
\documentclass[12pt]{article}
\usepackage{graphicx}

\begin{document}

\author{S. R. Gobira and M. C. Nemes \\
Departamento de F\'{i}sica, Instituto de Ci\^{e}ncias Exatas ICEX,\\
Universidade Federal de Minas Gerais,\\
U.F.M.G., CP 702, CEP 30161-970, \\
Belo Horizonte, M.G., Brasil.\\
e-mail: sgobira@fisica.ufmg.br and carolina@fisica.ufmg.br}
\title{Perturbative n-Loop Renormalization by an Implicit Regularization \
Technique }
\maketitle

\begin{abstract}
We construct a regularization independent procedure for implementing
perturbative renormalization. An algebraic identity at the level of the
internal lines of the diagrams is used which allows for the identification
of counterterms in a purely algebraic way. Order by order in a perturbative
expansion we obtain automatically in the process, finite contributions,
local and nonlocal divergences. The notorious complications introduced by
overlapping divergences never enter and since no subtractions are performed
(as in BPHZ) all the counterterms are readily displayed. We illustrate with $%
\phi _{6}^{3}$ theory to show that our framework renders a considerable
algebraic simplification as well as explicitates the connection between
renormalization and counterterms in the Lagrangian.
\end{abstract}

\vskip1truecm \noindent {PACS:11.10Gh, 11.25Db \newline
Keywords: Renormalization, Regularization\newline
hep-th/0102096v4}\newpage

\section{Introduction}

Quantum field theoretical predictions of physical quantities should in
principle be independent of the particular scheme used to renormalize the
theory. The renormalization program allows to get rid of the singularities
by the redefinition of the parameters in the Lagrangian in a consistent way
for a renormalizable model. In some methods like BPHZ [1], renormalization
may be carried out in one single step without intermediate regularization.
On the other hand , in this process we must make sure that the relevant
symmetries of the underlying theory are preserved and therefore avoid the
appearance of spurious anomalies which otherwise would have to be controlled
order by order in perturbation theory by imposing symmetry restoring
constraint equations. In this sense, in constructing proofs of
renormalizability to all orders care must be exercised with the BPHZ
technique: although it possesses the nice feature of being regularization
independent, gauge invariance is broken within the subtraction operations.

As for the existing regularization schemes whilst for the theories with low
symmetry content nearly all regulators do a good job, this is not the case
for most theories of particle interactions in which gauge symmetry,
supersymmetry etc play a fundamental role. Dimensional Regularization (DR)
[2-4] is an efficient and pragmatical method which explicitly preserves
gauge symmetry. However in the presence of dimension especific objects such
as $\gamma ^{5}$ matrices, a suitable generalization of the Dirac algebra
must be constructed to be compatible with the analytical continuation on the
space-time dimension. This is the case of the Electroweak sector of the
Standard Model. Since chiral symmetry is broken in this case, the
corresponding Ward-Slavnov-Taylor identities must be imposed order by order
what turns the computations beyond one loop order very hard.

For SUSY theories, the fact that the equality between Bose and Fermi degrees
of freedom only holds for especific values of the space time dimension, SUSY
is broken in DR. A naive scheme (Dimensional Reduction) in which the field
components are left unchanged while the loop integrals are performed in d
dimensions can be shown to be inconsistent, see reference.[5].

Similar problems arise in Chern-Simons field theories in which the
Levi-Civitta tensor is the three dimensional analog of the $\gamma ^{5}$
matrix [6,7].

A particularly interesting regularization independent framework is the
Diferential Renormalization program pioneered by Freedman et al [8]. The
basic ideia of this scheme is that renormalization comes from the fact that
products of propagators must be extended to be distributions so that a
Fourier transform is well defined. Working in (Euclidian) coordenate space
one write the amplitude as a derivative of a distribution less divergent at
coincident points. The derivatives are understood in the sense of
distribution theory, i.e. acting formally by parts. The amplitudes written
in this way are identical to the bare ones for separate points but behave
well at coincident points. An intrinsic arbitrary scale appears in this
process which as a Callan-Symanzik renormalization group parameter. The
advantage of this method is that it works in integer space-time dimension,
and it has been shown to yield satisfactory results where it was tested
[9-12]. However no procedure of Differential Renormalization beyond one loop
order such that gauge invariance is automatic has been constructed yet.

In this contribution we propose a perturbative renormalization framework
based on an implicit regularization technique. It bear some similarities to
both BPHZ and Differential Renormalization. For definiteness let us call it
Implicit Regularization Technique (IRT). Firstly it is essentially
regularization independent in the sense that a specific regulator needs
never be explicitate. A convenient identity at level of the integrand
enables us to rewrite the amplitude as a sum of three types of contributions
namely local divergences (basic divergent integrals which characterize the
divergent structure of theory), nonlocal divergences (typical of divergent
sub-structures contribitions) and finite contributions. Secondly, just like
Differential Renormalization arbitrare local terms can be duly parametrized
and properly adjusted on physical grounds. This is particularly important
for finite renormalization in order to clear the calculation from
regularization ambiguities. Finally our framework lives in the integer
space-time dimension which avoids the well-known problem with dimension
specific theories.

Now unlike BPHZ or Differential Renormalization which delives renormalized
amplitudes graph by graph (the former by effecting a subtraction in the
amplitude so as to render it finite and latter by extending a product of
propagators to become a well-defined distribution), we do not modify the
original amplitude at any step of the calculation but instead we isolate the
infinities as basic divergent integrals. As we shall see the counterterms
will naturally arise in our formulation in a systematic fashion. That is
because an important feature of our framework which is advantageous for
renormalizability proofs is that it can be effected in a essentially
algebraic fashion. As we will show there is no need to construct graphical
representations os subdivergences. Moreover the case of overlapping , nested
or even disjoint divergences can be treated precisely on the same footing:
the relevant counterterms appear naturally and systematically within our
procedure and there is no need to classify subdivergent contributions to
effect the proof of renormalizability for any renormalizable quantum field
theory.

In order to illustrate our method we study the renormalization of $\phi
_{6}^{3}$ theory to n-loop order and therefore show that it is a
renormalizable model.

\section{Renormalization by the Implicit Regularization Technique}

In this section we construct an extension of a technique firtly designed for
one[13,14] and two loop calculations [15] for performing a $n^{th}$ order
renormalizability proof.

In order to illustrate the procedure, consider the following divergent
amplitude, typical of one loop order: 
\begin{equation}
\int_{\Lambda }\frac{d^{4}k}{(2\pi )^{4}}\frac{1}{%
[(k+p)^{2}-m^{2}](k^{2}-m^{2})}\cdot  \label{1}
\end{equation}
The symbol $\Lambda $ under the integral sign presupposes, as discussed, an
implicit regularization. Now, in order to separate the logarithmic
divergence from the finite part, we use the following identity in the factor
involving the external momentum $p$: 
\begin{equation}
\frac{1}{[(k+p)^{2}-m^{2}]}=\sum_{j=0}^{N}\frac{\left( -1\right) ^{j}\left(
p^{2}+2p\cdot k\right) ^{j}}{\left( k^{2}-m^{2}\right) ^{j+1}}+\frac{\left(
-1\right) ^{N+1}\left( p^{2}+2p\cdot k\right) ^{N+1}}{\left(
k^{2}-m^{2}\right) ^{N+1}[\left( k+p\right) ^{2}-m^{2}]}\cdot  \label{2}
\end{equation}
In the above expression $N$ is chosen so that the last term is finite under
integration over $k$. Notice also that in the first term on equation (2),
the external momentum appears only in the numerator and thus after
integration they can yield at most polynomials in $p$ multiplied by
divergences. For our present example we need $N=0$, since we are dealing
with a logarithmic divergence. We can rewrite (\ref{1}) using (\ref{2}) as 
\begin{equation}
I=\int_{\Lambda }\frac{d^{4}k}{(2\pi )^{4}}\frac{1}{(k^{2}-m^{2})^{2}}-\int 
\frac{d^{4}k}{(2\pi )^{4}}\frac{p^{2}+2p\cdot k}{%
[(k+p)^{2}-m^{2}](k^{2}-m^{2})^{2}}\,\cdot  \label{3}
\end{equation}
Now only the first of these two integrals is divergent. The others can be
easily integrated out to yield 
\begin{equation}
I=I_{log}(m^{2})-\frac{i}{(4\pi )^{2}}Z_{0}(m^{2},p^{2})  \label{4}
\end{equation}
where 
\begin{equation}
I_{log}(m^{2})=\int_{\Lambda }\frac{d^{4}k}{(2\pi )^{4}}\frac{1}{%
(k^{2}-m^{2})^{2}}  \label{5}
\end{equation}
and 
\begin{equation}
Z_{0}(m^{2},p^{2})=\int_{0}^{1}dz\,\ln \Big(\frac{p^{2}z(1-z)-m^{2}}{-m^{2}}%
\Big).\cdot  \label{6}
\end{equation}
Note that, since no explicit form for the regulator has been used, one can
make immediate contact with other regularizations. Details of calculations
of several one loop amplitudes and their associated Ward identities by using
this method can be found in [14].

By convenience we divide the diagrams which contribute to a given order in
two classes: the first which do not contain diagrams which possess two point
functions as subdivergences and in the second class those which do.

Let us start with the first class of diagrams. To show how the procedure
works it is enough to consider a general Feynman amplitude with one external
momentum $p$ , one coupling constant $\lambda $ and one mass parameter $m$ .
We work in the $4$-dimensional space-time although the generalization to any
integer dimension is straightforward. We denote by $q$ a sum of internal
momenta $k_{i}$ . The amplitude in question can always be written as 
\begin{equation}
\Gamma =\prod_{i=1}^{n}\int_{\Lambda }\frac{d^{4}k_{i}}{(2\pi )^{4}}%
R(p,q,m,\lambda )\left[ \prod_{j=1}^{l}f_{j}(p,q_{j},m^{2})\right]  \label{7}
\end{equation}
where 
\begin{equation}
f_{j}(p,q_{j},m^{2})=\frac{1}{[(p-q_{j})^{2}-m^{2}]}  \label{8}
\end{equation}
and 
\[
\mbox{l}=\mbox{number of}\,\,\ f\,\,\,\mbox{structures} 
\]
\[
\mbox{n}=\mbox{number of loops.}\cdot 
\]

Note that we have explicitly separated the terms involving the external
momentum in the denominator, from which nonlocal divergent contributions can
arise after integration over the internal momenta. The structure $%
R(p,q,m,\lambda )$ contains all other ingredients of the amplitude such as
coupling constants, results of Dirac traces, and so on.

For simplicity we adopt the following notation 
\begin{equation}
\Gamma =(\Pi R)(\Pi f)  \label{9}
\end{equation}
where 
\begin{equation}
(\Pi R)=\prod_{i=1}^{n}\int_{\Lambda }\frac{d^{4}k_{i}}{(2\pi )^{4}}%
R(p,q,m,\lambda )  \label{10}
\end{equation}
and 
\begin{equation}
(\Pi f)=\prod_{j=1}^{l}f_{j}(p,q_{j},m^{2})\cdot  \label{11}
\end{equation}
As discussed before the source of all possible troubles in the
renormalization process will arise from the structure $(\Pi f).$ Our method
focus attention on these structures. In order to clearly separate finite,
``trivial'' divergences (whose dependence on the external momenta is only a
polynomial) from the nonlocal divergences we use a strategy which is
completely based on the identity (\ref{2})

Define the operator $T^{D}$ which acts on \textit{each} structure $f$ in the
following way 
\begin{equation}
T^{0}f=\frac{1}{q_{j}^{2}-m^{2}}+\frac{2p.q_{j}-p^{2}}{(q_{j}^{2}-m^{2})}%
\left\{ \frac{1}{[(p-q_{j})^{2}-m^{2}]}\right\}  \label{12}
\end{equation}
\begin{equation}
T^{1}f=\frac{1}{q_{j}^{2}-m^{2}}+\frac{(2p.q_{j}-p^{2})}{%
(q_{j}^{2}-m^{2})^{2}}+\frac{(2p.q_{j}-p^{2})^{2}}{(q_{j}^{2}-m^{2})^{2}}%
\left\{ \frac{1}{[(p-q_{j})^{2}-m^{2}]}\right\}  \label{13}
\end{equation}
\begin{eqnarray}
T^{2}f &=&\frac{1}{q_{j}^{2}-m^{2}}+\frac{(2p.q_{j}-p^{2})}{%
(q_{j}^{2}-m^{2})^{2}}+\frac{(2p.q_{j}-p^{2})^{2}}{(q_{j}^{2}-m^{2})^{3}} 
\nonumber \\
&&+\frac{(2p.q_{j}-p^{2})^{3}}{(q_{j}^{2}-m^{2})^{3}}\left\{ \frac{1}{%
[(p-q_{j})^{2}-m^{2}]}\right\} \cdot  \label{14}
\end{eqnarray}
Note that the action of the operator $T^{D}$ is equivalent to a Taylor
expansion around zero external momentum where the first terms are kept and
the rest of the series is resumed, yielding thus a convenient identity. Note
also that the degree of divergence of the various terms is decreasing.

The procedure we have in mind consists of applying the operation, in a
particular amplitude with the superficial degree of divergence $D$ , to 
\textit{each} function $f_{j}$ 
\begin{equation}
T^{D}\Gamma =(\Pi R)\prod_{j=1}^{l}T_{j}^{D}f_{j}(p,q_{j},m^{2})\cdot
\label{15}
\end{equation}
The result of the operation will always have the form 
\begin{equation}
T^{D}f(p,q,m^{2})=f^{div}(p,q,m^{2})+f^{fin}(p,q,m^{2})\cdot  \label{16}
\end{equation}
We define 
\begin{equation}
f^{div}(p,q,m^{2})=\sum_{i=0}^{D}f^{i}(p,q,m^{2})\cdot  \label{17}
\end{equation}
Let us exemplify. Take a quadratically divergent amplitude. To each
contribution of the form 
\[
\frac{1}{(p-q_{j})^{2}-m^{2}} 
\]
we associate

\begin{equation}
f^{0}(q,m^{2})=\frac{1}{q^{2}-m^{2}}  \label{18}
\end{equation}

\begin{equation}
f^{1}(p,q,m^{2})=\frac{2p.q-p^{2}}{(q^{2}-m^{2})^{2}}  \label{19}
\end{equation}
\begin{equation}
f^{2}(p,q,m^{2})=\frac{(2p.q)^{2}}{(q^{2}-m^{2})^{3}}  \label{20}
\end{equation}
and 
\begin{equation}
f^{fin}(p,q,m^{2})=\frac{p^{4}-4p^{2}(p.q)}{(q^{2}-m^{2})^{3}}+\frac{%
(2p.q-p^{2})^{3}}{(q^{2}-m^{2})^{3}[(p-q_{j})^{2}-m^{2}]}\cdot  \label{21}
\end{equation}
The definitions (\ref{18}),(\ref{19}),(\ref{20}),(\ref{21}) are not unique.
It is simply convenient for our purposes. Using these we rewrite the
amplitude as a sum of various contributions. According to our notation 
\begin{equation}
T^{D}\Gamma =(\Pi
R)\prod_{j=1}^{l}[f_{j}^{div}(p,q,m^{2})+f_{j}^{fin}(p,q,m^{2})]\cdot
\label{22}
\end{equation}
In this way we can identify three distinct contributions for the amplitude 
\begin{equation}
T^{D}\Gamma =\Gamma _{fin}^{1}+\Gamma _{local}+\Gamma _{nonlocal}  \label{23}
\end{equation}
where 
\begin{equation}
\Gamma _{fin}^{1}=(\Pi R)\prod_{j=1}^{l}f_{j}^{fin}(p,q,m^{2})\cdot
\label{24}
\end{equation}
The second contribution contains only local divergences and, for some
particular $(\Pi R)$ structure, it can contain finite contributions too. It
is identified as 
\begin{eqnarray}
\Gamma _{local} &=&(\Pi R)\prod_{j=1}^{l}f_{j}^{div}(p,q,m^{2})  \nonumber \\
&=&\Gamma _{fin}^{2}+\Gamma _{local}^{div}\cdot  \label{25}
\end{eqnarray}
These local divergences correspond to counterterms which are characteristic
of the order we are renormalizing. For example, they can have the form 
\begin{equation}
\int_{\Lambda }\frac{d^{4}k}{(2\pi )^{4}}\frac{1}{k^{2}-m^{2}}%
+p^{2}I_{log}(m^{2})+\,\,{\mbox{finite\,\,\, part}}\,\cdot  \label{26}
\end{equation}
The last term in equation (\ref{23}), namely the cross-terms, contain finite
contributions as well as ``nonlocal'' divergences. 
\begin{equation}
\Gamma _{nonlocal}=\Gamma _{fin}^{3}+\Gamma _{nonlocal}^{div}\cdot
\label{27}
\end{equation}
These nonlocal divergence contributions will always appear due to the
divergent subdiagrams (beyond two point functions) contained in the graph.
As we will show next in a particular example, the renormalization of
previous orders will always allow one to cancel these contributions if the
theory is renormalizable. In the present scheme the result is automatic and
follows from the operation we have just defined, in an algebraic manner.
There is no need for graphic representations of relevant contributions,
although it is possible.

The renormalized amplitude say, in $n^{th}$ loop order, can therefore be
defined as 
\begin{eqnarray}
\Gamma _{R}^{(n)} &=&T^{D}\Gamma ^{(n)}-\Gamma _{local}^{div(n)}-\Gamma
_{nonlocal}^{div(n)} \\
&=&\Gamma _{fin}^{1(n)}+\Gamma _{fin}^{2(n)}+\Gamma _{fin}^{3(n)}  \nonumber
\end{eqnarray}
where the contributions $\Gamma _{local}^{div(n)}$ and $\Gamma
_{nonlocal}^{div(n)}$ contain the counterterms typical of\ order n as well
as the counterterms coming from divergent subdiagrams of previous order as
will become clear in the examples. Notice from the equation above that our
framework automatically delivers the counterterms 
\begin{equation}
\Gamma _{CT}^{1}=-\Gamma _{local}^{div}-\Gamma _{nonlocal}^{div}
\end{equation}
and just as in BPHZ , by subtracting off the necessary counterterms leaves
with the finite part of the amplitude, the main difference being that here
the counterterms can be read out of the procedure.

Now we proceed to the second class of diagrams ,namely those which contain
two point functions as subdiagrams. Let us call U all the two point diagrams
contained in a given amplitude $\Gamma $. It is easy to see that that they
can be factored out inside of the total amplitude in the following sense 
\begin{equation}
\Gamma =\prod_{all\ \ \Sigma _{j}\ \in \ U}\mathcal{R}_{j}\Sigma
_{j}^{(l)}(q_{j}^{2})
\end{equation}
where $\mathcal{R}_{j}$ stands for the remaining pieces in the amplitude, $j$
characterizes a especific two point function, is one of the integration
momenta (but external to $\Sigma _{j}$ ). Now since the operation $%
T^{D}\Gamma $ is an identity, i.e.$T^{D}\Gamma =\Gamma $ we can define the
partially renormalized amplitude (with all two point function subdiagrams
properly renormalized ) as follows 
\begin{equation}
\bar{\Gamma}=\Gamma +\Gamma _{CT}^{2}
\end{equation}
therefore we have 
\begin{equation}
\Gamma _{CT}^{2}=\prod_{all\ \ \Sigma _{j}\ \in \ U}\mathcal{R}_{j}[\delta
_{j}^{(l)}m^{2}-A_{j}^{(l)}q_{j}^{2}]
\end{equation}
and $\Gamma _{CT}^{2}$ are all counterterms characteristic subdiagrams
involving two point functions.$\delta _{j}^{(l)}m^{2}$ stands for the mass
renormalization and $A_{j}^{(l)}$ for the wave function renormalization.
Explicit expressions for these objects will be given in the following
section where a specific example is worked out. In order to get the
renormalized amplitude of order n from $\bar{\Gamma}$ one proceeds in the
same way as for diagrams of class one defined above. We thus have 
\begin{eqnarray}
\Gamma _{R} &=&T^{D}\bar{\Gamma}-\bar{\Gamma}_{local}^{div}-\bar{\Gamma}%
_{nonlocal}^{div}  \nonumber \\
&=&\bar{\Gamma}_{fin}^{1}+\bar{\Gamma}_{fin}^{2}+\bar{\Gamma}_{fin}^{3}\cdot
\end{eqnarray}
The whole procedure will become apparent in the concrete example of
following section

\section{$\protect\lambda \protect\phi _{6}^{3}$ Theory as an example}

Consider the $\lambda \phi _{6}^{3}$ theory Lagrangian,

\begin{equation}
\mathcal{L}=\frac{1}{2}\left[ \left( \partial _{\mu }\phi _{0}(x)\right)
^{2}-m_{0}^{2}\phi _{0}^{2}(x)\right] -\frac{\lambda _{0}}{3!}\phi
_{0}^{3}(x)
\end{equation}
It is easy to show that a Feynman graph in this theory has the superficial
degree of divergence $D$ written as 
\begin{equation}
D=6-2N
\end{equation}
where $N$ is the number of external legs. This means that only Green's
functions with $N\leq 3$ are divergent. For the one-point functions we will
assume that we can impose the condition $\left\langle 0\right| \hat{\phi}%
\left| 0\right\rangle =0$ in all orders and we will not worry about
one-point diagrams. We will just work with the two and three-point Green's
functions which possess quadratic and logarithmic divergences.

We will effect the renormalization through the redefinition of the
Lagrangian parameters as: 
\begin{equation}
\phi _{0}=\sqrt{Z_{\phi }}\phi
\end{equation}
\begin{equation}
m_{0}^{2}=Z_{m}m^{2}
\end{equation}
\begin{equation}
\lambda _{0}=Z_{\lambda }\lambda
\end{equation}
which allow the Lagrangian to be rewritten as 
\begin{equation}
\mathcal{L}={\mathcal{L}_{F}}+{\mathcal{L}_{CT}}
\end{equation}
where 
\begin{equation}
{\mathcal{L}_{F}}=\frac{1}{2}\left[ \left( \partial _{\mu }\phi \right)
^{2}-m^{2}\phi ^{2}\right] -\frac{\lambda }{3!}\phi ^{3}
\end{equation}
and 
\begin{equation}
\mathcal{L}_{CT}=\frac{1}{2}\left[ (Z_{\phi }-1)\left( \partial _{\mu }\phi
\right) ^{2}-(Z_{\phi }Z_{m}-1)m^{2}\phi ^{2}\right] -(Z_{\phi
}^{3/2}Z_{\lambda }-1)\frac{\lambda }{3!}\phi ^{3}
\end{equation}
At the $n^{th}$ order one has 
\begin{equation}
\mathcal{L}_{CT}=\mathcal{L}_{CT}^{(1)}+\mathcal{L}_{CT}^{(2)}+\ldots 
\mathcal{L}_{CT}^{(n)}
\end{equation}
We effect the renormalization at each order imposing the conditions:

\begin{itemize}
\item  Relative to the propagator 
\begin{equation}
D_{R}^{-1}(0)=-m^{2}
\end{equation}
and 
\begin{equation}
\left| \frac{\partial }{\partial p^{2}}D_{R}^{-1}(p^{2})\right| _{p=0}=1
\end{equation}

\item  Relative to vertex function 
\begin{equation}
-iM_{R}(0)=-i\lambda (1+\mathrm{finite\ corrections})
\end{equation}
We can rewrite the bare Lagrangian\ (34) as 
\begin{equation}
\mathcal{L}=\frac{1}{2}\left[ (1+A)\left( \partial _{\mu }\phi \right)
^{2}-(m^{2}+\delta m^{2})\phi ^{2}\right] -(1+B)\frac{\lambda }{3!}\phi
^{3}\,.
\end{equation}
\end{itemize}

in order to identify the renormalization constants 
\begin{equation}
Z_{\phi }^{(n)}=1+A^{(n)}
\end{equation}
\begin{equation}
Z_{m}^{(n)}=\frac{1}{Z_{\phi }^{(n)}m^{2}}(m^{2}+\delta ^{(n)}m^{2})
\end{equation}
\begin{equation}
Z_{\lambda }^{(n)}=\frac{1+B^{(n)}}{(Z_{\phi }^{(n)})^{3/2}}
\end{equation}
at each order by the imposing renormalization conditions. Since, in pratice
we renormalize each diagram of the given order, the counterterms can be
written as 
\begin{equation}
B^{(n)}=\sum_{j=1}^{a}B_{j}^{(n)}
\end{equation}
\begin{equation}
A^{(n)}=\sum_{j=1}^{b}A_{j}^{(n)}
\end{equation}
\begin{equation}
\delta ^{(n)}m^{2}=\sum_{j=1}^{b}\delta _{j}^{(n)}m^{2}
\end{equation}
here $a,(b)$ is the number of three(two) point diagrams which contribute to
order $n$ .

At the $n^{th}$ order the inverse propagator function is written as 
\begin{eqnarray}
D_{R}^{-1}(p^{2}) &=&p^{2}-m^{2}-\Sigma _{R}^{(1)}(p^{2})-\Sigma
_{R}^{(2)}(p^{2})...  \nonumber \\
&&-\delta ^{(n)}m^{2}+A^{(n)}p^{2}-\Sigma ^{(n)}(p^{2})
\end{eqnarray}
and the vertex function as 
\begin{eqnarray}
-iM_{R}(p,p^{\prime }) &=&-i\lambda \{1+V_{R}^{(1)}(p,p^{\prime
})+V_{R}^{(2)}(p,p^{\prime })...  \nonumber \\
&&+V^{(n)}(p,p^{\prime })+B^{(n)}\}\cdot
\end{eqnarray}
Using the technique in each diagram contained in the $\Sigma ^{(n)}(p^{2})$
and in the $V^{(n)}(p,p^{\prime })$ amplitudes we separate the local
divergent part and identify all divergent substructures. Imposing
renormalization conditions we can always identify $A^{(n)}$, $\delta
^{(n)}m^{2}$ and $B^{(n)}$.

In order to identify the counterterms of the order in question and to write
the nonlocal ones in terms of divergences of lower orders, showing thus that
one need not worry about them, it is convenient to define the following
functions:

\begin{itemize}
\item  Relative to vertex correction counterterms(type $j$ diagrams) 
\begin{eqnarray}
iB_{j}^{(n)} &=&(-i\lambda )^{2n+1}(i)^{3n}I_{\log 1}^{(n)}(m^{2},\Lambda ) 
\nonumber \\
&=&\Gamma _{local}^{div(n)}
\end{eqnarray}
where 
\begin{equation}
I_{\log 1}^{(n)}(m^{2},\Lambda )=\prod_{i=1}^{n}\int_{\Lambda }\frac{%
d^{6}k_{i}}{(2\pi )^{6}}\Upsilon ^{(n)}(k_{1},k_{2},...k_{n},m^{2})
\end{equation}
with 
\begin{equation}
\Upsilon ^{(n)}(k_{1},k_{2},...k_{n},m^{2})=\frac{1}{(k_{1}^{2}-m^{2})^{3}}%
\left( \prod_{j=2}^{n}\frac{1}{(k_{j}^{2}-m^{2})^{2}}\right)
Q(k_{i},k_{i+1},m^{2})\cdot
\end{equation}
For $n=1$ 
\begin{equation}
Q=1,
\end{equation}
otherwise 
\begin{equation}
Q=\prod_{i=1}^{n-1}\left\{ \frac{1}{[(k_{i}-k_{i+1})^{2}-m^{2}]}\right\}
\cdot
\end{equation}
Notice that what we have defined here are generalizations of the simple
one-loop logarithmically divergent integral $I_{log}(m^{2})$ which we
encountered in our one-loop example.

\item  Relative to all finite contributions to vertex corrections(type $j$
diagram ), which corresponds to the renormalized diagram 
\begin{equation}
\Gamma _{R}^{(n)}=\Gamma _{fin}^{1(n)}+\Gamma _{fin}^{3(n)}=(-i\lambda
)^{2n+1}(i)^{3n}\prod_{i=1}^{n}\int \frac{d^{6}k_{i}}{(2\pi )^{6}}\Xi
^{(n)}(k_{1},...,k_{n},p,p^{\prime },m^{2})\cdot
\end{equation}

\item  Relative to the finite contribution, defined in equation (\ref{24})
for the overlapping diagrams 
\begin{eqnarray}
\Gamma _{fin}^{1(n)} &=&(\Pi R)\prod_{i=1}^{n}f_{i}^{fin}(p,k_{i},m^{2}) 
\nonumber \\
&=&\frac{(-i\lambda )^{2n}(i)^{3n-1}}{2}\prod_{i=1}^{n}\int \frac{d^{6}k_{i}%
}{(2\pi )^{6}}\Theta ^{(n)}(k_{1},k_{2},...k_{n},p,m^{2})\cdot
\end{eqnarray}
\end{itemize}

In each order there will appear new types of divergent integrals. Therefore
throughout the text we will define some new divergent integrals similar to
the ones above (eq.(56) ). These quantities are always independent of
external momenta. Next we apply the procedure to all diagrams up to two
loops in order to exemplify how the method works. To $n^{th}$ order it
suffices to treat four cases, the first related to the vertex function and
the others to the self-energy, which contain the overlapping divergences,
two point functions as subdivergences and nested two point functions.

\subsection{Three point functions}

\subsubsection{The one loop order}

The vertex correction has only one contribution at one loop level whose
diagram is depicted in figure 1. The corresponding amplitude is 
\begin{eqnarray}
\Gamma  &=&-iV^{(1)}(p,p^{\prime })  \nonumber \\
&=&\lambda ^{3}\int_{\Lambda }\frac{d^{6}k}{(2\pi )^{6}}\frac{1}{%
(k^{2}-m^{2})[(p-k)^{2}-m^{2}][(p^{\prime }-k)^{2}-m^{2}]}\cdot 
\end{eqnarray}
Using the notation introduced in section II we write 
\begin{equation}
-iV^{(1)}(p,p^{\prime })=\int_{\Lambda }\frac{d^{6}k}{(2\pi )^{6}}%
R(k,m^{2},\lambda )f(p,k,m^{2})f(p^{\prime },k,m^{2})
\end{equation}
with 
\begin{equation}
R(k,m^{2},\lambda )=\frac{\lambda ^{3}}{(k^{2}-m^{2})}\cdot 
\end{equation}
According to (IRT) we write, given that the divergence is logarithmic and
therefore $D=0$ 
\begin{equation}
-iT^{0}V^{(1)}(p,p^{\prime })=\Gamma _{local}^{div}+\Gamma _{fin}^{1}+\Gamma
_{fin}^{3}
\end{equation}
(recall that in this case $\Gamma _{fin}^{2}=\Gamma _{nonlocal}^{div}=0$ )
where 
\begin{eqnarray}
\Gamma _{local}^{div} &=&\int_{\Lambda }\frac{d^{6}k}{(2\pi )^{6}}%
R(k,m^{2},\lambda )f^{0}(k,m^{2})f^{0}(k,m^{2})  \nonumber \\
&=&\lambda ^{3}\int_{\Lambda }\frac{d^{6}k}{(2\pi )^{6}}\frac{1}{%
(k^{2}-m^{2})^{3}}  \nonumber \\
&=&\lambda ^{3}I_{\log 1}^{(1)}(m^{2},\Lambda )  \nonumber \\
&=&iB^{(1)}
\end{eqnarray}
and 
\begin{eqnarray}
\Gamma _{R} &=&\Gamma _{fin}^{1}+\Gamma _{fin}^{3}  \nonumber \\
&=&\lambda ^{3}\int \frac{d^{6}k}{(2\pi )^{6}}\Xi ^{(1)}(k,p,p^{\prime
},m^{2})
\end{eqnarray}
with 
\begin{eqnarray}
\lambda ^{3}\Xi ^{(1)}(k,p,p^{\prime },m^{2}) &=&R(k,m^{2},\lambda
)\{f^{fin}(k,p,m^{2})f^{fin}(k,p^{\prime },m^{2})  \nonumber \\
&+&f^{0}(k,m^{2})f^{fin}(k,p^{\prime },m^{2})  \nonumber \\
&&+f^{fin}(k,p,m^{2})f^{0}(k,m^{2})\}\cdot 
\end{eqnarray}

Notice that the finite part of this diagram contains the cross-terms $%
f_{0}\cdot f_{fin}$ since its integral is finite.

\subsubsection{The two loop order}

Three diagram types contribute to the vertex correction at two loops. The
total amplitude can be written as 
\begin{equation}
-iV^{(2)}(p,p^{\prime })=-3iV_{1}^{(2)}(p,p^{\prime
})-iV_{2}^{(2)}(p,p^{\prime })-iV_{3}^{(2)}(p,p^{\prime })\cdot
\end{equation}
In this order the counterterms will be identified as 
\begin{equation}
B^{(n)}=3B_{1}^{(2)}+B_{2}^{(2)}+B_{3}^{(2)}\cdot
\end{equation}

The first amplitude $-iV_{1}^{(2)}(p,p^{\prime })$ corresponds to the
diagram in figure 2. This diagram contains a quadratic divergent subdiagram
(a first order two-point function correction). It can be completely
separated in terms of the internal momentum $k_{1}$ as mentioned before.
Then 
\begin{eqnarray}
\Gamma &=&-iV_{1}^{(2)}(p,p^{\prime })  \nonumber \\
&=&\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi )^{6}}\frac{(-i\lambda )^{3}(i)^{4}%
}{(k_{1}^{2}-m^{2})^{2}[(p-k_{1})^{2}-m^{2}][(p^{\prime }-k_{1})^{2}-m^{2}]}
\nonumber \\
&&\times \left\{ i\Sigma ^{(1)}(k_{1}^{2})\right\}
\end{eqnarray}
where $i\Sigma ^{(1)}(k_{1}^{2})$ is the one loop self-energy amplitude. The
one loop renormalized self-energy is 
\begin{equation}
\Sigma _{R}^{(1)}(k_{1}^{2})=\Sigma _{CT}^{(1)}(k_{1}^{2})+\Sigma
^{(1)}(k_{1}^{2})
\end{equation}
where 
\begin{equation}
\Sigma _{CT}^{(1)}(k_{1}^{2})=\delta ^{(1)}m^{2}-A^{(1)}k_{1}^{2}\ \cdot
\end{equation}
Thus the amplitude containing no two point function substructure is directly
obtained as 
\begin{eqnarray}
\bar{\Gamma} &=&\Gamma +\Gamma _{CT}^{2}  \nonumber \\
&=&\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi )^{6}}\frac{(-i\lambda )^{3}(i)^{4}%
}{(k_{1}^{2}-m^{2})^{2}[(p-k_{1})^{2}-m^{2}][(p^{\prime }-k_{1})^{2}-m^{2}]}
\nonumber \\
&&\times \left\{ i\Sigma _{R}^{(1)}(k_{1}^{2})\right\} \cdot
\end{eqnarray}
All possible nonlocal divergences in this case will be canceled when we
consider the one loop renormalization. Next we use the IRT for the
logarithmic divergence. In our notation we obtain 
\begin{equation}
-iT^{0}\bar{\Gamma}=\bar{\Gamma}_{local}^{div}+\bar{\Gamma}_{fin}^{1}+\Gamma
_{fin}^{3}
\end{equation}
with 
\begin{equation}
\int \frac{d^{6}k_{2}}{(2\pi )^{6}}R(k_{1},k_{2},m^{2},\lambda )=\frac{%
(-i\lambda )^{3}(i)^{4}}{(k_{1}^{2}-m^{2})^{2}}\left\{ i\Sigma
_{R}^{(1)}(k_{1}^{2})\right\} =R(k_{1},m^{2},\lambda )\cdot
\end{equation}
The explicit expression for $i\Sigma _{R}^{(1)}(k_{1}^{2})$ will be given in
equation (107). Using the above expression for $R(k_{1},m^{2},\lambda )$ 
\begin{eqnarray}
\bar{\Gamma}_{local}^{div} &=&\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi )^{6}}%
R(k_{1},m^{2},\lambda )f^{0}(k_{1},m^{2})f^{0}(k_{1},m^{2})  \nonumber \\
&=&(-i\lambda )^{3}(i)^{4}I_{\log 2}^{(2)}(m^{2},\lambda ^{2},\Lambda
)=iB_{1}^{(2)}\cdot
\end{eqnarray}
We have just defined another logarithmic divergent quantity which is
characteristic of the two-loop order. Note that explicit appearence of
coupling constant. This should emphasize the fact that the amplitude depends
on a two point function subdiagram, which has been properly renormalized.
All counterterms possessing such type of subdiagram will look like this. 
\begin{equation}
I_{\log 2}^{(2)}(m^{2},\lambda ^{2},\Lambda )=\int_{\Lambda }\frac{d^{6}k_{1}%
}{(2\pi )^{6}}\frac{1}{(k_{1}^{2}-m^{2})^{4}}\left\{ i\Sigma
_{R}^{(1)}(k_{1}^{2})\right\}
\end{equation}
and the finite part is 
\begin{eqnarray}
\bar{\Gamma}_{fin}^{1}+\bar{\Gamma}_{fin}^{3} &=&\int \frac{d^{6}k_{1}}{%
(2\pi )^{6}}R(k_{1},m^{2},\lambda )\{  \nonumber \\
&&f^{0}(k_{1},m^{2})f^{fin}(p^{\prime },k_{1},m^{2})  \nonumber \\
&&+f^{fin}(p,k_{1},m^{2})f^{0}(k_{1},m^{2})  \nonumber \\
&&+f^{fin}(p,k_{1},m^{2})f^{fin}(p^{\prime },k_{1},m^{2})\}\cdot
\end{eqnarray}
It is not necessary to give explicit expressions for the finite part and
therefore we make explicit the divergent contributions only.

Now we consider the diagram corresponding to the second contribution $%
-iV_{2}^{(2)}(p,p^{\prime })$ which belong to class one (figure 3). The
amplitude reads 
\begin{eqnarray}
-iV_{2}^{(2)}(p,p^{\prime }) &=&\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi )^{6}}%
\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}R(k_{1},k_{2},m^{2},\lambda ) 
\nonumber \\
&&\times f(p,k_{1},m^{2})f(p^{\prime },k_{1},m^{2})f(p^{\prime },k_{2},m^{2})
\end{eqnarray}
with 
\begin{equation}
R(k_{1},k_{2},m^{2},\lambda )=\frac{(-i\lambda )^{5}(i)^{6}}{%
(k_{1}^{2}-m^{2})(k_{2}^{2}-m^{2})[(k_{1}-k_{2})^{2}-m^{2}]}\cdot
\end{equation}
Using the IRT we have 
\begin{equation}
-iT^{0}\,\,V_{1}^{(2)}(p,p^{\prime })=\Gamma _{local}^{div}+\Gamma
_{fin}^{1}+\Gamma _{nonlocal}
\end{equation}
where 
\begin{eqnarray}
\Gamma _{local}^{div} &=&(-i\lambda )^{5}(i)^{6}I_{\log
1}^{(2)}(m^{2},\Lambda )=iB_{2}^{(2)}  \nonumber \\
&=&\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi )^{6}}\int_{\Lambda }\frac{%
d^{6}k_{2}}{(2\pi )^{6}}\frac{(-i\lambda )^{5}(i)^{6}}{%
(k_{1}^{2}-m^{2})^{3}(k_{2}^{2}-m^{2})^{2}[(k_{1}-k_{2})^{2}-m^{2}]} 
\nonumber \\
&=&(-i\lambda )^{5}(i)^{6}\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi )^{6}}%
\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}\Upsilon
^{(2)}(k_{1},k_{2},m^{2})\cdot
\end{eqnarray}
In this type of structure (to all orders) the nonlocal contribution $\Gamma
_{nonlocal}$ will have the form 
\begin{equation}
\Gamma _{nonlocal}=\Gamma _{fin}^{3}+\Gamma _{nonlocal}^{div}
\end{equation}
and in this case we have 
\begin{equation}
\Gamma _{nonlocal}^{div}=(-i\lambda )^{5}(i)^{6}\int =\frac{d^{6}k_{1}}{%
(2\pi )^{6}}\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}\Xi
^{(1)}(k_{1},p,p^{\prime },m^{2})\Upsilon ^{(1)}(k_{2},m^{2})\cdot
\end{equation}
Note that this term is completely written in terms of one loop contributions
already considered. Therefore it poses no problem to renormalization. This
particular example illustrates a basic difference between the present method
and others: the subdivergences need not be previously identified. They
appear algebraically. In cases were it is simple to identify the
subdivergences , this is not necessarily a great advantage. However in
higher orders it might become considerably simpler to identify all divergent
substructures in an algebraic fashion. In fact, as will become clear in what
follows, the procedure is designed to explicitate all relevant (to
renormalization) subdivergences. The finite contributions can be written as 
\begin{eqnarray}
\Gamma _{R}^{(2)} &=&\Gamma _{fin}^{1}+\Gamma _{fin}^{3}  \nonumber \\
&=&(-i\lambda )^{5}(i)^{6}\int \frac{d^{6}k_{1}}{(2\pi )^{6}}\int \frac{%
d^{6}k_{2}}{(2\pi )^{6}}\Xi ^{(2)}(k_{1},k_{2},p,p^{\prime },m^{2})
\end{eqnarray}
with 
\[
\Xi =(-i\lambda )^{5}(i)^{6}\Xi ^{(2)}(k_{1},k_{2},p,p^{\prime },m^{2}) 
\]
\begin{eqnarray}
\Xi &=&R(k_{1},k_{2},m^{2},\lambda )  \nonumber \\
&&\times \{f^{fin}(k_{1},p,m^{2})f^{fin}(k_{1},p^{\prime
},m^{2})f^{fin}(k_{2},p^{\prime },m^{2})  \nonumber \\
&&\left. +f^{0}(k_{1},m^{2})f^{0}(k_{1},m^{2})f^{fin}(k_{2},p^{\prime
},m^{2})\right.  \nonumber \\
&&+f^{0}(k_{1},m^{2})f^{fin}(k_{1},p^{\prime },m^{2})f^{fin}(k_{2},p^{\prime
},m^{2})  \nonumber \\
&&+f^{fin}(k_{1},p,m^{2})f^{0}(k_{1},m^{2})f^{fin}(k_{2},p^{\prime },m^{2})\}
\nonumber \\
&&+\frac{(-i\lambda )^{5}(i)^{6}(2k_{1}.k_{2}-k_{1}^{2})\Xi
^{(1)}(k_{1},p,p^{\prime },m^{2})}{%
(k_{2}^{2}-m^{2})^{3}[(k_{1}-k_{2})^{2}-m^{2}]}\cdot
\end{eqnarray}
The last term in the above equation is obtained by using the operation (\ref
{12}) considering $k_{1}$ as external momentum. This is necessary to
identify the one loop structure.

The last two loop diagram $-iV_{3}^{(2)}(p,p^{\prime })$ is depicted in the
figure 4. The corresponding amplitude is 
\begin{eqnarray}
-iV_{3}^{(2)}(p,p^{\prime }) &=&\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi )^{6}}%
\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}\{R(k_{1},k_{2},m^{2},\lambda
)f(p,k_{1},m^{2})  \nonumber \\
&&\times f(p^{\prime },k_{2},m^{2})f(p-p^{\prime },k_{1}-k_{2},m^{2})\}
\label{68}
\end{eqnarray}
with 
\begin{equation}
R(k_{1},k_{2},m^{2},\lambda )=\frac{(-i\lambda )^{5}(i)^{6}}{%
(k_{1}^{2}-m^{2})(k_{2}^{2}-m^{2})[(k_{1}-k_{2})^{2}-m^{2}]}\cdot  \label{69}
\end{equation}
Using the IRT we have 
\begin{equation}
-iT^{0}V_{1}^{(2)}(p,p^{\prime })=\Gamma _{local}^{div}+\Gamma _{R}
\label{70}
\end{equation}
where 
\begin{eqnarray}
\Gamma _{local}^{div} &=&i\lambda ^{5}\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi
)^{6}}\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}\frac{1}{%
(k_{1}^{2}-m^{2})^{2}(k_{2}^{2}-m^{2})^{2}[(k_{1}-k_{2})^{2}-m^{2}]^{2}} 
\nonumber \\
&=&i\lambda ^{5}I_{\log 3}^{(2)}(m^{2},\Lambda )=iB_{3}^{(2)}\cdot
\label{71}
\end{eqnarray}
We defined above another logarithmic divergent quantity. This diagram type
is often called a primitively divergent diagram. Note that there are no
subdivergences.

\subsubsection{The $n$-loop order}

As discussed before we now consider only one contribution of each kind. The
vertex type contribution depicted in figure 5 is the first one. It will
appear as a substructure of the overlapping self-energy diagram which we
will also consider.

The amplitude corresponding to the vertex correction in figure 5 is 
\begin{equation}
-iV_{1}^{(n)}(p,p^{\prime })=(\Pi R)(\Pi f)  \label{72}
\end{equation}
with 
\begin{equation}
(\Pi R)=(-i\lambda )^{2n+1}(i)^{3n}\left\{ \prod_{j=1}^{n}\int_{\Lambda }%
\frac{d^{6}k_{j}}{(2\pi )^{6}}\frac{1}{k_{j}^{2}-m^{2}}\right\}
Q(k_{i},k_{i+1},m^{2})  \label{73}
\end{equation}
where $Q$ is the same function as defined in (58) and (59). The subscript $1$
in $V_{1}^{(n)}(p,p^{\prime })$ refers to the fact that only one diagram is
being considered (type $1$ ). The external momentum dependent part $(\Pi f)$
is given by 
\begin{equation}
(\Pi f)=\left\{ \frac{1}{(p-k_{1})^{2}-m^{2}}\right\} \prod_{j=1}^{n}\left\{ 
\frac{1}{(p^{\prime }-k_{j})^{2}-m^{2}}\right\} \cdot   \label{74}
\end{equation}
Using the IRT, we get 
\begin{eqnarray}
T^{0}(\Pi f) &=&\{f^{0}(k_{1},m^{2})+f^{fin}(k_{1},p,m^{2})\}  \nonumber \\
&&\times \prod_{j=1}^{n}\{f^{0}(k_{j},m^{2})+f^{fin}(k_{j},p^{\prime
},m^{2})\}\cdot   \label{75}
\end{eqnarray}
In the same way we have 
\begin{equation}
-iT^{0}V_{1}^{(n)}(p,p^{\prime })=\Gamma _{local}+\Gamma _{fin}^{1}+\Gamma
_{nonlocal}\cdot   \label{76}
\end{equation}
Since $\Gamma _{fin}^{2}=0$ we can write 
\begin{eqnarray}
\Gamma _{local}^{div} &=&iB_{1}^{(n)}  \nonumber \\
&=&(\Pi R)f_{1}^{0}(k_{1},m^{2})\prod_{j=1}^{n}f_{j}^{0}(k_{j},m^{2}) 
\nonumber \\
&=&(-i\lambda )^{2n+1}(i)^{3n}I_{\log 1}^{(n)}(m^{2},\Lambda )  \nonumber \\
&=&(-i\lambda )^{2n+1}(i)^{3n}\prod_{j=1}^{n}\int_{\Lambda }\frac{d^{6}k_{j}%
}{(2\pi )^{6}}\Upsilon ^{(n)}(k_{1},k_{2},...k_{n},m^{2})
\end{eqnarray}
and 
\begin{equation}
\Gamma _{nonlocal}=\Gamma _{fin}^{3}+\Gamma _{nonlocal}^{div}  \label{78}
\end{equation}
where 
\begin{eqnarray}
\Gamma _{nonlocal}^{div} &=&(-i\lambda
)^{2n+1}(i)^{3n}\prod_{j=1}^{n}\int_{\Lambda }\frac{d^{6}k_{j}}{(2\pi )^{6}}
\nonumber \\
&&\sum_{a=1}^{n-1}\Xi ^{(a)}(k_{1},..,k_{a};p,p^{\prime },m^{2})\Upsilon
^{(n-a)}(k_{a+1},..,k_{n};m^{2})\cdot 
\end{eqnarray}
Here we clearly see that the application of the method explicitates all the
subdivergencies in an algebraic way. Moreover it stresses the inductive
character of the method. If we assume that the theory is renormalized at $%
(n-1)^{th}$ order, the contribution at $n^{th}$ order will solely depend on
structures (finite and divergent) which have already played their role at
lower orders. Also it is noteworthy that \textit{all} divergencies and
finite parts of \textit{all} previous orders play an important role at $%
n^{th}$ order.

\subsection{Two-point functions}

\subsubsection{The one loop order}

The self-energy has only one diagram contribution at one loop level which we
identify in figure 6. It corresponds to the amplitude 
\begin{equation}
i\Sigma ^{(1)}(p^{2})=\frac{\lambda ^{2}}{2}\int_{\Lambda }\frac{d^{6}k}{%
(2\pi )^{6}}\frac{1}{(k^{2}-m^{2})[(p-k)^{2}-m^{2}]}  \label{80}
\end{equation}
where 
\begin{equation}
R(k,m^{2},\lambda )=\frac{\lambda ^{2}}{2}\frac{1}{(k^{2}-m^{2})}\cdot
\label{81}
\end{equation}
Using IRT we have 
\begin{equation}
(T^{2})i\Sigma ^{(1)}(p^{2})=\Gamma _{local}^{div}+\Gamma _{fin}^{1}
\label{82}
\end{equation}
with 
\begin{equation}
\Gamma _{local}^{div}=\frac{\lambda ^{2}}{2}\{I_{quad}^{(1)}(m^{2},\Lambda
)+p^{2}[g_{\mu \nu }\frac{4}{6}I_{\log }^{\mu \nu (1)}(m^{2},\Lambda
)-I_{\log 1}^{(1)}(m^{2},\Lambda )]\},  \label{83}
\end{equation}
where we have defined 
\begin{equation}
I_{quad}^{(1)}(m^{2},\Lambda )=\int_{\Lambda }\frac{d^{6}k}{(2\pi )^{6}}%
\frac{1}{(k^{2}-m^{2})^{2}}  \label{84}
\end{equation}
and 
\begin{equation}
I_{\log }^{\mu \nu (1)}(m^{2},\Lambda )=\int_{\Lambda }\frac{d^{6}k}{(2\pi
)^{6}}\frac{k^{\mu }k^{\nu }}{(k^{2}-m^{2})^{4}}\cdot  \label{85}
\end{equation}
The finite part is 
\begin{eqnarray}
\Gamma _{fin}^{1} &=&\Gamma _{R}=\frac{\lambda ^{2}}{2}\int \frac{d^{6}k}{%
(2\pi )^{6}}\Theta ^{(1)}(k,p,m^{2})  \nonumber \\
&=&\frac{\lambda ^{2}}{2}\left\{ \int \frac{d^{6}k}{(2\pi )^{6}}\frac{(p^{4})%
}{(k^{2}-m^{2})^{4}}\right.  \nonumber \\
&&\left. +\int \frac{d^{6}k}{(2\pi )^{6}}\frac{(2p.k-p^{2})^{3}}{%
(k^{2}-m^{2})^{4}[(p-k)^{2}-m^{2}]}\right\} \cdot  \label{86}
\end{eqnarray}
The explicit calculation of the integral in the above equation leads to 
\begin{equation}
\Gamma _{R}=i\Sigma _{R}^{(1)}(p^{2})=\frac{\lambda ^{2}}{4}\frac{i}{(4\pi
)^{3}}\left\{ (p^{2}-3m^{2})F(m^{2},p^{2})-\frac{p^{2}}{2}\right\}
\end{equation}
where $F(m^{2},p^{2}),$\ for $p^{2}<4m^{2}$ ,is given by 
\begin{equation}
F(m^{2},p^{2})=\frac{\sqrt{4m^{2}-p^{2}}}{\left| p\right| }\left[ 2\arctan
\left( \frac{\sqrt{4m^{2}-p^{2}}}{\left| p\right| }\right) +\pi \right] -2
\label{88}
\end{equation}
and for $p^{2}>4m^{2}$ , 
\begin{equation}
F(m^{2},p^{2})=-\frac{\sqrt{p^{2}-4m^{2}}}{\left| p\right| }\left[ \ln
\left( \frac{\left| p\right| -\sqrt{p^{2}-4m^{2}}}{\left| p\right| +\sqrt{%
p^{2}-4m^{2}}}\right) +i\pi \right] -2\ \cdot  \label{89}
\end{equation}
We now summarize the results obtained so far for one loop the
renormalization, 
\begin{equation}
A^{(1)}=\frac{i\lambda ^{2}}{2}\left\{ I_{\log 1}^{(1)}(m^{2},\Lambda )-%
\frac{4}{6}g_{\mu \nu }I_{\log }^{\mu \nu (1)}(m^{2},\Lambda )\right\}
\label{125}
\end{equation}
\begin{equation}
\delta ^{(1)}m^{2}=\frac{\lambda ^{2}}{2}iI_{quad}^{(1)}(m^{2},\Lambda )
\label{126}
\end{equation}
and 
\begin{equation}
B^{(1)}=-i\lambda ^{2}I_{\log 1}^{(1)}(m^{2},\Lambda )  \label{127}
\end{equation}
where $I_{\log 1}^{(1)}(m^{2},\Lambda )$ , $I_{\log }^{\mu \nu
(1)}(m^{2},\Lambda )$ and $I_{quad}^{(1)}(m^{2},\Lambda )$ \ are defined in
(56), (\ref{85}) and (\ref{84}), respectively.

\subsubsection{The two loop order}

Two types of diagram contribute to the self energy correction at two loops.
The total amplitude can be written as 
\begin{equation}
\Sigma ^{(2)}(p^{2})=2\Sigma _{1}^{(2)}(p^{2})+\Sigma _{2}^{(2)}(p^{2})\cdot
\label{90}
\end{equation}
Here the countertems to be identified are 
\begin{equation}
A^{(2)}=2A_{1}^{(2)}+A_{2}^{(2)}
\end{equation}
\begin{equation}
\delta ^{(2)}m^{2}=2\delta _{1}^{(2)}m^{2}+\delta _{2}^{(2)}m^{2}\cdot
\end{equation}
The first amplitude $i\Sigma _{1}^{(2)}(p^{2})$ corresponds to the diagram
in figure 7. This is the same case we have seen in equation (71).
Considering the one loop renormalization we can write 
\begin{equation}
\bar{\Gamma}=i\bar{\Sigma}_{1}^{(2)}(p^{2})=\frac{1}{2}\int_{\Lambda }\frac{%
d^{6}k_{1}}{(2\pi )^{6}}\frac{(-i\lambda )^{2}(i)^{3}}{%
(k_{1}^{2}-m^{2})^{2}[(p-k_{1})^{2}-m^{2}]}\{i\Sigma
_{R}^{(1)}(k_{1}^{2})\}\cdot  \label{91}
\end{equation}
Then we apply IRT and obtain 
\begin{equation}
iT^{2}\bar{\Sigma}_{1}^{(2)}(p^{2})=\bar{\Gamma}_{local}^{div}+\bar{\Gamma}%
_{fin}^{1}  \label{92}
\end{equation}
in which 
\begin{eqnarray}
\bar{\Gamma}_{local}^{div} &=&\frac{\lambda ^{2}}{2}i\left\{
I_{quad2}^{(2)}(m^{2},\lambda ^{2},\Lambda )\right.  \nonumber \\
&&\left. +p^{2}[\frac{4g_{\mu \nu }}{6}I_{\log 2}^{\mu \nu
(2)}(m^{2},\lambda ^{2},\Lambda )-I_{\log 2}^{(2)}(m^{2},\lambda
^{2},\Lambda )]\right\}  \label{93}
\end{eqnarray}
and 
\begin{equation}
I_{quad2}^{(2)}(m^{2},\lambda ^{2},\Lambda )=\int_{\Lambda }\frac{d^{6}k_{1}%
}{(2\pi )^{6}}\frac{1}{(k_{1}^{2}-m^{2})^{3}}\left\{ i\Sigma
_{R}^{(1)}(k_{1}^{2})\right\}  \label{94}
\end{equation}
\begin{equation}
I_{\log 2}^{\mu \nu (2)}(m^{2},\lambda ^{2},\Lambda )=\int_{\Lambda }\frac{%
d^{6}k_{1}}{(2\pi )^{6}}\frac{k^{\mu }k^{\nu }}{(k_{1}^{2}-m^{2})^{5}}%
\left\{ i\Sigma _{R}^{(1)}(k_{1}^{2})\right\}  \label{95}
\end{equation}
whereas 
\begin{eqnarray}
\bar{\Gamma}_{fin}^{1} &=&\frac{\lambda ^{2}}{2}i\int \frac{d^{6}k_{1}}{%
(2\pi )^{6}}\left\{ \frac{p^{4}}{(k_{1}^{2}-m^{2})^{5}}\right.  \nonumber \\
&&\left. +\frac{(2p.k_{1}-p^{2})^{3}}{%
(k_{1}^{2}-m^{2})^{5}[(p-k_{1})^{2}-m^{2}]}\right\} \left\{ i\Sigma
_{R}^{(1)}(k_{1}^{2})\right\} \cdot  \label{96}
\end{eqnarray}

The second amplitude $i\Sigma _{2}^{(2)}(p^{2})$ corresponds to the diagram
in figure 8. It reads 
\begin{equation}
i\Sigma _{2}^{(2)}(p^{2})=\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi )^{6}}%
\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}R(k_{1},k_{2},m^{2},\lambda
)f(p,k_{1},m^{2})f(p,k_{2},m^{2})  \label{97}
\end{equation}
with 
\begin{equation}
R(k_{1},k_{2},m^{2},\lambda )=\frac{1}{2}\{\frac{(-i\lambda )^{4}(i)^{5}}{%
(k_{1}^{2}-m^{2})(k_{2}^{2}-m^{2})[(k_{1}-k_{2})^{2}-m^{2}]}\}\cdot
\label{98}
\end{equation}
The same procedure enables us to write

\begin{eqnarray}
\Gamma _{local}^{div} &=&\frac{i\lambda ^{4}}{2}\left\{
I_{quad1}^{(2)}(m^{2},\Lambda )\right.  \nonumber \\
&&\left. +2p^{2}[\frac{4g_{\mu \nu }}{6}I_{\log 1}^{\mu \nu
(2)}(m^{2},\Lambda )-I_{\log 1}^{(2)}(m^{2},\Lambda )]\right\}
\end{eqnarray}
where 
\begin{eqnarray}
I_{quad1}^{(2)}(m^{2},\Lambda ) &=&\int_{\Lambda }\frac{d^{6}k_{1}}{(2\pi
)^{6}}\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}  \nonumber \\
&&\times \frac{1}{%
(k_{1}^{2}-m^{2})^{2}(k_{2}^{2}-m^{2})^{2}[(k_{1}-k_{2})^{2}-m^{2}]}
\label{100}
\end{eqnarray}
\begin{eqnarray}
I_{\log 1}^{\mu \nu (2)}(m^{2},\Lambda ) &=&\int_{\Lambda }\frac{d^{6}k_{1}}{%
(2\pi )^{6}}\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}  \nonumber \\
&&\times \frac{k_{2}^{\mu }k_{2}^{\nu }}{%
(k_{1}^{2}-m^{2})^{2}(k_{2}^{2}-m^{2})^{4}[(k_{1}-k_{2})^{2}-m^{2}]}
\end{eqnarray}
and the finite part coming from this contribution is 
\begin{equation}
\Gamma _{fin}^{2}=\frac{i\lambda ^{4}}{2}(I_{1}+2I_{2}+I_{3})  \label{102}
\end{equation}
with 
\begin{equation}
I_{1}=\int \frac{d^{6}k_{1}}{(2\pi )^{6}}\int \frac{d^{6}k_{2}}{(2\pi )^{6}}%
\frac{p^{4}}{%
(k_{1}^{2}-m^{2})^{3}(k_{2}^{2}-m^{2})^{3}[(k_{1}-k_{2})^{2}-m^{2}]}
\label{103}
\end{equation}
\begin{equation}
I_{2}=\int \frac{d^{6}k_{1}}{(2\pi )^{6}}\int \frac{d^{6}k_{2}}{(2\pi )^{6}}%
\frac{-4p^{2}(p.k_{2})^{2}}{%
(k_{1}^{2}-m^{2})^{3}(k_{2}^{2}-m^{2})^{4}[(k_{1}-k_{2})^{2}-m^{2}]}
\end{equation}
\begin{equation}
I_{3}=\int \frac{d^{6}k_{1}}{(2\pi )^{6}}\int \frac{d^{6}k_{2}}{(2\pi )^{6}}%
\frac{16(p.k_{1})^{2}(p.k_{2})^{2}}{%
(k_{1}^{2}-m^{2})^{4}(k_{2}^{2}-m^{2})^{4}[(k_{1}-k_{2})^{2}-m^{2}]}
\label{105}
\end{equation}

\begin{equation}
\Gamma _{nonlocal}=\Gamma _{fin}^{3}+\Gamma _{nonlocal}^{div}\cdot
\label{106}
\end{equation}
In terms of functions $\Theta ^{(1)}(k_{i},p,m^{2})$ and $\Upsilon
^{(1)}(k_{i},m^{2})$ we can write 
\begin{eqnarray}
\Gamma _{nonlocal}^{div} &=&\frac{(-i\lambda )^{4}(i)^{5}}{2}\int_{\Lambda }%
\frac{d^{6}k_{1}}{(2\pi )^{6}}\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}\{
\nonumber \\
&&\Theta ^{(1)}(k_{1},p,m^{2})\Upsilon ^{(1)}(k_{2},m^{2})  \nonumber \\
&&+\Theta ^{(1)}(k_{2},p,m^{2})\Upsilon ^{(1)}(k_{1},m^{2})\}\cdot
\label{107}
\end{eqnarray}
Note that $\Upsilon ^{(1)}$ is (the integrand of a) logarithmic divergence,
which, in DR would give us $1/\epsilon $ and when multiplied by the remainig
pieces of the amplitude would produce the celebrated term $\ln
p^{2}/\epsilon $ [16]. The (other) finite contributions are 
\begin{equation}
\Gamma _{fin}^{1}=\frac{(-i\lambda )^{4}(i)^{5}}{2}\int \frac{d^{6}k_{1}}{%
(2\pi )^{6}}\int \frac{d^{6}k_{2}}{(2\pi )^{6}}\Theta
^{(2)}(k_{1},k_{2},p,m^{2})  \label{108}
\end{equation}
and 
\begin{eqnarray}
\Gamma _{fin}^{3} &=&\frac{(-i\lambda )^{4}(i)^{5}}{2}\int \frac{d^{6}k_{1}}{%
(2\pi )^{6}}\int \frac{d^{6}k_{2}}{(2\pi )^{6}}\{  \nonumber \\
&&\Theta ^{(1)}(k_{1},p,m^{2})\frac{2k_{1}.k_{2}-k_{1}^{2}}{%
(k_{2}^{2}-m^{2})^{3}[(k_{1}-k_{2})^{2}-m^{2}]}  \nonumber \\
&&+\Theta ^{(1)}(k_{2},p,m^{2})\frac{2k_{1}.k_{2}-k_{2}^{2}}{%
(k_{1}^{2}-m^{2})^{3}[(k_{1}-k_{2})^{2}-m^{2}]}  \nonumber \\
&&+\int \frac{d^{6}k_{1}}{(2\pi )^{6}}\int \frac{d^{6}k_{2}}{(2\pi )^{6}}%
R(k_{1},k_{2},m^{2},\lambda )\{  \nonumber \\
&&f^{fin}(p,k_{1},m^{2})[f_{1}(k_{2},m^{2})+f_{2}(k_{2},m^{2})]  \nonumber \\
&&+f^{fin}(p,k_{2},m^{2})[f_{1}(k_{1},m^{2})+f_{2}(k_{1},m^{2})]\}\}\cdot
\label{109}
\end{eqnarray}
Summarizing the two loop renormalization constants obtained are, 
\begin{eqnarray}
A^{(2)} &=&\frac{\lambda ^{4}}{2}[\frac{4}{6}g_{\mu \nu }I_{\log 1}^{\mu \nu
(2)}(m^{2},\Lambda )-I_{\log 1}^{(2)}(m^{2},\Lambda )]  \nonumber \\
&&+\lambda ^{2}[\frac{4}{6}g_{\mu \nu }I_{\log 2}^{\mu \nu
(2)}(m^{2},\lambda ^{2},\Lambda )-I_{\log 2}^{(2)}(m^{2},\lambda
^{2},\Lambda )]  \label{128}
\end{eqnarray}
\begin{equation}
\delta ^{(2)}m^{2}=-[\frac{\lambda ^{4}}{2}I_{quad1}^{(2)}(m^{2},\Lambda
)+\lambda ^{2}I_{quad2}^{(2)}(m^{2},\lambda ^{2},\Lambda )]  \label{129}
\end{equation}
and 
\begin{equation}
B^{(2)}=\lambda ^{4}[I_{\log 1}^{(2)}(m^{2},\Lambda )+I_{\log
3}^{(2)}(m^{2},\Lambda )]+3\lambda ^{2}I_{\log 2}^{(2)}(m^{2},\lambda
^{2},\Lambda )  \label{130}
\end{equation}
where 
\[
I_{\log 1}^{(2)}(m^{2},\Lambda ),I_{\log 1}^{\mu \nu (2)}(m^{2},\Lambda
),I_{\log 2}^{(2)}(m^{2},\lambda ^{2},\Lambda ),I_{\log 2}^{\mu \nu
(2)}(m^{2},\lambda ^{2},\Lambda ),I_{\log 3}^{(2)}(m^{2},\Lambda ) 
\]
and 
\[
I_{quad1}^{(2)}(m^{2},\Lambda ),I_{quad2}^{(2)}(m^{2}.\lambda ^{2},\Lambda ) 
\]
are defined in equations (56), (126), (78), (\ref{95}), (\ref{71}), (\ref
{100}) and (\ref{94}) respectively.

\subsubsection{The $n$-loop order}

Let us first consider the overlapping self energy diagram of figure 9. It
corresponds to the amplitude 
\begin{equation}
i\Sigma _{1}^{(n)}(p^{2})=(\Pi R)(\Pi f)  \label{110}
\end{equation}
with 
\begin{equation}
(\Pi R)=\frac{(-i\lambda )^{2n}(i)^{3n-1}}{2}\left\{
\prod_{j=1}^{n}\int_{\Lambda }\frac{d^{6}k_{j}}{(2\pi )^{6}}\frac{1}{%
k_{j}^{2}-m^{2}}\right\} Q(k_{i},k_{i+1},m^{2})\cdot   \label{111}
\end{equation}
The external momentum dependent part is 
\begin{equation}
(\Pi f)=\prod_{j=1}^{n}\frac{1}{[(p-k_{j})^{2}-m^{2}]}\cdot   \label{112}
\end{equation}
Using the technique we have, as usual 
\begin{equation}
iT^{2}\Sigma _{1}^{(n)}(p^{2})=\Gamma _{fin}^{1}+\Gamma _{local}+\Gamma
_{nonlocal}  \label{113}
\end{equation}
where 
\begin{equation}
\Gamma _{fin}^{1}=\frac{(-i\lambda )^{2n}(i)^{3n-1}}{2}\prod_{j=1}^{n}\int 
\frac{d^{6}k_{j}}{(2\pi )^{6}}\Theta ^{(n)}(k_{1},k_{2},...k_{n};p,m^{2})
\label{114}
\end{equation}
and 
\begin{eqnarray}
\Gamma _{local} &=&\Gamma _{fin}^{2}+\Gamma _{local}^{div}  \nonumber \\
&=&(\Pi
R)\prod_{j=1}^{n}%
\{f^{0}(k_{j},m^{2})+f^{1}(k_{j},m^{2},p)+f^{2}(k_{j},m^{2},p)\}\cdot 
\end{eqnarray}
In this case we have $\Gamma _{fin}^{2}\neq 0$ . The counterterms
characteristic of the $n^{th}$ order are identified in the equation 
\[
\Gamma _{local}^{div}=i(\delta _{1}^{(n)}m^{2}-A_{1}^{(n)}p^{2})\cdot 
\]
The nonlocal part is 
\begin{equation}
\Gamma _{nonlocal}=\Gamma _{fin}^{3}+\Gamma _{nonlocal}^{div}  \label{116}
\end{equation}
where 
\begin{eqnarray}
\Gamma _{nonlocal}^{div} &=&\frac{(-i\lambda )^{2n}(i)^{3n-1}}{2}%
\prod_{j=1}^{n}\int_{\Lambda }\frac{d^{6}k_{j}}{(2\pi )^{6}}  \label{145} \\
&&\Bigg\{\sum_{a=1}^{n-1}\Theta
^{(a)}(k_{1},k_{2},..,k_{a};p^{2},m^{2})\Upsilon
^{(n-a)}(k_{a+1},..,k_{n};m^{2})+  \nonumber \\
&&\sum_{a=1}^{n-1}\Upsilon ^{(n-a)}(k_{1},k_{2},..,k_{n-a};m^{2})\Theta
^{(a)}(k_{n-a+1},..,k_{n};p^{2},m^{2})+  \nonumber
\end{eqnarray}
\[
\sum_{a,b=1}^{n-2}\Upsilon ^{(b)}(k_{1},k_{2},..,k_{b};m^{2})\Theta
^{(a)}(k_{b+1},..,k_{a};p^{2},m^{2})\Upsilon
^{(n-a-b)}(k_{a+1},..,k_{n};m^{2})\Bigg\}
\]
From the above equation it becomes clear that the renormalization of the
self energy to $n^{th}$-order requires all finite functions defined in
previous self energy diagrams (up to $(n-1)^{th}$-order) as well as all the
divergent contributions of the three-point functions also to the $(n-1)^{th}$%
-order. We may associate a graphical representation to the equation above
and, in this way, compare with the BPHZ results. The first term in the
equation (145) contains a sum of $n-1$ terms comprising $a$ finite functions
of the type $\Theta $ multiplied by the $n-a$ divergent vertex-type
functions. The second term is the symmetric to the first one (the vertex
functions and functions $\Theta $ swap sides). Finally the last term
contains vertex corrections to the left and to the right and finite
functions in the middle. This can be best visualized in the graph which
follows (figure 10).

Notice that in the present procedure no special treatment has been given to
the overlapping divergencies or to the nested ones, both appearing in the
self-energy. The reason is that the algebraic procedure produces only
disjoint divergent contributions.

In order to complete the renormalization of this theory we will still
consider two cases, both belonging to the second class defined previously.
Firstly we consider a specific case where two point functions explicitly
appear as subdivergences (see figure 11) and the other is an amplitude
containing an overlapping divergence diagram as substructure (figure 12). As
we mentioned before, the total intagral contains the two point function
substructures in factorized form. We therefore effect the renormalization of
the internal propagators directly using the counterterms of order. In this
way we immediately obtain $\bar{\Gamma}$ . Let us first consider the case in
figure 11. This diagram contain s subdiagrams involving nested two point
functions. Following the prescription which explicitates the renormalized
contributions of previous orders we get 
\begin{eqnarray}
\bar{\Gamma} &=&i\bar{\Sigma}_{2}^{(n)}(p^{2}) \\
&=&\frac{(-i\lambda )^{2}(i)^{3}}{2}\int_{\Lambda }\frac{d^{6}k_{n}}{(2\pi
)^{6}}\frac{1}{(k_{n}^{2}-m^{2})^{2}[(p-k_{n})^{2}-m^{2}]}  \nonumber \\
&&\times i\left\{ T_{k_{n}}^{2}\bar{\Sigma}_{2}^{(n-1)}(k_{n}^{2})+\delta
_{2}^{(n-1)}m^{2}-A_{2}^{(n-1)}k_{n}^{2}\right\}   \nonumber
\end{eqnarray}
with 
\begin{eqnarray}
i\bar{\Sigma}_{2}^{(n-1)}(k_{n}^{2}) &=&\frac{(-i\lambda )^{2}(i)^{3}}{2} \\
&&\int_{\Lambda }\frac{d^{6}k_{n-1}}{(2\pi )^{6}}\frac{1}{%
(k_{n-1}^{2}-m^{2})^{2}[(k_{n}-k_{n-1})^{2}-m^{2}]}  \nonumber \\
&&\times i\left\{ T_{k_{n-1}}^{2}\bar{\Sigma}_{2}^{(n-2)}(k_{n-1}^{2})+%
\delta _{2}^{(n-2)}m^{2}-A_{2}^{(n-2)}k_{n-1}^{2}\right\}   \nonumber
\end{eqnarray}
\begin{eqnarray*}
&&\bullet  \\
&&\bullet  \\
&&\bullet 
\end{eqnarray*}
\begin{eqnarray}
i\bar{\Sigma}_{2}^{(2)}(k_{3}^{2}) &=&\frac{(-i\lambda )^{2}(i)^{3}}{2}%
\int_{\Lambda }\frac{d^{6}k_{2}}{(2\pi )^{6}}\frac{1}{%
(k_{2}^{2}-m^{2})^{2}[(k_{3}-k_{2})^{2}-m^{2}]} \\
&&\times i\left\{ T_{k_{2}}^{2}\Sigma ^{(1)}(k_{2}^{2})+\delta
^{(1)}m^{2}-A^{(1)}k_{2}^{2}\right\}   \nonumber
\end{eqnarray}
\begin{equation}
i\Sigma ^{(1)}(k_{2}^{2})=\frac{(-i\lambda )^{2}(i)^{2}}{2}\int_{\Lambda }%
\frac{d^{6}k_{1}}{(2\pi )^{6}}\frac{1}{%
(k_{1}^{2}-m^{2})[(k_{2}-k_{1})^{2}-m^{2}]}\cdot 
\end{equation}
We can substitute the terms in brackets by renormalized function 
\begin{eqnarray}
\bar{\Sigma}_{2R}^{(n-1)}(k_{n}^{2}) &=&\Gamma _{R}^{(n-1)} \\
&=&T_{k_{n}}^{2}\bar{\Sigma}_{2}^{(n-1)}(k_{n}^{2})+\delta
_{2}^{(n-1)}m^{2}-A_{2}^{(n-1)}k_{n}^{2}\cdot   \nonumber
\end{eqnarray}
In this exemple the two point functions counterterms can be obtained for any
order $n>1$ as 
\begin{equation}
i\delta _{2}^{(n)}m^{2}=\frac{(-i\lambda )^{2}(i)^{3}}{2}\int_{\Lambda }%
\frac{d^{6}k_{n}}{(2\pi )^{6}}\frac{1}{(k_{n}^{2}-m^{2})^{3}}\left\{ i\bar{%
\Sigma}_{2R}^{(n-1)}(k_{n}^{2})\right\} 
\end{equation}
\begin{eqnarray}
iA_{2}^{(n)} &=&\frac{(-i\lambda )^{2}(i)^{3}}{2}\{ \\
&&\frac{4}{6}g^{\mu \nu }\int_{\Lambda }\frac{d^{6}k_{n}}{(2\pi )^{6}}\frac{%
k_{\mu }k_{\nu }}{(k_{n}^{2}-m^{2})^{5}}\left\{ i\bar{\Sigma}%
_{2R}^{(n-1)}(k_{n}^{2})\right\}   \nonumber \\
&&-\int_{\Lambda }\frac{d^{6}k_{n}}{(2\pi )^{6}}\frac{1}{%
(k_{n}^{2}-m^{2})^{4}}\left\{ i\bar{\Sigma}_{2R}^{(n-1)}(k_{n}^{2})\right\}
\}\cdot   \nonumber
\end{eqnarray}

Finally we will consider type 3 diagram (which contain a type 1 diagram)
show in figure 12. \ The corresponding amplitude reads 
\begin{equation}
i\Sigma _{3}^{(n)}(p^{2})=\frac{(-i\lambda )^{2}(i)^{3}}{2}\int_{\Lambda }%
\frac{d^{6}k_{n}}{(2\pi )^{6}}\frac{1}{%
(k_{n}^{2}-m^{2})^{2}[(p-k_{n})^{2}-m^{2}]}\left\{ i\Sigma
_{1}^{(n-1)}(k_{n}^{2})\right\} \cdot
\end{equation}
Note that since the structure $\Sigma _{1}^{(n-1)}(k_{n}^{2})$ can be
renormalized at $(n-1)^{th}$ order (see first example of order n), the $%
n^{th}$ order structure will also be renormalized by a $\Gamma
_{local}^{div} $. Its counterterms have the same form as in the previous
example. They are 
\begin{equation}
i\delta _{3}^{(n)}m^{2}=\frac{(-i\lambda )^{2}(i)^{3}}{2}\int_{\Lambda }%
\frac{d^{6}k_{n}}{(2\pi )^{6}}\frac{1}{(k_{n}^{2}-m^{2})^{3}}\left\{ i\Sigma
_{1R}^{(n-1)}(k_{n}^{2})\right\}
\end{equation}
\begin{eqnarray}
iA_{3}^{(n)} &=&\frac{(-i\lambda )^{2}(i)^{3}}{2}\{ \\
&&\frac{4}{6}g^{\mu \nu }\int_{\Lambda }\frac{d^{6}k_{n}}{(2\pi )^{6}}\frac{%
k_{\mu }k_{\nu }}{(k_{n}^{2}-m^{2})^{5}}\left\{ i\Sigma
_{1R}^{(n-1)}(k_{n}^{2})\right\}  \nonumber \\
&&-\int_{\Lambda }\frac{d^{6}k_{n}}{(2\pi )^{6}}\frac{1}{%
(k_{n}^{2}-m^{2})^{4}}\left\{ i\Sigma _{1R}^{(n-1)}(k_{n}^{2})\right\}
\}\cdot  \nonumber
\end{eqnarray}
Note that in this case the three point functions subdiagrams have been
renormalized together with the two point subdiagram, since it is contained
in the latter.

\section{Momentum routing independence}

In the exemples of the previous sections we have chosen the momentum routing
in such way as to obtain the simplest form for the final expressions. Of
course, the counterterms so obtained must be independent of the particular
routing one chooses. In order to exemplify this we consider the last exemple
given (type 3 diagram). One of the possible choices for the momentum routing
would be to arrange the labels in such a way that external momentum is
present in an internal line of the diagram as follows 
\begin{eqnarray}
i\Sigma _{3}^{(n)}(p^{2}) &=&\frac{(-i\lambda )^{2}(i)^{3}}{2} \\
&&\int_{\Lambda }\frac{d^{6}k_{n}}{(2\pi )^{6}}\frac{1}{%
(k_{n}^{2}-m^{2})[(p-k_{n})^{2}-m^{2}]^{2}}\left\{ i\Sigma
_{1}^{(n-1)}((p-k_{n})^{2})\right\}   \nonumber
\end{eqnarray}
and also, 
\begin{equation}
i\Sigma _{3}^{(n)}(p^{2})=\frac{(-i\lambda )^{2}(i)^{3}}{2}\int_{\Lambda }%
\frac{d^{6}k_{n}}{(2\pi )^{6}}\frac{1}{%
(k_{n}^{2}-m^{2})^{2}[(p-k_{n})^{2}-m^{2}]}\left\{ i\Sigma
_{1}^{(n-1)}(k_{n}^{2})\right\} \cdot 
\end{equation}
These two rotulations must be equivalent, so that the amplitude is momentum
routing independent as it should. Note that if the amplitude were finite,
this could immeaditely be accomplished through a shift $p-k_{n}=k_{n}^{%
\prime }$. However, since the amplitude is quadratically divergent, shifts
are not allowed without the inclusion of surface terms. This point has been
extensively discussed in our method (see refs. [13,14] ) and a similar
procedure can cure this problem in the present model. More difficult would
be theories with gauge symmetries and work along this line is in progress.
Note that in Dimensional Regularization the problem does not appear since
shifts are always allowed.

\section{Conclusion}

We have considered (in the self-energy) all possible complications which
usually appear in renormalization procedures: overlapping divergences,
nested divergencies and disjoint ones, all in the same graph at $n$-loops.
We have explicitly shown how these problems can be systematically resolved
order by order within our technique. A comparison with BPHZ at $n$-loops is
also included.

General aspects of the procedure that we have learned from this example is
that there will always be a divergent (local) order dependent contribution.
Also, there will always be a finite contribution composed by the product of
all finite parts of $f_{j}$%
%TCIMACRO{\UNICODE{0xb4}}%
%BeginExpansion
\'{}%
%EndExpansion
s. These two structures (divergent and finite) are typical of the $n^{th}$
order and poses no problem for renormalization.

As we have seen in the examples given, the identites we use in the integrand
leaves us then with crossed products of divergent and finite contributions.
All possible combinations will appear and all of them can either be
recognized as structures (finite or divergent) already encountered in lower
order amplitudes or they will give a finite contribution.

In summary we present a new perturbative renormalization procedure where an
algebraic identity at the level of the internal lines of the diagrams is
used. We have shown how the technique can be used to renormalize a scalar
theory at the $n$ order of the perturbative series. However no symmetry
aspect is mentioned here. Is this method gauge invariant? This question is
presently under investigation. At the one loop level we can preserve gauge
symmetry if use is made of relations involving divergent integrals of the
same degree of divergence \cite{14},\cite{13}. The difference among those
integrals is source of both ambiguities and symmetry violations. We are
working on the application of this method at the two loops level the Quantum
Electrodynamic (QED).

We see that the aplication of this method leads to a relatively simple
renormalization procedure. There is no need for a graphic representation of
the relevant contributions. When a diagram has divergent subdiagrams the
subdivergences need not be previously identified because they will appear in
an algebraic way. The procedure explicitate all relevant subdivergences.

\textbf{Acknowledgments}

The work of \ M.C. Nemes\ was partially supported by CNPq, FAPEMIG,
PRAXIS/BCC/4301/94, PRAXIS/FIS/12247/98 and POCTI/1999/FIS/35304. The work
of \ S.R. Gobira was supported by FAPEMIG.

We thank Prof. Marcelo Gomes\ for discussions on the subject of this
paper.\newpage

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\newpage

\begin{figure}[f]
\centering
\includegraphics[scale=1.0]{figure1.eps}
\caption{One-loop vertex correction $-iV^{(1)}(p,p^{\prime })$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=1.0]{figure2.eps}
\caption{The two-loop vertex correction contribution $-iV_{1}^{(2)}(p,p^{%
\prime })$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=1.0]{figure3.eps}
\caption{The two-loop vertex correction contribution $-iV_{2}^{(2)}(p,p^{%
\prime })$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=1.0]{figure4.eps}
\caption{The two-loop vertex correction contribution $-iV_{3}^{(2)}(p,p^{%
\prime })$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=1.0]{figure5.eps}
\caption{The n-loop vertex correction contribution $-iV_{1}^{(n)}(p,p^{%
\prime })$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=1.0]{figure6.eps}
\caption{One-loop self energy $i\Sigma ^{(1)}(p^{2})$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=1.0]{figure7.eps}
\caption{The two-loop self energy contribution $i\Sigma _{1}^{(2)}(p^{2})$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=1.0]{figure8.eps}
\caption{The two-loop self energy contribution $i\Sigma _{2}^{(2)}(p^{2})$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=1.0]{figure9.eps}
\caption{The n-loop self energy contribution $i\Sigma _{1}^{(n)}(p^{2})$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=0.9]{figure10.eps}
\caption{Graphic representation of equation (117)}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=0.9]{figure11.eps}
\caption{The n-loop self energy contribution $i\Sigma _{2}^{(n)}(p^{2})$}
\end{figure}

\begin{figure}[f]
\centering
\includegraphics[scale=1.2]{figure12.eps}
\caption{The n-loop self energy contribution $i\Sigma _{3}^{(n)}(p^{2})$}
\end{figure}

\end{document}

