Scattering cross section of the long gravitino wave in Schwarzschild spacetime

TL;DR

This work computes the differential scattering cross section for a gravitino ($s=\tfrac{3}{2}$) in Schwarzschild spacetime in the long-wavelength limit $M\omega\to0$ using a first-order perturbative approach tailored to Cartesian coordinates. By formulating a simulating wave function method—inspired by Archimedes’ buoyancy technique—the authors construct scattering and bounded states and connect simulation coefficients to the true gravitino wave, while carefully accounting for spin-1/2 contamination. The main result is that the gravitino cross section in this regime shares the same spin-dependent pattern as lower-spin fields and, explicitly, $\lim_{M\omega\to0} \frac{d\sigma}{d\Omega}= M^2 \cos^6(\theta/2)/\sin^4(\theta/2)$, mirroring the scalar, neutrino, and electromagnetic cases. The work demonstrates the equality of the simulating and true wave functions in this setting and suggests a deeper symmetry in how curved spacetime couples to massless fields across spins in the long-wavelength limit.

Abstract

We investigate the scattering of gravitino wave in a Schwarzschild gravitational field. Employing the simulating wave function method within the framework of perturbative techniques, we derive the differential scattering cross section for the gravitino wave in the long-wavelength limit. It is proven that the cross section of gravitino wave follows the same spin-dependent pattern as those of scalar, neutrino, and electromagnetic waves.

Figures (1)

• Figure 1: Flow chart for calculating differential scattering cross section of gravitino wave in the long-wavelength limit.