Quantum Gravity Constraints from Unitarity and Analyticity

TL;DR

This work derives rigorous positivity bounds on curvature corrections to Einstein gravity from unitarity and analyticity of graviton scattering in $D\ge 4$, constraining the coefficients of quartic and quadratic curvature terms in the low-energy action. By analyzing the forward four-graviton amplitude and exploiting dispersion relations, crossing symmetry, and the optical theorem, the authors obtain dimension-specific positivity conditions that must hold in any perturbative UV completion. They provide complete quartic-curvature bounds for $D=4$, $D=5$, and $D\ge 6$, extend the analysis to supersymmetric operators, and verify these bounds in weakly-coupled string theories; they also show that a primordial Gauss-Bonnet term with coefficient $\lambda$ is inconsistent unless new degrees of freedom appear at the natural cutoff $\Lambda \sim |\lambda \kappa^2|^{-1/2}$. Collectively, the results constrain the space of viable quantum-gravity theories and illuminate how UV completions imprint on low-energy gravitational dynamics.

Abstract

We derive rigorous bounds on corrections to Einstein gravity using unitarity and analyticity of graviton scattering amplitudes. In $D\geq 4$ spacetime dimensions, these consistency conditions mandate positive coefficients for certain quartic curvature operators. We systematically enumerate all such positivity bounds in $D=4$ and $D=5$ before extending to $D\geq 6$. Afterwards, we derive positivity bounds for supersymmetric operators and verify that all of our constraints are satisfied by weakly-coupled string theories. Among quadratic curvature operators, we find that the Gauss-Bonnet term in $D\geq 5$ is inconsistent unless new degrees of freedom enter at the natural cutoff scale defined by the effective theory. Our bounds apply to perturbative ultraviolet completions of gravity.