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Quantum Field Theory and Critical Phenomena$

Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Gauge Theories: Master Equation And Renormalization

Gauge Theories: Master Equation And Renormalization

Chapter:
(p.532) 21 GAUGE THEORIES: MASTER EQUATION AND RENORMALIZATION
Source:
Quantum Field Theory and Critical Phenomena
Author(s):

JEAN ZINN-JUSTIN

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198509233.003.0021

Abstract and Keywords

To discuss the renormalization of gauge theories in the non-abelian case in its full generality, it is necessary to use a rather abstract formalism, which allows one to understand the algebraic structure of the renormalization procedure without being overwhelmed by the notational complexity. There is, however, a price to pay: the translation of the general identities which then appear into usual and more concrete notation becomes a non-trivial exercise. This chapter is organized as follows. It first quantizes the theory in the temporal gauge. Using a simple identity, it shows the equivalence with a quantization in a general class of gauges. This identity automatically implies a BRS symmetry, and, therefore, a set of WT identities for correlation functions. It shows that WT identities are also direct consequences of a quadratic master equation satisfied the quantized action, equation in which gauge and BRS symmetries are no longer explicit. It shows that in the case of renormalizable gauges the master equation is stable under renormalization. This is solved to determine the structure of the renormalized action. The chapter verifies that the master equation encodes in a subtle way the gauge properties of the quantized action.

Keywords:   gauge theories, quantization, quadratic master equation, renormalization

To discuss the renormalization of gauge theories in the non-abelian case in its full generality, it is necessary to use a rather abstract formalism, which allows one to understand the algebraic structure of the renormalization procedure without being overwhelmed by the notational complexity. There is, however, a price to pay: the translation of the general identities which then appear into usual and more concrete notation becomes a non-trivial exercise.

Our strategy is the following. We first quantize the theory in the temporal gauge. Using a simple identity, we show the equivalence with a quantization in a general class of gauges. This identity automatically implies a BRS symmetry, and, therefore, a set of WT identities for correlation functions. We show that WT identities are also direct consequences of a quadratic master equation satisfied the quantized action, equation in which gauge and BRS symmetries are no longer explicit. We show that in the case of renormalizable gauges the master equation is stable under renormalization. We solve it to determine the structure of the renormalized action. We verify that the master equation encodes in a subtle way the gauge properties of the quantized action.

Physical observables in the theory should be independent of the dynamics of the gauge group degrees of freedom. Following the analysis of previous chapters, we first prove that the vacuum amplitude does not depend on this dynamics, that is, is gauge-independent. We argue that this property remains true if we add to the original action sources for gauge invariant operators of canonical dimension low enough, so that the total action remains renormalizable. As a consequence, correlation functions of these gauge invariant operators are independent of the gauge fixing procedure, and have, therefore, a physical meaning. A similar property holds for all gauge invariant operators but the discussion is more involved and will not be given. Finally, when a S-matrix can be defined, S-matrix elements also are gauge-independent.

21.1 Notation and Geometric Structure

Notation.

In what follows, we restrict ourselves to boson fields, the generalization to fermions being straightforward. All gauge fields and other scalar boson fields are combined into one vector denoted by A i, in which the index i stands for space variables x, Lorentz index μ, and group indices a, b:

                   Gauge Theories: Master Equation And Renormalization
A gauge transformation corresponding to an orthogonal representation of a compact Lie group G is written as
(21.1)                    Gauge Theories: Master Equation And Renormalization
in which ωα are the infinitesimal parameters of the group transformation. As in the example of equation (21.1), we reserve greek indices for the adjoint representation of the Lie algebra. Summation over repeated indices is always meant. It includes integration (p.533) over space variables. The index α also includes group indices and space variables. In more concrete form, equation (21.1) stands for
(21.2)                    Gauge Theories: Master Equation And Renormalization
where the matrices t b form a (in general reducible) representation of the Lie algebra of G and                    Gauge Theories: Master Equation And Renormalization                are the corresponding structure constants.

Finally, S(A) is the gauge invariant action and thus satisfies

(21.3)                    Gauge Theories: Master Equation And Renormalization

A group identity.

We have explained in Section 15.3 that, as a consequence of the group structure, quite generally the functionals                    Gauge Theories: Master Equation And Renormalization                satisfy equations which can be regarded as compatibility conditions for the system (21.3) considered as a set of differential equations for S(A):

(21.4)                    Gauge Theories: Master Equation And Renormalization
These equations are formally identical to equations (15.43) of Section 15.3 and (16.41) of Section 16.4.

Since here the functional                    Gauge Theories: Master Equation And Renormalization                is only affine, it is not difficult to verify the identities (21.4) by a direct calculation, using the commutation relations of the Lie algebra of G. Note, however, that because                    Gauge Theories: Master Equation And Renormalization                is a differential operator, it is necessary to carefully keep track of the δ-functions. In particular, since the indices α and i include space coordinates, the structure constants                    Gauge Theories: Master Equation And Renormalization                which are proportional to the numerical structure constants                    Gauge Theories: Master Equation And Renormalization                of the Lie algebra, have a non-trivial dependence on space variables:

(21.5)                    Gauge Theories: Master Equation And Renormalization
As a consequence, it is actually convenient to rewrite the identity (21.4) with the help of two functions                    Gauge Theories: Master Equation And Renormalization                and                    Gauge Theories: Master Equation And Renormalization                and the operator
(21.6)                    Gauge Theories: Master Equation And Renormalization
The identity (21.4) takes the form
(21.7)                    Gauge Theories: Master Equation And Renormalization
Let us for instance indicate here how this identity can be recovered for the gauge field part. We write Δ(ω) in explicit notation:
                   Gauge Theories: Master Equation And Renormalization
We then calculate the commutator explicitly
                   Gauge Theories: Master Equation And Renormalization
Using the antisymmetry of                    Gauge Theories: Master Equation And Renormalization                and the Jacobi identity one indeed obtains equation (21.7) in explicit form, with                    Gauge Theories: Master Equation And Renormalization                .

(p.534) 21.2 Quantization

In Chapter 19, we have first quantized in the temporal gauge and then shown how to pass from the temporal gauge to a covariant gauge. This is a strategy we now generalize and thus begin with a gauge invariant action S(A) quantized in the temporal gauge.

We thus introduce an equation for the space-dependent group elements g:

(21.8)                    Gauge Theories: Master Equation And Renormalization
where A g is the gauge transform of A by g, and να an arbitrary field belonging to the adjoint representation of the Lie algebra.

Gauge transformations define classes in field space, corresponding to fields and all their gauge transforms, that is, orbits of the gauge group. We assume that the equation intersects all gauge orbits once and thus has a unique solution for g (at least for A small so that the equation can be solved perturbatively).

We then calculate the variation of the equation in an infinitesimal gauge transformation of parameters ωα:

(21.9)                    Gauge Theories: Master Equation And Renormalization
(21.10)                    Gauge Theories: Master Equation And Renormalization
From the results derived in Sections 16.2–16.6, we infer the identity
(21.11)                    Gauge Theories: Master Equation And Renormalization
with
(21.12)                    Gauge Theories: Master Equation And Renormalization
The fields                    Gauge Theories: Master Equation And Renormalization                and C are spinless fermion fields (“ghost” fields) introduced to represent the determinant of M, and transforming under the adjoint representation of the group G.

We then insert the identity (21.11) into the representation of the partition function in the temporal gauge and change variables                    Gauge Theories: Master Equation And Renormalization                . This change of variables has the form of a gauge transformation and thus the gauge invariant action S(A) is unchanged. The dependence of S gauge in g disappears, and only the temporal gauge condition remains g-dependent. We have shown in Section 19.3.2 that the integration over g then yields a constant.

In the new gauge, the partition function then has the functional representation

(21.13)                    Gauge Theories: Master Equation And Renormalization
Since the result does not depend on the field ν (the noise in the terminology of Section 16.6) we integrate over ν with a measure dρ(ν), which we do not need specifying more precisely here, but which eventually will be chosen local and gaussian. Introducing w(λ) the generating functional of connected ν correlation functions,
(21.14)                    Gauge Theories: Master Equation And Renormalization
(p.535) we finally obtain
(21.15)                    Gauge Theories: Master Equation And Renormalization
with
(21.16)                    Gauge Theories: Master Equation And Renormalization
The interpretation of this expression is the following. Because the action is gauge invariant, the degrees of freedom associated with gauge transformations have no dynamics. To quantize these degrees of freedom and to give a meaning to the functional integral in the continuum, it is necessary to provide one. For the field and group element g, we thus have introduced a stochastic dynamics in the sense of Section 16.6. This dynamics is somewhat arbitrary and we shall have eventually to prove that “physical results” (we shall explain later what we mean by physical results) do not depend on its choice.

21.3 BRS Symmetry

We first recall that in the representation (21.15) the functional integral is invariant under gauge transformations which translate να (see Section 16.1):

(21.17)                    Gauge Theories: Master Equation And Renormalization
and thus
(21.18)                    Gauge Theories: Master Equation And Renormalization
This Slavnov–Taylor symmetry has played an essential role in the initial proof of the renormalizability of non-abelian gauge theories. However, the non-local character of this transformation explains the complexity of the proof.

It follows from the general analysis of Chapter 16 that the final action (21.16) has a BRS symmetry. Because the field that has a stochastic dynamics belongs to the gauge group, the BRS transformation for the field A i has the form of a gauge transformation (equation (17.39)). We have verified in Chapter 19 that, since we have provided a dynamics to a group element, the BRS symmetry is similar in form to the symmetry of the dynamic action of chiral models as described in Sections 16.4,17.3 (equations (16.46, 16.47) and (17.37–17.39)). Also, in these more abstract notations, the basic equation (21.4) is formally identical to the commutation relations in the case of non-linear representation of groups of Section 15.3, and to the relations (16.41) of Section 16.4. The corresponding BRS transformations for the A and C fields are thus given by equations (16.39). As we know from the general analysis of Chapter 16, the transformations of λ and C are independent of the dynamics. We conclude that the BRS transformations have the form

(21.19a)                    Gauge Theories: Master Equation And Renormalization
(21.19b)                    Gauge Theories: Master Equation And Renormalization
Using a more explicit notation one verifies that these transformations are identical to the BRS transformations derived in Section 19.3.2. It is convenient to set
(21.20)                    Gauge Theories: Master Equation And Renormalization

(p.536) The BRS operator.

To express the BRS symmetry, it is also useful to introduce the anticommuting differential operator

(21.21)                    Gauge Theories: Master Equation And Renormalization
The essential property of the BRS operator is that it has a vanishing square:
(21.22)                    Gauge Theories: Master Equation And Renormalization
as we know from general arguments and as one can again verify explicitly here. Hence, apart from contributions which are gauge invariant, BRS symmetric terms include contributions which are BRS exact, that is, of the form                    Gauge Theories: Master Equation And Renormalization                . We verify that the action (21.16) has exactly such a decomposition:
(21.23)                    Gauge Theories: Master Equation And Renormalization
This is not surprising because the property has been proven quite generally in Section 16.5 (see equation (16.59)). The quantized action is, therefore, BRS symmetric:
(21.24)                    Gauge Theories: Master Equation And Renormalization

Remark.

In all examples we consider here, the function w(λ) defined by equation (21.14) is quadratic in λ and thus the corresponding gaussian integral can be performed. After integration the new action is still BRS symmetric, the variation of                    Gauge Theories: Master Equation And Renormalization                now takes the form

                   Gauge Theories: Master Equation And Renormalization
ã being a constant matrix. The BRS operator is still nilpotent but its square only vanishes when the equation F(A) = 0 is satisfied. Therefore, the property (21.22) is not shared by all realizations of BRS transformations, and may require the introduction of additional auxiliary variables. The results which follow can be proven in all formulations.

21.4 WT Identities and Master Equation

Although we are interested only in A field correlation functions, to study the consequence of the BRS symmetry (21.19), it is necessary to introduce sources for all fields and all operators generated by BRS transformations:

(21.25)                    Gauge Theories: Master Equation And Renormalization
with
(21.26)                    Gauge Theories: Master Equation And Renormalization
(p.537) Under a BRS transformation, only the source terms for A,                    Gauge Theories: Master Equation And Renormalization                and C vary:
(21.27)                    Gauge Theories: Master Equation And Renormalization
This implies an equation for Z:
(21.28)                    Gauge Theories: Master Equation And Renormalization
The generating functional W of connected correlation functions satisfies the same equation:
(21.29)                    Gauge Theories: Master Equation And Renormalization
We then perform a Legendre transformation:
(21.30)                    Gauge Theories: Master Equation And Renormalization
with
(21.31)                    Gauge Theories: Master Equation And Renormalization
Equations (21.30, 21.31) imply as usual for the sources coupled to composite fields:
(21.32)                    Gauge Theories: Master Equation And Renormalization
Therefore, the generating functional of proper vertices Γ satisfies
(21.33)                    Gauge Theories: Master Equation And Renormalization
As in the case of the non-linear σ-model, because the transformations (21.19) involve composite operators, the WT identity satisfied by Γ takes the form of a quadratic equation. We thus expect that the transformations (21.19) will in general be renormalized.

Master equation.

In the classical limit, the 1PI functional Γ reduces to the action. We infer that the action also satisfies equation (21.33), a property that can be easily verified by a direct calculation:

(21.34)                    Gauge Theories: Master Equation And Renormalization
This master equation is the basic equation for the discussion of the renormalization of quantized gauge theories.

First, it implies without additional assumptions equation (21.33) as we now show. Equation (21.34) can be used in the form

(21.35)                    Gauge Theories: Master Equation And Renormalization
(p.538) We integrate by parts over A i,                    Gauge Theories: Master Equation And Renormalization                and C α using
                   Gauge Theories: Master Equation And Renormalization
Equation (21.35) then becomes
(21.36)                    Gauge Theories: Master Equation And Renormalization
In the regularized theory                    Gauge Theories: Master Equation And Renormalization                and                    Gauge Theories: Master Equation And Renormalization                , which are proportional to traces of matrices belonging to an orthogonal representation of the Lie algebra, vanish.

Note, however, that in the case of fermions and chiral gauge transformations the corresponding property is not be necessarily true and this may be the source of anomalies (Section 20.3).

Then, equation (21.36) can be rewritten as

(21.37)                    Gauge Theories: Master Equation And Renormalization
This equation directly leads to the WT identities (21.33).

The quantities                    Gauge Theories: Master Equation And Renormalization                and                    Gauge Theories: Master Equation And Renormalization                still vanish after renormalization if some group structure is preserved or quite generally if dimensional regularization has been used.

We now prove that the master equation (21.34), unlike the structure (21.26), is stable under renormalization.

21.5 Renormalization: General Considerations

We now prove the stability of the master equation following a method already explained in the example of the non-linear σ-model (see Chapter 14). We assume that the local action (21.26) has been regularized in a way compatible with gauge invariance. Perturbation theory then exhibits UV divergences which have to be removed by adding counter-terms to the action. The identities (21.33) imply relations among divergences. We use them to prove that equation (21.34) is stable under renormalization. In the next section, we then solve equation (21.34) to find the most general form of the renormalized action.

21.5.1 Counter-terms and master equation

As usual our analysis is based on a loop expansion of the regularized functional Γ, the first term being the unrenormalized action S:

(21.38)                    Gauge Theories: Master Equation And Renormalization
To simplify notations, we introduce an auxiliary field μα and add to the action (21.16) and to the 1PI functional Γ the combination −μαλα. The new functional then satisfies
(21.39)                    Gauge Theories: Master Equation And Renormalization
(p.539) and equations (21.33) and (21.34) become homogeneous quadratic equations which can be written symbolically as
(21.40)                    Gauge Theories: Master Equation And Renormalization
(21.41)                    Gauge Theories: Master Equation And Renormalization
The general arguments presented in Section 14.6 now prove that these equations are stable under renormalization. We recall here only the main steps.

Assuming as an induction hypothesis that we have been able to construct a renormalized action                    Gauge Theories: Master Equation And Renormalization                which satisfies equation (21.40) and renders Γ finite at ℓ −1 loop order, we write the consequences of equation (21.41) at loop order ℓ:

(21.42)                    Gauge Theories: Master Equation And Renormalization
The r.h.s. of the equation is finite by induction. We define                    Gauge Theories: Master Equation And Renormalization                as the sum of the divergent terms in the asymptotic expansion of Γ in terms of the regularizing parameter. The divergent part of Γ then satisfies
(21.43)                    Gauge Theories: Master Equation And Renormalization
Equation (21.43) shows that by defining the ℓ-loop renormalized action S by
(21.44)                    Gauge Theories: Master Equation And Renormalization
it is possible to render Γ ℓ-loop finite with a renormalized action still satisfying equation (21.40).

21.5.2 Solution of WT identities: general considerations

It now remains to solve equations (21.43) and (21.40) using power counting arguments to find the general form of the counter-terms and of the renormalized action. It is, however, useful to first exhibit several properties of these equations.

It is convenient to introduce some notation. We denote below by θi the set of all anticommuting fields K i,                    Gauge Theories: Master Equation And Renormalization                , C α and x i all commuting fields A i, L α, μα. As we see on the explicit expression (21.34), the master equation for the action S (and thus Γ) then takes the form

(21.45)                    Gauge Theories: Master Equation And Renormalization
Equation (21.43) reads
(21.46)                    Gauge Theories: Master Equation And Renormalization
where                    Gauge Theories: Master Equation And Renormalization                is the differential operator:
(21.47)                    Gauge Theories: Master Equation And Renormalization
or in explicit form
(21.48)                    Gauge Theories: Master Equation And Renormalization

(p.540) The nilpotency of the operator                    Gauge Theories: Master Equation And Renormalization                .

Quite remarkably the master equation (21.45) implies that                    Gauge Theories: Master Equation And Renormalization                vanishes as we now show. Since                    Gauge Theories: Master Equation And Renormalization                is of anticommuting type only the terms generated by the non-commutation of                    Gauge Theories: Master Equation And Renormalization                and                    Gauge Theories: Master Equation And Renormalization                with the differential operators ∂/∂θi and ∂/∂x i survive in                    Gauge Theories: Master Equation And Renormalization                :

(21.49)                    Gauge Theories: Master Equation And Renormalization
The expression can be rewritten as
(21.50)                    Gauge Theories: Master Equation And Renormalization
as a consequence of equation (21.45). The operator                    Gauge Theories: Master Equation And Renormalization                , which plays an essential role in solving WT identities, again is a BRS operator with vanishing square.

Canonical invariance of the master equation.

Equation (21.45) has properties analogous to the symplectic form                    Gauge Theories: Master Equation And Renormalization                of classical mechanics, it is invariant under canonical transformations. Let us change variables                    Gauge Theories: Master Equation And Renormalization                :

(21.51)                    Gauge Theories: Master Equation And Renormalization
(21.52)                    Gauge Theories: Master Equation And Renormalization
in which φ(x′, θ) is an anticommuting type function. We first eliminate x i in equation (21.45) using equation (21.51):
(21.53)                    Gauge Theories: Master Equation And Renormalization
We then eliminate θi using (21.52). We verify that we recover equation (21.45) in the new variables:
(21.54)                    Gauge Theories: Master Equation And Renormalization

Infinitesimal transformations.

We consider the infinitesimal form of canonical transformations, that is, expand the function φ to first order in a parameter ε:

(21.55)                    Gauge Theories: Master Equation And Renormalization
Then,
(21.56)                    Gauge Theories: Master Equation And Renormalization
We now calculate S(θ′, x′)
(21.57)                    Gauge Theories: Master Equation And Renormalization
(p.541) where the definition (21.47) has been used.

We thus find that any infinitesimal addition to S of a BRS exact term can be obtained by a canonical transformation acting on S.

The effect of the quantization procedure can be understood as such a transformation. In our original problem the dependence on μα, which is an artificial variable, cannot change. This imposes the dependence on μα of the function φ in equations (21.51, 21.52):

                   Gauge Theories: Master Equation And Renormalization
It follows that the general change of variables is equivalent to a change induced by the function                    Gauge Theories: Master Equation And Renormalization                on the restricted set                    Gauge Theories: Master Equation And Renormalization                with in addition the translation
(21.58)                    Gauge Theories: Master Equation And Renormalization
One verifies that the gauge invariant action with the K and L source terms, and the renormalized quantized action (21.76) are related by such a transformation with
(21.59)                    Gauge Theories: Master Equation And Renormalization

21.6 The Renormalized Action

We have shown that the renormalized action satisfies the master equation (21.34). To find the form of the renormalized action it is thus necessary to find the most general solution of equation (21.34) local in the fields and sources, and consistent with power counting. We work in four dimensions and assume a renormalizable gauge.

21.6.1 General gauges

We have to solve equation (21.34) taking into account the power counting, symmetries and locality. First, we note that in the original action only the product                    Gauge Theories: Master Equation And Renormalization                appears. This leads to ghost number conservation. If we assign a ghost number +1 to                    Gauge Theories: Master Equation And Renormalization                and −1 to C, then K i has a ghost number −1, and L α −2.

Power counting.

In four dimensions, the lagrangian density has dimension 4. We choose the gauge fixing term in such a way that the field A has the minimal dimension, that is, 1 and F(A) dimension 2 (we have exhibited such gauges in Chapters 18 and 19),

                   Gauge Theories: Master Equation And Renormalization
We impose [λ] = 2, by choosing w(λ) such λ has a constant propagator:
(21.60)                    Gauge Theories: Master Equation And Renormalization
The other dimensions follow. Since                    Gauge Theories: Master Equation And Renormalization                has a term linear in A and a constant part with one derivative it has dimension 1. The operator M αβ then has dimension 2. This implies that
                   Gauge Theories: Master Equation And Renormalization
By convention, we can choose                    Gauge Theories: Master Equation And Renormalization                , [C] = 2. This implies
                   Gauge Theories: Master Equation And Renormalization
(p.542) Summarizing,
(21.61)                    Gauge Theories: Master Equation And Renormalization
Since K and L have dimensions 3 and 4, respectively, the renormalized action can contain at most terms linear in K and L. Similarly, since λ has dimension 2, the renormalized action is at most a polynomial of second degree in λ, and the coefficient of λαλβ is a constant matrix.

The solution.

To solve equation (21.34), we now parametrize the solution in a way reminiscent of the initial action (21.26), but it should be kept in mind that the parameters which appear are renormalized and in general different from those parametrizing the action (21.26). The subscript “renormalized” is always implied, and is omitted only for notational simplicity.

Power counting implies that S is an affine function of K and L, and ghost number conservation implies that only the combinations                    Gauge Theories: Master Equation And Renormalization                and                    Gauge Theories: Master Equation And Renormalization                can appear. We thus set

                   Gauge Theories: Master Equation And Renormalization
Power counting tells us in addition that                    Gauge Theories: Master Equation And Renormalization                has dimension 0 and is thus field independent and                    Gauge Theories: Master Equation And Renormalization                has dimension 1 and can, therefore, only be an affine function of A.

The terms linear in L and K in equation (21.34) yield, respectively:

(21.62)                    Gauge Theories: Master Equation And Renormalization
(21.63)                    Gauge Theories: Master Equation And Renormalization
The first equation implies that the constants                    Gauge Theories: Master Equation And Renormalization                satisfy a Jacobi identity since the product                    Gauge Theories: Master Equation And Renormalization                is antisymmetric in (γ, δ, ε). They are, therefore, structure constants of a Lie algebra. It remains to show that the Lie algebra has not changed. This is straightforward when the gauge condition does not break the global symmetry. Otherwise, when the original algebra is semi-simple this follows from a continuity argument. Finally, in the general case one can still use gauge independence of physical observables as derived in Section 21.7. The structure constants are thus linearly related to the structure constants appearing in the initial action (21.26).

Equation (21.62) is also an integrability condition for equation (21.63) which implies the commutation relations:

(21.64)                    Gauge Theories: Master Equation And Renormalization
Therefore, the whole group structure is recovered. In terms of the BRS operator
(21.65)                    Gauge Theories: Master Equation And Renormalization
the two equations can be combined into a unique equation                    Gauge Theories: Master Equation And Renormalization                . The master equation then reduces to a condition of BRS symmetry. Defining also
(21.66)                    Gauge Theories: Master Equation And Renormalization
(p.543) we can write it as
(21.67)                    Gauge Theories: Master Equation And Renormalization
with
(21.68)                    Gauge Theories: Master Equation And Renormalization
Because D + and D have a different power of                    Gauge Theories: Master Equation And Renormalization                it is natural to expand                    Gauge Theories: Master Equation And Renormalization                in powers of                    Gauge Theories: Master Equation And Renormalization                , a situation we have already met in Section 16.5.2. Since the product                    Gauge Theories: Master Equation And Renormalization                has dimension 2, S is a polynomial of degree 2 in                    Gauge Theories: Master Equation And Renormalization                :
                   Gauge Theories: Master Equation And Renormalization
Moreover, λ has dimension 2 and, therefore, S (0) is a polynomial of degree 2 in λ, S (1) of degree 1 and S (2) is both λ and A independent. The                    Gauge Theories: Master Equation And Renormalization                independent term S (0) can be parametrized as
(21.69)                    Gauge Theories: Master Equation And Renormalization
The equation                    Gauge Theories: Master Equation And Renormalization                is automatically satisfied, and we note that S (0) can be written as
(21.70)                    Gauge Theories: Master Equation And Renormalization
with
(21.71)                    Gauge Theories: Master Equation And Renormalization
More generally one verifies for any function Σ polynomial in λ:
                   Gauge Theories: Master Equation And Renormalization
The other equations can be written as
                   Gauge Theories: Master Equation And Renormalization
The equation n = 0 for λ = 0 reduces to
(21.72)                    Gauge Theories: Master Equation And Renormalization
This equation implies that S(A) is gauge invariant.

Then, from equation (21.70),

                   Gauge Theories: Master Equation And Renormalization
The general solution is the sum of a special solution, and a D exact term
(21.73)                    Gauge Theories: Master Equation And Renormalization
where Φ(1) is proportional to                    Gauge Theories: Master Equation And Renormalization                :
(21.74)                    Gauge Theories: Master Equation And Renormalization
Power counting implies that g βγδ is a constant.

(p.544) The next equation becomes

                   Gauge Theories: Master Equation And Renormalization
whose solution can be written as
(21.75)                    Gauge Theories: Master Equation And Renormalization
because power counting prevents the addition of another D exact term. Finally, we note that the last equation D + S (2)is automatically satisfied. Summing all contributions, we find that the renormalized action can be written as
(21.76)                    Gauge Theories: Master Equation And Renormalization
where S(A) is gauge invariant and
(21.77)                    Gauge Theories: Master Equation And Renormalization
The renormalized action has a form similar to the bare action (21.26) except for an additional BRS exact term
(21.78)                    Gauge Theories: Master Equation And Renormalization
This term has exactly the form given in equation (21.57) and can be associated with a shift                    Gauge Theories: Master Equation And Renormalization                .

The quartic ghost term.

A comment now is in order: since in general the renormalized action is quartic in the ghost terms, in contrast to the initial action, the direct interpretation of the ghost integral as representing a determinant in local form is lost. However, the following result can be proven: if one adds to the gauge function F α a term linear in an auxiliary field transforming non-trivially under the gauge group, then the integration over this auxiliary field with an appropriate gaussian weight generates the quartic ghost terms in their most general form.

This property is expected from the general analysis of Chapter 16.

Renormalization of gauge invariant operators.

To generate correlation functions with operator insertions, one can add sources for them in the action. If the dimension of the gauge invariant operators is at most 4 the new action is still renormalizable. The general analysis is not modified; the only difference is that some coupling constants are now space-dependent. In the case of operators of higher dimensions, the action with sources is no longer renormalizable. It is still possible to renormalize it at any finite order by introducing enough renormalization constants. The determination of the general form of the renormalized action, that is, the solution of equation (21.34) is a non-trivial problem and requires more sophisticated cohomology techniques. In the case of compact Lie groups with semi-simple Lie algebras, the most general solution of equation (21.46) is the sum of gauge invariant terms and BRS exact contributions, that is, of the form Dφ. This result first conjectured has now been rigorously proven. The part concerning C, λ is simple but the difficulties come from the set                    Gauge Theories: Master Equation And Renormalization                . Note that the form of the renormalized operators, when inserted in field correlation functions, depends on the explicit gauge. Only the averages of products of gauge invariant operators, or the matrix elements between physical states, as we show in Section 21.7, are gauge-independent.

(p.545) 21.6.2 Linear gauges

For a special class of gauges, the preceding analysis can be simplified. This class is characterized by the property that the gauge fixing function F α(A) is linear in the field A, rather than quadratic as in the most general renormalizable case:

(21.79)                    Gauge Theories: Master Equation And Renormalization
In this case, the operator F α(A) is in general still of dimension 2, but its correlation functions are now directly related to the correlation functions of the A i field and, therefore, introduce no new independent renormalization.

To derive the consequences of equation (21.79), we use the λ-field equation of motion. We again explicitly parametrize w(λ) as

(21.80)                    Gauge Theories: Master Equation And Renormalization
Then, the λ-field equation of motion, which relies on the identity
                   Gauge Theories: Master Equation And Renormalization
with S given by equation (21.26), reads
(21.81)                    Gauge Theories: Master Equation And Renormalization
or in terms of the functional                    Gauge Theories: Master Equation And Renormalization                :
(21.82)                    Gauge Theories: Master Equation And Renormalization
Since F(A) is linear in A, the λ-field equation of motion is a first order differential equation and its implication for the generating functional of proper vertices Γ is simple:
(21.83)                    Gauge Theories: Master Equation And Renormalization
Equation (21.83) is satisfied by the action S and clearly is stable under renormalization. It implies that the quadratic and linear parts in λ of the action are unrenormalized. In particular, no term of the form                    Gauge Theories: Master Equation And Renormalization                can be generated. The action remains quadratic in the ghost fields. The renormalized action takes the simple form
(21.84)                    Gauge Theories: Master Equation And Renormalization
(D being defined by equation (21.67)) where in addition as stated above a αβ and F αi are unrenormalized.

Remark.

In the case of linear gauges another equation can be used to show that the gauge function is unrenormalized, the C ghost equation of motion. The equation

(21.85)                    Gauge Theories: Master Equation And Renormalization
(p.546) implies
(21.86)                    Gauge Theories: Master Equation And Renormalization
and thus, after Legendre transformation,
(21.87)                    Gauge Theories: Master Equation And Renormalization
Equation (21.85) is thus stable under renormalization.

We mention this property here for the following reasons: in the case of linear gauges, the introduction of the λ-field is not always useful, except for strict gauge conditions (generalized Landau gauges). If we integrate over λα and set l α to zero, then equation (21.83) disappears, while equation (21.87) remains and can be used to show that the gauge fixing term F(A) is not renormalized.

21.7 Gauge Independence

General correlation functions are gauge-dependent and, therefore, cannot be associated with physical observables. Using arguments similar to those given in Section 16.5.2, we show here that expectation values of gauge invariant operators and S-matrix elements are unaffected by infinitesimal changes of gauges. This establishes gauge independence at least for gauges which can be continuously connected, and confirms that expectation values of gauge invariant operators and S-matrix elements are physical observables.

Gauge invariant operators.

To generate correlation functions of gauge invariant operators, we add source terms for them to the action. The action with these sources is still gauge invariant (equation (21.3)) and the effective action BRS symmetric. Assuming a gauge invariant regularization, we examine how the vacuum amplitude is affected by an infinitesimal change of gauge δF α, before renormalization.

Since the non-gauge invariant part of the action has the general form D 0φ in which D 0 has been defined by equation (21.21) (see equation (21.23)), the variation δS of the action takes the form                    Gauge Theories: Master Equation And Renormalization                , and thus the variation of the vacuum amplitude is

(21.88)                    Gauge Theories: Master Equation And Renormalization
The operator D 0 is a differential operator and, therefore, we can integrate by parts. Using again the property that the traces                    Gauge Theories: Master Equation And Renormalization                and                    Gauge Theories: Master Equation And Renormalization                vanish we obtain
(21.89)                    Gauge Theories: Master Equation And Renormalization
and this expression vanishes as a consequence of the BRS symmetry.

We have, therefore, shown the bare correlation functions of gauge invariant operators are gauge-independent, at least within a class of gauges which can be continuously connected. There exists, therefore, a renormalization procedure which produces gauge-independent renormalized correlation functions of these operators.

These correlation functions contain the complete information about the physical properties of the gauge theory.

(p.547)

                   Gauge Theories: Master Equation And Renormalization

Fig. 21.1

S-matrix elements.

We now want to study the gauge independence of the perturbative S-matrix, when it exists. We first calculate the variation of renormalized correlation functions under an infinitesimal change of gauge. We assume that we have renormalized the theory in a given gauge, but not yet eliminated the regularizing parameter.

We then proceed as above. The variation of the action in an infinitesimal change of gauge has exactly the form Dδφ(D being now the renormalized BRS operator) considered in Section 21.6.1. We can still integrate by parts, but the resulting integrand does not vanish identically because we have introduced sources for non-gauge invariant fields:

(21.90)                    Gauge Theories: Master Equation And Renormalization
When we calculate correlation functions we obtain a sum of contributions in which one A i field has been replaced by                    Gauge Theories: Master Equation And Renormalization                , which is a linear combination of composite operators. When we go to the mass-shell, after amputation, we get a non-vanishing contribution only if there is a pole in each external momentum squared. For a composite operator, this happens only if the line is one particle reducible. Then, on the mass-shell, we get a contribution proportional to the matrix element of the field itself (see figure 21.1). This argument was presented for the first time in Appendix A7.2. The final result is that an infinitesimal change of gauge renormalizes multiplicatively the S-matrix elements. This corresponds to a field amplitude renormalization. Therefore, the S-matrix, properly normalized, is gauge-independent.

Bibliographical Notes

Renormalization of gauge theories is discussed in G. ’t Hooft, Nucl. Phys. B33 (1971) 173; ibidem B35 (1971) 167; A.A. Slavnov, Theor. Math. Phys. 10 (1972) 99; J.C. Taylor, Nucl. Phys. B33 (1971) 436; B.W. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3121, 3137, 3155; ibidem D7 (1973) 1049; G. ’t Hooft and M. Veltman, Nucl. Phys. B50 (1972) 318; D.A. Ross and J.C. Taylor, Nucl. Phys. B51 (1973) 125; B.W. Lee, Phys. Lett. 46B (1973) 214; Phys. Rev. D9 (1974) 933 .

The anticommuting type symmetry of the quantized action is exhibited in C. Becchi, A. Rouet and R. Stora, Comm. Math. Phys. 42 (1975) 127, where it is used to renormalize the abelian Higgs model. In H. Kluberg-Stern and J.-B. Zuber, Phys. Rev. D12 (1975) 467, quadratic WT identities generated by BRS symmetry appear and and the renormalization of a class of gauge invariant operators is discussed.

The first derivation of the master equation for the quantized action and the proof of renormalizability in an arbitrary gauge presented here, are found in the proceedings of the Bonn summer school 1974, J. Zinn-Justin in Trends in Elementary Particle Physics (Lectures Notes in Physics 37), H. Rollnik and K. Dietz eds. (Springer-Verlag, Berlin 1975).

(p.548)

For more details, see also J. Zinn-Justin in Proc. of the 12th School of Theoretical Physics, Karpacz 1975, Acta Universitatis Wratislaviensis 368; B.W. Lee in Methods in Field Theory, Les Houches School 1975, R. Balian and J. Zinn-Justin eds. (North-Holland, Amsterdam 1976); J. Zinn-Justin, Mod. Phys. Lett. A19 (1999) 1227 .

An alternative proof based on BRS symmetry and the BPHZ formalism can be found in C. Becchi, A. Rouet and R. Stora, Ann. Phys. (NY) 98 (1976) 287

Renormalization of background field gauges is discussed in H. Kluberg-Stern and J.-B. Zuber, Phys. Rev. D12 (1975) 482, 3159 .

A general study of gauge invariant operators can be found in S.D. Joglekar and B.W. Lee, Ann. Phys. (NY) 97 (1976) 160 .

The complete proof of the conjectured result is found in G. Barnich, and M. Henneaux, Phys. Rev. Lett. 72 (1994) 1588; G. Barnich, F. Brandt and M. Henneaux, Coram. Math. Phys. 174 (1995) 93 .

Attempts to quantize in quadratic gauges include G. ’t Hooft and M. Veltman, Nucl. Phys. B50 (1972) 318; S.D. Joglekar, Phys. Rev. D10 (1974) 4095 .

Consequences of BRS symmetry in the operator formalism are exhibited in T. Kugo and I. Ojima, Phys. Lett. B73 (1978) 459, Nucl. Phys. B144 (1978) 234 .

The problem of non-linear gauges has been discussed in its full generality in J. Zinn-Justin, Nucl. Phys. B246 (1984) 246 .