Next-to-leading order QCD calculations with parton showers I: collinear singularities

TL;DR

This work addresses the challenge of merging NLO QCD calculations with parton showering to yield realistic hadronic final states without sacrificing NLO accuracy. It develops a Coulomb-gauge framework for incorporating final-state collinear splittings via primary showers applied to Born-level graphs, backed by a subtraction-to-multiplication equivalence that preserves the $O(\alpha_s^{B})$ and $O(\alpha_s^{B+1})$ terms, while introducing Sudakov factors to model no-splitting probabilities. A series of constructions—primary and secondary splittings, virtuality cuts, and careful handling of real and virtual corrections—are demonstrated for the $e^+e^- \to 3$ jets process, with numerical tests validating the expected higher-order behavior. The approach enables interplay with standard MC event generators for hadronization and sets the stage for extending to soft-gluon effects in a companion work, thereby providing a practical path to realistic, NLO-accurate event generation. Overall, the paper offers a concrete scheme to couple NLO calculations with parton showers, enhancing predictive power for collider phenomenology.

Abstract

Programs that calculate observables in quantum chromodynamics at next-to-leading order typically suffer from the problem that, when considered as event generators, the events generated consist of partons rather than hadrons and just a few partons at that. Thus the events generated are completely unrealistic. These programs would be much more useful if the few partons were turned into parton showers, in the style of standard Monte Carlo event generators. Then the parton showers could be given to one of the Monte Carlo event generators to produce hadron showers. We show how to generate parton showers related to the final state collinear singularities of the next-to-leading order perturbative calculation for the example of e+ + e- --> 3 jets . The soft singularities are left for a separate publication.

Figures (10)

• Figure 1: Two cuts of one of the Feynman diagrams that contribute to $e^+e^- \to {\rm hadrons}$.
• Figure 2: Graphical representation of the three terms in Eq. (\ref{['pert']}). The third diagram is intended to represent the entire virtual contribution.
• Figure 3: Graphical representation of ${\cal I}[{\rm shower}]$ in Eq. \ref{['shower']}. The pair of square vertices represents the matrix element ${\cal M}$ and the Sudakov exponential.
• Figure 4: A cut Born graph. We depict here only the topology of the graph, leaving unspecified which lines are quarks and which are gluons. On the right, we show the graph after replacing the cut propagators by ${\cal I}[{\rm shower}]$, which is represented by a circle containing an $S$.
• Figure 5: Six order $\alpha_s^2$ cut Feynman graphs. (For a graph with a self-energy insertion with one adjacent propagator cut, we understand the complete virtual correction.) These graphs are the order $\alpha_s^2$ term in the expansion of the graph in Fig. \ref{['fig:born2']}.