Hilbert series for covariants and their applications to minimal flavor violation

• Published in: The journal of high energy physics Vol. 2024; no. 6; pp. 154 - 83
• Main Authors: Grinstein, Benjamín, Lu, Xiaochuan, Merlo, Luca, Quílez, Pablo
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 24.06.2024; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Effective Field Theories; Elementary Particles; Flavor (particle physics); Flavour Symmetries; Group theory; Hypotheses; Invariants; Modules; Particle physics; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Quarks; Regular Article - Theoretical Physics; Relativity Theory; Representations; Series (mathematics); SMEFT; String Theory; Effective Field Theories; SMEFT; Flavour Symmetries
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP06(2024)154

A bstract We elaborate how to apply the Hilbert series method to enumerating group covariants, which transform under any given representation, including but going beyond group invariants. Mathematically, group covariants form a module over the ring of the invariants. The number of independent covariants is given by the rank of the module, which can be computed by taking a ratio of two Hilbert series. In many cases, the rank equals the dimension of the group covariant representation. When this happens, we say that there is a rank saturation . We apply this technology to revisit the hypothesis of Minimal Flavor Violation in constructing Effective Field Theories beyond the Standard Model. We find that rank saturation is guaranteed in this case, leading to the important consequence that the MFV symmetry principle does not impose any restriction on the EFT, i.e. MFV SMEFT = SMEFT, in the absence of additional assumptions.