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Classical Gravitational Scattering from the Ultraviolet and the Absence of Calabi-Yau Integrals in the Conservative Sector at $O(G^5)$

Zvi Bern, Avery Jackman, Gareth Mansfield, Michael S. Ruf

Abstract

We explain why Calabi-Yau and complete elliptic integrals do not contribute to conservative observables at fifth post-Minkowskian order, despite appearing in intermediate steps. At even loop orders, conservative contributions are tied to terms proportional to the logarithm of the momentum transfer, which in dimensional regularization arise from singular regions. We show that in the classical limit, the integral classes responsible for Calabi-Yau and complete elliptic behavior are absent from the ultraviolet singular structures that generate the required logarithm. This perspective also suggests alternative strategies for analyzing the classical limit of multiloop integrals in the conservative sector at even loop orders.

Classical Gravitational Scattering from the Ultraviolet and the Absence of Calabi-Yau Integrals in the Conservative Sector at $O(G^5)$

Abstract

We explain why Calabi-Yau and complete elliptic integrals do not contribute to conservative observables at fifth post-Minkowskian order, despite appearing in intermediate steps. At even loop orders, conservative contributions are tied to terms proportional to the logarithm of the momentum transfer, which in dimensional regularization arise from singular regions. We show that in the classical limit, the integral classes responsible for Calabi-Yau and complete elliptic behavior are absent from the ultraviolet singular structures that generate the required logarithm. This perspective also suggests alternative strategies for analyzing the classical limit of multiloop integrals in the conservative sector at even loop orders.
Paper Structure (2 sections, 29 equations, 4 figures)

This paper contains 2 sections, 29 equations, 4 figures.

Table of Contents

  1. Family of two-loop cuspy integrals
  2. Integrating out subloops and numerators

Figures (4)

  • Figure 1: Examples of parent diagrams with only cubic vertices relevant for classical scattering at $\mathcal{O}(G^5)$. The double lines represent scalar matter, and the single lines represent gravitons.
  • Figure 2: Representative $\mathcal{O}(G^5)$ classical-scattering integral topologies. Diagrams \ref{['subfig:Ell']}--\ref{['subfig:Heun']} generate functions beyond GPLs: \ref{['subfig:Ell']} yields elliptic, \ref{['subfig:CY']} CY, and \ref{['subfig:Heun']} Heun integrals. In contrast, \ref{['subfig:Polylog']} and \ref{['subfig:ZigZag']} evaluate to GPLs.
  • Figure 3: Master integrals relevant for the evaluation of Eq. \ref{['eq:UVIntegral']}.
  • Figure 4: Propagators and loop-momentum labels for the elliptic integral topology.