The dimensional reduction of the spin-3 and spin-4 partially broken Fronsdal models

Here we perform a Kaluza-Klein dimensional reduction from $D+1$ to $D$ dimensions of a partially broken Fronsdal model for spin-3 and spin-4. The reduced models describe massive spin-3 and spin-4 particles without extra propagating modes. They differ from Singh-Hagen models by quadratic terms which vanish on shell, similarly to their higher dimensional massless parents. A covariant description of the reduced models in terms of gauge invariant field combinations naturally leads to higher derivative models, contrary to lower spin cases ($s\le 2$). For both $s=3$ and $s=4$ we have a rather peculiar massless limit. Although their particle content corresponds to the $\pm j$ helicities with $j=0,1,\cdots s$, as expected, they appear intertwined in a non trivial way preventing diagonalization through any local field redefinition.