An Analysis of Scheme Transformations in the Vicinity of an Infrared Fixed Point

TL;DR

This paper examines how scheme transformations affect the analysis of infrared fixed points in gauge theories, highlighting restrictions and their impact on the scheme dependence of fixed point calculations.

Contribution

It provides a detailed analysis of scheme transformation restrictions near infrared fixed points and constructs acceptable transformations to study their effects on fixed point scheme dependence.

Findings

Scheme transformations near IR fixed points face significant restrictions.

Acceptable scheme transformations can be constructed to analyze scheme dependence.

Finite-order series expansions may not accurately probe IR fixed points.

Abstract

We give a detailed analysis of the effects of scheme transformations in the vicinity of an exact or approximate infrared fixed point in an asymptotically free gauge theory with fermions. We list necessary conditions that such transformations must obey and show that, although these can easily be satisfied in the vicinity of an ultraviolet fixed point, they constitute significant restrictions on scheme transformations at an infrared fixed point. We construct acceptable scheme transformations and use these to study the scheme-dependence of an infrared fixed point, making comparison with our previous three-loop and four-loop calculations of the location of this point in the $\bar{MS}$ scheme. We also use an illustrative hypothetical exact $\beta$ function to investigate how accurately analyses of finite-order series expansions probe an infrared fixed point and the effect of a scheme transformation on these. Some implications of our work are discussed.