Researchers explain why certain very complicated integrals that appear during fifth post-Minkowskian (5PM) calculations do not affect the conservative part of classical gravitational scattering. In the classical limit and at even loop orders, the conservative contribution to the scattering amplitude is tied to a specific logarithm of the momentum transfer, ln(−q^2). In dimensional regularization this logarithm comes from expanding terms that are singular in the regulator, so the classical conservative piece is fixed entirely by those singular (1/ϵ) contributions.

The authors work in an amplitudes-based framework. They first build a classical integrand by expanding in soft graviton momenta and separate the relevant momentum regions with the method of regions. Instead of carrying integrals all the way through standard integration-by-parts (IBP) reduction and solving large systems of master integrals, they focus on the ultraviolet (UV) singular parts that produce the needed 1/ϵ poles. By examining those singular regions before the usual IBP mixing, they show that the integral families responsible for complete elliptic and Calabi–Yau (CY) three-fold behavior are simply not present in the UV singular structures that generate the ln(−q^2).

Some four-loop diagram topologies do generate elliptic, Heun, or Calabi–Yau special functions when evaluated fully. Other topologies evaluate to generalized polylogarithms (GPLs). The key point the paper makes is that the UV poles that control the classical conservative ln(−q^2) term at 5PM come only from regions tied to the simpler GPL-type topologies. The more exotic elliptic and CY integral classes live in finite regions of the integrals and therefore do not contribute to the singular piece that determines the conservative physics.