WKB Analysis of the Scattering of Massive Dirac Fields in Schwarzschild Black Hole Spacetimes

TL;DR

This paper develops WKB approximation formulas to analyze the transmission probability of massive Dirac fields scattering off Schwarzschild black holes, covering different energy regimes and parameter dependencies.

Contribution

It introduces new WKB-based approximation methods for calculating transmission probabilities of massive Dirac fields in Schwarzschild spacetime, considering various energy and parameter regimes.

Findings

Explicit transmission probability formulas for different energy regimes.

Dependence of transmission probability on mass and angular momentum.

Approximate solutions applicable to various parameter values.

Abstract

We analysis the radial equations for massive Dirac fields in Schwarzschild black hole spacetimes. Different approximation formulae under the WKB scheme are developed for the transmission probability ${\cal T}$ of the radial wavefunction with $E^{2}\gg V_{m}$, $E^{2}\approx V_{m}$, and $E^{2}\ll V_{m}$, where $E$ is the energy of the field and $V_{m}$ is the maximum value of the effective potential. Explicit results of ${\cal T}$ in these approximations are given for various values of $E$, the mass $m$, and the angular momentum parameter $\kappa$ of the fields. We also discuss the dependence of ${\cal T}$ on these parameters.