General Gauge-Yukawa-Quartic $\beta$-Functions at 4-3-2--Loop Order

TL;DR

This paper computes the most general 4-loop gauge and 3-loop Yukawa beta functions for four-dimensional theories, providing a comprehensive framework for understanding their renormalization group behavior.

Contribution

It derives the complete set of coefficients for the general 4-loop gauge and 3-loop Yukawa beta functions, extending previous results to the most general renormalizable theories.

Findings

Complete parametrization of 4-loop gauge beta functions

Complete parametrization of 3-loop Yukawa beta functions

Constraints from known theories and Weyl consistency conditions

Abstract

We determine the full set of coefficients for the completely general 4-loop gauge and 3-loop Yukawa $ \beta $-functions for the most general renormalizable four-dimensional theories. Using a complete parametrization of the $ \beta $-functions, we compare the general form to the specific $ \beta $-functions of known theories to constrain the unknown coefficients. The Weyl consistency conditions provide additional constraints, completing the determination.