Spin $3/2$ Fields Non-Minimal Coupling as Square Root of Linearized Gravity with Matter

TL;DR

This paper derives a non-minimal coupling for spin 3/2 fields by relating the Rarita-Schwinger equations to linearized gravity, revealing interactions with electromagnetic and Yang-Mills fields consistent with previous results.

Contribution

It introduces a novel approach to determine spin 3/2 interactions by interpreting the Rarita-Schwinger equations as the square root of linearized Einstein equations, connecting gravity and matter fields.

Findings

Electromagnetic interaction matches Ferrara et al.'s dipole term.

Interaction with gauge fields derived from linearized supergravity.

Consistent non-minimal couplings for spin 3/2 fields established.

Abstract

A non-minimal coupling for spin $3/2$ fields is obtained. We use the fact that the Rarita-Schwinger field equations are the square root of the full linearized Einstein field equations in order to investigate the form of the interaction for the spin $3/2$ field with gauge fields. We deduce the form of the interaction terms for the electromagnetic and non-Abelian Yang-Mills fields by implementing appropiate energy momentum tensors on the linearized Einstein field equations. The interaction found for the electromagnetic case happens to coincide with the dipole term found by Ferrara {\it et al} by a very different procedure, namely by demanding $g=2$ at the tree level for the electromagnetic interaction of arbitrary spin particles. The same interaction is found by using the resource of linearized Supergravity N=2. For the case of the Yang-Mills field Supergravity N=4 is linearized, providing the already foreseen interaction.