Particle number production and time variation with non-equilibrium quantum field theory

We study the particle number production and its time variation using non-equilibrium quantum field theory. We study the model proposed by Hotta et.al. \cite{Hotta:2012hi} for particle number production with a heavy neutral scalar and a light complex scalar. The interaction Lagrangian contains CP violating phase and particle number violating interaction among the scalars. The particle number violating mass term is also introduced, which splits a complex scalar into two real scalars with small non-degenerate mass. Therefore, the term generates particle and anti-particle mixing. We study the long time behavior of the particle number production rate.