Mass Scales and Their Relations in Symmetric Quantum Field Theory

TL;DR

This paper explores the role of mass scales in the linear sigma model, emphasizing their relation through Ward identities and the Implicitly Regularization method to streamline renormalization and the calculation of physical amplitudes.

Contribution

It demonstrates how to relate divergent and finite parts of amplitudes using mass scale identities within the Implicitly Regularization framework.

Findings

Mass scales are crucial for manipulating multi-mass graphs.

Finite and divergent parts can be expressed with a single renormalization scale.

The approach simplifies deriving symmetric counterterms and RG constants.

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.