Choosing integration points for QCD calculations by numerical integration

TL;DR

The paper tackles the challenge of efficiently sampling loop-momentum space in fully numerical next-to-leading order QCD calculations for three-jet-like observables in $e^+e^-$ annihilation. It presents a modular sampling framework built from multiple elementary methods, leveraging the geometry of scattering singularity surfaces (ellipsoids and spheres) and contour deformation to control integrand behavior. The approach assigns sampling densities to correspond to specific final-state cuts and scattering topologies (2→2 s/t, 2→3, 2→1), with special attention to soft and collinear regions and exceptional cases. While not claimed to be optimal, the method offers a systematic, adaptable recipe that improves convergence and flexibility for complex infrared-safe observables in QCD calculations.

Abstract

I discuss how to sample the space of parton momenta in order to best perform the numerical integrations that lead to a calculation of three jet cross sections and similar observables in electron-positron annihilation.

Figures (8)

• Figure 1: Two cuts of one of the Feynman diagrams that contribute to $e^+e^- \to {\it hadrons}$.
• Figure 2: Elementary scattering subdiagrams that occur at next-to-leading order.
• Figure 3: Singularity surfaces associated with the elementary scatterings in Fig. \ref{['fig:scatterings']}. In each case, the vectors $\vec{l}_2$ and $\vec{l}_3$ (or just $\vec{l}_2$ for 2 to 1 scattering) are indicated by arrows. We see the scattering singularity surface in the space of $\vec{l}_1$. For 2 to 2 (t) scattering and 2 to 2 (s) scattering, these surfaces are ellipsoids. For 2 to 3 scattering, the surface is a sphere, only part of which is shown. For 2 to 1 scattering, the surface reduces to a line segment. The integrand is typically not singular on the scattering singularity surface because of the contour deformation. However, the contour deformation vanishes along the heavy straight lines. Thus, in particular, in the 2 to 2 (t) case the integrand is actually singular at $\vec{l}_1 = 0$.
• Figure 4: Matching of scattering singularities to the structure of one loop virtual subdiagrams. A scattering singularity occurs when the energy of an intermediate state matches the energy of the final state. For each diagram, the relevant intermediate states are marked with a line through the graph. The label near the line indicates the type of the corresponding singularity. For some graphs, there is more than one scattering singularity, as indicated. The 2 to 1 singularity is marked only in the case of a self-energy subdiagram connected to a final state parton.
• Figure 5: Labelling of momenta for graphs of type (d) in Fig. \ref{['fig:virtualtypes']}.