Phase transitions, anomalous baryon number violation, and electroweak multiplet dark matter

We perform a comprehensive analysis of baryon number violation during an electroweak phase transition (EWPT) within the framework of a scalar electroweak multiplet extension of the Standard Model (SM). We classify the multiplet representations, topological properties, and corresponding thermal histories. Sphaleron or monopole topological field solutions emerge during the EWPT depending on the stage of the phase transition and the hypercharge of the new scalar multiplet. Furthermore, the monopole field solution pertains when the neutral component of the additional scalar multiplet is a viable dark matter candidate. We further analyze other formal considerations, including the construction of the “sphaleron matrix” for higher dimensional representations, computation of the sphaleron and monopole masses, and the choice of boundary conditions when solving the field equations of motion. We apply these considerations to the computation of sphaleron energy and monopole mass within the context of a multistep EWPT, employing the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>SU</a:mi><a:mo stretchy="false">(</a:mo><a:mn>2</a:mn><a:mo stretchy="false">)</a:mo></a:mrow><a:mrow><a:mi>L</a:mi></a:mrow></a:msub></a:mrow></a:math> septuplet scalar extension to the SM as a case of study from the minimal dark matter paradigm. For the first step of a two-step EWPT, we delineate the relationship between the monopole mass and the parameters relevant to dark matter phenomenology.