Antenna Showers with One-Loop Matrix Elements

TL;DR

The paper advances antenna-based parton showers by incorporating one-loop (NLO) matrix-element corrections up to ${\cal O}(\alpha_s^2)$, focusing on $e^+e^-$ initial states and the process $Z\to 3$ jets. It develops a detailed matching framework (tree-level and one-loop) and analyzes how different evolution-ordering schemes and renormalization-scale choices affect logarithmic structures, identifying $\mu_{\mathrm{PS}}$ proportional to $p_T$ as giving the best agreement with fixed-order results. Implemented in the VINCIA framework, the approach yields improved LEP-precision predictions for event shapes and fragmentation with a consistent per-event uncertainty treatment, and it demonstrates practical speed for extensive studies. The work lays groundwork for extending one-loop-accurate antenna showers to hadron collisions and sector-based schemes, and it highlights avenues for refining Sudakov factors in unordered regions to achieve higher-logarithmic accuracy.

Abstract

We consider the probability for a colour-singlet qqbar pair to emit a gluon, in strongly and smoothly ordered antenna showers. We expand to second order in alphaS and compare to the second-order QCD matrix elements for Z -> 3 jets, neglecting terms suppressed by 1/NC^2. We give a prescription that corrects the shower to the matrix-element result at this order for both soft and hard emissions, thereby explicitly reducing its dependence on evolution- and renormalization-scale choices. We confirm that the choice of pT for both of these scales absorbs all logarithms through order alphaS^2, and contrast this with various alternatives. We include these corrections in the VINCIA shower generator and study the impact on LEP event-shape and fragmentation observables. An uncertainty estimate is provided for each event, in the form of a vector of alternative weights.

Figures (18)

• Figure 1: Contours of constant value of $y_E = Q_E^2/m_{IK}^2$ for evolution variables linear ( top) and quadratic ( bottom) in the branching invariants, for virtuality-ordering ( left), $p_{\perp }$-ordering ( middle), and energy-ordering ( right). Note that the energy-ordering variables intersect the phase-space boundaries, where the antenna functions are singular, for finite values of the evolution variable. They can therefore only be used as evolution variables together with a separate infrared regulator, such as a cut in invariant mass, not shown here.
• Figure 2: Illustration of the regions of 3-parton phase space in which the subsequent evolution of the $qg$ and $g\bar{q}$ antennae is restricted (from above) by the strong-ordering condition. See the text for further clarification of this plot. Black: both antennae restricted. Dark Gray: one antenna restricted, the other unrestricted. Light Gray: both antennae unrestricted. Top/Bottom:$Q^2$ linear/quadratic in the branching invariants, for mass-ordering ( left), $p_{\perp }$-ordering ( middle), and energy-ordering ( right).
• Figure 3: The smooth-ordering factor ( solid) compared to a strong-ordering $\Theta$ function ( dashed).
• Figure 4: Illustration of scales and Sudakov factors involved in an unordered sequence of two $2\to 3$ branchings, representing the smoothly ordered shower's approximation to a hard $2\to 4$ process.
• Figure 5: Illustration of the evolution scales and Sudakov factors appearing in the exclusive 3-jet cross section, eq. \ref{['eq:ps3excl']}.