Particle abundance in a thermal plasma: Quantum kinetics versus Boltzmann equation

Abstract

We study the abundance of a particle species in a thermalized plasma by introducing a quantum kinetic description based on the nonequilibrium effective action. A stochastic interpretation of quantum kinetics in terms of a Langevin equation emerges naturally. We consider a particle species that is stable in the vacuum and interacts with heavier particles that constitute a thermal bath in equilibrium. Asymptotic theory suggests a definition of a fully renormalized single particle distribution function. Its real time dynamics is completely determined by the nonequilibrium effective action which furnishes a Dyson-like resummation of the perturbative expansion. The distribution function reaches thermal equilibrium on a time scale $\ensuremath{\sim}1/2{\ensuremath{\Gamma}}_{k}(T)$ with ${\ensuremath{\Gamma}}_{k}(T)$ being the quasiparticle relaxation rate. The equilibrium distribution function depends on the full spectral density as a consequence the fluctuation-dissipation relation. Such dependence leads to off shell contributions to the particle abundance. A specific model of a bosonic field $\ensuremath{\Phi}$ in interaction with two heavier bosonic fields ${\ensuremath{\chi}}_{1,2}$ is studied. The decay of the heaviest particle and its recombination lead to a width of the spectral function for the particle $\ensuremath{\Phi}$ and to off shell corrections to the abundance. We find substantial departures from the Bose-Einstein result both in the high temperature and the low temperature but high momentum region. In the latter the abundance is exponentially suppressed but larger than the Bose-Einstein result. We obtain the Boltzmann equation in renormalized perturbation theory and highlight the origin of the differences. Cosmological consequences are discussed: we argue that the corrections to the abundance of cold dark matter candidates are observationally negligible and that recombination erases any possible spectral distortions of the cosmic microwave background (CMB). However we expect that the enhancement at high temperature may be important for baryogenesis.