Two-loop renormalization of general bosonic effective field theories

TL;DR

The paper develops a general framework for renormalizing bosonic EFTs in $4-2\,oldsymbol{\epsilon}$ dimensions up to two loops and mass dimension six, using a Green's basis, the R$^*$ method, and a background-field approach. It provides complete template RGEs for arbitrary gauge and scalar content, with results organized for off-shell (Green's basis) and minimized (physical) operator sets, and demonstrates how to project these results onto concrete theories like the bosonic SMEFT. By reproducing known two-loop SMEFT results and applying to a scalar QCD-like model with singlets, the work validates the methodology and showcases broad applicability, including insights into operator mixing and mass-term effects via the dummy-field technique. The framework enables efficient extraction of anomalous dimensions across a wide class of theories and offers a foundation for future extensions to higher loops, higher dimensions, and fermionic sectors, with potential applications to conformal field theories and beyond.

Abstract

The renormalization of higher-dimensional operators in quantum field theory is essential for phenomenological analyses in particle physics, and plays a significant role in the study of critical phenomena. We present a framework for renormalizing general bosonic effective field theories beyond one loop, with arbitrary gauge symmetry and scalar field content. In particular, we calculate the renormalization group equations in such theories up to two loops and dimension six. When specialized to the bosonic sector of the Standard Model effective field theory (SMEFT) using simple replacement rules, our general expressions reproduce recent results from the literature. Due to the broad applicability of effective field theory, our general results can readily be applied to obtain the anomalous dimensions in extensions of the bosonic SMEFT and in a plethora of other theories. We also envision our results to provide useful data on the scaling dimensions of composite operators in conformal field theories without fermions.