Classifying and constraining local four photon and four graviton S-matrices

TL;DR

This paper systematically classifies all possible four photon and four graviton S-matrices polynomial in momenta, constrains their form using permutation invariance, and explores Regge growth bounds to identify physically acceptable theories across dimensions.

Contribution

It explicitly constructs modules over polynomial rings for all spacetime dimensions, enumerates local Lagrangians, and proposes a Regge growth bound that restricts polynomial modifications of Einstein gravity.

Findings

Classified all kinematically allowed four photon and four graviton S-matrices.

Identified a subset satisfying Regge growth bounds in photon case.

Found no polynomial modifications of Einstein gravity obey the bounds for D ≤ 6.

Abstract

We study the space of all kinematically allowed four photon and four graviton S-matrices, polynomial in scattering momenta. We demonstrate that this space is the permutation invariant sector of a module over the ring of polynomials of the Mandelstam invariants $s$, $t$ and $u$. We construct these modules for every value of the spacetime dimension $D$, and so explicitly count and parameterize the most general four photon and four graviton S-matrix at any given derivative order. We also explicitly list the 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 $t$. A four parameter subset of the polynomial photon S-matrices constructed above satisfies this Regge criterion. For gravitons, on the other hand, 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 $D\leq 6$. A preliminary analysis also suggests that every finite sum of pole exchange contributions to four graviton scattering also such violates our conjectured Regge growth bound, at least when $D\leq 6$, even when the exchanged particles have low spin.