General Quartic $\beta$-Function at Three Loops

TL;DR

This paper computes the most general three-loop quartic beta function in four-dimensional renormalizable theories, providing a comprehensive formula that encompasses specific known cases and includes related couplings.

Contribution

It presents the first complete three-loop quartic beta function for general theories, fixing all coefficients through comparison with known special cases.

Findings

Derived the general three-loop quartic beta function

Provided beta functions for cubic couplings and scalar masses

Made results available in software packages

Abstract

We determine the three-loop $\overline{\text{MS}}$ quartic $ \beta $-function for the most general renormalisable four-dimensional theories. A general parametrization of the $ \beta $-function is compared to known $ \beta $-functions for specific theories to fix all coefficients. Three-loop $ \beta $-functions for the cubic coupling and scalar mass terms also follow from the result, which is made available in software packages.