Conservative Black Hole Scattering at Fifth Post-Minkowskian and Second Self-Force Order

Abstract

Using the worldline quantum field theory formalism, we compute the conservative scattering angle and impulse for classical black hole scattering at fifth post-Minkowskian (5PM) order by providing the second self-force (2SF) contributions. This four-loop calculation involves non-planar Feynman integrals and requires advanced integration-by-parts reduction, novel differential-equation strategies, and efficient boundary-integral algorithms to solve a system of hundreds of master integrals in four integral families on high-performance computing systems. The resulting function space includes multiple polylogarithms as well as iterated integrals with a K3 period, which generate a spurious velocity divergence at $v/c=\sqrt{8}/3$. This divergence is present in the potential region and must be canceled by contributions from the radiative memory region, while its dimensional-regularisation pole should cancel against the radiative tail region. We find that the standard use of Feynman propagators to access the conservative sector fails to ensure this cancellation. We propose a conservative propagator prescription which realises both cancellations leading to a physically sensible answer. All available low-velocity checks of our result against the post-Newtonian literature are satisfied.

Figures (5)

• Figure 1: The 14 top-level sectors of the four-loop 2SF integral families: Planar P (a-g), extended planar PX1 (h-i) and PX2 (j), and the non-planar NP (k-m). Solid lines denote worldline propagators, $(v_{i}\cdot \ell+\mathrm{i} 0^{+})^{-1}$, wavy lines graviton propagators, $(\ell^{2}+\mathrm{i} 0^{+})^{-1}$, and the dotted lines may be interpreted as a cut worldline propagator, $\delta\!\!\!{}^-\!(\ell\cdot v_{i})$. In that sense, the top-level sectors have 13 propagators, and red graviton propagators may go on-shell. Finally, $q^{\mu}$ is the total momentum transfer.
• Figure 2: Non-zero entries of the $321\times 321$ DE matrix $A(x)$ of the planar family (P). The solid blocks on the diagonals determine the function spaces of CY3 (blue), K3 (orange), and polylogarithmic (green) type. The off-diagonal entries are in corresponding lighter colours.
• Figure 3: Active gravitons that become radiative (red) determine the three regions for a planar topsector and yield the required boundary integrals. In the tail and memory region, the $\mathrm{i} 0^{+}$ prescription is crucial.
• Figure 4: The 5PM-2SF contribution to the scattering angle, $\theta^{(5,2)}_{\text{cons}}(\gamma)$: Potential and memory contributions both diverge for $\gamma \rightarrow 3$. These divergences cancel for the full result if one uses Eq. \ref{['eq:I12']} irrespective of the value of $c_{M}$, which is set to 1 for this plot.
• Figure 5: The $\gamma$-$3$ prescription is given as the average of the two shown causality routings. In the first diagram retarded propagators meet in the symmetric point of the three-graviton interaction, and in the second, they are taken as advanced propagators.