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

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

Problem Statement
This work addresses the computation of conservative scattering observables (impulse and scattering angle) for non-spinning black holes in the general relativity two-body problem at the fifth post-Minkowskian (5PM) order and second self-force (2SF) order. While 5PM results at first self-force (1SF) order and lower PM orders have been established, the sub-sub-leading mass ratio contributions (2SF) at 5PM have remained out of reach. This order is crucial for high-accuracy models required by next-generation gravitational-wave observatories. The specific challenge lies in the complexity of the 2SF sector, which involves non-planar Feynman diagrams and requires the resolution of intricate cancellations between different spacetime regions (potential, tail, and memory) to yield finite, physical results.

Methodology
The authors employ the Worldline Quantum Field Theory (WQFT) formalism, modeling the black holes as point particles coupled to gravity. The calculation proceeds through the following technical stages:

1. Integrand Generation: Using recursive diagrammatic techniques, 651 Feynman diagrams are generated at 5PM order. These are organized into 14 top-level sectors based on Integration-by-Parts (IBP) identities, categorized into four integral families: Planar (P), Extended Planar (PX1, PX2), and Non-Planar (NP). The NP family captures non-linear memory effects.
2. IBP Reduction: The reduction of these integrals to master integrals (MIs) is performed using KIRA 3.0. This step is the computational bottleneck, consuming approximately $3 \times 10^6$ core hours. Key innovations include the automatic generation of symmetry relations for planar families and the manual derivation of symmetries for the non-planar family. A "pre-canonical" basis is chosen to minimize polynomial degrees, significantly accelerating the reduction.
3. Differential Equations (DEs): The 731 master integrals (321 in P, 144 in PX1, 46 in PX2, 220 in NP) are computed using differential equations in the kinematic variable $x$ (related to the Lorentz factor $\gamma$). The system is transformed into a canonical form $\partial_x J = \epsilon A(x) J$.
   • The function space involves multiple polylogarithms and iterated integrals.
   • Crucially, the geometry of the integrals reveals the presence of Calabi-Yau (CY) varieties, specifically a K3 surface and a three-dimensional CY variety.
   • After reduction, only the K3 sector contributes to the final physical result; the CY3 sector drops out.
4. Boundary Integrals and Prescription: The solution is constructed via expansion by regions (potential, tail, and memory). A critical issue arises: the potential region contains a spurious velocity divergence at $v/c = \sqrt{8/3}$ (corresponding to $\gamma=3$) and dimensional regularization poles. Standard Feynman propagator prescriptions fail to cancel the $\gamma=3$ divergence while maintaining finiteness.
   • The authors propose a "conservative propagator prescription" (termed the "$\gamma$-3" prescription). This involves using retarded propagators pointing toward the midpoint of the interaction for memory integrals, effectively averaging over causality routings. This prescription ensures the cancellation of both the $\epsilon$-poles and the spurious $\gamma=3$ divergence.

Key Contributions and Results

• First 5PM-2SF Conservative Result: The paper provides the first complete analytic description of the conservative 5PM-2SF scattering angle and impulse.
• Function Space: The result is expressed in terms of 36 linear combinations of iterated integrals. The function space includes the period of a K3 surface ($\varpi_{K3}$), its derivative, and iterated integrals with kernels involving square roots and the K3 period.
• Resolution of Divergences: The work identifies and resolves a novel spurious divergence at $\gamma=3$ that appears in the potential region. It demonstrates that this divergence must be canceled by contributions from the memory region, a cancellation that standard Feynman prescriptions fail to achieve.
• Analytic Expressions: The scattering angle $\theta^{(5,2)}_{\text{cons}}$ is presented as a sum of polynomial coefficients in $\gamma$ multiplied by the basis functions. The coefficients are provided explicitly in the supplementary material.
• Consistency Checks:
  • The result agrees with the Post-Newtonian (PN) expansion up to 4PN order in the low-velocity limit.
  • At 5PN order, the rational tail contribution matches previous literature, and the $\pi^2$ contribution confirms the conjecture that 5PN $\pi^2$ terms are purely potential.
  • The potential-only contribution matches recent low-velocity expansions found in other works.

Significance
The paper claims to close a significant gap in the perturbative description of black hole scattering, supplying crucial analytic input for high-accuracy waveform models. It highlights the necessity of re-evaluating the definition of the "conservative sector" in scattering problems, as standard Feynman propagator prescriptions are insufficient at this order to ensure physical finiteness. The emergence of K3 periods in the conservative sector marks a step up in complexity from previous 1SF calculations. The authors note that while their proposed prescription yields a physically sensible and finite answer, the physical motivation for this specific prescription remains opaque and requires further elucidation, potentially relying on operator definitions of the S-matrix. The work sets the stage for future computations of the fully dissipative 5PM-2SF sector, where retarded propagators are unambiguously defined.