We argue in favor of the independence on any scale, ultraviolet or infrared, in kernels of the effective action expressed in terms of dressed ${\cal N} =1$ superfields for the case of ${\cal N} =4$ super-Yang--Mills theory. Under ``finiteness'' of the effective action of dressed mean superfields we mean its ``scale independence''. We use two types of regularization: regularization by dimensional reduction and regularization by higher derivatives in its supersymmetric form. Based on the Slavnov--Taylor identity we show that dressed fields of matter and of vector multiplets can be introduced to express the effective action in terms of them. Kernels of the effective action expressed in terms of such dressed effective fields do not depend on the ultraviolet scale. In the case of dimensional reduction, by using the developed technique we show how the problem of inconsistency of the dimensional reduction can be solved. Using Piguet and Sibold formalism, we indicate that the dependence on the infrared scale disappears off shell in both the regularizations. \vskip 1cm \noindent Keywords: $R$-operation, gauge symmetry, ${\cal N} =4$ supersymmetry, Slavnov--Taylor identity.
hep-th0411150.tex
\documentstyle[11pt]{article}
%\topmargin -1cm
\textwidth 150mm
\textheight 210mm
\oddsidemargin.3in
\evensidemargin .3in
%\parindent 0pt

\def\a{\alpha}
\def\b{\beta}
\def\db{{\dot{\beta}}}
\def\t{\theta}
\def\bt{{\bar{\theta}}}
\def\yb{{\bar{y}}}
\def\Db{{\bar{D}}}
\def\Tr{{\rm Tr}}
\def\dis{\displaystyle}
\def\le{\left(}
\def\ri{\right)}
\def\da{{\dot{\alpha}}}
\def\no{\nonumber}
\def\del{\delta}
\def\bW{\bar{W}}
\def\G{\Gamma}
\def\rar{\rightarrow}
\def\fra1g2{\frac{1}{g^2}}
\def\dg{{\dagger}}
\def\Vt{\tilde{V}}
\def\Kt{\tilde{K}}
\def\SQa{\sqrt{\tilde{\a}}}
\def\e{\epsilon}
\def\sm{{\sigma_m}}
\def\f12{\frac{1}{2}}
\def\F{\Phi}
\def\pd{\partial}
\def\ve{\varepsilon}
\def\F{\Phi}
\def\bF{\bar{\Phi}}

\begin{document}
\begin{titlepage}
\flushright{USM-TH-161}\\
\vskip 2cm               
\begin{center}
{\Large \bf Finiteness of ${\cal N} =4$ super-Yang--Mills effective action
in terms of dressed  ${\cal N} =1$ superfields} \\
\vskip 1cm  
Igor Kondrashuk$^{a,b}$ and Ivan Schmidt$^a$ \\
\vskip 5mm  
{\it  (a) Departamento de F\'\i sica, Universidad T\'ecnica 
Federico Santa Mar\'\i a, \\
 Avenida Espa\~{n}a 1680, Casilla 110-V, Valparaiso, Chile} \\
\vspace{1cm}
{\it  (b) Departamento de Ciencias Basicas, 
Universidad del Bio-Bio, Campus 
Fernando May, Casilla 447, Avenida Andreas Bello, 
Chillan, Chile} \\
\end{center}
\vskip 20mm
\begin{abstract}
We argue in favor of the independence on any scale, ultraviolet or infrared, in  kernels 
of the effective action expressed in terms of  dressed  
${\cal N} =1$ superfields for the case of ${\cal N} =4$ super-Yang--Mills 
theory. Under ``finiteness'' of the effective action of dressed mean superfields 
we mean its ``scale independence''. We use two types of regularization:  regularization by dimensional reduction and 
regularization by higher derivatives in its supersymmetric form. Based on the Slavnov--Taylor 
identity we show that dressed fields of matter and of vector multiplets can be introduced to 
express the effective action in terms of them. Kernels of the effective action 
expressed in terms of such dressed effective fields do not depend on the ultraviolet scale.
In the case of  dimensional reduction, by using the 
developed technique we show how the problem of inconsistency of the dimensional reduction 
can be solved. Using Piguet and Sibold formalism, we indicate that the dependence on 
the infrared scale disappears off shell in both the regularizations. 


    
\vskip 1cm
\noindent Keywords: $R$-operation, gauge symmetry,  
${\cal N} =4$ supersymmetry, Slavnov--Taylor identity. 
\end{abstract}
\end{titlepage}




\def\a{\alpha}
\def\b{\beta}
\def\db{{\dot{\beta}}}
\def\t{\theta}
\def\bt{{\bar{\theta}}}
\def\yb{{\bar{y}}}
\def\Db{{\bar{D}}}
\def\Tr{{\rm Tr}}
\def\dis{\displaystyle}
\def\le{\left(}
\def\ri{\right)}
\def\da{{\dot{\alpha}}}
\def\no{\nonumber}
\def\del{\delta}
\def\bW{\bar{W}}
\def\G{\Gamma}
\def\rar{\rightarrow}
\def\fra1g2{\frac{1}{g^2}}
\def\dg{{\dagger}}
\def\Vt{\tilde{V}}
\def\Kt{\tilde{K}}
\def\SQa{\sqrt{\tilde{\a}}}
\def\e{\epsilon}
\def\sm{{\sigma_m}}
\def\f12{\frac{1}{2}}
\def\F{\Phi}
\def\pd{\partial}
\def\ve{\varepsilon}
\def\F{\Phi}
\def\bF{\bar{\Phi}}





The effective action is restricted by consequences of various symmetries of the 
classical action that at the quantum level take the form of specific identities.
One of them is the Slavnov--Taylor (ST) identity 
\cite{Slavnov:1972fg}-\cite{Zinn-Justin:1974mc}.   
This  generalizes the Ward--Takahashi identity of quantum electrodynamics 
to the non-Abelian case and can be derived starting from the property 
of invariance of the tree-level action with respect to BRST symmetry 
\cite{Becchi:1974md,Tyutin:1975qk}. The ST identity can be formulated  as 
equations involving  variational derivatives of the effective action. 
In the general ${\cal N} =1$  supersymmetric theory it can be 
written as \cite{Piguet:1996ys} 
\begin{eqnarray*}
& \Tr\left[\dis{\int~d^8z~\frac{\del \G}{\del V}\frac{\del \G}{\del K}
 - i \int~d^6y~\frac{\del \G}{\del c}\frac{\del \G}{\del L}
+ i \int~d^6\yb~\frac{\del \G}{\del \bar{c}}\frac{\del \G}{\del
\bar{L}}}\right. \no \\
& - \left. \dis{ \int~d^6y~\frac{\del \G}{\del b}\le\frac{1}{32}
\frac{1}{\a}\Db^2D^2V\ri
- \int~d^6\yb~\frac{\del \G}{\del
\bar{b}}\le\frac{1}{32}\frac{1}{\a}D^2\Db^2V\ri} \right] \\ 
& - \dis{i~\int~~d^6y~\frac{\del \G}{\del \F}~\frac{\del \G}{\del k} +
i~\int~d^6\yb~\frac{\del \G}{\del \bar{k}}~\frac{\del \G}{\del
\bF} = 0}. \no
\end{eqnarray*}
Here the standard definition of the measures in superspace is used 
\begin{eqnarray*}
d^8z \equiv d^4x~d^2\t~d^2\bt,~~~ d^6y \equiv d^4y~d^2\t,~~~
d^6\yb \equiv d^4\yb~d^2\bt.
\end{eqnarray*}
The effective action $\G$ generates one particle irreducible amplitudes of the quantum fields and contains all the information 
about the quantum behaviour of the theory. It is a functional of the effective fields $V,b,\bar{b},c,\bar{c},\F,\bF$ and 
external sources  $K,L,\bar{L},k,\bar{k},$ coupled at tree level to BRST-transformations of the corresponding classical 
fields \cite{Faddeev:1980be}. We use two types of 
regularization: regularization by higher derivatives \cite{Krivoshchekov:1978xg,West:1985jx} 
and regularization by dimensional reduction \cite{Siegel:1979wq,Capper:1979ns}. 







We will show that the actual variables of the effective action are dressed 
effective superfields, that is, they are effective superfields convoluted 
with some unspecified dressing functions that are parts of propagators. 
Kernels accompanying  the dressed effective fields in the effective action are related to the 
scattering amplitudes of the particles. A similar problem has been solved in 
component formalism \cite{Cvetic:2004kx,Cvetic:2006kk}. As has been argued in Refs. \cite{Cvetic:2004kx,Cvetic:2006kk} 
in that formalism the dressed mean fields appear to be the actual variables of the effective action,
leaving the kernels of the action independent of any scale, ultraviolet or infrared, in case of   ${\cal N} =4$ supersymmetric 
Yang--Mills theory. This statement has been confirmed by the explicit calculation 
in Ref. \cite{Cvetic:2006iu}. In the present paper this problem is considered by using ${\cal N} =1$ superfield formalism
which keep one of the supersymmetries apparent and conserve all the R-symmetries which we need to apply anomaly multiplet 
ideas \cite{Sohnius:1981sn}.  The important point of Refs. 
\cite{Cvetic:2004kx,Cvetic:2006kk,Cvetic:2002dx,Cvetic:2002in,Kondrashuk:2000br,Kang:2004cs,Kondrashuk:2003tw,Kondrashuk:2001yd} 
is the possibility to absorb the two point proper functions in the re-definition of the effective fields. 



We consider the {${\cal N}=4$ theory in {${\cal N}=1$ superfield 
formalism. This model has specific field contents. The Lagrangian of the model in terms of  ${\cal N}=1$ 
superfields is 
\begin{eqnarray*}
& S = \dis{\fra1g2\frac{1}{128}\Tr~\left[\int~d^6y~W_\a W^\a  +
\int~d^6\yb~\bW^\da \bW_\da \right.} \nonumber\\
& + \dis{\left.\int~d^8z~e^{-V}\bF_i~e^V~\F^i \right.}  \\
& + \dis{\left. \frac{1}{3!}\int~d^6y~i\e_{ijk}\F^i[\F^j,\F^k] 
+ \frac{1}{3!}\int~d^6\yb~i\e^{ijk}\bF_i[\bF_j,\bF_k]\right]}.
\end{eqnarray*}
For the {${\cal N}=1$ supersymmetry we use the notation of Ref. 
\cite{Kondrashuk:2000qb}. This Lagrangian 
for {${\cal N}=4$ supersymmetry is taken from  Ref. \cite{Gates:1983nr}. 
The flavor indices of the matter run in  
$i =1,2,3$ and the matter superfields are in the adjoint representation of the 
gauge group, $\F^i = \F^{ia}~T^a.$ (The vector superfield is in the same 
representation, $V = V^aT^a$). 



Consider for the beginning the general ${\cal N} = 1$ super-Yang--Mills whose 
classical action takes the form
\begin{eqnarray}
& S = \dis{\int~d^6y~\frac{1}{128}\fra1g2\Tr~ W_\a W^\a  +
\int~d^6\yb~\frac{1}{128}\fra1g2\Tr~ \bW^\da \bW_\da } \nonumber\\
& + \dis{\int~d^8z~\bF~e^V~\F}  \label{mattact}\\
& + \dis{\int~d^6y~\left[Y^{ijk}\F_i\F_j\F_k + M^{ij}\F_i\F_j \right]}
+ \dis{\int~d^6\yb~\left[\bar{Y}_{ijk}\bF^i\bF^j\bF^k \no
+ \bar{M}_{ij}\bF^i\bF^j \right]}.
\end{eqnarray}
We do not specify the representation of the matter fields here. It is some 
general reducible representation of the gauge group with a set of irreducible
representations. The Yukawa couplings $Y^{ijk}$ and  $M^{ij}$
appear at some general triple vertex and mass terms in four dimensions. 
The path integral describing the quantum theory is defined as
\begin{eqnarray}
& \dis{Z[J,\eta,\bar{\eta},\rho,\bar{\rho},j,\bar{j},K,L,\bar{L},k,\bar{k}] 
= \int~
dV~dc~d\bar{c}~db~d\bar{b}~d\F~d\bF~\exp i}\left[\dis{S}
\right.  \no \\
& \left. + \dis{2~\Tr\le\int~d^8z~JV + i\int~d^6y~\eta c
+ i\int~d^6\yb~\bar{\eta}\bar{c} + i\int~d^6y~\rho b
+ i\int~d^6\yb~\bar{\rho}\bar{b}\ri} \right. \no\\
& \left. + \dis{\le\int~d^6y~\F~j + \int~d^6\yb~\bar{j}~\bF\ri }
    \right. \label{pathRM}\\
& \left. + \dis{2~\Tr\le i\int~d^8z~K\del_{\bar{c},c}V + \int~d^6y~Lc^2 +
\int~d^6\yb~\bar{L}\bar{c}^2 \ri }\right. \no\\
& \left. + \dis{\int~d^6y~k~c~\F + \int~d^6\yb~\bF~\bar{c}~\bar{k}} \right], 
\no
\end{eqnarray}



The derivation of the ST identity in general supersymmetric theory can be 
found for example in Ref. \cite{Kondrashuk:2001yd}. The result is 
\begin{eqnarray}
& \Tr\left[\dis{\int~d^8z~\frac{\del \G}{\del V}\frac{\del \G}{\del K}
 - i \int~d^6y~\frac{\del \G}{\del c}\frac{\del \G}{\del L}
+ i \int~d^6\yb~\frac{\del \G}{\del \bar{c}}\frac{\del \G}{\del
\bar{L}}}\right. \no \\
& - \left. \dis{ \int~d^6y~\frac{\del \G}{\del b}\le\frac{1}{32}\frac{1}{\a}\Db^2D^2V\ri
- \int~d^6\yb~\frac{\del \G}{\del
\bar{b}}\le\frac{1}{32}\frac{1}{\a}D^2\Db^2V\ri} \right] \label{STrM}\\ 
& - \dis{i~\int~~d^6y~\frac{\del \G}{\del \F}~\frac{\del \G}{\del k} +
i~\int~d^6\yb~\frac{\del \G}{\del \bar{k}}~\frac{\del \G}{\del
\bF} = 0}. \no
\end{eqnarray}



Regularization is necessary to analyze the identities. First, we consider the regularization by dimensional reduction. 
Under this regularization the algebra of Lorentz indices is done in four dimensions but integration is done in $4-2\e$ dimensions
in the momentum or in the position space.  As  has been shown in Refs. \cite{Cvetic:2004kx,Cvetic:2006kk}, such a 
regularization is self-consistent at least in  
${\cal N}=4$ supersymmetric Yang--Mills theory in component formalism at all orders of the perturbation theory. In the next  
paragraphs we will show such a regularization procedure can be applied at {\it all} orders of the perturbation theory
in superfield formalism too if this regularization is combined with Piguet and Sibold formalism of 
Refs. \cite{Piguet:1984mv,Piguet:1984im}





Let's  analyse this identity in the following way. One can start by considering the monomial $Lcc$ of the effective action. 
Due to supersymmetry, superficial divergences are absent in chiral vertices \cite{Grisaru:1979wc,West:1986wu}. This theorem 
is a direct consequence of the Grassmanian integration and has been described, e.g., in Ref. \cite{Grisaru:1979wc}.  
However, there could be finite contributions.  They are scale-independent, in that sense they remain finite in the limit of 
removing the ultraviolet regularization. For example, at one loop level one can find among others the following kernel structure 
for the correlator $Lcc$  \cite{Cvetic:2004kx}:
\begin{eqnarray*}
& \dis{\int~d^4\t~d^4x_1d^4x_2d^4x_3~\frac{1}{(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^4}f^{bca}}\times \\
& \dis{ \times \le D^2~L^{a}(x_1,\t,\bt)\ri c^{b}(x_2,\t,\bt)c^{c}(x_3,\t,\bt).}
\end{eqnarray*}
 

Landau gauge is the specific case in gauge theories because we do not need to renormalize the gauge 
fixing parameter. Absence of the gauge parameter is enough condition to avoid this possible source of  appearance of 
the scale dependence in kernels of dressed mean superfields through the renormalization of the gauge parameter. From the effective 
action $\G$ we can extract the two point ghost proper correlator  $G(z-z'),$
\begin{eqnarray*}
G^{\dg} = G,
\end{eqnarray*}
and a two point connected ghost correlator $G^{-1}(z -z'),$ which is related to the previous one in the following way:
\begin{eqnarray*}
\int d^8z'~G(z_1 - z')~G^{-1}(z_2 - z') = \del^{(8)}(z_2 - z_1).
\end{eqnarray*}
This definition is valid in each order of perturbation theory. One can absorb this two point proper function into a non-local 
redefinition of the effective fields $K$ and $V$ in the following manner \cite{Kondrashuk:2000br}: 
\begin{eqnarray}
& \dis{\Vt \equiv  \int~d^8z'~V(z')~G^{-1}(z-z')}, \label{conv}\\
& \dis{\Kt \equiv \int~d^8z'~K(z')~G(z-z').} \no
\end{eqnarray}
One can see that the part of the ST identity without the gauge fixing term is covariant with respect to such a re-definition of 
the effective fields.  We will call the construction  (\ref{conv}) dressed effective (or mean) superfields. Proceeding at one 
loop level in terms of the dressed effective superfields, one can see from the ST identity that the divergence of the 
$\Kt c\Vt$ vertex must be canceled by the divergence of the $Lcc$ vertex. However, the ${\cal N} =1$ $Lcc$ vertex does 
not diverge at one loop 
level in the momentum space due to supersymmetry.\footnote{For simplicity we consider the Landau gauge.} 
This means that the ST identity clearly 
shows that the  $\Kt c\Vt$ is also finite \cite{Cvetic:2004kx,Cvetic:2006kk}, that is, it does not diverge in the limit of 
removing the regularization. The rest of the UV divergence in the propagator of the dressed gauge fields $\Vt\Vt$ can be 
removed by redefining of the gauge coupling constant. In theories with zero beta function, in case of this paper it is ${\cal N}=4$ 
supersymmetric Yang-Mills theory, this last divergence is absent. The other graphs are finite since proper correlators can be 
constructed from the $\Kt c\Vt$ (or $Lcc$), and $\Vt\Vt$ correlators by means of the ST identity. 
This is a direct consequence of the ST identity and $R$-operation \cite{Cvetic:2004kx,Bogolyubov:1980nc}. 
We can repeat this argument in each order of perturbation theory, which is  completely the same as in 
Refs. \cite{Cvetic:2004kx,Cvetic:2006kk} in component formalism. 





Thus, all the gauge part of the effective action can be 
considered as the functional of the effective fields $\Vt,$  $\Kt,$ $L,$ $c,$
$\bar{c},$ $\tilde{b}$ and $\tilde{\bar{b}}.$  Due to the antighost equation 
\cite{Kondrashuk:2000br}, the antighost field $b$ is always 
dressed in the same manner as the auxiliary field $K$ is dressed. Kernels of 
this effective action are functions of the gauge coupling, mutual distances and 
in general of the ultraviolet scale, because the divergences in subgraphs must be 
removed by the renormalization of the gauge coupling. But if the beta function is 
zero the kernels have no UV scale dependence. In the position space in component formalism the infrared divergences can be 
analysed in the same way like ultraviolet divergences were analysed in Ref.\cite{Bogolyubov:1980nc} in the momentum space 
by means of $R-$operation \cite{Cvetic:2006iu}. However, it is difficult to repeat this argument in the position space 
in superfield formalism since the propagator of vector superfield is dangerous in the infrared region. 
In view of this difficulty we use the formalism developed in Refs.  \cite{Piguet:1984mv,Piguet:1984im}, where 
the problem of off-shell infrared divergences of superfield formalism has been solved.



The infrared regulator has been introduced by means of the following trick of renormalization of the vector gauge field $V:$
\begin{eqnarray*}
V \rar V + \mu {\t^2}{\bt^2}V,
\end{eqnarray*} 
where $\mu$ is the infrared regulator mass. Propagators of the lowest components 
of the gauge superfield obtain a shift by the infrared regulator mass
which is enough to make the Feynman graphs safe in the infrared region of 
momentum space. It appears one can construct the classical action that satisfies 
a generalized ST identity which involves BRST counterparts of the gauge parameters. 
Then, a general solution to the generalized 
ST identity has been found at the classical level. As a consequence of 
that solution, the path integral that corresponds to this solution 
possesses the property of independence 
of v.e.v.s of gauge invariant quantities on the gauge parameters 
\cite{Piguet:1984im}. The independence of the physical quantities on the 
infrared scale was obtained 
by the same way. In addition, the new external superfield $u$ can be introduced 
so that a shift of its highest component is proportional to the infrared scale 
$\mu.$ That field also participates in the generalized ST identity. 






One can analyse the generalized ST identity of Ref. \cite{Piguet:1984im} which 
is obtained after the modification of standard one (\ref{STrM}) by including 
additional external fields and gauge parameters. 
Appearance of the additional insertions of the external field $u$ or 
spurions $\mu {\t^2}{\bt^2}$ into supergraphs does not change the nonrenormalization 
theorems. This property has been used in Refs. 
\cite{Yamada:1994id,Jack:1997pa,Avdeev:1997vx}
to derive the relation 
between softly broken and rigid renormalization constants in ${\cal N} =1$ 
supersymmetric 
theory. Thus, it cannot bring any changes for our conclusions about the 
 $Lcc$ vertex from the point of 
our analysis since all divergent subgraphs remain divergent and all convergence 
properties of subgraphs remain unchanged. In that sense subgraphs are 
finite after renormalization by the dressing functions but all the vertex as a whole 
is also finite superficially due to property of Grassmanian 
integration which is not 
broken by the insertions of the external superfields 
\cite{Yamada:1994id,Jack:1997pa,Avdeev:1997vx}. 



There is also another way to explain the independence of the physical quantities on the infrared scale $\mu.$ 
The point is that the the factors $(1 + \mu {\t^2}{\bt^2})$ coming from vertices 
will be canceled with factors $(1 + \mu {\t^2}{\bt^2})^{-1}$ coming from 
propagators. However, the propagators are IR-finite with the $\mu$ addition and thus 
the theory is regularized in the infrared. The same trick can be applied to demonstrate 
the independence of the correlators of dressed mean superfields in the 
Landau gauge of the infrared scale $\mu.$




In this paper we have significantly used the vanishing of the gauge beta function in ${\cal N}=4$ super-Yang-Mills theory. 
The vanishing of the beta function in first three orders of the perturbation theory has been established in Refs. 
\cite{Ferrara:1974pu,Jones:1977zr,Avdeev:1980bh}.  Originally, in the background field technique, it has been shown 
in Ref. \cite{Grisaru:1982zh} that in ${\cal N}=2$ supersymmetric Yang--Mills theory 
the beta function vanishes beyond one loop. The same result has been derived
in Ref. \cite{Howe:1983sr} by using the background field technique with unconstrained 
${\cal N}=2$ superfields. The arguments of \cite{Grisaru:1982zh,Howe:1983sr}  
are based on the fact that ${\cal N}=2$ supersymmetry prohibits any counterterms to the gauge coupling except 
for one loop contribution.  In Ref. 
\cite{West:1986wu}, the fact that ${\cal N}=2$ supersymmetric YM theory does 
not have contributions to the beta function beyond one loop has been 
argued based on currents of R-symmetry, which are in the same 
supermultiplet with the energy-momentum tensor. The  
proportionality of the trace anomaly of the energy momentum tensor to 
the beta function in general nonsupersymmetric Yang-Mills theories 
has been proved in Ref. \cite{Collins:1976yq}. Since R-symmetry does not have 
anomaly in ${\cal N}=4$ theory, the same is true for the anomaly of the 
conformal symmetry which is proportional to the beta function.        
At one loop level the beta function is zero with this field 
contents \cite{Ferrara:1974pu,Jones:1977zr}. Moreover, explicit calculation has 
been done at two loops in terms of ${\cal N}=1$ superfields \cite{Howe:1984xq},
and it has been shown that the beta function 
of ${\cal N}=2$ theory is zero at two loops.





The dimensional reduction was known to be inconsistent \cite{Siegel:1980qs}. We proposed solution to this
problem in component formalism in Ref. \cite{Cvetic:2006kk} for ${\cal N}=4$ super-Yang-Mills theory.
We can repeat similar arguments in case of superfield formalism. The only new feature here is the appearance 
of the infrared scale $\mu$ in subgraphs, as it has been explained above.  
The point is that the vertex $Lcc$ is always convergent superficially in superfield formalism. 
In ${\cal N} = 4$ super-Yang-Mills theory ultraviolet divergences in the subgraphs of 
the $Lcc$ vertex should cancel each other at the end. 
The insertion of the operator of the conformal anomaly into vacuum expectation values  of operators of 
gluonic fields at different points in spacetime is proportional to the beta function of the gauge coupling 
\cite{Collins:1976yq}. Due to the algebra of the four-dimensional supersymmetry the beta function should be zero 
\cite{Sohnius:1981sn}. Algebra of the supersymmetry operators in the Hilbert space created by dressed fields
can be considered as four-dimensional in Lorentz indices as well as in spinor indices since 
the limit $\e \rar 0$ is non-singular at one-loop order, two-loop order and higher orders 
as we have seen in the previous paragraphs. Thus, we can consider each correlator 
as pure four-dimensional, solving  order-by-order the problem of dimensional discrepancy of 
convolutions in Lorentz and spinor indices. The dependence on the infrared scale $\mu$ is canceled 
by itself from the contributions of vertices and propagators, as it has been shown above.




Another  regularization scheme for  ${\cal N}=1$  supersymmetric Yang--Mills theory exists
which is the higher derivatives regularization scheme \cite{West:1985jx,Krivoshchekov:1978xg}. 
According to Ref. \cite{West:1985jx}, ${\cal N}=2$ supersymmetry can be maintained by the regulator piece of the Lagrangian in HDR.
The scheme is discussed in detail in Ref. \cite{Faddeev:1980be} for the nonsupersymmetric case. A direct supersymmetric 
generalization of the regularization by higher derivatives  has been constructed in Ref. \cite{Krivoshchekov:1978xg}. 
This generalization has been considered in detail also in the paper \cite{West:1985jx}, in particular in the background 
field technique. As has been explicitly shown in Refs.  \cite{West:1985jx,Krivoshchekov:1978xg}, at one loop level HDR 
regularizes all the supergraphs in a gauge invariant manner and this repeats the 
corresponding construction  in the  nonsupersymmetric version of HDR \cite{Faddeev:1980be}.  However, when applied to
explicit examples, this approach is known to yield incorrect results in Landau gauge \cite{Martin:1994cg}. A number of 
proposals have been put forward to treat this problem \cite{Bakeyev:1996is,Asorey:1995tq}. As was shown in Ref.
\cite{Asorey:1995tq}, the contradiction, noticed in \cite{Martin:1994cg} is related
to the singular character of Landau gauge. In all other covariant gauges the method works and to include also the Landau 
gauge one has to add one more Pauli-Villars field to get the correct result \cite{Bakeyev:1996is,Asorey:1995tq}. 
Having used this regularization, new scheme has been proposed in  Refs. \cite{Slavnov:2003cx,Slavnov:2002kg,Slavnov:2001pu}. 
Calculations in the higher derivative regularization in terms of superfields can be found in 
Refs.\cite{Stepanyantz:2004sg,Soloshenko:2003nc,Soloshenko:2002np,Slavnov:2003cx,Slavnov:2002kg,Slavnov:2001pu}.   
To perform the calculation, it is proposed to break the gauge symmetry by using some HDR scheme with usual derivatives 
instead of covariant derivatives and then to restore the ST identity for the effective action by using some noninvariant 
counterterms. This problem has been solved in Refs. \cite{Slavnov:2003cx,Slavnov:2002kg,Slavnov:2001pu}.    
All the arguments, given above for the regularization by the dimensional reduction in favor of scale independence of the
kernels of dressed mean superfields are valid also for the 
regularization by higher derivatives. The only difference is that to remove the regularization by higher derivatives 
 we take the limit $\Lambda \rar \infty$ instead of $\e \rar 0$ as it was for the case of dimensional reduction, where
$\Lambda$ is the regularization scale of HDR. 




So far the pure gauge sector has been considered. Let us look now at the matter 
two point functions. Schematically, one can write the two point vertex as
\begin{eqnarray*}
\int d^8zd^8z'\bF(z)G_\F(z-z')\F(z').
\end{eqnarray*} 
The function $G_\F(z-z')$ can be divided into two equal parts $\tilde{G}_\F,$
\begin{eqnarray*}
\int d^8z'\tilde{G}_\F(z_1-z')\tilde{G}_\F(z'-z_2) = G_\F(z_1-z_2).
\end{eqnarray*} 
This is a product in the momentum space. 
Now we define the
new fields $\tilde{\F},$ 
\begin{eqnarray*}
\int d^8z'\tilde{G}_\F(z-z')\F(z') \equiv  \tilde{\F}(z),
\end{eqnarray*} 
and represent the effective action in terms of these fields. In particular,
the divergent part of the function  $\tilde{G}_\F$ can be absorbed into the 
redefinition of the Yukawa couplings and masses. However, in  ${\cal N}=4$
supersymmetric theory masses are absent and the Yukawa coupling (which coincides with 
the gauge coupling) is not renormalized due to the structure of the Yukawa terms
in the classical action. Thus, in terms of the dressed effective fields 
$\tilde{\F}$ and $\Vt$ the effective action does not have any dependence on the 
UV and IR scales for ${\cal N}=4$ supersymmetric theory. 





In conclusion, all the correlator $Lcc$ with all the possible contributions included turns out to be totally finite in 
${\cal N}=4$ super Yang-Mills theory in the Landau gauge and this property can be used to find it exactly in all the orders 
of the perturbation theory. The vertex $Lcc$ in spite of being scale independent cannot be found by conformal symmetry 
since the external auxiliary superfield $L$ does not propagate (it is not in the measure of the path integral). 
Three point connected Green functions of supermultiplets containing  physical fields (like vector 
supermultiplet or matter supermultiplet) could be fixed  by conformal symmetry up to some coefficient depending on the 
gauge coupling and number of colours.




\vskip 1cm
\noindent {\large{\bf{Acknowledgments}}}
\vskip 3  mm
The work of I.K. was supported by Ministry of Education (Chile) under grant 
Mecesup FSM9901 and by DGIP UTFSM, by  Fondecyt (Chile) grant \#1040368, and 
by Departamento de Investigaci\'on de la Universidad del Bio-Bio, Chillan 
(Chile). The work of I.S. was supported by Fondecyt (Chile) grants \#1030355.
We are grateful to Gorazd Cveti\v{c} for many useful discussions. 


   













\begin{thebibliography}{99}



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\bibitem{Slavnov:1972fg}
A.~A.~Slavnov,
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Theor.\ Math.\ Phys.\  {\bf 10} (1972) 99
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%%CITATION = TMPHA,10,99;%%


%\cite{Taylor:1971ff}
\bibitem{Taylor:1971ff}
J.~C.~Taylor,
``Ward Identities And Charge Renormalization Of The Yang-Mills Field,''
Nucl.\ Phys.\ B {\bf 33} (1971) 436.
%%CITATION = NUPHA,B33,436;%%



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A.~A.~Slavnov,
``Renormalization Of Supersymmetric Gauge Theories. 2. Nonabelian Case,''
Nucl.\ Phys.\ B {\bf 97} (1975) 155.
%%CITATION = NUPHA,B97,155;%%



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\bibitem{Faddeev:1980be}
L.~D.~Faddeev and A.~A.~Slavnov,
``Gauge Fields. Introduction To Quantum Theory,''
Front.\ Phys.\  {\bf 50}, 1 (1980)
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``Introduction to quantum theory of gauge fields'', Moscow, Nauka, (1988).




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%{SPIRES entry}
{\it Lectures given at Int. Summer Inst. for Theoretical Physics, 
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\end{thebibliography}
\end{document}






hep-th0411150.extracted_bib
\begin{thebibliography}{99}



%\cite{Slavnov:1972fg}
\bibitem{Slavnov:1972fg}
A.~A.~Slavnov,
``Ward Identities In Gauge Theories,''
Theor.\ Math.\ Phys.\  {\bf 10} (1972) 99
[Teor.\ Mat.\ Fiz.\  {\bf 10} (1972) 153].
%%CITATION = TMPHA,10,99;%%


%\cite{Taylor:1971ff}
\bibitem{Taylor:1971ff}
J.~C.~Taylor,
``Ward Identities And Charge Renormalization Of The Yang-Mills Field,''
Nucl.\ Phys.\ B {\bf 33} (1971) 436.
%%CITATION = NUPHA,B33,436;%%



%\cite{Slavnov:1974dg}
\bibitem{Slavnov:1974dg}
A.~A.~Slavnov,
``Renormalization Of Supersymmetric Gauge Theories. 2. Nonabelian Case,''
Nucl.\ Phys.\ B {\bf 97} (1975) 155.
%%CITATION = NUPHA,B97,155;%%



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\bibitem{Faddeev:1980be}
L.~D.~Faddeev and A.~A.~Slavnov,
``Gauge Fields. Introduction To Quantum Theory,''
Front.\ Phys.\  {\bf 50}, 1 (1980)
[Front.\ Phys.\  {\bf 83}, 1 (1990)];
%%CITATION = FRPHA,50,1;%% 
``Introduction to quantum theory of gauge fields'', Moscow, Nauka, (1988).




%\cite{Lee:1973hb}\cite{Zinn-Justin:1974mc}
\bibitem{Lee:1973hb}
B.~W.~Lee,
``Transformation Properties Of Proper Vertices In Gauge Theories,''
Phys.\ Lett.\ B {\bf 46} (1973) 214.
%%CITATION = PHLTA,B46,214;%%




%\cite{Zinn-Justin:1974mc}
\bibitem{Zinn-Justin:1974mc}
J.~Zinn-Justin,
``Renormalization Of Gauge Theories,''
SACLAY-D.PH-T-74-88
%\href{http://www.slac.stanford.edu/spires/find/hep/www?r=saclay-d.ph-t-74-88}
%{SPIRES entry}
{\it Lectures given at Int. Summer Inst. for Theoretical Physics, 
Jul 29 - Aug 9, 1974, Bonn, West Germany}.




%\cite{Becchi:1974md}
\bibitem{Becchi:1974md}
C.~Becchi, A.~Rouet and R.~Stora,
``Renormalization Of The Abelian Higgs-Kibble Model,''
Commun.\ Math.\ Phys.\  {\bf 42} (1975) 127.
%%CITATION = CMPHA,42,127;%%




%\cite{Tyutin:1975qk}
\bibitem{Tyutin:1975qk}
I.~V.~Tyutin,
``Gauge Invariance In Field Theory And Statistical Physics In Operator Formalism,''
LEBEDEV-75-39 (in Russian), 1975. 





%\cite{Piguet:1996ys}
\bibitem{Piguet:1996ys}
O.~Piguet, 
``Supersymmetry, supercurrent, and scale invariance,''
arXiv:hep-th/9611003.
%%CITATION = HEP-TH 9611003;%%\



%\cite{Krivoshchekov:1978xg}
\bibitem{Krivoshchekov:1978xg}
V.~K.~Krivoshchekov,
``Invariant Regularizations For Supersymmetric Gauge Theories,''
Teor.\ Mat.\ Fiz.\  {\bf 36} (1978) 291.
%%CITATION = TMFZA,36,291;%%



%\cite{West:1985jx}
\bibitem{West:1985jx}
P.~C.~West,
``Higher Derivative Regulation Of Supersymmetric Theories,''
Nucl.\ Phys.\ B {\bf 268} (1986) 113.
%%CITATION = NUPHA,B268,113;%%




%\cite{Siegel:1979wq}{Capper:1979ns}
\bibitem{Siegel:1979wq}
W.~Siegel,
``Supersymmetric Dimensional Regularization Via Dimensional Reduction,''
Phys.\ Lett.\ B {\bf 84}, 193 (1979).
%%CITATION = PHLTA,B84,193;%%




%\cite{Capper:1979ns}
\bibitem{Capper:1979ns}
D.~M.~Capper, D.~R.~T.~Jones and P.~van Nieuwenhuizen,
``Regularization By Dimensional Reduction Of Supersymmetric And
Nonsupersymmetric Gauge Theories,''
Nucl.\ Phys.\ B {\bf 167} (1980) 479.
%%CITATION = NUPHA,B167,479;%%



%\cite{Cvetic:2004kx}
\bibitem{Cvetic:2004kx}
  G.~Cvetic, I.~Kondrashuk and I.~Schmidt,
   ``Effective action of dressed mean fields for N = 4 super-Yang-Mills theory,''
  Mod.\ Phys.\ Lett.\ A {\bf 21} (2006) 1127
  [arXiv:hep-th/0407251].
  %%CITATION = HEP-TH 0407251;%%





%\cite{Cvetic:2006kk}
\bibitem{Cvetic:2006kk}
G.~Cveti\v{c}, I.~Kondrashuk and I.~Schmidt,
``On the effective action of dressed mean fields for N = 4 super-Yang-Mills 
theory,'' in {\it Symmetry, Integrability and Geometry: Methods and 
Applications,} SIGMA (2006) 002, arXiv:math-ph/0601002.
%%CITATION = MATH-PH 0601002;%%





%\cite{Cvetic:2006iu}
\bibitem{Cvetic:2006iu}
  G.~Cvetic, I.~Kondrashuk, A.~Kotikov and I.~Schmidt, ``Towards the two-loop Lcc vertex in Landau gauge,''
  arXiv:hep-th/0604112.
  %%CITATION = HEP-TH 0604112;%%








%\cite{Sohnius:1981sn}
\bibitem{Sohnius:1981sn}
M.~F.~Sohnius and P.~C.~West,
``Conformal Invariance In N=4 Supersymmetric Yang-Mills Theory,''
Phys.\ Lett.\ B {\bf 100} (1981) 245.
%%CITATION = PHLTA,B100,245;%%





%\cite{Cvetic:2002dx}
\bibitem{Cvetic:2002dx}
I.~Kondrashuk, G.~Cveti\v{c},    and I.~Schmidt,
``Approach to solve Slavnov-Taylor identities in nonsupersymmetric non-Abelian 
gauge theories,''
Phys.\ Rev.\ D {\bf 67} (2003) 065006
[arXiv:hep-ph/0203014].
%%CITATION = HEP-PH 0203014;%%


%\cite{Cvetic:2002in}
\bibitem{Cvetic:2002in}
G.~Cveti\v{c}, I.~Kondrashuk and I.~Schmidt,
``QCD effective action with dressing functions: Consistency checks in
perturbative regime,''
Phys.\ Rev.\ D {\bf 67} (2003) 065007
[arXiv:hep-ph/0210185].
%%CITATION = HEP-PH 0210185;%%



%\cite{Kondrashuk:2000br}
\bibitem{Kondrashuk:2000br}
I.~Kondrashuk,
``The solution to Slavnov-Taylor identities in D4 N = 1 SYM,''
JHEP {\bf 0011}, 034 (2000)
[arXiv:hep-th/0007136].
%%CITATION = HEP-TH 0007136;%%




%\cite{Kondrashuk:2003tw}
\bibitem{Kondrashuk:2003tw}
  I.~Kondrashuk,
  ``An approach to solve Slavnov-Taylor identity in D4 N = 1 supergravity,''
  Mod.\ Phys.\ Lett.\ A {\bf 19} (2004) 1291
  [arXiv:gr-qc/0309075].
  %%CITATION = GR-QC 0309075;%%




%\cite{Kang:2004cs}
\bibitem{Kang:2004cs}
  K.~Kang and I.~Kondrashuk, ``Semiclassical scattering amplitudes of dressed gravitons,''
  arXiv:hep-ph/0408168.
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\end{thebibliography}