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 1/(e^ωk/T-1) to a modified one given by a Gibbs formula. Integrated over momenta, the particle number density becomes roughly of order (coupling) X √T/MT³ 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 ≈6×10⁴ eV cm^-3 (M/GeV)^4/3. The bound from the closure density gives an upper WIMP mass bound roughly of order 1 GeV in the present model.