Beam Energy Scan at RHIC and z-Scaling

Beam Energy Scan at RHIC and z-Scaling

Available online at www.sciencedirect.com Nuclear Physics B (Proc. Suppl.) 245 (2013) 231–238 www.elsevier.com/locate/npbps Beam Energy Scan at RHIC...

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Available online at www.sciencedirect.com

Nuclear Physics B (Proc. Suppl.) 245 (2013) 231–238 www.elsevier.com/locate/npbps

Beam Energy Scan at RHIC and z-Scaling Mikhail Tokareva , Imrich Zborovsk´yb b

a Joint Instiutte for Nuclear Research, Joliot Curie 6, 141980 Dubna, Russia ˇ z, Czech Republic Nuclear Physics Institute,Academy of Sciences of the Czech Republic, 250 68 Reˇ

Abstract Beam Energy Scan (BES) data obtained at RHIC are briefly reviewed. Method of data analysis (z-scaling approach) based on self-similarity and locality of constituent interactions in hadron and nucleus collisions at high energy is described. The method is applied for analysis of BES data to search for signatures of phase transition and Critical Point (CP). Some results of analysis of hadron spectra measured in heavy ion collisions (HIC) at RHIC over a wide √ range of the energy sNN = 7.7 − 200 GeV are presented. Microscopic scenario of constituent interactions in the framework of this approach is discussed. Dependence of the energy loss on the momentum of the produced hadron, energy and centrality of the collision is studied. Self-similarity of the constituent interactions in terms of momentum fractions is used to characterize the nuclear medium by a ”specific heat” and the colliding nuclei by fractal dimensions. Kinematic regions which are assumed to be most preferable for search for signatures of phase transition of nuclear matter produced in HIC in BES are discussed. Keywords: nucleus-nucleus collisions, energy loss, scaling, phase transition

1. Introduction Experiments with nuclei at RHIC have provided evidence that a new state of nuclear matter exists [1, 2, 3, 4, 5]. This new state is characterized by a suppression of particle production at high pT [6], a large amount of elliptic flow (v2 ), constituent quark number scaling of v2 at intermediate pT [7, 8] and enhanced correlated yields at large Δη and Δφ  0 (the ridge effect) [9]. To understand the properties of the system in the framework of the Quantum Chromodynamics (QCD) is one of the main goals of high energy HIC experiments at RHIC and SPS. Calculations in lattice QCD (see [10, 11] and references therein) indicate that the energy density (3 − 5 GeV/fm3 ) and temperature (T  170 MeV) reached in central Au+Au collisions at RHIC are enough to observe signatures (enhancement of multiplicity, transEmail addresses: [email protected] (Mikhail Tokarev ), [email protected] (Imrich Zborovsk´y )

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verse momentum and particle ratios fluctuations, longrange correlations, strange-hadron abundances,...) of a possible phase transition from hadronic to quark and gluon degrees of freedom. Nevertheless, a clear indication of such a transition has yet to be observed. This has been widely discussed in the literature [12, 13, 14, 15]. The principal challenge remains localization of the Critical Point (CP) on the QCD phase diagram (Fig 1). Near the CP, several thermodynamic properties of the system such as the heat capacity, compressibility, correlation length are expected to diverge with a power-law behavior in the variable  = (T − T c )/T c , where T c is the critical temperature. The rate of the divergence can be described by a set of critical exponents. The critical exponents are universal in the sense that they depend only on degrees of freedom in the theory and their symmetry, but not on other details of the interactions. This scaling postulate is the central concept of the theory of critical phenomena [16]. An important step towards understanding the structure of the QCD phase diagram is systematic analysis

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of particle production as a function of collision energy, centrality and collisions species. Assuming the system is thermalized, temperature T and baryon chemical potential μB can be determined. A search for the location of a possible CP on the {T, μB } phase diagram, can be done by varying the beam energy. The proposed BES has been tasked to carry out this search [17].

Figure 2: The scaled elliptic flow versus the scale transverse kinetic √ energy in Au + Au collisions at sNN = 200 GeV [7, 8].

Figure 1: Phase diagram of nuclear matter.

The results from top RHIC energies suggest the existence of the QuarkGluon Plasma (QGP) [1, 2, 3, 4, 5]. The main task at hand now is to study the properties of the QGP and establish the QCD phase diagram. Lattice QCD calculations predict the transition between QGP and the hadronic gas as crossover at μB = 0, while at large μB they predict a first order phase transition. A point where the first order phase transition ends is called the CP. Experimentally, the QCD phase diagram can be studied by colliding heavy ions at varying beam energies that can provide a T − μB region for each energy. Then, one can look at the various signatures for the phase boundary and CP. The QCD phase diagram is the variation of temperature T and baryon chemical potential μB . These quantities can be extracted from the measured hadron yields. The particle ratios are used to obtain the chemical freeze-out a state when the yields of particles get fixed conditions using the statistical thermal model [18]. The two main extracted parameters are chemical freezeout temperature T and μB . 2. Some of BES results Among the evidence that collectively was established in the formation of the strong QGP in high energy HIC experiments at RHIC was the measurement of anisotropy v2 in particle momentum distributions [7, 8] and the nuclear modification factors RAA and RCP [19].

Figure 3: The scaled elliptic flow versus the scale transverse kinetic √ energy in Au + Au collisions at sNN = 7.7 − 62.4 GeV [19].

The dynamical charge correlation and fluctuation measurements have been also studied to search for CP [19, 20]. 2.1. Flow The significant elliptic flow v2 observed at RHIC together with the fact that flow apparently scales with quark number (Fig.2) prompted the conclusion that full energy RHIC collisions produce strongly coupled partonic matter. To determine if a strong QGP is produced √ in RHIC collisions at lower sNN , then a study of the behavior of the anisotrpic flow is important. The particle yields can be written in the following form   ∞  d2 N d3 N 1+2 νn cos([n(φ − Ψr )] ,(1) E 3 = 2πpT d pT dy dp n=1 where νn are Fourier components, pT is the transverse momentum, y is the rapidity, φ is the azimuthal angle and Ψr is the angle of the reaction plane.

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centrality. Besides charged hadrons, different probes (photons, hadrons, lepton pairs) with own specifics are used to investigate the produced matter in nuclear collisions. 2.3. Nuclear modification factor RCP

(a)

The nuclear modification factor RCP is defined as the ratio of yields at central collisions to peripheral ones, scaled by the corresponding number of binary colli√ sions found in the Glauber model. The ratio at sNN = 200 GeV was found to be about 0.2 for pT > 4 GeV/c. A strong enhancement of RCP is observed at lower energies (Fig.5). The ratio can be used to estimate the parton energy losses in the dense medium in a model dependent way.

(b) Figure 4: Spectra of charged hadrons produced in the cental (a) and √ peripheral (b) Au + Au collisions at sNN = 7.7 − 62.4 GeV [21].

Figure 3 shows the scaled strength of v2 over a wide √ range of sNN . The figure confirms that constituent quark number scaling continues to hold at 62.4 and 39 GeV. The flow data from the lower energy data sets support the conclusion that the strong QGP is produced at RHIC at these lower collision energies [19]. The data provides the possibility to gain information about the degree of thermalization of the hot, dense medium. The breaking of v2 number of quark scaling is assumd to indicate a transition from partonic to hadronic degrees of freedom. 2.2. Spectra Momentum spectra of produced particles as a function of the energy and centrality of collision give direct information on constituent interactions over a wide scale range. They have been also used for more sophisticated analysis (fluctuations, correlations, temperature, chemical potential). Figure 4 shows the charged particle spectra in Au+Au √ central (a) and peripheral (b) collisions at sNN = 7.7 − 62.4 GeV. Peripheral spectra show stronger dependence on the beam energy then the central ones. Exponential and power behavior of spectra in the low and high-pT range are observed, respectively. The high-pT component of the spectrum is sensitive to the collision

Figure 5: Nuclear modification factor RCP of charged hadrons pro√ duced in the central Au + Au collisions at sNN = 7.7 − 200 GeV [21].

Figure 6: Nuclear modification factor RCP of identified strange √ hadrons produced in the central Au + Au collisions at sNN = 7.7 − 39 GeV [21].

√ An enhancement trend of RCP ratio with sNN has been observed for strange hadrons (Fig. 6). It is assumed that in the absence of dense medium, there may not be suppression of high-pT particles, which can serve as an indication of ”turn-off” of the QGP signature. As seen from the figure the modification factor depends on the flavor content of detected particles.

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2.4. Phase diagram of nuclear matter The phase diagrams of the nuclear matter depends on chemical and kinetic freeze-out conditions. The particle ratios are used to study the chemical freeze-out conditions (a state when the yields of particles get fixed) in the framework of the statistical thermal model [18]. The two main extracted parameters are chemical freezeout temperature T ch and chemical potential μB . Figure 7 shows the dependence of the parameters on energies and centralities using the Grand-Canonical Ensemble (GCE) approach of THERMUS [18].

Figure 7: Phase diagram: chemical freeze-out conditions in the statistical thermal model [18].

and centralities. At a given collision energy, an anticorrelation between T kin and < β > was observed. For a given centrality, the freeze-out temperature at high energy is lower and the average collective velocity < β > is larger due to system expansion. 3. z-Scaling As it was emphasized in [16] the ”scaling” and ”universality” are concepts developed to understand critical phenomena. Scaling means that systems near CPs exhibiting self-similar properties are invariant under transformation of scale. According to universality, quite different systems behave in a remarkably similar fashion near the respective CPs. Therefore the concepts can be used to study critical phenomena in nuclear matter. We use z-scaling approach [23, 24] for analysis of data on inclusive spectra of hadrons produced in A + A collisions at high energies. A collision of nuclei is considered as an ensemble of individual interactions of their constituents. Structures of the colliding objects are characterized by parameters δ1 and δ2 . The constituents of the incoming objects with masses M1 , M2 and momenta P1 , P2 carry their fractions x1 , x2 . The inclusive particle with mass m1 carries the fraction ya of the momentum p/ya of the produced system fragmentation of which is characterized by a . The fragmentation of the recoil system is described by the parameter b and the momentum fraction yb . Multiple interactions of constituents are considered to be similar. This property reflects a self-similarity of the hadron interactions at a constituent level. The momentum conservation law of the constituent subprocess is subject to the relation (x1 P1 + x2 P2 − p/ya )2 = MX2

Figure 8: Phase diagram: kinetic freeze-out conditions in the Blast Wave model [22].

The kinetic freeze-out conditions are obtained from particle spectra in the framework of the Blast Wave model [22]. The model is used for simultaneous fits to π, K, p spectra with the extraction of the two parameters which are the kinetic freeze-out temperature T kin and the average flow velocity < β >. Figure 8 shows the variation of these parameters with different energies

(2)

where MX = x1 M1 + x2 M2 + m2 /yb is the recoil mass. The associate production of the particle with mass m2 ensures conservation of the additive quantum numbers. Equation (2) gives a kinematic constraint on the momentum fractions x1 , x2 , ya , and yb . The condition is an expression of the locality of the hadron interaction at a constituent level. The parameters characterizing structure (δ1 , δ2 ) of the colliding objects and fragmentation (a , b ) of the systems produced in the binary collisions are connected with the corresponding momentum fractions by the function Ω = (1 − x1 )δ1 (1 − x2 )δ2 (1 − ya )a (1 − yb )b .

(3)

The function Ω is a relative number of the configurations at the constituent level which include the binary

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subprocesses determined by the momentum fractions x1 , x2 , ya , yb . For p+ p interactions we have δ1 = δ2 ≡ δ. For A + A collisions we set δ1 = A1 δ and δ2 = A2 δ (A1 , A2 are atomic numbers). A fragmentation of the objects produced in constituent collisions in the scattered and recoil directions is described by the same parameter a = b ≡ F which depends on the type (F) of the inclusive particle. The parameters δ and F were found to have constant values in p + p collisions at high energies. They are interpreted as fractal dimensions in the corresponding space of the momentum fractions. The scaling variable z is given in terms of a single constituent subprocess determined by the momentum fractions x1 , x2 , ya , and yb . The fractions are obtained in a way to maximize the function Ω(x1 , x2 , ya , yb ) under condition (2). Having thus fixed the kinematics of the subprocess, we define z = z0 Ω−1



(4)

as a ratio of z0 = s⊥ /[(dNch /dη|0 )c m] and the maximal value of Ω. The variable z is proportional to the trans√ verse kinetic energy s⊥ of the binary subprocess consumed on the production of the inclusive particle with mass m1 and its counterpart with mass m2 . The quantity dNch /dη|0 is the corresponding multiplicity density of charged particles in the central interaction region at the pseudorapidity η = 0. The medium produced in the central region is characterized by the parameter c interpreted as a ”specific heat” of the matter created in the interaction [23, 24, 25]. The variable z has property of a fractal measure. It depends on the resolution Ω−1 at which the corresponding subprocess can be singled out of the inclusive reaction. The fractal variable z increases in the power like manner with the increasing resolution and diverges as Ω−1 → ∞. The parameters δ1 and δ2 are the corresponding fractal dimensions of the colliding objects. The parameter F stands for the fractal dimension of the fragmentation process. The scaling function ψ(z) is written as follows d3 σ πs J −1 E 3 . (5) ψ(z) = − (dN/dη) σin dp Here σin is the total inelastic cross section, dN/dη is the multiplicity density of inclusive particles and J is the Jacobian for the transformation from {p2T , y} to {z, η}. The normalization  ∞ ψ(z)dz = 1 (6) 0

allows to interpret the function ψ(z) as a probability density of production of the inclusive particle with the corresponding value of the variable z.

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√ 3.1. p+p collisions at s = 19 − 200 GeV In p + p collisions, the parameters δ, F , and c were found to be independent √ of kinematic variables over the wide energy range s = 19 − 200 GeV [23, 24]. The determination of the fractions x1 , x2 , ya , yb corresponding to the selected binary subprocesses allowed us to develop a microscopic scenario of the interaction at a constituent level. The dependencies of the momentum fractions and the recoil mass MX on the transverse momentum pT of the inclusive particle, as well as on the energy and centrality of the collisions represent important features of the constituent interactions. The universal shape of the scaling function ψ(z) and regular behavior of the characteristics of the subprocesses confirmed self-similarity of hadron production at a constituent level over a wide kinematical range. Figure 9 shows the scaling function of√identified hadron production in p + p collisions at s = 19 − 200 GeV. The symmetry transformation z → αF · z, ψ → α−1 F · ψ preserving the shape of the scaling function and the normalization (2) was used to compare ψ(z) for different hadrons. Flavor independence of the z-presentation of hadron spectra was found as illustrated in Fig. 9.

Figure 9: Spectra of hadrons produced in p + p collisions in zpresentation [23].

√ 3.2. Au + Au collisions at sNN = 62.4, 130, 200 GeV The phase transition into the QGP state produced in the central collisions of heavy nuclei is among most dramatic many body effects. This phenomenon is expected to be manifested as a change in the properties of the system with modification of the constituent interactions and subsequent fragmentation processes. At a microscopic level it is related with dissipative parton dynamics by formation of the observed hadrons. The energy dissipation significantly changes in ultradense nuclear environment characterized by high multiplicity of the secondary particles. It depends on the

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Figure 10: Scaling function in Au+Au collisions at



sNN = 62.4 GeV

Figure 12: Scaling function in Au + Au collisions at



sNN = 200 GeV

dent on the collision energy, one can obtain the identical ψ(z) for p+ p and Au+ Au interactions for all centralities in the considered energy range.

(7)

√ 3.3. Au+ Au collisions at sNN = 9.2 GeV and Pb+ Pb √ collisions at sNN = 17.3 GeV An analysis of the spectra in nuclear collisions in the framework of z-scaling showed indications that parameters corresponding to specific heat and nuclear fractal √ dimension can be dependent on energy at lower sNN [31]. A correlation between the parameters c and δ for √ Au + Au collisions at sNN = 9.2 GeV [32] and Pb + Pb √ collisions at sNN = 17.3 GeV [33] (h± = π± +K ± + p± ) was found. The restoration of the universal shape of ψ(z) at these energies can be reached in two scenarios: I - cAuAu = 0.23, δ = 0.5; cPbPb = 0.16, δ = 0.5; and II cAuAu = 0.11, δ = 0.15; cPbPb = 0.11, δ = 0.25. The first scenario (Fig.13(a)) corresponds to the large and the second one (Fig.13(b)) to the small (energy independent) values of the ”specific heat” c. In the I. case, the increase of the parameter c was obtained for the same value of δ = 0.5 as found from z-scaling at higher energies. In the II. case, the constancy of c requires diminishing of δ which can be indicative for the √ smeared nuclear fractal structure at small sNN . The self-similarity of particle production is reflected in both cases by the observation that z-presentation of the spec√ tra at sNN = 9.2 and 17.3 GeV are described by the same curve as for higher energies. The measured pT range of the spectra and the large errors at high pT do not allow us to discriminate between the both scenarios. Measurement of the hadron yields versus centrality for pT > 4 GeV/c is desirable to resolve the problem and to study the dependence of the energy loss in this region.

The value of pp = 0.2 has been obtained from z-scaling analysis of the charged hadrons produced in p + p collisions [23, 24]. Allowing the coefficient 0 to be depen-

3.4. Energy loss in particle production The increase of the fragmentation dimension AA with the multiplicity density (7) (collision centrality) is

Figure 11: Scaling function in Au + Au collisions at



sNN = 130 GeV

collision energy and can be sensitive to the type of the phase transition or location of CP on the phase diagram. It is expected that, in A + A collisions, the parameters δ, , and c should be changed due to modification of elementary processes in the produced medium [26, 27]. Figures 10,11 and 12 show the z-presentation of the spectra of charged hadrons [28, 29, 30] produced in √ Au + Au interactions at sNN = 62.4, 130 and 200 GeV, respectively. The collision centrality is characterized by the corresponding multiplicity density dNch /dη|0 of the charged particles produced in the mid-rapidity region. The fractal dimension of the gold nuclei was found to be δAuAu = 197δ where δ = 0.5. √The same value of δ was found in p + p collisions at s = 19 − 200 GeV. The independence of ψ(z) on the collision centrality for √ AuAu collisions at sNN = 62.4, 130 and 200 GeV is consistent with the constant value of the specific heat cAuAu = 0.11 provided the multiplicity dependence of the fragmentation dimension AA in the following form AA /dη) + pp . AA = 0 (dNch

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of small energy loss during formation of the inclusive hadrons with the production originating from the constituent subprocesses. A possible signal of the critical phenomena would be less smeared under such conditions. The preferable kinematic region corresponds to the measurements of hadron spectra at relatively low collision energies with high pT (region between green and blue points in Fig.15).

(a)

Figure 14: Analysis of STAR data: momentum fraction ya versus pT √ and centrality at sNN = 9.2 and 200 GeV [32].

(b) √ Figure 13: Scaling function in Au + Au collisions at sNN = 9.2 GeV √ and in Pb + Pb collisions at sNN = 17.3 GeV for two scenarios (see text): - I (a) and II (b). Data are taken from [32, 33].

connected with a decrease of the momentum fraction ya . This corresponds to larger energy loss ΔEa /Ea ∼ 1 − ya by the formation of the inclusive hadron. On the other hand, the energy loss depends on the traversed medium which converts it into the multiplicity of the associated particles. The larger AA the more energy loss consumed on production of secondary particles. In a such way the produced medium is via the multiplicity density characterized by the amount of the energy loss. As seen from Fig. 14, the energy loss decreases with the increasing momentum pT and increases with the collision energy. More energy is dissipated in the central collisions (full symbols) in comparison with the √ peripheral ones (empty symbols). At low sNN , the smaller energy loss is within I. scenario in comparison with the energy loss in II. scenario. The results of a MC data analysis of fractions ya in Au + Au collisions √ at the energy sNN = 7.7 GeV versus pT are shown in Fig. 15. As seen from the figure, the energy loss ( 1 − ya ) at low energies becomes smaller and less depends on centrality. We consider that a preferable region for search and study of a phase transition and CP in nuclear medium in A + A collisions is the region

Figure 15: MC data analysis: momentum fraction ya versus pT and √ centrality at sNN = 7.7 GeV.

4. Conclusion Search for clear signatures of the phase transtion of the nuclear matter in collisions of hadrons and nuclei is the main goal of the heavy ion experimental programms at the RHIC at BNL. At highest RHIC energies clear features of hadron production in Au + Au collisions have been found. Among them are suppression of hadron yields at high pT , constituent quark number scaling of v2 at intermediate pT , ridge effect and others. More detailed investigations could provide future analysis of

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the spectra obtained by the BES program at RHIC [17]. √ The BES over a range sNN = 7.7, 11.5, 19.6, 27 and 39 GeV is aimed to study the QCD phase diagram of nuclear matter over a wide T − μB region, to search for the possible QCD phase boundary and location of CP. The hypothesis of self-similarity of the hadron production is an important concept in searching for new physics. It is developed in the framework of z-scaling approach for p + p and A + A collisions at high collision energy. The STAR data on the inclusive spectra of the charged hadrons produced in central Au + Au collisions √ at sNN = 9.2, 62.4, 130, 200 GeV indicate similarity in z-presentation as a characteristic feature of mechanism of the hadron production. This property includes structure of the colliding objects, interactions of their constituents, and character of the fragmentation process. It was found that the universality of the shape of ψ(z) of the hadron production in Au + Au collisions √ at sNN = 62.4, 130, 200 GeV can be preserved for the constant value of the ”specific heat” cAuAu = 0.11 and δ = 0.5 for all centralities, provided the fragmentation dimension AuAu increases with multiplicity. At √ lower sNN , certain c − δ correlation was found. Two scenarios (with the large and small ”specific heat”) of the hadron production were discussed. We assume that discontinuity of the specific heat and fractal dimension connected with enhancement of c − δ correlation should be revealed at high pT . Location of a CP should be manifested more clearly in this region. Therefore, ex√ pected hadron spectra of the BES Program at sNN = 7.7, 11.5, 19.6, 27, 39 GeV for different centralities and high pT are of great interest. Acknowledgments The investigations have been partially supported by RVO61389005 and by the Ministry of Education, Youth and Sports of the Czech Republic grant 031303. References [1] I. Arsene et al. [BRAHMS Collaboration], Nucl. Phys. A 757 (2005) 1. [2] B. B. Back et al. [PHOBOS Collaboration], Nucl. Phys. A 757 (2005) 28. [3] J. Adams et al. [STAR Collaboration], Nucl. Phys. A 757 (2005) 102. [4] K. Adcox et al. [PHENIX Collaboration], Nucl. Phys. A 757 (2005) 184. [5] ”STAR Collaboration Decadal Plan”, Brookhaven National Laboratory, Relativistic Heavy Ion Collider, December, 2010. ”The PHENIX Experiment at RHIC, Decadal Plan 2011-2020”, Brookhaven National Laboratory, Relativistic Heavy Ion Collider, October, 2010.

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