Researchers explain why certain complicated integrals that appear during high‑order gravity calculations do not change the conservative part of the classical scattering result at fifth post‑Minkowskian order. In intermediate steps of the calculation one finds complete elliptic integrals and Calabi–Yau (CY) three‑fold integrals. The paper shows these integrals are absent from the singular pieces that produce the physical long‑distance, conservative effects, so they cancel out of the final conservative observables such as the scattering angle.

The team works in an amplitudes‑based framework. They build a classical integrand by expanding in the regime of small momentum transfer, then study the loop integrals that arise. To pick out the conservative contribution at even loop orders they focus on a specific kind of term: a logarithm of the momentum transfer, ln(−q2). In dimensional regularization — a common way to handle divergent integrals by shifting the number of dimensions — such a logarithm comes from poles in the regulator (1/ϵ) times a power of the momentum. That means the full conservative term is fixed by the divergent parts of the amplitude, not by the finite parts.

Different diagram shapes produce different classes of special functions when integrated. Some topologies give only generalized polylogarithms (GPLs), which are relatively well understood. Others have “maximal cuts” that probe richer geometries: one class yields elliptic behavior, a second leads to Heun functions, and a third to Calabi–Yau three‑fold integrals. The authors show that the integral classes responsible for the elliptic and CY behavior are not present in the ultraviolet (UV) singular structure that generates the 1/ϵ poles needed for the logarithm. In other words, the complicated functions appear in finite parts of integrals but not in the singular pieces that determine the conservative classical terms.