Energy and centrality dependence of particle multiplicity in heavy ion collisions from s NN − − − √ = 20 to 2760 GeV
EEnergy and centrality dependence of particle multiplicity inheavy ion collisions from √ s NN = 20 to 2760 GeV Leo Zhou ∗ and George S.F. Stephans † Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: November 9, 2018)The centrality dependence of midrapidity charged-particle multiplicities at a nucleon-nucleoncenter-of-mass energy of 2.76 TeV from CMS are compared to PHOBOS data at 200 and 19.6 GeV.The results are first fitted with a two-component model which parameterizes the separate contri-butions of nucleon participants and nucleon-nucleon collisions. A more direct comparison involvesratios of multiplicity densities per participant pair between the different collision energies. Theresults support and extend earlier indications that the influences of centrality and collision energyon midrapidity charged-particle multiplicities are to a large degree independent.
PACS numbers: 25.75.Dw
I. INTRODUCTION
With the start of operations at the Large Hadron Col-lider (LHC), the center-of-mass energies of data for heavyion collisions now extend up to √ s NN = 2.76 TeV. Oneof the most basic observables which can be used to un-derstand the hot and dense systems formed in such inter-actions of ultrarelativistic nuclei is the charged-particlemultiplicity density ( dN ch /dη ) in the midrapidity region.The pseudorapidity, defined as η = − ln[tan( θ/ θ is the polar angle with respect to the beam direction.Furthermore, the dependences of charged-particle den-sities on centrality (a percentile measure of the impactparameter of the collisions) as well as on collision energyare important in understanding how both “hard” and“soft” interactions contribute to particle production inthe collisions. Results for dN ch /dη in PbPb collisions at √ s NN = 2.76 TeV have been extracted at the LHC. Inthis paper, these new LHC data are compared with thosefrom AuAu collisions at √ s NN = 200 and 19.6 GeV usingresults from the Relativistic Heavy Ion Collider (RHIC).Similar comparisons of charged-particle multiplicitiesin heavy ion collisions have been performed using RHICdata from the PHOBOS detector at √ s NN values of 19.6,62.4, 130, and 200 GeV [1–3]. An overview of these andall other PHOBOS multiplicity results can be found inRef. [4]. Since the centrality dependence was found tobe essentially identical within uncertainties for all fourRHIC energies, only the two most extreme ones are in-cluded in the present analysis.These earlier analyses used two approaches to studythe dependence of dN ch /dη on centrality at different col-lision energies:1. Parameterizing the centrality dependence using a ∗ [email protected] † [email protected] two-component model proposed in Ref. [5]: dN ch dη = n pp (cid:16) (1 − x ) (cid:104) N part (cid:105) x (cid:104) N coll (cid:105) (cid:17) (1)where (cid:104) N part (cid:105) and (cid:104) N coll (cid:105) are respectively the aver-age numbers of participating nucleons and binarynucleon-nucleon collisions in bins of heavy ion inter-action centrality. In this model, the fit parameter n pp represents the average charged-particle multi-plicity in a single nucleon-nucleon collision, while x signifies the relative contribution of so-called “soft”and “hard” parton interactions. In heavy ion colli-sions, “hard” scatterings, characterized by the pro-duction of large transverse momentum ( p T ), are ex-pected to scale with N coll . In contrast, productionof lower p T particles is expected to scale with N part .The variation of the fit values could then be studiedas a function of collision energy. In a recent studyby the PHENIX Collaboration at RHIC, submittedafter this present work, the centrality dependencedescribed by Eq. (1) with x = 0 .
08 is found to bevery closely equivalent to a linear dependence usinga constituent-quark participant model [6].2. Computing the pairwise ratios between themultiplicity densities per participant pairs, dN ch /dη/ (cid:104) N part / (cid:105) , in corresponding centralitypercentile bins at the various energies.In the second approach, in order to address the factthat (cid:104) N part (cid:105) is different in respective percentile central-ity bins for the two energies, the ratios were calculatedin two ways. One was directly computing the ratios of dN ch /dη/ (cid:104) N part / (cid:105) in the same centrality bin at the twoenergies but attributing this ratio to the average of thetwo (cid:104) N part (cid:105) values. The other was rebinning the rawcollision data at one of the energies with new centralitypercentile cutoffs so that (cid:104) N part (cid:105) was close to the same forthe corresponding bins in both data sets. Reference [1]found that both types of ratio calculations “are in agree-ment, even within the significantly reduced systematicerrors.” a r X i v : . [ nu c l - e x ] J u l In this paper, both the first approach and a slightlymodified version of the second approach are employedfor the pairwise comparisons of collisions at three dif-ferent center-of-mass energies from the LHC and RHIC.It was deemed necessary to modify the second approachbecause the differences between (cid:104) N part (cid:105) in correspondingpercentile centrality bins at the LHC and RHIC energiesare quite large and rebinning to accommodate matching (cid:104) N part (cid:105) would require significant reanalysis of the rawdata sets. Instead, a more efficient method was chosen,in which multiplicity ratios were calculated by linearly in-terpolating the dN ch /dη/ (cid:104) N part / (cid:105) at one energy to the (cid:104) N part (cid:105) of each centrality bin in the other data set. II. METHOD OF ANALYSISA. Data Source
RHIC data for midrapidity charged-particle mul-tiplicity have been presented by the BRAHMS [7],PHENIX [8], PHOBOS [4, 9], and STAR [10] Collabo-rations. Since the PHOBOS Collaboration provides themost extensive set of published multiplicity results, itwas chosen for this analysis over the other RHIC exper-iments. The PHOBOS data for charged-particle mul-tiplicities in AuAu collisions at 19.6 and 200 GeV aretaken from Refs. [1] and [2]. These PHOBOS midrapiditycharged-particle multiplicity data are averaged over thepseudorapidity region | η | <
1. In addition, Ref. [1] liststhe charged-particle multiplicity data for inelastic p (¯ p )+ p collisions at 200 GeV ( dN ch /dη = 2 . ± .
08) and an in-terpolated value at 19.6 GeV ( dN ch /dη = 1 . ± . dN ch /dη for 200 GeV pp collisions directly measuredby PHOBOS [4] is consistent with 2.29 but has a largeruncertainty.The PbPb collision data at 2.76 TeV from CMS aretaken from Ref. [11]. While the ALICE [12, 13] andATLAS [14] Collaborations at the LHC also have heavyion collision data at the same energy, only CMS datawas used in this comparison with lower energy resultsfrom PHOBOS. The CMS heavy ion data extend tomore peripheral events, and the (cid:104) N part (cid:105) values from CMSwere extracted with a procedure similar to that used byPHOBOS. Furthermore, the ALICE, CMS, and ATLASheavy ion data for charged-particle multiplicity are con-sistent over the centrality range where they overlap [14],and so the conclusions presented here do not depend onthe choice of LHC data set. Notably, the qualitative sim-ilarity in the centrality dependence of multiplicity datafrom the LHC and the highest RHIC energy was firstmentioned by Ref. [13]. Both the ALICE and CMS pp data (discussed below), as well as the CMS PbPb datawere averaged over | η | < . pp colli-sions at 2.76 TeV. The pp data from CMS with theclosest energy are the measured non-single-diffractive (NSD) dN ch /dη = 4 . ± . ± . dN ch /dη = 3 . ± . +0 . − . (syst.) for inelastic pp collisions, and dN ch /dη = 4 . ± . +0 . − . (syst.) for NSD pp colli-sions [16]. These three data points were used to estimatea 2.36 TeV inelastic pp value from CMS, which was thenextrapolated to 2.76 TeV using the √ s NN dependence ofcharged-particle multiplicity density given in Ref. [17].Since the ALICE and CMS multiplicities for NSD eventsagree to better than 1% and the extrapolation from 2.36to 2.76 TeV is an increase of just over 3%, this procedurefor getting a comparable inelastic pp multiplicity valueat 2.76 TeV does not introduce significant additional un-certainty.The reported uncertainties in dN ch /dη/ (cid:104) N part / (cid:105) datacontain a combination of “slope” and “scale” uncertain-ties added in quadrature. The scale uncertainty is di-rectly proportional to the value of dN ch /dη/ (cid:104) N part / (cid:105) ,whereas the slope uncertainty accounts for variations ofindividual data points. The latter uncertainty wouldpredominantly shift all points by an amount that variessmoothly with centrality, although there is a small com-ponent that varies point to point. For the fits to Eq. (1),scale uncertainties enter into the systematic uncertaintyin n pp but have essentially no impact on the uncertaintyof x . For the ratio comparison, the scale uncertaintiescan be ignored when studying the shape of the centralitydependence. B. Two-Component Fit Comparison
In addition to (cid:104) N part (cid:105) , values of (cid:104) N coll (cid:105) are requiredfor comparison of PHOBOS and CMS using the two-component model given by Eq. (1). Reference [1] usedthe parameterization N coll = A × N αpart (found by fit-ting the results of a Glauber model calculation) with( A, α ) = (0 . , .
32) and (0 . , .
37) for 19.6 and 200GeV, respectively. The (cid:104) N coll (cid:105) values for 2.76 TeV werethe same as those used by CMS [11].The systematic uncertainties in the fit parameters, x (the fraction representing the relative contribution from“hard processes”) and n pp (the average multiplicity from pp collisions), are estimated by reperforming the fit withthe uncertainties added to or subtracted from the datapoints and looking at the resulting variations in fit re-sults. The first fit used the entire available data set ateach energy. In order to make the most direct possiblecomparison of fit results, a second fit was performed at200 GeV and 2.76 TeV with an N part range restricted tothat available in the 19.6 GeV data set. C. Ratio Comparison
In order to extract direct ratios of multiplicities, it isfirst necessary to interpolate the data for one of the en- part N h / d c h d N
10 (a)
RHIC 19.6 GeVRHIC 200 GeVCMS 2.76 TeV part N ) æ pa r t N Æ / ) ( h / d c h ( d N
110 (b)
RHIC 19.6 GeVRHIC 200 GeVCMS 2.76 TeV
FIG. 1 (Color online) (a) Midrapidity charged-particle multiplicity density dN ch /dη . (b) dN ch /dη divided by thenumber of participant pairs. In both panels, vertical and horizontal bars show systematic uncertainties in dN ch /dη and (cid:104) N part (cid:105) , respectively. Lines show the results of the two-component fit using Eq. (1). Only the fits restricted tothe common range of (cid:104) N part (cid:105) are shown. n pp x pp dN ch /dη Data/ n pp PHOBOS 19.6 GeV 1.357 ± ± ± ± ± ± ± ± N part ) 2.531 ± ± ± ± ± +0 . − . ± N part ) 4.686 ± ± ± TABLE I: Two-component model fit results, reported (or estimated) inelastic pp midrapidity dN ch /dη , and the ratioof pp dN ch /dη data over the fitted value of n pp . All uncertainties are systematic.ergies to values of N part which match those for the otherenergy. More specifically, the interpolations were per-formed on the values of dN ch /dη/ (cid:104) N part / (cid:105) , since thosevary much more slowly with centrality. For the currentanalysis, the CMS data at 2.76 TeV were interpolatedto the (cid:104) N part (cid:105) values for the PHOBOS 200 GeV data.For every PHOBOS point, the two CMS points with thenearest (cid:104) N part (cid:105) (one higher and one lower) were linearlyinterpolated to the value of (cid:104) N part (cid:105) for the PHOBOSpoint. Interpolating using more points and a higher or-der polynomial made essentially no difference in the fi-nal results. The uncertainty of each interpolated pointis calculated by also linearly interpolating the fractionalslope uncertainties of its two nearest neighboring CMSdata points. Scale uncertainties are ignored for the rea-sons discussed above. This interpolation method dependson the facts that the reported uncertainties are entirelysystematic and that the fractional slope uncertainties in dN ch /dη/ (cid:104) N part / (cid:105) vary slowly as a function of central-ity. After all the CMS data are properly interpolated, the ratios R are calculated by dividing the cor-responding dN ch /dη/ (cid:104) N part / (cid:105) values at each (cid:104) N part (cid:105) .Uncertainties in these ratios are calculated by adding thefractional uncertainties in quadrature.The same procedure was used to interpolate thePHOBOS 200 GeV data in order to take ratios withthe 19.6 GeV points. These newly extracted ratios( R ) were compared to those calculated using themethods described in Ref. [1] and were found to be consis-tent. The systematic uncertainties in the previous com-parisons of different PHOBOS energies were reduced bythe fact that some uncertainties were common to all en-ergies and could be factored out. A detailed evaluation ofthe methods used by CMS and PHOBOS did not revealany similar commonalities which could be eliminated.Therefore, for consistency of presentation, the (larger)systematic uncertainties given by the current analysisprocedure were used in comparing ratios of multiplici-ties for the various energies. Nevertheless, the differencein uncertainties of R / . between what is presentedhere and those in Ref. [1] were reduced by factoring outthe scale uncertainties. III. RESULTS
Results for the two-component fits of multiplicity forthe PHOBOS and CMS data are summarized in Table Iand Fig. 1. The two panels of Fig. 1 show the samedata and fits, but the multiplicities are divided by thenumber of participant pairs in the right panel to moreeasily compare the shapes at the various energies. Inorder to make the most meaningful comparison, only thefits restricted to the common range of (cid:104) N part (cid:105) are shown.In addition to the fit parameters, Table I lists published(or estimated) midrapidity charged-particle multiplicitiesfor inelastic pp collisions and the ratios of those data tothe fitted values of n pp .The two-component fits yield very similar values forthe x parameter at all three energies, especially for thefits restricted to a common (cid:104) N part (cid:105) range. This suggeststhat collision energy does not affect the division of par-ticle production between hard and soft scattering, evenover an energy range of more than two orders of mag-nitude. This observation is in contrast to the expecta-tion that hard scatterings, which scale with the numberof nucleon-nucleon collisions, should become increasinglyprevalent as the collision energy increases.It is important to note in this comparison that therelative increases in (cid:104) N part (cid:105) and (cid:104) N coll (cid:105) with increasingcenter-of-mass energy are dramatically different. For themost central collisions, the values of (cid:104) N part (cid:105) at 200 GeVand 2.76 TeV are roughly 3% and 8.5% larger, respec-tively, than that at 19.6 GeV. In contrast, the corre-sponding values of (cid:104) N coll (cid:105) are 24% and almost a factorof 2 larger. As a result, the relative inputs from thetwo terms in Eq. (1) towards the total multiplicity dif-fer somewhat at the different energies even though thefitted value of x is essentially unchanged. The stabilityof x indicates a surprising constancy of the relative con-tributions of each individual participant pair and eachindividual nucleon-nucleon collision, even when the colli-sions overall account for a greater proportion of the totalmultiplicity at higher energy.As expected, n pp increases significantly with increasingcollision energy. Table I compares these fit parameters topublished (or estimated) values of midrapidity charged-particle multiplicities for inelastic pp collisions at eachenergy. As shown in the last column of the table, the fit-ted values of n pp for all three energies are slightly abovethe actual pp multiplicities. Furthermore, the differencebetween the data and the fit parameters shows almost noenergy dependence, as all of the ratios are equal withintheir systematic uncertainties. Although there appearsto be a decrease in the ratio with increasing beam energy,this trend is quite weak and much less than the system-atic uncertainties in the individual points. Because manyof the systematic uncertainties are common between the two fits at 200 and 2760 GeV, the trend for n pp to ap-proach the pp multiplicity value as a larger range of N part is included may be more significant. part N X / Y R R R X æ part N Æ Y æ part N Æ h /d Y dN h /d X dN = X/Y R FIG. 2 (Color online) Double ratios of midrapidity dN ch /dη / (cid:104) N part (cid:105) . The lines are linear fits to guide theeye. Error bars show only the slope uncertainties. Thescale uncertainties are 8.3% for R and 10.7% for R / . .For a less model-dependent comparison, the multiplic-ity ratios R and R (where the higher en-ergy data are interpolated to the same N part values asthose for the lower energy) are plotted versus central-ity in Fig. 2. The combined slope uncertainties wouldpredominantly shift all points of a given set of ratios inthe same direction by a varying amount proportional tothe size of the error bars shown. The scale uncertainties(not shown) would move all points together by the samemultiplicative factor.It is entirely coincidental that the average values ofthese two ratios are almost identical in magnitude. Themore significant fact is that the two ratios are very simi-lar in shape. This is consistent with what was observedin Refs. [1–3] where multiplicity ratios over a narrowerrange of collision energies showed little or no centralitydependence. While there is some indication that the cen-trality dependence at the highest energy might be slightlysteeper, the slopes from linear fits of both ratios versus N part are consistent with each other within the uncer-tainties, and both slopes are close to zero. Thus, thisexpanded analysis supports and extends the claim thatmidrapidity charged-particle multiplicities “factor” intoindependent energy and centrality dependences.This factorization has also been explored at RHIC en-ergies by fitting all of the PHOBOS data using separateempirical functions of energy and N part alone [i.e. not asa function of both N part and N coll as in Eq.(1)] [4]. How-ever, this fitted centrality dependence for RHIC resultsis a very poor match to the CMS data, probably becausethe relationship between N part and N coll is very differentat the LHC energy. IV. CONCLUSION
Midrapidity charged-particle multiplicity data forheavy ion collisions at three broadly spaced center-of-mass energies were compared using two different ap-proaches. Both analyses show that the energy and cen-trality dependences of the multiplicity are largely inde-pendent. Increasing the collision energy does not ap-pear to alter the shape of the centrality dependence of multiplicity. In particular, the fractional contribution of“hard processes”, as parameterized by the value of x inthe two-component model, does not vary significantly,even over a range of more than two orders of magni-tude in collision energy. In a more direct comparison,ratios of midrapidity multiplicity densities show at mosta very weak dependence on collision energy or centrality.This rough “factoring” of the centrality and energy de-pendencies of midrapidity charged-particle multiplicity,as first proposed in Ref. [1] using PHOBOS data span-ning √ s NN = 19 . √ s NN = 2760 GeV.This work was partially supported by U.S. DOE GrantNo. DE-FG02-94ER40818 and by the Massachusetts In-stitute of Technology Undergraduate Research Opportu-nities Program. [1] B. B. Back et al. (PHOBOS Collaboration), Phys. Rev.C , 021902 (2004).[2] B. Alver et al. (PHOBOS Collaboration), Phys. Rev. C , 011901 (2009).[3] B. B. Back et al. (PHOBOS Collaboration), Phys. Rev.C , 021901 (2006).[4] B. Alver et al. (PHOBOS Collaboration), Phys. Rev. C , 024913 (2011).[5] D. Kharzeev and M. Nardi, Phys. Lett. B , 121(2001).[6] S. S. Adler et al. (PHENIX Collaboration), Phys. Rev.C , 044905 (2014).[7] I. Arsene et al. (BRAHMS Collaboration), Nucl. Phys.A , 1 (2005).[8] K. Adcox et al. (PHENIX Collaboration), Nucl. Phys. A , 184 (2005).[9] B. B. Back et al. (PHOBOS Collaboration), Nucl. Phys. A , 28 (2005).[10] J. Adams et al. (STAR Collaboration), Nucl. Phys. A , 102 (2005).[11] S. Chatrchyan et al. (CMS Collaboration), J. High En-ergy Phys. , 141 (2011).[12] K. Aamodt et al. (ALICE Collaboration), Phys. Rev.Lett. , 252301 (2010).[13] K. Aamodt et al. (ALICE Collaboration), Phys. Rev.Lett. , 032301 (2011).[14] G. Aad et al. (ATLAS Collaboration), Phys. Lett. B ,363 (2012).[15] V. Khachatryan et al. (CMS Collaboration), J. High En-ergy Phys. , 041 (2010).[16] K. Aamodt et al. (ALICE Collaboration), Eur. Phys. J.C , 89 (2010).[17] V. Khachatryan et al. (CMS Collaboration), Phys. Rev.Lett.105