Quantum kinetic equation and cosmic pair annihilation

Pair annihilation of heavy stable particles that occurs in the early universe is investigated, and a quantum kinetic equation for the momentum distribution of the annihilating particle is derived, using the influence functional method. A bosonic field theory model is used to describe the pair annihilation in the presence of decay product particles making up a thermal environment. A crossing symmetric Hartree approximation that determines self-consistently the equilibrium distribution is developed for an otherwise intractable theory. The time evolution equation and its Markovian approximation are derived to give a generalized Boltzmann equation including off-shell effects. The narrow width approximation to an energy integral in this equation gives the usual Boltzmann equation in a thermal bath of light particles. The off-shell effect is a correction to the Boltzmann equation at high temperatures, but is dominant at low temperatures. The effect changes the equilibrium distribution from the familiar (Formula presented) to a modified one given by a Gibbs formula. Integrated over momenta, the particle number density becomes roughly of order (coupling) (Formula presented) at low temperatures for S-wave annihilation. The relic mass density in the present universe is insensitive to the coupling strength in a large range of the mass and the coupling parameters, and scales with the WIMP mass as (Formula presented) The bound from the closure density gives an upper WIMP mass bound roughly of order 1 GeV in the present model.