On equivalence of gauge-invariant models for massive integer-spin fields

There are several approaches to formulate gauge-invariant models for massive integer-spin fields in $d$ dimensions including the following: (i) in terms of symmetric tensor fields ${\phi }_{{\mu }_1\dots {\mu }_k}$, with $k=s,s-1,\dots ,0$, restricted to be double traceless for $k\ge 4$; and (ii) in terms of a quartet of $traceful$ symmetric tensor fields ${\psi }_{{\mu }_1\dots {\mu }_k}$, of rank $k=s,s-1,s-2,s-3$. We demonstrate that these formulations in Minkowski space ${\mathbb{M}}^d$ are equivalent to the gauge-invariant theory for a massive integer-spin field proposed in 1989 by Pashnev. We also make use of the Klishevich-Zinoviev theory in ${\mathbb{M}}^d$ to derive a unique generalisation of the Singh-Hagen model for a massive integer-spin field in $d>4$ dimensions.