Minimizing Higgs Potentials via Numerical Polynomial Homotopy Continuation

TL;DR

This paper introduces the use of numerical polynomial homotopy continuation (NPHC) to efficiently find all stationary points of complex Higgs potentials, aiding the analysis of extended Higgs models.

Contribution

It applies NPHC to the most general two-doublet and five-singlet Higgs potential, surpassing traditional methods like Gr"obner bases in complexity.

Findings

Successfully finds all stationary points of complex Higgs potentials.

Reveals the structure of the potential, including maxima, minima, and saddle points.

Demonstrates applicability to highly involved Higgs potentials.

Abstract

The study of models with extended Higgs sectors requires to minimize the corresponding Higgs potentials, which is in general very difficult. Here, we apply a recently developed method, called numerical polynomial homotopy continuation (NPHC), which guarantees to find all the stationary points of the Higgs potentials with polynomial-like nonlinearity. The detection of all stationary points reveals the structure of the potential with maxima, metastable minima, saddle points besides the global minimum. We apply the NPHC method to the most general Higgs potential having two complex Higgs-boson doublets and up to five real Higgs-boson singlets. Moreover the method is applicable to even more involved potentials. Hence the NPHC method allows to go far beyond the limits of the Gr\"obner basis approach.