On the terms violating the custodial symmetry in multi-Higgs-doublet models

TL;DR

This paper proves that multi-Higgs-doublet models inherently include terms that violate custodial symmetry, which cannot be eliminated by symmetries and are expected to arise from higher-order corrections, affecting vacuum alignment and mass degeneracy.

Contribution

It demonstrates that custodial symmetry-violating terms are unavoidable in generic multi-Higgs-doublet models and cannot be removed by imposing symmetries on the Lagrangian.

Findings

O(4) violating terms cannot be excluded by symmetries.

Higher-order corrections will introduce custodial symmetry violation.

Vacuum alignment affects up-down quark mass degeneracy.

Abstract

We prove that a generic multi-Higgs-doublet model (NHDM) generally must contain terms in the potential that violate the custodial symmetry. This is done by showing that the O(4) violating terms of the NHDM potential cannot be excluded by imposing a symmetry on the NHDM Lagrangian. Hence we expect higher-order corrections to necessarily introduce such terms. We also note, in the case of custodially symmetric Higgs-quark couplings, that vacuum alignment will lead to up-down mass degeneration; this is not true if the vacua are not aligned.