Fermion scattering in a Bose-Einstein condensate

Abstract

In a recent work, we considered the propagation of fermions in the background of a scalar Bose-Einstein (BE) condensate. Using some illustrative Yukawa-type coupling models between the fermions and the scalar fields, we determined the dispersion relations of the fermions and the scalar modes in various models. To complement that work, here we consider the corresponding fermion spinors and propagators, which are required for the calculation of rates of processes involving the fermions, as well as the thermal, and/or higher order, corrections to such rates. We obtain and present here concise formulas which are useful for those applications. As an application and for illustrative purposes we specifically calculate the rate for a generic fermion (which we denote by $\chi$) with the fermions in the BE background, using commonly used models for the fermion interactions. Due to the fact that the background fermion dispersion relations are helicity dependent, the kinematics have some unique and non-standard features. For example, at a particular value of the momentum one of the fermion modes has zero group velocity, which leads to a singularity in the scattering rate of the type of the Van Hove singularity in the density of states of some condensed matter systems. The results of the framework presented here, besides their merit in their own right, can be useful in specific contexts and applications, such as cosmic-ray electron cooling through dark matter-electron scattering, and similar ones involving neutrino and/or electron propagation in a scalar Dark Matter background.