A numerical evaluation of the scalar hexagon integral in the physical region

TL;DR

This work introduces a numerically stable framework for evaluating scalar one-loop pentagon and hexagon integrals in the physical region by expressing them through $n$-dimensional triangle functions and $(n+2)$-dimensional box functions, which naturally separates IR-divergent and finite parts. The authors derive explicit reduction formulas for $N$-point functions (with $N\le 6$) and develop one- and two-dimensional integral representations for the atomic building blocks, followed by robust numerical integration strategies that combine deterministic and Monte Carlo methods. They validate the approach with pentagon and hexagon results across multiple kinematics, including threshold studies and symmetry checks, demonstrating accuracy around $10^{-4}$ and runtime ranging from seconds to hours. The work highlights the potential for fully numerical approaches to multi-particle one-loop processes, avoiding complex tensor reductions and large dilogarithm expressions, and provides a practical path toward scalable calculations in multi-scale, high-energy collider environments.

Abstract

We derive an analytic expression for the scalar one-loop pentagon and hexagon functions which is convenient for subsequent numerical integration. These functions are of relevance in the computation of next-to-leading order radiative corrections to multi-particle cross sections. The hexagon integral is represented in terms of n-dimensional triangle functions and (n+2)-dimensional box functions. If infrared poles are present this representation naturally splits into a finite and a pole part. For a fast numerical integration of the finite part we propose simple one- and two-dimensional integral representations. We set up an iterative numerical integration method to calculate these integrals directly in an efficient way. The method is illustrated by explicit results for pentagon and hexagon functions with some generic physical kinematics.

Figures (8)

• Figure 1: Analytic regions of the box and triangle integrands. The parabola defines the boundary $R=0$, the line the boundary $A+B+C=0$. Inside regions I,II and III the integrand has an imaginary part. The integrable square-root and logarithmic singularities are located at the boundaries of these regions as explained in the text. The line segments (a) and (b) stand for the integration regions of Figs. 2 and 3, respectively.
• Figure 2: The integrand of the function $S_{Tri}^{n=4}(6,4,1,1,1,1)$ is plotted for $t\in [0,1]$. The structure of the shown integrand is explained in the text.
• Figure 3: The integrand of the function $S_{Tri}^{n=4}(10,4,5/2,1,1,1)$ is plotted for $t\in [0.05,0.2]$. Integrable square-root and logarithmic singularities are visible as explained in the text.
• Figure 4: The diagrams containing the scalar integrals III and IV used as benchmarks in Table \ref{['table1']}.
• Figure 5: The graphs II and III defining the kinematics for the scalar integrals calculated in Table \ref{['table2']}.