An exact solution of the Dirac equation with CP violation

TL;DR

This paper derives an exact solution to the Dirac equation with a time-dependent, CP-violating fermion mass, revealing oscillatory effects in fermionic currents relevant for early Universe baryogenesis.

Contribution

It provides the first exact, helicity-conserving solution for a CP-violating, time-dependent fermion mass in Yukawa theory, applicable in both weak and strong CP violation regimes.

Findings

Fermionic currents can show large oscillatory magnification due to state squeezing.

Gradient approximation captures averaged currents well but misses oscillatory effects.

Agreement with semiclassical force is good in thick wall regimes, but discrepancies exist in thin wall regimes.

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 both in the case of weak and strong CP violation. Using this solution we then study the fermionic currents both in 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: 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.