NNuclear Physics A 00 (2020) 1–8
NuclearPhysics A / locate / procedia XXVIIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions(Quark Matter 2019)
Overview of fluctuation and correlation measurements
A. Rustamov a,b a GSI Helmholtzzentrum f¨ur Schwerionenforschung, Darmstadt, Germany b National Nuclear Research Center, Baku, Azerbaijan
Abstract
One of the ultimate goals of nuclear collision experiments at high energy is to map the phase diagram of stronglyinteracting matter. A very challenging task is the determination of the QCD phase structure including the search forcritical behavior and verification of the possible existence of a critical end point of a first order phase transition line.A promising tool to probe the presence of critical behavior is the study of fluctuations and correlations of conservedcharges since, in a thermal system, these fluctuations are directly related to the equation of state (EoS) of the systemunder the study. In this report an overview is given of several experimental measurements on net-proton multiplicitydistributions such as cumulants and multi-particle correlation functions.
Keywords:
Quark-gluon plasma, Fluctuations and correlations, Conservation laws
1. Introduction
The study of a phase structure of strongly interacting matter is the focus of many research activitiesworldwide. As the theory of strong interactions, Quantum Chromodynamics (QCD), is asymptotically free,in the realm of high temperature and / or density the fundamental degrees of freedom of the strong interactionscome into play. By colliding heavy-ions at di ff erent energies one hopes to heat and / or compress the matterto energy densities at which a transition from matter consisting of confined baryons and mesons to a state ofliberated quarks and gluons (deconfined phase) begins. However, liberated quarks and gluons are not whatone ultimately observes in experiments. The subsequent expansion and cooling of the deconfined phaseleads to formations of hadrons, which fly outwards, and get registered by the detectors. This process ofhadronization plays a key role in understanding what detectors see. The headway is to establish a bridgebetween the events which occur before the hadronization and the experimental outcome. The situation ismuch similar to reconstruction of the cosmological Big Bang from observables like Hubble expansion, thecosmic microwave background and the abundance of light atomic nuclei.Phase transitions are usually studied by looking to the response of the system to external perturbations.For example, the liquid gas phase transition can be probed by the response of the volume to a change in Email address: [email protected] (A. Rustamov) a r X i v : . [ nu c l - e x ] M a y A. Rustamov / Nuclear Physics A 00 (2020) 1–8 pressure, which is encoded in the isothermal compressibility. In the Grand Canonical Ensemble (GCE)formulation of statistical mechanics the latter contains fluctuations of liquid constituents from microstate tomicrostate. Hence, the objective is to relate macroscopic parameters of the system, which define its EoS,with its microscopic details encoded in fluctuations.In a similar way, phase transitions in strongly interacting matter can be addressed by investigating theresponse of the system to external perturbations via measurements of fluctuations of conserved charges suchas baryon number or electric charge [1, 2].For a thermal system of volume V and temperature T , within the Grand Canonical Ensemble, fluctuationsof a given net-charge ∆ N B = N B − N ¯ B are related to the corresponding reduced susceptibilities ˆ χ Bn [3]:1 VT κ n ( ∆ N B ) = ˆ χ Bn , (1)with ˆ χ Bn defined as n th derivative of the reduced thermodynamic pressure ˆ p ≡ pT with respect to the corre-sponding reduced chemical potential ˆ µ B ≡ µ B T and k n ( ∆ N B ) stands for cumulants of conserved net-chargedistributions.At LHC energies there would be, for vanishing light quark masses (u and d quarks), a temperature-driven second order phase transition between a hadron gas and a quark–gluon plasma [4]. For realistic quarkmasses this transition becomes a smooth cross over [5, 6]. Nevertheless, because of the small masses of thelight current quarks, one can still probe critical phenomena at LHC energies (vanishing baryon chemicalpotential) as reported in [7]. Indeed, recent LQCD calculations [5, 6] exhibit a rather strong signal for theexistence of a pseudo-critical chiral temperature of 156.5 ± µ B =
0. Moreover, this pseudo-criticaltemperature turns out to be in good agreement with the chemical freeze-out temperature as extracted by theanalysis of hadron multiplicities measured by the ALICE experiment [8, 9]. This implies that the stronglyinteracting matter created in central collisions of Pb nuclei at LHC energies freezes out near the chiral phasetransition line. Within statistical uncertainties, the pseudo-critical line is also consistent with freeze-outtemperatures determined by the STAR BES-I data [10].At larger values of µ B it is generally expected that a line of first order phase transition exists, which endsin a second order chiral critical point (cf. [4] and references therein).To increase sensitivity, it is better to exploit higher order cumulants because they are better messengersof long range correlations and large fluctuations in the proximity of the critical point.Two comments are in order here. First, the LQCD calculations are predicted for a thermal system in afixed volume. While the notion of the volume is not firmly defined in the experimental case, it is a commonpractice to use number of wounded nucleons as a proxy for the reaction volume (within the wounded nucleonmodel). In experiments, however, wounded nucleons always fluctuate from event to event, hence directcomparison with LQCD becomes challenging. Second, as mentioned above, the LQCD calculations areperformed within the GCE formulation of the statistical mechanics, where net-baryons are not conserved ineach microstate. However, in experiments baryon number is conserved in each event. While for the analysisof mean multiplicities the appropriate acceptance can be selected in order to fulfill the requirements of theGCE, for the higher moments there does not exist any a priory prescription for selecting the ”required”acceptance. Indeed, if the selected acceptance window is too small, possible dynamical correlations willbe washed out and net-baryons will be distributed according to the Skellam distribution originating fromPoisson distributions for single baryons due to small number statistics. For a larger acceptance, however,subtle contributions to measured second cumulants can become observable as for instance those comingfrom baryon number conservation. Hence the acceptance has to be selected large enough to avoid the smallnumber Poissonian limit and final results should be corrected for contributions originating from participantfluctuations and conservation laws.Interestingly, contributions from participant fluctuations to the second and third order cumulants of net-baryon distributions are found to vanish at mid-rapidity for LHC energies while higher cumulants of evenorder are non-zero even when the net-baryon number at mid-rapidity is zero [11, 12, 13]. . Rustamov / Nuclear Physics A 00 (2020) 1–8
2. Basic notations
The r th central moment of a discrete random variable X , with its probability distribution P ( X ), is gener-ally defined as µ r ≡ (cid:104) ( X − (cid:104) X (cid:105) ) r (cid:105) = (cid:88) X ( X − (cid:104) X (cid:105) ) r P ( X ) , (2)where (cid:104) X (cid:105) denotes the mean of the distribution (cid:104) X (cid:105) = (cid:88) X XP ( X ) . (3)In a similar way we introduce moments about the origin, thereafter referred to as raw moments (cid:104) X r (cid:105) = (cid:88) X X r P ( X ) . (4)The cumulants of X are defined as the coe ffi cients in the Maclaurin series of the logarithm of the char-acteristic function of X . The first four cumulants read κ = (cid:104) X (cid:105) ,κ = µ = (cid:68) X (cid:69) − (cid:104) X (cid:105) ,κ = µ = (cid:68) X (cid:69) − (cid:68) X (cid:69) (cid:104) X (cid:105) + (cid:104) X (cid:105) , (5) κ = µ − µ = (cid:68) X (cid:69) − (cid:68) X (cid:69) (cid:104) X (cid:105) − (cid:68) X (cid:69) + (cid:68) X (cid:69) (cid:104) X (cid:105) − (cid:104) X (cid:105) . In the following, the X quantity is replaced by the net-proton number ( n p − n ¯ p ) which is used as a proxyfor net-baryons. hD R = 2.76 TeV NN s Pb - ALICE, Pb - , centrality 0 c < 1.5 GeV/ p = 5 corr y D local conserv. = 2 corr y D local conserv. HIJINGratio, stat. uncert.syst. uncert.global conserv. ALI-PREL-339296
Fig. 1. ALICE Pb–Pb data.
Left panel : Pseudorapidity dependence of the normalized second cumulants of net-protons R at √ s NN = .
76 TeV. Global baryon number conservation is depicted as the pink band. The dashed lines represent the predictions from the modelwith local baryon number conservation [14]. The blue solid line, represents the prediction using the HIJING generator.
Right panel :Centrality dependence of the ratio of third to second order cumulants for net-protons at √ s NN = .
02 TeV. The ALICE data are shownby red markers, while the colored shaded areas indicate the HIJING and EPOS model calculations.
By their definition, cumulants are extensive quantities, i.e., are proportional to the system volume. Toremove the volume dependence normalized cumulants R , S σ and k σ are introducedR = κ ( n p − n p ) / (cid:68) n p + n p (cid:69) , S σ = κ /κ , k σ = κ /κ , (6) A. Rustamov / Nuclear Physics A 00 (2020) 1–8 where S and k denote the skewness and kurtosis of the distribution.In general all cumulants also depend on volume fluctuations. Before taking the ratios introduced in Eq. 6the contributions from volume fluctuations have to be accounted for. Only in this case these ratios dependneither on volume no on its fluctuations. In particular, at lower beam energies it is essential to removecontributions from volume fluctuations [11, 15].
3. Experimental results
In the left panel of Fig. 1 the acceptance dependence of the e ffi ciency corrected normalized cumulants R √ s NN = .
76 TeV, are presented [16, 13]. As already mentionedin the introduction, at LHC energies the R ff ected by volume and its fluctuations. Theanalysis is performed with the Indentity Method [17, 18, 19, 20] in eight pseudorapidity regions rangingfrom − . < η < . − . < η < .
8. The data exhibits linear approach to unity with decreasingacceptance, consistent with predictions based on the assumption of global baryon number conservation [21,11]. When imposing a finite acceptance cut the subtle correlations between baryons and anti-baryons,induced by the global baryon number conservation law, weakens. In the limit of small acceptance thesecorrelations become not visible anymore in the measured second order cumulants. However, the amountof correlation inside finite acceptance depends also on the correlation length ∆ y corr in the rapidity space.This local baryon number conservation [14] would lead to further suppression of the measured R values.Close inspection of Fig. 1, however, indicates that within experimental uncertainties the ALICE data arebest described with the large correlation length in the rapidity space, i.e., the observed correlations, to alarge extent, are induced by global baryon number conservation. The latter corresponds to the correlationlength of ∆ y corr = | y beam | . The HIJING results [22], on the other hand, underestimate the experimental dataand correspond to correlation length of ∆ y corr =
2. The large correlation length observed in the data takesplace before the time [23]
Fig. 2. STAR Au–Au data. The measured S σ ( left panel) and k σ ( right panel) values as a function of collision energy for net-protondistributions. The results are shown for 0-5% and 70-80% collisions within 0.4 < pT < / c and | y | < . Rustamov / Nuclear Physics A 00 (2020) 1–8 / SHINE results on κ /κ ( left panel ) and κ /κ ( right panel ) ratios on net-charge distribution at beammomenta of 150 / / c as a function of the mean number of wounded nucleons (cid:104) W (cid:105) . τ i ≤ τ f exp ( −| ∆ y | / , (7)where τ f denotes the hadron freeze-out time and is of order τ f = f m / c for central Pb–Pb collisions atLHC [24]. This implies that long range rapidity correlations ( ∆ y corr = | y beam | ), observed in ALICE data,can only be created at early times, shortly after the collision.For symmetry reasons all odd cumulants of net-proton distribution at LHC energies, measured at mid-rapidity, are vanishing. Indeed, the e ffi ciency corrected third cumulants of net-protons as measured by theALICE collaboration and presented in the right panel of Fig. 1 are, with the precision below 5%, consistentwith zero [25]. The consistency with the expected baseline indicates that all techniques and correctionprocedures applied to the experimental data are under control and establishes a solid approach to addressthe higher order cumulants. Indeed, the critical behavior at the LHC energies is predicted for higher ordercumulants of net-baryon distributions [26]. A factor of 10 more data, collected in 2018, are already su ffi cientto address the fourth order cumulans. From 2021 onward a factor of 100 more statistics will be recorded,which allows for precise measurements of six order cumulants. The excitation functions of the e ffi ciency corrected normalized cumulants ( S σ = κ /κ and k σ = κ /κ )in Au–Au collisions as measured by the STAR collaboration [27], are presented in Fig. 2. The measurementsare performed inside the sub-rage of the space space by imposing selection criteria on transverse momentumand rapidity of protons and anti-protons 0 . < p T ( GeV / c ) < . | y | < .
5. The values of κ /κ (cf. leftpanel of Fig. 2) are systematically below the ideal HRG baseline in the Boltzmann limit, both for central(0-5%) and peripheral (70-80%) collisions. This behaviour is in line with the baryon number conservatione ff ects as reported in [28]. On the other hand, the UrQMD [29] results, indicated by the hashed red lines,are above the experimental measurements. For energies higher than 20 GeV the UrQMD results overshootthe HRG baseline as well. In the right panel of Fig. 2 the κ /κ results are presented. For central collisionsat higher energies the experimental results approach the HRG limit of unity, while at around 27 GeV anevident dip emerges, rendering the non-monotonic energy dependence. The possible further increase of the κ /κ ratio, in the context of probing the proximity of the critical point, should be accompanied with thisbehaviour. Unfortunately the apparent increase of the measured κ /κ value at √ s NN = . A. Rustamov / Nuclear Physics A 00 (2020) 1–8
Fig. 4. HADES Au–Au data: E ffi ciency and volume corrected proton correlators C n ( n = , ,
4) as a function of the mean numberof protons (cid:68) N p (cid:69) within the selected phase-space bin, y ∈ y ± ∆ y ( ∆ y = ≤ p t ≤ / c, and for eight centralityselections. Error bars on data are statistical, cups delimit systematic uncertainties (shown, for clarity, only on the 0-5 % selection).Black dashed lines connect the data points in a given centrality selection and red solid curves are power-law fits C n ∝ (cid:68) N p (cid:69) α . It should be mentioned that, even if the dramatic increase of κ /κ at lower energies would be in place,it would still be not enough to claim the discovery of the critical point. In order to allow firm conclusions,further measurements at even lower energies would be necessary. With the upgraded STAR detector theapproved BES-II program aims at significantly improved statistics (about factor of 20), increased acceptancein both p T , y and extended measurements down to √ s NN = κ /κ √ s NN =
200 GeV the measured κ /κ √ s NN =
54 GeV they remain positive. This signchange between the two energies is at odds wit the latest LQCD calculations [3, 31, 32]. The experimentalmeasurements of 5 th order cumulants would shed light on these discrepancies. / SHINE
The NA61 / SHINE experiment aims at exploring the phase structure of strongly interacting matter byperforming 2-dimensional scan in beam momentum (13-150 / / c) and sizes of the colliding sys-tems. In the left and right panels of Fig. 3 the measured κ /κ and κ /κ ratios of net-charge distributionare presented, respectively. The consistency of both ratios with the corresponding EPOS results, shown bysolid black lines, indicates that the measurements are essentially driven by conservation laws. However, thecorresponding cumulant ratios for separate charges cannot be described with the EPOS model anymore [33]. . Rustamov / Nuclear Physics A 00 (2020) 1–8 This disagreement indicates that further systematic analysis is needed to fully understand the system sizedependence of fluctuation measurements performed by the NA61 / SHINE collaboration.
As mentioned in the previous section, it is essential to extend the fluctuation measurements towardslower energies. Recently the HADES experiment performed systematic studies of proton multiplicity dis-tribution in A–Au collisions at √ s NN = . ff erential properties of correlators. Indeed, as argued in [35], the functionalbehaviour of cumulants on selected rapidity range ∆ y allows to distinguish between long- and short-rangecorrelations. At HADES energies, however, it is more appropriate to use mean number of protons in-stead of ∆ y . In this representation short- and long-rage correlations lead to C n ∝ (cid:68) N p (cid:69) and C n ∝ (cid:68) N p (cid:69) n scalings, respectively. In Fig. 4 the e ffi ciency and volume corrected integrated multi-particle correlatorsare presented as measured by the HADES collaboration [15]. The data are fit with power-law functions C n ( (cid:68) N p (cid:69) ) = C (cid:68) N p (cid:69) α , where the exponent α and normalization constant C are fit parameters. For the mostcentral events (0-5%) the obtained values of α parameter are found to be close to the order of the corre-sponding multi-particle correlator, i.e, α ≈ n for all C n , with n =
2, 3, 4. This indicates that long rangecorrelations dominate the cumulants measured in Au–Au collisions by the HADES collaboration.
4. Summary
In summary, several measurements performed by the ALICE, HADES, NA61 / SHINE and STAR collab-orations are discussed. The correlations persistent in normalized second cumulant R , as measured by theALICE collaboration, are induced by collisions in the very early phase of the Pb–Pb interaction. After ac-counting for baryon number conservation, the ALICE data are in agreement with the corresponding secondcumulants of the Skellam distribution, consistent with the LQCD calculations at a pseudo-critical tempera-ture of about 156 MeV [3]. The power-law behaviour of multi-particle correlators, reported by the HADEScollaboration, suggest long range rapidity correlations. The STAR data on κ /κ exhibits non-monotonicbehaviour at around √ s NN =
20 GeV. However, the sign change of κ /κ , as reported by the STAR collabo-ration, is not consistent with the latest LQCD calculations. The measurements of net-charge fluctuations insmall systems performed by the NA61 / SHINE collaboration are in good agreement with the correspondingEPOS results. Near future challenges will be precision measurements of higher moments at RHIC, LHC,SIS as well as at facilities such as FAIR at GSI and NICA at JINR and their connection to fundamental QCDproperties.
Acknowledgments
This work is part of and supported by the DFG Collaborative Research Centre ”SFB 1225 (ISOQUANT)”.
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