Cumulants and Correlation Functions of Net-proton, Proton and Antiproton Multiplicity Distributions in Au+Au Collisions at energies available at the BNL Relativistic Heavy Ion Collider
STAR Collaboration, M. S. Abdallah, J. Adam, L. Adamczyk, J. R. Adams, J. K. Adkins, G. Agakishiev, I. Aggarwal, M. M. Aggarwal, Z. Ahammed, I. Alekseev, D. M. Anderson, A. Aparin, E. C. Aschenauer, M. U. Ashraf, F. G. Atetalla, A. Attri, G. S. Averichev, V. Bairathi, W. Baker, J. G. Ball Cap, K. Barish, A. Behera, R. Bellwied, P. Bhagat, A. Bhasin, J. Bielcik, J. Bielcikova, I. G. Bordyuzhin, J. D. Brandenburg, A. V. Brandin, I. Bunzarov, J. Butterworth, X. Z. Cai, H. Caines, M. Calderón de la Barca Sánchez, D. Cebra, I. Chakaberia, P. Chaloupka, B. K. Chan, F-H. Chang, Z. Chang, N. Chankova-Bunzarova, A. Chatterjee, S. Chattopadhyay, D. Chen, J. Chen, J. H. Chen, X. Chen, Z. Chen, J. Cheng, M. Chevalier, S. Choudhury, W. Christie, X. Chu, H. J. Crawford, M. Csanád, M. Daugherity, T. G. Dedovich, I. M. Deppner, A. A. Derevschikov, A. Dhamija, L. Di Carlo, L. Didenko, X. Dong, J. L. Drachenberg, J. C. Dunlop, N. Elsey, J. Engelage, G. Eppley, S. Esumi, O. Evdokimov, A. Ewigleben, O. Eyser, R. Fatemi, F. M. Fawzi, S. Fazio, P. Federic, J. Fedorisin, C. J. Feng, Y. Feng, P. Filip, E. Finch, Y. Fisyak, A. Francisco, C. Fu, L. Fulek, C. A. Gagliardi, T. Galatyuk, F. Geurts, N. Ghimire, A. Gibson, K. Gopal, X. Gou, D. Grosnick, A. Gupta, W. Guryn, A. I. Hamad, A. Hamed, Y. Han, et al. (292 additional authors not shown)
CCumulants and Correlation Functions of Net-proton, Proton and AntiprotonMultiplicity Distributions in Au+Au Collisions at RHIC
M. S. Abdallah, J. Adam, L. Adamczyk, J. R. Adams, J. K. Adkins, G. Agakishiev, I. Aggarwal, M. M. Aggarwal, Z. Ahammed, I. Alekseev,
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D. M. Anderson, A. Aparin, E. C. Aschenauer, M. U. Ashraf, F. G. Atetalla, A. Attri, G. S. Averichev, V. Bairathi, W. Baker, J. G. Ball Cap, K. Barish, A. Behera, R. Bellwied, P. Bhagat, A. Bhasin, J. Bielcik, J. Bielcikova, I. G. Bordyuzhin, J. D. Brandenburg, A. V. Brandin, I. Bunzarov, J. Butterworth, X. Z. Cai, H. Caines, M. Calder´on de la Barca S´anchez, D. Cebra, I. Chakaberia,
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P. Chaloupka, B. K. Chan, F-H. Chang, Z. Chang, N. Chankova-Bunzarova, A. Chatterjee, S. Chattopadhyay, D. Chen, J. Chen, J. H. Chen, X. Chen, Z. Chen, J. Cheng, M. Chevalier, S. Choudhury, W. Christie, X. Chu, H. J. Crawford, M. Csan´ad, M. Daugherity, T. G. Dedovich, I. M. Deppner, A. A. Derevschikov, A. Dhamija, L. Di Carlo, L. Didenko, X. Dong, J. L. Drachenberg, J. C. Dunlop, N. Elsey, J. Engelage, G. Eppley, S. Esumi, O. Evdokimov, A. Ewigleben, O. Eyser, R. Fatemi, F. M. Fawzi, S. Fazio, P. Federic, J. Fedorisin, C. J. Feng, Y. Feng, P. Filip, E. Finch, Y. Fisyak, A. Francisco, C. Fu, L. Fulek, C. A. Gagliardi, T. Galatyuk, F. Geurts, N. Ghimire, A. Gibson, K. Gopal, X. Gou, D. Grosnick, A. Gupta, W. Guryn, A. I. Hamad, A. Hamed, Y. Han, S. Harabasz, M. D. Harasty, J. W. Harris, H. Harrison, S. He, W. He, X. H. He, Y. He, S. Heppelmann, S. Heppelmann, N. Herrmann, E. Hoffman, L. Holub, Y. Hu, H. Huang, H. Z. Huang, S. L. Huang, T. Huang, X. Huang, Y. Huang, T. J. Humanic, D. Isenhower, W. W. Jacobs, C. Jena, A. Jentsch, Y. Ji, J. Jia,
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K. Jiang, X. Ju, E. G. Judd, S. Kabana, M. L. Kabir, S. Kagamaster, D. Kalinkin,
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K. Kang, D. Kapukchyan, K. Kauder, H. W. Ke, D. Keane, A. Kechechyan, Y. V. Khyzhniak, D. P. Kiko(cid:32)la, C. Kim, B. Kimelman, D. Kincses, I. Kisel, A. Kiselev, A. G. Knospe, L. Kochenda, L. K. Kosarzewski, L. Kramarik, P. Kravtsov, L. Kumar, S. Kumar, R. Kunnawalkam Elayavalli, J. H. Kwasizur, R. Lacey, S. Lan, J. M. Landgraf, J. Lauret, A. Lebedev, R. Lednicky, J. H. Lee, Y. H. Leung, C. Li, C. Li, W. Li, X. Li, Y. Li, X. Liang, Y. Liang, R. Licenik, T. Lin, Y. Lin, M. A. Lisa, F. Liu, H. Liu, P. Liu, T. Liu, X. Liu, Y. Liu, Z. Liu, T. Ljubicic, W. J. Llope, R. S. Longacre, E. Loyd, N. S. Lukow, X. Luo, L. Ma, R. Ma, Y. G. Ma, N. Magdy, R. Majka, ∗ D. Mallick, S. Margetis, C. Markert, H. S. Matis, J. A. Mazer, N. G. Minaev, S. Mioduszewski, B. Mohanty, M. M. Mondal, I. Mooney, D. A. Morozov, A. Mukherjee, M. Nagy, J. D. Nam, Md. Nasim, K. Nayak, D. Neff, J. M. Nelson, D. B. Nemes, M. Nie, G. Nigmatkulov, T. Niida, R. Nishitani, L. V. Nogach, T. Nonaka, A. S. Nunes, G. Odyniec, A. Ogawa, S. Oh, V. A. Okorokov, B. S. Page, R. Pak, A. Pandav, A. K. Pandey, Y. Panebratsev, P. Parfenov, B. Pawlik, D. Pawlowska, H. Pei, C. Perkins, L. Pinsky, R. L. Pint´er, J. Pluta, B. R. Pokhrel, G. Ponimatkin, J. Porter, M. Posik, V. Prozorova, N. K. Pruthi, M. Przybycien, J. Putschke, H. Qiu, A. Quintero, C. Racz, S. K. Radhakrishnan, N. Raha, R. L. Ray, R. Reed, H. G. Ritter, M. Robotkova, O. V. Rogachevskiy, J. L. Romero, L. Ruan, J. Rusnak, N. R. Sahoo, H. Sako, S. Salur, J. Sandweiss, ∗ S. Sato, W. B. Schmidke, N. Schmitz, B. R. Schweid, F. Seck, J. Seger, M. Sergeeva, R. Seto, P. Seyboth, N. Shah, E. Shahaliev, P. V. Shanmuganathan, M. Shao, T. Shao, A. I. Sheikh, D. Shen, S. S. Shi, Y. Shi, Q. Y. Shou, E. P. Sichtermann, R. Sikora, M. Simko, J. Singh, S. Singha, M. J. Skoby, N. Smirnov, Y. S¨ohngen, W. Solyst, P. Sorensen, H. M. Spinka, ∗ B. Srivastava, T. D. S. Stanislaus, M. Stefaniak, D. J. Stewart, M. Strikhanov, B. Stringfellow, A. A. P. Suaide, M. Sumbera, B. Summa, X. M. Sun, X. Sun, Y. Sun, Y. Sun, B. Surrow, D. N. Svirida, Z. W. Sweger, P. Szymanski, A. H. Tang, Z. Tang, A. Taranenko, T. Tarnowsky, J. H. Thomas, A. R. Timmins, D. Tlusty, T. Todoroki, M. Tokarev, C. A. Tomkiel, S. Trentalange, R. E. Tribble, P. Tribedy, S. K. Tripathy, T. Truhlar, B. A. Trzeciak, O. D. Tsai, Z. Tu, T. Ullrich, D. G. Underwood, I. Upsal,
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G. Van Buren, J. Vanek, A. N. Vasiliev, I. Vassiliev, V. Verkest, F. Videbæk, S. Vokal, S. A. Voloshin, F. Wang, G. Wang, J. S. Wang, P. Wang, Y. Wang, Y. Wang, Z. Wang, J. C. Webb, P. C. Weidenkaff, L. Wen, G. D. Westfall, H. Wieman, S. W. Wissink, R. Witt, J. Wu, Y. Wu, B. Xi, Z. G. Xiao, G. Xie, W. Xie, H. Xu, N. Xu, Q. H. Xu, Y. Xu, Z. Xu, Z. Xu, C. Yang, Q. Yang, S. Yang, Y. Yang, Z. Yang, Z. Ye, Z. Ye, L. Yi, K. Yip, Y. Yu, H. Zbroszczyk, W. Zha, C. Zhang, D. Zhang, S. Zhang, S. Zhang, X. P. Zhang, Y. Zhang, Y. Zhang, Y. Zhang, Z. J. Zhang, Z. Zhang, Z. Zhang, J. Zhao, C. Zhou, X. Zhu, Z. Zhu, M. Zurek, and M. Zyzak a r X i v : . [ nu c l - e x ] J a n (STAR Collaboration) Abilene Christian University, Abilene, Texas 79699 AGH University of Science and Technology, FPACS, Cracow 30-059, Poland Alikhanov Institute for Theoretical and Experimental Physics NRC ”Kurchatov Institute”, Moscow 117218, Russia Argonne National Laboratory, Argonne, Illinois 60439 American University of Cairo, New Cairo 11835, New Cairo, Egypt Brookhaven National Laboratory, Upton, New York 11973 University of California, Berkeley, California 94720 University of California, Davis, California 95616 University of California, Los Angeles, California 90095 University of California, Riverside, California 92521 Central China Normal University, Wuhan, Hubei 430079 University of Illinois at Chicago, Chicago, Illinois 60607 Creighton University, Omaha, Nebraska 68178 Czech Technical University in Prague, FNSPE, Prague 115 19, Czech Republic Technische Universit¨at Darmstadt, Darmstadt 64289, Germany ELTE E¨otv¨os Lor´and University, Budapest, Hungary H-1117 Frankfurt Institute for Advanced Studies FIAS, Frankfurt 60438, Germany Fudan University, Shanghai, 200433 University of Heidelberg, Heidelberg 69120, Germany University of Houston, Houston, Texas 77204 Huzhou University, Huzhou, Zhejiang 313000 Indian Institute of Science Education and Research (IISER), Berhampur 760010 , India Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India Indian Institute Technology, Patna, Bihar 801106, India Indiana University, Bloomington, Indiana 47408 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu 730000 University of Jammu, Jammu 180001, India Joint Institute for Nuclear Research, Dubna 141 980, Russia Kent State University, Kent, Ohio 44242 University of Kentucky, Lexington, Kentucky 40506-0055 Lawrence Berkeley National Laboratory, Berkeley, California 94720 Lehigh University, Bethlehem, Pennsylvania 18015 Max-Planck-Institut f¨ur Physik, Munich 80805, Germany Michigan State University, East Lansing, Michigan 48824 National Research Nuclear University MEPhI, Moscow 115409, Russia National Institute of Science Education and Research, HBNI, Jatni 752050, India National Cheng Kung University, Tainan 70101 Nuclear Physics Institute of the CAS, Rez 250 68, Czech Republic Ohio State University, Columbus, Ohio 43210 Institute of Nuclear Physics PAN, Cracow 31-342, Poland Panjab University, Chandigarh 160014, India Pennsylvania State University, University Park, Pennsylvania 16802 NRC ”Kurchatov Institute”, Institute of High Energy Physics, Protvino 142281, Russia Purdue University, West Lafayette, Indiana 47907 Rice University, Houston, Texas 77251 Rutgers University, Piscataway, New Jersey 08854 Universidade de S˜ao Paulo, S˜ao Paulo, Brazil 05314-970 University of Science and Technology of China, Hefei, Anhui 230026 Shandong University, Qingdao, Shandong 266237 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 Southern Connecticut State University, New Haven, Connecticut 06515 State University of New York, Stony Brook, New York 11794 Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Arica 1000000, Chile Temple University, Philadelphia, Pennsylvania 19122 Texas A&M University, College Station, Texas 77843 University of Texas, Austin, Texas 78712 Tsinghua University, Beijing 100084 University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan United States Naval Academy, Annapolis, Maryland 21402 Valparaiso University, Valparaiso, Indiana 46383 Variable Energy Cyclotron Centre, Kolkata 700064, India Warsaw University of Technology, Warsaw 00-661, Poland Wayne State University, Detroit, Michigan 48201 Yale University, New Haven, Connecticut 06520 (Dated: February 1, 2021)We report a systematic measurement of cumulants, C n , for net-proton, proton and antiproton,and correlation functions, κ n , for proton and antiproton multiplicity distributions up to the fourthorder in Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV. The C n and κ n are presented as a function of collision energy, centrality and kinematic acceptance inrapidity, y , and transverse momentum, p T . The data were taken during the first phase of theBeam Energy Scan (BES) program (2010 – 2017) at the Relativistic Heavy Ion Collider (RHIC)facility. The measurements are carried out at midrapidity ( | y | < < p T < c , using the STAR detector at RHIC. We observe a non-monotonic energydependence ( √ s NN = 7.7 – 62.4 GeV) of the net-proton C / C with the significance of 3.1 σ forthe 0-5% central Au+Au collisions. This is consistent with the expectations of critical fluctuationsin a QCD-inspired model. Thermal and transport model calculations show a monotonic variationwith √ s NN . For the multiparticle correlation functions, we observe significant negative values for atwo-particle correlation function, κ , of protons and antiprotons, which are mainly due to the effectsof baryon number conservation. Furthermore, it is found that the four-particle correlation function, κ , of protons plays a role in determining the energy dependence of proton C /C below 19.6 GeV,which cannot be solely understood by the negative values of κ for protons. I. INTRODUCTION
The main goal of the BES program at the RHIC isto study the QCD phase structure [1, 2]. This is ex-pected to lead to the mapping of the phase diagram forstrong interactions in the space of temperature ( T ) versusbaryon chemical potential ( µ B ). Both theoretically andexperimentally, several advancements have been madetowards this goal. Lattice QCD calculations have estab-lished that at high temperatures, there occurs a crossovertransition from hadronic matter to a deconfined state ofquarks and gluons at µ B = 0 MeV [3]. Experimentaldata from RHIC and the Large Hadron Collider (LHC)have provided evidence of this matter with quark andgluon degrees of freedom called the Quark-Gluon Plasma(QGP) [4–7]. The QGP has been found to hadronize intoa gas of hadrons, which undergoes chemical freeze-out(inelastic collisions cease) [8] at a temperature close tothe lattice QCD-estimated quark-hadron transition tem-perature at µ B = 0 MeV [9, 10]. A suite of interestingresults from the BES program indicate a change of equa-tion of state of QCD matter, with collision energy frompartonic-interaction-dominated matter at higher collisionenergies to a hadronic-interaction regime at lower ener-gies. These include the observations of breakdown in thenumber of constituent-quark scaling of the elliptic flow atlower √ s NN [11], non-monotonic variation of the slope ofthe directed flow for protons and net-protons at midra-pidity as a function of √ s NN [12], nuclear modificationfactor changing values from smaller than unity to largerthan unity at high p T as we go to lower √ s NN [13], and fi-nite to vanishing values of the three-particle correlationswith respect to the event plane [14] as we go to lower √ s NN .One of the most important studies of the QCD phase ∗ Deceased structure relates to the first-order phase boundary andthe expected existence of the critical point [15–21] at fi-nite baryon chemical potential. This is the end point ofa first-order phase boundary between quark-gluon andhadronic phases [22, 23]. Experimental confirmation ofthe CP would be a landmark of exploring the QCDphase structure. Previous studies of higher-order cu-mulants of net-proton multiplicity distributions suggestthat the possible CP region is unlikely to be below µ B = 200 MeV [24], which is consistent with the theoreticalfindings [18–21]. The versatility of the RHIC machinehas permitted the colliding energies of ions to be variedbelow the injection energy of √ s NN = 19.6 GeV [25], andthereby the RHIC BES program provides the possibilityto scan the QCD phase diagram up to µ B = 420 MeVwith the collider mode, and µ B = 720 MeV with thefixed-target mode [2, 26]. This, in turn, opens the possi-bility to find the experimental signatures of a first-orderphase transition and the CP [27, 28].Higher-order cumulants of the distributions of con-served charge, such as net-baryon ( B ), net-charge ( Q ),and net-strangeness ( S ) numbers, are sensitive to theQCD phase transition and CP [29–34]. The signaturesof conserved-charge fluctuations near the QCD criticalpoint have been extensively studied by various model cal-culations, such as the DSE method [35–37], PQM [38],FRG [39], NJL [40–42], PNJL [43, 44] and other effectivemodels [45–51]. However, these model calculations arebased on the assumption of thermal equilibrium with astatic and infinite medium. In heavy-ion collisions, finite-size and time effects will put constraints on the signifi-cance of the signals [52, 53]. A theoretical calculationsuggests the non-equilibrium correlation length ξ ≈ ξ ) [59, 60],it is proposed to study the higher moments – skewness ( S = (cid:10) ( δN ) (cid:11) /σ ) and kurtosis ( κ = (cid:10) ( δN ) (cid:11) /σ – 3) with δN = N – (cid:104) N (cid:105) , or cumulants C n (defined in Sec. II E)of distributions of conserved quantities. Both the magni-tude and the sign of the moments or C n [32, 61], whichquantify the shape of the multiplicity distributions, areimportant for understanding the phase transition and CPeffects. The aim is to search for signatures of the CP overa broad range of µ B in the QCD phase diagram [24].Furthermore, the products of the moments or ratios of C n can be related to susceptibilities associated with theconserved numbers. The product ( κσ ), or equivalently,the ratio ( C / C ) of the net-baryon number distributionis related to the ratio of fourth-order ( χ B4 ) to second-order( χ B2 ) baryon number susceptibilities [33, 62–65]. The ra-tio, χ B4 / χ B2 , is expected to deviate from unity near theCP. It has different values for the hadronic and partonicphases [65]. Similarly, the products S σ ( C / C ) and σ / (cid:104) N (cid:105) ( C / C ) are related to χ B3 / χ B2 and χ B2 / χ B , re-spectively. Experimentally, it is not possible to measurethe net-baryon distributions, however, theoretical calcu-lations have shown that net-proton multiplicity ( N p − N ¯ p = ∆ N p ) fluctuations reflect the singularity of the chargeand baryon number susceptibility, as expected at theCP [30]. Refs. [66, 67] discuss the effect of using net-proton as the approximation for the net-baryon distribu-tions and the acceptance dependence for the moments ofthe protons and antiprotons.In an early publication from the STAR experiment onthe higher moments of net-proton distributions, the se-lected kinematics of (anti)proton are | y | <
The data presented in the paper were obtained us-ing the Time Projection Chamber (TPC) [68] and theTime-of-Flight detectors (TOF) [72] of the SolenoidalTracker at RHIC (STAR) [68]. The event-by-event pro-ton ( N p ) and antiproton ( N ¯ p ) multiplicities are measuredfor Au+Au minimum-bias events at √ s NN = 7.7, 11.5,14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV for collisionsoccurring within a certain Z -position ( V z ) range of thecollision vertex (given in Table I) from the TPC centeralong the beam line. These data sets were taken with aminimum-bias trigger, which was defined using a coinci-dence of hits in the zero degree calorimeters (ZDCs) [73],vertex position detectors (VPDs) [74], and/or beam-beam counters (BBCs) [75]. The range of | V z | is cho-sen to optimize the event statistics and uniformity of theresponse of the detectors used in the analysis.In order to reject background events which involve in-teractions with the beam pipe, the transverse radius ofthe event vertex is required to be within 2 cm (1 cm for14.5 GeV) of the center of STAR [8]. We use two methodsto determine the V z : one from a fast scintillator-basedvertex position detector, and the other from the mostprobable point of common origin of the tracks, which arereconstructed from the hits measured in the TPC. To re-move pile-up events at energies above 27 GeV, we requirethe V z difference between the two methods to be within3 cm. Further, a detailed study of the TPC tracks as afunction of the TOF matched tracks with valid TOF in-formation is carried out and outlier events are rejected.To ensure the quality of the data, a run-by-run studyof several variables, such as the total number of uncor-rected charged particles measured in the TPC, averagetransverse momentum ( (cid:104) p T (cid:105) ) in an event, mean pseu-dorapidity ( η ) and azimuthal angle ( φ ) in an event, iscarried out. Outlier runs beyond ± σ , where σ corre-sponds to the standard deviation of run-by-run distribu-tions of a variable, are not included in the current analy-sis. In addition, the distance of closest approach (DCA)of the charged-particle track from the primary vertex, es-pecially the signed transverse DCA (DCA xy ) are studiedto remove bad events (The signed transverse DCA refersto the DCA with respect to the primary vertex in thetransverse plane. Its sign is the sign of the vector prod-uct of the DCA vector and the track momentum). Theseclasses of bad events are primarily related to unstablebeam conditions during the data taking and improperspace-charge calibration of the TPC. TABLE I. Total number of events for Au+Au collisions analysed for various collision energies ( √ s NN ) obtained after all of theevent selection criteria are applied. The Z -vertex ( V z ) range, the chemical freeze-out temperature ( T ch ) and baryon chemicalpotential ( µ B ) for 0-5% Au+Au collisions [8] are also given. √ s NN (GeV) No. of events (million) | V z | (cm) T ch (MeV) µ B (MeV)200 238 30 164.3 2862.4 47 30 160.3 7054.4 550 30 160.0 8339 86 30 156.4 10327 30 30 155.0 14419.6 15 30 153.9 18814.5 20 30 151.6 26411.5 6.6 30 149.4 2877.7 3 40 144.3 398FIG. 1. (Color online) Top left panel: The mass squared ( m ) versus rigidity for charged tracks in Au+Au collisions at √ s NN = 39 GeV. The rigidity is defined as momentum/z, where z is the dimensionless ratio of particle charge to the electron chargemagnitude. Bottom left panel: The specific ionization energy loss ( dE/dx ) as a function of rigidity measured in the TPC forthe same data set. Also shown as solid lines are the theoretical expectations for each particle species. Right panels: Rapidity( y ) versus transverse momentum ( p T ). The color reflects the relative yields of protons (top) and antiprotons (bottom) using theTPC PID for Au+Au collisions at √ s NN = 39 GeV. The dashed boxes represent the acceptance used in the current analysis.Two blobs at large rapidities are contaminated by particles other than (anti)protons. This contamination is rejected in latersteps of the analysis.TABLE II. Proton and antiproton track selection criteria at all energies. The N Fit and N
HitPoss represent the number of hitsused in track fitting and the maximum number of possible hits in the TPC. | y | p T (GeV/ c ) DCA (cm) N Fit N Fit / N HitPoss
No. of dE/dx points < < > > > Table I gives the total number of minimum-bias eventsanalyzed for each √ s NN and the corresponding chem-ical freeze-out temperature ( T ch ) and baryon chemicalpotential ( µ B ) values for central 0-5% Au+Au collisions.The beam energy values in the BES program are cho-sen so that the difference in µ B values is not larger than100 MeV between adjacent collision energies. B. Track selection, particle identification andacceptance
The proton and antiproton track selection criteria forall the √ s NN are presented in Table II. In order to sup-press contamination by tracks from secondary vertices, arequirement of less than 1 cm is placed on DCA betweeneach track and the event vertex. Tracks are requiredto have at least 20 points used in track fitting out ofa maximum of 45 possible hits in the TPC. To preventmultiple counting of split tracks, more than 52% of themaximum-possible fit points are required. A conditionis also placed on the number of points ( >
5) used to ex-tract the energy loss ( dE/dx ) values, which is used toidentify the (anti)protons from the charged particles de-tected in the TPC. The results presented here are withinkinematics | y | < < p T < c .Particle identification (PID) is carried out using theTPC and TOF by measuring the dE/dx and time offlight, respectively. Figure 1 (left top panel) shows a typ-ical plot of the square of the mass ( m ) associated with atrack measured in the TPC as a function of rigidity (de-fined as momentum/z, where z is the dimensionless ratioof particle charge to the electron charge magnitude) forAu+Au collisions at √ s NN = 39 GeV. The m is givenby: m = p (cid:18) c t L − (cid:19) , (1)where p , t , L , and c are the momentum, time-of-flight ofthe particle, path length, and speed of light, respectively.Protons and antiprotons can be identified by selectingcharged tracks for which 0.6 < m < /c .Figure 1 (left bottom panel) shows the dE/dx of mea-sured charged particles plotted as a function of the rigid-ity. The measured values of dE/dx are compared to theexpected theoretical values [77] (shown as solid lines inFig. 1) to select the proton and antiproton tracks. Aquantity called N σ,p for charged tracks in the TPC isdefined as: N σ,p = (1 /σ R ) ln (cid:18) (cid:104) dE/dx (cid:105)(cid:104) dE/dx (cid:105) th p (cid:19) , (2)where (cid:104) dE/dx (cid:105) is the truncated mean value of the trackenergy loss measured in the TPC, (cid:104) dE/dx (cid:105) th p is the cor-responding theoretical value for proton (or antiproton)in the STAR TPC [77] and σ R is the dE/dx resolu-tion which is momentum-dependent and of the order of 7.5% for the momentum range of this analysis. Assumingthat the N σ,p distribution in a given momentum range isGaussian, it should peak at zero for proton tracks andthe values represent the deviation from the theoreticalvalues for proton tracks in terms of standard deviations( σ R ). Momentum-dependent selection criteria are usedfor TPC tracks to select protons or antiprotons. For 0.4
Centrality selection plays a crucial role in the fluctu-ation analysis. There are two effects related to the cen-trality selection which need to be addressed. These are(a) the self-correlation [78, 79] and (b) centrality resolu-tion/fluctuations effects [78–81].One of the main self-correlation effects arises when par-ticles used for the fluctuation analysis are also used forthe centrality definition. This can be significantly re-duced by removing the particles used in the fluctuationanalysis from the centrality definition. Hence, we ex-clude protons and antiprotons from charged particles forthe centrality selection.The centrality resolution effect arises due to the factthat the number of participant nucleons and particle mul-tiplicities fluctuate even if the impact parameter is fixed.Through a model simulation it has been shown that thelarger the η acceptance used for centrality selection, the
200 400 600 800 - - - - - (a) 7.7 GeV
200 400 600 800 - - - - - (b) 11.5 GeV
200 400 600 800 - - - - - (c) 14.5 GeV
200 400 600 800 - - - - - (d) 19.6 GeV
200 400 600 800 - - - - - (e) 27 GeV
200 400 600 800 - - - - - (f) 39 GeV
200 400 600 800 - - - - - (g) 54.4 GeV
200 400 600 800 - - - - - (h) 62.4 GeV
200 400 600 800 - - - - - (i) 200 GeV
200 400 600 800 - - - - - Au+Au Col l isions Reference Charged Particle Multiplicity N o r m a li z ed N u m be r o f E v en t s FIG. 2. (Color online) The uncorrected reference charged particle multiplicity ( N ch ) distributions within pseudorapidity | η | < √ s NN = 7.7 - 200 GeV. These distributions are used for centralitydetermination. The shaded region at each √ s NN corresponds to 0-5% central collisions. The dashed line corresponds to MonteCarlo Glauber model simulations [76]. closer are the values of the cumulants to the actual val-ues [78]. This is because the centrality resolution is im-proved by increasing the number of particles for the cen-trality definition with wider acceptance. Therefore, tosuppress the effect of centrality resolution, one shoulduse the maximum available acceptance of charged parti-cles for centrality selection. In addition, it may be men-tioned that the choice of centrality definition also affectsthe way volume fluctuations (discussed later) contributeto the measurements.These are the driving considerations for the central-ity selection for net-proton studies presented in this pa-per and are discussed below. The basic strategy is tomaximize the acceptance window for the centrality de-termination as allowed by the detectors, and to not useprotons and antiprotons for the centrality selection. Inaddition, the centrality definition method given below isdetermined after several optimization studies using dataand models. These studies were carried out by varyingthe acceptances in η and charged particle types in or-der to understand the effect of the choice of centralitydetermination method on the analysis [79].In order to suppress the self-correlation, centrality res-olution and volume fluctuation effects with the avail-able STAR detectors, a new centrality measure is de-fined, and is different from other analyses reported bySTAR [8]. The centrality is determined from the uncor-rected charged particle multiplicity within pseudorapid-ity | η | < N ch ) after excluding the protons and antipro-tons. Strict particle identification criteria are used to re-move the proton and antiproton contributions. Chargedtracks with N σ,p < − TABLE III. The uncorrected number of charged particlesother than protons and antiprotons ( N ch ) within the pseu-dorapidity | η | < √ s NN = 7.7 – 200 GeV.% centrality N ch values at different √ s NN (GeV)200 62.4 54.4 39 27 19.6 14.5 11.5 7.70-5 725 571 621 522 490 448 393 343 2705-10 618 482 516 439 412 376 330 287 22510-20 440 338 354 308 289 263 231 199 15520-30 301 230 237 209 196 178 157 134 10530-40 196 149 151 136 127 116 103 87 6840-50 120 91 90 83 78 71 63 53 4150-60 67 51 50 47 44 40 36 30 2360-70 34 26 24 24 22 20 19 15 1170-80 16 12 10 11 10 9 13 7 5 which have TOF information an additional criterion, m < /c is applied. The resultant distribu-tion of charged particles is corrected for luminosity and V z dependence at each √ s NN . The corrected chargedparticle distribution is then fit to a Monte Carlo GlauberModel [25, 76] to define the centrality classes in the ex-periment (the percentage cross section and the associatedcuts on the charged-particle multiplicity). In the fittingprocess, a multiplicity-dependent efficiency has been ap-plied [25].Figure 2 shows the reference charged particle multi-plicity distributions after excluding protons and antipro-tons used for centrality determination for all of the √ s NN TABLE IV. The average number of participant nucleons ( (cid:104) N part (cid:105) ) for various collision centralities in Au+Au collisions at √ s NN = 7.7 – 200 GeV from a Monte Carlo Glauber Model. The numbers in parentheses are systematic uncertainties.% centrality (cid:104) N part (cid:105) values at different √ s NN (GeV)200 62.4 54.4 39 27 19.6 14.5 11.5 7.70-5 351 (2) 347 (3) 346 (2) 342(2) 343 (2) 338 (2) 340(2) 338 (2) 337 (2)5-10 299 (4) 294 (4) 292 (6) 294 (6) 299 (6) 289 (6) 289 (6) 291 (6) 290 (6)10-20 234 (5) 230 (5) 228 (8) 230 (9) 234 (9) 225 (9) 225 (8) 226 (8) 226 (8)20-30 168 (5) 164 (5) 161 (10) 162 (10) 166 (11) 158 (10) 159 (9) 160 (9) 160 (10)30-40 117 (5) 114 (5) 111 (11) 111 (11) 114 (11) 108 (11) 109 (11) 110 (11) 110 (11)40-50 78 (5) 76 (5) 73 (10) 74 (10) 75 (10) 71 (10) 72 (10) 73 (10) 72 (10)50-60 49 (5) 48 (5) 45 (9) 46 (9) 47 (9) 44 (9) 45 (9) 45 (9) 45 (9)60-70 29 (4) 28 (4) 26 (7) 26 (7) 27 (8) 26 (7) 26 (7) 26 (7) 26 (7)70-80 16 (3) 15 (2) 13 (5) 14 (5) 14 (6) 14 (5) 14 (6) 14 (6) 14 (4) - - - - - -
10 1 (a) 7.7 GeV - - - - - -
10 1 (b) 11.5 GeV - - - - - -
10 1 (c) 14.5 GeV - - - - - -
10 1 (d) 19.6 GeV - - - - - -
10 1 (e) 27 GeV - - - - - (f) 39 GeV - - - - - (g) 54.4 GeV - - - - - (h) 62.4 GeV - - - - - (i) 200 GeV - - - - - Au+Au Collisions < 2.0 (GeV/c) T ) p N D Net-Proton Multiplicity ( N o r m a li z ed N u m be r o f E v en t s FIG. 3. (Color online) Net-proton multiplicity (∆ N p ) distributions in Au+Au collisions at various √ s NN for 0-5%, 30-40% and70-80% collision centralities at midrapidity. The statistical errors are small and within the symbol size. The distributions arenot corrected neither for the finite-centrality-width effect nor for the reconstruction efficiencies of protons and antiprotons. studied here. The lower boundaries of each centralityclass based on N ch are given in Table III. Table IV givesthe average number of participant nucleons ( (cid:104) N part (cid:105) ) forvarious collision centralities for √ s NN = 7.7 - 200 GeVobtained from a Monte Carlo Glauber model simulation. D. Uncorrected net-proton multiplicitydistributions
Figure 3 shows the event-by-event net-proton multi-plicity (∆ N p ) distributions from Au+Au collisions at √ s NN = 7.7 – 200 GeV for 0-5%, 30-40% and 70-80% col-lision centralities. The ∆ N p distribution is obtained bycounting the number of protons and antiprotons withinthe y - p T acceptance on an event-by-event basis for agiven collision centrality and √ s NN . The distributionspresented in Fig. 3 are not corrected for the efficiency and acceptance effects. In general, the shape of the ∆ N p distributions is broader, more symmetric and closer toGaussian, for central collisions than for peripheral colli-sions. The shape of the distributions also changes with √ s NN . Cumulants ( C n ) up to the fourth order are ob-tained from these distributions for each collision central-ity and √ s NN . E. Definition of cumulants and integratedcorrelation functions
In this subsection, we give the definition of the cu-mulants used in this paper. Let N represent any entryin the data sample, its deviation from its mean value( (cid:104) N (cid:105) , referred to as the 1 st moment) is then given by δN = N − (cid:104) N (cid:105) . Any r th -order central moment is definedas: µ r = (cid:104) ( δN ) r (cid:105) . (3)The cumulants of a given data sample could be writtenin terms of moments as follows: C = (cid:104) N (cid:105) , (4) C = (cid:104) ( δN ) (cid:105) = µ , (5) C = (cid:104) ( δN ) (cid:105) = µ , (6) C = (cid:104) ( δN ) (cid:105) − (cid:104) ( δN ) (cid:105) = µ − µ , (7) C n ( n >
3) = µ n − n − (cid:88) m =2 (cid:18) n − m − (cid:19) C m µ n − m . (8)The relations between cumulants and various momentsare given as: M = C , σ = C , S = C ( C ) / , κ = C ( C ) . (9)where M , σ , S and κ are mean, variance, skewness andkurtosis, respectively. The products κσ and Sσ can beexpressed in terms of the ratio of cumulants as: σ /M = C C , Sσ = C C , κσ = C C . (10)With the above definition, we can calculate various or-der cumulants (moments) and cumulant ratios (momentproducts) from the measured event-by-event net-proton,proton and antiproton distributions for each centralityat a given √ s NN . For two independent variables X and Y , the cumulants of the probability distributionsof their sum ( X + Y ), are just the addition of cumu-lants of the individual distributions for X and Y i.e.C n,X + Y = C n,X + C n,Y for n th -order cumulant. Fora distribution of difference between X and Y , the cu-mulants are C n,X − Y = C n,X + ( − n C n,Y , where theeven-order cumulants are the addition of the individualcumulants, while the odd-order cumulants are obtainedby taking their difference. If the protons and antiprotonsare distributed as independent Poissonian distributions,the various order cumulants of net-proton, proton andantiproton distributions can be expressed as: C n,p = C ,p , C n, ¯ p = C , ¯ p ,C n,p − ¯ p = C ,p + ( − n C , ¯ p where the net-proton multiplicity distributions obeythe Skellam distribution and the Poisson base-line/expectation values of the net-proton, proton and an-tiproton cumulant ratios are:( σ /M ) p, ¯ p = ( Sσ ) p, ¯ p = ( κσ ) p, ¯ p = 1 , ( σ /M ) p − ¯ p = 1( Sσ ) p − ¯ p = C ,p + C , ¯ p C ,p − C , ¯ p , ( κσ ) p − ¯ p = 1 where C ,p and C , ¯ p are the mean values of proton andantiproton, respectively.On the other hand, it is expected that close to the CP,the three- and four-particle correlations are dominant rel-ative to two-particle correlations [59]. The various ordersintegrated correlation functions of proton and antipro-ton ( κ n , also known as factorial cumulants) are relatedto the corresponding proton and antiproton cumulants( C n ) through the following relations [82–84]: κ = C = (cid:104) N (cid:105) ,κ = − C + C ,κ = 2 C − C + C ,κ = − C + 11 C − C + C ,C = κ + κ ,C = κ + 3 κ + κ ,C = κ + 6 κ + 7 κ + κ , (11)where C and κ represent the mean values for protons orantiprotons. For proton and antiproton cumulant ratios C /C , C /C and C /C , they can be expressed in termsof corresponding normalized correlation functions κ n /κ ( n >
1) as: C C = κ κ + 1 , (12) C C = κ /κ − κ /κ + 1 + 3 , (13) C C = κ /κ + 6 κ /κ − κ /κ + 1 + 7 , (14)The higher-order integrated correlation functions κ n ( n >
1) are equal to zero when the distributions arePoisson. Thus, κ n can be used to quantify the deviationsfrom the Poisson distributions in terms of n -particle cor-relations. For simplicity, from here on, we refer to the κ n as correlation functions instead of integrated correlationfunctions.In the following subsections, we discuss correctionsthat are related to collision centrality bin width (Sec.F) and detection efficiency (Sec. G). This is followed bythe estimation of statistical and systematic uncertaintiesin sections H and I, respectively. F. Centrality bin width correction
The data are presented in this paper as a function ofvarious collision centrality classes for 0-5%, 5-10%, 10-20%, 20-30%, 30-40%, 40-50%, 50-60%, 60-70% and 70-80%. The finite size of centrality bins implies that theaverage number of protons and antiprotons varies evenwithin a centrality class. This variation has to be ac-counted for while calculating the cumulants in a broadcentrality class. In addition, it is known that calculatingcumulants in such broad centrality bins leads to a strongenhancement of cumulants and cumulant ratios due toinitial volume fluctuations [78, 85].0
Net-ProtonAu + Au Collisions - - |y| < 0.5 < 2.0 (GeV/c) T CBWC 10% D no CBWC 5% D no CBWC 2.5% D no CBWC C C C C æ part N Æ Average Number of Participant Nucleons
FIG. 4. (Color online) C n of net-proton distributions in Au+Au collisions at √ s NN = 7.7, 19.6 and 62.4 GeV as a function of (cid:104) N part (cid:105) . The results are shown for 10%, 5% and 2.5% centrality bins without CBWC and for 9 centrality bins (0-5%, 5-10%,10-20%,..., 70-80%) with CBWC. The bars are the statistical uncertainties. A Centrality Bin Width Correction (CBWC) is theprocedure used to take care of the measurements in awide centrality bin and is based on weighting the cumu-lants measured at each multiplicity bin by the number ofevents in the bin [78, 79, 85]. This procedure is mathe-matically expressed in the equation below: C n = (cid:80) r n r C rn (cid:80) r n r = (cid:88) r ω r C rn , (15)where the n r is the number of events at the r th multiplic-ity bin for the centrality determination, the C rn representsthe n th -order cumulant of particle number distributionsat r th multiplicity. The corresponding weight for the r th multiplicity bin is ω r = n r / (cid:80) r n r .Figure 4 shows the C n up to the fourth order as afunction of (cid:104) N part (cid:105) for three different collision energies: √ s NN = 7.7, 19.6 and 62.4 GeV. For each C n case, fourdifferent results are shown. One of them is the CBWCresult for 9 collision centrality bins, which correspond to0-5%, 5-10%, 10-20%, 20-30%,...,70-80%. For compar-ison, cumulants are also calculated for the other threecases, which are 10%, 5% and 2.5% centrality bin widthwithout CBWC. The higher-order cumulant results with10% centrality bins are found to have significant devi-ations compared to those with 5% and 2.5% centralitybins without CBWC. This finding means that it is im- (GeV) NN s sk Au+Au Collisions, 0-5% (GeV/c) < 2.0 T |y| < 0.5, 0.4 < p CBWCVolume fluctuation correction
FIG. 5. (Color online) κσ as a function of collision energyfor Au+Au collisions for 0-5% centrality. The data has beencorrected for volume fluctuation effects using CBWC, a datadriven approach, and a model-dependent volume fluctuationcorrection method. The bars are the statistical uncertainties. portant to correct for the CBW effect, as one normallyexpects that, irrespective of the centrality bin width, thecumulant values should exhibit the same dependence on (cid:104) N part (cid:105) . It is found that the results get closer to CBWCresults with narrower centrality bins and the results with2.5% centrality bins almost overlap with CBWC results,1 C C C C STAR Au+Au Collisions < p T < | y | < Average Number of Participant Nucleons h N part i Net-Proton Proton Anti-proton
Efficiency-uncorrected
FIG. 6. (Color online) Efficiency-uncorrected C n of net-proton, proton and antiproton multiplicity distributions in Au+Aucollisions at √ s NN = 7.7– 200 GeV as a function of (cid:104) N part (cid:105) . The results are CBW-corrected. The bars are the statisticaluncertainties. which indicates that the CBWC can effectively suppressthe effect of the volume fluctuations on cumulants (up tothe fourth order) within a finite centrality bin width.A different approach, volume fluctuation correction(VFC) method [86, 87], which assumes independent pro-duction of protons, has been also applied at √ s NN =7.7, 19.6 and 62.4 GeV for 0-5% Au+Au central col-lisions. The correction factors are determined by theGlauber model [87]. Figure 5 shows the comparison be-tween the results based on CBWC and VFC methods.As can be seen from the plot, for the 0-5% central colli-sions, the results of CBWC and VFC are found to be con-sistent within statistical uncertainties. However, follow-up UrQMD model studies reported in Ref. [81], indicatethat the VFC method (as discussed in Ref. [86]) does notwork, as the independent particle production model as-sumed in the VFC is expected to be broken. Therefore,we follow the data-driven method, CBWC, in this paper. G. Efficiency correction
Figure 6 shows the efficiency-uncorrected C n for pro-ton, antiproton and net-proton multiplicity distributionsin Au+Au collisions at √ s NN = 7.7 – 200 GeV as afunction of (cid:104) N part (cid:105) . This section discusses the methodof efficiency correction. One such method is called thebinomial-model-based method [67, 84, 88–90] and an-other is the unfolding method [91, 92]. The cumulantspresented in the subsequent sections are corrected for ef- ficiency and acceptance effects related to proton and an-tiproton reconstruction, unless specified otherwise.
1. Binomial model method
The binomial-based method involves two steps. Firstwe obtain the efficiency of proton and antiproton recon-struction in the STAR detector and then correct the cu-mulants for efficiency and acceptance effects using ana-lytic expressions. The former uses the embedding processand the latter invokes binomial model assumptions forthe detector response function for the efficiencies. Onecan find more details in Appendix A.The detector acceptance and the efficiency of recon-structing proton and antiproton tracks are determinedtogether by embedding Monte Carlo (MC) tracks, simu-lated using the GEANT [93] model of the STAR detectorresponse, into real events at the raw data level. One im-portant requirement is the matching of the distributionsof reconstructed embedded tracks and real data tracksfor quantities reflecting track quality and those used fortrack selection [8]. The ratio of the distribution of recon-structed to embedded Monte Carlo tracks as a functionof p T gives the efficiency × acceptance correction factor( ε TPC ( p T )) for the rapidity interval studied. We refer tothis factor as simply efficiency.The current analysis makes use of both the TPC andthe TOF detectors. While the TPC identifies low p T (0 . < p T < . c ) protons and antiprotons with2 æ part N Æ Averaged Number of Participant Nucleons D e t e c t o r E ff i c i en cy x A cc ep t an c e ( % ) Au+Au Collisions at RHICAu+Au Collisions at RHIC < 2.0 (GeV/c) T |y| < 0.5, 0.4 < p protonanti-proton < 0.8 (GeV/c) T T FIG. 7. (Color online) Efficiencies of proton and antiproton as a function of (cid:104) N part (cid:105) in Au+Au collisions for various √ s NN . Forthe lower p T range (0 . < p T < . c ), only the TPC is used. For the higher p T range (0 . < p T < c ), both theTPC and TOF are used. high purity, the TOF gives better particle identificationthan the TPC in the higher p T range (0 . < p T < . c ). However, not all TPC tracks have valid TOFinformation due to the limited TOF acceptance and themismatching of the TPC tracks to TOF hits. Thisextra efficiency is called the TOF-matching efficiency( ε TOF ( p T )). The TOF-matching efficiency is particle-species-dependent and can be obtained using a data-driven technique, which is defined as the ratio of the num-ber of (anti)proton tracks detected in the TOF to the to-tal number of (anti)proton tracks in the TPC within thesame acceptance [8]. Thus, the final average (anti)protonefficiency within a certain p T range can be calculated as: (cid:104) ε (cid:105) = p T (cid:82) p T ε ( p T ) f ( p T ) dp Tp T (cid:82) p T f ( p T ) dp T , (16)where the p T -dependent efficiency, ε ( p T ), is defined as ε ( p T ) = ε TPC ( p T ) for 0 . < p T < . c and ε ( p T ) = ε TPC ( p T ) × ε TOF ( p T ) for 0 . < p T < . c . Thefunction f ( p T ) is the efficiency-corrected p T spectrumfor (anti)protons [8].Figure 7 shows the average efficiency ( (cid:104) ε (cid:105) ) for protonsand antiprotons at midrapidity ( | y | < (cid:104) N part (cid:105) ). For 0 . < p T < c the efficiency is only from the TPC and for0 . < p T < c it is the product of efficienciesfrom the TPC and TOF. In Fig. 7, only statistical uncer-tainties are presented and a ±
5% systematic uncertaintyassociated with determining the efficiency is consideredin the analysis.
2. Unfolding method
In this section we discuss the effect of efficiency cor-rection on the C n measurement if the assumption of bi-nomial detector efficiency response breaks down due tosome of the reasons given in Refs. [94, 95]. The techniqueis based on unfolding of the detector response [91, 92].The response function is obtained by MC simulationscarried out in the STAR detector environment [93]. MCtracks are simulated through GEANT and embeddedin the real data, track reconstruction is performed asis done in the real experiment. Many effects can leadto non-binomial detector response in heavy-ion experi-ments. One of those effects could be track merging dueto the extreme environment of high particle multiplic-ity densities in the detector. Hence, we have performedthe embedding simulations using the real data for 0-5% Au+Au collisions at √ s NN = 200 GeV. The num-3 N o r m a li z ed C oun t s =10 pbar =10, N p N 00.10.2 N o r m a li z ed C oun t s =20 pbar =10, N p N 00.10.2 N o r m a li z ed C oun t s =40 pbar =10, N p N R a t i o R a t i o R a t i o N o r m a li z ed C oun t s =5 pbar =20, N p N 00.050.10.15 N o r m a li z ed C oun t s =10 pbar =20, N p N 00.050.10.15 N o r m a li z ed C oun t s =20 pbar =20, N p N 00.050.10.15 N o r m a li z ed C oun t s =40 pbar =20, N p N R a t i o R a t i o R a t i o R a t i o N o r m a li z ed C oun t s =10 pbar =40, N p N 00.050.1 N o r m a li z ed C oun t s =20 pbar =40, N p N 00.050.1 N o r m a li z ed C oun t s =40 pbar =40, N p N 00.050.1 N o r m a li z ed C oun t s =5 pbar =40, N p N R a t i o R a t i o R a t i o R a t i o Number of Protons = 200 GeV, 0-2.5% NN s < 2.0 (GeV/c) T |y| < 0.5, 0.4 < pEmbeddingBinomial fitBeta. fit Binomial/embeddingBeta./embedding FIG. 8. (Color online) Distributions of reconstructed protons (black circles) from embedding simulations in 200 GeV top2.5%-central Au+Au collisions. Red lines are fits to the binomial distribution, and green dotted lines represent the fit with thebeta-binomial distributions using the α that gives the minimum χ / ndf. Each panel presents results for a different combinationof the number of embedded protons and antiprotons as labeled in the legend. The ratio of the fits to the embedding data isshown for each panel at the bottom. ber of embedded tracks of N p and N ¯p are varied within5 ≤ N p(¯p) ≤
40. Since we are measuring the net-protonmultiplicity distributions, protons and antiprotons areembedded simultaneously. We have shown in Ref. [96]that for the event statistics in the current analysis, theefficiencies for kaon reconstruction follow binomial distri-butions.Figure 8 shows the reconstructed protons from theembedding data (black circles) of Au+Au collisionsat √ s NN = 200 GeV and 0-2.5% collision centrality.Each panel represents a different number of embedded(anti)protons. These distributions are fitted by a bino-mial distribution (red solid line) at a fixed efficiency ε .The ratios of the fitted function to the embedding dataare shown in the lower panels. The fitted χ / ndf rangesfrom 5.2 to 17.8 and the tails of the distributions arenot well described by the binomial distribution for sev-eral combinations of embedded N p and N ¯p tracks. Wefind that the embedding data is better described by abeta-binomial distribution given by: β ( n : N, a, b ) = (cid:90) dpB ( ε, a, b )B( n ; N, ε ) , (17)and with the beta distribution given as: β ( ε ; a, b ) = ε a (1 − ε ) b / B( a, b ) , (18)where B ( a, b ) is the beta function. The beta-binomialdistribution is given by an urn model. Let us consider −
10 110 C oun t s = 200 GeV, 0-2.5% NN s < 2.0 (GeV/c) T |y| < 0.5, 0.4 < p Binomial σ + α Beta-binomial, α Beta-binomial, σ - α Beta-binomial, − − p N ∆ B e t a ./ B i no m . FIG. 9. (Color online) Unfolded net-proton multiplicity dis-tributions for √ s NN = 200 GeV Au+Au collisions where thebinomial distribution (black circle), beta-binomial distribu-tions with α + σ (green triangle), α (red square) and α − σ (blue triangle) are utilised in response matrices. Ratios of thebeta-binomial unfolded distributions to that from binomialresponse matrices are shown in the bottom panel. TABLE V. Net-proton cumulant ratios and their statistical errors for 0-5% central Au+Au collisions at √ s NN = 200 GeV,(second column) from the conventional efficiency correction with the binomial detector response, and (third column) fromunfolding with the beta-binomial detector response. Systematic errors are also shown for the beta-binomial case. The lastcolumn shows the difference between two results normalized by total uncertainty, which is equal to the statistical and systematicuncertainties summed in quadrature.Cumulant ratio binomial ± statistical error beta ± statistical error ± systematical error significance C /C . ± neg . . ± neg . ± .
03 3 . C /C . ± .
01 0 . ± . ± neg . . × − C /C . ± .
21 0 . ± . ± .
08 4 . × − C /C . ± . − . ± . ± .
11 3 . × − C /C − . ± . − . ± . ± .
07 1 . N w white balls and N b black balls in the urn. Onedraws a ball from the urn. If it is white (black), re-turn two white (black) balls to the urn. This procedureis repeated with N times, then the resulting distributionof n white balls is given by the beta-binomial distribu-tions as β ( n ; N, N w , N b ). This is actually equivalent to β ( n ; N, α, ε ), where N w = αN with ε = N w / ( N w + N b ).A smaller α gives a broader distribution than the bino-mial, while the distribution becomes close to the binomialdistribution with a larger value of α .The beta-binomial distributions are numerically gener-ated with various values of α . These are compared to theembedding data to determine the best fit parameter valueof α . The green lines in Fig. 8 show the beta-binomialdistribution for the value of α that gives the minimum χ / ndf. It is found that χ / ndf ≈ N p , N ¯ p )combinations. With this additional parameter α , it isfound that the detector response is better described inthe tails by a beta-binomial distribution compared to abinomial distribution.From the embedding simulations as discussed above,the ε and α are parametrized as a function of N p and N ¯p . Using the parametrization, a 4-dimensional responsematrix between generated and reconstructed protons andantiprotons is generated with 1 billion events. The lim-ited statistics in the embedding simulations lead to un-certainties on the α values. Therefore, two more responsematrices are generated using α − σ and α + σ , where σ isthe statistical uncertainty on the α values determined bythe embedding simulation. Furthermore, the standardresponse matrices are also generated with the binomialdistribution as a reference using a multiplicity-dependentefficiency. These response matrices are used to correct forthe detector effects as a confirmation of this approach bycomparing to the binomial correction method describedin the previous section. The consistency of the unfoldingmethod has been checked through a detailed simulationand an analytic study.Figure 9 shows the unfolded net-proton distributionsfor 200 GeV Au+Au collisions at 0-2.5% centrality. Re-sults from four assumptions on the detector response areshown, one is the binomial detector response and theother three assume the beta-binomial distributions withdifferent non-binomial α values. The ratios of the beta- binomial unfolded distributions to the binomial unfoldeddistributions are shown in the bottom panel. The un-folded distributions with beta-binomial response matri-ces are found to be narrower with a decreasing value of α .Calculations are done for 0-2.5% and 2.5-5.0% central-ities separately and averaged to determine the C n valuesfor the 0-5% centrality. The C n values and their ratiosfrom data obtained using the binomial model method ofefficiency correction and those using the binomial detec-tor response matrix in the unfolding method are con-sistent. Table V summarizes the cumulant ratios andtheir errors. Results are also obtained from the unfold-ing method using the beta-binomial response functionwith non-binomial parameters in the range α ± σ . Thisrange in values of α is used to generate the systematicuncertainties associated with the unfolding method. Thedeviations of those non-binomial efficiency-corrected re-sults with respect to the conventional efficiency correc-tion with binomial detector response is found to be 3.1 σ for C /C and less than 1.0 σ for C /C and for C /C .The σ value is the statistical and systematic uncertaintiesadded in quadrature.These studies have been done for Au+Au collisions forthe highest collision energy of √ s NN = 200 GeV and top-most 5% centrality. This set of data provides the largestcharged-particle-density environment for the detectors,where we expect the maximum non-binomial detector ef-fects. Even in this situation, the differences in the twomethods of efficiency correction are at a level of less thanone σ . Thus, we conclude that the non-binomial detectoreffects on higher-order cumulant ratios presented in thiswork are within the uncertainties quoted for all of theBES-I energies. H. Statistical uncertainty
The higher-order cumulants are sensitive to the shapeof the distribution, and estimating their statistical un-certainty is crucial due to the limited available statistics.It has been shown that among the various methods ofobtaining statistical uncertainty on cumulants, the deltatheorem method [97] and the bootstrap method [78, 88,5 - - · C BootstrapDelta Theorem - - · C - C Au+Au Collisions = 19.6 GeV NN sNet-Proton|y|<0.5, <2.0 (GeV/c) T - C > part
0. The N and ε denote the number of events andthe particle-reconstruction efficiency, respectively. Thus,one can find that the statistical uncertainty strongly de-pends on the width ( σ ) of the distributions. For similarevent statistics, due to the increasing width of the net-proton distributions from peripheral to central collisions, the statistical uncertainties are larger in central collisionsthan those from peripheral. Furthermore, the reconstruc-tion efficiency increases the statistical uncertainties onthe cumulants compared to their corresponding uncor-rected case. A more detailed discussion can be found inAppendix B.The bootstrap method finds the statistical uncertain-ties on the cumulants in a Monte Carlo way by formingbootstrap samples. It makes use of a random selectionof elements with replacement from the original sampleto construct bootstrap samples over which the samplingvariance of a given order cumulant is calculated [99, 100].Let X be a random sample representing the experimen-tal dataset. Let µ r be the estimator of a statistic (suchas mean or variance etc.), on which we intend to find thestatistical error. Given a parent sample of size n , con-struct B number of independent bootstrap samples X ∗ , X ∗ , X ∗ , ..., X ∗ B , each consisting of n data points ran-domly drawn with replacement from the parent sample.Then evaluate the estimator in each bootstrap sample: µ ∗ r = µ r ( X ∗ b ) b = 1 , , , ..., B. (21)Then obtain the sampling variance of the estimator as:Var( µ r ) = 1 B − B (cid:88) b =1 (cid:16) µ ∗ r − ¯ µ r (cid:17) , (22)where ¯ µ r = B (cid:80) Bb =1 ( µ ∗ r ). The value of B is optimizedand in general, the larger the value of B the better theestimate of the error.Figure 10 shows the statistical uncertainties on vari-ous orders of C n obtained using the delta theorem andbootstrap methods for Au+Au collisions at √ s NN =19.6 GeV. The results are shown as a function of (cid:104) N part (cid:105) for each C n . The value of B is 200. Good agreementof the statistical uncertainties is seen from both meth-ods. The delta theorem method is used for obtaining thestatistical uncertainties on the results discussed below. I. Systematic uncertainty
Systematic uncertainties are estimated by varying thefollowing requirements for p (¯ p ) tracks: DCA, track qual-ity (as reflected by the number of fit points used intrack reconstruction), dE/dx and m for p (¯ p ) identifica-tion [70]. A ±
5% systematic uncertainty associated withdetermining the efficiency is also considered [8]. All of thedifferent sources of systematic uncertainty are added inquadrature to obtain the final systematic uncertaintieson the C n and its ratios. Figure 11 shows the variationsof the cumulants ratios with the changes in the above se-lection criteria for the net-proton distributions in Au+Aucollisions at √ s NN = 200 GeV.Table VI gives the systematic uncertainties on the C n of the net-proton distribution for 0-5% central Au+Aucollisions at √ s NN = 7.7 - 200 GeV. The statistical and6 dca < 0.8dca < 0.9dca < 1.0dca < 1.1dca < 1.2 nFitPts > 15nFitPts > 18nFitPts > 20nFitPts > 22nFitPts > 25 | < 1.6 p s |n | < 1.8 p s |n | < 2.0 p s |n | < 2.2 p s |n | < 2.5 p s |n ) /c < 1.10(GeV /c < 1.15(GeV /c < 1.20(GeV /c < 1.25(GeV /c < 1.30(GeV Au+Au200 GeVNet-Proton +5%-5%default T / C C / C C / C C æ part N Æ Average Number of Participant Nucleons
FIG. 11. (Color online) Ratios of cumulants ( C n ) as a function of (cid:104) N part (cid:105) , for net-protons distributions in Au+Au collisionsat √ s NN = 200 GeV obtained by varying the analysis criteria in terms of track selection criteria, particle identification criteriaand efficiency. Since variations with respect to default selection criteria are used to obtain the systematic uncertainties on themeasurements, the errors are shown only for the default case. C æ part N Æ Averaged Number of Participant Nucleons C C C C Au+Au Collisions at RHICAu+Au Collisions at RHIC < 2.0 (GeV/c) T |y| < 0.5, 0.4 < p Proton Anti-proton Net-proton
Statistical error Syst e matic error FIG. 12. (Color online) Collision centrality dependence of proton (open squares), antiproton (open triangles) and net-proton(filled circles) cumulants from (7.7 – 200 GeV) Au+Au collisions at RHIC. The data are from | y | < . . < p T < . c . Statistical and systematic uncertainties are shown as the narrow black and wide grey bands, respectively. Note thatthe net-proton and proton C from 0-5% and 5-10% central Au+Au collisions at 7.7 GeV have been scaled down by a factorof 2, indicated in the yellow box. TABLE VI. Total systematic uncertainty as well as the absolute uncertainties from individual sources, such as DCA andNhitsFit, for net-proton C n in 0-5% central Au+Au collisions at √ s NN = 7.7 - 200 GeV. The total systematic uncertainties areobtained by adding the uncertainties from individual sources in quadrature. √ s NN (GeV) Cumulant Total syst. DCA NhitsFit N σ,p m Efficiency C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C systematic uncertainties are presented separately in thefigures. III. RESULTS
In this section we present the efficiency-corrected cu-mulants and cumulant ratios of net-proton, proton andantiproton multiplicity distributions in Au+Au collisionsat √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and200 GeV. The cumulant ratios are related to the ratios ofbaryon number susceptibilities ( χ B ) computed in QCD-motivated models as: σ / M = χ B /χ B , S σ = χ B /χ B and κσ = χ B /χ B [33, 62–65]. Normalized correlation func-tions ( κ n /κ , n >
1) for proton and antiproton extractedfrom the measured C n are also presented. The statisticaluncertainties on κ n are obtained from the uncertainties on C n using standard error propagation method. Theseresults will be also compared to corresponding resultsfrom a hadron resonance gas (HRG) [101] and hadronic-transport-based UrQMD model calculations [102, 103].In the following subsections, the dependence of thecumulants and correlation functions on collision energy,centrality, rapidity and transverse momentum are pre-sented. The corresponding physics implications are dis-cussed. A. Centrality dependence
In this subsection, we show the (cid:104) N part (cid:105) (representingcollision centrality) dependence of the cumulants, cu-mulant ratios and normalized correlation functions inAu+Au collisions at √ s NN = 7.7 – 200 GeV. To un-8 C / C
27 GeV C / C C / C
39 GeV
200 GeV
STAR Au+Au Collisions < p T < | y | < Average Number of Participant Nucleons h N part i Net-ProtonProton Anti-proton
FIG. 13. (Color online) Collision centrality dependence of the cumulant ratios of proton, antiproton and net-proton multiplicitydistributions for Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV. The bars and caps representthe statistical and systematic uncertainties, respectively. derstand the evolution of the centrality dependence ofthe cumulants and cumulant ratios, we invoke the cen-tral limit theorem and consider the distribution at anygiven centrality i to be a superposition of several inde-pendent source distributions [24]. Assuming the averagenumber of sources for a given centrality is proportionalto the corresponding (cid:104) N part (cid:105) , the C n should have a lineardependence on (cid:104) N part (cid:105) and the ratios C /C , C /C and C /C should be constant as a function of (cid:104) N part (cid:105) .Figure 12 shows the (cid:104) N part (cid:105) dependence of C n for net-proton, proton and antiproton distributions in Au+Aucollisions at √ s NN = 7.7 – 200 GeV. Since the cumulantsare extensive quantities, the C n for net-proton, protonand antiproton increase with increasing (cid:104) N part (cid:105) for allof the √ s NN studied. The different mean values of theproton and antiproton distributions at each energy aredetermined by the interplay between proton-antiprotonpair production and baryon stopping effects. At the lower √ s NN , the effects of baryon stopping at midrapidity aremore important than at higher √ s NN , and therefore thenet-proton C n has dominant contributions from protons.The small mean values for antiprotons at lower √ s NN are due to their low rate of production. At higher √ s NN ,the pair production process dominates the production ofprotons and antiprotons at midrapidity. The ¯ p/p ratio for0-5% central Au+Au collisions at √ s NN = 200 GeV and7.7 GeV are 0.769 and 0.007, respectively [8, 104]. Largevalues of C and C also indicate that the net-proton,proton and antiproton distributions are non-Gaussian.To facilitate plotting, the net-proton and proton C fromthe 0-5% and 5-10% central Au+Au collisions at √ s NN = 7.7 GeV are scaled down by a factor of 2.Figure 13 shows the (cid:104) N part (cid:105) dependence of cumulantratios C / C , C / C and C / C for net-proton, protonand antiproton distributions measured in Au+Au colli-sions at √ s NN = 7.7 – 200 GeV. In terms of the moments of the distributions, they correspond to σ /M ( C / C ), S σ ( C / C ) and κσ ( C / C ). The volume effects arecancelled to the first order in these cumulant ratios. Itis found that both of the proton and antiproton cumu-lant ratios C / C and C / C show weak variations with (cid:104) N part (cid:105) . Based on the HRG model with Boltzmann ap-proximation, the orders of baryon number fluctuationscan be analytically expressed as C B /C B = C B /C B =tanh( µ B /T ) and C B /C B = 1, where µ B and T are thebaryon chemical potential and temperature of the sys-tem, respectively. The values of net-proton C / C showa monotonic decrease with increasing (cid:104) N part (cid:105) while thevalues of C / C show a slight increase with (cid:104) N part (cid:105) . Fora fixed centrality, both net-proton C / C and C / C show strong energy dependence, which can be understoodas C /C ∝ tanh( µ B /T ) and C /C ∝ / tanh( µ B /T ).At high √ s NN , the net-proton C / C ∝ tanh( µ B /T ) ≈ µ B /T → C / C ∝ / tanh( µ B /T ) ≈ T /µ B > µ B /T (cid:29) C /C and C / C approach unity. Dueto the connection between higher-order net-proton cu-mulant ratios and chemical freeze-out µ B and T , thosecumulant ratios have been extensively applied to probethe chemical freeze-out conditions and thermal nature ofthe medium created in heavy-ion collisions [105–107]. Fi-nally, the net-proton and proton C / C ratios have weak (cid:104) N part (cid:105) dependence for energies above √ s NN = 39 GeV.For energies below √ s NN = 39 GeV, the net-proton andproton C / C generally show a decreasing trend with in-creasing (cid:104) N part (cid:105) , except that, within current uncertain-ties, weak centrality dependences of C /C are observedin Au+Au collisions at √ s NN = 7.7 and 11.5 GeV.Figure 14 shows the variation of normalized correla-tion functions κ n /κ ( n >
1) with (cid:104) N part (cid:105) for protonsand antiprotons in Au+Au collisions at √ s NN = 7.7 –200 GeV. The values of κ are equal to mean C values9 − − κ / κ − − − − − − − −
27 GeV − −
39 GeV − − − − − −
200 GeV − κ / κ − − − − − − − − κ / κ κ / κ × STAR Au+Au Collisions < p T < | y | < Average Number of Participant Nucleons h N part i Proton Anti-proton
FIG. 14. (Color online) Collision centrality dependence of normalized correlation functions κ n /κ ( n = 2 , ,
4) for proton andantiproton multiplicity distributions in Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV. Thebars and caps represent the statistical and systematic uncertainties, respectively. For clarity, the X-axis values for protons areshifted and the values of proton and antiproton κ /κ at √ s NN = 7.7 GeV are scaled down by a factor of 2. for protons and antiprotons, and linearly increase with (cid:104) N part (cid:105) as shown in Fig. 12. The normalized two-particlecorrelation functions, κ /κ , for protons and antiprotonsare found to be negative and increase in magnitude withincreasing (cid:104) N part (cid:105) . The values of proton and antipro-ton κ /κ become comparable at √ s NN = 200 GeV butexhibit larger discrepancies at lower energies. This canbe understood as the interplay between baryon stoppingand pair production of protons and antiprotons as a func-tion of √ s NN . The centrality dependence of the normal-ized three and four particle correlation functions κ /κ , κ /κ of proton and antiproton do not show significantdeviation from zero within uncertainties for all centrali-ties and energies. The significances of proton κ /κ de-viating from zero are 1.04 σ , 0.05 σ , 1.27 σ , 0.90 σ , 0.95 σ ,0.40 σ , 2.91 σ , 1.43 σ , 0.11 σ in the 0-5% central Au+Aucollisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4,62.4 and 200 GeV, respectively. The σ is defined as thesum in quadrature of the statistical and systematic un-certainties.As shown in Eqs. (12 – 14), the proton and antiprotoncumulant ratios C /C , C /C and C /C can be ex-pressed in terms of corresponding normalized correlationfunction κ n /κ . Therefore, the results shown in Fig. 14provide important information on how different ordersof multiparticle correlation functions of protons and an-tiprotons contribute to the cumulant ratios. B. Acceptance dependence
In this subsection, we focus on discussing the accep-tance dependence of the proton, antiproton and net-proton cumulants ( C n ) and cumulant ratios in 0-5% cen-tral Au+Au collisions at √ s NN = 7.7 – 200 GeV. It was pointed out in Refs. [82, 83, 108, 109] that when the ra-pidity acceptance (∆ y ) is much smaller than the typicalcorrelation length ( ξ ) of the system (∆ y (cid:28) ξ ), the cu-mulants ( C n ) and correlation functions ( κ n ) should scalewith some power n of the accepted mean particle mul-tiplicities as C n , κ n ∝ (∆ N ) n ∝ (∆ y ) n . Meanwhile, inthe regime where the rapidity acceptance becomes muchlarger than ξ (∆ y (cid:29) ξ ), the C n and κ n scale linearlywith mean multiplicities or ∆ y . Thus, the rapidity accep-tance dependence of the higher-order cumulants and cor-relation functions of proton, antiproton and net-protondistributions are important observables to search for asignature of the QCD critical point in heavy-ion colli-sions. On the other hand, that acceptance dependenceof C n and κ n could be affected by the effects of non-equilibrium [110, 111] and smearing due to diffusion andhadronic re-scattering [112–115] in the dynamical expan-sion of the created fireball.
1. Rapidity dependence
Figure 15 shows the rapidity ( − y max < y < y max ,∆ y = 2 y max ) dependence of the C n for proton, antipro-ton and net-proton distributions in 0-5% central Au+Aucollisions at √ s NN = 7.7 – 200 GeV. The measurementsare made in the p T range of 0.4 to 2.0 GeV/ c . The rapid-ity acceptance is cumulatively increased and the C n val-ues for proton, antiproton and net-proton increase withincreasing rapidity acceptance. For √ s NN <
27 GeV,the proton and net-proton C n have similar values, aninevitable consequence of the small production rate ofantiproton at lower energies.Figure 16 shows the variation of normalized correla-tion functions κ n /κ with rapidity acceptance for pro-0 C
27 GeV
39 GeV
200 GeV C C C C × STAR Au+Au Collisions0-5% most central < p T < | y | < y max Rapidity Cut y max Net-Proton Proton Anti-proton
FIG. 15. (Color online) Rapidity acceptance dependence of cumulants of proton, antiproton and net-proton multiplicitydistributions in 0-5% central Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV. The bars andcaps represent statistical and systematic uncertainties, respectively. For clarity, the X-axis values for protons are shifted andthe values of proton, antiproton and net-proton C at √ s NN = 7.7 GeV are scaled down by a factor of 2. − − κ / κ − − − − − − − −
27 GeV − −
39 GeV − − − − − −
200 GeV − κ / κ − − − − − − − − − κ / κ κ / κ × −
101 0.2 0.4 −
101 0.2 0.4 −
101 0.2 0.4 −
101 0.2 0.4 −
101 0.2 0.4 −
101 0.2 0.4 −
101 0.2 0.4 − STAR Au+Au Collisions 0-5% most central < p T < | y | < y max Rapidity Cut y max Proton Anti-proton
FIG. 16. (Color online) Rapidity acceptance dependence of normalized correlation functions up to fourth order ( κ n /κ , n =2, 3, 4) for proton and antiproton multiplicity distributions in 0-5% central Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6,27, 39, 54.4, 62.4 and 200 GeV. The X-axis rapidity cut y max is applied as | y | < y max . The bars and caps represent statisticaland systematic uncertainties, respectively. For clarity, the X-axis values for protons are shifted and the values of proton andantiproton κ /κ at √ s NN = 7.7 GeV are scaled down by a factor of 2. ton and antiproton in 0-5% central Au+Au collisions at √ s NN = 7.7 – 200 GeV. The κ /κ values for protonsand antiprotons are negative and monotonically increasein magnitude when enlarging the rapidity acceptance upto y max =0.5 (∆ y = 1). For the antiproton, the valuesof κ /κ show stronger deviations from zero at higher √ s NN . As discussed in Fig. 14, the negative values of thetwo-particle correlation functions ( κ ) of protons and an-tiprotons are consistent with the expectation of the effectof baryon number conservation. Within current uncer-tainties, the acceptance dependence for the κ /κ and κ /κ of protons and antiprotons in Au+Au collisions at1 C / C
27 GeV C / C C / C
39 GeV
200 GeV
STAR Au+Au Collisions0-5% most central < p T < | y | < y max Rapidity Cut y max Rapidity Cut y max Net-Proton Proton Anti-proton
FIG. 17. (Color online) Rapidity-acceptance dependence of cumulant ratios of proton, antiproton and net-proton multiplicitydistributions in 0-5% central Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV. The bars andcaps represent statistical and systematic uncertainties, respectively. For clarity, the X-axis values for net-protons and protonsare shifted. different √ s NN are not significant.Figure 17 shows the rapidity acceptance dependence ofthe cumulant ratios C / C , C / C and C / C for pro-ton, antiproton and net-proton in 0-5% central Au+Aucollisions at √ s NN = 7.7 – 200 GeV. Based on Eqs. (12)to (14), the rapidity acceptance dependence of the cumu-lant ratios of proton and antiproton can be understood bythe interplay between different orders of normalized cor-relation functions ( κ n /κ ). The negative values of two-particle correlation functions ( κ ) for protons and an-tiprotons leads to a deviation of the corresponding C / C and C / C below unity. Due to low production rateof antiproton at low energies, the values of C / C and C / C for the net-proton distributions approach the cor-responding values for protons when the beam energy de-creases. The rapidity acceptance dependence of C / C , C / C and C / C values for protons and antiprotonsare comparable at √ s NN = 200 GeV. However, amongthese ratios, protons and antiprotons start to deviate atlower beam energies. This is mainly due to baryon stop-ping and the larger fraction of transported protons com-pared with proton-antiproton pair production at midra-pidity. The C / C values for proton, antiproton and net-proton distributions are consistent within uncertaintiesfor √ s NN = 39, 54.4, 62.4 and 200 GeV. Significant devi-ations from unity are observed for proton and net-proton C / C at √ s NN = 19.6 and 27 GeV, and the deviationdecreases with decreasing ∆ y acceptance, where the ef-fects of baryon number conservation plays an importantrole. For energies below 19.6 GeV, the rapidity accep-tance dependence of C / C for protons, antiprotons andnet-protons is not significant within uncertainties.
2. Transverse momentum dependence
Figure 18 shows the p T acceptance dependence for the C n of proton, antiproton and net-proton distributions atmidrapidity ( | y | < √ s NN = 7.7 – 200 GeV. We fix the lower p T cut at0.4 GeV/ c , and then the p T acceptance is increased byvarying the upper limit in steps between 1 and 2 GeV/ c .The average efficiency values used in the efficiency cor-rection for various p T acceptances are calculated basedon Eq. (16). By extending the upper p T coverage from1 GeV/ c to 2 GeV/ c , the mean numbers of protons in-creased about 50% and 80% at √ s NN = 7.7 and 200 GeV,respectively. It is found that the C n values for protons,antiprotons and net protons increase with increasing p T acceptance, except for a weak p T acceptance dependencefor C observed at energies below 39 GeV.Figure 19 shows the variation of normalized correla-tion functions κ n /κ with p T acceptance for protons andantiprotons at midrapidity ( | y | < √ s NN = 7.7 – 200 GeV. The κ /κ values for protons and antiprotons are found to be nega-tive and decrease with increasing p T acceptance at higher √ s NN . The κ /κ values for antiprotons approach zerowhen the beam energy is decreased, due to the small pro-duction rate of antiprotons at low energies. The negativevalues of κ /κ for protons observed at low energies aremainly dominated by the baryon stopping.Figure 20 shows the p T acceptance dependence of C / C , C / C and C / C for proton, antiproton andnet-proton distributions in 0-5% central Au+Au colli-sions at √ s NN = 7.7 – 200 GeV. In general, most of theratios show a weak dependence on p T acceptance for allof the √ s NN studied. The C / C ratios of proton andnet-proton distributions are similar for all √ s NN below2 C
27 GeV
39 GeV
200 GeV C C C C × STAR Au+Au Collisions0-5% most central < p T < p maxT (GeV/c), | y | < Transverse Momentum Cut p maxT (GeV/c) Net-Proton Proton Anti-proton
FIG. 18. (Color online) p T -acceptance dependence of cumulants of proton, antiproton and net-proton multiplicity distributionsfor 0-5% central Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV. The bars and caps representstatistical and systematic uncertainties, respectively. For clarity, the X-axis values for net-protons are shifted and the values ofproton, antiproton and net-proton C at √ s NN = 7.7 GeV are scaled down by a factor of 2. − − κ / κ − − − − − − − −
27 GeV − −
39 GeV − − − − − −
200 GeV − κ / κ − − − − − − − − κ / κ κ / κ × STAR Au+Au Collisions 0-5% most central < p T < p maxT (GeV/c), | y | < Transverse Momentum Cut p maxT (GeV/c) Proton Anti-proton
FIG. 19. (Color online) The p T -acceptance dependence of the normalized correlation functions up to fourth order ( κ n /κ , n =2, 3, 4) for proton and antiproton multiplicity distributions in 0-5% central Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6,27, 39, 54.4, 62.4 and 200 GeV. The bars and caps represent statistical and systematic uncertainties, respectively. For clarity,the X-axis values for protons are shifted and the values of proton and antiproton κ /κ at √ s NN = 7.7 GeV are scaled downby a factor of 2.
27 GeV. The C / C ratios for protons and antiprotonsare similar at higher beam energy. However, they differfrom each other at the lower √ s NN . From the above dif-ferential measurements, it is found that the baryon num-ber conservation strongly influences the cumulants andcorrelation functions in heavy-ion collisions, especially at low energies. It could be the main reason for the neg-ative two-particle correlation functions for protons andantiprotons [103].3 C / C
27 GeV C / C C / C
39 GeV
200 GeV
STAR Au+Au Collisions0-5% most central < p T < p maxT (GeV/c), | y | < Transverse Momentum Cut p maxT (GeV/c) Net-Proton Proton Anti-proton
FIG. 20. (Color online) p T -acceptance dependence of cumulant ratios of proton, antiproton and net-proton multiplicity distri-butions for 0-5% central Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV. The bars and capsrepresent statistical and systematic uncertainties, respectively. For clarity, the X-axis values for net protons are shifted. B1 / C B2 (1) C | < 0.5 η | B2 / C B3 (3) C (GeV/c) T p σ κ B2 / C B4 (5) C / C (2) C Net-b aryonNet-proton | < 0.5 η | (GeV/c) < 2.0 T / C (4) C σ κ / C (6) C C u m u l an t R a t i o s (GeV) NN sCollision Energy B1 / C B2 (1) C | < 0.5 η | (GeV/c) T p B2 / C B3 (3) C σ κ B2 / C B4 (5) C / C (2) C Net-proton + ResonancesNet-BaryonNet-proton | < 0.5 η | (GeV/c) < 2.0 T / C (4) C σ κ / C (6) C C u m u l an t R a t i o s (GeV) NN sCollision Energy FIG. 21. (Color online) Left panel: Collision energy dependence of C B2 / C B1 , C B3 / C B2 and C B4 / C B2 for various p T acceptances fromthe hadron resonance gas model. Right panel: The variation of net-proton and net-baryon C /C , C /C and C /C withinthe experimental acceptance [101]. Note: this simulation is done within a pseudorapidity window in order to make comparisonbetween baryons of different mass. C. Cumulants from models
Although our results can be compared to several mod-els [102, 116–127], we have chosen two models which do not have phase transition or critical point physics.They have contrasting physics processes to understandthe following: (a) the effect of measuring net-protonsinstead of net-baryons [66, 128], (b) the role of res-onance decay for net-proton measurements [129–132],4(c) the effect of finite p T acceptance for the measure-ments [103, 133], and (d) the effect of net-baryon numberconservation [128, 134, 135]. Models without a criticalpoint also provide an appropriate baseline for compari-son to data.
1. Hadron resonance gas model
The Hadron Resonance Gas model includes all the rel-evant degrees of freedom for the hadronic matter and alsoimplicitly takes into account the interactions that arenecessary for resonance formation [101, 136]. Hadronsand resonances of masses up to 3 GeV/ c are included.Considering a Grand Canonical Ensemble picture, thelogarithm of the partition function ( Z ) in the HRG modelis given as: ln Z ( T, V, µ ) = (cid:88) B ln Z i ( T, V, µ i )+ (cid:88) M ln Z i ( T, V, µ i ) , (23)where: ln Z i ( T, V, µ i ) = ± V g i π (cid:90) d p ln (cid:8) ± exp[( µ i − E ) /T ] (cid:9) , (24) T is the temperature, V is the volume of the system, µ i is the chemical potential, E is the energy, and g i is thedegeneracy factor of the i th particle. The total chemi-cal potential µ i = B i µ B + Q i µ Q + S i µ S , where B i , Q i and S i are the baryon, electric charge and strangenessnumber of the i th particle, with corresponding chemicalpotentials µ B , µ Q and µ S , respectively. The + and − signs in Eq. (24) are for baryons ( B ) and mesons ( M ),respectively. The n th -order generalized susceptibility forbaryons can be expressed as [136]: χ ( n ) x, baryon = x n V T (cid:90) d p ∞ (cid:88) k =0 ( − k ( k + 1) n (25)exp (cid:26) − ( k + 1) ET (cid:27) exp (cid:26) ( k + 1) µT (cid:27) , and for mesons: χ ( n ) x, meson = x n V T (cid:90) d p ∞ (cid:88) k =0 ( k + 1) n (26)exp (cid:26) − ( k + 1) ET (cid:27) exp (cid:26) ( k + 1) µT (cid:27) . The factor x represents either B , Q or S of the i th par-ticle, depending on whether the computed χ x representsbaryon, electric charge or strangeness susceptibility.For a particle of mass m with p T , η and φ , the vol-ume element ( d p ) and energy ( E ) can be written as d p = p T m T cosh( η ) dp T dηdφ and E = m T cosh η , where m T = (cid:112) p T + m . The experimental acceptance can beincorporated by considering the appropriate integrationranges in η , p T , φ and charge states by considering thevalues of | x | . The total generalized susceptibilities willthen be the sum of the contributions from baryons andmesons as in χ ( n ) x = (cid:80) χ ( n ) x, baryon + (cid:80) χ ( n ) x, meson .Figure 21 shows the variation of C B2 / C B1 , C B3 / C B2 and C B4 / C B2 as functions of √ s NN from a hadron resonancegas model [101]. The results are shown for different p T acceptances. The differences due to acceptance are verysmall, and the maximum effect is at the level of 5% for √ s NN = 7.7 GeV for C B4 / C B2 . The HRG results alsoshow that the net-proton results with resonance decaysare smaller compared to net baryons and larger than netprotons without the decay effect. Here also the effect is atthe level of 5% for the lowest √ s NN and smaller at higherenergies in the case of C B4 / C B2 . The corresponding effecton C B3 / C B2 and C B2 / C B1 is larger at the higher energiesand of the order of 17% for net protons without resonancedecay and net baryons, while the effect is 10% for net-proton with resonance decays and net-baryons.
2. UrQMD Model
The UrQMD (Ultra relativistic Quantum MolecularDynamics) model [137, 138] is a microscopic transportmodel where the phase space description of the reactionsare considered. It treats the propagation of all hadronsas classical trajectories in combination with stochasticbinary scattering, color string formation and resonancedecays. It incorporates baryon-baryon, meson-baryonand meson-meson interactions. The collisional term in-cludes more than 50 baryon species and 45 meson species.The model preserves the conservation of electric charge,baryon number, and strangeness number as expected forQCD matter. It also models the phenomenon of baryonstopping, an essential feature encountered in heavy-ioncollisions at lower beam energies. In this model, thespace-time evolution of the fireball is studied in terms ofexcitation and fragmentation of color strings and forma-tion and decay of hadronic resonances. Since the modeldoes not include the physics of the quark-hadron phasetransition or the QCD critical point, the comparison ofthe data to the results obtained from the UrQMD modelwill shed light on the contributions from the hadronicphase and its associated processes, baryon number con-servation and effect of measuring only net protons rela-tive to net baryons.In Fig. 22, the panels on the left present the energydependence of C n ratios of net-baryon distributions forvarious p T acceptance. It is observed that the larger the p T acceptance, the smaller the cumulant ratios. Further-more, with the same p T acceptance, the values of net-baryon C /C and C /C ratios decrease with decreas-ing energies. Figure 22 right also shows the comparisonof the cumulant ratios for net-baryon and net-proton dis-tributions within the experimental acceptance for various5 Net- b aryon < 0.5 (GeV/c) T T T T T T Net-protonNet-baryon T UrQMD Au+Au
5% Centrality h | - - / C C / C C / C C (GeV) NN sCollision Energy FIG. 22. (Color online) Left panel: UrQMD results on p T acceptance dependence of C / C , C / C and C / C ratio as afunction of √ s NN for net baryons. Right panel: Same ratios within the experimental acceptance for net protons and netbaryons. Note: similar to Fig 21, this simulation is done within a pseudorapidity window in order to make comparison betweenbaryons of different mass.
10 20 30 40 50 60 /ndf = 0.32) c poly3 ( /M s (1) |y| < 0.5 < 2.0 GeV/c T
10 20 30 40 50 60 /ndf = 0.72) c poly5 ( s (2) S
10 20 30 40 50 60 s + s = data s Poisson baseline/ndf = 1.3) c poly4 ( sk (3)
10 20 30 40 50 60 - s Monotonic: 3.4
10 20 30 40 50 60 - s Non-monotonic: 1.0 (GeV) NN sCollision Energy
10 20 30 40 50 60 - s Non-monotonic: 3.1 N e t - p r o t on M o m en t s D e r i v a t i v e (GeV) NN sCollision Energy FIG. 23. (Color online) Upper Panel: (1) σ /M , (2) Sσ and (3) κσ of net-proton distributions for 0-5% central Au+Aucollisions from √ s NN = 7.7 - 62.4 GeV. The error bars on the data points are statistical and systematic uncertainties added inquadrature. The black solid lines are polynomial fit functions which well describe the cumulant ratios. The legends also specifythe chi-squared per degree of freedom for the respective fits. The black dashed lines are the Poisson baselines. Lower Panel:Derivative of the fitted polynomial as a function of collision energy. The bar and the gold band on the derivatives representthe statistical and systematic uncertainties, respectively. √ s NN . The differences between results from different ac-ceptance are larger for UrQMD compared to the HRGmodel. In UrQMD the difference between net baryonsand net protons is larger at the lower beam energies for afixed p T and y acceptance. The negative C /C values of net-baryon distributions observed at low energies couldbe mainly due to the effect of baryon number conserva-tion. The effects of resonance weak decay and hadronicre-scattering on proton and net-proton number fluctua-tions in heavy-ion collisions have also been investigated6in Ref. [132] within the JAM model. It is importantto point out that in both the HRG model and UrQMDtransport model calculations, a suppression in C /C atlow collision energy is observed, as evident from the rightplots of Fig. 21 and Fig. 22, respectively. In the case ofthe transport results, the suppression is attributed to theeffect of baryon number conservation in strong interac-tions. However, the interpretation does not apply to theHRG calculation, since for the grand canonical ensem-ble (GCE), the event-by-event conservation is absent al-though, on average, the conservation law is preserved. Inaddition to the law of conservation, quantum effects andthe change of temperature and baryon chemical potentialcould play a role here.
3. Energy dependence
Figure 23 shows the collision-energy dependence of cu-mulant ratios (1) σ /M , (2) Sσ and (3) κσ of net-proton distributions for 0-5% central Au+Au collisionsfrom √ s NN = 7.7 - 62.4 GeV. As shown in Fig. 23, apolynomial of order four (five) well describes the plottedcollision-energy dependence of κσ ( Sσ ) of net-protondistributions for central Au+Au collisions with a χ /ndf= 1.3(0.72). The local derivative of the fitted polyno-mial function shown in the lower panel of Fig. 23 changessign, demonstrating the non-monotonic variation of themeasurements with respect to collision energy. The sta-tistical and systematic uncertainties on derivatives areobtained by randomly varying the data points at eachenergy within their statistical and systematic uncertain-ties.The significance of the observed non-monotonic depen-dence of κσ ( Sσ ) on collision energy, in the energy range √ s NN = 7.7 - 62.4 GeV, is obtained based on the fourth(fifth) order polynomial fitting procedure. This signif-icance is evaluated by randomly varying the κσ and Sσ data points within their total Gaussian uncertainties(statistical and systematic uncertainties added in quadra-ture) at each corresponding energy. This procedure is re-peated a million times for κσ and for Sσ . Out of 1 mil-lion trials, there are 1143 cases for κσ and 158640 casesfor Sσ where the signs of the derivative at all √ s NN arefound to be the same. Thus, the probability that at leastone derivative at a given √ s NN has a different sign fromthe derivatives at remaining energies among the 1 milliontrials performed is 0.99886 (0.84136), which correspondsto a 3.1 σ (1.0 σ ) effect for κσ ( Sσ ). Similarly, basedon the third-order polynomial fitting procedure, the cu-mulant ratio σ /M on the other hand ( χ /ndf = 0.32),exhibits a monotonic dependence on collision energy witha significance of 3.4 σ . Thus we find that the cumulantratios as a function of collision energy change from amonotonic variation to a non-monotonic variation with √ s NN as we go to higher orders. This is consistent withthe QCD-based model expectation that, the higher theorder of the moments, the more sensitive it is to physics C / C STAR Au+Au collisions | y | < < p T < C / C √ s NN (GeV) C / C UrQMD
HRG
GCEHRG
CEHRG
GCE+E.V. (R=0.5 fm)
FIG. 24. (Color online) Collision energy dependence of C / C , C / C and C / C for net-proton multiplicity distri-butions in 0-5% central Au+Au collisions. The experimen-tal net-proton measurements are compared to correspondingvalues from UrQMD and HRG models within the experimen-tal acceptances. The bars and caps represent the statisticaland systematic uncertainties of the experimental data, respec-tively. The widths of the bands reflect the statistical uncer-tainties for the model calculations. processes such as a critical point [59, 61].Figure 24 shows the collision-energy dependence of thecumulant ratios of net-proton multiplicity distributionsfor 0-5% central Au+Au collisions. The comparison hasbeen made between experimental measurements and thecorresponding results from the HRG and UrQMD mod-els. We observe that both models, which do not havephase transition effects, show monotonic variations of thecumulant ratios with beam energy. However, the experi-mental measurements of net-proton C / C ratios showa non-monotonic variation with √ s NN . On the otherhand, the net-proton C / C ( C / C ) in both model anddata show a smooth decrease (increase) trend with in-creasing √ s NN . Although both models show a smoothenergy dependence, the third-order ratios in the mid-dle panel are larger for UrQMD than for (GCE) HRGat collision energies above 14.5 GeV. At lower energy, asuppression relative to the results of GCE HRG is ob-served. On the other hand, the canonical ensemble (CE)7 TABLE VII. The right-tail p values of a chi-squared test between experimental data and various models (shown in Fig. 24) forthe energy dependence of the net-proton cumulant ratios in 0-5% central Au+Au collisions at two ranges of collision energy: √ s NN = 7.7 – 27 GeV and 7.7 – 62.4 GeV (the latter shown in the parenthesis). Those p values denote the probabilityof obtaining discrepancies at least as large as the results actually observed [139]. The right-tail p values are calculated via p = Pr( χ n > χ ), where χ n obeys the chi-square distribution with n independent energy data points and the χ values areobtained in the chi-squared test.Cumulant Ratios HRG GCE HRG CE HRG GCE+E.V. (R=0.5 fm) UrQMD C /C < < < < < < < < C /C < < < < < < < C /C - 1 C C1) - 1 C C3) - 1 C C5) k k
5% central
Au+Au Collisions at RHICAu+Au Collisions at RHIC < 2.0 (GeV/c) T |y| < 0.5, 0.4 < p pp Stat. errorSyst. error UrQMD k k k k (GeV) NN s (cid:214) Collision Energy C u m u l an t s and C o rr e l a t i on F un c t i on s FIG. 25. (Color online) Collision energy dependence of the scaled (anti)proton cumulants and correlation functions in 0-5%central Au+Au collisions at √ s NN = 7.7, 11.5, 14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV. The error bars and bands represent thestatistical and systematic uncertainties, respectively. The results from UrQMD model calculation are also shown for comparison. HRG, has presented a consistent suppression in all threepanels. In this approach, the baryon number conserva-tion is the main source of the suppression [140, 141].It is interesting to point out that GCE models incor-porating excluded volume effects (GCE E.V.) can alsoreproduce the suppression. The larger the repulsive vol-ume, the stronger the suppression. Since the repulsivevolume reflects the ‘baryon density’, the observed sup-pression GCE E.V. is due to the local density. For de-tails, see Refs. [127, 142, 143]. To quantify the levelof agreement between the experimental measurementsand the model calculations, the widely used χ testhas been applied for two energy ranges ( √ s NN = 7.7– 27 and 7.7 – 62.4 GeV). The χ value is calculated as χ ( R ) = (cid:80) √ s NN | R data − R model | error , where R denotes thecumulant ratios ( C /C , C /C , C /C ) and the ‘er-ror’ represents the statistical and systematic uncertain-ties of the data and the statistical uncertainties of themodel added in quadrature. In addition, the obtained χ value can be converted to the corresponding right-tail p -value, which is the probability of obtaining discrepanciesat least as large as the results actually observed [139].The resulting right tail p -values listed in Table VII arecalculated via p = Pr( χ n > χ ), where χ n obeys thechi-square distribution with n independent energy datapoints and the χ values are obtained in the chi-squaredtest. Usually, for the right tail p -value test, p < .
05 isthe commonly used standard to reject the null hypoth-8esis and claim a significant deviation between the dataand model results. It is found that the p -values from thethe χ test are smaller than 0.05 for all of the differentvariants of HRG and the UrQMD model at √ s NN = 7.7– 27 GeV, which means the deviations between data andmodel results are significant and cannot be explained bystatistical fluctuations. But, for the range √ s NN = 7.7– 62.4 GeV, the p -values of C /C for the HRG CE andUrQMD model cases are 0.128 and 0.0577, respectively.Clearly as far as these tests are concerned, all of theabove-mentioned models, showing monotonic energy de-pendences, do not fit the data in the most relevant energyregion, √ s NN ≤
27 GeV. This result will be further testedwith the high-precision data from the second phase of theRHIC beam energy scan program (BES-II).Based on Eq. (11), the cumulants can be expressed interms of the sum of various-order multiparticle correla-tion functions. In order to understand the contributionsto the cumulants, one can present different orders of cor-relation functions separately. Figure 25 shows the en-ergy dependence of the cumulants and correlation func-tions normalized by the mean numbers of protons andantiprotons in 0-5% central Au+Au collisions. By defini-tion and as shown in Fig. 25, the values of C /C − κ /κ . It is observed that the normalized sec-ond and third-order cumulants minus unity ( C /C − C /C −
1) are negative and show an increasing (de-creasing) energy dependence in magnitude for protons(antiprotons) with decreasing collision energies. Fromthe right panels in Fig. 25, the third-order normalizedcorrelation functions ( κ /κ ) of protons and antiprotonsshow flat energy dependence and are consistent with zerowithin uncertainties. Therefore, the energy dependencefor C /C is dominated by the negative two-particle nor-malized correlation functions ( κ /κ ), which is mainlydue to the effects of baryon number conservation. Thenormalized four-particle correlation functions ( κ /κ ) ofantiprotons show flat energy dependence and are con-sistent with zero within uncertainties. In panel 5) ofFig. 25, we observe a similar energy dependence trendfor the normalized fourth order cumulants ( C /C ) ofprotons as for the net-proton C /C in 0-5% centralAu+Au collisions shown in Fig. 24. For √ s NN ≥ C /C are dominated by thenegative two-particle correlation function ( κ ) of protons(see panel 2 in Fig. 25). For √ s NN < κ ) of protons playsa role in determining the energy dependence of proton C /C , which cannot be solely understood by the sup-pression effects due to negative values of κ for protons.As discussed in Refs. [82, 144], the observed large val-ues of the four-particle correlation function of protons( κ ) could be attributed to the formation of proton clus-ter and related to the signature of a critical point or afirst order phase transition. Therefore, it is necessary toperform precise measurements of the κ /κ of protonsbelow 19.6 GeV with high statistics data taken in thesecond phase of the beam energy scan at RHIC. In addi- tion, we compare the experimental data in Fig. 25 withUrQMD model calculations. The energy dependence ofthe second- and third-order normalized cumulants andcorrelation functions can be qualitatively described bythe UrQMD model. However, the non-monotonic energydependence observed in the proton C /C cannot be de-scribed by the UrQMD model. Furthermore, the three-and four-particle correlation functions ( κ and κ ) for(anti)protons from UrQMD show flat energy dependenceand are consistent with zero. Thus, it indicates that thehigher-order (anti)proton correlation functions κ and κ are not sensitive to the effect of baryon number conser-vation within the current acceptance, and therefore canserve as good probes of critical fluctuations in heavy-ioncollisions [103, 132]. IV. SUMMARY AND OUTLOOK
In summary, we report a systematic study of the cumu-lants of the net-proton, proton and antiproton multiplic-ity distributions from Au+Au collisions at √ s NN = 7.7 -200 GeV. The data have been collected with the STARexperiment in the first phase of the RHIC beam energyscan acquired over the period of 2010 - 2017. The energy,centrality and acceptance dependence of the correlationfunctions of protons and antiprotons are presented in thispaper. Both cumulants and correlation functions up tofourth order at midrapidity ( | y | < < p T < c in Au+Au collisions are presented to searchfor the signatures of a critical point and/or a first-orderphase transition over a broad region of baryon chemicalpotential.The protons and antiprotons are identified with greaterthan 97% purity using the TPC and TOF detectors ofSTAR. The centrality selection is based on midrapiditypions and kaons only to avoid self-correlation effects. Themaximum-allowed rapidity acceptance around midrapid-ity has been used for centrality determination to min-imize the effect of centrality resolution. The variationof the average number of protons and antiprotons in agiven centrality bin has been accounted for by applyinga centrality bin-width correction, which also minimizesvolume fluctuation effects. The cumulants are correctedfor the proton and antiproton reconstruction efficienciesusing a binomial response function. Study of the unfold-ing technique for efficiency correction of cumulants hasshown that, even in the 0-5% central Au+Au collisionsat √ s NN = 200 GeV, the case with the highest multiplic-ity, the results are consistent with the commonly-usedbinomial approach within current statistical uncertain-ties. The statistical errors on the cumulants are basedon the delta theorem method and are shown to be con-sistent with those obtained by the bootstrap method. Adetailed estimate of the systematic uncertainties is alsopresented. Results on cumulant ratios from differentvariants of the HRG and the UrQMD models are pre-sented to understand the effects of experimental accep-9tance, resonance decay, baryon number conservation, andnet-proton versus net-baryon analysis. The cumulant ra-tios show a centrality and energy dependence, which areneither reproduced by purely hadronic-transport-basedUrQMD model calculations, nor by different variants ofthe hadron resonance gas model. Specifically, the net-proton C / C ratio for 0-5% central Au+Au collisionsshows a non-monotonic variation with √ s NN , with a sig-nificance of 3.1 σ . This is consistent with the expecta-tions of critical fluctuations in a QCD-inspired model. A χ test has been applied to quantify the level of agree-ment between experimental data and model calculations.The resulting p -values suggest that the models fail to ex-plain the 0-5% Au+Au collision data at √ s NN ≤
27 GeV.The y and p T acceptance dependence of the cumulantsand their ratios provide valuable data to understand therange of the correlations and their relation to the accep-tance of the detector [82, 109]. Furthermore, the system-atic analysis presented here can be used to constrain thefreeze-out conditions in high-energy heavy-ion collisionsusing QCD-based approaches, and to understand the na-ture of thermalization in such collisions [105–107]. Fromthe analysis of multiparticle correlation functions, oneobserves significant negative values for κ of protons andantiprotons, which are mainly due to the effects of baryonnumber conservation in heavy-ion collisions. The valuesof κ of protons and antiprotons are consistent with zerofor all of the collision energies studied. Further, the en-ergy dependence trend of proton C / C below 19.6 GeVcannot be solely understood by the negative values of κ for protons, and the four-particle correlation function ofprotons ( κ ) is found to play a role, which needs to beconfirmed with the high statistics data taken in RHICBES-II, which began data-taking in 2018. Upgrades tothe STAR detector system have significantly improvedthe quality of the measurements [2]. Primarily the goalof BES-II is to make high-statistics measurements, withextended kinematic range in rapidity and transverse mo-mentum for the measurements discussed in this paper.The extended kinematic range in rapidity and transversemomentum are brought about by upgrading the innerTPC (iTPC) to extend the measurement coverage to | η | < p T acceptance down to 100 MeV/ c andimproved dE/dx resolution. Particle identification capa-bility will be extended to -1.6 < η < √ s NN to 3 GeV. With these upgrades,and with the benefits of extended kinematic coverage andthe use of sensitive observables, the RHIC BES Phase-II program will allow measurements of unprecedentedprecision for exploring the QCD phase structure within 200 < µ B (MeV) < TABLE VIII. Total number of collected/expected events inBES phase II for various collision energies ( √ s NN ) [2]. √ s NN (GeV) Year No. of events (million)27 2018 50019.6 2019 40014.5 2019 30011.5 2020 2309.2 2020 1607.7 2021 100 ACKNOWLEDGEMENT
We thank F. Karsch, M. Kitazawa, S. Gupta, D.Mishra, K. Rajagopal, K. Redlich, M. Stephanov, andV. Koch for stimulating discussions related to this work.We thank the RHIC Operations Group and RCF atBNL, the NERSC Center at LBNL, and the Open Sci-ence Grid consortium for providing resources and sup-port. This work was supported in part by the Officeof Nuclear Physics within the U.S. DOE Office of Sci-ence, the U.S. National Science Foundation, the Min-istry of Education and Science of the Russian Federa-tion, National Natural Science Foundation of China, Chi-nese Academy of Science, the Ministry of Science andTechnology of China and the Chinese Ministry of Educa-tion, the Higher Education Sprout Project by Ministryof Education at NCKU, the National Research Founda-tion of Korea, Czech Science Foundation and Ministryof Education, Youth and Sports of the Czech Republic,Hungarian National Research, Development and Innova-tion Office, New National Excellency Programme of theHungarian Ministry of Human Capacities, Departmentof Atomic Energy and Department of Science and Tech-nology of the Government of India, the National ScienceCentre of Poland, the Ministry of Science, Education andSports of the Republic of Croatia, RosAtom of Russia andGerman Bundesministerium fur Bildung, Wissenschaft,Forschung and Technologie (BMBF), Helmholtz Associ-ation, Ministry of Education, Culture, Sports, Science,and Technology (MEXT) and Japan Society for the Pro-motion of Science (JSPS).
Appendix A: Efficiency Correction
In order to correct the C n for efficiency effects, onehas to invoke a model assumption for the response of thedetector. The detector response is assumed to follow abinomial probability distribution function. The probabil-ity distribution function of measured proton number n p and antiproton number n ¯ p can be expressed as [67, 88]:0 p ( n p , n ¯ p ) = ∞ (cid:88) N p = n p ∞ (cid:88) N ¯ p = n ¯ p P ( N p , N ¯ p ) × N p ! n p ! ( N p − n p )! ( ε p ) n p (1 − ε p ) N p − n p × N ¯ p ! n ¯ p ! ( N ¯ p − n ¯ p )! ( ε ¯ p ) n ¯ p (1 − ε ¯ p ) N ¯ p − n ¯ p (A1)where the P ( N p , N ¯ p ) is the original joint probability dis-tribution of number of proton ( N p ) and antiproton ( N ¯ p ),and ε p , ε ¯ p are the efficiency of reconstructing the protons and antiprotons, respectively. In order to arrive at an ex-pression for efficiency-corrected cumulants or moments,the bivariate factorial moments are first defined as: F i,k ( N p , N ¯ p ) = (cid:28) N p !( N p − i )! N ¯ p !( N ¯ p − k )! (cid:29) = ∞ (cid:88) N p = i ∞ (cid:88) N ¯ p = k P ( N p , N ¯ p ) N p !( N p − i )! N ¯ p !( N ¯ p − k )! (A2) f i,k ( n p , n ¯ p ) = (cid:28) n p !( n p − i )! n ¯ p !( n ¯ p − k )! (cid:29) = ∞ (cid:88) n p = i ∞ (cid:88) n ¯ p = k p ( n p , n ¯ p ) n p !( n p − i )! n ¯ p !( n ¯ p − k )! (A3)The efficiency-corrected factorial moments are then givenas: F i,k ( N p , N ¯ p ) = f i,k ( n p , n ¯ p )( ε p ) i ( ε ¯ p ) k . (A4) Then the n th order efficiency-corrected moments ofnet-proton distributions are related to the efficiency-corrected factorial moments as: m n ( N p − N ¯ p ) = < ( N p − N ¯ p ) n > = n (cid:80) i =0 ( − i (cid:18) ni (cid:19) < N n − ip N i ¯ p > = n (cid:80) i =0 ( − i (cid:18) ni (cid:19) (cid:20) n − i (cid:80) r =0 i (cid:80) r =0 s ( n − i, r ) s ( i, r ) F r ,r ( N p , N ¯ p ) (cid:21) = n (cid:80) i =0 n − i (cid:80) r =0 i (cid:80) r =0 ( − i (cid:18) ni (cid:19) s ( n − i, r ) s ( i, r ) F r ,r ( N p , N ¯ p ) (A5)The Stirling numbers of the first ( s ( n, i )) and secondkind ( s ( n, i )), are defined as: N !( N − n )! = n (cid:88) i =0 s ( n, i ) N i (A6) N n = n (cid:88) i =0 s ( n, i ) N !( N − i )! (A7)where N , n and i are non-negative integer numbers. Theefficiency-corrected cumulants of net-proton distributionscan be obtained from the efficiency-corrected moments by using the recursion relation: C r ( N p − N ¯ p ) = m r ( N p − N ¯ p ) − r − (cid:88) s =1 (cid:18) r − s − (cid:19) C s ( N p − N ¯ p ) m r − s ( N p − N ¯ p ) (A8)where the C r denotes the r th -order cumulants of net-proton distributions.If the protons and antiprotons has the same efficiency, ε p = ε ¯ p = ε , the expressions for the first four efficiency-corrected cumulants can be explicitly written as:1 C X − Y = (cid:104) x (cid:105) − (cid:104) y (cid:105) εC X − Y = C x − y + ( ε − (cid:104) x (cid:105) + (cid:104) y (cid:105) ) ε C X − Y = C x − y + 3( ε − C x − C y ) + ( ε − ε − (cid:104) x (cid:105) − (cid:104) y (cid:105) ) ε C X − Y = C x − y − ε − C x + y + 8( ε − C x + C y ) + (5 − ε )( ε − C x + y ε + 8( ε − ε − C x + C y ) + ( ε − ε + 6)( ε − (cid:104) x (cid:105) + (cid:104) y (cid:105) ) ε (A9)where the ( X, Y ) and ( x, y ) are the numbers of ( p, ¯ p )produced and measured, respectively. The efficiency-corrected cumulants are sensitive to the efficiency anddepend on the lower order measured cumulants.In the current analysis, the proton and antiproton p T range is from 0.4 to 2 GeV/ c . This has been possibleby using particle identification information for the TPCin the p T range 0.4 to 0.8 GeV/ c and the TPC+TOFin the momentum range 0.8 to 2 GeV/ c . This resultsin two different efficiencies for proton reconstruction andtwo different values for antiprotons. Hence the above formulation which holds for one single value of efficiencyand ε = ε p = ε ¯ p has to be modified to take care offour different efficiency values, two each for the protonand antiproton corresponding to different p T ranges. Let ε p , ε p and ε ¯ p , ε ¯ p denote the efficiency for protons andantiprotons in the two sub-phase spaces, and denote thecorresponding number of protons and antiprotons in thetwo sub-phase spaces by N p , N p and N ¯ p , N ¯ p , re-spectively. Using analogous formulations as above, thebivariate factorial moments of protons and antiprotonsdistributions is given as: F r ,r ( N p , N ¯ p ) = F r ,r ( N p + N p , N ¯ p + N ¯ p ) = r (cid:88) i =0 r (cid:88) i =0 s ( r , i ) s ( r , i ) (cid:104) ( N p + N p ) i ( N ¯ p + N ¯ p ) i (cid:105) = r (cid:88) i =0 r (cid:88) i =0 s ( r , i ) s ( r , i ) (cid:104) i (cid:88) s =0 (cid:18) i s (cid:19) N i − sp N sp i (cid:88) t =0 (cid:18) i t (cid:19) N i − t ¯ p N t ¯ p (cid:105) = r (cid:88) i =0 r (cid:88) i =0 i (cid:88) s =0 i (cid:88) t =0 s ( r , i ) s ( r , i ) (cid:18) i s (cid:19) (cid:18) i t (cid:19) (cid:104) N i − sp N sp N i − t ¯ p N t ¯ p (cid:105) = r (cid:88) i =0 r (cid:88) i =0 i (cid:88) s =0 i (cid:88) t =0 i − s (cid:88) u =0 s (cid:88) v =0 i − t (cid:88) j =0 t (cid:88) k =0 s ( r , i ) s ( r , i ) (cid:18) i s (cid:19) (cid:18) i t (cid:19) × s ( i − s, u ) s ( s, v ) s ( i − t, j ) s ( t, k ) × F u,v,j,k ( N p , N p , N ¯ p , N ¯ p ) (A10)Similar to Eq. (A4) for the multivariate case, theefficiency-corrected multivariate factorial moments ofproton and antiproton distributions in the current caseare given as: F u,v,j,k ( N p , N p , N ¯ p , N ¯ p ) = f u,v,j,k ( n p , n p , n ¯ p , n ¯ p )( ε p ) u ( ε p ) v ( ε ¯ p ) j ( ε ¯ p ) k (A11)where f u,v,j,k ( N p , N p , N ¯ p , N ¯ p ) are the measured mul-tivariate factorial moments of proton and antiproton dis-tributions. By using Eq. (A5), (A8), (A10) and (A11),one can obtain the efficiency-corrected moments and cu-mulants of net-proton distributions for the case wherethe protons (antiprotons) have different efficiency in twosub-phase spaces. Through simulations as discussed in Refs. [88, 145], it has been shown that this formulationworks consistently. Another binomial-model-based effi-ciency correction method using track-by-track efficiencyis discussed in Ref. [90]. Appendix B: Statistical Uncertainties Estimation
According to Eqs. (A5), (A8) and (A10), the efficiency-corrected moments are expressed in terms of the facto-rial moments, and thereby the factorial moments are therandom variable X i in Eq. (19). The covariance of themultivariate moments can be written as:Cov( m r,s , m u,v ) = 1 n ( m r + u,s + v − m r,s m u,v ) (B1)2where n is the number of events, m r,s = (cid:104) X r X s (cid:105) and m u,v = (cid:104) X u X v (cid:105) are the multivariate moments, and the X and X are random variables. In this paper, X and X represent proton and antiproton number, respec-tively. Based on Eq. (B1), one can obtain the covariancefor the multivariate factorial moments as:Cov( f r,s , f u,v ) = Cov r (cid:88) i =0 s (cid:88) j =0 s ( r, i ) s ( s, j ) m i,j , u (cid:88) k =0 v (cid:88) h =0 s ( u, k ) s ( v, h ) m k,h = r (cid:88) i =0 s (cid:88) j =0 u (cid:88) k =0 v (cid:88) h =0 s ( r, i ) s ( s, j ) s ( u, k ) s ( v, h ) × Cov( m i,j , m k,h )= 1 n r (cid:88) i =0 s (cid:88) j =0 u (cid:88) k =0 v (cid:88) h =0 s ( r, i ) s ( s, j ) s ( u, k ) s ( v, h ) × ( m i + k,j + h − m i,j m k,h )= 1 n ( f ( r,u ) , ( s,v ) − f r,s f u,v ) (B2)where the f ( r,u ) , ( s,v ) is defined as: f ( r,u ) , ( s,v ) = (cid:68) X !( X − r )! X !( X − u )! X !( X − s )! X !( X − v )! (cid:69) = r (cid:80) i =0 s (cid:80) j =0 u (cid:80) k =0 v (cid:80) h =0 i + k (cid:80) α =0 j + h (cid:80) β =0 s ( r, i ) s ( s, j ) s ( u, k ) s ( v, h ) × s ( i + k, α ) s ( j + h, β ) f α,β (B3)The definition of the bivariate factorial moments f r,s , f u,v and f α,β can be found in Eq. (A3). The Equa-tion (B2) can be used in the standard error propaga- tion formula, Eq. (19), to obtain the statistical uncer-tainties of the efficiency-corrected cumulants. The de-tailed derivation of the analytical formulae for statisticaluncertainties on cumulants and moments exists in theliterature [88, 97]. If we put ε p = ε ¯ p = 1, the statisti-cal uncertainties on the cumulants and cumulant ratiosup to the eighth-order expressed in terms of central mo-ments ( µ n ) are given below, where the uncertainties arethe square root of the variances.3Var( C ) = µ /n Var( C ) = ( − µ + µ ) /n Var( C ) = (9 µ − µ µ − µ + µ ) /n Var( C ) = ( − µ + 48 µ µ + 64 µ µ − µ µ − µ µ − µ + µ ) /n Var( C ) = ( µ + 900 µ − µ µ − µ µ + 160 µ µ + 240 µ µ µ + 125 µ µ − µ µ + 200 µ µ − µ µ − µ µ − µ ) /n Var( C ) = ( − µ µ + µ − µ + 13500 µ µ + 39600 µ µ − µ µ − µ µ µ − µ µ + 405 µ µ − µ µ µ + 840 µ µ µ − µ + 216 µ µ + 510 µ µ µ + 440 µ µ + 1020 µ µ µ − µ µ + 225 µ − µ µ − µ µ − µ ) /n Var( C ) = (861 µ µ − µ µ − µ µ − µ µ + µ + 396900 µ − µ µ − µ µ + 79380 µ µ + 299880 µ µ µ + 176400 µ µ − µ µ + 558600 µ µ µ − µ µ µ − µ µ µ − µ µ + 137200 µ µ − µ µ µ − µ µ µ µ + 2310 µ µ µ − µ µ + 1890 µ µ µ + 966 µ µ µ + 343 µ µ − µ µ − µ µ + 1505 µ µ + 2590 µ µ µ + 2254 µ µ µ + 1715 µ µ + 1911 µ µ − µ µ − µ µ − µ ) /n Var( C ) = ( − µ µ + 5600 µ µ µ + 4256 µ µ − µ µ + 5376 µ µ µ − µ µ + 1624 µ µ − µ µ − µ µ − µ µ + µ − µ + 12700800 µ µ + 59270400 µ µ − µ µ − µ µ µ − µ µ + 322560 µ µ − µ µ µ + 1626240 µ µ µ + 1340640 µ µ µ + 677376 µ µ − µ µ + 2759680 µ µ µ + 5597760 µ µ µ µ − µ µ µ + 882000 µ µ − µ µ µ − µ µ µ − µ µ + 2007040 µ µ µ + 3684800 µ µ µ − µ µ µ − µ µ µ µ − µ µ µ µ − µ µ µ − µ µ µ + 3808 µ µ µ + 1680 µ µ µ + 512 µ µ + 940800 µ µ − µ µ − µ µ µ − µ µ − µ µ µ + 8960 µ µ µ + 6496 µ µ µ + 4480 µ µ µ − µ + 5040 µ µ + 9856 µ µ µ + 4704 µ µ + 6272 µ µ − µ µ − µ ) /n Var( C C ) = ( − µ (cid:104) N (cid:105) + µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + µ (cid:104) N (cid:105) ) /n Var( C C ) = (9 µ − µ µ + 6 µ µ + µ µ − µ µ µ + µ µ µ ) /n Var( C C ) = ( − µ + 9 µ + 40 µ µ − µ µ − µ µ µ + 6 µ µ + µ µ + 8 µ µ µ − µ µ µ + µ µ ) /n C C ) = ( µ (cid:104) N (cid:105) + 900 µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 160 µ µ (cid:104) N (cid:105) + 240 µ µ µ (cid:104) N (cid:105) + 125 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 200 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ (cid:104) N (cid:105) + 600 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 20 µ µ µ (cid:104) N (cid:105) + 30 µ µ µ (cid:104) N (cid:105) + 20 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 100 µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) + µ µ (cid:104) N (cid:105) ) /n Var( C C ) = ( − µ µ + µ µ − µ + 5400 µ µ + 30000 µ µ − µ µ − µ µ − µ + 345 µ − µ µ µ + 840 µ µ µ − µ µ µ + 216 µ µ + 2300 µ µ − µ µ µ + 240 µ µ µ µ − µ µ µ − µ µ µ + 30 µ µ − µ µ µ + 20 µ µ µ + 52 µ µ µ µ − µ µ µ + 100 µ µ µ − µ µ µ µ + µ µ µ ) /n Var( C C ) = ( 861 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + µ (cid:104) N (cid:105) + 396900 µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 79380 µ µ (cid:104) N (cid:105) + 299880 µ µ µ (cid:104) N (cid:105) + 176400 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 558600 µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 137200 µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ µ µ (cid:104) N (cid:105) + 2310 µ µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 1890 µ µ µ (cid:104) N (cid:105) + 966 µ µ µ (cid:104) N (cid:105) + 343 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 1505 µ µ (cid:104) N (cid:105) + 2590 µ µ µ (cid:104) N (cid:105) + 2254 µ µ µ (cid:104) N (cid:105) + 1715 µ µ (cid:104) N (cid:105) + 1911 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ (cid:104) N (cid:105) + 264600 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) + 1260 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 11760 µ µ µ (cid:104) N (cid:105) + 17640 µ µ µ (cid:104) N (cid:105) + 47040 µ µ µ (cid:104) N (cid:105) + 44100 µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) + 39200 µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) + 42 µ µ µ (cid:104) N (cid:105) + 56 µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 70 µ µ µ (cid:104) N (cid:105) + 112 µ µ µ (cid:104) N (cid:105) + 70 µ µ (cid:104) N (cid:105) − µ µ (cid:104) N (cid:105) + 44100 µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) + 420 µ µ µ (cid:104) N (cid:105) + 441 µ µ (cid:104) N (cid:105) + 1470 µ µ µ µ (cid:104) N (cid:105) − µ µ µ (cid:104) N (cid:105) + 1225 µ µ µ (cid:104) N (cid:105) − µ µ µ µ (cid:104) N (cid:105) + µ µ (cid:104) N (cid:105) ) /n C C ) = ( − µ µ + 4760 µ µ µ + 3136 µ µ µ + 112 µ µ µ µ + 70 µ µ µ − µ µ µ + 5376 µ µ µ − µ µ µ + 1624 µ − µ µ µ − µ µ µ − µ µ + µ µ − µ + 6747300 µ µ + 48686400 µ µ − µ µ − µ µ µ − µ µ + 282240 µ µ − µ µ µ + 1545600 µ µ µ + 664440 µ µ µ + 606816 µ µ − µ + 1881600 µ µ + 3974880 µ µ µ − µ µ + 102900 µ − µ µ − µ µ − µ − µ µ µ + 1764000 µ µ µ − µ µ µ − µ µ µ µ − µ µ µ µ − µ µ µ − µ µ µ + 3808 µ µ µ − µ µ µ + 512 µ µ − µ µ µ µ + 159936 µ µ µ + 3920 µ µ µ µ + 8960 µ µ µ µ + 224 µ µ µ µ + 896 µ µ µ µ + 28175 µ µ + 2100 µ µ µ + 9856 µ µ µ µ + 3136 µ µ µ − µ µ µ + 56 µ µ + 62720 µ µ µ µ + 39200 µ µ µ − µ µ µ µ − µ µ µ µ − µ µ µ µ − µ µ µ µ µ − µ µ µ + 128 µ µ µ µ − µ µ µ − µ µ µ + 140 µ µ µ µ + 112 µ µ µ + 3136 µ µ µ µ + 3920 µ µ µ µ − µ µ µ µ µ + 1225 µ µ − µ µ µ + µ µ µ ) /n [1] M. M. Aggarwal et al. (STAR Collaboration), (2010),arXiv:1007.2613 [nucl-ex].[2] BES-II White Paper (STAR Note):https://drupal.star.bnl.gov/STAR/starnotes/public/sn0598.[3] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K.Szabo, Nature , 675 (2006), arXiv:hep-lat/0611014[hep-lat].[4] I. Arsene et al. (BRAHMS Collaboration), Nucl. Phys. A757 , 1 (2005), arXiv:nucl-ex/0410020.[5] B. Back et al. (PHOBOS Collaboration), Nucl. Phys.
A757 , 28 (2005), arXiv:nucl-ex/0410022.[6] K. Adcox et al. (PHENIX Collaboration), Nucl. Phys.
A757 , 184 (2005), arXiv:nucl-ex/0410003.[7] J. Adams et al. (STAR Collaboration), Nucl. Phys.
A757 , 102 (2005), arXiv:nucl-ex/0501009.[8] L. Adamczyk et al. (STAR Collaboration), Phys. Rev.
C96 , 044904 (2017), arXiv:1701.07065 [nucl-ex].[9] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz,S. Krieg, C. Ratti, and K. K. Szabo (Wuppertal-Budapest Collaboration), JHEP , 073 (2010),arXiv:1005.3508 [hep-lat].[10] A. Bazavov et al. (HotQCD Collaboration), Phys. Lett. B795 , 15 (2019), arXiv:1812.08235 [hep-lat].[11] L. Adamczyk et al. (STAR Collaboration), Phys. Rev.Lett. , 142301 (2013), arXiv:1301.2347 [nucl-ex].[12] L. Adamczyk et al. (STAR Collaboration), Phys. Rev. Lett. , 162301 (2014), arXiv:1401.3043 [nucl-ex].[13] L. Adamczyk et al. (STAR Collaboration), Phys. Rev.Lett. , 032301 (2018), arXiv:1707.01988 [nucl-ex].[14] L. Adamczyk et al. (STAR Collaboration), Phys. Rev.Lett. , 052302 (2014), arXiv:1404.1433 [nucl-ex].[15] K. Fukushima and T. Hatsuda, Rept. Prog. Phys. ,014001 (2011), arXiv:1005.4814 [hep-ph].[16] M. A. Stephanov, K. Rajagopal, and E. V. Shuryak,Phys. Rev. D60 , 114028 (1999), arXiv:hep-ph/9903292[hep-ph].[17] M. A. Stephanov, Prog. Theor. Phys. Suppl. , 139(2004), arXiv:hep-ph/0402115.[18] Z. Fodor and S. D. Katz, JHEP , 050 (2004),arXiv:hep-lat/0402006 [hep-lat].[19] R. V. Gavai and S. Gupta, Phys. Rev. D78 , 114503(2008), arXiv:0806.2233 [hep-lat].[20] R. V. Gavai and S. Gupta, Phys. Rev.
D71 , 114014(2005), arXiv:hep-lat/0412035 [hep-lat].[21] S. Gupta, PoS
CPOD2009 , 025 (2009),arXiv:0909.4630 [nucl-ex].[22] S. Ejiri, Phys. Rev.
D78 , 074507 (2008),arXiv:0804.3227 [hep-lat].[23] E. S. Bowman and J. I. Kapusta, Phys. Rev.
C79 ,015202 (2009), arXiv:0810.0042 [nucl-th].[24] M. M. Aggarwal et al. (STAR Collaboration), Phys.Rev. Lett. , 022302 (2010), arXiv:1004.4959 [nucl- ex].[25] B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C81 , 024911 (2010), arXiv:0909.4131 [nucl-ex].[26] J. Adam et al. (STAR Collaboration), (2020),arXiv:2007.14005 [nucl-ex].[27] X. Luo and N. Xu, Nucl. Sci. Tech. , 112 (2017),arXiv:1701.02105 [nucl-ex].[28] A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov,and N. Xu, Phys. Rept. , 1 (2020), arXiv:1906.00936[nucl-th].[29] M. Asakawa, U. W. Heinz, and B. Muller, Phys. Rev.Lett. , 2072 (2000), arXiv:hep-ph/0003169 [hep-ph].[30] Y. Hatta and M. A. Stephanov, Phys. Rev.Lett. , 102003 (2003), [Erratum: Phys. Rev.Lett.91,129901(2003)], arXiv:hep-ph/0302002 [hep-ph].[31] V. Koch, A. Majumder, and J. Randrup, Phys. Rev.Lett. , 182301 (2005), arXiv:nucl-th/0505052 [nucl-th].[32] M. Asakawa, S. Ejiri, and M. Kitazawa, Phys. Rev.Lett. , 262301 (2009), arXiv:0904.2089 [nucl-th].[33] S. Gupta, X. Luo, B. Mohanty, H. G. Ritter, and N. Xu,Science , 1525 (2011), arXiv:1105.3934 [hep-ph].[34] H.-T. Ding, F. Karsch, and S. Mukherjee, Int. J. Mod.Phys. E24 , 1530007 (2015), arXiv:1504.05274 [hep-lat].[35] C. Shi, Y.-L. Wang, Y. Jiang, Z.-F. Cui, and H.-S.Zong, JHEP , 014 (2014), arXiv:1403.3797 [hep-ph].[36] F. Gao and Y.-x. Liu, Phys. Rev. D94 , 076009 (2016),arXiv:1607.01675 [hep-ph].[37] C. S. Fischer, Prog. Part. Nucl. Phys. , 1 (2019),arXiv:1810.12938 [hep-ph].[38] B. Friman, F. Karsch, K. Redlich, and V. Skokov, Eur.Phys. J.
C71 , 1694 (2011), arXiv:1103.3511 [hep-ph].[39] W.-J. Fu, J. M. Pawlowski, and F. Rennecke, Phys.Rev.
D101 , 054032 (2020), arXiv:1909.02991 [hep-ph].[40] Y. Lu, Y.-L. Du, Z.-F. Cui, and H.-S. Zong, Eur. Phys.J.
C75 , 495 (2015), arXiv:1508.00651 [hep-ph].[41] J.-W. Chen, J. Deng, H. Kohyama, and L. Labun, Phys.Rev.
D93 , 034037 (2016), arXiv:1509.04968 [hep-ph].[42] W. Fan, X. Luo, and H. Zong, Chin. Phys.
C43 , 033103(2019), arXiv:1702.08674 [hep-ph].[43] W.-J. Fu, Z. Zhang, and Y.-x. Liu, Phys. Rev.
D77 ,014006 (2008), arXiv:0711.0154 [hep-ph].[44] Z. Li, K. Xu, X. Wang, and M. Huang, Eur. Phys. J.
C79 , 245 (2019), arXiv:1801.09215 [hep-ph].[45] C. Herold, M. Nahrgang, Y. Yan, and C. Kobdaj, Phys.Rev.
C93 , 021902 (2016), arXiv:1601.04839 [hep-ph].[46] J.-W. Chen, J. Deng, and L. Labun, Phys. Rev.
D92 ,054019 (2015), arXiv:1410.5454 [hep-ph].[47] V. Vovchenko, D. V. Anchishkin, M. I. Gorenstein, andR. V. Poberezhnyuk, Phys. Rev.
C92 , 054901 (2015),arXiv:1506.05763 [nucl-th].[48] L. Jiang, P. Li, and H. Song, Phys. Rev.
C94 , 024918(2016), arXiv:1512.06164 [nucl-th].[49] A. Mukherjee, J. Steinheimer, and S. Schramm, Phys.Rev.
C96 , 025205 (2017), arXiv:1611.10144 [nucl-th].[50] H. Zhang, D. Hou, T. Kojo, and B. Qin, Phys. Rev.
D96 , 114029 (2017), arXiv:1709.05654 [hep-ph].[51] B. J. Schaefer and M. Wagner, Phys. Rev.
D85 , 034027(2012), arXiv:1111.6871 [hep-ph].[52] L. F. Palhares, E. S. Fraga, and T. Kodama, J. Phys.
G37 , 094031 (2010).[53] Z. Pan, Z.-F. Cui, C.-H. Chang, and H.-S. Zong, Int.J. Mod. Phys.
A32 , 1750067 (2017), arXiv:1611.07370[hep-ph]. [54] B. Berdnikov and K. Rajagopal, Phys. Rev.
D61 ,105017 (2000), arXiv:hep-ph/9912274 [hep-ph].[55] M. Stephanov and Y. Yin, Phys. Rev.
D98 , 036006(2018), arXiv:1712.10305 [nucl-th].[56] K. Rajagopal, G. Ridgway, R. Weller, and Y. Yin, Phys.Rev.
D102 , 094025 (2020), arXiv:1908.08539 [hep-ph].[57] X. An, G. Ba¸sar, M. Stephanov, and H.-U. Yee, Phys.Rev.
C102 , 034901 (2020), arXiv:1912.13456 [hep-th].[58] M. A. Stephanov, Phys. Rev.
D81 , 054012 (2010),arXiv:0911.1772 [hep-ph].[59] M. A. Stephanov, Phys. Rev. Lett. , 032301 (2009),arXiv:0809.3450 [hep-ph].[60] C. Athanasiou, K. Rajagopal, and M. Stephanov, Phys.Rev.
D82 , 074008 (2010), arXiv:1006.4636 [hep-ph].[61] M. A. Stephanov, Phys. Rev. Lett. , 052301 (2011),arXiv:1104.1627 [hep-ph].[62] S. Ejiri, F. Karsch, and K. Redlich, Phys. Lett.
B633 ,275 (2006), arXiv:hep-ph/0509051 [hep-ph].[63] M. Cheng et al. , Phys. Rev.
D79 , 074505 (2009),arXiv:0811.1006 [hep-lat].[64] B. Stokic, B. Friman, and K. Redlich, Phys. Lett.
B673 , 192 (2009), arXiv:0809.3129 [hep-ph].[65] R. V. Gavai and S. Gupta, Phys. Lett.
B696 , 459(2011), arXiv:1001.3796 [hep-lat].[66] M. Kitazawa and M. Asakawa, Phys. Rev.
C86 , 024904(2012), [Erratum: Phys. Rev.C86,069902(2012)],arXiv:1205.3292 [nucl-th].[67] A. Bzdak and V. Koch, Phys. Rev.
C86 , 044904 (2012),arXiv:1206.4286 [nucl-th].[68] K. H. Ackermann et al. (STAR Collaboration), Nucl.Instrum. Meth.
A499 , 624 (2003).[69] M. Anderson et al. , Nucl. Instrum. Meth.
A499 , 659(2003), arXiv:nucl-ex/0301015.[70] L. Adamczyk et al. (STAR Collaboration), Phys. Rev.Lett. , 032302 (2014), arXiv:1309.5681 [nucl-ex].[71] J. Adam et al. (STAR Collaboration), (2020),arXiv:2001.02852 [nucl-ex].[72] W. J. Llope (STAR Collaboration), Nucl. Instrum.Meth.
A661 , S110 (2012).[73] C. Adler, A. Denisov, E. Garcia, M. J. Murray, H. Stro-bele, and S. N. White, Nucl. Instrum. Meth.
A470 , 488(2001), arXiv:nucl-ex/0008005 [nucl-ex].[74] W. J. Llope et al. , Nucl. Instrum. Meth.
A522 , 252(2004), arXiv:nucl-ex/0308022 [nucl-ex].[75] F. S. Bieser et al. , Nucl. Instrum. Meth.
A499 , 766(2003).[76] M. L. Miller, K. Reygers, S. J. Sanders, and P. Stein-berg, Ann. Rev. Nucl. Part. Sci. , 205 (2007),arXiv:nucl-ex/0701025 [nucl-ex].[77] H. Bichsel, Nucl. Instrum. Meth. A562 , 154 (2006).[78] X. Luo, J. Xu, B. Mohanty, and N. Xu, J. Phys.
G40 ,105104 (2013), arXiv:1302.2332 [nucl-ex].[79] A. Chatterjee, Y. Zhang, J. Zeng, N. R. Sa-hoo, and X. Luo, Phys. Rev.
C101 , 034902 (2020),arXiv:1910.08004 [nucl-ex].[80] M. Zhou and J. Jia, Phys. Rev.
C98 , 044903 (2018),arXiv:1803.01812 [nucl-th].[81] T. Sugiura, T. Nonaka, and S. Esumi, Phys. Rev.
C100 , 044904 (2019), arXiv:1903.02314 [nucl-th].[82] B. Ling and M. A. Stephanov, Phys. Rev.
C93 , 034915(2016), arXiv:1512.09125 [nucl-th].[83] A. Bzdak, V. Koch, and N. Strodthoff, Phys. Rev.
C95 ,054906 (2017), arXiv:1607.07375 [nucl-th].[84] M. Kitazawa and X. Luo, Phys. Rev.
C96 , 024910 (2017), arXiv:1704.04909 [nucl-th].[85] S. He and X. Luo, Chin. Phys. C42 , 104001 (2018),arXiv:1802.02911 [physics.data-an].[86] V. Skokov, B. Friman, and K. Redlich, Phys. Rev.
C88 ,034911 (2013), arXiv:1205.4756 [hep-ph].[87] P. Braun-Munzinger, A. Rustamov, and J. Stachel,Nucl. Phys.
A960 , 114 (2017), arXiv:1612.00702 [nucl-th].[88] X. Luo, Phys. Rev.
C91 , 034907 (2015),arXiv:1410.3914 [physics.data-an].[89] T. Nonaka, M. Kitazawa, and S. Esumi, Phys. Rev.
C95 , 064912 (2017), arXiv:1702.07106 [physics.data-an].[90] X. Luo and T. Nonaka, Phys. Rev.
C99 , 044917 (2019),arXiv:1812.10303 [physics.data-an].[91] P. Garg, D. K. Mishra, P. K. Netrakanti, A. K. Mo-hanty, and B. Mohanty, J. Phys.
G40 , 055103 (2013),arXiv:1211.2074 [nucl-ex].[92] S. Esumi, K. Nakagawa, and T. Nonaka, Nucl. In-strum. Meth.
A987 , 164802 (2021), arXiv:2002.11253[physics.data-an].[93] V. Fine and P. Nevski, in
Proceedings CHEP 2000, 143. (2000).[94] A. Bzdak, R. Holzmann, and V. Koch, Phys. Rev.
C94 ,064907 (2016), arXiv:1603.09057 [nucl-th].[95] T. Nonaka, M. Kitazawa, and S. Esumi, Nucl. Instrum.Meth.
A906 , 10 (2018), arXiv:1805.00279 [physics.data-an].[96] L. Adamczyk et al. (STAR Collaboration), Phys. Lett.
B785 , 551 (2018), arXiv:1709.00773 [nucl-ex].[97] X. Luo, J. Phys.
G39 , 025008 (2012), arXiv:1109.0593[physics.data-an].[98] A. Pandav, D. Mallick, and B. Mohanty, Nucl. Phys.
A991 , 121608 (2019), arXiv:1809.08892 [nucl-ex].[99] B. Efron, The Annals of Statistics p1-26 (1979).[100] B. Efron, Computers and the Theory of Statistics :Thinking the Unthinkable (Society for Industrial andApplied Mathematics, 1979).[101] P. Garg, D. K. Mishra, P. K. Netrakanti, B. Mohanty,A. K. Mohanty, B. K. Singh, and N. Xu, Phys. Lett.
B726 , 691 (2013), arXiv:1304.7133 [nucl-ex].[102] J. Xu, S. Yu, F. Liu, and X. Luo, Phys. Rev.
C94 ,024901 (2016), arXiv:1606.03900 [nucl-ex].[103] S. He and X. Luo, Phys. Lett.
B774 , 623 (2017),arXiv:1704.00423 [nucl-ex].[104] B. I. Abelev et al. (STAR Collaboration), Phys. Rev.
C79 , 034909 (2009), arXiv:0808.2041 [nucl-ex].[105] A. Bazavov et al. , Phys. Rev. Lett. , 192302 (2012),arXiv:1208.1220 [hep-lat].[106] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti,and K. K. Szabo, Phys. Rev. Lett. , 062005 (2013),arXiv:1305.5161 [hep-lat].[107] S. Gupta, D. Mallick, D. K. Mishra, B. Mohanty, andN. Xu, (2020), arXiv:2004.04681 [hep-ph].[108] A. Bzdak and V. Koch, Phys. Rev.
C96 , 054905 (2017),arXiv:1707.02640 [nucl-th].[109] J. Brewer, S. Mukherjee, K. Rajagopal, and Y. Yin,Phys. Rev.
C98 , 061901 (2018), arXiv:1804.10215 [hep-ph].[110] S. Mukherjee, R. Venugopalan, and Y. Yin, Phys. Rev.Lett. , 222301 (2016), arXiv:1605.09341 [hep-ph].[111] S. Wu, Z. Wu, and H. Song, Phys. Rev.
C99 , 064902(2019), arXiv:1811.09466 [nucl-th].[112] Y. Ohnishi, M. Kitazawa, and M. Asakawa, Phys. Rev.
C94 , 044905 (2016), arXiv:1606.03827 [nucl-th].[113] M. Sakaida, M. Asakawa, H. Fujii, and M. Kitazawa,Phys. Rev.
C95 , 064905 (2017), arXiv:1703.08008 [nucl-th].[114] M. Nahrgang, M. Bluhm, T. Schaefer, and S. A. Bass,Phys. Rev.
D99 , 116015 (2019), arXiv:1804.05728 [nucl-th].[115] M. Asakawa, M. Kitazawa, and B. M¨uller, Phys. Rev.
C101 , 034913 (2020), arXiv:1912.05840 [nucl-th].[116] J. Li, H.-j. Xu, and H. Song, Phys. Rev.
C97 , 014902(2018), arXiv:1707.09742 [nucl-th].[117] Y. Lin, L. Chen, and Z. Li, Phys. Rev.
C96 , 044906(2017), arXiv:1707.04375 [hep-ph].[118] G. A. Almasi, B. Friman, and K. Redlich, Phys. Rev.
D96 , 014027 (2017), arXiv:1703.05947 [hep-ph].[119] Z. Yang, X. Luo, and B. Mohanty, Phys. Rev.
C95 ,014914 (2017), arXiv:1610.07580 [nucl-ex].[120] C. Zhou, J. Xu, X. Luo, and F. Liu, Phys. Rev.
C96 ,014909 (2017), arXiv:1703.09114 [nucl-ex].[121] A. Zhao, X. Luo, and H. Zong, Eur. Phys. J.
C77 , 207(2017), arXiv:1609.01416 [nucl-th].[122] V. Vovchenko, L. Jiang, M. I. Gorenstein, andH. Stoecker, Phys. Rev.
C98 , 024910 (2018),arXiv:1711.07260 [nucl-th].[123] M. Albright, J. Kapusta, and C. Young, Phys. Rev.
C92 , 044904 (2015), arXiv:1506.03408 [nucl-th].[124] K. Fukushima, Phys. Rev.
C91 , 044910 (2015),arXiv:1409.0698 [hep-ph].[125] P. K. Netrakanti, X. F. Luo, D. K. Mishra, B. Mohanty,A. Mohanty, and N. Xu, Nucl. Phys.
A947 , 248 (2016),arXiv:1405.4617 [hep-ph].[126] K. Morita, B. Friman, and K. Redlich, Phys. Lett.
B741 , 178 (2015), arXiv:1402.5982 [hep-ph].[127] S. Samanta and B. Mohanty, (2019), arXiv:1905.09311[hep-ph].[128] S. He, X. Luo, Y. Nara, S. Esumi, and N. Xu, Phys.Lett.
B762 , 296 (2016), arXiv:1607.06376 [nucl-ex].[129] M. Nahrgang, M. Bluhm, P. Alba, R. Bellwied,and C. Ratti, Eur. Phys. J.
C75 , 573 (2015),arXiv:1402.1238 [hep-ph].[130] D. K. Mishra, P. Garg, P. K. Netrakanti, andA. K. Mohanty, Phys. Rev.
C94 , 014905 (2016),arXiv:1607.01875 [hep-ph].[131] M. Bluhm, M. Nahrgang, S. A. Bass, and T. Schaefer,Eur. Phys. J.
C77 , 210 (2017), arXiv:1612.03889 [nucl-th].[132] Y. Zhang, S. He, H. Liu, Z. Yang, and X. Luo, Phys.Rev.
C101 , 034909 (2020), arXiv:1905.01095 [nucl-ex].[133] F. Karsch, K. Morita, and K. Redlich, Phys. Rev.
C93 ,034907 (2016), arXiv:1508.02614 [hep-ph].[134] A. Bzdak, V. Koch, and V. Skokov, Phys. Rev.
C87 ,014901 (2013), arXiv:1203.4529 [hep-ph].[135] P. Braun-Munzinger, A. Rustamov, and J. Stachel,(2019), arXiv:1907.03032 [nucl-th].[136] F. Karsch and K. Redlich, Phys. Lett.
B695 , 136 (2011),arXiv:1007.2581 [hep-ph].[137] S. A. Bass et al. , Prog. Part. Nucl. Phys. , 255 (1998),arXiv:nucl-th/9803035 [nucl-th].[138] M. Bleicher et al. , J. Phys. G25 , 1859 (1999), arXiv:hep-ph/9909407 [hep-ph].[139] R. L. Wasserstein and N. A. Lazar, American Statisti-cian , 129 (2016).[140] J.-H. Fu, Phys. Rev. C96 , 034905 (2017),arXiv:1610.07138 [nucl-th]. [141] P. Braun-Munzinger, B. Friman, K. Redlich, A. Rusta-mov, and J. Stachel, (2020), arXiv:2007.02463 [nucl-th].[142] J. Fu, Phys. Lett. B722 , 144 (2013).[143] A. Bhattacharyya, S. Das, S. K. Ghosh, R. Ray,and S. Samanta, Phys. Rev.
C90 , 034909 (2014), arXiv:1310.2793 [hep-ph].[144] A. Bzdak, V. Koch, and V. Skokov, Eur. Phys. J.
C77 ,288 (2017), arXiv:1612.05128 [nucl-th].[145] T. Nonaka, T. Sugiura, S. Esumi, H. Masui, and X. Luo,Phys. Rev.