A $QQ\to QQ$ planar doublebox in canonical form

TL;DR

This work tackles the two-loop planar doublebox with four external massive legs $p_i$ ($p_i^2=-m^2$) and two internal masses by deriving a system of differential equations for the master integrals and casting it into a canonical, uniform-transcendental form. The authors construct a canonical 25–master-integral basis and express the differential equations in a $d\log$ form with an alphabet of 16 base letters plus four additional ones, enabling solutions in terms of Chen iterated integrals, Goncharov polylogarithms, and harmonic polylogarithms up to depth four. Explicit results (in an ancillary file) are provided and checked against existing results and numerical methods, with particular attention to the analytic structure and limiting cases (e.g., $y\to 1$). The study advances NNLO computations for massive-quark scattering by delivering a complete analytic and efficiently evaluable description of this building block, and it outlines avenues to extend the method to other topologies and potential elliptic-function regimes.

Abstract

We consider a planar doublebox with four massive external momenta and two massive internal propagators. We derive the system of differential equations for the relevant master integrals, cast it in canonical form, write it as a $d\log$ form and solve it in terms of iterated integrals up to depth four.

Figures (2)

• Figure 1: Momentum labels of the propagators. The figure does not include the irreducible numerators $P_4$ and $P_6$. The thick internal lines are the massive propagators.
• Figure 2: Master integral topologies.