# Pion Transverse Momentum Spectrum, Elliptic Flow, and Interferometry in the Granular Source Model for RHIC and LHC Heavy Ion Collisions

Authors: Jing Yang; Yan-Yu Ren; Wei-Ning Zhang
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

We systematically investigate the pion transverse momentum spectrum, elliptic flow, and Hanbury-Brown-Twiss (HBT) interferometry in the granular source model for the heavy ion collisions of Au-Au atsNN=200 GeV and Pb-Pb at sNN=2.76 TeV with different centralities. The granular source model can well reproduce the experimental results of the heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). We examine the parameters involved in the granular source model. The experimental data of the momentum spectrum, elliptic flow, and HBT radii for the two collision energies and different centralities impose very strict constraints on the model parameters. They exhibit certain regularities for collision centrality and energy. The space-time structure and expansion velocities of the granular sources for the heavy ion collisions at the RHIC and LHC energies with different centralities are investigated.


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## Body

## 1. Introduction

The main purpose of relativistic heavy ion collisions is to probe the new matter, quark-gluon plasma (QGP), and study its properties. Because of the complexity of the nucleus-nucleus collisions, model investigation plays important roles in determining and characterizing the QGP. Single particle spectrum, elliptic flow, and two-particle Hanbury-Brown-Twiss (HBT) correlations are crucial final-state observables in relativistic heavy ion collisions [1–28]. They reflect the characteristics of the particle-emitting sources in different aspects at different stages. A combined investigation of these observables can provide very strong constraints for the source models. So far, much progress has been made in understanding the experimental data of the heavy ion collisions at the top energies of the Relativistic Heavy Ion Collider (RHIC) [29–57]. However, more detailed investigations of the physics beneath the data through multiobservable analyses are still needed. On the other hand, the experimental data of several TeV heavy ion collisions at the Large Hadron Collider (LHC) have been recently published [15–25, 58–63]. It is an ambitious goal for the models to explain consistently the data of the heavy ion collisions at the RHIC and LHC energies.In [64–67], Zhang et al. proposed and developed a granular source model of QGP droplets to explain the HBT data of the RHIC experiments [9–14]. Their investigations indicate that the short evolution lifetime and wide initial distribution of the QGP droplets in the granular source can lead to the result of the HBT radii, Rout~Rside. Here the labels “out” and “side” denote the transverse directions parallel and perpendicular to the transverse momentum of the pion pair [68–70]. The results of the pion transverse momentum spectrum [65, 66], elliptic flow [65], and HBT radii [65, 66] of granular source are well in agreement with the experimental data of the pion momentum spectrum and elliptic flow for the Au-Au collisions with minimum bias at sNN=200 GeV at the RHIC [2, 6] and the pion HBT measurements for the RHIC Au-Au collisions with 0–30% centrality [11, 12], respectively. They also find [66, 67] that the granular source model may reproduce the main characteristics of the two-pion source functions, extracted by the imaging techniques [71–75], in the RHIC experiments [13, 14]. In [76, 77], the fluctuating signatures of the single-event HBT correlation functions of granular sources are investigated. The detection of source inhomogeneity through the fluctuating single-event HBT correlation functions is discussed in [78], with the smoothed particle hydrodynamics (SPH) [79, 80]. Recently, the HBT analyses in the granular source model for the experimental data of the most central Pb-Pb collisions at sNN=2.76 TeV at the LHC [24] are performed [81]. The model parameters of the granular sources for the most central collisions at the RHIC and LHC energies are compared and discussed [81].Although the granular source model reproduced the experimental data of the pion HBT radii for the Au-Au collisions in a certain centrality region at the RHIC and explained the pion HBT radii in the most central Au-Au and Pb-Pb collisions at the RHIC and the LHC [65, 66, 81], it is still a challenge to models to explain consistently the experimental HBT measurements in different centrality regions of the collisions at the RHIC and the LHC energies. On the other hand, pion momentum spectrum and elliptic flow are very sensitive to collision centrality. A combined investigation of pion momentum spectrum, elliptic flow, and HBT interferometry in different centrality regions of the heavy ion collisions at the RHIC and the LHC energies, in the granular source model, is of great interest. In this work, we systematically investigate the pion transverse momentum spectrum, elliptic flow, and HBT interferometry in the granular source model for the heavy ion collisions at the RHIC and LHC energies with different centralities. In the combined investigations of pion momentum spectrum, elliptic flow, and HBT interferometry, we choose the experimental data in the centrality regions 0–5%, 10–20%, and 30–50% for the Au-Au collisions at sNN=200 GeV at the RHIC [1, 2, 7, 12] and in the centrality regions 10–20% and 40–50% for the Pb-Pb collisions at sNN=2.76 TeV at the LHC [17, 22, 25], because in these centrality regions the data of pion transverse momentum spectrum, elliptic flow, and HBT radii are available together. By comparing the granular source results of pion transverse momentum spectrum, elliptic flow, and HBT radii with the experimental data, we obtain the model parameters as a function of collision centrality and energy. We investigate the space-time structure and expansion velocities of the granular sources at the RHIC and LHC energies with the different centralities. Our investigations indicate that the granular source model can reproduce the experimental data of the pion transverse momentum spectra, elliptic flow, and HBT radii of the Au-Au collisions at the RHIC and the Pb-Pb collisions at the LHC. The parameters in the granular source model exhibit certain regularities for collision centrality and energy. The space-time structure and expansion velocities of the granular source are consistent with that reflected by the observables.The rest of this paper is arranged as follows. In Section2, we describe the basic ingredients of the granular source model used in this work. In Section 3, we present the pion transverse momentum spectrum, elliptic flow, and HBT results of the granular sources for the heavy ion collisions at the RHIC and LHC energies with different centralities. The regularities of the model parameters are also discussed in this section. In Section 4, we investigate the space-time structure and expansion velocities of the granular sources. Finally, the summary and discussions are given in Section 5.


## 2. Granular Source Model

In the heavy ion collisions at the RHIC top energies and the LHC energy, the created strong-coupled QGP (sQGP) systems in the central rapidity region may reach local equilibrium at a very short time and then expand rapidly along the beam direction (z-axis). Because of the random variations in the distribution of collision nucleons due to quantum fluctuations, the local equilibrium system is not uniform in the transverse plane (x-y plane) [82]. It may form many tubes along the beam direction during the subsequent fast longitudinal expansion and finally fragment into many QGP droplets with the effects of “sausage” instability, surface tension, and bulk viscosity [65, 83–85]. As a first-step idealized approximation, granular source model regards the whole source evolution as the superposition of the individual evolutions of the QGP droplets. Each droplet has a position-dependent initial velocity and evolves hydrodynamically.As in [65], we suppose the QGP droplets in the granular source initially distribute within a cylinder along z-axis by(1)dNddx0dy0dz0∝1-e-x02+y02/ΔRT2θRT-ρ0×θRz-z0,where ρ0=x02+y02 and z0 are the initial transverse and longitudinal coordinates of the droplet centers. The parameters RT and Rz describe the initial sizes of the source, and ΔRT is a shell parameter in the droplet frame [65].In [81], the Bjorken hypothesis [86] is used to describe the longitudinal velocity of droplet for the most central collisions, and the transverse velocity of droplet has a form of exponential power. Considering the longitudinal velocity of droplet varying with collision centrality, we also introduce a longitudinal power parameter, which will be determined by experimental data, to describe the longitudinal velocity phenomenologically. The initial velocities of the droplets in granular source frame are assumed as [65](2)vdi=signr0i·air0iRibi,i=1,2,3,where r0i is x0, y0, or z0 for i=1, 2, or 3, and sign(r0i) denotes the signal of r0i, which ensures an outward droplet velocity. In (2), Ri=(RT,RT,Rz), ai=(ax,ay,az), and bi=(bx,by,bz) are the magnitude and exponent parameters in x, y, and z directions, which are associated with the early thermalization and pressure gradients of the system at the fragmentation. It is convenient to use the equivalent parameters a¯T=(ax+ay)/2 and ΔaT=ax-ay instead of ax and ay. The parameters a¯T and ΔaT describe the transverse expansion and asymmetric dynamical behavior of the system at the fragmentation, respectively. For simplicity, we take bx=by=bT in calculations. The parameters bT and bz describe the coordinate dependence of exponential power in transverse and longitudinal directions.In the calculations of the hydrodynamical evolution of the droplet, we use the equation of state (EOS) of the S95p-PCE165-v0 [87], which combines the lattice QCD data at high temperature with the hadron resonance gas at low temperature. We assume systems fragment when reaching a certain local energy density and take the initial energy density of the droplets to be 2.2 GeV/fm3 for all considered collisions for simplicity [81]. The initial droplet radius is supposed satisfying a Gaussian distribution with the standard deviation σd=2.5 fm in the droplet local frame [81].The final identical pions are considered to be emitted out of the surfaces of droplets with momenta obeying the Bose-Einstein distribution in the local frame at freeze-out temperatureTf. To include the resonance decayed pions later as well as the directly produced pions at chemical freeze-out early, a wide region of Tf is considered with the probability [81](3)dPdTf∝fdire-Tchem-Tf/ΔTdir+1-fdir×e-Tchem-Tf/ΔTdec,Tchem>Tf>80MeV,where fdir is the fraction of the direct emission around the chemical freeze-out temperature Tchem; ΔTdir and ΔTdec are the temperature widths for the direct and decay emissions, respectively. In the calculations, we take fdir=0.75, ΔTdir=10MeV, and ΔTdec=90MeV as in [81]. The value of Tchem is taken to be 165 MeV as it is taken in the S95p-PCE165-v0 EOS [87].After fixing the parameters used in the calculations of hydrodynamical evolution and freeze-out temperature, the free model parameters are the three source geometry parameters (RT, ΔRT, and Rz) and the five droplet velocity parameters (a¯T, ΔaT, az, bT, and bz). They are associated with the initial size, expansion, and directional asymmetry of system and have significant influence on the observables of pion momentum spectra, elliptic flow, and HBT radii. We will combine the experimental data of these observables to investigate the parameters of the granular source as a function of the collision centrality and energy in the heavy ion collisions at the RHIC and LHC energies in next sections.


## 3. Pion Momentum Spectrum, Elliptic Flow, and HBT Results

In high energy heavy ion collisions, the invariant momentum distribution of final particles can be written in the form of a Fourier series [88, 89](4)Ed3Nd3p=12πd2NpTdpTdy1+∑n‍2vncosnϕ,where E is the energy of the particle, pT is the transverse momentum, y is the rapidity (it should not bring a mistake with coordinate from the context), and ϕ is the azimuthal angle with respect to the reaction plane. In (4), the first term on the right is the transverse momentum spectrum in the rapidity region dy, and the second harmonic coefficient v2 in the summation is called elliptic flow.At the RHIC and LHC energies, the spectators in nucleus-nucleus collisions depart from the reaction region rapidly after collision, and a very hot and dense fireball is formed in an almond shape perpendicular to the reaction plane. The particle spectra of transverse momentum at lowpT(pT≲3GeV/c) contain the information about the transverse expansion and thermal properties of the particle-emitting sources at freeze-out temperature [15–19]. By comparing the pion transverse momentum spectra of the granular sources with experimental data, we can constrain the velocity parameters a¯T and az of the granular sources.Choosingx axis on the reaction plane, elliptic flow v2 can be expressed as(5)v2pT=cos2ϕ=px2-py2pT2.Since the reaction plane orientation is hard to estimate exactly in experiment, an alternative technique for elliptic flow analysis is the measurement of the two-particle cumulant of azimuthal correlations, v222 [4, 5, 7, 8, 20–23, 90], which gives essentially the same results as the reaction-plane method [4, 7, 8, 91].In noncentral nucleus-nucleus collisions, the initial space-asymmetry of the system can bring anisotropic pressure gradients, which lead to the anisotropic transverse-momentum distributions of final particles and nonzerov2. For the granular source, the results of elliptic flow are very sensitive to the parameters ΔaT, bT, and bz [65]. The experimental data of the transverse momentum spectrum and elliptic flow impose strict constraints on the velocity parameters in the granular source model.Two-particle Hanbury-Brown-Twiss (HBT) correlation function is defined as the ratio of the two-particle momentum spectrumP(p1,p2) to the product of two single-particle momentum spectra P(p1)P(p2). It has been widely used to extract the space-time geometry and dynamic and coherence information of the particle-emitting source in high energy heavy ion collisions [92–96]. In the usual HBT analysis in high energy heavy ion collisions, the two-pion correlation functions are fitted by the Gaussian parameterized formula(6)Cqout,qside,qlong=1+λe-Rout2qout2-Rside2qside2-Rlong2qlong2,where qout, qside, and qlong are the Bertsch-Pratt variables, which denote the components of the relative momentum, q=p1-p2, in the “out” (parallel to the momentum of the particle pair in transverse plane), “side” (perpendicular to the momentum of the particle pair in transverse plane), and longitudinal directions [68–70], respectively. In (6) λ is chaoticity parameter of source, Rout, Rside, and Rlong are the HBT radii in out, side, and long directions. The results of HBT radii are related to the source geometry as well as expansion. We can finally determine the geometry parameters (RT, ΔRT, and Rz) and the velocity parameters (a¯T, ΔaT, az, bT, and bz) in the granular source model by the multiobservable analyses of the pion spectrum, elliptic flow, and HBT interferometry.In the multiobservable analyses, we choose the experimental data of the Au-Au collisions atsNN=200 GeV at the RHIC [1, 2, 7, 12] and the Pb-Pb collisions at sNN=2.76 TeV at the LHC [17, 22, 25] for determining the geometry and velocity parameters of the granular sources. These experimental data provide the identical pion pT spectrum, elliptic flow, and HBT radii simultaneously in the same centrality regions (for RHIC: 0–5%, 10–20%, and 30–50%; for LHC: 10–20% and 40–50%).In Figure1, we plot the pion transverse momentum spectra (a) and elliptic flow (b) of the granular sources for the heavy ion collisions at the RHIC and LHC energies with different centralities. The experimental data of the pion transverse momentum spectra and elliptic flow of the Au-Au collisions with 0–5%, 10–20%, and 30–50% centralities at sNN=200 GeV at the RHIC [1, 2, 7] and the Pb-Pb collisions with 10–20% and 40–50% centralities at sNN=2.76 TeV at the LHC [17, 22] are shown simultaneously. Complying with the experimental measurements, we use the rapidity cuts |y|<0.1 [1] and |y|<0.5 [16, 17] in the calculations of the pT spectra of the granular sources at the RHIC energy and the LHC energy, respectively. In the calculations of the elliptic flow of the granular sources, the pseudorapidity cuts |η|<1.0 and |η|<0.8 are adopted as the same as in the experimental analyses at the RHIC [7] and LHC [22], respectively.Figure 1
(Color online) (a) the pion transverse momentum spectra of the granular sources (solid lines) and the experimental data of negative pions in the Au-Au collisions atsNN=200 GeV [1, 2] and the Pb-Pb collisions at sNN=2.76 TeV [17]. (b) The elliptic flow of the granular source (solid lines) and the experimental data of negative pions in the Au-Au collisions at sNN=200 GeV [7] and the Pb-Pb collisions at sNN=2.76 TeV [22].
(a)
(b)As shown in Figure1, the pion transverse momentum spectra of the granular sources agree with the experimental data with different centralities at the RHIC and LHC energies simultaneously. The spectra at the LHC energy exhibit clear upwarp at pT>1.5 GeV/c as compared to those at the RHIC energy, because the average transverse velocities of the LHC sources are larger. The results of elliptic flow at the LHC energy almost match the elliptic flow results at the RHIC energy with the same and near centrality regions. The results of elliptic flow of granular sources exhibit clear centrality dependence as the same that the experimental data has [7, 22]. The v2 results decrease with increasing collision centrality. At pT>2.5 GeV/c, the granular source results of elliptic flow are a little higher than those of experimental data. It reflects the limitations of the model based on hydrodynamical evolution to describe the particles with high pT, which originate from the early stage with incomplete thermalization and have a relative small anisotropic characteristic in the transverse momentum space [97–99].In Figure2, we show the granular source results of the pion HBT radii and chaoticity parameter as a function of the transverse momentum of pion pair, kT=|p1+p2|/2, obtained by Gaussian parameterized formula fit in the longitudinally comoving system (LCMS) [94, 96]. The experimental data of STAR Au-Au collisions at sNN=200 GeV [12] and ALICE Pb-Pb collisions at sNN=2.76 TeV [25], which have the same centralities as the experimental data of pion pT spectra and v2 shown in Figure 1, are also plotted for comparing. In the HBT analyses of the granular sources for the RHIC and LHC energies, we applied the rapidity cut |y|<0.5 and the pseudorapidity cut |η|<0.8 as the same in the experimental analyses of STAR [12] and ALICE [25], respectively.Figure 2
(Color online) the results of pion HBT radii and chaoticity parameter of the granular sources (open circle) with different centralities for the highest RHIC energy (left column) and LHC energy (right column). The solid circle, star, and solid diamond symbols in the left column are the STAR data of the Au-Au collisions atsNN=200 GeV with 0–5%, 10–20%, and 30–50% centralities [12], respectively. The star and solid triangle-down symbols in the right column are the ALICE data of the Pb-Pb collisions at sNN=2.76 TeV with 10–20% and 40–50% centralities [25], respectively.From Figure2 it can be seen that the HBT radii Rout, Rside, and Rlong of the granular sources simultaneously agree with the experimental data at the RHIC and LHC energies. Both the transverse and longitudinal HBT radii increase with increasing collision centrality. At the LHC energy the results of the HBT radii Rout, Rside, and Rlong are larger than those at the RHIC energy, respectively. However, the results of the ratio of Rout/Rside are always about 1 and independent of the collision centrality and energy. In Figure 2(e), we plot the results of the chaoticity parameter λ of the granular sources and the RHIC experiments [12]. Because of absence of the experimental data of λ in the Pb-Pb collisions at the LHC [25], we plot in Figure 2(e′) only the results of the granular source. In experimental HBT measurements, many effects, such as the Coulomb interaction between the final particles, particle missing-identification, source coherence, and so forth, can influence the results of the chaoticity parameter λ [92–96]. Because these effects are not considered in the granular source model, the λ results of the granular sources are larger than the experimental data. The Coulomb interaction is greater for the particles with smaller relative momentum. It mainly influences the λ value which is related to the HBT correlation functions at zero relative momentum, and its effect on the HBT radii, which are related to the widths of the HBT correlation functions, is smaller. In experimental HBT analyses, the Coulomb interaction between final identical particles must be considered [92–96].In Table1 we present the values of the geometry and velocity parameters of the granular sources used in the multiobservable analyses. One can see that the value of RT for a certain collision centrality and energy is larger than that of Rz. So the initial geometry of the granular source is a short cylinder. However, the ratio of Rz to RT for the LHC sources becomes large for the noncentral collisions. It means the initial shape of the source changes from a short cylinder to a big cylinder with increasing collision energy. The initial transverse and longitudinal sizes of the granular sources RT and Rz increase with increasing collision centrality. The transverse shell parameter ΔRT also increases with increasing collision centrality. Because ΔRT≪RT, the shell effect is small and the initial distributions of droplets are almost volume distribution. For 10–20% centrality, the values of the geometry parameters RT and Rz of the granular source for the RHIC energy are smaller than that for the LHC energy, respectively. The QGP droplets in the granular sources initially distribute in larger transverse and longitudinal regions for the more central and higher energy collisions.Table 1
The geometry and velocity parameters of the granular sources.
Centrality
R
T
(
fm
)
Δ
R
T
(
fm
)
R
z
(
fm
)
a
¯
T
Δ
a
T
a
z
b
T
b
z
RHIC, 0–5%
5.8
0.7
3.9
0.469
0.066
0.593
0.76
0.13
RHIC, 10–20%
4.5
0.5
2.9
0.454
0.115
0.593
0.56
0.11
RHIC, 30–50%
2.5
0.3
0.5
0.437
0.156
0.593
0.37
0.06
LHC, 10–20%
6.0
0.9
5.5
0.431
0.092
0.592
0.35
0.13
LHC, 40–50%
2.5
0.4
1.8
0.407
0.131
0.590
0.23
0.03In transverse direction, the velocity parametersa¯T and bT increase with increasing collision centrality. For a fixed bT, the larger the parameter a¯T, the larger the average transverse velocity of droplet. Because the values of bT are less than one, the larger the bT, the smaller the average transverse velocity of droplet, if a¯T is fixed. In longitudinal direction, the parameter az is almost independent of collision centrality and energy. The values of the parameter bz are much smaller than those of bT, while bz increases with increasing collision centrality as bT. The large difference between the values of the transverse and longitudinal exponent parameters bT and bz and the different centrality dependence of a¯T and az reflect the different dynamical behaviors in transverse and longitudinal directions in the heavy ion collisions at the RHIC and LHC energies. In Figure 3 we plot the droplet velocities vdρ=a¯T(ρ/RT)bT and vdz=az(|z|/Rz)bz of the granular sources. One can see that the droplet transverse velocity of central collision is smaller than that of peripheral collision in the center region of the source, although the droplet transverse velocity of central collision is larger at the edge of the source. The average longitudinal velocity of droplet is larger than the average transverse velocity. At the LHC energy, the droplet longitudinal velocity for peripheral collision is almost a constant. And, it differs significantly for the different centralities. From Table 1, it can be seen that the velocity parameter ΔaT decreases with collision centrality. This leads to the increase of v2 with decreasing collision centrality.Figure 3
(Color online) the droplet transverse and longitudinal velocitiesvdρ=a¯T(ρ/RT)bT and vdz=az(|z|/Rz)bz.
(a)
(b)
(c)
(d)


## 4. Granular Source Space-Time and Expansion

In the granular source model, the results of single particlepT spectrum, elliptic flow, and HBT radii are strongly related to the source space-time and expansion properties. The successes of the granular source model in explaining the experimental data of pion pT spectrum, elliptic flow, and HBT radii in the heavy ion collisions at the RHIC and LHC inspire us to further study the granular source space-time and expansion features.In Figure4, we show the pictures of the distributions of pion source points in one granular source event with 4×105 pion pairs for the heavy ion collisions at the RHIC energy with 0–5% centrality. The z region for the x-y distributions (panels (a), (b), and (c)) is |z|<1 fm. The y region for the x-z distributions (panels (d), (e), and (f)) is |y|<1 fm. The time for the panels (a) and (d), (b) and (e), and (c) and (f) is t=4,8, and 12 fm/c, respectively. For each picture the exposure time is 0.5 fm/c. One can see that the distributions are inhomogeneous. There are separated “clumps,” which correspond to the separated droplets, in the distributions. The clump-structure exists in the whole duration of source evolution. Because the correlated two pions are taken from the same event in two-pion HBT analysis, the effect of the clump-structure on the HBT results will exist even for many-event analysis, although the clump-structure may smooth out instead of a continued distribution of the source point for many events. Because many droplets evolve simultaneously in the granular source, the source lifetime is smaller as compared to that of a continued big source, which evolves in whole and freezes out from the source surface [64, 65, 81, 100, 101]. It will be seen that the distributions of source points for many events are more volume distributions rather than surface distributions because of the contributions from the droplets in the central region of the granular source.Figure 4
The pictures of the distributions of pion source points in one granular source event for the RHIC central collision. Thez region for the panels (a), (b), and (c) is |z|<1 fm. The y region for the panels (d), (e), and (f) is |y|<1 fm. The time for the panels (a) and (d), (b) and (e), and (c) and (f) is t=4, 8, and 12 fm/c, respectively. The exposure time for these pictures is 0.5 fm/c.
(a)
(b)
(c)
(d)
(e)
(f)In Figure5, we plot the space-time distributions of the pion source points of the granular sources projected on t-rout, t-rside, and t-rlong planes, for five hundred events for the heavy ion collisions at the RHIC energy with 0–5% and 30–50% centralities. Here, the dashed lines are the average values of ri (i=out,side,long) and t for all of the source points, and the bullets are the average values of ri obtained by averaging over the same t bins. In the calculations the same rapidity cut |y|<0.5 as in the experimental HBT analyses [12] is used. We take the relative momentum cut |qi|<100MeV/c for pion pairs because most of the contributions in HBT correlation functions come from the particle pairs with small relative momenta [92–96]. One can see that the distributions for many events are smoothed. The widths of the ri-distributions for the central collision are wider than those for the peripheral collision. In side and long directions, the distributions are symmetric with respect to ri=0. However, one can observe a time-increased asymmetry for the distributions in out direction. It is because the coordinate-dependent source transverse expansion boosts particle momenta along the direction out of the source, and this coordinate-momentum correlation leads the result that the average angle between particle momentum and emitting coordinate trends to be smaller than isotropic emission. For the central and peripheral collisions, the average values of rout are 2.52 and 2.78 fm, respectively. The coordinate-momentum correlation for the peripheral is larger than that for the central collision. This is because the peripheral collision has a larger source transverse velocity as compared to the central collision (which will be seen in Figure 7(a)).Figure 5
(Color online) the source-point distributions of final identical pions inri-t plane (i=out,side,long), for the granular sources of the central and peripheral collisions at the RHIC energy. The dashed lines are the average values of ri and t for all of the source points. The bullets are the average values of ri obtained from the same t bins.
(a)
(b)
(c)
(d)
(e)
(f)In Figure6 we compare the space-time distributions of the pion source points of the granular sources for the collisions at the RHIC and LHC energies with the same centrality, 10–20%. As in Figure 5 the dashed lines are for the average values of ri and t for all of the source points, and the bullets are the average values of ri obtained from the same t bins. We take the relative momentum cut |qi|<100MeV/c in the calculations. The rapidity and pseudorapidity cuts |y|<0.5 and |η|<0.8 are taken for the RHIC and LHC collisions, respectively, as in the experimental HBT analyses [12, 25]. It can be seen that the widths of the rout, rside, and rlong distributions for the LHC source are wider as compared to those for the RHIC source with the same collision centrality. In side and long directions, the distributions are symmetric with respect to ri=0. But the distributions in out direction are asymmetric with respect to rout=0 because of the coordinate-momentum correlations arising from the source transverse expansion. The asymmetry effect is larger for the granular source at the LHC energy (〈rout〉=3.59 fm) than that at the RHIC energy (〈rout〉=2.64 fm). This is because the granular source for the LHC collision has a larger transverse velocity as compared to that for the RHIC collision.Figure 6
(Color online) the source-point distributions of final identical pions inri-t plane (i=out,side,long), for the granular sources of the RHIC and LHC collisions with 10–20% centrality. The dashed lines are the average values of ri and t for all of the source points. The bullets are the average values of ri obtained from the same t bins.
(a)
(b)
(c)
(d)
(e)
(f)Figure 7
(Color online) ((a)–(d)) the average transverse and longitudinal velocities of the pion source points,v¯ST and v¯SL, versus the emission time of the granular sources. ((e) and (f)) The average transverse and longitudinal droplet velocities v¯dT and v¯dL of the granular sources for the RHIC and LHC collisions.
(a)
RHIC
(b)
RHIC
(c)
LHC
(d)
LHC
(e)
LHC
(f)
RHICFigures7(a) and 7(b) display the average transverse and longitudinal velocities of the granular sources as a function of time at the RHIC energy. Figures 7(c) and 7(d) display the average transverse and longitudinal velocities of the granular sources as a function of time at the LHC energy. It can be seen that the average velocities increase with increasing collision centrality at both the energies. The larger average transverse velocities of sources at the LHC energy are consistent with the results that the pT spectra at the LHC energy exhibit upwarp at larger pT as compared to those at the RHIC energy (see Figure 1(a)). The differences of the transverse velocities for different centralities become small at the higher energy, and the differences of the longitudinal velocities for different centralities are larger at the LHC energy. In Figures 7(e) and 7(f) we present the average transverse and longitudinal droplet velocities for the granular sources at the RHIC and LHC energies. The average transverse and longitudinal droplet velocities increase with the decreasing collision centrality for both the energies. In our granular source model, the droplet evolution in the local frame is independent of the collision energy and centrality. However, the different droplet velocities lead to the difference between the average emission time in the source center-of-mass frame because of the different Lorentz time delays. The larger the droplet velocity the larger the emission time. For example, the average emission time of the granular sources for the RHIC central and peripheral collisions is 8.39 and 9.11 fm/c, respectively (see Figure 5). The average emission time for the granular source for LHC peripheral collision is the largest because the average droplet velocity is the largest in this case (see Figure 7(e)).In the two-pion interferometry in high energy heavy ion collisions, the difference between the transverse HBT radiiRout and Rside satisfies [94, 102, 103](7)Rout2kT-Rside2kT≈βTt~2-2βTr~outt~kT,where βT=|p1T+p2T|/(E1+E2) is the transverse velocity of the pair; t~=t-〈t〉 and r~out=rout-〈rout〉 are the deviations of source time and coordinate rout from their averages, respectively.In Figure8, we plot 〈βT2t~2〉, 2〈βTr~outt~〉, and the average transverse and longitudinal velocities of the pion pair as a function of kT. It can be seen that 〈βT2t~2〉 and 2〈βTr~outt~〉 increase as kT increases. At small kT, the values of 〈βT2t~2〉 are larger than those of 2〈βTr~outt~〉. But at large kT the values of 2〈βTr~outt~〉 are close and are even larger than the corresponding results of 〈βT2t~2〉. Both 〈βT2t~2〉 and 2〈βTr~outt~〉 decrease with collision centrality. The near values of 〈βT2t~2〉 and 2〈βTr~outt~〉 at a fixed kT lead to the HBT results Rout/Rside≈1 for the granular sources. From Figures 8(e) and 8(f) one can see that the average transverse velocities of the pair at the RHIC energy are higher than those at the LHC energy at large kT. However, the results of the average longitudinal velocities are opposite. For a fixed kT, a larger βL means a larger kL and therefore a larger Ek (Ek2=kT2+kL2+mk2) and smaller βT. It will be seen that the reason for the larger β¯L at large kT at the LHC energy is the larger longitudinal expansion of the granular source at the LHC energy as compared to that at the RHIC energy (see Figures 9(c) and 9(d)).Figure 8
(Color online) the granular sources〈βT2t~2〉, 2〈βTr~outt~〉, β¯T, and β¯L versus kT for the RHIC and LHC collisions.
(a)
RHIC
(b)
LHC
(c)
RHIC
(d)
LHC
(e)
RHIC
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LHCFigure 9
(Color online) thekT dependence of the average emission time t¯=〈t〉, time variance root Δt=[〈t~2〉]1/2, and average source transverse and longitudinal velocities v¯ST and v¯SL for the granular sources at the RHIC and LHC energies.
(a)
RHIC
(b)
LHC
(c)
RHIC
(d)
LHCIn Figure9, we show the average emission time t¯=〈t〉(kT), the time variance root Δt=[〈t~2〉]1/2(kT), and the average source transverse and longitudinal velocities v¯ST(kT) and v¯SL(kT) for the granular sources at the RHIC and LHC energies. The error bars for the results are statistical errors. One can see that the average emission time decreases with decreasing collision centrality and increases with increasing collision energy. It is because the average droplet velocity increases with decreasing collision centrality and increasing collision energy. The larger the droplet velocity, the larger time delay is. The values of Δt, which is also referred as the source lifetime, are almost independent of kT and smaller as compared to the values of t¯. For a fixed kT, the average source velocities increase with decreasing collision centrality. At large kT, the average longitudinal velocities of the granular sources at the LHC energy are larger than the corresponding values at the RHIC energy. It is because the pion pairs with larger kT at the LHC energy correspond to a larger average emission time in the source center-of-mass frame (see Figure 9(b)) and therefore have larger average longitudinal source velocities as compared to those at the RHIC energy (see Figures 7(b) and 7(d)). This result is consistent with the results of β¯L shown in Figures 8(e) and 8(f). The large longitudinal velocity of the granular source at the LHC energy boosts strongly the pair momentum of the particles with almost the same emission direction and leads to the large β¯L results.Finally, it should be noted that the emission time mentioned in the paper is the time recorded from the initial state of the granular source. The real emission time from the beginning of collision should also include the system preequilibrium timeτ0 and the breakup time t0, which should be different for different collision energies [81] and centralities. However, the lifetime of the granular source is independent of the time original point. The small source lifetime is a character of the granular source [64, 65, 81, 100, 101].


## 5. Summary and Discussion

We systemically investigate the pion transverse momentum spectrum, elliptic flow, and HBT interferometry in the granular source model for the heavy ion collisions at the RHIC highest energy and the LHC energy. The centrality and energy dependence of the observables at the two energies are examined. By comparing the granular source results with the experimental data of the Au-Au collisions atsNN=200 GeV at the RHIC and the Pb-Pb collisions at sNN=2.76 TeV at the LHC with different collision centralities, we investigate the geometry and velocity parameters in the granular source model as a function of collision centrality and energy. The space-time structure and expansion velocities of the granular sources at the RHIC and LHC energies with different centralities are examined. Our investigations indicate that the granular source model can well reproduce the experimental data of pion transverse momentum spectra, elliptic flow, and HBT radii in the Au-Au collisions at sNN=200 GeV with 0–5%, 10–20%, and 30–50% centralities [1, 2, 7, 12] and in the Pb-Pb collisions at sNN=2.76 TeV with 10–20% and 40–50% centralities [17, 22, 25]. The experimental data of pion momentum spectra, elliptic flow, and HBT radii impose very strict constraints on the parameters in the granular source model. They exhibit certain regularities for collision centrality and energy. The space-time structure and expansion velocities of the granular source are consistent with that reflected by the observables.In the granular source model we assume that the system created in the ultrarelativistic heavy ion collisions breaks up and forms the granular source of QGP droplets due to the dynamical instability in the fast expansion at the early stage and the surface tension of the strongly coupled QGP. We use ideal hydrodynamics to describe the droplet evolution and assume a Gaussian distribution for the droplet radius for simplicity. Because many droplets evolve simultaneously, the source lifetime is smaller as compared to that of a continued big source, which evolves in whole and freezes out from the source surface. For the granular source, the distribution of particle-emitting points of single event presents a clump-structure during the source evolution. However, the distribution of source points for many events is continued and presents a volume distribution because of the contribution from the droplets in the central region of the source. The short source lifetime, clump-structure of source points distribution of single event, and volume distribution of source points for many events are the characters of the granular source.The investigations for the granular source parameters indicate that the QGP droplets initially distribute in larger transverse and longitudinal regions for the more central and higher energy collisions. So, the distribution width of source points increases with increasing collision centrality and energy. In transverse direction, the droplet velocity of central collision is smaller than that of peripheral collision in the center region of the source, although the droplet transverse velocity of central collision is larger at the edge of the source. The average longitudinal velocity of droplet is larger than the average transverse velocity of droplet. The droplet longitudinal velocity increases with decreasing collision centrality. The larger droplet velocities in the peripheral collisions lead to larger average velocities of source. Because the difference of the droplet transverse velocities in and out of reaction plane decreases with collision centrality, the elliptic flow decreases with collision centrality. In HBT interferometry, the difference between the transverse HBT radiiRout and Rside is related to the transverse velocity of particle pair, the source lifetime, and the space-time correlation of source points. Both the quantities 〈βTt~2〉 and 2〈βTr~outt~〉 increase with the decreasing collision centrality and increasing transverse momentum of the pair, kT. However, the difference between the two quantities for a fixed kT is approximately equal to zero. This leads to the results of Rout(kT)≈Rside(kT).While one may argue on the details and the number of parameters used in the granular source model, the consistent explanation of a large number of measured one-particle and two-particle correlation quantities suggests that this model description captures some fundamental features of the space-time dynamics. After all, it is a challenge to a model to explain the experimental data of the momentum spectra, elliptic flow, and HBT radii simultaneously at the RHIC and LHC energies, and the final criterion for a model is experiments. It will be of interest to improve the granular source model and investigate the effects of QGP viscosity and droplet interaction on the granular source parameters. On the other hand, the studies of the forming mechanism and signals of the granular source will be of interest.



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*Source: 846154-2015-06-10.xml*