Exact solution of the Dirac equation withCPviolation

Abstract

We consider Yukawa theory in which the fermion mass is induced by a Higgs-like scalar. In our model the fermion mass exhibits a temporal dependence, which naturally occurs in the early Universe setting. Assuming that the complex fermion mass changes as a tanh-kink, we construct an exact, helicity-conserving, $CP$-violating solution for the positive- and negative-frequency fermionic mode functions, which is valid in both the weak and strong $CP$ violation cases. Using this solution we then study the fermionic currents in both the initial vacuum and finite-density/-temperature setting. Our result shows that---due to a potentially large state squeezing---fermionic currents can exhibit a large oscillatory magnification. Having in mind applications to electroweak baryogenesis, we then compare our exact results with those obtained in a gradient approximation. Even though the gradient approximation does not capture the oscillatory effects of squeezing, it describes quite well the averaged current, obtained by performing a mode sum. Our main conclusion is that while the agreement with the semiclassical force is quite good in the thick-wall regime, the difference is sufficiently significant to motivate a more detailed quantitative study of baryogenesis sources in the thin-wall regime in more realistic settings.