Dark matter Sommerfeld-enhanced annihilation and bound-state decay at finite temperature

Traditional computations of the dark matter (DM) relic abundance, for models where attractive self-interactions are mediated by light force-carriers and bound states exist, rely on the solution of a coupled system of classical on-shell Boltzmann equations. This idealized description misses important thermal effects caused by the tight coupling among force-carriers and other charged relativistic species potentially present during the chemical decoupling process. We develop for the first time a comprehensive ab initio derivation for the description of DM long-range interactions in the presence of a hot and dense plasma background directly from nonequilibrium quantum field theory. Our results clarify a few conceptional aspects of the derivation and show that under certain conditions the finite temperature effects can lead to sizable modifications of DM Sommerfeld-enhanced annihilation and bound-state decay rates. In particular, the scattering and bound states get strongly mixed in the thermal plasma environment, representing a characteristic difference from a pure vacuum theory computation. The main result of this work is a novel differential equation for the DM number density, written down in a form which is manifestly independent under the choice of what one would interpret as a bound or a scattering state at finite temperature. The collision term, unifying the description of annihilation and bound-state decay, turns out to have in general a nonquadratic dependence on the DM number density. This generalizes the form of the conventional Lee-Weinberg equation which is typically adopted to describe the freeze-out process. We prove that our number density equation is consistent with previous literature results under certain limits. In the limit of vanishing finite temperature corrections our central equation is fully compatible with the classical on-shell Boltzmann equation treatment. So far, finite temperature corrected annihilation rates for long-range force systems have been estimated from a method relying on linear response theory. We prove consistency between the latter and our method in the linear regime close to chemical equilibrium