Cone-based algorithms

The cone-based algorithm is still the most popular at hadron colliders. The jets are character-ized in term of variables that are invariant under boost along the beam axis. The variables are:

transverse momenta, p_T, or the transverse energy, E_T E sinθ; azimuthal angle around the beam direction, φ; and pseudo-rapidity, η^U lntanθ 2. The algorithm forms jets by associating together particles whose trajectories lie within a circle of specific radius R inη φspace.

The UA1 and UA2 cone-based algorithms

In the 80s, the UA1 and UA2 experiments were the firsts to use cone-based algorithms in a p ¯p col-lider. The UA1 algorithm started ordering in decreasing transverse ET the cells of the calorimeter with E_T V E_Tseed. The value used for E_Tseed was 2.5 GeV. The cell with the highest transverse energy initiated the first jet. The next cell was added to the first if it was within a distance R₀=1. If the cell was outside this radius then a new jet was initiated. This procedure was repeated until all cells above the E_Tseed threshold have been assigned to a jet. Finally, the cells with E_T

E_Tseed

are then added to each jet if R

R0. In contrast UA2 used another approach. In the UA2 algo-rithm the cluster did not have a limited size inη φ space. Once all the calorimeter cells were ordered in decreasing E_T, starting for the highest one, all the neighboring cells were joined into the cluster if the ETexceeded a given threshold. In that case, the threshold was 0.4 GeV.

The CDF cone-based algorithms: JetClu and MidPoint algorithms

Some years later, the CDF experiment started to use also a cone-based algorithm, the JetClu algorithm. This begins defining a list of seeds, calorimeter towers with E_T V 1 GeV. Each seed is the starting geometric center for a cone. Thus, for a specific geometric center for the cone (φCηC) the ‘particles’ i within the cone satisfy

∆RiC^GW ηi ηC

2^,

φi φC

2

J R (2.26)

and the energy of the jet, using the snowmass convention [28], is defined as:

E_T^C

∑

i^X Cone

E_Tⁱ (2.27)

Then the jet geometric center, defined as ηC ∑i^X ConeE_Tⁱηⁱ

E_T^C and φC ∑i^X ConeE_Tⁱφⁱ E_T^C

(2.28)

is calculated. This new point inη φ is then used as the center for a new trial cone. As this calculation is iterated the cone center flows until a “stable” solution is found, i.e., until the centroid of the energy depositions within the cone is aligned with the geometric axis of the cone.

Unfortunately, nothing prevents the resulting final stable cones from overlap as it is shown in figure 2.13. Therefore, a procedure must be included in the cone algorithm to specify how to split or merge overlapping cones. These cones are merged if their shared transverse energy is larger than a fixed fraction (e.g., f=75%) of the jet with smaller transverse energy; otherwise two jets are formed and the common towers are assigned to the jet closer inη φspace.

Figure 2.13:Lego plot of an event with reconstructed jets using a cone-based algorithm. This example illustrates an overlapping situation between two stable cones (red cluster).

After Run I, it was realized [29] that the iterative cone algorithm, as described above, is not infrared safe when applied to parton-level calculations. If two cones overlap in such a way that their centers can also be enclosed in one cone but there is little energy in the overlap region, then it turns out, as illustrated in figure 2.14, that the outcome is different depending on whether or not the overlap region contains a seed direction. This results in a logarithmic dependence on the seed cell threshold which would give a divergent cross section if the threshold was taken to zero for the purposes of making an idealized calculation. This divergence first shows up when there can be three nearby partons, which for jets in hadron collisions is NLO in the three-jet cross section and NNLO in the two-jet or inclusive one-jet cross sections. Moreover, this algorithm is sensitive to collinear radiation in the event, where the seed finding depends of calorimeter granularity. It is clear from figure 2.15 that the seed it is not considered anymore after its energy is split. In order to address these theoretical difficulties, an additional step, before the merging/splitting procedure, is included. The midpoint, in theη φ plane, between each pair of stable cones separated by less than 2R is added to the list of jets. The clustering algorithm is again iterated until stability is achieved. This step gives the name to the jet algorithm: MidPoint algorithm [29].

Figure 2.14: An illustration of infrared sensitivity in cone jet clustering. This example shows how the presence of a soft radiation between two jets may cause a merging of the jets that would not occur in the absence of the soft radiation.

Figure 2.15:An illustration of collinear sensitivity in jet reconstruction.

However, the merging/splitting procedure applied to resolve situation with overlapping cones in data introduces an element of arbitrariness when the set algorithm is applied to theoretical calculations. Current NLO inclusive jet cross section calculations require the addition of an ad hoc parameter R_sep[30]. The theoretical calculations use a cone algorithm with an enlarged cone size R^*O Rsep⁾ R, where Rsepis typically 1.3. In this way, theorists try to emulate effects of merging cones applied experimentally in the data, that results into jets with transverse size (η φ) larger than R. The use of different values for Rsepintroduces an uncertainty in the theoretical prediction of the order of 5%.