Higher-Spin Gauge Fields Interacting with Scalars: The Lagrangian Cubic Vertex

TL;DR

This work develops a BRST-based framework to construct the cubic Lagrangian vertex for higher-spin (HS) gauge-field triplets interacting with two massive scalars, in both flat and AdS spaces. A key result in flat space is a clean pattern: the spin-s triplet propagates a tower of irreducible modes (s, s-2, ..., 0/1) each coupling to its own conserved current built from the scalars, with the currents organized into W_p and J_{s-2p} structures. In AdS, curvature induces 1/L^2 deformations that modify both the currents and the couplings, and the authors provide explicit spin-2 and spin-3 examples along with an irreducible-field construction and an alternative derivation via gauging higher-derivative scalar symmetries. The results have direct implications for holography, including infinite sets of Ward identities for boundary scalar operators dual to HS theories, and set the stage for extending the cubic vertex to general spins in AdS and for exploring deeper HS symmetry structures.

Abstract

We apply a recently presented BRST procedure to construct the Largangian cubic vertex of higher-spin gauge field triplets interacting with massive free scalars. In flat space, the spin-s triplet propagates the series of irreducible spin-s, s-2,..,0/1 modes which couple independently to corresponding conserved currents constructed from the scalars. The simple covariantization of the flat space result is not enough in AdS, as new interaction vertices appear. We present in detail the cases of spin-2 and spin-3 triplets coupled to scalars. Restricting to a single irreducible spin-s mode we uncover previously obtained results. We also present an alternative derivation of the lower spin results based on the idea that higher-spin gauge fields arise from the gauging of higher derivative symmetries of free matter Lagrangians. Our results can be readily applied to holographic studies of higher-spin gauge theories.