Positivity bounds for massive spin-1 and spin-2 fields

We apply the recently developed positivity bounds for particles with spin, applied away from the forward limit, to the low energy effective theories of massive spin-1 and spin-2 theories. For spin-1 theories, we consider the generic Proca EFT which arises at low energies from a heavy Higgs mechanism, and the special case of a charged Galileon for which the EFT is reorganized by the Galileon symmetry. For spin-2, we consider generic (5) massive gravity theories and the special ghost-free' (3) theories. Remarkably we find that at the level of 2-2 scattering, the positivity bounds applied to (5) massive gravity theories, impose the special tunings which generate the (3) structure. For (3) massive gravity theories, the island of positivity derived in the forward limit appears relatively stable against further bounds.