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Two-Loop Hexa-Box Integrals for Non-Planar Five-Point One-Mass Processes

Samuel Abreu, Harald Ita, Ben Page, Wladimir Tschernow

TL;DR

This work advances two-loop QCD calculations for five-point, one-mass processes by constructing pure master integral bases for three non-planar hexa-box topologies and deriving canonical differential equations. It analyzes the symbol alphabet, including roots Δ_3, Δ_5, and Σ_5, and reveals that extended Steinmann relations do not universally hold in these topologies. The authors solve the differential equations numerically via generalized power series, providing high-precision boundary conditions in Euclidean and physical regions to enable robust amplitude evaluations. The results yield a compact, permutation-closed alphabet of 204 letters, and establish a practical framework for completing the full set of two-loop master integrals for massive-boson production with two jets at hadron colliders.

Abstract

We present the calculation of the three distinct non-planar hexa-box topologies for five-point one-mass processes. These three topologies are required to obtain the two-loop virtual QCD corrections for two-jet-associated W, Z or Higgs-boson production. Each topology is solved by obtaining a pure basis of master integrals and efficiently constructing the associated differential equation with numerical sampling and unitarity-cut techniques. We present compact expressions for the alphabet of these non-planar integrals, and discuss some properties of their symbol. Notably, we observe that the extended Steinmann relations are in general not satisfied. Finally, we solve the differential equations in terms of generalized power series and provide high-precision values in different regions of phase space which can be used as boundary conditions for subsequent evaluations.

Two-Loop Hexa-Box Integrals for Non-Planar Five-Point One-Mass Processes

TL;DR

This work advances two-loop QCD calculations for five-point, one-mass processes by constructing pure master integral bases for three non-planar hexa-box topologies and deriving canonical differential equations. It analyzes the symbol alphabet, including roots Δ_3, Δ_5, and Σ_5, and reveals that extended Steinmann relations do not universally hold in these topologies. The authors solve the differential equations numerically via generalized power series, providing high-precision boundary conditions in Euclidean and physical regions to enable robust amplitude evaluations. The results yield a compact, permutation-closed alphabet of 204 letters, and establish a practical framework for completing the full set of two-loop master integrals for massive-boson production with two jets at hadron colliders.

Abstract

We present the calculation of the three distinct non-planar hexa-box topologies for five-point one-mass processes. These three topologies are required to obtain the two-loop virtual QCD corrections for two-jet-associated W, Z or Higgs-boson production. Each topology is solved by obtaining a pure basis of master integrals and efficiently constructing the associated differential equation with numerical sampling and unitarity-cut techniques. We present compact expressions for the alphabet of these non-planar integrals, and discuss some properties of their symbol. Notably, we observe that the extended Steinmann relations are in general not satisfied. Finally, we solve the differential equations in terms of generalized power series and provide high-precision values in different regions of phase space which can be used as boundary conditions for subsequent evaluations.

Paper Structure

This paper contains 16 sections, 68 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Scattering Kinematics and Notation
  3. Hexa-box Topologies
  4. Canonical Differential Equation and Pure Basis
  5. Canonical Differential Equation
  6. Pure Basis
  7. Analytic Differential Equations
  8. The Symbol Alphabet
  9. Analytic Differential Equations from Numerical Samples
  10. Symbols of Non-Planar Hexa-Box Integrals
  11. Numerical Solution of Differential Equations
  12. Summary of the Approach
  13. Initial Value
  14. Numerical Evaluations in Physical Regions
  15. Validation
  16. ...and 1 more sections

Figures (13)

  • Figure 1: Two-loop five-point one-mass non-planar hexa-box topologies. The thick external line with label 1 denotes the massive external leg.
  • Figure 2: Propagator structures of two-loop five-point non-planar master integrals in the hexa-box topologies.
  • Figure 3: Penta-box with 6 master integrals.
  • Figure 4: Double-box with 4 master integrals.
  • Figure :
  • ...and 8 more figures