Correlation femtoscopy study at NICA and STAR energies within a viscous hydrodynamic plus cascade model
P. Batyuk, Iu. Karpenko, R.Lednicky, L.Malinina, K.Mikhaylov, O. Rogachevsky, D.Wielanek
aa r X i v : . [ nu c l - t h ] A ug Correlation femtoscopy study at energies available at the JINR Nuclotron-based IonCollider fAcility and the BNL Relativistic Heavy Ion Collider within a viscoushydrodynamic plus cascade model
P. Batyuk, ∗ Iu. Karpenko,
2, 3
R.Lednicky, L.Malinina,
1, 4, 5
K.Mikhaylov,
1, 6
O. Rogachevsky, and D.Wielanek Veksler and Baldin Laboratory of High Energy Physics, JINR Dubna, 141980 Dubna, Russia Bogolyubov Institute for Theoretical Physics, 03680 Kiev, Ukraine INFN - Sezione di Firenze, I-50019 Sesto Fiorentino (Firenze), Italy M. V. Lomonosov Moscow State University, Moscow, Russia D. V. Skobeltsyn Institute of Nuclear Physics, Moscow, Russia Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia Warsaw University of Technology, Faculty of Physics, Warsaw 00662, Poland
Correlation femtoscopy allows one to measure the space-time characteristics of particle produc-tion in relativistic heavy-ion collisions due to the effects of quantum statistics (QS) and final stateinteractions (FSI). The main features of the femtoscopy measurements at top RHIC and LHC en-ergies are considered as a manifestation of strong collective flow and are well interpreted withinhydrodynamic models employing equation of state (EoS) with a crossover type transition betweenQuark-Gluon Plasma (QGP) and hadron gas phases. The femtoscopy at lower energies was in-tensively studied at AGS and SPS accelerators and is being studied now in the Beam Energy Scanprogram (BES) at the BNL Relativistic Heavy Ion Collider in the context of exploration of the QCDphase diagram. In this article we present femtoscopic observables calculated for Au-Au collisions at √ s NN = 7 . − . vHLLE+UrQMD and their dependence onthe EoS of thermalized matter. PACS numbers: 25.75.-q, 25.75.GzKeywords: relativistic heavy-ion collisions, hydrodynamics, collective phenomena, Monte Carlo simulations,vHLLE, UrQMD
I. INTRODUCTION
One of the main motivations for heavy-ion collisionprograms is to study a new state of matter, the Quark-Gluon Plasma (QGP) which is defined as a deconfinedstate of quarks and gluons [1, 2]. The systematics oftransverse momentum spectra, elliptic and higher orderflow coefficients measured in heavy-ion collisions at BNLRelativistic Heavy Ion Collider (RHIC) and CERN LargeHadron Collider (LHC) energies confirmed the presenceof strong collective motion and hydrodynamic behaviorof the system [3–10]. While the hydrodynamic approachwas successful in reproduction of elliptic flow measuredat the top RHIC energies from the very beginning, it wasunable to reproduce the femtoscopic correlation measure-ments. The problem was solved several years ago alongwith substantial improvements in hydrodynamic model-ing [11–13]. The improvements comprise a presence ofpre-thermal transverse flow, an inclusion of shear vis-cous corrections to hydrodynamic evolution, an equa-tion of state compatible with recent lattice QCD cal-culations, and a consistent treatment of hadronic stage(hadronic cascade phase). As a result, existing state-of-the-art hydrodynamic models can reproduce, besides thetransverse momentum distributions and elliptic flow coef-ficients, also the pion femtoscopic measurements [14–16]. ∗ e-mail: [email protected] Recent lattice QCD calculations show that the tran-sition from QGP to a hadron gas at high temperatureand small µ B is a crossover [17–20], and there are nosigns of critical behavior in the region of µ B /T < √ s NN = 200 GeV) and LHC ( √ s NN = 2 .
76 TeV)data in viscous hydrodynamic + cascade model, wherean elaborate model-to-data comparison using Bayesianframework suggests that the speed of sound at all temper-atures cannot fall below the hadron resonance gas valueof ∼ .
15, and that the resulting posterior distributionover possible equations of state of matter is compatiblewith the lattice QCD results [22]. At the same time, thereare predictions inspired by the lattice QCD calculationson a possible change of existing regime to a first-orderphase transition occurring at lower energies and higherchemical potentials [23–29] and thus implying the exis-tence of a critical point on the QCD phase diagram ata moderate value of chemical potential [30]. These con-siderations motivated the BES program allowing one tostudy different parts of the QCD phase diagram at ex-isting accelerators like SPS and RHIC, and, in future, atNICA and FAIR facilities.It has been shown many years ago [31–33] that thelong duration of particle emission related to a first orderphase transition could reveal itself in the energy regionof the onset of deconfinement as a strong increase of theGaussian femtoscopic radius R out , measured along thepair transverse momentum, compared with the nearlyconstant radius R side , measured along the perpendicu-lar direction in the transverse plane. As a result, onemay expect a strong increase of the ratio R out /R side . Infact, a first order phase transition leads to a stalling ofthe mean expansion speed and a longer emission dura-tion ∆ τ manifested as an increase of the radius measuredalong the beam direction R long and of the ratio of trans-verse femtoscopy radii R out /R side , respectively. A largedata set of correlation functions of identical charged pionshas been recently obtained by the STAR Collaborationwithin the RHIC BES at √ s NN = 7.7, 11.5, 19.6, 27,39, 62.4 GeV. The study of R out /R side and R out − R side behavior as a function of √ s NN indicates a wide maxi-mum near √ s NN ∼
20 GeV, which is reported also bythe PHENIX Collaboration [34]. Could this wide max-imum be related to the expected change of the type ofphase transition?To answer this and other questions, we study thesensitivity of Bose-Einstein correlations of identical pi-ons to the EoS using a hybrid vHLLE+UrQMD model [35].The model combines the UrQMD approach [36, 37] forthe early and late stages of the evolution with numer-ical (3 + 1)-dimensional viscous hydrodynamical solu-tion [38] for the hot and dense expanding matter. Ahydrodynamic approach has an essential advantage forthe present analysis, since it allows one to simulate dif-ferent scenarios of hadron / quark-gluon transition bychanging the EoS and other transport coefficients input.The paper is organized as follows: the details of themodel are discussed in Section II; in Section III the fem-toscopy formalism is described; in Section IV the resultsare presented and discussed; Section V is dedicated toconclusions.
II. VHLLE+URQMD MODEL
The use of multi-component dynamical models for thedescription of dynamics of relativistic heavy-ion colli-sions at RHIC and higher energies is essential becausehydrodynamics alone cannot describe the entire reac-tion. At the early stage of collision, thermalization ofout-of-equilibrium quark-gluon system is assumed to oc-cur, which allows one to describe the subsequent complexmulti-particle dynamics using a relatively simple formal-ism of relativistic hydrodynamics. This requires an ap-proach to calculate initial conditions for the hydrody-namic evolution . As the matter expands, the charac-teristic mean free path of its constituents (quarks andgluons transforming into hadrons) becomes comparableto the system size. The interactions become less frequent,but they do not cease instantaneously, a process whichcan be simulated by switching from the hydrodynamical Note that for lower collision energies there exist one-, two-, andthree-fluid hydrodynamical models (e.g. [39]), which apply hy-drodynamical description already for incoming nuclei. evolution to a hadronic cascade, usually with the help ofCooper-Frye formula [40].For the early stage of collision different approaches (ormodels) have been used in literature, such as (MC-)KLN , IP-Glasma , EPOS , HIJING , Glauber model etc. Thoseapproaches have been developed for full RHIC or LHCenergies, and lose their applicability as the collision en-ergy decreases. As for the Glauber model, it can estimateinitial energy density profiles in transverse direction only.Therefore we choose to use UrQMD to simulate the ini-tial stage of the collision. We enforce a transition to hy-drodynamical description at a hyper-surface of constantlongitudinal proper time τ = √ t − z . The minimalvalue of the starting time τ is taken to be equal to theaverage time for the two colliding nuclei to completelypass through each other: τ = 2 R/ q ( √ s NN / m N ) − , (1)where R is average radius of the nucleus and m N is nu-cleon mass.At τ = τ energy, momentum and baryon/electriccharges of hadrons are distributed to fluid cells ijk around each hadron’s position according to Gaussian pro-files:∆ P αijk = P α · C · exp − ∆ x i + ∆ y j R ⊥ − ∆ η k R η γ η τ ! (2)∆ N ijk = N · C · exp − ∆ x i + ∆ y j R ⊥ − ∆ η k R η γ η τ ! , (3)where P α and N are 4-momentum and charge ofa hadron, { ∆ x i , ∆ y j , ∆ η k } are the distances betweenhadron’s position and center of a hydro cell ijk in each di-rection, γ η = cosh( y p − η ) is the longitudinal Lorentz fac-tor of the hadron as seen in a frame moving with the ra-pidity η , and C is a normalization constant. The normal-ization constant C is calculated so that the discrete sumof energy depositions to the hydrodynamic cells equalsto the energy of the hadron. The width parameters R ⊥ and R η control granularity of the produced initial state.For all collision energies in consideration, the resultinginitial energy density is large enough for the dense partsof the system to reside in the QGP phase.The following 3-dimensional viscous hydrodynamic ex-pansion is simulated with the vHLLE code [38]. Anotherinput to the hydrodynamic part is the EoS, for whichwe use chiral model EoS [41] or bag model EoS [42].Whereas the present version of the chiral model EoS hasa crossover type transition between QGP and hadronicphases for all baryon densities, the bag model EoS hasa first order phase transition between the phases alsofor all baryon densities. Therefore below we dub chiralmodel EoS as “XPT EoS”, and bag model EoS as “1PTEoS”. For both EoS there are publicly available tables, ] [GeV/fm ˛ ] p [ G e V / f m =0 B n FIG. 1. Thermodynamic pressure as a function of energydensity, evaluated at zero baryon density from the equationsof state used in the hydrodynamic stage: chiral model EoSwith crossover transition (XPT) and bag model EoS with firstorder phase transition (1PT). computed in full physically allowed T − µ B region, whichmakes them particularly useful for hydrodynamic com-putations with fluctuating initial conditions. Pressure asa function of energy density from both EoS is demon-strated on Fig. 1.Fluid to particle transition, or particlization, is set tohappen at a hypersurface of constant (hydrodynamic) en-ergy density ǫ sw = 0 . , when the hydrodynamicEoS corresponds to hadronic phase. The particlizationhypersurface is reconstructed with the CORNELIUS sub-routine [43]. At this hypersurface, individual hadrons aresampled using the Cooper-Frye formula including shearviscous corrections to the distribution functions. Thehadronic rescatterings and decays are treated with the UrQMD cascade.The initial state parameters R ⊥ , R η , hydrodynamicstarting time τ and shear viscosity over entropy ratio η/s in fluid phase are tuned for different collision ener-gies in order to approach basic experimental observablesin the RHIC Beam Energy Scan region: (pseudo)rapiditydistributions, transverse momentum spectra and ellipticflow coefficient [35]. The resulting values of the parame-ters are presented in Table I. The tuning has been madewith the XPT (chiral model) EoS, and in present workwe use the same set of parameter values (i.e. do not re-tune them) for the simulations with the 1PT (bag model)EoS. III. FEMTOSCOPY FORMALISM
Since the first demonstration of the sensitivity of theBose-Einstein correlations to the spatial scale of the emit-ting source done almost 60 years ago by G. Goldhaber,S. Goldhaber, W. Lee and A. Pais [44], the momentum √ s NN [GeV] τ [fm/c] R ⊥ [fm] R η [fm] η/s τ , initialstate granularity R ⊥ , R η and shear viscosity over entropyratio η/s adjusted for different collision energies in order toreproduce basic observables in the RHIC BES region. Anasterisk marks the values of τ which are adjusted instead ofbeing set directly from Eq. 1. correlation technique was successfully developed and isknown presently as a “correlation femtoscopy”. It wassuccessfully applied to the measurement of the space-timecharacteristics of particle production processes in highenergy collisions, especially in heavy-ion collisions [45–49]. Femtoscopy correlations are studied by means of atwo-particle correlation function. In a production processof a small enough phase space density, the correlations oftwo particles emitted with a small momentum k ∗ = | k ∗ | in the pair rest frame (PRF) are dominated by the ef-fects of their mutual final state interaction (FSI) andquantum statistics (QS), depending on the PRF temporal( t ∗ = t ∗ − t ∗ ) and spatial ( r ∗ = r ∗ − r ∗ ) separation of par-ticle emission points. Usually, one can neglect the tem-poral separation [50, 51] and in such equal-time approx-imation describe these effects by properly symmetrizedwave functions at a given total pair spin S , h ψ S ,α ′ α − k ∗ ( r ∗ ) i ∗ ,representing solutions of the scattering problem viewedin the opposite time direction. So the complex conju-gate, negative sign of the vector k ∗ = p ∗ = − p ∗ andthe detected channel α being the entrance one. Since theFSI factorization requires the FSI duration much largerthan the particle production time, the relative momen-tum should be small also in the intermediate channels α ′ ,so that one may consider the particles in these channelsbelonging to the same isospin multiplets as those in thedetected channel α . Particularly, for identical particles,the multi-channel problem reduces to the single elastictransition α → α only. Assuming further sufficientlysmooth behavior of single-particle spectra in a narrowcorrelation region (smoothness assumption) [45, 46], onecan neglect the space-time coherence and write the cor-relation function at a given k ∗ and pair three-momentum Calculations made in PRF are denoted by asterisk. P as C ( k ∗ , P ) = Z d r ∗ S α ( r ∗ , P ) (cid:12)(cid:12)(cid:12) ψ S ,α ′ α − k ∗ ( r ∗ ) (cid:12)(cid:12)(cid:12) , (4)where the overline describes the averaging over the totalpair spin S and summing over the intermediate channels α ′ . It is implied that particles are produced in a com-plex process with equilibrated spin and isospin projec-tions and so the separation distribution (source function) S α ( r ∗ , P ) is independent of S and α ′ .Experimentally, a two-particle correlation function isdefined as a ratio C ( q ) = A ( q ) /B ( q ). A ( q ) is a mea-sured distribution of the difference q = p − p , where p and p are three-momenta of two considered particlestaken from the same event, while B ( q ) is a reference dis-tribution of pairs of particles taken from different events.The momentum difference is usually calculated in the lon-gitudinally co-moving system (LCMS), where the longi-tudinal pair momentum vanishes. The vector q is usuallyexpressed in terms of q out , q side , q long , where the “long”axis is directed along the beam, “out” - along the pairtransverse momentum and “side” - perpendicular to thelatter one in the transverse plane.To perform a quantitative analysis of femtoscopic cor-relations, an analytical form of S is often used so thatthe result of the integration procedure in Eq. (4) can becompared with a correlation function C obtained from anexperiment. The source function is usually considered in-dependent of the relative momentum q and its Gaussianshape is assumed: S ( r ) ∼ exp − r ∗ ∗ − r ∗ ∗ − r ∗ ∗ ! . (5)The widths in three directions (out, side and long) arecalled “Gaussian femtoscopy radii”. In LCMS q out = γ t ( q ∗ out + β t ( m − m ) /m ), q side = q ∗ side , q long = q ∗ long ,where γ t and β t are the LCMS Lorentz factor and veloc-ity of the pair. Since the space-time separation in PRFand LCMS is related by the Lorentz boost in the out di-rection: r ∗ out = γ t ( r out − β t t ), r ∗ side = r side , r ∗ long = r long ,the PRF and LCMS Gaussian radii coincide except for R ∗ out = γ t R out . In present paper we consider the correla-tion function of two identical pions neglecting their FSI,so that | ψ − k ∗ ( r ∗ ) | = (cid:12)(cid:12) [exp( − i k ∗ r ∗ ) + exp( i k ∗ r ∗ )] / √ (cid:12)(cid:12) =(6)= 1 + cos(2 k ∗ r ∗ ) , and Eqs. (4, 5, 6) yield the 3-dim Gaussian form of thecorrelation function. This form is usually used to fit theLCMS Gaussian radii according to: C ( q ) = N (cid:0) λ exp( − R q − R q − R q ) (cid:1) . (7)where N is the normalization factor and λ is the corre-lation strength parameter, which can differ from unitydue to the contribution of long-lived emitters and a non-Gaussian shape of the correlation function; R out , R side , R long are the Gaussian femtoscopy radii in the LCMSframe. Eq. (7) assumes azimuthal symmetry of the pro-duction process, which forbids the presence of the cross-terms except for q out q long . We neglect the latter as-suming further the invariance under longitudinal boosts.Generally, in case of a correlation analysis with respect tothe reaction plane, all three cross-terms q i q j contribute.The described fitting procedure allows one to compareextracted femtoscopy radii from the model with existingexperimental data. This can be considered as a standardapproach. A disadvantage of this approach is due to thefact that the Gaussian parametrization can suppress im-portant information that could be derived from the longnon-Gaussian tails of source functions. The PHENIXand STAR collaborations have recently started to ap-ply a new “imaging technique” in order to extract di-rectly the source function [52]. In contrast to the stan-dard approach, the source imaging allows one to extracta real non-Gaussian source function, being in this sensea model-independent one.Here we will use the vHLLE+UrQMD model to study theeffect of a non-Gaussian shape of the source function andits dependence on the nature of the phase transition.This model provides the information on particle four-momenta and four-coordinates of the emission points al-lowing one to calculate the correlation function with thehelp of the weight procedure. For non-interacting identi-cal pions, the weight is given in Eq. (6). IV. RESULTS AND DISCUSSION
As mentioned above, the parameters of the model wereadjusted to approach experimental data for (pseudo-)rapidity distributions, transverse momentum spectraand elliptic flow coefficients with the XPT EoS, cor-responding to a crossover type transition [35]. Then,switching the EoS from XPT to 1PT leads to only smalldifferences in multiplicities and rapidity distributions ofthe produced hadrons in the model. At the same time,hydrodynamic evolution with the 1PT EoS leads to some-what decreased mean p ⊥ , elliptic and triangular flow co-efficients [53]. Those trends are explained by a less vio-lent transverse expansion with the 1PT EoS.In this section, we study the space-time characteristicsof the hadron emission in the model and present the re-sults for the Bose-Einstein correlations of identical pions,obtained with the two aforementioned EoS’s in a widecollision energy range √ s NN = 7.7 - 62.4 GeV coveredby the BES program at RHIC. A. Pion emission time distributions
In the model one has access to the space-time pointsof particle production in last collisions and resonance de-cays, in addition to their momenta. In Fig. 2 we visualizethe averaged time distributions of last interaction pointsof pions from the model simulations at the lowest, mid-dle and highest collision energies ( √ s NN = 7 .
7, 19 . . Eo1_77__1Entries 144530Mean 6.993RMS 2.407 [fm/c] t t d N / d Eo1_77__1Entries 144530Mean 6.993RMS 2.407 . . = . G e V NN s · (a) Eo1_77__1__1
Entries 449452Mean 15.45RMS 8.175 [fm/c] t t d N / d Eo1_77__1__1
Entries 449452Mean 15.45RMS 8.175 . . = . G e V NN s EoS: 1PTEoS: XPT · (b) FIG. 2. Pion emission times at the particlization surface (a)and the last interactions (b) in the center-of-mass system ofcolliding gold nuclei at different values of √ s NN . A detailed information on the pion emission times asa function of √ s NN for all simulated collision energiesand EoS’s is given in Table II. The time distributions ofmidrapidity pions have been obtained at the particliza-tion surface (points of their creation) and at the pointsof last interactions.From the Table II one can conclude an apparently weakdependence of the average pion creation time ¯ t at the par-ticlization surface on collision energy. It is an interplayof longer pre-thermal and shorter hydrodynamic stage atlower collision energies: at √ s NN <
39 GeV, the hydrostage starts at τ = 2 R/ ( γv z ) when the two colliding nu-clei have completely passed through each other, and thevalue of τ is as large as 3.2 fm/c at √ s NN = 7 . TABLE II. Extracted average pion emission times ¯ t as a func-tion of √ s NN in the center-of-mass system of colliding goldnuclei depending on the EoS used. √ s NN EoS particlization surface last interactions[
GeV ] ¯ t [fm/c] RMS [fm/c] ¯ t [fm/c] RMS [fm/c]7.7 1PT 7.24 2.84 13.15 6.56XPT 6.16 2.01 11.61 6.2611.5 1PT 7.33 2.31 13.09 6.92XPT 6.36 1.91 11.57 6.4119.6 1PT 6.88 2.16 13.18 7.56XPT 6.41 2.15 11.93 6.9327 1PT 6.85 2.37 13.38 8.07XPT 6.40 2.39 12.62 7.5739 1PT 7.17 2.75 13.98 8.30XPT 6.64 2.58 13.05 7.8562.4 1PT 7.00 2.82 14.11 8.50XPT 6.60 2.63 12.72 7.81 spond to larger values of ¯ t , which also depend weaklyon collision energy. The cascade somewhat smears therelative difference between the 1PT and XPT scenarios,both for ¯ t and RM S . B. Three-dimensional correlation radii in LCMS.
An example of one-dimensional projections of three-dimensional correlation function obtained with the vHLLE+UrQMD model using the 1PT EoS and XPT EoSis shown in Fig. 3. The analysis involved the simulationsperformed for gold-gold collisions at √ s NN = 7 . −
5% and thepair transverse momentum k T . The second one pertainsto the range of (0.15 - 0.25) GeV/c.One can see that the pion correlation functions at small q i (i = “out”, “side”, “long”) are not well described bythe Gaussian function in Eq. (7). The observed differencebetween correlation functions calculated with the 1PTand XPT EoS’s is noticeable in the “out” and “long”directions. In Fig. 4 this fact is demonstrated by theratios of individual projections. The ratios in the “out”and “long” directions reach values up to 1.03 at small q out and q long . A percent deviation from unity at small q side values appears due to the finite cuts on q out and q long .In Fig. 5 we present the m T -dependence of the three-dimensional femtoscopy LCMS radii calculated at √ s NN = 7.7, 11.5, 19.6, 27, 39, 62.4 GeV using the 1PT andXPT EoS’s, and a comparison of the obtained resultswith those ones obtained by the STAR collaboration [54].One can see that the model reasonably describes the m T -dependence of radii for all beam energies with bothEoS’s. As for the radii, they show different trends in q [GeV/c]0 0.05 0.1 ) ou t C ( q (a) EoS: 1PTEoS: XPTFit to 1PTFit to XPT EoS q [GeV/c]0 0.05 0.1 ) s i de C ( q (b) q [GeV/c]0 0.05 0.1 ) l ong C ( q (c) FIG. 3. One-dimensional projections of three-dimensionalcorrelation function (see Eq. (7)) of non-interacting identicalpion pairs onto “out” (a), “side” (b) and “long” (c) direc-tions. While projecting onto a direction, other two directionsare required to be within the range of (-0.03, 0.03) GeV/c.A fit with the Gaussian function is presented by dashed andsolid lines for the 1PT and XPT scenarios, respectively. the “out”, “side” and “long” directions. Whereas the R side using both EoS’s practically coincide, the R out withthe 1PT EoS is generally larger (however, not more than0.5 fm at any collision energy) than for the XPT EoS.This also leads to larger values of the R out /R side ratiousing the 1PT EoS. The difference comes from a weakertransverse flow developed in the fluid phase with the1PT EoS as compared with the XPT EoS . A longerlifetime of the fluid phase in the 1PT scenario also re-sults in a larger values of R long as compared with the A similar influence of the transverse flow on R out and R side hasbeen observed for the RHIC and LHC energies [16]. XPT scenario. Whereas one could expect that at lowercollision energies in the 1PT EoS a larger fraction of thefluid phase evolution occurs in the mixed phase with zerospeed of sound leading to an increase of evolution timeand R long , we did not observe such a trend in the model.The reason is that at lower collision energies in the modela sizable amount of radial flow is developed at pre-hydrostage. At the same time, the R out /R side ratio at lowestcollision energies shows a clear EoS dependence.The R out /R side and R out − R side as a function of √ s NN were studied at fixed m T by the STAR collaboration [54].A wide maximum near √ s NN ∼
20 GeV/c in both excita-tion functions was observed. This observation is howeveraccompanied by rather large systematic error bars. Wehave calculated the very same quantities in the modeland compared them with experimental data. The resultof comparison is shown in Fig. 6.One can see that due to large experimental error barsthe model calculations involving the XPT EoS are in astrong agreement with the data within the error bars atall energies, whereas the 1PT EoS overestimates the data.However, in the model taking into account the XPT EoSwe observe a monotonic increase in excitation functionsof both quantities, meanwhile the 1PT EoS results in anon-decreasing behavior of the quantities. The XPT EoS“works” better for lowest collision energies that mightbe seen earlier from a better description of individualradii in that energy region shown in Fig. 5. A study ofthe R out /R side ratio looks traditional in the modern fem-toscopy since the R out and R side radii are both reducedby flow, thus their ratio is a more robust against the floweffects.As mentioned above, the parameters of the model wereadjusted to approach the basic hadronic observables: ra-pidity, transverse momentum distributions and ellipticflow coefficients within the BES region, but not femto-scopic ones. No model tuning has been made for thefemtoscopy, therefore the obtained radii may be consid-ered as a free “prediction” even though the experimentaldata already exists. C. Source emission functions
In a Monte Carlo model one has an access to the space-time characteristics of produced particles, which allowsone to avoid the complicated procedure of solving the in-tegral equation (see Eq. 4) as it is done in experiment [52].The source emission function can be calculated directlyas: S ( r ∗ ) = P i = j δ ∆ ( r ∗ − r ∗ i + r ∗ j ) N ∆ (8)Here r i and r j are the particles space positions, r isthe particles separation in PRF; δ ∆ = 1 if | x | < p/ p is a size of the histogram bin. Thedenominator in Eq. (8) takes care for the normalization [GeV/c] long q XP T i P T / C i C (c) [GeV/c] side q XP T i P T / C i C (b) [GeV/c] out q R a t i o (a) FIG. 4. Ratios of one-dimensional projections of three-dimensional correlation functions for the two EoS. For each directionthe corresponding ratio is calculated as follows: C ( q i )( XP T ) /C ( q i )(1 P T ), where i denotes “out” (a), “side” (b) and “long”(c) directions, 1 P T and
XP T denote a type of the used EoS. by a product of the number of pairs N = P i = j .Fig. 7 demonstrates an example of one-dimensionalprojections of source emission function S ( r ∗ ) derivedfrom the model directly.One can see that calculations involving the 1PT EoSlead to longer visible tails in the projections as comparedwith the XPT EoS, especially for “out” direction. It isrelated to a weaker transverse flow developed in the fluidphase and a longer lifetime of the fluid phase taking placein the 1PT EoS. A similar observation has been reportedfor the “out” femtoscopic radii in the previous section.A set of functions consisting of a single Gaussian,double Gaussian, Gaussian + Exponential, Gaussian +Lorentzian, and Hump function [55] was tested for a de-scription of one-dimensional projections of source emis-sion functions. The best description was obtained withthe Hump-function and the double Gaussian function.The last one gives only slightly worse χ than the Hump-function, but allows one for a clear interpretation of pa-rameters and a more stable fit. The single Gaussian anddouble Gaussian fit functions are shown in Fig. 7. Theparameters extracted from these fits are presented in Ta-ble III. The fit of projections of source emission function witha single Gaussian gives a large value of χ /ndf . The fitwith a double Gaussian allows one to get much better val-ues of χ /ndf and obtain a better description of the tailsof projections of source emission functions until ∼
60 fmin “out” and ∼
25 fm in “side” and “long” directions,respectively. The radii extracted from the double Gaus-sian fit have a small component R smalli of 4-12 fm and alarge component R largei of 8-20 fm (as usual, i denotes“out”, “side” and “long” directions). It reflects the factthat pion source consists of direct particles (described bythe first component) and re-scatterings (the second one).The difference between radii extracted from the sourceemission functions obtained with the two EoS’s is seenfor both components - R smalli and R largei , but it is rathersmall, less than 0.5 fm. The radii are larger for the 1PTscenario being consistent with the three-dimensional fem-toscopic radii reported above.It is interesting to note that in case of the single Gaus-sian fit the values of the radii are approximately equalto the ones derived from the double Gaussian fit and av-eraged quadratically over relative contributions of smalland large radii. It means that the one-dimensional Gaus-sian radii roughly reflect the main features of double [GeV/c] T m [f m ] ou t R = 7.7 GeV NN s (a) 0.2 0.4 0.6 = 11.5 GeV NN s (e) 0.2 0.4 0.6 = 19.6 GeV NN s (i) 0.2 0.4 0.6 = 27 GeV NN s (m) 0.2 0.4 0.6 = 39 GeV NN s (q) 0.2 0.4 0.6 = 62.4 GeV NN s (u)0.2 0.4 0.6 [f m ] s i d e R (b) 0.2 0.4 0.6 (f) 0.2 0.4 0.6 (j) 0.2 0.4 0.6 (n) 0.2 0.4 0.6 (r) 0.2 0.4 0.6 (v)0.2 0.4 0.6 [f m ] l ong R (c) 0.2 0.4 0.6 (g) 0.2 0.4 0.6 (k) 0.2 0.4 0.6 (o) 0.2 0.4 0.6 (s) 0.2 0.4 0.6 (w) s i d e / R ou t R (d) 0.2 0.4 0.6 (h) [GeV/c]m (l) (p) 0.2 0.4 0.6 (t) 0.2 0.4 0.6 (x) FIG. 5. Comparison of the model three-dimensional LCMS femtoscopy radii fitted according to Eq. (7) with those measuredby the STAR collaboration at √ s NN = 7.7 ((a) - (d)), 11.5 ((e) - (h)), 19.6 ((i) - (l)), 27 ((m) - (p)), 39 ((q) - (t)), 62.4 ((u) -(x)) GeV. Open squares represents the STAR data. Triangles correspond to different types of EoS like they do in Fig. 3.TABLE III. Results of single (Eq. (5)) and double Gaussian fits of model source emission functions shown in Fig. 7. χ /ndf values in parenthesis correspond to the XPT EoS. All calculations are performed in PRF.Single Gaussian χ /ndf Radius 1PT XPT975 (1247) R out [fm] 11.10 ± ± R side [fm] 4.19 ± ± R long [ fm ] 4.59 ± ± χ /ndf Radius 1PT XPT51.6 (73.0) R smallout [fm] 8.91 ± λ smallout =0.66) 7.83 ± λ smallout =0.68)7.7 GeV 58.7 (66.6) R smallside [fm] 4.10 ± λ smallside =0.99) 4.08 ± λ smallside =0.99)130.3 (195.1) R smalllong [fm] 3.10 ± λ smalllong =0.64) 2.93 ± λ smalllong =0.74) R largeout [fm] 13.85 ± λ largeout =0.34) 12.88 ± λ largeout =0.32) R largeside [fm] 9.76 ± λ largeside =0.01) 9.45 ± λ largeside =0.01) R largelong [fm] 6.06 ± λ largelong =0.36) 6.24 ± λ largelong =0.26) Gaussian fits.Fig. 8 shows a √ s NN -dependence of the small and largeradii and their relative contributions extracted from thedouble Gaussian fit depending on EoS.The radii increase with increasing √ s NN for both typesof calculations. The visible difference between small andlarge radii in “out” direction (see Fig.8 (a)) decreaseswith increase of √ s NN . The relative contributions ofsmall and large radii to “out” direction are equal to ∼ .
65 and ∼ .
35 (see Fig.8 (d)) and practically donot depend on either √ s NN or type of EoS. The radii in “side” direction seem to be independent of √ s NN (seeFig.8 (b), (e)). The radii in “long” projection almostcoincide for both types of EoS (see Fig.8 (c)), but therelative contributions of them as a function of √ s NN demonstrate a difference depending on EoS. The relativecontribution of the large radii has a tendency to increasewith √ s NN and is larger in case of the 1PT scenario (seeFig.8 (f)).Of course, the best comparison with experiment is adirect comparison of source emission functions from themodel with the extracted ones experimentally. Neverthe-
10 20 30 40 50 60 s i d e / R ou t R (a) STAR dataEoS: 1PTEoS: XPT [GeV] NN s
10 20 30 40 50 60 ] [f m i d e - R ou t R (b) FIG. 6. Ratio of the “out” and “side” radii (a) and differ-ence of the radii squared (b) as a function of √ s NN derivedfrom the STAR data (0 . < k T < .
25 GeV/c, 0-5% central-ity) and compared with the model calculations using the twoEoS’s. less, this study shows us that the use of a double Gaussianfit also reflects a lot of interesting features of source emis-sion functions, while a single Gaussian fit used in manyexperiments can be a rather risky procedure due to poordescription of the source by this function. However, asit was demonstrated above, the radii extracted from thesingle Gaussian fit are equal to the properly averageddouble Gaussian radii, giving, in principle, a realistic in-formation on the source. This result is quite encouragingsince it is much easier to study the three-dimensionalradii than the source emission functions.
V. SUMMARY
We have presented the first study of pion femtoscopyin viscous hydro + cascade model vHLLE+UrQMD in theenergy range of the BES program at RHIC. It is shownthat the chiral model EoS [41] (XPT EoS), which has acrossover-type transition between QGP and hadron gasphases, in the fluid phase results in a quite reasonablereproduction of three-dimensional pion femtoscopic radiimeasured by the STAR collaboration.The “out” Gaussian femtoscopic radii obtained withthe bag model EoS (1PT EoS) are systematically largeras compared with the XPT EoS; the “side” radii coincidefor both types of EoS; the “long” radii are also somewhatlarger for the 1PT EoS.The 1PT EoS results in a systematically worse re-production of the data, however the differences betweentwo EoS’s are not so large. The R out /R side ratio and R out − R side are in agreement with the STAR resultswithin the error bars at all collision energies using theXPT EoS, but their energy dependences observed in themodel are quite monotonic as opposed to the broad max-imum around √ s NN = 20 GeV reported by STAR. Atthe same time, the 1PT EoS overestimates experimen-tal data points for both R out /R side and R out − R side .In particular, the latter EoS does not reproduce thefemtoscopic radii even at the lowest energy considered, √ s NN = 7 . √ s NN = 20 GeV and this should be an incentive forfuture experiments at NICA and FAIR facilities. It is anopen question whether a new set of parameters more suit-able for the femtoscopic radii description can be found.In addition to traditional femtoscopic radii, we havecalculated source emission functions of pion pairs. Weshow that it is possible to distinguish calculations withthe two different EoS. The projections of source emissionfunctions onto “out” direction are wider for the use of the1PT EoS. For “side” direction these projections coincidefor both scenarios; for “long” direction the projectionsobtained with the 1PT EoS are also wider in comparisonwith calculations using the XPT EoS. This observationis related to a weaker transverse flow developed in thefluid phase and a longer lifetime of the phase in case ofthe 1PT EoS used.In order to describe the source emission functionsquantitatively a set of different fitting functions has beentested. It is shown that the use of a double Gaussianfit to the source emission function gives a reasonable de-scription and allows one for a simple interpretation of theobtained small and large radii.So far we have performed femtoscopic analysis withvHLLE+UrQMD model only. As a next step we plan toextend the analysis using 3-fluid hydrodynamics-basedevent generator THESEUS [56]. In THESEUS the hy-drodynamical description of heavy ion reaction startsearlier, which results in different sensitivity to hydrody-namic EoS especially in the NICA energy range.0 (c)long r* [fm] (b)side ] - S (r * ) [f m (a)out EoS: 1PTDouble Gaussian fitSingle Gaussian fitEoS: XPTSingle Gaussian fitDouble Gaussian fit-7 · FIG. 7. One-dimensional projections of source emission functions of pions from the model (full simulation with cascade)obtained at √ s NN = 7.7 GeV for pion pairs satisfying the cut on transverse momentum: 0.2 < k T <
10 20 30 40 50 60 R [f m ] out (a)
10 20 30 40 50 60510152025 side (b)
10 20 30 40 50 60510152025 long (c)
10 20 30 40 50 60 l - (d) [GeV] NN s10 20 30 40 50 600.200.20.40.60.811.2 (e) EoS: 1PT, small radiiEoS: XPT, small radiiEoS: 1PT, large radiiEoS: XPT, large radii
10 20 30 40 50 600.200.20.40.60.811.2 (f)
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