Subtleties in the beta function calculation of N=1 supersymmetric gauge theories

TL;DR

This paper explores peculiarities and paradoxes in calculating the two-loop beta-function of N=1 supersymmetric gauge theories, highlighting issues related to different background field methods and the potential role of the rescaling anomaly.

Contribution

It identifies the origin of the anomaly puzzle in the covariant supergraph formalism and clarifies the differences between formalism approaches in beta-function calculations.

Findings

Covariant background field method yields a finite effective action but a non-zero beta-function.

Standard background field method shows divergences consistent with renormalization constants.

The anomaly puzzle may be an artifact of the rescaling anomaly in the formalism.

Abstract

We investigate some peculiarities in the calculation of the two-loop beta-function of $N=1$ supersymmetric models which are intimately related to the so-called "Anomaly Puzzle". There is an apparent paradox when the computation is performed in the framework of the covariant derivative background field method. In this formalism, it is obtained a finite two-loop effective action, although a non-null coefficient for the beta-function is achieved by means of the renormalized two-point function in the background field. We show that if the standard background field method is used, this two-point function has a divergent part which allows for the calculation of the beta-function via the renormalization constants, as usual. Therefore, we conjecture that this paradox has its origin in the covariant supergraph formalism itself, possibly being an artifact of the rescaling anomaly.