becchi c., rouet a., stora r. renormalization of gauge theories

Centre de Physique Théorique, CNRS 31, chemin Joseph Aiguier, F-13274 Marseille Cedex 2, France Received December 8, 1975 Gauge theories are characterized by the Slavnov identities w

ANNALS OF PHYSICS 98, 287-321 (1976)

Renormalization of Gauge Theories

C BECCHI,* A ROUET,' AND R Stora!

Centre de Physique Théorique, CNRS 31, chemin Joseph Aiguier,

F-13274 Marseille Cedex 2, France Received December 8, 1975

Gauge theories are characterized by the Slavnov identities which express their in-

variance under a family of transformations of the supergauge type which involve the Faddeev Popov ghosts These identities are proved to all orders of renormalized per-

turbation 1 they, within the BPHZ framework, when the underlying Lie ie algebra 1 is semi-

principle of Lowenstein [3] and Lam [4] allows to push the analysis of the algebraic

structure of gauge field models

Very few properties of the perturbation series are actually used here, namely,

———— lNothing more than the general consequences of locality [5], sharpened by the = =

theory of power counting [5],.which, through the fundamental theorem of re-

normalization theory [5] insure the existence of a basis [2, 6] of local operators

of given dimension and of imear relationships [2] between local operators of different dimensions More precisély, we shall never use the information contained

in the detailed structure of the coefficients involved in such relations, thus for-

t Present address: Max Planck Institut fiir Physik und Astrophysik, Miinchen

* Centre de Physique Théorique, CNRS, Marseille

287

Copyright ‘© 1976 by Academic Press, Inc

Ail rights of reproduction in any form reserved

of the phenomena which occur when abelian components are involved For instance,

we have noted that the fulfillment of discrete Symmetries allows favourable simp

illustrated on a specific example

The heart of the matter is to prove that one can fulfill to all orders of renormalized

perturbation theory a set of identities, the so-called Slavnov identities 1, 14], which express the invariance of the Faddeev Popov (®/7) [15] Lagrangian under a

set of nonlinear field transformations [1] which explicitly involve the Faddeev

whenever possible as an operator theory within a Fock space with indefinite

metric involving the ®/T ghost fields When this interpretation is possible, the Slavnov identities allow to define a ‘“‘physical” subspace of Fock space within which the norm is positive definite The restriction of the S operator to this subspace

is then both independent from the parameters which label the gauge function [11],

1 uni ntl bat

This article is divided into two main sections, and a number of appendices

devoted to some technical details

Section 2 covers the algebraic discussion of the Slavnov identities

Section 3 deals with a specific model (the SU2 Higgs Kibble model [16]) for

We start with a classical Lagrangian of the form:

L = (0a), {a„„}, {Ca} {Ca})

Š„ w)@a(X) = ỗ(x — y) f2z[@;(x) + Fi] thưt

(4)

Š„z(w) đ„(x) = 0, 8(x ~ y) ô, + of a’ Ay, (X) d(x ~~ y)

where {8} are the structure constants of }, {F,} some field translation parameters

The gauge function {G,} will be chosen to be linear in the fields and their derivatives:

g BQng

=- -ÌL œ@ Œ

<0 Sa “But ba Ya: œ (5` \v7

[ndices of b are raised and lowered by means of an invariant nondegenerate

symmetric tensor The role of the gauge term is to remove the degeneracy of the

quadratic part of Yay, which is connected with gauge invariance The field

The essential property of this Lagrangian is its invariance under the following

O ALTO W We 4 L1 C AVNOY ΠOTTIaHnons:

ðœ(x) = ðÀ Š„„œØ.(x) C(y) = ðÀ sợ),

Šđ„(x) = BA ð„s@Ø„„(x) Ca(y) = ŠÀ sđ„„(x), 6C,(x) = 6A G(x) = 6A sC,(),

where the summation over repeated indices and the integration over repeated

———————— &Pacetime variables are understood sSsesSsSsSsSsSsSsSsSSsSseese

dA is a spacetime independent infinitesimal parameter which commutes with

{po}, {4,,,}, but anticommutes with {C,}, {C,}, and, for two transformations labeled

by 6A, , dA,, dA, , and dA, anticommute

This invariance can be checked immediately, by using the composition law

for gauge transformations

LG) > Swalw) — FP ox — y) Buy) , (8)

Conversely, it is interesting to know whether & is up to a divergence the most

Pene o al Lag PTäAIDĐ old >1 Ũ Ð Oo an actio va ng AVITOV a 0 AtTO

and carrying no Faddeev Popov charge Q°":

Q°”C = C,

O°", — Q°"a„„ = 0

and given a functional F(¥), let us denote |

where 8Y — dAs¥ is the variation of Y under the Slavnov transformation of

parameter 5A (cf Eq (7)] A remarkable property of s is:

œ

5 L)= œ Cy (x) = J:

RENORMALIZATION OF GAUGE THEORIES 291

This property actually summarizes the group law as follows Let

Ờa = ỒÀ(Đyợ› + Ga) Cy = dA SPa 4 54a, = ŠÀ(0”a Ap be

(a) FEI + AEH + FFP = 0

(b) [6 Ø]„„ — /7'Ú, = 0, (c) [6 "le, — 7°83, = 9, (15)

Equation (15a) is the Jacobi identity If we choose a solution corresponding to

b, Eq (15b) and Eq (15Sc) assert that 6* is a representation of h If 5 is semisimple, then all solutions (cf Appendix A) of Eq (15d) are of the form

adjoint representation of Ð) In the semisimple case Ø, is thus equivalent to the

adjoint representation if {g,*} does not vanish identically, a requirement which ==~—~—~—~CSCS~S

belongs to the definition of {a} as a gauge field We may then choose in this case without any loss of generality Eq (13) to be identical with Eqs (3), (4), (7)

If ‹ is Slavnov invariant namely such that

a fortiori

s | P(x) dx = 0 (18)

Now, £# is of the form

L = Liq Epa}, (Gau}) + Ca(x) KexCa(y) + AL Epa}, (aut) + L( CeCe, Cox) (19)

where Yiny , invariant under gauge transformations, is by itself a solution of

Eq (17) By Eq (12) itis obvious from Eq (18) that

where J\* is a numerical symmetrical matrix If h has an abelian invariant part

of this is not the general solution, the @/Z mass term being left undetermined

Going back to Eq (17, 19) yields in the semisimple case:

= inv.UPas> Ayr a* a’ B} —— 314 a”

which is identical with Eq (2) modulo a redefinition of 6, and C, In the abelian case, there may arise an ambiguity unless the gauge function ©, , (a € ) contains

besides the 0#a,, terms a part which is invariant under %

For instance in quantum electrodynamics in the Stueckelberg gauge [11], there

may arise the Slavnov invariant photon-®IJ mass term:

A similar phenomenon, which occurs in the abelian Higgs Kibble model produces

quite spectacular complications [1] if one makes a comparison with its SU2 analog

(cf Section 3)! This is but one pathology associated with the abelian parts of h

which make it unstable under deformations

that b is semisimple (and compact!)

whose lowest order term in fi is given by Eq (2), assuming that all the parameters

in Eq (2) have been chosen in such a way that all mass parameters are strictly positive, and which is furthermore invariant in the renormalized sense [1] under the Slavnov transformation

and (ð,‡, Fe’, Tf, , Q,* are to be found as formal power series in # [1] whose lowest

fields: J‘ of the Bose type, &*, €* of the Fermi type

Performing the Slavnov transformation Eq (26), and using the quantum action

principle yield in terms of the Green functional [1] Z4( Z, m):

where Y stands for the sources and y for the external fields:

J = (i, & &},

(32)

nạ = {', 9

J S(x) dx is the most general dimension five insertion carrying [1] ¿z charge-l,

which is of the form:

[ dx d(x) = | dx Nel—s?(Son +P: + OP.) + 401 (33)

where s? is the naive transformation Eq (11) corresponding to Eq (26), and 70

lumps together all radiative corrections

Making explicit the external field dependence, we shall write

A = Ay + y'4; + GA,

QO — Q + vO; + “9 '

Introducing a new classical field 8 carrying ®/T charge +1 and linearly coupled

to A, the Lagrangian becomes:

and Z,( Z, 7, 8) is the Green functional corresponding to #{%’ Thus, computing

PLAS, 1) = — | dx [EG.8,1x) ZAF, 0) (39)

In terms of the vertex functional ['(¥, 7) = I'(¢;, C,, C,, 4) which is the

Legendre transform of Z-(¥, 4) with respect to ¥, Eq (40) reads:

Equation (42) leads to a consistency condition for 4 Indeed Eq (42) is a perturbed

version of the equation

We have seen in the previous section that Eq (44), as an equation for 3¢ Zeer

possesses the general solution

Recalling Eqs (28), (29), one has

We have thus obtained the consistency condition:

It is shown in Appendix B, by cohomological methods, that, as a result, the

external field dependent part of 4 is of the form

| 04: + EA2)G) đx = —s?7 [ [YP + TD, + ÊŒ + 1D,]G) đx + 004)

(59) where J7,, IZ, can be parametrized in terms of numerical coefficients 0;;, 2,7, 5° according to

IT; = 6?,j,C, + XS, ›

— f,E, ’

with

The coefficients @, 5, I can be chosen and will be chosen to be linear in 9, 2, F,

Now, we know from the quantum action principle that the external field depen-

dent part of A is of the form (cf Eqs (33), 34):

[ ax G4, + 4.)

— —s? dx (iP, + (P(x) +h Ị (y!O, (*Oj)@)dx — (62)

where Q;, Q, are formal power series in # and in the coefficients of ZH*!_ From

the previous observation that ÏÏ,, TT, are linear in 4,, A,, it follows that J7, , [7,

are formal power series of the same type as @,, Ó,

Let us now look at the external field independent terms in 4, which is O(4/4)

in the external fields The consistency condition now reads

Oolti, Lett) = sQolh, Lert) + OA) (72)

Since the equation

Lay - L —hOk, GM) — 0 (73)

is soluble for

Loy — & = Oh) (74)

its solution leaves us with

which we want to achieve

Now the consistency condition shows that

8*(%) = OE sono [D*?07ay"a,"a,Ƒ -} Ƒ*998a,7a 0a] (88)

C The Faddeev Popov Ghost Equation of Motion

It will be of interest in the following to write down the Faddeev Popov ghost

equation of motion in terms of the Slavnov identity It follows from Eq (42) that

once the Slavnovy identity has been proved,

and we know that the general solution of this equation is

Sctal = P2678, I (92) where I“°, is a symmetrical matrix Taking the Legendre transform of Eq (92) yields

which is not obvious since many ghost fields are involved (Ga, C, C, ) First one

should specify the connection between the parameters left arbitrary in the

fulfillment of suitable normalization conditions Then, once the theory has been

set up within the framework of a fixed Fock space, one has to specify a physical subspace within which the theory is reasonable, e.g., the S operator is unitary and independent from unphysical parameters among which the g,’ [cf Eq (26)]

In order to make this program explicit we shall treat in some details the SU2

——————— Hipzs Kibble model [l6] in a way which parallels our treatment of the abelian — —-

Higgs Kibble model [1]

A The Classical Theory

The basic fields are ¥ = {o, 7, , a,,, Cy, C,}, « = 1, 2, 3 At the classical level,

the Lagrangian is invariant under the Slavnov transformation:

595/98/2-2

Sa, = ðÀ [—(@/2) cm, (e2Xơ -+- F) C1 = 6 f,,

ôơ = SA[—(e/2) w*C,] = 8À Zo,

dy, = SA [6,C, — e€o’ag, C]= = OA Tas (95)

8C, = 8A [(2/2) °C, C,] = 6A T,,

ŠC, = A [e%a,, + pm] = 6A G,,

where e*2* are the SU(2) structure constants, e plays the role of a coupling constant, and F is the o field translation parameter This particular choice of transformation

C, C as vectors and leaving o invariant Even if this symmetry is to be preserved

Eq (95) is not the most general transformation law fulfilling the compatibility

conditions Eqs (14), (15), which depends on four parameters besides those which

label the gauge function: e, F, and wavefunction renormalizations for the o and C

~~ fields Given these parameters and introducing external fields {y} = {*, y°, yx*E*} ooooooooooo

coupled to {T} = {T,, 7, T.,,C,}, and sources {J} = {J*, J, I™, Ex, Ex

coupled to {¥} = {,, o, a, , €,, Cy} the Slavnov identity reads:

SZAS, n) = | AX [F(X) 8 aco) FIX) Soc + 78G) 8 uy,

— (x) ô„ư„ — Ễ'%)("8.„, pỗ„„)}Ze(Z, n) FHA EY je

—a, | 0

(99)

6 By {€ as —€x đu;

40.8 ¬ @ \ x8 ; apy fw Sy yy

ub ol Be — yj oO + CE a, AX) ON FD)

+ (pe2J(e*?7z(x) -L 8*%(ø(x) F)) BX — r)

law, one related with the field vacuum expectation value, five specifying the

external field independent part of the Lagrangian, constrained by conditions (14),

(15) One can alternatively specify the following physical parameters: m,, M, Maz, Which give the positions of the poles in the transverse photon, o, C C propagators respectively:

the value of the coupling constant

fix the values of Z,, Z,, 22, A, x, p, e, F, leaving free two parameters in the defini-

tion of the transformation laws For simplicity we shall of course choose

Z,=2Z,=1 (105)

Within the Fock space defined by the quadratic part of the Lagrangian, we _—_— §hall define the bare physical subspace generated by application on the vacuum —-

of the asymptotic fields a7", o!™ which explains that C, C, @a, = are considered

s “phosts.”

C According to this definition, it 1s easy to see [21] that the matrix elements of the

S operator between bare physical states do not depend on the gauge parameters x,

Maa *

B Radiative Corrections: Slavnov Identities, Normalization Conditions

Now, according to the analysis of Section 2, it is possible to find an effective

Lagrangian such that the Slavnov identity (96) holds to all orders (where now p is

to be determined as a formal power series in #) We also know that the Faddeev Popov equation of motion is (cf Eq (94)):

where « k is some formal power s series in h The theory depends c on ten formal power

of them can be fixed by imposing the normalization conditions (101), (102), (103)

(104) These normalization conditions are enough to © interpret the theory 1 in the

mass degeneracy still holds:

Expressing the Slavnov identity in terms of the vertex functional J° yields:

[ ax {ô„„@œ ` Baie dL" Oot + Salat 8 away

4 OG WL Bray lt + Segal [a(x + prix} = 0 (107)

In particular, one gets the following information on the two point functions:

(n3\ (n3 (n3) Ƒ (pỡ P ( n3) —

~ ¬ "LH HH (108) T„a,(PP) £0 p*) + £ (apXP*) f(D") — p*t cc(p*) = Y