Differential equations from unitarity cuts: nonplanar hexa-box integrals

  • Samuel Abreu
  • Ben Page
  • Mao ZengEmail author
Open Access
Regular Article - Theoretical Physics


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.


Perturbative QCD Scattering Amplitudes 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Supplementary material

ESM1 symbols.m (M 1649 kb)
ESM2 masters.m (M 95 kb)
ESM3 diffEqMatrices.m (M 546 kb)


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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Physikalisches InstitutAlbert-Ludwigs-Universitat FreiburgFreiburgGermany
  2. 2.Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and AstronomyUniversity of CaliforniaLos AngelesU.S.A.

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