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Mechanism of Chiral Symmetry Breaking for Three Flavours of Light Quarks and Extrapolations of Lattice QCD Results Guillaume Toucas

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Mechanism of Chiral Symmetry Breaking for Three Flavours of Light Quarks and Extrapolations of Lattice QCD Results Guillaume Toucas Mechanism of chiral symmetry breaking for three flavours of light quarks and extrapolations of Lattice QCD results Guillaume Toucas To cite this version: Guillaume Toucas. Mechanism of chiral symmetry breaking for three flavours of light quarks and extrapolations of Lattice QCD results. Other [cond-mat.other]. Université Paris Sud - Paris XI, 2012. English. NNT : 2012PA112260. tel-00754994 HAL Id: tel-00754994 https://tel.archives-ouvertes.fr/tel-00754994 Submitted on 20 Nov 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ED-517 Particules, Noyaux, Cosmos THESE` DE DOCTORAT Pr´esent´ee pour obtenir le grade de Docteur `es Sciences de l’Universit´eParis-Sud 11 Sp´ecialit´e: PHYSIQUE THEORIQUE´ par Guillaume TOUCAS M´ecanisme de brisure de sym´etrie chirale pour trois saveurs de quarks l´egers et extrapolation de r´esultats de Chromodynamique Quantique sur r´eseau Soutenue le 30 octobre 2012 devant le jury compos´ede: Prof. A. Abada Pr´esident du jury Dr. V. Bernard Directeur de th`ese Dr. S. Descotes-Genon Directeur de th`ese Prof. J.Flynn Rapporteur Dr. B. Kubis Rapporteur Dr. C. Smith Examinateur 2 3 Antoine se renverse la tˆete. Qui donc es-tu ? Hilarion Mon royaume est de la dimension de l’univers; et mon d´esir n’a pas de bornes. Je vais toujours, affranchissant l’esprit et pesant les mondes, sans haine, sans peur, sans piti´e, sans amour et sans Dieu. On m’appelle la Science. Antoine se rejette en arri`ere : Tu dois ˆetre plutˆot... le Diable ! Hilarion fixant sur lui ses prunelles : Veux-tu le voir ? Gustave Flaubert, La tentation de Saint-Antoine 4 Remerciements Je remercie tout d’abord Henk Hilhorst, le directeur du Laboratoire de Physique Th´eorique d’Orsay, de m’avoir acceuillit au sein de son unit´edurant ces trois ann´ees marqu´epar un travail complexe et difficile. Par ailleurs, celui-ci n’aurait pu s’´effectuer sans l’aide de mes deux directeurs de th`ese, S´ebastien Descotes-Genon et V´eronique Bernard, qui ont su trouver la patience infallible de mener `abien leur travail d’encadrement. Je les remercie de n’avoir jamais c´ed´eau d´ecouragement. Je voudrais maintenant adresser un grand merci `aBenoˆıt Blossier, qui partagea mon bureau ces trois ann´ees durant. Nos discussions politico-sociales passionn´ees sur l’actualit´e du monde seront pour moi un des meilleurs souvenirs humains de mes ann´ees de th`ese. Je remercie bien entendu tous les membres de mon jury, avec une mention particuli`ere pour Asmaa Abada, du Laboratoire, qui en accepta le role de pr´esidente, ainsi que pour ses pr´ecieux conseils concernant la tenue de la soutenance. Enfin, cette th´ese n’aurait pu se terminer sans l’aide d’Olivier P`ene, grˆace `alaquelle je su restreindre les inhibitions que j’entretenais `apropos de la terrible QCD sur r´eseau, qui hanta mes cauchemars durant l’une des p´eriodes les plus difficiles de l’´ecriture du manuscrit. Merci encore Olivier. Ces quelques mots ne sauraient ˆetre complˆets sans une adresse `atous mes camarades doctorants: C´edric, Julien, Joachim, Blaise, Charles, Andr´eas, Yannis, Antonin, Adrien, Antoine, Xavier, Bertrand, J´er´emie, Julien... que ceux qui seraient oubli´es hurlent tr`es fort !... H´elas, ma m´emoire commence serieusement `aflancher ! Je remercie enfin tous les autres membres du Laboratoire de Physique Th´eorique: Philippe, Samuel, Damir, Odile, Mireille, Philippe, Jean-Pierre, Patricia ainsi que tout ceux que j’oublie. Je voudrais aussi adresser mes plus sinc`eres salutations `aHagop Sazdjian, mon responsable d’enseignement des TDs de L1 que je dispensais chaque semaine avec grand plaisir. Enfin, un tout dernier mot `al’attention de toute la famille grˆace `ason aide, je pu malgr´e tout traverser cette ´epreuve qui se termine enfin. 6 Contents Introduction 11 1 QCD, Symmetries, and the Vacuum 13 1.1 Introduction.................................... 13 1.2 QCDandChiralSymmetry . .. .. .. .. .. .. .. .. 14 1.2.1 The QCD lagrangian and asymptotic freedom . 14 1.2.2 Chiralsymmetry .............................. 15 1.2.3 Currentsandcharges. 17 1.3 Spontaneous breaking of chiral symmetry . 18 1.3.1 Spontaneous symmetry breakdown and the QCD spectrum . 18 1.3.2 TheGoldstonetheorem .......................... 19 1.4 TheQCDvacuum ................................. 22 1.4.1 Orderparameters: overview . 22 1.4.2 Order parameters of the chiral spontaneous symmetry breaking . 23 1.4.3 The chiral limits Nf = 2 and Nf =3................... 24 1.4.4 Order parameters and the spectrum of the euclidean Dirac operator . 25 1.4.5 Thevacuumangle ............................. 28 1.5 Summary ...................................... 33 2 Lattice QCD, Effective Field Theories, and Chiral Perturbation Theory 35 2.1 Introduction.................................... 35 2.2 LatticeQCD .................................... 35 2.3 Some generalities about Effective Field Theories . .......... 39 2.3.1 DecouplingEFT .............................. 40 2.3.2 Non-decouplingEFT............................ 41 2.4 Chiral Perturbation Theory: foundations . ......... 42 2.4.1 Generatingfunctional . 42 2.4.2 WardIdentities............................... 42 2.4.3 Goldstonebosons.............................. 44 2.5 Chiral Perturbation Theory for three flavours . ......... 45 2.5.1 Powercounting............................... 45 2.5.2 Effective lagrangian at lowest order . 46 2.5.3 Higher Orders and Renormalization . 49 2.5.4 The generating functional at 1-loop . 51 2.6 Chiral Perturbation Theory for two flavours . 52 2.7 Review of numerical results for χPT low energy constants . 54 2.7.1 Nf =3 ................................... 54 2.7.2 Nf =2 ................................... 56 2.7.3 Some issues concerning the SU(3)theory ................ 57 2.8 Summary ...................................... 58 8 Contents 3 Resummed Chiral Perturbation Theory 61 3.1 Introduction.................................... 61 3.2 The problem of weak convergence . 61 3.3 Resummingchiralseries . 63 3.3.1 Procedure.................................. 63 3.3.2 Issueofunitarity .............................. 65 3.4 Massesanddecayconstants . 65 3.4.1 PionsandKaons .............................. 65 3.4.2 Handling Higher-Order remainders . 68 3.4.3 Resummingvacuumfluctuations . 69 3.4.4 The η .................................... 72 3.5 Electromagnetic form factors . 74 3.5.1 Definitionanddiscussion . 74 3.5.2 Pion electromagnetic square radius . 75 3.5.3 Kaon electromagnetic form factor . 76 3.5.4 Kaon electromagnetic square radius . 76 3.6 Kℓ3 formfactors .................................. 78 3.6.1 Definitionanddiscussion . 78 3.6.2 The Callan-Treiman points . 80 3.7 Alternative treatments of the unitarity part . .......... 81 3.8 Summary ...................................... 85 4 Analysis of Nf = 2 + 1 lattice data using Resummed Chiral Perturbation Theory 87 4.1 Introduction.................................... 87 4.2 Latticeinputs................................... 87 4.3 Fitstolatticedata ............................... 91 4.3.1 ObservablesandParameters. 91 4.3.2 Resultsanddiscussion . 94 4.3.3 Impact of alternative treatments of unitarity contributions. 99 4.4 Summary ...................................... 100 5 Topological observables 103 5.1 Introduction.................................... 103 5.2 Derivation through the effective potential . ......... 103 5.2.1 Structure of the one-loop generating functional . ......... 103 5.2.2 NLO expression of the topological susceptibility and the fourth cumulant106 5.3 Derivation using diagrammatic analysis . ........ 110 5.3.1 Combinatorics ............................... 110 5.3.2 Isolating the η propagator......................... 111 5.4 Expected sensitivity to the three-flavour chiral condensate........... 117 5.5 Thetwo-flavourcase ............................... 118 5.6 Topological observables on the lattice . ........ 120 5.6.1 Resummed framework (1): no η pole................... 120 5.6.2 Resummed framework (2): with η pole.................. 125 5.7 Summary ...................................... 128 6 Fits to lattice data on topological susceptibility 129 6.1 Introduction.................................... 129 6.2 Fitstolatticedata ............................... 129 6.2.1 Latticeinputs................................ 129 6.2.2 Finite-volume effects . 131 Contents 9 6.2.3 Parametersanddata............................ 132 6.2.4 Resultsanddiscussion . 133 6.3 Improved fits to pseudoscalar spectrum and topological susceptibility . 135 6.3.1 The dependence of Fπ onthelightestquarks . 135 6.3.2 Including the lattice spacings in the fits . 138 6.3.3 Resultsanddiscussion . 139 6.4 Summary ...................................... 142 Conclusion 143 Appendices 149 A Unitarity integrals and source terms from the generating functional 149 A.1 The one-loop unitarity integrals J¯, K, L and M ................ 149 A.2 Expression for the source terms Γˆµ andσ ¯ ....................150 B Tadpole propagator at finite volume 151 C “Total” fits including additional data points for higher masses and mo- menta 155 D Expressions of Feynman diagrams for topological
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