Regularizations: different prescriptions for identical situations

Abstract

We present a discussion where the choice of the regularization procedure and the routing for the internal lines momenta are put at the same level of arbitrariness in the analysis of Ward identities involving simple and well-known problems in quantum field theory. They are the complex self-interacting scalar field and two simple models where the scalar-vector-vector and axial-vector-vector process are pertinent. We show that, in all these problems, the conditions to symmetry relations preservation are put in terms of the same combination of divergent Feynman integrals, which are evaluated in the context of a very general calculational strategy, concerning the manipulations and calculations involving divergences. Within the adopted strategy, all the arbitrariness intrinsic to the problem are still maintained in the final results and, consequently, a perfect map can be obtained with the corresponding results of traditional regularization techniques. We show that, when we require an universal interpretation for the arbitrariness involved, in order to get consistency with all stated physical constraints, a strong condition is imposed for regularizations which automatically eliminates the ambiguities associated to the routing of the internal lines momenta of loops. The conclusion is clean and sound: the association between ambiguities and unavoidable symmetry violations in Ward identities cannot be maintained if an unique prescription is required for identical situations in the evaluation of divergent physical amplitudes.

Regularizations: a unique prescription for all situations

“…[2] we have considered the S → VV process in a simple model where the vector field is coupled to a 1/2 spin fermionic field. The corresponding amplitude can be written in the form…”

“…[2] also a simple model where a pseudoscalar and an axial field, in addition to a vector one, are coupled to a 1/2 spin fermionic field, such that the corresponding amplitude can be written as…”

“…In the present contribution we adopt the calculational method described in Refs. [1][2][3]9, 10] whose original reference is [4]. According to this strategy the Feynman integrals involved in the one-point function becomes…”

“…They suggest that there is only one reasonable way to maintain the Ward identities which is the consistency requirement (for detail see please Refs. [1][2][3])…”

“…Once it is unacceptable that the physical consequences of any theory become dependent on the involved arbitrariness, an ambiguities free and consistent treatment of the physical amplitudes must be searched. Having this in mind, in previous contributions [1][2][3] we have made detailed investigations on the question of arbitrariness and their possible associated ambiguities in the evaluation of perturbative (divergent) physical amplitudes and/or in the preservation of symmetry relations. A very general calculational strategy [4] has been used on the referred investigations.…”