Mass scales and their relations in symmetric quantum field theory

Abstract

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.

Divergent Integrals Treatment in Quantum Field Theory

“…Let functions Z k (λ 2 1 , λ 2 2 , p 2 ) and Y k (λ 2 1 , λ 2 2 , p 2 ) of the constants λ 2 1 and λ 2 2 represent two different masses and the module square of the moment-energy four-vector p 2 . Let us first define them in a general way and then study them in the particular cases where the results of amplitudes calculated in QFT can be expressed, as shown in [9] and [12].…”

“…The first fails to preserve symmetries, the second respects symmetries but presents difficulties with the spacetime dimensionality and a third that, if we take some care, respects symmetries and can be used without restriction in any space-time dimension. The methods are: the regularization of Pauli Villars [8], a pioneer of regularization methods; Dimensional Regularization (DR) that preserves the symmetries of QED; and the Implicit Regularization Method (IR) [9][10][11][12]. In the latter we say that the integral is regularized for the purposes of algebraic manipulations, but we do not need to use any regularization method explicitly.…”