Single mass scale diagrams: construction of a basis for the $ε$-expansion

TL;DR

The paper develops a constructive basis for the ε-expansion of single-mass-scale Feynman diagrams with two-particle massive cuts, guided by a Broadhurst-inspired sixth-root-of-unity approach. It combines polylogarithm splittings at $z=e^{\pm i\pi/3}$ with a weight-additive algebra, and validates the basis by expressing new two- and three-loop master integrals analytically in terms of basis elements. The authors provide explicit results up to weight 5, including a decomposition of a novel constant $V_{3,1}$ within the basis, and discuss the robustness of the approach via high-precision PSLQ calculations. They also outline conjectures and limitations, notably the mysterious role of the sixth root of unity and potential extensions to more complex massive-cut configurations.

Abstract

Exploring the idea of Broadhurst on the ``sixth root of unity'' we present an ansatz for construction of a basis of transcendental numbers for the epsilon-expansion of single mass scale diagrams with two particle massive cut. As example, several new two- and three-loop master integrals are calculated.

Figures (2)

• Figure 1: Examples of the simplest single mass scale integrals with two-particle massive cut. Bold and thin lines correspond to massive and massless propagators, respectively.
• Figure 2: Three-loop master-integrals. Bold and thin lines correspond to massive and massless propagators, respectively.