Electroweak sphalerons, scalar multiplets, and symmetry breaking patterns
In this study, we present a comprehensive analysis of the electroweak sphaleron formalism and its application to electroweak phase transition (EWPT) patterns in extensions of the Standard Model scalar sector with electroweak multiplets. We offer an equivalence proof for different choices for the form of sphaleron configurations; construct the previously unestablished high-dimensional $\text{SU}(2)$ sphaleron transformation matrix; and revisit the required boundary conditions needed for solving the sphaleron field equations. We then investigate the leading order sphaleron dynamics in the context of a multi-step EWPT. We showcase two distinct analytical approaches for extending the $\text{SU}(2)$ scalar multiplet to the standard model (SM) under differing EWPT scenarios, and perform an explicit calculation of the sphaleron energy using a septuplet example. In the context of a single-step EWPT leading to a mixed phase, we find that the additional multiplet's contribution to the sphaleron energy is negligible, primarily due to the prevailing constraint imposed by the $\rho$ parameter. Conversely, in a two-step EWPT scenario, the sphaleron energy can achieve significantly high values during the initial phase, thereby markedly preserving baryon asymmetry if the universe undergoes a first-order EWPT. In both cases, we delineate the relationship between the sphaleron energy and the parameters relevant to dark matter phenomenology.