Interaction of supersymmetric nonlinear sigma models with external higher spin superfields via higher spin supercurrents

We consider a four dimensional generalized Wess-Zumino model formulated in terms of an arbitrary K\"{a}hler potential $\mathcal{K}(\Phi,\bar{\Phi})$ and an arbitrary chiral superpotential $\mathcal{W}(\Phi)$. A general analysis is given to describe the possible interactions of this theory with external higher spin gauge superfields of the ($s+1,s+1/2$) supermultiplet via higher spin supercurrents. It is shown that such interactions do not exist beyond supergravity $(s\geq2)$ for any $\mathcal{K}$ and $\mathcal{W}$. However, we find three exceptions, the theory of a free massless chiral, the theory of a free massive chiral and the theory of a free chiral with linear superpotential. For the first two, the higher spin supercurrents are known and for the third one we provide the explicit expressions. We also discuss the lower spin supercurrents. As expected, a coupling to (non-minimal) supergravity ($s=1$) can always be found and we give the generating supercurrent and supertrace for arbitrary $\mathcal{K}$ and $\mathcal{W}$. On the other hand, coupling to the vector supermultiplet ($s=0$) is possible only if $\mathcal{K}=\mathcal{K}(\bar{\Phi}\Phi)$ and $\mathcal{W}=0$.