Hilbert series, the Higgs mechanism, and HEFT

• Published in: The journal of high energy physics Vol. 2023; no. 2; pp. 64 - 42
• Main Authors: Gráf, Lukáš, Henning, Brian, Lu, Xiaochuan, Melia, Tom, Murayama, Hitoshi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 07.02.2023; Springer Nature B.V; Springer Nature; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Effective Field Theories; Elementary Particles; Enumeration; Field theory; Gauge Symmetry; Gauge theory; Higgs Properties; High energy physics; Laboratories; Operators (mathematics); Partitions (mathematics); Physics; Physics and Astronomy; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; S matrix theory; Spontaneous Symmetry Breaking; String Theory; Symmetry; Higgs Properties; Gauge Symmetry; Effective Field Theories; Spontaneous Symmetry Breaking
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP02(2023)064

A bstract 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.