Evaluating multiloop Feynman integrals by Mellin-Barnes representation

TL;DR

The paper addresses the challenge of analytically evaluating double- and triple-box Feynman integrals in dimensional regularization by promoting Mellin-Barnes (MB) representations as a versatile tool for master integrals. It derives new MB representations for massive on-shell double boxes, including configurations with one leg off-shell and both planar and non-planar topologies, yielding multi-fold MB forms that enable analytic results in terms of polylogarithms and harmonic polylogarithms and cross-checks with differential equations and sector-decomposition numerics. The work demonstrates the MB approach's ability to handle general propagator powers and to resolve epsilon-singularities more readily, while offering pathways toward automation and broader applicability to complex multi-loop diagrams. Overall, MB representations complement traditional reduction and differential-equation methods, expanding the analytic and numerical toolkit for high-precision multi-loop calculations relevant to processes like Bhabha scattering and beyond.

Abstract

The status of analytical evaluation of double and triple box diagrams is characterized. The method of Mellin-Barnes representation as a tool to evaluate master integrals in these problems is advocated. New MB representations for massive on-shell double boxes with general powers of propagators are presented.

Figures (4)

• Figure 1: Planar triple box diagram.
• Figure 2: Planar massive on-shell double box of the first type
• Figure 3: Planar massive on-shell double box of the second type
• Figure 4: Non-planar massive on-shell double box