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Quantum Field Theory II

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Quantum Field Theory II Quantum Field Theory II FRANC¸ OIS GELIS Institut de Physique Theorique,´ CEA-Saclay MASTER HEP, E´ COLE POLYTECHNIQUE 2017–2018 Contents I Quantum Field Theory II1 1 Reminders of QFT I3 1.1 Special relativity.................................3 1.2 Free scalar fields, Mode decomposition.....................5 1.3 Interacting scalar fields, Interaction representation................8 1.4 LSZ reduction formulas.............................. 11 1.5 Generating functional............................... 13 1.6 Perturbative expansion and Feynman rules.................... 17 1.7 Calculation of loop integrals........................... 22 1.8 Kallen-Lehmann¨ spectral representation..................... 24 1.9 Ultraviolet divergences and renormalization................... 27 1.10 Spin 1/2 fields................................... 32 1.11 Spin 1 fields.................................... 37 1.12 Abelian gauge invariance, Quantum Electrodynamics.............. 41 1.13 Charge conservation and Ward-Takahashi identities............... 43 1.14 Perturbative Unitarity............................... 45 2 Functional quantization 57 2.1 Path integral in quantum mechanics....................... 57 2.2 Classical limit, Least action principle....................... 60 2.3 More functional machinery............................ 61 2.4 Path integral in scalar field theory........................ 66 2.5 Functional determinants.............................. 68 2.6 Quantum effective action............................. 70 2.7 Euclidean path integral and Statistical mechanics................ 74 i F. GELIS, 2017 3 Fermion path integral 77 3.1 Grassmann variables............................... 77 3.2 Path integral for fermions............................. 83 3.3 Path integral for photons............................. 84 3.4 Schwinger-Dyson equations........................... 86 3.5 Quantum anomalies................................ 89 4 Non-Abelian gauge symmetry 99 4.1 Non-abelian Lie groups and algebras....................... 99 4.2 Representations.................................. 105 4.3 Covariant derivative................................ 108 4.4 Field strength................................... 110 4.5 Non-Abelian gauge theories........................... 111 4.6 Classical equations of motion........................... 112 4.7 θ-term and strong-CP problem.......................... 113 4.8 Non-local gauge invariant operators....................... 117 5 Quantization of Yang-Mills theory 123 5.1 Introduction.................................... 123 5.2 Gauge fixing.................................... 124 5.3 Fadeev-Popov quantization and Ghost fields................... 125 5.4 Feynman rules for non-abelian gauge theories.................. 128 5.5 Ghosts and unitarity................................ 132 6 Renormalization of gauge theories 143 6.1 Ultraviolet power counting............................ 143 6.2 Symmetries of the quantum effective action................... 144 6.3 Renormalizability................................. 149 6.4 Background field method............................. 153 7 Renormalization group 159 7.1 Callan-Symanzik equations............................ 159 7.2 Correlators containing composite operators................... 162 7.3 Operator product expansion............................ 165 7.4 Application: QCD corrections to weak decays.................. 168 7.5 Non-perturbative renormalization group..................... 174 ii CONTENTS II Quantum Field Theory III 183 8 Effective field theories 185 8.1 EFT versus renormalization group........................ 185 8.2 Loops, renormalization and matching...................... 185 8.3 Standard model as an EFT............................ 185 8.4 Nonlinear sigma models............................. 185 8.5 Heavy quark effective theory........................... 185 8.6 Soft Collinear effective theory.......................... 185 9 Quantum anomalies 187 9.1 Axial anomalies in a gauge background..................... 187 9.2 Generalizations.................................. 197 9.3 Wess-Zumino consistency conditions....................... 201 9.4 ’t Hooft anomaly matching............................ 204 10 Localized field configurations 207 10.1 Domain walls................................... 207 10.2 Skyrmions..................................... 211 10.3 Monopoles.................................... 212 10.4 Instantons..................................... 220 11 Modern tools for amplitudes 233 11.1 Shortcomings of the usual approach....................... 233 11.2 Color ordering of gluonic amplitudes....................... 234 11.3 Spinor-helicity formalism............................. 238 11.4 Britto-Cachazo-Feng-Witten on-shell recursion................. 247 11.5 Gravitational amplitudes............................. 254 12 Lattice field theory 263 12.1 Discretization of space-time........................... 263 12.2 Scalar field theory................................. 263 12.3 Gluons and Wilson action............................. 265 12.4 Monte-Carlo sampling.............................. 267 12.5 Fermions..................................... 268 12.6 Hadron mass determination on the lattice.................... 272 12.7 Wilson loops and confinement.......................... 273 iii F. GELIS, 2017 13 Quantum field theory at finite temperature 277 13.1 Canonical thermal ensemble........................... 277 13.2 Finite-T perturbation theory........................... 278 13.3 Long distance effective theories......................... 291 13.4 Out-of-equilibrium systems............................ 297 14 Strong fields and semi-classical methods 305 14.1 Introduction.................................... 305 14.2 Expectation values in a coherent state...................... 306 14.3 Quantum field theory with external sources................... 312 14.4 Observables at LO and NLO........................... 313 14.5 Multi-point correlation functions at tree level.................. 318 iv Quantum Field Theory II 1 Chapter 1 Reminders of QFT I 1.1 Special relativity 1.1.1 Lorentz transformations Special relativity plays a crucial role in quantum field theories1. Various observers in frames that are moving at a constant speed relative to each other should be able to describe physical phenomena using the same laws of Physics. This does not imply that the equations governing these phenomena are independent of the observer’s frame, but that these equations transform in a constrained fashion –depending on the nature of the objects they contain– under a change of reference frame. Let us consider two frames F and F 0, in which the coordinates of a given event are respec- 0 tively xµ and x µ.A Lorentz transformation is a linear transformation such that the interval ds2 ≡ dt2 - dx2 is the same in the two frames2. If we denote the coordinate transformation by 0µ µ ν x = Λ ν x ; (1.1) the matrix Λ of the transformation must obey µν µ ν ρσ g = Λ ρΛ σ g (1.2) where gµν is the Minkowski metric tensor 0+1 1 B -1 C B C gµν ≡ B C : (1.3) @ -1 A -1 1An exception to this assertion is for quantum field models applied to condensed matter physics, where the basic degrees of freedom are to a very good level of approximation described by Galilean kinematics. 2The physical premises of special relativity require that the speed of light be the same in all inertial frames, which implies solely that ds2 = 0 be preserved in all inertial frames. The group of transformations that achieves this is called the conformal group. In four space-time dimensions, the conformal group is 15 dimensional, and in addition to the 6 orthochronous Lorentz transformations it contains dilatations as well as non-linear transformations called special conformal transformations. 3 F. GELIS, 2017 If we consider an infinitesimal Lorentz transformation, µ µ µ Λ ν = δ ν + ! ν (1.4) (with all components of ! much smaller than unity), this implies that !µν = -!νµ (1.5) (with all indices down). Consequently, there are 6 independent Lorentz transformations, three of which are ordinary rotations and three are boosts. Note that the infinitesimal transforma- tions (1.4) have a determinant3 equal to +1 (they are called proper transformations), and do 0 not change the direction of the time axis since Λ 0 = 1 ≥ 0 (they are called orthochronous). Any combination of such infinitesimal transformations shares the same properties, and their set forms a subgroup of the full group of transformations that preserve the Minkowski metric. 1.1.2 Representations of the Lorentz group More generally, a Lorentz transformation acts on a quantum system via a transformation U(Λ), that forms a representation of the Lorentz group, i.e. U(ΛΛ 0) = U(Λ)U(Λ 0) : (1.6) For an infinitesimal Lorentz transformation, we can write i U(1 + !) = I + ! Mµν : (1.7) 2 µν (The prefactor i=2 in the second term of the right hand side is conventional.) Since the !µν are antisymmetric, the generators Mµν can also be chosen antisymmetric. By using eq. (1.6) for the Lorentz transformation Λ-1Λ 0Λ, we arrive at -1 µν µ ν ρσ U (Λ)M U(Λ) = Λ ρΛ σM ; (1.8) indicating that Mµν transforms as a rank-2 tensor. When used with an infinitesimal transforma- tion Λ = 1 + !, this identity leads to the commutation relation that defines the Lie algebra of the Lorentz group Mµν;Mρσ = i(gµρMνσ - gνρMµσ)- i(gµσMνρ - gνσMµρ) : (1.9) In a fashion similar to eq. (1.8), we can obtain the transformation of the 4-impulsion Pµ, -1 µ µ ρ U (Λ)P U(Λ) = Λ ρP ; (1.10) which leads to the following commutation relation between Pµ and Mµν, Pµ;Mρσ = i(gµσPρ - gµρPσ) :
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