Searching for the QCD Critical Point through Fluctuations
at RHIC
Roli Esha
Department of Physics and Astronomy, University of California Los Angeles,
475 Portola Plaza, Los Angeles, CA 90095, USA
Fluctuations and correlations of conserved quantities (baryon number, strangeness, and charge) can be used to probe phases of strongly interacting QCD matter and the possible existence of a critical point in the phase diagram. The cumulants of the multiplicity distributions related to these conserved quantities are expected to be sensitive to possible increased fluctuations near a critical point and ratios of the cumulants can be directly compared to the ratios of the susceptibilities from Lattice QCD calculations. In these proceedings, the measurements of the cumulants of netproton multiplicity distributions from Au+Au collisions at = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 (up to fourth order) and 200 GeV (up to sixth order) as measured by the STAR experiment at RHIC will be presented. Multiparticle correlation functions will also be presented.
The measurement of higherorder cumulants are sensitive to experimental artifacts. Current efficiency correction methods are based on the assumption that the tracking efficiency is strictly binomial. An unfolding technique is used to account for multiplicity dependent detector responses and efficiency variations. A large sample of AMPT events will be used to check the validity of the unfolding technique. The comparison of the various correction approaches should provide important guidance towards a reliable experimental determination of the multiplicity cumulants.
PRESENTED AT
Thirteenth Conference on the Intersections of Particle and Nuclear Physics
Palm Springs, CA, USA, May 28–June 3, 2018
1 Introduction
The main goal of highenergy heavyion collisions has been to investigate QCD at high temperature and baryon density. At ordinary temperatures, the quarks and gluons are confined within hadrons, but at very high temperatures and densities, we have a deconfined phase of quarks and gluons, the Quark Gluon Plasma (QGP). Over the past years, evidence for the distinct phases of QGP and hadron gas has been established experimentally.
The Beam Energy Scan program at RHIC allows us to study the QCD phase diagram by varying the collision energy for heavy ions, hence, scanning baryon chemical potential () and temperature (). For vanishing baryon chemical potential, the transition from the QGP phase to the Hadron Gas phase is a smooth crossover [1], while QCDinspired models predict an existence of firstorder phase transition for large baryon chemical potential. Thermodynamic principles, hence, suggest the existence of a critical point in the QCD phase diagram.
Theoretically, eventbyevent fluctuations of conserved quantities, namely charge, baryon number and strangeness, is used to probe the critical phenomenon in the QCD phase diagram. Experimentally, this translates to the measurement of the cumulants of eventbyevent netparticle multiplicity distributions  netcharge, netproton (proxy for netbaryon) and netkaon (proxy for netstrangeness). Correlation function of various particles can also be obtained from the same. The cumulants () are related to the susceptibility () of the system [2], which is the derivative of free energy with respect to the chemical potential, in the following way
(1) 
where is the volume, is the temperature, is the pressure, and denotes the conserved quantity, that is, charge, baryon number or strangeness. The ratios of such cumulants as experimental observables cancel the volume and temperature dependence and can be directly compared to the ratios of susceptibilities from theoretical calculations. The higherorder cumulants were predicted to be more sensitive to the signatures of phase transition [3], and perhaps more susceptible to experimental artifacts as well.
In these proceedings, the cumulants for netproton and netkaon multiplicity will be presented from the data taken in the years 2010 and 2011 by the STAR experiment at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory. The measurement of cumulants of netcharge distribution will be presented by both the STAR and the PHENIX experiments.
2 Analysis details
The netparticle distributions are constructed from the eventbyevent difference of positively and negatively charged particles. The analysis is carried out using minimum bias events obtained after rejecting pileup and other background interactions. All the tracks in an event that are included in the analysis also undergo some quality checks.
In order to get the precise measurements of the eventbyevent fluctuations of conserved quantities and suppress background, a series of analysis techniques are applied [4]. These can be summarized as the following:

The centrality of the collision is determined by excluding the particle of interest. This helps avoid autocorrelation, which is a background effect and can reduce the magnitude of the signal in fluctuation measurements [5].

Centrality Bin Width Correction is applied to suppress volume fluctuation with a wide centrality range, which could lead to artificial centrality dependence for the observables. For this, the cumulants are obtained as the weighted average for the given centrality [6].
For the purpose of illustration, Figure 1 shows the raw, uncorrected distribution of the netcharge, netkaon and netproton multiplicity distributions for Au+Au collision at GeV for different centralities as measured by the STAR experiment [11]. The black circles are for 05% central collisions, the red squares for 3040% and blue stars for 7080% central collisions.
3 Results and discussions
The various cumulant ratios can be written as
(2) 
where is the mean, is the variance, is the skewness and is the kurtosis of the distribution.
Figure 2 shows the measurements of the cumulantratios for netcharge (left panel), netkaon (middle panel) and netproton (right panel) multiplicity distributions from Au+Au collisions at = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 and 200 GeV from the STAR experiment [11]. The black circles are the measurements for 7080%, green squares for 510% and red stars for 05% central collisions. The corresponding dashed lines are the Poisson baselines. The blue band is the result from 05% central collisions from the UrQMD model calculations. It should be noted that the kinematic range for the measurements are different. Results for netcharge include all charged particles within transverse momentum region of 0.2 to 2.0 GeV/c and pseudorapidity ranging from 0.5 to 0.5. The spallation protons with transverse momentum less than 0.4 GeV/c are excluded. For the cumulants of netkaons, (anti)kaons within transverse momentum region of 0.2 to 1.2 GeV/c and rapidity ranging from 0.5 to 0.5 are included. For the netproton analysis, (anti)protons within transverse momentum region of 0.4 to 2.0 GeV/c and rapidity ranging from 0.5 to 0.5 are included. Within large statistical error bars, we find a monotonic trend in the various cumulant ratios for netcharge and netkaon multiplicity distributions with the collision energy. However, higher cumulants of netproton multiplicity distributions show a nonmonotonic trend for central collisions.
Figure 3 shows the various cumulant ratios of the netcharge multiplicity distribution for 05% central collisions [12]. Within errors, the results are consistent with the expectations from Negative Binomial baseline. However, more statistics is needed at low beam energies.
In order to extract the multiparticle correlation functions () from the cumulants, we use the equations in Table 1. Here, is the average number of particles over the ensemble, and the order cumulants are expressed as a combination of lower order correlation functions (). The correlation functions follow the same powerlaw dependence on correlation lengths as the cumulants. Figure 4 shows the normalized fourthorder cumulant for protons for 05% central collisions with beam energy in the first panel and the normalized fourparticle correlation function in the second panel [13]. Both of these show nonmonotonic energy dependence. Upon subtracting from , we find a monotonic energy dependence, which indicated that the nonmonotonic behavior almost entirely arises from multiparticle correlations.
The sixthorder cumulants of netcharge and netbaryon distributions are predicted to be negative if the chemical freezeout is close enough to the phase transition [14]. Figure 5 shows the measurement of the ratio of the sixthorder cumulant to the secondorder cumulant for netcharge (left) and netproton (right) multiplicity distributions for Au+Au collisions at GeV at STAR [15]. We find that the ratio for netcharge is consistent with zero within large statistical errors. However, despite large error bars, negative values are observed for for netproton systematically from peripheral to central collisions. The systematic uncertainties have not yet been determined.
4 Unfolding for efficiency correction
Experimental effects like particle misidentification, track merging/splitting, etc. could lead to nonBinomial detector efficiencies. In addition, previous studies have shown that there could be noticeable consequences of multiplicitydependent behavior of detection efficiency on higherorder cumulants [16]. In order to understand these, unfolding techniques are being developed at STAR for reliable measurement of the cumulants.
The measured distribution is a convolution of original particle distribution and the detector response function. There are two key elements of the approach, namely, the correlation matrix and the response histogram. The correlation matrix contains the number correlation between the measured protons and antiprotons. The response histogram contains the distribution of produced particles for every detected number of particles. These are obtained using embedding. Two schemes for unfolding are being looked into  unfolding with initial (anti)proton distributions assumed to be Poisson distributions [17] and unfolding with iterations [18].
Cumulants for  True  Efficiency corrected  Efficiency corrected  Efficiency corrected 

netproton  distribution  (2D response  (1D response  (Factorial moment 
distribution  matrix)  matrix)  method)  
2.799 0.002  2.799 0.002  2.800 0.002  2.550 0.001  
31.44 0.01  31.43 0.01  49.78 0.02  12.63 0.01  
8.4 0.2  8.4 0.1  9.3 0.2  2.58 0.04  
91 1  91 2  89 3  12.5 0.3 
As an illustration, we use the eventbyevent distribution of (anti)protons given by the AMPT model. The efficiency for protons is assumed to be 0.8 – 0.0003 , while for antiprotons, it is 0.7 – 0.0003 respectively. The coefficient 0.0003 is the expected order of magnitude of the multiplicity dependence of efficiency in real data. We compare the results in Table 2. We find that the results from the datadriven method agree well with the cumulants of the true distribution. In the 2D response matrix situation, both protons and antiprotons are corrected simultaneously, that is, the response matrix is a twodimensional histogram containing the information for produced number of both protons and antiprotons for every measured number of protons and antiprotons. In the 1D response matrix approach, protons and antiprotons are corrected separately, that is, the response matrix is two onedimensional matrices; one for protons and the other for antiprotons. The corrected cumulants obtained from the factorial moment method, however, deviate from the true values considerably. This is because the factorial moment method assumes binomial efficiency correction. Centrality bin width correction is applied to the factorial moment method. Details of the efficiency correction appear to have a significant effect on the cumulants and STAR is in the process to determine a suitable analysis approach.
5 Summary and outlook
The cumulantratios for netcharge, netkaon and netproton multiplicity distributions were presented from both the STAR and PHENIX experiment from the Beam Energy Scan program at RHIC. We found that the ratios were mostly monotonic within error bars. Nonmonotonic energy dependences are observed for of protons and netproton multiplicity distributions for 05% central Au+Au collisions. We found that the fourparticle correlations contribute dominantly to the observed nonmonotonicity. The ratio is consistent with being negative as expected from theoretical calculations for Au+Au collisions at GeV within large statistical uncertainties.
Efficiency correction is an important ingredient in order to reliably calculate the higherorder cumulants. We are currently developing unfolding methods to account for various effects in detection efficiencies. Such an approach would be a significant improvement over the previous analyses under the assumption of binomial efficiency. AMPT simulations indicated that there could be a considerable effect on the cumulant calculations. Unfolding methods are being developed at STAR to account for these.
More data will be collected in BESII for Au+Au collisions at = 7.7 – 19.6 GeV in 2019 – 2020 with detector upgrades.
References
 [1] Y. Aoki et al, Nature, 443, 675 (2006).

[2]
M. A. Stephanov, Phys. Rev. Lett., 102, 032301 (2009).
M. Asakawa, S. Ejiri and M. Kitazawa, Phys. Rev. Lett., 103, 262301(2009).
M. A. Stephanov, Phys. Rev. Lett., 107, 052301 (2011). 
[3]
B. Ling, M. Stephanov, Phys. Rev. C93, 034915 (2016)
A. Bzdak, V. Koch, N. Strodthoff, arXiv:1607.07375.
A. Bzdak, V. Koch, V. Skokov, arXiv:1612.05128.  [4] X. Luo and N. Xu, Nucl. Sci. Tech., 28, 112 (2017).

[5]
STAR Collaboration, Phys. Rev. Lett., 105, 022302 (2010).
STAR Collaboration, Phys. Rev. Lett., 113, 092301 (2014).  [6] X. Luo, J. Xu, B. Mohanty, N. Xu, J. Phys. G40, 105104 (2013).
 [7] B. Efron et al, Chapman and Hill, 1993.
 [8] X. Luo, Phys. Rev. C 91, 034907 (2015).
 [9] A. Bzdak and V. Koch, Phys. Rev. C 91, 027901 (2015).
 [10] T. Nonaka, M. Kitazawa and S. Esumi, Phys. Rev C 95, 064912 (2017).
 [11] J. Thaeder (for the STAR Collaboration), Nucl. Phys. A 956, 320 (2016).
 [12] A. Adare et al. (PHENIX Collaboration) Phys. Rev. C 93, 011901 (2016).
 [13] X. Luo, Talk at INT Workshop 2017, Seattle, US.

[14]
F. Karsch and K. Redlich, Phys. Lett. B 695, 136 (2011).
B. Friman, F. Karsch, K. Redlich, V. Skokov, Eur. Phys. J. C 71, 1694 (2011). 
[15]
R. Esha [STAR Collaboration], Nucl. Phys. A 967, 457 (2017).
T. Nonaka (for the STAR Collaboration), Quark Matter 2018.  [16] A. Bzdak, R. Holzmann and V. Koch, Phys. Rev. C 94, 064907 (2016).
 [17] R. Esha [STAR Collaboration], PoS CPOD 2017, 003 (2018).
 [18] T. Nonaka, M. Kitazawa and S. Esumi, arXiv:1805.00279.