Gauge Theories: Master Equation And Renormalization
Gauge Theories: Master Equation And Renormalization
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.
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
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:
Finally, S(A) is the gauge invariant action and thus satisfies
A group identity.
We have explained in Section 15.3 that, as a consequence of the group structure, quite generally the functionals satisfy equations which can be regarded as compatibility conditions for the system (21.3) considered as a set of differential equations for S(A):
Since here the functional 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 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 which are proportional to the numerical structure constants of the Lie algebra, have a non-trivial dependence on space variables:
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:
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:
We then insert the identity (21.11) into the representation of the partition function in the temporal gauge and change variables . 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
21.3 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):
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
To express the BRS symmetry, it is also useful to introduce the anticommuting differential operator
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 now takes the form
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:
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:
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
The quantities and still vanish after renormalization if some group structure is preserved or quite generally if dimensional regularization has been used.
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:
Assuming as an induction hypothesis that we have been able to construct a renormalized action 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:
21.5.2 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, , 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
Quite remarkably the master equation (21.45) implies that vanishes as we now show. Since is of anticommuting type only the terms generated by the non-commutation of and with the differential operators ∂/∂θi and ∂/∂x i survive in :
Canonical invariance of the master equation.
Equation (21.45) has properties analogous to the symplectic form of classical mechanics, it is invariant under canonical transformations. Let us change variables :
We consider the infinitesimal form of canonical transformations, that is, expand the function φ to first order in a parameter ε:
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):
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 appears. This leads to ghost number conservation. If we assign a ghost number +1 to and −1 to C, then K i has a ghost number −1, and L α −2.
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),
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 and can appear. We thus set
The terms linear in L and K in equation (21.34) yield, respectively:
Then, from equation (21.70),
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 . 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.
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:
To derive the consequences of equation (21.79), we use the λ-field equation of motion. We again explicitly parametrize w(λ) as
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
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 , and thus the variation of the vacuum amplitude is
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.
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:
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).
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 .