Subjet Multiplicities at LHC Energies and the QCD Color Factor Ratio 𝐶_𝐴/𝐶_𝐹

Abstract

Subjet multiplicity distributions of two jets with highest transverse momenta are simulated in proton-proton collisions at LHC energies of 7 TeV and 14 TeV. Jets are defined via the software package FastJet version 3.0.3 which includes the implementation of 𝑘_𝑇 sequential recombination algorithm with a jet size of 𝑅=0.6. Subjets are resolved by rerunning the 𝑘_𝑇 algorithm on the jets and using the resolution cutoff of 𝑦_cut=10⁻³. The subjet multiplicity distributions and their average values are measured as a function of the jet momenta and upto rapidities |𝜂|<2. The results are used to calculate the ratio, 𝑟=(⟨𝑀_𝑔⟩−1)/(⟨𝑀_𝑞⟩−1).

1. Introduction

One of the main motivations of studying highly energetic proton-proton (p-p) collisions at Large Hadron Collider (LHC) at CERN has been the discovery of Higgs Boson(s). The aim has recently been achieved with the discovery of a Higgs Boson. However, many more studies are needed to establish its properties. In addition, searches for new particles will continue for many more years.

The proton-proton collisions at the LHC are dominated by jet production which constitute a large background to potential signals of new physics. Quantum chromodynamics (QCD) describes well the dynamics of jet production in terms of partons. In QCD, quarks and gluons have different color charges and hence different coupling strengths. The values for the color factors originate directly from the symmetry group SU(3) of QCD, calculated to be𝐶_𝐴=3 for the gluon and𝐶_𝐹=4/3 for the quark [1].

In hadron-hadron collisions, quarks and gluons fragment to produce collimated streams of particles conventionally called “jets.” A jet, characterized by its energy and momentum, is reconstructed according to a clustering prescription called a “jet algorithm” from the four vectors of constituents that may represent tracks, energy deposits in the calorimeter, reconstructed particle candidates, particles in a Monte Carlo (MC) event generator, or partons of a theory calculation. In dijet events, the two jets leading in transverse momentum 𝑃_𝑇can be associated with the two partons at leading order (LO) in perturbative QCD. The internal structure of these jets is then expected to depend mainly on the type of the primary parton, that is, either (anti-) quark or gluon, from which they originated. QCD predicts that gluons, because of their larger color factor, fragment more than quarks. Consequently, gluon-initiated jets become broader and exhibit a larger constituent multiplicity than quark jets. The ratio of the constituent multiplicity for gluon jets versus quark jets at LO is asymptotically given by the ratio of their color factor as𝐶_𝐴/𝐶_𝐹=9/4. Effects of higher orders in the strong coupling constant𝛼_𝑆, however, could change this theoretical prediction.

In order to study the jet substructure, an experimental observable called subjet multiplicity, 𝑀, is found to be very useful. It is defined as the number of subjets that can be resolved within a jet by reclustering the jet constituents with the same clustering algorithm. However, the spatial resolution is chosen to be finer than the one used in the jet reconstruction. In this way an infrared- and collinear-safe measure can be defined that is usable experimentally as well as in perturbative QCD. Various collinear and infrared safe jet algorithms exist [2].

Following a sequential recombination procedure, jets are defined by𝑘_𝑇algorithm which starts clustering first the softest objects, trying to undo the effects of parton showering [3–5] and is well suited to explore the jet substructure.𝑘_𝑇algorithm is implemented in the FastJet package [6–8].

In the present work, samples used to measure the subjet multiplicity at two center of mass energies (c.m.), 7 TeV and 14 TeV, corresponding to the current and future LHC energies, respectively, are generated from the event generator PYTHIA8 [9]. A sample of 7 million events at each of the two c.m. energies is generated. The dijet events are selected in which the average subjet multiplicity⟨𝑀⟩is studied as a function of the jet 𝑃_𝑇  in the range of 100 to 1100 GeV and in the pseudorapidity interval|𝜂|<2, where pseudorapidity𝜂=−lntan(𝜃/2)with𝜃as the azimuthal angle of the jet.

2. Jet Algorithm and the Subjet Multiplicity

The𝑘_𝑇algorithm [5] clusters all input objects𝑖,𝑖=1,…𝑛, according to the following iterative steps.(i)For each pair(𝑖,𝑗)of objects find 𝑑_𝑖𝑗=min𝑝²_𝑇𝑖,𝑝²_𝑇𝑗Δ𝑅²_𝑖𝑗𝑅², (1) where𝑅= 0.6 is the jet size in𝑦−𝜙space Δ𝑅_𝑖𝑗=𝑦_𝑖𝑗²+Δ𝜙_𝑖𝑗². (2)(ii)For each object calculate the beam distance 𝑑_𝑖𝑏=𝑝²_𝑇𝑖. (3)(iii)If the minimum𝑑_minof all possible𝑑_𝑖𝑗and𝑑_𝑖𝑏is𝑑_𝑖𝑗, then merge objects𝑖and𝑗into a single object by 4-momentum vector addition. If it is𝑑_𝑖𝑏then remove object𝑖from the list and define it to be a final jet.(iv)Repeat the above steps until no clustering objects are left.

2.1. Subjet Multiplicities

Subjets are resolved within a jet by repeating the application of𝑘_𝑇cluster algorithm described above. The clustering is terminated when all𝑑_𝑖𝑗,𝑑_𝑖𝑏are above the quantity 𝑑_cut=𝑦_cut𝑃²_𝑇(jet). All remaining objects are called subjets. While the stopping parameter 𝑑_cutdefines the hard scale of the process, the parameter𝑦_cutis known as the resolution parameter. The subjet structure depends upon the values chosen for the resolution parameter𝑦_cut. The mean subjet multiplicity⟨𝑀⟩is defined as the average number of subjets in a jet at a given value of𝑦_cut: ⟨𝑀⟩=1𝑁_jet^𝑁jet_𝑖=1𝑀𝑦_cut. (4)

By definition𝑀(𝑦_cut)≥1and0≤𝑦_cut≤1. The mean subjet multiplicity is measured for𝑦_cut=10⁻³and 𝑅=0.6, where𝑦_cutis the subjet resolution parameter. A study to optimise the value of𝑦_cutat 7 TeV was done in an earlier work, documented in [10, 11] and the same used here. This optimum value of𝑦_cutmatches with the one used for the definition of subjets in [12]. In addition two subjets within a jet are resolved if they are well separated in𝜂×𝜙space.

3. Determination of Ratio 𝑟

In the p-p collisions, the final state stable particles are arranged into jets by using a jet algorithm, as described above. These particles are produced in the hadronisation of hard partons. In a mixed sample of quark and gluon initiated particle jets,  𝑀 is the subjet multiplicity which can be written as a linear combination of subjet multiplicity in gluon jets 𝑀_𝑔and quark jets 𝑀_𝑞: 𝑀=𝑓𝑀_𝑔+1−𝑓𝑀_𝑞, (5) where𝑓is the fraction of gluon jets and (1−𝑓) is the fraction of quark jets in the mixed sample. The gluon jet fraction𝑓is the number of outgoing gluons that pass the selection cuts divided by the total number of outgoing partons that pass the selection cuts. Considering the above equation for two similar samples of jets at√𝑠  (√𝑠is the center of mass energy (c.m.))=7TeV and 14 TeV and assuming 𝑀_𝑔and 𝑀_𝑞to be independent of√𝑠, we get 𝑀⁷=𝑓⁷𝑀_𝑔+1−𝑓⁷𝑀_𝑞,𝑀¹⁴=𝑓¹⁴𝑀_𝑔+1−𝑓¹⁴𝑀_𝑞. (6) The solutions are 𝑀_𝑞=𝑓¹⁴𝑀⁷−𝑓⁷𝑀¹⁴𝑓¹⁴−𝑓⁷,𝑀_𝑔=1−𝑓⁷𝑀¹⁴−1−𝑓¹⁴𝑀⁷𝑓¹⁴−𝑓⁷, (7) where 𝑀¹⁴and 𝑀⁷are extracted from the total subjets in mixed samples of quark and gluon initiated jets in the data at√𝑠=14 and 7 TeV, respectively, and𝑓¹⁴and𝑓⁷are the gluon jet fractions at the two energies. This method relies on the extraction of quark and gluon fractions from the Monte Carlo information, obtained by tagging the particle jets with partons. This way the knowledge of the two gluon jet fractions is obtained. The subjet multiplicity distributions can be characterized by the mean values ⟨𝑀⟩such that⟨𝑀_𝑔⟩−1 and ⟨𝑀_𝑞⟩−1 gives the average number of subjets in a gluon and quark jet, respectively. The gluon jets are compared to the quark jets by having a ratio: 𝑀𝑟=^𝑔−1𝑀_𝑞.−1 (8)

4. Event Samples and Jet Selection

To study the subjet multiplicity in high𝑃_𝑇jets and to calculate the color factor ratio, events are generated using PYTHIA8 with CTEQ6L1 PDFs at√𝑠=7 TeV and 14 TeV. At each energy, 1 million events are generated in 4 bins of𝑝_𝑇in the pseudorapidity range|𝜂|<2. The bins in𝑝_𝑇are 50–300, 300–550, 550–800, and 800–1050 GeV. One million events are generated in each pseudorapidity range|𝜂|<1and1≤|𝜂|<2. One million events are also generated separately in each bin of jet pseudorapidity|𝜂|which are 0.0–0.5, 0.5–1.0, 1.0–1.5, and 1.5–2.0 for leading jets. Thus in total we have generated 7 million events for each energy for the present study.

Quark and gluon jets are unambiguously defined only in the leading order QCD where there are only two jets, that is, two partons in the final state. So a sample of dijet events in hard QCD 2→2 scattering events is defined by selecting the two jets leading in 𝑃_𝑇in each event with the following selection cuts:(i)𝑃^jet_𝑇>100 GeV,(ii)|Δ𝜙(𝑗₁,𝑗₂)−𝜋|<1.0(requiring the jets to be back-to-back in the azimuthal plane),(iii)|𝜂|<2.

For𝑘_𝑇algorithm, mean subjet multiplicity is measured for cone radius, 𝑅=0.6 and𝑦_cut=10⁻³.

5. Discrimination between Quark and Gluons Jets

To determine the ratio of color factor, jet substructure of gluon and quark initiated jets is studied in terms of subjet multiplicities. For the Monte Carlo PYTHIA8 events used here, the quark and gluon jets are identified using parton level information. The direction of each primary parton is determined after its perturbative evolution has terminated. The jet closest to the direction of the primary parton (quark/antiquark or gluon) tagged to be the quark/gluon jet, respectively. The two highest 𝑃_𝑇jets are identified as quark or gluon initiated jets via the tagging in the spatial coordinates and the kinematical cuts. Thus we implement these cuts by allowing the maximum distance in𝜂×𝜙space asΔ𝑅=(Δ𝜂)²+(Δ𝜙)²<0.4. The maximum relative deviation in transverse momentum between the jet and the associated parton with respect to the jet transverse momentum isΔ𝑃_𝑇/𝑃_𝑇(Δ𝑃_𝑇/𝑃_𝑇is abs (parton𝑃_𝑇−jet𝑃_𝑇)/jet𝑃_𝑇)<3.0. Figure 1 shows the ratio Δ𝑃_𝑇/𝑃_𝑇for quark and gluon jets matched, respectively, to partons for√𝑠=7 TeV and√𝑠=14 TeV. At each energy the matching of particle jets with the partons is found to be good as all the jets have the ratio less than 3.

Figure 1 

Distribution ofΔ𝑃𝑇/𝑃𝑇for the matched quark and gluon jets with𝑃jet𝑇>100 GeV at√𝑠=7 TeV compared to√𝑠=14 TeV for𝑘𝑇algorithm.

Figure 2 shows the fraction of gluon initiated jets via matching of the two particle jets leading in 𝑃_𝑇to the hard partons of the LO process as a function of jet 𝑃_𝑇for the two different pseudorapidity bins,|𝜂|<1and1≤|𝜂|<2. As expected the fraction of gluon jets decreases with increasing jet 𝑃_𝑇in both the cases. Also, quark jets make up a dominant fraction of the jets only at high jet transverse momenta.

Figure 2 

Fraction of gluon initiated jets differentiated via matching of the two particle leading jets to the hard partons of the leading-order process as a function of jet 𝑃𝑇from𝑘𝑇algorithm at√𝑠=7 TeV compared to√𝑠=14 TeV in the pseudorapidity bins of|𝜂|<1and1≤|𝜂|<2.

6. Systematic Uncertainties

The uncertainties in the subjet multiplicities are of two categories:(i)statistical uncertainties,(ii)systematic uncertainties.

For the present analysis using Monte Carlo samples, the uncertainties are mainly on account of the statistics which are found to be negligibly small. The systematic uncertainties arise due to the change in gluon fraction due to minimum 𝑃_𝑇imposed. This is measured as follows.

(i) Gluon Jet fraction: The uncertainty in gluon jet fraction introduces a systematic error on the subjet multiplicity. To calculate this, the gluon jet fraction is obtained as a function of minimum jet 𝑃_𝑇(GeV) by varying minimum jet 𝑃_𝑇from 90 to 110 GeV. This introduces an error of  ±0.02 on gluon fraction at√𝑠=7 TeV and±0.01 on gluon fraction at√𝑠=14 TeV. If the gluon fraction increases or decreases at both the energies, the change in ratio 𝑟 is not much effective. But the increase in gluon jet fraction at one energy and decrease at other or vice versa introduces a largest change in ratio 𝑟, giving minimum and maximum value of 𝑟. The difference of the calculated value 𝑟 given in Table 5 from the minimum value and maximum value of 𝑟 gives the systematic uncertainty due to gluon fraction as+0.08−0.05.

The same analysis for measuring 𝑟, when performed on the data from a detector such as CMS at LHC, would involve the following systematic errors also. In one such measurement with 36pb⁻¹data used for measuring the subjet multiplicity at√𝑠=7 TeV energy [10], we estimated the following uncertainties.

(ii) Jet-Energy calibration: The maximum systematic uncertainty in Jet-Energy calibration was estimated to be 5% depending upon jet 𝑃_𝑇and jet𝜂. This introduces an error of about 2% on the subjet multiplicity average value.

(iii) Jet-Energy resolution (JER): The uncertainty on the jet 𝑃_𝑇resolution (JER) was estimated to be less than 10% and this results in less than 0.25% uncertainty in the subjet multiplicity distributions.

(iv) Uncertainties due to the different physics models employed for the simulations: The uncertainty due to different physics models implemented in the Monte Carlo generators is estimated to be less than 3%.

Compounding these systematic errors (ii)–(iv), the maximum uncertainty in the subjet multiplicity of 3.28 at 7 TeV was estimated to be 0.10. The details can be found in [10]. We use the same percent value of each of the above mentioned systematic uncertainties in subjet multiplicities at the two energies which will be reflected in ratio 𝑟. The systematic errors on ratio 𝑟 are given in Table 6 and added in quadrature to have the total systematic uncertainty in the calculated value of the ratio 𝑟.

7. Results and Discussions

Figure 3 shows jet constituent multiplicity distributions for𝑘_𝑇algorithm at√𝑠=7 TeV and at√𝑠=14 TeV in the two pseudorapidity bins,|𝜂|<1and1≤|𝜂|<2. It may be observed that the average number of jet constituents is larger at 14 TeV. The subjet 𝑃_𝑇fractions for the two pseudorapidity regions and for the two energies are shown in Figure 4. In𝑘_𝑇algorithm, the procedure to resolve subjets is dependent on 𝑃_𝑇of jet and the 𝑃_𝑇of subjets in a particular jet is distributed in such a way that there is no sudden decrease. Thus as shown in Figure 4, we get monotonically decreasing values for subjet 𝑃_𝑇fraction.

Figure 3 

Distributions of the jet constituent multiplicity in particle jets having𝑃jet𝑇>100 GeV from𝑘𝑇algorithm at√𝑠=7 TeV compared to√𝑠=14 TeV in the pseudorapidity bins of|𝜂|<1and1≤|𝜂|<2.

Figure 4 

Distributions of the subjet 𝑃𝑇fraction in particle jets having𝑃jet𝑇>100 GeV from𝑘𝑇algorithm at√𝑠=7 TeV compared to√𝑠=14 TeV in the pseudorapidity bins of|𝜂|<1and1≤|𝜂|<2.

Figure 5 shows the subjet multiplicity distributions for jets matched to gluons in the two pseudorapidity bins and jets matched to quarks in the two pseudorapidity bins|𝜂|<1and for1≤|𝜂|<2. The mean values are given in Table 1. Average number of subjets in gluon jets is found to be larger than in the quark jets indicating a higher degree of collimation in quark jets.

Table 1 

Average subjet multiplicities for jets matched to gluons and quarks in the |𝜂| bins with 𝑃^jet_𝑇>100 GeV for 𝑘_𝑇 algorithm at √𝑠 = 7 TeV and √𝑠 = 14 TeV.

Figure 5 

Subjet multiplicity distributions for jets matched to gluons (top) and quarks (bottom), having𝑃jet𝑇>100 GeV, in the pseudorapidity bins of|𝜂|<1(left) and1≤|𝜂|<2(right) at√𝑠=7 TeV and√𝑠=14 TeV from𝑘𝑇algorithm.

Average subjet multiplicity,⟨𝑀⟩dependence on absolute jet pseudorapidity for√𝑠=7 TeV and at√𝑠=14 TeV for the whole𝑝_𝑇range, is shown in Figure 6. Figure 7 shows the subjet multiplicity distributions of jets as a function of the pseudorapidity bins (0.0–0.5, 0.5–1.0, 1.0–1.5, and 1.5–2.0) for four𝑝_𝑇intervals 50–300, 300–550, 550–800, and 800–1050 GeV. The values are given in Table 2. We observe that from inner to outer rapidities windows, with same rapidity window size, the subjet multiplicity tends to get smaller very slightly.

Table 2 

Average subjet multiplicities as a function of |𝜂| in different 𝑝_𝑇 bins, for jets with 𝑃^jet_𝑇> 100 GeV for 𝑘_𝑇 algorithm at √𝑠 = 7 TeV and √𝑠 = 14 TeV.

Figure 6 

Average subjet multiplicity⟨𝑀⟩as a function of absolute jet pseudorapidity|𝜂|for the whole range in𝑝𝑇at√𝑠=7 TeV and√𝑠=14 TeV for𝑘𝑇algorithm.

Figure 7 

Average subjet multiplicity⟨𝑀⟩as a function of absolute jet pseudorapidity|𝜂|in the four𝑝𝑇bins at√𝑠=7 TeV and√𝑠=14 TeV for𝑘𝑇algorithm.

Figure 8 shows the jet 𝑃_𝑇 versus ⟨𝑀⟩for pseudorapidity|𝜂|<2for quarks, gluons, and matched jets for the two energies under study. The mean values are given in Table 3.

Table 3 

Average subjet multiplicities as a function of 𝑃^jet_𝑇 (GeV) for mixed jets and jets matched to gluons and quarks, for 𝑘_𝑇 algorithm at √𝑠 = 7 TeV and √𝑠 = 14 TeV.

Figure 8 

Average subjet multiplicity⟨𝑀⟩as a function of jet 𝑃𝑇in the pseudorapidity range of|𝜂|<2at√𝑠=7 TeV and√𝑠=14 TeV for𝑘𝑇  algorithm.

The selected dijet sample consists of mixture of gluon and quark initiated jets as in the LO hard QCD 2→2 processes. Following the recipe from Section 3, we use (7) to separate these two types and obtain the average subjet multiplicities⟨𝑀⟩for gluon jets and quark jets in the five bins of jet𝑃_𝑇. The results are given in Table 5 (the weighted mean𝑀and the uncertainties are calculated by using𝑀=(∑𝑀_𝑖/error²_𝑖)/(∑1/error²_𝑖) and error²_𝑀=1/(∑1/error²_𝑖)). These values are then used to calculate the ratio 𝑟 from (8) as given in Table 5.

8. Summary

The subjet multiplicities in proton-proton collisions at√𝑠=7 TeV and at√𝑠=14 TeV are estimated in 14 million proton-proton collisions by selecting dijet samples simulated using PYTHIA8.

The jets and subjets have been resolved with the𝑘_𝑇jet algorithm for a jet size of 𝑅=0.6 and a subjet resolution cutoff𝑦_cut=10⁻³. The subjet multiplicity distribution and its average⟨𝑀⟩have been determined in the jet 𝑃_𝑇range of 100–1100 GeV for jet pseudorapidity|𝜂|<2. In the whole pseudorapidity range|𝜂|<2, the average subjet multiplicity⟨𝑀⟩for particle jets is found to decrease from 2.63±0.01  (stat.) down to 1.64±0.01  (stat.) for 7 TeV and from 2.76±0.01  (stat.) down to 1.75±0.01  (stat.) at 14 TeV with increasing jet 𝑃_𝑇. The average subjet multiplicity for jets matched to gluons, ⟨𝑀_𝑔⟩, is also found to decrease from 3.06±0.01  (stat.) down to 2.11±0.01  (stat.) for 7 TeV and from 3.07±0.01  (stat.) down to 2.11±0.01  (stat.) at 14 TeV with increase in jet 𝑃_𝑇from 100 GeV to 1100 GeV. The average subjet multiplicity for jets matched to quarks,⟨𝑀_𝑞⟩, is also found to decrease from 2.05±0.01  (stat.) down to 1.49±0.01 (stat.) for 7 TeV and from 2.13±0.01  (stat.) down to 1.52±0.01  (stat.) at 14 TeV with increase in jet 𝑃_𝑇.

This is a clear indication of a higher degree of collimation with rising jet 𝑃_𝑇. In Table 2, we can observe that from inner to outer rapidities the subjet multiplicity tends to get slightly smaller as well. Evidently, a direct conclusion on the gluon-quark fraction cannot be drawn since the subjet multiplicity not only depends on this fraction but also on details of the Multiple Parton Interactions (MPI), parton shower, and hadronization models. Exploiting data at two different center-of-mass energies, the model dependence can be potentially eliminated. This technique was applied for data at two center-of-mass energies at Tevatron, as described in [12] to measure the ratio 𝑟. This method has been exploited for present and future LHC energies to predict the value of ratio 𝑟. The values of average subjet multiplicities in quark and gluon jets in different jet 𝑃_𝑇bins are calculated from (7) using the values given in Tables 3 and 4 and the ratio from (8). The results are given in Table 5. The mean value of ratio 𝑟 is calculated by using maximum likelihood estimator of the subjet distributions as 𝑟=2.30±0.03  (stat.). The total systematic uncertainty from different sources comes out to be+0.28−0.27, given in Table 6. So the extracted value of ratio of subjet multiplicities of gluon jets and quark jets is 𝑟=2.30±0.03  (stat.)+0.28−0.27(sys.). This value converges to the ratio of the color factors,𝐶_𝐴/𝐶_𝐹=9/4. In [17], it is shown that the ratio of the multiplicities from quark and gluon jets should converge to the color factor ratio. The extracted value is in good agreement with the value predicted from the symmetry group SU(3) of QCD and also with the values measured at previous experiments listed in Table 7.

Table 4 

Fraction of gluon and quark jets in phase space |𝜂|<2 in different 𝑃^jet_𝑇 (GeV) bins for 𝑘_𝑇 algorithm.

Table 5 

Average subjet multiplicities (calculated using (7)) and ratio of subjet multiplicities (from (8)) in the phase space |𝜂|<2 in different 𝑃^jet_𝑇 (GeV) bins for 𝑘_𝑇 algorithm.

Table 6 

Systematic uncertainties on ratio of subjet multiplicities from different sources.

Table 7 

Color factor ratio for different experiments.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to acknowledge Klaus Rabbertz for his contributions in steering the work reported in [14, 15] from where the estimation of some errors has been adopted in the present paper.