Systematic analysis of 3HDM symmetries

TL;DR

The paper presents a systematic, brute-force classification of realisable symmetries in 3HDMs, focusing on HF and GCP transformations and extending the framework with GOOFy and T-GOOFy concepts. By iteratively imposing symmetry generators on the general 3HDM potential and analyzing their action in the $SU(2)$ bilinear space, it consolidates known results and uncovers new symmetry structures, including central-product liftings like $U(1) \circ V_4$ and CP4-type scenarios. The authors provide comprehensive, basis-invariant diagnostics (independent couplings and bilinear eigenvalue patterns) and tabulate the realisable symmetry groups, offering a practical reference for symmetry-based model building. They also highlight the nuanced role of basis choice, potential redundancies, and the possibility that GOOFy-type transformations open new avenues for decoupling sectors and exploring phenomenology, with future work extending these ideas to fermions and gauge interactions.

Abstract

Symmetries play a crucial role in shaping the structure and predictions of multi-Higgs-doublet models. In three-Higgs-doublet models considerable effort has been put into classifying possible symmetry groups and the conditions for their realisation, yet the completeness of existing classifications remains an open question. In this work, we revisit the problem of identifying realisable symmetries by re-examining conventional Higgs family and general CP transformations from an alternative perspective. Our analysis identifies certain limitations in previous approaches and introduces a clearer, more systematic framework for model builders. We expand our classification by investigating more generalised symmetry structures -- the recently identified GOOFy transformations, which act non-trivially on the Higgs doublets and their conjugates. Our analysis consolidates known results, uncovers previously overlooked structures, and expands the set of symmetries in three-Higgs-doublet models, offering both a clearer theoretical foundation and a practical reference for symmetry-based model building.