Generation of cosmological large lepton asymmetry from a rolling scalar field

We propose a new scenario to simultaneously explain a large lepton asymmetry and a small baryon asymmetry. We consider a rolling scalar field and its derivative coupling to the lepton number current. The presence of an effective nonzero time derivative of the scalar field leads to $CPT$ violation so that the lepton asymmetry can be generated even in thermal equilibrium as pointed out by Cohen and Kaplan. In this model, the lepton asymmetry varies with time. In particular, we consider the case where it grows with time. The final lepton asymmetry is determined by the decoupling of the lepton number violating interaction and can be as large as order unity. On the other hand, if the decoupling takes place after the electroweak phase transition, a small baryon asymmetry is obtained from the small lepton asymmetry at that time through sphaleron effects. We construct a model in which a rolling scalar field is identified with a quintessence field.