Multi-loop spectra in general scalar EFTs and CFTs

TL;DR

This paper advances the renormalisation of scalar EFTs to higher loops, providing new insights into their spectra and applications to various conformal field theories relevant for critical phenomena.

Contribution

It introduces an efficient renormalisation method for the most general scalar EFTs up to dimension six, enabling higher-loop calculations and broadening applications to CFTs.

Findings

Renormalised scalar EFTs up to six dimensions and rank two tensors.

Computed scaling dimensions for Ising, O(n), and hypercubic CFTs.

Results align with non-perturbative data and offer new predictions.

Abstract

We consider the most general effective field theory (EFT) Lagrangian with scalar fields and derivatives, and renormalise it to substantially higher loop order than existing results in the literature. EFT Lagrangians have phenomenological applications, for example by encoding corrections to the Standard Model from unknown new physics. At the same time, scalar EFTs capture the spectrum of Wilson--Fisher conformal field theories (CFTs) in $4-\varepsilon$ dimensions. Our results are enabled by a more efficient version of the $R^*$ method for renormalisation, in which the IR divergences are subtracted via a small-momentum asymptotic expansion. In particular, we renormalise the most general set of composite operators up to engineering dimension six and Lorentz rank two. We exhibit direct applications of our results to Ising ($Z_2$), $O(n)$, and hypercubic ($S_n \ltimes (Z_2)^n$) CFTs, relevant for a plethora of real-world critical phenomena. The computed scaling dimensions agree well with known non-perturbative results, and they lead to new predictions where such results do not yet exist. We thereby expand the understanding of generic EFTs and open new possibilities in diverse fields, such as the numerical conformal bootstrap.