Constraining tree-level gravitational scattering

We study the space of all kinematically allowed four graviton
S-matrices, having simple poles and polynomial in scattering momenta. To
classify pole exchanges, we enumerate all possible three point couplings
involving two gravitons and a massive spinning particle transforming in an
irreducible representation of the lorentz group. We demonstrate that the
space of analytic (i.e polynomial in momenta unlike pole exchanges)
S-matrices is the permutation invariant sector of a module over the ring of
polynomials of the Mandelstam invariants [image: s], [image: t] and [image:
u]. We construct these cubic couplings and modules for every value of the
spacetime dimension D,  and so explicitly count and parameterize the most
general four graviton S-matrix at any given derivative order. We also
explicitly list the cubic and quartic local Lagrangians that give rise to
these S-matrices. We then conjecture that the Regge growth of S-matrices in
all physically acceptable classical theories is bounded by [image: s^2]  at
fixed [image: t] (Classical Regge Growth conjecture). Using flat space limit
of AdS correlators and Chaos bound, we prove CRG in the context of a
certain class of interactions. We then use CRG to rule out modifications to
Einstein gravity- no polynomial addition to the Einstein S-matrix obeys
this bound for [image: Dleq 6]. For [image: Dgeq7] there is a single six
derivative polynomial Lagrangian consistent with our conjectured Regge
growth bound. Our conjecture thus implies that the Einstein four graviton
S-matrix does not admit any physically acceptable polynomial modifications
for [image: Dleq 6]. We also show that every finite sum of  pole exchange
contributions to four graviton scattering also violates our conjectured Regge
growth.