This paper presents a brief review on regularization methods and shows that the combination of two techniques could preserve symmetries in all orders of the Perturbation Theory. We will see, with two simple examples in the one-loop-order, what a regularization method needs to preserve symmetries. Furthermore, it will be shown that the problems with symmetry violations by regularization methods are related to the consequences of not being able to simply shift a variable in these divergent integrals. To facilitate the analysis between methods and provide analytical expressions for the finite parts of amplitudes, it was systematized in terms of a set of special functions. In the Appendix we introduce these functions and then cases of specific interest, functions continuity and some useful limits will be presented.
We illustrate the importance of mass scales and their relation in the specific case of the linear sigma model within the context of its one loop Ward identities. In the calculation it becomes apparent the delicate and essential connection between divergent and finite parts of amplitudes. The examples show how to use mass scales identities which are absolutely necessary to manipulate graphs involving several masses. Furthermore, in the context of the Implicitly Regularization, finite(physical) and divergent (counterterms) parts of the amplitude can and must be written in terms of a single scale which is the renormalization group scale. This facilitates, e.g., obtaining symmetric counterterms and immediately lead to the proper definition of Renormalization Group Constants.
We show that ambiguities and symmetry violations arising in one loop calculations can always be avoided provided the regularization scheme employed satisfies three consistency conditions. Our calculations are effected by assuming only implicitly the presence of a regulator in the integrand. We demonstrate in this way that there is a set of three relations involving divergent integrals of the same degree of divergence which are the source of both ambiguities and symmetry violations in Quantum Electrodynamics. Moreover we give analytical expressions for the finite parts of amplitudes off the mass shell systematized in terms of a set of special functions. Ward identities require hightly nom trivial relations involving those functions which we also derive.