excited nucleon spectrum therefore require large lattice volumes, with correspondingly large computational resources. Furthermore, the states are relatively massive, requiring a ﬁne lattice spacing, at least in the temporal direction. Recent advances in computational capabilities and more eﬃcient algorithms have enabled the ﬁrst dedicated lattice QCD simulations of the excited states of the nucleon to be undertaken. Of course, the calculation of the hadronic spectrum faces a formidable challenge in describing excited state decays, which is relevant in both full and quenched QCD. Lattice studies of excited hadrons are possible because at the current unphysically large quark masses and ﬁnite volumes used in the simulations many excited states are stable [20]. Contact with experiment can be made via extrapolations incorporating the nonanalytic behaviour predicted by chiral eﬀective ﬁeld theory. The rest of this review is laid out as follows. After brieﬂy reviewing in Sect. 2 the history of lattice calculations of the excited baryon spectrum, in Sect. 3 we outline the basic lattice techniques for extracting masses from correlation functions, including variational methods and Bayesian statistics. Section 4 describes the construction of a suitable basis of interpolating ﬁelds for any baryon we may wish to investigate. Section 5 outlines the simplest interpolating ﬁelds for spin-1/2 and spin-3/2 baryons, and how these can be used in the construction of lattice correlation functions. In Sect. 6, we present a survey of recent N ∗ results for both positive and negative parity baryons. Finally, in Sect. 7 we discuss conclusions of the existing studies, and speculate on future directions for the study of baryon spectra. Some technical aspects of the correlation matrix formalism for calculating masses, coupling strengths and optimal interpolating ﬁelds are described in the Appendix.

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2 History of Lattice N ∗ Calculations The history of excited baryons on the lattice is rather short, although recently there has been growing interest in ﬁnding new techniques to isolate excited baryons, motivated partly by the experimental N ∗ programme at Jeﬀerson Lab. An attempt to determine the ﬁrst positive-parity excitation of the nucleon using variational methods was made by the UKQCD collaboration [21]. A more detailed analysis of the positive parity excitation of the nucleon was performed by Leinweber [22] using Wilson fermions and an operator product expansion spectral ansatz. DeGrand and Hecht [23] used a wave function ansatz to access P wave baryons, with Wilson fermions and relatively heavy quarks. Subsequently, Lee and Leinweber [24] introduced a parity projection technique to study the − negative parity 12 states using an O(a2 ) tree-level tadpole-improved Dχ34 quark action, and an O(a2 ) tree-level tadpole-improved gauge action. Following this, Lee [25] reported results using a D234 quark action with an improved gauge ac+ − tion on an anisotropic lattice to study the 12 and 12 excitations of the nucleon. An anisotropic lattice with an O(a) improved quark action was also used by Nakajima et al. [26] to study excited states of octet and decuplet baryons. The RIKEN-BNL group [27] has stressed the importance of maintaining chiral symmetry on the lattice. At ﬁnite lattice spacing the Wilson fermion action is known to explicitly break the chiral symmetry of continuum QCD. A solution to this problem is provided through the introduction of a ﬁfth dimension, which allows chiral symmetry to be maintained even at non-zero lattice spacing [28]. The resulting domain wall fermion action was used in [27] to compute the masses − + of the N 12 and N 12 excited states. The analysis has recently been extended + by studying the ﬁnite volume eﬀects of the ﬁrst N 12 excited state extracted using maximum entropy methods [29]. A nonperturbatively O(a)-improved Sheikholeslami-Wohlert (SW) [30], or − clover, fermion action was used by Richards et al. [31] to study the N 12 and − ∆ 32 states. By appropriately choosing the coeﬃcient of the improvement term, all O(a) discretisation uncertainties can be removed, ensuring that continuumlike results are obtained at a ﬁnite lattice spacing. For more details on O(a)improvement, see the contribution by Zanotti et al. in this lecture series. − The BGR collaboration [32] has been investigating the masses of the N 12 + and N 12 excited states calculated with chirally improved (CI) and Fixed Point (FP) fermions. Both of these actions oﬀer the advantage of improved chiral properties over traditional Wilson-style fermions, without the cost associated with Ginsparg-Wilson fermions, which possess an exact analogue of chiral symmetry. The CSSM Lattice Collaboration has used an O(a2 ) improved gluon action and the O(a)-improved Fat Link Irrelevant Clover (FLIC) fermion action [33] to perform a comprehensive study of the spectrum of positive and negative parity baryons [34, 35]. Excited state masses in both the octet and decuplet multiplets ± ± ± ± ± ± have been computed, including the N 12 , N 32 , Σ 12 , Λ 21 , Ξ 12 , ∆ 12 and ± ∆ 32 baryons. The formulation of FLIC fermions and a review of its scaling and

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light quark mass phenomenology is reviewed by Zanotti et al. in this lecture series. Constrained-ﬁtting methods originating from Bayesian priors have also recently been used by Lee et al. [36] to study the two lowest octet and decuplet positive and negative parity baryons, using overlap fermions with pion masses down to ∼180 MeV. These authors have addressed the diﬃcult issue of contamination of the ﬁrst excited state of the nucleon with quenched η N artifacts [37]. While these authors claim to have observed the Roper in quenched QCD, it remains to be demonstrated that this conclusion is independent of the Bayesianprior assumed in their analysis [22]. It would be interesting to examine these correlation functions using correlation matrix techniques or alternative Bayesian techniques such as the Maximum Entropy Method. As mentioned in Sect. 1, we are currently seeing intense interest in exotic pentaquark spectroscopy, and recently the ﬁrst lattice studies of 5-quark states have appeared [38, 39, 40]. Here the unique advantage of lattice QCD can come to the fore, in predicting, for instance, the parity and spin of the lowest lying pentaquark state, which are as yet undetermined experimentally. The ﬁrst two of these early studies have used the standard Wilson gauge and fermion action, on lattice sizes of L = 1.2–2.7 fm [38], and L = 2.2 fm [39], while the third used domain wall fermions with a Wilson gauge action on a L = 1.8 fm lattice. The main challenge here has been the construction of lattice operators for states with 4 quarks and an antiquark, which can be variously constructed in terms of (qqq)(q¯ s) “nucleon-meson” [38] or (qq)(qq)¯ s “diquark-diquark-¯ s” [39, 40] operators. While two of these studies [38, 39] appear to favor negative parity for the lowest lying pentaquark state, it is not clear that this state is a new resonant combination of KN in S wave. The most recent study [40] reports an even parity ground state, and moreover identiﬁes the Roper resonance as a pentaquark state. The relatively small volumes, limited set of interpolating ﬁelds, and naive linear extrapolations means that these results must be regarded as exploratory at present. New studies using the FLIC fermion action with several diﬀerent interpolating ﬁelds are currently in progress [41]. We are clearly witnessing an exciting period in which the ﬁeld of baryon spectroscopy on the lattice is beginning to ﬂourish.

3 Lattice Techniques 3.1 Spectroscopy Recipe The computation of the spectrum of states in lattice QCD is in principle straightforward. The building blocks are the quark propagators ij Sαβ (x, y) = 0|ψαi (x)ψ¯βj (y)|0 ,

(1)

computed on an ensemble of gauge conﬁgurations, composed of gauge ﬁeld links Uµ (x) exp{iagAµ (x)}. The calculation then proceeds as follows:

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1. Choose an interpolating operator O that has a good overlap with P , the state of interest, 0 | O | P = 0 , and ideally a small overlap with other states having the same quantum numbers. 2. Form the time-sliced correlation function G(t, p) = O(x, t)O† (0, 0)e−ip·x , x

which can be expressed by a Wick expansion in terms of the elemental quark propagators of (1). 3. Insert a complete set of states between O and O† . The time-sliced sum puts the intermediate states at deﬁnite momentum, and one ﬁnds d3 k e−ip·x 0|O(x, t)|P (k)P (k)|O† (0, 0)|0 G(t, p) = 3 2E(k) (2π) x P

| 0 | O | P |2 e−iEP (p)t , = 2 EP (p) P

where the sum over P includes the contributions from two-particle and higher states. 4. Continue to Euclidean space t → −it, giving G(t, p) =

| 0 | O | P |2 e−EP (p)t . 2 EP (p)

(2)

P

At large Euclidean times, the lightest state dominates the spectral sum in (2), and we can extract the ground state mass. The determination of this ground state mass for the states of lowest spin has been the benchmark calculation of lattice QCD since its inception. However, our goal is to build up a more complete description of the baryon spectrum, ultimately determining not only the masses of some of the higher spin particles, but also the masses of the radial excitations. In the remainder of this section, we will address two issues that are crucial to attaining this goal: the application of variational techniques to isolate the higher excitations in (2), and the use of the appropriate statistical ﬁtting techniques to reliably extract the energies of those excitations. In the following section, we will describe the construction of the nucleon interpolating operators. 3.2 Fitting Techniques Variational Methods To conﬁdently extract other than the lowest-lying state in (2), it is crucial to have more than a single interpolating operator Oi in order to appeal to variational

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methods to determine the spectrum of states [42, 43]. Our aim is to measure every element of the matrix of correlators ' ( † Oi (x, t)Oj (0) . (3) Gij (t) = x

We now consider the eigenvalue equation G(t)u = λ(t, t0 )G(t0 )u ,

(4)

for some eigenfunction u of G−1 (t0 )G(t), with t0 ﬁxed. For the sake of illustration, we will consider a simple system with only two independent states, and a 2 × 2 matrix of correlators. Then the eigenvalues of (4) satisfy λ+ (t, t0 ) = e−(t−t0 )E0 , λ− (t, t0 ) = e−(t−t0 )E1 ,

(5)

and we have an exact separation between the energies of the two states, with coeﬃcients growing exponentially with t0 . For the physical case of more than two states, there are exponential corrections arising from the states of higher energy. Ideally, to suppress the contribution of these higher energy states we would wish to choose t0 to have as large a value as possible. However, increasing t0 comes at the price of increasing statistical noise, and therefore we are generally obliged to take t0 close to the origin. In practice, it is usual to adopt a variation of the above method, and not to attempt to diagonalise the transfer matrix at each time slice, but rather to choose a matrix of eigenvectors V (t0 ) for some t0 close to the source that diagonalises G(t0 )−1 G(t0 +1). For the case of a 2×2 matrix with only two states, the diagonal elements of the matrix V (t0 + 1)−1 G(t0 )−1 G(t)V (t0 + 1) are indeed equivalent to the eigenvalues of G(t0 )−1 G(t0 +1). A more pedagogical discussion of these concepts can be found in the Appendix. The eﬃcacy of this method relies on having a basis of correlators that can delineate the structure of the ﬁrst few states, together with suﬃciently high-quality statistics that the elements of the correlator matrix can be well determined. In the case of the glueball spectrum, the computational cost is dominated by the cost of generating the gauge conﬁgurations, and the overhead of measuring extra correlators is negligible. Furthermore, we are able to improve the statistical quality of the data by using translational invariance to average over the position of the source coordinate in (3.1). Variational techniques have been essential, and very successful, in the extraction of the glueball spectrum [44, 45]. In the case of operators containing quark ﬁelds, there is generally a considerable computational cost associated with the measurement of additional correlators. We will discuss the construction of such operators in the next section, but

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it is straightforward to see how this cost arises. Recall that the basic building blocks of spectroscopy are the quark propagators of (1), which satisfy ij jk (x, y)Sβγ (y, z) = δ ik δαγ δxz . Mαβ

(6)

where M is the fermion matrix of the fermion action ψ M ψ. The propagator is usually obtained by standard sparse matrix inversion methods, such as conjugate gradient. These methods are applicable to a ﬁxed source vector on the right hand side of (6), and additional correlators require the inversion of M for additional source vectors. A corollary of this is that the matrix of correlators is obtained for ﬁxed source coordinate, and therefore we cannot in general use translational invariance to average over the source coordinate in the manner of glueball calculations. Bayesian Statistics Given the computational cost of measuring additional correlators, it is important to extract the greatest possible information from the those that are measured. Furthermore, it is vital that the ﬁtting procedure be as reliable as possible, in particular by ensuring that the extracted masses are not adversely aﬀected by contamination from higher excitations. Historically, lattice calculations employed maximum likelyhood ﬁts to correlators such as (2), using a single or at most two states in the spectral sum. An acceptable χ2 /d.o.f. would only be obtained if the ﬁtted data were restricted to large temporal separations, in which one or two states did indeed dominate the spectral sum. Thus the ﬁts ignored the data at small temporal separations, which we have already seen has the largest signalto-noise ratio. It is natural to ask whether one can extract useful information from the data at small values of t, and a means of so doing is provided through the use of Bayesian Statistics. We will return to a further discussion of the the Bayesian approach in our survey of results.

4 Interpolating Fields The variational techniques and ﬁtting methods described above require a suitable basis of interpolating ﬁelds from which operators can be constructed which mimic the structure of each of the states to be extracted. In order to do so, we ﬁrstly have to consider the possible quantum numbers by which states are classiﬁed, and the extent to which they remain good quantum numbers in lattice calculations. We then have to consider the spatial structure of the states, and in particular their spatial extents, and whether we can construct interpolating operators that reﬂect the structure, through the use of, say, smeared interpolating ﬁelds. We will begin this section by detailing the quantum numbers with which states are classiﬁed, namely ﬂavour and parity, and the degree to which they are good quantum numbers on the lattice. We will then proceed to discuss the use of

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spatially extended interpolating operators, known as smearing. Finally, we will describe the operators that have been employed thus far in lattice calculations, and discuss these in terms of continuum quark model operators. 4.1 Continuum and Lattice Symmetries Baryon states are classiﬁed by their ﬂavour structure, either according to SU (2) (isospin), or SU (3) (strangeness), parity, and total spin. In nearly all lattice calculations of the hadron spectrum, exact isospin symmetry is imposed so the mu = md , and electromagnetic eﬀects are ignored. Thus the ﬂavour structure of baryon states composed of light (u, d, s) quarks is speciﬁed according to Total Isospin I, I3 , and the strangeness S, and the naming of baryon states follows from this labeling, as detailed in Table 1. Table 1. The table shows the ﬂavour classiﬁcation of baryon states constructed from light (u, d, s) quarks, together with a representative ﬂavour structure in terms of three valence quarks for the case I3 = I I 1/2 3/2 0 1 1/2 0

S

Baryon

0 0 −1 −1 −2 −3

N ∆ Λ Σ Ξ Ω

Flavour Structure (I3 = I) udu − | duu ) | uuu 1 √ (| uds − | dus ) 2 | uus | uss | sss

1 √ (| 2

The ﬂavour structure is straightforward to implement in the calculation of the correlation functions by only including the appropriate contractions in the Wick expansion of (3.1). 4.2 Angular Momentum and Lattice QCD More delicate is the identiﬁcation of the spin of particles in a lattice calculation. For the discussion of the spectrum, we are principally interested in the study of states at rest, for which the relevant continuum symmetry for classifying states is their properties under rotations, described by the group SU(2). The irreducible representations of SU(2) are labeled by the total spin J, and the projection of spin along a particular axis, say J3 , in the manner of isospin. We have already seen that we can impose exact isospin symmetry in our calculations, but the use of a hypercubic lattice has the consequence that we no longer have exact three-dimensional rotational symmetry, but rather the symmetries of the threedimensional cubic group of the three-dimensional spatial lattice, the octahedral group O [46].

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In contrast to the continuum rotation group, O is a ﬁnite group. It contains a total of 48 elements, and has a total of ﬁve single-valued irreducible representations (IR), corresponding to states of integer spin, and only three double-valued IRs, corresponding to states of half-integer spin. Thus each lattice IR is an admixture of diﬀerent values of J, listed for the double-valued representations in Table 2 [47]. Table 2. Irreducible representations of the octahedral group, listing the number of elements and the spin content Representation G1 G2 H

Dimension

Spin (J)

2 2 4

1/2, 7/2, 9/2, . . . 5/2, 7/2, 11/2, . . . 3/2, 5/2, 7/2, . . .

Thus in any lattice calculation at a ﬁxed value of the lattice spacing, we will aim to construct operators transforming irreducibly according to one of the IRs of Table 2, and extract the spectrum of states within each of these IRs. We will only be able to identify the angular momentum of the various states when we look for commonality between the masses extracted from the various IRs in the approach to the continuum limit. Thus, for example, the continuum state of spin 5/2 has two degrees of freedom associated with G2 , and the remaining four degrees of freedom associated with H. So far, most lattice calculations have employed local, S-wave propagators for the quarks, and as we shall see later, have employed operators transforming according to the IRs G1 , for spin 12 , or H, for spin 32 , and have implicitly assumed that these states with these spins are indeed the lightest states in their respective representations, as observed in the physical hadron spectrum. The technology required to construct general baryon interpolating operators transforming irreducibly under O has now be developed [48, 49], and a preliminary attempt at the spectrum using P -wave quark propagators has been made in [50]. 4.3 Parity The remaining symmetry that we must consider is parity, corresponding to the spatial-inversion operator Is . Clearly, this is a good symmetry on our hypercubic lattice, yielding the point group Oh , with 96 elements, and an additional subscript g or u on our irreducible representations corresponding to positive- and negativeparity representations respectively. One subtlety arises when one considers the determination of the spectrum on a lattice with either periodic or anti-periodic boundary conditions in the temporal direction. In that case, we have for identical source and sink.

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G(t) =

O+ (x, t)O+ (0)

x

−→

81

|AP+ |2 e−MP+ t ±

P+

|AP− |2 e−MP− (Nt −t) ,

(7)

P−

where O+ is an interpolating operator designed to isolate states of positive parity propagating in the forward time direction, Nt is the temporal extent of the lattice, AP is the amplitude of a state P , and the subscripts + and − denote contributions of states of positive or negative parity respectively. As discussed further in Sect. 5.1, a similar expression may be obtained for the opposite parity states. Thus we see that a complete delineation of the states on a periodic lattice only occurs as Nt → ∞. The superposition of results from periodic and antiperiodic boundary conditions can be used to eliminate the second term of (7). Alternatively, a ﬁxed boundary condition can be used to eliminate the second term of (7) by preventing states from crossing the temporal boundary of the lattice. 4.4 Smearing and Extended Interpolating Fields It is common to perform some smearing in the spatial dimensions at the source to increase the overlap of the interpolating operators with the ground states. Here we describe one such technique to do this: gauge-invariant Gaussian smearing [51]. The source-smearing technique [51] starts with a point source, ψ0 αa (x, t) = δ ac δαγ δx,x0 δt,t0

(8)

for source colour c, Dirac γ, position x0 and time t0 , and proceeds via the iterative scheme, ψi (x, t) = F (x, x ) ψi−1 (x , t) , x

where 1 F (x, x ) = (1 + α)

) δx,x

* 3 α

† Uµ (x, t) δx ,x+µ + Uµ (x − µ . + , t) δx ,x−µ 6 µ=1

Repeating the procedure N times gives the resulting fermion source, ψN (x, t) = F N (x, x ) ψ0 (x , t) .

(9)

x

The parameters N and α govern the size and shape of the smearing function. The propagator S is obtained from the smeared source by solving ab bc Sβγ = ψαa , Mαβ

(10)

for each colour, Dirac source c, γ respectively of (8) via a standard matrix inverter such as the BiStabilised Conjugate Gradient algorithm [52].

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5 Operators for Spin- 12 and Spin- 32 Baryons We will now see how the considerations described above apply to the construction of operators for the spin- 21 and spin- 23 baryons. Initially we will construct the operators based on analogy with the construction of the interpolating operators of the continuum, ﬁrst for spin 12 and then for spin 32 . We will then brieﬂy describe the construction of the operators that transform irreducibly under the lattice symmetries directly, without reference to the continuum discussion. 5.1 Spin- 12 Baryons: General Considerations Following standard notation, we deﬁne a two-point correlation function for a spin- 21 baryon B as GB (t, p) ≡ e−ip·x Ω |χB (x)χ ¯B (0)| Ω (11) x

where χB is a baryon interpolating ﬁeld and where we have suppressed Dirac indices. All formalism for correlation functions and interpolating ﬁelds presented in this article is carried out using the Dirac representation of the Dirac γ-matrices. The choice of interpolating ﬁeld χB is discussed in Sect. 5.2 below. The overlap of the interpolating ﬁeld χB with positive or negative parity states |B ± is parameterised by a coupling strength λB ± which is complex in general and which is deﬁned by + , MB + + Ω | χB (0) | B , p, s = λB + u + (p, s) , (12) EB + B + , MB − − Ω | χB (0) | B , p, s = λB − γ5 uB − (p, s) , (13) EB − where MB ± is the mass of the state B ± , EB ± = MB2 ± + p 2 is its energy, and β αβ uB ± (p, s) is a Dirac spinor with normalisation uα . For B ± (p, s)uB ± (p, s) = δ large Euclidean time, the correlation function can be written as a sum of the lowest energy positive and negative parity contributions

GB (t, p) ≈ λ2B +

(γ · p + MB + ) −E + t (γ · p − MB − ) −E − t e B + λ2B − e B , 2EB + 2EB −

(14)

when a ﬁxed boundary condition in the time direction is used to remove backward propagating states. The positive and negative parity states are isolated by taking the trace of GB with the operator Γ+ and Γ− respectively, where

1 MB ∓ Γ± = γ4 . (15) 1± 2 EB ∓ For p = 0, the energy EB ∓ = MB ∓ , so that Γ∓2 = Γ∓ and the Γ∓ are parity projectors In this case, positive parity states propagate in the (1, 1) and (2, 2) elements of the Dirac γ-matrix of (14), while negative parity states propagate in the (3, 3) and (4, 4) elements, and the masses of B ∓ may be isolated.

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5.2 Spin- 12 Baryons: Interpolating Fields There are two types of interpolating ﬁelds which have commonly been used in the literature. The notation adopted here is similar to that of [53]. To access the positive parity proton we use as interpolating ﬁelds T χp+ (16) 1 (x) = abc ua (x) Cγ5 db (x) uc (x) , and T χp+ 2 (x) = abc ua (x) C db (x) γ5 uc (x) ,

(17)

where the ﬁelds u, d are evaluated at Euclidean space-time point x, C is the charge conjugation matrix, a, b and c are colour labels, and the superscript T denotes the transpose. These interpolating ﬁelds transform as spinors under a parity transformation. That is, if the quark ﬁelds qa (x) (q = u, d, · · · ) transform as x) , (18) Pqa (x)P † = +γ0 qa (˜ where x ˜ = (x0 , −x), then Pχp+ (x)P † = +γ0 χp+ (˜ x) . For convenience, we introduce the shorthand notation abc a b c bb T cc (x, 0) tr S (x, 0)S (x, 0) G(Sf1 , Sf2 , Sf3 ) ≡ Sfaa f3 f2 1 (x, 0) Sfbb2 T (x, 0) Sfcc3 (x, 0) , + Sfaa 1

(19)

(x, 0) are the quark propagators in the background link-ﬁeld conﬁguwhere Sfaa 1−3 ration U corresponding to ﬂavours f1−3 . This allows us to express the correlation functions in a compact form. The associated correlation function for χp+ 1 can be written as ' ( $ # p+ −ip·x −1 , Su dC G (t, p; Γ ) = e tr −Γ G Su , CS , (20) 11

x

where · · · is the ensemble average over the link ﬁelds, Γ is the Γ± projection = Cγ5 . For ease of notation, we will drop the angled operator from (15), and C brackets, · · · , and all the following correlation functions will be understood to be ensemble averages. For the χp+ 2 interpolating ﬁeld, one can similarly write $ # −1 , γ5 Su γ5 dC , (21) Gp+ e−ip·x tr −Γ G γ5 Su γ5 , CS 22 (t, p; Γ ) = x

while the interference terms from these two interpolating ﬁelds are given by

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Gp+ 12 (t, p; Γ ) =

x

Gp+ 21 (t, p; Γ ) =

# $ −1 , Su γ5 dC e−ip·x tr −Γ G Su γ5 , CS ,

(22)

# $ −1 , γ5 Su dC e−ip·x tr −Γ G γ5 Su , CS .

(23)

x

The neutron interpolating ﬁeld is obtained via the exchange u ↔ d, and the strangeness –2, Ξ interpolating ﬁeld by replacing the doubly represented u or d quark ﬁelds in (16) and (17) by s quark ﬁelds. The Σ and Ξ interpolators are discussed in detail below. As pointed out in [22], because of the Dirac structure of the “diquark” in the parentheses in (16), in the Dirac representation the ﬁeld χp+ 1 involves both products of upper × upper × upper and lower × lower × upper components of = spinors for positive parity baryons, so that in the nonrelativistic limit χp+ 1 O(1). Here upper and lower refer to the large and small spinor components in the standard Dirac representation of the γ-matrices. Furthermore, since the “diquark” couples to total spin 0, one expects an attractive force between the two quarks, and hence better overlap with a lower energy state than with a state in which two quarks do not couple to spin 0. The χp+ 2 interpolating ﬁeld, on the other hand, is known to have little overlap with the nucleon ground state [22, 54]. Inspection of the structure of the Dirac γ-matrices in (17) reveals that it involves only products of upper × lower × lower 2 2 components for positive parity baryons, so that χp+ 2 = O(p /E ) vanishes in the nonrelativistic limit. As a result of the mixing of upper and lower components, the “diquark” term contains a factor σ · p, meaning that the quarks no longer couple to spin 0, but are in a relative L = 1 state. One expects therefore that two-point correlation functions constructed from the interpolating ﬁeld χp+ 2 are dominated by larger mass states than those arising from χp+ 1 at early Euclidean times. While the masses of negative parity baryons are obtained directly from the (positive parity) interpolating ﬁelds in (16) and (17) by using the parity projectors Γ± , it is instructive nevertheless to examine the general properties of the negative parity interpolating ﬁelds. Interpolating ﬁelds with strong overlap with the negative parity proton can be constructed by multiplying the previous p+ positive parity interpolating ﬁelds by γ5 , χp− 1,2 ≡ γ5 χ1,2 . In contrast to the posip− tive parity case, both the interpolating ﬁelds χp− 1 and χ2 mix upper and lower p− p− components, and consequently both χ1 and χ2 are O(p/E). − Physically, two nearby J P = 12 states are observed in the excited nucleon spectrum. In simple quark models, the splitting of these two orthogonal states is largely attributed to the extent to which the wave function is composed of scalar diquark conﬁgurations. It is reasonable to expect χp− 1 to have better overlap with scalar diquark dominated states, and thus provide a lower eﬀective mass in the moderately large Euclidean time regime explored in lattice simulations. If the correlator is larger, then this would be eﬀective mass associated with the χp− 2 1 p− evidence of signiﬁcant overlap of χ2 with the higher lying N 2 − states. In this

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event, a correlation matrix analysis (see Appendix) can be used to isolate these two states [32, 34]. Interpolating ﬁelds for the other members of the ﬂavour SU (3) octet are constructed along similar lines. For the positive parity Σ 0 hyperon one uses [53] / . T 1 T √ u (x) = (x) Cγ s (x) d (x) + d (x) Cγ s (x) u (x) , (24) χΣ abc 5 b c 5 b c 1 a a 2 . / 1 uTa (x) C sb (x) γ5 dc (x) + dTa (x) C sb (x) γ5 uc (x) . (25) χΣ 2 (x) = √ abc 2 Interpolating ﬁelds for accessing other charge states of Σ are obtained by d → u or u → d, producing correlation functions analogous to those in (20) through (23). Note that χΣ 1 transforms as a triplet under SU (2) isospin. An SU (2) singlet interpolating ﬁeld can be constructed by replacing “ + ” −→ “ − ” in (24) and (25). For the SU (3) octet Λ interpolating ﬁeld (denoted by “Λ8 ”), one has T T 8 1 χΛ 1 (x) = √ abc 2 ua (x) Cγ5 db (x) sc (x) + ua (x) Cγ5 sb (x) dc (x) 6 − dTa (x) Cγ5 sb (x) uc (x) , (26) . T T 8 1 χΛ 2 (x) = √ abc 2 ua (x) C db (x) γ5 sc (x) + ua (x) C sb (x) γ5 dc (x) 6 / (27) − dTa (x) C sb (x) γ5 uc (x) , which leads to the correlation function 8

GΛ 11 (t, p; Γ ) =

1 −ip·x e 6 x # $ # $ −1 + 2 G Ss , Su , CS −1 uC dC ×tr −Γ 2 G Ss , Sd , CS $ # $ # −1 + 2 G Su , Ss , CS −1 uC dC +2 G Sd , Ss , CS $ # $ # −1 − G Su , Sd , CS −1 sC sC − G Sd , Su , CS , (28) 8

8

8

Λ Λ and similarly for the correlation functions GΛ 22 , G12 and G21 . The interpolating ﬁelds for the SU (3) ﬂavour singlet (denoted by “Λ1 ”) are given by [53]

T T 1 χΛ 1 (x) = −2 abc − ua (x) Cγ5 db (x) sc (x) + ua (x) Cγ5 sb (x) dc (x) − dTa (x) Cγ5 sb (x) uc (x) , (29) . T Λ1 T χ2 (x) = −2 abc − ua (x) C db (x) γ5 sc (x) + ua (x) C sb (x) γ5 dc (x) / − dTa (x) C sb (x) γ5 uc (x) , (30)

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1

Λ where the last two terms are common to both χΛ 1 and χ1 . The correlation function resulting from this ﬁeld involves quite a few terms, 1

GΛ 11 (t, p; Γ ) = −ip·x cc T C −1 S bb γ5 + γ5 S aa CS cc T C −1 S bb γ5 e tr −Γ γ5 Ssaa CS d d u u s x cc T C −1 S bb γ5 + γ5 S aa CS cc T C −1 S bb γ5 + γ5 Ssaa CS d d s u u cc T C −1 S bb γ5 + γ5 S aa CS cc T C −1 S bb γ5 + γ5 Suaa CS d d u s s ucc T C −1 − γ5 Ssaa γ5 tr Sdbb CS dcc T C −1 − γ5 Suaa γ5 tr Ssbb CS aa bb cc T −1 − γ5 Sd γ5 tr Su CSs C .

(31)

In order to test the extent to which SU (3) ﬂavour symmetry is valid in the baryon spectrum, one can construct another Λ interpolating ﬁeld composed of the terms common to Λ1 and Λ8 , which does not make any assumptions about the SU (3) ﬂavour symmetry properties of Λ. We deﬁne . / c 1 uTa (x) Cγ5 sb (x) dc (x) − dTa (x) Cγ5 sb (x) uc (x) , (32) χΛ 1 (x) = √ abc 2 . / T c 1 T √ u χΛ (x) = (x) C s (x) γ d (x) − d (x) C s (x) γ u (x) , (33) abc b 5 c b 5 c 2 a a 2 to be our “common” interpolating ﬁelds which are the isosinglet analogue of χΣ 1 and χΣ 2 in (24) and (25). Such interpolating ﬁelds may be useful in determining the nature of the Λ∗ (1405) resonance, as they allow for mixing between singlet and octet states induced by SU (3) ﬂavour symmetry breaking. To appreciate the structure of the “common” correlation function, one can introduce the function abc a b c bb T cc G(Sf1 , Sf2 , Sf3 ) = (x, 0) tr S (x, 0)S (x, 0) Sfaa f3 f2 1 (x, 0) Sfbb2 T (x, 0) Sfcc3 (x, 0) , (34) −Sfaa 1 which is recognised as G in (19) with the relative sign of the two terms changed. c With this notation, the correlation function corresponding to the χΛ 1 interpolating ﬁeld is . # $ 1 −ip·x ΛC −1 , Su sC e tr −Γ G Sd , CS G11 (t, p; Γ ) = 2 x $/ # −1 +G Su , CSs C , Sd , (35) c

and similarly for the correlation functions involving the χΛ 2 interpolating ﬁeld.

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87

5.3 Spin- 32 Baryons In this section we extend the discussion from the previous section to the spin- 32 baryon sector. The mass of a spin- 32 baryon on the lattice is obtained from the two-point correlation function Gµν [35, 55], Gµν (t, p; Γ ) = tr [Γ Gµν (t, p)] , where αβ Gµν (t, p) =

β e−ip·x Ω| T χα µ (x) χν (0) |Ω ,

(36) (37)

x 3 and χα µ is a spin- 2 interpolating ﬁeld, Γ is a matrix in Dirac space with α, β Dirac indices, and µ, ν Lorentz indices. The interpolating ﬁeld operator used in the literature for accessing the isospin- 12 , spin- 32 , positive parity (charge +1) state is [35, 56, 57]

1 χpµ = abc uT a (x) Cγ5 γ ν db (x) gµν − γµ γν γ5 uc (x) . (38) 4

As pointed out in Sect. 5.1, all discussions of interpolating ﬁelds are carried out using the Dirac representation of the γ-matrices. This exact isospin- 12 interpolating ﬁeld has overlap with both spin- 23 and spin- 12 states and with states of both parities. The resulting correlation function will thus require both spin and parity projection. The charge neutral interpolating ﬁeld is obtained by interchanging u ↔ d. This interpolating ﬁeld transforms as a Rarita-Schwinger operator under parity transformations. That is, if the quark ﬁeld operators transform as in (18), then † N x) , PχN µ (x)P = +γ0 χµ (˜ and similarly for the Rarita-Schwinger operator Puµ (x)P † = +γ0 uµ (˜ x) ,

(39)

which will be discussed later. The computational cost of evaluating each of the Lorentz combinations in (38) is relatively high – about 100 times that for the ground state nucleon [24]. Consequently, in order to maximise statistics it is common to consider only the leading term proportional to gµν in the interpolating ﬁeld, χpµ −→ abc uT a (x) Cγ5 γµ db (x) γ5 uc (x) . (40) This is suﬃcient since we will in either case need to perform a spin- 32 projection. In order to show that the interpolating ﬁeld deﬁned in (40) has isospin 12 , we ﬁrst consider the standard proton interpolating ﬁeld given in (16) which we know to have isospin 12 . Applying the isospin raising operator, I + , on χp , one ﬁnds

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I + χp = abc (uT a Cγ5 ub )uc = abc (uT a Cγ5 ub )T uc = −abc (uT a Cγ5 ub )uc = 0. Similarly, for the interpolating ﬁeld deﬁned in (40), one has I + χpµ = abc (uT a Cγ5 γµ ub )γ5 uc = abc (uT a Cγ5 γµ ub )T γ5 uc = −abc (uT a Cγ5 γµ ub )γ5 uc =0, where we have used the representation-independent identities Cγµ C −1 = −γµT , Cγ5 C −1 = γ5T , and the identities which hold in the Dirac representation: C T = C † = C −1 = −C with C = iγ2 γ0 and γ5T = γ5 . We note that χ ¯pµ , corresponding to χpµ in (40), is ¯a γ5 (d¯b γµ γ5 C u ¯cT ) , χ ¯pµ = abc u

(41)

so that χpµ χ ¯pν = abc a b c (uTα a [Cγ5 γµ ]αβ dbβ )γ5 ucγ u ¯cγ γ5 (d¯bβ [γν γ5 C]β α u ¯Tαa ) T → γ5 Su γ5 tr γ5 Su γ5 (Cγµ Sd γν C)

T

+γ5 Su γ5 (Cγµ Sd γν C) γ5 Su γ5 ,

(42)

where the last line is the result after performing the Grassman integration over the quark ﬁelds with the quark ﬁelds being replaced by all possible pairwise contractions. In deriving the ∆ interpolating ﬁelds, it is simplest to begin with the state containing only valence u quarks, namely the ∆++ . The interpolating ﬁeld for the ∆++ resonance is given by [57] χ∆ µ

++

(x) = abc uT a (x) Cγµ ub (x) uc (x) ,

(43)

which transforms as a pseudovector under parity. The interpolating ﬁeld for a ∆+ state can be similarly constructed [55], + 1 abc T a 2 u (x) Cγµ db (x) uc (x) + uT a (x) Cγµ ub (x) dc (x) . χ∆ µ (x) = √ 3 (44) Interpolating ﬁelds for other decuplet baryons are obtained by appropriate substitutions of u, d → u, d or s ﬁelds. To project a pure spin- 32 state from the correlation function Gµν , one needs to use an appropriate spin- 32 projection operator [58],

Baryon Spectroscopy in Lattice QCD

1 1 3/2 Pµν (p) = gµν − γµ γν − 2 (γ · p γµ pν + pµ γν γ · p) . 3 3p

89

(45)

The corresponding spin- 12 state can be projected by applying the projection operator 1/2 3/2 = gµν − Pµν . (46) Pµν To use this operator and retain all Lorentz components, one must calculate the full 4×4 matrix in Dirac and Lorentz space. However, to extract a mass, only one pair of Lorentz indices is needed, reducing the amount of calculations required by a factor of four. The results from [35] which are summarised in Sect. 6.4 are calculated from the third row of the Lorentz matrix and using the projection Gs33 =

4

s G3µ g µν Pν3 ,

(47)

µ,ν=1

to extract the desired spin states, s = 12 or 32 . Following spin projection, the resulting correlation function, Gs33 , still contains positive and negative parity states. The interpolating ﬁeld operators deﬁned in (38) and (40) have overlap with both spin- 32 and spin- 12 states with positive and negative parity. The ﬁeld χµ transforms as a pseudovector under parity, as does the Rarita-Schwinger spinor, uµ . Thus the overlap of χµ with baryons can be expressed as M3/2+ 3 Ω|χµ |N 2 + (p, s) = λ3/2+ uµ (p, s) , (48) E3/2+ M3/2− 3 − γ5 uµ (p, s) , (49) Ω|χµ |N 2 (p, s) = λ3/2− E3/2− M1/2+ 1 γ5 u(p, s) , (50) Ω|χµ |N 2 + (p, s) = (α1/2+ pµ + β1/2+ γµ ) E1/2+ M1/2− 1 − u(p, s) , (51) Ω|χµ |N 2 (p, s) = (α1/2− pµ + β1/2− γµ ) E1/2− where the factors λB , αB , βB denote the coupling strengths of the interpolating ﬁeld χµ to the baryon B. For the expressions in (50) and (51), we note that the spatial components of momentum, pi , transform as vectors under parity and commute with γ0 , whereas the γi do not change sign under parity but anti-commute with γ0 . Hence the right-hand-side of (50) also transforms as a pseudovector under parity in accord with χµ . Similar expressions can also be written for χ ¯µ , M3/2+ 3 N 2 + (p, s)|χ ¯µ |Ω = λ∗3/2+ u ¯µ (p, s) , (52) E3/2+

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N

3 2−

N

1 2+

N

1 2−

(p, s)|χ ¯µ |Ω =

−λ∗3/2− -

(p, s)|χ ¯µ |Ω = − (p, s)|χ ¯µ |Ω =

M3/2− u ¯µ (p, s)γ5 , E3/2−

M1/2+ ∗ ∗ u ¯(p, s)γ5 (α1/2 + pµ + β1/2+ γµ ) , E1/2+

M1/2− ∗ ∗ u ¯(p, s)(α1/2 − pµ + β1/2− γµ ) . E1/2−

(53)

(54)

(55)

Note that we are assuming identical sinks and sources in these equations. However, calculations often use a smeared source and a point sink in which case λ∗ , α∗ and β ∗ are no longer complex conjugates of λ, α and β but are instead replaced by λ, α and β. We are now in a position to ﬁnd the form of (37) after we insert a complete set of intermediate states {|B(p, s)}. The contribution to (37) from each intermediate state considered is given by 3

3

¯ν |Ω Ω|χµ |N 2 + (p, s)N 2 + (p, s)|χ M3/2+ = +λ3/2+ λ3/2+ uµ (p, s)¯ uν (p, s) E3/2+ M3/2+ (γ · p + M3/2+ ) E3/2+ 2M3/2+ pµ γν − pν γµ + , 3M3/2+

= −λ3/2+ λ3/2+ −

2pµ pν 2 3M3/2 +

1 gµν − γµ γν 3

Ω|χµ |N 2 − (p, s)N 2 − (p, s)|χ ¯ν |Ω M3/2− = −λ3/2− λ3/2− γ5 uµ (p, s)¯ uν (p, s)γ5 E3/2− M3/2− (γ · p − M3/2− ) 1 = −λ3/2− λ3/2− gµν − γµ γν E3/2− 2M3/2− 3 2pµ pν pµ γν − pν γµ − − , 2 3M3/2 3M3/2− − 3

3

1

1

Ω|χµ |N 2 + (p, s)N 2 + (p, s)|χ ¯ν |Ω M1/2+ γ · p + M1/2+ =− (α1/2+ pµ + β1/2+ γµ )γ5 γ5 (α1/2+ pν + β 1/2+ γν ) , E1/2+ 2M1/2+ ¯ν |Ω Ω|χµ |N 2 − (p, s)N 2 − (p, s)|χ M1/2− γ · p + M1/2− = (α − pµ + β1/2− γµ ) (α1/2− pν + β 1/2− γν ) . E−/2+ 1/2 2M1/2− 1

1

To reduce computational expense, we consider the speciﬁc case when µ = ν = 3 and in order to extract masses we require p = (0, 0, 0). In this case we have the simple expressions

Baryon Spectroscopy in Lattice QCD 3

3

3 2−

3 2−

Ω|χ3 |N 2 + (p, s)N 2 + (p, s)|χ ¯3 |Ω = λ3/2+ λ3/2+

Ω|χ3 |N

(p, s)N

2 3

2 (p, s)|χ ¯3 |Ω = λ3/2− λ3/2− 3

γ0 M3/2+ + M3/2+ 2M3/2+ γ0 M3/2− − M3/2− 2M3/2−

91

,

(56) , (57)

γ0 M1/2+ + M1/2+ γ5 γ3 2M1/2+ γ0 M1/2+ + M1/2+ = +β1/2+ β 1/2+ , (58) 2M1/2+ γ0 M1/2− + M1/2− 1 1 ¯3 |Ω = β1/2− β 1/2− γ3 γ3 Ω|χ3 |N 2 − (p, s)N 2 − (p, s)|χ 2M1/2− γ0 M1/2− − M1/2− = +β1/2− β 1/2− . (59) 2M1/2− 1

1

Ω|χ3 |N 2 + (p, s)N 2 + (p, s)|χ ¯3 |Ω = −β1/2+ β 1/2+ γ3 γ5

Therefore, in an analogous procedure to that used in Sect. 5.1, where a ﬁxed boundary condition is used in the time direction, positive and negative parity states are obtained by taking the trace of the spin-projected correlation function, Gs33 , in (47) with the operator Γ = Γ± ,

± s (60) Gs± 33 = tr Γ G33 , where in this case (cf. (15)) Γ± =

1 (1 ± γ4 ) . 2

(61)

The positive parity states propagate in the (1,1) and (2,2) elements of the Dirac γ-matrix, while negative parity states propagate in the (3,3) and (4,4) elements for both spin- 21 and spin- 32 projected states. A similar treatment can be carried out for the ∆ interpolating ﬁelds but is left as an exercise for the interested reader. 5.4 Octahedral Group Irreducible Representations The procedure outlined above largely follows the continuum construction of interpolating operators used in, say, QCD sum-rule calculations. An alternative approach is to construct operators that lie in the irreducible representations (IRs) of the cubic group Oh of the lattice directly [48, 49]. To illustrate the latter method, we consider the construction of the I = 1/2, I3 = 1/2 nucleon interpolating operator, following the discussion of [49]. The starting point is the formation of a set of elemental baryon operators that are gauge invariant, and have the correct isospin, or ﬂavour, properties. In the case of point-like quark ﬁelds, these are given by

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Φαβγ = abc (uaα dbγ ucβ − daα ubγ ucβ ) ,

(62)

where the coordinate of the quark ﬁelds are suppressed, and α, β, γ are spinor indices. Examination of (62) reveals the constraints Φαβγ +Φγβα = 0 and Φαβγ + Φβγα + Φγαβ = 0 so that there are only 20 independent operators. The central result needed to ﬁnd operators transforming irreducibly under the IRs of Table 2 is the projection formula BiΛλF(x) =

dΛ (Λ)∗ Dλλ (R)UR BiF(x)UR† , gOh

(63)

R∈Oh

where Λ refers to an Oh IR, λ is the IR row, gOh is the number of elements (Λ) in Oh , dΛ is the dimension of the Λ IR, Dmn (R) is a Λ representation matrix corresponding to group element R, and UR is the quantum operator which implements the symmetry operations; the temporal argument is suppressed. Application of the formula requires explicit representation matrices for each group element. These are applied to the 20 linearly independent operators above, from which 20 linearly independent operators that transform irreducibly under Oh are identiﬁed. The procedure is Dirac-basis dependent, and linearly independent operators are shown in Table 3, using the DeGrand-Rossi representation of the Dirac γ-matrices [49]. There are three embeddings of G1 and a single embedding of H for both even parity (g) and odd parity (u); the two helicities of spin 1/2 correspond to the two rows of the two-dimensional representation G1g(1u) while the four helicities of spin 3/2 correspond to the four rows of the four-dimensional representation H. The three embeddings of G1 correspond to some linear combinations of the operators of (16)–(17) and the spin-1/2 projection of (38); the embedding of H corresponds to the spin-3/2 projection. Whilst the results in Table 3 may be less intuitive than their counterparts derived earlier, the procedure can easily extended to more complicated operators, including those with excited glue, or to pentaquark operators. Furthermore, we have seen explicitly that there are three independent spin-1/2 operators.

6 Survey of Results Now that we have provided a detailed description of the procedure for the calculation of excited baryons on the lattice, we will now present an overview of recent lattice calculations of the excited baryon spectrum. The emphasis of most calculations has been the demonstration that the excited nucleon spectrum is indeed accessible to lattice calculations, and most of the calculations share several features. Namely, they are in the quenched approximation to QCD, are obtained on lattice volumes of 2−3 fm, employ pseudoscalar masses of around 500 MeV, and make fairly limited investigations of the systematic uncertainties on the calculations.

Baryon Spectroscopy in Lattice QCD

93

Table 3. Combinations of the operators Φαβγ in (63) which transform irreducibly under Oh for the DeGrand-Rossi representation of the Dirac γ-matrices, employed by LHPC and MILC Collaborations (see [49]) Irrep

Row

Operators

G1g G1g

1 2

Φ112 + Φ334 −Φ221 − Φ443

G1g G1g

1 2

Φ123 − Φ213 + Φ314 Φ124 − Φ214 + Φ324

G1g G1g

1 2

2Φ114 + 2Φ332 − Φ123 − Φ213 + 2Φ134 − Φ314 −2Φ223 − 2Φ441 + Φ124 − Φ214 − 2Φ234 + Φ324

G1u G1u

1 2

Φ112 − Φ334 −Φ221 + Φ443

G1u G1u

1 2

Φ123 − Φ213 − Φ314 Φ124 − Φ214 − Φ324

G1u G1u

1 2

Hg Hg Hg Hg

1 2 3 4

Hu Hu Hu Hu

1 2 3 4

2Φ114 − 2Φ332 − Φ123 − Φ213 − 2Φ134 + Φ314 −2Φ223 + 2Φ441 + Φ124 + Φ214 + 2Φ234 − Φ324 √ 3(Φ113 + Φ331 ) Φ114 + Φ332 + Φ123 + Φ213 − 2Φ134 + Φ314 Φ √223 − Φ441 + Φ124 + Φ214 − 2Φ234 + Φ324 3(Φ224 + Φ442 ) √ 3(Φ113 − Φ331 ) Φ114 − Φ332 + Φ123 + Φ213 + 2Φ134 − Φ314 Φ √223 − Φ441 + Φ124 + Φ214 + 2Φ234 − Φ324 3(Φ224 − Φ442 )

The most comprehensive study of the excited nucleon spectrum has been obtained using FLIC fermions, detailed in [34, 35], and we will describe these calculations at length in these lectures. For some of the lowest-lying resonances, such as the Roper resonance and the odd-parity partner of the nucleon, there have been studies of the systematic uncertainties in the calculations, in particular arising from the ﬁnite lattice spacing, and the need to perform a chiral extrapolation [13]. For completeness, we will ﬁrst brieﬂy describe the gauge and fermion actions used in the FLIC fermion analysis. Additional details of the simulations can be found in [33, 34, 35]. 6.1 Lattice Actions for FLIC Calculation For the gauge ﬁelds, a mean-ﬁeld improved plaquette plus rectangle action is used. The simulations are performed on a 163 × 32 lattice at β = 4.60, which corresponds to a lattice spacing of a = 0.122(2) fm set by a string tension analysis

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√ incorporating the lattice Coulomb potential [59] with σ = 440 MeV. For the quark ﬁelds, a Fat-Link Irrelevant Clover (FLIC) [33] action is implemented. A detailed description of the FLIC fermion action can be found in the contribution by Zanotti et al. in this series. A ﬁxed boundary condition in the time direction is used for the fermions by setting Ut (x, Nt ) = 0 ∀ x in the hopping terms of the fermion action, with periodic boundary conditions imposed in the spatial directions. Gauge-invariant Gaussian smearing [51] in the spatial dimensions is applied at the source to increase the overlap of the interpolating operators with the ground states. Five masses are used in the calculations [33] and the strange quark mass is taken to be the second heaviest quark mass in each case. The analysis is based on a sample of 400 conﬁgurations, and the error analysis is performed by a thirdorder, single-elimination jackknife, with the χ2 per degree of freedom (NDF ) obtained via covariance matrix ﬁts. 6.2 Spin- 12 Baryons −

In Fig. 1 we show the nucleon and N ∗ ( 12 ) masses as a function of the pseudoscalar meson mass squared, m2π . The ﬁlled squares indicate the results for the FLIC action, and the stars for the Wilson action [34] (the Wilson points are obtained from a sample of 50 conﬁgurations). We note here that the spatial size of the lattice is L = 1.95 fm and that from the values of mπ given in Table 1 of [34] we have mπ L ≥ 5.52.

2090 -

N* (1/2 )

M (GeV)

2

1650

-

1.6 1535

-

FLIC NP Imp Clover (ani) Fixed Point CI Wilson DWF Wilson NP Imp Clover

1.2

N 939 +

0.8 0

0.2

0.4 2

0.6

0.8

1

mπ (GeV2 ) −

Fig. 1. Masses of the nucleon (N ) and the lowest J P = 12 excitation (“N ∗ ”). The FLIC and Wilson results are from [34], the DWF [28], Fixed Point [32], Chirally Improved (CI) Wilson [32], NP improved clover [31] and NP improved anisotropic clover − [26] results are shown for comparison. The empirical nucleon and low lying N ∗ ( 12 ) masses are indicated by the asterisks along the ordinate

Baryon Spectroscopy in Lattice QCD

95

For comparison, we also show results from earlier simulations with domain wall fermions (DWF) [28] (ﬁlled triangles), a nonperturbatively (NP) improved clover action on anisotropic lattices at three diﬀerent lattice spacings [26] (open diamonds), an NP improved clover action at β = 6.2 [31] (open squares, open circles and ﬁlled diamonds), and results using Fixed Point (FP) (crosses) and Chirally Improved (CI) (open inverted triangles) [32] fermion actions. The scatter of the diﬀerent NP improved results is due to diﬀerent source smearing and volume eﬀects: the open squares are obtained by using fuzzed sources and local sinks, the open circles use Jacobi smearing at both the source and sink, while the ﬁlled diamonds, which extend to smaller quark masses, are obtained from a larger lattice (323 × 64) using Jacobi smearing. The empirical masses of the − nucleon and the three lowest 12 excitations are indicated by the asterisks along the ordinate. In an unquenched calculation, the results may shift by the order of 10% compared with a quenched calculation [12]. There is excellent agreement between the diﬀerent improved actions for the nucleon mass, in particular between the FLIC [34], DWF [28], NP improved clover [26, 31], FP and CI [32] results. On the other hand, the Wilson results lie systematically low in comparison to these due to the large O(a) errors in this − action [33]. A similar pattern is repeated for the N ∗ ( 12 ) masses. Namely, the FLIC, DWF, NP improved clover, FP and CI masses are in good agreement with each other, while the Wilson results again lie systematically lower. A mass splitting of around 400 MeV is clearly visible between the N and N ∗ for all actions, including the Wilson action, despite its poor chiral properties. Furthermore, the − trend of the N ∗ ( 12 ) data with decreasing mπ is consistent with the approach to the mass of the lowest-lying physical negative parity N ∗ states. In the case of the NP-improved clover fermion action, with O(a2 ) discretisation errors, the calculation has been performed at three values of the lattice spacing, enabling a continuum extrapolation to be performed [31]. This is shown in Fig. 2, although it should be noted that a simple linear chiral extrapolation was performed in this calculation. There is a suggestion of a somewhat larger lattice spacing dependence for the higher excited resonance, emphasising the need to perform a careful analysis of systematic uncertainties in future calcu+ lations. Figure 3 shows the mass of the J P = 12 states (the excited state is denoted by “N (1/2+ )”). As is long known, the positive parity χ2 interpolating ﬁeld does not have good overlap with the nucleon ground state [22] and a correlation matrix analysis conﬁrms this result [34], as discussed below. It has + been speculated that χ2 may have overlap with the lowest 12 excited state, the N ∗ (1440) Roper resonance [28]. In addition to the FLIC and Wilson results from the present analysis, we also show in Fig. 3 the DWF results [28], and those from an earlier analysis with Wilson fermions together with the operator product ex+ pansion (OPE) [22]. The physical values of the lowest three 12 excitations of the nucleon are indicated by the asterisks. The most striking feature of the data is the relatively large excitation en+ ergy of the N ( 12 ), some 1 GeV above the nucleon. There is little evidence, therefore, that this state is the N ∗ (1440) Roper resonance. While it is possible

96

D.B. Leinweber et al.

5

4

mN r0

N(1535) 3

N(939)

2

Jacobi Fuzzed

1

0

0

0.01

0.02

(a/r0

0.03

0.04

)2

Fig. 2. The masses of the lowest-lying positive- and negative-parity baryons in units of r0−1 versus a2 in units of r02 , where r0 is Sommer’s scale [60]. The lines are linear ﬁts in a2 /r02 to the positive- and negative-parity baryon masses; diﬀerent plotting symbols correspond to diﬀerent types of smearing. Also shown are the physical values 2.8

M (GeV)

2.4

+ N ’(1/2 )

2100 +

2 1710 +

1.6 1440 +

FLIC Wilson WilsonOPE DWF

N

1.2 939+

0.8

0

0.2

0.4

0.6

0.8

1

1.2

2 m 2 (GeV ) π Fig. 3. Masses of the nucleon, and the lowest J P = 12 excitation (“N ”). The FLIC results [34] are compared with the DWF [28] and Wilson-OPE [22] analyses, as well as + with the Wilson results from [34]. The empirical nucleon and low lying N ∗ ( 12 ) masses are indicated by asterisks, with physical masses given in MeV +

that the Roper resonance may have a strong nonlinear dependence on the quark mass at m2π ∼ 0.2 GeV2 , arising from, for example, pion loop corrections, it is unlikely that this behaviour would be so dramatically diﬀerent from that of the

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N ∗ (1535) so as to reverse the level ordering obtained from the lattice. A more likely explanation is that the χ2 interpolating ﬁeld does not have good overlap with either the nucleon or the N ∗ (1440), but rather (a combination of) excited 1+ state(s). 2 Recall that in a constituent quark model in a harmonic oscillator basis, the mass of the lowest mass state with the Roper quantum numbers is higher than the lowest P -wave excitation. It seems that neither the lattice data (at large quark masses and with our interpolating ﬁelds) nor the constituent quark model have good overlap with the Roper resonance. Better overlap with the Roper is likely to require more exotic interpolating ﬁelds. As mentioned in Sect. 2, Lee et al. [36] have performed a calculation using overlap fermions with pion masses down to ∼180 MeV. Using new constrained curve ﬁtting techniques, they extract excited states from a single correlation function calculated with the standard nucleon interpolating ﬁeld in (16). The results from this calculation exhibit a dramatic drop in the mass of the ﬁrst + excited 12 state of the nucleon at light pion masses, reversing the level ordering + − of the ﬁrst 12 and 12 excited states. It is important, however, that this result be shown to be independent of the constrained curve ﬁtting techniques adopted for this analysis. A correlation matrix analysis involving several operators would shed considerable light on this issue. Recently there has also been speculation that the Roper resonance suﬀers from large ﬁnite volume errors [29]. To study this issue, the authors of [29] calculate the nucleon and its ﬁrst positive and negative parity excitations on three diﬀerent lattice volumes (La = 1.5, 2.2 and 3.0 fm). Using Maximum Entropy Methods, they ﬁnd that on large volume lattices (> ∼ 3.0 fm) the mass 1+ of the 2 excited nucleon state is reduced. A similar analysis remains to be − performed for the ﬁrst 12 nucleon state obtained from the same fermion action. + At present Wilson fermion scaling violations allow the 12 excited nucleon state − to sit lower than the ﬁrst 12 nucleon state obtained from the improved DWF action. It is essential to compare the masses of these states using the same fermion actions to remove systematic errors such as these. Similarly, a correlation matrix analysis involving several operators remains desirable. The BGR Collaboration has performed a calculation of the excited nucleon spectrum on two lattice volumes (La 1.8 and 2.4 fm) using Fixed Point and Chirally Improved Wilson fermions [32]. Using three diﬀerent operators to cre+ − ate the nucleon states, the lowest two states in both the 12 and 12 channels are extracted using a 3 × 3 correlation matrix. The eigenvectors and eigenvalues determined from the correlation matrix analysis should correspond to the solutions of (4). It is unclear, however, whether the constraints which have been employed in [32] are physical, and in particular whether the eigenvectors correspond to the optimal projections onto physical states deﬁned by the eigenvectors − u of (4). With this caveat, a splitting between the two 12 states is identiﬁed and + − the excited 12 state is observed to sit above the 12 states, in agreement with

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[34] (see next section). The results on two volumes do not exhibit any volume dependence. In the next section we present results from a 2 × 2 correlation matrix analysis using FLIC fermions on a single lattice volume. 6.3 Resolving the Resonances −

The mass splitting between the two lightest N ∗ ( 12 ) states (N ∗ (1535) & N ∗ (1650)) can be studied by considering the odd parity content of the χ1 and χ2 interpolating ﬁelds in (16) and (17). Recall that the “diquarks” in χ1 and χ2 couple diﬀerently to spin, so that even though the correlation functions built up from the χ1 and χ2 ﬁelds will be made up of a mixture of many excited states, they will have dominant overlap with diﬀerent states [22, 24]. By using the correlation-matrix techniques described in [34] (see also Appendix), two separate mass states are extracted from the χ1 and χ2 interpolating ﬁelds. The results from the correlation matrix analysis are shown by the ﬁlled symbols in Fig. 4, and are compared to the standard “naive” ﬁts performed directly on the diagonal correlation functions, χ1 χ1 and χ2 χ2 , indicated by the open symbols. − The results indicate that indeed the N ∗ ( 12 ) largely corresponding to the − χ2 ﬁeld (labeled “N2∗ ”) lies above the N ∗ ( 12 ) which can also be isolated via Euclidean time evolution with the χ1 ﬁeld (“N1∗ ”) alone. The masses of the corresponding positive parity states, associated with the χ1 and χ2 ﬁelds (labeled “N1 ” and “N2 ”, respectively) are shown for comparison. For reference, we also 2.8

M (GeV)

2.4 2100 + 2090

2

1710 + 1650

1.6

1535 1440 +

1.2 939 +

0.8

0

0.2

0.4

N1

N2

N1∗

N2∗

0.6

0.8

1

2 m 2 (GeV ) π +

−

Fig. 4. Masses of the J P = 12 and 12 nucleon states, for the FLIC action [34]. The positive (negative) parity states are labeled N1 (N1∗ ) and N2 (N2∗ ). The results from the projection of the correlation matrix are shown by the ﬁlled symbols, whereas the results from the standard ﬁts to the χ1 χ1 and χ2 χ2 correlation functions are shown by ± the open symbols (oﬀset to the right for clarity). Empirical masses of the low lying 12 states are indicated by the asterisks

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±

list the experimentally measured values of the low-lying 12 states. It is interesting to note that the mass splitting between the positive parity N1 and negative ∗ ∗ states (roughly 400–500 MeV) is similar to that between the N1,2 parity N1,2 and the positive parity N2 state, reminiscent of a constituent quark–harmonic oscillator picture. Turning to the strange sector, in Fig. 5 we show the masses of the positive and negative parity Σ baryons calculated from the FLIC action [34] compared with the physical masses of the known positive and negative parity states. The pattern of mass splittings is similar to that found in Fig. 4 for the nucleon. + Namely, the 12 state associated with the χ1 ﬁeld appears consistent with the + empirical Σ(1193) ground state, while the 12 state associated with the χ2 ﬁeld + lies signiﬁcantly above the observed ﬁrst (Roper-like) 12 excitation, Σ ∗ (1660). There is also evidence for a mass splitting between the two negative parity states, similar to that in the nonstrange sector. The spectrum of the strangeness 2.8

M (GeV)

2.4 2

2000 1880 +

1.6

1770 + 1750 1660 + 1620

1.2

1193+

Σ1

Σ2

Σ∗

Σ ∗2

1

0.8

0

0.2

0.4

0.6

0.8

1

2 m 2 (GeV ) π Fig. 5. As in Fig. 4 but for the Σ baryons

–2 positive and negative parity Ξ hyperons is displayed in Fig. 6. Once again, the pattern of calculated masses repeats that found for the Σ and N masses in Figs. 4 and 5, and for the respective coupling coeﬃcients [34]. The empirical masses of the physical Ξ ∗ baryons are denoted by asterisks. However, for all but the ground state Ξ(1318), the J P values are not known. Finally, we consider the Λ hyperons. In Figs. 7 and 8 we compare results obtained from the Λ8 and Λc interpolating ﬁelds, respectively, using the two different techniques for extracting masses. A direct comparison between the positive and negative parity masses for the Λ8 (open symbols) and Λc (ﬁlled symbols) states extracted from the correlation matrix analysis, is shown in Fig. 9. A similar pattern of mass splittings to that for the N ∗ spectrum of Fig. 8 is observed. In particular, the negative parity Λ∗1 state (diamonds) lies ∼400 MeV above the

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D.B. Leinweber et al. 2.8 2.4

M (GeV)

2250 ? 2120 ?

2

1950 ?

1690 ? 1620 ?

1.6

1318 +

1.2 0.8

0

0.2

0.4

Ξ1

Ξ2

Ξ 1∗

Ξ ∗2

0.6

0.8

1

2 m 2 (GeV ) π Fig. 6. As in Fig. 4 but for the Ξ baryons. The J P values of the excited states marked with “?” are undetermined 2.8

M (GeV)

2.4 2 1810+ 1800 1670 1600 +

1.6

1405

1.2 0.8

1116+

0

8

Λ2

8∗

Λ82∗

Λ1 Λ1 0.2

0.4

0.6

8

0.8

1

2 m 2 (GeV ) π Fig. 7. As in Fig. 4 but for the Λ states obtained using the Λ8 interpolating ﬁeld

positive parity Λ1 ground state (circles), for both the Λ8 and Λc ﬁelds. There is also clear evidence of a mass splitting between the Λ∗1 (diamonds) and Λ∗2 (squares). Using the naive ﬁtting scheme (open symbols in Figs. 7 and 8) misses the mass splitting between Λ∗1 and Λ∗2 for the “common” interpolating ﬁeld. Only after performing the correlation matrix analysis is it possible to resolve two separate mass states, as seen by the ﬁlled symbols in Fig. 8. As for the other baryons, there is little evidence that the Λ2 (triangles) has any signiﬁcant overlap with the ﬁrst positive parity excited state, Λ∗ (1600) (cf. the Roper resonance, N ∗ (1440), in Fig. 4).

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2.8

M (GeV)

2.4 2 1810+ 1800 1670 1600 +

1.6

1405

1.2 0.8

0

Λ2

c∗

Λ2∗

Λ1 0.2

0.4

c

c

Λ1

1116+

0.6

c

0.8

1

2 m 2 (GeV ) π Fig. 8. As in Fig. 4 but for the Λ states obtained using the Λc interpolating ﬁeld 2.8

M (GeV)

2.4 2 1810+ 1800 1670 1600 +

1.6

1405

1.2 0.8

1116+

0

Λc 0.2

Λ8 0.4

Λ1

Λ2

Λ∗1

Λ∗2

0.6

0.8

1

2 m 2 (GeV ) π Fig. 9. Masses of the positive and negative parity Λ states, for the octet Λ8 (open symbols) and “common” Λc (ﬁlled symbols) interpolating ﬁelds with the FLIC action [34]. The positive (negative) parity states labeled Λ1 (Λ∗1 ) and Λ2 (Λ∗2 ) are the two Λ states obtained from the correlation matrix analysis of the χΛ 1 and χ2 interpolating 1± ﬁelds. Empirical masses of the low lying 2 states are indicated by the asterisks

While it seems plausible that nonanalyticities in a chiral extrapolation [11] of N1 and N1∗ results could eventually lead to agreement with experiment, the situation for the Λ∗ (1405) is not as compelling. Whereas a 150 MeV pion-induced self energy is required for the N1 , N1∗ and Λ1 , 400 MeV is required to approach the empirical mass of the Λ∗ (1405). This may not be surprising for the octet ﬁelds, as the Λ∗ (1405), being an SU (3) ﬂavour singlet, may not couple strongly to an SU (3) octet interpolating ﬁeld. Indeed, there is some evidence of this in Fig. 9. This large discrepancy of 400 MeV suggests that relevant physics giving rise to a

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light Λ∗ (1405) may be absent from simulations in the quenched approximation. The behaviour of the Λ∗1,2 states may be modiﬁed at small values of the quark mass through nonlinear eﬀects associated with Goldstone boson loops including ¯ channels. While some of this the strong coupling of the Λ∗ (1405) to Σπ and KN coupling will survive in the quenched approximation, generally the couplings are modiﬁed and suppressed [12, 61]. It is also interesting to note that the Λ∗1 and Λ∗2 masses display a similar behaviour to that seen for the Ξ1∗ and Ξ2∗ states, which are dominated by the heavier strange quark. Alternatively, the study of more exotic interpolating ﬁelds may indicate the the Λ∗ (1405) does not couple strongly to χ1 or χ2 . Investigations at lighter quark masses involving quenched chiral perturbation theory will assist in resolving these issues. 6.4 Spin- 32 Baryons The interpolating ﬁeld deﬁned in (40) has overlap with spin- 12 and spin- 32 states of both parities. After performing appropriate spin projections on the correlation ± ± functions, the masses of the N 32 and N 12 states are extracted and displayed in ± Fig. 10 as a function of m2π . Earlier results for the N 12 states using the standard 1 spin- 2 interpolating ﬁeld [33, 34] from (16) are also shown with open symbols in Fig. 10 for reference. It is encouraging to note the agreement between the ± spin-projected 12 states obtained from the spin- 32 interpolating ﬁeld in (40) and ± the earlier 12 results from the same gauge ﬁeld conﬁgurations. We also observe

−

+

Fig. 10. Masses of the spin projected N 32 (ﬁlled triangles), N 32 (ﬁlled inverted + − triangles), N 12 (ﬁlled circles), and N 12 (ﬁlled squares) isospin- 12 states [35]. For + comparison, previous results from the direct calculation of the N 12 (open circles) and − N 12 (open squares) from Fig. 1 are also shown. The empirical values of the masses of + − − + the N 12 (939), N 12 (1535), N 32 (1520) and N 32 (1720) are indicated (in MeV) on the left-hand-side at the physical pion mass

Baryon Spectroscopy in Lattice QCD −

103 −

that the N 32 state has approximately the same mass as the spin-projected N 12 state which is consistent with the experimentally observed masses. The results − + for the N 32 state in Fig. 10 indicate a clear mass splitting between the N 32 and − N 32 states obtained from the spin- 23 interpolating ﬁeld, with a mass diﬀerence around 300 MeV. This is slightly larger than the experimentally observed mass diﬀerence of 200 MeV. + − Turning now to the isospin- 32 sector, results for the ∆ 32 and ∆ 32 masses + are shown in Fig. 11 as a function of m2π . The trend of the ∆ 32 data points with decreasing m± is clearly towards the ∆(1232), although some nonlinearity with − m2π is expected near the chiral limit [11, 12]. The mass of the ∆ 32 lies some 500 MeV above that of its parity partner, although with somewhat larger errors.

Fig. 11. Masses of the spin-projected ∆ 32

±

and ∆ 12

3/2+

±

isospin- 32 3− ∆ 2 (1700),

resonances [35]. −

(1232), ∆ 12 (1620) and The empirical values of the masses of the ∆ + ∆ 12 (1910) are shown (in MeV) on the left-hand-side at the physical pion mass ±

After performing a spin projection to extract the ∆ 12 states a discernible, but noisy, signal is detected. This indicates that the interpolating ﬁeld in (43) has only a small overlap with spin- 12 states. However, with 400 conﬁgurations we are able to extract a mass for the spin- 21 states at early times, shown in Fig. 11. ± Here we see the larger error bars associated with the ∆ 12 states. The lowest − excitation of the ground state, namely the ∆ 12 , has a mass ∼350–400 MeV + − + above the ∆ 32 , with the ∆ 32 possibly appearing heavier. The ∆ 12 state is found to lie ∼100–200 MeV above these, although the signal becomes weak at smaller quark masses. This level ordering is consistent with that observed in the empirical mass spectrum, which is also shown in the ﬁgure. − − The N 12 and ∆ 12 states will decay to N π in S-wave even in the quenched approximation [62]. For all quark masses considered here, with the possible

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exception of the lightest quark, this decay channel is closed for the nucleon. While there may be some spectral strength in the decay mode, we are unable to separate it from the resonant spectral strength. + + − − The N 32 and ∆ 12 states will decay to N π in P -wave, while N 32 and ∆ 32 states will decay to N π in D-wave. Since the decay products of each of these states must then have equal and opposite momentum and energy given by 2 2π 2 2 , E =M + aL these states are stable in the present calculations.

7 Conclusions The increasing eﬀort given to the study of the excited baryon and meson spectrum by the lattice community reﬂects the appreciation that the determination of the spectrum provides vital clues to the dynamics of QCD, and the mechanisms of conﬁnement. The N ∗ spectrum in particular has several features, such − as the anomalously light N ∗ (1440) Roper resonance and the J P = 12 Λ(1405), that defy a straightforward interpretation within the quark model, and whose understanding can help to resolve the competing pictures of hadron structure. The impetus to study the N ∗ resonances has also strengthened following the observation of the S = +1, Θ+ pentaquark state, whose properties remain essentially unknown, but whose various interpretations oﬀer vastly diﬀerent pictures of baryon structure. In these lectures, we have described the techniques required to compute the N ∗ spectrum in lattice QCD, and given an overview of the current status of lattice calculations. The lightest states of both parities for spin 1/2 and spin 3/2 have been successfully resolved in the quenched approximation to QCD in both the nucleon (isospin- 12 ) and ∆ (isospin- 32 ) sectors, and in general the level ordering, albeit at relatively large pseudoscalar masses, mπ ≥ 500 MeV, is in accord with that observed experimentally. At these large pseudoscalar masses the spectrum largely follows quark-model expectations. It is important to appreciate that most current spectroscopy calculations have employed essentially “S-wave” quark propagators. The measurement of a wider basis of interpolating operators will be an important element of future studies, and the technology to construct such operators with lattice symmetry properties has now been developed. An important by-product of such studies will be insight into the quark and gluon structure of such hadrons. The realisation that hadronic physics at physical values of the pion mass is very diﬀerent from that at mπ ≥ 300 MeV, and the consequent need to correctly account for the chiral properties of the theory, have been some of the most important developments in lattice QCD of recent years. FLIC fermions and the development of fermions having an exact analogue of chiral symmetry provide the means to attain such pion masses. There are suggestions that such radically

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diﬀerent behaviour at light quark masses has been seen in the excited nucleon sector. With the development of exact dynamical fat-link fermion algorithms [63, 64, 65], FLIC fermions provide tremendous promise for accessing the lightquark mass regime of full QCD. The continuation to physical values of the light quark masses poses extra challenges to the calculation of the resonance spectrum. The excited states are no longer stable under the strong interaction at suﬃciently light quark masses. Even in the quenched approximation, this instability is manifest through nonunitary behaviour in the correlators and through additional non-analytic terms in the chiral expansion of nucleon masses [13]. The means to study the full spectrum, including the scattering lengths of multi-particle states, is in principle known, relying on examining the ﬁnite-volume shift in the two-particle spectrum. However, the method is computationally very demanding, requiring the measurement of many operators. With the realisation of large-scale computing facilities expected over the next several years, we can expect many of these endevours to come to fruition. An exciting era for baryon spectroscopy on the lattice lays ahead.

Acknowledgements This work was supported in part by the Australian Research Council and by DOE contract DE-AC05-84ER40150 under which the Southeastern Universities Research Association (SURA) operates the Thomas Jeﬀerson National Accelerator Facility. Generous grants of supercomputer time from the Australian Partnership for Advanced Computing (APAC) and the Australian National Computing Facility for Lattice Gauge Theory are gratefully acknowledged.

A Appendix − Correlation Matrix Analysis In this section we outline the correlation matrix formalism for calculations of masses, coupling strengths and optimal interpolating ﬁelds. After demonstrating that the correlation functions are real, we proceed to show how a matrix of such correlation functions may be used to isolate states corresponding to diﬀerent masses, and also to give information about the coupling of the operators to each of these states (see also [34]). A.1 The U + U ∗ Method A lattice QCD correlation function for the operator χi χj , where χi is the i-th interpolating ﬁeld for a particular baryon (e.g. χp+ 2 in Sect. 5.2), can be written as

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+ , Gij ≡ Ω|T (χi χj )|Ω ¯ −S[U,ψ,ψ] ¯ χi χj DU DψDψe , = ¯ −S[U,ψ,ψ] ¯ DU DψDψe

(64)

where spinor indices and spatial coordinates are suppressed for ease of notation. ¯ ψ] = SG [U ] + The fermion and gauge actions can be separated such that S[U, ψ, ¯ ¯ ψM [U ]ψ. Integration over the Grassmann variables ψ and ψ then gives DU e−SG [U ] det(M [U ])Hij [U ] Gij = , (65) DU e−SG [U ] det(M [U ]) where the term Hij stands for the sum of all full contractions of χi χj . The pure gauge action SG and the fermion matrix M satisfy SG [U ] = SG [U ∗ ] ,

(66)

[U ∗ ]C −1 = M ∗ [U ] , CM

(67)

and

≡ Cγ5 . respectively, where C Using the result of (67), one has det (M [U ∗ ]) = det (M ∗ [U ]) ,

(68)

and since det(M [U ]) is real, this leads to det (M [U ∗ ]) = det (M [U ]) .

(69)

Thus, U and U ∗ are conﬁgurations of equal weight in the measure DU det(M [U ]) exp (−SG [U ]), in which case Gij can be written as * ) DU e−SG [U ] det(M [U ]) {Hij [U ] + Hij [U ∗ ]} 1 Gij = . (70) 2 DU e−SG [U ] det(M [U ]) Let us deﬁne

G± ij ≡ tr[Γ± Gij ] ,

(71)

where “tr” denotes the spinor trace and Γ± is the parity-projection operator

∗ [U ] , then G± deﬁned in (15). If tr [Γ Hij [U ∗ ]] = tr Γ Hij ij is real. This can be shown by ﬁrst noting that Hij will be products of Dirac γ-matrices, fermion propagators, and link-ﬁeld operators. In a γ-matrix representation which is −1 = γ ∗ . Fermion prop µC Hermitian, such as the Sakurai representation, Cγ µ −1 [U ∗ ]C −1 =M ∗ [U ], then agators have the form M , and recalling that since CM −1 [U ∗ ]C −1 =(M −1 [U ])∗ . For products of link-ﬁeld operators O[U ] we have CM contained in Hij , the condition O[U ∗ ] = O∗ [U ] is equivalent to the requirement that the coeﬃcients of all link-products are real. As long as this requirement

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C −1 inside the trace to is enforced, we can then simply proceed by inserting C ± show that the (spinor-traced) correlation functions Gij are real. If one chooses −1 = −γ ∗ and Cγ −1 = γ ∗ . Therefore, 0C kC the Dirac representation, then Cγ 0 k in the Dirac representation of the γ-matrices, if Hij contains an even number ± of spatial γ-matrices with real coeﬃcients, G± ij is purely real; otherwise Gij is purely imaginary. In summary, the interpolating ﬁelds considered here are constructed using only real coeﬃcients and have no spatial γ-matrices. Therefore, the correlation functions G± ij are real. This symmetry is explicitly implemented by including both U and U ∗ in the ensemble averaging used to construct the lattice correlation functions, providing an improved unbiased estimator which is strictly real. This is easily implemented at the correlation function level by observing M −1 ({Uµ∗ }) = [Cγ5 M −1 ({Uµ }) (Cγ5 )−1 ]∗ for quark propagators. A.2 Recovering Masses, Couplings and Optimal Interpolators Let us again consider the momentum-space two-point function for t > 0, Gij (t, p) = e−ip·x Ω|χi (t, x)χj (0, 0)|Ω .

(72)

x

At the hadronic level, Gij (t, p) =

e−ip·x

p ,s

x

Ω|χi (t, x)|B, p , s

B

× B, p , s|χj (0, 0)|Ω , where the |B, p , s are a complete set of states with momentum p and spin s |B, p , sB, p , s| = I . (73) p

s

B

We can make use of translational invariance to write 1 0 ˆ ˆ ˆ ˆ Gij (t, p) = Ω eHt e−iP·x χi (0)eiP·x e−Ht B, p , s e−ip·x p

x

s

B

, + × B, p , s χj (0) Ω + , = e−EB t Ω|χi (0)|B, p, sB, p, s|χj (0)|Ω . s

(74)

B

It is convenient in the following discussion to label the states which have the χ interpolating ﬁeld quantum numbers and which survive the parity projection as |Bα for α = 1, 2, . . . , N . In general the number of states, N , in this tower of excited states may be inﬁnite, but we will only ever need to consider a ﬁnite

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set of the lowest such states here. After selecting zero momentum, p = 0, the parity-projected trace of this object is then G± ij (t) = tr[Γ± Gij (t, 0)] =

N

α

e−mα t λα i λj ,

(75)

α=1 α

where λα i and λj are coeﬃcients denoting the couplings of the interpolating ﬁelds χi and χj , respectively, to the state |Bα . If we use identical source and sink interpolating ﬁelds then it follows from the deﬁnition of the coupling strength that α ± ± ∗ ∗ ± λj = (λα j ) and from (75) we see that Gij (t) = [Gji (t)] , i.e., G is a Hermitian matrix. If, in addition, we use only real coeﬃcients in the link products, then G± is a real symmetric matrix. For the correlation matrices that we construct, we have real link coeﬃcients but smeared sources and point sinks. Consequently, G is a real but non-symmetric matrix. Since G± is a real matrix for the inﬁnite number of possible choices of interpolating ﬁelds with real coeﬃcients, then we α can take λα i and λj to be real coeﬃcients without loss of generality. Suppose now that we have M creation and annihilation operators, where M < N . We can then form an M × M approximation to the full N × N matrix G. At this point there are two options for extracting masses. The ﬁrst is the standard method for calculation of eﬀective masses at large t described in Sect. 5.1. The second option is to extract the masses through a correlation-matrix procedure [44]. Let us begin by considering the ideal case where we have N interpolating ﬁelds with the same quantum numbers, but which give rise to N linearly independent states when acting on the vacuum. In this case we can construct N ideal interpolating source and sink ﬁelds which perfectly isolate the N individual baryon states |Bα , α

φ =

N

uα i χi ,

(76)

vi∗α χi ,

(77)

i=1

φα =

N i=1

such that α

Bβ | φ |Ω = δαβ z α u(α, p, s) , Ω| φα |Bβ = δαβ z α u(α, p, s) , α

(78) (79)

where z α and z α are the coupling strengths of φα and φ to the state |Bα . The ∗α coeﬃcients uα i and vi in (77) may diﬀer when the source and sink have diﬀerent smearing prescriptions, again indicated by the diﬀerentiation between z α and z α . For notational convenience for the remainder of this discussion repeated indices i, j, k are to be understood as being summed over. At p = 0, it follows that,

Baryon Spectroscopy in Lattice QCD

) α G± ij (t) uj =

109

*

tr Γ± Ω| χi χj |Ω uα j x

α −mα t = λα . i z e

(80)

The only t-dependence in this expression comes from the exponential term, which leads to the recurrence relationship α mα ± Gik (t + 1) uα G± k , ij (t) uj = e

(81)

which can be rewritten as ± α mα α uk . [G± (t + 1)]−1 ki Gij (t) uj = e

(82)

This is recognised as an eigenvalue equation for the matrix [G± (t + 1)]−1 G± (t) with eigenvalues emα and eigenvectors uα . Hence the natural logarithms of the eigenvalues of [G± (t + 1)]−1 G± (t) are the masses of the N baryons in the tower of excited states corresponding to the selected parity and the quantum numbers of the χ ﬁelds. The eigenvectors are the coeﬃcients of the χ ﬁelds providing the ideal linear combination for that state. Note that since here we use only real coeﬃcients in our link products, then [G± (t + 1)]−1 G± (t) is a real matrix and so uα and v α will be real eigenvectors. It also then follows that z α and z α will be real. These coeﬃcients are examined in detail in the following section. One can also construct the equivalent left-eigenvalue equation to recover the v vectors, providing the optimal linear combination of annihilation interpolators, mα ∗α ± vi Gij (t + 1) . vk∗α G± kj (t) = e

(83)

Recalling (80), one ﬁnds: α α α −mα t , G± ij (t) uj = z λi e

vi∗α

G± ij (t)

=

± α vk∗α G± kj (t)Gil (t) ul =

α z α λj e−mα t , α −2mα t z α z α λα i λj e

(84) (85) .

(86)

The deﬁnitions of (79) imply α α α −mα t , vi∗α G± ij (t) uj = z z e

(87)

indicating the eigenvectors may be used to construct a correlation function in which a single state mass mα is isolated and which can be analysed using the methods of Sect. 2. We refer to this as the projected correlation function in the following. Combining (86) and (87) leads us to the result vk∗α Gkj (t)Gil (t) uα α −mα t l = λα . i λj e vk∗α Gkl (t)uα l

(88)

By extracting all N 2 such ratios, we can exactly recover all of the real couplings α λα i and λj of χi and χj respectively to the state |Bα . Note that throughout

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this section no assumptions have been made about the symmetry properties of G± ij . This is essential due to our use of smeared sources and point sinks. In practice we will only have a relatively small number, M < N , of interpolating ﬁelds in any given analysis. These M interpolators should be chosen to have good overlap with the lowest M excited states in the tower and we should attempt to study the ratios in (88) at early to intermediate Euclidean times, where the contribution of the (N − M ) higher mass states will be suppressed but where there is still suﬃcient signal to allow the lowest M states to be seen. This procedure will lead to an estimate for the masses of each of the lowest M states in the tower of excited states. Of these M predicted masses, the highest will in general have the largest systematic error while the lower masses will be most reliably determined. Repeating the analysis with varying M and diﬀerent combinations of interpolating ﬁelds will give an objective measure of the reliability of the extraction of these masses. In our case of a modest 2 × 2 correlation matrix (M = 2) we take a cautious approach to the selection of the eigenvalue analysis time. As already explained, we perform the eigenvalue analysis at an early to moderate Euclidean time where statistical noise is suppressed and yet contributions from at least the lowest two mass states is still present. One must exercise caution in performing the analysis at too early a time, as more than the desired M = 2 states may be contributing to the 2 × 2 matrix of correlation functions. We begin by projecting a particular parity, and then investigate the eﬀective mass plots of the elements of the correlation matrix. Using the covariance-matrix based χ2 /NDF , we identify the time slice at which all correlation functions of the correlation matrix are dominated by a single state. In practice, this time slice is determined by the correlator providing the lowest-lying eﬀective mass plot. The eigenvalue analysis is performed at one time slice earlier, thus ensuring the presence of multiple states in the elements of the correlation function matrix, minimising statistical uncertainties, and hopefully providing a clear signal for the analysis. In this approach minimal new information has been added, providing the best opportunity that the 2 × 2 correlation matrix is indeed dominated by 2 states. The left and right eigenvectors are determined and used to project correlation functions containing a single state from the correlation matrix as indicated in (87). These correlation functions are then subjected to the same covariance-matrix based χ2 /NDF analysis to identify new acceptable ﬁt windows for determining the masses of the resonances.

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Hadron Structure and QCD: Eﬀective Field Theory for Lattice Simulations D.B. Leinweber1 , A.W. Thomas1 , and R.D. Young1 Special Research Centre for the Subatomic Structure of Matter, and Department of Physics, University of Adelaide, Adelaide SA 5005, Australia

Abstract. Chiral extrapolations will be essential for many years if one is to connect modern lattice QCD calculations with experiment. Given the enormous eﬀorts made to ensure that the lattice QCD simulations are a rigorous implementation of nonperturbative QCD, it is essential that the chiral extrapolation procedure should also be consistent with all known constraints of QCD. We review the enormous progress made on this problem over the past three or four years.

1 Introduction Central to the strong interaction within the Standard Model is the challenge of solving a strongly coupled ﬁeld theory with spontaneous symmetry breaking. Making a precise determination of the masses and properties of free mesons and baryons directly from QCD itself is a tremendous challenge. It is equally important that we learn the sort of qualitative lessons needed to develop an intuitive understanding or picture of hadron structure. Important issues include the content of non-valence quarks and their spin distribution, as well as the changes they induce in the nonperturbative QCD vacuum in relation to energy density and topology. There exist hints that the structure of individual hadrons may play a fundamental role in properties of dense matter, including its saturation properties [1, 2]. Changes of hadron properties in-medium [3, 4] should serve as precursors of the quark-gluon physics which one expects to appear explicitly at higher densities. This is a central issue for both nuclear and particle physics and a sound understanding of free hadron structure is a vital piece of that puzzle. It will be quite a few years before full QCD simulations can be performed at physical light quark masses. With computation time scaling like ∼1/mq3.6 [5], quark masses are typically restricted to above 50 MeV for accurate calculations with improved quark and gluon actions. The exception to this is to use staggered fermions – which have shown some results at 25 MeV [6] – but this approach has a number of technical problems, including a larger eﬀective lattice spacing for fermions and multiple pions (extra tastes). The latter, especially, threatens to complicate the chiral extrapolation problem which we explore next. Given the computational diﬃculty of making a calculation at low quark mass there are three options:

D.B. Leinweber, A.W. Thomas, and R.D. Young: Hadron Structure and QCD: Eﬀective Field Theory for Lattice Simulations, Lect. Notes Phys. 663, 113–129 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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1. Make no comparison with any physical hadronic data: while not leading to any errors of interpretation, such an approach is quite unsatisfying. Hadronic physics is an extremely active area of study, both experimental and theoretical and it is important that all of that work receives the best theoretical guidance possible. As the only known rigorous, nonperturbative method of solving the strong interaction problem, lattice QCD should be a vital component of this equation. 2. Use an appropriate method to extrapolate the properties calculated at large light-quark mass to the physical value: this is the chiral extrapolation problem, which is complicated by spontaneous chiral symmetry breaking in QCD. As we shall explain, this problem has recently been solved by a careful reformulation of the eﬀective ﬁeld theory. 3. One can use the lattice simulations, which do represent the rigorous consequences of non-perturbative QCD, as guidance for models of hadron structure. This simply requires that one treat the light quark masses as variable parameters in any model and compare the resulting hadron properties directly with lattice results.

2 Eﬀective Field Theory for QCD The power of eﬀective ﬁeld theory for strongly interacting systems has been apparent for more than 50 years, beginning with Foldy’s inclusion of the anomalous magnetic moment of the proton in a derivative expansion of the electromagnetic coupling of photons to baryons. Of course, the fact that nucleon form factors can be approximated by a dipole with mass parameter M 2 = 0.71 GeV2 means that the radius of convergence of the formal expansion is less than 0.84 GeV. In the last twenty years our understanding of the symmetries of QCD have led to a systematic expansion of hadron properties and scattering amplitudes about the limits mπ = 0 and q = 0, known as chiral perturbation theory. For the pseudoscalar mesons, which are rigorously massless in the limit where the u, d and s quark masses are zero (the chiral SU (3) limit), this approach has been especially successful. Although, even here, serious questions have been raised over whether the formal expansion is really convergent for masses as large as ms [7, 8, 9]. In the case of baryons the questions concerning convergence are even better developed. As our example we take the best studied case relevant to lattice QCD, namely the dependence of MN on light quark mass mq . Given the GellMann–Oakes–Renner (GOR) relation, m2π ∝ mq , we prefer to use m2π , rather than mq , as the measure of explicit chiral symmetry breaking in the expressions below1 . The formal chiral expansion for MN , in terms of mπ , about the SU (2) chiral limit in full QCD is: 1

We stress, however, that wherever the GOR relation fails, e.g. for m2π > 0.8 GeV2 or for mπ → 0 in QQCD (where there are logarithmic chiral corrections), one should revert to an expansion written explicitly in terms of mq .

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MN = a0 + a2 m2π + a4 m4π + a6 m6π + · · · +σNπ (mπ ) + σ∆π (mπ ) ,

(1)

where σBπ is the self-energy arising from a Bπ loop (with B = N or ∆). These N and ∆ loops generate the leading and next-to-leading non-analytic (LNA and NLNA) behaviour, respectively. These loop integrals contain ultraviolet divergences which require some regularisation prescription. The standard approach is to use dimensional regularisation to evaluate the self-energy integrals. Under such a scheme the N N π contribution simply becomes σNπ (mπ ) → cLNA m3π and the analytic terms, a0 and a2 m2π , undergo an inﬁnite renormalisation. The ∆ contribution produces a logarithm, so that the complete series expansion of the nucleon mass about mπ = 0 is: MN = c0 + c2 m2π + c4 m4π + cLNA m3π + cNLNA m4π log mπ + · · · ,

(2)

where the ai have been replaced by the renormalised (and ﬁnite) parameters ci . The coeﬃcients of the low-order, non-analytic contributions are known [12, 13, 14]: 32 2 3 3 3 g . (3) cLNA = − g2 , cNLNA = 32 π fπ2 A 32 π fπ2 25 A 4 π ∆ Although strictly one should use values in the chiral limit, we take the experimental numbers with gA = 1.26, fπ = 0.093 GeV, the N − ∆ mass splitting, ∆ = 0.292 GeV and the mass scale associated with the logarithms will be taken to be 1 GeV. We stress that (2) was derived in the limit mπ /∆ 1. At just twice the physical pion mass this ratio approaches unity. Mathematically the region mπ ≈ ∆ is dominated by a square root branch cut which starts at mπ = ∆. Using dimensional regularisation this takes the form [15]: 2 mπ −3 32 2 1 3 2 g (2∆ − 3mπ ∆) log 32 π fπ2 25 A 2π µ2 ) * ∆ − ∆2 − m2π 3 − 2(∆2 − m2π ) 2 log ∆ + ∆2 − m2π

(4)

for mπ < ∆, while for mπ > ∆ the second logarithm becomes an arctangent. Clearly, to access the higher quark masses in the chiral expansion, currently of most relevance to lattice QCD, one requires a more sophisticated expression than that given by (2). 2.1 Naive Convergence Forgetting for a moment the issues surrounding the ∆π cut, we note that the formal expansion of the N → N π → N self-energy integral, σNπ , has been

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shown to have poor convergence properties. Using a sharp, ultra-violet cut-oﬀ, Wright showed [16] that the series expansion, truncated at O(m4π ), diverged for mπ > 0.4 GeV. This already indicated that the series expansion motivated by dimensional regularisation would have a slow rate of convergence. A similar conclusion had been reached somewhat earlier by Stuckey and Birse [10]. In considering the convergence of the truncated series, (2), it is helpful to return to the general form from which it was derived, namely (1). The dimensionally regulated approach requires that the pion mass should remain much lighter than every other mass scale involved in the problem. This requires that mπ /ΛχSB 1 (and mπ /∆ 1 if we use the simple logarithm in (2) rather than the full cut structure of (4)). An additional mass scale, which we address in detail below, is set by the physical extent of the source of the pion ﬁeld [17]. This scale, which is of order −1 RSOU RCE , corresponds to the transition between the rapid, non-linear variation required by chiral symmetry and the smooth, constituent-quark like mass behaviour observed in lattice simulations at larger quark mass. An alternative to dimensional regularisation is to regulate (1) with a ﬁnite ultra-violet cut-oﬀ in momentum space. Figure 1 displays results for a dipole-vertex regulator.

1.8

MN GeV

1.6 1.4 1.2 1 0

0.2

0.4 0.6 mΠ 2 GeV2

0.8

1

Fig. 1. Fits to lattice data with dipole regulator for the self-energy loop N → N π (neglecting the N → ∆π term for simplicity). The dashed lines show the expansion up to successive orders in mπ , from m2π up to m8π – these alternate about the solid curve as we go through even and odd terms

2.2 Renormalization and Eﬀective Field Theory In this section we draw extensively from the pedagogic introduction to eﬀective ﬁeld theories given a few years ago by LePage [11]. In that case the problem considered was to develop an eﬀective ﬁeld theory for the solution of the Schr¨ odinger equation for an unknown potential. The key physical idea of the eﬀective ﬁeld

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theory is to introduce an energy scale, λ, above which one does not attempt to understand the physics. For example, the energy region above that cut-oﬀ may involve new physics – e.g., in the very high energy limit, physics beyond the Standard Model. Because one does not pretend to control physics above the scale λ, one should not include momenta above λ in computing radiative corrections. Instead, one introduces renormalization constants which depend on (or “run with”) the choice of cut-oﬀ λ so that, to the order one works, physical results are independent of λ. As LePage points out, “it makes little sense to reduce a [i.e., increase λ ∼ 1/a] below the range” in which we understand the physics, because the “structure they see there is almost certainly wrong.” When it comes to speciﬁc problems with the development of an eﬀective theory of NN scattering, this line of argument leads him to conclude that “the problem is with dimensional regularization – and not with eﬀective ﬁeld theory.” In particular, dimensional regularization involves integrating loop momenta in the eﬀective ﬁeld theory over momenta all the way to inﬁnity – way beyond the scale where the eﬀective theory has any physical signiﬁcance. In the context of our present problem of chiral extrapolation, the essential lesson to draw from LePage’s investigation is that one should not take the cutoﬀ to ∞, that is one should not use dimensional regularization. Nor should one attempt to determine the “true” cut-oﬀ of the theory. Rather, one should choose a cut-oﬀ scale somewhat below the place where the eﬀective theory omits essential physics and use data to constrain the renormalization constants of the theory and hence eliminate the dependence on that cut-oﬀ as far as possible. The issue is then what mass scale determines the upper limit beyond which the eﬀective ﬁeld theory is applicable. This is commonly taken to be ΛχP T ∼ 4πfπ ∼ 1 GeV. Unfortunately this is incorrect for baryons. In the context of nuclear physics it has long been appreciated that nuclear sizes could never be derived from naive dispersion relation considerations of nearest t-channel poles – anomalous thresholds associated with the internal structure of nuclei dominate. Similarly, the size of a baryon is determined by nonperturbative QCD beyond the scope of chiral perturbation theory. The natural scale associated with the −1 size of a nucleon is the inverse of its radius or a mass scale RSOU RCE ∼ 0.2 to 0.5 GeV – far below ΛχP T . In the context of eﬀective ﬁeld theory it is inconsistent to keep loop contributions from momenta above this scale! As a result of these considerations, we are led to regulate the radiative corrections which give rise to the leading and next-to-leading non-analytic contributions to the mass of the nucleon using a ﬁnite range regulator with a mass −1 in the range RSOU RCE . As far as possible residual dependence on the speciﬁc choice of mass scale (and form of regulator function) will be eliminated by ﬁtting the renormalization constants to nonperturbative QCD – in this case data obtained from lattice QCD simulations. The quantitative success of applying the method is to be judged by the extent to which physical results extracted from the analysis are indeed independent of the regulator (over a physically sensible range). We refer to this approach as ﬁnite range regularization, FRR.

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3 Chiral Expansion of the Nucleon Mass Having established the philosophical framework for our analysis we now investigate the eﬀective rate of convergence of the chiral expansions obtained using diﬀerent functional forms for the regulator. For comparison we also make comparisons with the results obtained using a dimensionally regulated approach. To analyse the merits of various regularisation schemes, at best, we would require exact knowledge of how the nucleon mass varies with quark mass. Having only one value for the experimental nucleon mass we cannot determine the parameters that govern the quark mass dependence of MN without taking some information from alternate sources. Through lattice QCD we have a reliable, non-perturbative method for studying the variation of MN with mπ . In principle, this allows one to ﬁx the parameters of the chiral expansion using data obtained in simulations performed at varying quark mass. Lattice simulations of full QCD are restricted to the use of relatively heavy quarks and hence it is not clear, a priori, whether the eﬀective ﬁeld theory expansion is capable of linking to even the lightest simulated quark mass where mπ ∼ 500 MeV. We take as input both the physical nucleon mass and recent lattice QCD results of the CP-PACS Collaboration [18] and the JLQCD Collaboration [19]. This enables us to constrain an expression for MN as a function of the quark mass. The lattice results of [18] have been obtained using improved gluon and quark actions on ﬁne, large volume lattices with high statistics. These simulations were performed using an Iwasaki gluon action [20] and the mean-ﬁeld improved clover fermion action. In this work we concentrate on only those results with msea = mval and the two largest values of β (i.e., the ﬁnest lattice spacings a ∼ 0.09– 0.13 fm). We use just the largest volume results of [19], where simulations were performed with non-perturbatively improved Wilson quark action and plaquette gauge action. The lattice volumes and spacings are similar for the two data sets. We set the physical scale at each quark mass, using the UKQCD method [21]. That is, we use the Sommer scale r0 = 0.5 fm [22, 23]. This choice is ideal in the present context because the static quark potential is insensitive to chiral physics. This ensures that the results obtained represent accurate estimates of the continuum, inﬁnite-volume theory at the simulated quark masses. The lattice data lies in the intermediate mass region, with m2π between 0.3 and 1.0 GeV2 . To remove the bias implicit in choosing any particular regularization scheme, we allow each scheme to serve as a constraint curve for the other methods. In this way we generate six (one for each regularisation) diﬀerent constraint curves that describe the quark mass dependence of MN . The ﬁrst case corresponds to the truncated power series obtained through the dimensionally regulated (DR) approach, (2). We work to analytic order m6π , which is necessary in order to remove incorrect short distance physics arising from the non-analytic contribution at order m4π log mπ . The second procedure (labeled “BP”) takes a similar form but the branch point obtained from the dimensionally regularised N → ∆π transition is retained in its full functional form. That is, the logarithm in (2) is

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replaced by the full expression, (4), which ensures the correct non-analytic structure where the logarithm converts to an arctangent above the branch point. We refer to this form as the dimensionally-regulated branch-point (BP) approach. Finally, based on (1), we evaluate the loop integrals σNπ ≡ cLN A Iπ (mπ , Λ) and σ∆π ≡ cN LN A Iπ∆ (mπ , Λ), where: 2 ∞ k 4 u2 (k) Iπ = dk 2 (5) π 0 k + m2 and Iπ∆ = −

8∆ 3

0

∞

dk √

k 4 u2 (k) √ . k 2 + m2 (∆ + k 2 + m2 )

(6)

We use four diﬀerent functional forms for the ﬁnite-ranged, ultra-violet vertex regulator, u(k) – namely the sharp-cutoﬀ (SC), θ(Λ − k); monopole (MON), Λ2 /(Λ2 + k 2 ); dipole (DIP), Λ4 /(Λ2 + k 2 )2 ; and Gaussian (GAU), exp(−k 2 /Λ2 ). Closed expressions for these integrals in the ﬁrst three cases are given in the Appendix of [24]. Provided one regulates the eﬀective ﬁeld theory below the point where new short distance physics becomes important, the essential results will not depend on the cut-oﬀ scale [11]. We use knowledge learned in [24] as a guide for the appropriate scales for each of the regulator forms. In particluar, we choose regulator masses of 0.4, 0.5, 0.8 and 0.6 GeV for the sharp, monopole, dipole and Gaussian regulators respectively. For all six regularisation prescriptions, both DR-based and FRR, we ﬁt four free parameters to constrain the expansion of the nucleon mass. With the values of the non-analytic contributions ﬁxed to their model-independent values, the coeﬃcients of analytic terms up to m6π are allowed to vary to best ﬁt the lattice simulation results and to reproduce the physical nucleon mass. The resultant curves are displayed in Fig. 2. To the naked eye all of the curves are very much in agreement with each other. All of them are able to give an accurate description of the lattice data and match the physical value of MN . We now pose the question: if any one of these curves were presumed to produce an exact description of the mass of the nucleon as a function of mπ , how well could an alternate regularisation scheme match it? All forms have been based on the same, low-energy eﬀective ﬁeld theory and therefore will be equivalent in the limit mq → 0. Hence, an additional question of more direct practical importance is: over what range can a particular regularisation scheme match an alternate scheme? Within the radius of convergence, physical conclusions must be independent of regularisation and renormalisation. The best ﬁt parameters for our constraint curves, M (mπ ), are shown in Table 1. We stress that the parameters listed in this table are bare quantities and hence renormalisation scheme dependent. If one is to rigorously compare the parameters of the eﬀective ﬁeld theory, the self-energy contributions need to be Taylor expanded about mπ = 0 in order to yield the renormalisation for each of the coeﬃcients in the quark mass expansion about the chiral limit. A comparison of the resulting quark-mass expansion for each of the regularisation schemes is

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1.8

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Fig. 2. Various regularisation schemes providing constraint curves for the variation of MN with pion mass. The short dash curve corresponds to the simple dimensional regularisation scheme (DR) and the long dash curve to the more sophisticated dimensionally regulated approach (BP), which keeps the correct non-analytic structure at the ∆π branch point. The four ﬁnite-range regulators (solid curves) are indistinguishable at this scale. Oscillations of the dimensionally regulated schemes (dashed curves) about the solid lines are apparent Table 1. We show the bare ﬁt parameters obtained for the constraint curves, M (mπ ) (in appropriate powers of GeV). The regularization schemes are explained in the text Regulator DR BP SC MON DIP GAU

a0

a2

a4

a6

0.877 0.821 1.02 1.58 1.22 1.11

4.06 4.57 1.15 0.850 0.925 1.03

5.57 8.92 −0.359 −0.162 −0.177 −0.265

−3.24 −2.02 0.055 −0.007 −0.009 0.024

shown in Table 2. The most remarkable feature of Table 2 is the very close agreement between the values of the renormalised coeﬃcients, especially for the ﬁnite-range regularisation (FRR) schemes. For example, whereas the variation in a0 between all four FRR schemes is 50%, the variation in c0 is a fraction of a percent. For a2 the corresponding ﬁgure is 30% compared with less than 9% variation in c2 . If one excludes the sharp cut-oﬀ (SC) regulator, the monopole, dipole and Gaussian results for c2 vary by only 2%. Finally, for c4 the agreement is good for the latter three schemes. A comparison between a4 and c4 is especially important in order to understand why the FRR schemes are so eﬃcient. Whereas the renormalised coeﬃcients are consistently very large for the three smooth FRR schemes, the bare coeﬃcients of the residual expansion are of order 100 times smaller! That is, once one incorporates the eﬀect of the ﬁnite size of the nucleon by smoothly

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Table 2. The renormalized chiral expansion parameters for the constraint curves, M (mπ ). All quantities are given in appropriate powers of GeV. The regularisation schemes are explained in the text Regulator DR BP SC MON DIP GAU

c0

c2

c4

0.877 0.881 0.895 0.898 0.897 0.897

4.10 3.84 3.02 2.78 2.83 2.83

5.57 7.70 14.0 23.5 21.7 21.2

suppressing pion loops for pion masses above 0.4–0.5 GeV, the residual series expansion has vastly improved convergence properties. In contrast, in order to ﬁt the nucleon mass data over an extended range of pion mass, the dimensional regularisation schemes require bare expansion coeﬃcients which are much larger, 30 times larger in the case of a4 . Still these coeﬃcients are not large enough to reach the consistent values of the smooth FRR results reported in Table 2. Given the constraint curves, M (mπ ), we wish to test how well an alternative regularisation technique can reproduce the same curve. For any well-deﬁned quantum ﬁeld theory, all physical results should be independent of the regularisation and renormalisation schemes. By doing a one-to-one comparison over diﬀerent ranges of pion mass we are able to determine the eﬀective convergence range of each scheme. For each constraint curve, we have ﬁve alternate curves, containing free parameters, which may be adjusted to ﬁt this constraint. We obtain a best ﬁt of each particular regularisation scheme to our constraint curve over some window, mπ ∈ (0, mW ). This curve ﬁtting is achieved by considering a dense set of points in the window and minimising a chi-square between the curves [24]. Of course, all of the regularisation prescriptions have precisely the same structure in the limit mπ → 0. As our test window moves out to larger pion mass the accurate reproduction of the low-energy constants serves as a test of the eﬃciency of the regularisation. We are concerned with the chiral expansion properties about the limit of vanishing quark mass. It is important to test how well the low-energy expansion parameters are reproduced as the curve-matching window is increased. We show in Fig. 3 the renormalised expansion parameters, c0 , of the other regulators constrained to the DIP curve, as a function of the curve-ﬁtting window. A similar plot for the coeﬃcient c2 is shown in Fig. 4. From Fig. 3 it is clear that all the FRR curves can reproduce the dipole curve exceptionally well over the range considered. The DR-motivated curves fail to ﬁt the dipole constraint and reproduce c0 to an accuracy of 1% beyond a pion mass of 0.5 GeV2 . Our 1% cut is deﬁned such that the variation in the contribution to the physical nucleon mass is less than 1% (e.g. in the case of c2 , this requires

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c 0 GeV DIP

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Fig. 3. Recovery of the physical coeﬃcient c0 from various regularisation schemes constrained to the dipole regularised curve. c0 is plotted as a function of the size of the curve-ﬁtting window (0, m2W ). The dashed horizontal line indicates the maximum deviation of c0 in order to determine the physical nucleon mass to within 1%

4.2

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1.0

Fig. 4. Recovery of the physical coeﬃcient c2 from various regularisation schemes constrained to the dipole regularised curve. c2 is plotted as a function of the size of the curve-ﬁtting window (0, m2W ). The dashed horizontal line indicates the maximum deviation of c2 in order to determine the nucleon mass to within 1% at the physical point

that the error in the quantity c2 m2π /MN < 1% at the physical point). Similar features are observed in the case of the coeﬃcient c2 . Once again the agreement between the FRR curves are remarkable, the DR curves break down at the order of m2π ∼ 0.4 GeV2 . Consistent results are found for the renormalised coeﬃcients c4 over the entire ﬁt range 0 ≤ m2π ≤ 0.8 GeV2 , provided a smooth ﬁnite-range regulator is used

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to regulate the eﬀective ﬁeld theory. c4 is compromised in the dimensionallyregulated schemes as it attempts to simulate higher-order terms absent in the truncated expansion. These exercises make it quite clear that one cannot hope to use a series expansion based upon dimensional regularisation to analyse data over a window wider than 0.4 GeV2 . That means one would need to have suﬃcient, accurate lattice QCD data in this mass region to ﬁx four ﬁtting parameters before one could hope to trust the chiral coeﬃcients so obtained. Within this region, it would also be necessary to have data very near the physical pion mass. This is certainly beyond the likely computational capacity of current collaborations in the next 5–10 years. On the other hand, it is apparent that by using the improved convergence properties of the FRR schemes one can use lattice data in the region up to 0.8 GeV2 . This is a regime where we already have impressive data from CP-PACS [18], MILC [6] and JLQCD [19]. In particular, the FRR approach oﬀers the ability to extract reliable ﬁts where the low-mass region is excluded from the available data. The consideration of this practical application to the extrapolation of lattice data will be discussed further below. Indeed, these results suggest that given data from the next generation of (10 Teraﬂops) lattice QCD machines currently under construction, the FRR schemes should allow us to extract c0 and c2 , independent of any model, at the 1% level and c4 at the level of a few percent. 3.1 Extrapolation and Lambda Dependence In this section we remove the physical nucleon mass as a constraint to examine the chiral extrapolation of the lattice data. We then ﬁt the same set of lattice data, as discussed above, with the chiral expansions based on six diﬀerent regularisation prescriptions. In all schemes we allow the coeﬃcients of the analytic terms up to m6π to be determined by the lattice data. The regulator masses are once again ﬁxed to their preferred values. The best ﬁts, shown in Fig. 5, serve to highlight the remarkable agreement between all ﬁnite-range parameterisations. This demonstrates that the extrapolation can be reliably performed using FRR with negligible dependence on the choice of regulator. In view of the general discussion in Sect. 2.2, it is important to check the residual dependence on the choice of cutoﬀ scale Λ. The resulting variation of the extrapolated nucleon mass, at the physical pion mass, for dipole masses ranging from 0.6 to 1.0 GeV is shown in Fig. 6. We see that the residual uncertainty introduced by the cutoﬀ scale is less than 2%, which is insigniﬁcant compared to the statistical error in extrapolating such a large distance. With the present data this statistical error is found to be around 13%.

4 Other Applications The principles of FRR chiral eﬀective ﬁeld theory have been applied to a range of hadronic observables. In addition to the nucleon mass as discussed above, the

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MN GeV

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1

mΠ GeV 2

2

Fig. 5. Extrapolation of lattice data for various regularisations prescriptions. The four indistinguishable, solid curves show the ﬁts based on the four diﬀerent FRRs. The short dash-dot curve corresponds to the DR case and the long dash-dot that of BP

1.1

MN phys GeV

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Fig. 6. The extrapolated nucleon mass for varying dipole mass, Λ

chiral extrapolation of the ∆ baryon and ρ-meson have also been studied [26, 27]. By use of a FRR one automatically includes the correct physics associated with the opening of the decay channels ∆ → N π or ρ → ππ. With regard to the lattice simulations on a ﬁnite volume, it was demonstrated that a more realistic account of these p-wave decays could be achieved through the use of a FRR. At the moderate quark masses simulated on the lattice the chiral eﬀects are minimal. In the absence of the rapid nonanalytic chiral variation, hadron properties act very like the constituent quark model. In the case of the masses already discussed it has been observed that the data is quite linear in quark mass. In fact the rho-meson has a linear quark mass dependence that is 2/3 the average slope of the nucleon and ∆ baryons [28].

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Further insight has come through studies of the quark mass dependence of electromagnetic properties of baryons. Early work provided an extrapolation of the nucleon and octet baryon magnetic moments by incorporating the LNA behaviour of chiral EFT [29, 30]. Lattice calculations have not exhibited the rapid nonanlytic variation due to the large quark masses simulated [31]. The observed variation is proportional to 1/mq , as to be expected of a particle comprised of constituent quarks with a Dirac moment. Recognising this fact provided a consistent extrapolation incorporating both the correct chiral structure and this heavy quark behaviour. This extrapolation was applied to the octet baryon charge radii with similar success [32]. More recently an extensive lattice study of the momentum dependence of the nucleon’s magnetic form factor has been completed [33]. The Q2 -dependence of the form factor is described well by a dipole at all quark masses. The extracted dipole mass shows a linear variation in the quark mass. As the quark masses approach zero the magnetic and electric radii are know to diverge – the pion cloud extends to inﬁnity. A simple extrapolation has been proposed which interpolates between the chiral and heavy quark regime [34]. Despite the simulation being restricted to the quenched approximation with relatively heavy quarks, this extrapolation brings the lattice data into remarkable agreement with experiment, see Fig. 7.

Fig. 7. Extrapolated proton electric form factor for varying lattice spacings – from [34]

4.1 Hadron Spectroscopy As a consequence of improved computing resources and the development of novel spin and parity projection methods, there has been enormous progress in the calculation of the masses of excited baryons in quenched lattice QCD [35]–[43].

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Established ﬁtting techniques have successfully enabled the extraction of resonance masses for pion masses in the range m2π > 0.3 GeV2 . The diﬃculty lies in maintaining a clear signal for the eﬀective mass of the baryon resonance as the quarks become light. In this region the masses of the ﬁrst negative and positive parity excited states obtained from standard interpolating ﬁelds follow the expectations of the naive harmonic oscillator based quark model, with roughly equal spacing between the ground state nucleon, the ﬁrst 1/2− and the ﬁrst 1/2+ excited states. Given the enormous experimental eﬀort to disentangle the details of baryon spectroscopy and ﬁnally resolve issues of missing or exotic states it is important to have as much of a contribution from lattice QCD as possible. As should be obvious by now, that requires that the chiral extrapolation of the masses of the excited baryons should also be understood. This is a far more challenging problem than the extrapolation of the nucleon mass which has dominated our discussion until now. The complications stem from the large number of nearby states which can contribute important chiral behaviour and the fact that their coeﬃcients cannot be constrained in any model independent way. Nevertheless, the problem is of such interest that recent attempts have been made to generalise the chiral extrapolation techniques to those baryon excited states currently accessible on the lattice [47]. We begin with the formal expansion of the mass of a baryon about the chiral limit σBB π (mπ , Λ) + · · · (7) MB = a0 + a2 m2π + a4 m4π + B

where ai are the coeﬃcients of the analytic terms, Λ is the FRR scale and σBB π are pion-induced self-energy contributions. We shall restrict our consideration to self-energy loops involving pion-baryon pairs in relative P -wave, focussing on the most important baryon states as determined by the strength of the mesonbaryon coupling constants – see [46]. In Fig. 8 we show the extrapolation of CSSM Lattice collaboration data [42, 43, 49] for several low-lying excited baryon states [47]. The dashed lines represent naive linear extrapolations while the solid lines incorporate the non-analytic variation with mq arising from coupling to nearby resonances. The experimental masses are shown as ﬁlled ellipses. The analysis of excited state masses shown here is clearly in its early stages. The eﬀect of chiral non-analytic behaviour is to reduce the mass of the physical resonance below what would be given by naive linear extrapolation. Of course, we have to realize that we are adding full QCD chiral corrections to lattice data that includes QQCD chiral corrections, but this does not present a problem of principle [44, 48]. It is interesting to note that a small downward shift in the lattice results upon unquenching (as suggested in [49]) would provide remarkable agreement with experiment. We note that the incorporation of chiral − − non-analytic behavior inverts the N 12 (1535) and N 32 (1520) ordering when compared to the naive linear extrapolation, in accord with experiment.

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Fig. 8. Extrapolation of lattice data for selected baryon resonances. The dashed lines represent naive linear extrapolations while the solid lines incorporate the non-analytic variation with mq arising from coupling to nearby resonances. The experimental masses are shown as ﬁlled ellipses. The ﬁgure is taken from [46]

Encouraged by these early successes, work is now in progress to include additional self-energy contributions. S-wave contributions are of particular interest due to the non-analytic structure of their contributions. Ultimately, quenched chiral perturbation theory will be formulated providing the opportunity to quantitatively estimate the predictions of full QCD.

5 Conclusion In this brief review it has been possible to give only an introduction to the exciting developments in this very important and topical area of hadron physics. The next few years will see tremendous advances in the quantity and quality of lattice data for masses and other hadron properties. If one is to maximise the information extracted from this data, the techniques of ﬁnite range regularisation which have been explained here will be crucial.

Acknowledgements We wish to thank J. Ashley, B. Crouch and D. Morel for helpful discussions concerning many aspects of the work presented here. This work was supported by the Australian Research Council.

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Lattice Chiral Fermions from Continuum Defects H. Neuberger1,2 1 2

School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 Rutgers University, Department of Physics and Astronomy, Piscataway, NJ 08855, USA

Abstract. We consider whether defects of co-dimension two could produce new lattice chiral fermions.

1 Introduction A vector like gauge theory with an inﬁnite number of ﬂavors can have a mass matrix that has an index equal to unity, but otherwise is of the order of the ultra-violet cutoﬀ [1]. This produces a strictly massless charged Weyl particle accompanied by an inﬁnite number of heavy Dirac particles. This setup has been used on the lattice with a domain-wall [2] realization of the mass matrix where it leads to the overlap [3]. Here we ask whether a string realization of inﬁnite ﬂavor space might achieve a substantially diﬀerent form of lattice chirality. The question is left unanswered, but there is some indication that employing a variant of Baxter’s corner transfer matrix (CTM) [4] a more general construction of lattice chiral fermions might be possible. It is hoped that the present discussion will be a motivator to develop the idea further.

2 Inﬁnite Flavor Space Many reviews have been written about the new way of putting chiral fermions on the lattice – for one example see [3]. Therefore, after a very brief overview, this section will mention only facts essential for what follows later. The overlap construction was based on two independent suggestions about regularizing chiral gauge theories, one by D. B. Kaplan [2] and the other by S. Frolov and A. Slavnov [5]. The ﬁrst of these was, in turn, motivated by an earlier paper by C. Callan and J. Harvey [6]. This earlier paper is also the basis of the present article. The overlap path of development was completed in the summer of 1997 with the publication of [7]. One very often hears in this context about a paper by Ginsparg and Wilson [8], published in 1982, which fell into oblivion until after the publication of the ﬁrst paper in [7]. In the second paper in [7] the overlap line of development was merged with the Ginsparg-Wilson proposal. This conﬂuence played an important

Permanent Address.

H. Neuberger: Lattice Chiral Fermions from Continuum Defects, Lect. Notes Phys. 663, 131–145 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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psychological role in convincing the majority of workers in the ﬁeld that indeed the problem of lattice chirality was solved by the overlap. At the moment there exists only one construction of lattice chirality that truly works. To understand this construction as well as our attempt to go beyond it there is no need to be familiar with [8] (although this paper is strongly recommended on general grounds) nor with the many papers re-deriving the overlap with the GinspargWilson relation as a a starting point. At the practical level too, it is fair to say, little concrete gain has been produced by the many Ginsparg-Wilson based papers appearing from 1998 onwards, except a sobering recognition that the problem of lattice chirality could have been solved in 1982, or soon after, had we focused on this prescient paper. In this article we shall not diverge very much from what has proven to be successful. We simply go to another example in the paper of Callan and Harvey, and using yet another observation of D. B. Kaplan [9] try to follow the procedure undertaken when the overlap was developed to see where this ends up leading us. To do this it is necessary to only review the basic logical steps in the overlap construction. The general framework that unites the ideas of D. B. Kaplan [2] and of S. Frolov and A. Slavnov [5] is to think about a formally vector-like gauge theory which contains an inﬁnite number of ﬂavors [1]. The main new object in the Lagrangian is a mass matrix that has an analytic index equal to unity and whose non-zero spectrum is separated from zero by a gap of order the cutoﬀ energy: ¯ µ Dµ ψ + ψ¯R M ψL + ψ¯L M † ψR . (1) Lψ = ψγ The conditions on M are: – M has exactly one zero mode. – M † has no zero mode. (The adjoint of M is deﬁned so that the Euclidean fermion propagators have the right formal adjointness properties.) – The spectrum of M M † is separated from zero by a ﬁnite gap. If these condition hold for M , they will also hold for M + δM , if δM is small enough. This property protects the mechanism from being destroyed by radiative corrections. These conditions ensure that there will be one Weyl fermion and an inﬁnite number of heavy Dirac fermions. To make this into a well deﬁned construction – one that can be implemented on a ﬁnite computer – the following steps were carried out in the context of the domain-wall/overlap construction: – Four dimensional space-time is replaced by a ﬁnite lattice. Invariably one chooses a torus shape. All ordinary inﬁnities are eliminated by this. – In order to deal with the single remaining inﬁnity (in ﬂavor space) one makes as simple a choice as one can for M and the space it acts on. – One next formally integrates out all fermions. This step is formal because the number of fermions is inﬁnite. Nevertheless, for special choices of M , one can interpret the result as something ﬁnite and well deﬁned times an inﬁnite factor that looks, intuitively, harmless. The harmless inﬁnite factor

Lattice Chiral Fermions from Continuum Defects

133

is discarded and the remaining structure is a well deﬁned candidate for the chiral determinant line bundle over gauge conﬁguration space. – The candidate is subjected to several tests: it has to reproduce instanton physics and the associated fermion number conservation violating processes; it has to reproduce anomalies of global and local type, etc. (It goes without saying that ordinary Feynman diagrams must be reproduced.) – In the vector-like case one can next try to derive as simple a form as possible for the eﬀective action and see directly how continuum chiral Ward identities are reproduced by this action. We shall go some way on this path in the case of a string defect3 .

3 Two Dimensional Flavor Space 3.1 Axial Symmetric Case Let us start by reviewing Callan and Harvey’s paper, [6] – and add some further details, mainly for pedagogical reasons. First we need to set our conventions for the Gamma matrices in 6 Euclidean dimensions. The various directions will be labeled as: a = 1, 2, 3, 4, 5, 6, µ = 1, 2, 3, 4, α, β = 5, 6. Γµ = σ1 ⊗ γµ

Γ5 = σ1 ⊗ γ5

As a result,

Γa =

0 ρ†a

ρa 0

Γ6 = σ 2 ⊗ 1

(3)

Γ7 = −iΓ1 Γ2 Γ3 Γ4 Γ5 Γ6 = −i(1 ⊗ γ5 )(iσ3 ⊗ γ5 ) = σ3 ⊗ 1

or

1 Γ7 = 0

0 −1

(2)

(4)

(5)

The Weyl matrices are: ρµ = γµ ,

ρ5 = γ5 ,

ρ6 = i

(6)

Thus, for 6D chiral fermions we can use four component spinors with action ¯ † Da )ψ Lψ = ψ(ρ a

(7)

From the 4D point of view one can think about the ψ’s as Dirac fermions consisting of two 4D Weyl fermions of opposite handedness ψL,R : γ5 ψL = ψL , γ5 ψR = −ψR , ψ¯R γ5 = ψ¯R , ψ¯L γ5 = −ψ¯L 3

(8)

Although we do not get very far, we already ﬁnd ourselves in disagreement with a recent paper by Nagao.

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Each one of the above four fermion ﬁelds has two unrestricted components. ¯ µ Dµ ψ + ψ¯R (iD6 + D5 )ψL + ψ¯L (iD6 − D5 )ψR Lψ = ψγ

(9)

With m = iD6 + D5 and m† = iD6 − D5 this can then be written in the generic form: / 4 ψ + ψ¯R mψL + ψ¯L m † ψR (10) Lψ = ψ¯D m, viewed as an operator on the 2D space of x5 and x6 , has a hermitian conjugate, m † . The xα , α = 5, 6 are viewed as continuous indices in ﬂavor space. In the Callan-Harvey set-up we have, in six dimensions, one Dirac fermion, comprising of a four component ψ as above, as well as another ﬁeld, a four component χ, of opposite handedness. / 4ψ + χ / 4χ ¯D Lψ,χ = ψ¯D

mΦ ψL ¯ ¯R + ψR χ χL Φ∗ − m † †

mΦ ψR ¯L + ψ¯L χ χR Φ∗ − m †

(11)

We see that the structure has the generic form of (1) with a mass matrix M given by m Φ M= (12) Φ∗ −m † ψ and, in terms of the ﬂavor doublet Ψ ≡ , a fermionic Lagrangian of the form χ / 4 )Ψ + ΨR (M ⊗ 1)ΨL + ΨL (M † ⊗ 1)ΨR LΨ = Ψ¯ (1 ⊗ D

(13)

Above, ψ and χ are viewed as two four dimensional Euclidean Dirac fermions and are distinguished by the discrete portion of a ﬂavor index. This discrete portion can take one of two possible values. The unit matrices in the direct / 4 is four by four and M is two by two. In this products are all two by two while D dimension-counting we have ignored all color indices and the continuum labels. M is arranged to have unit analytical index by picking Φ = eiφ f (r)

(14)

where x5 = r cos φ and x6 = r sin φ. The natural inner product has an extra factor of r in the measure. As a result, ∂r† = −∂r − 1r . Taking the A5 and A6 gauge ﬁeld components to vanish we have # i $ m = ∂5 + i∂6 = eiφ ∂r + ∂φ r # i $ † −iφ m = −∂5 + i∂6 = e −∂r + ∂φ r

(15)

Lattice Chiral Fermions from Continuum Defects

The matrix M now becomes iφ e M= 0

0

e−iφ

∂r + ri ∂φ f (r)

f (r) ∂r − ri ∂φ

135

(16)

leading to M† =

−∂r − 1r + ri ∂φ f (r)

f (r) −∂r − 1r − ri ∂φ

e−iφ 0

0 eiφ

(17)

M has a Single Zero Mode We now look for zero eigenstates of M . The zero mode equation is ∂r − kr f (r) η˜1k =0 η˜2k f (r) ∂r + kr

(18)

Zero modes of M will be two component functions of x5 and x6 , denoted by η. In the r, φ polar coordinates, in order to get a zero mode, both components of η need to carry the same angular momentum k ∈ Z. Thus, the angular dependence of η is of the form eikφ η˜ where η˜ is a function of r only. A zero mode of M is also a zero mode of M † M . Acting on an η˜ ﬁeld carrying angular momentum k, M † M has the form −∂r − 1r − kr f f ∂r − kr † M M= (19) f −∂r − 1r + kr f ∂r + kr A bit of algebra gives 2 −∂r2 − 1r ∂r + kr2 + f 2 † M M= −f − fr

−f − fr 2 2 −∂r − 1r ∂r + kr2 + f 2

(20)

We can go to the sum and diﬀerence ﬁelds √12 (ψ ± χ) which diagonalizes M † M to: 1 1 1 1 1 † = (21) M M 2 1 1 −1 −1 2 −∂r2 − 1r ∂ + f 2 − f − fr + kr2 0 2 0 −∂r2 − 1r ∂ + f 2 + f + fr + kr2 This can also be written as: 1 1 1 1 1 † M = M 2 1 1 −1 −1 (∂r + f )† (∂r + f ) 0 k2 1 + r2 0 0 (∂r − f )† (∂r − f )

0 1

(22)

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H. Neuberger

The two terms above are both positive semi-deﬁnite operators and, for a zero mode, they must both vanish on η˜. This requires k = 0. The zero mode will have to solve the following equation: η˜1 + η˜2 0 ∂r + f (r) =0 (23) η˜1 − η˜2 0 ∂r − f (r) Only the operator in the left-upper entry has a zero mode, given by e− We conclude that M has a single zero modes in all cases of interest.

r 0

f (ρ)dρ

.

M † has no Zero Mode We would like now to prove that M † has no zero modes. We ﬁrst go to M M † hoping to show it is strictly positive. ψ MM† = (24) χ (f − fr )e2iφ −∂r2 − 1r ∂r − r12 ∂φ2 + f 2 ψ f 1 1 2 −2iφ 2 2 χ (f − r )e −∂r − r ∂r − r2 ∂φ + f Expanding in angular momentum modes, ψ= einφ ψn (φ), χ= einφ χn (φ) n

(25)

n

we obtain for the zero mode equation 1 (n + 1)2 2 2 + f ψn+1 + (f − −∂r − ∂r + r r2 1 (n − 1)2 2 2 −∂r − ∂r + + f χn−1 + (f − r r2

f )χn−1 = 0 r f )ψn+1 = 0 r

(26)

Note that the ordered pair (χ−n−1 , ψ−n+1 ) satisﬁes the same equations as the ordered pair (ψn+1 , χn−1 ), so one could restrict the analysis to only n ≥ 0. In matrix notation, we are dealing with the operator 2 + f2 f − fr −∂r2 − 1r ∂r + (n+1) 2 r (27) O= 2 f − fr −∂r2 − 1r ∂r + (n−1) + f2 r2 Consider the identity 1 1 f (∂r + f )(∂r + f )† = −∂r2 − ∂r + 2 + f − + f 2 r r r Let us take n = 0 ﬁrst. Then we ﬁnd, with 1 1 1 K=√ , 2 1 −1

(28)

(29)

Lattice Chiral Fermions from Continuum Defects

KOK =

(∂r + f )(∂r + f )† 0

0 (∂r − f )(∂r − f )†

137

(30)

It is easy to see that KOK has no zero modes. For |n| ≥ 2 the operator O can be written as the operator O for n = 0 plus a non-negative operator. Hence, we have established that M M † has no zero modes with n = 0 and with |n| ≥ 2. The case |n| = 1 needs special analysis. Due to the symmetry between n = +1 and n = −1 we know that M † either has no zero modes, or that it has two zero modes. In either case, M has a nonzero index. With some mild additional assumptions on f we can deal with the n = 1 case too. These assumptions are not necessary; in other words, one can eliminate the case n = 1 also with diﬀerent assumptions. We assume that f satisﬁes:

f 1 f = f ≥ 0 , f − ≤0, f (∞) = v > 0 (31) r r r For example, an f given by f (r) =

vr , r+a

a>0

(32)

satisﬁes these assumptions. Actually, f (0) = 0 should hold always, since the phase of the ﬁeld Φ winds around the origin and |Φ| = f . Another example would be 1 − 1 sinh 2r f r (33) f = v tanh , f − = a 2r 2 r a a r cosh a 3

The expression is indeed negative because sinh x = x + x3 + · · · The assumptions on f (31) are satisﬁed for all non-zero values of the dimensionless parameter λ, deﬁned by: λ = va

(34)

Consider now the identities: 1 1 f (∂r + f )(∂r + f )† = −∂r2 − ∂r + f 2 + f − + 2 r r r 1 f † 2 2 (∂r + f ) (∂r + f ) = −∂r − ∂r + f − f − r r Using them we write −∂r2 − 1r ∂r + r22 + f 2 f − fr f − fr −∂r2 − 1r ∂r + f 2 1 + (∂r + f )(∂r + f )† − f + fr f − fr = r2 f † f −r (∂r + f ) (∂r + f ) + f + 2f 0 0 f −1 1 1 0 1 + = 2 + (f − ) 1 −1 r 0 1 r r 0 1 † 0 (∂r + f )(∂r + f ) + 0 (∂r + f )† (∂r + f )

(35)

f r

(36)

138

H. Neuberger

Since the matrix

−1 1

1 −1

(37)

is negative semi-deﬁnite, every term in the above additive decomposition is positive semi-deﬁnite. For a zero mode they all have to have zero expectation value and this is impossible. A similar decomposition should handle the case ( fr ) ≥ 0 which would correspond to an f which blows up at inﬁnity. Spectral Gap Let f (r) = vgλ (x), with lim gλ (x) = 1 λ→0

(38)

where x = vr and λ is the positive dimensionless parameter introduced earlier. To establish the existence of a gap we need to show that the spectrum of M † M is separated by a gap from its single zero eigenstate. It suﬃces to show this for wave-functions with no φ dependence, see (20). The non-zero portion of the spectrum of M † M also is the joint spectrum of (∂r + f )(∂r + f )† and (∂r − f )† (∂r − f ). We use (∂r + f )(∂r + f )† rather than (∂r + f )† (∂r + f ) to eliminate the zero mode. ∂r ∂r† = ∂r† ∂r +

1 r2

1 f + 2 r r f † † 2 (∂r − f ) (∂r − f ) = ∂r ∂r + f − f + r (∂r + f )(∂r + f )† = ∂r† ∂r + f 2 + f −

(39)

So, all we need is to ﬁnd lower bounds for 1 gλ + 2 x x gλ h2 = gλ2 − gλ + x h1 = gλ2 + gλ −

(40)

For λ → 0 gλ (x) → 1 and it is easy to check that h1,2 are bounded by positive numbers: h1 →

1 x2

−

1 x

h2 = 1 +

+1≥ 1 x

3 4

(41)

≥1

Making λ positive nonzero and small cannot change these bounds by much, for any x away from zero. It is easy to see that at x = 0, where smoothness in λ breaks down, these positive lower bounds still hold. Thus, keeping λ small enough we are sure we achieved all our objectives.

Lattice Chiral Fermions from Continuum Defects

139

Eﬀective Low Energy Theory We could restrict the ﬁelds ψL , χL , ψ¯L , χ ¯L to have no dependence on φ, the φ ¯R to be given by eiφ and the φ dependence of χR , ψ¯R to be dependence of ψR , χ ¯R , χR to vanish given by e−iφ . This requires us to constrain the ﬁelds ψ¯R , ψR , χ at the origin, where the angle φ is not deﬁned. After extracting the angular dependence we can drop the φ integral in the action and again rotate the new ψ,χ ﬁelds to ψ±χ. The diﬀerence combination can also be dropped. All the ﬁelds we have dropped are “heavy” and dropping them is consistent with eﬀective ﬁeld theory logic. The scale of their masses is given by the asymptotic value of f at inﬁnity. The remaining heavy and light ﬁelds (from the four dimensional point of view) correspond to a single, four component, complex, fermion ﬁeld living in a ﬁve dimensional world with a boundary at r = 0, where r is the ﬁfth dimension. The single zero mode we have is left-handed from the four dimensional point of view. We end up with a set-up similar to the domain wall case, only that the extra dimension is now [0, ∞) and there is an r-factor in the internal measure factor. Because of this similarity it is unlikely that a truly new may of regularizing chiral fermions could be obtained pursuing this any further. 3.2 A Less Symmetric Arrangement David Kaplan [9] had the idea to make a Cartesian arrangement in which the phase of Φ changes in jumps between consecutive constant values in each quarter of plane. The length of Φ is frozen to v. One way to do this is to take the scalar ﬁeld as: (42) Φ(x5 , x6 ) = v[(x5 ) + i(x6 )] The sign functions can be smoothed out at the origin. This would also smooth out the discontinuities in the phase – Φ would become a smooth function. Now i∂6 + ∂5 v[(x5 ) + i(x6 )] M= (43) −i∂6 + ∂5 v[(x5 ) − i(x6 )] The zero mode equations become [i∂6 + ∂5 ]η1 + v[(x5 ) + i(x6 )]η2 = 0 [−i∂6 + ∂5 ]η1 + v[(x5 ) − i(x6 )]η2 = 0

(44)

The zero mode is unique, and given by η1 = η2 = Ae−v[|x5 |+|x6 |]

(45)

This mode is localized at the origin, which is the point at which the two lines on which f5 and f6 vanish intersect. In the original Callan-Harvey setting the two lines which intersect are lines on which the real part and, respectively, the imaginary part of the complex frozen Higgs ﬁeld, vanish.

140

H. Neuberger

If we smooth out the sign functions, we shall have a diﬀerentiable zero mode. This may not be necessary when we go to a lattice. The arrangement allows us to go to a regular square lattice in the x5 –x6 plane. The lattice will have to be inﬁnite. If we truncate it, on its boundary we expect to ﬁnd a partner to the chiral fermion at the origin, of opposite chirality. Our considerations so far put us in disagreement with Nagao’s paper [10]: We do not ﬁnd extra zero modes in the continuum beyond the one four dimensional Weyl fermion at the string defect and we see no need for an axially symmetric lattice structure which is necessarily inhomogeneous. Only the winding of the φ phase is necessary for the Callan-Harvey mechanism to work, and that we have shown can be achieved by a Cartesian arrangement. 3.3 Smooth Cartesian Arrangement We now eliminate all jumps in the Cartesian case: v(xα ) → fα (xα )

(46)

The real functions fα depend on a single real argument on the inﬁnite real line. They are monotonically increasing and asymptotically constant, going from −vα− at −∞ to vα+ at ∞, with vα± > 0 and large, of the order of a 4D ultraviolet cutoﬀ. Also, we choose fα (0) = 0. The scalar ﬁeld Φ is: Φ(x5 , x6 ) = f5 (x5 ) + if (x6 )

(47)

Deﬁne two operators on functions of a xα : dα = ∂α + f (xα ),

d†α = −∂α + f (xα ),

[d5 , d6 ] = 0

(48)

The dα ’s have zero modes but the d†α do not. ¯ χ. ¯ Change again the ﬁeld basis by ψ, χ → √12 (ψ ± χ) and the same for ψ, The Lagrangian changes by changing the matrix M : 1 1 −id†6 1 1 1 d5 M→ (49) M = 1 −1 2 1 −1 id6 −d†5 We shall continue to denote the (new) mass matrix by M . Properties of M in the Cartesian Case We now need to prove that indeed M has a single zero mode and that M † has no zero mode. This is very easy: † d5 d5 + d†6 d6 0 † M M= (50) 0 d5 d†5 + d6 d†6

Lattice Chiral Fermions from Continuum Defects

MM† =

d5 d†5 + d†6 d6 0 † 0 d5 d5 + d6 d†6

141

(51)

This not only proves our statement, but also shows that the zero eigenvalue of M † M is separated by a large gap from the the rest of the spectrum. Hence, the arrangement is stable under deformations smaller than the cutoﬀ scale. Note that the above construction is a general prescription of combining two commuting operators with index into a new operator with the same index acting on a doubled product space. This procedure can be iterated, doubling the number of extra dimensions at each step. Spectral Gap in the Cartesian Case Here we deal with the two functions f1,2 which obey fα2 + fα → v 2 as λ, approaches zero and the same logic as in the previous gap analysis, for the axially symmetric case, again applies.

4 Compactiﬁcations If the 2D x5 −x6 plane is compactiﬁed the chiral fermion will get a partner of opposite chirality and the index will be lost. Actually, one can generate more Dirac fermions than the minimum needed, as we shall see. Consider ﬁrst the axially symmetric case: To obtain a compactiﬁcation we need to replace f by a periodic function. f (r) would go from zero at r = 0 to zero at r = R. It would grow to a positive large value, stay at it for a long while, and then drop back to zero at R. It is natural to compactify now to a sphere. Each half sphere contains one of the poles of the original sphere and a circle on the boundary where f is in the middle of its range. At each pole we have a chiral fermion, the chiralities at the two poles being opposite. In total, we have a Dirac fermion. Turn now to the Cartesian case: fα is made periodic as follows: it would go from −vα− at zero to vα+ and continue back after a long while to −vα− at some Lα . This naturally leads to a two torus of sides L5 , L6 . Now, there are chiral fermions at ( L45 , L46 ), ( 3L4 5 , L46 ), ( L45 , 3L4 6 ) and ( 3L4 5 , 3L4 6 ). There are two pairs of Weyl fermions, each pair has two members of opposite chirality. Together, they make up something like a u and d quark. The behavior is similar to that of naive fermions where momentum space has been traded for x5 –x6 and we have squared the 2D Dirac operator. To get open boundaries, like those leading to the simples form of the overlap, we take, for example, vα+ to inﬁnity. The boundary conditions are imposed by this limit like in the case of the MIT bag model. This procedure certainly works in the domain wall case. Here, one would guess that an open square of sides L25 , L6 2 with four massless Weyl fermions located at the corners of the square will be produced. We still would have two massless Dirac fermions.

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H. Neuberger

4.1 MIT-Bag Boundary Conditions As just mentioned, additional (but inessential) simpliﬁcations are possible in the vector-like case. These simpliﬁcations amount to taking the lattice fermion mass parameter to inﬁnity where-ever this is possible. As a result, fermions are excluded from some portion of ﬂavor space. This exclusion leaves behind a boundary and the issue is to determine what boundary condition has been induced by this process. This problem has been solved many years ago: Whenever one uses mass-like terms to exclude a region of space-time from being visited by fermions one is doing an MIT-bag-like construction [11]. (It is not necessary that the fermion motion be constrained to a bounded region – the bag can be inﬁnite in some directions.) We need to slightly generalize from the original construction to mass matrices that are not necessarily diagonal. One has a surface of co-dimension 1 and one focuses on the region close to the surface. The normal to the surface is nµ . One wants to exclude fermions from one side of the surface by taking a mass parameter to inﬁnity there. On the other side of the surface we want to maintain ordinary propagation. Call the excluded region “outside” and the other “inside”. In the inﬁnite limit of the mass parameter, a boundary condition is generated on the inside fermions at the surface. Near the surface we can pick coordinates τ and yi where i = 1, 2, . . . , d − 1. The yi are along the surface and the τ is perpendicular to it. xµ = τ nµ + ξµi yi

(52)

where y · n = 0,

ξµi nµ = 0, ξµi ξµj = δ ij

(53)

Now, ∂µ = ∂τ nµ + ξµi ∂yi

(54)

The (Euclidean) equation of motion for the fermions is of the form (Aµ ∂µ + B)ψ = 0

(55)

˜ Aµ and B are matrices in ﬂavor and spinor space. B can be written as B = v B ˜ with B a matrix with entries of order unity while v is the scale that will go to ˜ is invertible. Deﬁne inﬁnity. One assumes that B ˜ −1 Aµ A˜µ = B

(56)

For the purpose of determining the boundary conditions we only care about the derivative terms in the direction perpendicular to the surface. Hence we need to look only at (57) (A˜ · n∂τ + v)ψ = 0 Writing the solution as

1

ψ = e z vτ φ

(58)

Lattice Chiral Fermions from Continuum Defects

143

we see that z has to be an eigenvalue of −A˜ · n. By convention, τ vanishes at the surface and increases to positive values outside it. Solutions that tend to diverge need to be eliminated by the boundary condition. They correspond to eigenvectors of −A˜ · n whose real part is positive. The boundary condition is that ψ be orthogonal to the subspace spanned by these eigenvectors at the boundary. Note that A˜ · n will in general not be hermitian or anti-hermitian and we are talking about its right eigenvectors.

5 Generalization of the Overlap Consider again the Cartesian set-up on the inﬁnite x5 −x6 plane. Following the overlap construction, our task is to ﬁnd a formal way to integrate out all fermions with a ﬁxed four-dimensional gauge background. This formal way should produce a candidate non-perturbative regularization of the chiral determinant line bundle we should be familiar with from continuum ﬁeld theory. In the overlap case the formal construction admitted a trivial renormalization and produced a well deﬁned object that was subsequently proven to share many topological and diﬀerential features with the continuum chiral determinant line bundle. The trivial renormalization in the overlap case amounted to dropping an inﬁnite multiplicative factor from the partition function, reducing it to an overlap of two states in a ﬁnite dimensional Hilbert space. We conjecture that here something similar happens, only that the role of the states is taken by analogues of Baxter’s corner transfer matrices. It is plausible that the partition function again contains inﬁnities only as multiplicative constants that are exponentials of local functionals of the gauge ﬁeld background with diverging coeﬃcients. We use Wilson mass terms to implement the (43) structure on an inﬁnite two dimensional square lattice. This lattice is naturally divided into four semiinﬁnite quadrants. Keeping the variables on the two inﬁnite lines bounding a given quadrant ﬁxed we sum over all internal fermions. This produces a corner transfer matrix K, for each quadrant, depending on the background gauge ﬁeld and on the quadrant mass structure. In several cases Baxter has shown that K normalizes simply to an operator with a discrete spectrum. The spectrum is discrete since K is, roughly, the exponent of a rotation generator in the plane, and the rotation is by a ﬁnite angle given by π2 . The generator of rotations surely has a discrete spectrum. Each K in itself does not “know” that the mass parameters undergo jumps. The existence of jumps is encoded in the fact that we have four diﬀerent K-matrices. Extracting the harmless normalization constant which are inﬁnite, but proportional to the free energy of the very massive fermions in the gauge ﬁeld background, we should be left with an equation of the form: chiral determinant = T rK1 {U }K2 {U }K3 {U }K4 {U }

(59)

The trace operation performs the integration over all remaining fermion ﬁelds on the boundaries between the quadrants. This trace is conjectured to produce the bundle we are after and requires that an overall phase be left undetermined

144

H. Neuberger

in the process of constructing the K-operators. The variables {U } represent the collection of gauge ﬁeld link variables that live on the four dimensional ﬁnite toroidal lattice that replaces the continuum spanned by xµ , µ = 1, 2, 3, 4.

6 What Next? To proceed along this path one would need to do the following: – Show that indeed the matrices K can be constructed in a natural fashion. The situation is slightly diﬀerent from that of Baxter in that the fermions are massive, very far from being 2D conformally invariant. However, the entries of K are, after all, just the inverses of some massive propagators in a gauge background, so should be essentially local. – Understand precisely how a natural phase indeterminacy enters and why (59) should be interpreted as deﬁning a line bundle over gauge orbit space. Note that if a spectral decomposition for the K matrices were to apply, one has a nice structure of rings of states: ψ1 |ψ2 ψ2 |ψ3 ψ3 |ψ4 ψ4 |ψ1

(60)

– Find out by what mechanism anomalies are reproduced. – Take the vector-like case and extract as simple a limit as possible. In the vector-like case the phase ambiguity should disappear. – If there is a simple eﬀective action for the Dirac case, does it obey the Ginsparg-Wilson relation [8]? We end here expressing the hope that some progress on the above lines will occur in the future.

Acknowledgements I wish to acknowledge useful discussions with David Kaplan, Seif RandjbarDaemi and Sasha Zamolodchikov. I also wish to acknowledge partial support at the Institute for Advanced Study in Princeton from a grant in aid by the Monell Foundation, as well as partial support by the DOE under grant number DE-FG02-01ER41165 at Rutgers University.

References 1. 2. 3. 4.

Rajamani Narayanan, Herbert Neuberger, Phys. Lett. B 302 (1993) 62. 131, 132 David B. Kaplan, Phys. Lett. B 288 (1992) 342. 131, 132 H. Neuberger, Ann. Rev. Nucl. Part. Sci 51 (2001) 23. 131 R. J. Baxter, J. Stat. Phys 15 (1976) 485; R. J. Baxter, J. Stat. Phys 19 (1978) 461; R. J. Baxter, J. Stat. Phys 17 (1977) 1. 131

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5. S. A. Frolov, A. A. Slavnov, Phys. Lett. B 309 (1993) 344. 131, 132 6. C. G. Callan Jr. and J. A. Harvey, Nucl. Phys. B 250 (1985) 427. 131, 133 7. H. Neuberger, Phys. Lett. B434 (1998) 99; H. Neuberger, Phys. Lett. B 437 (1998) 117. 131 8. P. Ginsparg, K. Wilson, Phys. Rev. D 25 (1982) 2649. 131, 132, 144 9. D. B. Kaplan, private communication. 132, 139 10. K. Nagao, Nucl. Phys. B 636 (2002) 264. 140 11. A. Chodos, R. L. Jaﬀe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D 9 (1974) 3471. 142

Computing the η and η Mesons in Lattice QCD K. Schilling1 , H. Neﬀ2 , and T. Lippert1 1 2

Wuppertal University, D42097 Wuppertal, Germany Department of Physics, Boston University, Boston MA02215, USA

Abstract. It has been known for a long time that the large experimental singlet-octet mass gap in the pseudoscalar meson mass spectrum originates from the anomaly of the axial vector current, i.e. from nonperturbative eﬀects and the nontrivial topological structure of the QCD vacuum. In the Ncolour → ∞ limit of the theory, this connection elucidates in the famous Witten-Veneziano relation between the η -mass and the topological susceptibility of the quenched QCD vacuum. While lattice quantum chromodynamics (LQCD) has by now produced impressive high precision results on the ﬂavour nonsinglet hadron spectrum, the determination of the pseudoscalar singlet mesons from direct correlator studies is markedly lagging behind, due to the computational complexity in handling observables that include OZI-rule violating diagrams, like the η propagator. In this article we report on some recent progress in dealing with this numerical bottleneck problem.

1 Introduction Long before the advent of QCD as the ﬁeld theory of strong interactions, the π-meson has been allocated the rˆ ole of the Goldstone Boson in a near-chiral world of light quarks while its ﬂavour singlet partner, the η meson, was thought to acquire its nearly baryonic mass of 960 MeV through ultraviolet quantum ﬂuctuations that prevent the ﬂavour singlet axial vector current µ(0)

j5

(x) = q¯(x)γ5 γ µ q(x)

(1)

from being conserved, even in the massless theory. In this scenario the breaking of SU (3)L × SU (3)R chiral symmetry occurs as a renormalization eﬀect within a µ(0) Wilson expansion of the operator ∂µ j5 that suﬀers operator mixing with the topological charge density D (x) Q(x) = (g 2 /32π 2 ) trF (x)αβ Fαβ

in the form

µ(0)

∂µ j5

˜ (x) = 2Nf Q(x) := Q(x) .

(2) (3)

The pseudoscalar operator on the right hand side of this ABJ anomaly [1] equation is proportional to the number of active quark ﬂavours, Nf and contains the gluonic ﬁeld tensor, F , and its dual, F D . Its Nf dependence is indicative of “disconnected” quark loop contributions as they appear in the socalled triangle K. Schilling, H. Neﬀ, and T. Lippert: Computing the η and η Mesons in Lattice QCD, Lect. Notes Phys. 663, 147–175 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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disconnected insertion of γ_5 γ_µ

quark sum over loop N_fflavour quark loops i

gluon exchange

valence quark

Fig. 1. Perturbative triangle diagram of ﬂavour singlet axial current as disconnected diagram with γ5 γµ insertion

diagrams of perturbation theory, see Fig. 1. Disconnected diagrams are peculiar to matrix elements of ﬂavour singlet operators and can be characterized by intermediate states free of quarks, i.e. by quark-antiquark annihilation into gluons with subsequent pair creation processes. Their phenomenological rˆ ole has been studied e.g. in various strong interaction fusion processes where light quark antiquark pairs annihilate strongly into charm (or bottom) quark antiquark pairs: u+u ¯ → c + c¯ or

b + ¯b .

(4)

These processes, as depicted in Fig. 2, are said to violate the OZI-rule [2]. Inspection of the renormalization procedure for this triangle diagram reveals a clash between quantum physics and symmetry requirements: the renormalizazion program poses the alternative of either giving up gauge invariance or chiral symmetry. Accordingly, (3) reﬂects nonconservation of chiral symmetry, for the sake of preserving gauge invariance, in form of the ABJ-anomaly (for an excellent review on the subject, see the Schladming lectures of Crewther [3]). disconnected insertions of γ_5 :

u

x

c, b

x’

Fig. 2. OZI rule violating quark fusion process with disconnected quark loops and γ5 insertions

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149

The lesson to be learnt is that pseudoscalar ﬂavour singlet mesons are deeply aﬀected by and thus indicative for the topological structure of the QCD vacuum. Needless to say that the time averaged vacuum expectation value of ˜ Q = Q/(2N f ) itself is zero, 0|Q(x)|0 = 0, by translation invariance and parity conservation of strong interactions3 . As a consequence, the net topological charge of the QCD vacuum state is zero. The key quantity describing topological ﬂuctuations of the QCD vacuum is the topological susceptibility, χ, which is deﬁned by χ = d4 x0| Q(x)Q(0) |0 . (5) In the so-called ’t Hooft limit of the theory, when the number of colours, Nc is taken to inﬁnity (Nc → ∞ at ﬁxed value for the number of active ﬂavours, Nf , in the weak gauge coupling limit g 2 ∼ 1/Nc ), Witten and Veneziano were able to derive a simple relation for the η mass in the chiral limit [4] m2η = lim

Nc →∞

2Nf χ|quenched , Fπ2

(6)

where Fπ denotes the pion decay constant. Within the Witten-Veneziano model assumption, that the real world is well described by the above limit relation, LQCD can predict the η mass by computing the topological susceptibility in quenched QCD. The lattice implications of (6) have been discussed in quite some detail in a recent paper of GIUSTI et al. [5]. In this contribution we will focus on those problems that one encounters in the direct approach to extract physics from fermionic two-point correlators in the singlet sector.

2 Prolegomena: Pseudoscalars in LQCD In this section we shall consider a symmetric world with three ﬂavours to enter the discussion on the computational tasks to be met. 2.1 Flavour Octet Sector In the ﬂavour octet sector of QCD (with mass degenerate u, d, s quarks), the nonconservation of the axial vector current is determined by the PCAC relation: µ(8)

∂µ j5

(x) = 2m¯ q γ5 q ,

(7)

which is much simpler in structure than its ﬂavour singlet analogue, (3). In LQCD, the computation of the π-meson proceeds by hitting the vacuum state |0 with the quark bilinear pseudoscalar operator as given on the right hand side 3

There is no spontaneous breaking of parity in strong interaction physics.

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of (7). This creates both pions and excited π-like states. Let us consider such an operator carrying negative charge u ¯(x)γ5 d(x)|0 = |π − x + excited states

(8)

with up and down quark operators, u and d respectively. The propagator in the pseudoscalar meson channel from x to x , comprising the π − -meson and its excited friends, would hence read like this: (8) ¯ )γ5 u(x )¯ GPS (x, x ) = 0|d(x u(x)γ5 d(x)|0 .

(9)

This vacuum expectation value is most easily computed by Wick contracting the two valence quark operators with matching ﬂavour, and the result can be simply expressed in terms of the two quark propagators, Pu = Mu−1 (x, x ) and Pd = Md−1 (x, x ) as follows: GPS (x, x ) = Tr[Pu∗ (x, x )Pd (x, x )] , (8)

(10)

where the trace is to be taken both in spin and colour space. In LQCD, the quark propagators (from source x0 to sink x ) can be computed in the background ﬁeld of the QCD ground state (the vacuum) by solving the lattice Dirac equations MD (x )P (x, x ) = δ(x − x0 ) ,

(11)

where MD is the discretized form of the Dirac operator including the gluon interaction and the r.h.s. denotes a point source located at lattice site x0 , see Fig. 3. This is done by applying modern iterative solvers, like BiCGstab [6]. Note that

x

x’

Fig. 3. The connected propagator G (x, x ) for pseudoscalar meson states with negPS ative charge, with a quark d and an antiquark u ¯ running from source x to sink x in the unquenched QCD background ﬁeld, with γ5 insertions (8)

– MD {U } depends on the gluonic ﬁeld conﬁguration (i.e. the QCD vacuum conﬁguration {U } that has been produced independently by a suitable sampling process such as Hybrid Monte Carlo [7, 8]) – the eﬃciency of the solver is governed by the (ﬂuctuating) condition number of the large sparse matrix MD {U } [6],[9]. The simulation of the QCD vacuum on the lattice being a nonperturbative computation, the gluonic lattice vacuum conﬁgurations comprise the entirety of vacuum polarization eﬀects, i.e. any conceivable ﬂuctation from creation and

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151

quark

gluon

gluon antiquark

Fig. 4. Gluon propagators including quark loop self energy diagrams

annihilation of quarks and antiquarks (sea quark eﬀects). So in Fig. 3 all vacuum polarization diagrams as depicted in Fig. 4 are understood to be included. We note in passing, that the numerical computation of the complete inverse of the lattice Dirac operator is prohibitively expensive: its computational eﬀort grows with lattice volume, which in todays simulations is anywhere between 164 and 644 space-time lattice sites! So, in practical calculations of the π-meson, one restricts the source, x0 , to just a few space points in a particular time slice, say in t = 0. The task for calculation is then the pseudoscalar two-point function: 0 1 (8) GPS (x0 , x ) = Tr[Pu∗ (x0 , x )Pd (x0 , x )] , (12) where the brackets denote the average over an ensemble of some hundred gauge ﬁeld vacuum states. Note that it is enough in practice to compute but a few columns of the entire inverse Dirac matrix per conﬁguration for achieving suﬃcient accuracy of the estimate, (12). Once this pseudoscalar two-point function has been calculated the pion contribution is obtained by removing the excited states from the pseudoscalar channel. For this purpose, one ﬁrst extracts the zero momentum contributions by summing over all (spatial) x in time slice t : ' ( (8) ∗ Tr[Pu (x0 , x )Pd (x0 , x )] . (13) CPS (t = 0, t ) := x

This latter two-point function at large enough time separations t between source and sink will be dominated by the ground-state of the channel, i.e. the pion. Prior to that, it contains a superposition of ground and excited states: t→∞ (8) CPS (0, t ) = ci exp(−Ei t ) −→ c0 exp(−mπ t ) . (14) i

It is obvious that the strict localization of the source at the very lattice site x0 overly induces excited state contaminations; hence a suitable smearing of the source around this site will deemphasize such pollutions and will help to achieve precocious ground state dominance. This is welcome because the twopoint correlator decreases exponentially and is prone to suﬀer large statistical ﬂuctuations at large values of t . The numerical task in exploiting (14) is then to optimize the smearing such as to achieve an early onset of a plateau: the latter is deﬁned as the t -range

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over which GPS (0, t ) decays with a single exponential. This is tantamount to requiring the “local pseudoscalar mass”, mPS (t ), to remain constant: (8)

mPS (t ) := −∂t ln[CPS (0, t )] = const. = mπ . (8)

!

(15)

Smearing of sources or sinks is achieved by a diﬀusive iteration [8] process: think of replacing pointlike sources or sinks by δ(x0 ) = φ(0) (x0 ) −→ φ(N +1) (x) , where φ(N ) (x) is peaked around x0 and is constructed by the recursion 1 (i+1) (i) (i) p.t. φs (x) = φs (x + µ) . φ (x, t) + α 1 + 6α s µ

(16)

(17)

The index “p.t.” stands for “parallel transported”, and the sum extends over the six spatial neighbours of x. As a result the entire series of smeared “wave functions” φ(i) , i = 1, 2, . . . , N − 1 is gauge covariant under local gauge transformations. Quark operators are computed after N = 25 such smearing steps, with the value α = 4.0. The smearing procedure is applied to meson sources as well as to sinks for connected and disconnected diagrams. 2.2 Flavour Singlet Sector In the case of SU (3) ﬂavour symmetry the appropriately normalized ﬂavour singlet pseudoscalar operator reads 1 q¯i (x)γ5 qi (x) . O(0) (x) = √ 3 i

(18)

Here the sum extends now equally over all (mass degenerate u, d, and s) quark ﬂavours. The ﬂavour singlet pseudoscalar meson propagator is estimated from the average 0 1 (0) GPS (x, x ) = O(0)† (x )O(0) (x) . (19) In this case the Wick contraction now induces both connected and disconnected (two-loop) diagrams, D, 0 1 (20) D(x, x ) = tr[O(0)† (x )]tr[O(0) (x)] , such that ﬁnally GPS (x, x ) = GPS (x, x ) − Nf D(x, x ) , (0)

(8)

(21)

as depicted in Fig. 5. Note that at this stage the trace operations in (20) refer to spin and colour space only and the Wick contraction procedure leads to the

The Computation of η and η in LQCD

x

153

x’

-N_f

x’

x

Fig. 5. The connected and OZI-rule violating (“disconnected”) contributions to the (0) η propagator G from (19). Note that these diagrams keep only track of the valence PS quark lines: all gluon exchanges due to the vacuum background ﬁeld have been omitted for clarity of the graph. It goes without saying that in unquenched QCD, the gluon propagators implicitly induce sea quark loops as in Fig. 4

weight Nf = 3 of the disconnected relative to the connected contribution, the latter being identical to the expression given in (13). The relative minus sign is due to Fermi statistics of the quark operators. We note in passing, that connected and disconnected correlators come along with intrinsic positive sign such that the above minus sign leads to a steepening of C (0) (t ) w.r.t. C (8) (t ), iﬀ D shows a soft drop in t . Let us mention that the terminology “two-loop correlator” might be misleading: it refers to the fact that we are dealing here with two (valence quark) loop insertions of the γ5 operator. The unquenched QCD vacuum includes of course a host of sea quark loops that are implicitly taken into account when sampling unquenched QCD vacua in a LQCD simulation. They would enter in form of vacuum polarization eﬀects on the very gluon propagators as depicted in Fig. 4. It is illuminating to consider the mechanism for the origin of the singlet octet mass gap in a model to real QCD: let us start in the Nc = ∞ world, where the UA (1) symmetry holds and ﬂavour octet and singlet mesons are degenerate, m0 = m8 . This degeneracy is only broken by the higher quark loop contributions which are suppressed by inverse powers in Nc . It is hence straightforward to approximate the η propagator, starting out from this scenario, as an explicit multiloop expansion of quark antiquark annihilation diagrams in the pure gluonic background ﬁeld of quenched QCD, as illustrated in Fig. 6. The actual calculation can be greatly simpliﬁed by introducing an eﬀective loop-loop coupling, µ20 . In momentum space the η -propagator then takes the form of a geometric series:

-

+

+

-

....

Fig. 6. Pictorial receipe for setting up the η s propagator in quenched QCD as an explicit multiloop expansion

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Pη

) l * ∞ −µ20 1 = 2 1+ . p + m28 p2 + m28

(22)

l=1

The series is readily summed up with the result: Pη =

p2

1 , + (m28 + µ20 )

(23)

which reveals that the eﬀective loop-loop coupling is nothing but the mass gap between singlet and octet pseudoscalar masses: ∆m2 =: m20 − m28 = µ20 ,

(24)

We emphasize that in this eﬀective model it is only by summation of the entire series to all orders in l that we arrive at a simple pole expression for the singlet propagator in momentum space and hence at a simple exponential falloﬀ in time of the two-point correlator. Note that this summation happens automatically in an unquenched QCD situation where all multiquarkloop eﬀects are already implicitly included in the very gauge ﬁeld vacuum conﬁguration!

3 The Real World with n¯ n s¯ s Mixing So far we have been leading a rather academic discussion in an SU (3) symmetric world. We would like next to describe the η/η problem in a more realistic setting. This opens up a pandora box of four parameters for decay matrix elements plus the mixing angle among isosinglet particle states. Two degrees of freedom in the decay constants can be characterized as mixing angles [10]. 3.1 Phenomenological Approach: Alignment Hypothesis In order to ease the interpretation of ﬂavour breaking eﬀects in the isosinglet sector, Feldmann et al. [11, 12] have proposed some time ago an intuitive scheme to describe the η−η phenomenology by use of a quark state Fock state representation in conjunction with an alignment assumption. The latter reduces the number of parameters to two couplings plus a single mixing angle of the particle states in that Fock space. Let us brieﬂy review their scenario: they proceed from the two isosinglet states √ ¯ u + dd/ 2 + ··· (25) |ηn = Ψn |u¯ |ηs = Ψs |s¯ s + · · · , (26) where ψn,s denote light cone wave functions4 . The key assumption of their phenomenological approach is their hypothesis of alignment of the current matrix elements: 4

There dots in (25) represent higher Fock states, including |gg and components from the heavy quark sectors, such as c¯c .

The Computation of η and η in LQCD

0|¯ qi γ5 γµ qi |ηj = δij f i pµ , i, j = n, s .

155

(27)

Due to the QCD interactions the physical η and η states will turn out to be mixtures of these Fock states:

|ηn cos φ − sin φ |ηn |η = U (φ) : = . (28) sin φ cos φ |ηs |ηs |η As a consequence of the alignment hypothesis, (27), the four decay constants, fPi

i 0|jµ5 |P = 0|¯ qi γ5 γµ qi |P ≡ fPi pµ n

i = n, s;

P = η, η

(29)

s

can be expressed in terms of f , f and the rotation angle φ [11]:

fηn fηn

fηs fηs

= U (φ)

fn 0

0 fs

.

(30)

By feeding the anomalous PCAC relation ˜ ∂ µ q¯i γ5 γµ qi = 2mi q¯i γ5 qi + Q

(31)

into the four decay matrix elements i 0|∂ µ jµ5 |η = fηi m2η ,

i 0|∂ µ jµ5 |η = fηi m2η ;

(i = n, s) ,

(32)

the authors of [11, 12, 13] arrive at the mass matrix mixing relation ) * √ 2

1 ˜ n ˜ µ2nn + f n2 0|Q|η 0 mη s 0|Q|ηn † f √ = U (φ) U (φ) , (33) 2 ˜ ˜ s 0 m2η µss2 + 1s 0|Q|η n 0|Q|ηs f

f

with a single mixing angle that is in accord with Fock state mixing. In (33) the abbreviations µ2 stand for the explicit chiral symmetry breaking terms from nonvanishing quark masses, mi : 2mi 0|q¯i γ5 qi |ηi ; i = n, s . (34) µ2ii := fn For consistency of the approach, they have to postulate the symmetry of the mass ˜ i ; i = n, s. matrix, which sets a constraint among the matrix elements 0|Q|η The l.h.s. of (33) being a continuum relation, great care must be exercised in renormalizing the operators. Since the underlying anomalous PCAC relation, (31) is actually to be seen as a Ward-Takahashi identity from chiral rotations, the standard Wilson lattice fermions provide a problematical scheme for regularizing the matrix entries in (33), with the Wilson lattice fermions lacking chiral symmetry at ﬁnite lattice spacing [14]. This complex situation would be largely improved when dealing with Ginsparg-Wilson fermions in unquenched QCD, whose lattice chiral symmetry prevents undesired divergencies from operator mixings [5]. On the lattice, the mass2 entries to the matrix (33) are best accessible from propagator studies in the singlet channels [15, 16, 17, 18]. We would like to explain next, that the idea of Fock state mixing of eigenstates can in principle be pursued very naturally from a variational approach on the lattice, once we have fully unquenched QCD vacuum conﬁgurations in the future, without making reference to an alignment hypothesis.

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3.2 Mixing Singlet Mesons on the Lattice On the lattice, we can prepare the the Fock states |ηn and |ηs by hitting the vacuum at some (smeared) source location, say x = (0, 0, 0, 0), with the bilinear isosinglet operators, Oi (0)|0 = q¯i (0)γ5 qi (0)|0,

i = n, s

(35)

and by waiting long enough to see the ground states emerge. In the sense of a variational method, one can next work with a superposition of these operators to achieve earlier exponential decay of the correlator and/or lowering of the ground state, the η-meson: Oα = cos α On − sin α Os .

(36)

Suppose this variational ansatz delivers an immediate plateau for some value of the variational parameter, α = α∗ : in that case we have actually hit an eigenmode of the interacting system and the Fock state mixing angle is φ = α∗ ! Since one expects little mixing with higher Fock states, one could then recover the η ground state from the orthogonal combination Oα⊥ = sin α On + cos α Os .

(37)

In general the state mixing information is encoded more deeply in the correlation matrix (still in operator quark content basis) √

Cnn√ (t) − 2Dnn (t) 2Dns (t) C(t) = . (38) 2Dsn (t) Css (t) − Dss (t) √ Note that the weight factors 2 and 2 in this relation originate from the combi¯ quark natorics when Wick contracting the degenerate u (with u ¯) and d (with d) operators contained On 1 ¯ 5 d) . uγ5 u + dγ q¯i γ5 qn := √ (¯ 2

(39)

If plateau formation does not occur immediatley, at tp = 0, one needs to follow the time evolution up to a later time step when the higher excitations have died out, and ascertain the plateau condition within the time interval tp → tp + ∆t in form of a generalized eigenvalue problem [19]: C(tp + ∆t)

+ cos α − sin α

= exp(−mη ∆t) C(tp )

+ cos α − sin α

.

(40)

The problem can be recast into a symmetric form C −1/2 (tp )C(tp + ∆t)C −1/2 (tp ) · ηα = exp(−mη ∆t) · ηα

(41)

The Computation of η and η in LQCD

with the deﬁnition

ηα (tp ) = C +1/2 (tp )

+ cos α − sin α

157

(42)

According to the symmetric plateau condition, (41), the direction of the ground state vector, ηα (tp ), remains ﬁxed under further time displacements ∆t through the transfer operator, T (∆t) = C −1/2 (tp )C(tp +∆t)C −1/2 (tp ). Hence the η mixing angle in the quark state basis is given by the direction of eigenvector ηα (tp ) at the onset of the plateau and might diﬀer appreciably from the variational angle α. The problem (41) being symmetric, the η is retrieved as the second, perpendicular eigenstate of T (∆t). It might be illuminating to put (41) into perspective with the previous plateau condition, (15): by Taylor expanding of C(tp + ∆t) to lowest order in ∆t one can readily recast (41) into a local condition, with a “logarithmic derivative” matrix construct, C −1/2 (tp )C (tp )C −1/2 (tp ): C −1/2 (tp )C (tp )C −1/2 (tp ) · ηα (tp ) = −mη · ηα (tp ) .

(43)

In principle, the phenomenological alignment postulate of (27) can be tested on the lattice by computing another, nonsymmetric set of correlators C˜ij (t) = 0|∂ µ q¯i (t)γ5 γµ qi (t)¯ qj (0)γ5 qj (0)|0 ,

(44)

which allows to extract the various decay matrix elements. Let us emphasize in concluding this section that the scenario presented here relates to fully unquenched three ﬂavour QCD; according to the discussion in 2.2 the mass plateau conditions can only be anticipated if the quark contents of the bilinear operators in (35) are equally present in the quark sea5 .

4 The Three Computational Bottlenecks In this section we shall focus in more detail on the numerical obstacles that one encounters when carrying out the above program. These diﬃculties are computationally much more severe than in nonsinglet spectroscopy and hence require the development of advanced algorithms and numerical techniques: 1. Present limitations in sea quark ﬂavours. It was only in the second half of the nineties that LQCD had developed the means to tackle ﬁrst semirealistic large scale simulations of QCD beyond the valence (or “quenched”) approximation. The reason is that Wilson-like discretizations of the Dirac operator are mandatory once we wish to deal with strong interaction situations sensitive to ﬂavour symmetry. But such Wilson-like lattice fermions need extremely large supercomputer resources when it comes to actually simulate the eﬀects of dynamical sea quarks, i.e. to sample unquenched CCD vacuum conﬁgurations. The hybrid Monte Carlo algorithm (HMC) is a nonlocal sampling 5

See the discussion on the partially quenched approach in Sect. 5.2

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technique which was developed and improved ever since the late eighties [7]; it continues to be the workhorse in all of the major QCD simulation projects with Wilson-like fermions [8, 20, 16, 21]. For technical reasons, however, HMC has the shortcoming of being limited to even numbers of sea quark ﬂavours. As a consequence, all the above projects neglected dynamical s-quarks in working with mass degenerate u and d sea quarks only, i.e. considered an SU (2) symmetric sea of quarks only. It is to be hoped that the next generation of large-scale QCD simulation with Teracomputers will overcome this limitation with new sampling algorithms. 2. Trace computations. We have seen that the peculiarity of ﬂavour singlet objects lies in the annihilation of the valence quark lines into the ﬂavour blind gluonic soup which implies the need to compute disconnected diagrams. But inspection of (20) shows that the x -summation from momentum zero projection requires the evaluation of the trace of the inverse Dirac operator. This task is deﬁnitely beyond the reach of modern linear equation solvers for matrices of rank 106 and more. Traditionally stochastic estimator techniques (SET) have been the popular workaround [22, 23]. 3. Noisy signals. The singlet spectrum being inherently determined by the physics of the quantum ﬂuctuations of the QCD vacuum, the two-loop correlators are bound to suﬀer from a serious noise level. This calls for the use of noise reduction methods, in order to circumvent the need for overly large ensembles of vacuum ﬁeld conﬁgurations. With this motivation low mode expansion methods have been proposed recently with considerable success to replace stochastic estimator techniques [15, 24]. 4.1 Stochastic Estimate of Fermion Loops The basic idea of stochastic estimation of the momentunm zero projected loop operator in (20), x tr γ5 M −1 (x, t) , is very simple: instead of attempting to solve (11) V on delta-like sources, one introduces socalled stochastic volume sources which are completely delocalized, on the entire lattice volume V: ξ = (ξ1 , ξ2 , . . . , ξV ) .

(45)

The components xi are chosen to be real random numbers sampled from a normal √ distribution N (xi ) = 1/ 2π exp(−x2i /2) or socalled Z2 noise which is nothing but equally distributed random numbers ±1. One solves the Dirac lattice equation on an ensemble of Ns such random source vectors, {ξ α }, for each individual gauge conﬁguration: −1 α ξ . (46) ζ α = γ5 MD and computes the ensemble average of the scalar products 0 1 (ξ, ζ) := s

Ns V 1 −1 . ξi∗α ζiα TR γ5 MD Ns V α=1 i=1

(47)

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159

One might phrase this procedure as an all-at-one-stroke method, at the expense −1 with of nonclosed loops sneaking in, in form of contributions from γ5 MD ij i = j. These undesired contributions are suppressed, however, due to the “componentwise orthonormality” relation 0

ξi∗ ξj

1 s

=

Ns 1 Ns →∞ ξ ∗α ξ α −→ δij . Ns α=1 i j

(48)

Note that (47) readily allows for restricting the grand trace operation over the entire space-time lattice, TR, to any particular time slice, simply by truncating the i-summation to the appropriate three-dimensional subvolume, such as the estimator for the momentum zero expression on time slice t, to yield −1 tr γ M (x, t) . 5 x In principle, these considerations hold for all kinds of insertions, not just γ5 like in our example. In any case the “sneakers” are suppressed as O(Ns−1 ) terms, but with a strength that depends on the insertion. The practical advantage of this stochastic procedure is, that in the instance of γ5 insertions one needs to compute only some few hundred solutions to (46) which is much less than O(V )! But we should keep in mind that this is achieved only at the expense of injecting additional noise into the problem6 . 4.2 Low Eigenmode Approximation to Fermion Loops We start with the eigenvalue problem for the socalled Hermitian Dirac operator, Q5 := γ5 MD ,

(49)

Q5 ψi = λi ψi .

(50)

which reads The quark loop with γ5 insertion at time slice t is simply expressed in terms of this eigensystem: 1 ψi (t)|ψi (t) . (51) Q−1 5 (t) = λi ψi |ψi i The disconnected two-loop correlator from Fig. 5 then has the form 6

We would like to point out that some authors [25] have refrained from using noisy relying on sources altogether by using a single “solid” wall source with ξi = 1 and −1 , with i = j. gauge symmetry to suppress the unwanted nonclosed loops, γ5 MD ij Actually this argument relies on Elitzur’s theorem according to which the vacuum expectation value O vanishes for any such non gauge invariant operator O! However, if the sample of vacuum gauge ﬁelds is limited to some few hundred independent conﬁgurations only this procedure oﬀers too little control over these systematic errors; nevertheless one could reﬁne this variant by applying the “solid” wall source idea on an ensemble of stochastic gauge copies of each individual vacuum conﬁguration. To our knowledge this has never been attempted.

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D(t) =

1 ψi (t0 )|ψi (t0 ) 1 ψj (t0 + t)|ψj (t0 + t) , λi ψi |ψi λj ψj |ψj t i j

(52)

0

where t represents the time separation between source and sink and we have exploited translational invariance by summing over all time locations of the source, t0 . This summation is for free, once the eigenfunctions are available7 . The determination of the entire spectrum being of course prohibitively expensive, we proceed by arranging the spectrum in ascending order of |λi | attempting to saturate the eigenmode expansion (52), with its lowest modes. The intuitive reasoning behind such a strategy is, that the underlying physics is expected to be encoded in the infrared rather than ultraviolet modes. After all, it is well known that for Wilson fermions 15/16 of the spectrum is related to the unphysical doublers that become frozen to order O(a−1 ) as the lattice spacing a is going to zero. 0.07

κ=0.156 κ=0.1575

Modulus of Eigenvalue

0.06 0.05 0.04 0.03 0.02 0.01 0 0

50

100 150 200 Index of Eigenvalue

250

300

Fig. 7. Cumulative spectral distribution of the lowest eigenmodes at the smallest (lower curve) and largest (upper curve) sea quark masses of the SESAM project [24]

Figure 7 oﬀers an impression on the cumulative distribution of the lowest modes of the Dirac operator in the situation of the SESAM simulation, at their smallest and largest sea quark masses. Note that the masses of valence quarks (inserted into the Dirac operator) and sea quarks (inserted into the hybrid Monte Carlo sampling of gauge ﬁeld conﬁgurations) are chosen to coincide; Fig. 7 therefore reﬂects the feedback mechanism from dynamical eﬀects onto the spectral representation as quark masses are diminished. Figure 7 suggests that the important spectral activity is mostly within the lowest 50 modes. 7

We mention that there might be other opportunities for spectral methods in LQCD with their potential to deal with all-to-all propagators: like the long standing string breaking problem or heavy-light meson scattering [26].

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But how many modes do suﬃce for saturating (52)? For the saturation of the two point correlators, one would anticipate the need for larger values for the cutoﬀ, λc , as one diminishes the time separations towards a few lattice spacings. This expectation is indeed conﬁrmed by the numerical results for the pion corrrelator as displayed in Fig. 8: The diagram shows a sequence of curves which – in descending order – refer to C(t = 1), C(t = 2), . . . , C(t = 16), which are plotted versus the spectral cutoﬀ. For small values of t the curves keep rising with λc , while asymptotic saturation is found to be better at larger t-values. 0.016

κ=0.1575

Correlation Function

0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0

50

100 150 200 # of Eigenpairs

250

300

Fig. 8. Pion correlator in spectral approximation, plotted versus spectral cutoﬀ, λc . Shown is a sequence of trajectories, as obtained on ﬁxed values of t: the curves represent C (8) (t = 1), C (8) (t = 2), C (8) (t = 3) etc, from top to bottom [24]. In this representation the justiﬁcation of truncation is signaled by a ﬂat λc -dependency

In Fig. 9, we plotted the pion correlator from a truncated eigenmode approximation (TEA) with cutoﬀ λc = 300 and from standard iterative solvers. There remain noticeable diﬀerences in the two curves for this value of the cutoﬀ. Therefore, one would refrain from using TEA for connected hadron propagators unless really needed. The situation w.r.t. λc turns out to be much more favourable with the disconnected piece, D(t), as shown in Fig. 10. How well the idea of saturation actually works out quantitatively is illustrated in Fig. 11, where we compare the two-loop correlator in the truncated eigenmode approximation (TEA) against the results from a previous comprehensive study with the estimation by an ensemble of stochastic sources [23]. We ﬁnd that the contributions from the lowest 300 eigenmodes agree remarkably well with the SET computations which are based here on 400 source vectors, on a 16 × 32 lattice in the sea quark mass range covered used by the SESAM project! The

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Correlation Function

0.1

standard method TEA Fit κ=0.1575 l=300

0.01

0.001

0

5

10

15 ∆t

20

25

30

Fig. 9. Quality of the truncated eigenmode approximation at the smalles SESAM sea quark mass for the pion correlator, with a spectral cutoﬀ, λc = 300. Satisfactory agreement with the standard result from using iterative solvers is only reached at t 10

5.5e-14 κ=0.1575 SS

5e-14 Two-loop Function

4.5e-14 4e-14 3.5e-14 3e-14 2.5e-14 2e-14 1.5e-14 1e-14 5e-15 0

50

100 150 200 # of Eigenvalues

250

300

Fig. 10. Quality of low eigenmode approximation as function of time separation for the OZI rule violating term, D(t). The curves represent (from top to bottom) C(1), C(2), . . . , C(16) [24]

net computational eﬀort behind the two sets of data points in Fig. 11 is about equal, yet the TEA data among themselvesa appear to be less ﬂuctuating. This positive message from Fig. 11 bears substantial promise for future lattice studies with lighter quark masses, as SET is doomed to degrade when penetrating more deeply into the more chiral regime while the truncated eigenmode

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0.004

SET TEA

0.0035 Two-loop Function

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κ=0.1575 LL l=300

0.003 0.0025 0.002 0.0015 0.001 0.0005 0

0

5

10

15 ∆t

20

25

30

Fig. 11. The two-loop correlator, D(∆t), as obtained from SET [23] (circles) and TEA (triangles) in comparison. Local sources and sinks have been used and Ns = 300 and λc = 300 were chosen [24]

approximation (TEA) will thrive: (a) the lowest modes will dominate all the more as |λmin | approaches zero; (b) modern Arnoldi eigensolvers have been made very eﬃcient in present day LQCD simulations [27] and will not deteriorate for lighter quarks as long as one sticks to ﬁxed lattice volumes. Conversely, iterative solvers on stochastic sources will suﬀer substantial losses in convergence rate, once the condition number of the Dirac matrix, c : = |λmax /λmin | starts exploding. 4.3 Navigating between Scylla and Charybdis There is another lesson from Fig. 11: we ﬁnd the noise level in D(t) to be rather independent of t, i.e. of signal size. This is of course in line with the fact that we are hunting for vacuum ﬂuctuations proper. It therefore appears impracticable to achieve ground state projection in the very standard fashion by increasing t: the η correlator being the imbalance between connected and two-loop signals one is to keep track of a signal in form of a rapidly diminishing diﬀerence between two quantities, with one being at ﬁxed noise level. Given this situation we are challenged by complying to two conﬂicting needs: on one hand large t values are needed to deﬂate excited state contaminations and on the other hand the struggle for an acceptable signal-to-noise ratio impedes working in the large-t regime. Our strategy to deal with this conundrum of the η correlator, (21) is to contrive a procedure for circumventing this conﬂicting requirements. The receipe is the following: (8)

– suppress the excited states from CP S (∆t) by using an exponential ﬁt from the window of its mass plateau;

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– deemphasize the short range noise level inside D by means of the truncated eigenmode expansion (52). This method could also be characterized through an intermediate “synthetic data stage” since the actual ground state analysis for the singlet correlator is organized as a sequence of three steps: (a) extract (8)

an exponential ﬁt to the lattice data for E(∆t) = CP S (∆t), (b) combine this ﬁt curve, E(∆t), with the lattice data for D(∆t) into the “synthetic” data (0) set for CP S (∆t) := E(∆t) − Nf D(∆t), in the spirit of (21), (c) establish (0)

(0)

a mass plateau in CP S (∆t) and perform another exponential ﬁt to CP S (∆t) within the plateau region in order to determine the singlet pseudoscalar mass. We ﬁnd this dual ﬁltering approach to operate very well in the two-ﬂavour case, on the SESAM QCD conﬁgurations. This is illustrated in Fig. 12 which manifests very clearly a plateau in the local mass plot of the η -correlator. The onset of plateau formation is precocious in the sense that we still maintain very precise signals. Therefore the statistical accuracy is suﬃciently high to resolve the mass gap between the singlet and nonsinglet masses, whose lattice values for

Local Masses

0.5 0.45

κ=0.1575 LL κ=0.1575 SS κ=0.156 SS

l=300

π mass

0.4 0.35 0.3 0.25

π mass 2

4

6

8

10

12

∆t Fig. 12. Mass plateau formation in the η (singlet) correlator after dual ﬁltering with TEA, for two diﬀerent values of hopping parameters (sea quark masses) versus time separation between source and sink, ∆t [24]. The plot shows the respective lattice values for the pion (nonsinglet) mass for comparison. The eﬀect of smearing onto the signal is also illustrated: SS and LL refer “to smeared source and sink” and “local source and sink”, respectively

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these sea quark masses are also included in the plot, being indicated with the symbol “π”. Let us emphasize that we consider the upshot of precocious asymptotia in the η -correlator as an essential outcome of our analysis; it corroborates that the nonperturbative lattice simulations take eﬀect in implicitly creating the mass gap between singlet and nonsinglet states, much in line with the perturbative reasoning from (22): indeed, the simulation data for the two-loop correlator, D(t), turn out to be the diﬀerence between just two exponentials rather than a multiexponential superposition. This will enable us to perform a rather detailed physics analysis in the singlet pseudoscalar channel. One might still worry about possible systematic errors from spectral truncation in the singlet mass plateaus contained in Fig. 12. In this context it is reassuring to observe nice agreement within errors with the results obtained from stochastic sources, as illustrated in Fig. 13. SET κ=0.156 SS SET κ=0.1575 SS TEA SS 0.5

Local Masses

0.55

0.45

l=300

π mass

0.4 0.35 0.3 0.25

π mass 2

4

6

8

10

12

∆t Fig. 13. Comparing mass plateaus of the singlet correlator obtained with SET and TEA [24]. Smeared sources and sinks are used throughout. Notations as in Fig. 12

It is interesting to compare the plateau formation from dual ﬁltering (Fig. 12) with the state-of-the-art results from optimized smearing as obtained in the most recent comprehensive studies of the CP-PACS collaboration [20] Fig. 14. The comparison teaches us that dual ﬁltering is indeed remarkably eﬀective in unravelling the mass plateaus in the pseudoscalar singlet mesons.

5 Towards Realistic Physics Results 5.1 Unquenched Two Flavour World Given the good quality of signals within the SESAM setting, we are prepared to compute the entire mass trajectory of singlet pseudoscalars. In Table 1 we list

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1.1 1.0

=0.1375

eff

m η/(t)

0.9 =0.139 0.8

=0.140

0.7 =0.141 0.6 0.5

0

1

2

3 t

4

5

6

Fig. 14. State-of-the-art mass plateau formation from optimized smearing as obtained in a comprehensive study by the CP-PACS group, at their smallest lattice spacing, a = .1076 fm, on a 243 × 48 lattice, for various sea quark masses. Quotation from [20]

the essential run parameters of the SESAM project, which is based on vacuum ﬁeld conﬁgurations from standard Wilson fermions on a 163 × 32 lattice [8], with two active sea quark ﬂavours, at one value of the coupling, β = 5.6. Four diﬀerent sea quark masses have been used. The most interesting control parameter is the ratio between the pion and ρ-meson masses as determined on the lattice; it is quoted in the second column of Table 1. Table 1. Simulation parameters used at β = 5.6 and numbers of stochastic sources Ns (see (47) and Fig. 13). Last column: numbers of available decorrelated vacuum ﬁeld conﬁgurations, Nconf κsea 0.1560 0.1565 0.1570 0.1575

mπ /mρ

L3 ∗ T

Ns

Nconf

0.834(3) 0.813(9) 0.763(6) 0.692(10)

16 ∗ 32 163 ∗ 32 163 ∗ 32 163 ∗ 32

400 400 400 400

195 195 195 195

3

This setting allows for a chiral mass extrapolation of the singlet pseudoscalar meson composed of u and d quarks, as plotted in Fig. 15. The plot contains the mass trajectory for the singlet channel as well as the nonsinglet extrapolation, marked by “π”. We conclude that the data allow for a safe chiral extrapolation in both channels, though there is deﬁnite need to lower the sea quark masses in the simulations. The precision is suﬃcient to resolve the mass gap between singlet and nonsinglet pseudoscalars.

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Quadratic chiral extrapolation, 2 flavours 0.2 958 MeV 0.15 0.1 0.05

715 MeV 290 MeV

Pion eta’ masses from TEA experimental eta’ mass pseudoexperimental eta’ mass

0

0.01

0

0.02

0.03

0.04

0.05

0.06

quark mass Fig. 15. Plot of m2η versus quark mass, all given in lattice units [28]

As we have excluded strange quarks altogether, we can of course not expect to hit the experimental η mass: the extrapolated singlet mass of 290 MeV should be compared to the η rather than to the η mass. It is evident, that we must address the issue of treating strange quarks in addtion to u and d quarks. Another shortcoming of SESAM is its restriction to one coupling β which does not allow for a continuum extrapolation. Very recently, the CP-PACS collaboration has found in their Nf = 2 study with an improved action that this extrapolation tends to induce a considerable increase in the pseudoscalar singlet mass [20]. 5.2 Partially Quenched Scenario for the Three Flavour World One of the main restrictions of todays unquenched QCD simulations is their limitation to two mass degenerate dynamical sea quark ﬂavours. This presents a serious shortcoming when dealing with the real physics of the η-η system. Nevertheless, the partially quenched approach [30] oﬀers a crutch to include the strange quark sector into the analysis. In the partially quenched scenario, strange quarks might appear as valence quarks only and not contribute at all dynamically to the quark sea. For the sake of discussion, let us depart from the fully quenched situation. In this setting, the s-loop s-loop correlator, Dss in (38), amounts to the occurence of a double-pole: m20,ss /(p2 + m2s )2 in momentum space. In this expression, ms denotes the valence approximation mass estimate for the pseudoscalar s¯s meson that can be determined on the lattice from the connected piece of the Css correlator. As we have seen in Sect. 2.2, the double pole impedes an exponential decay in t and hence will require a diﬀerent strategy of analysis: it can readily be rewritten as a derivative on the single

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pole expression m20,ss /(p2 + m2s )2 = −

m20,ss ∂ . ∂m2s p2 + m2s

(53)

Iﬀ m20,ss is a constant w.r.t. p2 , (53) is readily Fourier transformed and predicts a linear t-dependence in the ratio of disconnected to connected correlators to look for: Dss (t) ! 2 = m0,ss /(2ms ) × t . (54) R(t) := Css (t) Note that prior to the exploitation of this ratio relation on actual data, ground state ﬁltering the Css -data is highly advisable, as explained in Sect. 4.3. The eﬀective strength of the double pole, m20,ss , can be extracted from (54) and is interpreted as the mass shift due to OZI-rule violation dynamics in channel 2, much in the spirit of (24): m2ss = m2s + m20,ss .

(55)

Let us turn now to the partially quenched setting. We wish to consider a situation with broken SU (3) ﬂavour symmetry, where alle valence and sea quarks can diﬀer in masses. According to [30] partial quenching of a particular quark species is achieved by introduction of scalar pseudoquark partners (of equal mass) that freeze those quark degress of freedom inside the determinant. In this context, one introduces a basis of states with Nq quarks and k additional pseudoquarks. In our situation, we have Nq = 5 (three valence and two sea quarks) and k = 3, if we allow for diﬀerent masses of sea and valence quarks. The correlator has to make reference to these degrees of freedom, i, j. In Fourier space, it is modeled by the momentum space ansatz [15] Cij =

δij i m20 , − 2 2 p 2 + mi (p2 + mi )(p2 + m2j )F (p2 )

i, j = 1, 2, . . . Nq + k .

(56)

The quantity i is equal to +1 (−1) for quark (pseudoquark = determinant eater) channels. The function F (p2 ) can be viewed as to model the implicit multiloop contributions from sea quarks to the disconnected pole terms: F (p2 ) = 1 +

sea quarks

σ

m20 . p2 + m2σ

(57)

mi , mj and mσ denote neutral pseudoscalar nonsinglet meson masses, while m0 refers to the eﬀective singlet interaction that might depend on the quark species involved. F reads in the case of the SESAM simulation with degenerate dynamical u and d quarks 2m2 (58) F (p2 ) = 1 + 2 0 2 . p + mσ

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Note that the model incorporates both the fully quenched limit, where F (t) ≡ 1, as well as the consistently Nf = 2 unquenched situation, where u and d valence and dynamical quark masses coincide, such that m1 = m2 = mσ : in this latter instance, the double pole term in the nonstrange sector is lifted and turned into a displaced single pole term, in accordance with the considerations presented in Sect 4.3. For practical matters, a partially quenched lattice analysis of (56) is carried out in two steps as follows 1. The pseudoscalar meson masses, m1 , m2 , mσ , are determined ﬁrst by LQCD from connected pseudoscalar correlators with the appropriate valence quark settings (in the case of mσ and m0,σ , sea and valence quark masses are identiﬁed), 2. the singlet correlator matrix contains a set of mass gap parameters, m0,ij . They can be computed ﬁnally from a ﬁt of the Fourier transform of the ansatz (56), to the lattice data for C(t). The latter ﬁtting can be done time-slicewise, yielding ﬁt parameters for “local” eﬀective mass gaps which should exhibit mass plateau formation. It goes without saying that throughout all the numerical analysis steps one should use the dual ﬁltering method with “synthetic” data in the sense of Sect. 4.3. 5.3 Partially Quenched Results In a comprehensive study [29] we analyzed some 800 independent SESAM conﬁgurations on 163 × 32 lattices at β = 5.6 [31], with four diﬀerent sea quark masses. In Fig. 16 we have plotted the set of hopping parameters of valence quarks in perspective with their critical values, for our diﬀerent sea quark settings. The full QCD situation with 2 quark ﬂavours in the sense of Sect. 5.1 is indicated by the dotted line, κsea = κvalence and the thin line connects the hopping parameters for strange quarks. We analyse the data according to (56). In doing so we proceed to study effective mass gaps as determined from the hairpin diagrams, Dij (t), by ﬁtting, w.r.t. m0ij (t): = µ at each value of t, the zero-momentum Fourier transforms of ansatz (56) to our data. In broken SU (3) with two active quark ﬂavours, we introduce two isosinglet pseudoscalars, designated by the indices n (for “light”, nonstrange) and s (for strange). In this manner, we can study three types of hairpin diagrams, Dnn , Dns , Dss , with the strange quark mass determined on the lattice from the kaon mass (for κ locations, see Fig. 16). As a result, we ﬁnd satisfying plateau formations in m0ij (t) as illustrated in Fig. 17. In fact, the numerical quality of our signals comfortably allows for the consecutive extrapolations (i) to light valence and then (ii) to light sea quark masses; the chiral sea quark extrapolation is exhibited in Fig. 18. From the consecutive chiral extrapolations, we obtain for the quadratic mass matrix in the quark-ﬂavour basis

2 m20,ns mnn , (59) M := m20,sn m2ss

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Kappa 0.162

critical kappas light kappas Simulated kappas 2 flavour full QCD strange kappas

0.161

kappa_valence

0.16 0.159 0.158 0.157 0.156 0.155 0.1555

0.156

0.1565

0.157 0.1575 kappa_sea

0.158

0.1585

Fig. 16. Our points of simulation in the hopping parameter plane of valence/sea quarks

m_0,nn mass plateaus, kappa_sea=0.1575, alpha=0.0 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04

kappa_val=0.1555 kappa_val=0.15608 kappa_val=0.1565 kappa_val=0.15687 kappa_val=0.1570 kappa_val=0.1575 1

2

3

4

5

6

7

8

time Fig. 17. First evidence of plateau formation in a partially quenched setting, illustrated on the eﬀective mass gap, m0nn (t), in the disconnected two loop correlator, Dnn (t). The data refer to the lightest sea quark mass of SESAM, i.e. κsea = .1575, for various isosinglet valence quark masses

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Chiral extrapolation for elements of quadratic mass matrix 0.3

m_nn m_0,ns m_ss

0.25 0.2 0.15 0.1 0.05 0 0

0.05

0.1

0.15

0.2

(pion mass)^2 Fig. 18. Extrapolations of the quadratic mass matrix elements of light (n) and strange (s) isosinglet valence quarks to QCD with chiral sea quarks (59), at α = 0 2 where m2nn := Mnn + m20nn . With the ρ-meson mass scale [8] we ﬁnd √

(306 ± 91)2 2(201 ± 34)2 M= √ MeV2 . 2(201 ± 34)2 (680 ± 15)2

Diagonalization of M renders Mη = 292 ± 31 MeV ,

Mη = 686 ± 31 MeV .

(60)

These numbers from Nf = 2 look promising: the individual mass values – while being low w.r.t. the real three ﬂavour world – yield a mass splitting that compares very well to phenomenology. 5.4 Data in Accord to χPT? So far we have neglected that the partially quenched scenario has been devised in the framework of chiral perturbation theory (χPT) where we expect to deal with a limited number of eﬀective couplings. In fact, Bernard and Golterman [30] showed that the eﬀective chiral symmetry violation (due to the chiral anomaly) is most simply described by the following contribution to the chiral Lagrangian: L0 = +

Nf 2 2 µ (η ) + α(∂µ η )2 , 2

(61)

i.e. in terms of just two constants, µ (mass gap in the chiral limit) and α (interaction parameter) only! If the tree approximation of χPT to their partially

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quenching scenario holds, the simulation data should be ﬁtted in terms of these two constants, irrespective of the (supposedly light!!) valence quark masses chosen. We will now address the question whether this is indeed the case. To this end we have to ﬁrst rewrite the correlators (56), by including the parameter α. Actually, the Lagrangian translates into two-point correlation functions (propagators) for the neutral pseudoscalar quark bilinears Φii (x) = q¯i (x)γ5 qi (x) ,

(62)

where the index i runs over the quark and pseudoquark d.o.f. [30]. In momentum space the disconnected part of these ps-correlators again has a compact form [32] with the infamous double pole structure Dij (p) = N

(p2

(µ2 + αp2 )(p2 + Md2 ) 2 )(p2 + M 2 ) . + Mii2 )(p2 + Mjj η

(63)

The prefactor, N = 1/(1 + Nf α) and the singlet mass, Mη2 = (Md2 + Nf µ2 )/(1 + Nf α) are seen to carry explicit α-dependencies. Rememeber that the octet masses (Mii , Mjj , and Md ) from valence and sea quarks, respectively, are readily computed from connected diagrams on the lattice. Our present concern is the determination of the mass gap, µ, and α from our lattice data. We remark that the outcome of the analysis at this stage seems not to be in accord with the tree approximation to the Lagrangian, (62), as the latter predicts the mass plateaus to be independent of the valence quark mass, mv , contrary to the apparant ﬁndings in Fig. 17. Therefore, we repeated the above mass plateau study on a set of nonvanishing α-parameters, in an attempt to verify the compliance of our data with such independence: inspecting the plot of m0nn vs. mv (Fig. 19) it becomes obvious (i) that m0nn (mv ) shows a simple, linear behaviour and (ii) that α can indeed be adjusted to produce a zero slope of m0nn (mv )8 ! In this way it is straightforward to ﬁnd an optimal value, αopt , that eliminates (for a given sea quark mass) the mv -dependence from m0nn , resulting in a universal plateau level. This collapse of data into universal plateaus (of heights µ ˜) is exempliﬁed in Fig. 20. The comparision with the situation encountered with α = 0 (Fig. 17) provides clear evidence for our sensitivity in determining αopt . After chiral sea quark extrapolation we arrive at a ﬁrst estimate of α: α = 0.028 ± 0.013

µ = 203 ± 34 MeV .

(64)

Note that the numerical value for the mass gap at the chiral point is robust w.r.t. the two diﬀerent approaches presented here. One might phrase it like this: the allowance for a nonvanishing α parameter is tantamount to the anticipation of the respective chiral valence quark limit in the mass gap! 8

The situation is less favourable for the gap values extracted from Dns [33].

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m_0,nn, kappa_sea=0.1575 0.016 0.014 0.012 0.01 0.008 0.006 alpha=-0.01 alpha=0 alpha=0.01 alpha=0.03 alpha=0.05

0.004 0.002 0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

valence quark mass

Fig. 19. Minimizing the valence quark dependence of the mass gap by variation of α m_0,nn mass plateaus, kappa_sea=0.1575, alpha=0.0328 0.15

0.1

0.05

0

kappa_val=0.1555 kappa_val=0.15608 kappa_val=0.1565 kappa_val=0.15687 kappa_val=0.1570 kappa_val=0.1575 1

2

3

4

5

6

7

8

time

Fig. 20. The analogue of Fig. 17: Universal plateau formation, at the lightest sea quark mass, when adjusting α to the value 0.0328

6 Conclusion The application of spectral techniques provides unprecedentedly high accuracy in the study of hairpin diagrams. Spectral methods thus provide access to detailed studies of the ﬂavour singlet mesons. In this work, this has been demonstrated in two ﬂavour QCD as well as in the Nf = 2 partially quenched scenario, with standard Wilson fermions and in the regime of medium mass sea quarks (see

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K. Schilling et al.

Fig. 16. Since spectral methods are at their best in the truly chiral regime, this looks very promising in regard to future simulations of ﬂavour singlet mesons with realistically light sea quarks, that we expect to come up within the overlap fermion formulation on the lattice.

References 1. S. Adler, Phys. Rev. 177, 2426 (1969), J. Bell and R. Jackiw, Nuovo Cim. 60 A, 47 (1969), S. Adler and W.A. Bardeen, Phys. Rev. 182, 1517 (1969). 147 2. S. Okubo, Phys. Lett. 5, 165 (1963), G. Zweig, CERN preprint TH412 (1964), J. Iizuka, Progr. Theor. Phys. Suppl. 37–38 (1966) 21. 148 3. R. Crewther, Phys. Lett. 70 b, 349 (1977), Riv. Nuov. Cim. 2, 63 (1979), Acta. Phys. Austr. Suppl XIX, 47 (1978). 148 4. E. Witten, Nucl. Phys. B 156, 269 (1979), G. Veneziano Nucl. Phys. B 159, 213 (1979) and Phys. Lett 95 B, 90 (1980). 149 5. L. Giusti, G.C. Rossi, M. Testa, and G. Veneziano, Nucl. Phys. B 628, 234 (2002) and references quoted therein. 149, 155 6. A. Frommer, V. Hannemann, B. N¨ ockel, Th. Lippert, and K. Schilling, Int. Journ. Mod. Phys. 5, 6 1073 (1994). 150 7. S. Duane, A.D. Kennedy, B.J. Pendleton, and D. Roweth, Phys. Lett. B 195 216 (1987). 150, 158 8. N. Eicker N. Eicker, P. Lacock, K. Schilling, A. Spitz, U. Gl¨ assner, S. G¨ usken, H. Hoeber, Th. Lippert, T. Struckmann, P. Ueberholz, J. Viehoﬀ, and G. Ritzenh¨ ofer (SESAM Coll.), Phys. Rev. D 59, 14509 (1999). 150, 152, 158, 166, 171 9. S. Fischer, A. Frommer, U. Gl¨ assner, Th. Lippert, G. Ritzenh¨ ofer, and K. Schilling, Comp. Phys. Comm. 98, 20 (1996). 150 10. H. Leutwyler, Nucl. Phys. Proc. Suppl. 64 223 (1998). 154 11. Th. Feldmann, P. Kroll, and B. Stech, Phys. Rev. D 58 114006 (1998). 154, 155 12. Th. Feldmann and P. Kroll, Eur. Phys.J C 5 327 (1998), Th. Feldmann, P. Kroll, and B. Stech, Phys. Lett. B 449 339 (1999). 154, 155 13. P. Kroll and Th. Feldmann, Phys. Scripta T 99 13 (2002). 155 14. M. Bochicchio, G. Martinelli, G. Rossi, and M. Testa, Nucl. Phys. B 262 275 (1985). 155 15. L. Venkataraman and G. Kilcup, hep-lat79711006. 155, 158, 168 16. UKQCD C. McNeile et al. (UKQCD Collaboration) Phys. Lett. B 491 123 (2000), Erratum ibid. B 551 391 (2003). 155, 158 17. C. McNeile, C. Michael, and K.J. Sharkey, Phys. Rev. D 65, 014508 (2002). 155 18. C. Michael, Phys.Scripta T 99,7 (2002). 155 19. M. L¨ uscher and U. Wolﬀ, Nucl. Phys. B 339 222 (1990). 156 20. V.I Lesk, S. Aoki, R. Burkhalter, M. Fukugita, K.-I. Ishikawa, N. Ishizuka, Y. Iwasaki, K. Kanaya, Y. Kuramashi, M. Okawa, Y. Taniguchi, A. Ukawa, T. Umeda, and T. Yoshie, (CP-PACS Collaboration) Phys. Rev. D 67 074503 (2003). 158, 165, 166, 167 21. QCDSF M. G¨ ockeler et al. (QCDSF collaboration) Phys. Lett. B 545 112 (2002. 158 22. See e.g. W. Wilcox, Talk given at Interdisciplinary Workshop on Numerical Challenges in Lattice QCD, Wuppertal, Germany, 22–24 Aug 1999. Published in Wuppertal 1999, Numerical challenges in lattice quantum chromodynamics, heplat/9911013. 158

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23. T. Struckmann, K. Schilling, G. Bali, N. Eicker, S. G¨ usken, T. Lippert, H. Neﬀ, B. Orth, W. Schroers, J. Viehoﬀ, and P. Ueberholz (SESAM and TχL Coll.), Phys.Rev. D 63 074503 (2001). 158, 161, 163 24. H. Neﬀ, N. Eicker, Th. Lippert, J.W. Negele, and K. Schilling, Phys. Rev. 64 114509 (2001). 158, 160, 161, 162, 163, 164, 165 25. Y. Kuramashi, M. Fukugita, H. Mino, M. Okawa, and A. Ukawa, Phys. Rev. Lett. 72 3448 (1994). 159 26. C. Michael and J. Peisa, Phys. Rev. D 58 034506 (1998). 160 27. H. Neﬀ, Nucl. Phys. B 106 (Proc. Suppl.) 1055 (2002), hep-lat/0110076. 163 28. H. Neﬀ, T. Lippert, J.W. Negele, and K. Schilling, Nucl. Phys. B 119 (Proc.Suppl.) 251 (2003), hep-lat/0209117. 167 29. H. Neﬀ, Th. Lippert, J. Negele, and K. Schilling, contribution LATTICE 2003. 169 30. C.W. Bernard and M. F.L. Golterman, Phys. Rev. D 49 486 (1994). 167, 168, 171, 172 31. N. Eicker et al. (SESAM coll.) Phys. Letts. B 407 (1997) 219. 169 32. M. Golterman et al, JHEP 0008 (2000) 023. 172 33. H. Neﬀ et al., in preparation. 172

Strong and Weak Interactions in a Finite Volume M. Testa Dip. di Fisica, Univ. di Roma “La Sapienza” and INFN, Sezione di Roma, Piazzale Aldo Moro 2, I-00185 Rome, Italy.

Abstract. In these lectures we review the basic ideas of ﬁnite volume quantization of two-body hadronic systems below the inelastic threshold and their relevance to lattice computations of weak hadron decays.

1 Introduction Lattice QCD oﬀers a natural opportunity to compute non-leptonic weak decays matrix elements from ﬁrst principles. The main diﬃculties are related to the continuum limit of the regularized theory (the ultra-violet problem) and to the relation between matrix elements computed in a ﬁnite Euclidean space-time volume and the corresponding physical amplitudes (the infrared problem). The ultra-violet problem, which deals with the construction of ﬁnite matrix elements of renormalized operators constructed from the bare lattice ones, has been addressed in a series of papers [1]–[3] and we will not consider it further. The infrared problem arises from two sources: – the unavoidable continuation of the theory to Euclidean space-time and – the use of a ﬁnite volume in numerical simulations. One of the main obstacles in the extraction of physical amplitudes from lattice simulations stems from the rescattering of ﬁnal state particles in Euclidean space. The formalization of this problem, in the inﬁnite-volume case, was considered in [4]. An important step towards the solution of the infrared problem has recently been achieved by Lellouch and L¨ uscher [5] (denoted in the following by LL), who derived a relation between the K → ππ matrix elements in a ﬁnite volume and the physical kaon-decay amplitudes. This relation is valid up to exponentially vanishing ﬁnite-volume eﬀects. The LL proposal is based on the following relation between ﬁnite and inﬁnite volume matrix elements: |ππ, E = mK | HW (0) |K|

2

= V 2 |V ππ, E| HW (0) |KV | ×8π[qφ (q) + kδ (k)]

2

# m $3 K

k (1)

In (1) |ππ, EV denotes a ﬁnite volume two pion state with zero total momentum and “angular momentum” and energy E, while |KV denotes a single ﬁnite M. Testa: Strong and Weak Interactions in a Finite Volume, Lect. Notes Phys. 663, 177–197 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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volume kaon state with zero momentum. Both states are normalized to 1. |ππ, E and |K denote the corresponding inﬁnite volume states covariantly normalized according to the usual convention which, for single particle states reads p|p = (2π)3 2ωp δ (3) (p − p ) .

(2)

q is deﬁned by q≡

Lk 2π

(3)

2

and

E2 − m2π (4) 4 denotes the relative momentum of the two pion system with total center of mass energy E. The function φ(q) is deﬁned in (41) below and δ(k) is the ππ s-wave phase shift. The LL strategy consists in tuning the volume V so that the ﬁrst excited twopion state (n = 1) is degenerate in energy with the kaon state (L ≈ 5 ÷ 6 F m) and compute the ﬁnite volume Green’s function d3 xd3 y σ(x, t)HW (0)K(y, t )V ≈ k(E) =

t →−∞

V

= emK t V K| K(0) |0 V 2 × 0| σ(0) |ππ, nV n

×V ππ, n| HW (0) |KV e−En t

= emK t V K| K(0) |0 V 2 × |0| σ(0) |ππ, nV | n

× |V ππ, n| HW (0) |KV | e−En t .

(5)

The last equality in (5) is justiﬁed by the cancellation of the ﬁnal state interactions phases in 0| σ(0) |ππ, nV and V ππ, n| HW (0) |KV . Then, from 2 d3 x σ(x, t)σ(0) = V |0| σ(0) |ππ, nV | e−En t (6) V

n

one can compute |0| σ(0) |ππ, 1V | and, ﬁnally, |V ππ, 1| HW (0) |KV |. The LL formula has been derived for a large enough volume, n = 1 ÷ 7 and ∆E = ∆P = 0 [5]. In these lectures we will present an approach to ﬁnite volume quantization and the LL formula, based on [8], [9]. The plan is the following. – In Sect. 2 we present a general technique which allows to derive summation theorems, needed to relate ﬁnite volume sums to inﬁnite volume integrals,

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– in Sect. 3 we present an approach to the L¨ uscher quantization condition [7] based on these summation theorems, both in Non-Relativistic Quantum Mechanics (Sect. 3.1) and in Relativistic Quantum Field Theory (RQFT) (Sect. 3.2). In particular we will examine the general consequences of inelasticity on the L¨ uscher quantization formula. – From the results of Sect. 3 we show in Sect. 4 that the matrix elements of a scalar operator between the vacuum and a two pion state at rest in a ﬁnite volume is proportional, up to exponentially small corrections, to the corresponding inﬁnite volume matrix element through a constant that only depends on the state and not from the operator. – In Sect. 5 we discuss an heuristic method to derive the previous constant and therefore the LL formula, while – in Sect. 6 a more careful derivation is presented, which allows to extend the LL formula to all elastic states under the inelastic threshold and the case when the inserted scalar operator carries a non-zero energy-momentum. Some technical details are relegated to the appendices.

2 Summation Theorems In this section we address the question of how fast a sum converges to an integral and we introduce a general strategy to solve this problem, based on Fourier transforms. It is well known [6] that, if we denote by pn the discrete set of momenta allowed, inside a volume V , by periodic boundary conditions, i.e. 2π n (7) L where n is a vector with integer components, we have, for “slowly varying functions”, f (pn ) and g(pn ), pn ≡

+∞ 1 1 ∗ f (pn )g (pn ) −→ f (p)g ∗ (p)dp V →∞ (2π)3 V p n

(8)

−∞

We want to know how fast is the asymptotic convergence of (8). We will illustrate our summation technique showing that, for a very general class of functions, the rate of convergence of (8) is exponential. We will also ﬁnd, a particular class of functions, of interest in physical applications, for which (8) is exact for given, ﬁxed, ﬁnite volumes, i.e. +∞ 1 1 ∗ f (pn )g (pn ) = f (p)g ∗ (p)dp . V p (2π)3 n

(9)

−∞

We start considering two functions, f˜(x) and g˜(x), which decrease exponentially when x is outside a given support V , and consider their Fourier transforms, f (p) and g(p)

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f˜(x)eip·x dx ≈

f (p) =

g(p) =

f˜(x)eip·x dx

(10)

g˜(x)eip·x dx .

(11)

V

g˜(x)eip·x dx ≈ V

In the computation of the Fourier transforms (10), (11), we kept the option to extend the x integral over the full space or over V . Since f˜(x) and g˜(x) decrease exponentially outside V , the two options diﬀer by exponentially small terms, as V → ∞. If the support of f˜(x) and of g˜(x) is entirely contained inside V , (10) and (11) become exact equalities. We have, with exponential precision in V , 1 ∗ ˜ (12) f (p)g ∗ (p)dp f (x)˜ g (x)dx = (2π)3 1 f (pn )g ∗ (pn ) (13) g ∗ (x)dx = ≈ f˜(x)˜ V p V

n

where we performed the Fourier transform in (12) and the Fourier series in (13). Equations (12) and (13) show that 1 1 f (p)g ∗ (p)dp ≈ f (pn )g ∗ (pn ) (14) 3 (2π) V p n

holds with exponential precision in V . If the supports of f˜(x) and g˜(x) are entirely contained inside V (14) is exact, and not only exponentially approximated, as can easily be checked from the above derivation. In the following we will also need to relate sums and integrals involving functions which are not “smooth”, i.e. when they are particular kinds of distributions. As an example we start illustrating a one dimensional problem. In this case we want to relate, again with exponential precision in the 1-dimensional volume L, 2π n) the sum L1 n kf2(p −p2n , (with pn ≡ L n), to some deﬁnition of the corresponding +∞ dp f (p) integral, for example P −∞ 2π k2 −p2 . To do this, we start by solving an auxiliary problem: the computation of the Green function GL (x − y) in the 1-dimensional volume L

2 ∂ 2 + k GL (x − y) = δ(x − y) (15) ∂x2 with periodic boundary conditions. A particular solution of the diﬀerential (15) is given by any one of the inﬁnite volume Green functions1 . dp eipx sin k|x − y| . (16) = G∞ (x − y) = P 2π k 2 − p2 2k 1

In inﬁnite volume we have several possible Green functions, diﬀering by terms proportional to e±ikx .

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It is easy to check that the required GL (x − y) is given by cos k(x − y) 2k tan kL 2 sin k|x − y| cos k(x − y) + . = 2k 2k tan kL 2

GL (x − y) = G∞ (x − y) +

(17) (18)

We now consider consider a function f˜(x) exponentially vanishing outside the volume L. We have + L2 1 f (pn ) GL (x)f˜(x)dx = (19) L n k 2 − p2n −L 2 +∞ +∞ dp f (p) f (k) ≈ GL (x)f˜(x)dx = P + (20) kL 2 − p2 2π k k tan −∞ −∞ 2 where, as before, we used the fact that the integral extended over the interval (− L2 , L2 ) diﬀers from the one extended over (−∞, +∞) by exponentially small corrections. In (19) we used the Fourier series, while in (20) we used the Fourier transform. Equations (19) and (20) give +∞ dp f (p) 1 f (pn ) f (k) = P + , (21) 2 2 2 2 L n k − pn k tan kL −∞ 2π k − p 2 which provides the required relation between the sum and integral, valid up to exponential corrections in L2 . Equation (21) shows an interesting feature: f (pn ) 1 n k2 −p2n has no limit L → ∞ for ﬁxed k. The reason is that, as L varies, L n) some of the pn get close to k, so that the sum L1 n kf2(p −p2n has an erratic behavior with L. However, if we consider a sequence of L’s such that, for instance, kL = n) lπ, with l a positive integer, then, with exponential precision, L1 n kf2(p −p2n = +∞ dp f (p) P −∞ 2π k2 −p2 . This is intuitively clear. In fact for such a sequence of L’s, k falls exactly at the midpoint of two successive pn ’s, thus reproducing the principal value deﬁnition of the integral. We now turn to the technically more diﬃcult case of a three dimensional summation formula analogous to (21). We follow the same strategy and start with the ﬁnite volume Green function (∆ + k 2 )GV (x − x ) = δ(x − x ) , with periodic boundary conditions, for k 2 =

4π 2 2 L2 n

for any n. We have

1 eipn ·(x−x ) , GV (x − x ) = V k 2 − p2n

(22)

(23)

{pn }

2

Equation (21) becomes exact if the support of f˜(x) is entirely contained in (− L2 ,

L ). 2

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where the pn are deﬁned in (7). For reasons which will become clear in a moment, it is convenient to consider the expansion of GV (x) in spherical harmonics [7]. In particular we will be interested in the s-wave projection of GV (x) 1 sin pn r 1 1 (0) GV (r) ≡ , (24) dΩ GV (x) = 2 2 4π Ω V k − pn pn r {pn }

(0)

where pn ≡ |pn |. From (22), we see that, for r = 0, GV (r) satisﬁes the rotationally invariant part of the Helmholtz equation and therefore has the form (0)

GV (r) = −

sin kr cos kr +c . 4πr kr

(25)

The ﬁrst term in (25) satisﬁes

cos kr (∆ + k 2 ) − = δ(x − x ) 4πr

(26)

so that the additional term must be regular at r = 0 and is therefore proportional to sinkrkr . A possible way to compute c is d # (0) $ rGV (r) , r→0 dr

(27)

1 cos pn r . r→0 V k 2 − p2n

(28)

c = lim so that, formally,

c = lim

{pn }

Equation (28) together with the deﬁnition [7] 1 1 Z00 (s, q 2 ) ≡ √ 4π {n} (n2 − q 2 )s

(29)

gives c=−

Z00 (1, q 2 )

. (30) 3 2π 2 L There is a point of caution here. In fact the sum appearing in (28) is the same which enters in GV (0) and is, therefore, divergent. The limit in (28) could provide the necessary process of analytic continuation (analogous to s-analytic continuation in L¨ uscher’s paper [7]). We will not discuss this point any further, because (25) and (30) coincide with (3.29) in [7]. We now consider, for illustration, the particular case of a spherically symmetrical function f (p). We have, in analogy with (21), ∞ 1 f (pn ) 1 f (p) = P d 3p 2 + cf (k) , (31) V k 2 − p2n (2π)3 k − p2 −∞ {pn }

Strong and Weak Interactions in a Finite Volume

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up to exponentially small corrections, provided that f˜(r), the Fourier transform of f (p), ∞ 1 ˜ d 3 p e−ip·x f (p) , (32) f (r) ≡ (2π)3 −∞ vanishes exponentially with r ≡ |x|. To prove (31) we start, as before, with the ﬁnite-volume integral (0) d 3 xf˜(r)GV (x) = d 3 xf˜(r)GV (r) V V

sin(kr) cos(kr) +c = d 3 xf˜(r) − . 4πr kr V

(33)

The assumption of the exponential decrease of f˜(r) at large r allows us to extend the integrals in (33) to inﬁnite volume with an exponentially small error. On the other hand we have 1 f (pn ) , (34) d 3 xf˜(r)GV (x) = V k 2 − p2n {pn }

∞ cos(kr) f (p) 1 3 ˜ d xf (r) − P d 3p 2 , (35) = 4πr (2π)3 k − p2 −∞ sin(kr) = f (k) . (36) d 3 xf˜(r) kr Equations (33), (34), (35) and (36) prove (31). We end this section remarking that, as before, both (21) and (31) become exact, and not only exponentially approximated, when the support of f˜(x) is entirely contained inside the given volume.

3 The L¨ uscher Quantization Condition In this section we will review some aspects of the quantization in a cubic box [7], following the derivation presented in [9]. We will be interested in the structure of the zero-momentum, ﬁnite volume, energy eigenstates, |ππ, nV , which, when analyzed in terms of the angular momentum, look like a complicated superposition [7] ∞ +l (n) αl, m |ππ, n; l, mV . (37) |ππ, nV = l=0 m=−l

In the following we will deal with cubically invariant states. In the framework of non relativistic quantum mechanics, the ﬁnite-volume quantization formulas become exact for ﬁnite-range potentials and in the presence of an angular momentum cut-oﬀ, which, in its simplest form, assumes that only s-waves interact. In order to be deﬁnite, we start from the center of mass Schr¨ odinger equation

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M. Testa

∆ + k 2 Ψk (x) = U Ψk (x) ,

(38)

˜ (r)P(s−wave) , explicitly contains an swhere the interaction potential, U ≡ U wave projector, thus guaranteeing that only s-waves interact. Moreover we will ˜ (r) vanishes for r > R. treat the case of a ﬁnite range potential, i.e. U We parametrize the energy eigenvalues in a cubic box with periodic boundary conditions, through the “radial” relative momentum k of the two particle state. k is related to the center of mass energy E by (39) E = 2 m2π + k 2 As we will show, the allowed values of k belong to two classes [7] 1. either k obeys the equation h(k, L) ≡

φ(q) + δ(k) =n π

(40)

where n is a non-negative integer. δ(k), in (40), denotes the inﬁnite volume s-wave phase-shift, q ≡ kL 2π and π 3/2 q Z00 (1; q 2 ) 1 2 Z00 (s; q 2 ) = √ (n − q 2 )−s , 4π n∈Z 3 tan φ(q) = −

(41) (42)

2 2 n (free spectrum), if at least two pn and pn exist with 2. or k 2 = p2n = 2π L 2 pn = p2n which are not related by a cubic transformation (pn = R4 pn ). An 2π example of such vectors is provided by p = 2π L (3 0 0) and p = L (2 2 1). These states are non physical (non interacting) and their existence is a consequence of the angular momentum cut-oﬀ assumption. In the following we will refer to them as “spurious states”. As shown below, states of type 1 contain a non-zero s-wave component [8] in the expansion given in (37). Furthermore, this s-wave component is undistorted by the presence of the boundary, as compared to the inﬁnite volume s-wave function. As for the states of type 2, they cannot simply be plane-waves of the form eipn ·x , because of the presence of the interaction. However the combination φ(x) = eipn ·x − eipn ·x is a solution of the Schr¨ odinger equation even in the presence of the potential, because φ(x) does not contain any s-wave component and all other angular momenta are not interacting [8]. The spurious states φ(x) have a non-zero cubically-invariant projection, given by R4 (eipn ·x − eipn ·x ) , (43) R4

and do not project on s-wave. We will now proceed to prove (40), both in Quantum Mechanics and in RQFT.

Strong and Weak Interactions in a Finite Volume

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3.1 Quantum Mechanics We start considering (38) in an inﬁnite volume and transform it into an integral equation [6] 1 eik·(x−y) ˜ ik·x Ψk (x) = e U (|y|)P(s−wave) Ψk (y) − (44) dy 4π |x − y| 1 M(p) eip·x , = eik·x + (45) dp 2 (2π)3 k − p2 + iε

where M(p) =

˜ (|y|)P(s−wave) Ψk (y)e−ip·y . dy U

(46)

The disconnected contribution eik·x in (44) and (45) is allowed because it is a solution of the free (U = 0) Schr¨ odinger equation (38). For large |x| the integral over p in (45) is dominated by the singular region p ≈ k. On the other hand, for large |x| we know that [6] Ψk (x)

≈

|x|→∞

eik·x +

1 i2δ(k) eikr (e − 1) 2i kr

(47)

so that we can identify3 M(k) =

4π i2δ(k) (e − 1) . ik

(48)

Projecting over the s-wave, which by assumption is the only interacting one, we get dp M(p) sin pr (e2iδ(k) + 1) sin kr Ψk |s−wave (r) = +P , (49) 2 kr (2π)3 k 2 − p2 pr where we used the identity 1 1 = iπ δ(x) + P . x − iε x

(50)

We now repeat the same steps for a system enclosed inside a ﬁnite volume V , with periodic boundary conditions. In this case (45) is replaced by ˜ (|y|)P(s−wave) Ψ V (y) ΨkV (x) = dyGV (x − y)U k V

1 MV (pn ) ipn ·x = e V p k 2 − p2n

(51)

n

where ΨkV (x) denotes the ﬁnite volume eigenfunction subject to periodic boundary conditions. In (51), MV (pn ) is deﬁned by 3

By assumption, only the s-wave shift, δ(k) is diﬀerent from zero.

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M. Testa

MV (p) =

˜ (|y|)P(s−wave) Ψ V (y)e−ip·y . dy U k

(52)

V 2iδ(k)

We notice that the term (e 2 +1) sinkrkr in (49), which comes from the disconnected contribution and the δ-function of (50), is not present in (51). In fact disconnected contributions are not allowed in (51) because, although solution of the free Scr¨ odinger diﬀerential equation, they do not satisfy the periodic boundary conditions, due to k = pn . On the other hand, the δ-function contribution comes from the iε in (50), and therefore does not appear in the ﬁnite volume Green function (23). As we will show in a moment, both these terms are brought back through the summation formula (31). By construction, both Ψk (x) and ΨkV (x) satisfy (38). The diﬀerent boundary conditions result in a diﬀerent angular momentum content. However both solutions, when projected, say on the s-wave, should give rise to the same function of r, up to a normalization constant. In formulas we have (53) ΨkV s−wave (r) ∝ Ψk |s−wave (r) where, from (51), 1 MV (pn ) sin pn r . ΨkV s−wave (r) = V p k 2 − p2n pn r

(54)

n

For short range potentials, entirely contained inside V , we have, from (46) and (52), that (55) MV (p) ∝ M(p) . Both M(p) and MV (p) depend on p = |p|, because, as seen from (46) and (52), they are Fourier transforms of rotationally invariant functions (with support contained inside V ). The proportionality constant in (53) and (55) depends on the relative normalization of ΨkV (y) and Ψk (y): its value will be determined later. Equations (46) and (52) also show that MV (pn ) is the Fourier transform of a function which vanishes identically outside a volume. We are therefore allowed to use the summation formula (31), which in this case is exact, and not only exponentially exact, provided the interaction range is contained inside V . Equation (31) then gives 1 M(pn ) sin pn r (56) ΨkV s−wave (r) = V p k 2 − p2n pn r n dp M(p) sin pr sin kr =P + cM(k) , (57) 3 2 2 (2π) k − p pr kr for a particular arbitrary choice of the absolute normalization constant. Equations (30), (48) and (53) show that, adjusting the value of k, we can make (57) identical to (49). The values of k for which this happens are easily checked to be those identiﬁed by (40). In particular this derivation shows that the disconnected terms, together with the ones coming from the iε in (50) are reconstructed, as V grows, by the c contribution in (31).

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3.2 Field Theory The validity of (40) for quantized levels with energies below the inelastic threshold in RQFT, has been discussed in [7]. In this section we describe a diﬀerent approach, very close to the one used in Sect. 3.1, which also helps to clarify the size of the corrections to (40) due to the presence of an inelastic threshold, Ein [8], [9]. The concept of wave function in RQFT is an approximate one. The object closest to a wave function is the Bethe-Salpeter (BS) wave function. In inﬁnite volume, for an incoming state with zero total momentum, the BS wave function is deﬁned as Φk (x) = 0|T (φ(x)φ(0))|k, −kin

(58)

where φ(x) is an appropriately normalized pion ﬁeld. In the following we will consider the t = 0 BS wave function4 Φk (x, 0) = 0|φ(x, 0)|nn|φ(0)|k, −kin , (59) n

In (59) we separate the contribution of single pion states, Ψk (x) and that of multi-pion states, Ik (x), as Φk (x) = Ψk (x) + Ik (x) , where

(60)

d3 p p| φ(0) |k, −kin eip·x (2π)3 2Ep 0|φ(0)|3π3π|φ(0)|k, −kin eip3π ·x + · · · Ik (x) =

Ψk (x) =

(61) (62)

{3π}

The sum in (62) runs over three pion states and the ellipses represent the contribution from states with a higher number of particles. We can further separate, in Ψk (x), the connected part Ψk (x) = eik·x + d3 p conn + p| φ(0) |k, −kin eip·x (2π)3 2Ep

(63)

and parametrize p|φ(0)|k, −kconn in , according to reduction formulas, as = p|φ(0)|k, −kconn in

M 1 , 4Ek Ep − Ek − iε

(64)

where the oﬀ-shell scattering amplitude, M, becomes the physical one5 , M(k → p), when p = k, and is parametrized as 4 5

The extension to a generic time t = 0 is straightforward. As before we keep the interaction only for the s-wave.

188

M. Testa

M(k → p)|p=k =

4π 2Ek i2δ(k) (e − 1) , i k

(65)

for energies below the inelastic threshold, Ein ≡ 4mπ . We have Ψk (x) = eik·x M d3 p eip·x + (2π)3 2Ep 4Ek (Ep − Ek − iε) sin kr ik = eik·x + M(k → p)|p=k 16πEk kr d3 p M +P eip·x , (2π)3 2Ep 4Ek (Ep − Ek )

(66)

where use has been made of (50). As in Sect. 3.1, the relative momenta k allowed in a box are determined by the condition that the s-wave projection of the ﬁnite volume BS wave function, ΦVk |s−wave (r), is not deformed by the presence of the boundary. ΦVk (x) is given by an expression similar to (59), where the sum over intermediate states runs over ﬁnite volume energy eigenstates. The s-wave projection of Φk (x) has an expression similar to (49) ik sin kr Φk |s−wave (r) = (1 + M(k → p)|p=k ) 16πEk kr d3 p (Ep + Ek )M sin pr + Ik |s−wave (r) +P (2π)3 2Ep 4Ek (p2 − k 2 ) pr i2δ(k) e + 1 sin kr = 2 kr (Ep + Ek )M sin pr d3 p + Ik |s−wave (r) , +P (2π)3 2Ep 4Ek (p2 − k 2 ) pr

(67)

where we used the elastic amplitude parametrization given in (65). In a similar way the s-wave projection of ΦVk (x) is given by ΦVk |s−wave (r) ≡ 1 1 (Epn + Ek )M sin pn r + IkV |s−wave (r) . V 2Epn 4Ek (p2n − k 2 ) pn r

(68)

{pn }

The sum {pn } in (68) is over the single particle momenta deﬁned in (7). The quantization condition will again choose those values of k for which (67) and (68) become identical. We stress once more that in the ﬁnite volume expression (68), disconnected terms and iε’s are missing, because the eigenmomenta k do not coincide with any of the pn . These terms will appear as V grows, as already discussed in Sect. 3.1. In order to implement the quantization condition, we must discuss how fast ﬁnite volume sums converge to inﬁnite volume integrals. As outlined in Sect. 2,

Strong and Weak Interactions in a Finite Volume

189

the rate of convergence depends from the nature of the singularities of the summand functions and of their derivatives. We start with IkV |s−wave (r). According to (62), Ik (x) is a Fourier transform Ik (x) = dp ρk (p)eip·x , (69) where ρk (p) ≡

0|φ(0)|3π3π|φ(0)|k, −kin δ(p3π − p) ;

(70)

{3π}

its asymptotic behavior, according to the Riemann-Lebesgue lemma [10], is determined by the diﬀerentiability properties of ρk (p). The matrix element 3π|φ(0)|k, −kin , in (70), has a pole in the q 2 variable of the φ(0) channel, when the virtual pion described by φ(0) reaches the mass-shell. This pole is present both when 2Ek is under or above the inelastic threshold. In fact, when 2Ek < Ein , the residue of such a pole describes the allowed process ππ +π → πππ, while, when 2Ek > Ein , it also gets a contribution from the 3456 2Ek

process 3456 ππ → ππππ. The three particle phase space integration, understood 2Ek

in the sum over intermediate states (70), softens these pole singularities and transforms them into a cut. Therefore ρk (p) is ﬁnite for any value of p, but, in general, not inﬁnitely diﬀerentiable. In order to decide its inﬂuence on the asymptotic behavior of ρk (p), we must identify the position of the tip of this cut: if it lies on the real integration axis, Ik (x) will decrease as a power of 1r , otherwise its decrease will be exponential. The analysis, presented in appendix A, shows that, for 2mπ ≤ 2Ek ≤ 4mπ , the tip of the cut does not reach the real integration path, and therefore, within this range of energies, Ik (x) decreases exponentially. It must be recalled at this point that in Ψk (x), (63), there is a contribution, analogous to the one discussed above, corresponding to a crossed singularity in which φ(0) creates a three pion state. A similar analysis can be performed with the same conclusions. Since Ik (x), and therefore Ik |s−wave (r), is exponentially decreasing, when 2Ek < Ein , we conclude, through an argument similar to the one given in Sect. 2, that the approach of IkV |s−wave (r) to Ik |s−wave (r) is exponentially fast in V , for this range of energy levels. In order to get the eigenvalue condition we must now discuss the volume dependence of the ﬁrst term in (68). Here the strategy will follow closely the one adopted in Sect. 3.1. As for Ik (x), it is easy to show [8] that also M is the Fourier transform of an exponentially vanishing function in x space. We can therefore transform the sum in (68) into an integral, using (31), which, as discussed in Sect. 2, is valid up to exponentially small corrections in the volume

190

M. Testa

1 1 (Epn + Ek )M sin pn r V 2Epn 4Ek (p2n − k 2 ) pn r {pn } +∞ 1 dp (Ep + Ek )M sin pr M sin kr = +c . P 3 2 2 (2π) 4Ek kr −∞ 2Ep 4Ek (k − p ) pr

(71)

(72)

Equations (68) and (67) then give at once the L¨ uscher quantization condition (40), with exponential precision in V . The numerical size of the corrections is determined by the precision with which the sum in (68) is able to reproduce the corresponding integral. In particular, a possible source of concern is the (practically relevant) situation in which the quantization volume allows only very few (perhaps two or three) elastic states under the inelastic threshold. In this case, even though the ﬁnite-volume eﬀects are exponentially small, one is working in a ﬁxed volume and, in order to establish that the corrections are indeed negligible, one needs an estimate of such eﬀects. As it should be clear from the previous discussion, there are essentially two types of exponentially small corrections. The ﬁrst type is analogous to the corrections which would be present, even in Quantum Mechanics, in the presence of an exponentially decreasing potential, rather than a ﬁnite range one. The second type of corrections, characteristic of RQFT, are due to the existence of inelasticity and are responsible for the failure of (40) above Ein . As discussed above they come from a singularity of ρk (p) which is complex for 2Ek < Ein . When 2Ek approaches Ein this singularity is closer to the real axis, and the range of the exponential increases, so that the rate with which the ﬁnite volume expressions approach the inﬁnite volume ones, although still exponential, gets smaller and smaller. When 2Ek > Ein the volume corrections become power-like, rather than exponential: in this way the quantization condition is, in general, aﬀected by the presence of inelasticity. In the case of pions there are indications that inelasticity is negligible up to around 1 Gev [11]. This fact should shift the singularity further from the real axis, and allow the L¨ uscher quantization formula to be applicable also to situations in which there are only few states under thew inelastic threshold, without sizeable corrections.

4 Matrix Elements of Scalar Operators As discussed in Sects. 3.1 and 3.2, cubically invariant states with total energy below Ein belong generically to two classes. Referring to (37) we have states which – contain an s-wave component and are classiﬁed according to the L¨ uscher quantization condition (40): these are the “physical” states, – do not contain an s-wave component. These states are called “spurious” because they originate from the simplifying assumption that only s-waves are interacting. In the following we will be interested in relating the matrix elements of a local scalar operator, say σ(x), computed in a ﬁnite volume V , 0| σ(0) |ππ, nV , to

Strong and Weak Interactions in a Finite Volume

191

the corresponding matrix element, 0| σ(0) |ππ, En , computed in inﬁnite volume at the same energy. Since “spurious” states do not contain an s-wave component, we have 0| σ(0) |ππ, n; SpuriousV = 0 ,

(73)

and 0| σ(0) |ππ, nV = 0 only for the physical states, classiﬁed according to (40). In Sect. 3.2 we have shown that physical states are characterized by the proportionality of Φk |s−wave (x) and ΦVk |s−wave (x)6 : Φk |s−wave (x) = α(En )ΦVk |s−wave (x) , (74) where α(En ) is a coeﬃcient which takes into account the diﬀerent normalizations of ﬁnite and inﬁnite volume states. Given the s-wave projection of the BS wave function, it is possible, through operations of diﬀerentiation and taking x → 0, while renormalizing at same time, build the matrix elements of any σ(0) between the vacuum and a two pion state. From (74) we have (75) σ(En ) ≡ 0| σ(0) |ππ, En = α(En ) 0| σ(0) |ππ, En V . Equation (75) is valid with exponential accuracy for the two pion states at rest, under Ein . In the next sections we will show how to compute the (operator independent) coeﬃcient α(En ).

5 The Nature of the LL Relation In order to ﬁnd the relative normalization of the states at ﬁnite and inﬁnite volume, we start by considering the correlator d3 x σ(x, t)σ(0)V −→ V →∞

V

(2π)3 2(2π)6

dp1 dp2 δ(p1 + p2 )e−(ω1 +ω2 )t 2ω1 2ω2

×|0|σ(0)|p1 , p2 |2 1 2 = dEe−Et |0| σ(0) |ππ, E| 2(2π)3 dp1 dp2 × δ(p1 + p2 )δ(E − ω1 − ω2 ) 2ω1 2ω2 ∞ dE −Et π 2 e = |0| σ(0) |ππ, E| k(E) . 2(2π)3 E¯ E

(76)

On the other hand we could proceed diﬀerently 6

We include again the time dependence as an argument of the BS wave function.

192

M. Testa

d3 x σ(x, t)σ(0) = V

|0| σ(0) |ππ, nV | e−En t 2

n

V

∞

dEρV (E) |0| σ(0) |ππ, EV | e−Et , 2

−→ V

V →∞

(77)

¯ E

¯ is the where, due to (73), only physical states contribute. In (76) and (77) E energy threshold and ρV (E) denotes a function to be determined, which provides the correspondence between ﬁnite volume sums and inﬁnite volume integrals. In many cases, for example in one dimension, ρV (E) can be identiﬁed as the density of states at energy E. In Sect. 6 we show that, also in the presence of cubic boundary conditions, ρV (E) is given by ρV (E) =

qφ (q) + kδ (k) dn = E, dE 4πk 2

(78)

with exponential precision in the volume. The expression in (78) is the one which would be heuristically derived interpreting ρV (E) as the density of states, as seen from (40). Comparing (76) and (77) we get the correspondence V EρV (E) |ππ, EV . (79) |ππ, E ⇔ 4π k(E) In a similar way one gets |K, p = 0 ⇔

√

2mV |K, p = 0V ,

(80)

from which it is easy to recover the LL relation, (1), without any restriction on the four-momentum transfer. Although the present approach appears superﬁcially to be equivalent to the one of [5], there is an important diﬀerence in the two derivations. The result of [5] was obtained at a ﬁxed value of n and therefore at a ﬁxed volume V , tuned so that mK = En , with n < 8. We, on the other hand, derived the same result at ﬁxed energy E, for asymptotically large volumes V . This implies that, as V → ∞, we must simultaneously allow n → ∞. A question which arises naturally at this point is what is the relation between the two approaches? The answer requires a more detailed discussion, developed in the following section, where it will be shown that the constraints of locality allow us to establish (78) with exponential accuracy for elastic states under the inelastic threshold.

6 Summation Theorems, Locality and the LL Formula Locality has been an important ingredient in establishing (75), through the summation formula, (31). In this section we will discuss another important rˆ ole of locality in the actual computation of the coeﬃcient α(En ).

Strong and Weak Interactions in a Finite Volume

193

Our approach to the LL formula, outlined in the previous section, is based on the identiﬁcation

2 −En t

|0| σ(0) |ππ, nV | e

n

∞

dE | σ(E)| e−Et 2

≈

(81)

¯ E

of ﬁnite and inﬁnite volume correlators. We are therefore naturally led to the question of how well (81) is satisﬁed. Since σ(x) is a local operator, σ(0) |0 is a localized state which does not diﬀer much from the vacuum state, away from 0. This is a consequence of clustering, which guarantees that, in the absence of massless particles, the probability of ﬁnding particles at a distance r away from the origin in the state σ(0) |0 decreases exponentially like e−2mπ r . As a consequence, if V is greater that the localization volume, we can write, at t = 0, d3 xσ(x, 0)σ(0)V = V |0|σ(0)|ππ, nV |2 V

n ∞

=

dE|σ(E)|2 =

d3 xσ(x, 0)σ(0)

(82)

¯ E

with exponential accuracy in V . The second and third equalities in (82) are obtained by inserting the complete set of energy eigenstates on a ﬁnite and inﬁnite volume respectively. In view of clustering the result is the same apart from the exponentially small perturbation at the boundary. Equations (82) and (74) express the important property that the matrix elements of a local operator are smooth in energy, in the sense that the sum over discrete energy levels reproduces, with exponential accuracy, the corresponding integral ∞ |σ(En )|2 . (83) dE|σ(E)|2 = V α(En ) ¯ E n Introduction of time dependence does not substantially change this result ([8], [9]) and we also have ∞ |σ(En )|2 e−En t dE| σ(E)|2 e−Et = V (84) α(E ) ¯ n E n with exponential precision in V . The basic tool for relating integrals to sums is provided by the Poisson identity [10] +∞ +∞ δ(n − x) = e2πilx . (85) n=−∞

l=−∞

Equation (85), together with the substitution x → h(E, L), where h(E, L) is deﬁned in (40), gives

194

M. Testa

∞

dE |σ(E)| e−Et = 2

E

|σ(En )|2 e−En t − Q(L, t) , ∂h(E,L) n ∂E

(86)

En

where Q(L, t) ≡

l=0

∞

dE|σ(E)|2 e−Et e2ilδ(k) e2ilφ(q) .

(87)

¯ E

Let us consider the large L behavior of Q(L, t). The techniques of asymptotic analysis [12], together with the fact that φ(q) → ∞ when L → ∞ at ﬁxed k, suggest that the large L behavior of each term of the sum over l in (87) is dominated by the critical points of φ(q), i.e. the points at which φ(q) has a vanishing derivative or the points where φ(q) and σ(E) are not continuous or diﬀerentiable. q = 0 is a critical point of φ(q) as follows from φ(q) ≈ 2π 2 q 3 q≈0

(88)

so that, in absence of further critical points we expect that Q(L, t) −→ 0 L→∞

(89)

as a power of L1 depending on the threshold behavior of σ(E) which, for local observables, may be make arbitrarily small [8]. A comparison between (84) and (86) then leads to the identiﬁcation ∂h(E, L) (90) α(En ) ⇔ V ∂E En and gives the LL relation with exponential accuracy, for energies between the elastic and inelastic thresholds. Although a rigorous mathematical proof that q = 0 is the only critical point of φ(q) in the complex q-plane is lacking, in Appendix B we provide a simple, indirect argument which strongly points towards this conclusion.

7 Conclusions We have reviewed ﬁnite-volume eﬀects in matrix elements of local operators, examining the conditions under which the LL formula is valid. We have also presented an alternative simple derivation of the L¨ uscher quantization condition in Quantum Field Theory, which allows to examine the general consequences of inelasticity on the quantization formula.

Acknowledgements This work was supported by European Union grant HTRN-CT-2000-00145. I thank the organizers of LHP2001 for the hospitality in Cairns and the Institute for Nuclear Theory for the hospitality at the University of Washington,

Strong and Weak Interactions in a Finite Volume

195

where part of this work was done. I also thank the Department of Energy for partial support during the completion of this work. The material presented in these lectures was developed in collaboration with D. Lin, G. Martinelli and C. Sachrajda to whom goes my gratitude for countless discussions. Many thanks for discussions are also due to L. Lellouch, M. L¨ uscher and G. C. Rossi.

Appendix A The tip of the cut in the p integration in (69) is determined by the minimum value of (91) p23π ≡ (p1 + p2 + p3 )2 , subject to the constraint that the virtual pion, described by φ(0) in (70), is on-shell (92) (E1 + E1 + E1 − Eππ )2 − (p1 + p2 + p3 )2 − m2π = 0 . In (92) we denoted 2Ek by Eππ . This problem is easily solved by the Lagrange multiplier method which amounts to minimize the unconstrained expression F (p1 , p1 , p1 ) ≡ (p1 + p2 + p3 )2 +λ[(E1 + E1 + E1 − Eππ )2 − (p1 + p2 + p3 )2 − m2π ] , (93) where λ is the Lagrange multiplier. We have, for example, ∂F = 2(p1 x + p2 x + p3 x ) ∂p1 x +λ[2(E1 + E2 + E3 − Eππ )

p1 x − 2(p1 x + p2 x + p3 x )] = 0 E1

(94)

and other similar relations. From (94) we easily get p1 = p2 = p3 = p ,

(95)

where p is determined by (92), which gives E≡

mπ + p 2 =

4 m2π Eππ . + 3 Eππ 6

(96)

It follows from (96) that p is real, and therefore the tip of the cut reaches the integration path, only when Eππ > 4mπ .

Appendix B In this appendix we present simple argument, based on a summation theorem, which strongly suggests that the q = 0 critical point of φ(q), described in (88), is the only one in the complex q-plane.

196

M. Testa

We start applying (14) to the case of two functions f (p) and g(p) which only depend on p ≡ |p| ∞ 1 1 f (p)g ∗ (p)p2 dp = f (pn )g ∗ (pn ) 2 2π 0 V {pn }

1 = νn f (pn )g ∗ (pn ) , V

(97)

{pn }

where νn is the number of integer vectors with given |n|. The last sum in (97) is performed over the distinct values of pn 7 . On the other hand the pn ’s are the solutions of φ(q) = nπ (98) because, from (41), we have tan φ(q) = 0 ⇒ q = |n| ,

(99)

where n is any vector with integer components. We are now in a position to use the Poisson Identity (85). Putting x → φ(q)/π, taking into account that φ (|n|)| =

4π 2 2 n νn

(100)

and after multiplication by p2 /2π 2 f (p)g ∗ (p) and integration over p (85) becomes 2 1 f (0)g ∗ (0) + 3V V

νn f (pn )g ∗ (pn )

{pn =0}

1 1 f (0)g ∗ (0) + νn f (pn )g ∗ (pn ) 3V V {pn } 1 = f (p)g ∗ (p)dp + (2π)3 1 ∞ Lp p2 f (p)g ∗ (p)e2ilφ( 2π ) dp . + 2π 2 0

=−

(101)

l=0

Equation (101), together with (97) shows that 1 ∞ Lp p2 f (p)g ∗ (p)e2ilφ( 2π ) dp 2 2π 0 l=0 1 1 νn f (pn )g ∗ (pn ) − f (p)g ∗ (p)dp = V (2π)3 {pn }

1 f (0)g ∗ (0) − 3V 7

(102)

We remind the reader that (97) is valid up to exponentially small corrections for large volumes.

Strong and Weak Interactions in a Finite Volume

197

is indeed exponentially small with power corrections concentrated at threshold. This result is in complete agreement with the heuristic analysis presented after (87) and strongly suggests that the only critical point of φ(q) is at q = 0. In fact if other critical points of φ(q) were present in the complex q-plane, these would show up in (102), through a large L behavior diﬀerent from the one just found. This argument would be completely rigorous if one could show that the behaviour of the sum over l, in the l.h.s. of (102) is not better than the behaviour of the individual addends. In other words, one should be able to exclude the possibility that other hypothetical critical points of φ(q), diﬀerent from 0, could give contributions cancelling after the sum over l is performed. If the possibility of this cancellation is excluded, then each term of the sum over l in (87) would have the behaviour discussed in Sect. 6.

References 1. M. Bochicchio et al., Nucl. Phys. B 262 331 (1988); L. Maiani, G. Martinelli, G. C. Rossi and M. Testa, Phys. Lett. 176 B 445 1986 and Nucl. Phys. B 289 505 (1987). 177 2. C. Bernard, T. Draper, G. Hockney and A. Soni, Nucl. Phys. B. Proc. Suppl. 4 483 (1988). 3. C. Dawson et al., Nucl. Phys. B 514 313 (1998). 4. L. Maiani and M. Testa, Phys. Lett. B 245 585 (1990). 177 5. L. Lellouch and M. L¨ uscher, Commun. Math. Phys. 219 31 (2001). 177, 178, 192 6. L. Schiﬀ: Quantum Mechanics 3, 66 (1988). 179, 185 7. M. L¨ uscher: Nucl. Phys. B 354, 531 (1991) 179, 182, 183, 184, 187 8. C. -J. D. Lin, G. Martinelli, C. T. Sachrajda, M. Testa, hep-lat/0104006, Nucl. Phys. B 619, 467 (2001). 178, 184, 187, 189, 193, 194 9. C. -J. D. Lin, G. Martinelli, C. T. Sachrajda, M. Testa, hep-lat/0111033, Nucl. Phys. Suppl. 109 A, 218 (2002). 178, 183, 187, 193 10. M. J. Lighthill, Introduction to Fourier analysis and generalised functions (Cambridge Univ. Press 1958). 189, 193 11. See for example D. Morgan and M. R. Pennington, in The second Daphne Physics Handbook, edited by L Maiani, G. Pancheri and N. Paver, INFN-LNF. 190 12. N. Bleistein, R. A. Handelsman Asymptotic expansions of integrals (Dover Editions, 1986). 194

Hadron Properties with FLIC Fermions J.M. Zanotti1,2 , D.B. Leinweber1 , W. Melnitchouk3 , A.G. Williams1 , and J.B. Zhang1 1

2

3

Department of Physics and Mathematical Physics and Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, 5005, Australia John von Neumann-Institut f¨ ur Computing NIC, Deutsches ElektronenSynchrotron DESY, D-15738 Zeuthen, Germany Jeﬀerson Lab, 12000 Jeﬀerson Avenue, Newport News, VA 23606

Abstract. The Fat-Link Irrelevant Clover (FLIC) fermion action provides a new form of nonperturbative O(a)-improvement in lattice fermion actions oﬀering near continuum results at ﬁnite lattice spacing. It provides computationally inexpensive access to the light quark mass regime of QCD where chiral nonanalytic behaviour associated with Goldstone bosons is revealed. The motivation and formulation of FLIC fermions, its excellent scaling properties and its low-lying hadron mass phenomenology are presented.

1 Introduction The origin of the masses of light hadrons represents one of the most fundamental challenges to QCD. Despite the universal acceptance of QCD as the basis from which to derive hadronic properties, there has been slow progress in understanding the generation of hadron mass from ﬁrst principles. Solving the problem of the hadronic mass spectrum would allow considerable improvement in our understanding of the nonperturbative nature of QCD. The only available method at present to derive hadron masses directly from QCD is a numerical calculation on the lattice. The high computational cost required to perform accurate lattice calculations at small lattice spacings has led to increased interest in quark action improvement. In this article we present results of simulations of the spectrum of light mesons and baryons using an O(a) improved fermion action [1, 2, 3, 4]. In particular, we will start with the standard clover action and replace the links in the irrelevant operators with APE smeared [5, 6], or fat links. We shall refer to this action as the Fat-Link Irrelevant Clover (FLIC) action. Although the idea of using fat links only in the irrelevant operators of the fermion action was developed here independently, suggestions have appeared previously [7]. In Sect. 2, we provide the reader with some background information on lattice fermion actions. In particular, we start with the basic lattice discretisation of the derivative appearing in the continuum Dirac action, followed by improvements suggested ﬁrst by Wilson [8] and then by Sheikholeslami and Wolhert [9]. Section 3 contains the procedure for creating the FLIC fermion action while in Sect. 4 we describe the gauge conﬁgurations used in our lattice simulations. The results of an investigation of the scaling of this action at ﬁnite lattice spacing are J.M. Zanotti, D.B. Leinweber, W. Melnitchouk, A.G. Williams, and J.B. Zhang: Hadron Properties with FLIC Fermions, Lect. Notes Phys. 663, 199–225 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

200

J.M. Zanotti et al.

presented in Sect. 5. In Sect. 6 we investigate the problem of exceptional conﬁgurations by performing simulations of hadron masses at light quark masses corresponding to mπ /mρ = 0.35. Section 7 discusses the evidence for enhancement in octet-decuplet mass splittings as one approaches the chiral limit in the quenched approximation and ﬁnally in Sect. 8 we summarise the results.

2 The Lattice Quark Action 2.1 The Naive Fermion Action In Euclidean space-time, the continuum Dirac action is written as ψ¯ (D / + m) ψ ,

(1)

where the covariant derivative is deﬁned as Dµ = ∂µ + i g Aµ . Wilson [8] discretised the continuum Dirac action by replacing the derivative with a symmetrised ﬁnite diﬀerence. Gauge links are included to not only encode the gluon ﬁeld, Aµ , but to also maintain gauge invariance

1 ¯ ψ(x) γµ Uµ (x) ψ(x + aˆ µ) − Uµ† (x − aˆ µ) ψ(x − aˆ µ) . ψ¯ D /ψ = 2a µ

(2)

Our notation uses the Pauli (Sakurai) representation of the Dirac γ-matrices deﬁned in Appendix B of Sakurai [10]. In particular, the γ-matrices are hermitian, γµ = γµ† and σµν = [γµ , γν ]/(2i) such that γµ γν = δµν + i σµν . The gauge link variables at space-time position x are deﬁned as a Aµ (x + λˆ µ) dλ . (3) Uµ (x) = P exp i g 0

Here the operator P path-orders the Aµ ’s along the integration path, a is the lattice spacing, and g is the coupling constant. The continuum Dirac action is recovered in the limit a → 0 by Taylor exµ) in powers of the lattice spacing a. Keeping only panding the Uµ and ψ(x + aˆ the leading term in a (and for ease of notation we write aˆ µ→µ ˆ), (2) becomes # # $ 1 ¯ µ ˆ$ + . . . (ψ(x) + a ψ (x) + . . .) ψ(x) γµ 1 + i a g Aµ x + 2a 2 # $ $ # µ ˆ$ + . . . (ψ(x) − a ψ (x) + . . . − 1 − i a g Aµ x − 2

¯ γµ (∂µ + O(a2 ))ψ(x) + i g ψ(x) ¯ γµ Aµ + O(a2 ) ψ(x), = ψ(x)

(4)

which is the kinetic part of the standard continuum Dirac action in Euclidean space-time to O(a2 ). Hence we arrive at the simplest (“naive”) lattice fermion action,

Hadron Properties with FLIC Fermions

S N = mq

201

¯ ψ(x) ψ(x)

x

1 ¯ ˆ) − Uµ† (x − µ ˆ) ψ(x − µ ˆ) ψ(x) γµ Uµ (x) ψ(x + µ 2a x,µ ¯ M N [U ] ψ(y) , ≡ ψ(x) xy

+

(5)

x

where the interaction matrix M N is 1 † N [U ] = mq δij + [γµ Ui,µ δi,j−µ − γµ Ui−µ,µ δi,j+µ ] . Mi,j 2a µ

(6)

The Taylor expansion in (4) shows that the naive fermion action of (5) has O(a2 ) errors. It also preserves chiral symmetry. However, in the continuum limit it gives rise to 2d = 16 ﬂavours of quark rather than one. This is the famous doubling problem and is easily demonstrated by considering the inverse of the free ﬁeld propagator (obtained by taking the fourier transform of the action with all Uµ = 1) i S −1 (p) = mq + γµ sin pµ a , (7) a µ which has 16 zeros within the Brillouin cell in the limit mq → 0. e.g., pµ = (0, 0, 0, 0), (π/a, 0, 0, 0), (π/a, π/a, 0, 0), etc. Consequently, this action is phenomenologically not acceptable. There are two approaches commonly used to remove these doublers. The ﬁrst involves adding operators to the quark action which scale with the lattice spacing and thus vanish in the continuum limit. These operators are chosen to drive the doublers to high energies and hence are suppressed. This technique for improving fermion actions proceeds via the improvement scheme proposed by Symanzik [11] and is discussed in more detail in the following sections. The second method for removing doublers involves “staggering” the quark degrees of freedom on the lattice. This procedure exploits the fact that the naive fermion action has a much larger symmetry group, UV (4)⊗UA (4), to reduce the doubling problem from 2d = 16 → 16/4 while maintaining a remnant chiral symmetry. This approach is not used in this article so the details of the action will not be discussed here. Details of the derivation of staggered fermions can be found in most texts (e.g. [12, 13]). 2.2 Wilson Fermions In order to avoid the doubling problem, Wilson [8] originally introduced an irrelevant (energy) dimension-ﬁve operator (the “Wilson term”) to the standard naive lattice fermion action (5), which explicitly breaks chiral symmetry at O(a).

1 ¯ SW = ψ(x) γµ ∇µ − r a ∆µ + m ψ(x) , (8) 2 µ

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where r is the “Wilson coeﬃcient,” ∇µ ψ(x) =

1 [Uµ (x) ψ(x + µ ˆ) − Uµ† (x − µ ˆ) ψ(x − µ ˆ)] , 2a

(9)

and

1 [Uµ (x) ψ(x + µ ˆ) + Uµ† (x − µ ˆ) ψ(x − µ ˆ) − 2 ψ(x)] . (10) a2 The Wilson action in (8) has no doublers for r > 0 as the Wilson term gives the extra ﬁfteen species at pµ = π/a a mass proportional to r/a. Also, if r = 1 then the Wilson action has no ghost branches in its dispersion relation (see, for example [14]). In terms of link variables, Uµ (x), the Wilson action can be written

4r ¯ 1 ¯

ˆ) S W = mq + ψ(x) ψ(x) + ψ(x) (γµ − r) Uµ (x) ψ(x + µ a 2a x,µ x − (γµ + r) Uµ† (x − µ ˆ) ψ(x − µ ˆ) , (11) W L ψy , (12) ψ¯xL Mxy ≡ ∆µ ψ(x) =

x,y

where the interaction matrix for the Wilson action, M W , is usually written † W (r − γµ ) Ux,µ δx,y−µ + (r + γµ ) Ux−µ,µ Mxy [U ] a = δxy − κ δx,y+µ , (13) µ

with a ﬁeld renormalisation κ = 1/(2 mq a + 8 r) , √ ψ L = ψ/ 2 κ . We take the standard value r = 1 and the quark mass is given by

1 1 1 mq a = − . 2 κ κc

(14)

(15)

In the free theory the critical value of kappa, κc , where the quark mass vanishes, is 1/8r. In the interacting theory, κc is renormalised away from 1/8r. The quark mass has both multiplicative and additive renormalisations due to the explicit breaking of chiral symmetry by the Wilson term. In the continuum limit, one ﬁnds ) * ar 2 4 ¯ /+m− D ψ(x) + O(a) . (16) SW = d x ψ(x) D 2 µ µ By lifting the mass of the unwanted doublers with a second derivative, O(a) discretisation errors have been introduced into the fermion matrix. In contrast, the Wilson gauge action [8] has only O(a2 ) errors. Hence there is enormous interest in applying Symanzik’s improvement program [11] to the fermion action to remove O(a) errors by adding additional higher dimension terms.

Hadron Properties with FLIC Fermions

203

2.3 Improving the Fermion Action The addition of the Wilson term to the fermion action introduces large O(a) errors which means that in order to extrapolate reliably to the continuum limit, simulations must be performed on ﬁne lattices, which are therefore very computationally expensive. The scaling properties of the Wilson action at ﬁnite a can be improved by introducing any number of irrelevant operators of increasing dimension whose contributions vanish in the continuum limit. The ﬁrst attempt at removing these O(a) errors was by Hamber and Wu [15] who added a two link term to the Wilson action

1 1 ˆ) ψ L (x + 2ˆ µ) SHW = SW + κ ψ¯L (x) − r + γµ Uµ (x) Uµ (x + µ 4 8 x,µ

1 1 + ψ¯L (x + 2ˆ µ) − r − γµ Uµ† (x + µ ˆ) Uµ† (x) ψ L (x) . (17) 4 8 The removal of the classical O(a) terms is easily observed through a Taylor expansion. While this action also removes O(a2 ) errors at tree-level, it has only received a small amount of interest due to the computational expense in evaluating the double hopping term. Calculations that have been done with this action show that it works well at coarse lattice spacings and has the added bonus that it has an improved dispersion relation [16]. A more systematic approach [11] to O(a) improvement of the lattice fermion action in general [9] is to consider all possible gauge invariant, local dimensionﬁve operators, respecting the symmetries of QCD i g a CSW r ¯ ψ σµν Fµν ψ , . 4 ← − ← − / O2 = c2 a ψ¯ Dµ Dµ ψ + ψ¯ D µ D µ ψ , O1 = −

bg a mq tr {Fµν Fµν } , 2 . / ← − O4 = c4 mq ψ¯ γµ Dµ ψ − ψ¯ D µ γµ ψ , O3 =

(18)

O5 = −bm a m2q ψ¯ ψ . Note that the operator D /D / is linearly related to O1 and O2 as D /D / = Dµ Dµ −

g σµν Fµν . 2

(19)

Operator O1 is a new local operator in the lattice fermion action and must be included. On the other hand, O3 and O5 of (18) act to simply renormalise the coeﬃcients of existing terms in the lattice action, removing O(a mq ) terms from the relation between bare and renormalised quantities. For example, the renormalisation of the quark mass mq → mq (1 − bm a mq ) incorporates O5 . Similarly, O3 introduces a mass dependence in the gauge coupling g 2 → g 2

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(1 − bg a mq ) such that the lattice spacing remains constant for constant g as mq is varied [17]. For the quenched approximation, bg = 0. The key observation to eﬃcient O(a) improvement is that the O(a) improvement aﬀorded by two-link terms of the fermion action [18] may be incorporated to O(a) into the standard Wilson fermion action complemented by O1 though the following transformation of the fermion ﬁelds /) ψ , ψ → ψ = (1 + bq r a mq ) (1 − cq r a D ← − ¯ ¯ ¯ ψ → ψ = (1 + bq r a mq ) ψ (1 + cq r a D /) ,

(20)

where ψ represents the physical fermion ﬁeld recovered in the continuum limit, while ψ is the lattice fermion ﬁeld used in the numerical simulations. The Jacobian of the transformation is 1 to O(a) [9]. At tree-level, bq = cq = 1/4 correctly incorporates the O(a) corrections of O2 and O4 into the fermion action. Note that this ﬁeld transformation renormalises the fermion mass of the simulations

1 (21) m → m 1 + ram . 2 For the spectral quantities investigated herein, it is suﬃcient to work with the lattice fermion-ﬁeld operators alone. Here the fermion operators, act only as interpolators between the QCD vacuum and the state of interest and do not aﬀect the eigenstates of the QCD Hamiltonian. However, for matrix elements of fermion operators, hadron decay constants, or oﬀ-shell quantities such as the quark propagator, it is important to take this ﬁeld redeﬁnition into account to connect the lattice ﬁeld operators to the continuum ﬁeld operators incorporating important O(a) contributions. In summary, O1 , the “clover” term, is the only dimension-ﬁve operator explicitly required to complement the Wilson action to obtain O(a) improvement. This particular action is known as the Sheikholeslami-Wohlert fermion [9] action SSW = SW −

i g a CSW r ¯ ψ(x) σµν Fµν ψ(x) , 4

(22)

where CSW is the clover coeﬃcient which can be tuned to remove O(a) artifacts to all orders in the gauge coupling constant g. 1 at tree-level , CSW = (23) 1/u30 mean-ﬁeld improved , with u0 the tadpole improvement factor correcting for the quantum renormalisation of the operators (see deﬁnition in Sect. 4). Nonperturbative (NP) O(a) improvement [19] uses the axial Ward identity to tune CSW and remove all O(a) artifacts provided one simultaneously improves the coupling g 2 , the quark mass mq , and the currents [19]. The advantage of the clover action is that it is local and is only a small overhead on Wilson fermion simulations. Further details of the improvement the clover action provides at ﬁnite lattice spacing is given in Sect. 5.

Hadron Properties with FLIC Fermions

205

Uµ (x) (1) Oµν(x)

x

=

ν µ Fig. 1. Loops required to construct Fµν

The name “clover” is associated with the SW fermion action due to the lattice discretisation of the ﬁeld strength tensor, Fµν . An expression for Fµν is obtained by considering the sum of the four 1 × 1 link paths surrounding any lattice site in the µ − ν plane as shown in Fig. 1. Using the expansion for the elementary link product (see for example [12, 13, 20]), we obtain the lattice expression for Fµν $ 1 # (1) (1)† Oµν (x) − Oµν g a Fµν (x) = (x) 8i $ 1 # (1) (1)† − Tr Oµν (x) − Oµν (x) , 3 2

(24)

where Fµν is made traceless by subtracting 1/3 of the trace from each diagonal element and (1) (x) = Uµ (x) Uν (x + µ ˆ) Uµ† (x + νˆ) Uν† (x) Oµν

ˆ) Uν† (x − µ ˆ) Uµ (x − µ ˆ) + Uν (x) Uµ† (x + νˆ − µ + Uµ† (x − µ ˆ) Uν† (x − µ ˆ − νˆ) Uµ (x − µ ˆ − νˆ) Uν (x − νˆ) ˆ − νˆ) Uµ† (x) . + Uν† (x − νˆ) Uµ (x − νˆ) Uν (x + µ

(25)

Substantial progress has been made to improve Fµν to O(a6 ) by adding terms constructed using larger loops [20].

3 Fat-Link Irrelevant Fermion Action The established approach to nonperturbative (NP) improvement [19] tunes the coeﬃcient of the clover operator to all powers in g 2 . Unfortunately, this formulation of the clover action is susceptible to the problem of exceptional conﬁgurations as the quark mass becomes small. Chiral symmetry breaking in the clover fermion action introduces an additive mass renormalisation into the Dirac

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operator that can give rise to singularities in quark propagators at small quark masses. In practice, this prevents the simulation of small quark masses and the use of coarse lattices (β < 5.7 ∼ a > 0.18 fm) [21, 22]. Furthermore, the plaquette version of Fµν , which is commonly used in (22), has large O(a2 ) errors, which can lead to errors of the order of 10% in the topological charge even on very smooth conﬁgurations [23]. The idea of using fat links in fermion actions was ﬁrst explored by the MIT group [24] and more recently has been studied by DeGrand et al. [21, 25, 26], who showed that the exceptional conﬁguration problem can be overcome by using a fat-link (FL) clover action. Moreover, the renormalisation of the coeﬃcients of action improvement terms is small. In principle it is acceptable to smear the links of the relevant operators. The symmetry of the APE smearing process ensures that eﬀects are O(a2 ). The factors multiplying the link and staple ensure the leading order term is eiagAµ , an element of SU (3). Issues of projecting the smeared links to SU (3) are O(a2 ) eﬀects and therefore correspond to irrelevant operators [27]. However, the net eﬀect of APE smearing the links of the relevant operators is to remove gluon interactions at the scale of the cutoﬀ. While this has some tremendous beneﬁts, the short-distance quark interactions are lost. As a result decay constants and vector-pseudoscalar mass splittings of heavy mesons, which are sensitive to the wave function at the origin, are suppressed [28]. The solution to this is to work with two sets of links in the fermion action. In the relevant dimension-four operators, one works with the untouched links generated via Monte Carlo methods, while the smeared fat links are introduced only in the higher dimension irrelevant operators. The eﬀect this has on decay constants and vector-pseudoscalar mass splittings of heavy mesons is under investigation and will be reported elsewhere. Fat links [21, 25] are created by averaging or smearing links on the lattice with their nearest neighbours in a gauge covariant manner (APE smearing). The smearing procedure [5, 6] replaces a link, Uµ (x), with a sum of the link and α times its staples Uµ (x) → Uµ (x) = (1 − α) Uµ (x) +

4 α Uν (x) Uµ (x + νˆ) Uν† (x + µ ˆ) 6 ν=1 ν=µ

+

Uν† (x

− νˆ) Uµ (x − νˆ) Uν (x − νˆ + µ ˆ) ,

(26)

followed by projection back to SU (3). We select the unitary matrix UµFL which maximises # $ Re tr UµFL Uµ† , by iterating over the three diagonal SU (2) subgroups of SU (3). Performing eight iterations over these subgroups gives gauge invariance up to seven signiﬁcant ﬁgures. We repeat the combined procedure of smearing and projection n times. We create our fat links by setting α = 0.7 and comparing n = 4 and 12 smearing sweeps. The mean-ﬁeld improved FLIC action now becomes

Hadron Properties with FLIC Fermions FL FL SSW = SW −

i g CSW κ r ¯ ψ(x) σµν Fµν ψ(x) , 4 2 (uFL 0 )

207

(27)

where Fµν is constructed using fat links, uFL 0 is calculated in an analogous way to (33) with fat links, and where the mean-ﬁeld improved Fat-Link Irrelevant Wilson action is

Uµ† (x − µ ˆ) Uµ (x) FL ¯ ψ(x) + κ ¯ = ψ(x + µ ˆ) − ψ(x − µ ˆ) ψ(x) ψ(x) γµ SW u0 u0 x x,µ FL

Uµ (x) UµFL† (x − µ ˆ) −r ψ(x + µ ˆ) + ψ(x − µ ˆ) . (28) uFL uFL 0 0 As reported in Table 1, the mean-ﬁeld improvement parameter for the fat links is very close to 1. Hence, the mean-ﬁeld improved coeﬃcient for CSW is accurate.4 It is in this way that the extensive task of non-perturbatively calculating the renormalisations of the improvement coeﬃcients discussed in Sect. 2.3 is avoided. APE smearing the links of dimension ﬁve operators suppresses the renormalisation, allowing the precise matching of improvement coeﬃcients with only tree-level knowledge of their values. Table 1. The value of the mean link for diﬀerent numbers of APE smearing sweeps, n, at α = 0.7 on a 163 × 32 lattice at β = 4.60 which corresponds to a lattice spacing of 0.122(2) fm set by the string tension n

uFL 0

4 (uFL 0 )

0 4 12

0.88894473 0.99658530 0.99927343

0.62445197 0.98641100 0.99709689

In addition, one can now use highly improved deﬁnitions of Fµν (involving terms up to u12 0 ), which give impressive near-integer results for the topological charge [20]. In particular, we employ the 3-loop O(a4 )-improved deﬁnition of Fµν in which the standard clover-sum of four 1 × 1 loops lying in the µ, ν plane is combined with 2 × 2 and 3 × 3 loop clovers. Bilson-Thompson et al. [20] ﬁnd

3 1×1 −i 3 1 2×2 3×3 W − W + W (29) − h.c. g Fµν = 8 2 µν 20u40 µν 90u80 µν Traceless where W n×n is the clover-sum of four n × n loops and Fµν is made traceless by subtracting 1/3 of the trace from each diagonal element of the 3 × 3 colour matrix. This deﬁnition reproduces the continuum limit with O(a6 ) errors. On approximately self-dual conﬁgurations, this operator produces integer topological charge to better than 4 parts in 104 . We also consider a 5-loop improved 4

Our experience with topological charge operators suggests that it is advantageous to include u0 factors, even as they approach 1.

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Fµν for the 203 × 40 lattice at β = 4.53. Since the results for the 5-loop operator agree with the 3-loop version to better than 4 parts in 104 [20], we are eﬀectively using the same action as far the scaling analysis is concerned. The use of thin links in (28) ensures that the relevant dimension-four operators see all the dynamics of the Monte-Carlo generated gauge ﬁelds. Upon expanding the thin links in terms of the gauge potential, O(a2 ) contributions of energy-dimension six are revealed, which, ideally, should be removed via the fat-link irrelevant operator procedure. Fortunately, actions with many irrelevant operators (e.g. the D234 action) can now be handled with conﬁdence as treelevel knowledge of the improvement coeﬃcients is suﬃcient. However, as we will see, the scaling of the O(a)-improved FLIC fermion action of (27) and (28) is excellent already, and the added computational expense of the two-link hopping terms required in O(a2 )-improvement is not well motivated at present. Work by DeForcrand et al. [29] suggests that 7 cooling sweeps are required to approach topological charge within 1% of integer value. This is approximately 16 APE smearing sweeps at α = 0.7 [30]. However, achieving integer topological charge is not necessary for the purposes of studying hadron masses, as has been well established. To reach integer topological charge, even with improved deﬁnitions of the topological charge operator, requires signiﬁcant smoothing and associated loss of short-distance information. Instead, we regard this as an upper limit on the number of smearing sweeps. Using unimproved gauge ﬁelds and an unimproved topological charge operator, Bonnet et al. [23] found that the topological charge settles down after about 10 sweeps of APE smearing at α = 0.7. Consequently, we create fat links with APE smearing parameters n = 12 and α = 0.7. This corresponds to ∼2.5 times the smearing used in [21, 25]. Further investigation reveals that improved gauge ﬁelds with a small lattice spacing (a = 0.122 fm) are smooth after only 4 sweeps. Hence, we perform calculations with 4 sweeps of smearing at α = 0.7 for and consider n = 12 as a second reference. Table 1 lists the values of uFL 0 n = 0, 4 and 12 smearing sweeps. We also compare our results with the standard Mean-Field Improved Clover (MFIC) action. We mean-ﬁeld improve as deﬁned in (27) and (28) but with thin links throughout. For this action, the standard 1-loop deﬁnition of Fµν is used.

4 Lattice Simulations The simulations are performed using the Luscher-Weisz [31] mean-ﬁeld improved, plaquette plus rectangle, gauge action on 123 × 24 and 163 × 32 lattices with lattice√spacings of 0.093, 0.122 and 0.165 fm determined from the string tension with σ = 440 MeV. We deﬁne SG =

5β 1 β 1 Re tr(1 − Usq (x)) − Re tr(1 − Urect (x)) , 3 sq 3 12u20 rect 3

where the operators Usq (x) and Urect (x) are deﬁned as

(30)

Hadron Properties with FLIC Fermions

Usq (x) = Uµ (x) Uν (x + µ ˆ) Uµ† (x + νˆ) Uν† (x) ,

209

(31)

Urect (x) = Uµ (x) Uν (x + µ ˆ) Uν (x + νˆ + µ ˆ) † † † × Uµ (x + 2ˆ ν ) Uν (x + νˆ) Uν (x) + Uµ (x) Uµ (x + µ ˆ) Uν (x + 2ˆ µ) × Uµ† (x + µ ˆ + νˆ) Uµ† (x + νˆ) Uν† (x) .

(32)

The link product Urect (x) denotes the rectangular 1 × 2 and 2 × 1 plaquettes, and for the tadpole improvement factor we employ the plaquette measure u0 =

1/4 1 Re trUsq . 3

(33)

Initial studies of FLIC, mean-ﬁeld improved clover and Wilson quark actions were made using 50 conﬁgurations. The scaling analysis of FLIC fermions was performed with a total of 200 conﬁgurations at each lattice spacing and volume. In addition, for the light quark simulations, 94 conﬁgurations are used on a 203 × 40 lattice with a = 0.134(2) fm. Gauge conﬁgurations are generated using the Cabibbo-Marinari pseudo-heat-bath algorithm with three diagonal SU (2) subgroups looped over twice. Simulations are performed using a parallel algorithm with appropriate link partitioning [32], and the error analysis is performed by a third-order, single-elimination jackknife, with the χ2 per degree of freedom (NDF ) obtained via covariance matrix ﬁts. A ﬁxed boundary condition is used for the fermions by setting Ut (x, nt) = 0

and

UtFL (x, nt) = 0

∀x

(34)

in the hopping terms of the fermion action. The fermion source is centered at the space-time location (x, y, z, t) = (1, 1, 1, 3), which allows for two steps backward in time without loss of signal, for all simulations except those on the 203 × 40 lattice at β = 4.53 which has the fermion source located at (x, y, z, t) = (1, 1, 1, 8). Gauge-invariant Gaussian smearing [33] in the spatial dimensions is applied at the source to increase the overlap of the interpolating operators with the ground states. The source-smearing technique [33] starts with a point source, ψ0 αa (x, t) = δ ac δαγ δx,x0 δt,t0

(35)

for source colour c, Dirac γ, position x0 = (1, 1, 1) and time t0 and proceeds via the iterative scheme, F (x, x ) ψi−1 (x , t) , ψi (x, t) = x

where 1 F (x, x ) = (1 + α)

) δx,x

* 3 α

Uµ (x, t) δx ,x+µ + Uµ† (x − µ . + , t) δx ,x−µ 6 µ=1

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Repeating the procedure N times gives the resulting fermion source ψN (x, t) = F N (x, x ) ψ0 (x , t) .

(36)

x

The parameters N and α govern the size and shape of the smearing function. We simulate with N = 20 and α = 6, except on the 203 × 40 lattice which has N = 35. The propagator, S, is obtained from the smeared source by solving ab bc Sβγ = ψαa , Mαβ

(37)

for each colour, Dirac source c, γ respectively of (35) via the BiStabilised Conjugate Gradient algorithm [34].

5 Scaling of FLIC Fermions Hadron masses are extracted from the Euclidean time dependence of the calculated two-point correlation functions. For baryons the correlation functions are given by +

, G(t; p, Γ ) = e−i p·x Γ βα Ω|T χα (x)χ ¯β (0) |Ω , (38) x

where χ are standard baryon interpolating ﬁelds, Ω represents the QCD vacuum, Γ is a 4 × 4 matrix in Dirac space, and α, β are Dirac indices. At large Euclidean times one has G(t; p, Γ )

Z 2 −Ep t e tr [Γ (−iγ · p + M )] , 2Ep

(39)

where Z represents the coupling strength of χ(0) to the baryon, and Ep = (p 2 + M 2 )1/2 is the energy. Selecting p = 0 and Γ = (1 + γ4 )/4, the eﬀective baryon mass is then given by M (t) = log[G(t)] − log[G(t + 1)] .

(40)

Meson masses are determined via analogous standard procedures. The critical value of κ, κcr , is determined by linearly extrapolating m2π as a function of mq to zero. Figure 2 shows the nucleon eﬀective mass plot for the FLIC action on a 163 × 32 lattice at β = 4.60 which corresponds to a lattice spacing of 0.122(2) fm set by the string tension. The fat links are created with 4 APE smearing sweeps at α = 0.7 (“FLIC4”). The eﬀective mass plots for the other hadrons are similar, and all display acceptable plateau behavior. Good values of χ2 /NDF are obtained for many diﬀerent time-ﬁtting intervals as long as one ﬁts after time slice 8. All ﬁts for this action are therefore performed on time slices 9 through 14. For the Wilson action and the FLIC action with n = 12 (“FLIC12”), the eﬀective mass plots look similar to Fig. 2 and display good plateau behavior. The ﬁtting regimes used for these actions are 9–13 and 9–14, respectively.

Hadron Properties with FLIC Fermions

211

Fig. 2. Eﬀective mass plot for the nucleon for the FLIC action from 200 conﬁgurations on a 163 × 32 lattice at β = 4.60 which corresponds to a lattice spacing of 0.122(2) fm set by the string tension. The fat links are created with 4 sweeps of smearing at α = 0.7. The ﬁve sets of points correspond to the κ values listed in Table 2, with κ increasing from top down Table 2. Values of κ and the corresponding π, ρ, N and ∆ masses for the FLIC action with 4 sweeps of smearing at α = 0.7 on a 163 × 32 lattice at β = 4.60. The value for κcr is provided in Table 3. A string tension analysis incorporating the lattice coulomb √ term provides a = 0.122(2) fm for σ = 440 MeV κ

mπ a

mρ a

mN a

m∆ a

0.1260 0.1266 0.1273 0.1279 0.1286

0.5797(23) 0.5331(24) 0.4744(27) 0.4185(30) 0.3429(37)

0.7278(39) 0.6951(45) 0.6565(54) 0.6229(65) 0.5843(97)

1.0995(58) 1.0419(64) 0.9709(72) 0.9055(82) 0.8220(102)

1.1869(104) 1.1387(121) 1.0816(152) 1.0310(194) 0.9703(286)

The values of κ used in the simulations for all quark actions are given in Table 3. We have also provided the values of κcr for these fermion actions when using our mean-ﬁeld improved, plaquette plus rectangle, gauge action at β = 4.60. We have mean-ﬁeld improved our fermion actions so we expect the values for κcr to be close to the tree-level value of 0.125. Improved chiral properties are seen for the FLIC and MFIC actions, with FLIC4 performing better than FLIC12. The behavior of the ρ, nucleon and ∆ masses as a function of squared pion mass are shown in Fig. 3 for the various actions. The ﬁrst feature to note is the excellent agreement between the FLIC4 and FLIC12 actions. On the other hand, the Wilson action appears to lie somewhat low in comparison. It is also reassuring that all actions give the correct mass ordering in the spectrum. The value of the squared pion mass at mπ /mρ = 0.7 is plotted on the abscissa for

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Table 3. Values of κ and κcr for the four diﬀerent actions on a 163 × 32 lattice at β = 4.60 which corresponds to a lattice spacing of 0.122(2) fm set by the string tension

κ1 κ2 κ3 κ4 κ5 κcr

Wilson

FLIC12

FLIC4

MFIC

0.1346 0.1353 0.1360 0.1367 0.1374 0.1390

0.1286 0.1292 0.1299 0.1305 0.1312 0.1328

0.1260 0.1266 0.1273 0.1279 0.1286 0.1300

0.1196 0.1201 0.1206 0.1211 0.1216 0.1226

Fig. 3. Masses of the nucleon, ∆ and ρ meson versus m2π for the FLIC4, FLIC12 and Wilson actions on a 163 × 32 lattice at β = 4.60 which corresponds to a lattice spacing of 0.122(2) fm set by the string tension

the three actions as a reference point. This point is chosen in order to allow comparison of diﬀerent results by interpolating them to a common value of mπ /mρ = 0.7, rather than extrapolating them to smaller quark masses, which is subject to larger systematic and statistical uncertainties. The scaling behaviour of the diﬀerent actions is illustrated in Fig. 4. The present results for the Wilson action agree with those of [35]. The ﬁrst feature to observe in Fig. 4 is that actions with fat-link irrelevant operators perform extremely well. For both the vector meson and the nucleon, the FLIC actions perform signiﬁcantly better than the mean-ﬁeld improved clover action. It is also clear that the FLIC4 action performs systematically better than the FLIC12. This suggests that 12 smearing sweeps removes too much short-distance information from the gauge-ﬁeld conﬁgurations. On the other hand, 4 sweeps of smearing combined with our O(a4 ) improved Fµν provides excellent results, without the ﬁne tuning of CSW in the NP improvement program.

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Fig. 4. Nucleon and vector meson masses for the Wilson, NP-improved, mean-ﬁeld clover and FLIC actions. Results from the present simulations, based on 50 conﬁgurations and indicated by the solid points, are obtained by interpolating the results of Fig. 3 to mπ /mρ = 0.7. The fat links are constructed with n = 4 (solid squares) and n = 12 (stars) smearing sweeps at α = 0.7. The solid triangles are results for the FLIC4 action when 200 conﬁgurations are used in the analysis. The FLIC results are oﬀset from the central value for clarity. Our MF clover result at a2 σ ∼ 0.075 lies systematically low relative to the FLIC actions

Notice that for the ρ meson, a linear extrapolation of previous mean-ﬁeld improved clover results in Fig. 4 passes through our mean-ﬁeld improved clover result at a2 σ ∼ 0.075 which lies systematically low relative to the FLIC actions. However, a linear extrapolation does not pass through the continuum limit result, thus conﬁrming the presence of signiﬁcant O(a) errors in the mean-ﬁeld improved clover fermion action. While there are no NP-improved clover plus improved glue simulation results at a2 σ ∼ 0.075, the simulation results that are available indicate that the fat-link results also compete well with those obtained with a NP-improved clover fermion action. Having determined FLIC4 is the preferred action, we have increased the number of conﬁgurations to 200 for this action. As expected, the error bars are halved and the central values for the FLIC4 points move to the upper end of the error bars on the 50 conﬁguration result, further supporting the promise of excellent scaling. In order to further test the scaling of the FLIC action at diﬀerent lattice spacings, we consider four diﬀerent lattice spacings and three diﬀerent volumes. String tensions, volumes and hadron masses are given in Table 4 and the results are displayed in Fig. 5. The two diﬀerent volumes used at a2 σ ∼ 0.075 indicate a small ﬁnite volume eﬀect, which increases the mass for the smaller volume at

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Table 4. String tensions, β, volumes and results for the vector meson and nucleon masses interpolated to mP /mV = 0.7. The scale for the small β = 4.60 lattice estimates are taken from the large β = 4.60 lattice √ √ √ β Volume Nconﬁgs a σ mv / σ mN / σ u0 4.38 4.53 4.60 4.60 4.80

163 × 32 203 × 40 123 × 24 163 × 32 163 × 32

200 94 200 200 200

0.371 0.299 0.274 0.274 0.210

2.378(25) 2.318(18) 2.434(26) 2.336(22) 2.427(23)

3.450(35) 3.408(26) 3.554(33) 3.400(26) 3.538(61)

0.8761 0.8859 0.8889 0.8889 0.8966

a2 σ ∼ 0.075 and ∼ 0.045. Examination of points from the small and large volumes separately indicates continued scaling toward the continuum limit. While the ﬁnite volume eﬀect will produce a diﬀerent continuum limit value, the slope of the points from the smaller and larger volumes agree, consistent with errors of O(a2 ). Focusing on simulation results from physical volumes with extents ∼2 fm and larger, we perform a simultaneous ﬁt of the FLIC, NP-improved clover

Fig. 5. Nucleon and vector meson masses for the Wilson, Mean-Field (MF) improved clover, NP-improved clover and FLIC actions obtained by interpolating simulation results to mπ /mρ = 0.7. For the FLIC action (“FLIC4”), fat links are constructed with n = 4 APE-smearing sweeps with smearing fraction α = 0.7, except for the point at a2 σ ∼ 0.09 which has n = 6. Results from the current simulations are indicated by the solid symbols; those from earlier simulations by open or hatched symbols. The solid-lines illustrate ﬁts, constrained to have a common continuum limit, to FLIC, NPimproved clover and Wilson fermion action results obtained on physically large lattice volumes

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and Wilson fermion action results. The ﬁts are constrained to have a common continuum limit and assume errors are O(a2 ) for FLIC and NP-improved clover actions and O(a) for the Wilson action. An acceptable χ2 per degree of freedom is obtained for both the nucleon and ρ-meson ﬁts. These results indicate that FLIC fermions provide a new form of nonperturbative O(a) improvement. The FLIC fermion results display nearly perfect scaling indicating O(a2 ) errors are small for this action.

6 Search for Exceptional Conﬁgurations Chiral symmetry breaking in the Wilson action allows continuum zero modes of the Dirac operator to be shifted into the negative mass region. This problem is accentuated as the gauge ﬁelds become rough (a → large). Local lattice artifacts at the scale of the cutoﬀ (often referred to as dislocations) give rise to spurious near zero modes. The quark propagator can then encounter singular behaviour as the quark mass becomes light. Exceptional conﬁgurations are a severe problem in quenched QCD (QQCD) because instantons are low action ﬁeld conﬁgurations which appear readily in QQCD. These instanton conﬁgurations give rise to approximate zero modes which should be suppressed at light quark masses by det M which is present in the link updates in full QCD. This determinant is not present in QQCD and as a result, near-zero modes are overestimated in the ensemble. The addition of the clover term to the fermion action broadens the distribution of near-zero modes. As a result, the clover action is notorious for revealing the exceptional conﬁguration problem in QQCD. The FLIC action is expected to reduce the number of exceptional conﬁgurations by smoothing the gauge ﬁelds of the irrelevant operator via APE smearing [5, 6]. The smoothing procedure has the eﬀect of suppressing the local lattice artifacts and narrowing the distribution of near-zero modes, enabling simulations to be performed at light quark masses not currently accessible with the standard mean-ﬁeld or non-perturbative improved clover fermion actions. In order to access the light quark regime, we would like our preferred action to be eﬃcient when inverting the fermion matrix. Figure 6 compares the convergence rates of the diﬀerent actions on a 163 × 32 lattice at β = 4.60 by plotting the number of stabilised biconjugate gradient [34] iterations required to invert the fermion matrix as a function of mπ /mρ . For any particular value of mπ /mρ , the FLIC actions converge faster than both the Wilson and mean-ﬁeld improved clover fermion actions. Also, the slopes of the FLIC lines are smaller in magnitude than those for Wilson and mean-ﬁeld improved clover actions, which provides great promise for performing cost eﬀective simulations at quark masses closer to the physical values. Problems with exceptional conﬁgurations have prevented such simulations in the past. The ease with which one can invert the fermion matrix using FLIC fermions (also see [36]) leads us to attempt simulations down to light quark masses corresponding to mπ /mρ = 0.35. Previous attempts with Wilson-style fermion

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Fig. 6. Average number of stabilised biconjugate gradient iterations for the Wilson, FLIC and mean-ﬁeld improved clover (MFIC) actions plotted against mπ /mρ . The fat links are constructed with n = 4 (solid squares) and n = 12 (stars) smearing sweeps at α = 0.7 on a 163 × 32 lattice at β = 4.60 which corresponds to a lattice spacing of 0.122(2) fm set by the string tension

actions on conﬁgurations with lattice spacing ≥0.1 fm have only succeeded in getting down to mπ /mρ = 0.47 [37]. In order to search for exceptional conﬁgurations, we follow the technique used by Della Morte et al. [37] and note that in the absence of exceptional conﬁgurations, the standard deviation of an observable will be independent of the number of conﬁgurations considered in the average. Exceptional conﬁgurations reveal themselves by introducing a signiﬁcant jump in the standard deviation as the conﬁguration is introduced into the average. In severe cases, exceptional conﬁgurations can lead to divergences in correlation functions or prevent the matrix inversion process from converging. The simulations are on a 203 × 40 lattice with a lattice spacing of 0.134(2) fm set by a string tension analysis incorporating the lattice coulomb term. The physical length of the lattice is ∼2.7 fm. We have used an initial set of 100 conﬁgurations, using n = 6 sweeps of APE-smearing and a ﬁve-loop improved lattice ﬁeld-strength tensor. Figure 7 shows the standard deviation of the pion mass for eight quark masses on subsets of 30 (consecutive) conﬁgurations with a cyclic property enforced from conﬁguration 100 to conﬁguration 1. At ﬁrst glance, it is obvious that the error blows up for several quark masses at N = 12 and drops again at N = 42. As conﬁgurations 12 through 41 are included in the average at N = 12, this indicates that conﬁguration number 41 is a candidate for an exceptional conﬁguration. An inspection of the pion mass in Fig. 8 shows that the pion mass for the third lightest quark mass decreases signiﬁcantly more than the second or fourth lightest quark masses. This indicates that κcr for this conﬁguration lies somewhere between κ6 and κ7 . A solution to this problem would be to use the the modiﬁed quenched approximation (MQM) from [22] and move κcr on this conﬁguration back to the ensemble average for κcr . However, since

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Fig. 7. The standard deviation in the error of the π mass for eight quark masses (with the star symbols being the lightest quark mass) calculated on 30 conﬁgurations plotted against the starting conﬁguration number for the FLIC-fermion action on a 203 × 40 lattice with a = 0.134(2) fm

Fig. 8. The π mass calculated for eight quark masses (with the star symbols being the lightest quark mass) on 30 conﬁgurations plotted against the starting conﬁguration number for the FLIC-fermion action on a 203 × 40 lattice with a = 0.134 fm

the movement of κcr is largely a quenched artifact and would be suppressed in a full QCD simulation we prefer to simply identify and remove such conﬁgurations from the ensemble. Obviously, if we ﬁnd that a signiﬁcant percentage of our conﬁgurations are having trouble at a particular quark mass, then it would make no sense to proceed with the simulation. We would then have to conclude

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that we have reached the light quark mass limit of our action and simply step back to the next lightest mass. Now let us return to Fig. 7. In addition to the highly exceptional conﬁguration number 41, we also notice a large increase in error in the lightest quark mass for conﬁguration numbers 2, 13, 30, 34 and 53. Upon removal of these conﬁgurations, we see in Fig. 9 a near-constant behaviour of the standard deviation for the remaining conﬁgurations. This means that our elimination rate for our FLIC6 action on a lattice with a spacing 0.134 fm is about 6%. So for the 100 conﬁgurations used in this analysis, we are able to use 94 of them to extract hadron masses.

Fig. 9. The standard deviation in the error of the π mass for eight quark masses (with the star symbols being the lightest quark mass) calculated on 30 conﬁgurations plotted against the starting conﬁguration number for the FLIC-fermion action on a 203 × 40 lattice with a = 0.134 fm. Conﬁguration numbers 2, 13, 30, 34, 41 and 53 have been omitted

A similar analysis on a 163 × 32 lattice at β = 4.60 providing a ﬁner lattice spacing of 0.122(2) fm reveals a much smaller exceptional conﬁguration rate. In a sample of 200 conﬁgurations, 4 were identiﬁed as exceptional. The increase from 2% to 6% in going from a 0.125 to 0.135 fm suggests that the coarser lattice spacing is near the limit of applicability for FLIC fermions in the light quark mass regime.

7 Octet-Decuplet Mass Splittings The results presented in this section are based on an initial sample of 94 gaugeﬁeld conﬁgurations of an anticipated 400 conﬁgurations. Table 5 reports simulation results for non-strange low-lying hadrons. Figure 10 shows the N and ∆

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Table 5. Values of κ and the corresponding π, ρ, N and ∆ masses on a 203 × 40 lattice for the FLIC action with 6 sweeps of smearing at α = 0.7. A string tension analysis √ incorporating the lattice coulomb term provides a = 0.134(2) fm for σ = 440 MeV κ 0.1278 0.1283 0.12885 0.1294 0.1299 0.13025 0.1306 0.1308

mπ a

mρ a

mN a

m∆ a

0.5400(30) 0.4998(31) 0.4521(34) 0.3990(38) 0.3434(43) 0.2978(47) 0.2419(54) 0.1972(69)

0.7304(55) 0.7053(58) 0.6774(63) 0.6491(72) 0.6228(87) 0.6040(107) 0.5845(143) 0.5812(213)

1.0971(80) 1.0522(84) 1,0006(91) 0.9465(101) 0.8944(116) 0.8562(134) 0.8172(171) 0.7950(215)

1.2238(98) 1.1899(102) 1.1528(108) 1.1162(115) 1.0841(125) 1.0630(135) 1.0443(154) 1.0380(189)

masses as a function of m2π for the FLIC-fermion action on 203 ×√40 lattices with a = 0.132 fm (which corresponds to a string tension scale with σ = 450 MeV) such that the nucleon extrapolation passes through the physical value for clarity. An upward curvature in the ∆ mass for decreasing quark mass is observed in the FLIC fermion results. This behaviour, increasing the quenched N − ∆ mass spitting, was predicted by Young et al. [38, 39] using quenched chiral perturbation theory (QχPT) formulated with a ﬁnite-range regulator. A ﬁt to the

Fig. 10. Nucleon and ∆ masses for the FLIC-fermion action on a 203 × 40 lattice. Here √ we select a = 0.132 fm (which corresponds to the string tension with σ = 450 MeV) such that the nucleon extrapolation passes through the physical value for clarity. The solid curves illustrate ﬁts of ﬁnite-range regularised quenched chiral perturbation theory [38, 39] to the lattice QCD results. The dashed curves estimate the correction that will arise in unquenching the lattice QCD simulations [38, 39]. Stars at the physical pion mass denote experimentally measured values

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FLIC-fermion results is illustrated by the solid curves. The dashed curves estimate the correction that will arise in unquenching the lattice QCD simulations [38, 39]. We note that after we have corrected for the absence of sea quark loops, our results agree simultaneously with the physical values for both the nucleon and ∆. We also calculate the light quark mass behaviour of the octet and decuplet hyperons. The strange quark mass is chosen in order to reproduce the physical strange quark mass according to the phenomenological value of an s¯ s pseudoscalar meson, (41) m2ss = 2m2K − m2π . Upon substitution of the physical masses for the π and K mesons, this corresponds to an s¯ s pseudoscalar meson mass of ∼ 0.470 GeV2 which occurs at our third heaviest quark mass in the 203 × 40 lattice analysis. The results from this calculation are given in Table 6 and are illustrated in Fig. 11. The results show the correct ordering and in particular, we notice a mass splitting between the strangeness = −1 (I = 1) Σ and (I = 0) Λ baryons becoming evident in the light quark mass regime. Table 6. Values of κ, the octet Λ, Σ, Ξ and decuplet Σ ∗ , Ξ ∗ masses on a 203 × 40 lattice for the FLIC action with 6 sweeps of smearing at α = 0.7. A string tension √ analysis provides a = 0.134(2) fm for σ = 440 MeV κ

mΛ a

mΣ a

mΞ a

mΣ∗ a

mΞ∗ a

0.1278 0.1283 0.12885 0.1294 0.1299 0.13025 0.1306 0.1308

1.0696(84) 1.0376(86) 1.0006(91) 0.9615(97) 0.9235(106) 0.8955(117) 0.8667(137) 0.8544(154)

1.0616(83) 1.0328(86) 1.0006(91) 0.9680(97) 0.9383(106) 0.9178(116) 0.8980(132) 0.8919(152)

1.0381(87) 1.0206(88) 1,0006(91) 0.9799(94) 0.9603(98) 0.9462(102) 0.9323(109) 0.9254(114)

1.2002(101) 1.1776(104) 1.1528(108) 1.1284(113) 1.1070(118) 1.0930(124) 1.0806(132) 1.0772(142)

1.1765(104) 1.1652(106) 1.1528(108) 1.1406(110) 1.1299(113) 1.1229(115) 1.1166(118) 1.1145(120)

Just as we saw the non-analytic behaviour of quenched chiral perturbation theory in the ∆-baryon mass in Fig. 10 leading to an enhancement of the quenched N − ∆ mass spitting, Fig. 12 shows a similar enhancement for the decuplet-octet mass splittings in Σ and Ξ baryons respectively. The quark model predicts that the hyperﬁne splittings should approximately satisfy [40] Ξs∗ − Ξs = µs µq = Σs∗ − Σs ,

(42)

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Fig. 11. Octet (top) and decuplet (bottom) baryon masses for the FLIC-fermion action on a 203 × 40 lattice with a = 0.134 fm

where the baryon label denotes the hyperon mass and µs (µq ) denotes the magnetic moment of the strange (light) constituent quark. Figure 13 shows that even though the quenched approximation enhances the splitting between octet and decuplet baryons, the splittings for the Σ and Ξ baryons still satisfy (42). Agreement of the quenched QCD results with the quark model prediction is not surprising since both have a suppressed meson cloud. Similarly, one expects further suppression of the meson cloud when two (heavy) strange quarks are present in a baryon.

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Fig. 12. Octet and decuplet baryon masses for Σ (top) and Ξ (bottom) for the FLICfermion action on a 203 × 40 lattice with a = 0.134 fm

Fig. 13. Decuplet (MD ) – octet (MO ) baryon mass splittings for the FLIC-fermion action on a 203 × 40 lattice with a = 0.132 fm

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8 Summary We have calculated hadron masses to test the scaling of the Fat-Link Irrelevant Clover (FLIC) fermion action, in which only the irrelevant, higher-dimension operators involve smeared links. One of the main conclusions of this work is that the use of fat links in the irrelevant operators provides a new form of nonperturbative O(a) improvement. This technique competes well with O(a) nonperturbative improvement on mean ﬁeld-improved gluon conﬁgurations, with the advantage of a reduced exceptional conﬁguration problem. Quenched simulations at quark masses down to mπ /mρ = 0.35 have been successfully performed on a 203 × 40 lattice with a lattice spacing of 0.134(2) fm on 94 out of 100 conﬁgurations. Simulations at such light quark masses reveal the non-analytic behaviour of quenched chiral perturbation theory and provide for an interesting analysis of the hyperﬁne splittings between octet and decuplet baryons.

Acknowledgements We thank Ross Young for contributing the ﬁts of ﬁnite-range regularized quenched chiral perturbation theory to the FLIC fermion results illustrated in Fig. 10. Generous grants of supercomputer time from the Australian Partnership for Advanced Computing (APAC) and the Australian National Computing Facility for Lattice Gauge Theory are gratefully acknowledged. This work was supported in part by the Australian Research Council and by DOE contract DE-AC05-84ER40150 under which the Southeastern Universities Research Association (SURA) operates the Thomas Jeﬀerson National Accelerator Facility.

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Subject Index

algorithm 30, 49, 81, 157, 209, 210 angular momentum 72, 80, 135, 136, 183, 184, 186 anisotropic 74, 94, 95 anomaly 147, 171 APE-smearing 67, 68, 214, 216 Asqtad 49 Asqtad action 29, 30, 32–35, 37, 43, 44, 49, 50, 53, 56 asymptotic behaviour 18, 32, 34, 55, 57 axial vector 147, 149 bag 141, 142 baryons 10–12, 15, 71–75, 82, 84, 88, 89, 96, 99, 100, 109, 113, 114, 117, 125, 199, 210, 221, 223 Bayesian prior 75 Bayesian techniques 75 Bethe-Salpeter (BS) 187 BGR collaboration 74, 97 boundary condition 80–82, 91, 94, 141–143, 184–186, 209 branch cut 115 Callan-Harvey 134, 139, 140 CEBAF 71 chiral condensate 18, 53 chiral expansion 105, 114, 115, 118, 121, 123 chiral extrapolation 10–12, 16, 45, 59, 95, 101, 113, 114, 117, 123, 124, 126, 166 chiral Lagrangian 171 chiral limit 5, 18, 45, 47–49, 52–54, 56–61, 103, 114, 115, 119, 126, 200 chiral perturbation theory 5, 55, 114, 117, 127, 219, 223 chiral symmetry breaking 17, 27, 47, 55, 60, 114, 215

chirally improved 74, 94, 95, 97 CLEO 71 clover action 3, 5, 60, 95, 199, 204–206, 212 coarse lattice 203 condensate 17, 18, 52–55 conﬁnement 47, 60, 104 connected diagrams 172 constituent quark 57, 72, 99, 221 continuum extrapolation 167 continuum limit 27, 40, 51, 80, 177, 201–204, 207, 213–215 convergence 2, 33, 114, 116, 118, 119, 121, 123, 163, 179, 189, 215 cooling 208 correlation function 15, 76, 82, 83, 85–89, 91, 97, 105, 107, 109, 110 correlation matrix 73, 75, 85, 95, 97–101, 105, 110 coupling 2–4, 14, 18, 20, 21, 29, 30, 42, 72, 73, 82, 89, 99, 102, 105, 108, 114, 126, 127, 149, 153, 154, 166, 167, 200, 203, 204, 210 covariance matrix 94, 209 CP-PACS collaboration 5, 6, 8, 10, 12, 118, 128 CSSM Lattice Collaboration 112 cut-oﬀ 14, 116, 117, 119, 120, 183, 184 decay constants 15, 154, 155, 204, 206 decuplet 6, 10, 74, 75, 88, 220, 221, 223 DeGrand-Rossi representation 92 determinant 1, 14, 22, 25, 133, 143, 168, 215 dimensional regularization 117 diquark 72, 84 Dirac operator 21, 39, 66, 141, 150, 151, 157–160, 206, 215 disconnected diagrams 148, 152

228

Subject Index

domain wall fermion 74 dynamical fermions 11, 63, 169 Dyson–Schwinger equation 18, 33, 49 Edinburgh plot 11 Eﬀective Field Theory 114, 116 eﬀective ﬁeld theory 4, 73, 114, 116–119, 123, 139 elastic states 179, 190, 192 Elitzur’s theorem 159 η mesons 5 η mesons 5, 147 exceptional conﬁgurations 200, 215, 216 excited baryon 71, 73, 74, 92, 104, 125, 126 excited nucleon 73, 84, 92, 93, 97, 105 exotic 72, 75, 97, 102, 126 expectation value 1, 22, 138, 149, 150, 159 fat links 68, 199, 206–208, 210, 211, 213, 214, 216, 223 fermion doubling 2 ﬁfth dimension 74, 139 ﬁnite range regularization (FRR) 117, 127 ﬁnite volume 11, 15, 33, 36, 47, 60, 74, 97, 124, 177–179, 181, 183, 185, 186, 188, 192, 213, 214 Fixed Point 95 ﬁxed point 74, 94 ﬂavour 2, 7, 29, 30, 78, 79, 86, 91, 101, 147–150, 152–154, 157, 158, 173 ﬂavour structure 79 ﬂavour symmetry breaking 30, 86 FLIC 65–69, 74, 75, 93–96, 98, 99, 101, 104, 105, 199, 206, 208–216, 218, 219, 223 Fourier transform 28, 39, 169, 179–181, 183, 186, 189, 201 gap

132, 138, 141, 147, 153, 154, 164–166, 170–173 Gauge Dependence 50 gauge ﬁxing 18 Gell-Mann-Oakes-Renner (GOR) relation 114 generating functional 19, 21, 22 Ginsparg-Wilson relation 132, 144 gluon propagator 18, 32, 50

Goldstone boson 28, 72, 102, 147, 199 Gribov copies 17, 24, 25 Gribov copy 25 GSI 71 Haar measure 21 Hadron Spectrum 1, 12 hadron spectrum 4, 6, 7, 15, 17, 80, 147 heavy-hadron 7 heavy-mesons 5, 15 Higgs ﬁeld 139 hybrid 150, 157, 160 hyperﬁne 5, 6, 72, 220, 223 hyperon 72, 85, 221 inelastic threshold 179, 187–190, 192 instanton 58–60 interpolating ﬁelds 71, 73, 75, 78, 82–86, 88, 91, 97–99, 101, 102, 105, 107, 108, 110, 126, 210 irreducible representations 80, 91 irrelevant operators 67, 68, 199, 203, 206, 208, 212, 223 J-parameter 9, 12 jackknife 55, 94, 209 Jacobi smearing 95 Jeﬀerson Lab 71, 74 JLQCD Collaboration

118

K-mesons 3, 12, 220 kaon-decay 177 kappa 202 kernel 39, 42 Kogut–Susskind 17, 27, 30, 32 Kogut–Susskind action 32 Landau gauge 18, 23, 24, 33, 34, 47 Laplacian gauge 18, 24, 25, 49, 51 lattice 1–3, 5–15, 17–24, 26–28, 30–36, 38–40, 42–47, 49, 53–56, 58–61, 65, 71–76, 78–82, 84, 87, 91–95, 97, 98, 104, 105, 107, 113–119, 123–127, 131, 132, 140, 142–144, 147, 149–151, 155–167, 169, 172, 174, 177, 199–201, 203–220, 222, 223 lattice artefacts 18, 33 Lellouch-L¨ uescher (LL) 177, 195 Lepage term 30 light-baryons 10

Subject Index link

2, 20, 21, 23, 65–68, 74, 83, 108, 109, 144, 200, 202, 203, 205, 206, 209, 215 LNA 125 logarithms 5, 109, 115 low energy 139 magnetic moment 114, 221 mass function 18, 33–37, 43, 45, 47–49, 51–53, 55, 56, 58–60 mass generation 47 mass splittings 5, 6, 11, 72, 99, 200, 206, 220, 222 matrix elements 15, 154, 155, 157, 177, 179, 190, 191, 193, 194, 204 maximum entropy method 74, 75, 97 mean-ﬁeld improvement 40, 68, 207 MILC 93, 123 mixing 8, 72, 84, 86, 147, 154–157 molecule 72 MOM 53 Monte Carlo 2, 21, 22, 150, 157, 160, 206 Naik term 29 naive lattice action 2 NLNA 115 noisy sources 159 nonanalytic behaviour 73, 199 nonperturbatively (NP) improved 95 nonrelativistic limit 84 nucleon 10, 71, 73–76, 84, 87, 91, 93–95, 97, 99, 104, 105, 114, 115, 117–121, 123–126, 210–212, 215, 219, 220 octahedral group 79, 80 octet 10, 74, 75, 85, 86, 101, 125, 149, 153, 154, 172, 220, 221, 223 Overlap action 44, 46, 48 overlap actions 17, 61 overlap fermions 75, 97 OZI 162, 168 parallel transport 20 parity 71–75, 78–84, 87–89, 91, 92, 95, 98–101, 103, 109, 110, 125, 126, 149 partially quenched 157, 167–171, 173 path integral 1, 18 path ordering 20 PCAC relation 155 pentaquark 73, 75, 92, 104 π-mesons 3, 12, 147, 149–151, 220

229

pion decay constant 5, 149 plaquette 2, 20, 68, 93, 118, 206, 208, 209, 211 point group 80 precision 50, 71, 147, 166, 180, 181, 190, 192, 193 projection 25, 50, 65, 67, 68, 74, 79, 83, 87–89, 92, 98, 103, 107, 125, 158, 163, 182, 184, 188, 191, 206 pseudoscalar meson 3, 8, 10, 94, 147, 150, 152, 166, 169, 220 quantization 177–179, 183, 188, 190, 194 quantum ﬁeld theory 1, 2, 19, 121, 179, 194 quark model 14, 72, 79, 97, 104, 124, 220, 221 quark propagator 17, 18, 20, 22, 23, 30–32, 35, 36, 41–45, 47, 49, 51, 56, 57, 59, 60, 215 quenched 3–8, 10, 12–17, 22, 30, 55, 59–61, 71, 73, 75, 92, 95, 102–105, 125, 127, 147, 149, 153, 167, 169, 200, 204, 215–217, 219–221, 223 quenched approximation 22, 30, 59 Rarita-Schwinger 87, 89 renormalization 117, 143, 147, 148 resonances 71, 72, 93, 103, 104, 110, 126, 127 ρ-mesons 124, 166, 171, 212, 213, 215 RIKEN-BNL group 74 Roper 71, 75, 93, 95–97, 100 rotational symmetry 29, 33, 35, 43–45, 49, 59, 79 running coupling constant 63 running mass 18, 32, 34, 42, 48, 53, 55 running quark mass 17, 53, 55, 60 sea quark 7–10, 13, 151, 153, 157, 158, 160, 161, 165–170, 172, 173, 220 SESAM 160–162, 164–170 Sheikholeslami-Wohlert (SW) action 74, 204 short-distance physics 68 Σ baryons 99, 220, 221 sign function 40, 42, 65 singlet 85, 86, 101, 147–149, 152–155, 158, 164–169, 172–174 smoothing 208, 215

230

Subject Index

source-smearing 81, 209 spectroscopy 71–73, 75, 78, 104, 105, 126, 157 spin-projection 65, 68 split-link 65, 67, 68 spontaneous symmetry breaking 113 spurious states 184 Staggered action 27–29 staggered actions 17 staggered fermions 113, 201 staples 30, 68, 206 static quark potential 13, 14, 32 stochastic estimate 158 string model 13, 14 string tension 93, 207, 208, 210–212, 216, 219, 220 sum-rule 72, 91 summation 154, 158, 160, 178, 179, 181, 186, 192, 195 Symanzik improved action 35, 202 tadpole-improvement 30, 31, 68, 74, 204, 209 taste 28 topological charge 39, 147, 149, 206–208 topological susceptibility 147, 149 transfer matrix 77, 131, 143

tree-level correction 32, 42, 45 truncated eigenmode approximation (TEA) 161 UKQCD collaboration

8, 13, 74, 129

vacuum 3, 15, 108, 113, 147, 149–151, 153–159, 163, 166, 179, 191, 193, 204, 210 valence quark 5, 8, 10, 71–73, 79, 150, 158, 169, 170, 172, 173 vector-meson 63, 212–214 Ward identities 133 Ward-Takahashi identity 155 Weyl fermion 132, 140 Wilson action 5, 7, 65–67, 69, 94, 95, 202, 203, 207, 211, 212, 215 Wilson doublers 39 Wilson loop 20 Wilson mass parameter 40 Wilson-Dirac operator 39, 40, 42 Witten-Veneziano relation 147 Z function 46 zero mode 132, 135, 136, 138–140 zero momentum 108, 151, 178 Zolotarev 42