Quantum Field Theoryweb.phys.ntnu.no/~mika/lect1.pdftime. In the formulation of quantum theory we...

Quantum Field Theory:Lecture notes for FY3464 and FY3466 and a bit more

M. Kachelrieß

M. Kachelrieß

Institutt for fysikkNTNU, TrondheimNorwayemail: [email protected]

If you find typos (almost sure, if you read carefully enough), conceptional errors, strangeexplanations, inconsistencies or have any suggestions, send me please an email!

Copyright c© M. Kachelrieß 2014; cover figure from [Rav05].Last up-date May 29, 2014, available from http://web.phys.ntnu.no/~mika/cpp.php

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Contents

1. Introduction 11.1. Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1. Hamilton’s principle and Lagrange’s equations . . . . . . . . . . . . . 21.1.2. Palatini’s formalism and Hamilton’s equations . . . . . . . . . . . . . 51.1.3. Green functions and the response method . . . . . . . . . . . . . . . . 71.1.4. Relativistic particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2. Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.1. Manifolds and tensor fields . . . . . . . . . . . . . . . . . . . . . . . . 121.2.2. Covariant derivative and the geodesic equation . . . . . . . . . . . . . 151.2.3. Integration and Gauss’ theorem . . . . . . . . . . . . . . . . . . . . . . 20

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.bis Quantum mechanics 241.1. Reminder of the operator approach . . . . . . . . . . . . . . . . . . . . . . . . 241.2. Path integrals in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . 271.3. Generating functional for Green functions . . . . . . . . . . . . . . . . . . . . 311.4. Oscillator as an 1–dimensional field theory . . . . . . . . . . . . . . . . . . . . 34Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2. Scalar fields 392.1. Lagrange formalism and path integrals for fields . . . . . . . . . . . . . . . . . 392.2. Generating functional for a scalar field . . . . . . . . . . . . . . . . . . . . . . 402.3. Green functions for a free scalar field . . . . . . . . . . . . . . . . . . . . . . . 472.4. Vacuum energy and the Casimir effect . . . . . . . . . . . . . . . . . . . . . . 512.5. Perturbation theory for interacting fields . . . . . . . . . . . . . . . . . . . . . 532.6. Green functions for the λφ4 theory . . . . . . . . . . . . . . . . . . . . . . . . 552.7. Loop diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.7.1. Self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.7.2. Vacuum energy density . . . . . . . . . . . . . . . . . . . . . . . . . . 632.7.3. Vertex correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.7.4. Basic idea of renormalisation . . . . . . . . . . . . . . . . . . . . . . . 68

2.A. Appendix: Evaluation of Feynman integrals . . . . . . . . . . . . . . . . . . . 70Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3. Global symmetries and Noether’s theorem 743.1. Internal symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.2. Noether’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

v

Contents

3.3. Symmetries of a general space-time . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4. Outlook: Classical versus quantum symmetries . . . . . . . . . . . . . . . . . 81

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4. Spin-1 and spin-2 fields 84

4.1. Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2. Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3. Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4. Source of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.A. Appendix: Large extra dimensions and massive gravity . . . . . . . . . . . . . 96

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5. Fermions and the Dirac equation 100

5.1. Spinor representation of the Lorentz group . . . . . . . . . . . . . . . . . . . . 100

5.2. Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3. Dirac, Weyl and Majorana fermions . . . . . . . . . . . . . . . . . . . . . . . 115

5.4. Spin-statistic connection and Graßmann variables . . . . . . . . . . . . . . . . 118

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6. Scattering processes 127

6.1. Unitarity of the S-matrix and its consequences . . . . . . . . . . . . . . . . . 127

6.2. LSZ reduction formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.3. Specific processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3.1. Trace method: Compton scattering and the Klein-Nishina formula . . 134

6.3.2. Helicity method and polarized QED processes . . . . . . . . . . . . . . 139

6.4. Soft photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.A. Appendix: Decay widths and cross sections . . . . . . . . . . . . . . . . . . . 145

6.A.1. Decay widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.A.2. Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7. Gauge theories 152

7.1. Electromagnetism as abelian gauge theory . . . . . . . . . . . . . . . . . . . . 152

7.2. Non-abelian gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.2.1. Gauge invariant interactions . . . . . . . . . . . . . . . . . . . . . . . . 153

7.2.2. Gauge fields as connection . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.2.3. Curvature of space-time . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.3. Quantization of gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.3.1. Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.3.2. Non-abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.A. Appendix: Feynman rules for an unbroken gauge theory . . . . . . . . . . . . 165

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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Contents

8. Renormalisation 168

8.1. Anomalous magnetic moment of the electron . . . . . . . . . . . . . . . . . . 169

8.2. Power counting and renormalisability . . . . . . . . . . . . . . . . . . . . . . . 175

8.3. Renormalisation of the λφ4 theory . . . . . . . . . . . . . . . . . . . . . . . . 177

8.3.1. Renormalisation and regularisation . . . . . . . . . . . . . . . . . . . . 177

8.3.2. The λφ4 theory at one-loop . . . . . . . . . . . . . . . . . . . . . . . . 180

8.4. Vacuum polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.4.1. Vacuum polarisation in QED . . . . . . . . . . . . . . . . . . . . . . . 187

8.4.2. Vacuum polarisation in QCD . . . . . . . . . . . . . . . . . . . . . . . 191

8.5. Effective action and Ward identities . . . . . . . . . . . . . . . . . . . . . . . 194

8.5.1. Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

8.5.2. Ward-Takahashi identities . . . . . . . . . . . . . . . . . . . . . . . . . 196

8.6. Renormalisation and critical phenomena . . . . . . . . . . . . . . . . . . . . . 199

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

9. Thermal field theory 208

9.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

9.2. Scalar gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

9.2.1. Free scalar gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

9.2.2. Interacting scalar gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

9.A. Appendix: Equilibrium statistical physics in a nut-shell . . . . . . . . . . . . 217

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

10.Symmetries and symmetry breaking 221

10.1. Symmetry breaking and Goldstone’s theorem . . . . . . . . . . . . . . . . . . 221

10.2. Renormalisation of theories with SSB . . . . . . . . . . . . . . . . . . . . . . 227

10.3. Abelian Higgs model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

11.GSW model of electroweak interactions 239

11.1. Gauge sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

11.2. Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

11.3. Higgs sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

12.Phase transitions and topological defects 251

12.1. Phase transitions in the abelian Higgs model . . . . . . . . . . . . . . . . . . 251

12.2. Decay of the false vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

12.3. Topological defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

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Contents

13.Anomalies, instantons and axions 26913.1. Axial anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26913.2. Instantons and the strong CP problem . . . . . . . . . . . . . . . . . . . . . . 27513.3. Axions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

14.Hadrons, partons and QCD 28614.1. Trace anomaly and hadrons masses . . . . . . . . . . . . . . . . . . . . . . . . 286

14.2. Corrections to e+e− annihilation . . . . . . . . . . . . . . . . . . . . . . . . . 28914.3. DGLAP equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

15.Gravity as a gauge theory 295

15.1. Vielbein formalism and the spin connection . . . . . . . . . . . . . . . . . . . 29515.2. Action of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29915.3. Linearized gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30215.4. Wess-Zumino model and supersymmetry . . . . . . . . . . . . . . . . . . . . . 306Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

16.Cosmological models for an homogeneous, isotropic universe 31116.1. Friedmann-Robertson-Walker metric . . . . . . . . . . . . . . . . . . . . . . . 31116.2. Friedmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

16.3. Evolution of simple cosmological models . . . . . . . . . . . . . . . . . . . . . 32316.4. Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

17.Thermal relics 329

17.1. Time-line of important dates in the early universe . . . . . . . . . . . . . . . 32917.2. Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33117.3. Thermal relics as dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 33417.4. WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

18.Baryogenesis 34218.1. Sakharov conditions for baryogenesis . . . . . . . . . . . . . . . . . . . . . . . 34218.2. Sphaleron transitions and B − L violation . . . . . . . . . . . . . . . . . . . . 345

18.3. Electroweak baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34618.4. Leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

19.Quantum fields in curved space-time 348

19.1. Quantisation in curved space-times . . . . . . . . . . . . . . . . . . . . . . . . 348

viii

Notation and conventions

We use natural units with ~ = c = 1, but keep Newton’s gravitational constant GN 6= 1.Maxwell’s equations are written in the Lorentz-Heaviside version of the cgs system. Thusthere is a factor 4π in the Coulomb law, but not in Maxwell’s equations. Sommerfeld’sfine-structure constant is α = e2/(4π) ≈ 1/137.

We choose as signature of the metric −2, thence the metric tensor in Minkowski space isηµν = ηµν = diag(1,−1,−1,−1). If not otherwise specified, Einstein’s summation conventionis implied.

The d’Alembert or wave operator is ≡ ∂µ∂µ = ∂2

∂t2−∆, while the four-dimensional nabla

operator has the components ∂µ ≡ ∂∂xµ =

(∂∂t ,

∂∂x ,

∂∂y ,

∂∂z

)

.

A boldface letter denotes the components of a three-vector V = Vx, Vy, Vz = Vi, i =1, 2, 3 or the three-dimensional part of a contravariant vector with components V µ =V 0, V 1, V 2, V 3 = V 0,V ; a covariant vector has in Minkowski space the componentsVµ = (V0,−V ).Vectors and tensors in index free notation are also denoted by boldface letters, V = V µ∂µ org = gµνdx

µ ⊗ dxν .Greek indices α, β, . . . encompass the range α = 0, 1, 2, . . . D, latin indices i, j, k, . . . encom-pass i = 1, 2, . . . D, where D denotes the dimension of the space-time. In chapter 15, latinindices a, b, c, . . . denote tensor components with respect to the vielbein field eaµ.

Our nomenclature for disconnected, connected, and one-particle irreducible (1PI) n-pointGreen functions and their corresponding generating functionals is as follows:

Green function generating functional

disconnected G(x1, . . . , xn) Z[J, . . .]connected G(x1, . . . , xn) W [J, . . .]1PI Γ(x1, . . . , xn) Γ[φ, . . .]

We normalise Dirac spinors as u(p, s)u(p, s) = 2m.

We use as covariant derivative Dµ = ∂µ + igAaµTa with coupling g > 0, field strength F aµν =

∂µAaν−∂νAaµ−fabcAbµAcν and generators T a satisfying [T a, T b] = ifabcT c for all gauge groups.

Special cases used in the SM are the groups U(1), SU(2) and SU(3) with g = e, g, gs andT a = 1, τa/2, λa/2 in the fundamental representation. In particular, the electric charge ofthe positron is q = e > 0.

Employing dimensional regularisation, we change the dimension of loop integrals from n = 4to n = 4− ε or d = 2ω = 4− 2ε.

The results of problems marked by ♣ are used later in the text, those marked by ♥ requiremore efforts and time than average ones.

xviii

1. Introduction

This chapter introduces material from classical mechanics and differential geometry. Depend-ing on the background of the reader, the degree of familiarity with these topics may varyconsiderably. As a hopefully for most easy starter, we review the formulation of classicalmechanics using action principles in the first section. We illustrate also the use of the Greenfunction method using the example of harmonic oscillator and recall the mechanics of a rela-tivistic particle. Then we introduce the concept of tensor fields on Riemannian manifolds andthe tools necessary to do physics in curved space-times. The major two concepts introducedare “covariant derivative” which describes how a vector is transported in a curved space-time.

1.1. Classical mechanics

We start by recalling briefly the Lagrangian and Hamiltonian formulation of classical me-chanics which are both important for the transition to a quantum theory. Either formulationof classical mechanics can be derived using an action principle as starting point.

Variational principles Fundamental laws of Nature as Newton’s axioms or Maxwell equa-tions were discovered as differential equations. Starting from Leibniz and Euler, physicistsrealised that one can re-express differential equations in the form of variational principles:In this approach, the evolution of a physical system is described by the extremum of an ap-propriately chosen functional. Various versions of such variational principles exist, but theyhave in common that the functionals used have the dimension of “energy times time”, i.e. thefunctionals have the same dimension as Planck’s constant ~. A quantity with this dimensionis called action S. An advantage of using the action as main tool to describe dynamical sys-tems is that this allows us to implement easily both space-time and internal symmetries. Forinstance, choosing as action a Lorentz scalar leads automatically to relativistically invariantfield equations. Moreover, the action S as a scalar quantity summarises economically theinformation contained typically in a set of various coupled (partial) differential equations.

If the variational principle is formulated as an integral principle, then the functional S willdepend on the whole path q(t) described by the system between the considered initial and finaltime. In the formulation of quantum theory we will pursue, the propagator as probabilityamplitude for the time evolution of a particle from the point q(t) to the point q′(t′) willplay a central role. In the usual approach to quantum mechanics, we reinterpret the classicalHamilton functionH(q, p) as an operator imposing canonical commutation relations, [q, p] = i.In contrast, we will look for a direct connection from the classical action S[q] along the path[q(t): q′(t′)] to the transition amplitude 〈q′, t′|q, t〉. Thus the use of the action principle willnot only allow us an easy discussion of the symmetries of a physical system, but lies also atthe heart of the approach to quantum theory we will follow.

1

1. Introduction

1.1.1. Hamilton’s principle and Lagrange’s equations

A map F [f(x)] from a certain space of functions f(x) into the real or complex numbers iscalled a functional. We will consider functionals from the space C2[a : b] of (at least) twicedifferentiable functions between fixed points a and b. More specifically, Hamilton’s principleuses as functional the action S defined by

S[qi] =

∫ b

adt L(qi, qi, t) , (1.1)

where L is a function of the 2n independent functions qi and qi = dqi/dt as well as of theparameter t. In classical mechanics, we call L the Lagrange function of the system, qi areits generalised coordinates, qi the corresponding velocities and t is the time. The extremaδS[q] = 0 of this action give those paths q(t) from a to b which are solutions of the equationof motions for the system described by L.

How do we find those paths that extremize the action S? First of all, we have to prescribewhich variables are kept constant, which are varied and which constraints the variations haveto obey. Depending on the variation principle we choose, these conditions and the functionalform of the action will differ. Hamilton’s principle corresponds to an infinitesimal variationof the path, qi(t) → qi(t) + δqi(t) with δqi(t) = εηi(t), that keeps the endpoints fixed but isotherwise arbitrary. Moreover, the time t is not varied in Hamilton’s principle, δt = 0 and thescale factor ε determining the magnitude of the variation δqi(t) is time-independent. Thusvarying the action S requires only to calculate the variation δL of the Lagrangian L. Thenotation S[qi] stresses that we consider the action as a functional only of the coordinates qi:The velocities qi are not varied independently because ε is time-independent, and thus theirchange δ(qi) is determined by the change of the coordinates, δ(qi) = d/dt(δqi).

Following this prescription, the resulting infinitesimal variation δS[qi] of the action,

δS[qi] ≡ S[qi + δqi]− S[qi] +O(ε2) , (1.2)

can be obtained by a Taylor expansion of L(qi + δqi, qi + δqi, t) up to linear terms in ε.Performing the expansion, we arrive at

δS[qi] =

∫ b

adt δL(qi, qi, t) =

∫ b

adt

(∂L

∂qiδqi +

∂L

∂qiδqi)

, (1.3)

where we applied—as always in the following—Einstein’s convention to sum over a repeatedindex pair. Thus e.g. the first term in the bracket equals

∂L

∂qiδqi ≡

n∑

i=1

∂L

∂qiδqi

for a system described by n generalised coordinates. We can eliminate the dependent variationδqi of the velocities, integrating the second term by parts using δ(qi) = d/dt(δqi),

δS[qi] =

∫ b

adt

[∂L

∂qi− d

dt

(∂L

∂qi

)]

δqi +

[∂L

∂qiδqi]b

a

. (1.4)

The boundary term [. . .]ba vanishes, because we required that the variations δqi are zero at theendpoints a and b. Since the variations are otherwise arbitrary, the sum in the first bracket

2


has to be zero for an extremal curve, δS = 0. Paths that satisfy δS = 0 are classically allowed.The equations resulting from the condition δS = 0 are called the (Euler-) Lagrange equationsof the action S,

δS

δqi=∂L

∂qi− d

dt

∂L

∂qi= 0 , (1.5)

and give the equations of motion of the system specified by L.Hamilton’s principle is often also called the principle of least action. Note that the last

name is a misnomer, since the extremum can be also a maximum or saddle point of the action.

Example: Show that the variation δ and the time-derivative d/dt acting on q commute:We consider a variation δq = q − q0 = εη(t) with ε time-independent and denote the original pathwith q0. Then

dq

dt= q = q0 + εη

or

δ(q) ≡ q − q0 = εη =d

dt(δq) ,

since we keep ε constant.

The Lagrangian L is not uniquely fixed: Adding a total time-derivative, L → L′ = L +df(q, t)/dt, does not change the resulting Lagrange equations,

S′ = S +

∫ b

adt

df

dt= S + f(q(b), tb)− f(q(a), ta) , (1.6)

since the last two terms vanish varying the action with the restriction of fixed endpoints aand b.

Lagrange function We illustrate now how one can use symmetries to constrain the possibleform a Lagrangian L. As example, we consider the case of a free non-relativistic particlewith mass m subject to the Galilean principle of relativity. More precisely, we use that thehomogeneity of space and time forbids that L depends on x and t, while the isotropy of spaceimplies that L depends only on the norm of the velocity vector v, but not on its direction.Thus the Lagrange function of a free particle can be only a function of v2,

L = L(v2).

Let us consider two inertial frames moving with the infinitesimal velocity ε relative to eachother. Then a Galilean transformation connects the velocities measured in the two framesas v′ = v + ε. The Galilean principle of relativity requires that the laws of motion have thesame form in both frames, and thus the Lagrangians can differ only by a total time-derivative.Expanding the difference δL in ε gives with δv2 = 2vε

δL =∂L

∂v2δv2 = 2vε

∂L

∂v2. (1.7)

The difference δL has to be a total time-derivative. Since v = q, the derivative term ∂L/∂v2

has to be independent of v. Hence, L ∝ v2 and, to be consistent with usual notation, we callthe proportionality constant m/2, and the total expression kinetic energy T ,

L = T =1

2mv2 . (1.8)

3

1. Introduction

For a system of non-interacting particles, the Lagrange function L is additive, L =∑

a12mav

2a.

If there are interactions (assumed for simplicity to dependent only on the coordinates), thenwe subtract a function V (r1, r2, . . .) called potential energy. One confirms readily that thischoice for L reproduces Newton’s law of motion.

Energy The Lagrangian of a closed system depends, because of the homogeneity of time,not on time. Its total time derivative is

dL

dt=∂L

∂qiqi +

∂L

∂qiqi . (1.9)

Replacing ∂L/∂qi by (d/dt)∂L/∂qi, it follows

dL

dt= qi

d

dt

∂L

∂qi+∂L

∂qiqi =

d

dt

(

qi∂L

∂qi

)

. (1.10)

Hence the quantity

E ≡ qi ∂L∂qi− L (1.11)

remains constant during the evolution of a closed system. This holds also more generally, e.g.in the presence of static external fields, as long as the Lagrangian is not time-dependent.

We have still to show that E coincides indeed with the usual definition of energy. Usingas Lagrange function L = T (q, q) − V (q), where the kinetic energy T is quadratic in thevelocities, we have

qi∂L

∂qi= qi

∂T

∂qi= 2T (1.12)

and thus E = 2T − L = T + V .

Conservation law’s In a general way, we can derive the connection between a symmetry ofthe Lagrangian and a corresponding conservation law as follows: Let us assume that under achange of coordinates qi → qi + δqi, the Lagrangian changes as total time derivative,

L→ L+ δL = L+dδF

dt. (1.13)

In this case, the equation of motions are unchanged and the coordinate change qi → qi + δqi

is a symmetry of the Lagrangian. But the change dδF/dt has to equal δL induced by thevariation δqi,

∂L

∂qiδqi +

∂L

∂qiδqi − dδF

dt= 0 . (1.14)

Replacing again ∂L/∂qi by (d/dt)∂L/∂qi and applying the product rule gives as conservedquantity

Q =∂L

∂qiδqi − δF . (1.15)

Thus any continuous symmetry of a Lagrangian system results in a conserved quantity. Inparticular, energy conservation follows for a system invariant under time-translations withδqi = qiδt. Other conservation laws are discussed in problem 1.3.

4


1.1.2. Palatini’s formalism and Hamilton’s equations

Legendre transformation and the Hamilton function In the Lagrange formalism, we de-scribe a system by specifying its generalized coordinates and velocities using the Lagrangian,L = L(qi, qi, t). An alternative is to use generalized coordinates and their canonically conju-gated momenta pi defined as

pi =∂L

∂qi. (1.16)

The passage from qi, qi to qi, pi is a special case of a Legendre transformation1: Startingfrom the Lagrangian L we define a new function H(qi, pi, t) called Hamiltonian or Hamilton

function viaH(qi, pi, t) = piq

i − L(qi, qi, t) . (1.17)

Here we assume that we can invert the definition (1.16) and are thus able to substitutevelocities qi by momenta pi in the Lagrangian L.

The physical meaning of the Hamiltonian H follows immediately comparing its definingequation with the one for the energy E. Thus the numerical value of the Hamiltonian equalsthe energy of a dynamical system; we insist however that H is expressed as function ofcoordinates and their conjugated momenta.

A coordinate qi that does not appear explicitly in L is called cyclic. The Lagrange equationsimply then ∂L/∂qi = const., so that the corresponding canonically conjugated momentumpi = ∂L/∂qi is conserved.

Palatini’s formalism and Hamilton’s equations Previously, we considered the action S as afunctional only of qi. Then the variation of the velocities qi is not independent and we arriveat n second order differential equations for the coordinates qi. An alternative approach is toallow independent variations of the coordinates qi and of the velocities qi. We trade the latteragainst the momenta pi = ∂L/∂qi and rewrite the action as

S[qi, pi] =

∫ b

adt[piq

i −H(qi, pi, t)]. (1.18)

The independent variation of coordinates qi and momenta pi gives

δS[qi, pi] =

∫ b

adt

[

piδqi + qiδpi −

∂H

∂qiδqi − ∂H

∂piδpi

]

. (1.19)

The first term can be partially integrated, and the resulting boundary terms vanishes byassumption. Collecting then the δqi and δpi terms and requiring that the variation is zero,we obtain

0 = δS[qi, pi] =

∫ b

adt

[

−(

pi +∂H

∂qi

)

δqi +

(

qi − ∂H

∂pi

)

δpi

]

. (1.20)

As the variations δqi and δpi are independent, their coefficients in the round brackets have tovanish separately. Thus we obtain in this formalism directly Hamilton’s equations,

qi =∂H

∂pi, pi = −

∂H

∂qi. (1.21)

1The concept of a Legendre transformation should be familiar from thermodynamics.

5

1. Introduction

This approach can be applied also to field theories, treating e.g. the vector potential Aµ andthe field-strength Fµν in electrodynamics as independent variables.

Consider now an observable O = O(qi, pi, t). Its time-dependence is given by

dO

dt=∂O

∂qiqi +

∂O

∂pipi +

∂O

∂t=∂O

∂qi∂H

∂pi− ∂O

∂pi

∂H

∂qi+∂O

∂t, (1.22)

where we used Hamilton’s equations. If we define the Poisson brackets A,B between twoobservables A and B as

A,B = ∂A

∂qi∂B

∂pi− ∂A

∂pi

∂B

∂qi, (1.23)

then we can rewrite Eq. (1.22) as

dO

dt= O,H+ ∂O

∂t. (1.24)

Note that we have obtained a formal correspondence between classical and quantum mechan-ics: The time-evolution of an observable O in the Heisenberg picture is given by the sameequation as in classical mechanics, if the Poisson bracket is changed against a commutator.

Since the Poisson bracket is antisymmetric, we find

dH

dt=∂H

∂t. (1.25)

Hence the Hamiltonian H is a conserved quantity, if and only if H is time-independent.

Liouville’s theorem We consider now the evolution of a many-particle system in 6n-dimensional phase space ωi ≡ (qi, pi) with i = 1, . . . , 3n. The phase space density f(qi, pi, t)determines the probability to find the system in the state ωi = (qi, pi) at time t. Conservationof probability leads to a conservation law valid for f ,

∂f

∂t+

∂

∂ωi(fωi) = 0 . (1.26)

We use first Hamilton’s equations (1.21) to replace ω,

∂

∂ωi(fωi) =

∂

∂qi(f qi)+

∂

∂pi(f pi) =

∂

∂qi

(

f∂H

∂pi

)

− ∂

∂pi

(

f∂H

∂qi

)

, (1.27)

and then that the mixed second derivatives of H commute, ∂qi∂pi = ∂pi∂qi,

∂

∂ωi(fωi) =

∂f

∂qi∂H

∂pi− ∂f

∂pi

∂H

∂qi= f,H (1.28)

= qi∂f

∂qi+ pi

∂f

∂pi. (1.29)

Inserting these results into the conservation law (1.26), Liouville’s theorem follows

df

dt=∂f

∂t+ f,H = ∂f

∂t+∂f

∂qiqi +

∂f

∂pipi = 0 . (1.30)

Thus the time evolution of a Hamiltonian system in phase-space is divergence-free. In analogyto the dynamics of a fluid, one rephrases this often as “a Hamiltonian flow is incompressible.”

6


1.1.3. Green functions and the response method

We can test the internal properties of a physical system, if we impose an external force J(t)on it and compare its measured to its calculated response. If the system is described bylinear differential equations, then the superposition principle is valid: We can reconstructthe solution x(t) for an arbitrary applied external force J(t), if we know the response toa normalised delta-function like kick J(t) = δ(t − t′). Mathematically, this corresponds tothe knowledge of the Green function G(t − t′) for the differential equation D(t)x(t) = J(t)describing the system. Even if the system is described by a non-linear differential equation,we can often use a linear approximation in case of a sufficiently small external force J(t).Therefore the Green function method is extremely useful and we will apply it extensivelydiscussing quantum field theories.

We illustrate this method for the example of the harmonic oscillator which is the prototypefor a quadratic, and thus exactly solvable, action. In classical physics, causality implies thatthe knowledge of the external force J(t′) at times t′ < t is sufficient to determine the solutionx(t) at time t. We define two Green functions G and GR by

x(t) =

∫ t

−∞dt′G(t− t′)J(t′) =

∫ ∞

−∞dt′GR(t− t′)J(t′) , (1.31)

where the retarded Green function GR satisfies GR(t− t′) = G(t− t′)ϑ(t− t′). This definitionis motivated by the trivial relation J(t) =

∫dt′ δ(t − t′)J(t′): An arbitrary force J(t) can be

seen as a superposition of delta functions δ(t − t′) with weight J(t′). If the Green functionG(t− t′) determines the response of the system to a delta function-like force, then we shouldobtain the solution x(t) integrating G(t− t′) with the weight J(t).

We convert the equation of motion mx +mω2 x = J of a forced harmonic oscillator intothe form D(t)x(t) = J(t) by writing

D(t)x(t) ≡ m(

d2

dt2+ ω2

)

x(t) = J(t) . (1.32)

Inserting (1.31) into (1.32) gives

∫ ∞

−∞dt′D(t)GR(t− t′)J(t′) = J(t) . (1.33)

For an arbitrary external force J(t), this relation can be only valid if

D(t)GR(t− t′) = δ(t− t′) . (1.34)

Thus a Green function G(t− t′) is the inverse of its defining differential operator D(t). Per-forming a Fourier transformation,

G(t− t′) =∫

dΩ

2πG(Ω)e−iΩ(t−t′) and δ(t− t′) =

∫dΩ

2πe−iΩ(t−t′) , (1.35)

the differential equation (1.34) is transformed into an algebraic one,

GR(Ω) =1

m

1

ω2 − Ω2. (1.36)

7

1. Introduction

Im(Ω)

Re(Ω)

C+

−ω − iε×

ω − iε×

Figure 1.1.: Poles and contour in the complex Ω plane used for the integration of the retardedGreen function.

For the back-transformation with τ = t− t′,

GR(τ) =

∫dΩ

2πm

e−iΩτ

ω2 − Ω2, (1.37)

we have to specify how the poles at Ω2 = ω2 are avoided. We will use Cauchy’s residuetheorem,

∮dz f(z) = 2πi

∑resz0 f(z), to calculate the integral. This requires to close the

integration contour adding a path which gives a vanishing contribution to the integral. Thisis achieved, when the integrand G(Ω)e−iΩτ vanishes fast enough along the added path. Thuswe have to choose for positive τ the contour C− in the lower plane, e−iΩτ = e−|ℑ(Ω)|τ → 0for ℑ(Ω) → −∞, while we have to close the contour in the upper plane for negative τ . Ifwe want to obtain the retarded Green function GR(τ) which vanishes for τ > 0, we have toshift therefore the poles Ω1/2 = ±ω into the lower plane by adding a small negative imaginarypart, Ω1/2 → Ω1/2 = ±ω − iε, or

GR(τ) = −1

2πm

∫

dΩe−iΩτ

(Ω− ω + iε)(Ω + ω + iε), (1.38)

The residue resz0f(z) of a function f with a single pole at z0 is given by

resz0 f(z) = limz→z0

(z − z0)f(z) . (1.39)

Thus we pick up at Ω1 = −ω − iε the contribution 2πi e+iωτ/(−2ω), while we obtain2πi e−iωτ/(2ω) from Ω2 = ω − iε. Combining both contributions, we arrive at

GR(τ) =i

2mω

[e−iωτ − eiωτ

]ϑ(τ) =

1

m

sin(ωτ)

ωϑ(τ) (1.40)

as result for the retarded Green function of the forced harmonic oscillator.We can now obtain a particular solution solving (1.31). For instance, choosing J(t) =

δ(t− t′), results inx(t) =

1

m

sin(ωτ)

ωϑ(τ) . (1.41)

Thus the oscillator was at rest for t t < 0, got a kick at t = 0, and oscillates according x(t)afterwards. Note that the following two points: First, the fact that the kick proceeds themovement is the result of our choice of the retarded (or causal) Green function. Second,the particular solution (1.41) for a oscillator initially at rest can generalised by adding thesolution to the homogeneous equation x+ ω2 x = 0.

8


1.1.4. Relativistic particle

In special relativity, we replace the Galilean transformations as symmetry group of spaceand time by Lorentz transformations. The latter are all those coordinate transformationsxµ → xµ = Λµνxν that keep the squared distance

s212 ≡ (t1 − t2)2 − (x1 − x2)2 − (y1 − y2)2 − (z1 − z2)2 (1.42)

between two space-time events xµ1 and xµ2 invariant. The squared distance of two infinitesi-mally close space-time events is called line-element. In Minkowski space, it is given by

ds2 = dt2 − dx2 − dy2 − dz2 (1.43)

using a Cartesian inertial frame. We can interprete the line-element ds2 as a scalar product,if we introduce the metric tensor ηµν with elements

ηµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

(1.44)

and a scalar product asaµb

µ = ηµνaµbν . (1.45)

Since the metric ηµν is indefinite, the norm of a vector aµ can be

aµaµ > 0, time-like, (1.46)

aµaµ = 0, light-like or null-vector, (1.47)

aµaµ < 0, space-like. (1.48)

The cone of all light-like vectors starting from a point P is called lightcone, cf. Fig. 1.2.The time-like region inside the light-cone consists of two parts, past and future. Only eventsinside the past light-cone can influence the physics at point P , while P can influence only hisfuture light-cone. The line x = x(σ) describing the position of an observer is called worldline.The proper-time τ is the time displayed by a clock moving with the observer. With ourconventions—negative signature of the metric and c = 1—the proper-time elapsed betweentwo space-time events equals the integrated line-element between them,

τ12 =

∫ 2

1ds =

∫ 2

1dτ =

∫ 2

1[dt2 − (dx2 + dy2 + dz2)]1/2 (1.49)

=

∫ 2

1[ηµνdx

µdxν]1/2 . (1.50)

If we parameterise the worldline traced by an observer passing through 1 and 2 by the pa-rameter σ, e.g. such that σ(τ = 1) = 0 and σ(τ = 2) = 1, then

τ12 =

∫ 1

0dσ

[

ηµνdxµ

dσ

dxν

dσ

]1/2

. (1.51)

Note that τ12 is invariant under a reparameterisation σ = f(σ).

9

1. Introduction

t

x

xb

(x− y)2 > 0 time-like

(x− y)2 = 0 light-like

(x− y)2 < 0 space-like

Figure 1.2.: Lightcone at the point x generated by light-like vectors. Contained in the light-cone are the time-like vectors, outside the space-like ones.

The only invariant differential we have at our disposal to form an action for a free point-likeparticle is the line-element, or equivalently the proper-time,

S0 = α

∫ b

ads = α

∫ b

adσ

ds

dσ(1.52)

with L = αds/dσ. We check now if this choice which implies the Lagrangian

L = αdτ

dσ= α

[

ηµνdxµ

dσ

dxν

dσ

]1/2

(1.53)

for a free particle is sensible: The action has the correct non-relativistic limit,

S0 = α

∫ b

ads = α

∫ b

adt√

1− v2 =∫ b

adt

(

−m+1

2mv2 +O(v4)

)

, (1.54)

if we set α = −m. The mass m corresponds to a potential energy in the non-relativisticlimit and has therefore a negative sign in the Lagrangian. The time t enters the relativisticLagrangian in a Lorentz invariant way as one of the dynamical variables, xµ = (t,x), whileσ assumes now t’s purpose to parameterise the trajectory, xµ(σ). Since a moving clock goesfaster than a clock at rest, solutions of this Lagrangian maximize the action.

The Lagrange equations ared

dσ

∂L

∂(dxα/dσ)=

∂L

∂xα. (1.55)

Consider e.g. the x1 component, then

d

dσ

∂L

∂(dx1/dσ)=

d

dσ

(1

L

dx1

dσ

)

= 0 . (1.56)

10


Since L = dτ/dσ, Newton’s law follows for the x1 coordinate after multiplication with dσ/dτ ,

d2x1

dτ2= 0 , (1.57)

and similar for the other coordinates. An equivalent, but often more convenient form for theLagrangian of a free particle is

L = −mηµν xµxν , (1.58)

where we set x = dx/dσ. If there are no interactions (except gravity), we can neglect themass m of the particle and one often sets m → −1. Since the Lagrangian of a free particledoes not depend explicitely on the evolution parameter σ, there exists at least one conservedquantity. This conservation law, H = m, expresses the fact that the tangent vector xµ has aconstant norm.

Next we want to add an interaction term Sem between a charged particle and an electromag-netic field. The simplest possible action is to integrate the potential Aµ along the world-linexµ(σ) of the particle,

Sem = −q∫

dxµAµ(x) = −q∫

dσdxµ

dσAµ(x) . (1.59)

In the last expression, we can view qxµ as the current jµ induced by the particle and thusthe interaction has the form jµAµ. Any candidate for Sem should be invariant under a gaugetransformation of the potential,

Aµ(x)→ Aµ(x) + ∂µΛ(x) . (1.60)

This is the case, since the induced change in the action,

δΛSem = −q∫ 2

1dσ

dxµ

dσ

∂Λ(x)

∂xµ= −q

∫ 2

1dΛ = −q[Λ(2)− Λ(1)] , (1.61)

depends only on the endpoints. Thus the variation of this contribution to the action vanishes,δ(δΛSem) = 0, keeping the endpoints fixed.

Assuming that the Lagrangian is additiv,

L = L0 + Lem = −m[ηµνdxµ/dσ dxν/dσ]1/2 − qdx

µ

dσAµ(x) (1.62)

the Lagrange equations give now

d

dσ

[mdxα/dσ

[ηµνdxµ/dσ dxν/dσ]1/2+ qAα

]

= qdxλ

dσ

∂Aλ(x)

∂xα. (1.63)

Performing then the differentiation of A(x(σ)) with respect to σ and moving it to the RHS,we find

md

dσ

[dxα/dσ

dτ/dσ

]

= q

(dxλ

dσ

∂Aλ∂xα

− dxλ

dσ

∂Aα∂xλ

)

= qdxλ

dσFαλ , (1.64)

where we introduced the electromagnetic field-strength tensor Fµν = ∂µAν −∂νAµ. Choosingσ = τ we obtain the covariant version of the Lorentz equation,

md2xα

dτ2= q Fαλ u

λ . (1.65)

You should work through problem 1.4, if this equation and the covariant formulation of theMaxwell equations are not familiar to you.

11

1. Introduction

1.2. Differential geometry

We develop in this section the techniques required to do analysis in spaces which look onlylocally as Rn or Minkowski space R(1, 3). Perhaps the most important application of thesemethodes is Einstein’s theory of general relativity, where Minkowski space is replaced by acurved space-time. However, most of the mathematical structures we introduce have also aclose analogue in gauge theories which are theories we will use later on to describe electroweakand strong interactions.

Equivalence principle The idea underlying the equivalence principle emerged in the 16thcentury, when among others Galileo Galilei found experimentally that the acceleration g ofa test mass in a gravitational field is universal. Because of this universality, the gravitatingmass mg and the inertial mass mi are identical in classical mechanics. While mi = mg

can be achieved for one material always by a convenient choice of units, there should be ingeneral deviations for test bodies with differing compositions. Current limits for departuresfrom universal gravitational attraction for different materials are however extremly tight,|∆gi/g| < 10−12.

As a result, gravity has compared to the three other known fundamental interactions theunique property that it can be switched-off locally: Inside a freely falling elevator, one doesnot feel any gravitational effects except tidal forces. The latter arise if the gravitational field isnon-uniform and tries to deform the elevator. Inside a sufficiently small freely falling system,also tidal effects plays no role. Einstein promoted the equivalence of inertial and gravitatingmass to the postulate of the “strong equivalence principle:” In a small enough region aroundthe center of a freely falling coordinate system all physics is described by the laws of specialrelativity.

In general relativity, the gravitational force of Newton’s theory that accelerates particlesin an Euclidean space is replaced by a curved space-time in which particles move force-freealong geodesic lines. In particular, photons move still as in special relativity along curvessatisfying ds2 = 0, while all effects of gravity are now encoded in the non-Euclidean geometryof space-time which is determined by the line-element ds2 or the metric tensor gµν ,

ds2 = gµνdxµdxν . (1.66)

Switching on a gravitational field, the metric tensor gµν can be tranformed only locally by acoordinate change into the form ηµν = diag(1,−1,−1,−1). Thus we should develop the toolsnecessary to do analysis on a curved manifold M which geometry is described by the metrictensor gµν .

1.2.1. Manifolds and tensor fields

Manifolds A manifold M is any set that can be continuously parametrized. The number ofindependent parameters needed to specify uniquely any point of M is its dimension n, theparameters x = x1, . . . , xn are called coordinates. Locally, a manifold with dimension ncan be approximated by R

n. Examples are Lie groups, the phase space (qi, pi) of classicalmechanics with dimension 2n, and space-time.

We require the manifold to be smooth: the transitions from one set of coordinates to anotherone, xi = f(xi, . . . , xn), should be C∞. In general, it is impossible to cover all M with onecoordinate system that is well-defined on all M . An example are spherical coordinate (ϑ, φ)

12


on a sphere S2, where φ is ill-defined at the poles. Instead one has to cover the manifold withpatches of different coordinates that partially overlap.

Vector fields A vector field V (xa) on (a subset S of)M is a set of vectors associating toeach space-time point xa ∈ S exactly one vector. The paradigm for such a vector field is thefour-velocity u(τ) = dx/dτ which is the tangent vector to the world-line x(τ) of a particle.Since the differential equation dx/dσ =X(σ) has locally always a solution, we can find for anygiven X a curve x(σ) which has X as tangent vector. Although the definition u(τ) = dx/dτcoincides with the one familiar from Minkowski space, there an important difference: In a

general manifold, we can not imagine a vector V as an “arrow”−−→PP ′ pointing from a certain

point P to another point P ′ of the manifold. Instead, the vectors V generated by all smoothcurves through P span a n-dimensional vector space at the point P called tangent space TP .We can visualise the tangent space for the case of a two-dimensional manifold embedded inR3: At any point P , the tangent vectors lie in a plane R2 which we can associate with TP .

In general, TP 6= TP ′ and we cannot simply move a vector V (xµ) to another point xµ. Inparticular, we cannot add V (xµ) + V (xµ) for xµ 6= xµ. Therefore we cannot differentiatea vector field without introducing an additional mathematical structure which allows us totransport a vector from one tangent space to another.

If we want to decompose the vector V (xµ) into components V ν(xµ), we have to introducea basis eµ in the tangent space. There are two natural choices for such a basis: First, wecould use four Cartesian basis vectors as in a Cartesian inertial system in Minkowski space.We will follows this approach later, when we discuss gravity as a gauge theory in chapter15. Now, we will use the more conventional approach and use as basis vectors the tangentialvectors along the coordinate lines xµ in M ,

eµ =∂

∂xµ≡ ∂µ . (1.67)

Note that the index µ with value i in eµ denotes the i.th basis vector eµ = (0, . . . , 1, . . . 0), withan one at the i.th position, not a component. Using this basis, a vector can be decomposedas

V = V µeµ = V µ∂µ . (1.68)

A coordinate change

xµ = f(x1, . . . , xn) , (1.69)

or more briefly xµ = xµ(xν) changes the basis vectors as

eµ =∂

∂xµ=∂xν

∂xµ∂

∂xν=∂xν

∂xµeν . (1.70)

Therefore the vector V will be invariant under general coordinate transformations,

V = V µ∂µ = V µ∂µ = V , (1.71)

if its components transform opposite to the basis vectors eµ = ∂µ, or

V µ =∂xµ

∂xνV ν . (1.72)

13

1. Introduction

Covectors or 1-forms In quantum mechanics, we use Dirac’s bracket notation to associateto each vector |a〉 a dual vector 〈a| and to introduce a scalar product 〈a| b〉. If the vectors |n〉form a basis, then the dual basis 〈n| is defined by 〈n|n′〉 = δnn′ .

Similiarly, we define a basis eµ dual to the basis eµ in TP by

eµ(eν) = δµν . (1.73)

This basis can be used to form a new vector space T ∗P called the cotangent space which is

dual to TP . Its elements ω are called covectors or one-forms,

ω = ωµeµ . (1.74)

Combining a vector and an one-form, we obtain a map into the real numbers,

ω(V ) = ωµVνeµ(eν) = ωµV

µ . (1.75)

The last equality shows that we can calculate ω(V ) in component form without reference tothe basis vectors. In order to simplify notation, we will use therefore in the future simplyωµV

µ; we also write eµeµ instead of eµ(eν).Using a coordinate basis, the duality condition (1.73) is obviously satisfied, if we choose

eµ = dxµ. Then the 1-form becomes

ω = ωµdxµ . (1.76)

Thus the familiar “infinitesimals” dxµ are actually the finite basis vectors of the cotangentspace T ∗

P .We require again that the transformation of the components ωµ of a covector cancels the

transformation of the basis vectors,

ωµ =∂xν

∂xµων . (1.77)

This condition guarantees that the covector itself is an invariant object, since

ω = ωµdxµ =

∂xν

∂xµων∂xµ

∂xσdxσ = ωµdx

µ = ω . (1.78)

Covariant and contravariant tensors Next we want to generalise the concept of vectors andcovector. We call a vector X also a contravariant tensor of rank one, while we call a covectoralso a covariant vector or covariant tensor of rank one. A general tensor of rank (n,m) is amultilinear map

T = T µ,...,να,...,β ∂µ ⊗ . . .⊗ ∂ν︸︷︷︸

n

⊗ dxα ⊗ . . .⊗ dxβ︸︷︷︸

m

(1.79)

which components transforms as

T µ,...,να,...,β (x) =∂xµ

∂xρ. . .

∂xν

∂xσ︸︷︷︸

n

∂xγ

∂xα. . .

∂xδ

∂xβ︸︷︷︸

m

T ρ,...,σγ,...,δ (x) (1.80)

under a coordinate change.

14


Metric tensor A Riemannian manifold is a differentiable manifold containing as additionalstructure a symmetric tensor field gµν which allows us to measure distances and angles. Wedefine the scalar product of two vectors a(x) and b(x) which have the coordinates aµ and bµ

in a certain basis eµ as

a · b = (aµeµ) · (bνeν) = (eµ · eν)aµbν = gµνaµbν . (1.81)

Thus we can evaluate the scalar product between any two vectors, if we know the symmetricmatrix gµν composed out of the N2 products of the basis vectors,

gµν(x) = eµ(x) · eν(x) , (1.82)

at any point x of the manifold. This symmetric matrix gµν is called metric tensor.In the same way, we define for the dual basis eµ the metric gµν via

gµν = eµ · eν . (1.83)

A comparison with Eq. (1.75) shows that the metric gµν maps covariant vectors Xµ intocontravariants vectors Xµ, while gµν provides a map into the opposite direction. In the sameway, we can use the metric tensor to raise and lower indices of any tensor.

Next we want to determine the relation of gµν with gµν . We multiply eρ with eµ = gµνeν ,

obtainingδρµ = eρ · eµ = eρ · gµνeν = gρνgµν (1.84)

orδρµ = gµνg

νρ . (1.85)

Thus the components of the covariant and the contravariant metric tensors, gµν and gµν , areinverse matrices of each other. Moreover, the mixed metric tensor of rank (1,1) is given bythe Kronecker delta, gνµ = δρµ. Note that the trace tr(gµν) of the metric tensor is therefore not1− 1− 1− 1 = −2, but

tr(gµν) = gµµgµµ = δµµ = 4 , (1.86)

because we have to contract an upper and a lower index.

1.2.2. Covariant derivative and the geodesic equation

Covariant derivative In an inertial system in Minkowski space, taking the partial derivative∂µ maps a tensor of rank (n,m) into a tensor of rank (n,m+1). Additionally, this map obeyslinearity and the Leibniz product rule. We will see that the partial derivative in a curvedspace does not satisfy in general anymore these rules. Therefrore, we have to introduce amodified derivative which we call covariant derivative.

We start by considering the gradient ∂µφ of a scalar φ. Since a scalar by definition doesnot depend on the coordinate system, φ(x) = φ(x), its gradient transforms as

∂µφ→ ∂µφ =∂xν

∂xµ∂νφ . (1.87)

Thus the gradient is a covariant vector. Simililarly, the derivative of a vector V transformsas a tensor,

∂µV → ∂µV =∂xν

∂xµ∂νV , (1.88)

15

1. Introduction

because V is an invariant quantity. If we consider however its components V µ = eµ ·V , thenthe moving coordinate basis in curved space-time, ∂µe

ν 6= 0, leads to an additional term inthe derivative,

∂µVν = eν · (∂µV ) + V · (∂µeν) . (1.89)

The term eν · (∂µV ) transforms as a tensor, since both eν and ∂µV are tensors. This impliesthat the combination of the two remaining terms has to transform as tensor too, wich wedefine as covariant derivative

∇µV ν ≡ eν · (∂µV ) = ∂µVν − V · (∂µeν) . (1.90)

The first equality tells us that we can view the covariant derivative ∇µV ν as the projectionof ∂µV onto the direction eν .

We expand now the partial derivatives of the basis vectors as a linear combination of thebasis vectors,

∂ρeµ = −Γµρσeσ . (1.91)

The n3 numbers Γµρσ are called (affine) connection coefficients or symbols, in order to stressthat they are not the components of a tensor. You are asked to derive the transformationproperties of the connection coefficients in problem 1.10. Introducing this expansion into(1.90) we can rewrite the covariant derivative of a vector field as

∇µV ν = ∂µVν + ΓνσµV

σ . (1.92)

Using ∇σφ = ∂σφ and requiring that the usual Leibniz rule is valid for φ = XµXµ leads to

∇σXµ = ∂σXµ − ΓνµσXν (1.93)

and to∂ρeµ = Γσµρeσ . (1.94)

For a general tensor, the covariant derivative is defined by the same reasoning as

∇σT µ...ν... = ∂σTµ...ν... + ΓµρσT

ρ...ν... + . . .− ΓρνσT

µ...ρ... − . . . (1.95)

Note that it is the last index of the connection coefficients that is the same as the index ofthe covariant derivative. The plus sign goes together with upper (superscripts), the minuswith lower indices.

Parallel transport We say a tensor T is parallel transported along the curve x(σ), if itscomponents T µ...ν... stay constant. In flat space, this means simply

d

dσT µ...ν... =

dxα

dσ∂αT

µ...ν... = 0 . (1.96)

In curved space, we have to replace the normal derivative by a covariant one. We define thedirectional covariant derivative along x(σ) as

D

dσ=

dxα

dσ∇α . (1.97)

Then a tensor is parallel transported along the curve x(σ), if

D

dσT µ...ν... =

dxα

dσ∇αT µ...ν... = 0 . (1.98)

16


Metric compatibility Relations like ds2 = gµνdxµdxν or gµνp

µpν = m2 become invariantunder parallel transport only, if the metric tensor is covariantly constant,

∇σgµν = ∇σgµν = 0 . (1.99)

A connection satisfying Eq. (1.99) is called metric compatible and leaves lengths and anglesinvariant under parallel transport. This requirement gurantees that we can introduce locally inthe whole space-time Cartesian inertial coordinate systems where the laws of special relativityare valid. Moreover, these local inertial systems can be consistently connected by paralleltransport using an affine connection satisfying the constraint (1.99).

Note that we already build in this constraint into our definition of the covariant derivative:If the length of a vector would not be conserved under parallel transport, then we shoulddifferentiante in (1.89) also the scalar product in V µ = eµ ·V , leading to additional terms in(1.90).

Geodesic equation Which one of the infinitly many affine connection should we use to cal-culate covariant derivatives on a general space-time? Ultimately, the combined action forgravity and matter should select the correct connection—an approach we resume in Chap-ter 15. For the moment, we use a simple workaround which does not require the knowledgeof the action of gravity: In flat space, we know that the solution to the equation of motionsof a free particle is a straight line. Such a path is characterized by two properties: It isthe shortest curve between the considered initial and final point, and it is the curve whosetangent vectors remains constant if they are parallel transported along it. Both conditionscan be generalised to curved space and the curves satisfying either one of them are calledgeodesics.

Using the definition of a geodesics as the “straightest” line on a manifold requires as math-ematical structure only the possibility to parallel transport a tensor and thus the existenceof an affine connection. In contrast, the concept of an “extremal” (shortest or longest) linebetween two points on a manifold relies on the existence of a metric. Requiring that thesetwo definitions agree fixes uniquely the connection to be used in the covariant derivative.

We start by defining geodesics as the “straightest” line or an autoparallel curve on amanifold—the case which is almost trivial: The tangent vector along the path x(σ) isuµ = dxµ/dσ. Then the requirement (1.98) of parallel transport for uµ becomes

D

dσ

dxµ

dσ=

d2xµ

dσ2+ Γµρσ

dxρ

dσ

dxσ

dσ= 0 . (1.100)

Introducing the short-hand xµ = dxµ/dσ, we obtain the geodesic equation in its usual form,

xµ + Γµρσxρxσ = 0 . (1.101)

Note that the antisymmetric part of the connection Γµρσ drops out of the geodesic equation,because xρxσ is symmetric: Contracting a symmetric tensor Sµν with an antisymmetric tensorAµν , we find

SµνAµν = −SµνAνµ = −SνµAνµ = −SµνAµν , (1.102)

where we used first the antisymmetry of Aµν , then the symmetry of Sµν , and finally exchangedthe dummy summation indices. Thus the contraction is zero, SµνA

µν = 0.

17

1. Introduction

Next we derive the defining equation for a geodesics as the extremal curve between twopoints on a manifold. The Lagrangian of a free particle in Minkowski space, Eq. (1.58), isgeneralized to a curved space-time manifold with the metric tensor gµν by replacing ηµν withgµν (we set also m = −1),

L = gµν xµxν . (1.103)

The Lagrange equations ared

dσ

∂L

∂(xλ)− ∂L

∂xλ= 0 . (1.104)

Only the metric tensor gµν depends on xµ and thus ∂L/∂xλ = gµν,λxµxν . Here we introduced

also the short-hand notation gµν,λ = ∂λgµν for partial derivatives. Now we use ∂xµ/∂xν = δµνand apply the chain rule for gµν(x(σ)), obtaining first

gµν,λxµxν = 2

d

dσ(gµλx

µ) = 2(gµλ,ν xµxν + gµλx

µ) (1.105)

and then

gµλxµ +

1

2(2gµλ,ν − gµν,λ)xµxν = 0 . (1.106)

Next we rewrite the second term as

2gλµ,ν xµxν = (gλµ,ν + gλν,µ)x

µxν , (1.107)

multiply everything by gκµ and arrive at our desired result,

xκ +1

2gκλ(gµν,λ + gµλ,ν − gµν,λ)xµxν = xκ + κµνxµxν = 0 . (1.108)

Here we defined in the last step the Christoffel symbols

µ

νλ

=

1

2gµκ(∂νgκλ + ∂λgνκ − ∂κgνλ) . (1.109)

They are also called Levi-Civita or Riemannian connection. Comparison with Eq. (1.101)shows that our two geodesic equations agree, if we choose as connection the Christoffel sym-bols. Moreover, the Christoffel symbol is symmetric in its two lower indices and, as we willshow next, compatible to the metric tensor.

We define2

Γµνλ = gµκΓκνλ . (1.110)

Thus Γµνλ is symmetric in the last two indices. Then it follows

Γµνλ =1

2(∂νgµλ + ∂λgνµ − ∂µgνλ) . (1.111)

Adding 2Γµνλ and 2Γνµλ gives

2(Γµνλ + Γνµλ) = ∂bgµλ + ∂λgνµ − ∂µgνλ (1.112)

+ ∂µgνλ + ∂λgµν − ∂νgµλ = 2∂λgµν (1.113)

2We showed that the metric tensor can be used to raise or to lower tensor indices, but the connection Γ isnot a tensor.

18


or∂λgµν = Γµνλ + Γνµλ . (1.114)

Applying the general rule for covariant derivatives, Eq. (1.95), to the metric,

∇λgµν = ∂λgµν − Γκµcgκν − Γκνλgµκ = ∂λgµν − Γµλν − Γνλµ , (1.115)

and inserting Eq. (1.114) shows that

∇λgµν = ∇λgµν = 0 . (1.116)

Hence ∇λ commutes with contracting indices,

∇λ(XµXµ) = ∇λ(gµνXµXµ) = gµν∇λ(XµXν) (1.117)

and conserves the norm of vectors as announced. Thus the Levi-Civita connection is symmet-ric and compatible with the metric. These two properties specifiy uniquely the connection.

The fact that we obtained as connection the Christoffel symbols when we derived thegeodesic equation for a classical spin-less point particle justifies the use of a torsionless con-nection which is metric compatible: Although e.g. a star consists of a collection of individualparticles carrying spin si, its total spin sums up to zero,

∑

i si ≈, 0, because the si are uncor-related. Thus we can describe macrosopic matter in general relativity as a classical spinless3

point particle (or fluid if its extension is important) leading to a symmetric connection. In theremainder of this section, we will use always as affine connection the Levi-Civita connection.Following standard practise, we will denote these connection coefficients also with Γabc.

Example: Sphere S2. Calculate the Christoffel symbols of the two-dimensional unit sphere S2.The line-element of the two-dimensional unit sphere S2 is given by ds2 = dϑ2 + sin2 ϑdφ2. A fasteralternative to the definition (1.109) of the Christoffel coefficients is the use of the geodesic equation:From the Lagrange function L = gabx

axb = ϑ2 + sin2 ϑφ2 we find

∂L

∂φ= 0 ,

d

dt

∂L

∂φ=

d

dt(2 sin2 ϑφ) = 2 sin2 ϑφ+ 4 cosϑ sinϑϑφ

∂L

∂ϑ= 2 cosϑ sinϑφ2 ,

d

dt

∂L

∂ϑ=

d

dt(2ϑ) = 2ϑ

and thus the Lagrange equations are

φ+ 2 cotϑϑφ = 0 and ϑ− cosϑ sinϑφ2 = 0 .

Comparing with the geodesic equation xκ+Γκµν x

µxν = 0, we can read off the non-vanishing Christoffel

symbols as Γφϑφ = Γφ

φϑ = cotϑ and Γϑφφ = − cosϑ sinϑ. (Note that 2 cotϑ = Γφ

ϑφ + Γφφϑ.)

We can use also the Hamiltonian formulation for a relativistic particle. From the LagrangianL = 1

2gµν xµxν we determine first the conjugated momenta pµ = ∂L/∂xµ = xµ and perform

then a Legendre transformation,

H(xµ, pµ, τ) = pµxµ − L(xµ, xµ, τ) = 1

2gµνpµpν . (1.118)

3The curious student may wonder if orbital angular momentum leads to torsion: The answer is no, because onecannot define an orbital angular momentum density which transforms properly as a tensor, cf. Eq. (3.24f).

19

1. Introduction

Hamilton equations give then

xµ =∂H

∂pµ= gµνpν (1.119)

and

pµ = − ∂H∂xµ

= −1

2

gαβ

∂xµpαpβ . (1.120)

This is a useful alternative to the standard geodesic equation: First, it makes clear that themomentum component pµ is conserved, if the metric tensor is independent of the coordi-nate xµ. Second, we can calculate pµ directly from the metric tensor, without knowing theChristoffel symbols. Combining the Eqs. (1.119) and (1.120) one can rederive the standardform of the geodesic equation, cf. problem 1.9.

1.2.3. Integration and Gauss’ theorem

Having defined the covariant derivative of an arbitrary tensor field, it is natural to ask howthe inverse, the integral over a tensor field, can be defined. The short answer is that this isin general impossible: Integrating a tensor field requires to sum tensors at different points inan invariant way, which is only possible for scalars.

If we attempt to generalise an integral like

I =

∫

d4xφ(x)

valid in an Cartesian intertial frame xµ in Minkowski space to a general space-time withcoordinates x, we have to take into account that the Jacobi determinant J = det(∂xµ/∂xρ)of the coordinate transformation can deviate from one. We can express the Jacobian J bythe determinant g ≡ det(gµν) of the metric tensor as follows: Applying the tranformation lawof the metric tensor,

gµν(x) =∂xµ

∂xρ∂xν

∂xσgρσ(x) (1.121)

to the case where the xρ are inertial coordinates, we obtain with g = det(gµν) = det(ηµν) = −1that

det(g) = J2 det(g) = −J2 (1.122)

or J =√

|g|. Thus I =∫d4x√

|g| φ is an invariant definition of an integral over a scalarfield which agrees for inertial coordinates with the one known from special relativity. Now wechoose as scalar φ the divergence of a vector field, φ = ∇µXµ, or

I =

∫

d4x√

|g| ∇µXµ . (1.123)

Our aim is to derive a generalised Gauss’ theorem. Let us recall that this theorem allows usto convert a n dimensional volume integral over the divergence of a vector field into a n − 1dimensional surface integral of the vector field,

∫

Ωd4x ∂µX

µ =

∫

∂ΩdSµX

µ . (1.124)

The only way how (1.124) may be reconciled with (1.123) is to hope that the contractedChristoffel symbols can be expressed as a partial derivative of

√

|g| in such a way that theunwanted terms in (1.123) disappear. In order to check this possibility, we calculate thereforenow the partial derivative of the metric determinant g.

20


Useful formula for derivatives We start considering the variation of a general matrix Mwith elements mij(x) under an infinitesimal change of the coordinates, δxµ = εxµ. It isconvenient to look at the change of ln detM ,

δ ln detM ≡ ln det(M + δM)− ln det(M)

= ln det[M−1(M + δM)] = ln det[1 +M−1δM ] =

= ln[1 + tr(M−1δM)] +O(ε2) = tr(M−1δM) +O(ε2) . (1.125)

In the last step, we used ln(1 + ε) = ε+O(ε2). Expressing now both the LHS and the RHSas δf = ∂µfδx

µ and comparing then the coefficients of δxµ gives

∂µ ln detM = tr(M−1∂µM) . (1.126)

Applied to derivatives of√

|g|, we obtain

1

2gµν∂λgµν =

1

2∂λ ln g =

1√

|g|∂λ(√

|g|) . (1.127)

This expression coincides with contracted Christoffel symbols,

Γµµν =1

2gµκ(∂µgκν + ∂νgµκ − ∂κgµν) =

1

2gµκ∂νgµκ =

1

2∂ν ln g =

1√

|g|∂ν(√

|g|) . (1.128)

Next we consider the divergence of a vector field,

∇µXµ = ∂µXµ + ΓµλµX

λ = ∂µXµ +

1√

|g|(∂µ√

|g|)Xµ =1

√

|g|∂µ(√

|g|Xµ) , (1.129)

and of an anti-symmetric tensor of rank 2,

∇µAµν = ∂µAµν + ΓµλµA

λν + ΓνλµAµλ = ∂µA

µν . (1.130)

In the latter case, the Christoffel symbols cancel. This generalises to completely anti-symmetric tensors of all orders. In contrast, we find for a symmetric tensor of rank 2,

∇µSµν = ∂µSµν + ΓµλµS

λν + ΓνλµSµλ =

1√

|g|∂µ(√

|g|Sµν) + ΓνλµSµλ . (1.131)

We can express in the second term the Christoffel symbol as derivative of the metric tensor,

∇µSµν =1

√

|g|∂µ(√

|g|Sµν)−1

2(∂νgµλ)S

µλ . (1.132)

Thus we can perform the covariant derivative of Sµν without the need to know the Christoffelsymbols.

Gauss’ theorem for the divergence of a vector field follows now directly from Eq. (1.129),∫

Ωd4x√

|g| ∇µXµ =

∫

Ωd4x ∂a(

√

|g|Xµ) =

∫

∂ΩdSµ

√

|g|Xµ . (1.133)

If the fields Xµ vanishes on the boundary, terms like ∇µXµ can be dropped in the action.Similiarly, this equation allows us to derive global conservation laws from ∇µXµ = 0 in thesame way as in Minkowski space. In contrast, the divergence (1.132) of a symmetric tensor ofrank two contains an additional term (∂νgµλ)S

µλ which prohibits the use of Gauss’ theorem.

21

1. Introduction

Summary

Lagrange’s and Hamilton’s equations follow extremizing the action S[qi] =∫ ba dt L(q

i, qi, t)

and S[qi, pi] =∫ ba dt

[piq

i −H(qi, pi, t)], respectively.

In a curved space-time we require a connection to compare vectors at different point. Theunique connection which is symmetric and compatible with the metric is the Levi-Civitaconnection.

Further reading

Volume 1 of the Landau-Lifshitz’s series treats classical mechanics in concise way. [Car03]contains a clear introduction to differential geometry on a level accessible for physicists.

Problems

1.1 Higher derivatives.Find the Lagrange equations for a Lagrange

function containing higher derivatives, L =L(qi, qi, qi, . . .).

1.2 Oscillator with friction.Consider an one-dimensional system described bythe Lagrangian

L = e2αt[1

2mq2 − V (q)

]

.

a.) Show that the equation of motion correspondsto an oscillator with friction term.b). Derive the energy lost per time dE/dt of theoscillator, with E = 1

2mq2 + V (q).

c.) Show that the result in b.) agrees with theone obtained from the Lagrange equations of thefirst kind,

d

dt

∂L

∂qi− ∂L

∂qi= Qi ,

where the generalised forces Qi perform the workδA = Qiδq

i.d.) Derive the Green function for a harmonic os-cillator with friction.

1.3 Conservation laws.Discuss the symmetries of the Galelean transfor-mations and the resulting conservation laws, fol-lowing the example of time translation and energyconservation.

1.4 Electromagnetism.

Compare Eq. (1.65) to the usual F = q(E + v ×B) and derive thereby the elements of the field-strength tensor Fµν . What is the meaning of thezero component of the Lorentz force?

1.5 Little group

Find the Lorentz transformations Λ νµ which keep

the momentum of a particle invariant, Λ νµ p

µ = pν ,for a a) massive particle and b) massless particle,considering an infinitesimal transformation ωµν .By how many parameters is ωµν determined inthe two cases?

1.6 Inertial coordinates.

The equivalence principle implies that we can findin a curved space-time locally coordinates withxµ = 0 and gµν = ηµν . Show that this require-ment holds in a Riemannina manifold choosing asconnection the Christoffel symbols.

1.7 Lie bracket.

Show that the commutator [V ,W ] of two vectorfields V = V µ∂µ and W =Wµ∂µ is again a vectorfield.

1.8 Torsion

Derive the transformation properties of a connec-tion. Show that the difference of two connectionsis a tensor.

1.9 Geodesic equation from H.

Show that the Eqs. (1.119) and (1.120) imply the

22


standard form of the geodesic equation.

1.10 Affine parameter.

1.11 Hamiltonian of relativistic particle.

23

1.bis Quantum mechanics

We introduce Feynman’s path integral as an alternative to the standard operator approach toquantum mechanics. Most of our discussion of quantum fields will be based on this approach,and thus becoming familiar with this technique using the simpler case of quantum mechanicsis of central importance. Instead of using the path integral or the propagator directly, we willuse as basic tool the ground-state persistence amplitude 〈0,∞|0,−∞〉J : This quantity is theprobability amplitude that a system under the influence of an external force J stays in itsground-state. Since we can apply an arbitrary force J , it contains all information about thesystem. Moreover, it is relatively easy to calculate and therefore it will serve us as main toolto study quantum field theories later on.

1.1. Reminder of the operator approach

A classical system described by a Hamiltonian H(qi, pi, t) can be quantised promoting qi andpi to operators which satisfy the canonical commutation relations

[xi, pj

]= iδij . The latter are

the formal expression of Heisenberg’s uncertainty relation. Apart from ordering ambiguities,the Hamilton operator H(qi, pi, t) can be directly read from the Hamiltonian H(qi, pi, t).

General principles A physical system in a pure state is fully described by a probabilityamplitude

ψ(a, t) = 〈a|ψ(t)〉 ∈ C , (1.1)

where a is a set of quantum numbers specifying the system and the states |ψ(t)〉 form acomplex Hilbert space. The probability to find the specific value a∗ in a measurement is givenby p(a∗) = |ψ(a∗, t)|2. The possible values a∗ are the eigenvalues of hermitian operators Aand the eigenvectors |a〉 form an orthogonal, complete basis. In Dirac’s bra-ket notation,

A|a〉 = a|a〉 , 〈a|a′〉 = δ (a− a′) ,∫

da|a〉〈a| = 1 , (1.2)

In general, operators do not commute. Their commutation relations can be obtained by thereplacement A,B → i[A,B] in (1.23).

The state of a particle moving in one dimension in a potential V (q) can be described eitherby the eigenstates of the position operator q or of the momentum operator p. Both form acomplete, orthonormal basis, and they are connected by a Fourier transformation which wechoose to be asymmetric. This asymmetry is reflected in the completeness relation of thestates,

∫

dq |q〉〈q| = 1 and

∫dp

2π|p〉〈p| = 1 . (1.3)

When there is the danger of an ambiguity, operators will be written with a “hat”; otherwisewe drop the hat.

24

1.1. Reminder of the operator approach

Time-evolution Since the states |ψ(t)〉 form a complex Hilbert space, the superpositionprinciple is valid: If ψ1 and ψ2 are possible states of the system, then also

ψ(α, t) = c1ψ1(α, t) + c2ψ2(α, t) , ci ∈ C . (1.4)

In quantum mechanics, a stronger version of this principle holds which states that if ψ1(α, t)and ψ2(α, t) are describing the possible time-evolution of the system, then it does also thesuperposed state ψ(α, t). This implies that the time-evolution is described by a linear, homo-geneous differential equation. Choosing it as first-order in time, we can write the evolutionequation as

i∂t|ψ(α, t)〉 = D|ψ(α, t)〉 , (1.5)

where the differential operator D on the RHS has to be still determined.

We call the operator describing the evolution of a state from ψ(t) to ψ(t′) the time-evolutionoperator U(t′, t). This operator is unitary, U−1 = U †, in order to conserve probability andforms a group, U(t3, t1) = U(t3, t2)U(t2, t1) with U(t, t) = 1. For an infinitesimal time stepδt,

|ψ(t+ δt)〉 = U(t+ δt, t) |ψ(t)〉 , (1.6)

we can set with U(t, t) = 1

U(t+ δt, t) = 1− iHδt . (1.7)

Here we introduced the generator of infinitesimal time-translations H. The analogue toclassical mechanics suggests that H is the operator version of the classical Hamilton functionH(q, p). Hence

|ψ(t+ δt)〉 − |ψ(t)〉δt

= −iH |ψ(t)〉 . (1.8)

Comparing (1.5) and (1.8) reveals that the generator of infinitesimal time-translations H isequal to the operator D on the RHS of (1.5). We call a time-evolution equation of this typefor an arbitrary Hamiltonian H Schrodinger equation.

Plugging ψ(t) = U(t, 0)ψ(0) in the Schrodinger equation gives

[

i∂U(t, 0)

∂t−HU(t, 0)

]

ψ(0) = 0 . (1.9)

Since this equation is valid for an arbitrary state ψ(0), we can rewrite it as an operatorequation,

i∂t′U(t′, t) = HU(t′, t) . (1.10)

Integrating gives as formal solution

U(t′, t) = 1− i

∫ t′

tdt′′H(t′′)U(t′′, t) . (1.11)

If H is time-independent, we obtain simply by iteration

U(t, t′) = exp(−iH(t− t′)) . (1.12)

25


Propagator We insert the solution of U for a time-independentH into |ψ(t′)〉 = U(t′, t)|ψ(t)〉and multiply from the left with 〈q′|,

ψ(q′, t′) = 〈q′|ψ(t′)〉 = 〈q′| exp[−iH(t′ − t)]|ψ(t)〉 . (1.13)

Then we insert 1 =∫d3q|q〉〈q|,

ψ(q′, t′) =∫

d3q 〈q′| exp[−iH(t′ − t)]|q〉〈q|ψ(t)〉 =∫

d3q K(q′, t′; q, t)ψ(q, t) . (1.14)

In the last step we introduced the propagator or Green function K in its coordinate repre-sentation,

K(q′, t′; q, t) = 〈q′| exp[−iH(t′ − t)]|q〉 . (1.15)

The Green function K equals the probability amplitude for the propagation between twospace-time points; K(q′, t′; q, t) is therefore also called more specifically two-point Greenfunction. We can express the propagator K by the solutions of the Schrodinger equation,ψn(q, t) = 〈q|n(t)〉 = 〈q|n〉 exp(−iEnt) as

K(q′, t′; q, t) =∑

n,n′

〈q′|n〉〈n| exp(−iH(t′ − t))|n′〉︸︷︷︸

δn,n′ exp(−iEn(t′−t))

〈n′|q〉

=∑

n

ψn(q′)ψ∗

n(q) exp(−iEn(t′ − t)) . (1.16)

Let us compute the propagator of a free particle in one dimension, described by the Hamil-tonian H = p2/2m. We write with τ = t′ − t

K(q′, t′; q, t) =⟨q′∣∣ e−iHτ |q〉 =

⟨q′∣∣ e−iτ p2/2m

∫dp

2π|p〉〈p| q〉

=

∫dp

2πe−iτp2/2m

⟨q′∣∣ p〉〈p| q〉 =

∫dp

2πe−iτ(p2/2m)+i(q′−q)p .

The integral is Gaussian, if we add an infinitesimal factor exp(−εp2) to the integrand toensure the convergence of the integral. Thus the physical value of the energy E = p2/(2m)seen as a complex variable is approached from the negative imaginary plane, E → E − iε.Taking afterwards the limit ε→ 0, we obtain

K(q′, t′; q, t) =( m

2πiτ

)1/2eim(q′−q)2/2τ . (1.17)

Example: Calculate the three Gaussian integralsA =∫dx exp(−x2/2), B =

∫dx exp(−ax2/2+bx),

and C =∫dx · · · dxn exp(−xAx/2 + Jx) for a symmetric matrix A.

a.) We square the integral and calculate then A2 introducing polar coordinates, r2 = x2 + y2,

A2 =

∫ ∞

−∞dx

∫ ∞

−∞dy exp(−(x2 + y2)/2) = 2π

∫ ∞

0

dr re−r2/2 = 2π

∫ ∞

0

dt e−t = 2π ,

where we substituted t = r2/2 in the last step. Thus the result for the basic Gaussian integral isA =

√2π. All others solvable variants of Gaussian integrals can be reduced to this result.

b.) We complete the square in the exponent,

−a2

(

x2 − 2b

ax

)

= −a2

(

x− b

a

)2

+b2

2a,

26

1.2. Path integrals in quantum mechanics

and shift then the integration variable to x′ = x− b/a. The result is

B =

∫ ∞

−∞dx exp(−ax2/2 + bx) = eb

2/2a

∫ ∞

−∞dx′ exp(−ax′2/2) =

√

2π

aeb

2/2a . (1.18)

c.) The notation xAx = xTAx and bx = bTx means now matrix multiplication. We complete againthe square,

(X −A−1J)TA(X −A−1J) = XTAX −XTAA−1J − JTA−1AX + JTA−1AA−1J

= XTAX − 2JTX + JTA−1J ,

and shift the integration vector, X ′ = X −A−1J . Then we obtain

C =

∫

dx1 · · ·dxn exp(−XTAX/2 + JTx) = exp(JTA−1J/2)

∫

dx′1 · · · dx′n exp(−X ′TAX ′/2) .

Since the matrix A is symmetric, we can diagonalize A via an orthogonal transformation D = OTAO.This corresponds to a rotation of the integration variables, Y = OX ′. The Jacobian of this transfor-mation is one, and thus the result is

C = exp(JTA−1J/2)

∫

dy1 · · · dynn∏

i=1

exp(−aiy2i /2)

=

√

(2π)n

detAexp

(1

2JTA−1J

)

. (1.19)

In the last step we expressed the product of eigenvalues ai as the determinant of the matrix A.


In problem 1.1 you are asked to calculate the classical action of a free particle and of aharmonic oscillator and to compare them to the corresponding propagators found in quantummechanics. Surprisingly, you will find that in both cases the propagator can be written asK(q′, t′; q, t) = N exp(iS) where S is the classical action along the path [q′ : q] and N anormalisation constant. This suggests that we can reformulate quantum mechanics, replacingthe standard operator formalism used to evaluate the propagator (1.15) “somehow” by theclassical action.

To get an idea how to proceed, we look at the famous double-slit experiment: Accordingto the superposition principle, the amplitude A for a particle to move from the source at q1to the detector at q2 is the sum of the amplitudes Ai for the two possible paths,

A = K(q2, t2; q1, t1) =∑

paths

Ai . (1.20)

Clearly, we could add in a Gedankenexperiment more and more screens between q1 and q2,increasing at the same time the number of holes. Although we replace in this way continuousspace-time by a discrete lattice, the differences between these two descriptions should vanishfor sufficiently small spacing τ . Moreover, for τ → 0, we can expand U(τ) = exp(−iHτ) ∼1− iHτ . Applying then H = p2/(2m) + V (q) to eigenfunctions |q〉 of V (q) and |p〉 of p2, wecan replace the operator H by its eigenvalues. In this way, we hope to express the propagatoras a sum over paths, where the individual amplitudes Ai contain only classical quantities.

27


q1

q2

q

t

q0

0

q1

τ

q2

2τ

q3

3τ

q3

3τ

qN−1

τ

qN

Figure 1.1.: Left: The double slit experiment. Right: The propagator K(qN , τ ; q0, 0) ex-pressed as a sum over all N -legged paths.

We will apply now this idea a particle moving in one dimension in a potential V (q). Thetransition amplitude A for the evolution from the state |q, 0〉 to the state |q′, t′〉 is

A ≡ K(q′, t′; q, 0) =⟨q′∣∣ e−iHt′ |q〉 . (1.21)

This amplitude equals the matrix-element of the propagator K for the evolution from theinitial space-time point q(0) to the final point q′(t′).

Let us split the time evolution into two smaller steps, writing e−iHt′ = e−iH(t′−t1)e−iHt1 .The amplitude becomes

A =⟨q′∣∣ e−iH(t′−t1)e−iHt1 |q〉 .

Inserting unity expressed as a sum over the position eigenstates gives

A =⟨q′∣∣ e−iH(t′−t1)

∫

dq1 |q1〉〈q1|︸︷︷︸

=1

e−iHt1 |q〉

=

∫

dq1K(q′, t′; q1, t1)K(q1, t1; q, 0). (1.22)

This formula expresses simply the group property, U(t′, 0) = U(t′, t1)U(t1, 0), of the time-evolution operator U evaluated in the basis of the continuous variable q. More physically, wecan view this equation as an expression of the quantum mechanical rule for combining ampli-tudes: If the same initial and final states can be connected by various ways, the amplitudesfor each of these processes should be added. A particle propagating from q to q′ must besomewhere at the intermediate time t1. Labelling this intermediate position as q1, we com-pute the amplitude for propagation via the point q1 as the product of the two propagators inEq. (1.22) and integrate over all possible intermediate positions q1.

We continue to divide the time interval t′ into a large numberN of time intervals of durationτ = t′/N , with the final aim of performing the limit of infinitesimal small time steps τ . Thenthe propagator becomes

A =⟨q′∣∣ e−iHτe−iHτ · · · e−iHτ︸︷︷︸

N times

|q〉 . (1.23)

28


We insert again a complete set of states |q〉i between each exponential, obtaining

A =

∫

dq1 · · · dqN−1

⟨q′∣∣ e−iHτ |qN−1〉〈qN−1| e−iHτ |qN−2〉 · · · 〈q1| e−iHτ |q〉

≡∫

dq1 · · · dqN−1KqN ,qN−1KqN−1,qN−2

· · ·Kq2,q1Kq1,q0 , (1.24)

where we have defined q0 = q and qN = q′. Note that these initial and final positions are fixedand therefore are not integrated over. Figure 1.1 illustrates that we can view the amplitudeA as the integral over the partial amplitudes Apath of the individual N -legged paths.

We ignore the problem of defining properly the limit N →∞, keeping N finite. We rewritethe amplitude as sum over the amplitudes for all possible paths,

A =∑

paths

Apath ,

where

∑

paths

=

∫

dq1 · · · dqN−1, Apath = KqN ,qN−1KqN−1,qN−2

· · ·Kq2,q1Kq1,q0 .

Let us look at the last expression in detail. We can expand the exponential in each propagatorKqj+1,qj = 〈qj+1| e−iHτ |qj〉 for a single sub-interval, because τ is small,

Kqj+1,qj = 〈qj+1|(

1− iHτ − 1

2H2τ2 + · · ·

)

|qj〉

= 〈qj+1| qj〉 − iτ 〈qj+1|H |qj〉+O(τ2) . (1.25)

The first term is a delta function, which we can express as

〈qj+1| qj〉 = δ(qj+1 − qj) =∫

dpj2π

eipj(qj+1−qj) . (1.26)

In the second term of (1.25), we insert a complete set of momentum eigenstates between Hand |qj〉. This gives

− iτ 〈qj+1|(p2

2m+ V (q)

)∫dpj2π|pj〉〈pj | qj〉

= −iτ∫

dpj2π

(

p2j2m

+ V (qj+1)

)

〈qj+1| pj〉〈pj | qj〉

= −iτ∫

dpj2π

(

p2j2m

+ V (qj+1)

)

eipj(qj+1−qj) , (1.27)

where we used 〈q| p〉 = exp(ipq). In the first line, we view the operator p as operating to theright, while V (q) operates to the left.

The expression (1.27) is not symmetric in qj and qj+1. The reason for this asymmetry isthat we could insert the factor 1 either to the right or to the left of the Hamiltonian H in thesecond term of (1.25). In the latter case, we would have obtained pj+1 and V (qj) in (1.27).Since the difference [V (qj+1)− V (qj)]τ ∼ V ′(qj)(qj+1 − qj)]τ ∼ V ′(qj)qjτ2 is of order τ2, the

29


ordering problem should not matter in the continuum limit which we will take eventually.Combining (1.26) and (1.27), the propagator for the step qj → qj+1 is

Kqj+1,qj =

∫dpj2π

eipj(qj+1−qj)[

1− iτ

(pj

2

2m+ V (qj)

)

+O(τ2)]

. (1.28)

Since we work only at O(τ), we can exponentiate the factor in the square bracket,

1− iτ H(pj , qj) +O(τ2) = e−iτH(pj ,qj) . (1.29)

Next we rewrite the exponent in the first factor using qj = (qj+1 − qj)/τ , such that we canfactor out the time-intervall τ . The amplitude Apath consists of N such factors. Combiningthem, we obtain

Apath =N−1∏

j=0

∫dpj2π

exp iτN−1∑

j=0

[pj qj −H(pj , qj)] . (1.30)

We recognise the argument of the exponential as the discrete approximation of the actionS[q, p] in the Palatini form of a path passing through the points q0 = q, q1, · · · , qN−1, qN = q′.The propagator becomes then

K =

∫

dq1 · · · dqN−1Apath

=

N−1∏

j=1

∫

dqj

N−1∏

j=0

∫dpj2π

exp iτ

N−1∑

j=0

[pj qj −H(pj , qj)] . (1.31)

Note that there is one momentum integral for each of the N intervals, while there is oneposition integral for each of the N − 1 intermediate positions.

If N → ∞, this approximates an integral over all functions p(t), q(t) consistent with theboundary conditions q(0) = q, q(t′) = q′. We adopt the notation DpDq for the functional orpath integral over all functions p(t) and q(t),

K ≡∫

Dp(t)Dq(t)eiS[q,p] =∫

Dp(t)Dq(t) exp(

i

∫ t′

0dt (pq −H(p, q))

)

. (1.32)

This result is known as the path integral in phase-space. It allows us to obtain for any classicalsystem which can be described by a Hamiltonian the corresponding quantum dynamics.

If the Hamiltonian is of the form H = p2/2m+V (q), as we have assumed in our derivation,we can carry out the quadratic momentum integrals in (1.31). We can rewrite this expressionas

K =

N−1∏

j=1

∫

dqj exp−iτN−1∑

j=0

V (qj)

N−1∏

j=0

∫dpj2π

exp iτ

N−1∑

j=0

(pj qj − p2j/2m

).

The p integrals are all uncoupled Gaussians. One such integral gives

∫dp

2πeiτ(pq−p

2/2m) =

√m

2πiτeiτmq

2/2 , (1.33)

where we added again an infinitesimal factor exp(−εp2) to the integrand.

30

1.3. Generating functional for Green functions

The propagator becomes

K =

N−1∏

j=1

∫

dqj exp−iτN−1∑

j=0

V (qj)

N−1∏

j=0

(√m

2πiτexp iτ

mq2j2

)

=( m

2πiτ

)N/2N−1∏

j=1

∫

dqj exp iτN−1∑

j=0

(

mq2j2− V (qj)

)

. (1.34)

The argument of the exponential is again a discrete approximation of the action S[q] of apath passing through the points q0 = q, q1, · · · , qN−1, qN = q′, but now seen as functional ofonly the coordinate q. As above, we can write this in the more compact form

K = 〈qf , tf |qi, ti〉 =∫

Dq(t)eiS[q] =∫

Dq(t) exp(

i

∫ tf

ti

dt L(q, q)

)

, (1.35)

where the integration includes all paths satisfying the boundary condition

q(ti) = qi , q(tf ) = qf . (1.36)

This is the main result of this section, and is known as the path integral in configuration

space. It will serve us as starting point discussing quantum field theories, although it is lessgeneral than the previous path integral in phase space.

Knowing the path integral and thus the propagator is sufficient to solve scattering problemsin quantum mechanics. In a relativistic theory, the particle number during the course of ascattering process is however not fixed, since energy can be converted into matter. In orderto prepare us for such more complex problems, we will generalise in the next section the pathintegral to a generating functional for n-point Green functions. For instance, 4-point Greenfunctions will be the essential ingredient to calculate 2→ 2 scattering processes.


Having re-expressed the transition amplitude 〈qf , tf |qi, ti〉 of a quantum mechanical systemas a path integral, we want generalize first this result to the matrix elements of an arbitrarypotential V (q) between energy eigenstates |n〉 and |n′〉. For all practical purposes, we canassume that we can expand V (q) as a power-series in q; thus it is sufficient to consider thematrix elements 〈n′|qm|n〉. In quantum field theory, the initial and final states are generallyfree particles which are described mathematically as harmonic oscillators. In this case, weare able to reconstruct all excited states |n〉 from the ground-state,

|n〉 = 1√n!

(a†)n |0〉 .

Therefore it will be sufficient to study matrix elements between the ground-state |0〉. Aim ofthis subsection is to define a generating functional for these quantities.

31


Time-ordered products of operators and the path integral In a first step, we try to includethe operator qm into the transition amplitude 〈qf , tt|qi, ti〉. We can reinterpret our result forthe path integral as follows,

〈qf , tt|1|qi, ti〉 =∫

Dq(t) 1 eiS[q(t)] . (1.37)

Thus we can see the LHS as matrix element of the unit operator 1, while the RHS correspondsto the path integral average of the classical function f(q) = 1. Now we want to generalizethis trivial statement to two operators A(q(ta)) and B(q(tb)) given in the Heisenberg picture.In evaluating ∫

Dq(t) A(q(ta))B(q(tb)) eiS[q(t)] = 〈qf , tt|?|qi, ti〉 , (1.38)

we go back to Eq. (1.24) and choose ta and tb as two of the intermediate times ti,

=

∫dq1 · · · dqN−1 · · · 〈qa+1, ta+1| A |qa, ta〉 · · · 〈qb+1, tb+1| B |qb, tb〉 . . . for ta > tb .

∫dq1 · · · dqN−1 · · · 〈qb+1, tb+1| B |qb, tb〉 · · · 〈qa+1, ta+1| A |qa, ta〉 . . . for ta < tb .

(1.39)Since the time along a classical path increases, the matrix-elements of the operators A(ta)and B(tb) are also ordered with time increasing from the right to the left. If we define thetime-ordered product of two operators as

TA(x1)B(x2) = A(x1)B(x2)ϑ(t1 − t2) +B(x2)A(x1)ϑ(t2 − t1) , (1.40)

then the path integral average of the classical quantities A(ta) and B(tb) corresponds to thematrix-element of the time-ordered product of these two operators,

〈qf , tf |TA(ta)B(tb)|qi, ti〉 =∫

Dq(t)A(ta)B(tb) eiS[q(t)] , (1.41)

and similar for more than two operators.

External sources We want to include in our formalism the possibility that we can changethe state of our system applying an external driving force or source term J(t). In quantummechanics, we could imagine e.g. a harmonic oscillator in the ground-state |0〉, making atransition under the influence of an external force J to the state |n〉 at time t and back tothe ground-state |0〉 at time t′ > t. Including such transitions, we can mimick the relativisticprocess of creation and annihilation of a particle as follows: We identify the vacuum (i.e. thestate containing zero real particles) with the ground-state of quantum mechanical system,and the creation and annihilation of n particle with the (de-) excitation of the n.th energylevel by an external source J .

Schwinger realized that adding a coupling to an external source, viz.

L→ L+ J(t)q(t) , (1.42)

will lead to an efficient way to calculate the matrix elements of an arbitrary polynomial ofoperators q(tn) · · · q(t1): If the source J(t) would be a simple number instead of a time-dependent function in the augmented path-integral

〈qf , tf |qi, ti〉J ≡∫

Dq(t)ei∫ tfti

dt(L+Jq) , (1.43)

32


then we could obtain 〈qf , tf |qm|qi, ti〉J simply by differentiating 〈qf , tf |qi, ti〉J m-times withrespect to J . However, the LHS is a functional of J(t) and thus we need to perform insteada functional derivative with respect to J(t). We define the concept of a functional derivativeby analogy with the normal differentiation of functions,

∂

∂xk1 = 0 ,

∂xl

∂xk= δlk ,

∂

∂xk[F (x)G(x)] = F (x)

∂G(x)

∂xk+G(x)

∂F (x)

∂xk. (1.44)

For a continuous index, the Kronecker delta becomes a delta function,

δ

δJ(x)1 = 0 ,

δJ(x)

δJ(x′)= δ(x− x′) , (1.45)

and the Leibniz rule becomes

δ

δJ(x)F [J ]G[J ] = F [J ]

δG[J ]

δJ(x)+G[J ]

δF [J ]

δJ(x). (1.46)

Now we are able to differentiate 〈qf , tf |qi, ti〉J with respect to the source J . Starting from

1

δJ(t1)

∫

Dq(t) ei∫∞

−∞dtJ(t)q(t) = i

∫

Dq(t) q(t1)ei∫∞

−∞dtJ(t)q(t) , (1.47)

we obtain

〈qf , tf |Tq(t1) · · · q(tn)|qi, ti〉 = (−i)n δn

δJ(t1) · · · δJ(tn)〈qf , tf |qi, ti〉J

∣∣∣∣J=0

. (1.48)

Thus the source J(t) is a convinient tool to obtain the functions q(t1) · · · q(tn) in front ofexp(iS). Having performed the functional derivatives, we set the source J(t) to zero, comingback to the usual path integral.

Ground-state persistence amplitude As last step, we want to convert the transition ampli-tude 〈qf , tf |qi, ti〉J into the probability amplitude that a system which was in the ground-state|0〉 at ti → −∞ remains there at tf →∞ in the presence of the source J(t). Inserting a com-plete set of energy eigenstates, 1 =

∑

n |n〉〈n|, into the propagator, we obtain

〈q′, t′|q, t〉 =∑

n

ψn(q′)ψ∗

n(q) exp(−iEn(t′ − t)) . (1.49)

We can isolate the ground-state n = 0 by adding either to the energies En or to the timedifference τ = (t′ − t) a small negative imaginary part. In this case, all terms would beexponentially damped in the limit τ → ∞, and the ground-state as state with the smallestenergy would more and more dominate the sum. Alternatively, we can add a term +iεq2

to the Lagrangian. Instead of adding an infinitesimal small negative imaginary part to thetime, we can do a somewhat more drastic change, rotating the time axis by 90 degrees inthe complex time plane, τ → −iτ . This procedure called Wick rotation corresponds to thetransition from Minkowski to Euclidean space,

x2 = t2 − x2 → x2E = −[t2 + x2] ,

resulting in an exponentially damped path integral.

33


Finally, we have only to connect our results, Eqs. (1.41) and (1.48), we obtained so far.Adding a coupling to an external source J(t) and an damping factor +iεq2 to the Lagrangiangives us the ground-state persistence amplitude

Z[J ] ≡ 〈0,∞|0,−∞〉J =

∫

Dq(t) ei∫∞

−∞dt(L+Jq+iεq2) (1.50)

in the presence of a classical source J . Taking derivatives w.r.t. the external sources J , andsetting them afterwards to zero, we obtain

δnZ[J ]

δJ(t1) · · · δJ(tn)

∣∣∣∣J=0

= in∫

Dq(t) q(t1) · · · q(tn)ei∫∞

−∞dt(L+iεq2) . (1.51)

The RHS corresponds to the path integral in Eqs. (1.41), except for the iεq2 factor. Butthis factor damps in the limit of large t everything except the ground state. Thus we foundthat Z[J ] is the generating functional of the expectation value of the time-ordered product ofoperators q(ti),

(−i)n δnZ[J ]

δJ(t1) · · · δJ(tn)

∣∣∣∣J=0

= 〈0,∞|Tq(t1) · · · q(tn)|0,−∞〉 = G(t1, . . . , tn) . (1.52)

In the last step, we defined also the n-point Green function G(t1, . . . , tn). These functionswill be the main building block we will use to perform calculations in quantum field theory,and the formula corresponding to Eq. (1.52) will be our master formula in field theory.

1.4. Oscillator as an 1–dimensional field theory

Canonical quantisation An one-dimensional harmonic oscillator can be viewed as a freequantum field theory in one time and zero space dimensions. In order to stress this equiva-lence, we rescale the usual Lagrangian

L(x, x) =1

2mx2 − 1

2mω2x2 , (1.53)

where m is the mass of the oscillator and ω its frequency as

φ(t) ≡ √mx(t) . (1.54)

We call the variable φ(t) a “scalar field,” and the Lagrangian now reads

L(φ, φ) =1

2φ2 − 1

2ω2φ2 . (1.55)

After the rescaling, the kinetic term φ2 is “canonically normalised,” i.e. it has the dimension-less coefficient 1/2.

We derive the corresponding Hamiltonian, determining first the conjugate momentum π asπ(t) = ∂L/∂φ = φ(t). Thus the (classical) Hamiltonian follows as

H(φ, π) =1

2π2 +

1

2ω2φ2 . (1.56)

34


The transition to quantum mechanics is performed by promoting φ and π to operators whichsatisfy the canonical cammutation relations [φ, π] = i.

The harmonic oscillator is solved most efficiently introducing creation and annihilationoperators, a† and a. They are defined by

φ =1√2ω

(

a† + a)

, and π = i

√ω

2

(

a† − a)

, (1.57)

and satisfy[a, a†

]= 1. The Hamiltonian follows as

H =ω

2

(

aa† + a†a)

=

(

aa† +1

2

)

ω . (1.58)

We interprete N ≡ aa† as the number operator, counting the number of quanta having eachthe energy ω in a given state |n〉.

We now work in the Heisenberg picture where the operators are time dependent. The timeevolution of the operator a(t) can be found from the Heisenberg equation,

ida

dt= [a,H] = ωa , (1.59)

from which we deduce that

a(t) = a(0)e−iωt = a0e−iωt . (1.60)

As a consequence, the field operator can be expressed in terms of the creation and annihilationoperators as

φ(t) =1√2ω

(

a0e−iωt + a†0e

iωt)

. (1.61)

Path integral approach We solve now the same problem, the rescaled Lagrangian (1.55),in the path integral approach. Using this method, it is convenient to include a coupling toan external force J . Morover, it is also useful to recall how we solved the classical forcedoscillator in section 1.1.3 applying the Green function method.

Let us define the effective action Seff as the sum of the classical action S, the coupling to theexternal force J and a small negative imaginary part to make the path integral well-defined,

Seff = S +

∫

dt(Jφ+ iεφ2

)=

∫

dt

[1

2φ2 − 1

2ω2φ2 + Jφ+ iεφ2

]

. (1.62)

This function is the integrand of the path integral. We start our work by massaging Seff intoa form such that the path integral can be easily performed. The first two terms in Seff canbe viewed again as the action of differential operator D(t) on φ2(t),

D(t)φ2(t) =1

2

(d2

dt2− ω2

)

φ2(t) .

We can evaluate this operator, if we go to Fourier space,

φ(t) =

∫dE

2πe−iEt φ(E) and J(t) =

∫dE

2πe−iEt J(E) . (1.63)

35


To keep the action real, we have to write all bilinear quantities as φ(E)φ(−E′), etc. Then onlythe phases dependent on time, and the t integration gives a factor 2πδ(E − E′), expressingenergy conservation,

Seff =1

2

∫dE

2π

[φ(E)(E2 − ω2 + iε)φ(−E) + J(E)x(−E) + J(−E)x(E)

](1.64)

In the path integral, this expression corresponds to a Gaussian integral of the type of (1.18),where we should “complete the square.” Shifting the integration variabel to

φ(E) = φ(E) +J(E)

E2 − ω2 + iε,

we obtain

Seff =1

2

∫dE

2π

[

φ(E)(E2 − ω2 + iε)φ(−E)− J(E)1

E2 − ω2 + iεJ(−E)

]

. (1.65)

Here we see that the “damping rule” for the path integral makes also energy integral overthe denominator well-defined. The physical interpretation of this way of shifting the poles—which differs from our treatement of the retarded Green function in the classical case—willbe postponed to the next chapter, where we will discuss this issue in detail.

We are now in the position to evaluate the generating functional Z[J ]. The path integralmeasure is invariant under a simple shift of the integration variable, Dφ = Dφ, and we omitthe tilde from now on. Furthermore, the second term in Seff does not depend on φ and canbe factored out,

Z[J ] = exp

(

− i

2

∫dE

2πJ(E)

1

E2 − ω2 + iεJ(−E)

)

(1.66)

×∫

Dφ expi

2

∫dE

2π

[φ(E)(E2 − ω2 + iε)φ(−E)

](1.67)

Setting the external force to zero, J = 0, the first factor becomes one and the generatingfunctional Z[0] becomes equal to the path integral in the second line. But for J = 0, theoscillator remains in the ground-state and thus Z[0] = 〈0,∞|0,−∞〉 = 1. Therefore

Z[J ] = exp

(i

2

∫dE

2πJ(E)

1

E2 − ω2 + iεJ(−E)

)

(1.68)

Inserting the Fourier transformed quantities, we arrive at

Z[J ] = exp

(i

2

∫

dt

∫

dt′ J(t)G(t− t′)J(t′))

. (1.69)

where the Feynman propagator

GF (t− t′) =∫

dE

2πe−iE(t−t′) 1

E2 − ω2 + iε(1.70)

differs from the retarded propagator GR (1.38) by the position of its poles.

36


This result allows us to calculate arbirary matrix elements between oscillator states. Forinstance, we obtain the expectation value 〈0|φ2 |0〉 from

〈0| |Tφ(t2)φ(t1) |0〉 = (−i)2 δ2Z[J ]

δJ(t1)δJ(tn)

∣∣∣∣J=0

= iGF (t2 − t1) =1

2ωeiω|t−t

′| . (1.71)

Here, we used in the last step the explicit expression for GF which you should check inproblem 1.6. Taking the limit t2 ց t1 and replacing φ2 → mx2, we reproduce the standardresult 〈0| x2 |0〉 = 1/(2mω). Matrix elements between excited states |n〉 = (n!)−1/2(a†)n |0〉are obtained by expressing the creation operator a† using π(t) = φ(t) as

a† =

√ω

2

(

1− i

ω

d

dt

)

φ(t) . (1.72)

Summary

Using Feynman’s path integral approach, we can express a transition amplitude as a sum overall paths, 〈qf , tf |qi, ti〉 =

∫Dq(t) exp(iS[q]). Adding a linear coupling to an external source

J and a damping term to the Lagrangian, we obtain the ground-state persistence amplitude〈0,∞|0,−∞〉J . This quantity serves as the generating functional for n-point Green functionsG(t1, . . . , tn) which are the time-ordered vacuum expectation values of operators q1, . . . , qn.

Further reading

Our presentation of the path integral follows the one of [Mac99] which contains as well as[Das06] useful additional material. Reading original papers like Feynman’s presentation ofthe path integral in [Fey48] can be still very informative, particularly about the sometimesurprising initial motivations of the authors.

Problems

1.1 Classical action.

Calculate the classical action S[q] for a free par-ticle and an harmonic oscillator. Compare theresults with the expression for the propagatorK = 〈x′, t′|x, t〉 = N exp(iφ) of the correspond-ing quantum mechanical system and express bothφ and N through the action S.

1.2 Propagator as Green function.

Show that the Green function or propagatorK(x′, t′;x, t) = 〈x′| exp[−iH(t′ − t)]|x〉 of theSchrodinger equation is the inverse of the differ-

ential operator (i∂t −H).

1.3 Statistical mechanics.

Derive the connection between the partition func-tion Z = Tr e−βH of statistical mechanics and thepath integral in Euclidean time.

1.4 Feynman propagator.

Find the explicit expression for the Feynmanpropagator used in (1.71) from a.) its its defi-nition as time-ordered product of fields φ, and b.)

37


evaluating (1.70) using Cauchy’s theorem.

1.5 Matrix elements from Z[J ].Evaluate the matrix element 〈0| |φ2 |1〉 of an har-monic oscillator from Z[J ].

1.6 Classical driven oscillator.Consider an harmonic oscillator satisfying

q(t) + ω2q(t) = 0 , for t < 0 and t > T

q(t)− Ω2q(t) = 0 , for 0 < t < T

where ω and Ω are real constants. a.) Show that

for q(t) = A1 sin(ωt) for t < 0 and ΩT ≫ 1, thesolution q(t) = A2 sin(ω0t + α) with α = const.satisfies

A2 ≈1

2(1 + ω2/Ω2)1/2 exp(ΩT ).

b.) If the oscillator was in the ground-state att < 0, how many quanta are created? Find the ex-plicit expression for the Feynman propagator usedin (1.71) from its its definition as time-orderedproduct of fields φ.

38

2. Scalar fields

We extend in this chapter the path integral approach from quantum mechanics to the simplestfield theory, a real scalar field φ(x). In particular, we will introduce the generating functionalZ[J ] = 〈0 + |0−〉J of n-point Green functions as our main tool to calculate the time-orderedvacuum expecation value of a product of fields φ(x1) · · · φ(xn). After having gained someexperience with the case of a free field in sections 2.1–2.4, we will use perturbation theory todiscuss a scalar field with a λφ4 self-interaction in the last two sections.

2.1. Lagrange formalism and path integrals for fields

A field is a map which associates to each space-time point x a k-tupel of values φa(x),a = 1, . . . , k. The space of field values φa(x) can be characterised by its transformationproperties under Poincare transformations and internal symmetry groups. The latter are inpractically all physical applications Lie groups like R, U(1), SU(n) or SO(n). Except for a realscalar field φ, these fields have several components. Thus we have to generalise Hamilton’sprinciple to a collection of fields φa(x), where the index a includes all internal as well asspace-time indices. Moreover, the Lagrangian for a field φa(x) will contain not only time butalso space derivatives.

To ensure Lorentz invariance, we consider a scalar Lagrange density L that may, anal-ogously to L(q, q), depend on the fields φa and their first derivatives ∂µφa. We include noexplicit time-dependence, since “everything” should be explained by the fields φa and theirinteractions. The Lagrangian L(φa, ∂µφa) is obtained by integrating the density L over agiven space volume V . The action S is thus the four-dimensional integral

S[φa] =

∫ b

adt L(φa, ∂µφa) =

∫

Ωd4xL (φa, ∂µφa) (2.1)

with Ω = V × [ta : tb].

A variation δφa(x) of the fields leads to a variation of the action,

δS =

∫

Ωd4x

[∂L

∂φaδφa +

∂L

∂(∂µφa)δ(∂µφa)

]

, (2.2)

where we have to sum over field components (a = 1, . . . , k) and the Lorentz index µ = 0, . . . , 3.The correspondence q(t)→ φ(x) implies that the parameter ε in Hamilton’s principle dependsnot on x. We can therefore eliminate again the variation of the field gradients ∂µφ

a by a partialintegration using Gauß’ theorem,

δS =

∫

Ωd4x

[∂L

∂φa− ∂µ

(∂L

∂(∂µφa)

)]

δφa = 0 . (2.3)

39

2. Scalar fields

The surface term vanishes, since we require that the variation is zero on the boundary ∂Ω.Thus the Lagrange equations for the fields φa are

∂L

∂φa− ∂µ

(∂L

∂(∂µφa)

)

= 0 . (2.4)

If the Lagrange density L is changed by a four-dimensional divergence, δL = ∂µKµ, and

surface terms can be dropped, the same equations of motion result. Note also that it is inmost applications in field theory more efficient to perform directly the variation δφa of theaction S[φa] than to use the Lagrange equations.

Path integrals for fields The path integrals becomes now a functional integral over the nfields φa,

K =

∫

Dφ1 · · · Dφn eiS[φ] =∫

Dφ1 · · · Dφn ei∫Ωd4xL (φa,∂µφa) (2.5)

A major problem we have to address later is that the n fields φa are often not independent: Forinstance, in electrodynamics all potentials Aµ connected by a gauge transformation describethe same physical situation. This redundancy makes the path integral ill-defined. In suchcases, we need to factor out these gauge degrees of freedom from the integration measureDA0 · · · DA3.

We start therefore with the simplest case of a scalar field φ, discussing first the case of afree field and then the case of a scalar field with self-interactions.

2.2. Generating functional for a scalar field

Lagrangian The Klein-Gordon equation as a relativistic wave equation can be derived anal-ogously to the (free) Schrodinger equation,

i∂tψ = H0ψ , (2.6)

which is obtained by substituting

ω → i∂t k→ −i∇x (2.7)

in the non-relativistic energy-momentum relation ω = k2/(2m). With the same replacements,the relativistic ω2 = m2 + k2 becomes

(+m2)φ = 0 with = ηµν∂µ∂ν = ∂µ∂

µ . (2.8)

The relativistic energy-momentum relation implies that the solutions to the free Klein-Gordon

equation consist of plane-waves with positive and negative energies ±√

k2 +m2. For thestability of a quantum system it is essential that its energy eigenvalues are bounded frombelow. Otherwise, we could generate e.g. in a scattering process φ + φ → nφ an arbitrarilyhigh number of φ particles with sufficiently low energy, and no stable form of matter couldexist. Interpreting the Klein-Gordon equation as a relativistic wave equation for a singleparticle can be therefore not fully satisfactory, since the energy of its solutions is not boundedfrom below.

40


How do we guess the correct Lagrange density L ? Plane waves can be seen as a collection ofcoupled harmonic oscillators at each space-time point. The correspondence q ↔ ∂µφ meansthat the “kinetic” field energy is quadratic in the field derivatives. Relativistic invarianceimplies that the Lagrangian is a scalar, leaving as the only two possible terms containingderivatives

ηµν(∂µφ)(∂νφ) and φφ .

Using the action principle to derive the equation of motions, we can however drop boundaryterms performing partial integrations. Thus these two terms are equivalent, up to a minussign. The Klein-Gordon equation φ = −m2φ suggests that the mass term is also quadraticin the field φ. Therefore we try as Lagrange density

L =1

2ηµν (∂

µφ) (∂νφ)− 1

2m2φ2 ≡ 1

2ηµν∂

µφ∂νφ− 1

2m2φ2 . (2.9)

From now, we will drop the parenthesis around (∂µφ) and it should be understood from thecontext that the derivative ∂µ acts only on the first field φ. Even shorter alternative notationsare (∂µφ)

2 and finally the concise (∂φ)2. With the short-cut φ,α ≡ ∂αφ and

∂

∂φ,α(ηµνφ,µφ,ν) = ηµν

(δαµφ,ν + δαν φ,µ

)= ηανφ,ν + ηµαφ,µ = 2φ,α , (2.10)

the Lagrange equation becomes

∂L

∂φ− ∂α

(∂L

∂φ,α

)

= −m2φ− ∂α∂αφ = 0 . (2.11)

Thus the Lagrange density (2.9) leads indeed to the Klein-Gordon equation. We can un-derstand the relative sign in the Lagrangian splitting the relativistic kinetic energy into the“proper” kinetic energy (∂tφ)

2/2 and the gradient energy density (∇φ)2/2,

L =1

2φ2 − 1

2(∇φ)2 − 1

2m2φ2 . (2.12)

The last two terms correspond to a potential energy and carry therefore the opposite sign ofthe first one.

Instead of guessing, we can derive the correct Lagrangian L as follows: We multiply thefree field equation for φ by a variation δφ that vanishes at the endpoints ta and tb. Then weintegrate over Ω = V × [ta : tb], perform a partial integration of the kinetic energy, use theLeibniz rule and ask that the variation vanishes,

A

∫

Ωd4x δφ ( +m2)φ = A

∫

Ωd4x

[−δ(∂µφ)∂µφ+ δφφm2

]= (2.13)

= A

∫

Ωd4x δ

[

−1

2(∂µφ)

2 +1

2φ2m2

]

= 0 . (2.14)

The term in the square brackets agrees with our guess (2.9), taking into account that thesource-free field equation fixes the Langrangian only up to the overall factor A. In analogywith a quantum mechanical oscillator, we want that the coefficients of the two terms are ±1/2and thus we set |A| = 1.

We can determine the correct overall sign of L by calculating the energy density ρ of thescalar field and requiring that it is bounded from below and stable against small perturbations.

41

2. Scalar fields

We identify the energy density ρ of the scalar field with its Hamiltonian density H , and usethe connection between the Lagrangian and the Hamiltonian known from classical mechanics.The transition from a system with a finite number of degrees of freedom to one with an infinitenumber of degrees of freedom proceeds as follows,

pi =∂L

∂qi⇒ πa =

∂L

∂φa, (2.15)

H = piqi − L ⇒ H =

∑

a

πaφa −L . (2.16)

The canonically conjugated momentum π of a real scalar field is

π =∂L

∂φ= φ . (2.17)

Thus the Hamilton density is

H = πφ−L = π2 −L =1

2φ2 +

1

2(∇φ)2 +

1

2m2φ2 ≥ 0 (2.18)

and thus obviously positive definite. Moreover, generating fluctuations δφ costs energy andthus the system is stable against small perturbations. Hence the transition from a singleparticle interpretation of the Klein-Gordon equation to a field theoretic interpretation issufficient to cure the problem of the negative energy solutions. Note also that we couldsubtract a constant ρ0 from the Langragian which would drop out of the equation of motions.From Eq. (2.18) we see that such a constant corresponds to a uniform energy density in space.Such a term would give a contribution to the cosmological constant in the gravitational fieldequation, but would be otherwise unobservable.

Next we generalise the Lagrangian by subtracting a polynomial in the fields, V (φ), subjectto the stability constraint discussed above. Hence the potential should be bounded frombelow, and we can expand it around its minimum at φ ≡ v,

dV

dφ

∣∣∣∣φ=v

= 0 ,d2V

dφ2

∣∣∣∣φ=v

≡ m2 > 0 . (2.19)

The term V ′′(v) acts as mass term, while we will see that terms φn with n ≥ 3 generateinteractions between n particles, as expected from the analogy of a quantum field to coupledquantum mechanical oscillators discussed in problem ??. The field has the non-zero valueφ = v everywhere, if the minimum v of V (φ) is not at zero, v 6= 0. If the value of V (φ) at theminimum v is not zero, V (v) 6= 0, there is a non-zero energy density ρ = −V (v) everywhere.

Generating functional In quantum mechanics a particle cannot be created or destroyedand thus the main question to ask in a scattering problem is how likely it is that a particlemoves from x to x′. In quantum field theory, we address a more complex problem: We want tocalculate how likely it is that a particle is created by an external source J(x), propagates fromx to x′, disappearing there in an interaction with another source J(x′). As we have alreadyseen in the previous chapter, such questions can be conviniently addressed considering thefunctional Z[J ], where we add a coupling between the field and an external source,

Z[J ] = 〈0 + |0−〉J = N∫

Dφ exp i∫

Ωd4x

(1

2∂µφ∂

µφ− 1

2m2φ2 + Jφ

)

. (2.20)

42


To ensure the convergence of the integral, we add an infinitesimal small imaginary part tothe squared mass of the particle, m2 → m2 − iε. Next we perform one integration by part ofthe first term, exploiting the fact that the boundary term vanishes,

Z[J ] = N∫

Dφ exp i∫

Ωd4x

(

−1

2φ(+m2)φ+ Jφ

)

. (2.21)

The first two terms, A = −(+m2)/2, are quadratic and symmetric in the field φ,

− 1

2

∫

d4x φ(x)(x +m2)φ(x) =

∫

d4xd4x′ φ(x)A(x, x′)δ(x − x′)φ(x′) . (2.22)

Note that the operator A is local, A(x, x′) ∝ δ(x− x′): Since special relativity forbids actionat a distance, non-local terms like φ(x′)( + m2)φ(x) should not appear in a relativisticLagrangian.

If we discretise continuous space-time xµ into a lattice, we can use Eq. (1.19) to performthe path integral,

Z[J ] = N((2πi)N

det[A]

)1/2

exp

(

−1

2iJA−1J

)

≡ NZ[0] exp(iW [J ]) . (2.23)

The pre-factor of the exponential function does not depend on J and is thus given by NZ[0] =〈0 + |0−〉. The vacuum should be stable and normalised to one in the absence of sources,〈0+ |0−〉 = 1. Therefore we should ensure that Z[J ] is properly normalised what implies thatN−1 = Z[0].

In the last step of Eq. (2.23), we defined a new functional W [J ] that depends only quadrat-ically on the source J ; therefore it should be much easier to calculate than Z[J ]. Going forN →∞ back to continuous space-time, the matrix multiplications become integrations,

Z[J ] = exp(iW [J ]) = exp

(

− i

2

∫

d4xd4x′J(x)A−1(x, x′)J(x′)

)

(2.24)

and

W [J ] = −1

2

∫

d4xd4x′J(x)A−1(x, x′)J(x′) . (2.25)

Propagator In order to evaluate the functional W [J ] we have to find the inverse ∆(x, x′) ≡A−1(x, x′) of the differential operator A, defined by

− (+m2)∆(x, x′) = δ(x− x′) . (2.26)

Because of translation invariance, the Green function ∆(x, x′) can depend only on the differ-ence x − x′. Therefore it is advantageous to perform a Fourier transformation and to go tomomentum space,

−∫

d4k

(2π)4(+m2)∆(k)e−ik(x−x′) =

∫d4k

(2π)4e−ik(x−x′) , (2.27)

or

∆F (k) =1

k2 −m2 + iε, (2.28)

43

2. Scalar fields

where the pole at k2 = m2 is avoided by the iε prescription. Thus the m2 → m2 − iεprescription introduced to ensure the convergence of the path integral tells us also how tohandle the poles of the Green function. The index F specifies that the propagator ∆F is theGreen function obtained with the m2− iε prescription proposed by Feynman. (Some authorsuse instead DF for the propagator of massive bosons and ∆F f for the propagator of masslessbosons.)

Note that the four components kµ are independent, i.e. that ∆(k) describes the propagationof a virtual particle that has—in contrast to a real or external particle—not to be on “mass-shell:” in general

k0 6= ±ωk ≡√

k2 +m2 − iε .

We can evaluate the k0 integral in the coordinate representation of ∆F (x− x′) explicitly,

∆F (x− x′) =∫

d4k

(2π)4e−ik(x−x′)

k20 − k2 −m2 + iε(2.29)

=

∫d3k

(2π)3

∫dk02π

e−ik0(t−t′)eik(x−x′)

(k0 − ωk + iε)(k0 + ωk − iε), (2.30)

using Cauchy’s theorem. The integrand has two simple poles at +ωk − iε and −ωk + iε,cf. Fig. 2.1. For negative τ = t − t′, we can close the integration contour C+ on the upperhalf-plane, including the pole at −ωk,

∫

dk0e−ik0τ

(k0 − ωk + iε)(k0 + ωk − iε)= 2πi res−ωk

= 2πieiωkτ

−2ωkfor τ < 0 . (2.31)

For positive τ , we have to choose the contour C− in the lower plane, picking up2πi e−iωkτ/(2ωk) and an additional minus sign since the contour is clockwise. Combiningboth results, we obtain

i∆F (x) =

∫d3k

(2π)32ωk

[e−iωktϑ(x0) + eiωktϑ(−x0)

]eikx , (2.32)

or after shifting the integration variable k→ −k in the second term,

i∆F (x) =

∫d3k

(2π)32ωk

[

e−i(ωkt−kx)ϑ(x0) + ei(ωkt−kx)ϑ(−x0)]

. (2.33)

Comparing this expression with the spectral representation (1.16) of the non-relativisticpropagator, we note three important properties of the relativistic propagator: First, thepropagator contains positive and negative frequencies, as expected from the existence of solu-tions to the Klein-Gordon equation with positive and negative frequencies. Second, positivefrequencies propagate forward in time, while negative frequencies propagate backward. If weimagine that the propagating particle carries a conserved charge, then we can associate thepositive frequencies to the propagation of a particle (with charge q) and the negative frequen-cies to the propagation of an anti-particle (with charge −q). In this way, the resulting currentis frame-independent. Third, the normalisation of plane waves include a factor 1/

√2ωk, or

〈k|k′〉 = 2ωk(2π)3δ(k − k′) , (2.34)

to ensure a relativistically invariant normalisation. In order to check this statement, youshould work through problem 2.3.

44


Im(k0)

Re(k0)

C−

C+

−ω + iε×

ω − iε×

Figure 2.1.: Poles and contours in the complex k0 plane used for the integration of the Feyn-man propagator.

Remark: Other Green functions are obtained, if we choose another prescription for the handling

of the poles. For positive τ = t− t′, we have to close the circle on the upper half-plane. Shifting both

poles to the lower half-plane, ±ωk − iε, gives thus the retarded propagator ∆ret(x) vanishing for all

τ > 0. In the opposite case, we shift both poles to the upper half-plane, ±ωk + iε, and obtain the

advanced propagator ∆adv(x). Both propagators are real-valued, propagating a real solution of the

wave equation into another real one at a different time, as required in classical physics. Moreover, both

Green functions have support inside the light-cone, the retarded in the forward and the advanced in the

backward part of the light cone. This behaviour should be contrasted with the Feynman propagator

∆F which is complex-valued and non-zero in R(1, 3).

Another way to handle the singularities is to use Cauchy’s Principal value prescription, obtaining

∆(x) = 12 [∆adv(x) + ∆ret(x)]. This choice corresponds to an action-at-distance which seems to have

no relevance in physics. Finally, we can shift one pole up and the other one down. The choice ±(ωk+iε)

opposite to the one used for the Feynman propagator leads first to the wrong non-relativistic limit,

where the waves e+i(ωkt) propagate forward in time. Second, performing a Wick rotation, we have to

rotate k0 → +ik0 which leads to an Eucledian action which is bounded from above instead from below.

The Feynman prescription is thus the only way to handle the singularities of the propagator which

guraentees that the correlation functions of statistical physics and the Green function of quantum field

theory are analytically connected.

We are now in the position to evaluate the generating functional

W [J ] = −1

2

∫

d4xd4x′J(x)∆F (x− x′)J(x′) . (2.35)

in Fourier space. Inserting the Fourier transformations for the propagator as well as for thesources J gives

W [J ] = −1

2

∫

d4xd4x′∫

d4k

(2π)4d4k′

(2π)4d4k

(2π)4J(k)∗eikx

e−ik(x−x′)

k2 −m2 + iεJ(k′)e−ik′x′ . (2.36)

45

2. Scalar fields

Exchanging the integration order and performing the space-time integrations leads to theconservation of the four-momenta entering and leaving the two interaction points, (2π)8δ(k−k)δ(k− k′): The source J(k)∗ produce a scalar particle with momentum k, and thus only theFourier component k of the scalar propagator contributes. This is a very general behavior,based solely on the translation invariance of the free particle states we are using. In thefinal step, we cancel two of the three momentum integrations with the two momentum deltafunctions and are left with

W [J ] = −1

2

∫d4k

(2π)4J(k)∗

1

k2 −m2 + iεJ(k) . (2.37)

The closer the particles emitted by the source J(k) are on-shell, k2 → m2, the larger istheir contribution to W [J ]. For k2 = m2, the propagator diverges finally. Reason for thisunphysical result is that our formalism assumes that the exchanged particle is stable, and areal particle can thus travel an infinite distance. If we would take the finite life-time τ1/2 ofthe exchanged particle into account, the infinitesimal iε would be replaced by a finite quantitydetermined by τ1/2.

The functional W [J ] has the same structure as the one for the harmonic oscillator found inthe last chapter. We will see that it contains, as in 1-dimensional case, all information abouta free scalar field, not only about its ground state.

Attractive Yukawa potential by scalar exchange From our macroscopic experience, weknow the two cases of electromagnetism, where equal electric charges repel each other, and ofgravity where two masses attract each other. The first physics question we want to answer withour newly developped formalism is if the scalar field falls into the category of a fundamentallyattractive or repulsive interaction.

In order to address this question, we consider two static point charges as external sources,J = J1(x1) + J2(x2) with Ji = δ(x−xi), in W [J ]. Multiplying out the terms in J(x)∆F (x−x′)J(x′) gives four contributions, Wij ∝ JiJj : The terms W11[J ] and W22[J ] correspondto the emission and re-absorption of the particle by the same source Ji. They are examplesfor self-interactions that we neglect for the moment. The interaction between two differentcharges is given by

W12[J ] =W21[J ] = −1

2

∫

d4xd4x′∫

d4k

(2π)4J1(x)

e−ik(x−x′)

k2 −m2 + iεJ2(x

′) (2.38)

= −1

2

∫

dtdt′∫

d4k

(2π)4e−ik0(t−t′) eik(x1−x2)

k2 −m2 + iε. (2.39)

Performing one of the two time integrals, e.g. the one over t′, gives 2πδ(k0). Hence ourassumption of static sources implies that the virtual particle carries zero energy and is space-like, k2 = −k2 < 0. Eliminating then the k0 integral with the help of the delta function, weobtain next

W12[J ] =1

2

∫

dt

∫d3k

(2π)3eikr

k2 +m2(2.40)

with r = x1 − x2. The denominator is always positive, and we can therefore omit the iε.Before we can go on, we have to make sense out of the infinite time integral: Looking at

Z[J ] = 〈0| exp(−iH[J ]τ |0〉 = exp(iW [J ]) , (2.41)

46

2.3. Green functions for a free scalar field

we see thatW [J ] = −Eτ , where τ = t−t′ is the considered time interval. Hence the potentialenergy V of two static point charges separated by the distance r is

V = −(W12 +W21)/τ = −∫

d3k

(2π)3eikr

k2 +m2= −e−mr

4πr< 0 . (2.42)

Thus the potential energy V of two equal charges is reduced by the exchange of a scalarparticle, which means that the scalar force between them is attractive. If the exchangedparticle is massive, the range of the force is of order 1/m. These two observations were thebasic motivation for Yukawa to suggest pion-exchange as model for the nuclear force. Notealso that we obtain in the limit m → 0 a 1/r potential as in Newton’s and Coulomb’s law.Thus we learned the important fact that the only two known forces of infinite range, theelectromagnetic and the gravitational force, are transmitted by massless particles, the photonand the graviton, respectively.

The result V ∝ 1/r for m = 0 and n = 4 space-time dimensions, or more generallyV ∝ 1/rn−3 for n ≥ 4, follows from simple dimensional analysis. For m = 0, the onlyremaining dimensionfull parameter after the integration over k is the distance r. As thepotential energy V has the dimension [V ] = mn−3, it has to scale as r−n+3.


We have used up to now the definition of a Green function as the inverse of the correspondingdifferential operator. This definition is sufficiently general for quantum mechanics, where wewant to describe the propagation of a single particle from x to x′. In quantum field theory,we are interested in more complicated processes where many particles can be created anddestroyed. For instance, two protons scattering at the LHC in Geneva can produce dozensof particles in the final state. Such processes are described by n-point functions Green’sfunctions G(x1, . . . , xn). Moreover, we will introduce two types of Green functions, namelydisconnected n-point functions G(x1, . . . , xn) and connected n-point functions G(x1, . . . , xn),

Consider the expansion of the exponential in Eq. (2.79),

Z[J ] = exp(iW [J ]) =

∞∑

n=0

in

n!W n =

∞∑

n=0

in

n!

∫

d4x1 · · · d4xnGn(x1, · · · , xn)J(x1) · · · J(xn) .

(2.43)For the moment, we assume that Z[J ] is normalised properly so that Z[0] = 1. We will discussthe normalisation later in the case of an interacting theory. The RHS serves as definition of thedisconnected n-point Greens function G(x1, · · · , xn). They can be calculated as the functionalderivatives of Z[J ],

G(x1, . . . , xn) =1

inδn

δJ(x1) · · · δJ(xn)Z[J ]

∣∣∣∣J=0

. (2.44)

For n = 2 we should rederive the Feynman propagator. Starting from

1

i

δZ[J ]

δJ(x)=

1

i

δ

δJ(x)exp

(

− i

2

∫

d4x1d4x2J(x1)∆F (x1 − x2)J(x2)

)

= −∫

d4x1∆F (x− x1)J(x1) exp(iW [J ]) , (2.45)

47

2. Scalar fields

we obtain

1

i

δ

δJ(y)

1

i

δ

δJ(x)Z[J ] = i∆F (x− y) exp(iW [J ])

+

(∫

d4x1∆F (x− x1)J(x1))(∫

d4x1∆F (y − x1)J(x1))

exp(iW [J ]) . (2.46)

Setting J = 0 gives the desired result for the 2-point function as

G(x, y) = i∆F (x− y) . (2.47)

It is straightforward to continue: Another functional derivative gives the 3-point function,

δ

iδJ(x1)

δ

iδJ(x2)

δ

iδJ(x3)Z[J ] =

−(∫

d4x∆F (x1 − x)J(x))(∫

d4x∆F (x2 − x)J(x))(∫

d4x∆F (x3 − x)J(x))

exp(iW [J ])

− i∆F (x2 − x3)∫

d4x∆F (x1 − x)J(x) exp(iW [J ])

− i∆F (x2 − x1)∫

d4x∆F (x3 − x)J(x) exp(iW [J ])

− i∆F (x3 − x1)∫

d4x∆F (x2 − x)J(x) exp(iW [J ]) . (2.48)

For n odd, we obtain always a source J in the pre-factor because W [J ] is an even polynomialin J . Hence all odd n-point functions are zero. We continue with the 4-point function: Aftertaking another derivative and setting J = 0, only terms linear in J of (2.48) contribute andthus

G(x1, x2, x3, x4) =− [∆F (x1 − x2)∆F (x3 − x4)+ ∆F (x1 − x3)∆F (x2 − x4)+ ∆F (x1 − x4)∆F (x2 − x3)] . (2.49)

We see that the 4-point function is the sum of all permutations of the product of two 2-pointfunctions. For instance, the first term ∆F (x1−x2)∆F (x3−x4) in the 4-point function describesthe independent propagation of a scalar particle from x1 to x2 and of another one from x3to x4. Thus our approach leads indeed to a many-particle theory. Since we did not includeinteractions, particles are propagating independently and the n-point function factorises intoproducts of two 2-point functions. Thus the functional Z[J ] generates disconnected Greenfunctions. The statement that the n-point function is the sum of all permutations of theproduct of two 2-point functions holds for all n and is called “Wick’s theorem”.

We consider next the connected n-point functions G(x1, . . . , xn). Their generating func-tional is W [J ],

G(x1, . . . , xn) =1

inδn

δJ(x1) · · · δJ(xn)iW [J ]

∣∣∣∣J=0

. (2.50)

For a free theory, W is quadratic in the sources J . Hence, all connected n-point functionsG(x1, . . . , xn) with n > 2 vanish and the only non-zero one is the two-point function with

G(x, y) = i∆F (x− y) = G(x, y) . (2.51)

48


To summarise: There exists only one non-zero connected n-point function in a free the-ory which is determined by the Feynman propagator, G(x, y) = i∆F (x − y). All non-zerodisconnected n-point functions can be obtained by permuting the product of n/2 two-pointfunctions (“Wick’s theorem”). Hence any higher-order Green’s function can be constructedout of a single building block, the Feynman propagator.

In perturbation theory, we will recast the interacting theory – loosely speaking – in “inter-action vertices times free propagators”. This enables us to derive simple Feynman rules thattell us how one constructs an arbitrary Green function out of vertices and propagators.

Causality and the Feynman propagator We expect that a relativistic field theory auto-matically implements the requirement of causality: No signal using our φ particles as carriershould travel with a speed larger than the one of light. On the other hand, the Feynmanpropagator i∆F (x1 − x2) does not vanish outside the light-cone, but goes only exponentiallyto zero, i∆F (x1 − x2) ∝ exp(−m|x1 − x2|) for space-like distances (x1 − x2)2 < 0, cf. prob-lem 2.8. One may wonder, if this means that the uncertainty principle makes the light-cone”fuzzy” and thus the axiom of special relativity that no signal can be transmitted with v > cis violated on scales smaller <∼ 1/m.

This problem is addressed best following the usual approach of quantum mechanics: Weconsider the classical field φ(x) as operator φ(x) and ask when a measurement of φ(x) influ-ences φ(x′). We have shown that the Feynman propagator equals the 2-point Green function.From our discussion in Sec. 1.3 we know that a n-point Green function corresponds to thevacuum expectation value of the time-ordered product of n field operators. Combining allthis gives

G(x1, x2) =1

i2δ2

δJ(x1)δJ(x2)Z[J ]

∣∣∣∣J=0

= 〈0|Tφ(x1)φ(x2)|0〉 = i∆F (x1 − x2) . (2.52)

The property ∆F (x − y) = ∆F (y − x) implies that the field operators φ(x1) and φ(x2)commute,

〈0|Tφ(x1)φ(x2)|0〉 = 〈0|φ(x1)φ(x2)|0〉ϑ(t1 − t2)+ 〈0|φ(x2)φ(x1)|0〉ϑ(t2 − t1) . (2.53)

Using the analogy of a free quantum field to an infinite set of oscillators, we try to expressthe field operator φ(x) through annihilation and creation operators. Comparing

〈0|Tφ(x)φ(0)|0〉 = i∆F (x) =

∫d3k

(2π)32ωk

[

e−i(ωkt−kx)ϑ(x0) + ei(ωkt−kx)ϑ(−x0)]

(2.54)

with the Fourier decomposition of a classical, free scalar field,

φ(x) =

∫d3k

√

(2π)32ωk

[

a(k)e−ikx + a†(k)e+ikx]

, (2.55)

suggests to reinterpret the classical Fourier coefficients a(k) and a†(k) as annihilation andcreation operators,

a†(k) |0〉 = |k〉 , a(k)∣∣k′⟩= (2π)32ωkδ(k − k′) , and a(k) |0〉 = 0 . (2.56)

49

2. Scalar fields

Hence the vacuum state |0〉 is defined by a(k) |0〉 = 0 for all k. Imposing the commutationrelations

[a(k), a†(k′)] = δ(k − k′) , (2.57)

we rederive the RHS of Eq. (2.52). Moreover, these relations lead to canonical commutationrelations between the field φ and its canonically conjugated momentum density π = φ atequal times,

[φ(x, t), π(x′, t)] = iδ(x − x′) . (2.58a)

[φ(x, t), φ(x′, t)] = [π(x, t), π(x′, t)] = 0 . (2.58b)

You may compare the (missing) details of this derivation to problem ??, where the analogouscalculation is done in 1 + 0 dimensions.

We come back to the question if the commutator of two fields vanishes for space-like sepa-ration. We evaluate first

[φ(x), φ(x′)] =∫

d3k

∫

d3k′[

a(k)e−ikx + a†(k)e+ikx, a(k′)e−ik′x′ + a†(k′)e+ik′x′]

=

=

∫

d3k(

e−ik(x−x′) − e+ik(x−x′))

= ∆(x− x′)−∆(x′ − x) ≡ D(x− x′) .(2.59)

For equal times, t = t′, exchanging the dummy variable k→ −k in the second term shows thatthe contribution from positive and negative energies cancel. Thus the equal-time commutatorof two fields is zero, as claimed in (2.58b).

For space-like distances, (x−x′)2 < 0, we can find a Lorentz transformation which changesthe ordering of the space-time events,

x− x′ → −(x− x′) .

(This is impossible for time-like separated events, since otherwise different observers couldnot agree on the flow of time.) Since the Green function ∆(x1 − x2) is Lorentz invariant,the value of the scalar function D(x1 − x2) has to be the same in all inertial frames. But forspace-like distances, we can transform D(x) into −D(x), and therefore D(x) has to vanish ifx is outside the light-cone of x′ and vice versa. Thus we have shown that also the commutatorof two space-like separated fields vanishes,

[φ(x), φ(x′)] = 0 for (x− x′)2 < 0 , (2.60)

which is the condition for causality.How do we reconcile this result with the fact that the the Feynman propagator does not

vanish outside the light-cone? There are two main differences between these two quantities:First, [φ(x), φ(x′)] is an operator, while the Feynman propagator i∆(x1 − x2) is a vacuumexpectation value. But the quantum vacuum fluctuates, and these fluctuations are correlatedalso on space-like distances, similiar to the ERP correlations in quantum mechanics. Second,in [φ(x), φ(x′)] we subtract the contribution of positive and negative frequencies, while we addthem in the Feynman propagator. As a result, the contributions from a particle travelling thedistance x and from an anti-particle travelling the distance −x cancel in the commutator,while they add up in the Feynman propagator. Since causality relies on the cancelationbetween positive and negative energy modes in [φ(x), φ(x′)], we conclude that a relativisticquantum theory has to incorporate anti-particles.

50

2.4. Vacuum energy and the Casimir effect

2.4. Vacuum energy and the Casimir effect

Vacuum energy We now aim at calculating the energy of the vacuum state of a free scalarquantum field. The energy density ρ of the field φ is given by the vacuum expectation valueof its Hamiltonian density H ,

ρ = 〈0|H |0〉 = ρ0 +1

2〈0|π2 + (∇φ)2 +m2φ2|0〉 = ρ0 + ρ1 . (2.61)

Here we added the constant energy density ρ0 to (2.18) and used that the vacuum is nor-malised, 〈0|0〉 = 1. For the calculation of ρ1, we can recycle our result for the propagator ofa scalar field by considering φ2(x) as the limit of two fields at nearby points,

〈0|φ(x′)φ(x)|0〉x′ցx =

∫d3k

(2π)32ωke−ik(x′−x)

∣∣∣∣x′ցx

=

∫d3k

(2π)32ωk. (2.62)

We perform first the differentiation in 〈π2〉 = 〈φ2〉 and 〈(∇φ)2〉 and send then x′ ց x. Thusπ2 and (∇φ)2 add a ω2

k and k2 term, respectively,

ρ = 〈0|H |0〉 = ρ0 +

∫d3k

(2π)32ωk

[1

2(ω2k + k

2 +m2)

]

= ρ0 +

∫d3k

(2π)31

2ωk . (2.63)

If we insert ~ and c into this expression, we see that ρ0 as a classical contribution to the energydensity of the vacuum is ∝ ~

0, while the second term ρ1 ∝ ~ωk/V as a quantum correctionis linear in ~. The total energy density ρ of the vacuum state of a free scalar field has avery intuitive interpretation: Additionally to the classical energy density ρ0, it sums up thezero-point energies of all individual modes k of a free field. Despite its simplicity, we cannotmake sense out of this result: Since both the density of modes and their energy increases with|k|, the integral diverges. This is the first example that momentum integrals in quantumfield theories are often ill-defined and require some care and cure. In problem 2.3 you shouldrederive our result for ρ1 expressing the field φ by annihilation and creation operators, toconvince yourself that this divergence is not caused by the point-splitting method employedin Eq. (2.62).

Let us now consider the case that the Hamiltonian (2.18) describes the physics correctlyonly up to the energy scale Λ, while the modes with |k| >∼ Λ do not contribute to ρ1. Such apossibility exists e.g. in supersymmetric theories where the contributions of different particletypes cancel each other above the scale ΛSUSY where supersymmetry is broken. Integratingthe contribution to the vacuum energy density by field modes up to the cutoff scale Λ, wefind

ρ1 =

∫ Λ

0

dk k2

2π21

2ωk ∼ Λ4 (2.64)

in the limit Λ≫ m. Since only the total energy density ρ is observable, the unknown ρ0 can bealways chosen such that ρ0+ρ1 agrees with observations, even if |ρ0|, |ρ1| ≫ |ρ|. Nevertheless,the strong sensitivity of ρ1 to the value of the cutoff scale Λ is puzzling for two reasons: First,cosmological observations determine the total vacuum energy density ρΛ to which all typesof fields contribute as ρΛ ∼ (meV)4. On the other hand, accelerator experiments give noindications that a cancellation mechanism as supersymmetry works at energy below few TeV.Thus we should expect at least ρΛ ∼ (ΛSUSY)

4 >∼ (fewTeV)4, which is 60 orders of magnitudelarger than observed, if there is not a strong cancellation of the various contributions to ρΛ

51

2. Scalar fields

taking place. Second, the behavior ρ1 ∼ Λ4 implies that all scalar particles with mass m <∼ Λcontribute equally to ρ. In other words, we cannot calculate ρ without knowing the physicsat energy scales much larger than those accessible experimentally to us. Such a behaviourshould be truly exceptional, otherwise nothing like chemistry or solid state physics would bepossible before knowing the “theory of everything”.

Casimir effect Although we cannot calculate ambiguously the vacuum energy, we candetermine the energy difference of different vacua. As a concrete example, we consider now theCasimir effect. Between two parallel, perfectly conducting plates at distance d electromagneticwaves have the form sin(nπx/d) with discrete energy ωn = nπ/d. Thus the vacuum energyinside the box of volume dLyLz is

E =

∞∑

n=1

∫dkydkz

(2π)2LyLz

√(nπ

d

)2+ k2y + k2z . (2.65)

Clearly, the change from a continuous to a discrete spectrum of energy eigenvalues has notimproved the convergence. Compared to the case of a massless scalar field, there is anadditional factor two due to the two spin degrees of freedom of the photon. Moreover, we usea box normalisation in the perpendicular direction.

To simplify the calculations, we consider a 1+1 dimensional system of two plates separatedby the distance d. Then the energy density ρ = E/d of a massless scalar field inside the platesis

ρ(d) =π

2d2

∞∑

n=1

n . (2.66)

Next we introduce a cutoff function f(a) = exp(−anπ/d) which suppresses the high-energymodes,

ρ(d)→ ρ(a, d) =π

2d2

∞∑

n=1

ne−anπ/d . (2.67)

This procedure is called regularisation: For a > 0, we obtain a well-defined mathematical sumwhich we can manipulate following the usual rules of analysis, while we recover for a → 0our old divergent result. We have chosen as argument of the exponential anπ/d, because thephysically relevant quantities are the energy levels ωn = nπ/d of the system. Now we canevaluate the regularised sum, rewriting it as a geometrical sum,

ρ(a, d) =π

2d2

∞∑

n=1

ne−anπ/d = − 1

2d

∂

∂a

∞∑

n=0

e−anπ/d (2.68)

= − 1

2d

∂

∂a

1

1− e−aπ/d=

π

2d2e−aπ/d

(1− e−aπ/d)2. (2.69)

Next we use ex(1− e−x)2 = 4 sinh2(x/2) and expand ρ(a, d) for small a in a Laurent series,

ρ(a, d) =π

8d21

sinh2(aπ/2d)=

1

2πa2− π

24d2+O(a2d−4) . (2.70)

Note that we isolated the divergence into a term which does not depend on the distance dof the plates. Thus the divergence cancels in the difference of the vacuum energy with andwithout plates,

ρCas(d) ≡ lima→0

[ρ(a, d) − ρ(a, d→∞)

]= − π

24d2. (2.71)

52

2.5. Perturbation theory for interacting fields

This final step in order to obtain a finite result is called renormalisation. One can verify thatthe result is not only independent of the cutoff parameter a, but also from the shape of thecutoff function f(a). In contrast, the individual terms in Eq. (2.70) may depend on the cutofffunction.

The quantity measured in actual experiments is the force F with which the plates attract(or repel) each other. This force is given by

− F =∂E

∂d=∂(dρCas)

∂d=

π

24d2. (2.72)

Thus two parallel plates attract each other. The experimentally relevant case of electromag-netic waves between two parallel plates in 3+1 dimensions can be calculated analogously. Thetheoretical prediction has been confirmed with a precision on the 1% level.

We conclude this discussion of the Casimir effect with two remarks:1. If the main motivation for the introduction of the cutoff function would be physics (e.g.“X-rays do not see perfectly conducting plates,” then we would have to worry about the exactform of the cutoff function f(a): the transition from f = 1 (low frequencies) to f = 0 (highfrequencies) would have a physical meaning and should show up in the result.2. We shall come back to the vacuum energy in Section 2.6, examining the connection tovacuum diagrams generated by Z[0] and the cosmological constant.

2.5. Perturbation theory for interacting fields

We know already from quantum mechanics that adding an anharmonic term to an oscillatorforces us to use either perturbative or numerical methods. The same happens in field theory:No analytic solution for a realistic interacting theory is at present known in d = 4. Thereforewe will develop in the remaining part of this chapter perturbative methods.

General formalism The Lagrange density L in the functional Z[J ] for the scalar field con-sidered up to now was at most quadratic in the fields and its derivatives. On one hand, thisallowed us to evaluate the path integral, while on the other hand this means that the fieldhas no interactions: Two wave packets described by the free propagator just pass each otherwithout interaction, as the superposition principle prescribes. As next step we add thereforean interaction term LI to the free Lagrangian L0, i.e. we set L = L0 + LI . Then thegenerating functional Z[J ] for a real scalar field φ with mass m becomes

Z[J ] = N∫

Dφ exp i∫

d4x

(1

2∂µφ∂

µφ− 1

2m2φ2 + LI + Jφ

)

, (2.73)

where we introduced also a normalization constant N . Since the vacuum state should benormalised to one, 〈0 + |0−〉 = 1, this normalization constant is determined also for aninteracting theory by

N−1 = Z[0] =

∫

Dφ eiS . (2.74)

We assume that the interaction term LI is a polynomial P(φ) in the field φ and containsan expansion parameter λ which is small in the considered kinematic regime, LI = λP(φ)with λ≪ 1. This suggests to use perturbation theory, i.e. to expand

exp i

∫

d4xLI(φ) = 1+ iλ

∫

d4xP(φ(x)) + (iλ)2

2!

∫

d4x1d4x2P(φ(x1))P(φ(x2)) + . . . (2.75)

53

2. Scalar fields

Since

iφ(x)ei∫d4x′(L0+Jφ) =

δ

δJ(x)ei

∫d4x′(L0+Jφ) , (2.76)

we can perform the replacement

LI(φ(x)))→ LI

(1

i

δ

δJ(x)

)

. (2.77)

Then the interaction LI does not depend longer on φ and can be pulled out of the functionalintegral,

Z[J ] = N exp i

∫

d4xLI

(1

i

δ

δJ(x)

)∫

Dφ exp i∫

d4x (L0 + Jφ)

= N exp i

∫

d4xLI

(1

i

δ

δJ(x)

)

Z0[J ] = N exp i

∫

d4xLI

(1

i

δ

δJ(x)

)

eiW0[J ] . (2.78)

Here we denoted the free functionals discussed earlier with Z0 and W0. Their solution wasgiven in Eq. (2.23),

Z0[J ] = Z0[0] exp

(

− i

2

∫

d4xd4x′J(x)∆F (x− x′)J(x′))

= Z0[0]∞∑

n=0

in

n!W n

0 . (2.79)

Perturbation theory consists in a double expansion of the two exponentials in Z[J ]: One inthe coupling constant λ and one in the number of external sources J . The latter is fixed bythe number of external particles in a scattering process, while the former is chosen accordingto the desired precision of the calculation.

Choosing the interaction term Even restricting possible interactions to be polynomials inthe field, we have still an infinite number of choices. Surprisingly, the simple argument ofdimensional analysis will be an important guide line in the selection of “good” interactionterms. We first remember that using natural units, ~ = c = 1, the dimension of all physicalquantities can be expressed by powers of one basic unit which we choose as mass m.

Then we use that the action has dimension zero, [S] = m0, and thus the Lagrangian[L ] = m4 in four space-time dimensions. From the free Lagrangian, we conclude that thedimension of a bosonic field in four dimension is [φ] = m1. Thus we can order possibleself-couplings of a scalar field according to their dimension as

LI = c2Mφ3 + c4φ4 +

c5Mφ5 + . . . , (2.80)

where the coupling constants ci are dimensionless and we introduced the mass scale M toensure [L ] = m4. We call φn a dimension-n operator. Similar as in the case of the Fermiconstant, GF =

√2g2/(8m2

W ), the scale M could be connected to the exchange of a heavyparticle. If the scale M is large compared to the center-of-mass (cms) energy

√s of the

process considered, the interaction probability contains typically factors like (√s/M)n−4.

Thus for√s ≪ M the interaction is the stronger suppressed the higher the dimension of

the corresponding operator is. If on the other hand√s ≫ M , then we expect that the

effective expansion parameter becomes λ(√s/M))n−4 ≫ 1 and perturbation theory becomes

thus unreliable.

54

2.6. Green functions for the λφ4 theory

In a first approach, we neglect therefore operators of dimension five and higher. Simplifyingfurther LI , we want to include only one interaction term. In this case, a φ3 term would leadto an unstable vacuum. Therefore our choice for the scalar self-interaction is LI = −λφ4/4!,where the factor 1/4! was added for later convenience. If this choice of interaction is realizedin Nature for a specific particle has to be decided by experiment.


We start now with the perturbative evaluation of Eq. (2.78) for a λφ4/4! interaction. From

Z[J ] =

(

1− iλ

4!

∫

d4xδ4

δJ(x)4+ . . .

)

Z0[J ] = Z0[J ]−iλ

4!

∫

d4xδ4Z0[J ]

δJ(x)4+ . . .

= Z0[J ](1 + λz1[J ] + λ2z2[J ] + . . .

)(2.81)

we see that we will generate a series of the type free Green function plus correction O(λ)plus higher order corrections. The calculation is very similar to the calculation of the freefour-point function, with the difference that now the four sources are at the same point. Wefind

(δ

iδJ(x)

)4

exp(iW0[J ]) =

[

3(i∆(0))2 + 6i∆(0)

(∫

d4y∆F (x− y)J(y))2

+

(∫

d4y∆F (x− y)J(y))4]

exp(iW0[J ]) . (2.82)

Next we introduce a graphical representation for the various terms in Eq. (2.82). EachFeynman propagator ∆F (x− y) is represented by

i∆F (x− y) = x

y , (2.83)

a source term J(x) by

i

∫

d4x J(x) =

(2.84)

and an interaction vertex by

− iλ

∫

d4x = • (2.85)

Each source and vertex has its own coordinates and an integration over all coordinates isimplied. In case of a φ4 interaction, a vertex connects four lines. Using this notation, wecan express Z1 as

Z1[J ] =1

4!

3

+ 6

+

exp

(1

2

)

. (2.86)

A graph consists of lines and dots, i.e. vertices or sources. We distinguish internal andexternal lines: A line which ends on boths side at a dot with at least two lines attached iscalled internal; otherwise it is an external line. The three graphs contained in Z1[J ] differ bythe number of loops, i.e. by the number of closed lines. A connected graph with n lines and

55

2. Scalar fields

V dots has the loop number l = n− V + 1. Thus increasing the number of dots reduces thenumber of loops. A graph with loop number l = 0 is called a tree graph.

Note that the first and the second term can be derived from the third one by joining two andone lines, respectively. There are six ways to join one line, and three ways to join two lines.Thus the prefactors of the various terms could be derived by simple symmetry arguments.

Now we can derive disconnected Green functions valid at O(λ) by performing functionalderivatives,

G(n)(x1, . . . , xn) =1

inδn

δJ(x1) · · · δJ(xn)Z0[J ] (1 + λz1[J ])

∣∣∣∣J=0

. (2.87)

In the graphical notation, differentiating with respect to J(x) amounts to replace the opendot denoting the source i

∫d4yJ(y) by its position x,

1

i

δ

δJ(x)

= x

(2.88)

Vacuum diagrams We call terms in the perturbative evaluation of Z[J ] which contain nosources J vacuum diagrams. Setting J = 0 eliminates all graphs containing at least one source.Thus these diagrams correspond to loops without external lines and the corresponding Greenfunctions are the “zero-point” Green functions G(0).

Let us assume that the integration measure of the free theory is normalized such that thefree “zero-point” function is one, Z0[0] = 1. Interactions induce however a change in thisnormalization. Setting J = 0 in our result for Z[J ] at lowest order perturbation theory,Eq. (2.86), gives

G(0) ≡ Z[0] = 1− iλ

8

∫

d4x(i∆F (0))2 . (2.89)

Since vacuum diagrams do not contribute to scattering processes, one often prefers to elim-inate these diagrams by an appropriate normalization of Z[J ]. We now show that choosingN−1 = Z[0] as normalisation condition eliminates all these graphs.

Calling the normalized functional Z[J ] with Z[J ] = Z[J ]/Z[0], expanding nominator anddenominator up to O(λ), we have at lowest order perturbation theory

Z[J ] =Z[J ]

Z[0]=

1 + λz1[J ] +O(λ2)1 + λz1[0] +O(λ2)

Z0[J ] = 1 + λ(z1[J ]− z1[0]) Z0[J ] +O(λ2) . (2.90)

Thus dividing Z[J ] by the source-free functional eliminates indeed all vacuum graphs. Itbecomes obvious that this procedure works at any order perturbation theory, if we look atthe generating functional for connected graphs, W [J ]. As dividing Z[J ] by the source-freefunctional Z[0] corresponds to

iW [J ] = ln Z[J ] = lnZ[J ]− lnZ[0] , (2.91)

it is clear that this procedure eliminates indeed all vacuum graphs. Note that because ofN = exp ln(N ), a normalization different from one is equivalent to adding a constant termto the Lagrangian,

L → L + ln(N )/(V T ) = L − ρ , (2.92)

where V T is the four-dimensional integration volume in the action. The normalised generatingfunctional at O(λ) is thus obtained by dropping the two-loop diagram ∝ (i∆F (0))

2 andcorresponds to a vacuum with zero vacuum energy density, i.e. a vanishing cosmologicalconstant.

56


2-point functions We start by taking one derivative of the normalised generating functional,

1

i

δ

δJ(x1)

[

1 +1

4!

(

6

+

)]

exp

(1

2

)

= (2.93)

=

[1

4!

(

6 · 2

+ 4

)

+

(

1 +1

4!

(

6

+

))

]

× exp

(1

2

)

=

=

[

+1

4!

(

12

+ 4

+ 6

Æ

(2.94)

+

)]

exp

(1

2

)

.

Every term in this expression contains at least one source J , and the one-point functionG(1)(x) vanishes therefore. If we proceed to the two-point function G(2)(x1, x2), we have todifferentiate only those terms with one source,

1

i2δ2

δJ(x1)δJ(x2)Z[J ] = (2.95)

=1

i

δ

δJ(x2)

[

+1

4!

(

12

+vanishing termsfor J=0

)]

exp

(1

2

)

(2.96)

=

(

+1

2

)

exp

(1

2

)

. (2.97)

Setting the sources J to zero, the exponential factor becomes one. Converting the graphicalformula back into standard notation, we find the 2-point function G(2)(x1, x2) at order O(λ)as the sum of the free 2-point function G(2)0 (x1, x2) and a correction,

G(2)(x1, x2) = G(2)0 (x1, x2)−iλ

2

∫

d4xi∆F (x1 − x)i∆F (x− x)i∆F (x− x2) . (2.98)

This correction is called the self-energy Σ of the scalar particle. Note that the pre-factorcombines to −1/2iλ, so there appears an extra factor 1/2. Such factors are called symmetryfactors. They appear because we included a factor 1/4! in LI to compensate for the 4!permutations of four sources. If a diagram has a certain symmetry, i.e. if it can be rearrangedby permutating propagators and/or vertices giving the same expression, the cancellation isonly partially. In such cases a symmetry factor appears.

4-point functions The disconnected 4-point function G(x1, x2, x3, x4) is shown graphicallyin Fig. 2.2. The last diagram corresponds to the connected 4-point function G(x1, x2, x3, x4).Next we want to derive G(x1, x2, x3, x4) from its generating functional W [J ]. We insert Z[J ]

57

2. Scalar fields

x1 x2 x1 x3 x1 x4

x3 x4 x2 x4 x2 x3

x1 x2 x1 x3 x1 x4

x3 x4 x2 x4 x2 x3

x1 x2 x1 x3 x1 x4

x3 x4 x2 x4 x2 x3

x1

x4x3

x2

Figure 2.2.: Graphs contributing to the disconnected four-point function G(x1, x2, x3, x4).

into

iW [J ] = ln(Z[J ]) = ln exp

(1

2

)

+ ln

1 +

1

4!

6

+

+O(λ2)

=1

2 +

1

4!

6

+

+O(λ2) (2.99)

where we expanded the logarithm, ln(1 + x) ∼ x. Taking four derivatives with respect to J ,only the last term survives,

G(x1, x2, x3, x4) = −iλ∫

d4xi∆F (x1 − x)i∆F (x2 − x)i∆F (x3 − x)i∆F (x4 − x) . (2.100)

Symmetry factors We illustrate by an example how one can determine the sym-metry factor in more complicated cases. Consider the so-called “sunrise diagram”,

x1 x2y1 y2

58


which is a second order diagram, corresponding to the term

1

2!

(−iλ4!

)2 ∫

d4y1d4y2〈0|T [φ(x1)φ(x2)φ4(y1)φ4(y2)]|0〉 + (y1 ↔ y2)

in the perturbative expansion. The exchange graph y1 ↔ y2 is identical to the original one,canceling the factor 1/2! from the Taylor expansion. We have to count the number of possibleways to combine the fields in the time-ordered product into five propagators. As shorthandnotation, we mark a possible combination as φ(x1)φ(y1).

We have four possibilities to combine φ(x1) with φ(y1), φ(x1)φ(y1). Similiarly, there are

four possibilities for φ(x2)φ(y2). The remaining six fields can be combined in 3! ways into

pairs, φ(y1)φ(y1)φ(y1)φ(y2)φ(y2)φ(y2). Thus the symmetry factor of this diagram is

S =

(1

2!× 2

)(1

4!

)2

(4× 4× 3!) =1

3!

Feynman rules for the λφ4 theory in coordinate space We can summarize our results infew simple rules which allows us to write down Green functions directly, without the needto derive them from their generating functional. The rules refer to connected diagrams afterintegration over internal variables.

1. Draw all topologically different diagrams for the chosen order O(λn) and number ofexternal coordinates or particles.

2. To each line connecting the points x and x′ we associate a propagator i∆F (x− x′).3. Each vertex has a factor −iλ and connects n lines for a λφn interaction.

4. Integrate over all intermediate points.

5. Add a symmetry factor counting the number of possible identical rearrangements of thediagram.

6. The rules above give a n-point Green function. A scattering process is described bythe transition amplitude iA between a fixed inital and final state which contain real,on-shell particles. Thus the propagators of the external lines, which describe virtualparticles, should be replaced by on-shell wave functions—this rule will be derived inchapter 6.2. Moreover, we will show later that we can ignore all loop corrections inexternal lines.

Feynman rules in momentum space The integration over intermediate points is trivial,because all propagators (and possible wave function on external lines) depend like exp(±ikx)on the position of the interaction point. Hence the space integrations result in 4-momentumconservation at each vertex. Therefore the integration over the momenta of V −1 propagatorcan be trivially performed in a Green function with V vertices, whereas the remaining four-momenta delta function expresses overall momentum conservation. As a result, in a Greenfunction with loop number l = n− V + 1 remain l non-trivial four-momenta integrations.

The Feynman rules in momentum space have thus the following changes:

2. To each line we associate a propagator i∆F (k) = i/(k2 −m2 + iε).

59

2. Scalar fields

3. Fix the external momenta and impose 4-momentum conservation at each vertex.

4. Integrate over all unconstrained momenta with∫d4k/(2π)4. The number of indepen-

dent momenta we have to integrate over equals the loop number of the graph.

6. If the diagram should represent the transition ampliude iA of a scattering process,propagators on the external lines should be replaced by on-shell wave functions. It isadvantagous to omit the normalisation factors N = [(2π)3(2ωk)]

−1/2 from the matrixelement and to include them later into a flux (for the inital state) and phase space factor(for the final state). In this way, the amplitude iA is Lorentz invariant and easier tomanipulate. Thus for scalar particles, the Feynman rule for external particles is simplyto write “1”.

2.7. Loop diagrams

Aim of this section is to illustrate the regularization and renormalization procedure used inquantum field theory calculating three one-loop diagrams of the λφ4 theory. We will havetime to digest these examples, before we will come back to the problem of renormalisation inChapter 8. The basic steps in the evaluation of simple Feynman integrals are summarised inthe appendix 2.A.

2.7.1. Self-energy

We consider first the only one-loop diagram contained in Z1[J ], the 2-point function of ascalar particle at O(λ),

G(2)(x1, x2) = i∆F (x1 − x2)−λ

2∆F (0)

∫

d4x∆F (x1 − x)∆F (x− x2) . (2.101)

We start concentrating on the correction term, the self-energy Σ(x1−x2) of the scalar particle,and insert the Fourier representation of the two propagators into the integral,

Σ(x1 − x2) = −λ

2∆F (0)

∫

d4xd4p

(2π)4d4p′

(2π)4eip(x1−x)

p2 −m2 + iε

eip′(x−x2)

p′2 −m2 + iε. (2.102)

The d4x integration produces (2π)4δ(p − p′), then one of the momentum integrations can beperformed. Together this gives

Σ(x1 − x2) = −λ

2∆F (0)

∫d4p

(2π)4eip(x1−x2)

(p2 −m2 + iε)2. (2.103)

Inserting also for the free Green function its Fourier representation, we arrive at

G(2)(x1, x2) =∫

d4p

(2π)4eip(x1−x2)

[

i

p2 −m2 + iε−

λ2∆F (0)

(p2 −m2 + iε)2

]

. (2.104)

The Green function in momentum space is thus

G(2)(p) = i

p2 −m2 + iε

[

1 +iλ2 ∆F (0)

p2 −m2 + iε

]

. (2.105)

60

2.7. Loop diagrams

Assuming that perturbation theory is justified, the second term in the parenthesis should besmall. Thus [1 + λa] = [1− λa]−1 +O(λ2) and

G(2)(p) = i

p2 −m2 − iλ2 ∆F (0) + iε

. (2.106)

The residue of the free propagator defined the “bare” particle mass at zero order in λ. Tomake this clearer, we write now m0 instead of m. The physical (or renormalized) mass of thescalar particle is at order λ given by

m2phys = m2

0 + δm2 = m20 +

iλ

2∆F (0) . (2.107)

Hence interactions shift the “bare” mass m0 used initially in the classical Lagrangian L .Theorists tend to call this procedure “renormalization”, although the scalar mass was neverproperly normalized. It is important to realize that renormalization happens in any interactingtheory, independently of the question if δm2 is finite or infinite. A familiar example in aclassical context is the Debye screening of an electric charge in a plasma.

As next step, we have to calculate (and to interpret properly)

i∆F (0) =

∫d4k

(2π)4i

k2 −m2 + iε. (2.108)

Since the mass correction is δm2 = iλ∆F (0)/2, the Feynman propagator at coincident points∆F (0) has to be purely imaginary. Otherwise the λφ4 theory would contain no stable particles.

Wick rotation The integrals appearing in loop graphs can be easier integrated, if one per-forms a Wick rotation from Minkowski to Euclidean space: Rotating the integration contouranti-clockwise to −i∞ : +i∞ avoids both poles in the complex k0 plane and is thus admissible.Introducing as new integration variable ik4 = k0, it follows

∫ ∞

−∞dk0

1

k2 −m2 + iε=

∫ i∞

−i∞dk0

1

k2 −m2 + iε= i

∫ ∞

−∞dk4

1

k2 −m2 + iε. (2.109)

We next combine kE = (k, k4) into a new four-vector. Since

k2E = −(|k|2 + k24) (2.110)

we work now (apart from the overall sign) in an Euclidean space. In particular, the denom-inator never vanishes and we can omit the iε. Moreover, the integrand is now sphericallysymmetric. Thus we have

i∆F (0) =

∫d4kE(2π)4

1

k2E +m2. (2.111)

As required by our interpretation δm2 = i∆F (0), the propagator ∆F (0) is imaginary. Becausethe relative sign of the momenta and the mass term indicates, if we work in the Euclideanor Minkowski space, we will omit the index E in the following. Introducing furthermorespherical coordinates, we see that ∆F (0) diverges quadratically for large k,

λi∆F (0) ∝∫ Λ

0dk k3

1

k2 +m2∝ Λ2 . (2.112)

61

2. Scalar fields

Dimensional regularization Using the integral representation

1

k2 +m2=

∫ ∞

0ds e−s(k

2+m2) (2.113)

and interchanging the integrals, we can reduce the momentum integral to a Gaussian integral.Manipulations like interchanging the order of integrations or a change of integration variablesin divergent expressions as Eq. (2.112) are however ambigious. Before we can proceed, wehave to “regularize” therefore the integral, similar as we did introducing a cutoff function intothe expression of the zero-point energy.

We will use dimensional regularization, i.e. we will calculate integrals for n = 4− ε dimen-sions where they are finite. Then we find

i∆F (0) =

∫ ∞

0ds

∫dnk

(2π)ne−s(k

2+m2) =1

(4π)n/2

∫ ∞

0ds s−n/2e−sm

2. (2.114)

The substitution x = sm2 transforms the integral into one of the standard representations ofthe gamma function (see the appendix 2.A for some useful formula),

i∆F (0) =(m2)

n2−1

(4π)n/2Γ(

1− n

2

)

. (2.115)

This expression diverges for n = 2, 4, 6, . . ., but is as announced finite for n = 4− ε and smallε. In the next step, we would like to expand the expression in a Laurent series, separatingpole terms in ε and a finite reminder.

Appearance of a dimensionfull scale As the expression stands, we can not expand theprefactor of the Gamma function for small ε, because it is dimensionfull. In order to makethe factor mn−2 dimensionless, we should supply a new mass scale.

More physically, we can understand the need for an additional dimensionfull scale by therequirement that the interaction

∫dnxLI remains dimensionless if we deviate from n = 4

dimensions. In order to obtain the correct dimension of the interaction term, keeping thedimensions of physical quantities fixed, the coefficient of the φ4 term has to aquire the massdimension −n + 4. In order to achieve this, we introduce a mass scale µ (with [µ] = 1) asfollows,

SI =

∫

d4xLI = −∫

d4xλ

4!φ4 → −µ4−n

∫

dnxλ

4!φ4 . (2.116)

Adding the factor µ4−n to our previous result, we obtain

λµ4−ni∆F (0) =iλ

(4π)2m2

(4πµ2

m2

)2−n/2Γ(1− n/2) . (2.117)

We expand the dimensionless last two factors in this expression around n = 4 using (A.45)for the Gamma function,

Γ(1− n/2) = Γ(−1 + ε/2) = −2

ε− 1 + γ +O(ε) (2.118)

anda−ε/2 = e−(ε/2) lna = 1− ε

2ln a+O(ε2) . (2.119)

62

2.7. Loop diagrams

Note that we require the expansion of the prefactor up to O(ε2) because of the pole term in(2.118). Thus the mass correction is given by

λµ4−ni∆F (0) ∝ m2

[

−2

ε− 1 + γ +O(ε)

] [

1 +ε

2ln

(4πµ2

m2

)

+O(ε)]

. (2.120)

This expansion allows us to separate the correction into a divergent term ∝ 1/ε and a finiteterm that depends on the scale µ. We will pursue these topics later in more detail, for themoment we note simply that i) the divergence can be hidden via Eq. (2.107) in m2

phys, andii) the scale dependence µ will lead to the “running” of masses and coupling constants.

2.7.2. Vacuum energy density

We can generate out of the self-energy diagram new one-loop graphs by adding or subtractingtwo external lines. Subtracting two lines generates an one-loop graph without external lines1,the “zero-point” Green function G(0) at order λ0.

One way to calculate this quantity is to evaluate directly Z0[0] using

detA = exp ln detA = exp tr lnA (2.121)

which gives

Z0[0] = −1

2exp tr ln( −m2) . (2.122)

We postpone the question how such an expression can be evaluated and use instead anotherapproach, recycling our result for the self-energy. Vacuum diagrams are generated by thefunctional Z[J ] setting J = 0,

〈0 + |0−〉 = Z[0] =

∫

Dφ exp i∫

Ωd4x

(1

2∂µφ∂

µφ− 1

2m2φ2

)

. (2.123)

We saw in Eq. (2.62) that the zero-point energy is related to the propagator at coincidentpoints. Since we suspect a connection between vacuum diagrams and the zero-point energy,we try to relate Z[0] and ∆F (0). Taking a derivative with respect to m2 gives

∂

∂m2〈0 + |0−〉 = − i

2

∫

d4x 〈0 + |φ(x)2|0−〉 = − i

2

∫

d4x i∆F (0)〈0 + |0−〉 . (2.124)

The additional factor 〈0 + |0−〉 = N−1 on the RHS takes into account that we defined theFeynman propagator with respect to a normalised vacuum. Translation invariance impliesthat 〈0 + |0−〉 does not depend on x. Thus we obtain

∂

∂m2ln〈0 + |0−〉 = − i

2

∫

d4x i∆F (0) = −i

2V T i∆F (0) (2.125)

with V T as the four-dimensional integration volume. Next we use our result (2.115) for thepropagator, i∆F (0) = C(m2)n/2−1, to perform the integration over m2,

ln〈0 + |0−〉 = −iV T[

Cmn

n− ρ0

]

, (2.126)

1Although G(0) is often represented as a closed loop, it has also no internal line; this is in agreement with ourgeneral formula l = n− V + 1.

63

2. Scalar fields

where we introduced the integration constant ρ0. Exponentiating this expression,

〈0 + |0−〉 = 〈0 + | exp (−iHT ) |0−〉 = exp (−iρ V T ) , (2.127)

we see that we should associate ρ with the energy density of the vacuum (at order O(λ0)).Replacing C, we find

ρ =mn

(4π)n/2nΓ(

1− n

2

)

− ρ0 . (2.128)

The energy density given by Eq. (2.128) diverges for n = 2, 4, 6 . . . as

Γ(

1− n

2

)

∼ 2

n− 4. (2.129)

We can make ρ finite and equal to the observed value ρΛ, if we choose ρ0 as

ρ0 = µn−4

[1

2

m4

(4π)21

n− 4− ρΛ

]

. (2.130)

The prefactor µn−4 ensures that the action remains dimensionless also for n 6= 4 (keeping thedimension of physical quantities constant).

Note that this implies that we should start off with L −ρ0 instead of L . Even if we dismissa cosmological constant in the classical Lagrangian, it will appear automatically by quantumcorrections. More generally, every possible term that is not forbidden by a symmetry in L

will show up calculating loop corrections.We have seen that we can absorb the vacuum energy density ρ into the normalisation of

the path integral, ∫

Dφe−∫d4xρ = N

∫

Dφ→∫

Dφ .

Therefore, one may wonder if ρ has a real physical meaning. The crucial difference betweenthe vacuum energy and a simple normalisation constant is that ρ depends on the parameters(masses, coupling constants) of the considered theory. Since we should define the path integralindependent of the Langrangian we integrate, we cannot eliminate the contribution of thevacuum diagrams to the cosmological constant in the gravitational field equations by a simpleredefinition of the integration measure.

Remark: Equivalence to the zero-point energy:Performing the k0 integral in Eq. (2.108) or using 2.33

i∆F (0) =

∫d3k

(2π)32ωk=

∫d3k

(2π)32√

m2 + k2(2.131)

and integrating then with respect to m2,

i

∫

dm2∆F (0) =

∫d3k

(2π)3

√

m2 + k2 , (2.132)

shows that the present expression for the vacuum energy agrees with the one in Eq. (2.63). However,

the resulting expression for ρ differ: While Eq. (2.64) shows that ρ ∝ Λ4 we found ρ ∝ m4 in the case

of dimensional regularization. Thus in this scheme a massless particle as the photon would give a zero

contribution to the cosmological constant.

This disturbing difference has to be caused by the regularization scheme. We will discuss later which

prediction is more trustable.

64

2.7. Loop diagrams

2.7.3. Vertex correction

For our last example we add two external lines to the self-energy diagram. This generatesdiagrams which describe e.g. 2 → 2 scattering at O(λ2). At tree-level, the same processwas given by the four-point function (2.100). The corresponding Feynman amplitude inmomentum space is simply iA = iλ. Thus we suspect that the loop process leads to therenormalisation of the coupling constant λ.

Determining the Feynman amplitude Instead of calculating the order λ2 term in the per-turbative expansion of the generating functional Z[J ] we use directly the Feynman rules toobtain the Feynman amplitude for this process. According to these rules, the first steps inthe calculation of the are to draw all Feynman diagrams, to find the symmetry factor and toassociate then the right mathematical expressions to the symbols.

In coordinate space, we have to connect four external points (say x1, . . . , x4) with the helpof two vertices (say at x and y) which combine each four lines. An example is shown here

x1 x3

x2 x4

x

y

Two other diagrams are obtained connecting x1 with x2 or x4. In order to determine thesymmetry factor, we consider the expression for the four-point function corresponding to thegraph shown above,

1

2!

(

−i λ4!

)2 ∫

d4xd4y〈0|Tφ(x1)φ(x2)φ(x3)φ(x4)φ4(x)φ4(y)|0〉 + (x↔ y) , (2.133)

and count the number of possible contractions: We can connect φ(x1) with each one of thefour φ(x), and then φ(x3) with one of the three remaining φ(x). This gives 4× 3 possibilities.Another 4×3 possibilities come by the same reasoning from the upper part of the graph. Theremaining pairs φ2(x) and φ2(y) can be combined in two possibilities. Finally, the factor 1/2!from the Taylor expansion is cancelled by the exchange graph. Thus the symmetry factor is

S =1

2!2!

(4× 3

4!

)2

2 =1

2. (2.134)

Next we associate the right mathematical expressions to the symbols of the graphs: Wereplace internal propagator by i∆(k), external lines by 1 and vertices by −iλ. Imposing four-momentum conservation at the two vertices leaves one free loop momentum, which we call p.The momentum of the other propagator is then fixed to p − q, where q2 = s = (p1 + p2)

2,q2 = t = (p1− p3)2, and q2 = u = (p1− p4)2 for the three graphs shown in Fig. 2.3. Thus theFeynman amplitudes in n = 4 are at order O(λ2)

iA(2)i =

1

2λ2∫

d4p

(2π)41

[p2 −m2 + iε]

1

[(p− q)2 −m2 + iε]. (2.135)

65

2. Scalar fields

k2 k3

k1 k4 k1 k3

k2 k4

k1 k4

k2 k3

iAs iAt iAu

Figure 2.3.: Three graphs contributing to φ(k1)φ(k2)→ φ(k3)φ(k4) at order λ2.

The squared cms energy s and the two variables describing the momentum transfer t and uare called Mandelstam variables. For 2 → 2 scattering, they are connected by s + t + u =m2

1 +m22 +m2

3 +m24, see problem 2.11. According to the value of q2 one calls the diagrams

the s, t and u channel.

Performing again a naive power-counting analysis, we find that the amplitude is logarith-mically divergent,

A(2)i ∝

∫d4p

p4∝ ln(Λ) . (2.136)

If we consider the infinite number of one-loop graphs characterised by n = V ≥ 0, then wesee that adding two external lines increases the number of propagators in the loop by one.As a result, the convergence of the loop integral improves from a quartic divergence (vacuumenergy), over a quadratic divergence (self-energy energy) to a logarithmic divergence for thevertex correction. Adding two or more external lines to the vertex correction would thereforeproduce a finite diagram. At one-loop level, the λφ4 theory contains thus only a finite numberof divergent Feynman graphs. Going beyond one-loop, this interesting behavior persists andas a result a renormalisation of the three parameters contained in the classical Lagrangian,m2, λ and ρ, is sufficient to eliminate all divergencies.

Calculating the loop integral The path to be followed in the evaluation of simple loopintegrals as (2.135) can be sketched schematically as follows: Regularise the integral (andadd a mass scale if you use DR). Combine then the denominators, and shift the integrationvariable to eliminate linear terms in the denominator by completing the square. Performingthe same shift of variables in the nominator, the linear terms can be dropped as the vanishafter integration. Finally, Wick rotate the integrand, and reduce the integral to a known oneby a suitable variable substitution. We do the last steps once in the appendix 2.A where wederive a list of useful Feynman integrals which we simply look up in the future.

We start by rewriting the integral for n = 4− ε dimensions as

iA(2)i =

1

2λ2(µ2)4−n

∫dnp

(2π)n1

D, (2.137)

where we introduced also the short-cut D for the denominator in the integrand. Next we use

1

ab=

∫ 1

0

dz

[az + b(1− z)]2(2.138)

66

2.7. Loop diagrams

to combine the two denominators, setting a = p2 −m2 and b = (p− q)2 −m2,

D ≡ az + b(1− z) = p2 −m2 − 2pq(1 − z) + q2(1− z) . (2.139)

Then we eliminate the term linear in p substituting p′2 = [p− q(1− z)]2,D = p′2 −m2 + q2z(1 − z) . (2.140)

Since dnp = dnp′, we can drop the primes and find

iA(2)s =

1

2λ2(µ2)4−n

∫ 1

0dz

∫dnp

(2π)n1

[p2 −m2 + q2z(1 − z)]2 . (2.141)

Performing a Wick rotation requires that q2z(1 − z) < m2 for all z ∈ [0 : 1], or q2 < 4m2.The integral is of the type I(ω, 2) calculated in the appendix and equals

I(ω, 2) = i1

(4π)ωΓ(2− ω)Γ(2)

1

[m2 − q2z(1 − z)]2−ω . (2.142)

Inserting the result into the Feynman amplitude gives

A(2)s =

1

2λ2(µ2)4−n

Γ(2− n/2)(4π)n/2

∫ 1

0dz [m2 − q2z(1 − z)]n/2−2 (2.143)

=λ2

32π2(µ2)2−n/2Γ(2− n/2)

∫ 1

0dz

[m2 − q2z(1− z)

4πµ2︸︷︷︸

f

]n2−2

. (2.144)

In the last step, we made the function f dimensionless. Now we take the limit ε = 4−n→ 0,expanding both the Gamma function

Γ(2− n/2) = Γ(ε/2) =2

ε− γ +O(ε)

andf−ε/2 = 1− ε

2ln f +O(ε2) . (2.145)

Fromλ2

32π2µε(2

ε− γ +O(ε)

)[

1− ε

2

∫ 1

0dz ln f +O(ε2)

]

(2.146)

we see that all diagrams give the same divergent part, while we have to replace q2 by thevalue s, t, u appropriate for the three diagrams,

A = A(1) +A(2)s +A(2)

t +A(2)u +O(λ3)

= −λµε/2 + 3λ2µε

16π2ε− λ2µε

32π2[3γ + F (s,m, µ) + F (t,m, µ) + F (u,m, µ)

]. (2.147)

with

F (q2,m, µ) =

∫ 1

0dz ln

[m2 − q2z(1− z)

4πµ2

]

. (2.148)

Note that t and u are in the physical region negative and thus the condition q2 < 4m2 isalways satisfied for these two diagrams. By contrast, for the s channel diagram the relationq2 = s > 4m2 holds: In this case, we have to continue analytically the result (6.16) into thephysical region. We will postpone this task to chapter 6 and note for the moment only thatthereby the argument of the logarithm in (6.16) changes sign. Additionally, an imaginarypart of the scattering amplitude is generated.

67

2. Scalar fields

2.7.4. Basic idea of renormalisation

The regularisation of loop integrals has introduced as a new unphysical parameter the renor-malisation scale µ. As we perform perturbation theory at order λn, we have to connect theparameters mn, λn, ρn of the truncated theory with the physical ones of the full theory.This process is called renormalisation and will eliminate the unphysical parameter µ.

Renormalisation of the coupling We try to connect the amplitude iA to a physical mea-surement. We assume that experimentalists measured φφ → φφ scattering. It is sufficientthat they provide us with a single value, e.g. with the value of the differential cross sectiondσ/dΩ at zero-momentum transfer close to threshold s = 4m2. Then

F (4m2,m, µ) = Cλ2 ln

[4m2

µ2

]

+ const , (2.149)

while for s≫ m2 we find

F (s,m, µ) = Cλ2 ln

[s

µ2

]

+ const . (2.150)

Subtracting the infinite parts (and the constant term), we obtain for s, t, u≫ m2,

A = −λ− Cλ2[ln(s/µ2) + ln(t/µ2) + ln(u/µ2)

]+O(λ3) ≡ −λ−Cλ2L(s/µ2) , (2.151)

where we introduced also the sloppy notation L for the three log terms in the square bracket.This expression for A is finite but still arbitrary since it contains µ.

We use now the experimental measurement at the scale s = 4m2 to connect via

A = −λ− Cλ2L(4m2/µ2) (2.152)

the measured value λ0 of the coupling to our calculation,

− λ0 = −λ− Cλ2L(4m2/µ2) +O(λ3) . (2.153)

Now we solve for λ,

−λ = −λ0 + Cλ2L(4m2/µ2) +O(λ3)= −λ0 + Cλ20L(4m

2/µ2) +O(λ30) . (2.154)

In the second line, we could replace λ2 by λ20, because their difference is of O(λ3). Next weinsert λ back into the matrix element A for general s and replace then again λ2 by λ20,

A = −λ− Cλ2L(s/µ2) +O(λ3)= −λ0 + Cλ20L(4m

2/µ2)− Cλ20L(s/µ2) +O(λ30)= −λ0 − Cλ20L(s/4m2) +O(λ30) . (2.155)

Combining the log’s, the scale µ has canceled and we find

A = −λ0 −λ20

32π2[ln(s/4m2) + ln(t/4m2) + ln(u/4m2)

]+O(λ30) . (2.156)

Thus the matrix element is finite and depends only on the measured value λ0 of the couplingconstant and the kinematical variables s and t.

68

2.7. Loop diagrams

Running coupling We look now from a somewhat different point of view at the problem ofthe apparent µ dependence of physical oservables. Assume that we have subtracted the infiniteparts (and the constant term) of the amplitude A, obtaining Eq. (2.151). We now demandthat the scattering amplitude A as a physical observable is independent of the arbitrary scaleµ, dA/dµ = 0. If we did a perturbative calculation up to O(λn), the condition dA/dµ = 0can hold only up to terms O(λn+1). The explicit µ dependence of the amplitude, ∂A/∂µ 6= 0,can be only cancelled by a corresponding change of the parameters m and λ contained in theclassical Lagrangian, converting them into “running” parameters m(µ) and λ(µ). Then thecondition dA/dµ = 0 becomes

(∂

∂µ+∂m2

∂µ

∂

∂m2+∂λ

∂µ

∂

∂λ

)

A(s, t,m(µ), λ(µ), µ) = 0 . (2.157)

The only explicit µ dependence of A is contained in the F (q2,m, µ) functions, giving in thelimit ε→ 0

∂

∂µA(s, t,m(µ), λ(µ), µ) = −3 λ2

32π2∂

∂µF (q2,m, µ) =

3λ2

16π2µ. (2.158)

Since the change of m(µ) and λ(µ) is given by loop diagrams, it includes at least an additionalfactor λ. Therefore the action of the derivatives ∂m2 and ∂λ on the 1-loop contribution A(2)

leads to a term of O(λ3) which can be neglected. Thus the only remaining term will be givenby ∂µ acting on the tree-level term A(1). Note that this holds also at higher orders and ensuresthat ∂µλ at O(λn+1) is determined by the parameters calculated at O(λn). Combining thetwo contributions we find

µ∂λ

∂µ=

3λ2

16π2+O(λ3) . (2.159)

Thus the scattering amplitude A is independent of the scale µ, if we transform the cou-pling constant λ into a scale dependent “running” coupling λ(µ) whose evolution is given byEq. (2.159). If we truncate the perturbation series at a finite order, the cancellation of thescale dependence is however incomplete.

Separating variables in Eq. (2.159), we find

λ(µ) =λ0

1− 3λ0/16π2 ln(µ/µ0)(2.160)

with λ0 ≡ λ(µ0) as initial condition. Thus the running coupling λ(µ) increases logarithmicallyfor increasing µ.

Comparing (8.86) to our result for the scattering amplitude (2.156),

A = −λ0(

1 +λ0

32π2[ln(s/4m2) + ln(t/4m2) + ln(u/4m2)

])

+O(λ30) , (2.161)

we see that we can rewrite the amplitude using a symmetric point q2 = s = t = u andµ20 = 4m2 as

A = −λ0(

1 +3λ032π2

ln(q2/µ20)

)

= −λ(µ) . (2.162)

This shows that the q2 dependence of the amplitude A in the limit q2 ≫ m2 is determinedcompletely by the µ2 dependence of the running coupling λ(µ).

69

2. Scalar fields

2.A. Appendix: Evaluation of Feynman integrals

Combination of propagators The standard strategy in the evolution of loop integrals is thecombination of the n propagator denominators into a single propagator-like denominator ofhigher power. One uses either Schwinger’s proper-time representation

i

p2 −m2 + iε=

∫ ∞

0ds e−is(p2−m2+iε) (2.163)

or the Feynman parameter integral

1

x1 · · · xn= Γ(n)

∫ 1

0dx1 · · ·

∫ 1

0dxn δ(1 −

∑

i

xi) [x1 · · · xn]−n

=

∫

triangledx1 · · · dxn [x1 · · · xn]−n . (2.164)

In order to derive this formula for n = 2, consider

1

b− a

∫ b

a

dx

x2=

1

b− a

(1

a− 1

b

)

=1

ab(2.165)

for a, b ∈ C. Setting x = az + b(1− z) and changing the integration variable, we obtain

1

ab=

∫ 1

0

dz

[az + b(1− z)]2. (2.166)

The cases n > 2 can be derived by induction.

Evaluation of Feynman integrals We want to calculate integrals of the type

I(ω,α) =

∫d2ωk

(2π)2ω1

[k2 −M2 + iε]α(2.167)

defined in Minkowski space. Performing a Wick rotation to Euclidean space and introducingspherical coordinates results in

I(ω,α) = i(−1)α∫

d2ωk

(2π)2ω1

[k2 +M2 + iε]α= i

(−1)α(2π)2ω

Ω2ω

∫ ∞

0dk

k2ω−1

[k2 +M2]α, (2.168)

where we denoted the volume vol(S2ω−1) of a unit sphere in 2ω dimensions2 by Ω2ω. Youare asked in problem 2.10 to show that Ω2ω = 2πω/Γ(ω) and thus Ω4 = 2π2. Substitutingk = m

√x and using the integral representation (A.31) for Euler’s beta function allows us to

express the k integral as a product of Gamma functions,

∫ ∞

0dk

k2ω−1

[k2 +M2]α=

1

2M2ω−2α

∫ ∞

0dx

xω−1

[1 + x]α=

1

2M2ω−2αB(ω,α−ω) = 1

2M2ω−2αΓ(ω)Γ(α− ω)

Γ(α).

(2.169)

2A n − 1-dimensional sphere Sn−1(R) encloses the n-dimensional volume x21 + . . .+ x2

n ≤ R2, while its ownn − 1-dimensional volume is given by x2

1 + . . . + x2n = R2. The volume of a unit 1-sphere is a length,

vol(S1) = 2, of a unit 2-sphere an area, vol(S2) = 4π, and of a unit 3-sphere a volume, vol(S3) = 2π2. If wesay that the volume of a sphere is 4πR3/3, we mean in fact the volume of the 3-ball B3(R), x2

1+x22+x

23 ≤ R2

which is enclosed by the 2-sphere S2(R).

70


Combining this result with Ω2ω = 2πω/Γ(ω) we obtain

I(ω,α) =

∫d2ωk

(2π)2ω1

[k2 −M2 + iε]α= i

(−1)α(4π)ω

Γ(α− ω)Γ(α)

[M2 + iε]ω−α . (2.170)

We can generate additional formulas by adding first a dependence on a external momentumpµ, shifting the integration variable k → k + p,

∫d2ωk

(2π)2ω1

[k2 + 2pk −M2 + iε]α= i

(−1)α(4π)ω

Γ(α− ω)Γ(α)

[M2 + p2 + iε]ω−α . (2.171)

Taking then derivatives with respect to the external momentum pµ results in

Iµ(ω,α) =

∫d2ωk

(2π)2ωkµ

[k2 + 2pk −M2 + iε]α= −pµI(ω,α) (2.172)

and

Iµν(ω,α) =

∫d2ωk

(2π)2ωkµkν

[k2 + 2pk −M2 + iε]α= (2.173)

= i(−π)ω(2π)2ω

Γ(α− ω − 1)

Γ(α)

pµpν(α− ω − 1)− 12ηµν(M

2 + p2)

[M2 − p2 + iε]α−ω. (2.174)

Contracting both sides with kµkν and using ηµνηµν = 2ω gives

I2(ω,α) =

∫d2ωk

(2π)2ωk2

[k2 + 2pk −M2 + iε]α= i

(−π)ω(2π)2ω

Γ(α− ω − 1)

Γ(α)

(α− 2ω − 1)p2 − ωM2

[M2 − p2 + iε]α−ω.

(2.175)Special cases often needed are

I(ω, 2) =i

(4π)ωΓ(2− ω) (M2 − p2 + iε)ω−2 , (2.176)

I2(ω, 2) = −i

(4π)ωωΓ(1− ω) (M2 − p2 + iε)ω−1 , (2.177)

and

I(2, 3) =i

32π21

M2 − p2 + iε. (2.178)

Summary

The Feynman propagator obtained by the m2 − iε prescription is the unique Green functionwhich can be analytically continued to an Euclidean Green function. It propagates particles(with positive frequencies) forward in time, while anti-particles (with negative frequencies)propagate backward in time. The exchange of time-like quanta with zero energy between twostatic sources leads to the Yukawa potential. The corresponding force mediated by a scalarfield is attractive. The three loop diagrams we calculated in the λφ4 theory were infiniteand had to be regularised. Renormalising the three parameters contained in the classicalLagrangian of the λφ4 theory, ρ0, m

2, and λ, eliminated the divergencies and converted theminto “running” quantities.

71

2. Scalar fields

Further reading

The quantisation of fields using both canonical quantisation and the path integral approach isdiscussed extensively in [GR96]. For an treatement of the λφ3 theory in n = 6 dimensions—atheory which resembles a bit more QED than the λφ4 theory we discussed—see [Sre07].

Problems

2.1 Complex scalar field.

Derive the Lagrangian and the Hamiltonian fora complex scalar field, considering φ = (φ1 +iφ2)/

√2 and φ∗ = (φ1− iφ2)/

√2 as the dynamical

variables. Find then the conserved current of thiscomplex field, proceeding similiar as in the case ofthe Schrodinger equation.

2.2 Maxwell Lagrangian.

Derive the Lagrangian for the photon field Aµ

from the source-free Maxwell equation ∂µFµν = 0

following the steps from (2.13) to (2.14) in thescalar case. What is the meaning of the unusedset of Maxwell equations?

2.3 Lorentz invariant integration measure.

Argue that d3k/(2ωk) is a Lorentz invariant inte-gration measure by showing that

∫

d4k δ(k2−m2)ϑ(k0)f(k0,k) =

∫d3k

2ωkf(ωk,k)

(2.179)for any function f .

2.4 Yukawa potential.

Show that the Yukawa potential V (r) =e−mr/(4πr) is the Fourier transform of (k2 +m2)−1, cf. Eq. (2.42).

2.5 Vacuum energy.

Rederive (2.63) expressing φ by annihilation andcreation operators. Show that rewriting all cre-ation operators on the left of the annihilation op-erators results in ρ1 = 0. (This prescription iscalled “normal ordering”.)

2.6 ζ function regularisation.

a.) The function f(t) = t/(et − 1) is the generat-ing function for the Bernoulli numbers Bn, i.e.

f(t) =t

et − 1=

∞∑

n=0

Bn

n!tn .

Calculate the first Bernoulli numbers up to B2.b.) The Riemann ζ function can be defined asζ(s) =

∑∞n=1

1ns for s > 1 and then analytically

continued into the complex s plane. The Bernoullinumbers are connected to the Riemann ζ functionwith negative odd argument as ζ(1−2n) = −B2n

2n .show using

1

a

a

ea − 1=

1

a

∞∑

n=0

Bn

n!an

that you can split the sum into a divergent and afinite part. The divergent term will cancel in thedifference of the vacuum energy with and withoutplates, and the remaining finite term is determinedby B2/2.

2.7 Casimir effect.

Repeat the calculation of the Casimir effect for ascalar field in 1+1 dimensions using dimensionalregularization.

2.8 Feynman propagator.

Discuss the behavior of the scalar Green function∆F (0, r) for large r = |x|.

2.9 Green functions.

Show that the connected and the unconnectedn-point Green functions are identical for n = 2,while they differ in general for n ≥ 3.

2.10 Volume of n dimensional sphere.

a.) Calculate the volume of the unit sphere Sn−1

defined by x21 + . . .+ x2n = 1 in Rn.b.) Generalize the result to arbitrary (not neces-sarily integer) dimensions and show that it agreeswith the familiar results for n = 1, 2 and 3.

2.11 Mandelstam variables.

Show that for 2 → 2 scattering s + t + u =m2

1+m22+m

23+m

24 is valid. Express t as function

of the scattering angle between p1 and p3, derive

72


the lower and upper limits of t and the relationbetween dσ/dΩ and dσ/dt.

2.12 Renormalisation invariance of thepropagator.Derive analogously to Eq. (2.157ff) an equationdm2(µ)/dµ = f(m2) requiring that the propaga-tor (2.106) is independent of the scale µ.

2.13 Renormalisation with an Euclidean

momentum cutoff.Calculate the self-energy (mass correction) andvertex correction using an Euclidean momen-tum cutoff Λ. Derive the RGE equationsdm2(Λ)/dΛ = f(m2), dλ(Λ)/dΛ = f(λ) anddρ(Λ)/dΛ = f(ρ) and compare them to the onesderived using DR.

73

3. Global symmetries and Noether’s theorem

Emmy Noether showed 1917 that any global continuous symmetry of a classical system de-scribed by a Lagrangian L leads to a locally conserved current. We can divide such sym-metries into two classes: Symmetries of the space-time manifold and internal symmetries ofa groups of fields. Prominent examples are the Poincare symmetry of Minkoswki space andglobal symmetries that lead to the conservation laws of electic charge or baryon number,respectively.

A locally conserved vector current leads classically always to global charge conservation,while this does not hold for tensors of higher rank: In the latter case, global charge conser-vation requires additionally the existence of a Killing vector field which in turn indicates acorresponding symmetry of space-time. The non-existence of such a field in a general space-time is the reason why e.g. the energy of a photon is not conserved in an expanding Universe.

3.1. Internal symmetries

We have used up-to now mainly space-time symmetry of Minkowski space, namely the re-quirement of Lorentz invariance, to deduce possible terms in the action. If we allow for morethan one field, e.g. several scalar fields, the new possibility of internal symmetries arise. Forinstance, we can look at a theory of two massive scalar fields,

L =1

2(∂µφ1)

2 − 1

2m2

1φ21 −

1

4λ1φ

41 +

1

2(∂µφ2)

2 − 1

2m2

2φ22 −

1

4λ2φ

42 −

1

2λ3φ

21φ

22 . (3.1)

As it stands, the theory contains five arbitrary parameters, two masses mi and three couplingconstants λi. For arbitrary values of these parameters, no new additional symmetry results.In nature, we find however often a set of particles with nearly the same mass and (partly) thesame couplings. One of the first examples was suggested by Heisenberg after the discovery ofthe neutron, which has a mass very close to the one of the proton, mn ∼ mp: With respectto strong interactions, it is useful to view the proton and neutron as two different “isospin”states of the nucleon, similiar as an electron has two spin states. An example of an exactsymmetry are particles and their anti-particles, as e.g. the charged pions π± which can becombined into one complex scalar field.

If we set in our case m1 = m2 and λ2 = λ1, the Lagrangian becomes invariant under theexchange φ1 ↔ φ2. Adding the further condition that λ3 = λ1 = λ2, we arrive at

L =1

2

[(∂µφ1)

2 + (∂µφ2)2]− 1

2m2(φ21 + φ22)−

λ

4(φ21 + φ22)

2 . (3.2)

Now any orthogonal transformation O ∈ O(2) in the two-dimensional field space φ1, φ2leads to the same Langrangian L . In particular, the Lagrangian is invariant under a rotationR(α) ∈ SO(2) which mixes φ1, φ2 as

(φ′1φ′2

)

=

(cosα − sinαsinα cosα

)(φ1φ2

)

. (3.3)

74

3.2. Noether’s theorem

The fields transform as a vector φ = φ1, φ2 and a rotation leaves the length of the vector φinvariant. Generalising this to n scalar fields, we can write down immediately a theory thatis invariant under transformations φa → Rabφb where R is an element of SO(n),

L =1

2(∂µφ)

2 − 1

2m2φ2 − λ

4(φ2)2 . (3.4)

Note that the Lagrangian is only invariant under global, i.e. , space-time independant rota-tions.

An interaction vertex connects two particles of type a and two particles of type b, soit is of the type −iλ(δabδcd + δacδbd + δadδbc). Since all particles have the same mass, allRabφb are eigenstates of the free Hamiltonian. Thus the propagator Dab(x− x′) is diagonal,Dab(x− x′) ∝ δab.


From our experience in classical and quantum mechanics, we expect that global continuoussymmetries lead also in field theory to conservation laws for the generators of the symmetry.In order to derive such a conservation law, we assume that our collection of fields φa has acontinuous symmetry group. Thus we can consider an infinitesimal change δφa of the fieldsthat keeps L (φa, ∂µφa) invariant,

0 = δL =δL

δφaδφa +

δL

δ∂µφaδ∂µφa . (3.5)

Now we exchange δ∂µ = ∂µδ in the second term and use then the Lagrange equations,δL /δφa = ∂µ(δL /δ∂µφa), in the first one. Then we can combine the two terms usingthe product rule,

0 = δL = ∂µ

(δL

δ∂µφa

)

δφa +δL

δ∂µφa∂µδφa = ∂µ

(δL

δ∂µφaδφa

)

. (3.6)

Hence the invariance of L under the change δφa implies the existence of a conserved current,∂µj

µ = 0, with

jµ =δL

δ∂µφaδφa . (3.7)

If the transformation δφa leads to change in L that is a total four-divergence, δL = ∂µKµ,

and boundary terms can be dropped, then the equations of motion remain invariant. Theconserved current jµ is changed to

jµ =δL

δ∂µφaδφa −Kµ . (3.8)

In Minkowski space, we can convert this differential form of a conservation law into a globalone, obtaining a globally conserved charge

Q =

∫

Vd3x j0 . (3.9)

Often (but not always) this charge has a profound physical meaning.

75


Internal symmetries As an example, we can use our n scalar fields invariant under thegroup1 SO(n). We need the infinitesimal generators Ti of rotations,

φ′a = Rabφb ≈ (1 + αiTi +O(α2i ))abφb . (3.10)

SO(n) has an anti-symmetric Lie algebra with n(n−1)/2 generators. Thus a theory invariantunder SO(n) has n(n− 1)/2 conserved currents. The special case n = 2 has as an importantapplication the π± system. We combine the two real fields φ1 and φ2 into the complex fieldφ = (φ1 + iφ2)/

√2, then the Lagrangian becomes

L = ∂µφ†∂µφ−m2φ†φ− λ(φ†φ)2 . (3.11)

Under the combined phase transformations φ→ eiϑφ and φ† → e−iϑφ†, the Lagrangian L isclearly invariant. With δφ = iϑφ, δφ† = −iϑφ†, the conserved current follows as

jµ = iq[

φ†∂µφ− (∂µφ†)φ]

, (3.12)

where we absorbed the parameter ϑ in the definition of the charge q. The conserved chargeQ =

∫d3xj0 can take any value in R. Therefore we cannot interpret j0 as the probability

density to observe a φ particle. Instead, we should associate Q with a conserved additivequantum number as the electric charge. In the similar case of the proton and antiproton,we could view the conserved charge also as nucleon number, or in modern language, baryonnumber.

Space-time symmetries of Minkowski space The Poincare group as symmetry group ofMinkowski space has ten generators. If the Lagrangian does not depend explicitely on space-time coordinates, i.e. L = L (φa, ∂µφa), ten conservation laws for the fields φa follow.We consider first the behaviour of the fields φa and the Langrangian under an infinitesimaltranslation xµ → xµ + εµ. As in the case of internal symmetries, we consider only globaltransformations and thus ε does not depend on x. From

φa(x)→ φa(x+ ε) ≈ φa(x) + εµ∂µφa(x) (3.13)

we find the change

δφa(x) = εµ∂µφa(x) = ∂µ(εµφa(x)) .

If the Lagrange density L contains no explicit space-time dependence, it will change simplyas L (x)→ L (x+ ε) or

δL (x) = εµ∂µL (x) = ∂µ(εµL (x)) . (3.14)

Thus Kµ = εµL (x) and inserting both in the Noether current gives

jµ =∂L

∂(∂µφa)[εν∂νφa]− εµL = εν

[∂L

∂(∂µφa)

∂φa∂xν− ηµνL

]

= ενTµν (3.15)

1Although the Langrangian is invariant under the larger group O(n), we consider only the subgroup SO(n)which is continously connected to the identity. The additional discrete transformations contained in O(n)can be used to classify solutions of the Lagrangian, but do not lead to additional conservation laws.

76


with T µν as (energy-momentum) stress tensor and the four-momentum pν as four conservedNoether charges,

pν =

∫

d3x T 0ν . (3.16)

The conserved energy-momentum stress tensor that is defined by Eq. (3.15) is called canonical.The definition (3.15) does not guaranty that T µν is symmetric. A symmetric stress tensor,T µν = T νµ, is however the condition for the conservation of total angular momentum, aswill show in the next paragraph2. Since the Lagrange density is only determined up to afour-divergence, we can symmetrize always T µν adding a four-dimensional divergence.

Example: The general expression (3.15) for the canonical stress tensor becomes for a free complexscalar field

T µν = 2∂µφ†∂νφ− ηµνL . (3.17)

Thus the canonical stress tensor of a scalar field is already symmetric. Its 00 component agrees withtwice the result for the energy-density ρ and the Hamilton density H of a single real scalar field,

T 00 = 2 |∂tφ|2 −L = |∂tφ|2 + |∇φ|2 +m2|φ|2 = ρ = H .

We consider now plane-wave solutions to the Klein-Gordon equation, φ = N exp(ikx). If we insert∂µφ = ikµφ into L , we find L = 0 ad thus

T 00 = 2N2k0k0 . (3.18)

Changing from the continuum normalisation to a box of size V = L3 amounts to replace (2π)3 by L3.Thus the normalisation constant N−2 = (2π)32ω becomes for finite volume N−2 = 2ωV . Thence theenergy-density T 00 = ω/V agrees with the expectation for one particle with energy ω per volume V .The remaining components of T µν are fixed by its tensor structure,

T µν = 2N2kµkν =kµkν

ωV. (3.19)

Since the stress tensor T µν is symmetric, we can find a frame in which T µν is diagonal with T ∝diag(ω, vxpx, vypy, vzpz)/V . Note that the spatial part of the stress tensor corresponds to the pressure

tensor, cf. Eq. (9.54). Thus a scalar field can be described by the stress tensor of an ideal fluid,

T µν = diag(ρ, Px, Py, Pz), with energy density ρ and pressure P .

Angular momentum If the tensor T µν is symmetric, we can construct six more conservedquantities. Define

Mµνλ = xν T µλ − xλ T µν , (3.20)

then Mµνλ is conserved with respect to the index µ,

∂µMµνλ = δνµ T

µλ − δλµT µν = T νλ − T λν = 0 , (3.21)

provided that T νλ = T λν . In this case,

Jµν =

∫

d3xM0µν =

∫

d3x[xµT 0ν − xνT 0µ

], (3.22)

2Another reason to require a symmetric stress tensor T µν is that it serves as source term for the symmetricgravitational field.

77


is a globally conserved tensor. The antisymmetry of Jµν implies that there exist six conservedcharges. The three charges

J ij =

∫

d3x[xiT 0j − xjT 0i

], (3.23)

correspond to the conservation of total angular momentum, since T 0j is the momentumdensity. The remaining three charges J0i express the fact that the center-of-mass moves withconstant velocity.

While Jµν transforms as expected for a tensor under Lorentz transformations, it is notinvariant under translations xµ → xµ + εµ. Instead, the angular momentum changes as

Jµν → Jµν + εµpν − ενpµ . (3.24)

Clearly, this is a consequence of the definition of the orbital angular momentum with respectto the center of rotation. We want therefore to split the total angular momentum Jµν intothe orbital angular momentum Lµν and an intrinsic part connected to a non-zero spin of thefield. The latter we require to be invariant under translations. We set

Sα =1

2εαβγδJ

βγuδ , (3.25)

where uα is the four-velocity of the center-of-mass system (cms). Because of the antisymmetryin βγ of εαβγδ , the change in (3.24) induced by a translation drops out in Sα. In the cms,uα = (1, 0, 0, 0) and thus S0 = 0 and Sαu

α = 0. The other components are S1 = J23, S2 = J31,and S3 = J12. Thus the vector Sα describes as desired the intrinsic angular momentum of afield. It is called the Pauli-Lubanski spin vector.

3.3. Symmetries of a general space-time

In the case of a Riemanian space-time manifold (M ,g), we say the space-time posses asymmetry if it looks the same as one moves from a point P along a vector field ξ to adifferent point P . More precisely, we mean with “looking the same” that the metric tensortransported along ξ remains the same.

Such symmetries may be obvious, if one uses coordinates adapted to these symmetries: Forinstance, the metric may be independent from one or several coordinates. Let us assume thatthe metric is e.g. independent from the time coordinate x0. Then x0 is a cyclic coordinate,∂L/∂x0 = 0, of the Lagrangian L = dτ/dσ of a free test particle moving in M . WithL = dτ/dσ, the resulting conserved quantity ∂L/∂x0 = const. can be written as

∂L

∂x0= g0β

dxβ

Ldσ= g0β

dxβ

dτ= ξ · u (3.26)

with ξ = e0 and u as the four-velocity. Hence the quantity ξ · u = p0/m is conserved alongthe solutions xα(σ) of the Lagrange equation of a free particle on M , i.e. along geodesics:The motion of all test particles in the corresponding space-time conserve energy. The vectorfield ξ that points in the direction in which the metric does not change is called Killing vectorfield.

78

3.3. Symmetries of a general space-time

Since we allow arbitrary coordinate systems, space-time symmetries are in general notevident by a simple inspection of the metric tensor. We say the metric is invariant movingalong the Killing vector field ξ, when the resulting change δgµν of the metric is zero,

δξgµν = gµν(x)− gµν(x) = 0 . (3.27)

Here we make an active coordinate transformation, moving from x to x which we assume forsimplicity to be separated by an infinitesimal distance. Then we can work in the approxima-tion

xi = xi + εξi(xk) +O(ε2) , ε≪ 1 , (3.28)

and neglect all terms quadratic in ε. Equation (3.27) expresses the condition that the metrictensor after being transported along ξ from x to x has the same functional dependence onthe coordinates as the original metric tensor.

We start recalling the tranformation law for a rank two tensor as the metric under anarbitrary coordinate transformation,

gµν(x) =∂xµ

∂xα∂xν

∂xβgαβ(x) . (3.29)

Applied to the transport along ξ defined in (3.28), we obtain

gµν(x) =∂xµ

∂xα∂xν

∂xβgαβ(x)

= (δµα + εξµ,α)(δνβ + εξν,β)g

αβ(x)

= gµν(x) + ε(ξµ,ν + ξν,µ) +O(ε2) . (3.30)

In order to be able to compare the new gµν(x) with gµν(x), we have to express gµν(x) asfunction of x. A Taylor expansion gives

gµν(x) = gµν(x+ εξ) = gµν(x) + εξα∂αgµν(x) +O(ε2) . (3.31)

Setting equal Eqs. (3.30) and (3.31), we obtain

gµν(x) + ε(ξµ,ν + ξν,µ) = gµν(x) + εξα∂αgµν(x) . (3.32)

Thus the metric is kept invariant, if the condition

ξµ,ν + ξν,µ − ξk∂kgµν = 0 (3.33)

or

gµα∂αξν + gνα∂αξ

µ − ξα∂αgµν = 0 (3.34)

is satisfied. Expressing partial derivatives as covariant ones, the terms containing connectioncoefficients cancel and we obtain the Killing equations

∇µξν +∇νξµ = 0 . (3.35)

The vectors ξ are called Killing vectors of the metric. Moving along a Killing vector field, themetric is kept invariant.

79


We now check that Eq. (3.35) leads indeed to a conservation law, as required by our initialdefinition of a Killing vector field. We multiply the equation for geodesic motion,

Duµ

dτ= 0, (3.36)

by the Killing vector ξµ and use Leibniz’s product rule together with the definition of theabsolut derivative (1.97),

ξµDuµ

dτ=

d

dτ(ξµu

µ)−∇µξνuµuν = 0. (3.37)

The second term vanishes for a Killing vector field ξµ, because the Killing equation impliesthe antisymmetry of ∇µξν . Hence the quantity ξµuµ is indeed conserved along any geodesics.

Example: Find all ten Killing vector fields of Minkowski space and specifiy the correspondingsymmetries and conserved quantities.The Killing equation ∇µξν +∇νξµ = 0 simplifies in Minkowski space to

∂µξν + ∂νξµ = 0 . (3.38)

Taking one more derivative and using the symmetry of partial derivatives, we arrive at

∂ρ∂µξν + ∂ρ∂νξµ = 2∂µ∂ρξν = 0 . (3.39)

Integrating this equation twice, we find

ξµ = ω µν xν + aµ . (3.40)

The matrix ωµν has to be antisymmetric in order to satisfy Eq. (3.38). Thus the Killing vector fields sare determined by ten integration constants. They agree with the infinitesimal generators of Lorentztransformations, cf. appendix B.3.The four parameters aµ generate translations, xµ → xµ + aµ, described by four Killing vector fieldswhich can be chosen as the Cartesian basis vectors of Minkowski space,

T 0 = ∂t, T 1 = ∂x, T 2 = ∂y, T 3 = ∂z, .

For a particle with momentum pµ = muµ moving along xµ(λ), the existence of a Killing vector T µ

impliesd

dλ(T µ · u) = d

mdλ(T µ · p) = 0 ,

i.e. the conservation of the four-momentum component pµ.Consider next the ij (=spatial) components of the Killing equation. Three additional Killing vectorsare

J1 = y∂z − z∂y , cyclic permutations. (3.41)

The existence of Killing vectors J i implies that J i · p is conserved along a geodesics of particle. But

J1 · p = ypz − zpy = Lx

and thus the angular momentum around the origin of the coordinate system is conserved.The other three components satisfying the 0α component of the Killing equations (ω 0

1 = ω 10 ),

K1 = t∂z + z∂t , cyclic permutations. (3.42)

The conserved quantity tpz − zE = const. now depends on time and is therefore not as popular as

the previous ones. Its conservation implies that the center of mass of a system of particles moves with

constant velocity vα = pα/E.

80

3.4. Outlook: Classical versus quantum symmetries

Global conservation laws An immediate consequence of Eq. (1.129) is a covariant form ofGauß’ theorem for vector fields. In particular, we can conclude from local current conser-vation, ∇µjµ = 0, the existence of a globally conserved charge. If the conserved current jµ

vanishes at infinity, then we obtain also in a general space-time

∫

Ωd4x√

|g| ∇µjµ =

∫

Ωd4x∂µ(

√

|g|jµ) = 0 . (3.43)

For a non-zero current, the volume integral over the charge density j0 remains constant,

∫

Ωd4x√

|g| ∇µjµ =

∫

V (t2)d3x√

|g|j0 −∫

V (t2)d3x√

|g|j0 = 0 . (3.44)

Thus the conservation of Noether charges of internal symmetries as the electric charge, baryonnumber, etc., is not affected by an expanding universe.

Next we consider the energy-momentum stress tensor as example for a locally conservedsymmetric tensors of rank two. Now, the second term in Eq. (1.132) prevents us to convertthe local conservation law into a global one. If the space-time admits however a Killing fieldξ, then we can form the vector field Pµ = T µνξν with

∇µPµ = ∇µ(T µνξν) = ξν∇µT µν + T µν∇µξν = 0 . (3.45)

Here, the first term vanishes since T µν is conserved and the second because T µν is symmet-ric, while ∇µξν is antisymmetric. Therefore the vector field Pµ = T µνξν is also conserved,∇µPµ = 0, and we obtain thus the conservation of the component of the energy-momentumvector in direction ξ.

In summary, global energy conservation requires the existence of a time-like Killing vectorfield. Moving along such a Killing field, the metric would be invariant. Since we expect in anexpanding universe a time-dependence of the metric, a time-like Killing vector field does notexist and the energy contained in a “comoving” volume changes with time.


Anomalies Our discussion of symmetries has been purely classical. A classical symmetryholds also in quantum theory, if the path integral (i.e. functionals like Z[J ]) remains invariantunder the symmetry transformation δφa. This requires however not only the invariance of L

but also of the path integral measure Dφa. If the latter is not invariant, then the classicalsymmetry is broken by quantum effects and one speaks of an “anomaly”. The two mostimportant example are

• the chiral anomaly: Weak interactions of left and right chiral fermions differ. Theintegration measure in the path integral for fermions is not invariant under electroweakgauge transformations, thus a consistent quantum theory seems impossible. Only wayout is that the anomalous terms of all matter fields cancels. This happens if the electriccharge of all fermions vanishes,

∑

f Qf = 0, or in one generation

∑

f

Qf = Qνl +Ql + 3QU + 3QD = 0 .

81


First, we note (without knowing why) that the condition is indeed fulfilled for theparticle content of the standard model. Second, that the faith in this relation lead tothe prediction of the top quark, twenty years before its discovery.

• the breaking of conformal invariance in string theory. In this case, the anomalous termvanishes for a definite number of space-time dimensions, D = 10 or 26.

Ward identities The generating functional for a single, real scalar field given in Eq. (2.73)can be generalized in a straightforward way to a complex scalar field. We treat φ and φ∗ asthe two independent degrees of freedom, and add therefore also two independent sources Jand J∗. Coupling them as Ls ≡ Jφ∗+J∗φ to the fields keeps the Lagrangian real. We denotethe total Lagrangian as

Leff = Lcl + Ls , (3.46)

with Lcl = L0 + Lint as the Lagrangian used to derive to the classical equation of motions.Thus the generating functional is

Z[J, J∗] =∫

DφDφ∗ exp i∫

d4x (Lcl + Jφ∗ + J∗φ) . (3.47)

We assume now that the classical Langrangian Lcl is invariant under an internal globalsymmetry. To be concrete, we consider again the U(1) symmetry with δφ = −iεφ, δφ∗ =iεφ∗. If we keep the external sources J and J∗ constant, the source term Ls breaks thesymmetry. This seems to imply that the generating functional Z[J, J∗] and thus also the Greenfunctions and physical observables are not invariant under the U(1) symmetry of the classicalLagrangian. On the other hand, the fields φ and φ∗ are integrated out in the generatingfunctional Z[J, J∗] which therefore can not depend on a change of the fields generated by asymmetry of L . As a result, identities between different Green functions can be derived.

We sketch now the idea how these so-called Ward identities are derived. For δφ = iεφ,δφ† = −iεφ†, the change of the action is

δ

∫

d4xLeff = δ

∫

d4xLs =

∫

d4x iε(Jφ∗ − J∗φ) (3.48)

The resulting change of the generating functional is thus

δZ[J, J∗] =∫

DφDφ∗∫

d4x iε (Jφ∗ + J∗φ) exp i∫

d4x′ Leff . (3.49)

The variation δZ[J, J∗] has to vanish. Moreover, we can as usually substitute fields by func-tional derivatives of sources. Then the part from Eq. (8.153) can be moved outside thefunctional integral, leading to the equation

0 =

∫

d4x

[

Jδ

δJ− J∗ δ

δJ∗

]

expiW . (3.50)

Taking additional functional derivatives we can generate identities between connected Greenfunctions.

82


Summary of chapter

Noether’s theorem shows that continuous global symmetries lead to conservation laws. Sinceany local symmetry O(x) contains as a special case global transformations O, local symmetrieslead to the same conservation laws as the corresponding global symmetries. In the cases ofthe conservation of electric and colour charge, the global symmetry is a consequence of anunderlying local gauge symmetry which we will study later in chapter 7 in detail. In mostother cases however, as e.g. the conservation of baryon or lepton number, no underlying localsymmetry exist and one speaks therefore of accidental symmetries.

Further reading

Scale invariance which is briefly introduced in problem 3.5 is discussed in more detail in[Pok87]. For the analogue of the classical conserved currents in quantum theory see [DGH94].

Problems

3.1 Lagrangian for N scalar fields.

The most general expression for the Lagrangedensity L of N scalar fields φi which is Lorentzinvariant and at most quadratic in the fields is

L =1

2Aij∂

µφi∂µφj −1

2Bijφiφj − C .

Argue why the coefficient matrices A,B,C have tobe real. Show that L can be recast into “canoni-cal form”

L =1

2∂µφi∂µφi −

1

2biφiφi − C

by linear field redefinitions.

3.2 Stress tensor for point particles.

Find the stress tensor for an ensemble of N non-relativistic a.) point particles, ii) On scales L suchthat ∆N/L3 ≫ 1, we can describe the phase spacedensity by a smooth function f(x,p). b.) in theapproximation n≫ 1, where the distributuion canbe approximated Discuss the physical meaning ofthe different elements of T µν.

3.3 Stress tensor for an ideal fluid.

a.) Find first the stress tensor of dust, i.e. ofpressure-less matter.b.) Generalise this result to an ideal fluid. (Hint:The state of an ideal fluid is completely deter-

mined by its energy density ρ and its pressure P ;in the rest-frame if the fluid, Pij = Pδij .)

3.4 Stress tensor for the electromagneticfield.Determine the canonical energy-momentum stresstensor T µν of the free Maxwell field. SymmetrizeT µν and determine the energy-density ρ = T 00.Find the trace of T µν.

3.5 Scale invariance.Consider the effect of a scale transformationx→ eαx on a scalar field with Lagrange density

L =1

2∂µφ∂

µφ− 1

2m2φ2 − 1

4λφ4

assuming that it acts linearly on the fields,

φa(x)→ eDαφa(eαx).

Here α is a number and D a matrix in field space.We write down first the infinitesimal version of thescale transformation, with the aim to show that L

can be made invariant for a specific value of D, ifm = 0.ii) Find the corresponding conserved current sµ.iii) Show that the current sµ can be written assµ = xν Tµν , where Tµν is an “improved” energy-momentum tensor. Hint: Proceed similar as inthe case of the angular momentum tensor. Derive

83

4. Spin-1 and spin-2 fields

The electric as well as the gravitational potential follow a 1/r law. According to our discussionof the Yukawa potential, we expect therefore that the two forces are mediated by masslessparticles. Since they exist as macroscopic fields, these particles should follow Bose-Einsteinstatistics. Such particles have integer spin and are described by tensor fields.

The quantisation of fields with higher spin becomes increasingly difficult—for the interact-ing spin-2 case no satisfactory solution has been found up to present. Our aim in this chapteris therefore much more restricted than in the scalar case: we derive the (linear) field equationsdescribing the propagation of electromagnetic and gravitational perturbations, their solutionsand Feynman propagators.

4.1. Tensor fields

Massive fields can be boosted to their rest-frame, kµ = (m,0). Wigner showed that thetransformation properties of a massive field under Poincare transformations are completelyspecified knowing its transformation properties under rotations in its rest-frame. As we willsee, this implies that tensor fields of rank n correspond to particles with spin s = n.

Lorentz invariance requires that each component of a free field ψa, where a represents acollection of tensor and spinor indices, satisfies the Klein-Gordon equation. Additionally, wehave to impose constraints which eliminate undesired components,

(+m2

)ψa(x) = 0 , and fi(ψa(x)) = 0 . (4.1)

For instance, massive fields with spin s have 2s+1 components, i.e. a massive spin-1 field hasthree and a massive spin-2 field has five polarisation states. On the other hand, a vector fieldAµ has four components, and a symmetric 2.rank tensor field hµν has ten components. Thusthere exist already in the case of massive spin-1 and spin-2 fields too many degrees of freedomthat have to be projected out. The reason for this mismatch is that in general a tensor ofrank n is reducible, i.e. it contains components of rank < n. For instance, the trace hµµ of asecond rank tensor transforms clearly as a scalar. Therefore we should choose the constraintsfor massive fields with spin s such that all components with spin < s are eliminated.

Example: An object which contains invariant subgroups with respect to a symmetry operation iscalled reducible. Two examples for reducible tensors are:a) The reducible subgroups of an arbitrary tensor of rank two with respect to general coordinatetransformations.We can split any tensor of rank two in its symmetric and anti-symmetric part,

T µν =1

2(T µν + T νµ) +

1

2(T µν − T νµ) = Sµν +Aµν .

As the (anti-)symmetrisation is linear, it is clearly invariant under the change T µν(x) = ∂xµ

∂xρ∂xν

∂xσ Tρσ(x).

Moreover, the trace of a tensor is an invariant. Thus we can break up Sµν into its trace S = Sµµ and

84

4.1. Tensor fields

its traceless part Sµν − Sδµν/d in d dimensions.b) The reducible subgroups of a symmetric tensor hµν of rank two with respect to spatial rotations.We split hµν into a scalar h00, a vector h0i and a reducible tensor hij ,

hµν =

(h00 h0i

hi0 hij

)

.

Then we decompose hij again into its trace hii and its traceless part hij − hδij/(d − 1). The latter

has 6− 1 = 5 degrees of freedom in d = 4, as required for a massive spin-2 field.

This problem is more severe for massless fields, where only two physical degrees of freedomexist1. In this case, even the irreducible tensors contain too many degrees of freedom. Theycan be consistently eliminated only, if some redundancy of the field variables exists which inturn leads to a symmetry of the fields. In this chapter, we discuss the consequences of thisredundancy called gauge symmetry on the level of the wave equations and their solutions forthe photon and the graviton.

Tensor structure of the propagator We can gain some insight into the general tensor struc-ture of the Feynman propagator with spin s > 0 using the definition of the 2-point Greenfunction as the time-ordered vacuum expectation values of fields. In general, we can expressan arbitrary solutions of a free spin s = 0, 1 and 2 field by its Fourier components as

φ(x) =

∫d3k

√

(2π)32ωk

[

a(k)e−i(ωkt−kx) + h.c.]

, (4.2)

Aµ(x) =∑

r

∫d3k

√

(2π)32ωk

[

ar(k)εµr e

−i(ωkt−kx) + h.c.]

, (4.3)

hµν(x) =∑

r

∫d3k

√

(2π)32ωk

[

ar(k)εµνr e−i(ωkt−kx) + h.c.

]

, (4.4)

where the momentum is on-shell, kµ = (ωk,k) and r labels the spin or polarisation states. Theconstraints fi(ψ(x)) = 0 are now conditions on the polarisation vector and tensor, respectively,which depend on k.

Proceeding as in the scalar case, (2.54), we expect that e.g. the propagator for a vectorfield is

iDµνF (x) = 〈0|TAµ(x)Aν∗(0)|0〉 = (4.5)

=∑

r

∫d3k

(2π)32ωk

[

εµr (k)e−i(ωkt−kx)ϑ(x0) + εµ∗r (k)ei(ωkt−kx)ϑ(−x0)

]

(4.6)

=

∫d4k

(2π)4Pµν(k)

k2 −m2 + iε. (4.7)

The expression (4.6) is in line with the interpretation of the propagator as the probabilityfor the creation of a particle at x with any momentum k and polarisation r, its propagationto x′ followed by its annihilation. In the last step, we assumed that the tensor Pµν(k) whichcorresponds to a sum over the polarisation states εµr (k) of the particle introduces no additionalsingularities.

1See the appendix B.4 for a brief discussion.

85


In the following, we will assume that the propagators for s > 0 can be derived from thescalar one, ∆F (k), knowing the polarisation states as

DµνF (k) =

∑

r

εµr (k)εν∗r (k)∆F (k) , (4.8)

DµνρσF (k) =

∑

r

εµνr (k)ερσ∗r (k)∆F (k) . (4.9)

Later on we will come back to this topic deriving the propagators directly from the Lagrangedensity for spin-1 and spin-2 fields.

4.2. Vector fields

Proca and Maxwell equations A massive vector field Aµ has four components in d = 4space-time dimensions, while it has only 2s + 1 = 3 independent spin components. Corre-spondingly, a 4-vector Aµ transforms under a rotation as (A0,A), i.e. it contains a scalarand a three-vector. Therefore we have to add to the four Klein-Gordon equations for Aµ oneconstraint which eliminates A0: The only linear, Lorentz invariant possibility is

(+m2

)Aµ(x) = 0 and ∂µA

µ = 0 . (4.10)

In momentum space, this translates into (k2 −m2)Aµ(k) = 0 and kµAµ(k) = 0. In the rest

frame of the particle, kµ = (m,0), and the constraint becomes A0 = 0. Hence a field satisfying(4.10) has clearly only three components as required for a massive s = 1 field. We can choosethe three polarisation vectors which label the three degrees of freedom in the rest frame e.g.as the Cartesian unit vectors, εi ∝ ei.

The two equations can be combined into one equation called Proca equation,

(ηµν− ∂µ∂ν)Aν +m2Aµ = 0 . (4.11)

To show the equivalence of this equation with (4.10), act with ∂µ on it,

(∂ν−∂ν)Aν +m2∂µAµ = m2∂µA

µ = 0 . (4.12)

Hence, a solution of the Proca equation fulfils automatically the constraint ∂µAµ = 0 for

m2 > 0. On the other hand, we can neglect the second term in (4.11) for ∂νAν = 0 and

obtain the Klein-Gordon equation.We now go over to the case of a massless spin-1 field which is described by the Maxwell equa-

tions. In classical electromagnetism, the field-strength tensor Fµν is an observable quantity,while the potential Aµ is merely a convenient auxiliary quantity. From the definition

Fµν = ∂µAν − ∂νAµ , (4.13)

it is clear that Fµν is invariant under the transformations

Aµ(x)→ A′µ(x) = Aµ(x) + ∂µΛ(x) . (4.14)

Thus A′µ(x) is for any Λ(x) physically equivalent to Aµ(x), leading to the same field-strength

tensor and thus e.g. to the same Lorentz force on a particle. The transformations (4.14) are

86

4.2. Vector fields

called gauge transformations. Note that the mass term m2Aµ in the Proca equation breaksgauge invariance.

If we insert into the Maxwell equation the definition of the potential,

∂µFµν = ∂µ(∂

µAν − ∂νAµ) = Aν − ∂µ∂νAµ = jν , (4.15)

we see that this expression equals the m = 0 limit of the Proca equation. Gauge invarianceallows us to choose a potential Aµ such that ∂µA

µ = 0. Such a choice is called fixing thegauge, and the particular case ∂µA

µ = 0 is denoted as Lorenz gauge. In the Lorenz gauge,the wave equation simplifies to

Aµ = jµ . (4.16)

Additionally, we can add to the potential Aµ any function χ satisfying χ = 0. We canuse this freedom to set A0 = 0. Inserting then a plane-wave Aµ ∝ εµeikx into the free waveequation, Aν = 0, we find that k is a null-vector and that εµkµ = −ε · k = 0. Thus thephoton propagates with the speed of light, is transversely polarised and has two polarisationstates as expected for a massless particle.

Closely connected to the gauge invariance of electrodynamics is the fact that its source, theelectromagnetic current, is conserved. The antisymmetry of Fµν , which is the basis for thesymmetry (4.14), leads also to ∂µ∂νF

µν = 0. Thus the Maxwell equation ∂µFµν = jν implies

the conservation of the electromagnetic current jµ,

∂µ∂νFµν = ∂νj

ν = 0 . (4.17)

Propagator for massive spin-1 fields The propagator Dµν for a massive spin-1 field isdetermined by

[ηµν( +m2)− ∂µ∂ν

]Dνλ(x) = δµλδ(x) . (4.18)

Inserting the Fourier transformation of the propagator and of the delta function gives

[(−k2 +m2

)ηµν + kµkν

]Dνλ(k) = δµλ . (4.19)

We will apply the tensor method to solve this equation: In this approach, we use first alltensors available in the problem to construct the required tensor of rank 2. In the case athand, we have at our disposal only the momentum kµ of the particle—which we can combineto kµkν—and the metric tensor ηµν . Thus the tensor structure of Dµν(k) has to be of theform

Dµν(k) = Aηµν +Bkµkν (4.20)

with two unknown scalar functions A(k2) and B(k2). Inserting this ansatz into (4.19) andmultiplying out, we obtain

[(−k2 +m2)ηµν + kµkν

][Aηνλ +Bkνkλ] = δµλ ,

−Ak2δµλ +Am2δµλ +Akµkλ +Bm2kµkλ = δµλ ,

−A(k2 −m2)δµλ + (A+Bm2)kµkλ = δµλ . (4.21)

In the last step, we regrouped the LHS into the two tensor structures δµλ and kµkλ. Acomparison of their coefficients gives then A = −1/(k2 −m2) and

B = − A

m2=

1

m2(k2 −m2).

87


Thus the massive spin-1 propagator follows as

DµνF (k) =

−ηµν + kµkν/m2

k2 −m2 + iε. (4.22)

Next we check our claim that the propagator DabF (k) of spin s > 0 fields can be obtained

as sum over their polarisation states ε(r)a times the scalar propagator ∆(k). As the theory is

Lorentz invariant, we can choose the frame most convenient for this comparison which is therestframe of the massive particle. Then kµ = (m,0) and the three polarisation vectors canbe chosen as the Cartesian basis vectors,

− ηµν + kµkν/m2 =

0 0 0 00 1 0 00 0 1 00 0 0 1

=∑

r

εµ(r)εν(r) . (4.23)

Thus both methods agree and allow us to derive the Feynman propagator. In the latterapproach, working from the RHS to the LHS of (4.23), we derive first the expression valid forthe Feynman propagator in a specific frame. Then we have to rewrite the expression in aninvariant way using the relevant tensors, here ηµν and kµkν .

Propagator for massless spin-1 fields As we have seen, we can set m = 0 in the Procaequation and obtain the Maxwell equation. The corresponding limit in (4.22) leads howeverto an ill-defined result. As we know that the number of degrees of freedom differs betweenthe massive and the massless case, this is not too surprising. If we try next the limit m→ 0in (4.21), then we find

−Ak2δµλ +Akµkλ = δµλ . (4.24)

The last equation has for arbitrary k with A = −1/k2 and A = 0 no solution. Moreover,the function B is undetermined. We can understand this physically, since for a massless fieldcurrent conservation holds. But ∂µJ

µ(x) = 0 implies kµJµ(k) = 0 and thus the kµkν term

does not influence physical quantities: In physical measurable quantities, as e.g. W [J ], thepropagator is always matched between conserved currents, and the longitudinal part kµkν

drops out.We now try to construct the photon propagator from its sum over polarisation states.

First we consider a linearly polarised photon with polarisation vectors ε(r)µ lying in the plane

perpendicular to its momentum vector k. If we perform a Lorentz boost on ε(1)µ , we will find

ε(1)µ = Λνµε(1)ν = a1ε

(1)µ + a2ε

(2)µ + a3kµ , (4.25)

where the coefficients ai depend on the direction β of the boost. Thus, in general the po-larisation vector will not be anymore perpendicular to k. Similiarly, if we perform a gaugetransformation

Aµ(x)→ A′µ(x) = Aµ(x) + ∂µΛ(x) (4.26)

withΛ(x) = iλ exp(−ikx) + h.c. , (4.27)

thenA′µ(x) = ε′µ exp(−ikx) + h.c. (4.28)

88

4.2. Vector fields

with ε′µ = εµ + λkµ. Since λ is arbitrary, only the components of εµ transverse to k canhave physical significance. Using εµk

µ = 0, we can eliminate both the time-like and thelongitudinal component choosing ε0 = −ε3.

We will use now current conseration, kµJµ(k) = 0, to derive a convenient expression for

the propagator of a massless vector particle. The two polarisation vectors of a photon should

satisfy the normalisation ε(a)∗µ εµ(b) = δab. For a linearly polarised photon propagating in z

direction, kµ = (ω, 0, 0, ω), the polarisation vectors are ε(1)µ = δ1µ and ε

(2)µ = δ2µ. If we perform

the sum over the two polarisation states, we find∑

r

ε(r)∗µ ε(r)ν = diag0, 1, 1, 0 . (4.29)

If we try to rewrite this expression in an invariant way using ηµν and kµkν/k2, we fail: We

cannot cancel at the same time η00 = +1 and η33 = −1 by kµkν/k2. We introduce therefore

additionally the momentum vector kµ = (ω, 0, 0,−ω) obtained by a spatial reflection fromkµ. This allows us to write the polarisation sum as an invariant tensor expression,

∑

r

ε(r)∗µ ε(r)ν = −ηµν +kµkν + kµkν

kk≡ −λµν . (4.30)

Current conservation, kµJµ(k) = 0, implies that the second term in the polarisation sum does

not contribute to physical observables. For the same reason, we can add an arbitrary termξkµkν . We use this freedom to eliminate the k dependence and to set

Jµ∗(∑

r

ε(r)∗µ ε(r)ν

)

Jν = Jµ∗(

−ηµν + (1− ξ)kµkνk2

)

Jν . (4.31)

Now we can read off the photon propagator in a form suitable for loop calculations as

DµνF (k) =

−ηµν + (1− ξ)kµkν/k2k2 + iε

. (4.32)

A specific choice of the parameter ξ called gauge fixing parameter corresponds to the choiceof a gauge in Eq. (4.15). In particular, the Feynman gauge ξ = 1, which leads to a formof the propagator often most convenient in calculations, correspond to the Lorenz gauge

in Eq. (4.15). In this gauge,∑3

r=0 ε(r)∗µ ε

(r)ν = diag−1, 1, 1, 1: the propagator contains

nonphysical degrees of freedom, time-like and longitudinal photons, which contributions cancelin physical observables. Similarly, for all other values ξ the propagator is explicitly Lorentzinvariant but contains unphysical degrees of freedom. We will see later that it is a generalfeature of gauge theories as electromagnetism that we have to choose between a covariantgauge which introduces unphysical degrees of freedom and a gauge which contains only thetransverse degree of freedom but selects a specific frame.

Asymptotic behaviour The behaviour of a propagator for large (Euclidean) momenta k de-termines the degree of divergence of loop integrals. The massless spin-1 propagator decreaseslike Dµν

F (k) ∝ 1/k2 for large k, while the massive propagator behaves as DµνF (k) ∝ const. This

implies that the divergences in loop diagrams are more severe for massive vector particles thanfor massless ones. In particular, inserting additional massive propagators into a loop graphdoes not improve its convergence and thus a theory with massive spin-1 particles contains aninfinite number of divergent diagrams at each loop order.

89


Repulsive Coulomb potential by vector exchange We consider again two static pointcharges as external sources, but using now a vector current Jµ = Jµ1 (x1) + Jµ2 (x2). SinceJµ = (ρ, j), only the zero component, Jµi = δµ0 δ(x − xi), contributes for a static source toW [J ]. Moreover, we can neglect the longitudinal part kµkν/m2 of the propagator. This isjustified, since the concept of a potential energy makes only sense in the non-relativistic limit,i.e. for V ≪ m or equivalently r ≫ 1/m. Hence

W12[J ] = −1

2

∫

d4xd4x′∫

d4k

(2π)4Jµ1 (x)

−ηµν e−ik(x−x′)

k2 −m2 + iεJν2 (x

′) (4.33)

=1

2

∫

dtdt′∫

d4k

(2π)4e−ik(x−x′)

k2 −m2 + iε. (4.34)

Comparing with our earlier result for scalar exchange in Eq. (2.42), it becomes clear withoutfurther calculation that spin-1 exchange between equal charges is repulsive. In the limitm→0, we obtain the Coulomb potential with the correct sign for electromagnetic interactions.

Lorentz transformations on A(r)µ Before we move on, let us examine how the quantity

A(r)µ (x) describing the two physical states of a photon transform under a Lorentz transforma-

tion. We expect a solution of (4.15) to transform as a four-vector, but in determing A(r)µ (x)

we have also to fix a gauge in order to eliminate the two unphysical degree of freedoms.Let us assume that an observer in the frame x has chosen A0(x) = 0 and ∇ · A(x) = 0.Lorentz invariance implies that a second observer in the frame x′ can chose A′0(x′) = 0 and∇′ · A′(x′) = 0 and observes the same physics. However, if the two frames are connected

by a boost, then the A(r)0 cannot be zero in both frames, if A

(r)µ transforms as a four-vector:

Performing a Lorentz transformation changes the gauge, and therefore the transformationlaw of a photon state is

Aµ = ΛµνAν + ∂µΛ , (4.35)

where Λ is such that in both frames the same gauge can be used. You are asked to examine

the Lorentz transformation properties of A(r)µ in more detail in problem 4.3.

4.3. Gravity

Wave equation From Newton’s law we know that gravity is fundamentally attractive andof long-range. Thus the gravitational force has to be mediated by a massless particle whichcan not be a spin s = 1 particle.

Analog to the electric field E = −∇φ we can introduce a classical gravitational field g asthe gradient of the gravitational potential, g = −∇φ. We obtain then ∇ · g(x) = −4πGρ(x)and as Poisson equation

∆φ(x) = 4πGρ(x) , (4.36)

where ρ is the mass density, ρ = dm/d3x.Special relativity gives us two hints how we should transfer this equation into a relativistic

framework: First, the Laplace operator ∆ on the LHS is the c → ∞ limit of minus thed’Alembert operator . Second, the RHS is the v/c → 0 limit of something incorporatingnot only the mass density but all types of energy densities. To proceed, consider first howthe mass density ρ transforms under a Lorentz transformation: An observer moving with the

90

4.3. Gravity

speed β relative to the rest frame of the matter distribution ρ measures the energy densityρ′ = γdm/(γ−1dV ) = γ2ρ, where γ =

√

1− β2. This is the transformation law of the 00component of a tensor of rank 2, alas the energy-momentum stress tensor T µν .

Thus the field equations for a purely scalar theory of gravity would be

φ = −4πGT µµ . (4.37)

Such a theory predicts no coupling between photons and gravitation, since T µµ = 0 for theelectromagnetic field, and is in contradiction to the observed gravitational lensing of light.A purely vector theory for gravity fails too, since it predicts not attraction but repulsion oftwo masses. Hence we are forced to consider a symmetric spin-2 field hµν as mediator of thegravitational force; its source is the energy-momentum stress tensor

hµν = −2κT µν . (4.38)

Let us consider as a warm-up first the massive case: A symmetric, massive spin-2 field hasten independent component, but only 2s+ 1 = 5 physical spin degrees of freedoms. Thus wehave to impose five constraints additional to the source-free equation

(+m2)hµν = 0 . (4.39)

Proceeding as in the s = 1 case, (4.10), we use as constraint ∂µhµν = 0 which provides now

four constraints. These constraints we can use to set h0µ = 0. We obtain the missing fifthconstraint subtracting the trace hii which transforms as a scalar from hij .

We move now to the massless case considering a plane wave hµν = εµν exp(ikx). In analogyto the photon case, we expect that also the graviton has only two, transverse degrees offreedom. If we choose the plane wave propagating in the z direction, k = kez, then we expectthat the polarisation tensor can be expressed as

εµν =

0 0 0 00 ε11 ε12 00 ε12 −ε11 00 0 0 0

. (4.40)

Here we used that the polarisation tensor has to be symmetric and traceless. The choice(4.40) is called the transverse traceless (TT) gauge.

Metric perturbations as a tensor field In the case of the photon, we could reduce the de-grees of freedom from four to two, because of the redundancy implied by the gauge symmetryof electromagnetism. Moreover, the gauge symmetry lead to the conservation of the electro-magnetic current. The two obvious questions to address next are which symmetry and whichconservation law are connected to gravitation.

The second part of the questions is the simpler one, since we know already that in flatspace ∂µT

µν = 0 holds. Thus for gravity energy-momentum conservation will play the roleof current conservation, implying that a gravitational wave is transverse, kµT

µν(k) = 0. Inorder to answer the first part of the question, we have to consider the properties of hµν .

A unique characteristique of gravitation compared to all other known interactions is thatits “charge”, the gravitational mass, is equal to the inertial mass. The equivalence principleimplies that test-particles with arbitrary properties move along the same world-line, if they

91


are released at the same initial point and move only under the influence of the gravitationalforce. This universality motivated Einstein to describe the effect of gravity by the curvatureof space-time.

We associate therefore the symmetric tensor field hµν with small perturbations around theMinkowski2 metric ηµν ,

gµν = ηµν + λhµν , λhµν ≪ 1 . (4.41)

(Here we dropped the bar, anticipating that hµν may be a function of hµν ; in particular,the metric tensor we use to describe space-time is dimensionless, while the bosonic field hµνshould have mass dimension one. This change of dimension implies that the coupling strength

λ is dimensionfull, λ ∝ G−1/2N , as suggested by [GN ] = m−2.)

We choose a Cartesian coordinate system x and ask ourselves which transformations arecompatible with the splitting (4.41) of the metric. If we consider global (i.e. space-time

independent) Lorentz transformations ΛβαΛβα, then x

′α = Λαβxβ. The metric tensor transform

as

g′αβ = ΛραΛσβgρσ = ΛραΛ

σβ(ηρσ + hρσ) = ηαβ + ΛραΛ

σβhρσ = η′αβ + ΛραΛ

σβhρσ . (4.42)

Since h′αβ = ΛραΛσβhρσ , we see that (global) Lorentz transformations respect the splitting(4.41). Thus the perturbation hµν transforms as a rank-2 tensor under global Lorentz transfor-mations. We can view therefore hµν as a symmetric rank-2 tensor field defined on Minkowskispace that satisfies the wave equation (4.38), similar as the photon field is a rank-1 tensorfield fulfilling Maxwell’s equations.

The splitting (4.41) is however clearly not invariant under general coordinate transfor-mations, as they allow e.g. the rescaling gµν → Ωgµν . We restrict therefore ourselves toinfinitesimal coordinate transformations,

xµ = xµ + εξµ(xν) (4.43)

with ε≪ 1. Then the Killing equation (3.34) simplifies to

h′µν = hµν − ∂µξν − ∂νξµ , (4.44)

because the term ξρ∂ρhµν is quadratic in the small quantities hµν and εξµ and can be neglected.

It is more fruitful to view this equation not as a coordinate but as a gauge transformationanalogous to (4.14): Both h′µν and hµν describe the same physical situation, since the(linearized) Einstein equations do not fix uniquely hµν for a given source. In momentumspace, (4.44) becomes a condition on the polarisation tensor,

ε′µν = εµν − ξµkν − ξνkµ. (4.45)

We can use this freedom to eliminate four components of εµν . After that, we can still addfour functions ξµ with ξµ = 0, eliminating four additional components. This justifies theuse of the TT gauge.

2The same analysis could be performed for small perturbations around an arbitrary metric g(0)µν , adding

however considerable technical complexity.

92

4.3. Gravity

Graviton propagator We follow now the same approach as in the derivation of the photonpropagator. For a graviton propagating in z direction, kµ = (ω, 0, 0, ω), we choose as the

two polarisation states ε(1)µµ setting ε11 = 1/

√2 and ε12 = 0 and ε

(2)µµ setting ε11 = 0 and

ε12 = 1/√2, respectively. They satisfy the normalisation ε

(a)µν εµν(b) = δab.

Now we should perform the sum over the two polarisation states,∑

r ε(r)µν ε

(r)ρσ , and express

the result as a linear combination of ηµν and kµkν + kµkν . A straightforward way to do this isto combine first the ten independent quantities of the symmetric tensors in ten-dimensionalvectors, εµν → Ea, ηµν → Na, and kµkν + kµkν → Ka, to calculate the tensor products ofthese vectors and to compare then the resulting 10 ⊗ 10 matrices. An alternative, shorter

way is to use the requirement that the propagator is transverse in all indices, kµ∑

r ε(r)µν ε

(r)ρσ =

. . . = kσ∑

r ε(r)µν ε

(r)ρσ = 0, because of energy-momentum conservation, ∂µT

µν(x) = 0. Thisimplies that the graviton propagator should be composed of the projection operators λµνused for the photon (cf. Eq. (4.30)) as follows

∑

r

ε(r)µν ε(r)ρσ = Aλµνλρσ +B [λµρλνσ + λµσλνρ] . (4.46)

The last two terms have a common coefficient, since the LHS is invariant under exchangesof µ ↔ ν or ρ ↔ σ. We fix A and B by evaluating this expression for two sets of indices.Since the only non-zero elements of λµν are λ11 = λ22 = −1, we obtain choosing e.g. 1212as indices

∑

r

ε(r)12 ε

(r)12 =

1

2= Bλ11λ22

and thus B = 1/2. Similiarly, it follows A = −1/2 choosing e.g. as indices 1111. Thus wefound—with surprising ease—the polarisation sum required for the graviton propagator,

∑

r

ε(r)µν ε(r)ρσ = −1

2λµνλρσ +

1

2λµρλνσ +

1

2λµσλνρ . (4.47)

We continue to proceed in the same way as for the photon. Energy-momentum conservation,kµT

µν(k) = 0, implies that the kµkν + kµkν term in λµν does not contribute to physicalobservables. We can therefore drop again all terms proportional to the graviton momentumkµ,

T µν∗(∑

r

ε(r)∗µν ε(r)ρσ

)

T ρσ = T µν∗(

−1

2ηµνηρσ +

1

2ηµρηνσ +

1

2ηµσηνρ

)

T ρσ. (4.48)

Thus the graviton propagator in the Feynman gauge is given by

Dµν;ρσF (k) =

12(−ηµνηρσ + ηµρηνσ + ηµσηνρ)

k2 + iε. (4.49)

Other Rξ like gauges are obtained by the replacement ηµν → ηµν − (1 − ξ)kµkν/k2. In thecase of gravity, the Feynman gauge ξ = 1 is most often called harmonic gauge, but also thenames Hilbert, Loren(t)z, de Donder and confusingly many others are in use.

93


Attractive potential by spin-2 exchange We consider now the potential energy created bytwo point masses as external sources interacting via a tensor current T µν = T µν1 (x1)+T

µν2 (x2).

Specialising to static sources, only the zero-zero component, T µνi = δµ0 δν0δ(x−xi), contributes

to W [J ]. Hence

W12[J ] = −1

2

∫

d4xd4x′∫

d4k

(2π)4T 001 (x)DF 00;00(k) e

−ik(x−x′) T 002 (x′) . (4.50)

Looking at the nominator of the graviton propagator, we find −1 + 1 + 1 = 1 > 0. Thusspin-2 exchange is attractive, as required for the force mediating gravity. In problem 4.4,you are asked to determine the coupling λ in L = λhµνTµν comparing (4.50) to Newton’sgravitational potential.

Helicity We determine now how a metric perturbation hµν transforms under a rotation withthe angle α. We choose the wave propagating in z direction, k = kez, the TT gauge, and therotation in the xy plane. Then the general Lorentz transformation Λ ν

µ becomes

Λ νµ =

1 0 0 00 cosα sinα 00 − sinα cosα 00 0 0 1

. (4.51)

Since k = kez and thus Λ νµ kν = kµ, the rotation affects only the polarisation tensor. We

rewrite ε′µν = Λ ρµΛ σ

ν ερσ in matrix notation,

ε′ = ΛεΛT . (4.52)

It is sufficient in TT gauge to perform the calculation for the xy sub-matrices. The resultafter introducing circular polarisation states ε± = ε11 ± iε12 is

ε′µν± = exp(∓2iα)εµν± . (4.53)

The same calculation for a circularly polarised photon gives ε′µ± = exp(∓iα)εµ±.Any plane wave ψ which is transformed into ψ′ = e−ihαψ by a rotation of an angle α around

its propagation axis is said to have helicity h. Thus if we say that a photon has spin 1 anda graviton has spin 2, we mean more precisely that electromagnetic and gravitational planwaves have helicity one and two, respectively. Doing the same calculation in an arbitrarygauge, one finds that the remaining, unphysical degrees of freedom transform as helicity oneand zero (problem 4.6).

4.4. Source of gravity

The dynamical energy-momentum tensor If we compare the wave equation for a photonand a graviton, then there is an important difference: The former is in the classical limitexact. The photon carries no charge and does not contribute to its source term. As a result,the wave equation is linear. In contrast, a gravitational wave carrying energy and momentumacts as its own source. The LHS of (4.38) should be therefore the limit of a more complicatedequation, which we write symbolically as Gµν = −2κTµν . In the absence of matter, Gµν = 0,

94

4.4. Source of gravity

and the LHS reduces to the yet unknow (exact) field equations for the gravitational field gµν .These field equations should be given as the variation of an appropriate action of gravity,called the Einstein-Hilbert action SEH, with respect to the metric tensor gµν .

Even without knowing the action of gravity, we can derive an important conclusion. Theyet unknown Gµν on the LHS should be given as the variation of the action SEH of gravitywith respect to the metric tensor gµν . If the total action is the sum of SEH and the action2κSm including all relevant matter fields, then the variation of the matter action Sm shouldgive the energy-momentum tensor as the source of the gravitational field,

2√

|g|δSmδgµν

= Tµν . (4.54)

Since the presence of gravity implies a curved space-time, the replacements ∂µ, ηµν ,d4x →∇µ, gµν , d4x

√

|g| have to performed in Sm before the variation is performed.The tensor Tµν defined by this equation is called dynamical energy-momentum stress ten-

sor. In order to show that this rather bold definition makes sense, we have to prove that∇µ(δSm/δgµν ) = 0, and we have to convince ourselves that this definition reproduces thestandard results we know already.

Conservation of the stress tensor We start by proving that the dynamical energy-momentum tensor defined by Eq. (19.1) is locally conserved. We consider the change ofthe matter action under a variation of the metric,

δSm =1

2

∫

Ωd4x√

|g| Tµνδgµν = −1

2

∫

Ωd4x√

|g| T µνδgµν . (4.55)

We allow infinitesimal but otherwise arbitrary coordinate transformations,

xµ = xµ + ξµ(x) . (4.56)

For the resulting change in the metric δgµν we can use Eqs. (3.30) and (3.31),

δgµν = ∇µξν +∇νξµ . (4.57)

We use that T µν is symmetric and that general covariance guarantees that δSm = 0 for acoordinate transformation,

δSm = −∫

Ωd4x√

|g| T µν∇µξν = 0 . (4.58)

Next we apply the product rule,

δSm = −∫

Ωd4x√

|g| (∇µT µν)ξν +∫

Ωd4x√

|g| ∇µ(T µνξν) = 0 . (4.59)

The second term is a four-divergence and thus a boundary term that we can neglect. Theremaining first term vanishes for arbitrary ξµ only, if the energy-momentum tensor is con-served,

∇µT µν = 0 . (4.60)

Hence the local conservation of energy-momentum is a consequence of the general covariance ofthe gravitational field equations, in the same way as current conservation follows from gaugeinvariance in electromagnetism. You should convince yourself that the dynamical energy-momentum stress tensor evaluated for the examples of the Klein-Gordon and the Maxwellfield agrees with the symmetrized canonical stress tensor, cf. problem 4.10.

95


4.A. Appendix: Large extra dimensions and massive gravity

Large extra dimensions As mentioned in chapter 3, quantum corrections break the con-formal invariance of string theory except we live in a world with n = 10 or 26 space-timedimensions. There are two obvious answers to this result: First, one may conclude that stringthoery is disproven by reality or, second, one may adjust reality. Consistency of the secondapproach with experimental data could be achieved, if the n− 4 dimensions are compactifiedwith a sufficiently small radius R, such that they are not visible in experiments sensible towave-lengths λ≫ R.

Let us check what happens to a scalar particle with mass m, if we add a fifth, compactdimension y. The Klein-Gordon equation for a scalar field φ(xµ, y) becomes

(5 +m2)φ(xµ, y) = 0 (4.61)

with the five-dimensional d’Alembert operator 5 = − ∂2y . The equation can be separated,φ(xµ, y) = φ(xµ)f(y), and since the fifth dimension is compact, the spectrum of f is discrete.Assuming periodic boundary conditions, f(x) = f(x+R), gives

φ(xµ, y) = φ(xµ) cos(nπy/R) . (4.62)

The energy eigenvalues of these solutions are ω2k,n = k2 + m2 + (nπ/R)2. From a four-

dimensional point of view, the term (nπ/R)2 appears as a mass term, m2n = m2 + (nπ/R)2.

Since we usually consider states with different masses as different particles, we see the five-dimensional particle as a tower of particles with mass mn but otherwise identical quantumnumbers. Such theories are called Kaluza-Klein theories, and the tower of particles Kaluza-Klein particles. If R≪ λ, where λ is the length-scale experimentally probed, only the n = 0particle is visible and physics appears to be four-dimensional.

Since string theory includes gravity, one often assumes that the radius R of the extra-dimensions is determined by the Planck length, R = 1/MPl = (8πGN )1/2 ∼ 10−34 cm. In thiscase it is difficult to imagine any observational consequences of the additional dimensions.Antoniadis asked in 1990 what happens if some of the extra dimensions are large,

R1,...,δ ≫ Rδ+1,...,6 = 1/MPl .

Since the 1/r2 behaviour of the gravitational force is not tested below d∗ ∼mm scales, onecan imagine that large extra dimensions exists that are only visible to gravity: Relating thed = 4 and d > 4 Newton’s law F ∼ m1m2

r2+δ at the intermediate scale r = R, we can derivethe “true” value of the Planck scale in this model: Matching of Newton’s law in 4 and 4 + δdimensions at r = R gives

F (r = R) = GNm1m2

R2=

1

M2+δD

m1m2

R2+δ. (4.63)

This equation relates the size R of the large extra dimensions to the true fundamental scaleMD of gravity in this model,

G−1N = 8πM2

Pl = RδM δ+2D , (4.64)

while Newton’s constant GN becomes just an auxiliary quantity useful to decribe physicsat r >∼ R. (You may compare this to the case of weak interactions where Fermi’s constant

96


GF ∝ g2/m2W is determined by the weak coupling constant g and the mass mW of the

W -boson).

Next we ask, if MD ∼ TeV is possible, i.e. if one may test such theories at accelerators asLHC? Inserting the measured value of GN and plus MD = 1TeV in Eq. (4.64) we find therequired value for the size R of the large extra dimension as

δ 1 2 6

R/cm 1013 0.1 10−12

Thus the case δ = 1 is excluded by the agreement of the dynamics of the solar system with4-dimensional Newtonian physics. The cases δ ≥ 2 are possible, because Newton’s law isexperimentally tested only for scales r >∼ 1mm.

Massive gravity Theories with extra dimensions contain often from our 4-dimensional pointof view a Kaluza-Klein tower of massive gravitons. Such modified theories of gravity havefound large interest since one may hope to find an alternative explanation for the acceleratedexpansion of the Universe.

A striking difference between the spin-2 and the spin-0 and 1 cases is that the limit m→ 0of the massive spin-2 propagator and thus of the potential energy V12 is not smooth: In prob-lem 4.5, you are asked to derive the massive spin-2 propagator. As result, you should find

Dµν;ρσF (k) =

1

2

−23G

µνGρσ +GµρGνσ +GµσGνρ

k2 −m2 + iε, (4.65)

where

Gµν(k) = −ηµν + kµkν/m2 (4.66)

is the polarisation tensor for a massive spin-1 particle. Thus the nominator in the massivespin-2 propagator is as in the massless case a linear combination of the tensor products oftwo spin-1 polarisation tensors. However, the coefficients of the first term differ and thus them→ 0 limit of the massive propagator does not agree with the massless case. In particular,the difference cannot be compensated by a rescaling of the coupling constant GN , because itis not an overall factor: Imagine for instance that we determine the value of GN by calculatingthe potential energy of two non-relativistic sources like the Sun and the Earth. This requiresin massive gravity—for an arbitrarily small graviton mass—a gravitational coupling constantGN a factor 3/4 smaller than in the massless case. Having fixed GN , we can predict thedeflection of light by the Sun. Since the first term in the propagator couples the traces T µµof two sources, it does not contribute to the deflection of light. As a result of the reducedcoupling strength, the deflection angle of light by the Sun decreases by the same factor andany non-zero graviton mass would be in conflict with observations.

When this result was first derived in 1970, its authors explained this discontinuity bythe different number of degrees of freedom in the two theories: Even if the Compton wave-length of a massive graviton is larger than the observable size of the Universe, and thus theYukawa factor exp(−mr) indistinguishable from one, the additional spin states of a massivegraviton may change physics. Two years later, Vainshtein realised that perturbation theorymay break down in massive gravity and thus a calculation using one-graviton exchange isnot reliable. More precisely, a theory of massive gravity contains an additional length scaleRV = (GM/m4)1/5 and for distances r ≪ RV the theory has to be solved exactly.

97


Summary of chapter

Tensor fields satisfy second-order differential equations and have mass dimension one. Abosonic field with spin s has a polarisation tensor of rank 2s and thus their propagators areeven in the momentum k. For a massive bosonic field, the polarisation tensors contains stensor products of kµkν and therefore its propagator scales as Dµ1,··· ,µs;ν1,··· ,νs(k) ∝ k2s−2.This implies that the divergences of loop diagrams aggravate for higher spin fields.

Massless spin one and two fields have only two, transverse degrees of freedom. A Lorentzinvariant description for such fields is only possible, if the remaining number of non-physicaldegrees of freedom is redundant. This redundancy implies that fields connected by a gaugetransformation are equivalent and describe the same physical system. In the case of photons,the gauge symmetry implies that they couple to a conserved current, in the case of gravitonsthat they couple to a conserved energy-momentum conservation tensor.

Further reading

Problems

4.1 Polarisation of massive spin-1 particle.Determine the polarisation vectors εµr (k) of a mas-sive spin-1 particle for arbitrary kµ.

4.2 Feynman propagator as sum over solu-tions.Follow the steps from (2.29) to (2.33) in the scalarcase for a massive spin-1 propagator.

4.3 Constrained systems ♥.Calculate the Hamiltonian density of the pho-ton field. Apply canonical quantisation and derivethe Lorentz transformation property of the photonfield operator.

4.4 Gravitational coupling.Determine the coupling λ in L = λhµνTµν com-paring (4.50) to Newton’s gravitational potential.

4.5 Massive spin-2 propagator.Derive the propagator (4.65) of a massive spin-2 particle: Use the tensor method to write downthe most general combination of tensors built fromηµν and kµ (or from Gµν and kµ), compatible withthe symmetries and constraints on the polarisationtensor.

4.6 Helicity.Show that the unphysical degrees of freedom of

an electromagnetic wave transform as helicity 0,and of a a gravitational wave as helicity 0 and 1.

4.7 Helicity.

Show that h in ψ′ = e−ihαψ for a plane wave isthe eigenvalue of the helicity operator J ·p/|p|.

4.8 Scalar QED ♣.a.) Introduce in the Lagrangian (3.11) of a com-plex scalar field the interaction with photons viathe minimal substitution, ∂µ → Dµ = ∂µ + ieAµ,and read off the interaction vertices.b. Calculate the matrix element for “scalar Comp-ton scattering”, φγ → φγ and show that it isgauge invariant. [Try to simplify the matrix el-ement choosing a “good” frame and polarisationvectors.]

4.9 The φφhµν vertex.

Expand the action of a scalar particle coupled togravity,

S =

∫

d4x√

|g|(

gµν∂µφ∂νφ−1

2m2φ2 − ρΛ

)

vin gµν = ηµν + λhµν to first order in λ. Find theFeynman rule for the φφhµν vertex.

4.10 Dynamical stress tensor.

98


Show that the formula (19.1) for the stress tensorcan be simplified to Tµν = 2∂L /∂gµν − gµνL .

Derive the dynamical energy-momentum stresstensor Tαβ of a real scalar field φ and of the photonfield Aµ.

99

5. Fermions and the Dirac equation

Up to now we have discussed fields which transform as tensors under Lorentz transformationsand have even spin or helicity. Since we can boost massive particles into their rest-frame, wecan use our knowledge of non-relativistic quantum mechanics to anticipate that additionalrepresentations of the rotation group SO(3) and thus also of the Lorentz group SO(1, 3) existwhich correspond to particles with half-integer spin. Such particles are described by anti-commuting variables which is the fundamental reason for the Pauli principle, or the stabilityof matter.

5.1. Spinor representation of the Lorentz group

In order to introduce spinors we have to find the corresponding representation of the Lorentzgroup. As always it is simpler to work at “linear order,” which is in this case the Lie algebra.

The Lie algebra of the Poincare group1 contains ten generators, the three generators J ofrotations, the three generators K of Lorentz boosts and the four generators for translations.The Killing vector fields V of Minkowski space generate these symmetries, and therefore thegenerators T at each point are given by the Killing vector fields. Thus we can use (3.41) and(3.42) to calculate their commutation relations as (cf. problem 5.1)

[Ji, Jj ] = iεijkJk , (5.1a)

[Ji,Kj ] = iεijkKk , (5.1b)

[Ki,Kj ] = −iεijkJk . (5.1c)

Here we followed physicist’s convention and identified iT = V , so that the generators arehermetian. Moreover, we restrict our attention to the Lorentz group which is sufficient toderive the concept of a Weyl spinor. Note that the algebra of the boost generators K is notclosed. Thus in contrast to rotations, boosts do not form a subgroup of the Lorentz group.The fact that the comutator of two (not parallel) boosts involves a rotation,

eiKxδψ eiKyδη e−iKxδψ e−iKyδη = 1− [Kx,Ky]δψδη + . . . = 1 + iJzδψδη + . . . (5.2)

is the origin of the Thomas precession.

The commutation relations above are satisfied by ±iσ/2, suggesting that we can rewritethe Lorentz group as a product of two SU(2) factors. We try to decouple the two sets ofgenerators J and K by introducing two non-hermitian ladder operators

J± =1

2(J ± iK) . (5.3)

1See Appendix B.3 for a brief review of the Lorentz group.

100

5.1. Spinor representation of the Lorentz group

Their commutations relations are

[J+i , J

+j ] = iεijkJ

+k , (5.4a)

[J−i , J

−j ] = iεijkJ

−k , (5.4b)

[J+i , J

−j ] = 0 i, j = 1, 2, 3 . (5.4c)

Thus J− and J+ commute with each other and generate each a SU(2) group. The Lorentzgroup is2 therefore ∼ SU(2) ⊗ SU(2), and states transforming in a well-defined way arelabbeled by a pair of two angular momenta, (j−, j+), corresponding to the eigenvalues ofJ−z and J+

z , respectively. From our knowledge of the angular momentum algebra in nonrel-ativistic quantum mechanics, we conclude that the dimension of the representation (j−, j+)is (2j− + 1)(2j+ + 1). Because of J = J− + J+, the representation (j−, j+) contains allpossible spins j in integer steps from |j− − j+| and j− + j+.

The representation (0, 0) has dimension one, transforms trivially, J = K = 0, and cor-responds therefore to the scalar representation. The smallest non-trivial representations areJ+ = 0, i.e. (j−, 0) with J (1/2) = iK(1/2), and J− = 0, i.e. (0, j+) with J (1/2) = −iK(1/2).Both representations have spin 1/2 and dimension two. We define therefore two types oftwo-component spinors,

φL : (1/2, 0), J (1/2) = σ/2, K(1/2) = +iσ/2 , (5.5)

φR : (0, 1/2), J (1/2) = σ/2, K(1/2) = −iσ/2 , (5.6)

which we call left-chiral and right-chiral Weyl spinors. These Weyl spinors form the fun-damental representation of the Lorentz group: All higher spin states can obtained as ten-sor products involving these two spinors. Their transformation properties under a finiteLorentz transformation with parameters α and η follow by exponentiating their generatorsas exp(−iJα+ iKη) (compare B.3 for our choice of signs),

φL → φ′L = exp

[

− iσα

2− ση

2

]

φL ≡ SLφL , (5.7)

φR → φ′R = exp

[

− iσα

2+ση

2

]

φR ≡ SRφR . (5.8)

We defined in section 4.3 the helicity h of a particle as the phase e−ihα gained by a planewave which is rotated the angle α around its propagation axis. Thus both Weyl spinors havehelicity h = 1/2. While the two types of Weyl spinors transform the same way under rotationstheir behavior under Lorentz boosts is opposite.

We ask now if can convert a a left- into a right-chiral spinor and vice versa. Thus weshould find a spinor φL constructed out of φL which transforms as SRφL. Changing SL intoSR requires reversing the relative sign between the rotation and the boost term, which weachieve by complex conjugating φL,

φ∗′L =

[

1 +iσ∗α2− σ

∗η2

+ . . .

]

φ∗L . (5.9)

2More precisely, they have the same Lie algebra and are thus locally isomorphic but differ globally.

101


Because of σ∗1 = σ1, σ∗2 = −σ2, σ∗3 = σ3, and σ1σ2 = −σ2σ1, σ2σ3 = −σ3σ2, we obtain the

desired transformation property multiplying φ∗′L with σ2,

σ2φ∗′L = σ2

[

1 +i(σ1,−σ2, σ3)α

2− (σ1,−σ2, σ3)η

2

]

φ∗L (5.10)

=

[

1− iσα

2+ση

2

]

σ2φ∗L = SRσ2φ

∗L . (5.11)

Thus the two representations are inequivalent, i.e. they are not connected by a unitary trans-formation, and therefore φL and φR describe different physics. Obviously, we can add to σ2φ

∗L

an arbitrary phase eiδ without changing the transformation properties. We define

φcR ≡ −iσ2φ∗L , (5.12)

which ensures that (φcL)c = φL. When we discuss later the coupling of a fermion to an external

field, we will see that φcL is the charge conjugated spinor of φL.The transformations SL and SR that belong to the orthochronous Lorentz group do not mix

the L and R Weyl spinors. Consider however the effect of parity, Px = −x, on the generatorsK and J . The velocity changes sign, v → −v, i.e. transforms as a polar vector, while theangular momentum J as axial vector remains invariant. Thus parity interchanges (1/2, 0)and (0, 1/2) and hence φL and φR, as one would expect from a left- and right-chiral object.If parity is a symmetry of the theory examined, one can therefore not consider separately thetwo spinors φL and φR. Instead, it proves usefule to combine them into a four-spinor (Diracor bi-spinor)

ψ =

(φLφR

)

. (5.13)

For the construction of a Lagrangian we need for the mass term scalars and for the kineticenergy vectors built out of the Weyl fields; they should be real to provide a real Lagrangian. Incontrast to the real Lorentz transformation Λµν acting on tensor fields, the matrices SL/R arehowever complex and thus the Weyl fields are complex too. This suggests together with thefact that a measurement device should be the same after a rotation by 2π that observablesare bilinear quantities in the fermion fields, such that they transform tensorial and theireigenvalues are real.

Out of the two Weyl spinors, we can form four different products φ†L/RφL/R leading to the

combinations S†LSL, S

†RSR, S

†LSR, and S

†RSL. The rotation iσα/2 cancels in all four products,

since it enters with the same sign in SL and SR, and the Pauli matrices are hermitian, σ† = σ.By contrast, the cancelation of the boost ση/2 requires a combination of a L and R field,

φ′†Lφ′R = φ†L

[

1 + iσα

2− ση

2+ . . .

] [

1− iσα

2+ση

2+ . . .

]

φR = φ†LφR , (5.14)

and similiarly for φ†RφL. Thus φ†LφR and φ†RφL transform as Lorentz scalars, but not φ†LφLand φ†RφR. So what are the transformation properties of the latter two products? Performingan infinitesimal boost along the z axis, we find

φ′†Rφ′R = φ†R

[

1 +σ3η

2+ . . .

] [

1 +σ3η

2+ . . .

]

φR = φ†RφR + ηφ†Rσ3φR . (5.15)

This looks like an infinitesimal Lorentz transformation of the time-like component j0 = φ†RφRof a four-vector jµ, if we can associate the spatial part j with φ†RσφR. Checking how j

102

5.2. Dirac equation

transforms, we find using σiσj = δij+iεijkσk that j1 and j2 are invariant, while j3 transformsas

φ′†Rσ3φ′R = φ†R

[

1 +σ3η

2+ . . .

]

σ3

[

1 +σ3η

2+ . . .

]

φR = ηφ†RφR + φ†Rσ3φR . (5.16)

Thus φ†RσµφR with σµ ≡ (1,σ) transforms as a four-vector, j0 → j0+ ηj3 and j3 → ηj0 + j3.

Performing the same calculation for the L fields reproduces the same result except for anopposite sign of η. The wrong sign can be remedied setting now σµ ≡ (1,−σ), so that a

four-vector bilinear in L fields is φ†LσµφL.

5.2. Dirac equation

We can obtain the spinor φL/R(p) describing a particle with momentum p by boosting theone describing a particle at rest, φL/R(0),

φR(p) = exp[ση

2

]

φR(0) = exp[ησn

2

]

φR(0) = [cosh(η/2) + σn sinh(η/2)] φR(0) . (5.17)

If we replace the boost parameter η by the Lorentz factor3 γ = cosh η and use the identitiescosh(η/2) =

√

(cosh η + 1)/2 and sinh(η/2) =√

(cosh η − 1)/2, we can express the spinor as

φR(p) =

[(γ + 1

2

)1/2

+ σp

(γ − 1

2

)1/2]

φR(0) . (5.18)

Here p = p/|p| is the unit vector in direction of p. Inserting γ = E/m and combining thetwo terms in the angular bracket, we arrive at

φR(p) =E +m+ σp√

2m(E +m)φR(0) . (5.19)

Similiarly, we find

φL(p) =E +m− σp√

2m(E +m)φL(0) . (5.20)

Thus the L and R spinors differ only by the sign of the helicity operator σp which measuresthe projection of the spin σ on the momentum p of the particle. For a particle at rest, thisdifference disappears and we set therefore φL(0) = φR(0). This allows us to eliminate thezero momentum spinors, giving

φRL(p) =E ± σpm

φLR(p) . (5.21)

In matrix form, these two equations correspond to(

−m E − σpE + σp −m

)(φL(p)φR(p)

)

= 0 . (5.22)

We introduce4 the 4× 4 matrices

γµ =

(0 σµ

σµ 0

)

. (5.23)

3Recall the relations E = m cosh η and p = m sinh η connecting E, p, and the rapidity η.4Some authors [Ryd96] use γi → −γi and exchange φL and φR in the four-spinor.

103


Then we arrive with γµpµ = γ0E − γp at the compact expression

(γµpµ −m)ψ(p) = 0 . (5.24)

Identifying pµ = i∂µ we obtain the Dirac equation.The representation used for the Dirac spinor and the gamma matrices is called chiral or

Weyl representation. Others can be obtained by a unitary transformation, UγµU † = γµ andUψ = ψ.

Dirac’s way towards the Dirac equation The Klein-Gordon equation was historically thefirst wave equation derived in relativistic quantum mechanics. Applied to the hydrogen atom,it failed to reproduce the correct energy spectrum. Dirac tried to derive as an alternativean equation linear in the derivatives ∂µ. Since Lorentz invariance requires that ∂µ has to becontracted with another four-vector, a first order equation has the form

(iγµ∂µ −m)ψ = 0 . (5.25)

Main task for Dirac was to uncover the nature of the quantities γµ in this equation. Theycannot be normal numbers, since then they would form a four-vector, specify one directionin space-time and thus break Lorentz invariance.

We will follow now Dirac’s approach, which will lead us to a definition of the gammamatrices and their properties which is independent of the considered representation. We startmultipling the Dirac equation with −(iγµ∂µ+m) and compare the result to the Klein-Gordonequation,

− (iγµ∂µ +m)(iγν∂ν −m)ψ = (γµγν∂µ∂ν +m2)ψ = (+m2)ψ = 0 . (5.26)

Using the symmetry of partial derivatives, we can rewrite

γµγν∂µ∂ν =1

2γµ, γν∂µ∂ν . (5.27)

Remembering next the definition of the d’Alembert operator, = ηµν∂µ∂ν , we see that theγµ satisfy

γµ, γν = 2ηµν . (5.28)

These anti-commutation relations define a Clifford algebra, implying

(γ0)2

= 1 ,(γi)2

= −1 and γµγν = −γνγµ (5.29)

for µ 6= ν. The last condition shows again that the γµ cannot be normal numbers. Note thatγ0 is hermetian, while the γi are anti-hermetian,

γ0 = γ0† and γi = −γi† . (5.30)

One can readily check that the γµ matrices in the Weyl representation satisfy these relations.The definition (5.28) implies that we can apply in the usual way the metric tensor to raise

or to lower the indices of the gamma matrices, γµ = ηµνγν . Thus we can write γν∂ν = γν∂

ν .Since the contraction of the gamma matrices γµ with a four-vector Aµ will appear frequently,we introduce the so-called Feynman slash,

A/ ≡ Aµγµ , (5.31)

104

5.2. Dirac equation

as useful shortcut. This notation also stresses that the gamma matrices γµ allow us to map afour-vector Aµ on an element A/ of the Clifford algebra which then can be applied on a spinorψ. Although we suppress the spinor indices, you should keep in mind that the matrices(γab)

µ carry both tensor and spinor indices. Thus the gamma matrices transform not onlyas a vector under Lorentz transformations, but additionally the two indices a and b shouldtransform according to the spinor representation of the Lorentz group.

Clifford algebra and bilinear quantities We now determine the minimal matrix represen-tation for the Clifford algebra defined by Eq. (5.28). Thus we should find the number ofindependent products that we can form out of the four gamma matrices. Five obvious ele-ments are the unit matrix 1 = (γ0)2 and the four gamma matrices γµ themselves. Because of(γµ)2 = ±1, the remaining products should consist of γµ matrices with different indices. Thusthe only product of four γµ matrices that we have to consider is γ0γ1γ2γ3. This combinationwill appear from now on very often and deserves therefore a special name. Including theimaginary unit to make it hermetian, we define

γ5 ≡ γ5 ≡ iγ0γ1γ2γ3 . (5.32)

Because the four gamma-matrices in γ5 anti-commute, we can rewrite its definition introduc-ing the completely anti-symmetric tensor εαβγδ in four dimensions as

γ5 =i

24εαβγδ γ

αγβγγγδ . (5.33)

This suggests that bilinear quantities containing one γ5 matrix transform as pseudo-tensors,i.e. change sign under a parity transformation x→ −x. Two important properties of the γ5

matrix are (γ5)2 = 1 and γµ, γ5 = 0.Next we consider products of three γµ matrices. For instance,

γ1γ2γ3 = γ0γ0︸︷︷︸

1

γ1γ2γ3 = −iγ0γ5 . (5.34)

Hence these products are equivalent to γµγ5, giving us four more basis elements.Finally, we are left with products of two γµ matrices. We start from the defining equa-

tion (5.28),γµγν = −γνγµ + 2ηµν , (5.35)

divide by two, add γµγν/2 and introduce the commutator of two gamma matrices,

γµγν =1

2γµγν − 1

2γνγµ + ηµν =

1

2[γµ, γν ] + ηµν . (5.36)

Adding again for later convenience an imaginary unit, we define the anti-symmetric tensorσµν as

σµν ≡ i

2[γµ, γν ] . (5.37)

The six matrices σµν are the remaining basis elements of the four-dimensional Clifford algebra.All together, the basis has dimension 16,

Γ = 1, γ5, γµ, γ5γµ, σµν , (5.38)

105


as the 4 × 4 matrices. Hence an arbitrary 4 × 4 matrix can be decomposed into a linearcombination of the 16 basis elements of the Clifford algebra. Moreover, the smallest matrixrepresentation of the Clifford algebra are 4 × 4 matrices. Some useful properties of gammamatrices are collected in the Appendix A.3.

Knowing the dimension of the γ matrices, we can count the number of degrees of freedomrepresented by a Dirac spinor ψ. As the γ matrices and the Lorentz transformation actingon spinors are complex, the field ψ is complex too and has thus four complex degrees offreedom. We know already that the Dirac equation describes spin 1/2 particles, which comewith 2s+1 = 2 spin degrees of freedom for a particle plus 2 for its anti-particle. Thus in thiscase the number of physical states matches the four components of the fields ψ. Note also thedifference to the case of a complex scalar or vector field: There we introduced two complexfields φ± = (φ1± iφ2)/

√2, which are connected by (φ±)∗ = φ∓. The real fields φ1 and φ2 are

not mixed by Lorentz transformations and thus we count two real degrees of freedom.We come now to the construction of bilinear quantities out of the Dirac spinors. It is

tempting to try e.g. as scalar ψ†ψ and as vector current jµ = ψ†γµψ. Using the chiralrepresentation, we find however

ψ†ψ =(

φ†L, φ†R

)( φLφR

)

= φ†LφL + φ†RφR ,

while we learnt thatφ†LφR + φ†LφR = ψ†γ0ψ

is Lorentz invariant. Similiarly, ψ†γµψ is neither Lorentz invariant nor real. The latterproblem arises, because γ0 is hermetian and the γi are anti-hermetian. We can express the(anti-) hermitian property of the gamma matrices succinctly as

γµ† = γ0γµγ0 =

(γ0)2γ0 = γ0 ,−γi(γ0)2 = −γi . (5.39)

Both problems are solved, if bilinear quantities are constructed as

ψ†γ0Γψ ≡ ψΓψ , (5.40)

where ψ = ψ†γ0 is called the adjoint spinor to ψ and Γ is any of the 16 basis elements givenin Eq. (5.38). In this way, the complex conjugated of a bilinear becomes

(ψΓψ′)∗ = (ψΓψ′)† = ψ′†Γ†γ0ψ = ψ′†γ0γ0Γ†γ0ψ = ψ′Γψ (5.41)

withΓ = γ0Γ†γ0 . (5.42)

The analogue ψ†ψ to the probability density ψ∗ψ of the Schrodinger equation is thus thezero-component of a four-current, ψ†ψ = ψ†γ0γ0ψ = ψγ0ψ = j0, as one should expect in arelativistic theory.

Solutions We search for plane wave solutions ue−ipx and ve+ipx of the Dirac equation withm > 0 and E = p0 = |p| > 0. The algebra is simplified, if we construct the solutions first inthe rest frame of the particle. Then p/ = mγ0, and thus the use of the Dirac representation,

γ0 = 1⊗ τ3 =(

1 00 −1

)

, and γi = σi ⊗ iτ2 =

(0 σi

−σi 0

)

, (5.43)

106

5.2. Dirac equation

γ5 = 1⊗ τ1 =(

0 11 0

)

, (5.44)

where γ0 is diagonal is most convenient. Here σi and τi are the Pauli matrices, ⊗ denotes thetensor product, 0 and 1 are 2× 2 matrices. The Dirac equation becomes

(p/ −m)u = m(γ0 − 1)u = 0 (5.45a)

(p/+m)v = m(γ0 + 1)v = 0 . (5.45b)

Since (γ0 ± 1)2 = 2(1± γ0), we see that (1± γ0)/2 are projection operators P±, satisfying

P 2± = P±, P±P∓ = 0, and P+ + P− = 1 .

Inserting γ0 into (5.45), the four solutions follow as

u(m,+) = N

1000

, u(m,−) = N

0100

, v(m,−) = N

0010

, v(m,+) = N

0001

.

(5.46)The additional s = ±1 label should be the quantum number of a suitable operator labellingthe two spin states of a Dirac particle. Note the opposite order of the spin label in the vspinor compared to u. We will see later that this choice is required by the structure of therelativistic spin operator sµ. As an intuitiv argument, we add that this labelling correspondsto our interpretation of antiparticles as particles moving backwards in time: The spinorv describes two states with negative energy, negative 3-momemtum p and negative spin srelative to u.

The solutions are orthogonal,

u(p, s)u(p, s′) = N 2δs,s′ and v(p, s)v(p, s′) = −N 2δs,s′ , (5.47)

but not normalised to one. Note also the minus sign introduced in vv by the correspondingminus in the (3,4) corner of γ0. Since we know that ψ†ψ is the zero component of a four-vector,the normalisation of the corresponding spinor products is

u†(p, s)u(p, s′) = N 2Epmδs,s′ and v†(p, s)v(p, s′) = −N 2Ep

mδs,s′ . (5.48)

Summing over spins, we obtain in the rest frame

∑

s

ua(m, s)ub(m, s) =

(1 00 0

)

ab

N 2 =1

2(γ0 + 1)abN 2 , (5.49)

∑

s

va(m, s)vb(m, s) =

(0 00 −1

)

ab

N 2 =1

2(γ0 − 1)abN 2 . (5.50)

We saw that γ0 ± 1 corresponds in an arbitrary frame to (p/ ±m)/m. Thus in general theserelations become

Λ+ ≡∑

s

ua(p, s)ub(p, s) = N 2

(p/+m

2m

)

ab

, (5.51)

Λ− ≡ −∑

s

va(p, s)vb(p, s) = N 2

(−p/+m

2m

)

ab

, (5.52)

107


where we defined Λ± as the projection operator on states with positive and negative energy,respectively.

The two most common normalisation conventions for the Dirac spinors are N =√2m

and N = 1. We will use the former, N =√2m, which has two advantages: First, spurious

singularities in the limit m → 0 disappear. Second, the normalisation of fermion states andthus also the phase space volume becomes identical to the one of bosons.

The solutions of the Dirac equation for an arbitrary frame can be simplest obtained re-membering (p/−m)(p/ +m) = p2 −m2 = 0, i.e.

u(p,±) = p/+m√

2m(m+ E)u(0,±) and v(p,±) = −p/+m

√

2m(m+ E)v(0,±) . (5.53)

Here, the normalisation was fixed using (5.47).

Spin We have seen that the Dirac equation describes particle with spin s = 1/2. Thus the ±degeneracy of the u and v spinors should correspond to the different spin states of a s = 1/2particle. We introduce the spin operator

Σ =

(σ 00 σ

)

, (5.54)

as an obvious generalisation of the non-relativistic spin matrices. This operator has theeigenvalues Σzu(m,±) = ±u(m,±) and Σzv(m,±) = ±v(m,±) and can therefore be used toclassify the spin states of a Dirac particle in the rest frame, where [HD,Σz] ∝ [γ0,Σz] = 0.Note however that [HD,Σz] 6= 0 for p2 6= m2, and thus the eigenvalue of Σz is not conservedfor a moving particle. This comes not as a surprise, because the total angular momentumL+ s and not only the spin s should be conserved.

We are looking now for the relativistic generalisation of the three-dimensional spin operatorΣ. It should be a product of gamma matrices which contains in the rest frame of the particleσi in the diagonal. We note first that γ5γi =

(0 11 0

)(0 σi

−σi 0

)has the required structure. Then

we define the spin vector sµ with the properties s2 = −1, sµ = (0, s)|p=m and thus s · p = 0.Since

γ5s/|p=m = −γ5siγi =(σs 00 −σs

)

, (5.55)

we see that γ5s/ measures in the rest frame the projection of the spin along the chosen axiss. Moreover, γ5s/ commutes with the Dirac Hamiltonian, [γ5s/, p/] = 0, and has, because of(γ5s/)2 = 1, as eigenvalues ±1. If we apply γ5s/ on the spinor v in the rest frame,

γ5s/ v(m, s) =

(σs 00 −σs

)(0χ

)

=

(0

−σsχ

)

= sj

(0χ

)

, (5.56)

we see that χ1 has the eigenvalue s1 = −1, while χ2 has the eigenvalue s2 = +1. Thisexplains the “wrong” order of the two spin states of v(p, s) in (5.46). Finally, we can definea projection operator on a definite spin state by

Λs =1

2(1 + γ5s/) . (5.57)

Thus we can obtain from an arbitrary Dirac spinor ψ a state with definite sign of the energyand spin by applying the two projection operators Λ± and Λs.

108

5.2. Dirac equation

Hamiltonian form The Dirac equation can be transformed into Hamiltonian form by mul-tiplying with γ0,

i∂tψ = HDψ = (−iγ0γi∂i + γ0m)ψ . (5.58)

Looking back at the (anti-) hermiticity properties (5.30) of the γµ matrices, we see that theycorrespond to the one required to make the Dirac Hamiltonian hermitian. By tradition, onere-writes HD often with β = γ0 and αi = γ0γi as

i∂tψ = HDψ = (α · p+ βm)ψ . (5.59)

Considering the semi-classical limit, one sees that the matrix α has the meaning of a velocityoperator, see problem ??.

Lagrange density For a scalar field, we could rewrite after a partial integration the Lagrangedensity as L = −φ†(+m2)φ. More generally, the definition of the propagator as two-pointfunction requires that the Lagrange density can be expressed as L ∝ ΨDΨ for a field Ψsatisfying DΨ = 0 with the differential operator D. This suggests to try for the Dirac field

L = ψ(iγµ∂µ −m)ψ (5.60)

with ψ and ψ as independent variables. The Lagrange equation for ψ gives trivially the Diracequation,

∂µδL

δ∂µψ− δL

δψ= (iγµ∂µ −m)ψ = 0 , (5.61)

while the Lagrange equation for ψ gives the adjoint equation for ψ.Using the gamma matrices and the Dirac spinor in the chiral representation it is straight-

forward to express the Dirac Lagrangian (5.60) by Weyl fields,

L = iφ†Rσµ∂µφR + iφ†Lσ

µ∂µφL −m(φ†LφR + φ†RφL) . (5.62)

This implies that the Dirac equation is invariant under Lorentz transformations, because wehave already checked that all ingredients of (5.62) are invariant. Note also that out of the twopossible combinations of the two Lorentz scalars we found, only the one invariant under parity,PφL = φR, entered the mass term. Moreover, P (σµ∂µ) = σµ∂µ, and thus the combination ofthe kinetic energies of φL and φR is also invariant under parity.

Keeping in mind that [L ] = m4, we see that the dimension of a fermion field in fourdimension is [ψ] = m3/2. Thus we can order possible couplings of a fermion to spin-1 particlesaccording to their dimension as

LI = c1Aµψγµψ + c2Aµψγ

5γµψ +c3MFµν ψσ

µνψ +c4MFµν ψγ

5σµνψ + . . . , (5.63)

where the coupling constants ci are dimensionless and we introduced the mass scale M .

Feynman propagator The Green functions of the Dirac equation are defined by

(i∂/ −m)S(x, x′) = δ(x − x′) . (5.64)

Translation invariance implies S(x, x′) = S(x−x′) and thus it is again convenient to performa Fourier transformation. Then the Fourier components S(p) have to obey

(p/−m)S(p) = 1 . (5.65)

109


After multiplication with p/+m, we can solve for the propagator in momentum space,

iSF (p) = ip/+m

p2 −m2 + iε=

i

p/−m+ iε, (5.66)

where the last step is only meant as a symbolical short-cut. Here, we chose again with the+iε prescription the causal or Stuckelberg-Feynman propagator for the electron and, moregenerally, for spin 1/2 particles. Note also the connection to the scalar propagator ∆F ,

iSF (x) = −(i∂/ −m)i∆F (x) . (5.67)

Example: Feynman propagator as sum over solutions:We follow the steps from (2.29) to (2.33) in the scalar case, finding now

SF (x) = i

∫d3p

(2π)3

∫dp02π

(p/ +m)e−ip0teipx

(p0 − Ep + iε)(p0 + Ep − iε)(5.68)

=

∫d3p

(2π)3

[

−ip/+m

2Epe−iEptϑ(x0) + i

−Epγ0 − pγ +m

−2EpeiEptϑ(−x0)

]

eipx . (5.69)

Now we change as in the bosonic case the integration variable as p→ −p in the second term,

iSF (x) =

∫d3p

2Ep(2π)3

[

(p/ +m)e−i(Ept−px)ϑ(t) + (−p/+m)ei(Ept−px)ϑ(−t)]

. (5.70)

Using finally (5.51), we arrive at

iSF (x) =

∫d3p

2Ep(2π)3

∑

s

[

u(p, s)u(p, s)e−i(Ept−px)ϑ(x0)− v(p, s)v(p, s)ei(Ept−px)ϑ(−x0)]

. (5.71)

We see that our choice for the normalisation of the Dirac spinors means that fermionic states arenormalised as bosonic states, 〈p|p′〉 = 2Ep(2π)

3δ(p− p′). The minus sign between the positive energysolution propagating forward in time and the negative energy solution propagating backward in timeis a direct consequence of our −iε prescription. This sign implies that fermionic fields anti-commute,and explains the Pauli exclusion principle and thus the stability of matter.Hence this sign is clearly one of the more important ones. Let us have therefore a look back tounderstand what causes ultimately the minus sign in the fermion propagator. To simplify the discusion,we neglect the inessential mass term. In the positive frequency term (p0γ0 − pγ)eipx, we pick up aminus from the residuum, p0 → −Ep, and a minus from the variable change, p → −p, resulting inp/→ −p/, which gives combined

SF (x) ∝ p/ e−ipxϑ(t)− p/ eipxϑ(−t) .

Thus the relative minus sign has its origin in the fact that the numerator of the fermion propagator is

odd in the momentum, while a bosonic propagator is even. In turn, the fermion propagator is linear

in the momentum, because the fermion wave equation is a first-order equation and thus the mass

dimension of the fermion field 3/2.

110

5.2. Dirac equation

Axial and vector U(1) symmetries Out of the 16 billinear forms, two transform as vectorsunder proper Lorentz transformations, jµ = ψγµψ and jµ5 = ψγ5γµψ. We now want to checkif these two currents are conserved. In the first case, we expect from non-relativistic quantummechanics that the global phase transformation,

ψ(x)→ ψ′(x) = eiφψ(x) and ψ(x)→ ψ′(x) = e−iφψ(x) , (5.72)

leads to the conserved current jµ = ψγµψ. Inspection of the Lagrange density shows imme-diately that δL = 0.

In the second case, the underlying symmetry is using γ5, γµ = 0,

ψ′(x)→ eiφγ5ψ(x) and ψ(x)→ ψ′(x) = (eiφγ

5ψ(x))†γ0 = ψ(x)eiφγ

5. (5.73)

The resulting (infinitesimal) change is

δL = 2miψγ5ψ . (5.74)

Thus the axial or chiral symmetry UA(1) is broken by the mass term, leading to the non-conservation of the current jµ5 .

Chirality To understand this better we re-express the Dirac Langrangian using eigenfunc-tions of γ5. We can split any solution ψ of the Dirac equation into

ψL =1

2(1− γ5)ψ ≡ PLψ and ψR =

1

2(1 + γ5)ψ ≡ PRψ . (5.75)

PL and PR are projection operators satisfying PL + PR = 1, P 2L/R = PL/R and PRPL =

PLPR = 0. Since γ5ψL = −ψL and γ5ψR = ψR, ψL,R are eigenfunctions of γ5 with eigenvalue±1. Expressing the mass term through these fields as

ψψ = ψ(P 2L + P 2

R

)ψ = ψ† (PRγ

0PL + PLγ0PR

)ψ = ψLψR + ψRψL (5.76)

and similiarly for the kinetic term,

ψ∂/ψ = ψ(P 2L + P 2

R

)∂/ψ = ψ† (PRγ

0γµPR + PLγ0γµPL

)∂µψ = ψL∂/ψL + ψR∂/ψR , (5.77)

the Dirac Lagrange density becomes

L = iψL∂/ψL + iψR∂/ψR −m(ψLψR + ψRψL) . (5.78)

Comparing this expression to (5.62), we see that we can identify the Dirac fields ψL/R in thechiral representation with the Weyl fields φL/R as follows,

ψL =

(φL0

)

and ψR =

(0φR

)

. (5.79)

Thus the projection operators (5.75) allows us to define the left- and right-chiral componentsof a Dirac field in an arbitrary representation.

The two kinetic terms which are invariant under chiral transformations connect left- to left-chiral and right- to right-chiral fields, while the mass term mixes left- and right-chiral fields.Such a mass term is called Dirac mass. The distinction between left- and right-chiral fields isLorentz invariant: In terms of Weyl spinors, we saw that the Lorentz transformations SL andSR do not mix φL and φR—which qualified them to form the irreducible representation of theLorentz group. In terms of Dirac spinors, the relation [γ5, σµν ] = 0 gurantees that left andright chiral fields transform separately under a Lorentz transformation, ψ′

L/R = S(Λ)ψL/R.However, the mass term of a massive Dirac particle will mix left- and right-chiral fields asthey evolve in time.

111


Helicity The helicity operator s · p/|p| measures the projection of the spin Σ/2 on themomentum p of a particle,

Σ · p2|p| ψ = hψ . (5.80)

The helicity operator and the Dirac Hamiltonian commute, [HD,Σ · p] = 0, because there isno orbital angular momentum in the direction of p. Therefore common eigenfunctions of HD

and h called helicity states can be constructed. Positive helicity particles are called right-handed, negative helicity particles are called left-handed. For a massive particle, helicity is aframe-dependent quantity: If we choose e.g. a frame with β‖p and β > p, then the particlesmoves in the opposite direction and h changes sign. Since we cannot “overtake” a masslessparticle, helicity becomes in this case a Lorentz invariant quantity.

Helicity and chirality eigenstates can be seen as complimentary states. The former one isa conserved, frame-dependent quantum number, while the latter is frame-independent, butnot conserved. Thus helicity states are e.g. useful to describe scattering processes wherethe detector measures spin in a definite frame. If on the other hand the interactions of afermion are spin-dependent, then one should choose chiral fields, since the Lagrangian shouldbe Lorentz invariant.

Lorentz transformations Our derivation of the Weyl spinors as the fundamental represen-tation of the Lorentz group provided automatically their transformation properties under afinite Lorentz transformation. Using the Weyl representation, the tranformation law for aDirac spinor follows as

ψ(x)→ ψ(x′) = S(Λ)ψ(x) =

(φL(x

′)φR(x

′)

)

=

(SL 00 SR

)(φL(x)φR(x)

)

. (5.81)

We want to express the transformation matrix S(Λ) by gamma matrices, such that it isrepresentation independent. We set

S(Λ) = exp (−iXµνωµν/4) , (5.82)

where the antisymmetric matrix ωµν parametrises the Lorentz transformation and the sixgenerators (Xab)µν have to be determined. Then we split ωµν into boosts and rotations,

1

2ωµνX

µν = ω0iX0i + ω12X

12 + ω13X13 + ω23X

23 . (5.83)

Parameterising the rotation by αi = εijkωjk and the boost by ηi = ω0i it follows that thegenerators Xµν coincide with the σµν matrices, which are given in the Weyl representationexplicitely as

σ0i = i

(−σi 00 σi

)

, (5.84)

and

σij = εijk

(σk 00 σk

)

. (5.85)

In contrast to (5.81), the epxression S(Λ) = exp (−iσµνωµν/4) is valid in any representationof the gamma matrices (see problem 5.5).

112

5.2. Dirac equation

Finally, we want to derive the transformation law of the gamma matrices. A straightforwardmethod is to use our knowledge that the Dirac equation is Lorentz invariant. The connectionbetween spinors in different frames is given by

ψ′(x′) = ψ′(Λx) = S(Λ)ψ(x) = S(Λ)ψ(Λ−1x′) (5.86)

andψ(x) = S(Λ−1)ψ′(x′) = S(Λ−1)ψ(Λx) . (5.87)

Inserting the underlined part of Eq. (5.86) for ψ′(x′) in Eq (5.87), we obtain the expectedresult

S−1(Λ) = S(Λ−1) . (5.88)

We insert ψ(x) = S−1(Λ)ψ′(x′) in the Dirac equation valid in the inertial system x andmultiply from the left with S(Λ),

[iS(Λ)γµS−1(Λ)∂µ −m

]ψ′(x′) = 0 . (5.89)

Next we use ∂µ = Λνµ∂′ν and obtain

[iS(Λ)γµS−1(Λ)Λνµ︸︷︷︸

γ′′ν

∂ ′ν −m

]ψ′(x′) = 0 . (5.90)

Hence the Dirac equation is form invariant under Lorentz transformations, if

Sab(Λ)γµbcS

−1cd (Λ)Λ

νµ = γ′νad (5.91)

orS(Λ)γµS−1(Λ) = Λ µ

ν γν . (5.92)

Since we have not used the condition det(Λ) = 1, this equation should hold both for discreteand proper Lorentz transformations. You are asked in problem 5.6 to confirm that (5.92)is true using the properites of the Clifford algebra which is done easiest for an infinitesimalLorentz transformation.

Example: Lorentz transformation properties of bilinear quantities:Because of (σµν )

† = − i2 [γ

ν†, γµ†] = γ0σµνγ0, we find S† = γ0S−1γ0. Hence under a Lorentz trans-

formation x′µ = Λµνx

ν , the spinor transforms as ψ′(x′) = S(Λ)ψ(x) and the adjoint Dirac spinoras

ψ′(x′) = ψ′†(x′)γ0 = ψ†(x)S†γ0 = ψ†(x)γ0S−1(γ0)2 = ψ(x)S−1 .

Thus ψψ transforms as a scalar under Lorentz transformations,

ψ′(x′)ψ(x′) = ψ(x)S−1Sψ(x) = ψ(x)ψ(x) ,

and using (5.92) the current

jµ(x′) = ψ′(x′)γµψ′(x′) = ψ(x)S−1γµSψ(x) = Λ µν j

ν(x)

as a four-vector.

Equation (5.91) illustrates nicely the transformation properties of the gamma matrices γµab:The index a transforms with S(Λ) as ψ, the index b transforms with S−1(Λ) as ψ, while theindex µ transforms with Λ.

113


Parity A process seen in a mirror is described by a coordinate system xµ → x′µ = (x0,−x).If the Dirac equation is mirror symmetric, or invariant under parity P, then both processesare possible solutions of it.

Since γ0 anti-commutes with γi and commutes with itself,

(γ0γ0 + γ0γi)ψ = (γ0 − γi)γ0ψ , (5.93)

we can change the derivative from ∂µ to ∂′µ,

γ0(iγµ∂µ −m)ψ(x) = (iγµ∂′µ −m) γ0ψ(x)︸︷︷︸

ψ′(x′)

= 0 (5.94)

after multiplication of the Dirac equation with γ0. More generally, we can allow for anunobservable phase factor η = eiφ, defining the action of the parity operation on a Diracspinor as

ψ′(x′) = Pψ(x) = ηγ0ψ(x) . (5.95)

The parity operation P applied to left and right chiral fields exchanges them,

PψL = ηγ01

2(1− γ5)ψ =

1

2(1 + γ5)ηγ0ψ = PRψ

′ , (5.96)

as we noticed already from P (SL) = SR discussing the fundamental representation of theLorentz group.

Example: Find the transformation property of ψ(x)ψ(x) and ψ(x)γ5ψ(x) under parity.With ψ(x′) = Pψ(x) = ηγ0ψ(x) and ψ(x′) = η∗ψ(x)γ0, we find

ψ′(x′)ψ′(x′) = ηη∗ψ(x)γ0γ0ψ(x) = ψ(x)ψ(x) .

andψ′(x′)γ5ψ′(x′) = ηη∗ψ(x)γ0γ5γ0ψ(x) = −ψ(x)γ5ψ(x) .

Thus the bilinear ψψ transforms as a scalar, while ψγ5ψ transforms as a pseudo-scalar.

Analogously, the axial vector current ψγµγ5ψ and the axial tensor current ψσµνγ5ψ transform odd

under parity. Hence such interaction terms can be omitted in theories that respect parity (like QED

and QCD), but have to be included in weak interactions.

Charge conjugation We already discussed possible couplings between a Dirac fermion andthe photon field. The only one with a dimensionless coupling e > 0 that also respects parityis

LI = −ejµAµ = −eψγµψAµ . (5.97)

Solving the Lagrange equations for L0 +LI gives the Dirac equation including a coupling tothe electromagnetic field as

[iγµ(∂µ + ieAµ)−m]ψ(x) = 0 . (5.98)

This corresponds to the “minimal coupling” prescription known from quantum mechanics.Having defined the coupling to an external electromagnetic field, we can ask ourselves how

the Dirac equation for a charged conjugated field ψc should look like. In the case of a scalar

114

5.3. Dirac, Weyl and Majorana fermions

particle, complex conjugation transformed a positively charged particle into a negative oneand vice versa. We try the same for the Dirac equation,

[−iγµ∗(∂µ − ieAµ)−m]ψ∗(x) = 0 . (5.99)

The matrix γµ∗ satisfies also the Clifford algebra. Hence we should find the unitary transfor-mation U−1γµU = −γµ∗ or setting U ≡ Cγ0

(Cγ0)−1γµCγ0 = −γµ∗ . (5.100)

If it exists, then the charge-conjugated field ψc ≡ Cγ0ψ∗ satisfies the Dirac equation withq = −e,

[iγµ(∂µ − ieAµ)−m]Cγ0ψ∗(x) = 0 . (5.101)

Explicit calculation shows that we may choose C = γ2γ0, see problem 5.8.

In the chiral representation, ψc = Cγ0ψ∗ = iγ2ψ∗ becomes

ψc =

(0 iσ2

−iσ2 0

)(φ∗Lφ∗R

)

=

(iσ2φ∗R−iσ2φ∗L

)

, (5.102)

which is in agreement with φcL = iσ2φ∗R and φcR = −iσ2φ∗L found earlier.

Example: Since the γ2 has the same from in the Dirac and the chiral representation, we findapplying C on the spinors u(p,±) and v(p,±) immediately that

uc(p, s) = Cγ0u∗(p, s) = iv(p, s)

vc(p, s) = Cγ0v∗(p, s) = −iu(p, s) .

From

iSF (x) =

∫d3p

(2π)32ωp

∑

s

[

u(p, s)u(p, s)e−i(Ept−px)ϑ(x0)− v(p, s)v(p, s)ei(Ept−px)ϑ(−x0)]

(5.103)

this implies that

StF (x) = CSt

F (−x)C−1 , (5.104)


Up to now we have discussed the Dirac equation, having in mind a massive particle carryinga conserved U(1) charge as e.g. lepton number that allows us to distinguish particles andanti-particles. We call such particles Dirac fermions. In this section we consider the casewhere one of these two conditions is not fullfilled. In the SM all particles except the neutrinoscarry an electric charge and are thus Dirac fermions.

115


Weyl fermions, q 6= 0 and m = 0: The (massive) Dirac equation in the chiral representa-tion (

−m i(∂0 + σ∇)i(∂0 − σ∇) −m

)(φL(p)χR(p)

)

= 0 (5.105)

decouples for m = 0 into two equations called Weyl equations,

i(∂0 − σ∇)φL(p) = 0 (5.106a)

i(∂0 + σ∇)χR(p) = 0 . (5.106b)

A fermion described by the Weyl equations is called a Weyl fermion. In particular, until the1990’s neutrino masses were experimentally consistent with zero and neutrinos were incorpo-rated into the SM as Weyl fermions.

For a plan-wave χR(p) = χRe−iεpx and p0 = |p| = E, we recover (5.22) in the limit m = 0,

(E − σp)χRe−iεpx = 0 . (5.107)

Thus the dispersion relation of a right-chiral Weyl fermion is E = σ|p| for both positive andnegative energy solutions, ε = ±1. Since σ has the eigenvalues ±1, only the solution withpositive helicity is allowed for χR. Remember also that helicity is frame independent fora massless particle; in this case positive helicity agrees with right chirality5. Thus a Weylfermion (m = 0, q 6= 0) has two degrees of freedom, consisting of the left-chiral 2-spinorχL(σ = −1) with negative helicity and the right-chiral 2-spinor χR(σ = +1) with positivehelicity.

Majorana fermions, m > 0 and q = 0: The Dirac field ψ has to be complex, because ittransforms under the complex representation S(Λ) of the Lorentz group. However a neutralfermion field ψM , where we cannot distinguish particles and antiparticles, should have onlyhalf of the degrees of freedom of a charged Dirac field. By analogy with the scalar case, weexpect that we can halve the number of degrees of freedom by imposing a reality condition,ψM = ψ∗

M . But this condition can be Lorentz invariant only in a special representation ofthe gamma matrices where S(Λ) is real. More generally, we halve the number of degrees offreedom by using a self-conjugated field ψc = ψ. A fermion described by a self-conjugatedfield ψM is called a Majorana fermion and the corresponding spinor a Majorana spinor.

We can replace any Dirac field ψD by a pair of self-conjugated fields,

ψM,1 =1√2(ψD + ψcD) , (5.108a)

ψM,2 =1√2(ψD − ψcD) . (5.108b)

and vice versa inverting these relations. Expressed by Weyl spinors, a Majorana spinorbecomes

ψc = ψ =

(iσ2φ∗L−iσ2φ∗L

)

. (5.109)

The Majorana 4-spinor is composed of only one Weyl spinor, which shows again that itdescribes 2 degrees of freedom. Thus a Majorana fermion (m > 0, q = 0) has two degrees of

5Most authors call ψL/R and φL/R not left and right-chiral but left and right-handed, although this identifi-cation holds only for massless particles.

116


freedom, consisting either of a left-chiral 2-spinor χR(σ) with positive and negative helicityor a right-chiral 2-spinor χR(σ) with both helicities.

The condition ψM = ψ∗M is Lorentz invariant only, if σ∗µν = −σµν . This defines the

Majorana representation of the γ matrices in which all γµ and thus σµν are imaginary, thecharge conjugation matrix C is the unity matrix, C = 1, and Majorana spinors can bechosen to be real. Since the spinors are real in this representation, no phase invarianceψ(x) → ψ′(x) = exp(iφ)ψ(x) as in (5.72) can be implemented for a Majorana fermion6 andthus they cannot carry any conserved U(1) charge.

Dirac versus Majorana mass terms Charge conjuated spinors were defined by

ψc = Cγ0ψ∗ = Cψt, ψc = ψtC .

We define

ψcL ≡ (ψL)c =

1

2(1 + γ5)ψc = (ψc)R ,

which is consistent with our previous definition for Weyl spinors.As we saw, a Dirac mass term

−LD = mDψψ = mD(ψLψR + ψRψL)

connects L and R components of the same field and ψ = ψL+ψR is a mass eigenstate. We nowuse the observation that (ψL)

c = (ψc)R allows us to obtain new mass terms7 called Majoranamass terms,

−LL = mL(ψcLψL + ψLψ

cL)

−LR = mR(ψcRψR + ψRψ

cR)

which connect the L and R components of conjugated fields.The mass eigenstates χ and ω are the self-conjugated fields

χ = ψL + ψcL = χc

ω = ψR + ψcR = ωc

with−LL = mLχχ, −LR = mRωω

In the general case we expect that both Dirac and Majorana mass terms are present

−LDM = mDψLψR +mLψcLψL +mRψ

cRψR + h.c.

=1

2mD(χω + ωχ) +mLχχ+mRωω

or

−LDM = (χ, ω)

(mL mD/2mD/2 mR

)(χω

)

6Note that this argument does not forbid that a Majorana fermion carries conserved charges which transformunder a real representation of a summetry group: An example are gluinos, the supersymmetric partners ofthe gluons, which are Majorana fermions and transform under the adjoint representation of SU(3).

7Note that the terms ψcLψL = ψt

LCψL, etc., vanish because of Ct = −C, if one does not already assumes onthe classical level that fields are anticommuting variables.

117


The eigenvalues of this matrix are

m1,2 =1

2

(mL +mR)±√

(mL −mR)2 +m2D

and the eigenvectors

η1 = cos ϑχ− sinϑω

η2 = sinϑχ+ cos ϑω

with tan 2ϑ = mD/(mL −mR).

Seesaw model The seesaw model tries to explain why neutrinos have much smaller massesthan all other particles in the standard model. Let us assume that there exist both left- andright-chiral neutrinos and that they obtain Dirac masses as the other fermions, say of ordermD ∼ 100GeV. The right-chiral νR does not participate in any SM interaction and suffersthe same fate as a scalar particle: its mass will be driven by quantum corrections to a valueclose to the validity of the effective theory (the SM) we are using, so mR ≫ mD. Expanding

m1,2 ≈1

2

mR ±mR

√

1 +m2D/m

2R

the two eigenvalues are m1 ≈ m2D/(2mR) and m2 ≈ mR. For mR ∼ 1014 GeV, we find light

neutrino mass in the eV or sub-eV range, as required by experimental data.

5.4. Spin-statistic connection and Graßmann variables

Spin-statistic connection We have noted at several places that fermionic fields should anti-commute: Expressing the Feynman propagator as sum of solutions or as time-ordered vacuumexpectation value of the fields, and writing down a Majorana mass term required that thequadratic form ψaψb is antisymmetric. In a relativistic quantum field theory the spin and thestatistics of a field is coupled:

• The wave equations of bosons are second-order differential equations. Therefore bosonicfields have mass dimension 1 and their propagators ∆(p) are even in the momentum p.As a result, bosonic fields commute and satisfy Bose-Einstein statistics.

• In contrast, fermions satisfy first-order differential equations and have mass dimension3/2. Therefore the fermion propagator SF (p) is linear in p and SF (x) is an antisym-metric function. This implies that fermions satisfy Fermion-Dirac statistics and aredecribed by anti-commuting classical spinors or operators.

This leads to a practical and a principal question: First, the practical one: How do weimplement that classical functions which enter the path integral do anti-commute? Andsecond, does the anticommutation of fermionic variables lead to a consistent picture? Inparticular is the Hamiltonian of such a theory bounded from below?

We will start to address the latter question calculating the energy density ρ = H of theDirac field assuming that the classical spinors u and v do commute. We determine first thecanonically conjugated momenta as

π =∂L

∂ψ= iψγ0 = iψ† (5.110)

118


and π = 0. Thus the Hamilton density is

H = πψ −L = iψ†∂tψ − ψ(iγµ∂µ −m)ψ = iψ†∂tψ , (5.111)

where we used the Dirac equation in the last step. To make this expression more explicit, weexpress now ψ by plane wave solutions,

ψ(x) =∑

s

∫d3p

√

(2π)32Ep

[

bs(p)us(p)e−ipx + d†s(p)vs(p)e

+ipx]

, (5.112a)

ψ†(x) =∑

s

∫d3p

√

(2π)32Ep

[

b†s(p)u†s(p)e

+ipx + ds(p)v†s(p)e

−ipx]

. (5.112b)

Inserting these expressions into (5.111) gives schematically (b†+d)(b−d†), where the relativeminus sign comes from ∂t acting on ψ. Since the spinors u and v are orthonormal, cf. (5.48),only the diagonal terms survive, (b† + d)(b − d†) → b†b − dd†. Hence the energy of a Diracfield is

H =

∫

d3x H =∑

s

∫

d3pEp

[

b†s(p)bs(p)− ds(p)d†s(p)]

. (5.113)

If d and d† would be normal Fourier coefficients of an expansion into plan waves, the secondterm would be negative and the energy density of a fermion field could be made arbitrarilynegative.

This conclusion is avoided if the fermion fields anticommute: In canonical quantisation, onepromotes the Fourier coefficients to operators. Requiring then anti-commutation relationsbetween the creation and annihilation operators,

ds(p), d†s′(p′) = δs,s′δ(p − p′) , (5.114)

compensates the sign in the second term.

If we restore units, then we have to add a factor ~ on the RHS. Thus the RHS vanishesin the classical limit, ~→ 0, which implies that classical spinors should anti-commute. Thuswe should perform the path integral of fermionic fields over anti-commuting numbers whichare called Graßmann numbers. Either way, we find the amazing result that in a relativisticquantum field theory the spin determines the statistics in such a way the Hamiltonian isbounded from below.

Graßmann variables We now proceed to the practical question how the mathematics ofanticommuting numbers works. We define a Graßmann algebra G requiring that for a, b ∈ Gthe anticommutation rules

a, a = a, b = b, b = 0 (5.115)

and thus a2 = b2 = 0 are valid. Then any smooth function f of a and b can be expanded intopower-series as

f(a, b) = f0 + f1a+ f1b+ f2ab

= f0 + f1a+ f1b− f2ba . (5.116)

119


Defining the derivative as acting to the right, ∂ ≡→∂ , it is

∂f

∂a= f1 + f2b (5.117)

∂f

∂b= f1 − f2a (5.118)

and∂2f

∂a∂b= − ∂2f

∂b∂a= −f2 . (5.119)

Defining integration for Graßmann variables, we require linearity and that the infinitesimalsda, db are also Graßmann variables,

⇒a,da = b,db = a,db = da, b = da,db = 0 . (5.120)

Multiple integrals are iterated,

∫

dadbf(a, b) =

∫

da

(∫

dbf(a, b)

)

. (5.121)

We have to determine the value of∫da and

∫daa. For the first, we write

(∫

da

)2

=

(∫

da

)(∫

db

)

=

∫

dadb = −∫

dbda = −(∫

da

)2

(5.122)

and thus∫da = 0. We are left with

∫daa: Since there is no intrinsic scale—states are empty

or occupied—we are free to set ∫

daa = 1 . (5.123)

This implies also that there is no difference between definite and indefinite integrals forGraßmann variables. Moreover, differentiation and integration is equivalent for Graßmannvariables.

Assume now that η1 and η2 are real Graßmann variables and A ∈ R. Then∫

dη1dη2 eη2Aη1 =

∫

dη1dη2 (1 + η2Aη1) =

∫

dη1dη2 η2Aη1 =

∫

dη1 Aη1 = A . (5.124)

Next we consider a two-dimensional integral with an anti-symmetric matrix A and η = (η1, η2).Then

A =

(0 a−a 0

)

(5.125)

and ηTAη = 2aη1η2. Using an arbitrary matrix would lead to the same result, since itssymmetric part cancels. Expanding again the exponential gives

∫

d2η exp

(1

2ηTAη

)

= a = (det(A))1/2. (5.126)

One can show that an arbitrary antisymmetric matrix can be transformed into block diagonalform, where the diagonal is composed of matrices of the type (5.125). Thus the last formulaholds for arbitrary n.

120


Finally, we introduce complex Grassmann variables η = (η1, . . . , ηn) and their complexconjugates η = (η1, . . . , ηn). For any complex matrix A,

∫

dnηdnη exp(

η†Aη)

=

n∏

i=1

ai = det(A) . (5.127)

We can compare this to the result over commuting complex variables, zi = (xi+ iyi)/√2 and

zi = (xi − iyi)/√2, with dxdy = dzdz∗ and

∫

dnzdnz∗ exp(

−z†Az)

=(2π)n

det(A). (5.128)

Thus for Graßmann variables the determinant appearing in the evaluation of a Gaussianintegral is in the nominator, while it is in the denominator for real or complex valued functions.

Path integral for fermions The path integral for a free Dirac fermion without externalsources is

Z[0] =

∫

DψDψ eiS[ψ,ψ] =

∫

DψDψ ei∫d4xψ(i∂/−m)ψ . (5.129)

For its evaluation, we use (5.127) in the limit n → ∞. Since it is quadratic in the fields, wecan perform it formally,

Z[0] = det(i∂/ −m) = exp tr ln(i∂/ −m) . (5.130)

Using the cyclic property of the trace, we write

tr ln(i∂/ −m) = tr ln γ5(i∂/ −m)γ5 = tr ln(−i∂/ −m) =

=1

2[tr ln(i∂/ −m) + tr ln(−i∂/ −m)] =

1

2[tr ln( +m2)] . (5.131)

Thus Z[0] = exp[+tr ln(+m2)/2]: We have found the remarkable result that the zero-pointenergy of fermions has the opposite sign compared to the one of bosons. We arrive at thesame conclusion, using anti-commutation relations

ds(p), d†s′(p′) = δs,s′δ(p − p′) (5.132)

in Eq. (5.113),

H =∑

s

∫

d3pEp

[

b†b+ d†d− δ(3)(0)]

. (5.133)

With δ(3)(0) =∫d3x/(2π)3 we see that the last term corresponds to the negative zero-point

energies of a fermion.Note that this opens the possibility that the zero-point energies of (groups of) bosons and

fermions cancel exactly, provided that i) the degrees of freedom of fermions and bosons agree.(For instance, the trace in Eq. (5.131) includes the trace over the 4x4 matrix in spinor space,leading to a factor four larger results than for a single scalar.) ii) Their masses are the same,mf = mb. iii) Their interactions match, so that also higher-order corrections are identicalfor fermions and bosons. The corresponding symmetry that guarantees automatically thatthe conditions i)-iii) are satisfied is called “supersymmetry.” In an unbroken supersymmetrictheory, the cosmological constant would be zero.

Clearly, condition ii) is most problematic, since e.g. no bosonic partner of the electron hasbeen found (yet). Hence supersymmetry must be a broken symmetry, but as long as the massspitting m2

f −m2b between fermions and bosons is not too large, it might be still “useful.”

121


Feynman rules Next we add Graßmannian sources η and η to the action, S[ψ, ψ]+ ηψ+ ψη.Then we have to complete the square,

ψAψ + ηψ + ψη = (ψ + ηA−1)A(ψ +A−1η)− ηA−1η , (5.134)

obtaining

Z[η, η] =

∫

DψDψ ei∫d4xψAψ+ηψ+ψη = Z[0] e−iηA−1η

= Z[0] exp

(

−i∫

d4x d4x′ η(x)SF (x− x′)η(x′))

. (5.135)

Here, A−1(x, x′) = SF (x− x′) = −STF (x′ − x) which corresponds to the fact that the matrixA is antisymmetric.

The propagator of a Dirac fermion is a line with an arrow representing the flow of theconserved charge which distinguishes particles and anti-particles. Thus a fermion line cannotsplit, and the arrow cannot change direction. In contrast, the arrow along a Majorana fermionline can change direction, since there is no difference between a particle and anti-particle inthis case.

= SF (p) =i

p/−m+ iε

We look at possible interaction terms of a Dirac fermion with scalars and photons, restrict-ing ourselves to dimensionless coupling constants,

LI = −gsSψψ − gaPψγ5ψ − eψγµψAµ . (5.136)

Both interaction terms of the fermion with the scalars respect parity, if the field S is atrue scalar and the field P a pseudo-scalar. Analogous to the −iλ coupling in the case of ascalar self-interactions, we read off from the Lagrangian the following interaction vertices inmomentum space,

S−igs

P−igaγ5

ǫµ−ieγµ

Fermion loops A closed fermion loop with n propagators corresponds to

ψ†(x1)ψ(x1)ψ†(x2)ψ(x2)...ψ

†(xn)ψ(xn)

Thus we have to anticommute ψ†(x1) and ψ(xn), generating a minus sign. Another wayto understand the minus sign of fermion loops is to look at the generating functional forconnected graphs setting the sources to zero, iW [0] = lnZ[0] = ln detA. The generated

122


graphs are single-closed loops with n Feynman propagators. The change from 1/detA in Zfor bosonic fields to detA in Z for fermionic fields implies an additional minus sign for closedfermion loops.

More formally, we can compare a complex bosonic field φ and fermionic field ψ coupled toan external field σ. In the bosonic case, we find

Z[σ] =

∫

DφDφ∗ ei∫d4xφ∗(B−σ)φ =

1

det(B − σ) (5.137)

with B = −(−m2), while we obtain for fermions

Z[σ] =

∫

DψDψ ei∫d4xψ(A−σ)ψ = det(A− σ) (5.138)

with A = i∂/ −m. We rewrite now the generating functional for connected graphs for zerosources as

iW [σ] = − ln det(B − σ) = −Tr ln(B − σ) = −Tr ln(1−B−1σ) + const. (5.139)

=∞∑

n=1

1

nTr(B−1σ)n + const. (5.140)

Evaluating the trace in the coordinate space results in

Tr(B−1σ) =

∫

d4x〈x|B−1σ|x〉 =∫

d4xi∆F (0)σ(x) (5.141a)

Tr(B−1σ)2 =

∫

d4xd4y〈x|B−1σ|y〉〈y|B−1σ|x〉

=

∫

d4xd4yi∆F (x− y)σ(y)i∆F (y − x)σ(x) . (5.141b)

Tr(B−1σ)3 = . . .

Thus the expansion generates the closed single loops with n propagators and symmetry factor1/n!. The same holds for the fermionic case, except that we obtain an additional overall minussign since we start from iW [σ] = ln det(A− σ).

Furry’s theorem What is the relation between diagrams containing fermion loops with op-posite orientation in QED? A fermion loop with n external photons attached corresponds toa trace over n fermion propagators separated by gamma matrices,

G1 = tr[γµ1SF (y1, yn)γµnSF (y2, y3) · · · γµ2SF (y2, y1)] . (5.142)

If we insert CC−1 = 1 between all factors in the trace, use CγµC−1 = −γµT andCSF (−x)C−1 = STF (x), then

G1 = (−1)ntr[γTµ1STF (yn, y1)γTµnSTF (y3, y2) · · · γTµ2STF (y1, y2)] (5.143)

= (−1)ntr[γµ1SF (y1, y2) · · · γµnSF (yn, y1)] = (−1)nG2 , (5.144)

where we used BTAT = (AB)T in the last step. Except for the factor (−1)n, the lastexpression corresponds to the loop G2 with opposite orientation. Hence for an odd number

123


x2

x1

x3

x4

y2

y1

y3

y4<

>

x2

x1

x3

x4

y2

y1

y3

y4>

<

Figure 5.1.: Two topologically distinct diagrams contributing to the photon four-point func-tion.

of propagators, the two contributions cancel, while they are equal for an even number ofpropagators. Alternatively, we can use a symmetry argument to convince us that all diagramswith an odd number of photons are zero in vacuum: Because the QED Langrangian is invariantunder charge conjugation, but ψ → ψc = CψT and Aµ → Acµ = −Aµ, all Green functionswith an odd number of photons have to vanish.

Symmetry factors in QED The last issue we want to address in this chapter is the questionif the interactions in (5.136) lead to symmetry factors. We recall that drawing Feynmandiagrams we should include only diagrams which are topologically distinct after the integrationover internal coordinates. For instance, the two diagrams

x1 x2 x1 x2

are not topologically distinct, because a rotation around the x1–x2 axis interchanges them.Therefore, the corresponding two-point function G(x1, x2) has to be invariant under a changeof the orientation of the fermion loop – as it is guarantied by the Furry theorem. The two-pointfunction G(x1, x2) consists of two identical diagrams obtained by exchanging the integrationvariables y1 and y2. Thus the factor 2! compensates the 1/2! from the Taylor expansion ofexp(−Lint), if we draw only one diagram, and its symmetry factor is one.

Next we consider the four-point function G(x1, x2, x3, x4) which describes photon-photonscattering. Two diagrams describing this process are shown in Fig. 5.1. We can specify thesediagrams by the order of the external lines connected to the fermion loop: Then the diagramon the LHS is denoted by (1, 2, 3, 4)+ , where the subscript specifies the orientation of theloop, while the diagram on the RHS is given by (1, 4, 3, 2)−. A rotation around the x1–x3axis converts (1, 2, 3, 4)+ into (1, 2, 3, 4)− and hence the orientation of the fermion loop playsagain no role. However, now topologically distinct diagrams exist, as we can not transformsmoothly the diagram (1, 2, 3, 4)+ into (1, 4, 3, 2)− .

The four-point function G(x1, x2, x3, x4) contains 4!×3! diagrams, obtained by permutatingy1, y2, y3, y4 and x2, x3, x4. After integration over the free yi variables, the factor 4! compen-sates the 1/4! from the Taylor expansion of exp(−Lint). In configuration space, the 3! = 6topologically distinct diagrams shown in Fig. 5.2 remain which carry no additional symmetryfactor. The upper diagrams are permutation of the LHS diagram, the lower three diagramsof the RHS diagram.

Thus the resulting rule for QED is very simple: We do not need symmetry factors, if we drawall diagrams which are toplogically distinct after the integration over internal coordinates.

124


x2

x1

x3

x4

<

>x2

x1

x4

x3

<

>x3

x1

x2

x4

<

>

x3

x1

x4

x2

<

>x4

x1

x2

x3

<

>x4

x1

x3

x2

<

>

Figure 5.2.: The six topologically distinct diagrams contributing to the photon four-pointfunction.

Fermion loops with an odd number of fermions are zero and can be omitted. Independent ofthe type of interaction, any fermion loop leads to an additional minus sign.

Summary

The fundamental representation of the orthochronous Lorentz group for massive particles isgiven by left and right-chiral Weyl spinors. These two two-spinors are mixed by parity andthus one combines them into a Dirac four-spinor for parity conserving theories like electro-magnetic and strong interactions.

Fermions satisfy first-order differential equations and have mass dimension 3/2. Thereforethe fermion propagator SF (p) is linear in p and thus SF (x) is an antisymmetric function in x.As a result, fermions satisfy Fermion-Dirac statistics and are decribed either by Grassmannvariables or by anti-commuting operators.

A Weyl fermion has m = 0 and q 6= 0 and satisfies the Weyl equation; its solution hastwo degrees of freedom, a left-chiral field with positive helicity and a right-chiral field withnegative helicity. A Majorana fermion is a self-conjugated field with m 6= 0 and q = 0 whichhas therefore also only two degrees of freedom. It is decribed fully either by a left-chiral or aright-chiral 2-spinor χR(σ) with both helicities. In the Majorana representation, the spinorcan be chosen to be real.

Further reading

The symmetries of the Dirac equation as well as of other relativistic wave equations areextensively discussed in W. Greiner Relativistic Quantum Mechanics. Two-component Weyland Mojorana spinor are discussed in more detail in Srednicki who introduces also the notationof dotted and undotted spinor indices.

125


Problems

5.1 Lie algebra of the Poincare group.Derive the commutation relation (5.1) plus themissing ones involving translation using that the(hermetian) generator T a are connected to theKilling fields V a by iT a = V a. Rewrite the ex-pressions in an Lorentz invariant form.

5.2 Pauli equation.Linearise the Schrodinger equation following thelogic in Sec. 5.1 and show that the Pauli equationfollows.

5.3 gs-factor of the electron.Square the Dirac equation with an external fieldAµ and show that the spin of the electron leads tothe additional interaction term σµνF

µν comparedto the Klein-Gordon equation. Show that theDirac equation predicts for the interaction of non-relativistic electron with a static magnetic fieldHint = (L+ 2s)eB, i.e. a gs equal to two for theelectron.

5.4 Spin vector.i) Show that the Pauli-Lubanski spin vector

(3.25), setting Jµν = σµν , becomes in the rest-frame

Wi = −mΣi . (5.145)

ii) Show thatWµsµ can be used as relativistic spin

operator.

5.5 Condition (5.82).Show that S(Λ) = exp (−iσµνωµν/4) is valid inany representation of the gamma matrices.

5.6 Condition (5.92).Show that (5.92) is true for an infinitesimal

Lorentz transformation.

5.7 Rotation of Dirac spinor.Introducing the matrix (IX)νµ, parametrise asS(Λ) = exp (−iωIµνn σµν/4). Using [(IX)νµ]

3 =(IX)νµ and show that...

5.8 Properties of C.a.) Prove the following properties of the chargeconjugation matrix C, CT = −C and C−1γµC =−γTµ . b.) Show that iγ2γ0 has the required prop-erties.

5.9 Majorana flip properties ♣.Derive the properties of Majorana bilinears

ξΓη = ±ηΓξ under an exchange ξ ↔ η.

5.10 Fermionic vacuum energy.Calculate the contribution of a Dirac fermion

to the vacuum enrgy density, following the steps(2.123) to (2.128) for a scalar.

126

6. Scattering processes

Most information about the properties of fundamental interactions and particles is obtainedfrom scattering experiments. In a scattering process, the initial and final state contains widelyseparated particles which can be treated as free, real particles which are on mass-shell. Bycontrast, n-point Green functions describe the propagation of virtual particles. In order tomake contact with experiments, we have to find therefore the link between Green functions andexperimental results from scattering experiments. We introduce first the scattering matrixS which is an unitary operator mapping the initial state at t = −∞ on the final stateat t = +∞. The unitarity of the S-matrix restricts the analytic structure of Feynmanamplitudes, in particular it implies the optical theorem. Then we derive the connectionbetween n-point Green functions and scattering amplitudes, justifying our Feynman rules forFeynman amplitudes, before we perform some explicit calculations of few tree-level processes.Finally, we consider the emission of additional soft particles and show that this process canbe described by an universal factor.

The relation between Feynman amplitudes and cross sections or decay widths is essentiallythe same as in non-relativistic quantum mechanics; it is reviewed in appendix 6.A.

6.1. Unitarity of the S-matrix and its consequences

A scattering process is fully described in the Schrodinger picture by the knowledge how initialstates |i, t〉 at t → −∞ are transformed into final states |f, t〉 at t → ∞. This knowledge isencoded in the S-matrix elements

|f, t =∞〉 = Sfi |i, t = −∞〉 . (6.1)

An intuitive, but mathematically delicate definition of the scattering operator S is the t→∞limit of the time-evolution operator U(t,−t)

S = limt→∞

U(t,−t) . (6.2)

Thus the scattering operator S evolves an eigenstate |n, t〉 of the Hamiltonian from t = −∞to t = +∞,

S |n,−∞〉 = |n,∞〉 . (6.3)

The unitarity of the scattering operator, S†S = SS† = 1, expresses the fact that we (should)use a complete set of states for the initial and final states in a scattering process,

1 =∑

n

|n,+∞〉〈n,+∞| =∑

n

S |n,−∞〉〈n,−∞|S† = SS† (6.4)

=∑

n

|n,−∞〉〈n,−∞| =∑

n

S† |n,+∞〉〈n,+∞|S = S†S . (6.5)

127


Optical theorem We split the scattering operator S into a diagonal part and the transitionoperator T , S = 1 + iT , and thus

1 = (1 + iT )(1− iT †) = 1 + i(T − T †) + TT † (6.6)

oriTT † = T − T † . (6.7)

Note that in perturbation theory the LHS is O(g2n), while the RHS is O(gn). Hence thisequation implies a non-linear relation between the transition operator evaluated at differentorder. At lowest order perturbation theory, the LHS vanishes and T is real, T = T †. Thiscorresponds to our observation that 2-point functions are imaginary for physical values of theenergy.

We consider now matrix elements between the initial and final state,

〈f |T − T † |i〉 = Tfi − T ∗if = i 〈f |TT † |i〉 = i

∑

n

TfnT∗in . (6.8)

If we set |i〉 = |f〉, we obtain a connection between the forward scattering amplitude Tii andthe total cross section σtot called the optical theorem,

2ℑTii =∑

n

|Tin|2 . (6.9)

The optical theorem informs us that the attenuation of a beam of particles in the state i,dNi ∝ −|ℑTii|2Ni, equals the total probabiilty that they scatter into states n.

The RHS is given by the total cross section σtot up to a factor depending on the flux ofinitial particles and possible symmetry factors. For the case of two particles in the initialstate, comparison with (6.172,6.173) shows that

ℑAii =1

2pcms√sσtot . (6.10)

Note also that the forward scattering amplitude Tii means scattering without change in anyconserved quantum number, since we have separated already the identity part, Tii = (Sii −1)/i.

Imaginary part of the amplitude Consider the Feynman amplitude A as a complex functionof the squared center-mass energy s. The threshold energy

√s0 in the cms equals the minimal

energy for which the reaction is kinematically allowed. For s < s0 and s ∈ R, s and, becauseof the optical theorem, also A are real, s = s∗ and A(s) = [A(s)]∗. Thus

A(s) = [A(s∗)]∗ for s < s0 . (6.11)

If A(s) is an analytic function, then also the RHS is analytic and we can continue the relationinto the complex s plane. In particular, along the real axis for s > s0

ℜA(s+ iε) = ℜA(s− iε) and ℑA(s+ iε) = −ℑA(s− iε) . (6.12)

Thus starting from s0, the amplitudeM has a branch cut along the real s axis. How does animaginary part in a Feynman diagram arise? From

1

x± iε= P

(1

x

)

∓ iπδ(x) , (6.13)

128

6.2. LSZ reduction formula

we see that virtual particles which propagate on-shell lead to poles and to imaginary termsin the amplitude.

The second relation in (6.12) allows us to obtain the imaginary part of a Feynman amplitudecalculating its discontinuity,

disc(A) ≡ A(s+ iε)−A(s− iε) = 2ℑA(s+ iε) . (6.14)

The prototype of a function having a discontinuity and a branch cut (along R−) is the loga-rithm,

Ln(z) = Ln(reiϑ) = ln(reiϑ) + 2kπi = ln(r) + (ϑ+ 2kπ)i (6.15)

with ℑ ln(x+ iε) = −π.

Example: Verify the optical theorem for φφ→ φφ scattering at O(λ2):The logarithmic terms in the scattering amplitude (2.147) for φφ → φφ scattering at one-loop havethe form

f(q2,m) =

∫ 1

0

dz ln[m2 − q2z(1− z)

](6.16)

with q2 = s, t, u. In the physical region, the relation q2 > 4m2 holds only for the s chanel diagram.The argument of the logarithm becomes negative for

x1/2 =1±

√

1− 4m2/s2

2=

1

2± 1

2β (6.17)

with β =√

1− 4m2/s2. Using now Im[ln(−x± iε)] = ±π, the imaginary part follows as

ℑ(A) = πλ2

32π2

∫ 1

2+ 1

2β

1

2− 1

2β

dx =λ2

32πβ . (6.18)

The optical theorem implies thus that the total cross section σtot(φφ→ all) at O(λ2) equals

σtot =ℑAii

2pcms√s=

λ2

32πs, (6.19)

where we used 2pcms =√sβ. On the other hand, the Feynman amplitude at tree level is simply A = λ

and thus the elastic cross section for φφ → φφ scattering follows as σel = λ2/(32πs). At O(λ2), theonly reaction contributing to the total cross section is elastic scattering, and thus the two cross sections

agree.

Note also the treatement of the symmetry factors: In the loop diagram, the symmetry factor S = 1/2!

is already included, while the corresponding factor for the two identical particles in the final state is

added only integrating the cross section.


We defined the generating functional Z[J ] as the vacuum-vacuum transition amplitude in thepresence of a classical source J . Thus the path integral

〈0,∞|0,−∞〉J = Z[J ] =

∫

Dφ exp i∫

Ωd4x

(1

2∂µφ∂

µφ− 1

2m2φ2 + Jφ

)

(6.20)

129


contains the boundary condition φ(x) → 0 for t → ±∞. We can either try to connect theGreen functions derived from Z[J ] to S-matrix elements or define a new functional with theappropriate boundary conditions. We choose the first way, restricting ourselves for simplicityto the case of a real scalar field.

Let us start with the case of a 2→ 2 scattering process. We can generate a free two-particlestate composed of plane-waves by

|k1,k2〉 = a†(k1)a†(k2) |0〉 . (6.21)

We obtain localized wave packets defining new creation operators with

a†1 =∫

d3k f1(k)a†(k) , (6.22)

where f1(k) is e.g. a Gaussian centered around k1,

f1(k) ∝ exp[−(k − k1)2/(2σ2)] . (6.23)

We assume that our initial state at t = −∞ can be described by freely propagating wavepackets,

|i〉 = limt→−∞

a†1(t)a†2(t) |0〉 = |k1,k2;−∞〉 , (6.24)

and similarly the final state as

|f〉 = limt→∞

a†1′(t)a†2′(t) |0〉 = |k1′ ,k2′ ; +∞〉 . (6.25)

Green functions are the time-ordered vacuum expectation value of field operators. The firstproperty, time-ordering, is automatically satisfied for the transition amplitude 〈f |i〉, since wecan write

〈f |i〉 = limt→∞

〈0| a1′(t)a2′(t)a†1(−t)a†2(−t) |0〉

= limt→∞

〈0|Ta1′(t)a2′(t)a†1(−t)a†2(−t) |0〉 . (6.26)

Thus we only have to re-express the creation and annihilation operators as (projected) fieldoperators. We define a scalar product for solutions of the Klein-Gordon equation as follows,

(φ, χ) = i

∫

d3x φ∗(x)←→∂0χ(x) ≡ i

∫

d3x

[

φ∗(x)∂χ(x)

∂t− ∂φ∗(x)

∂tχ(x)

]

. (6.27)

Comparing this definition to Eq. (3.12), we see that the scalar product is the zero componentof the conserved current Jµ. Thus the value of the scalar product (φ, χ) is time-independentand corresponds to the number of particles minus the number of anti-particles.

For plane-wave components with definite momentum,

φk(x) =1

√

(2π)32ωke−ikx = Nk e−ikx , (6.28)

the scalar product is given by

(φk, φk′) = δ(k − k′), and (φ∗k, φ∗k′) = −δ(k − k′) (6.29)

130


and zero otherwise,

(φk, φ∗k′) = (φ∗k, φk′) = 0.

Thus we can invert the free field

φ(x) =

∫d3k

√

(2π)32ωk

[

a(k)e−ikx + a†(k)e+ikx]

, (6.30)

to obtain

a†(k) = −(φk, φ) = −iNk

∫

d3x e−ikx←→∂0φ(x) . (6.31)

Using the identity

a†(k,∞)− a†(k,−∞) = −i∫ ∞

−∞dt

∂

∂ta†(k, t) (6.32)

we insert first (6.31) assuming a wave-package localized around k1 and perform then the timedifferentiation,

a†(k1,∞)− a†(k1,−∞) = −i∫

d3kf1(k)

∫

d4x ∂t

(

e−ikx←→∂0 φ(x))

(6.33)

= −i∫

d3kf1(k)

∫

d4x(

e−ikx∂2t φ(x)− φ(x)∂2t e−ikx)

,

where the two terms linear in ∂t canceled. Then we use that the field is on-shell, k2 = m2,for the replacement

∂2t e−ikx = (∇2 −m2)e−ikx.

Since the field is localized, we can perform two partial integrations moving thereby ∇2 to theleft, obtaining

a†(k1,∞)− a†(k1,−∞) = −i∫

d3kf1(k)

∫

d4x e−ikx(+m2

)φ(x) . (6.34)

In a free theory, φ(x) satisfies the Klein-Gordon equation and the RHS would vanish. Since weaim to solve an interacting theory with e.g. LI = λφ4/4!, then

(+m2

)φ(x) = λφ3/3! 6= 0.

Now we can forget the wave-packet, σ → 0, and write simply

a†(k,−∞) = a†(k,∞) + iNk

∫

d4x e−ikx(+m2

)φ(x) . (6.35)

Taking the hermetian conjugate, we obtain for the annihilation operator

a(k,∞) = a(k,−∞) + iNk

∫

d4x eikx(+m2

)φ(x) . (6.36)

When we insert these expressions into 〈f |i〉, we obtain a four-point function combining thesecond terms from the RHS of (6.35) and (6.36). Including the terms a†(k,∞) and a(k,−∞)generates particles propagating from t = −∞ to t = +∞ with momenta unchanged, i.e.to terms corresponding to disconnected graphs. Hence we do not need to consider these

131


contributions, restricting ourselves to connected Green functions1. For n particles in theinitial and m particles in the final state, we obtain

〈k′1, . . . ,k′m; +∞|k1, . . . ,kn;−∞〉 = (i)n+mn∏

i=1

∫

d4xi Nkie−ikixi

(xi +m2

)

m∏

j=1

∫

d4yj Nkieik

′

jyj(yj +m2

)〈0|Tφ(x1) · · · φ(ym+n) |0〉 . (6.37)

This is the reduction formula of Lehmann, Symanzik and Zimmermann (LSZ): For eachexternal particle we obtained the corresponding plan wave component and a Klein-Gordonoperator. Since the latter is the inverse of the free 2-point functions, we can re-phrase thecontent of the LZS formula simply as follows: Replace the 2-point functions of external linesby appropriate wave functions, e.g. φk(x) for scalar particles in the initial state and φ∗k(x) forscalar particles in the final state.

Since we started from field operators in the Heisenberg picture, the matrix element is inthe Heisenberg picture, too. In the Schrodinger picture, it is

〈k1, . . . ,km|iT |k1, . . . ,kn〉 ≡ iTfi ,

where we used also S = 1 + iT and the fact that we neglected disconnected parts.Finally, we define the Fourier transformed n-point function as

G(x1, . . . , xn) =

∫ n∏

i=1

d4ki(2π)4

exp

(

−i∑

i

kixi

)

G(k1, . . . , kn) . (6.38)

Then we obtain the LSZ reduction formula in momentum space,

iTfi = (i)n+mNk1 · · ·NknNk′

1· · ·Nk′

m(k21 −m2) · · · (k2n −m2)(k′21 −m2) · · · (k′2m −m2)

G(k1, · · · , kn,−k1, . . . ,−km) . (6.39)

The Green function G(k1, · · · , kn,−k1, . . . ,−km) is multiplied by zeros, since the exter-nal particles satisfy k2 = m2. Thus Tfi vanishes, except when poles 1/(k2 − m2) ofG(k1, · · · , kn,−k1, . . . ,−km) cancel these zeros. In the case of external scalar particles, onlytheir normalization factors are left. As they are not essential for the calculation of the tran-sition amplitudes, one include these normalization factors into the phase space of final stateparticle and in the flux factor of initial particles. This explains our Feynman rule to replacethe scalar propagator by one for amplitudes in momentum space.

The derivation of the LSZ formula for particles with spin s > 0 proceeds in the sameway. Their wave-functions contain additionally polarization vectors εµ(k), tensors εµν(k), orspinors u(p) and u(p) and their charge conjugated states. In the case of a photon (graviton),we have to add εµ(k) (εµν(k)) in the initial state and the complex conjugated ε∗µ(k) (ε∗µν(k))in the final state. In the case of Dirac fermions, we have to assign four different spinors to thefour possible combinations of particle and anti-particles in the initial and final state. Havingchosen u(p) as particle state with an arrow along the direction of time, the simple rule that a

1You may wonder that we eliminate in this way also the forward scattering amplitudes Tii: Analyticity ofscattering amplitudes guaranties that the amplitude calculated for, e.g., scattering angle ϑ > 0 containsalso correctly the case ϑ = 0.

132


fermion line corresponds to a complex number ψ · · ·ψ fixes the designation of the other spinorsas shown in Fig. 6.1: Connecting the upper fermion lines, u(p′) · · · u(p), corresponds to thescattering e−(p) + X → e−(p′) + X ′, while v(q) · · · u(p) describes the annihilation processe+(q)+e−(p)→ X+X ′. Connecting the lower fermion lines, v(q) · · · v(q), corresponds to thescattering e+(q) + X → e+(q′) + X ′, while u(p′) · · · v(q) describes the pair creation processX + X ′ → e+(q′) + e−(p′). Note that the Feynman amplitude A is defined omitting thenormalisation factor Np = [2ωp(2π)

3]−1/2 from all wave-functions.

v(q)

εµ(k)

u(p)

v(q)

ε∗µ(k′)

u(p′)

Figure 6.1.: Feynman rules for external particles in momentum space; initial state on the left,final state on the right.

Wave function renormalization Up to now we have pretended that we can describe the fieldsin the initial and final state as free particles. Although e.g. Yukawa interactions between two,by assumption, widely separated particles at t = ±∞ are negligibly small, self-interactionspersist. These interactions lead to a renormalization of the external wave-functions.

We can rephrase the problem as follows: If the creation operator a†(k,−∞) corresponds tothe one of a free theory. then it can only connect one-particle states with the vacuum. Thusthe expectation value between the vacuum and any many-particle state is zero. In contrast,the interacting field can also connect many-particle states to the vacuum and therefore itsoverlap with single-particle states is reduced,

a†(k,−∞) |0〉 =√Z |k〉+

√1− Z

∣∣k′,k′′,k′′′

⟩+ . . .

(6.40)

=√Za†0(k1) |0〉+

√1− Z

∣∣k′,k′′,k′′′

⟩+ . . .

. . (6.41)

We are lead to conclude that the free and the interacting fields are connected by

φ(x)→√Zφ0(x) (6.42)

for t→ ±∞. More precisely, this relation holds only for matrix elements of the fields, not forthe field operators themselves.

As a result, we should include in all formulas derived above appropriate factors of thewave-function normalization constant Z. However, we will show later that these factors arecanceled. A simpler short-cut is therefore to exclude all corrections in external legs and touse the LSZ formula as derived.

133


6.3. Specific processes

We consider now in detail a few specific processes. First, we derive the Klein-Nishina for-mula for the Compton scattering cross section using the standard “trace method”. Then wecalculate polarized e+e− → µ+µ− and e+e− → γγ scattering using helicity methods.

6.3.1. Trace method: Compton scattering and the Klein-Nishina formula

Matrix element The Feynman amplitude A of Compton scattering e−(p)+γ(k)→ e−(p′)+γ(k′) at O(e2) consists of two diagrams,

iA = −ie2u(p′)[

ε/′p/+ k/ +m

(p+ k)2 −m2ε/+ ε/

p/− k/′ +m

(p− k′)2 −m2ε/′]

u(p) . (6.43)

Since the denominator is a non-zero light-like vector, we have omitted the iε. Note thatthe two amplitudes can be transformed into each other replacing ε ↔ ε∗′ and k ↔ −k′.This symmetry called crossing symmetry relates processes where a particle is replaced by ananti-particle with negative momentum on the other side of the reaction.

We evaluate the process in the rest-frame of the initial electron. Then p = (m,0) andchoosing εµ = (0, ε) as well as ε′µ = (0, ε′), it follows

p · ε = p · ε′ = 0 . (6.44)

Moreover, the photons are transversely polarized,

k · ε = k′ · ε′ = 0 , (6.45)

and we choose real polarization vectors. We anti-commute p/ in the nominator to the right,p/ε/′ = 2p · ε′ − ε/′p/ = −ε/′p/ and use the Dirac equation, p/u(p) = mu(p). Moreover, we cansimplify also the denominator with p2 = m2 and obtain

A = e2u(p′)

[ε/′k/ε/2p · k +

ε/k/′ε/′

2p · k′]

u(p) . (6.46)

Often the initial electron target is not polarized, and the spin of the final electron is notmeasured. Thus we sum the squared matrix element over the final and average over the initialelectron spin,

∣∣A∣∣2=

1

2

∑

s,s′

|A|2 = e4

2

∑

s,s′

∣∣∣∣u(p′)

(ε/′k/ε/2p · k +

ε/k/′ε/′

2p · k′)

u(p)

∣∣∣∣

2

. (6.47)

Calculating |A|2 = AA∗ requires the knowledge of A∗ = A†. For a general amplitude Acomposed of spinors,

A = u(p′)Γu(p) , (6.48)

with Γ denoting a product of the basis elements given in Eq. (5.38), we have with u = u†γ0

A∗ = u(p)γ0Γ†γ0u(p′) ≡ u(p)Γu(p′) (6.49)

134


and thus

Γ = γ0Γ†γ0 (6.50a)

γµ = γµ (6.50b)

γ5 = −γ5 (6.50c)

a/b/ · · · z/ = z/ · · · b/a/ . (6.50d)

We now write out the spinor indices,

∣∣A∣∣2=e4

2

∑

s,s′

ua(p′)

[ε/′k/ε/2p · k +

ε/k/′ε/′

2p · k′]

ab

ub(p)uc(p)

[ε/k/ε/′

2p · k +ε/′k/′ε/2p · k′

]

cd

ud(p′) . (6.51)

Using the property (5.51) of the Dirac spinors,

∑

s

ua(p, s)ub(p, s′) = (p/+m)ab (6.52)

we obtain

∣∣A∣∣2=e4

2

[p/′ +m

]

da

[ε/′k/ε/2p · k +

ε/k/′ε/′

2p · k′]

ab

[p/+m]bc

[ε/k/ε/′

2p · k +ε/′k/′ε/2p · k′

]

cd

. (6.53)

Since p/+m combines a spinor inM and one inM∗, the result is a trace over gamma matrices,

∣∣A∣∣2=e4

8tr

(p/′ +m)

[ε/′k/ε/p · k +

ε/k/′ε/′

p · k′]

(p/ +m)

[ε/k/ε/′

p · k +ε/′k/′ε/p · k′

]

. (6.54)

Useful identities for the evaluation of such traces are given in the appendix.We simplify this trace by anti-commuting identical variables, such that they become neigh-

bours. Then we can use a/a/ = a2 and reduce thereby the number of gamma matrices in eachstep by two. Multiplying out the terms in the trace, we obtain three contributions that wedenote by

tr = S1

(p · k)2+

S2

(p′ · k)2+

2S3(p · k) (p · k′) . (6.55)

We consider only the first term S1 in detail. Starting from

S1 = tr(p/′ +m

)ε/′ε/k/ (p/+m) k/ε/ε/′

(6.56)

= trp/′ε/′ε/ k/p/

︸︷︷︸

2kp−p/k/

k/ε/ε/′+m2tr

ε/′ε/ k/k/︸︷︷︸

k2=0

ε/ε′

(6.57)

= 2 (k · p) trp/′ε/′ε/k/ε/ε/′

− tr

p/′ε/′ε/p/k/k/ε/ε/′

(6.58)

we arrived at an expression with only six gamma matrices. We continue the work,

S1 = 2 (k · p) trp/′ε/′ε/k/ε/ε/′

= −2 (k · p) tr

p/′ε/′k/ ε/ε/

︸︷︷︸

ε2=−1

ε/′

(6.59)

= 2 (k · p) trp/2ε/

′ k/ε/′︸︷︷︸

2k·ε′−ε/′k/

= 2 (k · p)

[2(k · ε′

)tr(p/′ε/′)− tr

p/′ε/′ε/′k/

](6.60)

= 8 (k · p)[2(k · ε′

) (p′ · ε′

)+(p′ · k

)]. (6.61)

135


We want to eliminate as next step the two scalar products that include p′. Because of four-momentum conservation p′ + k′ = p+ k, it is

(p′ − k

)2=(p− k′

)2(6.62)

and thusp′ · k = p · k′ . (6.63)

Multiplying the four-momentum conservation equation by ε′, it follows moreover

p1 + k1 = p2 + k2 ⇒ ε′ · p︸︷︷︸

0

+ε′ · k = ε′ · p′ + ε′ · k′︸︷︷︸

0

. (6.64)

Thus our final result for S1 is

S1 = 8 (k · p)[

2(k · ε′

)2+ k′ · p

]

. (6.65)

S2 can be obtained observing the crossing symmetry of the amplitude by the replacementsε↔ ε′ and k ↔ −k′. The cross term S3 has to be calculated and we give here only the finalresult for the combination of the three terms, where some terms cancel

∣∣A∣∣2= e4

[ω′

ω+ω

ω′ + 4(ε · ε′)2 − 2

]

. (6.66)

Cross section To obtain the cross section, we have to calculate the flux factor and to performthe integration over the phase space of the final state,

dσ =1

4I(2π)4 δ(4)(Pi − Pf )|Afi|2

n∏

f=1

d3pf2Ef (2π)3

=1

4I|Afi|2 dΦ(n) , (6.67)

with the final state phase space dΦ(n). The flux factor I in the rest system of the electron issimply

I ≡ vrel p1 · p2 = mω . (6.68)

Using Eq. (2.179),d3p′

2E′ = d4p′δ(4)(p′2 −m2)ϑ(p′0) , (6.69)

the phase space integration becomes

dΦ(2) =1

(2π)2

∫

dΩk′|k′|2dk′2|k′|

∫d3p′

2E′ δ(4)(p′ + k′ − p− k

)= (6.70)

=1

8π2

∫

dΩk′|k′|dk′ δ((p+ k − k′

)2 −m2)

. (6.71)

The argument of the delta function is

(p+ k − k′

)2 −m2 = m2 + 2p · k − 2p · k′ − 2k · k′ −m2 (6.72)

= 2m|k| − 2m|k′| − 2|k||k′| (1− cosϑ) (6.73)

= 2m(ω − ω′)− 2ωω′ (1− cos ϑ) ≡ f(ω′) . (6.74)

136


In order to use ∫

dxδ [f(x)] g(x) =g(x)

|f ′(x)|

∣∣∣∣f(x0)=0

(6.75)

we have to determine the derivative f ′(ω′),

f ′(ω′) = −2m− 2ω′ (1− cos ϑ) , (6.76)

and the zeros of f(ω′),0 = 2m

(ω − ω′)− 2ωω′ (1− cos ϑ) . (6.77)

Solving for ω′ givesω′ [ω (1− cos ϑ) +m] = mω (6.78)

andω′ =

ω

1 + ωm (1− cos ϑ)

. (6.79)

This is the famous relation for the frequency shift of a photon found first experimentallyin the scattering of X-rays on electrons by Compton 1921. The observed energy change ofphotons was crucial in accepting the quantum nature (“particle-wave duality”) of photons.Combining everything, we obtain

dΦ(2) =1

8π2

∫

dΩk′|ω′|dω′ δ(2m(ω − ω′)− 2ωω′(1− cos ϑ)

)(6.80)

=1

8π2

∫

dΩk′ω′

2m[1 + ω

m (1− cosϑ)] =

1

16π2mω

∫

dΩk′ ω′2 (6.81)

and thus as differential cross section (Klein, Nishina 1928)

dσ

dΩ=

1

4mω

ω′2

16π2mω

∣∣A∣∣2=

α2

4m2

ω′2

ω2

[ω′

ω+ω

ω′ + 4(ε · ε′)2 − 2

]

(6.82)

with α ≡ e2/(4π). For scatterings in the forward direction, ϑ → 0 and thus ω′ → ω,the scattered photon retains (in the lab frame) its energy even in the ultra-relativistic limitω ≫ m. The same holds in the classical limit, ω ≪ m, but now for all directions. Thus weobtain as classical limit of the Klein-Nishina formula the polarized Thomson cross section

dσ

dΩ≈ α2

m2

(ε · ε′

)2= r20

(ε · ε′

)2, (6.83)

with r20 = α/m as the classical electron radius.Averaging and summing over the photon polarization vectors is simplest, if we choose the

angle between ε and ε′ as ϑ. Then∑

r,r′

(ε · ε′

)2= 1 + cos2 ϑ . (6.84)

The integration over the scattering angle ϑ can be done analytically. We use x = cos ϑ andset ω ≡ ω/m,

σ =πα2

m2

∫ 1

−1dx

[1

[1 + ω(1− x)]3 +1

1 + ω(1− x) −1− x2

[1 + ω(1− x)]2]

(6.85)

=πα2

2m2

1 + ω

ω3

[2ω(1 + ω)

(1 + 2ω− ln(1 + 2ω)

]

+ln(1 + 2ω)

2ω− 1 + 3ω

(1 + 2ω)2

. (6.86)

137


Since in the electron rest frame s = (p + k)2 = m2 + 2mω = m2(1 + 2ω), we can useω = (s/m2 − 1)/2 to express σ in an explicit Lorentz invariant form.

Approximations for the non-relativistic and the ultra-relativistic limit are

σ = σTh ×

1− 2ω +O(ω2) for ω ≪ 1 ,38ω

(ln(2ω) + 1

2

)+O(ω−2) for ω ≫ 1 ,

(6.87)

where the Thomson cross section is given by σTh = 8πα2/(3m2). These approximations areshown together with the exact result in the left panel of Fig. 6.2.

In the ultra-relativistic limit s ≫ m2, the total cross section for Compton scattering de-creases as σ ∝ 1/s. On the other hand, the differential cross section in the forward directionis constant. As a result, the relative importance of the forward region ϑ ∼ 0 increases forincreasing s even faster than required by unitarity: While dσ/dx is symmetric around x = 0in the classical limit ω → 0, it becomes more and more asymmetric with a a shrinking peakaround the forward region at ϑ ∼ 0, cf. the right panel of Fig. 6.2.

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1 10 100 1000

σ/σ 0

ω/m

0

1

2

3

4

5

6

7

8

-1 -0.5 0 0.5 1

σ/σ 0

x

Figure 6.2.: Left: The total cross section σ/σTh as function of ω together with the classicaland ultra-relativistic limits given in Eq. (6.87). Right: The differential crosssection dσ/dx as function of x = cos ϑ for ω = 0.01, 0.1, 1, and 10 (from top todown).

Crossing symmetry We noticed that the two amplitudes in Compton scattering can betransformed into each other replacing ε↔ ε∗′ and k ↔ −k′. This is an example of a generalsymmetry of relativistic quantum field theories called crossing symmetry. Using the Feynmanrules for in and out particles, it follows that matrix elements where an in-going particle isreplaced by an out-going anti-particle or vice versa are related by the following substitutions,

• exchange the momentum k ↔ −k′;• exchange particle and anti-particle wave functions; thus in momentum space, 1↔ 1 for

spinless particles, ε↔ ε∗′ for spin-1 and u↔ v for fermions.

• multiply by −1 for each exchanged fermion pair.

The additional minus for fermions is required, because the replacement p ↔ p′ in a fermionspin sum gives

∑

s

u(p, s)u(p, s) = (p/+m) = −(p/′ −m

)= −

∑

s

v(p′, s)v(p′, s) . (6.88)

138


Note that this symmetry allows us to obtain the matrix elements of different processes: Forinstance, we can relate the processes e−e+ → µ−µ+ with e−µ− → e−µ− and µ−µ+ → e−e+.

In a more formal approach the crossing symmetry is derived not by relying on perturbationtheory and the Feynman rules, but using the analytical properties of S-matrix elements in arelativistic quantum field theory. The LSZ reduction formula distinguishes in and out particlesonly by the sign of the momenta used in the Fourier transformation. If one can analyticallycontinue the residue of a pole in an S-matrix element from p0 to −p0, then one converts theS-matrix for a particle with φ(p) into the one for an anti-particle with φ∗(−p). Remarkably,the basic properties of a relativistic quantum field theory, locality and causality, are sufficientto proof that this analytical continuation is possible.

6.3.2. Helicity method and polarized QED processes

The number of terms that have to calculated grows in the trace method as ∼ n2/2 with thenumber n of diagrams. For large n, it should be therefore favorable to calculate the amplitudeA(s1, . . . , sf ) for fixed polarizations of the external particles: The amplitude is a complexnumber and can be trivially squared. An efficient way to calculate polarized amplitudes useshelicity wave functions, an approach used also in most modern computer programs as e.g.Madgraph.

Massless fermions We restrict our short introduction into helicity methods to masslessparticles. In the case of fermions, we know that then the use of Weyl spinors in the chiralrepresentation is most convenient,

uL(p) =

(φL(p)0

)

and uR(p) =

(0

φR(p)

)

. (6.89)

We do not need to consider vL,R(p), since they correspond to particle spinors of oppositehelicity, uR,L(p). Moreover, two out of the fours possible scalar products involving uL,R arezero for massless fermions,

uL(p)uL(q) = uR(p)uR(q) = 0 . (6.90)

This motivated Xu, Zhang and Chang to introduce a bracket notation for the helicity spinorsas follows

uL(p) = 〈p , uR(p) = [p , uL(p) = p] , uR(p) = p〉 . (6.91)

Then the only non-zero Lorentz-invariant spinor products are given by a pair of brackets ofthe same type,

uL(p)uR(q) = 〈pq〉 and uR(p)uL(q) = [pq] . (6.92)

We call the quantities on the RHS angle and square brackets. Next we consider the tensorproduct of the spinors,

p〉[p = uR(p)uR(p) = PRp/ , and p]〈p = uL(p)uL(p) = PLp/ . (6.93)

These identities connect the massless spinors p〉 and [p to the light-like four-vector pµ.We are now in position to derive some basic properties of the brackets. First, we can

connect the two types of spinor products as

〈pq〉 = uL(p)uR(q) = [uR(p)uL(q)]∗ = [qp]∗ . (6.94)

139


p2

p1

p4

p3

Figure 6.3.: Feynman diagrams for the process e−e+ → µ+µ−.

Next, multiplying p〉[p and q]〈q and taking the trace gives

〈pq〉[qp] = tr[q/p/PR] = 2p · q , (6.95)

so that

|〈pq〉|2 = |[qp]|2 = 2p · q . (6.96)

Finally, we express the spinor products through Weyl spinors and use uR(p) = iσ2u∗L(p),

〈pq〉 = φ†L(p)φR(q) = φ∗La(p)(iσ2)abφ

∗Lb(q) . (6.97)

Then the antisymmetry of (iσ2)ab = εab implies

〈pq〉 = −〈qp〉 and [pq] = −[qp] . (6.98)

Thus the brackets are square roots of the corresponding Lorentz vector products which areantisymmetric in their two arguments. Finally, we note that the Fierz identity applied to thesigma matrices,

(σµ)ab(σµ)cd = 2(iσ2)ac(iσ2)bd , (6.99)

allows the simplification of contracted spinor expressions,

〈pγµq]〈kγµℓ] = 2〈pk〉[ℓq] , 〈pγµq][kγµℓ〉 = 2〈pℓ〉[kq] . (6.100)

e−e+ → µ−µ+ scattering It is now time to apply our new “bracket” formalism. Weconsider the tree-level amplitude for, e.g., e−L (1)e

+R(2) → µ−L (3)µ

+R(4) in QED, given by the

single diagram shown in Fig. 6.3. considering all momenta as outgoing. Then the amplitudeis

iA = (−ie)2−iq2

uL(3)γµuL(4) uL(2)γµuL(1) (6.101)

=ie2

q2〈3γµ4]〈2γµ1] =

2ie2

q2〈32〉[14] , (6.102)

where we have used the Fierz identity (6.99) in the last step. Now, 〈32〉 and [14] are bothsquare roots of

s23 = (k2 + k3)2 = (k1 + k4)

2 (6.103)

140


which is just the Mandelstam invariant u. In the e+e− center of mass frame, u = −2E2(1 +cos ϑ), and q2 = s = 4E2. Then

|iA|2 = e4(1 + cos ϑ)2 . (6.104)

You should rederive this result using the more familiar trace formalism and compare theamount of algebra required in the two approaches (problem 6.4).

Massless gauge bosons In the next step, we incorporate massless gauge bosons as thephoton into this framework. We claim that the polarization vectors for a final-state masslessvector boson of definite helicity can be represented as

ǫ∗µR (k) =1√2

〈rγµk]〈rk〉 , ǫ∗µL (k) = − 1√

2

[rγµk〉[rk]

. (6.105)

Here k is the momentum of the vector boson, and r is a fixed lightlike 4-vector, called thereference vector. The only requirement on r is that it cannot be collinear with k.

Now we show that this definition makes sense: First, we note that the vectors satisfy[ε∗R(k)]

∗ = ε∗L(k). One can also check that the polarization vectors are correctly normalized.Moreover, the Dirac equation, k/ k] = 0, guaranties that the polarization vectors (6.105) aretransverse,

kµε∗µR,L(k) = 0 . (6.106)

Finally, we have to show that a change from one reference vector r to another lightlikevector s corresponds to a gauge transformation and thus does not affect physics. The changeof a polarization vector under a change of reference vector r → s is

ε∗µR (k; r)− ε∗µR (k; s) =1√2

(〈rγµk]〈rk〉 −

〈sγµk]〈sk〉

)

(6.107)

=1√2

1

〈rk〉〈sk〉− 〈rγµk]〈ks〉 + 〈sγµk]〈kr〉

. (6.108)

Now we use first the tensor products (6.93), and then the antisymmetry of the brackets,

ε∗µR (k; r)− ε∗µR (k; s) =1√2

1

〈rk〉〈sk〉− 〈rγµk/s〉+ 〈sγµk/r〉

(6.109)

=1√2

1

〈rk〉〈sk〉〈s(k/γµ + γµk/)r〉

(6.110)

=1√2

1

〈rk〉〈sk〉〈sr〉 2kµ . (6.111)

The last line finally follows from the Clifford algebra of Dirac matrices. Thus the difference ofthe polarization vectors induced by a change of the reference vector is a function proportionalto the photon momentum,

ε∗µR (k; r)− ε∗µR (k; s) = f(r, s)kµ . (6.112)

Dotted into an on-shell photon amplitude, this expression will give zero by current conserva-tion. Thus we can use the most convenient reference vector s which can be chosen differentlyin any gauge-invariant set of Feynman diagrams.

141


p1

p2

p3

p4

p1

p2

p4

p3

Figure 6.4.: Feynman diagrams for the process e−e+ → γγ.

e−e+ → γγ scattering As second example, we consider now a scattering process withphotons as external particles, e−e+ → γγ, as illustration for the use of the polarizationvectors. We label the momenta as in Fig. 6.4, taking all momenta as outgoing.

Then the amplitude for this process is

iA = (−ie)2〈2(

ε/(4)i(2/ + 4/)

s24ε/(3) + ε/(3)

i(2/ + 3/)

s23ε/(4)

)

1] , (6.113)

where we use the shorthand (2/+ 4/) for (k/2 + k/4) and define sij = (i+ j)2.

There are four possible choices for the photon polarizations. Exchanging the momenta 3and 4 relates the cases γRγL and γLγR, while parity connects γRγR and γLγL. We startshowing that the latter two amplitudes are zero in the massless limit we consider. For thecase of γRγR, we choose as reference vector r = 2 for both polarization vectors,

εµ(3) =1√2

[2γµ3〉[23]

, εµ(4) =1√2

[2γµ4〉[24]

. (6.114)

When these choices are used in (6.113), we find, with the use of the Fierz identity (6.99),

〈2γµεµ(4) ∝ 〈2γµ[2γµ4〉 = 2〈22〉[4 = 0 , (6.115)

which vanishes because of 〈22〉 = 0. A similar cancellation occurs with ε(3). So the entirematrix element vanishes. The amplitude for the case γLγL must then also vanish by par-ity; alternatively, we can find the same cancellation for that case by using r = 1 in bothpolarization vectors.

To compute the amplitude for the case γRγL, choose

εµ(3) =1√2

[2γµ3〉〈23〉 and εµ(4) = − 1√

2

[1γµ4〉[14]

. (6.116)

Then the second diagram in Fig. 6.4 vanishes because of (6.115). Using the Fierz identity,the first diagram gives

iA =−ie2s24

2 · 2(−2)〈23〉[14] 〈24〉[1(2/+ 4/)2〉[31] . (6.117)

142

6.4. Soft photons

Now we use the Dirac equation, 2/2〉 = 0, and replace the vector 4/L by an angle bracket,

iA =2ie2

s13〈23〉[14]〈24〉[14]〈42〉[31] (6.118)

=2ie2

〈13〉[31]〈23〉〈24〉〈42〉[31] = 2ie2〈24〉2

〈23〉2. (6.119)

Here we used also four-momentum conservation to convert s24 into such brackets that we endup with an expression containing only Mandelstam variables, s23 = u, s13 = s24 = t,

|iA|2 = 4e4t

u= 4e4

1− cosϑ

1 + cosϑ. (6.120)

6.4. Soft photons

The addition of an additional vertex introduces typically a factor α/π ∼ 0.2% into a QEDcross section. Thus one may hope that perturbation theory in QED converges, at leastinitially, reasonably fast. An exception to this rule is the emission of an additional soft orcollinear photon from an external line shown in Fig. 6.5. The denominator of the additionalpropagator goes for k → 0 to

1

(p+ k)2 −m2→ 1

2p · k ∼1

2Eω(1 − cosϑ), (6.121)

where we assumed |p| ≫ m in the last step. The denominator can blow up in two differentlimits: Firstly, in case of emission of soft photons, ω → 0. Secondly, in case of collinearemission of photons, ϑ → 0, if the mass of the emitting particle can be neglected. We haveseen in the example of Compton scattering that both cases correspond to the classical limit.

p+kp

k

Figure 6.5.: Emission of an additional soft or collinear photon from an external line in thefinal state.

Universality and factorization The fact that a photon sees in the soft limit k → 0 a classicalcurrent should lead to considerable simplifications: In particular, interference effects shoulddisappear and the amplitude An+1 for the emission of an additional soft photon should fac-tories into an universal factor εµSµ and the amplitude An for the original process.

143


Let us start considering the emission of a soft photon by a spinless particle. If a scalar inthe final state with momentum p and charge q emits a photon with momentum k → 0, then2

An+1 = qεµ(2p

µ + kµ)

(p− k)2 −m2 + iεAn → −q

ε · pp · k − iε

An . (6.122)

For the emission of a soft photon from an initial state particle, the corresponding factor is+q ε · p/(p · k + iε). In the case of an internal line, in general no factor (p · k)−1 appears fork → 0, since the virtual particle is off-shell.

For a spin-1/2 particle in the initial state, the emission of a soft photon adds the factor

qεµu(p, s)γ

µ(p/ + k/ −m)

(p− k)2 −m2 + iε→ −q εµu(p, s)γ

µ(p/ −m)

−2p · k + iε(6.123)

to the amplitude An. Now we replace p/+m by the spin sum∑

s u(p, s)u(p, s), and use

u(p, s)γµu(p, s′) = 2mpµ

p0δs,s′ = 2pµδs,s′ . (6.124)

This relation can be checked by direct calculation, or by noting that the currentqu(p, s)γµu(p, s) should become q(ρ, ρv) in the classical limit k → 0. Thus we obtain thesame universal factor describing the emission of a soft photon,

Sµ = −q pµ

p · k − iε, (6.125)

as in the case of a scalar. Moreover, we confirmed that the amplitude indeed factorizes,An+1 = εµS

µAn ≡ εµAµn+1. If we allow for the emission of m soft photons from external

particles with charge qi, then

Aµ1···µmn+m →m∑

i=1

siqi pµm

p · k + isiεAn , (6.126)

where the signs are si = −1 for an initial and si = +1 for a final state particle.We have seen that the polarization vector εµ(k) of a photon does not transform as a four-

vector, cf. Eq. (4.25), but acquires a term proportional to kµ. As we exploited already atvarious places, amplitudes containing polarization vectors εµ(k) of external photons thereforehave to vanish when contracted with kµ. Thus Eq. (6.126) implies in the limit k → 0

kµjAµ1···µmn+m →m∑

i=1

siqiAn = 0 . (6.127)

The prefactor of An is the total charge in the final state minus the total charge in the initialstate. In order to obtain a Lorentz invariant matrix element for the soft emission of masslessspin-1 particles, we have therefore to require that they couple to a conserved charge3. ThusLorentz invariance is sufficient to guaranty the conservation of the electromagnetic current inthe low-energy limit. While this argument does not rely on gauge invariance it does not sayanything about the impact of “hard” photons.

2We use the Feynman rule for a φφAµ vertex derived in problem 4.8.3 Weinberg who developped this argument could show additionally the following [Wei65]: Massless spin-2particles have to couple with universal strength to its source, the energy-momentum stress tensor. Thusany consistent theory of massless spin-2 particles implies a low-energy theory satisfying the equivalenceprinciple. Consistent low-energy theories of interacting massless particles with s > 2 are not possible—thussuch particle should decouple in the classical limit.

144

6.A. Appendix: Decay widths and cross sections

Bremsstrahlung We discuss now as a concrete example the case of bremsstrahlung, i.e.the emission of a real photon in the scattering of a charged particle in the Coulomb fieldA0 = −Ze/(4π|x|) of a static charge. The S-matrix element of this process is

iSfi = 2πδ(E′ + ω − E′)−Ze3|q|2 u(p′)

[

ε/p/′ + k/ +m

2p′ · k γ0 + γ0p/− k/ +m

−2p · k ε/

]

u(p) , (6.128)

where 1/|q|2 is the Fourier transform of A0. Note that the external field breaks translationinvariance and the momentum is not conserved. We commute now the momenta,

iSfi ∝ e2u(p′)[2ε · p′ − (p/′ −m)ε/+ k/ε/

2p′ · k γ0 + γ02ε · p− ε/(p/−m) + k/ε/

−2p · k

]

u(p) , (6.129)

such that we can use in the next step the Dirac equation. Neglecting additionally in the softlimit the k/ term in the nominator, we find

iSfi ∝ e2u(p′)γ0u(p)[ε · p′p′ · k −

ε · pp · k

]

. (6.130)

As we have shown in the previous paragraph in general, the amplitude factorizes into theamplitude describing the “hard” process and the universal correction term. The latter consistsof the two terms expected for the emission of a soft photon from an initial line with momentump and a final line with momentum p′. The probability for the emission of an additional softphoton is given integrating the square bracket over the phase space,

dPn+1 =dσn+1

dσn=

[ε · p′p′ · k −

ε · pp · k

]2 d3k

(2π)32ωk∝ dωk

ωk. (6.131)

This probability diverges for ω → 0 and therefore the process is called infrared (IR) divergent.

The resolution to this IR problem lies in the fact that soft photons with energy belowthe energy resolution Eth of the used detector are not detectable. Therefore the amplitudefor the emission of n real soft photons with E < Eth are indistinguishable from amplitudesincluding virtual photons. The IR divergences in the real and virtual corrections cancel,leading to a finite result. We will discuss a detailed example for how this cancellation worksin chapter 14.2.


We establish first the connection between the correctly normalised transition matrix elementM and the Feynman amplitude A where we do not include the normalisation factors ofexternal particles. Then we derive decay widths and cross sections describing 1 → n and2→ n processes.

Normalisation We have split the scattering operator S into a diagonal part and the transitionoperator T , S = 1 + iT . Taking matrix elements, we obtain

Sfi = δfi + (2π)4 δ(4)(Pi − Pf )iMfi (6.132)

145


where we set also Tfi = (2π)4 δ(4)(Pi − Pf )Mfi.

We use as relativistic normalisation for all types of one-particle states

⟨p|p′

⟩= 2Ep(2π)

3δ(p − p′) . (6.133)

For a finite normalisation volume, which makes defining decay widths and cross sectionseasier, this becomes

⟨p|p′

⟩= 2EpV δp,p′ , (6.134)

The Feynman amplitude A neglects all normalisation factors of external particles. Thus thetransition between the matrix elementMfi and the Feynman amplitude A for a process withn particles in the initial and m in the final states is given by

Mfi =n∏

i=1

(2EiV )−1/2m∏

f=1

(2EfV )−1/2Afi . (6.135)

6.A.1. Decay widths

We consider the decay of a particle into n particles in the final state. Squaring the scatteringamplitude Sfi for i 6= f using (2π)4 δ(4)(0) = V T gives as differential transition probability

dWfi = (2π)4 δ(4)(Pi − Pf )V T |Mfi|2n∏

f=1

V d3pf(2π)3

. (6.136)

The decay rate dΓ is the transition probability per time,

dΓfi = limT→∞

dWfi

T= (2π)4 δ(4)(Pi − Pf )V |Mfi|2

n∏

f=1

V d3pf(2π)3

. (6.137)

Going over to the Feynman amplitude A eliminates the volume factors V ,

dΓfi = (2π)4δ(4)(Pi − Pf )1

2Ei|Afi|2

n∏

f=1

d3pf2Ef (2π)3

. (6.138)

Moreover, the phase space integrals in the final state are now Lorentz invariant, d3pf/(2Ef ).Introducing the n-particle phase space volume

dΦ(n) = (2π)4 δ(4)(Pi − Pf )n∏

f=1

d3pf2Ef (2π)3

, (6.139)

the decay rate becomes

dΓfi =1

2Ei|Afi|2 dΦ(n) . (6.140)

Since both |Mfi|2 and the phase space dΦ(n) are Lorentz invariant, the decay rate Γ ∝ 1/Ei =1/(γimi) shows explicitly the time dilation effect for a moving particle.

146


Two-particle decays We evaluate the two particle phase space dΦ(2) in the rest frame ofthe decaying particle,

dΦ(2) = (2π)4 δ(M − E1 − E2) δ(3)(p1 + p2)

d3p12E1(2π)3

d3p22E2(2π)3

(6.141)

We perform the integration over d3p1 using the momentum delta function. In the resultingexpression,

dΦ(2) =1

(2π)21

4E1E2δ(M − E1 − E2) d

3p2 , (6.142)

E1 is now a function of p2, E21 = p22+m

21. Introducing spherical coordinates, d3p2 = dΩp22dp2,

dΦ(2) =1

(2π)2dΩ

∫ ∞

0δ(M − E1 − E2)

p22dp24E1E2

, (6.143)

and evaluating the delta function with M − E1 −E2 =M − x and

dp2dx

=p2x

E1E2(6.144)

gives

dΦ(2) =|p′cms|4π2

dΩ , (6.145)

where

p2cms =1

4M2

[M2 − (m1 +m2)

2] [M2 − (m1 −m2)

2]

(6.146)

is the cms momentum of the two final state particles.

Three-particle decays The three particle phase space dΦ(3) is

dΦ(3) = (2π)4 δ(M −E1−E2−E3) δ(3)(p1+p2+p3)

d3p12E1(2π)3

d3p22E2(2π)3

d3p32E3(2π)3

. (6.147)

We can use again the momentum delta function to perform the integration over d3p3,

dΦ(3) =1

(2π)5δ(M −E1 − E2 − E3)

d3p1d3p2

8E1E2E3, (6.148)

To proceed we have to know the dependence of the matrix element on the integration variables.If there is no preferred direction (either for scalar particles or spin averaged), we obtain

dΦ(3) =1

8(2π)54πp21dp1 2πd cos ϑdp2

E1E2E3δ(M −E1 − E2 − E3)

=1

32π3p1dp1 (p1p2d cosϑ)(p2dp2)

E1E2E3δ(M − E1 − E2 − E3) . (6.149)

We rewrite next the momentum integrals as energy integrals. The energy-momentum relationE2i = m2

i + p2i gives EidEi = pidpi for i = 1, 2. Furthermore,

E23 = (p1 + p2)

2 +m23 = p21 + p22 + 2p1p2 cos ϑ+m2

3 (6.150)

147


and thus E3dE3 = p1p2d cosϑ for fixed p1,p2. Performing the angular integral, we obtain

dΦ(3) =1

32π3dE1dE2dE3δ(M − E1 − E2 − E3) , (6.151)

and finally

dΦ(3) =1

32π3dE1dE2 . (6.152)

The last step is only valid, if the argument of the delta function is non-zero. Thus theremaining task is to determine the boundary of the integration domain. Introducing theinvariant mass of the pair (i, j)

m223 = (p− p1)2 = (p2 + p3)

2 =M2 − 2ME1 (6.153)

m213 = (p− p2)2 = (p1 + p3)

2 =M2 − 2ME2 (6.154)

m212 = (p− p3)2 = (p1 + p2)

2 =M2 − 2ME3 , (6.155)

where the last column is valid in the rest frame of the decaying particle with mass M . WithE1 + E2 + E3 = M one finds m2

23 +m213 +m2

12 = M2 +m22 +m2

3. Therefore only two out ofthe three variables are independent. Let’s choose m2

23 and m213 as integrations variables, with

m223 as the outer one. Then

(m2 +m3)2 ≤ m2

23 ≤ (M −m1)2 . (6.156)

For given value of m223, we have now to determine the allowed range of m2

13. Inserting energyand momentum conservation into E2

3 = p23 +m23, we obtain

(M − E1 − E2)2 = m2

3 + p21 + p

22 + 2p1 · p2 . (6.157)

The extrema correspond to

p1 · p2 = ±p1p2 = ±√

(E21 −m2

1)(E22 −m2

2) . (6.158)

6.A.2. Cross sections

We consider now the interaction of two particles in the rest system of either particle 1 or 2.For simplicity, we consider two uniform beams. They may produce n final state particles.The total number of such scatterings is

dN ∝ vMøln1n2dV dt (6.159)

where ni is the density of particle type i = 1, 2. The Møller velocity vMøl is a quantity whichcoincides in the rest frame of particle 1 or 2 with the norm of |v2| and |v1|, respectively.Therefore it is often denoted simply as their relative velocity vrel. The proportionality con-stant has the dimension of an area and is called cross section σ. We define in the rest systemof either particle 1 or 2

dN = σvMøln1n2dV dt , (6.160)

while we set in an arbitrary frame

dN = An1n2dV dt . (6.161)

148


We determine now A. Since both dN and dV dt = d4x are Lorentz invariant, the expressionAn1n2 has to be Lorentz invariant too. Since the densities transform as

ni = ni,0γ = ni,0Eimi

, (6.162)

the expression

AE1E2

p1 · p2(6.163)

is also Lorentz invariant. In the rest system of particle 1, it becomes

AE1E2

E1E2 − p1p2= A = σvMøl . (6.164)

Thus we found that A in an arbitrary frame is given by

A = σvMølp1 · p2E1E2

. (6.165)

A more handy expression for A is obtained as follows: In the rest frame 1, we have

p1 · p2 = m1E2 = m1m2

√

1− v2Møl

. (6.166)

Thus the Møller velocity is given in general by

vMøl =

√

1− m21m

22

(p1 · p2)2. (6.167)

Since this expression is Lorentz invariant, we see that the notation of the Møller velocity asrelative velocity is misleading.

Next we define

I ≡ vMøl p1 · p2 =√

(p1 · p2)2 −m21m

22 . (6.168)

Inserting (6.165) together with the definition of I into (6.161), we obtain

dN = σI

E1E2V(n1V )(n2dV )dt . (6.169)

Here, we re-grouped the terms to make clear that after integration the total number N ofscattering events is proportional to the number N1 = n1V and N2 =

∫n2dV of particles of

type 1 and 2, respectively. The number N of scattering events per time and per particles 1and 2 is however simply the transition probability per time,

dWfi

T=

dN

N1N2T= dσ

I

E1E2V. (6.170)

Inserting the expression (6.137) for dWfi, we find

dσ =E1E2V

2

I(2π)4 δ(4)(Pi − Pf )|Mfi|2

n∏

f=1

V d3pf(2π)3

. (6.171)

149


Changing from the complete matrix element M to the Feynman amplitude A introduces afactor (2E1V )−1(2E2V )−1 for the initial state and

∏

f (2EfV )−1 for the final state. Thus thearbitrary normalisation volume V cancels and we obtain

dσ =1

4I(2π)4 δ(4)(Pi − Pf )|Mfi|2

n∏

f=1

d3pf2Ef (2π)3

=1

4I|Afi|2 dΦ(n) (6.172)

with the final state phase space dΦ(n). The three pieces composing the differential crosssection, the flux factor I, the Feynman amplitude A, and the final state phase space dΦ(n),are each Lorentz invariant.

A symmetry factor S = 1/n! has to be added to the total cross section, if there are nidentical particles in the final state.

2–2 scattering The flux factor becomes in the cms

I2 = (p1 · p2)2 −m21m

22 = p2cms(E1 + E2)

2 (6.173)

or I = pcms√s. Adding also the known expression for the 2-particle phase space gives

dσ

dΩ=

1

64π2s

p′cms

pcms|Afi|2 . (6.174)

The momenta pcms and p′cms are given by (6.146) with the replacement M → s.

Summary of chapter

The LSZ reduction formula shows that S-matrix elements are obtained from connected Greenfunctions by a replacement of the propagators on external lines with the corresponding wave-functions times the wavefunction renormalisation constant

√Z. Cross sections and decay

rates are calculated from squared Lorentz-invariant Feynman amplitudes A, the Lorentz-invariant final state phase space dΦ(n) and a flux factor I. Squared Feynman amplitudes canbe obtained using “Casimir’s trick”. If the number of diagrams increases, it is more convinientto calculate directly the amplitude using helicity methods.

Further reading/sources

The derivation of Boltzmann H-theorem follows closely the one in Weinberg. The LSZ formulafor particles with spin s > 0 is derived in Greiner. For additional information about thehelicity formalisms see Haber [Hab94] and Peskin[Pes11] from which our examples are taken.

Problems

150


6.1 Muon decay ♥.Derive the differential decay rate of the processµ−(p1)→ e−(p4)νµ(p3)νe(p2), via the exchange ofa W -boson described by the vertex

− ig√2fγµ(1− γ5)f Wµ .

You can neglect all terms of order m2e/m

2W ,

m2µ/m

2W .

6.2 Optical theorem.Consider the theory of two light scalar fields φ1and φ2 with mass m coupled to one heavy scalarΦ with mass M > 2m,

L = L0 + gφ1φ2Φ

where L0 is the free Lagrangian.a.) Calculate the width Γ of the decay Φ→ φ1φ2.b.) Draw the Feynman diagram(s) and write downthe Feynman amplitude iM for the scattering pro-cess φ1(p1)φ2(p2) → φ1(p

′1)φ2(p

′2). What is your

interpretation of the behaviour of the amplitudefor s = (p1 + p2)

2 → M2? c.) Consider the one-loop correction iMloop to the mass of Φ,

1

Write down iMloop first for an arbitrary momen-tum p of the external particle Φ, then for its restframe, p = (M, 0). Find the poles of the integrandand use the theorem of residues to perform the q0

part of the loop integral. Finally, use the identity

1

x± iε= P

(1

x

)

∓ iπδ(x)

to find the imaginary part of the amplitude. Youshould find MΓ(Φ→ φ1φ2) = ImMloop, a specialcase of the optical theorem.

6.3 Identities for gamma matrices.

Derive the following identities γµa/γµ = −2a/,tr[a/b/] = 4 a · b, tr[a/b/c/d/] = 4[(a · b)(c · d) − (a ·c)(b · d) + (a · d)(b · c)].

6.4 Polarised cross sections in the trace for-malism.

Calculate the squared matrix element of the pro-cess e−L(1)e

+R(2) → µ−

L (3)µ+R(4) using the trace

formalism. (Hint: Use the spin projection oper-ators (5.57) to obtain the required polarisations.)

6.5 Emission of two soft photons.

Consider the emission of two soft photons fromthe same external line and show that the matrixelementMn+2 contains a product of the same fac-tor as in the emission of single photons from twoseparate lines.

6.6 Light deflection in gravity ♥.

Consider the scattering of a scalar particle anda photon via graviton exchange. Calculate thescattering amplitude and the cross section in thestatic limit pµ = (M⊙,0) for small-angle scatter-ing (k2 = ϑ = 0 in the nominator) and show thatit agrees with Einstein’s prediction for light deflec-tion by the Sun.

151

7. Gauge theories

We discuss in this chapter field theories in which the Lagrangian is invariant under a con-tinuous group of local transformations in internal field space. The symmetry group of thesetransformations is called the gauge group and the vector fields associated to the generatorsof the group the gauge fields. We introduce as first step unbroken gauge theories, i.e. theorieswith massless gauge bosons, and defer the more complex case of broken gauge symmetries tothe chapters 10 and 11. The Standard Model (SM) of particle physics contains with quantumelectrodynamics (QED) with the abelian gauge group U(1) and quantum chromodynamics(QCD) as an non-abelian gauge theory with group SU(3) two examples for unbroken gaugetheories. Non-abelian gauge theories were first studied by Yang and Mills and are thereforealso often called Yang-Mills theories. The structure of Yang-Mills theories has many similiar-ities with gravity. We use this property to introduce curvature of a space-time as analogonto the field-strength in Yang-Mills theories.

7.1. Electromagnetism as abelian gauge theory

In classical electromagnetism, the field-strength tensor F is an observable quantity, while thepotential A is merely a convenient auxiliary quantity. From the definition

Fµν = ∂µAν − ∂νAµ , (7.1)

it is clear that F is invariant under the transformations

Aµ(x)→ A′µ(x) = Aµ(x)− ∂µΛ(x) . (7.2)

Thus A′µ(x) is for any Λ(x) physically equivalent to Aµ(x), leading to the same field-strength

tensor and thus e.g. to the same Lorentz force on a particle. The transformations (4.14) arecalled gauge transformations. Remember that a mass term m2AµA

µ of the photon wouldbreak gauge invariance.

Consider now e.g. a free Dirac field ψ(x) with electric charge q. We saw already that thisfield is invariant under global phase transformations exp[iqΛ] ∈ U(1), implying a conservedcurrent via Noether’s theorem. Can we promote this global U(1) symmetry to a local one,

ψ(x)→ ψ′(x) = U(x)ψ(x) = exp[iqΛ(x)]ψ(x) (7.3)

by making the phase U space-time dependent as in (4.14)? The partial derivatives in theDirac Lagrangian will lead to an additional term ∝ ∂µU(x), destroying the invariance of thefree Lagrangian. However, if we add a field Aµ(x) which transforms as defined in (4.14),the two gauge-dependent terms will cancel. Thus local U(1) gauge invariance requires theexistence of a massless gauge boson and fixes its interaction with matter as

∂µ → Dµ = ∂µ + iqAµ . (7.4)

152

7.2. Non-abelian gauge theories

Moreover, the gauge field has to transform as

Aµ(x)→ A′µ(x) = Aµ(x)− ∂µΛ(x) = Aµ(x)−

i

qU(x)∂µU

†(x) , (7.5)

where we expressed the change δAµ(x) through the group elements U(x).To summarise: The invariance of complex (scalar or Dirac) fields under global phase trans-

formations exp[iqΛ] ∈ U(1) implies a conserved current, promoting it to a local U(1) symmetryrequires the existence of a massless U(1) gauge boson coupled via gauge-invariant derivativesto other fields.


7.2.1. Gauge invariant interactions

We want to generalise now electromagnetism, using as symmetry group instead of the abeliangroup U(1) larger groups like SO(n) or SU(n). A group like SU(n) will describe the interac-tions of n2 − 1 gauge bosons with matter, using as a single parameter the gauge coupling g.The gauge transformations will moreover mix fermions living in the same representation ofthe group, requiring that these fermions have the same interactions and the same mass if thesymmetry is unbroken. In this way, non-abelian gauge theories lead to a partial unificationof matter fields and, separately, of interactions. Note the difference to an abelian symmetry:The emission of a photon does not change any quantum number (apart from the momentum)and thus does not “mix” different particles. Therefore there is also no connection betweenthe electric charge of different particles.

The two non-abelian groups used in the SM are SU(2) for weak and SU(3) for strong inter-actions. A matrix representation for the fundamental representation of these two groups arethe Pauli matrices, T a = σa/2, and the Gell-Mann matrices, T a = λa/2, respectively. Un-der the fundamental representation the fermions transform as doublets for SU(2), as tripletsfor SU(3), etc. Since the number of generators is m = n2 − 1 for SU(n), the groups SU(2)contains three gauge bosons, while SU(3) contains eight bosons carrying strong interactions.The most important difference of these non-abelian groups compared to U(1) is that the gen-erators T a ≡ T aij of such groups do not commute with each other. As a result, we may expectthat both the expression for the field-strength tensor, Eq. (4.13), and the transformation lawfor the gauge field, Eq. (4.14), becomes more complicated. In contrast, we postulate that theinteraction law jµA

µ remains valid, with the sole difference that now Aµ = AaµTa. Thus Aµ

is a Lorentz vector with values in the Lie algebra of the gauge group.

Example: SU(2) as gauge group for the weak interactions.

Charged current interactions lead e.g. in beta-decay to the transmutations n ↔ p and e− ↔ νe.

This suggests to consider e− and νe as different states of a doublet, (e−, νe) and SU(2) (or SO(3)) as

symmetry group, if weak interactions are described by a gauge interaction. These groups have three

generators, and predict therefore the existence of neutral currents. The success of the Fermi theory at

energies s ≪ G−1F ∼ m2

W /g2 tells us that these gauge bosons are massive. Thus the gauge symmetry

is broken, an effect we will have to understand latter.

We derive now the transformation laws and structure of the gauge sector, requiring that thetransformation of the fermions and their interaction with the gauge field are locally invariant.

153

7. Gauge theories

A local gauge transformation

U(x) = exp[ig

m∑

a=1

ϑa(x)T a] ≡ exp[igϑ(x)] (7.6)

changes a vector of fermion fields ψ with components ψ1, . . . , ψn as1

ψ(x)→ ψ′(x) = U(x)ψ(x) . (7.7)

Here we assumed that the fermions transform under the fundamental representation of thegroup, i.e. as doublets for SU(2), triplets for SU(3), etc. Already global gauge invariance ofthe fermion mass term requires m1 = m2 = . . . = mn and for simplicity we set mi = 0. Wecan implement local gauge invariance, if derivatives transform in the same way as ψ. Hencewe define a new covariant derivative Dµ requiring

Dµψ(x)→ [Dµψ(x)]′ = U(x)[Dµψ(x)] . (7.8)

The gauge field should compensate the difference between the normal and the covariantderivative,

Dµψ(x) = [∂µ + igAµ(x)]ψ(x) . (7.9)

In the non-abelian case, the gauge field Aµ is a matrix that is connected to its componentfields by

Aµ = AaµTa . (7.10)

We now determine the transformation properties of Dµ and Aµ demanding that (7.7) and(7.8) hold. Combining both requirements gives

Dµψ(x)→ [Dµψ]′ = UDµψ = UDµU

−1Uψ = UDµU−1ψ′ , (7.11)

and thus the covariant derivative transforms as D′µ = UDµU

−1. Using its definition (7.9), wefind

[Dµψ]′ = [∂µ + igA′

µ]Uψ = UDµψ = U [∂µ + igAµ]ψ . (7.12)

We compare now the second and the fourth term, after having performed the differentiation∂µ(Uψ). The result

[(∂µU) + igA′µU ]ψ = igUAµψ (7.13)

should be valid for arbitrary ψ and hence we arrive after multiplying from the right with U−1

at

Aµ → A′µ = UAµU

−1 +i

g(∂µU)U−1 (7.14a)

= UAµU−1 − i

gU∂µU

−1 . (7.14b)

Here we used also ∂µ(UU−1) = 0. In most cases, the gauge transformation U is an unitary

transformation and one sets U−1 = U †. A term changing as U(x)Dµ(x)U†(x) is called to

transform homogenously, while the potential Aµ is said to transforms inhomogenously.

1We suppress in the following most indices; writting them out gives e.g. ψ′

i(x) = Uij(x)ψj(x) with Uij(x) =exp[ig

∑ma=1 ϑ

a(x)T aij ].

154


Example: We can determine the transformation properties of Aµ also by demanding that (7.9)defines the interaction term in a gauge invariant way. Replacing ∂µ → Dµ in the free Lagrange densityof fermions and inserting then U−1U = 1 gives

Lf + LI = iψγµDµψ = iψγµ∂µψ − gψγµAaµT

aψ =

= iψU−1Uγµ∂µU−1Uψ − gψU−1UγµAa

µTaU−1Uψ . (7.15)

Using then ψ′ = Uψ, we obtain

Lf + LI = iψ′γµU∂µU−1ψ′ − gψ′γµUAa

µTaU−1ψ′

= iψ′γµ∂µψ′ − gψ′γµ

UAaµT

aU−1 − i

gU(∂µU

−1)

ψ′ . (7.16)

The Lagrange density Lf +LI is thus invariant, if the gauge field transforms as given in Eqs. (7.14).

Specialising to infinitesimal transformations,

U(x) = exp(igϑa(x)T a) = 1 + igϑ(x) +O(ϑ2) , (7.17)

it followsAµ(x)→ A′

µ(x) = Aµ(x)− ig[Aµ(x), ϑ(x)] − ∂µϑ(x) . (7.18)

In the abelian U(1) case, the commutator term is not present and the transformation lawreduces to the known Aµ → Aµ − ∂µϑ. For a (semi-simple) Lie group one defines

[T a, T b] = ifabcT c (7.19)

with structure constants fabc that can be chosen to be completely antisymmetric. Thus

Aaµ(x)→ Aa′µ (x) = Aaµ(x) + gfabcAbµ(x)ϑc(x)− ∂µϑa(x)

= Aaµ(x)− [δac∂µ − gfabcAbµ(x)]ϑc(x)≡ Aaµ(x)−Dac

µ ϑc(x) , (7.20)

where we introduced in the last line the covariant derivative acting on the gauge fields whichtransform according the adjoint representation of the gauge group2. In this way, the generalsubstitution rule ∂µ → Dµ can be used not only to derive the coupling of a gauge field toother fields but can be also applied to its self-interactions.

Finally, we have to derive the field strength tensor F and the Lagrange density LYM of thegauge field. The quantity F 2 requires now additionally a summation over the group index a,

LYM = −1

4F aµνF

aµν = −1

2trFµνF

µν , (7.21)

where the standard normalisation trT aT b = δab/2 for the group generators T a was assumed.The last equation shows that it is sufficient for the gauge invariance of the action that thefield-strength tensor transforms homogeneously,

F (x)→ F ′(x) = U(x)F (x)U †(x) . (7.22)

2The n complex fermion and n2 − 1 real gauge fields of SU(n) live in different representations of the group,as already the mismatch of their number indicates, see also Appendix B. Note also that the gauge trans-formations of the gauge fields have to be real, in contrast to the ones of the fermion fields.

155

7. Gauge theories

There are several ways to derive the relation between F and A. The field-strength tensorshould be anti-symmetric. Thus we should construct it out of the commutator of gauge invari-ant quantities that in turn should contain A. An obvious try is Fµν ∝ [Dµ,Dν ]. Requiringthat we reproduce the standard result for U(1), we obtain

Fµν = F aµνTa =

1

ig[Dµ,Dν ] = ∂µAν − ∂νAµ + ig[Aµ, Aν ] . (7.23)

In components, this equation reads explicitly

F aµν = ∂µAaν − ∂νAaµ − gfabcAbµAcν . (7.24)

Differential forms:A surface in R3 can be described at any point either by its two tangent vectors e1 and e2 or bythe normal n. They are connected by a cross product, n = e1 × e2, or in index notation,

ni = εijke1,jek,2 . (7.25)

In four dimensions, the ε tensor defines a map between 1-3 and 2-2 tensors. Since ε is an-tisymmetric, the symmetric part of tensors would be lost; hence the map is suited only forantisymmetric tensors.Antisymmetric tensors of rank n can be seen also as differential forms: Functions are forms oforder n = 0; differential of functions are an example of forms of order n = 1,

df =∂f

∂xidxi . (7.26)

Thus the dxi form a basis, and one can write in general

A = Ai dxi . (7.27)

For n > 1, the basis has to be antisymmetrized. Hence, a two-form as the field-strength tensoris given by

F =1

2Fµνdx

µ ∧ dxν (7.28)

with dxµ ∧ dxν = −dxν ∧ dxµ. Looking at df , we can define a differentiation of a form ω withcoefficients w and degree n as an operation that increases its degree by one to n+ 1,

dω = dwα,...,βdxn+1 ∧ dxα ∧ . . . ∧ dxβ (7.29)

Thus we have F = dA. Moreover, it follows d2ω = 0 for all forms. Hence for an abelian gaugetransformation F′ = d(A+ dχ) = F.

Since the field-strength tensor is anti-symmetric, we can use also the language of differentialforms: There are two possibilities to form a two-form out of the one-form A = Aµdx

µ

(which has values in a non-abelian Lie group): Via exterior differentiation dA and via A2 =[Aµ, Aν ]dx

µ ∧ dxν. Hence we have to fix the coefficients a and b in F = adA + bA2. (Webecome also for a moment mathematicians, using their normalisation igA→ A.)

We apply d to A→ A′ = UAU † + UdU †, obtaining

dA→ dA′ = dUAU † + UdAU † − UAdU † + dUdU † . (7.30)

156


Here we used d2 = 0 and the Leibniz rule for forms: Because of dxi ∧ dxj = −dxj ∧ dxi, therule is d(ω ∧ σ) = (dω) ∧ σ + (−1)nω ∧ dσ, if ω is a form of degree n. Thus there is a minuscommuting d with the one-form A, but no minus commuting d with U that as a function isa form of degree zero. Next we consider the transformation of A2,

A2 → A′2 = (UAU †+UdU †)(UAU †+UdU †) = UA2U †+UAdU †−dUAU †−dUdU † , (7.31)

where we used again d(UU †) = 0. Adding the two equations using a = b = 1, we see that sixterms cancels, and the two remaining ones are

dA+A2 → U(dA+A2)U † . (7.32)

Thus F = dA+A2 transforms homogeneously as required but is not gauge invariant.

7.2.2. Gauge fields as connection

There is a close analogy between the covariant derivative ∇µ introduced for a space-timecontaining a gravitational field and the gauge invariant derivative Dµ required for a space-time containing a gauge field. In the former case, the moving coordinate basis in curvedspace-time, ∂µe

ν 6= 0, introduces an additional term in the derivative of vector componentsV µ = eµ ·V . Analogously, a non-zero gauge field Aµ leads to a rotation of the basis vectors eiin group space which in turn produces an additional term ψ · (∂µei) performing the derivativeof a ψi = ψ · ei.

Let us rewrite our formulas such that the analogy between the covariant gauge derivativeDµ to the covariant space-time derivative ∇µ becomes obvious. The vector ψ of fermion fieldswith components ψ1, . . . , ψn transforming under the fundamental representation of a gaugegroup can be written as

ψ(x) = ψi(x)ei(x) . (7.33)

We can pick out the component ψj by multypling with the corresponding basis vector ej,

ψj = ψ · ej(x) . (7.34)

If the coordinate basis in group space depends on xµ, then the partial derivative of ψi acquiresa second term,

∂µψi = (∂µψ) · ei +ψ · (∂µei) . (7.35)

We can argue as in chapter 1 that (∂µψ) · ei is an invariant quantity, defining therefore asgauge invariant derivative

Dµψi = (∂µψ) · ei = ∂µψi −ψ · (∂µei) . (7.36)

The change ∂µei of the basis vector in group space should be proportional to gAµ. Setting

∂µei = −ig(Aµ)ijej (7.37)

we are back to our old notation.

157

7. Gauge theories

dx

1

4dy 3

x x+ dx

x+ dy x+ dx+ dy

2

Figure 7.1.: Parallelogram used to calculate the rotation of a test field ψi moved along a closedloop in the presence of a non-zero gauge field Aµ.

Gauge loops The correspondance between the derivatives ∇µ and Dµ suggests that we canuse the gauge field Aµ to transport fields along a curve xµ(σ). In empty space, we can usethe partial derivative ∂µψ(x) to compare fields at different points,

∂µψ(x) ∝ ψ(x+ dxµ)− ψ(x) . (7.38)

If there is an external gauge field present, the field ψ is additionally rotated in group spacemoving it from x to x+ dx,

ψ(x+ dx) = ψ(x+ dx) + igAµ(x)ψ(x)dxµ (7.39)

= ψ(x) + ∂µψ(x)dxµ + igAµ(x)ψ(x)dx

µ . (7.40)

Then the total change is

ψ(x+ dx)− ψ(x) = [∂µ + igAµ(x)]ψ(x)dxµ = Dµψ(x)dx

µ . (7.41)

Thus we can view3

Pdx(x) = 1− igAµ(x)dxµ (7.42)

as an operator which allows us to transport a gauge-dependent field the infinitesimal distancefrom x to x+ dx.

We ask now what happens to a field ψi(x), if we transport it along an infinitesimal paral-lelogram, as shown in Fig. 7.1. Calculating the path 2, we find

Pdy(x+ dx) = 1− igAν(x+ dx)dyν

= 1− igAν(x)dyν − ig∂µAν(x)dx

µdyν , (7.43)

where we Taylor expanded Aν(x+ dx). Combining the paths 1 and 2, we arrive at

Pdy(x+ dx)Pdx(x) = [1− igAν(x)dyν − ig∂µAν(x)dx

µdyν ][1− igAµ(x)dxµ]

= 1− igAµ(x)dxµ − igAν(x)dy

ν − ig∂µAν(x)dxµdyν

− g2Aν(x)Aµ(x)dyνdxµ +O(dx3) . (7.44)

Instead of performing the calculation for a round trip 1→2→3→4, we evaluate next 4→3which we then subtract from 1→ 2. In this way, we can re-use our result for 1→ 2 afterexchanging labels, Aµdx

µ ↔ Aνdyν , obtaining

Pdx(x+ dx)Pdy(x) = 1− igAν(x)dyν − igAµ(x)dx

µ − ig∂νAµ(x)dxµdyν

− g2Aµ(x)Aν(x)dxµdyν +O(dx3) . (7.45)

3Note the sign change compared to the covariant derivative where we pull-back the field from x+ dx to x.

158


The first three terms on the RHS’s of (7.44) and (7.45) cancel in the result P () for theround-trip, leaving us with

P () ≡ Pdy(x+ dx)Pdx(x)− Pdx(x+ dy)Pdy(x) =

− ig ∂µAν − ∂µAν + ig[Aµ, Aν ]dxµdyν . (7.46)

Maxwell’s equations inform us that the line integral of the vector potential equals the enclosedflux: The area of the parallelogram corresponds to dxµdyν , and the prefactor has to betherefore the field-strength tensor. If the enclosed flux is non-zero, then P ()ψi 6= ψi andthus the field is rotated.

7.2.3. Curvature of space-time

Curvature and the Riemann tensor We continue to work out the analogy between Yang-Mills theories and gravity further. Both the gauge field Aµ and the connection Γµκρ transforminhomogenously. Therefore we can not use them to judge if a gauge or gravitational field ispresent. In the gauge case, we introduced therefore the field-strength: It transforms homoge-nously and thus the statement Fµν(x) = 0 holds in any gauge. This suggests to transform(7.23) into a definition for a tensor measuring the curvature of space-time,

(∇α∇b −∇b∇α)T µ...ν... = [∇α,∇β]T µ...ν... 6= 0 , (7.47)

Thus the curvature of space-time should be proportional to the area of a loop and the amounta tensor is rotated.

For the special case of a vector V α we obtain with

∇ρV α = ∂ρVα + ΓαβρV

β (7.48)

first

∇σ∇ρV α = ∂σ(∂ρVα + ΓαβρV

β) + Γακσ(∂ρVκ + ΓκβρV

β)− Γκρσ(∂κVα + ΓαβκV

β) . (7.49)

The second part of the commutator follows from the simple relabelling σ ↔ ρ,

∇ρ∇σV α = ∂ρ(∂dVα + ΓaβσV

β) + Γακρ(∂σVκ + ΓκbσV

β)− Γκσρ(∂κVα + ΓαβκV

β) (7.50)

Now we subtract the two equations using that ∂ρ∂σ = ∂σ∂ρ and Γαβρ = Γαρβ,

[∇ρ,∇σ]V α =[∂ρΓ

αβσ − ∂σΓαβρ + ΓακρΓ

κβσ − ΓακσΓ

κβρ

]V β ≡ RαβρσV β . (7.51)

The tensor Rαβρσ is called Riemann or curvature tensor. In problem 7.5, you are asked to

show that the tensor Rαβρσ = gαγRγβρσ is antisymmetric in the indices ρ↔ σ, antisymmetric

in α↔ β and symmetric against an exchange of the index pairs (αβ)↔ (ρσ). Therefore, wecan construct out of the Riemann tensor only one non-zero tensor of rank two, contractingα either with the third or fourth index, Rραρβ = −Rραβρ. We define4 the Ricci tensor for

a pseudo-Riemannian metric by Rαβ = Rραβρ, while we set Rab = Rcabc for a Riemannian

4For a comparison of various sign conventions see (A.6) in Appendix A.

159

7. Gauge theories

metric (e.g. for the spatial part of the metric ds2 = dt2 − dl2). Then the the Ricci tensor isgiven by

Rαβ = Rραβρ = −Rραρβ = ∂βΓ

ραρ − ∂ρΓραβ + ΓσβρΓ

ρασ − ΓραβΓ

σρσ . (7.52)

A further contraction gives the curvature scalar,

R = Rαβgαβ . (7.53)

Example: Sphere S2. Calculate the Ricci tensor Rαβ and the scalar curvature R of the two-dimensional unit sphere S2.We have already determined the non-vanishing Christoffel symbols of the sphere S2 as Γφ

ϑφ = Γφφϑ =

cotϑ and Γϑφφ = − cosϑ sinϑ. We will show later that the Ricci tensor of a maximally symmetric

space as a sphere satisfies Rab = Kgab. Since the metric is diagonal, the non-diagonal elements of theRicci tensor are zero too, Rφϑ = Rϑφ = 0. We calculate with

Rab = Rcacb = ∂cΓ

cab − ∂bΓc

ac + ΓcabΓ

dcd − Γd

bcΓcad

the ϑϑ component,

Rϑϑ = 0− ∂ϑ(Γφϑφ + Γϑ

ϑϑ) + 0− ΓdϑcΓ

cϑd = 0 + ∂ϑ cotϑ− Γφ

ϑφΓφϑφ

= 0− ∂ϑ cotϑ− cot2 ϑ = 1

From Rab = Kgab, we find Rϑϑ = Kgϑϑ and thus K = 1. Hence Rφφ = gφφ = sin2 ϑ.The scalar curvature is (diagonal metric with gφφ = 1/ sin2 ϑ and gϑϑ = 1)

R = gabRab = gφφRφφ + gϑϑRϑϑ =1

sin2 ϑsin2 ϑ+ 1× 1 = 2 .

Note that our definition of the Ricci tensor guaranties that the curvature of a sphere is also positive,

if we consider it as subspace of a four-dimensional space-time.

Comparison We can push the analogy further by remembering that the field-strength definedin (7.23) is a matrix. Writing out the implicit matrix indices of Fµν in Eq. (7.23) gives

(Fµν)ij = ∂µ(Aν)ij − ∂ν(Aµ)ij + ig (Aµ)ik(Aν)kj − (Aν)ik(Aµ)kj . (7.54)

Comparing this expression to

Rαβµν = ∂µΓαβν − ∂νΓαβµ + ΓαρµΓ

ρβν − ΓαρνΓ

ρβµ (7.55)

we see that the first two indices of the Riemann tensor, α and β, correspond to the groupindices ij in the field-strength tensor. This is in line with the relation of the potential (Aµ)ijand the connection Γαβν implied by (7.37).

160

7.3. Quantization of gauge theories


7.3.1. Abelian case

We discussed already in Sec. 4.2 that we can derive the photon propagator only fixing a gauge.Now we reconsider this problem, and ask how we should modify the Lagrange density in orderto be able to derive the photon propagator. The Lagrange density that leads to the Maxwellequation is

Lem = −1

4FµνF

µν = −1

2(∂µA

ν∂µAν − ∂νAµ∂µAν)

=1

2(Aν∂µ∂

µAν −Aµ∂µ∂νAµ) =1

2Aµ [η

µν− ∂µ∂ν ]Aν

where we made as usual a partial integration dropping the surface term. Deriving the photonpropagator requires to invert the term in the square bracket. In particular, we should findthe inverse of the operator

Pµν(k) = ηµν − kµkν/k2 . (7.56)

We have already seen that this operator projects any four-vector on the three-dimensionalsubspace orthogonal to k. This ensures that the physical degrees of the photon contain nolongitudinal component.

More formally, we see that Pµν(k) is a projection operator,

PµνP λν = Pµλ , (7.57)

and has thus as only eigenvalues 0 and 1. Since Pµν(k) does not look like the the unitoperator, it has at least one zero eigenvalue and is thus not invertible. Since its trace is

Pµµ = ηµνPµν = δµµ − 1 = 3 , (7.58)

we see that three eigenvalues are one and one eigenvalue is zero. The latter eigenvaluecorresponds to kµP

µν = 0.

We can invert the operator Pµν(k), if we choose a gauge such that the subspace parallel tok is included. The simplest choice is the Lorenz gauge. Imposing this gauge on the level ofthe Lagrangian means adding the term

L → Leff = L + Lgf = L − 1

2(∂µAµ)

2 . (7.59)

More generally, we can add the term

Lgf = −1

2ξ(∂µAµ)

2 (7.60)

that depends on the arbitrary parameter ξ. This group of gauges is employed in the proofof the renormalizability of gauge theories and is therefore called Rξ gauge. The combinedeffective Lagrange density is thus

Leff = Lem + Lgf = −1

4FµνF

µν − 1

2ξ(∂µAµ)

2 =1

2Aν

[

ηµν−(

1− 1

ξ

)

∂µ∂ν]

Aµ . (7.61)

161

7. Gauge theories

To find the propagator we have to invert the term in the square brackets. Denoting this termin momentum space again by Pµν(k), we find

Pµν = −k2ηµν + (1− ξ−1)kµkν . (7.62)

This expression can be split into a transverse and a longitudinal part by factoring out termsproportional to the corresponding projection operators: Denoting now with PµνT = ηµν −kµkν/k2 the original transverse projection operator from Eq. (7.56), we find

Pµν =− k2(

PµνT +kµkν

k2

)

+ (1− ξ−1)kµkν

=− k2PµνT − ξ−1k2PµνL (7.63)

where the reminder is proportional to the longitudinal projection operator PµνL = kµkν/k2.Since PµνT and PµνL project on orthogonal subspaces, we can invert now the two terms sepa-rately, and the photon propagator becomes

iDµνF (k2) =

−iPµνTk2 + iε

+−iξPµνLk2 + iε

(7.64)

=−i

k2 + iε

[

ηµν − (1− ξ) kµkνk2 + iε

]

. (7.65)

Special cases are the Feynman gauge ξ = 1, the Landau gauge ξ = 0, while ξ →∞ correspondsto the unitary gauge. Gauge invariance implies current conservation, ∂µJ

µ(x) = 0 orkµJ

µ(k) = 0. Thus the arbitrary, ξ dependent part of the photon propagator vanishes inphysical quantities, because it is always matched between two conserved currents. For finiteξ we see that the propagator is proportional to k−2 and we expect that less problems arise inloop calculations compared to the unitary gauge.

7.3.2. Non-abelian case

In the non-abelian case, the gauge transformation (7.20) adds not only a term ∂µϑ but mixesalso the fields via the term fabcAbµϑ

c. Therefore, we cannot expect that the unphysicaldegrees of freedom simply decouple and the quantisation of non-abelian theories becomesmore challenging.

We consider first as a toy model for the generating functional of a Yang-Mills theory thetwo-dimensional integral

Z ∝∫

dxdy eiS(x) . (7.66)

Since the integration extends from −∞ to ∞, the y integration does not merely change thenormalisation of Z but makes the integral ill-defined. We can eliminate the dangerous yintegration by introducing a delta function,

Z ∝∫

dxdy δ(y)eiS(x) . (7.67)

Since the value of y in the delta function plays no role, we can replace δ(y) by δ(y − f(x))with an arbitrary function f(x). If y = f(x) is the solution of g(x, y) = 0, we obtain with

δ(g(x, y)) =δ(y − f(x))|∂g/∂y| (7.68)

162


assuming that ∂g/∂y > 0

Z ∝∫

dxdy∂g

∂yδ(g)eiS(x) . (7.69)

Generalising this to n dimensions, we need n delta functions and have to include the Jacobian,

Z ∝∫

dnxdny det

(∂gi∂yj

)∏

i

δ(gi)eiS(x) . (7.70)

We now translate this toy example to the Yang-Mills case. The functions g are the gaugefixing conditions that we choose as

ga(x) = ∂µAaµ(x)− ωa(x) , (7.71)

where the ωa(x) are arbitrary functions. The discrete index i corresponds to x, a, explainingwhy the gauge freedom results in an infinity: Although the integration measure of a compactgauge group is finite, the summation over R4 gives an infinite answer. Finally, we see thatthe infinitesimal generators ϑa correspond to the redundant coordinates yi.

The generating functional for a Yang-Mills theory is thus with DA ≡∏3i=0DAi as short-cut

Z ∝∫

DA det

(δga

δϑb

)∏

x,a

δ(ga)eiSYM , (7.72)

where we omit for the moment the source term Ls.Our task is to evaluate first δga/δϑb and then to transform the determinant and the delta

functionals into terms of similar form as our starting point, the action exp(iSYM): Only ifwe succeed to convert the determinant and the delta functionals into auxiliary fields, we canhope to use perturbation theory in the usual way.

From the connection between an infinitesimal gauge transformation and the covariantderivative given in Eq. (7.20), we obtain

ga(x)→ ga(x)− ∂µDabµ ϑ

b(x) . (7.73)

Thus the required functional derivative is

δga(x)

δϑb(y)= −∂µDab

µ δ(x− y) . (7.74)

We can eliminate the determinant remembering∫dηdη eηAη = detA from Eq. (5.127), ex-

pressing the Jacobian as a path integral over Graßmann variables ca and ca,

det

[δga(x)

δϑb(y)

]

∝∫

DcDc eiSFP . (7.75)

The corresponding Lagrangian is

LFP = −ca∂µDabµ c

b = (∂µca)(Dabµ c

b) (7.76)

= ∂µca∂µca − gfabcAcµ∂µcacb , (7.77)

where we made a partial integration and inserted the definition of the covariant derivativesacting on the gauge field, Eq. (7.20).

163

7. Gauge theories

As a result, we have recast the determinant as the kinetic energy of complex scalar fieldsca that interact with the gauge fields. Since we had to use for the scalar fields Graßmannvariables ca, their statistics is fermionic. Clearly, such fields are acceptable only as virtualparticles in loop diagrams. They should never show up as real external particles and aretherefore called Faddeev-Popov ghosts. In an abelian theory as U(1), the interaction term inEq. (7.77) is absent and ghost fields decouple. Since they change then only the normalisationof the path integral, they can be omitted all together in QED.

Next we have to eliminate the δ(ga(x)). They contain the arbitrary functions ωa(x), butthe path integral does not dependent on them. Thus we have the freedom to multiply withan arbitrary function f(ωa), thereby changing only the normalisation. Our aim is to generateafter integrating over the delta functions a term exp(iSgf), as in the case of electrodynamics.Choosing

Z → exp

(

− i

2ξ

∫

d4xωa(x)ωa(x)

)

Z , (7.78)

integrating∏

x,a δ(ga) exp

(

− i2ξ

∫d4xωa(x)ωa(x)

)

with the help of δ(ga) and (7.73), we obtain

as gauge-fixing term the desired

Lgf = −1

2ξ∂µAaµ∂

νAaν . (7.79)

The complete Lagrange density Leff of a non-abelian gauge theory consists thus of fourpieces,

Leff = LYM + Lgf + LFP + Ls . (7.80)

where the last one couples sources linearly to the fields,

Ls = JµAµ + ηc+ cη . (7.81)

We break both LYM and LFP into a piece of O(g0) defining the free propagator, and piecesof O(g) corresponding to a three gluon and a two ghost-gluon vertex, respectively, and a fourgluon vertex of O(g2). After a partial integration of the free part, we obtain

LYM + LFP =1

2Aaν (η

µν− ∂µ∂ν)Aaµ − gfabcAaµAbν∂µAcν −

1

4g2fabef cdeAaµA

bνA

cµAdν

− caca − gfabcAcµ∂µcacb . (7.82)

The Feynman rules can now be read off after Fourier transforming into momentum space, cf.problem ??. Combining the resulting expression with Lgf , we see that the gluon propagatoris diagonal in the group indices and otherwise identical to the photon propagator in Rξ gauge.The ghost propagator is the one of a mass-less scalar particle,

∆ab(k) =δab

k2 + iε. (7.83)

Being a fermion, a closed ghost loop introduces however a minus sign.

164

7.A. Appendix: Feynman rules for an unbroken gauge theory

Non-covariant gauges The introduction of ghost fields can be avoided, if one uses non-covariant gauges which depend on an arbitrary vector nµ. An example used often in QED isthe Coulomb or radiation gauge,

∂µAµ − (nµ∂

µ)(nµAµ) = 0 (7.84)

with nµ = (1, 0, 0, 0). In QCD, one employs often the axial gauge,

nµAµa = 0, a = 1, . . . , 8 (7.85)

with n2 = 0. Then the Fadeev-Popov determinant does not dependent on Aµa , and can beabsorbed in the normalisation of the path integral. While non-covariant gauges bypass theintroduction of unphysical particles in loop graphs, the resulting propagators are unhandy.Moreover, they contain spurious singularities which require care. Therefore in most applica-tions the use of the Rξ gauge is advantageous.


The Feynman rules for a non-broken Yang-Mills theory as QCD are given; for the abeliancase of QED set the structure constants fabc = 0, T = 1 and replace gs → eqf , where qf isthe electric charge of the fermion in units of the elementary charge e > 0.

Propagators

− iδab

[gµν

k2 + iǫ− (1− ξG)

kµkν(k2)2

]

(7.86)µ, a ν, b

g

δabi

k2 + iǫ(7.87)

ωa b

Triple Gauge Interactions

−gsfabc[ gµν(p1 − p2)ρ + gνρ(p2 − p3)µ

+gρµ(p3 − p1)ν ]

p1 + p2 + p3 = 0

(7.88)

µ, a ν, b

ρ, c

p1

p2

p3

165

7. Gauge theories

Quartic Gauge Interactions

−ig2s[

feabfecd(gµρgνσ − gµσgνρ)

+feacfedb(gµσgρν − gµνgρσ)

+feadfebc(gµνgρσ − gµρgνσ)]

p1 + p2 + p3 + p4 = 0

(7.89)

µ, a ν, b

ρ, cσ, d

p1 p2

p3p4

Fermion Gauge Interactions

− i gsγµT aij (7.90)

µ, a

jip1

p2

p3

Ghost Interactions

−gs fabcpµ1

p1 + p2 + p3 = 0(7.91)

µ, c

a bp1

p2

p3

Summary of chapter

Requiring local symmetry under a gauge group as SU(n) or SO(n) specifies the self-interactionsof massless gauge boson as well as their couplings to fermions and scalars. The presenceof self-interactions implies that a pure Yang-Mills theory is non-linear. The gauge invariantderivative Dµ is the analogon to the covariant derivative ∇µ of gravity, while the field-strengthcorresponds to the Riemann tensor: Both measure the rotation of a vector parallel-transportedalong a closed loop.

The quantisation of Yang-Mills theories in the covariant Rξ gauge leads to ghost particles—fermionic scalars—which compensate the unphysical degrees of freedom contained in thevector fields Aµ even after gauge fixing ∂µA

µ = 0.

Further reading

The Feynman rules are taken from [RS12]. This article contains all Standard Model Feynmanrules in a convention independent notation which allows an easy comparison of references withdiffering conventions.

166


Problems

7.1 Non-abelian Maxwell equations.Derive the non-abelian analogue of the Maxwellequations. Do gauge invariant currents andcharges exist i) locally and ii) globally?

7.2 Palatini approach.Consider the Yang-Mills action as a functional ofthe potential and the field-strength,

SYM[Aµ, Fµν ] = (7.92)

− 1

4

∫

d4xF aµν

(∂µA

aν − ∂νAa

µ − gfabcAbµA

cν

).

Derive the non-abelian Maxwell equations in thePalatini approach, i.e. by varying Aµ and Fµν in-dependently.

7.3 Adjoint representation.a.) Derive the Jacobi identity [[A,B], C] =

[A, [B,C]] − [B, [A,C]] and show that the struc-ture constants ifabc of a Lie algebra satisfy theLie algebra by themselves.b.) Insert (T a

A)bc = −ifabc for the adjoint repre-sentation in the general definition (7.9) for the co-variant derivative and show that the result agreeswith (7.20).

7.4 Hyperbolic plane H2.The line-element of the Hyperbolic plane H2 isgiven by ds2 = y−2(dx2 + dy2) with y ≥ 0.

a.) Write out the geodesic equations and deducethe Christoffel symbols Γa

bc.b.) Calculate the Riemann (or curvature) tensorRa

bcd and the scalar curvature R.

7.5 Riemann tensor.Derive the symmetry properties of the Riemanntensor and the number of its independent com-ponents for an arbitrary number of dimensions.(Hint: Simplest using an inertial system.)

7.6 Three and four gauge boson vertex.Derive the tensor structure Vrst(kρ1 , k

σ2 , k

τ3 ) of the

three-gluon vertex by Fourier-transforming thepart of LI containing three gluon fields to mo-mentum space,

F =

∫

d4p1d4p2d

4p3 (2π)4δ(p1 + p2 + p2)

× f(Aaµ(p1)A

bµ(p2)Acν(p3))

and then eliminating the fields by functionalderivatives with respect to them

V rst(kρ1 , kσ2 , k

τ3 ) =

δ3F

δArρ(k1) δA

sσ(k2) δA

tτ (k3)

Similiarly, derive the tensor structureVrst(kρ1 , k

σ2 , k

τ3 , , k

λ4 ) of the four-gluon vertex. Use

alternatively symmetry arguments, if possible.

167

8. Renormalisation

We have encountered already three examples of divergent loop integrals discussing the λφ4

theory. In these cases, it was possible to subtract the infinities in such a way that we obtainedfinite and physically sensible observables. Main aim of this chapter is to obtain a betterphysical understanding of this renormalisation procedure. We will learn that the λφ4 theoryas well as the electroweak and strong interactions of the SM are examples for renormalisabletheories: For such theories, the renormalisation of the finite number of parameters containedin the initial classical Lagrangian is sufficient to make all observables finite in any orderperturbation theory. While most of our discussion stays within our standard perturbativetreatement, we introduce in the last section a non-perturbative approach based on ideasdevelopped in solid-state physics and the renormalisation group.

Why renormalisation at all? We are using perturbation theory with free, non-interactingparticles as asymptotic states as tool to evaluate non-linear quantum field theories. Interac-tions change however the parameters of the free theory, as we know already both from classicalelectrodynamics and quantum mechanics. In the former case, Lorentz studied 1904 the con-nection between the observed electron mass mphy, its mechanical or inertial mass m0 and itselectromagnetic self-energy mel in a toy model. He described the electron as a sphericallysymmetric uniform charge distribution with radius re, obtaining

mphy = m0 +mel = m0 +4e2

5re. (8.1)

Special relativity forces us to describe the electron as a point particle: Taking thus the limitre → 0, classical electrodynamics implies an infinite “renormalisation” of the “bare” electronmass m0 by its electromagnetic self-energy mel.

Another familiar example for renormalisation appears in quantum mechanics. Perturbationtheory is possible, if the Hamilton operator H can be split into a solvable part H(0) and aninteraction λV ,

H = H(0) + λV , (8.2)

and the parameter λ is small. Using then as starting point the normalised solutions |n(0)〉 ofH(0),

H(0)|n(0)〉 = E(0)n |n(0)〉 , (8.3)

we can find the eigenstates |n〉 of the complete Hamiltonian H as a power-series in λ,

|n〉 = |n(0)〉+ λ|n(1)〉+ λ2|n(2)〉+ . . . (8.4)

Since we started with normalised states, 〈n(0)|n(0)〉 = 1, the new states |n〉 are not longercorrectly normalised. Thus going from free (or “bare”) to interacting states requires to re-normalise the states,

R 〈n|n〉R = 1 ⇒ |n〉R ≡ Z1/2 |n〉 . (8.5)

168

8.1. Anomalous magnetic moment of the electron

q

p

p′

=q

p

p′

+ kq

p

p′

1

Figure 8.1.: The general vertex for the interaction of a fermion with a photon and its pertur-bative expansion up to O(e3) within QED.

A very similar problem we encountered introducing the LSZ formalism. In the parlanceof field theory, we continue often to call this procedure wave-function renormalisation, al-though Z renormalises field operators. Note also that in non-degenerate perturbation theory,〈n(0)|n(1)〉 = 0 , so we see that Z = 1 +O(λ2).

The familiar process of renormalisation in quantum mechanics becomes for quantum fieldtheories more mysterious by the fact that the renormalisation constants are infinite. Thus wehave to regularise as first step, i.e. employing a method which makes our expressions finite,before we perform the renormalisation.


We start the chapter with the calculation of the anomalous magnetic moment of the electron.Apart from being the first successful loop calculation in the history of particle physics, thisprocess illustrates the case that quantum corrections introduce new types of interactions:While an electron on the classical level interacts via Lint = eψγµψAµ with an electromagneticfield, loop graphs add an (e3/m)ψσµνψFµν interaction. Characteristic for renormalisabletheories is that the loop integrals associated with these new interactions are finite.

Vertex function We want to write down the most general form Λµ of the coupling termbetween an external electromagnetic field and an on-shell Dirac fermion, consistent withLorentz invariance, current conservation and parity. Since p2 = p′2 = m2, the only non-trivialscalar variable in the problem is p ·p′. We choose to use the equivalent quantity q2 = (p−p′)2as the variable on which the arbitrary scalar functions in our ansatz for Λµ depend. Imposingparity conservation forbids the use of γ5. Hence the most general ansatz compatible withLorentz invariance and parity is

Λµ(p, p′) = A(q2)γµ +B(q2)pµ + C(q2)p′µ +D(q2)σµνpν +E(q2)σµνp′ν . (8.6)

Current conservation requires qµΛµ(p, p′) = 0 and leads to C = B and E = −D. Hence

Λµ(p, p′) = A(q2)γµ +B(q2)(pµ + p′µ) +D(q2)σµνqν . (8.7)

Hermeticity finally requires that A,B are real and D is purely imaginary.

169

8. Renormalisation

Gordon decomposition We derive now an identity that allows us to eliminate one of thethree terms in Eq. (8.7), if we sandwich Λµ between two spinors which are on-shell. Weevaluate

Fµ = u(p′)[p/′γµ + γµp/

]u(p) (8.8)

first using the Dirac equation for the two on-shell spinors, finding

Fµ = 2mu(p′)γµu(p) . (8.9)

Secondly, we can use γµγν = ηµν − iσµν , obtaining

Fµ = u(p′)[(p′ + p)µ + iσµν(p′ − p)ν

]u(p) . (8.10)

Equating (8.9) and (8.10) gives the Gordon identity: It allows us to separate the Dirac currentinto a part proportional to (p + p′)µ, i.e. with the same structure as a scalar current, and apart vanishing for q = p′ − p→ 0 which couples to the spin of the fermion,

u(p′)γµu(p) = u(p′)

[(p′ + p)µ

2m+

iσµν(p′ − p)ν2m

]

u(p) . (8.11)

Moreover, the Gordon identity shows that the three terms in Eq. (8.7) are not independent.Depending on the context, we can eliminate therefore the most annoying term. We followconventions and introduce the (real) form-factors F1(q

2) and F2(q2) by

Λµ(p, p′) = F1(q2)γµ + F2(q

2)iσµνqν2m

= (8.12)

= F1(q2)(p′ + p)µ

2m+ [F1(q

2) + F2(q2)]

iσµνqν2m

. (8.13)

The form-factor F1 is the coefficient of the electric charge, eF1(q2)γµ, and should thus go to

one for small momentum transfer, F1(0) = 1. Therefore the magnetic moment of an electronis shifted by 1 + F2(0) from the tree-level value g = 2 you derived in problem 5.3. Thedeviation a ≡ (g− 2)/2 is called anomalous magnetic moment, the two form-factors are oftencalled electric and magnetic form-factors.

Note the usefulness of the procedure to express the vertex function using only generalsymmetry requirements but not a specific theory for the interaction: Equation (8.12) allowsexperimentalists to present their measurements using only two scalar functions which in turncan be easily compared to predictions of specific theories.

Anomalous magnetic moment After having discussed the general structure of the electro-magnetic vertex function, we turn now to its calculation in perturbation theory for the caseof QED. The Feynman diagrams contributing to the matrix element at O(e3) with wave-functions as external lines are shown in Fig. 8.1. We separate the matrix element into thetree-level part and the one-loop correction, −ieu(p′) [γµ + Γµ]u(p). Using the Feynman gaugefor the photon propagator, we obtain

Γµ(p, p′) =∫

d4k

(2π)4−i

k2 + iε(−ieγν) i

p/′ − k/ −m+ iεγµ

i

p/− k/ −m+ iε(−ieγν) . (8.14)

This integral is logarithmically divergent for large k,∫ Λ

dkk3

k2k2∝ ln Λ . (8.15)

170


Before we perform the explicit calculation, we want to understand if this divergence is con-nected to a specific kinematical configuration of the momenta. We split therefore the vertexcorrection into an on-shell and an off-shell part,

Γµ(p, p′) = Γµ(p, p) + [Γµ(p, p′)− Γµ(p, p)] ≡ Γµ(p, p) + Γµoff(p, p′) . (8.16)

Next we rewrite1 the first fermion propagator for small p′ − p as

1

p/′ − k/ −m =1

p/− k/ −m+ (p/′ − p/) = (8.17)

=1

p/− k/ −m −1

p/− k/ −m(p/′ − p/) 1

p/ − k/ −m + . . . (8.18)

The first term of this expansion leads to the logarithmic divergence of the loop integral forlarge k. In contrast, the reminder of the expansion that vanishes for p′ − p = q → 0 containsadditional powers of 1/k and is thus convergent. Hence the UV divergence is contained solelyin the on-shell part of the vertex correction, while the function Γµoff(p, p

′) = Γµ(p, p′)−Γµ(p, p)is well-behaved. Moreover, we learn from Eq. (8.12) that the divergence is confined to F1(0),while F2(0) is finite. This is good news: The divergence is only connected to a quantityalready present in the classical Lagrangian, the electric charge. Thus we can predict thefunction Γµ(p, p′) for all values p′ 6= p, after we have renormalised the electric charge in thelimit of zero momentum transfer.

We now calculate the vertex function (8.14) explicitly. We set

Γµ(q) = −ie2∫

d4k

(2π)4N µ(k)

D(8.19)

withN µ = γν(p/′ + k/ +m)γµ(p/+ k/ +m)γν (8.20)

andD =

[(p′ + k)2 −m2

] [(p+ k)2 −m2

]k2 . (8.21)

Then we combine the propagators introducing Feynman parameter integrals,

Γµ(q) = −ie2∫

d4k

(2π)4

∫ 1

0dα

∫ 1−α

0dβ

2N µ(k)

k2 + α[(p′ + k)2 −m2 − k2] + β[(p + k)2 −m2 − k2]3 .(8.22)

The complete calculation of the vertex function (8.14) for arbitrary off-shell momenta isalready quite cumbersome. In order to shorten the calculation, we restrict ourselves thereforeto the part contributing to the magnetic moment F2(0). Because of

Λµ(p, p′) =[F1(q

2) + F2(q2)]γµ − F2(q

2)(p′ + p)µ

2m(8.23)

we can simplify the calculation of N µ(k), throwing away all terms proportional to γµ whichdo not contribute to the magnetic moment. Moreover, we can consider the limit that theelectrons are on-shell and the momentum transfer to the photon vanishes.

1The identity1

A+B=

1

A−B

1

AB + . . .

can be checked by multiplying with A+B.

171

8. Renormalisation

Using the on-shell condition, p2 = p′2 = m2, the two square brackets simplify to 2p′ · k and2p · k, respectively,

D =k2 + 2k · (αp′ + βp)

3. (8.24)

Next we eliminate the term linear in k completing the square,

D =

(k + αp′ + βp︸︷︷︸

l

)2 − (αp′ + βp)23

=[l2 − (α2m2 + β2m2 + 2αβp′ · p)

3. (8.25)

Since the momentum transfer to the photon vanishes, q2 = 2m2 − 2p′ · p→ 0, we can replacep′ · p→ m2 and obtain as final result for the denominator

D =l2 − (α+ β)2m2

3. (8.26)

Now we move on to the evaluation of the nominator N µ(k). Performing the change of ourintegration variable from k = l − (αp′ + βp) to l, the nominator becomes

N µ(l) = γν(P/ ′ + l/+m)γµ(P/ + l/+m)γν (8.27)

with P/ ′ ≡ (1− α)p/′ − βp/ and P/ ≡ (1− β)p/− αp/′.Multiplying out the two brackets and ordering the result according to powers of m, we

observe first that the term ∝ m2 leads to ∝ γµ and thus does not contribute to F2(0). Nextwe split further the term linear in m according to powers of l: The term linear in l vanishesafter integration, while the term ml/0 results in

m(γνP/ ′γµγν + γνγµP/γν) = 4m(P ′µ + Pµ) = 4m[(1 − 2α)p′µ + (1− 2β)pµ] . (8.28)

Using the symmetry in the integration variables α and β, we can rewrite this expression as

→ 4m[(1− α− β)(p′µ + pµ)] . (8.29)

We split the m0 term in the same way according to the powers of l/. The m0 l/2 term gives aγµ term, the m0 l/ vanishes after integration, and the m0 l/0 gives after some work

γνP/ ′γµP/γν → 2m[α(1 − α) + β(1− β)](p′ + p)µ . (8.30)

Finally, the m0 term contributes to the anomalous magnetic moment

→ −2m(p′ + p)µ[2(1 − α)(1− β)] . (8.31)

Combining all terms, we find

N µ = 4m(1− α− β)(p′ + p)µ + 2m[α(1 − α) + β(1− β)](p′ + p)µ

− 4m(1− α)(1 − β)(p′ + p)µ =

= 2m[(1 − α− β)(α + β)](p′ + p)µ . (8.32)

Thus

Γµ2 (0) = −2ie2∫

dαdβ

∫d2ωl

(2π)2ωN µ

[l2 − (α+ β)2m2]3, (8.33)

where the subscript 2 indicates that we account only for the contribution to the anomalousmagnetic moment.

172


We can perform now the l integral using the formula (??) for I(ω, a) with ω = 2 and a = 3,

I(2, 3) =i

32π21

(α+ β)2m2 + iε, (8.34)

obtaining as expected a finite result. As last step, we do the integrals over the Feynmanparameters α and β,

∫ 1

0dα

∫ 1−α

0dβ

1− α− βα+ β

=1

2(8.35)

and find thus

Γµ2 (0) =e2

8π21

2m(p′ + p)µ . (8.36)

We have reproduced the result of the first successful calculation of a loop correction in aquantum field theory, performed by Schwinger 1948, F2(0) = α/2π. Together with Bethe’sprevious estimate of the Lamb shift in the hydrogen energy spectrum, this stimulated theview that a consistent renormalisation of QED is possible.

The currently most precise experimental value for the electron anomalous magnetic momentae ≡ F2(0) is

aexpe = 0.001 159 652 180 73(28)[0.24 ppb] . (8.37)

The calculation of the universal (i.e. common to all charged leptons) QED contribution hasbeen completed up to fourth order. There exists also an estimate of the dominant fifth ordercontribution,

auniℓ = 0.5(α

π

)

− 0.328 478 965 579 193 78 . . .(α

π

)2

+1.181 241 456 587 . . .(α

π

)3− 1.9144(35)

(α

π

)4+ 0.0(4.6)

(α

π

)5

= 0.001 159 652 176 30(43)(10)(31) · · · (8.38)

The three errors given in round brackets are: The error from the uncertainty in α, thenumerical uncertainty of the α4 coefficient and the error estimated for the missing higherorder terms [Jeg07].

Comparing the measured value and the prediction using QED, we find an extremely goodagreement. First of all, this is strong support that the methods of perturbative quantum fieldtheory we developed so far can be successfully applied to weakly coupled theories as QED.Secondly, it means that additional contributions to the anomalous magnetic moment of theelectron have to be tiny.

Electroweak and other corrections The lowest order electroweak corrections to the anoma-lous magnetic moment contain in the loop virtual gauge bosons (W±, Z) or a higgs boson hand are shown in Fig. 8.2. We will consider the electroweak theory describing these diagramsonly later; for the present discussion it is sufficient to know that the weak coupling constant isg ∼ 0.6 and that the scalar and weak gauge bosons are much heavier than leptons, M ≫ m.

The second diagram corresponds schematically to the expression

∼ g2∫

d4k

(2π)41

k2 −M2

A(m2, k)

[(p − k)−m2]2. (8.39)

173

8. Renormalisation

As in QED, this integral has to be finite and we expect that it is dominated by momentaup to the mass M of the gauge bosons, k <∼M . Therefore its value should be proportionalto g2m2/M2 (times a possible logarithm ln(M2/m2)) and electroweak corrections to theanomalous magnetic moment of the electron are suppressed by a factor (m/M)2 ∼ 10−10

compared to the QED contribution. The property that the contribution of virtual heavyparticles to loop processes is suppressed in the limit |q2| ≪M2 is called “decoupling”. Notethe difference to the case of the mass of a scalar particle or the cosmological constant: In theseexamples, the loop corrections are infinite and we cannot predict these quantities. However,we consider it as unnatural that the measured value ρvac is so much smaller than the expectedminimal value Λ4 >∼ (TeV)4. In contrast, the anomalous magnetic moment is finite but, aswe include loop momenta up to infinity, depends in principal on all particles coupling tothe electron, even if they are arbitrarily heavy. Only if these heavy particles “decouple,” wecan calculate ae without knowing e.g. the physics at the Planck scale. Thus the decouplingproperty is a necessary ingredient of any theory of high energy physics, otherwise nothing likechemistry or solid state physics would be possible before knowing the “theory of everything”.

Clearly, the contribution of heavy particles (either electroweak gauge and higgs bosonsor other not yet discovered particles) is more visible in the anomalous magnetic momentof the muon than of the electron. Moreover, a relativistic muon lives long enough that ameasurement of its magnetic moment is feasible. This is one example how radiative corrections(here evaluated at q2 = 0) are sensitive to physics at higher scales M : If an observable canbe measured and calculated with high enough precision, one can be sensitive to suppressedcorrections of order g2m2/M2. Other examples are rare processes like µ→ e+γ or Bs → µ+µ−

which are suppressed by a specific property of the SM which one does not expect to hold ingeneral. The achieved precision in measuring and calculating such processes is high enoughto probe generically scales of M ∼ 100TeV, i.e. much higher than the mass scales that canbe probed directly at current accelerators as LHC. For an overview see [Jeg07].

W W

νµ Z Hµ

γa) b) c)

Figure 8.2.: Lowest order electroweak corrections to the anomalous magnetic moment offermions.

Finite versus divergent parts of loop corrections We found that the vertex correction couldbe split into two parts

Λµ(p, p′) = F1(q2)γµ + F2(q

2)iσµνqν2m

, (8.40)

where the form factor F2(q2) is finite for all q2, while the form factor F1(q

2) diverges forq2 → 0. The important observation is that F2(q

2) corresponds to a Lorentz structure thatis not present in the original Lagrangian of QED. This suggests that we can require from a“nice” theory that

174

8.2. Power counting and renormalisability

• all UV divergences are connected to structures contained in the original Lagrangian,all new structures are finite. The basic divergent structures are also called “primitive”divergent graphs.

• If there are no anomalies, then loop corrections respect the original (classical) symme-tries. Thus, e.g., the photon propagator should be at all orders transverse, respectinggauge invariance. We will see that as consequence the high-energy behaviour of thetheory improves.

In such a case, we are able to hide all UV divergences in a renormalisation of the originalparameters of the Lagrange density.

8.2. Power counting and renormalisability

We try to make the requirements on a “nice” theory a bit more precise. Let us consider theset of λφn theories in d = 4 space-time dimensions and check which graphs are divergent.We define the superficial degree D of divergence of a Feynman graph as the difference be-tween the number of loop momenta in the nominator and denominator of a Feynman graph.We can restrict our analysis to those diagrams called 1P irreducible (1PI) which cannot bedisconnected by cutting an internal line: All 1P reducible diagrams can be decomposed into1PI diagrams which do not contain common loop integrals and can be therefore analysedseparately. Moreover, we are only interested in the loop integration and define therefore the1PI Green functions as graphs where the propagators on the external lines were stripped off.In d = 4 space-time dimensions, the degree D of divergence of a 1PI Feynman graph is thus

D = 4L− 2I , (8.41)

where L is the number of independent loop momenta and I the number of internal lines.The former contributes a factor d4p, while the latter corresponds to a scalar propagator with1/(p2 −m2) ∼ 1/p2 for p→∞.

Momentum conservation at each vertex leads for an 1PI-diagram to

L = I − (V − 1) , (8.42)

where V is the number of vertices and the −1 takes into account the delta function leadingto overall momentum conservation. Thus

D = 2I − 4V + 4 . (8.43)

Each vertex connects n lines and any internal line reduces the number of external lines bytwo. Therefore the number E of external lines is given by

E = nV − 2I . (8.44)

As result, we can express the superficial degree D by the order of perturbation theory (V ),the number of external lines E and the degree n of the interaction polynomial φn,

D = (n− 4)V + 4− E . (8.45)

From this expression, we see that

175

8. Renormalisation

• for n = 3, the coefficient of V is negative. Therefore only a finite number of terms inthe perturbative expansion are infinite. Such a theory is called super-renormalisable,the corresponding terms in the Hamiltonian are also called relevant.

• For n = 4, we find D = 4 − E. Thus the degree of divergence is independent of theorder of perturbation theory being only determined by the number of external lines.Such theories contain an infinite number of divergent graphs, but they all correspond toa finite number of divergent structures—the so-called primitive divergent graphs. Suchinteractions are also called marginal and are candidates for a renormalisable theory.

• Finally, for n > 4 the degree of divergence increases with the order of perturbation the-ory. As result, there exists an infinite number of divergent structures, and increasing theorder of perturbation theory requires more and more input parameter to be determinedexperimentally. Such a theory is called non-renormalisable, the interaction irrelevant.

In particular, the λφ4 theory as an example for a renormalisable theory has only threedivergent structures: i) the case E = 0 and D = 4 corresponding a contribution to thecosmological constant, ii) the case E = 2 and D = 2 corresponding to the self-energy, andiii) the case E = 4 and D = 0, i.e. logarithmically divergence, to the four-point function. Aswe saw in chapter 2, the three primitive divergent diagrams of the λφ4 theory correspond tothe following physical effects: Vacuum bubbles renormalise the cosmological constant. Theeffect of self-energy insertions is twofold: Inserted in external lines it renormalises the field,while self-energy corrections in internal propagators lead to a renormalisation of its mass.The vertex correction finally renormalises the coupling strength λ.

Let us move to the case of QED. Repeating the discussion, we obtain the analogue toEq. (8.45), but accounting now for the different dimension of fermion and bose fields,

D = 4−B − 3

2F , (8.46)

where B and F count the number of external bosonic and fermionic lines, respectively. Thereare six different superficially divergent primitive graphs in QED: The photon and the fermioniccontribution to the cosmological constant (D = 4), the vacuum polarisation (D = 2), thefermion self-energy (D = 1), the vertex correction (D = 0) and light-by-light scattering(D = 0).

In a gauge theory, the true degree of divergence can be smaller than the superficial one.For instance, light-by-light scattering corresponds to a term L ∼ A4 that violates gaugeinvariance. Thus either the gauge symmetry is violated by quantum corrections or such aterm is finite.

Because of the correspondence of the dimension of a field and the power of its propagator,we can connect the superficial degree of divergence of a graph to the dimension of the couplingconstants at its vertices. The superficial degree D(G) of divergence of a graph is connectedto the one of its vertices Dv by

D(G)− 4 =∑

v

(Dv − 4) (8.47)

which in turn depends as

Dv = δv +3

2fv + bv = 4− [gv] (8.48)

176

8.3. Renormalisation of the λφ4 theory

on the dimension of the coupling constant g at the vertex v. Here, fv and bv are the numberof fermion and boson fields at the vertex, while δv counts the number of derivatives.

Thus the dimension of the coupling constant plays a crucial role deciding if a certain theoryis “nice” in the naive sense defined above. Clearly D = 0 or [g] = 0 is the border-line case:

• If at least one coupling constant has a negative mass dimension, [g] < 0 and Dv > 4,the theory is non-normalisable. Examples are the Fermi theory of weak interactions,[GF ] = −2, and gravitation, [GN ] = −2.

• If all coupling constants have positive mass dimension, [g] > 0 and Dv < 4, the theoryis super-normalisable. An example is the λφ3 theory in D = 4 with [λ] = 1.

• The remaining cases, with all [gi] = 0, are candidates for renormalisable theories. Ex-amples are Yukawa interactions, λφ4, Yang-Mills theories that are unbroken (QED andQCD) or broken by the Higgs mechanism (electroweak interactions).

Theories with massive gauge bosons We have assumed that the propagators of bosonsbehave as 1/(p2 − m2) ∼ 1/p2 for p → ∞. This is true for scalars and for massless spin-1particles like the photon and gluons. In contrast, the longitudinal part in the propagator ofa massive spin-1 particles leads to a worse asymptotic behaviour. Including a mass term forgauge bosons breaks gauge invariance and leads to a non-renormalisable theory. A solutionto this problem is the introduction of gauge boson masses via interactions that respect thegauge symmetry as in the Higgs mechanism.

Furry’s theorem We have not included in our list of primitive divergent graphs of QEDloops with an odd number of fermions, i.e. the tadpole (B = 1 and D = 3) and “photonsplitting” graph (B = 3 and D = 1). Such diagrams vanish in vacuum, as fact known asFurry’s theorem. Since the momentum flow in a loop graph plays no rule, we can write aloop as “1/2(clockwise+ anti-clockwise)”. Then we insert CC−1 = 1 between all factors inthe trace, use CγµC−1 = −γµT and CSF (−x)C−1 = STF (x). Hence for an odd number ofpropagators, the two contributions cancel.

Alternatively, we can use a symmetry argument to convince us that all diagrams withan odd number of photons are zero: Because the QED Langrangian is invariant under C,ψ → ψc = CψT and Aµ =→ −Aµ, all Green functions with an odd number of photonsvanish.


We have argued that a theory with dimensionless coupling constant is renormalisable, i.e.that a multiplicative shift of the parameters contained in the classical Lagrangian is sufficientto obtain finite Green functions. The simplest theory of this type in D = 4 is the λφ4 theoryfor which we will discuss now the renormalisation procedure at one loop level. As starter, weexamine the general structure of the divergences and review different regularisation options.

8.3.1. Renormalisation and regularisation

Structure of the divergences We learnt that the degree of divergence decreases increasingthe number of external lines, since the number of propagators increases. The same effect has

177

8. Renormalisation

taking derivatives wrt external momenta p,

∂

∂p/

1

k/ + p/−m = − 1

(k/ + p/−m)2.

This means that

1. we can Taylor expand loop integrals, confining the divergences in the lowest order terms.Choosing e.g. p = 0 as expansion point in the fermion self-energy,

Σ(p) = A0 +A1p/+A2p2 + . . .

with

An =1

n!

∂n

∂p/nΣ(p) ,

we know that A0 is (superficially) linear divergent. Thus A1 can be maximally loga-rithmic divergent, while all other An are finite.

2. We could choose a different expansion point, leading to different renormalisation con-ditions (within the same regularisation scheme).

3. The divergences can be subtracted by local operators, i.e. polynomials of the fields andtheir derivatives.

The last statement requires that non-local terms as e.g. ln(p2/µ2) which can be generated by sub-loops in a diagram of order gn are cancelled by counter-terms of order n′ < n. A sketch why thisshould be true goes as follows:Green functions become singular for coinciding points, i.e. when the convergence factor e−kx in theEuclidean Green function becomes one. In the simplest cases (separate divergences in one diagram)as

〈0|φ(x′)φ(x)|0〉x′→x =

∫d3k

(2π)32ωke−ik(x′−x)

∣∣∣∣x′→x

=

∫d3k

(2π)32ωk, (8.49)

the infinities are eliminated by normal ordering. More complicated are overlapping divergences wheretwo or more divergent loops share a propagator. Wilson suggested to expand the product of two fieldsas the sum of local operator Oi times coefficient functions Ci(x− y) as

φ(x)φ(y) =∑

i

Ci(x− y)Oi(x) ,

where the whole spatial dependence is carried by the coefficients. For a massless scalar field, dimen-sional analysis dictates that Ci(x) ∝ x−2+di , if the local operator Oi has dimension di. Note thatonly the unity operator has a singular coefficient function 1/x2 corresponding to the massless scalarpropagator. Similarly we can expand product of operators,

On(x)Om(y) =∑

i

Cinm(x− y)Oi(x) ,

where now Cinm(x) ∝ x−dn−dm+di . Thus we can use this operator product expansion (or briefly

“OPE”) to rewrite the overlapping divergences in terms of coefficient functions and finite local oper-

ators. Moreover, the sub-divergence occurring at order k, when p < k points coincide, have the form

found at order p. We conclude that non-local terms due to overlapping divergencies are cancelled by

the counter terms found at lower order.

178


Regularisation schemes Mathematical manipulations as shifting the integration variable indivergent loop integral are only well-defined, if we convert first these integrals into convergentones. This process is called regularisation and can be done in a number of ways. In general,one reparametrises the integral in terms of a parameter Λ (or ε) called regulator such thatthe integral becomes finite for finite regulator while the limit Λ → ∞ (or ε → 0) returns tothe original integral.

• We can avoid UV divergences evaluating loop-integrals introducing an (Euclidean) mo-mentum cutoff,

I → IΛ ≡∫ Λ

d4k F (k) = A (Λ) +B + C

(1

Λ

)

.

Somewhat more sophisticated, we could introduce instead of a hard cutoff a smoothfunction which suppresses large momenta. Using Schwinger’s proper-time representation

1

p2 +m2=

∫ ∞

0ds e−s(p

2+m2) (8.50)

we can cut-off large momenta setting

1

p2 +m2→ e−(p2+m2)/Λ2

p2 +m2=

∫ ∞

Λ−2

ds e−s(p2+m2) . (8.51)

Although conceptual very easy, both regularisation schemes violate generically all sym-metries of our theory. This is not a principal flaw, since we should recover these sym-metries in the limit Λ → ∞. But calculations become much thougher since we cannotuse the symmetries at intermediate steps, and therefore these schemes are in practisenot useful except for the simplest cases.

• Pauli-Villars regularisation is a scheme respecting gauge invariance of QED. Its basicidea is to add heavy particles coupled gauge invariantly to the photon,

1

k2 −m2 + iε→ 1

k2 −m2 + iε+∑

i

aik2 −M2 + iε

.

For k2 ≪ M2, physics is unchanged, while for k2 ≫ M2 and ai < 0 the combinedpropagators scale as M2/k4 and the convergence of loop integrals improves. Since theheavy particles enter with the wrong sign, they are unphysical and serve only as amathematical tool to regularise loop diagrams.

• Dimensional Regularisation (DR) is one of the most useful and least intuitive regulators.The reason for its usefulness is that it preserves gauge invariance. In DR, we replaceour integrals with

I → Id ≡∫ ∞

0ddkF (k)

where d is the dimension of our measure. As the example∫ddk k−2 = 0 which we

will discuss later shows the “measure” we implement by physical requirements with DRis not a positive measure—as a mathematician would require. Nevertheless, we can

179

8. Renormalisation

perform integrals in d 6= 4 space-time dimensions. Then we can replace d → 4 − ε inthe result and take the limit ε→ 0, splitting the result into poles and finite parts.

Evaluating fermion traces in the denominator using DR, we have to extend the Cliffordalgebra to d dimensions. A natural choice is tr(γµγν) = dηµν . Problematic is howeverthe treatment of γ5 ≡ iγ0γ1γ2γ3 relying heavily on d = 4. As a result DR breaks chiralsymmetry. The same is true for supersymmetry.

• In lattice regularisation one replaces the continuous space-time by a discrete lattice.The finite lattice spacing a introduces a momentum cutoff, eliminating all UV diver-gences. Moreover the (Euclidean) path-integral becomes well-defined and can be calcu-lated numerically without the need to do perturbation theory. Thus this approach isparticularly useful in the strong-coupling regime of QCD. It has been very successful inpredicting many results of low-energy QCD, including hadron masses and form-factors,which otherwise are only calculable in low-energy approximations to QCD. Note thatlattice regularisation for finite a respects the gauge symmetry, but spoils translationand Lorentz symmetry of the underlying QFT. Nevertheless, one recovers in the limita → 0 a relativistic QFT. A longstanding problem of lattice theory was how to imple-ment correctly chiral fermions. This question has been recently solved and thus the SMcan be now in a mathematically consistent, non-perturbative way defined as a latticetheory.

• Various other regularisation schemes as e.g. zeta function regularisation or point split-ting methods exists.

Even fixing a regularisation scheme, e.g. DR, we can choose various renormalisation condi-tions. Three popular choices are

• on-shell renormalisation. In this scheme, we choose the subtraction such that the on-shell masses and couplings coincide with the corresponding values measured in processeswith zero momentum transfer. For instance, we define the renormalised electric chargevia the Thomson limit of the Compton scattering amplitude. While this choice is veryintuitive, it is not practical for QCD.

• the minimal subtraction (MS) scheme, where we subtract only the divergent 1/ε poles.

• the modified minimal subtraction MS (read em-es-bar) scheme, where we subtract ad-ditionally the ln(γE) + 4π term appearing frequently. This scheme gives more compactexpressions than the others and is most often used in theoretical calculations. Asdrawback, quantities like mMS

e have to be translated into the physical mass me.

8.3.2. The λφ4 theory at one-loop

There are two equivalent ways to perform perturbative renormalisation. In the one we usefirst called often “conventional” perturbation theory we use the “bare” (unrenormalised)parameters in the Lagrangian,

L = L0 + Lint =1

2(∂µφ0)

2 − 1

2m2

0φ20 −

λ04!φ40 . (8.52)

Then we introduce a renormalised field φR = Z−1/2φ φ0 and choose the parameters Zφ,m0 and

λ0 as function of the regularisation parameter (ε, Λ, . . .) such that the field φR has finite

180


Green functions. In the following we discuss the renormalisation procedure at one-loop levelfor the Green functions of the λφ4 theory in this scheme. Since any 1P reducible diagramcan be decomposed into 1PI diagrams which do not contain common loop integrals, we canrestrict our analysis again to 1PI Green functions.

Mass and wave-function renormalisation The Feynman propagator in momentum space is

i∆F (p) =

∫

d4x eipx 〈0|Tφ0(x)φ0(0) |0〉 =i

p2 −m20 + iε

(8.53)

at lowest order perturbation theory. The full propagator i∆F (p) is the sum of an infinitechain of self-energy insertions,

i∆F (p) =i

p2 −m20 + iε

+i

p2 −m20 + iε

(−iΣ(p2)

) i

p2 −m20 + iε

+ . . . (8.54)

This expression looks like an infinite tower of independently propagating particle with differentmasses. But if we sum up the terms of this binomial series, we obtain

i∆F (p) =i

p2 −m20 + iε

1

1 + iΣ(p2) ip2−m2

0+iε

=i

p2 −m20 − Σ(p2) + iε

. (8.55)

Now we see that the effect of the interactions results in a shift of the particle mass, m20 →

m20 + Σ(p2). Next we have to show that Σ(p2) is finite after renormalisation. The 1-loop

expression is

− iΣ(p2) =−iλ02

∫d4k

(2π)4i

k2 −m20 + iε

, (8.56)

i.e. quadratically divergent. As a particularity of the φ4 theory, the p2 dependence of theself-energy Σ shows up only at the 2-loop level.

We perform a Taylor expansion of Σ(p2) around the arbitrary point µ,

Σ(p2) = Σ(µ2) + (p2 − µ2)Σ′(µ2) + Σ(p2) , (8.57)

where Σ(µ2) ∝ Λ2, Σ′(µ2) ∝ ln Λ and Σ(p2) is the finite reminder. A term linear in Λ isabsent, since we cannot construct a Lorentz scalar out of pµ. Note also Σ(µ2) = Σ′(µ2) = 0.

Now we insert (8.57) into (8.55),

i

p2 −m20 − Σ(p2) + iε

=i

p2 −m20 − Σ(µ2)

︸︷︷︸

p2−µ2

−(p2 − µ2)Σ′(µ2)− Σ(p2) + iε, (8.58)

where we see that we can identify µ with the physical mass given by the pole of the propagator.

We aim at rewriting the remaining effect for p2 → µ2 of the self-energy insertion, Σ(µ2), asa multiplicative rescaling. In this way, we could remove the divergence from the propagatorby a rescaling of the field. At leading order in λ, we can write

Σ(p2) =[1− Σ′(µ2)

]Σ(p2) +O(λ20) (8.59)

181

8. Renormalisation

and thus

i∆F (p) =1

1− Σ′(µ2)i

p2 − µ2 − Σ(p2) + iε=

iZφ

p2 − µ2 − Σ(p2) + iε(8.60)

with the wave-function renormalisation constant

Zφ =1

1− Σ′(µ2)= 1 + Σ′(µ2) . (8.61)

Close to the pole, the propagator is the one of a free particle with mass µ,

i∆F (p) =iZφ

p2 − µ2 + iε+O(p2 − µ2) . (8.62)

Thus the renormalisation constant Zφ is the same Z appearing in the LSZ formula. As faras S-matrix elements are concerned, the renormalisation of external lines has therefore thesole effect of cancelling the Z−1 factor from the LSZ formula. Therefore we can obtain S-matrix elements replacing propagators on external lines directly with on-shell wave-functions,excluding at the same time all diagrams where external lines are renormalised.

We define the renormalised field φ = Z−1/2φ φ0 such that the renormalised propagator

i∆R(p) =

∫

d4x eipx 〈0|Tφ(x)φ(0) |0〉 = Z−1φ i∆(p) =

i

p2 − µ2 − Σ(p2) + iε(8.63)

is finite. Similarly, we define renormalised n-point functions by

G(n)R (x1, . . . , xn) = 〈0|Tφ(x1) · · ·φn(xn) |0〉 = Z

−n/2φ G

(n)0 (x1, . . . , xn) . (8.64)

Since the 1PI n-point Green functions miss n field renormalisation constant compared toconnected n-point Green functions, the connection between renormalised and bare 1PI n-pointfunctions becomes

Γ(n)R (x1, . . . , xn) = Z

n/2φ Γ

(n)0 (x1, . . . , xn) . (8.65)

This induces a corresponding sign change in the RGE for 1PI Green functions.

Coupling constant renormalisation We can choose an arbitrary point inside the kinematicalregion, s + t + u = 4µ2 and s ≥ 4µ2, to define the coupling. For our convenience and lesswriting work, we choose instead the symmetric point

s0 = t0 = u0 =4µ2

3.

The bare four-point 1PI Green function is (see section 2.7.3)

Γ(4)0 (s, t, u) = −iλ0 + Γ(s) + Γ(t) + Γ(u) , (8.66)

the renormalised four-point function at (s0, t0, u0) is

Γ(4)R (s0, t0, u0) = −iλ . (8.67)

182


Next we expand the bare 4-point function around s0, t0, u0,

Γ(4)0 (s, t, u) = −iλ0 + 3Γ(s0) + Γ(s) + Γ(t) + Γ(u) (8.68)

where the Γ(x) are finite and zero at x0. Now we define a vertex (or coupling constant)renormalisation constant by

− iZ−1λ λ0 = −iλ0 + 3Γ(s0) (8.69)

Inserting this definition in (8.68) we obtain

Γ(4)0 (s, t, u) = −iZ−1

λ λ0 + Γ(s) + Γ(t) + Γ(u) (8.70)

which simplifies at the renormalisation point to

Γ(4)0 (s, t, u) = −iZ−1

λ λ0 . (8.71)

We use now the connection between renormalised and bare Green functions,

Γ(4)R (s, t, u) = Z2

φΓ(4)0 (s, t, u) (8.72)

and thus

− iλ = Z2φZ

−1λ (−iλ0) . (8.73)

The relation between the renormalised and bare coupling in the λφ4 theory is thus

λ = Z2φZ

−1λ λ0 . (8.74)

Now we have to show that Γ(4)R (s, t, u) is finite.

Γ(4)R (s, t, u) = Z2

φΓ(4)0 (s, t, u)

= −iZ2φZ

−1λ λ0 + Z2

φ[Γ(s) + Γ(t) + Γ(u)]

= −iλ+ Z2φ[Γ(s) + Γ(t) + Γ(u)] (8.75)

Since Zφ = 1 +O(λ2), this is equivalent to

Γ(4)R (s, t, u) = −iλ+ [Γ(s) + Γ(t) + Γ(u)] +O(λ3) (8.76)

consisting only of finite expressions. This completes the proof that at one-loop order allGreen functions in the λφ4 theory are finite, renormalising the field φ, its mass and couplingconstant as

φ = Z−1/2φ φ0 (8.77a)

λ = Z2φZ

−1λ λ0 , (8.77b)

µ2 = m20 +Σ(µ2) = m2

0 + δm2 . (8.77c)

183

8. Renormalisation

Renormalised perturbation theory We can use the conditions (8.77) in order to formulateperturbation theory using directly only renormalised quantities. We set L0 = L +Lct whereL has the same structure as L0 but is expressed through renormalised quantities,

L =1

2(∂µφ)

2 − 1

2m2φ2 +

λ

4!φ4 (8.78)

and thus

Lct =1

2(Zφ − 1)(∂µφ)

2 − 1

2δm2φ2 + (Zλ − 1)

λ

4!φ4 . (8.79)

The term Lct is called counter term Lagrangian and contains the divergent renormalisationconstants. The latter are all of order λ and thus we can treat Lct as a perturbation. Applyingrenormalised perturbation theory consists of the following steps:

1. Starting from (8.78), one derives propagator and vertices.

2. One calculates 1-loop 1PI diagrams, finds the divergent parts which determine the

counter terms in L(1)ct at order O(λ).

3. The new Lagrangian L + L(1)ct is used to generate 2-loop 1PI diagrams and L

(2)ct .

4. The procedure is iterated moving to higher orders.

Renormalisation group equations Consider two renormalisation schemes R and R′. In

the two schemes, the renormalised field will in general differ, being φR = Z−1/2φ (R)φ0 and

φR′ = Z−1/2φ (R′)φ0, respectively. Hence the connection between the two renormalised fields

is

φR′ =Z

−1/2φ (R′)

Z−1/2φ (R)

φR ≡ Z−1/2φ (R′, R)φR . (8.80)

As both φR and φR′ are finite, also Zφ(R′, R) is finite. The transformations Z

−1/2φ (R′, R)

form a group, called the renormalisation group.

If we consider

G(n)0 (x1, . . . , xn) = Z

n/2φ G

(n)R (x1, . . . , xn) , (8.81)

we know that the bare Green function is independent of the renormalisation scale µ. Takingthe derivative with respect to µ, the LHS thus vanishes. For the simplest case of a masslesstheory, we find thus

0 =d

d lnµ

[

Zn/2φ G

(n)R (x1, . . . , xn)

]

=

[∂

∂ lnµ+

∂λ

∂ lnµ

∂

∂λ+n

2

∂ lnZφ∂ lnµ

]

G(n)R (x1, . . . , xn)

≡[

∂

∂ lnµ+ β

∂

∂λ+n

2γ

]

G(n)R (x1, . . . , xn) . (8.82)

Here we introduced in the last step the beta function

β(λ) = µ∂λ(µ)

∂µ, (8.83)

184


g

β(g)

gc g

β(g)

gc

IR

UV

UV

IR

Figure 8.3.: Left: Example of a beta function with a perturbative IR and an UV fixed point.Right: General classification of UV and IR stable fixed points of the RGE flow.

which determines the logarithmic change of the coupling constant and the anomalous dimen-sion

γ(µ) = µ∂ lnZφ(µ)

∂µ(8.84)

of the field φ. The physical meaning of γ(µ) is the topic of problem 8.6. Knowing the twofunctions β(λ) and γ(µ), we can calculate the change of any Green function under a changeof the renormalisation scale µ. The general solution of (8.82) can be found by the method ofcharacteristics or by the analogy of d/d lnµ with a convective derivative, cf. problem 8.7.

Equations of the type (8.82) are called generically renormalisation group equations orbriefly RGE. They come in various flavors, carrying the name of their inventors: Stuckelberg–Petermann, Callan–Symanzik, Gell-Man–Low, . . .

Asymptotic behaviour of the beta-function The behaviour of the beta-function β(µ) in thelimit µ → 0 and µ → ∞ provides a useful classification of quantum field theories. Considere.g. the example shown in the left panel of Fig. 8.3: This beta function has a trivial zero atzero coupling, as we expect it in any perturbative theory, and an additional zero at gc. Howdoes the beta-function β(µ) evolve in the UV limit µ→∞?

• Starting in the range 0 ≤ g(µ) < gc implies β > 0 and thus dg/dµ > 0. Therefore ggrows with increasing µ and the coupling is driven towards gc.

• Starting with g(µ) > gc implies β < 0 and thus dg/dµ < 0. Therefore g decreases forincreasing µ and we are driven again towards gc.

Fixed points gc approached in the limit µ → ∞ are called UV fixed points, while IR fixedpoints are reached for decreasing µ. The range of values [g1 : g2] which is mapped by theRGE flow on the fixed point is called its basin of attraction.

To see what happens for µ → 0, we have only to reverse the RGE flow, dµ → −dµ, andare thus driven away from gc: If we started in 0 ≤ g(µ) < gc, we are driven to zero, whilethe coupling goes to infinity for g(µ) > gc. Thus g = 0 is an IR fixed point. The distinctionbetween IR and UV fixed points is sketched in the right panel of Fig. 8.3. If the beta function

185

8. Renormalisation

has several zeros, the theory consists of different phases which are not connected by the RGflow.

Working through problem 8.5, you should show that the beta-function of the λφ4 theoryat one loop is given by

β(λ) = b1λ2 + b2λ

3 + . . . =3λ2

16π2+O(λ3) , (8.85)

so that the running coupling satisfies

λ(µ) =λ0

1− 3λ0/16π2 ln(µ/µ0). (8.86)

Thus the theory has λ = 0 as an IR fixed point and the use of free particles as asymptoticstates is sensible (but requires renormalisation). On the other hand, the coupling increasesfor µ → ∞ formally as λ → ∞. Clearly, we cannot predict the behaviour of λ(µ) in thestrong-coupling limit, because perturbation theory breaks down. The solution (8.86) suggestshowever that the coupling explodes already for a finite value of µ: The beta function has apole for a finite value of µ called Landau pole where the denominator of (8.86) becomes zero.

The large logs Although we derived the beta functions only at O(λ2), the running couplingcontains arbitrary powers of [λ0 ln(µ

2/µ20)]n, since λ(µ) = λ0

∑

n[b1λ0 ln(µ2/µ20]

n/n!. Ifthe expansion parameter b1λ0 is small, we managed to sum up the leading terms from higherorder corrections. If on the other hand the expansion parameter b1λ0 is large, our perturbativeapproach fails anyways in this regime.

8.4. Vacuum polarisation

We now turn from the λφ4 theory to the case of unbroken gauge theories as QED and QCD.There are several complications compared to the simpler case of a scalar theory: First, thecomplexity of the calculations will increase together with the number of vertices. Second,we have to ensure that the gauge symmetry is not spoiled by loop corrections. From a moretechnical point of view, we note that in processes involving gauge bosons small errors canhave drastic consequences, because the leading terms in a gauge-invariant set of Feynmandiagrams often cancel. Finally, theories containing massless spin 1 or 2 particles have IRdivergencies additional to UV divergencies, showing up in the emission of real or virtual softphotons, gluons or gravitons. Our discussion aims not to be complete. Instead we pick out asingle process, the vacuum polarisation from which we can deduce the salient point of gaugetheories: asymptotic freedom.

Using “conventional” perturbation theory, we define wave-function renormalisation con-stants for the electron and photon as

ψ0 ≡ Z1/22 ψ (8.87)

Aµ0 ≡ Z1/23 Aµ (8.88)

Analogous to Eq. (8.74), we expect that the electric coupling is renormalised by

e(µ) =Z2Z

1/23

Z1e0 , (8.89)

186


where Z1 is the charge renormalisation constant, and Z2 and Z1/23 take into account that two

electron fields and one photon field enter the 3-point function.As it stands, the renormalisation condition (8.89) creates two major problems: First, the

factor Z1/22 will vary from fermion to fermion. For instance, the wave-function renormalisation

constant of a proton includes strong interactions while the one of the electron does not. Asa result, it is difficult to understand why the electric charge of an electron and an protonare renormalised such that they have the same value for q2 → 0, while they would disagreefor q2 6= 0. In particular, we would expect that the universe is not electrically neutral, evenassuming the same number density of both particles, ne = np. Second, we see that therenormalised covariant derivative

Dµ = ∂µ + iZ1

Z2e(µ)Aµ (8.90)

remains only gauge-invariant, if Z1 = Z2. Clearly, this condition would also ensure theuniversality of the electric charge. We postpone the proof that Z1 = Z2 holds in suitablerenormalisation schemes to section 8.5.2, where we will derive the Ward-Takahashi identities.In a non-abelian theory as QCD, where we have to ensure that the gauge coupling in all termsof Eq. (7.82) remains after renormalisation the same, several constraints of the type Z1 = Z2

arise.

8.4.1. Vacuum polarisation in QED

We calculate next the one-loop correction to the photon propagator, the so called vacuumpolarisation tensor. Using the Feynman rules for QED, we obtain for the contribution of onefermion species with mass m

iΠµν(q) = −(−ie)2∫

d4k

(2π)4tr[γµi(k/ +m)γν i(k/ + q/+m)]

(k2 −m2)[(k + q)2 −m2]= −e2

∫d4k

(2π)4N µν

D. (8.91)

Our first task is to show that the vacuum polarisation tensor respects gauge invariance, i.e. hasthe structure Πµν(q) = (q2gµν−qµqν)Π(q2). For this it is sufficient to show that qµΠ

µν(q) = 0.We write

q = (q + k −m)− (k −m) (8.92)

and obtain

qµN µν = tr[(q/ + k/ −m)− (k/ −m)](k/ +m)γν(k/ + q/+m) (8.93)

= [(q + k)2 −m2]tr(k/ +m)γν − (k2 −m2)trγν(k/ + q/+m) (8.94)

where we used the cyclic property of the trace. Employing dimensional regularisation (DR)with d = 4− 2ε in order to obtain well-defined integrals,

qµiΠµν(q) = −e2µ2ε

∫ddk

(2π)d

tr[(k/ +m)γν ]

(k2 −m2)− tr[γν(k/ + q/+m)]

(k + q)2 −m2

, (8.95)

we are allowed to shift the integration variable in one of the two terms. Thus qµΠµν(q) = 0

and hence the vacuum polarisation tensor at order O(e2) is transverse as required by gaugeinvariance. In Eq. (8.159), we will extend this result to full photon propagator, i.e. we willshow that it holds in particular at any order perturbation theory.

187

8. Renormalisation

Let us pause a moment and summarise before we start with the evaluation of Π(q2): In ourpower-counting analysis we found as superficial degree of divergence D = 2. This result wasbased on the assumption that the numerator N behaves as a constant. But the only constantavailable is m2 which would lead to a mass term of the photon, e2AµΠ

µνAν ∝ e2m2AµηµνAν .

Thus the transversality of Πµν implies that the m2 term in the numerator will disappear atsome step of our calculation. Thereby the convergence of the polarisation tensor improves,becoming a “mild logarithmic” one.

We proceed with the explicit evaluation of Πµν . Taking the trace of (8.91) and using itstransversality, we find in d = 2ω dimensions

Πµµ(q) = (q2δµµ − q2)Π(q2) = (d− 1)q2Π(q2)

and

(d−1)q2iΠ(q2) = −e2µ2ε∫

ddk

(2π)dtr[γµ(k/ +m)γµ(k/ + q/+m)]

(k2 −m2)[(k + q)2 −m2]= −e2µ2ε

∫ddk

(2π)dND. (8.96)

We combine the two propagators introducing the Feynman parameter integral

1

ab=

∫ 1

0

dx

[ax+ b(1− x)]2(8.97)

with a = (k + q)2 −m2 and b = k2 −m2. Thus the denominator becomes

D = ax+ b(1− x) = k2 + 2kqx+ q2x−m2 = (k + qx)2 + q2x(1− x)−m2 . (8.98)

Next we introduce as new integration variable K = k + qx,

D = K2 + q2x(1− x)−m2 = K2 − a , (8.99)

and a = m2 − q2x(1− x) > 0 as new short-cut.

Evaluating the trace in the denominator using DR, we have to extend the Clifford algebrato d = 2ω dimensions. A natural choice is tr(γµγν) = dηµν , giving with γµγµ = d andγµq/γµ = (2− d)q/ as result for the trace

N = N µµ = d[(2− d)k · (k + q) + dm2] = d(2− d)[K2 − q2x(1− x)] + dm2 . (8.100)

In the last step, we performed the shift k → K = k − qx omitting linear terms in K thatvanish after integration. Combining our results for N and D we arrive at

(d− 1)q2iΠ(q2) = −e2µ2εd∫ 1

0dx

∫ddK

(2π)d(2− d)K2 + dm2 − (2− d)q2x(1− x)]

(K2 − a)2 . (8.101)

Finally, we want to use our results for the Feynman integrals I0(ω,α) and I2(ω,α) whichwere obtained performing a Wick rotation. The latter is possibel as long as we do not pass asingularity. Since the prefactor x(1− x) of q2 has as maximum 1/4, this requires q2 < 4m2.

188


We start to look for the first two terms where we expect a cancellation of the m2 term inthe nominator,

∫ddK

(2π)d(2− d)K2 + dm2

(K2 − a)2 = (2− d)I2(ω, 2) + dm2I(ω, 2) (8.102)

=i

(4π)ωΓ(2)

[−2ω(1− ω)Γ(1− ω) aω−1 + 2ωm2Γ(2− ω) aω−2

](8.103)

=i

(4π)ω2ωΓ(2− ω)(−a+m2)aω−2 (8.104)

=4 i

(4π)2Γ(ε) (4π)ε

q2x(1− x)[m2 − q2x(1− x)]ε . (8.105)

Hence the m2 term dropped indeed out of the nominator and the whole expression is propor-tional to q2, as required by the LHS of (8.101). Evaluating the third term in the same waywe obtain

−∫

ddK

(2π)d(2− d)q2x(1− x)

(K2 + a)2= −(2− d)q2x(1− x)I(ω, 2) (8.106)

=2 i

(4π)2Γ(ε) (4π)ε

q2x(1− x)[m2 − q2x(1− x)]ε . (8.107)

Adding the two contributions, we arrive at

Π(q2) = − 8e2

(4π)2Γ(ε)

∫ 1

0dxx(1− x)

[4πµ2

m2 − q2x(1− x)

]ε

, (8.108)

where the factor iq2 cancelled. We included also the factor (4πµ2)ε into the last term, whichbecomes thereby dimensionless. Now we expand the Gamma function, Γ(ε) = 1

ε − γ +O(ε),around2 d = 4− 2ε and because of the resulting 1/ε term all other ε dependent quantities,

Π(q2) = − e2

12π2

1

ε− γ + ln(4π)− 6

∫ 1

0dx x(1− x) ln

[m2 − q2x(1− x)

µ2

]

. (8.109)

The prefactor x(1 − x) has its maximum 1/4 for x = 1/2. Thus the branch cut of thelogarithm starts at q2 = 4m2, i.e. when the virtuallity of the photon is large enough that iscan decay into a fermion pair of mass 2m. This is a nice illustration of the optical theorem:The polarisation tensor is real below the pair creation threshold, and acquires an imaginarypart above (which equals the pair creation cross section of a photon with mass m2 = q2, cf.problem 8.8).

The x integral can be integrated by elementary functions, but we display the result onlyfor the two limiting cases of small and large virtualities, q2/m2 → 0 and |q|2/m2 → ∞. Inthe first case, we obtain with ln(1− x) ∼ x

Π(q2) = − e2

12π2

[1

ε− γ + ln(4π) + ln(µ2/m2) +

q2

5m2+ . . .

]

, (8.110)

while the opposite limit gives

Π(q2) = − e2

12π2

[1

ε− γ + ln(4π)− ln(|q2|/µ2) + . . .

]

, (8.111)

2Note we use here the conventions d = 2ω = 4− 2ε.

189

8. Renormalisation

In the MS scheme, we obtain the renormalisation constant Z3 for the photon field as thecoefficient of the pole term,

Z3 = 1− e2

12π2ε. (8.112)

More often the on-shell renormalisation scheme is used in QED. Here we require that quantumcorrections to the electric charge vanish for q2 = 0, i.e. we choose Z3 such that Πon(q2 = 0) =0. This is obviously achieved setting

Πon(q2) = Π(q2)−Π(0) =e2

60π2q2

m2+ . . . , (8.113)

for |q2| ≪ m2. This q2 dependence leads to a modification of the Coulomb potential, whichcan be measured e.g. in the Lamb shift.

Beta function We can derive the scale dependence of the renormalised electric charge from

e0 =µε

Z1/23

e , (8.114)

where we used Z1 = Z2. Then the beta function is given as

β(e) ≡ µ ∂e∂µ

= µ∂

∂µ

(

µ−εZ1/23 e0

)

. (8.115)

Since the bare charge e0 is independent of µ, we have to differentiate only µ and Z3,

β(e) = µ∂e

∂µ= −εµ−εZ1/2

3 e0 + µµ−ε1

2Z

−1/23

∂Z3

∂µe0 (8.116)

= −εe+ µ

2Z3

∂Z3

∂µe . (8.117)

Inserting Z3 = 1− e2/(12π2ε) and thus

∂Z3

∂µ= − 1

12π2ε

2e∂e

∂µ(8.118)

gives

β(e) = −εe− µ

12π2εZ3

∂e

∂µe2 = −εe− 1

12π2εZ3β(e) e2 . (8.119)

Note that Z3 is scheme-dependent, while the beta-function remains scheme independent upto two loop (problem 8.5). Solving for β and neglecting higher order terms in e2, we find inthe limit ε→ 0

β(e) = −εe(

1− e2

12π2ε+O(e4)

)

=e3

12π2. (8.120)

Thus the beta function contains indeed the renormalised charge on the RHS. Finally we notethat the beta function can be re-expressed as

β(e2) ≡ µ2 ∂e2

∂µ2= eβ(e) =

e4

12π2. (8.121)

190


1

Figure 8.4.: Feynman diagrams describing vacuum polarisation in QCD at one-loop.

Its solution,

e2(µ) =e2(µ0)

1− e2(µ0)6π2 ln

(µµ0

) (8.122)

shows not only explicitly the increase of e2 with µ, but moreover that the electric couplingdiverges for a finite value of µ. This singularity called Landau pole happens at

µ = µ0 exp(6π2/e2(µ0)) = me exp(3π/2α(me)) ∼ 1056 GeV≫MPl (8.123)

and has therefore no physical relevance.

8.4.2. Vacuum polarisation in QCD

We only sketch the calculations in QCD, stressing the new or different points compared toQED. In Fig. 8.4, we show the various 1-loop diagrams contributing to the vacuum polarisationtensor in a non-abelian theory as QCD. Most importantly, the three-gluon vertex allows inaddition to the quark loop now also a gluon loop. Since a fermion loop has an additional minussign, we expect that the gluon loop gives a negative contribution to the beta function. Thisopens the possibility that non-abelian gauge theories are in contrast to QED asymptoticallyfree, if the number of fermion species is small enough.

Quark loop The vertex changes from −ieγµ in QED to −igsT aγµ in QCD. Since the quarkpropagator is diagonal in the group index, a quark loop contains for each flavor additionallythe factor

trT aT a = 1

2δaa = 4 . (8.124)

Thus we have only to replace e → 4nfgs in the QED result, where nf counts the number ofquark flavors. For the three light quarks, u, d and s, it is an excellent approximation to setm = 0. In contrast, the masses of the other three quarks (c, b and t) can not be neglected.As the calculation for massive quarks shows is the effect of particle masses well approximatedincluding in the loop only particles with mass 4m2 < µ2, making nf scale dependent.

191

8. Renormalisation

Loop with three-gluon vertex Since the three-gluon vertex connects identical particles, wehave to take into account symmetry factors similar as in the case of the λφ4 theory. Welearnt that the imaginary part of a Feynman diagram corresponds to the propagation of realparticles. Thus the imaginary part of the gluon vacuum polarisation can be connected to thetotal cross section of g → gg scattering. This cross section contains a symmetry factor 1/2!to account for two identical particles in the final state. Therefore the same symmetry factorshould be associated to the vacuum polarisation with a gluon loop3. Applying the Feynmanrule for the three-gluon and using the Feynman-t’Hooft gauge, we find

iΠµνab 2(q2) =

1

2(−ig)2

∫d4k

(2π)4N µνab

(k2 + iε)[(k + q)2 + iε](8.125)

with

N µνab = fbcd[−ηνρ(q + k)σ + ηρσ(2k − q)ν + ησν(2q − k)ρ] (8.126)

× facd[−ηµρ(q + k)σ + ηρσ(q − 2k)µ + ηµσ(k − 2q)ρ] . (8.127)

Evaluating the colour trace,facdf cdb = Ncδ

ab (8.128)

and extracting in the usual way the pole part using DR one obtains for d = 4− ε

iΠµνab 2(q2) = −i Ncg

2

16π2ε

(µ2

q2

)ε/2(11

3qµqν − 19

6ηµνq2

)

+O(ε0) . (8.129)

Thus the contribution from the three-gluon vertex alone is not transverse—demonstratingagain that a covariant gauge requires the introduction of ghost particles.

Ghost loop This diagram has the same dependence on the structure constants as the pre-vious one,

iΠµνab, 3(q2) = −(−ig)2

∫d4k

(2π)4fbdc(k − q)νfacdkν

(k2 + iε)[(k + q)2 + iε](8.130)

and can thus be combined with the three-gluon loop. Evaluating the integral results in

iΠµνab, 3(q2) = −i Ncg

2

16π2ε

(µ2

q2

)ε/2(1

3qµqν +

1

6ηµνq2

)

+O(ε0) (8.131)

and summing the three-gluon and ghost loops gives the expected gauge-invariant expression.Moreover, the sum has the opposite sign as the quark loop and can thus lead to the oppositebehaviour of the beta function as in QED.

Four-gluon loop and tadpole diagrams The loop with the four-gluon vertex contains amassless propagator and does not depend on external momenta,

iΠµνab,4(q2) ∝

∫ddk

k2 + iε. (8.132)

Our general experience with DR tells us that this graph is zero, as gluons are massless.However, this loop integral is also in DR ambiguous: For any space-time dimension d, the

3This argument does not apply to the quark loop, since cutting leads in this case to a distinguishable qq state

192


integral is either UV or IR divergent. To proceed, we split therefore the integrand introducingthe arbitrary mass M as

1

k2 + iε=

1

k2 −M2 + iε− M2

(k2 + iε)(k2 −M2 + iε). (8.133)

Now the IR and UV divergence is separated, and we can use d < 2 in the first term and2 < d < 4 in the second. By dimensional analysis, both terms have to be proportional to apower of the arbitrary mass M . As the LHS is independent of M , the only option is that thetwo terms on the RHS cancel, as an explicit calculation confirms. The two remaining tadpolediagrams (5 and 6 of Fig. 8.4) vanish by the same argument.

Asymptotic freedom Deriving the beta function in QCD, we should evaluate

g(µ) =Z2Z

1/23

Z1g0 , (8.134)

where Z1 is the charge renormalisation constant, and Z2 and Z3 are quark and gluon renor-malisation constants, respectively. Combining all contributions to the 1/ε poles as 1-loopcontribution b1 to the beta function of QCD gives

β = µ2∂g2

∂µ2= −α

2s

4π

(b1 + b2αs + b3α

2s + . . .

)(8.135)

with αs ≡ g2s/(4π),b1 =

11

3Nc −

2

3nf (8.136)

and nf as number of quark flavors. For nf < 16, the beta function is negative and therunning coupling decreases as µ → ∞. Asymptotic freedom of QCD explains the apparentparadox that protons are interacting strongly at small Q2, while they can be described indeep-inelastic scattering as a collection of independently moving quarks and gluons.

Let us consider now the opposite limit, µ→ 0. The solution of (8.135) at one loop,

1

αs(µ2)=

1

αs(µ20)+ b1 ln

(µ2

µ20

)

(8.137)

shows that the QCD coupling constant becomes formally infinite for a finite value of µ.We define ΛQCD as the energy scale where the running coupling constant of QCD diverges,α−1s (Λ2

QCD) = 0. Experimentally, the best measurement of the strong coupling constant has

been performed at the Z resonance at LEP, giving αs(m2Z) ∼ 0.1184. Thus at one-loop level,

ΛQCD = mZ exp

(1

2b1αs(m2Z)

)

. (8.138)

ΛQCD depends on the renormalisation scheme and, numerically more importantly, on the

number of flavors used in b1: For instance, ΛMSQCD ∼ 220MeV for nf = 3. The fact that the

running coupling provides a characteristic energy scale is called dimensional transmutation:Quantum corrections lead to the break-down of scale-invariance of classical QCD with masslessquarks and to the appearence of massive bound-states, the mesons and baryons with masses oforder ΛQCD. Note also that we are able to link exponentially separated scales by dimensionaltransmutation.

193

8. Renormalisation

Coupling constant unification While the strong coupling αs ≡ α3 decreases with increasingµ2, the electromagnetic coupling αem ≡ α1 increases. Since two lines in a plane meet at onepoint, there is a point with α1(µ∗) = α3(µ∗) and one may speculate that at this point atransition to a “unified theory” happens. Since the running is only logarithmic, unificationhappens at exponentially high scales, µ∗ ∼ 1016 GeV, but interestingly still below the Planckscale MPl. The problems becomes more challenging, if we add to the game the third, theweak coupling α2. The situation in 1991 assuming the validity of the SM is shown in theleft panel of Fig. 8.5. The width of the lines indicates the experimental and theoretical error,and the three couplings clearly do not meet within these errors. The right panel of the samefigure assume the existence of supersymmetric partners to all SM particles with an “averagemass” of around MSUSY ∼ 200GeV. As a result, the running changes above µ = 200GeV,and now the three couplings meet for 2× 1016 GeV.

60

50

40

30

20

10

103 105 107 109 1011 1013 1015 10170

World average 91

Q (GeV)

α1 (Q)-1

α i (

Q)

-1

α2 (Q)-1

α3 (Q)-1

(a)

60

50

40

30

20

10

0102 104 106 108 1010 1012 1014 1016 1018

α–1 (µ)1

µ [GeV]

α–1 (µ)2

α–1 (µ)3

(b)

Figure 8.5.: The measurements of the gauge coupling strengths at LEP do not evolve towardsa unified value in the SM (left), but meet at 2 × 1016 GeV assuming low-scalesupersymmetry (right).

8.5. Effective action and Ward identities

In this section we introduce first the effective action as the generating functional of 1PI Greenfunctions. Then we will use the developed formalism to derive the Ward identities which implye.g. that the exact photon propagator is transverse and that the renormalisation of the electriccharge is universal. Later on in Sec.10.2 we will see that the discussion of the renormalisationof theories with spontaneous symmetry breaking simplifies using the effective action.

8.5.1. Effective action

To start, recall the definition for the generating functionals of a real scalar field,

Z[J ] =

∫

Dφ expi∫

d4x(L (x) + J(x)φ(x)) = eiW [J ] . (8.139)

194


Next we define the classical field φc(x) as φc(x) = δW [J ]/δJ(x). Performing the functionalderivative in its definition, we see immediately why this definition makes sense,

φc(x) =δW [J ]

δJ(x)=

1

iZ

δZ[J ]

δJ(x)=

1

Z

∫

Dφφ(x) exp i∫

d4y(L + Jφ) (8.140a)

=〈0|φ(x)|0〉J〈0|0〉J

= 〈φ(x)〉J . (8.140b)

Thus the classical field φc(x) is defined as the vacuum expectation value of the field φ(x) inthe presence of the source J(x).

Now we define the effective action Γ[φc] as the Legendre transform of W [J ],

Γ[φc] =W [J ]−∫

d4xJ(x)φc(x) , (8.141)

where (8.140a) should be used to replace J(x) by φc(x) on the RHS. This procedure iscompletely analogous to the construction of the Hamilton function from the Lagrange functionin classical mechanics: It will allow us to answer the question “which source J(x) producesa given φc(x),” because we can use φc as independent variable in Γ[φc]. We compute thefunctional derivative w.r.t. φc of this new quantity,

δΓ[φc]

δφc(y)=

∫

d4xδJ(x)

δφc(y)

δW

δJ(x)−∫

d4xδJ(x)

δφc(y)φc(x)− J(y).

Using the definition δWδJ(x) = φc(x), the first and second term cancels and we end up with

δΓ[φc]

δφc(y)= −J(y) . (8.142)

This is the desired relation determing the source J(x) producing a given classical field φc(x).To see why Γ[φc] is called effective action, we consider a free scalar field. The free generating

functional for connected Green functions is

W0[J ] = −1

2

∫

d4xd4x′ J(x)∆F (x− x′)J(x′) , (8.143)

and the classical field becomes

φc(x) =δW

δJ(x)= −

∫

d4x′∆F (x− x′)J(x′) . (8.144)

If we apply the Klein-Gordon operator to the classical field,

(+m2)φc(x) = −∫

d4x′(+m2)∆F (x− x′)J(x′) (8.145a)

=

∫

d4x′ δ(x − x′)J(x′) = J(x) , (8.145b)

we see that φc is a solution of the classical field equation. Now we have all we need to writedown an explicit expression for the free effective action,

Γ0[φc] =W0[J ]−∫

d4xJ(x)φc(x) (8.146a)

=1

2

∫

d4xd4x′ J(x)∆F (x− x′)J(x′) . (8.146b)

195

8. Renormalisation

Inserting the expression (8.145) for J(x) and integrating partially twice, we obtain

Γ0[φc] =1

2

∫

d4xd4x′[(+m2)φc(x)

]∆F (x− x′)

[(′ +m2)φc(x

′)]

= −1

2

∫

d4xφc(x)( +m2)φc(x) = S0[φc] .

Thus the effective action of a free field equals the classical action of the field, justifying itsname.

To gain some more information on the meaning of Γ[φ], we can evaluate it perturbativelyand expand it in φc, giving us a new set of n-point Green functions,

Γ[φc] =∑

n

1

n!

∫

d4x1 · · · d4xnΓ(n)(x1, ..., xn)φc(x1) · · · φc(xn) . (8.147)

The functions Γ(n) are the one-particle-irreducible Green functions. Hence, the effectiveaction Γ0[φc] is the generating functional for 1PI Green functions, the proof we postpone tosection 10.2.

Example: Show that a) Γ(2) is equal to the inverse propagator or inverse 2-point function, and b)derive the connection of Γ(3) to the connected 3-point function.a.) We write first

δ(x1 − x2) =δφ(x1)

δφ(x2)=

∫

d4xδφ(x1)

δJ(x)

δJ(x)

δφ(x2)

using the chain rule. Next we insert φ(x) = δW/δJ(x) and J(x) = −δΓ/δφ(x) to obtain

δ(x1 − x2) = −∫

d4xδ2W

δJ(x)δJ(x1)

δ2Γ

δφ(x)δφ(x2).

Setting J = φ = 0, it follows∫

d4x iG(x, x1) Γ(2)(x, x2) = −δ(x1 − x2) (8.148)

or Γ(2)(x1, x2) = iG−1(x, x1).b.) The connected 3-point function G(3)(x1, x2, x3) is obtained by appending propagators to theirreducible 3-point vertex function Γ(3)(x1, x2, x3), one for each external leg,

G(3)(x1, x2, x3) = i

∫

d4x′1d4x′2d

4x′3G(2)(x1, x

′1)G

(2)(x2, x′2)G

(2)(x3, x′3)Γ

(3)(x1, x2, x3) .

This generalises to n > 3 and therefore one calls the Γ(n) also amputated Green functions.

8.5.2. Ward-Takahashi identities

The redundancy implied by gauge invariance leads us to believe that not all of Green func-tions of a gauge theory are independent: Gauge invariance implies relations between Greenfunctions called Ward or Ward-Takahashi identities in QED and Slavnov-Taylor identities inthe non-abelian case. We use now the formalism of the effective action we developed to derivethese for QED: Consider the generating functional

Z[Jµ, η, η] =

∫

DADψDψ expi∫

d4xLeff, (8.149)

196


where Jµ is a four-vector source, η and η are Graßmannian sources and the effective La-grangian is composed of a classical term, a source term and a gauge fixing term,

Leff = Lcl + Ls + Lgf , (8.150a)

Lcl = −1

4FµνF

µν + iψγµDµψ −mψψ , (8.150b)

Ls = JµAµ + ψη + ηψ , (8.150c)

Lgf = −1

2ξ(∂µA

µ)2 . (8.150d)

We consider now the renormalised version of the Lagrangian Leff , where the renormalisedcovariant derivative Dµ is given by Eq. (8.90) with Z1 6= Z2 in general. This implies that aninfinitesimal gauge transformation has the form

Aµ → A′µ = Aµ + ∂µΛ (8.151a)

ψ → ψ′ = ψ − ieΛψ with e ≡ Z1

Z2e . (8.151b)

As Lcl is gauge invariant by construction, the variation of Leff under an infinitesimal gaugetransformation consists only of

δ

∫

d4x (Lgf + Ls) =

∫

d4x

[

−1

ξ(∂µA

µ)Λ + Jµ∂µΛ + ieΛ(ψη − ηψ)]

. (8.152)

Now we integrate partially the first term twice and the second term once, to factor out thearbitrary function Λ,

δ

∫

d4xLeff =

∫

d4x

[

−1

ξ(∂µA

µ)− ∂µJµ + ie(ψη − ηψ)]

Λ . (8.153)

Thus the variation of the generating functional Z[Jµ, η, η] is

δZ[Jµ, η, η] =

∫

DADψDψ exp

(

i

∫

d4xLeff

)

iδ

∫

d4xLeff . (8.154)

The fields Aµ, ψ and ψ are however only integration variables in the generating functional.The gauge transformation (8.151) is thus merely a change of variables which cannot affectthe functional Z[Jµ, η, η]. Thus its variation has to vanish, δZ[Jµ, η, η] = 0.

If we substitute fields by functional derivatives of their sources, the part from Eq. (8.153)can be moved outside the functional integral, leading to

0 =

[

−1

ξ∂µ

δ

δJµ− ∂µJµ + ie

(

ηδ

δη− η δ

δη

)]

expiW

= −1

ξ

(

∂µδW

δJµ

)

− ∂µJµ + ie

(

ηδW

δη− η δW

δη

)

. (8.155)

Differentiating this equation with respect to Jν(y) and setting then the sources Jµ, η and ηto zero gives us our first result,

− 1

ξ

(

∂µδ2W

δJµ(x)δJν(y)

)

= ∂µgµνδ(x − y) . (8.156)

197

8. Renormalisation

The second derivative of W w.r.t. to the vector sources Jµ is the full photon propagatorDµν(x− y). If we go to momentum space, we have

i

ξk2kµDµν(k) = kν . (8.157)

Splitting the propagator into a transverse part and a longitudinal part as in (7.64), thetransverse part immediately drops out and we find

i

ξk2kνDL(k

2) = kν . (8.158)

Thus the longitudinal part of the exact propagator agrees with the longitudinal part of thetree level propagator,

iDL(k2) = iDL(k

2) =ξ

k2. (8.159)

This implies that higher order corrections do not affect the longitudinal part of the photonpropagator. Since we can expand all relations as power series in the coupling constant e, thisholds also at any order in perturbation theory.

Let us go back to the constraint for the variation of the generating functional Z undergauge transformations, Eq. (8.155). We aim to derive identities between 1PI Green functionsand want therefore to transform it into a constraint for the effective action Γ. If we Legendretransform W [J, η, η] into Γ[A, ψ, ψ],

Γ[ψ, ψ, Aµ] =W [η, η, Jµ]−∫

d4x(JµAµ + ψη + ηψ) , (8.160)

we can replace the functional derivatives of W with classical fields, and the sources withfunctional derivatives of Γ, i.e.

δW

δJµ= Aµ,

δW

δη= ψ,

δW

δη= ψ

δΓ

δAµ= −Jµ, δΓ

δψ= −η, δΓ

δψ= −η

with all the fields classical. This transforms Eq. (8.155) into

1

ξ(∂µA

µ(x))− ∂µδΓ

δAµ(x)+ ie

(

ψδΓ

δψ(x)− ψ δΓ

δψ(x)

)

= 0 , (8.161)

a master equation from which we can derive relations between different types of Green frunc-tion. Differentiating with respect to ψ(x1) and ψ(x2) and setting then the fields to zero givesus the most important one of the Ward-Takahashi identities, relating the 1PI 3-point functionΓ(3)(x, x1, x2) to the 2-point function Γ(2)(x1, x2) of fermions,

∂µΓ(3)µ (x, x1, x2) = ie

(Γ(2)(x, x1)δ(x− x2)− Γ(2)(x, x2)δ(x − x1)

)(8.162)

or, after Fourier transforming

kµΓ(3)µ (p, k, p + k) = eS−1

F (p + k)− eS−1F (p) . (8.163)

Taking the limit kµ → 0, the usual Ward identity Γ(3)µ (p, 0, p) = e∂µS

−1F (p) follows.

The Green functions in this equation are finite, renormalised quantities and thus Z1/Z2 hasto be finite in any consistent renormalisation scheme too. In any scheme where we identifydirectly e(0) with the measured electric charge, also the finite parts of Z1 and Z2 agree, i.e.the measured electric charge is universal.

198

8.6. Renormalisation and critical phenomena


Overview The behaviour thermodynamical systems exhibit close to the critical points intheir phase diagram is called “critical phenomenon.” For a fixed number of particles, we cancharacterise thermodynamical systems using the free energy F = U − TS. Ehrenfest intro-duced the classification of phase transitions according to the order of the first discontinuousderivative of F with respect to any thermodynamical variable φ. Hence a phase transitionwhere at least one derivative ∂nF/∂φn is discontinuous while all ∂n−1F/∂φn−1 are continuousis called a n.th order phase transition.

Critical phenomena are for a particle physicist interesting by at least three reasons:

• We can learn about symmetry breaking: We should look out for ideas how we cangenerate mass terms without violating gauge invariance. Systems like ferromagnetsshow that symmetries as rotation symmetry can be broken at low energies although theHamiltonian governing the interactions is rotation symmetric.Another example is a plasma: Here the screening of electric charges modifies theCoulomb potential to a Yukawa potential; the photon has three massive degrees offreedom, still satisfying gauge invariance, kµΠ

µν(ω,k) = 0, but with ω2 − k2 6= 0.

• Experimentally one finds that close a critical point, T → Tc, the correlation length ξdiverges, while otherwise correlations are exponentially suppressed. Thus the 2-pointfunction of a certain order parameter φ scales as

〈φ(x)φ(0)〉 ∝ exp(−|x|/ξ) .

Comparing this to the 2-point function of an Euclidean scalar field φ,

〈φ(x)φ(0)〉 → 4π

|x|2 exp(−m|x|)

in the limit m|x| ≫ 1, we find the correspondence ξ ∝ m−1. Considering a statisticalsystem on a lattice with spacing a, i.e. m|x| = na/ξ, we see that the continuum limita→ 0 corresponds to ξ →∞ for finite m.Thus the correlation functions of a statistical system correspond for non-zero a to bareGreen functions and a finite value of the regulator of the corresponding quantum fieldtheory. The connection to renormalised Green functions (a → 0 or Λ → ∞) is onlypossible when the statistical system is at a critical point.

• Near a critical point, T → Tc, thermodynamical systems show a universal behaviour.More precisely, they fall in different universality classes which unify systems with verydifferent microscopic behaviour. The various universality classes can be characterised bycritical exponents, i.e. by the exponents γi with which characteristic thermodynamicalquantities Xi diverge approaching Tc, i.e. Xi = [b(T − Tc)]−γi .This phenomenon is similar to our realisation that e.g. two λφ4 theories, one withλ = 0.1 and another one with λ = 0.2, are not fundamentally different but connectedby a RGE transformation.

Landau’s mean field theory Landau suggested that close to a second-order phase transitionthe free energy can be expanded as an even series in the order parameter. Considering e.g.

199

8. Renormalisation

the magnetisation M , we can write for zero external field H the free energy as

F = A(T ) +B(T )M2 + C(T )M4 + . . . (8.164)

We can find the possible value of the magnetisation M by solving

0 =∂F

∂M= 2B(T )M + 4C(T )M3 . (8.165)

The variable C(T ) has to be positive in order that F is bounded from below. If also B(T )is positive, only the trivial solution M = 0 exists. If however B(T ) is negative, two solutionswith non-zero magnetisation appear. Let us use a linear approximation, B(T ) ≈ b(T − Tc),and C(T ) ≈ c valid close to Tc. Then

M =

0 for T > Tc,

±[b2c(Tc − T )

]1/2for T < Tc.

(8.166)

Note also that the ground-state breaks theM → −M symmetry of the free energy for T < Tc.Representing the thermodynamical quantity M as integral of the local spin density,

M =

∫

d3x s(x) , (8.167)

we can rewrite the free energy in a way resembling the Hamiltonian of a stationary scalarfield,

F =

∫

d3x[(∇s)2 + b(T − Tc)s2 + cs4 −H · s

]. (8.168)

Here, (∇s)2 is the simplest ansatz leading to an alignment of spins in the continuous language.Minimising F will give us the ground-state of the system for a prescribed external field H(x)and temperature T . For small s, we can ignore the s4 term. The spin correlation function〈s(x)s(0)〉 is found as response to a delta function-like disturbance H0δ(x) as

〈s(x)s(0)〉 =∫

d3k

(2π)3H0e

ik·x

k2 + b(T − Tc)(8.169)

as it follows immediately by analogy with the Yukawa potential, m2 → b(T − Tc). Thus thecorrelation function is

〈s(x)s(0)〉 = H0

4πre−r/ξ (8.170)

withξ = [b(T − Tc)]−1/2 . (8.171)

Hence Landau’s theory reproduces the experimentally observed behaviour ξ →∞ for T → Tc.Moreover, the theory predicts as critical exponent 1/2. Notice that the value of the expo-nent depends only on the polynomial assumed in the free energy, not on the underlyingmicro-physics. Thus another prediction of Landau’s theory is an universal behaviour of ther-modynamical systems close to their critical points in the dependence on T − Tc.

Experiments show that this prediction is too strong: Thermodynamical systems fall intodifferent universality classes, and we should try to include some micro-physics into the de-scription of critical phenomena.

200


sN−1

s0

s1

s2

s3

s4

s5

s′0

s′1

f

s′2

Figure 8.6.: One block spin transformation a→ 2a for an one-dimensional lattice model.

Kadanoff’s block spin transformation Close to a critical point, collective effects play adecisive role even in case of short-range interactions. In d dimension, a particle is coupled bycollective effects to (ξ/a)d particles and standard perturbative methods will certainly fail forξ →∞.

Kadanoff suggested to remove the short-wave length fluctuations by the following procedure:Each step of a block spin transformation consists of i) dividing the lattice into cells of size(2a)d, ii) assigning a common spin variable to the cell, iii) rescaling 2a→ a.

At each step, the number of strongly correlated spins is reduced. After n transformations,the correlation length decreases as ξn = ξ/(2n). When the correlation length becomes ofthe order of the lattice spacing, collective effects play no role: All the physics can be readoff from the Hamiltonian. If the procedure is not trivial, this implies that in each step theHamiltonian changes. In particular, the coupling constant K is changed as

K2 = f(K) , K3 = f(K2) = f(f(K)) , . . . (8.172)

One-dimensional Ising model We illustrate the idea behind Kadanoff’s block spin transfor-mation using the example of the one-dimensional Ising model. This model consists of spinswith value si = ±1 on a line with spacing a, interacting via nearest neighbour interactions.

We consider only the piece of six spins shown in Fig. 8.6. The corresponding partitionfunction is

Z6 =∑

sN−1,s0,s1,...,s4

exp [K(sN−1s0 + s0s1 + . . . s3s4)] (8.173)

=∑

s′0,s′

1,s′

2

∑

sN−1,s1,s3

exp[K(sN−1s

′0 + s′0s1 + . . . s3s4)

]. (8.174)

The step a → 2a requires to perform the sums over the unprimed spins. Expanding theexponentials using (sisj)

2n = 1 gives terms like

exp[K(s′0s1)] = 1 +Ks′0s1 +K2

2!+K3

3!s′0s1 + . . . (8.175)

= cosh(K) + s′0s1 sinh(K) (8.176)

= cosh(K)[1 + s′0s1 tanh(K)] . (8.177)

201

8. Renormalisation

The terms linear in s1 cancel in the sum and we obtain

∑

s1

exp[K(s′0s1)] exp[K(s1s′1)] = 2 cosh2(K)[1 + tanh2(K) s′0s

′1] . (8.178)

Thus the summation over the unprimed spins changes the strength of the nearest neighbour-hood interaction and generates additionally a new spin-independent interaction term. We trynow to rewrite the last expression in a form similar to the original one,

2 cosh2(K)[1 + tanh2(K) s′0s′1] = exp[g(K) +K ′s′0s

′1)] . (8.179)

Using (8.177) to replace exp(K ′s′0s′1), we find

tanh(K ′) = tanh2(K) . (8.180)

This determines the function g as

g(K) = ln

(2 cosh2K

coshK ′

)

. (8.181)

The summation over the other spins s3, s5, . . . can be performed in the same way. Thusthe partition function on a lattice of size 2a has the same nearest-neighbour interactionswith a new coupling K ′ ≡ K1 determined by (8.180). Iterating this procedure generates arenormalisation flow with

tanh(Kn) = tanh2n(K) . (8.182)

Fixed point behaviour In general, we will not be able to calculate the transformation func-tion f(K). But even without the knowledge of f(K), we can draw some important insightfrom general considerations. First, the exact RGE equation4 is of the type of a heat ordiffusion equation,

∂tX = ∇2X ,

on the space of functionals X = exp(−S). Its flow is therefore a gradient flow (“Fick’s law”)which has only two possible asymptotics: a runaway solution to infinity and the approach toa fixed point defined by Kc = f(Kc).

With ξn+1 = ξn/2 and labeling the n dependence implicitly vis ξn = f(Kn) we can write

ξ(f(K)) =1

2ξ(K) . (8.183)

At a fixed point Kc = f(Kc) only two solutions exist,

ξ(Kc) = 0 and ξ(Kc)→∞ . (8.184)

The second possibility corresponds to the approach of a critical point, allowing the limit a→ 0and thus the continuum limit necessary for the transition to a QFT. This point is called acritical fixed point, while the fixed point with zero correlation length is called trivial.

We can now easily generalise our previous discussion of the fixed point behaviour for thebeta function from one to n-dimensions. The general behaviour of the RGE flow can be

4A brief derivation for the curious is given in Section ??

202


ξ =∞

ξ = 10

ξ = 1

A

B

C

D

Figure 8.7.: A two-dimensional illustration of the RGE flow: A, B and C are three fixedpoints on the critical surface ξ = ∞. The fixed points A and C have a stabledirection along the critical surface, while B has two unstable directions. Thetrivial fixed point D is a stable fixed point, attracting all points starting not onthe critical surface.

understood from the two-dimensional example in Fig. 8.7. The dashed lines show surfacesof constant correlation length, including a critical surface ξ = ∞. Also shown are threecritical fixed points (A, B, C) and a trivial one (D). We remember that in each RGE stepthe correlation length decreases. Thus the trivial fixpoint is an attractor, i.e. inside a smallenough neighbourhood all points will flow towards it. On the other hand, the critical line hasat least one unstable direction, the one orthogonal to its surface: Even points infinitesimalclose to the surface will flow away and eventually end in a trivial fixpoint. Moreover, we seethat also inside the critical surface stable and unstable directions exist: The fixpoint B willattract all points in-between A and C (“its basin of attraction”).

We can identify universality classes of QFTs with stable critical fixpoints and their basinof attraction.

Effective action, RGE flow and irrelevant operators The RGE flow generates all kind ofcouplings compatible with the symmetries of the fundamental Hamiltonian. The task ofunderstanding the RGE flow in an infinite-dimensional space of couplings is simplified by thefollowing observation: Relevant and marginal interactions are typically already included inour starting Hamiltonian. For instance, in the case of the λφ4 theory in d = 4, we includea constant term5 (d = 0, leading to a cosmological constant ρΛ), the mass term m2φ2 withd = 2, and the interaction λφ4. Hence, the RGE flow will renormalise the values ρΛ, m

2 andλ and introduce an infinite set of irrelevant operator Od,i with dimension d ≥ 6.

It is now time to explain the reason why the non-renormalisable operators are called ir-relevant in this context. Let us start from the Euclidean generating functional restricted to

5If we do not, the normalisation constant of the path integral will do the job.

203

8. Renormalisation

wave-numbers below a cutoff, k ≤ Λ,

Z[J ] =

∫

DφΛ exp−∫

Ωd4x

(1

2∂µφ∂

µφ+1

2m2φ2 +

λ

4!φ4)

, (8.185)

where DφΛ = Dφ|k|≤Λ. Now we want to integrate out the fields with wave-numbers betweensΛ ≤ k ≤ Λ from the generating functional. We express φ(x) as Fourier modes, obtaining e.g.

Sint =1

24

∫d4k1(2π)4

· · · d4k4

(2π)4(2π)4δ(k1 + . . .+ k4)φ(k1) · · · φ(k4) . (8.186)

Then we introduce the field φ(k) which coincides with the original field φ in the range wewant to integrate out and vanishes otherwise,

φ(k) =

φ(k) for sΛ ≤ |k| ≤ Λ,

0 otherwise.(8.187)

We set now φ = φ + φ in Eq. (8.185) and call afterwards the integration variable φ againφ. Then

Z[J ] =

∫

DφsΛDφ exp[

−∫

d4x

(1

2(∂µφ+ ∂µφ)

2 +1

2m2(φ+ φ)2 +

λ

4!(φ+ φ)4

)]

=

∫

DφsΛe−S[φ]Dφ exp[

−∫

d4x

(1

2(∂µφ)

2 +1

2m2φ2 +

λ

6φ3φ+

λ

4φ2φ2 +

λ

6φφ3 +

λ

24φ4)]

=

∫

DφsΛe−Seff [φ] , (8.188)

where we used that the mixed quadratic terms vanish (being orthogonal). To proceed, wewould have to expand the exponential and to evaluate it perturbatively. Apart from generatingterms which renormalises the original parameters, we would obtain an infinite number ofhigher-dimension operators Od,i with dimension d ≥ 6. For instance O2n,1 = φ2n, O2n,2 =φ2n−1

φ, etc. Thus the effective Lagrangian including fluctuations up to Λ1 ≡ sΛ can bewritten as

Leff =1

2∂µφ∂

µφ+1

2m2

Λ1φ2 +

λΛ1

4!φ4 +

∑

d≥6

∑

i

Cd,i(Λ1)Od,i . (8.189)

The coefficients C2n,1 are given by a loop integral over n propagators,

C2n,1 ∝ λn∫ Λ

Λ1

d4k

(2π)4

(1

k2 +m2

)n

∝ λn(

1

Λ2n−4− 1

Λ2n−41

)

. (8.190)

We see now that relevant operators are determined by the high-energy cutoff, marginal op-erators (∝ ln(Λ/Λ1) are influenced in the same way by each decade in k, while irrelevantoperators are determined by the low-energy cutoff.

As a side remark, we note that both directions of the RGE flow—towards the UV or theIR—are useful discussing QFTs. The point of view of a RGE flow towards the IR is useful,if we want to connect a theory at high-scales to a simpler theory at lower energy scales.An example for this approach is chiral perturbation theory where one connects QCD to aneffective theory of mesons and baryons at low energies. In the opposite view, we may look

204


e.g. at the SM as an effective theory known to be valid up to scales around TeV and ask whathappens if we increase the cutoff.

You may have noticed that we have rescaled the effective Lagrangian in Eq. (8.189) suchthat the kinetic term maintained its canonical normalisation. According to step iii) in theKadanoff-Wilson prescription, we have to rescale Z after each step a→ fa (or Λ→ Λ/f). Ifwe rescale distances by x→ x′ = x/f , the functional integral is again over modes φ(x′) withx > a. Keeping the kinetic term invariant,

∫

d4x (∂µφ)2 =

∫

d4x′ (∂′µφ′)2 =

1

f2

∫

d4x (∂µφ′)2 (8.191)

requires thus a rescaling of the field as φ′ = fφ. Let us consider now an irrelevant interaction,e.g. g6φ

6. Then

g6

∫

d4xφ6 =g6f2

∫

d4x′ φ′ 6 (8.192)

shows that the new coupling g′6 is rescaled as g′6 = g6/f2: As f grows and the cutoff scale Λ

decreases, the value of an irrelevant coupling is driven to zero. Clearly, a relevant operator asthe cosmological constant ρ or a mass term m2φ2 shows the opposite behaviour and grows. Asresult, irrelevant couplings are in our low-energy world suppressed and as first approximationa renormalisable theory emerges at low energies.

We can generalise now our earlier discussion of the two-dimensional RGE flow, Fig. 8.7. TheRGE flow stops at fixed points on critical surfaces of low dimension. All initial values on thecritical surface (or basin of attraction) flow to the critical fixed point. Directions perpendicularto the critical surface are controlled by the irrelevant interactions; flows beginning off thesurface are driven to the trivial fixed point. On the way towards ξ →∞ only the relevant andmarginal interactions survive. Insisting not on ξ →∞ and keeping a finite cutoff (somewherebetween TeV and MPl, depending on the limit of validity of the theory we assume) we cankeep irrelevant interaction but may require some fine-tuning of their parameters.

Summary of chapter

Using a power counting argument for the asymptotic behaviour of the free Green functions,we singled out theories with dimensionless coupling constants: Such theories with marginalinteractions are renormalisable, i.e. are theories with a finite number of primitive divergentdiagrams. Consequently, the multiplicative renormalisation of the finite number of parameterscontained in the classical (effective) Lagrangian is sufficent to obtain finite Green functionsat any order perturbation theory.

The scale dependence of renormalised Green functions can be interpreted as a running ofcoupling constants and masses. The use of a running coupling sums up the leading logarithmsof type lnn(µ2/µ20), and a suitable choice of the renormalisation scale in a specific problemsreduces the remaining scale dependence of any perturbative result.

Different interactions can be characterised by the asymptotic behavior of their couplingconstants. Gauge theories with a sufficiently small number of fermions are the only renormal-isable interactions which are asymptotically free, i.e. their running coupling constant goes tozero for µ→∞.

The non-perturbative approach of Wilson provides an argument why the SM as descriptionof our low-energy world is renormalisable: Integrating out high-energy degrees of freedom,

205

8. Renormalisation

irrelevant couplings are driven to zero and thus it is natural that a renormalisable theoryemerges at low energies.

Further reading

Our discussion of the renormalisation of non-abelian gauge theories left out most details.For instance, we did not introduce the non-abelian analogue of the Ward-Takahshi identifieswhich are easiest derived using the formalism of the BRST symmetry. I recommend thoseinterested to fill the gaps to start with the books of Ramond and Pokorski. Banks gives aextremely concise and lucid discussion of renormalisation.

Problems

8.1 gs-factor of gauge bosons.Derive the gs-factor of Yang-Mills gauge bosonsfrom the non-abelian Maxwell equations.

8.2 Effective vertex for µ→ eγ.Derive the effective vertex Γµ for the tranitionµ → eγ where all three particles are on-shell andthe process violates parity. Use current conserva-tion and that the photon is on-shell to show thatΓµ = iqνσµν([A(q

2) + B(q2)γ5], where qν is four-momentum of the photon.

8.3 Comparison of cutoff and DR.Recalculate the three basic primitive diagramsof a scalar λφ4 theory using as regularisation acutoff Λ. Find the correspondence between thecoefficients of poles in DR and divergent terms inΛ.

8.4 Primitive divergent diagrams of scalarQED.Find the basic primitive diagrams of scalar QED

L =1

2(Dµφ)

†Dµφ− 1

2m2φ†φ− 1

4FµνF

µν

and their superficial degree of divergence.

8.5 β function of the λφ4 theory.The β function determines the logarithmic changeof the coupling constants. a) Show that the βfunction can be written in DR as

β(λ) = −ελ− µ

Z

dZ

dµλ

with Z−1 = Z−1λ Z2

φ. b.) Show that Z−1λ =

1−3λ/(16π2ε) in one loop approximation and find

the β-function. c.) Up to which order is the βfunction scheme independent? d.) Solve the dif-ferential equation for λ(µ).

8.6 Anomalous dimension.

8.7 General solution of RGE equation.

Find the general solution of the RGE equa-tion (8.82) using the method of characteristics orthe following analogue: Coleman suggested to con-sider the growth rate g(x) of bacteria in an one-dimensional flow as analogue to the RGE.a.) Show that the density ρ(x, t) of bacteria satis-fies

[∂

∂t+ v(x)

∂

∂x− g(x)

]

ρ(x, t) = 0

b.) The position of a fluid element is described byx = x(x, t) with the initial condition x(x, 0) = x.Then x(x, t) satisfies

d

dtx(x, t) = v(x)

Show that for ρ(x, 0) = ρ0(x) at later time ρ(x, t)is given by

ρ(x, t) = ρ0(x(x, t)) exp

(∫ t

0

dt′ g(x(x, t′))

)

.

8.8 Imaginary part of the photon polarisa-tion tensor.

Derive the imaginary part of the photon polari-sation tensor at one loop and show that it equals

206


the pair creation cross probability of a photon withvirtuality q2.

8.9 A toy model for the effective action ap-proach.Consider

Z =

∫

dxdy exp([−(x2 + y2 + λx4 + λx2y2)]

as a toy model for the generating functional oftwo coupled scalar fields. Integrate out the fieldy, assume then that λ is small: Expand first theresult and then rewrite it as an exponential. Showthat this process results in a.) a renormalisationof the mass term x2, b.) a renormalisation of thecoupling term x4, and c.) the appearance of new(“irrelevant”) interactions xn with n ≥ 6.

207

9. Thermal field theory

9.1. Overview

Introduction The Green functions we have considered so far were all defined as expectationvalue of products of fields in a pure state, the vacuum in the absence of real particles, |0〉. Outof these Green functions, we could built up our quantities of prime interest, decay or scatteringamplitudes for 1→ n and 2→ n particles via the LSZ formalism. In this chapter, we discusshow Green functions should be calculated, if the vacuum is non-empty and described by adensity matrix ρ. Examples for such systems are the early Universe or the dense, hot interiorof a star. The simplest and at the same time most important cases of thermal systems arethose in equilibrium.

In equilibrium statistical physics, the partition function is of central importance as all otherthermodynamic quantities can be calculated from it. In the grand canonical ensemble (whereboth particles and energy may be exchanged between system and reservoir) it takes the form,

Z(V, T, µ1, µ2, . . . ) =∑

n

〈n| e−βH−µiNi |n〉 = e−βΩ , (9.1)

where β ≡ 1/T is the inverse temperature of the system and n denotes a complete set ofquantum numbers. Finally, the Landau or grand canonical free energy

Ω(U,S,Ni) = −T lnZ = U − TS − µiNi = PV (9.2)

connects the microscopic partition function to thermodynamics. We already saw that thepartition function is closely related to the corresponding field theory in Euclidean space, whilewe can derive from Ω all relevant thermodynamical quantities: For instance, we obtain thepressure P from Ω as P = ∂Ω/∂V |T,µi . In addition the expectation value of any observable,O, is given (setting in the following always µi = 0) as

〈O〉 = Z−1Tr[

O e−βH]

. (9.3)

Calculational approaches Two main approaches to calculations are used in thermal fieldtheory:

• In the real-time formalism one applies the formula (9.3) valid for any observable directlyto Green functions, that is

G(x1, . . . , xn) = 〈T (φ(x1) · · · φ(xn))〉 (9.4)

= Z−1Tr[

e−βH T (φ(x1) · · · φ(xn))]

. (9.5)

The main advantage of this approach is that it can be extended to the non-equilibriumcase. In particular, one can investigate the time evolution of a system towards thermal

208

9.1. Overview

equilibrium. As disadvantage, we note that the proper definition of the contours forthe propagators becomes more complicated and one has to introduce therefore matrix-valued 2× 2 propagators.

• In the imaginary time formalism, we rotate from Minkowski to Euclidean space, t→ −it,so that the transition amplitude from an initial state, |q(ti)〉, to a final state, |q(tf )〉, isgiven by

〈q(tf )| e−(tf−ti)H |q(ti)〉 =∫ q(tf )

q(ti)Dq e−S , (9.6)

where S is now the Euclidean action. If we set the evolution time, tf − ti, equal to theinverse temperature β and integrate over all periodic paths q(ti) = q(tf + β), we have

∑

q

〈q| e−βH |q〉 =∫ q(t+β)

q(t)Dq e−S = Z . (9.7)

We now see that we have formally connected the path integral formulation of quantummechanics (in Euclidean space) to the partition function of statistical mechanics.

In contrast to T = 0 field theory, Green functions and the partition function in theEuclidean are not not merely a mathematical tool but our main objects of interest.Since the Euclidean Green functions do depend on temperature instead of time, we arenot able to describe time-dependent phenomena in this approach.

Thermal Green functions The trace in the partition function of statistical physics impliesthat we have to sum over configurations connecting the same physical state at t and t + β.In the path integral corresponding to statistical mechanics, the periodicity condition q(t) =q(t + β) for the real coordinate q is clearly the only possible choice. In contrast, fields mayonly be observable through bilinear quantities, as e.g. ψΓψ for a fermion field. This raisesthe question, if we should require periodic or anti-periodic boundary conditions for fermionicfields.

We consider first thermal Green functions Gβ for a free scalar field. From the Heisenbergequation for the field operator,

φ(t,x) = eiHtφ(0,x)e−iHt , (9.8)

we find1

G+β (t

′,x′; t,x) = tr [e−βHφ(t′,x′)φ(t,x)]/Z

= tr [e−βHφ(t′,x′)eβHe−βHφ(t,x)]/Z

= tr [φ(t′ + iβ,x′)e−βHφ(t,x)]/Z

= tr [e−βHφ(t,x)φ(t′ + iβ,x′)]/Z = G−β (t

′ + iβ,x′; t,x) . (9.9)

Hence for bosonic fields,

G±β (t

′,x′; t,x) = G∓β (t

′ + iβ,x′; t,x) , (9.10)

1We set G(t) = G+ϑ(t) +G−ϑ(−t) for bosonic and S(t) = S+ϑ(t)− S−ϑ(−t) for fermionic propagators.

209


implying periodic boundary condition for the thermal propagators,

Gβ(t′,x′; t,x) = Gβ(t

′ + iβ,x′; t,x) . (9.11)

The derivation (9.9) goes through unchanged for fermionic fields,

S±β (t

′,x′; t,x) = S∓β (t

′ + iβ,x′; t,x) . (9.12)

But now the anti-commuting nature of fermionic fields—more specifically the minus sign in(5.71)—leads to anti-periodic boundary condition for their thermal propagators,

Sβ(t′,x′; t,x) = −Sβ(t′ + iβ,x′; t,x) . (9.13)

Both periodicity conditions discretise the frequency spectrum of the wave-functions. Thusthe Fourier transform of thermal fields contained in a box of size V × β is given by

φ(t,x) =1√βV

∞∑

n=−∞

∑

p

φn,p e−i(ωnt+p·x) (9.14)

with ωn = 2nπT for bosonic and ωn = (2n + 1)πT for fermionic fields, respectively, andn ∈ Z. The frequencies ωn are called Matsubara frequencies. Similarly, the Green functionsfor non-interacting fields are given in the limit V →∞ by

Gβ(t,x) =1

β

∞∑

n=−∞

∫d3p

(2π)3Gn(ωn,p) e

−i(ωnt+p·x) (9.15)

with

Gn(ωn,p) =1

ω2n + p

2 +m2. (9.16)

Thermal Green functions and vertices come without imaginary units, because we have trans-formed the path integral to Euclidean time.

9.2. Scalar gas

We will illustrate the basics of thermal field theory considering the simplest example, a gasof scalar particles.

9.2.1. Free scalar gas

The free energy density F of a free scalar field obtained performing the path integral in thefree partition function Z is given by

βF =1

2

∞∑

n=−∞

∫d3k

(2π)3ln[ω2n + k

2 +m2]. (9.17)

In order to evaluate the sum, it is sufficient to consider

A = T∑

n

ln[−(iωn)2 + E2

k

](9.18)

210

9.2. Scalar gas

where we set also E2k = k2 +m2. We eliminate first the logarithm by differentiating A w.r.t.

Ek,dA

dEk

= −2TEk

∑

n

1

(iωn)2 − E2k

. (9.19)

The function dA/dEk has in the complex ω plane poles along the imaginary axis at ω =iωn = 2πnT i plus two poles on the real axis at ω = ±Ek∓ iε, cf. Fig. 9.1. We convert the suminto a contour integral using Cauchy’s theorem in “reverse order:” Since cot(x) has poles atx ∈ Z with residua 1, and thus coth(βω/2) at iωn with residua 2/β, we find

dA

dEk

= −2TEk

β

2

∑

n

resωn

coth(βω/2)

ω2n − E2

k

=Ek

2πi

∮

Cdω

coth(βω/2)

ω2 − E2k

. (9.20)

The integrand vanishes as 1/|ω|2 for |ω| → ∞, and thus we can break up the contour C intotwo pieces which we close at ±i∞. Note that we pick up thereby a minus sign, since we changethe orientation of the integration path. Now we can use Cauchy’s theorem in the “normalorder” to evaluate the residua of the two enclosed poles at ±Ek ∓ iε,

dA

dEk

= Ek

∑

±res±Ek

coth(βω/2)

ω2 −E2k

= cosh(βEk/2) = 1 +2

eβEk − 1. (9.21)

Integration gives

F =1

2

∫d3k

(2π)3

[

Ek + 2T ln(1− e−βEk) + c]

(9.22)

The integration constant c cancels against the normalization constant of the path integral;dropping also the T = 0 vacuum part and taking the high-temperature limit T ≫ m gives

F = T

∫d3k

(2π)3ln(1−e−βEk) ≈ T 4

2π2

∫ ∞

0dxx2 ln

(

1− exp−√

x2 + (βm)2)

= −π2T 4

90+m2T 2

24.

(9.23)

Example: Derive the high-temperature expansion T ≫ m of the free energy F given in Eq. (9.23).We perform a Taylor expansion of ln(1 − exp−[x2 + (βm)2]1/2) around βm = 0,

∫ ∞

0

dxx2 ln(1− e−[x2+(βm)2]1/2) =

∫ ∞

0

dxx2 ln(1− e−x) +(βm)2

2

∫ ∞

0

dxx

ex − 1+O(βm)4 .

In the first integral we expand the logarithm,

∫ ∞

0

dxx2 ln(1− e−x) = −∞∑

n=1

∫ ∞

0

dxx2e−nx = −2∞∑

n=1

1

n4= −2ζ(4) = −2 π

4

90,

while we use in the second integral

1

ex − 1=

e−x

1− ex=

∞∑

n=1

e−nx .

With the definition (??) for the Gamma function, the second integral results in Γ(2)ζ(2) = π2/6.

211


Im(ω)

Re(ω)−ω + iε

+ω − iε+

+++++++++

Im(ω)

Re(ω)−ω + iε

+ω − iε+

+++++++++Figure 9.1.: Poles and contours in the complex ω plane used for the evaluation of the free

energy F .

9.2.2. Interacting scalar gas

As main application of the formalism of thermal field theory we will calculate the first quan-tum correction to the equation of state of a gas of scalar particles interacting with a λφ4

interaction. This O(λ) correction is given by a two-loop vacuum diagram and serves us alsoas demonstration how mixed UV divergences are cancelled by counter terms found at lowerorder. After that we discuss the IR behaviour of the massless a λφ4 theory. We will find thatthe plasma generates a thermal mass for the scalar particle, an effect we will try to imitategenerating masses for the particles of the SM.

Equation of state of a scalar gas We aim now at the first-quantum correction to theequation of state of gas of massive scalar particles interacting with a λφ4 interaction. Thecorresponding Hamiltonian is given by

H =

∫

d3x

[1

2π2 +

1

2(∇φ)2 +

1

2m2φ2 +

λ

4!φ4]

(9.24)

= H0 +Hint , (9.25)

where H0 =∑

k ωk(1/2 + a†kak). We expand also the Landau free energy in λ,

Ω = Ω0 + λΩ1 + λ2Ω2 + . . . (9.26)

The interaction term produces to first order in λ the following vacuum diagram,

=λ

4!

⟨φ4⟩=

3λ

4!

⟨φ2⟩⟨φ2⟩=λ

8

(

∆(t = 0,x = 0))2,

where the factor three has come from the possible ways of joining the lines to form the twoloops seen in the diagram. (Alternatively, we can use that 〈φn〉 is just the expectation value

212

9.2. Scalar gas

of φn times a Gaussian.) We could now insert into the formula the Green function derived inthe imaginary time formalism. Instead, we apply a more instructive argument following thesteps in Eqs. (2.62-2.63): At T = 0, the vacuum is empty and the a†kak term in H0 gives zerocontribution,

∆(t = 0, x = 0) =

∫d3k

(2π)31

2ωk

. (9.27)

For T > 0, the expectation value of the number operator Nk = a†kak is just the numberdistribution nk of φ particles,

∆(t = 0, x = 0) =

∫d3k

(2π)31

ωk

(1

2+⟨

a†kak⟩)

(9.28)

=

∫d3k

(2π)31 + 2nk2ωk

. (9.29)

In thermal equilibrium the number density of a scalar field follows Bose-Einstein statistics,that is

nk =1

eβωk − 1. (9.30)

More formally, this result can be derived directly from the periodicity condition of the Greenfunctions (problem 9.2). Note that we can view the propagator as the sum of a vacuum part(“1/2”) and a thermal part (“nk”). In the latter, the high energy modes are exponentiallysuppressed and thus no UV divergences should appear in the temperature dependent partsof physical observables. Thus our standard renormalisation program at T = 0 should applyin the same way at T > 0.

Continuing the derivation of the first order correction to Ω we have

λ

4!

⟨φ4⟩=λ

8

[∑

k

1 + 2nk2ωk

]2(9.31)

=λ

8

[(∑

k

1

2ωk

)2

︸︷︷︸

vacuum term

+

(∑

k

nkωk

)2

︸︷︷︸

finite T-dependent term

+ 2

(∑

k

1

2ωk

)(∑

k

nkωk

)

︸︷︷︸

infinite vacuum × T-dependent term

]

. (9.32)

The mixed term gives rise to a temperature dependent UV divergence, which from the previousargument should not appear in our calculation. Thus we expect that this term will be cancelledby an one-loop counter term.

We consider therefore the self-energy Σ, which is given by

Σ =λ

2∆(t = 0,x = 0) =

λ

2

∑

k

1 + 2nk2ωk

, (9.33)

which contains the usual T = 0 divergence coming from the unsuppressed sum over frequen-cies. We use physical perturbation theory, adding an additional interaction term of the formδm2φ2 to the Lagrangian, where δm2 is chosen such that m corresponds to the physical mass.Thus δm2 is determined by

δm2 +λ

2

∑

k

1

2ωk

= 0 . (9.34)

213


Because the product δm2∆(0) is of O(λ), we have to add the following vacuum diagram toΩ1,

1

2δm2∆(0, 0) = =

1

2

(

−λ2

∑

k

1

2ωk

)(∑

k

1 + 2nk2ωk

)

. (9.35)

Comparing this expression to the troublesome mixed term, we see that they agree but havethe opposite sign. Thus the one-loop subdivergence cancels the temperature dependent UVdivergence at two-loop in Ω1. As result, we obtain the consistent expression

λ

4!

⟨φ4⟩=λ

8

(∑

k

nkωk

)2

+ vac , (9.36)

which we can now calculate explicitly. For simplicity, we restrict ourselves to the high-temperature limit,

∑

k

nkωk

=

∫d3k

(2π)31

ω

1

eβω − 1=

T 2

2π2

∫ ∞

0dx

x

ex − 1︸︷︷︸

π2/6

=T 2

12, (9.37)

where we have used that m/T → 0 at high T . So our final result for Ω1 is

Ω1 =λ

8

(T 2

12

)2

=λ

1152T 4 , (9.38)

which we may compare to the non-interacting result, Ω0 = π2T 4/90: The ratio Ω1/Ω0 ≈10−3λ seems to indicate a fast convergence of the perturbative expansion of the pressure forany reasonable value of the coupling.

We simply quote the result to three loops or second order in λ from the literature,

P =π2T 4

9

[1

10− 1

8

λ

16π2+

1

8

(

3 lnµ

4πT+

31

35+ C

)(λ

16π2

)2]

. (9.39)

The parameter µ in Eq. (9.39) is not the chemical potential but as usually the renormalisationscale. As the pressure is a physical quantity, it should not depend upon such a parameter,but the truncation used in the expansion causes this µ dependence, which then leads to a µdependence at O(λ3). Examining this µ dependence of P we have

µdP

dµ=π2T 4

9

[

− 1

8

1

16π2µdλ

dµ︸︷︷︸

β=3λ2/16π2

+3

8

(λ

16π2

)2

+O(λ3)]

= O(λ3) . (9.40)

Thus the dependence on the renormalisation scale µ is of higher order in λ than the orderof perturbation theory we are working with. Still we can try to minimise the remainingdependence by a suitable choice of µ: Clearly, a sensible choice in this case is µ = 4πT , forwhich the logarithm vanishes. Still, any other choice is mathematically as correct as this one.Instead of being worried about this dependence on the renormalisation scale, we may takeadvantage of it as follows: Varying the renormalisation scale µ in a “reasonable range”, saybetween µ = 2πT and µ = 8πT , we obtain an error estimate for the missing higher-ordercorrections.

Finally, we note that µ ∝ T implies that a QCD plasma becomes in the large tempera-ture limit an asymptotically free gas of quarks and gluons, while at T = O(ΛQCD) a phasetransition to colourless mesons and baryons happens.

214

9.2. Scalar gas

Figure 9.2.: Left: A second order correction to the mass; Right: Ring diagram, each externalbubble corresponds to an insertion of m2

D.

IR behaviour We have found a fast convergence of the perturbative expansion from theresults for the pressure of a scalar gas. For applications particularly interesting is the case ofa massless particle and we consider now as a toy-model for QCD a λφ4 theory with m = 0.

Looking back at our previous result for the self-energy, Eq. (9.33), setting m → 0 anddropping the vacuum term, we have

Σ =λ

2∆(t = 0,x = 0)→ λ

2

∫d3k

(2π)3nkωk

=λ

2

T 2

12. (9.41)

Thus the thermal part of the self-energy induces at first order in λ a thermal or Debye mass,

m2D =

λ

2

T 2

12. (9.42)

Switching to the covariant form of the thermal propagator, Eq. (9.15), the contribution shownin the left panel of figure 9.2 at second order is

Σ2 = −λ

2T∑

n

∫d3k

(2π)3m2D

(ω2n + k

2)2. (9.43)

While in the terms with n 6= 0 the ωn act as a IR cutoff, we see that for n = 0 and thusω0 = 0 the integral is proportional to

∫dk/k2 and thus IR divergent. If we go to higher terms

in the expansion and add additional loops to the ‘primary loop’ as shown in the right panelof figure 9.2, then the IR divergence only increases in power.

The solution to this problem is to account for the thermal mass of the scalar particleproperly. If we use an effective propagator which includes the Debye mass of the particle,

1

ω2n + k

2 →1

ω2n + k

2 +m2D

, (9.44)

then the n = 0 term of Eq. (9.43) with ω0 = 0 is given by

Σ2,n=0 = −λ

2T

∫d3k

(2π)3m2D

(k2 +m2D)

2= −λ

2

4

(T 2

12

)(T

8πmD

)

(9.45)

215


and thus finite.

Since the Debye mass scales as mD ∝ λ1/2, the contribution of Σ2,n=0 in the perturbativeexpansion is not of order λ2 but of order λ3/2. Thus we obtained a term which is non-analyticin the coupling—which can not happen, if we sum a finite number of terms. As explanation,we have to look at the expansion of the effective propagator for smallmD (restricting ourselvesagain to the n = 0 term),

1

k2 +m2D

=1

k2− m2

D

k4+m4D

k6+ . . . (9.46)

Here we can view e.g. the m4D/k

6 term as a ring diagram with three massless propagatorsk−2 and two factors m2

D produced by self-energy insertions. Thus including the thermal masscorresponds to summing up the infinite sum of diagrams shown in the right panel of figure 9.2.

We can formalise the inclusion of the Debye mass in the originally massless scalar theoryas follows: We reorganise perturbation theory by adding a mass term to the free Lagrangianand subtracting it from the the interaction term,

L0 =1

2(∂µφ)

2 − 1

2m2φ2 (9.47)

Lint = −λ

4!φ4 +

1

2m2φ2 . (9.48)

Here we may set m2 = m2D or keep it as a free parameter to be determined by, for example,

that the free energy is independent of this parameter

dFdm2

= 0 . (9.49)

This reformulation of the perturbative expansion in thermal field theories is called screenedor optimised perturbation theory.

Symmetry restoration at high temperature Let us reconsider the Hamiltonian (9.24). Wehave chosen the sign of λ such that H is bounded from below, and m2 > 0 ensures that theminimal energy is obtained for 〈φ〉 = 0. If we add in this case the thermal mass,

V (φ, T ) =1

2

(

m2 +λ

24T 2

)

φ2 +λ

4!φ4 , (9.50)

we increase simply the effective mass of the φ particle. Something more drastic happens, ifwe switch the sign of the vacuum mass term and set m2 < 0. Then for T = 0 the minimalenergy is obtained for φ0 = ±

√

−6m2/λ. Thus we expect that the vacuum is non-empty andfilled with a non-zero, classical field φ, having either the value +φ0 or −φ0. Picking out oneof the two breaks the φ→ −φ symmetry of the Hamiltonian. Increasing the temperature, thepositive thermal mass grows. Above some critical temperature, the effective mass becomestherefore again positive and the minimal energy configuration becomes φ0 = 0. This is showngraphically in figure 9.3.

This behaviour resembles the one we know from a ferromagnet: Below some critical tem-perature, spontaneous magnetisation breaks rotation invariance, which is above Tc restored.

216

9.A. Appendix: Equilibrium statistical physics in a nut-shell

-10

-5

0

5

10

15

20

-10 -5 0 5 10

V(φ

,T)/

µ4

φ/µ

T=0

T=9

T=15

Figure 9.3.: Potential with finite temperature with µ < 0 and T = 0, 0 < T < Tc and T = Tcshowing restoration of symmetry.


The distribution function f(p) of a free gas of fermions or bosons in kinetic equilibrium are

f(p) =1

exp[β(E − µ)]± 1(9.51)

where β = 1/T the inverse temperature, E =√

m2 + p2, and +1 refers to fermions and-1 to bosons, respectively. As we will see later, photons as massless particles stay also inan expanding universe in equilibrium and may serve therefore as a thermal bath for otherparticles. A species X stays in kinetic equilibrium, e.g., if in the reaction X + γ → X + γ theenergy exchange with photons is fast enough.

The chemical potential µ is the average energy needed, if an additional particles is added,dU =

∑

i µdNi. If µ is zero, If the species X is also in chemical equilibrium with other species,e.g. via the reaction X + X ↔ γ+ γ with photons, then their chemical potentials are relatedby µX + µX = 2µγ = 0.

The number density n, energy density ρ and pressure P of a species X follows as

n =g

(2π)3

∫

d3p f(p) (9.52)

ρ =g

(2π)3

∫

d3p Ef(p) (9.53)

P =g

(2π)3

∫

d3pp2

3Ef(p) (9.54)

The factor g takes into account the internal degrees of freedom like spin or colour. Thus fora photon, a massless spin-1 particle g = 2, for an electron g = 4, etc.

217


Derivation of the pressure integral Eq. (9.54:)a. Intuitive for a classical gas: old ex.b. Comparing the 1. law of thermodynamics, dU = TdS − PdV , with the total differentialdU = (∂U/∂S)V dS + (∂U/∂V )SdV gives P = −(∂U/∂V )S .Since U = V

∫Ef(p) and S ∝ ln(V f(p)), differentiating U keeping S constant means P =

−V∫(∂E/∂V )f(p).

We write ∂E/∂V = (∂E/∂p)(∂p/∂L)(∂L/∂V ). To evaluate this we note that ∂E/∂p = p/E,that from V = L3 it follows ∂L/∂V = 1/3L2 and that finally the quantisation conditions of freeparticles, pk = 2πk/L implies ∂p/∂L = −p/L. Combined this gives ∂E/∂V = −p2/(3EV ).

In the non-relativistic limit T ≪ m, eβ(m−µ) ≫ 1 and thus differences between bosons andfermions disappear,

n =g

2π2e−β(m−µ)

∫ ∞

0dp p2e−β

p2

2m = g

(mT

2π

)3/2

exp[−β(m− µ)] (9.55)

ρ = mn (9.56)

P = nT ≪ ρ (9.57)

These expressions correspond to the classical Maxwell-Boltzmann statistics2. The numberof non-relativistic particles is exponentially suppressed, if their chemical potential is small.Since the number of protons per photons is indeed very small in the universe (cf. Ex. 4), andtherefore also the number of electron (the universe should be neutral), the chemical potentialµ can be neglected in cosmology at least for protons and electron.

In the relativistic limit T ≫ m with T ≫ µ all properties of a gas are determined by itstemperature T ,

n =g T 3

2π2

∫ ∞

0dx

x2

ex ± 1= ε1

ζ(3)

π2gT 3 (9.58)

ρ =g T 4

2π2

∫ ∞

0dx

x3

ex ± 1= ε2

π2

30gT 4 (9.59)

p = ρ/3 (9.60)

where for bosons ε1 = ε2 = 1 and for fermions ε1 = 3/4 and ε2 = 7/8, respectively.

Since the energy density and the pressure of non-relativistic species is exponentially sup-pressed, the total energy density and the pressure of all species present in the universe canbe well-approximated including only the relativistic ones,

ρrad =π2

30g∗T

4 (9.61)

prad = ρrad/3 =π2

90g∗T

4 (9.62)

where

g∗ =∑

bosons

gi

(TiT

)4

+7

8

∑

fermions

gi

(TiT

)4

. (9.63)

2Integrals of the type∫

∞

0dxx2ne−ax2

can be reduced to a Gaussian integral by differentiating with respectto the parameter a.

218


Entropy Rewriting the first law of thermodynamics, dU = TdS − PdV , as

dS =dU

T+P

TdV =

d(V ρ)

T+p

TdV =

V

T

dρ

dTdT +

ρ+ P

TdV (9.64)

and comparing this expression with the total differential dS(T, V ), one derives

∂S

∂V T=ρ+ P

T. (9.65)

Since the RHS is independent of V for constant T , we can integrate and obtain

S =ρ+ P

TV + f(T ) . (9.66)

The integration constant f(T ) has to vanish to ensure that S is an extensive variable, S ∝ V .The total entropy density s ≡ S/V of the universe can again approximated by the rela-

tivistic species,

s =2π2

45g∗ST

3 (9.67)

where now

g∗,S =∑

bosons

gi

(TiT

)3

+7

8

∑

fermions

gi

(TiT

)3

(9.68)

The entropy S is an important quantity because it is conserved during the evolution of theuniverse. Conservation of S implies that S ∝ g∗,SR3T 3 = const. and thus the temperatureof the Universe evolves as

T ∝ g−1/3∗,S R−1 . (9.69)

When g∗ is constant, the temperature T ∝ 1/R. Consider now the case that a particlespecies, e.g. electrons, becomes non-relativistic at T ∼ me. Then the particles annihilate,e++e− → γγ, and its entropy is transferred to photons. Formally, g∗,S decreases and thereforethe temperature decreases for a short period less slowly than T ∝ 1/R.

Since s ∝ R−3 and also the net number of particles with a conserved charge, e.g. nB ≡nB − nB ∝ R−3 if baryon number B is conserved, the ratio nB/s = const.

Summary of chapter

Thermal Green functions are (anti-) periodic functions in imaginary time, leading to discreteenergies with ωn = 2nπT for bosonic and ωn = (2n + 1)πT for fermionic fields, respectively.No new UV divergences appear for T > 0, since the thermal distribution function vanishexponentially for E/T → ∞. In a plasma, even massless particles can aquire a temperaturedependent (Debye) mass and symmetries of the Lagrangian may be hidden at low temperau-res.

Further reading

Blaizot’s lecture notes [Bla11] are a useful starting point before one turns to text booksdedicated to thermal field theory as [KG11].

219


Problems

9.1 Free energy of a free scalar field.

Calculate the free energy of a free scalar fieldusing the Matsubara/imaginary time formalism:Express first the free energy density as

F =1

2T∑

n

∫d3p

(2π)3ln(ω2

n + p2 +m2) .

Use then the residue theorem and that coth(βω/2)has poles at the right position to convert thesum into a contour integral. Finally, discard theT = 0 part and calculate the integral in the high-temperature limit T/m≫ 1.

9.2 Bose-Einstein distribution.Derive (9.30) from (9.11).

220

10. Symmetries and symmetry breaking

We have seen in the last chapter that the discrete Z2 symmetry of our standard λφ4 La-grangian could be hidden at low temperatures, if we choose a negative mass term in the zerotemperature Lagrangian. Although such a choice seems at first sight unnatural, we will inves-tigate this case in the following in more detail. Our main motivation is the expectation thathiding a symmetry by choosing a non-invariant ground-state retains the “good” properties ofthe symmetric Lagrangian. Coupling then such a scalar theory to a gauge theory, we hopeto break gauge invariance in a “gentle” way which allows e.g. gauge boson masses withoutspoiling the renormalisability of the unbroken theory.

As additional motivation we remind that couplings and masses are not constants but dependon the scale considered. Thus it might be that the parameters determining the Lagrangianof the Standard Model at low energies originate from a more complete theory at high scales,where the mass parameter µ2 is originally still positive. In such a scenario, µ2(Q2) maybecome negative only after running it down to the electroweak scale Q = mZ .

10.1. Symmetry breaking and Goldstone’s theorem

Let us start classifying the possible destinies of a symmetry:

• Symmetries may be exact. In the case of local gauge symmetries as U(1) or SU(3) forthe electromagnetic and strong interactions, we expect that this holds even in theoriesbeyond the SM. In contrast, there is no good reason why global symmetries of the SMas B −L should be respected by higher-dimensional operators originating from a morecomplete theory valid at higher energy scales.

• A classical symmetry may be broken by quantum effects. As a result, the correspondingNoether currents are non-zero and the Ward identities of the theory are violated. Ifthe anomalous symmetry is a local gauge symmetry, the theory becomes thereby non-renormalisable. Moreover, we would expect e.g. in case of QED that the universality ofthe electric charge does not hold exactly.

• The symmetry is explicitly broken by some “small” term in the Lagrangian. An examplefor such a case is isospin which is broken by the mass difference of the u and d quarks.

• The Lagrangian contains an exact symmetry but the ground-state is not symmetricunder the symmetry. In field theory, the ground-state corresponds to the mass spectrumof particles. As a result, the symmetry of the Lagrangian is not visible in the spectrumof physical particles. If the ground-state breaks the original symmetry because oneor several scalar fields acquire a non-zero vacuum expectation value, one calls thisspontaneous symmetry breaking (SSB). As the symmetry is not really broken on theLagrangian level, a perhaps more appropriate name would be “hidden symmetry.”

221


In this and the following chapter, we discuss the case of spontaneous symmetry breaking, firstin general and then applied to the electroweak sector of the SM. Since the breaking of aninternal symmetry should leave Poincare symmetry intact, we can give only scalar quantities φa non-zero vacuum expectation value 〈0|φ|0〉 6= 0. This excludes non-zero vacuum expectationvalues (vev) for tensor fields, which would single out a specific direction. On the other hand,we can construct scalars as 〈0|φ|0〉 = 〈0|ψψ|0〉 6= 0 out of the product of multiple fieldswith spin. In the following, we will always treat φ as an elementary field, but we shouldkeep in mind the possibility that φ is a composite object, e.g. a condensate of fermion fields,〈φ〉 = 〈ψψ〉, similar to the case of superconductivity.

Spontaneous breaking of discrete symmetries We will first consider the simplest exampleof a theory with a broken symmetry: A single scalar field with a discrete reflection symmetry.Consider the familiar λφ4 Lagrangian, but with a negative mass term which we include intothe potential V (φ),

L =1

2(∂µφ)

2 +1

2µ2φ2 − λ

4φ4 =

1

2(∂µφ)

2 − V (φ) . (10.1)

The Lagrangian is invariant under the symmetry operation φ → −φ for both signs of µ2.The field configuration with the smallest energy is a constant field φ0, chosen to minimise thepotential

V (φ) = −1

2µ2φ2 +

λ

4φ4, (10.2)

which has the two minimaφ0 = ±v ≡

√

µ2/λ . (10.3)

In quantum mechanics, we learn that the wave-function of the ground state for the potentialV (x) = −1

2µ2x2+λx4 will be a symmetric state, ψ(x) = ψ(−x), since the particle can tunnel

through the potential barrier. In field theory, such tunnelling can happen in principle too.However, the tunnelling probability is proportional to the volume L3, and vanishes in thelimit L→∞: In order to transform φ(x) = −v into φ(x) = +v we have to switch an infinitenumber of oscillators, which clearly costs an infinite amount of energy.

Thus in quantum field theory, the system has to choose between the two vacua ±v and thesymmetry of the Lagrangian is broken in the ground state. Had we used our usual Lagrangianwith a positive mass term, the vacuum expectation value of the field would have been zero,and the ground state would respect the symmetry.

Quantising the theory (10.1) with the negative mass around the usual vacuum, |0〉 with〈0|φ|0〉 = φc = 0, we find modes behaving as

φk ∝ exp(−iωt) = exp(−i√

−µ2 + |k|2 t) , (10.4)

which can grow exponentially for |k|2 < µ2. More generally, exponentially growing modesexists, if the potential is concave at the position of φc, i.e. for

m2eff(φc) = V ′′(φc) = −µ2 + 3λφ2c < 0 (10.5)

or |φc| <√

µ2/(3λ).Clearly, the problem is that we should, as always, expand the field around the ground-state

v. This requires that we shift the field as

φ(x) = v + ξ(x) , (10.6)

222


splitting it into a classical part 〈φ〉 = v and quantum fluctuations ξ(x) on top of it. Then weexpress the Lagrangian as function of the field ξ,

L =µ4

4λ+

1

2(∂µξ)

2 − 1

2(2µ2)ξ2 − µ

√λξ3 − λ

4ξ4 . (10.7)

In the new variable ξ, the Lagrangian describes a scalar field with positive mass mξ =√2µ >

0. The original symmetry is no longer apparent: Since we had to select one out of thetwo possible ground states, a term ξ3 appeared and the φ → −φ symmetry is broken. Thenew cubic interaction term rises now the question, if our scalar λφ4 theory becomes non-renormalisable after SSB: As we have no corresponding counter-term at our disposal, therenormalisation of µ and λ has to cure also the divergences of the interaction.

Finally, we note that the contribution µ4/(4λ) to the energy density of the vacuum is incontrast to the vacuum loop diagrams generated by Z[0] classical and finite. Thus it is unlikelythat we can use the excuse that “quantum gravity” will solve this problem. Moreover, we seelater that symmetries will be restored at high temperatures or at early times in the evolutionof the Universe. Even if we take the freedom to shift the vacuum energy density, we haveeither before or after SSB an unacceptable large contribution to the cosmological constant(problem 10.1).

Spontaneous breaking of continuous symmetries Our main aims to understand the SSBof the electroweak gauge symmetry. As next step we look therefore at a system with a globalcontinuous symmetry. We discussed already in section 3.1 the case of N real scalar fieldsdescribed by the Lagrangian

L =1

2

[(∂µφ)

2 + µ2φ2]− λ

4(φ2)2 . (10.8)

Since φ = φ1, . . . , φN transforms as a vector under rotations in field space,

φi → Rijφj (10.9)

with Rij ∈ O(n), the Lagrangian is clearly invariant under orthogonal transformations.

Before we consider the general case of arbitrary N , we look at the case N = 2 for which thepotential is shown in Fig. 10.1. Without loss of generality, we choose the vacuum pointingin the direction of φ1: Thus v = 〈φ1〉 =

√

µ2/λ and 〈φ2〉 = 0. Shifting the field as in thediscrete case gives

L =1

2

µ4

λ+

1

2(∂µξ)

2 − 1

2(2µ2)ξ21 + Lint , (10.10)

i.e. the two degrees of freedom of the field φ split after SSB into one massive and one masslessmode.

Since the mass matrix consists of the coefficients of the terms quadratic in the fields, thegeneral procedure for the determination of physical masses is the following: Determine firstthe minimum of the potential V (φ). Expand then the potential up to quadratic terms,

V (φ) = V (φ0) +1

2(φ− φ0)i(φ− φ0)j

∂2V

∂φi∂φj︸︷︷︸

Mij

+ . . . (10.11)

223


The term of second derivatives is a symmetric matrix with elements Mij ≥ 0, because weevaluate it by assumption at the minimum of V . Diagonalising Mij gives as eigenvaluesthe squared masses of the fields. If the potential has n > 0 flat directions, the vacuum isdegenerated and n massless modes appear. The eigenvectors of the mass matrix Mij arecalled the mass eigenstates or physical states. Propagators and Green functions describe theevolution of fields with definite masses and should be therefore build up on these states.

Looking at Fig. 10.1 suggests to use polar instead of Cartesian coordinates in field space.In this way, the rotation symmetry of the potential and the periodicity of the flat directionis reflected in the variables describing the scalar fields. Introducing first the complex fieldφ = (φ1 + iφ2)/

√2, the Lagrangian becomes

L = ∂µφ†∂µφ+ µ2φ†φ− λ(φ†φ)2 . (10.12)

Next we setφ(x) = ρ(x)eiϑ(x) (10.13)

and use ∂µφ = [∂µρ+ iρ∂µϑ]eiϑ to express the Lagrangian in the new variables,

L = (∂µρ)2 + ρ2(∂µϑ)

2 + µ2ρ2 − λρ4 . (10.14)

Shifting finally again the fields as ρ = v + ξ with v =√

µ2/2λ, we find

L =µ4

4λ+µ2

2λ(∂µϑ)

2 + (∂µξ)2 − 2µ2ξ2 − 2µ

√2λξ3 − λξ4

+

[√

2µ2

λξ + ξ2

]

(∂µϑ)2 .

(10.15)

The phase ϑ which parametrises the flat direction of the potential V (ϑ, ξ) remained massless.This mode is called Goldstone (or Nambu-Goldstone) boson and has derivative couplings tothe massive field ξ, given by the last term in Eq. (10.15). This is a general result, implyingthat static Goldstone bosons do not interact. Another general property of Goldstone bosonsis that they carry the quantum number of the corresponding symmetry generator. Thus theyare scalar or pseudo-scalar particles. The only exception is the case when a supersymmetrywhich has fermionic generators is spontanously broken.

Let us now discuss briefly the case of general N for the Lagrangian (10.8). The lowestenergy configuration is again a constant field. The potential is minimised for any set of fieldsφ0 that satisfies

φ20 =

µ2

λ. (10.16)

This equation only determines the length of the vector, but not its direction. It is convenientto choose a vacuum such that φ0 points along one of the components of the field vector.Aligning φ0 with its Nth component,

φ0 =

(

0, . . . , 0,

√

µ2

λ

)

, (10.17)

we now follow the same procedure as in the previous example. First we define a new set offields, with the Nth field expanded around the vacuum

φ(x) = (φk(x), v + ξ(x)) , (10.18)

224


Figure 10.1.: Scalar potential symmetric under O(2)

where k now runs from 1 to N − 1. Then we insert this, and the value v =√

µ2/λ for thevacuum expectation value into the Lagrangian, and obtain

L =1

2(∂µφ

k)2 +1

2(∂µξ)

2 − 1

2(2µ2)ξ2 +

1

4

µ4

λ

−√λµξ3 −

√λµξ(φk)2 − λ

2ξ2(φk)2 − λ

4

[(φk)2

]2 − λ

4ξ4 .

(10.19)

This Lagrangian describes N − 1 massless fields and a single massive field ξ, with cubic andquartic interactions. The O(N) symmetry is no longer apparent, leaving as symmetry groupthe subgroup O(N−1), which rotates the φk fields among themselves. This rotation describesmovements along directions where the potential has a vanishing second derivative, while themassive field corresponds to oscillations in the radial direction of V . This can be visualisedfor N = 2, where we get the ”Mexican hat” potential shown in figure 10.1.

Goldstone’s theorem The observation that massless particles appear in theories with spon-taneously broken continuous symmetries is a general result, known as Goldstone’s theorem.The first example for such particles was suggested by Nambu in 1960: He showed that a mass-less quasi-particle appears in a magnetised solid, because the magnetic field breaks rotationinvariance. Goldstone applied soon after that this idea to relativistic QFTs and showed thatmassless scalar elementary particles appear in theories with SSB. Since no massless scalarparticles are known to exist, this theorem appeared to be a dead end for the application ofSSB to particle physics. So our task is two-fold: First we should derive Goldstone’s theoremand then we should find out how we can bypass the theorem applying it to our case of interest,gauge theories.

The theorem is obvious at the classical level: Consider a Lagrangian with a symmetry Gand a vacuum state invariant under a subgroup H of G. For instance, choosing a Lagrangianinvariant under G = O(3) and picking out a vacuum along φ3, the subgroup H = O(2) ofrotation around φ3 keeps the vacuum invariant. Let us denote with U(g) a representationof G acting on the fields φ and with U(h) a representation of H, respectively. Since we

225


consider constant fields, derivative terms in the fields vanish and the potential V alone hasto be symmetric under G, i.e.

V (U(g)φ) = V (φ) . (10.20)

Moreover, we know that the vacuum is kept invariant for all h, φ′0 = U(h)φ0, but changes

for some g, φ′0 6= U(g)φ0. Using the invariance of the potential and expanding V (U(g)φ0)

for an infinitesimal group transformation gives

V (φ0) = V (U(g)φ0) = V (φ0) +1

2

∂2V

∂φi∂φj

∣∣∣∣0

δφiδφj + . . . , (10.21)

where δφi denotes the resulting variation of the field. Equation (10.21) implies that

Mijδφiδφj = 0 . (10.22)

The variation δφi depends on whether the transformation belong to U(h) or not: In the formercase, the vacuum φ0 is unchanged, δφi = 0 and (10.22) is automatically satisfied. If on theother hand g does not belong to H, i.e. is a member of the left coset G/H, then δφi 6= 0,implying that the mass matrix Mij has a zero eigenvalue. It is now clear that the number ofmassless particles is simply determined by the dimensions of the two groups G and H: Thenumber of Goldstone bosons is equal to the number of symmetries spontaneously broken, orto the dimension of the left coset G/H.

Quantum case The previous discussion was based on the classical potential. Thus we shouldaddress the question if this picture survives quantum corrections.

Noether’s theorem tells us that every continuous symmetry has associated to its generatorsgi conserved charges Qi. On the quantum level this means the operators Qi commute withthe Hamiltonian, [H,Qi] = 0. Subtracting the cosmological constant, we have H |0〉 = 0. Ifthe vacuum is invariant under the symmetry Q, then exp(iϑQ) |0〉 = |0〉. For the infinitesimalform of the symmetry transformation, exp(iϑQ) ≈ 1 + iϑQ, and we conclude that the chargeannihilates the vacuum,

Q |0〉 = 0 . (10.23)

Or, in simpler words, the vacuum has the charge 0.

Now we came to the case we are interested in, namely that the symmetry is spontaneouslybroken and thus Q |0〉 6= 0. We first determine the energy of the state Q |0〉. From

HQ |0〉 = (HQ−QH) |0〉︸︷︷︸

H|0〉=0

= [H,Q] |0〉 = 0 , (10.24)

we see that at least another state Q |0〉 exists which has as the vacuum |0〉 zero energy.

We represent the charge operator as the volume integral of the time-like component of thecorresponding current operator,

Q =

∫

d3xJ0(t,x) . (10.25)

The state

|s〉 =∫

d3x eik·xJ0(t,x) |0〉 → Q |0〉 for k → 0 (10.26)

226

10.2. Renormalisation of theories with SSB

becomes in the zero-momentum limit equal to the state Q |0〉 we are searching for. Moreover,applying P on |s〉 gives (problem 10.5)

P |s〉 = p |s〉 . (10.27)

Thus the SSB of the vacuum, Q |0〉 6= 0, implies excitations of the system with a frequencythat vanishes in the limit of long wavelengths. In the relativistic case, Goldstone’s theorempredicts massless states, while in the non-relativistic case relevant for solid states the theorempredicts collective excitations with zero energy gap.


When we went through the SSB of the scalar field, we saw that new φ3 interactions wereintroduced. The question then arises, are new renormalisation constants needed when asymmetry is spontaneously broken? This would make these theories non-renormalisable.

We can address this questions in two ways. One possibility is to repeat our analysis of therenormalisability of the scalar theory in section 8.3.2, but now for the broken case with µ2 < 0.Then we would find that the φ3 term becomes finite, renormalising fields, mass and couplingas in the unbroken case. This is not unexpected, because shifting the field φ → φ = φ − v,which is an integration variable in the generating functional, should not affect physics. On theother hand, such a shift reshuffles the splitting L = L0 + Lint in our standard perturbativeexpansion in the coupling constant. To avoid this problem, we analyse SSB in the following ina different way: We develop as a new tool the loop expansion which is based on the effectiveaction formalism. Additionally of being not affected by a shift of the fields, this formalismallows us to calculate the potential including all quantum corrections in the limit of constantfields.

First we recall the definition of the classical field,

φc(x) =δW

δJ(x)=

1

Z

∫

Dφφ(x) expi[S +

∫

d4x′J(x′)φ(x′)] (10.28)

and of the effective action1

Γ[φc] =W [J ]−∫

d4x′J(x′)φc(x′) ≡W [J ]− 〈Jφ〉 , (10.29)

which lead to the converse relation for J ,

J(x) = − δΓ[φ]δφ(x)

. (10.30)

Effective potential In general we will not be able to solve the effective action. StudyingSSB, we can however make use of a considerable simplification: The fields we are interested inare constant, and it should be therefore useful to perform a gradient expansion of the effectiveaction Γ[φ],

Γ[φ] =

∫

d4x

[

−Veff(φ) +1

2F2(φ)(∂µφ)

2 + . . .

]

. (10.31)

1We will suppress the subscript c on the classical field from now on and use brackets 〈Jφ〉 to indicateintegration.

227


Here, we introduced also the effective potential Veff(φ) as the zeroth order term of the expan-sion in (∂µφ)

2, i.e. the only term surviving for constant fields.

If we now choose the source J(x) to be constant, the field φ(x) has to be uniform too,φ(x) = φ, by translation invariance. Together this implies that

Γ = −ΩVeff , (10.32a)

−J =δΓ[φ]

δφ= −Ω ∂Veff(φ)

∂φ, (10.32b)

where Ω is the space-time volume. Hence, as announced, we only have to calculate the effectivepotential, not the full effective action. In the absence of external sources, J = 0, Eq. (10.32b)simplifies to V ′

eff(φ) = 0. Thus this is the quantum version of our old approach where weminimised the classical potential V (φ) in order to find the vacuum expectation value of φ.Note that Eq. (10.32b) contains all quantum corrections to the classical potential, the onlyapproximation made so far neglecting gradients of the classical field.

In order to proceed, we use that we know the classical potential and we assume thatquantum fluctuations are small. Then we can perform a saddle-point expansion around theclassical solution φ0, given by the solution to

δS[φ] + 〈Jφ〉δφ

∣∣∣∣φ0

= 0 (10.33)

or,

φ0 + V ′(φ0) = J(x) . (10.34)

We write the field as φ = φ0+ φ, i.e. as a classical solution with quantum fluctuations on top.Then we can approximate the path integral by

Z = exp i~W ≈ exp i

~

[S[φ0] + 〈Jφ0〉

]∫

Dφ expi

~

∫

d4x1

2

[

(∂µφ)2 − V ′′(φ0)φ

2]

,

(10.35)where V ′′ is the second derivative of the potential term of the theory. Planck’s constant ~

has been restored to indicate that what we are doing here is an expansion in ~, or a loopexpansion, which we will show later. The functional integral over φ is quadratic and can beformally solved directly, it is equal to det( + V ′′)−

12 . Using the identity ln detA = tr lnA,

we find

W = S[φ0] + 〈Jφ0〉+i~

2tr ln[+ V ′′(φ0)] +O(~2) . (10.36)

To evaluate the operator trace, we write out the definition and insert two complete sets ofplane waves,

tr ln[+ V ′′] =∫

d4x〈x| ln[+ V ′′]|x〉

=

∫

d4xd4k

(2π)4d4q

(2π)4〈x|k〉〈k| ln[ + V ′′]|q〉〈q|x〉

=

∫

d4xd4k

(2π)4ln[−k2 + V ′′] . (10.37)

228


Performing the Legendre transform and putting everything together, we obtain for theeffective potential including the first quantum corrections

Veff(φ0) = V (φ0)−i~

2

∫d4k

(2π)4ln[k2 − V ′′(φ0)

]+O(~2) . (10.38)

As an example we can use the λφ4 theory, with

V ′′(φ0) = µ2 +1

2λφ20 , (10.39)

where we see that V ′′(φ0) can be interpreted as an effective mass, consisting of the µ2 andthe contribution 1

2λφ20 due to the constant background field φ0. The total effective potential

at order O(~2) consists of the classical potential V (φ), i.e. the classical energy density of ascalar field with vacuum expectation value φ, while the first quantum correction is given bythe zero-points energies2 of a scalar particle with effective mass V ′′(φ0).

Not surprisingly, the effective potential is divergent and we have to introduce counter-termsthat eliminate the divergent parts. Our effective potential is then

Veff(φ0) = V (φ) +~

2

∫d4k

(2π)4ln

(k2E + V ′′(φ0)

k2E

)

+Bφ20 + Cφ40 +O(~2) . (10.40)

Here, we Wick rotated the integral to Euclidean space and subtracted an infinite constant inorder to make the logarithm dimensionless. (Equivalently we could have added an additionalconstant counter-term A renormalising the vacuum energy density.) The integral can besolved in different regularisation schemes. Here we will expand the logarithm,

ln

(

1 +V ′′

k2E

)

=

∞∑

n=1

1

n

(V ′′

k2E

)n

, (10.41)

and cutoff the integral at some large momenta Λ. The first two terms of the sum will dependon the cutoff, being proportional to Λ2 and ln(Λ2/V ′′), respectively. Performing the integraland neglecting terms that vanish for large Λ, we obtain

Veff(φ0) = V (φ0) +Λ2

32π2V ′′(φ0) +

V ′′(φ0)2

64π2ln

(V ′′(φ0)Λ2

)

. (10.42)

Now we see that if we start out with a massless λφ4, our cutoff-dependent terms are

V ′′ =1

2λφ20, and (V ′′)2 =

λ2

4φ40, (10.43)

which both can be absorbed into the counter-terms B and C by imposing appropriate renor-malisation conditions.

Let us stress the important point in this result: The renormalisation of the λφ4 theoryusing the effective potential approach is not affected at all by a shift of the field: We are freeto use both signs of µ2 and any value of the classical field φ0 in Eq. (10.42). Independentlyof the sign of µ2, we need only symmetric counter-terms, as a cubic term does not appear atall.

2Integrating i∆F (0) w.r.t. m2 reproduces the one-loop term.

229


= + + . . .

1

Figure 10.2.: Perturbative expansion of the one-loop effective potential V(1)eff for the λφ4 the-

ory; all external legs have zero momentum.

We can rephrase this point as follows: If we renormalise before we shift the fields, we knowthat we obtain finite renormalised Green functions. But shifting the fields does not changethe total Lagrangian. Thus the effective action and the effective potential are unchanged too.Consequently the theory has to stay renormalisable after SSB.

Let us now discuss what happens with a non-renormalisable theory in the effective potentialapproach. Including e.g. a φ6 term leads to (V ′′)2 ∝ φ8 which requires an additional counter-term Dφ8, generating in turn even higher order terms and so forth. Thus in this case an

infinite number of counter-terms is needed for the calculation of V(1)eff . How does this finding

goes together with our statement that non-renormalisable theories are predictive below acertain cutoff scale Λ? The reason for this apparent contradiction becomes clear, if we look

again at the series expansion of the logarithm in the one loop contribution V(1)eff ,

V(1)eff = i

∞∑

n=1

∫d4k

(2π)41

2n

[V ′′(φ0)k2

]n

. (10.44)

This contribution is an infinite sum of single loops with progressively more pairs of externallegs with zero-momentum attached, see Fig. 10.2 for the case of V ′′(φ0) =

12λφ

20. (We added

the factor i, because we returned to Minkowski space; the symmetry factor 2n appearingautomatically in this approach accounts for the symmetry of a graph with n vertices underrotations and reflection.) As we saw, the superficial degree of divergence increases with thenumber of external particles for a λφn theory and n > 4. Hence every single diagram in the

infinite sum contained in V(1)eff diverges and requires a counter-term of higher order in φn.

Therefore the effective potential approach is not useful for non-renormalisable theories.

Expansion in ~ as a loop expansion To show that the expansion in ~ is really a loopexpansion, we introduce artificially a parameter a into our Lagrangian so that

L (φ, ∂µφ, a) = a−1L (φ, ∂µφ) . (10.45)

Now we want to determine the power P of a in an arbitrary 1PI Feynman graph, aP : Apropagator is the inverse of the quadratic form in L and contributes thus a positive powera, while each vertex ∝ Lint adds a factor a−1. The number of loops in an 1PI diagram isgiven by L = I − V + 1, cf. Eq. (8.42), where I is the number of internal lines and V is thenumber of vertices. Putting this together we see that

P = I − V = L− 1 (10.46)

230


and it is clear that the power of a gives us the number of loops.We should stress that using a loop expansion does not imply a semi-classical limit, S ≫ ~:

Our fictitious parameter a is not small; in fact, it is one. The loop expansion is not affectedby a shift in the fields, since a multiplies the whole Lagrangian. Thus this procedure isparticularly useful discussing the renormalisation of theories with SSB.

Effective action as generating functional for 1PI Green functions We have now all thenecessary tools in order to show that the tree-level graphs generated by the effective actionΓ[φ] correspond to the complete scattering amplitudes of the corresponding action S[φ]. Wecompare our familiar generating functional

Z[J ] =

∫

Dφ expiS + 〈Jφ〉 = eiW [J ] , (10.47)

with the functional Va[J ] of a fictitious field theory whose action S is the effective action Γ[φ]of the theory (10.47) we are interested in,

Va[J ] =

∫

Dφ exp

i

aΓ[φ] + 〈Jφ〉

= eiUa[J ] . (10.48)

Additionally, we introduced the parameter a with the same purpose as in (10.45): In the limita → 0, we can perform a saddle-point expansion and the path integral is dominated by theclassical path. From (10.35), we find thus

lima→0

aUa[J ] = Γ[φ] + 〈Jφ〉 =W [J ] , (10.49)

where we used the definition of the effective action, Eq. (10.29), in the last step. The RHSis the sum of all connected Green functions of our original theory. The LHS is the classicallimit of the fictitious theory Va[J ], i.e. it is the sum of all connected tree graphs of this theory.Equation (8.147) shows the vertices of this theory are given by Γ(n)(x1, ..., xn), i.e. the 1PIGreen functions of our original theory. Thus we can represent the connected graphs of Was tree graphs whose effective vertices are the sum of all 1PI graphs with the appropriatenumber of external lines.

Another proof of the Goldstone theorem With the help of the effective potential we cangive another simple proof of the Goldstone theorem. We know that the zero of the inversepropagator determines the mass of a particle. From Eq. (8.148), the exact inverse propagatorin momentum space for a set of scalar fields is given by

∆−1ij (p2) =

∫

d4x eip(x−x′) δ2Γ

δφi(x)δφj(x′). (10.50)

Massless particles correspond to zero eigenvalues of this matrix equation for p2 = m2. If weset p = 0, the fields are constant. But differentiating the effective action w.r.t. to constantfields is equivalent to differentiating simply the effective potential,

∂2Veff∂φi(x)∂φj(x′)

= 0 . (10.51)

Thus our previous analysis of Goldstone’s theorem using the classical potential holds also inthe quantum case, if we simply replace the classical by the effective potential.

231


Coleman-Weinberg Problem Sidney Coleman and Erick Weinberg [CW73] used this for-malism to investigate if quantum fluctuations could trigger SSB in an initially massless theory.Rewriting the effective potential a bit we have

Veff(φ) =

[Λ2

64π2λ+B

]

φ2 +

[λ

4!+

λ2

(16π)2lnφ2

Λ2+ C

]

φ4 . (10.52)

Now we impose the renormalising conditions, first

d2Veffdφ2

∣∣∣∣φ=0

= 0 , (10.53)

which implies that

B = − λΛ2

64π2. (10.54)

When renormalising the coupling constant, we have to pick a different point than φ = 0, dueto the logarithm being ill-defined there. This means that we have to introduce a scale µ.Taking the fourth derivative and ignoring terms that are independent of φ, we find

d4Veffdφ4

∣∣∣∣φ=µ

= λ = 24λ2

(16π)2lnµ2

Λ2. (10.55)

We can convince ourselves that this expression gives the correct beta function,

β(µ) = µ∂λ

∂µ=

3

16π2λ2 +O(λ3) . (10.56)

Using the complete expression for Eq. (10.55), we can determine C and obtain for therenormalised effective potential (problem 10.6)

Veff(φ) =λ(µ)

4!φ4 +

λ2(µ)

(16π)2φ4[

lnφ2

µ2− 25

6

]

+O(λ3) . (10.57)

This potential has two minima outside of the origin, so it seems that SSB does indeed happen.These minima lie however outside the expected range of validity of the one loop approximation:Rewriting the potential as Veff(φ) = λφ4/4!(1 + aλ ln(φ2/µ2) + . . . suggest that we can trustthe one-loop approximation only as long as (3/32π2)λ ln(φ2/µ2)≪ 1.

10.3. Abelian Higgs model

After we have shown that the renormalisability is not affected by SSB, we now try to applythis idea to a the case of a gauge symmetry. First of all, because we aim to explain themasses of the W and Z bosons as consequence of SSB. Secondly, we saw that SSB of globalsymmetries leads to massless scalars which are however not observed. As SSB cannot changethe number of physical degrees of freedom, we hope that each of the two diseases is thecure of the other: The Goldstone bosons which would remain massless in a global symmetrydisappear becoming the required additional longitudinal degrees of freedom of massive gaugebosons in case of the SSB of a gauged symmetry.

232


The Abelian Higgs model, which is the simplest example for this mechanism, is obtainedby gauging a complex scalar field theory. Introducing in the Lagrangian (10.12) the covariantderivative

∂µ → Dµ = ∂µ + ieAµ (10.58)

and adding the free Lagrangian of an U(1) gauge field gives

L = −1

4FµνF

µν + (Dµφ)†(Dµφ) + µ2φ†φ− λ(φ†φ)2 . (10.59)

The symmetry breaking and Higgs mechanism is best discussed changing to polar coordinatesin field-space, φ = ρ expiϑ. Then we insert

Dµφ =[∂µρ+ iρ(∂µϑ+ eAµ)

]eiϑ (10.60)

into the Lagrangian, obtaining

L = −1

4FµνF

µν + ρ2(∂µϑ+ eAµ)2 + (∂µρ)

2 + µ2ρ2 − λρ4 . (10.61)

The only difference to the ungauged model is the appearance of the gauge field in the prospec-tive mass term ρ2(∂µϑ+ eAµ)

2. This allows us to eliminate the angular mode ϑ which showsup nowhere else by performing a gauge transformation on the field Aµ: The action of a U(1)gauge transformation Aµ → A′

µ = Aµ−∂µΛ on the original field φ is just a phase shift, henceρ is unchanged and ϑ is shifted by a constant, ϑ → ϑ′ = ϑ + eΛ. This means that if weconsider the gauge invariant combination

Bµ = Aµ +1

e∂µϑ (10.62)

as new variable, we eliminate ϑ completely, as Fµν(Aµ) = Fµν(Bµ) is gauge invariant,

L = −1

4FµνF

µν + e2ρ2(Bµ)2 + (∂µρ)

2 + µ2ρ2 − λρ4 . (10.63)

It is now evident that the Goldstone mode ϑ has disappeared, while the new gauge invariantfield Bµ obtained a mass and an interaction term eρ. Thus the U(1) symmetry is hiddenusing the massless Aµ field as variable, while the field Bµ produces a gauge invariant massterm in (10.63).

Eliminating the field ρ in favour of fluctuations χ around the vacuum v =√

µ2/λ, i.e.shifting as usually the field as

ρ =1√2(v + χ) , (10.64)

we find after some algebra as new Lagrangian

L =− 1

4FµνF

µν +1

2M2(Bµ)

2 + e2vχ(Bµ)2 +

1

2e2χ2(Bµ)

2

+µ4

4λ+

1

2(∂µχ)

2 − 1

2(√2µ)2χ2 −

√λµχ3 − λ

4χ4 .

(10.65)

As in the ungauged model we obtain a χ3 self-interaction and a contribution to the vacuumenergy density. But the gauge field Bµ acquired the massM = ev, therefore having now three

233


spin degrees of freedom. The additional longitudinal one has been delivered by the Goldstoneboson which in turn disappeared: The gauge field has eaten the Goldstone boson, so to speak.We also see that the number of degrees of freedom before SSB (2 + 2) matches the numberafterwards (3+1). The phenomenon that breaking spontaneously a gauge symmetry does notlead to massless Goldstone bosons because they become the longitudinal degree of freedomof massive gauge bosons is called the Higgs effect.

The gauge transformation we used to eliminate the ϑ field corresponds to the Higgs modelin the unitary gauge, where only physical particles appear in the Lagrangian. The massivegauge boson is decribed by the Procca Lagrangian and we know that the resulting propagatorbecomes constant for large momenta. Hence, this gauge is convenient for illustrating theconcept of the Higgs mechanism, but not suited for loop calculations.

A different way to consider the model is to keep the Cartesian fields φ = 1√2(φ1 + iφ2).

Then the Lagrangian is

L =− 1

4FµνF

µν +1

2

[(∂µφ1 − eAµφ2)2 + (∂µφ2 + eAµφ1)

2]

+µ2

2(φ21 + φ22)−

λ

4(φ21 + φ22)

2 . (10.66)

Performing the shift due to the SSB, φ1 = v + φ1 and φ2 = φ2, the Lagrangian becomes

L =− 1

4FµνF

µν +1

2M2A 2

µ − evAµ∂µφ2

+1

2

[

(∂µφ1)2 − 2µ2φ21

]

+1

2(∂µφ2)

2 + . . . ,(10.67)

where we have omitted interaction and vacuum terms not relevant to the discussion. As wesee, the Goldstone boson φ2 does not disappear and it couples to the gauge field Aµ. On theother hand, the mass spectrum of the physical particles is the same as in the unitary gauge.The degrees of freedom before and after breaking the symmetry do not match, hence there isan unphysical degree of freedom in the theory, namely that corresponding to φ2.

Gauge fixing and gauge boson propagator In order to make the generating functionalZ[Jµ, J, J∗] of the abelian Higgs model well-defined, we have to remove the gauge freedomof the classical Lagrangian. Using the Faddeev-Popov trick to achive this implies to add agauge-fixing and a Faddeev-Popov ghost term to the classical Lagrangian,

Leff = Lcl + Lgf + LFP = Lcl −1

2ξG2 − c∂G

∂ϑc . (10.68)

Here G = 0 is a suitable gauge condition, ϑ are the generators of the gauge symmetry andc, c are grassmannian ghost fields.

In the unbroken abelian case we used as gauge condition G = ∂µAµ. With the gauge

transformation Aµ → Aµ − ∂µϑ the ghost term becomes simply LFP = c(−)c. Thus theghost fields completely decouple from any physical particles, and the ghost term can beabsorbed in the normalisation.

In the present case of a theory with SSB, we want to use the Fadeev-Popov term to cancelthe mixed Aµ∂µφ2 term. Therefore we include the Goldstone boson φ2 in the gauge condition,

G = ∂µAµ − ξevφ2 = 0 . (10.69)

234


From

φ2 =∂µA

µ

ξev, (10.70)

we see that the unitary gauge corresponds to ξ →∞. We calculate first G2,

G2 = (∂µAµ)(∂νA

ν)− 2ξevφ2∂µAµ + ξ2e2v2φ22 , (10.71)

integrate partially the cross term and insert the result into Lgf ,

Lgf = −1

2ξG2 = − 1

2ξ(∂µA

µ)2 − evAµ∂µφ2 −1

2ξ(ev)2φ22 . (10.72)

Now we see that the second term cancels the unwanted mixed term in Lcl, while a ξ dependentmass term ξM2 for φ2 appeared.

If we write out the terms in Leff quadratic in Aµ and φ2,

Leff,2 = −1

4FµνF

µν +1

2M2A2

µ −1

2ξ(∂µA

µ)2 +1

2(∂µφ2)

2 − 1

2ξM2φ22 , (10.73)

we can find the boson propagator. Using the antisymmetry of Fµν and a partial integration,we transform F 2 into standard form,

−1

4FµνF

µν = −1

2

(∂µA

ν∂µAν − ∂νAµ∂µAν)

=1

2

(Aν∂µ∂

µAν −Aµ∂µ∂νAµ)

=1

2Aµ(g

µν− ∂µ∂ν)Aν .

The part of the Lagrangian quadratic in Aµ then reads

LA =1

2Aµ[gµν− ∂µ∂ν

]Aν +

1

2Aµg

µνM2Aν +1

2ξAµ∂

µ∂νAν

=1

2Aµ[gµν(+M2)− (1− ξ−1)∂µ∂ν

]Aν .

To find the propagator we want to invert the term in the bracket, denote this term by Pµν(k).If we go to momentum space, then

Pµν = −(k2 −M2)gµν + (1− ξ−1)kµkν . (10.74)

This can be split into a transverse and a longitudinal part by factoring out terms proportionalto PµνT = gµν − kµkν

k2,

Pµν = −(k2 −M2)(PµνT +

kµkν

k2)+ (1− ξ−1)kµkν

= −(k2 −M2)PµνT −(k2 −M2

k2− 1 + ξ−1

)

kµkν

= −(k2 −M2)PµνT − ξ−1(k2 − ξM2)PµνL , (10.75)

235


where the longitudinal part is PµνL = kµkν

k2. Since PµνT and PµνL as projection operators are

orthogonal to each other, we can invert the two parts separately and obtain

iDµνF (k2) =

−iPµνTk2 −M2 + iε

+−iξPµνL

k2 − ξM2 + iε

=−i

k2 −M2 + iε

[

gµν − (1− ξ) kµkν

k2 − ξM2 + iε

]

. (10.76)

As we see, the transverse part propagates with mass M2, while the longitudinal part propa-gates with mass ξM2. The limit ξ → ∞ corresponds again to the unitary gauge and ξ = 1corresponds to the easier Feynman-’t Hooft gauge. For finite ξ we see that the propagator isproportional to k−2 and no problems arise in loop calculations, as it did in the unitary gauge.

The Goldstone boson φ2 has the usual propagator of a scalar particle, however with gauge-dependent mass ξM2.

Ghosts Using the Faddeev-Popov ansatz for the gauge introduces ghosts field through theterm

LFP = −c δGδϑ

c (10.77)

into the Lagrangian. To calculate δG/δϑ, we have to find out how the gauge fixing conditionG changes under an infinitesimal gauge transformation. Looking first at the change of thecomplex field,

φ→ φ = φ+ ieϑφ = φ+ ieϑ1√2(v + φ1 + iφ2) , (10.78)

we see that the fields φ1 and φ2 are mixed under the gauge transformation.

Aµ → Aµ = Aµ + ∂µϑ (10.79)

φ1 → φ1 = φ1 − eϑφ2 (10.80)

φ2 → φ2 = φ2 + eϑ(v + φ1) . (10.81)

Inserting this into the gauge fixing condition (10.69) and differentiating with respect to thegenerator, we obtain

δG

δϑ=

δ

δϑ

(

∂µAµ + ξevφ2

)

= + ξe2v(v + φ1) . (10.82)

Thus after spontaneous symmetry breaking the ghost particles receive a ξ-dependent massand interact with the Higgs field φ1. To see this explicitly we insert δG/δϑ into the ghostLagrangian,

LFP = −c[+ ξe2v(v + φ1)

]c

= (∂µc)(∂µc)− ξM2cc− ξe2vφ1cc .(10.83)

The second term corresponds to the mass ξe2v2 = ξM2 for the ghost field, while the thirdone describes the ghost-ghost-Higgs interaction.

To sum this up, we have the following propagators in the Rξ gauge, where we follow commonpractise and denote with h the physical Higgs boson and with φ the Goldstone boson:

236


Gauge boson Aµ with mass MA = ev

µ ν = i

k2−M2+iǫ

[

−gµν + (1 − ξ)kµkν

k2−ξM2+iǫ

]

Higgs boson h with mass squared M2h = 2µ2

= i

k2−M2

h+iǫ

Goldstone boson φ with mass squared ξM2A

= i

k2−ξM2

A+iǫ

Ghost c with mass squared ξM2A

= i

k2−ξM2

A+iǫ

1

Before we finish this chapter, we should answer why the Goldstone theorem does not applyto the case of the Higgs model. The characteristic property of gauge theories that no man-ifestly covariant gauge exists which eliminates all gauge freedom is also responsible for thefailure of the Goldstone theorem: In the first version of our proof, we may either choose agauge as the Coulomb gauge. Then only physical degrees of freedom of the photon propagate,but the potential A0(x) drops only as 1/|x| and the charge Q defined in (10.25) becomes ill-defined. Alternatively, we can use a covariant gauge as the Lorentz gauge. Then the chargeis well-defined, but unphysical scalar and longitudinal photons exist. The Goldstone theoremdoes apply, but the massless Goldstone bosons do not couple to physical modes.

In the second version of our proof, the effective potential for the scalar and for the gaugesector do not decouple and mix by the same reason after SSB. This invalidates our analysisincluding only scalar fields.

Summary of chapter Examining spontaneous symmetry breaking of internal symmetries,we found three qualitatively different types of behaviours: For a broken global continuoussymmetry, Goldstone’s theorem predicts the existence of massless scalars. In the case ofbroken approximate symmetries, this can explain the existence of light scalar particles—anexample are pions. The case of broken global continuous symmetry which are exact seemsto be not realised in nature, since no massless scalar particles are observed. If we gauge thebroken symmetry, the would-be massless Goldstone bosons become the longitudinal degreesof freedom required for massive spin-1 bosons. Finally, neither Noether’s nor Goldstone’s the-orems apply to the case of discrete symmetries; therefore the breaking of discrete symmetriesdoes not change the mass spectrum of the theory.

237


We developed the effective potential as tool to study the renormalisability of spontaneouslybroken theories: This approach allows the calculation of all quantum corrections to the clas-sical potential in the limit of constant fields and is invariant under a shift of fields. Therebywe could establish that renormalisability is not affected by SSB.

Further reading

Our discussion of the effective potential is based on the 1966 Erice lecture “Secret Symmetry”of S. Coleman [Col88].

Problems

10.1 Contribution to the vacuum energydensity from SSB.Calculate the difference in the vacuum energy

density before and after SSB in the SM usingv = 256GeV andm2

h = 2µ2 = (125)2GeV2. Com-pare this to the observed value of the cosmologicalconstant.

10.2 Scalar Lagrangian after SSB.Derive Eq. (10.10) and write down the expliciteform of Lint.

10.3 Quantum corrections to 〈φ〉.We implicitely assumed that quantum correctionsare small enough to that the field stays at thechosen classical minimum. Calculate 〈φ(0)2〉 ford space-time dimensions and show that this as-sumption is violated for d ≤ 2.

10.4 Instability of 〈φ〉.

Calculate the imaginary part of the self-energyfor a scalar field with the Lagrangian (10.1), i.e.with a negative squared mass µ2 < 0. Discuss thephysical interpretation.

10.5 Goldstone mode as zero mode.

Show that the state |s〉 defined in Eq. (10.26) haszero energy for k → 0.

10.6 Coleman-Weinberg problem.

Derive Eq. (10.57), find the minima of the po-tential and discuss the validity of the one-loop ap-proximation.

10.7 Effective potential at finite tempera-ture.

Find the effective potential at finite temperatureT > 0 for a scalar with mass m and a λφ4 self-interaction.

238

11. GSW model of electroweak interactions

Fermi introduced a current-current interaction between four fermions as explanation for thenuclear beta-decay. After the discovery of parity violation in weak interactions, it was re-alised that the weak currents have a V − A form, LFermi = GF /

√2Jµ(x)Jµ(x) with e.g.

Jµ(x) = ψeγµ(1− γ5)ψν + h.c for the leptonic current. Using this Lagrangian, all experimen-tal data about weak interactions known until 1973 could be explained. Being a dimensionsix interaction, the Fermi theory belongs however according to our power counting analysisto the class of non-renormalisable theories. Attempts to develop a more complete theory ofweak interactions started therefore already in the late 1950ies. A first step towards a renor-malisable gauge theory for weak interactions was the introduction of “intermediate vectorbosons” W± with mass mW , leading to an interaction JµW±

µ of similiar type as in QED andto a dimensionless coupling constant g ∝ GF /m2

W .Using these new vector bosons, we can write the charged current interaction in a more

economical way introducing doublets of left-handed fermions,(νee

)

L

and

(ud

)

L

.

Associating then the bosons W± with the ladder operators τ± = (τ1 ± iτ2)/√2 (we denote

the Pauli matrices in this context with τi), the charged current interaction becomes

LFermi =GF

2√2

(νe e

)

L(τ+W+

µ + τ−W−µ )γµ

(νee

)

L

.

The doublet structure of fermions in the Fermi interaction suggests to use SU(2) as gaugegroup for weak interactions. The gauge group is often called weak isospin and denoted bySU(2)L in order to stress that only left-handed fermions participate in this gauge interaction,while right-handed fermions tranform as singlets. If it would be possible to identify τ3 withthe photon, a unification of the weak and electromagnetic forces would have been achieved.There are three major obstacles defeating this attempt:

• The fermion mass term m(ψRψL + ψLψR) is not gauge invariant.

• The generators of SU(2) are traceless, which means that the multiplets must have zeronet charge.

• The currents generated by τ± should have a vector minus axial vector (V −A) structure,while the electromagnetic current has to be a pure vector current.

The last argument seems to be impossible to overcome using a single gauge group. Glashowwas the first to realise that Nature may not always choose the most economical solution:He suggested that the gauge group of weak and electromagnetic interactions is the productof two groups, SU(2)L ⊗ U(1)Y , where the Y stands for hypercharge. Salam and Weinbergadded to this model the idea of SSB, using the Higgs effect to break SU(2)L⊗U(1)Y down toU(1)em. In this form, the Glashow-Salam-Weinberg (GSW) model of electroweak interactionssurvived all experimental tests until present.

239


11.1. Gauge sector

A SU(2) ⊗ U(1) gauge theory contains four gauge bosons, of which only one should remainmassless. If we use as scalar field to break the gauge symmetry a complex SU(2) doublet,then we add four real degrees of freedom. Three of them will become the longitudinal degreesof freedom for the three massive gauge bosons, so that just one physical Higgs field remainsin this most economical version of electroweak symmetry breaking.

We choose the complex scalar SU(2) doublet as

Φ =

(φ+

φ0

)

=1√2

(φ1 + iφ2φ3 + iφ4

)

. (11.1)

The corresponding scalar Lagrangian

L = (∂µΦ)†(∂µΦ) + µ2Φ†Φ− λ(Φ†Φ)2 (11.2)

is invariant under both SU(2) and U(1) transformations of Φ,

Φ→ exp

iα · τ2

Φ

Φ→ expiϑΦ .

We avoid an electrically charged vacuum by choosing the vacuum expectation value in the φ0

direction,

〈0|Φ|0〉 =(

0v√2

)

. (11.3)

Electroweak symmetry breaking should leave Uem(1) invariant. With this choice of the vev,we see that the combination 1+ τ3 keeps the ground state invariant,

δΦ = iε(1+ τ3)Φ = iε

(2 00 0

)(

0v√2

)

= 0 . (11.4)

Since the hypercharge generator is the identity in the weak isospin space, we should associatetherefore the combination Y + τ3 with the electric charge, Q ∝ Y + τ3.

Now we gauge the model, introducing covariant derivatives,

∂µΦ→ DµΦ =

(

∂µ +ig

2τ ·W µ +

ig′

2Bµ

)

Φ , (11.5)

where we suppressed a unit matrix in isospin space in front of ∂µ and Bµ. The field-strengths

Fµνa = ∂µW νa − ∂νW µ

a − gεabcW µb W

νc (11.6)

Gµν = ∂µBν − ∂νBµ , (11.7)

correspond to the three SU(2)L gauge fields W and to the UY (1) field Bµ. The couplingsof the two groups are g and g′, respectively, and the structure constants of SU(2) are thecompletely antisymmetric tensor εabc. The Lagrangian describing the Higgs-gauge sector isthen

L = −1

4F 2 − 1

4G2 + (DµΦ)

†(DµΦ)− V (Φ) . (11.8)

240

11.1. Gauge sector

We now spontaneously break this theory using the unitary gauge, since we are mainlyinterested in the mass spectrum of physical particles. We separate φ0 into the vev v andfluctuations h(x) as

Φ =

(

01√2(v + h(x))

)

=v + h√

2

(01

)

≡ v + h√2χ . (11.9)

Inserting the part containing v into Eq. (11.8) gives as mass terms for the gauge bosons

Lm =v2

2χ†(g

2τ ·W µ +

g′

2Bµ

)(g

2τ ·W µ +

g′

2Bµ

)

χ . (11.10)

Multiplying out the brackets and using (τ ·W )2 =W 2 and χ†τ ·W µχ = −W µ3 , we find

Lm =v2

2

[g2

4(W 2

1 +W 22 +W 2

3 ) +g′2

4B2 − gg′

2W µ

3 Bµ

]

. (11.11)

Apparently the neutral fields W µ3 and Bµ are connected, and therefore we rewrite the mass

term as

Lm =g2v2

8(W 2

1 +W 22 ) +

v2

8(gW µ

3 − g′Bµ)2 . (11.12)

The two fields in the first bracket are the gauge bosons we know from the charged currentinteractions in the Fermi theory,

W± =1√2(W1 ∓ iW2) . (11.13)

The second bracket corresponds to a new neutral massive gauge boson called Z boson, whichis a mixture of the B and the W3 fields,

Zµ =1

√

g2 + g′2

(gW µ

3 − g′Bµ)= cos ϑWW

µ3 − sinϑWB

µ . (11.14)

The mixing of the B and the W 3 field is parametrised by ϑW called Weinberg angle: ForϑW = 0, the mixing disappears and hypercharge and electric charge become identical.

The combination of W3 and B orthogonal to Z does not show up in the mass Lagrangian,and should therefore correspond to the massless photon,

Aµ =1

√

g2 + g′2

(g′W µ

3 + gBµ)= sinϑWW

µ3 + cos ϑWB

µ . (11.15)

We can now write Lm in terms of physical fields,

Lm =1

2m2WW

+µ W

−µ +1

2m2ZZµZ

µ , (11.16)

reading off the gauge boson masses, mW = gv/2, mZ = (g2+ g′2)1/2 v/2 = mW/(cos ϑW ) andmA = 0, as function of the still unknown values of the coupling constants g, g′ and the vevv. Thus in the GSW theory, the mass ratio of the gauge bosons is fixed at tree level by theWeinberg angle, mW/mZ = cos ϑW .

241


Couplings The new coupling constants g and g′ should be related to the electromagneticcoupling e and the Weinberg angle ϑW . We can find the connection between these parameters,if we replace the original fieldsW µ

3 and Bµ with the physical fields Zµ and Aµ in the covariantderivative (11.5) by inverting Eqs. (11.14) and (11.15),

gW µ3 τ3 + g′Bµ1 = g(cos ϑWZ

µ + sinϑWAµ)1+ g′(− sinϑWZ

µ + cos ϑWAµ)τ3

= (g sinϑW τ3 + g′ cos ϑW1)Aµ + (g cos ϑW τ3 − g′ sinϑW1)Zµ . (11.17)

Using then tanϑW = g′/g, we obtain

gW µ

3 τ32

+ g′Bµ1

2=

1

2g sinϑW (τ3 + 1)Aµ +

g

2 cos ϑW

(τ3 − sin2 ϑW (τ3 + 1)

)Zµ . (11.18)

If we assign weak isospin 1/2 to a lepton doublet, then the electric charge of the chargedlepton is

e = g sinϑW = g′ cos ϑW , (11.19)

while the neutrino stays as desired neutral. Remarkably, the electric coupling is smaller thanthe weak one: Weak interactions are weak, because the mass of mW implies a short range,not because the coupling is small.

We can use the remaining piece of the covariant derivative to derive the connection betweenthe Fermi constant GF and the vev v (problem 11.1), finding

GF√2=

g2

8m2W

=1

2v2(11.20)

or v ≈ 246GeV. Thus a measurement of the gauge boson masses determines the gaugecouplings g and g′, which in turn fix ϑW .

11.2. Fermions

Fermion masses Our remaining task discussing the electroweak mass spectrum at the tree-level is to generate masses for the fermions. The fermion mass termm(ψRψL+ψLψR) becomesgauge invariant, if we can promote the parameter m into a SU(2) doublet. This is easiestacomplished introducing a Yukawa coupling yf = m/v between the lepton doublet L, singleteR and the scalar doublet Φ,

LY = −yf(

LΦeR + eRΦ†L)

. (11.21)

This term generates fermion masses as well as Yukawa interactions between fermions and theHiggs: Inserting the vacuum expectation value of Φ, the mass term is

Lm = −yfv√2

[(νe e

)

L

(01

)

eR + eR(0 1

)(νee

)

L

]

= −mf (eLeR + eReL) = −mf ee ,

i.e. it is a normal Dirac mass term.In this way we can generate masses for the down-like fermions with τ3 = −1/2 like the

electron. In order to generate masses for the other half, we must use the charge conjugatedHiggs doublet, iτ2φ

∗.

242

11.2. Fermions

To see why this works, we look at the action of an SU(2)L transformation on the quarkand anti-quark doublets. The quarks transform as

q → q′ = Uq = exp

iα · τ2

q with q =

(ud

)

, (11.22)

while the charge conjugated version of this relation is

q∗′

= U∗q∗ = exp

− iα · τ ∗

2

(ud

)

. (11.23)

Now the bar denotes antiparticles. The complex conjugated Pauli matrices are

τ∗1 = τ1, τ∗2 = −τ2, τ∗3 = τ3 . (11.24)

This means thatτ12 ,− τ2

2 ,τ32

satisfy the same Lie algebra as the original matrices, and is an

appropriate basis for the antiquarks. This representation is called 2∗, while the original oneis called the 2 representation. These two representations must be unitary equivalent, that isfor some unitary matrix V we have

V exp

− iα · τ ∗

2

V −1 = exp

iα · τ2

. (11.25)

For infinitesimal |α|, this equation becomes

V (−τ ∗)V −1 = τ (11.26)

implying V = iτ2. We then have

V

(ud

)

=

(d−u

)

. (11.27)

This doublet transforms as the 2 representation. This means that we can use the charge con-jugated field iτ2φ

∗ to generate Dirac masses for up-like quarks and, if desired, for neutrinos.As the Yukawa couplings yf are not predicted, the fermion masses are arbitrary in the GSW

theory. Theoretical prejudice suggests as “natural” values of the couplings around O(0.1), asin the case of the gauge couplings g and g′. Even excluding neutrinos, there is instead a largehierarchy in the values of the Yukawa couplings, ranging from ye =

√2me/v ∼ 10−5 for the

electron to yt =√2mt/v ∼ 1 for the top quark.

Flavour mixing Accounting for the second and third generation of leptons and quarks, thecoupling yf introduced in Eq. (11.21), and hence also the mass mf , become arbitrary 3 × 3matrices, y and m. In particular, there is no reason for y to be either diagonal or hermitian.It can still be diagonalized by a biunitary transformation, i.e. by

S†mT =mD , (11.28)

where S and T are two different unitary matrices and mD is diagonal and positive. Thismeans that the weak eigenstates ψ = ψe, ψµ, ψτ , , which are diagonal in the interactionbasis but have indefinite masses, can be tranformed into mass eigenstates ψ′ as

ψLmψR = ψLSS†mTT †ψR = ψ

′Lmdψ

′R . (11.29)

243


These new eigenstates will of course no longer be diagonal in the interaction basis, which willlead to flavour mixing.

If we look at the weak current Jµ, and insert the mass eigenstates,

Jµ = νLγµeL = ν ′

LγµS†

νSee′L , (11.30)

we see that only the product S†νSe ≡ U will be observable. This means that we have some

freedom of choice, and it is convention to choose mixing only for the neutrinos, Se = 1 andU = Sν . In the same way, we shift the mixing in the quark sector completely to the down-likequarks. One denotes the mass eigenstates as (d, s, b) and the weak eigenstates as (d′, s′, b′),respectively, while (u, c, t) are unmixed by definition. The two matrices U which arise due tothis phenomenon are known as the MNS-matrix in the neutrino case and the CKM-matrix inthe quark case.

Note that the mixing matrices cancel in the case of the neutral current: Because the neutralcurrents couples e.g. neutrinos to neutrinos, products like S†

νSν appear which are equal tounity due to unitarity.

Conservation laws

11.3. Higgs sector

All particles of the SM except the Higgs particle have been firmly established. The searchfor the Higgs has been the major goal of the Large Hadron Collider at CERN which reportedin 2012 the discovery of scalar resonance with mass ∼ 125GeV. At the time of writing, theproduction and decay modes of this new scalar are compatible with that of a SM Higgsparticle.

Since the Higgs mass m2H = 2λv2 depends on the unknown self-coupling λ, it is a priori a

free parameter. In this section, we will sketch how one can find theoretical bounds for λ andthus the Higgs mass. Vice versa, assuming that the scalar resonance is indeed the SM Higgs,we can use its mass 125GeV to investigate until which energy scale the SM is valid.

Lower bound for mH Calculating the beta function of the Higgs self-coupling, we have to

add to the λ(φ†φ

)2term we examined already the effects from all other massive SM particles.

In practise, it is often sufficient to include only the massive gauge bosons, W± and Z, andthe top quark as the only fermion with a Yukawa coupling of order one. Then one finds forthe beta function (note the different normalization of V (Φ))

dλ

dt=

1

16π2

12λ2 + 12λy2t − 12y4t −3

2λ(3g2 + g′2

)+

3

16

[2g4 + (g2 + g′2)2

]

(11.31)

with t = ln(µ2/µ20

). Note the signs of the different terms: the gauge boson loops and the

scalar self-energy contribute positively, while the fermion loop has a negative sign. Thecoupling λ must be positive for the vacuum state to be stable. We are looking first for a lowerbound on mh =

√2λv. For small values of λ, we neglect all terms of O(λ),

dλ

dt≈ 1

16π2

[

− 12y4t +3

16

(2g4 + (g2 + g′2)2

)]

. (11.32)

244

11.3. Higgs sector

In general, we should solve the matrix of RGEs for λ, yt, g and g′ simultanously, which isonly numerically possible. For simplicity, we neglect therefore the scale dependence of theRHS and obtain

λ(Λ)− λ(v) = 1

16π2

[

− 12y4t +3

16

(2g4 + (g2 + g′2)2

)]

ln

(Λ2

v2

)

. (11.33)

Imposing λ(Λ) > 0, we get a lower bound for m2H ,

m2H >

v2

8π2

[

− 12y4t +3

16

(2g4 + (g2 + g′2)2

)]

ln

(Λ2

v2

)

. (11.34)

Upper bound formH An upper bound for mH corresponds to the case of large λ. Thereforewe need to keep now only the self-coupling of the Higgs in the beta-function,

β(λ) =dλ

dt=

3λ2

4π2, (11.35)

leading to a Landau pole at

µL = µ0 exp

4π2

3λ0

. (11.36)

We need λ to be small enough to do perturbation theory. Including a cutoff, Λ, µL > Λ, andasking for λ <∞, we obtain the inequality

3λ(µ0)

4π2ln

(Λ2

µ20

)

< 1 . (11.37)

Using m2H = 2λ(v)v2 and v = µ0, we find as upper bound for mH as function of the cutoff

scale,

m2H <

8π2v2

3 ln (Λ2/v2). (11.38)

Combining the two bounds, it is clear that the possible range of Higgs masses shrinks forincreasing Λ, and disappears for a large but finite value of the cutoff. This is a sign that theelectroweak theory is a trivial theory: We can keep an interacting theory only for a finitevalue of the cutofff, while in the limit Λ→∞ only the trivial solution λ = 0 is possible.

Now we turn this argument around: If the scalar particle with mass 125GeV measured atthe LHC is indeed the SM Higgs, at which scale Λ new physics has latest to appear, changingthe SM RGE flow? In figure 11.1, the bounds obtained using the full RGE of the SM areshown as green band (lower limit) and black lines (upper limit for λ < π and λ < 2π): Thegreen band suggests Λ < 108 GeV, implying that the SM cannot be a complete theory validup to the Planck scale. We will come back to this plot, discssuing the other, less restrictiveversions of the lower bound later.

Perturbative unitarity In the SM, all particle masses are given by the product of the vev vand some coupling constant. Particle masses much larger than the electroweak scale requiretherefore large coupling constants, leading to the breakdown of perturbation theory. Moreprecisely, a properly normalised amplitude A has to be bounded, |A| ≤ 1, reflecting theunitarity of the S-matrix.

245


GeV) / Λ(10

log4 6 8 10 12 14 16 18

[GeV

]H

M

100

150

200

250

300

350

LEP exclusionat >95% CL

Tevatron exclusion at >95% CL

Perturbativity bound Stability bound Finite-T metastability bound Zero-T metastability bound

error bands, w/o theoretical errorsσShown are 1

π = 2λπ = λ

GeV) / Λ(10

log4 6 8 10 12 14 16 18

[GeV

]H

M

100

150

200

250

300

350

Figure 11.1.: The scale Λ at which the two-loop RGEs drive the quartic SM Higgs couplingnon-perturbative (upper curves), and the scale Λ at which the RGEs create aninstability in the electroweak vacuum (lower curves). The absolute vacuum sta-bility bound is displayed by the light shaded [green] band, while the less restric-tive finite-temperature and zero-temperature metastability bounds are medium[blue] and dark shaded [red], respectively. Figure taken from [?].

To see how too heavy particles within the SM violate perturbative unitarity we considerthe case of 2→ 2 elastic scattering. The differential cross section is

dσ

dΩ=

1

64π2s|M|2 . (11.39)

Using a partial wave decomposition we can express the Feynman amplitude for the scatteringof spinless particles as

M(s, ϑ) = 16π

∞∑

l=0

(2l + 1)Pl(cos ϑ)Tl , (11.40)

where Tl is the l.th partial wave and Pl are the Legendre polynomials. Inserting the paertialwave expansion into (11.39) and integrating gives for the total cross section

σ =8π

s

∞∑

l=0

∞∑

l′=0

(2l + 1)(2l′ + 1)TlT∗l

∫ 1

−1d(cos ϑ)Pl(cos ϑ)Pl′(cos ϑ)

︸︷︷︸

= 2(2l+1)

δll′

=16π

s

∞∑

l=0

(2l + 1)|Tl|2 , (11.41)

where we have used that the Pl’s are orthogonal polynomials. The optical theorem gives withPl(1) = 1 on the other hand

σ =ℑ [M(ϑ = 0)]

2pcms√s

=8π

pcms√s

∞∑

l=0

(2l + 1)ℑ(Tl) . (11.42)

246

11.3. Higgs sector

Comparing the two expressions for the total cross section σ in the ultrarelativistic limit, usings ∼ p2cms/4, immediately yields the unitarity requirement

|Tl|2 = ℑ(Tl) , (11.43)

which in turn implies |Tl| ≤ 1. Thus the partial amplitude Tl lies within a circle of radiusone centered at (0, i/2) in the complex plane. For the real part of the amplitude, relevant forelastic scattering, the stronger bound

|ℜ(Tl)| <1

2(11.44)

applies.The simplest possible application is the elastic scattering1 of Higgs bosons: In lowest order

perturbation theory, the Feynman amplitude of this process is simply iA(hh→ hh) = 6iλ. Asthe amplitude is independent of the scattering angle, it corresponds to pure s-wave scattering,l = 0. Thus

T0 =A16π

=3λ

8π<

1

2(11.45)

or mh =√2λv <∼ 800GeV. For larger values of mh, the non-sensical result Tl > 1/2 indicates

that perturbation theory fails. Similiar findings are obtained, if one calculates the scatteringof longitudinal gauge bosons as function of mh. These considerations lead to the belief thatone would either find the Higgs boson at a multi-TeV collider as the LHC or discover a newstrongly interacting sector coupled to the gauge bosons.

Comparison with electroweak corrections The fermion masses are widely spread in the SM,

mν , . . . ,mb ≪ mW ,mZ ≪ mt

Therefore it is often useful to integrate out heavy degrees of freedom. The decoupling theoremderived by Appelquist and Corrazone states that heavy degrees of freedom decouple at lowenergies,

L (Φ, g, φ, g′)intgr. out Φ−−−−−−−→ Leff(φ,Zg

′) +O(E

mΦ

)

,

if L and Leff are renormalizable. As an example for this behavior, we have seen that theeffect of heavy particles in the anomalous magnetic moment of leptons is suppressed.

In the case of the electroweak sector, decoupling does not happen in several phenomenlog-ically important cases: For example sending mt → ∞, but keeping mb finite breaks SU(2)invariance. This makes Leff non-renormalizable, and therefore decoupling does not apply.Similarly, mh →∞ results in an higgless effective theory which is not renormalisable.

Example: Higgs production via gluon fusion.The Higgs couples to the massless gluons only at the one-loop level via a quark loop. Apart frombeing the most important production process of Higgs bosons in a proton-proton collider like LHC,this process illustrates also the non-decoupling of heavy fermions.We calculate the Feynman amplitude for the processs using dimensional regularization in n = 4 − 2ǫdimensions: Although we know that we will obtain a finite result, individual terms in intermediate

1In this case, a symmetry factor 1/S = 1/2! has to be included both in (11.39) and (11.41) which thereforedrops out from the unitarity bound (11.44).

247


steps are divergent (cf. Eq. (11.51)).For a fermion of mass mf in the loop, the Feynman rules give

iA = −(−igs)2tr(TATB)(−imf

v

)∫dnk

(2π)n(i)3Nµν

Dεµ(p)εν(q) (11.46)

The denominator is D = (k2 −m2f )[(k + p)2 −m2

f ][(k − q)2 −m2f ]. We use first the usual Feynman

parameterization to combine the denominators,

1

ABC= 2

∫ 1

0

dx

∫ 1−x

0

dy

[Ax+By + C(1 − x− y)]3 . (11.47)

We find1

D→ 2

∫

dxdy1

[k′ 2 −m2f + xym2

h]3. (11.48)

We evaluate the numerator assuming transverse gluons, i.e. dropping terms proportional to pµ or qν ,obtaining

Nµν = 4m

[

ηµν(

m2f − k2 −

m2h

2

)

+ 4kµkν + pνqµ]

. (11.49)

Now we shift momenta in the integration, drop terms linear in k′ from the numerator and use therelation ∫

dnk′k′µk′ν

(k′2 − C)m =1

nηµν

∫

dnk′k′2

(k′2 − C)m . (11.50)

The integrals can be done using the standard formulas of dimensional regularization,

∫dnk′

(2π)nk′2

(k′2 − C)3 =i

32π2(4π)ε

Γ(1 + ε)

ε(2− ε)C−ε (11.51)

∫dnk′

(2π)n1

(k′2 − C)3 = − i

32π2(4π)εΓ(1 + ε)C−1−ε. (11.52)

The amplitude is

A(gg → h) = −αsm

2f

2πvδAB

(

ηµνm2

h

2− pνqµ

)∫

dxdy

(

1− 4xy

m2f − xym2

h

)

εµ(p)εν(q). (11.53)

Evaluating the two integrals over the Feynman parameters in the limit m≫ mh, we find

A(gg → h) = − αs

6πvδAB

(

ηµνm2

h

2− pνqµ

)

εµ(p)εν(q). (11.54)

Thus gluon fusion gg → h is independent of the mass of the heavy fermion in the loop in the limit

m≫ mh. Thus we can use the decay width Γ(h→ gg) as a counter for all fermions with an arbitrary

high mass m—as long as their mass is generated by the SM Higgs effect. In this case, the suppression

of heavy particles in loops is compensated by their increasing Yukawa couplings yf ∝ mf to the Higgs

h.

Electroweak radiative corrections In our theory so far we have introduced all in all 24 (26)parameters (problem 11.2). In the Higgs and gauge sector, we can predict with only fourparameters (v, λ, g, g′) the masses of four particles as well as the electroweak interactions ofall fermions. Much less satisfactory is the description of the fermion sector, which requires asexperimental input eight (or ten, if neutrinos are Majorana particles) elements of the mixing

248

11.3. Higgs sector

matrices. Additionally, the twelve fermion masses have to be known in order to determinethe fermion-Higgs interactions.

It is convenient to choose αem, GF , mZ , which are experimentally most precisely known,to be the parameters for the gauge sector instead of g, g′, v. This set of parameters maybe defined on-shell. As a typical example for the structure of the electroweak radiativecorrections, we write down the expression for mW with this set of parameters,

m2W =

παemGF√2 sin2(ϑW )(1 −∆r)

. (11.55)

The quantity ∆r is the one-loop correction to the W mass, and its dominant contributionsare

∆r = ∆r0 −1− sin2(ϑW )

sin2(ϑW )∆ρ+∆rrem (11.56)

with

∆r0 = 1− αem(0)

αem(mZ)(11.57)

∆ρ =3GF (m

2t −m2

b)

8π2√2

(11.58)

∆rrem =

√2GFm

2W

16π211

3

[

ln

(m2H

m2h

)

− 5

6

]

. (11.59)

The first term ∆r0 accounts simply for the running of αem from q2 = m2e ∼ 0 to the electroweak

scale q2 = m2Z . The second contribution, ∆ρ, depends quadratically on the mass difference

between members of the same fermion doublet. We recognise such a behavior again as anexample for non-decoupling due to the breaking of SU(2) in the limit mt/mb → ∞. Finally,the reminder is dominated by the effects of the Higgs, but is only logaritmically dependenton mh. Thus even a precise determination of the electroweak parameters leaves a relativelylarge range of Higgs masses open.

Goldstone boson equivalence theorem One might expect that in processes where the energyis much higher than the electroweak scale, s ≫ m2

W , the unbroken theory becomes a validdescription. In particular, we should be able to replace a massive gauge boson by a masslesstransverse boson plus a massive Goldstone boson. More precisely, the equivalence theoremstates that amplitudes involving longitudinal gauge bosons are equal to those of the unphysicalGoldstone bosons up to corrections of O(mW /E),

A(W+L ,W

−L , ZL) = A(φ+, φ−, φ3) +O(mW /E) . (11.60)

An example for this correspondence which allows to simplify calculations in the high-energylimit is discussed in problem 11.5. The Higgs-Goldstone Lagrangian is

L =1

2(∂µh)

2 +1

2∂µw

+∂µw− +1

2(∂µz)

2 − λvh2 (11.61)

− λvh(2w+w− + z2 + h2)− 1

4λ(2w+w− + z2 + h2)2 . (11.62)

249


Summary of chapter

The electroweak sector of the SM is described by a SUL(2) ⊗ UY (1) gauge symmetry whichis broken spontanously to Uem(1). Three out of the four degrees of the complex scalar SU(2)doublet are pseudo Goldstone bosons (w±, z) which become the longitudinal degrees of free-dom of the three massive gauge bosons, W± and Z. The remaining scalar degree of freedombecomes a physical Higgs boson.

The SM does not explains why SSB happens. Also the Yukawa sector and the replicationof three families remain mysteries within this theory.

Further reading

An excellent source especially for phenomenological aspects of the SM is [DGH94]. The Feyn-man rules for the electroweak model in Rξ gauge are derived e.g. in Romao and Silva (2012);this reference provides also conversion rules between various sign conventions commonly useddiscussing gauge theories.

Problems

11.1 Fermi constant GF .Find the connection between the charged currentin the Fermi and the GSW theory, and determinethereby the numercial value of v.

11.2 Mixing matrices.Derive the number of free parameters in the quarkand neutrino mixing matrices as the sum of realmixing angles ϑi and phases eiφi for n genera-tion of Dirac and Majorana fermions, respectively.Take into account that not all phases of a fermionfield need to be observable.

11.3 Feynman rules for Higgs-gauge sector.Derive the Feynman rules for the vertices betweenthe scalar and the vector sector.

11.4 Higgs decay h→ γγ ♥.Calculate the matrix element for the processh→ γγ following the example h→ gg.

11.5 Higgs decay h→W+W−.Calculate the matrix element and the decay widthfor Higgs decay into transverse and longitudi-nal W-bosons, h → W+

T W−T and h → W+

L W−L .

Consider then the decay into Goldstone bosons,h → w+w−, and show that M(h → W+

L W−L ) =

M(h→ w+w−) +O(mW /mh).

11.6 Scattering of longitudinal gaugebosons.Consider the scattering of longitudinal gauge

bosons, W+L W

−L → W+

L W−L , which can be found

toO(M2W /s) from the Goldstone boson scattering.

Find the s-wave amplitude in the limit mh ≪ sand mh ≫ s. Apply the unitarity condition(11.44) and derive the limits where perturbativeunitarity breaks down and the Higgs-gauge sectorof the SM becomes a strongly interacting theory.

11.7 Goldstone boson fermion couplings.Derive the Feynman rules for the vertices

of fermions with charged or neutral Goldstonebosons G in Rξ gauge. Use

φ(x) =

(G+(x)

1√2(v + h(x) + iG0(x))

)

(11.63)

to read off the vertices of Goldstone bosons withfermions (assuming no mixing/one generation).

250

12. Phase transitions and topological defects

We introduced at the end of chapter 9 the idea that spontaneously broken symmetries offield theories could be restored at high temperatures. In this case the hot, early Universewould be in a symmetric phase, followed by transitions to phases with more and more brokensymmetries. Phase transitions in the early Universe can lead to observable consequences formainly two reasons: First, the state of the Universe deviates from thermodynamical equilib-rium during a first-order phase transition. This is in an important pre-requisite that processeslike baryogenesis which require out-of-equilibrium conditions can work successfully. Second,phase transitions lead generically to the formation of topological defects. These defects arezero-, one- or two-dimensional extended solutions of the classical equation of motions whichcontain in their core the unbroken vacuum 〈φ〉 = 0. Depending on the symmetry breakingscale 〈φ〉 = 0 and their dimensionality, they lead to (un-) desirable cosmological consequencesand can thus be used to constrain particle physics models beyond the SM.

12.1. Phase transitions in the abelian Higgs model

Effective potential at T > 0 We can include quantum and thermal fluctuations by studyingthe temperature-dependent effective potential at the one-loop level. We obtain the latterreplacing vacuum expectation values with thermal averages in the definition of classical fields,the effective action and effective potential at non-zero temperatures. In particular, we cantransform the T = 0 effective potential (10.38) defined in Euclidean space,

Veff(φ) = V (φ) +~

2

∫d4k

(2π)4ln[k2 + V ′′(φ)

]+O(~2) , (12.1)

into the temperature-dependent effective potential replacing the integration over the contin-uous energy k0 by a summation over discrete Matsubara frequencies ωn,

βVeff (φ) = βV (φ) +1

2

∑

n

∫d3k

(2π)3ln[ω2n + k

2 + V ′′(φ)]+O(~2) . (12.2)

The sum over n is performed in the same way as in the calculation of the free energy F of afree scalar gas in section 9.2.1,

βV(1)eff (φ) =

1

2

∫d3k

(2π)3

[

βEk + 2 ln(1− e−βEk)]

, (12.3)

where we replaced also V ′′(φ) = m2. Next we split the one-loop contribution βV(1)eff (φ) into

the T = 0 vacuum part V(1)eff (φ, 0) and a temperature dependent thermal part V

(1)eff (φ, T ),

V(1)eff (φ) =

1

2

∫d4k

(2π)4ln[k2 +m2

]+ T

∫d3k

(2π)3ln(

1− e−βEk

)

, (12.4)

251


where we dropped a φ-independent constant.In order to simplify the discussion we consider separately the two limiting cases that quan-

tum or thermal fluctuations dominate. In the latter case, V(1)eff (φ, 0)≪ V

(1)eff (φ, T ), we evaluate

V(1)eff (φ, T ) in the high-temperature limit (compare to the derivation of (9.23)),

V(1)eff (φ, T ) = T

∫d3k

(2π)3ln(1−e−βEk) =

T 4

2π2

∫ ∞

0dxx2 ln(1−e−

√x2+a2) = −π

2T 4

90+a2T 4

12+. . . .

(12.5)with a = βV ′′(φ). For our standard choice V ′′(φ) = µ2+ 1

2λφ2, the effective potential becomes

V (φ) + V(1)eff (φ, T ) = −π

2T 4

90− 1

2µ2φ2 +

1

24λφ2T 2 +

λ

12φ4 . (12.6)

The terms quadratic in the field φ agree with the thermal or Debye mass m2D = λ

2T 2

12 wefound in (9.42). The minimum of Veff(φ) moves smoothly away from φ = 0 below the criticaltemperature, no barrier is formed which implies a second order phase transition.

An example for a first order phase transition is given by the abelian Higgs model. In thelimit e4 ≫ λ, the T = 0 part of the radiative corrections due to the vector boson is important.Neglecting the T > 0 correction, the effective potential is

Veff(φ) = −µ2φ2

2(1− ξ) + λφ4

4!

(

1− 3

2ξ

)

+ξλ

4φ4[

lnφ2

µ2

]

, (12.7)

where ξ = 3e4

16π2λ parametrises the relative size of the loop corrections due to scalar and gaugebosons. For ξ ≥ 2, a barrier between the true and the false vacuum is formed and a first-orderphase transition results.

12.2. Decay of the false vacuum

When the Universe goes through a first order phase transition cooling down, the minimum atφ = 0 in the potential of a set of scalar fields may change from being the global to be a (false)local minimum. Since there is a barrier between the two minima, the field φ has to tunnelfrom the false to the true minimum. The true vacuum nucleates at a localized position inspace-time, when a quantum (or thermal) fluctuation tunnels through (or above) the barrier.It forms an extending bubble which contains the energetically favored ground-state. In thecase of thermal fluctuations, this phenomenon is known to everybody from boiling water.

The equivalent problem in quantum mechanics is the tunneling through the barrier in adouble-well potential V (x) depicted in the left panel of Fig. 12.1. The tunneling probabilityP can be calculated using the WKB method as

P ∼ exp

(

−i∫ b

adx√

2m[V (x)− E]

)

, (12.8)

where a and b are the beginning and end point of the tunneling trajectory. In order to translatethis prescription to a field theory, we should rewrite the tunneling probability P first as anEuclidean path integral. The exponent is the integral over the (imaginary) momentum of theparticle, which we can express as

i

∫

dx p = i

∫

dt px = i

∫

dt (E + L) =

∫

dtE (−LE) . (12.9)

252


In the last step we assumed that the energy of the particle is normalized to zero and changedto Euclidean time tE = it. Now we can rewrite the tunneling probability as an Euclideanpath integral,

P =

∫

Dx exp(−SE [x]) (12.10)

with

SE [x] =

∫

dx

[1

2

d2x

dt2E+ V (x)

]

. (12.11)

Since the potential V changes sign performing a Wick rotation, a classically forbidden pathin Minkowski space corresponds to an allowed Euclidean path shown in the right panel ofFig. 12.1. Thus finding the tunneling probability between two vacuua amounts to findingsolutions in the Euclidean theory which connect the two vacua and minimize the action.

Figure 12.1.: Tunneling through a double-hill potential in Minkowski space corresponds to theclassically allowed path from one maxima to another one in Euclidean space.

Bounce solution We now turn to the case we are interested in, namely the tunneling of ascalar field φ sitting in the false, metastable vacuum a into the true vacuum b, cf. Fig. 12.2. InEuclidean space, we should find solutions of the classical action starting from b which bounceat the classical turning point a back to b. These solutions are called “bounce” or instantons,since the tunneling happens instantanously in Minkowski space. Moreover, these solutionsshould have a finite action such that they give a non-zero contribution to the semi-classicallimit of the path integral. In principle, we should find all solution with a finite action andthen integrate their contribution. Since the tunneling probability is however exponentiallysuppressed, the integral is dominated by the solution which minimizes the action.

Solutions which are symmetric under rotations minimize the gradient energy. We usetherefore an O(4) symmetric ansatz, where the real scalar field φ is only a function of theradial coordinate r2 = x2 + t2E. Then the Euclidean action for a real scalar field becomes

S = 2π2∫ ∞

0dr

[

1

2

(dφ

dr

)2

+ V (φ)

]

(12.12)

and the field equation simplifies to (problem 12.2)

d2φ

dr2+

3

r

dφ

dr− dV

dφ= 0 . (12.13)

253


b

a

φ

V (φ) V (φ)

φb a

Figure 12.2.: Left: Transition from the false vacuum (a) to the true one (b) requires inMinkowski space tunneling through a barrier. Right: In Euclidean space, aclassically allowed path starts from b and bounces at a back to b; the zero pointof the potential is chosen to agree with a.

The boundary conditions φ → 0 for r → ∞ and dφ/dr|r=0 = 0 ensure a regular solution atr = 0 and a finite action. Viewing (12.13) as a mechanical problem, it describes the motionof a particle in a potential with the friction term 3

rdφdr . Thus the two parts a→ b and b→ a

of the bounce are not equivalent: We should choose the starting position φ(t = −∞) uphillof b such that at t = +∞ the field comes to rest at a.

We can proceed analytically, if we consider the limiting case that either the potentialdifference between the true and the false vacuum is much smaller or much larger than thepotential barrier which separates them. We will consider here the first case, which correspondsto the so called thin-wall approximation. This allows us to approximate the potential as

V (φ) = V0(φ) +O(ε) , (12.14)

where V0(φ) is the symmetric potential and the difference ε = V (φ+) − V (φ−) between thefalse and the true minima is a small parameter. For small ε, the volume energy ∝ εr4 of abubble dominates the surface energy ∝ r3 only, if the bubble is sufficiently large. In this case,the thickness of the bubble wall, i.e. the region where dφ/dr deviates significantly from zero,is also much smaller than the bubble size. Thus we can neglect the 3

rdφdr term in (12.13), and

the field equation simplifies to the one of an one-dimensional problem,

φ′′ ≡ d2φ

dr2=

dV0dφ

. (12.15)

Using the chain rule, dV0/dφ = V ′0/φ

′ and (φ′ 2)′ = 2φ′′φ′, we find after one x integration

1

2φ′ 2 − V0 = c . (12.16)

We determine the integration constant by asking that the contribution to the action S goesto zero for r →∞. With φ′(∞) = 0, this gives c = −V (φ+). Separating variables in (12.16)leads to

r =

∫dφ

√

2[V0(φ)− V (φ+)]. (12.17)

254


To gain more insight, we consider now a specific potential. We choose our favorite λφ4

potential,

V0(φ) =λ

4

(φ2 − η2

)2. (12.18)

Compared to chapter 10, we shifted the potential such that the two minima at φ = ±η =±√

µ2/λ have as required the value zero, V0(φ±) = 0. Inserting this potential into (12.17),we find

r = ∓√2

η√λarctanh (φ/η) + r0 . (12.19)

The integration constant r0 determines the position of the bubble wall. Inverting (12.19)gives as field profile

φ∓(r) = ∓η tanh(

η√λ√2

(r − r0))

. (12.20)

The solution φ∓ interpolate between the two vaccua ±η of the symmetric potential V0(φ).The thickness of the bubble wall is determined by the argument of the tanh and is given byδ ∼√2/(η√λ).

Knowing the solution φ(r), we can calculate the action. For r ≫ r0, the field is constantφ ≈ η and V (φ+) = 0. Therefore, the contribution S> from this region to the action is zero.For r ≪ r0, the field φ ≈ −η is again constant, but the potential contributes V (φ−) ≈ −ε or

1

2(φ′)2 +

dV

dr≈ −ε

and thus

S< = −1

2π2εr40 . (12.21)

Finally, the contribution from the bubble wall is with (12.16)

S = 2π2r30

∫

dr

[1

2

(φ′)2

+ V (φ)

]

=

∫

dx 2V (φ) = (12.22)

= 2π2r30

∫ φ2

φ1

dφ√

2V (φ) = 2π2r30I , (12.23)

with φ1/2 ≈ φ(r0 ∓ ξδ) and ξ = O(1). The quantity

I =

∫ φ2

φ1

dφ√

2V (φ) (12.24)

has the interpretation of the surface tension of the bubble. Adding the two contributions, thetotal action follows as

S = −1

2π2εr40 + 2π2r30I . (12.25)

We find the solution which gives the largest tunneling probability minimizing the action Sw.r.t. r0,

r0 =3I

ε. (12.26)

255


For our choice (12.17) for the potential, the surface tension is I = 2µ3/(3λ) and the action

S =27π2I4

2ε3=

16π2µ12

3λ4ε3. (12.27)

For ε→ 0, the action becomes infinite. Equivalently, the transformation from one to anotherground-state ±η of the symmetric potential V0(φ), costs an infinite amount of energy, as wehave argued in chapter 10.

The tunneling probability per time and volume follows then as

p = A

∫

Dφ exp(−S[φ]) , (12.28)

where the pre-factor A is determined by Det( − V ′′(φ))−1/2 and its zero-modes [CC77]. Inpractice, the result is rather insensitive to the exact value of A, and setting A ∼ V ′′(φ0) issufficient for simple estimates.

We can now justify the assumptions made: Our starting assumption that ε is small impliesthat r0 is large, r0 ∝ 1/ε. Moreover, a small ε implies that the bubble of true vacuumis separated by a thin wall from the metastable vacuum, δ ∼ (εr0)

−1/3. Therefore, thisapproximation is called the thin-wall approximation.

In order to describe the time evolution of a bubble, we have to continue analytically theO(4) symmetric solution r20 = x2 + t2E back to Minkoswki space. From r20 = x2 − t2, we seethat the bubble extends close to the speed of light for t≫ r0. The resulting O(3,1) symmetrymeans that all observers1 in inertial systems will measure the same expansion law.

Scalar instantons For the case of a massless λφ4 theory we can find a class of exact solutionsfor the bounce (12.13), which are often called scalar instantons. For a massless particle,the classical solution should fall off as a power-law. Then dimensional analysis tell us thatφ(r) ∼ 1/r for r → ∞. Since the massless theory is scale-invariant, the solutions should beparametrised by an arbitrary parameter ρ characterising their size. This suggests to insert asansatz φ(r) ∝ ρ/(r2 + ρ2) into (12.13) what results in (problem 12.3)

φ(r) =

(8

λ

)1/2 ρ

r2 + ρ2. (12.29)

Thus the action of the bounce becomes

S =27π2I4

2ε3=

8π2

3λ. (12.30)

Although the SM vacuum is metastable, the probability that inside the past-light cone of theobserved universe a tunneling happened is practically zero.

Example: Application to the metastable SM vacuum:We use V = λeffφ

4/4 as crude approximation for the scalar potential of the SM, where λeff is therunning coupling at the scale µ = φ. We can check using the thin-wall approximation that the exactresult (12.30) makes sense: The surface tension is I ∼ (|λeff |/2)1/2φ3/3 and ε ∼ |λeff |φ4/4. Thus theaction of the bounce becomes

S =27π2I4

2ε3=

8π2

3|λeff |1The facts that the bubble extends with v ∼ 1 and that the bubble is thin imply (un?)-fortunately thatobservers will be dissolved without having the time to notice.

256

12.3. Topological defects

agreeing with (12.30). An analysis of the RGE of the SM gives for the effective Higgs boson couplingat the Planck scale λeff(MPl) ∼ −0.01. For the estimate of the probability P that a bubble of thetrue vacuum has nucleated in the past-light cone of an observer, we can set V T ∼ t40 with t0 as thepresent age of the universe. Setting the pre-factor A by dimensional reasons equal to A ∼ φ4 ∼ M4

Pl,the tunneling probability is

P ∼ (t0MPl)4 exp−8π2/(3|λeff |) ∼ 10−11000 . (12.31)

Despite of the enormous volume factor, (t0MPl)4 ∼ (8×1060)4 , the tunneling probability is exceedingly

small.


A rather generic consequence of phase transitions is the formation of extended solutions tothe field equations which are stable by virtue of a topological quantum number. The latterarises if the space of possible field configurations is given a non-trivial topological structureby the condition that a functional F [φ] of the fields φ as the potential energy or the Euclideanaction is finite.

Two field configurations φ1 and φ2 are topologically equivalent, if they can be transformedcontinuously into each other keeping the functional F [φ] finite during the transformation.Thus topologically equivalent field configurations form equivalence classes which can char-acterized by a topological quantum number and are separated by an infinite energy barrier.Within each equivalence class, the minimal energy solution will be stable. The spatially uni-form ground-state we have assumed up to now as our vacuum in perturbative calculations isthus disconnected from other stable solutions which cannot spread out to become spatiallyuniform due to their non-trivial topology.

As a system crosses its critical temperature cooling down, it performs a transition from itssymmetric state to a state with broken symmetry. While the correlation length ξ becomesformally infinite at the phase transition, the order parameter in the broken phase can take thesame value only in a finite volume, restricted by the finite propagation speed of the relevantwaves. In the case of the expanding Universe, we expect the formation of order one topologicaldefect per causally connected region.

Domain walls are two-dimensional topological defects which separate three-dimensionalvolumes containing a different vacua. Examples are ferromagnetes where domains of uniformmagnetization exist. Depending on the gauge group and the pattern of symmetry breaking,topological defects with one and zero dimensions are also possible: In the first case, an one-dimensional line or string contains a vacua different from the surrounding, while it is thesecond case a point-like object called monopole.

We will proceed as in chapters 10 and 11, moving from a Higgs models with a single scalarfield and increasing then their number. At the same, the dimensionality of the topologicaldefects formed will decrease. As start, we will consider however the Sine-Gordon model whichis defined in 1+1 space-time dimensions.

Sine-Gordon solitons We have chosen in general as potential V (φ) a polynomial in the fieldφ. In the case of SSB, the periodicity of the angular variable in φ = ηeiϑ leads however toa periodicity of the potential. An example for a Lagrangian with a periodic potential is the

257


Sine-Gordon model defined by

L =1

2(∂µφ)

2 − V (φ) =1

2(∂µφ)

2 − a

b2[1− cos(bφ)] (12.32)

with µ = 0, 1 and two real parameters a and b. Expanding the potential V (φ) for small φ,

L =1

2(∂µφ)

2 − 1

2aφ2 − ab2

4!φ4 + . . . (12.33)

and comparing to our usual λφ4 potential, we see that they agree for small φ if we identify2

a = m2 and b2 = λ. The potential of the Sine-Gordon model is periodic, and has zeros forφ = 2πn/

√λ.

From the Lagrangian (12.32), the wave equation

∂2φ

∂t2− ∂2φ

∂x2+a

bsin(bφ) = 0 (12.34)

follows. It admits static and traveling wave solutions, φ(x, t) = f(±(x− vt)) ≡ f(±ξ). Youcan check that

f(ξ) =4

barctan[exp(±γ

√

a/bξ)] (12.35)

is a solution with γ = (1 − v2)−1/2 as usual Lorentz factor, problem 15.1. The solution withthe plus sign is called a kink, the one with minus an anti-kink. The kink interpolates betweenthe n = 0 ground-state at x→ −∞ and n = 1 (φ = 2π/

√λ) at x→∞. The extension ℓ of the

soliton is determined by the argument of the arctan and is given by ℓ ∼√b/(γ√a) = λ/(γm).

0

1

2

3

4

5

6

7

-6 -4 -2 0 2 4 6

φ(ξ)

ξ/l

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-6 -4 -2 0 2 4 6

φ(ξ)

ξ/l

Figure 12.3.: Left: Kink solution φ(ξ) for a = b = 1; Right: its derivative dφ(ξ)/dξ.

The energy (or mass) of a static solution interpolating between n = 0 and n = 1 is

E =

∫ ∞

−∞dx

[

1

2

(∂φ

∂x

)2

+ V (φ)

]

=

∫ ∞

−∞dx 2V (φ) =

∫ 2π/b

0dφ

∂x

∂φV (φ) = (12.36)

=

∫ 2π/b

0dφ√

2V (φ) =

√2a

b2

∫ 2π

0dϑ [1− cos(ϑ)]1/2 =

8m

λ. (12.37)

2Recall that a scalar field has dimension [φ] = (n− 2)/2 in n space-time dimensions. Thus the argument ofcos(bφ) is dimensionless while the φ4 interaction requires a m2 pre-factor.

258


Here we used in the second step the requirement that E < ∞, i.e. Eq. (12.16) with c = 0.More generally, a static solution connecting n1 and n2 has the energy density E/A = 8|n1 −n2|m/λ. Thus the energy spectrum of these solutions is discrete and inverse proportionallyto the coupling constant. The latter property implies that these solutions are not accessiblein a perturbative calculation which is a power series in the coupling constant λ. In theweak coupling regime, the mass 8m/λ of the soliton is much larger than the mass m of theelementary field φ, while in the strong couplig limit the soliton is the lightest excitation.

We can imagine the Sine-Gordon model as a chain of arrows, which the force F = −∂xVtries to orient e.g. downwards. Then a kink is a solution which points from x = −∞ . . . − ℓdownwards, turning at ξ ∼ 0 by 2π, and pointing at x >∼ ℓ again downwards. To untwist thechain, we would have to turn the arrows on one of the two side of ξ which would cost aninfinite amount of energy. Therefore the winding number ν ≡ n1−n2 is a conserved quantumnumber and the solutions are stable. We can understand why the winding number is calleda topological quantum numbers as follows: We can map R on a compact interval by identifythe points x = −∞ and x = ∞. Then the kink becomes a Mobius band, which cannot besmoothly transformed into on a circle S1.

A conserved quantum number implies the existence of a conserved current: The conditionE <∞ requires that the field approaches for x±∞ one of the vacuum states, and thus

φ(∞)− φ(−∞) =2π√λν . (12.38)

We can rewrite this is an integral,∫ ∞

−∞dx ∂xφ =

2π√λν . (12.39)

Since the solution is two-dimensional, we can set as current

jµ =

√λ

2πεµν∂νφ . (12.40)

The antisymmetry of εµν ensures then that ∂µjµ = 0. The conserved charge follows with

ε01 = 1 as

Q =

∫

dx j0 =

√λ

2π

∫

dx ∂xφ =

√λ

2π[φ(∞)− φ(−∞)] = n1 − n2 , (12.41)

i.e. equals the winding number ν. In contrast to the conserved charges we have encounteredup to now, the current is not the Noether current of a global symmetry and we did not haveto use the equation of motions in its derivation. Instead, the charge has a topological origin.

Solitons mantain their shape, although they are the solution of a non-linear equation. Thedistinctive property of the Sine-Gordon solitons in 1+1 space-time dimensions is that thisholds also for the asymptotic regions of a scattering event: In particular, two solitons emergefrom a collision unchanged except possibly for a phase shift, although the superpositionprinciple is not valid.

Domain walls We now move to topological defects in 3 + 1 dimensions considering two-dimensional topological defects called domain walls. Assuming that the domain wall is uni-form in the y − z plane, its action is

S[φ] =

∫ ∞

−∞dx

[1

2(φ′)2 + V (φ)

]

. (12.42)

259


Instead of integrating the equation of motions, as in Eqs. (12.15), to (12.17) we will use nowan argument due to Bogomolnyi: He suggested to rewrite the action “completing the square”as

S[φ] =1

2

∫ ∞

−∞dx[

φ′ ∓√

2V (φ)]2±∫ φ(∞)

φ(−∞)dφ

√

2V (φ) . (12.43)

The second integral depends only on the boundary values of the field at infinity. For fieldconfigurations which connect the same ground-state at x = −∞ and +∞, this boundaryterm vanishes, and the minimum S[φ] = 0 of the action is attained for constant fields. If thefield configurations connect however different ground-states, then we can associate the secondintegral to a non-zero topological charge. The winding number in the Sine-Gordon case is aspecific example for such a topological charge.

A lower bound for the action for fields with non-zero topological charge is

S[φ] ≥∫ η

−ηdφ

√

2V (φ) (12.44)

which is attained when the first integral vanishes. In this case, we arrive separating againvariables at

x = ±∫

dφ√

2V (φ). (12.45)

To proceed, we have to choose a definite potential. For the choice (12.18), we come back tothe solution

φ∓(x) = ∓η tanh(

η√λ√2

(x− x0))

(12.46)

we have used in the calculation of the bounce. The solution φ∓ interpolates between the twovaccua ±η of the symmetric potential V0(φ) and contains at x = x0 a two-dimensional planewith the unbroken vacuum φ = 0.

More generally, two-dimensional surfaces of finite size which separate domains with opposi-tite values of η are formed during a phase transition. They are called domain walls; theircosmological implications are discussed later on.

Global cosmic strings We continue our way through possible topological defects looking atthe consequences of a global continuous symmetry. Thus we replace the single real field by aset of two real or one complex field φ = φ1 + iφ2 with potential

V (φ) =λ

4

(

φ†φ− η2)2

. (12.47)

We use cylinder coordinates ρ, ξ, z, and search for static solutions using as ansatz

φ = ηeinϑf(ρ) . (12.48)

The phase einϑ can be chosen arbitrarily at ρ = 0, unless the field is zero at the origin. Inorder to ensure a single-valued field, we have therefore to impose the boundary conditionf(ρ) → 0 for ρ → 0. In the opposite limit, ρ → ∞, the field should approach one of itsminima,

φ→ ηeinϑ and f(ρ)→ 1 , (12.49)

260


in order to minimize the energy.For the ansatz (12.48), the field equations are separable and we find from ∆φ = λ(φ2−η2)φ

as differential equation for f

d2f

dξ2+

1

ξ

df

dξ− n2

ξ2f = ξ(f2 − 1)f . (12.50)

Here we have introduced also the new dimensionless variable ξ = λ1/2ηρ. This differentialequation together with the two boundary conditions for ρ → 0 and ρ → ∞ has to be solvednumerically.

We can however estimate the extension of a cosmic string by dimensional arguments. Thescale of the problem is set by ρ/ξ = λ−1/2η−1. Thus the core containing the unbrokenvacuum φ = 0 has the radius O(λ−1/2η−1), while for larger distances ρ≫ λ−1/2η−1 the fieldapproaches the broken phase |φ| = η.

The extended solution should have again a finite energy E per length L,

E

L=

∫ ∞

0dρρ

∫ 2π

0dϑ[

|∇φ|2 + V (φ)]

(12.51)

=

∫ ∞

0dρρ

∫ 2π

0dϑ

[

∂ρφ†∂ρφ+

1

ρ2∂φ†

∂ϑ

∂φ

∂ϑ+ V (φ)

]

. (12.52)

The first and the last term give a finite contribution, since ∂ρf(ρ) → 0 and V (φ) → 0 for

ρ→∞. In contrast, the middle term contributes ∝∫ R0 dρρ−1f(ρ), i.e. a logarithmic diverging

term ln(ρ/δ) to the linear energy density. The scale δ has to be determined by the typicalextension of the string, and thus the energy density inside the radius R around the string isE/L ∼ ln[R/(λ1/2η)]. If we consider instead of the idealization of an isolated global string therealistic case of a string network, then R should be given by the typical distance of strings.

Local cosmic strings We can avoid the (formal) problem of the infinite energy associatedwith a string if we gauge the model. In this case we obtain the abelian Higgs model,

L = −1

4FµνF

µν + (Dµφ)†(Dµφ) + µ2φ†φ− V (φ) . (12.53)

The kinetic term Dµφ = ∂µφ + ieAµφ contains now two contributions which can cancel forρ→∞. We require therefore that Aµ is a pure gauge field for ρ→∞ with

Aµ ∼ −i

e∂µ ln(φ/η) . (12.54)

Then Dµφ and Fµν approach zero for ρ → ∞, and the energy density per length of a localstring is finite.

A local string carries magnetic flux of the corresponding gauge field. Integrating (12.54)around a circle in a plane with z = const. and large ρ gives

Φ =

∫

B · dS =

∮

A · dl = − i

e

∫ 2π

0dϑ∂ϑ(inϑ) =

2πn

e. (12.55)

Thus the magnetic flux carried by local strings is quantised in units of 2π/e. If we go backto cgs units, then the unit of magnetic flux becomes hc/e, determined completely by the

261


defect n scalars dimension homotopy group

domain wall 1 2 π0(M0)cosmic string 2 1 π1(M0)monopole 3 0 π2(M0)texture 4 - π3(M0)

Table 12.1.: The number n of scalars determines the dimension of the vacuum manifoldM0

(= Sn−1 for the potential (12.56)) and the dimension 3 − n of the hypersurfacecontaining the unbroken vacuum.

three fundamental constants of a relativistic quantum theory. A local cosmic string withwinding number n carries n flux quanta, analogous to quantized tubes of magnetic flux in asuperconductor.

The coupled field equations for φ and Aµ can be solved in general only numerically andtherefore we only estimate the extension and the energy of a local string. At the core of thestring, the vacuum is unbroken, 〈φ〉 = 0, leading to an potential energy density V (0) = λη4/4.The extension of the string is controlled by the Compton wave-lengths

λA =1

mA=

1

evand λφ =

1

mφ=

1√2λv

of the two massive fields Aµ and φ. Thus the scalar field contributes to the energy of thestring

Eφ/L ∼ (λη4/4)× πλ2φ ∼ η2 .The energy density of the magnetic field contributes B2/(8π) × πλ2A. From (12.55), we findΦ = πλ2AB = 2π/e for a string with winding number n = 1 and thus

EA/L ∼ (eλA)−2 ∼ η2.

As a result, the magnetic and scalar contributions to the energy density are of the same order,E/L ∼ η, given by the vev η of the scalar field.

Global monopoles Let us try to condense our results: If we split a complex scalar fieldsintro real ones, φ = φ1 + iφ2, then we considered in d = 4 potentials of the type

V (φ) =λ

4

(n∑

i=1

φ2i − η2)2

. (12.56)

For n = 1, the possible ground-states φ2 = η2 correspond to the zero-dimensional sphereS0. The single constraint φ(x1, x2, x3) = 0 defines a two-dimensional surface in R

3, whichcorresponds to a domain wall. The abelian Higgs model with one complex scalar doublet hasn = 2. Now φ21 + φ22 = η2 defines an one-dimensional sphere S1 as manifold of the possibleground-states, while a topological defect defined by φ1 = φ2 = 0 is as section of two surfacesa line.

The logical next step is to consider three scalar fields which transform under SO(3). Thenthe vacuum manifold is the sphere S2 and we expect zero-dimensional topological defects

262


which are called monopoles. We sketch only the physical picture and discuss first globalmonopoles. We use spherical coordinates and search for static solutions using as ansatz

φi = ηh(r)xir, (12.57)

which satisfies the requirement

φiφi = η2 for |x| → ∞ , (12.58)

if h(r) → ±1 for r → ∞. Additionally, the function h(r) should satisfy the boundarycondition h(r) → 0 for r → 0 to ensure a non-singular φ(0). The energy density at large r,where h(r) ≈ η is given by

ρ ∼ 1

2(∂iφ)

2 ∼ 3

2

η

r2(12.59)

and thus the energy contained inside a sphere of radius R diverges linearly, E ≈ 3ηR/2. Incontrast to the mild logarithmic divergence in case of globel strings, the behavior E ≈ 3ηR/2is desastrous, leading to a fast collapse of the universe.

The natural way-out is to go on to local monopoles and to check if the gauge fields cancompensate the scalar gradient energy. Before we do so, we ask however if there is a wayto ensure that the total energy of global monopoles is finite. Imagine a monopole (h(r) = 1for r → ∞) and an anti-monopole (h(r) = −1 for r → ∞) pair separated by the distance d:Their fields φi(r) will cancel for r ≫ d, leaving over only higher multipole moments φi ∼ r−2.Thus the energy density of a monopole-antimonopole pair scales as ρ ∼ ηr−4 and thereforetheir total energy E is finite. Separating a monopole-antimonopole pair would cost an ininiteamount of energy—instead a new monopole-antimonopole pair will be created as soon as thepotential energy between the pair exceeds a certain threshold. We can view this behavior asa simple model for the confinement of colored particles as quarks and gluons in QCD, wherethe potential energy V (r) for distances r >∼ Λ−1

QCD scales also linearly.

’t Hooft-Polyakov monopoles The simplest model exhibiting local monopoles is the Georgii-Glashow model. It contains a SO(3) gauge field Aaµ supplemented by a triplet of real Higgsscalars φa. Choosing a uniform vev as φa = (0, 0, v) leaves a residual U(1) symmetry unbroken,corresponding to rotations around the 3-axis in isospin space. Thus one can view the Georgii-Glashow model as a toy model for the electroweak sector of the SM, where the gauge fieldA3µ plays the role of the photon and the Z boson is missing.

’t Hooft and Polyakov showed first that this model contains extended classical solutionswhich have finite energy and correspond to local magnetic monopoles. Using the adjointrepresentation (T aA)bc = −ifabc for the scalar triplet of real Higgs scalars φa, , the covariantderivative becomes

Diφa = ∂iφ

a − eεabcAbiφc (12.60)

or in vector notation Diφ = ∂iφ− eAi ×φ.We fix again the asumptotic behavior of the gauge fields by requiring that the kinetic term

Diφa vanishes for r →∞. From

∂iφa ∼ η δ

ai − xixar2

,

263


we conclude that Abi should be constant, while (12.60) implies that Ai is perpendicular to φ.Evaluating Diφ

a using εaijεakl = δikδjl − δilδjk shows that

Aai =v

e[1− f(r)]εaij x

j

r(12.61)

with f(r)→ 0 for r →∞ leads to the desired asymptotic behavior of the covariant derivative.Now the regularity of the solution Aai at r = 0 requires f(0) = 1.

Inserting the two Ansatze (??) and (12.61) into the energy functional of the Georgii-Glashow model amd minimizing the energy results in two coupled differential equations forthe functions H(r) and K(r). In general, they have to be solved numerically. Therefore weconsider only the limit λ/g2 = m2

h/(2m2W )→ 0. Then the potential energy V (φ) is negligible,

V (φ) =λ

4

(3∑

a=1

φ2i − η2)2

→ 0 for λ→ 0 , (12.62)

if the fields satisfy the constraint (12.58). Thus the boundary conditions (12.58) remain validin the limit considered. Neglecting V (φ), we can use Bogomolnyi’s trick to derive an exactsolution. We express the static energy as

E =1

2

∫

d3x[(Ba

i )2 + (Dµφ

a)2]=

1

2

∫

d3x[(Ba

i ∓Diφa)2 ± 2Ba

iDiφa]≥ 0 , (12.63)

where we assumed for simplicity that the monopole carries no electric charge, Aa0 = Eai = 0.Then we obtain the bound

E ≥∫

d3xBaiDiφ

a (12.64)

which is attained, if the fields satisfy Diφa = ±Ba

i . Now we have to solve only the simplerfirst-order equation Diφ

a = 12εijkF

ajk. Its solution is called a Bogomolnyi-Parad-Sommerfield

(BPS) monopole and, generalising, all solitons which minimise the classical action are denotedas BPS states or BPS solitons.

Finally, we want to determine the charge of a BPS monopole. While the massive fieldsfall off exponentially for r → ∞, the component of F aij connected to the massless photondecrease as a power-law. Moreover, the fields (??) and (12.61) are practically uniform atlarge distances r from the center of the monopole. Thus in a laboratory at x, the field xF ij

corresponds to the massless photon, while the two states orthogonal to F ij are the massiveweak gauge bosons. This implies that the part of F aij which corresponds to electromagnetismis parallel to φa,

Fij = F aijφa

v. (12.65)

The magnetic part of the SU(2) field-strength tensor is

Bi = Bai

φa

v=

1

2εijkF

ajk = εijk∂jA

ak −

1

2eεijkεabcA

bjA

ck (12.66)

where we inserted the definition of F ajk. If we are far away from the center of the soliton, wecan use the asymptotic expression for the fields, setting f(r) = 0 and h(r) = 1. Performingthe differentiations (problem 12.8) we arrive at

Bai =

1

g

ninar2

. (12.67)

264


Thus the U(1) magnetic field is given by

Bi =1

g

nir2. (12.68)

Comparing this to the expression qni/(4πr2) for the field of a monopole, we conclude that

the magnetic charge of the BPS monopole is qm = 4π/g.

Textures Finally we should comment on the case n = 4 which corresponds to the elec-troweak model. The four equation φi(x1, x2, x3) = 0 have in general no solution in R

3. Thusregions which contain the unbroken vacuum φi = 0 will be not formed during the electroweakphase transition. Still, correlations can not exist beyond the horizon scale and thus non-zerogradients ∂µφi can be produced. In case of global textures, static solutions with positiveenergy density can exist. In case of of a broken gauge symmetry, gauge fields will compen-sate the ∂µφi term in Dµφi. Thus local textures as predicted by the SM have no observableconsequences.

Homotopy groups and winding numbers The topological quantum numbers we meet dis-cussing extended classical solutions can be associated to winding numbers of maps betweenthe ground states of a field theory and its configuration space. Let us define the ground-stateor vacuum manifold M0 of a theory as the set of all global minima V (φ) = 0 of its potential,

M0 = φ : V (φ) = 0 . (12.69)

The condition that the potential energy is finite requires that

lim|x|→∞

φ(x) = φ ∈M0 . (12.70)

We can compactify Rn (where n is the number of spatial dimensions) to the sphere Sn using

e.g. a stereographic projection as shown in Fig. 12.4. Then we can view the condition thatthe potential energy is finite as a mapping Sn → M0. We ask now the question when twosuch mappings are topologically distinguished?

We consider only the mappings S1 → S1 ≈ R2\0 which are easiest to visualise. Two

closed loops with base point x0, i.e. curves x(t) with t ∈ [0 : 1] and x(0) = x(1) = x0, areshown in Fig. 12.5. While the black loop is contractable, the red one is wrapped once around0 and therefore not contractable to its base-point x0 by continuous transformations. Wedefine the maps

U (ν) = eiϑν =[

U (1)]ν, (12.71)

which count the number of times a loop is wrapped around the origin. The integer ν is calledthe winding number of the map and can be rewritten as an integral,

ν =i

2π

∫ 2π

0dϑU∂ϑU

† . (12.72)

Clearly, this formula reproduces the correct values for the mappings defined above. Moreover,it is useful for the proof that ν is invariant under continuous transformations. It is sufficientto investigate an infinitesimal change δU . Unitarity UU † = 1 implies that δU †U + δU †U = 0

265


N

P

P ′

Rn

Sn

Figure 12.4.: A stereographic projection maps points P ∈ Rn onto points P ′ ∈ Sn. The northpole N of the sphere Sn corresponds to the sphere Sn−1

R at spatial infinity,x2i + · · ·+ x2n = R2 →∞, of Rn.

ϑ

x0x0

Figure 12.5.: Two loops in R2\0 with common base point x0: the black one is contractable

and has winding number ν = 0, the red loop is wrapped once around 0 andhas winding number ν = −1.

and δU † = −U †δUU †. Since we will interested later in the non-abelian version of the windingnumber, we will not use that the U are commuting complex numbers.

We calculate the variation of the integrand in the integral formula (12.72) for ν,

δ(U∂ϑU†) = δU∂ϑU

† + U∂ϑδU† (12.73)

= δU∂ϑU† − U∂ϑU †δUU † − UU †∂ϑδUU

† − UU †δU∂ϑU† (12.74)

= −U[

∂ϑU†δU + U †∂ϑδU

]

U † = −U∂ϑ[

U †δU]

U † . (12.75)

Here, we inserted δU † = −U †δUU † and performed the differentiations. Then the first andfourth terms cancel, and finally we combined the remaining two terms using the product rule.In case of the abelian winding number ν of (12.72),

δν =i

2π

∫ 2π

0dϑ δ(U∂ϑU

†) = − i

2π

∫ 2π

0dϑ∂ϑ

[

U †δU]

= 0 . (12.76)

266


Thus the winding number ν is an integer which is invariant under infinitesimal deformationsof the loop. Since any continuous transformation can be built up out of infinitesimal ones,the maps S1 → S1 can be divided into different equivalence classes, each characterised byone value of ν ∈ Z . Only maps within each class can be continuously transformed into eachother. Mathematicians say that such maps are homotopic. The theory of homotopy groupsaddresses the question into how many homotopy classes πn(X) the set of maps Sn → X canbe divided.

Using two results from the theory of homotopy groups one can show that a grand uni-fied theory implies the existence of magnetic monopoles. First, the second homotopy groupπ2(G/H) of the quotient group G/H equals the first homotopy group π1(H). In our case,H = SU(3)⊗ U(1) and π1(U(1) = Z is non-trivial. Thus the sequence

π2(G/H) = π1(H) = π1(SU(3) ⊗ U(1)) = Z (12.77)

shows that, if Uem(1) is unified at higher scales within a larger semi-simple group, thenmagnetic monopoles should be produced at the GUT phase transition.

Summary of chapter

Classical static solutions of theories with SSB can fall into different equivalence classes whichare separated by an infinite potential energy barrier. Depending on the number of Higgsfields, this leads to two-, one or zero-dimensional solitons containing the unbroken vacuum.Instantons or bounces are solutions of the classical field equation in Euclidean space whichevolve in Euclidean time between two different vacua. If their action is finite, they describein Minkowski space instantanous tunneling from the false to the true vacuum.

Further reading/sources

Topologically non-trivial solutions of the classical field equations are discussed in more detailin [Rub02]. For a calculation of the effective potential of the SM and the resulting tunnelingprobability see [Esp13] and the references therein.

Problems

12.1 Order of the phase transition.

Determine the critical temperature Tc of the λφ4

theory from (12.6). Calculate the pressure andthe heat capacity above and below Tc and con-firm thereby that the phase transition is of secondorder.

12.2 Field equation for the bounce solution.

Show that (12.13) agrees with the Klein-Gordonequation ∆φ = V ′(φ) for a radial symmetric φ.

(Use (1.129) to evaluate ∆φ = ∇a∇aφ.)

12.3 Scalar instantons.

Show that the ansatz φ(r) = Aρ/(r2 + ρ2) solves(12.13). Determine A and the action S.

12.4 Soliton solution of the Sine-Gordonequation.

Show that (12.35) solves the Sine-Gordon equa-tion and discuss the behavior under Lorentz trans-

267


formations.

12.5 Domain wallEstimate the extension of domain wall from di-mensional analysis.

12.6 Derrick’s theorem.Consider the behavior of the energy functionalE[φ] = T [φ] + V [φ] of scalar field φ under scaletransformation x → λx in D space dimensions.Show that for D ≥ 2 no stable solutions with fi-nite energy exist.

12.7 Duality of Maxwell

Show that the source-free Maxwell equations areinvariant under the duality transformation

E′ = E cosα+B sinα

B′ = −E sinα+B cosα .

Show that the invariance of the Maxwell equationswith sources requires the existence of magneticmonopoles.

12.8 Magnetic field of a monopole

Derive Eq. (12.67).

268

13. Anomalies, instantons and axions

We have seen that any global continuous symmetry of a system described by a Lagrangian L

leads to a locally conserved current on the classical level. In order to decide if these classicalsymmetries do survive quantisation, we have to study if the generating functional Z[J ] retainsthe symmetries of the classical Lagrangian L .

Various examples where a quantised system does not share the symmetries of its classicalcounter-part have been found. As this behaviour came as a surprise, it was called “anomalous”and the non-zero terms violating on the quantum level the classical conservation laws werecalled “anomalies”. A case we encountered already is the breaking of scale invariance (orconformal symmetry) in the process of renormalising massless QED and QCD. The onlyother example of anomalous theories in d = 4 space-time dimensions are models containingchiral fermions, i.e. theories where left and right-handed fermions interact differently. Thisanomaly is called axial or chiral anomaly and shows up in loop graphs containing fermionscoupled to gauge fields or gravitons.

The fact that a classical symmetries is broken by quantum effects is often described as“the anomaly breaks the classical symmetry”. One should keep in mind that on the quantumlevel there is no symmetry to start with. Thus there is no Goldstone boson associated to asymmetry broken by an anomaly.

13.1. Axial anomalies

Anomaly from non-invariance of the path integral measure Anomalies were first discov-ered performing diagrammatic calculations using canonical quantisation. At a first glance, itlooks like the path integral formalism predicts that all classical symmetries of the Lagrangianalso hold at the quantum level—and the “failure” to reproduce the anomalies found usingcanonical quantisation deemed the path integral approach untrustworthy. It was only re-alised by Fujikawa in 1979 that the integration measure in the path integral may transformnon-trivially under a symmetry transformation, leading to an anomaly.

The simplest model exhibiting an axial anomaly is axial electrodynamics, i.e. a fermioncoupled via its vector and axial current to two different gauge fields. The Lagrangian for thissystem reads

L = ψiγµ(∂µ + iqVµ + igAµγ5)ψ − 1

4F 2 − 1

4G2 (13.1)

with

Fµν = ∂µAν − ∂νAµ and Gµν = ∂µVν − ∂νVν . (13.2)

Performing a combined UV (1)⊗UA(1) gauge transformation induces the following change of

269


the fields,

ψ(x)→ ψ′(x) = exp iqΛ(x) + igα(x)γ5ψ(x) , (13.3a)

ψ(x)→ ψ′(x) = ψ(x) exp −iqΛ(x) + igα(x)γ5 , (13.3b)

Vµ → V ′µ = Vµ − ∂µΛ(x) , (13.3c)

Aµ → A′µ = Aµ − ∂µα(x) . (13.3d)

Since the fermionic field ψ is described by a Graßmann variable, the integration measure Dψtransforms like

Dψ →[Det exp (iqΛ(x) + igα(x)γ5)

]−1Dψ . (13.4)

Thus the product of the fermionic measure changes as

DψDψ → DψDψ[Det exp (2igα(x)γ5)

]. (13.5)

As result, the variation introduced by a vector UV (1) gauge transformation cancels in thefermionic measure DψDψ, while the corresponding change under the axial UA(1) gauge trans-formation adds up. We can rewrite this non-zero variation of the integration measure as aneffective shift δL in the Lagrangian using the identity Det(expB) = expTr(B),

δL = 2α(x)g∑

n

φ∗n(x)γ5φn(x) = 2α(x)g

∑

n

A(x) , (13.6)

where Tr(B) =∫d4x tr(iδL ). For the wave-functions φn(x), we can use any complete set of

solutions of the Dirac equation,

iD/φn = λnφn and∑

n

φx(x)φ†n(y) = δ(x− y)I , (13.7)

where D is the gauge-invariant derivative with respect to the two gauge fields. The sum overthe eigenfunctions will be UV divergent, and we have therefore to regularise it. A convenientregulator to add is the function

f = expλ2n/M

2= exp

−D/ 2/M2

, (13.8)

which approaches fast zero for λ2n → −∞. The limit f → 1 of the regulator corresponds toM →∞, which will allow us later an expansion of our result for

A(x) =∑

n

φ†n(x)γ5 exp

−D/ 2/M2

φn(x) (13.9)

in powers of 1/M2.As the vector UV (1) gauge transformation keeps the integration measure invariant, we

ignore it in the following calculations. Thus we need to calculate D/ 2 including only the axialgauge field A,

D/D/ = γµγνDµDν = D2 +

g

2σµνF

µν (13.10)

with

D2 = (∂µ + igAµ)(∂µ + igAµ) = ∂2 + 2igAµ∂µ + ig

(∂µA

µ)− g2AµAµ . (13.11)

270


Since the trace is invariant to the choice of our wave functions we choose plane-waves φn =e−ikx

I. Then we can replace the differentiations by momenta,

exp

(

−D2

M2

)

φn(x) = exp

(

−(kµ + gAµ)2

M2+

ig(∂µAµ)

M2

)

φn(x) . (13.12)

Taking the continuum limit of the sum and writing out the full regulator, we obtain

A(x) =

∫d4k

(2π)4tr[

eikxγ5 exp

−D/ 2/M2

e−ikx]

=

∫d4k

(2π)4tr

[

γ5 exp

−(kµ + gAµ)2

M2+

ig∂µAµ

M2+g

2

σµνFµν

M2

]

. (13.13)

The second term in · · · is zero because it does not depend on kµ and tr[γ5] = 0. Shiftingvariables to the dimensionless MKµ = kµ + gAµ we obtain

A(x) =M4

∫d4K

(2π)4e−K

2tr

[

γ5 exp

g

2

σµνFµν

M2

]

. (13.14)

If we expand the exponential in powers of 1/M2, only the terms up to O(M−4) will survivein the limit M →∞. Using the antisymmetry of Fµν , we can also replace σµν by iγµγν andfind then

exp. . .= 1 +

ig

2γµγνF

µν 1

M2+

1

2

(ig

2

)2

γµγνγαγβFµνFαβ

1

M4+O

(1

M6

)

. (13.15)

The trace properties of the gamma matrices inform us that the first two terms vanish, whilethe third results in a term proportional to the totally antisymmetric tensor ǫµναβ . Introducingthe dual tensor Fαβ = 1

2ǫαβµνFµν and then performing the Gaussian integral over K we are

left with

Tr(B) = 2ig2

16π2

∫

d4xα(x)FµνFµν . (13.16)

In a classical theory without gauge fields, a local axial gauge transformation leads to thechange

L → L + g(∂µα)ψγµγ5ψ . (13.17)

Thus the non-invariance of the measure in the path integral leads to violation of the classicallyconserved axial current,

∂µjµ5 = ∂µ(ψγ

µγ5ψ) = ∂µ(jµR − j

µL) =

g2

8π2FµνF

µν . (13.18)

This equation is also known as the Adler-Bell-Jackiw anomaly equation.The extra term introduced in the Lagrangian by an axial gauge transformation is pro-

portional to FµνFµν and thus odd under CP. This interaction term is gauge invariant, has

dimension four and corresponds therefore to a renormalisable interaction. The reason whywe have not considered it earlier is that it is a total derivative,

Fµν Fµν =

1

2ǫµνρσ(∂

µAν − ∂νAµ)(∂ρAσ − ∂σAρ)= 2ǫµνρσ∂

µAν∂ρAσ = ∂µ(2ǫµνρσAν∂ρAσ) = ∂µKµ . (13.19)

Since the four-divergence Kµ is not gauge invariant, it cannot be an observable. Therefore itis not excluded that Kµ is singular, leading after integration to non-zero effects in the action.Before we discuss in more detail when it is justified to neglect a total derivative term, we willrederive the anomaly using a diagrammatic approach.

271


Perturbative appearance of anomalies Historically, the chiral anomaly was first encoun-tered calculating the process π0 → γγ using perturbation theory. Describing the neutral pionwithin a non-relativistic quark picture as a π0 = (uu+ dd)/

√2 state, we can view the process

as

k + p2

kk − p1

p2

p1

γµγ5

γλ

γκ

+

k + p1

kk − p2

p1

p2

γµγ5

γκ

γλ

Here, the γ5 matrix accounts for the fact that the pion is a pseudoscalar particle.The two diagrams are connected by the crossing symmetry κ ↔ λ, p1 ↔ p2. The total

matrix element of this process is thus given by the sum

Aκλµ(p1, p2) = Sκλµ(p1, p2) + Sλκµ(p2, p1) , (13.20)

where the matrix element Sκλµ describing the first diagram (neglecting coupling constants)is given by

Sκλµ = −(−i)3∫

d4k

(2π)4tr

[

γκi

k/ − p/1γµγ5

i

k/ + p/2γλ

1

k/

]

. (13.21)

It is sufficient to consider only massless fermions: The anomaly is connected to the UVdivergences of these diagrams and in this limit masses play no role.

We check now, if the classical conservation law for the vector and the axial current hold alsoat the one-loop level. Current conservation implies the following three relations in momentumspace,

pκ1Aκλµ = pλ2Aκλµ = 0 (13.22a)

(p1 + p2)µAκλµ = 0 , (13.22b)

where (13.22a) are the equations for the conservation of the vector current, and Eq. (13.22b)for the conservation of the axial current. Crossing symmetry implies that these relations mustalso hold for the two individual amplitudes S.

We check first, if the axial current is conserved. In evaluating

Iκλ ≡ (p1 + p2)µSκλµ = −

∫d4k

(2π)4tr[γκ(k/ − p/1)(p/1 + p/2)γ

5(k/ + p/2)γλk/]

(k − p1)2(k + p2)2k2(13.23)

we use(p/1 + p/2)γ

5 = p/1γ5 − γ5p/2 = −(k/ − p/1)γ5 − γ5(k/ + p/2)

and (k/ ± p/i)2 = (k ± pi)2, so that we can divide the integral into two parts,

Iκλ =

∫d4k

(2π)4tr[γκγ

5(k/ + p/2)γλk/]

(k + p2)2k2︸︷︷︸

Aκλ(p2)

+

∫d4k

(2π)4tr[γκ(k/ − p/1)γ5γλk/

]

(k − p1)2k2︸︷︷︸

Bκλ(p1)

. (13.24)

Observe now that the LHS is a pseudo-tensor of rank 2, while the RHS is the sum of the twoquantities Aκλ and Bκλ which each only depend on one momentum. As it is not possible to

272


create a pseudo-tensor of rank 2 from that, the RHS must be zero. We have thus shown thatthe axial current is conserved.

Next we verify the conservation of the vector current, following the same line of argumentas in the case of the axial current. Starting from

Jλµ ≡ pκ1Sκλµ = −∫

d4k

(2π)4tr[p/1(k/ − p/1)γµγ5(k/ + p/2)γλk/

]

(k − p1)2(k + p2)2k2, (13.25)

we shift the integration variable k′ = k + p2 and reorder the terms in the trace, obtaining

Jλµ = −∫

d4k′

(2π)4tr[(k/′ − p/1 − p/2)γµγ5k/′γλ

]

(k′ − p1 − p2)2k′2︸︷︷︸

Aλµ(q=p1+p2)

+

∫d4k′

(2π)4tr[(k/′ − p/2γµγ5k/′γλ

]

k′2(k′ − p2)2︸︷︷︸

Bλµ(p2)

. (13.26)

Thus we conclude that Jλµ also vanishes. For the pλ2Sκλµ part, we follow the same argument,but we shift the integration variable by k′ = k − p1 instead.

We have now shown that both the axial current and the vector current are conserved. Usingthese results in the calculation for the decay width of a π0 leads however to a width whichvanishes in the limit of a massless pion. Including the small pion mass leads still to life-timemuch longer than observed.

Looking back at our calculation, we should check therefore if all our manipulations werelegitimate, although we have not regularised the divergent loop integrals. In particular, thesuperficial divergence of the diagrams Sκλµ is worse than the logarithmic divergences we havebecome accustomed to,

Sκλµ = −∫

d4k

(2π)4tr[γκk/γµγ

5k/γλk/]

k6+ subleading terms (13.27)

producing a linear divergent term.An essential ingredient for the derivation of the current conservation was the shift of our

integration variable. Such a shift can only be done if the integral is properly convergent (afterregularisation, if required) or only logarithmic divergent. By contrast, in the case of a lineardivergent integral, the shift k′ = k − a

∫

d4k′f(k′)?=

∫

d4kf(k − a) =∫

d4k

f(k)− aµ∂f

∂kµ+ . . .

(13.28)

changes the value of the integral: Using Gauss’ theorem, we can convert the gradient terminto a surface integral which will lead to a finite change of the integral because of Ω ∝ k3 andf ′ ∝ k−3. You should show in problem 13.1 that the shift kµ − aµ in (13.27) changes Sκλµ as

Sκλµ → S′κλµ = Sκλµ +

1

8π2ǫκλµνa

ν (13.29)

where ǫκλµν is again the totally anti-symmetric tensor.We can look back at the corresponding proof of gauge invariance for the vacuum polar-

isation, Eqs. (8.92–8.95). There we used dimensional regularisation which respects gaugesymmetry. Since γ5 is not well-defined for d = 4 − ε, we cannot apply this regularisationmethod here. Using Pauli-Villar regularisation as an alternative would break axial symmetry,since it consists of adding massive particles. Thus, from a technical point of view, anomalies

273


arise, if no regularisation and renormalisation procedures exists which respects the classicalsymmetry.

Since the shift of the integration variables required to obtain current conservation differ forthe vector and axial current, we can absorb only one of the resulting boundary terms (13.29)by a suitable chosen counter-term. Clearly, we will choose the vector current to be conserved:Otherwise electric charge conservation would be violated, while the axial current is anywaybroken by mass effects. So if we add to T a counter term,

Aκλµ = Sκλµ(p1, p2) + Sλκµ(p2, p1) +1

4π2ǫκλµν(p

ν1 + pν2) , (13.30)

our vector current will still be conserved, but the axial current will not

(p1 + p2)µAκλµ =

1

2π2ǫκλµνp

µ2pν1 . (13.31)

The added contribution gives a non-zero contribution to the divergence of the axial currentwhich is identical to our previous non-perturbative result,

∂µjµ5 =

e2

8π2Fµν F

µν . (13.32)

This implies that the perturbative one-loop result for the anomaly is not changed by higher-order corrections. In particular, processes which are dominated by the anomaly like π0 → 2γcan be calculated reliably in lowest-order perturbation theory, although for Q2 = m2

π ∼ Λ2QCD

the strong coupling constant αs(Q2) is certainly not small and corrections of the type shown

in are naively expected to be large.The corresponding result for a non-abelian theory like QCD is

∂µjµ5 =

nfg2

16π2tr(

Fµν Fµν)

, (13.33)

where nf denoted the number of quarks flavors.

Cancellations of anomalies Anomalies may be useful when they can break ”accidental”global symmetries like baryon and lepton number. As we will see later, this opens the possi-bility to explain why the universe consists mainly of matter. Similarly, the explicit breakingof global symmetries in the quark Lagrangian can explain why in certain cases no Goldstonebosons are observed.

In contrast, symmetries and the resulting Ward identities between Green functions arecrucial for the renormalisability of gauge theories. If these identitites are not satisfied, physicalobervables in renormalisable gauges depend on the gauge-fixing parameter ξ, while Lorentzinvariance is violated in unitary gauges. The excellent agreement of electroweak precisiondata with experiment is a strong argument that the underlying theory is renormalisable. Asthe V-A structure of the electroweak interactions is however similar to our toy model of axialelectrodynamics, we can only expect that the anomalies of individual loop diagrams cancelafter summing over all contributions.

As an example we analyse an one-loop VVA diagram, i.e. a triangle diagram with twovector and one axial vertices. The anomaly written out for the left-handed fermions is

Aabc =[

tr(MLaM

Lb M

Lc

)+ tr

(MLaM

Lc M

Lb

)]

= tr(MLa

MLb M

Lc

)(13.34)

274

13.2. Instantons and the strong CP problem

where the trace is over SU(2) doublets. TheM ’s are the coupling matrices for all of the threeinteractions in the loop. If we identify the interactions at the vertex a with Z, b with W−

and c with W+ and recall the coupling matrices, then

MLa =

g

cos ϑW

(τ32

+ sin2 ϑWQ)

(13.35a)

MLb = gT+ =

g

2τ+ (13.35b)

MLc = gT− =

g

2τ− . (13.35c)

Requiring that the trace vanishes gives us the following constraint,

0!= tr

[(τ32

+ sin2 ϑWQ)

τ+, τ−]

=1

4tr(τ32

+ sin2 ϑWQ)

=∑

i

QLi , (13.36)

where we used τ+, τ− = 1/4 and tr(τ3) = 0. Thus the anomaly vanishes if the electriccharges of all left-handed fermions sum to zero. If we now only look at the leptons andquarks separately, then

Qe +Qν = −1 6= 0 and Qu +Qd =2

3− 1

3=

1

36= 0 .

Including both leptons and quarks where we account by the factor three for their colourquantum number we find

Qe +Qν + 3(Qu +Qu) = −1 + 31

3= 0 .

Therefore, chiral anomalies are canceled in the SM within each full fermion generation. Inother words, if a single member of a hypothetical fourth generation of fermions would befound, anomaly cancellation would require the existence of a complete set of quarks andleptons. Within the SM, there is no explanation for the miraculous cancellation between thequark and lepton sector, and this has been one of the major motivations to consider GUTs.

The AAA anomaly is automatically zero, since

Aabc ∝ tr[

τa

τ b, τ c]

= 2δbctrτa = 0 . (13.37)

We have restricted our analysis of the anomaly to an abelian model. In the non-abelian case,the field-strength contains term linear and quadratic in the gauge fields. As a result, additionalanomalies in square and pentagon diagrams appear. However, the absence of anomalies inthe triangle diagrams guaranties also their absence in square and pentagon diagrams.


The effective Lagrangian induced by the chiral anomaly is a surface term. This suggeststhat a term of the same structure can be obtained as the topological charge of a Yang-Millstheory, following the argument of Bogomol’nyi in Eqs. (12.43). If such a topological chargecharge exist, then the unbroken vacuum of a Yang-Mills theory has a non-trivial vacuumstructure. Such a non-trivial vacuum structure has however only physical consequences if thecorresponding classical tunneling solutions have a finite action. We should therefore searchfor classical solutions of a pure, Euclidean Yang-Mills theory with SYM <∞. These solutionsare the non-abelian analogue of the bounce solution we have considered earlier.

275


Instantons We define an Euclidean Yang-Mills theory by the action

S =1

2

∫

d4x trFµνFµν

, (13.38)

where Fµν = F aµνTa = ∂µAν − ∂νAµ + ig[Aµ, Aν ] and derivatives and integrations are with

respect to Euclidean coordinates (x1, x2, x3, x4 = ix0) with xi ∈ R. Since the metric tensoris ηµν = δµν , we do not need to distinguish between lower and upper indices. Therefore, inEuclidean space the dual of the dual field strength tensor is again the field strength tensor,˜Fµν = Fµν , while in Minkowski space it is the negative of the field strength tensor, ˜Fµν =−Fµν .

We first examine QED, i.e. the case of an abelian YM theory. A finite action,∫

τd4x FF <∞ (13.39)

requires that F decreases faster than τ−2 in the limit τ2 = x2+x24 →∞. But as a classical YMtheory contains no scale, FF must be a polynomial in τ : Thus F ∼ O(τ−3) and A ∼ O(τ−2)and then the total derivative (13.19) behaves as

Kµ = 2ǫµνρσAν∂ρAσ ∼ O(τ−5) . (13.40)

As a result, the surface term in∫

Ωd4 FF =

∫

∂ΩdΩµKµ → 0 (13.41)

vanishes and we see that in an abelian theory FF does not influence physical quantities. Thisargument justifies our usual practise to neglect surface terms in QED.

We now turn to the non-abelian case. Then we can express tr FµνFµν again as a four-divergence,

∂µKµ = 2 tr FµνFµν (13.42)

where now

Kµ = 2εµνσρtr

(

AνFσρ +2

3igAµAσAρ

)

. (13.43)

Choosing a pure gauge field,

Aµ =i

g(∂µU)U−1 , (13.44)

at spatial infinity results in Fµν = 0 for τ → ∞ and ensures that the action is finite. Onthe other hand, a gauge transformation U which becomes constant for τ → ∞, i.e. dependsin this limit only on the angles, gives A ∼ O(τ−1) and thus K ∼ O(τ−3). As a result, thesurface integral may become non-zero in the limit τ →∞. One may wonder if we can gaugeaway Aµ ∝ ∂µUU

−1 on τ = ∞ by performing a suitable gauge transformation U . Since Uhas to be regular in all R4, it must be constant a τ = 0 and independent of the angles. ThusU is continously connected to the identity and can be used only to gauge away Aµ in thesame homotopy class. Thus the surface term FF has physical significance, if the gauge fieldsat τ →∞ are split into non-trivial topological classes.

We now turn to the specific case of SU(2). Any SU(2) matrix can be written as U = a+ib·σwith a2 + |b|2 = 1. Thus SU(2) is isomorphic to S3. Compactifying R3 to S3, (13.44) defines

276


the map S3 → S3. The question if non-trivial instanton solutions exist is thus equivalent tothe existence of topologically non-trivial mappings S3 → S3, that is to say if π3(S

3) is notthe identity.

The generalisation of the winding number (12.72) from S1 → S1 to the case S3 → S3 is

ν = − 1

24π2

∫

dx1dx2dx3 εijktr[(U∂iU

†)(U∂jU†)(U∂kU

†)]

, (13.45)

where we used the Cartesian coordinates of R3. Since dxi∂i = dxi∂i, the expression is equallyvalid using the three angles specifying a point on S3. In order to show that the windingnumber is invariant under continuous transformations, δν = 0, it is sufficient to consider thevariation of a single factor in the trace. Now we can profit from (12.72) where we derivedalready the variation of this factor in the non-abelian case as

δ(U∂iU†) = −U∂i

(U †δU

)U † .

Inserting this relation into the integrand results in

E ≡ εijktr[(U∂iU

†)(U∂jU†)δ(U∂kU

†)]

= −εijktr[

∂iU†U∂jU

†U∂k(U †δU

)]

. (13.46)

Then we perform a partial integration and use that terms in ∂k∂iU† vanish contracted with

εijk,

E = εijktr[

∂iU†∂kU∂jU

†δU + ∂iU†U∂jU

†∂kUU†δU

)]

, (13.47)

In the second term, we use U∂jU† = −∂jUU † and ∂kUU

† = −U∂kU † to symmetrize theexpression in j and k,

E = εijktr[

∂iU†∂kU∂jU

†δU + ∂iU†∂jU∂kU

†δU]

= 0 . (13.48)

Thus the winding number ν is invariant under infinitesimal and thus under continous trans-formations.

Next we try to express the winding number as a volume integral over tr FµνFµν. Wewrite (13.45) as a surface integral

ν =1

24π2

∫

dΩµ εµνστ tr[(U∂νU

†)(U∂σU†)(U∂τU

†)]

= − ig3

24π2

∫

dΩµ εµνστ tr [AνAσAτ ] .

(13.49)where we used that Aµ is a pure gauge field for τ →∞, (13.44). Since then also F aσρ = 0, wecan use (13.43)

ν =g2

32π2

∫

dΩµ Kµ =

g2

16π2

∫

d4x tr FµνFµν . (13.50)

Mathematicians call the winding number of the mapping S3 → S3 the Pontryagin index,while mathematical physicists use often the term Chern-Simon number.

We define instantons as self-dual and anti-self-dual solutions,

Fµν = Fµν and Fµν = −Fµν , (13.51)

277


of the classical Yang-Mills equations. Using Bogomol’nyi’s trick, we can show that theycorrespond to the topologically non-trivial solutions with the lowest energy. We write first

tr(Fµν − Fµν)2

= tr

F 2µν + F 2

µν − 2Fµν Fµν≥ 0 . (13.52)

For the calculation of F 2 we use ǫµνρσǫµνκλ = 2(δρκδσλ − δρλδσκ) and end up with F 2 = F 2.Thus

trFµνFµν

≥ tr

Fµν Fµν

0 , (13.53)

which is minimized if F is self-dual. We obtain the same bound for an antiself-dual solution,if we choose a plus sign in Eq. (13.52). Integrating over space, we find on the LHS twice theEuclidean action, while the RHS equals 16π2ν/g2. Thus the Euclidean action is bounded by

S ≥ 8π2|ν|g2

, (13.54)

and instantons as the non-trivial solutions with the lowest energy have the action S = 8π2/g2.Our remaining task is to write down an explicit form of the mappings S3 → S3 and to show

that the definition (13.45) results in a integer winding number. We choose to write U(x) as

U (0)(x) = 1 (13.55)

U (1)(x) =x4 + ix · σ

τ(13.56)

U (n)(x) =

(x4 + ix · σ

τ

)n

(13.57)

where xµ is the unit vector xµ = (sinχe, cos χ). Evaluation of (13.45) for U (1)(x) confirmsthen ν = 1.

We used for our discussion of instantons in non-abelian Yang-Mills theories the specificexample of SU(2). A theorem of Brott states that for any simple Lie group G containingSU(2) the maps S3 → G can be deformed continously to the ones of S3 → SU(2). Thusall our results apply identically to the case of strong interactions, SU(3). In the remainingpart of this chapter, we will discuss the impact of instantons on the QCD vacuum, while wepostpone the electroweak case to chapter 18.

Tunneling interpretation We consider the four-dimensional cylinder defined by x4 ∈ [−T :+T ] and |x| ≤ R and sketched in Fig. 13.1 in the limit T,R → ∞. At x4 = −T , we chooseAµ as a pure gauge field with winding number ν1, and at x4 = T with winding number ν2.On the boundary |x| = R, we choose U constant and thus Aµ is zero. Calculating the totalwinding number, we find ν = ν1 − ν2.

g2

16π2

∫

Ωd4x tr

(FF)=

g2

32π2

∫

∂Ωd3σµK

µ =g2

32π2

∫

d3x[K0(t = −∞)−K0(t = +∞)

]

= ν1 − ν2 = ν (13.58)

The minus sign appears, because of the opposite orientations of the two caps. Thus theclassical solutions we have determined interpolate between a vacua with winding number ν1at time t = −∞ and a vacua with winding number ν2 at time t = +∞. The two vacua areseparated by a finite energy barrier, and thus the solutions decribe the quantum tunnelingbetween different vacua.

278


x4

R3

−T

T

Figure 13.1.:

The ϑ vacuum The tunneling interpretation indicates that the true vacuum of a pure Yang-Mills theory is the superposition of all vacua with fixed winding number ν. Let us call thesevacua |ν〉 and the true one |ϑ〉. Applying a gauge tranformation U ≡ U (1) with unit windungnumber results in

U |ν〉 = |ν + 1〉 . (13.59)

On the other hand, the Yang-Mills Hamiltonian is invariant under gauge transformations,

UHU † = H , (13.60)

or [H,U ] = 0. Thus the true vacuum |ϑ〉 is a common eigenstate of H and U . Since thevacuum is normalised, the eigenvalue of U has to be a phase,

U |ϑ〉 = eiϑ |ϑ〉 . (13.61)

Its arguments ϑ is a conserved quantum number. Thus a pure Yang-Mills theory is charac-terised by two numbers, the coupling g and the new parameter ϑ.

The true vacuum |ϑ〉 is a linear superposition of the vacua with fixed winding number givenby

|ϑ〉 =∞∑

n=−∞e−inϑ |n〉 , (13.62)

since

U |ϑ〉 =∞∑

n=−∞e−inϑ |n+ 1〉 = eiϑ

∞∑

n=−∞e−inϑ |n〉 . (13.63)

The matrix elements 〈n′|H |n〉 depend only on the difference ν = n′ − n, since (13.60) gives

⟨n′ + 1

∣∣H |n+ 1〉 =

⟨n′∣∣H |n〉 . (13.64)

Using also the parity properties, P |n〉 = − |n〉 and PHP−1 = H, we obtain

⟨n′∣∣H |n〉 =

⟨−n′

∣∣H |−n〉 . (13.65)

279


Hence the matrix elements 〈n′|H |n〉 depend only on the absolut value |ν| = |n′ − n|. Weconclude that instantons lead to an effective potential Veff(ϑ) which is periodic and even in ϑ

H |ϑ〉 = L3Veff(ϑ) |ϑ〉 , (13.66)

with Veff(ϑ) = Veff(ϑ+2π) and Veff(ϑ) = Veff(−ϑ), where L3 is the considered volume. A gen-eral argument due to Weinberg shows that points of enhanced symmetry are stationary pointsof the action. Thus we expect the minimum of Veff(ϑ) to coincide with the CP conservingpoint ϑ = 0.

We included into our definition of the path integral for gauge theories with the help of theFadeev-Popov trick only gauge fields which are continously connected with the identity. Thusour next task is to add the effect of the ϑ vacuum to the path integral. The path integral inthe presence of external sources is identical to the vacuum peristence amplitude,

〈ϑ′|ϑ〉J =∑

n,n′

ei(n′ϑ′−nϑ)〈n′|n〉J . (13.67)

Now we introduce the difference ν = n′ − n so that we can rewrite the phase as n′ϑ′ − nϑ =n(ϑ′ − ϑ) + νϑ′. But 〈n′|n〉J depends only on ν and thus we can perform the sum over n.This leads to a factor δ(ϑ′ − ϑ), which expresses the fact that ϑ is conserved. Thus

〈ϑ|ϑ〉J =∑

ν

eiνϑ∫

DA(ν) e−S+JA (13.68)

where DA(ν) denotes the integration over all gauge field configurations with the windingnumber ν. Replacing ν with the help of Eq. (13.50) and introducing DA ≡∑ν DA(ν) gives

〈ϑ|ϑ〉J =

∫

DA e−S+JA exp

ig2

16π2ϑtr(FF )

. (13.69)

Thus instantons induce an additional term Lϑ to the classical Yang-Mills Lagrangian,

Leff = L +ϑg2

16π2tr(FF ) , (13.70)

which depends on the arbitrary parameter ϑ ∈ [0, 2π[. In order to discuss observable effectsof this additional term, we have to add fermions to the pure Yang-Mills theory we discussedup to now.

Fermionic contribution to ϑ The axial anomaly leads to the additional term in the effectiveLagrangian

Leff = L +αnfg

2

16π2tr(FF ) , (13.71)

if we perform a chiral UA(1) tranformation qL,R → eiαγ5qL,R on the nf quark fields. This

term has the same structure as the instanton contribution, and if we choose

α = − ϑ

2nf(13.72)

the two terms cancel. Thus for massless quarks the ϑ parameter is unphysical and we canchoose the “simple” ϑ = 0 vacuum.

280


We can understand this if we consider the effect of an instanton transition. For zero masses,we know that

∂µjµ5 = ∂µ(j

µR − j

µL) =

g2

16π2tr(Fµν F

µν) . (13.73)

Integrating gives as change of the axial charge Q5 = NR −NL

∆Q5 = Q5(t = −∞)−Q5(t = +∞) = 2ν . (13.74)

Thus an instanton process ν = ±1 changes the axial quark number by two units, creatinga left-chiral and destroying a right-chiral quark and vice versa. As chirality is a conservedquantum number for massless particles, at least one of the two states connected by theinstanton process can therefore not correspond to the vacuum. By contrast, for m > 0 themass term mixes left- and right-chiral fields: the quark-antiquark pair can annihilate via themass term and the states can be identified with the vacuum.

We now consider massive quarks. Then the axial current is additionally to the axial anomalyexplicitely broken by the quark masses, since a Dirac mass term transform as

mqq → me2iαγ5qq = cos(2α)mqq + i sin(2α)mqγ5q (13.75)

under a chiral tranformation qL,R → eiαγ5qL,R. The second term violates T (CP) invariance,

and as a result the cancellation (13.72) will fail. If we consider nf flavor of quarks, then themass matrix Mij will change as Mij → e2iαMij and thus

arg detM → arg detM + 2nfα . (13.76)

Therefore only the combinationϑ ≡ ϑ+ arg detM (13.77)

is an observable quantity: It is a question of convenience, if we choose real mass matrices androtate all CP violation via the chiral anomaly into the ϑ term. Or if we eliminate Lϑ andtransfer its effect into CP violating complex mass matrices.

We look now for observable consequences of the ϑ vacuum in the low-energy interactionsof hadrons. Since the ϑ term (and the axial anomaly) is a flavor singlet, the change c in thequark mass,

c ≡ sin(2αq)mq , (13.78)

is the same for all quark flavours. In order to shift the effects of the ϑ term completely intothe mass matrix of the quarks, we need also

nf∑

q=1

2αq = −ϑ . (13.79)

Solving for small ϑ gives

c = − ϑ∑

qm−1q

, (13.80)

and thus the ϑ dependent, CP violating part of the mass term becomes

L(ϑ)m = −iϑ

(∑

q

m−1q

)−1∑

q

qγ5q . (13.81)

281


For light nucleon and mesons which consist only of u and d quarks this simplifies to

L(ϑ)m = −iϑ mumd

mu +md

(uγ5u+ dγ5d

). (13.82)

This mass term generates an CP violating effective pion-nucleon interaction which in turnleads to an electric dipole moment dn of the neutron. The corresponding limit on dn boundsthe value of ϑ as |ϑ| ≤ 2× 10−10. Although the value of ϑ is a free parameter within the SM,it seems natural to ask for for an explanation why ϑ is so small.

The most straight-forward explanation would be that one current quark mass is zero, i.e.that one quark has no Yukawa coupling to the SM Higgs. Then (13.82) shows for nf = 2clearly that the CP violating effect disappear. This holds also for nf > 2, because arg detM =0 if one mass eigenvalue is zero. While it has been debatable if the mu ∼ 5MeV deduced fromchiral pertrubation theory for the current u quark mass might lowered down to mu = 0, thispossibility was closed by lattice data around 2005. The question why ϑ is so small is calledthe strong CP problem of the SM.

13.3. Axions

Peceei and Quinn proposed to promote the parameter ϑ to a dynamical variable which settlesautomatically at its minimum zero. The basic ingredient of this proposal is a new masslesspseudo-scalar field a called axion which couples to gluons with an interaction of the samestructure as the ϑ term,

La =1

2(∂µa)

2 − g2

16π2a

faFF . (13.83)

Since a is a pseudo-scalar, the interaction term aF F conserves CP. Adding La to the effectiveLagrangian (13.70) of QCD means that observables depend only on the combination ϑ−a/fa.If La is invariant under the shift a→ a+const., then we can use this arbitrariness to absorbthe ϑ parameter into a redefinition of the field a. Such a shift symmetry is typical for aGoldstone boson which has only derivative couplings. Thus the pseudo-scalar particle ashould be the Goldstone boson of a spontanously broken global symmetry, and the aF Finteraction suggests to choose this symmetry as a chiral U(1) symmetry. This symmetry iscalled Peceei-Quinn symmetry UPQ(1), its Goldstone boson a axion.

Weinberg and Wilzcek realised that there is an additional twist in this “simple” proposal:The effective potential (13.66) generated via the FF term has two effects: First, the instantoneffects break the shift symmetry, generating a masss term m2

a = ∂2Veff/∂a2 for the axion.

Second, the vev of the axion field will relax to the minumum of the potential, ∂Veff/∂a = 0.But we have argued above that the minmum of Veff(ϑ) is situated at ϑ = 0. Thus also in thecase of a massive axion the strong CP problem is solved.

Let us illustrate the key points of this idea with a concrete model. We add a complex scalarfield φ and a set of heavy fermions ψ to the SM,

L = iψ∂/ψ +1

2∂µφ

†∂µφ− yi(ψLψRφ+ h.c.)− V (φ) . (13.84)

While φ is a SM singlet, the fermions are charged under SU(3) and U(1). The Lagrangian isinvariant under the global chiral UPQ(1) gauge transformation

φ→ eiαφ and ψL/R → e±iα/2ψL/R . (13.85)

282

13.3. Axions

We now break spontanously the Peccei-Quinn UPQ(1) symmetry, choosing as usually for V (φ)a Mexican hat potential. Splitting φ into its vacuum expectation value and the fluctuatingfields,

φ =fa + ρ√

2eia/fa

(01

)

, (13.86)

we generate a mass term for ρ while a remains massless. We assume that fa is much largerthan the energy scale E we are interested in, and thus we can neglect the field ρ,

L =1

2(∂µa)

2 −miψeiγ5a/faψ − V (φ) +O(E2/f2a ) . (13.87)

The combined expression is clearly invariant under UPQ(1) transformations a → a + αfa.Expanding the exponential, we generate mass terms mi = yifa/

√2 for the heavy fermions

plus fermion-axion interactions,

Lint = −miψψ − imi

faaψγ5ψ − . . . (13.88)

The latter lead to a AV V triangle graph for the process a→ 2g which in turn induces via thechiral anomaly the desired FF term. In the same way, an effective coupling a→ 2γ couplingis generated. Thus the characteristic feature of an axion is its two-gluon and two-photoncoupling.

Finally, we mention that the parameters of the pion and axion sector are connected by

mafa ≈ mπfπ , (13.89)

since the mass of both Goldstone bosons is generated by instantons and they mix a↔ 2g ↔π0.

Summary of chapter

The CP-odd term FµνFµν is a gauge invariant renormalisable interaction. Terms of this

type are produced by instanton transitions between Yang-Mills vacua with different windingnumbers and by the chiral anomaly. While we can rotate classically all CP violating phasescontained in the quark mass matrices into the single CP violating phase of the CKM matrix,the chiral anomaly leads additionally to the change ϑ→ ϑ = ϑ+arg detM in the coefficient ofthe FµνF

µν term. Since the physics origin of both contributions seem to be disconnected, it ispuzzling that they sum up to |ϑ| <∼ 10−10. A possible solution is the Peceei-Quinn symmetrywhich promotes the parameter ϑ to the field a = faϑ which settles automatically at theminimum at ϑ = 0 of the instanton potential Veff .

Further reading

Instantons are discussed in more detail in [Rub02].Wess-Zumino consistency condition, gravitational anomalies.

Problems

283


13.1 Surface term.

Derive (13.29): Use Gauss’ theorem to convertthe gradient term into a surface integral. Derivethe trace formula for 6 gamma’s and one γ5 andevaluate the trace. Transform the result into Eu-clidean space and evaluate it.

13.2 Pion decay width.

Calculate the decay width of π0 → 2γ taking intoaccount the chiral anomaly. Compare the value to

the measured one.

13.3 Arg det M in the SM.Determine arg(det(M)) in the SM and show thatit is independent of a possible phase of the vev ofthe Higgs.

13.4 Euclidean YM.Derive the connection bwteen Aµ in Minkowskiand Euclidean space. Show that this tranforma-tion law also leads to a real action for the gaugedscalar field theory.

284

15. Gravity as a gauge theory

We introduce the action and the field equations of gravity, proceeding in a way which stressesthe similarity of gravity as a gauge theory with the group GL(4) to Yang-Mills theories. As abonus, this way allows us to describe also the gravitational interactions of fermions as well asto understand how gravity selects among the many mathematically possible connections on aRiemannian manifold. The linearised Einstein equations can be used to decribe the emissionand propagation of weak gravitational waves.

15.1. Vielbein formalism and the spin connection

The equivalence principle postulates that in a small enough region around the center of a freelyfalling coordinate system all physics is described by the laws of special relativity. At largerdistances, gravity manifests itself as curvature of space-time. Thus physical laws involvingonly quantities transforming as tensors on Minkowski space are valid on a curved space-timeperforming the replacement

∂µ, ηµν ,d4x → ∇µ, gµν , d4x√

|g| . (15.1)

Here, the covariant derivative ∇µ was defined using as connection the Christoffel symbols(or Levi-Civita connection) from Eq. (1.109). We recall that the two requirements ∇ρgµν =0 (“metric connection”) and Γαβγ = Γαγβ (“torsionless connection”) select uniquely thisconnection. In the following, we want to understand if these conditions are a consequence ofEinstein gravity or necessary additional constraints.

A useful framework to address these questions is the Vielbein formalism which is alsonecessary to include spinors into the framwork of general relativity. Since the Levi-Civitaconnection can be applied only to objects with tensorial indices, the substitution rule ∂µ → ∇µcannot be applied to the case of spinor representations of the Lorentz group.

Vielbein formalism We apply the equivalence principle as physical guide line to obtain thephysical laws including gravity: More precisely, we use that we can find at any point P alocal inertial frame for which the physical laws become those known from Minkowski space.

We demonstrate this first for the case of a scalar field φ. The usual Lagrange densitywithout gravity,

L =1

2∂µφ∂

µφ− V (φ) , (15.2)

is still valid on a general manifold M(xµ), if we use in each point P locally free-fallingcoordinates, ξa(P ). In order to distinguish these two sets of coordinates, we label inertialcoordinates by latin letters (a, b, . . .) while we keep greek indices µ, ν, . . . for arbitrary coor-dinates. We choose the locally free-falling coordinates ξa to be ortho-normal. Thus in thesecoordinates the metric is given by ds2 = ηabdξ

adξb with η = diag(1,−1,−1,−1). Then the

295


action of a scalar field including gravity is

S[φ] =

∫

d4ξ

[1

2ηab∂aφ∂bφ− V (φ)

]

, (15.3)

where ∂a = ∂/∂ξa. This action looks formally exactly as the one without gravity – howeverwe have to integrate over the manifold M(xµ) and all effects of gravity are hidden in thedependence ξa(xµ).

We introduce now the vierbein (or tetrad) fields eaα by

dξm =∂ξm

∂xµdxµ ≡ emµ (x) dx

µ . (15.4)

Thus we can view the vierbein emµ (x) both as the transformation matrix between arbitrarycoordinates x and inertial coordinates ξ or as a set of four vectors in T ∗

xM . In the absence ofgravity, we can find in the whole manifold coordinates such that emµ (x) = δmµ .

We define analogously the inverse vierbein eµm by

dxµ =∂xµ

∂ξmdξm ≡ eµm(x) dξ

m . (15.5)

The name is justified bydξm = emµ dxµ = emµ e

µn dξ

n , (15.6)

i.e. emµ eµn = δmn . Moreover, we can view the vierbein as a kind of square-root of the metric

tensor, sinceds2 = ηmndξ

mdξn = ηmnemµ e

nνdx

µdxν = gµνdxµdxν (15.7)

and thusgµν = ηmne

mµ e

nν . (15.8)

Taking the determinant of this equation, we see that the volume element is

d4ξ =√

|g|d4x = det(emµ )d4x ≡ Ed4x . (15.9)

For later use, we note that latin indices are raised and lowered by the flat metric. It is alsopossible to construct mixed tensors, having both latin and greek indices. For instance, wecan rewrite the energy-momentum tensor as Tµν = emµ Tmν = emµ e

nνTmn.

We have now all the ingredients needed to express (15.3) in arbitrary coordinates xµ of themanifoldM. We first change the derivatives,

L =1

2ηmne

mµ e

nν∂

µφ∂νφ− V (φ) =1

2gµν∂

µφ∂νφ− V (φ) (15.10)

and then the volume element in the action,

S[φ] =

∫

d4x√

|g|[1

2gαβ∂αφ∂βφ− V (φ)

]

. (15.11)

As it should, we reproduced the usual action of a scalar field including gravity. Note that thesole effect of gravitational interactions is contained in the metric tensor and its determinant,while the connection plays no role since ∇µφ = ∂µφ. Similarly, the connection drops out ofthe Lagrangian of a Yang-Mills field, since the field-strength tensor is antisymmetric.

296

15.1. Vielbein formalism and the spin connection

Fermions and the spin connection We now proceed to the spin-1/2 case. Without gravity,we have

L = ψ(iγµ∂µ −m)ψ (15.12)

with γµ, γν = 2ηµν . Performing a Lorentz transformation, xa = Λbaxb, the Dirac spinor ψ

transforms as

ψ(x) = S(Λ)ψ(x) = exp

(

− i

4ωµνσµν

)

ψ(x) (15.13)

with ωµν = −ωνµ as the six parameters and σµν = i2 [γ

µ, γν ] as the six infinitesimal generatorsof these transformations.

Switching on gravity, we replace xµ → ξm and γµ → γm. General covariance reduces now tothe requirement that we have to allow in an inertial system arbitrary Lorentz transformations.The condition that the Dirac equation is invariant under local Lorentz transformations Λ(x)allows us to derive the correct covariant derivative, in a manner completely analogous to theYang-Mills case: We have to compensate the term introduced by the space-time dependenceof S(ξ) in

∂aψ → ∂aψ(ξ) = Λba∂b[S(ξ)ψ(ξ)] (15.14)

by introducing a “latin” covariant derivative

∇a = eαa (∂α + iωα) , (15.15)

and requiring an inhomogeneous transformation law

ωα → ωα = SωαS† − iS∂αS

† (15.16)

for ω. As a result,∇a → ∇a = ΛbaS∇aS† (15.17)

and the Dirac Lagrangian is invariant under local Lorentz transformations.The connection ωα is a matrix in spinor space. Expanding it in the basis σµν , we find as a

more explicit expression for the covariant derivative

∇a = ∂a +i

2ωµνa σµν = eαa

(

∂α + i1

2ωµνα σµν

)

= eαa

(

∂α +1

2ωµνα Xµν

)

. (15.18)

In the second step we replaced the infinitesimal generators σµν specific for the spinor repre-sentation by the general generators Xµν of Lorentz transformations chosen appropriate forthe representation the ∇a act on. The Lie algebra of the Lorentz group implies that theconnection ωµνa is antisymmetric in the indices µν, if they are both up or down, ωµνa = −ωνµa .

The transformation law (15.16) of the spin connection ω under Lorentz transformations Sis completely analogous to the transformation properties (7.14) of a Yang-Mills field Aµ undergauge transformations U . One should keep in mind however two important differences: First,a vector lives in a tangent space which is naturally associated to a manifold: In particular,we can associate a vector in TPM with a trajectory x(σ) through P . Therefore we have thenatural coordinate basis ∂i in TPM and can introduce vielbein fields. In contrast, matterfields ψ(x) lives in their group manifold which is attached arbitrarily at each point of themanifold, and the gauge fields act as a connection telling us how we should transport ψ(x) to

297


ψ(x′). However, there is no coordinate basis in the group manifold and nothing like torsionfor the gauge fields. Second, we associate physical particles with spin s to the correspondingrepresentations of the Poincare group: Thus we identify fluctuations of the gauge field Aµ

as the quanta of the vector field, while it is in the case of gravity not the fluctuation of theconnection but of the metric tensor gµν .

Transition to the standard notation We now establish the connection between the vierbeinand the standard formalism, using that for tensor fields the two formalisms have to agree.Inserting into the definition of the covariant derivative for the components Aa of a vector A,

∇µAa = ∂µAa + ω a

µ bAb , (15.19)

the decomposition Aa = eaνAν and requiring the validity of the Leibnitz rule gives

∇µAa = (∇µeaν)Aν + eaν(∇µAν)= (∇µeaν)Aν + eaν

(

∂µAν + ΓνµλA

λ)

. (15.20)

Using∂µA

a = ∂µ(eaνA

ν) = eaν(∂µAν) + (∂µe

aν)A

ν (15.21)

to eliminate the second term in (15.20), we obtain

∇µAa = (∇µeaν)Aν + ∂µAa − (∂µe

aν)A

ν + eaνΓνµλA

λ . (15.22)

Comparing to (15.19) we can read off how the covariant derivative acts on an object withmixed indices,

∇µeaν = ∂µeaν − Γλνµe

aλ + ω a

µ bebν . (15.23)

More generally, covariant indices are contracted with the usual connection Γλνµ while vierbeinindices are contracted with ω a

µ b. In a moment, we will show that the covariant derivative ofthe vielbein field is zero, ∇µeaν = 0. Sometimes this property is called tetrade “postulate,”but in fact it follows naturally from the definition of the vierbein field.

In order to derive an explicit formula for the spin connection ω aµ b we compare now the the

covariant derivative of a vector in the two formalisms. First, we write in a coordinate basis

∇A = (∇µAν)dxµ ⊗ ∂ν = (∂µAν + ΓνµλA

λ)dxµ ⊗ ∂ν . (15.24)

Next we compare this expression to the one using a mixed basis,

∇A = (∇µAm) dxµ ⊗ em = (∂µAm + ω m

µ nAn) dxµ ⊗ em (15.25)

= [∂µ(emν A

ν) + ω mµ ne

nλA

λ] dxµ ⊗ (eσm∂σ) . (15.26)

Moving eσm to the left and using the Leibniz rule as well as eσmemν = δσν , it follows

∇A = eσm[emν ∂µA

ν +Aν∂µemν + ω m

µ nenλA

λ] dxµ ⊗ ∂σ (15.27)

= [∂µAσ + eσm(∂µe

mν )A

ν + eσmenλω

mµ nA

λ] dxµ ⊗ ∂σ . (15.28)

ThusΓνµλ = eσm∂µe

mν + eσme

nλω

mµ n = emµ ∇λeνm (15.29)

298

15.2. Action of gravity

or multiplying with two vierbein fields,

ω mµ n = eσme

nλΓ

νµλ − eσm∂µemν . (15.30)

Clearly, the connection Γνµλ is in general not symmetric, Γνµλ 6= Γνλµ.

Next we show that the covariant derivative of the vierbein vanishes. We start from thecovariant derivative of a general mixed two-tensor,

∇ρXνp = ∂ρXνp + ΓλνµXνp + ω aµ bXνp . (15.31)

Setting now Xνp = ηpqeqν and expressing the connection via Eq. (15.29), it follows

∇ρXνp = ∇ρeνp = ∂ρeνp + ωρpν + eσpemν ∇ρeσm = (15.32)

= ∂ρeνp − ∂ρeνp + ωρpν + ωρνp = 0 , (15.33)

because of the antisymmetry of the spin connection. Similarly, one shows that ∇ρeνp = 0.


Einstein-Hilbert action The analogy to the Yang-Mills case suggests as Lagrange densityfor the gravitational field

L =√

|g|RµνστRµνστ . (15.34)

This Lagrange density has mass dimension 4 and would thus lead to a dimensionless grav-itational coupling constant and a renormalisable theory of gravity. However, such a theorywould be in contradiction to Newton’s law. Hilbert chose instead the curvature scalar whichhas the required mass dimension d = 6,

LEH =√

|g|R . (15.35)

As we know, we can always add a constant term to the Lagrangian, R → R − 2Λ, whichwould act as cosmological constant. The Lagrangian is a function of the metric, its first andsecond derivatives1, LEH(gµν , ∂ρgµν , ∂ρ∂σgµν). The resulting action

δSEH[gµν ] =

∫

Ωd4x√

|g| R− 2Λ (15.36)

is a functional of the metric tensor gµν , and a variation of the action with respect to the metricgives the field equations for the gravitational field. If we consider gravity coupled to fermions,we have to use the spin connection ωµ in the matter Lagrangian as well as expressing

√

|g|and R through latin quantities,

√

|g| → E and R = Rµνgµν → Rmnη

mn.

We derive the resulting field equations for the metric tensor gab directly from the actionprinciple

δSEH = δ

∫

Ωd4x√

|g|(R− 2Λ) = δ

∫

Ωd4x√

|g| (gµνRµν − 2Λ) = 0 . (15.37)

1Recall that the Lagrange equations are modified in the case of higher derivatives which is one reason whywe directly vary the action in order to obtain the field equations.

299


We allow for variations of the metric gµν restricted by the condition that the variation of gµνand its first derivatives vanish on the boundary ∂Ω.

δSEH =

∫

Ωd4x

√

|g| gµνδRµν +√

|g|Rµνδgµν + (R− 2Λ) δ√

|g|

. (15.38)

Our task is to rewrite the first and third term as variations of δgµν or to show that they areequivalent to boundary terms.

Let us start with the first term. Choosing normal coordinates, the Ricci tensor at theconsidered point P becomes

Rµν = ∂ρΓρµν − ∂νΓρµρ . (15.39)

Hence

gµνδRµν = gµν(∂ρδΓρµν − ∂νδΓρµρ) = gµν∂ρδΓ

ρµν − gµρ∂ρδΓνµν , (15.40)

where we exchanged the indices ν and ρ in the last term. Since ∂ρgµν = 0 at P , we canrewrite the expression as

gµνδRµν = ∂ρ(gµνδΓρµν − gµρδΓνµν) = ∂ρX

ρ . (15.41)

The quantity Xρ is a vector, since the difference of two connection coefficients transformsas a tensor. Replacing in Eq. (15.41) the partial derivative by a covariant one promotes ittherefore in a valid tensor equation,

gµνδRµν =1√

|g|∂µ(√

|g|Xµ) . (15.42)

Thus this term corresponds to a surface term that vanishes.

Next we rewrite the third term using

δ√

|g| = 1

2√

|g|δ|g| = 1

2

√

|g| gµνδgµν = −1

2

√

|g| gµνδgµν (15.43)

and obtain

δSEH =

∫

Ωd4x√

|g|

Rµν −1

2gµνR+ Λ gµν

δgµν = 0 . (15.44)

Hence the metric tensor fulfils in vacuum the equation

1√

|g|δSEHδgµν

= Rµν −1

2Rgµν + Λgµν ≡ Gµν + Λgµν = 0 , (15.45)

where we introduced the Einstein tensor Gµν . The constant Λ is called the cosmologicalconstant.

We consider now the combined action of gravity and matter, as the sum of the Einstein-Hilbert Lagrange density LEH/2κ and the Lagrange density Lm including all relevant matterfields,

L =1

2κLEH + Lm =

1

2κ

√

|g|(R− 2Λ) + Lm . (15.46)

We will determine the value of the coupling constant κ in the next section, demanding thatwe reproduce Newtonian dynamics in the weak-field limit. We have already argued that

300


the source of the gravitational field is the energy-momentum stress tensor which lead to thedefinition

2√

|g|δSmδgµν

= Tµν . (15.47)

Einstein’s field equation follows then as

Gµν + Λgµν = −κTµν . (15.48)

Palatini action We start from the Einstein-Hilbert Lagrangian (15.35), but considerit now as a function of the metric tensor, the connection and its first derivatives,LEH(gµν ,Γ

σµν , ∂τΓ

σµν), while we allow an independent variation of the metric tensor and

the connection in the action. We obtain the desired dependence of the Lagrangian expressingthe Ricci tensor through the connection and its derivatives,

LEH =√

|g|gµνRµν =√

|g|gµν(∂νΓ

σµσ − ∂σΓσµν + ΓτµσΓ

στσ − ΓτµνΓ

στσ

). (15.49)

The variation with respect to the metric,

δgSEH =

∫

Ωd4x δ

√

|g| gµν

Rµν = 0 (15.50)

gives Rµν = 0, i.e. the usual Einstein equations in vacuum.For the variation with respect to the connection we use first the Palatini equation,

δΓSEH =

∫

Ωd4x√

|g| gµνδRµν =

∫

Ωd4x√

|g| gµν[∇ν(δΓσµσ)−∇σ(δΓσµν)

]. (15.51)

Applying then the Leibniz rule and relabelling some indices, we find

δΓSEH =

∫

Ωd4x√

|g|∇ν[gµνδΓσµσ − gµσδΓσµσ

]

+

∫

Ωd4x√

|g|[(∇νgµν)δΓσµσ − (∇νgµσ)δΓσµσ

]. (15.52)

We kept the second line, because we consider an arbitrary connection. Thus we do not knowyet if the covariant derivative of the metric vanishes.

Next we perform a partial integration of the first two terms, converting it into a surfaceterm which we can drop. In the remaining part we relabel indices so that we can factor outthe variation of the connection,

δΓSEH = −∫

Ωd4x√

|g|[δνρ∇σgµσ −∇ρgµν

]δΓρµν . (15.53)

We use now that the connection is symmetric in the absence of fermion. Then also thevariation δΓρµν is symmetric and the antisymmetric part in the square bracket drops out.Asking that δΓSEH = 0 gives therefore

1

2δνρ∇σgµσ +

1

2δµρ∇σgνσ −∇ρgµν = 0 (15.54)

or ∇σgµν = ∇σgµν = 0. Thus the Einstein-Hilbert action implies the metric compatibility ofthe connection.

301


Performing the same exercise with the Einstein-Hilbert plus the matter action consideredas functional of the vierbein eµm and the connection ωµ, one finds the following: From thevariation δωSEH one obtains automatically a metric connection which is however in generalnot symmetric. The torsion is sourced by the spin-density of fermions. The variation δeSEHgives the usual Einstein equation.

15.3. Linearized gravity

We are looking for small perturbations hµν around the Minkowski metric ηµν ,

gµν = ηµν + hµν , hµν ≪ 1 . (15.55)

These perturbations may be caused either by the propagation of gravitational waves or by thegravitational potential of a star. In the first case, current experiments show that we shouldnot hope for h larger than O(h) ∼ 10−22. Keeping only terms linear in h is therefore anexcellent approximation. Choosing in the second case as application the final phase of thespiral-in of a neutron star binary system, deviations from Newtonian limit can become large.Hence one needs a systematic “post-Newtonian” expansion or even a numerical analysis todescribe properly such cases.

Linearized Einstein equations in vacuum From ∂µηνρ = 0 and the definition

Γµνλ =1

2gµκ(∂νgκλ + ∂λgνκ − ∂κgνλ) . (15.56)

we find for the change of the connection linear in h

δΓµνλ =1

2ηµκ(∂νhκλ + ∂λhνκ − ∂κhνλ) =

1

2(∂νh

µλ + ∂λh

µν − ∂µhνλ) . (15.57)

Here we used η to raise indices which is allowed in linear approximation. Remembering thedefinition of the Riemann tensor,

Rµνλκ = ∂λΓµνκ − ∂dΓµνλ + ΓµeλΓ

λνκ − ΓµρκΓ

ρνλ , (15.58)

we see that we can neglect the terms quadratic in the connection terms. Thus we find for thechange

δRµνλκ = ∂κδΓµνκ − ∂κδΓµνλ

=1

2

∂λ∂νh

µκ + ∂λ∂κh

µν − ∂λ∂µhνκ − (∂κ∂νh

µλ + ∂κ∂λh

µν − ∂κ∂µhνλ)

=1

2

∂λ∂νh

µκ + ∂d∂

µhνλ − ∂λ∂µhνκ − ∂κ∂νhµλ. (15.59)

The change in the Ricci tensor follows by contracting a and c,

− δRνκ = δRλνλκ =1

2

∂λ∂νhλκ + ∂κ∂

λhνλ)− ∂λ∂λhνκ − ∂κ∂νhλλ

. (15.60)

Next we introduce h ≡ hµµ, = ∂µ∂µ, and relabel the indices,

− δRµν =1

2

∂µ∂ρh

ρν + ∂ν∂ρh

ρµ −hµν − ∂µ∂νh

. (15.61)

302


We now rewrite all terms apart from hµν as derivatives of the vector

ξµ = ∂νhνµ −

1

2∂µh , (15.62)

obtaining

− δRµν =1

2−hµν + ∂µξν + ∂νξµ . (15.63)

Looking back at the properties of hµν under gauge transformations, Eq. (4.44), we see that wecan gauge away the second and third term. Thus the linearized Einstein equation in vacuumbecomes simply

hµν = 0 (15.64)

if the harmonic gauge

ξµ = ∂νhνµ −

1

2∂µh = 0 , (15.65)

is chosen. Thus the familiar wave equation holds for all ten independent components of hµν ,and the perturbations propagate with the speed of light c. Inserting plane waves hµν =εµν exp(ikx) into the wave equation, one finds immediately that k is a null vector.

TT gauge We can recover our old result for the polaristion tensor describing the physicalstates contained in a gravitational perturbation. We consider a plane wave hµν = εµν exp(ikx).After fixing the harmonic gauge ∂µhµν = 0, we can still add four function ξµ with ξµ = 0.We can choose them such that four components of hµν vanish. In the TT gauge, we set(i = 1, 2, 3)

h0i = 0, h = 0 . (15.66)

The harmonic gauge condition becomes ξα = ∂βhβα or

ξ0 = ∂βhβ0 = ∂0h

00 = −iωε00eikx = 0 , (15.67)

ξa = ∂βhβa = ∂bh

ba = −ikbεabeikx = 0 . (15.68)

Thus ε00 = 0 and the polarization tensor is transverse, kbεab = 0. If we choose the planewave propagating in z direction, k = kez, the z raw and column of the polarization tensorvanishes too. Accounting for h = 0 and εαβ = εβα, only two independent elements are left,and we recover our old result,

ε =

0 0 0 00 ε11 ε12 00 ε12 −ε11 00 0 0 0

. (15.69)

In general, one can construct the polarization tensor in TT gauge by setting first the non-transverse part to zero and then subtracting the trace. The resulting two independent ele-ments are (again for k = kez) then ε11 = 1/2(εxx − εyy) and ε12.

303


Linearized Einstein equations with sources We rewrite first the Einstein equation in analternative form where the only geometrical term on the LHS is the Ricci tensor. Because of

R µµ −

1

2g µµ (R − 2Λ) = R− 2(R− 2Λ) = −R+ 4Λ = −κT µ

µ (15.70)

we can perform with T ≡ T µµ the replacement R = 4Λ − κT in the Einstein equation andobtain

Rµν = −κ(Tµν −1

2gµνT ) + gµνΛ . (15.71)

This form of the Einstein equations is often useful, when it is easier to calculate T that R.We note also that (15.71) implies that an empty universe with Λ = 0 is characterized by avanishing Ricci tensor Rµν = 0.

Now we move on the determination of the linearized Einstein equations with sources. Wefound 2δRµν = hµν . By contraction follows 2δR = h. Combining both terms gives

(

hµν −1

2ηµνh

)

= 2(δRµν −1

2ηµνδR)

= −2κδTµν . (15.72)

Since we assumed an empty universe in zeroth order, δTµν is the complete contribution tothe energy-momentum tensor. We omit therefore in the following the δ in δTµν .

We introduce as useful short-hand notation the “trace-reversed” amplitude as

hµν ≡ hµν −1

2ηµνh . (15.73)

The harmonic gauge condition becomes then

∂µhµν = 0 (15.74)

and the linearized Einstein equation in the harmonic gauge follow as

hµν = −2κTµν . (15.75)

Because of ¯hµν = hµν and Eq. (15.71), we can rewrite this wave equation also as

hµν = −2κTµν (15.76)

with the trace-reversed energy-momentum tensor Tµν ≡ Tµν − 12ηµνT .

Newtonian limit The Newtonian limit corresponds to v/c → 0 and thus the only non-zeroelement of the energy-momentum tensor becomes T tt = ρ. We compare the metric

ds2 = (1 + 2Φ)dt2 − (1− 2Φ)(dx2 + dy2 + dz2

)(15.77)

to Eq. (15.55) and find as metric perturbations

htt = 2Φ hij = 2δijΦ hti = 0 . (15.78)

In the static limit → −∆ and v = 0, and thus

−∆

(

h00 −1

2η00h

)

= −4∆Φ = −2κρ . (15.79)

Hence the linearized Einstein equation has the same form as the Newtonian Poisson equation,and the constant κ equals κ = 8πG.

304


Detection principle of gravitational waves Consider the effect of a gravitational wave ona free test particle that is initially at rest, ua = (1, 0, 0, 0). As long as the particle is at rest,the geodesic equation simplifies to ua = Γa00. The four relevant Christoffel symbols are inlinearized approximation, cf. Eq. (15.57),

Γa00 =1

2(∂0h

a0 + ∂0h

a0 − ∂ah00) . (15.80)

We are free to choose the TT gauge in which all component of hab appearing on the RHSare zero. Hence the acceleration of the test particle is zero and its coordinate position isunaffected by the gravitational wave: The TT gauge defines a “comoving” coordinate system.

The physical distance l of e.g. two test-particles is given by integrating

dl2 = gαβdξαdξβ = (hαβ − δαβ)dξαdξβ , (15.81)

where gαβ is the spatial part of the metric and dξ the spatial coordinate distance betweeninfinitesimal separated test particles. Hence the passage of a periodic gravitational wave,hab ∝ cos(ωt), results in a periodic change of the separation of freely moving test particles.The relative size of this change, ∆L/L, is given by the amplitude h of the gravitational wave.

Linearized action of gravity We have derived the linearized wave equation describing grav-itational waves directly from the Einstein equation. Now we want to obtain the linearizedaction of gravity. A straightforward but lengthy approach would be to expand the Einstein-Hilbert action to O(h3). Instead, we profit from our knowledge of the graviton propagator,which we derived in chapter 4 in Eq. (4.49),

Dµν;ρσF (k) =

12(−ηµνηρσ + ηµρηνσ + ηµσηνρ)

k2 + iε=Pµν;ρσ(k)

k2 + iε. (15.82)

The corresponding Lagrangian is as usually quadratic in the fields, 12h

µνPµν;ρσhρσ. Per-

forming partial integrations and the contractions with the metric tensors gives us as thecorresponding action

S =1

32πG

∫

d4x

[1

2(∂αhβγ)

2 − 1

4(∂αh)

2

]

. (15.83)

We know that the propagator of a massless particles with helicity h > 0 can be invertedonly, if either the gauge freedom is completely fixed or a gauge-fixing term is added. Incontrast to the TT gauge, the harmonic gauge contains still unphysical degrees of freedom.Thus the expression obtained corresponds to the quadratic Einstein-Hilbert Lagrangian inthe harmonic gauge plus a gauge-fixing term,

L(2)EH + Lgf =

1

32πG

[1

2(∂αhβγ)

2 − 1

4(∂αh)

2

]

= L(2)EH +

(

∂µhµν −1

2∂µh

)2

. (15.84)

Specializing (15.83) to the TT gauge, we obtain

SEH =1

32πG

∫

d4x1

2(∂µhij)

2 . (15.85)

305


We can express an arbitrary polarisation state as the sum over the polarisation tensors forcircular polarised waves,

hµν =∑

a=+,−h(a)ε(a)µν . (15.86)

Inserting this decomposition into (15.87) and using ε(a)µν εµν(b) = δab, the action becomes

STTEH =1

32πG

∑

a

∫

d4x1

2

(

∂µh(a))2

. (15.87)

Thus the gravitational action in the TT gauge consists of two scalar degree of freedom, h+ andh−, which determine the contribution of left- and right-circular polarised waves. Apart fromthe prefactor, the action is the same as the one of two scalar fields. This means that we canshort-cut many calculations involving gravitational waves by using simply the correspondingresults for scalar fields.

We can understand this equivalence by recalling that the part of the action action quadraticin the fields just enforces the relativistic energy-momentum relation via a Klein-Gordon equa-tion for each field component. The remaining content of (15.83) is just the rule how the un-physical components in hµν have to be eliminated. In the TT gauge, we have applied alreadythis information, and thus the two scalar wave equations for h(±) summarize the Einsteinequations at O(h2).

15.4. Wess-Zumino model and supersymmetry

We have noted already that a symmetry relating bosons and fermions could lead to a cancella-tion of zero-point energies and radiative corrections. However, the symmetry transformationswe considered up to now cannot achieve this since they do not change the spin of a particle.The way-out is to consider symmetries generated by Grassman variables.

We will consider the simplest case of single massless fermion and some scalar fields. Anon-shell massless fermion has two degrees of freedom, and we will therefore start to add twoscalar fields to match the on-shell fermionic degrees of freedom. Our initial Lagrangian is

L0 =1

2(∂µS)

2 +1

2(∂µP )

2 +1

4χγµ

↔∂ µ χ (15.88)

where we use Majorana spinors, i.e. the χ are Grassmann variables. For the following, it isusefeul to recall the flip properties of Majorana spinors derived in problem 5.9. They implyin particular that the vector current χγµχ vanishes. Thus the model has a chiral symmetryof χ and the standard phase symmetry in S and P :

χ→ eiβγ5χ , S + iP → eiβ(S + iP ) . (15.89)

A transformation that interchanges χ with S and P , χ → S,P , must be parametrized by aMajorana spinor parameter α. Moreover, the transformation has to account for the differentnumber of derivatives of bosonic and fermionic fields. In the ansatz

δ(S,P ) = αΓχ , (15.90)

the 4×4-matrix Γ in spinor space can be either 1 or γ5. In order to match the two real scalarfields to the complex spinor, we made the ansatz S + iP in (15.89). This suggests that S is a

306

A. Conventions and useful formula

A.4. Gamma function

Definition The Gamma function Γ(z) is defined as the generalization of the factorial n! withΓ(n) = (n− 1)!. Thus Γ(1) = 1 and zΓ(z) = Γ(z + 1). As first step of this generalization, weshow that

Γ(z) =

∫ ∞

0dt e−ttz−1 (A.29)

is an integral representation of the Gamma function valid in the positive half-plane ℜ z > 0:First, the RHS equals Γ(1) = 1. Second, we can rewrite this integral representation as

Γ(z + 1) = −∫ ∞

0dt

(d

dte−t)

tz =

∫ ∞

0dt ze−ttz−1 + e−ttz

∣∣∞0. (A.30)

The boundary term vanishes and thus the integral representation also satisfies the recurrencerelation Γ(z + 1) = zΓ(z).

Next we extend Γ(z) into the half-plane ℜ z > −1 by setting Γ1(z) = Γ(z + 1)/z. Thefunction Γ1(z) has a simple pole at z = 0. In the second step, we set Γ2(z) = Γ(z+2)/[z(z−1)],defining thereby the function Γ2(z) valid in the half-plane ℜ z > −2 with simple poles atz = 0 and -1. By induction, we can extend thus the Gamma function as an analytic functionin the whole complex plane except for simple poles at z = 0,−1,−2, . . ..

Euler’s beta function is defined by

B(a, b) ≡ Γ(a)Γ(b)

Γ(a+ b)=

∫ 1

0dt ta−1(1− t)b−1 =

∫ ∞

0dt

ta−1

(1 + t)a+b. (A.31)

The integral representations are valid for ℜ(a) > 0,ℜ(b) > 0. We can use the first one toderive the values of the Gamma function for half-integer arguments: Substituting t = sin2 ϑgives

Γ(a)Γ(b)

Γ(a+ b)= 2

∫ π/2

0dϑ (sinϑ)2a−1(cos ϑ)2b−1 , (A.32)

what results in Γ(1/2) =√π for a = b = 1/2. Using then the recurrence relation, we can

generate from Γ(1) = 1 and Γ(1/2) =√π the values for all (positive) integer and half-integer

arguments: Γ(n) = (n− 1)!, Γ(3/2) =√π/2, Γ(5/2) = 3

√π/4, etc.

Logarithmic derivative The (first) logarithmic derivative of the Gamma function is called

ψ1(z) =d ln Γ(z)

dz=

Γ′(z)Γ(z)

. (A.33)

We define moreover the special value

ψ1(1) =Γ′(1)Γ(1)

≡ −γ (A.34)

as the Euler-Mascheroni constant γ. Differentiating zΓ(z) = Γ(z+1) we obtain Γ(z)+zΓ′(z) =Γ′(z + 1) or

1 +zΓ′(z)Γ(z)

=Γ′(z + 1)

Γ(z)=zΓ′(z + 1)

Γ(z + 1). (A.35)

424

A.4. Gamma function

This gives us a recurrence relation for the psi function,

ψ1(z + 1) =1

z+ ψ1(z) , (A.36)

which becomes for the special case of integers n = 0, 1, 2, . . .

ψ1(n+ 1) =1

n+ ψ1(n) =

1

n+

1

n− 1+ . . . + ψ1(1) . (A.37)

Thus we can express the psi function for integer values through the Euler-Mascheroni constantand a finite harmonic serie,

ψ1(n+ 1) = −γ +

n∑

k=1

1

k. (A.38)

What is left to determine is the value of γ. Consider the asymptotic behavior of ψ1(x) forx→∞ using Stirling’s formula,

ln Γ(x+ 1) =

(

x+1

2

)

lnx− x+1

2ln 2π +O(x−1) . (A.39)

Differentiating this expression results in

ψ1(x+ 1) = lnx+1

2x+O(x−2) . (A.40)

In the limit x → ∞, we find thus ψ1(x + 1) = lnx. Inserting this equality into (A.38), weobtain an expression for the Euler-Mascheroni constant,

γ = limn→∞

(n∑

k=1

1

k− ln(n)

)

= 0.5772 . . . (A.41)

(Suprisingly it is not known, if this number is rational or not.)

Expansion near a pole We are now in the position to derive formula like (2.118) for theexpansion of the Gamma function around its poles. We start expanding Γ(1+ ε) around one,

Γ(1 + ε) = Γ(1) + εΓ′(1) +O(ε2) = 1 + εΓ(1)ψ1(1) +O(ε2) = 1− εγ +O(ε2) . (A.42)

From zΓ(z) = Γ(z + 1) it follows

Γ(ε) =Γ(1 + ε)

ε=

1

ε− γ +O(ε) . (A.43)

Applying zΓ(z) = Γ(z + 1) again, we obtain

Γ(−1 + ε) =−Γ(ε)1− ε = −(1 + ε− ε2 + . . .)

[1

ε− γ +O(ε)

]

= −1

ε− 1 + γ +O(ε) . (A.44)

For general n, the formula

Γ(−n+ ε) =(−1)nn!

[1

ε+ ψ1(n+ 1) +O(ε)

]

, (A.45)

holds for the expansion of the Gamma function near a pole.

425

Index

ΛQCD, 193ϑ vacuum, 2791P irreducible (1PI), 175

action principle, 3Adler-Bell-Jackiw anomaly, 271anomalous dimension, 185anomaly, 81, 175asymptotic freedom, 191, 193

basin of attraction, 185beta function, 185, 190Birkhoff theorem, 403biunitary transformation, 243black hole

entropy, 407ergosphere, 406Kerr, 404Schwarzschild, 395

Bogomolnyi bound, 260, 264Bogomolnyi-Parad-Sommerfield (BPS)

soliton, 264Bogulyubov coefficients, 350, 381Boltzmann equation, 331, 384

collisionless, 332, 384in FRW universe, 332, 385photons, 370

bounce solution, 253bracket

angle, 139square, 139

bremsstrahlung, 145

canonically normalisation, 34Casimir effect, 52causality, 49chirality, 111, 116Christoffel symbols, 18Clifford algebra, 104confinement, 263

conformal time, 314conformal transformation, 354

conformally flat, 352, 354

connection coefficients, 16

coordinatecyclic, 5

coordinates

Boyer-Lindquist, 404

comoving, 311Eddington-Finkelstein, 398

Kruskal, 401

Schwarzschild, 396

cosmic censorship, 400, 401, 407cosmological constant, 42, 300, 320, 321

cosmological principle, 311

covariant derivative, 15

CP problemstrong, 282

critical density, 321

cross section

Klein-Nishina, 137

Thomson, 137cross sections, 148

crossing symmetry, 134, 138

curvature scalar, 160

curveautoparallel, 17

dark matter

cold, 335, 387

hot, 335, 387Debye mass, 215

deceleration parameter, 318

decoupling limit, 174

degree of divergencesuperficial, 175

dimensional regularization, 62

dimensional transmutation, 193

Dirac representation, 107

445

Index

distanceangular diameter, 319coordinate, 318luminosity, 318proper, 318

domain wall, 259

effective potential, 228temperature-dependent, 251

Einstein-Hilbert action, 299energy

center-of-mass (cms), 54energy-momentum tensor

canonical, 77dynamical, 95

equivalence principle, 12, 91, 144, 295ergosphere, 406event horizon, 405

factorisation, 143fermion

Dirac, 115Majorana, 116Weyl, 116

Feynman parameter, 70Feynman slash, 104Fierz identities, 140, 423Fierz relations, 423fix point, 185fixed point critical, 202fixed point trivial, 202flip properties

Majorana spinor, 126, 306, 422formfactor, 170fragmentation function, 291Friedmann equation, 320functional integral, 30Furry’s theorem, 123, 177

gaugeRξ, 161, 164Coulomb, 165Feynman, 93, 162harmonic, 93, 303, 305Landau, 162Lorenz, 87radiation, 165transverse traceless, 91, 303

unitary, 162, 191

gauge transformationabelian, 86

generating functional

1PI, 196, 231connected, 47

disconnected, 47

Goldstone’s theorem, 225Gordon decomposition, 170

graph

loop, 56tree, 56

Green function

1PI, 196, 231connected, 47

disconnected, 47, 48

ground-state persistence amplitude, 33

Hamilton’s equations, 5, 20

Hamilton’s principle, 2

Harisson-Zel’dovich spectrum, 378Hawking radiation, 408

helicity, 94, 112

helicity spinor, 139Higgs effect, 234

homotopy group, 265

horizonevent, 325, 395

particle, 325

Hubble parameter, 314

Hubble’s law, 314

ideal fluid, 319

infinite redshift surface, 397inflaton, 363

infrared divergence, 145

instanton, 253, 346

scalar, 256interaction

irrelevant, 176, 177

marginal, 177non-normalisable, 176, 177

normalisable, 177

relevant, 176, 177super-normalisable, 176, 177

irreducible tensor, 84

Ising model, 201

446

Index

Jeans length, 331

Kaluza-Klein particles, 96Killing equation, 79Killing vector, 78

Klein-Gordon equation, 40in a FRW background, 356, 375

ΛCDM model, 324Landau pole, 186, 191Landau’s mean field theory, 199Legendre transformation, 5, 19

Levi-Civita connection, 18Liouville’s theorem, 6local operator, 43loop expansion, 230

loop number, 56LSZ formula, 182LSZ reduction formula, 132

Møller velocity, 148, 334, 386Majorana fermions, 116Mandelstam variables, 66, 421

matrix representation, 427Matsubara frequencies, 210, 251metric

Friedmann-Robertson-Walker, 313metric compatibility, 17

metric tensor, 15minimal coupling, 353misalignement mechanism, 393monopole

’t Hooft-Polyakov, 263

Bogomolnyi-Parad-Sommerfield(BPS), 264

magnetic, 262, 393

operatordimension n, 54

optical theorem, 128

Palatini formalism, 5

parallel transport, 16partial wave analysis, 246path integral, 30

fermion, 121scalar, 40

Yang-Mills, 163

Pauli-Lubanski spin vector, 78

Peceei-Quinn symmetry, 283, 392perturbation theory

optimised, 216

screened, 216photons

soft, 143

plus prescription, 291polarisation

CMB, 373

potentialCoulomb, 90

gravitational, 94

Yukawa, 46, 90power spectrum, 378

primitive divergent, 176

primitive graph, 175principle of equivalence, 12

principle of relativity, 431

Proca equation, 86propagator

fermion, 110

ghost, 164

graviton, 93non-abelian, 164

photon, 162

Proca, 88scalar, 43

recombination, 330

redshiftcosmological, 315

reducible tensor, 84

regularisation, 52renormalisation, 53

renormalization conditions, 180

representationadjoint, 428

fundamental, 428

Ricci tensor, xix, 159Riemann tensor, 159, 302

scattering

φφ→ φφ, 65e+e− → γγ, 142

e+e− → µ+µ−, 140Compton, 134

447

Index

Schrodinger equation, 25Schwarzschild metric, 396seesaw model, 118, 347singularity

coordinate, 395physical, 395

space-timestatic, 395stationary, 395

spacesmaximally symmetric, 312

sphaleron, 346spinor

adjoint, 106Dirac, 102Majorana, 116Weyl, 101

stationary limit surface, 406strings

cosmic, 260, 261supersymmetry, 51, 121symmetry

Peceei-Quinn, 282shift, 282

symmetry breaking, 222spontaneous, 221–237

symmetry factor, 57, 124, 150, 192

tensorreducible, 84

tensor method, 87thin-wall approximation, 254, 256time-ordered product, 32topological charge, 260

unitarityperturbative, 245

vacuum diagrams, 56vacuum energy density

renormalisation, 63

Ward-Takahashi, 198Ward-Takahashi identities, 196Weinberg angle, 241Weyl fermions, 116Weyl’s postulate, 311white hole, 400

Wick rotation, 33Wick’s theorem, 48winding number, 265

zero-point energyfermions, 121scalar, 51

448

Quantum Field Theoryweb.phys.ntnu.no/~mika/lect1.pdftime. In the formulation of quantum theory we...

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