On the uniqueness of minimal coupling in higher-spin gauge theory

We address the uniqueness of the minimal couplings between higher-spin fields and gravity. These couplings are cubic vertices built from gauge non-invariant connections that induce non-abelian deformations of the gauge algebra. We show that Fradkin-Vasiliev's cubic 2 - s - s vertex, which contains up to 2s - 2 derivatives dressed by a cosmological constant Lambda, has a limit where: (i) Lambda -> 0; (ii) the spin-2 Weyl tensor scales non-uniformly with s; and (iii) all lower-derivative couplings are scaled away. For s = 3 the limit yields the unique non-abelian spin 2-3-3 vertex found recently by two of the authors, thereby proving the uniqueness of the corresponding FV vertex. We extend the analysis to s = 4 and a class of spin 1 - s - s vertices. The non-universality of the flat limit high-lightens not only the problematic aspects of higher-spin interactions with Lambda = 0 but also the strongly coupled nature of the derivative expansion of the fully nonlinear higher-spin field equations with Lambda not equal 0, wherein the standard minimal couplings mediated via the Lorentz connection are subleading at energy scales root vertical bar Lambda vertical bar << E << M-p. Finally, combining our results with those obtained by Metsaev, we give the complete list of all the manifestly covariant cubic couplings of the form 1 - s - s and 2 - s - s, in Minkowski background.