A Mean-Field Approach to the Dielectric Response of Bulk Superconductors for Light Dark Matter Direct Detection

The dielectric function is central to describing many-body screening effects in dark matter (DM) direct detection with condensed matter targets. Current superconducting detector analyses employ the free-electron Lindhard dielectric function to model in-medium effects, an approximation whose validity in the superconducting state remains untested. We derive the electronic dielectric function for bulk superconductors within the Bardeen-Cooper-Schrieffer (BCS) framework, incorporating the full Bogoliubov quasiparticle coherence factors in the random-phase approximation. A systematic comparison with the Lindhard function for aluminum and tungsten silicide (WSi) reveals good agreement for energy depositions $ω\gtrsim5Δ$, establishing the Lindhard function as a robust approximation for superconducting DM detectors operating in this regime.