Efficient numerical integration of thermal interaction rates

Abstract

In many problems in particle cosmology, interaction rates are dominated by 2 ↔ 2 scatterings, or get a substantial contribution from them, given that 1 ↔ 2 and 1 ↔ 3 reactions are phase-space suppressed. We describe an algorithm to represent, regularize, and evaluate a class of thermal 2 ↔ 2 and 1 ↔ 3 interaction rates for general momenta, masses, chemical potentials, and helicity projections. A key ingredient is an automated inclusion of virtual corrections to 1 ↔ 2 scatterings, which eliminate logarithmic and double-logarithmic IR divergences from the real 2 ↔ 2 and 1 ↔ 3 processes. We also review thermal and chemical potential induced contributions that require resummation if plasma particles are ultrarelativistic.

Smooth interpolation between thermal Born and LPM rates

“…These have recently been analysed in some generality in ref. [31]. On the other hand, there are real 1 + n ↔ 2 + n processes, with n ≥ 0, that need often to be summed to all orders in n, through LPM resummation [1][2][3][4].…”

“…A method to determine virtual corrections was worked out in ref. [31]. The starting point is to determine the real 1 → 3 rate, given in its eq.…”

“…(2.37) of ref. [31]. The virtual corrections contain thermal 1-loop integrals of various types, of which only a subclass can lead to a HTL contribution, proportional to T 2 [32].…”

“…(2.37) of ref. [31], the only one is the mixed fermion-boson loop, weighted by a loop momentum, denoted by B(P ˜ ; , γ) E • P . Specifically, carrying out the angular integral in eq.…”

“…Therefore the methods of ref. [31] are of little help here, and a dedicated analysis has become necessary.…”