Net-proton number fluctuations and the Quantum Chromodynamics critical point
STAR Collaboration, J. Adam, L. Adamczyk, J. R. Adams, J. K. Adkins, G. Agakishiev, 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, K. Barish, A. Behera, R. Bellwied, A. Bhasin, J. Bielcik, J. Bielcikova, L. C. Bland, I. G. Bordyuzhin, J. D. Brandenburg, A. V. Brandin, J. Butterworth, 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, D. Chen, J. H. Chen, X. Chen, Z. Chen, J. Cheng, M. Cherney, M. Chevalier, S. Choudhury, W. Christie, H. J. Crawford, M. Csanád, M. Daugherity, T. G. Dedovich, I. M. Deppner, A. A. Derevschikov, L. Didenko, X. Dong, J. L. Drachenberg, J. C. Dunlop, T. Edmonds, N. Elsey, J. Engelage, G. Eppley, R. Esha, S. Esumi, O. Evdokimov, A. Ewigleben, O. Eyser, R. Fatemi, S. Fazio, P. Federic, J. Fedorisin, C. J. Feng, Y. Feng, P. Filip, E. Finch, Y. Fisyak, A. Francisco, L. Fulek, C. A. Gagliardi, T. Galatyuk, F. Geurts, A. Gibson, K. Gopal, D. Grosnick, W. Guryn, A. I. Hamad, A. Hamed, J. W. Harris, S. He, W. He, X. He, S. Heppelmann, S. Heppelmann, N. Herrmann, E. Hoffman, L. Holub, Y. Hong, S. Horvat, Y. Hu, H. Z. Huang, S. L. Huang, et al. (259 additional authors not shown)
NNet-proton number fluctuations and the Quantum Chromody-namics critical point
Observations from collisions of heavy-ion at relativistic energies have established theformation of a new phase of matter, Quark Gluon Plasma (QGP), a deconfined state of quarksand gluons in a specific region of the temperature versus baryonic chemical potential phasediagram of strong interactions. A program to study the features of the phase diagram, suchas a possible critical point, by varying the collision energy ( √ s NN ), is performed at the Rela-tivistic Heavy-Ion Collider (RHIC) facility. Non-monotonic variation with √ s NN of momentsof the net-baryon number distribution, related to the correlation length and the susceptibil-ities of the system, is suggested as a signature for a critical point . We report the first evi-dence of a non-monotonic variation in kurtosis × variance of the net-proton number (proxyfor net-baryon number) distribution as a function of √ s NN with 3.1 σ significance, for head-on (central) gold-on-gold (Au+Au) collisions measured using the STAR detector at RHIC.Non-central Au+Au collisions and models of heavy-ion collisions without a critical point showa monotonic variation as a function of √ s NN . One of the fundamental goals in physics is to understand the properties of matter when sub-jected to variations in temperature and pressure. Currently, the study of the phases of stronglyinteracting nuclear matter is the focus of many research activities worldwide, both theoreticallyand experimentally
6, 7 . The theory that governs the strong interactions is Quantum Chromody-namics (QCD), and the corresponding phase diagram is called the QCD phase diagram. From1 a r X i v : . [ nu c l - e x ] J u l ifferent examples of condensed-matter systems, experimental progress in mapping out phase dia-grams is achieved by changing the material doping, adding more holes than electrons. Similarly itis suggested for the QCD phase diagram, that adding more quarks than antiquarks (the energy re-quired is defined by the baryonic chemical potential, µ B ), through changing the heavy-ion collisionenergy, enables a search for new emergent properties and a critical point in the phase diagram. Fig-ure 1 shows a conjectured QCD phase diagram. It has at least two distinct phases: a QGP at highertemperatures, and a state of confined quarks and gluons at lower temperatures called the hadronicphase . It is inferred from lattice QCD calculations that the transition is consistent with beinga cross over at small µ B , and that the transition temperature is about 155 MeV . An importantpredicted feature of the QCD phase structure is a critical point
2, 15 , followed at higher µ B by a firstorder phase transition. Attempts are being made to locate the critical point both experimentallyand theoretically. Current theoretical calculations are highly uncertain about the location of thecritical point. Lattice QCD calculations at finite µ B face numerical challenges in computing
16, 17 .Within these limitations, the current best estimate from Lattice QCD is that there is a critical pointlocated above µ B ∼
300 MeV
16, 17 . The goal of this work is to search for the possible signatures ofthe critical point by scanning the temperature ( T ) versus µ B in the QCD phase diagram by varyingthe collision energy √ s NN of the heavy-ion collisions .Another key aspect of investigating the QCD phase diagram is to determine whether the sys-tem has attained thermal equilibrium. Several theoretical interpretations of experimental data havethe underlying assumption that the system produced in the collisions should have come to localthermal equilibrium during its evolution. Experimental tests of thermalization for these femto-2cale expanding systems are non-trivial. However, the yields of produced hadrons and fluctuationsof multiplicity distributions related to conserved quantities have been studied and shown to havecharacteristics of thermodynamic equilibrium for higher collision energies
16, 18–23 .3 B /T = 2 C he m i c a l f r ee z e - ou t Gas-Liquid
Baryonic Chemical Potential µ B (MeV) Quark-Gluon PlasmaHadron Gas T e m pe r a t u r e T ( M e V ) LHC SPS AGS SIS
RHIC RHIC FXTFAIRNICA
Figure 1: Conjectured QCD phase diagram. The phase boundary (solid line) between the hadronicgas phase and the high-temperature quark-gluon phase is a first-order phase transition line, whichbegins at large µ B and small T and curves towards smaller µ B and larger T . This line ends at theQCD critical point whose conjectured position, indicated by a square, is uncertain both theoret-ically and experimentally. At smaller µ B there is a cross over indicated by a dashed line. Theregion of µ B / T ≤ µ B / T ≤
16, 17 . Thered-yellow dotted line corresponds to the chemical freeze-out (where inelastic collisions amongthe constituents of the system cease) inferred from particle yields in heavy-ion collisions using athermal model. The liquid-gas transition region features a second order critical point (red-circle)and a first-order transition line (yellow line) that connect the critical point to the ground state ofnuclear matter ( T ∼ µ B ∼
925 MeV) . The regions of the phase diagram accessed by past(AGS and SPS), ongoing (LHC, RHIC, SPS and RHIC operating in fixed target mode), and future(FAIR and NICA) experimental facilities are also indicated.4pon approaching a critical point, the correlation length diverges and thus renders, to alarge extent, microscopic details irrelevant. Hence observables like the moments of the conservednet-baryon number distribution, which are sensitive to the correlation length, are of interest whensearching for a critical point. A non-monotonic variation of these moments as a function of √ s NN has been proposed as an experimental signature of a critical point
2, 3 . However, considering thecomplexity of the system formed in heavy-ion collisions, signatures of a critical point are de-tectable only if they can survive the evolution of the system, including the effects of finite size andtime . Hence, it was proposed to study higher moments of distributions of conserved quantities( N ) due to their stronger dependence on the correlation length . The promising higher momentsare the skewness, S = (cid:10) ( δ N ) (cid:11) / σ , and kurtosis, κ = [ (cid:10) ( δ N ) (cid:11) / σ ] – 3, where δ N = N – M , M is the mean and σ is the standard deviation. The magnitude and the sign of the moments, whichquantify the shape of the multiplicity distributions, are important for understanding the criticalpoint
3, 25 . An additional crucial experimental challenge is to reconstruct, on an event-by-eventbasis, all of the baryons produced within the acceptance of a detector . However, theoreticalcalculations have shown that the proton-number fluctuations can also reflect the baryon-numberfluctuations at the critical point .The measurements reported here are from Au+Au collisions recorded by the STAR detector at RHIC from the years 2010 to 2017. The data is presented for √ s NN = 7.7, 11.5, 14.5, 19.6,27, 39, 54.4, 62.4 and 200 GeV as part of phase-I of the Beam Energy Scan (BES) program atRHIC . These √ s NN values correspond to µ B values ranging from 420 MeV to 20 MeV atchemical freeze-out . All valid Au+Au collisions occurring within 60 cm (80 cm for √ s NN =5.7 GeV) of the nominal interaction point, having vertex position in the transverse plane within 2cm (1 cm for √ s NN = 14.5 GeV) of the beam axis, and having signals in trigger detectors abovea noise threshold (called minimum bias) are included in the analysis . For the results presentedhere, the number of minimum bias Au+Au collisions ranges between 3 million for √ s NN = 7.7GeV and 585 million at √ s NN = 54.4 GeV. These statistics are found to be sufficient in order tocompute the moments of the net-proton distributions up to the fourth order . The collisions arefurther characterised by their impact parameter, which is indirectly determined from the measuredmultiplicity of charged particles other than protons and anti-protons in the pseudo-rapidity range | η | <
1, where η = − ln [ tan ( θ / )] , with θ being the angle between the momentum of the particleand the positive direction of the beam axis. We exclude protons and anti-protons specifically toavoid self-correlation effects . The effect of self-correlation is found to be negligible from a studycarried out using standard heavy-ion collision event generators, HIJING and UrQMD . Theeffect of resonance decays and the pseudo-rapidity range for centrality determination have beenunderstood and optimised using model calculations
34, 35 . The results presented here correspond totwo event classes: central collisions (impact parameters ∼ ∼ √ s NN = 7.7 GeV and 33 million at √ s NN = 54.4 GeV.The protons ( p ) and anti-protons ( ¯ p ) are identified, along with their momentum, by recon-structing their tracks in the Time Projection Chamber (TPC) placed within a solenoidal magnetic6eld of 0.5 Tesla, and by measuring their ionization energy loss ( dE / dx ) in the sensitive gas-filledvolume of the chamber. The selected kinematic region for protons covers all azimuthal angles forthe rapidity range | y | < .
5, where rapidity is the arctanh of the component of speed parallel to thebeam direction in units of the speed of light, with full azimuthal angle. The precise measurementof dE / dx with a resolution of 7% in Au+Au collisions allows for a clear identification of protonsup to 800 MeV/ c in transverse momentum ( p T , the component of momentum perpendicular tothe beam direction). The identification for larger p T (up to 2 GeV/ c , with purity above 97%) wasmade by a Time Of Flight detector (TOF) (see Methods) having a timing resolution of better than100 ps. A minimum p T threshold of 400 MeV/ c and a maximum distance of closest approach tothe collision vertex of 1 cm for each p ( ¯ p ) candidate track is used to suppress contamination fromsecondaries (for example protons from interactions of energetic particles produced in the collisionswith detector materials)
18, 36 . This p T acceptance accounts for approximately 80% of the total p + ¯ p multiplicity at mid-rapidity. This is a significant improvement from the results previously re-ported which only had the p + ¯ p measured using the TPC (0.4 < p T (GeV/ c ) < p T acceptance is increased upto 2 GeV/ c . The observation of non-monotonic variation of the kurtosis × variance is much moresignificant with the increased acceptance. The increased fluctuations are found to have contribu-tions from protons and anti-protons in the entire p T range studied. For the rapidity dependence ofthe observable see Fig. 7 in Methods.Figure 2 shows the event-by-event net-proton ( N p − N ¯ p = ∆ N p ) distributions obtained bymeasuring the number of protons ( N p ) and anti-protons ( N ¯ p ) at mid-rapidity ( | y | < − NN s Au+Au Collisions T N o r m a li z ed N u m be r o f E v en t s ) p - N p = N p N ∆ Net-Proton (
Figure 2: Event-by-event net-proton number distributions for head-on (0-5% central) Au+Aucollisions for nine √ s NN values measured by the STAR detector at RHIC. The distributions arenormalized to the total number of events at each √ s NN . The statistical uncertainties are smallerthan the symbol sizes and the lines are to guide the eye. The distributions in this figure are notcorrected for proton and anti-proton detection efficiency. The deviation of the distribution for √ s NN = 54.4 GeV from the general energy dependence trend is understood to be due to the reconstructionefficiency of protons and anti-protons being different compared to other energies.transverse momentum range 0.4 < p T (GeV/ c ) < √ s NN . Tostudy the shape of the event-by-event net-proton distribution in detail, cumulants ( C n ) of variousorders are calculated, where C = M , C = σ , C = S σ and C = κσ .Figure 3 shows the variation of net-proton cumulants ( C n ) as a function of √ s NN for cen-tral and peripheral Au+Au collisions. The cumulants are corrected for the multiplicity variations8 (1) C (3) C STAR FXT (2) C (4) C Stat. uncertaintySyst. uncertainty N e t - p r o t on C u m u l an t s (GeV) NN sCollision Energy Au+Au Collisions at RHICAu+Au Collisions at RHIC
Net-proton < 2.0 (GeV/c) T |y| < 0.5, 0.4 < p Figure 3: Cumulants ( C n ) of the net-proton distributions for central (0-5%) and peripheral (70-80%) Au+Au collisions as a function of collision energy. The transverse momentum ( p T ) range forthe measurements is from 0.4 to 2 GeV/ c and the rapidity ( y ) range is ± . These corrections suppressthe volume fluctuations considerably
34, 37 . A different volume fluctuation correction method hasbeen applied to the 0-5% central Au+Au collision data and the results were found to be consistent.The cumulants are also corrected for finite track reconstruction efficiencies of the TPC and TOF9etectors. This is done by assuming a binomial probability distribution to reconstruct particles outof those produced
36, 39 . A cross-check using a different method based on unfolding (see Methods)of the distributions for central Au+Au collisions at √ s NN = 200 GeV has been found to give valuesconsistent with the cumulants shown in Fig. 3. Further, the efficiency correction method used hasbeen verified in a Monte Carlo closure test. Typical values for the efficiencies in the TPC (TOF) forthe momentum range studied in 0-5% central Au+Au collisions at √ s NN = 7.7 GeV are 83%(72%)and 81%(70%) for the protons and anti-protons, respectively. The corresponding efficiencies for √ s NN = 200 GeV collisions are 62%(69%) and 60%(68%) for the protons and anti-protons, respec-tively. The statistical uncertainties are obtained using both a bootstrap approach
30, 39 and the Deltatheorem
30, 39, 40 method. The systematic uncertainties are estimated by varying the experimentalrequirements to reconstruct p ( ¯ p ) in the TPC and TOF. These requirements include the distance ofthe proton and anti-proton tracks from the primary vertex position, track quality reflected by thenumber of TPC space points used in the track reconstruction, the particle identification criteria,and the uncertainties in estimating the reconstruction efficiencies. The systematic uncertainties atdifferent collision energies are uncorrelated.The large values of C and C for central Au+Au collisions show that the distributions havenon-Gaussian shapes ‡ , a first possible indication of enhanced fluctuations arising from a possiblecritical point
15, 25 . The corresponding values for peripheral collisions are small and close to zero.For central collisions, the C and C monotonically decrease with √ s NN , while the C and C showa non-monotonic variation, with a possible minimum between √ s NN of 11.5 and 39 GeV. ‡ Data in Fig. 4 shows deviation from a Skellam expectation.
10e employ ratios of cumulants in order to cancel volume variations to first order. Further,these ratios of cumulants are related to the ratio of baryon-number susceptibilities ( χ B n = d n Pdµ nB , where n is the order and P is the pressure of the system) at a given T and µ B , computed in QCD and QCD-based models as: C / C = S σ = ( χ B3 / T ) / ( χ B2 / T ) and C / C = κσ = ( χ B4 ) / ( χ B2 / T ) . Close tothe critical point, QCD-based calculations predict the net-baryon number distributions to be non-Gaussian and susceptibilities to diverge, causing moments, especially higher-order quantities like κσ , to have non-monotonic variation as a function of √ s NN
4, 41 .Figure 4 shows the variation of S σ (or C / C ) and κσ (or C / C ) as a function of √ s NN for central and peripheral Au+Au collisions. In central collisions a non-monotonic variation withbeam energy is observed for κσ . The κσ values go below unity (statistical baseline) and then risetowards values above unity with decrease in beam energy. The peripheral collisions on the otherhand show a monotonic variation with √ s NN and κσ values are always below unity. It is worthnoting that in peripheral collisions, the system formed may not be hot and dense enough to undergoa phase transition or come close to the QCD critical point. The central Au+Au collision data for κσ ( S σ ), in the collision energy range of 7.7 – 62.4 GeV, are well described by a polynomialfunction of order four (five) in √ s NN , with χ /NDF ∼ √ s NN thereby indicating a non-monotonic variation ofthe measurement with the collision energy. The uncertainties of the derivatives are obtained byvarying the data points randomly at each energy within the statistical and systematic uncertaintiesseparately. The overall significance of the change in the sign of the slope for C / C versus √ s NN ,based on the fourth order polynomial function fitting procedure, from √ s NN = 7.7 to 62.4 GeV is11
10 20 50 100 2000.00.20.40.60.81.0 (1) s S Au+Au Collisions at RHICAu+Au Collisions at RHIC
Net-proton < 2.0 (GeV/c) T |y| < 0.5, 0.4 < p UrQMD 0-5%HRG 0-5%
STAR Data
70 - 80%Stat. uncertaintySyst. uncertaintyProjected BES-IIStat. uncertainty S T A R F X T sk (2) N e t - p r o t on H i gh M o m en t s (GeV) NN sCollision Energy Figure 4: S σ (1) and κσ (2) as a function of collision energy for net-proton distributions measuredin Au+Au collisions. The results are shown for central (0-5%, filled circles ) and peripheral (70-80%, open squares) collisions within 0.4 < p T (GeV/ c ) < | y | < and atransport model calculation (UrQMD ) for central collisions (0-5%) are shown as black and goldbands, respectively. These model calculations utilize the experimental acceptance, and incorporateconservation laws for strong interactions, but do not include a phase transition or a critical point.3.1 σ . This significance is obtained by generating one million sets of points, where for each set, themeasured C / C value at a given √ s NN is randomly varied within the total Gaussian uncertainties(systematic and statistical uncertainties added in quadrature). Then for each new C / C versus12 s NN set of points, a fourth order polynomial function is fitted and the derivative values calculatedat different √ s NN (as discussed above). Out of the one million set of points, 1143 are found tohave the same derivative sign at all √ s NN . The probability that at least one derivative at a given √ s NN has a different sign from the remaining ones in the one million set of points is found tobe 0.998857, which corresponds to 3.1 σ . The significances are calculated using statistical andsystematic uncertainties of the derivatives, added in quadrature.The expectations from an ideal statistical model of hadrons assuming thermodynamical equi-librium, called the Hadron Resonance Gas (HRG) model , calculated within the experimentalacceptance, are also shown in Fig. 4. The HRG results are similar to those from a system of to-tally uncorrelated and statistically random particle production. The HRG results are close to unityfor κσ without any dependence on √ s NN . For S σ , HRG calculations deviate significantly frommeasurements for Au+Au collisions at 0-5% centrality below √ s NN = 62.4 GeV. Corresponding κσ ( S σ ) results for 0-5% Au+Au collisions from a transport-based UrQMD model calcula-tion, which incorporates conservation laws and most of the relevant physics apart from a phasetransition or a critical point, and which is calculated within the experimental acceptance, show amonotonic decrease (increase) with decreasing collision energy. Similar conclusions are obtainedfrom JAM , another microscopic transport model. Neither of the model calculations explains themeasured dependence of the κσ and S σ of the net-proton distribution on √ s NN for central Au+Aucollisions.In conclusion, we have presented measurements of net-proton cumulant ratios with the STAR13etector at RHIC over a wide range in µ B (20 to 420 MeV) which are relevant to a QCD criticalpoint search in the QCD phase diagram. We have observed a non-monotonic behaviour, as afunction of √ s NN , in net-proton κσ in central Au+Au collisions with a significance of 3.1 σ . Incontrast, monotonic behaviour with √ s NN is observed for the statistical hadron gas model, andfor a nuclear transport model without a critical point, as observed experimentally in peripheralcollisions. The deviation of the measured κσ from several baseline calculations with no criticalpoint, and its non-monotonic dependence on √ s NN , are qualitatively consistent with expectationsfrom a QCD-based model which includes a critical point
3, 15 . Our measurements can also becompared to the baryon-number susceptibilities computed from QCD to understand various otherfeatures of the QCD phase structure as well as to obtain the freeze-out conditions in heavy-ioncollisions. Higher event statistics, which will allow for a more differential measurement of theseexperimental observables in y - p T along with comparison to theoretical QCD calculations whichincludes the dynamics associated with heavy-ion collisions, will help in establishing the criticalpoint. Acknowledgements
We thank P. Braun-Munzinger, S. Gupta, F. Karsch, M. Kitazawa, V. Koch,D. Mishra, K. Rajagopal, K. Redlich, and M. Stephanov for several stimulating discussions. Wethank the RHIC Operations Group and RCF at BNL, the NERSC Center at LBNL, and the OpenScience Grid consortium for providing resources and support. This work was supported in part bythe Office of Nuclear Physics within the U.S. DOE Office of Science, the U.S. National ScienceFoundation, the Ministry of Education and Science of the Russian Federation, National NaturalScience Foundation of China, Chinese Academy of Science, the Ministry of Science and Tech-14ology of China and the Chinese Ministry of Education, the Higher Education Sprout Project byMinistry of Education at NCKU, the National Research Foundation of Korea, Czech Science Foun-dation and Ministry of Education, Youth and Sports of the Czech Republic, Hungarian NationalResearch, Development and Innovation Office, New National Excellency Programme of the Hun-garian Ministry of Human Capacities, Department of Atomic Energy and Department of Scienceand Technology of the Government of India, the National Science Centre of Poland, the Ministryof Science, Education and Sports of the Republic of Croatia, RosAtom of Russia and GermanBundesministerium fur Bildung, Wissenschaft, Forschung and Technologie (BMBF), HelmholtzAssociation, Ministry of Education, Culture, Sports, Science, and Technology (MEXT) and JapanSociety for the Promotion of Science (JSPS).1. Shuryak, E.V., Quantum Chromodynamics and the Theory of Superdense Matter,
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Methods(a) Event selection and Proton and anti-proton identification in STAR detector:
To reject pile-up and other background events, information from the fast detectors, scintil-lator based vertex postion detector (VPD) and the time-of-flight (TOF) detector
1, 2 and the timeprojection chamber (TPC) are used. To further ensure a good quality of data, run (a set of mini-mum bias data taken in a certain time interval) by run study of several variables were carried out toremove bad events. The variables used includes the total number of uncorrected charged particles,average transverse momentum in an event, mean pseudorapidity and azimuthal angle in an eventetc. In addition, the distance of closest approach (DCA) of the charged particle track from the pri-mary vertex, especially the signed transverse average DCA and its stability, are studied to remove20 (GeV/c) · p -2 -1 0 1 200.51 p + K + p p - K - p Rapidity -1 -0.5 0 0.5 1 ( G e V / c ) T p Proton ) / c ( G e V m Au+Au 39 GeV
Figure 5: Left panel: Square of the mass of the charged particles, requiring timing informationfrom the TOF, as a function of the product of the momentum ( p ) and the ratio of the particlescharge to the elementary charge e ( q ), both measured using the TPC in Au+Au collisions at √ s NN = 39 GeV. The white dashed lines correspond to the expected square of the mass of each particlespecies. Right panel: The transverse momentum ( p T ) versus the rapidity ( y ) for protons measuredin the STAR detector for Au+Au collisions.the bad events. These classes of bad events are primarily related to the unstable beam conditionsduring the data taking and improper space-charge calibration of TPC.Figure 5 (left panel) shows a typical distribution of the square of the mass associated witheach track in an event obtained from the TOF
1, 2 as a function of the product of the momentum andthe charge of the track determined by the TPC . The proton candidates are well separated fromother hadrons like kaons and pions. The right panel of Fig. 5 shows p T versus y for protons in theSTAR detector. The white dashed rectangular box is the region selected for the results presentedhere. It may be noted that STAR, being a collider experiment, has a p T versus y acceptance near21id-rapidity that is uniform across all beam energies studied. Uniform acceptance allows for theresults to be directly compared across all the √ s NN .The constant p T versus y acceptance near mid-rapidity raises the issue of contribution ofbackground protons to the analysis. This can be gauged by looking at the DCA of the protontrack from the primary vertex and comparing it to the corresponding results for the anti-protons.A DCA criterion of less than 1 cm is used in the analysis reported here. This criterion reducesthe background protons contributions in the momentum range of the study to less than 2-3%. Thissmall effect across all beam energies is added to the systematic uncertainties obtained by varyingthe DCA criteria between 1.2 and 0.8 cm. (b) Efficiency corrections using unfolding of net-proton multiplicity distributions: The unfolding method was applied to a data set that provides the most dense charged parti-cle environment in the detectors (0-5% central Au+Au collisions at √ s NN = 200 GeV), where oneexpects the maximum non-binomial detector effects. Detector-response matrices were determinedbased on detector simulations with respect to generated and measured protons and anti-protons .All possible non-binomial effects, including multiplicity dependent efficiency, were corrected byutilizing the response matrices. The detector response in such cases was found to be best de-scribed by a beta-binomial distribution. Even in this situation, the differences in the binomial andunfolding methods of efficiency correction were at a level of less than one σ of the uncertainties.Cumulants and their ratios up to the fourth order, corrected for the detector efficiencies using22 C u m u l an t s C C C C = 200 GeV, 0 5.0% NN s < 2.0 (GeV/c) T ∆ CBWC, eff. corrunfolded, binomial σ + α unfolded, beta. α unfolded, beta. σ α unfolded, beta. C u m u l an t R a t i o s /C C x1/5 /C C /C C Figure 6: Cumulants and their ratios up to the fourth order, corrected for proton and anti-protonreconstruction efficiencies in √ s NN = 200 GeV Au+Au collisions at 0-5% centrality. Results fromthe conventional efficiency correction are shown as black filled circles, results from the unfoldingwith the binomial detector response are shown as black open circles, and results from beta-binomialdetector response with α + σ , α and α − σ are shown as green triangles, red squares and bluetriangles, respectively. The parameter α quantifies the deviation from binomial effects, obtainedfrom simulation. C / C is scaled by a constant factor.23he unfolding method, are shown in Fig. 6 for 0-5% central Au+Au collisions at √ s NN = 200 GeV.The results are obtained by using centrality bin width correction (CBWC) at 2.5% bin width. Foreach column, the first point is efficiency corrected using the binomial model method (as employedin the present analysis), the next point is the result corrected for the binomial detector responseusing the unfolding technique, and the last three points are from unfolding using the beta-binomialresponse with three values of the non-binomial parameter. The results are ordered from left to rightin terms of increasing deviations of the response function compared to the binomial distribution.Checks using unfolding of the distributions for central Au+Au collisions have been found to yieldvalues consistent with the cumulants obtained using the default binomial method of efficiencycorrection, within the current statistics of the measurements. An alternate approach called themoment expansion method was used for efficiency correction and found to be consistent with theunfolding method. (c) Rapidity dependence of C / C for 0-5% central Au+Au collisions: The cumulant ratio C / C of net-proton multiplicity distributions for 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 is shown in Fig. 7.The C / C value is close to unity for all collision energies for the smallest rapidity acceptance. At √ s NN = 200 GeV, the C / C values remain close to unity as rapidity acceptance is increased, whilefor √ s NN = 7.7 GeV, the C / C values increase as rapidity acceptance is increased. The C / C values decrease as rapidity acceptance is increased at the intermediate collision energies of √ s NN = 19.6 and 27 GeV. 24 (1) 7.7GeV (2) 11.5GeV (3) 14.5GeV (4) 19.6GeV (5) 27GeV (6) 39GeV (7) 54.4GeV (8) 62.4GeV (9) 200GeV / C N e t - p r o t on C max Rapidity Accceptance Cut y
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Zyzak (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 PhysicsNRC ”Kurchatov Institute”, Moscow 117218, Russia Argonne National Laboratory, Argonne,Illinois 60439 American University of Cairo, New Cairo 11835, New Cairo, Egypt BrookhavenNational Laboratory, Upton, New York 11973 University of California, Berkeley, California94720 University of California, Davis, California 95616 University of California, Los Ange-les, California 90095 University of California, Riverside, California 92521 Central ChinaNormal University, Wuhan, Hubei 430079 University of Illinois at Chicago, Chicago, Illi-nois 60607 Creighton University, Omaha, Nebraska 68178 Czech Technical University inPrague, FNSPE, Prague 115 19, Czech Republic Technische Universit¨at Darmstadt, Darmstadt354289, Germany ELTE E ¨otv¨os Lor´and University, Budapest, Hungary H-1117 Frankfurt In-stitute for Advanced Studies FIAS, Frankfurt 60438, Germany Fudan University, Shanghai,200433 University of Heidelberg, Heidelberg 69120, Germany University of Houston, Hous-ton, Texas 77204 Huzhou University, Huzhou, Zhejiang 313000 Indian Institute of ScienceEducation and Research (IISER), Berhampur 760010 , India Indian Institute of Science Educa-tion and Research (IISER) Tirupati, Tirupati 517507, India Indian Institute Technology, Patna,Bihar 801106, India Indiana University, Bloomington, Indiana 47408 Institute of ModernPhysics, Chinese Academy of Sciences, Lanzhou, Gansu 730000 University of Jammu, Jammu180001, India Joint Institute for Nuclear Research, Dubna 141 980, Russia Kent State Univer-sity, Kent, Ohio 44242 University of Kentucky, Lexington, Kentucky 40506-0055 LawrenceBerkeley National Laboratory, Berkeley, California 94720 Lehigh University, Bethlehem, Penn-sylvania 18015 Max-Planck-Institut f¨ur Physik, Munich 80805, Germany Michigan State Uni-versity, East Lansing, Michigan 48824 National Research Nuclear University MEPhI, Moscow115409, 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 25068, Czech Republic Ohio State University, Columbus, Ohio 43210 Panjab University, Chandi-garh 160014, India Pennsylvania State University, University Park, Pennsylvania 16802 NRC”Kurchatov Institute”, Institute of High Energy Physics, Protvino 142281, Russia Purdue Uni-versity, West Lafayette, Indiana 47907 Rice University, Houston, Texas 77251 Rutgers Uni-versity, 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,36ingdao, Shandong 266237 Shanghai Institute of Applied Physics, Chinese Academy of Sci-ences, Shanghai 201800 Southern Connecticut State University, New Haven, Connecticut 06515 State University of New York, Stony Brook, New York 11794 Temple University, Philadel-phia, Pennsylvania 19122 Texas A&M University, College Station, Texas 77843 Universityof 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, Kolkata700064, India Warsaw University of Technology, Warsaw 00-661, Poland Wayne State Uni-versity, Detroit, Michigan 4820162