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

Authors: S. Zhang; Y. G. Ma; J. H. Chen; C. Zhong
Journal: Advances in High Energy Physics
 (2016)

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## 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 atsNN = 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.


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## Body

## 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 Section2, 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 functionS(x,p) in heavy-ion collisions used in this study is similar as in [44] (1)Sx,p=mTcosh⁡η-YΩr,ϕse-τ-τ02/2Δτ21eK·u/Tkin±1=mTcosh⁡η-YΩr,ϕse-τ-τ02/2Δτ2∑n=1∞∓n+1e-K·u/Tkin≃mTcosh⁡η-YΩr,ϕse-τ-τ02/2Δτ2e-K·u/Tkin.In cylindrical coordinates, source moving four-velocity and momentum can be written, respectively, as(2)uμx=cosh⁡ηcosh⁡ρr,ϕs,sinh⁡ρr,ϕscos⁡ϕb,sinh⁡ρr,ϕssin⁡ϕb,sinh⁡ηcosh⁡ϕb,Kμ=mTcosh⁡Y,pTcos⁡ϕp,pTsin⁡ϕp,mTsinh⁡Y.And the flow rapidity is given by(3)ρr,ϕs=r~ρ0+ρ2cos⁡2ϕb;here the normalized elliptical radius is as follows:(4)r~≡rcos⁡ϕs2Rx2+rsin⁡ϕs2Ry2,with(5)tan⁡ϕs=RyRx2tan⁡ϕb.In (1), spatial weighting of source elements is selected as a simple pattern [44]:(6)Ωr,ϕs=1,r~<10,r~>1.Here are the main parameters in this model, the kinetic freeze-out temperatureTkin, 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: (7)Tkin=Tlim11+exp⁡8.559-ln⁡sNN/0.093/sNN0.057/0.846,β=βlim11+exp⁡5.666-ln⁡sNN/0.124sNN0.065,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 energysNN. Data are from [41, 42].Figure 2
The averaged radial flowβ as a function of centre-of-mass energy sNN. Data are from [41, 42].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](8)CK→,q→=1+∫d4xeiq→·x→-β→tSx,K∫d4xSx,K,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→=K→/K→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] (9)Rside2=12x~2+y~2-12x~2-y~2cos⁡2ϕp-x~y~sin⁡2ϕp,Rout2=12x~2+y~2+12x~2-y~2cos⁡2ϕp+x~y~sin⁡2ϕp-2βTt~x~cos⁡ϕp+t~y~sin⁡ϕp+βT2t~2,Rlong2=z~2-2βlt~z~+βl2t~2,where(10)fxK≡∫d4xfxSx,K∫d4xSx,K,x~μ=xμ-x~μK.In the calculation, observables are related to integrals of emission function (1) over phase space d4x=dxdydzdt=τdτdηrdrdϕs, weighted with some quantities B(x,K). If B(x,K)=B′(r,ϕs,K)τisinhj⁡ηcoshk⁡η, then the integrals can be written as in [44](11)∫02πdϕs∫0∞rdr∫-∞∞dη∫-∞∞τdτSx,KBxK=mTHiB′j,kK,and some useful integrals(12)Hi≡∫-∞∞dττi+1e-τ-τ02/2Δτ2,Gj,kx,K≡∫-∞∞dηe-βcosh⁡ηsinhj⁡ηcoshk+1⁡η,B′j,kK≡∫02πdϕs∫0∞rdrGj,Kx,KB′x,Keαcos⁡ϕb-ϕpΩr,ϕs,where we define(13)α≡pTTsinh⁡ρr,ϕs,β≡mTTcosh⁡ρr,ϕs.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:(14)dNpTdpT=∫dϕp∫d4xSx,K.Figure3 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 atsNN = 200 GeV [43] and the data in central Pb+Pb collisions at sNN = 2.76 TeV [14].


## 3. Energy Dependence of HBT Radii

The parameters are configured as following. The kinetic freeze-out temperatureTkin 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:(15)Rsideth-Rsideexp=δs,Routth-Routexp=δo,Rlongth-Rlongexp=δl.Actually from (9) and the algorithm in [44, 51], one can find the HBT radii parameter dependence as follows:(16)Rside2=Rside2Tkin,ρ0,R0,Rout2=Rout2Tkin,ρ0,R0,τ0,Δτ,Rlong2=Rlong2Tkin,ρ0,R0,τ0,Δτ.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.Figure4 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 ofRside, 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 5
R 0 as a function of centre-of-mass energy sNN with different kinetic temperature parametrisation.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 Figure4 but for τ0 and Δτ. Experimental data (shadowed area) is taken from [20].AfterRout 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 Figure4 but for the Rout2-Rside2. Experiment data is taken from [20].


## 4. Transverse Momentum Dependence of HBT Radii

With the above parameter configuration, we also calculated the transverse momentum dependence of HBT radii atsNN=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 atsNN=200 GeV. Markers are experimental data from [20].Figure 9
The same as Figure8 but for the central Pb+Pb collisions at sNN=2760 GeV. Markers are experimental data from [39].


## 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.



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