Conformal Behavior at Four Loops and Scheme (In)Dependence

TL;DR

This paper investigates the infrared behavior of SU(N) gauge theories at four loops across different schemes, finding consistent patterns in the anomalous dimension and implications for conformal versus chiral symmetry breaking phases.

Contribution

It provides a detailed four-loop analysis of the beta function and anomalous dimensions in multiple schemes, comparing results to previous two- and three-loop studies, and assesses the conformal window.

Findings

Anomalous dimension decreases at higher loops across schemes.

Estimated anomalous dimensions suggest some theories are conformal.

Scheme dependence is minimal at four loops, supporting robust conclusions.

Abstract

We search for infrared zeros of the beta function and evaluate the anomalous dimension of the mass at the associated fixed point for asymptotically free vector-like fermionic gauge theories with gauge group SU(N). The fixed points of the beta function are studied at the two, three and four loop level in two different explicit schemes. These are the modified regularization invariant, RI', scheme and the minimal momentum subtraction, mMOM, scheme. The search is performed in Landau gauge where the beta function of the gauge parameter vanishes. We then compare our findings to earlier identical investigations performed in the modified minimal subtraction, $\bar{\text{MS}}$, scheme. It is found that the value of the anomalous dimension of the mass is smaller at three and four loops than at two loops. This seems to be a generic pattern that is observed in all three different schemes. We then estimate the value of the anomalous dimension to be $\gamma \sim 0.225-0.375$ for twelve fundamental flavors and three colors, $\gamma \sim 0.500 - 0.593$ for two adjoint flavors and two colors and finally $\gamma \sim 1.12-1.70$ for two two-indexed flavors and three colors with the lower and upper bound set by the minimum and maximum value respectively over all three schemes and at three and four loops. Our analysis suggests that the former two theories lie in the conformal window while the latter belongs to the chirally broken phase.