Generalized Scheme Transformations for the Elimination of Higher-Loop Terms in the Beta Function of a Gauge Theory

TL;DR

This paper introduces a new class of scheme transformations that can eliminate higher-loop terms in the beta function of gauge theories, improving the analysis of infrared zeros across a broader range of fermion numbers.

Contribution

It develops a generalized one-parameter class of scheme transformations that remove loop terms up to a certain order, extending the applicability of such transformations in gauge theory analysis.

Findings

The new scheme transformations satisfy physical acceptability criteria over a wider fermion range.

They effectively eliminate higher-loop terms in the beta function.

A modified transformation removes the three-loop term specifically.

Abstract

We construct and study a generalized one-parameter class of scheme transformations, denoted $S_{R,m,k_1}$ with $m \ge 2$, with the property that an $S_{R,m,k_1}$ scheme transformation eliminates the $\ell$-loop terms in the beta function of a gauge theory from loop order $\ell=3$ to order $\ell=m+1$, inclusive. These scheme transformations are applied to the higher-loop calculation of the infrared zero of the beta function of an asymptotically free gauge theory with multiple fermions. We show that scheme transformations in this generalized class satisfy a set of criteria for physical acceptability over a larger range of numbers of fermions than previously studied scheme transformations. We also present an interesting modification of a different type of scheme transformation that removes the three-loop term in the beta function.