Out of equilibrium O(N) linear-sigma system: Construction of perturbation theory with gap and Boltzmann equations

We establish from first principles a perturbative framework that allows us to compute reaction rates for processes taking place in nonequilibrium O(N) linear-sigma systems in a broken phase. The system of our concern is a quasiuniform system near equilibrium or a nonequilibrium quasistationary system. We employ the closed-time-path formalism and use the so-called gradient approximation. No further approximation is introduced. In the course of the construction of the framework, we obtain the gap equation that determines the effective masses of π and of σ, and the generalized Boltzmann equation that describes the evolution of the number-density functions of π and of σ.