#  Renormalized Schwinger-Dyson functional

Enore Guadagnini${}^{\,a}$ and Vittoria Urso${}^{\,b}$

${}^{a\,}$Dipartimento di Fisica E. Fermi dell’Università di Pisa, and INFN
Sezione di Pisa,
Largo B. Pontecorvo 2, 56127 Pisa, Italy.
${}^{b\,}$Dipartimento di Matematica e Fisica Ennio De Giorgi, Università del
Salento,
and IIT di Lecce, Piazza Tancredi 7, 73100 Lecce, Italy

###### Abstract

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.

##  1 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 $S[\phi]$ be a function of a
set of fields denoted by $\phi(x)$. The basic Schwinger-Dyson equation takes
the form

$\left\langle\frac{\delta
S[\phi]}{\delta\phi(x)}\,\phi(y_{1})\phi(y_{2})\cdots\phi(y_{n})\right\rangle=i\sum_{j=1}^{n}\,\delta(x-y_{j})\left\langle\phi(y_{1})\cdots\phi(y_{j-1})\phi(y_{j+1})\cdots\phi(y_{n})\right\rangle\;,$
(1.1)

where the vacuum expectation value $\langle{\cal P}[\phi]\rangle$ of a field
operator ${\cal P}[\phi]$ is given by

$\left\langle{\cal P}[\phi]\right\rangle=\frac{\int D\phi\;e^{iS[\phi]}\;{\cal
P}[\phi]}{\int D\phi\;e^{iS[\phi]}}\;.$ (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 $\delta S[\phi]/\delta\phi(x)$,

$F(x_{1},x_{2},...,x_{n})=\left\langle\frac{\delta
S[\phi]}{\delta\phi(x_{1})}\,\frac{\delta
S[\phi]}{\delta\phi(x_{2})}\cdots\frac{\delta
S[\phi]}{\delta\phi(x_{n})}\right\rangle\;.$ (1.3)

The generating functional $Z_{SD}[B]$ of the expectation values (1.3) is
called the Schwinger-Dyson functional and is defined by

$Z_{SD}[B]=\left\langle e^{i\int dx\,B(x)\,\delta
S[\phi]/\delta\phi(x)}\right\rangle\;,$ (1.4)

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 $2\pi k\int A\wedge dA$ and the variation of
the action with respect to the fields is proportional to the curvature
$F_{A}=dA$. By introducing the coupling $\int B\wedge dA$ of the curvature
with an external classical source $B=B_{\mu}(x)dx^{\mu}$, one finds

$\widetilde{Z}_{SD}[B]\equiv\left\langle e^{2\pi i\int B\wedge
dA}\right\rangle=e^{-i\,\pi/2k\int B\wedge B}\;,$ (1.5)

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
$M$— whereas 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 $\\{H_{\gamma}[A]\\}$ of the
holonomies,

$H_{\gamma}[A]={\rm P}e^{i\int_{\gamma}A}\;,$

which are associated with all the possible closed oriented paths
$\\{\gamma\\}$ in $M$ with a given base-point $x_{0}\in M$, which represents
the starting/final point of each closed path $\gamma$. This correspondence is
denoted by

$\\{A(x)\\}\;\leftrightarrow\;\\{H_{\gamma}[A]\\}\;.$ (1.6)

In turn, the value of the holonomies $H_{\gamma}[A]$ 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 $\gamma_{0}$, with
parametrization $x^{\mu}(\tau)$ in which $0\leq\tau\leq 1$. Let the holonomy
$\widehat{H}_{\gamma_{0}}(s)$ with $0\leq s\leq 1$ be defined by

$\widehat{H}_{\gamma_{0}}(s)={\rm P}e^{i\int_{0}^{s}d\tau\,A(\tau)}\;,$ (1.7)

in which $A(\tau)=A_{\mu}(x(\tau))(dx^{\mu}(\tau)/d\tau)$. Note that
$H_{\gamma_{0}}[A]=\widehat{H}_{\gamma_{0}}(1)$. An infinitesimal deformation
$\gamma_{0}+\delta\gamma$ of the path $\gamma_{0}$ can be described by the
parametrization $x^{\mu}(\tau)+\epsilon^{\mu}(\tau)$ with
$\epsilon^{\mu}(\tau)\ll 1$. At first order in $\epsilon^{\mu}$, one has
$H_{\gamma_{0}+\delta\gamma}\simeq H_{\gamma_{0}}+\Delta H_{\gamma_{0}}$,
where the infinitesimal modification $\Delta H_{\gamma_{0}}$ of the holonomy,

$\Delta
H_{\gamma_{0}}=i\int_{0}^{1}ds\,\widehat{H}_{\gamma_{0}}(s)\,\epsilon^{\mu}(s)\,\dot{x}^{\nu}(s)F_{\mu\nu}(x(s))\,\widehat{H}^{-1}_{\gamma_{0}}(s)\,H_{\gamma_{0}}[A]\,,$
(1.8)

is specified by $\widehat{H}_{\gamma_{0}}(s)$ and by the values
$\\{F_{A}(x)\\}$ of the curvature. If the path $\gamma$ is homotopically
equivalent to $\gamma_{0}$, the value of the associate holonomy
$H_{\gamma}[A]$ is expected to be determined by $H_{\gamma_{0}}[A]$ with the
help of a set of infinitesimal transformations of the path. As shown in
equation (1.8), the corresponding infinitesimal modifications of the holonomy
can be fixed by means of $\widehat{H}_{\gamma_{0}}(s)$ and the local values
$\\{F_{A}(x)\\}$ of the curvature. If the 3-manifold $M$ is simply connected,
any closed path $\gamma$ is homotopic with $\gamma_{0}$. If $M$ is not simply
connected, for each generator $g$ of the fundamental group $\pi_{1}(M)$ one
can choose a representative path $\gamma_{g}$ and, in agreement with equation
(1.7), one can compute the corresponding holonomy
$\widehat{H}_{\gamma_{g}}(s)$. For each set of homotopic paths, the associated
holonomies can be determined by combining $\widehat{H}_{\gamma_{g}}(s)$ with
the local values $\\{F_{A}(x)\\}$ of the curvature. Thus relation (1.6) should
develop into

$\\{A(x)\\}\;\leftrightarrow\;\\{H_{\gamma}[A]\\}\;\leftrightarrow\;\\{\widehat{H}_{\gamma_{1}}(s),\widehat{H}_{\gamma_{2}}(s)\,,\dots,\widehat{H}_{\gamma_{g}}(s),\dots\,;F_{A}(x)\\}\;.$
(1.9)

Consequently, in the computation of the functional integration, the sum over
the configurations $\\{A(x)\\}$ can then be envisaged to be decomposed into a
sum over the values $\\{F_{A}(x)\\}$ of the curvature and a sum over the
values of the holonomies associated with the paths which represent the
generators of the fundamental group $\pi_{1}(M)$. The values of the curvature
should correspond to the “purely local” degrees of freedom which are
independent of the topology of the manifold $M$. Whereas the values of the
holonomies
$\\{\widehat{H}_{\gamma_{1}}(s),\widehat{H}_{\gamma_{2}}(s)\,,\dots,\widehat{H}_{\gamma_{g}}(s)\\}$
should describe the effects of the nontrivial topology of $M$.

How to carry out the precise disentanglement of the “purely local” degrees of
freedom —described 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 $\delta S[\phi]/\delta\phi(x)$ 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 $\delta
S[\phi]/\delta\phi(x)$. 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 $\delta
S[\phi]/\delta\phi(x)$ 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 short-distance 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 $\phi^{3}$ and $\phi^{4}$ 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 $\delta S[\phi]/\delta\phi(x)$ —that in $d$ dimensional spacetime has
dimension $(d-1)$— is really determined by the ultraviolet properties of
operators of dimensions $(d-2)$ and/or $(d-3)$. For instance, when the
interacting lagrangian entering the action $S[\phi]$ is a cubic function of a
scalar field $\phi(x)$, $Z_{SD}[B]$ can be related to the ordinary generating
functional $Z[J]$ of correlation functions of the field $\phi(x)$. In general,
it turns out that the connected component of $Z_{SD}[B]$ is the union of a
local functional of $B(x)$ and a non-local part which is specified by the
expectation values of field components $\phi(x)$ and possibly $\phi^{2}(x)$.

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 ${\mathbb{R}}^{3}$, 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.

##  2 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

$S[\phi]=\int d^{4}x\,\left({\textstyle{\raise 0.8pt\hbox{$\scriptstyle
1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}\partial_{\mu}\phi\,\partial^{\mu}\phi-{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}m^{2}\phi^{2}+{\textstyle{\raise 0.8pt\hbox{$\scriptstyle
1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle 3$}}}}g\phi^{3}\right)\;,$ (2.1)

where $\phi(x)$ 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 $\phi(x)$ is defined by

$Z[J]=\left\langle e^{i\int d^{4}x\,J(x)\,\phi(x)}\right\rangle\;.$ (2.2)

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 $\Gamma[\varphi]$ be the effective action
which corresponds to the sum of the one-particle-irreducible diagrams with
external legs represented by $\varphi(x)$. In agreement with the structure of
the lagrangian (2.1), the additional normalization condition is given by
$\left(\delta\Gamma/\delta\varphi(x)\right)|_{\varphi=0}=0$. Note that, in the
case of the $\phi^{4}$ model, the vanishing of the proper vertices which are
linear and cubic in powers of the fields is a consequence of the symmetry
$\varphi\rightarrow-\varphi$ which is imposed to the effective action. In the
case of gauge fields, the analogue of the condition
$\left(\delta\Gamma/\delta\varphi(x)\right)|_{\varphi=0}=0$ 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

$\left\langle{\cal P}[\phi]\right\rangle=\frac{\left\langle\,{\cal
P}[\phi]\,e^{iS_{I}[\phi]}\,\right\rangle_{0}}{\left\langle\,e^{iS_{I}[\phi]}\,\right\rangle_{0}}\;,$
(2.3)

where $S_{I}[\phi]$ denotes the integral of the interaction lagrangian

$S_{I}[\phi]={\textstyle{\raise 0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower
0.8pt\hbox{$\scriptstyle 3$}}}}g\int d^{4}x\,\phi^{3}(x)\;,$ (2.4)

and the vacuum expectation value of the time-ordered product of the fields

$\left\langle\,{\cal
P}[\phi]\,e^{iS_{I}[\phi]}\,\right\rangle_{0}\equiv\langle 0|\,{\rm
T}\left({\cal P}[\phi]\,e^{iS_{I}[\phi]}\right)|0\rangle$ (2.5)

corresponds to the sum of the Feynman diagrams which are obtained by means of
the Wick contractions [32] of the fields. The set of the connected diagrams is
denoted by $\left\langle\,{\cal
P}[\phi]\,e^{iS_{I}[\phi]}\,\right\rangle_{0}^{c}$. The Feynman propagator
reads

$\langle 0|\,{\rm
T}\left(\phi(x)\,\phi(y)\right)|0\rangle=\mathop{\vbox{\halign{#\cr\kern
3.0pt\cr$\,\,\vrule depth=4.0pt,width=0.4pt\leaders\vrule
depth=0.0pt,height=0.4pt\hfill\vrule depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{\phi(x)\,\phi}\hfil$\crcr}}}\limits(y)=i\,\langle
x|\,\frac{1}{-\partial^{2}-m^{2}+i\epsilon}\,|y\rangle\;.$ (2.6)

The composite operator $\delta S[\phi]/\delta\phi(x)$ takes the form

$\frac{\delta S}{\delta\phi(x)}\equiv
E_{\phi}(x)=\left[-\partial^{2}-m^{2}\right]\phi(x)+g\phi^{2}(x)=\nabla\phi(x)+g\phi^{2}(x)\;,$
(2.7)

where we have introduced the simplifying notation
$\nabla\phi(x)\equiv\left[-\partial^{2}-m^{2}\right]\phi(x)$. The Schwinger-
Dyson functional (1.4) can be written as $Z_{SD}[B]=\exp(iW_{SD}[B])$ where
$iW_{SD}[B]$ is given by the sum of the connected diagrams

$iW_{SD}[B]=\sum_{n}\frac{i^{n}}{n!}\int d^{4}x_{1}\cdots
d^{4}x_{n}B(x_{1})\cdots B(x_{n})\,\left\langle E_{\phi}(x_{1})\cdots
E_{\phi}(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}\;.$ (2.8)

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 (2.8) contributing to $Z_{SD}[B]$. Indeed, as a consequence of the
normalisation —shown in equations (1.2) and (2.3)— of the generating
functional (1.4), the multiplying factor of the numerator simplifies —or
cancels— with the same factor of the denominator.

Let us examine the perturbative evaluation of $\left\langle
E_{\phi}(x_{1})\cdots
E_{\phi}(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$. In agreement with
the Wick Theorem, let us first consider all the possible Wick contractions of
the operator $\nabla\phi(x)$. Since
$\left[-\partial^{2}-m^{2}\right]\mathop{\vbox{\halign{#\cr\kern
3.0pt\cr$\,\,\vrule depth=4.0pt,width=0.4pt\leaders\vrule
depth=0.0pt,height=0.4pt\hfill\vrule depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{\phi(x)\,\phi}\hfil$\crcr}}}\limits(y)=i\delta^{4}(x-y)$,
the Wick contraction of $\nabla\phi(x)$ with the fields contained in
$e^{iS_{I}[\phi]}$ gives

$\left\langle\,\nabla\phi(x)\,e^{iS_{I}[\phi]}\,\right\rangle_{0}=i\left\langle\nabla\mathop{\vbox{\halign{#\cr\kern
3.0pt\cr$\,\,\vrule depth=4.0pt,width=0.4pt\leaders\vrule
depth=0.0pt,height=0.4pt\hfill\vrule depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{\phi(x)S_{I}}\hfil$\crcr}}}\limits[\phi]\,e^{iS_{I}[\phi]}\,\right\rangle_{0}=\left\langle(-g\phi^{2}(x))\,e^{iS_{I}[\phi]}\,\right\rangle_{0}\;,$
(2.9)

and consequently

$\left\langle
E_{\phi}(x)\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=\left\langle\left(\nabla\phi(x)+g\phi^{2}(x)\right)\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=0\;.$
(2.10)

Let us now consider $\left\langle
E_{\phi}(x_{1})E_{\phi}(x_{2})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$.
Because of equation (2.9), the contraction of $\nabla\phi(x_{1})$ with the
fields contained in $e^{iS_{I}[\phi]}$ gives a vanishing result as a
consequence of the sum with the term $g\phi^{2}(x_{1})$, as shown in equation
(2.10) for the case of $\left\langle
E_{\phi}(x)\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$. So we must consider the
contraction of $\nabla\phi(x_{1})$ with $E_{\phi}(x_{2})$, which produces

${\nabla\mathop{\vbox{\halign{#\cr\kern 3.0pt\cr$\,\,\vrule
depth=4.0pt,width=0.4pt\leaders\vrule depth=0.0pt,height=0.4pt\hfill\vrule
depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{\phi(x_{1})E}\hfil$\crcr}}}\limits}_{\phi}(x_{2})=i\left[-\partial^{2}-m^{2}\right]\delta^{4}(x_{1}-x_{2})+2ig\,\phi(x_{2})\,\delta^{4}(x_{1}-x_{2})\;.$
(2.11)

Thus one finds

$\displaystyle\left\langle
E_{\phi}(x_{1})E_{\phi}(x_{2})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$
$\displaystyle=$ $\displaystyle
i\left[-\partial^{2}-m^{2}\right]\,\delta^{4}(x_{1}-x_{2})$ (2.12)
$\displaystyle\qquad+2ig\delta^{4}(x_{1}-x_{2})\left\langle\phi(x_{2})\,e^{iS_{I}[\phi]}\right\rangle_{0}^{c}\;.$

The normalization condition
$\left(\delta\Gamma/\delta\varphi(x)\right)|_{\varphi=0}=0$ on the absence of
the tadpole implies
$\left\langle\phi(x_{2})\,e^{iS_{I}[\phi]}\right\rangle_{0}^{c}=0$. Therefore

$\left\langle
E_{\phi}(x_{1})E_{\phi}(x_{2})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=i\left[-\partial^{2}-m^{2}\right]\,\delta^{4}(x_{1}-x_{2})\;.$
(2.13)

The same arguments illustrated above give

$\left\langle
E_{\phi}(x_{1})E_{\phi}(x_{2})E_{\phi}(x_{3})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=-4\,g\,\delta^{4}(x_{1}-x_{2})\,\delta^{4}(x_{2}-x_{3})\;.$
(2.14)

The structure of the diagrams associated with $\left\langle
E_{\phi}(x_{1})E_{\phi}(x_{2})\cdots
E_{\phi}(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$, for generic $n\geq
4$, can be obtained by first considering all the Wick contractions of the
field operators of the type $\nabla\phi$. The combinatoric of these
contractions can easily be obtained by taking into account the symmetric role
of the operators $E_{\phi}(x_{1})E_{\phi}(x_{2})\cdots E_{\phi}(x_{n})$ in the
computation of $W_{SD}[B]$, as shown in equation (2.8). For the connected
diagrams, we find

$\left\langle E_{\phi}(x_{1})E_{\phi}(x_{2})\cdots
E_{\phi}(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=0\quad,\quad\hbox{ if
$n$ is odd}\;,$ (2.15)

and

$\displaystyle\left\langle E_{\phi}(x_{1})E_{\phi}(x_{2})\cdots
E_{\phi}(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$ $\displaystyle=$
$\displaystyle\left\langle\phi(x_{2})\,\phi(x_{4})\cdots\phi(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}\times$
(2.16)
$\displaystyle{\hskip-170.71652pt}\times(n-1)!!\,(2ig)^{n/2}\,\delta^{4}(x_{1}-x_{2})\cdots\delta^{4}(x_{n-1}-x_{n})\quad,\quad\hbox{
if $n$ is even}\;.$

Equations (2.15) and (2.16) show that the expectation value $\left\langle
E_{\phi}(x_{1})E_{\phi}(x_{2})\cdots
E_{\phi}(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$ is completely
specified by the expectation value
$\left\langle\phi(x_{2})\,\phi(x_{4})\cdots\phi(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$.
Therefore the standard renormalization procedure for the correlation functions
of the field $\phi(x)$ canonically defines the renormalization for
$\left\langle E_{\phi}(x_{1})E_{\phi}(x_{2})\cdots
E_{\phi}(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$. Equation (2.10)
implies

$i\int d^{4}x\,B(x)\,\left\langle
E_{\phi}(x)\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=0\;,$ (2.17)

The contact terms (2.13) and (2.14) give origin to a local contribution to
$W_{SD}[B]$. In particular, we find

$\frac{i^{2}}{2!}\int d^{4}x_{1}d^{4}x_{2}\,B(x_{1})B(x_{2})\,\left\langle
E_{\phi}(x_{1})E_{\phi}(x_{2})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=i\int
d^{4}x\,\left\\{-{\textstyle{\raise 0.8pt\hbox{$\scriptstyle
1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}\partial_{\mu}B\partial^{\mu}B+{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}m^{2}B^{2}\right\\}\;,$ (2.18) $\frac{i^{3}}{3!}\int
d^{4}x_{1}d^{4}x_{2}d^{4}x_{3}\,B(x_{1})B(x_{2})B(x_{n})\,\left\langle
E_{\phi}(x_{1})E_{\phi}(x_{2})E_{\phi}(x_{3})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=i\int
d^{4}x\,\left[{\textstyle{\raise 0.8pt\hbox{$\scriptstyle 2$}\over\hbox{\lower
0.8pt\hbox{$\scriptstyle 3$}}}}gB^{3}\right]\;.$ (2.19)

The expectation values
$\left\langle\phi(x_{2})\,\phi(x_{4})\cdots\phi(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$,
with $n\geq 4$, correspond to non-local amplitudes. By collecting all the
results on $\left\langle E_{\phi}(x_{1})E_{\phi}(x_{2})\cdots
E_{\phi}(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}$ with $n\geq 4$, we
get

$\displaystyle\sum_{n=4}^{\infty}\frac{i^{n}}{n!}\int d^{4}x_{1}\cdots
d^{4}x_{n}B(x_{1})\cdots B(x_{n})\,\left\langle E_{\phi}(x_{1})\cdots
E_{\phi}(x_{n})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=$
$\displaystyle\quad=\sum_{p=2}^{\infty}\frac{i^{(2p)}(2p-1)!!\,(2ig)^{p}}{(2p)!}\int
d^{4}x_{1}\cdots d^{4}x_{2p}B(x_{1})\cdots B(x_{2p})\,\times$
$\displaystyle\qquad\qquad\qquad\times\left\langle\phi(x_{2})\,\phi(x_{4})\cdots\phi(x_{2p})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}\delta^{4}(x_{1}-x_{2})\cdots\delta^{4}(x_{2p-1}-x_{2p})=$
$\displaystyle\qquad=\sum_{p=2}^{\infty}\frac{(-ig)^{p}}{p!}\int
d^{4}x_{1}\cdots d^{4}x_{p}B^{2}(x_{1})\cdots
B^{2}(x_{p})\left\langle\phi(x_{1})\,\phi(x_{2})\cdots\phi(x_{p})\,e^{iS_{I}[\phi]}\,\right\rangle^{c}_{0}=$
$\displaystyle\qquad=\left\langle e^{i\int
d^{4}x(-g)B^{2}(x)\phi(x)}\,e^{iS_{I}[\phi]}\>\right\rangle^{c}_{0}=Z\left[\left(J=-gB^{2}\right)\right]\;.$
(2.20)

The sum of the contributions (2.17)-(2.20) shows that the renormalized
Schwinger-Dyson functional for the $\phi^{3}$ model satisfies

$Z_{SD}[B]=e^{iR[B]}\,Z[\widetilde{J}[B]]\;,$ (2.21)

where

$R[B]=\int d^{3}x\,\left\\{-{\textstyle{\raise 0.8pt\hbox{$\scriptstyle
1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}\partial_{\mu}B\partial^{\mu}B+{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}m^{2}B^{2}+{\textstyle{\raise 0.8pt\hbox{$\scriptstyle
2$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle 3$}}}}gB^{3}\right\\}\;,$ (2.22)

and

$\widetilde{J}[B](x)=-gB^{2}(x)\;.$ (2.23)

The expectation values (2.13) and (2.14) determine the local functional $R[B]$
of equation (2.22). As shown in equation (2.21), the renormalization of
$Z_{SD}[B]$ is specified by the standard renormalization of $Z[J]$.

The structure of the results for the $\phi^{3}$ model admits appropriate
generalizations which depend on the form of the lagrangian of each field
theory. Let us consider for instance the $\phi^{4}$ model which is specified
by the action

$S[\phi]=\int d^{4}x\,\left({\textstyle{\raise 0.8pt\hbox{$\scriptstyle
1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}\partial_{\mu}\phi\,\partial^{\mu}\phi-{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}m^{2}\phi^{2}-{\textstyle{\raise 0.8pt\hbox{$\scriptstyle
1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle 4$}}}}g\phi^{4}\right)\;.$
(2.24)

One has

$\frac{\delta
S}{\delta\phi(x)}=\left[-\partial^{2}-m^{2}\right]\phi(x)-g\phi^{3}(x)\;.$
(2.25)

The perturbative expansion of the Schwinger-Dyson functional $Z_{SD}[B]$ can
be examined by means of the method described above. We find

$Z_{SD}[B]=e^{iQ[B]}\left\langle\exp\left\\{i\int
d^{4}x\,\left[-2gB^{3}(x)\,\phi(x)+{\textstyle{\raise 0.8pt\hbox{$\scriptstyle
3$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}gB^{2}(x)\,\phi^{2}(x)\right]\right\\}\right\rangle\;,$ (2.26)

in which

$Q[B]=\int d^{4}x\left[-{\textstyle{\raise 0.8pt\hbox{$\scriptstyle
1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}\partial_{\mu}B(x)\partial^{\mu}B(x)+{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}m^{2}B^{2}(x)+{\textstyle{\raise 0.8pt\hbox{$\scriptstyle
3$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle 4$}}}}gB^{4}(x)\right]\;.$
(2.27)

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 $\phi(x)$, a coupling
with the operator $\phi^{2}(x)$ is also present. Note that the short distance
behaviour of the composite operator $\phi^{2}(x)$ is taken into account by the
standard renormalization of the generating functional $Z[J]$ of the
correlation functions because $\phi^{2}(x)$ has canonical dimension $2$. For
instance, the one-loop correlation $\langle\phi^{2}(x)\phi^{2}(y)\rangle$ is
described by the diagram of Figure 1 (with removed external legs), which
enters the ordinary renormalization of the $\phi^{4}$ theory. Thus, for this
model also, the renormalization of $Z_{SD}[B]$ can be specified by the
standard renormalization procedure.

Figure 1. One loop diagram of the $\phi^{4}$ model contributing to
$\langle\phi^{2}(x)\phi^{2}(y)\rangle$.

As these examples have shown, the connected component $W_{SD}[B]$ of the
renormalized Schwinger-Dyson functional contains a local part, expressions
(2.22) and (2.27), which is determined by the form of the lagrangian of each
specific model, and a non-local contribution which is related to the vacuum
expectation values of field operators of dimension 1 or 2, i.e. the fields
$\phi(x)$ and $\phi^{2}(x)$. This universal feature can be used to specify the
renormalized values of the parameters of each model by the introduction of
appropriate normalization conditions written in terms of the operator
$\left[\delta S/\delta\phi(x)\right]$. Equivalently, some of the vacuum
expectations values of the products of $\left[\delta S/\delta\phi(x)\right]$
are uniquely specified and do not receive perturbative changes. For example,
in the $\phi^{3}$ model, relations (2.13) and (2.14) are exact and are not
modified by loop corrections.

##  3 Topological models

In this section we consider gauge theories of topological type in
${\mathbb{R}}^{3}$. The action of the $SU(N)$ quantum Chern-Simons theory [33,
34, 24] in the Landau gauge is given by

$\displaystyle S$ $\displaystyle=$ $\displaystyle{k\over 4\pi}\int
d^{3}x\,\Big{\\{}\epsilon^{\mu\nu\tau}\left[{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}A^{a}_{\mu}\partial_{\nu}A^{a}_{\tau}-{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
6$}}}}\,f^{abc}A^{a}_{\mu}A^{b}_{\nu}A^{c}_{\tau}\right]$ (3.1)
$\displaystyle\qquad\qquad-M^{a}\partial^{\mu}A^{a}_{\mu}+\partial^{\mu}{\overline{c}}^{a}\left(\partial_{\mu}c^{a}-f^{abc}A^{b}_{\mu}c^{c}\right)\Big{\\}}\;,$

and then

$\frac{\delta S}{\delta A^{a}_{\mu}(x)}=\left({k\over
4\pi}\right)\left[\epsilon^{\mu\nu\tau}\left(\partial_{\nu}A_{\tau}^{a}(x)-{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}\,f^{abc}A^{b}_{\nu}(x)A^{c}_{\tau}(x)\right)+\partial^{\mu}M^{a}(x)+\partial^{\mu}{\overline{c}}^{b}(x)f^{abd}c^{d}(x)\right]\;.$
(3.2)

The Schwinger-Dyson functional $Z_{SD}[L^{a}_{\mu}]$ is defined by

$Z_{SD}[L^{a}_{\mu}]=\left\langle\exp\left(i\int
d^{3}x\,L^{a}_{\mu}(x)\left[\delta S/\delta
A_{\mu}^{a}(x)\right]\right)\right\rangle\;,$ (3.3)

and the generating functional $Z[J^{a}_{\mu}]$ of the correlation functions
for the gauge field $A^{a}_{\mu}(x)$ is given by

$Z[J^{a}_{\mu}]=\left\langle\exp\left(i\int
d^{3}x\,J^{a\mu}(x)A_{\mu}^{a}(x)\right)\right\rangle\;.$ (3.4)

In order to examine the diagrams entering the vacuum expectation values of the
product of fields $\delta S/\delta A^{a}_{\mu}(x_{1})\cdots\delta S/\delta
A^{b}_{\nu}(x_{n})$, one needs to use the following relationship between the
propagators of the fields

$\epsilon^{\mu\sigma\lambda}\partial_{\sigma}\mathop{\vbox{\halign{#\cr\kern
3.0pt\cr$\,\,\vrule depth=4.0pt,width=0.4pt\leaders\vrule
depth=0.0pt,height=0.4pt\hfill\vrule depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{A^{a}_{\lambda}(x)A}\hfil$\crcr}}}\limits\\!{}^{b}_{\nu}(y)+\partial^{\mu}\\!\mathop{\vbox{\halign{#\cr\kern
3.0pt\cr$\,\,\vrule depth=4.0pt,width=0.4pt\leaders\vrule
depth=0.0pt,height=0.4pt\hfill\vrule depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{M^{a}(x)A}\hfil$\crcr}}}\limits\\!{}^{b}_{\nu}(y)=i\left(\frac{4\pi}{k}\right)\delta^{ab}\,\delta^{\mu}_{\nu}\,\delta^{3}(x-y)\;,$
(3.5)

which can be derived from the action (3.1), or it can be checked directly by
means of the expressions

$\mathop{\vbox{\halign{#\cr\kern 3.0pt\cr$\,\,\vrule
depth=4.0pt,width=0.4pt\leaders\vrule depth=0.0pt,height=0.4pt\hfill\vrule
depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{A^{a}_{\mu}(x)A}\hfil$\crcr}}}\limits\\!{}^{b}_{\nu}(y)=\delta^{ab}\left(\frac{4\pi}{k}\right)\int{d^{3}p\over(2\pi)^{3}}\,e^{ip(x-y)}\,\epsilon_{\mu\nu\lambda}{p^{\lambda}\over
p^{2}}\;,$ (3.6) $\mathop{\vbox{\halign{#\cr\kern 3.0pt\cr$\,\,\vrule
depth=4.0pt,width=0.4pt\leaders\vrule depth=0.0pt,height=0.4pt\hfill\vrule
depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{A^{a}_{\mu}(x)M}\hfil$\crcr}}}\limits\\!{}^{b}(y)=-\delta^{ab}\left(\frac{4\pi}{k}\right)\int{d^{3}p\over(2\pi)^{3}}\,e^{ip(x-y)}\,{p_{\mu}\over
p^{2}}\;.$ (3.7)

The presence of the operator
$\partial^{\mu}{\overline{c}}^{b}(x)f^{abd}c^{d}(x)$ in the equation (3.2)
does not modify the perturbative relations between the expectation values of
the operators $\delta S/\delta A^{a}_{\mu}(x_{1})\cdots\delta S/\delta
A^{b}_{\nu}(x_{n})$ —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 $A^{a}_{\mu}(x)$ and the auxiliary field $M^{a}(x)$.
Thus, by means of the arguments presented in Section 2 , one gets

$Z_{SD}[L^{a}_{\mu}]=e^{iF[L^{a}_{\mu}]}\,Z[\,\widehat{J}[L]^{a}_{\mu}\,]\;,$
(3.8)

where

$F[L^{a}_{\mu}]=-{k\over 4\pi}\int
d^{3}x\,\epsilon^{\mu\nu\tau}\left\\{{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}L^{a}_{\mu}\partial_{\nu}L^{a}_{\tau}+{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
3$}}}}f^{abc}L^{a}_{\mu}L^{b}_{\nu}L^{c}_{\tau}\right\\}\;,$ (3.9)

and

$\widehat{J}[L]^{a}_{\mu}(x)={k\over
8\pi}\epsilon^{\mu\nu\tau}\,f^{abc}L_{\nu}^{b}(x)L^{c}_{\tau}(x)\;.$ (3.10)

In addition to the local component which is described the function
$F[L^{a}_{\mu}]$ of equation (3.9), the connected component of
$Z_{SD}[L^{a}_{\mu}]$ contains powers of $L^{a}_{\mu}$ greater than (or equal
to) four. In particular, for each connected diagram $D$ contributing to
$Z[J^{a}_{\mu}]$ there is a corresponding connected diagram $D^{\prime}$
contributing to $Z_{SD}[L^{a}_{\mu}]$. In agreement with equation (3.8),
$D^{\prime}$ can be obtained from $D$ by the introduction of the “effective
classical vertex” (3.10) in each external leg of $D$, as depicted in Figure 2.
Clearly, the renormalization of the amplitude associated with each diagram $D$
defines a canonical renormalization of the amplitude corresponding to
$D^{\prime}$. So, equation (3.8) provides a definition of the renormalized
Schwinger-Dyson functional $Z_{SD}[L^{a}_{\mu}]$.

$D$$D^{\prime}$

Figure 2. Diagram $D$ and the associated diagram $D^{\prime}$ contributing to
$Z_{SD}$.

Equation (3.8) also specifies the leading term of the operator product
expansion [4]

$\left[\delta S/\delta A^{a}_{\mu}(0)\right]\,\left[\delta S/\delta
A^{b}_{\nu}(x)\right]\rightarrow\sum_{n}C_{n}(x)\,{\cal O}_{n}\;.$ (3.11)

Indeed the value of the coefficient function $C_{1}(x)$ for the identity
operator ${\cal O}_{1}=1$ is determined by expression (3.9)

$C_{1}(x)=\left(\frac{k}{4\pi}\right)\delta^{ab}\epsilon^{\mu\nu\lambda}\partial_{\lambda}\delta^{3}(x)\;,$
(3.12)

and it does not receive perturbative corrections.

The Schwinger-Dyson functional for the abelian Chern-Simons theory can be
obtained in the $f^{abc}\rightarrow 0$ limit. In this case, equation (3.8)
becomes

$Z_{SD}[L_{\mu}]=\exp\left\\{-i{k\over 4\pi}\int
d^{3}x\,\epsilon^{\mu\nu\tau}\,{\textstyle{\raise 0.8pt\hbox{$\scriptstyle
1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}L_{\mu}\partial_{\nu}L_{\tau}\right\\}\;.$ (3.13)

When $L_{\mu}(x)$ coincides with the de Rham-Federer current [35, 36, 37, 38]
associated with a link $\cal L$ (with support on a Seifert surface [25] which
bounds the link $\cal L\,$), expression (3.13) represents precisely the
exponent of the linking matrix corresponding to $\cal L$. 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
${\mathbb{R}}^{3}$, with fields $A^{a}_{\mu}(x)$ and $B^{a}_{\mu}(s)$ and
gauge-fixed action in the Landau gauge

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{3}x\,\epsilon^{\mu\nu\lambda}\left\\{{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}\,B^{a}_{\mu}F^{a}_{\nu\lambda}(A)+g\left[{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}A^{a}_{\mu}\partial_{\nu}A^{a}_{\lambda}-{\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
6$}}}}\epsilon^{abc}\,A^{a}_{\mu}A^{b}_{\nu}A^{c}_{\lambda}\right]\right\\}$
(3.14) $\displaystyle\qquad+\int
d^{3}x\Bigl{\\{}M^{a}\partial^{\mu}A^{a}_{\mu}+N^{a}\partial^{\mu}B^{a}_{\mu}+\partial^{\mu}{\overline{c}}^{a}(\partial_{\mu}c^{a}-\epsilon^{abc}A_{\mu}^{b}c^{c})$
$\displaystyle{\hskip
51.21504pt}+\partial^{\mu}{\overline{\xi}}^{a}(\partial_{\mu}\xi^{a}-\epsilon^{abc}A_{\mu}^{b}\xi^{c}-\epsilon^{abc}B_{\mu}^{b}c^{c})\Bigr{\\}}\;,$

where the real parameter $g$ denotes a coupling constant and

$F^{a}_{\mu\nu}(A)=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}-\epsilon^{abc}A^{b}_{\mu}A^{c}_{\nu}\;.$
(3.15)

The non-vanishing propagators for the components of the connection and the
auxiliary fields are given by [43]

$\displaystyle\mathop{\vbox{\halign{#\cr\kern 3.0pt\cr$\,\,\vrule
depth=4.0pt,width=0.4pt\leaders\vrule depth=0.0pt,height=0.4pt\hfill\vrule
depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{A^{a}_{\mu}(x)B}\hfil$\crcr}}}\limits\\!{}^{b}_{\nu}(y)$
$\displaystyle=$
$\displaystyle\delta^{ab}\int{d^{3}k\over(2\pi)^{3}}\,e^{ik(x-y)}\,\epsilon_{\mu\nu\lambda}{k^{\lambda}\over
k^{2}}\;,$ (3.18) $\displaystyle\mathop{\vbox{\halign{#\cr\kern
3.0pt\cr$\,\,\vrule depth=4.0pt,width=0.4pt\leaders\vrule
depth=0.0pt,height=0.4pt\hfill\vrule depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{B^{a}_{\mu}(x)B}\hfil$\crcr}}}\limits\\!{}^{b}_{\nu}(y)$
$\displaystyle=$
$\displaystyle-g\,\delta^{ab}\int{d^{3}k\over(2\pi)^{3}}\,e^{ik(x-y)}\,\epsilon_{\mu\nu\lambda}{k^{\lambda}\over
k^{2}}\;,$ (3.21)

and

$\displaystyle\mathop{\vbox{\halign{#\cr\kern 3.0pt\cr$\,\,\vrule
depth=4.0pt,width=0.4pt\leaders\vrule depth=0.0pt,height=0.4pt\hfill\vrule
depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{A^{a}_{\mu}(x)M}\hfil$\crcr}}}\limits\\!{}^{b}(y)$
$\displaystyle=$
$\displaystyle\delta^{ab}\int{d^{3}k\over(2\pi)^{3}}\,e^{ik(x-y)}\,{k_{\mu}\over
k^{2}}\;,$ (3.24) $\displaystyle\mathop{\vbox{\halign{#\cr\kern
3.0pt\cr$\,\,\vrule depth=4.0pt,width=0.4pt\leaders\vrule
depth=0.0pt,height=0.4pt\hfill\vrule depth=4.0pt,width=0.4pt\,\,$\crcr\kern
0.8pt\nointerlineskip\cr$\hfil\displaystyle{B^{a}_{\mu}(x)N}\hfil$\crcr}}}\limits\\!{}^{b}(y)$
$\displaystyle=$
$\displaystyle\delta^{ab}\int{d^{3}k\over(2\pi)^{3}}\,e^{ik(x-y)}\,{k_{\mu}\over
k^{2}}\;.$ (3.27)

In this model one has

$\frac{\delta S}{\delta
A^{a}_{\mu}(x)}=\epsilon^{\mu\nu\lambda}\left[\left(\partial_{\nu}B^{a}_{\lambda}-\epsilon^{abc}A^{b}_{\nu}B_{c}^{c}\right)+{\textstyle{\raise
0.8pt\hbox{$\scriptstyle g$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}F^{a}_{\nu\lambda}(A)\right]-\partial^{\mu}M^{a}-\epsilon^{abc}\left[\partial^{\mu}{\overline{c}}^{a}c^{c}-\partial^{\mu}{\overline{\xi}}^{a}\xi^{c}\right],$
(3.28)

and

$\frac{\delta S}{\delta B^{a}_{\mu}(x)}={\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}\epsilon^{\mu\nu\lambda}F^{a}_{\nu\lambda}(A)-\partial^{\mu}N^{a}-\epsilon^{abc}\partial^{\mu}{\overline{\xi}}^{a}c^{c}\;.$
(3.29)

The Schwinger-Dyson functional $Z_{SD}[L^{a}_{\mu},H^{a}_{\mu}]$ is defined by

$Z_{SD}[L^{a}_{\mu},H^{a}_{\mu}]=\left\langle e^{i\int
d^{3}x\left[L^{a}_{\mu}\left(\delta S/\delta
A_{\mu}^{a}\right)+H^{a}_{\mu}\left(\delta S/\delta
B_{\mu}^{a}\right)\right]}\right\rangle\;,$ (3.30)

where $L^{a}_{\mu}(x)$ and $H^{a}_{\mu}(x)$ are classical sources. The
generating functional $Z[J^{a}_{\mu},K^{a}_{\mu}]$ of the correlation
functions for the gauge field $A^{a}_{\mu}(x)$ and $B_{\mu}^{a}(x)$ is given
by

$Z[J^{a}_{\mu},k^{a}_{\mu}]=\left\langle e^{i\int
d^{3}x\left[J^{a}_{\mu}A_{\mu}^{a}+K^{a}_{\mu}B^{a}_{\mu}\right]}\right\rangle\;.$
(3.31)

With several field components, the construction and the sum of the Feynman
diagrams becomes rather laborious. We get

$Z_{SD}[L^{a}_{\mu},H^{a}_{\mu}]=e^{iG[L^{a}_{\mu},H^{a}_{\mu}]}\,Z[\widetilde{J}^{a}_{\mu},\widetilde{K}^{a}_{\mu}]\;,$
(3.32)

in which

$G[L^{a}_{\mu},H^{a}_{\mu}]=-\int
d^{3}x\,\epsilon^{\mu\nu\lambda}\Bigl{\\{}H^{a}_{\mu}\partial_{\nu}L^{a}_{\lambda}+\epsilon^{abc}H^{a}_{\mu}L^{b}_{\nu}L^{c}_{\lambda}+{\textstyle{\raise
0.8pt\hbox{$\scriptstyle g$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}L^{a}_{\mu}\partial_{\nu}L^{a}_{\lambda}+{\textstyle{\raise
0.8pt\hbox{$\scriptstyle g$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
3$}}}}\epsilon^{abc}L^{a}_{\mu}L^{b}_{\nu}L^{c}_{\lambda}\Bigr{\\}},$ (3.33)

and

$\widetilde{J}^{a}_{\mu}=\epsilon^{\mu\nu\lambda}\epsilon^{abc}\left[H^{b}_{\nu}L^{c}_{\lambda}+{\textstyle{\raise
0.8pt\hbox{$\scriptstyle g$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}L^{b}_{\nu}L^{c}_{\lambda}\
\right]\qquad,\qquad\widetilde{K}^{a}_{\mu}={\textstyle{\raise
0.8pt\hbox{$\scriptstyle 1$}\over\hbox{\lower 0.8pt\hbox{$\scriptstyle
2$}}}}\epsilon^{\mu\nu\lambda}\epsilon^{abc}L^{b}_{\nu}L^{c}_{\lambda}\;.$
(3.34)

In addition to the contact terms, which are specified by the local functional
$G[L^{a}_{\mu},H^{a}_{\mu}]$, 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 (3.32) one derives the
following relation for the connected mean value

$\displaystyle\int
d^{3}xd^{3}yd^{3}zd^{3}t\,L^{a}_{\mu}(x)L^{b}_{\nu}(y)L^{c}_{\lambda}(z)H^{d}_{\rho}(t)\left\langle\frac{\delta
S}{A^{a}_{\mu}(x)}\frac{\delta S}{A^{b}_{\nu}(y)}\frac{\delta
S}{A^{c}_{\lambda}(z)}\frac{\delta S}{B^{d}_{\rho}(t)}\right\rangle^{c}=$
$\displaystyle\;\;=-i\frac{3}{4\pi}\int
d^{3}x\,d^{3}y\,\frac{(x-y)^{\lambda}}{|x-y|^{3}}\epsilon_{\lambda\mu\nu}\epsilon^{\mu\rho\tau}\epsilon^{\nu\sigma\xi}\epsilon^{abc}\epsilon^{ade}\,H^{b}_{\rho}(x)L^{c}_{\tau}(x)L^{d}_{\sigma}(y)L^{e}_{\xi}(y)\;.$
(3.35)

Actually in both the Chern-Simons and the BF theories, the connected
components of the Schwinger-Dyson functional containing up to six powers of
the external classical sources are exhibited in closed form because of the
vanishing of the loop corrections to the two-points and three-points
correlation functions of the gauge fields [29, 34, 43].

##  4 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 $\delta
S[\phi]/\delta\phi(x)$ 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 $\phi^{3}$ and $\phi^{4}$ 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
(3.32) 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.

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