# A Description of Pseudorapidity Distributions of Charged Particles Produced in Au+Au Collisions at RHIC Energies

Authors: Z. J. Jiang; Dongfang Xu; Yan Huang
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

In heavy ion collisions, charged particles come from two parts: the hot and dense matter and the leading particles. In this paper, the hot and dense matter is assumed to expand according to the hydrodynamic model including phase transition and decouples into particles via the prescription of Cooper-Frye. The leading particles are as usual supposed to have Gaussian rapidity distributions with the number equaling that of participants. The investigations of this paper show that, unlike low energy situations, the leading particles are essential in describing the pseudorapidity distributions of charged particles produced in high energy heavy ion collisions. This might be due to the different transparencies of nuclei at different energies.


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## Body

## 1. Introduction

The BNL Relativistic Heavy Ion Collider (RHIC) accelerates nuclei up to the center-of-mass energies from a dozen GeV to 200 GeV per nucleon. In the past decade, the measurements from such collisions have triggered an extensive research for the properties of matter at extreme conditions of very high temperature and energy densities [1–33]. One of the most important achievements from such research is the discovery that the matter created in nucleus-nucleus collisions at RHIC energies is in the state of strongly coupled quark-gluon plasma (sQGP) exhibiting a clear collective behavior nearly like a perfect fluid with very low viscosity [10–33].The best approach for describing the space-time evolution of fluid-like sQGP is the relativistic hydrodynamics. However, owing to the formidable complexities of hydrodynamic equations, the most analytical work so far is mainly limited to 1+1 expansion for a perfect fluid with simple equation of state, which can be found as an important application in the analysis of the pseudorapidity distributions of charged particles in high energy physics. In this paper, combining the effects of leading particles, we will discuss such distributions in the framework of 1+1 hydrodynamic model including phase transition [10].


## 2. A Brief Introduction to the Model

Here, for the purpose of completeness and applications, we will list the key ingredients of the hydrodynamic model [10].(
1
) The expansion of fluid is subject to the conservation of energy and momentum. This is reflected in continuity equation(1)∂Tμν∂xν=0,μ,ν=0,1,where Tμν is the energy-momentum tensor. For a perfect fluid(2)Tμν=ε+puμuν-pgμν,where(3)uμ=u0,u1=cosh⁡yF,sinh⁡yF,uμuμ=1,is the 4-velocity of fluid and yF is its rapidity. ε and p in (2) are the energy density and pressure of fluid, which meet the thermodynamical relations(4)ε+p=Ts,dε=Tds,dp=sdT,where T and s are the temperature and entropy density of fluid, respectively. To close (1), another relation, namely, the equation of state(5)dpdε=sdTTds=cs2,is needed, where cs is the sound speed of fluid, which takes different values in sQGP and in hadronic phase.(
2
) Project (1) to the direction of uμ and the direction perpendicular to uμ, respectively. This leads to equations(6)∂suν∂xν=0,(7)∂Tsinh⁡yF∂t+∂Tcosh⁡yF∂z=0.Equation (6) is the continuity equation for entropy conservation. Equation (7) means the existence of a scalar function ϕ satisfying relations(8)∂ϕ∂t=Tcosh⁡yF,∂ϕ∂z=-Tsinh⁡yF.From ϕ and Legendre transformation, Khalatnikov potential χ is introduced via relation(9)χ=ϕ-tTcosh⁡yF+zTsinh⁡yF.In terms of χ, the variables t and z can be expressed as(10)t=eθT0∂χ∂θcosh⁡yF+∂χ∂yFsinh⁡yF,z=eθT0∂χ∂θsinh⁡yF+∂χ∂yFcosh⁡yF,where T0 is the initial temperature of fluid and θ=ln⁡T0/T. Through the above equations, the coordinate base of t,z is transformed to that of θ,yF, and (6) is translated into the so-called telegraphy equation(11)∂2χ∂θ2-2β∂χ∂θ-1cs2∂2χ∂yF2=0,β=1-cs22cs2.(
3
) Along with the expansions of matter created in collisions, it becomes cooler and cooler. As its temperature drops from the initial T0 to the critical Tc, phase transition occurs. The matter transforms from sQGP state to hadronic state. The produced hadrons are initially in the violent and frequent collisions. The major part of these collisions is inelastic. Hence, the abundances of identified hadrons are changing. Furthermore, the mean free paths of these primary hadrons are very short. The movement of them is still like that of a fluid meeting (11) with only difference being the value of cs. In sQGP, cs=c0=1/3, which is the sound speed of a massless perfect fluid, being the maximum of cs. In the hadronic state, cs=ch<c0. At the point of phase transition, that is, as T=Tc, cs is discontinuous.(
4
) The solution of (11) for the sector of sQGP is [10](12)χ0θ,yF=q0c02eβ0θI0β0c0y02θ-yF2,where q0 is a constant determined by tuning the theoretical results to experimental data and I0 is the 0th-order modified Bessel function of the first kind, and(13)β0=1-c022c02=1,y0θ=θc0.In the sector of hadrons, the solution of (11) is [10](14)χhθ,yF=q0c02BθI0λθ,yF,where(15)Bθ=eβhθ-θc+β0θc,λθ,yF=βhchyh2θ-yF2,βh=1-ch22ch2,yhθ=θ-θcch+θcc0,θc=ln⁡T0Tc.


## 3. The Pseudorapidity Distributions of Charged Particles

(1
) The Invariant Multiplicity Distributions of Charged Particles Frozen Out from sQGP. From Khalatnikov potential χ, the rapidity distributions of charged particles frozen out from fluid-like sQGP read [34](16)dNsQGPdyF=q0c02Abcosh⁡ydzdyF-sinh⁡ydtdyF,where Ab is the area of overlap region of collisions, being the function of impact parameter b or centrality cuts. Inserting (10) into the above equation, the part in the round brackets becomes(17)cosh⁡ydzdyF-sinh⁡ydtdyF=1Tc2∂∂θχ+∂χ∂θcosh⁡y-yF-1T∂∂yFχ+∂χ∂θsinh⁡y-yF.With the expansions of hadronic matter, it continues becoming cooler. According to the prescription of Cooper-Frye [34], as the temperature drops to the freeze-out temperature TFO, the inelastic collisions among hadrons cease. The yields of identified hadrons remain unchanged becoming the measured results in experiments. The invariant multiplicity distributions of charged particles equal [10, 15, 34](18)d2NsQGP2πpTdydpT=12π3∫dNsQGPdyFmTcosh⁡y-yFexpmTcosh⁡y-yF-μB/T+δT=TFOdyF,where mT=m2+pT2 is the transverse mass of produced charged particle with rest mass m. μB in (18) is the baryochemical potential. For Fermi charged particles, δ=1 in the denominator of (18), and for Bosons, δ=-1. The meaning of (18) is evident. It is the convolution of dNsQGP/dyF with the energy of the charged particles in the state with temperature T.The integral interval ofyF in (18) is -yhθf,yhθf. The integrand is evaluated with T=TFO. At this moment, the fluid freezes out into the charged particles. Replacing χ in (17) by χh of (14), it becomes(19)cosh⁡ydzdyF-sinh⁡ydtdyFT=TFO=1TFOβhch2BθFOSθFO,yFsinh⁡y-yF+CθFO,yFcosh⁡y-yF,where(20)SθFO,yF=βhyFλθFO,yFβhchyhθFOλθFO,yFI0λθFO,yF+βh+1βh-2βhchyhθFOλ2θFO,yFI1λθFO,yF,CθFO,yF=βh+1βh+βhchyhθFO2λ2θFO,yFI0λθFO,yF+1λθFO,yFyhθFOch+1-2βhchyhθFO2λ2θFO,yFI1λθFO,yF,where I1 is the 1st-order modified Bessel function of the first kind.(2
) The Invariant Multiplicity Distributions of Leading Particles. Investigations have shown that the leading particles are formed outside the overlap region of collisions [35, 36]. The generation and movement of them are therefore beyond the scope of hydrodynamic description and should be treated separately.In our previous work [24–26], we once argued that the rapidity distributions of leading particles take the Gaussian form(21)dNLeadb,sNN,ydy=NLeadb,sNN2πσexp⁡-y-y0b,sNN22σ2,where y0b,sNN and σ are, respectively, the central position and width of distributions. It can be expected that y0b,sNN should increase with increasing energies and centrality cuts, while σ should not, at least not apparently, depend on the energies, centrality cuts, and even colliding systems. The specific values of them can be determined by comparing the theoretical results with experimental data. NLeadb,sNN in (21) represents the number of leading particles, which, for an identical nucleus-nucleus collision, equals half of the number of participants.The investigations have shown that [37], for certain rapidity, the invariant multiplicity distributions of leading particles possess the form(22)d2NLead2πpTdydpT∝exp⁡-apT2,where a is a constant. Then, as a function of rapidity, the invariant multiplicity distributions of leading particles can be written as(23)d2NLead2πpTdydpT=dNLeaddyaπexp⁡-apT2,which is normalized to NLead.(3
) The Pseudorapidity Distributions of Charged Particles. Writing invariant multiplicity distributions in terms of pseudorapidity, we have(24)d2N2πpTdηdpT=1-m2mT2cosh2yd2N2πpTdydpT,where(25)d2N2πpTdydpT=d2NsQGP2πpTdydpT+d2NLead2πpTdydpT.To fulfill the transformation of (24), another relation(26)y=12ln⁡pT2cosh2⁡η+m2+pTsinh⁡ηpT2cosh2⁡η+m2-pTsinh⁡ηis in order.Substituting (18) and (23) into (24) and carrying out the integration of pT, we can get the pseudorapidity distributions of charged particles produced in high energy heavy ion collisions. Figures 1 and 2 show such distributions in Au+Au collisions at sNN=200 and 62.4 GeV, respectively. The solid dots in the figures are the experimental measurements [5]. The dashed, dashed-dotted, and dotted curves are, respectively, the contribution from pions, kaons, and protons got from the hydrodynamic result of (18). The dotted-star curves are the components of leading particles obtained from (23). The solid curves are the sums of the four types of curves. χ2/NDF for each curve is listed in Table 1. It can be seen that the combined contribution from both hydrodynamics and leading particles matches up well with experimental data.Table 1
χ
2
/
N
D
F, initial temperature T0, and central position y0 in different centrality Au+Au collisions at sNN=200, 62.4, and 19.6 GeV, respectively.
Centrality cuts (%)
0–3
3–6
6–10
10–15
15–20
20–25
25–30
30–35
35–40
40–45
45–50
χ
2
/
N
D
F
200 GeV
0.496
0.693
0.402
0.260
0.310
0.217
0.354
0.151
0.161
0.123
0.089
62.4 GeV
0.696
0.519
0.300
0.192
0.188
0.079
0.266
0.376
0.444
0.481
0.405
19.6 GeV
0.816
0.533
0.359
0.614
0.802
0.309
0.559
0.496
0.229
0.577
---
T
0 (GeV)
200 GeV
0.950
0.949
0.948
0.947
0.945
0.941
0.922
0.909
0.885
0.874
0.860
62.4 GeV
0.680
0.676
0.675
0.671
0.660
0.650
0.623
0.607
0.599
0.593
0.585
19.6 GeV
0.551
0.549
0.547
0.543
0.539
0.533
0.529
0.518
0.510
0.493
---
y
0
200 GeV
2.86
3.03
3.13
3.23
3.46
3.54
3.55
3.57
3.58
3.59
3.60
62.4 GeV
2.80
2.88
3.05
3.21
3.35
3.39
3.42
3.46
3.50
3.51
3.54Figure 1
The pseudorapidity distributions of charged particles produced in different centrality Au+Au collisions atsNN=200 GeV. The solid dots are the experimental measurements [5]. The dashed, dashed-dotted, and dotted curves are, respectively, the contribution from pions, kaons, and protons got from the hydrodynamic result of (18). The dotted-star curves are the components of leading particles obtained from (23). The solid curves are the sums of the four types of curves.Figure 2
The pseudorapidity distributions of charged particles produced in different centrality Au+Au collisions atsNN=62.4 GeV. The solid dots are the experimental measurements [5]. The dashed, dashed-dotted, and dotted curves are, respectively, the contribution from pions, kaons, and protons got from the hydrodynamic result of (18). The dotted-star curves are the components of leading particles obtained from (23). The solid curves are the sums of the four types of curves.Experiments have shown that the overwhelming majority of charged particles produced in Au+Au collisions atsNN=200 GeV consists of pions, kaons, and protons with proportions of about 84%, 12%, and 4%, respectively [38], which are roughly independent of energies, centrality cuts, and colliding systems. In calculations, the ratios of these three kinds of particles take about the same as these values. ch in (15) takes the values of ch=0.45 and 0.42 for sNN=200 and 62.4 GeV from the investigations of [15, 39–41]. Tc in (15) takes the well-recognized value of Tc=180 MeV. The freeze-out temperature TFO takes the values of TFO=120MeV from the studies of [6], which also shows that the baryochemical potential μB in (18) is about equal to 20 and 50 MeV for sNN=200 and 62.4 GeV, respectively. For the most central collisions at these two different energies, T0 in (15) takes the values of T0=0.95 and 0.68 GeV referring to those given in [15]. This allows us to determine the constant q0 in (16) to be q0=7.38×10-4, 6.01×10-4, and 4.50×10-3 for pions, kaons, and protons, respectively. Keeping q0 unchanged, T0 is fixed for the rest centrality cuts by making theoretical results fit in with experimental data. The results are listed in Table 1. It can be seen that T0 decreases slowly with increasing centralities especially in the first four cuts. Table 1 also lists the central position y0 in (21). As addressed above, it increases with increasing energies and centralities. The width parameter σ in (21) takes the value of a constant of σ=0.90, being independent of energies and centrality cuts. The parameter a in (23) takes the value of a=0.92 for the two different energies.Figure3 shows the pseudorapidity distributions of charged particles produced in Au+Au collisions at sNN=19.6 GeV. χ2/NDF for each curve is listed in Table 1. The meanings of different types of curves are the same as those in Figures 1 and 2. It can be seen that, in the absence of leading particles, the hydrodynamics alone can give a good description to the experimental observations. This is different from Figures 1 and 2, where leading particles are essential in fitting experimental data. This difference might be caused by the different transparencies of nuclei in different energies. As the analyses given in [42], in central Au+Au collisions at sNN=200 GeV, the leading particles are located at about y0=2.91. This position is far away from the mid-rapidity region, where, relative to the low yields of charged particles frozen out from sQGP, the effect of leading particles is evident which should be considered separately. On the contrary, in case of Au+Au collisions at sNN=19.6 GeV, y0=1.28. This position is so close to the mid-rapidity region that the effect of leading particles is hidden by the large yields of charged particles generated from the freeze-out of sQGP. Therefore, there is no need to consider the contribution of leading particles separately.Figure 3
The pseudorapidity distributions of charged particles produced in different centrality Au+Au collisions atsNN=19.6 GeV. The solid dots are the experimental measurements [5]. The dashed, dashed-dotted, and dotted curves are, respectively, the contribution from pions, kaons, and protons got from the hydrodynamic results of (18). The solid curves are the sums of the three types of curves.In drawing Figure3, T0 takes the values as those listed in Table 1. ch=0.40 and μB=210MeV. The other parameters, such as Tc, TFO, and q0, are the same as those used in drawing Figures 1 and 2.


## 4. Conclusions

By taking into consideration the effect of leading particles, the hydrodynamic model incorporating the phase transition is used to analyze the pseudorapidity distributions of charged particles produced in Au+Au collisions at RHIC energies.The hydrodynamic model contains rich information about transport coefficients of sQGP, such as the sound speedc0 in sQGP, the sound speed ch in hadronic phase, the phase transition temperature Tc, the chemical freeze-out temperature TFO, the baryochemical potential μB, and the initial temperature T0. With the exception of T0, the other five coefficients take the values either from the well-known theoretical results or from experimental measurements. As for T0, there are no widely accepted results so far. In our calculations, T0 in the most central Au+Au collisions at sNN=200 and 62.4 GeV takes the value referring to those given by other investigations, which enables us to ascertain the constant q0 in (16). In the rest centrality cuts and in Au+Au collisions at sNN=19.6 GeV, T0 is determined by maintaining q0 unchanged and comparing the theoretical results with experimental data.The leading particles, by conventional definition, are the particles carrying on the quantum numbers of colliding nucleons and taking away the most part of incident energy. They are separately in projectile and target fragmentation region. The present investigations show that the importance of leading particles in describing the pseudorapidity distributions of charged particles produced in heavy ion collisions is related to the incident energy. At high energy, owing to the high transparency of nuclei, the contribution of leading particles is evident and indispensable, while, at low energy, as a result of poor transparency of nuclei the effect of leading particles is integrated with the results of freeze-out of sQCD. It does not need to be dealt with separately.



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