Local Interactions of Higher-Spin Potentials That are Gauge Invariant in Linear Approximation

We study connected Wightman functions of $N$ conserved currents, each of which is formed from a scalar field and has even spin $l_i$. The UV divergence of this vertex function is regularized by the analytic continuation in the space dimension $D⟶D-ϵ$. We evaluate the residue of ${ϵ}^{-1}$ only, which is a local interaction Lagrangian density and gauge invariant in linear