Differential equations from unitarity cuts: nonplanar hexa-box integrals

TL;DR

This paper derives and solves differential equations for nonplanar hexa-box integrals relevant to 2-loop 5-point QCD amplitudes, using unitarity cuts and algebraic geometry to streamline calculations.

Contribution

It presents a complete set of pure integrals and a novel, efficient method to derive and solve differential equations for complex multi-loop integrals.

Findings

Derived epsilon-factorized differential equations for nonplanar hexa-box integrals.

Developed a fast IBP system solution in 8 hours on a single CPU core.

Reconstructed analytic differential equations from numerical phase-space points.

Abstract

We compute $\epsilon$-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.