General scalar renormalisation group equations at three-loop order

TL;DR

This paper derives three-loop renormalisation group equations for general scalar quantum field theories in four dimensions, providing a comprehensive framework applicable to various models, including explicit examples like $U(n) imes U(n)$ theories.

Contribution

It presents the first general three-loop RG equations for scalar QFTs, avoiding model-specific loop integrals and enabling broad application.

Findings

Derived three-loop beta functions for scalar interactions and masses.

Provided explicit RG equations for $U(n) imes U(n)$ scalar theories.

Reiterated known four-loop expressions for consistency.

Abstract

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 $\overline{\text{MS}}$ 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 derive $\beta$ functions for scalar masses and cubic interactions from it. As an example, the results are applied to compute all renormalisation group equations in $U(n) \times U(n)$ scalar theories.