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

Abstract

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 Λ, has a limit where: (i) Λ → 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 Λ = 0 but also the strongly coupled nature of the derivative expansion of the fully nonlinear higher-spin field equations with Λ = 0, wherein the standard minimal couplings mediated via the Lorentz connection are subleading at energy scales |Λ| < < 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.

Higher-spin fermionic gauge fields and their electromagnetic coupling

“…For cubic deformations S 1 = a, it is indeed impossible to construct an object with agh > 2 [9]. The result is however more general and holds in fact also for higher order deformations, as it follows from the results of [34][35][36][37][38][39].…”

“…Now, if one makes an antighost-number expansion of the local form a, it stops at agh = 2 [9,[34][35][36][37][38],…”

“…The latter include one non-abelian (2s − 2)-derivative vertex, a 2s-derivative abelian one that is gauge invariant up to a total derivative and exists in D ≥ 5, and a Born-Infeld type abelian one containing 2s + 2 derivatives. This could be seen by using either the light-cone method [10] or the cohomological methods [8,9].…”

“…For these fields, we employ the powerful machinery of BRST-BV cohomological methods [20,21] to construct systematically consistent interaction vertices, 2 with the underlying assumptions of locality, Poincaré invariance and conservation of parity, and without relying on other methods. The would-be off-shell 1 − s − s cubic vertices will complement their bosonic counterparts constructed in [9].…”

“…Indeed, N = 2 SUGRA [6,7] allows massless gravitini to have dipole and higher-multipole couplings, but forbids a non-zero U(1) charge in flat space. Gravitational and EM multipole interactions also show up, for example, as the 2 − s − s and 1 − s − s trilinear vertices constructed in [8,9] for bosonic fields. 1 These cubic vertices are but special cases of the general form s − s ′ − s ′′ , that involves massless fields of arbitrary spins.…”