Gravitational and higher-derivative interactions of a massive spin $5/2$ field in (A)dS space

Using an on shell gauge invariant formulation of relativistic dynamics I study the interaction vertices for a massive spin $5/2$ Dirac field propagating in (anti–)de Sitter [(A)dS] space of dimension greater than or equal to four. The gravitational interaction vertex for the massive spin $5/2$ field and all cubic vertices for the massive spin $5/2$ field and massless spin 2 field with two and three derivatives are obtained. In dimension greater than four, I demonstrate that the gravitational vertex of the massive spin $5/2$ field involves, in addition to the standard minimal gravitational vertex, contributions with two and three derivatives. I find that for the massive spin $5/2$ and massless spin 2 fields one can build two higher-derivative vertices with two and three derivatives. Limits of massless and partial massless spin $5/2$ fields in (A)dS space and limits of massive and massless spin $5/2$ fields in flat space are discussed.