Electroweak Baryogenesis with \texttt{BARYONET}: a self-contained review of the WKB approach

Abstract: We present a comprehensive, self-contained pedagogical computation of the baryon asymmetry of the Universe within electroweak baryogenesis (EWBG), from the derivation of the semiclassical, CP-dependent force to the formulation and solution of the transport equations obtained from the Boltzmann equations$-$all implemented in the open-source code \texttt{BARYONET}. Our analysis follows the semiclassical WKB approach, where spatially varying complex masses across expanding bubble walls feel CP-violating forces that bias plasma transport. Starting from the stationary Boltzmann equation in the wall frame and projecting onto a hierarchy of velocity moments, we derive a compact, fluid-like system of coupled differential equations for chemical potentials and velocity perturbations. After obtaining the solutions, one can define the left-handed baryon chemical potential, which acts as the source term for the weak sphalerons. These processes generate the baryon asymmetry in front of the wall, which is subsequently frozen once it passes through it. We validate the framework against established formalisms and provide benchmarks in representative scenarios, including singlet extensions of the Standard Model, two-Higgs-doublet models, and Higgs$-\phi^6$ constructions. The resulting \texttt{BARYONET} implementation delivers an automated, reproducible pipeline for WKB-based baryogenesis studies, connecting formal derivations with phenomenological applications. In parallel, we revisit standard EWBG ingredients$-$\emph{diffusion constants}, \emph{Yukawa/helicity-flip rates}, and \emph{strong/weak sphaleron rates}$-$to clarify conventions, update numerical inputs, and present a pedagogical derivation, ensuring transparent reproducibility.