Gravitating (bi-)sphalerons

TL;DR

This paper investigates how sphaleron and bisphaleron solutions in the electroweak model respond to gravity, revealing complex bifurcation patterns and critical phenomena depending on the Higgs mass and gravitational coupling.

Contribution

It provides a detailed analysis of the bifurcation structures and critical behavior of sphaleron solutions in the coupled electroweak-gravity system, highlighting new solution branches and termination points.

Findings

Multiple solution branches exist for different Higgs masses.

Bifurcation points depend on the ratio of vector-boson mass to Planck mass.

Solutions terminate at cusp catastrophes or connect to known solutions like Bartnik-McKinnon.

Abstract

The standard model of electroweak interactions is minimally coupled to gravity and the response of the spherically symmetric solutions -the sphaleron and the bisphaleron- to gravity is emphasized. For a given value of the Higgs mass $M_H$, several branches of solutions exist which terminate into cusp-catastrophy at some ($M_H$-depending) critical value of the parameter $\alpha$ defined by the ratio of the vector-boson mass to the Planck mass. A given branch either bifurcates from another one at an intermediate value of $\alpha$ or persists in the limit $\alpha \to 0$ where it terminates into a flat sphaleron or bisphaleron or into a Bartnik-McKinnon solution. These bifurcation patterns are studied in some details.