Two-Loop Vertices in Quantum Field Theory: Infrared and Collinear Divergent Configurations

TL;DR

The paper develops a comprehensive numerical framework for evaluating infrared- and collinear- divergent two-loop three-point functions. It combines Landau-equation classifications, sector decomposition, Bernstein-Sato-Tkachov relations, hypergeometric-function techniques, and Mellin-Barnes transforms to extract IR poles and compute finite parts, yielding multivariate integral representations that generalize Nielsen-Goncharov polylogarithms. The authors demonstrate the method on a wide range of two-loop vertex topologies, validate results against known analytic expressions, and apply the approach to electroweak observables, confirming its accuracy and utility. This work enables reliable NNLO computations in the presence of multiple mass scales and threshold effects, bridging analytical insight with robust numerical evaluation. The framework promises broad applicability to QED/QCD corrections within the Standard Model and beyond, where infrared and collinear effects are essential.

Abstract

A comprehensive study is performed of two-loop Feynman diagrams with three external legs which, due to the exchange of massless gauge-bosons, give raise to infrared and collinear divergencies. Their relevance in assembling realistic computations of next-to-next-to-leading corrections to physical observables is emphasised. A classification of infrared singular configurations, based on solutions of Landau equations, is introduced. Algorithms for the numerical evaluation of the residues of the infrared poles and of the infrared finite parts of diagrams are introduced and discussed within the scheme of dimensional regularization. Integral representations of Feynman diagrams which form a generalization of Nielsen - Goncharov polylogarithms are introduced and their numerical evaluation discussed. Numerical results are shown for all different families of multi-scale, two-loop, three-point infrared divergent diagrams and successful comparisons with analytical results, whenever available, are performed. Part of these results has already been included in a recent evaluation of electroweak pseudo-observables at the two-loop level.

Figures (15)

• Figure 1: The scalar one-loop, three-point Green function with one massless internal line. All momenta are flowing inwards.
• Figure 2: A two-loop diagram contribution to the $W$-boson self-energy.
• Figure 3: The irreducible two-loop vertex diagrams $V^{{{E}}}$. External momenta are flowing inwards.
• Figure 4: The irreducible two-loop vertex diagrams $V^{{{I}}}$. External momenta are flowing inwards.
• Figure 5: The $V^{{{I}}}$ infrared configurations. The photon line represents a general massless particle while the dashed and the continuous lines represent different massive particles. The mass of the two particles in the bubble are $m_1$ and $m_2$.