Geometric minimization of highly symmetric potentials

TL;DR

This paper introduces a geometric method for efficiently finding the global minimum of highly symmetric Higgs potentials, improving upon traditional approaches and providing new insights into vacuum structure and CP-violation.

Contribution

It presents a novel geometric approach to minimize symmetric Higgs potentials, applicable to complex models like S_4 and A_4-symmetric three-Higgs-doublet models.

Findings

The method reliably finds the global minimum for various potentials.

Recent analyses may have used local minima, affecting phenomenological conclusions.

Identifies a new symmetry linking different minima and discusses geometrical CP-violation.

Abstract

In non-minimal Higgs mechanisms, one often needs to minimize highly symmetric Higgs potentials. Here we propose a geometric way of doing it, which, surprisingly, is often much more efficient than the usual method. By construction, it gives the global minimum for any set of free parameters of the potential, thus offering an intuitive understanding of how they affect the vacuum expectation values. For illustration, we apply this method to the S_4 and A_4-symmetric three-Higgs-doublet models. We find that at least three recent phenomenological analyses of the A_4-symmetric model used a local, not the global minimum. We discuss coexistence of minima of different types, and comment on the mathematical origin of geometrical CP-violation and on a new symmetry linking different minima.