Analytical Results for Dimensionally Regularized Massless On-shell Double Boxes with Arbitrary Indices and Numerators

TL;DR

This work addresses the analytic evaluation of dimensionally regulated massless on-shell two-loop double box diagrams with arbitrary indices and numerators. By solving IBP recurrence relations, the authors reduce any diagram to two master integrals $K_{(1)}^{(d)}$ and $K_{(2)}^{(d)}$ plus boundary diagrams, and express $K_{(2)}$ in terms of $K_{(1)}$ via differential relations. They provide explicit analytic results for the first master box in terms of polylogarithms ${ m Li}_{a}(-t/s)$ up to $a=4$ and generalized polylogarithms ${S}_{a,b}(-t/s)$ with $a=1,2$, $b=2$, and obtain a complete $oldsymbol{\epsilon}$-expansion for the second master box, including an extensive set of boundary contributions. The authors validate their results through asymptotic expansions in the limits $t/s o0$ and $s/t o0$ using the strategy of regions and cross-checks against the IBP relations, providing a robust analytic framework for two-loop massless amplitudes with arbitrary numerators.

Abstract

We present an algorithm for the analytical evaluation of dimensionally regularized massless on-shell double box Feynman diagrams with arbitrary polynomials in numerators and general integer powers of propagators. Recurrence relations following from integration by parts are solved explicitly and any given double box diagram is expressed as a linear combination of two master double boxes and a family of simpler diagrams. The first master double box corresponds to all powers of the propagators equal to one and no numerators, and the second master double box differs from the first one by the second power of the middle propagator. By use of differential relations, the second master double box is expressed through the first one up to a similar linear combination of simpler double boxes so that the analytical evaluation of the first master double box provides explicit analytical results, in terms of polylogarithms $\Li{a}{-t/s}$, up to $a=4$, and generalized polylogarithms $S_{a,b}(-t/s)$, with $a=1,2$ and $b=2$, dependent on the Mandelstam variables $s$ and $t$, for an arbitrary diagram under consideration.

Figures (3)

• Figure 1: Planar double box diagram.
• Figure 2: A box with a diagonal.
• Figure 3: A box with a one-loop insertion.