Finite-size effects and the search for the critical endpoint of QCD
Eduardo S. Fraga, Takeshi Kodama, Letícia F. Palhares, Paul Sorensen
aa r X i v : . [ h e p - ph ] J un Finite-size effects and the search for the criticalendpoint of QCD
Eduardo Souza Fraga ∗ Instituto de Física, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, BrazilE-mail: [email protected]
Takeshi Kodama
Instituto de Física, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, BrazilE-mail: [email protected]
Letícia F. Palhares
Instituto de Física, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, BrazilInstitut de Physique Théorique, CEA/DSM/Saclay,Orme des Merisiers, 91191 Gif-sur-Yvette cedex, FranceE-mail: [email protected]
Paul Sorensen
Physics Department, Brookhaven National Laboratory,Upton, NY 11973-5000, USAE-mail: [email protected]
Taking into account the finiteness of the system created in heavy ion collisions, we show sizableresults for the modifications of the chiral phase diagram at volume scales typically encounteredin current experiments and demonstrate the applicability of finite-size scaling as a tool in theexperimental search for the critical endpoint. Using data from RHIC and SPS and assumingfinite-size scaling, we find that RHIC data from 200 GeV down to 19 . m &
510 MeV. We also present predictions for the fluctuations at lowerenergies currently being investigated in the Beam Energy Scan program.
The many faces of QCDNovember 1-5, 2010Gent, Belgium ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ inite-size effects and the search for the critical endpoint of QCD
Eduardo Souza Fraga
Relativistic heavy ion collision experiments have entered a new era of investigations of thephase diagram of strong interactions. The Beam Energy Scan program at RHIC [1] is probing arelatively wide window in center-of-mass energy, which translates into higher values of the bary-onic chemical potentials and lower values of the temperature associated with the plasma created inthe collision process. The hope is that the scan will able to reach the region where a second-ordercritical point is expected to exist [2]. Therefore, pragmatic theoretical tools that can be phenomeno-logically applied to data analysis are demanded.Identifying the presence of a critical point, and even of a first-order transition coexistence line,is not a simple task in heavy ion experiments. One has to uncover, from different sorts of complexbackgrounds, signatures related to large fluctuations that would stem from the critical behavior ofthe order parameter of the chiral transition, the chiral condensate. In a system in equilibrium and inthe thermodynamic limit, those are a consequence of the unlimited growth of the correlation length[3], and in the case of heavy ions would lead to non-monotonic behavior [4, 5] or sign modifications[6] of particle correlation fluctuations.Although often compared to the case of the early universe at the time of the primordial quark-hadron transition, the space-time scales present in the dynamics of the quark-gluon plasma formedin heavy ion collisions differ from the cosmological ones by almost twenty orders of magnitude.So, realistically, this system is usually small, short-lived, and part of the time out of equilibrium.The finite (short) lifetime of the plasma state and critical slowing down could severely constrain thegrowth of the correlation length, as shown by estimates in Refs. [7, 8]. The finite size of the systemcould dramatically modify the phase structure of strong interaction, as shown using lattice simula-tions [9, 10] and different effective model approaches [11, 12, 13, 14], and also affect significantlythe dynamics of phase conversion [15, 16]. As a consequence, all signatures of criticality based onnon-monotonic behavior of particle correlation fluctuations will probe a pseudocritical point thatcan be significantly shifted from the genuine (unique) critical endpoint by finite-size correctionsand will be sensitive to boundary effects. m ( MeV) T ( M e V ) L = infinityL = 5 fmL = 3 fmL = 2 fmAPCPBCL = infinity
Figure 1:
Displacement of the pseudocritical end-point in the T − m plane as the system size is de-creased for different boundary conditions. -1 ) T c ( m = ) ( L ) / T c ( m = ) ( i n f i n it y ) PBCAPC
Figure 2:
Normalized crossover temperature at m = / L for the cases withPBC and APC. The latter effect was demonstrated using the linear sigma model coupled to quarks with twoflavors [17] as an effective theory for the chiral transition in Ref. [14]. Here we illustrate the rele-2 inite-size effects and the search for the critical endpoint of QCD
Eduardo Souza Fraga vance the effects coming from the finite size of the system, of typical linear size L , and the natureof the boundary might have in the investigation of the phase diagram of strong interactions usingheavy ion collisions in Figs. 1 and 2. Fig. 1 shows the displacement of the pseudocritical point,comparing periodic boundary conditions (PBC) and anti-periodic boundary conditions (APC): bothcoordinates of the critical point are significantly modified, and m CEP is about 30% larger for PBC.For m =
0, the crossover transition is also affected by finite-size corrections, increasing as the sys-tem decreases, as shown in Fig. 2. Again, PBC generate larger effects: up to ∼
80% increase in thecrossover transition temperature at m = L = L is motivated by the estimated plasma size presumably formed in high-energy heavy ion colli-sions at RHIC [18]. The upper limit is essentially geometrical, provided by the radius of the nucleiinvolved, whereas the lower limit is an estimate for the smallest plasma observed.Nevertheless, the finiteness of the system in heavy ion collisions also brings a bright side: thepossibility of using finite-size scaling (FSS) analysis [19, 20, 21]. FSS is a powerful statisticalmechanics technique that prescinds from the knowledge of the details of a given system; instead,it provides information about its criticality based solely on very general characteristics. And sincethe thermal environment corresponding to the region of quark-gluon plasma formed in heavy ioncollisions can be classified according to the centrality of the collision, events can be separatedaccording to the size of the plasma that is created. So, heavy ion collisions indeed provide anensemble of differently-sized systems.Although it is clearly not simple to define an appropriate scaling variable in the case of heavyion collisions, the flexibility of the FSS method allows for a pragmatic approach for the use ofscaling plots in the search for the critical endpoint as was delineated in Ref. [14] and performedin Ref. [22]. The essential point is that although the reduced volume of the plasma formed inhigh-energy heavy ion collisions will dissolve a possible critical point into a region and make theeffects from criticality severely smoothened, as discussed above, the non-monotonic behavior ofcorrelation functions for systems of different sizes, given by different centralities, must obey FSSnear criticality [3, 23]. [MeV] m n / x g - L æ T p Æ T p s = 3374 +/- 636 MeV crit m =0.67 n =1.00 x g =19.6 GeV nn s =62.4 GeV nn s =130 GeV nn s =200 GeV nn s Figure 3:
Scaled s p T / h p T i vs m for different sys-tem sizes, and with n = / g x =
1. Data ex-tracted from RHIC collisions at energies √ s NN = . , . , [MeV] m n / x g - L æ T p Æ T p s = 509 +/- 55 MeV crit m =0.67 n =1.00 x g =19.6 GeV nn s =62.4 GeV nn s =130 GeV nn s =200 GeV nn s Figure 4:
Scaled s p T / h p T i vs m for different systemsizes. Again, n = / g x =
1. Data extracted fromRHIC collisions at energies √ s NN = . , . , inite-size effects and the search for the critical endpoint of QCD Eduardo Souza Fraga
Using data from RHIC and SPS and defining appropriate scaling variables, we can generatescaling plots for data sets with √ s NN = . , . , . , ,
200 GeV. For finite L , crossover effectsbecome important. If the correlation length diverges as x ¥ ∼ t − n at criticality, where n is thecorresponding critical exponent, in the case of L − t − n ≫ L limits the growth of the correlation length, rounding all singularities[3, 23]. If L is finite, x is analytic in the limit t →
0, and one can draw scaling plots of L / x vs. somecoupling for different values of L to find that all curves cross at a given value in this limit, which isa way to determine its critical value. The critical temperature and so on can also be determined inthis fashion, since the curves will also cross at t = t c . This scaling plot technique can be extended,taken to its full power for other quantities, such as correlation functions. An observable X in a finitethermal system can be written, in the neighborhood of criticality, in the following form [21]: X ( t , L ) = L g x / n f ( tL / n ) , (1)where g x is the bulk (dimension) exponent of X and { g } dimensionless coupling constants. Thefunction f ( y ) is universal up to scale fixing, and the critical exponents are sensitive essentially todimensionality and internal symmetry, which will give rise to the different universality classes [23].Using the appropriate scaling variable, all curves should collapse into one single curve if one is notfar from the critical point. So, this technique can be applied to the analysis of observables that aredirectly related to the correlation function of the order parameter of the transition, such as fluctua-tions of the multiplicity of soft pions [4]. The correct scaling variable should measure the distancefrom the critical point, thereby involving both temperature and chemical potential in the case of theQCD phase diagram. This would produce a two-dimensional scaling function and make the anal-ysis of heavy ion data highly nontrivial. Phenomenologically, we adopt a simplification motivatedby results from thermal models for the freeze-out region, connecting temperature and chemical po-tential. We can parametrize the freeze-out curve by √ s NN , and build our one-dimensional scalingvariable from this quantity and the size of the system. For details, we refer the reader to Ref. [22].To search for scaling, we consider the correlation measure s p T / h p T i [24] scaled by L − g x / n ,according to Eq. 1. We consider the p T fluctuations s p T scaled by h p T i to obtain a dimensionlessvariable. We use the correlation data measured in bins corresponding to the 0 − − − − − − − −
70% most central collisions. Weestimate the corresponding lengths L to be 12 .
4, 11 .
1, 9 .
6, 8 .
0, 6 .
8, 5 .
6, 4 .
5, and 3 . n = / g x around 1 (ignoring small anomalous dimension corrections). We also varied the value of g x from0 . . g x within this range does not improve the scaling behavior.In Figs. 3 and 4 we plot s p T / h p T i scaled by L − g x / n vs m for different system sizes, using dataextracted from collisions at √ s NN = . , . , ,
200 GeV. If there is a critical point at m = m crit ,the curves for different sizes of the system should cross at this value of m . However, since thecurrently available data is restricted to not so large values of the chemical potential, one has toperform extrapolations using fits. The scaling function f in Eq. (1) is expected to be smoothlyvarying around the critical point, so we fit the data corresponding to a given linear size L to apolynomial, but constraining the polynomials to enforce the condition that all the curves cross atsome m = m crit , where m crit is an adjustable parameter in the fit. This clearly assumes the existenceof a critical point. In Fig. 3 we use a linear fit. The approximate energy independence of s p T / h p T i inite-size effects and the search for the critical endpoint of QCD Eduardo Souza Fraga along with the linear fit, leads to a very large m value where the curves can cross ( m ∼ f allows the curves fromdifferent system sizes to cross at a much smaller value of m . Based on this fit, we find that the datais consistent with a critical point at m ∼
510 MeV corresponding to a √ s NN of 5.75 GeV. part N æ T p Æ T p s = . G e V NN s = . G e V NN s = . G e V NN s = 509 MeV crit m =0.67 n =1.00 x g Figure 5:
The expected measurement of s p T / h p T i as a function of the number of participants at lowerenergies assuming the critical point is at 509 MeV as extracted from the quadratic polynomial fit of STARdata. RHIC has also run at lower energies in order to search for a critical point in the Beam EnergyScan program. Using the quadratic polynomial fit of STAR data (Fig. 4) and assuming the criticalpoint is at 509 MeV we can make predictions for s p T / h p T i at lower energies. We show thisexpectation as a function of the number of participants, N part , for three proposed beam energies:11 . , . inite-size effects and the search for the critical endpoint of QCD Eduardo Souza Fraga
We are grateful to M. Chernodub, S.L.A. de Queiroz and Á. Mócsy for fruitful discussions.This work was partially supported by CAPES-COFECUB (project 663/10), CNPq, FAPERJ andFUJB/UFRJ.
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