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Two-loop non-planar hexa-box integrals with one massive leg

Adam Kardos, Costas G. Papadopoulos, Alexander V. Smirnov, Nikolaos Syrrakos, Christopher Wever

TL;DR

This work advances the analytic and semi-analytic calculation of two-loop, one-mass hexabox master integrals by applying the Simplified Differential Equations approach to the non-planar sector. It establishes a canonical differential-equation framework for the $N_1$ family with a reduced alphabet and GPL representations up to weight 4, and provides weight-2 GPLs plus one-dimensional integral representations for the $N_2$ and $N_3$ families, aided by a new boundary-term computation strategy and a region-based alternative for challenging sectors. The results enable efficient, cross-checked representations suitable for NNLO QCD predictions of $W$, $Z$, and Higgs production with two jets, and they lay groundwork for completing the remaining five-point one-mass master integrals with robust numerical evaluation. The combination of analytic GPL structures and practical one-dimensional integrals offers a scalable path toward precise phenomenology at NNLO in the five-point, one-mass sector.

Abstract

Based on the Simplified Differential Equations approach, we present results for the two-loop non-planar hexa-box families of master integrals. We introduce a new approach to obtain the boundary terms and establish a one-dimensional integral representation of the master integrals in terms of Generalised Polylogarithms, when the alphabet contains non-factorisable square roots. The results are relevant to the study of NNLO QCD corrections for $W,Z$ and Higgs-boson production in association with two hadronic jets.

Two-loop non-planar hexa-box integrals with one massive leg

TL;DR

This work advances the analytic and semi-analytic calculation of two-loop, one-mass hexabox master integrals by applying the Simplified Differential Equations approach to the non-planar sector. It establishes a canonical differential-equation framework for the family with a reduced alphabet and GPL representations up to weight 4, and provides weight-2 GPLs plus one-dimensional integral representations for the and families, aided by a new boundary-term computation strategy and a region-based alternative for challenging sectors. The results enable efficient, cross-checked representations suitable for NNLO QCD predictions of , , and Higgs production with two jets, and they lay groundwork for completing the remaining five-point one-mass master integrals with robust numerical evaluation. The combination of analytic GPL structures and practical one-dimensional integrals offers a scalable path toward precise phenomenology at NNLO in the five-point, one-mass sector.

Abstract

Based on the Simplified Differential Equations approach, we present results for the two-loop non-planar hexa-box families of master integrals. We introduce a new approach to obtain the boundary terms and establish a one-dimensional integral representation of the master integrals in terms of Generalised Polylogarithms, when the alphabet contains non-factorisable square roots. The results are relevant to the study of NNLO QCD corrections for and Higgs-boson production in association with two hadronic jets.
Paper Structure (9 sections, 38 equations, 1 figure)

This paper contains 9 sections, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hexabox integral families
  3. Pure bases and simplified canonical differential equations
  4. Boundary terms
  5. Integral representation
  6. Weight 3:
  7. Weight 4:
  8. Implementation:
  9. Conclusions

Figures (1)

  • Figure 1: The five non-planar families with one external massive leg. The first row corresponds to the so-called hexabox topologies, whereas the diagrams of the second row are known as double-pentagons. We label them as follows: $N_1$ (top left), $N_2$ (top middle), $N_3$ (top right), $N_4$ (bottom left), $N_5$ (bottom right). All diagrams have been drawn using JaxodrawBinosi:2008ig.