  Pair annihilation of heavy stable particle that occurs in the early universe
is investigated, and 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 is 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 $1/(e^{\omega_{k}/T} - 1)$ to a modified one given by a Gibbs formula.
Integrated over momenta, the particle number density becomes roughly of order
(coupling) $\times \sqrt{T/M}\cdot T^{3}$ at low temperatures for the 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 (\approx 6 \times 10^{4} 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.
