General Quartic $\beta$-Function at Three Loops

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.

Comments: 30 pages