Techniques for QCD calculations by numerical integration

TL;DR

This paper develops a fully numerical framework for perturbative QCD calculations of infrared-safe observables, focusing on $e^+e^- \to hadrons$ and three-jet-like quantities at order ${\cal O}(\alpha_s^2)$. It combines all integrations numerically, employing contour deformation to handle scattering singularities and a Monte Carlo integration scheme equipped with carefully designed sampling densities to manage collinear and soft divergences. The work demonstrates automatic cancellation of infrared singularities across cuts, provides a detailed contour-deformation strategy, and validates the method with a numerical example and comparisons to established results, using the beowulf code. The approach shows promise for flexible, fully numerical treatment of complex QCD calculations beyond traditional mixed analytic-numeric methods.

Abstract

Calculations of observables in quantum chromodynamics are typically performed using a method that combines numerical integrations over the momenta of final state particles with analytical integrations over the momenta of virtual particles. I describe the most important steps of a method for performing all of the integrations numerically.

Figures (16)

• Figure 1: Two cuts of one of the Feynman diagrams that contribute to $e^+e^- \to {\it hadrons}$.
• Figure 2: Diagram for a simple calculation. All two and three parton cuts of this diagram in $\phi^3$ theory are used, with a measurement function that gives the average transverse energy in the final state.
• Figure 3: The two and three parton cuts of the simple $\phi^3$ diagram.
• Figure 4: The eight contributions to the sample diagram after performing the energy integrations. The line through a propagator in a loop indicates that this propagator is put on shell, with positive energy flowing in the direction of the arrow. The direction for positive energy flow around the loop depends on whether the contour over loop energy is closed in the upper or the lower half plane.
• Figure 5: Space of loop momentum $\vec{\ell}_2$ for the virtual loop in the graph of Fig. \ref{['fig:cuts']}(a) for a representative choice of $\vec{q}$, $\vec{\ell}_4$, and $\vec{\ell}_5 = \vec{q} - \vec{\ell}_4$.