Extremal black hole scattering at O(G^3): graviton dominance, eikonal exponentiation, and differential equations

TL;DR

This work computes the conservative classical dynamics of hyperbolic scattering for two extremal black holes in $\mathcal N=8$ supergravity at ${\mathcal O}(G^3)$ using both eikonal and EFT methods. The authors develop a novel velocity-differential-equation framework to evaluate loop integrals with exact velocity dependence, isolating the potential (conservative) region and enabling a complete, relativistic PM treatment. They demonstrate exponentiation of the eikonal phase up to two loops, extract the scattering angle, and show agreement with EFT-derived Hamiltonians, including a high-energy limit consistent with graviton-dominated Einstein gravity results. The findings highlight graviton dominance at this order and suggest universality of high-energy massive scattering, while providing a flexible computational toolkit for future extensions to more general gravity theories and higher PM orders.

Abstract

We use $\mathcal N=8$ supergravity as a toy model for understanding the dynamics of black hole binary systems via the scattering amplitudes approach. We compute the conservative part of the classical scattering angle of two extremal (half-BPS) black holes with minimal charge misalignment at $\mathcal O(G^3)$ using the eikonal approximation and effective field theory, finding agreement between both methods. We construct the massive loop integrands by Kaluza-Klein reduction of the known $D$-dimensional massless integrands. To carry out integration we formulate a novel method for calculating the post-Minkowskian expansion with exact velocity dependence, by solving velocity differential equations for the Feynman integrals subject to modified boundary conditions that isolate conservative contributions from the potential region. Motivated by a recent result for universality in massless scattering, we compare the scattering angle to the result found by Bern et. al. in Einstein gravity and find that they coincide in the high-energy limit, suggesting graviton dominance at this order.

Figures (17)

• Figure 1: Example of KK reduction with massless exchange. The diagram in (a) reduces to the pair of diagrams in (b) and (c). The thin and thick lines denote massless and massive momenta respectively.
• Figure 2: One-loop topologies.
• Figure 3: Massless two-loop topologies
• Figure 4: Two loop integrals that are of the double-box type.
• Figure 5: Two loop integrals that are not of the double-box type.