The greybody factor for the Proca field in the spherically symmetric black hole spacetimes

Abstract

In this work, we study the greybody factor for a massive spin-1 field (Proca field) in the Schwarzschild black hole spacetime. We used vector spherical harmonics (VSH) to separate angular components, and we found a one-dimensional decoupled wave equation for the odd-parity mode and three coupled equations for the even-parity mode, including the monopole modes. For the even-parity case, we apply the Frolov–Krtouš–Kubizňák–Santos (FKKS) method to decouple the equations, which separates the Proca field into the even-parity vector mode and the even-parity scalar mode. We then compute the greybody factor using the rigorous bound method and the WKB approximation. We also compare the results from these two approaches. The results indicate that among the three polarization modes of the Proca field, the even-parity vector mode has the highest transmission probability, followed by the odd-parity mode, with the even-parity scalar mode having the lowest transmission probability. Moreover, the greybody factor for the small mass limit of even-parity vector polarization presented a higher transmission probability than the Maxwell particle, which interests us and requires further studies