Eikonal scattering and asymptotic superluminality of massless higher spin fields

We consider scattering of massless higher-spin particles in the eikonal regime in four dimensions. By demanding the absence of asymptotic superluminality, corresponding to positivity of the eikonal phase, we place constraints on the possible cubic couplings which can appear in the theory. The cubic couplings come in two types: lower-derivative non-Abelian vertices and higher-derivative Abelian vertices made out of gauge-invariant curvature tensors. We find that the Abelian couplings between massless higher spins lead to an asymptotic time advance for certain choices of polarizations, indicating that these couplings should be absent unless new states come in at the scale suppressing the derivatives in these couplings. A subset of non-Abelian cubic couplings are consistent with eikonal positivity, but are ruled out by consistency of the four-particle amplitude away from the eikonal limit. The eikonal constraints are, therefore, complementary to the four-particle test, ruling out even trivial cubic curvature couplings in any theory with a finite number of massless higher spins and no new physics at the scale suppressing derivatives in these vertices.