Advanced Quantum Field Theory : Renormalization, Non-Abelian Gauge Theories and Anomalies Lecture notes Amsterdam-Brussels-Geneva-Paris Doctoral School \Quantum Field Theory, Strings and Gravity" Brussels, October 2016 Adel Bilal Laboratoire de Physique Th´eoriquede l'Ecole´ Normale Sup´erieure PSL Research University, CNRS, Sorbonne Universit´es,UPMC Paris 6 24 rue Lhomond, F-75231 Paris Cedex 05, France Abstract These are lecture notes of an advanced quantum field theory course intended for graduate students in theoretical high energy physics who are already familiar with the basics of QFT. The first part quickly reviews what should be more or less known: functional inte- gral methods and one-loop computations in QED and φ4. The second part deals in some detail with the renormalization program and the renormalization group. The third part treats the quantization of non-abelian gauge theories and their renormalization with spe- cial emphasis on the BRST symmetry. Every chapter includes some exercices. The fourth part of the lectures, not contained in the present notes but based on arXiv:0802.0634, discusses gauge and gravitational anomalies, how to characterise them in various dimen- sions, as well as anomaly cancellations. Contents 1 Functional integral methods 1 1.1 Path integral in quantum mechanics . .1 1.2 Functional integral in quantum field theory . .2 1.2.1 Derivation of the Hamiltonian functional integral . .2 1.2.2 Derivation of the Lagrangian version of the functional integral . .3 1.2.3 Propagators . .5 1.3 Green functions, S-matrix and Feynman rules . .7 1.3.1 Vacuum bubbles and normalization of the Green functions . .7 1.3.2 Generating functional of Green functions and Feynman rules . 10 1.3.3 Generating functional of connected Green functions . 12 1.3.4 Relation between Green functions and S-matrix . 14 1.4 Quantum effective action . 16 1.4.1 Legendre transform and definition of Γ[']......................... 16 1.4.2 Γ['] as quantum effective action and generating functional of 1PI-diagrams . 17 1.4.3 Symmetries and Slavnov-Taylor identities . 20 1.5 Functional integral formulation of QED . 21 1.5.1 Coulomb gauge . 21 1.5.2 Lorentz invariant functional integral formulation and α-gauges . 23 1.5.3 Feynman rules of spinor QED . 26 1.6 Exercises . 28 1.6.1 Linear operators on L2(R4)................................ 28 1.6.2 Green's function and generating functional in 0 + 1 dimensions . 29 1.6.3 Z[J], W [J] and Γ['] in φ4-theory . 30 1.6.4 Effective action from integrating out fermions in QED . 30 2 A few results independent of perturbation theory 31 2.1 Structure and poles of Green functions . 31 2.2 Complete propagators, the need for field and mass renormalization . 34 2.2.1 Example of a scalar field φ ................................. 34 2.2.2 Example of a Dirac field . 37 2.3 Charge renormalization and Ward identities . 38 2.4 Photon propagator and gauge invariance . 40 2.5 Exercices . 43 2.5.1 Two-particle intermediate states and branch cut of the two-point function . 43 2.5.2 A theory of two scalars, one charged and one neutral . 44 3 One-loop radiative corrections in φ4 and QED 45 3.1 Setup . 45 3.2 Evaluation of one-loop integrals and dimensional regularization . 46 3.3 Wick rotation . 50 3.4 Vacuum polarization . 52 3.5 Electron self energy . 55 3.6 Vertex function . 57 3.6.1 Cancellation of the divergent piece . 58 3.6.2 The magnetic moment of the electron: g − 2 ....................... 59 3.7 One-loop radiative corrections in scalar φ4 ............................. 61 3.8 Exercices . 64 3.8.1 The simplest (non-trivial) one-loop integral . 64 3.8.2 Vacuum polarization in QED . 65 Adel Bilal : Advanced Quantum Field Theory -2 Lecture notes - September 29, 2016 4 General renomalization theory 66 4.1 Degree of divergence . 66 4.2 Structure of the divergences . 69 4.3 Bogoliubov-Parasiuk-Hepp-Zimmermann prescription and theorem . 72 4.4 Summary of the renormalization program and proof . 73 4.5 The criterion of renormalizability . 74 4.6 Exercices . 75 4.6.1 Yukawa theory . 75 4.6.2 Quantum gravity . 75 4.6.3 Non-renormalizable interactions from integrating out a massive field . 76 5 Renormalization group and Callan-Szymanzik equations 77 5.1 Running coupling constant and β-function: examples . 77 5.1.1 Scalar φ4-theory . 77 5.1.2 QED . 80 5.2 Running coupling constant and β-functions: general discussion . 82 5.2.1 Several mass scales . 82 5.2.2 Relation between the one-loop β-function and the counterterms . 83 5.2.3 Scheme independence of the first two coefficients of the β-function . 85 5.3 β-functions and asymptotic behaviors of the coupling . 86 5.3.1 case a : the coupling diverges at a finite scale M ..................... 86 5.3.2 case b : the coupling continues to grow with the scale . 86 5.3.3 case c : existence of a UV fixed point . 87 5.3.4 case d : asymptotic freedom . 88 5.3.5 case e : IR fixed point . 89 5.4 Callan-Symanzik equation for a massless theory . 90 5.4.1 Renormalization conditions at scale µ ........................... 90 5.4.2 Callan-Symanzik equations . 92 5.4.3 Solving the Callan-Symanzik equations . 94 5.4.4 Infrared fixed point and critical exponents / large momentum behavior in asymptotic free theories . 95 5.5 Callan-Symanzik equations for a massive theory . 96 5.5.1 Operator insertions and renormalization of local operators . 96 5.5.2 Callan-Symanzik equations in the presence of operator insertions . 97 5.5.3 Massive Callan-Symanzik equations . 98 5.6 Exercices . 99 5.6.1 β-function in φ3-theory in 6 dimensions . 99 5.6.2 3-point vertex function in an asymptotically free theory . 99 6 Non-abelian gauge theories: formulation and quantization 100 6.1 Non-abelian gauge transformations and gauge invariant actions . 100 6.2 Quantization . 104 6.2.1 Faddeev-Popov method . 105 6.2.2 Gauge-fixed action, ghosts and Feynman rules . 107 6.2.3 BRST symmetry . 108 6.3 BRST cohomology . 111 6.4 Exercices . 115 6.4.1 Some one-loop vertex functions . 115 6.4.2 More general gauge-fixing functions . 115 6.4.3 Two ghost - two antighost counterterms . 115 Adel Bilal : Advanced Quantum Field Theory -1 Lecture notes - September 29, 2016 7 Renormalization of non-abelian gauge theories 116 7.1 Slavnov-Taylor identities and Zinn-Justin equation . 116 7.1.1 Slavnov-Taylor identities . 116 7.1.2 Zinn-Justin equation . 117 7.1.3 Antibracket . 118 7.1.4 Invariance of the measure under the BRST transformation . 118 7.2 Renormalization of gauge theories theories . 120 7.2.1 The general structure and strategy . 120 7.2.2 Constraining the divergent part of Γ . 122 7.2.3 Conclusion and remarks . 127 7.3 Background field gauge . ..