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We derive CP-violating transport equations for fermions for electroweakbaryogenesis from the CTP-formalism including thermal corrections at theone-loop level. We consider both the VEV-insertion approximation (VIA) and thesemiclassical (SC) formalism. We show that the VIA-method is based on an {\emassumption} that leads to an ill-defined source term containing a pinchsingularity, whose regularisation by thermal effects leads to ambiguitiesincluding spurious ultraviolet and infrared divergences. We then carefullyreview the derivation of the semiclassical formalism and extend it to includethermal corrections. We present the semiclassical Boltzmann equations forthermal WKB-quasiparticles with source terms up to the second order ingradients that contain both dispersive and finite width corrections. We alsoshow that the SC-method reproduces the current divergence equations and that acorrect implementation of the Fick's law captures the semiclassical source termeven with conserved total current $\partial_\mu j^\mu = 0$. Our results showthat the VIA-source term is not just ambiguous, but that it does not exist.Finally, we show that the collisional source terms reported earlier in thesemiclassical literature are also spurious, and vanishes in a consistentcalculation.

CP-violating transport theory for Electroweak Baryogenesis with thermal corrections