Renormalization and non-renormalization of scalar EFTs at higher orders

TL;DR

This paper advances the understanding of scalar EFTs by performing higher-order renormalization, constructing operator bases with Hilbert series, and discovering new non-renormalization and selection rules affecting operator mixing.

Contribution

It introduces a systematic method for higher-loop renormalization of scalar EFTs using Hilbert series and uncovers new selection rules for operator mixing at multiple loops.

Findings

Full one-loop results at mass dimension twelve.

Five-loop calculations at mass dimension six.

Identification of new operator mixing selection rules.

Abstract

We renormalize massless scalar effective field theories (EFTs) to higher loop orders and higher orders in the EFT expansion. To facilitate EFT calculations with the R* renormalization method, we construct suitable operator bases using Hilbert series and related ideas in commutative algebra and conformal representation theory, including their novel application to off-shell correlation functions. We obtain new results ranging from full one loop at mass dimension twelve to five loops at mass dimension six. We explore the structure of the anomalous dimension matrix with an emphasis on its zeros, and investigate the effects of conformal and orthonormal operators. For the real scalar, the zeros can be explained by a `non-renormalization' rule recently derived by Bern et al. For the complex scalar we find two new selection rules for mixing $n$- and $(n-2)$-field operators, with $n$ the maximal number of fields at a fixed mass dimension. The first appears only when the $(n-2)$-field operator is conformal primary, and is valid at one loop. The second appears in more generic bases, and is valid at three loops. Finally, we comment on how the Hilbert series we construct may be used to provide a systematic enumeration of a class of evanescent operators that appear at a particular mass dimension in the scalar EFT.