Hilbert Series, the Higgs Mechanism, and HEFT

TL;DR

This paper extends Hilbert series methods to include massive particles and applies them to systematically enumerate operators in the Higgs Effective Field Theory, revealing the Higgs mechanism's manifestation at the level of the Hilbert series.

Contribution

It introduces new techniques for incorporating massive particles and spurion fields into Hilbert series calculations, applied specifically to HEFT.

Findings

Hilbert series can now include massive, spinning particles.

The Higgs mechanism is reflected in the Hilbert series framework.

A comprehensive operator basis for HEFT is systematically constructed.

Abstract

We expand Hilbert series technologies in effective field theory for the inclusion of massive particles, enabling, among other things, the enumeration of operator bases for non-linearly realized gauge theories. We find that the Higgs mechanism is manifest at the level of the Hilbert series, as expected for the partition function of an $S$-matrix that is subject to the Goldstone equivalence theorem. In addition to massive vectors, we detail how other massive, spinning particles can be studied with Hilbert series; in particular, we spell out the ingredients for massive gravity in general spacetime dimensions. Further methodology is introduced to enable Hilbert series to capture the effect of spurion fields acquiring vevs. We apply the techniques to the Higgs Effective Field Theory (HEFT), providing a systematic enumeration of its operator basis. This is achieved both from a direct and a custodial symmetry spurion-based approach; we compare and contrast the two approaches, and our results to those appearing in previous literature.