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

Abstract

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 Λ 5 massive gravity theories and the special 'ghost-free' Λ 3 theories. Remarkably we find that at the level of 2-2 scattering, the positivity bounds applied to Λ 5 massive gravity theories, impose the special tunings which generate the Λ 3 structure. For Λ 3 massive gravity theories, the island of positivity derived in the forward limit appears relatively stable against further bounds.

Positivity constraints on interacting spin-2 fields

“…However, higher order terms do arise suppressed by the scale Λ 3 . The Λ 3 -EFT was discussed for instance in [25,31] and the higher order operators take the form…”

“…Whether or not any single or multiple interacting massive spin-2 fields could have a standard UV completion remains as yet unclear [16]. By assuming the UV completion to be Lorentz invariant, causal and unitary it is possible to derive particular bounds on the scattering amplitudes of the low energy EFTs, so called positivity bounds, which restrict the allowed parameter space of the EFT [17][18][19][20][21][22][23][24][25][26]. We stress that a failure to satisfy these bounds does not imply the EFT is inconsistent but only that it could not have a standard UV completion, but this does not preclude a non-standard UV completion [23].…”

“…In this paper we apply these bounds to effective field theories of one or more interacting pseudo-linear massive spin-2 fields. The particular case of a single pseudo-linear spin-2 field was considered already in [21] where it was shown that for the leading 'ghost-free' interactions some of the definite helicity positivity bounds have to be marginal, unlike its close cousin 'ghost-free massive gravity' [20,23,25,26]. Another interesting example where positivity bounds are marginal is in the case of massless Galileons [32].…”

“…Interestingly, massless Galileons arise as the decoupling limit of massive gravity [33], which would seem to suggest that massive gravity cannot have a standard UV completion. However, it was shown in [20] that there exists a compact region in the parameter space for which the scattering amplitudes of this theory are compatible with forward limit positivity bounds and this has been further considered in [23][24][25][26]. This is because taking a decoupling limit is distinct to considering the low-energy EFT, and while Galileons are massless in the decoupling limit of massive gravity (where the mass is sent to zero), those modes remain massive in the low-energy limit of massive gravity.…”

UV constraints on massive spinning particles: lessons from the gravitino

“…two incoming LH spin-3/2 particles scattering into two outgoing LH spin-3/2 particles), the only vertices from (30) which contribute are g 7 and g 8 (since g 1 , g 2 do not contain the necessary XXXX terms and g 3−6 contain traces of Ψ, which vanish on-shell since physical polarisations are always traceless, c.f. (55) and (56)).…”