Measurements of jet rates with the anti-kT and SiScone algorithms at LEP with the OPAL detector

The strong interactions in the Standard Model are described with Quantum Chromodynamics (QCD). The verification of the predictions of QCD is essential for the understanding of Standard Model and is instrumental for searches of the physics beyond the Standard Model. One of the precise QCD predictions, which depends strongly on the only theory parameter, the constant of strong interaction αs, is the topology of the e +e− → hadrons events, i.e. multi-jet events. Recent theoretical developments allow to make predictions for this process even more precise than before. In the same time, these events can be studied for the first time with the recently developed jet clustering algorithms. The presented analysis aims to provide such measurements.


Introduction
The strong interactions in the Standard Model are described with Quantum Chromodynamics (QCD). The verification of the predictions of QCD is essential for the understanding of Standard Model and is instrumental for searches of the physics beyond the Standard Model. One of the precise QCD predictions, which depends strongly on the only theory parameter, the constant of strong interaction α s , is the topology of the e + e − → hadrons events, i.e. multi-jet events.
Recent theoretical developments allow to make predictions for this process even more precise than before. In the same time, these events can be studied for the first time with the recently developed jet clustering algorithms. The presented analysis aims to provide such measurements.

Jet algorithms
A jet clustering algorithm is a way to simplify the high energy collision event topology and exhibit the underlying physics at the parton level. The main goal of such a procedure is to reconstruct the kinematic variables of the partons produced during the primary hard interaction. The energy and the momenta of the partons are reconstructed by combining momenta and energy of the charged and neutral particles which are clustered into jets. Several jet algorithms are used to perform the combination in different environments -e + e − , pp or e ± p collisions. A detailed overview of their properties can be found elsewhere 1 . Basically, the contemporary algorithms can be divided in two groups: the sequential recombination algorithms and the cone algorithms.
The jets clustered with the sequential recombination algorithms are defined with the clustering procedure. The objects in the event are sequentially recombined into jets taking into account a pairwise distance measure between them. The definition of the distance measure, the order of recombination, the recombination scheme and the criteria to stop the procedure are the intrinsic properties of the algorithm.
The cone algorithms use a concept of cone object, which is defined as a set of considered objects with some given properties, e.g. having the jet energy above the defined threshold E cut .
In this analysis two new jet algorithms, SiScone 2,3,4 and anti-k T 5,3 are applied to e + e − → hadrons events. For both algorithms the used value of the R parameter, the angular jet size (radius) was equal to 0.7 for the basic studies. The maximal fraction of the energy shared between jets was set to 0.75. For a comparison with previously published results a PxCone algorithm was used 6 .

The OPAL detector
A detailed description of the OPAL detector can be found elsewhere 7,8 . Briefly OPAL is a multi-purpose detector with 93 % solid angle coverage. Tracking of charged particles was performed by a central detector, enclosed in a solenoid which provided a uniform axial magnetic field. The solenoid coil was surrounded by a time-of-flight counter array and a barrel lead-glass electromagnetic calorimeter with a presampler. Outside the electromagnetic calorimetry, the magnet return yoke was instrumented with streamer tubes to form a hadronic calorimeter. The detector was completed with muon detectors outside the magnet return yoke. The integrated luminosity was evaluated using small angle Bhabha scattering events observed in the forward calorimeters 9 .

Data and Monte Carlo samples
The analysis is based on the data samples obtained between 1995 and 2001 at the centre-ofmass energies close to √ s = 91, 130,136,161,172,183,189,192,196,200,202,205,207 GeV(see a summary elsewhere 10 ). For presentation purposes the results were combined into four samples with the average energies √ s = 91, 133, 177, 197 GeV. For the correction of the reconstructed events to the hadron (particle) level Monte Carlo samples with simulation of e + e − → hadrons events were produced with KK2f 11 and hadronised with Pythia 6.1 12 or Herwig 6.2 13 generators. For the reconstruction of the events at the particle level all undecayed particles were used.
For the estimation of background from e + e − → W + W − → qqqq and e + e − → W + W − → qqll processes, the Monte Carlo samples were produced with the grc4f2.1 14 and KORALW1.42 15 generators and hadronised with JETSET 15 . The samples above were passed through the Geant3based 16 OPAL detector simulation 17 and reconstructed in the same way as the data.

Event selection and reconstruction
The performed event selection is identical to the one used in the previous jet analyses 10 . Briefly, the events where rejected if the tracking system, electromagnetic calorimeter or the trigger system were inoperational, less than five tracks were reconstructed or the event had energetic initial state radiation 9,10 .
The events were required to have | cos θ| < 0.95, where the θ is the polar angle of the thrust axis calculated from all track and cluster objects. The events with a high probability to contain e + e − → W + W − → qqqq or e + e − → W + W − → qqll processes 18 were rejected.
The input for the reconstruction procedure at the (simulated) detector level was a set of tracks and the electromagnetic calorimeter clusters. To avoid double counting of the clustered objects an energy-flow algorithm 19,20 , matching tracks to the clusters in the electromagnetic calorimeter was applied.
The measured distributions of the jet rates were obtained in the following way. To take into account the presence of e − e + → W − W + background events the simulated detector level distributions of the e + e − → W + W − events weighted to the same luminosity as the data were subtracted from the detector level distributions. The obtained distributions were corrected bin-by-bin to the detector effects. The corrections were obtained as a ratio of the simulated detector level distributions and the particle level distributions of the same events. Finally, the distributions were normalised to the number of weighted entries.
The systematic uncertainties on the obtained results had several sources. The procedure of systematic uncertainties estimation is exactly the same as in the previous analyses 10 . The dominant systematic effects are the hadronisation modelling uncertainties.

Results and discussion
The results obtained with the SiScone and anti-k T jet algorithms are shown a in Fig. 1. As the results of the clustering procedures strongly depend on the value of cone radius R this dependence was studied, see Fig. 2.

Summary
The jet rate observables in e + e − annihilations to hadrons were measured with various clustering procedures. The new results obtained with clustering algorithms used earlier are consistent with the previously published measurements. The results obtained with the recently developed clustering algorithms provide an input for the validation of QCD and possibly precise estimation of α s .