Hilbert Series, the Higgs Mechanism, and HEFT

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.

33 pages + an appendix, 1 ancillary Mathematica file

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• 23, 381, 6242, 103268, 1743183, ...: Hilbert series for CP-violating operators in SMEFT
• Primary Observables for Top Quark Collider Signals
• Walls, bubbles and doom -- the cosmology of HEFT
• Constructing Operator Basis in Supersymmetry: A Hilbert Series Approach
• Double Higgs Production in Vector Boson Fusion at NLO QCD in HEFT
• Counting Operators in $N=1$ Supersymmetric Gauge Theories
• Positively Identifying HEFT or SMEFT
• The Art of Counting: a reappraisal of the HEFT expansion
• $h\to gg$ and $h\to \gamma \gamma$ with Anomalous Couplings at Next-to-Leading Order in QCD
• Primary Observables for Electroweak Gauge Boson Collider Signals
• General Signals for Charged Lepton Flavor Violating Decays
• Assessing (H)EFT theory errors by pitting EoM against Field Redefinitions