Four-Loop Gauge and Three-Loop Yukawa Beta-Functions in a General Renormalizable Theory

TL;DR

This paper calculates the four-loop gauge and three-loop Yukawa beta-functions for general renormalizable quantum field theories, providing detailed formulas applicable to the Standard Model and its extensions.

Contribution

It introduces a method to determine unknown coefficients in beta-functions through direct calculations in simplified models, extending to complex theories with multiple Higgs doublets.

Findings

Derived four-loop gauge beta-functions with matrix Yukawa interactions.

Provided explicit formulas for beta-functions in the Standard Model extensions.

Established a systematic approach for high-loop beta-function calculations.

Abstract

We present the beta functions of gauge and Yukawa couplings in general four-dimensional quantum field theory, at four and three loops, respectively. The essence of our approach is fixing unknown coefficients in the most general ansatz for beta-functions by direct calculation in several simplified models. We apply our results to the Standard Model and its extension with an arbitrary number of Higgs doublets and provide expressions for all four-loop gauge couplings beta-functions with matrix Yukawa interactions.