Platonic Sphalerons

TL;DR

This paper constructs new sphaleron solutions in Weinberg-Salam theory with discrete symmetries, linking their properties to rational maps and baryon number, and provides explicit examples with various symmetries.

Contribution

It introduces a novel class of sphaleron solutions characterized by discrete symmetries and their relation to rational maps and baryon number in Weinberg-Salam theory.

Findings

Constructed N=3 sphaleron with tetrahedral symmetry

Constructed N=4 sphaleron with cubic symmetry

Constructed N=5 sphaleron with octahedral symmetry

Abstract

We construct sphaleron solutions in Weinberg-Salam theory, which possess only discrete symmetries. Related to rational maps of degree N, these sphalerons carry baryon number Q_B=N/2. The energy density of these sphalerons reflects their discrete symmetries. We present an N=3 sphaleron with tetrahedral energy density, an N=4 sphaleron with cubic energy density, and an N=5 sphaleron with octahedral energy density.