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.

arXiv comment33 pages + an appendix, 1 ancillary Mathematica file

References in corpus (36)

• Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching
• Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO
• The Chiral Approach to the Electroweak Interactions
• Heavy Majorana Neutrinos in the Effective Lagrangian Description: Application to Hadron Colliders
• Operators up to Dimension Seven in Standard Model Effective Field Theory Extended with Sterile Neutrinos
• Operator bases, $S$-matrices, and their partition functions
• Low-derivative operators of the Standard Model effective field theory via Hilbert series methods
• Hilbert Series for Constructing Lagrangians: expanding the phenomenologist's toolbox
• A Geometric Formulation of Higgs Effective Field Theory: Measuring the Curvature of Scalar Field Space
• Geometry of the Scalar Sector
• Hilbert series and operator bases with derivatives in effective field theories
• Dimension Seven Operators in Standard Model with Right handed Neutrinos
• Is SMEFT Enough?
• Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation
• The complete HEFT Lagrangian after the LHC Run I
• Which EFT
• The electroweak effective field theory from on-shell amplitudes
• An explicit construction of the dimension-9 operator basis in the standard model effective field theory
• The Geometric Standard Model Effective Field Theory
• Effective Field Theory of Gravity to All Orders
• Hilbert Series for Flavor Invariants of the Standard Model
• Constructing effective field theories via their harmonics
• The Higgs Trilinear Coupling and the Scale of New Physics
• Constructing massive on-shell contact terms
• Unitarity Violation and the Geometry of Higgs EFTs
• 2, 12, 117, 1959, 45171, 1170086, ...: A Hilbert series for the QCD chiral Lagrangian
• Determining the Shape of Higgs Potential at Future Colliders
• Hilbert Series and Operator Basis for NRQED and NRQCD/HQET
• Non-Decoupling New Particles
• Amplitude bases in generic EFTs
• Flavor Invariants and Renormalization-group Equations in the Leptonic Sector with Massive Majorana Neutrinos
• On-shell Higgsing for EFTs
• CP violation and flavor invariants in the seesaw effective field theory
• Reparameterization Invariant Operator Basis for NRQED and HQET
• Spelling Out Leptonic CP Violation in the Language of Invariant Theory
• Conformal Structure of the Heavy Particle EFT Operator Basis

Cited by in corpus (17)

• The Standard Model effective field theory at work
• Complete NLO Operators in the Higgs Effective Field Theory
• An EFT hunter's guide to two-to-two scattering: HEFT and SMEFT on-shell amplitudes
• Complete NNLO Operator Bases in Higgs Effective Field Theory
• Primary Observables for Indirect Searches at Colliders
• 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