Notes on Quantum Field Theory Marco Serone SISSA, via Bonomea 265, I-34136 Trieste, Italy with contributions by Andrea Gambassi and Roberto Iengo Last update: December 13, 2018 1 Contents 1 Introduction 5 2 General Properties of QFT 7 2.1 The K¨all´en-Lehmann Spectral Representation . .......... 7 2.1.1 AsymptoticTheory: aBriefReminder . .. 7 2.1.2 Spectral Representation . 10 2.1.3 Spectral Representation for Fermions∗∗ ................ 13 2.2 The Cluster Decomposition Principle and the Connected S-Matrix . 16 2.3 The Reduction Formula for Connected S-Matrix Elements . ......... 18 2.4 TheOpticalTheorem .............................. 21 2.4.1 PerturbativeUnitarity . 23 2.5 UnstableParticles ............................... 26 2.6 Causality and Analyticity . ... 28 2.7 Bound States and Resonances∗∗ ......................... 34 3 Renormalization Theory 39 3.1 Superficial Degree of Divergence . .... 39 3.2 Cancellation of divergences and Local Counterterms . .......... 44 3.3 Regularization and Renormalization: QED Case . ....... 46 3.4 Dimensional Regularization . .... 53 4 External Fields and Generating Functionals 57 4.1 The1PI EffectiveAction............................. 58 4.2 The Coleman-Weinberg Effective Potential . ...... 60 4.3 A Subtlety about Effective Potentials∗∗ .................... 62 4.4 Functional Relations . .. 64 4.4.1 Schwinger-Dyson Equation . 64 4.4.2 Symmetries and Ward-Takahashi Identities . ..... 65 2 4.4.3 WTIdentitiesinQED.......................... 69 5 The Renormalization Group 74 5.1 Relevant, Marginal and Irrelevant Couplings . ......... 74 5.2 The Sliding Scale and the Summation of Leading Logs . ....... 79 5.3 Asymptotic Behaviours of β-Functions ..................... 82 5.4 The Callan-Symanzik RG Equations . ... 85 5.5 MinimalSubtraction .............................. 88 5.6 SchemeDependence ............................... 89 5.7 Leading Logs and Callan-Symanzik Equations . ....... 92 5.8 “Irrelevant” RG Flow of Dimensionful Couplings . ........ 93 5.9 “Relevant” RG Flow of Dimensionful Couplings and Renormalization of CompositeOperators............................... 96 5.10 RG Improved Effective Potential . ...101 5.11 Anomalous Dimension of the Photon and QED β-function . 103 6 Non-Abelian Gauge Theories 105 6.1 Introduction and Classical Analysis . .......105 6.2 Quantum Treatment: the Faddeev-Popov Method . ......111 6.3 BRSTSymmetry................................. 119 6.4 TheBackgroundFieldMethod . 124 6.4.1 Themethod................................ 125 6.4.2 Two-Point Function of the Background Field: Feynman Rules . 127 6.5 Proof of the Renormalizability of Non-Abelian Gauge Theories∗∗ ......130 6.5.1 The Master Equation∗∗ .........................130 6.5.2 Structure of divergences⋆⋆ ........................131 7 Effective Field Theories 137 7.1 1PI vs Wilsonian Actions . 138 7.2 TwoScalars.................................... 139 7.3 Yukawa Theory I: Heavy Scalar, Light Fermion . ......142 7.4 Yukawa Theory II: Heavy Fermion, Light Scalar . ......143 7.5 Naturalness and the Hierarchy Problem . .....144 7.6 Non-LeptonicDecays.............................. 145 7.6.1 Useful Color and Spinor identities . ...149 7.7 (Ir)relevance of Higher Dimensional Operators . .........151 7.8 RedundantOperators .............................. 153 3 8 Spontaneously Broken Symmetries 157 8.1 Why Spontaneous Symmetry Breaking? . ...158 8.2 TheGoldstoneTheorem ............................. 160 8.3 Vacuum Alignement and Pseudo-Goldstone Bosons . .......162 8.4 Spontaneously Broken Gauge Symmetries: the Higgs Mechanism . 164 8.5 The Goldstone Boson Equivalence Theorem⋆⋆ .................168 8.6 Effective Field Theories for Broken Symmetries⋆ ...............170 8.6.1 Adding Gauge Fields⋆ ..........................174 8.7 SU(3) SU(3) SU(3) :MesonsinQCD⋆ ...............175 V × A → V 8.8 SO(5) SO(4): A Composite Higgs?⋆⋆ ....................179 → 8.9 Effective Field Theories for Broken Symmetries: Generale Case⋆⋆ ......184 9 Anomalies 188 ⋆ 9.1 The U(1)A Chiral Anomaly from One-Loop Graphs .............189 9.2 Gauge Anomalies⋆ ................................194 9.3 A Relevant Example: Cancellation of Gauge Anomalies in the SM⋆ .....198 9.4 Path Integral Derivation of the Chiral Anomaly⋆⋆ ...............200 9.5 The Wess-Zumino Consistency Conditions⋆⋆ ..................203 9.6 ’t Hooft Anomaly Matching and the Wess-Zumino-Witten Term⋆ ......205 9.7 Anomalous Breaking of Scale Invariance⋆ ...................207 9.8 The Strong CP Problem and a Possible Solution: Axions . .......211 10 Some Formal Developments⋆ 215 10.1 Asymptotic Nature of Perturbation Theory⋆ ..................215 10.1.1 Asymptotic Series and Optimal Truncation⋆ ..............217 10.1.2 Borel Summation⋆ ............................218 10.2 Vacuum Decay in the Presence of External Fields⋆ ..............220 10.2.1 Landau Levels by Path Integral⋆ ....................221 10.2.2 Vacuum Instability for a Constant Electric Field⋆ ...........224 10.2.3 Instability of a Scalar Field Vacuum⋆ ..................225 10.2.4 Instability of a Fermion Field Vacuum⋆⋆ ................231 11 Final Project: The Abelian Higgs Model⋆ 235 11.1 One-loop Effective Potential⋆ ..........................236 11.2 The Quantum Effective Action⋆ .........................238 11.3 RG equations and Their Solutions⋆ .......................242 4 Chapter 1 Introduction Quantum Field Theory (QFT) is the fundamental tool that is currently used for the description of physics at very short distances and high energies. Since high energy implies relativistic motion, QFT has to nicely combine special relativity with quantum mechanics. The description in terms of “fields”, indeed, arise to have a manifestly relativistic invariant description, where time and space are treated (almost) on equal footing. One might think that including special relativity in quantum mechanics should be possible without drastic consequences. This is not true, and its reason is intuitivelyclear.Atveryshorttimescales, the energy-time uncertainty principle tells us that particles with energy E mc2 could ≥ be created from the vacuum for a time t !/E (virtual particles), before disappearing ∼ again in the vacuum. This effect is totally negligible in studying physical systems at low energies and long time scales, but it becomes relevant for physical processes whose time scale is t !/E or less. A quantum description in terms of single-particle wave function ∼ is then inadequate and a more powerful description is needed.Thisformulationisin fact Quantum Field Theory. It leads to striking consequences, such as the prediction of anti-particles and an understanding of the spin-statistic relation between particles. The experimental successes of QFT are impressive, in particularwhenappliedtothedescription of electrodynamics, giving rise to the Quantum Electro Dynamics (QED). Most of the considerations in these lectures are devoted to the study of fields which are weakly interacting, namely in which the interactions canbestudiedinaperturbative fashion, starting from the description of free fields. These notes do assume that the reader has a basic knowledge of QFT. Quantizations of spin 0, spin 1/2 and abelian spin 1 fields are assumed, as well asbasicnotionsofthepath integral formulation of QFT (including Berezin integrationforfermions),Feynmanrules, basic knowledge of renormalization and the notion of functional generators of disconnected, 5 connected and one-particle irreducible (1PI) Green functions. In writing these notes we have often consulted refs. [1]andespecially[2]. Some parts of these notes follow in part either ref. [1]orref.[2]. When this is the case, we warn the reader with a footnote. Notice that in these notes the metric convention is mostly minus, like in ref. [1], in contrast to ref.[2], where it is mostly plus. This implies a multitude of sign changes with respect to ref. [2]. Moreover, we warn the reader that we do not always follow the same notation as refs. [1, 2], in order to have a common and light notation throughout the notes. These notes cover most of the QFT course given at SISSA by M.S.,incollaboration with Roberto Iengo until the academic year 2010-2011, in collaboration with Andrea Gam- bassi until 2014-2015, and in collaboration with Joan Elias-Mir´oin the years 2016-2017 and 2017-2018. The tutorials and exercises, essential partsoftheQFTcourse,arenot reported in these notes. This implies that some technical topics closely associated with the exercises, such as details of the renormalization of QED, Yukawa and other theories, are not currently included in these lecture notes. The notes are far from being comprehensive. Due to lack of time, several important topics are not covered at all, or just mentioned. Infra-red divergences, the deep inelastic scattering and the operator product expansion, connection to critical phenomena and statistical field theory are examples of important topics currently missing. Phenomenological applications are limited to a minimum, since they are more systematically considered in the Standard Model course. The topics marked with a in the text are optional for students in the Astroparticle ∗ curriculum. The topics marked with are optional for all the students. ∗∗ These notes are preliminary and surely contain many typos, imprecisions, etc. We hope that the students will help us in improving the notes and in spotting the many mistakes in there. Acknowledgments IthankRobertoIengoandAndreaGambassifortheircontribution in the elaboration of part of these lecture notes. I thank Roberto Iengo,