Off-shell construction of some trilinear higher spin gauge field interactions

TL;DR

The paper develops off-shell, Lagrangian trilinear couplings among higher spin gauge fields in flat space for configurations with two equal spins $s$ and a higher even spin $s'\ge 2s$. Using Noether's procedure together with corrected gauge transformations, it derives explicit vertices: (i) 1-1-2 and 1-1-4 for spin-1 couplings; (ii) 2-2-4 and 2-2-6 expressed through deWitt-Freedman curvatures, showing automatic inclusion of lower-spin sectors via curvature contractions; and (iii) a general s-s-2s vertex formulated in a generating-function formalism with the curvature-based current $\\Psi^{(2s)}_{(\\Gamma)}$. Depending on the derivative structure, the framework yields either Bell-Robinson-type currents or a ladder of lower even-spin couplings, linking to flat-space limits of Fradkin–Vasiliev theory and AdS/CFT considerations. The results provide a concrete, curvature-based method to construct and analyze HS trilinear interactions with clear gauge and field-redefinition consistency, strengthening the bridge between HS physics and holography.

Abstract

Several trilinear interactions of higher spin fields involving two equal ($s=s_{1}=s_{2}$) and one higher even ($s_{3}\geq 2s$) spin are presented. Interactions are constructed on the Lagrangian level using Noether's procedure together with the corresponding next to free level fields of the gauge transformations. In certain cases when the number of derivatives in the transformation is $2s-1$ the interactions lead to the currents constructed from the generalization of the gravitational Bell-Robinson tensors. In other cases when the number of derivatives in the transformation is more than $2s-1$ we obtain the finite tower of interactions with smaller even spins less than $s_{3}$ in full agreement with our previous results for the interaction of the higher even spins field with a conformal scalar [1,2].