Effects of centrality fluctuation and deuteron formation on proton number cumulant in Au+Au collisions at s NN − − − √ = 3 GeV from JAM model
Arghya Chatterjee, Yu Zhang, Hui Liu, Ruiqin Wang, Shu He, Xiaofeng Luo
EEffects of centrality fluctuation and deuteron formation on proton number cumulantin Au+Au collisions at √ s NN = 3 GeV from JAM model Arghya Chatterjee, Yu Zhang, Hui Liu, Ruiqin Wang, Shu He, and Xiaofeng Luo ∗ Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan 430079, China
We studied the effects of centrality fluctuation and deuteron formation on the cumulants ( C n ) andcorrelation functions ( κ n ) of protons up to sixth order in most central ( b < √ s NN = 3 GeV from a microscopic transport model (JAM). The results are presented as afunction of rapidity acceptance within transverse momentum 0 . < p T < c . We comparedthe results obtained by centrality bin width correction (CBWC) using charged reference particlemultiplicity with CBWC done using impact parameter bins. It was found that at low energies thecentrality resolution for determining the collision centrality using charged particle multiplicities isnot good enough to reduce the initial volume fluctuations effect for higher-order cumulant analysis.New methods need to be developed to classify events with high centrality resolution for heavy-ion collisions at low energies. Finally, we observed that the formation of deuteron will suppressthe higher-order cumulants and correlation functions of protons and is found to be similar to theefficiency effect. This work can serve as a noncritical baseline for the QCD critical point search atthe high baryon density region. I. INTRODUCTION
Quantum Chromodynamics (QCD) is the fundamen-tal theory of the strong interaction. One of the majorgoals of high energy heavy-ion collision experiments is toexplore the phase structure of the strongly interactingnuclear matter. The QCD phase structure can be dis-played in the phase diagram, which is represented as afunction of baryon chemical potential ( µ B ) and tempera-ture ( T ) [1]. QCD based model calculations predict thatat large µ B the transition from hadronic matter to Quark-Gluon Plasma (QGP) is of first order. The end point ofthe first order phase transition boundary is known asQCD critical point (CP), after which there is no genuinephase transition but a smooth crossover from hadronicto quark-gluon degrees of freedom [2–4]. Many effortshave been made to find the signature of the CP, theoret-ically [5–16] and experimentally [17–21]. However, thelocation of the CP and even the existence is still not con-firmed yet [22]. The experimental confirmation of theQCD critical point would be a landmark in exploring theQCD phase structure.In heavy-ion collisions, one of the foremost methodsfor the critical point search is through measurementsof higher-order cumulants of conserved quantities, suchas net-baryon, net-charge and net-strangeness number.Theoretically, it is expected that the higher-order cumu-lants of conserved charges are sensitive to the correlationlength ( ξ ) of the system, which will diverge near the crit-ical point [23–26]. As a result non-monotonic variationof higher-order cumulant ratios from its baseline valuesare expected in existence of critical point. Furthermore,theoretical calculation suggests that the ratio of sixth tosecond order cumulant ( C /C ) is sensitive to the phase ∗ xfl[email protected] transition and will become negative when the chemicalfreeze-out is close to the chiral phase transition bound-ary [27, 28]. Thus, the sixth order fluctuation could serveas a sensitive probe of the signature of the QCD phasetransition [29]. Experimentally, due to the detection in-efficiency of neutral particles and multi-strange baryons,the net-proton and net-kaon are used as an experimen-tal proxy of net-baryon and net-strangeness respectively.In the last few years, the measurement of second, thirdand fourth order cumulants of net-charge [20, 30], net-proton [17–19], and net-kaon [21] multiplicity distribu-tions have been conducted by the STAR and PHENIXexperiment in the first phase of beam energy scan (BES-I, 2010-2017) program at Relativistic Heavy Ion Collider(RHIC). The measurement of second order mixed cumu-lant have also been reported [31]. Recently, the HADESexperiment published the proton number fluctuations infixed target Au+Au collisions at √ s NN = 2.4 GeV [32].Within current statistical uncertainties, the cumulants ofnet-charge and net-kaon distributions are found to haveeither modest or monotonic dependence on the beam en-ergy, while the fourth order cumulant ratio ( C /C ) ofthe net-proton distributions exhibit non-monotonic be-haviors as a function of √ s NN , with a 3.1 σ signifi-cance [33]. To further confirm this non-monotonic be-haviours, it is important to perform high precision fluc-tuation measurements at higher µ B region. To fulfill thisgoal, RHIC has started the second phase of beam energyscan program (BES-II) since 2018, focusing on the colli-sion energies below 27 GeV. From 2018 to 2020, STARexperiment has taken the data of high statistics Au+Aucollision at √ s NN = 9.2, 11.5, 14.6, 19.6 and 27 GeV(collider mode) and √ s NN = 3.0 – 7.7 GeV (fixed tar-get mode). On the other hand, to understand variousbackground contributions from different physics process,model (without CP) studies are important to providebaselines for the experimental search of the QCD crit-ical point. These background contributions may arise a r X i v : . [ nu c l - e x ] S e p ifrom the limited detector acceptance/efficiency, initialvolume fluctuation, autocorrelation and centrality reso-lution, centrality width, baryon number conservation andresonance decay. Some of those effects have been studiedpreviously [34–42] and needs to be understood properlybefore making solid physics conclusions.In this paper, we studied the effects of centrality fluc-tuation and deuteron formation on the proton cumulantand correlation functions up to sixth order in most cen-tral Au+Au collisions at √ s NN = 3 GeV using JAMmodel. The paper is organized as follows. In sectionII, we briefly discuss the JAM model used for this analy-sis. In section III, we introduce the observables used forthe present study. In section IV, we present the cumu-lants up to sixth order of proton multiplicity distributionat √ s NN = 3GeV with JAM model and discuss the effectof centrality fluctuation and deuteron formation. Thearticle is summarised in section V. II. THE JAM MODEL
JAM (Jet AA Microscopic Transport Model) is a non-equilibrium microscopic transport model contracted onresonance and string degrees of freedom [43, 44]. Hadronsand their excited states have explicit space and timepropagation by the cascade method. Inelastic hadron-hadron collisions with resonance are applied at low en-ergy whereas the string picture and hard parton-partonscattering are modeled at intermediate and high-energyrespectively. The nuclear mean-field is applied based onthe simplified version of the relativistic quantum molec-ular dynamics (RQMD) approach [45]. Previously, JAMmodel has been used to compute several cumulants andstudied different effects on particle number fluctuation inheavy-ion collision phenomenology [40, 46]. More detailsabout JAM model can be found in reference [44, 46, 47].In this study, we have analyzed around 25 million centralevents for Au+Au system at √ s NN = 3 GeV generatedusing JAM model. Using the simulated events we calcu-lated up to sixth order cumulants and correlation func-tions of event-by-event proton multiplicity distribution.The light nuclei like deuteron is not directly generatedin JAM model, rather it is produced with an afterburnercode along coalescence of nucleons with the phase spaceobtained from the JAM model [48]. The coalescence con-ditions are constrained by relative distance (∆ R ) and rel-ative momentum (∆ P ) in two body centre of mass frame.When the relative distance and momentum of any twonucleons are less than the given parameters ( R , P ), thelight nuclei are considered to be formed [48–51]. Basedon the charge rms radius of wave function for deuteron,we fixed the coalescence parameters of deuteron ∆ R = 4fm and ∆ P = 0 . c , respectively. III. OBSERVABLES AND METHODS
Higher-order multiplicity fluctuations can be charac-terized by different order cumulants ( C n ). The n th ordercumulant are expressed via generating function [52] as, C n = ∂ n ∂α n K ( α ) | α =0 , (1)where K ( α ) is the cumulant generating functions, whichis logarithm of moment generating function ( K ( α ) =ln( M ( α ))). From event-by-event multiplicity distribu-tions, the various order cumulants can be expressed interms of central moment as follows: C = (cid:104) N (cid:105) , (2) C = (cid:104) ( δN ) (cid:105) , (3) C = (cid:104) ( δN ) (cid:105) , (4) C = (cid:104) ( δN ) (cid:105) − (cid:104) ( δN ) (cid:105) , (5) C = (cid:104) ( δN ) (cid:105) − (cid:104) ( δN ) (cid:105)(cid:104) ( δN ) (cid:105) , (6) C = (cid:104) ( δN ) (cid:105) − (cid:104) ( δN ) (cid:105)(cid:104) ( δN ) (cid:105) − (cid:104) ( δN ) (cid:105) (7)+ 30 (cid:104) ( δN ) (cid:105) , where N is the event-by-event particle number and δN = N −(cid:104) N (cid:105) represents the deviation of N from its mean. (cid:104) ... (cid:105) represents an average over the event sample. The n -thorder cumulant C n is connected to thermodynamic num-ber susceptibilities of a system at thermal and chemicalequilibrium. C n = V T χ n , (8)where V is the system volume, which is difficult to bemeasured in heavy-ion collisions. To cancel out the vol-ume dependence different order cumulant ratios are mea-sured as experimental observables which are related tothe ratios of thermodynamic susceptibilities [24, 53]. C C = χ χ = σ M , C C = χ χ = Sσ,C C = χ χ = κσ , C C = χ χ , C C = χ χ (9)where M , σ , S and κ are mean, sigma, skewness andkurtosis of the multiplicity distribution respectively. Be-sides, the multi-particle correlation function κ n (or fac-torial cumulant) can also be expressed in terms of singleparticle cumulants [54, 55], κ = C , (10) κ = − C + C , (11) κ = 2 C − C + C , (12) κ = − C + 11 C − C + C , (13) κ = 24 C − C + 35 C − C + C , (14) κ = − C + 274 C − C + 85 C (15) − C + C , In case of Poisson distribution, higher-order correlationfunction κ n ( n >
2) are equal to zero. Thus, κ n can beiialso used to quantify the deviations from the Poissondistributions.In this study, we have analyzed around 25 million cen-tral events ( b < N ch ) around mid-rapidity in which the small-est centrality bin is a single multiplicity value. To avoidautocorrelation effect, protons have been excluded from N ch within | η | < th order cumulants ( C n ) are calculated in eachbin i and then weight it by the number of events in eachbin ( n i ), C in = (cid:80) i n i C in (cid:80) i n i , (16)where C in is the n th order cumulant in i -th bin (eitherin b=0.1 fm bin or in each Refmult3 bin) and ( (cid:80) i n i )represents the total number of events. The uncertaintiesreported in the results are statistical due to finite size ofevent sample and obtained using standard error propa-gation method, called Delta theorem [56–58]. Generallythe uncertainty on cumulant measurement is inverselyproportional to the number of events and proportionalto the certain power of the width of the proton multi-plicity distributions. IV. RESULTS
In this section, we will start the discussion with pro-ton dN/dy distribution. Figure 1 shows the dN/dy dis-tribution of proton and deuteron in most central Au+Aucollisions at √ s NN = 3 GeV in JAM model. The cen-tral collision are chosen with impact parameter less than3 fm. The transverse momentum is set to be within0 . < p T < . c for proton selection. The deuteronformation probability is proportional to the initial pro-ton yield in that event according to coalescence after thekinetic freeze-out [59, 60], i.e. λ d = Bn p i , (17) y d N / d y JAM model0.4 < p T < 2.0 GeV/ c b < 3.0 fmAu+Au p (no d-formation)p (d-formation)deuteron (d) FIG. 1. (Color online) Rapidity ( dN/dy ) distributions forproton with and without deuteron formation in most central( b < √ s NN = 3 GeV in JAM model.
20 40 60 80 100 120
Number of Proton N o r m a li z e d N u m b e r o f E v e n t p T < 2.0 (GeV/ c )b < 3.0 fm|y| < 0.5JAM modelAu+Au no d-formationd-formation FIG. 2. (Color online) Normalized event-by-event proton mul-tiplicity distributions in most central ( b < √ s NN = 3 GeV with and without deuteron formationin JAM model. here we assume that the neutron yield is proportionalto the proton yield in each event. B and n p i representsthe coalescence parameter and initial proton number re-spectively. The above assumption is valid where volumefluctuation is minimum [61]. So the initial proton number(p without d-formation) can be approximated by addingobserved proton and deuteron number as shown in Fig. 1, dN p i dy = n p i = n p + n d (18)Figure 2 shows the event-by-event proton number dis-tribution in most central Au+Au collision at √ s NN =3 GeV with and without deuteron formation in JAMv × 10 b (fm) N u m b e r o f e v e n t s ×10 (b)Refmult3 = 67Refmult3 = 60 FIG. 3. (a) (Color online) Correlation between Refmult3(charged particles excluding protons within | η | < b < √ s NN = 3 GeV. (b) (Color online) Impact parameter distri-butions for fixed Refmult3 values. model. The distributions are obtained by counting pro-tons within 0 . < p T < . c . The distributionspresented in Fig. 2 are not corrected by centrality binwidth as described in previous section.Let us first discuss the validity of centrality bin widthcorrection using Refmult3 at √ s NN = 3 GeV. As we dis-cussed in the previous section, at very low energies, evena single multiplicity bin corresponds to a wide initial vol-ume fluctuation. This can be demonstrated in Fig. 3.Figure 3-(a) shows the two-dimension correlation plot be-tween Refmult3 and impact parameter at √ s NN = 3 GeV.We can observe that at 3 GeV, no such strong negativecorrelation is found between charged particles at mid-rapidity region (Refmult3) and impact parameter as ob-served in higher energies [37]. It indicates at low energiesthe charged particles at mid-rapidity region are insensi-tive to the initial collision geometry and have poor cen-trality resolution. Figure 3-(b) shows the b-distributions for two different fixed Refmult3 values, 67 (peak valueof Refmult3 distribution, have maximum weight) and 60.We can clearly see that even a fixed Refmult3 correspondsto all the impact parameter values from 0-3 fm with analmost similar weight as unbiased b-distribution.Figures 4 and 5 show the rapidity acceptance depen-dence for the cumulants and cumulant ratios of protonmultiplicity distributions for two different centrality def-inition in Au+Au collisions at √ s NN = 3 GeV from theJAM model simulation. The results are also comparedwith the cumulants calculated without CBWC. We ob-served that both the cumulants and cumulant ratios ob-tained from Refmult3-CBWC have a large deviation fromimpact parameter CBWC at √ s NN = 3 GeV. We used 0.1fm bin for the impact parameter based CBWC. From theabove comparison we can conclude that unlike higher col-lision energies, the CBWC using charged-particle multi-plicity bin cannot effectively suppress initial volume fluc-tuations in Au+Au collisions at √ s NN = 3 GeV [37].Thus, new methods for classifying events at low energyheavy-ion collisions are needed to determine the collisioncentralities. Recently, in Refs. [62, 63], machine learninghave been proposed to determine the collision central-ity with high resolution in heavy-ion collisions. Thosecould be used to address the centrality fluctuation ef-fect on cumulant analysis at low energies. In the subse-quent sections, we use b-CBWC to understand the effectof deuteron formation on proton number cumulant andcorrelation functions.Theoretically, it was predicted that the rapidity win-dow dependence of proton cumulants are important ob-servables to search for the QCD critical point and under-stand the non-equilibrium effects of dynamical expansionon the fluctuations in heavy-ion collisions [55]. It is ex-pected that the proton cumulant and correlation func-tions will shows power law dependence with the rapidityacceptance and number of protons as C n , κ n ∝ (∆ y ) n ∝ ( N p ) n due to the long range correlation close to the crit-ical point. This relationship will holds if the rapidity ac-ceptance is less than the typical correlation length nearcritical point (∆ y < ξ ) [40, 55]. On the other hand,if the rapidity acceptance is large enough comparing tocorrelation length (∆ y (cid:29) ξ ), the proton cumulant andmulti-particle correlation function will be dominated bystatistical fluctuation as C n , κ n ∝ ∆ y ∝ N p . However,if the rapidity acceptance is further enlarged the baryonnumber conservation effect will dominate over statisticalfluctuation.Figure 6 shows the variation of cumulants C n withthe rapidity acceptance ( − y max < y < y max , ∆ y = 2 y max ) of proton multiplicity distributions in most cen-tral Au+Au collisions at √ s NN = 3 GeV. The measure-ments are done within the transverse momentum range0.4 to 2.0 GeV/ c . All cumulants are saturate around∆ y ∼ .
2, which is the acceptance up to beam rapid-ity ( y beam = 1 .
039 at √ s NN = 3GeV) [37]. C and C linearly increase as a function of rapidity acceptance upto 2 y beam due to increase in proton number with accep- c u m u l a n t ( C n ) C Au+AuJAM modelb < 3.0 fm0.4 < p T < 2.0 GeV/ c C without-CBWCrefmult3-CBWCb-CBWC 0204060 C C C y C FIG. 4. (Color online) Rapidity acceptance dependence cumulants of proton multiplicity distributions in top central ( b < √ s NN = 3 GeV. The centrality bin width correction is done with (a) each Refmult3-bin (black square)and (b) 0.1 fm impact parameter bin (red circle). The results also compared with the cumulants calculated without CBWC(green triangle). C u m u l a n t r a t i o s C / C C / C C / C C / C C / C y p T < 2.0 GeV/ c b < 3.0 fm without-CBWCrefmult3-CBWCb-CBWC FIG. 5. (Color online) Rapidity acceptance dependence cumulant ratios of proton multiplicity distributions in top centralAu+Au collisions at √ s NN = 3 GeV. The centrality bin width correction is done with (a) each Refmult3-bin (black square)and (b) 0.1 fm impact parameter bin (red circle). The results also compared with the cumulants calculated without CBWC(green triangle). tance. We observed about 7% reduction in the meanvalue of protons for the case of deuteron formation. C increases in low rapidity acceptance and shows a peakaround ∆ y ∼ C ) values are negative above ∆ y ∼ C and C ) are consistent with zero with large statisticaluncertainties. To obtain further understanding on effectof deuteron formation, we randomly reduced 7% of totalproton number in each event using binomial sampling.Although the values are not identical, we observed theeffect of 7% random dropping of protons are very closei c u m u l a n t ( C n ) C Au+AuJAM modelb < 3.0 fm0.4 < p T < 2.0 GeV/ c C no d-formationd-formation7% p drop 0102030 C C C y C FIG. 6. (Color online) Rapidity acceptance dependence cumulants ( C ∼ C ) of proton multiplicity distributions in top central( b < √ s NN = 3 GeV. The results are obtained with/without deuteron formation in JAM model.The blue cross marker represents the 7% random dropping of protons in each event C o rr . F un c . ( n ) Au+AuJAM modelb < 3.0 fm0.4 < p T < 2.0 604020020 no d-formationd-formation7% p drop 0 1 2 3 y FIG. 7. (Color online) Rapidity acceptance dependence of correlation functions ( κ ∼ κ ) of proton multiplicity distribution intop central ( b < √ s NN = 3 GeV with/without deuteron formation. The blue cross marker representsthe 7% random dropping of protons in each event. to the case of deuteron formation.Figure 7 shows the rapidity acceptance dependence ofcorrelation function κ n of protons in most central Au+Aucollisions at √ s NN = 3 GeV within the p T range 0.4 to2.0 GeV/ c . Different orders of correlation function valuesare saturate around ∆ y ∼ κ in-creases as a function of rapidity window. The two particlecorrelation function ( κ ) of protons are found to be nega- tive and decreases monotonically up to ∆ y ∼ .
2. Threeparticle correlation function ( κ ) of proton increases with∆ y ∼ acceptance. Fourth, fifth and sixth order correla-tion function of protons ( κ , κ and κ ) are found to beclose to zero up to ∆ y ∼
1, and start to deviate from zerowhen further enlarging the rapidity acceptance. Interest-ingly, the odd order correlation functions are found to bepositive while even order correlation functions show neg-ii C u m u l a n t r a t i o s C / C C / C C / C C / C C / C y p T < 2.0 GeV/ c b < 3.0 fm no d-formationd-formation7% p drop FIG. 8. (Color online) Rapidity acceptance dependence cumulant ratios ( C /C , C /C , C /C , C /C and C /C ) of protonmultiplicity distributions in most central ( b < √ s NN = 3 GeV. The results are obtained with/withoutdeuteron formation in JAM model. The blue cross marker represents the 7% random dropping of protons in each event. ative values up to sixth order at large rapidity acceptanceat √ s NN = 3 GeV. Those strong rapidity acceptance de-pendence are mainly attributed to the effects of baryonnumber conservations [64–67]. In addition, we observedthat if we randomly drop 7% protons in each event thenthe cumulants are close to the deuteron formation case.Figure 8 shows rapidity acceptance dependence of cu-mulant ratios C /C , C /C , C /C , C /C and C /C of proton multiplicity distributions in Au+Au collisionat √ s NN = 3 GeV. At small rapidity acceptance, thecumulant ratios follow statistical fluctuations (Poisson)baseline and the values are close to unity ( C m /C n ∼ C /C and C /C decrease smoothly with ∆ y and satu-rate around ∆ y ∼
2. The values of C /C and C /C arepositive in small rapidity acceptance, then change signaround ∆ y ∼ . y ∼ C /C is close to 1 within statistical uncertainty atsmaller rapidity acceptance and shows negative valuesat larger acceptance. We observed a good agreementin cumulant ratios for deuteron formation and randomdropping of proton case. V. SUMMARY
In this work, we studied the effects of centrality fluctu-ation and deuteron formation on the cumulant and cor-relation function of protons up to sixth order in mostcentral Au+Au collisions at √ s NN = 3 GeV using JAMmodel. We presented the results as a function of therapidity acceptance within transverse momentum 0 .
. √ s NN = 3 GeV. This is mainly dueto the effects of baryon number conservation in heavy-ioncollisions. The results obtained by centrality bin widthcorrection (CBWC) using charged reference particle mul-tiplicity are compared with the CBWC done using finerimpact parameter bins. It was observed that the cen-trality resolution for determining the collision centralityusing charged particle multiplicities cannot effectively re-duce the centrality fluctuations in heavy-ion collisions atlow energies. It brings challenges for us to carry out cu-mulant measurements in low energy heavy-ion collisions.New methods, such as machine learning techniques, needto be built up and applied to determine the collision cen-trality with high resolution, which is crucial for preciselymeasuring the higher-order cumulant in heavy-ion colli-sions at low energies. On the other hand, we discussedthe effect of deuteron formation on cumulant and cor-relation functions of protons and found it is similar tothe binomial efficiency effect due to the loss of protonsvia deuteron formation. This work can serve as a non-critical baselines for the future QCD critical point searchin heavy-ion collisions at high baryon density region.iii VI. ACKNOWLEDGEMENT
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