Exploration of jet substructure using iterative declustering in pp and Pb-Pb collisions at LHC energies

The ALICE collaboration at the CERN LHC reports novel measurements of jet substructure in pp collisions at $\sqrt{s}$= 7 TeV and central Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Jet substructure of track-based jets is explored via iterative declustering and grooming techniques. We present the measurement of the momentum sharing of two-prong substructure exposed via grooming, the $z_{\rm{g}}$, and its dependence on the opening angle, in both pp and Pb-Pb collisions. We also present the first measurement of the distribution of the number of branches obtained in the iterative declustering of the jet, which is interpreted as the number of its hard splittings. In Pb-Pb collisions, we observe a suppression of symmetric splittings at large opening angles and an enhancement of splittings at small opening angles relative to pp collisions, with no significant modification of the number of splittings. The results are compared to predictions from various Monte Carlo event generators to test the role of important concepts in the evolution of the jet in the medium such as color coherence.


Introduction
The objective of the heavy-ion jet physics program at the LHC is to probe fundamental, microscopic properties of nuclear matter at high densities and temperatures. Jets provide well-calibrated probes of the dense medium created in heavy-ion collisions. In pp collisions, the production of jets and their substructure have been measured extensively and these measurements are well-reproduced by theoretical calculations based on perturbative QCD (pQCD) (see Refs. [1,2] and citations therein). Jets are produced in high-momentum transfer processes, which occur on time scales much shorter than the formation time of the Quark-Gluon Plasma (QGP) generated in heavy-ion collisions; the production rates of jets in heavy-ion collisions can therefore be calculated accurately using the same pQCD approaches as for pp collisions, after taking into account the effects of nuclear geometry and nuclear modification of parton distribution functions (PDFs) [3].
Jets traversing the QGP will interact via elastic and radiative processes which modify the reconstructed jet cross section and structure relative to jets in vacuum ("jet quenching") [4]. Jet quenching effects have been extensively observed in nuclear collisions at RHIC and LHC in measurements of inclusive production and correlations of high-p T hadrons and jets, including correlations of high-energy triggers (hadrons, photons, W and Z bosons, and jets) and reconstructed jets [5][6][7][8] as well as in the measurement of jet shapes [9][10][11][12][13][14]. Comparisons of these measurements to theoretical jet quenching calculations enable the determination of dynamical properties of the QGP, notably the transport parameterq [15].
More recently, the modification of the jet substructure due to jet quenching has been explored in heavyion collisions using tools developed for the measurement of jet substructure in pp collisions for QCD studies and Beyond Standard Model searches [2,16]. A key tool is iterative declustering, which subdivides jets into branches or splittings that can be projected onto the phase space of such splittings, called the Lund plane [17][18][19]. While the splitting map contains kinematic information of all splittings, techniques like grooming [20,21] can be applied to isolate a specific region of the splitting map according to different criteria such as mitigation of non-perturbative effects, enhancement of the jet quenching signal or simplification of perturbative calculations.
In this work we focus on Soft Drop (SD) grooming [21]. This technique selects the first splitting in the declustering process for which the subleading prong carries a fraction z of the momentum of the emitting prong larger than some value z cut , which in this analysis is set to z cut = 0.1. Note that this criterion selects a subset of the splittings. The grooming procedure removes soft radiation at large angles to expose a twoprong structure in the jet. The shared momentum fraction of those prongs is called z g , the groomed subjet momentum balance. The measurement of z g in vacuum is closely related to the Altarelli-Parisi splitting functions [21].
Theoretical considerations of the in-medium modification of z g can be found in [22][23][24][25]. A key physics ingredient in the theoretical calculations is color coherence [26]. This is the effect by which a color dipole cannot be resolved by the medium as two independent color charges if the opening angle of the dipole is small compared to a fundamental medium scale. If the dipole cannot be resolved, it will propagate through the medium as a single color charge. If color coherence is at work, there will be parts of the jet substructure that won't be resolved, leading to a reduced effective number of color charges and thus a reduced amount of energy loss in medium.
With the grooming technique we select a hard two-prong substructure. Then we inspect the dependence on the opening angle of the rate of such two-prong objects in medium relative to vacuum. We are interested in understanding whether large-angle splittings are more suppressed relative to vacuum than small-angle splittings, as one would expect if large-angle splittings are resolved by the medium and radiate in the medium incoherently. Previous measurements of z g by the CMS collaboration [27] show a modification in central Pb-Pb collisions relative to the pp reference whilst measurements performed at RHIC by the STAR collaboration showed no modification [28]. Those measurements did not scan the ∆R dependence and cover different intervals of the subleading prong energies that can bias towards different typical splitting formation times.
This work reports the measurement by the ALICE collaboration of z g , the shared momentum fraction of two-prong substructure, its dependence on the opening angle and n SD , the number of splittings satisfying the grooming condition obtained via the iterative declustering of the jet, in pp collisions at √ s = 7 TeV and central (0-10%) Pb-Pb collisions at √ s NN = 2.76 TeV.

Data sets and event selection
A detailed description of the ALICE detector and its performance can be found in Refs. [29,30]. The analysed pp data were collected during Run 1 of the LHC in 2010 with a collision centre-of-mass energy of √ s = 7 TeV using a minimum bias (MB) trigger. The MB trigger configuration is the same as described in Ref. [31]. The data from heavy-ion collisions were recorded in 2011 at √ s NN = 2.76 TeV. This analysis uses the 0-10% most-central Pb-Pb collisions selected by the online trigger based on the hit multiplicity measured in the forward V0 detectors [32]. The datasets and event selection are identical to Refs. [5,9]. After offline selection, the pp sample consists of 168 million events, while the Pb-Pb sample consists of 19 million events.
The analysis uses charged tracks reconstructed by the Inner Tracking System (ITS) [33] and Time Projection Chamber (TPC) [34] which both cover the full azimuth and pseudo-rapidity |η| < 0.9. Tracks are required to have transverse momentum 0.15 < p T < 100 GeV/c. The track selection is slightly different in the analysis of the 2010 and the 2011 data. The former uses a subclass of tracks with worse momentum resolution that is excluded from the latter [35].
In pp collisions, the tracking efficiency is approximately 80% for tracks with p T > 1 GeV/c, decreasing to roughly 56% at p T = 0.15 GeV/c, with a track momentum resolution of 1% for p T = 1 GeV/c and 4.1% for p T = 40 GeV/c [30,31,36]. In Pb-Pb collisions, the tracking efficiency is about 2 to 3% worse than in pp. The track p T resolution is about 1% at p T = 1 GeV/c and 2.5% for p T = 40 GeV/c. As a vacuum reference for the Pb-Pb measurements we use simulated pp collisions at √ s = 2.76 TeV, calculated using PYTHIA 6.425 (Perugia Tune 2011) [27] and embedded into real central Pb-Pb events at the detector level, to take into account the smearing by the background fluctuations. We use the embedding of PYTHIA-generated events instead of the embedding of real pp data measured at √ s = 2.76 TeV due to the limited size of the data sample. The PYTHIA MC describes well vacuum intrajet distributions [2]. In this paper, we validate the PYTHIA calculation by comparing it to jet substructure measurements in pp collisions at √ s = 7 TeV.

Jet reconstruction
Jets are reconstructed from charged tracks using the anti-k T algorithm [37] implemented in FastJet [38] with a jet resolution parameter of R = 0.4. The four-momenta of tracks are combined using the E-scheme recombination [38] where the pion mass is assumed for all reconstructed tracks. In order to ensure that all jet candidates are fully contained within the fiducial volume of the ALICE detector system, accepted jets were required to have their centroid constrained to |η jet | < 0.5.
The jet finding efficiency is 100% in the measured kinematic ranges. The jet energy instrumental resolution is similar for pp and Pb-Pb collisions, varying from 15% at p ch T,jet = 20 GeV/c to 25% at p ch T,jet = 100 GeV/c. The Jet Energy Scale (JES) uncertainty is dominated by the tracking efficiency uncertainty which is 4%.
In pp collisions, no correction for the underlying event is applied. In Pb-Pb collisions, the jet energy is partially adjusted for the effects of uncorrelated background using the constituent subtraction method [39]. Constituent subtraction corrects individual jet constituents by modifying their four-momentum. The momentum that is subtracted from the constituents is determined using the underlying event density, ρ, which is calculated by clustering the event into R = 0.2 jets using the k T algorithm [40,41] and taking the median jet p T density in the event. The two leading k T jets are removed before calculating the median, to suppress the contribution of true hard jets in the background estimation. The correction is applied such that the total momentum removed from the jet is equal to ρ × A j , where A j is the jet area. This background subtraction is applied both to the measured data and to the embedded PYTHIA reference.

Observables
Jet constituents are reclustered using the physical Cambridge/Aachen (CA) metric [42], leading to an angle-ordered shower. The declustering process consists of unwinding the clustering history step by step, always following the hardest branch. The first declustering step identifies the final subjet pair or branch that was merged. The second declustering step identifies the subjet pair that was merged into the leading subjet of the final step, etc. The coordinates of the subleading prong in the Lund Plane (log(z∆R), log(1/∆R)) are registered at each declustering step, where z is the fraction of momentum carried by the subleading prong z = min(p T,1 ,p T,2 ) p T,1 +p T,2 , with p T,1 and p T,2 being the momenta of the leading and subleading prongs, respectively, and ∆R the opening angle of the splitting.
The observable n SD is obtained by counting the number of splittings in the declustering process that satisfy the Soft Drop selection z > z cut , z cut = 0.1. The observable z g corresponds to the subjet momentum balance, z, of the first splitting satisfying the SD selection. Jets with n SD = 0 are labelled "untagged jets". The z g distribution is absolutely normalised, including the untagged jets in the normalisation. This choice of normalisation, used here for the first time, provides crucial information for quantitative comparison of jet substructure measurements in Pb-Pb and pp collisions since it allows the results to be interpreted in terms of not only a change of shape in the distribution but also in terms of net enhancement/suppression of the yield of splittings satisfying the SD condition in a given jet transverse momentum range.
The tracking system enables the measurement of subjets with angular separation smaller than 0.1 radians and a scan of the z g distribution in ranges of ∆R: ∆R < 0.1, ∆R > 0.1 and ∆R > 0.2.
For data from pp collisions, the correction of the detector effects was performed via unfolding. The results are presented in the jet momentum interval of 40 < p ch T,jet < 60 GeV/c, chosen to balance statistical precision and detector effects. In Pb-Pb collisions, the results are presented at detector-level, with the uncorrelated background subtracted on average from the jet p T and from the substructure observable. The vacuum reference is thus smeared by background fluctuations and instrumental effects. The Pb-Pb results are presented in the jet momentum range of 80 < p ch T,jet < 120 GeV/c, where uncorrelated background is negligible.

Corrections and systematic uncertainties
For data from pp collisions, the unfolding of instrumental effects is carried out using a four-dimensional response matrix that encodes the smearing of both jet p ch T and the substructure observable (shape part,ch , p part,ch T,jet , shape det,ch , p det,ch T,jet ), where "shape" denotes either z g or n SD . The upper index "part" refers to particle-level and "det" refers to detector-level quantities, obtained from simulations in which pp collisions are generated by PYTHIA (particle-level) and then passed through a GEANT3-based model [43] of the ALICE detector. We note that the particle-level jet finding is performed using the true particle masses so the unfolding corrects for the pion mass assumption at detector level.
To generate vacuum reference distributions for comparison to Pb-Pb results, which are not fully corrected, we superimpose detector-level PYTHIA events onto real Pb-Pb events. Consequently, no two-  Table 2: Relative systematic uncertainties on the measured distributions in Pb-Pb collisions for three selected jet shape intervals and one ∆R selection in the jet p ch T,jet interval of 80-120 GeV/c.
Tracking efficiency(%) 4.9 2.8 11.4 11.2 7.9 11.1 track effects are present, however their impact in data is negligible due to the large required number of clusters per track. The matching of particle-level and embedded jets is performed as described in [9]. The matching efficiency is consistent with unity for jets with p T above 30 GeV/c.
For pp collisions, Bayesian unfolding in two dimensions as implemented in the RooUnfold package [44] is used. The prior is the two-dimensional distribution (p part,ch T,jet , shape part,ch ) generated with PYTHIA. The default number of iterations chosen for z g and n SD is 4, which corresponds to the first iteration for which the refolded distributions agree with the corresponding raw distributions within 5%. A closure test was also carried out, in which two statistically independent Monte Carlo (MC) samples are used to fill the response and the pseudo-data. For this test, the unfolded solution agrees with the MC truth distribution within statistical uncertainties.
Unfolding of the distributions was attempted for the Pb-Pb case, but no convergence on a mathematically consistent solution was obtained. The reason is that for Pb-Pb collisions data, the response is strongly non-diagonal due to the presence of sub-leading prongs at large angles that are not correlated to particlelevel prongs. Strategies to suppress such uncorrelated secondary prongs are beyond the scope of this analysis.
The systematic uncertainties are determined by varying key aspects of the correction procedures for instrumental response and background fluctuations. The most significant components of the systematic uncertainties for z g and n SD are tabulated in Table 1 and 2. For pp collisions, the tracking efficiency uncertainty is ±4% [13]. The effect of this uncertainty on the substructure measurement is assessed by applying an additional track rejection of 4% at detector-level prior to jet finding. A new response is built and the unfolding is repeated, with the resulting variation in the unfolded solution symmetrised and taken as the systematic uncertainty. This is the largest contribution to the JES uncertainty. To estimate the regularisation uncertainty, the number of Bayesian iterations is varied by ±1 with respect to the default analysis value. The prior is varied by reweighting the response such that its particle-level projection matches the unfolded solution. The detector-level intervals in p T and the substructure observables are modified to determine what in the table is referred to as truncation uncertainty. The uncertainty labelled "Binning" in the tables corresponds to a variation in binning of both p T and substructure observables, subject to the constraint of at least 10 counts in the least populous bin to ensure the stability of the unfolding procedure.
In the case of Pb-Pb collisions, the evaluation of the uncertainty due to tracking efficiency is carried out similarly to the pp case. The z g measurement is done differentially in ranges of ∆R. The limits of the ∆R ranges were varied by ±10%, which corresponds approximately to the width of the distribution of the relative difference of particle-level and embedded-level ∆R in Pb-Pb collisions. The differences between PYTHIA and the unfolded pp distributions are taken into account when using PYTHIA as a reference for Pb-Pb measurements. This is done by reweighting the embedded PYTHIA reference so that its particle-level projection matches the unfolded pp p T,jet vs z g (or p T,jet vs n SD ) correlation. The difference between the reference smeared with the default and the reweighted response is assigned as the corresponding uncertainty.
In both the pp and Pb-Pb analyses, the uncertainties are added in quadrature. All the contributions to the overall uncertainty produce changes in a given interval of the distribution that are strongly anti-correlated with changes in a different interval, i.e., they induce changes in the shape of the observable. Figures 1 and 2 show fully corrected distributions of z g and n SD measured in pp collisions at √ s = 7 TeV for charged jets in the interval 40 < p ch T,jet < 60 GeV/c. The results are compared to distributions obtained from PYTHIA 6 (Perugia Tune 2011), from PYTHIA 6 + POWHEG [45], to consider the impact of NLO effects, and from the newer PYTHIA 8 (Tune 4C) [46].

Results
The z g distribution is well-described within systematic and statistical uncertainties by all the MC generators considered. As discussed above, untagged jets contribute to the normalisation of the distributions. The untagged contribution is not shown in Fig. 1, due to the suppressed zero on the horizontal axis, but is shown in Fig. 2 in the bin representing n SD = 0. Table 3 shows the tagged fraction for data and simulations. For pp (rightmost column), the untagged fraction is about 2%. The Monte Carlo distributions in Fig. 2 disagree with the data in the tails of the distribution. They have a significantly lower fraction of jets with no splittings (n SD = 0) than observed in data. The addition of POWHEG corrections to PYTHIA 6 shifts the distribution towards a larger number of splittings, as expected from NLO corrections which introduce an additional hard parton in the final state. Figure 3 shows z g distributions measured in central Pb-Pb collisions for various ranges of angular separation ∆R. The results are presented in the uncorrected transverse momentum range 80 ≤ p ch T,jet < 120 GeV/c and compared to the distribution of PYTHIA jets embedded into real 0-10% central Pb-Pb events. Figure 3 shows a larger difference between the measured Pb-Pb and embedded reference distributions for larger values of ∆R, indicating a relative suppression in the rate of symmetric splittings (z g ≈ 0.5) in central Pb-Pb collisions. However, due to the steeply falling z g distribution, the fraction of all jets that exhibit symmetric splittings is small, and this strong suppression corresponds to a suppression of only a few percent in the total rate of jets passing both the SD and angular cuts (c.f. Tab. 3). Conversely, in the small ∆R limit a small excess of splittings is observed in the data. Figure 3 also shows comparisons to predictions from the JEWEL event generator [47] and Hybrid model [48] calculations. The JEWEL simulations include the medium response from jet-medium interactions [49]. The theoretical predictions must be smeared to account for the detector effects as well as fluc-    tuations due to uncorrelated background. This smearing is performed by constructing a 6-dimensional response matrix by superimposing PYTHIA events at detector level to real 0-10% central Pb-Pb events. The 6-dimensional matrix maps every embedded jet from a given bin of (z part g , ∆R part , p part T,jet ) to (z det g , ∆R det , p det T,jet ). The smearing of the distributions significantly modifies the predictions and is essential for quantitative comparison of the measurements and calculations.
The models capture the qualitative trends of the data, namely the enhancement of the number of smallangle splittings and the suppression of the large-angle symmetric splittings. The fraction of jets not passing the SD selection is similar in the models and data. However discrepancies are observed in the angular selection. For instance the number of SD splittings that pass the angular cut of ∆R > 0.2 is the  The suppression of splittings at large opening angles is qualitatively expected from vacuum formation time and colour coherence arguments [24]. The wider the opening angle, the shorter the formation time of the splitting. This makes it more likely that the splitting propagates through, and is modified by, the medium. If coherence effects are at play in the medium then it is expected that splittings that are separated by more than the coherence angle will be more suppressed since they radiate energy independently. Figure 4 shows the comparison of n SD distributions from Pb-Pb measurements and the embedded PYTHIA reference. The data exhibit a shift towards lower number of splittings. The discrepancies between the distributions from PYTHIA and from pp collisions are incorporated as a part of the reference uncertainty via the reweighting procedure described above. The corresponding curves for the Hybrid model and JEWEL are also shown in the plot.
To explore the dependence of the n SD distribution on the fragmentation pattern, we also show a calculation in which the pp reference distribution is based solely on light-quark fragmentation. Since the quark fragmentation is harder, we see that the number of splittings peaks at lower values, in line with what we observe in the data. The smeared JEWEL and Hybrid model calculation agree with the qualitative trend of the data.
The trends indicate that the larger the opening angle, the more suppressed the splittings are, and this is qualitatively consistent with large-angle prongs being more resolved by the medium and thus more suppressed. The same process could lead to a reduction in the number of hard splittings as observed in Figure 4. However, it is worth noting that both the Hybrid and JEWEL models, in spite of their capturing of the general trends of the data, they do not incorporate the physics of color coherence and all the prongs in the jet lose energy incoherently. This points to a simpler interpretation of the results for instance in terms of formation times of the splittings and their interplay with the medium length. The vacuum formation time t f ≈ ω/k 2 T ≈ 1/(ω∆R 2 ), with ω and k T being the energy and relative transverse momentum of the radiated prong, is shorter for large-angle splittings, meaning that vacuum, large-angle splittings, will be produced mostly in the medium and their resulting prongs will be further modified by the medium. At large angles, the component of vacuum splittings that propagate in vacuum is less than at small angles, resulting in an enhanced contribution of medium-modifications compared to small-angle splittings.

Summary
This Letter presents the measurement of jet substructure using iterative declustering techniques in pp and Pb-Pb collisions at the LHC. We report distributions of n SD , the number of branches passing the soft drop selection, and z g , the shared momentum fraction of the two-prong substructure selected by the mass drop condition, differentially in ranges of splitting opening angle.
Generally, good agreement between distributions for pp collisions and vacuum calculations is found except for the fraction of untagged jets, which is underestimated by the models. In Pb-Pb collisions, a suppression of the z g distribution is observed at large angles relative to the vacuum reference whilst at low opening angles there is a hint of an enhancement. These observations are in qualitative agreement with the expected behaviour of two-prong objects in the case of coherent or decoherent energy loss [24] in the BMDPS-Z [50,51] framework. However, the models that are compared to the data do not implement color coherence and yet they capture the qualitative trends of the data. This suggests that other effects might drive the observed behaviour, for instance the interplay between formation time of the splittings and medium length.
The number of splittings obtained by iteratively declustering the hardest branch in the jet, n SD , is shifted towards lower values in Pb-Pb relative to the vacuum reference. This suggests that medium-induced radiation does not create new splittings that pass the SD cut. On the contrary, there is a hint of fewer splittings passing the SD cut, pointing to a harder, more quark-like fragmentation in Pb-Pb compared to pp collisions, in qualitative agreement with the trends observed for other jet shapes [9].
With these measurements, we have explored a region of the Lund plane delimited by the Soft Drop cut z > 0.1. Other regions of the phase space of splittings will be scanned systematically in the future with larger data samples.

Acknowledgements
The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The