#### Alladi Ramakrishnan Hall

#### Constraining tree-level gravitational scattering

#### Subham Dutta Chowdhury

##### University of Chicago

*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 s, t and 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 $s^2$ at fixed $t$ (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 $D \leq 6$. For $D\geq 7$ 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 $D \leq 6$. We also show that every finite sum of pole exchange contributions to four graviton scattering also violates our conjectured Regge growth.*

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