Emergence of Calabi-Yau manifolds in high-precision black hole scattering

TL;DR

The paper advances a high-precision, classical gravitational two-body scattering analysis to 5th order in Newton's constant ($G^5$) within a worldline quantum-field-theory framework, extracting the impulse, scattering angle, radiated energy, and recoil. A key finding is the emergence of Calabi–Yau three-fold (CY3) period structures in the radiative sector, signaling a novel geometric layer in high-loop classical gravity observables. The authors develop and deploy a comprehensive toolbox—IBP reduction, canonical differential equations, boundary matching via velocity-region analysis, Schwinger parametrization, and explicit CY3/K3 function bases—to express observables in terms of iterated integrals over CY3 and K3 periods. Validation is performed against numerical relativity in the perturbative regime, and the results are poised to inform next-generation gravitational-wave templates and high-precision EFT approaches in gravity and beyond.

Abstract

Using the worldline quantum field theory formalism, we compute the radiation-reacted impulse, scattering angle, radiated energy and recoil of a classical black hole (or neutron star) scattering event at fifth post-Minkowskian and sub-leading self-force orders (5PM-1SF). This state-of-the-art four-loop computation employs advanced integration-by-parts and differential equation technology, and is considerably more challenging than the conservative 5PM-1SF counterpart. As compared with the conservative 5PM-1SF, in the radiation sector Calabi-Yau three-fold periods appear and contribute to the radiated energy and recoil observables. We give an extensive exposition of the canonicalization of the differential equations and provide details on boundary integrations, Feynman rules, and integration-by-parts strategies. Comparisons to numerical relativity are also performed.

Figures (8)

• Figure 1: Gravitational two-body scattering event: Two black holes (or neutron stars) with masses $m_{i}$ and incoming velocities $v_{i}$, impact parameter $b$ and resulting relative scattering angle $\theta$, radiated gravitational wave energy $E_{\text{rad}}$ and recoil.
• Figure 2: Graphical representation of the Calabi-Yau (CY) $n$-folds emerging in the black hole scattering: The elliptic curve (topologically a torus), and two-dimensional projections of the K3 surface and CY3 reflecting their symmetries. Red and blue lines are (projections of) the real $n$-dimensional cycles $\Gamma_n$. The corresponding periods over the $n$-form $\Omega_n(x)$, i.e. $\int_{\Gamma_n} \Omega_n(x)$, depend on the so-called modulus $x$ (related to the relative velocity of the black holes $v_{1}\cdot v_{2}/c^{2}=(x+x^{-1})/2)$ parametrizing the shape of CYs and yield master integrals in our problem.
• Figure 3: Non-zero entries of the odd parity $232\times 232$ differential equation matrix $\mathbf{\hat{M}(x, D)}$: The blocks on the diagonals determine the function spaces of the multiple sub-sectors. The not-magnified diagonal sectors give rise to multiple polylogarithms.
• Figure 3: The $M$, $M_1$, and $M_2$ families, with zero, one, and two jumps, respectively.
• Figure 4: The scattering angle $\mathbf{\theta}$: Plotted as a function of the impact parameter in units of the Schwarzschild radius, $bc^{2}/GM$, up to order $G^{5}$ for an equal-mass scenario with initial relative velocity $v=0.5125c$. The black dots are existing numerical relativity (NR) simulations Rettegno:2023ghr. The $G^5$ curve follows from (\ref{['theta5']}) (excluding the unknown $\nu^2 \theta^{(5,2)}$ contribution). The dotted line is the exact in $G$ ($\nu=0$) probe limit result for geodesic motion in a Schwarzschild background. The inset plot depicts the relative differences to the numerical relativity data. Larger values of $bc^{2}/GM$ correspond to the perturbative regime. We find agreement with NR within the error for $bc^{2}/GM>12.5$. The monotonically falling corrections to the consecutive $G^{n}$ orders yield an intrinsic error estimate of our $G^{5}$ results: they are more precise than the NR data for $bc^{2}/GM>14$.