Time Dependent CP-even and CP-odd Signatures of Scalar Ultra-light Dark Matter in Neutrino Oscillations

Scalar ultra-light dark matter (ULDM) interacting with neutrinos can induce, under certain conditions, time-dependent modifications to neutrino oscillation probabilities. The limit in which the ULDM perturbation can be treated as constant throughout the neutrino propagation time has been addressed by several previous works. We complement these by systematically analyzing the opposite limit -- accounting for the temporal-variations of the ULDM potential by solving time-dependent Schr\"odinger equations. In particular, we study a novel two-generations-like CP violating (CPV) signature unique to rapidly oscillating ULDM. We derive the leading order, time-dependent, corrections to the oscillation probabilities, both for CP conserving (CPC) and CPV couplings, and explain how they can be measured in current and future experiments.