Massive gravitino scattering amplitudes and the unitarity cutoff of the new Fayet-Iliopoulos terms

We compute the $2\rightarrow 2$ gravitino scattering amplitudes at tree level in supergravity theories where supersymmetry is spontaneously broken. In the unitary gauge, the gravitino becomes massive (of mass $m_{3/2}$) by absorbing the Goldstino, and the scattering amplitudes of its longitudinal polarisations grow with energy as $\kappa^2 E^4/m_{3/2}^2$, signaling a potential breakdown of unitarity at a scale $\Lambda^2\sim m_{3/2}/\kappa\sim M_{\mathrm{SUSY}}^2$. As we show explicitly in the Polonyi model, this leading term is cancelled by the contributions coming from the scalar partner of the Goldstino (sgoldstino), restoring perturbative unitarity up to the Planck scale. This is expected since supersymmetry is spontaneously broken, in analogy with the situation occuring in the Standard Model, where massive gauge bosons scattering preserves unitarity at high energy once we consider the contributions from the Higgs boson. However, when supersymmetry is broken by the new Fayet-Iliopoulos $D$-term, with ungauged R-symmetry, the above cancellation does not occur. In this case, the unbroken phase is singular and there is no contribution able to cancel the quartic divergences of the amplitudes, leading to a cutoff $\Lambda\sim M_\mathrm{SUSY}$ where the effective theory breaks down. The same behaviour is obtained when supersymmetry is non-linearly realised.