System-size dependence of the charged-particle pseudorapidity density at $\sqrt{s_{\rm NN}} = 5.02$ TeV for pp, p-Pb, and Pb-Pb collisions

We present the first systematic comparison of the charged-particle pseudorapidity densities for three widely different collision systems, pp, p-Pb, and Pb-Pb, at the top energy of the Large Hadron Collider ($\sqrt{s_{\rm NN}} = 5.02$ TeV) measured over a wide pseudorapidity range (${-3.5<\eta<5}$), the widest possible among the four experiments at that facility. The systematic uncertainties are minimised since the measurements are recorded by the same experimental apparatus (ALICE). The distributions for p-Pb and Pb-Pb collisions are determined as a function of the centrality of the collisions, while results from pp collisions are reported for inelastic events with at least one charged particle at midrapidity. The charged-particle pseudorapidity densities are, under simple and robust assumptions, transformed to charged-particle rapidity densities. This allows for the calculation and the presentation of the evolution of the width of the rapidity distributions and of a lower bound on the Bjorken energy density, as a function of the number of participants in all three collision systems. We find a decreasing width of the particle production, and roughly a smooth ten fold increase in the energy density, as the system size grows, which is consistent with a gradually higher dense phase of matter.


Introduction
The number of charged particles produced in energetic nuclear collisions is an important indicator for the strong interaction processes that determine the particle production at the sub-nucleonic level.In particular, the production of charged particles is expected to reflect the number of quark and gluon collisions occurring during the initial stages of the reaction.The total number of particles produced also provides information on the energy transfer available from the initial colliding beams to particle production, as a consequence of nuclear stopping [1].In order to help unravel this complex scenario it is important to compare the particle production amongst collision systems of different sizes over a wide kinematic range.
We present the measured charged-particle pseudorapidity density, dN ch /dη , for pp, p-Pb, and Pb-Pb (previously published [2]) collisions at the same collision energy of √ s NN = 5.02 TeV in the nucleonnucleon centre-of-mass reference frame.This is, at present, the maximum available energy at CERN's Large Hadron Collider (LHC) for Pb-Pb collisions.The measurements were carried out using ALICE at LHC (for earlier dN ch /dη results see for example Refs.[3,4,5]).The three studied reactions have different characteristics probing widely different particle production yields and mechanisms.In Pb-Pb collisions, the total particle yield for central collisions is of the order 10 4 [2], and a strongly coupled plasma of quarks and gluons (sQGP) is formed [6,7,8,9], whose collective and transport properties are currently under intense study.On the other hand, pp collisions represent the simplest possible nuclear collision system, where the average total particle production is much smaller (≈ 80, by integrating the measured distributions), and is to first approximation much less subject to collective effects [10].The p-Pb system is intermediate to the other reactions, corresponding to the situation where a single nucleon probes the nucleons in a narrow cylinder of the target nucleus.The extent to which p-Pb is governed by the initial state cold nuclear matter of the lead ion or whether collective phenomena in the hot and dense medium play an important role is, at present, a matter under scrutiny by the community [10,11].
In this letter, we compare the three reactions and present the ratios of the charged-particle pseudorapidity density distributions (dN ch /dη ) of the more complex reactions to the pp distribution.Owing to ALICE's unique large acceptance in pseudorapidity, and using simple and robust assumptions, we transform the measured charged-particle pseudorapidity density distributions into charged-particle rapidity density distributions (dN ch /dy).This allows us to calculate the width of the rapidity distributions as a function of the number of participating nucleons.The parameters of the transformation also allow us to estimate a lower bound on the energy density using the well-known formula from Bjorken [12].An energy density exceeding the critical energy density of roughly 1 GeV/fm 3 [13] is a necessary condition for the formation of deconfined matter of quarks and gluons, and thus it is of the utmost interest to understand the development of these energy densities across different collision systems.
2 Experimental set-up, data sample, analysis method, systematic uncertainties A detailed description of the ALICE detector and its performance can be found elsewhere [14,15].The present analysis uses the Silicon Pixel Detector (SPD) to determine the pseudorapidity densities in the range −2 < η < 2 and the Forward Multiplicity Detector (FMD) in the ranges −3.5 < η < −1.8 and 1.8 < η < 5.The V0, comprised of two plastic scintillator discs covering −3.7 < η < −1.7 (V0C) and 2.8 < η < 5.1 (V0A), and the ZDC, two zero-degree calorimeters located 112.5 m from the interaction point, measurements determine the collision centrality and are used for offline event selection [2].
The results presented are based on data from collisions at a centre-of-mass energy per nucleon pair of √ s NN = 5.02 TeV as collected by ALICE during LHC Run 1 (2013) for p-Pb, and during Run 2 (2015) for pp and Pb-Pb.The FMD suffered high levels of background noise during the 2016 p-Pb campaign, due to the high collision rate, and this data is therefore not used for the present analysis.About 10 5 events with a minimum bias trigger requirement [2] were analysed in the centrality range from 0% to 90% and System-size dependence of dN ch /dη at √ s NN = 5.02 TeV ALICE Collaboration 0% to 100% of the visible cross section for Pb-Pb and p-Pb collisions, respectively.The minimum bias trigger for p-Pb and Pb-Pb collisions in ALICE was defined as a coincidence between the V0A and V0C sides of the V0 detector.
The data from the p-Pb collisions were taken in two beam configurations: one where the lead ion travelled toward positive pseudorapidity and one where it travelled toward negative pseudorapidity.The results from the latter collisions are mirrored around η = 0.The centre-of-mass frame in p-Pb collisions does not coincide with the laboratory frame, due to the single magnetic field in the LHC, and thus the rapidity of the centre-of-mass is y CM = ±0.465for the two directions, respectively, in the laboratory frame.For this reason, pseudorapidity, calculated with respect to the laboratory frame, is denoted η lab whenever p-Pb results are presented.
Likewise, for the pp collisions, about 10 5 events with coincidence between V0A and V0C and at least one charged particle in |η| < 1 were analysed.By requiring at least one charged particle at midrapidity, the so-called INEL>0 event class, the systematic uncertainty, related to the absolute normalisation to the full inelastic cross section, is reduced, while still sampling a large fraction (> 75%) of the hadronic cross section [16,17].
The standard ALICE event selection [18] and centrality estimator based on the V0 amplitude [19,20] are used in this analysis.The event selection consists of: a) exclusion of background events using the timing information from the ZDC (for Pb-Pb and p-Pb, e.g., beam-gas interactions) and V0 detectors, b) verification of the trigger conditions, and c) a reconstructed position of the collision (primary vertex).
In Pb-Pb collisions, centrality is obtained from the sum amplitude in both V0 detector arrays (V0M).For p-Pb only the amplitude in the array on the lead-going side (V0A or V0C) is used.In Pb-Pb collisions, the 10% most peripheral collisions have substantial contributions from electromagnetic processes and are therefore not included in the results presented here [19].
A primary charged particle is defined as a charged particle with a mean proper lifetime τ larger than 1 cm/c, which is either a) produced directly in the interaction, or b) from decays of particles with τ smaller than 1 cm/c [21].All quantities reported here are for primary, charged particles, though "primary" is omitted in the following for brevity.
The analysis method is identical to that of previous publications [2]: the measurement of the chargedparticle pseudorapidity density at midrapidity is obtained from counting particle trajectories determined using the two layers of the SPD.The SPD has a lower transverse momentum acceptance of 50 MeV/c, and the yield is extrapolated down to p T = 0 MeV/c via simulations.In the forward regions, the measurement is provided by the analysis of the deposited energy signal in the FMD and a statistical method is employed to calculate the inclusive number of charged particles.A data-driven correction [22], based on separate measurements exploiting displaced collision vertices, is applied to remove the background from secondary particles.
Systematic uncertainty estimations for the midrapidity measurements are detailed elsewhere [2,16,20], and are from background suppression, transverse momentum extrapolation, weak decays, and simulations.The estimates are obtained through variation of thresholds and simulation studies.For pp (p-Pb), the total systematic uncertainty amounts to 1.5% (2.7%) over the whole pseudorapidity range; while for Pb-Pb the total systematic uncertainty is 2.6% at η = 0 and 2.9% at |η| = 2.The systematic uncertainty is mostly correlated over pseudorapidity for |η| < 2, and largely independent of centrality.The uncertainty in the forward region, estimated via variations of thresholds and simulation studies, is the same for all collision systems and is uncorrelated across η, amounting to 6.9% for η > 3.5 and 6.4% elsewhere within the forward regions [22].centrality class definition in Pb-Pb, is estimated to vary from 0.6% for the most central to 9.5% for the most peripheral class [23].The 80% to 90% centrality class has residual contamination from electromagnetic processes as detailed elsewhere [19], which gives rise to an additional 4% systematic uncertainty in the measurements.No overall systematic uncertainty has been estimated for p-Pb collisions, as the centrality selection in that collision system is inherently difficult to map to the underlying dynamics of the collisions [20].

Results
Figure 1 shows the measured pseudorapidity densities in pp, and in central p-Pb, and the previously published results for Pb-Pb [2] collisions at √ s NN = 5.02 TeV for primary particles.
Figure 2 shows, as a function of centrality, the measured charged-particle pseudorapidity densities for p-Pb collisions at √ s NN = 5.02 TeV.The strategy of centrality selection for proton on nucleus reactions is explained elsewhere [20].The ALICE Collaboration has previously presented dN ch /dη for Pb-Pb collisions at this energy [2].
In Fig. 3, the charged-particle pseudorapidity densities in p-Pb and Pb-Pb reactions are divided by the pp distributions corresponding to the INEL>0 trigger class.The ratio is r X = (dN ch /dη| X )/(dN ch /dη| pp ), where X labels p-Pb and Pb-Pb collisions, in centrality classes, as a function of pseudorapidity.In the ratios, systematic uncertainties, of common origin, are partially cancelled, and, as an estimate, the magnitude of the resulting systematic uncertainties are given only by the uncertainties in the dN ch /dη| X measurements, since the uncertainties are independent of the collision system.In p-Pb collisions the rapidity of the centre-of-mass is non-zero, which is not taken into account in the ratios.2) to the data (discussed in Section 4).rapidity, which is not possible to perform reliably with the ALICE apparatus.
The ratio of the p-Pb relative to the pp distributions increases with pseudorapidity from the p-going to the Pb-going direction for central collisions, which Brodsky et al and Adil et al [25,26] suggest is a sign of scaling of the pp distribution with the increasing number of participants as the lead nucleus is probed by the incident proton, and thus independent proton-nucleon scatterings on the lead-ion side.A similar scaling, however, does not hold for the Pb-Pb reaction.The ratios cannot be obtained by simple scaling of the elementary pp distributions.Instead, the ratio of the Pb-Pb relative to the pp distributions exhibits an enhancement of particle production around midrapidity for the more central collisions which is indicative of the formation of the sQGP [7].Likewise, r pPb increases for all but the two most peripheral centrality classes as η lab → 3.In Pb-Pb collisions it is seen that the various mechanisms behind the pseudorapidity distributions are more transversely directed than in pp collisions by the increase of r PbPb as |η| → 0 4 Rapidity and energy-density dependence on system size and discussion It has been shown that the charged-particle rapidity density (dN ch /dy) in Pb-Pb collisions, to a good accuracy, follows a normal distribution over the considered rapidity interval (|y| 5) [2,27].Those results relied on calculating the average Jacobian dN ch /dy = J = β using the full p T spectra, at midrapidity, of charged pions and kaons as well as protons and antiprotons.Here, we use the approximation where ϑ is the polar angle of emission, and identify a = p T /m with an effective ratio of transverse momentum over mass.With this, the effective Jacobian can be written as We further make the ansatz that dN ch /dy is normal distributed for symmetric collision systems (pp and Pb-Pb), so that dN ch /dη can be parameterised as  where A and σ are the total integral and width of the distribution, respectively, and y the rapidity in the centre-of-mass frame.Motivated by the observed approximate linearity of r pPb (see lower panel of Fig. 3), we replace A with (αy + A) for the asymmetric system (p-Pb) and parameterise dN ch /dη lab as The functions f and g defined in Eq. ( 1) and Eq. ( 2), respectively, describe the measurements within the measured region with χ 2 per degrees of freedom (ν) in the range of 0.1 to 0.5.The small χ 2 /ν values are a consequence of the relatively large uncorrelated systematic uncertainties on the measurements.That is, the charged-particle distributions for pp, p-Pb, and Pb-Pb collisions at √ s NN = 5.02 TeV follow a normal distribution in rapidity, with free parameters A, a, σ , and α in the asymmetric case.
The top panel of Fig. 4 shows the best-fit parameter values of the normal width (σ dN ch /dy ) for all three collision systems as a function of the average number of participating nucleons ( N part ) calculated using a Glauber model [28].The best-fit parameters are found taking statistical and uncorrelated systematic uncertainties into account.The result using the above procedure, for the most central Pb-Pb collisions, is found to be compatible with previous results extracted by unfolding with the mean Jacobian estimated from transverse momentum spectra [2].The open points (crosses) and dashed lines on the figure are from evaluations of Eq. ( 1) and Eq. ( 2), and direct calculations of σ dN ch /dy , respectively, using model System-size dependence of dN ch /dη at √ s NN = 5.02 TeV ALICE Collaboration N part calculations with EPOS-LHC [29].EPOS-LHC was chosen as it provides predictions for all three collision systems.The parameterisation, in terms of the two functions, of this model calculation generally reproduces the widths of the charged-particle rapidity densities, except in the asymmetric case where a direct evaluation of the standard deviation is less motivated.
The general trend is that the widths decrease as N part increases, consistent with the behaviour of the r PbPb ratios.Notably, the width of the dN ch /dy distributions in p-Pb and Pb-Pb, for low number of participant nucleons in the collisions, approaches the width of the pp distribution, which, presumably, is dominated by kinematic and phase space constraints.
The lower panel of Fig. 4 shows the dependence of a on the average number of participants.The righthand ordinate is the same, but multiplied by the average mass m = (0.215 ± 0.001) GeV/c 2 estimated from measurements of identified particles in Pb-Pb collisions at √ s NN = 2.76 TeV [30].To better understand the parameter a, this parameter extracted from the EPOS-LHC calculations, using the above procedure, is also shown in the figure.The dotted lines show the average p T /m predicted by EPOS-System-size dependence of dN ch /dη at √ s NN = 5.02 TeV ALICE Collaboration LHC [29].The EPOS-LHC calculations indicate that the extracted effective transverse momentum to mass ratio a is consistently smaller than the ratio of the average transverse momentum to the average mass.Thus a gives a lower bound on p T / m .
We can estimate the energy density that is reached in the collisions as a function of the number of participants for the three systems.A conventional approach is to use the model originally proposed by Bjorken [12] in which the energy density (ε Bj ) depends on the rapidity density of particles and the volume of a longitudinal cylinder with cross sectional area determined by the overlap between the colliding partners and length determined by a characteristic particle formation time Here, S T ≈ πR 2 ≈ πN 2/3 part is the transverse area spanned by the participating nucleons, dE T /dy is the transverse-energy rapidity density, and τ is the formation time.While a formation time of τ = 1 fm/c is often assumed, it is left as a free parameter here.With m T = m 1 + ( p T / m ) 2 , the transverseenergy rapidity density can be approximated by where f total = 0.55 ± 0.01, the ratio of charged particles to all particles [31], accounts for neutral particles not measured in the experiment, and is assumed the same for all collision systems.Substituting the derived dN ch /dy and the effective a = p T /m p T / m results in a lower bound estimate for the Bjorken energy density (ε LB ) where a and m are as in the top panel of Fig. 4.
The transverse area S T is estimated in a numerical Glauber model [32,33] as shown in Fig. 5.We consider two extremes for the transverse area spanned by the participating nucleons: a) the exclusive (or direct) overlap between participating nucleons, ∩ and open markers in Fig. 5, and b) the inclusive (or full) area of all participating nucleons, ∪ and full markers in Fig. Figure 6 shows the lower-bound energy density estimate, ε LB τ ≤ ε Bj τ, as a function of the number of participants, which reaches values between 10 and 20 GeV/(fm 2 c) in the most central Pb-Pb collisions.The uncertainties are from standard error propagation of Eq. ( 3) of uncertainties on the best-fit parameter values, the number of participants, mean mass, and f total .A rise from roughly 1 GeV/(fm 2 c) to over 10 GeV/(fm 2 c) is observed if the transverse area is assumed to be the inclusive area of participating nucleons.This trend is illustrated by a power-law (CN p part ) fit to the data in the figure, with the parameter values C = (0.8 ± 0.3) GeV/(fm 2 c) and p = 0.44 ± 0.08.On the other hand, if the transverse area is assumed to be the smaller exclusive overlap area, we observe a substantially larger lower bound on the energy density, but a less dramatic increase with increasing number of participating nucleons.Also shown in the figure are estimates of the Bjorken energy density ε Bj τ for Pb-Pb reactions at √ s NN = 2.76 TeV [31].
These results where obtained from measurements of the transverse energy in the collisions and using the inclusive estimate of the transverse area S T .The trend of the √ s NN = 5.02 TeV results are similar to these earlier results.Bearing in mind that for the largest LHC collision energy we show a lower bound estimate of the energy density in Fig. 6, we find a likely overall increase in the energy density from √ s NN = 2.76 TeV to 5.02 TeV.

Summary and conclusions
We have measured the charged particle pseudorapidity density in pp, p-Pb, and Pb-Pb collisions at √ s NN = 5.02 TeV over the widest possible pseudorapidity range available at the LHC.The distributions where determined using the same experimental apparatus and methods, and systematic uncertainties have been minimised to within the capabilities of the set-up.While the particle production in central Pb-Pb collisions clearly exhibits an enhancement as compared to pp collisions, particle production in p-Pb collisions is consistent with dominantly incoherent nucleon-nucleon collisions.By transforming the measured pseudorapidity distributions to rapidity distributions we have obtained systematic trends for the width of the rapidity distributions and a lower bound on the energy density, which shows a clear scaling behaviour as a function of the average number of participant nucleons.The decreasing width of the deduced rapidity distributions with increasing participant number suggests that the kinematic spread of particles, including longitudinal degrees of freedom, is reduced due to interactions in the early stages of the collisions.This is also reflected in the accompanying growth of the energy density.

Figure 1 :
Figure1: Charged-particle pseudorapidity density in Pb-Pb[2] and p-Pb for the 5% most central collisions, and for pp collisions with INEL>0 trigger class.For symmetric collision systems (Pb-Pb and pp) the data has been symmetrised around η = 0 and points for η > 3.5 have been reflected around η = 0.The boxes around the points and to the right reflect the uncorrelated and correlated, with respect to pseudorapidity, systematic uncertainty, respectively.The relative correlated, normalisation, uncertainties are evaluated at dN ch /dη| η=0 .The lines show fits of Eq. (1) (Pb-Pb and pp) and Eq.(2) (p-Pb) to the data (discussed in Section 4).Please note that the ordinate has been cut twice to accommodate for the very different ranges of the charged-particle pseudorapidity densities.

Figure 2 :
Figure 2: Charged-particle pseudorapidity density in p-Pb collisions at √ s NN = 5.02 TeV in seven centrality classes based on the V0A and V0C estimators.The lines are obtained using a fit of a scaled, normal distribution in rapidity Eq. (2) to the data (discussed in Section 4).

Figure 3 :
Figure 3: Ratio r X of the charged-particle pseudorapidity density in Pb-Pb (top) and p-Pb (bottom) in different centrality classes to the charged-particle pseudorapidity density in pp in the INEL>0 event class.Note, for Pb-Pb η lab is the same as the centre-of-mass pseudorapidity.

Figure 4 :
Figure 4: The width (top) and effective p T /m (bottom) fit parameters as a function of the mean number of participants in pp, p-Pb, and Pb-Pb collisions at √ s NN = 5.02 TeV.Vertical uncertainties are the standard error on the best-fit parameter values, while horizontal uncertainties reflect the uncertainty on N part from the Glauber calculations.Also shown are similar fit parameters from the same parameterisation of EPOS-LHC calculations as well as direct calculations of the standard deviation of the dN ch /dy distributions and the p T / m ratio from the EPOS-LHC calculations.

Figure 5 :
Figure 5: The transverse area S T as calculated in a numerical Glauber model for two extreme cases: a) only the exclusive overlap of nucleons is considered (∩, open markers) and b) the inclusive area of participating nucleons contribute (∪, closed markers) in both p-Pb and Pb-Pb at √ s NN = 5.02 TeV.
5. Estimate of the lower bound on the Bjorken transverse energy density in pp, p-Pb, and Pb-Pb collisions at √ s NN = 5.02 TeV, considering the exclusive (∩, open markers) and inclusive (∪, full markers) overlap area S T of the nucleons.The expression CN part p is fitted to case ∪, and we find C = (0.8 ± 0.3) GeV/(fm 2 c) and p = 0.44 ± 0.08.Also shown is an estimate, via dE T /dy, of ε Bj from Pb-Pb collisions at √ s NN = 2.76 TeV (stars with uncertainty band) [31].
Both observations System-size dependence of dN ch /dη at √ s NN = 5.02 TeV ALICE Collaboration are consistent with the gradual establishment of a high-density phase of matter with increasing size of the collision domain.System-size dependence of dN ch /dη at √ s NN = 5.02 TeV ALICE Collaboration of Nuclear Physics (DOE NP), United States of America.In addition, individual groups or members have received support from: Marie Skłodowska Curie, Strong 2020 -Horizon 2020 (grant nos.824093, 896850), European Union; Academy of Finland (Center of Excellence in Quark Matter) (grant nos.346327, 346328), Finland.