Beam Energy Dependence of Hanbury-Brown-Twiss Radii from a Blast-Wave Model

Abstract

The beam energy dependence of correlation lengths (the Hanbury-Brown-Twiss radii) is calculated by using a blast-wave model and the results are comparable with those from RHIC-STAR beam energy scan data as well as the LHC-ALICE measurements. A set of parameters for the blast-wave model as a function of beam energy under study are obtained by fit to the HBT radii at each energy point. The transverse momentum dependence of HBT radii is presented with the extracted parameters for Au+Au collision at [subscript]sNN[/subscript] = 200 GeV and for Pb+Pb collisions at 2.76 TeV. From our study one can learn that particle emission duration cannot be ignored while calculating the HBT radii with the same parameters. And tuning kinetic freeze-out temperature in a range will result in system lifetime changing in the reverse direction as it is found in RHIC-STAR experiment measurements.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

S. Zhang 1 and Y. G. Ma 1,2 and J. H. Chen 1 and C. Zhong 1

Academic Editor:Ming Liu

1, Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China

2, ShanghaiTech University, Shanghai 200031, China

Received 25 May 2016; Revised 9 September 2016; Accepted 9 October 2016; 22 November 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³ .

1. Introduction

The Quark-Gluon-Plasma (QGP) predicted by quantum chromodynamics (QCD) [1] can be formed in relativistic heavy-ion collisions. It is believed that this kind of new state of matter is produced in the early stage of central Au+Au collisions at the top energy in the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory [2-5]. It was concluded that the hot-dense matter is a strongly interacting partonic matter named as sQGP under extreme temperature and energy density with sufficient experimental evidences [6-10]. Recently, many results in Pb+Pb and p+Pb collisions at _sNN = 2.76 TeV in the Large Hadron Collider (LHC) were also reported for exploring properties of the hot-dense quark-gluon matter [11-14].

Mapping the QCD phase diagram and locating the phase boundary and possible critical end point become hot topic in the field [1, 15-18]. The properties inherited from QGP will imprint signal on observables which can reflect phase transition information. The geometry of the system shall undergo phase space evolution from QGP stage to hadron kinetic freeze-out stage, which can be considered as an observable that is sensitive to the equation of state [19, 20]. Hanbury-Brown-Twiss (HBT) technique invented for measuring sizes of nearby stars [21] was extended to particle physics [22] and heavy-ion collisions [23-31]. The HBT technique can also be applied to extract the precise space-time properties from particle emission region at kinetic freeze-out stage in heavy-ion collisions. Furthermore this technique has been evolved to search for new particles and to measure particle interactions [32-34].

Experimental results on HBT study in high energy nuclear reaction were reported by STAR [35, 36] and PHENIX [37, 38] at RHIC top energy in Au+Au collisions, as well as by ALICE [39] at _sNN = 2.76 TeV in Pb+Pb collisions. Recently STAR and PHENIX collaborations have also presented beam energy dependence of HBT radii [20, 40] and a nonmonotonic changing behaviour for the square difference between outward radius and sideward radius (^Rout2 -^Rside2 ) with increase of beam energy was found. This behaviour could be sensitive to equation of state and was considered as a probe related to the critical end point of QGP phase transition [19]. A finite-size scaling (FSS) analysis of experimental data was performed in [19] and the analysis suggested that a second-order phase transition took place with a critical end point located at a chemical freeze-out temperature of ~165 MeV and a baryon chemical potential of ~95 MeV.

In this paper we present beam energy dependence of HBT radii calculated from a blast-wave model. Firstly, experimental data of HBT radii from RHIC-STAR and LHC-ALICE are fitted and parameters for the blast-wave model are configured as a function of beam energy. The transverse momentum dependence of HBT radii is calculated at RHIC top energy and LHC energy with these parameters. From the results, it was found that particle emission duration is important for calculating transverse momentum dependence of HBT radii and changing of kinetic freeze-out temperature will result in system lifetime changing in reverse direction as that in the RHIC-STAR experimental analysis [20].

The paper is organised as follows. In Section 2, blast-wave model and HBT correlation function are briefly introduced. Some kinetic parameters are presented as a function of beam energy. Section 3 presents energy dependence of extracted HBT radii with various kinetic temperatures, system lifetime, particle emission duration, and so forth. Transverse momentum dependence of HBT radii is discussed in Section 4. Finally Section 5 gives the summary.

2. Blast-Wave Model and HBT Correlation Function

The particle emission function S(x,p) in heavy-ion collisions used in this study is similar as in [44] [figure omitted; refer to PDF] In cylindrical coordinates, source moving four-velocity and momentum can be written, respectively, as [figure omitted; refer to PDF] And the flow rapidity is given by [figure omitted; refer to PDF] here the normalized elliptical radius is as follows: [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

In (1), spatial weighting of source elements is selected as a simple pattern [44]: [figure omitted; refer to PDF]

Here are the main parameters in this model, the kinetic freeze-out temperature _Tkin , the radial flow parameter _ρ0 , the "elliptic flow parameter" _ρ2 which controls second-order oscillation of transverse rapidity by the relation as in (5), the system lifetime _τ0 , and the particle emission durations Δτ, _Rx , and _Ry related to system size and space asymmetry. In this calculation we assume that the system is in most central heavy-ion collisions and thus set the _Rx =_Ry =_R0 , _ρ2 =0. In experimental measurement, hadron spectra can be fitted by the blast-wave model with integrating the emission function except _pT and Y. The kinetic freeze-out temperature _Tkin and the averaged radial flow (β) were extracted from the fit. For detailed technique information, one may refer to [41]. The averaged radial flow is related to the flow rapidity ρ=^tanh-1 [...]β, from which the radial flow parameter _ρ0 is calculated. Figures 1 and 2 present the measured _Tkin and (β) at a wide beam energy range, respectively. The data come from [41, 42]. The kinetic freeze-out temperature _Tkin and the averaged radial flow (β) can be parametrised as a function of _sNN by empirical formula: [figure omitted; refer to PDF] where _Tlim = 169.171 MeV and _βlim = 0.399. And then free parameters in the blast-wave model will be _R0 , _τ0 , and Δτ, which are all related to expanding characters of the collision system. And it will be determined by the HBT correlation calculation which will be discussed below in detail.

Figure 1: Kinetic freeze-out temperature as a function of centre-of-mass energy _sNN . Data are from [41, 42].

[figure omitted; refer to PDF]

Figure 2: The averaged radial flow (β) as a function of centre-of-mass energy _sNN . Data are from [41, 42].

[figure omitted; refer to PDF]

In our previous works, the blast-wave model was coupled with thermal equilibrium model to describe the hadron production and its spectra with a range of thermal parameters [45] and with coalescence mechanism to calculate the light nuclei production and to predict the di-baryons production rate [46, 47]. In addition, the DRAGON model [48] and the THERMINATOR2 [49, 50] model have also been developed as event generator to study the phase space distribution of hadrons at freeze-out stage. It is also successfully applied in experimental data analysis [41, 42] to extract the kinetic freeze-out properties and to provide the phase space distribution to calculate the HBT correlation in theory [44, 51].

The identical two particle HBT correlation function can be written as [26, 51] [figure omitted; refer to PDF] here K is average momentum for the two particles, K=(1/2)(_p1 +_p2 ), q denotes relative momentum between two particles, q=_p1 -_p2 , and β[arrow right]=K[arrow right]/_{K[arrow right]0} . From [44, 51-53], the "out-side-long" coordinates system is used in this calculation, in which the long direction _Rlong is parallel to the beam, the sideward direction _Rside is perpendicular to the beam and total pair momentum, and the outward direction _Rout is perpendicular to the long and sideward directions. After expanding angular dependence of C(K,q) in a harmonic series with the "out-side-long" coordinates system, the HBT radii can be written as [44, 51] [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

In the calculation, observables are related to integrals of emission function (1) over phase space ^d4 x=dx dy dz dt=τ dτ dηr dr d_[varphi]s , weighted with some quantities B(x,K). If B(x,K)=^{B[variant prime]} (r,_[varphi]s ,K)^τisinhj [...]η^coshk [...]η, then the integrals can be written as in [44] [figure omitted; refer to PDF] and some useful integrals [figure omitted; refer to PDF] where we define [figure omitted; refer to PDF]

Retière and Lisa [44] have provided a systematic analysis of parameter range for the blast-wave model and investigated the _pT spectra, the collective flow, and the HBT correlation of hadrons produced in heavy-ion collisions. In this calculation we will use the algorithm developed in [44, 51] to study the energy and transverse momentum dependence of pion HBT correlation radii. Based on the discussion above, the free parameters will be _R0 , _τ0 , and Δτ which can be determined by fitting experimental data by (9). Before the study of energy dependence on HBT radii, we calculated pion's spectra by using this algorithm in the blast-wave model: [figure omitted; refer to PDF]

Figure 3 presents pion's spectra which are comparable with experimental data from STAR at _sNN = 200 GeV in central Au+Au collisions [43] and ALICE at _sNN = 2.76 TeV in central Pb+Pb collisions [14], respectively.

Figure 3: Comparison of pion's spectra from blast-wave model (lines) and the data in central Au+Au collisions at _sNN = 200 GeV [43] and the data in central Pb+Pb collisions at _sNN = 2.76 TeV [14].

[figure omitted; refer to PDF]

3. Energy Dependence of HBT Radii

The parameters are configured as following. The kinetic freeze-out temperature _Tkin and the averaged radial flow (β) are from (7) as a function of _sNN , but in some cases _Tkin are fixed to 90, 100 and 120 MeV for comparison. In numerical calculation, the particle emission duration Δτ is set to zero and in another case the energy dependence of Δτ will be extracted by fit on the data. The _R0 will also be given by fit the data at each energy point. The experimental results of HBT radii are taken from the STAR and the ALICE collaborations [20, 39] at centre-of-mass energy _sNN points, 7.7, 11.5, 19.6, 27, 39, 62.4, 200, and 2760 GeV. The difference between calculated radii results and the experimental data should reach a minimum value (_δs ,_δo ,_δl ) for each energy point: [figure omitted; refer to PDF] Actually from (9) and the algorithm in [44, 51], one can find the HBT radii parameter dependence as follows: [figure omitted; refer to PDF] So _R0 can be determined directly by fit on ^Rside2 . And _τ0 and Δτ can be extract by fit on ^Rout2 and ^Rlong2 simultaneously. We then learnt that the difference of ^Rout2 -^Rside2 depends not only on the system lifetime _τ0 but also on the particle emission duration Δτ.

Figure 4 presents our calculation on HBT radii for identical charged pion-pion correlation with the configured parameters. The HBT radii show an increasing trend with the increasing of centre-of-mass energy _sNN . In the case of Δτ≠0, the results can describe experimental data successfully. However, for Δτ = 0.0, _Rout cannot be fitted despite the fact that _Rlong can be well matched by the calculation. Since _Tkin and _ρ0 are taken from experimental results, _Rside will only depend on parameter _R0 , which reflects the system size where particles are emitted. Figure 5 displays the extracted _R0 as a function of _sNN . It demonstrates a similar trend of energy dependence as _Rside . With fixed temperature of _Tkin (90, 100, and 120 MeV), it is found that a large _R0 is needed to fit the data while _Tkin sets to small value. This is consistent with evolution of the fireball created in heavy-ion collisions, where temperature becomes lower while system size increases.

Figure 4: The extracted values of _Rside , _Rout , and _Rlong as a function of centre-of-mass energy _sNN . Different types of lines represent different kinetic temperature parametrisation. Markers are experimental data from [20, 39]. Left panels are results with finite particle emission duration (Δτ) and right panels for the cases of Δτ=0.

[figure omitted; refer to PDF]

Figure 5: _{R 0} as a function of centre-of-mass energy _sNN with different kinetic temperature parametrisation.

[figure omitted; refer to PDF]

_{R out} and _Rlong depend not only on _τ0 but also on Δτ. Figure 6 shows _τ0 and Δτ as a function of centre-of-mass energy _sNN from fit to the data. Δτ slightly depends on the _sNN . From Figure 6 one can see that _τ0 generally increases with the increasing of _sNN in trends but there exists a minimum value at _sNN ~39 GeV. It may imply that the system in higher energy (such as at LHC) will undergo a longer time evolution than in lower energy before hadron rescattering ceases (the kinetic freeze-out status). With fixed temperature of _Tkin (90, 100, and 120 MeV), the system lifetime _τ0 and the particle emission duration Δτ are all in reverse order to the temperature _Tkin . This suggests that a system expanding with a long lifetime and a broad duration will result in a lower temperature, which is consistent with the behaviour of _R0 as discussed above. We learnt that our results are comparable with the experimental results with Δτ≠0. With the system lifetime and HBT radii calculation all taken into account, it can be concluded that the particle emission duration cannot be ignored while fitting the HBT radii (_Rside , _Rout , and _Rlong ) at the same time.

Figure 6: The same as Figure 4 but for _τ0 and Δτ. Experimental data (shadowed area) is taken from [20].

[figure omitted; refer to PDF]

After _Rout and _Rside are all calculated, difference of ^Rout2 -^Rside2 as a function of centre-of-mass energy _sNN can be obtained as shown in Figure 7. In the case of Δτ≠0, the calculated results can describe the data very well. However, it is unsuccessful to fit the data with Δτ=0 for the current parameter configuration. Energy dependence of the difference of ^Rout2 -^Rside2 demonstrates a nonmonotonic increasing trend with the increasing of _sNN . The peak of experimental results locates at _sNN ~17.3 GeV [20] and the calculated results give a very similar behaviour for the peak emerging. And in [19], the theoretical work proposes the critical end point (CEP) for deconfinement phase transition at _sNN = 47.5 GeV by applying FSS. Anyway other observables, such as elliptic flow and fluctuations, should be considered together and other basic theoretical calculations are awaiting for comparison, which contribute to locate the CEP and understand underlying physics around this energy region.

Figure 7: The same as Figure 4 but for the ^Rout2 -^Rside2 . Experiment data is taken from [20].

[figure omitted; refer to PDF]

4. Transverse Momentum Dependence of HBT Radii

With the above parameter configuration, we also calculated the transverse momentum dependence of HBT radii at _sNN =200 GeV and 2760 GeV in central heavy-ion collisions. Figures 8 and 9 show the HBT radii as a function of transverse momentum in central Au+Au collisions at _sNN =200 GeV and in central Pb+Pb collisions at _sNN =2760 GeV, respectively. The experimental data is from [20, 39]. _Rside , _Rout , and _Rlong decrease with the increasing of transverse momentum _pT as shown in Figure 8, which indicates that high _pT particles are emitted from near the centre of the fireball. It is found that the calculated results fit the STAR data in the case of Δτ≠0 but fails to describe the _Rout with Δτ = 0. The similar _pT dependence trend is found in central Pb+Pb collisions at _sNN = 2760 GeV as shown in Figure 9. In the Δτ≠0 case, the calculated results reproduce the _Rside and _Rout exactly but slightly underestimate the value of _Rlong . Again, a reasonable parameter configuration cannot be found for fitting ALICE data in the case of Δτ = 0. These results suggest that the system lifetime and particle emission duration should be taken into account at the same time while describing _Rside , _Rout and _Rlong with the same parameter configuration in the blast-wave model.

Figure 8: The transverse momentum dependence of HBT radii in central Au+Au collisions at _sNN =200 GeV. Markers are experimental data from [20].

[figure omitted; refer to PDF]

Figure 9: The same as Figure 8 but for the central Pb+Pb collisions at _sNN =2760 GeV. Markers are experimental data from [39].

[figure omitted; refer to PDF]

5. Summary

The HBT radii (_Rside , _Rout , and _Rlong ) are calculated from the blast-wave model in the "out-side-long" (osl) coordinates system. In comparison with the experimental data [20, 39], we found that, in the case of Δτ≠0, the parameter configuration for blast-wave model can successfully describe the experimental results of collision energy and transverse momentum dependence of _Rside , _Rout , and _Rlong . Since the collision system has different temperature at each centre-of-mass energy point, the configured parameters can be considered as the preferred values with a case of _Tkin as a function of _sNN and Δτ≠0 as shown in Figures 5 and 6. However, it cannot be configured for the blast-wave parameter to fit the experimental data while setting Δτ to zero. This may imply that the particle emission duration plays an important role to describe the system expanding and cannot be ignored while calculating the _Rside , _Rout , and _Rlong to fit the data at the same time. And the difference of ^Rout2 -^Rside2 presents a nonmonotonic increasing trend with the increasing of _sNN as seen in the experimental analysis [20], which is sensitive to the equation of state and might be related to the critical end point with other observables taken into account.

Acknowledgments

This work was supported in part by the Major State Basic Research Development Program in China under Contract no. 2014CB845400, the National Natural Science Foundation of China under Contract nos. 11421505, 11220101005, 11105207, 11275250, 11322547, and U1232206, and the CAS Project Grant no. QYZDJ-SSW-SLH002.

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