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UV Constraints on Massive Spinning Particles: Lessons from the Gravitino

Self-interacting massive particles with spin $\geq 1$ unavoidably violate unitarity; the question is at what scale. For spin-$1$ the strong coupling scale (at which perturbative unitarity is lost) cannot be raised by any finite tuning of the interactions, while for spin-$2$ there exists a special tuning of the Wilson coefficients which can raise this scale (and enjoys numerous special properties such as ghost-freedom). Here, we fill in the missing piece by describing how the self-interactions of a massive spin-$3/2$ field, or "massive gravitino", become strongly coupled at high energies. We show that while several different structures appear in the leading order potential, the strong coupling scale cannot be raised (in the absence of additional fields). At the level of the off-shell Lagrangian, it is always the non-linear symmetries of the longitudinal Stuckelberg mode that dictate the strong coupling, and we show that in general it is only possible to parametrically raise the strong coupling scale if Wess-Zumino structures exist. We complement this off-shell approach with a first analysis of positivity bounds for a massive spin-$3/2$ particle, showing that any potential self-interaction which contributes to an on-shell 2-to-2 elastic process at tree level must vanish if this low-energy theory is to have a standard UV completion. We identify the mixing between the longitudinal mode and the transverse modes as the main obstacle to positivity, and clarify how the non-Abelian nature of non-linear (dRGT) massive gravity allows it to satisfy positivity where all known spin $\geq 3/2$ Abelian theories fail. Our results imply that a massive gravitino cannot appear alone in a controlled EFT---it must be accompanied by other particles, e.g.~as part of a supermultiplet. Together with the spin-$1$ and spin-$2$ cases, we suggest features which will persist in even higher spin massive theories.