Foundations of Quantum Field Theory [3 ed.] 9811221928, 9789811221927

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Foundations of Quantum Field Theory

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World Scientific Lecture Notes in Physics ISSN: 1793-1436 Published titles* Vol. 67: Quantum Scaling in Many-Body Systems M A Continentino Vol. 69: Deparametrization and Path Integral Quantization of Cosmological Models C Simeone Vol. 70: Noise Sustained Patterns: Fluctuations and Nonlinearities Markus Loecher Vol. 71: The QCD Vacuum, Hadrons and Superdense Matter (2nd ed.) Edward V Shuryak Vol. 72: Massive Neutrinos in Physics and Astrophysics (3rd ed.) R Mohapatra and P B Pal Vol. 73: The Elementary Process of Bremsstrahlung W Nakel and E Haug Vol. 74: Lattice Gauge Theories: An Introduction (3rd ed.) H J Rothe Vol. 75: Field Theory: A Path Integral Approach (2nd ed.) A Das Vol. 76: Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (2nd ed.) J A Gonzalo Vol. 77: Principles of Phase Structures in Particle Physics H Meyer-Ortmanns and T Reisz Vol. 78: Foundations of Quantum Chromodynamics: An Introduction to Perturbation Methods in Gauge Theories (3rd ed.) T Muta Vol. 79: Geometry and Phase Transitions in Colloids and Polymers W Kung Vol. 80: Introduction to Supersymmetry (2nd ed.) H J W Müller-Kirsten and A Wiedemann Vol. 81: Classical and Quantum Dynamics of Constrained Hamiltonian Systems H J Rothe and K D Rothe Vol. 82: Lattice Gauge Theories: An Introduction (4th ed.) H J Rothe Vol. 83: Field Theory: A Path Integral Approach (3rd ed.) Ashok Das Vol. 84: Foundations of Quantum Field Theory Klaus Dieter Rothe

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World Scientific Lecture Notes in Physics – Vol. 84

Foundations of Quantum Field Theory

Klaus D Rothe University of Heidelberg, Germany

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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World Scientific Lecture Notes in Physics — Vol. 84 FOUNDATIONS OF QUANTUM FIELD THEORY Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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To my wife, Neusa Maria, and my son, Thomas

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PREFACE Quantum Field Theory (QFT) emerged in the 1930’s as a natural extension of Quantum Mechanics to include Special Relativity and particle creation in its second quantized formulation. Over the years QFT has gone through an extensive evolution with regard to the role of the quantum fields involved, the range of applicability (elementary particles, phase transitions in solid state physics), the treatment of infinities (renormalization) resulting from its local structure (microcausality), the asymptotic behaviour of Green functions (Callan–Symanzik equation, asymptotic freedom), and the analyticity property of transition amplitudes (S-matrix theory). In particular one has learned why Quantum Electrodynamics (QED) is so successful at low energies, whereas perturbative Quantum Chromodynamics (QCD), the theory of the strong interactions, is successful at high energies (asymptotic freedom, deep inelastic scattering). One has further learned that phenomena such as spontaneous symmetry breaking observed in solid state physics (ferromagnet) also plays a role in the theory of weak interactions in particle physics, where it is referred to as “Higgs mechanism”. The present lectures essentially represent the content of a two-semester course held by the author at the University of Heidelberg, and thus provides an adequate time-frame for the lecturer and student. As such it was intended to be a compact book providing a bird’s eye view of the very basic foundations of QFT, including the traditional operator, as well as the more modern path integral approach, and should serve as a good basis for post-graduate students, and as orientation for lecturers. Very extensive treatises of the subject can be found in the still excellent book of Bjorken and Drell, as well as in more up-to-date books, such as by C. Itzykson and J.-B. Zuber, E. Peskin and D.V. Schroeder, Lewis H. Ryder and S. Weinberg, which have also served as a basis for these lectures.1 Aside from the author’s point of view in presenting, choosing and arranging the material, most of it can be found in some or other way in the existing literature. We have tried to present the material in reasonable detail, with emphasis on transparency and repeated cross references, at the expense of being sometimes pedantic. We therefore believe that the reader will be able to follow the material without engaging in detailed calculations, which are cumbersome at times. Though we have exemplified various regularization procedures (Pauli–Villars, Dimensional, Taylor-subtraction), we have dominantly used the traditional Pauli–Villars regularization as being the most intuitive one. We paid much attention in Chapter 2 to the Lorentz group and its representations in Hilbert-space, since they play a fundamental role in Chapters 3 and 5, where some knowledge of Group Theory on the part of the student is assumed. Much of these particular chapters is based on a series of remarkable articles by Steven Weinberg in Physical Review 1964, which underline the fundamental ideas behind the construction of a Quantum Field Theory from the operator point of view. 1 J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields (McGraw-Hill, 1965); C. Itzikson, and J.-B. Zuber Quantum Field Theory (McGraw-Hill, 1980); E. Peskin and D.V. Schroeder, Frontiers in Physics, 1995; Lewis H. Ryder, Quantum Field Theory (Cambridge University Press, 1985 and 1996); S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1996).

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Preface

Many interesting topics could not be covered in these lectures. Thus subjects such as the conformal group and analyticity of scattering amplitudes have only been marginally touched. In particular, Quantum Chromodynamics and the Weak Interactions were not included. They would have exceeded the intended size of this book, and have been left to other more extensive treatises. In turn, we left much room for the Callan–Symanzik equation, which has received considerable attention in the seventies, and will also be revisited in Chapter 18 with regard to the renormalization group. Although the renormalization group actually preceded chronologically the work of Callan and Symanzik, we have preferred to first present the latter, since it connects directly with the chapter on renormalization in Chapter 16. As for the figures, they were drawn with the aid of the program “METAFONT” developed by Thorsten Ohl and others.2 Although the diagrams are perhaps not as professional as those of publishing companies, the procedure to generate them with “METAFONT” is nevertheless of remarkable simplicity in limited cases. This is the reason for having restricted ourselves to presenting only examples of diagrams directly related to the text. Furthermore, only references related directly to the text were quoted. An extensive list of references can be found in the above cited books. I would like to thank Dr. Elmar Bittner for being always ready to help me with his expertise to solve computer related problems, and to Thorsten Ohl for helping me with some more complicated diagrams.

2 Thorsten

1996.

Ohl, CERN Computer Newsletter 220 April 1995, 221 October 1995 and December

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Contents Preface

vii

1

The Principles of Quantum Physics 1.1 Principles shared by QM and QFT . . . . . . . . . . . . . . . . . 1.2 Principles of NRQM not shared by QFT . . . . . . . . . . . . . .

2

Lorentz Group and Hilbert Space 2.1 Defining properties of Lorentz transformations . . . . . 2.2 Classification of Lorentz transformations . . . . . . . . . 2.3 Lie algebra of the Lorentz group . . . . . . . . . . . . . 2.4 Finite irreducible representation of L↑+ . . . . . . . . . . 2.5 Transformation properties of massive 1-particle states . 2.6 Transformation properties of zero-mass 1-particle states

. . . . . .

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9 9 12 14 17 20 23

Search for a Relativistic Wave Equation 3.1 A relativistic Schr¨ odinger equation . . . . . . . . . . . . 3.2 Difficulties with the wave equation . . . . . . . . . . . . 3.3 The Klein–Gordon equation . . . . . . . . . . . . . . . . 3.4 KG equation in the presence of an electromagnetic field

. . . .

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. . . .

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27 27 29 32 33

The Dirac Equation 4.1 Dirac spinors in the Dirac and Weyl representations 4.2 Properties of the Dirac spinors . . . . . . . . . . . . 4.3 Properties of the γ-matrices . . . . . . . . . . . . . . 4.4 Zero-mass, spin = 12 fields . . . . . . . . . . . . . . . 4.5 Majorana fermions . . . . . . . . . . . . . . . . . . .

. . . . .

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36 36 44 45 47 51

5

The Free Maxwell Field 5.1 The radiation field in the Lorentz gauge . . . . . . . . . . . . . . 5.2 The radiation field in the Coulomb gauge . . . . . . . . . . . . .

54 54 56

6

Quantum Mechanics of Dirac Particles 6.1 Probability interpretation . . . . . . . . . . . . . . . . . . . . . . 6.2 Non-relativistic limit . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Negative-energy solutions and localization . . . . . . . . . . . . .

58 58 61 62

3

4

ix

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1 1 3

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Contents 6.4 6.5

7

8

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The Klein Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . Foldy–Wouthuysen Transformation . . . . . . . . . . . . . . . . .

Second Quantization 7.1 Fock-space representation of fields 7.2 Commutation relations . . . . . . 7.3 P, C, T from equations of motion . 7.4 P, C, T in second quantization . .

. . . .

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64 67

. . . .

74 74 82 85 88

Canonical Quantization 8.1 Lagrangian formulation and Euler–Lagrange equations 8.2 Canonical quantization: unconstrained systems . . . . 8.3 Canonical quantization: constrained systems . . . . . 8.4 QED as a constrained system . . . . . . . . . . . . . .

. . . .

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. . . .

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. . . .

90 90 95 98 101

Global Symmetries and Conservation Laws 9.1 Noether’s Theorem . . . . . . . . . . . . . . 9.2 Internal symmetries . . . . . . . . . . . . . 9.3 Translational invariance . . . . . . . . . . . 9.4 Lorentz transformations . . . . . . . . . . .

. . . .

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. . . .

107 107 110 112 114

10

The Scattering Matrix 10.1 The S-matrix and T-matrix . . . . . . . . . . . . . . . . . . . . . 10.2 Differential cross-section . . . . . . . . . . . . . . . . . . . . . . . 10.3 LSZ reduction formula . . . . . . . . . . . . . . . . . . . . . . . .

117 118 122 127

11

Perturbation Theory 11.1 Interaction picture and U-matrix . . . . . . . . . . . 11.2 Interaction picture representation of Green functions 11.3 Wick theorems . . . . . . . . . . . . . . . . . . . . . 11.4 2-point functions . . . . . . . . . . . . . . . . . . . . 11.5 Feynman Diagrams for QED . . . . . . . . . . . . . 11.6 Furry’s theorem . . . . . . . . . . . . . . . . . . . . 11.7 Going over to momentum space . . . . . . . . . . . . 11.8 Momentum space Feynman rules for QED . . . . . . 11.9 Moeller scattering . . . . . . . . . . . . . . . . . . . 11.10 The Moeller differential cross-section . . . . . . . . . 11.11 Compton scattering . . . . . . . . . . . . . . . . . .

134 134 136 139 142 148 153 154 156 157 160 163

9

12

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Parametric Representation of a General Diagram 166 12.1 Cutting rules for a general diagram . . . . . . . . . . . . . . . . . 166 12.2 An alternative approach to cutting rules . . . . . . . . . . . . . . 172 12.3 4-point function in the ladder approximation . . . . . . . . . . . 175

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Contents 13

xi

Functional Methods 13.1 The Generating Functional . . . . . . . . . . . . . . . . . . . 13.2 Schwinger’s Construction of Z[j] . . . . . . . . . . . . . . . . 13.3 Feynman Path-Integral . . . . . . . . . . . . . . . . . . . . . 13.4 Path-integral representation of correlators in QM . . . . . . . 13.5 Feynman path-integral representation in QFT . . . . . . . . . 13.6 Path-Integral for Grassman-valued fields . . . . . . . . . . . . 13.7 Extension to Field Theory . . . . . . . . . . . . . . . . . . . . 13.8 Mathews–Salam representation of QED generating functional 13.9 Faddeev–Popov quantization and α-gauges . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

179 180 180 184 189 194 198 204 206 209

14

Dyson–Schwinger Equation 213 14.1 Classification of Feynman Diagrams . . . . . . . . . . . . . . . . 213 14.2 Basic building blocks of QED . . . . . . . . . . . . . . . . . . . . 215 14.3 Dyson–Schwinger Equations . . . . . . . . . . . . . . . . . . . . . 218

15

Regularization of Feynman Diagrams 15.1 Pauli–Villars and dimensional regularization 15.1.1 Electron self-energy . . . . . . . . . 15.1.2 Photon vacuum polarization . . . . 15.1.3 The vertex function . . . . . . . . .

16

17

. . . .

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220 220 222 226 231

Renormalization 16.1 The principles of renormalization . . . . . . . . . . . . 16.2 Renormalizability of QED . . . . . . . . . . . . . . . . 16.2.1 Fermion 2-point function . . . . . . . . . . . 16.2.2 Photon 2-point function . . . . . . . . . . . . 16.2.3 Vertex function . . . . . . . . . . . . . . . . . 16.3 Ward–Takahashi Identity and overlapping divergences 16.4 1-loop renormalization in QED . . . . . . . . . . . . . 16.5 Composite operators and Wilson expansion . . . . . . 16.6 Criteria for renormalizability . . . . . . . . . . . . . . 16.7 Taylor subtraction . . . . . . . . . . . . . . . . . . . . 16.8 Bogoliubov’s recursion formula . . . . . . . . . . . . . 16.9 Overlapping divergences . . . . . . . . . . . . . . . . . 16.10 Dispersion relations: a brief view . . . . . . . . . . . .

. . . . . . . . . . . . .

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236 237 240 242 244 245 246 251 255 256 260 263 264 267

Broken Scale Invariance and Callan–Symanzik Equation 17.1 Scale transformations . . . . . . . . . . . . . . . . . . . . 17.2 Unrenormalized Ward identities of broken scale invariance 17.3 Broken scale invariance and renormalized Ward identities 17.4 Weinberg’s Theorem . . . . . . . . . . . . . . . . . . . . . 17.5 Solution of CS equation in the deep euclidean region . . . 17.6 Asymptotic behaviour of Γ and zeros of the β-function . . 17.7 Perturbative calculation of β(g) and γ(g) in φ4 theory . . 17.8 QED β-function and anomalous dimension . . . . . . . . 17.9 QED β-function and leading log summation . . . . . . . .

. . . . . . . . .

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273 274 277 280 283 286 290 293 298 299

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Contents 17.10 Infrared fix point of QED and screening of charge . . . . . . . . 301

18

Renormalization Group 303 18.1 The Renormalization Group equation . . . . . . . . . . . . . . . 303 18.2 Asymptotic solution of RG equation . . . . . . . . . . . . . . . . 309

19

Spontaneous Symmetry Breaking 19.1 The basic idea . . . . . . . . . . . . . . . . . 19.2 More about spontaneous symmetry breaking 19.3 The Goldstone Theorem . . . . . . . . . . . . 19.4 Realization of Goldstone Theorem in QFT .

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311 311 313 316 318

Effective Potentials 20.1 Generating functional of proper functions 20.2 The effective potential . . . . . . . . . . . 20.3 The 1-loop effective potential of φ4 -theory 20.4 WKB approach to the effective potential 20.5 The effective potential and SSB . . . . . .

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321 321 324 325 327 329

20

Index

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335

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Chapter 1

The Principles of Quantum Physics Quantum Field Theory is a natural outgrowth of non-relativistic Quantum Mechanics, combining it with the Principles of Special Relativity and particle production at sufficiently high energies. We therefore devote this introductory chapter to recalling some of the basic principles of Quantum Mechanics which are either shared or not shared with Quantum Field Theory.

1.1

Principles shared by QM and QFT

We briefly review first the principles which non-relativistic Quantum Mechanics (NRQM), relativistic Quantum Mechanics (RQM) and Quantum Field Theory (QFT) have in common. (1) Physical states Physical states live in a Hilbert space Hphys and are denoted by |Ψi. (2) Time development In the Schr¨ odinger picture, operators OS are independent of time and physical states |Ψ(t)iS obey the equation, i¯ h

∂ |Ψ(t)iS = H|Ψ(t)iS , ∂t

∂t OS = 0

with H the Hamiltonian. In the Heisenberg picture physical states |ΨiH are independent of time and operators O(t)H obey the Heisenberg equation i¯ h∂t OH (t) = [OH (t), H], 1

∂t |Ψ >H = 0 .

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The Principles of Quantum Physics

The states in the two pictures are related by the unitary transformation |Ψ(t) >S = e−iHt |Ψ >H . (3) Completeness Eigenstates |Ψn > of H, H|Ψn >= En |Ψn > are assumed to satisfy the completeness relation X |Ψn >< Ψn | = 1 n

with n standing for a discrete or continuous label. (4) Observables To every observable corresponds a hermitian operator; however, not every hermitian operator corresponds to an observable. (5) Symmetries Symmetry transformations are represented in the Hilbert space H by unitary (or anti-unitary) operators. (6) Vector space The complete system of normalizable states |Ψi ∈ H defines a linear vector space. (7) Covariance of equations of motion: If S and S 0 denote two inertial reference frames, then covariance means that the equation ∂ i¯ h |Ψ(t)i = H|Ψ(t)i ∂t implies ∂ i¯ h 0 |Ψ0 (t0 )i = H|Ψ0 (t0 )i. ∂t Furthermore, there exists a unitary operator U which realizes the transformation S → S 0: U |Ψ(t)i = |Ψ0 (t)i (8) Physical states All physical states can be gauged to have positive energy1 E = hΨp |H|Ψp i, 1 In

|Ψp i ∈ Hphys

a relativistic theory, H can also contain negative energy states, which then require a particular interpretation or must decouple altogether from the “physical sector” of the theory.

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1.2 Principles of NRQM not shared by QFT

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3

(9) Space and time translations Space-time translations are realized on |Ψi respectively by2 ~

i

US (~a) = e+ h¯ ~a·P

resp.

i

UT (t) = e− h¯ tH

where P~ is the momentum operator, and rotations are realized on |Ψi by i

~

Urot (~ α) = e+ h¯ α~ ·L

[Li , Lj ] = i¯hijk Lk ,

123 = 1 ,

~ the generator of rotations with L ~ = ~r × p~ . L

1.2

Principles of NRQM not shared by QFT

The following principles of non-relativistic quantum mechanics must be abandoned in the case of QFT: (1) Probability amplitude In NRQM we associate with the state |Ψ(t)i a wave function ψ(~r, t) = h~r|Ψ(t)i . 2

|Ψ(~r, t)| d3 r then represent the probability of finding a particle in the interval [~r, ~r +d~r] at time t. (Notice the treatment of space and time on unequal footing.) In QFT we can have particle production, that is, we are dealing with “many-particle” physics. Hence notions linked to a one particle picture must be abandoned in the relativistic case. (2) Galilei transformations For a scalar function ψ(~r, t) and a Galilei transformation ~r0 = ~r − ~v t, t0 = t we must have ψ 0 (~r0 , t) = phase × ψ(~r, t) = phase · ψ(~r0 + ~v t, t). 2 In

Quantum Mechanics i

~

h~ r|e+ h¯ ~a·P |~ pi = h~ r + ~a|~ pi , i

¯~ h ~ |~ h~ r |P r0 i = ∇ r−~ r0 ) r δ(~ i

~

e+ h¯ ~a·P |~ ri = |~ r − ~ai h~ r|e

iα ~ ~ ·L h ¯

|Ψi =

Z

i

~

~ ·L d3 r0 h~ r|e+ h¯ α |~ r 0 ih~ r 0 |Ψi

¯~ h ~ r 0 i = (~ h~ r|L|~ r× ∇ r−~ r 0) r )δ(~ i i

~

~ ·L |~ ri = |R−1 (~ α)~ ri, e+ h¯ α

(R−1 ~ r)i = (R−1 )ij rj .

In this chapter we use everywhere lower indices, repeated indices being summed over.

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The Principles of Quantum Physics

Consider in particular a plane wave ψp~ (~r, t) in S as seen by an observer in S 0 moving with a velocity ~v with respect to S: i

0

p ~02

0

ψp~0 (~r0 , t) = e h¯ (~p ·~r − 2m t) , where We have

~r0 = ~r − ~v t,

p~0 = p~ − m~v . i

ψ 0 (~r0 , t) = ψp~−m~v (~r0 , t) = η(t)e− h¯ m~v·~r ψp~ (~r, t)

with

~ v2 2

i

η(t) = e h¯ m

t

.

(1.1) (1.2)

We seek an operator U~v (t) with the property U~v (t)|Ψp~ (t)i = |Ψp~−m~v (t)i. From and

ψp~−m~v (~r0 , t) = h~r0 |U~v (t)|Ψp~ (t)i ~

h~r0 |eit~v·P = h~r0 + ~v t| = h~r|

we conclude, by comparing with (1.1),

i

~

i

~

U~v (t) = η(t)e h¯ t~v·P e− h¯ m~v·R

(1.3)

~ is the position operator. Now where R [Ri , Pj ] = i¯hδij . Hence we may write U~v (t) in the form i

~

~

U~v (t) = e h¯ [~v·(tP −mR)] where we have used

(1.4)

1

eA eB = e 2 [A,B] eA+B . Denoting by |~ pi the eigenstates of the momentum operator P~ |~ pi = p~|~ pi we have For the solution

i

~

e− h¯ m~v·R |~ pi = |~ p − m~v i.

(1.5)

~2 i p

|Ψp~ (t)i = |~ pie− h¯ 2m t

of the “free” Schr¨ odinger equation i¯h∂t |ψp~ (t) = H0 |ψp~ (t)i we obtain from (1.3), i

~2 i p

~

U~v (t)|Ψp~ (t)i = η(t)e h¯ t~v·P |~ p − m~v ie− h¯ 2m t p−m~ v )2 i (~ 2m

= |~ p − m~v ie− h¯

t

,

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1.2 Principles of NRQM not shared by QFT

5

or U~v (t)|Ψp~ (t)i = |Ψp~−m~v (t)i .

(1.6)

Furthermore, making use of e−B AeB = A + [A, B], we have i

~

e+ h¯ m~v·R

for

[A, B] = c − number,

P~ 2 − i m~v·R~ (P~ − m~v )2 e h¯ = 2m 2m

or U~v−1 (t)H0 U~v (t) = with H0 =

(P~ − m~v )2 2m

P~ 2 . 2m

Correspondingly we have from (1.7) E00 = hΨp~0 (t)|H0 |Ψp~0 (t)i = hΨp~ (t)|U~v−1 (t)H0 U~v (t)|Ψp (t)i (P~ − m~v )2 (~ p − m~v )2 |Ψp~ (t)i = , = hΨp~ (t)| 2m 2m in accordance with expectations. (3) Covariance of equations of motion From H0 |Ψp~ (t)i = i¯h∂t |Ψp~ (t)i follows U~v (t)H0 |Ψp~ (t)i = i¯hU~v (t)∂t |Ψp (t)i , which we rewrite as
(U~v (t)H0 U~v−1 (t))U~v (t)|Ψp (t)i = i¯h U~v (t)∂t U~v−1 (t) U~v (t)|Ψp (t)i + i¯h∂t (U~v (t)|Ψp (t)i) .

Noting from (1.3) and (1.2) that i¯ hU~v (t)∂t U~v−1 (t) =

1 ~ m~v 2 + ~v · P~ 2

we obtain, using (1.6), 1 [U~v (t)H0 Uv−1 (t) − ( m~v 2 + ~v · P~ )]|Ψp~−m~v (t)i = i¯h∂t |Ψp~−m~v (t)i , 2 or, recalling (1.7), P2 |Ψp~−m~v (t)i = i¯h∂t |Ψp~−m~v (t)i. 2m

(1.7)

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The Principles of Quantum Physics

This equation expresses on operator level the covariance of the free-particle equation of motion: If |ψp~ (t)i is a solution of the equations of motion, then |ψp~−m~v(t) i is also a solution. Group-property As Eq. (1.4) shows, a Galilei transformation is represented by the unitary operator ~

U~v (t) = ei~v·K with

~ = (tP~ − mR) ~ K

the generators of boosts. We have [Ki , Kj ] = [tPi − mRi , tPj − mRj ] = 0 , so that different “boosts” commute with each other. The Galilei transformations thus correspond to an abelian Lie group. In particular U~v0 (t)U~v (t) = U~v0 +~v (t) . Boosts Let S and S 0 be two inertial frames whose clocks are synchronized in such a way, that their respective origins coincide at time t = 0. Then we have for an eigenstate of the momentum operator, as seen by observers O and O0 in S and S 0 , O : |~ pi

i

~

pi = |~ p − m~v i O0 : U~v (0)|~ pi = e− h¯ m~v·R |~ respectively. In particular, for a particle at rest in system S we obtain, from the point of view of O0 , O0 : U~v (0)|~0i = | − m~v i . Define

i

~

U−~v (0) =: U [B(~ p)] = e h¯ p~·R ,

where B(~ p) stands for a Galilei transformation taking p~ = 0 → p~ = m~v and E = p ~2 0 → E = 2m = 12 m~v 2 . U [B(~ p)] is thus an operator which takes a particle at rest into a particle with momentum p~. One refers to this as a “boost” (active point of view). We have the following property of Galilei transformations not shared by Lorentz transformations (compare with (2.23)): boosts and rotations separately form a group. Indeed one easily checks that [Ki , Kj ] = 0 ,

[Ki , Lj ] = 0 .

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1.2 Principles of NRQM not shared by QFT

page 7

7

(4) Causality We next want to show that NRQM violates the principle of causality. We have for any interacting theory, i

0

|Ψ(t0 )i = e− h¯ (t −t)H |Ψ(t)i.

Let |En i be a complete set of eigenstates of H:3

H|En i = En |En i X |En ihEn | = 1 . n

Then h~r0 |Ψ(t0 )i = =

X 0 i h~r0 |En ihEn |Ψ(t)ie− h¯ (t −t)En n

XZ n

Define

i

0

d3 rh~r0 |En ihEn |~rih~r|Ψ(t)ie− h¯ (t −t)En .

h~r|En i = ϕn (~r),

X n

ϕn (~r0 )ϕ∗n (~r) = δ 3 (~r0 − ~r)

h~r|Ψ(t)i = ψ(~r, t) , as well as

X

i

0

ϕn (~r0 )ϕ∗n (~r)e− h¯ En (t −t) := K(~r0 , t0 ; ~r, t).

(1.8)

n

The kernel K(~r, t; ~r0 , t0 ) satisfies a heat-like equation: (i¯h∂t − H)K(~r, t; ~r0 , t0 ) = 0

K(~r, t; ~r0 , t0 )|t=t0 = δ 3 (~r − ~r0 ). In terms of this kernel we have from above, Z 0 0 ψ(~r , t ) = d3~r K(~r0 , t0 ; ~r, t)ψ(~r, t) .

(1.9)

(1.10)

We now specialize to the case of a free point-like particle. In that case  p ~2  En → E(~ p) = 2m P 2 ~ P 3 H0 = ⇒ n →d p i  2m 1  h~r|ϕn i → ϕp (~r) = e h¯ p~·~r (2π¯ h)3/2

and correspondingly we have with (1.8), Z ~2 0 0 i i p 1 K0 (~r0 , t0 ; ~r, t) = d3 pe h¯ p~·(~r −~r) e− h¯ 2m (t −t) 3 (2π¯h)  3/2 r 0 −~ r )2 i m (~ m h ¯ 2 t0 −t = . e 2π¯hi(t0 − t) 3 For

notational simplicity we suppose the spectrum to be discrete.

(1.11)

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The Principles of Quantum Physics

Notice that the kernel K0 satisfies the desired initial condition (1.9). From (1.10), for the initial condition ψ(~r, t0 ) = δ 3 (~r − ~r0 ), we get ψ(~r, t) =



m 2πi¯h(t − t0 )

or in particular P(~r, t) = |ψ(~r, t)|2 =

3/2



r −~ r0 )2 i m (~ 2 (t−t0 )

e h¯

m 2π¯h(t − t0 )

3

,

,

that is, for an infinitesimal time after t0 one already finds the particle with equal probability anywhere in space; this violates obviously the principle of relativity, as well as causality.

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page 9

Chapter 2

Lorentz Group and Hilbert Space In this chapter we first discuss the realization of the homogeneous Lorentz transformations in four-dimensional space-time, as well as the corresponding Lie algebra. From here we obtain all finite dimensional representations, and in particular the explicit form of the matrices representing the boosts for the case of spin = 1/2, which will play a fundamental role in Chapter 3. The Lorentz transformation properties of massive and zero-mass 1-particle in Hilbert space (and their explicit realization in Chapter 9) lie at the heart of the Fock space representation (second quantization) in Chapter 9. It is assumed that the reader is already familiar with the essentials of the Special Theory of Relativity and of Group Theory.

2.1

Defining properties of Lorentz transformations

Homogeneous Lorentz transformations are linear transformations on the space-time coordinates,1 xµ → x0µ = Λµν xν ,

(xµ ) = (ct, ~r),

µ = 0, 1, 2, 3

(2.1)

leaving the quadratic form ds2 = c2 dt2 − d~r2

(2.2)

invariant. Any 4-tuplet transforming like the coordinates in (2.1) is called a contravariant 4-vector. In particular, energy and momentum of a particle are components of a 4-vector p (pµ ) = (ω(~ p), p~) = ( p~2 + m2 c2 , p~), p~ = γ(v)m~v (2.3)

1 The inhomogeneous Lorentz transformations including the space-time translations will be discussed in Chapter 9. We adopt the convention that repeated upper and lower indices are to be summed over.

9

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Lorentz Group and Hilbert Space

with γ(v) = q

1 1−

v2 c2

.

The same transformation law defines 4-vector fields at a given physical point: µ

V 0 (x) = (Λ−1) )µν V ν (Λx) .

(2.4)

Note that the quadruple (Λx)µ referred to S is the same point as the quadruple xµ referred to S 0 . Thus, alternatively V 0µ (x0 ) = Λµν V ν (x) .

(2.5)

Notice that (2.4) and (2.5) represent inverse transformations of the reference frame, respectively. Examples are provided by the 4-vector current j µ = (cρ, ~j) and the ~ of electrodynamics in a Lorentz-covariant gauge. vector potential Aµ = (φ, A) The differential element dxµ transforms like  0µ  ∂x dxν . dx0µ = ∂xν Hence it also transforms like a contravariant 4-vector, since ∂x0µ = Λµν . ∂xν The partial derivative ∂x∂ µ , on the other hand, transforms differently. The usual chain rule of differentiation gives  ν ∂x ∂ ∂ = . 0µ 0µ ∂x ∂x ∂xν From the inversion of (2.1) it follows that  ν ∂x = (Λ−1 )νµ . ∂x0µ Hence for the partial derivative ∂µ =

∂ ∂xµ

(2.6)

we have the transformation law

∂µ0 = ∂ν (Λ−1 )νµ .

(2.7)

Four-tuples which transform like the partial derivative are called covariant 4-vectors. Contravariant and covariant 4-vectors are obtained from each other by raising and lowering the indices with the aid of the metric tensors g µν and gµν , defined by2   1  −1   , g2 = 1 ||gµν || = ||g µν || =    −1 −1 2 This

chapter is largely based on the papers by S. Weinberg in Physical Review. Note that Weinberg uses the metric g µν = (−1, 1, 1, 1).

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2.1 Defining properties of Lorentz transformations

page 11

11

respectively, in terms of which the invariant element of length (2.2) can be written in the form ds2 = gµν dxµ dxν . The requirement that ds2 be a Lorentz invariant gµν dx0µ dx0ν = gλρ dxλ dxρ now implies

gµν Λµλ Λνρ = gλρ .

(2.8)

Thus the metric gµν is said to be a Lorentz-invariant tensor. It is convenient to write this equation in matrix notation by grouping the elements Λµν into a matrix as follows:  0 0 0 0  Λ0Λ1Λ2Λ3  Λ10 · · ·   Λ=  · · · · . · · · Λ33

Defining the elements of the transpose matrix ΛT by (ΛT )µν = Λνµ we can write (2.8) as follows: ΛT gΛ = g . From here we obtain for the inverse Λ−1 , Λ−1 = gΛT g , or in terms of components

(Λ−1 )µν = g µλ Λρλ gρν = Λνµ .

(2.9)

Define the dual to a contravariant 4-vector v µ by vµ = gµν v ν .

(2.10)

Thus gµν (g µν ) serve to lower (raise) the Lorentz indices. In particular g µλ gλν = g µν = δ µν . We have after a Lorentz transformation, upon using (2.9) vµ0 = vν (Λ−1 )νµ

(2.11)

or we conclude that vµ defined by (2.10) does indeed transform like a covariant 4-vector. In particular we see that the following 4-tuplets transform like covariant and contravariant 4-vectors, respectively: ~ , (∂µ ) = (∂0 , ∇) where ~ = ∇



~ , (∂ ν ) = (∂0 , −∇)

∂ ∂ ∂ , , ∂x1 ∂x2 ∂x3



.

(2.12)

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Lorentz Group and Hilbert Space

2.2

Classification of Lorentz transformations

The Lorentz invariance of the scalar product vµ v µ = v 2 allows us to divide the 4-vectors into three classes which cannot be transformed into each other by a Lorentz transformation: (a) v µ time-like (v 2 > 0) (b) v µ space-like (v 2 < 0) (c) v µ light-like (v 2 = 0) This means in particular that space-time separates, as far as Lorentz transformations are concerned, into three disconnected regions referring to the interior and exterior of the light cone x2 = 0, as well as to the surface of the light cone itself. The trajectory of a point particle localized at the origin of the light cone at time t = 0 lies within the forward light cone; Moreover, if we attach a light cone to the particle at the point where it is momentarily localized, the tangent to the trajectory at that point does not intersect the surface of that light cone. The Lorentz invariant r r r ds2 d~r2 v2 2 = dt − = dt 1 − dτ = c2 c2 c2 taken along the trajectory of the particle is just the proper time, measured in the rest frame of the particle. From (2.8) follow two important properties of Lorenz transformations: (i) det Λ = ±1 X (ii) (Λ00 )2 − (Λi0 )2 = 1 , i

implying Λ00 ≥ 1

or

Λ00 ≤ −1 .

Note that this allows for four types of transformations which cannot be smoothly connected by varying continuously the parameter labelling the transformation. We thus have four possibilities characterizing the Lorentz invariance of the differential element (2.2): (a)

L↑+ : det Λ = 1, Λ00 ≥ 1

(b)

L↑− : det Λ = −1, Λ00 ≥ 1

(improper, orthochrone LT )

(c)

L↓+ : det Λ = 1, Λ00 ≤ −1

(proper, non-orthochrone LT )

(d)

L↓− : det Λ = −1, Λ00 ≤ −1

(proper, orthochrone LT )

(improper, non-orthochrone LT )

Only the first set of transformations is smoothly connected to the identity and hence form a Lie group. The remaining transformations do not have the group property.

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2.2 Classification of Lorentz transformations

page 13

13

They are obtained by adjoining to the transformations in L↑+ space reflections, spacetime reflections and time inversion, respectively, as represented by the matrices       1 0 −1 0 −1 0 ΛP = , ΛP T = , ΛT = . 0 −1 0 −1 0 1 Only L↑+ represents an exact symmetry of nature. General form of a Lorentz boosts From (2.3) one has for a boost in the z-direction, taking the particle from rest to a momentum p~ = γ(v)mvˆ e3 ,3   γ(v) 0 0 βγ(v)  0 1 0 0   (2.13) L(mvˆ e3 ) =   0 0 1 0  , βγ(v) 0 0 γ(v)

where4

1 ω(¯ p) γ(v) = p = , mc 1 − β2

Since

β=

v |~ p| = . c ω(~ p)

γ 2 − (βγ)2 = 1 p ω(~ p) = p~2 + m2 c2 ,

we may parametrize γ and βγ as follows: γ(v) =

ω(~ p) = cosh θ, mc

βγ =

|~ p| = sinh θ, mc

tanh θ =

|~ p| ω(~ p)

(2.14)

or (2.13) now reads 

cosh θ  0 L(|p|ˆ e3 ) =   0 sinh θ

3 Lorentz 4 We

 0 sinh θ 0 0   . 1 0  0 cosh θ

(2.15)

transformations representing a boost to momentum p ~ we denote by L(~ p). have ω(p) = mc =

p

p

p2 + m2 c2 mc

=

γ 2 (v)β 2 + 1 =

p

γ 2 (v)m2 v 2 + m2 c2

r

mc β2 +1= 1 − β2

|~ p| |~ p| γ(v)m|~v | v = = = ω(p) mcγ(v) mcγ(v) c .

0 1 0 0

p

1 1 − β2

= γ(v)

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Lorentz Group and Hilbert Space

Note that this is a hermitian matrix! For a boost in an arbitrary direction one can show that the corresponding matrix elements are given by (note that g ij = δij , pˆj = −ˆ pj ) L00 (~ p) = cosh θ Li0 (~ p) = L0i (~ p) = pˆi sinh θ i i L j (~ p) = g j − pˆi pˆj [cosh θ − 1]

2.3

(2.16)

Lie algebra of the Lorentz group

Consider an infinitesimal Lorentz transformation in 3+1 dimensions. It is customary to parametrize its matrix elements as follows. In the case of L↑+ we are dealing with a six-parameter group parametrized by the “velocities” associated with boosts, and the Euler angles associated with the rotations. This is just the number of independent components of an antisymmetric second rank tensor. In analogy to the rotation group it is customary to write for the matrix elements of an infinitesimal Lorentz transformation Λµν ' g µν + ω µν ,

ωµν = −ωνµ .

Using the metric tensor as raising and lowering operators for the indices, we further have ω ij = −ω ji = −ωij

ω i0 = ω 0i = −ωi0 = ω0i . We may rewrite this transformation in matrix form as follows i Λ ' 1 + ωµν Mµν , 2

(2.17)

where Mµν denote the generators associated with the transformations in L↑+ , with the property Mµν = −Mνµ and the matrix elements5 (Mµν )λρ = −i g µλ g νρ − g νλ g µρ




.

(2.18)

Here again it is implied that Lorentz indices can be raised and lowered with the aid of the metric tensor g µν = gµν . Expression (2.17) is just the infinitesimal expansion of µν i Λ = e 2 ωµν M . (2.19)
ωµν 1 5 λ µλ ν λ λ νλ µ λ λ λ λ Λ

ρ

'g

ρ

+

2

g

g

ρ

−g

g

ρ

'g

ρ

+ 2 (ω

ρ

− ωρ ) = g

ρ

+ ω ρ.

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page 15

2.3 Lie algebra of the Lorentz group

15

Explicitly we have

M01



 0100 1 0 0 0  = −i  0 0 0 0 , 0000

M02



 0010 0 0 0 0  = −i  1 0 0 0 , 0000

M03



 0001 0 0 0 0  = −i  0 0 0 0 . 1000

Hence for a pure Lorentz transformation in the −z-direction (boost in the +z direction)   0001 0 0 0 0  (2.20) Λ ' 1+θ 0 0 0 0 , 1000 where we have set

ω03 = θ . Equation (2.20) just represents the first two leading terms of the expansion of the finite Lorentz boost (2.15) in the x3 -direction around θ = 0. We thus identify M0i with the generators for pure velocity transformations along the z-axis. Notice that these matrices are anti-hermitian. On the other hand, the hermitian matrices Mij generate rotations in the (ij)-plane: M12

 0 0 00 0 0 1 0  = −i   0 −1 0 0  , 0 0 00 

M23

 00 0 0 0 0 0 0  = −i  0 0 0 1 , 0 0 −1 0 

M31

 000 0  0 0 0 −1   = −i  0 0 0 0  010 0 

the generators of rotations about the z-, x- and y-axis respectively. One verifies from (2.18) that the generators of Lorentz transformations satisfy the Lie algebra [Mµν , Mλρ ] = −i(g µλ Mνρ + g νρ Mµλ − g νλ Mµρ − g µρ Mνλ ) .

(2.21)

Explicitly [Mij , Mkl ] = −i(g ik Mjl + g jl Mik − g il Mjk − g jk Mil ) [Mi0 , Mj0 ] = −iMij [Mj0 , Mkl ] = −i(g jk M0l − g jl M0k ) . This algebra simplifies if we define the generators Ji =

1 ijk Mjk , 2

Ki = Mi0 ,

J~† = J~ ,

~ † = −K ~ K

(2.22)

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Lorentz Group and Hilbert Space

which now satisfy the simple commutation relations6,7 [Ji , Jj ] = iijk Jk [Ji , Kj ] = iijk Kk

(2.23)

[Ki , Kj ] = −iijk Jk reminiscent of the rotation group in four dimensions (except for the minus sign). This is not surprising, since the Lorentz transformations are connected to the four-dimensional rotation group by an analytic continuation in the parameters parametrizing the boosts, to pure imaginary values. The underlying Minkowski character of the space-time manifold is hidden in the anti-hermiticity of the operators Mi0 generating the boosts. Correspondingly we have for a finite transformation L↑+ in space-time ~

~ ~

Λ = ei~α·J−iθ·K ,

Λ† 6= Λ−1 ,

(2.24)

where we have made the identifications ωij = ijk αk ,

ω0i = θpˆi .

(2.25)

with θ the angle labelling the boost (2.16). For a pure rotation in the ij-plane, 

 00 0 0 0 0 0 0  Mij = i   0 ijk Jk  , 0

(2.26)

where Jk is the generator of rotations around the k-axis (i, j, k taken cyclicly), with the explicit realization     0 0 −1 0 0 0 J1 = −i  0 0 1  , J2 = −i  0 0 0  0 −1 0 10 0   0 10 J3 = −i  −1 0 0  . 0 00

Next, it will be our aim to obtain higher dimensional representations of the generators of the Lorentz group. 6 These commutation relations do not fix the sign of the generator K of boosts. We follow in i (2.22) the convention adopted by S. Weinberg. 7 The minus sign in the last commutation relations reflects the fact that the Lorentz group SO(3, 1) can be considered as the complexification of the rotation group in four dimensions, SO(4). The fact that the commutator of two generators of the boost is given as a linear combination of the generators of rotations is related to the phenomenon of the Thomas precession. From the group theoretic point of view it expresses the fact that, if we perform the sequence of infinitesimal boosts ~ − 1 (δ θ~ · K) ~ 2 + · · ·, the effect is just an g(−δθ2 )g(−δθ1 )g(δθ2 )g(δθ1 ) with g(δθ) = 1 + iδ θ~ · K 2 infinitesimal rotation in the frame we started from.

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2.4 Finite irreducible representation of L↑+

2.4

page 17

17

Finite irreducible representation of L↑+

In order to obtain a characterization of the irreducible representation L↑+ , we define the new operators ~ = 1 (J~ + iK) ~ , A 2 ~ , ~ = 1 (J~ − iK) B 2

~ = (A1 , A2 , A3 ) , A ~ = (B1 , B2 , B3 ) . B

(2.27)

Note that A†i = Ai and Bi† = Bi (see (2.22)). It then follows from the commutation relations (2.23) that [Ai , Aj ] = iijk Ak [Bi , Bj ] = iijk Bk [Ai , Bj ] = 0 .

(2.28)

~ and B ~ thus satisfy two Lie algebras which are decoupled from The operators A ~ 2 and B ~ 2 , which each other. Correspondingly, there exist two Casimir operators, A commute with all elements of the group. Their eigenvalues thus serve to classify the irreducible representations of L↑+ . In the irreducible basis, the matrix elements of ~ and B ~ are evidently given by the well-known expressions known the generators A from the rotation group: (A) ~ hA, a0 ; B, b0 |A|A, a; B, bi = δb0 b J~a0 a (B)

~ hA, a0 ; B, b0 |B|A, a; B, bi = δa0 a J~b0 b , where a and b take the values 1 A = 0, , 1, ... 2 1 b = −B, −B + 1, ..., B ; B = 0, , 1, ... 2

a = −A, −A + 1, ..., A ;

In particular, we have (A)

(B)

(J3 )a0 a = aδa0 a , (J3 )b0 b = bδb0 b p (A) (A) (J1 ± iJ2 )a0 a = A(A + 1) − a(a ± 1) δa0 ,a±1 p (B) (B) (J1 ± iJ2 )b0 b = B(B + 1) − b(b ± 1) δb0 ,b±1 .

~ these matrix elements read from (2.27) For the operators J~ and iK (A) (B) ~ a; B, bi = : J~(A,B) 0 ~ 0 ~ hA, a0 ; B, b0 |J|A, a0 b0 ;ab = δb b Ja0 a + δa a Jb0 b .

(A) (B) ~ ~ (A,B) 0 ~ 0 ~ hA, a0 ; B, b0 |iK|A, a; B, bi = : iK a0 b0 ;ab = δb b Ja0 a − δa a Jb0 b .

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Lorentz Group and Hilbert Space

From these matrix elements we then obtain the corresponding representations of finite Lorentz transformations L↑+ by simple exponentiation, in a way analogous to (2.24), ~ ~ ~ (A,B) Da0 b0 ,ab [Λ] =< A, a0 ; B, b0 |ei~α·J−iθ·K |A, a; B, b > , or compactly

~ (A,B) } , D(A,B) [Λ] = exp{i~ α · J~(A,B) − iθ~ · K

with

θ~ = θpˆ ,

where pˆ denotes the direction of the boost. The matrices D(A,B) exhaust all possible finite dimensional representations of L↑+ . They are non-unitary except for the trivial representation D(0,0) . Example 1 ~ Consider the representation (A, B) = (j, 0). In that representation the operator B ~ = −iJ. ~ This possibility is allowed by the commutation is realized by zero or K relations (2.28). One has ~ ~(j,0)

D(j,0) [Λ] = ei(~α+iθ)·J with

(2.29)

(j,0) ~ a; 0, 0i . J~a0 a = hj, a0 ; 0, 0|J|j,

Example 2 ~ = 0 or K ~ = iJ, ~ and the Consider the representation (A, B) = (0, j). In that case A Lorentz transformations are realized by ~ ~(0,j)

D(0,j) [Λ] = ei(~α−iθ)·J

(2.30)

~ with the corresponding matrix elements for J: (0,j) ~ 0; j, bi . J~b0 b = h0, 0; j, b0 |J|0,

The above representations will play a central role in the chapters to follow. We observe that D(j,0) [Λ]† = D(0,j) [Λ−1 ] . We introduce the following notation8 D(j) [Λ] := D(j,0) [Λ] , ¯ (j) [Λ] := D(0,j) [Λ] . D

(2.31)

¯ (j) [L(~ D p)] = D(j) [L(−~ p)] ,

(2.32)

We then have in particular

8 We

follow closely the notation of S. Weinberg, Phys. Rev. 133 (1964) B1318.

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2.4 Finite irreducible representation of L↑+ as well as

19

¯ (j) [Λ] = D(j) [Λ−1 ]† . D

For a pure rotation

The case j =

page 19

(2.33)

¯ (j) [R] = D(j) [R] . D

1 2

For the case of j = 1/2 it is easy to give an explicit expression for the matrices 1 ¯ ( 12 ) [L(~ D( 2 ) [L(~ p)] and D p)] representing a boost, since in that case the matrices representing J~ are just one half of the Pauli matrices (4.4): θ

1

ˆσ D( 2 ) [L(~ p)] = e− 2 p·~ ,

θ ˆσ ¯ ( 12 ) [L(~ D p)] = e 2 p·~ .

(2.34)

Using (ˆ p · ~σ )2 = 1 we have ˆσ e−θp·~ = cosh θ − (ˆ p · ~σ ) sinh θ ,

(2.35)

or recalling (2.14), we may write this also as ˆσ e−θp·~ =

ω(~ p) − p~ · ~σ . mc

We thus conclude that θ

ˆσ ˆ σ 1/2 e− 2 p·~ = (e−θp·~ ) =



ω(~ p) − p~ · ~σ mc

1/2

.

Defining tµ = (1, ~σ ) ,

t¯µ = (1, −~σ )

we may thus write the matrices (2.34) in the compact form D

( 21 )

[L(~ p)] =



t·p mc

1/2

,

¯ ( 21 ) [L(~ D p)] =



t¯ · p mc

1/2

.

(2.36)

These matrices will play a central role in our discussion of the Dirac equation in Chapter 4. Their explicit form is most easily obtained by returning to (2.35) and noting that θ θ θ ˆσ e− 2 p·~ = cosh − (ˆ p · ~σ ) sinh . 2 2 Now, r r θ 1 ω(~ p) + mc cosh = (cosh θ + 1) = 2 2 2mc r r r θ ω(~ p) − mc ω(~ p) + mc |~ p| 2 θ sinh = (cosh − 1) = = . 2 2 2mc 2mc ω(~ p) + mc

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Lorentz Group and Hilbert Space

Hence D

( 12 )

r

[L(~ p)] =

Similarly ¯ ( 12 )

D

[L(~ p)] =

ω(p) + mc 2mc

r

ω(p) + m 2m

 1−

p~ · ~σ ω(p) + mc



p~ · ~σ ω(p) + mc

1+



.

(2.37)



.

(2.38)

These explicit expressions will prove useful in our discussion of the Dirac equation in Chapter 4. For the rest of these lectures we set c = 1.

2.5

Transformation properties of massive 1-particle states

In analogy to the Galilei transformations discussed in Chapter 1, we take U [L(~ p)] to be the unitary operator taking the state |s, σ > of a particle of spin s, sz = σ at rest into a 1-particle state of momentum p~.9 r p m (2.39) U [L(~ p)]|s, σi, ω(~ p) = p~2 + m2 |~ p, s, σi = ω(~ p) where with normalization

|s, σi := |~ p = 0, s, σi

(2.40)

hs, σ 0 |s, σi = δσ0 σ .

(2.41)

Note that the spin of a particle at rest is a well defined quantity, whereas for a moving relativistic particle this is not the case. The kinematical factor introduced in (2.39) compensates for the non-covariant normalization of the 1-particle states: h~ p, s, σ 0 |~ p, s, σi = δσ0 σ δ 3 (~ p 0 − p~).

(2.42)

The form of this kinematical factor can be motivated in the following way: The normalization (2.42) of the 1-particle states corresponds to the completeness relation XZ d3 p|~ p, s, σih~ p, s, σ| = 1 . σ

Now, d3 p is not a relativistically invariant integration measure, whereas d3 p/ω(~ p) is. Indeed, making use of the usual properties of the Dirac delta-function we have Z Z Z d3 p 3 f (ω(~ p), p~) = d p dp0 θ(p0 )δ((p0 )2 − ω 2 (~ p))f (p0 , p~) . 2ω(~ p) 9 We follow again closely the notation of S. Weinberg, Phys. Rev. 133 (1964) B1318 and Phys. Rev. 134 (1964) B882. It is to be kept in mind that, unlike us, S. Weinberg uses the metric gµν = (−1, 1, 1, 1).

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21

The delta-function insures the proper energy momentum relation for a free particle, p2 = m2 , while the theta-function insures that the vector pµ is time-like, that is, the particle has positive energy. Both properties are preserved by Lorentz transformations in L↑+ . Furthermore, d4 p is a Lorentz-invariant measure since 0µ ∂p 4 0 d p = ν d4 p ∂p

and

0µ ∂p ∂pν = det Λ = 1 .

We thus conclude that

d3 p0 d3 p = . 2ω(~ p 0) 2ω(~ p)

(2.43)

This explains roughly the origin of the kinematical factor in (2.39).10 Now let U [Λ] be the unitary operator inducing a Lorentz transformation on the 1-particle state |~ p, s, σ 0 i. Using the group property of Lorentz transformations L↑+ , we have U [Λ]|~ p, s, σi = =

 

m ω(~ p) m ω(~ p)

1/2 1/2

U [Λ]U [L(~ p)]|s, σi ~ ~ U [L(Λp)]U [L−1 (Λp)ΛL(~ p)|s, σi .

(2.44)

It is easy to see that the matrix ~ RW = L−1 (Λp)ΛL(~ p)

(2.45)

is not equal to one unless Λ represents a pure boost colinear with p~. In general RW represents a pure rotation — the so-called Wigner rotation — in the rest frame of the particle. We may thus make use of the completeness relation X |s, σihs, σ| = 1 , σ

valid in the rest frame of the particle in order to write (2.44) in the form U [Λ]|~ p, s, σi = = 10 For





m ω(~ p) m ω(~ p)

1/2 X σ0

1/2 X σ0

~ U [L(Λp)]|s, σ 0 ihs, σ 0 |U [RW ]|s, σi (s)

~ U [L(Λp)]|s, σ 0 iDσ0 σ [RW ] ,

a more detailed analysis see E. P. Wigner, Ann. Phys. Math. 40 (1939) 149.

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Lorentz Group and Hilbert Space

where we have made the identification (s)

hs, σ 0 |RW |s, σi = Dσ0 σ [RW ] with D(s) [RW ] a (2s + 1)-dimensional irreducible representation of the rotation group. We thus finally have U [Λ]|~ p, s, σi =

s

~ X ω(Λp) ~ s, σ 0 iD(s) |Λp, σ 0 σ [RW ] . ω(~ p) 0

(2.46)

σ

At this point we can now firmly establish the correctness of our choice of normalization factor in (2.46). To this end we start from the completeness relation σ=s X

σ=−s

Z

d3 p |~ p, s, σ >< p~, s, σ| = 1 ,

(2.47)

and multiply this relation from the left with U [Λ], and from the right with U −1 [Λ]: σ=s X

σ=−s

Z

d3 p U [Λ]|~ p, s, σ >< p~, s, σ| U −1 [Λ] = 1 .

We now make use of (2.46) in order to rewrite this relation as X Z

σ 0 ,σ 00

d3 p

s X ~ ω(Λp) (s)? (s) ~ s, σ 00 | = 1 . Dσ0 σ [RW ]Dσ00 σ [RW ] < Λp, |Λ~ p, s, σ 0 > ω(~ p) σ=−s

Making use of the unitarity of the matrix representation of the rotation group, we have X (s) (s)? Dσ0 σ [RW ]Dσ00 σ [RW ] = δσ0 σ00 . σ

Hence we obtain from above XZ

d3 p

σ

~ ω(Λp) ~ s, σ >< Λp, ~ s, σ| = 1 . |Λp, ω(~ p)

Recalling the transformation property of the integration measure, Eq. (2.43), the above expression reduces to XZ

~ Λp, ~ s, σ >< Λp, ~ s, σ| = 1 , d3 (Λp)|

σ

showing that our choice of normalization is consistent with the Lorentz covariance of the completeness relation.

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2.6 Properties of zero-mass 1-particle states

2.6

23

Transformation properties of zero-mass 1-particle states

We are now in the position of discussing the Lorentz transformation properties of zero-mass particle states. The transformation rules have been completely worked out by E. Wigner.11 In the case of zero mass particles we can no longer go into the rest frame of the particle to define a general state in terms of a Lorentz boost. In fact, it is well known that a massless particle of spin j is polarized either along or opposite to its direction of motion, corresponding to two possible helicity states. If parity is not conserved, there may exist but one helicity state, as is exemplified by the neutrino (anti-neutrino) with negative (positive) helicity. Correspondingly we expect these helicity states to transform under a one-dimensional representation, independent of the spin of the particle. Following Wigner, we choose for our “standard” state a particle moving in the positive z-direction with four-momentum k˜µ = |κ, 0, 0, κ >, ˜ λi. These states replace the states |s, σi in the massive (k˜2 = 0) and helicity λ : |k, case. Whereas the states |s, σi belong to a representation of the rotation group, ˜ λi furnish a representation of the little group, a subgroup of the helicity states |k, the Lorentz group consisting of all homogeneous proper Lorentz transformations leaving our standard 4-vector k˜ invariant. In analogy to the massive case, we define the state of a massless particle of ˜ λ > into the desired arbitrary momentum p~ by “boosting” the standard state |k, new state: r µ ˜ λi , U [L(~ p)]|k, (2.48) |~ p, λi = |~ p| where U [L(~ p)] is the unitary operator corresponding to the Lorentz transformation L(p) which takes our standard four-momentum k˜µ into pµ , pµ = Lµν (~ p)k˜ν ,

(2.49)

and µ in (2.48) is an arbitrary parameter with the dimensions of a mass. There are various ways of defining L(~ p); we shall make the choice12 Lµν (~ p) = Rµλ (ˆ p)B λν (|~ p|) .

(2.50)

Here B λν (|p|) is a “boost” along the z-axis with non-zero components (compare with (2.16) B 11 (|~ p|) = B 22 (|~ p|) = 1

B 33 (|~ p|) = B 00 (|~ p|) = cosh φ(|~ p|)

B 30 (|~ p|) = B 03 (|~ p|) = sinh φ(|~ p|) . 11 E.P. 12 We

Wigner, Theoretical Physics (International Atomic Energy Vienna, 1963) p. 59. follow again the notation of S. Weinberg, Phys. Rev. 134 (1964) B882.

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Lorentz Group and Hilbert Space

To determine φ(|~ p|) we observe that

so that

    1 |~ p|     0  0 Bνµ (|~ p|)k˜ν = κeφ  0 =  0  1 |~ p| φ(|~ p|) = ln



|~ p| κ



.

(2.51)

We choose R(ˆ p) as the rotation (say, in the plane containing p~ and the z-axis) into the unit vector pˆ. The kinematic factor (µ/|~ p|)1/2 in (2.48) is inserted because of our choice of non-relativistic normalization of the states h~ p 0 , λ0 |~ p, λi = δ 3 (~ p − p~ 0 )δλλ0 . In order to obtain the transformation law of the states (2.48) under a general Lorentz transformation, we now proceed in a way analogous to that followed in Section 2.5. We thus have r µ ˜ λi U [Λ]U [L(~ p)]|k, (2.52) U [Λ]|~ p, λ > = |~ p| r µ ˜ λi . ~ ~ = U [L(Λp)]U [L−1 (Λp)ΛL(~ p)]|k, |~ p| ~ But the transformation L−1 (Λp)ΛL(~ p) leaves our standard vector (κ, 0, 0, κ) invariant and hence belongs to the little group. Indeed, ~ ~ L−1 (Λp)ΛL(~ p)k˜ = L−1 (Λp)Λp = k˜ .

(2.53)

Hence the four-by-four matrix ˜ = L−1 (Λp)ΛL(~ ~ Λ p)

(2.54)

belongs to the “little group” of the Lorentz Group. The corresponding unitary ˜ λi, and ˜ thus also does not change the momentum of the state |k, operator U [Λ] must induce the following linear transformation on the massless 1-particle state ˜ λ >: |k, X ˜ λi = ˜ λ0 idλ0 λ [Λ], ˜ k, ˜ ˜ µν k˜ν = k˜µ , U [Λ]| |k, Λ λ0

˜ is an irreducible representation of the little group (compare with (2.46)). where d[Λ] Correspondingly we have from (2.52) r X µ ~ λ0 > dλ0 λ [Λ] ˜ . U [Λ]|~ p, λ >= |Λp, (2.55) |~ p| 0 λ

In order to obtain the representation matrices of the little group, we must examine the nature of the transformations, leaving our standard vector k˜µ = (κ, 0, 0, κ)

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25

invariant. Since the little group is a subgroup of the Lorentz group, the representa˜ are obtained as a special case of the representation matrices for tion matrices d[Λ] a general Lorentz transformation, discussed in Section 2.4. It suffices to look at an infinitesimal transformation of the form ˜ µ ≈ gµ + ω Λ ˜ µν ν ν

ω ˜ µν = −˜ ω νµ ,

,

where the infinitesimal parameters are now required to satisfy ω ˜ µν k˜ν = 0 .

(2.56)

Inspection of (2.56) shows that Ωµν is in general a function of three parameters θ, χ1 and χ2 with the non-zero components given by ω ˜ 12 = θ

or

ω ˜ 01 = −˜ ω31 = χ1 , ω ˜ 02 = ω ˜ 23 = χ2

(2.57)

 0 χ1 χ2 0   ˜ µν ) ≈  χ1 0 −θ −χ1  . (Λ  χ2 θ 0 −χ2  0 χ1 χ2 0

(2.58)



The unitary operator acting on the states is correspondingly given by i

µν

˜ = e 2 ω˜ µν M U [Λ]

,

where Mµν are the generators of Lorentz transformations satisfying the Lie algebra (2.21). Using (2.22) we then have for an infinitesimal transformation, ˜ ≈ 1 + iθJ3 + iχ1 (K1 − J2 ) + iχ2 (K2 + J1 ) , U [Λ] or using (2.27), ˜ ≈ 1 + iθJ3 − (χ1 + iχ2 )A− + (χ1 − iχ2 )B+ , U [Λ]

(2.59)

where J3 = A3 + B3 , and where A− and B+ stand for A− = A1 − iA2

,

B+ = B1 + iB2 .

Since [J3 , A− ] = −A−

,

[J3 , B+ ] = B+

we see that B+ and A− act as raising and lowering operators for the eigenvalues of J3 , respectively. The eigenvalues of the helicity operator J3 = A3 + B3 of a general state |A, a; B, b > are λ = a + b. In nature a massless spin j particle only exists in two helicity states with helicity λ = +j (right handed) or λ = −j (left handed). For a transformation of the little group not to change this helicity we therefore demand for such an helicity state A− |λ >= 0 ,

B+ |λ >= 0 .

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Lorentz Group and Hilbert Space

This leads to the identification |λ >= |A, −A; B, B >

with

λ=B−A .

A spin-j massless particle thus transforms under the (0, j), ( 12 , j + 12 ), · · · (right handed) or (j, 0), (j + 12 , j), · · · (left handed) representation of the little group. It then follows from (2.59) by exponentiation, that (A− and B+ annihilate the state) −1 ˜ λ0 |U [L−1 (Λp)Λ ˜ λi = δλ0 λ exp{iλΘ[L−1 (Λp)ΛL(~ ˜ = hk, ~ ~ dλλ0 [Λ] L(~ p)]|k, p)]} , (2.60) the phase Θ(θ, χ1 , χ2 ) being some more or less complicated function of the little group parameters, which reduces to θ in (2.57) for infinitesimal transformations. It must satisfy the group property

˜ 1Λ ˜ 2 ] = Θ[Λ ˜ 1 ] + Θ[Λ ˜ 2] . Θ[Λ Hence finally we have from (2.55), s ~ |Λp| ˜ ~ eiλΘ[Λ] |Λp, λi, U [Λ]|~ p, λi = |~ p|

˜ = L−1 (Λp)ΛL(~ ~ Λ p)) .

(2.61)

(2.62)

This result will play an important role when we proceed to discuss the quantization of the electromagnetic field in Chapter 7.

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Chapter 3

Search for a Relativistic Wave Equation In this chapter we engage in the search for a relativistic wave equation for a spin zero particle moving in an external potential, reducing to the ordinary Schr¨odinger equation in the non-relativistic limit. We begin by looking at the case of a free particle. We adhere to the familiar quantum mechanical principles, as long as we can. We shall encounter a number of difficulties which will lead us to eventually abandon the usual probability interpretation.

3.1

A relativistic Schr¨ odinger equation

As discussed in Chapter 1, the time evolution of a quantum mechanical state is governed by the equation of motion (¯h = c = 1) i∂t |Ψ(t) >= H|Ψ(t) > ,

where H is the Hamiltonian operator of the system obtained from the corresponding classical Hamiltonian by representing the canonically conjugate dynamical variables q and p in H(q, p) by operators satisfying canonical commutation relations. In this and the following two sections we consider the case of a free particle. Since in a relativistic theory the relation between the energy of a free particle and its p momentum is given by E = p~2 + m2 , it is natural to take for the Hamiltonian p H = p~2 + m2 .

In the coordinate representation, the canonical commutation relations are as usual realized by ¯h ~ p~ → ∇ . i Defining the wave function associated with the state |Ψ > by φ(~r, t), one arrives thus at the wave equation q ~ 2 + m2 φ(~r, t) . i∂t φ(~r, t) = −∇ (3.1) 27

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Search for a Relativistic Wave Equation

This equation in turn implies that φ(~r, t) is also a solution of the Klein–Gordon (KG) equation ( + m2 )φ(~r, t) = 0 , (3.2) where denotes the D’Alembert operator. Since this operator is manifestly Lorentzinvariant, covariance of the physical laws demand that under a Lorentz transformation the wave function transforms like a scalar: φ0 (~r 0 , t0 ) = φ(~r, t) .

(3.3)

This is also implied by Eq. (3.1), since on the space of solutions of the Klein–Gordon equation (3.2) the operator appearing on the left- and right-hand sides of Eq. (3.1) transform in the same way: q q ~ 2 + m2 = ( + m2 )2 − ∂ 2 → i∂0 . −∇ 0

Note that in contrast to the non-relativistic case, where the wave function does not transform like a scalar, but rather picks up a phase under Galilei transformations, such a phase is absent in the relativistic case, rendering the covariance of the equation of motion manifest in this case. The operator appearing on the rhs of Eq. (3.1) is defined in terms of its Taylor expansion: q ~2 ~ 2 + m2 = m − ∇ + · · · −∇ 2m ~

Hence its eigenfunctions are given by the plane waves eip·~r , with the corresponding p 2 2 eigenvalues p~ + m . As a consequence, the action of this operator on any absolute integrable function f (~r, t) in R3 is defined via its Fourier transform 1 φ(~r, t) = (2π)3/2 as

q

~ 2 + m2 φ(~r, t) = −∇

Z

Z

d3 p ˜ φ(~ p, t)ei~p·~r 2ω(~ p)

d3 p p 2 ˜ p, t)ei~p·~r . p~ + m2 φ(~ 2ω(~ p)

The factor 1/2ω(~ p) has been included in the measure to make it relativistically invariant. In order for φ(~r, t) to be a solution of Eq. (3.1), we must further have ˜ p, t) = a(~ φ(~ p)e−iω(~p)t where ω(p) = form

p p~2 + m2 . Hence, the most general solution to Eq. (3.2) has the Z 1 d3 p φ(~r, t) = a(~ p)ei(~p·~r−ω(~p)t) . (3.4) 2ω(~ p) (2π)3/2

In order for φ(~r, t) to be a Lorentz scalar, the Fourier coefficients will themselves have to be Lorentz scalars: a0 (~ p 0 ) = a(~ p) . (3.5)

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3.2 Difficulties with the wave equation

29

In the non-relativistic limit ω(~ p) = m +

p~2 + O(~ p4 ) , 2m

so that we recover the Schr¨ odinger 1-particle wave function of the non-relativistic theory. Since the argument of the exponential appearing in the Fourier integral is not Galilei-invariant, the corresponding non-relativistic field φnr (~r, t) does not transform as a scalar. In fact, each Fourier component will pick up a phase such as witnessed in Eq. (1.1) of Chapter 1. We therefore see that in this respect the transformation law in the relativistic case is simpler than in the case of Galilei transformations.

3.2

Difficulties with the wave equation

As we now show, the wave equation (3.1) presents a number of problems which will eventually lead us to abandon it. Locality The wave equation (3.1) has an unwanted property: In order to determine the change in the solution at the point ~r in an infinitesimal time interval (t, t + dt), we must know the function for time t at all points ~r. This property is contained in the Taylor expansion of the Hamilton operator in powers of the Laplacian and is referred to as non-locality. It is also evident from the form of the solution (3.4) by noting that Z d3 p p 2 idt p~ + m2 a(~ p)ei(~p·~r−ω(~p)t) + · · · φ(~r, t + dt) ' φ(~r, t) − 3/2 2ω(~ p) (2π) which upon using the Taylor expansion in p2 becomes " # ~2 ∇ φ(~r, t + dt) ' φ(~r, t) − idt m − + · · · φ(~r, t) + · · · 2m

Probability interpretation In order for the wave function to have the interpretation of a probability amplitude, it should have at least the following properties: (i) Its normalization with respect to some suitable integration measure should be independent of time, as well as of the choice of inertial frame. (ii) The associated probability density should be positive semi-definite. The second condition is satisfied by defining the probability density by P(~r, t) = |φ(~r, t)|2 .

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Search for a Relativistic Wave Equation

Furthermore, adopting the following definition for the scalar product Z (ϕ, χ)t = d3 rϕ∗ (~r, t)χ(~r, t) , and making use of the equation of motion (3.1), one finds that the normalization of the wave function is preserved in time: d (φ, φ)t = 0 . dt The probability of finding a particle anywhere in space should, however, also be independent of the choice of inertial frame. Since the wave function φ(~r, t) transforms like a scalar (see Eq. (3.3)) this is not the case due to the Lorentz non-invariance of the measure d3 r. For the case of a free particle this defect is easily repaired by adopting a new definition for the scalar product: Z Z ↔ (ϕ, χ) = i d3 r(ϕ∗ ∂0 χ − ∂0 ϕ∗ · χ) =: i d3 rϕ∗ ∂0 χ . (3.6) Indeed, making use of the Fourier decomposition (3.4) for a free particle wave function, one computes Z d3 p |a(~ p)|2 . (φ, φ) = 2ω(~ p) Because of the invariance property (2.43) of the integration measure, and the scalar property (3.5) of the Fourier amplitude, we conclude that the normalization of the wave function is a Lorentz invariant with respect to the scalar product (3.6). In fact, one easily shows that it has all the properties expected from a scalar product: (ϕ, χ)t = (ϕ, χ)∗t (ϕ1 + ϕ2 , χ)t = (ϕ1 , χ)t + (ϕ2 , χ)t , and (φ, φ)t > 0 for solutions of the equation of motion. The scalar product (φ, φ)t is also timeindependent, as one easily checks using the equation of motion (3.2): Z
d (φ, φ) = i d3 r φ∗ ∂02 φ − (∂02 φ∗ )φ dt ZV   ~ 2 φ − (∇ ~ 2 φ∗ )φ =i d3 r φ∗ ∇ ZV   ~ · φ∗ ∇φ ~ − (∇φ ~ ∗ )φ =i d3 r∇ ZV   ~ − (∇φ ~ ∗ )φ → 0 =i df~ · φ∗ ∇φ ∂V

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31

for ∂V → ∞ and φ → 0 at infinity. In fact, the probability density satisfies a continuity equation analogous to that in non-relativistic Quantum Mechanics: ~ · ~j(~r, t) = 0 , ∂0 P(~r, t) + ∇ where

↔

P = iφ∗ ∂0 φ ,

~ − (∇φ ~ ∗ )φ) . ~j = −i(φ∗ ∇φ

We may in fact collect P(~r, t) and j(~r, t) into a 4-vector as follows: ↔

j µ (~r, t) = (P(~r, t), ~j(~r, t)) = i(φ∗ ∂ µ φ) ↔

←

where ∂ µ = ∂~ µ − ∂ µ . Indeed, under a Lorentz transformation we have (compare with (2.5)) ↔

j 0µ (x0 ) = iφ0∗ (x0 ) ∂ 0µ φ0 (x0 ) ↔

= iφ∗ (x)Λµν ∂ ν φ(x) = Λµν j ν (x) . Although we have succeeded in satisfying the requirement (i), this is not the case as far as requirement (ii) is concerned. As a simple example shows, the probability density is not a positive semi-definite quantity. To see this, consider the superposition of two plane waves for two free particles of mass m: φ = ce−ik·x + de−iq·x ,

k 2 = q 2 = m2

with k · x = kµ xµ , etc. Let c and d be real. One then has ↔

φ∗ i ∂ 0 φ = 2c2 ωk + 2d2 ωq + 2cd(ωk + ωq ) cos((k − q) · x) . Now, for every 4-vector (k − q)µ there exists a vector xµ such that (k − q) · x = 0. For this vector (~k − ~q) · ~x . x0 = ωk − ωq

We may then rewrite the above expression as    ↔ 2ωk ωk + ωq c φ∗ i ∂ 0 φ = (c, d) . ωk + ωq 2ωq d

The matrix appearing here is hermitian, and may thus be diagonalized, its eigenvalues being q λ± = (ωk + ωq ) ± (ωk − ωq )2 + (ωk + ωq )2 .

Hence, though the total probability is positive, the density in space-time is not. ↔ Since λ− < 0, there always exist coefficients such that φ∗ i ∂ 0 φ < 0, which proves our claim.

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3.3

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Search for a Relativistic Wave Equation

The Klein–Gordon equation

One tentative way out to save Lorentz covariance in the presence of interaction would be to treat space and time from the outset on equal footing by working with a differential equation of second order in space as well as time. This equation should nevertheless contain the solution discussed previously. Noting that q q ~ 2 + m2 )(−i∂t + −∇ ~ 2 + m2 ) = + m2 (3.7) (i∂t + −∇ one is thus led in the absence of interaction, to the Klein–Gordon equation (

+ m2 )φ = 0 .

(3.8)

The general solution of this equation is now given by φ(x) = φ(+) (x) + φ(−) (x)

(3.9)

where

Z 3 d k (±) ~ i(~k·~r∓ωk t) 1 φ (~r, t) = a (k)e . 3/2 2ω (2π) k The solution (3.9) thus represents in general two wave packets moving away from each other with time. If we choose the Fourier amplitudes a(+) (k) and a(−) (k) to be concentrated around ~k = ~k0 , then these two wave packets will separate from each other with twice the group velocity ! ∂ω(~k) |~k0 | vg = = , ∂|~k| ω(~k0 ) (±)

~ k=~ k0

which could be interpreted as a two-particle state. Since q (−) (−) ~ 2 + m2 , i∂t φ = −H0 φ , H0 = −∇

we may regard the solution φ(−) as solutions of the Schr¨odinger equation for negative energy. These negative energy solutions do not fit into our probabilistic interpretation, since with the scalar product (3.6), Z 3 Z ↔ d k (−) (−) (−) 3 (−)∗ (−) (φ , φ ) = d rφ (~r, t) ∂ 0 φ (~r, t) = − |a (k)|2 < 0 . 2ωk

The negative energy solutions of the KG equation thus carry negative norm with respect to the scalar product (3.6). In the free case we may nevertheless ignore their existence, since they satisfy the orthogonality property (φ(+) , φ(−) ) = 0 and as a result we have no mixing of positive and negative energy solutions: Z 3 d k (+) ~ 2 (φ, φ) = (|a (k)| − |a(−) (~k)|2 ) . 2ωk

Thus we may restrict ourselves to the positive energy sector of the theory. This will, however, no longer be true if we allow for interactions with an external potential, which will induce transitions between positive and negative energy states.

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3.4

page 33

33

KG equation in the presence of an electromagnetic field

The interaction of a charge q with an external electromagnetic field is introduced in the Klein–Gordon equation by the usual minimal substitution ∂µ → ∂µ + iqAµ ,

(3.10)

~ is related to the electric field E in the usual way: where Aµ = (Φ, A) ~ + ∂t A ~ = −∇Φ ~ . E This leads us to consider the equation of motion (Dµ Dµ + m2 )φ = 0

(3.11)

Dµ = ∂µ + iqAµ .

(3.12)

with the covariant derivative

It is important to realize that unlike the free particle case, this equation can no longer be factorized in the form (3.7): (Dµ Dµ + m2 ) 6= (H + i∂0 )(H − i∂0 ) , where H is the Hamiltonian for a relativistic particle moving in an external electromagnetic field: q ~ 2 + m2 + qΦ H = −(∂~ + iq A)

Eq. (3.11) is covariant under the following gauge transformation Aµ → A0µ = Aµ + ∂µ Λ ,

(3.13)

where Λ(x) denotes an arbitrary function of x. Indeed, under this transformation Dµ → Dµ0 = ∂µ + iqAµ + iq∂µ Λ , or equivalently Dµ → Dµ0 = e−iqΛ(x) Dµ eiqΛ(x) .

(3.14)

In particular D0µ Dµ0 = e−iqΛ(x) Dµ Dµ eiqΛ(x) . Hence defining the gauge-transformed wave function φ0 (x) by φ(x) → φ0 (x) = e−iqΛ(x) φ(x)

(3.15)

Eq. (3.14) implies 0

(D µ Dµ0 + m2 )φ0 (x) = 0 .

(3.16)

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Search for a Relativistic Wave Equation

The transformation law (3.15) can be restated in the following way: The wave function φ(x) is a functional of the vector potential Aµ (x): φ(x) = φ(x; Aµ ) . The transformation law (3.15) for the covariant derivative then implies that under the gauge transformation (3.13) the functional φ(x; Aµ ) transforms as follows: φ(x; Aµ + ∂µ Λ) = e−iqΛ(x) φ(x; Aµ ) .

(3.17)

The gauge covariance of the equation of motion (3.11) allows us to choose in par~ ticular the covariant Lorentz gauge ∂·A = 0. In this gauge the 4-tuplet Aµ = (A0 , A) transforms like a 4-vector. This demonstrates the manifest Lorentz covariance of the equation of motion (3.16) in the Lorentz gauge. Negative energy solutions and antiparticles Consider the case where the vector potential is independent of time. In that case there exist stationary solutions φ(±) (x) = ϕ(±) (~r)e∓iEt ,

E>0

suggesting again a two-particle interpretation for a general wave packet. Labelling these solutions by the charge q appearing in the covariant derivative (3.12), substitution into Eq. (3.11) leads to the equations ~ 2 + 2qEA0 + iq(∇ ~ · A) ~ + 2iq A ~·∇ ~ − q 2 Aµ Aµ ]ϕ(+) (~r) = 0 [m2 − E 2 − ∇ ~ 2 − 2qEA0 + iq(∇ ~ · A) ~ + 2iq A ~·∇ ~ − q 2 Aµ Aµ ]ϕ(−) (~r) = 0 , [m2 − E 2 − ∇ where we have set Φ = A0 . From here we see that φ(−) (~r; q) = φ(+)∗ (~r; −q) .

(3.18)

This suggests the identification of the “negative energy” solution φ(−) with the respective antiparticle. The transformation (3.18) for scalar fields is referred to as charge conjugation. Probability interpretation In the free-particle case, we attempted to identify the probability density with the time component of a conserved 4-vector-current. It was found to satisfy all requirements, provided we restricted ourselves to positive energy solutions from the outset. As we now show, this will no longer be possible in the presence of an electromagnetic interaction, which will invariably lead to transitions to states involving negative energy solutions. Following the general line of approach adopted in the free particle case, we note to begin with that we can define again a conserved current by jµ (x; A) = i[φ∗ (x)Dµ φ(x) − (Dµ φ(x))∗ φ(x)] .

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35

This current is gauge-invariant jµ (x; A + ∂A) = i[φ0∗ (x)Dµ0 φ0 (x) − (Dµ0 φ0 (x))∗ φ0 (x)] = jµ (x; A)

and thus defines a Lorentz covariant observable. With the aid of the equation of motion (3.11) one easily checks that this current is conserved: ∂µ j µ (x; A) = 0 . Since j µ transforms like a 4-vector density, we take its zero component to define the probability density: ↔

P(x; A) ≡ j 0 (x; A) = iφ∗ (x)(∂ 0 + 2iqA0 )φ(x) .

(3.19)

Unfortunately this definition of the probability density already violates positivity for “positive energy” solutions; indeed, consider the stationary wave function φ(x) = ϕ(+) (~r)e−iEt . Substitution into (3.19) yields P(x; A) = 2(E − qΦ)|ϕ(+) (~r)|2 . Hence, even if E > 0, this density is not positive semi-definite, since the sign of the Coulomb potential can be either positive or negative. The above considerations lead us to abandon at this stage our search for a relativistic scalar wave equation conforming to the principles of non-relativistic quantum mechanics. We shall, however, return to the field equation (3.11) after having learned in Chapter 7 to interpret φ(x) as an operator-valued field acting on a Hilbert space of Fock states.

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Chapter 4

The Dirac Equation We begin this chapter by obtaining the relativistic “Schr¨odinger” equation for a free spin-1/2 field by following first the historical approach, and then presenting a derivation based on Lorentz covariance and space-time parity alone. This leads us to a four-component wave equation which is first-order in space and time coordinates. We present the solution of this equation for three choices of basis: The Dirac, Weyl (or chiral), and Majorana representations. The latter representation is shown to be particularly useful for the case of Majorana fermions, i.e. fermions which are their own anti-particles. We show that the Dirac equation allows for the notion of a probability density after suitable interpretation of the negative energy states.

4.1

Dirac spinors in the Dirac and Weyl representations

In this section we present the derivation of the Dirac equation by following the historical path, as well as a purely group-theoretical approach relying on Lorentz transformation properties alone. We then obtain the general solution of these equations in terms of the four independent Dirac spinors. Dirac equation: historical derivation Since in a manifestly Lorentz-covariant wave equation, space and time variables should appear on equal footing, Dirac demanded that the hamiltonian in the equation ∂t ψ(~r, t) = Hψ(~r, t) (4.1) should depend linearly on the momentum p~ canonically conjugate to ~r. This led him to the Ansatz1 ~ + mβ)ψ . i∂t ψ = (−i~ α·∇ (4.2) 1 P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edn. (Oxford University Press, Oxford 1958).

36

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37

The triple α ~ = (α1 , α2 , α3 ) and β in this equation cannot be just numbers, since this would already be inconsistent with rotational covariance. Hence they are expected to be given by matrices. These matrices must be hermitian, in order to warrant the hermiticity of the Hamilton operator. Furthermore, Eq. (4.2) should lead to the correct relation between energy and momentum for free particles. In order to see what this implies, we differentiate Eq. (4.2) with respect to time, thus obtaining ~ 2 + m2 β 2 − im(βαi + αi β)∂i ]ψ −∂t2 ψ = [−(~ α · ∇) 1 = [− {αi , αj }∂i ∂j + m2 β 2 − im{β, αi }∂i ]ψ . 2 Here the bracket {A, B} denotes the anticommutator of two objects: {A, B} = AB + BA . In order to get the desired energy momentum relation, this equation has to reduce to the Klein–Gordon equation, which is the case if {αi , αj } = 2δij ,

{β, αi } = 0,

β2 = 1 .

(4.3)

From here we deduce the following properties of the matrices: Tracelessness Since β 2 = αi2 = 1, it follows from {β, αi } = 0 that trαi = trβ 2 αi = tr(βαi β) = −trαi , trβ = trαi2 β = tr(αi βαi ) = −trβ ,

or tr αi = 0 ,

tr β = 0 .

Dimensionality Since αi2 = β 2 = 1, the eigenvalues of αi and β are either +1 or −1. From the tracelessness of the matrices it then follows that the dimension of the matrices must be even. Minimal dimension The Pauli matrices σ1 =



01 10



,

σ2 =



0 −i i 0



,

σ3 =



1 0 0 −1



,

(4.4)

together with the identity matrix 1 represent a complete basis for 2 × 2 hermitian matrices. Of these, the Pauli matrices satisfy the first of the conditions (4.3); however, the identity matrix cannot be identified with β, since trβ = 0. Since the dimension of the matrices must be even, we conclude that the dimension of these matrices must be at least four.

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The Dirac Equation The following 4 × 4 matrices satisfy all the requirements (4.3):     0 σi 1 αi = , β= . σi 0 −1

(4.5)

The same applies of course to matrices obtained from the above ones via a unitary transformation (unitary, in order to preserve the hermiticity of the matrices). For the choice of basis (4.5), the equation reads   ~ m −i~σ · ∇ i∂t ψ = ψ. ~ −m −i~σ · ∇ We can compactify the notation by introducing the definitions 0 γD = β,

i γD = βαi ,

µ ν {γD , γD } = 2g µν ,

(4.6)

where the subscript D stands for “Dirac representation”. Explicitly we have     1 0 0 σi 0 i γD = , γD = , i = 1, 2, 3 . (4.7) 0 −1 −σi 0 µ . This notation is justified since We may collect these matrices into a 4-tuplet γD we shall show later that these matrices “transform” (in a sense to be made precise later) under Lorentz transformations as a “4-vector”. In terms of the matrices (4.6) the Dirac equation takes the compact form2

iγ 0 ∂0 ψ = (m − iγ i ∂i )ψ , (iγ µ ∂µ − m)ψ = 0 ,

{γ µ , γ ν } = 2g µν .

(4.8)

This equation implies that ψ(r, t) is also a solution of the Klein–Gordon equation (3.8). We thus have the following Fourier decomposition into positive and negative energy solutions, Z r m X 1 3 d p [Uα (~ p, σ)a(~ p, σ)e−ip·x + Vα (¯ p, σ)b∗ (~ p, σ)eip·x ] , ψα (x) = 3/2 ω(~ p) σ (2π) (4.9) where pµ = (ω(~ p), p~), and the sum in σ extends over the two spin orientations in the rest-frame q of the particle, as we shall see. The reason for displaying explicitly m the factor ω(~ p) will become clear from the transformation (7.15) and canonical

normalization (7.40) in Chapter 7. For ψ(x) to be a solution of the Dirac equation (4.8), the (positive and negative energy) Dirac spinors U (p, σ) and V (p, σ) must satisfy the equations (γ µ pµ − m)U (~ p, σ) = 0

(γ µ pµ + m)V (~ p, σ) = 0 .

(4.10)

2 Here and in what follows: in formulae which hold generally, without reference to a particular basis such as the Dirac representation, we omit the subscript D.

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39

µ Recalling the explicit form (4.7) of the γD -matrices, we obtain for the independent solutions in the Dirac representation,   p~·~σ c   χ(σ) χ (σ) (D) (D) ω+m , V (~ p, σ) = N− U (~ p, σ) = N+ , (4.11) p ~·~ σ χc (σ) ω+m χ(σ)

where ω(~ p) = ω, and N± are normalization constants to be determined below. χ(σ) and χc (σ) denote the spinor and its conjugate in the rest frame of the particle,         1 1 1 0 χ = = , χ − , χc (σ) = c−1 χ(σ) , 0 1 2 2 with c the “charge conjugation” matrix defined by 1

χc (σ) = c−1 χ(σ) = (−1) 2 +σ χ(−σ) . Notice that c=



0 −1 1 0



= −iσ2 ,

(4.12)

c2 = −1.

The matrix c has the fundamental property3 c~σ c−1 = −~σ ∗ .

(4.13)

The following algebraic relations will turn out to be useful: Cγ µ C −1 = −γ µ∗ , where C = Cγ 5 γ 0 = iγ 2 , with C= and



c0 0c



,

(4.14)

C2 = 1 , C† = C

(4.15)

C † = −C

(4.16)

C 2 = −1,

γ 5 = iγ 0 γ 1 γ 2 γ 3 . We further have Cγ µ C −1 = γ µT

Cγ 0∗ C −1 = γ 0 ,

Cγ i∗ C −1 = −γ i .

In the Dirac representation, γ 5 is the off-diagonal 4 × 4 matrix   01 5 γD = . 10

(4.17)

(4.18)

In order to fix the normalization constants in (4.11), we need to choose a scalar product. To this end we observe that 㵆 = γ 0 γµ γ 0 . 3 The

reason for introducing c will become clear in Chapter 7, Eq. (7.15).

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The Dirac Equation

Hence the Dirac operator iγ µ ∂µ − m is hermitian with respect to the “Dirac” scalar product Z ¯ 0 , ψ¯ = ψ † γ 0 . (ψ, ψ 0 ) = d4 xψψ (4.19) Correspondingly we normalize the Dirac spinors by requiring4 ¯ (~ ¯ = U †γ0 , U p, σ)U (~ p, σ 0 ) = δσσ0 , U V¯ (~ p, σ)V (~ p, σ 0 ) = −δσσ0 , V¯ = V † γ 0 ,

(4.20)

which finally leads in the Dirac representation to the normalized Dirac spinors r r     p~·~σ c χ(σ) ω+m ω + m ω+m χ (σ) (D) (D) . , V (~ p, σ) = U (~ p, σ) = p ~·~ σ χc (σ) 2m 2m ω+m χ(σ) (4.21) Dirac spinors in the Weyl representation We now present a derivation of the Dirac equation based on group-theoretical arguments alone. Our fundamental requirement will be that the solution of the “relativistic Schr¨ odinger equation” should belong to a representation of the Lorentz group. In particular consider the irreducible representations (1/2,0) and (0,1/2) in (2.28). Denoting the wave functions in the respective representations by ϕ(x) and ϕ(x), ¯ this means that in analogy to (2.4), under Lorentz transformations, ϕσ (x)−→ϕ0σ (x) =

X

(1)

Dσσ2 0 [Λ−1 ]ϕσ0 (Λx)

(4.22)

σ0

ϕ¯σ (x)−→ϕ¯0σ (x) =

X

1

¯ ( 2 )0 [Λ−1 ]ϕ¯σ0 (Λx) . D σσ

(4.23)

σ0

The operators acting on these fields are thus given by 2 × 2 matrices. A complete set of such matrices is given by the identity and the three Pauli matrices. As we next show the set of four matrices5 tµ = (1, ~σ ) ,

t¯µ = (1, −~σ )

(4.24)

transform as a “4-vector” in the following sense: 1

1

¯ ( 2 ) [Λ] = Λµ tν D( 2 ) [Λ−1 ]tµ D ν

(4.25)

¯ ( 12 ) [Λ−1 ]t¯µ D( 21 ) [Λ] = Λµν t¯ν . D

(4.26)

Note that these transformation laws cannot be interpreted as a change in basis. They are easily verified for an infinitesimal Lorentz transformation Λµν ' g µν + ω µν , 4 The 5 We

ω 0i = ω i0 = θi ;

ωij = −ijk αk ,

minus sign is a consequence of our Dirac scalar product. follow in general the notation of S. Weinberg, Phys. Rev. 133 (1964) B1318.

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4.1 Dirac and Weyl representations

41

and the corresponding expression for the (1/2,0) and (0,1/2) representations (2.29) and (2.30) with 1 1 ~σ J~( 2 ,0) = J~(0, 2 ) = . 2 We have ~σ 2 1 ~ σ ) ( ~ · . ¯ 2 [Λ] ' 1 + i(~ α − iθ) D 2 1

~ · D( 2 ) [Λ] ' 1 + i(~ α + iθ)

Notice that the rotational part does not care about which of the two representations we are in. Keeping only terms linear in the parameters we have (i) for t0 = 1, 1 ~ − i(~ ~ · ~σ ¯ ( 12 ) [Λ] ' 1 + i[(~ α − iθ) α + iθ)] D( 2 ) [Λ−1 ]t0 D 2 ' 1 + θ~ · ~σ + · · ·

' 1 + θi ti + · · ·

(ii) for ti = σi , 1 ¯ ( 21 ) [Λ] ' σi + i σi σj (αj − iθj ) − i σj σi (αj + iθj ) D( 2 ) [Λ−1 ]ti D 2 2 i 1 ' σi + [σi , σj ]αj + {σi , σj }θj 2 2 ' σi − ijk αj σk + θi

' ti − ijk αj tk + θi .

On the other hand we have for an infinitesimal Lorentz transformation Λµν , Λ0ν tν ' t0 + ω 0 i ti

Λiν tν ' ti + ω ij tj + ω i0 t0 .

In accordance with our previous parametrization we have Λ0ν tν ' 1 + θi ti

Λiν tν ' ti − ijk αj tk + θi ,

where we have set t0 = 1. This establishes our claim (4.25) for tµ . In a similar way one demonstrates the transformation law (4.26). It now follows from (4.25) and (4.26) that the equations −mϕ(x) + itµ ∂µ ϕ(x) ¯ =0

(4.27)

−mϕ(x) ¯ + it¯µ ∂µ ϕ(x) = 0

(4.28)

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The Dirac Equation

transform covariantly under Lorentz transformations.6 Indeed, multiplying the first 1 equation from the left with D( 2 ) [Λ−1 ] and making the replacement x → x0 = Λx, one has 1

1

¯ ¯ −1 ]ϕ(x −mD( 2 ) [Λ−1 ]ϕ(x0 ) + (D( 2 ) [Λ−1 ]itµ ∂µ0 D[Λ]) D[Λ ¯ 0) = 0 , or recalling that ∂µ0 = ∂ν (Λ−1 )νµ , we have from (4.25) ¯ D[Λ−1 ]tµ ∂µ0 D[Λ] = tν ∂µ (Λ−1 )µν = tν ∂ν . Together with (4.22) this implies mϕ0 (x) + itµ ∂µ ϕ¯0 (x) = 0 , which proves the covariance of Eq. (4.27). In the same way, one also proves the covariance of the second equation. Equations (4.27), (4.28) represent a coupled set of equations, which only decouple in the case of zero-mass fermions. They may be collected into a single equation by defining the 4 × 4 matrices   0 tµ µ γW = ¯µ , (4.29) t 0 where the subscript W stands for “Weyl”.7 One explicitly checks that they satisfy the anticommutation relations {γ µ , γ ν } = 2g µν . Writing ψ(x) in the form ψ(x) :=



ϕ(x) ϕ(x) ¯



(4.30)

,

(4.31)

the above coupled set of equations takes the form µ (iγW ∂µ − m)ψ = 0 .

Multiplying this equation from the left with the operator (iγ µ ∂µ + m) and using the anticommutation relations (4.30) we see that ψ is also a solution of the Klein– Gordon equation: ( + m2 )ψ(x) = 0 describing the propagation of a free particle with the correct energy-momentum relation. By further defining the 4 × 4 matrices (in the Weyl-representation)   (1) 1 D 2 0 D( 2 ) = (4.32) 1 ¯(2) , 0 D 6 Substituting (4.27) into (4.28) yields (m2 + tµ t¯ν ∂ ∂ )ϕ = 0. Noting that with our definitions µ ν (4.24) for tµ and t¯µ , we have tµ t¯ν +tν t¯µ = 2g µν , we recover the Klein–Gordon equation, as desired. 7 H. Weyl, The Theory of Groups and Quantum Mechanics (Dover Publications, Inc. New York, 1931).

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43

the transformation laws (4.25), (4.26) can be collected to read 1

and

1

µ ν D( 2 ) [Λ] = Λµν γW D( 2 ) [Λ−1 ]γW

(4.33)

ψ(x) → ψ 0 (x) = D[Λ−1 ]ψ(Λx).

(4.34)

On this level we now have manifest Lorentz covariance of the Dirac equation (4.8). Note also that the inverse of the matrix D is now equivalent to the corresponding “Dirac” adjoint (recall (2.33)) 1

1

0 0 D( 2 ) [Λ−1 ] = γW D( 2 ) [Λ]† γW .

(4.35)

This will play an important role when we come to define scalar products. (W ) We now decompose the solution to the Dirac equation as in (4.9). For Uα (~ p, σ) (W ) and Vα (~ p, σ) the Dirac equation then reads (γW · p − m)U (W ) (~ p, σ) = 0 ,

(γW · p + m)V (W ) (~ p, σ) .

(4.36)

Recalling the explicit form of the (1/2,0) and (0,1/2) representations (2.36) of boosts, we conclude that     1 p)])2 0 t·p 0 m(D( 2 ) [L(~ . (4.37) γW · p = ¯ = 1 ¯ ( 2 ) [L(~ t·p 0 p)])2 0 m(D ¯ p)] = D−1 [L(~ Recalling from (2.32) that D[L(~ p)], we can solve the set of algebraic equations (4.36) for the four independent Dirac spinors, to give  (1)  1 D 2 [L(~ p)]χ(σ) (W ) U (~ p, σ) = √ ¯ ( 12 ) [L(~ p)]χ(σ) 2 D   (1) 1 p)]χc (σ) D 2 [L(~ V (W ) (~ p, σ) = √ , (4.38) ¯ ( 12 ) [L(~ p)]χc (σ) 2 −D where U (W ) (~ p, σ) and V (W ) (~ p, σ) have been normalized with respect to the scalar product (4.19). Making use of the explicit form (2.37) and (2.38) of the 2 × 2 matrices representing the boosts, one can rewrite the expressions (4.38) in the explicit form ! r p ~·~ σ )χ(σ) 1 ω + m (1 − ω+m (W ) U (~ p, σ) = √ , p ~·~ σ 2m (1 + ω+m )χ(σ) 2 ! r p ~·~ σ (1 − ω+m )χc (σ) 1 ω+m (W ) V (~ p, σ) = √ . (4.39) p ~·~ σ 2m )χc (σ) −(1 + ω+m 2 Comparing with (4.21), we seem to be arriving at different results. In fact, these results can be shown to be unitarily equivalent. Indeed, the γ µ -matrices (4.29) and (4.7) are related by the unitary transformation µ µ −1 γD = SγW S

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The Dirac Equation

with

  1 1 1 , S −1 = S † . S=√ 2 −1 1 Correspondingly we have for the Dirac spinors

(4.40)

U (D) (~ p, σ) = SU (W ) (~ p, σ) V (D) (~ p, σ) = SV (W ) (~ p, σ) ,

(4.41)

which are readily seen to coincide with the spinors (4.21). The basis in which the γ-matrices take the form (4.7) is referred to as the Dirac representation. The basis in which the γ-matrices take the form (4.29) is referred to as the Weyl representation. The same applies to the Dirac spinors (4.21) and (4.39), respectively. The choice of representation is a matter of taste and depends on the specific problem and question one wants to address. Thus, to discuss the non-relativistic limit of the Dirac equation, it is convenient to work in the Dirac representation. If one is dealing with massless charged fermions, it is more convenient to work in the Weyl representation, since the Dirac equation reduces to two uncoupled equations in this case. We shall have the opportunity to work in still another basis, the so-called Majorana representation, which turns out to be particularly suited if the fermions are massless and charge neutral (neutrinos, for example).

4.2

Properties of the Dirac spinors

One easily proves the following results for both representations: (a) Orthogonality relations ¯ (~ U p, σ 0 )U (~ p, σ) = χ† (σ 0 )χ(σ) = δσ0 σ , V¯ (~ p, σ 0 )V (~ p, σ) = −χ†c (σ 0 )χc (σ) = −δσ0 σ .

(4.42)

The positive (negative) energy spinors are seen to have positive (negative) norm and to be orthogonal, respectively. One furthermore has ¯ (~ U p, σ 0 )V (~ p, σ) = 0 ,

V¯ (~ p, σ 0 )U (~ p, σ) = 0 .

We thus conclude that the “positive” and “negative” energy solutions8 for halfintegral spin are also mutually orthogonal with respect to the “Dirac scalar product”. (b) Projectors on positive and negative energy states According to (a) the matrices Λ+ (p)αβ =

X σ

Λ− (p)αβ = − 8 See

Chapter 5 for this terminology.

¯β (~ Uα (~ p, σ)U p, σ) ,

X σ

Vα (~ p, σ)V¯β (~ p, σ)

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4.3 Properties of the γ-matrices

45

have the properties of projectors on the positive and negative energy solutions, respectively. In particular, the property Λ+ (p) + Λ− (p) = 1 follows from the completeness relation X ¯β (~ (Uα (~ p, σ)U p, σ) − Vα (~ p, σ)V¯β (~ p, σ)) = δαβ σ=± 12

for the spinors. We have for both representations Λ± (p) =

4.3

m±γ·p . 2m

(4.43)

Properties of the γ-matrices

We next list some useful properties of the γ-matrices which are independent of the choice of representation. (a) The trace of an odd number of γ-matrices vanishes Proof: tr[γ µ1 ...γ µ` ] = tr[γ 5 γ µ1 ...γ µ` γ 5 ] = (−1)` tr[γ µ1 ...γ µ` ] ,

(4.44)

where we have used the cyclic property of the trace, as well as γ52 = 1. (b) Reduction of the trace of a product of γ-matrices In general it follows, by repeated use of the anticommutator (4.30) of γ-matrices, that tr(γµ1 ...γµ2n ) = 2gµ1 µ2 tr(γµ3 ...γµ2n ) − 2gµ1 µ3 tr(γµ2 γµ4 ...γµ2n )

(4.45)

2n

+ ... + 2(−1) gµ1 µ2n tr(γµ2 γµ3 ...γµ2n−1 ) − tr(γµ1 ...γµ2n ) ,

or tr(γµ1 ...γµ2n ) = gµ1 µ2 tr(γµ3 ...γµ2n ) − gµ1 µ3 tr(γµ2 γµ4 ...γµ2n ) + ... + (−1)2n gµ1 µ2n tr(γµ2 γµ3 ...γµ2n−1 ) .

As a Corollary to this we have the “contraction” identity tr[γ λ γ µ γ ν γ ρ γλ ] = −2tr[γ ρ γ ν γ ρ ]

(4.46)

tr(γ µ γ ν ) = 4g µν , tr(a/b/) = 4a · b tr(γ µ γ ν γ ρ γ σ ) = 4(g µν g ρσ − g µρ g νσ + g µσ g νρ ) ,

(4.47)

as well as

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The Dirac Equation

where we followed the Feynman convention of writing a/ := γ µ aµ . Notice that the factor 4 arises from tr1 = 4, the dimension of space-time. We further have the contraction identities γ λ γ µ γ ν γλ = 4g µν γ λ γ µ γ ν γ ρ γλ = γ µ γ ν γ ρ λ µ ν ρ σ

(4.48)

σ µ ν ρ

ρ ν µ σ

γ γ γ γ γ γλ = 2(γ γ γ γ + γ γ γ γ ) which will prove useful in Chapters 15 and 16. (c) The γ 5 -matrix In the Weyl representation the upper and lower components of the Dirac spinors are referred to as the positive and negative chiality components, corresponding to the eigenvalues of the matrices9     01 −1 0 5 5 γD = , γW = . 10 0 1 As one easily convinces oneself, one has (from here on we follow the convention of Itzykson and Zuber and of most other authors, and choose 0123 = 1) γ5 = −

i µνλρ γ µ γ ν γ λ γ ρ = iγ 0 γ 1 γ 2 γ 3 , 4!

γ52 = 1 .

(4.49)

This expression defines the γ5 matrix in both representations. (d) Lorentz transformation properties of γ 5 For Λ a Lorentz transformation, we have the algebraic property 0

0

0

0

D[Λ−1 ]µνλρ γ µ γ ν γ λ γ ρ D[Λ] = µνλρ Λµµ0 Λνν 0 Λλλ0 Λρρ0 γ µ γ ν γ λ γ ρ . Now µνλρ Λµµ0 Λνν 0 Λλλ0 Λρρ0 = det Λ · µ0 ν 0 λ0 ρ0 . Hence we conclude that γ 5 “transforms” in particular like a pseudoscalar under space reflections, and in general as D[Λ−1 ]γ 5 D[Λ] = γ 5 det Λ . 9 Note

that 5 −1 5 SγD S = γW ,

0 −1 SγD S =



01 10



0 = γW

where S is given by (4.40). This agrees with the usual V − A coupling of neutrinos in the weak interactions.

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47

(e) Traces involving γ 5 tr(γ 5 γ µ γ ν ) = 0 , tr(γ 5 γ µ γ ν γ λ γ ρ ) = −4iµνλρ = 4iµνλρ . Here the first relation follows from the fact that there exists no Levi–Civita tensor with two indices in four dimensions. The second relation follows from the fact that the right-hand side should be a Lorentz invariant pseudotensor of rank four, for which µνλρ is the only candidate, and choosing the indices as in (4.49) to fix the constant.

4.4

Zero-mass, spin =

1 2

fields

In the extreme relativistic limit we expect the mass of the fermion to be negligible. For m = 0 the Dirac Hamiltonian operator commutes with γ5 . Hence we may classify the eigenfunctions of the Hamiltonian by the eigenvalues of γ5 . It is thus desirable to work in the Weyl representation, where γ5 is diagonal. In this representation the Dirac operator becomes off-diagonal in the large momentum limit, and the Weyl equations (4.27) and (4.28) reduce to the form itµ ∂µ ϕ(x) ¯ =0 , µ it¯ ∂µ ϕ(x) = 0 ,

(4.50)

(i∂0 + i~σ · ∇)ϕ(x) ¯ =0,

(4.52)

(4.51)

or (i∂0 − i~σ · ∇)ϕ(x) = 0 .

(4.53)

The 2 × 2 matrix ~σ · n ˆ /2 represents the projection of the angular momentum operator on the direction of motion of the particle and is called the helicity operator. Correspondingly one refers to its eigenvalues ±1/2 as helicity. Equations (4.52) and (4.53) are just the Weyl equations for a massless particle. If parity is not conserved, we may confine ourselves to either one of the two equations, that is to either particles polarized in the direction of motion (positive helicity) or opposite to the direction of motion (negative helicity). This is the case for neutrinos (antineutrinos) participating in the parity-violating weak interactions, which carry helicity −1/2 (+1/2). If parity is conserved, both helicity states must exist. The fact that the massive Dirac equation turns into Weyl equations in the “infinite momentum frame” shows that at high energies massive particles are polarized “parallel” or “anti-parallel” to the direction of motion. However, whereas the helicity of a massless particle is a Lorentz invariant, this is not the case for a massive particle: If a massive particle is polarized in the direction of motion in one inertial frame, its polarization will be a superposition of all possible spin projections in a different inertial system. Phrased in a different way: If the particle is massive one can always catch up with it and ultrapass it, so that the particle appears moving “backwards”, while continuing to be polarized in the original direction. With a zero mass particle you can never catch up since it is moving at the speed of light.

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The Dirac Equation

Chirality As the last argument above shows, the m = 0 case has to be treated separately, and cannot be obtained as the zero-mass limit of massive case discussed so far, which was based on the existence of a rest frame of the particle. According to our discussion in the chapter on Lorentz transformations, zero-mass 1-particle states indeed transform quite differently from the massive ones. In the zero mass case, the Dirac equations for the U and V spinors reduce to one and the same equation: γ · pU (~ p, λ) = 0,

γ · pV (~ p, λ) = 0.

(4.54)

Let us define the “spin” operator ~ = Σ



~σ 0 0 ~σ



.

(4.55)

In terms of the Gamma matrices (Dirac or Weyl basis) this operator reads ~ = γ 5 γ 0~γ . Σ

(4.56)

The Dirac equation may then be written in the form ~ ·n (Σ ˆ )U (~ p, λ) = γ 5 U (~ p, λ) ~ ·n ˆ is just the helicity operator in the 4-component reprewhere n ˆ = |~pp~| . Thus 12 Σ sentation. The helicity operator (4.56) commutes with the free Dirac Hamiltonian. The same applies to γ5 , if the mass of the particle is zero. Since furthermore ~ ·n [γ5 , Σ ˆ] = 0 , we may classify the eigenstates of the zero-mass Dirac Hamiltonian according to their helicity and chirality, the latter being defined as the corresponding eigenvalue ±1 of γ 5 . Such states are obtained from the solutions U to the Dirac equation with the aid of the projection operator P± =

1 (1 ± γ5 ), 2

P±2 = P± .

We have γ5 U± = ±U± ,

~ ·n Σ ˆ 1 U± = ± U± 2 2

where U± = P± U,

U (~ p) =



ω(~ p, − 12 ) ω ¯ (~ p, 12 )



.

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49

Recalling that in the Weyl representation     −1 0 00 (W ) 5 = , P+ = γW , 0 1 01 we have (W )

U+

(~ p) =

page 49



0 ω ¯ (~ p, 21 )



,

(W )

U−

(~ p) =

P−W = 



10 00

ω(~ p, − 12 ) 0





.

The eigenvalue of γ5 thus coincides with twice the eigenvalue of the helicity operator: particles of positive (negative) chirality, carry helicity +1/2(−1/2). Solution of Weyl equations Experiment shows that neutrinos (antineutrinos) only occur with negative (positive) helicity. One thus refers to ν (¯ ν ) as being left (right) handed. This is reflected by the so-called V − A (vector minus axial vector) coupling of the neutrino sector. Since parity is violated, the absence of right-handed neutrinos and left-handed antineutrinos is admissible. The 4-component Dirac field (4.9) of the massive case is thus replaced in this case by   ϕσ (x) ψα (x) = , ϕ¯σ¯ (x) with 1 ϕσ (x) = (2π)2/3 ϕ¯σ¯ (x) =

Z

1 (2π)2/3

Z

3

d p

r

  m 1 1 −ip·x 1 ip·x ∗ ωσ (~ p, − ) a(~ p, − )e + b (~ p, )e , (4.57) |~ p| 2 2 2

  r m 1 1 1 ω ¯ σ¯ (~ p, ) a(~ p, )e−ip·x + b∗ (~ p, − )eip·x , d3 p |~ p| 2 2 2

(4.58)

where the spin projection now refers to helicity. The Weyl equations thus reduce to solving the eigenvalue problems         ~σ · n ˆ 1 1 1 ~σ · n ˆ 1 1 1 w p~, − = − w p~, − , w ¯ p~, = w ¯ p~, . (4.59) 2 2 2 2 2 2 2 2 For the momentum pointing in the z-direction, the eigenvalue problems are solved by     1 1 0 1 ˜ ˜ ω(k, − ) = , ω ¯ (k, ) = (4.60) 1 0 2 2 with k˜ = (κ, 0, 0, κ). Now perform the following transformation: First boost the momentum k˜µ to the momentum pµ = (|~ p|, 0, 0, |~ p|) with the matrix (see (2.15))   cosh θ 0 0 sinh θ  0 1 0 0   . B(|~ p|) =   0 0 1 0  sinh θ 0 0 cosh θ

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The Dirac Equation

Hence

    |~ p| 1   0 µ ν θ 0 ˜  . B ν (|~ p|)k = κe   =   0 0 |~ p| 1

This determines θ as a function of |~ p|:

θ = ln



|~ p| κ



.

We now rotate the vector pµ thus obtained in the desired direction of the final vector pµ with the rotation matrix R(ˆ p)µν ,   00 0 0 0   . R(ˆ p) =   0 R(ˆ p)  0

The result is

pµ = Lµν (~ p)k˜ν

p) is the matrix (2.50): where Lµν (~ L(~ p)µν = (R(ˆ p)B(|~ p|))µν =



|~ p| κ



µ

R(ˆ p) ν .

Here L(~ p) plays the role of the boost L(~ p) in the massive case. Correspondingly we have from (4.60) and (2.34) for the 2-component spinors 1 1 p)]σ¯ ,− 21 = ω ¯ σ¯ (~ p, ) = D( 2 ) [L(~ 2



|~ p| κ

 12

p)]σ¯ ,− 21 . D( 2 ) [R(ˆ



|~ p| κ

 12

p)]σ¯ , 21 . D( 2 ) [R(ˆ

1

(4.61)

Similarly we have 1 1 p)]σ¯ , 21 = ω ¯ σ¯ (~ p, ) = D( 2 ) [L(~ 2

1

(4.62)

The fields (4.57) and (4.58) now take the form       Z r 1 m (1) 1 −ip·x 1 ip·x ∗ 3 2 [L(~ 1 ϕσ (x) = d p D p )] a p ~ , − e + b p ~ , e σ,− 2 |~ p| 2 2 (2π)2/3 (4.63)       Z r 1 m ¯(1) 1 −ip·x 1 ip·x 3 ∗ 2 ϕ¯σ¯ (x) = d p D [L(~ p)]σ¯ , 12 a p~, e + b p~, − e . |~ p| 2 2 (2π)2/3 In this form, the Fourier decomposition resembles closely that of a massive field except for the fact that in the massless case U = V . Alternatively we have using

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page 51

51

(4.61) and (4.62), 1 ϕσ (x) = (2π)2/3

ϕ¯σ¯ (x) =

1 (2π)2/3

      1 ip·x 1 −ip·x ∗ d pD [R(~ e + b p~, e p)]σ,− 21 a p~, − 2 2 (4.64)       Z 1 1 1 ¯ ( 2 ) [R(~ d3 p D p)]σ¯ , 21 a p~, e−ip·x + b∗ p~, − eip·x 2 2 Z

3

( 12 )

where we have chosen κ = m.

4.5

Majorana fermions

So far we have considered the Dirac representation, particularly suited for discussing the non-relativistic limit as we shall see, and the Weyl (or chiral) representation particularly suited for discussing the relativistic limit, or the case of zero mass particles. There exists another choice of basis for the Gamma matrices called the Majorana representation which is particularly suited for the case of charge selfconjugate fermions, referred to as Majorana fermions. Spin zero, charge neutral particles are called “self-conjugate”, and are described by real fields satisfying the Klein–Gordon equation, which itself is real. In the case of “self-conjugate” spin one-half particles, the analogon is provided by the “Majorana” representation. The Majorana representation is, however, also useful in the case where fermions and anti-fermions are distinct particles if the symmetry group in question is for instance the orthogonal group O(N ) rather than the unitary group U (N ). The reason is that in the Majorana representation the Dirac equation is real. We therefore discuss separately the notion of Majorana representation and Majorana fermions. Majorana representation There exists a choice of basis in which all Dirac matrices are purely imaginary. This is called the Majorana representation of the Gamma-matrices. They are obtained from the corresponding ones in the Dirac representation via the unitary transformation µ µ −1 γM = SγD S with

and have the property

  1 1 σ2 S=√ 2 σ2 − 1 µ∗ µ γM = −γM .

One explicitly computes       0 σ2 iσ3 0 0 −σ2 0 1 2 γM = , γM = , γM = , σ2 0 0 iσ3 σ2 0     −iσ1 0 σ2 0 3 5 γM = , γM = . 0 −iσ1 0 −σ2

(4.65)

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The Dirac Equation

Now, the Dirac wave function and its charge conjugate (in the Dirac and Weyl representations) are related by (see Chapter 7, Eq. (7.48)), ψc (x) = Cψ ∗ (x) .

(4.66)

Applying the unitary operator (4.65) to both sides of the equation, we obtain for the Dirac wave function in the Majorana representation ψc(M ) = Γψ (M )∗ where Γ = SCS −1∗ = −i. We thus conclude that ψc(M ) = −iψ (M )∗ .

(4.67)

Summarizing we have in the Dirac, Weyl and Majorana representations ψc = Γψ ∗ where Γ=



(4.68)

iγ 2 (Dirac, Weyl) . −i (Majorana)

(4.69)

Majorana spinors Applying the unitary operator (4.65) to the Dirac spinors in the Dirac representation (4.21) (we use c−1 = iσ2 )

U

(D)

V

(D)

(~ p, σ) = (~ p, σ) =

r

r

ω+m 2m ω+m 2m

 

χ(σ) ~ σ ·~ p ω+m χ(σ)



 p ~·~ σ ω+m iσ2 χ(σ) iσ2 χ(σ)

,

we obtain for the corresponding spinors in the Majorana representation U

V

(M )

(M )

1 (~ p, σ) = √ 2

i (~ p, σ) = √ 2

r

r

  ~ σ ·~ p ω + m (1 + σ2 ω+m )χ(σ) ~ σ ·~ p 2m (σ2 − ω+m )χ(σ)

(4.70)

  p ~·~ σ ( ω+m σ2 + 1)χ(σ) ω+m . p ~·~ σ 2m (σ2 ω+m σ2 − σ2 )χ(σ)

(4.71)

Recalling that σ2~σ σ2 = −~σ ∗ we see from (4.70) and (4.71) that U (M )∗ (~ p, σ) = iV (M ) (~ p, σ) .

(4.72)

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53

In the case of Majorana fermions, it is convenient to redefine the phase of the Dirac spinors via the replacements iπ

U (M ) −→ e− 4 U (M ) ,

iπ

V (M ) −→ e− 4 V (M ) .

For this new choice of phase, Eq. (4.72) is replaced by U (M )∗ (~ p, σ) = V (M ) (~ p, σ) .

(4.73)

Self-conjugate Dirac fields Fields describing fermions which are their own anti-particles are said to be selfconjugate and have the Fourier representation Z o 1 d3 p Xn (M ) −ip·x ∗ (M ) ip·x p ψ (M ) (x) = a(~ p , σ)U (~ p , σ)e + +a (~ p , σ)V (~ p , σ)e , (2π)3/2 2ωp σ with the property (4.73). These fields are real,

ψ (M ) (x) = ψ (M )∗ (x) , and play the role of real scalar fields in the case of spin 1/2 fields.

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Chapter 5

The Free Maxwell Field As is well known, the electromagnetic field can be interpreted on the quantum level as a flux of quanta, called photons. In fact, this interpretation first arose in connection with Planck’s formula describing the spectrum of black-body radiation. As Maxwell’s equations show, these quanta propagate with the velocity of light in all inertial frames, so there exists no rest frame we can associate with them. Photons can thus be viewed as “massless” particles. According to our discussion in Section 6 of Chapter 2, the respective 1-particle states should thus transform according to a one-dimensional representation of the little group. The Aharonov Bohm effect shows that it is the vector potential Aµ which plays the fundamental role in quantum mechanics. This vector potential however trans1 1 forms under the D( 2 , 2 ) representation of the Lorentz group and involves a priori four degrees of freedom. Of these, A0 is associated with the “Coulomb potential” and thus corresponds to non-radiative degrees of freedom which are only present if there is a matter source. This leaves us with three degrees of freedom. One of these is not observable (on classical level) as a result of the underlying gauge invariance of physical quantities. For a pure radiation field one is thus left with only two degrees of freedom, corresponding to the two helicity states of a photon. These statements become obvious in the Coulomb gauge, which is thus also called the “physical” gauge. We are, however, not limited to this choice of gauge which in practical calculations complicates matter considerably, due to the fact that it breaks manifest Lorentz covariance. We shall thus review the solutions of Maxwell’s equations in two different gauges — the Lorentz gauge and the (non-covariant) Coulomb gauge.

5.1

The radiation field in the Lorentz gauge

In the absence of a source, the Maxwell equations (Coulomb’s and Amp`ere’s law) become ∂µ F µν = 0 , (5.1) with the usual identification 1 B i = − ijk F jk , E i = F i0 2 54

(5.2)

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55

for the magnetic and electric fields, respectively. The electromagnetic field tensor Fµν can be written in the form Fµν = ∂µ Aν − ∂ν Aµ and is evidently invariant under the gauge transformation Aµ → Aµ + ∂µ Λ .

(5.3)

In terms of the vector potential Aµ the homogeneous Maxwell equations (5.1) read Aµ − ∂ µ ∂ν Aν = 0 .

(5.4)

By choosing Λ in (5.3) to be given by Λ(x) = −

Z

d4 yG(x − y)∂µ Aµ (y) ,

G(x) = δ 4 (x)

we arrive at the Lorentz gauge ∂µ Aµ = 0 .

(5.5)

In this gauge the equations of motion (5.4) read Aµ (x) = 0 . The general solution of this equation is well known from the course in electrodynamics and is given by 1 A (x) = (2π)3/2 µ

Z

i d3 k X h µ ~ q  (k, ν)a(~k, ν)e−ik·x + µ∗ (~k, ν)a∗ (~k, ν)eik·x , 2|~k| ν=0 3

where a(k, ν), ν = 0...3 are the Fourier coefficients and µ are the corresponding polarization tensors playing a role analogous to the Dirac spinors. In the Lorentz gauge (5.5) we must have k · (~k, ν) = 0,

k2 = 0 ,

ν = 1, 2, 3, 4 .

With Aµ real, we choose the polarization tensors to be real. The choice of the Fourier coefficients is then dictated by the reality of the electromagnetic field. In particle language it corresponds to the fact that the photon is its own antiparticle! The Fourier coefficients a∗ (~k, ν) (a(~k, ν)) will eventually be identified with the creation (destruction) operators of 1-particle states. However, only two of these states can correspond to photons of helicity +1 and −1. In the Coulomb gauge this becomes manifest. This gauge, thus often referred to as the “physical gauge”, has however the drawback of not being manifestly Lorentz invariant.

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5.2

page 56

The Free Maxwell Field

The radiation field in the Coulomb gauge

The Coulomb gauge

~ ·A ~=0 ∇

(5.6)

can be reached by performing the gauge transformation (5.3) with Z −1 ~ 0 · A(~ ~ r 0 , t) . Λ(~r, t) = d3 r0 ∇ 4π|~r − ~r 0 | In this gauge, the free Maxwell equations for the vector potential read: Aµ − ∂ µ ∂0 A0 = 0 .

(5.7)

Setting µ = 0, it follows from here that ~ 2 A0 = 0 . ∇ This is Laplace’s equation; it only has the trivial solution if we require that the vector potential tends to zero at infinity. In the Coulomb gauge we thus have in the absence of sources, A0 = 0. This shows that in this gauge the vector potential possesses only two degrees of freedom, corresponding to a radiation field, in agreement with our general considerations in Section 6 of Chapter 2, showing that a zero mass particle can exist only in two helicity states. Correspondingly we have for the general solution of (5.7) in the gauge (5.6) Z 1 d3 k X ~ ~ q A(~r, t) = ~(k, λ)[a(~k, λ)e−ik·x + a∗ (~k, −λ)eik·x ] (5.8) (2π)3/2 ~ 2|k| λ=±1 with ~k · ~(~k, λ) = 0, and λ the helicity, where we have used (5.13). We are allowing for both helicity states, since parity is conserved. We now show how to choose these ~ a real field. polarization tensors for A Define the two orthogonal vectors 1 (µ± ) = √ (0, 1, ±i, 0) . 2

(5.9)

Introducing as in Section 6 of Chapter 2 the light-like standard 4-momentum k˜ by k˜µ = |κ|nµ ,

(nµ ) = (1, 0, 0, 1) = (1, n ˆ)

(5.10)

describing the motion of a photon in z-direction with energy |~κ|, we see that the standard polarization tensors (5.9) are in fact eigenvectors of the helicity operator     0 10 0 0 J~ · n ˆ= , J3 = −i  −1 0 0  . 0 J3 0 00 Indeed,

J~ · n ˆ µ± = ±µ± .

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5.2 The radiation field in the Coulomb gauge

57

Inspired by our considerations in Section 6 of Chapter 2 on the Little group, we now define the polarization tensors for a photon with general momentum ~k by µ (~k, ±1) = Lµν (~k)ν± ,

(5.11)

where Lµν (k) are the elements of the Little group taking the standard vector k˜µ into the final vector k µ (see Chapter 2), Lµν (~k) = Rµλ (~k)Bνλ (|~k|). ~ The boost matrix B µν represents a boost of k˜ along the 3-direction to k µ = (|κ|, 0, 0, |κ|), and leaves the standard polarization tensors unchanged since it only affects the 0 and 3 elements, leaving the (1, 2) elements unchanged. Hence   0 µ ~ µ ~ ν µ ˆ ν  (k, ±1) = L ν (k)± = R ν (k)± = , (5.12) ˆ j Rij (k) ± where we have used R=



10 0R



,

B µν ν± = ν± ,

with R the matrix rotating the standard momentum into the direction of ~k. Noting that (R−1 )µν = Rνµ , we easily verify the following properties of the polarization tensors: ∗µ (~k, λ0 )µ (~k, λ) = δλλ0 , 0 (~k, λ) = 0 , ∗µ (~k, λ) = µ (~k, −λ), k · (~k, λ) = 0 .

(5.13)

The last property follows from the following manipulations: k˜µ µ± = 0 ⇒ k˜λ (R−1 R)λν ν± = 0 ⇒ k˜λ (R−1 )λρ Rρν ν± = 0 ⇒ k · (~k, λ) = 0 , where we have used the Lorentz transformation property (2.6) of a covariant 4-vector. The polarization tensor satisfies the following completeness relation: X

ki kj = δ ij . i (~k, λ)∗j (~k, λ) + ~k 2 λ=±1

(5.14)

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Chapter 6

Quantum Mechanics of Dirac Particles In the case of the scalar field we have seen that there was no possibility for a probability interpretation in the sense of non-relativistic Quantum Mechanics. In the case of Dirac spin one-half particles, such a probabilistic interpretation is possible if their electromagnetic interaction is restricted to external electromagnetic fields.

6.1

Probability interpretation

In order for P (~r, t) to describe a probability density, we must require Z d d3 rP (~r, t) = 0 . dt This property is satisfied if P (~r, t) satisfies a continuity equation ~ · ~j(~r, t) = 0 . ∂t P (~r, t) + ∇

(6.1)

Now, in order that the normalization Z d3 rP (~r, t) = 1 be independent of the choice of the inertial system, d3 rP (~r, t) must transform like a scalar. Claim: If P (~r, t) is the zero component of a conserved 4-vector field j µ (~r, t),   P (x) j (x) = , ~j(x) µ

58

∂µ j µ (x) = 0 ,

(6.2)

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Z

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59

3

d rP (~r, t) =

Z

d3 r0 P 0 (~r0 , t0 ) ,

(6.3) µ

where x and x0 are related by a Lorentz transformation: x0 = Λµν xν . Proof: Since the probability density satisfies a continuity equation (6.1), it follows that the integrals in (6.3) are independent of time. It is therefore convenient to choose t = t0 = 0. We now convert the three-dimensional integrals in (6.3) into four-dimensional integrals by noting that Z Z d3 xP (x) = d4 xδ(x0 )j 0 (x) x0 =0 Z = d4 xδ(n(0) · x)(n(0) · j(x)) , (6.4) where nµ (0) is the unit time-like vector nµ (0) = nµ (0) = (1, 0, 0, 0) . The integration in (6.4) is thus restricted to the plane “perpendicular” to the unit 4-vector nµ (0). We now rewrite the above integral in terms of Lorentz-transformed variables. By assumption we have under a Lorentz transformation xµ → x0µ = Λµν xν the transformation law (see (2.5)) j 0µ (x0 ) = Λµν j ν (x) . Making use of the Lorentz invariance of the integration measure d4 x and defining the Lorentz-transformed unit 4-vector n0µ by n(0) · x = n0 (0) · x0 , we have

Z

d3 rP (~r, 0) =

Z

d4 x0 δ(n0 (0) · x0 )(n0µ (0)j 0µ (x0 )) .

(6.5)

We next want to convert the right-hand side of this equation again into a threedimensional integral with respect to the Lorentz-transformed variables. In order to achieve this, we shall have to do some work. Note that the right-hand side involves an integral in t0 over the hyperplane n0 · x0 = 0 where n0µ = nν (Λ−1 )νµ . Noting that xµ can also be written in the form X xµ = xρ nµ (ρ) , ρ

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where nµ (ρ), ρ = 0, 1, 2, 3 are the 4-unit basis 4-vectors nµ (ρ) = δ µρ , we now do the analogous thing with x0µ . Expanding x0µ in terms of the corresponding Lorentz-transformed basis vectors, n0µ (ρ) = Λµν nν (ρ) we have

x0µ =

X

ξ ρ n0µ (ρ) .

ρ

Since

n0 (ρ) · n0 (λ) = g λρ

we also have

x0 · n0 (0) = ξ 0 ,

Now

x0 · n0 (i) = ξ i .

(6.6)

0µ ∂x J = ν = detΛ = 1 . ∂ξ

Hence we have from (6.5), Z Z 3 d rP (~r, t) = d4 ξδ(ξ 0 )(n0µ (0)j 0µ (x0 (ξ)) Z = d3 ξ(n0µ (0)j 0µ (x0 (ξ)) .

(6.7)

ξ 0 =0

Because of (6.6), the d3 ξ integration extends over the surface σ 0 “perpendicular” to n0µ (0). We want to convert this integral into an integral over the surface “perpendicular” to nµ (0). We thus need some kind of “Gauß theorem” in Minkowski space. Such a theorem exists and states Z Z d4 x∂µ F µ (x) = dσ(x)nµ (x)F µ (x) , (6.8) V

S(V )

where S(V ) is the closed oriented surface composed of ξ 0 = 0 and x0 = 0, closed at infinity, bounding the volume V . According to Gauß’ law (6.8) we have Z Z Z d3 ξn0µ (0)j 0µ (x0 (ξ)) − d3 x0 nµ (0)j 0µ (x0 ) = d4 x0 ∂µ0 j 0µ (x0 ) = 0 ξ 0 =0

x00 =0

V

where we have used current conservation, and n0µ and nµ are oriented unit vectors “orthogonal” to the surfaces ξ 0 = 0 and x0 = 0, respectively. Hence Z Z d3 ξn0µ (0)j 0µ (x0 (ξ)) = d3 r0 j 00 (~r0 , t0 ) . From (6.7) we thus conclude that (6.3) is satisfied. This establishes the independence of the normalization integrals (6.3) of the choice of Lorentz frame.

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61

It remains to find a suitable candidate for the probability density. Now, since the Dirac field transforms as (see (4.34) and (4.35)) ψ(x) −→ ψ 0 (x) = Dαβ [Λ−1 ]ψβ (Λx) , ¯ ψ(x) −→ ψ¯0 (x) = ψ¯β (Λx)Dβα [Λ] under a Lorentz transformation Λ, it follows, using (4.33), that the bilinear µ ¯ j µ (x) = ψ(x)γ ψ(x)

transforms like a 4-vector field. One also easily establishes that it is conserved. Indeed, from the Dirac equation (i∂/ − m)ψ = 0 follows

←

←

where ∂ is defined by

¯ ∂/ −m) = 0 , ψ(−i

(6.9)

(6.10)

←

ψ¯ ∂µ := ∂µ ψ¯ , and use has been made of βγ µ† β = γ µ

(β = γ 0 ) .

Multiplying (6.9) with ψ¯ from the left and (6.10) with ψ from the right, and subtracting one equation from the other, we finally have ¯ µ ψ) = 0 , ∂µ (ψγ which proves our claim. We thus identify the probability density with 0 ¯ P (x) = ψ(x)γ ψ(x) .

6.2

(6.11)

Non-relativistic limit

In order to discuss the non-relativistic limit of the Dirac equation, one preferably works in the Dirac representation of the Gamma matrices. Of course such a limit can only be discussed for massive particles. From (4.21) we have1    (1)  r χ(σ) ω+m U (~ p, σ) . (6.12) U (~ p, σ) = = p ~·~ σ 2m U (2) (~ p, σ) ω+m χ(σ) Noting that p~ p~ 1 ' ' v, ω+m 2m 2 1 J.D.

Bjorken and S.D. Drell, Relativistic Quantum Fields (McGraw-Hill, 1965).

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we see from (6.12) that U (2) (~ p, σ) '

~v · ~σ (1) U (~ p, σ) . 2

Hence one refers to the upper (lower) components of the positive energy spinor U (p, σ) as the “large” (“small”) components. For the case of the negative energy spinor V (p, σ) the situation is just the reverse. Hence in the non-relativistic limit the positive and negative energy solutions have the form   p ~2 χ(σ) ψ (+) (x) ∼ e−imt ei~p·~x−i 2m t , 0   p ~2 0 ψ (−) (x) ∼ eimt e−i~p·~x+i 2m t c . (6.13) χ (σ) They evidently satisfy i∂t ψ (±) (x)n.r. = ±Hn.r. ψ (±) (x)n.r. , with

~2 ∇ +m . (6.14) 2m This equation shows that only the solutions of positive energy lead to the Schr¨odinger equation in the non-relativistic limit. The mass dependent exponential in (6.13) is irrelevant, and merely corresponds to a shift in the “zero point energy”; it accounts for the rest-mass energy mc2 of a particle at rest. Hn.r. = −

6.3

Negative-energy solutions and localization

If at time t = 0 we have a Gaussian wave packet localized within a distance 2d around r = 0, such as      1  2 1 1 1 χ( 2 ) − r2 2d U e ψ(~r, t = 0) = 0, , U 0, = , (6.15) 2 2 0 (πd2 )3/4 then this corresponds to a Gaussian momentum distribution with uncertainty ∆p ' ¯h/d in the momentum. To make this explicit, we recall that the general solution to the Dirac equation at any time is of the form Z r X   1 m 3 ψ(~r, t) = d p U (~ p, σ)a(~ p, σ)e−ip·x + V (~ p, σ)b∗ (~ p, σ)eip·x , ωp σ (2π)3/2 p). Since the Fourier coefficients a(p, σ) and b∗ (p, σ) are indewhere p0 = ωp = ω(~ pendent of time, they are determined by the initial condition (6.15) via the orthogonality relations ωp U † (~ p, σ)U (~ p, σ 0 ) = δσσ0 m

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63

ωp δσσ0 m U † (~ p, σ)V (−~ p, σ 0 ) = 0. V † (~ p, σ)V (~ p, σ 0 ) =

(6.16)

One has    Z r2 1 1 2m † U d3 re− 2d2 e−i~p·~r , (~ p , σ)U 0, 3/2 2 3/4 2 (2π) (πd )    Z r2 1 2m 1 † d3 re− 2d2 ei~p·~r . V b∗ (~ p, σ) = (~ p , σ)U 0, 3/2 2 3/4 2 (2π) (πd ) a(~ p, σ) =

The remaining integral can be computed by the usual procedure of rewriting the exponent as a quadratic form and making a distortion of the integration contour in the complex plane, or by using the formulae Z ∞ Z ∞ 4π ∂ dr3 f (|~r|)e−i~p·~r = − drf (r) cos pr p ∂p 0 −∞ and

Z

r2

dre− 2d2 cos pr =

r

π − p2 d2 e 2 , 2

where p = |~ p|. Either procedure gives Z |~ p|2 d2 r2 d3 re− 2d2 ei~p·~r = (2πd2 )3/2 e− 2 . Hence we have a(~ p, σ) = 2m

∗

b (~ p, σ) = 2m





d2 π

d2 π

3/4

3/4

e−

e

−

|~ p|2 d2 2

|~ p|2 d2 2

U † (~ p, σ)U

†

V (~ p, σ)U

  1 0, 2 

1 0, 2



.

Now, using (4.21) and (6.15) we have   r 1 ω+m δ 1 U † (~ p, σ)U ~0, = 2 2m σ, 2 r      p~ · ~σ 1 1 ω+m † σ+ 21 † ~ = (−1) χ (−σ) χ . V (~ p, σ)U 0, 2 2m ω+m 2 Hence 

d2 π

3/4 r

ω + m − |~p|2 d2 2 δσ, 12 , e 2m  2 3/4 r    d 1 ω + m − |~p|2 d2 |~ p| ∗ σ+ 21 † 2 2m χ (−σ)~σ · n ˆχ b (~ p, σ) = (−1) e , π 2m 2 ω+m

a(~ p, σ) = 2m

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where n ˆ=

p~ , |~ p|

ω = ωp .

Hence the Fourier coefficients associated with the negative energy solutions are of order |~ p|/(ω + m) smaller than those associated with the positive energy solutions. They only become relevant at momenta comparable with the rest mass of the particle. In particular this means that a non-relativistic Schr¨odinger limit can only exist provided the uncertainty in the momentum is smaller than the rest mass of the particle. Since the uncertainty in the momentum for the wave packet in question is ∆pd ' ¯h this means that for a non-relativistic limit to exist, we must demand that d>

¯ h = λc . mc

This condition is of course satisfied for plane waves, which correspond to d = ∞. In that case a non-relativistic limit exists for sufficiently small momenta. In fact, a positive energy plane wave at time t = 0 will remain a positive energy plane wave at all times. On the other hand, if the particle is localized initially within a Compton wave length, negative energy components will contribute significantly. Such a localization can for instance be achieved by the application of sufficiently strong external fields. The result of applying such fields is beautifully exemplified by the Klein paradox, which we now discuss.

6.4

The Klein Paradox

As we have seen in the previous section, negative energy solutions are expected to play a role if we try to localize a particle within a Compton wave length by applying for instance a sufficiently strong field. As we shall see later on, the effect of such a field can be understood in terms of electron-positron pair creation. This picture, however, also prevails if we consider the scattering of an electron by a static Coulomb potential in the absence of a radiation field. In that case we can always ~ vanishes. The Dirac Hamiltonian choose a gauge in which the vector potential A correspondingly takes the form ~ + βm + V (~r) H = −i~ α·∇ with V (~r) = eφ(~r), A0 = φ . We look for stationary solutions of the corresponding Dirac equation, i∂t ψ = Hψ having the form ψ(~r, t) = e

−iEt

  ψ1 (~r) . ψ2 (~r)

(6.17)

(6.18)

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65

Working in the Dirac representation, substitution of (6.18) into (6.17) leads to the coupled set of equations ~ 2 + (m + V )ψ1 Eψ1 = −i¯ σ · ∇ψ ~ 1 + (−m + V )ψ2 . Eψ2 = −i¯ σ · ∇ψ From the second equation we have ψ2 (~r) =

−i ~ 1 (~r) . ~σ · ∇ψ E + m − V (~r)

(6.19)

Substitution of this expression into the first equation leads to an equation for the “large” component ψ1 (~r): ~ [E − V (~r) − m]ψ1 (~r) = (−i~σ · ∇)

1 ~ 1 (~r) . (−i~σ · ∇)ψ E − V (~r) + m

(6.20)

We now specialize to the case of one spatial dimension. Suppose V (~r) to be given by the simple potential barrier in one spatial z-dimension depicted in the figure below,

V

V0 I

II z

where the momentum of the plane waves takes the following values in regions I and II:  2 kI = E 2 − m 2 2 . (6.21) k = 2 kII = (E − V0 )2 − m2

For an incident spin 1/2 particle which is polarized in the z-direction we have from (6.20) and (6.19) for the incident and reflected waves in region I   χ(↑) (I) ψin (z) = a↑ eikI z kI E+m χ(↑) and (I) ψref l (z)

(  = e−ikI z b↑

  ) χ(↑) χ(↓) + b↓ kI . −kI E+m χ(↑) E+m χ(↓)

Since we have no reflection in region II, the solution ψ (II) (x) has in this case the form (    ) χ(↑) χ(↓) (II) ikII z ψ (z) = e d↑ + d↓ . kII −kII E+m−V0 χ(↑) E+m−V0 χ(↓)

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Continuity at the potential barrier (current conservation) requires (we choose the potential barrier to lie at z = 0) a↑ + b↑ = d↑ ; b↓ = d↓ kII E + m (a↑ − b↑ ) = d↑ kI E − V0 + m

;

b↓ = −

It is thus convenient to define

r := We then have to solve 2a↑ = (1 + r)d↑ ,

kII E + m d↓ . kI E + m − V0

kII E + m . kI E − V0 + m

2b↑ = (1 − r)d↑ ,

b↓ = rd↓ ,

(6.22)

b↓ = d↓ .

Hence b↓ = d↓ = 0, i.e. there is no spin flip. Now, according to (6.21) kII is real for E > V0 + m, E < V0 − m. Hence free particle propagation in region II occurs for these values. Only the case E < V0 − m corresponding to free propagation in a classically forbidden region is of interest to us. The inequality states that for a potential strength of at least twice the mass of the particle there will be registered a current flow corresponding to the propagation of free particles. ¯ 3 ψ can be expressed in terms The ratio of transmitted to incident current J = ψγ of this parameter r: Jtr 4r = Jin (1 + r)2 Jref l (1 − r)2 Jtr =1− = . Jin (1 + r)2 Jin For the classically allowed energy range E − V0 > m we have 0 < r < 1, and correspondingly the usually expected result 0
= −i dtHI (t) −∞ < 0|e |0 >

By expanding the exponential in powers of HI (t) we obtain a perturbative representation of the Green functions: R n n P ∞ d4 yHI (y) |0 > < 0|T ϕI (x1 )...ϕI (xn ) n=0 (−i) n! −∞ < Ω|T ϕ(x1 )...ϕ(xn )|Ω >= < 0|U (∞, −∞)|0 > (11.21) where the normalization factor has the corresponding perturbative expansion ( n ) X (−i)n Z ∞ 4 d yHI (y) |0 > . (11.22) T < 0|U (∞, −∞)|0 >=< 0| 1 + n! −∞ n=1 Here HI (y) is the interaction Hamiltonian density as defined by Z HI (t) = d3 yHint (ϕI (~y , t), πI (~y , t)) .

(11.23)

Hence each term in the expansion (11.21) is explicitly computable, since it involves only the (free) interaction picture fields. The interaction Hamiltonian density HI (y) is understood to be normal ordered with respect to these fields in order to allow for a lower bound of the ground-state energy (for an appropriate sign of the coupling constant). The computation of each term in the expansion (11.21) is then based on three theorems due to Wick, which we discuss next.

11.3

Wick theorems

For a formulation of the Wick theorems we need to introduce the notion of a simple contraction as well as time-ordered contraction of two free operators. Simple contraction Consider the product A(x)B(y) of two free field operators. We write this product as the sum of a “normal-ordered” product and a “simple contraction”, which we denote by a lower square bracket: A(x)B(y) =: A(x)B(y) : +A(x)B(y).

(11.24)

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The first term stands here for the “normal ordered product” : A(x)B(y) := A(+) (x)B (+) (y) + A(−) (x)B (−) (y) + A(−) (x)B (+) (y) + (−1)(A)(B) B (−) (y)A(+) (x) , where A(+) (x) and A(−) (x) (etc.) denote the positive and negative frequency parts of A(x) in the decomposition A(x) = A(+) (x) + A(−) (x) in terms of destruction and creation operators, and the positive frequency parts are arranged to stand to the right of the negative energy parts, taking account of statistics. The second term is a c-number. We have : A(x)B(y) := (−1)(A)(B) : B(y)A(x) : ,

(11.25)

where (Q) denotes the Grassman signature of the operator Q. Since < 0| : A(x)B(y) : |0 >= 0 it follows that < 0|A(x)B(y)|0 >=< 0|A(x)B(y)|o > , and since A(x)B(y) is a c-number, we further have, A(x)B(y) =< 0|A(x)B(y)|0 > . In particular, since A(−) (x)B (+) (x) is already in normal-ordered form, we have A(−) (x)B (+) (y) = 0. On the other hand, A(+) (x)B (−) (y) = (−1)(A)(B) B (−) (y)A(+) (x) + A(+) (x)B (−) (y) , or A(+) (x)B (−) (y) = [A(+) (x), B (−) (y)] where the bracket stands for the generalized commutator including statistics, [A(x), B(y)] = A(x)B(y) − (−1)(A)(B) B(y)A(x) .

(11.26)

Time-ordered contraction: We repeat the above discussion for the case of the time-ordered product of two field operators, defined in general by T A(x)B(y) = θ(x0 − y 0 )A(x)B(y) + (−1)(A)(B) θ(y 0 − x0 )B(y)A(x) ,

(11.27)

where (A) and (B) denote again the Grassman signature of the operators A and B. From (11.27), (11.25) and θ(x) + θ(−x) = 1, we have T (A(x)B(y)) =: A(x)B(y) : +A(x)B(y) ,

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141

i.e. A(x)B(y) =< 0|T A(x)B(y)|0 > .

(11.28)

It is thus convenient to define a new contraction, the time-ordered contraction of two operators, represented by an upper bracket: A(x)B(y) = θ(x0 − y 0 )A(x)B(y) + (−1)(A)(B) θ(y 0 − x0 )B(y)A(x) . Time-ordered contractions thus behave with respect to the exchange of the two operators like T -products. We are now ready to state Wick’s theorems:3 Wick theorem No. 1 The product of (free) operators is equal to the corresponding normal-ordered product plus the normal products including all possible contractions. X A1 ...An = : A1 ...An : + : A1 ...Ai ...Aj ...An : A1 ...Ai ...Aj ...An : +

X

i,j

: A1 ...Ai ...Aj ...Ak ...Al ...An :

i,j,k,l

Here the terms involving contractions are understood to be computed by first bringing the respective pairs next to each other by taking due account of the signature of the fields. This applies as well to the following theorems. Corollary to Wick theorem No. 1 If an operator product is itself a normal-ordered product of operators, then all contractions of the operators within this normal product are to be omitted. Wick theorem No. 2 The T -product of (free) operators is equal to the sum of the corresponding normalordered product and the normal ordered products including all possible time-ordered contractions: X T (A1 ...An ) = : A1 ...An : + : A1 ...Ai ...Aj ...An : A1 ...Ai ...Aj ...An : i,j

+

X

: A1 ...Ai ...Aj ...Ak ...Al ...An :

(11.29)

i,j,k,l

Corollary to Wick’s theorem No. 2 If one of the operators in the T -product of (free) operators is itself a normal product of operators, then all time-ordered contractions of the operators within this normal ordered product are to be omitted. 3 G.C.

Wick, Phys. Rev. 80 (1950) 208.

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Wick theorem No. 3 The vacuum expectation value of the time-ordered product of n + 1 free operators A, B1 , B2 , ...Bn is equal to the sum of the n possible vacuum expectation values of the time-ordered products which one obtains by contracting the operator A with each of the n operators Bi , i = 1, ...n: < 0|T AB1 ...Bn |0 >=

n X i=1

< 0|T AB1 ...Bi ...Bn |0 > .

The proof of this theorem is based on the fact that in a vacuum expectation value of such an operator product only those terms survive, in which all of the operators, including A, have been contracted. Theorem 3 implies that in the computation of the individual terms in the expansion (11.21) of a general Green function, all of the fields will have to be contracted in order to give a non-vanishing contribution. Hence each term in the expansion (11.21) will in general involve a multiple integral over a product of pairwise time-ordered contractions represented by the vacuum expectation value of two free time-ordered fields according to (11.28). The following section is thus devoted to a discussion of the various 2-point functions associated with the bosonic, fermionic, and radiation fields.

11.4

2-point functions

As we have seen, the simple and time-ordered Wick contractions are c-number functions, generally referred to as 2-point functions. They represent the basic building blocks of any perturbative calculation, and hence will be discussed in the following sequel. We do this in detail for the case of the real scalar free field, limiting ourselves to a statement of the results for the case of fermions and the radiation field. Basic definitions In the case of a real scalar field φ(x), the following 2-point functions play a role in perturbative QFT:4 < 0|[φ(x), φ(0)]|0 >, = i∆(x) < 0|φ(+) (x)φ(−) (0)|0 >, = i∆(+) (x) −θ(x0 ) < 0[φ(x), φ(0)]|0 > = i∆ret (x) = −iθ(x0 )∆(x) 0

(11.30)

0

θ(−x ) < 0|[φ(x), φ(0)]|0 > = i∆adv (x) = iθ(−x )∆(x) < 0|T φ(x)φ(0)|0 > = i∆F (x) . Of these the first two 2-point functions are evidently solutions of the (homogeneous) Klein–Gordon equations: ( 4 We

+ m2 )∆(x) = 0,

have used translational invariance.

(

+ m2 )∆(+) (x) = 0 .

(11.31)

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143

The remaining three are the Green functions corresponding to different boundary conditions: (

+ m2 )∆ret/adv (x) = δ 4 (x)

(

+ m2 )∆F (x) = −δ 4 (x).

(11.32)

Equations (11.32) follow from the commutation relations [φ(x), φ(0)]ET = 0 ,

[φ(x), ∂0 φ(y)]ET = iδ 3 (x − y)

and the property ∂0 θ(x0 ) = δ(x0 ) of the Heaviside step function. For ∆F (x), ∂0 < 0|T φ(x)φ(0)|0 > = < 0|T ∂0 φ(x)φ(0)|0 > ∂02 < 0|T φ(x)φ(0)|0 > = < 0|[∂0 φ(x)φ(0)]|0 > δ(x0 )+ < 0|T ∂02 φ(x)φ(0)|0 > ~ 2 − m2 )φ(x)φ(0)|0 > , = −iδ 4 (x)+ < 0|T (∇ ~ 2 + m2 )φ(x) = 0. where we have used (∂02 − ∇ Fourier representations We now seek the Fourier representations of the various 2-point functions. For the function ∆(x) we had already found (see Eq. (7.34)) Z 3 h i d q −iωq x0 i iωq x0 e − e ei~q·~x (2π)3 2ωq Z i d4 q(q 0 )δ(q 2 − m2 )e−iq·x . =− (2π)3

∆(x) = −

Recalling the definition of the epsilon function (q 0 ) = θ(q 0 ) − θ(−q 0 ) , we may separate ∆(x) into its positive and negative frequency parts ∆(x) = ∆(+) (x) + ∆(−) (x) , where Z i d4 qθ(q 0 )δ(q 2 − m2 )e−iq·x (2π)3 Z i (−) ∆ (x) = d4 qθ(−q 0 )δ(q 2 − m2 )e−iq·x . (2π)3 ∆(+) (x) = −

Note that ∆(−) (x) = −∆(+) (−x) .

(11.33)

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We further have from (11.33) ∆

(+)

i (x) = − (2π)3

Z

d3 q −iωq x0 +i~q·~x e . 2ωq

Recalling further the following property of the step function θ(x) + θ(−x) = 1 we also have the following decomposition into an advanced and retarded part: ∆(x) = ∆adv (x) − ∆ret (x) . Making use of the Fourier representation 1 θ(x ) = − 2πi 0

Z

∞

dq

0e

−∞

−iq 0 x0

q 0 + i

,

1 θ(−x ) = 2πi 0

Z

∞

0

dq 0

−∞

0

e−iq x q 0 − i

(11.34)

of the θ-function, where  is an infinitesimal parameter, we have, ∆adv (x) = θ(−x0 )∆(x) Z 3  Z 0 0  −i d q −iωq x0 dq 0 e−iq x iωq x0 × e − e ei~q·~x = 2πi q 0 − i (2π)3 2ωq  Z 4  1 d q 1 1 =− − e−iq·x , (2π)4 2ωq q 0 − ωq − i q 0 + ωq − i where a shift in the q 0 -integration has been made. Proceeding in the same way for ∆ret , and combining denominators we may summarize the results in the compact form Z

e−iq·x d4 q 4 2 (2π) q − m2 − iq 0  Z e−iq·x d4 q . ∆ret (x) = − 4 2 (2π) q − m2 + iq 0 

∆adv (x) = −

Note that the epsilon prescriptions correspond to the integration contours shown in Fig. 11.1.

−ωq ·

· ωq

CA

−ωq ·

· ωq

CR

Fig. 11.1. Integration contour and pole location for ∆adv (x) and ∆ret (x).

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In order to get the Fourier representation of the time-ordered 2-point function ∆F (x), we write it as ∆F (x) = θ(x0 )∆(+) (x) + θ(−x0 )∆(+) (−x) , and proceed as before. Making use again of the Fourier representation (11.34) of the step function, we have " # Z ∞ Z 3 −i(q 0 +ωq )x0 +i~ q ·~ x −i(q 0 −ωq )x0 +i~ q ·~ x 1 i d q e e dq 0 ∆F (x) = − , 2πi −∞ (2π)3 2ωq q 0 + i q 0 − i or making a shift in the q 0 -integration variable ∆F (x) =

Z

d4 q 1 (2π)4 2ωq



1 1 − q 0 − ωq + i q 0 + ωq − i



e−iq

0

x0 +i~ q ·~ x

.

An algebraic combination of the two terms results in ∆F (x) =

Z

e−iqx d4 q . (2π)4 q 2 − m2 + i

We check the result by observing that indeed ( + m2 )∆F (x) = −δ 4 (x). The epsilon in the denominator again tells us that the integration in the q 0 -plane is to be performed along the contour CF with pole locations shown in the figure below.

−ωq · · ωq

CF

Fig. 11.2. Integration contour and pole locations for ∆F (x). It is evident from the Fourier representations above that the 2-point functions ∆ret (x), ∆adv (x) and ∆F (x) satisfy the equations for a Green function of the Klein– Gordon operator. The function ∆F (x) is called the Feynman propagator for a scalar field and will play a central role in the perturbative expansion of a general n-point “Green function”, as is evident from the Wick theorems 2 and 3 of the previous section. The above functions are singular on the light cone and exhibit logarithmic cuts in the variable x2 . The respective Fourier integrals may be computed to give5 the following explicit representations of the 2-point functions ∆(x), ∆(±) (x) and ∆F (x) 5 N.N.

Bogoljubov and O.V. Sirkov, Quantenfelder (Physik Verlag, 1984), p. 281.

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√ 1 m (x0 )δ(x2 ) + √ (x0 )θ(x2 )J1 (m x2 ) 2 2π 4π x 1 (±) 0 2 ∆ (x) = (x )δ(x ) 8π   p √ √ mθ(x2 ) imθ(−x2 ) 0 2 2 2 √ √ (x )J1 (m x ) ± iN1 (m x ) ± K1 (m −x ) + 4π 2 −x2 8π x2 √ im K1 (m −x2 + i) √ ∆F (x) = . (11.35) 4π 2 −x2 + i ∆(x) = −

The corresponding expressions for the retarded and advanced Green function then follow from their definition (11.30). The functions J1 (x), N1 (x) and K1 (x) are the Bessel, Neumann and Bessel functions of the second kind (Hankel function for imaginary argument), respectively. The explicit expressions (11.35) show that the 2-point functions are singular on the light-cone. Indeed, we have for small argument6 z + O(z 3 ) 2  21 z z N1 (z) = − + ln + 1 + O(z 3 ) πz π 2  1 z z K1 (z) = + ln + 1 + O(z 3 ) . z 2 2 J1 (z) =

Introducing these expansions into (11.35) we have in particular for ∆(x), ∆(x) = −

1 m2 (x0 )δ(x2 ) + (x0 )θ(x2 ) + O(x3 ) . 2π 8π

The functions (11.35) belong to the class of generalized functions or distributions and are strictly defined only under an integral when acting on sufficiently smooth test functions ζ(x) Z f [ζ] = dxf (x)ζ(x) . Quite generally the behaviour of Green functions on the light cone determines the behaviour of scattering processes at high energies. The singular behaviour for x2 → 0 also lies at the root of the need for renormalization of quantum field theories, as we shall see in Chapter 16. Causality We have seen that the 2-point function ∆(x) is causal in the sense that it vanishes for space-like distances, which cannot be connected by a light signal: ∆(x) = 0,

x2 < 0 .

6 See Magnus, Oberhettinger and Soni, “Formulas and Theorems for the Special Functions of Mathematical Physics”.

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This property we had already established on the basis of Lorentz invariance and equal-time commutation relations. It no longer holds for the 2-point functions ∆(±) (x) and ∆F (x). We nevertheless have that both of these functions are exponentially damped for large space-like distances if the boson is massive: ∆F (x) =

im K1 (m|x|) , 4π 2 |x|

x2 < 0 .

Pauli has given an interpretation for this acausal behaviour: Suppose we measure the position of an electron at the point x at time t with a shutter. At the same time another observer at another position y makes a measurement to see whether he finds the electron there. According to the acausal behaviour of ∆F (x − y) he will find an electron with a certain probability. This, however, reflects the fact that in the course of the measurement observer 2 creates an electron-positron pair from the vacuum, of which the positron annihilates the electron measured by observer 1. Table of 2-point functions of free scalar, fermion and Maxwell fields Proceeding as in Chapter 7 we further obtain the 2-point functions for the fermion field and the gauge fields of QED in the Coulomb gauge. All results are summarized in the following table. (I) Scalar field < 0|[φ(x), φ(0)]|0 >= i∆(x) < 0|φ(x)φ(0)|0 >= φ(x)φ(0) = i∆(+) (x) < 0|T φ(x)φ(0)|0 >= φ(x)φ(0) == i∆F (x) with ∆F (x) =

Z

e−iqx d4 q , 4 2 (2π) q − m2 + i

(

+ m2 )∆F (x) = −δ 4 (x)

(II) Fermion field < 0|[ψα (x), ψ¯β (0]|0 >= iS(x)αβ = (i∂/ + m)αβ i∆(x) (11.36) (+) (+) < 0|ψα (x)ψ¯β (0)|0 >= ψα (x)ψ¯β (0) = iS (x)αβ = (i∂/ + m)αβ i∆ (x) < 0|T ψα (x)ψ¯β (0)|0 >= ψα (x)ψ¯β (0) = iSF (x)αβ = (i∂/ + m)αβ i∆F (x) with SF (x) =

Z

d4 p (p/ + m) e−ipx 4 2 (2π) p − m2 + i

(i∂/ − m)SF (x) = δ 4 (x) . Here [,] denotes the generalized commutator (7.27).

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(III) Gauge-field in Coulomb gauge   ∂i∂j < 0|[Ai (x), Aj (0)]|0 >= δ ij − iD(x) ∆   ∂i∂j ij i j ij < 0|T A (x)A (0)|0 >= iDF (x) = δ − iDF (x) ∆ where DFij (x)

Z

i j

k k ij d4 k δ − ~k2 −ikx e = , ∂i DFij (x) = 0 (2π)4 k 2 + i   ∂i∂j ij ij DF (x) = − δ − δ 4 (x) . ∆

(11.37)

Notice that in the Coulomb gauge ∂i Ai (x) = 0, which is reflected by the equations above.

11.5

Feynman Diagrams for QED

According to formulae (10.25), the calculation of the differential cross-section for electron-electron (Moeller) scattering, electron-positron (Bhabha) scattering, and electron-photon (Compton) scattering, as well as generalizations thereof involving arbitrary numbers of electrons, positrons and photons in the final state, require the calculation of the transition amplitude from the LSZ reduction formula (10.48). A general 2 → 2 scattering process is represented diagrammatically in the figure below.

Fig. 11.3. A general 2 → 2 scattering process.

The arrows indicate the flow of momentum. With every ingoing electron (out¯ going positron) is associated a field ψ(x), whereas with every outgoing electron (ingoing positron) is associated a field ψ(x). Note that there is the same number n ¯ of fields ψ(x) and ψ(x), reflecting conservation of charge. We see from (11.21) that a general n-point Green function G can be written as the ratio of two vacuum expectation values of time ordered products. For the sake of clarity, let us write symbolically G=

ˆ G , V

where V stands for “vacuum”. We have from (11.21) and (11.22), ( " n #) X (−i)n Z ∞ 4 ˆ G =< 0|T ϕI (x1 )...ϕI (xn ) [1 + d zHI (z) |0 > n! −∞ n=1

(11.38)

(11.39)

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(

n ) X (−i)n Z ∞ 4 d zHI (z) 1+ |0 > . n! −∞ n=1

149

(11.40)

We now give a diagrammatic interpretation of V for the case of QED. Diagrammatic representation of V For Electrodynamics (note that HI is normal ordered) HI (x) = −LI (x) = e0 : ψ¯I (x)γµ ψI (x)AµI (x) : ,

(11.41)

where e0 is the (unrenormalized, negative) bare electron charge. Consider V to lowest order O(e20 ): Z Z (−ie0 )2 4 d z1 d4 z2 V ' 1+ 2! h0|T : ψ¯I (z1 )γµ ψI (z1 )AµI (z1 ) :: ψ¯I (z2 )γν ψI (z2 )AνI (z2 ) : |0i or doing the contractions R R V (2) = 1 − 21 (−ie0 )2 d4 z1 d4 z2 tr[γ µ iSF (z1 − z2 )γ ν iSF (z2 − z1 )]iDF (z1 − z2 )µν .

Note that an extra minus sign emerged as a result of the reordering of the fermion fields when computing the contractions. Note further that the fermionic part could be written as a trace. Both are characteristic for every fermion loop. Diagrams which are not connected to any external line are called vacuum diagrams. Examples are shown in the figures below.

(11.42) Fig. 11.4. Second order vacuum graphs in φ3 and φ4 theory. We now show that these vacuum diagrams cancel to all orders in the definition (11.38) of G. Cancellation of vacuum diagrams ¯ 1 )ψ(y ¯ 2 )|Ω) > in Consider as an example the 4-point function < Ω|T ψ(x1 )ψ(x2 )ψ(y QED. According to (11.39), R 4 n o ˆ 1 , x2 ; y1 , y2 ) = h0|T ψI (x1 )ψI (x2 )ψ¯I (y1 )ψ¯I (y2 )e−i d zHI (z) |0i G(x

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or, conveniently written using Wick’s theorems,   p ∞ X X (−i)p p ˜ 1 , x2 ; y1 , y2 ) = G(x (11.43) p! l p=0 l=0 Z × d4 z1 ...d4 zl h0|T ψI (x1 )ψI (x2 )ψ¯I (y1 )ψ¯I (y2 )HI (z1 )...HI (zl )|0iP C Z × dz`+1 . . . dzp h0|T HI (zl+1 )...HI (zp )|0i , where the subscript P C stands for “partially connected”, i.e. diagrams involving no (disconnected) vacuum diagrams. The binomial coefficient   p p! = `!(p − `)! l is the number of ways one can select l interaction Hamiltonians from a total of p. We may rewrite the summation on the r.h.s. of (11.43) as follows: rhs =

Z ∞ X (−i)l l=0

l!

×

d4 z1 ...d4 zl h0|T ψI (x1 )ψI (x2 )ψ¯I (y1 )ψ¯I (y2 )HI (z1 )...HI (zl )|0iP C Z ∞ X (−i)p−l p=l

(p − l)!

dz`+1 . . . dzp h0|T HI (zl+1 )...HI (zp )|0i .

The last factor can be rewritten as (p = l + r) Z ∞ X (−i)r r=0

r!

dz1 . . . dzr h0|T HI (z1 )...HI (zr )|0i = h0|T e−i

R

d4 zHI (z)

|0i = V .

ˆ and G in (11.38) becomes Hence V factorizes out from G, Z X (−i)l Z d4 z1 ... d4 zl G(x1 , x2 ; y1 , y2 ) = l! l

×h0|ψI (x1 )ψI (x2 )ψ¯I (y1 )ψ¯I (y2 )HI (z1 )...HI (zl )|0iP C .

This explicitly demonstrates the cancellation of vacuum graphs to all orders. We now take account of this from the outset in order to compute the fermion and photon propagator in second order of QED. Second order Fermion propagator of QED Making use of Wick’s theorems we have to second order in the coupling constant for the fermion 2-point function 2 ¯ β |Ω >'< 0|T ψI (x)α ψ¯I (y)β |0 > + (−ie0 ) < 0|T ψI (x)α ψ¯I (y)β < Ω|T ψ(x)α ψ(y) 2! Z Z 4 4 ¯ ¯ × d z1 : ψI (z1 )A / I (z1 )ψI (z1 ) : d z2 : ψI (z2 )A/ I (z1 )ψI (z2 ) : |0 > . (11.44)

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151

In order to write down an explicit expression, we first reorder the fields by bringing contracted pairs adjacent to each other in standard form, keeping track of the number of permutations involved, and the corresponding Grassman signature. All fields must be contracted in order to give a non-vanishing result. Representing the time-ordered contractions by ψαI (x)ψ¯βI (y)

= α, x

β, y

(11.45)

AµI (x)AνI (y)

= µ, x

ν, y

(11.46)

Fig. 11.5. Graphical representation of contractions. we thus obtain < Ω|T ψα (x)ψ¯β (y)|Ω >≡ iSF0 (x − y) = iSF (x − y) + (11.47) Z (−ie0 )2 d4 z1 d4 z2 [iSF (x − z1 )γµ iSF (z1 − z2 )γν iSF (z2 − y)]αβ iDFµν (z1 − z2 ) + · · · as represented by the sum of diagrams

x, α

y, β '

+

(11.48)

Fig. 11.6. Electron propagator to second order. 1 Notice that the combinatorial factor 2! in (11.21) has been omitted since there are 2! contractions leading to identical results.

Second order photon propagator of QED In a similar way we have to second order for the 2-point function of the gauge field, after cancellation of the contribution of the vacuum graphs, (−ie0 )2 < Ω|T Aµ (x)Aν (y)|Ω >=< 0|AµI (x)AνI (y)|0 > + < 0|T AµI (x)AνI (y) 2! Z × d4 z1 d4 z2 : ψ¯I (z1 )A / I (z1 )ψI (z1 ) :: ψ¯I (z2 )A/ I (z2 )ψI (z2 ) : |0 > + · · · (11.49)

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or explicitly < Ω|T Aµ (x)Aν (y)|Ω >≡ iDF0 (x − y) ' iDFµν (x − y) + (11.50) Z Z + (−ie0 )2 d4 z1 d4 z2 iDFµλ (x − z1 )tr [γλ iSF (z1 − z2 )γρ iSF (z2 − z1 )] iDFρν (z2 − y). The result has the diagrammatic representation

x, µ

y, ν '

+

(11.51)

Fig. 11.6. Photon propagator to second order.

Topologically equivalent diagrams The perturbative results obtained by performing Wick contractions keeping track of the Grassman signature of the fields, performing the space-time integrations dictated by the LSZ reduction formula, and attaching the respective momentumspace wave functions associated with the external states involved in the process can be resumed by a set of rules, called Feynman rules. They consist in first drawing all diagrams contributing to the perturbative expansion of the relevant Green function in configuration space up to the desired order in the coupling constant e0 , taking care of including only topologically inequivalent diagrams. In order to decide whether two diagrams are topologically equivalent or not, we must first label the fermion lines by arrows indicating the direction of flow of (negative) electron charge. Conservation of charge requires that the direction of flow of this charge is always the same as we follow through a fermionic “path”. Since the gauge field is electrically neutral, there is no arrow to be associated with a photon line. We then label all external lines in configuration space. Thus, for example, the diagrams

x

y x

y

(11.52)

Fig. 11.7(a). Topologically equivalent diagrams. are topologically equivalent since they are the same under reflection about the xyaxis. On the other hand, the diagrams

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153

y2

x2

x2

y2

(11.53)

x1

y1

x1

y1

Fig. 11.7(b). Topologically inequivalent diagrams. are topologically inequivalent. Note that in these diagrams we have not labeled the corresponding vertices. For such an n-point 1-loop diagram there exist n! possible reorderings of these vertex labels for a diagram of order n. Since the corresponding coordinates are integrated over, every reordering will give the same result. Remembering that from the expansion of the exponential in (11.20) we also have to include a factor 1/n!, we can ignore this factor if we do not count diagrams which only differ in their labeling of the vertices. As one easily convinces oneself, every closed fermion loop involves an odd number of permutations of the fermion fields in the computation of the contractions so that a minus sign will have to be associated with every fermion loop. Gauge field loops do not occur because of the linear dependence on Aµ (x) of the interaction Hamiltonian.

11.6

Furry’s theorem

The following theorem allows us to discard from the outset an in principle infinite set of Feynman diagrams. Furry’s theorem All Feynman diagrams containing a fermion loop with an odd number of vertices can be omitted. Proof: Furry’s theorem is a consequence of the charge conjugation invariance of QED. Under charge conjugation the gauge field transforms as follows: Uc Aµ (x)Uc−1 = −Aµ (x) . Making use of the charge conjugation invariance of the physical vacuum, we thus have < Ω|T Aµ1 (x1 )...Aµn (xn )|Ω >= (−1)n < Ω|T Aµ1 (x1 )...Aµn (xn )|Ω > , or < Ω|T Aµ1 (x1 ) · · · Aµn (xn )|Ω >= 0,

n odd .

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It will be useful to understand this result diagrammatically. Recall that we had introduced in (4.13) a matrix c with the property, c~σ c−1 = −~σ ∗ . Correspondingly we defined a matrix C in (4.17) with the property Cγ µ C −1 = γ µT , where T denotes the transpose. With the definition C | = γ 5 C, it then follows that C | γµC | −1 = −γ µT ,

(11.54)

implying for the fermion propagator C | SF (x)C | −1 = SFT (−x) .

(11.55)

Now, for every fermion loop contributing to a general diagram there will be a corresponding fermion loop with the fermion charge flowing in the opposite direction. Isolating the loop-part, we have for a loop with n vertices, Z   loop1 = tr γ µn SF (zn − zn−1 )γ µn −1 SF (zn−1 − zn−2 ) · · · γ µ1 SF (z1 − zn ) , and for the charge flow in the opposite direction, Z loop2 = tr [SF (zn − z1 )γ µ1 · · · SF (zn−2 − zn−1 )γ µn−1 SF (zn−1 − zn )γ µn ] ,

where the integrals stand for the integration over the space-time coordinates labelling the vertices. Rewriting the trace in the second integrand as tr C | [· · ·]C | −1 will not change its value. Making use of (11.54) and (11.55) and adding the two loop contributions, we have (1 + (−1)n )tr[γ µn SF (zn − zn−1 )γ µn−1 SF (zn−1 − zn−2 ) · · · γ µ1 SF (z1 − zn )] , showing that the two contributions add to zero for n odd.

11.7

Going over to momentum space

Having written down all topologically inequivalent diagrams with their respective labelling of electron charge flow, we go over to momentum space by replacing the space-time 2-point functions by their respective Fourier transforms. For each argument of the 2-point functions there will be a momentum in Fourier space associated with the particles participating in the interaction. Translational invariance will imply an energy-momentum conservation delta function at each vertex. We thus label the internal lines of the diagram by the momentum to be integrated over. There will survive an overall energy-momentum conserving delta function of the form G(p1 · · · pn ) = (2π)4 δ 4 (p1 + p2 + ... + pn )G0 (p1 · · · pn )

(11.56)

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155

where we have taken all external momenta to be incoming. We now exemplify the transition to momentum space for the case of the electron 2-point function. Electron and photon 2-point function in momentum space Consider the 2-point function (11.47). Taking the Fourier transform we have Z Z d4 x d4 yeip·x eiq·y SF0 (x − y) = (2π)4 δ 4 (p + q)SF0 (p) where SF0 (p) =

Z

d4 zeip·z SF0 (z) .

Hence SF0 (p) corresponds in this case to G0 in (11.56), where the overall deltafunction has been factored out. Going to momentum space we have from (11.47) for the fermion propagator in second order perturbation theory Z Z d4 p −ip(x−y) d4 p −ip(x−y) 0 e iS (p) = e iSF (p) + F 4 (2π) (2π)4 Z Z Z 4 Z d q1 d4 q2 d4 p d4 k + d4 z1 d4 z2 4 4 4 (2π) (2π) (2π) (2π)4 × e−iq1 (x−z1 ) e−iq2 (z1 −z2 ) e−ip(z2 −y) e−ik(z1 −z2 )

× [iSF (q1 )(−ie0 γµ )iSF (q2 )(−ie0 γν )iSF (p)] iDFµν (k) . The z1 and z2 integrations implement 4-momentum conservation at each vertex: Z Z d4 z1 d4 z2 ⇒ (2π)4 δ 4 (q1 − q2 − k)(2π)4 δ 4 (p − q2 − k)) = (2π)8 δ 4 (q1 − p)δ 4 (p − q2 − k) .

This leaves us with the exponential factor e−iq1 x eipy . Doing the integration over q1 and q2 we are left with Z Z d4 p −ip(x−y) n d4 p −ip(x−y) 0 e iS (p) = e iSF (p) + F (2π)4 (2π)4 Z o d4 k µν [iS (p)(−ie γ )iS (p − k)(−ie γ )iS (p)] iD (k) + ··· , F 0 µ F 0 ν F F (2π)4

or equivalently

Z

d4 k [iSF (p)(γµ )iSF (p − k)(γν )iSF (p)] iDFµν (k) + · · · (2π)4 (11.57) For (11.50) we obtain correspondingly Z d4 p iDµλ (k)tr[γλ iSF (k−p)γρ iSF (p)]iDρν (k)+· · · iDF0µν (k) = iDFµν (k)+(−ie0 )2 (2π)4 (11.58)

iSF0 (p) = iSF (p) + (−ie0 )2

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Momentum space Feynman rules for QED

There exist two ways of formulating the Feynman rules. (A) Write down the Feynman diagram in question in momentum space, labelling all (oriented) lines by their momenta, taking account of energy-momentum conservation at each vertex. In the case of fermions, the direction of the flow of electron (positron) charge is taken to be along (opposite to) the direction of the momentum flow. An example is given by the self-energy diagrams

k p

p p−k

Fig. 11.8. Second and fourth order electron self-energy diagrams. The Feynman rules for evaluating such a diagram are given by 1. With each oriented fermion propagator associate a factor   i iSF (p)αβ = p/ − m + i αβ 2. With each photon propagator associate a factor (see (13.98)), iDFµν (k) = −

ig µν + gauge terms k 2 + i

(11.59)

3. With each vertex associate a factor µ −ie0 γαβ

4. For each loop momentum ` perform an integration with the measure Z d4 l (2π)4 5. Introduce a minus sign for every independent fermion loop. The results are the momentum-space Green functions G0 in (11.56), with G stripped off the factor expressing overall energy-momentum conservation. (B) A second possibility is to proceed as follows: 1. Label all internal and external lines by a momentum pi , without taking account of energy-momentum conservation.

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2. With each oriented fermion propagator associate a factor   i iSF (p)αβ = p/ − m + i αβ 3. With each photon propagator associate a factor iDFµν (k) = −

ig µν + gauge terms k 2 + i

4. With each vertex associate a factor µ −ie0 γαβ (2π)4 δ(p0 + k − p)

5. For each internal momentum pi perform the integration with measure Z d4 pi (2π)4 6. Introduce a minus sign for every independent fermion loop. The results are the momentum-space Green functions G in (11.56) including the factor expressing overall energy-momentum conservation.

11.9

Moeller scattering

We now show for the case of electron–electron elastic scattering, how to compute the corresponding cross-section using the LSZ formalism of Chapter 10. Moeller scattering in space-time We first return to a space-time description of the Green function for electron– electron elastic scattering, in order to exemplify how to take account of the Fermi statistics involved. Let us compute explicitly the T-matrix for electron–electron scattering in second order of perturbation theory. This requires us to calculate G(2) (x01 , x02 ; x1 , x2 )C , where the subscript stands for “connected”. We have (we omit the subscript “I” for “interaction picture”) Z Z G(2) (x01 , x02 ; x1 , x2 )β1 β2 α1 α2 = d4 z1 d4 z2 h0|T ψβ1 (x01 )ψβ2 (x02 )ψ¯α1 (x1 )ψ¯α2 (x2 ) ¯ 1 )(−ie0 )A ¯ 2 )(−ie0 )A/ (z2 )ψ(z2 ) : |0iC . × : ψ(z / (z1 )ψ(z1 ) :: ψ(z

Taking account of the Fermi statistics and performing the time-ordered contractions we have two distinct contractions (see Fig. 11.9) Z Z (2) 0 0 4 G (x1 , x2 ; x1 , x2 )β1 β2 α1 α2 = d z1 d4 z2 iDF (z2 − z1 )µ1 µ2 × [iSF (x02 − z2 )(−ie0 γ µ2 )iSF (z2 − x2 )]β2 α2

× [iSF (x01 − z1 )(−ie0 γ µ1 )iSF (z1 − x1 )]β1 α1 − (x01 ↔ x02 , β1 ↔ β2 ) ,

(11.60)

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where the minus sign in the “exchange” term arises as a result of the Fermi statistics.

x02 , β2

x01 , β1

x01 , β1

x02 , β2 (11.61)

γ

γ

x2 , α2

x2 , α2

x1 , α1

x1 , α1

Fig. 11.9. Moeller scattering in space-time. The T-matrix Recalling that SF (x) =

Z

d4 q SF (q)e−iqx , (2π)4

D

µν

(z) =

Z

d4 k µν D (k)e−ik·z , (2π)4 F

we obtain for the connected Green function G0(2) Z Z Z Z 4 0 4 0 Z 4 d q1 d4 q2 d4 k d q1 d q2 0 0 4 4 G(x1 , x2 ; x1 , x2 )β1 β2 α1 α2 = d z1 d z2 (2π)4 (2π)4 (2π)4 (2π)4 (2π)4 0

0

0

0

× [iSF (q20 )(−ie0 )γ µ2 iSF (q2 )]β2 α2 e−iq2 ·(x2 −z2 ) e−iq2 ·(z2 −x2 )

× [iSF (q10 )(−ie0 )γ µ1 iSF (q1 )]β1 α1 e−iq1 ·(x1 −z1 ) e−iq1 ·(z1 −x1 )

× iDF (k)µ1 µ2 e−ik·(z2 −z1 )

− (x01 ↔ x02 , β1 ↔ β2 ) .

(11.62)

According to the LSZ substitution rules (10.52) and (10.55) this Green function is to be multiplied from the right by −1

←

−1

←

(r) Kα1 β1 = [iZψ 2 (i ∂/ 1 +m)u(x1 ; p~1 , σ1 )]α1 [iZψ 2 (i ∂/ 2 +m)u(x2 ; p~2 , σ2 )]α2

and from the left by 1

→0

−1

→0

(`) Kβ1 β2 = [−iZψ2 u ¯(x01 ; p~10 , σ10 )(i ∂/ 1 − m)]β1 ][−iZψ 2 u ¯(x02 ; p~20 , σ20 )(i ∂/ 2 −m)]β2 .

Integration over z1 and z2 leads to the delta functions Z Z d4 z1 : (2π)4 δ 4 (k + p01 − p1 ) , d4 z2 : (2π)4 δ 4 (k + p2 − p02 ) . Subsequent integration over the space-time coordinates as implied by (10.52) and (10.55), will then produce four additional delta functions: Z Z d4 x01 : (2π)4 δ 4 (q10 − p01 ) , d4 x02 (2π)4 δ 4 (q20 − p02 ) Z Z d4 x1 : (2π)4 δ 4 (q1 − p1 ) , d4 x2 (2π)4 δ 4 (q2 − p2 ).

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page 159

159

There remain five momentum integrations, leaving us with a δ function expressing overall energy-momentum conservation. The T -matrix is defined in (10.12). Taking out the δ-function we obtain  4 r r r r p m m m m −i 0 0 iDµ1 µ2 (p01 − p1 ) −iT (~ p1 , p~2 , p~1 , p~2 ) = [ Zψ ]−4 3 ωp1 ωp2 ωp01 ωp02 F (2π) 2

0 ¯ (~ ×U p10 , σ10 )(p/1 − m) [iSF (p01 )(−ie0 γµ1 )iSF (p1 )] (p/1 − m)U (~ p1 , σ1 ) 0 0 0 0 ¯ (~ ×U p2 , σ2 )(p/2 − m) [iSF (p2 )(−ie0 γµ )iSF (p2 )] (p/2 − m)U (~ p2 , σ2 ) 2

− (~ p10 ↔ p~20 , σ10 ↔ σ20 ).

(11.63)

We see that the LSZ rules correspond in the present case to the following substitution rules for the external legs of the Green function: - for every incoming external fermion line: iSF (p)αβ

−1 Zψ 2 r m Uβ (~ p, σ) . →i 3 (2π) 2 ωp

(11.64)

- for every outgoing external fermion line: 0

iSF (p )αβ

−1 Zψ 2 r m ¯α (~ →i U p 0 , σ0 ) , 3 (2π) 2 ωp0

(11.65)

and completing these rules for the photon, - for every incoming photon: −1

iDFµν (k)

Z 2 ν →i A 3 2 , (2π) 2 2k0

iDFµν (k)

Z 2 ∗µ →i A 3 2 . (2π) 2 2k0

(11.66)

- for every outgoing photon: −1

(11.67)

To order O(e20 ) we may set Zψ = 1 (see Chapter 16). The result (11.63) thus reads7 r m m m m Tf i = (−ie0 )2 0 3 3 3 (2π) ω1 (2π) ω2 (2π) ω1 (2π)3 ω20   gµν ¯ (~ ¯ (~ + gauge U p20 , σ20 )γ ν U (~ p2 , σ2 ) × [U p10 , σ10 )γ µ U (~ p1 , σ1 ) (p01 − p1 )2 −(p01 ↔ p02 , σ10 ↔ σ20 )] . 7 See

(13.98) for the choice of photon propagator.

(11.68)

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Perturbation Theory

The gauge-dependent terms in the photon propagator are proportional to either k µ or k ν or k µ k ν , where k = (p01 − p1 ) = −(p02 − p2 ) is the momentum associated with the photon line. Since we have the on-shell properties ¯ (~ U p10 , σ1 )(p/01 − p/1 )U (p1 , σ1 ) = 0 ¯ (p20 , σ20 )(p/02 − p/1 )U (p1 , σ1 ) = 0 , U it follows that the gauge-dependent terms in the photon propagator do not contribute, as promised. The result is represented in the figure below.

p02 , β2

p01 , β1

p01 , β1

p02 , β2 (11.69)

γ

γ

p2 , α2

p1 , α1

p2 , α2

p1 , α1

Fig. 11.10. Moeller scattering in momentum space.

11.10

The Moeller differential cross-section

To order O(e20 ) we can set the bare charge e0 = e, α = e2 /4π = 1/137 and m0 = m, the physical mass. For the corresponding differential cross-section we need to compute the absolute square of this amplitude. We shall do this for the case of unpolarized electrons, by averaging over the spins of the initial state, and summing over the spins of the final state. Recalling our definition of the Stapp M-functions (10.22), we have |T |2 = =

1 X X | < p~10 σ10 , p~20 σ20 |T |~ p1 σ1 , p~2 σ2 > |2 4 σ ,σ 0 0 1

2

σ1 ,σ2

1 1 1 1 |M|2 (2π)3 2ω1 (2π)3 2ω2 (2π)3 2ω10 (2π)3 2ω20

and recalling (4.43), X σ

¯β (~ Uα (~ p, σ)U p, σ) = Λ(p)αβ =



p/ + m 2m



, αβ

one finds n 1 1 (2m)4 e4 tr(γ µ Λ(p1 )γ ν Λ(p01 ))tr(γµ Λ(p2 )γν Λ(p02 )) 0 4 [(p1 − p1 )2 ]2 o 1 0 0 −tr(γ µ Λ(p1 )γ ν Λ(p02 )γµ Λ(p2 )γν Λ(p01 )) 0 + (p ↔ p ) . 1 2 (p1 − p1 )2 (p02 − p1 )2

|M|2 =

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11.10 The Moeller differential cross-section

161

The traces are easily evaluated. We have8 tr(γ µ Λ(p1 )γ ν Λ(p01 )) =

1 µ 0ν 2 µν (p p − g µν p1 · p01 + pν1 p0µ 1 +m g ) , m2 1 1

implying tr(γ µ Λ(p1 )γ ν Λ(p01 ))tr(γµ Λ(p2 )γν Λ(p02 )) 2 = 4 [(p1 · p2 )2 + (p1 · p02 )2 + 2m2 (p1 · p02 − p1 · p2 )] m and 2 [(p1 · p02 )2 − 2m2 (p1 · p2 )] , m4 where we have used p1 + p2 = p01 + p02 , implying p1 · p2 = p01 · p02 , and p1 · p01 = p2 · p02 . Therefore n (p · p )2 + (p · p0 )2 + 2m2 (p · p0 − p · p ) 1 2 1 1 1 2 2 2 |M|2 = 8e4 (11.70) [(p01 − p1 )2 ]2 (p1 · p2 )2 + (p1 · p01 )2 + 2m2 (p1 · p01 − p1 · p2 ) (p1 · p2 )2 − 2m2 (p1 · p2 ) o + + 2 . [(p02 − p1 )2 ]2 (p01 − p1 )2 (p02 − p1 )2 tr[γ µ Λ(p1 )γ ν Λ(p02 )γµ Λ(p2 )γν Λ(p01 )] = −

Differential cross-section in center of mass (cm)-system The differential cross-section has the form given by (10.25). We shall compute it in the cm system of the initial particles as represented in the figure below. p01 = (ω, p~0 )

p1 = (ω, p~)

p2 = (ω, −~ p)

p02 = (ω, −~ p)

We may express the above invariant quantities in terms of the total cm energy and the cm scattering angle θ: 2 (p1 + p2 )2 ≡ s = Ecm = 4ω 2 (ω1 = ω2 = ω10 = ω20 = ω) s p1 · p2 = − m2 = 2ω 2 − m2 2 p1 · p01 = ω 2 − p~ 2 cos θ = ω 2 (1 − cos θ) + m2 cos θ 8

p1 · p02 = ω 2 (1 + cos θ) − m2 cos θ .

tr(γµ γν ) = 4gµν → tr(γµa/) = 4aµ

tr(γµ γν γλ γρ ) = 4(gµν gλρ − gµλ gνρ + gµρ gνλ )

tr(γµa/γν b/) = 4(aµ bν − gµν (a · b) + bµ aν )

(11.71)

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Perturbation Theory

Noting that

p

p (s − 2m2 )2 − 4m2 = 2 (p1 · p2 )2 − m4

we have for the differential cross-section dσ =

(2π)4 δ(p01 + p02 − p1 − p2 ) d3 p01 d3 p02 p |M|2 . 0 (2π)3 2ω1 (2π)3 2ω20 4(p1 · p2 − m4 )

(11.72)

In the cm system, Z Z d3 p02 d3 p02 4 0 0 δ (p + p − p − p ) → δ 3 (~ p10 + p~20 )δ(2ω10 − 2ω1 ) 1 2 1 2 (2π)3 2ω20 (2π)3 2ω20 1 → δ(ω 0 − ω) . (2π)3 4ω In order to do the p~10 integration, we note that it follows from ω 2 = p~ 2 + m2 that ωdω = |~ p|d|~ p| . Hence

|~ p 0 |2 d|~ p 0| =

or

Furthermore p

d3 p01 = (2π)3 2ω10 [(p1 · p2 )2 − m4 ] =

p

√

p

ω 02 − m2 ω 0 dω 0 ,

ω 02 − m2 ω 0 dω 0 0 dΩ . (2π)3 2ω 0

(2ω 2 − m2 )2 − m4 = 2ω

p

ω 2 − m2 .

The remaining integration over |~ p01 | and ω 0 can now be trivially performed, and we obtain from (11.72) 1 1 |M|2 dΩ . dσ = 64 (2π)2 ω 2 Finally, making use of (11.71) in (11.70) one obtains the Moeller formula (α = e2 /4π)    dσ α2 (2ω 2 − m2 )2 4 3 (ω 2 − m2 )2 4 = − + 1+ . dΩ 4ω 2 (ω 2 − m2 )2 sin4 θ sin2 θ (2ω 2 − m2 )2 sin2 θ In the ultrarelativistic limit, where m/ω → 0,   dσ α2 4 2 1 → 2 − + , dΩ ω sin4 θ sin2 θ 4 whereas in the nonrelativistic limit we obtain9 the Mott (1930) differential cross9

θ θ cos 2 2 4 2 1 1 − − = sin4 θ sin2 θ 4 sin4 θ2 cos4 z θ2 2 sin2 θ2 cos2

sin θ = 2 sin

=

1 − 2 sin2 4 sin4 θ2

θ 2

cos2

cos4 θ2

θ 2

=

θ 2 (cos2 θ2 + sin2 θ2 ) − 2 sin2 θ2 4 sin4 θ2 cos4 θ2

cos2

θ 2

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11.11 Compton scattering section.10

 α 2 1 dσ → dΩ m 16v 4

page 163

163

1 sin4

θ 2

1 + cos4

θ 2

1 − 2 θ sin 2 cos2

θ 2

!

p| with v = |~ m . In the forward direction θ → 0 this expression reduces to the Rutherford cross-section,    α 2 1 dσ 1 (Rutherf ord X-section) . (11.73) −→ dΩ n.rel m 16v 2 sin4 θ2

11.11

Compton scattering

As one further example, we consider the scattering of light (photon) off a free electron, as shown in Fig. 11.11. Topologically these diagrams are the same as in Fig. 11.10:11 kf kf ki ki

pf

p i + ki

pi

pf

pi − kf

pi

Fig. 11.11. Lowest order Compton scattering. The corresponding differential cross-section now reads (see (10.25)) dσf i =

(2π)4 δ 4 (pf + kf − pi − ki )| < p~f , σf , ~kf |M|~ pi , σi , ~ki > |2 d3 kf d3 pf 2(s − m2 ) (2π)3 2kf0 (2π)3 2p0f

with M the Stapp M-function. The transition amplitude is obtained from the LSZ formalism via the substitutions (11.64) to (11.67) in the corresponding Green function. For the Stapp-functions these substitutions take the simpler form: - for every incoming external fermion propagator: −1

iSF (pi )αβ → iZψ 2 Uβ (~ pi , σi ) . - for every outgoing external fermion propagator: 1

− ¯ iSF (pf )αβ → iZψ 2 U pf , σf ) . α (~

- for every incoming external photon propagator: −1

iDFµν (ki ) → iZψ 2 ν (~ki ) . 10 N.F.

Mott and H.S.W. Messey, Theory of Atomic Collisions (Oxford, 1956). technical reasons the photons are represented by dashed lines; the metafont program does allow for wiggly lines with arrows for aesthetical reasons. 11 For

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Perturbation Theory

- for every outgoing external photon propagator: −1

iDFµν (kf ) → iZA 2 ∗µ (kf ) . Hence we have from Fig. 11.11, using the above rules,12 1 1 ¯ (pf , σf ) 6 ∗f    6 i + 6 i  6 ∗ M = e20 U pi + k i − m pi − k f − m f

!

U (pi , σi ).

Rationalizing the denominators; using a/b/ = 2b.a − b/a/ and the Dirac equation of motion; going to the laboratory frame p~i = 0 where pi .i = pi .∗i = pi .f = pi .∗f = 0; and, setting ki2 = kf2 ki · i = kf · f = 0 using (5.13), one obtains  !  6 i k f 6 ∗f 6 ∗f k i 6 i 2¯ M = e0 U (~ pf , σf ) + U (~ pi , σi ). (11.74) 2pi · ki 2pi · kf Note that unlike the amplitude (11.63) for Moeller scattering, this amplitude is symmetric under the exchange (6 i , ki ) ↔ (6 f , kf ) corresponding to the Bose statistics of photons. Taking the absolute square of M and summing over the initial and final spins using (4.43) one obtains   " # ! !   X 6 i k f 6 ∗f p + m 6 ∗i k 6 f 6 f k f 6 ∗i p + m 6 ∗f k i 6 i 2 4 i i i |M| = e0 tr . + + 2pi · ki 2pi · kf 2m 2pi · ki 2pi · kf 2m σ ,σ f

i

(11.75) The trace can be computed using repeatedly the formula (4.45). The computation is very tedious and is just a matter of patience. We further simplify the calculation by taking the photons to be linearly polarized. The polarization tensors are then real, and the result of taking the trace is,13   X e40 ωf ωi 2 2 |M| = + + 4(f · i ) − 2 . (11.76) 2m2 ωi ωf σ ,σ f

i

It remains to take care of the integration measure. We seek the differential crosssection for scattering a photon off an electron at rest into a solid angle dΩ. Hence d3 kf = ωf2 dωf dΩf 3

d pf . Integration where ωf = |~kf |. It remains to work out δ 4 (pf + kf − pi − ki ) (2π)3 2ω f over p~f fixes the spatial momentum as a function of the incoming and outgoing photon-momentum: p~f = ~ki − ~kf

In the lab frame this leaves the remaining energy delta-function in the form δ(p0f + ωf − m − ωi ) = δ(F (ωf ; ωi , cos θ)) 12 A

factor −i has been taken out because of our definition of T . Itzikson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, 1980); S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1996). 13 C.

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165

with F (ωf ; ωi , cos θ) =

q

ωi2 − 2ωi ωf cos θ + ωf2 + m2 + ωf − m − ωi

where θ is the angle between ~ki and ~kf . Using the (familiar) formula, δ(F (ωf ; ωi , cos θ)) =

| dωdf

δ(ωf − ω0 ) F (ωf ; ωi , cos θ)|ωf =ω0



(11.77)

with ωf = ω0 the (only) zero of F (ωf ; ωi , cos θ), ω0 =

1+

ωi , − cosθ)

ωi m (1

we have δ(p0f + ωf − m − ωi ) =

p0f ω0 δ(ωf − ω0 (θ)) . mωi

From (11.77) we see that there is a shift in the frequency of the scattered photon, which carries the name of A.H. Compton, who discovered this shift in his X-ray studies in 1923. Putting things together we have with (11.74), ! ! ! p0f ωf (2π)4 |M|2 ωf2 dωf dΩ 1 δ(ωf − ω0 ) , (11.78) dσ = 2(s − m2 ) (2π)3 2ωf mωi (2π)3 2p0f or integrating over ωf , α2 dσ = 02 dΩ 2m



ωf ωi

2 

 ωf ωi 2 + + 4(f · i ) − 2 ωi ωf

(11.79)

where we have made use of s defined in (10.24), which reduces in the laboratory frame to s − m2 = 2mωi . This is the Klein–Nishina scattering cross-section for scattering of photons off electrons at rest. If the incident photons are unpolarized and the polarization of the final photons remains undetected, then we must average over the two helicities of the incoming and outgoing photons and divide by 2. We do the same for the electrons. Using (5.14) of Chapter 6 we have XX (~(kf , λf ) · ~(ki , λi ))2 = [(kˆf · kˆi )2 − kˆf2 − kˆi2 + 3] = (cos2 θ + 1) , λf

or

λi

 2   dσ 1 X X dσ α2 ωf ωf ωi 2 = = + − sin θ . dΩ 4 dΩ 2m2 ωi ωi ωf σ ,σ λf ,λi

f

i

(11.80)

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Chapter 12

Parametric Representation of a General Diagram In the previous chapter we have repeatedly used the representation of 1-loop diagrams in terms of integrals over the Feynman parameters ranging over the finite domain (0,1). The resulting integrals had the virtue of exhibiting explicitly the Lorentz-covariant structure with respect to the external momenta, the result having been of the form ! Z Y X N (α; p) I(p) = dαi δ 1 − αi , (12.1) D(α; p) i i where N (α; p) was given by a (matrix valued) polynomial resulting from the spin of the electron. We now show how to obtain such a representation for a general Feynman diagram involving an arbitrary number of loops.1

12.1

Cutting rules for a general diagram

We shall avoid the complications introduced by the spin of a particle, by restricting ourselves to the case of spin-zero particles. Taking all external momenta to be incoming, a general Feynman diagram will be given in terms of the integral !   Y Z Y I  4 V I X d ki i 4 4 IG (p) = (2π) δ Pv − vi ki (12.2) (2π)4 ki2 − m2i + i v=1 i=1 l=1

where our notation is as follows:

1 We

follow here C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill Inc., 1980).

166

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page 167

167

V = Number of vertices I = Number of internal lines L = Number of independent loops Pv = External momentum entering a vertex v kl = Momentum associated with the ll th internal line (

1, if the momentum ki leaves the vertex v vi −1, if the momentum ki enters the vertex v 0, if the momentum ki neither leaves nor enters the vertex v The “incidence matrix” vi evidently satisfies V X

vi = 0 .

(12.3)

v=1

The delta-function in (12.2) implements energy-momentum conservation at each vertex. From (12.3) and the delta function in (12.2) follows V X

Pv = 0 .

v=1

Since the delta functions at the vertices reduce the number of integrations by V − 1 (the last integration leaves an overall δ-function) we also have L = I − (V − 1) .

(12.4)

In order to obtain the desired representation for (12.2) we begin by representing the individual Feynman propagators as Z ∞ 2 2 i = dβi eiβi (ki −mi +i) . (12.5) ki2 − m2i + i 0 For the energy-momentum conserving delta function we use the Fourier representation Z I P X 4 4 (2π) δ (Pv − vi ki ) = d4 yv e−iyv ·(Pv − i vi ki ) . (12.6) i=1

Introduce (12.5) and (12.6) into (12.2) and perform the individual ki -integrations by noting that P 2 yv vi Z  v Z
P 2βi d4 ki iβi ki2 + β1 d4 ki0 iβi ki02 −iβi yv vi ·ki v i e = e e . (2π)4 (2π)4 With

Z

∞

−∞

4

d ke

iβi (k02 −k12 −k22 −k32 )

=



π −iβi

1/2 

π iβi

3/2

=

−iπ 2 βi2

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Parametric Representation of a General Diagram

we have for (12.2),

IG (p) =

Z Y

d4 yv e−i

v=1

P

v

yv ·Pv

Z

0

I ∞Y

i=1



e  dβi

P

−i[βi m2i +(

yv vi )2 /4βi ]



  .

v

i(4πβi )2

This integral is expected to contain an overall delta function expressing the conservation of the energy and momentum associated with the external momenta. In order to make this explicit, we perform the following shift in the integration variables y1 ...yv−1 : y1 = z1 + y y2 = z2 + y · ·

yv−1 = zv−1 + y yv = y. The Jacobian of this transformation is unity. Making use of (12.3) we then have ! X (12.7) IG (p) = (2π)4 δ 4 Pv I˜G (p) , v

where

I˜G (p) =

Z VY −1 v=1

4

d zv

Z

0

I ∞Y

i=1

"

2

#

e−iβi mi dβi e i(4πβi )2

−i

PI

i=1

PV −1

zv vi v=1 4βi


2

PV −1

+

v=1

zv ·Pv

!

.

(12.8) The exponent in the integrand involves a quadratic form q(z; β) in the integration variables zv . Defining a (V − 1) × (V − 1) dimensional matrix dG (β) with matrix elements I X 1 v1 i v2 i , [dG (β)]v1 v2 = βi i=1

we may write the expression in brackets in the exponential as q(z, β) =

V −1 V −1 V −1 X 1 X X dG (β) zv1 zv2 + zv · Pv . 2 v =1 v =1 2 v=1 1

2

We now make use of the fact that the matrix dG (β) is symmetric, and hence may be diagonalized. Denoting the diagonal form of a symmetric matrix Aij by a superscript D, we have quite generally (xi → yi , Ji → J˜i ) P Z Y Z Y P P ˜ n n xi Aij xj −iJi xi − 2i −i y AD y −i Ji yi i,j i dxi e = dyi e 2 i,j i ii i i=1

i=1

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12.1 Cutting rules for a general diagram

=

Y i

s

2π iAD ii

!

i

˜

e 2 J(A

D −1

)

J˜

page 169

169

−1 i √ e 2 JA J . = ( −i2π)n √ det A

Making use of this formula, and taking account of the fact that we are dealing with a four-dimensional integration,2 we obtain from (12.8), after proper identifications,
PI ! Z ∞ Y I i QG (p;β)− βi m2i +i i=1 e (12.9) I˜G (p) = dβi  L 0 i(4π)2 PG (β)2 l=1 where

QG (p; β) =

V −1 X

v1 ,v2

d−1 G (β)v1 v2 (Pv1 · Pv2 )

(12.10)

and PG (β) = β1 · · · βI det[dG (β)] .

(12.11)

The computation of det dG (β) requires the introduction of the notion of the tree graph corresponding to a given Feynman diagram. Definition: A tree graph corresponding to a general diagram G is a connected subdiagram (no vertex is isolated) of G which contains all vertices of G but no loops. Example 1 With the box diagram

Fig. 12.1. Box diagram. are associated four tree diagrams such as, for example,

Fig. 12.2. Tree diagram associated with box diagram. Note that if one were to cut on further internal line, the tree diagram would separate into two disconnected pieces. If we would reintroduce Planck’s constant, the tree graphs would correspond to contributions of zero order in h ¯ . Making use of (12.4) we see that G involves 2n

= V − 1 and we have four copies (y1 , · · · y4 ). Hence, with A = dG (β) √  V −1 ( −i2π)n 1 → (4π)2 √
2 . det A det dG (β)

Use further L = I − (V − 1).

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I − L = V − 1 internal lines, as one readily checks for the above diagrams. Since a tree graph has no loops, L = I − (V − 1) = 0, or the corresponding “incidence” matrix (vi ) for a tree diagram is thus a V × (V − 1) matrix of rank (V − 1). We are now in the position of stating3 (without proof) the result for the determinant appearing in (12.11): ! X Y 1 . (12.12) det[dG (β)] = βi T ree

i∈T ree

Note that the result is given in terms of a sum over all tree diagrams associated with a given Feynman diagram, each term in the sum involving the product of Feynman parameters associated with the internal lines of the respective (connected) tree diagrams. It follows that ! X Y PG (β) := β1 · · · βI det dG [β] = βi (12.13) T ree

i∈T / ree

“i ∈ / Tree” refers to the internal lines of G to be cut in order to arrive at the tree diagram. Note that PG (β) is a homogeneous polonomial of degree I − (V − 1) = L, with L the number of independent loops. Next, one can show that (12.10) has the form ! Y 1 X sc βi , (12.14) QG (p; β) = PG (β) C

i∈C

where the sum runs over all possible cuts C of L + 1 internal lines of the diagram separating this diagram into two connected tree diagrams G1 (C) and G2 (C), and where sc denotes the 4-vector square of the sum of external momenta entering G1 (C)(or, by energy-momentum conservation, equivalently G2 (C)): 

sc = 

X

v∈G1 (C)

2



Pv  = 

X

v∈G2 (C)

2

Pv  .

Finally, making use of the homogeneity properties of QG (p, β) and PG (β) QG (p, λβ) = λQG (p, β)

(12.15)

L

PG (λβ) = λ PG (β) , we may reduce the integration in (12.9) to the range (0,1). To this end we introduce again the identity ! Z ∞ I dλ 1X 1= δ 1− βi λ λ i=1 0 3 N.

Nakanishi, Graph Theory and Feynman Integrals (Gordon and Breach, New York, 1970).

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page 171

179

and12.1 rescale the integration follows: Cutting rules for avariables general as diagram

171

βi = λαi .

(12.16)

and rescale the integration variables as follows:

We then obtain from (12.9) and (12.15), βi = λαi .

(12.16)

We then obtain from (12.9) and (12.15), P

2V −I−2

z }| { Z P δ(1 − i αi ) ∞ dλ 2V I− 2L eiλ[QG (p,α)− i αi m2i +i] . −I−2 ˜ IG (p) =  λ dα   P i L Z 1 Y }| { z Z I 2 λ I − 2L iλ[QG (p,α)−P αi m2 +i] δ(1 −(α)i αi ) 0 ∞ dλ 0 PG i(4π)21 i i . I˜G (p) =  λ e 2 L i=1 dαi  λ (12.17) 0 i=1 0 PG (α) i(4π)2 Since L = I − (V − 1) = I − L = V − 1 it follows that 4 (12.17) 1

Z

I 1Y

Since L = I − (V − 1) = I − L = V − 1 it follows that4

Z 1Y I (i)3V3V−2I−3 Z 1Y I ˜ −2I−3 IG˜(p) = (i) 2 L Γ(2V − I − 2) dαi IG (p) = [(4π) ]2 L Γ(2V − I − 2) 0 i=1dαi [(4π) ] PI 0 i=1 δ(1 − PI1 αi ) δ(1 − 1 αi ) ×    −I−2. . ×P  G (α) 22 QG (p, α) − P PII αi m22 + i2V2V−I−2 P (α) Q (p, α) − i=1 α m i + i G

G

i=1

i

(12.18)

(12.18)

i

The divergenceofofΓ(2V Γ(2V−−I I−−2) 2) for for vanishing vanishing argument The divergence argument reflects reflectsthe thelogarithmic logarithmic divergence theoriginal originalintegral integral (12.17) (12.17) for 4L − divergence of ofthe − 2I 2I == 2(I 2(I−−2(V 2(V−−1)). 1)).InIncase case of such a divergence we must return to (12.17) and regularize the integral in λ by of such a divergence we must return to (12.17) and regularize the integral in λ by introducing a cutoff. Notice α[0, becauseofofthe theδ-function. δ-function. itroducing a cutoff.Notice thatthat nownow α[0, 1] 1] because Example 2

Example 2

Consider the vertex diagram of φ3 theory:

Consider the vertex diagram of

p0 α2 q

α3 α1 p

There are 3 tree graphs associated with it: There are three tree graphs associated with it:

Fig.12.3: Tree graphs of vertex diagram Fig. 12.3. Tree graphs of vertex diagram.

4 We

4 Weuse usethe the definition definition

ZZ ∞

∞

Γ(n) Γ(n)==

n−1 −x dxx dxxn−1 ee−x ..

00

Notice that Γ[2V I− for2V 2V−−II− −22 = = −n(n −n(n = 0, of of Notice that Γ[2V −− I− 2]2]==∞∞for 0,1, 1,2...). 2...). This Thisreflects reflectsthe thedivergence divergence λ-integration the lowerendpoint, endpoint,which which in in turn turn is of of thethe thethe λ-integration at at the lower is aa reflection reflectionofofthe theUV UVdivergence divergence original integral! original integral!

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Parametric Representation of a General Diagram

We have correspondingly P(α) = α1 + α2 + α3

Q(α) = q 2 α1 α2 + m2 α1 α3 + m2 α2 α3 , so that we have the integral representation Z Y 1 1 δ(α1 + α2 + α3 − 1) ˜ Ivertex = . dαi 2 (4π)2 0 0 [q α1 α2 + m2 α1 α3 + m2 α2 α3 − m2 + i]

(12.19)

Example 3 Consider again the box-graph in Feynman parameter space (the labelling is chosen for later comparison) p01 p02 β

α1 p1

α2 β0

p2

Fig. 12.4. Box-graph in Feynman-parameter space. The corresponding integral is convergent, so that (12.18) applies. Pbox (p) is computed from the sum of connected tree graphs of Example 1 to be Pbox (α, β) = α1 + α2 + β + β 0

(12.20)

and Qbox (p) is computed from the (disconnected) tree graphs obtained by cutting one further internal line, to be Qbox (p; α, β, β 0 ) = where

α1 α2 s + ∆(t; α, β, β 0 ) Pbox (α, β)

s = (p1 + p2 )2 , and

(12.21)

t = (p01 − p1 )2

∆(t; α, β, β 0 ) = ββ 0 t + m20 (α1 + α2 )(β + β 0 ).

From (12.18) we thus have Z 1Y 2 δ(α1 + α2 + β + β 0 − 1) i 0 ˜ Ibox = Γ(2) dα (dβdβ ) i (4π)2 [α1 α2 s + ∆(t; α, β, β 0 ) − m20 + i] 0 i=1

(12.22)

(12.23)

where we have set Pbox (α, β) = 1 because of the delta function.

12.2

An alternative approach to cutting rules

The results (12.17) and (12.18) are well suited for getting a quick analytic expression for (12.1) when N (α; p) = 1. The following approach also allows to include a polynomial N (α; p). The idea is based on the analogy to electrical circuits.5 5 J.D.

Bjorken and S.D. Drell, Relativistic Quantum Fields (McGraw-Hill Inc., 1965).

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12.2 An alternative approach to cutting rules

173

Consider the diagram

Fig. 12.5. Two current loops. Let R be the number of independent loops (rungs) in a diagram. With the rth rung we associate a loop momentum lr , which is eventually to be integrated over. Making a change of variable ki = Ki +

R X

ηir lr

(12.24)

r=1

with Ki a function of the external variables p, p0 , and ηir constants to be suitably chosen, we obtain for the integral (12.2), Z d4 l1 · · · d4 lR IG (p) ∼ 2 (k1 − m21 ) · · · (kI2 − m2I )   Z Z 1 I X = d4 l1 · · · d4 lR αj  dα1 · · · dαI δ 1 − 0

j=1

(I − 1)! P P × P [ i (Ki2 − m2i )αi + 2 j,r Kj αj ηjr lr + jrr0 αj ηjr ηjr0 lr lr0 ]I

where use has been made of the Feynman formula (15.4) (see Chapter 15 for a proof): P Z 1 I Y δ(1 − j αj ) 1 = (I − 1)! dα1 ...dαI P . (12.25) a + i ( j αj aj + i)I 0 j=1 j Eliminating the mixed term by requiring I X

Kj αj ηjr = 0,

j=1

r = 1···R

one finds, after integration over the loop momenta6 R1 PI dα1 · · · dαI δ(1 − j=1 αj ) 0 P IG (P ) ∼ ∆2 [ j (Kj2 − m2j )αj ]I−2R

where with

∆ = det ||z|| zrr0 =

I X

ηjr ηjr0 αj .

j=1

6 J.D.

Bjorken and S.D. Drell, Relativistic Quantum Fields (McGraw-Hill, Inc.).

(12.26)

(12.27)

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Parametric Representation of a General Diagram

Since R = L the result (12.27) agrees with (12.18). In order to get a more detailed understanding of this result, let us return to our example of the triangle graph represented on shell, p2 = p02 = m2 , by the integral Z 1 d4 k I(p) = 4 2 2 (2π) (k − 2p · k)(k − 2p0 · k)(k 2 − m2 ) + i R1 R1 R1 Z dα1 0 dα2 0 dα3 δ(1 − α1 − α2 − α3 ) d4 k 0 . = 2! (2π)4 [α1 (k 2 − 2p · k) + α2 (k 2 − 2p0 · k) − α3 (k 2 − m2 ) + i]3 Our general result (12.27) evidently corresponds to a block-diagonalization of the denominator D, which, taking account of the delta function, we rewrite as D = [k − (α1 p + α2 p0 )]2 − (α1 p + α2 p0 )2 − α3 m2 = k 02 − α12 m2 − α22 m2 − 2α1 α2 p · p0 − α3 m2

= k 02 − α12 m2 − α22 m2 + α1 α2 (p − p0 )2 − 2α1 α2 m2 − α3 m2 = q 2 α1 α2 + α1 α3 m2 + α2 α3 m2 − m2 + k 02

where (k 0 corresponds to `) k 0 = k − (α1 p + α2 p0 ) .

(12.28)

Integrating this result over k 02 yields the result (12.19). Let us take another look at this result from the point of view of (12.27).

p0 k2 = p 0 − k q

k

k1 = p − k

p

The above calculation suggests what change of variable (12.24) is to be performed: k1 ≡ p − k = K1 (q, p, p0 ) − k 0

k2 ≡ p0 − k = K2 (q, p, p0 ) − k 0 0

k3 ≡ K3 (q, p, p ) + k

(12.29)

0

with K1 (q, p, p0 ) = p − α1 p − α2 p0

K2 (q, p, p0 ) = p0 − α1 p − α2 p0

(12.30)

K3 (q, p, p0 ) = α1 p + α2 p0

and η = 1, ` = k 0 . One checks that the mixed term vanishes, as required by (12.26): X αj Kj (q, p, p0 ) = 0. i

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175

Some further algebra shows that 3 X (Kj2 (q, p, p0 ) − m2 )αi = α1 p2 + α2 p02 − (α1 p + α2 p0 )2 − m2 j=1

= α1 α2 q 2 + α1 α3 m2 + α2 α3 m2 − m2 .

Putting things together and integrating over k (factor 2! cancels) we obtain (12.19). The example demonstrates that, in general, the change of variable (12.24) one seeks may not be trivial. We shall make use of this change of variable when discussing the regularization and renormalization of the QED vertex function in Chapters 15 and 16, respectively.

12.3

4-point function in the ladder approximation

We now apply the results obtained in this chapter to the calculation of the asymptotic behaviour of the 4-point function of a φ3 -theory in the so-called ladder approximation, as given by the sum of diagrams

p01 α1

α2

p1

p02

βn−1

β1

αn−1

β10

αn p2

0 βn−1

Fig. 12.6. Ladder diagram. These diagrams are UV convergent, so that we can write for a general diagram the integral representation (12.18) (n)

Glad (p) = (−ig)2n I˜G (p)

(12.31)

with (recall definition (12.7)) (n) I˜lad (p) =

iΓ(n) [(4π)2 ]n−1

Z Y n i

dαi

n−1 Y j−1

dβj dβj0




P P P δ( αi + βj + βj0 − 1)  n , × [Plad (α; β, β 0 )]2 Qlad (p; α; β, β 0 ) − m20 + i

(12.32)

where we have used V = 2n, I = 3n − 2, L = n − 1. Let us introduce again the two Lorentz invariants, s = (p1 + p2 )2 , t = (p01 − p1 )2 .

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Parametric Representation of a General Diagram

We shall be interested in studying the limit s → ∞ of (12.32). For this, we shall need to exhibit only the s-dependence of the homogeneous polynomial (12.14); it is obtained by considering the cut

p01

β2

βn−1

p02

p1

β20

0 βn−1

p2

Fig. 12.7. Leading diagram for s → ∞. and has the form Qlad (p; α, β, β 0 ) =

α1 α2 · · · αn s + ∆(t; α; β, β 0 ) . Plad (α; β, β 0 )

The limit s → ∞ is determined by the integration region around αi = 0. Hence, it is sufficient to consider the integral (n) I˜lad (p)

i

→ n−1 (n − 1)! (4π)2

Z

0

1

  Y
 dβj dβj0  Plad (0; β, β 0 ) n−2 j

  Z Y X X 0   ×δ βj + βj − 1 dαi j

with

0

j

i

1 ¯ β, β 0 )]n , [α1 ...αn s + ∆(t;

∆(t; β, β 0 ) = ∆(t; 0; β, β 0 ) − m20 Plad (0; β, β 0 ). Note that the delta function no longer involves the α-parameters. One can do the α-integrations explicitly,7 obtaining for (12.31) and s → ∞, in the leading log approximation 1 (ln s)n−1 (n) Glad (p) → −ig 2 Kn (t) , (12.33) s (n − 1)! where Kn (t) =

7 See



−g 2 (4π)2

n−1

(n−2)!

Z

0

1 n−1 Y j=1

dβj dβj0

n−2  P P δ( j βj + βj0 − 1) Plad (0; β, β 0 .  n−1 ∆(t; β, β 0 (12.34)

R.J. Eden, P. Landshoff, D.I. Olive and J.C. Polkinghorne, The Analytic S-Matrix.

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177

For the computation of PG (0, β, β 0 ) we only need the connected tree diagrams obtained from Fig. 12.6 by cutting n − 1 internal β lines: with a cut of n − 1 internal β-lines: n−1 n−1 Y Y Plad (0, β, β 0 ) = (βj + βj0 ) = Pj (0, β, β 0 ) . j=1

j=1

0

The computation of ∆(t; 0, β, β ) entering QG involves the sum of diagrams resulting from one further cut, which necessarily implies the presence of simultaneous cuts of a βi and βi0 line. It is not hard to see that ∆(t, 0; β, β 0 ) has the form ∆(t, 0; β, β 0 ) =

n−1 X

∆j (t; 0, β, β 0 )

j=1

where

Y

k6=j

Pk (0, β, β 0 ) ,

∆j (t; 0, β, β 0 ) = tβj βj0 .

(12.35)

(12.36)

Regge behaviour We now show that Kn (t) factorizes. Make the change of variable (scale transformation) βj = ρj β¯j , βj0 = ρj (1 − β¯j ) =: ρj β¯j0 . The Jacobian of the transformation is Jj = ρj . Hence dβj dβj0 = ρj dρj dβ¯j dβ¯j0 δ(β¯j + β¯j0 − 1) ∆j (t; 0, βj , βj0 ) = ρ2j ∆j (t, 0; β¯j , β¯j0 ) = tρ2j β¯j β¯j0 Plad (0, β, β 0 ) =

n−1 Y

ρk ,

k=1

X ∆(t, 0; β, β 0 ) = ρj ∆j (t, 0; β¯j , β¯j0 ) . Plad (0, β, β 0 ) j We thus obtain for the integral in (12.34) P P δ( βj + βj0 − 1) h in−1 0) 2 0 j=1 Plad (0; β, β 0 ) P∆(t;0;β,β 0 ) − m0 (0;β,β lad # Qn−1 ¯ Z 1 "n−1 X  ¯0 Y j=1 δ(βj + βj − 1) 0 ¯ ¯ = dβk dβk dρk δ ρj − 1 hP in−1 . n−1 0 k=1 ρj ∆j (t; 0; β¯j , β¯j0 ) − m20 j

I=

Z

1 n−1 Y

(dβj dβj0 )

Making use of the inverse of the Feynman formula (12.25) in Chapter 15, we have P Z 1 n−1 Y δ( j ρj − 1) dρk hP in−1 n−1 0 k=1 ρj ∆j (t; 0; β¯j , β¯j0 ) − m20 j

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Parametric Representation of a General Diagram n−1 Y 1 = (n − 2)! j=1

i ¯ ∆j (t; 0; βj , β¯j0 ) − m20

!

,

and we finally arrive at 

−g 2 Kn (t) = (4π)2

( n−1 n−1 Y Z

1

0

j=1

dβ¯j dβ¯j0

δ(β¯j + β¯j0 − 1) ∆j (t; 0; β¯j , β¯0 ) − m2 j

)

.

Hence Kn (t) = K n−1 (t) with



−g 2 K(t) = (4π)2

 (Z

1

0

dβ¯j dβ¯j0

δ(β¯j + β¯j0 − 1) tβ¯j β¯0 − m2 j

0

)

.

(12.37)

Thus we have from (12.33) in the limit s → ∞,


n−1 K(t)ln s (n) I˜lad (s, t) → −g 2 s−1 . (n − 1)!

In the leading log-approximation, the sum of the ladder diagrams is therefore given by the exponential series (n) Glad (s, t)

2 −1

∼ −g s

∞ X


n−1 1 K(t)ln s , (n − 1)! n=1

which can be summed to give (n)

Glad (s, t) ∼ −g 2 sα(t) ,

(12.38)

where α(t) = −1 + K(t) . The function α(t) is called Regge trajectory. This sample calculation thus provides an explicit example for what has been termed Regge behaviour of scattering amplitudes.8 Such Regge behaviour, and the associated Regge poles, have been the subject of intensive studies in the sixties, and developed into a field of research of its own, referred to as S-matrix theory.

8 T. Regge, Nuovo Cimento 14, (1959) 951; P.D.B. Collins, An Introduction to Regge Theory and High Energy Physics (Cambridge University Press, Cambridge, 1977).

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Chapter 13

Functional Methods The derivation of the Feynman rules in Chapter 11 has been based on operator methods in the interaction picture. The rules allow for the perturbative computation of the Green functions. The corresponding perturbative series is known to be at best an asymptotic series in the coupling constant with zero radius of convergence around the origin of the complex coupling-constant plane. It thus reflects the true content of the theory only as long as it provides a good approximation in the sense of an asymptotic series for sufficiently small coupling constant. It furthermore presupposes the existence of a unique vacuum given by the Fock vacuum. This appears to be the case for Quantum Electrodynamics (QED), where perturbation theory provides an excellent description of the experimental results. As we know however, this reflects only partially, in general, a much more complex situation. Thus it is known that the QCD vacuum is highly non-trivial, and is in fact a condensate of quark–antiquark pairs, very much reminiscent of the superconducting ground state in superconductors known to be populated by spin-zero electron–electron (Cooper) pairs. Many of the modern developments in Quantum Field Theory relating to such non-perturbative phenomena would have been missed had one adhered to perturbation theory in the formulation of Chapter 11. In fact, for a very small class of models in 1+1 dimensions (Thirring model, QED2 ) exact solutions have been constructed using operator methods; they exhibit non-trivial properties lying outside the scope of perturbation theory.1 From here one learned that Quantum Field Theories in 3+1 dimensions can be expected to exhibit highly non-trivial non-perturbative features. It was only the functional approach to QFT which opened the possibility for studying such non-perturbative phenomena in “realistic” theories. This approach, dispensing of the notion of an underlying “Fock space”, will be the subject of this chapter. We shall show how one recovers the perturbative series embodied by Eq. (11.21), and prepare the ground for new approximation schemes paying tribute to “non-perturbative” phenomena, as exemplified in Chapter 20. 1 E. Abdalla, M. Christina B. Abdalla and K.D. Rothe, Non-Perturbative Methods in 2 Dimensional Quantum Field Theory (World Scientific, 1991, and extended version 2001).

179

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13.1

The Generating Functional

For the sake of simplicity, we shall restrict our discussion in this section to the case of a self-interacting real scalar field φ(x). We define2 R 4 Z[j] =< Ω|T ei d xφ(x)j(x) |Ω > , (13.1)

where j(x) is an external c-number source, and Z[0] = 1. We define R the functional derivative δ/δj(x) as an operation commuting with the integration d4 x and having the property   δ , j(y) = δ 4 (x − y) . (13.2) δj(x) Thus

δ δj(x)

Z

d yj(y)φ(y) =

From (13.1) we thus have,



 δ d y , j(y) φ(y) = φ(x) . δj(x)

Z

4

4

δ Z[j]|j=0 =< Ω|φ(x)|Ω > , iδj(x) or more generally 

δ δ ... Z[j] iδj(xn ) iδj(x1 )



=< Ω|T φ(xn )...φ(x1 )|Ω > .

(13.3)

j=0

Hence Z[j] plays the role of a generating functional for the n-point Green functions of QFT. Our next task thus consists in obtaining an explicit representation for this generating functional. We shall discuss two constructions, due to J. Schwinger and R. Feynman, respectively. Of these, the second one will provide the basis for the chapters to follow.

13.2

Schwinger’s Construction of Z[j]

Schwinger’s construction of the generating functional is based on the (Euler–Lagrange) equations satisfied by the fields in question. We exemplify the construction for the case of a real scalar field satisfying the Euler–Lagrange equations of motion (

+ m2 )φ(x) = J(φ(x))

(13.4)

associated with a lagrangian density L=

1 1 ∂µ φ∂ µ φ − m2 φ2 + LI (φ) , 2 2

with J(φ(x)) = 2 J.

∂LI , ∂φ(x)

Schwinger, Quantum Electrodynamics (Dover Press, New York, 1958).

(13.5)

(13.6)

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181

some polynomials of the fields acting as the source of the interaction. From (13.1) we have (note that this time we have not set j = 0) 1 δ x0 |Ω > , Z[j] =< Ω|Tx∞0 φ(x)T−∞ i δj(x) where we have introduced Tab [j] = T e with Lj (t) =

Z

i

Rb

d3 xLj (x),

a

dtLj (t)

,

Lj (x) = φ(x)j(x)

and the time-ordering operation defined by ∞ n Z tb X i dt1 ...dtn T (Lj (t1 )...Lj (tn )) n! ta n=0 Z tb Z t1 Z tn−1 ∞ X in =1+ dt1 dt2 ... dtn Lj (t1 )Lj (t2 )...Lj (tn ) .

Tab [j] =

ta

n=1

ta

ta

It follows from here that ∂ ∞ T 0 = Tx∞0 (−iLj (x0 )) ∂x0 x and

∂ x0 x0 T = iLj (x0 )T−∞ . ∂x0 −∞

We thus obtain ∂ x0 x0 < Ω|Tx∞0 φ(x)T−∞ |Ω > = < Ω|Tx∞0 ∂0 φ(x)T−∞ |Ω > ∂x0 x0 + i < Ω|Tx∞0 [φ(~x, x0 ), Lj (x0 )]T−∞ |Ω > . Now, 0

[φ(x), Lj (x )] = so that

Z

d3 y[φ(~x, x0 ), φ(x0 , ~y )]j(~y , x0 ) = 0 ,

∂ x0 x0 < Ω|Tx∞0 φ(x)T−∞ |Ω >=< Ω|Tx∞0 ∂0 φ(x)T−∞ |Ω > . ∂x0 Differentiating once again and making use of the ETC [Lj (x0 ), ∂0 φ(~x, x0 )] = ij(x) we similarly obtain ∂2 x0 x0 ∞ < Ω|Tx∞0 φ(x)T−∞ |Ω >=< Ω|Tx∞0 ∂02 φ(x)T−∞ |Ω > +j(x) < Ω|T−∞ |Ω > , ∂x20

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Functional Methods

or (

x

+ m2 )

δ Z[j] =< Ω|Tx∞0 [( iδj(x)

0

x

x + m2 )φ(x)]T−∞ |Ω > +j(x)Z[j] .

(13.7)

We thus finally have with (13.4), Kx

δ Z[j] = J iδj(x)



δ iδj(x)



Z[j] + j(x)Z[j]

(13.8)

where Kx =

x

+ m2 .

The non-interacting case In the absence of an interaction, Eq. (13.8) reads Kx

δ Z (0) [j] = j(x)Z (0) [j] . iδj(x)

(13.9)

The form of this equation suggests making the exponential Ansatz Z (0) [j] = eiW

(0)

[j]

with the “initial condition” W (0) [0] = 0 . This leads us to solve the equation Kx

δ W (0) [j] = j(x) , δj(x)

(13.10)

having the solution (see (11.32)) δ W (0) [j] = − δj(x)

Z

d4 y∆F (x − y; m2 )j(y) .

Note that we have chosen for the Green function the Feynman prescription. The functional differential Eq. (13.10) has the solution Z 1 W (0) [j] = − d4 xd4 y j(x)∆F (x − y; m2 )j(y) . 2 We thus conclude that for the non-interacting case R 4 R 4 −1 i Z (0) [j] = e 2 d x d y j(x)K (x−y)j(y) ,

where K −1 (z) is the inverse of the Klein–Gordon operator. Note that Z (0) [0] = 1 in accordance with < Ω|Ω >= 1. We now turn to the case where an interaction is present.

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183

The interacting case For the case where LI (φ) does not vanish, we make the Ansatz δ

Z[j] = N −1 eF [ iδj ] Z (0) [j] , where N is a normalization constant to be determined below. Using (13.9) we obtain δ δ δ Kx Z[j] = N −1 eF [ iδj ] Kx Z (0) [j] iδj(x) iδj(x) i h δ = N −1 eF [ iδj ] , j(x) Z (0) [j] + j(x)Z[j] .

Comparing the right-hand side of this expression with the right-hand side of Eq. (13.8), we conclude that we must require   h i δ δ δ eF [ iδj ] , j(x) Z (0) [j] = J (13.11) eF [ iδj ] Z (0) [j] . iδj(x)

Setting

δ = πj (x), iδj(x)

we may rewrite (13.11) as h i eF [πj ] , j(x) Z (0) [j] = J(πj )eF [πj ] Z (0) [j] . Noting with (13.2) that we require

[j(x), πj (y)] = iδ 4 (x − y) [eF [πj ] , j(x)] = J(πj )eF [πj ] .

Expanding the exponential and F [πj ] into a Taylor series, we see that this means, −i

δF [πj ] = J(πj ) , δπj (x)

or recalling (13.6) we conclude,       Z δ 1 δ δ F = i d4 zLI ≡ iSI . iδj i δj(z) iδj Hence

Z[j] = N −1 eiSI [ iδj ] Z (0) [j] . δ

(13.12)

The so-far arbitrary normalization constant N is fixed by requiring Z[0] = 1:   δ . (13.13) N = eiSI [ iδj ] Z (0) [j] j=0

Note that for an interaction lagrangian not involving any derivatives in the fields, we have LI = −HI . By expanding the exponential in (13.12) in powers of j and performing the differentiations with respect to the external source as in (13.3), one verifies that one generates the perturbative series (11.21) with N given by the sum over vacuum graphs.

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13.3

Feynman Path-Integral

Schwinger’s construction of the generating functional provides a closed formula from which one recovers the perturbative series representing a general n-point Green function by differentiating the generating functional n times with respect to the external source and then setting the source equal to zero, according to Eq. (13.3). We now derive an explicit representation of this generating functional in terms of an infinite dimensional (functional) integral. We shall do this first for the case of Quantum Mechanics of one degree of freedom and then extend the result to the case of an infinite number of degrees of freedom.3 Path-integral representation of free propagation kernel in QM Consider the Schr¨ odinger wave function in the coordinate representation, ψ(q, t) =< q|Ψ(t) >, where |q > is an eigenstate of the position operator Q: Q|q >= q|q >. For simplicity we shall consider here the space to be one-dimensional. Using completeness of the states |q >, Z dq|q >< q| = 1 , and the fact that (t0 ≥ t; We introduce here h ¯ .) i

0

|Ψ(t0 ) >= e− h¯ (t −t)H0 ) |Ψ(t) > , we have the identity ψ(q 0 , t0 ) =

Z

dqK0 (q 0 , t0 ; q, t)ψ(q, t) ,

where K0 (q 0 , t0 ; q, t) stands for i

0

K0 (q 0 , t0 ; q, t) =< q 0 , t0 |q, t >=< q 0 |e− h¯ (t −t)H0 |q >

(13.14)

with H0 the Hamiltonian of a free particle H0 =

p2 . 2m

The computation of the integral kernel proceeds by introducing a complete set of eigenstates of the momentum operators, and gives (see Chapter 1, Eq. (1.11)) K0 (q 0 , t0 ; q, t) =



m 2π¯hi(t0 − t)

1/2

0 2 i m (q −q) 2 t0 −t

e h¯

.

(13.15)

The kernel (13.14) has the interesting property of satisfying the following composition law: Z 0 0 K0 (q , t ; q, t) = dq 00 K0 (q 0 , t0 ; q 00 , t00 )K0 (q 00 , t00 ; q, t) . (13.16) 3 R.P.

Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

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185

We make use of this composition law in order to obtain a functional representation of this kernel. To this end we subdivide the time interval ∆t = t0 − t into N equal time intervals  = (t0 − t)/N , where we shall eventually let N tend to infinity. Making repeated use of the completeness of the states |q >, we have 0

i

< q 0 |e− h¯ H0 (t −t) |q > =

Z NY −1 `=1

i

dq (`) < q 0 |e− h¯ H0  |q (N −1) >

(13.17)

i

i

× < q (N −1) |e− h¯ H0  |q (N −2) > · · · < q (1) |e− h¯ H0  |q > , where, according to (13.15), we have, exactly, i

< q (`+1) |e− h¯ H0  |q (`) >=

 m 1/2 i m (q(`+1) −q(`) )2  e h¯ 2 2π¯hi

(13.18)

for the individual factors. Using H0 = −L0 , the following expression is short hand for the multiple integral (13.17) in the limit where  → 0 with N  = t0 − t (it is now convenient to do some relabelling): K0 (qb , tb ; qa , ta ) =

Z

q(tb )=qb

i

Dqe h¯

q(ta )=qa

R tb ta

dt L0 (q(t),q(t)) ˙

,

(13.19)

where we have made the identifications 1 (q (`+1) − q (`) )2 m → L0 (q (`) , q˙(`) ), 2  Z tb N −1 X L0 (q (`) , q˙(`) ) → dtL0 (q(t), q(t)) ˙ , ta

`=0

and where Dq is shorthand for the integration measure Dqi :=

−1  m  N2 NY dq (`) 2πi¯h

(13.20)

`=1

in the limit N → ∞,  → 0,

N  = ta − tb = f ixed .

(13.21)

Checking the use of the completeness relation In order to create some confidence in our procedure, let us verify that we recover from (13.17) the formerly derived result for the kernel (13.15). We have Z PN  m 1/2 Z i m [N dxN −1 ]... [N dx1 ]e h¯ 2 i=1 →0 2πi¯ h

K(b; a) = lim

(xi −xi−1 )2 

(13.22)

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with xN = xb ,

x0 = xa

and

 m  12 . 2πi¯h Consider the exponential factors involving x1 . We have (y = x1 − x0 ), Z Z 2 2 2 2 im im [N dx1 ]e 2¯h [(x2 −x1 ) +(x1 −x0 ) ] = N dye 2¯h {[(x2 −x0 )−y] +y } Z 2 2 im = N dye 2¯h {2y −2y(x2 −x0 )+(x2 −x0 ) } Z 2 2 im 1 1 = N dye 2¯h {2[y− 2 (x2 −x0 )] + 2 (x2 −x0 ) } Z 2 2 im im = N dze 2¯h [2z ] e 2¯h2 (x2 −x0 ) . N =

Now

Hence

Z Z

[N dx1 ]e

ibz 2

dze

=



2 2 im 2¯ h [(x2 −x1 ) +(x1 −x0 ) ]

iπ b

 12

.

(13.23)

  12 2 im 1 e 2¯h·2 (x2 −x0 ) . = 2

Repeating this process with the next adjacent integration variable x2 , we have Z Z 2 2 2 2 im im im [N dx2 ]e 2¯h· (x3 −x2 ) e 2¯h·2 (x2 −x0 ) = [N dy]e 2¯h·2 [3y −4y(x3 −x0 )+2(x3 −x0 ) ] Z 2 2 i3m 2 2 = [N dy]e 2¯h·2 [(y− 3 (x3 −x0 )) + 9 (x3 −x0 ) ] " 1 # 2 im 2iπ¯h2 2 = N e 2¯h·3 (x3 −x0 ) ] , 3m or finally   12 Z   21 2 2 2 im im im 1 1 [N dx2 ]e 2¯h· (x3 −x2 ) e 2¯h·2 (x2 −x0 ) = e 2¯h·3 (x3 −x0 ) ] , 2 3 where we have again used (13.23). Repeating this procedure N − 1 times, we finally obtain for the r.h.s. of (13.22),    1/2 im 1 2 exp (xb − xa ) . N 2¯h · N  Hence, multiplying this result with the remaining factor N , we finally obtain, after identifying N  with tb − ta ,  1/2 2 im (xb −xa ) m (13.24) K(b; a) = e 2¯h tb −ta , 2πi¯h(tb − ta )

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187

where we have set N  = tb − ta . We have thus recovered the propagation kernel for the free particle motion. Notice that the singular  dependence in (13.22) has been completely absorbed by the integration. Semi-classical derivation for the free-particle kernel In the case where an interaction is present, the above explicit calculation can in general no longer be performed. Approximate methods reminiscent of the WKB method in Quantum Mechanics may be of interest. Let us illustrate this for the case of the free particle with the lagrangian 1 mx˙ 2 . 2

L0 =

Let xcl be the solution of the associated Lagrange equation x ¨cl = 0. We make the change of variable x = xcl + y subject to the boundary conditions xcl (ta ) = xa ,

xcl (tb ) = xb .

Then y(ta ) = y(tb ) = 0 . Now xcl (t) = xa + We further have S[xcl + y] =

Z

xb − xa (t − ta ) . tb − ta

tb

dtL(x˙ cl + y) ˙    #  Z tb " 1 2 ∂2L ∂L + y˙ = dt L(x˙ cl ) + y˙ ∂ x˙ xcl 2 ∂ x˙ 2 xcl ta Z m tb = Scl + dt y˙ 2 2 ta ta

since




∂L ∂ x˙ cl

= const and y(ta ) = y(tb ) = 0. Hence we have from (13.19), i

K(b; a) = F (tb , ta )e h¯ Scl

with i

F (tb , ta ) = e h¯ Scl

Z

y(tb )=0

y(ta )=0

Now Lcl =

i

Dy e h¯

R tb ta

dt

m 2 m (xb − xa )2 x˙ cl = , 2 2 (tb − ta )2

m 2 ˙ 2 y

.

(13.25)

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or Scl = Hence

m (xb − xa )2 . 2 tb − ta 2 im (xb −xa ) tb −ta

K(b, a) = F (tb , ta )e 2¯h

to be compared with (13.24). To compute F (tb , ta ), we nevertheless need to do an explicit calculation. In many cases, one is however only interested in the limit ta → −∞, tb → ∞. In that case, F (tb , ta ) just tends to an irrelevant number, which divides out after proper normalization of the ground state. Path integral representation of kernel with interaction Although in the interacting case we can no longer obtain a closed expression for the propagation kernel, we can still obtain for it a functional representation of the form of Eq. (13.19), with the free action replaced by the action of the fully interacting theory: R Z qb

i

< qb |e− h¯ H(tb −ta ) |qa >=

i

Dqe h¯

qa

tb ta

dtL(q(t),q(t)) ˙

.

(13.26)

From the mathematical standpoint, one considers the kernel (13.26) for time analytically continued to imaginary time, t = −iτ : R τb Z qb 1 1 −h dτ LE (q(τ ),q(τ ˙ )) H(τb −τa ) −h ¯ τa ¯ |qa >= Dq e , (13.27) < qb |e qa

with 1 LE (q, q) ˙ = m 2



dq dτ

2

+ V (q) .

This leaves in (13.27) an exponentially damped exponential. We now provide a systematic derivation of this result. Corresponding to the composition property (13.16) one splits the (euclidean) time interval into N segments. Using the completeness of the eigenstate |q > of the position operator, one then has analogous to (13.17), 1

< q 0 |e− h¯ H(τ

0

−τ )

|q > =

Z NY −1 `=1

1

dq (`) < q 0 |e− h¯ H |q (N −1) > × 1

(13.28) 1

< q (N −1) |e− h¯ H |q (N −2) > · · · < q (1) |e− h¯ H |q > .

We take the Hamiltonian operator H to have the form H=

Pˆ 2 ˆ , + V (Q) 2m

ˆ Making use with the interaction part only depending on the position operator Q. of the Baker–Campbell–Hausdorff formula 1

eA eB = eA+B+ 2 [A,B]+... ,

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189

we may factorize exp(− h¯1 H) for  small as follows ˆ2  P

1



ˆ

e− h¯ H ≈ e− h¯ 2m e− h¯ V (Q) , where we have neglected terms of order O(2 ) in the exponent. We thus have ˆ2  P





< q (`+1) |e− h¯ H |q (`) >≈< q `+1) |e− h¯ 2m |q (`) > e− h¯ V (q

(`)

)

.

The matrix element for real time has already been given in (13.18). We thus find (make substitution i → ) 

< q (`+1) |e− h¯ H |q (`) >≈

 m 1/2 m (q(`) −q(`−1) )2 1 (`)  e− h¯ V (q ) . e− 2¯h 2π¯h

Substituting this expression into (13.28), we thus obtain 1

< q 0 |e− h¯ H(τ

0

−τ )

|q >≈

−1 PN −1  m N/2 Z NY (`) (`) 1 dq (`) e− h¯ `=0 LE (q ,q˙ ) , 2π¯h `=1

where LE (q

(`)

, q˙

(`)

m )= 2



q (`+1) − q (`) 

2

+ V (q (`) ) .

Set q (`) → q(t` ) . Taking the limit (13.21), we obtain with the measure (13.20) (i → ) after suitable relabelling, the announced result (13.27). Continuing back to real time we end up with (13.26).4

13.4

Path-integral representation of correlators in QM

In quantum field theory, the vacuum expectation value of time-ordered products of field operators (Green functions) G(x1 , x2 ..xn ) :=< Ω|T φ(x1 )φ(x2 )...φ(xn )|Ω >

(13.29)

are of interest. These are known to be analytic functions in the complex t-plane, allowing for a continuation t → −iτ in the time variables. The operators φ(xi ) appearing in (13.29) are related to the corresponding operators φ(~x, 0) in the Schr¨odinger picture by φ(~x, t) = eiHt φ(~x, 0)e−iHt . 4 For a non-trivial application to the (i) unharmonic oscillator, see I. Bender, D. Gromes and K.D. Rothe, Nucl. Phys. B136 (1978) 259. (ii) Coulomb potential, see I.H. Duru and H. Kleinert, Phys. Lett. 84B (1979) 185.

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Continuing these operators formally to imaginary time, we define the “euclidean” operators ˆ x, τ ) = eHτ φ(~x, 0)e−Hτ . φ(~ The QM analogon of the field φ(~x, t) is the infinite set of coordinates Q~x (t) labelled ~x. In the following, we shall restrict ourselves to just one degree of freedom Q(t). The generalization to an infinite number is then done in the following section. In the euclidean formulation, we then have −Hτ ˆ ) = eHτ Q(0)e ˆ Q(τ .

Actually these operators, taken by themselves, are ill defined if the spectrum of H is unbounded above (we shall always assume that the spectrum is bounded from below). However, for an operator product ordered according to descending times from left to right, the (euclidean) correlator ˆ 1 )Q(τ ˆ 2 )...Q(τ ˆ n )|q, τ >, < q 0 , τ 0 |Q(τ

τ 0 > τ1 > τ2 > ...τn > τ

is well defined. Inserting a complete set of energy eigenstates to the left and right of the operators, we obtain from here X 0 ˆ 1 )...Q(τ ˆ n )|q, τ >= ˆ 1 )...Q(τ ˆ n )|E` > , < q 0 , τ 0 |Q(τ e−E`0 τ eE` τ ψ`0 (q 0 )ψ`∗ (q) < E`0 |Q(τ `,`0

where we have set < q, τ |E` >=< q|e−Hτ |E` >=: ψ` (q)e−E` τ with ψ` (q) eigenfunctions of the Hamiltonian corresponding to the eigenvalues E` . Assuming that there exists an energy gap between the ground state and the first excited state, we conclude that in the limit τ 0 → ∞ and τ → −∞ ˆ 1 )...Q(τ ˆ n )|q, τ >−→ e−E0 (τ 0 −τ ) ψ0 (q 0 )ψ0∗ (q) < E0 |Q(τ ˆ 1 )...Q(τ ˆ n )|E0 > . < q 0 , τ 0 |Q(τ (13.30) ˆ ) by unit operators, we have In particular, replacing the operators Q(τ < q 0 , τ 0 |q, τ >−→ e−E0 (τ

0

−τ )

ψ0 (q 0 )ψ0∗ (q) .

(13.31)

We thus conclude that (τ 0 > τ1 > τ2 > ...τn > τ ) ˆ 1 )...Q(τ ˆ n )|q, τ > < q 0 , τ 0 |Q(τ ˆ 1 )...Q(τ ˆ n )|E0 > . −→< E0 |Q(τ < q 0 , τ 0 |q, τ >

(13.32)

Notice that for the formula (13.32) to be valid we only require the existence of an energy gap, and a non-vanishing projection of the states |q > and |q 0 > on the ground state |E0 >. In order to appreciate the importance of our euclidean discussion it is instructive to interrupt for a moment our discussion to consider as example the Minkowski functional integral for the quantum mechanical harmonic oscillator.

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191

Digression: Harmonic oscillator We wish to calculate < xb , T |xa , −T > for the case of the harmonic oscillator described by the lagrangian L=

1 m(x˙ 2 − ω 2 x2 ) . 2

Since this is the interacting case, we shall treat it using the semiclassical method described previously. In order to implement the boundary conditions it is again convenient to make the change of variable x = xcl + y , where xcl is a solution of the classical equation of motion x ¨cl + ω 2 xcl = 0 .

(13.33)

The most general solution of (13.33) is given by xcl = A sin ωt + B cos ωt . Imposing the boundary conditions x(−T ) = xa , x(T ) = xb , this fixes A and B to be xb − xa , 2 sin ωT

A=

B=

xb + xa . 2 cos ωT

Making use of this, we obtain for Lcl ,  2   1 (xa + x2b ) cos(2ωT ) − 2xa xb (x2b − x2a ) 2 Lcl = mω cos(2ωt) − sin(2ωt) . 2 sin(2ωT ) sin2 (2ωT ) Integrating over time from −T to +T, we obtain for the corresponding action Scl =

mω [(x2 + x2b ) cos(2ωT ) − 2xa xb ] . 2 sin(2ωT ) a

We thus conclude, imω

2

2

< xb , T |xa , −T >= F (T ; ω)e 2¯h sin(2ωT ) [(xa +xb ) cos(2ωT )−2xa xb ] ,

(13.34)

where F (T, ω) =

Z

y(T )=0

y(−T )=0

i

Dy e h¯

RT

−T

dt m ˙ 2 −ω 2 y 2 ) 2 (y

=

y(T )=0

y(−T )=0

with D the differential operator D=−

Z

d2 − ω2 . dt2

i

Dy e h¯

RT

−T

dt m 2 yDy

,

(13.35)

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Note again that the linear term in y is absent since Scl corresponds to an extremum of the action. Expand y in a complete set of orthonormal eigenfunctions of the operator D, Dun = λn un , satisfying the boundary condition un (−T ) = un (T ) = 0 . We have for the normalized eigenfunctions and eigenvalues nπt 1 un (t) = √ sin T T Make the expansion

λn =

∞ X

y(t) =

 nπ 2 T

− ω2 .

an un (t) .

n=0

With the change of variable

Dy → J

Y

dan ,

n

where J is the Jacobian of the transformation, we have for (13.35) Z

y(T )=0

y(−T )=0

Dy e

m i 2¯ h

RT

−T

dtyDy

→J

Z Y N

im

dan e 2¯h

n=1

= J

1 QN

m n=1 2¯ hλn

where J 0 is independent of ω. Noting that lim

N →∞

N  Y

n=1

ω2 T 2 1− 2 2 n π

−1/2

we finally obtain 0

F (T, ω) = J (T )



P

λn a2n

1/2 = J 

0

(sin ωT ) = ωT

ωT sin ωT

1/2

N  Y

n=1

ω2 T 2 1− 2 2 n π

− 21

−1/2

.

The function J 0 (T ) is determined by taking the limit ω → 0 and comparing with the free-particle case (1.11): J 0 (T ) = Hence F (T, ω) =





m 1/2 . 2πi¯hT

1/2 mω . 2πi¯h sin 2ωT

(13.36)

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193

Summarizing we finally have for (13.34), < xb , T |xa , −T >=



mω 2πi¯ h sin(2ωT )

1/2

2

mω

2

ei sin(2ωT ) [(xa +xb ) cos(2ωT )−2xa xb ] . (13.37)

We now observe that this transition amplitude has no definite limit for T → ∞. However, for imaginary time t = −iT , this result tends to 1/2 mω 2 2 mω e− 2¯h (xa +xb ) 2πi¯h sin 2ωT → e−ωT ϕ0 (xb )ϕ0 (xa ) 

< xb , −iT |xa , iT > → with

ϕ0 (x) =

 mω  14 π¯h

mω

e− 2¯h x

2

the oscillator ground state wave function with ground state energy 12 ω. One checks R 2 that φ0 = 1 corresponds to < E0 |E0 >= 1. This result is thus of the form (13.30) ˆ = 1. with Q Back to (13.32) We now return to our original discussion. In order to obtain a path-integral representation for the left-hand side of (13.32), we write ˆ 1 ) · · · Q(τ ˆ n )|q, τ >= < q 0 , τ 0 |Q(τ 0 ˆ −H(τ1 −τ2 ) · · · e−H(τn−1 −τn ) Qe ˆ −H(τn −τ ) |q > , < q 0 |e−H(τ −τ1 ) Qe ˆ = Q(0). ˆ where Q We proceed as before and repeatedly insert a complete set of ˆ thus obtaining eigenstates of the operator Q, Z Y n

ˆ 1 )...Q(τ ˆ n )|q, τ >= < q , τ |Q(τ 0

h

0

0

0

(1)

< q , τ |Q

(1)

, τ1 > Q

dQ(`)

`=1

(1)

Q(2) ...Q(n) < Q(n) , τn |q, τ > .

Making use of the path integral representation for the individual propagation kernels, and following the steps leading to (13.27), we may write the final result in the compact form < q 0 , τ 0 |Q(τ1 )...Q(τn )|q, τ >=

Z

Q(τ 0 )=q 0

Q(τ )=q

DQ Q(τ1 )...Q(τn )e−SE [Q;τ

where SE [Q; τ 0 , τ ] stands for the euclidean action 0

SE [Q; τ , τ ] =

Z

τ0

˙ 00 )) . dτ 00 LE (Q(τ 00 ), Q(τ

τ

The path integral is calculated as implied by (13.38):

0

,τ ]

,

(13.38)

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(i) Split the time interval [τ, τ 0 ] into n sub-intervals [τi+1 , τi ] corresponding to the ˆ i ) at “times” τ1 · · · τn . n “insertions” of the Operators Q(τ (ii) Divide each of the sub-intervals [τi+1 , τi ] into ni further sub-intervals. (iii) Consider all paths Q(τ ) starting at q at time τ and ending at q 0 at time τ 0 . (iv) Weigh each path with the factor Q(τ1 )...Q(τn ) exp(−SE [Q; τ 0 , τ ]). Sum the contributions over all paths by integrating over all possible values of the coordinates at the intermediate times, with the integration measure (13.20). (v) Take the limit i = (τi+1 − τi )/ni → 0, ni → ∞, keeping the product ni i = (τi+1 − τi ) fixed in each sub-sub-interval. We may relax the condition τ1 > τ2 > ... > τn . In that case the r.h.s. represents ˆ i ), i = 1, · · · n: the matrix element of the time-ordered product of the n operators Q(τ Z Q(τ 0 )=q0 0 ˆ 1 )...Q(τ ˆ n )|q, τ >= DQ Q(τ1 )...Q(τn )e−SE [Q;τ ,τ ] . < q 0 , τ 0 |T Q(τ Q(τ )=q

The right-hand side exhibits the symmetry with respect to the operators Q(τi ) of the time-ordered product of the left-hand side. Taking now the limit τ → −∞, τ 0 → ∞, we obtain (see (13.32)) R DQQ(τ1 )...Q(τn )e−SE [Q] ˆ ˆ R , (13.39) < E0 |T Q(τ1 )...Q(τn )|E0 >= DQ e−SE [Q]

where S[Q] is now the euclidean action Z ∞ ˙ )) . SE [Q] = dτ LE (Q(τ ), Q(τ −∞

This is our final result. Note that the normalization factor of the measure in (13.20) drops out in the ratio (13.39).

13.5

Feynman path-integral representation in QFT

The extension of the above result to field theory is now obvious. To this end we first extend the result (13.39) to the case of a quantum-mechanical system of K degrees of freedom, with coordinates Q → {Qk } = (Q1 ....QK ) . We evidently have ˆ k (τ1 )...Q ˆ k (τn )|E0 >= < E0 |T Q 1 n

R

DQQk1 (τ1 )...Qkn (τn )e−SE [Q] R , DQe−SE [Q]

where the integration measure now stands for DQ :=

K N −1 Y Y

k=1 l=1

dQk (τl ) .

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195

We now consider the limit where this number K of degrees of freedom tends to infinity. Replacing the label k by a continuous label ~x, ˆ ~x) , ˆ k (τ ) → φ(τ, Q

Qk (τ ) → φ(τ, ~x) ,

we thus arrive at the following expression for the euclidean correlator in a scalar field theory: R Dφφ(x1 )...φ(xn )e−SE [φ] ˆ ˆ R , < Ω|T φ(x1 )...φ(xn )|Ω >= Dφe−SE [φ]

where, in order to conform to QFT notation, we have set |E0 >= |Ω >. The corresponding n-point Green function in Minkowski space is recovered by performing an analytic continuation in “time” to the real time axis (τ → it) with the (formal) result R )...φ(xn )eiS[φ] ˆ 1 )...φ(x ˆ n )|Ω >= Dφ φ(x R 1 < Ω|T φ(x (13.40) DφeiS[φ]

with

S[φ] = where

Z

d4 yL(y) ,

L(y) ≡ L(φ(y), ∂µ φ(y)) is now the Minkowski–Lagrange density defining the (scalar) field theory of interest. The right-hand side of (13.40) represents the “partially connected” n-point Green function. Perturbation Theory recovered We now illustrate how one recovers from (13.40) the perturbative series defined by (11.21). To this end we define the generating functional Z R 4 Z[J] = Dφ ei d y(L(y)+J(y)φ(y)) . (13.41) In particular, Z[0] =

Z

Dφ ei

R

d4 yL(y)

.

The ratio Z[J] Z[0] is the path-integral version of Schwinger’s generating functional Z[J], Eq. (13.1). The partially connected n-point Green functions of the theory are correspondingly obtained by functional differentiation of ln Z[J] with respect to the external source J(x) and subsequently setting J = 0. Separating the action associated with the Lagrange density into its free and interacting part, S[φ] = S0 [φ] + SI [φ]

(13.42)

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and in (13.41) making the expansion eiS[φ] = eiS0 [φ]

∞ n X i n (SI [φ]) , n! n=0

we may evidently write5  n ∞ n  X δ i Z (0) [J] , SI Z[J] = n! iδJ n=0 where Z (0) [J] =

Z

DφeiS0 [φ]+i

R

d4 yJ(y)φ(y)

(13.43)

.

(13.44)

This is to be compared with the Schwinger result (13.12). For the case of a real scalar field, the free part of the action is purely quadratic in the fields having the form Z Z 1 1 S0 [φ] = d4 z (∂µ φ∂ µ φ − m2 φ2 ) = − d4 x φ( + m2 )φ (13.45) 2 2 Z Z 1 d4 x d4 yφ(x)K(x, y)φ(y) . =− 2 K(x, y) = (

x

+ m2 )δ 4 (x − y) ,

Z (0) [J] is easily evaluated by rewriting the exponent in (13.44) as a quadratic form, leaving us with: Z R 4 R 4 01 −1 0 −1 0 −1 0 0 1 Z (0) [J] = Dφ e−i d z d z [ 2 (φ−JK )(z)K(z,z )(φ−K J)(z )− 2 J(z)K (z,z )J(z )] , where

K −1 (x, y) = (

+ m2 )−1 xy = −∆F (x − y) .

By making a shift, the φ integration will contribute a constant which drops out in the ratio Z[J] = Z[J] Z[0] . We thus have  n ∞ R i Z[J] X in δ = SI [ ] e− 2 Z[0] n! iδJ n=0

d4 z

R

d4 z 0 J(z)∆F (z−z 0 )J(z 0 )

.

(13.46)

Performing the functional differentiations with respect to the external source and setting this source equal to zero at the end, reproduces the perturbative series of the scalar field theory in question. 5 From

δ

R

classicalR mechanics we are familiar with the computation of δS[q] ≡ S[q + δq] − S[q]. Thus dtf (q(t)) ≡ dt[f (q(t) + δq(t)) − f (q(t))]. With the “rule”

we correspondingly have δij , i.e.

∂ ∂xi

P

δ δJ(x)

x k = ki . j j j

R



δ J(y) δJ(x)



= δ 4 (x − y)

d4 yJ(y)φ(y) = φ(x). This is a natural generalization of

∂ x ∂xi j

=

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197

Example: φ3 theory In this case SI [φ] = or SI



1 δ i δJ



=

where F

(2)



1 δ i δJ



Z

Z

d4 z

λ 3!

To second order Z[J] = F

λ 3!

(2)



d4 zφ3 (z) 

1 δ i δJ(z)

3

.

 1 δ Z (0) [J] + ... i δJ

    2 i 1 δ i2 1 δ = 1 + SI + SI . 1! i δJ 2! i δJ

Now < Ω|T φ(x)φ(y)|Ω > = =



δ δ iδJ(x) iδJ(y) Z[J]

Z[0] 



J=0

   δ δ 1 δ + ··· . F (2) Z (0) [J] Z[0] iδJ iδJ(x) iδJ(y) J=0

We have Z  Z δ 1 Z (0) [J] = − d4 ξJ(ξ)∆F (ξ − y) + d4 ξ∆F (y − ξ)J(ξ) Z (0) [J] iδJ(y) 2 Z  =− d4 ξJ(ξ)∆F (ξ − y) Z (0) [J] and

δ δ Z (0) [J] = i∆F (x − y)Z (0) [J] iδJ(x) iδJ(y) Z  Z  4 4 0 0 0 + d ξJ(ξ)∆F (ξ − y) d ξ J(ξ )∆F (ξ − x) Z (0) [J] .

Now apply F (2) , divide by Z[0] and set J = 0 at the end. The result is represented schematically by the following set of diagrams:     δ δ δ Z (0) [J] = F (2) iδJ iδJ(x) iδJ(y) J=0 

 ⊗ 1 +



 +

+

(13.47)

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and (13.48)

Z[0] = 1 +

The last term in the sum (tad -pole) is eliminated by normal ordering. Regarding these diagrams as a power series in λ we see that the contribution of the vacuum diagrams (second term in the bracket) cancels in the ratio (13.46).

13.6

Path-Integral for Grassman-valued fields

So far we have considered quantum mechanical systems involving only bosonic degrees of freedom. But the fundamental matter fields in nature carry spin 1/2. In contrast to the bosonic case these fields anticommute in the limit h ¯ → 0, and hence become elements of a Grassmann algebra in this limit. We therefore expect that the path integral representation of Green functions built from fermion fields will involve the integration over anticommuting (Grassmann) variables.6 We therefore begin with the introduction of some basic notions regarding the differentiation and integration of functions of Grassmann variables. The integration rules are then applied to calculate specific integrals, which will play an important role throughout this book. The results we obtain will give us a strong hint regarding the path integral representation of fermionic Green functions in theories of interest for elementary particle physics. We begin our discussion with some basic definitions. Grassmann Algebra The elements η1 , ..., ηN are said to be the generators of a Grassmann algebra, if they anticommute among each other, i.e. if {ηi , ηj } = ηi ηj + ηj ηi = 0,

i, j = 1, ..., N.

(13.49)

From here it follows that ηi2 = 0.

(13.50)

A general element of a Grassmann algebra is defined as a power series in the ηi ’s. Because of (13.50), however, this power series has only a finite number of terms: X X f (η) = f0 + fi η i + fij ηi ηj + ... + f12...N η1 η2 ...ηN . (13.51) i

i6=j

As an example consider the function g(η) = e

−

PN

i,j=1

ηi Aij ηj

.

6 This section has been taken over from Lattice Gauge Theories — An Introduction by Heinz J. Rothe (World Scientific), with the author’s permission. For a comprehensive discussion of the functional formalism for fermions the reader may consult the book by F. Berezin, The Method of Second Quantization (Academic Press, New York, London, 1966).

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199

It is defined by the usual power series expansion of the exponential. Since the terms appearing in the sum — being quadratic in the Grassmann variables — commute among each other, we can also write g(η) as the product g(η) =

Y

e−ηi Aij ηj ,

i,j

or, making use of (13.50), g(η) =

N Y

i,j=1 i6=j

(1 − ηi Aij ηj ).

Next we consider the following function of a set of 2N -Grassmann variables which we denote by η1 , .., ηN , η¯1 , ..., η¯N : P − η¯i Aij ηj ij h(η, η¯) = e . Proceeding as above, we now have that h(η, η¯) =

N Y

(1 − η¯i Aij ηj ).

i,j=1

Notice that in contrast to the previous case, this expression also involves diagonal elements of Aij . Integration over Grassmann variables We now state the Grassmann rules for calculating integrals of the form Z Y N

dηi f (η),

i=1

where f (η) is a function whose general structure is given by (13.51). Since a given Grassmann variable can at most appear to the first power in f (η), the following rules suffice to calculate an arbitrary integral: Z Z dηi = 0, dηi ηi = 1. (13.52) When computing multiple integrals one must further take into account that the integration measures {dηi } also anticommute among themselves, as well as with all ηj ’s {dηi , dηj } = {dηi , ηj } = 0, ∀i, j. These integration rules look indeed very strange. But, as we shall see soon, they are the appropriate ones to allow us to obtain a path-integral representation of

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fermionic Green functions. As an example, let us apply these rules to calculate the following integral: Z Y PN N − η¯i Aij ηj i,j=1 I= . (13.53) d¯ η` dη` e `=1

To evaluate (13.53), we first write the integrand in the form e

−

P

i,j

η¯i Aij ηj

=

N Y

e

−¯ ηi

i=1

PN

j=1

Aij ηj

.

Since η¯i2 = 0, only the first two terms in the expansion of the exponential will contribute. Hence ! ! ! P X X X − η¯i Aij ηj i,j = 1 − η¯1 e A1i1 ηi1 1 − η¯2 A2i2 ηi2 . . . 1 − η¯N AN iN ηiN . i1

i2

iN

(13.54) Now, because of the Grassmann integration rules (13.52), the integrand of (13.53) must involve the product of all the Grassmann variables. We therefore only need to consider in (13.54) the term X K(η, η¯) ≡ (ηi1 η¯1 )(ηi2 η¯2 ) . . . (ηiN η¯N )A1i1 A2i2 . . . AN iN , (13.55) i1 ,...,iN

where we have set η¯k ηik = −ηik η¯k to eliminate the minus signs appearing in (13.54). The summation clearly includes only those terms for which all the indices i1 , . . . , iN are different. Now, the product of Grassmann variables in (13.55) is antisymmetric under the exchange of any pair of indices i` and i`0 . Hence we can write the expression (13.55) in the form X K(η, η¯) = η1 η¯1 η2 η¯2 . . . ηN η¯N i1 i2 ...iN A1i1 A2i2 . . . AN iN , i1 ...iN

where i1 i2 ...iN is the Levi–Civita-tensor in N -dimensions. Recalling the standard formula for the determinant of a matrix A, we therefore find that K(η, η¯) = (det A)η1 η¯1 η2 η¯2 . . . ηN η¯N . We now replace the exponential in (13.53) by this expression and obtain I=

"N Z Y

#

d¯ ηi dηi ηi η¯i det A = det A.

i=1

Let us summarize our result for later convenience: Z

D(¯ η η)e

−

PN

i,j=1

η¯i Aij ηj

= det A,

D(¯ η η) =

N Y

`=1

d¯ η` dη` .

(13.56)

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201

There is another important formula we shall need. It will allow us to calculate integrals of the type Z PN − η¯i Aij ηj i,j=1 . (13.57) η η)ηi1 ...ηi` η¯i01 . . . η¯i0` e Ii1 ···i` i01 ···i0` = D(¯ Consider the following generating functional Z P P − η¯i Aij ηj + (¯ ηi ρi +ρ¯i ηi ) i,j i Z[ρ, ρ¯] = D(¯ η η)e ,

(13.58)

where all indices are understood to run from 1 to N , and where the “sources” {ρi } and {¯ ρi } are now also anticommuting elements of the Grassmann algebra generated by {ηi , η¯i , ρi , ρ¯i }. To evaluate (13.58) we first rewrite the integral as follows:  P Z P 0 − η¯i Aij ηj0 ρ¯ A−1 ρ i,j D(¯ η η)e e i,j i ij i , Z[ρ, ρ¯] = where

ηi0 = ηi − −1

X k

A−1 ik ρk ,

η¯i0 = η¯i −

X

ρ¯k A−1 ki ,

k

and A is the inverse of the matrix A. Making use of the invariance of the integration measure under the above transformation7 and of (13.56), we find that P ρ¯ A−1 ρ Z[ρ, ρ¯] = det A e i,j i ij j . (13.59) Notice that in contrast to the bosonic case, this generating functional is proportional to det A. Differentiation of Grassmann Variables We now complete our discussion on Grassmann variables by introducing the concept of a partial derivative on the space of functions defined by (13.51). Suppose we want to differentiate f (η) with respect to ηi . Then the rules are the following: 1. If f (η) does not depend on ηi , then ∂ηi f (η) = 0 2. If f (η) depends on ηi , then the left derivative → ∂/∂ηi is performed by first bringing the variable ηi (which never appears twice in a product!) all the way to the left, using the anticommutation relations (13.49), and then applying the rule ∂~ ηi = 1. ∂ηi ← − Correspondingly, we obtain the right derivative ∂ /∂ηi by bringing the variable ηi all the way to the right and then applying the rule ← − ∂ = 1. ηi ∂ηi 7 This

is ensured by the Grassmann integration rules.

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Thus for example ∂ ηj ηi = −ηj ∂ηi or η¯i ηj

(i 6= j),

← − ∂ = −ηj . ∂ η¯i

Notice that, because of the peculiar definition of Grassmann integration, we have that Z ∂ f (η). dηi f (η) = ∂ηi Hence integration over ηi is equivalent to partial differentiation with respect to this variable! Another property, which can be easily proved, is that   ∂ ∂ , f (η) = 0. ∂ηi ∂ηj Let us apply these rules to some cases of interest. Consider the function P ρ¯ η E(¯ ρ) = e j j j ,

where {ηi , ρ¯i } are the generators of a Grassmann algebra. If they were ordinary c-numbers then we would have that ∂ E(¯ ρ) = ηi E.(¯ ρ) . ∂ ρ¯i

(13.60)

This result is in fact correct. To see this, let us write E(¯ ρ) in the form E(¯ ρ) =

Y (1 + ρ¯j ηj ). j

Applying the rules of Grassmann differentiation, we have that Y ∂ E(¯ ρ) = ηi (1 + ρ¯j ηj ). ∂ ρ¯i j6=i

But because of the appearance of the factor ηi we are now free to extend the product to include i = j. Hence we arrive at the naive result (13.60). It should, however, P be noted, that the order of the Grassmann variables in i ρ¯i ηi was important. By reversing this order we get a minus sign, and the rule is not the usual one! By a similar argument one finds that e

P

j

η¯j ρj

← − P ∂ η¯ ρ = η¯i e j j j . ∂ρi

Let us now return to the generating functional defined in (13.58). Proceeding as

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203

above one can easily show that " Ii1 ...i` ;i01 ...i0`

← − ← − # ∂ ∂ ∂ ∂ = ... Z[ρ, ρ¯] ... ∂ ρ¯i1 ∂ ρ¯i` ∂ρi01 ∂ρi0`

,

(13.61)

ρ=ρ=0 ¯

where the left-hand side has been defined in (13.57). By making use of the explicit expression for Z given in (13.59), one can calculate the right-hand side of (13.61). Since we shall need this expression in later chapters, we will derive it here. To this effect we first rewrite (13.59) as follows ! Y ρ¯i P A−1 ρj X −1 ij j Ai1 k1 ρk1 Z[ρ, ρ¯] = det A e = (det A) 1 + ρ¯i1 (13.62) i

× 1 + ρ¯i2

X

A−1 i2 k2 ρk2

k2

!

k1

... 1 + ρ¯i`

X

A−1 i` k` ρk`

k`

!

[...],

where the indices k1 , ..., k` are summed, and [· · ·] stands for the remaining factors not involving the variables ρ¯i1 , ..., ρ¯i` . The only terms which contribute to the left derivatives in (13.61) are those involving the product ρ¯i1 ...¯ ρi` . Furthermore since we will eventually set all “sources” ρi and ρ¯i equal to zero, we can replace [· · ·] by 1. The contribution in (13.62), which is relevant when computing (13.61), is therefore given by X ˜ ρ¯] = det A Z[ρ, (¯ ρi1 A−1 ρi` A−1 i1 k1 ρk1 )...(¯ i` k` ρk` ), {ki }0

where all ki ’s are different, and the “prime” on {ki }0 indicates that the ki ’s take only values in the set (i01 , i02 , ..., i0` ), labelling the right derivatives in (13.61). Thus we can write the above expression in the form X −1 −1 ˜ ρ¯] = det A Z[ρ, A−1 ¯i1 ρi0P ρ¯i2 ρi0P ...¯ ρi` ρi0P , (13.63) i1 i0 Ai2 i0 ...Ai` i0 ρ P

P1

P2

1

P`

where the sum extends over all permutations  0 i1 i02 ... P : 0 iP1 i0P2 ...

i0` i0P`



2

.

`

(13.64)

Each of the products of Grassman variables appearing in the sum (13.63) can be put into the form i0 i0 ...i0 Fi11i22...,i`` ≡ ρ¯i1 ρi01 ρ¯i2 ρi02 ...¯ ρi` ρi0` by using the anticommutation rules for Grassmann variables. It follows that " # X i0 ...i0 −1 −1 σ P Z˜ = (det A) (−1) A 0 ...A 0 F 1 ` (ρ, ρ¯), P

i1 iP

1

i` iP

`

i1 ...i`

where (−1)σP is the signum of the permutation (13.64). We now apply the left and right derivatives indicated in (13.61) to the above expression and obtain the

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following important result: Z P X − η¯i Aij ηj −1 i,j = ξ` (det A) (−1)σP A−1 D(¯ η η)ηi1 ...ηi` η¯i01 ...¯ ηi0` e i1 i0 ..Ai` i0 , P

P1

P`

(13.65)

where ξ` = (−1)`(`−1)/2 . As a particular case of (13.65) we have that Z P − η¯i Aij ηj i,j = (det A)A−1 D(¯ η η)ηi η¯j e ij .

(13.66)

Let us define the 2-point correlation function < ηi η¯j >=

R

P − η¯i Aij ηj i,j D(¯ η η)ηi η¯j e P . R − η¯i Aij ηj i,j D(¯ η η)e

Then it follows from (13.56) and (13.66) that

< ηi η¯j >= A−1 ij .

(13.67)

We shall refer to (13.67) as a contraction. The generalization of (13.67) to arbitrary “correlation” function is then given by P R − η¯i Aij ηj i,j D(¯ η η)ηi1 ...ηi` η¯i01 · · · η¯i0` e P < ηi1 ...ηi` η¯i01 ...¯ ηi0` >= . (13.68) R − η¯i Aij ηj i,j D(¯ η η)e Note that the euclidean correlation function contains no time-ordering operation.

13.7

Extension to Field Theory

From our discussion in Section 13.5, the formal extension of the generating functional (13.58) to Field Theory is obvious. We have for euclidean time τ , Z R 4 R 4 0 R 4 0 0 ¯ η (z)ρ(z)) Z[ρ, ρ¯] = D(¯ η η)e− d z d z η¯(z)K(z,z )η(z ) e d z(ρ(z)η(z)+¯ . One way of making the connection with a system with a discrete number of degrees of freedom is to suppose K to be an operator defined on a compact manifold M, with a discrete eigenvalue spectrum and a complete set of eigenfunctions: Kun = λn [K]un . For the time being we shall assume that K has no zero eigenvalues. Starting from Z R Z[0] = D¯ η Dη e− η¯Kη

(13.69)

(13.70)

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205

we expand the fields η(x), η(x) in terms of the above eigenfunctions: η(x) =

X

an un (x),

X

an u†n (x)

n

η(x) =

(13.71) ,

n

with an , an , bn , bn Grassmann-valued c-number coefficients. Choosing un and vm to be orthonormal with respect to a measure dµ(x) on M, Z

dµ(x)u†n (x)un0 (x) = δnn0

(13.72)

we make the change of variable DηDη = J

Y

dan dan ,

(13.73)

n

where J is the Jacobian of the transformation. Introducing the expansions (13.71) in (13.70) and using (13.72), we obtain Z[0] = J

Z Y

dan dan e−

n

Now, e−

P

λn (K)an an

=

Y

P

e−λn [K]an an =

n

λn [K]an an

.

Y (1 − λn [K]an an ) . n

The last equality only holds, because of the Grassmann properties a2n = a2n = 0. Furthermore, according to the Berezin integration rules of the previous section, Z Y P Y dan dan e− λn [K] = λn [K] . (13.74) n

Hence we conclude, Z[0] = J

n

Y

λn [K] = J det K.

(13.75)

n

The unknown constant J arising from the change of variable (13.73) will eventually drop out when calculating correlators such as (13.68). Equation (13.75) corresponds to the identification made in (13.56). If there are zero modes present, then these should be omitted in (13.75), resulting in Z[0] = J

Y0 n

λn [K] = det0 K ,

(13.76)

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where the “prime” indicates omission of zero modes. Hence det0 K is just the subdeterminant corresponding to the non-vanishing eigenvalues of K. The existence of “zero-modes” is not seen in perturbation theory.8 The divergencies occurring in the loop-integrations have their reflection in the non-existence of the product (13.75) because of the (in general unbounded) growth of the eigenvalues with n → ∞. Thus again, a regularization procedure has to be given for defining the determinant via (13.75).

13.8

Mathews–Salam representation of QED generating functional

We now make use of the results above in order to arrive at a representation of the generating functional of QED in terms of a path integral over gauge-field configurations alone. Returning to Minkowski space via the formal substitutions τ → ix0 , we have for the generating functional of the n-point Green-functions of fermions interacting with a gauge field Aµ :9 Z Z R 4 µ R 4 R 4 Z[η, η; J] = [DA] [Dψ][Dψ] ei d zL(z) ei d z(ηψ+ψη)(z) ei d zJ (z)Aµ (z) , (13.77)

where L(x) is the QED Lagrange density ¯ / − m)ψ, L = LG (A) + ψ(iD

1 LG (A) = − Fµν F µν , 4

(13.78)

with Dµ the covariant derivative iDµ = i∂µ − e0 Aµ . Integrating over the fermions we have Z R 4 Z[η, η¯, J] = [DA]Det(iD / − m)ei d zLG (A(z)) R 4 R 4 R 4 0 0 0 µ × ei d z d z η(z)G(z,z ;A)η(z ) ei d zJ(z)µ A (z) ,

(13.79)

(13.80)

where G(x, y; A) is the inverse of the Dirac operator (fermionic Green function in an external field Aµ ): (iD / − m)G(x, y; A) = δ (4) (x − y) .

(13.81)

Let us consider the individual terms in (13.80). In order to simplify the presentation, we set m = 0. Then formally i 6 ∂G = 1 + e0 A 6 G, 8 E.

Abdalla, M. Christina B. Abdalla and K.D. Rothe, Non-Perturbative Methods in 2 Dimensional Quantum Field Theory (World Scientific, 1991, and extended version 2001). 9 P.T. Mathews and A. Salam, Phys. Rev. 94 (1954) 185.

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13.8 Mathews–Salam representation of QED generating functional 207 that is, G=

1 1 + e0 A 6 G. i 6∂ i 6∂

Iterating this equation   1 1 1 1 1 1 + e0 A 6 + e0 A 6 e0 A 6 + ··· G(z, z 0 ; A) = i 6∂ i 6∂ i 6∂ i 6∂ i 6∂ i 6∂ z,z 0 where



1 i 6∂



zz 0

= SF (z − z 0 )

is the Feynman propagator satisfying i 6 ∂SF (z) = δ 4 (z) . In terms of Feynman diagrams

A G(x, y; A) =

x

y

+

A

A

+

+ · · · (13.82)

Fig. 13.1. Green function in external gauge-field. The diagrammatic sum represents the effect of repeated scatterings of virtual fermions in the external field Aµ , and cannot generally be performed in closed form (except in two space-time dimensions!). Explicitly it is represented by Z iG(x, y; A)αβ = iSF (x − y)αβ + d4 z1 [iSF (x − z1 )(−ie0 A 6 (z1 ))iSF (z1 − y)]αβ +

Z

4

d z1

Z

d4 z2 [iSF (x−z1 )(−ie0 A 6 (z1 ))iSF (z1 −z2 )(−ie0 A 6 (z2 ))iSF (z2 −y)]αβ +· · ·

In a similar spirit we now show that the logarithm of a functional determinant has a simple interpretation in terms of Feynman diagrams: it may be represented as an infinite sum of properly weighted one fermion-loop diagrams. This is easily seen on a formal level. Thus consider in general an hermitian operator D which we decompose into a “free” (D0 ) and an “interacting” part (DI ) as follows: D = D0 + DI . Borrowing familiar formulas from matrix algebra, we write: ωD ≡ ln det D = Tr ln (D0 + DI ) , ω

D D0

 n ∞ X 1 det D (−1)n+1 −1 = Tr ln (1 + D0 DI ) = Tr DI = ln det D0 n D0 n=1

(13.83)

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where D0−1 DI is to be regarded as a (non-local) operator, and where Tr denotes the trace with respect to “internal” indices as well as space-time. The functional ωD possesses the obvious property, ω D = ωD − ωD0 . D0

The sum (13.83) has a simple graphical interpretation: it is given by the sum of (connected) 1-loop graphs as shown in Fig. 13.2, with the following “Feynman” rules: (a) −DI for each vertex,

(b) D0−1 for each propagator, (c) (−1) overall factor for being a fermion loop, (n−1)! n!

(d)

=

1 n

combinatorial factor for a nth order diagram.

As an example consider the euclidean Dirac operator of QED, iD 6 = (i 6 ∂ − e0 A 6 ) with Aµ (x) some arbitrarily chosen “external” gauge-field configuration. According to (13.83) we have ln

∞ i X h det(i 6 ∂ − e A (−1)n+1 n −e0 on 0 6 ) = Tr A 6 det(i 6 ∂) n i 6∂ n=1 Z = − d4 z1 tr iSF (0)(−ie0 A 6 (z1 )) (13.84) Z Z 1 − d4 z1 d4 z2 tr [iSF (z1 − z2 )(−ie0 A 6 (z2 ))iSF (z2 − z1 )(−ie0 A 6 (z1 ))] + · · · 2

This series has the graphical representation

+

+ ···

(13.85)

Fig. 13.2. Functional determinant. The functional W [A] = −i ln det D / is called the 1-loop effective action. As we have learned, W [A] is given by the sum of 1-loop connected diagrams shown in the figure above, whereas eiW [A] will also involve disconnected diagrams such as

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209

(13.86)

⊗ Fig. 13.3. Disconnected 1-loop graphs in external field.

The effect of the Aµ integration in (13.80) is then to turn the external Aµ lines into photon propagators connecting these loops in all possible ways. This will lead to diagrams such as

Fig. 13.4. Second order vacuum graph. The divergences occurring in the loop-integrations implicit in (13.84) have their reflection in the non-existence of the product (13.75) because of (in general, unbounded) growth of the eigenvalues with n → ∞. Thus again, a regularization procedure has to be given for defining these integrals.

13.9

Faddeev–Popov quantization and α-gauges

We have seen in Sections 3 and 4 of Chapter 8 that gauge theories are first class systems requiring a choice of gauge for their quantization. The Faddeev–Popov method10 provides a systematic procedure for implementing a particular gauge in a gauge theory. We illustrate it for the case of QED. Starting point is the functional integral (we go to Minkowski space) Z[0] =

Z

DA

Z

¯

iS[A,ψ,ψ] ¯ DψDψe ,

(13.87)

with ¯ = SG [A] + SF [A, ψ, ψ] ¯ , S[A, ψ, ψ] where SF is the fermionic action, and SG is the Maxwell action which, after a partial integration, can be written in the form SG [A] = 10 L.D.

Z

d4 zAµ (z)(gµν

− ∂µ ∂ν )Aν (z) .

Faddeev and V.N. Popov, Phys. Lett. B167 (1986) 225.

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The differential operator appearing in SG [A] is not invertible, since it is transversal to the gradiant of any function, thus having an infinite set of zero-modes of the form ∂ µ Λ(x). In fact, the corresponding Lagrange density is invariant under the simultaneous U (1) gauge transformations Aµ (x) → Aθµ (x) = Aµ (x) + ∂µ θ(x) ψ(x) → ψ θ (x) = e−ie0 θ(x) ψ(x) .

(13.88)

All configurations Aθµ , ψ θ and ψ¯θ are said to belong to the orbit of Aµ , ψ, ψ and are physically equivalent. The integration measure (Haar measure) DA is, however, ¯ gauge-invariant, and so is DψDψ. All configurations belonging to a given orbit thus contribute to the functional integral with the same weight, with the result that the Aµ -integral diverges. To gain control of this infinity, we shall have to fix the gauge. The Faddeev– Popov method provides a systematic procedure for achieving this. To do this, we first have to choose a condition which selects just one configuration on each orbit. Familiar conditions are: ~ ·A ~=0 ∇

(Coulomb gauge)

µ

∂µ A = 0

(Lorentz gauge) .

In general, we suppose such a condition to be local in xµ : f (A(x)) = 0. Let ∆[A] be defined by 1 = ∆[A] where

Z

δ[f (Aθ )] =

Dθ δ[f (Aθ )] ,

Y x

(13.89)

(13.90)


δ f (Aθ (x)) .

It follows from the invariance of Dθ under the shift θ → θ + θ0 , that ∆[A] = ∆[Aθ ] , so that the identity (13.90) can also be written in the form Z 1 = Dθ ∆[Aθ ]δ[f (Aθ )] . Introduce this identity into the functional integral and make use of the gauge invariance of the integration measure and of the action in the functional integral (13.87), ¯ = S[Aθ , ψ θ , ψ¯θ ] S[A, ψ, ψ] in order to write this functional integral in the form Z Z Z θ θ ¯θ Z[0] = Dθ DAθ ∆[Aθ ]δ[f (Aθ )] Dψ¯θ Dψ θ eiS[A ,ψ ,ψ ] .

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13.9 Faddeev–Popov quantization and α-gauges ¯ we are left with After renaming the variables Aθ → Aµ , ψ θ → ψ, ψ¯θ → ψ, Z R 4 Z[0] = ΩU (1) [DA]F P ei d zL where [DAµ ]F.P. = DAµ ∆[A] and ΩU (1) =

Z

Y

211

(13.91)

δ(f (A(x))

x

Dθ

is the (infinite) “volume” of the gauge-group U (1). We have thus succeeded in factorizing the infinite part of the originally undefined functional integral. ∆[A] may be calculated as follows. With Aµ (x) chosen such that f (A(x)) = 0, we have to first order in θ,   Z δf (A(x)) ∂ µ θ(y) + · · · f (Aθ (x)) = f (A(x) + ∂θ(x)) = d4 y δAµ (y) Z = d4 yM(x, y)θ(y) + · · · where M(x, y) = −∂yµ

δf (A(x)) . δAµ (y)

(13.92)

Hence it follows from (13.90) that 1 = ∆[A]

Z

 Y Z 4 Dθ δ d yM(x, y)θ(y) . x

Using the usual rule for a change of variable in δ-functions, as well as (13.56), we have Z R ∆[A] = det M = D¯ cDc e− c¯Mc , (13.93) with c, c¯ Grassmann-valued spin-zero (ghost) fields. Notice that unlike Dirac fermions, these ghost fields carry no spin index. ∆[A] is called the Faddeev–Popov determinant. Aµ -propagator in α-gauges The so-called α-gauges were first introduced by ’t Hooft.11 Choose the gauge fixing function f (x) to be some local function of the form f (A(x)) = h(A(x)) − κ(x) . 11 G.’t

Hooft, Nucl. Physics., Ser. B (1971) 173.

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Functional Methods κ2

Now average in (13.91) over all “κ-gauges” with Gaussian weight e−i 2α : Z R 4 2 Z R 4 i − 2α d zκ (z) Z[0] → Z[0] = Dκe DA∆[A]δ[h(A) − κ] ei d zL Z R 4 2 R 4 i = DA∆[A]e− 2α d zh (A) ei d zL .

Choose in particular

h(A(x)) = ∂µ Aµ (x) . Then we have from (13.92) and (13.93), ∆[A] = det Hence Z[0] =

Z

)ei

DA(det

R

.

d4 z[− 14 Fµν F µν −

(∂µ Aµ )2 2α

] i

e

R

d4 zLF

,

where LF stands for the fermionic part of the Lagrangian (13.78). Write   Z   Z 1 (∂µ Aµ )2 1 1 d4 z − Fµν F µν − = d4 z − (∂µ Aν − ∂ν Aµ )∂ µ Aν − (∂µ Aµ )2 4 2α 2 2α     Z 1 1 = d4 zAµ gµν − 1 − ∂µ ∂ν Aν . 2 α Hence we have for the Maxwell part, Z R i ZG [0] = DA(det )e 2

d4 zAµ [gµν

1 −(1− α )∂µ ∂ν )]Aν

.

(13.94)

Note that the ghost part completely decouples. This is no longer the case in QCD. From (13.94) we see that the Feynman propagator of the gauge field is given in compact notation (compare (13.94) with (13.45) and recall that ∆F = 1/ ),    −1 1 µν DF = gµν − 1 − ∂µ ∂ν α =

g µν − (1 − α) ∂

µ

∂ν

.

(13.95)

Two choices of α are of particular interest: DFµν

g µν −

∂µ∂ν

(Lorentz gauge)

(13.96)

(Feynman gauge)

(13.97)

In momentum space, these propagators read, respectively  µν  µ ν g −(1−α) k 2k   k   − (α gauges) 2 +i   kµ ν µν k k µν g − DF (k) = − k2 (Lorentz gauge)  .  k2µν +i     g − k2 +i (Feynman gauge)

(13.98)

α=0: α=1:

=

DFµν =

g

µν

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Chapter 14

Dyson–Schwinger Equation The fully dressed propagators and vertex-functions form the building blocks of a general diagram in Quantum Field Theory. Their respective perturbation series as obtained from (11.21) can be summed graphically into the form of highly nonlinear integral equations, which only admit a perturbative solution, as we show in this chapter.

14.1

Classification of Feynman Diagrams

The following classification of Feynman Diagrams will be of fundamental importance in the subsequent development of the subject: Tree graphs Diagrams which do not involve loops have a tree-like structure, and are called treegraphs. Examples are

Fig. 14.1. Tree diagrams. A general Feynman diagram in QED is constructed in terms of the 2- and 3-point functions suitably integrated over the internal space-time variables. The skeleton diagram of a general diagram is obtained by stripping it off all corrections to the bare vertices, photon and fermion lines. An example is: 213

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Dyson–Schwinger Equation

Fig. 14.2. Skeleton diagram. Conversely we obtain from a given skeleton diagram a general diagram by adding the “meat” to the “bones” via the substitutions.

−→

−→

−→

Fig. 14.3. Adding “meat” to the “bones”. Connected diagrams We have seen in Section 5 of Chapter 11 that disconnected “vacuum” diagrams cancel in the computation of a general Green function. This nevertheless leaves us with disconnected diagrams such as

Fig. 14.4. Disconnected diagrams of 2-particle scattering.

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215

contributing to a fermion-4-point function. Such diagrams we call “partially connected”. Connected Green functions are defined to contain no such disconnected diagrams. 1-particle irreducible (1PI) diagrams Among the connected diagrams there exist diagrams with the property that they can be separated into two disconnected pieces by cutting a single intermediate line. A simple example is provided by

Fig. 14.5. 1-particle reducible diagrams. Connected diagrams which cannot be separated into two disconnected pieces by cutting a single internal line are called 1-particle irreducible (1PI). An example of such a diagram is

Fig. 14.6. 1-particle irreducible (1PI) diagram.

14.2

Basic building blocks of QED

The classical lagrangian of QED reads 1 L = − Fµν F µν + ψ(i∂/ − e0 A/ − m0 )ψ . 4

(14.1)

This Lagrange density serves as basis for the perturbative expansion. It contains terms quadratic and cubic in the fields. It is customary to treat the “mass term” mψψ, being quadratic in the fields, as part of the kinetic terms associated with the 2 fermion field. The absence of an analogous term µ2 A2µ for the gauge field reflects the masslessness of the photon. We claim that the interaction of the fields will not induce such a term either. This is consistent with the experimental fact that the photon is a massless quantum. We similarly claim that the interaction of the fields will not induce other types of interactions on quantum level. It can then be shown that the theory satisfies the basic renormalization criteria to be given later. The 2-point functions of the photon and fermion,1 1 Fully “dressed” 2-point functions are denoted by a “prime”. Blobs which are (are not) 1PI are represented by hatched (shaded) circles respectively.

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Dyson–Schwinger Equation hΩ|T Aµ (x)Aν (y)|Ωi = iDF0µν (x − y) =

µ

ν

(14.2)

< Ω|T ψα (x)ψ¯β (y)|Ω >= iSF0 (x − y)αβ =

α

β

(14.3)

α

as well as the 3-point function < Ω|T ψα (x)ψ¯β (y)Aµ (z)|Ω >amp. = −ie0 Γµαβ (x, y, z) =

(14.4)

µ

β form the basic building block of any Feynman diagram of

by

form the basic building block of any Feynman diagram of QED. They are given by the iterative solution of the non-linear diagrammatic equations

=

+

=

+

Fig. 14.7. Non-linear Dyson–Schwinger equations. In algebraic form these diagrammatic equations are given in momentum space by iSF0 (p) = iSF (p) + iSF (p)(−iΣ(p))iSF0 (p)

(14.5)

iDF0µν (k) = iDFµν (k) + iDFµλ (k) (−iΠλρ (k)) iDF0ρν (k) ,

(14.6)

where Σ(p) and Π(k) are given by the sum of 1-particle irreducible (1PI) diagrams of the type −iΣ(p) =

+ ‘

+

+ · · · (14.7)

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−iΠλρ (p) =

page 217

217

+ ···

+

(14.8)

Fig. 14.8. Perturbative expansion of electron self-energy and vacuum polarization. with

1 SF (p) =  p − m0 + i

the Feynman propagator of the fermion, and DFµν (k) = −

P µν (k) + “gauge” , k 2 + i

P µν (k) = g µν −

kµ kν k2

(14.9)

the Feynman propagator of the gauge-field in a general gauge. The term abbreviated by “gauge” stands for an expression of the form gauge = k µ f ν (k) + k ν f µ (k) ≡ DFµν (k)long , where the function f µ (k) depends on the choice of gauge. In the α-gauges (see (13.98)) kµ kν gauge = (1 − α) 2 . (14.10) (k + i)2 P µν (k) is the projector with the property kµ P µν = 0,

P µλ Pλν = P µν .

(14.11)

Σ(p) and Π(k) are the full 1PI 2-point functions associated with the fermion and the photon respectively. They are referred to as the electron self-energy and vacuum polarization, respectively. It follows from Lorentz invariance, that they have the following algebraic decomposition  Σ(p) = Σ1 (p2 ) + (p − m)Σ2 (p2 ) , Πµν (k) = (−k 2 g µν + k µ k ν )π(k 2 ) = −P µν (k)k 2 π(k 2 ) .

(14.12)

The fact, that the “polarization tensor” Π(k)µν is “transversal”, i.e. kµ Πµν (k) = 0

(14.13)

follows from the gauge invariance of the lagrangian (14.1), as we shall see later on. It is easily seen that the algebraic Eq. (14.5) has the solution 1 SF0 (p) =  . p − m − Σ(p)

(14.14)

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Dyson–Schwinger Equation

This expression formally stands for the inverse of the 1PI 2-point function  SF0−1 (p) = p − m0 − Σ(p) . (14.15) Substituting (14.12) into (14.6), and using (14.13) we arrive at the algebraic equation iDF0µν (k) = iDFµν (k) + P µρ (k)π(k 2 )iDF0ρν (k) ,

or DF0µν (k) = −

P µν (k) + DFµν (k)long + P µρ (k)π(k 2 )DF0ρν (k) . k 2 + i

(14.16)

Make the Ansatz DF0µν (k) = −P µν (k)DF0 (k 2 ) + DF0µν (k)long and use P µρ (k)P ρν (k) = P ρν (k),

P µρ DFρν (k)long = 0 .

We then have from (14.16) P µν (k) + DFµν (k)long k 2 + i − P µν (k)π(k 2 )DF0 (k 2 ) .

−P µν (k)DF0 (k 2 ) + DF0µν (k)long = −

From the tensorial structure we conclude, DF0µν (k) =

−P µν (k)   + DFµν (k)long , 2 2 k 1 − π(k )

(14.17)

i.e. the longitudinal part of the propagator is not affected by the interaction. As for the full 1PI 3-point function, it may be written in the form   −ie0 Γµ (p0 , p; k) = −ie0 γ µ + Λµ (p0 , p) .

The fermion self-energy Σ(p), photon polarization tensor Πµν (k), and 1PI irreducible 3-point function Γµ (p0 , p; k) represent the basic building blocks of any n-point function in QED.

14.3

Dyson–Schwinger Equations

We still need to compute Σ(p) und π(q) appearing in (14.14) and (14.17). They are given by the IPI 2-point functions of the fermion and gauge field, respectively. It is not hard to see that they satisfy the following non-linear integral equations involving the 2- and 3-point functions, which themselves involve Σαβ (p) and π µν (k 2 ): " # Z d4 k µ 0 0 ν −iΣαβ (p) = (−ie0 )γ iDF (k)µν iSF (p − k)(−ie0 )Γ (p − k, p) (14.18) (2π)4 αβ

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µν

−iΠ (k) = −

Z

219

" # d4 p 0 µ 0 ν tr iSF (p)(−ie0 )γ iSF (p + k)(−ie0 )Γ (p, p − k) (14.19) (2π)4

and having the diagrammatic representation

= Fig. 14.9. Dyson–Schwinger equation for electron-self-energy.

=

Fig. 14.10. Dyson–Schwinger equations for vacuum polarization.

=

+

Fig. 14.11. Dyson–Schwinger equation for vertex function. Notice that one of the vertices on the right-hand side of the first two equations must be a “naked” one, in order to avoid double counting of diagrams such as

Fig. 14.12. Diagrams included only once in DS equations. Equations (14.5), (14.6), together with (14.18) and (14.19) are called (unrenormalized) “Dyson–Schwinger equations”. They evidently are highly non-linear, and can only be solved perturbatively. The vertex function Γµ (p0 , p) entering in these equations must be computed perturbatively or iteratively from Fig. 14.11 in terms of the 4-point function of the fermion field.

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Chapter 15

Regularization of Feynman Diagrams In the preceding chapter we have obtained the non-linear integral equations satisfied by the (unrenormalized) 2- and 3-point functions of QED. We proceed now to calculate these quantities in second order of perturbation theory. This will involve as a first step the regularization of divergent integrals and the use of some technical tools. For didactic purposes we shall illustrate these tools as we go along. It should also be kept in mind that the results for the 2- and 3-point functions are gauge dependent, since under a gauge transformation the photon and fermion fields transform as in (13.88). We shall use the so-called Feynmann gauge (see (13.98)), for which the photon-propagator takes the particularly simple form DFµν (k) =

−gµν . k 2 + i

In this chapter we shall illustrate two regularization procedures in terms of simple examples: (i) the traditional Pauli–Villars regularization, and (ii) the dimensional regularization. Another procedure based on “Taylor subtraction” will be discussed in Chapter 16, Sections 7 and 9. In the case of QCD the procedure (ii) is particularly suited in order to respect gauge invariance.

15.1

Pauli–Villars and dimensional regularization

Feynman diagrams containing loops involve, in general, divergent integrals. In order to obtain a control of these divergencies we must introduce counter terms involving new parameters turning these integrals finite. Various approaches have been practised, such as Pauli–Villars,1 Dimensional,2 and heat-kernel 3 -regularization. Each 1 W.

Pauli and F. Villars, Rev. Mod. Phys. 21 (1949) 434. Giambiagi and C.G. Bollini, Il Nuovo Cimento B, 12 (1972) 20; G’t Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. 3 E. Abdalla, M.C.B. Abdalla and K.D. Rothe, Non-perturbative Methods in 2-Dimensional Quantum Field Theory (World Scientific, Second edition, 2001). 2 J.J.

220

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221

one has its merits, the first two being the most popular ones. Pauli–Villars regularization By making the replacement 1 1 1 → 2− 2 k2 k k − M2

we improve the asymptotic behaviour by 2 powers in k. This procedure is referred to as Pauli–Villars regularization, and M as the Pauli–Villars mass. It plays the role of a cutoff giving control of the divergence of a Feynman integral. The simple introduction of a cutoff as upper limit in an integral may have the drawback of destroying Lorentz covariance and gauge invariance of a gauge theory. The latter has in fact been the motivation for introducing the so-called “dimensional regularization” in the context of Quantum Chromodynamics (QCD). Dimensional regularization We replace the Pauli–Villars regularization by first Wick rotating p0 to ip4 , and then continuing in the dimension from space-time D = 4 to D = 4 − ,  > 0 using the following prescription: d4 p → d4 P = µ ˜4−D P D−1 dP dΩD and the rule

Z

dΩD =

2π D/2 , Γ(D/2)

(15.1)

(15.2)

where d4 P is the euclidean integration measure associated with P = (p4 , p~). The parameter µ ˜ with dimensions of a mass (not to be confused with the IR regulating mass µ of the photon in the sequel) has been introduced in order to keep the dimension of the integration measure unchanged. This parameter will play the role of the Pauli–Villars regulator mass in the new scheme. We shall make use of the following result:
Z Γ α− D 1 1 D 4−D D 2 d P 2 . (15.3) =µ ˜ π2 α α− (P + a) Γ(α) [a] D2 This result follows by using the above rules and the definition of the Γ-function.4 The following calculations are facilitated Q by making use of the following Feynman parameter representation of a product j 1/aj : P Z 1 n Y δ(1 − j αj ) 1 = (n − 1)! dα1 ...dαn P . (15.4) a + i ( j αj aj + i)n 0 j=1 j To prove this formula, we start from the obvious representation Z ∞ P n Y i i β (a +i) = dβ1 , ...dβn e j j j , a + i 0 j=1 j

4 We

remark that dimensional regularization does not work if γ5 is involved.

(15.5)

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where  is taken to be positive in order to ensure convergence of the integral. Introducing in (15.5) the identity   Z ∞ Z ∞ X 1X  dλ  δ 1− βj , 1= dλδ(λ − βj ) = λ λ j 0 0 j and making the change of variable

βj = αj , λ Eq. (15.5) takes the form n Y

i = a + i j=1 j Recalling

Z

∞

dα1 ...dαn

0

Z

∞

0

Z

∞

X dλ n λ(i λ δ(1 − αj )e λ j

dx xn−1 e−xA =

0

and setting A = −i

X

αj aj + 

j

P

j

αj aj −

P

j

αj )

.

Γ(n) , An

X

αj ,

j

the above equation takes the form n Y

i = Γ(n) a + i j=1 j

Z

0

which proves our assertion (15.4).

15.1.1

1

P δ(1 − αj ) P P dα1 ...dαn , [−i αj aj +  αj ]n

Electron self-energy

To second order in the (bare) coupling constant e0 the electron self-energy in the Feynman-gauge is represented by the integral Z d4 k −igµν µ i −iΣ(2) (p) = (−ie0 )2 γ   γν , (15.6) (2π)4 k 2 + i (p − k ) − m0 + i corresponding to the diagram

k p

p p−k

Fig. 15.1. Second order electron self-energy.

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223

It is useful to write (15.6) in the form

where I(p) = (−ie0 )

2

−iΣ(2) (p) = gµν γ µ I(p)γ ν ,

(15.7)

Z

(15.8)

1 d4 k (p/ − k/) + m0 . (2π)4 k 2 − µ2 + i (p − k)2 − m20 + i

We have introduced a mass µ for the photon, in order to prevent possible infrared divergences. We shall further comment on this point at the end of this section. Pauli–Villars regularization The integral (15.8) is superficially linearly ultraviolet divergent. We shall first adopt the Pauli–Villars regularization by making the substitution k2

X 1 1 −→ C` 2 . 2 − µ + i k − M`2 + i `

In order for the integral in (15.8) to be finite with this substitution, we must choose C0 = 1, C1 = −1; M0 = µ, M1 = Λ; C` = 0 for ` > 1 , where Λ is a regulating mass which will eventually be taken to infinity. The objective is to have control of the divergence. Let α0 = e20 /4π be the “bare” fine structure constant. Equation (15.8) then takes the form Z 1 X α0 C` p/ − k/ + m0 I(p) = − 3 d4 k . (15.9) 4π k 2 − M`2 + i (p − k)2 − m20 + i `=0

For a finite Pauli–Villars mass Λ the regularized integrand in (15.9) behaves asymptotically like k −5 . This power fall-off is sufficient to render the integral finite. For Λ → ∞ we obtain of course a divergent result. The purpose of introducing the Pauli–Villars regulator is to be able to control this divergence. Making use of (15.4) we now rewrite (15.9) in the form Z Z 1 Z 1 α0 d4 k dα dβδ(1 − α − β) (15.10) I(p) = − 3 4π 0 0   X (p − k + m0 ) × C` ( )2 . i h ` α(k 2 − M`2 ) + β (p − k)2 − m20 + i Using α + β = 1 we now rewrite the denominator in the integrand as follows: h i α(k 2 − M`2 ) + β (p − k)2 − m20 = k 2 + β(p2 − m20 − 2p · k) − (1 − β)M`2 .

Making the change

k = k 0 + βp

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in the integration variable, I(p) takes the form  0 (1 − β) p − k +m0 dβ C` 02 . d k {k − β 2 p2 + β(p2 − m20 ) − (1 − β)M`2 + i}2 0 ` (15.11) Since the denominator in I(p) only depends quadratically on k 0 , terms in the numerator which are linear in k 0 will not contribute to the integral, and we are left with (k 0 → k) α0 I(p) = − 3 4π

Z

4 0

Z

1

X

α0 I(p) = − 3 4π

Z

4

d k

Z

1

dβ

0

 N (β; p) = (1 − β) p +m0

X

C`

`

N (β; p) D` (k 2 ; β; p)

D` (k 2 ; β; p) = (k 2 − A` + i)2 ,

(15.12)

where A` = β 2 p2 − β(p2 − m20 ) + (1 − β)M`2 . Since the integral in k is absolutely convergent, we may exchange the order of the integrations, and perform first the integration over the momenta. In order to avoid the singularities of the integrand, it is convenient to “rotate” the k 0 integration contour to the positive imaginary axis (Wick rotation). This corresponds to the change of integration variables k 0 → ik 4 with the corresponding change in integration measure (Kα = (k 4 , ~k)): Z i dΩ4 = 2π 2 . (15.13) d4 k → id4 K = K 2 dK 2 dΩ4 , dΩ4 = sin2 ΘdΘdΩ , 2 Note that we cannot do a continuation to the negative imaginary k 0 axis, since the integrand possesses singularities in the lower half of the complex k 0 -plane, due to the i prescription. We thus obtain with (15.12) (K 2 = x) −iα0 I(p) = 4π

Z

0

1

dβ

Z

0

∞

dx

X `

C`

xN (β; p) . D`2 (−x; β; p)

We may evaluate the integrals by making use of the formula  X   X + A`  x = lim C ln − 1 ` X→∞ X→∞ (x + A` )2 A` 0 ` `      X + A0 X + A1 A0 = lim ln − ln = −ln X→∞ A0 A1 A1 P where we have used C` = 0. Making use of this result we have lim

X

iα0 I(p) = 4π

Z

0

C`

1

Z

X

dx

h i  β 2 p2 − β(p2 − m2 ) + (1 − β)µ2 − i   0 dβ (1 − β) p +m0 ln β 2 p2 − β(p2 − m20 ) + (1 − β)Λ2 − i

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225

or taking Λ → ∞ we obtain, after rescaling,

Z i  iα0 1 h dβ (1 − β) p +m0 ln f (β; p2 , m20 , µ2 ) 4π 0  2 α0 1 Λ − i (m0 + p)ln 4π 2 m20

I(p) →

where f (β; p

2

, m20 , µ2 )

= ln



(β 2 p2 − β(p2 − m20 ) + (1 − β)µ2 − i (1 − β)m20



(15.14)

.

(15.15)

We have chosen the branch of the logarithm such that iI(p) is real for p2 below the threshold for electron–photon production. Indeed, for p2 ≤ (m0 + µ)2 we have in the numerator of the argument of the logarithm, βm20 + (1 − β)µ2 − β(1 − β)p2 ≥ βm20 + (1 − β)µ2 − β(1 − β)(m0 + µ)2 . (15.16) The rhs has an extremum for β = µ/(m0 + µ), or 1 − β = m0 /(m0 + µ). At this value, the rhs takes the value zero. It also corresponds to a minimum. Hence f (β; p2 , m20 ) ≥ 0 is below the electron–photon production threshold. Above this threshold the electron self-energy is seen to become complex, corresponding to the fact that for p2 > (m0 +µ)2 electron–photon pairs can be produced. We thus finally have for the second order regularized electron self-energy,5  2  Λ α0 (2) Σreg (p; e0 , m0 ; M ) = (4m0 − p)ln (15.17) 4π m20 Z 1  α0 − dβ[2m0 − (1 − β)p]ln f (β; p2 , m20 , µ2 ) . 2π 0

Notice that the superficial linear UV divergence turned out to be a logarithmic one. We shall see in Chapters 16 and 17 the role played by the (for Λ → ∞) divergent terms in the renormalization program. This completes the calculation of the electron self-energy to second order in the coupling constant using PV regularization. Dimensional regularization We wish now to confront the PV calculation with the dimensional regularization. To this end we return now to (15.12), perform again a Wick rotation, continue in space-time to D = 4 −  as described at the beginning of this chapter, and set all 5 We

recall the following properties of γ-matrices: Sp(γµ γν ) = 4gµν Sp (γµ γν γλ γρ ) = 4gµρ gνλ − 4gµλ gνρ + 4gµν gλρ









γλ p γ λ = −γλ γ λ p + γλ {p , γ λ } = −2 p .

| {z } −4

| {z } 2pλ

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Pauli–Villars coefficients to zero, except for c0 = 1. Integrating over K using (15.3), and making use of the expansion   D 1 2 Γ 2− ∼ −γ = −γ (15.18) 2  2− D 2 with we then have

 = 4 − D,

γ = Euler constant

 Z 1 (1 − β)p/ + m0 α0 D 4−D D 2 µ π dβ ˜ Γ 2 − 3 α− D 4π 2 0 A0 2 Z 1  α0  dβ[(1 − β)p/ + m0 ]e− 2 ln A0 = −i e− 2 ln π eln µ˜ Γ 4π 2 0   Z  2     α0 1 − γ (1 + ln µ ˜) 1 − ln π = −i 2 dβ[(1 − β)p/ + m0 ] 1 − ln A0 4π 0 2  2   Z 1 α0 2 A0 =i dβ[(1 − β)p/ + m0 ] ln 2 − + γ + ln π . 4π 0 µ ˜ 

I(p) = −i

This can be written in the form Z α0 1 2 I(p) = i dβ((1 − β)p/ + m0 )f (β; p2 , m20 , µ2 ) − + finite constants 4π 0  in agreement with (15.14). We see that  plays the role of the Pauli–Villars mass.

15.1.2

Photon vacuum polarization

We now repeat the analogous calculation for the photon 2-point function. This will show that the dimensional regularization simplifies considerably the calculation. For second order in the coupling constant the photon polarization tensor is represented by the diagram

p+k

k

k p

Fig. 15.2. Second order photon vacuum polarization. with the corresponding 1-loop integral (note the extra minus sign for the fermion loop) ! Z d4 p i i µν 2 µ ν  −iΠ (k) = −(−ie0 ) tr γ γ . (15.19) (2π)4 p − m0 + i p + k − m0 + i

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227

This integral is superficially quadratically divergent. On the formal level we observe that ! Z d4 p i i µ 2 ν ν −ik Πµν (k) = −(−ie0 ) tr  =0, γ −γ   (2π)4 p − m0 + i p + k − m0 + i

provided that the shift p + k → p in the integration variable is performed. Such a shift is however only allowed if the integral is convergent. It nevertheless reflects the underlying gauge invariance of the theory. Since we do not want to break this symmetry, our regularization procedure should respect this symmetry. Pauli–Villars regularization We first consider again the PV regularization by replacing (15.19) by Z d4 p X 2 −iΠµν (k)reg = −e0 C` Iµν (p, k; M` ) , (2π)4

(15.20)

`

with6

 tr[γ µ (p + M` )γ ν (p/ + k/ + M` )] , Iµν (p; k, M` ) = 2 (p − M`2 + i)[(p + k)2 − M`2 + i]

(15.21)

where the constants C` and masses M` (l = 0, 1, 2) are chosen such that the integrand tends to zero faster than 1/p4 . This is realized by requiring7 2 X

C` = 0 ,

`=0

2 X

C` M`2 = 0 , C0 = 1 , M0 = m0 ; C` = 0, ` > 2 .

(15.22)

`=0

These equations have the solution C1 = 1 , C2 = −2; M12 = m20 + 2Λ2 , M22 = m20 + Λ2 .

(15.23)

Now,    tr{γµ (p + M` )γν (k + p + M` )} =: Pµν (p, k; M` )   = 4gµν M`2 − p · (k + p) + 4[pµ (k + p)ν + (k + p)µ pν ] .

6 Note that in contrast to the previous case of the fermion self-energy, we have made the replacement m0 → M` everywhere. The reason is that we want to guarantee gauge invariance, i.e. kµ Πµν (k) = 0. 7 We establish these conditions by making a Taylor expansion of the denominators in powers of  

p2 /M`2 and p.k/M`2 and requiring that for p → ∞,

P

`

C` Iµν (p; k; M` ) → 0

1 p5

:

[p2 − M`2 ][(p + k)2 − M`2 ] = p4 + 2p2 (p · k) + p2 k2

− 2p2 M`2 − 2M`2 (p · k) − M`2 k2 + M`4

n

' p4 1 + [p2

−

1 1 ' 4 p + k)2 − M`2 ]

M`2 ][(p

n

1−



1 k2 + 2(p · k) − 2M`2 p2



o

1 2 (k + 2(p · k) − 2M`2 ) p2

+ ···

+0



o

1 p6



.

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Making again use of the Feynman parametrization (15.4), expression (15.20) then takes the following form: Z d4 p X 2 C` Pµν (p; k, M` ) (15.24) −iΠµν (k)reg = −e0 (2π)4 ` Z 1 Z 1 δ(1 − α − β) dβ dα × . 2 2 {α(p − M` ) + β[(k + p)2 − M`2 ] + i}2 0 0 Now, Z

1

dα

0

Z

1

δ(1 − α − β) {α(p2 − M`2 ) + β[(k + p)2 − M`2 ] + i}2

dβ

0

=

Z

1

dβ

0

{(p +

βk)2

1 . + β(1 − β)k 2 − M`2 + i}2

Setting p0 = p + βk, Eq. (15.24) takes the form h i M`2 − (p0 − βk) · (p0 + (1 − β)k) X Z d4 p 0 Z 1 −iΠµν (k)reg = −e20 C` dβ gµν (2π)4 0 {p02 + β(1 − β)k 2 − M`2 + i}2 ` h i (p0 − βk)µ (p0 + (1 − β)k)ν + (µ ↔ ν) X Z d4 p0 Z 1 − 4e20 C` dβ . (2π)4 0 {p02 + β(1 − β)k 2 − M`2 + i}2 `

After symmetric integration in p0 this reduces to  X Z d4 p0 Z 1 n 2 2 02 2 dβ M − p + β(1 − β)k −iΠµν (k)reg = −4e0 C` gµν ` (2π)4 0 ` o 1 . +2p0µ p0ν − 2β(1 − β)kµ kν 02 {p + β(1 − β)k 2 − M`2 + i}2 Now, it follows from Lorentz invariance that Z Z 1 d4 p pµ pν f (p2 ) = gµν d4 p p2 f (p2 ) . 4 Hence Πµν (k)reg = (−gµν k 2 + kµ kν )π(k 2 ) + gµν π ˆ (k 2 ) , where (α0 = e20 /4π) 2

α0 =i 3 π

Z

d p

α0 = −i 3 π

Z

4

π(k )reg

4

Z

1

dβ

0

X

C`

X

C`

`

{p2

2β(1 − β) + β(1 − β)k 2 − M`2 + i}2

and 2

π ˆ (k )reg

d p

Z

0

1

dβ

`

M`2 − 21 p2 − β(1 − β)k 2 . {p2 + β(1 − β)k 2 − M`2 + i}2

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15.1 Pauli–Villars and dimensional regularization

229

Notice that we have separated the vacuum polarization tensor into a longitudinal (gauge dependent) and a transversal (gauge independent) part. We perform again a Wick rotation p0 → iP4 in the integration variable (see (15.13)). This leaves us with Z Z 1 X β(1 − β) α0 2 2 2 P dP dβ C` 2 (15.25) π(k )reg = −2 π {P − β(1 − β)k 2 + M`2 − i}2 0 `

and 2

π ˆ (k )reg

α0 = π

Z

2

P dP

2

Z

1

dβ

0

X `

C`

M`2 + 12 P 2 − β(1 − β)k 2 . {P 2 − β(1 − β)k 2 + M`2 − i}2

(15.26)

To evaluate the integrals we first introduce a cutoff X in the P -integration. Because of (15.22) the result of the P -integrations will be finite in the limit where X → ∞. Using   Z X X +a x2 = a − 2aln +X dx (x + a)2 a 0 the longitudinal piece (15.26) reduces to Z 1  o α0 X n 2 π ˆ (k )reg = − C` 2 dβ β(1 − β)k 2 − M`2 . 2π 0 `

Because of (15.22) we see that π ˆ (k 2 )reg vanishes, so that the result for Πµν (k)reg is, in fact, transversal : µν 2 µ ν 2 Πµν reg (k) = (−g k + k k )π(k )reg .

(15.27)

This reflects the gauge-invariance of the polarization tensor already referred to previously. To compute the transversal piece π(k 2 )reg we make use of   Z X X +a x = ln −1 , dx (x + a)2 a 0 and find 2

π(k )reg

X 2α0 =− lim C` π X→∞ `

Z

0

1

(

dβ 2β(1 − β) ln



X + M`2 − β(1 − β)k 2 M`2 − β(1 − β)k 2



or explicitly (Z   1 2α0 X + m20 − β(1 − β)k 2 π(k )reg − lim dβ 2β(1 − β)ln π X→∞ m20 − β(1 − β)k 2 0   Z 1 X + m20 + 2Λ2 − β(1 − β)k 2 + dβ 2β(1 − β)ln m20 + 2Λ2 − β(1 − β)k 2 0  ) Z 1 X + m20 + Λ2 − β(1 − β)k 2 −2 dβ 2β(1 − β)ln . m20 + Λ2 − β(1 − β)k 2 0 2

)

−1

,

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Using again (15.22) one finds that the limit X → ∞ is finite. Letting subsequently Λ tend to infinity we are left with V π(k 2 )P reg

α0 = − ln 3π



Λ2 2m20



where we have used

2α0 + π Z

0

Z

1

0

dββ(1 − β)ln

1

dβ β(1 − β) =



 m20 − β(1 − β)k 2 − i , m20 (15.28)

1 . 6

(15.29)

This concludes the PV calculation for the photon 2-point function. Dimensional regularization Keeping only the ` = 0 term in the expansion (15.25) and using (15.3) we obtain 2

π(k )reg

α0 4−D D/2 Γ(2 − D 2) = 2 3µ ˜ π π Γ(2)

Z

dβ

β(1 − β) . [−β(1 − β)k 2 + m20 − i0)]2−D/2

Making use of the expansion (15.18)   D 1 2 Γ 2− ∼ −γ = −γ D 2  2− 2 with  = 4 − D,

γ = Euler constant

we obtain for the leading terms in , which plays here the role of the cutoff Λ, π(k 2 )Dim reg

Z 1  2 2  α0  ln˜µ2 −  2 2 2 −γ = 2 (e )π dβ β(1 − β)e− 2 ln (−β(1−β)k +m0 −i0) π  0    2 α0    2 '2 1 + ln µ ˜ 1 − ln π −γ π 2 2  Z 1 h
i  × dβ β(1 − β) 1 − ln −β(1 − β)k 2 + m20 − i0 . 2 0

Using (15.29) we obtain to order O(α0 ),  2 µ ˜2 − γ − ln π + ln 2  m0   2 Z 1 α0 m0 − β(1 − β)k 2 − i0 −2 , dββ(1 − β)ln π 0 m20

π(k 2 )Dim reg '

α0 3π



to be compared with the corresponding PV result (15.28).

(15.30)

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15.1.3

231

The vertex function

To order e20 the unrenormalized vertex function is given by the sum of diagrams

p0

p0 −ie0 Γµ (p0 , p) = q, µ

+ q, µ p

k

(15.31)

p

Fig. 15.3. Second order vertex function. Writing

Γµ (p0 , p) = γ µ + Λµ (p0 , p) ,

we have, introducing an IR cutoff µ, Z −ig λρ i i d4 k µ 0 2 γλ   γµ   γρ . Λ (p , p) = (−ie0 ) 4 2 (2π) k − µ2 + i p0 − k − m0 + i p − k − m0 + i Pauli–Villars regularization

This integral is logarithmically ultraviolet divergent. We regularize it by making the following replacement in the photon-propagator: 1

X −g µν −g µν −→ C ` k 2 − µ2 + i k 2 − M`2 + i `=0

with C0 = 1, C1 = −1, M0 = µ, M1 = Λ .

We have introduced here a mass for the photon, although the integral appears to converge for µ = 0 in the infrared region k 2 ' 0. It turns out, however, that this is not the case if the electrons are on their mass shell. The calculation is rather cumbersome. It nevertheless simplifies if one limits oneself to the respective mass shell of the electrons. On shell we have Z d4 k X C` Λµreg (p0 , p)|p02 =p2 =m20 = −ie20 (2π)4 k 2 − M`2 + i ` 0    γλ (p − k + m)γ µ (p − k + m)γ λ × 2 . (k − 2p0 · k + i)(k 2 − 2p · k + i) Making use of the Feynman parameter representation (12.25) one has Z Z 1 Z 1 Z 1 d4 k X µ 0 2 Λreg (p , p) = −ie0 2! C` dα1 dα2 dα3 (2π)4 0 0 0

(15.32)

`

×

δ(1 − α1 − α2 − α3 )N µ (p, p0 , k) , {α1 (k 2 − 2p0 · k) + α2 (k 2 − 2p · k) + α3 (k 2 − M`2 ) + i}3

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  0  N µ (p, p0 , k) = γλ (p − k + m0 )γ µ (p − k + m0 )γ λ .

We now make the change of variable (12.28):

k = k 0 + α1 p + α2 p0 .

(15.33)

This turns the denominator D` in the integrand of (15.32) into a form quadratic in k 02 : D` = [q 2 α1 α2 − m2 (α1 + α2 )2 − α3 M`2 + k 02 ]3 (15.34)

where q = p0 − p. Performing this change of variable also in the numerator, and separating it in its γ-even and odd parts, µ µ N µ = Neven + Nodd , µ , we have for Neven   µ Neven = m0 γλ p/ 0 − α1p/ − α2p/ 0 − k/0 )γ µ + γ µ (p/ − α1p/ − α2p/ 0 − k/0 ) γ λ , (15.35)

µ and for Nodd n o µ Nodd = γλ [(1 − α2 )p/0 − α1p/ − k/0 ]γ µ [(1 − α1 )p/ − α2p/0 − k/0 ] + m20 γ µ γ λ . (15.36)

Using the contraction identity

γλ γ µ γ ν γ λ = 4g µν µ we obtain for Neven , µ Neven = 4m0 [(p + p0 )µ − 2α1 pµ − 2α2 p0µ ] ,

0 where we have dropped the term linear in k , which will eventually drop out after symmetric integration in k 0 . Since the denominator function is symmetric under the exchange of α1 and α2 , under the α-integral we can make the replacement µ Neven → 4m0 (1 − α1 − α2 )(p + p0 )µ → 4m0 α3 (p + p0 )µ .

(15.37)

µ The calculation of Nodd is a bit more involved. Making use of the contraction identities γλ γ σ γ µ γ ρ γ λ = −2γ ρ γ µ γ σ , γλ γ µ γ λ = −2γ µ

we obtain for (15.36), µ Nodd = −2[(1 − α1 )p/ − α2p/0 − k/0 ]γ µ [(1 − α2 )p/0 − α1p/ − k/0 ] − 2m20 γ µ .

Note that the factors now appear inverted! To implement later the renormalization condition (16.77), we shall need the ¯ (p0 )N µ U (p). This means matrix elements of N µ taken between the Dirac spinors: U

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233

that we have to move p/ and p/0 to the right and left respectively, until we can make use of their equations of motion: (p/ − m)U (p) = 0,

¯ (p0 )(p/0 − m) = 0 . U

This requires some care and patience. Keeping only terms quadratic in k 0 (odd terms in k 0 will cancel in the k 0 -integral) one obtains after some algebra (the Dirac spinors are understood),  n m2 q2 µ 2 (1 − α1 )(1 − α2 ) − 0 (15.38) Nodd = 4 m0 − 2 2 m2 + 0 [(1 − α1 )(1 − α2 ) − α1 (1 − α1 ) − α2 (1 − α2 ) − α1 α2 ] γ µ 2 o − m0 [α3 (1 − α1 )pµ + α3 (1 − α2 )p0µ ] − 2k/0 γ µk/0 , where we have used

q 2 = (p0 − p)2 = 2m20 − 2(p · p0 )2

as well as the δ-function δ(1 − α1 − α2 − α3 ). µ It is remarkable that the γ-“odd” term Nodd contains a γ-“even” term proporµ in tional to pµ and p0µ when sandwiched between Dirac spinors. Joining Neven µ (15.37) and Nodd in (15.38), we find    k2 m2 q2 (1 − α1 )(1 − α2 ) + 0 [(α12 + α22 ) − 2(α1 + α2 )] − 2 γµ Nµ = 4 m20 − 2 2 4 (p + p0 )µ , (15.39) + 4m20 α3 (α1 + α2 ) 2m0 where we have used (under the k 0 -integral) k/0 γ µk/0 = kσ0 k 0ρ γ σ γ µ γρ → k 02 gσρ γ σ γ µ γρ = −2



k 02 4



γµ ,

as well as the symmetry of the α-integral under the substitutions α1 → α2 , α2 → α1 . Between spinors Λµ (p’,p) is thus of the form   0 µ ¯ (p0 )Λµ (p0 , p)U (p) = U ¯ (p0 ) γ µ F (q 2 ) + (p + p ) G(q 2 ) U (p) U (15.40) 2m0 P with the form factor G(q 2 ) given by the finite integral (we can set ` C` → 1 and M` → µ in (15.32)) ! Z 1Y 3 3 X m20 α3 (α1 + α2 ) α dαi δ 1 − αj . G(q 2 )reg = − π 0 i=1 [m20 (α1 + α2 )2 + α3 µ2 − α1 α2 q 2 ] 1 (15.41) The form factor F (q 2 ) involves a finite piece, plus a divergent piece arising from the k 02 term in the k 0 -integration: Z 1 −α Y F (q)reg = dαi δ(α1 + α2 + α3 − 1)[I(q 2 ; α)f in + Idiv (q 2 ; α)] (15.42) π 0 0

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with the finite part of the k 0 -integral m2

2

2

I(q ; α)f in

(1 − α1 )(1 − α2 )(m20 − q2 ) + 20 [(α12 + α22 ) − 2(α1 + α2 )] = [m20 (α1 + α2 )2 + α3 µ2 − q 2 α1 α2 ]

(15.43)

and I(q 2 ; α)div the divergent part of the integral arising from the k 0 2 term in the numerator. Performing a Wick rotation k 0 → iK4 and integrating, we have
X Z dxx −x 2 4 (15.44) I(q ; α)div = 2 C` [A(q 2 ; α) + α3 M`2 + x]3 `

where x = K 2 and A(q 2 ; α) stands for A(q 2 ; α) = m20 (α1 + α2 )2 − q 2 α1 α2 .

(15.45)

Explicitly we have, Z

x 

X→∞

−1 −1 − 2 ; α) + α µ2 + x]3 2 ; α) + α Λ2 + x]3 2 [A(q [A(q 3 3 0 1 1 2 2 2 2 + ln [A(q ; α) + α3 µ ] − ln [A(q ; α) + α3 Λ ]. 2 2

I(q 2 ; α)div =

dxx



Putting things together we have for Λ → ∞, F (q 2 )reg ∼ −

α π

Z

1

0

Y

dαi δ(1 −

X

αi )

n (1 − α )(1 − α )(m2 − q2 ) + m20 [(α2 + α2 ) − 2(α + α )] 1 2 1 2 0 1 2 2 2 × 2 2 2 2 [m0 (α1 + α2 ) + α3 µ − q α1 α2 ]  2  1 m0 (α1 + α2 ) + α3 µ2 − q 2 α1 α2 1 Λ2 o + ln − ln . (15.46) 2 α3 m20 2 m20 Dimensional regularization We return to the computation of the singular term I(q 2 )div in (15.42) by dimensional regularization. We wish to use formula (15.3) to compute (15.44). Observing that Z

0

∞

dxx =

Z

∞

0

we can write I(q 2 ; α)div = 2

Z

dK 2 K 2 →

1 π2

Z

d4 K

 2 −K 4 dD K 2 2 π (K + a(q 2 ; α))3

where a(q 2 ; α) = A(q 2 ; α) + α3 µ2 .

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235

We have, using (15.3) and (15.18), Z D K2 1 d K 2 2 2 π (K + a(q 2 ; α))3 Z D Z D 1 1 a(q 2 ; α) 1 d K d K + =− 2 2 2 2 2 2 2 π (K + a(q ; α)) 2 π (K + a(q 2 ; α))3 1  −  Γ( 2 ) 1 1 =− µ ˜ π 2 ,  + 2 Γ(2) a(q 2 ; α) 2 4

I(q 2 ; α))div = −

or    2  1 1   I(q ; α)div = − (1 + ln µ ˜) 1 − ln π −γ 1 − ln a(q 2 ; α) + 2 2  2 4 1 a(q 2 ; α) n 1 1 γ 1o = ln + − + ln π + + , 2 µ ˜2  2 2 4 2

2

Λ where the terms in brackets now play the role of ln m 2. 0 The parallel presentation of the Pauli–Villars and dimensional regularization should serve to appreciate the advantage or disadvantage of these two regularization procedures.

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Chapter 16

Renormalization The individual terms contributing to the perturbative expansion of a general Green function are represented by Feynman diagrams. As we have seen in Chapter 15, the integrals representing diagrams involving at least one loop are generally divergent, and are thus a priori meaningless. We may nevertheless isolate the finite part of a diagram by subtracting a suitable polynomial involving infinite constants. Since this procedure does not define a priori a unique finite part, there remains an ambiguity in the choice of this polynomial. If we do this for each loop-diagram separately, we appear to generate, in general, an infinite number of undetermined constants as we consider diagrams to all orders of perturbation theory. If this were so, the theory would be meaningless. There exists, however, a class of theories, for which the number of arbitrary constants is not only finite, but can also be absorbed into the bare parameters describing the classical theory, such as couplings constants, masses and the wave function normalization constant Z appearing in the asymptotic condition (10.16). Such theories are called renormalizable. There exist simple criteria to decide whether a theory is renormalizable or not, provided the interaction of the fields does not induce new types of vertices. The process of renormalization — that is, given a unique meaning to the theory, essentially involves two steps: (i) In the first step one introduces a suitable cutoff in the Feynman integrals, in order to be able to associate with them a finite expression. The result will of course be cutoff dependent. This step is referred to as regularization (see previous chapter). (ii) In the second step one eliminates this cutoff dependence by trading the “bare” parameters for physical parameters, which are fixed by comparison with experiment, and cannot be intrinsically calculated from the theory. This “normalization” procedure is referred to as renormalization. In the following sections we shall consider, in particular, the case of electrodynamics. The Feynman rules for this case were given in Section 8 of Chapter 11. 236

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16.1

237

The principles of renormalization

In this section we shall consider the general principles of renormalization leading to unambiguous and finite results for an arbitrary Green function in Quantum Field Theory. We shall illustrate these principles for QED, described by the Lagrange density 1 (16.1) L = − Fµν F µν + ψ(i∂/ − e0 A/ − m0 )ψ , 4 where e0 and m0 are “bare” parameters having the dimensions of charge and mass respectively, and the fields stand for unrenormalized fields. The renormalization program consists of essentially three steps: (i) Regularization As we have seen, Feynman diagrams involving at least one loop are generally divergent, and hence are a priori meaningless and required regularization by effectively introducing a cutoff. As we have exemplified in Chapter 15, there exist many ways of doing this: Pauli–Villars, analytic, dimensional, Taylor subtraction, subtracted dispersion relations and heat-kernel regularization, just to mention the most important ones. This procedure generally leads to the introduction of a parameter carrying the dimensions of a mass. We shall denote this parameter generically by Λ. It is the necessity of introducing some dimensional parameter to regularize the theory, which results in the breaking of scale invariance on quantum level, even if the classical lagrangian is scale invariant, as is the case for QED with massless fermions. Thus after regularization we are left with Green functions which depend on this “cutoff parameter” as well as on the bare constants appearing in the defining lagrangian: G(n,n,`) (x1 ...; y1 ...; z1 ...; m0 , e0 ; Λ) reg

(16.2) µ1

µ`

= hΩ|T ψα1 (x1 )...ψαn (xn )ψ β1 (y1 )...ψ βn (yn )A (z1 )...A (z` )|Ωireg . (ii) Renormalization The result obtained for the regularized Green functions will depend on the regularization scheme one employs. Physical results should, of course, not depend on the choice of such a scheme. In a so-called renormalizable theory, we can nevertheless obtain an unambiguous and unique result by imposing suitable normalization conditions. The procedure for achieving this is called renormalization. In a renormalizable theory the number of normalization conditions to be imposed is at most equal to the number of fields (fermions, gauge field) and parameters (electron charge and mass) in the theory. With the aid of these normalization conditions we can eliminate the cutoff dependence, trading the bare (meaningless) parameters entering in the classical lagrangian for physically meaningful parameters, which can be determined from experiment. (iii) Renormalization group equations It is up to us to choose a suitable set of parametrizations. We thus obtain different results for different parametrizations. All of these results should, however,

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describe the same physics. The equations relating these different results are called renormalization group equations. Steps (i) and (ii) can be summarized in an algorithm to be described next. Algorithm Starting point of our algorithm is the classical lagrangian (16.1). The Feynman diagrams computed from this lagrangian are in general divergent, and thus need to be first defined via some suitable regularization procedure, which will introduce a cutoff Λ into the theory (step (i)). Renormalizability of the theory now means to introduce new parameters via the substitutions1 −1 m0 = Zm m , e0 = Ze−1/2 e,

(16.3)

1/2 1/2 ψ = Zψ ψ˜ , Aµ = ZA A˜µ ,

(16.4)

and to choose the renormalization constants to be suitable functions of the cutoff ˜ of such as to turn the (still) cutoff dependent expressions for all Green functions G ˜ the theory into finite expressions in the limit Λ → ∞: Greg → G → Gren . Since Zm , Ze , Zψ and ZA are dimensionless, they can be chosen to depend on the renormalized parameters of the theory in the form   Λ , Z = Z e, m

(16.5)

where e and m are the electric charge and mass as taken from experiment. Renor˜ defined by malizability then means that the regularized Green functions G     Λ Λ −`/2 −n (n,n,` ˜ G (...; e, m; Λ) = ZA e, Zψ e, (16.6) m m       Λ Λ −1 × G(n,n,`) ...; Ze−1/2 e, e, Zm m; Λ e, reg m m has a finite limit for suitably chosen renormalization constants, as we let Λ go to “infinity”, where the limit Λ → ∞ stands symbolically for “removing the cutoff”.2 If the limit exists, then (16.6) defines in this limit the so-called renormalized Green functions.3 It is very important to realize that this supposes that finiteness is achieved in this way for all Green functions of the theory.4 This requirement does not fix the constants (16.5) uniquely since different cutoff procedures will lead to different cutoff dependent results. Uniqueness of the Green 1 Our notation for the renormalization constants does not follow convention in order to associate them directly with the fields and parameters involved. 2 We denote generally by a “tilde” the pre-renormalized Green functions. 3 Notice that formally the rhs of (16.6) is just the rhs of (16.2) with ψ and Aµ now replaced by the “tilde” fields. We generally denote pre-renormalized Green functions by a tilde. 4 For later notational convenience we shall frequently omit the dependence of G reg (· · · ; e, m; Λ) ˜ · · ; e, m; Λ) on the parameters., and write Greg (· · ·) and G(· ˜ · ·), respectively. and G(·

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239

functions is achieved by imposing suitable conditions. To be specific, we take QED as an example.5 (a) If m is to have the meaning of a physical mass, we must have for the inverse of the fermion 2-point function: (for notation see table of Section 4 in Chapter 11) S˜F−1 (p; e, m; Λ)|p/=m = 0.

(16.7)

(b) We require the residue of the pole of the fermion 2-point function to be one, in accordance with the asymptotic condition (10.16) for the (pre) renormalized field defined in (16.6): (p/ − m)S˜F (p; e, m; Λ)|p/=m = 1. (16.8) (c) For the photon 2-point function it turns out that we need not impose a condition analogous to (a), requiring the photon 2-point function to have the pole at zeromass. This happens to be guaranteed by the gauge invariance of the theory, as we shall see. (d) We require the residue at the (zero-mass) pole of the photon 2-point function to be “one”, again in accordance with the asymptotic condition: ˜ µν (k; e, m; Λ)tr |k2 ≈0 ≈ −g µν + k2 D F

kµ kν . k2

(16.9)

(e) The Fourier transform of the“amputated” and regularized 3-point function, µ ¯ ieΓµreg (x, y, z)) =< Ω|T ψ(x)ψ(y)A (z)|Ω >amp,reg

stripped off an overall charge obeys the following relation6 1

˜ µ (p0 , p; e, m, Λ) = Z1 Γµ (p0 , p; Ze− 2 e, Z −1 m, Λ) , Γ m reg with

1

− 12

Z1 = Zψ ZA2 Ze

(16.10) (16.11)

and we require that at zero momentum transfer we have for the on-shell 3-point function ˜ µ (p, p) = γ µ . Γ (16.12) This amounts to requiring e to be the electron charge measured in e− e− scattering in the forward direction, a process described exactly by the diagrams of Fig. 11.11. 5 In the literature, the renormalization constants are denoted by Z , Z , Z , the relation to the 1 2 3 ones introduced above being −1/2

ZA = Z3 , Zψ = Z2 , Ze

1/2

ZA Zψ = Z1 .

6 Notice that unlike (16.6), the renormalization constants appear with positive powers, since the (fully dressed) external legs have been amputated.

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For QED the above normalization conditions fix the renormalization constants as a function of the physical parameters e, m and the cutoff. In the limit Λ → ∞, ˜ (n,n,`) (...; e, m; Λ) → G(n,n.`) G (...; e, m) ren

(16.13)

now defines unique functions called the “renormalized Green functions”, in all orders of perturbation theory.

16.2

Renormalizability of QED

The substitutions (16.3) and (16.4) can be viewed as the introduction of counterterms properly chosen to accommodate the renormalization conditions. The fields and parameters (charge and mass) appearing in the lagrangian (16.1) refer to unrenormalized quantities. A perturbative scheme referring from the start to renormalized Greens functions in terms of renormalized parameters is obtained by making the substitutions (16.3) and (16.4) in the bare Lagrangian (16.1) and rewriting it in the form L = L0 + L0I , where and

1 L0 = − Fµν F µν + ψ(i∂/ − m0 )ψ , 4 ¯ / ψ − 1 δZA Fµν F µν + δZψ ψ(i∂/ − m0 )ψ L0I = −Z1 eψA 4

(16.14)

with Aµ , ψ, ψ¯ now renormalizd fields, and δZA , δZψ , Z1 given by δZA = ZA − 1 ,

δZψ = Zψ − 1

1

− 21

Z1 = Zψ ZA2 Ze

.

(16.15)

Note that this amounts to working with Feynman rules given in terms of the Feynman propagators 1 , SF (p) = p/ − m0 + i DFµν (k) =

−P µν (k) + DFµν (k)long k 2 + i

with the projector (14.9) and a modified interaction L0I adjusted to satisfy the normalization conditions (a) through (e). In a perturbative approach we must think of the renormalization constants as given by a power series in the fine structure constant:     ∞ X Λ Λ n (n) −1= α δzi . δZi := Zi e, m m n=1

(16.16)

The expansion coefficients of these renormalization constants will be determined to every given order in perturbation theory by the normalization conditions (16.7)– (16.12).

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241

The quadratic counterterms The terms (F )

δL0 (G)

δL0

= δZψ ψ(i∂/ − m0 )ψ

1 1 = − δZA Fµν F µν = ˆ δZA Aµ (g µν 4 2

(16.17)

− ∂ µ ∂ ν )Aν

in L0I play a special role since they are quadratic in the fields. Indeed, their role is just to provide the necessary “counterterms” allowing one to implement the normalization conditions (16.8) and (16.9) for the renormalized fields ψ˜α , A˜µ . Let us examine how this works. Let us suppose for the moment that the trilinear interaction term in (16.14) is absent. We then obtain for the fermion 2-point function

−→

+

+ ···

+

with

β = iδZψ (p/ − m0 )αβ

α

(16.18)

Hence i i i + iδZψ (p/ − m0 ) + ... p/ − m0 p/ − m0 p/ − m0

iSF (p) −→ =

iZψ−1 i 1 = = iZψ−1 SF (p) . p/ − m0 1 + δZψ p/ − m0

Similarly we have,

−→

+

+

+ ···

with

λ

ρ = iδZA (−g ρλ k 2 + k λ k ρ ) = −iδZA k 2 P λρ (k)

(16.19)(16.19)

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Hence −1/2

DFµν (k) −→ ZA

DFµν (k)tr + DFµν (k)long .

(16.20)

This means that we may equivalently work with the interaction Lagrangian 1 L = − Fµν F µν + ψ(i∂/ − m0 )ψ + Z1 eψγ µ ψAν 4 and new Feynman rules given in terms of the Feynman propagators SF (p) −→

Zψ−1

(16.21)

p/ − m0 + i

DFµν (k)tr −→ −

−1 ZA P µν (k) k 2 + i

(16.22)

kµ kν . k2

(16.23)

with P µν the projector P µν (k) = g µν −

We are now ready to implement the rest of the (formal) renormalization program by including the trilinear interaction term.

16.2.1

Fermion 2-point function

The 1PI 2-point function Σ(p) (fermion self-energy) is given by an infinite sum of diagrams such as represented for example by the diagrams −1 ZA

Zψ−1 Fig. 16.1. Fermion self-energy diagrams. 1

With the new propagators (16.21), (16.22), and vertex Z1 e = Zψ ZA2 e0 γ µ from (16.14), all fermion self-energy contributions factorize, conform to our notation (16.6), as Σ(p)reg −→ Zψ Σ(p)reg .

(16.24)

Correspondingly the Dyson–Schwinger Eq. (14.5) is replaced by the pre-renormalized Dyson–Schwinger equation S˜F (p) =

Zψ−1 p/ − m0

+

Zψ−1 

p/ − m0

 Zψ Σ(p)reg S˜F (p)

(16.25)

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243

Eq. (16.25) has the solution7 S˜F (p) =

Zψ−1 p/ − m0 − Σ(p)reg

≡ Zψ−1 SF0 (p) .

(16.26)

It is a peculiarity of the proper self-energy, that a multiplicative renormalization (16.24) is not sufficient to render it finite, but requires also a subtractive renormalization. To this effect we write Σ(p)reg in the form of a Taylor expansion in p/ around p/ = m, where m is the physical mass:   ∂Σ(p)reg (p/ − m) + Σ0 (p) . (16.27) Σ(p)reg = Σ(p)reg |p/=m + ∂p/ p/=m Note that the calculation of the second term on the rhs is performed after first making the replacement p2 = (p/)2 . It contributes to Zψ below. Note also that the third term Σ0 (p) is of order (p/ − m)2 , which will be important for satisfying ˜ the normalization condition. In order for S(p) to have a pole at the physical mass m = m0 + δm we must choose δm = Σ(p)reg |p/=m . Set further B=



(16.28)

 ∂Σ(p)reg . ∂p/ p/=m

(16.29)

Then we are left with ˜ S(p) =

Zψ−1 (p/ − m)(1 − B) − Σ0 (p)

.

(16.30)

We now require that Zψ = We then have S˜F (p) =

1 . 1−B

(16.31)

1 ˜ p/ − m − Σ(p)

where ˜ Σ(p) = Zψ Σ0 (p) . ˜ Removing the cutoff, we finally have with Σ(p) → Σ(p)ren , SF (p)ren =

1 . p/ − m − Σ(p)ren

Note that since Σ0 (p) ≈ (p/ − m)2 for p/ → m, the residue at the pole is one. 7 Note

that the “prime” here, and in general, stands for the subscript “reg”.

(16.32)

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Photon 2-point function

We proceed as above. The 1PI 2-point function is this time given by the sum of 1PI diagrams such as for example the diagrams

Zψ−1 (16.33)

Zψ−1 Fig. 16.2. Second order vacuum polarization diagrams. 1

With the new propagators (16.21), (16.22) and vertex Z1 e = Zψ ZA2 e0 γ µ from (16.14), all vacuum polarization diagrams factorize, conform to our notation (16.6), as µν Πµν reg (k) −→ ZA Πreg (k) . Again this generalizes to a diagram of arbitrary order. The pre-renormalized Dyson– Schwinger equation (14.6) is correspondingly replaced by, ˜ µν (k) = Z −1 Dµν (k)tr + Dµν (k)long + Dµλ (k)Πλρ (k)reg D ˜ ρν (k) . D F A F F F F Using (see (15.27)), µν 2 µ ν 2 Πµν reg (k) = (−g k + k k )π(k )reg

we obtain from here, in correspondence to (16.26), ˜ µν (k)tr = D F and

−1 µν −ZA P (k) −1 0µν   = ZA DF (k)tr 2 2 k 1 − π(k )reg + i

˜ µν (k)long = Dµν (k)long , D F F

(16.34)

(16.35)

with P µν (k) the projector (14.9). Expanding π(k 2 ), in analogy to (16.27), around k 2 = 0, the physical photon mass, π(k 2 )reg = π(0)reg + π 0 (k 2 ) ,

(16.36)

we may rewrite (16.34) as µν

˜ µν (k)tr = D F

k2



−1 −ZA P (k)  . 1 − π(0)reg − k 2 π 0 (k 2 ) + i

(16.37)

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245

Notice that π 0 (0) = 0. Hence in order to implement the normalization condition (16.9) we must choose 1 . (16.38) ZA = 1 − π(0)reg We thus have

˜ µν (k)tr = D F where

k2



−P µν (k)  1−π ˜ (k 2 ) + i

π ˜ (k 2 ) = ZA π 0 (k 2 ) .

(16.39)

We expect again π ˜ (k 2 ) to tend to finite limit π(k)ren for suitably chosen renormalization constants, as we remove the cutoff. We thus finally obtain for the renormalized transversal photon a 2-point function DFµν (k)ren =



−P µν (k)

 . k 2 1 − π(k 2 )ren + i

(16.40)

It is important to note that only the transversal part of the propagator gets renormalized by the interaction! Notice again that π(0)ren = 0, so that the residue at the pole is one.

16.2.3

Vertex function

Similar considerations based on an inspection of the sum of diagrams such as

Zψ−1 −1 ZA

q Zψ−1

Fig. 16.3. 1-loop QED vertex diagram. 1

show that with the new set of Feynman rules and vertex Z1 e = Zψ ZA2 eo , we have e0 Λµ (p0 , p)reg −→ Z1 eΛµ (p0 , p)reg or,

(16.41)

˜ µ (p0 , p) = Z1 [γ µ + Λµ (p0 , p)reg ] Γ

where Z1 is the renormalization constant defined in (16.11) and it is conventional to drop an overall charge factor e. More precisely, ˜ µ (p0 , p) = Z1 [γ µ + Λµ (p0 , p; S 0 , D0 , Γ; e0 , m0 )] Γ F F skel where the subscript “skel” is short for “skeleton”.

(16.42)

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It is by no means trivial to see that for a suitable choice of renormalization constants Ze , Zm , Z1 as a function of the cutoff, (16.42) will define a quantity which is finite in the limit where the cutoff is taken to “infinity”. There persist two problems in this renormalization program: • infrared divergences to cope with, as we have witnessed in the case of Σ(p)reg and Γµ (p0 , p)reg . The origin of this divergence lies in the fact that we have represented the photon propagator by an isolated pole. This ignores the fact that we can always produce an arbitrary number of soft real photons. • Overlapping divergences, such as exemplified by the diagram of Fig. 16.9. Diagrams exhibiting overlapping divergences pose, in general, a serious problem in the proof of renormalizability of a Quantum Field Theory. In this respect QED is simpler than φ4 -theory as J.C. Ward has shown.8

16.3

Ward–Takahashi Identity and overlapping divergences

Consider the 1-particle-reducible 3-point function G(x, y, z)µαβ = hΩ|T ψα (x)ψ¯β (y)Aµ (z)|Ωi

(16.43)

having the diagrammatic representation

Fig. 16.4. 3-point Green function. As a first step we wish to express the gauge field in terms of the electromagnetic µ ¯ current j µ (x) = e0 ψ(x)γ ψ(x). As is well known this relationship is not unique, and requires a choice of gauge. We shall work in the so-called “alpha-gauges”. As we have seen in Section 9 of Chapter 13, these gauges can be implemented by adding 1 the gauge breaking term − 2α (∂ · A)2 to the QED lagrangian. This corresponds to working with the lagrangian 1 ¯ ∂/ − e0 A/ )ψ − 1 (∂ · A)2 L = − Fµν F µν + ψ(i 4 2α implying the modified Maxwell equation     1 µ µ ¯ µψ . gλ − 1 − ∂ ∂λ Aλ = j µ = e0 ψγ α 8 J.C.

Ward, Phys. Rev. 77 (1950) 182.

(16.44)

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247

For α 6= 0 the differential operator possesses an inverse obtained by solving the differential equation     1 µ µ gλ − 1 − ∂ ∂λ DFλν (x − y; α) = g µν δ 4 (x − y) . α The result is (13.95), DFµν (x; α)

=

g µν −

∂µ∂ν

+α

∂µ∂ν . ( )2

DFµν (x) is nothing but the Feynman propagator of the photon field in the “alpha gauges”. In momentum space, it reads µ ν

DFµν (k; α)

−g µν + k kk2 kµ kν = − α . k 2 + i (k 2 )2

The solution of (16.44) is given by Z Aµ (z) = d4 z 0 DFµν (z − z 0 ; α)jν (z 0 ) . We may thus write for the 3-point function (16.43), Z G(x, y, z)µ = d4 z 0 DFµν (z − z 0 ; α)Vν (x, y, z 0 ) , where

Vµ (x, y, z)αβ = hΩ|T ψα (x)ψ¯β (y)jµ (z)|Ωi .

(16.45)

(16.46)

Making use of the involution property of Fourier transforms Z Z  d4 q −iq·z  ˜ d4 z 0 f (z − z 0 )g(z 0 ) = e f (q)˜ g (q) , 4 (2π)

we obtain from (16.45) in momentum space (k = p0 − p), after extracting a fourdimensional δ-function, G(p0 , p, k; α)µ = DFµν (k; α)Vν (p0 , p) ,

(16.47)

where Vµ (p0 , p) is defined by the three-fold Fourier transform   FT Vν (x, y, z) = (2π)4 δ 4 (p0 − p − k)Vµ (p0 , p) . On the other hand, we have from Fig. 16.4,9 µν

G(p0 , p, k; α)µαβ = iD0 F (k; α)iSF0 (p0 )(−ie0 )Γν (p0 , p)iSF0 (p) .

(16.48)

We have thus two representations of the 3-point function (16.43). Note that contrary to the representation (16.47), the representation (16.48) requires the 9 In

0 (p) = S (p) this section, SF reg . F

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knowledge of the full photon propagator. We recall, however, that we have for the “divergence” of this propagator, the remarkable property µν

kµ D0 F (k; α) = kµ DFµν (k; α) = −α

kν . k2

We thus obtain from (16.47) and (16.48) kµ µ 0 V (p , p) k2

(16.49)

kµ 0 0 µ 0 S (p )Γ (p , p)SF0 (p) k2 F

(16.50)

kµ G(p0 , p, k; α)µ = −α and kµ G(p0 , p, k; α)µ = e0 α respectively. We conclude that

(p0 − p)µ V µ (p0 , p) = −e0 (p0 − p)µ SF0 (p0 )Γµ (p0 , p)SF0 (p).

(16.51)

We now rewrite the left-hand side of this equation by observing that the current j µ (x) is the Noether current associated with the global U (1) symmetry of the lagrangian (16.1), as represented by the transformation ψ(x) → eiθ ψ(x) .

(16.52)

As a result, this current is conserved ¯ µψ ∂µ j µ = 0 , j µ = e0 ψγ and the corresponding charge Q=

Z

d3 xj 0 (x)

is the generator of the local symmetry transformation (16.52). The corresponding infinitesimal transformation implies the commutation relations  0  j (~x, t), ψα (~y , t) = −e0 δ 3 (~x − ~y )ψα (~y , t)   Q, ψα (y) = −e0 ψα (y) , (16.53)  0  3 j (~x, t), ψ¯β (~y , t) = e0 δ (~x − ~y )ψ¯β (~y , t) ,

which may also be verified explicitly by making use of the canonical commutation relations of the fermion field. Making use of (16.53) we find from (16.46), h i ¯ ¯ ∂µz V µ (x, y, z) = −e0 δ 3 (z − y)hΩ|T ψ(x)ψ(y)|Ωi − δ 3 (x − z)hΩ|T ψ(x)ψ(y)|Ωi h i = −e0 δ 3 (z − y)iSF0 (x − y) − δ 3 (x − y)iSF0 (x − y) , (16.54) or going to momentum space, this reads

  (p0 − p)µ V µ (p0 , p) = −e0 SF0 (p) − SF0 (p0 ) .

(16.55)

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249

Replacing the lhs in (16.51) by this expression, we finally have,10  0  0 (p0 − p)µ Γµ (p0 , p) = SF−1 (p0 ) − SF−1 (p) .

(16.56)

This is the so-called Ward–Takahashi identity. It follows only from the invariance of the QED lagrangian under global U (1) transformations. Differentiating (16.56) with respect to p0 , and setting p = p0 , one obtains the identity originally obtained by J.C. Ward:11 ∂ 0 (16.57) Γµ (p, p) = µ SF−1 (p) . ∂p Recalling (16.27) and (16.28) we can rewrite (16.57) as ∂Σ0 (p) = −Λµ (p, p)reg . ∂p/µ

(16.58)

The requirement that this identity be satisfied also by the corresponding renormalized quantities leads to the identification Z1 = Zψ , as we show next. Renormalized Ward–Takahashi identity Recalling from (16.6), Zψ−1 SF0 = S˜F → Sren ,

˜ µ → Γµ , Z1 Γµ = Γ ren

(16.59)

we have from (16.56),   ˜ µ (p0 , p) = Z −1 S˜−1 (p0 ) − S˜−1 (p) . (p0 − p)µ Z1−1 Γ F F ψ

(16.60)

It follows from here that the form of the Ward identity is preserved under renormalization, provided that in the limit Λ → ∞, Z1 = Zψ .

(16.61)

Ze = ZA .

(16.62)

Together with (16.11) this implies

Taking the limit of infinite cutoff, recalling that SF−1 (p)ren = p/ − m − Σ(p)ren , and setting

Γµren (p0 , p) = γ µ + Λµren (p0 , p)

we can equally write for (16.60),

10 Y.


(p/ − p/) + (p0 − p)µ Λµren (p0 , p) = − Σ(p0 )ren − Σ(p)ren + (p/0 − p/) ,

Takahashi, Il Nuovo Cimento 6 (1957) 371. Ward, Phys. Rev. 77 (1950) 293.

11 J.C.

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or
(p0 − p)µ Λµren (p0 , p) = − Σren (p0 ) − Σren (p) .

From here we obtain, by differentiating with respect to p0µ and setting p0µ = pµ , Λµren (p, p) = −

∂ Σ(p)ren . ∂pµ

(16.63)

The problem of overlapping divergences in QED Consider the diagram of Fig. 16.5 with an overlapping, linear divergence.

p

p

Fig. 16.5. Diagram with overlapping divergence.

As we see from, 1 1 ∂ 1 =− γµ , ∂pµ k/ − p/ − m k/ − p/ − m k/ − p/ − m differentiation with respect to the external momentum p, amounts to inserting a zero momentum photon line in every fermion propagator as shown in Fig. 16.6:12

∂Σ(p) = ∂pµ

p

p

p p

+

p

+

p

(16.64)

Fig. 16.6. Ward identity for self-energy diagram of Fig. 16.5. 12 For

the sake of visualization, we have replaced one of the photon lines by a dashed line.

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251

This is precisely what is done in the Ward identity (16.63). What has one gained? It turns out that the vertex function Λµ (p0 , p) does NOT involve overlapping divergences, and can thus be renormalized in a conventional way. This was the observation of J.C. Ward. By integration of (16.63), we then obtain the renormalized electron self-energy: Σ(p0 )ren − Σ(p)ren = −

Z

p

p0

dq µ Λ(q, q)µren .

Now, remembering that Σ(p)ren |p/=m = 0, we finally have Σ(p)ren = −

Z

p

p/=m

dqµ Λµ (q, q)ren .

A similar procedure can be followed in the case of the photon vacuum polarization. Let us write in analogy to (16.63), −

∂ π(k 2 )ren = ∆µ (k, k)ren ∂kµ

(16.65)

where π(k 2 )ren is given by (16.39), and the right-hand side involves again zero momentum photon insertions in the vacuum polarization diagram in question. Again, ∆µ does not involve overlapping divergences, and hence can be renormalized in a conventional way. Integrating (16.65), and remembering that π(0)ren = 0, we obtain, Z k π(k 2 )ren = dkµ0 ∆µ (k 0 , k 0 )ren , 0

or using the parametrization kµ (x) = xkµ + (1 − x)kµ0 , we alternatively have, π(k 2 )ren =

Z

0

1

dx(k − k 0 )µ ∆µ (k(x), k(x))ren .

For more details see L.H. Ryder.13

16.4

1-loop renormalization in QED

Equations (16.31) and (16.38) define the renormalization constants Zψ and ZA to all orders of perturbation theory. As we shall see, these constants play a central role in the Callan–Symanzik equations to be discussed in Chapter 17. We now proceed to calculate the renormalized 2- and 3-point functions, as well as the corresponding renormalization constants, in 1-loop order of perturbation theory. Computation of Zψ and Σren (p2 ) The calculation of the renormalization constant Zψ and Σren (p2 ) requires the computation of δm and B from (16.28) and (16.29) respectively. To second order we 13 L.H.

Ryder, Quantum Field Theory (Cambridge University Press, 1996, second edition).

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can set α0 = α, m0 = m. Replacing p2 in the argument of ln f in (15.17) by (p/)2 , and expanding Σreg (p) in the Taylor series (16.27), we then obtain for the electron self-energy,  2 3α Λ + δm0 (16.66) δm ' mln 4π m2 with

δm0 ' − and

α 2π

Z

1

dβm(1 + β)ln

0

α B ' − ln 4π

with B0 '

α 2π

Z

0

1





Λ2 m2

m2 β 2 + (1 − β)µ2 m2 (1 − β) 



,

+ B0

(16.67)

  2 2   m β + (1 − β)µ2 2m2 β(β 2 − 1) dβ (1 − β)ln − . (16.68) m2 (1 − β) β 2 m2 + (1 − β)µ2

Subtracting from (15.17) δm + B(p/ − m), the Λ dependent terms cancel, and we obtain with (16.27) and (16.29) for the renormalized electron self-energy to order O(α) (no wave function renormalization to this order: Σren (p) ∼ Σ0 (p)), Z 1 α Σ(p)ren ' dβ[(1 − β)p/ − 2m]ln f (β; p2 , m2 , µ2 ) − B 0 (p/ − m) − δm0 . (16.69) 2π 0 One checks explicitly that Σ(p)ren is of order (p/ − m)2 , as expected. We notice that the threshold for the cut in the logarithmic function in (16.69) coincides in the limit µ → 0 with the mass shell p2 = m2 , so that in a physical process, we cannot isolate photons in this limit from physical fermions. We also recall that the above results are gauge dependent, the fermion 2-point function being itself a gauge-dependent quantity. As was shown by Yennie and Fried,14 this singular divergence can be avoided by choosing the ’t Hooft gauge α = 3 (see (13.98)). However, this does not solve the problem in general, the proper solution being that of averaging over soft photons taking account of the experimental resolution.15 Computation ZA and πren (k 2 ) From the result (15.30) for the dimensional regularization,   2α0 2 µ ˜2 dim 2 π (k )reg ' − − γ − ln π + ln 2 3π  m0 Z 1  α0 m20 − β(1 − β)k 2 − i0  +2 dβ β(1 − β)ln π 0 m20

and (16.38),

ZA = 14 H.M. 15 F.

1 ≈ 1 + π(0)reg 1 − π(0)reg

Fried and D.R. Yennie, Phys. Rev. 112 (1958) 1391. Bloch and A. Nordsieck, Phys. Rev. 52 (1957) 54.

(16.70)

(16.71)

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253

it follows to second order in dimensional regularization    µ  α 2 µ ˜2 dim ZA α, ,  ' 1 − − γ − ln π + ln 2 . m 3π  m

(16.72)

On the other hand, from the result (15.28) for the Pauli–Villars regularization, with M = Λ, we have  2    Z m − β(1 − β)k 2 − i0 α0 Λ2 2α0 1 PV 2 dββ(1 − β)ln , π (k )reg ≈ − ln + 3π 2m20 π 0 m2 (16.73) and  2  α0 Λ PV ZA '1− . (16.74) ln 3π 2m2 In both cases, we have for the proper renormalized vacuum polarization to order O(α),  2  Z m − β(1 − β)k 2 − i0 2α 1 dββ(1 − β)ln π(k 2 )ren ' . (16.75) π 0 m2 Computation of Z1 and Γµ (p, p0 )ren We first recall (see (15.40)) that on the fermion mass-shell, to second order Γµ (p0 , p) had the form (p + p0 )µ ¯ (~ ¯ (~ U p 0 , σ 0 ))Γµ (p0 , p)U (~ p, σ) = U p 0 , σ 0 )[γ µ + γ µ F (q 2 ) + G(q 2 )]U (~ p, σ) . 2m (16.76) At zero momentum transfer we must require from (16.12) that (we simplify our notation) ¯ (~ ˜ µ (p, p)U (~ ¯ (~ ¯ (~ U p)Γ p) = Z1 U p)Γµreg (p, p)U (~ p) = U p)γ µ U (~ p) .

(16.77)

In order to implement the renormalization condition (16.77) we make use of the Gordon relation 1 ¯ 0 ¯ (~ U (~ p )[(p + p0 )µ + iσ µν (p0 − p)ν ]U (~ p) U p 0 )γ µ U (~ p) = 2m

(16.78)

where

i µ ν [γ , γ ] . 2 Hence we have from (16.76) equivalently, σ µν =

(16.79)

¯ (~ ¯ (~ U p 0 )Γµ (p0 , p)U (~ p) = U p 0 )[(1 + F (q 2 ) + G(q 2 ))γ µ − iσ µν qν G(q 2 )]U (~ p) . We define new form factors F1 (q 2 ) and F2 (q 2 ), F1 (q 2 ) = F (q 2 ) + G(q 2 ) ,

F2 (q 2 ) = G(q 2 )

(16.80)

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in terms of which ¯ (~ ˜ µ (p0 , p)U (~ ¯ (~ U p 0 )Γ p) = Z1 U p 0 )[γ µ (1 + F1 (q 2 )reg ) − iσ µν qν F2 (q 2 )reg ]U (~ p) . (16.81) Writing Z1 in the form of the expansion (16.16), Z1 = 1 + αz1 + · · · , we require from (16.77) to order O(α), αz1 + F1 (0)reg = 0 .

(16.82)

This leads (15.46) and (15.41) to the form factors (Z1 − 1 = −F1 (0)ren ) Z  X  α 1Y 2 dαi δ 1 − αi (16.83) F1 (q )ren = − π 0 ( 2 m2 (1 − α1 )(1 − α2 )(m20 − q2 ) − 20 [(α1 + α2 )2 + 2α1 α2 ] × m20 (α1 + α2 )2 + α3 µ2 − q 2 α1 α2  1 + ln (m20 (α1 + α2 ) + α3 µ2 − q 2 α1 α2 ) − {q2 = 0} 2 ! Z 1Y 3 X α m2 α3 (α1 + α2 ) 2 F2 (q )ren = − dαi δ 1 − αj . (16.84) π 0 i m2 (α1 +α2 )2 +α3 µ2 − α1 α2 q 2 1 Note that this form factor diverges logarithmically as we let the mass of the photon tend to zero. This was the reason for introducing an infrared cutoff to begin with. The origin of this divergence lies in the fact that we have represented the photon propagator by an isolated pole. This ignores the fact that we can always produce an arbitrary number of soft real photons, which leads to the replacement of this isolated pole by a cut in the complex k-plane. These soft photons are excited whenever charged particles suffer a change in velocity, that is, for non-vanishing momentum transfer. As has been shown16 by Bloch and Nordsieck this infrared divergence is removed upon averaging over such photons, taking account of the experimental resolution. Finally, the anomalous magnetic moment is determined by F2 (0)ren , whose value is easily calculated. For q = 0, Z Z (1−α1 ) α 1 (1 − α1 − α2 ) F2 (0)ren = − dα1 π 0 (α1 + α2 ) 0   1 α 1 α =− α1 + α12 = . (16.85) π 2 2π 0

Using the Gordon relation (16.78) we have from (16.81), with F1 (0)ren = 0, to lowest order in q µ , " #   0 µ α q (p + p ) ν 0 µ 0 0 µν ¯ (~ + 1+ iσ U (~ p) . U (~ p )[Γ (p , p)U (~ p) ≈ U p ) 2m 2π 2m 16 F.

Bloch and A. Nordsieck, Phys. Rev. 52 (1957) 54.

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255

α with the Comparing with Section 6.5 of Chapter 6, we identify F2 (0) = 1 + 2π anomalous magnetic moment of the electron, first calculated by Schwinger in 1948.17

16.5

Composite operators and Wilson expansion

Formally, local composite operators are obtained as the product of local operators in the limit where the arguments tend to the same point. As we know already from the behaviour of 2-point Green functions, these limits are in general singular, so that the corresponding composite operator cannot be defined in this naive sense. By looking at the behaviour of composite operators of local free fields, Wilson proposed for an interacting theory the existence of an expansion for x → y of the form18   X x+y → C0 (x−y)+C1 (x−y)O1 (x)+· · · (16.86) A(x)B(y) = CN (x−y)ON 2 N =0

where ON are local and finite operators with O0 (x) = 1, and CN (x−y) are c-number functions, of which some are singular in the limit x → y. As a simple illustration, we consider a free scalar field theory. Wick’s theorem tells us that ϕ(x)ϕ(y) = h0|ϕ(x)ϕ(y)|0i+ : ϕ(x)ϕ(y) : = i∆(+) (x − y)+ : ϕ(x)ϕ(y) :

= i∆(+) (x − y)+ : ϕ(x)2 : +O(x − y).

This expansion is of the form (16.86) with A(x) = B(x) = ϕ(x), and C0 (z) = i∆(+) (z), C1 (z) = 1 . O0 (z) = 1, O1 (z) =: ϕ2 (z) : . The fact that the coefficient C1 is finite reflects that the scale dimension of the normal-ordered product :ϕ(x)ϕ(x) : is exactly two, and thus identical with its engineering (canonical) dimension. Let us denote the engineering dimension of a general operator Q by [Q].19 Perturbatively we then have in nth order for z → 0, after renormalization, (n)

AB

AB

CN (z) ∼ g n z −δON (ln µ|z|)ρON ,

(16.87)

where, for a strictly renormalizable theory, AB δO = [A] + [B] − [ON ] , if [g] = 0 . N

Hence on the perturbative level the coefficients CN of strictly renormalizable theories tend to zero for terms in the operator expansion involving composite operators ON with engineering dimension [ON ] > [A] + [B] . 17 J.

Schwinger, Phys. Rev. 73 (1948) 416. Wilson, Phys. Rev. 179 (1969) 1499; and Phys. Rev. Ser. D, 3 (1971) 1818. 19 One measures dimensions in units of “energy” = “mass”.

18 K.

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Note that the arguments of the logarithm have to be dimensionless. The parameter µ appearing in (16.87) carries the dimensions of “mass” and is introduced by the need of regularization. It may for instance be the mass parameter introduced in dimensional regularization, or may refer to the point at which the subtraction is performed in a Taylor subtraction scheme (see Section 7). Non-perturbatively we have seen in Section 2 of Chapter 12, that the logarithms can sum up to a power behaviour, so that after summation to all orders we may have for z → 0, AB AB CN (z) → z −δON (µ|z|)γ0N with γ0AB = γA + γB − γON . N Here γQ is referred to as the anomalous dimension of the operator Q (see Chapter 17). Following Wilson, the c-cumber functions CN should be universal and characteristic of the operator product in question. According to this hypothesis, we can determine them by considering suitable n-point functions. With the notation GAB (x, y; z1 . . . zn ) =< Ω|T A(x)B(y) ϕ(z1 ) · · · ϕ(zn )|Ω > we have from (16.86), X   CN ()GON (x; z1 . . . zn ) , GAB (x + , x − ; z1 . . . zn ) = 2 2 N

where GON (x; z1 , . . . zn ) =< Ω|T N [A(x)B(x)] ϕ(z1 ) · · · ϕ(zn )|Ω > , defines the normal product N [A(x)B(x)] of the two (interacting) fields at the same point. For free fields A and B, N [A(x), B(x)] =: A(x)B(x) :.

16.6

Criteria for renormalizability

In order to obtain simple criteria for the renormalizability of a Quantum Field Theory we need to introduce the notion of the superficial degree of divergence of a Feynman diagram. One obtains a superficial estimate of the degree of divergence of a Feynman diagram G, by simply scaling all internal momenta with a common factor λ, and examining the behaviour of the corresponding Feynman integral IG in the limit of λ → ∞: ω(G) IG →λ IG . One calls ω(G) the superficial degree of divergence of the diagram G. We consider, in general, a theory involving spin 0, spin 1/2 and spin 1 fields. Under a “dilatation”, the respective propagators scale as λ−2 , λ−1 and λ−2 . We adopt the following notation:

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257

EF = Number of external fermion lines EB = Number of external boson lines IF = Number of internal fermion lines IB = Number of internal boson lines V = Number of vertices L = IF + IB − (V − 1) = Number of independent loops EFv = Number of external fermion lines connected to vertex v EBv = Number of external boson lines connected to vertex v IFv = Number of internal fermion lines connected to vertex v IBv = Number of internal boson lines connected to vertex v δv = Number of derivatives contained in vertex v gv = Coupling constant associated with the vertex v We then have (D = Dimension of space-time) X ω(G) = DL + δv − 2IB − IF v

= (D − 1)IF + (D − 2)IB +

(16.88) X (δv − D) + D . v

We want to express these quantities in terms of the number of external fermion and boson lines, and the dimension (in units of mass) of the coupling constants labelling the vertices. Denoting the dimension of a quantity A by [A], we observe that in 4 space-time dimensions 3 [φ] = [Aµ ] = 1 , [ψ] = . 2
Correspondingly we define [v] = dimension of vertex, including [gv ] [v/gv ] =

Noting that

X

3 3 EF + EBv + IFv + IBv + δv . 2 v 2

IFv = 2IF ,

v

we then have

or using (16.88),

X

IBv = 2IB ,

v

X X v  3 = EF + EB + 3IF + 2IB + δv , gv 2 v v X v

([v/gv ] − 4) =


3 EF + EB + ω(G) − 4 . 2

Now, from [d4 xLI ] = 0 we have, [v/gv ] + [gv ] = 4

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or it follows from above ω(G) = 4 −

X v

3 [gv ] − EF − EB , 2

(16.89)

which is the desired result. Equation (16.89) shows that if there exist couplings for which [gv ] < 0, then the superficial degree of divergence will increase indefinitely with the order of the diagram, independent of the number of (unrenormalized) constants, and the theory is thus called unrenormalizable. One is thus led to the following classification of Quantum Field Theories: (i) Unrenormalizable QFT According to our above comments these are theories for which ∃v : [gv ] < 0 so that ω(G) > 0 for an infinite set of diagrams. Examples are ¯ ρ ψ∂ρ φ, g ψγ ¯ ρ γ 5 ψ∂ρ φ, ψγ ¯ ρ ψ ψγ ¯ ρ ψ, φd>4 LI = g ψγ (ii) Renormalizable QFT (in the extended sense) These are theories for which [gv ] ≥ 0 , ∀v .. Under these conditions there exists an upper limit of the superficial degree of divergence for any diagram of arbitrary order, so that we can achieve finiteness of the Green functions by adding a finite number of counterterms. It may, however, not be possible to absorb all of these counter terms into a redefinition of the unrenormalized parameters of the lagrangian defining the QFT in question. An example is ¯ 5 ψφ. The UV divergence of the 4-point function LI = g ψγ

requires the introduction of a counterterm of the form δL = δZg 4 φ4 corresponding to a local interaction not contained in the above Lagrangian. (iii) Renormalizable QFT (in the strict sense) These are theories for which [gv ] ≥ 0, ∀v again holds, but for which all the counter terms can be absorbed into a redefinition of the bare parameters and fields of the lagrangian. Examples are ↔

¯ + λφ4 , gp ψγ ¯ 5 ψφ + λφ4 , eψA ¯ / ψ , eψγ ¯ 5 A/ ψ , (eφ† ∂µ Aµ + e2 φ2 A2 ) . LI : gs ψψφ µ

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259

(iv) Superrenormalizable QFT These are theories for which [gv ] ≥ 1, ∀v. In this case, there exist only a few diagrams which are divergent. The fact that ω(G) < 0 does not exclude the possibility that G contains a subdiagram g with ω(g) ≥ 0. Thus consider, for example, the diagram in gφ4 theory ([g] = 0, EF = 0, EB = 6). We have ω(G) = 4 −

X

3 [gv ] − EF − EB = −2 . 2

Fig. 16.6. ω(G) = −2 diagram involving logarithmicallly divergent subdiagram. This diagram nevertheless contains the subdiagram

Fig. 16.7. ω(g) = 0 subdiagram of Fig. 16.6. with ω(g) = 0. For what follows, we recall the definition of a subdiagram (restricted definition): g is a subdiagram of G(g ⊂ G) if all lines which connect two vertices in g, also lie in g. We have Theorem Let G be a (connected) 1P I diagram; Let g be a (connected) 1P I subdiagram of G; Let Fg be the family of all (connected) 1P I subdiagrams of G. Then the (euclidean) Feynman integral I (E) (G) is absolutely convergent, provided ω(g) < 0, ∀g ∈ Fg and ω(G) < 0 . We have the following two corollaries.

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Corollary 1. In a scalar QFT with derivative couplings, or a theory with non-scalar particles, the presence of polynomials in the numerator of the integrand of a Feynman integral can lead to cancellations reducing the degree of divergence of the integral. Example: As we have seen in Chapter 15, Figs. 15.1 and 15.2 are only logarithmically divergent. Corollary 2. If ω(g) < 0, ∀g ∈ Fg but ω(G) ≥ 0 , then, in the limit Λ → ∞, (E)

IG (p; m; Λ) → I (E) (p; m) + P(p; m; Λ) , where PΛ (p, m; Λ) is a cutoff dependent polynomial of degree less or equal to ω(G) in the external momenta p, and internal masses mi . Proof: Since ω(G) measures the degree of homogeneity of IG in the external momenta and internal masses, we have ∂ ω+1 (E) I (λp, λm; Λ) ∼ λω−(ω+1) = λ−1 , ∂λω+1 G and hence this expression is absolutely convergent. Corollary 2 then follows upon using the above theorem.

16.7

Taylor subtraction

We have considered so far the renormalization based on two possible regularization procedures — Pauli–Villars and dimensional regularization. There exist a number of other possibilities for renormalization. In the following, we shall consider that obtained by performing a Taylor subtraction of the integrand of a Feynman integral. This leads to the so-called Bogoliubov recursion formula, whose general solution is given by Zimmermann’s forest formula. Renormalization by Taylor subtraction We shall restrict again our discussion to the case of scalar particles. We suppose the Feynman integral in question to be given in the Feynman-parametric form of Eq. (12.17). Correspondingly we write Z Y IG (p) = dλ dαi JG (p; α, λ) , (16.90) i

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261

where p stands for all the external momenta. According to (12.17) the integrand will be a function of the Lorentz invariants pi · pj , the Feynman parameters αi and λ. The convergence of the integral is controlled by the behaviour of the integrand at λ = 0. We observe that repeated differentiation with respect to the external momenta improves this behaviour. The procedure thus consists in subtracting from J (p; α) a truncated Taylor expansion around a conveniently chosen “point” in the space of the Lorentz invariants. Thus consider for instance the diagram (φ3 -theory)

l1 p

p l2

Fig. 16.8. φ3 self-energy. The corresponding integrand will be a function of the Lorentz invariant p2 . We perform the following Taylor subtraction around p2 = 0: RG = (1 − TG )JG (p2 ; α, λ) ,

(16.91)

with ω(G) terms

z  }| ∂JG (p2 ; α, λ) 2 2 TG JG (p ; α) = JG (0; α)) + p ∂p2 p2 =0

+

{

···.

(16.92)

The number of terms to be kept in the Taylor series depends on the degree of divergence of the diagram. Since the diagram in question is logarithmically divergent, it suffices to consider the first term in this Taylor series (considering more terms would result in “oversubtraction”). Note that we could have expanded also around another point, other than p2 = 0. The above subtraction for ω(G) = 0 improves the λ → 0 behaviour for the example in question sufficiently, to render the λ integration finite. Indeed, making use of the cutting rules of Chapter 12, we have P(α) = α1 + α2 corresponding to the tree diagrams

`1 `2 For QG (P, α) one correspondingly obtains from (12.14), QG (α1 , α2 ) =

p2 α1 α2 . α1 + α2

(16.93)

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Now, [g] = 1, EF = 0, EB = 2 , or ω(G) = 4 −

X 3 [gv ] − EF − EB = 0 . 2 v

Hence the integral is logarithmic divergent. From (12.17) we have for λ → 0 in the integrand of the λ-integral, after a single Taylor subtraction, i 2 iα1 α2 p2 1 h iλp2 α1 α2 /(α1 +α2 ) e − 1 e−im λ(α1 +α2 ) → . λ (α1 + α2 ) The λ-integration is now finite. Let us exemplify this by calculating the Feynman integral associated with the diagram of Fig. 16.8. Returning to (12.17) we must compute Z ∞ i 2 dλ h iλ(Q(p;α)−m20 ) e − e−iλm0 . I(p; α) = λ ε

It is convenient to take p2 < 0 and to deform the integration contour along the real axis to one along the negative imaginary axis, by performing the “rotation” λ = −iρ

in the complex λ-plane. The sign in this change of variable reflects the fact that this rotation is to be performed counterclockwise. Choosing p2 < 0, that is, below the threshold for particle production, we encounter no singularities of the integrand, so that in this case Z ∞ i 2 dρ h −ρ(m20 −Q(p;α)) I(p; α) = e − e−ρm0 , p2 < 0 . (16.94) ρ ε This leaves us with (12.17) for the second order φ3 -selfenergy, (−ig0 )2 −iΣreg (p) = i(4π)2

Z

0

2 1Y 1

δ(1 − Σαi ) dαi  2 I(p; α) . P(α)

(16.95)

There are several ways of evaluating this integral: (a) Setting  = 0 and using Pauli–Villars regularization. (b) We evaluate the integral (16.94) in the limit ε → 0 by performing the following manipulations:  Z ∞  i 2 dln ρ h −ρ(m20 −Q(p;α)) I(p, α) = dρ e − e−ρm0 dρ ε Z ∞ h i i d d h −ρ(m20 −Q(p;α)) −ρ(m20 −Q(p;α)) −ρm20 −ρm20 = dρ ln ρ e −e − ln ρ e −e dρ dρ ε Z ∞ h i h i 2 2 2 2 2 = −ln ε e−ε(m0 −Q(p;α)) − e−εm0 + dρln ρ (m20 − Q2 )e−ρ(m0 −Q ) − m20 e−ρm0   2  m0 − Q(p; α) = −ln (16.96) m20

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where we made the change of variable (m0 − Q)ρ → ρ0 , (m0 ρ) → ρ0 , respectively. Hence finally we have to second order, 2 Σ(2) ren (p )

g2 = 02 16π

Z

0

1

dαln



m2 − α(1 − α)p2 m2



,

(16.97)

where we have made the replacements g0 → g and m0 → m.

16.8

Bogoliubov’s recursion formula

Differentiation with respect to the external momenta of a Feynman graph may not reduce the degree of divergence since the divergent subgraphs may not involve the variable of the differentiation. Bogoliubov, Parasiuk,20 and Hepp,21 have shown how to include the divergent subgraphs in the Taylor approach. Consider a Feynman diagram IG with ω(G) ≥ 0 and let Fγ be the family of all superficially divergent, connected 1P I subdiagrams γ of G:22 Fγ = (γ : ω(γ) ≥ 0) , (Fγ , we have X [< Ω|φ(0)|α >< α|φ(0)|Ω > eipα ·(x−y) − (x → y)] < Ω|[φ(x), φ(y)]|Ω >= α

with the sum running over a complete set of positive energy states, which can be rewritten as Z d4 q ρ(q)(e−iq·(x−y) − eiq·(x−y) ) < Ω|[φ(x), φ(y)]|Ω >= (2π)3 where ρ(q 2 ) is the density ρ(q 2 ) = (2π)3

X α

δ 4 (q − pα )| < Ω|φ(0)|α > |2 .

This density is a positive Lorentz invariant and vanishes outside the forward light cone. It can thus be written as ρ(q 2 ) = θ(q 0 )σ(q 2 ),

with

σ(q 2 ) = 0

if

q2 < 0 .

We have then < Ω|[φ(x), φ(y)]|Ω >= i

Z

0

∞

dm02 σ(m02 )∆(x − y; m02 ).

(16.108)

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The LSZ asymptotic condition (10.16) implies < Ω|[φ(x), φ(y)]|Ω >= iZφ ∆(x − y; m2 ) + i

Z

∞

m02 >m2

dm02 σ(m02 )∆(x − y; m02 ) .

Taking the time-derivative on both sides, we have with (7.36) Z 1=Z+ dm02 σ(m02 ) m02 >m2

or positivity of σ(m2 ) implies Z≤1, with Zφ = 0 being reached only for a free field. In momentum space, Eq. (16.108) takes the form of a dispersion relation:25 Z ∞ σ(s) . (16.109) Σ(p2 ) = ds s − p2 + i s>m2 The analytic approach The analyticity properties of Feynman amplitudes have been the subject of extensive studies documented in a number of books. A nice account has been given by Bjorken and Drell.26 These studies have played a central role in the so-called “S-matrix theory” of the 1960’s.27 The “dispersive” approach was initiated in 1954 with the work of M. Gell-Mann, M.L. Goldberger and W. Thirring.28 The analytical approach based on the study of perturbative expansions has received most of the attention. In this section we only briefly touch on this subject, by providing a representation of the renormalized self-energy in φ3 theory as a dispersion relation. The analytic structure of the corresponding Feynman diagrams in lowest non-trivial order of perturbation theory is particularly simple, but serves to illustrate some points, such as the Taylor expansion in (16.91). Consider a general self-energy diagram of φ3 -theory as shown in Fig. 16.11:

p

p

Fig. 16.11. General self-energy diagram. Its analyticity properties on the physical sheet in the external momentum are determined by the vanishing of the argument of the exponential in (12.17), or the 25 G.

K¨ allan, Helv. Phys. Acta 25 (1952) 417; H. Lehmann, Nuovo Cimento 11 (1954) 342. Bjorken and S.D. Drell, Relativistic Quantum Fields (McGraw-Hill, 1965). 27 G.F. Chew, S-Matrix Theory of Strong Interactions (W.A. Benjamin Inc., New York, 1962); S. Mandelstam, Rept. Progr. Phys. 25. 28 M. Gell-Mann, M.L. Goldberger and W. Thirring, Phys. Rev. 95 (1954) 1612. 26 J.D.

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denominator in (12.18) (PG (α) plays no role) in the multidimensional parameter space (p, α): X J ≡ QG (p; α) − αi m2i = 0 , i

supplemented by the condition that the solution α(0) satisfies either

∂J =0, ∂αi α(0) i

(0)

or αi

=0.

These are the Landau conditions. As an example we have for the diagram of Fig. 16.8, taking account of the δ-function in α, J(p2 ; α) = p2 α(1 − α) − m2 , ∂ J = p2 (1 − 2α) = 0 ∂α implying α(0) =

1 2

and a singularity at p2 − 4m2 = 0. p

p

Fig. 16.12. Self-energy diagram of φ3 theory. Indeed, the diagram is represented by the integral (see Eq. (16.97)), 2 (2) 2 Σ(2) ren (p ) − Σren (−µ ) =

g2 16π 2

Z

1

dβln

0



m2 − β(1 − β)p2 − i m2 + β(1 − β)µ2



.

(16.110)

We see that the integrand exhibits a logarithmic cut in p2 extending to the right in the complex p2 plane from p2 = m2 /β(1 − β) to p2 = ∞. The discontinuity across the cut is given by ln (m2 − β(1 − β)(p2 + i)) − ln (m2 − β(1 − β)(p2 − i)) = −2πiθ(−m2 + β(1 − β)p2 ) or we obtain for the discontinuity of Σ(p2 ) discΣ

(2)

g2 (p ) = (−2πi) 16π 2 2

Z

1

dβθ(−m2 + β(1 − β)p2 )    Z 1 1 m2 g2 1 2 − 2 = (−2πi) dβθ −(β − ) + , 16π 2 0 2 4 p s g2 4m2 = −i 1 − 2 θ(p2 − 4m2 ) . (16.111) 8π p 0

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Let us repeat this calculation starting from the basic integral Z i2 d4 q . −iΣ(2) (p2 ) = (−ig0 )2 (2π)4 [q 2 − m20 + i0][(q − p)2 − m20 + i0] This integral is ultraviolet divergent, but its regularized expression only differs from the renormalized expression by a cutoff dependent additive constant which does not contribute to the discontinuity to be calculated. The calculation is simplified by going into the frame p~ = 0: Z Z dq 0 d3 q 2 2 −iΣ(p0 ) = g0 2π (2π)3 1 × 0 , [q − ωq + i0][q 0 + ωq − i0][(q 0 − p0 ) − ωq + i0][(q 0 − p0 ) + ωq − i0] p where ωq = ~q2 + m20 . The location of the poles in the complex q 0 -plane and the contour of integration are depicted in the figure below. Imq 0

CF

−ωq · p0 − ωq · 0

Req 0

ωq · p + ωq ·

Fig. 16.13. q 0 -integration contour and singularities for Σ(2) (q02 ). We have • p0 < 2m0

The q 0 -contour is never “pinched” by the singularities in q 0 as ~q varies, so that there is no singularity in p0 for p0 < 2m0 .

• p0 ≥ 2m0 For p0 = 2m0 the q0 -contour is pinched at q 0 = m as ~q → 0, and there will be a cut from p0 = 2m0 to ∞. In order to compute the discontinuity across this cut we distort the contour across the singularity at q 0 = ωq , keeping track of this pole, which contributes with an infinitesimal clockwise contour C around it:   Z Z ∞ dq 0 (−2πi) 0 1 d3 q (2) 2 2 δ(q − ωq ) . (16.112) ΣC (p0 ) = g0 (2π)3 −∞ 2π 2p0 ωq [p0 − 2ωq + i0] The discontinuity of Σ(p) arises from here, and is discΣ(p0 ) = ΣC (p0 + i0) − ΣC (p0 − i0)

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16.10 Dispersion relations: a brief view Z ∞ dq 0 d3 q 3 (2π) −∞ 2π   (−2πi) 0 1 1 δ(q − . − ω ) × q 2p0 ωq [p0 − 2ωq + i0] [p0 − 2ωq − i0] = g02

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271

Z

(16.113)

Recalling that

1 1 − = −2iπδ(x) x + i0 x − i0 the remaining integral is easily done: Z 1 d3 q 0 2 discΣ(p ) = g0 (−2πi) δ(p0 − 2ωq ) 3 0 (2π) 2p ωq   Z Z ∞ ω − 12 p0 −ig02 2 2 21 dωω(ω − m ) dΩ δ = 0 (2π)2 2p20 0 s g2 4m2 = −i 0 1 − 2 0 θ(p0 − 2m0 ). 8π p0

(16.114)

Using relativistic invariance, and setting now m0 = m and g0 = g, this result agrees with (16.111). Starting from the usual Cauchy integral Z 1 Σ(s) Σ(p2 ) = ds 2πi C s − p2 with C given by the contour closed at infinity, enclosing the cut, we cannot discard the contribution at ∞. Hence we write the Cauchy integral for Σ(p2 )/(p2 − µ2 ). Taking account of the pole at p2 = µ2 we obtain this way the subtracted dispersion relation Z (p2 − µ2 ) ∞ discΣ(s) 2 2 Σ(p ) = Σ(µ ) + (16.115) 2 + i0)(s − µ2 + i0) 2πi (s − p 2 4m q 2 Z g 2 1 − 4m p2 − µ2 ∞ s ds . (16.116) = Σ(µ2 ) − 8π (s − p2 + i0)(s − µ2 + i0) 4m2 This corresponds to a single, final Taylor subtraction in (16.91) for ω(G) = 0. Bogoliubov’s recursion formula provides the generalization thereof to arbitrary diagrams with divergent subgraphs. Cutcosky rules Let us rewrite the result (16.113) as follows: Z d4 q (−2πi)δ(q 0 − ωq )θ(q 0 )(−2πi)δ(p0 − q 0 − ωq )θ(p0 − q 0 ) 2 2 −idiscΣ(p ) = g0 (2π)4 2p0 ωq Z 4 d q = g02 (−2πi)2 δ(q 2 − m20 )θ(q 0 )δ((p − q)2 − m20 )θ(p0 − q 0 ). (2π)4

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This result provides an example of the so-called Cutkoski rules which state, that the discontinuity of Σ(p2 ) across the cut in the physical region p2 ≥ m20 is obtained in the present case by putting the “particles” in the intermediate state on their respective mass shell, with the rule29 1 −→ (−2πi)θ(q 0 )δ(q 2 − m20 ) q 2 − m20 + i0 1 −→ (−2πi)θ(q 0 − p0 )δ((q − p)2 − m20 ) , (q − p)2 − m20 + i0 as represented in the following diagram.

p

q−p

p −→

(16.117)

q Fig. 16.14. Cutkosky rule discontinuity. This observation establishes the link with the K¨allan–Lehmannn representation.

29 R.E.

Cutkosky, J. Math. Phys. 1 (1960) 429.

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Chapter 17

Broken Scale Invariance and Callan–Symanzik Equation If there were no dimensional parameters in the theory, then dimensional arguments alone (plus translational invariance) would uniquely fix, up to a normalization constant, the form for the 2-point function of a scalar field to be hΩ|φ(x)φ(y)|Ωi =

1 1 . 2 2π (x − y)2

(17.1)

The function (17.1) evidently scales with the canonical dimension of the scalar field under the dilation x → λx of the coordinates, hΩ|φ(λx)φ(λy)|Ωi = λ−2dφ hΩ|φ(x)φ(y)|Ωi,

dφ = 1 .

(17.2)

If we take the vacuum to be invariant under such a scale transformation, then we deduce from (17.2) the transformation law φ(x) → U [λ]φ(x)U −1 [λ] = φ0 (x) = λφ(λx) .

(17.3)

The corresponding Lagrange density describing the dynamics of the scalar field would thus scale as follows under a dilatation L(x) → L0 (x) = λ4 L(λx) . Hence in this case the corresponding action is left invariant by the unitary operator inducing the dilatation (17.3). According to the theorem of Noether, we expect a conserved current corresponding to this continuous symmetry, called the dilatation current S µ . In analogy to electrodynamics we further expect a Ward identity to be associated with the conservation of this current. De factum there exists no non-trivial Quantum Field Theory exhibiting such dilatation invariance. Even if the defining classical lagrangian does not involve any dimensioned parameter (such as would be the case for electrodynamics of massless fermions), the need of renormalization will require the introduction of dimensioned 273

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Broken Scale Invariance and Callan–Symanzik Equation

parameters, which break the dilatation invariance of the classical theory. One speaks of broken scale invariance, and the modified Ward Identities reflecting this fact are called the Callan–Symanzik equations. One would nevertheless be tempted to argue that despite the appearance of dimensioned “mass” parameters in the theory, these parameters should not play a role at high energies. This “intuitive feeling” turns out to be incorrect. It turns out to be however almost correct in the case of so-called asymptotically free theories, where the violation of the “canonical” scaling law is soft in the sense that it is only broken by logarithmic factors. An important example of such theories is Quantum Chromodynamics (QCD). In general, however, the Green functions of the theory will exhibit a powerlike (or logarithmic) deviation from the canonical behaviour at large (euclidean) momenta (short distances). Thus for the 2-point function of a scalar field one would have for example for (x − y)2 → 0, 1 1  γ hΩ|φ(x)φ(y)|Ωi ≈  2 2 (x − y) µ (x − y)2 φ

(17.4)

with µ a parameter with the dimension of a mass introduced by the renormalization procedure. Here γφ is called anomalous dimension of the scalar field φ(x). As we shall see, such anomalous dimensions can be calculated in perturbation theory from the knowledge of the wave function renormalization constants. They are the analogue of the critical exponents of systems at criticality in solid state physics. This is the reason, why solid state physicists have made extensive use of field-theoretic methods in order to compute such critical exponents. The anomalous scaling behaviour also holds if the lagrangian contains massive particles. In order to see this, we shall have to examine more closely the behaviour of Feynman diagrams in the so-called deep euclidean region, where all the external momenta in the diagram, continued analytically to the euclidean domain, tend to infinity. The general result runs under the name of Weinbergs’s theorem. It will eventually imply that the Callan–Symanzik equation, which is an inhomogeneous equation, reduces to an homogeneous one in the deep-euclidean region.

17.1

Scale transformations

We begin by recalling Noether’s Theorem. (See Section 1 of Chapter 9.) Let φα (x), α = 1, ..., n stand for n complex fields whose dynamics is described by a Lagrange density
L(x) = L {φα (x), ∂µ φα (x)} with

S[φ] =

Z

d4 xL(x)

(17.5)

the corresponding action. Consider a linear infinitesimal transformation φα (x) → φα (x) + δ φα (x) with a δ φα (x) = a Dαβ φβ (x) ,

(17.6)

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275

where a are A infinitesimal parameters characterizing the transformation. If it is a symmetry transformation, then it leaves the action (17.5) invariant, implying that the corresponding change in the Lagrange density must be representable as the divergence of a 4-vector field: δ L(x) = a ∂µ F µ,a (x) . Noether’s theorem then states that the following A “currents” J µ,a (a = 1 · · · , A) are conserved:  X  ∂L µ,a a J = D φβ + h.c. − F µ,a , ∂µ J µ,a = 0 . (17.7) ∂∂µ φα αβ α,β

If, on the other hand, the transformation (17.6) does not represent a symmetry transformation of the action (17.5), then these conservation laws will be replaced by ∂µ J µ,a = ∆a , (17.8) where the rhs is so far unspecified. We shall be interested in particular in the Noether current S µ associated with dilatations or scale transformations xµ −→ λxµ . Like all other transformations of physical interest, scale transformations can be implemented in a linear or non-linear way. We shall be interested in a linear implementation on the fields: −1 ~ ~ ~ 0 (x) = λd φ(λx), ~ −→ U [λ]φ(x)U φ(x) [λ] = φ U

or in infinitesimal form (λ = 1 + ) ~ ~ δ φ(x) = (d + 1xµ ∂µ )φ(x) .

(17.9)

~ Here d is in general a matrix acting on the vector field φ(x). In a classical field theory involving no dimensioned parameters, d would be a diagonal matrix multiplying all Bose fields by one and all Fermi fields by 3/2, according to their canonical (engineering) dimensions. In models involving a mass term, this symmetry will be explicitly broken. In a QFT this symmetry will further be broken by renormalization, since it requires the introduction of new dimensioned scale parameters. To give an example consider the theory of interacting bosons and fermions, L = LS + LSB where

and LSB

1 λ0 LS = ψi∂/ψ + ∂µ φ∂ µ φ + g0 ψγ5 ψφ − φ4 2 4! is the symmetry breaking term 1 LSB = −m0 ψψ − µ20 φ2 . 2

(17.10)

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In the absence of LSB , the transformations φ(x) → φ0 (x) = λφ(λx) , would result in

ψ(x) → ψ 0 (x) = λ3/2 ψ(λx)

(17.11)

LS (x) → L0S (x) = λ4 L(λx) ,

thus implying the invariance of the action (17.5) under the transformation (17.11). For LSB we however have 1 LSB (x) −→ L0SB (x) = −m0 λ3 ψψ(λx) − µ20 λ2 φ2 (λx) , 2 so that scale invariance will be broken by this term. Infinitesimally we have from (17.11) (we omit the -parameter)   3 µ + x ∂µ ψ, δφ = (1 + xµ ∂µ )φ , δψ = 2 or δLS = (4 + xµ ∂µ )LS = ∂µ (xµ LS ) δLSB = −(3 + xµ ∂µ )m0 ψψ − (2 + xµ ∂µ )

(17.12) µ20 2

φ2

= ∂µ (xµ LSB ) + m0 ψψ + µ20 φ2 .

(17.13)

We see that (17.12) gives a vanishing contribution to the action, while Z δSSB [φ] = d4 x∆ , ∆ = m0 ψψ + µ20 φ2 . Retracing the derivation of the Noether theorem, we see that this means that the corresponding Noether current is no longer conserved, ∂µ S µ = ∆ ,

(17.14)

which is the local version of the statement that masses break scale invariance. It is important to realize that what we have done is not “dimensional analysis”. Indeed, a dimensional analysis requires to also scale the masses as m → λ−1 m, and is always an exact symmetry of the action. Example of a scale current Consider the Lagrange density L=

λ 1 ∂µ φ∂ µ φ − φ4 . 2 4!

We have (see (17.12)) F µ = xµ L S ,

(17.15)

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so that according to (17.7) the corresponding scale (dilatation) current, now denoted by S µ , is given by S µ = ∂ µ φ(1 + xν ∂ ν )φ − xµ L = φ∂ µ φ + xν J µ,ν ,

(17.16)

where J µ,ν is the canonically constructed energy momentum tensor J µ,ν =

∂L ∂ ν φ − g µν L = ∂ µ φ∂ ν φ − g µν L . ∂(∂µ φ)

Note that this energy-momentum tensor is in general not symmetric, and is conserved only with respect to the first index: ∂µ J µ,ν = 0 . Using the equations of motion φ=

∂L ∂φ

one easily checks that the current (17.16) is conserved, as expected. In QFT the need for renormalization will force us to introduce dimensioned parameters, even if such parameters were absent in the classical action. Accordingly the fields φ(x) will transform with anomalous scale dimensions. These anomalous dimensions turn out to be computable in perturbation theory. The basic idea has its roots in Eq. (17.14), and the Ward identities for renormalized n-point functions implied by it.

17.2

Unrenormalized Ward identities of broken scale invariance

We now consider the Ward identities for the unrenormalized n-point functions following from broken scale invariance. For the sake of concreteness the reader may think of the theory of a massive, real scalar field described by a Lagrange density L=

1 1 g0 (∂µ φ)2 − m20 φ2 − φ4 . 2 2 4!

(17.17)

The divergence of the dilatation current is in this case given by (17.14) with ∆ = m20 φ2 , and S µ = ∂ µ φ(1 + xν ∂ ν )φ − xµ L − m20 φ2 . Although the corresponding charge is not conserved, we still have the equal time commutation relations  0  S (y), φ(x) ET = −iδ 3 (~x − ~y )δφ(x) (17.18)

for the unrenormalized fields. We define as usual the time-ordered product of n fields A(x),   X T A(x1 )...A(xn ) = δp θ(x0i1 − x0i2 )...θ(x0in−1 − x0in )A(xi1 )...A(xin ) P

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where the sum runs over all permutations in time of the fields, and δp is the “signature” of the permutation; i.e. 1(−1) for even (odd) permutation of commuting (anticommuting) fields. We then have from (17.14) and (17.18) the Ward Identity
∂ hΩ|T S µ (y)φ(x1 )...φ(xn ) |Ωi = hΩ|T ∆(y)φ(x1 )...φ(xn )|Ωi µ ∂y n X −i δ 4 (y − xi )hΩ|T φ(x1 )...φ(xi−1 )δφ(xi )φ(xi+1 )...φ(xn )|Ωi . i=1

Integrating both sides of the equation, and dropping the surface term, we arrive at Z
d4 yhΩ|T ∆(y)φ(x1 )...φ(xn ) |Ωi = (17.19) i

n X i=1

hΩ|T φ(x1 )...φ(xi−1 )δφ(xi )φ(xi+1 )...φ(xn )|Ωi .

These equations are called zero-energy theorems, since the integral over ∆ carries zero energy and momentum in Fourier space. (The soft-pion theorems of current algebra are special cases of (17.19).) Identifying δφ with δφ(x) = (dφ + xµ ∂µ )φ(x) ,

dφ = 1 ,

(17.20)

and pulling the derivative in (17.20) past the time-ordering operation in (17.19) (since the scalar fields commute at equal times, this does not give rise to “contact” or “seagull” terms), we can write the Ward identity in the form  Z n  X ∂ dφ + xµi hΩ|T φ(x )...φ(x )...φ(x )|Ω = −i d4 yhΩ|∆(y)φ(x1 )...φ(xn )|Ωi. 1 i n µ ∂x i i=1 (17.21) Taking the Fourier transform of both of these equations, and making use of translational invariance, we obtain from here for the lhs,1 ! Z n X ∂ µ d4 x1 ...d4 xn eip1 ·x1 ...eipn ·xn ndφ + xi hΩ|T φ(x1 − xn )...φ(0)|Ωi ∂xµi i=1 ! Z n−1 Pn Z n−1 Y X µ ∂ iy·( pi ) ip ·y ip ·y 1 = dye dyi e 1 1 ...e n−1 n−1 ndφ + yi ∂yiµ 1 i=1 ×hΩ|T φ(y1 )...φ(yn−1 )φ(0)|Ωi .

Making use of  Z Z Z ∂ d4 yeip·y y µ ∂µ f = − d4 y(4 + iy µ pµ )eip·y f (y) = − 4 + pµ d4 yeip·y f (y) ∂pµ 1 Make

change of variable xi = yi + y , i = 1 · · · n − 1 , xn = y, so that n X i=1

xi

X X ∂ X ∂ ∂ ∂ → (yi + y) − y = yi . ∂xi ∂yi ∂yi ∂yi n−1

n−1

n−1

i=1

i=1

1

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17.2 Unrenormalized Ward identities of broken scale invariance we arrive at the momentum-space version of the Ward identity (17.21): " # n−1 X (n) µ ∂ ndφ − 4(n − 1) − pr µ G(n) (p1 ...pn ) = −iG∆ (0; p1 ...pn ) . ∂p r r=1

page 279

279

(17.22)

Here G(n) (p1 · · · pn ) stands for the n-point Green function after separation of the overall momentum-conservation delta function: Z Y n Pn dxr ei 1 pr ·xr hΩ|T φ(x1 )...φ(xn )|Ωi = (2π)4 δ(Σpi )G(n) (p1 ...pn ) , r=1

(n)

and G∆ is the corresponding function with a “mass insertion”: Z Y n 1

dxr ei

P

pr ·xr

(n) 4 ˜ hΩ|T ∆(0)φ(x 1 )...φ(xn )|Ωi = (2π) δ(Σpr )G∆ (0; p1 ...pn ) ,

˜ with ∆(0) the Fourier transform of ∆(x) evaluated at zero momentum: Z ˜ ∆(0) = d4 y∆(y) . The first two terms on the lhs of (17.22) just reflect the engineering dimension (ndφ − 4(n − 1)) of the n-point function G(n) . Making use of ordinary dimensional analysis, we may write G(n) in the form   ndφ −4(n−1) S pi · pj (n) (n) 2 G (p1 ...pn ) = S , , g0 F (17.23) m20 S where we have introduced the Lorentz invariant variable !2 n n−1 n−1 X X X 2 2 S= pr = pr + pr . r=1

r=1

(17.24)

1

Note that we have explained the dependence of F (n) on the unrenormalized mass m0 and (dimensionless) coupling constant g0 . Returning to (17.23) and (17.22), we note that n−1 n−1 X X ∂ ∂ pµr µ (pi · pj ) = 2(pi · pj ) . pµr µ S = 2S , ∂pr ∂pr 1 1 We see from here that

n−1 X 1

Hence we can write n−1 X 1

pµr

pµr

∂ (pi · pj ) =0. ∂pµr S

∂ (n) ∂ = −m0 F (n) µF ∂pr ∂m0



S pi · pj , , g0 m20 S



,

(17.25)

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or we obtain from (17.23), n−1 X 1

pµr

∂ (n) G = ∂pµr

ndφ − 4(n − 1) − m0



∂ ∂m0



g0 ,λ

!

G(n) .

We conclude that on the formal level we can write the Ward identity (17.22) in the form2   ∂ (n) m0 G(n) (p1 ...pn ) = −iG∆ (0; p1 ...pn ) . (17.26) ∂m0 g0 ,Λ After all this work, this is a surprisingly simple, but not unexpected result. Indeed, in this form we recognize that the unrenormalized Ward identity (17.21) essentially states that differentiation of a general Green function with respect to the bare mass m0 , and subsequent multiplication with m0 , simply amounts to a zero-momentum “mass insertion” in each internal boson line. Thus we have for a φ-field propagator m0

i i i ∂ = 2 (−i2m20 ) 2 2 2 2 ∂m0 p − m0 p − m0 p − m20

with the corresponding diagrammatic representation

m20

∂ ∂m20

=

(17.27)

Fig. 17.1. Dot represents mass insertion −i2m20 .

17.3

Broken scale invariance and renormalized Ward identities

In the case of the Ward identities associated with gauge invariance of QED or QCD, renormalization can be performed consistent with this gauge invariance, and the corresponding Ward identity remains form-invariant, as we have seen. This is quite different from the case of the Ward identities associated with scale transformations. Even if we start from a scale invariant lagrangian on classical level, there exists no regularization procedure which will respect this invariance. The need for introducing dimensioned parameters into the theory will destroy this symmetry already on the 1-loop level. As a result, the Ward identity for the unrenormalized proper functions will change in form under renormalization. The result is known as the Callan– Symanzik equation, to be discussed next. In order to conform to usage, we first define the 1PI (amputated) Green functions n h i−1
Y
(n) iΓ(n) G(2) Greg {pi } . reg {pi } = reg (pj ) j=1

2 S.

Coleman, Selected Erice Lectures (Cambridge University Press).

(17.28)

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The factor of “i” has been introduced to conform to convention (see Chapter 20, Eq. (20.12)). From Fig. 17.1 we conclude that in terms of the proper Green functions, Eq. (17.26) reads   ∂ (n) m0 Γ(n) (p1 ...pn )reg = Γ∆ (0; p1 ...pn )reg . (17.29) ∂m0 g0 ,Λ We now derive from (17.29) the corresponding Ward identities for the renormalized proper functions. To this end, we review briefly the basic principles of renormalization. Starting point is the unrenormalized Lagrange density (17.17). The Green functions calculated in terms of this lagrangian are given by divergent integrals, and thus need to be regularized by a suitable cutoff procedure. This leads to cutoff dependent Green functions. In a renormalizable theory, the cutoff dependence of an n-point unrenormalized Green function can be eliminated by (i) a judicious choice of the bare parameters as functions of the cutoff, accompanied by (ii) a multiplication of the regularized Green function with a suitably chosen cutoff dependent constant 1 Zφ2 for each of the n fields in the corresponding vacuum expectation value of the time-ordered product. It is convenient to write     Λ Λ − 12 −1 g, g, m0 = Zm g, m. g0 = Zg m m Renormalizability then means that for a proper choice of the renormalization constants, the function 1

e (n) (p1 ...pn ; g; m; Λ) = Z n/2 Γ(n) (p1 ...pn ; Zg− 2 g, Z −1 m; Λ) Γ m reg φ

(17.30)

possesses a finite limit as we remove the cutoff (Λ → ∞), keeping g and m fixed. Note that for the proper Green functions the wave function rernormalization constant Zφ appears with inverse powers as compared to (16.6). The multiplicative renormalization amounts to the substitution 1/2

φ = Zφ φren where φren is referred to as the renormalized field. In a practical calculation the renormalization constants will naturally emerge as functions of the cutoff Λ, bare coupling constant g0 and the renormalized or bare mass. In line with (16.3) and (16.4) we define them by     1 Λ Λ − 12 2 g = Zg g0 , g0 , φren = Zφ g0 , φ. (17.31) m0 m0 Unless required, we shall not distinguish between the different functional dependencies, and denote all of these functions by one and the  same symbol, Zσ .  ∂ We now apply the “bare mass insertion operator” m0 ∂m0 to g0 ,Λ



−n 2 e (n) Γ(n) {p}; g, m; Λ , reg {p}; g0 , m0 ; Λ = Zφ Γ

(17.32)

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or (17.29) becomes,    n  n
∂ − e (n) −1 − 2 ˜ (n) m0 Zφ 2 Γ ({p}; g, m; Λ) = Z∆ Zφ Γ∆ ({p}; g, m; Λ . ∂m0 g0 ,Λ

Note that the partial differentiation with respect to the bare mass is to be performed, holding the cutoff and bare coupling constant fixed. Making use of the chain rule of differentiation, we obtain for the lhs, upon multiplication by Z∆ , # "      ∂ ∂g ∂ n ∂ln Zφ ∂m e (n) . + m0 − m0 Γ Z∆ m0 ∂m0 g0 ,Λ ∂m ∂m0 g0 ,Λ ∂g 2 ∂m0 g0 ,Λ (17.33) In perturbation theory, Z∆ is given by the expansion X (`)  Λ  g0` . Z∆ = 1 + Z∆ m0 `

The cutoff dependent terms are defined only up to finite additive constants. We can make use of this freedom to fix these finite parts by requiring for Z∆ the normalization   ∂m =m. Z∆ m 0 ∂m0 g0 ,Λ Note that this normalization is consistent with lowest order perturbation theory, where Z∆ = 1 and m = m0 . Having thus fixed Z∆ we further define, together with (17.33),       ∂ Λ ∂ Λ Λ =m g g0 , = −Λ g g0 , (17.34) β˜ g, m ∂m m ∂Λ m and



Λ γ˜ g, m



    1 ∂ 1 ∂ Λ Λ = m ln Zφ g0 , =− Λ ln Zφ g0 , . 2 ∂m m 2 ∂Λ m

(17.35)

Note that we have used dimensional analysis to replace the derivative with respect to the mass by a derivative with respect to the cutoff. This shows that we only need the knowledge of the (divergent) cutoff dependent terms in order to calculate the β-function and anomalous dimension. We now suppose that (17.33) tends to a finite limit as we let the cutoff go to infinity. Hence each of the three terms in (17.33) must also have this property. Accordingly, as a function of g and m, the cutoff dependent constants β˜ and γ˜ must have a finite limit as we let the cutoff tend to infinity:     Λ Λ ˜ β(g) = lim β g, , γ(g) = lim γ˜ g, . Λ→∞ Λ→∞ m m In the limit Λ → ∞ we thus finally obtain from (17.33) the differential equation for the renormalized proper n-point functions   ∂ ∂ (n) m + β(g) − nγφ (g) Γ(n) ({p}) = Γ∆ ({p}). (17.36) ∂m ∂g

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This is the Callan–Symanzik equation replacing the naive Ward identity (17.29). Several remarks are in order:3 1. The generalization to more complicated field theories involving different types of interactions, masses and coupling constants is obvious, with every type of field being associated its own anomalous dimension: " # N X X X ∂ ∂ (n ···n ) + − m` nk γφk Γ(n1 ...nN ) ({p}) = Γ∆ 1 N ({p}) . βr ∂m` ∂g r r `

k=1

2. The Callan–Symanzik equation for Green functions involving other types of operators such as partially conserved currents, for instance, will involve different anomalous dimensions, but the β-functions, which make only reference to the underlying dynamics will remain unchanged. 3. From a practical point of view these equations will only be useful, provided we are allowed to calculate the beta function and anomalous dimension perturbatively. We shall see in Section 8 that this will be the case for QED in the low energy domain (and QCD in the high energy domain). However, before we can demonstrate this, we make an excursion to the deep euclidean behaviour of Feynman graphs.

17.4

Weinberg’s Theorem

The Callan–Symanzik equation (17.36) can straightforwardly be continued to the euclidean region, where scalar products will be given in terms of the (euclidean) metric of R4 . We shall argue in the following section that in the deep euclidean region, where all the components of the momenta tend to infinity, the right-hand side of the CS equation can be neglected, thus turning this inhomogeneous differential equation into a homogeneous one. Let us consider a scalar field theory without derivative couplings. At first we assume that the Feynman integral is convergent. As we have seen in Section 1 of Chapter 12, such an integral can be put into the form (ω(G) < 0), Γ(− ω(G) 2 ) × 2 [(4π) ]L ! PI ω(G) Z Y I I X [QG (p, α) − 1 α` m` ] 2 dα` δ 1 − α` , PG (α)2 1

IG (p) = i1−V −ω(G)/2

(17.37)

`=1

where ω(G) is the superficial degree of divergence of the diagram G (see (16.88)),
ω(G) = 4L − 2I = 2 I − 2(V − 1) ,

3 K. Symanzik, Comm. Math. Phys. 18 (1970) 227; C.G. Callan, Phys. Rev. Ser. D 2 (1970) 1541; R.J. Crewther, Asymptotic Behaviour in Quantum Field Theory (Coference Carg` ese, 1975) p. 345.

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and where PG (α) and QG (p, α) are the homogeneous polynomials of the Feynman parameters defined in (12.13) and (12.14), respectively. The theorem to be stated below applies to Feynman amplitudes continued to euclidean momenta. Performing the continuation p = (p0 , p~) → P = (ip4 , p~) in (17.37) with (E) QG (p; α) → QG (P ; α) , (E)

and scaling all external momenta as P −→ λP , QG (P ; α) scales as (E)

(E)

QG (λP ; α) = λ2 QG (P ; α) . Correspondingly we have,
i1−V −ω(G)/2 Γ − ω(G) 2 λω(G) (17.38) [(4π)2 ]L P ω(G) Z Y I  X  [Q(E) (P, α) − 12 α` m2` ] 2 G λ × dα` δ 1 − α` . [PG (α)]2

(E) IG (λP ) =

`=1

Definition Euclidean momenta tending to infinity in all directions we call nonexeptional. Statement 1 Provided that the zero mass limit of (17.38) exists for nonexceptional momenta, i.e. Z Y I

`=1



dα` δ 1 −

X

it follows that (E)

IG

α`

 [Q(E) (P ; α)] ω(G) 2 [P(α)]2

< ∞,


{λP } ∼ λω(G) .

(17.39)

Note that this is a non-trivial assumption. Indeed, infrared (IR) divergences can invalidate this result. In the case where the Feynman diagram is not UV convergent, the above result is modified by logarithmic powers, as the following examples illustrate. Some Examples: Consider the following diagram:

Fig. 17.2. Second order 4-point function in φ4 -theory.

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This diagram has superficial degree of divergence ω(G) = 0, and thus requires regularization. The result after renormalization can be read off from Eq. (16.97):   Z 1 P2 g2 (E) 2 dα ln 1 + α(1 − α) 2 . I (−P ) = 2(4π)2 0 m The asymptotic behaviour is given by I (E) (P ) →

g2 P2 ln 2(4π)2 m2

(17.40)

and corresponds to a behaviour (17.39) with logarithmic corrections. R Consider now the above diagram with n mass insertions −im20 d4 y : φ2 (y):

1

n

Fig. 17.3. Diagram with mass insertions. This diagram can be viewed as an 4 + n-point function with n (zero-momentum) external lines, with a factor −im20 for each proper of a mass insertion. It is represented by the (Minkowski) Feynman integral 2 Z d4 k (i)n+2 2 2 n (−ig0 )   . In∆ (p ) = (−im0 ) 2 (2π)4 (k 2 − m20 + i)n+1 (p + k)2 − m20 + i For n ≥ 1 this integral is UV convergent. Doing a Wick rotation, k 0 → ik 4 , and p0 → ip4 , we have Z 4 i d K (−1)n+2   In∆ (−P 2 ) = (m20 )n g02 2 (2π)4 (K 2 + m20 )n+1 (P + K)2 + m20 (E)

where P = (p4 , p~), etc. For n > 1 the asymptotic behaviour of In∆ is easily read off: Z ∞ π2 x i g02 m20 i g02 m2n 0 n+2 (−1) dx = , In∆ (−P 2 ) ' 2 2 2 n+1 2 (4π) P (x + m0 ) n(n − 1) (2π)4 P 2 0

where we have used (15.1) and (15.2) with D = 4, and have set K 2 = x. For n = 1 we have, using (15.4) and setting K = K 0 − αP , K 02 = x, Z Z 1 Z 1 Z ∞ i (m20 g02 ) 1 δ(α1 + α2 + α − 1) 2 I∆ (−P ) ' dα1 dα2 dα xdx 3 2 (4π)2 0 [x + α(1 − α)P 2 + m20 ] 0 0 0 Z Z 1 Z 1 i (m20 g02 ) 1 1 δ(α1 + α2 + α − 1) ' dα dα dα 1 2 2 (4π)2 0 2 α(1 − α)P 2 + m20 0 0 Z i (m20 g02 ) 1 α ' dα . 2 (4π)2 0 α(1 − α)P 2 + m20

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The asymptotic behaviour of this integral is determined by the integration region around β = 1. We have for n = 1, that is, for one mass insertion, Z α ig02 m20 1 2 dα I∆ (−P ) = 2 2 m2 2(4π) P 0 α(1 − α) + P 20   1 ig02 m20 P 2 ln . +O ≈ 2(4π)2 P 2 m20 (P 2 )2 In particular, we see that (m0 → m) I∆ (−P 2 ) im2 ' 2 . 2 I(−P ) P

(17.41)

The reason for the power behaviour for n > 1 is that in Fig. 17.3 the inclusion of mass insertions amounts to the introduction of superrenormalizable quadratic interactions in the lagrangian, whereas in the case of Fig. 17.2, one is dealing with a renormalizable interaction, characterized by one dimensionless coupling constant. The picture behind this is that for n = 0 (Fig. 17.2) momentum flows equally through both branches connecting the vertices with the non-exceptional external lines, whereas for n ≥ 1 the momentum prefers to flow along the “upper branch” in Fig. 17.3. This observation is at the heart of Weinberg’s Theorem: Renormalized Feynman diagrams, in which only strictly renormalizable couplings are involved, scale in the case of all momenta being non-exceptional is as follows, (E)

IG (λP ) ≈ λω(G) (ln λ)n(G) , where n(G) is some integer.4

17.5

Solution of CS equation in the deep euclidean region

The CS equation may evidently be continued analytically in all the momenta to the euclidean domain, by simply performing the substitutions p0 → ip4 , the scalar product now being positive, semi-definite. Having done this, we now consider the solution of the CS equation in the deep euclidean region for non-exceptional momenta that is, in the limit where all components of the momenta grow large. Since the mass operator φ2 carries the dimension [φ2 ] = 2, we have for a mass insertion in the proper functions, in the deep euclidean region (see (17.41)),5
  Γ∆ {λp}; ... λ −2
∼ λ P ln (17.42) m Γ {λp}; ... 4 S.

Weinberg, Phys. Rev. 118 (1960) 838. here on, Γ denotes the renormalized proper Green function as defined in (20.12).

5 From

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in every order of perturbation theory. Correspondingly the euclidean asymptotic n-point proper function satisfies the homogeneous CS equation  
∂ ∂ (17.43) m + β(g) − nγφ (g) Γ(as) {p}; g, m = 0 . ∂m ∂g Although Weinberg’s theorem only holds in every given order of perturbation theory, we now assume that (i) the CS equation also holds for the complete proper function, and (ii) Weinberg’s theorem continues to hold for non-exceptional momenta in the deep Minkowskian region. We now seek the solution of the asymptotic CS equation (17.43). To this end it is convenient to rewrite the CS equation in a slightly different form. The asymptotic CS equation states that a small change in the mass-parameter m can be compensated by a corresponding change in the coupling constant g and a suitable rescaling of the fields associated with the wave function renormalization constant. Now, a change in the mass parameter corresponds to a change of the scale in which we measure momenta and other dimensional quantities. Hence let us consider what happens under a rescaling of the mass. Let D be the engineering dimension of the proper function in question. Ordinary dimensional analysis then allows us to write (g assumed dimensionless) 
m Γ {λp}; g, m = λD Γ {p}; g, . (17.44) λ Hence, we can view a scaling m → m/λ of the mass as a scaling p → λp in the momentum. Noting that ∂ m m m ∂ m m 0 m m f = f ; λ f = − f0 , (17.45) ∂m λ λ λ ∂λ λ λ λ we thus have

m


∂ Γ {λp}... = ∂m

 
∂ −λ + D Γ {λp}, ... . ∂λ

(17.46)

t = ln λ .

(17.47)

It is convenient to define a new variable t by

We may then rewrite the asymptotic CS equation in the form  
∂ ∂ − + β(g) + D − γ(g) Γ(as) {et p}; g, m = 0 , ∂t ∂g

(17.48)

with γ(g) = nγφ (g) and t → ∞. Generally, we must keep in mind, that ordinary dimensional analysis tells us that Γ({et pi }, g, m) = eDt Γ({pi }, g, e−t m) .

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The behaviour of the proper function in the deep euclidean limit is thus controlled by the limit m → 0. The first and third terms in the CS equation take care of dimensional analysis. The other two terms take care of anomalous behaviour in m introduced by the mass dependence of the renormalization constants. We distinguish three cases: (i)

β(g) = 0,

γ(g) = 0

This is the case, if the theory is finite to begin with, so that in the limit Λ → ∞, the Zi do not depend on m0 (or m):,   Λ → Zi (g0 ) . Zi g0 , m In that case the asymptotic CS equation reads  
∂ − + D Γ(as) {et p}; g, m = 0 , ∂t

implying for t → ∞ the asymptotic behaviour

Γ({et p}; g, m) → eDt Φ({p}; g, e−t m). The CS equation thus tells us that the proper function scales in the deep euclidean region with the canonical dimension, including the possibility of a broken scale invariance such as (17.4). (ii)

β(g) = 0,

γ(g) 6= 0

There is no coupling constant renormalization; the mass dependence only enters via the cutoff dependent logarithms ln (Λ2 /m2 ) appearing in the wave function renormalization constant in perturbation theory (and the possible failure of the m → 0 limit). We have in this case, for t → ∞,
Γ {et p}; g, m → et(D−γ(g)) Φ {p}; g, e−t m) ,

and one says that the proper function scales with an anomalous dimension D −γ(g). (iii)

β(g) 6= 0,

γ(g) 6= 0

In this case the solution of Eq. (17.48) is more involved. Consider the auxiliary equation   ∂ ∂ − + β(g) g¯(g, t) = 0 (17.49) ∂t ∂g satisfying the “initial” condition

g¯(g, 0) = g . Making use of (17.49), we now observe that  R g¯(g,t) 0 γ(g0 )  R g¯(g,t) 0 γ(g0 ) ∂ ∂ − dg β(g0 ) − dg β(g0 ) e g = γ(g)e g . − + β(g) ∂t ∂g

(17.50)

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Hence we conclude that in this case,
− Γ {et p}, g, m → etD Φ {pi }; g¯(g, t), e−t m)e

R g¯(g,t) g

γ(g 0 )

dg 0 β(g0 )

(17.51)

where the additional dependence on m in the deep euclidean region (see (17.58)) now arises from the dependence of t on the running coupling constant. The function g¯(g, t) is referred to as the running coupling constant, which together with γ(¯ g (g, t)) now controls the asymptotic behaviour. Alternative form of the solution It is convenient to rewrite Eq. (17.49) for g¯(g, t) in a different form. To this end, consider the function f defined by β(g)

∂ g¯(g, t) =: f (g, t) . ∂g

(17.52)

f (g, 0) = β(g) .

(17.53)

From (17.50) we see that One readily checks that f (g, t) is itself again a solution of (17.49), though with a different boundary condition. Indeed, using Eq. (17.52),   ∂ ∂ ∂ ∂ − f (g, t) = −β(g) g¯(g, t) = −β(g) f (g, t) ∂t ∂g ∂t ∂g so that we have



∂ ∂ − + β(g) ∂t ∂g



f (g, t) = 0 .

From here and (17.53) we infer
f (g, t) = β g¯(g, t) .

Hence we may write (17.52) also in the form


∂ g¯(g, t) = β g¯(g, t) . ∂t

(17.54)

We now use (17.54) in order to put the solution (17.51) into an alternative form. To this end we make the change of variable g 0 → t0 in the exponential of (17.51), leaving us with Z

g

g ¯(g,t)

0

) = dg β(g 0 ) 0 γ(g

Z

0

t

0

dt



∂¯ g (g, t0 ) ∂t0



γ(¯ g (g, t0 ))
= β g¯(g, t0 )

Z

0

t


dt0 γ g¯(g, t0 ) .

(17.55)

We may thus write (17.51) in the alternative form



− R t dt0 γ Γ(as) {et p}; g, m = etD Φ {p}; g¯(g, t), e−t m e 0

g ¯(g,t0 )




.

(17.56)

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Asymptotic behaviour of Γ and zeros of the β-function

From perturbation theory we know that β(0) = 0. We also know the behaviour of the beta function β(g) in the neighbourhood of g = 0. The behaviour of the betafunction “far away” from g = 0 is in general not known. For the sake of discussion let us nevertheless assume that the beta-function vanishes at a discrete set of values gi∗ of g. Depending on the sign of β near g = 0 we distinguish two situations, as depicted in the figures below.

Fig. 17.4(a). β-function for case (a).

Fig. 17.4(b). β-function for case (b). It then follows from (17.49) and the figures above that Case (a); −→

g¯(g, t) t→∞

(

−→

0 , g ∈ (0, g1∗ ) g2∗ , g ∈ (g1∗ , g2∗ )

g1∗ , g ∈ (0, g2∗ )

g¯(g, t) t→−∞ Case (b); −→

g¯(g, t) t→∞ −→

g¯(g, t) t→−∞

(

g1∗ , g ∈ (0, g2∗ )

0 , g ∈ (0, g1∗ ) g2∗ , g ∈ (g1∗ , g2∗ )

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This means: In case (a) are the “even” ( “odd”) zeros of β “UV(IR) attractors” (fix points). In case (b) are the “odd” (“even”) zeros of β “UV (IR) attractors” (fix points). Suppose that g ∗ is such a relevant fix point. In that case we have from (17.56)


∗ Γ({et pi }, g, m −→ et D−γ(g ) Φ {pi }, g¯(g, t), e−t m) e−(t) ,

where

(t) =

Z

t

0

Introduce t=

  dt0 γ(¯ g (g, t0 )) − γ(g ∗ ) .

S 1 ln 2 , 2 m

with

We distinguish different possibilities:

S=

n X

(17.57)

p2r .

1

(1) (t) is finite in the limit t → ∞. In this case we have √ Γ({et pi }, g, m) −→ ( S)D



S m2

− 21 γ(g∗ )

  m Φ {pi }; g ∗ , √ e−(∞) . S

(17.58)

Let us examine under what condition this is realized. We consider two examples: (a) Assume that the beta-function behaves like β(g) ' b(g1∗ − g) for 0 < g < g1∗ with b > 0 (case (b)). In that case we have from (17.54) ∂ g¯ = β(¯ g ) ' −b(¯ g − g1∗ ) ∂t near the fix-point. We may solve this differential equation by separation of variables: Z g¯ dg 0 ∗ t=− ∗ , g close to g1 0 g b(g − g1 ) or

g¯ − g1∗ ' (g − g1∗ )e−bt = ce−bt .

Hence g¯ is driven toward g1∗ as t → ∞:
γ g¯(g, t) ≈ γ(g1∗ + ce−bt ) ' γ(g1∗ ) + γ 0 (g1∗ )ce−bt

where ‘prime’ denotes ‘derivative’. Substitution into (17.57) thus yields (∞) < ∞ . Hence Γ(n) scales with anomalous dimension γ(g1∗ ). (b) An example of a relevant IR fixed point at g = 0 is provided by a φ4 theory (see (17.69)): 3g 2 β(g) ' = bg 2 . (17.59) (4π)2

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Hence

∂¯ g = b¯ g2 ∂t

has the solution g¯(g, t) =

g . 1 − bgt

With γ = cg 2 one finds in the IR limit t → −∞, (t = −∞) = −

cg . b

A particularly interesting case is, where (2) (t) does not have a finite limit for |t| → ∞.

An example of immediate interest is provided by QCD,6 where β(g) ' −bg 3 , b > 0 . Hence g∗ = 0 is an UV fixed point. We may then solve (17.54) near the fixed point, i.e. ∂ g¯ = −b¯ g3 ∂t to give   1 1 1 g2 2 t= − . (17.60) → g ¯ (g, t) ' 2b g¯2 g2 1 + 2bt

In order to compute (t) for t → ∞ we again need only the (known) leading term in the perturbative expansion of the anomalous dimension: γ(¯ g (g, t)) ∼ c¯ g (g, t)2 . Noting that γ(g ∗ ) = 0, we have from (17.57), Z t
(t) ∼ dt0 γ g¯(g, t0 ) 0 Z t c 1 2 = ln (1 + 2bg 2 t) ∼ cg dt0 2 t0 1 + 2bg 2b 0

or

e

(t)

c  S 4b ∼ ln 2 . m

We thus obtain in this case logarithmic corrections to the naive asymptotic behaviour:   − 4bc   √ D
S m t Φ {pi }, g∗, √ (17.61) Γ {e p}; g, m → ( S) ln Λ2s S with Λs being some strong interaction scale.

6 D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; H.D. Politzer, Phys. Rev. Lett. 30 (1973) 1346.

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293

Perturbative calculation of β(g) and γ(g) in φ4 theory

Before engaging in an explicit perturbative calculation of the anomalous dimension and beta-function in φ4 -theory, it is convenient to rewrite the normalization conditions in a slightly different form. The wave function renormalization constant and coupling constant renormalization constant are calculated from the knowledge of (2) (4) Γreg (p) (see (20.11)) and Γreg (p), respectively. To second order we have

−iΣ(p2 )reg '

2 iΓ(4) reg (p ) ' −ig0 +

(17.62)

Fig. 17.5. Proper diagrams contributing to β(g) and γ(g). In terms of Σ(p2 )reg the normalization conditions state (compare with (16.29) and (16.31))
m2 − m20 − Σreg (m2 ) = 0   ∂Σreg (p2 ) |p2 =m2 = 1 . (17.63) Zφ 1 − ∂p2 Furthermore (see (20.12) for convention) 1

(Zφ2 )4 Γ(4) reg (0) = −g .

(17.64)

φ4 beta-function The β-function is calculated from (17.64). There exist three topologically distinct 1-loop graphs contributing to the 4-point function in order g 2 : 1

3

1

2

1

2

2

4

3

4

4

3

Fig. 17.6. Topologically distinct diagrams. (Note that the relabelling of the integration variables z1 ↔ z2 is taken care of by the factor 1/2! arising from the expansion of the exponential in the formula (11.21). Consider the sum of (amputated) 1PI diagrams in momentum space: I(p2 ) = 3

(4!) 2!

2



− 2

ig0 4!

2 Z

d4 k i i 2 4 2 2 (2π) [k − m0 − i] [(p − k) − m20 + i]

(17.65)

where the factor (4!) is a combinatorial factor arising from the (4!2 /2) possible 2! contractions with HI = g4!0 φ4 .

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Regularizing a la Pauli–Villars we have for the sum of the three graphs Ireg (p2 ) =

3g02 2

Z

1 d4 k X 1 C` 2 , (2π)4 [k − M`2 + i] [(p − k)2 − m20 + i]

(17.66)

`

where C0 = 1, M0 = m0 C1 = −1, M1 = Λ. Making use of the Feynman parameter representation (15.4), we may write this in the form Z 3g 2 d4 k X C` Ireg (p2 ) = 0 2 (2π)4 ` Z 1 1 × dα  2 . 0 (k − αp)2 + α(α − 1)p2 − (1 − α)M`2 − αm20 + i

Making the change of variable k 0 = k − αp, and performing the Wick rotation k 0 = ik 4 , p0 = ip4 , we obtain (see (15.13)) 3ig02 /2 Ireg (−P ) = (4π)2 2

Z

1

dα

0

X

C`

Z

0

∞

x dx  2 , 2 x + α(1 − α)P + αm20 + (1 − α)M`2

where we have set K 2 = x. Making use of the above expressions for C0 and C1 , we obtain from here Z Z X→∞ 3ig02 /2 1 2 Ireg (−P ) = dα dx x · (4π)2 0 0 ) ( 1 1 ,  2 −  x + α(1 − α)P 2 + αm20 + (1 − α)Λ2 ]2 x + α(1 − α)P 2 + m20 where we have introduced a cutoff X for the x-integration, since each term, taken separately, diverges logarithmically. Noting that Z

0

X

 B X x+B + B x+B 0 X B −→ ln −1+ , B X

x dx = ln (x + B)2



we obtain from above −3ig02 /2 Ireg (−P 2 ) = (4π)2

Z

0

1



dαln 

2

P α(1 − α) m 2 + 1 0

2

2

0

0

P Λ α(1 − α) m 2 + α + (1 − α) m2



 ,

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295

where we have taken the limit X → ∞. Retaining only the terms which do not vanish as we let the cutoff Λ → ∞ go to infinity, we have for the complete second order Minkowski 1-particle irreducible 4-point function, now including also the point-like interaction (see (20.12) for definition of Γ(n) ),     p2 Z 1 2 2 1 − α(1 − α) 2 3g0 /2  m0 2  − ln Λ  . dαln  Γ(4) (17.67) reg (p ) = −g0 − (4π)2 1 − α m20 0 Setting pµ = 0, we have with (17.64) for the renormalized coupling, Z 1  Λ2 3 g02 dαln (1 − α) − ln . g = g0 + 2 (4π)2 0 m20

(17.68)

From here we obtain for the β-function in 1-loop order, β (2) (g) =

3g 2 . (4π)2

In terms of g the renormalized 4-point function takes the form   Z 3g 2 /2 1 p2 (4) 2 Γ (p ) ' −g − dαln 1 − α(1 − α) 2 . (4π)2 0 m

(17.69)

(17.70)

Note that to order g 2 there is no contribution from the wave function renormalization constant. φ4 Anomalous dimension As we have seen in Section 16.2, the calculation of Zφ , and consequently of γφ to order g 2 requires the calculation of the proper self-energy diagram of Fig. 17.7. This 2-loop diagram with superficial degree of divergence ω = 2 involves an overlapping divergence. Following closely the reasoning taking one from the expansion (16.27) to (16.31) in the case of QED, we see that to second order g 2 ,   ∂Σ(q 2 )reg Zφ ≈ 1 + . (17.71) ∂q 2 q 2 =0 From here and (17.35) we then calculate the anomalous dimension γφ (g). Since we are dealing with a diagram with overlapping divergences, we shall use the BPH prescription (16.101) already applied in the context of the diagram of Fig. 16.9. We label the self-energy diagram as shown in Fig. 17.7. The diagram involves two overlapping subgraphs γ1 (upper half) and γ2 (lower half) sharing one common propagator. We have for the full Feynman diagram (we suppress the i prescriptions),  2 Z d 4 p 0 Z d 4 p i i i 2 2 g0 −iΣ(q ) = (4!) 4! (2π)4 (2π)4 p2 − m20 (p + q − p0 )2 − m20 p02 − m20 (17.72) where (4!)2 is a combinatorial factor.

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p

q

q 0

p+q−p p0

Fig. 17.7. φ4 self-energy diagram with overlapping divergence. We shall follow the notation of (16.100) and define J ≡ Σ(q 2 ). According to the BPH prescription we expand the integrand of the upper half (lower half) subgraphs γ1 (γ2 ) in a Taylor series around p0 = q = 0 (p = q = 0), keeping only the first leading term: Z 1 d4 p Tγ1 Jγ1 = g02 (2π)4 (p2 − m20 )2 and Tγ2 Jγ2 =

g02

Z

1 d4 p0 . (2π)4 (p02 − m20 )2

Following the recipe (16.101) we then have Z Z d4 p0 d4 p 1 1 1 Σ(q 2 ) = g02 4 (2π) (2π)4 p2 − m20 (p + q − p0 )2 − m20 p02 − m20 Z Z 1 1 d4 p 0 d4 p − 2 4 02 (2π) (p − m0 ) (2π)4 (p2 − m20 )2  Z Z 1 1 d4 p d4 p0 − . (2π)4 (p2 − m20 ) (2π)4 (p02 − m20 )2 The same reasoning as in the case of Fig. 16.9 shows that the divergences of the subgraphs have been taken care of. These cancellations are realized by the second integral in the second and third lines at fixed p0 and p respectively. The remaining integration implies an overall quadratic divergence, resulting in a second order cutoff dependent polynomial δLφ of the form δZφ (q 2 − m20 ) (compare with (16.17)) from where the wave function renormalization constant can be read off. More concretely: Doing a Wick rotation, continuing q 0 to iQ4 , and making use of the Feynman parameter formula (12.25), we obtain, using dimensional regularization, g02 J − Tγ1 J = i (4π)2 2

Z

d4 P 0 −1 4 02 (2π) P + m20

Z

0

1

dα

ln [α(1 − α)(P 0 − Q)2 + m20 ] m20

where, as expected, the singularity in  = D − 4 was cancelled by the counterterm Tγ1 J , and the limit  → 0 has been taken. Comparing with (17.70) we see that this is nothing but the corresponding figure in Fig. 17.5 with a pair of external legs closed to a loop. Taking the derivative with respect to Qµ , the remaining

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297

counterterm Tγ2 J gives a vanishing contribution and we obtain, g02 ∂Σ(−Q2 ) = −Σ0 (−Q2 )2Qµ = µ ∂Q (4π)2

Z

1 −2α(1 − α)(P 0 − Q)µ d4 P 0 . 2 4 02 (2π) (P + m0 ) [α(1 − α)(P 0 − Q)2 + m20 ]

Using Lorentz covariance, Σ0 (−Q2 ) = −

g02 (4π)4

Z

Λ2

P 02 dP 02

0

(P 02

1 α(1 − α) , 2 + m0 ) [α(1 − α)(P 0 − Q)2 + m20 ]

or setting Q=0, g2 Σ (0) = − 0 4 (4π) 0

Z

Λ2

P 02 dP 02

0

(P 02

α(1 − α) 1 . 2 + m0 ) [α(1 − α)P 02 + m20 ]

From (17.71) we thus obtain for Λ → ∞, Zφ ' 1 −

Λ2 g02 ln 2 + const. 4 (4π) m0

and γφ (g0 ) '

g02 . (4π)4

Our result for the renormalization constant and anomalous dimension differs from the one quoted by Itzykson and Zuber.7 Checking the results Let us check our results. From (17.70) we obtain for the leading asymptotic behaviour for euclidean momentum,  2 3g 2 P Γ(4) (−P 2 ) ' −g − ln . 2 2(4π) m2 Differentiating this expression with respect to the mass and (renormalized) coupling constant we have with (17.69), m or

∂ (4) 3g 2 ∂ 3g 2 Γ (−P 2 ) ' , β(g) Γ(4) (−P 2 ) ' − , γφ (g)Γ(4) = 0(g 3 ) 2 ∂m (4π) ∂g (4π)2   ∂ ∂ 2 3 m + β(g) − γφ (g) Γ(4) as (−P ) = O(g ) . ∂m ∂g

Since the β-function (17.69) is positive, g = 0 is an IR fixed point, and we cannot draw any conclusion concerning the high-energy behaviour of the φ4 -theory. Notice that this holds for all Green functions. The same is true for QED as we show next. 7 C.

Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, 1980), p. 652.

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QED β-function and anomalous dimension

The success of QED at low energies (Lambshift, etc.) can be understood in terms of the behaviour of the β-function near zero coupling and the smallness of the electromagnetic coupling itself (α = 1/137). Let us examine therefore the behaviour of the QED β-function near zero coupling. The β-function and anomalous dimension of QED requires the computation of     ∂ ∂ ln Ze = −α0 Λ ln Ze β(α) = α0 m ∂m ∂Λ m0 ,α0 ,Λ m0 ,α0 ,Λ and 1 γA (α) = 2



∂ m ln ZA ∂m



m0 ,α0 ,Λ

=−



Λ ∂ ln ZA 2 ∂Λ



.

m0 ,α0 ,Λ

Because of the QED-Ward identity (16.61), Z1 = Zψ . Hence Ze = ZA , or we have to arbitrary order β(α) = 2γA (α). (17.73) From the result (15.30) for the dimensional regularization  Z  m2 − β(1 − β)k 2  µ2 2 α0 − γ − ln π + ln 2 +2 dβ β(1−β)ln  m π m2 (17.74) we have from (16.38) to order O(α2 ), dim 2 πreg (k )

α0 '− 3π



dim ZA



  α 2 µ2 µ  − γ − ln π + ln 2 α, ,  ' 1 − m 3π  m

(17.75)

and from the result (15.28) for the Pauli–Villars regularization PV πreg (k 2 ) ' −

α0 ln 3π



Λ2 2m2



+

2α0 π

Z

0

1

dββ(1 − β)ln



m2 − β(1 − β)k 2 m2



, (17.76)

we have correspondingly, PV ZA

  2   α Λ Λ '1− ln . α, m 3π 2m2

(17.77)

Hence we obtain for both regularizations β(α) '

2α2 , 3π

γA (α) '

α . 3π

(17.78)

Notice that in lowest order of perturbation theory, β(α) is positive. Hence unlike QCD, α = 0 is not a “relevant UV fix-point” for QED, but rather an infrared (IR) fix-point. This explains the success of QED at low energies.

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299

QED β-function and leading log summation

The asymptotic behaviour (17.56) of a general proper function shows that it deviates, in general, from that expected from dimensional analysis. Let us examine in some detail what this precisely means from the point of view of perturbation theory. We shall illustrate the ideas for QED. Consider the renormalized photon 2-point function in the Lorentz gauge: DFµν (k; α, m) =

k2

where P µν is the usual projector

−P µν (k)
, 1 − πren (k 2 ; α, m)

P µν (k) = g µν −

kµ kν . k2

The corresponding 1PI function, that is, 1 − πren (k 2 ) of (16.75) satisfies asymptotically the Callan–Symanzik equation   ∂ ∂ + β(α) − 2γA (α) (1 − πren (k 2 )) ≈ 0 . (17.79) m ∂m ∂α Let us check this in 1-loop order. From (16.75) we have for π(k 2 )ren , and K euclidean,  2  Z 2α 1 m + β(1 − β)K 2 dββ(1 − β)ln π 0 m2  2 α K ' ln . 3π m2

(1−loop) πren (−K 2 ) =

Recalling that 2α β(α) = 2γA (α) = , α 3π

(17.80)

and performing the differentiations, one verifies that (17.79) is satisfied asymptotically to order O(α2 ). We now take the converse point of view. Instead of computing the beta-function from the coupling constant renormalization constant via (17.34), we can compute it from the knowledge of the asymptotic behaviour of π(k 2 ) using the asymptotic CS equation (17.79). To this end it is useful to rewrite this equation in terms of the variable introduced in (17.47), and using the Ward identity (17.80): 

−

∂ ∂ + β(α) ∂t ∂α



π(e2t k 2 ) β(α) = . α α2

(17.81)

The merit of writing the CS equation in this form is that it no longer involves the anomalous dimension. We can thus use the knowledge of an asymptotic expansion of

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π(e2t k 2 )ren in powers of ln t (leading log summation) in order to obtain conversely a perturbative expansion for the β-function. Introducing the ansatz X π(e2t k 2 ) = αn (an + bn t + cn t2 + dn t3 + · · ·) n=1

β(α) = α2

X

(β1 + β2 α + β3 α2 + · · ·)

n=1

and equating separately terms in powers of α and t = ln u, with u =   1 2t 2 π(e k ) = α(b1 ln u + a1 ) + O ln u u

k2 m2 ,

we find

+α2 (b2 ln u + a2 )

+α3 (c3 ln 2 u + b3 ln u + a3 ) +α4 (d4 ln 3 u + c4 ln 2 u + b4 ln u + a4 ) +··· with the coefficients constrained by the relations c1 = d1 = · · · = 0 c2 = d2 = · · · = 0 2c3 + b1 b2 = 0,

d3 = 0,

etc.

and β1 = −b1

β2 = −b2

β3 = −b3 − b1 a2 ,

etc.

This result is in agreement with Weinberg’s theorem. Notice that in order αn there are no higher terms than ln n−1 u. By performing a 3-loop calculation of π(k)ren , de Raf¨ ael and Rosner obtained in this way8 the following result for the QED-β function up to fourth order in α: β(α) =

α3 121α4 2α2 + 2− + O(α5 ) . 3π 2π 144π 3

(17.82)

The result is represented in Fig. 17.8 below.9 It is interesting that there has been discussion in the literature about the uniqueness of the beta function. Noting that the renormalized proper functions are unique under renormalization, the above procedure should also guarantee the uniqueness of the beta function. Notice that this beta-function is “universal” in the sense, that it applies to all Green functions of the theory. 8 Eduardo

de Raf¨ ael and Jonathan L. Rosner, Ann. of Phys. 82(2), (1974) pp. 301–608. third term in this expression disagrees in sign with the one quoted in N.N. Bogoliubov and N.N. Shirkov, Introduction to the Theory of Quantized Fields, (1959). 9 The

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17.10 Infrared fix point of QED and screening of charge

301

Fig. 17.8. E. de Raf¨ael and J.L. Rosner.

17.10

Infrared fix point of QED and screening of charge

We have seen that the β-function of QED in 1-loop approximation is given by β(α) =

e2 2α2 + 0(α3 ) , α = . 3π 4π

Hence QED is not asymptotically free, and as a consequence its behaviour in the deep euclidean region is unknown. Nevertheless, QED has been remarkably successful in the low energy region. This success, usually attributed to the smallness of the fine structure constant (α = 1/137) is at least consistent with the fact that α = 0 is an infrared (IR) fixed point of QED. Let us therefore assume that the value α = 1/137 lies to the left of the first non-trivial zero of the QED β-function. In view of its smallness the running coupling constant may be taken to satisfy the equation 2¯ α2 ∂α ¯ ' . ∂t 3π Separating variables, integration of this equation yields α ¯ (α, t) Z

0

t

0

dt =

Z

α(α,t) ¯

α

3π 1 dα 2 02 = − 2 α 3π 0

or α ¯ (α, t) '

α . 1 − 2α 3π t



1 1 − α ¯ α



,

(17.83)

This is the case considered in (17.60). The pole on the r.h.s. is referred to as the “Landau pole”. The IR limit corresponds to t → −∞. The result shows that significant deviations from α = 1/137 will occur only for (t is negative!) “microscopic” √ energy values of S: √  3π  S = O e− 2α . me Rigorously speaking, this result applies only to the deep infrared region. From a phenomenological point of view one may nevertheless think of the running coupling constant (17.83) as replacing α in Thomson-scattering at (very) small momentum transfer q 2 , with S replaced by q 2 . Note however that in this picture we are actually

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far away from the non-exceptional region since the external momenta are restricted in this case to the electron mass shell p2i = m2e . Since a small momentum transfer corresponds in Thompson scattering to a large separation R of the charges, we may equally well think of the running coupling constant as a function of R. From the phenomenological point of view we could thus be led to replace the classical Coulomb potential by V (R) =

α ¯ (R) = R R(1 +

α 2α 3π ln (me R))

.

Note that expansion of the denominator in powers of α yields ! α 2α V (R) ' 1− ln (me R) , R 3π

(17.84)

which is just the famous Uhlenbeck formula. The potential (17.84) can be viewed as expressing the effect of screening of the 2-point charges by virtual electron-positron pairs. The creation of such pairs should set in at separations R of the order of the inverse electron mass, as brought also in evidence by the argument of the logarithmic factor. Notice that this “vacuum polarization” leads to an enhancement (reduction) of the effective charge inside (outside) the radius of the electrons.

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Chapter 18

Renormalization Group In Section 1 of Chapter 16 we imposed suitable conditions in order to define the wave function renormalization constant, as well as the renormalized mass and coupling as a function of the cutoff and the bare parameters. We have done this without introducing further dimensioned parameters into the theory. This is an option we are free to take, but need not be the most adequate one from the point of view of perturbation theory. Thus from the point of view of the convergence of perturbation theory it could turn out to be advantageous to define the coupling constant in φ4 -theory, instead of (17.64), by Γ(4) (p · · · p ) = −¯ g (µ) , (18.1) 1 4 ren 1 2 pi ·pj =−µ (δij − 4 )

where µ is a new parameter with the dimensions of a mass, and Γ(n) (p1 , · · ·) stands for the 1PI renormalized proper n-point Green function. Similarly, we could choose for the mass parameter a value not being identical with the physical mass associated with the field φ(x). To be concrete let us take the coupling constant parameter to be parametrized by µ.1 The physical consequences should of course be independent of how we choose our parametrization. This means that the vertex functions computed for different choices of parametrizations must be related in a definite way. The Renormalization Group Equation (RGE) expresses in differential form this relationship as a function of µ.2

18.1

The Renormalization Group equation

The renormalization group equation expressing the independence of the renormalized 1PI functions of the normalization point µ is a homogeneous equation taking the form   ∂ µ ¯ g , µ ) ∂ − γ¯ (¯ µ + β(¯ g , ) Γ(n) (p; g¯, m, µ) = 0 (18.2) ∂µ m ∂¯ g m 1 We

shall not parametrize the mass to simplify the discussion. Stueckelberg and A. Peterman, Helv. Phys. Acta 26 (1953) 499; M. Gell-Mann and F. Low, Phys. Rev. 95 (1954), 1300; N.N. Bogoliubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience, New York, 1959). 2 E.C.G.

303

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304

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Renormalization Group

reminiscent of the asymptotic Callan–Symanzik equation (17.43), where the nomenclature is the same except for the overhead “bar” distinguishing the beta-function and anomalous-dimension from those of the CS equation. We begin by considering some examples. Example 1: Coupling constant in φ4 -theory The coupling constant g need not be defined at p2 = 0 as in (17.64), but can be chosen to be defined at some euclidean point p2 = −µ2 . From (17.70) we have for the renormalized 4-point 1PI function, Z  p2  3g 2 /2 1 (4) 2 dβln 1 − β(1 − β) + O(g 4 ) (18.3) Γ (p ) = −g − (4π)2 0 m2 where g is the usual renormalized coupling constant, normalized at zero momentum. Defining a new coupling constant by g¯(µ) = −Γ(4) (−µ2 ) we have g¯(µ) ' g +

3g 2 /2 (4π)2

Z

0

1

dβln



1 + β(1 − β)

µ2 m2



+ O(g 4 ) .

(18.4)

In terms of this parametrized coupling constant the renormalized 4-point function (18.3) now takes the form  2  Z 1 m − β(1 − β)p2 3¯ g2 (4) 2 ¯ dβln Γ (p ; g¯, m, µ) = −¯ g− + O(¯ g3 ) 2(4π)2 0 m2 + β(1 − β)µ2 where the possibility of inverting the relation between g and g¯ has been assumed in the second term on the right. Viewed as a function of g¯(µ) and µ, the independence of µ of (18.3) is thus expressed by the differential equation   d ¯ (4) ∂ µ ∂ ¯ (4) 2 ¯ µ Γ (p; g¯, m, µ) = µ + β(g, ) Γ (p ; g¯, m, µ) + O(¯ g3 ) , (18.5) dµ ∂µ m ∂¯ g where β¯ is the function3  µ ∂  µ β¯ g, ≡ µ g¯ g, , m ∂µ m

(18.6)

and we have made explicit the dependence of g¯ on g, m and µ. Notice that here the right-hand side defines the left-hand side. Instead of considering the coupling g = g¯(0) and g¯ = g¯(µ) it is useful to consider them at neighbouring points µ1 and µ2 . According to (18.4) we then have  2  Z 1 3¯ g12 m + β(1 − β)µ22 g¯2 = g¯1 + dβln + O(¯ g13 ) . (18.7) 32π 2 0 m2 + β(1 − β)µ21 3 This β ¯ function is not to be confused with the β function of the Callan–Symanzik equation, except for the observation made later on.

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page 305

305

Notice that in going from (18.4) to this expression we have to order O(¯ g13 ) make the replacement g → g¯1 in the factor multiplying the integral. This is important in the ¯ (4) (p2 ; g2 , m, µ2 ) of µ2 then requires4 sequel. Independence of Γ     µ2 µ1 d ¯ (4) 2 Γ p ; g¯2 g¯1 , , , m, µ2 = 0 . µ2 dµ2 µ1 m Performing the differentiation and setting µ2 = µ1 , we have   µ  ∂  ∂ 1 ¯ ¯ (4) (p2 ; g¯1 , m, µ1 ) = 0 , µ1 + β g¯1 , Γ ∂µ1 m ∂¯ g1 where

   µ   ∂ µ2 µ1 1 = µ2 β¯ g¯1 , g¯2 g¯1 , , m ∂µ2 µ1 m µ2 =µ1

(18.8)

is referred to as the “Beta function”. Explicitly we have to order O(¯ g 3 ) (we now set µ1 = µ, g¯1 = g¯) Z  µ  3¯ g 2 /2 1 2β(1 − β)µ2 = dβ 2 + O(¯ g3 ) . (18.9) β¯ g¯, 2 m (4π) 0 m + β(β − 1)µ2 Notice that for µ → ∞, this β-function tends to

g2 ¯ g , µ ) → 3¯ β(¯ , m (4π)2

(18.10)

which is the CS value (17.69). Observe also that the limit µ → ∞ can also be seen as the limit m → 0. So far, Eq. (18.6) merely defines the “Beta” function as a function of the parameter µ. We now define a “running coupling constant” as the solution to the differential equation  µ d µ g¯ = β¯ g¯, . (18.11) dµ m It is interesting to compare the solution for this “running” coupling constant with the “parametrized” coupling constant (18.4). For the latter we have for large µ,   µ  3g g¯(µ) → g 1 + ln , (4π)2 m

whereas with (18.10), Eq. (18.11) asymptotically solves the differential equation µ

d¯ g 3¯ g2 = dµ (4π)2

with the solution analogous to (17.83), g¯(µ) ' 4 S.

g 1−

3g (4π)2 ln

µ M


,

Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1996).

(18.12)

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where M is some large scale at which g¯ = g is defined. We have in the present case an IR fix-point at g¯ = 0. From above, we see that this amounts to significant 2 deviations from g only for µ of the order µ ≈ M exp( (4π) 3g ). Example 2 Recall the calculation of the QED renormalization constant ZA from (16.38). The Bogoliubov recursive renormalization and the Taylor subtraction procedure of Chapter 16 make no reference to the “normalization conditions” to be imposed on the 2and 3-point functions. Similarly, a “minimal subtraction” procedure in dimensional regularization consists of simply subtracting the divergent “1/” term in (16.70) also making no such reference. By the same token we could have chosen to expand πreg (k 2 ) around k 2 = −µ2 , instead of the photon mass k 2 = 0 in (16.36): π(k 2 )reg = π(−µ2 )reg + π 0 (k 2 ; µ) .

(18.13)

We then have for the (correctly normalized, transversal) photon 2-point function, instead of (16.37), ˜ µν (k) = D F

−1 µν −ZA P (k) , k 2 (1 − π(−µ2 )reg ) − k 2 π 0 (k 2 ; µ)

where ZA is given by (16.38): ZA =

1 . 1 − π(0)reg

(18.14)

1 (1 − π(−µ2 )reg )

(18.15)

˜ µν (k) is independent of µ. Define D F ZA (µ) = and

ZA (µ) 1 − π(0)reg Z¯A (µ) = = . ZA 1 − π(−µ2 )reg

(18.16)

α ¯ (µ) = Z¯A (µ)α = ZA (µ)α0 .

(18.17)

Then Z¯A (µ) is a finite, cutoff independent constant for Λ → ∞, and

Notice that we have used the Ward identity Zα = ZA . This leaves us with the µ independent expression DFµν (k)ren =

−Z¯A (µ)P µν (k) k 2 (1 − π(k 2 ; α ¯ , µ)ren )

(18.18)

with −1 π(k 2 ; α ¯ , µ)ren = ZA (µ)π 0 (k 2 , ZA (µ)¯ α(µ), m − δm, µ) .

(18.19)

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307

By construction this propagator is correctly normalized at k 2 = 0. Hence we also have α ¯ (µ) = (1 − π(0; α ¯ , µ)ren ) . Z¯A (µ) = α Equation (18.18) is a special case of the statement

k 2 (1

Z¯A (µ1 )P µν (k) Z¯A (µ2 )P µν (k) = 2 . 2 − π(k ; α ¯ 1 , µ1 )ren ) k (1 − π(k 2 ; α ¯ 2 , µ2 )ren )

(18.20)

The above example shows that a change in the subtraction procedure requires a change in the multiplicative renormalization constant in order to keep the physics ¯ unchanged. This multiplicative constant Z(µ) is cutoff independent, but depends on µ. Since the lhs in (18.18) is independent of µ we have again a differential equation expressing this independence of µ of the corresponding 1PI function. The requirement   d 1 − π(k; α ¯ , µ)ren =0 (18.21) dµ Z¯A (µ) implies   µ ∂  µ  ∂ ¯ +β α ¯, − 2¯ γA α (1 − π(k; α ¯ , µ)ren ) = 0 ¯, µ ∂µ m ∂α ¯ m

(18.22)

with the running coupling constant satisfying the differential equation, µ

 µ d α ¯ = β¯ α ¯, dµ m

(18.23)

µ and γ¯A (¯ α, m ) defined in an analogous way to (18.8) by

  µ µ d µ ln Z¯A α γ¯A α ¯, = ¯ (µ), . m 2 dµ m

(18.24)

Notice that (18.24) involves a total derivative. This equation may be formally integrated to give   Z µ 0 0 1 dµ 0 µ ln Z¯A2 (µ) = γ ¯ α ¯ (µ ), . A µ0 m 0 Equation (18.22) is the renormalization group equation for the vacuum polarization in QED. Explicit expressions to order O(¯ α2 ) From (18.14), (18.15) and (17.76) we have ZA (µ) = ZA +

2¯ α π

Z

0

1

dββ(1 − β)ln



1 + β(1 − β)

µ2 m2



+ O(¯ α2 ) .

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From here, (18.16) and (18.17) follow, to get,   Z µ2 2α ¯ 1 dββ(1 − β)ln 1 + β(1 − β) 2 + O(¯ α2 ) , Z¯A (µ) = 1 + π 0 m   Z 2α2 1 µ2 α ¯ (µ) ' α + dββ(1 − β)ln 1 + β(1 − β) 2 . π 0 m And from (18.23) and (18.24) we further have,
Z µ µ2  µ β¯ α ¯, m [β(1 − β]2 m α ¯ 1 2 =4 dβ + O(¯ α2 ) = 2¯ γ α ¯, µ2 α ¯ m π 0 1 + β(1 − β) 2

(18.25)

m

and from (18.13) and (18.19) finally follows,   2 Z 2α ¯ 1 m − β(1 − β)k 2 2 + O(¯ α2 ) . π(k ; α ¯ , m, µ)ren = dββ(1 − β)ln π 0 m2 + β(1 − β)µ2 Note that the Ward-identity relation (17.80) is preserved. Notice also that, analogous to (18.10) in the φ3 case, we have in the µ → ∞ (or zero mass) limit  µ 2¯ α2 , β¯ α ¯, → m 3π

(18.26)

which is the CS value (17.78), and

α ˜ (µ) ≈

α(M ) 1−

2α(M ) µ 3π ln ( M )

,

(18.27)

where the singularity is referred to as Landau singularity. The increase of α(µ) with µ (decrease with distance) has its analogue in electrostatics in a dielectric medium, where the presence of a charge polarizes the medium, such as to screen the charge. Example 3 Consider the photon propagator (18.18). Using (18.17) we have µν αDren (k 2 ) = α ¯ (µ)

−P µν (k) . ¯ , m, µ)ren ] k 2 [1 − π(k 2 ; α

Since the lhs does not depend on µ, the right-hand side does not depend on µ either; nor does it involve a multiplicative wave function renormalization constant in the numerator. We thus have the RG equation without anomalous dimension, or    µ  ∂   1 − π(k 2 ; α ∂ ¯ , m, µ)ren µ + β¯ α ¯, =0. (18.28) ∂µ m ∂α ¯ α ¯ Notice that since the Ward identity Zα = ZA , this is nothing but Eq. (18.21). Equation (18.28) is a convenient starting point for the calculation of the beta function (17.82) of de Raf¨ ael and Rosner.

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18.2 Asymptotic solution of RG equation

18.2

309

Asymptotic solution of RG equation

Let Γ(n) (p; g, m) be a conventionally renormalized 1PI n-point function in φ4 theory. As we have seen, normalizing the coupling constant at a different point µ, will require for a proper normalization of the vertex function the introduction of a finite renormalization constant Z¯φ (µ), implying the appearance of a term γ¯ (¯ g , µ) in the RG equation. The generalization of (18.20) to an arbitrary proper function reads n

− ˜ ¯ Z¯φ 2 (µ1 )Γ

(n)

(n) −n ¯˜ ({pi }; α ({pi }; α ¯ 1 , m, µ1 ) = Z¯φ 2 (µ2 )Γ ¯ 2 , m, µ2 ).

(18.29)

In different terms: For the proper functions, Eq. (16.6) reads in compact notation in the present case,     Λ Λ ˜ (n) Γ ...; e¯, µ, = Γ(n) Z −n e¯, µ, reg (...; e0 , m0 , Λ) m m

or

d µ dµ



Z

−n

    Λ Λ ˜ (n) Γ ...; e¯, µ, e¯, µ, =0. m m

From here, we arrive at   ∂ ∂ ¯ ¯ (n) ({pi }; g¯, m, µ) = 0 , µ +β − n¯ γ Γ ∂µ ∂¯ g

(18.30)

(18.31)

with β¯ and γ¯ defined as in (18.8) and (18.24), respectively. ¯ (n) (pi ; g¯, m, µ), but with the moConsider now the renormalized 1PI function Γ menta scaled by a factor et . It satisfies the RG equation (18.31) with pi replaced by et pi .   
∂ ∂ ¯ µ +β − n¯ γ Γ(n) {et pi }; g¯, m, µ = 0 . ∂µ ∂¯ g We now follow arguments as already used in Chapter 17. By dimensional analysis we have

¯ (n) {et pi }; g¯, m, µ = eDt Γ ¯ (n) {pi }; g¯, e−t m, e−t µ Γ

or

∂ ¯ t Γ({e p}, g¯, m, µ) = ∂t



−µ

 ∂ ∂ ¯ t pi }, g¯, m, µ) . −m + D Γ({e ∂µ ∂m

We use this equation in order to exchange the µ-differentiation for the t-differentiation in the RG-equation, which now reads:
DΓ(n) {et p}; g¯, m, µ = 0 ,

where

D=



∂ ∂ ∂ − −m + β¯ + D − n¯ γ ∂t ∂m ∂¯ g



.

Correspondingly we have   µ ∂  µ  ∂ ∂ ¯ ¯ t pi }; g¯, m, µ) = 0 . − −m + β g¯, + D − n¯ γ g¯, Γ({e ∂t ∂m m ∂¯ g m

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Consider now this equation in the limit µ → ∞, g¯ fixed. This limit is equivalent to m → 0. From (18.9) and (18.25) it is interesting to observe that in our examples of ¯ φ4 -theory and QED, the second order β-function approached respectively their CS value in the µ → ∞ limit. Indeed, from (18.26) we have for QED,   µ(t) 2¯ α2 β¯ α ¯, → βCS (α ¯) = + O(¯ α3 ) m 3π   µ(t) α ¯ γ¯ α ¯, → γCS (¯ α) = + O(¯ α3 ) m 3π and from (18.10) for φ4 -theory,   3¯ g2 µ(t) + O(¯ g3 ) . → β¯CS (¯ g) = β g¯, m (4π)4 This could be expected to be generally true since the µ → ∞ limit corresponds, by dimensional analysis, to the m → 0 limit. This would reconcile the generally valid homogeneous RG equation with the asymptotically valid homogeneous CS equation. µ From the parallel treatment leading to (17.56) we infer for t → ∞ and m → ∞,


− R t dt0 γ g¯(g,t0 ) Γ {et pi }; g, m → eDt Φ {pi }; g¯(g, t), e−t m e 0 ,

where g¯(g, t) is the running coupling constant satisfying (17.54).

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Chapter 19

Spontaneous Symmetry Breaking The so-called spontaneous breaking of a continuous symmetry plays a central role in systems with an infinite number of degrees of freedom such as condensed matter, where it is responsible for phase transitions, as well as in QFT, where it is known as the Higgs mechanism in the Weak Interactions. A fundamental theorem due to Goldstone plays here a central role.

19.1

The basic idea

The perturbative expansion as given by (11.21) implicitly contains the assumption of a vanishing vacuum expectation value of the fields in question. In the case of fermionic fields this property is guaranteed by the Lorenz invariance of the vacuum. In the case of the scalar field, this is not sufficient. There may exist, however, further symmetries which may be invoked to guarantee a vanishing expectation value h0|φ(x)|0i = 0 in this case. Thus consider for instance the Lagrangian of a real scalar field, 1 L = (∂µ φ)2 − Vef f (φ) 2 with the “effective potential” Vef f (φ) =

1 2 2 µ φ + gφ4 2

(19.1)

as represented in Fig. 19.1. We see that the potential (as well as the lagrangian) is symmetric under the global and discrete transformations φ(x) → −φ(x) . Let U be the unitary operator inducing this transformation Uφ(x)U −1 = −φ(x) . 311

(19.2)

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Fig. 19.1. Vef f = 21 µ2 φ2 + gφ4 for µ = g = 1. If the ground state is invariant under this transformation, then U|Ωi = |Ωi , and hence, hΩ|φ(x)|Ωi = hΩ|U † φ(x)U|Ωi = −hΩ|φ(x)|Ωi , implying the vanishing of the vacuum expectation value of the field.

Fig. 19.2. Vef f = − 12 µ2 φ2 + gφ4 for µ = g = 1. If we however change the sign of the mass term in L, then the Vef f (φ) will become of the form of Fig. 19.2. Though the potential still respects the symmetry under the transformation (19.2), the now two-fold degenerate ground state no longer does, corresponding to the non-vanishing vacuum expectation values hΩ|φ(x)|Ωi = ±a where p to zeroth order of ¯h, the parameter a represents the location of the minima ± µ2 /2g of Vef f . One says that the symmetry (19.2) has been spontaneously broken. A negative mass term in the lagrangian would correspond to a “tachyon”. In order to avoid it, we must now expand φ(x) about one of the potential minima. With respect to the new field, the resulting mass term in the effective lagrangian √ will have the correct sign to represent a physical particle of mass 2µ. On quantum level spontaneous symmetry breakdown may occur through quantum fluctuations which may turn the effective potential at tree (¯h = 0) level with a non-degenerate vacuum, into an effective potential with degenerate vacua. Effective

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313

potentials to 1-loop order and spontaneous symmetry breakdown will be the subject of the following chapter. At this point, it should be emphasized that we have considered an example exhibiting SSB of a discrete symmetry. A very interesting new phenomenon occurs if we have a SSB of a continuous symmetry. A well-known example is provided by the Heisenberg ferromagnet, to be discussed in the next section. If we have the temperature as an additional parameter, then a symmetry preserving effective potential may turn into a symmetry breaking potential as we tune the temperature past some critical value. In this case one speaks of a phase transition. In the Heisenberg ferromagnet this transition is accompanied by the appearance of socalled “spin-waves”, zero mass excitations, the analogon of “Goldstone bosons”, to be discussed in Section 3.

19.2

More about spontaneous symmetry breaking

A symmetry of the Hamiltonian is spontaneously broken, if the ground state of the system does not exhibit this symmetry. There exist many examples of spontaneous symmetry breakdown (SSB) in nature. Our interest will now concentrate on the spontaneous breakdown of continuous symmetries. In classical physics this turns out to imply the existence of a continuum of degenerate ground states. The bending of the Euler stick provides an example: Exerting a force on one end of the stick, directed along it, the stick will suddenly bend, once the force exceeds a certain critical value when subject to a small perturbation. The new configuration now represents the state of lowest energy. The direction in which the stick will bend is, however, chosen “spontaneously”. All these directions are energetically equivalent and may be rotated into each other without the expenditure of energy. On the other hand, in quantum mechanics with a finite number of degrees of freedom the ground state turns out to be unique due to tunneling, and there exists no SSB. In order to witness SSB we must turn to quantum mechanical systems with an infinite number of degrees of freedom. Ferromagnetism provides here an instructive example. The ground state of a ferromagnet is characterized by the fact, that there ex~ also in the absence of an externally applied ists a non-vanishing magnetization M magnetic field. This is, however, the case only for temperatures below a critical temperature Tc , called the Curie-temperature, i.e. ~ = 0 , T > Tc ; M

~ 6= 0 , T < Tc . M

The macroscopic state of this ferromagnet is described by the free energy F [T, V ; M], which is a function of the temperature and volume, and a functional of the magnetization M(~r). It is convenient to define the corresponding free energy density F by Z ~ ~ r), T ). F [T, V, M] = d3 rF(M(~ V

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We now assume that near the critical temperature, the ferromagnet exhibits a weak ~ and nearly constant magnetization M(r). Assuming rotational invariance of the free energy density (though not of the ground state) one is led to make the Ansatz ~ · M(~ ~ r) F(~(~r), T ) = ∇


2


~ r)|2 + b(T ) M(~ ~ r) · M(~ ~ r) 2 + a(T )|M(~

(19.3)

where the coefficients a and b are allowed to depend on the temperature. It turns out that this expansion in powers of the magnetization provides a good qualitative description near the Curie temperature, provided we attribute the following properties to the coefficients: b(T ) > 0 , T ' Tc

a(T ) > 0 , T > Tc a(T ) < 0 , T < Tc

In order to determine the properties of the ground state, we need to minimize ~ ·M ~ )2 is a the free energy (19.3) with respect to the magnetization. Since (∇ semi-positive quantity, this minimum will correspond to a constant magnetization. Under this condition, we have the two possible situations, depending on whether the temperature lies above or below the critical one: For T < Tc we have the free ~ representing a cut of a three-dimensional energy depicted below with ρ = |M|, picture obtained by rotating the figure about the vertical axis.

q Fig. 19.3. Vef f with minima at ρ = ± −a 2b . The figure thus actually represents the bottom of a bottle of champagne (Mexican hat), and hence describes a ground state in which the magnetization can take any one of the orientations related by rotations about the vertical axis. They all correspond to the same energy, and hence we have an infinite set of degenerate ground states associated with this SSB. Are these ground states all unitarily equivalent, i.e. does there exist a unitary operator turning one vacuum state into another? The spontaneous magnetization of a ferromagnet corresponds to a state in which the spin-components at each of the lattice sites do not add to zero when averaged over a infinitesimal, though macroscopic region of space. Taking the volume V of the ferromagnet for the time being to be finite, we imagine a situation of maximal magnetization, in which all of the spins sj are oriented along a given axis, which we take to be the z-axis.

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315

↑s1 ↑s2 · · · ↑si · · · ↑sj · · · ↑sN For the corresponding Hamiltonian we take the (rotationally invariant) expression H=−

X

i,jV

~i · S ~j Jij S

corresponding to an (isotropic) Heisenberg ferromagnet. The fact that there exists a spontaneous magnetization means in quantum mechanical terms that the ground state expectation value of the spin-operator at the ith site does not vanish: ~i (t)|ΩiV 6= 0 , hΩ|S where V denotes the volume of the sample. Of course, this can only be the case if the ground state does not respect the rotational symmetry, since otherwise the expectation value would vanish according to the Wigner–Eckart theorem. Now consider the operator X ~S ~ ¯ = eiθ· ~= ~i . U(θ) , S S iV

ˆ This operator induces a global rotation of the spins by an angle θ about the θ-axis: U(θ)|ΩiV = |Ωθ iV ,

~ V 6= 0 . S|Ωi

Nevertheless ¯ H] = 0 , [U(θ), that is, no energy needs to be expended in order to arrive at the new state. An alternative view will be helpful: We need not turn all spins at once; thus we could begin by first turning just N/2 of the spins through an angle; this will of course require the input of energy. This amount of energy will be recovered, once we subsequently turn the remaining N/2 spins. ↑↑ · · · ↑ %% · · · % | {z } | {z } N/2

N/2

For N finite, the different ground states thus obtained are unitarily equivalent. Thus choose the rotation to be about the z-axis. We then have for the ith spin, ~ ~ (1) hΩ0 |eiθ·Si |Ω0 iV = d 1 21 (θ) 2 2

and for the total number of N spins, N N   1 θ ¯~ ( ) hΩ0 |Ωθ iV = hΩ0 |eiθ·S |Ω0 iV = d 1 21 (θ) . = cos 2 2 2

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That is, for a ferromagnet with a finite number of sites, the old and new ground state have a non-vanishing projection. As this number, however, tends to infinity, we find in the limit N → ∞, hΩ0 |Ωθ iV →∞ = 0 ,

0 < θ < 2π .

(19.4)

Hence, in the limit of an infinite number of degrees of freedom, the different ground states are orthogonal to each other, so that they define unitarily inequivalent Hilbert spaces (sectors) of the theory. Physically this reflects the fact that the attempt to flip the spins will require an infinite amount of energy, no matter how we try to realize it. We conclude that the symmetry of the Hamiltonian under joint rotation of the spins by an angle θ has been broken, as witnessed by the appearance of a continuous set of inequivalent ground states. This is only possible for quantum mechanical systems with infinite degrees of freedom.

19.3

The Goldstone Theorem

The above observation has dramatic consequences, which are the content of the following theorem:1 Goldstone Theorem: Given causality, Poincar´e invariance and a translationally invariant ground-state with positive norm, the spontaneous breakdown of a continuous symmetry implies the existence of zero modes with the property ω(k) → 0 as k → 0, the number of such zero modes being equal to the dimension of the group associated with the broken symmetry. For the ferromagnet, these zero modes are the well-known spin-waves. In the context of QFT, such excitations would correspond to zero mass particles. Since up to date the photon and neutrinos are the only observed zero mass particles in nature, the theorem seems to imply at first sight the absence of SSB in particle physics. This is not quite true, as we shall indicate later on. Let us prove the above theorem. Starting point is the assumption that there exist operators Qa such that U −1 HU = H, and

~ ~

U = eiθ·Q

~ ~

In different terms, Since however

eiθ·Q |Ωi = |Ωiθ 6= |Ωi . Qa |Ωi = 6 0. [H, Qa ] = 0 ,

this implies a continuous degeneracy of the ground states. 1 J.

Goldstone, Nuovo Cimento 19 (1961) 154.

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317

Now, according to Noether’s theorem there exists a conserved current j µ,a for every generator Qa of a symmetry of a lagrangian, Z a Q (t) = d3 rj 0,a (~r, t) (19.5) where ∂µ j µ,a (x) = 0 . We now show that2 for a translationally invariant ground state the generator defined by (19.5) cannot exist, if the symmetry is spontaneously broken. To this end, we consider in particular the norm Z ||Qa (t)|Ωi|| = hΩ|Qa (t)Qa (t)|Ωi = d3 rhΩ|j0a (~r, t)Qa (t)|Ωi . Because of the translational invariance of the vacuum, we can translate away the space and time-dependence, thus obtaining Z a a a ¯ a hΩ|Q (t)Q (t)|Ωi = hΩ|j0 (0, 0)Q (0)|Ωi d3 r . V →∞

a

If the generator Q does not annihilate the vacuum, then Qa |Ωi is a state of infinite norm, and hence does not lie in the Hilbert space of the theory. This is a restatement of our observation, that the ground states in (19.4) are orthogonal to each other, and hence inequivalent. In order to gain control of the situation, we define the operator Z QaR (t) = d3 rj0a (~r, t) . |~ r |≤R

Because of the assumed locality (microscopic causality) we nevertheless have that for the commutator [QaR (t), A(0)], with A(x) some (non-singlet) operator local with respect to the Noether currents, that a non-vanishing limit R → ∞ exists: lim [QaR (t), A(t, ~r)] = B a (t, ~r) .

R→∞

(19.6)

From the conservation of the current it further follows that the right-hand side in (19.6) will not depend on the time t in the limit R → ∞. We now take the vacuum expectation value of both sides of this equation with R → ∞: hΩ|Qa (t)A(t, ~r)|Ωi − hΩ|A(t, ~r)Qa (t)|Ωi = hΩ|B a (t, ~r)|Ωi .

(19.7)

Since, by assumption, Qa does not annihilate the vacuum, the right-hand side of (19.7) need not vanish. We are thus led to suppose that there exists an operator A(x) in the theory such that ∃A : hΩ|B a |Ωi = 6 0. 2 J.

Bernstein, Rev. Mod. Phys. 46 (1974) 7.

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Spontaneous Symmetry Breaking

We now assume as usual that the eigenstates |ni of the generator of translations (4-momentum) are complete P µ |ni = pµn |ni , Σ|nihn| = 1 , and in particular, that P µ |Ωi = 0 . We make use of this completeness in order to rewrite (19.7) as X  hΩ|Qa (t)|nihn|A(t, ~r)|Ωi − hΩ|A(t, ~r)|nihn|Qa (t)|Ωi = hΩ|B a (t, ~r)|Ωi . (19.8) n

Now, the generator Qa cannot change the spatial momentum of the states. Hence the sum only extends over states with vanishing 3-momentum: p~n = 0. Translational invariance in time further allows us to explicitly extract the time dependence as follows: X   hΩ|Qa (0)|nihn|A(0)|Ωie−iEn t −hΩ|A(0)|nihn|Qa (0)|ΩieiEn t = hΩ|B a (0)|Ωi . {~ pn =0}

Since the right-hand side is, however, time-independent, only states of zero energy can contribute to the sum, and since the rhs does not vanish, these states must lie in the spectrum of the operator A. They evidently correspond to zero-mass excitations, the so-called Goldstone bosons! This concludes the proof.3

19.4

Realization of Goldstone Theorem in QFT

Let us consider a simple field-theoretical example realizing the above Goldstonephenomenon. To this end, consider the Lagrangian L = ∂µ ϕ† ∂ µ ϕ + µ2 ϕ† ϕ − g(ϕ† ϕ)2

(19.9)

describing the self-interaction of charged scalar fields. We take g to be positive, so that the spectrum is bounded from below. The Lagrangian evidently is invariant under the global phase transformation ϕ(x) → ϕ(x)eiα .

(19.10)

The Euler–Lagrangian equations read   2 ~ 2 − ∂ + µ2 ϕ = 2gϕ|ϕ|2 . ∇ ∂t2 For g → 0 this equation states that −~ p2 + E 2 + µ2 = 0. Note that this is not the usual Einstein relation between energy and momentum of a particle with mass µ, since the µ2 term has the wrong sign! Hence, when quantizing the field around 3 If Qa is a spin zero operator (this excludes supersymmetry), then the Goldstone bosons must also carry spin zero.

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319

ϕ = 0 we have a tachyon, and we cannot regard the field ϕ as interpolating a physical particle. Now, including the µ2 ϕ∗ ϕ term in the potential, we have from (19.9) L = ∂µ ϕ† ∂ µ ϕ − Vef f (|ϕ|) with the effective potential 

µ2 Vef f (|ϕ|) = g |ϕ| − 2g 2

2

−

µ4 . 4g

(19.11)

This potential shows a continuum of minima parametrized in zeroth order of h ¯ as follows: s µ2 iα hΩ|ϕ|Ωi = e . 2g Hence we have SSB, and the ground state is no longer invariant under the phase transformation (19.10), so that we have a continuous infinity of classical ground states. It is convenient to parametrize the field ϕ as follows, i 1 ϕ(x) = √ ρ(x)e λ θ(x) . 2

(19.12)

With respect to ρ the potential has the form of Fig. 19.3. 2 We expanded ρ around the local minimum of Vef f ( ρ2 ), ρ(x) = λ + φ(x) ,

λ=

s

µ2 g

(19.13)

so that we have now, hΩ|φ(x)|Ωi = 0 . We make the following remarkable observations: (i) φ describes an excitation with the physical mass grangian

√

2µ, as seen from the la-

 
µ
i 1 ρ − λi θ(x) θ(x) λ L = ∂µ ρe ∂ ρe − Vef f √ 2 2 √ 2 1 ( 2µ) 1 = (∂µ φ)2 − φ2 + (∂µ θ)2 2 2 2 1 2 µ4 g 1 + 2 φ (∂µ θ)2 + φ(∂µ θ)2 + − λgφ3 − φ4 . 2λ λ 4g 4 (ii) θ describes a zero-mass boson. It is the Goldstone boson associated with the spontaneously broken U (1) symmetry.

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Spontaneous Symmetry Breaking

Now, as we already remarked, zero-mass, spin-zero particles have not been found in nature. Does this exclude spontaneous symmetry breaking in high-energy physics? The following lagrangian provides an important example for circumventing Goldstone’s theorem. Evasion of the Goldstone Theorem How can we have a spontaneously broken symmetry without zero mass scalar particles? The massless degree of freedom θ appeared above as a phase of the scalar field ϕ. Hence let us gauge the Lagrangian (19.9) by minimally coupling the scalar field (tachyon) to a gauge field and adding a Maxwell term. We thus obtain 1 L = − Fµν F µν + |Dµ ϕ|2 + µ2 |ϕ|2 − g(ϕ† ϕ)2 , 4

(19.14)

where Dµ = ∂µ + ieAµ .

Make again the change of variable (19.12), the phase exp( λi θ) now is, however, no longer observable, since it may be gauged away by the transformation i ρ(x) ϕ(x) → ϕ(x)e− λ θ(x) ≡ √ 2

1 ∂µ θ(x) ≡ Bµ . eλ Only ρ and Bµ thus survive as physical degrees of freedom. We again set ρ(x) = λ + φ(x) in the lagrangian, which now reads Aµ (x) → Aµ (x) +

1 1 L = − (∂µ Bν − ∂ν Bµ )2 + e2 λ2 Bµ2 4 2 1 √ 2 2 1 2 + (∂µ φ) − ( 2µ) φ + LI 2 2 with

e2 2 2 g Bµ φ − λgφ3 − φ4 + const. 2 4 Note again the change in sign of the φ-mass term as compared to (19.14). We recognize that the√above lagrangian now describes the dynamics of a physical scalar particle φ of mass 2µ and a massive vector field B of mass mB = eλ. There is no longer any vestige of the “would be Goldstone boson”, which has been “eaten” by the gauge field, thereby turning it “fat”. This evasion of the Goldstone boson through the “back door” of a gauge field interaction was discovered by Higgs–Kibble– Guralnik and Brout, and is generally referred to in high-energy physics as the Higgs mechanism.4 It plays a fundamental role in the Weak Interactions. The reason for the breakdown of Goldstone’s theorem lies in the existence of negative metric states associated with the longitudinal photons, which are the ones that became fat in the course of this “dinner”. LI = e2 λBµ2 φ +

4 P. Higgs, Phys. Lett. 12 (1964) 132; F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 132; G. Guralnik, C. Hagen and T. Kibble, Phys. Rev. Lett. 13 (1964) 1964.

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Chapter 20

Effective Potentials In the past chapters we have concentrated on the vacuum expectation value of the product of field operators, and in particular on connected Green functions. The basic constituents of a general Feynman diagram are however the proper Green functions. Since they are interlaced by simple propagators within a general Feynman diagram, their associated momentum-space integrals can be carried out independently. The generator of the proper Green functions will provide the definition of the effective potential to arbitrary order in ¯h, as we shall see.

20.1

Generating functional of proper functions

We have seen in Chapter 13 that the functional defined by (13.1) is the generating functional of the Green functions (connected and disconnected) of the QFT in question. It thus formally has the expansion ∞ n Z X i Z[j] = 1 + d4 x1 . . . d4 xn G(n) (x1 , . . . , xn )j(x1 ) . . . j(xn ) n! n=1 with

G(n) (x1 , · · · xn ) =< Ω|T φ(x1 ) · · · φ(xn )|Ω >

and |Ω > the normalized physical vacuum. It will be convenient to define a new functional W [j] via Z[j] = eiW [j] . It is well known that ln Z[j] is the generating functional of the connected Green functions, i.e.   δ δ ··· W [j] = −ihΩ|T φ(x1 ) . . . φ(xn )|ΩiC . iδj(x1 ) iδj(xn ) j=0

We conclude that the generating functional W [j] formally also has the expansion ∞ n Z X i iW [j] = d4 x1 . . . d4 xn G(n) c (x1 · · · xn )j(x1 ) · · · j(xn ). n! n=1 321

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Generating functional of proper Green Functions The amputated Green functions have already been defined in (17.28). They are obtained by stripping off the fully dressed propagators (including the wave function renormalization constants) associated with the external lines of a connected Green function. They have already been seen to be the entities from which we obtain the so-called proper (1-particle irreducible (1PI), connected) Green functions. We have represented these functions by doubly shaded (hatched) blobs. As was shown by Jona-Lasinio,1 the generating functional of proper Green functions is the Legendre transform of the generating functional of connected diagrams. We obtain this generating functional as follows: Consider the (classical) field ϕc (x) defined by δW [j] . (20.1) ϕc (x) = δj(x) The right-hand side of this equation can be viewed as the vacuum expectation value of the field operator φ(x) in the presence of the external source j(x): ϕc (x) = hΩ[j]|φ(x)|Ω[j]i .

(20.2)

Hence for a φ4 -type potential with a minimum at φ = 0 we expect this expectation value to vanish when turning off the external source j. We shall make use of this in the sequel. Following a procedure familiar from Thermodynamics, we trade the external source j for ϕc by performing the Legendre transformation   Z δ , (20.3) Γ[ϕc ] = W [j] − d4 xj(x) W [j] δj(x) j=J[ϕc ] where j(x) = J(x|ϕc ) stands for the solution of Eq. (20.1) for j in terms of ϕc . Note that this procedure is analogous to the transition from the Lagrangian to the Hamiltonian formulations, with   Z δL −H[ϕ, π] = L[ϕ, ϕ] ˙ − d3 xϕ(x) ˙ , δ ϕ(x) ˙ ϕ=F ˙ [π] where ϕ(x) ˙ is a solution of π(x) =

δL . δ ϕ(x) ˙

As we now demonstrate, Γ[ϕc ] defined by (20.3) is the generating functional of proper Green functions Γ(n) (x1 , . . . , xn ), defined by the expansion (notice the absence of in ) Γ[ϕc ] = 1 G.

Z ∞ X 1 d4 x1 . . . d4 xn Γ(n) (x1 , . . . , xn )ϕc (x1 ) . . . ϕc (xn ) . n! n=2

Jona-Lasinio, Nuovo Cimento 34 (1964) 1790.

(20.4)

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20.1 Generating functional of proper functions

323

The Legendre transformation (20.3) can be written in the form   Z 4 Γ[ϕc ] = W [j] − d xj(x)ϕc (x) .

(20.5)

j=J[ϕc ]

From here we obtain δ Γ[ϕc ] = δϕc (x)

Z

4

d y

(

)  δW [j] δJ(y|ϕc ) − ϕc (y) · − J(x|ϕc ) . δj(y) δϕc (x) j=J[ϕc ]

Making use of (20.1), this reduces to δΓ[ϕc ] = −J(x|ϕc ) . δϕc (x)

(20.6)

From (20.1) and (20.6) we have the identity δW [j] ϕc (x) ≡ δj(x) j=J[ϕc ]

and

δ 2 Γ[ϕc ] δJ[x|ϕc ] =− δϕc (x)δϕc (y) δϕc (y)

(20.7)

(20.8)

respectively. Differentiating identity (20.7) with respect to ϕc (y), we get   δ δ 4 W [j] j=J[ϕ ] δ (x − y) = c δϕc (y) δj(x) Z 2 δJ(u|ϕc ) δ W [j] · , ≡ d4 u δj(u)δj(x) j=J[ϕc ] δϕc (y)

or making use of (20.8), 4

δ (x − y) = − Hence the kernel

Z

δ 2 Γ[ϕc ] δϕc (z)δϕc (y)

δ 2 Γ[ϕc ] δ 2 W [j] d u . · δϕc (u)δϕc (y) δj(u)δj(x) j=J[ϕc ] 4

(20.9)

is the inverse of

δ 2 W [j] . iδj(z)iδj(y) j=J[ϕc ]

Now, in the absence of spontaneous symmetry breaking we have ϕc = 0 for j = 0. Hence, with the identifications δ 2 Γ[ϕc ] (2) Γ (y − x) = , δϕc (y)δϕc (x) ϕc =0 δ 2 W [j] = −i < Ω|φ(z)φ(y)|Ω >= ∆F (z − y) iδj(z)iδj(y) j=0

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Effective Potentials

we have in this case Z

d4 zΓ(2) (y − z)∆F (z − x) = δ 4 (y − x) ,

(20.10)

or going to momentum-space, ˜ (2) (p) = p2 − m2 − Σ(p) . Γ

(20.11)

The interpretation of a general Γ(n) as a proper Green function may be obtained by successive differentiation at ϕc = 0 of the “master identity” (20.9). Thus, for instance, one further differentiation of this identity with respect to ϕc (z) and setting ϕc = 0 leads to Z Z Z 4 (3) (2) 4 (2) d uΓ (u, y, z)Gc (u, z) = i d u d4 vΓ(2) (u, y)G(3) (v, z) c (x, u, v)Γ i.e. iΓ

(3)

(x, y, z) =

Z

4 0

d x

Z

4 0

d y

Z

0 0 0 (2) 0 d4 z 0 G(3) (x , x)Γ(2) (y 0 , y)Γ(2) (z 0 , z) c (x , y , z )Γ

or using (20.10), iΓ(3) (x, y, z) = hΩ|T ϕ(x)ϕ(y)ϕ(z)|Ωiamp . Thus iΓ(3) coincides with the amputated 3-point function, which happens to be also 1-particle-irreducible. In general one can show that it holds for 1PI Green functions. iΓ(n) (x1 , · · · , xn ) = hΩ|T ϕ(x1 ) · · · ϕ(xn )|Ωi1P I .

20.2

(20.12)

The effective potential

Consider the Lagrange density L=

1 1 (∂µ φ)2 − m20 φ2 − V (φ) 2 2

with V (φ) =

g0 4 φ . 4!

In the tree graph approximation we have from (20.11), Γ(2) (x − y) =

Z

d4 p ˜ (2) Γ (p)eip(x−y) = (2π)4

Z

= −(

d4 p 2 (p − m20 )eip(x−y) (2π)4 + m20 )δ 4 (x − y) ,

(20.13)

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20.3 The 1-loop effective potential of φ4 -theory

325

and from (20.12), (4) iΓtree (x1

. . . x4 ) = −ig

Z

d4 z

4 Y

i=1

δ 4 (z − xi )

(20.14)

as the only contributions to Γ[ϕc ]tree . Hence, recalling (20.4), Γ[ϕ]tree = Scl [ϕ]

(20.15)

we conclude that the generating functional Γ[ϕ] coincides to lowest order in ¯h with the classical action. It can thus be regarded as providing a generalization of the classical action to all orders of h ¯ . One refers to it as the effective action: Γ[ϕ] = Sef f [ϕ] . Expanding the integrand of the effective action in powers of derivative of the fields, one has in general, making use of translational invariance, "Z n−1 # Y X 1 Z 4 4 (n) d zi Γ (z1 · · · zn−1 , 0)ϕ(z1 + x) · · · ϕ(zn−1 + x)ϕ(x) , Γ[ϕ] = d x n! n=2 i=1 which we can write in the form   Z 1 4 2 Γ[ϕ] = d x −Vef f (ϕ) + Z(ϕ)(∂µ ϕ) + · · · . 2

(20.16)

Comparing with (20.15), we see that Vef f (ϕ) generalizes the classical (tree graph) potential, including the mass term, to all orders in ¯h. It is thus referred to as the effective potential. It is evidently obtained from the integrand of the effective action, by considering constant field configurations σ. In that case we have from (20.4) Vef f (σ) = −

∞ X X 1 1 n ˜ (n) ˜ (n) (0, · · · , 0)amp , σ Γ (0, · · · , 0) = i σn G n! n! n=2 n=2

(20.17)

˜ (n) (0, · · · , 0) is the momentum-space 1PI n-point function defined via the where Γ Fourier transform Z ˜ (n) (p1 · · · pn )(2π)4 δ 4 (p1 +· · ·+pn ) . d4 x1 · · · d4 xn ei(p1 ·x1 +···pn ·xn ) Γ(n) (x1 · · · xn ) = Γ The effective potential plays a central role in the discussion of spontaneous symmetry breaking.

20.3

The 1-loop effective potential of φ4 -theory

1-loop approximation In this approximation, the generating functional Γ[φc ] of 1PI-irreducible Green’s functions involves an infinite number of terms, as expressed diagrammatically below:

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Effective Potentials

σ

σ

σ

σ + ···

+ σ

(20.18)

σ

Fig. 20.1. Sum of 1-loop diagrams for Vef f . Although this series is in general not summable, it may be summed up for vanishing external momenta (constant field configurations σ), where the vertices represent simply zero-momentum insertions. In this case the above series formally reduces in momentum space to the sum, (1−loop) Vef f (σ; Λ)

1 = m20 σ 2 + V (σ) − 2

Z

 n ∞ d4 K X (−1)n V 00 (σ) , (20.19) (2π)4 n=1 2n K 2 + m20

Λ

0

where a Wick rotation has been performed, a UV cutoff Λ for the momentum integration has been introduced, and use has been made of (20.17). In arriving at this expression the following combinatoric factors have been taken into account: 1. A factor n1 : There exist (n − 1)! permutations of the n pairs of external lines leading to topologically inequivalent graphs (the n cyclic permutations are topologically equivalent). 2. The factor 21 in g20 σ 2 arises from the 4! 2! possible contractions of the Bose fields 1 in V (ϕ) and leaves 12 . with the remaining fields. This cancels the factor 4! 3. An overall factor of 12 . It is a consequence of an incomplete cancellation of the factor 1/n! in the perturbative expansion, since diagrams with labeled vertices differing by a flow of “virtual” charge in the loops are identical since the field φ(x) is neutral (unlike the case of electrons). Notice that for n ≥ 2 the terms in the sum above exhibit for m0 = 0 an ever increasing logarithmic divergence at k 2 = 0. Formal summation however reduces this divergence to a soft one of the logarithmic type: (1−loop)

Vef f

(σ; Λ) =

1 2 2 1 m σ + V (σ) + 2 0 2

Z

0

Λ2

K 2 dK 2 ln 16π 2



1+

 g0 2 2 σ K 2 + m20

, (20.20)

where use has been made of (15.1) and (15.2) with D = 4 and a cutoff has been introduced. The quadratic UV divergence already present in (20.19) continues however to exist, and requires renormalization. Since, as seen from (20.19) the divergence occurs only for the first two terms in the series, it can be cured on tree-graph level by the addition of suitable counter-terms to the Lagrangian (20.13).

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20.4 WKB approach to the effective potential

20.4

327

WKB approach to the effective potential

For our forthcoming discussion it will be useful to develop a slightly different view of the effective potential. To this end we begin with a consideration which appears to be unrelated to our discussion in the preceding section, leaving to the end of this section the demonstration that we obtain in this way two equivalent approaches to the calculation of the effective potential. We shall consider again the Lagrange density (20.13). The corresponding action in the presence of an external source j(x) is given by   Z 1 1 (∂µ φ)2 − m20 φ2 − V (φ) + jφ . S[φ, j] = d4 x (20.21) 2 2

We search for the minimum of this action. It corresponds to a field configuration satisfying the classical equation of motion (

+ m20 )φc` (x) + V 0 (φc` (x)) = j(x) .

(20.22)

Notice that φc` (x) is a functional of j. Expanding a general field configuration about this classical field, φ(x) = φc` (x) + ξ(x) , substituting this into (20.21), and expanding in powers of ξ, we obtain using (20.22), Z Z X
1 ξp S[φ; j] = Sc` [j] − d4 x ξ + m20 + V 00 (φc` ) ξ − d4 x V (p) (φc` ) , (20.23) 2 p! p≥3

with



 1 2 Sc` [j] = d x − φc` ( + m0 )φc` − V (φc` ) + jφc` . 2 The expansion (20.23) corresponds to an expansion in powers of the Planck constant h ¯ . This is easily seen by absorbing the h¯1 in ln Z = iS/¯h into the ξintegration by making the change of variable 1 ξ→ξ 1/2 ¯h in the corresponding Feynman path integral, which leads to h i
P∞ p ξp R Z −i d4 x ξ +m20 +V 00 (φc` ) ξ− h ¯ 2 p! V (p) (φc` ) i p≥3 Z[j] = e h¯ Sc` [j] Dξe . Z

4

The WKB approximation in quantum mechanics corresponds to dropping the contribution from the last term describing the non-Gaussian fluctuations. Making use of 2 the fundamental formula Z R 4
−1/2 1 Dξe−i d ξ(x)D(x)ξ(x) = Det D(x) = e− 2 T rln D , (20.24) 2 This

formula generalizes the N -dimensional integral

Z Y N 0

to the continuum.

dan e

i

P

n

λn a2 n

=

Y n

λn

!− 12

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we have in this approximation (we assume that for j → 0, φc` → 0)
!− 12 + m20 + V 00 (φc` ) . ( + m20 )

i Z[j] Z[j] = ' e h¯ Sc` [j] Det Z[0]

(20.25)

We thus have for the generating functional of connected Green functions,   h ¯ 1 00 W [j] ' Sc` [j] + i T rln 1 + V (φc` ) + θ(¯h2 ) . (20.26) 2 + m20 In order to obtain the connection with the effective potential discussed in Section 5, we now proceed to obtain from here the generating functional of the proper functions via the Legendre transform (20.3). We need to express W [j] as a function of ϕc (x) defined in (20.1). From (20.26) we have ϕc (x) =

δSc` [j] + O(¯ h) . δj(x)

Now, Z δSc` [j] δφc` (z) δφc` (z) δφc` (z) = d4 z[− ( + m20 )φc` (z) − V 0 (φc` ) + j(x) δj(x) δj(x) δj(x) δj(x) + φc` (x)δ(z − x)] = φc` (x) , where use has been made of the equation of motion (20.22). Hence, with (20.26) and (20.1) we have, ϕc (x) = φc` (x) + θ(¯h) . (20.27) Since φc` represents a stationary configuration of S[φc` , j], we have, setting ϕc (x) = φc` (x) + ¯hξ(x), S[ϕc , j] = S[φc` + ¯hξ, j] = S[φc` , j] + O(¯h2 ) , so that we also have W [j] = S[ϕc , j] +

i¯h T rln 2

  V 00 (ϕc ) 1+ + θ(¯ h2 ) . + m20

(20.28)

We thus have achieved our first goal, to order h ¯ . Performing the subtraction of the R integral in (20.3), the jϕc term φj contained in S[ϕ, j] in (20.28) cancels so that to this order h ¯ Γ[ϕc ] = S[ϕc , 0] + i T rln  2 Z 1 = d4 x (∂µ ϕc )2 − 2


+ m20 + V 00 (ϕc ) + O(¯h2 )

1 2 2 i¯h m ϕ − V (ϕc ) + hx|ln 2 0 c 2



1+

V 00 (ϕc ) + m20



(20.29)  |xi .

Hence Γ[φ] is only a function of ϕc as guaranteed by the Legendre transformation.

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329

As we now show, this is identical with the effective potential (20.20). Indeed, expanding the logarithm, we have   X (−1)n+1 Z 1 1 00 00 Trhx|ln 1 + V (ϕ ) |xi = d4 xhx| c 2 2 V (ϕc )|xi + m0 n + m 0 n=1 Z Z X (−1)n+1 Z d4 k 0 d4 k 4 d x hx|k 0 ihk 0 | = 4 4 n (2π) (2π) n=1

1 V 00 (ϕc )|kihk|xi + m20

X (−1)n+1 Z d4 k  V 00 (ϕc ) n = n (2π)4 −k 2 + m20 n=1  00 n Z X (−1)n+1 V (ϕc ) d4 K =i n (2π)4 K 2 + m20 n=1   Z 4 V 00 (ϕc ) d K ln 1 + 2 , =i (2π)4 K + m20 or with (20.16), we have, introducing a cutoff, m2 1 Vef f (σ) = 0 σ 2 + V (σ) + 2 2

Z

0

Λ

K 2 dK 2 ln 32π 2

  V 00 (ϕc ) 1+ 2 K + m20

(20.30)

in agreement with (20.20). Notice that the overall factor of 21 arose here from the determinant in (20.24) appearing with the power − 12 (unlike + 12 in the fermionic case), which in turn is linked to the fact, that φ(x) is a real (charge neutral) bosonic field. Our alternative derivation has shown that the effective potential is nothing but the effective action one obtains by expanding the classical action about the constant field configuration minimizing the action, dropping all terms of higher order than quadratic in the fluctuations, and then performing the remaining gaussian functional integration. We now give an example of SSB for the case of φ4 theory.

20.5

The effective potential and SSB

Let us examine the effective potential (20.30) with regard to spontaneous symmetry breaking. This will give us also the opportunity to gain insight into the renormalization program, as well as the renormalization group aspects. The m 6= 0 effective potential We rewrite (20.20) in the form (1−loop)

Vef f

(σ; Λ) =

    Z Λ2 1 2 2 g0 4 xdx g0 σ 2 2 2 m0 σ + σ + ln x + m + − ln (x + m ) . 0 0 2 4! 32π 2 2 0 (20.31)

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The integrals are elementary. We have for Λ → ∞, Z

0

Λ2

Λ2 +a y2 y2 ln y − − a(yln y − y) xdxln (x + a) = 2 4 a 

(20.32)

1 1 1 Λ2 1 ≈ Λ4 ln Λ2 − a2 ln − Λ4 + aΛ2 − a2 . 2 2 a 4 4

We thus find for the effective potential, 1 (20.33) Vef f (σ) = m20 σ 2 + V (σ) 2 "    2 # a2 Λ2 g0 σ 2 2 m20 g0 σ 2 1 m40 Λ2 1 g0 σ 2 − ln + + ln 2 + Λ − − 32π 2 2 a 2 m0 2 2 2 4 2 where a = m20 +

g0 σ 2 . 2

To simplify the notation set z≡

g0 σ 2 , 2

a = m20 + z .

To renormalize this potential we take the point of view of Section 16.2 of Chapter 16 by taking g0 and m0 to be physical parameters g and m, respectively, and adding the counter terms  1  B(Λ)z + C(Λ)z 2 . (20.34) δV (σ; Λ) = 2 32π We can then write the (pre-) renormalized (see (16.6)) effective potential in the form    2   1 2 2 1 z  m 1 2 2 ˜ Vef f (σ; Λ) = m σ + V (σ) + (m + z) ln 1 + 2 + ln 2 32π 2 2 m Λ2  2  m 1 1 1 − m4 ln + zΛ2 − m2 z − z 2 2 Λ2 2 4 1 + [B(Λ)z + C(Λ)z 2 ] . (20.35) 32π 2 The coefficients B(Λ) and C(Λ) are associated with the mass and coupling constant renormalizations. We fix the arbitrary constants by imposing the mass-shell condition ˜ (2) (p)|p2 =0 = −m2 , Γ or equivalently (see (20.17)) d2 V˜ef f dσ 2

= m2 . σ=0

(20.36)

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20.5 The effective potential and SSB This gives, 1 B(Λ) = − 32π 2

  2  m 2 2 m ln +Λ Λ2

331

(20.37)

and leaves only the mass term quadratic in σ. The usual normalization condition for the 4-point function at vanishing external 4-momenta requires (see (20.14) for sign) ˜ (4) (0, 0, 0, 0) = −g Γ

(20.38)

or (see (20.17) for sign) d4 V˜ef f =g . (20.39) dσ 4 σ=0 The easiest way to evaluate the left-hand side is to expand the logarithm in powers of z. A simple calculation yields,  2   m 1 1 3 ln . (20.40) C(Λ) = − + 32π 2 2 Λ2 4 Putting things together, we are left with the renormalized effective potential gσ 4 1 (20.41) Vef f (σ) = m2 σ 2 + 2 4! " #   2    1 m4 gσ 2 gσ 2 gσ 2 3 gσ 2 2 + + m . 1 + ln 1 + − 32π 2 2 2m2 m2 2 2 2 The immediate question arises: is there spontaneous symmetry breakdown? A plot of Vef f shows that one has to go to extremely high values of g to see a vestige of a spontaneous symmetry breakdown. The m = 0 effective potential As the result (20.40) shows, the implementation of normalization condition (20.38) for m = 0 is not possible. As before, we take m0 and g0 to be physical parameters g and m in (20.33), respectively. Setting m0 = m in (20.33), we then obtain,  2   2   g 1 gΛ 2 g 2 gσ 1 Vef f (σ, Λ) = σ 4 + σ + ln − σ4 . (20.42) 4! 32π 2 2 8 2Λ2 2 The UV divergence is again cured by the addition of suitable counterterms   1 1 1 2 4 δVef f (Λ) = B (Λ)σ + C (Λ)σ 0 0 32π 2 2 4! which in turn are fixed by imposing conditions on Vef f ,  σ2 g 1  V˜ef f (σ; Λ) = σ 4 + B0 (Λ) + gΛ2 2 4!  32π   22!  4 1 gσ 1 σ 2 + C0 (Λ) + 3g ln − . 32π 2 2Λ2 2 4!

(20.43)

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The coefficients B0 (Λ) and C0 (Λ) are associated with the mass and coupling constant renormalization. We try to fix the arbitrary constants by requiring the massshell condition ˜ (2) (p)|p2 =0 = 0 , Γ or equivalently (see (20.17)) d2 V˜ef f dσ 2

This gives,

=0

!

(m = 0) .

(20.44)

σ=0

B0 (Λ) = −gΛ2 .

As we already observed, the usual normalization condition for the 4-point function at vanishing external 4-momenta cannot be implemented due to the logarithmic dependence of (20.42) on σ. We thus need to choose a different normalization point σ = µ. A straightforward calculation yields !    1 11 d4 V˜ef f gµ2 2 =g+ . (20.45) g¯(µ) ≡ C0 (Λ) + 3g ln 2 − dσ 4 32π 2 2Λ 3 σ=µ

The identification of g with g(µ) now requires   11 gµ2 2 , C0 (Λ) = −3g ln 2 − 2Λ 3 which is in fact is the choice of counterterm made by S. Coleman and E. Weinberg. Replacing g by g(µ) in (20.43) one finally obtains for the renormalized effective potential   σ4 g¯(µ)2 σ2 25 σ 4 Vef f (σ, g¯, µ) = g¯(µ) + 3 ln 2 − (20.46) 4! 32π 2 µ 6 4! which is the result obtained by Coleman and Weinberg.3 Is there SSB? The question of immediate interest is, whether spontaneous symmetry breakdown (SSB) occurs. The location σex of the extrema is obtained by solving the equation   3 2 3 g 3 g 2 σex σex 25 g 2 σex 0 Vef (σ , µ) = σ + ln − + =0. ex f ex 2 2 3! 64π µ 6 264π 2 σM = 0 corresponds to a maximum, while σm , given by ln

2 σm 32π 2 = − + O(1) , µ2 3g

that is 2 σm ∼ µ2 e− 3 S.

32π 2 3g

Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888.

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20.5 The effective potential and SSB

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corresponds to two minima with Vef f (±σm ) = −e−

64π 2 3g

g 2 25