# Advances towards the systematization of calculations with Implicit
Regularization

B. Z. Felippe(a,b<EMAIL_ADDRESS>A. P. Baêta Scarpelli(a)
<EMAIL_ADDRESS>A. R. Vieira(c<EMAIL_ADDRESS>J. C. C.
Felipe(d<EMAIL_ADDRESS>(a)Centro Federal de Educação Tecnológica
- MG
Avenida Amazonas, 7675 - 30510-000 - Nova Gameleira - Belo Horizonte-MG -
Brazil (b) Universidade Federal de Itajubá - Campus Itabira
Rua Irmã Ivone Drumond, 200 - Distrito Industrial II, Itabira, Minas Gerais,
Brazil (c) Universidade Federal do Triângulo Mineiro, Campus Iturama, Iturama
- MG - Brasil (d) Instituto de Engenharia, Ciência e Tecnologia, Universidade
Federal dos Vales do Jequitinhonha e Mucuri, Avenida Um, 4050 - 39447-790 -
Cidade Universitária - Janaúba - MG- Brazil

###### Abstract

There is currently a high demand for theoretical predictions for processes at
next-to-next-to-leading order (NNLO) and beyond, mainly due to the large
amount of data which has already been collected at LHC. This requires
practical methods that meet the physical requirements of the models under
study. We develop a new procedure for applying Constrained Implicit
Regularization which simplifies the calculation of amplitudes, including
finite parts. The algebraic identities to separate the divergent parts free
from the external momenta are used after the Feynman parametrization. These
algebraic identities establish a set of scale relations which are always the
same and do not need to be calculated in each situation. This procedure
unifies the calculations in massive and non-massive models in an unique
procedure. We establish a systematization of the calculation of one-loop
amplitudes and extend the procedure for higher-loop orders.

###### pacs:

11.10.Gh, 11.30.-j, 11.15.-q

##  I Introduction

The success of Quantum Field Theory is highly attached to the principles of
renormalization, which allow, through suitable renormalization conditions, the
redefinition of the quantities of interest in order to obtain physical
meaningful results. This is because the perturbative calculations of
transition amplitudes involve divergences. In the high energy regime, one has
to deal with ultraviolet (UV) divergences coming from the locality of
fundamental interactions. The calculation of Feynman amplitudes, thus, is
intricate with the technical problem of regularizing integrals in a consistent
way such that the finite part presents the expected physical content of the
amplitude.

The regularization technique to be used should meet some important features of
the $S$-matrix, like the preservation of unitarity and causality, the
preservation of gauge symmetry and supersymmetry, among others. Besides, the
renormalization procedure can be more involved depending on the regularization
applied. The usual textbook technique is Dimensional Regularization DimReg ,
which is powerful due to gauge symmetry preservation in all orders of
perturbation theory. The dimensional extension, however, causes difficulties
when the model under investigation encompasses dimensional dependent
mathematical objects, like the $\gamma_{5}$ matrix. This is the case of
topological and supersymmetric theories. Dimensional Reduction DimRed is a way
out of this problem for extending only the dimension of the Feynman integrals,
but preserving the symmetry group algebra in the original space-time
dimension. However, there are some mathematical inconsistencies in the
procedure when the calculation is performed beyond one-loop order. The quantum
action principle has been used in order to consistently apply Dimensional
Reduction beyond one-loop order in supersymmetric models Stockinger . These
difficulties with dimensional extension are the motivating fact for several
developments of regularization procedures which are carried out in the proper
dimension of the model (good discussions on this topic are carried out in
Marcos-1 , Marcos-2 and NNLO ).

Differential Renormalization (DR) is one of these approaches DR . It works in
the proper dimension of the theory in coordinate space, and has been proved to
be simple and powerful in many applications DR2 -DR5 . It consists in the
manipulation of singular distributions attributing to them properties of the
regular ones. In the end, these singularities are substituted by renormalized
functions and several mass parameters are introduced in the results. Relations
between these parameters are established in order to preserve symmetries. A
further development in order to automatically satisfy symmetries came with the
constrained version of Differential Renormalization (CDR) cdf1 -cdf8 .

Implicit Regularization ir1 -ir3 , on the other hand, is carried out in
momentum-space, also in the physical dimension of the theory. The basic idea
of the Implicit Regularization (IReg) procedure of a Feynman integral is to
consider, before manipulating the integrands, the presence of some implicit
regularization scheme or function. The scheme composes the originally
divergent integral and allows the separation of its part dependent on the
regularization from the finite part, which must be independent of the
regularization used. The separation can be done by applying a simple algebraic
identity to the integrand, such that the divergent parts are written only in
terms of the internal momentum in the loops and do not need to be evaluated.
The independence of the divergent integrals from the external momentum is a
highly desirable feature, since we will only need local counterterms in the
Lagrangian of the model in order to eliminate any divergences that arise in
the perturbative calculus. Furthermore, these divergent integrals can be
written as functions of an arbitrary mass parameter that characterizes the
freedom of separating of the divergent part of an amplitude and plays the role
of scale in the renormalization group equation. Symmetries of the model or
phenomenological requirements determine arbitrary parameters which are surface
terms and arise from the procedure. There is a special choice for the
parameters that automatically delivers symmetric amplitudes in anomaly free
cases. Fixing these parameters at the beginning of the calculation
considerably simplifies the application of the method. This results in a
constrained version of Implicit Regularization (CIReg). Implicit
Regularization has been successfully applied to a wide variety of problems,
including non-abelian and supersymmetric models, and calculations beyond the
one-loop order ir4 -irn . There is a class of four dimensional regularization
techniques, however, CIReg included, for which the difficulties with
mathematical objects like the $\gamma_{5}$ matrix and the Levi-Civita tensor
remain. One deals, for example, with ambiguities in the Dirac trace involving
the $\gamma_{5}$ matrix. This problem was discussed in Bruque , in which a
consistency procedure was proposed.

It is important to note that, in its traditional formulation, the procedure of
IReg is applied after performing the algebra of the symmetry group in the
amplitude, which is decomposed into a combination of integrals. Each integral,
then, is separated into finite and divergent parts. The Feynman
parametrization process is applied only to finite integrals. However, for a
single Feynman integral, the expansion to separate the divergences generates a
set of finite integrals, which can be large in case of high degrees of
divergence. In addition, expansions generate high powers of momenta in the
numerator and denominator, which can complicate the calculations quite a bit.

The technique known as Loop Regularization (LORE) Lore-1 -Lore-3 has some
aspects in common with Constrained Implicit Regularization, such as the use of
consistency conditions, which, in practice, eliminate surface terms that cause
violation of symmetries. LORE prescribes that Feynman parametrization is
applied to the amplitude as a whole, like in Dimensional Regularization.
Afterwards, the algebra of the symmetry group is performed and then the
integration in momenta. In the case of divergent parts, consistency conditions
are used to write the amplitude in terms of scalar loop integrals. Finally,
these scalar integrals are regularized similarly to the Pauli-Villars
procedure and then calculated.

In this paper, we propose a new approach for implementing Implicit
Regularization which greatly simplifies the calculation of the amplitudes,
mainly the finite parts. Assuming the action of a regularization, we apply, as
in LORE, Feynman parametrization in the complete amplitude and eliminate the
surface terms through what we call consistency relations, so as to have only
scalar divergent integrals. We then make use of algebraic identities to expand
the integrands to obtain basic divergences which are free from the external
momenta. These algebraic identities establish a set of scale relations which
are always the same and do not need to be calculated in each situation. As a
byproduct, the scale relations allow to introduce an arbitrary mass scale that
will be useful in the process of renormalization. The results of this new
approach are the same of Constrained Implicit Regularization. However, the
procedure also permits the introduction of local arbitrary parameters for the
models they are needed. This approach is also extended for multiloop
calculations, since there is currently a high demand for theoretical
predictions for processes at next-to-next-to-leading order (NNLO) and beyond,
mainly due to the large amount of data which has already been collected at
LHC. Therefore, practical methods for higher order calculations are being
intensively investigated.

The paper is divided as follows: in section II we present a brief review of
the traditional procedure for Implicit Regularization, with examples to be
compared with the new ones; in section III, the new for implementing
constrained IReg is presented, with discussions and comparisons with the cases
of the previous section; in section IV, we carry out a systematization of the
calculation of one-loop amplitudes; in section V, we present the extension for
higher-loop calculations; discussions on implicit regularization of infrared
divergences are carried out in section VI; section VII is left for conclusions
and perspectives.

##  II A brief review of Implicit Regularization

Implicit Regularization (IReg) can be formulated by a set of rules. The first
thing to be done is to assume a regularization is applied to the complete
amplitude, so as algebraic manipulations can be carried out in the integrand.
We then perform the group algebra and write the momentum-space amplitude as a
combination of basic integrals, multiplied by polynomials of the external
momentum and typical objects of the symmetry group. We give below examples of
basic integrals:

$I,I_{\mu},I_{\mu\nu}=\int^{R}\frac{d^{4}k}{(2\pi)^{4}}\frac{1,k_{\mu},k_{\mu}k_{\nu}}{(k^{2}-m^{2})[(p-k)^{2}-m^{2}]},$
(1)

in which the index $R$ in the integrals is to indicate they are regularized.
Each one of these basic integrals can be treated following a set of rules. So,
a table with their results can be used whenever a new calculation is being
performed. The rules of Constrained Implicit Regularization (CIReg) for
calculations at one-loop order can be stated as:

  1. 1.

an amplitude is assumed to be regularized with a 4D technique which is
maintained implicit;

  2. 2.

to obtain the divergent part of a basic integral, we apply recursively the
identity,

$\displaystyle\frac{1}{(p-k)^{2}-m^{2}}=\frac{1}{(k^{2}-m^{2})}-\frac{p^{2}-2p\cdot
k}{(k^{2}-m^{2})\left[(p-k)^{2}-m^{2}\right]},$ (2)

so as the divergent part do not have the external momentum $p$ in the
denominator. This will assure local counterterms. The remaining divergent
integrals have the form

$\int_{k}^{R}\frac{k_{\mu_{1}}k_{\mu_{2}}\cdots}{(k^{2}-m^{2})^{\alpha}},$ (3)

in which we use $\int_{k}$ as a simplification of $\int d^{4}k/(2\pi)^{4}$;

  3. 3.

The divergent integrals with Lorentz indices must be expressed in terms of
divergent scalar integrals and surface terms. For example:

$\int_{k}^{R}\frac{k_{\mu}k_{\nu}}{(k^{2}-m^{2})^{3}}=\frac{1}{4}\left\\{\eta_{\mu\nu}\int_{k}^{R}\frac{1}{(k^{2}-m^{2})^{2}}-\int_{k}^{R}\frac{\partial}{\partial
k^{\nu}}\left(\frac{k_{\mu}}{(k^{2}-m^{2})^{2}}\right)\right\\}.$ (4)

The surface terms, that vanish for finite integrals, depend here on the method
of regularization to be applied. They are symmetry-violating terms, since the
possibility of making shifts in the integrals needs the surface terms to
vanish. Non-null surface terms imply that the amplitude depend on the momentum
routing choice. In practice, setting them zero from the beginning is
equivalent to canceling these surface terms by means of local symmetry-
restoring counterterms;

  4. 4.

Finally, the divergent part of the integrals is written as a combination of
the basic divergences

$I_{log}(m^{2})=\int_{k}^{R}\frac{1}{(k^{2}-m^{2})^{2}}\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\
I_{quad}(m^{2})=\int_{k}^{R}\frac{1}{(k^{2}-m^{2})},$ (5)

which will require local counterterms in the process of renormalization.

Let us comment on the first item above. Ideally, the regularization technique
that is maintained implicit should have the properties of not modifying
neither the integrand nor the dimension of spacetime. The former property is
to preserve the finite part and the latter is a requirement in order to not
violate supersymmetry. In previous works, the cutoff regularization had been
cited as a possibility, although it is known that it causes the violation even
of simple symmetries. This should be understood in the following sense. When
surface terms are fixed to a given value (zero, for example) in order to
obtain a symmetric result, this is completely equivalent to add symmetry
restoring counterterms to the amplitude. If we expand the Feynman integral as
prescribed by IReg and then solve the divergent parts by using cutoff
regularization, the symmetry violating terms will come from the tensorial
divergent basic integrals. Thus, the procedure of IReg allows us to identify
all the symmetry-violating terms. We know exactly which part should be
subtracted. In this sense, as long as the implicit technique is defined in the
proper dimension of the model and do not modify the integrand, it will not get
in the way of the procedure. In practice, we do not make use of any specific
regularization. It was shown in papers as bphz1 and bphz2 , the connection of
the method with the Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) theorem.
Therefore, the IReg can also be seen as a map of Feynman amplitudes into
results with desirable properties.

Let us now present an example to show how the traditional procedure of IReg
applies to a complete amplitude in order to compare, in the next section, with
the new proposed approach. We consider below the vacuum polarization tensor of
spinorial QED, which after the use of Feynman rules, is given by

$-i\Pi^{\mu\nu}=q^{2}\int_{k}^{R}\frac{\mbox{tr}\left\\{\gamma^{\mu}(\hbox{$k\\!\\!\\!{/}$}-\hbox{$p\\!\\!\\!{/}$}+m)\gamma^{\nu}(\hbox{$k\\!\\!\\!{/}$}+m)\right\\}}{(k^{2}-m^{2})[(k-p)^{2}-m^{2}]}.$
(6)

Following the steps listed above, we first calculate the trace and write the
amplitude as a combination of basic integrals. We end up with

$-i\Pi^{\mu\nu}=4q^{2}\left\\{2I^{\mu\nu}-p^{\mu}I^{\nu}-p^{\nu}I^{\mu}-\frac{1}{2}\eta^{\mu\nu}\left[I_{quad}(m^{2})+I_{1}-p^{2}I\right]\right\\},$
(7)

where $I$, $I_{\mu}$ and $I_{\mu\nu}$ are defined in eq. (1),
$I_{quad}(m^{2})$ is given in (5) and

$I_{1}=\int_{k}^{R}\frac{1}{[(k-p)^{2}-m^{2}]}.$ (8)

The next step is to calculate each one of the integrals. Let us give the
directions of the calculations of $I_{\mu\nu}$. It is a divergent integral and
the integrand need to be expanded using (2) in order to separate the finite
part from the regularization dependent one. We have, after discarding the null
integrals,

$\displaystyle
I_{\mu\nu}=\int_{k}^{R}\frac{k_{\mu}k_{\nu}}{(k^{2}-m^{2})^{2}}-p^{2}\int_{k}^{R}\frac{k_{\mu}k_{\nu}}{(k^{2}-m^{2})^{3}}+4p^{\alpha}p^{\beta}\int_{k}^{R}\frac{k_{\mu}k_{\nu}k_{\alpha}k_{\beta}}{(k^{2}-m^{2})^{4}}+$
$\displaystyle+p^{4}\int_{k}\frac{k_{\mu}k_{\nu}}{(k^{2}-m^{2})^{4}}-\int_{k}\frac{[p^{2}-2(p\cdot
k)]^{3}k_{\mu}k_{\nu}}{(k^{2}-m^{2})^{4}[(k-p)^{2}-m^{2}]}.$ (9)

The first term is quadratically divergent and the second and third terms are
logarithmically divergent. We use the procedure of (4) to obtain scalar basic
divergences and follow the notation from reference A-Vieira :

$\int_{k}^{R}\frac{k_{\mu}k_{\nu}}{(k^{2}-m^{2})^{2}}=\frac{\eta_{\mu\nu}}{2}\left[I_{quad}(m^{2})-\upsilon_{2}\right],$
(10)
$\int_{k}^{R}\frac{k_{\mu}k_{\nu}}{(k^{2}-m^{2})^{3}}=\frac{\eta_{\mu\nu}}{4}\left[I_{log}(m^{2})-\upsilon_{0}\right]$
(11)

and

$\int_{k}^{R}\frac{k_{\mu}k_{\nu}k_{\alpha}k_{\beta}}{(k^{2}-m^{2})^{4}}=\frac{\eta_{\mu\nu\alpha\beta}}{24}\left[I_{log}(m^{2})-\xi_{0}\right],$
(12)

in which
$\eta_{\mu\nu\alpha\beta}\equiv\eta_{\mu\nu}\eta_{\alpha\beta}+\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}$
and where $\upsilon_{0}$, $\upsilon_{2}$ and $\xi_{0}$ are surface terms.
These terms are arbitrary and regularization dependent because they are
differences between two integrals with the same degree of divergences, as
shown in eq. (4). In this case, the indices $0$ and $2$ corresponds to
logarithmic and quadratic divergences, respectively.

The last two integrals in (9) are finite and can be solved. The Feynman
parametrization can be applied when necessary, like in the last one. Note that
the high power on the momenta in the numerator and in the denominator make
calculations longer. The final result for $I_{\mu\nu}$ is given by

$\displaystyle I_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\eta_{\mu\nu}}{2}I_{quad}(m^{2})+\frac{1}{12}\left(-p^{2}\eta_{\mu\nu}+4p_{\mu}p_{\nu}\right)I_{log}(m^{2})-\frac{\eta_{\mu\nu}}{12}(6\upsilon_{2}-3p^{2}\upsilon_{0}+2p^{2}\xi_{0})-\frac{p_{\mu}p_{\nu}}{3}\xi_{0}+$
(13) $\displaystyle+$
$\displaystyle\frac{i}{(4\pi)^{2}}\left\\{\frac{1}{12p^{2}}\left[(p^{2}-4m^{2})p^{2}\eta_{\mu\nu}-4(p^{2}-m^{2})p_{\mu}p_{\nu}\right]Z_{0}(p^{2},m^{2},m^{2},m^{2})+\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{18}(p_{\mu}p_{\nu}-p^{2}\eta_{\mu\nu})\right\\},$

where

$Z_{k}(p^{2},m_{1}^{2},m_{2}^{2},m_{3}^{2})=\int_{0}^{1}dz\,\,z^{k}\ln{\left\\{\frac{p^{2}z(1-z)+(m_{1}^{2}-m_{2}^{2})z-m_{1}^{2}}{(-m_{3}^{2})}\right\\}}.$
(14)

The same procedure is used for calculating the other Feynman integrals. We
obtain, for the vacuum polarization tensor,

$\displaystyle-i\Pi^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{4}{3}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\left\\{I_{log}(m^{2})-\frac{i}{(4\pi)^{2}}\left[\frac{(p^{2}+2m^{2})}{p^{2}}Z_{0}(p^{2},m^{2})+\frac{1}{3}\right]\right\\}+$
(15) $\displaystyle-$ $\displaystyle
4\upsilon_{2}\eta^{\mu\nu}+\frac{4}{3}\left\\{\upsilon_{0}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})-(2p^{\mu}p^{\nu}+p^{2}\eta^{\mu\nu})(\xi_{0}-2\upsilon_{0})\right\\},$

in which, for economy, we used $Z_{k}(p^{2},m^{2},m^{2},m^{2})\equiv
Z_{k}(p^{2},m^{2})$. Note that the amplitude is transversal if
$\upsilon_{2}=0$ and $\xi_{0}=2\upsilon_{0}$. This approach of Implicit
Regularization, in which the surface terms are parametrized and fixed in the
end is useful when the model under investigation presents ambiguities as in
the cases of chiral anomalies or topological field theories Viglioni . The
Constrained Implicit Regularization, on the other hand, fixes the surface
terms to zero from the beginning. This automatically delivers symmetric
results, as it has been shown for Abelian and non-Abelian gauge theories and
for supersymmetric models.

##  III A new approach for implementing Constrained Implicit Regularization

The basic idea of Constrained Implicit Regularization, as we already
discussed, is to assume the presence of a regularization with the aim of using
mathematical identities in order to separate the regularization dependent from
the finite part. The divergent part is a combination of scalar basic
divergences, which are obtained after the use of consistency relations that
eliminate surface terms. Here, we propose a procedure for applying Constrained
Implicit Regularization which simplifies enormously the process of
calculation. The steps are listed below:

  1. 1.

as in the original procedure, a regularization scheme is assumed to be acting
in the complete amplitude, with the same desirable characteristics already
presented;

  2. 2.

Feynman parametrization is applied to the Feynman integrals. It can be applied
to the complete amplitude. When it is carried out in the complete amplitude,
the needed shift in the momentum of integration is just a modification in the
loop momentum;

  3. 3.

the algebra of the group of symmetry is carried out. In the case the integrals
are treated separately, the algebra is performed before Feynman
parametrization;

  4. 4.

the integrals in the momenta are separated by degree of divergence, all with
even powers of the integration momentum in the numerator, of the type

$\int^{R}\frac{d^{4}k}{(2\pi)^{4}}\frac{1,k_{\mu}k_{\nu},k_{\mu}k_{\nu}k_{\alpha}k_{\beta},\cdots}{(k^{2}+H^{2})^{n}},$
(16)

being $H^{2}$ function of the external momenta, of the masses and of the
Feynman parameters. If factors of $k^{2}$ appear in the numerator, they should
be canceled with factors in the denominator by adding and subtracting $H^{2}$;

  5. 5.

for the divergent parts, the surface terms are eliminated by means of the
consistency relations, in order to obtain scalar integrals. For one-loop
logarithmically and quadratically divergent integrals, we set, respectively,

$\int^{R}\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\mu_{1}}k_{\mu_{2}}\cdots
k_{\mu_{n}}}{(k^{2}+H^{2})^{2+\frac{n}{2}}}=\frac{\eta_{\mu_{1}\mu_{2}\cdots\mu_{n}}}{2^{\frac{n}{2}}(\frac{n}{2}+1)!}\left\\{\int^{R}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}+H^{2})^{2}}-\alpha_{\frac{n}{2}}\right\\}$
(17)

and

$\int^{R}\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\mu_{1}}k_{\mu_{2}}\cdots
k_{\mu_{n}}}{(k^{2}+H^{2})^{1+\frac{n}{2}}}=\frac{\eta_{\mu_{1}\mu_{2}\cdots\mu_{n}}}{2^{\frac{n}{2}}(\frac{n}{2})!}\left\\{\int^{R}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}+H^{2})}-\beta_{\frac{n}{2}}\right\\},$
(18)

in which $n$ is even and $\eta_{\mu_{1}\mu_{2}\cdots\mu_{n}}$ is the symmetric
combination of the products of metric tensors,
$\eta_{\mu_{1}\mu_{2}}\cdots\eta_{\mu_{n-1}\mu_{n}}$, with coefficient $1$. In
the expression above, we left the parameters for the surface terms
$\alpha_{\frac{n}{2}}$ and $\beta_{\frac{n}{2}}$ only for completeness, since
in the constrained version of IReg they are fixed null. In order to obtain the
relations above, we use recursively the relation,

$\int_{k}^{R}\frac{\partial}{\partial
k^{\mu_{n}}}\left(\frac{k_{\mu_{1}}\cdots
k_{\mu_{n-1}}}{(k^{2}+H^{2})^{m-1}}\right)=\int_{k}^{R}\frac{{\cal
S}[\eta_{\mu_{1}\mu_{n}}k_{\mu_{2}}\cdots
k_{\mu_{n-1}}]}{(k^{2}+H^{2})^{m-1}}-2(m-1)\int_{k}^{R}\frac{k_{\mu_{1}}\cdots
k_{\mu_{n}}}{(k^{2}+H^{2})^{m}},$ (19)

until the first integral of the second member of the equation is scalar. Apart
two integrals, including the scalar one, all the others will be surface terms
and are gathered in one parameter which will be fixed null. In the equation
above, we define ${\cal S}[T_{\mu_{1}\cdots\mu_{n}}]$ as the minimal
symmetrization of the tensor $T$, in the sense that only distinct terms are
considered and all of them have coefficient one. For example, ${\cal
S}[k_{\mu}k_{\nu}p_{\alpha}]=k_{\mu}k_{\nu}p_{\alpha}+k_{\mu}k_{\alpha}p_{\nu}+k_{\nu}k_{\alpha}p_{\mu}$.
Important to note that $m=\frac{n}{2}+2$ for logarithmic divergences and
$m=\frac{n}{2}+1$ for quadratic divergences. By the definitions of equation
(5), the remaining scalar divergences above are $I_{log}(-H^{2})$ and
$I_{quad}(-H^{2})$.

  6. 6.

in complement to the previous step, since in CIR surface terms are set to
zero, shifts in divergent integrals are allowed;

  7. 7.

the next step, which is one of the basic ideas of IReg, is the use of
algebraic identities in order to get the divergent integrals free from the
external momenta. Here, we use recursively a simpler expansion,

$\frac{1}{(k^{2}+H^{2})}=\frac{1}{(k^{2}-\lambda^{2})}-\frac{\lambda^{2}+H^{2}}{(k^{2}-\lambda^{2})(k^{2}+H^{2})}.$
(20)

We have the advantage of obtaining closed expressions to be used in any
calculation:

$I_{log}(-H^{2})=I_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\ln{\left(-\frac{H^{2}}{\lambda^{2}}\right)}$
(21)

and

$I_{quad}(-H^{2})=I_{quad}(\lambda^{2})-(\lambda^{2}+H^{2})I_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\left[\lambda^{2}+H^{2}-H^{2}\ln{\left(-\frac{H^{2}}{\lambda^{2}}\right)}\right].$
(22)

These are called scale relations, which, as a byproduct, introduce an energy
scale for the renormalization group, $\lambda^{2}$, that can be simply one of
the masses of the model. The basic divergences are now factorized out of the
integrals in the Feynman parameters, which can be computed. As in the
traditional formulation of IReg, the divergent part of the amplitudes is
written in terms of these basic divergences: $I_{log}(\lambda^{2})$,
$I_{quad}(\lambda^{2})$, etc.

The implementation of the above steps simplifies a lot the calculations of the
finite parts. An additional advantage is related to models which present
fields with different masses or non-massive fields. In non-massive models, as
long as the off-shell amplitude is infrared finite, the traditional procedure
requires that a fictitious mass is introduced in the propagator so as the
expansion in the integrand can be carried out. At the end of the calculation,
the scale relations are used to remove the fictitious mass from the divergent
part so that the limit of the mass going to zero can be taken. Here, the
procedure is unified, since all the mass dependence is inside $H^{2}$.

It is important to carry out a discussion on the possibility of applying
Feynman parametrization before the use of the identities of Implicit
Regularization. Normally, in regularization processes, the amplitude is
regularized and, then, considering that the integral is well defined, the
Feynman parametrization is carried out. In Dimensional Regularization, for
example, the amplitude is extended to dimension $d$ and, after this procedure,
the parametrization is promptly applied. The key point in Feynman
parametrization is the shift, $k\to k+f(p_{i},x_{i})$ ($f(p_{i},x_{i})$ is a
linear function of the external momenta $p_{i}$ and of the parameters
$x_{i}$), which is carried out to solve the $k$ integral. When the integral is
assumed to be finite, this shift is allowed, since it will not generate any
surface term. In the case of IReg, there is not an explicit regularization
applied. So, in the more general approach of IReg, care is taken that surface
terms are not discarded. Therefore, expansions are performed on the integrand
in order to separate the divergent parts from the finite ones. The Feynman
parametrization is then applied only to the finite part, which will not
produce surface terms. The surface terms are thus identified and parametrized
in the consistency relations used to reduce the tensorial divergent basic
integrals to scalar ones. Part of the surface terms that are maintained and
parametrized in the more general IReg is related with the momentum routing in
the loops. When Feynman parametrization is performed in the whole amplitude
before the expansion of IReg is carried out, this part is lost, as Feynman’s
post-parametrization shift just eliminates the routing dependency, as it is
evident in the following calculation. Therefore, the steps described above
exactly implement the Constrained Implicit Regularization, in which all the
surface terms are set to zero. The identities used in constrained IReg, then,
commute with the Feynman parametrization of divergent integrals.

Just as an example, we show below, for a simple linearly divergent integral,
the equivalence of applying the Feynman parameterization before or after the
expansion of the integrals. Let us consider the massless integral $I_{\mu}$
and its known result within Implicit Regularization,

$I_{\mu}=\int_{k}^{R}\frac{k_{\mu}}{k^{2}(p-k)^{2}}=\frac{p_{\mu}}{2}\left\\{I_{log}(\lambda^{2})+\upsilon_{0}+\frac{i}{16\pi^{2}}\left[2-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]\right\\},$
(23)

being $\upsilon_{0}$ the surface term defined in (11). We now solve this same
integral by applying Feynman parametrization before. We have

$I_{\mu}=\int_{0}^{1}\,dx\,\int_{k}^{R}\frac{k_{\mu}}{\left[(k-px)^{2}+H^{2}\right]^{2}},$
(24)

with $H^{2}=p^{2}x(1-x)$, and in which we avoided to carry out the traditional
shift. Now the integral in momentum should be compared with the also known
result within Implicit Regularization,

$\displaystyle
I_{\mu}(p_{1},p_{2},m^{2})=\int_{k}^{R}\frac{k_{\mu}}{[(k-p_{1})^{2}-m^{2}][(k-p_{2})^{2}-m^{2}]}=\frac{(p_{1}+p_{2})_{\mu}}{2}\left\\{I_{log}(m^{2})+\upsilon_{0}+\right.$
$\displaystyle-\left.\frac{i}{16\pi^{2}}\int_{0}^{1}\,du\,\ln\left[\frac{(p_{1}-p_{2})^{2}u(1-u)-m^{2}}{(-m^{2})}\right]\right\\}.$
(25)

We then identify the $k$ integral of (24) with

$\displaystyle I_{\mu}(xp,xp,-H^{2})$ $\displaystyle=$ $\displaystyle
xp_{\mu}\left\\{I_{log}(-H^{2})+\upsilon_{0}-\frac{i}{16\pi^{2}}\int_{0}^{1}\,du\,\ln\left(\frac{H^{2}}{H^{2}}\right)\right\\}$
(26) $\displaystyle=$ $\displaystyle
xp_{\mu}\left[I_{log}(-H^{2})+\upsilon_{0}\right].$

Substituting this in (24), we have

$\displaystyle I_{\mu}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\,dx\,xp_{\mu}\left[I_{log}(-H^{2})+\upsilon_{0}\right]=\int_{0}^{1}\,dx\,xp_{\mu}\left[I_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\ln\left(-\frac{H^{2}}{\lambda^{2}}\right)+\upsilon_{0}\right]$
(27) $\displaystyle=$
$\displaystyle\frac{p_{\mu}}{2}\left\\{I_{log}(\lambda^{2})+\upsilon_{0}+\frac{i}{16\pi^{2}}\left[2-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]\right\\},$

that is the same result as in the calculation in which the Feynman
parametrization is carried out only after the expansion. It is important to
note that the effect of making the shift $k\to k+px$ in (24) is the automatic
discard of the surface term $\upsilon_{0}$.

Let us now carry out an example of calculation which is a little more involved
than the QED vacuum polarization tensor and consider the vector-field self-
energy in which the two fermions in the loop have different masses. It is the
case of the influence of the heavy quarks, the doublet $(t,b)$, in the
corrections to the $W$-boson mass (see, for example, Hollik ). The
corresponding Feynman graph is depicted in Figure 1. Let us also, for
pedagogical reasons, assign an arbitrary distribution of momenta in the
internal lines. The amplitude is proportional to

Figure 1: Diagram which contributes to the self-energy of a massive vector-
field. The fermionic fields in the loop have different masses and the momentum
routing in the amplitude reads $k+(\alpha-1)p$ for the propagator with mass
$m_{1}$ and $k+\alpha p$ for the propagator with mass $m_{2}$. The wavy and
solid lines represent the vectorial and fermionic fields, respectively.

$\Pi^{\mu\nu}(p,m_{1},m_{2})=\int_{k}^{R}\frac{\mbox{tr}\left\\{\gamma^{\mu}[\hbox{$k\\!\\!\\!{/}$}+(\alpha-1)\hbox{$p\\!\\!\\!{/}$}+m_{2}]\gamma^{\nu}[\hbox{$k\\!\\!\\!{/}$}+\alpha\hbox{$p\\!\\!\\!{/}$}+m_{1}]\right\\}}{[(k+\alpha
p)^{2}-m_{1}^{2}]\\{[k+(\alpha-1)p]^{2}-m_{2}^{2}\\}},$ (28)

which after Feynman parametrization, with the shift $k\to k+(x-\alpha)p$,
reads

$\Pi^{\mu\nu}(p,m_{1},m_{2})=\int_{0}^{1}dx\,\int_{k}^{R}\frac{\mbox{tr}\left\\{\gamma^{\mu}[\hbox{$k\\!\\!\\!{/}$}+(x-1)\hbox{$p\\!\\!\\!{/}$}+m_{2}]\gamma^{\nu}[\hbox{$k\\!\\!\\!{/}$}+x\hbox{$p\\!\\!\\!{/}$}+m_{1}]\right\\}}{(k^{2}+H^{2})^{2}},$
(29)

with $H^{2}=p^{2}x(1-x)+(m_{1}^{2}-m_{2}^{2})x-m_{1}^{2}$. We note that the
entire dependence on the $\alpha$ parameter disappeared. In other words, when
the total amplitude is Feynman parametrized together, the amplitude is
automatically momentum-routing invariant. After calculating the trace, with
only even terms in $k^{\mu}$ remaining, and canceling terms in $k^{2}$ by
adding and subtracting $H^{2}$, we stay with

$\displaystyle\Pi^{\mu\nu}(p,m_{1},m_{2})=4\int_{0}^{1}dx\,\left\\{-\eta^{\mu\nu}\int_{k}^{R}\frac{1}{k^{2}+H^{2}}+2\int_{k}^{R}\frac{k^{\mu}k^{\nu}}{(k^{2}+H^{2})^{2}}+\right.$
$\displaystyle+\left.\left\\{2(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})x(1-x)+(m_{1}-m_{2})[(m_{1}+m_{2})x-m_{1}]\eta^{\mu\nu}\right\\}\int_{k}^{R}\frac{1}{(k^{2}+H^{2})^{2}}\right\\}.$
(30)

The first two integrals, which are quadratically divergent, cancel out if we
use equation (10) and fix $\upsilon_{2}=0$, as it is prescribed by constrained
IReg. For the remaining $I_{log}(-H^{2})$, we use the scale relation (21), so
that the amplitude is written as

$\displaystyle\Pi^{\mu\nu}(p,m_{1},m_{2})$ $\displaystyle=$ $\displaystyle
4\int_{0}^{1}dx\,\left\\{2(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})x(1-x)+(m_{1}-m_{2})[(m_{1}+m_{2})x-m_{1}]\eta^{\mu\nu}\right\\}$
(31) $\displaystyle\times$
$\displaystyle\left\\{I_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\ln{\left(-\frac{H^{2}}{\lambda^{2}}\right)}\right\\}.$

Finally, we obtain

$\displaystyle\Pi^{\mu\nu}(p,m_{1},m_{2})=8(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\left\\{\frac{1}{6}I_{log}(\lambda^{2})-\frac{i}{(4\pi)^{2}}(\tilde{Z}_{1}-\tilde{Z}_{2})\right\\}$
$\displaystyle+4(m_{1}-m_{2})\eta^{\mu\nu}\left\\{-\frac{1}{2}(m_{1}-m_{2})I_{log}(\lambda^{2})-\frac{i}{(4\pi)^{2}}[(m_{1}+m_{2})\tilde{Z}_{1}-m_{1}\tilde{Z}_{0}]\right\\},$
(32)

where $\tilde{Z}_{k}$ is a short for
$Z_{k}(p^{2},m_{1}^{2},m_{2}^{2},\lambda^{2})$. Since the vector field is
massive, the polarization tensor is not transverse. Rather, the amplitude
obeys the relation

$\displaystyle p_{\nu}\Pi^{\mu\nu}$ $\displaystyle=$ $\displaystyle
4(m_{1}-m_{2})p^{\mu}\left\\{-\frac{1}{2}(m_{1}-m_{2})I_{log}(\lambda^{2})-\frac{i}{(4\pi)^{2}}[(m_{1}+m_{2})\tilde{Z}_{1}-m_{1}\tilde{Z}_{0}]\right\\}$
(33) $\displaystyle=$ $\displaystyle(m_{1}-m_{2})T^{\mu},$

where

$T^{\mu}=\int_{k}^{R}\frac{\mbox{tr}\left\\{\gamma^{\mu}(\hbox{$k\\!\\!\\!{/}$}-\hbox{$p\\!\\!\\!{/}$}+m_{2})(\hbox{$k\\!\\!\\!{/}$}+m_{1})\right\\}}{(k^{2}-m_{1}^{2})[(k-p)^{2}-m_{2}^{2}]}.$
(34)

This is a very direct and compact calculation. Note that the procedure would
be identical in the case of a off-shell calculation in non-massive QED, with
the obvious modification of $H^{2}$, with only the transverse part remaining.

The Ward identity in eq. (33) is a particular case of a diagrammatic relation,
as depicted in Figure 2. This diagrammatic relation is respected as long as
the regularization applied is compatible with shifts in the momentum of
integration (see a good discussion in Bruque ). In the abelian case, QED for
example, gauge invariance is fulfilled if and only if momentum routing
invariance is as well, as we can easily see in Figure 2$(a)$. In the approach
we present in this work, the shifts we perform in Feynman parametrization is
already an assumption of momentum routing invariance. This already delivers
gauge invariant results. Furthermore, when considering the electroweak theory
as a whole, we have an additional relation as in Fig. 2$(b)$ due to the change
of flavors and we recover QED when $m_{1}=m_{2}=m$. In this case, mass terms
have already broken the larger gauge symmetry $SU(2)\otimes U(1)$ into simply
$U(1)$.

Figure 2: (a) Diagrammatic representation of the gauge and momentum routing
invariance relation for QED. (b) The same diagrammatic representation for
fermions with different masses.

It is important to comment on the fact that some aspects of our approach are
similar to procedures adopted in Loop Regularization (LORE). In the case of
LORE, after the amplitude is Feynman parametrized, consistency conditions are
applied which, in practice, discard surface terms. Such conditions, which are
the same of IReg, were determined, in that case, by the requirement that
symmetries be respected in specific amplitudes Lore-1 and then generalized.
The remaining scalar divergent loop integrals, however, are calculated using a
procedure which is similar to the Pauli-Villars regularization Pauli-Villars .
On the other hand, Implicit Regularization is based on the elimination of
surface terms and in the expansion of the integrand so as the renormalization
needs only local counterterms. The remaining divergent integrals do not need
be explicitly calculated.

In the next section, we present a systematization for calculations of general
one-loop divergent amplitudes.

##  IV A systematization for the calculation of one-loop integrals

We present now a systematization of the calculation of one-loop Feynman
integrals in the framework of this new approach for implementing Implicit
Regularization. The procedure is very interesting, since the results encompass
the finite parts. The methodology we carry out in this section applies to
integrals rather than in the complete amplitude. It is not a problem, since
different Feynman parametrization in the integrals that are part of the
amplitude has implications only in the surface terms which are fixed zero in
constrained IReg.

Let us begin with a general one-loop integral with logarithmic degree of
divergence, which is written as

$I^{(0)}_{\mu_{1}\cdots\mu_{n}}=\int_{k}^{R}\frac{k_{\mu_{1}}\cdots
k_{\mu_{n}}}{(k^{2}-m^{2})[(p_{1}-k)^{2}-m_{1}^{2}]\cdots[(p_{r}-k)^{2}-m_{r}^{2}]},$
(35)

with $r=1+\frac{n}{2}$. The first step is to carry out the Feynman
parametrization. We use

$\frac{1}{a_{1}\cdots
a_{r}b}=r!\int_{0}^{1}dx_{1}\int_{0}^{1-x_{1}}dx_{2}\cdots\int_{0}^{1-\sum_{i=1}^{r-1}x_{i}}dx_{r}\frac{1}{\,\,\,\Big{[}\sum_{k=1}^{r}(a_{k}-b)x_{k}+b\Big{]}^{r+1}},$
(36)

with $a_{k}=[(p_{k}-k)^{2}-m_{k}^{2}]$ and $b=(k^{2}-m^{2})$. Considering the
denominator is given by $D^{r+1}$, it is possible to rearrange $D$ in order to
write

$D=\left[k-\sum_{k=1}^{r}p_{k}x_{k}\right]^{2}+Q^{2},$ (37)

with

$Q^{2}=\sum_{k=1}^{r}\left[p_{k}^{2}x_{k}(1-x_{k})+(m^{2}-m_{k}^{2})x_{k}\right]-\sum_{k\neq
l}(p_{k}\cdot p_{l})x_{k}x_{l}-m^{2}$ (38)

We now perform a shift in the integral in the momentum: $k\to k+q$, with
$q_{\mu}=\left[\sum_{k=1}^{r}p_{k}x_{k}\right]_{\mu}$, and we get, in the
numerator,

$N_{\mu_{1}\cdots\mu_{n}}=\prod_{i=1}^{n}(k+q)_{\mu_{i}},$ (39)

from which only the even powers of $k$ survive, let us call it
$\tilde{N}_{\mu_{1}\cdots\mu_{n}}$. Our logarithmic divergent amplitude is
then given by

$I^{(0)}_{\mu_{1}\cdots\mu_{n}}=r!\int
dX\int_{k}^{R}\frac{\tilde{N}_{\mu_{1}\cdots\mu_{n}}}{\,\,\,[k^{2}+Q^{2}]^{r+1}},$
(40)

where $\int dX$ stands for all the integrals in the Feynman parameters. The
higher power in $k$ in $\tilde{N}_{\mu_{1}\cdots\mu_{n}}$ is responsible for
the logarithmic divergence. All the other terms are finite. For the divergent
part, we have

$\displaystyle I^{(0)\Lambda}_{\mu_{1}\cdots\mu_{n}}$ $\displaystyle=$
$\displaystyle r!\int dX\,\int_{k}^{R}\frac{k_{\mu_{1}}\cdots
k_{\mu_{n}}}{\,\,\,(k^{2}+Q^{2})^{r+1}}$ (41) $\displaystyle=$
$\displaystyle\frac{\eta_{\mu_{1}\cdots\mu_{n}}}{2^{\frac{n}{2}}}\int
dX\,\int_{k}^{R}\frac{1}{(k^{2}+Q^{2})^{2}},$

in which we have used the consistency relation of (17) and the fact that
$r=\frac{n}{2}+1$. Next, we use the the scale relation (21) to obtain

$\displaystyle
I^{(0)\Lambda}_{\mu_{1}\cdots\mu_{n}}=\frac{\eta_{\mu_{1}\cdots\mu_{n}}}{2^{\frac{n}{2}}}\int
dX\,\left\\{I_{log}(m^{2})-\frac{i}{16\pi^{2}}\ln{\left(-\frac{Q^{2}}{\lambda^{2}}\right)}\right\\}.$
(42)

The unique part which is dependent of the Feynman parameters is the one that
contains $Q^{2}$. The other can be factorized out of the integral. The final
result is given by

$I^{\Lambda}_{\mu_{1}\cdots\mu_{n}}=\frac{\eta_{\mu_{1}\cdots\mu_{n}}}{2^{\frac{n}{2}}}\left\\{\frac{1}{r!}I_{log}(m^{2})-\frac{i}{16\pi^{2}}Z^{(r,0)}\right\\},$
(43)

in which we define the function

$Z_{\mu_{1}\cdots\mu_{i}}^{(r,k_{1},\cdots,k_{r})}=Z_{\mu_{1}\cdots\mu_{i}}^{(r,k_{1},\cdots,k_{r})}(p_{1},\cdots,p_{r},m_{1}^{2},\cdots,m_{r}^{2},m^{2})\equiv\int
dX\,q_{\mu_{1}}\cdots q_{\mu_{i}}x_{1}^{k_{1}}\cdots
x_{r}^{k_{r}}\ln{\left(-\frac{Q^{2}}{m^{2}}\right)}.$ (44)

We now have to take care of the other finite terms, which appears if $n\geq
2$. A typical finite term of (40) is

$F^{(l)}_{\mu_{1}\cdots\mu_{n}}=r!\int dX\,{\cal
S}\left[A_{\mu_{1}\cdots\mu_{l}}q_{\mu_{l+1}}\cdots q_{\mu_{n}}\right],$ (45)

with $l<n$ even and

$A_{\mu_{1}\cdots\mu_{l}}=\int_{k}\frac{k_{\mu_{1}}\cdots
k_{\mu_{l}}}{\,\,\,(k^{2}+Q^{2})^{r+1}}=\frac{i}{16\pi^{2}}\frac{\Gamma(\frac{n-l}{2})}{2^{\frac{l}{2}}\Gamma(r+1)}\frac{1}{\,\,\,(Q^{2})^{\frac{n-l}{2}}}\eta_{\mu_{1}\cdots\mu_{l}}.$
(46)

We, then, stay with

$\displaystyle F^{(l)}_{\mu_{1}\cdots\mu_{n}}$ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}\frac{\Gamma(\frac{n-l}{2})}{2^{\frac{l}{2}}}\,{\cal
S}\left[\eta_{\mu_{1}\cdots\mu_{l}}\,\int dX\,\frac{q_{\mu_{l+1}}\cdots
q_{\mu_{n}}}{(Q^{2})^{\frac{n-l}{2}}}\right]$ (47) $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}\frac{\Gamma(\frac{n-l}{2})}{2^{\frac{l}{2}}}{\cal
S}\left[\eta_{\mu_{1}\cdots\mu_{l}}Y_{\mu_{l+1}\cdots\mu_{n}}^{\left(r,\frac{n-l}{2}\right)}\right],$

where

$Y_{\mu_{1}\cdots\mu_{n}}^{(r,l)}=Y_{\mu_{1}\cdots\mu_{n}}^{(r,l)}(p_{1},\cdots,p_{r},m_{1}^{2},\cdots,m_{r}^{2},m^{2})\equiv\int
dx_{1}\cdots dx_{r}\,\frac{q_{\mu_{1}}\cdots q_{\mu_{n}}}{(Q^{2})^{l}}.$ (48)

Finally, we can write the general result as

$\displaystyle I^{(0)}_{\mu_{1}\cdots\mu_{n}}$ $\displaystyle=$
$\displaystyle\frac{\eta_{\mu_{1}\cdots\mu_{n}}}{2^{\frac{n}{2}}}\left\\{\frac{1}{r!}I_{log}(m^{2})-\frac{i}{16\pi^{2}}Z^{(r,0)}\right\\}+$
(49) $\displaystyle+$
$\displaystyle\frac{i}{16\pi^{2}}\sum_{i=0}^{\frac{n-2}{2}}\frac{\Gamma(\frac{n-2i}{2})}{2^{i}}{\cal
S}\left[\eta_{\mu_{1}\cdots\mu_{2i}}Y_{\mu_{2i+1}\cdots\mu_{n}}^{\left(r,\frac{n-2i}{2}\right)}\right];\,\,\,\,\,\,\,\,\,\,\,\,r=\frac{n}{2}+1,$

in which we substituted $l$ by $2i$, since $l$ is even.

For a linearly divergent integral,

$I^{(1)}_{\mu_{1}\cdots\mu_{n}}=\int_{k}^{R}\frac{k_{\mu_{1}}\cdots
k_{\mu_{n}}}{(k^{2}-m^{2})[(p_{1}-k)^{2}-m_{1}^{2}]\cdots[(p_{r}-k)^{2}-m_{r}^{2}]},$
(50)

with $n$ odd and $r=\frac{n+1}{2}$, the procedure is very similar, with the
result

$\displaystyle I^{(1)}_{\mu_{1}\cdots\mu_{n}}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{\frac{n-1}{2}}}{\cal
S}\left[\sum_{k=1}^{r}p_{k\mu_{1}}\eta_{\mu_{2}\cdots\mu_{n}}\right]\frac{1}{(r+1)!}I_{log}(m^{2})-\frac{i}{16\pi^{2}}\frac{1}{2^{\frac{n-1}{2}}}{\cal
S}\left[Z_{\mu_{1}}^{(r,0)}\eta_{\mu_{2}\cdots\mu_{n}}\right]+$ (51)
$\displaystyle+$
$\displaystyle\frac{i}{16\pi^{2}}\sum_{i=0}^{\frac{n-3}{2}}\frac{\Gamma(\frac{n-2i-1}{2})}{2^{(2i+1)/2}}{\cal
S}\left[\eta_{\mu_{1}\cdots\mu_{2i}}Y_{\mu_{2i+1}\cdots\mu_{n}}^{\left(r,\frac{n-2i-1}{2}\right)}\right];\,\,\,\,\,\,\,\,\,\,\,\,r=\frac{n+1}{2},$

in which the last term only appears for $n\geq 3$.

In order to complete the systematization of the calculation of one-loop
amplitudes, we turn our attention now to quadratically divergent integrals,
$I^{(2)}_{\mu_{1}\cdots\mu_{n}}$. Since the calculation is a little more
involved, we will show below the main steps. The integral to be calculated has
the same form as (35), but with $r=\frac{n}{2}$, $n$ even. After Feynman
parametrization, we obtain the expression of (40), which will be broken in
three parts: the higher power in $k$ in the numerator is quadratically
divergent; the $(n-2)$-th power in $k$ in the numerator is logarithmically
divergent; and the $(n-4)$-th power and lower, if they exist, are finite. It
is important to note that even the two divergent peaces contribute to the
finite result, as it is evident from the calculations of the last section.

Let us begin with the quadratic divergence:

$\displaystyle I^{(2),1}_{\mu_{1}\cdots\mu_{n}}=\left(\frac{n}{2}\right)!\int
dX\,\int_{k}^{R}\frac{k_{\mu_{1}}\cdots
k_{\mu_{n}}}{\,\,\,(k^{2}+Q^{2})^{\frac{n}{2}+1}}=\frac{\eta_{\mu_{1}\cdots\mu_{n}}}{2^{\frac{n}{2}}}\int
dX\,\int_{k}^{R}\frac{1}{(k^{2}+Q^{2})},$ (52)

in which we made use of (18). Next, we resort to the scale relation of
equation (22) to obtain

$\displaystyle
I^{(2),1}_{\mu_{1}\cdots\mu_{n}}=\frac{\eta_{\mu_{1}\cdots\mu_{n}}}{2^{\frac{n}{2}}}\int
dX\,\left\\{I_{quad}(m^{2})-(m^{2}+Q^{2})\left[I_{log}(m^{2})+\frac{i}{16\pi^{2}}\right]+\frac{i}{16\pi^{2}}Q^{2}\ln{\left(-\frac{Q^{2}}{m^{2}}\right)}\right\\}$
$\displaystyle=\frac{\eta_{\mu_{1}\cdots\mu_{n}}}{2^{\frac{n}{2}}}\left\\{\frac{1}{\left(\frac{n}{2}\right)!}I_{quad}(m^{2})-\frac{1}{\left(\frac{n}{2}+2\right)!}\left\\{\sum_{k=1}^{r}\left[\frac{n}{2}p_{k}^{2}+\left(\frac{n}{2}+2\right)(m^{2}-m_{k}^{2})\right]+\right.\right.$
$\displaystyle\left.\left.-\sum_{k\neq l}(p_{k}\cdot
p_{l})\right\\}\left[I_{log}(m^{2})+\frac{i}{16\pi^{2}}\right]+\frac{1}{16\pi^{2}}\int
dX\,Q^{2}\ln{\left(-\frac{Q^{2}}{m^{2}}\right)}\right\\},$ (53)

where the last integral can be written in terms of
$Z^{(r,k_{1},\cdots,k_{r})}$ functions.

For the second part, which is logarithmically divergent, we have

$\displaystyle I^{(2),2}_{\mu_{1}\cdots\mu_{n}}=\left(\frac{n}{2}\right)!\int
dX\,{\cal S}\left[q_{\mu_{1}}q_{\mu_{2}}\int_{k}\frac{k_{\mu_{3}}\cdots
k_{\mu_{n}}}{\,\,\,(k^{2}+Q^{2})^{\frac{n}{2}+1}}\right]$
$\displaystyle=\frac{1}{2^{\frac{n-2}{2}}}\int dX\,{\cal
S}[q_{\mu_{1}}q_{\mu_{2}}\eta_{\mu_{3}\cdots\mu_{n}}]\left\\{I_{log}(m^{2})-\frac{i}{16\pi^{2}}\ln{\left(-\frac{Q^{2}}{m^{2}}\right)}\right\\}$
$\displaystyle=\frac{1}{2^{\frac{n-2}{2}}}{\cal
S}\left[\eta_{\mu_{3}\cdots\mu_{n}}\left\\{\frac{1}{\left(\frac{n}{2}+2\right)!}\left[2\sum_{k=1}^{n/2}p_{k\mu_{1}}p_{k\mu_{2}}+\sum_{k\neq
l}p_{k\mu_{1}}p_{l\mu_{2}}\right]I_{log}(m^{2})-\frac{i}{16\pi^{2}}Z^{(n/2,0)}_{\mu_{1}\mu_{2}}\right\\}\right].$
(54)

The remaining finite part is given by

$\displaystyle
I^{(2),3}_{\mu_{1}\cdots\mu_{n}}=\left(\frac{n}{2}\right)!\sum_{l=4}^{n}\int
dX\,{\cal S}\left[q_{\mu_{1}}\cdots
q_{\mu_{l}}\int_{k}\frac{k_{\mu_{l+1}}\cdots
k_{\mu_{n}}}{\,\,\,(k^{2}+Q^{2})^{\frac{n}{2}+1}}\right]$
$\displaystyle=\frac{i}{16\pi^{2}}\sum_{l=4}^{n}\frac{\Gamma\left(\frac{l}{2}-1\right)}{2^{(n-l)/2}}\int
dX\,{\cal S}\left[q_{\mu_{1}}\cdots
q_{\mu_{l}}\eta_{\mu_{l+1}\cdots\mu_{n}}\right]\frac{1}{(Q^{2})^{\frac{l}{2}-1}}$
$\displaystyle=\frac{i}{16\pi^{2}}\sum_{i=2}^{n/2}\frac{\Gamma(i-1)}{2^{(n-2i)/2}}{\cal
S}\left[\eta_{\mu_{2i+1}\cdots\mu_{n}}Y_{\mu_{1}\cdots\mu_{2i}}^{(m,i-1)}\right].$
(55)

The total result for the quadratically divergent integral is given by
$I^{(2)}_{\mu_{1}\cdots\mu_{n}}=I^{(2),1}_{\mu_{1}\cdots\mu_{n}}+I^{(2),2}_{\mu_{1}\cdots\mu_{n}}+I^{(2),3}_{\mu_{1}\cdots\mu_{n}}$.

##  V Directions on higher order calculations

There is currently a high demand for theoretical predictions for processes at
next-to-next-to-leading order (NNLO) and beyond, mainly due to the large
amount of data which has already been collected at LHC. With this aim, new
calculation techniques have been developed in recent years, which seek, as far
as possible, to preserve the physical dimension of the spacetime NNLO .
Higher-loop calculations are usually very long and intricate and friendly
procedures are welcome. There are very nice techniques to implement multiloop
calculations in quantum field theory, mainly with the use of Dimensional
Regularization (see, for example, Grozin ). In the context of this new
approach for implementing constrained IReg, we give some directions for future
systematization of multi-loop calculations. Before starting, we note that it
is not always possible, in higher-order calculations, to make a single Feynman
parametrization for the entire amplitude, as in the case of non-planar graphs.
However, as long as shifts in the momenta of integration are allowed in all
steps, momentum routing invariance will be ensured.

In the last section, we obtained general finite parts which are integrals, in
the Feynman parameters, which contain factors of $\ln{[Q^{2}/(-m^{2})]}$ or
powers of $1/Q^{2}$. The challenging part is the one which includes the
logarithm. With the aim of applying Feynman parametrization, we will make use
of the following mathematical identity:

$\ln{a}=\lim_{\varepsilon\to 0}\frac{1}{\varepsilon}(a^{\varepsilon}-1).$ (56)

In a simple view, we say that $\ln{a}$ is equal to the first order coefficient
of the expansion of $a^{\varepsilon}$ in the limit of small $\varepsilon$.
This trick in the context of Feynman integrals at higher loop orders, as far
as we know, was first used in ir11 and irn to calculate finite parts.

So, let us give an example in which the one-loop finite part is an integral of
a function with a factor of a logarithm. We consider the two-loop nested self-
energy of the electron in QED, which is depicted in Figure 3.

Figure 3: Diagrammatic representation of the nested contribution to the
electron self-energy at two-loop order. The wavy and solid lines represent the
photon and fermion propagators, respectively.

The finite part of the subgraph is given by

$i\tilde{\Sigma}^{(1)}=2q^{2}\frac{i}{16\pi^{2}}\int_{0}^{1}dx\,[2m-\hbox{$p\\!\\!\\!{/}$}(1-x)]\ln{\left(-\frac{H^{2}}{m^{2}}\right)},$
(57)

with $H^{2}=H^{2}(p^{2},m^{2})=p^{2}x(1-x)-m^{2}x$. The integral for the two-
loop graph is then written as

$i\Sigma^{(2)}(p)=-2iq^{4}\frac{i}{16\pi^{2}}\int_{0}^{1}dx\,\int_{k}^{R}\frac{\gamma^{\alpha}(\hbox{$k\\!\\!\\!{/}$}+m)[2m+(x-1)\hbox{$k\\!\\!\\!{/}$}](\hbox{$k\\!\\!\\!{/}$}+m)\gamma_{\alpha}}{(k^{2}-m^{2})^{2}(p-k)^{2}}\ln{\left(-\frac{H^{2}(k^{2},m^{2})}{m^{2}}\right)}.$
(58)

Let us define

$F=-2iq^{4}\frac{i}{16\pi^{2}}\int_{0}^{1}dx\,\int_{k}^{R}\frac{\gamma^{\alpha}(\hbox{$k\\!\\!\\!{/}$}+m)[2m+(x-1)\hbox{$k\\!\\!\\!{/}$}](\hbox{$k\\!\\!\\!{/}$}+m)\gamma_{\alpha}}{(k^{2}-m^{2})^{2}(p-k)^{2}}\left(-\frac{H^{2}}{m^{2}}\right)^{\varepsilon},$
(59)

such that $i\Sigma^{(2)}(p)=F_{\varepsilon}$, being $F_{\varepsilon}$ the
first order coefficient in the expansion of $F$ in powers of $\varepsilon$. We
then write

$\left(-\frac{H^{2}}{m^{2}}\right)^{\varepsilon}=\left[\frac{x(1-x)}{(-m^{2})}\right]^{\varepsilon}\,(k^{2}-\tilde{m}^{2})^{\varepsilon};\,\,\,\,\tilde{m}^{2}=\frac{m^{2}}{(1-x)},$
(60)

to obtain

$F=-2iq^{4}\frac{i}{16\pi^{2}}\int_{0}^{1}dx\,\left[\frac{x(1-x)}{(-m^{2})}\right]^{\varepsilon}\int_{k}^{R}\frac{\gamma^{\alpha}(\hbox{$k\\!\\!\\!{/}$}+m)[2m+(x-1)\hbox{$k\\!\\!\\!{/}$}](\hbox{$k\\!\\!\\!{/}$}+m)\gamma_{\alpha}(k^{2}-\tilde{m}^{2})}{(k^{2}-m^{2})^{2}(k^{2}-\tilde{m}^{2})^{1-\varepsilon}(p-k)^{2}}.$
(61)

In the equation above, we multiplied the numerator and the denominator by a
factor of $(k^{2}-\tilde{m}^{2})$ for convenience to obtain a well defined
Feynman parametrization, from which, we obtain

$F=-2iq^{4}\frac{i}{16\pi^{2}}\frac{\Gamma(4-\varepsilon)]}{\Gamma(1-\varepsilon)}\int_{0}^{1}dx\,\int_{0}^{1}du\,\int_{0}^{1-u}dv\,v\left[\frac{x(1-x)}{(-m^{2})(1-u-v)}\right]^{\varepsilon}\int_{k}^{R}\frac{\tilde{N}}{(k^{2}+Q^{2})^{4-\varepsilon}},$
(62)

where $Q^{2}=p^{2}u(1-u)-\tilde{m}^{2}(1-u-v)-m^{2}v$ and $\tilde{N}$ is the
even part in $k$ of

$N=\gamma^{\alpha}(\hbox{$k\\!\\!\\!{/}$}+\hbox{$p\\!\\!\\!{/}$}u+m)[2m+(x-1)(\hbox{$k\\!\\!\\!{/}$}+\hbox{$p\\!\\!\\!{/}$}u)](\hbox{$k\\!\\!\\!{/}$}+\hbox{$p\\!\\!\\!{/}$}u+m)\gamma_{\alpha}[(k+pu)^{2}-\tilde{m}^{2}].$
(63)

The divergent part of $F$ is the one with quartic terms in $k$ in the
numerator. Let us carry out explicitly the calculation of this part. After
performing the Dirac algebra, the quartic part of the numerator is written as

$\displaystyle N^{(4)}$ $\displaystyle=$ $\displaystyle
2k^{4}[4mx-u(x-1)\hbox{$p\\!\\!\\!{/}$}]-8k^{2}u(x-1)(p\cdot
k)\hbox{$k\\!\\!\\!{/}$}$ (64) $\displaystyle=$ $\displaystyle
2(k^{2}+Q^{2})^{2}[4mx-u(x-1)\hbox{$p\\!\\!\\!{/}$}]-4(k^{2}+Q^{2})Q^{2}[4mx-u(x-1)\hbox{$p\\!\\!\\!{/}$}]$
$\displaystyle+$ $\displaystyle
2Q^{4}[4mx-u(x-1)\hbox{$p\\!\\!\\!{/}$}]-8(k^{2}+Q^{2})u(x-1)(p\cdot
k)\hbox{$k\\!\\!\\!{/}$}+8Q^{2}u(x-1)(p\cdot k)\hbox{$k\\!\\!\\!{/}$},$

in which we added and subtracted $Q^{2}$ to the factors of $k^{2}$. The first
and fourth terms will result in logarithmically divergent integrals. For the
first, we have

$\displaystyle
F^{(4)}_{1}=-4iq^{4}\frac{i}{16\pi^{2}}\frac{\Gamma(4-\varepsilon)]}{\Gamma(1-\varepsilon)}\int_{x,u,v}v\left[\frac{x(1-x)}{(-m^{2})(1-u-v)}\right]^{\varepsilon}[4mx-u(x-1)\hbox{$p\\!\\!\\!{/}$}]\int_{k}^{R}\frac{1}{(k^{2}+Q^{2})^{2-\varepsilon}},$
(65)

with $\int_{x,u,v}$ representing the integrals in the Feynman parameters. We
then expand the above expression for small $\varepsilon$ to get the
coefficient of the first order term

$\displaystyle F^{(4)}_{1\varepsilon}$ $\displaystyle=$
$\displaystyle-4iq^{4}\frac{i}{16\pi^{2}}\int_{x,u,v}v[4mx-u(x-1)\hbox{$p\\!\\!\\!{/}$}]\left\\{6I_{log}^{(2)}(-Q^{2},m^{2})+\right.$
(66) $\displaystyle+$
$\displaystyle\left.\left[6\ln{\left[\frac{x(1-x)}{(1-u-v)}\right]}-11\right]I_{log}(-Q^{2})\right\\},$

in which, we use of the definition of the basic logarithmic divergence typical
of two-loop order,

$I_{log}^{(2)}(m^{2},\lambda^{2})=\int_{k}^{R}\frac{1}{(k^{2}-m^{2})^{2}}\ln{\left[\frac{(k^{2}-m^{2})}{(-\lambda^{2})}\right]}.$
(67)

Besides the scale relations (21) and (22) we have already used for the one-
loop calculations, we can easily obtain, through the same algebraic
manipulations, the corresponding two-loop one,

$I^{(2)}_{log}(m^{2},\lambda^{2})=I^{(2)}_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\left\\{\ln{\left(\frac{m^{2}}{\lambda^{2}}\right)}+\frac{1}{2}\ln^{2}{\left(\frac{m^{2}}{\lambda^{2}}\right)}\right\\},$
(68)

where $I^{(2)}_{log}(\lambda^{2})\equiv
I^{(2)}_{log}(\lambda^{2},\lambda^{2})$. This relation will be used to obtain
a divergent part free from the external momenta. But let us first get the
result of the other divergent term,

$F^{(4)}_{4}=16iq^{4}\frac{i}{16\pi^{2}}\frac{\Gamma(4-\varepsilon)]}{\Gamma(1-\varepsilon)}\int_{x,u,v}vu(x-1)\left[\frac{x(1-x)}{(-m^{2})(1-u-v)}\right]^{\varepsilon}p^{\alpha}\gamma^{\beta}\int_{k}^{R}\frac{k_{\alpha}k_{\beta}}{\,\,(k^{2}+Q^{2})^{3-\varepsilon}}.$
(69)

For the integral in $k$, we can write

$\int_{k}^{\Lambda}\frac{k_{\alpha}k_{\beta}}{(k^{2}+Q^{2})^{3-\varepsilon}}=\frac{1}{2(2-\varepsilon)}\left\\{\int_{k}^{R}\frac{\eta_{\alpha\beta}}{\,\,(k^{2}+Q^{2})^{2-\varepsilon}}-\int_{k}^{R}\frac{\partial}{\partial
k^{\beta}}\frac{k_{\alpha}}{\,\,(k^{2}+Q^{2})^{2-\varepsilon}}\right\\},$ (70)

from which we discard the surface term as prescribed by constrained IReg.
After substituting the above relation in $F^{(4)}_{4}$ and picking the first
order coefficient of the expansion in powers of $\varepsilon$, we get

$F^{(4)}_{4\varepsilon}=8iq^{4}\frac{i}{16\pi^{2}}\hbox{$p\\!\\!\\!{/}$}\int_{x,u,v}vu(x-1)\left\\{3I_{log}^{(2)}(-Q^{2},m^{2})+\left[3\ln{\left[\frac{x(1-x)}{(1-u-v)}\right]}-4\right]I_{log}(-Q^{2})\right\\}.$
(71)

We now get together the two divergent integrals, use the scale relations (21)
and (68) and integrate the coefficients of the basic divergences to obtain

$\displaystyle F^{(4)}_{1\varepsilon}+F^{(4)}_{4\varepsilon}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}q^{4}\frac{i}{16\pi^{2}}\left\\{2(\hbox{$p\\!\\!\\!{/}$}+8m)I_{log}^{(2)}(m^{2})-(3\hbox{$p\\!\\!\\!{/}$}+32m)I_{log}(m^{2})\right\\}+$
(72) $\displaystyle+$ $\displaystyle
4iq^{4}\left(\frac{i}{16\pi^{2}}\right)^{2}\int_{x,u,v}v\left\\{\left\\{4mx\left[6\ln{\left[\frac{x(1-x)}{(1-u-v)}\right]}-5\right]+\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.u(x-1)\hbox{$p\\!\\!\\!{/}$}\left[12\ln{\left[\frac{x(1-x)}{(1-u-v)}\right]}-7\right]\right\\}\ln{\left(-\frac{Q^{2}}{m^{2}}\right)}+\right.$
$\displaystyle+$
$\displaystyle\left.6[2mx-u(x-1)\hbox{$p\\!\\!\\!{/}$}]\ln^{2}{\left(-\frac{Q^{2}}{m^{2}}\right)}.\right\\}$

There are still three finite integrals considering the quartic part of the
numerator. Besides, we have finite integrals coming from the quadratic and
zeroth order in $k$ terms in the numerator. These calculations are
straightforward: the integration in $k$ is performed, resulting in
$\varepsilon$-depending powers of $Q^{2}$ in the denominator; the expansion in
powers of $\varepsilon$ is performed to take the first order coefficient. The
final result for the two-loop amplitude is given by

$\displaystyle i\Sigma^{(2)}(p)$ $\displaystyle=$
$\displaystyle-\frac{i}{2}q^{4}\frac{i}{16\pi^{2}}\left\\{2(\hbox{$p\\!\\!\\!{/}$}+8m)I_{log}^{(2)}(m^{2})-(3\hbox{$p\\!\\!\\!{/}$}+32m)I_{log}(m^{2})+\frac{1}{72}\frac{i}{16\pi^{2}}(5\hbox{$p\\!\\!\\!{/}$}+352m)\right\\}+$
(73) $\displaystyle+$ $\displaystyle
4iq^{4}\left(\frac{i}{16\pi^{2}}\right)^{2}\int_{x,u,v}v\left\\{A\ln{\left(-\frac{Q^{2}}{m^{2}}\right)}\ln{\left[-\frac{Q^{2}}{m^{2}}\left(\frac{x(1-x)}{(1-u-v)}\right)^{2}\right]}+\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{Q^{2}}\left[B\ln{\left[-\frac{Q^{2}}{m^{2}}\frac{x(1-x)}{(1-u-v)}\right]}+C\right]+\frac{D}{Q^{4}}\left[\ln{\left[-\frac{Q^{2}}{m^{2}}\frac{x(1-x)}{(1-u-v)}\right]}-1\right]\right\\},$

with

$\displaystyle A$ $\displaystyle=$ $\displaystyle
6[2mx-u(x-1)\hbox{$p\\!\\!\\!{/}$}],$ (74) $\displaystyle B$ $\displaystyle=$
$\displaystyle
2\left\\{u\hbox{$p\\!\\!\\!{/}$}[5u^{2}p^{2}(x-1)+2m^{2}(x+4))]-2m[2u(u+3)xp^{2}+3(m^{2}-x\tilde{m}^{2})]\right\\},$
(75) $\displaystyle C$ $\displaystyle=$ $\displaystyle
4m(2uxp^{2}+m^{2}-\tilde{m}^{2}x)-u\hbox{$p\\!\\!\\!{/}$}[2u^{2}p^{2}(x-1)+m^{2}(x+4)],$
(76) $\displaystyle D$ $\displaystyle=$
$\displaystyle-(u^{2}p^{2}-\tilde{m}^{2})\left\\{4m(xu^{2}p^{2}+m^{2})-u\hbox{$p\\!\\!\\!{/}$}[u^{2}(x-1)p^{2}+m^{2}(x+3)]\right\\}.$
(77)

There are some interesting comments about the two-loop calculation above.
First, the basic divergence $I_{log}^{(2)}(m^{2})$ appears naturally, even for
massive models, when the expansion in $\varepsilon$ is carried out. Second,
the momentum integration of the finite part is easily performed with the help
of Feynman parametrization. And, last but not least, the elimination of the
surface terms is simpler than the traditional procedure adopted in IReg irn ,
which needs new relations for each loop-order.

###  V.1 Massless models

The calculation above is just an example of how the present approach can be
applied to a two-loop massive model, which is a calculation difficult to be
implemented in the usual procedure of IReg. It is interesting to have some
higher order results to be compared with the ones obtained before. Let us then
resort to non-massive spinorial electrodynamics. For the two-loop graph
treated above, the procedure is not simply the limit $m\to 0$, since the
divergence of the one-loop subgraph was subtracted with the physical mass as
the parameter. However, it is simple to carry out this calculation.

For the one-loop subgraph, we have

$i\Sigma^{(1)}=-q^{2}\int_{k}^{R}\frac{\gamma^{\alpha}\hbox{$k\\!\\!\\!{/}$}\gamma_{\alpha}}{k^{2}(p-k)^{2}}=q^{2}\hbox{$p\\!\\!\\!{/}$}\left\\{I_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\left[\ln{\left(-\frac{p^{2}}{\lambda^{2}}\right)}-2\right]\right\\}$
(78)

Subtracting out the one-loop subdivergence, we have for the two-loop nested
diagram

$i\Sigma^{(2}(p)=-2iq^{4}\frac{i}{16\pi^{2}}\int_{k}^{R}\frac{\hbox{$k\\!\\!\\!{/}$}}{k^{2}(p-k)^{2}}\left[\ln{\left(-\frac{k^{2}}{\lambda^{2}}\right)}-2\right]=A_{1}+A_{2}.$
(79)

For $A_{1}$, the part with contains the logarithm, we use
$A_{1}=F_{\varepsilon}$, as before, with

$\displaystyle F$ $\displaystyle=$
$\displaystyle-2iq^{4}\frac{i}{16\pi^{2}}(-\lambda^{2})^{-\varepsilon}\int_{k}^{R}\frac{\hbox{$k\\!\\!\\!{/}$}}{(k^{2})^{1-\varepsilon}(p-k)^{2}}$
(80) $\displaystyle=$
$\displaystyle-2iq^{4}\frac{i}{16\pi^{2}}(-\lambda^{2})^{-\varepsilon}\frac{\Gamma(2-\varepsilon)}{\Gamma(1-\varepsilon)}\int_{0}^{1}dx\,(1-x)^{-\varepsilon}\int_{k}^{R}\frac{\hbox{$k\\!\\!\\!{/}$}+\hbox{$p\\!\\!\\!{/}$}x}{\,\,\,(k^{2}+H^{2})^{2-\varepsilon}}$

After expanding for $\varepsilon\to 0$, collecting the first order coefficient
and using the scale relations as prescribed, one obtains

$\displaystyle
A_{1}=iq^{4}\frac{i}{16\pi^{2}}\hbox{$p\\!\\!\\!{/}$}\left\\{-I_{log}^{(2)}(\lambda^{2})-\frac{1}{2}I_{log}(\lambda^{2})+\frac{1}{2}\frac{i}{16\pi^{2}}\left[\ln^{2}{\left(-\frac{p^{2}}{\lambda^{2}}\right)}-\ln{\left(-\frac{p^{2}}{\lambda^{2}}\right)}-3\right]\right\\}$
(81)

The term $A_{2}$ is a simple one-loop integral. When the terms are put
together, we get

$i\Sigma^{(2}(p)=iq^{4}\frac{i}{16\pi^{2}}\hbox{$p\\!\\!\\!{/}$}\left\\{-I_{log}^{(2)}(\lambda^{2})+\frac{3}{2}I_{log}(\lambda^{2})+\frac{1}{2}\frac{i}{16\pi^{2}}\left[\ln^{2}{\left(-\frac{p^{2}}{\lambda^{2}}\right)}-5\ln{\left(-\frac{p^{2}}{\lambda^{2}}\right)}+5\right]\right\\},$
(82)

which can be compared with the result of irn , where higher order calculations
with IReg in massless gauge theories have been implemented. In that paper, the
complete renormalization of spinorial QED to two-loop order has been
performed.

For a more significant comparison, let us look at the photon self-energy. The
diagrams which contribute are exhibited in Figure 4.

$(a)\,\,\,\,-i\Pi^{\mu\nu}_{a}(p)$$(b)\,\,\,\,-i\Pi^{\mu\nu}_{b}(p)$

FIG. 4. Contributions to the two-loop photon self-energy in spinorial QED.

The amplitude for the graph of Figure 4$(a)$, corresponding to the nested
contribution, can be computed by simply using the finite part of the subgraph,
which is the electron self-energy. We have

$\displaystyle-i\Pi^{\mu\nu}_{a}=-2i\frac{i}{16\pi^{2}}q^{4}\int_{k}^{R}\frac{\mbox{tr}\left\\{\gamma^{\nu}\hbox{$k\\!\\!\\!{/}$}\gamma^{\mu}(\hbox{$k\\!\\!\\!{/}$}-\hbox{$p\\!\\!\\!{/}$})\right\\}}{k^{2}(k-p)^{2}}\left[\ln{\left(-\frac{p^{2}}{\lambda^{2}}\right)}-2\right],$
(83)

where the factor $2$ is to take into account the graph with the insertion in
the inferior line of the loop. The second term is proportional to the one-loop
vacuum polarization tensor. For the first one, which contains a logarithm, we
write

$-i\Pi^{\mu\nu}_{a1F}=-2i\frac{i}{16\pi^{2}}\frac{q^{4}}{(-\lambda^{2})^{\varepsilon}}\int_{k}^{R}\frac{\mbox{tr}\left\\{\gamma^{\nu}\hbox{$k\\!\\!\\!{/}$}\gamma^{\mu}(\hbox{$k\\!\\!\\!{/}$}-\hbox{$p\\!\\!\\!{/}$})\right\\}}{(k^{2})^{1-\varepsilon}(k-p)^{2}},$
(84)

from which we will extract the coefficient of the first power in $\epsilon$.
We Feynman parametrize the complete integral to obtain

$-i\Pi^{\mu\nu}_{a1F}=-8i\frac{i}{16\pi^{2}}q^{4}\frac{\Gamma(2-\varepsilon)}{\Gamma(1-\varepsilon)}\int_{0}^{1}dx[-(1-x)\lambda]^{-\varepsilon}\int_{k}^{R}\frac{\left[2k^{\mu}k^{\nu}-k^{2}\eta^{\mu\nu}+x(1-x)(p^{2}\eta^{\mu\nu}-2p^{\mu}p^{\nu})\right]}{[k^{2}+H^{2}]^{2-\varepsilon}},$
(85)

with $H^{2}=p^{2}x(1-x)$. It is possible to reorganize the terms so as to
have, in the numerator of the $k$ integral,

$2k^{\mu}k^{\nu}-(k^{2}+H^{2})\eta^{\mu\nu}+2x(1-x)(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu}).$
(86)

The two first terms will originate quadratically divergent integrals. Since

$\frac{\partial}{\partial
k_{\nu}}\left(\frac{k^{\mu}}{(k^{2}+H^{2})^{1-\varepsilon}}\right)=\frac{\eta^{\mu\nu}}{(k^{2}+H^{2})^{1-\varepsilon}}-2(1-\varepsilon)\frac{k^{\mu}k^{\nu}}{(k^{2}+H^{2})^{2-\varepsilon}},$
(87)

we can reduce the tensorial integral to a scalar one by discarding the surface
term, so that we have, for these two terms,

$-8i\frac{i}{16\pi^{2}}q^{4}\frac{\varepsilon\Gamma(2-\varepsilon)}{(1-\varepsilon)\Gamma(1-\varepsilon)}\int_{0}^{1}dx[-(1-x)\lambda]^{-\varepsilon}\int_{k}^{R}\frac{\eta^{\mu\nu}}{\;\;\;[k^{2}+H^{2}]^{1-\varepsilon}}.$
(88)

The coefficient of the first power in the expansion in $\varepsilon$ is then
given by

$-8i\frac{i}{16\pi^{2}}q^{4}\eta^{\mu\nu}\int_{0}^{1}dx\;I_{quad}(-H^{2}).$
(89)

For the other part, we stay with

$-16i\frac{i}{16\pi^{2}}q^{4}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\frac{\Gamma(2-\varepsilon)}{\Gamma(1-\varepsilon)}\int_{0}^{1}dx[-(1-x)\lambda]^{-\varepsilon}x(1-x)\int_{k}^{R}\frac{1}{\;\;\;[k^{2}+H^{2}]^{2-\varepsilon}},$
(90)

which furnishes, for the first order coefficient,

$-16i\frac{i}{16\pi^{2}}q^{4}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\int_{0}^{1}dx\;x(1-x)\left[I_{log}^{(2)}(-H^{2})-[\ln(1-x)+1]I_{log}(-H^{2})\right].$
(91)

Using the scale relations for one- and two-loop, we obtain

$\displaystyle-16i\frac{i}{16\pi^{2}}q^{4}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\int_{0}^{1}dx\;x(1-x)\left\\{I_{log}^{(2)}(\lambda^{2})-\frac{1}{2}\frac{i}{16\pi^{2}}\ln^{2}\left(-\frac{H^{2}}{\lambda^{2}}\right)-\frac{i}{16\pi^{2}}\ln\left(-\frac{H^{2}}{\lambda^{2}}\right)+\right.$
$\displaystyle\left.-[\ln(1-x)+1]\left[I_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\ln\left(-\frac{H^{2}}{\lambda^{2}}\right)\right]\right\\}$
$\displaystyle=-\frac{8}{3}i\frac{i}{16\pi^{2}}q^{4}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\left\\{I^{(2)}_{log}(\lambda^{2})-\frac{1}{6}I_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\left[\frac{1}{2}\ln^{2}\left(-\frac{p^{2}}{\lambda^{2}}\right)+\frac{5}{6}\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]\right\\}.$
(92)

We still have the second term of eq. (83), which gives

$-i\Pi^{\mu\nu}_{a2}=-\frac{8}{3}i\frac{i}{16\pi^{2}}q^{4}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\left\\{-2I_{log}(\lambda^{2})+\frac{i}{16\pi^{2}}\left[\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)-\frac{10}{3}\right]\right\\},$
(93)

to be added to the first one to give, for the nested graph,

$\displaystyle-i\Pi^{\mu\nu}_{a}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}i\frac{i}{16\pi^{2}}q^{4}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\left\\{I^{(2)}_{log}(\lambda^{2})-\frac{13}{6}I_{log}(\lambda^{2})+\frac{i}{16\pi^{2}}\left[-\frac{1}{2}\ln^{2}\left(-\frac{p^{2}}{\lambda^{2}}\right)+\right.\right.$
(94) $\displaystyle+$
$\displaystyle\left.\left.\frac{1}{6}\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)-\frac{10}{3}\right]\right\\}-8i\frac{i}{16\pi^{2}}q^{4}\eta^{\mu\nu}\int_{0}^{1}dx\;I_{quad}(-H^{2}).$

Note that the unique term which violates the transversality of the amplitude
is the one in $I_{quad}(-H^{2})$. This part, if the appropriated scale
relations is used, encompasses several contributions in addition to the
quadratic divergence. We opt to maintain them gathered in this way, since they
must be cancelled out when added to the result for the overlapped diagram.

The amplitude for the overlapped graph of Fig. 4$(b)$, with the addition of
the counterterms to cancel out the subdivergences, is given by

$\displaystyle-i\Pi^{\mu\nu}_{b}$ $\displaystyle=$ $\displaystyle
2iq^{4}\int_{k}^{R}\int_{l}^{R}\frac{\mbox{tr}\\{\gamma^{\nu}\hbox{$l\\!\\!\\!{/}$}(\hbox{$k\\!\\!\\!{/}$}-\hbox{$p\\!\\!\\!{/}$})\gamma^{\mu}\hbox{$k\\!\\!\\!{/}$}(\hbox{$l\\!\\!\\!{/}$}-\hbox{$p\\!\\!\\!{/}$})\\}}{k^{2}l^{2}(k-l)^{2}(p-k)^{2}(p-l)^{2}}+$
(95) $\displaystyle+$
$\displaystyle\frac{8}{3}iq^{2}I_{log}(\lambda^{2})\left[I_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)+\frac{5}{3}\frac{i}{16\pi^{2}}\right](p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu}).$

It is important to note that, usually, the counterterms to cancel the
subdivergences are calculated as different graphs to be added to the
amplitude, as prescribed by the forest formula of BPHZ. Here, equivalently, we
simply defined a quantity which is already free from the subdivergences, just
to give a treatment similar to the one given to the nested diagram, for which
we used only the finite part of the subgraph.

For the overlapped graph, it is simpler to break the amplitude in a
combination of integrals and then apply, to these integrals, the proposed
approach. After the calculation of the trace of the Dirac matrices, we obtain,
for the numerator (using the symmetry in exchanging $k$ and $l$),

$\displaystyle N^{\mu\nu}$ $\displaystyle=$ $\displaystyle
8iq^{4}\left\\{k^{2}l^{2}\eta^{\mu\nu}-4k^{2}l^{\mu}l^{\nu}+4(k\cdot
l)k^{\mu}k^{\nu}+4k^{2}l^{\nu}p^{\mu}+2p^{2}l^{\mu}k^{\nu}-2k^{2}(l\cdot
p)\eta^{\mu\nu}+\right.$ (96) $\displaystyle-$ $\displaystyle\left.4(k\cdot
l)l^{\nu}p^{\mu}-4(k\cdot p)l^{\mu}p^{\nu}-4(k\cdot p)k^{\mu}l^{\nu}+4(k\cdot
p)l^{\mu}l^{\nu}+2(k\cdot p)(l\cdot p)\eta^{\mu\nu}+\right.$ $\displaystyle-$
$\displaystyle\left.p^{2}(k\cdot l)\eta^{\mu\nu}+2(k\cdot
l)p^{\mu}p^{\nu}\right\\}.$

We will focus, in the above amplitude, in discussing some sensible points,
like the cancellation of the quadratic divergences. The first three terms in
eq. (96) originate quadratically divergent integrals, which should give us the
exact term in order to cancel out the gauge violating remaining term in the
nested diagram. Let us calculate the first one. We have

$J^{\mu\nu}=8iq^{4}\eta^{\mu\nu}\int_{k}^{R}\int_{l}^{R}\frac{k^{2}l^{2}}{k^{2}l^{2}(k-l)^{2}(p-k)^{2}(p-l)^{2}},$
(97)

in which we cancel only the factor $k^{2}$, that is sufficient to make easy
the solution of the integral. It is important to note that, by performing
appropriated shifts, the dependence of $J^{\mu\nu}$ on the external momentum
$p$ is completely eliminated. However, we intend to maintain this dependence
in different parts of the result (obviously, this dependence cancels out when
the parts are put together) The integration in $k$ gives

$I_{log}(\lambda^{2})+\frac{i}{16\pi^{2}}\left[2-\ln\left[-\frac{(p-l)^{2}}{\lambda^{2}}\right]\right].$
(98)

We then stay with

$J^{\mu\nu}=8iq^{4}\eta^{\mu\nu}\left\\{\left[I_{log}(\lambda^{2})+2\frac{i}{16\pi^{2}}\right]\int_{l}^{R}\frac{l^{2}}{l^{2}(p-l)^{2}}-\frac{i}{16\pi^{2}}\int_{l}^{R}\frac{l^{2}}{l^{2}(p-l)^{2}}\ln\left[-\frac{(p-l)^{2}}{\lambda^{2}}\right]\right\\}.$
(99)

For the second integral, we use the procedure to reduce the logarithm to
powers of $\epsilon$. For both integrals, we apply Feynman parametrization
and, only after this step, we cancel the factors of $l^{2}$ by adding and
subtractig $H^{2}$. In the end, we obtain

$\displaystyle J^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[I_{log}(\lambda^{2})+2\frac{i}{16\pi^{2}}\right]\int_{0}^{1}dx\,I_{quad}(-H^{2})+$
(100) $\displaystyle-$
$\displaystyle\frac{i}{16\pi^{2}}\int_{0}^{1}dx\left\\{I_{quad}^{(2)}(-H^{2},\lambda^{2})-[\ln(1-x)+1]I_{quad}(-H^{2})\right\\}+\cdots,$

with

$I_{quad}^{(2)}(m^{2},\lambda^{2})=\int_{k}^{R}\frac{1}{(k^{2}-m^{2})}\ln{\left[\frac{(k^{2}-m^{2})}{(-\lambda^{2})}\right]}$
(101)

and in which we only displayed the terms that involve quadratic divergences.
The second quadratically divergent integral reads

$\displaystyle A^{\mu\nu}$ $\displaystyle=$
$\displaystyle-32iq^{4}\int_{k}^{R}\int_{l}^{R}\frac{k^{2}l^{\mu}l^{\nu}}{k^{2}l^{2}(k-l)^{2}(p-k)^{2}(p-l)^{2}}$
(102) $\displaystyle=$
$\displaystyle-16iq^{4}\left\\{\left[I_{log}(\lambda^{2})+2\frac{i}{16\pi^{2}}\right]\int_{0}^{1}dx\,I_{quad}(-H^{2})\right.+$
$\displaystyle-$
$\displaystyle\left.\frac{i}{16\pi^{2}}\int_{0}^{1}dx\left\\{I_{quad}^{(2)}(-H^{2},\lambda^{2})-\ln(1-x)I_{quad}(-H^{2})\right\\}\right\\}+\cdots,$

and, the third,

$\displaystyle B^{\mu\nu}$ $\displaystyle=$ $\displaystyle
32iq^{4}\int_{k}^{R}\int_{l}^{R}\frac{k^{\mu}l^{\nu}(k\cdot
l)}{k^{2}l^{2}(k-l)^{2}(p-k)^{2}(p-l)^{2}}$ (103) $\displaystyle=$
$\displaystyle
8iq^{4}\left\\{\left[I_{log}(\lambda^{2})+2\frac{i}{16\pi^{2}}\right]\int_{0}^{1}dx\,I_{quad}(-H^{2})\right.+$
$\displaystyle-$
$\displaystyle\left.\frac{i}{16\pi^{2}}\int_{0}^{1}dx\left\\{I_{quad}^{(2)}(-H^{2},\lambda^{2})-\ln(1-x)I_{quad}(-H^{2})\right\\}\right\\}+\cdots.$

Again, we only show the terms that contribute to quadratic divergences. The
sum of these three terms gives

$J^{\mu\nu}+A^{\mu\nu}+B^{\mu\nu}=8i\frac{i}{16\pi^{2}}q^{4}\eta^{\mu\nu}\int_{0}^{1}dx\,I_{quad}(-H^{2})+\cdots,$
(104)

which is exactly the necessary to cancel out the quadratic divergence coming
from the nested diagram. We should comment on a notable feature of this new
procedure with regard to quadratic divergences. Usually in IReg, when dealing
with massles models, quadratic divergences are disregarded on the grounds that
it is possible to construct a parametrization in which they are null. Here it
is not even necessary to use the scale relations for such divergences, as the
cancellation occurs for the packages gathered in $I_{quad}(-H^{2})$ and
$I_{quad}^{(2)}(-H^{2},\lambda^{2})$.

The other integrals originated from the terms of (96) are easily calculated
with the approach presented in this paper. The final result for the sum
$-i\Pi^{\mu\nu}_{a}-i\Pi^{\mu\nu}_{b}$ agrees perfectly with the one of irn
and is given by

$\displaystyle-i\Pi^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{8}{3}iq^{4}\frac{i}{16\pi
2}(p^{\mu}p^{\nu}-\eta^{\mu\nu}p^{2})\left\\{\frac{3}{2}I^{2}_{log}(\lambda^{2})+\right.$
(105) $\displaystyle+$
$\displaystyle\left.\frac{i}{16\pi^{2}}\left[-3I^{(2)}_{log}(\lambda^{2})+\frac{31}{6}I_{log}(\lambda^{2})-\frac{3}{2}\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)+\frac{3}{2}\right]-p^{2}I^{O}\right\\},$

with

$I^{O}=\int_{k}\int_{l}\frac{1}{k^{2}l^{2}(k-l)^{2}(p-k)^{2}(p-l)^{2}}.$ (106)

As a final comment in this section, it is important to note that the use of
Implicit Regularization beyond one-loop order is not new. For example, in ir8
the procedure was used to calculate the beta-function of the Wess-Zumino model
in order of three loops. In the paper multiloop , procedures were developed
for the application of the technique in non-massive models in n-loop order,
with the determination of scale and consistency relations for an arbitrary
loop order.

We believe that the procedure we discuss in this section improves the
methodology of calculation and can be systematized and be of great help in
multiloop phenomenological calculations.

##  VI Comment on Implicit Regularization of Infrared Divergent Amplitudes

Although the present study is focused in the regularization of ultraviolet
divergences, we carry out below a brief discussion on the application of
Implicit Regularization in the treatment of infrared divergences. This was the
subject of the paper IRed , in which one- and higher-loop orders applications
were discussed. The procedure was used in the analysis of the origin of the
two-loop contributions to $N=1$ super Yang-Mills beta-function 2-loop .

The essential steps of Implicit Regularization, when used in the case of
ultraviolet divergences, have their counterpart for infrared ones. Moreover, a
new scale appears, typically an infrared scale which is completely independent
of the ultraviolet one. In order to perform all the algebraic manipulations
which are necessary in IReg, we assume the presence of a infrared regulator,
which will be indicated by the upper index $\tilde{R}$. Considering that the
integral is under this regularization, all the steps indicated in section III
are allowed, such as using algebraic identities to separate the divergent
part, performing shifts in the integration coordinate, with the consequent
elimination of surface terms, and the cancellation of common factors in the
numerator and denominator.

First, we consider the following ultraviolet divergent massless integral and
its result within Implicit Regularization,

$I=\int^{R}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}(p-k)^{2}}=I_{log}(\lambda^{2})-\frac{i}{16\pi^{2}}\left[\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)-2\right],$
(107)

and then proceed with the calculation of

$U=\int^{\tilde{R}}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}(p-k)^{2}},$ (108)

which, by power counting, is infrared divergent and ultraviolet finite. In
order to be able to use all the tools developed for ultraviolet divergent
integrals we firstly note that

$\frac{1}{k^{4}}=-\int d^{4}u\
e^{iku}\int^{\tilde{R}}\frac{d^{4}z}{(2\pi)^{4}}\frac{1}{z^{2}(z-u)^{2}},$
(109)

in which $z$ and $u$ are configuration variables.

Note the striking similarity between the above $z$ integral and equation
(107). We can thus write the result immediately:

$\displaystyle
I(u^{2})=\int^{\tilde{R}}\frac{d^{4}z}{(2\pi)^{4}}\frac{1}{z^{2}(z-u)^{2}}=I_{log}(\tilde{\lambda}^{-2})-\frac{i}{16\pi^{2}}\left[\ln\left(-u^{2}\tilde{\lambda}^{2}\right)-2\right].$
(110)

But now $I_{log}(\tilde{\lambda}^{-2})$ is an infrared basic divergent
integral and a scale relation has been used to introduce the infrared scale
$l^{2}=1/\tilde{\lambda}^{2}$.

Using this result and (109) in (108) we have

$\displaystyle U=-\int_{k}^{\tilde{R}}\frac{1}{(p-k)^{2}}\int d^{4}u\
e^{iku}I(u^{2})=-\frac{i}{(4\pi)^{2}}\int_{k}^{\tilde{R}}\int d^{4}u\int
d^{4}x\frac{e^{i(p-k)x}}{x^{2}}e^{iku}I(u^{2})$
$\displaystyle=-\frac{i}{(2\pi)^{2}}\int d^{4}u\
\frac{e^{ipu}}{u^{2}}\left\\{\tilde{I}_{log}(\tilde{\lambda}^{-2})-\frac{i}{16\pi^{2}}\left[\ln\left(-u^{2}\tilde{\lambda}^{2}\right)-2\right]\right\\}$
$\displaystyle=-\frac{1}{p^{2}}\left\\{I_{log}(\tilde{\lambda}^{-2})+\frac{i}{16\pi^{2}}\left[\ln\left(-\frac{p^{2}}{\bar{\tilde{\lambda}}^{2}}\right)+2\right]\right\\},$
(111)

with
$\bar{\tilde{\lambda}}^{2}\equiv\frac{4}{e^{2\gamma}}\tilde{\lambda}^{2}$,
where $\gamma=0,5772...$ is the Euler-Mascheroni constant.

The technique also allows the treatment of integrals in which infrared and
ultraviolet divergences are present at the same time. Let us consider, just as
a toy example,

$G=\int^{R,\tilde{R}}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}.$ (112)

The key point in order to separate these two kinds of divergence is to
multiply the numerator and the denominator by a factor $(p-k)^{2}$. We stay
with

$G=p^{2}U+\int^{R}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}(p-k)^{2}}-2p^{\mu}\int^{\tilde{R}}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}(p-k)^{2}},$
(113)

in which the first, already calculated, term is only infrared divergent, the
second one is only ultraviolet divergent and the last one is finite. The final
result is given by

$G=I_{log}(\lambda^{2})+I_{log}(\tilde{\lambda}^{-2})+\frac{i}{16\pi^{2}}\left[\ln{\left(\frac{\lambda^{2}}{\bar{\tilde{\lambda}}^{2}}\right)}+2\right].$
(114)

The examples discussed above are cases which involves off-shell infrared
divergences. Implicit Regularization has not a systematized prescription for
the case of on-shell infrared divergences. Usually, since this kind of
singularity should be cancelled out in physical calculations, these cases are
dealt by provisionally assigning a small mass to the particle,
$p^{2}=\mu^{2}$.

There are some difficult situations like the one of

$F=\int_{k}\frac{1}{k^{2}(k-p_{1})^{2}(k-p_{1}-p_{2})^{2}(k-p_{1}-p_{2}-p_{3})^{2}},$
(115)

with $p_{1}^{2}=p_{2}^{2}=p_{3}^{2}=(p_{1}+p_{2}+p_{3})^{2}=0$. This integral
is off-shell finite, but encompasses an unavoidable infrared divergence in the
on-shell limit. We illustrate below the difficulty of treating this kind of
integral using the approach of Implicit Regularization.

First, let us Feynman parametrize the factor,

$\frac{1}{k^{2}(k-p_{1})^{2}}=\int_{0}^{1}dx\frac{1}{[(k-p_{1}x)^{2}+H^{2}]^{2}},$
(116)

with $H^{2}=p_{1}^{2}x(1-x)$. Since $p_{1}^{2}=0$, we have

$\frac{1}{k^{2}(k-p_{1})^{2}}=\int_{0}^{1}dx\frac{1}{(k-p_{1}x)^{4}},$ (117)

which, in $F$, gives us

$F=\int_{0}^{1}dx\int_{k}^{\tilde{R}}\frac{1}{(k-p_{1}x)^{4}(k-p_{1}-p_{2})^{2}(k-p_{1}-p_{2}-p_{3})^{2}}.$
(118)

We then perform the shift $k\to k+p_{1}x$ and use the inverse Fourier
transform of $1/k^{4}$ to write

$F=-\int_{0}^{1}dx\int_{k}^{\tilde{R}}\int
d^{4}ze^{ikz}I(z^{2})\frac{1}{(k-q)^{2}(k-q-
p_{3})^{2}};\;\;\;\;q=(1-x)p_{1}+p_{2}.$ (119)

The new shift $k\to k+q$ gives us

$F=-\int_{0}^{1}dx\int_{k}^{\tilde{R}}\int
d^{4}ze^{i(k+q)z}I(z^{2})\frac{1}{k^{2}(k-p_{3})^{2}}.$ (120)

As we have done before for $p_{1}$, since $p_{3}^{2}=0$, we write

$\frac{1}{k^{2}(k-p_{3})^{2}}=\int_{0}^{1}dy\frac{1}{(k-p_{3}y)^{4}}$ (121)

and, then, make the shift $k\to k+p_{3}y$ in the $k$-integral to obtain

$\displaystyle F$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}dx\int_{0}^{1}dy\int
d^{4}z\int_{k}^{\tilde{R}}\frac{e^{i(k+q+p_{3}y)z}}{k^{4}}I(z^{2})$ (122)
$\displaystyle=$ $\displaystyle\int_{0}^{1}dx\int_{0}^{1}dy\int d^{4}z\int
d^{4}u\int_{k}^{\tilde{R}}e^{i(k+q+p_{3}y)z}e^{iku}I(z^{2})I(u^{2})$
$\displaystyle=$ $\displaystyle\int_{0}^{1}dx\int_{0}^{1}dy\int d^{4}z\int
d^{4}u\,\,\delta^{(4)}(u+z)e^{i(q+p_{3}y)z}I(z^{2})I(u^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\int_{0}^{1}dy\int
d^{4}z\,\,e^{i(q+p_{3}y)z}[I(z^{2})]^{2}.$

The integrals in the Feynman parameters $x$ and $y$ are easily solved. We have

$\displaystyle\int_{0}^{1}dx\int_{0}^{1}dy\,\,e^{i(q+p_{3}y)z}=\int_{0}^{1}dx\int_{0}^{1}dy\,\,e^{i[(1-x)p_{1}+p_{2}+p_{3}y]z}$
$\displaystyle=e^{ip_{2}z}\left\\{\int_{0}^{1}dx\,\,e^{i(1-x)p_{1}z}\right\\}\left\\{\int_{0}^{1}dy\,\,e^{iyp_{3}z}\right\\}$
$\displaystyle=-\frac{e^{ip_{2}z}}{(p_{1}z)(p_{3}z)}\left[e^{ip_{1}z}-1\right]\left[e^{ip_{3}z}-1\right].$
(123)

Finally, we can write

$\displaystyle F=-\int
d^{4}z\frac{e^{ip_{2}z}}{(p_{1}z)(p_{3}z)}[e^{ip_{1}z}-1][e^{ip_{3}z}-1]\left\\{I_{log}(\tilde{\lambda}^{-2})-\frac{i}{16\pi^{2}}\left[\ln\left(-z^{2}\tilde{\lambda}^{2}\right)-2\right]\right\\}^{2}.$
(124)

In the equation above, we have a set of Fourier transforms calculated in
$p_{2}$, $p_{1}+p_{2}$, $p_{1}+p_{2}+p_{3}$ and $p_{2}+p_{3}$. It is important
to note that the infrared divergent content is enclosed in the
$I_{log}(\tilde{\lambda}^{-2}))$ which is free from the external momentum.
Nevertheless, it is clear that this approach of using integrals in the
configuration space is not the best one to deal with on-shell infrared
divergences. There are good strategies to treat on-shell infrared divergences,
including higher order loops, like, for example, the one described in
Anastasiou .

##  VII Concluding comments

In this paper, we have established a new procedure for the application of
Implicit Regularization in its constrained version. The constrained version of
Implicit Regularization, which fixes all the surface terms to zero,
automatically delivers symmetric amplitudes as already demonstrated in a wide
variety of articles. This is due to the fact that symmetries such as gauge-
invariance are related, in the context of Feynman integrals, to momentum
routing invariance in the loops A-Vieira . This new approach uses this fact
with the aim of Feynman parametrize the complete amplitude after a
regularization is assumed to be implicitly acting in the divergent integral.
As it is well known, the usefulness of Feynman parametrization lies in the
possibility of making a shift in the momentum of integration. A shift in a
divergent integral would have to be compensated with a surface term if the
degree of divergence is at least linear. This is why the amplitude should be
regularized before Feynman parametrization is carried out. In addition, the
regularization prescription must be such that these surface terms are null.
This is the case of Dimensional Regularization, which turns the amplitude
finite in the extended dimension and thus forces the surface terms to vanish.
In the case of the constrained version of Implicit Regularization, this is
accomplished with the help of the consistency relations.

The procedure which is presented in this paper enforces momentum routing
invariance by shifting the momentum of integration after the Feynman
parametrization of the complete amplitude. It also fixes other remaining
surface terms to zero. The great simplification in the approach occurs in
consequence of the way the divergent part is separated from the finite one.
While in the traditional application of IReg the integrand is expanded before
Feynman parametrization, which is carried out only in the finite part, here
this separation takes place after this step, by using scale relations that are
always the same. We then avoid to deal with finite integrals with high powers
in the momenta in the numerator and the denominator. Another advantage is the
unification of the procedure to be adopted in massive and non-massive models,
since the scale relations are in charge of introducing the mass parameter for
the basic divergences and for the renormalization group equations. The great
simplification in one-loop calculation for this new approach extends for
higher-loop orders, as demonstrated in section V. The procedure above can be
systematized and be of great help in multiloop phenomenological calculations.
This is part of a future work.

##  VIII Acknowledgements

This work was partially supported by Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq).

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