Hawking radiation for a Proca field in D-dimensions
We study the wave equation of a massive vector boson in the background of a $D$-dimensional Schwarzschild black hole. The mass term introduces a coupling between two physical degrees of freedom of the field, and we solve the resulting system of ordinary differential equations numerically, without decoupling. We show how to define decoupled transmission factors from an $\mathbf{S}$ matrix and compute them for various modes, masses, and space-time dimensions. The mass term lifts the degeneracy between transverse modes, in $D=4$, and excites the longitudinal modes, in particular, the $s$ wave. Moreover, it increases the contribution of waves with larger $\ensuremath{\ell}$, which can be dominant at intermediate energies. The transmission factors are then used to obtain the Hawking fluxes in this channel. Our results alert for the importance of modeling the longitudinal modes correctly, instead of treating them as decoupled scalars as they are in current black hole event generators; thus, they can be used to improve such generators for phenomenological studies of TeV gravity scenarios.