The invariant space of multi-Higgs doublet models

TL;DR

This paper uses invariant theory and partition analysis to characterize the physical parameter space of multi-Higgs doublet models, providing a systematic, basis-invariant method for counting parameters and describing the model's structure.

Contribution

It introduces a novel application of Hilbert series and Omega calculus to the 3HDM, enabling a full description of its physical parameters and a basis-invariant counting technique for models with multiple doublets.

Findings

Computed the Hilbert series for 3HDM using partition analysis.

Provided a general counting method for models with N scalar doublets.

Developed a basis-invariant parameter counting technique for complex Lagrangians.

Abstract

In a model with more than one scalar doublet, the parameter space encloses both physical and unphysical information. Invariant theory provides a detailed description of the counting and characterization of the physical parameter space. The Hilbert series for the 3HDM is computed for the first time using partition analysis, in particular Omega calculus, giving rise to the possibility of a full description of its physical parameters. A rigorous counting of the physical parameters is given for the full class of models with $N$ scalar doublets as well as a decomposition of the Lagrangian into irreducible representations of $\mathrm{SU}(N)$. For the first time we derive a basis-invariant technique for counting parameters in a Lagrangian with both basis-invariant redundancies and global symmetries.