Higher-Loop Structural Properties of the $\beta$ Function in Asymptotically Free Vectorial Gauge Theories

TL;DR

This paper analyzes higher-loop properties of the beta function in asymptotically free gauge theories, focusing on the IR zero and its implications for quasiconformal dynamics, with calculations up to four-loop order.

Contribution

It provides new analytic and numerical insights into the behavior of the beta function at higher loops, including inequalities and effects of supersymmetry.

Findings

Calculated the beta function's minimum and slope at IR zero at multiple loop orders.

Derived inequalities relating loop order dependence of beta function properties.

Extended results to supersymmetric gauge theories.

Abstract

We investigate some higher-loop structural properties of the $\beta$ function in asymptotically free vectorial gauge theories. Our main focus is on theories with fermion contents that lead to an infrared (IR) zero in $\beta$. We present analytic and numerical calculations of the value of the gauge coupling where $\beta$ reaches a minimum, the value of $\beta$ at this minimum, and the slope of $\beta$ at the IR zero, at two-, three-, and four-loop order. The slope of $\beta$ at the IR zero is relevant for estimates of a dilaton mass in quasiconformal gauge theories. Some inequalities are derived concerning the dependence of the above quantities on loop order. A general inequality is derived concerning the dependence of the shift of the IR zero of $\beta$, from the $n$-loop to the $(n+1)$-loop order, on the sign of the $(n+1)$-loop coefficient in $\beta$. Some results are also given for gauge theories with ${\cal N}=1$ supersymmetry.