Extremal black hole scattering at $$ \mathcal{O} $$(G3): graviton dominance, eikonal exponentiation, and differential equations

Abstract

We use $$ \mathcal{N} $$ 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 $$ \mathcal{O} $$ O (G3) 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.

Post-Minkowskian effective field theory for conservative binary dynamics

“…[109], can be then translated to our framework when pursuing higher PM orders. (Similar integrals appear in the soft expansion discussed in [90].) As we mentioned, notice that the term proportional to (4.27) does not enter in the scattering angle at 2PM.…”

“…Therefore, in order to properly isolate the relevant (conservative) region of integration, we need a prescription to account for each one of them. We will adapt to our framework the procedure first introduced in [3], and later elaborated further in [63,64,90] to extend to higher PM orders.…”

“…As it was discussed in [3], in this case the procedure boils down to evaluating the k 0 energy integral by an (oriented) average over poles in the upper/lower half complex plane, that is lines appear in the EFT computation, a given prescription can in principle be applied to any type of integral. Therefore, by adapting the rules from [64,90] to our case we can evaluate all the energy integrals in the EFT approach at any PM order. For instance, by averaging over graviton permutations [126].…”

“…some hybrid approach between the two (plausibly in combination with the classical double copy [111][112][113][114][115][116][117][118][119][120][121], on-shell methods [103,104], soft limits [90], and simplifications of the Einstein-Hilbert action [110]). In either case, the B2B dictionary [1,2] provides a direct (analytic) connection between scattering data and bound states, neatly demonstrating the need to push the computations to (even) higher orders, to fully incorporate the power of the PM expansion in the dynamics of binary systems.…”

“…At higher orders, however, the relationship is a bit more subtle, among other things because of the mismatch between the impact parameter entering in (A.8) and the one in the stationary phase approximation in the eikonal approach, e.g. [90]. We can, on the other hand, attempt to connect directly the impulse rather than the scattering angle.…”