Two Loop Renormalization of Scalar Theories using a Geometric Approach

TL;DR

This paper develops a geometric method to compute two-loop counterterms in scalar effective field theories, enabling broad applications including SMEFT and chiral perturbation theory.

Contribution

It introduces a general geometric formula for two-loop counterterms applicable to a wide class of scalar EFTs with high-dimensional operators.

Findings

Derived a universal two-loop counterterm formula using geometric quantities.

Applied the formula to SMEFT, O(n) EFT, and chiral perturbation theory.

Provided explicit two-loop renormalization group equations for these theories.

Abstract

We derive a general formula for two-loop counterterms in Effective Field Theories (EFTs) using a geometric approach. This formula allows the two-loop results of our previous paper to be applied to a wide range of theories. The two-loop results hold for loop graphs in EFTs where the interaction vertices contain operators of arbitrarily high dimension, but at most two derivatives. We also extend our previous one-loop result to include operators with an arbitrary number of derivatives, as long as there is at most one derivative acting on each field. The final result for the two-loop counterterms is written in terms of geometric quantities such as the Riemann curvature tensor of the scalar manifold and its covariant derivatives. As applications of our results, we give the two-loop counterterms and renormalization group equations for the O(n) EFT to dimension six, the scalar sector of the Standard Model Effective Field Theory (SMEFT) to dimension six, and chiral perturbation theory to order $p^6$.