Journal of Modern Physics, 2012, 3, 1408-1449
http://dx.doi.org/10.4236/jmp.2012.310178 Published Online October 2012 (http://www.SciRP.org/journal/jmp)
A Systematization for One-Loop 4D Feynman
Integrals-Different Species of Massive Fields
O. A. Battistel1, G. Dallabona2
1Departamento de Fsica, Universidade Federal de Santa Maria, Santa Maria, Brazil
2Departamento de Ciências Exatas, Universidade Federal de Lavras, Lavras, Brazil
Email: orimar.battistel@gmail.com, gilson.dallabona@gmail.com
Received August 20, 2012; revised September 20, 2012; accepted October 2, 2012
ABSTRACT
A systematization for the manipulations and calculations involving divergent (or not) Feynman integrals, typical of the
one loop perturbative solutions of Quantum Field Theory, is proposed. A previous work on the same issue is general-
ized to treat theories and models having different species of massive fields. An improvement on the strategy is adopted
so that no regularization needs to be used. The final results produced, however, can be converted into the ones of rea-
sonable regularizations, especially those belonging to the dimensional regularization (in situations where the method
applies). Through an adequate interpretation of the Feynman rules and a convenient representation for involved propa-
gators, the finite and divergent parts are separated before the introduction of the integration in the loop momentum. On-
ly the finite integrals obtained are in fact integrated. The divergent content of the amplitudes are written as a combina-
tion of standard mathematical object which are never really integrated. Only very general scale properties of such ob-
jects are used. The finite parts, on the other hand, are written in terms of basic functions conveniently introduced. The
scale properties of such functions relate them to a well defined way to the basic divergent objects providing simple and
transparent connection between both parts in the assintotic regime. All the arbitrariness involved in this type of calcula-
tions are preserved in the intermediary steps allowing the identification of universal properties for the divergent inte-
grals, which are required for the maintenance of fundamental symmetries like translational invariance and scale inde-
pendence in the perturbative amplitudes. Once these consistency relations are imposed no other symmetry is violated in
perturbative calculations neither ambiguous terms survive at any theory or model formulated at any space-time dimen-
sion including nonrenormalizable cases. Representative examples of perturbative amplitudes involving different species
of massive fermions are considered as examples. The referred amplitudes are calculated in detail within the context of
the presented strategy (and systematization) and their relations among other Green functions are explicitly verified. At
the end a generalization for the finite functions is presented.
Keywords: Feynman Integrals; Perturbative Amplitudes
1. Introduction
Given the fact that exact solutions for Quantum Field
Theories (QFT) are rarely possible, almost all knowledge
constructed through this formalism about the phenome-
nology of fundamental interacting particles has been ob-
tained within the context of perturbative techniques. In
order to get the predictions in such framework, many
nontrivial mathematical difficulties must be circum-
vented due to the presence of infinities or divergences in
the perturbative series for the elementary process. We
have to find a consistent prescription to handle the ma-
thematical indefiniteness involved, which means to avoid
the breaking of global and local symmetries as well as
simultaneously to avoid ambiguities in the produced re-
sults. By ambiguities we understand any dependence on
the final results on possible arbitrary choices involved
in intermediary steps of the calculations. If they exist,
undoubtedly, the predictive power of the formalism it is
destroyed. The first and most immediate of such ambi-
guities are those associated with the choices of the labels
for the momenta carried by the internal lines of loop per-
turbative amplitudes. They naturally appear when the
divergence degree is higher than the logarithmic one. The
result for such amplitudes may be dependent on the par-
ticular choices for the routings due to the fact that in this
case the amplitudes are not invariant under shifts in the
loop momentum. A second and important type of choice
is the regularization prescription. Two different choices
for the regularization can lead to different results for the
calculated amplitudes. These two kinds of ambiguities
are very well-known in the corresponding literature. A
third and more general one has been recently considered
C
opyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1409
in the context of perturbative calculations, which is the
denominated scale ambiguities [1]. They are related to
the choice for a common scale for the finite and diver-
gent parts when they are separated in a Feynman integral.
There is an arbitrariness involved in the separation of
these terms in a summation when they have different
divergence degrees. The scale properties of the pertur-
bative amplitudes are the most general guides for the
consistency of the procedures. There are situations in
which a symmetry violating is non-ambiguous relative to
the choice for the labels of the internal lines momentum
but it is ambiguous relative to the choice for the common
scale. In addition to the difficulties coming from the di-
vergences we frequently have also those coming from the
extension of the mathematical expressions involved. Apart
from a few number of simple amplitudes, the mathe-
matical complexity of the obtained expression, not rarely,
makes prohibitive any analysis of the obtained results.
Considering these aspects of the perturbative calcula-
tions in QFT it would be desirable to get a procedure to
manipulate and calculate divergent physical amplitudes
without compromising the results with a particular regu-
larization scheme. In addition to this, we would like to
make the calculations preserving all the possible choices
for the arbitrariness involved like those related to the
choice of routings for the internal momenta and for the
common scale for the finite and divergent parts. To com-
plete such adequate calculational strategy it would be
desirable to get also a systematization for the finite parts
of the amplitudes in a way that the mathematical expres-
sions become simple allowing the required analysis and
algebraic operations related to the renormalization pro-
cedures, among others.
If one agrees with this line of reasoning the present
work may constitute a contribution on this direction. We
present in this paper a calculational strategy which ful-
fills the requirements stated above. We start by formu-
lating the steps involved in the calculation of perturbative
amplitudes, through the corresponding Feynman rules, in
such a way that no regularization needs to be specified.
The calculations are made by using arbitrary choices for
the internal lines of loop amplitudes and an arbitrary
scale parameter is introduced in the separation of terms
associated with different degrees of divergences. Through
the procedure no divergent integral is really calculated.
They are reduced to standard forms which are then un-
touched. The finite parts are not contaminated with any
type of modification and a systematization through
structure functions is introduced. The result is a com-
pletely algebraic procedure where no limits or expan-
sions are taken. All the procedures like Ward identities
verifications, renormalization procedures and so on, are
made by using properties of the finite functions and basic
divergent objects. In addition to this, the important aspect
of the procedure is its general character; all the ampli-
tudes in all theories and models are treated in an abso-
lutely identical way. We treat amplitudes in renormaliz-
able and non renormalizable theories formulated in even
and odd space-time dimension within the same strategy.
Symmetry violating terms as well as ambiguous ones
may be simultaneously eliminated in a consistent way.
Anomalous amplitudes are consistently described with-
out the presence of ambiguities in any (even) space-time
dimension.
The material we present in this work may be consid-
ered as an extension of that presented in [2]. The ques-
tions considered here are not new. In the literature there
are many works about this issue and certainly many oth-
ers continue to be done nowadays. In particular, the re-
duction of tensor integrals to scalar ones, made in the
present work through the properties of the introduced
finite functions, has been studied by Passarino and Velt-
man [3] as well as other authors [4-12]. The scalar inte-
grals has been considered by G.’t Hooft and Veltman
[13]. Recently, new works have been produced specially
involving massless propagators like in [14-29] (and ref-
erences therein). The present systematization for the per-
turbative calculations must be understood as a contribu-
tion to this type of investigation. The very general char-
acter of the procedure and the absence of restrictions of
applicability may represent some advantages which can
be useful for some users of the perturbative solutions of
QFT’s. With the material presented here any self-energy,
decay amplitude and elastic scattering of two fields can
be calculated in fundamental theories.
The work is organized as follows. In the Section 2 we
define the set of basic one-loop 4D Feynman integrals
which we will discuss in future sections. In the Section 3
we explain the strategy adopted to handle the diver-
gences as well as we define the basic divergent objects
used to write the divergent content of the perturbative
amplitudes. The basic functions (and some of their useful
properties) used to systematize the finite parts of the am-
plitudes are introduced in the Section 4. The solution of
the basic one-loop integrals is considered in the Section 5
and the explicit calculation of perturbative amplitudes in
the Section 6. In the Section 7 we consider the explicit
verification of the relations among the Green functions
for the calculated amplitudes and in the Section 8 the
questions related to the ambiguities and symmetry rela-
tions are discussed. A generalization for the finite func-
tions and their useful properties are presented in the Sec-
tion 9 and, finally, in the Section 10 we present our final
remarks and conclusions.
2. Basic One-Loop Feynman Integrals
First of all we call the attention to the fact that in pertur-
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1410
bative calculations, independently of the specific theory
or model, in loop amplitudes, we have to take the inte-
gration over the unrestricted momentum. We can con-
sider such an operation as the last Feynman rule. Pre-
cisely at this step all the one-loop perturbative ampli-
tudes will become combinations of a relatively small
number of mathematical structures, the Feynman inte-
grals. Some of such structures are undefined quantities
because they are divergent integrals. Given this situation
we have at our disposal two distinct but, in principle,
equivalent attitudes to adopt. We can perform the calcu-
lation of the desired amplitudes one by one, within the
context of a chosen regularization prescription or equiva-
lent philosophy, ignoring any type of possible systemati-
zation of the procedures or identifying the set of opera-
tions we have to repeat in calculating different ampli-
tudes considering such required operations in a separat-
edly way. In adopting the second option, the immediate
systematization of the perturbative calculations is to con-
sider the study of the set of Feynman integrals we need to
solve in order to calculate all the one-loop amplitudes.
Here we will restrict our attention to the fundamental
theories but this attitude can always be followed.
In this linere of asoning we first separate the ampli-
tudes by the number of internal lines or propagators.
Thus the one propagator amplitudes in fundamental theo-
ries will be reduced, in some step of the calculations, to a
combination of the integrals



4
4
1;
d
;.
2πi
k
k
D

22
ii
km

 


11
II
(1)
Here we introduced the definition
i
Dk . Such structures are the most sim-
ple ones but are also those having the most severe degree
of divergences: the cubic one 1
I
. The one-loop am-
plitudes having two internal propagators, on the other
hand, will be written as a combination of the structures



4
22 24
d
;;
2π
k
II I

1; ;.
ij
kkk
D

.DDD
(2)
Here ijij The highest degree of divergence
here is the quadratic one occurring in 2
I
. In calculating
amplitudes having three internal propagators we need to
evaluate the integrals


4
33 334
1; ;
d
;; ;;
2πijl
kkkkkk
k
II IID


DDD
.
(3)
Here we have defined ijlij l
. The higher degree
of divergence involved in the above set of integrals is the
linear one in 3
I

. Two of them are finite structures. We
can introduce also the ingredients required to calculate
amplitudes having four internal lines, the four propaga-
tors Feynman integrals


44 444
4
4
;; ;;
1; ;;;
d.
2πijlm
II III
kkkkkkkkkk
k
D


DDD
(4)
Now ijlmijl m
. Only one of such structures is
divergent which is the logarithmically divergent structure
4
I

kk
k
.
In the above definitions i and i
m are the arbi-
trary momentum carried by an internal propagator and its
mass, respectively. The arbitrary internal momenta i
are related to the external ones through the relations of
energy-momentum conservation in vertices connecting
the internal lines with the external ones. The adoption of
arbitrary routing for the internal lines momenta is of cru-
cial importance due to the divergent character of the
Feynman integrals involved, in particular for those hav-
ing degree of divergence higher than the logarithmic one
just because in this case the result may be dependent on
the chosen routing. In adopting such general arbitrary
routing for the internal lines we can identify possible
ambiguous terms arising in a certain calculation which
are undefined combinations of the internal lines momenta
(not related to the external ones). This aspect will be-
come clear in a moment.
When we find a combination of divergent Feynman
integrals in a certain step of the calculation of a pertur-
bative amplitude, in order to give an additional step we
have to specify the prescription we will adopt to handle
the mathematical indefinitions involved. Usually this
means adopting a regularization prescription or an equiv-
alent philosophy. All the results, after this, will be com-
promised with the particular aspects of the chosen regu-
larization. The so obtained results will represent only the
consequences of the arbitrary choice made for the regu-
larization. Even if there are elements of the calculations
which are independent of the regularization scheme em-
ployed, certainly, there are parts of the result which will
be specific of the particular regularization used.
In the present work we will follow an alternative pro-
cedure. We will not compromise the results with a par-
ticular choice in any step of the calculation. The choice
for the regularization will be avoided. The routing of the
internal lines momenta will be taken as arbitrary and the
most important and new aspect specially for calculations
involving different species of massive fields, the com-
mon scale for the finite and divergent parts, will be as-
sumed also as being arbitrary. With this attitude all the
possibilities for such choices will still remain in the final
results. Thus, it will be possible to make a very general
analysis of the results searching for the universal condi-
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1411
tions which are necessary to be preserved in order to get
consistent results in perturbative calculations. This means
to obtain results which are simultaneously free from am-
biguous and symmetry violating terms. In order to fulfill
this program, in the next section, we will describe the
strategy to be adopted in the manipulations and calcula-
tions of divergent Feynman integrals.
3. The Strategy to Handle Divergent
Feynman Integrals and the Basic
Divergent Structures
When we use the Feynman rules to construct the pertur-
bative amplitudes there are two distinct steps. First, with
propagators, vertex operators, combinatorial factors,
traces over Dirac matrices, traces over internal symme-
tries operators and so on, we construct the amplitudes for
one value of the loop momentum k. The next step is to
take a summation over all values for such momentum,
since it is not restricted by the energy momentum con-
servation at all vertices of the corresponding diagram.
This means integrating over the loop momentum. It is
possible to use these two distinct moments of the calcu-
lation to formulate a strategy to handle the divergences
present in perturbative calculation of QFT which may
avoid the use of a regularization [30]. The idea is very
simple and does not involve any kind of magic. Only an
adequate interpretation of the usual procedures is re-
quired. The first step is the same described above: to
construct the amplitude corresponding to one value of the
unrestricted momentum. Then before taking the integra-
tion, the last Feynman rule, we make a counting in the
power of loop momentum in order to get the superficial
degree of divergence of the amplitude in the space-time
dimension we are working. Having this at hand we adopt
the following representation for the involved propagators






22
1
22
=0
1
1
22
11
12
12
iii
j
Nii
j
j
N
ii
N
Dkk m
kkk
k
kkk
kk






22
2
1
222
22
,
j
i
N
i
ii
m
m
km




(5)
taking N in the summation as equal or major than the
superficial degree of divergence. Here
is an arbitrary
parameter having dimension of mass which plays the role
of a common scale to both finite and divergent parts of
the corresponding Feynman integral. Through this pa-
rameter a precise connection between the finite and di-
vergent parts is stated. Note that (as must be required) the
expression above is an identity and in addition the right
hand side is really independent of the arbitrary parameter
2
. After the adoption of the adequate representation for
the propagators and making all the convenient algebraic
reorganizations, we take the integration over the loop
momentum k. Then we note that the internal momenta
dependent parts of the Feynman integrals are located
only in finite integrals. On the other hand, the divergent
parts will reside in standard forms of divergent integrals,
after a convenient reorganization, where no physical pa-
rameter is present. Then we can perform the integration
of the finite integrals obtained and in the divergent ones
we need not to make any additional operation.
In order to allow a compactation of some expressions
in future sections it is convenient to introduce the defini-
tion ii i
222
2i
A
kkk m
 , so that we can write the
above expression as






1
1
11
22 22
=0
11
1jN
jN
Nii
jN
j
ii
AA
DkkD






. (6)
The steps above described, required to implement the
procedure, can be formulated within the context of the
language of regularizations. In such formulation we take
the integration over the loop momentum and then the
divergences are stated. We adopt then a regularization in
an implicit way in all Feynman integrals. It is required of
such regularization distribution only very general proper-
ties. In addition to rendering the integral convergent we
require that such distribution is even in the loop momen-
tum in order to be consistent with the Lorentz symmetry
and that a “connection limit” exists. Schematically
  

 
2
44
22
44
4
4
dd
lim ,
2π2π
d.
2π
ii
i
kk
fkfkG k
kfk






where the i
s
are parameters of the distribution
22
,Gki, and the limits which allow to remove the
distribution in the finite integrals
2
22
lim, 1,
ii
i
Gk
 
must be well-known. By assuming the presence of this
very general regularization we can manipulate the inte-
grand through algebraic identities just because the inte-
grals are then finite. Next, the identity (5) is used to re-
write the propagators in the Feynman integrals. In the so
obtained finite integrals we take the connection limit
eliminating the regularization and performing then the
integration. In the divergent integrals so obtained no ad-
ditional modifications are made. Only a convenient reor-
ganization in the form of standard objects is promoted.
There are no practical differences in both procedures
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1412
described above. The only difference is the presence of
the subscript in the divergent integrals indicating
that a regularization was assumed in an implicit way. The
first formulation however represents the evolution of the
second one proposed and developed by O. A. Battistel
and denominated as implicit regularization, just because
it allows us to perform all the necessary calculations
without mentioning the word regularization in perturbat-
ive calculation for any purposes, as we shall see in what
follows when representative examples of amplitudes
calculations will be considered in detail.
The terms which will be converted in divergent inte-
grals, when the integration over the loop momentum is
taken, can be conveniently organized so that all the di-
vergent content is present in the standard objects (at the
one-loop level in fundamental theories)









4
2
44
22
44
43
22
24
dd
2π2π
4
dd
2π2π
kkkk
k
k
4
43
22
43
22
4
4
,
g kk
k
k
g
kk
kk
k
g kk
k
 

  





 
(7)





44
2
43
22
4
d
Δ
2π2π
kk
k
k



42
22
d
,
g
k
k

(8)





44
4
22
2d
,
kk g
k


2
42
22
d
2π2π
kk
k


 
(9)


4
2
42
22
d1
,
k
k


log2π
I
(10)

4
2
quad 422
d1k
Ik
.
2π

(11)
In nonrenormalizable theories or in two or more loops
calculations new objects analogous to these can be de-
fined. Note that all the steps performed are perfectly va-
lid within reasonable regularization prescriptions, in-
cluding the dimensional regularization technique. This
means that it is possible to make contact with the results
corresponding to the ones belonging to such methods. To
do this it is only necessary to evaluate the divergent
structures obtained according to the specific chosen re-
gularization prescription just because the finite parts
must be the same due to the fact that, in all acceptable
regularization the connection limit must exist. As a con-
sequence, finite integrals must not be modified. More
details about the procedure will be presented in a mo-
ment when examples of perturbative (divergent) ampli-
tudes are considered.
4. Basic Structure Functions for the Finite
Parts
Once the procedure described above is adopted, finite
Feynman integrals must be solved. In general, to solve
such integrals is not a problematic task. However, fre-
quently, the obtained result is a very large mathematical
expression making difficult any type of analysis. The
experience, in realizing such type of calculations, re-
vealed that it is possible to identify basic functions to
systematize the results for the finite parts of the pertur-
bative Green functions so that the results became very
simplified and all the analysis required became simple
and transparent. Such basic functions will emphasize, in
a natural way, many important aspects typical of the per-
turbative physical amplitudes like, for example, unitarity.
Further required manipulations, in renormalization pro-
cedures, in the verification of relations among Green
functions or Ward identities, can be completely simpli-
fied in terms of simple properties of such basic functions.
It is possible to show that the finite parts of amplitudes
having a certain number of internal propagators can be
reduced to a unique function written, in an integral form,
in terms of Feynman parameters. Our next task will be to
define the referred basic structures and to explicit their
useful properties to be used in posterior sections where
we will consider the evaluation of the divergent Feynman
integrals defined in the first subsection above. The prop-
erties considered for such basic functions will be used in
future sections, when we will consider explicit examples
of amplitudes evaluation and in the verification of rela-
tions among Green functions.
4.1. Basic Two-Point Structure Functions
After the adoption of the procedure described in the Sec-
tion 3 above, when we are considering a calculation in-
volving amplitudes having two internal propagators the
finite parts so obtained can be always written in terms of
the following functions
1
22 22
12 2
0
;,;dln
k
kQ
Zmpmxx



m m
.
(12)
In the expression above, p is a momentum carried by
an internal line or a combination of them, 1 and 2
are masses carried by the propagators,
is a parameter
with dimension of mass which plays the role of a com-
mon scale for all the involved physical quantities and

22222 2
;, ,1Qmpm xpxxmmx m
12 12 1
. The role
of the masses can be inverted through a simple change in
the integration variable. In intermediary steps of pertur-
bative calculations it is enough to maintain the integral
representation but if one wants to solve the integration in
the Feynman parameter this operation can be easily per-
formed. For the first component of the above set of func-
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1413
tions we will obtain
 


22 22
01 2
22
22
22 22
12
2
;,;
2ln 2
;,; ,
2
Zmpm
pm
hmp m
p


 


2
22
12
22
2
1
ln
m
mm
pm




22 22
12
;,;p m


where hm
possesses three representa-
tions:
1) for . In this region of values for
we have

2
12
pmm
2
2
p

 
 
22
12 12
2
2
12
2
2
12
2
ln
hmmpmm
mm pm
mm pm



22
2
2
12
2
2
12
.
p
m p
m p


 
22
12
mm
2) for . In this case we get

2
1
2
mm p



22
22
12
2
2
mmp
m
p
 






2
12
.pmm
12
2
12
2
12
4
arctan
hmmp
pm
mm
 


3) for In this region we write
2

 
 

22
2
22
12
22
12
22
12
.
m
mm
mm
mm






22 22
12
;,;pm
22
12 1
22
12
22
12
22
12
2
ln
2π
hpmmpm
pmm p
pmm p
ip mmp
 
 
 

We can note then that the function
k
Zm
acquires an imaginary part in the region ,

2
12
pmm
2
as required by unitarity. It is possible to state relations
among the functions corresponding to different values for
k. Examples of such relations are




22
12
2
22 22
2
2
;,; ,
mm
p
pm


22 22
11 2
2222
2211
22 22
222
12
01
2
;,;
ln ln
22
2
Zmpm
mmmm
pp
pmmZm
p



(13)

22 22
21 2
22 22
12 22
222
222
12 2222
11 2
2
2
22 22
1
012
2
;,;
1ln
18 63
2;,;
3
;,; .
3
Zmpm
mm mm
pp
pmmZmpm
p
mZmpm
p
 
 



k
0k
(14)
Through such relations all components of the set can
be reduced to that having the number of reduced in
one unity and successively to finally be reduced to only
the
function. These type of reduction is very
useful in verifications of symmetry relations as we shall
see in a moment.
4.2. Basic Three-Point Structure Functions
In evaluating the finite parts of Feynman integrals asso-
ciated with amplitudes having three internal propagators,
Equation (3), we can obtain considerable simplification if
the results are written in terms of the following functions

1
11
22212
12312
00
;, ;,dd,
nm
x
nm
x
x
mpmqmxx Q

p q
(15)
where and are momenta of the internal lines or a
combination of them and,
 

22 2
12132
22
112212
2222 2
1221311
;,, ;,,
112
.
QQmpmxqmx
pxx qxxpqxx
mmxmmxm

 

If the considered amplitude possesses two or more
Lorentz indexes it is useful to define another set of
auxiliary functions. They are defined as
1
11
2222
1231 2122
00
;, ;,;ddln.
xnm
nm Q
mpmqmx xxx





(16)
The elements of the above set of functions can be re-
duced to nm
and k
Z
functions if useful or necessary.
However, in intermediary steps of calculations it is fre-
quently convenient to maintain the presence of nm
function to give a compactation of the results and opera-
tions. Now we consider useful properties for the func-
tions nm and
nm .
The first aspect is relative to the reduction of all the
elements of the set having a certain value for nm
to
that having 1.nm
1nm
We now show such reduction
firstly considering those for . We start by con-
sidering 01
. After some algebraic effort, which involves
only basic mathematical operations like integration by
parts, we can write the expression
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
Copyright © 2012 SciRes. JMP
1414
 




2
2
2 222222
1
010230 120 1
2222 2
222
222
13
12
00
222
1
;,;;,;
2
,
qpq pq
CZm pqmZmpmZm
pqpq p
pq qmm
pmm
ppq


 
 2222
3
;,;qm


 




















where we have defined
22
2
22 .
pq
pq pq
10
1
C
can be written as Through the same type of manipulations the function
 




2
2
222 2222
1
100230 130
2222 2
222 222
13 12
00
222
1
;,; ;,;
2
.
ppq pq
CZm pqmZmqmZ
pqpq q
pq
qmm pmm
qqp
2222
1 2
;,;mpm


 










 








10
and In the last two equations above, we can note that both functions 01
may be related through a set of simul-
taneous transformations.
The reduction of the functions 20
and 02
can be written as
 








2
22
2 222 22
1
020231 231
22 22
222 222
12 13
00 01
2222
;,; ;,;
2
1,
pq qpq
CZmpq mZm pqmZ
pq pq
pmm qmm
pq
pppq
22 22
1 2
;,;mpm



 








 

 


and
 






2
2
222
1
20123113
22 22
222 222
13 13
10
222
;,
; ;
2
.
pq ppq
CZm pq mZmqm
pq pq
qmm qmm
pq
qpq
22
00
2
1
,;
q

11

 








 




on the other hand, it is interesting to obtain two alternative forms. First we write For the component
 






2
2
222 2
1
111231 13
22 2
222 222
12 13
10
222
1
;,; ;,
2
.
pq q
CZmpq mZmqm
pq p
pmm qmm
pq
ppq
2
00
22
;
pq
pq













The second form is
 








2
22
222222
1
1102 312 31
22 2
222 222
13 12
00 10
2222 2
1
;,; ;,;
2
.
pq p
CZm pqmZm pqmZ
pq q
qmmpmm
pq pq
pqpp q
22
1 2
;,;mpm












 


 


2,nm
The explicit expressions for the nm
functions, corresponding to
can be completed if we develop the
00
in terms of nm
and k
Z
functions. Such function can be written as



222
13 01
1
.
2
m m
2
2222222
000231 001210
111
;,;
222
Zm pqmmpm mq
 








O. A. BATTISTEL, G. DALLABONA 1415
3nmThe expressions corresponding to the first reduction of the nm
functions having are






22
222 22
1
302232 13
22 22
222 222
13 13
20
222
;,
; ;,
2
.
Cpqp pq
Zm pq mZmqm
pq pq
qmm pmm
pq
qqp

22
10
2
1
; 2
q


 







 




and
 










2
22
222222
1
0302 312 322
22
222 222
12 13
22 22
21 2010
22222 2
;,;2;,; ;
2
1
;,; 2
pq q
CZm pq mZmpqmZm p
pq
pmmqmm
pq
pqZmpm
pqppp q




2
222
3
2
,;
.
q m



12 .



 




The two different forms for the function
are written as
 





2
2
222 22
1
21223213
22 2
222 222
12 13
20
222
1
;,; ;,;
2
,
pqq Zm pqmZmqm
pq p
pmm qmm
pq
ppq

22
10
222
Cpq
pq













and
 








2
22
222222
1
2122 312 301
22
222 222
13 12
11
222
1
;,;;,; 2
2
.
pqp
CZm pq mZm pq m
pq q
qmmpmm
pq
qqp

10
222
2
pq
pq











 






12 .
Firstly the form Finally we consider the expressions for the function
 











2
22
222222
1
120231 232 2
22
222 222
13 12
22 22
21 2010
22222
;,;2;,; ;
2
1;,; 2
pqp
CZm pq mZm pq mZm pq
pq
qmmpmm
pq pq
Zmpm
qpqqqp




2
222
3
2
2
,;
,
m






 





and then a second form can be obtained
 










2
22
2 222 22
1
1212 322 321
22 2
222 222
12 13
222 2
21 301
22222
1
;,; ;,;
2
1;, ;2
pq q
CZm pqmZm pqmZm
pq p
pmmqmm
pq pq
Zmqm
ppqpp





22 22
3
11
2
;,;
.
qm
q




nm




 







For the
used in the above expressions we have the following expressions



2
2222222
101231 101320
11
;,;2
36
Zm pq mmqm mp






222
1211 ,m m





and





22
222222 2222
0102 31231011202
11
;,; ;,;2
36
Zm pqmZmpq mmpm mq









222
1311 .m m


Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
Copyright © 2012 SciRes. JMP
1416
With these expressions we can write the functions
nm
corresponding to completely in terms of
of functions
3nm
k
Z
and nm
with 2.nm
1nm
The reductions present above are very useful in par-
ticular to allow the identification of important properties
of the basic functions associated to amplitudes having
three internal propagators. These referred properties are
required when relations among Green functions or Ward
identities are verified. They are particular combinations
of a couple of elements of the set of functions which can
be constructed directly from the reductions presented
above. The usefulness of these properties will become
very clear in future sections. They are
1)

:



2
222222 22222
011002301213 00
111
;,; ;,;,
222
qpqZmpqmZmpmqmm

 

(17)



2
22222222222
10010230121200
111
;,; ;,;.
222
ppqZmpqm Zmqmpmm

 
2nm

(18)
2) :



2
2222222
0211023001301
111
;,; ,
422
qpqZmpqmqmm

  (19)
 



22 222 2 2
1121310
11
;,; ,
22
Zmpmqm m



22
2222222
11201 23023
1;,; ;,;
2
qpqZmpqm Zmpqm
 

 



(20)



2
2222222
12 10
111
,
2
pmm
20 110 2300
;,
;
42
ppqZmpqm

 

 

(21)



2
222222 22222
31201
111
;,; ;,;.
2
ppqZmpqm Zmqmpmm
11 021231 1
22

 
3nm

(22)
3) :





22 222 2 2
2121 32
11
;,;
22
Zmpmq m m

222
22 222 222 22
21300 231232 23
0
1;,;2;,; ;,;
2
,
qpqZmpqm ZmpqmZmpqm
 
 

(23)
 


2
222222
03122230113 02
;,
; ,
22
qpqZmpqmqmm
2
11

  

(24)




22
2222222222
122122 312 3101311
111
;,;;,;,
2
q pqZmpqmZmpqmqmm
22




 


(25)
 


2
222222 22222
12032232 1312
111
;,; ;,;
222
ppqZmpqm Zmqmpmm
02 ,

 

(26)





222
1012 20
2
1,
2pmm

222
22 222 222 22
302102 312 322 3
1;,;2;,; ;,;ppqZmpqmZmpqmZmpqm

 
 
(27)
 



222
12 11
1
.
2
pmm
22
2222222
21122231 2301
11
;,; ;,;
22
ppqZmpqm Zmpqm






  (28)
functions, It is also useful to note similar properties involving the nm





 



2
2
22222
3
2
11
m
m


 

  

 
22222
01102321 2112
22
22 2
2222222222
22312321112
222 2
3212
1
ln1ln;, ;,
22
1
;, ;,;,
2
1;
2
qpqmmpZmpmZmpm
pqZm pq mZm pqmp m mZmpm
pqm mZmpq



 
 

 




  




 


(29)


22222
31300
1
,.
2
mqmm




O. A. BATTISTEL, G. DALLABONA 1417




 





22
2222222
32
10013221 311 3
22
22 2
222222
23 213231
222 2
2313
11
1ln1ln;,;,
22
1
;, ;,
2
1;
2
mm
ppqmmqZmqmZmqm
pqZm pq mZm pqmqm mZ
pqm mZm p
 


  
  


  
 
 


 



 



222
2222
11 3
;,mqm




22222
21200
1
,.
2
qm pmm




(30)
Furthermore, note that when on the left hand side we
have nm
for what , on the right hand side we
will have only functions with nm , and so on.
Such type of structures are precisely the expected ones
when the Ward identities are considered. It is clear that
other functions corresponding to higher values of and
, and analogous relations among them, can be obtained.
In the final Section 9 we will show how to generalize all
above functions and their relations to an arbitrary number
of points. At the present purposes the
will be enough.
3nm
2
n
m
nm
4.3. Basic Four-Point Structure Functions
The finite parts of four-point functions calculations admit
a systematization analogous to the three-point functions.
The basic functions are defined as
given above

112
11 1123
12 3
2
00 0
dd d,
ijk
xxx
ijk
x
xx
xx x
Q



(31)
where

 

22 22222
12132431122 3312
222222 2
1323 121 132 1431
;, ,;, ,;, ,1112
22 .
QQmpmxqmxrmxpxx qxxrxxpqxx
prxxqrxxmmxmmxmmxm

 
If the considered amplitude possesses at least two Lo-
rentz indexes it is useful to define another set of auxiliary
functions
112
1123
3
0d ,
ijk
xxx
11
12
00
dd
ijk
x
xx
x
Q
xx

  (32)
and if four or more Lorentz indexes are involved it is
convenient to define also the functions
112
11 1
123123 2
00 0
dddln.
xxx
ijk
ijk Q
xx xxxx
 


  (33)
The elements of the set of functions ijk
and ijk
defined above can be reduced to functions ijk
if useful
or necessary. However, in order to give a compactation
of the results and operations, in intermediary steps of
calculations, frequently, it is convenient to maintain the
ijk
and ijk
in the corresponding expressions. All the
functions of the set ijk
can be, at the final, reduced to
the most simple ones 000
. As examples of such reduc-
tions let us consider those corresponding to 1ijk

ijk
.
They can be written as
: 1) Functions



  



2
22
222 222222
1000023400 13412000
2
2
222 222222
0032400 12413000
00
2
111
;,;, ;,;,
222
111
;,;, ;,;,
222
1
2
qr qrm p qm p rmmqmrmpmm
C
qr rprqpmpqmrqmmpmrmq mm
pm
C
 
 

 














2
2
C
qp rqq r


222222222
4230012314000
11
;,;, ;,;,,
22
prmqrmmpmqmrm m











 
2
222 222222
0100023400 13412000
2
2
22
222222222
32400 12413000
2
2
00
2
111
;,;, ;,;,
222
111
;,;, ;,;,
222
1
2
pr rqrpqmpqmprmm qm rmpmm
C
pr prmpqmrqmmpmrmq mm
C
pq rppm
C
 
 

 






 





 


00
r q


222222222
4230012314000
11
;,;, ;,;,,
22
prmqrmmpmqmrm m


 


Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1418



 



2
2222222
00100234001341
2
2
222 2222
00324001241
2
2
22
00
2
11
,,;, ;,;,
22
11
;,;, ;,,;,
22
1
2
pq qrqprmpqmprmm qm rmpm
C
pr qppqrmpqm rqmmpmrmqm
C
pq pq
C



 










22
2 000
22
3 000
1
2
1
2
m
m









222222222
4320012314
11
;,;, ;,;,
22
m p rmq rmmpmqmrmm


 

000
,



 
222
2
2.r pq
,pq
where we have defined
 
22
222
2
Cpqrpqpr qrp qrqpr
Note that and

2 3
m m
010 100


2 4
,prm m
001 100

 .
ijk
2) Functions :




 



2
13
000
2
2
m
2
22 222
12
222 222
1000023400 134000
2
22
2
222222
0032400 124
2
11
;,;,;,;,
22
11
;,;, ;,;,
22
qr qrpmm
mpqmprmm qm rm
C
qm
qr rprqpm qpm qrmmpmrm
C
 


 


 

 









2
14
000 ,
2
m
22
2
222 222
0023400 123
2
11
;,;, ;,;,
22
rm
qp rqqrpmrpm rqmmpm qm
C

 



 








2
12
000
0
2
2
m
22
2
222 222
0100023400 134
2
2
22 222
13
22 2222
0023400 12400
2
11
;,;, ;,;,
22
11
;,; ,;,;,
22
pm
pr rqrpqmpqmprmm qm rm
C
pr prqmm
mqrmq pmmpmrm
C


 






 




 



2
14
000 ,
2
m
22
2
222222
0023400 123
2
11
;,;, ;,;,
22
rm
pq rpp rqmrpm rqmmpm qm
C

 
 


 





 



2
12
000
2
13
000
2
2
m
m

22
2
222 222
0010023400 134
2
22
2
22 2222
0023400 124
2
11
;,;, ;,;,
22
11
;,; ,;,;,
22
pm
pq qrqprmpqmprmm qm rm
C
qm
pr qppqrmqrmq pmmpmrm
C


 
 














000 .
2
2
22 222
14
222222
0023400 123
2
11
;,;, ;,;,
22
pq pqrmm
m rpm rqmmpmqm
C

ijk

 



 

3) Functions
:





 
222222
00000 23412100
2222
140011 000
11
;,;,
33
121
,
336
mpqmprmpm mq
rmmm
 








222
13 010
1
3
m m







 
222
3 4
2
100
,;,
11
,
224
mprm




222 222
100002341023 4 012
222222 222
1 22001 3110141011
111
;,;,;,;, ;
444
111
444
mpqmprm mpqmprm mpq
pmmqmm rmmm
 





Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
JMP
1419





 
222222 2
01010 23412110
2222
140111 010
11
;,;,
44
111
,
4224
mpqmprmpm mq
rmmm
 





 


22
13 020
1
4
m m






 
222222
00101 23412101
222 2
140021 001
11
;,;,
44
111
.
4224
mpqmprmp m mq
rmmm
222
13 011
1
4
m m
 





 



The systematization obtained through the functions
ijk
, ijk
and ijk
is enough to write all four-point
amplitude. In order to verify relations among Green
functions or Ward identities some properties of those
functions are useful too. In our case it is sufficient the
following properties:
i) :
1ijk
Copyright © 2012 SciRes.




2
100010 001
222 222222
002340013412000
11
;,;, ;,;,,
22
ppqpr
mpqmprmm qm rmpmm

 
 

 





222
12000 .mm
ijk
2
100010001
222 222
0023400 134
11
;,;,;,;,
22
ppqpr
mpqmprmm qm rmp






2) :
2






000 12100 ,
22
pmm



2
200110 101
222222222
0023 4102 3 4012 3 4
222
11
;,;,;,;,;,;,
22
11
ppqpr
mpqmprmmpqmprmmpqmprm










2
110020 011
222 222222
1023410 13412010
11
;,;,;,;,,
22
ppqpr
mpqmprmm qm rmpmm


 

 





2
22
2001,
ppqpq
m



101 011 002
222 2222
01234011341
11
;,;, ;,;,
22
mpqmprmm qm rmpm



 







22
2 22
0023410234
222
00012 100
11
;,;,;,;,
22
11 ,
22
mpqmprm mpqmprm
pmm



  


2
200110 101
2 222
01 234
;,;,
ppqpr
mpqmprm

 
 




2
110020011
222 222222
1023410 13412010
11
;,;, ;,;,,
22
ppqpr
mpqmprmmqmrmp m m










2
101011 002
22
2001 .
ppqpr
m


22
2 2222
0123401 1341
11
;,;,;,;,
22
mpqmprmm qm rmpm


  

O. A. BATTISTEL, G. DALLABONA
1420
3) 3ijk:





2
300210201
2222222
00 23410 2340123
222 22222
20 2341123402 2
2
100
1;,;, ;,;, ;,
2
11
;,;, ;,;;;,
22
1
2
ppqpr
22
4
2
3 4
;,
;,
pqmprm mpqmprm mpqmprm
mpqmprm mpqmprmmpqmprm
p


 
 









22
12200 ,mm

m




22
2020,m

2
120030 021
222 2222
20234201341
11
;,;; ;,;,
22
ppq pr
mpqmprmmqmrmpm










2
22
2002,m






102 012 003
222 2222
02234021341
11
;,;, ;,;,
22
ppq pr
mpqmprmmqmrmpm











2222
22
2 110
1;,;,
1
,
2
ppqpr mpqmprm
m


  

210 12011110 234
222 2222
20234112 3 40101
2
11
;,
;,;,;,
22
mpqmprmmpqmprmpm












22
22222
01 23411 23402 2
222
00112 101
11
;,;,;,;,;
22
11 ,
22
mpqmprmmpqmprmmpq
pmm






2
201111 102
22
3 4
,;,
ppqpr
mprm

 





2
111021 012
22
2011.
ppqpr
m

 

22
22222
1123411 1341
11
;,;,;,;,
22
mpqmprmmqm rmpm



 

4) 4ijk:





2
400310 301
2
3 41023 401234
222 222222
2023 41123 4 0223 4
3
133
;,;, ;,;,;,;,
222
33
;,; ,3;,; ,;,; ,
22
1
2
ppqpr
pqmprm mpqmprmmpqmprm
mpqmprm mpqmprmmpqmprm



 







22
2 22222
00 2
m





222222 222
02 3 4212 34122 3 4
222 222
03 23420012300
33
;,;,;,;,;,;,
22
131
;,;, ,
222
mpqmprm mpqmprmmpqmprm
mpqmprmpmm



  




(34)




2
130040 031
22
2 030
11
,
ppq pr
m

 

(35)
22
2 2222
30234301341
;,;, ;,;,
22
mpqmprmmqmrmp m









2
103013 004
222 222222
2 003
11
,
ppqpr
m

 
 


03 23403 1341
;,;, ;,;,
22
mpqmprmm qm rmpm

(36)
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
Copyright © 2012 SciRes. JMP
1421






2
310220211
222222 222
;,mprm
10 23 420234112 3 4
222 222
3023 42123 4
222 2
12 234110
1;,;,;,;, ;,
2
1;,;, ;,;,
2
11
;,;,
22
ppqpr
mpqmprmmpqmprm mpq
mpqmprm mpqmprm
mpqmprmp
















22
12210 ,mm

(37)





2
301211 202
2222222
01 23411 2340223
22222222
2123412234032
2
101
1;,;, ;,;,;,;
2
11
;,;, ;,;,;,
22
1
2
ppqpr
mpqmprm mpqmprmmpqm
mpqmprm mpqmprmmpqm
p



 









22
12201 ,mm

22
4
2
34
,
;,
prm
prm

(38)







2
22
34
22
2 111
;,
1
,
2
prm
m

(39)
211121 112
222 2
11234212
222 2
12 2340111
11
;,;,;,
22
11
;,;,
22
ppqpr
mpqmprm mpqm
mpqmprmpm

















2
22
4
22
2 120
;,
1
,
2
ppqpr
prm
m



(40)
220 130 121
222 2
20 23430 23
222 2
21 2340201
11
;,;, ;,
22
11
;,;,
22
mpqmprm mpqm
mpqmprmp m















2
202112 103
22
4
22
2 102
;,
1
,
2
ppqpr
prm
m

 

(41)
22
2 2
02 23412 23
2222
03 2340021
11
;,;, ;,
22
11
;,;,
22
mpqmprm mpqm
mpqmprmpm












2
130040 031
22
2030,
ppq pr
m

 

(42)
22
2 2222
30234301341
11
;,;, ;,;,
22
mpqmprmmqmrmp m









2
103013 004
22
2003.
ppqpr
m

 

Similar relations can be obtained for others compo-
nents of the set by exploring the properties relating these
functions which are the interchanges pq, pr
22
2 2222
0323403 1341
11
;,;,;,;,
22
mpqmprmm qm rmpm





(43)
,
mm, and mm (analogously to the
23 24 ij
func-
tio
rts of a four
pwill be nl
ns). The systematization allows us to treat the pertur-
bative four-point amplitudes in an exact way. By succes-
sive reductions all the content of finite-
ction ritten in terms of oy 000
pa
oint funw
(more 00
and 0
Z
). Let us now consider the evaluation
of the integrals (1)-(4) in terms of the systematization
After introducing the strategy to be adopted to hand
with thivergens in perturbative calculations of QFT,
as well as to state the standard divergent structures in
terms of which th
introduced.
5. Manipulations and Calculations of the
One-Loop Feynman Integrals
le
e dce
e divergent parts will be written and to
define the set of basic functions in terms of which the
O. A. BATTISTEL, G. DALLABONA
1422
finite parts will be written, we can consider the solu
the divergent Feynman integrals presen
tion
ofted in (1)-(4).
3D
5.1. One-point Feynman Integrals
If we want to solve the Feynman integral

1
I
defined
in (1), by using the procedure described in previous sec-
tions, first we identify the divergence degree
.
After this we have to adopt the adequate representation
for the propagator. This means taking in the ex-
pression (5) to get
3N







23 4
11 1
34 4
2 2222
1
.
AA
kkD



(44)
Next we reorganize inenient way in order to get the basic divergent structures defined in Section 3. Then we
1
1
kA
A
2
22 22 2
1
k
Dkkk

a conv
write the above expression in the form


 





22 2
2kkk


  111
1111
234
2222 22
1even
4
211
22 2
11 44
424
3
2
3,
kkkkkk
kkk
kkmk
Dkkk
kkkAk
km

 


 
 
 
 

 
 
 
 
t
o
organizations are made to get
completely in combinations
nd then we get
22 22
1
kk
D


where we have written only the terms which are even in
he loop momentum k by simplicity just because the odd
nes will be ruled out after the introduction of the inte-
gration sign. Convenient re
the divergent terms written
of the five objects (7)-(11) a

 
 








4
22222
1111 111111
4
1
22 222222
11 11quadlog
4
d111
322
2π
k
k
Ikkkkk kkkk
D
kmkkI mI
  

 


 
 
4
44
211
22 2
11
444
22 22
1
2
dd
3.
2π2π
kk kA k
kk
km kkD


 
 
 
Only finite terms will be integrated in the next step and no additional modification will be made. The result is the
expression

 
 



   
 



4
222
1111 111111
4
1
22 2222222
11 11quad1log1
d11
32
2π
i
4π
k
k
Ikkkkkkk
D
kmkkI mIm


 



 

22
2
2 21
1
22
1
2
ln .
kk
m
m




 
 

 

s for the definition of the divergent objects
precisely on this form will become clear in future sec-
tions. It is possible to show that for any value of N in the
involved divergent objects a regularization must be as-
sumed and the integration made. However, as we shall
see in a moment, this is not necessary in any situation.
ivergent i
follow the same
eaking, the same
in (44) can be
n be avoided by
The reason
expression (44) major than 3 the result can be put in the
above form. Note that, following our strategy, no men-
tion needs to be made to regularization techniques until
this step. On the other hand, the above result can be con-
veed to any regularization prescription since all the
ste
Now we can consider the quadratically d
gral defined in (1). For this purpose we
procedure applied above. Strictly sp
representation for the propagator used
adopted. However, algebraic effort cart
ps performed are perfectly valid in the presence of all
regularization distribution. Such eventually adopted regu-
larization, in this case, will be present only in the basic
divergent objects just because it can be removed from the
finite integrals by taking the connection limit. If, on the
other hand, we want to attribute a definite value for the
taking the value N = 2 in the expression (5) just because
the obtained expression may be put in the same form for
any superior value. Having this in mind in all situations
where we have to calculate the integral 1
nte-
I
we will have
to integrate the expression (omitting an odd term in the k
loop momentum)
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1423


 





2
22 23
11 1
1333
22 22 22
1
4.
km
kk A
k
kkkD



 

ion, we will get

22 2
11 2
22 22
1even
11 1
km
Dkk
 



So, taking the integration after some convenient reorg
1
k
anizat

 

2
22 2
4
22222
quad1 log
4
1
d1
2π
k
ImIkk
D


 
  
 
 




411
143
22
3
4
1
43
2
22 2
11
d
2π
d.
2π
km
k
k
A
k
kkkm


 




1
Solving the finite terms we obtain
 


2
.k





(45)
Again note the general character of the expression.
Only mathematical operations free from choices have
been made.
5. s
Now we consider the integrals having two propagators.
First we take the simplest one: the 2
2
222 2222
1
1quad1log1111
22
ln
4π
m
i
IImIm mk
 


 
 


 



where we have used the definition (6) in order to write
the expressions in a more compact way. Now we intro-
duce the integration sign to get



2. Two-Point Feynman Integral

I
integral. When
is integral needs to lved, as a consequence of the
used in (44wever, given the divergence
n be
r both propaga-
th
ap
be so
). Ho
plication of Feynman rules, we first adopt the repre-
sentation (5) for the propagators. If one wants to use an
unique representation for the propagators the expression
may be that
degree involved, some algebraic simplification ca
obtained assuming the value =1N fo
tors. We have to integrate the summation of terms

2
12
222
22 2222
1
12 12
11 ,
i
ii
AAA
DkkDkD

 



44
2
2
2log
442
22
1
12
4
12
42
22
d1 d
2π2π
d.
2π
i
i
A
kk
II
DkD
AAk
kD






12
The finite ones can be integrated by using usual tools
to yield
i

22222
2log012
2;,; ,
4π
i
II Zmpm




(46)
where we have introduced the definition 21
kk p.
The same procedure can be adopted when the integral
2
I
needs to be solved. In our procedure, before taking
the integration, we first write



 







12
2 222 2
21 1
44
22 22
2
43
22 22
11
2 2
i
ij i
ji
kkkkkk
m km
kk
Ak
kDkD
2
21 2
3
22
12 even
2
22
4
1
2
ij
kkk
kk k
Dk
AAk
22
A Ak
12
4
22
12
.
kD


 
 





 


 






Note that odd terms have been omitted. After some reorgan solving the finite integrals
btained to get
ization, we take the integration
o
 




22
2log
22 22
01 2
22 2222 22
11 201 2
22
;,;
;,;;,; .
22
PP
II
Zmpm P
mpm Zmpm




 

2
i
4πpZ






(47)
Here we have defined 12
Pk k.
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1424
Next, we can follow strictly the same procedure to get the expression for the integral 2
I
in our procedure. The
first step is to write






22 2222
2211
2 3
22 22
12 even
1122 124
22
22
22 222222 222 2
22112211
24
11
24
24
1
6
kk kkkk
kmkm
Dkk
kkkk
kk kk kkk
4
22
km
kmkmkm
k
 

 




 
  


 
 
 
 
 











kk











3
23
22
222
12
56 45
22222222
111
3
233
212
66
22 22
112
.
ij
ii i
iji i
ij
ij
ij
AA kk
A AkkAAkkAkk
kk kDkD
AAkk AAkk
kDkD

 
 
 

 





 



Now we take the integration, after a convenient reorganization of the terms to write the divergent terms as a mbination
of the basic divergent structures, and perform the integration in the finite terms by using standard techniques, to get
co
 



22 222
2222111
222
221122 2111
2
22 211211
2
12
2
11 1
2
26 4
11
22
12 12
122
12
I kkkkkkmm
kkk kgkkkkkk
kk kkkkkk
  
 
  
 





 

 
 
 



22 21
12
12 kkkk k




22
121 1quad
1
22
kkk gI












2
222 2 2
12log21log
2
2222
2
2221 2111log222
22222 22
21 211
11
2
41
2
11
22 ln
62
1
;,; ;
2
gmmIgkkI
m
kkkkkkkkIg mm
gpZmpm Zmp
 
 



 
 
 


 





 



222222
22112
2222 2222 22
121121111 2
22 22
110 12
,; ,,;
1;,; ;,;
2
;,; ,
mppZpmm
gmmZmpm kpkpZmpm
kkZ m pm

 







 



(48)
which completes the calculation of the Feynman inte-
grals having two internal propagators.
5.3. Three-Point Feynman Integrals
Now we evaluate the integrals having three propagators.
The first element of the set (3) is finite and may be cal-
culated by taking any value for N in the expression (5).
We write the result as


22 22 2
300123
42
123
d1 ;,,, ,
(2 π)4π
ki
I mpmqm
D



(49)
4
where we adopted the definitions 31
kkq and
21
kk p. The definition (15) for the nm
functions
has been used. The same comment applies to the second
element of the set (3). The result can be written as

 

4
34
123
01101 00
22
d
2π
i.
4π4π
k
k
ID
i
pq k
 
 

 

(50)
By simplicity, we will omit the arguments of three-
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1425
after taking the integration we have to adopt the adequate
representation for the propagators. In this case we can
first write
point functions nm
and nm
whenever it is not in-
volved four-point structures. The next integral of the set
(3), which is 3
I
, is logarithmically divergent. Then
 
33 123
33
22 2
1
4
1
4
ij
i
i
AAk k
kk AkkAAAkk
Dk

 






can
33
22 22
1123
.
i
j
iij
DkDkD


 

ter we take the integration. Solving the finite integrals we
2
k
123 even 
rted in a dect af
put the
kk

Only the first term will be conveivergent obj
results in the form
 



 

22
3 log
11
44
i
IgI
 
20
1
qpppqqpg
 
0211 1100
2
10110101101 100
22 2
2
4π
.
4π4π4π
q
ii i
kp qkpqkk
 
 


 


 
  




(51)
Now let us consider the linearly divergent structure, the integral 3
I
. The first step is to rewrite it using (5), as we
did above, and next we solve the finite integrals to write the result as




 







123 even
22
22
2
33 3
65
22 222
,, 11,, 1
ij ijl
ij l
ijli jijl
lij
2
2
333
321123
45645
2222 222222
111
24
12
ij
ij i
ijij ij
ij
AA
AA
kkkk
kkk A
AAA
DkkkkDkD
AA AAA
AA A
kDkDk
   
 
 
 
 

 
 


  







kkk




2
22
123
66
222
123
.
jl
AAA
Dk D
ten
ations in the finite terms the result can be put into
the form
By reorganizing in a convenient way the first term so that it is writas a combination of the basic divergent objects
(7)-(11), and after this taking the integration and performing the oper



  

 

 


2 222
3123123
22 22
123 log123 log
11
12 24
1111
122 122
11
Ikkkkkkg gg
kkkg Ikkkg I


  
 
 
 
 
  
 
 


2
12 3 log
12 2
kkkg I

 



 





21 21 121212
1010 10010101
111 001101
2
11
22
4π
i
qpq pqqpqpqppppq
gqgqgqgpg pgp
ikkkkk pq

  
 
 










2
300321
2
4πqqqppp qqp







101 10110
1 1011010220111100
10220111100
11111 022000
1
2
1
2
1
2
kk pq
kk pqkppqqqppqg
kppqqpq qpg
kpqqpppqqg

 

 

   

 








 






. (52)
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1426
In fundamental theories the considered integrals are
enough to evaluate the one-loop amplitudes having three
internal propagators.
5.4. Four-Point Feynman Integrals
Finally, we consider the four-point function integrals.
Only one of them is a divergent structure which makes
the job easy. The first, the scalar one, can be written as


2
000
4π,
(53)
where we have identified the four-point structure func-
tions previously defined in the Equation (31) and also the
external momentum 41
rk k
. Next, one can imme-
diatly see that, for the vector integral, we can write
I
4414
,JkI

 (54)

 
2
4100010 001
4π,Ji pqr


 (55)
and that for the one having two Lorentz indexes, we have
I
441414114
,
4
IiJkIkIkkI
 

 
 (56)
where

22222 222
4123423 24
;, ;,;,;;,mpmqmrmJpqmmJprmm
 




(57)


JJ


2
2222
3 4000200110101
1
;,;,4π
6
mqmrm igpppqpr
 

12
;,Jm
p



(58)
On the other hand,
441414141 141 141141 114
,
I
JkI
 



kIkIkkIkkIkkIkkkI
  
 
 
 (59)
where

22222 22
23423 24
, ;,;;qmrmJpqmmJprmm
 




2
41
;, ;JJmpm
 
(60)



2
10
1
4π
2
Ji gpgpgp
 
 

0
.
 
300
ppp
 
(61)


210 201
ppqpqpqppppr prp rpp
  


 
 
The last one we consider is the logarithmically divergent one, which we write as
 
 
 



222 2
log
11
24
g
gggg gg ggI
  
 
 
 


48
gggggggggggg



2 2
44
11
24 48
IJ g
 
 

 

22
g
  


 

g


gggg
  
k

411114,
222
Ikk Ikk IkkIkkk IkkkkI
 
  

 
  
 
(62)
where

1411411 4114111
111
  
222222 22
123423 24
;, ;,;,;;,mpmqmrmJp qmmJp rmm




(63)
4
J J








 
 
2
000 200
011 400
031 013
11
4π
12 2
11
23
11
+
33
Ji gg gppgpp
gqrrqg qr rqpppp
rqqqrqqqqrrr qrrrqrq
  
 
  



  




 


JJJJ






022
211 .
r
pqprprpq qprp rpqp

   

(64)

r
qrq
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1427
ith the above results for the Feynman integrals at hand a
kkW
we can perform all the one-loop amplitudes for one, two,
three and four fermionic propagators in the context of
fundamental gauge theories. In the next section we eva-
luate some representative amplitudes involving vector
vertexes.
6. Physical Amplitudes
In the preceding sections we have considered the evalua-
tion of the Feynman integrals introduced in the Section 2,
crucial for the one-loop calculation in the con-
text of fundamental gauge theories like QED. All the
integrals have been written in terms of the set of diver-
which are
gent objects;

,
,
, I
and in
log and quad
I, de-
t11) terms of the func-fined in
tions
he Equations (7)-(
k
Z
, nm
and, nml
o, th
ing prop
p am
three
defi the Equations (12),
wree anur-point functions,
eg the above cited
oplitbe reduced to a
pieces: 0
ned in
d fo
rties relatin
udes can
(15) an
respectiv
functio
comb
d, (31)
ely. B
ns, all
ination of only
for t
y us
one-lo
basic
Z
, 00
and,
000
.
In the present section we will evaluate some represen-
tative amplitudes of the perturbative calculations by us-
ing the systematization introduced in the preceding sec-
Ward identities. We choose for this purpose
n functions of the Standard
nctions having only fer-
perat anonwe stat
qntiti
tions. We will consider an example for each number of
points taking the amplitude corresponding to the higher
degree of divergence. With this attitude we will have an
opportunity to use all the ingredients we have introduced
in our proposed systematization. In next sections we will
consider the relations among Green functions, ambigui-
ties and
simple but representative Gree
model; the one-loop Green fu
mionic internal lines. It is simple to state relations among
these structures as well as to state Ward identities to be
obeyed by them.
In the construction of such Green functions through
the Feynman rules, apart from coupling constants, inter-
nal symmetry oorsd so , have toe the
amplitudes for one value of the loop momentum k, which
are theuaes


;;
;.
ij liFa ajFbb
lFdd
tTr SkkmSkkm
Skkm


(65)
heT quantities are vertice operators belonging to
the set
55
1, ,,,
i



appearing in the coupling of fermionic currents to the
bosonic fields in the Lagrangian. After defining the op-
erators corresponding Lorentz indexes are attached to
ij l
t . The quantities
F
S are fermionic propagators
carrying momentum
and mass a
m which we
will write as
,
aa
Fa
kkm
SD

where through the quantity

2
aaa
Dkkm

 

state a connection with the procedure described in the
proceeding sections. The corresponding one-loop ampli-
tudes are obtained by taking the integration of the t
structures in the loop momentum k;

2 we
4
4
d.
2π
ij lij l
k
Tt
 

In the present work we will consider the cases where
the structures above correspond to divergent amplitudes
for one, two, three and four-point functions. They are all
connected due to relations among Green functions and
Ward identities as we will see.
6.1. One-Point Functions
We start by taking the cases having the highest diver-
gence degrees; the one-point functions. First, we write
for the one value of the k momentum, the quantities

;,
iiFaa
tTrSkkm
 
or

 
11
111
11
1.
kk
tTrmTr
DD
 (66)
The corresponding one-loop amplitudes, obtained by
integrating the above structures in the loop momentum,

4
11
4
d,
2π
k
Tt

are divergent quantities. The superficial degree of diver-
gence is cubic. Now, taking two different possibilities to
the vertice operators we can construct the one
functions which will be useful in future developm
First we take the scalar one-point function which means
to e
-point
ents.
assum 11
. We get then

 
1
1
11
1
1,
Skk
tTr mT
DD

r
or, solving the Dirac traces,
1
1
1
4.
S
tm
D



At this point we adopequate reptation for
the propagator
t the adresen
, as we have made when we discussed the
solution of the 1
I
integral. Then we get
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1428

 



4
2
11 1
4
22 2
1 log
d4
2π
SS
k
Ttmkk
m I







divergent objects as
well as the presence of a potentially ambiguous term, the
lastnce here 1
k
2
1quad
4mI



2
22 2
11
22
1
ln .
4π
imm
m








Note the presence of the basic
one, si
is arbitrary.
king 1
Now ta
 in the expression (66) we get the
vector one-point function



1
1
11
1.
Vkk
tTr mTr
DD

 

Using the results for the Dirac traces involved we get
1
1.k
grals
1
I
11
DD

Adopting the adequate representation for the propaga-
tor as we have made in the calculation of the inte
4
Vk
t

and 1
I
we get
 

 


2
111
2
4
3kkk






4
2
11
4
222
111
111
d4
2π
222
2.
VV
k
Tkt k
kmk
kkk







 


Note that the result is completely potentially ambigu-
ous since all the quantities involved are arbitrary (the
momentum 1
k and the scale 2
). Let us now consider
an example of two-point functions.
6.2. Two-Point Function
If one wants to consider a representative Green function
of
nic
e write them from the defini-
tion (65) as


2
the perturbative calculation, concerning the consis-
tency in the manipulations and calculations involving
divergent Feynman integrals, certainly there is no better
onrmioe than the fetwo-point functions. We will con-
sider three of such amplitudes related among them
through Ward identities. W



 
12 12
1
212
12
kk kk
tT
kk
mTr D


 


Firstly we consider the scalar-scalar where 1
12
12
rD

2
11
2 1212
12 12
1.
kk
mTr mmTr
DD

 
21
(SS). For this case we get first (after taking the
Dirac traces)
 
22
1212
12 12
11 1
SS
tkkmm
DD D

 

Now when the integration is taken the problems we
have to solve are the integrals (45) and (46). Following
the procedure we have adopted we get
.

 

 
 


22
2
quad1 log
4
222 2
quad2 log2
2
d
22
2π
4π
Tkt
ImI
i
ImI
ipm




 
 


 


4
2
22
2 2
21 2log
2
222
1 1
22
1
ln
4π
i
mm
22 2
22
ln
m pm
m
 


2
22
22
12 012
2;,
4π
SS SS
m
m mI
m Zmpm















2
22
pp PP
 







g 11
2
;,


Next, we consider the amplitude scalar-vector (SV) by takin
and 2
, we get


21
12
22
SV k
tmm
D
 
e we have to solve the integrals (46) and (47). We get then
12 21
12
1.
mkmk
D
To calculate the corresponding amplitud
 



 


22
21log
22 2222 22
2111 21012
2
1
2
;,;;,; .
4π
pmmI
ipmmZmpmmZmpm



4
21
4
d1
442
2π
SV SV
Tt
kmmP




 
 
 
(67)
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1429
Now we consider the most complex and interesting
case; the vector-vector (VV) amplitude. It is obtained
from the general definition (65)ming 1
by assu
,
2
 . We get the the expression
,kgt




where we have adopted the definitions


212
2,
VV
ttk
 


PP






2,
sij
tkk

12
1,
ij ij
kk kkskk kkD



  (68)
and

2
2
12
11 1
,
PP
tpmm
DD D

 

assumes the val-
ues 1
12 12
which is precisely the pseudoscalar scalar (PP) two-point
function. In the definition (68) above s


. After taking the integration in these expressions
we have to solve the integrals (45), (46), (47) and, (48).
Substituting the obtained results we get



 


4
22222 2
2121 2
4
2 2
22 12 11
2 2
2212 121122 122111
d,4 2
24
33
22
22 22
33
k
Ttkk mm
kk kk kk
kk kk kkkkkk kk kkkk
 
  
 
 
 
 


 

 
 
 
 
 
 
 
22
21 21
2π
22kk kk 
 
 


2
22222 1
1 lo112
2
222 2222
2
quad2log222
log 2
2ln
2ln
2
34π
m
I m
m
gImIm m
gppp I

  
 
 
22 2
22 12 11quadg
2
3gk
kkkkkg Imm
  
 
 

 
 
22
2i

 

 




 


 









 


2
21 2112
2 2 222222 2222
1201 21211 2
2 2
8;,;;,;
2;,; 4;,
4π4π
gpppZmpmZmpm
ii
gpmmZmpmgmmZmpm
 
 





 
 
 
an
2
; ,
22222222
d

 

 




4
222
quad 1
4
d2
2π
PP PP
k
TtImI

 
 
 
2
2 222
log11
22
1
2
222 2222
quad2 og
2
22
12 log0
l
n
4π
2l
4π
2
i
mm
m
i
Im
l
I22
22
2
nmm
2
4π
m
i
pm
m IZ
 












 


 


 


m







22 222
12
;,; .pmpp PP
 






Then








22 2
log1 2log
222222222
21 2112
2
2222 2222 22
1201 211 2
222 22
12 012
2
3
24 ;, ;;, ;
4π
;,; 2;,;
;,; ,
VV
TgpppI gmmI
igpppZmpmZmpm
gmmZmpm Zmpm
gmmZmpmA
 
 

 



 
 
 

 


 


 

where we have defined the quantity
2
4
(69)
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1430
 
 
 
22
2
5.








Note the presence in the above expression of poten-
tially ambiguous terms since the quantity
12
Pk k
is depences for arbitrary quantities as well as
the presence of terms dependent on physical combination
of the arbitrary intern 21
pk k which
are not dependent on tor the routing of the
internal lines momenta of the loop amplitude but are de-
pendent on the arbitrary choice for the common scale.
6.3. Three-Point Functions
Now we consider the case of three-point functions. In
this case the higher degree of divergence involved is the
linear one. We will take three related amplitudes in order
to exploit the potentiality of the proposed systematization.
From the definition (65) we get first the expression



22222222
12
2
1
422 35
6
11
33
33
11
33
36
AmmPp
PPpPpP ppPP pp
PPp pgP Pp PpPpp
 
  
 
 
 
 

 







 
  
 

dent on choi
al momentum
he choices f









 



123 123 23
1231 123
123 123
13 12
21233123
123 123
32
121231312 3
123 123
kk kkkkkkkk
tTr mTr
DD
kk kkkk kk
mTr mTr
DD
kk kk
mmTr mmTr
DD
1
23 1 23
123
kk
mmTrD
 


 

 
 

  
 
 
 
 



123 123
123
1.mmmTr D

(70)
e take in all vertice scalar operators

123
ˆ
1
So if w
  we get



 
 
 
22
131223132 32
13 1223
22 22
2313 13212 1
123
111
222 2
1.
SSS
tmm mmmmmkkmm
DDD
mkkmmmkkmmD
 





By using the developments made in solving the integrals (46) and (49) we get the expression














 

4
2
log
4
2
2
22 2222 22
101 30123200
2
2
2
22222222
20230123100
2
2
22222
d
2;,;;,;
4π
2;,;;,;
4π
2;,;;,
SSS SSS
k
Tt mI
imZmqmZmpmpq mm
imZmpqm Zmpmqmm
imZmpqm Zmq
 
 

 


 




 



 



2
22 2
;mpmm




1
123
4mm

30
2 3
2
4π
On the other hand, taking
01 32
100

2π
,
21 and, 31
is obtainin
study of integrals (46), (47) and, () we get
n Equation (70) and by using the resultg in the
49
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1431














023012
2
22
322110
222 22222
11 301 3
2
π
2
22 ;,;;,;
4π
2
i
pp
qmmmm qm
iqZmqmZmqm

2
2222222
2;,;;,;
pZmpqmZmpm



31 00
2
222
0 23
; ,;
m
Zm pq m
22
31 log
222
4
VSS
Tk
k qpI


 
 
 
2


 






22
3221 01
ppqmmmm pm

 


Having two vector inde get the SVV amplitude
,,
SPP
gt






2
2100
.m
xes we






23 13133233 12
,,
SVV
tmtkk mtkkmtkk
 
 


where we have defined








3
sij i ji
t kkkkkkskk
 
  
123
1
,,
j
kk D




with 1s , and



 
 
 
13 1223
13 1223
22 22
13232231313212
11 1
22 2
2
SPP
tmmmmmm
DDD
mkkmmmkk mmmkk m
 



22
1
123
1.
mD


We get then

 




 

 

22
13log1300 1
2
1 320101 311011 210 100
1 311011210 100
24
4π
22
2,
SVV
SPP
i
TmmgIgmm qqm
ppmmpq mmmmm
qpm mm mmgT
 
 
 

302101
2
mm


 

 












(71)


2SPP 













123log
22
22 2222 22
101 3012323200
2
22
2
2222222
202301 2313100
2
2
222222
302301 3
2
4
2;,;;,;
4π
2;,;;,;
4π
2;,;;,;
4π
TmmmI
imZmqmZmpmkk mm
imZmpqmZmpmkk mm
imZmpqm Zmqm
 
 

  


 




 



 

 

22
2
212 100
.kk mm






(72)
Finally, let us consider the case of triple vector operators. First we get
,
PPV
gt
 






31233213 3312
4,,4,,4,,
VVVVPP PVP
ttkkktkkktkkkgtgt
 







wh ave ben introdu

ere the following definitions heced



1232321
31212
123
313
1,
kkmmkkkkkkmm
kk kkkkmm D


 

 


4
VPP
tkk kk
 






1232321
31212
123
4
1,
tkkkkkkmm kkkk
kkkkkkmm D


 



313
PVP kkmm
 
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1432



1232321
31212
123
4
1,
PPV
tkkkkkkmm kkkk
kkkkkkmm D


 

 

313
kkmm

and







3,,
sijlij lj
tkkkkkkkkkskkk
 

  
123
1,
l
kD



with 1s . With the aid of the integrals (49), (50), (51) and, (5


,,
sijl
tkkk

may be written
explicitly by
2) the tensors 3














 

  
22
1231 231 23
22
123 123
2
123
132
11
,, 11
12 24
11
24
11515
24
11515
24
s
Tkkkkkk sgkkk s
gkkk sgkkk s
sk sksk
sk sksk
   
  
 
 


 

 
 

 






11
24



 

 




 


22
123
22
log log
2
log100001 00
10 00
151
24
11
11
12 12
11
111
12 2
11
2
kkks
gqsps qpIgpsqsqpI
gpqsI gpssqs
gp sq

  

 
 










 




 






 









 
01 001001
30200302212011 10
212011 102111
12 11
1
111
2
11 1
111
11
ssgps qs
ppp sqqq sppqs
pqps sqppss
pqqss qpq
 
 
  
 
 
 
 

 












1202 11 01
120211 01
1
1.
s
qqps s
 








On the other hand, the exVPP
T
pressions for ,
P
VP
T
P
PV
T
may be written as and,








 








22222 222
31log11 301 3
2
2 2
2222 2
0232132012100
2
2
2
222 22;,;;,
4π
;,;2
2
VPP i
TkkpqI qZmqmZmqm
Zmpqmppqmm mmpmm
i
ppqm
 

22 22222
01 2023
22;
,;;,;
4πpZ
mpmZmpqm
2 2
;


 
 
 


 








2
2
132 103100
,mmmq mm



 




 




















222222
21log01 202
2
22 222
1123 1 3210
22
222222
31 00013023
2
222 2;,;
4π
2;,;2
2;,;;,;
4π
2
PVP i
TkkqpI pZmpmZm
Zmpmqpqmmm m
i
qm mqZmqmZmpqm
 
 


2
222
3
22
;,;
pqm
 
 
 








 

 





2
22
31 3 2012 100,qpqmmmm pmm



  


Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1433




















22 2222
32log01 202
2
2
2
2222
123312 1103 100
2
222222 2
0130231
2
22;
4π
2; ,;2
2;,; ;,;2
4π
PPV i
TkkpqIpZmpm Zmp
Zmpqmpqmm mmqmm
iqZmqm ZmpqmZ

 


2
222
3
,;;,;
qm
 
 
 
6.4. The Four-Vector Four-Point Function
four fermioni
ur-vector four-point function, given by
 

 

 













2
222
23
2
2
3121 012100
;,;
2.
mpqm
pqmmm mpm m









Finally, in the next section we perform the calculation of four-point functions.
As an example of calculation of a Green function of the perturbative calculations havingc propagators, we
consider the fo
11223 3
11 1
VVVV
tTr
kk mkkmkkm
 

 4 4
1
,
kk m




or













3
12 4
123 4
12 3 4
1234
12
34
12 13
121 2
1234
2
1
23 1
14 23
121 2
1234
VVVV kk kkkkkk
tTr D
kk kkkk
mmTr mmTr
kk kkkk
m mTrmm Tr
D
 
 

12
24
1234
kk
DD


  
  
  
 
 
 










12
4
1234
12
12
1234
kk
D
kk
DD
2
1
13
24 34
12 12
1234
1234
1234
1.
kk kkkk
mm Trmm Tr
mmmmTr D

After performing the Dirac traces we identify the following structure
VVPP
gt
 
 
 
  




4
.
VVVVPPVVPVPVPVVP VPPV VPVP
PPPP
t tgt gt gt gtgt
gg gg ggt
 
 






(73)
d usl tenstemced

;
1234
,,;,kkkk


(74)
where
In the above expression a convenient anefusorial syatization was introdu











;;;
;1234; 1234;1234;
;;
; 1234; 1234
,;,,;,,;
,;,,;,,
ttkkkk tkkkktkkkk t
tkkkk tkkkk
  
  
  
 










;
12
4; 1
2
,;,
ijlni ji
ln l
tkkkkkkkk kkk
kk kkkk

 
 
  

1234
1.
j
n
k
kk D






(75)
1
Here and 2
assum the values 1. We also see that the coefficients of the metric tensor are four-point
amwith vector aneudoscalar vertices defined as
e
d psplitudes
12 3 4
123
11 22 33
111
tTr
kk mkkmkkm

   
 4
44
1
.
kkm




Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1434
After performing the Dirac traces, the four-point amplitudes with vector and pseudoscalar vertices acquire the form



2
1 3
1 4
2
kk
kk
kk
k
 

 


 
 





 


12 3 4
13 4341221
32 424134
52 323146
71 414238
4
4
4
4
ts kkkkmmkkkkskk
s kkkkmmkkkkskk
s kkkkmmkkkkskk
skkkkmmkkkkskk
 

 

 

 


 





3
2 4
4
k
kk
kk
 



 







91 313 2410
11121 234123
13
4
4
4,
PPPP
skkkkmmkkkkskk
skkkkmmkkkks kk
sg t




 

 


(76)
i


where

5,
 and 1
i
s and
55 5
112233
111
PPPP
tTr
kkm kkmkkm
 
 5
4 4
1
.
kkm




Below we identify the values of i
s
according to the corresponding amplitude
7 8 912341234 5 6
1011 1213
s
sssss
ss
PVPV
PVVP
VPPV
VPVP
VVPP
 sss

ss

PPVV

 









 





 





 





 





Some algebraic effort is necessary in order to obtain an expression for the above amplitudes. This is a tedious task,
although easy, because the number of external momenta and Lorentz indexes involved produce very large mathematical
expressions. Consider first the t=2j=3 and =4l. From the results (53), (54), (56), (59)
and, (62) we get
ensor (74) for =1i, , k



  
  





12
;2 222
222
2
12
1111
22
111
222
1
41
ss g g
ggg
srsqJ
  
  






  

 

 













221
1221
212 1
41
41 4
44,
qsrJspJ spJ
sqrsqr JrsqpJspJ
qsrpJspJqrsqrpJ spJ

 
  
4; 1 2
11
62
Tss g
 

 
 
 
log12
41 1
6gg ggggIss J
   

 



 

 
 
 
 
 
 
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1435
J
J
J
where J

,

,
and
are given by in Equations (55), (57), (60) and (63). Replacing the above result
(with appropri1
ate values for the symbols and 2) in Equation (74) gives

  
  


222 2
111 4
222
ggg ggggggI
  
 

 
  


2 222
log
4111
322 2
3
32 16
Tggg
JrJpJqrJqp
  
 


 

 
 
 
 
 








8
888
4
4
JprpJ
rq qrJpq qpJrpprpq qpJ
pqr qrpqr qrpqrqrJ
p
 
   
  
r



 
88qr rqrp prJqrqrpqqp prrp J
  

 
 
 












4
4.
qr qrpqr qrpqrqrJ
pqr qrpqr qrpqrqrJ
pqr qrpqr qrpqrqrJ
  
  

 




For the amplitudes listed in the table above we may write
 
 
  
 

 


1234 22
1log0123
678913 ,
PPPP
TsgIgFppFqqFrrFp
pqFqpF qrFrqFsgT
 
 

 
  






45
rFrpF
 
where

000
6 000
1
s
s


  




 
2
2
02 003 004005 004312
2
2
22
4178 00031910 000
2
2
1
11
Fs ss srqmms
rmms sqmms s
pm



  
 


 

2
22 2
4234 000325
1rpmm ssqpmm s


  





    
 
 


2
11112 000
1,mss













1220 320 52002 511191291010
177 81210111112797 891010
111112 7 9 78 910011112 00
2
1
22242
2222
2222 1
2
Fss ssssssss
sssssssss ssssss
ssssssssss s
srqm
 


   


  
 
 


 




 

222
4322002100342
2
2 2
2
53262007418200
22
2 2
9311020010 100112112
112
2121
2112
ms ssrpmm
sqpmmssr mms
sq mmsssp mms



 
 





 
 
 


 

 





4 200
8 100
1
1
1
s
s



200 ,








230242052073017 83 40111123
2
2
2222 22
22 2
Fss ssssssssss
ssssssssssrq mm


 
 
 

 







0
2
2 2
2
326 0207418 020
22
22
93110 020112112020
12
212 1
qpmmssrmms
sq mmssp mms








 

 




10
1s



 
11 34 1011127 11 78 101432020
22
3424020401
21
1srp mmss





8 010
12 010
11,
1
s
s





 

5
2s



Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1436









32024025029 5 01910 5601115
11 12560111911 1291001
2
22
1432 002342
2
53
2222 22
22
212
2
Fsss sssssss
ss ssssss ss
srqmmssrp mm
sqpm
 

  
 
 
 
 
 




 

 
 
2
26 0026 001
2
2
22
7418 00293110 002
2
2
112112 00212 001
11
2121
211,
ms s
srmm ssqmm s
sp mmss










 
 
 
 






01
2
4 002
1
s
s




 
10 001
1s



 


2021195 109 120111 9 1112910 11 78 01
2
2
119109 0000143
2 22
222
sss ssssssssssss
sssrq mm
 
 
 

 
 



 
4115
1121012 001
22
3424 101
2 1
21
ssss
srp mms
 






 

2
2
5326101
2
2
22
7418 10100193110 101100
2
2
112112 101100
21
2121
21,
sqp mms
srmm ssqmm s
sp mms
 






 
 
 
 





22F s

  
 
100
001 000









 


 
11 9791011 12011112910109100011 1200
2
222
1432 101001342
22
222
212
sssss ssssssssss
srqmmssrp mm


 
 
 
 







 

2
2 2
2
53261016 1007418
2
2
93110 10110 001100000
2
2
11211210112 100
1
212
21
21,
sqpmmsssr mms
sqm mss
sp mmss





 

   





 







5211502119 101019 105610

4 101
1018 001
1
s
s





22 22Fs ssssssss
 
  


63115201173 107 120111127 11 78
22 2222Fs sssssssssss
 
 
  

 



 
2
2
11 127017001112001432110
22
3424 1101
22
22 1
21
sssssssr qmms
srp mms
  


 
 







 

2
2
005326
2
2
2 2
7418 11010001000093110
2
2
11211211012 100
21
21 2
21,
sqpmms
srmm ssqmm s
sp mmss












 





9 1010
2 010
sss
s



110
110 010
1




 


7311520117 8101783 410111271178
22 222Fssssssssss sssss

 




117 8017 80011001432110
22
34241104
22
22 1
21
sssssssrq mms
srp mmss
11 9 10
2
2
010
s s
 
 






6 110
s




 

  








 


2
2
100532
2
2
7418 1108 010100000
2
2
9311011010 010
2
2
112112 110100
21
21
21
21,
sqpmm
srm mss
sqmmss
sp mms















 







Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
Copyright © 2012 SciRes. JMP
1437



 


841151111 5 1011 9 1011 340111 78 01
2
222
1432 0113424 0114
2
2
532
2222 22
212 1
21
Fs sssssssssss
srqmmssrpmmss
sqpmms
11 0000
001
2s
 

  
  
 
 











2
2
6 01101074180118
22
22
93110 011010112112 011010
21
212 1
srm mss
sqmmsspm ms
 
 

 
 
 

 

  

 
 
001
001000 ,







 



 
941151111 1230111 1270111 125610
2
2
11 129101011 1200001432011
22 2
34240110015
222 2 2
222 1
212
F ssssssssssss
sssssssr qmms
srp mmssqp
 
 

  
 

 
 









 


2
3 26
2
2
22
74180110019311001110 010
2
2
112112 01112 001010000
1
2121
21,
mm s
srmmssqm mss
sp mmss







 
 
 
 





0116 010
s






00
2
2
000
2
.
rp
p




 






2
222222
log024012
22
00 00
22
222
00
42;,;,
22
22
PPPP
TI ZmrpmZmqm
qrqrpqp rrq
pqp prqqrprq




 




 
 


 





Above, the following compact definitions were also
used

222
123
;, ;,,mqm

222
124
;, ;,,
nmnm mpmrm


222
134
;, ;,,mrm

222
234
;,;,,pmr pm



112 34 56
01112
1
1,
s ss
nmnm mp


nmnm mq


nmnm mq


and

11sss s
78
91
11
s
ss s


2125
11sssss

s s

 
 
69 10
1,s s


4 78
11 1,

312 3
s
sss sss


434 56 1112
1,


11
s
ssss
ss

11 12
1.

5 78910
11
s
sss
ss s 
is point, fulfilled
oposed systematization
mplitudes. However,
another important aspect involved in perturbative calcu-
lations can be also considered which, within the context
of our procedure, became very simple and transparent,
that is the verification of relations among the Green
functions and, consequently, of the associated Ward
identities. We perform such task in the next section.
7. Relations among Green Functions
In the preceding sections we have described in details a
procedure to handle the divergences typical of the per-
turbative calculations in QFT. The procedure is very
general since all the choices involved have been pre-
served; the internal momenta were taken as arbitrary so
that all possible choices can be made in the final results,
the choice of regularization is avoided since all the steps
performed are allowed in the context of all reasonable
regularization prescription and an arbitrary scale was
adopted in the separation of terms having different de-
grees of divergent and finite ones. We can ask ourselves
at this point about the consistency of the perrmed op-
erations as usual in such type of manipulations and cal-
culations. In order to verify this aspect we can make a
minimal test of consistency by verifying if the relations
among the calculated Green functions remain preserved
after the realized operations. The required consistency is
to verify such identities without assuming particular
fo
Our main purpose has been, at th
which is to show how the pr
works in the calculation of physical a
O. A. BATTISTEL, G. DALLABONA
1438
choices for the involved arbitrariness, which means that
the relations need to be satisfied in the presence of poten-
tially ambiguous and symmetry violating terms. Essen-
tially, what we want to know is if the performed opera-
tions have preserved the property of linearity of the inte-
gration which seems to be a trivial task but, given the
mathematical indefinitions involved, it is not. Only if the
operations realized until this point possess the desired
consistency we can give an additional step which is to
verify if the potentially ambiguous and symmetry violat-
ing terms can be eliminated in a consistent way. Let us
consider this aspect in detail now.
We start by considering the VV two-point function
whose calculation we have considered in detail in the Sec.
(VI). In order to state a relation with other calculated
amplitudes it is enough to note the identity bellow

 




21
11
11
21
11
11
kk kk m
kk mk
mm kk m












 
2 2
2 2
2 2
11
11
.
kk m
k m
kk m












After taking the Dirac traces in both sides we can
identify that
 
2 1,
VS
mmt

(77)
The above relation means that it is expected that if we
integrate both sides in the loop momentum k the corre-
sponding relation among the loop amplitudes remain
valid, i.e.,
 
2 1.
VS
mmT

(78)
This means that by calculating all the involved ampli-
tudes in a separated way and after this contracting the VV
amplitude the reorganization of the terms must allow the
identification of the amplitudes in the specific combina-
tion of the right hand side. This type of identity is highly
ontrivial to be preserved in traditional regularization
r
 
2 1,
SS
mmT
(79)
which implies that




21 21
2111 22
21 1122
2
21
,,
,,
.
VV
VV
SS
SS
kk kkT
kk TkmTkm
mmTkm Tkm
mmT




 
 

(80)
We can note from the above expressions that all am-
plitudes of the perturbative calculations are related
among them. In particular, the above considered relations
involve the amplitudes: VV, VS, SS, PP, V and, S.
For the calculated three-point function structures we
can verify the relations



31
21233 1
,, ,
VVV
VV VVSVV
kkT
Tkk TkkmmT

 
 

 
(81)

 

21
13232 1
,, ,
SVV
SV SVSSV
kkT
Tkk Tkk mmT





(82)


 
32
12133 2
,,.
SSV
SS SSSSS
kkT
TkkTkk mmT



(83)
Now we can note that all the three, two and one-point
calculated functions are in fact related among them
through precise relations. In the above considered rela-
tions the following structures are involved: VVV, VVS,
VSS, VV, VS and, SS plus the ones which appear as sub-
structures: VPP, SPP, PP and S.
If we consider four-point functions, the same will oc-
cur. To evaluate the VVVV function all the above men-
tioned structures will appear as well as other four-point
structures. This is a very crucial point. We can start from
a finite amplitude and by successive contractions we can
relate such amplitude with the cubically divergent one-
point function. The challenge is then to evaluate all the
perturbative amplitudes within a certain prescription
maintaining all the relations among them preserved in a
simultaneous way. Within the context of our procedure
we will show that all the relations presented above can be
verified in the presence of all remaining arbitrariness. We
emphasize that such type of verifications are very non-
trivial for all traditional techniques.
Let us start by the property (78). Taking the expression
for the VV amplitude, Equation (, and contracting
with 21
kkp


21
112 2
,,
VV
VV
kkt
tkmtkm





21
112 2
,,
VV
VV
kkT
Tkm Tkm



n
pescriptions. A similar procedure allows us to state that

 
21
112 2
,,
VS
SS
kkT
TkmTkm
 69)
we get
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1439




12
21 21
211
22
101
12
,
2
4;
4;
,.
VV
pTk k
mm mm
mmpZm
mpZmpm
pAkk


  






2
log
22 22
1 2
22
2
,;
,;
pI
pm






By comparing to the resu) for the VS mplitude
we can identify

2

lt (67 a



12 12
22
,,
2
VV
pTk kpAk k
mmkk

 


 

21
21
21 12
,.
VS
mmT kk





In order to complete the verification of the property
(78), the last term in the above equation must be identi-
fied with the one-point vector functions. It is simple to
note that if an
is added and subtracted in the ex-
pression for
A
, a reorganization allows us to identify
 
VV
pAT kT k




.
22 2
2121
2mmk k

 
12
  

So, the relation (78) is obtained preserved by our
calculation.
The relation (80) is, on the other hand, emblematic to
explain many aspects of our procedure and we will make
the discussion in details. First we note that by contracting
the expression (67) for the VS amplitude it is obtained







2
12 1122
22
21 log
22222
12112
22222
101 2
2
2
4;,;
4;,;.
VS
pT
mm k kkk
mmpI
mmpZmpm
mpZmp m








 

 


(84)
We know that this result needs to be related to the SS
amplitude as well as with S amplitudes having different
masses. This means that quadratic divergences need to
appear from the right hand side in a non-cancelling way.
At first sight it seems that it is not possible to satisfy the
relation. However, we note that on the left hand side of
the identity (84) we have the function Z1 and in the right
hand side only Z0 must appear. Let us consider the reduc-
tion of Z1 to Z0 through the property (13) in order to
adequate the right hand side of the Equation (84). The
referred reduction is the property (13) which allows us to
write
 
22222
11 2
2;,;pZ mpm
2222222
1201 2
22
22 22
12
11 22
22
;,;
lnln .
pmmZmpm
mm
mm mm


 
 
 
 
(85)
Now consider the result obtained for the 1
I
integral
at the value 0,
i
k
which is nothing more than a scale
property of the basic quadratic divergent object
2
quad
I,


22222
quad 1quad1log
2
22 2
11
22
1
ln .
4π
ImIm I
imm
m








(86)
We get then

 



2
22221
quad 1ln m
i
ImImmI




quad21log22
2
22 2
2log22
22
4π
ln
4π
m
i
mI













 
the scale properties of the divergent objects
22 22
12
11 22
222
ln ln .
4π
mm
immmm


 

 

 

Now note that we can relate the reduction of the finite
functions to
 
 

22222
11 2
222 2222
1201 2
22
2;,;
;,;
pZmpm
pmmZmpm






quad 1quad 2
2
22 1
1log 22
2
22 2
2log 22
ln
4π
ln ,
4π
Im Im
m
i
mI
m
i
mI
Substituting in the expression for VS amplitude we
will identify the relation (79) among the Green f
VS, SS, and, S. Note that the precise connection
the finite functions and the basic divergent object allows
exact way the considered relation
ions. It is not necessary to emphasize
that the same procedure is nontrivial within the context
of traditional regularization methods.
Let us now consider the relations among the three-
point functions calculated in the previous sect
tracting the VVV amplitude, calculated in
with q




















unctions
between
us to verify in an
among Green funct
ion. Con-
last section,
and using the properties (23), (24), (25) and
(29) in order to eliminate the nm
functions having
3nm in favor of those having 2nm we get
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
Copyright © 2012 SciRes. JMP
1440









 

log
222222
212 112
2
222
1121 2101
4
2;,;,
π
;,
I
igp ZmpmZmpm
mpmmmmZm


2
22
log
4
3
g pqpq pqI



22
21
4
mmZ
2
12 23
2
222
1223 log
4
,,
3
2
VVV
qTA kkA kkgppp
gm mmmI
 







 
 
 


 


















2
22222
332123
222
22 3
2222
02 3
;,
8 ;,;
; ;2
m mZm pq m
ppZmpqm
Zm pqq



 


 




22 22
2
22 2
22
223 1
22
222
23202 32
22
22222
12321
;,;
2;, ,
;,;
4π
2; ,;;,,
pm
pqZm pq mZmq m
i
mmmZm pq m
Zm pq mZmpm





 




 

2
2
;p
22
2
;m2
1120 10
pq q


 











 



22
22
22222
22 312 30211
2
2
2
3 101002
1
8;,; ;,;
2
4π
11
22
4π
iqqZm pqmZm pq mqpq
qmm p








22
22
2222
22 3123
;
,;;,;Zmpqm Zmpqm
 
8
iq

 

 

2
22 2222
01 2023
1;,;;,;
4ZmpmZm pq m








  
 





2
3 01
2
222
23123
2
;,;
m
mp qm
2
22 2222
01 2023
1;,;;,;
4ZmpmZm pqm

2
2222
1230211
22
1 0000
1;,;
2
1
2
Zm pq mqpq
m q








 
2
31 2 110
2
222
2
2
11
24
8;
,;
4π
pqmm mmq
ipqZmpqmZ


  








 






 




22
11 20 0110
2
2
003 1003 1
130013
2
1
2
11
24
48
4π
qp
q qpqq
qmm mm
igmmmmp









 

 




13 11 1311
88
p
mm
qp mmpq
 

 
Given the obtained result, we now use the properties (1
22 222
11 20211
2
2
3 1312 110
22
1
1;,;
2
2
Zmpmqpq
pqmmmmmm
mm gI
 






 

 
g
9) a
3 log
 




201 302
8
.
SPP
mmqq
T




nd (20) to eliminate the nm
functions having 2nm
in favor of those having 1nm. We get then










2
2 2
log log
2222222
4
3
2;,;,
g pqpq pqI
gp ZmpmZmpm
 



 

2
22
12 23log21211 2
2
22 222
2111221101
24
4π
;,
I
i
gm mmmI
mmZmpm mmmZm

2
12 23
4
,,
3
VVV
qTA kkA kkgppp
 







 


 
 


 


 












22 22
2
22 2
2222
22323
22
2222 222
321233 22023
2
2222
22312
2
;,;
2;,;,
;,;,;
8;,;;
4π
pm
pqZm pqmZmpqm
mmZmpqm mmmZmpqm
ippZmp qmZ m





 


 
 
 
 




1

2
22
3
,;pqm
O. A. BATTISTEL, G. DALLABONA 1441

 




 







22 2222 2
21 211
;,; ;,;Zmpm Zmpm

2
2
22
222222
223123
2
22
222222
223123
2
2
22 222222
0 120230110
8;
,;;,;
4π
8;,;;,;
4π
1
;,;;,;
42
iqqZmpqmZm pq m
iqpZm pq mZm pq m
ZmpmZm pqmqpq




















1



 











 
2
2
31 00
22
222222
22 3123
2
2
22 2222
0 12023
22
31 13log
1
4
8;,;;,;
4π
1
;,;;,;
4
24
qmm
ipqZm pq mZmpqm
ZmpmZm pqm
mmmmgI

 












 

 

02
1 311012 110
42pq m mmmg
 
 
2
2 2
01103 100
2
1
1
24
4π
qpqqm m
ig






 




1300
1 011311012110
42
mm
mqp mmmm


 
2

1320 1013
88p
pmmqq mm
 

.
SPP
T
 

 



Finally, using relation (17) we write

 

 






222
12 log
222222 2
21 211 221112
2
2
22 222
2110121223 log
4
23
8;,, ,
4π
4;,;,2
4
3
VVV
qTgmmIgpp pI
igpppZmpmZmmm
gmmmZmpmAkkgmmI
gp
 
 
 


 
 
 

 

 

 

 




2
log
2222
4;
gm mZmp

2
;
p
 









  
 
22
log
222
2222
22 312 3
2
22
22 22222
321233 22023
22
233 11 3log
8;,;,
4π
4;,4 ;,;
,2
qpqpqI
igpqpqpqZmpqm Zmpqm
gmmZmpqm gmmmZmpqm
AkkmmmmgI

 
 
 



 



 










 




 
 

 

1 3001320101 302 101
2
13110121101 00
1 311012 110 100
488
4π
42
42 ,
SPP
igmmppmmqqmm m
qpmmmmm
pqmmmmmg T
 

 

 
 

 


 






If we consider the results for thplitudes VV and
SVV , Equations (69) and (71) to note that
the expression above may be id as being the
relation (81). It is not difficult to verify the relations (82)
and (83) by performing the same sequence of steps.
The procedure used above can also be adopted to state
analogous constraints to the four-point Green function.
As an example of such constraint we have

 

41
123 234
41
,, ,,
.
VVVV
VVV VVV
SVVV
kkT
TkkkTkkk
mmT
 
 



e am
, it is now easy
dentifie
 

 



In order to show that the calculated four-point ampli-
tude VVVV satisfies this relation, at first we contract Eq-
uation (73) with

41
kk
and eliminate the ijk
Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA
1442
having i + j + k = 1 in favor of those having i + j + k = 0.
The next step is to use the properties (34)-(43) in order to
eliminate the ijk
and ijk
functions having i + j + k =
favohose having i + j + k = 3 and so on. The
latioeasy but involves a lot of algebra, therefore
we will now it explicitly. All the required ingredi-
have given in the preceding sections.
8. Ambiguities and Symmetry relations
e Sect6 we have evaluated, within the systema-
on ped, Green’s functions which are typical of
erturb tive calculations. In particular, all the con-
d amudes appear in the context of Standard
. the evaluated Green’s functions, having
ee ofrgences higher than the logarithmic one, it
is possible tnote the presence of terms where the de-
nce he internal momenta appear as arbitrary
tities (th summations of them). This is expected
ce a sh in the integrating momentum generates sur-
faces terms which implies that different choices for the
ernal lines momenta lead to different am-
udes.possible dependence on the choices for the
ls of tnternal lines momenta characterizes what
enoe as ambiguities. This situation is not ac-
in this case, the power of predict-
of thry is destroyed. In addition, fundamental
tries like the space-time homogeneity
are not prese ved in the perturbative calculations. It will
sung to find global and local gauge symme-
tries as well as internal symmetries violated in physical
itudes ving the space-time homogeneity broken.
ere is onone possibility to save such type of calcula-
oinate the ambiguous terms in a consistent
niv way. Within the context of the adopted
gy tbiguous terms are automatically separated
d preservso that it is easy to identify them.
one-point function it is simple to identify

2
1
4,kk






4 in
calcu
ents
In th
tizati
the p
sidere
Model
degr
pende
quan
sin
label of t
plit
labe
we d
tion
space-tim
not be
ampl
Th
tions: t
and u
strate
an
In th
r of t
n is
t sho
been
ion
ropos
a
plit
In all
dive
o
on t
e
ift
he int
This
he i
minat
ceptable just because,
e theo
e symme
r
rprisi
ha
ly
elim
ersal
he am
ed
e case of
ambiguous 11
S
Tm










22 2 2
111
2
111
2
111
22
2
4
3
2.
kmk
kkk
kkk








 






In the tnt functions we get
2
1
ambiguous4
V
Tk
 

wo-poi
2,P





2
2,
SV mP




2
ambiguous,
PP
TPP


ambiguous
ambiguous
  



 

22
ambiguous
2
22
2
1
2
13
3
13.
6
VV
TP
PP pP pP
PP PP
gPPpPpP
 
 


 
 
 


 
 






 






In the case of three-point functions we found


2
31
ambiguous2,
VSS
Tkk


  
 


2
31
ambiguous 2,
VPP
Tkk


 
 


2
21
ambiguous2,
PVP
Tkk


 
 


2
32
ambiguous 2,
PPV
Tkk


  
 














2
123
amb
2
231
2
123
2
132
2
123
2
123
2
123
4
3
222
3
222
3
222
3
222
3
222
3
222 .
3
VVV
Tkkk
gkkk
gkkk
gkkk
kkk
kkk
kkk

 
 
 





 
 













 


 

In all the above listed ambiguous terms it can be noted
that they invariably appear as multiplying the objects ,
and, . All these terms present simultaneously
scale ambiguities because such objects are dependent on
. This is due to the fa
SS
TP




21
Tm
 

the arbitrary mass scale ct that in
all amplitudes the obtained expression is independent of
the parameter 2 if the terms containing the objects ,
and, are absent. This statement can be verified
directly by differentiating the expression or changing the
scale to another one, like for example one of the involved
fermionic masses, through the scale properties of the
finite function and of the basic divergent objects
2
quad
I
2
log
I and . The referred properties are
 

22222
quad 1quad1log
2
22 2
11
22
1
ln ,
4π
ImIm I
imm
m








Copyright © 2012 SciRes. JMP
O. A. BATTISTEL, G. DALLABONA 1443
,
 

2
1
22
ln ,
π
m
i




22
log 1log4
Im I

22 2222 22
121
2
1
2
2
;,;
,
pm







12
2
;,;
1ln
1
kk
Zmpm Zm
k


2222 2222
1232 1231
2
1
2
2
;,;,;;,;,;
ln .
11
nm nm
mpm qmmpmqm
1
nm





xpressions for
the perturbative amplitudes which are nonambiguous
relative to the choice for the internal lines momenta, but
are ambiguous relative to the choice for the common
scale for the finite and divergent parts. This aspect can be
easily noted in the considered amplitudes. In the VV
two-point function
 
This means that there are terms in the e

 
 

scale_amb
222
12
22
2
422
55
66
11
33
1.
3
mm
pg
pp pp
pp


 

 




  


 






In the SVV amplitude
22
2
22
VV
T
pp
















2
13
2,
SVV
Tmm
 

 
 
scale_amb
and in the VVVV amplitude
 
 
 

scale_amb
2
42
33
44
33
24
33
4.
3
g
gg
gg
g
 
 
 
 












In such examples the listed term
22
22
22
VVVV
T


 
 












s are independent of
the choices for the internal momenta. They can be con-
verted in ambiguities through their evaluation in inter-
mediary steps within the context of traditional regulari-
zation techniques. Again we can note that all the poten-
tially scale ambiguous terms are combinations of the
objects
and, .
Let us now consider the symmetry relations. It is easy
to see that the situation is completely similar to the ques-
tion of ambiguities considered above. There are two
types of impositions coming from the symmetries for the
ampl