Renormalized Schwinger-Dyson functional

We consider the perturbative renormalization of the Schwinger-Dyson functional, which is the generating functional of the expectation values of the products of the composite operator given by the field derivative of the action. It is argued that this functional plays an important role in the topological Chern-Simons and BF quantum field theories. It is shown that, by means of the renormalized perturbation theory, a canonical renormalization procedure for the Schwinger-Dyson functional is obtained. The combinatoric structure of the Feynman diagrams is illustrated in the case of scalar models. For the Chern-Simons and the BF gauge theories, the relationship between the renormalized Schwinger-Dyson functional and the generating functional of the correlation functions of the gauge fields is produced.


Introduction
The Schwinger-Dyson equations [1,2] of quantum field theory can be derived [3,4] from the invariance of the functional integration under field translations. The structure of the Schwinger-Dyson equations is determined by the action functional, which is involved in the computation of the vacuum expectation values of the fields. Let the action S[φ] be a function of a set of fields denoted by φ(x). The basic Schwinger-Dyson equation takes the form δS [φ] δφ(x) φ(y 1 )φ(y 2 ) · · · φ(y n ) = i n j=1 δ(x−y j ) φ(y 1 ) · · · φ(y j−1 )φ(y j+1 ) · · · φ(y n ) , (1.1) where the vacuum expectation value P[φ] of a field operator P[φ] is given by Dφ e iS [φ] . (1.2) Recently, developments of the Schwinger-Dyson equations have been applied in the study of various subjects like, for instance, the renormalization theory [5,6], condensed matter investigations [7,8], and bound states and strong interactions [9,10,11,12,13,14,15,16,17,18]. Standard Schwinger-Dyson equations have been used also in the case of topological quantum field theories with and without matter [19,20,21]. We are interested in a particular generalisation of equation (1.1) which concerns the computation of the expectation values of the products of the composite operator δS[φ]/δφ(x), δφ(x n ) . where B(x) denotes a classical source. The functional (1.4) plays an important role in the low-dimensional gauge field theories of topological type, like the Chern-Simons and BF quantum field theories [22,23,24]. In these models, the derivative of the action with respect to the components of the connection is proportional to the curvature (plus possible additional contributions which are related to the gauge-fixing lagrangian terms), that combined with the topology of 3-manifolds determines the values of the Wilson line observables [22].
In facts, when the gauge structure group of these topological models is abelian, the Schwinger-Dyson functional provides the complete solution for the gauge invariant observables [22,23].
For instance, when the first homology group [25] H 1 (M) of the 3-manifold M is trivial, one can compute [22] the observables of the abelian Chern-Simons theory defined in M by means of perturbation theory. The action for the connection A is given by 2πk A ∧ dA and the variation of the action with respect to the fields is proportional to the curvature F A = dA. By introducing the coupling B ∧ dA of the curvature with an external classical source B = B µ (x)dx µ , one finds that specifies the expectation values of the Wilson lines associated with links in M. Quite remarkably, an appropriate generalization [22] of this procedure furnishes the solution of the abelian Chern-Simons theory in a generic closed and oriented 3-manifold M. Indeed, when the first homology group H 1 (M) is not trivial, for each element of the torsion subgroup [25] of H 1 (M) one can introduce a corresponding classical background connection. Then one needs to take the sum of the Schwinger-Dyson functionals that are computed in the presence of each background connection. Somehow, in the functional integration, the values of the curvature correspond to the local degrees of freedom -which do not depend on the topology of the manifold Mwhereas the effects of topology are taken into account by the background connections.
In the case of the non-abelian SU(N) Chern-Simons theory, the structure of the gauge orbits, which are associated with the SU(N) connections, does not admit [26] a simple description based on the homology group H 1 (M). Yet, in the characterization of the local degrees of freedom which are not related with the manifold topology, the non-abelian curvature F A = 2dA+i[A, A] appears to play a fundamental role. In fact, the value F A (x) of the curvature in the point x is specified [4,27] by the value of the gauge holonomy associated an infinitesimal loop centered in x, and each infinitesimal loop does not depend on the topology of M. Let us present a rough sketch of a possible argument that can be used to make this statement more precise.
SU(N) gauge connections can be described by one-forms defined in M with values in the SU(N) Lie algebra. The local value {A(x)} of each configuration A can also be specified by the set {H γ [A]} of the holonomies, which are associated with all the possible closed oriented paths {γ} in M with a given basepoint x 0 ∈ M, which represents the starting/final point of each closed path γ. This correspondence is denoted by In turn, the value of the holonomies H γ [A] as a function of the paths can be determined by combining the local values {F A (x)} of the curvature with smooth deformations of the paths. In order to illustrate this point, let us consider a nontrivial reference path γ 0 , with parametrization . An infinitesimal deformation γ 0 + δγ of the path γ 0 can be described by the parametrization x µ (τ ) + ǫ µ (τ ) with ǫ µ (τ ) ≪ 1. At first order in ǫ µ , one has H γ 0 +δγ ≃ H γ 0 + ∆H γ 0 , where the infinitesimal modification ∆H γ 0 of the holonomy,  How to carry out the precise disentanglement of the "purely local" degrees of freedomdescribed by the curvature-and the "topology dependent" degrees of freedom in the functional integration is an open problem. In order to investigate this issue in the case of the nonabelian Chern-Simons and BF theories, in the present article we analyze a preliminary question which is related to the perturbative computation of the renormalized generating functional of the vacuum expectation values of the products of the nonabelian curvature F A (x) in different points of spacetime. Indeed, for the topological abelian gauge theories the renormalization is trivial, whereas in the non-abelian case the renormalization task is not trivial. The main purposes of our article is to show how the renormalization of the corresponding Schwinger-Dyson functional Z SD [B] is canonically determined by the standard renornalization procedure [28,29] for the correlation functions of the gauge connections.
We demonstrate that, in the Chern-Simons and BF theories, the renormalized Schwinger-Dyson functional is related with the generating functional Z[J] of the correlation functions of the gauge connections by some kind of duality transformation. Therefore the standard perturbative procedure called "renormalized perturbation theory" [4] provides a canonical renormalization for Z SD [B]. Note that we are not interested in the matrix elements the composite operator δS[φ]/δφ(x) between generic states; this issue can be studied by means of standard techniques [4,29,30]. Motivated by the results of the topological models with an abelian gauge group, we shall concentrate on the vacuum expectation values of products of operators δS[φ]/δφ(x). In this case, the relationship that we derive between Z SD [B] and Z [J] shows that the standard technique [4,29,30] for the study of the renormalization properties of the composite operator δS[φ]/δφ(x) greatly simplifies.
Let us remember that the renormalization of the lagrangian field theory models is expected [31,32] to be independent of the global aspects of the manifold that do not modify the shortdistance behaviour of the theory. Therefore, since the nonabelian curvature F A (x) describes degrees of freedom which do not depend on the topology of the manifold, we shall consider the renormalization properties of Z SD [B] in flat spacetime.
The combinatoric structure of the Feynman diagrams -entering the perturbative computation of Z SD [B]is illustrated in the simple case of the field theory models φ 3 and φ 4 in four dimensions in Section 2. By means of the Wick contractions [3,4] of the field operators, we examine the Feynman diagrams which are associated with the expectation values (1.3). We demonstrate that the short distance behaviour of the products of the composite operator Applications and extensions of the results of Section 2 are presented in Section 3, where low dimensional gauge theories of topological type are considered. For the nonabelian SU(N) Chern-Simons model and the ISU(2) BF gauge theory in R 3 , the relationship between the renormalized Schwinger-Dyson functional and the generating functional of the correlation functions of the gauge fields is produced. Section 4 contains the conclusions.

Structure of the Feynman diagrams
The case of a cubic interaction lagrangian is relevant for the topological gauge theories in low dimensions. So, let us first consider the field theory model which is defined by the action where φ(x) is a real scalar field and the real parameter g denotes the coupling constant. The generating functional Z[J] of the correlation functions of the field φ(x) is defined by The renormalization of Z[J] is obtained by means of the standard procedure denominated "renormalized perturbation theory" [4]. In this scheme, the lagrangian parameters assume their renormalized values and, in order to maintain the validity of the normalization conditions at each order of perturbation theory, local counterterms are introduced, which cancel exactly all the contributions to these parameters which are obtained in the loop expansion. The normalization conditions for the model defined be the action (2.1) concern the values of the mass, of the coupling constant and the wave function normalization. Finally, in order to complete the list of the normalization conditions, one needs to require the absence of a proper vertex which is linear in the field. Let Γ[ϕ] be the effective action which corresponds to the sum of the one-particleirreducible diagrams with external legs represented by ϕ(x). In agreement with the structure of the lagrangian (2.1), the additional normalization condition is given by (δΓ/δϕ(x)) | ϕ=0 = 0. Note that, in the case of the φ 4 model, the vanishing of the proper vertices which are linear and cubic in powers of the fields is a consequence of the symmetry ϕ → −ϕ which is imposed to the effective action. In the case of gauge fields, the analogue of the condition (δΓ/δϕ(x)) | ϕ=0 = 0 is automatically satisfied. Let us now consider the perturbative computation of the mean values (1.3). The perturbative expansion [3,4] of a generic expectation value (1.2) can be written as where S I [φ] denotes the integral of the interaction lagrangian The composite operator δS[φ]/δφ(x) takes the form where we have introduced the simplifying notation ∇φ( Note that the overall multiplying factor, which is given by the sum of the vacuum-to-vacuum diagrams, is not included in the set of the connected diagrams ( and consequently Because of equation (2.9), the contraction of ∇φ(x 1 ) with the fields contained in e iS I [φ] gives a vanishing result as a consequence of the sum with the term gφ 2 (x 1 ), as shown in equation (2.10) for the case of E φ (x) e iS I [φ] c 0 . So we must consider the contraction of ∇φ(x 1 ) with E φ (x 2 ), which produces (2.11) Thus one finds The normalization condition (δΓ/δϕ(x)) | ϕ=0 = 0 on the absence of the tadpole implies φ(x 2 ) e iS I [φ] c 0 = 0. Therefore The same arguments illustrated above give (2.14) The structure of the diagrams associated with c 0 , for generic n ≥ 4, can be obtained by first considering all the Wick contractions of the field operators of the type ∇φ. The combinatoric of these contractions can easily be obtained by taking into account the symmetric role of the operators , as shown in equation (2.8). For the connected diagrams, we find and

Equations (2.15) and (2.16) show that the expectation value
0 . Therefore the standard renormalization procedure for the correlation functions of the field φ(x) canonically defines the renormalization for The contact terms (2.13) and (2.14) give origin to a local contribution to W SD [B]. In particular, we find c 0 , with n ≥ 4, correspond to non-local amplitudes. By collecting all the results on The sum of the contributions (2.17)- (2.20) shows that the renormalized Schwinger-Dyson functional for the φ 3 model satisfies The perturbative expansion of the Schwinger-Dyson functional Z SD [B] can be examined by means of the method described above. We find In this case, Z SD [B] is related to the expectation value of a term in which, in addition to a coupling with the field operator φ(x), a coupling with the operator φ 2 (x) is also present. Note that the short distance behaviour of the composite operator φ 2 (x) is taken into account by the standard renormalization of the generating functional Z[J] of the correlation functions because φ 2 (x) has canonical dimension 2. For instance, the one-loop correlation φ 2 (x)φ 2 (y) is described by the diagram of Figure 1 (with removed external legs), which enters the ordinary renormalization of the φ 4 theory. Thus, for this model also, the renormalization of Z SD [B] can be specified by the standard renormalization procedure.

Topological models
In this section we consider gauge theories of topological type in R 3 . The action of the SU(N) quantum Chern-Simons theory [33,34,24] in the Landau gauge is given by and then δS δA a µ (x) and the generating functional Z[J a µ ] of the correlation functions for the gauge field A a µ (x) is given by In order to examine the diagrams entering the vacuum expectation values of the product of fields δS/δA a µ (x 1 ) · · · δS/δA b ν (x n ), one needs to use the following relationship between the propagators of the fields which can be derived from the action (3.1), or it can be checked directly by means of the expressions The presence of the operator ∂ µ c b (x)f abd c d (x) in the equation (3.2) does not modify the perturbative relations between the expectation values of the operators δS/δA a µ (x 1 ) · · · δS/δA b ν (x n ) -that have been derived in Section 2 by using the Wick contractions of the fields-because this additional term has no contractions with the gauge fields A a µ (x) and the auxiliary field M a (x). Thus, by means of the arguments presented in Section 2 , one gets and Equation (3.8) also specifies the leading term of the operator product expansion [4] δS/δA a Indeed the value of the coefficient function C 1 (x) for the identity operator O 1 = 1 is determined by expression (3.9) and it does not receive perturbative corrections. The Schwinger-Dyson functional for the abelian Chern-Simons theory can be obtained in the f abc → 0 limit. In this case, equation (3.8) becomes When L µ (x) coincides with the de Rham-Federer current [35,36,37,38] associated with a link L (with support on a Seifert surface [25] which bounds the link L ), expression (3.13) represents precisely the exponent of the linking matrix corresponding to L. For this reason, in the abelian Chern-Simons theory the Schwinger-Dyson functional provides the solution [22] for the link observables.
Finally, let us consider the ISU(2) BF theory [39,40,41,42,43] in R 3 , with fields A a µ (x) and B a µ (s) and gauge-fixed action in the Landau gauge where the real parameter g denotes a coupling constant and The non-vanishing propagators for the components of the connection and the auxiliary fields are given by [43] A a µ (x)B b ν (y) = δ ab and In this model one has (3.18) and δS δB a µ (x) The Schwinger-Dyson functional Z SD [L a µ , H a µ ] is defined by where L a µ (x) and H a µ (x) are classical sources. The generating functional Z[J a µ , K a µ ] of the correlation functions for the gauge field A a µ (x) and B a µ (x) is given by With several field components, the construction and the sum of the Feynman diagrams becomes rather laborious. We get and In addition to the contact terms, which are specified by the local functional G[L a µ , H a µ ], in the BF theory there are additional expectation values that can be displayed in their exact form. For instance, since the vacuum polarization vanishes, from equation (3.22) one derives the following relation for the connected mean value

Conclusions
For renormalizable quantum field theories we have shown that, in the perturbative computation of the corresponding Schwinger-Dyson functional Z SD , the use of standard renormalized perturbation theory provides a canonical renormalization procedure for Z SD . In facts, the short distance behaviour of the products of the composite operator δS[φ]/δφ(x) turns out to be determined by the ultraviolet properties of the field operators of dimensions 1 (and possibly 2). The explicit combinatoric of the Wick contractions of the field operators and the resulting structure of the Feynman diagrams have been illustrated in the simple cases of the φ 3 and φ 4 models. We have shown that the connected component of Z SD is the union of a local functional of the classical source and a non-local part which is specified by the expectation values of field components. The arguments that have been presented in these scalar models naturally extend to a generic theory.
In order to study possible applications of the Schwinger-Dyson functional in gauge field theories of topological type, for the non-abelian Chern-Simons and BF gauge theories, the relationship between the renormalized Schwinger-Dyson functional and the generating functional of the correlation functions of the gauge fields has been derived. In these cases, the vanishing of the loop corrections for the two-points and three-points correlation functions implies that the connected components of the renormalized Schwinger-Dyson functional containing up to six powers of the external classical sources have been produced in closed form. In these topological models, the derivative of the action with respect of the gauge fields is proportional to the curvature of the connection (plus gauge-fixing contributions). So relations (3.8) and (3.22) could possibly be used for the introduction of appropriate field variables -similar to the local gauge invariant variables decomposition [44]-which simplify the functional integration when the theory is defined in topological non-trivial manifolds.