Triple collinear emissions in parton showers

TL;DR

The paper develops a framework to augment parton showers with triple-collinear splittings by constructing a local subtraction scheme that enables fully differential, four-dimensional Monte Carlo implementation of the NLO DGLAP kernels Pqq' for both timelike and spacelike evolution. It derives explicit fixed-order expressions for Pqq'^(T)(z) and Pqq'^(S)(z), and implements the approach in two independent MC environments, confirming numerical consistency and cross-framework agreement. The study shows that, for the tested observables, the flavor-changing triple-collinear contributions have a modest impact (~1%), but the method establishes a path toward on-the-fly computation of complete NLO kernels and extension to broader triple-collinear functions. This work thus advances high-precision shower simulations and paves the way for fuller NLO accuracy in fully exclusive event generation.

Abstract

A framework to include triple collinear splitting functions into parton showers is presented, and the implementation of flavor-changing NLO splitting kernels is discussed as a first application. The correspondence between the Monte-Carlo integration and the analytic computation of NLO DGLAP evolution kernels is made explicit for both timelike and spacelike parton evolution. Numerical simulation results are obtained with two independent implementations of the new algorithm, using the two independent event generation frameworks Pythia and Sherpa.

Figures (6)

• Figure 1: Validation of the simulation of triple-collinear parton splittings in final-state (top row) and initial-state (bottom row) branchings with final-state (left panels) and initial-state (right panels) spectator. We show Durham $k_T$-jet rates in $e^+e^-\to$hadrons at LEP I, $k_T$-jet rates in neutral current DIS at HERA II with $Q^2>100~{\rm GeV}^2$, and $k_T$-jet rates in $pp\to e^+\nu_e$ at the 8 TeV LHC (top left to bottom right).
• Figure 2: Impact of the simulation of triple-collinear parton splittings on final-state (top row) and initial-state (bottom row) evolution with final-state (left panels) and initial-state (right panels) spectator. Top panels show the ratio between the leading-order result and the leading-order simulation including triple-collinear branchings. Middle and bottom panels show a comparison between the simulation of up to one triple-collinear splitting and arbitrarily many (both not including the leading-order result). For details, see Fig. \ref{['fig:validation']}.
• Figure 3: Kinematics mapping for final-state splittings with final-state spectator.
• Figure 4: Kinematics mapping for final-state splittings with initial-state spectator.
• Figure 5: Kinematics mapping for initial-state splittings with final-state spectator.