Pentagon functions for massless planar scattering amplitudes
T. Gehrmann, J. M. Henn, N. A. Lo Presti
TL;DR
This paper tackles the challenge of computing massless planar two-loop five-point amplitudes by constructing a minimal, analytically tractable function space—the pentagon functions—via a canonical differential-equations framework. It classifies planar pentagon functions up to weight four, develops one-fold integral representations, and provides a public numerical code to evaluate both the pentagon functions and the associated 61 master integrals in the UT basis. The authors validate their analytic solutions against known results, ensure correct analytic properties across Euclidean and physical regions, and deliver a robust toolkit for all-planar two-loop five-point amplitudes, with clear pathways to non-planar and higher-loop extensions. The work thus advances both the theoretical understanding of multi-loop multi-leg integrals and the practical computation of two-loop five-particle scattering processes.
Abstract
Loop amplitudes for massless five particle scattering processes contain Feynman integrals depending on the external momentum invariants: pentagon functions. We perform a detailed study of the analyticity properties and cut structure of these functions up to two loops in the planar case, where we classify and identify the minimal set of basis functions. They are computed from the canonical form of their differential equations and expressed in terms of generalized polylogarithms, or alternatively as one-dimensional integrals. We present analytical expressions and numerical evaluation routines for these pentagon functions, in all kinematical configurations relevant to five-particle scattering processes.