Vetoed jet clustering: The mass-jump algorithm

A new class of jet clustering algorithms is introduced. A criterion inspired by successful mass-drop taggers is applied that prevents the recombination of two hard prongs if their combined jet mass is substantially larger than the masses of the separate prongs. This"mass jump"veto effectively results in jets with variable radii in dense environments. Differences to existing methods are investigated. It is shown for boosted top quarks that the new algorithm has beneficial properties which can lead to improved tagging purity.


Introduction
The Large Hadron Collider (LHC) will restart in 2015 with unprecedented center-of-mass energy, pushing the window to find new particles beyond the Standard Model (SM) wide open. Practically all processes -SM or hypothetical -contain quarks or gluons in the final state and it is important that they can be reconstructed reliably. These coloured partons undergo parton showering and hadronization before they leave a signal in the detector. In order to make sense of experimental data it is therefore necessary to collect nearby radiation into jets, which are then assumed to correspond to their initiating (hard) partons.
Whenever jets are used as input for an analysis, the significance of the results crucially depends on the validity of this kinematic correspondance. Hence there has been ongoing effort to construct new and improved jet algorithms that are infrared and collinear safe, most of which proceed via sequential recombination [1,2,3,4,5,6] or cones [7,8,9,10], or follow completely different original ideas [11,12,13]. In the majority of these algorithms, jets are constructed with fixed angular size R, defined between two particles as ∆R = ∆η 2 + ∆φ 2 where ∆η and ∆φ are the distances in pseudorapidity and azimuthal angle, respectively.
Despite this splendour of algorithms to select from, choosing the optimal jet radius is always a compromise [14] as it may be different for jets of different energy or position in the detector.
Ref. [15] consequently proposes to employ a variable clustering radius instead, which in this case is taken inversely proportial to the jet transverse momentum, R ∝ 1/p ⊥ . An entirely different approach is taken by mass-drop tagging algorithms [16,17,18]. They address heavy resonances which are so highly boosted that their subsequent decay products cannot reasonably be resolved with conventional jet algorithms. Due to the high center-of-mass energy of the LHC, boosted top quarks, Higgs bosons, etc. are expected to be copiously produced during the upcoming run. To tag these resonances, it is possible to capture all decay products in a large-radius fat jet and apply substructure methods. The basic idea states that a jet should be broken up into two separate subjets if the jet mass experiences a significant drop in the procedure. These algorithms allow to identify hard substructure without referring to a fixed (sub)jet radius and turned out to perform very well in Higgs boson and top quark tagging (see e.g. [19,20,21] for reviews). Implicitly, a p ⊥ -dependent subjet radius is given by the mass cut, as the characteristic separation between the daughters of an energetic resonance is ∆R daughters 2m mother /p ⊥ .
In this paper, we supplement existing jet algorithms with a recombination veto which may prevent further clustering at a jet radius smaller than the given R. The working principle is similar to mass-drop tagging: if the recombination of two jet candidates leads to a significant mass jump, they should be resolved separately. In contrast to algorithms with variable radius, the veto is a property of two jets, i.e. the effective clustering radius now also depends on the jet's vicinity. This way well-separated jets are clustered conventionally with only small deviations, whereas on the other hand the merging of two hard prongs into a heavy resonance is vetoed. This paper is organized as follows. In Section 2, the mass-jump algorithm is motivated and 1 described in detail. Throughout the paper, we focus on consequences of the recombination veto in comparison to both classic jet algorithms as well as mass-drop taggers. In Section 3, we first evaluate the peformance for well-separated jets, and then turn to the highly boosted regime.
Beneficial properties for top quark tagging are pointed out. Conclusions are drawn in Section 4.  [16] established the family of mass-drop tagging (MDT) algorithms. It targets the 2-prong substructure of a large jet to identify the decay of a heavy resonance. Its modified 3-prong variant, the HEPTopTagger [17], enforces the following iterative procedure to act on a given fat jet clustered with the Cambridge/Aachen jet algorithm [4,5].
• Un-do the last clustering of the jet j into j 1 , j 2 with m j 1 > m j 2 .
• If a significant mass drop occurred, m j 1 < θ · m j , both j 1 and j 2 are kept as candidate subjets. Otherwise discard j 2 . 1 • Repeat these steps for the kept subjets unless m j i < µ, in which case j i is added to the set of output subjets.
The mass-drop (MD) procedure thus serves two purposes: It grooms the jet from (largeangle) soft radiation and applies a criterion to identify separate prongs based on jet mass. In the HEPTopTagger, the set of output subjets is then further processed and cuts applied. The default values of the two free parameters are chosen as θ = 0.8 and µ = 30 GeV [17,23].
Note that the unclustering algorithm is designed to follow the cascade decay chain of the top quark, At parton level the successive mass drops τ = m j 1 m j are given by hence the parameter θ has to be chosen sufficiently large to incorporate the first decay. In case the unclustering proceeds via t → j (bj) → j bj one obtains 1 It has been pointed out [22] that following the heavier prong leads to a (small) wrong-branch contribution.
This can be avoided by discarding the subjet candidate with smaller transverse mass m 2 ⊥ ≡ m 2 + p 2 ⊥ instead. As this modification is irrelevant for the remainder of this paper, we do not distinguish between the MDT and this modified mass-drop tagger (mMDT).
which is typically smaller than τ 1 . ∆R bj = ∆φ 2 + ∆η 2 is the R-distance between the subjets b and j.

The mass-jump clustering algorithm
Commonly used sequential jet clustering algorithms define an infrared and collinearly safe procedure to merge particles into jets step by step. Termination of this sequential recombination is given (in the inclusive algorithms) in terms of a minimum jet separation R. All input particles (calorimeter towers) are labelled as jet candidates and a distance measure betweens pairs of two is defined, where n = 1 corresponds to the k T algorithm [1,2,3], n = 0 to the Cambridge/Aachen algorithm [4,5], and n = −2 to the anti-k T algorithm [6]. Sequential recombination then proceeds as follows: 1. Find the smallest d jaj b among the jet candidates. If it is given by a beam distance, d jaB , label j a a jet and repeat step 1.
2. Otherwise combine j a and j b by summing their four-momenta, see e.g. Ref. [8]). In the set of jet candidates, replace j a and j b by their combination and go back to step 1.
Clustering eventually ends when all particles are merged into jets. The measure d serves two purposes here: First, it determines the order of recombination given by the pair with the smallest distance d jaj b at each step. Second, it acts as an upper bound on the jet radius, because a minimal beam distance d jaB implies ∆R jajn > R ∀ jet candidates j n .
We present a modification to these jet clustering algorithms which we call mass-jump (MJ) clustering. In the spirit of a reverse mass-drop procedure as outlined in the previous paragraph, "sub"jets are directly constructed by applying a veto at each recombination step, 2 where the parameter θ now acts as a mass-jump threshold. After all input particles / calorimeter towers are labelled as active jet candidates, the recombination algorithm is defined as follows: 1. Find the smallest d jaj b among active jet candidates; if it is given by a beam distance, d jaB , label j a passive and repeat step 1.
2. Combine j a and j b by summing their four-momenta, p ja+j b = p ja + p j b (E-scheme). If the new jet is still light, m ja+j b < µ, replace j a and j b by their combination in the set of active jet candidates and go back to step 1.
Otherwise check the mass-jump criterion: If all these criteria for the veto are fulfilled, label j a passive. Do the same for j b . If either of j a or j b turned passive, go back to step 1.

4.
No mass-jump has been found, so replace j a and j b by their combination in the set of active jet candidates. Go back to step 1.
Clustering terminates when there are no more active jet candidates left. Passive candidates are then labelled jets. Note that for θ = 0 or µ = ∞ this algorithm is identical to standard sequential clustering without veto, which in this case can be reduced to steps 1 and 4.

Properties
The mass-jump veto only has an impact on jet candidates which are separated by ∆R < R and whose combined mass would be above the (arbitrary) scale µ. It is designed to resolve close-by jets (which could come from a boosted resonance decay such as W ± , Z, H, ...) separately. As the vetoed jets are excluded from further clustering, their jet radius is smaller than the parameter R, which now gives an upper bound. A lower bound is effectively induced by a finite threshold scale µ.
There are several similarities and differences compared to MD unclustering. prongs become passive. Active jet candidates continue clustering (d) unless a veto is called, which can also act against a (hypothetical) recombination with a passive jet (e). Jet clustering continues for the remaining particles, giving additional jets (f).
In the idealized case, the output jets of both algorithms are comparable but differ in two aspects. First, MDT subjets are groomed even after a mass drop until they reach m < µ whereas MJ jets continue collecting radiation in the regime between m > µ and the mass jump. Although this effect is expected to be absent for reasonable large values of µ, if undesired it is straightforward to apply MDT-like grooming on the MJ jets. Second, the MJ clustering algorithm also returns jets which did not experience mass jumps (f) that are absent among MDT subjets (1,3,5). These can be desirable (well-separated jets for finite R) or can be considered junk; in the latter case it is again straightforward to remove them as these are the only jets turned passive by the upper bound on the jet radius instead of a mass jump. 3 Performance

Sparse environment: QCD dijets
We compare the MJ clustering algorithm to its standard counterparts. In events with low multiplicity, the effect of the veto should be only limited. 100 QCD dijet events are simulated with Pythia8 [24] where the minimum parton transverse momentum at matrix element level is chosenp min ⊥ = 150 GeV. The analysis is implemented as a Rivet [25] plugin. For jet clustering parameters R = 0.8 and p ⊥ ≥ 50 GeV, the three standard algorithms (antik T , Cambridge/Aachen and k T algorithms as provided by FastJet [26]) agree very well in the number of jets n std , which is 2 (in roughly one in two events) or above. We perform a parameter scan for the MJ clustering arguments θ and µ. Fig. 2 (bottom panel) shows the difference in the average number of jets per event (∆n =n MJ −n std ). The mutual leading jets (i.e. the min [n MJ , n std ] jets with largest p ⊥ ) in each event are matched, and differences between the MJ and standard algorithms are investigated on a jet-by-jet basis. For each pair (j MJ , j std ), we obtain the R-distance (∆R j MJ ,j std ) and relative difference in transverse momentum (∆p ⊥ = The upper two panels of Fig. 2 show results of these two observables averaged over all matched jet pairs. larger. In particular for the k T algorithm these differences can be substantial, namely ∆R ∼ 0.35 and ∆p ⊥ ∼ 0.25 for the considered setup. The C/A and especially the anti-k T algorithm behave much more moderately under the MJ veto. For the latter, deviations only reach ∆R ∼ 0.1 and ∆p ⊥ ∼ 0.1 even in the strong-veto region, and are almost absent in the bulk of parameter space. Generally the differences between MJ vetoed and standard clustering are smallest for the antik T algorithm and largest for the k T algorithm, with the C/A algorithm taking an intermediate position. This characteristic is directly related to the ordering of the cluster sequence, which is crucial in the MJ algorithm. If soft particles are clustered first (k T ), is it very likely to induce fake substructure which will fulfill the mass-jump condition at the stage when these soft clusters are recombined. The anti-k T algorithm on the other hand clusters around hard prongs and therefore is much more robust, while the purely angular-based C/A algorithm is moderately prone to vetoing fake soft clusters.
The number of jets is naturally equal or larger in the vetoed algorithms compared to the standard algorithms with equal jet clustering radius (Fig. 2 lower panels). If however the veto acts too strong, hard jets are split and may not pass the p ⊥ ≥ p min ⊥ cut any more, resulting in a decreasing number of jets again. For large minimum transverse momentum, say p ⊥ > 100 GeV for our sample, ∆n ultimately becomes negative.
Also for other jet clustering radii and p ⊥ thresholds, results are qualitatively very similar to the ones described above, so we omit further plots.

Busy environment: boosted top quarks
Tagging boosted top quarks is an important target in many current experimental studies and also an ideal playground to investigate the performance of MJ clustering in busy environments. In order to probe the moderately boosted energy regime and illustrate the algorithm, we simulate top pair production via a hypothetical heavy vector boson, respectively. Those fat jets are fed to the HEPTopTagger [17] which performs the following three-step procedure.
1. Subjets are obtained from the fat jet via mass-drop unclustering.
3. Cuts on subjet mass ratios determine whether or not the candidate is tagged as top.
For comparison with our veto algorithm, we apply the same algorithm but where the subjets are now obtained directly with MJ clustering starting from the fat jet's constituent particles.
Steps 2 and 3 remain unchanged such that the difference in tagging performance can be directly compared. We take R = ∞ and scan the parameter space in θ and µ. Results are based on each 10,000 signal and background events (QCD dijets withp min ⊥ = 150 GeV) generated with Pythia8 and analyzed within Rivet. The resulting tagging efficiencies = #tagged #fat jets are shown in Fig. 3. 4 Indeed the peak tagging efficiencies are equal for both algorithms. However, as argued in Sec. 2.3, MJ jet finding allows for well-performing top tagging in a much wider parameter range in θ. The reason for this behaviour lies in the absence of an equivalent to the cascade mass drops experienced in MDT's (such as t → bW + → bjj ). This feature can also be directly seen in

Conclusions
We developed and investigated a new jet clustering algorithm which includes a recombination veto based on jet mass. In this mass-jump (MJ) procedure, the clustering radius R now acts as an upper limit on jet size and the merging of two hard prongs is prevented. We showed that in sparse events with well-separated jets, the effect of the veto is very limited in a large range of parameter space. Also the anti-k T clustering algorithm is more robust against fake twoprong substructure than the Cambridge/Aachen and k T algorithms. In the dense environment of hadronically decaying boosted top quarks, MJ clustering is comparable to mass-drop taggers (MDT) by which the veto was inspired in the first place; the main difference being that cascade mass drops as present in MDT's are avoided which allows for stricter threshold parameters. The larger parameter space then leads to improved ROC curves for the new MJ clustering in the context of the HEPTopTagger algorithm.
Vetoed clustering algorithms are a promising tool for collider experiments as they make room for more flexibility. The optimal jet clustering radius depends on various parameters such as the type of initiating particle, its energy or transverse momentum, and the surrounding topology of the event. The MJ veto automatically adjusts the jet radius such that hard substructure is separated into isolated jets. This feature may prove helpful in a variety of events where jets are not well-separated.