QQCD Critical Point and High Baryon Density Matter ∗ B. Mohanty , and N. Xu , School of Physical Sciences, National Institute of Science Education andResearch, HBNI, Jatni 752050, India, Institute of Modern Physics, 509Nanchang Road, Lanzhou 730000, China and Nuclear Science Division,Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAWe report the latest results on the search for the QCD critical pointin the QCD phase diagram through high energy heavy-ion collisions. Themeasurements discussed are based on the higher moments of the net-protonmultiplicity distributions in heavy-ion collisions. A non-monotonic varia-tion in the product of kurtosis times the variance of the net-proton distri-bution is observed as a function of the collision energy with 3 σ significance.We also discuss the results of the thermal model in explaining the mea-sured particle yield ratios in heavy-ion collisions and comparison of thedifferent variants of hardon resonance gas model calculation to the data onhigher moments of net-proton distributions. We end with a note that theupcoming programs in high baryon density regime at various experimen-tal facilities will complete the search for the QCD critical point throughheavy-ion collisions.PACS numbers: 25.75.-q,25.75.Nq, 12.38.Mh, 12.38.-t,25.75.Gz
1. Introduction
Relativistic heavy-ion collisions at varying center of mass energy ( √ s NN )allows for the study of the phase diagram of nuclear matter [1]. The un-derlying theory is the one that governs the strong interactions - QuantumChromodynamics (QCD). The conjectured phase diagram of QCD is shownin Fig. 1. The current status of the phase diagram is as follows. There aretwo distinct phases in the phase structure: de-confined state of quarks andgluons called the quark gluon plasma (QGP) and the confined state of gasof hadrons and resonances (HRG). The phase boundary (shown as a solidline in Fig. 1) between the hadronic gas phase and the high-temperature ∗ Presented at workshop on ”Criticality in QCD and the Hadron Resonance Gas”,Wroclaw (online), July 29-31, 2020 (1) a r X i v : . [ nu c l - e x ] J a n QCD-Critical-Point-HRG printed on January 25, 2021 µ B /T = 2 µ B /T = 3 Gas-Liquid C he m i c a l f r ee z e - ou t 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 CSR
RHIC RHIC FXTFAIRNICA HIAF
Fig. 1. Conjectured QCD phase diagram of temperature ( T ) versus baryonic chem-ical potential ( µ B ). See text for details. quark-gluon phase is a first-order phase transition line, which begins atlarge baryon chemical potential ( µ B ) and small temperature ( T ) and curvestowards smaller µ B and larger T . This line ends at the QCD critical pointwhose conjectured position, indicated by a square, is uncertain both theo-retically and experimentally. At smaller µ B there is a cross over indicatedby a dashed line. The region of µ B / T ≤ µ B / T ≤ T ∼ µ B ∼
925 MeV) [4].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.In this proceeding, we discuss the success and tests of the hadron res-onance gas model using the particle ratios and fluctuations in net-protonnumber produced in heavy-ion collisions. We also discuss the status of thesearch for the QCD critical point and future experimental directions in this
CD-Critical-Point-HRG printed on January 25, 2021 connection at the upcoming facilities.
2. Particle ratio and thermal model
Thermal models, assuming approximate local thermal equilibrium, havebeen successfully applied to matter produced in heavy-ion collisions. Mostpopular variant of such a model employs Grand Canonical Ensemble (GCE),hence uses chemical potentials to account for conservation of quantum num-bers on an average [5]. For systems created via elementary collisions (smallsystem) or via low energy heavy-ion collisions, the Canonical Ensemble (CE)approach is used. In the large volume limit, the GCE and the CE formalismsshould be equivalent. In heavy-ion collisions at energies spanning from fewGeV to few TeV it may be worthwhile to ask at what collision energy atransition from GCE to CE occurs [6] ?
Figure 2 (1) in the upper panel shows the energy dependence of K / π particle yield ratio produced in heavy-ion collisions at AGS [7, 8, 9], SPS [10,11] and RHIC [12]. The thermal model calculation explains the K / π ratiosthat reflect the strangeness content relative to entropy of the system formedin heavy-ion collisions. This can be treated as a success of the application ofthermal model to heavy-ion collisions. A peak in the energy dependence of K + / π + could be due to associated production dominance at lower energiesas the baryon stopping is large. The peak is consistent with the calculatednet baryon density reaching a maximum [13] has been suggested to be asignature of a change in degrees of freedom (baryon to meson [14] or hadronsto QGP [15]) while going from lower to higher energies. The K − / π − ratioseems unaffected by the changes in the net-baryon density with collisionenergy and shows a smooth increasing trend. Figure 2 (2) in the lower panel shows the energy dependence of φ / K − yield ratio measured in heavy-ion collisions [16, 17, 18]. As one moves fromhigher to lower collision energy, the φ / K − ratio changes rapidly from aconstant value to larger values. The transition happens below the collisionenergy where the freeze-out net-baryon density peaks (see upper panel).Thermal model calculations with GCE explains the measurements up tocollision energy of 5 GeV. At lower energies the GCE model expectationis that the φ / K − ratio should decrease in contrast to that observed in ex-periments. On the other hand, the increase in φ / K − at lower energies is QCD-Critical-Point-HRG printed on January 25, 2021 (GeV) NN s (cid:214) Collision Energy ) ratios - )/N(K f (2) N( Data
GCE, thermal model (fm) sc r 2.2 3.2 4.2 6.2 } StrangenessCE ) ratios p (1) N(K)/N( + p / + K - p / - K Data Thermal fitBaryon densityat freeze-out S t r ange P a r t i c l e R a t i o s ) - ( f m B r F r ee z e - ou t Fig. 2. (1) Ratio of yields of kaon to pion ( K + /π + (circles) and K − /π − (triangles)produced in central heavy-ion collisions at mid-rapidity as a function of √ s NN .Thermal fits are also shown as bands (yellow band for K + / π + and green bandfor K − / π − ) in the plot. Dot-dashed line represents the net-baryon density at thechemical freeze-out. The dot-dashed line represents the net-baryon density at theChemical Freeze-out as a function of collision energy, calculated from the thermalmodel [13]. (2) Ratio of yields of φ -meson to kaon ( φ/K − ) produced in centralheavy-ion collisions at mid-rapidity as a function of √ s NN . The various bandsshows the thermal model expectation from grand canonical ensemble (GCE) andcanonical ensemble (CE) formulations in the HRG model. explained by thermal model with CE framework for strangeness produc-tion. The results are also sensitive to the choice of the additional controlparameter, r sc , in CE framework, which decides the typical spatial size of s ¯ s correlations. Hence, we find that a high statistics and systematic mea-surement of φ / K − yield ratio can be used to test the transition of GCE toCE in thermal models. As the size of the s ¯ s correlations depends on themedium properties, such studies will provide valuable data for estimationof the volume in which open strangeness is produced. CD-Critical-Point-HRG printed on January 25, 2021
3. Net-proton number fluctuations and QCD critical point
The QCD critical point is a landmark on the QCD phase diagram. Ex-perimental signatures for critical point is enhanced fluctuations coupled tothe critical modes. In this respect the baryon number fluctuations are sen-sitive to the criticality [19]. At the critical point, generally, the correlationlength takes large values, and that leads to non-Gaussian fluctuations [20].Higher-order fluctuations are more sensitive to the criticality, the third or-der ( Sσ ) and the fourth order ( κσ ) are common measures for the QCDcritical point search, where σ , S and κ are called the standard deviation,skewness and the kurtosis of the distribution, respectively. Experimentally,net-proton distribution is considered as a proxy for net-baryon distributions. (1) s S Au+Au CollisionsAu+Au Collisions
Net-proton < 2.0 (GeV/c) T |y| < 0.5, 0.4 < p UrQMD 0-5%HRG GCEHRG CEHRG EV (r=0.5fm)
STAR
STAR FXT H A D ES sk (2) N e t - p r o t on H i gh M o m en t s (GeV) NN sCollision Energy Fig. 3. (1) Sσ and (2) κσ of net-proton distributions for 70-80% peripheral(open squares) and 0-5% central (filled-circles) Au+Au collisions as a functionof √ s NN [21]. Projected statistical uncertainty for the second phase of the RHICBES program is shown by the green-band and the blue arrow shows the region of √ s NN to be covered by the STAR experiments fixed-target program. Results ofcalculations are shown for different variants (Ideal GCE [23], excluded volume [24]and CE [25]) of HRG model and transport model (UrQMD). The solid red and thedashed blue line in (2) is a schematic representation of expectation from a QCDbased model calculation in presence of a critical point. Figure 3 shows the most relevant measurements over the widest range in µ B (20 −
450 MeV) to date for the critical point search [21]. As we go fromobservables involving lower order moments ( Sσ ) to higher order moments( κσ ), deviations between central and peripheral collisions for the measuredvalues increases. Central collisions κσ data show a non-monotonic variation QCD-Critical-Point-HRG printed on January 25, 2021
Table 1. The p values of a χ test between data and various models for the √ s NN dependence of S σ and κσ values of net-proton distributions in 0-5% centralAu+Au collisions. The results are for the √ s NN range 7.7 to 27 GeV [21] whichis the relevant region for the physics analysis presented here. Moments HRG GCE HRG EV HRG CE UrQMD(r = 0.5 fm) S σ < < < κσ κσ = 1 ata significance of ∼ σ [21]. The deviations of κσ below the baseline arequalitatively consistent with theoretical considerations including a criticalpoint [22]. In addition, experimental data show deviation from heavy-ioncollision models without a critical point. This can be seen from the table 1which shows values of a χ test between the experimental data and variousmodels. In all cases, within 7.7 < √ s NN (GeV) <
27, the χ tests return p -values that are less than 0.05. This implies that the monotonic energydependence from all of the models are statistically inconsistent with thedata. Although a non-monotonic variation of the experimental data withcollision energy looks promising for the QCD critical point search, a morerobust conclusion can be derived when the uncertainties get reduced andsignificance above 5 σ is reached. This is the plan for the RHIC Beam EnergyScan Phase-II program. In the previous sub-section we have seen that the data deviates from theexpectations based on UrQMD and HRG models. Figures 4 and 5 showthat several features of the data are qualitatively consistent with LQCDcalculations of net baryon-number fluctuations up to NLO in µ B /T [2].Specifically, (a) M/σ > Sσ , where M is the mean of the net-proton distri-bution; C /C is smaller than unity and tending to decrease with increasing M/σ ; and with increasing M/σ , the cumulant ratio C /C departs furtheraway from unity than the ratio C /C for √ s NN ≥ . C /C = p + p ( C /C ) ; C /C = p + p ( C /C ) and C /C = p C /C + p ( C /C ) . Where p , p , p , and p are fit parameters and we have used the equivalence between product ofthe moments and ratios of cumulants as C /C = M/σ ; C /C = Sσ /M and C /C = κσ . The good agreement between data and LQCD inspiredfits for √ s NN range between 200 to 19.6 GeV, suggests that the heavy-ioncollisions have produced a strongly interacting QCD matter. CD-Critical-Point-HRG printed on January 25, 2021 /C (1) C Au + Au Collisions0-5% Au + Au Collisions at RHIC Statistical uncertaintySystematical uncertainty
200 62.4 39 27 19.6 7.7 NN s /C (2) C B m GCE CEGCE EV (r=0.5fm)
UrQMDLQCD inspired fit N e t - p r o t on C u m u l an t R a t i o s s Measured M/
Fig. 4. Net-proton cumulant ratios as a function of
M/σ . Also shown are theexpectations from different variants of HRG model (lines), UrQMD (yellow band)and LQCD inspired fits (green bands) [2].
4. Experimental programs for high baryon density
As seen from the measurements discussed in previous section, to com-plete the critical point search program a high statistics phase - II of thebeam energy scan program at RHIC is needed. In addition, future new ex-periments, which are all designed with high rates, large acceptance, and thestate-of-the-art particle identification, at the energy region where baryondensity is high, i.e., 500 MeV < µ B <
800 MeV, see Fig. 6, will be needed.The new facilities for studying high baryon density matter includes (a)Nuclotron-based Ion Collider fAcility (NICA) at the Joint Institute forNuclear Research (JINR), Dubna, Russia [27], (b) Compressed BaryonicMatter (CBM) at Facility for Antiproton and Ion Research (FAIR), Darm-
QCD-Critical-Point-HRG printed on January 25, 2021 σ M/ σ S |y| < 0.5 < 2.0 GeV/c T stat σ = (UrQMD) σ S σ σ + σ = (STAR) σ S σ ) σ STAR (S σ = M/ σ S ) σ HRG (S ) σ UrQMD (SLQCD inspired fit
Fig. 5. Sσ versus the M/σ of net-proton distribution in high energy heavy-ioncollisions. Also shown are the expectation from HRG, UrQMD and LQCD inspiredfits [2]. stadt, Germany [28], and (c) CSR External-target Experiment (CEE) atHigh Intensity heavy-ion Accelerator Facility (HIAF), Huizhou, China [29].
5. Summary and Outlook
The workshop dealt with two topics: Criticality and hadron resonancegas models.
Criticality:
A robust and vibrant research program is now establishedboth experimentally (several facilities) and theoretically to study the QCDphase structure [30] and seeking for the QCD critical point in the phasediagram. The observables are well established and the results from a firstsystematic measurements are promising.
Thermal models:
Another success story has been use of hadron res-onance gas models to extract freeze-out dynamics, provide evidences forlocal thermalisation in heavy-ion collisions and act as baseline for severalmeasurements in heavy-ion collisions. This can be extended further to testthe details of the model, like GCE vs. CE, and applications to higher orderfluctuations to probe true thermal nature of the system formed in heavy-ion
CD-Critical-Point-HRG printed on January 25, 2021 Collision Energy √ s NN (GeV) Heavy Ion CollisionInteraction Rates (Hz)
STAR BES-II
NICA
HIAF
FAIR SIS100
HADES
STAR FXT ALICEsPHENIX F i x ed - T a r ge t C o lli de r B
750 500 375 200 100 25
Fig. 6. Interaction rates (in Hz) for high-energy nuclear collision facilities as afunction of √ s NN [26]. Accelerators in collider mode are shown by blue symbols(ALICE, sPHENIX, RHIC BES-II and NICA) and those operating in fixed targetmode by red symbols (STAR fixed traget (FXT), FAIR (CBM, SIS), HADES, andHIAF). collisions [31]. High baryon density:
Gradual shift of attention of the heavy-ion com-munity is expected towards a return to the low energy collisions, wherestate-of-art accelerator facility with large luminosity and much advancesdetector systems with excellent particle identification will allow us to un-ravel the physics of a rotating high baryon density QCD matter subjectedto magnetic field, similar to the neutron stars.
A¯ cknowledgments
F. Karsch, V. Koch, A. Pandav, and K. Redlich forexciting discussions. We also thank the colleagues from STAR and ALICEcollaborations. B.M. was supported in part by the Chinese Academy ofSciences President’s International Fellowship Initiative and J C Bose Fel-lowship from Department of Science of Technology, Government of India.N.X. was supported in part by the Chinese NSF grant No.11927901 and theUS DOE grant No.KB0201022.REFERENCES [1] STAR Internal Note - SN0493, 2009.0
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