Derivation of transport equations for a strongly interacting Lagrangian in powers of $\hbar$ and $1/N_c$

Abstract

Transport theory for an interacting fermionic system is reviewed and applied to the chiral Lagrangian of the Nambu-Jona-Lasinio model. Two expansions must be applied: an expansion in the inverse number of colors, $1/N_c$, due to the nature of the strong coupling theory, and a semiclassical expansion, in powers of $\hbar$. The quasiparticle approximation is implemented at an early stage, and spin effects are omitted. The self-energy is evaluated, self-consistently only in the Hartree approximation, and semi-perturbatively in the collision integral. In the Hartree approximation, $O((1/N_c)^0)$, the Vlasov equation is recovered to $O(\hbar^1)$, together with an on-mass shell constraint equation, that is automatically fulfilled by the quasiparticle ansatz. The expressions for the self-energy to order $O((1/N_c))$ lead to the collision term. Here one sees explicitly that particle-antiparticle creation and annihilation processes are suppressed that would otherwise be present, should an off-shell energy spectral function be admitted. A clear identification of the $s$, $t$ and $u$ channel scattering processes in connection with the self-energy graphs is made and the origin of the mixed terms is made evident. Finally, after ordering according to powers in $\hbar$, a Boltzmann-like form for the collision integral is obtained.