Differential equations from unitarity cuts: nonplanar hexa-box integrals
Samuel Abreu, Ben Page, Mao Zeng
TL;DR
This work develops and applies unitarity-compatible IBP reduction to derive $\epsilon$-factorized differential equations for the nonplanar 2-loop, 5-point hexabox topology. It constructs a pure basis of master integrals, enabling canonical form DEs and symbol-level solutions, and validates a second approach that reconstructs the analytic equations from a finite set of phase-space samples using a conjectured alphabet. The authors obtain 73 master integrals, solve the differential equations at symbol level, and demonstrate that the computation is feasible on modest hardware (about 8 hours on a single CPU core). They also provide explicit DE matrices, pure integrals, and symbol data as supplementary material, advancing analytic progress for nonplanar 5-point amplitudes beyond planar results.
Abstract
We compute $ε$-factorized differential equations for all dimensionally-regularized integrals of the nonplanar hexa-box topology, which contribute for instance to 2-loop 5-point QCD amplitudes. A full set of pure integrals is presented. For 5-point planar topologies, Gram determinants which vanish in $4$ dimensions are used to build compact expressions for pure integrals. Using unitarity cuts and computational algebraic geometry, we obtain a compact IBP system which can be solved in 8 hours on a single CPU core, overcoming a major bottleneck for deriving the differential equations. Alternatively, assuming prior knowledge of the alphabet of the nonplanar hexa-box, we reconstruct analytic differential equations from 30 numerical phase-space points, making the computation almost trivial with current techniques. We solve the differential equations to obtain the values of the master integrals at the symbol level. Full results for the differential equations and solutions are included as supplementary material.