The Differential equation method: calculation of vertex-type diagrams with one non-zero mass

TL;DR

The paper addresses analytic evaluation of two-loop vertex diagrams with one non-zero mass $m^2$ using the differential equation method (DEM). DEM derives differential equations in the mass parameter by IBP identities, reducing complex diagrams to simpler one-loop structures and yielding one-dimensional integral representations and Taylor expansions in the external momentum squared $q^2$. Analytic results are presented for diagrams contributing to $Z -> b bbar$ and $H -> gg$, including a massless case and a one-mass case, with explicit integral forms, threshold behavior, and high-order Taylor coefficients that enable high-precision numerical evaluation. The results are cross-checked against prior semi-analytic work and demonstrate DEM's utility for precision Standard Model calculations of vertex corrections with zero thresholds.

Abstract

The differential equation method is applied to evaluate analytically two-loop vertex Feynman diagrams. Three on-shell infrared divergent planar two-loop diagrams with zero thresholds contributing to the processes Z --> bb bar (for zero b mass) and/or H --> gg are calculated in order to demonstrate a new application of this method.