Pentagon Functions for One-Mass Planar Scattering Amplitudes

TL;DR

This work constructs a complete basis of one-mass pentagon functions to express all planar two-loop, five-particle scattering amplitudes with a single massive external leg. By solving canonical differential equations in epsilon form and representing master integrals via Chen's iterated integrals, the authors obtain weight-four functions that are algebraically independent and permutation-closed, with explicit weight-1 to weight-4 representations and one-fold/two-fold integral forms for higher weights. A public C++ library provides fast, stable numerical evaluation across the full physical phase space, enabling direct phenomenological use and cross-checks against existing master-integral bases. The methodology is designed to be robust to square-root structures and spurious singularities, and the authors outline clear paths to extensions to non-planar cases and other decay regions.

Abstract

We present analytic results for all planar two-loop Feynman integrals contributing to five-particle scattering amplitudes with one external massive leg. We express the integrals in terms of a basis of algebraically-independent transcendental functions, which we call one-mass pentagon functions. We construct them by using the properties of iterated integrals with logarithmic kernels. The pentagon functions are manifestly free of unphysical branch cuts, do not require analytic continuation, and can be readily evaluated over the whole physical phase space of the massive particle production channel. We develop an efficient algorithm for their numerical evaluation and present a public implementation suitable for direct phenomenological applications.

Figures (3)

• Figure 1: Representative diagrams for the planar five-particle integral families with a single external massive leg at one and two loops. The arrows denote momentum flow. The thick line represents the massive external leg.
• Figure 2: One-loop squared integral family (${\rm 1L}^2$) appearing in the computation of two-loop five-particle amplitudes with a single external massive leg.
• Figure 3: Distribution of logarithmic relative error of the one-mass pentagon functions (see \ref{['eq:mindigits']}) sampled on $10^5$ kinematical points from a typical physical phase space (solid orange line). The cumulative distribution is displayed by the dashed line. The average evaluation time of all one-mass pentagon functions in double precision on a single thread is estimated in a run where kinematical points are evaluated in parallel on a server with Intel(R) Xeon(R) Silver 4216 CPU @ 2.10GHz.