Pentagon Functions for Scattering of Five Massless Particles

TL;DR

The paper tackles the challenge of computing all two-loop massless five-point scattering integrals by constructing a minimal, canonical set of transcendental functions, the pentagon functions, via a unified differential equations approach. By solving DEs in the canonical form across all permutations and organizing results by transcendental weight, the authors achieve analytic, branch-cut-free representations for planar and non-planar master integrals and provide practical, public numerical implementations. They classify the function space up to weight four, present one-fold integral representations for higher-weight functions, analyze boundary behavior at $\Delta=0$, and validate the framework through multiple cross-checks, enabling immediate NNLO phenomenology for five-petal processes. The work significantly extends planar pentagon results to the non-planar sector, delivering robust tools (Mathematica and C++ libraries) for high-precision computations of five-particle amplitudes in collider physics.

Abstract

We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of pentagon functions suitable for immediate phenomenological applications.

Figures (4)

• Figure 1: All integral topologies with the maximal number of denominators from each integral family considered in this work. Momenta and propagator indices are shown for the standard permutation $\sigma_0$. All lines are massless, and all external momenta are incoming. The numbering of the denominators corresponds to \ref{['eq:denominators']}.
• Figure 2: Definition of the single-valued logarithms of the parity odd letters in the physical region $\Delta < 0$ (shaded). The discontinuity at $a_{k} = 0$ is added or subtracted depending on the direction of the branch-cut crossing.
• Figure 3: Section of the physical region $P_0$ by the plane $v_1 = 3$, $v_3 =v_4 = 1$. The physical region is inside the orange ellipse depicting $\Delta = 0$. The four gray lines, representing $a_{k} = 0$ with $k=1,3,4,5$ (see \ref{['a_def']}), split the surface $\Delta = 0$ in six arcs. The functions $g^{(1)}_{3,k}$, $k=1,3,4,5$, are constant on each arc.
• Figure 4: Histogram of minimal logarithmic relative error of pentagon functions (see \ref{['eq:mindigits']}) sampled on 90000 kinematical points of a generic five-particle physical phase space. The average evaluation time in double precision is obtained from running 64 parallel jobs on a server with Intel(R) Xeon(R) Silver 4216 CPU @ 2.10GHz.