Sphalerons and the Electroweak Phase Transition in Models with Higher Scalar Representations

TL;DR

This paper studies sphaleron solutions in an extended gauge theory with higher scalar representations, analyzing their energy, profiles, and implications for the electroweak phase transition, revealing that higher representations relax the conditions for a strong first order transition.

Contribution

It provides the first detailed analysis of sphalerons in models with higher scalar representations, including their energy scaling and phase transition criteria.

Findings

Sphaleron profiles are similar across scalar representations.

Sphaleron energy scales linearly with scalar VEV.

Higher representations relax the strong first order phase transition condition.

Abstract

In this work we investigate the sphaleron solution in a $SU(2)\times U(1)_X$ gauge theory, which also encompasses the Standard Model, with higher scalar representation(s) ($J^{(i)},X^{(i)}$). We show that the field profiles describing the sphaleron in higher scalar multiplet, have similar trends like the doublet case with respect to the radial distance. We compute the sphaleron energy and find that it scales linearly with the vacuum expectation value of the scalar field and its slope depends on the representation. We also investigate the effect of $U(1)$ gauge field and find that it is small for the physical value of the mixing angle, $\theta_{W}$ and resembles the case for the doublet. For higher representations, we show that the criterion for strong first order phase transition, $v_{c}/T_{c}>\eta$, is relaxed with respect to the doublet case, i.e. $\eta<1$.