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\markboth{\small \quad \em A.A.~Slavnov\hfill}{\hfill\small \em Renormalization \quad}

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\begin{document}
\sloppy
\medskip
\rightline{\sl To memory of my friend Misha Marinov}
\medskip

\title{RENORMALIZATION BY ENFORCING A SYMMETRY}

\author{ A.A.~SLAVNOV}

\address{
%\thanks{E-mail:$~~$ slavnov@mi.ras.ru}
Steklov Mathematical Institute, Russian Academy of Scienies\\Gubkina St 8, GSP-1, 117966 Moscow\\
E-mail:$~~$ slavnov@mi.ras.ru} \maketitle

\vspace*{0.4cm}

\abstracts{
A new renormalization scheme for theories with nontrivial internal 
symmetry is proposed. The sheme is regularization independent and 
respects the symmetry requirements.
}

\vspace*{0.6cm}

\tableofcontents

\newpage

\section {Introduction}


Renormalization procedure in the theories with nontrivial internal 
symmetries, like gauge invariant models, is complicated by the necessity 
to provide the symmetry of the renormalized theory. The crucial role in 
this procedure is played by the relations between Green functions which 
are the quantum analogue of a classical symmetry. In the case of Quantum 
Electrodynamics (QED) these relations are Ward-Takahashi identities 
(WTI),\cite{W,T} which connect three- and two-point Green functions. 
For non-Abelian gauge theories corresponding relations (STI) were obtained 
in Refs.~\protect\citebk{Sl} and \citebk{Tl}.

There are essentially two approaches to renormalization of gauge 
invariant theories. The first one uses some intermediate gauge invariant 
regularization, e.g. dimensional regularization,\cite{H.V,B.M} 
higher covariant derivatives,\cite{Sl2}$^-$\cite{B.S} or 
lattice regularization.\cite{Wil} Using these regularizations one can 
prove that the counterterms needed to eliminate ultraviolet divergencies 
preserve the gauge invariant structure of the renormalized Lagrangian.

In the second approach, known as algebraic renormalization, one firstly 
defines a finite renormalized theory following the Bogoliubov-Parasiuk 
R-operation\,\cite{B.P} with some particular subtractions.\cite{Z} 
This procedure in general breaks gauge invariance, however one can use a 
finite renormalization freedom to restore STI for renormalized Green 
functions\,\cite{BRS,P.R} (for recent development and more 
complete references see Ref.~\protect\citebk{Gr}).

Both these approaches have some advantages and disadvantages. Using 
invariant regularizations allows to prove in a rather simple way gauge 
independence of observables. Moreover, the dimensional regularization 
proved to be an efficient way for calculations of Feynman diagrams. On 
the other hand in this case the procedure is explicitely regularization 
dependent and, more important, is not directly applicable to some 
theories including the Standard Model and supersymmetric theories. The 
invariant regularization procedure for anomaly free theories including 
chiral fermions, like Salam-Weinberg or unified models was constructed 
recently,\cite{F.S,D.S} and for general Feynman diagrams is 
still rather complicated.

The algebraic renormalization uses only the algebraic structure of the
underlying theory and does not require any particular regularization. 
So it may be applied to any anomaly free model. However to provide the 
validity of STI for renormalized Green functions in this approach is a 
nontrivial problem and calculations are rather clumsy. Note that it 
refers also to the models without chiral fermions.

In the present paper I propose a new renormalization procedure which 
shares the advantages of both these approaches. The procedure is 
regularization independent and at the same time provides automatically 
STI for renormalized Green functions.\cite{Sl0}

To formulate the main idea of the method I remind some general properties
of renormalization procedure which may be found in many textbooks (see
e.g.\ Ref.\,\protect\citebk{B.Sh}).

In any 4d renormalizable theory ultraviolet divergencies may occur in
diagrams with the number of external lines less than five. The vertex
functions with three and four external lines may diverge linearly or
logarithmically. According to the R-operation\,\cite{B.P,B.Sh}
these divergencies are removed by recursive subtractions of local
counterterms (polynomials in momentum space), which may be identified with
the values of the vertex functions at some (infrared safe) external
momenta. Making these subtractions one firstly introduce some
intermediate regularization which makes the integral as a whole convergent
by power counting. Such a regularization which is removed after
making all necessary subtractions is assumed in the following. Finite
renormalized functions obtained in this way have an ambiguity which is a
polynomial in external momenta of the degree given by power counting (see
Ref.\,\protect\citebk{B.Sh}).

In what follows we assume that if some intermediate regularization is
introduced, it satisfies some natural requirements which usually are
tacitly assumed. The most important properties are the following.
Regularization does not change the value of integrals convergent by power
counting (i.e., in the limit when the regularization is removed one gets
the same result as without regularization). Global symmetries are
respected by the regularization.

After these preliminaries I formulate the general idea of the method. It
follows from the discussion above that ultraviolet divergencies of the
vertices with three and four external lines may be removed not necessarily
by subtracting the vertex at some fixed external momenta but also by
subtracting the values of the corresponding vertices with only one
external momentum fixed, in particular the fixed momentum may be equal to
zero.  In the case of linear divergencies one
has to use a properly symmetrized subtraction. Sometimes symmetrization is
also useful for subtracting logarithmically divergent integrals.

The subtraction procedure described above makes vertex functions finite
but obviously violates locality. To restore the locality one can use the
fact that in gauge theories the value of a vertex function at zero
momentum of one of the external gauge fields is related to the values of
other correlators having at least one external gauge field less. In QED
this relation is given by WTI and expresses the electron photon vertex at
zero photon momentum in terms of electron polarization. In non-Abelian
models the corresponding relations express the vertex functions with three
and four external gauge fields in terms of polarization operators and
ghost-gauge field vertices. Having this in mind one can define the
renormalized proper vertex functions as follows:
\begin{equation}
\Gamma^r(p,q,k)=\Gamma(p,q,k)- \Gamma(p,q,0)+F^r(p,q)\,.
\label{1a}
\end{equation}
Here $\Gamma(p,q,k)$ denotes some proper vertex function with indices
suppressed and $F^r$ is a combination of other renormalized correlators,
such that for a proper choice of renormalization freedom 
\begin{equation}
\Gamma^r(p,q,0)-F^r(p,q)=0
\label{2a}
\end{equation}
by virtue of gauge invariance. In other words, Eq.\,(\ref{2a}) is
the renormalized WTI in QED or renormalized STI in non-Abelian models. The
notation $\Gamma^r$ means a renormalized vertex function.

The subtraction (\ref{1a}) involves the value of three or four point
vertex functions at one of the external momenta equal to zero. Obviously
if at least one of the remaining external momenta is different from zero,
it serves as the infrared regulator and this function is infrared finite.
So our procedure does not meet any infrared problems.

The renormalization procedure works loopwise. One firstly makes arbitrary
(infrared safe) subtractions of one loop polarization operators and (in
non-Abelian models) ghost-gauge field vertices. Then one defines the
renormalized three and four gauge field one loop correlators by the
relations of the type (\ref{1a}), where $F^r$ depends only on finite
renormalized correlators. The r.h.s.\ of the
Eq.\,(\ref{1a}) is ultraviolet finite.
The WTI and STI are obviously satisfied by the renormalized functions
(\ref{1a}). Finally, as was discussed above, the difference
\begin{equation}
\Gamma(p,q,0)-F^r(p,q)
\label{3a}
\end{equation}
in any anomaly free model in the limit of an intermediate regularization
removed is a polynomial in $p,q$ (with
coefficients divergent in the limit when a regularization is removed). It
follows from the fact that $\Gamma^r(p,q,0)$ is obtained from $
\Gamma(p,q,0)$ by subtracting a polynomial. It proves the locality of our
procedure. Note that if one applies the same procedure to an anomalous
model, the difference (\ref{3a}) will not be local anymore, which makes
the model inconsistent.

After defining renormalized one loop correlation functions one proceeds
defining multiloop correlators according to the standard R-operation. With
subgraph divergencies being removed one needs to make a subtraction
corresponding to the superficial divergency of the diagram, which is done
by the same procedure. There is nothing specific in our method at this
point and we refer for further details to textbooks (e.g.\ Ref.\,\protect\citebk{B.Sh}).


 In the second section I illustrate the method by applying it to QED.  In the
third section the renormalization of the Yang-Mills theory is constructed.
In the fourth section I discuss a simple anomaly free model with chiral
fermions.

\section{Invariant regularization of QED}


In Quantum Electrodynamics formal WT identities for proper Green 
functions may be written in the form:
\begin{equation}
 \Gamma_{\mu}(p,0)=e \partial_{\mu} \Sigma(p)\,,
 \label{1}
 \end{equation}
where $ \Gamma_{\mu}$ is the proper photon-electron vertex and 
$ \Sigma$ is the electron self energy. We shall define the 
renormalized Green functions which satisfy automatically this identity.

The renormalization procedure works loopwise. We start by defining the
renormalized one-loop electron self energy. The most general structure 
of $\Sigma$ is
\begin{equation}
 \Sigma(p)=( \hat{p}-m) \Sigma_1(p^2)+ \Sigma_2(p^2)\,.
\label{2}
\end{equation}
Renormalized electron self energy is given by the folllowing equation,
\begin{equation}
\Sigma^r(p)=( \hat{p}-m)\left[ \Sigma_1(p^2)- \Sigma_1(a^2)\right]+ \Sigma_2(p^2)
- \Sigma_2(m^2)\,.
\label{3}
\end{equation}
Here $a$ is some arbitrary (infrared finite) normalization point. Equation
(\ref{3}) guarantees that the electron propagator has a pole at $p^2
=m^2$.

The renormalized vertex function is defined by the equation:
\begin{equation}
\Gamma^r_{\mu}(p,q)= \Gamma_{\mu}(p,q)- \Gamma_{\mu}(p,0)+e \partial
_{\mu} \Sigma^r(p)\,.
\label{4}
\end{equation}
Here $\Sigma^r$ is the finite renormalized function defined by the 
Eq.\,(\ref{3}). The vertex function diverges logarithmically, so 
$\Gamma^r_{\mu}$  is obviously finite and hence regularization 
independent. By construction it satisfies WTI:
\begin{equation}
\Gamma^r_{\mu}(p,0)=e \partial_{\mu} \Sigma^r(p)
\label{5}
\end{equation}
At first sight the subtraction (\ref{4}) may seem to be nonlocal. 
However it is not. Locality of our procedure follows from the fact 
that in any regularization scheme the difference
\begin{equation}
\Gamma_{\mu}(p,0)-e \partial_{\mu} \Sigma^r(p)
\label{6}
\end{equation}
is a local polynomial. (See the discussion in the
Introduction).
In a gauge invariant regularization scheme one can rewrite the
Eq.\,(\ref{4}) in the form 
\begin{equation} \Gamma^r_{\mu}(p,q)\!=\!
\Gamma^{inv}_{\mu}(p,q)+(z_2-1) \gamma_{\mu}\! -\!\left[
\Gamma^{inv}_{\mu}(p,0)\!+\!(z_2-1) \gamma_{\mu}\!-\!e \partial_{\mu}
\Sigma^{inv,r}(p)\right].
\label{7}
\end{equation}

In an invariant regularization scheme the term in square bracketts 
vanishes by virtue of WTI, and $ \Gamma^r_{\mu}(p,q)= 
\Gamma^{inv,r}_{\mu}(p,q)$. In a gauge
invariant regularization the Green functions $\Gamma^{inv,r}_{\mu}$ 
and $\Sigma^{inv,r}$, renormalized by the counterterms obeying the 
relation $z_1=z_2$, satisfy WT identity. The locality of the procedure 
is manifest.

Gauge invariant renormalization of the vacuum polarization makes no 
problems. The polarization tensor has the structure:
\begin{equation}
\Pi_{\mu \nu}(p)=(g_{\mu \nu}p^2-p_{\mu}p_{\nu})\, \Pi_1(p^2)+ 
g_{\mu \nu} \Pi_2(p^2)\,.
\label{8}
\end{equation}
The renormalized polarization tensor is defined by the equation
\begin{equation}
\Pi^r_{\mu \nu}= P_{\mu \alpha}\left[\Pi_{\alpha \nu}(p)- \Pi_{\alpha \nu}
(0)\right]-p^2 P_{\mu \nu}\, \Pi_1(b^2)\,.
\label{9}
\end{equation}
Here $b$ is again some arbitrary normalization point and $P_{\mu \nu}
=g_{\mu \nu}-p_{\mu}p_{\nu}p^{-2}$ is the transversal projection 
operator. The function $ \Pi^r_{\mu \nu}$ is finite and
obviously satisfies the transversality 
condition
\begin{equation}
p_{\mu} \Pi^r_{\mu \nu}(p)=0
\label{10}
\end{equation}
required by gauge invariance.

To complete the one-loop renormalization we have to consider the 
diagram with four photon external lines. The differential WTI for 
the function $\Pi_{\mu \nu \rho \sigma}(p,q,k)$ reads:
\begin{equation}
\Pi_{\mu \nu \rho \sigma}(p,q,0)=0\,.
\label{11}
\end{equation}
This equation is preserved in any gauge invariant regularization 
scheme. As was discussed in the introduction the result obtained in
any other regularization scheme may differ by a polynomial. As $\Pi_{\mu
\nu \rho \sigma}$ is presented by a logarithmically divergent integral in
arbitrary scheme after removal of an intermediate regularization 
$ \Pi_{\mu \nu \rho \sigma}(p,q,0)$ is 
a zero order polynomial (constant tensor).

We define the renormalized four-point function by the equation:
\begin{equation}
\Pi^r_{\mu \nu \rho \sigma}(p,q,k)= \Pi_{\mu \nu \rho \sigma}
(p,q,k)- \Pi_{\mu \nu \rho \sigma}(p,0,0)\,.
\label{12}
\end{equation}
As the function $\Pi_{\mu \nu \rho \sigma}$ diverges logarithmically, 
the renormalized function defined by the Eq.\,(\ref{12}) is obviously 
finite. The external momentum $p$ serves as the infrared regulator, so
no infrared problems arise. It also satisfies WTI (\ref{11}). Locality of 
the procedure 
holds for the same reasons as above.

Renormalization of the Green functions according to Eqs.\,(\ref{3}, 
\ref{4}, \ref{9}, \ref{12}) is equivalent to introducing the 
following one-loop counterterms to the QED Lagrangian:
\begin{eqnarray}
&& L^r=- \frac{z_3}{4}F_{\mu \nu}F_{\mu \nu}+iz_2 \bar{\psi}(\hat
{\partial}-m) \psi \nonumber \\
&&\qquad ~+ \,\delta m \,\bar{\psi} \psi+ez_1 \bar{\psi} \gamma_{\mu} \psi A_
{\mu}+z_4(A_{\mu}A_{\mu})^2+ \delta \mu\, A_{\mu}^2\,.
\label{13}
\end{eqnarray}
Here
\begin{eqnarray}
&&\!\!\!\!\!\!\!\!\! 1-z_2= \Sigma_1(a^2); \quad \delta m=- \Sigma_2(m^2); \quad (1-z_1) 
\gamma_{\mu}=\Gamma_{\mu}(p,0)-e \partial_{\mu} \Sigma^r(p); \nonumber \\
&&\!\!\!\!\!\!\!\!\! 1-z_3= \Pi_1(b^2); \quad \delta \mu=- \Pi_2(0); \quad z_4=- \frac{1}{3}
\Pi_{0000}(p,0,0)\,.
\label{14}
\end{eqnarray}
In an invariant regularization scheme
\begin{equation}
\delta \mu=0; \quad z_4=0; \quad z_1=z_2\,.
\label{15}
\end{equation}

Assuming that the one loop counterterms are introduced according to 
Eqs.\,(\ref{13}, \ref{14}), we define the renormalized two loop functions 
by Eqs.\,(\ref{3}, \ref{4}, \ref{9}, \ref{12}). The 
divergencies in subgraphs of the diagrams corresponding to these 
functions are killed by 
the one-loop counterterms (\ref{13}), therefore they may diverge only 
superficially. Repeating literally the discussion given above, we 
prove the locality and gauge invariance of our procedure at two-loop 
level. Extension to higher loops is straightforward. To avoid
misunderstanding I emphasize that our renormalization is
formulated as the recursive substraction procedure and does not require
explicit introduction of counterterms. Equations (\ref{13}-\ref{15}) are
presented to establish the connection with more conventional approach.


\section{Renormalization of Yang-Mills theory}

In this section we apply the same idea to renormalization of Yang-
Mills theory. The two-point Green functions will be renormalized 
with the help of local subtractions compatible with gauge invariance, 
and renormalized vertex functions will be defined by means of nonlocal 
subtractions supplemented by addition of explicitly known finite 
terms restoring the locality and gauge invariance.

In a covariant $ \alpha$-gauge the effective action of Yang-Mills 
theory looks as follows
\begin{equation}
S= \int d^4x\left[- \frac{1}{4}F^a_{\mu \nu}F^a_{\mu \nu}+ \frac{1}{2 
\alpha}(\partial_{\mu}B_{\mu})^2+ 
+ \bar{c}^a \partial_{\mu}( \delta^{ab} \partial_{\mu}-gt^{abc}B^c_
{\mu})c^b\right].
\label{16}
\end{equation}
We want to define renormalized Green functions so that they satisfy
automa\-ti\-cally ST identities. For the Yang-Mills self energy these 
identities reduce to the condition of transversality . So the 
one-loop renormalized Yang-Mills field self energy may be defined in 
the same way as in QED, by the Eq.\,(\ref{9}).

Next identity relates the three point function for the Yang-Mills 
field with the ghost-gauge interaction vertex and ghost and gauge 
fields propagators.

It follows from the structure of ghost interaction vertex and Lorentz
invariance, that the ghost field self energy has the form
\begin{equation}
\Pi^{ab}_G(p^2)= \delta^{ab}p^2 \Pi(p^2)\,.
\label{17}
\end{equation}
We renormalize it by making a subtraction at arbitrary infrared finite 
point
\begin{equation}
\Pi^r(p^2)= \Pi(p^2)- \Pi(c^2)\,.
\label{18}
\end{equation}

The ghost-gauge vertex also may be renormalized at will. It diverges
logarithmically, and it follows from the structure of interaction that 
the corresponding local structure is proportional to $t^{abc}k_{\mu}$, 
where $k_{\mu}$ is the ghost field momentum. Therefore we can perform 
the renormalization by the following prescription:
\begin{eqnarray}
&& \Gamma^{abc,r}_{\mu}(k,p)= \Gamma^{abc}_{\mu}(k,p)-k_{\mu} \Gamma^
{abc}(b^2)\,, 
\label{19}\\[1mm]
&&\Gamma^{abc}_{\mu}(k,p)_{p=0}=k_{\mu} \Gamma^{abc}(k^2)\,.
\label{20}
\end{eqnarray}
In this equation $p$ stands for the gauge field momentum.

The subtractions (\ref{9}, \ref{18}, \ref{19}) are equivalent to 
introducing to the Lagrangian the following counterterms
\begin{eqnarray}
&& - \,\frac{z_2-1}{4}\left( \partial_{\mu}B^a_{\nu}-
\partial_{\nu}B^a_{\mu}\right)\left( \partial_{\mu}B^a_{\nu}-
\partial_{\nu}B^a_{\mu}\right)\nonumber\\
&& +\,(\tilde{z}_2-1)\, \bar{c}^a\, \Box\, c^a-( \tilde{z}_1-1)\,g\,t^
{acb}\, \bar{c}^a \,\partial_{\mu}(B^c_{\mu}c^b)\; .
\label{21}
\end{eqnarray}

Now we should define the renormalized three-point Yang-Mills field 
vertex, shown in Fig.\,\ref{fig1}, in such a way it satisfies the ST 
identities. 
\begin{figure}[h]
\epsfxsize=0.3\textwidth
\centerline{\epsfbox{fig1.eps}}
\caption{Three-point vertex function for Yang-Mills fields.}
\label{fig1}
\end{figure}
We firstly assume that some gauge invariant 
regularization  is introduced. Then one can get in a usual 
way the following identity for renormalized correlators
\begin{eqnarray}
&& \frac{i}{\alpha}\left\langle B^a_{\mu}(x)B^b_{\nu}(y) \partial_{\rho}B^c_{\rho}
(z)\right\rangle^r= \tilde{z}_2\left\langle \partial_{\mu}c^a(x) \bar{c}^c(z)B^b_{\nu}(y)\right\rangle^r \nonumber\\[1mm] 
&& - \,\tilde{z}_2\, \tilde{g}\,t^{ade} \left\langle B^d_{\mu}(x)c^e(x)
 \bar{c}^c(z)B^b_{\nu}(y)\right\rangle^r+(x \rightarrow y, a 
\rightarrow b, \mu 
 \rightarrow \nu)\,.
 \label{22}
 \end{eqnarray}
 Here the parameter $ \tilde{g}$ is the effective coupling constant 
 which enters the gauge transformations in the renormalized theory
\begin{equation}
\delta B^a_{\mu}= \partial_{\mu} \alpha^a- \tilde{g}t^{abc}B^b_{\mu}
\alpha^c\,.
\label{23}
\end{equation}
I recall that in this derivation we assume that some gauge invariant
regularization is present, so that Eq.\,(\ref{22}) and the following
equations make sense. However the final definition of the renormalized
vertices will be regularization independent.

Our definition of the renormalized three-point Yang-Mills vertex is 
based on the identity (\ref{22}). So we firstly transform the Eq.\,(\ref{22}) 
to a more convenient form. In this transformation we shall 
use the quantum equations of motoin for the ghost field $c$. These 
equations look as follows
\begin{equation}
\int\left[ \Box c^a(x)- \tilde{g}t^{aed} \partial_{\mu}(B^e_{\mu}(x)c^d(x))+
\tilde{z}_2^{-1} \eta^a(x)\right]e^{iL_R}\,dB_{\mu}d \bar{c}dc=0\,.
\label{26a}
\end{equation}
Differentiating this equation with respect to $ \eta^b(y)$ and
$J^c_{\mu}(z)$ we have
\begin{eqnarray}
\left\langle \Box c^a(x)\, \bar{c}^b(y)\right\rangle^r- \tilde{g}\,t^{aed}
\left\langle \partial_{\mu}
(B^e_{\mu}(x)c^d(x))\, \bar{c}^b(y)\right\rangle^r %\nonumber\\
+ \tilde{z}_2^{-1}\, \delta^{ab} \delta (x-y)=0
\label{27}
\end{eqnarray}
and
\begin{equation}
\left\langle \Box c^a(x)\, \bar{c}^c(z)B^b_{\nu}(y)\right\rangle^r- 
 \tilde{g}\, t^{aed} \left\langle \partial_{\mu}(B^e_{\mu}(x)c^d(x))\, \bar{c}^c(z)
B^b_{\nu}(y)\right\rangle^r=0
\label{28}
\end{equation}

Note that Eqs.\,(\ref{26a}, \ref{27}, \ref{28}) are valid also in the 
``partially renormalized" theory, described by the Lagrangian, including 
the counterterms (\ref{21}) (no counterterms for three- and four-point 
gauge field vertices). It follows from the Eq.\,(\ref{28}) that the sum 
of the first two terms in the l.h.s.\  of identity (\ref{22}) is 
transversal with respect to differentiation over $x_{\mu}$ and remaining 
terms are transversal with respect to differentiation over $y_{\nu}$. 
Using this observation we can rewrite the identity (\ref{22}) as follows. 
We firstly perform a Fourier transformation of Eq.\,(\ref{22}). Then 
cutting the Yang-Mills field propagators 
corresponding to  the external lines of the three-point correlator at the 
l.h.s.\ of the Eq.\,(\ref{22}) and taking into account the transversality of 
Yang-Mills field self energy, we can write for the proper vertex function 
$ \Gamma^{abc,r}_{\mu \nu \rho}(p,q)$ the following identity:
\begin{eqnarray}
q_{\rho} \Gamma^{abc,r}_{\mu \nu \rho}(p,q)-q^2\left[(G^{-1}_{tr})^r_{\mu
\alpha}(p)G^r(q) \Gamma^{abc,r}_{\alpha \nu}(p,q) \right. \nonumber\\[1mm]
 \left.\qquad +( \mu \rightarrow \nu,\; a \rightarrow b,\; p \rightarrow -p-q)\right]=0
\,.
\label{36}
\end{eqnarray}
In deriving this equation we used the transversality of the r.h.s.\ of
Eq.\,(\ref{22}) discussed above,  and replaced the inverse gauge field
propagator $G_{\mu \nu}$ by its transversal part. After such projection 
the first term in the r.h.s.\ of Eq.\,(\ref{22}), which is longitudinal, 
does not contribute. In this equation
\begin{eqnarray}
&& \delta^{ab}(G^{-1}_{tr})^r_{\mu \nu}=P_{\mu \alpha}(G^{-1})^{ab,r}_{\alpha
\nu}\,,\nonumber\\[1mm]
&& \delta^{ab}G^r(p)=G^{ab,r}(p)\,,
\label{25}
\end{eqnarray}
$G^{ab,r}(p)$ is the renormalized ghost field propagator.
The function $\Gamma^{abc,r}_{\mu \nu}(p,q)$ is the Fourier transform of
the expectation value of the composite operator
\begin{equation}
\tilde{\Gamma}^{abc,r}_{\mu \nu}=(G^{-1})^r_{\nu \alpha}(y)(G^{-1})^r(z)
\left\langle
\tilde{z}_2 \tilde{g}t^{aed}B^e_{\mu}(x)c^d(x) \bar{c}^c(z)B^b_{\nu}(y)\right\rangle^r\,.
\label{25a}
\end{equation}
Differentiating the equality (\ref{36}) with respect to $q_{\rho}$ and 
putting $q=0$ we get the differential identity, which will be used for 
the definition of renormalized three-point Yang-Mills field vertex
\begin{eqnarray}
\Gamma^{abc,r}_{\mu \nu \rho}(p,0)- \frac{\partial}{ \partial q_{\rho}}
\big[q^2(G^{-1}_{tr})^r_{\mu \alpha}(p)G^r(q) \Gamma^{abc,r}_{\alpha \nu}(p,q)
\nonumber\\ +( \mu \rightarrow \nu,\; a \rightarrow b,\; p \rightarrow -p-q)\big]
_{q=0}=0\,.
\label{37}
\end{eqnarray}
The renormalized proper vertex function for gauge fields is defined by 
the equation
\begin{eqnarray}
&&\!\!\!\!\!\!\!\! \Gamma^{abc,r}_{\mu \nu \rho}(p,q)= \Gamma^{abc}_{\mu \nu \rho}(p,q)- 
\Big\{\Gamma^{abc}_{\mu \nu \rho}(p,0)+\Gamma^{abc}_{\mu \nu \rho}(0,q)
\label{24}\\
&&\!\!\!\!\!\!\!\!- \frac{ \partial}{ \partial q^{\rho}}\left[q^2(G^{-1}_{tr})^r_{\mu
\alpha}(p)G^r(q) \Gamma^{abc,r}_{\alpha \nu}(p,q)+ 
(\nu \rightarrow \mu,\; a \rightarrow b,\;
p \rightarrow -p-q)\right]_{q=0}\nonumber\\ 
&&\!\!\!\!\!\!\!\!  -\frac{ \partial}{ \partial p^{\mu}}[p^2(G^{-1}_{tr})^r_{\rho \beta}(q)G^r
(p) \Gamma^{cba,r}_{\beta \nu}(q,p)+(\nu \rightarrow \rho,\; c \rightarrow 
b,\; q \rightarrow -p-q)]_{p=0} \Big\}\,.
\nonumber
\end{eqnarray}

The divergent part of $ \Gamma^{abc}_{\mu \nu \rho}(p,q)$ is the first 
order polynomial in $p,q$. Therefore the sum of the three first terms in 
the r.h.s.\ of Eq.\,(\ref{24}) is finite. Below we shall prove that the 
remaining terms in the r.h.s.\ of Eq.\,(\ref{24}) are also finite.

As it was mentioned above the r.h.s.\ of Eq.\,(\ref{24}) includes the 
expectation value of the composite operator $ \Gamma^{abc}_{\mu \nu}$. 
So we have to prove that the counterterms (\ref{21}) make it finite. In 
this proof we again shall use the quantum equations of motion for gauge 
fields (\ref{26a}, \ref{27}, \ref{28}).
\begin{figure}[h]
\epsfxsize=0.5\textwidth
\centerline{\epsfbox{fig2.eps}}
\caption{Decomposition of $ \Gamma_{\mu \nu}$ into one-particle reducible and
irreducible parts.}
\label{fig2}
\end{figure}
The function $\tilde{\Gamma}_{\mu \nu}$ may be separated into one particle
reducible and irreducible parts as follows (see Fig.\,\ref{fig2}):
\begin{equation}
\tilde{\Gamma}^{abc,r}_{\mu \nu}(x,y,z)= \int \tilde{G}^{ad}_{\mu}(x-x')
\tilde{\Gamma}^{dbc,r}_{\nu}(x',y,z)dx'
+( \tilde{\Gamma}^{ir})^{abc}_{\mu \nu}(x,y,z)\,.
\label{29}
\end{equation}
Here $ \tilde{\Gamma}^{ir}_{\mu \nu}$ denotes the strongly connected part 
of $ \tilde{\Gamma}_{\mu \nu}$, and $ \Gamma^r_{\nu}$ is the renormalized
ghost-gauge field vertex. Finally
\begin{equation}
\tilde{G}^{ab}_{\mu}(x-x')=\left\langle \tilde{z}_2 \tilde{g}t^{aed}B^e_{\mu}(x)c^d(x)
\bar{c}^b(x')\right\rangle  \,.
\label{30}
\end{equation}
By virtue of Lorentz invariance $G^{ab}_{\mu}(p)$ is proportional to
$p_{\mu}$. Then the Eq.\,(\ref{27}) gives
\begin{equation}
G^{ab}_{\mu}(p)= \tilde{z}_2p_{\mu}G^{ab}(p)+\delta^{ab}p_{\mu}p^{-2}
\label{32}
\end{equation}
Substituting the decomposition (\ref{29}, \ref{32}) into Eq.\,(\ref{28}) we
get
\begin{equation} 
\Gamma^{abc,r}_{\nu}(p,q)+ ip_{\mu} (\Gamma^{ir})^{abc}_{\mu \nu}(p,q)=0
\label{33}
\end{equation}
This equation shows that the function $p_{\mu}(\Gamma^{ir})
^{abc}_{\mu \nu}$ is finite. As the corresponding integral diverges 
logarithmically, Lorentz invariance implies that $( \Gamma^{ir})^{abc}_{\mu 
\nu}(p,q)$ is also finite. The Eq.\,(\ref{33}) expresses the well known fact,
firstly established in Ref.\,\protect\citebk{Sl}, that renormalization of the
composite operator (\ref{25a}) is related to the renormalization of the
ghost-gauge field vertex. It can be taken as a definition of the
renormalized composite operator (\ref{25a}) in terms of the renormalized
ghost-gauge field vertex, or vice versa.

The function $\Gamma^{abc}_{\mu \nu}(p,q)$ enters into Eq.\,(\ref{24}) being
multiplied by the transverse projector. As the Eqs.\,(\ref{29}-\ref{32})
show the reducible part of $\Gamma_{\mu \nu}$ is longitudinal and hence
only irreducible part of $ \Gamma_{\mu \nu}$ contributes to Eq.\,(\ref{24}).
It proves the finiteness of the r.h.s.\ of Eq.\,(\ref{24}).
Therefore the Eq.\,(\ref{24}) indeed defines the finite function
$\Gamma^{abc,r}_{\mu \nu \rho}(p,q)$ which does not depend on a particular 
regularization scheme used for its calculation provided this
regularization satisfies the natural conditiona formulated earlier.


Now we shall prove that the definition (\ref{24}) corresponds to 
subtraction from $ \Gamma^{abc}_{\mu \nu \rho}(p,q)$ a local polynomial 
and that $ \Gamma^{abc,r}_{\mu \nu \rho}(p,q)$ satisfies ST identities.

In arbitrary regularization preserving a global symmetry of the model the
divergent part of the three point vertex has the following structure
\begin{equation}
i(z_1-1)t^{abc}\left[(p-k)_{\rho}g_{\mu \nu}+(k-q)_{\mu}g_{\nu
\rho}+(q-p)_{\nu}g_{\mu \rho}\right].
\label{34}
\end{equation}
Having this in mind we rewrite the sum of the three first terms in the
r.h.s.\ of Eq.\,(\ref{24}) in the form
\begin{eqnarray}
&&\Big(\Gamma^{abc}_{\mu \nu \rho}(p,q)+i(z_1-1)t^{abc}[(p-k)_{\rho}g_{\mu
\nu}+(k-q)_{\mu}g_{\nu \rho}+(q-p)_{\nu}g_{\mu \rho}]\Big) \nonumber\\[1mm]
&&-\Big( \Gamma^{abc}_{\mu \nu \rho}(p,0)+i(z_1-1)t^{abc}[2p_{\rho}g_{\mu
\nu}-p_{\mu}g_{\nu \rho}-p_{\nu}g_{\mu \rho}]\Big) \nonumber\\[1mm]
&&-\Big( \Gamma^{abc}_{\mu \nu \rho}(0,q)+i(z_1-1)t^{abc}[q_{\rho}g_{\mu
\nu}-2q_{\mu}g_{\nu \rho}+q_{\nu}g_{\mu \rho}]\Big)\,.
\label{35a}
\end{eqnarray}
The constant $ z_1 $ is defined by the condition
\begin{eqnarray}
&&\!\!\!\!\!\!\!\! \Gamma^{abc}_{\mu \nu
\rho}(p,0)+i(z_1-1)t^{abc}(2p_{\rho}g_{\mu \nu}-p_{\mu}g_{\nu
\rho}-p_{\nu}g_{\mu \rho}) \label{35b}\\[1mm]
&&\!\!\!\!\!\!\!\!-\frac{\partial}{\partial q_{\rho}} \left\{q^2(G^{-1}_{tr})^r_{\mu
 \alpha}(p)G^r(q) \Gamma^{abc,r}_{\alpha \nu}+( \mu \rightarrow \nu,\; a
\rightarrow b,\; p \rightarrow -p-q)\right\}_{q=0}=0\,.
\nonumber
\end{eqnarray}
Equation (\ref{35b}) is nothing but the differential STI, hence in any
anomaly free model in the limit when intermediate regularization is
removed it may be satisfied for a proper choice of $z_1$.  After such
transformation the expression in the curly bracketts in the r.h.s.\ of
Eq.\,(\ref{24}) vanishes in the limit when intermediate regularization is
removed and we get 
\begin{eqnarray} 
&& \Gamma^{abc,r}_{\mu \nu \rho}(p,q)=
\Gamma^{abc}_{\mu \nu \rho}(p,q)\nonumber\\[1mm]
&& +\,i(z_1-1)t^{abc}[(p-k)_{\rho}g_{\mu \nu}+(k-q)_{\mu}g_{\nu
\rho}+(q-p)_{\nu}g_{\mu \rho}]\,. \label{35c} 
\end{eqnarray} 
So we proved
that in the limit when intermediate regularization is removed our
renormalization procedure coincides with the usual local subtraction. The
validity of STI is obvious.

The last one-loop diagram to be analyzed is the proper four-point
Yang-Mills vertex. It can be analyzed in exact analogy with the three
point function. However it also can be done in an easier way, which does
not require the analysis of composite operators. Below we present the
corresponding construction.

In any regularization this function has the following
structure
\begin{equation}
\Pi^{abcd}_{\mu \nu \rho \sigma}(p,q,k)=P\left[At^{abe}t^{cde}g_{\mu
\rho}g_{\nu \sigma}+B \delta^{ab} \delta^{cd}g_{\mu \nu}g_{\rho
\sigma}\right]+O(p,q,k)\,.
\label{38}
\end{equation}
Here $P$ is the symmetrization operator with respect to the pairs $(a,
\mu;b, \nu; c, \rho; d, \sigma)$, $A$ and $B$ are some (divergent when
regularization is removed) constants, $O(p,q,k)$ is a finite function.

It follows from Eq.\,(\ref{38}) that 
\begin{equation}
B= \frac{1}{3}p^{-4}p_{\mu}p_{\nu}p_{\rho}p_{\sigma}
\Pi^{aaaa}_{\mu \nu \rho \sigma}(p,0,0)+O_1
\label{39}
\end{equation}
(no summation over $a$),
\begin{equation}
A= \frac{1}{3Np^2}P_{\nu \rho}(p)p_{\mu}p_{\sigma} \Pi^{abab}_{\mu \nu
\rho \sigma}(p,0,0)+O_2
\label{40}
\end{equation}
(no summation over $a$, and $a \neq b$).
Here $N=\sum_{b,e}t^{abe}t^{abe}$,  $\;P_{\mu \nu}$ is the
transversal projector operator and $O_1, O_2$ are some finite functions.

We define the renormalized four-point function as follows:
\begin{eqnarray}
&& \Pi^{abcd,r}_{\mu \nu \rho \sigma}(p,q,k)=\Pi^{abcd}_{\mu \nu \rho \sigma}
(p,q,k)\nonumber\\ [1mm]
&& -P(t^{abe}t^{cde}g_{\mu \rho}g_{\nu \sigma}) 
(3Np^2)^{-1}P_{\beta \gamma}(p)p^{\alpha}p^{\delta}
(\Pi_{con})^{mnmn}_{\alpha \beta \gamma \delta}(p,0,0)\nonumber\\[1mm]
&&-P(g_{\mu \nu}g_{\sigma \rho} \delta^{ab}
\delta^{cd})3^{-1}p^{-4}p^{\alpha}p^{\beta}p^{\gamma}p^{\delta}
(\Pi_{con})^{mnmn}_{\alpha \beta \gamma \delta}(p,0,0)\,.
\label{41}
\end{eqnarray}
Note that the subtracted terms at the r.h.s.\ of this equation include the
functions $\Pi_{con}(p,0,0)$ which represent all connected one loop 
diagrams with four external gauge field lines. They include apart from the 
proper four-point vertex also weakly connected diagrams shown at Fig.\,\ref
{fig3}. The divergent subgraphs in these diagrams are assumed to be 
renormalized as described above and do not introduce either ultraviolet or
infrared singularities.

\begin{figure}[h]
\epsfxsize=0.3\textwidth
\centerline{\epsfbox{fig3.eps}}
\caption{Weakly connected diagrams contributing to the l.h.s.\ of Eq.\,(\ref{41}).}
\label{fig3}
\end{figure}

By virtue of Eqs.\,(\ref{39}, \ref{40}) the renormalized function $
\Pi^{abcd,r}_{\mu \nu \rho \sigma}(p,q,k)$ defined by the Eq.\,(\ref{41}) 
is finite.

To prove the locality and gauge invariance we use again 
differential STI. The differential STI for the four point function imply the
relation
\begin{eqnarray}
&&\!\!\!\!\!\!\!\!\left\langle \partial_{\mu}B^a_{\mu}(x) \partial_{\nu}B^b_{\nu}(y)
\partial_{\rho}B^c_{\rho}(z) \partial_{\sigma}B^d_{\sigma}(u)\right\rangle 
\nonumber\\[1mm]
&&\!\!\!\!\!\!\!\! = \delta(x-u) \delta(y-z) \delta^{ad} \delta^{bc}+\left(\{x,a\} \rightarrow \{y,b\}\right)+
\left(\{x,a\} \rightarrow \{z,c\}\right)\,.
\label{42}
\end{eqnarray}
This equation shows that the connected part of the correlator (\ref{42}) 
is equal to zero. The corresponding differential identity for the Fourier
transform of the four point vertex looks as follows
\begin{equation}
p_{\mu}p_{\nu}(\Pi^r_{con})^{abcd}_{\mu \nu \rho \sigma}(p,0,0)=0\,.
\label{44}
\end{equation}

Let us replace all the functions $ \Pi^{abcd}_{\mu \nu \rho \sigma}$ at
the r.h.s.\ of Eq.\,(\ref{41}) by the functions renormalized with the help of
local subtractions
\begin{equation}
\Pi^{abcd}_{\mu \nu \rho \sigma} \rightarrow
\Pi^{abcd}_{\mu \nu \rho \sigma}%\nonumber\\
+(z_4-1)P(t^{abe}t^{cde}g_{\mu \nu}g_{\rho \sigma})+(\tilde{z_4}-1)P\left(g_{\mu
\nu}g{\rho \sigma} \delta^{ab} \delta^{cd}\right).
\label{45a}
\end{equation}
The constants $z_4$ and $ \tilde{z}_4$ are defined by the condition that
 in the limit when a regularization is removed the functions (\ref{45a})
satisfied the differential identity (\ref{44}). It is always possible in
an anomaly free model. Such a substitution does not change the
Eq.\,(\ref{41}) as the terms $ \sim (z_4-1)$ and $ \sim (\tilde{z}_4-1)$ in
the first the second and the third lines cancel. We see that the function
renormalized according to our prescription in the limit when a
regularization is removed coincides with the corresponding function
renormalized with the help of local subtractions. The differential STI are
fullfilled by construction.

Extension to the higher loops is done exactly as in the case of QED.
Introducing the one-loop counterterms we are left with the two-loop
diagrams which diverge only superficially. Using the standard
R-operation we can define the 
renormalized two- and higher loop correlation functions by the same equations as 
above. They are obviously finite. The 
proof of locality is identical to the one loop-case.


\section{A model with chiral fermions}


As the last example we consider a simple model which includes chiral
fermions. We choose the Abelian sector of Salam-Weinberg model with 
the Higgs interaction switched off. This model is known to be anomaly 
free but not a vectorlike one.

The Lagrangian looks as follows:
\begin{eqnarray}
&& L=- \frac{1}{4}( \partial_{\mu}B_{\nu}- \partial_{\nu}B_{\mu})^2+i 
\sum_{q \pm} \bar{\psi}_{q \pm} \gamma_{\mu}( \partial_{\mu}-ig_{q 
\pm}B_{\mu})\psi_{q \pm}\nonumber\\ 
&& \qquad +\,i \sum_{l \pm} \bar{\psi}_{l 
\pm} \gamma_{\mu}(\partial_{\mu}-ig_{l \pm}B_{\mu}) \psi_{l \pm}\,.
\label{47}
\end{eqnarray}
Here $ \psi_{q \pm}$ and $\psi_{l \pm}$ represent left (right) handed
fields of quarks and leptons respectively. The corresponding charges 
are denoted by $g_{q \pm}$ and $g_{l \pm}$. Due to the condition
\begin{equation}
\sum_{q \pm}g^3_{q \pm}+ \sum_{l \pm}g^3_{l \pm}=0
\label{48}
\end{equation}
the triangle anomaly is absent.

Renormalized gauge field and fermion propagators are defined by the
Eqs.\,(\ref{3}, \ref{9}) of Sec.\,2. The renormalized gauge
field-fermion vertices are defined as in Eq.\,(\ref{4}),
\begin{equation}
\Gamma^{\mu,r}_{q \pm,l \pm}= \Gamma^{\mu}_{q \pm,l \pm}(p,q)-
\Gamma^{\mu}_{q \pm,l \pm}(p,0)+g_{q \pm,l \pm} \partial_{\mu} 
\Sigma^r_{q \pm,l \pm}(p)\;.
\label{49}
\end{equation}
The four-point gauge field correlator can be renormalized according to
Eq.\,(\ref{12}), but in this case the subtracted term has to include all
connected diagrams with four external gauge field lines.

The only essentially new moment is a possible renormalization of the
three-point gauge field vertex. In QED it is zero due to Furry theorem, 
but in our case it does not vanish.

WTI for this function have the same form as for the four-point 
function
\begin{equation}
p_{\mu} \Gamma_{\mu \nu \rho}(p,q)=0; \quad \Gamma_{\mu \nu \rho}(0,q)
=0\,.
\label{50}
\end{equation}
Accordingly we choose the following definition of renormalized vertex
function
\begin{equation}
\Gamma^r_{\mu \nu \rho}(p,q)= \Gamma_{\mu \nu \rho}(p,q)-
\Gamma_{\mu \nu \rho}(p,0)-\Gamma_{\mu \nu \rho}(0,q)\,.
\label{51}
\end{equation}
This function is finite and can be calculated in any regularization 
scheme. In a gauge invariant scheme one has due to WTI (\ref{50})
\begin{equation}
 \Gamma^r_{\mu \nu \rho}(p,q)= \Gamma^{inv}_{\mu \nu \rho}(p,q)\,.
\label{53}
\end{equation}

\section{Discussion}

The renormalization procedure presented above combines the advantages 
of algebraic renormalization and invariant regularization schemes. It 
is regularization independent and may be applied to any anomaly free 
model. At the same time it preserves gauge invariance at all stages 
and does not require additional fine tuning of counterterms to restore 
the symmetry. It also avoids the problem of infrared singularities in 
renormalization of Green functions. 

From the point of view of practical 
calculations it may be more complicated than dimensional regularization 
if one is interested only in some particular diagram. Our procedure is 
recurrent and it requires knowleadge of lower order diagrams for some 
particular configurations of external momenta. However it seems well 
suited for systematic calculation of Feynman diagrams up to a given 
order. It is important to emphasise that our renormalization procedure 
may be applied directly to the models where dimensional regularization 
fails, in particular to the Standard Model and supersymmetric theories.

It is worth to mention that to make practical calculations easier one
can combine our renormalization procedure with the dimensional 
regularization. For example in chiral fermion models one may 
define\,\cite{H.V,B.M} the 
$ \gamma_5$ matrix as the product $ \gamma_5= \gamma_0 \gamma_1 
\gamma_2 \gamma_3$, and then apply the dimensional 
regularization. For diagrams with virtual fermion lines this definition 
is known to break chiral invariance. However the symmetry may be easily 
restored with the help of subtractions described above. 

It is worth to
stress that our renormalization is defined as the subtraction procedure
using the combinatorics of the standard R-operation. It does not require
explicit calculation of counterterms. The expressions for counterterms
presented in the paper serve the goal to establish the connection with
more standard approaches and to facilitate the proofs.


\section*{Acknowledgements}
\addcontentsline{toc}{section}{\numberline{}Acknowledgements}
This work was partially done while the author was
 visiting Max-Planck-Institute for Physics in Munich. I wish to thank
 D.~Maison and J.~Wess for hospitality and Humboldt Foundation for a 
 generous support. My thanks to the members of Theoretical group for 
 helpful discussion. This researsh was supported in part by Russian 
 Basic Research Fund under grant 99-01-00190 and by the president 
 grant for support of leading scientific schools 0015-96046.

\section*{References}
\addcontentsline{toc}{section}{\numberline{}References}

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