Centrality dependence of the charged-particle multiplicity density at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV

The centrality dependence of the charged-particle multiplicity density at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV is presented. The charged-particle density normalized per participating nucleon pair increases by about a factor 2 from peripheral (70-80%) to central (0-5%) collisions. The centrality dependence is found to be similar to that observed at lower collision energies. The data are compared with models based on different mechanisms for particle production in nuclear collisions.

f ) × N coll , where N part is the number of participating nucleons, N coll is the number of binary nucleonnucleon collisions and f quantifies their relative contributions. The number of particles produced by each source is distributed according to a Negative Binomial Distribution, parametrized with µ and κ, where µ is the mean multiplicity per source and κ controls the large multiplicity tail.
In the Glauber calculation [16], the nuclear density for 208 Pb is modeled by a Woods-Saxon distribution for a spherical nucleus with a radius of 6.62 fm and a skin depth of 0.546 fm, based on data from low energy electron-nucleus scattering experiments [17]. A hard-sphere exclusion distance of 0.4 fm between nucleons is employed. Nuclear collisions are modeled by randomly displacing the two colliding nuclei in the transverse plane. Nucleons from each nucleus are assumed to collide if the transverse distance between them is less than the distance corresponding to the inelastic nucleon-nucleon cross section, estimated from interpolating data at different centre-of-mass energies [18] to be 64 ± 5 mb at √ s = 2.76 TeV.
The values of f , µ and κ are obtained from a fit to the measured VZERO amplitude distribution. The fit is restricted to amplitudes above a value corresponding to 88% of the hadronic cross section. In this region the trigger and event selection are fully efficient, and the contamination by electromagnetic processes is negligible. Centrality classes are determined by integrating the measured distribution above the cut, as shown in Fig. 1.
The determination of dN ch /dη is performed for each centrality class. The primary vertex position is extracted by correlating hits in the two SPD layers. All events in the sample corresponding to 0-80% of the hadronic cross section are found to have a well-defined primary vertex. To minimize edge effects at the limit of the SPD acceptance, we require |z vtx | < 7 cm for the reconstructed vertex, leading to a sample of about 49 000 events.
The measurement of the charged-particle multiplicity is based on the reconstruction of tracklets [2]. A tracklet candidate is defined as a pair of hits, one in each SPD layer. Using the reconstructed vertex as the origin, differences in azimuthal (∆ϕ, bending plane) and polar (∆θ , non-bending direction) angles for pairs of hits are calculated [19]. Tracklets are defined by hit combinations that satisfy a selection on the sum of the squares (δ 2 ) of ∆ϕ and ∆θ , each normalized to its estimated resolution (60 mrad for ∆ϕ and 25 sin 2 θ mrad for ∆θ ). The tolerance in ∆ϕ for tracklet reconstruction effectively selects charged particles with transverse momentum above 50 MeV/c. If multiple tracklet candidates share a hit, only the combination with the smallest δ 2 is kept.
The charged-particle pseudo-rapidity density dN ch /dη in |η| < 0.5 is obtained from the number of tracklets by applying a correction α × (1 − β ) in bins of pseudo-rapidity and z-position of the primary vertex. The factor α corrects for the acceptance and efficiency of a primary track to form a tracklet, and β reflects the fraction of background tracklets from uncorrelated hits. The fraction β is estimated by matching the tails of the data and background δ 2 distributions. The latter is obtained by selecting combinatorial tracklets from a sample of simulated events with similar SPD hit multiplicities generated with HIJING [20] and a GEANT3 [21] model of the detector response. The estimated background fraction varies from 1% in the most peripheral to 14% in the most central class.
The correction α is obtained as the ratio of the number of generated primary charged particles and the number of reconstructed tracklets, after subtraction of the combinatorial background. Thus, α includes the corrections for the geometrical acceptance, detector and reconstruction inefficiencies, contamination by weak decay products of strange particles, photon conversions, secondary interactions, and undetected particles with transverse momentum below 50 MeV/c. The correction is about 1.8 and varies little with centrality. Its magnitude is dominated by the effect of tracklet acceptance: the fraction of SPD channels active during data taking was 70% for the inner and 78% for the outer layer.
Systematic uncertainties on dN ch /dη are estimated as follows: for background subtraction, from 0.1% in the most peripheral to 2.0% in the most central class, by using an alternative method where fake hits are injected into real events; for particle composition, 1%, by changing the relative abundances of protons, pions, kaons by up to a factor of two; for contamination by weak decays, 1%, by changing the relative contribution of the yield of strange particles by a factor of two; for extrapolation to zero transverse momentum, 2%, by varying the estimated yield of particles at low transverse momentum by a factor of two; for dependence on event generator, 2%, by using quenched and unquenched versions of HIJING [20], as well as DPMJET [22] for calculating the corrections. The systematic uncertainty on dN ch /dη due to the centrality class definition is estimated as 6.2% for the most peripheral and 0.4% for the most central class, by using alternative centrality definitions based on track or SPD hit multiplicities, by using different ranges for the Glauber model fit, by defining cross-section classes integrating over the fit rather than directly over the data distributions, by changing the N part dependence of the particle production model to a power law, and by changing the nucleon-nucleon cross section and the parameters of the Woods-Saxon distribution within their estimated uncertainties and by changing the inter-nucleon exclusion distance by ±100%. All other sources of systematic errors considered (tracklet cuts, vertex cuts, material budget, detector efficiency, background events) were found to be negligible. The total systematic uncertainty on dN ch /dη amounts to 7.0% in the most peripheral and 3.8% in the most central class. A large part of this uncertainty, about 5.0% for the most peripheral and 2.5% for the most central class, is correlated among the different centrality classes. The dN ch /dη values obtained for nine centrality classes together with their systematic uncertainties are given in Table 1. As a cross-check of the centrality selection the dN ch /dη analysis was repeated using centrality cuts defined by slicing perpendicularly to the correlation between the energy deposited in the ZDC and the VZERO amplitude. The resulting dN ch /dη values differ by 3.5% in the most peripheral (70-80%) and by less than 2% in all the other classes from those obtained by using the VZERO selection alone, which is well within the systematic uncertainty. Independent cross-checks performed using tracks reconstructed in the TPC and ITS instead of tracklets yield compatible results.
In order to compare bulk particle production in different collision systems and at different energies, the charged-particle density is divided by the average number of participating nucleon pairs, N part /2, determined for each centrality class. The N part values are obtained using the Glauber calculation, by classifying events according to the impact parameter, without reference to a specific particle production   Table 1. The systematic uncertainty in the N part values is obtained by varying the parameters entering the Glauber calculation as described above. The geometrical N part values are consistent within uncertainties with the values extracted from the Glauber fit in each centrality class, and agree to better than 1% except for the 70-80% class where the difference is 3.5%. Theoretical descriptions of particle production in nuclear collisions fall into two broad categories: twocomponent models combining perturbative QCD processes (e.g. jets and mini-jets) with soft interactions, and saturation models with various parametrizations for the energy and centrality dependence of the saturation scale. In Fig. 3 we compare the measured (dN ch /dη) / N part /2 with model predictions. A calculation based on the two-component Dual Parton Model (DPMJET [10], with string fusion) exhibits a stronger rise with centrality than observed. The two-component HIJING 2.0 model [25], which has been tuned [11] 1 to high-energy pp [19,23] and central Pb-Pb data [2], reasonably describes the data. This model includes a strong impact parameter dependent gluon shadowing (s g ) which limits the rise of particle production with centrality. The remaining models show a weak dependence of multiplicity on centrality. They are all different implementations of the saturation picture, where the number of soft gluons available for scattering and particle production is reduced by nonlinear interactions and parton recombination. A geometrical scaling model with a strong dependence of the saturation scale on nuclear mass and collision energy [12] predicts a rather weak variation with centrality. The centrality dependence is well reproduced by saturation models [13] and [14] 1 , although the former overpredicts the magnitude.
In summary, the measurement of the centrality dependence of the charged-particle multiplicity density at mid-rapidity in Pb-Pb collisions at √ s NN = 2.76 TeV has been presented. The charged-particle density normalized per participating nucleon pair increases by about a factor 2 from peripheral (70-80%) to central (0-5%) collisions. The dependence of the multiplicity on centrality is strikingly similar for the data at √ s NN = 2.76 TeV and √ s NN = 0.2 TeV. Theoretical descriptions that include a moderation of the multiplicity evolution with centrality are favoured by the data.