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Analytical and Geometric Approches of Non-Perturbative Quantum Field Theories

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Analytical and Geometric Approches of Non-Perturbative Quantum Field Theories Analytical and Geometric approches of non-perturbative Quantum Field Theories Pierre J. Clavier1,2 1 Sorbonne Universités, UPMC Univ Paris 06, UMR 7589, LPTHE, 75005, Paris, France 2 CNRS, UMR 7589, LPTHE, 75005, Paris, France Defended the 3rd of July 2015 arXiv:1511.09190v1 [math-ph] 30 Nov 2015 A la mémoire de André Clavier. Si l’eau pouvait éteindre un brasier amoureux, Ton amour qui me brûle est si fort douloureux, Que j’eusse éteint son feu de la mer de mes larmes. Pierre de Marbeuf. Et la mer et l’amour ont l’amer pour partage. Contents Remerciements v Résumé long vii Introduction 1 1 The Hopf algebra of renormalization 7 1.1 Algebraic preliminaries . .7 1.1.1 Algebra, coalgebra . .7 1.1.2 Bialgebra, Hopf algebra . .9 1.1.3 From bialgebras to Hopf algebras . 11 1.1.4 Characters group . 14 1.2 Connes–Kreimer Hopf algebra . 15 1.2.1 Elements of graph theory . 15 1.2.2 The Hopf algebra structure of Feynman graphs . 17 1.2.3 Birkhoff decomposition . 20 1.2.4 Regularized and renormalized Feynman rules . 23 1.3 Renormalization Group . 24 1.3.1 Renormalization group equation . 24 1.3.2 Correlation functions . 27 1.3.3 RGE for the correlation functions . 28 2 Linear Schwinger–Dyson equations 31 2.1 The massless Yukawa model . 31 2.1.1 The Schwinger–Dyson equation . 31 2.1.2 Parametric solution . 33 2.1.3 Lessons from the massless Yukawa model . 37 2.2 The massive Yukawa model . 37 2.2.1 The Schwinger–Dyson equation . 37 2.2.2 Solution in the ultraviolet limit . 40 2.2.3 Solution in the infrared limit . 42 2.3 Massive linear Wess–Zumino model . 44 2.3.1 The Schwinger–Dyson equation . 45 2.3.2 Parametric solution . 47 i ii CONTENTS 3 The massless Wess–Zumino model I: the physical plane 51 3.1 The equations . 51 3.1.1 The renormalization group equation . 51 3.1.2 The Schwinger–Dyson equation . 53 3.1.3 The one-loop Mellin transform . 54 3.2 Asymptotic solution . 57 3.2.1 Contributions from individual poles . 57 3.2.2 Derivation of the asymptotic solution . 60 3.3 Corrections to the asymptotics . 62 3.3.1 Change of variables . 62 3.3.2 Taking care of every poles: a guideline . 65 3.3.3 Solutions up to the fifth order . 67 3.3.4 Comparison to numerical results . 75 4 The massless Wess–Zumino model II: the Borel plane 77 4.1 Elements of Borel summation theory . 77 4.1.1 The need of Borel summation theory . 77 4.1.2 The Borel transform . 78 4.2 Mapping to the Borel plane . 79 4.2.1 Renormalization group equation . 79 4.2.2 Schwinger–Dyson equation . 81 4.2.3 Formal series vs. singularities . 83 4.3 Truncated Schwinger–Dyson equation . 87 4.3.1 Justification of the truncation . 87 4.3.2 Truncated equation . 88 4.3.3 Link to MZVs . 91 4.3.4 Numerical analysis . 92 4.4 Singularities of the solution . 94 4.4.1 Localization of the singularities . 94 4.4.2 Study of the negative singularities . 96 4.4.3 Study of the positive singularities . 98 4.4.4 Transcendental Content of the Borel transform . 99 4.4.5 Weight of the odd Zetas . 101 5 BV formalism 105 5.1 Foreword . 105 5.1.1 Open and closed symmetries . 106 5.1.2 Some geometric concepts . 108 5.1.3 A motivational introduction . 110 5.1.4 From configuration space to odd symplectic manifolds . 111 5.2 BV formalism as a theory of integration . 112 5.2.1 BV laplacian . 112 5.2.2 BV integral and a Stokes theorem . 113 5.2.3 On the advantage of the BV formalism . 115 5.3 Gauge-fixing in BV formalism . 116 5.3.1 An informative example . 116 CONTENTS iii 5.3.2 Lagrangian submanifold . 120 5.3.3 Gribov ambiguities . 122 5.4 Master equations . 124 5.4.1 BV laplacians in coordinate . 124 5.4.2 Quantum master equations . 127 5.4.3 Classical limit . 129 Conclusion 131 Appendices 135 A Geometric BRST formalism 135 A.1 BRST differential . 135 A.1.1 Set-up . 135 A.1.2 Koszul–Tate resolution . 136 A.1.3 Chevalley–Eilenberg complex . 137 A.1.4 BRST bi-complex . 137 A.2 Poisson superalgebra . 139 A.2.1 Poisson superalgebra . 139 A.2.2 BRST Poisson superalgebra . 141 A.3 QED . 142 A.3.1 Physical states of a theory . 142 A.3.2 Action of BRST differential on fields . 143 A.3.3 Field content of QED . 145 B From BV to BRST 147 B.1 Non-degenerate polyvectors . 147 B.2 Darboux–Morse theorem . 148 B.3 Quantum BV . 149 B.4 equivariante cohomology . 149 B.4.1 Polyvectors: BV formalism . 150 B.4.2 Differential forms: BRST model . 150 C Feynman integrals 153 C.1 Two-points integrals . 153 C.1.1 A new method for two-points integrals . 153 C.1.2 Sanity check . 155 C.1.3 Integral with numerator . 156 C.2 Three-point integral . 157 C.2.1 Formula and method . 157 C.2.2 First computations . 159 C.2.3 Mellin–Barnes . 159 C.2.4 Contour integrals and series . 161 C.2.5 Results . 163 Bibliography 169 iv CONTENTS Remerciements En premier lieu, bien sur, je voudrais remercier celui qui a fait des trois ans de ce doctorat une expérience incroyablement enrichissante, mon directeur de recherche, Marc Bellon. Merci Marc d’avoir été aussi présent, à l’écoute et toujours près à m’aider quand j’en avais besoin, mais également d’avoir accepté que je travaille seul sur certains projets. J’ai le sentiment d’avoir beaucoup appris sous votre égide et espère que nous continuerons à collaborer : il me reste encore tellement à apprendre ! Il me faut également exprimer ma reconnaissance à l’égard des rapporteurs, les professeurs Dominique Manchon et Dirk Kreimer, pour le temps qu’ils ont passé à lire ce manuscrit, et pour leurs commentaires qui ont largement contribué à améliorer la qualité globale de ce texte. Enfin, merci à tous les autres membres du jury d’avoir accepté d’en faire partie. Merci donc à Bernard Julia, Loïc Foissy, Fidel Schaposnik et Christian Brouder pour être présent le jour de la soutenance, j’espère ne pas leur donner de raison de regretter le déplacement. Merci également à ceux qui ont passé du temps à relire cette thèse, qui ont permis de lui donner une finition bien plus lisse que la première version. Merci donc à Nguyen Viet Dang, Marc Bellon,Astrid Cadet, Annick Clavier et Sophie Dorfman pour leurs efforts. Il va sans dire que je suis l’unique responsable de toutes les erreurs, imprécisions, bêtises, fautes et autres coquecigrues qui ont survécu à toutes ces lectures. La recherche n’est pas une activité solitaire. Je voudrais donc très chaleureusement remercier tout ceux avec qui j’ai eu la chance de discuter et d’échanger et qui ont tous permis à ma réflexion de se construire. En premier lieu bien sur, merci à Nguyen Viet Dang, Christian Brouder et Frédéric Hélein de m’avoir convaincu de travailler sur le for- malisme de Batalin—Vilkovisky, le cinquième chapitre de ce manuscrit n’aurait pas ex- isté sans eux. Et merci également d’avoir accepter de prendre le temps d’expliquer tant de choses à un pauvre physicien intéressé par des choses un peu formelles. Merci égale- ment, pêle-mêle, à Camille Laurent—Gengoux, Serguei Barannikov, Sylvain Lavau, Sofian Teber, Olivier Bouillot, Lucien Heurtier, Guillaume Bossard, Robin Zegers, Vincent Rivasseau... Je souhaite également remercier les membres du LPTHE, et en premier lieu ses directeurs successifs Olivier Babelon et Benoit Douçot pour avoir créé un tel climat favorable à la recherche. Merci également aux secrétaires Isabelle Nicolai, Françoise Got et Annie Richard pour leur patience vis-à-vis du thésard ignorant des choses administratives que j’ai été. Et, bien sur, merci à tous les chercheurs et à toutes les chercheuses du labo pour toute les discussions autour de la machine à café. Je suis également très reconnaissant aux anciens doctorants du labo de m’avoir v vi Remerciements accueilli comme ils l’ont fait, et d’avoir pris sous leurs ailes le jeune thésard (si si, thésard!) effrayé que j’étais. Merci donc à Emmanuele, Thomas (le grand) et Thibault. Un merci tout particulier à Gautier, sans qui tant d’heure auraient stupidement été perdues à travailler. Merci également à Bruno pour être le grand frère spirituel de tous les thésards du labo. Une pensée aussi pour les compagnons de galère, ceux qui ont commencés leurs doctorats en même temps que moi. Courage Sacha, Trésa, Harold et Tianhan, c’est bientôt à votre tour de passer à la casserole ! Merci aussi aux p’tits jeunes pour avoir rendu plus agréable l’année où j’étais le vieux du groupe. Merci aux bizuts Luc et Frédéric, merci à Matthieu et Oscar pour être de tels sujets d’amusement à travers leurs thèmes de recherches, merci à Thomas (le petit) pour avoir repris les séminaires des doctorants du LPTHE, et un grand merci à Hugo pour avoir été la hotline mathematica/LaTeX pendant quasiment un an. Je me permets maintenant de remercier tous les copains du M2. Les collocs d’abord : merci Pierre, Ced, Laetitia et Nico pour ces.
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