  The future space-borne gravitational wave detector, Laser Interferometer
Space Antenna (LISA), has the potential of detecting the fundamental fields,
such as the charge and mass ultra-light scalar field. In this paper we study
the effect of lighter vector field on the gravitational waveforms from extreme
mass-ratio inspirals (EMRI) system, consisting of a stellar-mass object and the
massive black hole (MBH) in the Einstein-Proca theory of a massive vector field
coupling to gravity. Using the perturbation theory, we compute the energy
fluxes including the contributions of the Proca field and the gravitational
field, then obtain the adiabatic inspiraling orbits and corresponding
waveforms. Our results demonstrate that the vector charge and mass carried by
the secondary body lead to detectable effects on EMRI waveform, and LISA has
the potential to measure the mass of the Proca field with greater precision.
