UV complete me: positivity bounds for particles with spin

Abstract

For a low energy effective theory to admit a standard local, unitary, analytic and Lorentz-invariant UV completion, its scattering amplitudes must satisfy certain inequalities. While these bounds are known in the forward limit for real polarizations, any extension beyond this for particles with nonzero spin is subtle due to their non-trivial crossing relations. Using the transversity formalism (i.e. spin projections orthogonal to the scattering plane), in which the crossing relations become diagonal, these inequalities can be derived for 2-to-2 scattering between any pair of massive particles, for a complete set of polarizations at and away from the forward scattering limit. This provides a set of powerful criteria which can be used to restrict the parameter space of any effective field theory, often considerably more so than its forward limit subset alone.

Positivity constraints on interacting spin-2 fields

“…Thus the more accurate statement is that if the leading operators are marginally ruled out, then it means that they must be suppressed in such away that higher derivative operators contribute equally or dominantly to the desired bounds. This nevertheless has a profound effect on the assumed structure of the effective field theory expansion, and it is quite possible that the application of more general positivity bounds, such as for example the non-forward limit bounds [47,48] exclude the EFTs entirely.…”

“…where λ 1 , λ 2 stand for the polarization states of the ingoing and outgoing particles which are assumed to be equal in elastic scattering. We refer the reader to [26,47] for derivation of (3.1) in our current notations.…”

“…Since the latter is less common in the literature we present it in more detail. This basis was first used in the modern context in [47] and was shown to be a convenient basis for positivity bounds away from the forward limit.…”

“…The polarization vectors in the transversity basis in the frame where the momentum of the corresponding particle is p µ = (E, 0, 0, p) are defined as follows [47]: As in the SVT basis in (3.2), the entire configuration of transversities can be specified by ten (real) numbers, α 1 , .., α 5 , β 1 , ...β 5 , as…”

UV constraints on massive spinning particles: lessons from the gravitino

“…However, the disadvantage of the helicity basis is that crossing symmetry is rather complicated at finite t (crucially, the u-channel is not always a positive image of the s-channel, which makes positivity bounds difficult to construct). This was resolved in [54] by instead using the transversity basis (i.e. scattering particular superpositions of helicity states) which enjoys a far simpler crossing relation.…”