General scalar renormalisation group equations at three-loop order

For arbitrary scalar QFTs in four dimensions, renormalisation group equations of quartic and cubic interactions, mass terms, as well as field anomalous dimensions are computed at three-loop order in the MS¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\mathrm{MS}} $$\end{document} scheme. Utilising pre-existing literature expressions for a specific model, loop integrals are avoided and templates for general theories are obtained. We reiterate known four-loop expressions, and from those derive β functions for scalar masses and cubic interactions. As an example, the results are applied to compute all renormalisation group equations in U(n) × U(n) scalar theories.