The Two-Loop Four-Graviton Scattering Amplitudes

TL;DR

The authors compute analytic expressions for the two-loop four-graviton scattering amplitudes in Einstein gravity, including ultraviolet counterterms from ${\cal L}_{\text{GB}}$ and ${\cal L}_{\text{R}^3}$ to renormalize the theory. They employ a two-loop numerical unitarity framework that bypasses explicit integrand construction and reduces the amplitude to a minimal basis of master integrals. Analytic results are reconstructed from exact numerical evaluations using finite-field arithmetic and Thiele interpolation, yielding remainders ${\cal R}^{(2)}_{\vec{h}}(s,t)$ expressed as linear combinations of a basis of polylogarithms ${h_i\in B}$. A key finding is that four-graviton observables depend on fewer EFT couplings than naively expected: for helicities ${\{-,-,+,+\}}$ the remainder is independent of ${c}_{GB}(\mu)$ and ${c}_{R^3}(\mu)$, while for ${\{\pm,+,+,+\}}$ it depends on a single combination ${c(\mu)=c_{\text{R}^3}(\mu)-\tfrac{1}{2} c_{\text{GB}}(\mu)}$. In the $s$-channel Regge limit ($s \gg -t>0$) the ${\{-,-,+,+\}}$ remainder scales as $s^3$ with a structure involving ${\pi}$, logarithms $L=\log(-s/t)$ and zeta values, and is independent of the renormalization scale ${\mu}$, illustrating a decoupling pattern that links the amplitude to classical gravitational dynamics.

Abstract

We present the analytic form of the two-loop four-graviton scattering amplitudes in Einstein gravity. To remove ultraviolet divergences we include counterterms quadratic and cubic in the Riemann curvature tensor. The two-loop numerical unitarity approach is used to deal with the challenging momentum dependence of the interactions. We exploit the algebraic properties of the integrand of the amplitude in order to map it to a minimal basis of Feynman integrals. Analytic expressions are obtained from numerical evaluations of the amplitude. Finally, we show that four-graviton scattering observables depend on fewer couplings than naively expected.

Figures (1)

• Figure 1: Topologically inequivalent propagator structures for 2-to-2 scattering, including non-planar contributions.