Local Multiplicity Fluctuations as a Signature of Critical Hadronization at LHC
aa r X i v : . [ nu c l - t h ] N ov Local Multiplicity Fluctuations as a Signature of Critical Hadronizationat LHC
Rudolph C. Hwa and C. B. Yang , Institute of Theoretical Science and Department of PhysicsUniversity of Oregon, Eugene, OR 97403-5203, USA Institute of Particle Physics, Central China Normal University, Wuhan 430079, P. R. China (Dated: April 29, 2018)In central Pb-Pb collisions at LHC the multiplicity of particles produced is so high that it shouldbecome feasible to examine the nature of transition from the deconfined quark-gluon state to theconfined hadron state by methods that rely on the availability of high multiplicity events. Weconsider four classes of the transition process ranging from critical behavior to totally randombehavior, depending on whether or not there is clustering of quarks and on whether or not there iscontraction of dense clusters due to confinement. Fluctuations of bin multiplicities in each event arequantified, and then the event-by-event fluctuations of spatial patterns are analyzed. A sequenceof measures are proposed and are shown to be effective in capturing the essence of the differencesamong the classes of simulated events. It is demonstrated that a specific index has a low value forcritical transition but a larger value if the hadronization process is random.
PACS numbers: 24.60.Ky, 25.75.-q, 25.75.Nq
I. INTRODUCTION
Phase transition has always been a subject of greatinterest in many fields. The possibility of observing ev-idences for the critical point in heavy-ion collisions hasinvigorated extensive experimental programs at variouslaboratories [1, 2]. The physics of QCD critical pointconcerns a dense system of strongly-interacting matterthat is in thermal and chemical equilibrium and is at theend of the phase boundary between quark and hadronphases [3]. The search for signals of that boundary in-volves experiments at energies where high baryon densitycan be produced. At the Large Hadron Collider (LHC)where the collision energy is much higher, the physics ofhadronization involves issues that are different from thatprobed at lower energies where high chemical potential isexpected. Since a hot and dense plasma is produced atLHC, the deconfined state persistes for a long time beforethe system is dilute enough to undergo transition to thehadron phase. That transition may or may not be rec-ognizable as a critical phenomenon, since hadronizationtakes place on the surface over a period of time while thesystem expands. The accumulation of hadrons emittedover that period can smear out any signal of interest evenin the best circumstance for critical transition, which isnot assured on theoretical grounds. Nevertheless, or per-haps particularly because of the difficulty in detectingrevealing signals, it is of interest to investigate whetherappropriate measures exist. It is the aim of this work tofind the most effective way to extract dynamical signalsindicative of collective behavior of a system transition-ing from quarks to hadrons. Our search does not relyon the validity of any theoretical view on the possibilityof critical behavior in heavy-ion collisions at very highenergy.Despite the fact that a large amount of data on Pb-Pbcollisions at LHC has been produced in the past year [4], so far no spectacular departure from earlier expectations[5] by extrapolation from the Relativistic Heavy-Ion Col-lider (RHIC) has been reported. It is not certain whetherit is because there is no unexpected new physics, or be-cause unanticipated physics is not revealed in the con-ventional observables. We venture here to ask whetherthere may be an area of investigation of the LHC datathat has thus far been neglected but may prove to befruitful when suitable measures are used.An obviously outstanding feature of the data obtainedat LHC is that the total multiplicity N ch of charged par-ticles produced is unprecedentedly high, around 6 × in central collisions at 2.76 TeV [6]. Observables that relyon large N ch can thus be exploited in ways that could nothave been possible at lower energies, thereby forming afrontier that has not been explored. Another aspect ofcentral Pb-Pb collisions is that there is surely a decon-fined system of quarks and gluons, which must undergosome kind of transition to the confined state of hadrons.Theoretically, the use of Cooper-Frye scheme [7] to cal-culate the hadronic properties avoids the issue about thenature of the deconfinement-to-confinement transition.In our investigation we leave open the question of whatthe nature of that transition is. We propose four sce-narios that range from critical behavior on one end tonon-critical on the other. The focus is on the discoveryof measures that can possibly distinguish those cases. Totest whether the proposed measures are feasible experi-mentally, we simulate the events in each case and examinethe effectiveness of the measures.The theme of our investigation is about the fluctua-tions of spatial patterns during the quark-hadron transi-tion. The momentum of each particle is usually expressedin terms of ( p T , η, φ ), where p T is the transverse momen-tum, η the pseudo-rapidity, and φ the azimuthal angle.For any interval of p T one can study the ( η, φ ) distribu-tion in any given event, called the lego plot when shownin some finite-size binning. For convenience, we refer tothat distribution as a spatial pattern, which changes fromevent to event. The basic question is whether the fluctua-tions of those patterns contain any information about thenature of the quark-hadron transition. To study thosepatterns one needs good resolution from the experimen-tal side and efficient description from the theoretical side.Furthermore, one should not integrate over all p T becausethe superposition of different patterns at different ∆ p T intervals can smear out all recognizable features. Thus tobe able to have high resolution in all ( p T , η, φ ) variablesas well as to have enough particles in small bins in eachvariable requires high event multiplicities. That is wherethe LHC data become extremely useful.There are recent reports from ALICE on the event-by-event fluctuations of global observables in Pb-Pb colli-sions [8, 9]. Non-statistical fluctuations of mean p T arefound to be lower than a scaling behavior in event mul-tiplicity as the collisions become more central. No ex-planation of the phenomenon is given in terms of knownmodels. Our study here is on local multiplicity fluctua-tions in precisely those central collisions.There are two kinds of fluctuations that we shall con-sider. One is the fluctuation of bin-multiplicity from binto bin in an event, and the other is the event-to-eventfluctuation of the spatial patterns. On the former it canbe just random fluctuations, but it can also be in the formof clusters of all sizes, as in second-order phase-transition[10]. Both types of patterns will be generated to initiatequark configurations in the ( η, φ ) space before and dur-ing time evolution throughout the period of hadroniza-tion for two distinct classes of models. We shall constructan algorithm for simulating the global effect of color con-finement through contractions of dense clusters betweentime steps in the hadronization period and the oppositeeffect of thermal agitation by randomization right aftereach contraction. Pion formation according to chosencriteria can occur throughout the process, resulting in anevent distribution of pions in ( p T , η, φ ). That distribu-tion is then analyzed by factorial moments to filter outstatistical fluctuations. Repeating the simulation overmany times generates event-by-event fluctuations whichturn out to be crucial to the finding of revealing signa-tures. Large deviations from the average event structureare possible but very rare; however, they can make sig-nificant influence on the measure that we shall propose. II. LOCAL MULTIPLICITY FLUCTUATIONSWITH CRITICAL CLUSTERING
In search for ways to simulate configurations that havea wide range of characteristics without theoretical preju-dices, ranging from critical to non-critical cases, we focusin this section on finding a simple procedure to generateconfigurations that correspond to critical behavior. Wefirst review a number of related investigations on the localbehaviors of multiplicity fluctuations of systems under- going a second-order phase transition describable by theGinzburg-Landau (GL) theory. They are then to be con-nected to a distribution of cluster production that canreadily be used to simulate initial configurations beforehadronization begins. Only one parameter is needed todescribe the clustering. Later, for the non-critical caseit is only necessary to change that parameter so as toobtain random distributions.It was suggested that the normalized factorial moments F q can be used as a quantitative measure of local fluctu-ations [11]. F q is defined by F q ( δ ) = h n ! / ( n − q )! ih n i q , (1)where n is the multiplicity in a bin of size δ d in a d-dimensional phase space, and the averages are performedover many events. A power-law behavior F q ( δ ) ∝ δ − ϕ q (2)over a range of small δ is referred to as intermittency, andhas been observed in many systems of collisions [12]. Thevirtue of F q is that it filters out statistical fluctuations,as can be seen as follows. If the multiplicity distribu-tion P n can be written as a convolution of the Poissondistribution P n ( n ) and a dynamical distribution D ( m ),i.e., P n = Z ∞ dm m n n ! e − m D ( m ) , (3)then the numerator of F q is ∞ X n = q n !( n − q )! P n = Z ∞ dmm q D ( m ) , (4)which is a simple moment of D ( m ). Thus, if the dynamicsis trivial, i.e., D ( m ) = δ ( m − ¯ n ), then F q = 1 for all q .Any deviation from 1 is then a measure of non-trivialdynamical fluctuations, and a power-law behavior in Eq.(2) would suggest a dynamics that is not characterizedby a particular scale [11]. The footprint of a PT that hasfluctuations of all scales may then be possibly observedin the measurement of intermittency [13].To obtain a theoretical quantification of PT in terms of F q , a study of second-order PT in the Ginzburg-Landautheory was carried out in Ref. [14], in which the orderparameter is identified with the multiplicity density. Itis found that to a high degree of accuracy F q satisfies thepower-law behavior F q ∝ F β q , (5)where β q = ( q − ν , ν = 1 . , (6)essentially independent of the details of the GL param-eters. Such a behavior was experimentally verified bythe study of photon number fluctuations of a single-modelaser at the threshold of lasing [15], confirming that it is aPT problem describable by GL theory [16]. On the theo-retical side, it has also been found that using the 2D Isingmodel to simulate quark-hadron PT the resulting scalingbehavior of F q is in agreement with Eqs. (5) and (6) [17].It does not mean, however, that an analysis for F q in thecurrent data from heavy-ion collisions can verify or falsifythe connection between hadronization and second-orderPT because of the complications that are present in suchsystems but absent in the optical system. The followingsections are aimed at addressing such complications. Wenote that in the discussion above concerning Eqs. (1) to(6) no mention is made of the specifics about multiplici-ties, whether hadrons or quarks. In the remainder of thissection we shall use particles as a generic term that canrefer to either hadrons or quarks. The result to be ob-tained will be used in the following sections to generateconfigurations of quarks (and/or antiquarks) just beforehadronization.From a different perspective it is of interest to studythe problem of clustering, since critical phenomenon isknown to be characterized by clusters of all sizes. It is asubject relevant to our investigation here because we shallanalyze the 2D spatial distribution of particles in the η - φ space. Cluster formation in the context of hadronizationhas been considered before [18], but from the point ofview of self-organized criticality instead of second-orderPT. Since the latter exhibits the behavior of Eqs. (5) and(6), it is not difficult to see how such a behavior can beachieved by an adjustment of the scaling property of thecluster formation.As a generic problem on clusters of particles, let C bethe number of particles in a cluster, and P ( C ) be theprobability distribution in C , which we assume to havethe scaling form P ( C ) ∝ C − γ , γ > . (7)We let the center of a cluster be distributed randomlyin a 2D space which we take to be 1 unit of length oneach side. We further let a particle in a cluster C to bedistributed randomly around its center with a Gaussianwidth of σ = 0 . C − / (8)so that there can be high density of particles, thoughwith decreasing probability at high C . Such spikes ofmultiplicity are what can give rise to intermittency.To investigate the multiplicity fluctuations in small binsizes, we divide the unit square into M × M bins, M vary-ing from 8 to 70. We let N particles be distributed in theunit square in accordance to Eq. (7) and (8), and allow N to be approximately 100, since the sum of all clustersmay not be exactly the same for every event. With n being the number of particles in a bin, we perform thecalculation for F q ( δ ) according to Eq. (1), by averagingover all bins first, and then averaging over all (5 × ) events for any fixed δ = 1 /M . Only bin multiplicitieswith n ≥ q are counted in F q ( M ). The result for γ = 2 isshown in Fig. 1(a) that exhibits non-trivial dependenceon M ; it is not strictly linear in the log-log plot. How-ever, when F q is plotted against F for q = 3 , ,
5, verygood linearity is found, as shown in Fig. 1(b) replicat-ing the behavior found for GL [14]. Similar power-lawbehavior is obtained for other values of γ . We show inFig. 1(c) the dependence of the exponent β q , defined inEq. (5), on q −
1, and find in accordance to Eq. (6) that ν = 1 . , . . γ = 1 . , . , .
5, respec-tively.
10 20 4010 M 〈 F q 〉 (a) γ =2 q=5q=4q=3q=2 〈 F 〉 (b) q=5q=4q=3 q−1 β q (c) γ =2.5 γ =2.0 γ =1.5 FIG. 1: Intermittency analysis of clustering model for γ = 2in Eq. (7): (a) scaling in M , (b) F -scaling in F q vs F , and(c) power-law behavior of β q in Eq. (6). What we have done above is to find a quick way torelate a particle distribution in a 2D space to a GL-type F q -scaling behavior through the use of a cluster distri-bution. Since a range of values of ν has been obtained,the relationship between γ and ν has no specific dynam-ical significance. It merely demonstrates that large localmultiplicity fluctuations can generate large F q with scal-ing properties. In the case of γ = 2 . β = 1, thus showingslight deviation from Eq. (6). The special value ν = 1 . γ between 2.0 and 2.5. Although nodynamics has been introduced to establish that connec-tion, our objective here is accomplished by having founda simple procedure to simulate configurations that maybe relevant to a quark and antiquark system that is at theedge of PT to hadrons. This is only the starting point ofa much more complicated problem in heavy-ion collisionsthat we shall describe next. III. SIMULATION OF EVENTS WITH SPATIALFLUCTUATIONS IN HEAVY-ION COLLISIONS
If there is no dynamical structure in the spatial pat-tern of soft hadrons in the lego plot (of η and φ ), thenthere should be nothing interesting in whatever measurethat is used to analyze the data. The reverse is, however,not necessarily true. Unless a measure is sensitive to theconsequence of certain dynamics, one may not observewhat is interesting. To provide motivation for an experi-mental effort to search for an unconventional signal, it isnecessary to demonstrate the worthiness of such an effortunder the best of circumstances. The optimal scenariois that the quark-gluon system undergoes a second-orderPT in forming hadrons so that there are large multiplicityfluctuations. It is not known whether such a phenomenonoccurs at LHC. Our aim here is to simulate events thatbelong to different classes of dynamical characteristics,ranging from robust criticality to mundane randomness.For each class of hadronization the measure to be usedin the following section should exhibit distinguishing fea-tures in the hope that analysts of the real data wouldhave the incentive to pursue the more difficult task ofextracting worthwhile information from what is actuallyobserved.The physics we aim to simulate is for the duration be-tween the end of the quark phase and the beginning ofthe hadron phase. It is assumed that the density is solow that the confining forces among the quarks and theantiquarks begin to redistribute them spatially. Gluonsare assumed to have been converted to q and ¯ q that arethe basic units prior to hadronization. We restrict ourattention to the central rapidity region with | η | ≤ ≤ φ ≤ π for central collisions. The surface of theplasma cylinder with those values of η and φ is mappedto the unit square S . Since the fluctuation propertiesto be discussed below do not depend on the precise areathat S corresponds to, a portion of the ( η, φ ) region inthe actual data should suffice to serve as a workable ba-sis for analysis. The cylinder hadronizes at the surfaceonly, layer by layer. Thus we have to consider many timesteps t i , i = 1 , , · · · . Between adjacent time steps the q and ¯ q adjust their positions in S in accordance to aprocedure that we impose to simulate confinement andpion emission. The quarks have thermal p T distributionwhose inverse slope T i decreases incrementally with t i .At a new time step a new set of q and ¯ q are introducedto represent the movement of the next layer of quarks tothe surface. The confinement procedure starts over againuntil most of the quarks (not necessarily all) are pionizedbefore another time step is taken. This is a general out-line of the algorithm to convert q and ¯ q to pions, whosecoordinates in ( p T , η, φ ) are registered for event-by-eventanalysis later.The main part of our physics input is the deconfine-ment to confinement transition between time steps. Theprincipal characteristic of a critical phenomenon is thetension between the ordered and the disordered motions of a system. In the Ising model of a magnetic system thenear-neighbor interaction between spins tends to alignall spins in the same direction, but the thermal motiontends to randomize them. If the quark-hadron PT is inthe same class of criticality, then the tension is betweenconfinement that draws the q ¯ q into an ordered pair andthe disordered thermal agitation that tends to keep thequark system in the deconfined state. On the surfaceof the plasma cylinder the q and ¯ q may start to clusteras the density and temperature get close to the transi-tion point, thereupon a local region of high density con-tracts under the confinement force to improve the likeli-hood of a q ¯ q pair to fall within a confinement distance.In principle, we should keep track of the colors of thequarks, but that would complicate the procedure muchmore. The simple algorithm we adopt mimics the generalidea of color confinement and generates large multiplic-ity fluctuations. One of the chief objectives of this workis to see whether many such local fluctuations at differ-ent time steps can survive the superposition at the endof the hadronization process, and still be detected by aneffective measure. The feasibility issue is addressed atthe cost of precision in QCD, which is difficult in the softregime over an extended period.Let us describe first the case of critical hadronization,followed later by other cases that are less critical. A. Critical
1. Initial Configuration
We start by seeding the unit square S with 500 pairsof q ¯ q . They are clustered according to the probabilitydistribution P ( C ), given in Eq. (7). That is, C pairs of q ¯ q are placed in a cluster centered at a random pointin S ; they are Gaussian distributed around the centerwith a width specified by Eq. (8). As a result of thestudy in Sec. II that relates the exponents γ and ν , wechoose γ = 2, corresponding to ν not quite up to 1.3.As the cluster multiplicity C is summed over all clusters,we stop the seeding process when the total just exceeds500. Inside each cluster the q and ¯ q are not correlated;they are independently distributed in ( η, φ ) as well as in p T , for which the thermal distribution is the exponential,exp( − p T /T ), with T set at 0.4 GeV.The above procedure is for setting up an initial con-figuration at t = 0, counting from just before the firstpions are emitted, but long after the collision time. Thevalue of T is not set at T c which is lower, but at a valueabove the observed average h p T i , since the bulk of the pi-ons will be produced at later time when T will be lower.The clustering is put in to generate a spatial configu-ration that is most likely to represent a system movingtoward a critical transition.
2. Pionization
In the above configuration if a q and ¯ q are within a dis-tance d from each other that is less than d = 0 .
03, thenwe regard the pair as effectively a pion and let it be takenaway from the configuration, but registering the pion po-sition in a separate ( η, φ ) space at the midpoint betweenthe pair. We assign a value of p T to that pion that isequal to the sum of the p iT values of the q and ¯ q . This isbased on the recombination model, where the recombi-nation function has a momentum-conserving δ -function: δ ( p T − p T − p T ) [19–21]. The thermal distributions ofquarks and pions have the same T . We ignore the colorand flavor of the quarks without losing the essence of lo-cal fluctuation, since if the q and ¯ q had color and flavorlabels, it would take longer in the iteration process forthe q ¯ q pair in a cluster to pionize without changing the( η, φ ) coordinates appreciably.
3. Contraction
The probability that a q ¯ q pair is within d apart issmall, so the majority of the quarks remain in S . Theyare under the influence of the color forces to move towardconfinement. To describe that movement in a collectiveway rather than in terms of pair-wise interaction that isunrealistic, we adopt the contraction procedure as fol-lows.Let S be divided into 5 × q and ¯ q in each bin. Separate the bins into twotypes: dense bins have more q and ¯ q than the averagebin multiplicity, and the dilute bins have less. The dif-ference between q and ¯ q is ignored here. If adjacent densebins share a common side, they are grouped together asmembers of a cluster of dense bins. Let D refer to such acluster of dense bins, which may have an irregular shape,but are connected. Let N D denote the number of bins in D . Define ~r D to be the coordinates in S that is the centerof mass of D . Now, we do a contraction of D . That is,we redistribute all the q and ¯ q in D , centered at ~r D , butwith a Gaussian width σ D = σ N / D , (9)where σ is a parameter that characterizes the degree ofcontraction. We use σ = 0 . q and ¯ q that are spread out originally in N D binsare drawn closer together to be located mostly within theGaussian peak, resulting in a contraction. Since there are5 × . × .
2, to which σ D should be compared after contraction. This is how wemodel the effect of the ordered motion due to confine-ment. We repeat step to allow q ¯ q pairs to pionize inthis new configuration.
4. Randomization
The disordered motion that counter-balances the or-dered motion is the thermal randomization. We modelthat by requiring all the q and ¯ q in the dilute bins to beredistributed randomly throughout S , resulting in a newconfiguration. We then repeat step to have anotherround of contraction and pionization. Each time in theiteration process more q ¯ q pairs are converted to pions, aswe alternate contraction and randomization until around95% of the q ¯ q system is depleted. We regard that as theend of one time step in which the quarks on the cylin-der surface are hadronized. We then proceed to the nexttime step when a new layer of quarks move up to thesurface.We note that before the next time step is taken, pi-onization becomes increasingly difficult when fewer andfewer q and ¯ q remain to find their partners to coalesce.Decreasing σ to 0.05 speeds up the process near theend, but does not change the pion distribution in ( η, φ )very much. Physically, it is not necessary that all q ¯ q in alayer hadronize before the next layer of quarks moves up.Leaving roughly 5% to be mixed with the next layer of q ¯ q seems reasonable in our attempt to model a continuousprocess of hadronization by discrete steps.
5. Subsequent Time Steps
To advance to the next time step we introduce 100 newpairs of q ¯ q according to the distribution P ( C ) and addthem to the existing q and ¯ q that remain from the pre-vious step. We then follow the same procedure as aboveto contract, pionize, and randomize repeatedly until 5%of q and ¯ q remains. At the i th time step all q and ¯ q havethe p T distribution with an inverse slope T i = T i − − .
02 GeV . (10)That is the T i that the pions emitted at t i will also have.This is carried out 10 times so that in total we intro-duce 1500 q ¯ q paris, most of which turn into pions. Tohave approximately 1400 pions produced in | η | < p T , η, φ ) for the event is then stored for later analysis oflocal fluctuations. B. Quasi-critical
Consider now the case where clustering does not occurin the initial configuration, nor when new layers of q and¯ q move to the surface. Thus the initial 500 pairs of q ¯ q are seeded randomly in S , and so are the subsequent 100pairs at each time step. There is then no correspondencewith the second-order PT discussed in Sec. II. Techni-cally, this corresponds to the exponent γ in Eq. (7) beingvery large, because the probability for having a large clus-ter C is then very small. Specifically, we choose γ = 5 forno clustering. One may regard this case as being in corre-spondence to a cross-over in the phase diagram where nodistinct boundary between the quark and hadron phasescan be identified. However, we require that the confine-ment forces to be still at work to turn quarks to hadrons.So we use the contraction-randomization procedure de-scribed in subsection A to carry out pionization. Sucha procedure simulates the tension between confinementand deconfinement, but cannot be regarded as what isneeded for a critical transition. The coordinates of eachpion in ( p T , η, φ ) are recorded as before for later analysis. C. Pseudo-critical
Suppose now that we seed the configurations with clus-tering, but do not impose contraction between time steps.Thus the q and ¯ q configurations, as each layer reaches thesurface, are close to the critical condition, when we set γ = 2 as we have done above. However, the hadronizationprocess is carried out by letting q ¯ q pairs form pions when-ever a pair gets close together. Without contraction wecannot require the distance between pairs to be less than d as in Subsec. A.2, since the probability for that to oc-cur is low. Without new dynamics during hadronizationwe simply let each quark to search for the nearest anti-quark within a distance d = 0 . q ¯ q pair. Then wego to another quark and repeat the process. When nomore pairs can be found within that distance, we pro-ceed to the next time step whatever numbers of q and ¯ q are left. With more q ¯ q pairs supplied from the next layer,the probability of pionization is increased. This processwill not convert all q ¯ q pairs to pions since the quarks canbe far apart from antiquarks without contraction. Tohave quarks left over at the end of ten time steps doesnot matter as far as the local multiplicity fluctuation isconcerned. We shall see that there is still interestingstructure that can be extracted from the accumulatedevents simulated that way. D. Non-critical
To the other extreme situation away from the abovethree cases we consider the non-critical case of no orga-nized dynamics at all, i.e., random configurations and nocontraction. The result should not have any content ofinterest. We have nevertheless carried out the simulationand will show the result that is non-trivial and thereforeinstructive.A summary of the four cases can be expressed as amatrix shown below.clustering no clusteringcontraction critical quasi − critical no contraction pseudo − critical non − critical (11) IV. MOMENTS FOR EVENT-BY-EVENTFLUCTUATIONS
We now consider the method of analysis of the manyevents either as measured at LHC or as generated inthe models described in the preceding section. Becauseof the high multiplicity of particles produced in cen-tral collisions, we make cuts in p T , such as in an in-ternal ∆ p T around p T = 1 GeV/c. We have considered∆ p T = 0 . , .
07 and 0.1 GeV/c. In such small intervalsthe particle multiplicities are significantly reduced andspatial patterns in ( η, φ ) begin to appear because of thepossibility of empty bins.To find an effective measure of the fluctuations we needit to be sensitive to both the spatial variations from binto bin and the event-by-event fluctuations. We refer tothe former as horizontal and the latter as vertical. Differ-ent moments are to be taken for horizontal and vertical averages so as to allow spatial and event-wise fluctuationsto manifest separately.We divide the unit square into M bins with M beingnot more than 70, depending on the multiplicity in the∆ p T interval, just so that the important part of the M dependence is captured. For the spatial fluctuations weuse the horizontal factorial moments for event e , definedas F eq ( M ) = f eq ( M ) / [ f e ( M )] q , (12)where f eq ( M ) = h n ( n − · · · ( n − q + 1) i h . (13)The average h i h is performed over all M bins, n be-ing the multiplicity in a bin. Only n ≥ q is counted in f eq ( M ). Clearly, if M is very large, f eq ( M ) may be zero,for q ≥
2, although f e ( M ) is never zero. However, theremay be an event where F eq ( M ) may not vanish at a large M ; then it would imply sharp spikes of multiplicity insome bins. If the fluctuations among the bins are Poisso-nian, then we have F eq ( M ) = 1 for any M , as is the casewith vertical fluctuations discussed in Sec. II. Interestingspatial fluctuations are, however, not Poissonian.For each event we can calculate F eq ( M ). We note that F eq ( M ) is a simple characterization of spatial pattern,but is not the only one possible. Any alternative descrip-tion can also be used in the study below on event-by-event fluctuations. One may therefore regard F q ( M ) asa generic symbol.If h F q ( M ) i v denotes the vertical average of F eq ( M ) overall events, then the fluctuation of F eq ( M ) from h F q ( M ) i v is what we want to quantify. To that end we consider the p th-order moments ofΦ q ( M ) = F eq ( M ) / h F q ( M ) i v , (14)i.e., C p,q ( M ) = (cid:10) Φ pq ( M ) (cid:11) v , (15)which is a double moment introduced earlier for the studyof chaotic behavior of particle production in branchingprocesses [22]. Whereas q must be an integer, p need notbe. In fact, the derivative of C p,q ( M ) at p = 1, i.e.,Σ q ( M ) = ddp C p,q ( M ) (cid:12)(cid:12)(cid:12)(cid:12) p =1 = h Φ q ln Φ q i v , (16)has been related to an entropy defined in the event space[22], but by itself it is not very useful because it dependson M . Simplification can occur if C p,q ( M ) has a power-law behavior in MC p,q ( M ) ∝ M ψ q ( p ) . (17)Then an (entropy) index can be defined as µ (1) q = d Σ q ( M ) d ln M = ddp ψ q ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p =1 , (18)which is independent of M . It was found that µ (1) q cancharacterize the fluctuations of spatial patterns so wellthat they are as useful as the Lyapunov exponents inproviding a quantitative measure of classical chaos [23].The power-law behavior of C p,q ( M ) in Eq. (17) hasbeen referred to as erraticity [24, 25]. It was proposedas the next logical step to take beyond the intermittencyanalysis. Attempts have been made to find experimen-tally the erratic fluctuations of F eq from event to event inmultiparticle production [12]. In meson-proton collisionsat 250 GeV/c beam momentum, it was found that theerracticity measures are dominated by statistical fluctu-ations at such low energy [26]. In nucleus-nucleus colli-sions interesting signals have been found both in mod-els [27] and in emulsion experiments [28] even at lowenergies. At high energy where we can examine the deconfinement-to-confinement transition, it will becomeclear in the next section, where model calculations cangenerate concrete local multiplicity fluctuations, that itis more comprehensive to consider the range 1 ≤ p ≤ C p,q ( M ). If in that range ψ q ( p ) has a lineardependence on p , it is better to define the slope in thatwider range as µ q = dψ q ( p ) /dp. (19)It is this quantity µ q , referred to as erraticity indices,that will become an effective measure of the criticalityclasses that is independent of p and M . A large valueof µ q means that there are influential contributions to C p,q ( M ) from large F eq ( M ) weighted more heavily atlarge p , which in turn implies that very erratic fluctua-tions of the spatial patterns are involved to render F eq ( M )non-vanishing at large q and M . FIG. 2: Examples of bin multiplicity fluctuations in ( η, φ ) forthe four cases arranged in the matrix form of (11), i.e., (a)critical, (b) quasi-critical, (c) pseudo-critical, (d) non-critical.
V. RESULTS OF MODEL CALCULATIONS
We have simulated in the order of 10 events for thefour models ranging from critical to non-critical casesdescribed in Sec. III. To give visual images of their qual-itative differences, we select one event from each case toillustrate their behaviors in the lego plots. For an eventto contribute to F q at q ≥
2, it cannot be typical if theaverage bin multiplicity h n i is small. We show untypi-cal events that have large n in some bins and thus cancontribute to non-trivial F at M = 30 and ∆ p T = 0 . p T = 1 GeV/c, for which h n i is approximately0.1. In Fig. 2 the four lego plots are arranged as a ma-trix in the classification given in (11). Each of the plotscorresponds to an event in a given class that contribute tothe largest value of F ( M ) at M = 30; thus there shouldbe at least one bin that has a bin multiplicity n ≥
10 20 40 8010 M 〈 F q 〉 (a) q=5q=4q=3q=2 〈 F 〉 (b) q=5q=4q=3 2 3 4248 ν =1.41 (c) q−1 β q
10 20 40 8010 M 〈 F q 〉 (a) q=5q=4q=3q=2 〈 F 〉 (b) q=5q=4q=3 2 3 4248 ν =1.41 (c) q−1 β q FIG. 3: Intermittency analysis for the critical case. Panelsare the same as in Fig. 1.
10 20 4010 M 〈 F q 〉 (a) q=5q=4q=3q=2
12 16 20 〈 F 〉 (b) q=5q=4q=3 2 3 4248 ν =1.33 (c) q−1 β q
10 20 4010 M 〈 F q 〉 (a) q=5q=4q=3q=2
12 16 20 〈 F 〉 (b) q=5q=4q=3 2 3 4248 ν =1.33 (c) q−1 β q FIG. 4: Intermittency analysis for the quasi-critical case.Panels are the same as in Fig. 3.
To see the scaling behavior, we examine the verticalaverages h F q ( M ) i v vs M in log-log plots for q = 2 , · · · , p T = 0 . v will be omitted for brevity. For the critical casethe results are shown in Fig. 3 where in (a) there is an
10 20 40 8010 M 〈 F q 〉 (a) q=5q=4q=3q=2 〈 F 〉 (b) q=5q=4q=3 2 3 4248 ν =1.26 (c) q−1 β q
10 20 40 8010 M 〈 F q 〉 (a) q=5q=4q=3q=2 〈 F 〉 (b) q=5q=4q=3 2 3 4248 ν =1.26 (c) q−1 β q FIG. 5: Intermittency analysis for the pseudo-critical case.Panels are the same as in Fig. 3.
10 20 40124 M 〈 F q 〉 (a) q=5q=4q=3q=2 〈 F 〉 (b) q=5q=4q=3 2 3 41.52.53.5 (c) q−1 β q
10 20 40124 M 〈 F q 〉 (a) q=5q=4q=3q=2 〈 F 〉 (b) q=5q=4q=3 2 3 41.52.53.5 (c) q−1 β q FIG. 6: Intermittency analysis for the non-critical case. Pan-els are the same as in Fig. 3. increase with M before M gets larger than 20, but theyall have similar behavior for different q , so when h F q i isplotted against h F i , we see in (b) a simple straight-linebehavior for q = 3 , ,
5. Using Eqs. (5) and (6) to de-scribe the power β q , we find in (c) that ν crit = 1 . ν quasi = 1 .
33, as shown in Fig. 4. In the pseudo-critical cases, shown in Fig. 5, we see robust scaling be-havior in (a). F -scaling behavior in (b) yields the value ν pseudo = 1 .
26 shown in (c). Finally, in the non-scalingcase we find the opposite situation where h F q ( M ) i de-creases with increasing M , as exhibited in Fig. 6 (a). Itmeans that bin multiplicities do not get large enough de-viations from h n i through random fluctuations so that,when the bin size gets small, h F q ( M ) i approaches 1 thatone expects from Poissonian fluctuations, as stated justbelow Eq. (4). In (b) we still see regularity in h F q i plot-ted against h F i , but it is important to recognize thatthe low end corresponds to high M with all h F q i around1, while the high end is for low M , quite contrary tothe three cases in Figs. 3-5. One can extract the valuesof β ( q ) as in (c), but there is no sensible value of ν tobe assigned to this case. In short, the non-critical casedoes not yield any interesting result from the study ofthis kind. We have to go beyond simple intermittencyanalysis in order to find a suitable description that canrender quantitative comparison between the critical andnon-critical cases.Although the results shown in Figs. 3-5 are clear andeasily quantifiable in terms of ν , a great deal of infor-mation is lost by calculating those averages. To exhibitthe degree of fluctuations of the spatial patterns fromthe average, let us use P (Φ q ) to denote the probabilitydistribution of event Φ q ( M ) at fixed M , where Φ q ( M ) isdefined in Eq. (14). In Fig. 7 we show P (Φ q ) for q = 2and 4, and for clarity only for M = 8 and 30. The fourclasses of criticality are again in the matrix format of(11). We see that in all cases the distributions for q = 2and M = 8 (solid lines) are peaked at Φ q = 1. But forother values of q and M , the four cases differ in differ-ent ways. In the other extreme situation correspondingto q = 4 and M = 30 (lines with crosses), we see that P (Φ ) is peaked at Φ = 0 in all cases. That is becausein small bins the average bin multiplicity h n i is much lessthan 4, so the values of Φ for many events are 0. In fact,for (c) pseudo-critcial and (d) non-critical, only a smallfraction of events have large enough bin fluctuations torender F e non-zero, so P (Φ ) has a δ function peak atΦ = 0, whose positions are shifted in Fig. 7 for visibil-ity’s sake. In the intermediary values of q = 4 , M = 8,the dashed lines in all four cases are all very broad, signi-fying wide fluctuations. For q = 2 , M = 30 (dash-dottedlines) the peaks are around Φ = 1, similar to the solidlines.In the insets of Fig. 7 (c) and (d) we show that for q = 4 and M = 30 there are contributions to P (Φ )at extremely large Φ (in the order of 10 ). The verti-cal axes have the scale factor 10 − . They balance the δ functions at Φ = 0 so that the average is h Φ i v = 1, bydefinition. This irregular behavior reveals the nature offluctuations from event to event. When the average binmultiplicity h n i is about 0.03, it is difficult to find eventsin which there is a bin with n ≥
4, unless there are dy-namical effects (such as confinement contraction) to in-troduce large fluctuations. In cases (c) and (d) almostall events have n < = 0, except for some veryrare events that make non-trivial contribution to non-zero Φ , whose values are therefore very large because F ( M = 30) is exceedingly large (due to the smallnessof f ) even though Φ is normalized by h F ( M = 30) i which is proportional to the rarity of such events.The probability distributions P (Φ q ) contain too muchinformation that cannot easily be conveyed. We learnfrom Fig. 7 that for q = 2 there are no drastic differ- P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) Φ q P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) Φ q P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) Φ q P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) × Φ q P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) × −6 Φ q × −6 P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) × −6 Φ q × −6 P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) × −6 Φ q × −6 P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) × −6 Φ q × −6 P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) × −6 Φ q × −6 P ( Φ q ) (a)q=2,M=8q=2,M=30 (b)q=4,M=8q=4,M=300 1 2 30123 (c) × −6 Φ q × −6 FIG. 7: Probability distributions of Φ q ( M ) for various valuesof q and M . The panels correspond to the criticality casesexpressed in (3.3) and as shown in Fig. 2. The insets in (c)and (d) are for large values of Φ q as explained in the text.The vertical axes have a common scale factor of 10 − . ences among the four cases when M is increased from8 to 30. It means that bin multiplicities in each casecan fluctuate sufficiently to exceed n = 2 and generatea modest width of the peaks in P (Φ ) around Φ = 1.We therefore should not expect a good measure at q = 2to distinguish the criticality classes. For q = 4, however,we see significant differences between the cases (a,b) withcontraction and (c,d) without contraction. To quantifytheir differences we consider the moments C p,q ( M ) de-fined in Eq. (15). For p = 2 , q = 4, we show in Fig. 8 the M dependence for three intervals of ∆ p T : 0 . , .
07 and0.1 GeV/c. The cases a, b, c, and d in the legend corre-spond to the panels in Fig. 7 arranged in the matrix formof (11). Some of the open symbols are displaced slightlyfrom M = 8 ,
16 and 30 in order to avoid them from beingcovered by the filled symbols. We see that they satisfypower-law behavior very well in all cases, validating themeaningfulness of the erraticity exponents ψ q ( p ) definedin Eq. (17). Similar study can be done for p = 1 . , . ψ ( p ) are shown in Fig. 9. In that fig-ure the cases (a) and (b) are depicted collectively by opensymbols, while the cases (c) and (d) are by filled symbols.The values of ψ ( p ) in the (c,d) group are all larger thanthose in the (a,b) group. Evidently, there are good lineardependencies on p between 1.25 and 2 in all cases. Thestraight lines drawn through them are extended only to p = 1 .
1. At p = 1 it is necessary that ψ q (1) = 0 because C ,q ( M ) = 1 for any q and M . The lines, if extrapolatedfurther to p = 1, would all miss the origin (1 ,
0) by littlebits. There are some reason for that to happen at thepoint p = 1, which we discuss below. For now, we con-centrate on the linear portion in Fig. 9 and determine0 ∆ p T a b c d0.040.070.10 M C , FIG. 8: Power-law behavior of C , ( M ) exhibiting erraticity.The labels a, b, c, d correspond to the four panels in Figs.2 and 7 in the matrix format of (11). Pairs of symbols areapproximated by one straight line, so 12 types of symbols arerepresented by 6 lines. ∆ p T are in the units of GeV/c andare around the value p T = 1 GeV/c. ∆ p T (a,b) (c,d)0.040.070.10 p ψ ( p ) ∆ p T (a,b) (c,d)0.040.070.10 p ψ ( p ) FIG. 9: Linear dependencies of the erraticity exponents ψ ( p )on p for the (a,b) cases in open symbols and for the (c,d) casesin the solid symbols. Their slopes give the values of the indices µ . The units of ∆ p T are in GeV/c. the slopes µ , as defined in Eq. (19). It is clear that the(c,d) group has essentially no dependence on ∆ p T . Evenin the (a,b) group the spread due to different ∆ p T cuts isnot severe enough to make it unreasonable for us to givethe three lines in each group an average slope as follows: µ a,b = 1 . ± . , µ c,d = 4 . ± . . (20)It is remarkable that we can obtain numerical summaryof the different cases independent of p and M and onlymildly dependent on ∆ p T . The values of these indices mean that for the models (a,b) that have contraction dueto confinement there are spikes in bin multiplicity locally,making them easier to have bins with n ≥ µ or less erratic) than for the models (c,d) thathave no contraction. In the latter cases there are notenough multiplicity fluctuations to have n ≥ F at high M is extremely low, but in those rareevents the value of F is so high (thus more erratic) thatthe p th moment with p ≥ .
25 give them such a highweight as to raise ψ ( p ) and µ significantly above thosefor (a,b) cases.Turning now to the complication at the point p = 1,we note that if the straight lines in Fig. 9 go through theorigin (1 ,
0) exactly, then the slopes are constant through-out 1 ≤ p ≤ p = 1 exclusively. In that case we could returnto Eqs. (16) and (18) and determine µ (1)4 directly fromΣ ( M ). Unfortunately, we have found that Σ ( M ) doesnot depend on ℓnM linearly, so we are unable to makeuse of the Eq. (18) to calculate the slope at p = 1. Thisis consistent with the fact that the straight lines in Fig.9 do not cross the origin precisely, and that some bend-ing of those lines near the point (1 ,
0) reveals the lack ofuniversality of the slopes for all p . Stated differently, thepower-law behavior of Eq. (17) is not true for all p ; thelocal multiplicity fluctuations can be so severe and areso different among the four cases that the indices µ (1) q at p = 1 cannot summarize their differences.Returning to the results expressed in Eq. (20), we canconclude that what separates (a,b) from (c,d) is the dom-inating effect of contraction over clustering. Recall from(11) that the two columns are for cases where initial andreseeded configurations between time steps are with clus-tering (a,c), and without clustering (b,d), while the tworows are for cases where the configurations between timesteps undergo contraction (a,b), and no contraction (c,d).Since there are 10 substeps within each time step, con-tractions rearrange the configurations sufficiently so thatthe erraticity moments C p,q retain essentially no mem-ory of the clustering effects in the input. Physically, itmeans that as the quark-gluon plasma approaches lowdensity near the end of its expansion, whether or notthere is critical clustering, the confinement forces thatact on the quarks near the surface exert the dominanteffect on drawing the q ¯ q pairs together in order to pio-nize, despite the opposing tendency to randomize due tothermal activities that persist. The same process is re-peated each time a new layer moves to the surface. Thetension between confinement and deconfinement is whatleads to large local fluctuations evidenced by the low val-ues of the erraticity indices µ a , b4 . Without that tensionthe bin-multiplicity fluctuations have no dynamical pushbeyond randomness, so it is highly erratic to have somerare events to contribute to non-trivial C p, , hence highervalues of µ c , d4 .1 VI. CONCLUSION
The purpose of this work is mainly to describe an un-conventional method to analyze the LHC data in thehope that some experimentalists may find it adventure-some. The new frontier opened up by the high multiplic-ity events provides a fertile ground for exploration that isnot feasible at lower energies. That is why the subject isnot among those predictions that could be extrapolatedfrom RHIC. If there is any hint of critical behavior in thequark-hadron transition, that would be a new discoveryat LHC. Even if nothing critical is found, analysis alongthe line suggested here should lead to deeper understand-ing of the hadronization process.Our models of the four classes of criticality may notturn out to be realistic, but they have been useful intesting the effectiveness of the erraticity moments andindices. The basic issue is, of course, whether the pro-posed measures can be applied to the real data to uncoverinteresting physics. There exist other physical processesthat are totally ignored in this study. Chief among themis minijet production, which is understood to be copiousin Pb-Pb collisions at 2.76 TeV [29]. Usual jet study at LHC is for p T very large, e.g. >
50 GeV/c. Our analysishere is for p T ≈ p T enhancement by minijets can lead to multiplic-ities greater than 4 in very small bins, but rare eventswith large fluctuations are what the erraticity moments C p,q ( M ) are sensitive to. Thus at this point we cannotrule out contamination of the phase transition effects byminijets. Such possibilities perhaps would encourage theexperimentalists to investigate the subject, either to findresolution of ambiguities or to gain new perspective onan aspect of physics that is not well understood. Acknowledgment
This work was supported in part, by the U. S. Depart-ment of Energy under Grant No. DE-FG02-96ER40972and by the National Natural Science Foundation of Chinaunder Grant No. 11075061, and the Program of Introduc-ing Talents of Discipline to Universities under Grant No.B08033. [1] M. Bleicher, J. Phys. G: Nucl. Part. Phys. , 124035(2011).[2] B. Mohanty, Nucl. Phys. A , 899c (009); J. Phys. G:Nucl. Part. Phys. , 124023 (2011).[3] M. Stephanov, Prog. Theor. Phys. Suppl. , 139(2004); J. Phys. G: Nucl. Part. Phys. , 124147 (2011).[4] For reviews, see F. Antinori, J. Phys. G: Nucl. Part. Phys. , 124038 (2011); J. Schukraft (for ALICE Collabora-tion), ibid, 124003 (2011); P. Steinberg (for ATLAS Col-laboration), ibid, 124004 (2011).[5] N. Armesto, in Quark-Gluon Plasma 4 , edited by R. C.Hwa and X. N. Wang (World Scientific, Singapore, 2010),p. 375.[6] K. Aamodt et al. (ALICE Collaboration), Phys. Rev.Lett. , 252301 (2010).[7] F. Cooper and G. Frye, Phys. Rev. D , 186 (1974).[8] S. Heckel (for ALICE Collaboration), J. Phys. G: Nucl.Part. Phys. , 124095 (2011).[9] P. Christakoglou (for ALICE Collaboration), arXiv:1111.4506.[10] J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E.J. Newman, The Theory of Critical Phenomena (Claren-don, Oxford, 1992).[11] A. Bialas and R. Peschanski, Nucl. Phys.
B 273 ,703(1986);
B 308 , 857 (1988).[12] W. Kittel and E. A. De Wolf,
Soft Multihadron Dynam-ics , (World Scientific, Singapore, 2005), p. 429.[13] A. Bialas and R. C. Hwa, Phys. Lett. B , 436 (1991).[14] R. C. Hwa and M. T. Nazinov, Phys. Rev. Lett. , 741 (1992); R. C. Hwa, Phys. Rev. D , 2773 (1993).[15] M. R. Young, Y. Qu, S. Singh and R. C. Hwa, OpticsComm. , 325 (1994).[16] H. Haken, Rev. Mod. Phys. , 67 (1975).[17] Z. Cao, Y. Gao and R. C. Hwa, Z. Phys. C , 661 (1996).[18] R. C. Hwa, C. S. Lam and J. Pan, Phys. Rev. Lett. ,820 (1994); R. C. Hwa and J. Pan, Phys. Rev. C , 2516(1994).[19] R. C. Hwa and C. B. Yang, Phys. Rev. C , 034902(2003); , 024905 (2004).[20] V. Greco, C. M. Ko, and P. L´evai, Phys. Rev. Lett. ,202302 (2003); Phys. Rev. C , 034904 (2003).[21] R. J. Fries, B. M¨uller, C. Nonaka, and S. A. Bass, Phys.Rev. Lett. , 202303 (2003); Phys. Rev. C , 044902(2003).[22] Z. Cao and R. C. Hwa, Phys. Rev. Lett. , 1268 (1995);Phys. Rev. D , 6608 (1996).[23] Z. Cao and R. C. Hwa, Phys. Rev. E , 326 (1997).[24] R. C. Hwa, Acta Phys. Polon. B , 1789 (1996).[25] Z. Cao and R. C. Hwa, Phys. Rev. D , 074011 (2000).[26] M. R. Atayan et al. (EHS/NA22 Collaboration) Phys.Lett. B , 22 (2003).[27] F. M. Liu, H. Liao, M. Liu, F. Liu, and L. S. Liu, Phys.Lett. B , 293 (2001).[28] D. Ghosh, A. Deb, M. Mondal and J. Ghosh, Phys. Lett.B540