Kinetic Heating from Contact Interactions with Relativistic Targets: Electrons Capture Dark Matter in Neutron Stars

Dark matter can capture in neutron stars from scattering off ultra-relativistic electrons. We present a method to calculate the capture rate on degenerate targets with ultra-relativistic momenta in a compact astronomical object. Our treatment accounts for the target momentum and the Fermi degeneracy of the system. We derive scaling relations for scattering with relativistic targets and confirm consistency with the non-relativistic limit and Lorentz invariance. The potential observation of kinetic heating of neutron stars has a larger discovery reach for dark matter--lepton interactions than conventional terrestrial direct detection experiments. We map this reach onto a set of bosonic and fermionic effective contact interactions between dark matter and leptons as well as nucleons. This completes the study of spin-0 and spin-1/2 dark matter kinetic heating of neutron stars through contact operators with Standard Model fermions up to dimension-6. Our method is generalizable to dark matter scattering in any degenerate medium where the Pauli exclusion principle leads to relativistic targets with a constrained phase space for scattering.