Evaluating double and triple (?) boxes
V. A. Smirnov
TL;DR
The paper addresses the analytic evaluation of massless double-box and on-shell planar triple-box Feynman integrals for four-point scattering, surveying the methodological toolkit for multi-loop, multi-scale diagrams. It emphasizes IBP-based reduction to master integrals for double boxes, together with Mellin–Barnes representations and differential equations; for triple boxes, it presents a sevenfold MB representation and analyzes Regge-limit behaviour via expansion by regions. A key contribution is the explicit epsilon- expansion of the triple-box master integral, with coefficients $c_i(x,L)$ (e.g., $c_6=16/9$) expressed through polylogarithms and GenPolyLog, and an outline of partial analytic reductions and numerical checks that pave the way for a full analytic solution. The work demonstrates progress toward analytic control of higher-loop, multi-scale Feynman integrals and highlights the role of harmonic polylogarithms and related function bases in expressing such results, with implications for three-loop BFKL analyses and beyond.
Abstract
A brief review of recent results on analytical evaluation of double-box Feynman integrals is presented. First steps towards evaluation of massless on-shell triple-box Feynman integrals within dimensional regularization are described. The leading power asymptotic behaviour of the dimensionally regularized massless on-shell master planar triple-box diagram in the Regge limit $t/s \to 0$ is evaluated. The evaluation of the unexpanded master planar triple box is outlined and explicit results for coefficients at $1/\ep^j$, j=2,...,6, are presented.