Measurement of higher cumulants of net-charge multiplicity distributions in AuAu collisions at –200 GeV
We report the measurement of cumulants () of the net-charge distributions measured within pseudorapidity ( 0.35) in AuAu collisions at =7.7–200 GeV with the PHENIX experiment at the Relativistic Heavy Ion Collider. The ratios of cumulants (e.g. , ) of the net-charge distributions, which can be related to volume independent susceptibility ratios, are studied as a function of centrality and energy. These quantities are important to understand the quantum-chromodynamics phase diagram and possible existence of a critical end point. The measured values are very well described by expectation from negative binomial distributions. We do not observe any nonmonotonic behavior in the ratios of the cumulants as a function of collision energy. The measured values of and can be directly compared to lattice quantum-chromodynamics calculations and thus allow extraction of both the chemical freeze-out temperature and the baryon chemical potential at each center-of-mass energy. The extracted baryon chemical potentials are in excellent agreement with a thermal-statistical analysis model.
One of the main goals in the study of relativistic heavy ion collisions is to map the quantum chromodynamics (QCD) phase diagram at finite temperature () and baryon chemical potential () Stephanov et al. (1998). Although the exact nature of the phase transition at finite baryon density is still not well established, several models suggest that, at large and low , the phase transition between the hadronic phase and the quark-gluon-plasma (QGP) phase is of first order Alford et al. (1998); Stephanov (1996) and that at high and low there is a simple cross over from the QGP to hadronic phase Aoki et al. (2006); Pisarski and Wilczek (1984); Stephanov (2004); Fodor and Katz (2004); Ejiri (2008). The point at which the first-order phase transition ends in the plane is called the QCD critical end point (CEP), which is one of the central targets of the Relativistic Heavy Ion Collider (RHIC) beam-energy-scan program. Several calculations also reported the possible existence of the CEP in the phase diagram Fodor and Katz (2004); Stephanov (2004); Stephanov et al. (1999).
RHIC at Brookhaven National Laboratory has provided a large amount of data from AuAu collisions at different colliding energies, which gives us a unique opportunity to scan the plane and investigate the possible existence and location of the CEP. In the thermodynamic limit, the correlation length () diverges at the CEP Stephanov et al. (1998). Event-by-event fluctuations of various conserved quantities, such as net-baryon number, net-charge, and net-strangeness are proposed as possible signatures of the existence of the CEP Koch et al. (2005); Asakawa et al. (2000, 2009). It has been shown in lattice QCD that with a next-to-leading-order Taylor series expansion around vanishing chemical potentials, the cumulants of charge-fluctuations are sensitive indicators for the occurrence of a transition from the hadronic to QGP phase Ejiri et al. (2006); Bazavov et al. (2012). Typically, the variances of net-baryon, net-charge, and net-strangeness distributions are proportional to as (=)= Stephanov et al. (1999), where is the multiplicity, and (=) is the mean of the distribution.
Recent calculations reveal that higher cumulants of the fluctuations are much more sensitive to the proximity of the CEP than earlier measurements using second cumulants () Stephanov (2009); Asakawa et al. (2009). The skewness () and kurtosis () are related to the third and fourth moments (= and (=. The ratio of the various order () of cumulants () and conventional values (, , and ) can be related as follows: = , = , = , and = . Because diverges at the CEP, the ratios of cumulants and should rise rapidly when approaching the CEP Gavai and Gupta (2011); Cheng et al. (2009). The cumulants of conserved quantities of net-baryon, net-charge, and net-strangeness obtained from lattice QCD calculations Ejiri et al. (2006); Bazavov et al. (2012); Cheng et al. (2009) and a hadron resonance gas (HRG) model Karsch and Redlich (2011) are related to the generalized susceptibilities of -th order () associated with the conserved quantum numbers as , , , and . One advantage of measuring , , , and is that the volume dependence of , , , and cancel out in the ratios, hence theoretical calculations can be directly compared with the experimental measurements. These cumulant ratios can also be used to extract the freeze-out parameters and the location of the CEP Bazavov et al. (2012). Net-electric charge fluctuations are more straightforward to measure experimentally than net-baryon number fluctuations, which are partially accessible via net-proton measurement Aggarwal et al. (2010). While net-charge fluctuations are not as sensitive as net-baryon fluctuations to the theoretical parameters, both measurements are desirable for a full understanding of the theory.
We report here precise measurements of the energy and centrality dependence of higher cumulants of net-charge multiplicity ( = ) distributions measured by the PHENIX experiment at RHIC in AuAu collisions at = 7.7, 19.6, 27, 39, 62.4, and 200 GeV. These measurements cover a broad range of in the QCD phase diagram.
The PHENIX detector is composed of two central spectrometer arms, two forward muon arms, and global detectors Adcox et al. (2003). In this analysis, we use the central arm spectrometers, which cover a pseudorapidity range of 0.35. Each of the two arms subtends /2 radians in azimuth and is designed to detect charged hadrons, electrons, and photons. For data taken at = 62.4 and 200 GeV in 2010 and 2007, respectively, the event centrality is determined using total charge deposited in the beam-beam counters (BBC), which are also used for triggering and vertex determination. For lower energies ( = 39 GeV and below) the acceptance of the BBCs (3.0 3.9) are within the fragmentation region, so alternate detectors must be employed. For data taken at = 39 and 7.7 GeV in 2010, centrality is determined using the total charge deposited in the outer ring of the reaction plane detector (RXNP), which covers Richardson et al. (2011). For data taken at = 19.6 and 27 GeV in 2011, the RXNP was absent, so centrality is determined using the total energy of electromagnetic calorimeter (EMCal) clusters to minimize the correlation with the charge of the tracks measured in the same acceptance. More details on the procedure are given in Adler et al. (2005). The analyzed events for the above mentioned energies are within a collision vertex of 30 cm. The number of analyzed events are 2M, 6M, 21M, 154M, 474M, and 1681M for = 7.7, 19.6, 27, 39, 62.4, and 200 GeV AuAu collisions, respectively.
The number of positively charged () and negatively charged () particles measured on an event-by-event basis are used to calculate the net-charge () distributions for each collision centrality and energy. The charged-particle trajectories are reconstructed using information from the drift chamber and pad chambers (PC1 and PC3). A combination of reconstructed drift-chamber tracks and matching hits in PC1 are used to determine the momentum and charge of the particle. Tracks having a transverse momentum () between 0.3 and 2.0 GeV/ are selected for this analysis. The ring imaging Čerenkov detector is used to reduce the electron background resulting from conversion photons. To further reduce the background, selected tracks are required to lie within a 2.5 matching window between track projections and PC3 hits, and a 3 matching window for the EMCal.
Figure 1(a) and (b) show distributions in AuAu collisions for central (0%–5%) and peripheral (55%–60%) collisions at different collision energies. These distributions are not corrected for reconstruction efficiency. The centrality classes associated with the average number of participants () are defined for each 5% centrality bin. These classes are determined using a Monte-Carlo simulation based on Glauber model calculations with the BBC, RXNP, and EMCal detector response taken into account Adler et al. (2005); Miller et al. (2007).
The distributions are characterized by cumulants and related quantities such as , , , and , which are calculated from the distributions. The statistical uncertainties for the cumulants are calculated using the bootstrap method Efron and Tibshirani (1994). Corrections are then made for the reconstruction efficiency, which is estimated for each centrality and energy using the hijing1.37 event generator Wang and Gyulassy (1991) and then processed through a geant simulation with the PHENIX detector setup. For all collision energies, the average efficiency for detecting the particles within the acceptance varies between 65%–72% and 76%–85% for central (0%–5%) and peripheral (55%–60%) events, respectively with 4%–5% variation as a function of energy. The efficiency correction applied to the cumulants is based on a binomial probability distribution for the reconstruction efficiency Bzdak et al. (2013). The efficiency corrected , , , and as a function of are shown in panels (c-f) of Fig. 1.
The and for net-charge distributions increase with increasing , while and decrease with increasing for all collision energies. At a given value, , , and of net-charge distributions decrease with increasing collision energy. However, the width () of net-charge distributions increases with increasing collision energy indicating the increase of fluctuations in the system at higher .
The systematic uncertainties are estimated by: (1) varying the cut to less than 10 cm; (2) varying the matching parameters of PC3 hits and EMCal clusters with the projected tracks to study the effect of background tracks originating from secondary interactions or from ghost tracks; (3) varying the centrality bin width to study nondynamical contributions to the net-charge fluctuations due to the finite width of the centrality bins Adare et al. (2008); Adler et al. (2007); Konchakovski et al. (2006); and (4) varying the lower cut. The total systematic uncertainties estimated for various cumulants for all energies are: 10%–24% for , 5%–10% for , 25%–30% for , and 12%–19% for . The systematic uncertainties are similar for all centralities at a given energy and are treated as uncorrelated as a function of . For clarity of presentation, the systematic uncertainties are only shown for central (0%–5%) collisions.
Figure 2 shows the dependence of , , , and extracted from the net-charge distributions in AuAu collisions at different . The results are corrected for the reconstruction efficiencies. Statistical uncertainties are shown along with the data points. The systematic uncertainties are constant fractional errors for all centralities at a particular energy, hence they are presented for the central (0%–5%) collision data point only. The systematic uncertainties on these ratios across different energies varies as follows: 20%–30% for , 15%–34% for , 12%–22% for , and 17%–32% for . It is observed in Fig. 2 that the ratios of the cumulants are weakly dependent on for each collision energy; the values of and decrease from lower to higher collision energies, while the and values are constant as a function of within systematic uncertainties.
The collision energy dependence of , , and of the net-charge distributions for central (0%–5%) AuAu collisions are shown in Fig. 3. The statistical and systematic uncertainties are shown along with the data points. The experimental data are compared with negative-binomial-distribution (NBD) expectations, which are calculated by computing the efficiency corrected cumulants for the measured and distributions fit with NBD’s respectively, which also describe total charge () distributions very well Adare et al. (2008); Adler et al. (2007). The various order ( = 1, 2, 3 and 4) of net-charge cumulants from NBD are given as = , where and are cumulants of and distributions, respectively Barndorff-Nielsen et al. (2012); Tarnowsky and Westfall (2013).
The and values in Fig. 3(a) and Fig. 3(b), respectively both decrease with increasing . The NBD expectation agrees well with the data. The values in Fig. 3(c) remain constant and positive, between at all the collision energies within the statistical and systematic uncertainties. However, there is 25% increase of values at lower energies compared to higher energies above = 39 GeV, which is within the systematic uncertainties. These data are in agreement with a previous measurement Adamczyk et al. (2014a), but provide a more precise determination of the higher cumulant ratios, verified by the NBD method of correcting for efficiency, which is simple and analytical for all cumulant ratios with the standard binomial correction Bzdak et al. (2013). The values in Fig. 3(d) remain constant at all collision energies within the uncertainties and are well described by the NBD expectation. From the energy dependence of , , , and , no obvious nonmonotonic behavior is observed. Although both previous measurements by STAR Adamczyk et al. (2014a, b) use the pseudorapidity range , compared to the present measurement spanning , these measurements are all within the central rapidity region and are expected to be valid for comparison to lattice QCD calculations. The efficiency corrected results for the cumulant ratios , , and remain the same within statistics whether each single arm of the PHENIX central spectrometer (azimuthal aperture ) or both arms () are used. This is a clear verification of the insensitivity of measured cumulant ratios to volume effects.
|PHENIX + Ref. Bazavov et al. (2012); Mukherjee (2014)||PHENIX + Ref. Borsanyi et al. (2013)||STAR + Ref. Borsanyi et al. (2014)|
The precise measurement of both and in the present study allow both and to be determined, unlike a previous calculation in Ref. Borsanyi et al. (2014, 2013), which was only able to use the measurement from Ref. Adamczyk et al. (2014a). The comparison of for different with the lattice calculations (Fig. 3(b) in Ref. Bazavov et al. (2012); Mukherjee (2014)) enables us to extract the chemical freeze-out temperature (). Furthermore, can be extracted by comparing the measured ratios with the lattice calculations of (Fig.3(a) in Ref. Bazavov et al. (2012); Mukherjee (2014)). The extracted and values are listed in Table 1. The and extracted using the lattice calculations in the continuum limit from Ref. Borsanyi et al. (2013) are also depicted in Table 1. The extracted freeze-out parameters using different lattice results agree very well. However, the extracted are 2-4 MeV lower using Ref. Borsanyi et al. (2013) than with Ref. Bazavov et al. (2012); Mukherjee (2014), which is well within the stated uncertainties. The detailed freeze-out parameter extraction procedure is given in Ref. Bazavov et al. (2012); Borsanyi et al. (2013, 2014). This is a direct combination of experimental data and lattice calculations to extract physical quantities. The dependence of shown in Fig. 4 is in agreement with the thermal-statistical analysis model of identified particle yields Cleymans et al. (2006). The extracted in the present net-charge measurement and the values reported in Borsanyi et al. (2014) are in agreement within stated uncertainties, with some tension at = 27 GeV. Available lattice results allow extraction of and from = 27 GeV and higher using the present net-charge experimental data. Other recent calculations Alba et al. (2014); Bazavov et al. () have used both net-proton and net-charge measurements to estimate the freeze-out parameters.
In summary, fluctuations of net-charge distributions have been studied using higher cumulants (, , , and ) for 0.35 with the PHENIX experiment in AuAu collisions ranging from = 7.7 to 200 GeV. The ratios of cumulants (, , , and ) have been derived from the individual cumulants of the distributions studied as a function of and . The and values decrease with increasing collision energy and are weakly dependent on centrality, whereas and values remain constant over all collision energies within uncertainties. The efficiency corrected values from the NBD expectation reproduce the experimental data. These data are in agreement with a previous measurement Adamczyk et al. (2014a), but provide more precise determination of the higher cumulant ratios and . In the present study we do not observe any significant nonmonotonic behavior of , , , and as a function of collision energies. Comparison of the present measurements together with the lattice calculations enables us to extract the freeze-out temperature and baryon chemical potential () over a range of collision energies. The extracted values are in excellent agreement with the thermal-statistical analysis model Cleymans et al. (2006).
We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions. We thank F. Karsch and S. Mukherjee for providing us with tables of their calculations and for helpful discussions. We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (People’s Republic of China), Ministry of Science, Education, and Sports (Croatia), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l’Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), National Science Fund, OTKA, Károly Róbert University College, and the Ch. Simonyi Fund (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), Basic Science Research Program through NRF of the Ministry of Education (Korea), Physics Department, Lahore University of Management Sciences (Pakistan), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation (Sweden), the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the Hungarian American Enterprise Scholarship Fund, and the US-Israel Binational Science Foundation.
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