Three-mass triangle integrals and single-valued polylogarithms

TL;DR

The authors develop a principled, function-space framework for evaluating two-loop three-mass triangle integrals without internal masses, showing that region I results can be written in terms of a finite basis of single-valued polylogarithms in one complex variable. They construct this basis, including new indecomposable elements like $\mathcal{Q}_3(z)$, and prove that the master integrals up to weight four admit compact analytic expressions. The analytic continuation to all same-sign kinematics is streamlined by the basis’ symmetry properties, and boundary-case region V is handled via controlled limits. Comparisons with differential equations and numerical checks validate the results and highlight improvements over previous representations that used more complicated iterated integrals or region-specific expressions.

Abstract

We study one and two-loop triangle integrals with massless propagators and all external legs off shell. We show that there is a kinematic region where the results can be expressed in terms of a basis of single-valued polylogarithms in one complex variable. The relevant space of single-valued functions can be determined a priori and the results take strikingly a simple and compact form when written in terms of this basis. We study the properties of the basis functions and illustrate how one can easily analytically continue our results to all kinematic regions where the external masses have the same sign.

Figures (2)

• Figure 1: The master integrals for the two-loop three-mass triangle integrals.
• Figure 2: Different regions in $(u,v)$ space. The parabola represents the curve $\lambda(1,u,v)=0$.