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

There are several approaches to formulate gauge-invariant models for massive integer-spin field in d dimensions including the following: (i) in terms of symmetric tensor fields φμ₁ μₖ, with k = s, s-1, , 0, restricted to be double traceless for k≥ 4; and (ii) in terms of a quartet of $traceful$ symmetric tensor fields ψμ₁ μₖ, of rank $k=s,s-1,s-2, s-3$. We demonstrate that these formulations in Minkowski space Mᵈ 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 Mᵈ to derive a generalisation of the Singh-Hagen model for a massive integer-spin field in $d>4 $ dimensions.