General trilinear interaction for arbitrary even higher spin gauge fields

TL;DR

The paper solves the problem of general trilinear couplings for Fronsdal higher-spin fields in flat space using Noether's procedure, yielding a complete cubic vertex for spins $s_1\ge s_2\ge s_3$ with minimal derivative count $\Delta_{min}=s_1+s_2-s_3$ and extending to higher-order corrections via $\mathcal{L}_{I}^{(i,j)}$ for $i+j\le 3$. It provides a detailed classification of gauge-invariant cubic vertices by Weyl/Riemann content, and demonstrates the vertex is unique up to partial integration and field redefinitions for a fixed derivative number, with deDonder gauge simplifying to the leading column where traces decouple. The work also discusses gauge transformations and the possibility of a Lie algebra structure for nonlinear gauge transformations in even spins, deriving derivative-count constraints for quartic and higher-order interactions: quartic terms carry $2s-2$ derivatives and general $n$th-order terms carry $(s-2)(n-2)+2$ derivatives, consistent with a scale length $L$ and a dimensionless coupling. These results advance the understanding of consistent interactions among higher-spin fields in flat space and bear on holographic higher-spin theories, while outlining avenues for completing the higher-order HS gauge algebra.

Abstract

Using Noether's procedure we present a complete solution for the trilinear interactions of arbitrary spins $s_{1},s_{2}, s_{3}$ in a flat background, and discuss the possibility to enlarge this construction to higher order interactions in the gauge field. Some classification theorems of the cubic (self)interaction with different numbers of derivatives and depending on relations between the spins are presented. Finally the expansion of a general spin $s$ gauge transformation into powers of the field and the related closure of the gauge algebra in the general case are discussed.