(A)dS exchanges and partially-massless higher spins

We determine the current exchange amplitudes for free totally symmetric tensor fields ${\text{vf}}_{{\mu }_1\mathrm{.}\mathrm{.}\mathrm{.}{\mu }_s}$ of mass $M$ in a $d$-dimensional $dS$ space, extending the results previously obtained for $s=2$ by other authors. Our construction is based on an unconstrained formulation where both the higher-spin gauge fields and the corresponding gauge parameters $\Lambda_{\mu_1

… \mu_{s-1}}$ are not subject to Fronsdal’s trace constraints, but compensator fields ${\alpha }_{{\mu }_1\mathrm{.}\mathrm{.}\mathrm{.}{\mu }_{s-3}}$ are introduced for $s>2$. The free massive $dS$ equations can be fully determined by a radial dimensional reduction from a $(d+1)$-dimensional Minkowski space time, and lead for all spins to relatively handy closed-form expressions for the exchange amplitudes, where the external currents are conserved, both in $d$ and in $(d+1)$ dimensions, but are otherwise arbitrary. As for $s=2$, these amplitudes are rational functions of $(ML{)}^2$, where $L$ is the $dS$ radius. In general they are related to the hypergeometric functions ${}_3{F}_2(a,b,c;d,e;z)$, and their poles identify a subset of the “partially-massless” discrete states, selected by the condition that the gauge transformations of the corresponding fields contain some non-derivative terms. Corresponding results for $AdS$ spaces can be obtained from these by a formal analytic continuation, while the massless limit is smooth, with no van Dam-Veltman-Zakharov discontinuity.