A Systematization for One-Loop 4 D Feynman Integrals-Different Species of Massive Fields

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 generalized 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 reasonable 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 propagators, the finite and divergent parts are separated before the introduction of the integration in the loop momentum. Only the finite integrals obtained are in fact integrated. The divergent content of the amplitudes are written as a combination of standard mathematical object which are never really integrated. Only very general scale properties of such objects 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 calculations are preserved in the intermediary steps allowing the identification of universal properties for the divergent integrals, which are required for the maintenance of fundamental symmetries like translational invariance and scale independence 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 dimension 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.


Introduction
Given the fact that exact solutions for Quantum Field Theories (QFT) are rarely possible, almost all knowledge constructed through this formalism about the phenomenology of fundamental interacting particles has been obtained within the context of perturbative techniques.In order to get the predictions in such framework, many nontrivial mathematical difficulties must be circumvented 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 mathematical indefiniteness involved, which means to avoid the breaking of global and local symmetries as well as simultaneously to avoid ambiguities in the produced results.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 ambiguities are those associated with the choices of the labels for the momenta carried by the internal lines of loop perturbative amplitudes.They naturally appear when the divergence degree is higher than the logarithmic one.The result for such amplitudes may be dependent on the particular 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 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 divergent 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 perturbative 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 divergences we frequently have also those coming from the extension of the mathematical expressions involved.Apart from a few number of simple amplitudes, the mathematical complexity of the obtained expression, not rarely, makes prohibitive any analysis of the obtained results.
Considering these aspects of the perturbative calculations in QFT it would be desirable to get a procedure to manipulate and calculate divergent physical amplitudes without compromising the results with a particular regularization 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 complete 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 expressions become simple allowing the required analysis and algebraic operations related to the renormalization procedures, 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 fulfills the requirements stated above.We start by formulating 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 untouched.The finite parts are not contaminated with any type of modification and a systematization through structure functions is introduced.The result is a completely algebraic procedure where no limits or expansions 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 amplitudes in all theories and models are treated in an absolutely identical way.We treat amplitudes in renormalizable 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 without the presence of ambiguities in any (even) space-time dimension.
The material we present in this work may be considered as an extension of that presented in [2].The questions considered here are not new.In the literature there are many works about this issue and certainly many others continue to be done nowadays.In particular, the reduction 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 Veltman [3] as well as other authors [4][5][6][7][8][9][10][11][12].The scalar integrals has been considered by G.'t Hooft and Veltman [13].Recently, new works have been produced specially involving massless propagators like in [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] (and references therein).The present systematization for the perturbative calculations must be understood as a contribution to this type of investigation.The very general character 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 we explain the strategy adopted to handle the divergences 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 amplitudes are introduced in the Section 4. The solution of the basic one-loop integrals is considered in the Section 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 relations are discussed.A generalization for the finite functions and their useful properties are presented in the Section 9 and, finally, in the Section 10 we present our final remarks and conclusions.

Basic One-Loop Feynman Integrals
First of all we call the attention to the fact that in pertur-bative calculations, independently of the specific theory or model, in loop amplitudes, we have to take the integration over the unrestricted momentum.We can consider such an operation as the last Feynman rule.Precisely at this step all the one-loop perturbative amplitudes will become combinations of a relatively small number of mathematical structures, the Feynman integrals.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 calculation of the desired amplitudes one by one, within the context of a chosen regularization prescription or equivalent philosophy, ignoring any type of possible systematization of the procedures or identifying the set of operations we have to repeat in calculating different amplitudes considering such required operations in a separatedly way.In adopting the second option, the immediate systematization of the perturbative calculations is to consider 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 amplitudes by the number of internal lines or propagators.Thus the one propagator amplitudes in fundamental theories will be reduced, in some step of the calculations, to a combination of the integrals Here we introduced the definition i D k .Such structures are the most simple 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 .
Here ij i j 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       Here we have defined ijl ij 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 propagators Feynman integrals

In the above definitions
i and i m are the arbitrary 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 crucial importance due to the divergent character of the Feynman integrals involved, in particular for those having 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 become clear in a moment.
When we find a combination of divergent Feynman integrals in a certain step of the calculation of a perturbative 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 equivalent philosophy.All the results, after this, will be compromised with the particular aspects of the chosen regularization.The so obtained results will represent only the consequences of the arbitrary choice made for the regularization.Even if there are elements of the calculations which are independent of the regularization scheme employed, 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 procedure.We will not compromise the results with a particular 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 common scale for the finite and divergent parts, will be assumed 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-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 ambiguous 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 calculations of divergent Feynman integrals.

The Strategy to Handle Divergent Feynman Integrals and the Basic Divergent Structures
When we use the Feynman rules to construct the perturbative amplitudes there are two distinct steps.First, with propagators, vertex operators, combinatorial factors, traces over Dirac matrices, traces over internal symmetries 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 conservation 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 calculation 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 required.The first step is the same described above: to construct the amplitude corresponding to one value of the unrestricted momentum.Then before taking the integration, 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 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 parameter a precise connection between the finite and divergent 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 parameter 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- , so that we can write the above expression as 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 properties.In addition to rendering the integral convergent we require that such distribution is even in the loop momentum in order to be consistent with the Lorentz symmetry and that a "connection limit" exists.Schematically where the i s  are parameters of the distribution , and the limits which allow to remove the distribution in the finite integrals must be well-known.By assuming the presence of this very general regularization we can manipulate the integrand through algebraic identities just because the integrals are then finite.Next, the identity ( 5) is used to rewrite 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 additional modifications are made.Only a convenient reorganization in the form of standard objects is promoted.
There are no practical differences in both procedures 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 perturbative 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 integrals, when the integration over the loop momentum is taken, can be conveniently organized so that all the divergent content is present in the standard objects (at the one-loop level in fundamental theories) In nonrenormalizable theories or in two or more loops calculations new objects analogous to these can be defined.Note that all the steps performed are perfectly valid within reasonable regularization prescriptions, including 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 regularization 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 consequence, finite integrals must not be modified.More details about the procedure will be presented in a moment when examples of perturbative (divergent) amplitudes are considered.

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, frequently, the obtained result is a very large mathematical expression making difficult any type of analysis.The experience, in realizing such type of calculations, revealed that it is possible to identify basic functions to systematize the results for the finite parts of the perturbative 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 perturbative physical amplitudes like, for example, unitarity.Further required manipulations, in renormalization procedures, in the verification of relations among Green functions or Ward identities, can be completely simplified 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 properties 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 relations among Green functions.

Basic Two-Point Structure Functions
After the adoption of the procedure described in the Section 3 above, when we are considering a calculation involving amplitudes having two internal propagators the finite parts so obtained can be always written in terms of the following functions 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 common scale for all the involved physical quantities and   tions we will obtain where h m  possesses three representations: 1) for .In this region of values for we have We can note then that the function  acquires an imaginary part in the region ,   ; , ; 1 ln 18 6 3 2 ; , ; 3 ; , ; . 3 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.

Basic Three-Point Structure Functions
In evaluating the finite parts of Feynman integrals associated with amplitudes having three internal propagators, Equation (3), we can obtain considerable simplification if the results are written in terms of the following functions where and are momenta of the internal lines or a combination of them and, .
If the considered amplitude possesses two or more Lorentz indexes it is useful to define another set of auxiliary functions.They are defined as The elements of the above set of functions can be reduced to nm  and k Z functions if useful or necessary.However, in intermediary steps of calculations it is frequently convenient to maintain the presence of nm  function to give a compactation of the results and operations.Now we consider useful properties for the functions nm and 

nm . 
The first aspect is relative to the reduction of all the elements of the set having a certain value for n m  to that having 1.
We now show such reduction firstly considering those for .We start by considering 01  .After some algebraic effort, which involves only basic mathematical operations like integration by parts, we can write the expression where we have defined , ; ; , ; 2 .p p q p q C Z m p q m Z m q m Z p q p q q p q q m m p m m q q p 2 2 2 2 1 2 ; , ; In the last two equations above, we can note that both functions 01  may be related through a set of simultaneous transformations.
The reduction of the functions 20  and 02  can be written as , ; , ; ; 2 .p q p p q C Z m p q m Z m q m p q p q q m m q m m p q q p q  on the other hand, it is interesting to obtain two alternative forms.First we write For the component , ; ; , 2 .p q q C Z m p q m Z m q m p q p p m m q m m p q p p q 2 00 2 2

;
p q The second form is Z m p q m Z m p q m Z p q q q m m p m m p q p q p q p p q 2 2 1 2 ; , ;

2, n m
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 The expressions corresponding to the first reduction of the nm  functions having ; , ; ; , 2 .

C
p q p p q Z m p q m Z m q m p q p q q m m p m m p q q q p   ; , ; 2 ; , ; ; 2 1 ; , ; 2 p q q C Z m p q m Z m p q m Z m p p q p m m q m m p q p q Z m p m p q p p p q .
The two different forms for the function  are written as , ; ; , ; 2 , p q q Z m p q m Z m q m p q p p m m q m m p q p p q . p q p C Z m p q m Z m p q m p q q q m m p m m p q q q p    Firstly the form Finally we consider the expressions for the function ; , ; ; , ; 2 p q p C Z m p q m Z m p q m Z m p q p q q m m p m m p q p q Z m p m q p q q q p and then a second form can be obtained ; , ; 2 For the  used in the above expressions we have the following expressions With these expressions we can write the functions nm  corresponding to completely in terms of of functions The reductions present above are very useful in particular 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) : functions, It is also useful to note similar properties involving the nm 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 n m , 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.

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 where ; , , ; , , ; , , .
If the considered amplitude possesses at least two Lorentz indexes it is useful to define another set of auxiliary functions and if four or more Lorentz indexes are involved it is convenient to define also the functions They can be written as q r q r m p q m p r m m q m r m p m m C q r r p r q p m p q m r q m m p m r m q m m p m C p r r q r p q m p q m p r m m q m r m p m m C p r p r m p q m r q m m p m r m q m m C p q r p p m C  p q q r q p r m p q m p r m m q m r m p m C p r q p p q r m p q m r q m m p m r m q m C p q p q where we have defined

C
p q r p q p r q r p q r q p r Note that and q r q r p m m m p q m p r m m q m r m C q m q r r p r q p m q p m q r m m p m r m C r m q p r q q r p m r p m r q m m p m q m C p m p r r q r p q m p q m p r m m q m r m C p r p r q m m m q r m q p m m p m r m C p m p q q r q p r m p q m p r m m q m r m C q m p r q p p q r m q r m q p m m p m r m C 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:  p pq pr m p q m p r m m p q m p r m m p q m p r m ; , ; ,  ; , ; , p q m p r m m p q m p r m m p q m p r m m p q m p r m m p q m p r m m p q m p r m p , ; ; ; , ; , 2 , ; , ; , ; , 2 , ; , p q m p r m m p q m p r m m p q m p r m m p q m p r m m p q m p r m m p q m p r m  p pq pr m p q m p r m m p q m p r m m p q m p q m p r m m p q m p r m m p q m p r m p Similar relations can be obtained for others components of the set by exploring the properties relating these functions which are the interchanges p q  , p r  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 th ivergen s in perturbative calculations of QFT, as well as to state the standard divergent structures in terms of which th introduced.

Manipulations and Calculations of the
One-Loop Feynman Integrals le e d ce e divergent parts will be written and to define the set of basic functions in terms of which the finite parts will be written, we can consider the solu the divergent Feynman integrals presen tion of ted in ( 1)-( 4). 3 D

One-point Feynman Integrals
If we want to solve the Feynman integral   1 I  defined in (1), by using the procedure described in previous sections, first we identify the divergence degree  .After this we have to adopt the adequate representation for the propagator.This means taking in the expression (5) to get Next we reorganize in enient way in order to get the basic divergent structures defined in Section 3. Then we organizations are made to get completely in combinations nd then we get 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 Only finite terms will be integrated in the next step and no additional modification will be made.The result is the expression s for the definition of the divergent objects precisely on this form will become clear in future sections.It is possible to show that for any value of N in the involved divergent objects a regularization must be assumed 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 mention needs to be made to regularization techniques until this step.On the other hand, the above result can be conve ed 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 ca rt ps performed are perfectly valid in the presence of all regularization distribution.Such eventually adopted regularization, 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) ion, we will get So, taking the integration after some convenient reorg Solving the finite terms we obtain Again note the general character of the expression.Only mathematical operations free from choices have been made.

s
Now we consider the integrals having two propagators.First we take the simplest one: the 2 where we have used the definition (6) in order to write the expressions in a more compact way.Now we introduce the integration sign to get When is integral needs to lved, as a consequence of the used in (44 wever, given the divergence n be r both propaga-th ap be so ). Ho plication of Feynman rules, we first adopt the representation (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 = 1 N fo tors.We have to integrate the summation of terms The finite ones can be integrated by using usual tools to yield where we have introduced the definition 2 1 k k 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 Note that odd terms have been omitted.After some reorgan solving the finite integrals btained to get    ization, we take the integration ; , ; ; , ; ; , ; . 2 2 Here we have defined 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 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 , ; ; , ; 2 ; , ; , which completes the calculation of the Feynman integrals having two internal propagators.

Three-Point Feynman Integrals
Now we evaluate the integrals having three propagators.The first element of the set (3) is finite and may be calculated by taking any value for N in the expression (5).We write the result as 4π where we adopted the definitions 3 1 k k q   and By simplicity, we will omit the arguments of three-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 involved four-point structures.The next integral of the set (3), which is 3 .
ter we take the integration.Solving the finite integrals we Only the first term will be conve ivergent obj results in the form 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 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 writ as a combination of the basic divergent objects ( 7)- (11), and after this taking the integration and performing the oper 4π i q p q p q q p q p q p p p p q g q g q g q g p g p g p k k p q k k p q k p p q q q p p q g k p p q q p q q p g k p q q p p p q q g In fundamental theories the considered integrals are enough to evaluate the one-loop amplitudes having three internal propagators.

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 where we have identified the four-point structure func-tions previously defined in the Equation ( 31) and also the external momentum 4 1 r k k   .Next, one can immediatly see that, for the vector integral, we can write and that for the one having two Lorentz indexes, we have where ; , ; , ; , ; ; , On the other hand, where     210 201 p p q p q p q p p p p r p r p r p p The last one we consider is the logarithmically divergent one, which we write as where ; , ; , ; , ; ; , J i g g g p p g p p g q r r q g q r r q p p p p r q q q r q q q q r r r q r r r q r q r p q p r p r p q q p r p r p q p   r q r q   ith the above results for the Feynman integrals at hand a k k W 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 evaluate some representative amplitudes involving vector vertexes.

Physical Amplitudes
In the preceding sections we have considered the evaluation of the Feynman integrals introduced in the Section 2, crucial for the one-loop calculation in the context of fundamental gauge theories like QED.All   .
In the present section we will evaluate some representative amplitudes of the perturbative calculations by using the systematization introduced in the preceding sec-Ward identities.We choose for this purpose n functions of the Standard nctions having only ferperat an on we stat q ntiti  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, ambiguities 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, internal symmetry o ors d so , have to e the amplitudes for one value of the loop momentum k, which are the ua es 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.

One-Point Functions
We start by taking the cases having the highest divergence degrees; the one-point functions.First, we write for the one value of the k momentum, the quantities The corresponding one-loop amplitudes, obtained by integrating the above structures in the loop momentum, are divergent quantities.The superficial degree of divergence 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.
or, solving the Dirac traces, At this point we adop equate rep tation for the propagator t the ad resen , as we have made when we discussed the solution of the 1 I integral.Then we get  divergent objects as well as the presence of a potentially ambiguous term, the last nce here 1 k Note the presence of the basic in the expression (66) we get the vector one-point function Using the results for the Dirac traces involved we get Adopting the adequate representation for the propagator as we have made in the calculation of the inte Note that the result is completely potentially ambiguous 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.

Two-Point Function
If one wants to consider a representative Green function of nic e write them from the definition (65) as the perturbative calculation, concerning the consistency in the manipulations and calculations involving divergent Feynman integrals, certainly there is no better on rmio e than the fe two-point functions.We will consider three of such amplitudes related among them through Ward identities.W Firstly we consider the scalar-scalar where 1 For this case we get first (after taking the Dirac traces) 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 .
Next, we consider the amplitude scalar-vector (SV) by takin   and 2 e we have to solve the integrals (46) and (47).We get then  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   where we have adopted the definitions and 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 ; , ; .p m p p P P

T g p p p I g m m I i g p p p Z m p m Z m p m g m m Z m p m Z m p m g m m Z m p m A
where we have defined the quantity (69) Copyright © 2012 SciRes.
Note the presence in the above expression of potentially ambiguous terms since the quantity is depen ces for arbitrary quantities as well as the presence of terms dependent on physical combination of the arbitrary intern p k k   which are not dependent on t or the routing of the internal lines momenta of the loop amplitude but are de-pendent on the arbitrary choice for the common scale.

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 e take in all vertice scalar operators   By using the developments made in solving the integrals ( 46) and (49) we get the expression On the other hand, taking  and, 3 1   i s obtainin study of integrals (46), (47) and, ( ) we get n Equation (70) and by using the result g in the 49 Having two vector inde get the SVV amplitude , , where we have defined We get then Finally, let us consider the case of triple vector operators.First we get ere the following definitions h e ced With the aid of the integrals (49), (50), (51) and, (5  may be written explicitly by 2) the tensors 3  s s g p s q s p p p s q q q s p p q s p q p s s q p p s s p q q s s q p q s q q p s s On the other hand, the ex VPP T  pressions for , PVP T  PPV T  may be written as and, ; 2 2 2 ; , ; 4π 2 ; , ; 2 2 ; , ; ; , ; 4π

The Four-Vector Four-Point Function
four fermioni ur-vector four-point function, given by 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 having c propagators, we consider the fo After performing the Dirac traces we identify the following structure where In the above expression a convenient an efu sorial sy atization was introdu  , ; ,  , ;   , ; ,  , ; , , Here and 2

  assum the values 1
 .We also see that the coefficients of the metric tensor are four-point am with vector an eudoscalar vertices defined as e d ps plitudes Copyright © 2012 SciRes.JMP After performing the Dirac traces, the four-point amplitudes with vector and pseudoscalar vertices acquire the form Below we identify the values of i s according to the corresponding amplitude 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 = 2 j = 3 and = 4 l .From the results (53), ( 54), ( 56), (59) and, (62) we get ensor (74 q s r J s p J s p J s q r s q r J r s q p J s p J q s r p J s p J q r s q r p J s p J where J  ,  ,  and  are given by in Equations ( 55), ( 57), ( 60) and (63).Replacing the above result (with appropri 1 ate values for the symbols and 2 ) in Equation ( 74) gives r q q r J p q q p J r p p r p q q p J p q r q r p q r q r p q r q r J p 8 8 q r r q r p p r J q r q r p q q p p r r p J q r q r p q r q r p q r q r J p q r q r p q r q r p q r q r J p q r q r p q r q r p q r q r J For the amplitudes listed in the table above we may write              1    s s PPPP T I Z m r p m Z m q m q r q r p q p r r q p q p p r q q r p r q Above, the following compact definitions were also used ; , ; , , is point, fulfilled oposed systematization mplitudes.However, another important aspect involved in perturbative calculations 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.

Relations among Green Functions
In the preceding sections we have described in details a procedure to handle the divergences typical of the perturbative calculations in QFT.The procedure is very general since all the choices involved have been preserved; 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 degrees of divergent and finite ones.We can ask ourselves at this point about the consistency of the per rmed operations as usual in such type of manipulations and calculations.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 choices for the involved arbitrariness, which means that the relations need to be satisfied in the presence of potentially ambiguous and symmetry violating terms.Essentially, what we want to know is if the performed operations have preserved the property of linearity of the integration 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 violating 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 After taking the Dirac traces in both sides we can identify that The above relation means that it is expected that if we integrate both sides in the loop momentum k the corresponding relation among the loop amplitudes remain valid, i.e.,   .
This means that by calculating all the involved amplitudes 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 combination of the right hand side.This type of identity is highly ontrivial to be preserved in traditional regularization r which implies that .
We can note from the above expressions that all amplitudes 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 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 relations the following structures are involved: VVV, VVS, VSS, VV, VS and, SS plus the ones which appear as substructures: VPP, SPP, PP and S.
If we consider four-point functions, the same will occur.To evaluate the VVVV function all the above mentioned 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 onepoint 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 nontrivial for all traditional techniques.
Let us start by the property (78).Taking the expression for the VV amplitude, Equation ( , and contracting with 2 , . By comparing to the resu ) for the VS mplitude we can identify , , In order to complete the verification of the property (78), the last term in the above equation must be identified with the one-point vector functions.It is simple to note that if an   is added and subtracted in the expression for A  , a reorganization allows us to identify 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 .
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 Z 1 and in the right hand side only Z 0 must appear.Let us consider the reduction of Z 1 to Z 0 through the property (13) in order to adequate the right hand side of the Equation (84).The referred reduction is the property (13) 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   We get then the scale properties of the divergent objects Now note that we can relate the reduction of the finite functions to 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 threepoint 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.Conlast section,  and using the properties ( 23), ( 24), ( 25) and (29) in order to eliminate the nm  functions having ; ; 2 ; , ; , ; ; , , 2 ; m 2 11 20 10 i q q Z m p q m Z m p q m q p q q m m p ; , ; ; , ; Z m p q m Z m p q m   Given the obtained result, we now use the properties (1 4 3 2 ; , ; ,  i q q Z m p q m Z m p q m i q p Z m p q m Z m p q m Z m p m Z m p q m q p q Finally, using relation (17) we write , 2 q p q p q I i g p q p q p q Z m p q m Z m p q m g m m Z m p q m g m m m Z m p q m If we consider the results for th plitudes VV and SVV , Equations ( 69) and (71) to note that the expression above may be i d 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 , , , , .
In order to show that the calculated four-point amplitude VVVV satisfies this relation, at first we contract Equation (73) with   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 = favo hose having i + j + k = 3 and so on.The latio easy but involves a lot of algebra, therefore we will no w it explicitly.All the required ingredihave given in the preceding sections.

Ambiguities and Symmetry relations
e Sect 6 we have evaluated, within the systemaon p ed, Green's functions which are typical of erturb tive calculations.In particular, all the cond am udes appear in the context of Standard .
the evaluated Green's functions, having ee of rgences higher than the logarithmic one, it is possible t note the presence of terms where the dence he internal momenta appear as arbitrary tities (th summations of them).This is expected ce a sh in the integrating momentum generates surfaces terms which implies that different choices for the ernal lines momenta lead to different amudes.
possible dependence on the choices for the ls of t nternal lines momenta characterizes what eno e as ambiguities.This situation is not acin this case, the power of predictof th ry is destroyed.In addition, fundamental tries like the space-time homogeneity are not prese ved in the perturbative calculations.It will su ng to find global and local gauge symmetries as well as internal symmetries violated in physical itudes ving the space-time homogeneity broken.ere is on one possibility to save such type of calculao inate the ambiguous terms in a consistent niv way.Within the context of the adopted gy t biguous terms are automatically separated d preserv so that it is easy to identify them.
one-point function it is simple to identify In the case of three-point functions we found 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 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   ; , ; , ; , ; ; , ; , ; ; , ; , ; 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 s are independent of the choices for the internal momenta.They can be converted in ambiguities through their evaluation in intermediary steps within the context of traditional regularization techniques.Again we can note that all the potentially 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 question of ambiguities considered above.There are two types of impositions coming from the symmetries for the amplitudes.The general ones, coming from Lorentz and CPT, present in the Furry's theorem, whose implication states that all amplitude which has an odd number of external vectors and only one species of fermion at the internal lines must vanish identically, and that coming from the divergence of the fermionic vector current which states a precise relation with the corresponding scalar current.The first of the impositions mentioned above implies that the amplitude

□
Due to the same reasons, the theorem states that the amplitude for the process V VV  , which is the VVV amplitude symmetrized in the final state, , , , , , must vanish for the case of equal masses.The arbitrary internal momenta for the second channel obey, q l l   and p l l  .This means that it is required Concerning the symmetry relations coming from the proportionality of the divergence of the fermionic vector current with the scalar current, we note that in the VV two-point function we get By comparing to the resu ) for the VS amplitude we can identify which means that the symmetry relation is broken by the terms which are all combination jects , .
of the ob  ,  and, □ .In fact this result requir e as the Furry's theorem, a vanishing value fo tor one point function.Following this line of reasoning we note that the SVV amplitude possesses a symmetry violating term which is independent of the choice for the internal lines m enta es the sam r the vec om The same occurs for the VV VV  process where the violating term is proporcional to with a nonanbiguous coefficient.
In view of the above comments and others omitted, it is very simple to conclude that all these unwanted problems can be removed from the amplitudes in a consistent way.There are simple but powerful arguments.If we consider that a perturbative solution for the amplitudes of a QFT must be compatible with the space-time homogeneity or it does not make any sense, if we cannot admit that the scale independence can be broken by any method or tion nm gra that we denominated them as Consistency Relations.Such conditions can be easily understood.In fact the definition of the objects ,   and, □ has been conveniently made in order to get clean and sound clarifications.First note that so that we can identify   strategy adopted to give some meaning for the perturbative amplitudes and if we also cannot admit that an acceptable interpretation for the perturbative solu breaks symmetry relations of the underlying theory, then it becomes necessary to impose a set of properties for the divergent Fey an inte ls in order to recover these symmetries, due to the fact that the perturbative series is not automatically translational and scale invariant and symmetry preserving.Fortunately all these problems can be solved simultaneously.It is enough to impose 0.

     □
We can look at these conditions as a set of properties required to a regularization method in order to produce consistent results or we can think that this is the set of properties required to the perturbative series in order to get the space-time homogeneity maintained in the calculated expressions (among others).Due to these reasons The factor 4 in the last condition is justified by the symmetrization in the Lorentz indexes.In order to give symmetrical role to all indexes four terms need to be introduced in the left hand side given the factor 24 to the fourlinear in loop momentum integral.Frequently it is convenient to write such integral in symmetrized form.We adopted the definition of the object □ in a non- simmetrized way only to reduce the mathematical expressions.Note that through the Gauss theorem these quantities are identified as surfaces terms.It becomes clear now that if these conditions are not imposed the perturbative calculations simply does not make any sense.It is on the other hand simple to verify that these conditions are satisfied in the presence of any distribution.W and gauge symmetries are violated as well as the amplitudes may be ambiguous quantities.T prescription is universal since in other dimensions as well as in theories or models where higher degree of divergences are present analogous conditions can be identified.This interpretation of the perturbative calculations provides us the required consistency.The calculated amplitudes are ambiguities free and symmetry preserving.ithout these conditions being fulfilled space-time, local he If one agrees with the arguments put above then the adoption of a regularization become completely unnecessary for any purposes in the perturbative calculations.All the required manipulations and calculations, incluing the renormalization, can be performed, following o r strategy, without any mention to the word regularization.And, which is better, the results are so consistent as desirable and no restrictions of applicability exist.

Generalizations of the Finite Functions and Their Relationship
Through the proposed method to manipulate and calculate divergent integrals, in the above section we have been learning how to systematize the finite parts of the one, two, three, and four-point integr which are present in the relevant amplitudes belonging to fundamental theories.It is not hard to see that this systematization could be generalized to amplitudes with an arbitrary number of points.In this section we discuss some aspects of this generalization.We begin by defining the et of functions ; , ; ; , ;  representing a Kronecker delta symb l.All finite parts of the one-loop Feynman integrals with an arbitrary number of points, handled by the proposed approach, can be systematized through this set of functions.We recognize that Equation ( 93) is the generalization of definitions ( 12), ( 16) and (33) and Equation (94 s the generalization of Equations ( 15), ( 32) and (31).
In the preceding sections we have systematically evaluated the one, two, three, and four-point vector amplitudes and verified their Ward identities.Within our approach, the verification of the Ward identities is greatly simplified by using a set of identities characteristic   , like those given by Equations ( 17)-(29).In order to obtain such identities for an arbitrary number of points first we note that where 0 n  and 1 2 , , , 0,1,2, , , , , After an integration by parts, the first term on the right hand side of the above equation may be rewritten as x The first term is a total derivativ .So, performing the integral over 1 x , we write the above expression as , , , , , 0 0 0 ... , 0 0 0 1 1 1 The symmetry of   with the j-esimo term given by   , the solution of the above system of linear equations can be written in a formal way by

Conclusions
In the present work we considered general aspects involved in the calculations of perturbative ampli QFT's.A very general procedure is presented for this purpose.The work can be considered as an extension of a previous one where only one species of fermion has been considered [2].In addition, the calculations in the present contribution have been done by adopting an arb scale parameter putting the calculations in the most general way.All the arbitrariness involved in the calculations were preserved in intermediary steps.The adoption oided, the internal momenta are assumed as arbitrary and the common scale for the finite and divergent parts was taken as arbitrary t integrals were not really evaluated.Only properties of such quantities were used.This became possible through a convenient interpretation of the Feynman rules.The perturbative amplitudes for one value of the loop (unrestricted) momentum are not integrated before a convenient representation for the propagators is assumed.When the integration is taken all the dependence on the internal arbitrary momen sent in finite integrals.In the divergent ones no physical quantity is present.Only the arbitrary scale appears there.ta is pre-e divergent parts are written as a combination of standard mathematical objects which are never really integrated and the finite parts are written, after the integration is performed, in terms of finite structure functions.So, two very general types of systematization are proposed; 1) Divergent parts.The divergent conten of one loop amplitudes perturbative amplitudes belonging to fundamental theories can be written as a combination of five objects;   All self energies, decays and elastic scattering of two fields can be calculated by using the results presented here as well as their symmetry relations can be verified.The results written in terms of the systematization above can be used in the context of regularizations since all the operations performed are valid in the presence of any reasonable regularization distribution.All we need to evaluate is the standard divergent objects.
As a last comment we argue that if we want to give some meaning to the perturbative calculations we have to impose that the space-time homogeneity and the scale independence need to be recovered.Otherwise, the amplitudes become completely arbitrary quantities as well as local and gauge symmetries may be violated (invariably by the ambiguous terms).If we agree with this arguis □ , ment, our procedure makes this job easy.All we need to impose that the conveniently defined objects   and   become identically vanishing.This assumption can be viewed as completely reasonable since these objects can be identified as surfaces terms which are really vanishing quantities in the presence of any distribution.The same will occur by assuming the analytic continuation of the integrals to a continuum and complex dimension which is the ingredient of the dimensional regularization.So, in any consistent interpretation of the perturbative amplitudes only the basic divergences   will remain in a calculated divergent amplitude.They need not to be calculated since they will be absorbed in the renormalization of physical parameters.The calculation of beta functions can be done by using the scale properties of such objects.
All these comments allow us to conclude that within the context of our strategy the amplitudes are automatically ambiguities free and symmetry preserving and no regularization method needs to be used for any purp The strategy, in addition, is universal since it can be applied to any theory or model, renormalizable or not, and formulated in odd and even space-time dimensions in an ose.absolutely identical way.And, which is still better, the results are as consistent as desirable.Investigatio volving higher space-time dimensions (odd and even) as well as nonrenormalizable theories in four dimensions are presently under way and the obtained results are in accordance with our best expectations.
In addition, other authors have been made investigations by using the procedure adopted in the present work.In particular in [31] the authors explored some very interesting aspects of the systematization proposed in [2] concluding that there are important advantages relative to the traditional ones.


can be written as Through the same type of manipulations the function The systematization allows us to treat the perturbative four-point amplitudes in an exact way.By successive reductions all the content of finite ction ritten in terms of o y 000 pa oint fun w  (more 00 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 in the coupling of fermionic currents to the bosonic fields in the Lagrangian.After defining the operators corresponding Lorentz indexes are attached to with the procedure described in the proceeding sections.The corresponding one-loop amplitudes are obtained by taking the integration of the t structures in the loop momentum k;

2 4
structures saving, in this way, considerable computational time.
By recursive use of the above relation it is possible to reduce all functions  reduction is useful, for example, in applications whe e w are interested in numerical results because within this procedure we have to manipulate only

2 )
Finite parts.The finite content can be written as a combination of only three functions for amplitudes having two, three and four internal propagators.
the integrals have been written in terms of the set of diver-