Energy Dependence of Moments of Net-proton Multiplicity Distributions at RHIC
We report the beam energy ( = 7.7 - 200 GeV) and collision centrality dependence of the mean (), standard deviation (), skewness (), and kurtosis () of the net-proton multiplicity distributions in Au+Au collisions. The measurements are carried out by the STAR experiment at midrapidity () and within the transverse momentum range 0.4 0.8 GeV/ in the first phase of the Beam Energy Scan program at the Relativistic Heavy Ion Collider. These measurements are important for understanding the Quantum Chromodynamic (QCD) phase diagram. The products of the moments, and , are sensitive to the correlation length of the hot and dense medium created in the collisions and are related to the ratios of baryon number susceptibilities of corresponding orders. The products of moments are found to have values significantly below the Skellam expectation and close to expectations based on independent proton and anti-proton production. The measurements are compared to a transport model calculation to understand the effect of acceptance and baryon number conservation, and also to a hadron resonance gas model.
The Beam Energy Scan (BES) program at the Relativistic Heavy-Ion Collider (RHIC) facility aims at studying in detail the QCD phase structure. This enables us to map the phase diagram, temperature () versus baryonic chemical potential (), of strong interactions. Important advancements have been made towards the understanding of the QCD phase structure at small . Theoretically, it has been found that at high temperatures, there occurs a cross-over transition from hadronic matter to a de-confined state of quarks and gluons at = 0 MeV (1). Experimental data from RHIC and the Large Hadron Collider have provided evidence of the formation of QCD matter with quark and gluon degrees of freedom (2). Several studies have been done to estimate the quark-hadron transition temperature at = 0 (3). Interesting features of the QCD phase structure are expected to appear at larger (4). These include the QCD critical point (CP) (5); (6) and a first order phase boundary between quark-gluon and hadronic phases (7).
Previous studies of net-proton multiplicity distributions suggest that the possible CP region is unlikely to be below = 200 MeV (8). The versatility of the RHIC machine has permitted the center of mass energy () to be varied below the injection energy ( = 19.6 GeV), thereby providing the possibility to scan the QCD phase diagram above 250 MeV. The value is observed to increase with decreasing (9). The goal of the BES program at RHIC is to look for the experimental signatures of a first order phase transition and the critical point by colliding Au ions at various (10).
Non-monotonic variations of observables related to the moments of the distributions of conserved quantities such as net-baryon, net-charge, and net-strangeness (11) number with are believed to be good signatures of a phase transition and a CP. The moments are related to the correlation length () of the system (12). The signatures of phase transition or CP are detectable if they survive the evolution of the system (13). Finite size and time effects in heavy-ion collisions put constraints on the significance of the desired signals. A theoretical calculation suggests a non-equilibrium 2-3 fm for heavy-ion collisions (14). Hence, it is proposed to study the higher moments (like skewness, = and kurtosis, =  – 3 with = – ) of distributions of conserved quantities due to a stronger dependence on (12). Both the magnitude and the sign of the moments (15), which quantify the shape of the multiplicity distributions, are important for understanding phase transition and CP effects. Further, products of the moments can be related to susceptibilities associated with the conserved numbers. The product of the net-baryon number distribution is related to the ratio of fourth order () to second order () baryon number susceptibilities (16); (17). The ratio / is expected to deviate from unity near the CP. It has different values for the hadronic and partonic phases (17).
This Letter reports measurements of the energy dependence of higher moments of the net-proton multiplicity ( = ) distributions from Au+Au collisions. The aim is to search for signatures of the CP over a broad range of in the QCD phase diagram. Theoretical calculations have shown that fluctuations reflect the singularity of the charge and baryon number susceptibility, as expected at the CP (18). The measurements presented here are within a finite acceptance range and only use the protons among the produced baryons. Refs. (19); (20) discuss the advantages of using net-baryon measurements and effects of acceptance on which the measurements depend intrinsically (e.g. conservation laws and other finite statistical fluctuations dominate near full and small acceptance respectively).
The data presented in the paper were obtained using the Time Projection Chamber (TPC) of the Solenoidal Tracker at RHIC (STAR) (21). The event-by-event proton () and anti-proton () multiplicities are measured for Au+Au minimum-bias events at = 11.5, 19.6, 27, 39, 62.4, and 200 GeV for collisions occurring within = 30 cm from the TPC center along the beam line. For 7.7 GeV, is 50 cm. The 19.6 and 27 GeV data were collected in the year 2011 and the other energies were taken in 2010. Interactions of the beam with the beam pipe are rejected by choosing events with a radial vertex position in the transverse plane of less than 2 cm. The numbers of events analyzed are 3, 6.6, 15, 30, 86, 47, and 238 for = 7.7, 11.5, 19.6, 27, 39, 62.4, and 200 GeV, respectively. Similar studies have also been carried out in + collisions with 0.6 and 7 events at = 62.4 and 200 GeV, respectively. The centrality selection utilizes the uncorrected charged particle multiplicity other than identified protons and anti-protons within pseudorapidity 1.0 measured by the TPC. It is found that the measured net-proton moment values depend on the choice of the pseudorapidity range for the centrality selection. However the values of the moments do not change if the centrality selection range is further increased to the full acceptance of the TPC (which leads to a 15% increase in charged particle multiplicity). In the UrQMD (22) studies, after increasing the range used for centrality selection to two units, it is observed that the maximum decrease of moments is 2.5% and 35% for = 200 and 7.7 GeV, respectively (23). There is minimal change for central collisions compared to other centralities. For each centrality, the average number of participants () is obtained by Glauber model calculations. The measurements are carried out at midrapidity ( 0.5) in the range 0.4 0.8 GeV/. Ionization energy loss () of charged particles in the TPC is used to identify the inclusive () (24). The minimum cut and a maximum distance of closest approach (DCA) to the collision vertex of 1 cm for each candidate track suppress contamination from secondaries (24). To have a good purity of the proton sample (better than 98%) for all beam energies, the maximum is taken to be 0.8 GeV/. This interval accounts for approximately 50% of the total uncorrected multiplicity at midrapidity. The average proton reconstruction efficiency for the range studied is between 70-78% and 83-86%, for central and peripheral collisions, respectively, at different .
distributions from 70-80%, 30-40%, and 0-5% Au+Au collision centralities are shown in Fig. 1. The is not corrected for reconstruction efficiency. The distributions are also not corrected for the finite centrality width effect (23). The subsequent analysis in this Letter is corrected for the centrality width effect. The corresponding Skellam distributions are also shown, , where is a modified Bessel function of the first kind, and and are the measured mean multiplicities of proton and anti-protons (25). The data seems to closely follow the Skellam distributions. To study the detail shape of the distribution, we discuss the various order cumulants (), where = , = , = and = . For both proton and anti-proton distributions being Poissonian, the distribution will be a Skellam and have = and = 1.
The four cumulants that describe the shape of distributions at various collision energies are plotted as a function of in Fig. 2. We use the Delta theorem approach to obtain statistical errors (26). The typical statistical error values for , , and for central Au+Au collisions at 7.7 GeV are 0.3%, 2.5% and 2.5% respectively, and those for high statistics 200 GeV results are 0.04%, 1.2% and 2.0% respectively. Most of the cumulant values show a linear variation with . The values increase as decreases, in accordance with the energy and centrality dependence of baryon transport. and have similar values as a function of for a given . and follow each other closely as a function of at any given . The differences between these groupings decrease as decreases. The decrease in the values with increasing beam energy indicates that the distributions become symmetric for the higher beam energies. The particle production at any given centrality can be considered a superposition of several identically distributed independent sources the number of which is proportional to (8). For the cumulants, this means a linear increase with as the system volume increases. This reflects that the cumulants are extensive quantities that are proportional to system volume. The lines in Fig. 2 are linear fits to the cumulants, which provide a reasonable description of the centrality dependence. This indicates that the volume effect dominates the measured cumulants values. The between the linear fit and data are smaller than 3.2 for all cumulants presented. The slight deviation of some cumulants in most central collisions from the fit line are due to the corresponding proton distributions.
In order to cancel the volume effect to first order and to understand the collision dynamics, we present the ratios of the cumulants (= ) and (= ) as a function of for all collision energies, in Fig. 3. The are normalized to the corresponding Skellam expectations. Results with correction for the reconstruction efficiency are presented. The correction for a finite track reconstruction efficiency is done by assuming a binomial distribution for the probability to reconstruct particles out of produced (27); (20). These observables are related to the ratio of baryon number susceptibilities () at a given temperature () computed in QCD motivated models as: = and = (16); (17). Close to the CP, QCD based calculations predict the net-baryon number distributions to be non-Gaussian and susceptibilities to diverge, causing and to have non-monotonic variations with and/or (6); (12).
We observe in Fig. 3 the and the normalized to Skellam expectations are below unity for all of the Au+Au collision data sets presented. The deviations below unity of the order of 1-3% (28) as seen for the central collisions for energies above 27 GeV are expected from quantum statistical effects. The measured and are compared to expectations in which the cumulants of distributions are constructed by considering independent production of protons and anti-protons. For independent production, the various order ( = 1, 2, 3 and 4) net-proton cumulants are given as = + , where and are cumulants of the measured distributions of and , respectively. This approach breaks intra-event correlations between and . The results from independent production are found to be in good agreement with the data. However, for 39 GeV, the of net-protons are dominated by the corresponding values from the proton distributions. The assumption that and have independent binomial distributions (29) also leads to a good description of the measurements (similar to independent production, but not plotted in Fig. 3).
Systematic errors are estimated by varying the following requirements for tracks: DCA, track quality reflected by the number of fit points used in track reconstruction, and the selection criteria for identification. The typical systematic errors are of the order 4% for and , 5% for and 12% for . A 5% uncertainty in reconstruction efficiency estimation is also considered. The statistical and systematic (caps) errors are presented separately in the figures.
Figure 4 shows the energy dependence of and for for Au+Au collisions for two collision centralities (0-5% and 70-80%), corrected for () reconstruction efficiency. The values normalized to the corresponding Skellam expectations are shown in the bottom panel of Fig. 4. The Skellam expectations reflect a system of totally uncorrelated, statistically random particle production. The corresponding results from the + collisions are also shown and found to be similar to peripheral Au+Au collisions for = 62.4 and 200 GeV within the statistical errors. For below 39 GeV, differences are observed between the 0-5% central Au+Au collisions and the peripheral collisions. The results are closer to unity for = 7.7 GeV. Deviations of 0-5% Au+Au data from Skellam expectations, () are found to be most significant for 19.6 GeV and 27 GeV, with values of 3.2 and 3.4 for , and 4.5 and 5.6 for , respectively. The deviations for 5-10% Au+Au data are smaller for with values of 2.0 and 0.6 and are 5.0 and 5.4 for , for 19.6 GeV and 27 GeV, respectively. A reasonable description of the measurements is obtained from the independent production approach. The data also show deviations from the hadron resonance gas model (31); (32) which predict and /Skellam to be unity. To understand the effects of baryon number conservation (33) and experimental acceptance, UrQMD model calculations (a transport model which does not include a CP) (22) for 0-5% Au+Au collisions are shown in the middle and bottom panels of Fig. 4. The UrQMD model shows a monotonic decrease with decreasing beam energy (23).
The current data provide the most relevant measurements over the widest range in (20 to 450 MeV) to date for the CP search, and for comparison with the baryon number susceptibilities computed from QCD to understand the various features of the QCD phase structure (6); (16); (17). The deviations of and below Skellam expectation are qualitatively consistent with a QCD based model which includes a CP (34). However the UrQMD model which does not include a CP also shows deviations from the Skellam expectation. Hence conclusions on the existence of CP can be made only after comparison to QCD calculations with CP behavior which include the dynamics associated with heavy-ion collisions, such as finite correlation length and freeze-out effects.
In summary, measurements of the higher moments and their products ( and ) of the net-proton distributions at midrapidity ( 0.5) within 0.4 0.8 GeV/ in Au+Au collisions over a wide range of and have been presented to search for a possible CP and signals of a phase transition in the collisions. These observables show a centrality and energy dependence, which are neither reproduced by non-CP transport model calculations, nor by a hadron resonance gas model. For 39 GeV, and values are similar for central, peripheral Au+Au collisions and + collisions. Deviations for both and from HRG and Skellam expectations are observed for 27 GeV. The measurements are reasonably described by assuming independent production of and , indicating that there are no apparent correlations between the protons and anti-protons for the observable presented. However at the lower beam energies, the net-proton measurements are dominated by the shape of the proton distributions only. The data presented here also provides information to extract freeze-out conditions in heavy-ion collisions using QCD based approaches (35); (36).
We thank M. Asakawa, R. Gavai, S. Gupta, F. Karsch, K. Rajagopal, K. Redlich and M. A. Stephanov for discussions related to this work. We thank the RHIC Operations Group and RCF at BNL, and the NERSC Center at LBNL, the KISTI Center in Korea and the Open Science Grid consortium for providing resources and support. This work was supported in part by the Offices of NP and HEP within the U.S. DOE Office of Science, the U.S. NSF, CNRS/IN2P3, FAPESP CNPq of Brazil, Ministry of Ed. and Sci. of the Russian Federation, NNSFC, CAS, MoST, and MoE of China, the Korean Research Foundation, GA and MSMT of the Czech Republic, FIAS of Germany, DAE, DST, and CSIR of the Government of India, National Science Centre of Poland, National Research Foundation (NRF-2012004024), Ministry of Sci., Ed. and Sports of the Rep. of Croatia, and RosAtom of Russia. Finally, we gratefully acknowledge a sponsored research grant for the 2006 run period from Renaissance Technologies Corporation.
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