Most general EFTs from spurion analysis: Hilbert series and Minimal Lepton Flavor Violation

TL;DR

This paper establishes a saturation theorem for EFTs using spurion analysis, demonstrating that the EFT Lagrangian with spurions captures all invariants under the unbroken subgroup, and applies this to Minimal Lepton Flavor Violation scenarios.

Contribution

The paper proves a saturation theorem for EFTs constructed via spurion analysis and computes Hilbert series for various MLFV scenarios, linking spurion invariants to unbroken subgroup invariants.

Findings

EFT Lagrangian with spurions saturates the non-spurion EFT restricted to unbroken subgroup invariants.

Hilbert series calculations match the number of invariants in MLFV scenarios.

Provided computational tools for Hilbert series of invariants and covariants.

Abstract

We derive a saturation theorem for general Effective Field Theories (EFTs) constructed using spurion analysis. Let $S$ be a set of spurion fields introduced to organize the breaking of a global symmetry $G_f$, and $H_S$ be the subgroup of $G_f$ that remains unbroken under a generic vacuum expectation value $\langle S\rangle$; we show that the EFT Lagrangian constructed from the spurion analysis $saturates$ the EFT Lagrangian without the spurions but restricted to $H_S$ invariance, provided that arbitrary powers of the spurion fields are allowed. As examples, we study several implementations of the Minimal Lepton Flavor Violation (MLFV) principle, corresponding to various origins of the neutrino masses. In each scenario, we compute the Hilbert series to obtain the numbers of independent lepton flavor covariants that appear in the corresponding EFT at mass dimension 6. These numbers agree with the number of $H_S$ invariants in the EFT without the spurions, demonstrating the saturation theorem. Motivated by phenomenological connections, we provide linearly independent spurion polynomials for selected lepton flavor covariants. An ancillary file is supplied at https://github.com/HilbertSeries/Group_Invariants_and_Covariants , which is a Mathematica notebook that provides functions for computing general Hilbert series of invariants and covariants of compact classical groups. It presents examples demonstrating the use of the code, including the Hilbert series for our MLFV scenarios.