The on-shell massless planar double box diagram with an irreducible numerator

TL;DR

This paper tackles the evaluation of the on-shell massless planar two-loop double-box integral with an irreducible numerator, a key piece in two-loop scattering amplitudes. Using a Mellin-Barnes representation, the authors derive a direct calculation of the irreducible-numerator master integral $I_{Irr}$, including its full $\epsilon$-expansion up to ${\cal O}(\epsilon^0)$, and express the result in terms of logarithms and polylogarithms. They construct a differential-equation framework for the double-box master integrals in a new basis $\{I_1,I_{Irr}\}$, revealing structure and singularities and demonstrating consistency with known relations and reductions. Furthermore, they show that in $d=6$ dimensions the master integrals are finite and obtain explicit finite expressions, verified by numerical integration. Collectively, the work resolves a reduction-induced $1/\epsilon$ issue and provides a robust, analytically tractable master integral for tensor two-loop massless amplitudes in QCD-like theories.

Abstract

Using a Mellin-Barnes representation, we compute the on-shell massless planar double box Feynman diagram with an irreducible scalar product of loop momenta in the numerator. This diagram is needed in calculations of two loop corrections to scattering processes of massless particles, together with the double box without numerator calculated previously by Smirnov. We verify the poles in epsilon of our result by means of a system of differential equations relating the two diagrams, which we present in an explicit form. We verify the finite part with an independent numerical check.

Figures (2)

• Figure 1: The planar double box diagram
• Figure 2: Master integrals with less than six propagators