Abstract
In a weakly coupled ultrarelativistic plasma, 1 + n ↔ 2 + n scatterings, with n ≥ 0, need sometimes to be summed to all orders, in order to determine a leading-order interaction rate. To implement this “LPM resummation”, kinematic approximations are invoked. However, in cosmological settings, where the temperature changes by many orders of magnitude and both small and large momenta may play a role, such approximations are not always justified. We suggest a procedure to smoothly interpolate between LPM-resummed 1 + n ↔ 2 + n and Born-level 1 ↔ 2 results, rendering the outcome applicable to a broader range of masses and momenta. The procedure is illustrated for right-handed neutrino production from a Standard Model plasma, and dilepton production from a QCD plasma.
Highlights
It is typical of resummations that their consistent implementation requires the presence of a scale hierarchy
In the case of LPM resummation, the scale hierarchy is that characterizing the UR regime, i.e. with masses small compared with momenta, the latter of which are of order πT
We start by working out the kinematics of Born-level 1 ↔ 2 processes in a special coordinate system which permits us to put the expression in a form similar to that appearing in LPM resummation
Summary
We start by recalling the derivation of the phase space average for thermal 1 ↔ 2 reactions at the Born level, arriving at an expression (cf. eq (2.25)) which can subsequently be interpolated (cf. section 4) to the result obtained from LPM resummation (cf. section 3). For determining the interaction rate originating from 1 ↔ 2 reactions at the Born level (this is generically denoted by ΓB1↔or2n, remarking that an overall normalization factor is needed for obtaining the physical rate, cf footnotes 2 and 3), it is sufficient to compute the functional form originating from 1 → 2 decays of a would-be non-equilibrium particle. The longitudinal momentum components, appearing inside Θ, satisfy eq (2.16), viz. As guaranteed by the Dirac-δ in eq (2.19), a and p2⊥ are not independent; their relation can be expressed as p2a. The latter condition can be completed into a square, and is always satisfied This implies that no upper bound needs to be imposed on p⊥. Making an effort to write the argument of the Dirac-δ in a more transparent form, we end up with. For a polynomial Θ, the remaining integrals could be carried out in terms of polylogarithms, for us it is advantageous to leave them unintegrated
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