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

TL;DR

This paper analyzes gravitino scattering amplitudes in supergravity, showing how unitarity is preserved in some models but breaks down at a certain scale in models with Fayet-Iliopoulos terms, indicating a cutoff for the effective theory.

Contribution

It demonstrates the conditions under which unitarity is maintained or violated in gravitino scattering, especially highlighting the role of Fayet-Iliopoulos terms in supersymmetry breaking.

Findings

Goldstino and sgoldstino contributions cancel leading divergences in certain models.

Unitarity is preserved up to the Planck scale in models with spontaneous SUSY breaking.

Fayet-Iliopoulos terms lead to a lower cutoff scale where the effective theory breaks down.

Abstract

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.