Positivity in Massive Spin-3/2 EFTs and the Planck-Suppressed Neighbourhood of Supergravity

Abstract

It is well known that a strictly massless spin-$3/2$ particle can interact consistently only within supergravity. Recently, positivity arguments have shown that an effective field theory of a massive Majorana spin-$3/2$ particle admits a smooth $m \to 0$ limit only if a graviton is present and the four-fermion contact interactions are tuned to the values dictated by $\mathcal{N}=1$ supergravity. In this work, we investigate how this limit is approached at finite mass. Assuming that the graviton $t$-channel pole can be discarded, we derive non-forward, tree-level dispersive bounds on massive spin-3/2 contact operators and determine the region of effective couplings consistent with unitarity and analyticity. For sufficiently small $m$, we find that the allowed parameter space forms a bounded, Planck-suppressed neighbourhood of the supergravity point, defined by the supergravity values of the four-fermion couplings. The supergravity point lies on the boundary of this region. In the regime $m \ll M_{\rm Pl}$, the volume of the allowed region scales parametrically as \[ \mathrm{Vol} \sim \frac{m^{6}}{M_{\rm Pl}^{6}} \, , \] and shrinks to zero as $m \to 0$, smoothly reproducing the massless-limit results. The allowed region becomes unbounded when mass approaches the Planck scale. We further analyze the effect of including additional light scalar and pseudo-scalar degrees of freedom, motivated by the Polonyi model, and find that their couplings are also bounded in a way similar to the contact couplings and that it doesn't enlarge the allowed contact coupling space.