Constraint on massive vector field with extreme-mass-ratio inspirals around a slowly rotating black hole

We study the influence of a massive vector (Proca) field on the energy fluxes from extreme-mass-ratio inspirals (EMRIs) around a slowly rotating Kerr black hole. The secondary compact object, carrying a Proca hair, emits additional dipolar radiation that alters total energy flux relative to general relativity (GR). These modifications induce a secular drift in the orbital evolution of circular geodesic orbits, leading to measurable dephasing in the resulting EMRIs waveforms. By evaluating waveform mismatches between the Einstein-Proca framework and its GR counterpart, we show that the Laser Interferometer Space Antenna (LISA) can distinguish the signatures of a light Proca field when black hole rotation is included. Furthermore, using a Fisher information matrix analysis, we forecast LISA's capability to place stringent constraints on the Proca mass with EMRIs signal from slowly rotating Kerr black holes. For representative EMRIs configurations, we find that LISA can detect or constrain Proca masses down to $\mu_v\sim 10^{-20}$eV, with typical fractional uncertainties at the level of tens percent, depending on the black-hole spin.