On a two-loop crossed six-line master integral with two massive lines

TL;DR

This paper analytically computes a complex two-loop Feynman integral with two massive lines, deriving boundary conditions via Mellin-Barnes representation and revealing relations among harmonic polylogarithms.

Contribution

It provides the first analytical calculation of this specific two-loop integral and introduces new relations among harmonic polylogarithms at the sixth root of unity.

Findings

Analytical expression for the two-loop crossed six-line master integral.

Boundary conditions derived from Mellin-Barnes representation.

New relations among harmonic polylogarithms of sixth root of unity.

Abstract

We compute the two-loop crossed six-line vertex master integral with two massive lines in dimensional regularisation, and give the result up to the finite part in D-4. We focus in particular on the purely analytical calculation of the boundary condition which we derive from a three-fold Mellin-Barnes representation. We also describe how the computation of the boundary condition is used to derive three non-trivial relations among harmonic polylogarithms of the sixth root of unity.