Two studies on conformal and strongly coupled quantum field theories in d > 2 dimensions Matthijs Hogervorst To cite this version: Matthijs Hogervorst. Two studies on conformal and strongly coupled quantum field theories in d > 2 dimensions. High Energy Physics - Theory [hep-th]. ENS Paris - Ecole Normale Sup´erieurede Paris, 2015. English. <tel-01171444> HAL Id: tel-01171444 https://tel.archives-ouvertes.fr/tel-01171444 Submitted on 3 Jul 2015 HAL is a multi-disciplinary open access L'archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destin´eeau d´ep^otet `ala diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publi´esou non, lished or not. The documents may come from ´emanant des ´etablissements d'enseignement et de teaching and research institutions in France or recherche fran¸caisou ´etrangers,des laboratoires abroad, or from public or private research centers. publics ou priv´es. Two studies on conformal and strongly coupled quantum field theories in d > 2 dimensions Deux essais sur les th´eoriesquantiques des champs conformes et fortement coupl´eesen d > 2 dimensions Matthijs Hogervorst Ecole´ normale superieure´ Ecole´ Doctorale 564: Physique en ^Ile-de-France 2015 Two studies on conformal and strongly coupled quantum field theories in d > 2 dimensions Deux essais sur les th´eoriesquantiques des champs conformes et fortement coupl´eesen d > 2 dimensions Matthijs Hogervorst Th`esede doctorat de l'Ecole´ normale sup´erieure,Paris Ecole´ Doctorale 564 : Physique en ^Ile-de-France Soutenue le 29 juin 2015 devant le jury compos´ede : Miguel Costa examinateur Giuseppe Mussardo rapporteur Hugh Osborn examinateur Vyacheslav Rychkov directeur de th`ese Marco Serone rapporteur Jan Troost examinateur Jean-Bernard Zuber pr´esident du jury Abstract This thesis investigates two aspects of Conformal Field Theories (CFTs) in d dimensions. Its first part is devoted to conformal blocks, special functions that arise in the partial wave expansion of CFT four-point functions. We prove that these conformal blocks admit an expansion in terms of polar coordinates and show that the expansion coefficients are determined by recursion relations. Conformal blocks are naturally defined on the complex plane: we study their restriction to the real line, and show that they obey a fourth-order differential equation there. This ODE can be used to efficiently compute conformal blocks and their derivatives in general d. Several applications to the conformal bootstrap program are mentioned. The second half of this thesis investigates RG flows that are defined by perturbing a CFT by a number of relevant operators. We study such flows using the Truncated Conformal Space Approach (TCSA) of Yurov and Zamolodchikov, a numerical method that allows for controlled computations in strongly coupled QFTs. Two different RG flows are considered: the free scalar field deformed by a mass term, and φ4 theory. The former is used as a benchmark, in order to compare numerical TCSA results to exact predictions. 4 TCSA results for φ theory display spontaneous Z2 symmetry breaking at strong coupling: we study the spectrum of this theory both in the Z2-broken and preserved phase, and we compare the critical exponents governing the phase transition to known values. In a separate chapter, we show how truncation errors can be reduced by adding suitable counterterms to the bare TCSA action, following earlier work in d = 2 dimensions. R´esum´e Cette th`eseexamine deux aspects des th´eoriesconformes des champs (TCC) en d di- mensions. Sa premi`erepartie est d´edi´eeaux blocs conformes, des fonctions sp´ecialesqui contribuent au d´eveloppement en ondes partielles des fonctions `aquatre points dans les TCC. On montre que ces blocs admettent un d´eveloppement en coordonn´eespolaires dont les coefficients se calculent par une r´ecurrence.Les blocs conformes sont naturellement d´efinis sur le plan complexe : on consid`erealors leur restriction `al'axe r´eel,afin de montrer qu'ils ob´eissent `aune ´equationdiff´erentielle sur ce domaine, ce qui m`ene`aun algorithme efficace pour calculer les blocs conformes et leurs d´eriv´eespour tout d. Quelques applications au programme de bootstrap sont d´evelopp´ees.La seconde partie de cette th`eseexamine les perturbations d'une TCC par des op´erateurs pertinents. On ´etudiede tels flots du groupe de renormalisation en utilisant la M´ethode de Troncature Conforme (MTC) de Yurov et Zamolodchikov, une m´ethode num´eriquequi permet de faire des calculs non-perturbatifs en th´eoriequantique des champs. Deux th´eoriesdiff´erentes sont consid´er´ees: le boson libre avec un terme de masse, et la th´eorie φ4. Pour le dernier cas, les r´esultats de la MTC mettent en ´evidencela brisure de sym´etrie Z2. Finalement, on d´eveloppe une m´ethode pour r´eduireles erreurs de troncature en ajoutant des contre-termes `al'action \nue" de la MTC, suivant des travaux ant´erieursen d = 2 dimensions. Contents Abstract . .i R´esum´e................................................... ii Introduction iii Citations to published work . viii R´esum´esubstantiel . ix Conventions and special functions . xii 1 Elements of conformal field theory1 1.1 Conformal group . .2 1.2 Conformal algebra . .3 1.3 Local operators . .6 1.3.1 Tensor operators . .8 1.3.2 Descendants . .9 1.4 Constraints on correlation functions . 10 1.4.1 Scalar correlators . 10 1.4.2 Spinning correlators . 12 1.4.3 Example: free scalar boson . 13 1.5 Weyl invariance . 14 1.5.1 The free boson revisited . 15 1.6 Radial quantization . 16 1.6.1 State-operator correspondence . 18 1.6.2 Adjoint states and matrix elements . 19 1.6.3 Unitarity constraints on CFTs . 21 1.7 Operator product expansion . 22 1.7.1 Conformal block decomposition . 24 1.7.2 Conformal bootstrap . 27 1.8 Casimir differential equations . 29 2 Conformal blocks in radial coordinates 32 2.1 Conformal blocks in the Dolan-Osborn coordinates . 33 2.1.1 Dolan-Osborn kinematics in radial quantization . 34 2.1.2 Expansion coefficients from the Casimir equation . 38 2.1.3 Decoupling of descendants for the leading twist . 40 2.2 Conformal blocks in the ρ coordinate . 41 2.2.1 Comparison between the z and ρ expansions . 45 2.3 Outlook: potential applications to the conformal bootstrap . 48 2.3.1 Inexpensive derivative evaluation for all ∆ and ` ..................... 50 2.3.2 Truncated bootstrap equation with an error estimate . 51 2.4 Summary . 52 3 Conformal blocks in the diagonal limit 54 3.1 Introduction . 54 3.2 Differential equations on the diagonal . 56 3.3 Frobenius' method . 60 3.4 Computing conformal blocks and their derivatives efficiently . 61 3.5 Summary . 64 4 TCSA for scalar fields d > 2 dimensions 65 4.1 Introduction . 65 4.2 Truncated Conformal Space Approach: general setup . 65 4.2.1 A case study for TCSA in d dimensions . 68 4.3 Free scalar in d dimensions . 69 4.3.1 Constructing the Hilbert space . 70 4.3.2 Primaries and descendants . 71 4.3.3 Gram matrix . 72 4.3.4 Null states in integer d .................................... 73 4.3.5 Non-unitarity at fractional d ................................ 75 4.4 TCSA eigenvalue problem . 76 4.4.1 Simple versus generalized eigenvalue problem . 76 4.4.2 Working in the presence of null states . 77 4.4.3 Matrix elements via the OPE method . 78 5 The φ2 flow in TCSA 80 5.1 Canonical quantization on the cylinder . 80 5.1.1 Casimir energy . 82 5.1.2 Massive states on the cylinder . 84 5.2 Intermezzo: computing observables using conformal perturbation theory . 85 5.2.1 Casimir energy . 85 5.2.2 Excited states . 87 5.3 TCSA setup . 87 5.4 Numerical results . 88 5.4.1 Casimir energy . 89 5.4.2 Massive excitations . 89 5.4.3 Discussion . 90 6 Cutoff dependence and renormalization 92 6.1 Warm-up: improving φ4 theory in flat space . 93 6.2 Cutoff dependence in TCSA . 97 6.2.1 General remarks . 97 6.2.2 Computation of ∆H via two-point functions . 99 6.2.3 RG improvement . 101 6.2.4 Other treatments of renormalization . 103 6.3 Renormalization for the φ2 flow................................... 105 6.3.1 Renormalization details . 105 6.3.2 Numerical results . 107 7 The Landau-Ginzburg flow 109 7.1 Theoretical expectations . 110 7.2 Numerical results . 111 7.3 Non-unitarity and complex energy levels . ..