Abstract
At very initial stage of relativistic heavy ion collisions a wave of quark-gluon matter is produced from the break-up of the strong color electric field and then thermalizes at a short time scale (1 fm/c). However, the quark-gluon plasma (QGP) system is far out of chemical equilibrium, especially for the heavy quarks which are supposed to reach chemical equilibrium much late. In this paper a continuing quark production picture for strongly interacting QGP system is derived, using the quark number susceptibilities and the equation of state; both of them are from the results calculated by the Wuppertal-Budapest lattice QCD collaboration. We find that the densities of light quarks increase by 75% from the temperature MeV to MeV, while the density of strange quark annihilates by 18% in the temperature region. We also offer a discussion on how this late production of quarks affects the final charge-charge correlations.
1. Introduction
The ultrarelativistic heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) provide a unique opportunity for scientists to investigate the properties of the quark-gluon plasma (QGP) in laboratory, which is believed to be a form of early universe after the big bang. In these collisions, the heavy nuclei can be depicted in the center-of-mass frame (CMF) of system as two Lorenz contracted pancakes. Most of the baryons are expected to be carried away by the receding pancakes, while in the region of the collisions large energy is deposited in the form of quarks and gluons [1, 2]. A large amount of quark-antiquark pairs will be produced from the break-up of the strong color electric field established between two receding pancakes, when the energy in the field is larger than the pair production threshold [3–5]. The collisions between the newborn constituents are violent enough to drive initial collision products to a new deconfined quark-gluon plasma at a short time scale, (~1 fm/c) [6]. The initial QGP is gluon rich and thermalizes very rapidly. But complete chemical equilibrium will take much longer time. For quarks, the rate of chemical equilibrium is much lower than that of gluon, because the relevant QCD cross sections are 2-3 times smaller [7, 8]. Another process of late production of quarks is the strange-quark production, which will take place up to 5–10 fm/c when the local temperature drops to near 160 MeV [9].
The quark late stage production can place a significant impact on the correlations between charges and be investigated through the balance functions [10]. The late production of quarks causes the widths of balance functions smaller, because the later the charges produce, the closer the collective flow will constrain charges in momentum space. In [11], Pratt extends the theory of balance function to the scenario involving two-wave production of quarks. The first wave takes place in the first fm/c of the collisions when gluons thermalize into quark-gluon plasma (QGP), and the second wave ensues at hadronization, 5–10 fm/c, after a roughly isentropic expansion that roughly conserves the number of quarks [11]. However this canonical picture of the evolution of the QGP in high-energy heavy-ion collisions is still oversimplified. In principle, the number of quark-antiquark pairs could adjust continually up to the time when full chemical equilibrium reaches. This time may occur late; for instance, for strangeness pairs, the saturation of number happens at 5–10 fm/c after collision [9].
The experimental results available so far show that the hot QCD matter exhibits robust flow phenomena, which are well and consistently described by near-ideal relativistic hydrodynamics [1, 12]. This arouses the suspicion of applicability of perturbative QCD to the hot and dense matter at the region of temperature . To account for nonperturbative effect, lattice QCD provides a reliable method. Numerical simulations of QCD thermodynamics on the lattice are reaching unprecedented level of accuracy. The Wuppertal-Budapest lattice QCD collaboration computed the QCD equation of state with 2 + 1 staggered flavor [13]. Their results of the large trace anomaly hints at a strongly interacting system instead of the weakly interacting parton gas system. For the strongly interacting system, it is a big challenge to calculate the number densities of quarks directly. However, lattice QCD provide a technology to calculate the quark number susceptibilities with high accuracy [14]. In the classical weakly interacting gas limit, the diagonal quadratic susceptibilities are related to the number densities of quarks. This provides a possibility to infer the chemical composition of quark-gluon matter at strongly interacting region. Although the interaction between constituents in the quark-gluon system at temperature is not weak enough, the hard thermal loop (HTL) perturbation theory motivates and justifies a picture of weakly interacting quasiparticle at this region of temperature. In quasiparticle model, the interaction between constituents is encoded in the quasiparticle masses and the quasiparticles behave like a free gas of massive constituents. Plumari et al. further obtained that quasiparticle model can reproduce the susceptibilities by using the quark and gluon masses that come from the HTL formalism [15].
In this paper, we try to use the susceptibilities calculated from lattice QCD to investigate the evolution of the number densities of quarks. For the expansion of quark-gluon matter, the combination of boost invariant longitudinal expansion and cylindrical symmetric transverse expansion is considered. The equation of state (EOS) from lattice QCD results [13] is used in our hydrodynamic calculations. The rest of this paper is organized as follows. In Section 2, we will describe the calculation method used in this work. In Section 3, we will present our results for a continuing, late quark production picture and make discussions. Finally, we will give the summary and conclusions in Section 4.
2. Calculation Method
A system in the thermal equilibrium (for grand-canonical ensemble) is characterized by its partitions function, where is the Hamiltonian of the system and and are the conserved charges and the chemical potentials, respectively. In case of three flavor quarks, these are strangeness, baryon number, and electric charge or, equivalently, the three quark flavors: up, down, and strange. In this work, we focus on the quark number susceptibilities. So denotes the (net) number of quarks and is the corresponding chemical potential. The mean and the covariance are then expressed in terms of derivatives of the partition function with respect to the appropriate chemical potentials, where . The susceptibilities is introduced by The diagonal susceptibilities, , are a measure of the fluctuation of the system, whereas the off-diagonal susceptibilities, , characterize the correlation between the conserved charges and .
We assume the quark-gluon matter can be described by the weakly interacting quark-gluon quasiparticle gas and further neglect small correlations due to quantum statistics which are of the order of one percent for the temperatures under consideration [16, 17]. Thus, the fluctuation of , the (net) number of quarks, is given by Since and for an ideal gas, is the sum of the number of quarks and antiquarks for flavor . In case of flavor , can be for the light quark and for the strange quark. By substituting (4) to (2), we can relate to the corresponding diagonal susceptibilities defined by (3): As mentioned above, the susceptibilities, , are investigated a lot in lattice QCD simulation. In [14], Borsányi et al. computed the at the region of temperature 125–400 MeV for flavor with the physical quark mass and performed the continuum extrapolation to remove artificial cut-off effect. Their results are presented in Figure 1.
To derive the evolution of chemical composition, we need to overcome the problem with how to describe the expansion of quark-gluon system. In the heavy ion collisions at the RHIC and the LHC, the amount of net charges is negligibly small in the central rapidity of the collisions. For simplicity, we use the hydrodynamics with zero net-baryon densities to describe the system expansion, which means neglecting the effect of those small net charges on the temperature and velocity fields of hydrodynamics. We assume the longitudinal boost invariant expansion and combine with cylindrically symmetric expansion in transverse direction, which is the Bjorken picture [18] in ultrarelativistic heavy-ion collisions. In this scheme, we can have two related equations written in cylindrical coordinates which govern the expansion of a neutral system (free of net charge) of QGP:
where we have two independent variables, and , and three dependent unknown quantities, temperature , entropy density , and transverse flow velocity . To solve these three unknown quantities, we need to supplement EOS.
We implement the EOS in [13], where the related quantities are computed with the same physics setup as ones which are used to calculate the susceptibilities [14]. In [13] a parametrization of the trace anomaly as a function of the temperature is given by where dimensionless variable is defined as . The parameters are listed in Table 1. The consistency between the calculated results and the parametrization is shown in Figure 2. From this parametrization the normalized pressure can be obtained by the definite integral:
With the help of the definition of the trace anomaly, , we can have energy density, , and the squared velocity of sound, : The numerical results of the squared velocity of sound and the pressure are presented in Figures 3(a) and 3(b).
(a)
(b)
In Figures 4(a) and 4(b) we present the hydrodynamic solution of the transverse expansion (at ) for the QGP system with Bjorken cylinder. In the calculation, we assume the initial time and take the following initial temperature and velocity profiles as in [19]: where and are the initial temperature and transverse radius of the system. They are taken to be 400 MeV and 7 fm in our calculations, respectively.
(a)
(b)
Once the evolution the quark-gluon system is available, we can derive the evolution of the rapidity densities of quark numbers, . For the Bjorken cylinder, we have from (5) where is Lorenz factor for transverse flow. In this paper, we only care about the chemical evolution of QGP, so the integral range of is determined by for each time. Here is selected to be 150 MeV as in [13].
3. Results
We first show in Figure 5 the rapidity densities of quark numbers, , for light and strange quarks. The time dependence of the number density for the light quark deviates from that for the strange quark at very early stage, when starts to deviate from slightly. arises with time rapidly from to 14.6 fm/c by 75%, while drops by 18% at that time after a rise during the first 1.5 fm/c. This observable is cut-off at fm/c, because at the time the temperature reaches the critical temperature MeV defined by Wuppertal-Budapest group [13].
The ratio of number of strange quark to the number of light quark is an important indication of chemical composition, which can be calculated using the measured ratio, in the experiments: The approximation is valid for the system which most comprises of , and has the same amount of antiparticles as particles. Our results on are shown in Figure 6. The value of is 0.22 at MeV when the quark-gluon system terminates in our calculation. This value is about twice bigger than the values 0.09 and 0.1, calculated from the measured in the experiments at the LHC [2] and at the RHIC [20], respectively. This indicates that the chemical composition still alters slightly during the hadron gas through strange particle annihilation, decay, and baryon annihilation [21, 22].
The late production mechanism of quarks can play a significant impact on the charge-charge correlation of QGP. The motion of the balancing pair (quark and antiquark) which are produced at the same point of space-time coordinate can be separated into a collective mode, due to the collective expansion of the system, and a diffusive mode, due to the collisions with the surrounding medium. One can study the total motion of quarks by overriding the diffusive motion on a collective flow , which can be determined by hydrodynamic calculations. In [23], by overriding the diffusive motion on a longitudinal Bjorken flow, the following equation of the charge-charge correlation, , is derived: Here is the relaxation time for diffusion, is the relative rapidity, is the average rapidity of the two particles, , and is a parameter of local scattering rate. One can find , the rapidity width of [23]: by multiplying (13) by and integrating over both variables. Here is given by The width is determined by the time at which charges are created, the time at which the diffusion of charges freezes, and the local scattering rate .
The local scattering rate characterizes the ability of restoring local equilibrium for a source cell. Generally a system created by a nuclei-nuclei collision with a higher CMF energy has a larger because it has a larger local densities and a higher temperature. The samples of for three selected are shown as functions of in Figure 7. The charges created at a later time diffuse with a smaller width . One of the reasons for that is that the later the charges are created, the more they are constrained by the collective flow. On the other hand, the diffusion time for the charges created later is smaller. For a fixed , the value of for a larger is smaller. This is because the balancing pair in a source cell with a larger local scattering rate is harder to diffuse.
One can expect that has the following integral formulism with respect to time because of continuous creation or annihilation of charges: where describes the normalized probability of charges diffusing away from one another by a distance at time given they were created at . Generally the can be determined by solving the diffusion equation. Here we assume the diffusion of balancing charges has a Gaussian-like structure: where is calculated by (14).
The rapidity width of is defined: The systems with different local scattering rate have different rapidity width of . In Figure 8, we plot the of as a function of local scattering rate, . The symbols triangle and square are for the correlations of strange quarks and light quarks. The width of has a much larger dependence on the local scattering rate than that of . This is because strange quark annihilates by 18% while light quark increases by 75% during the evolution of QGP.
The different dependences of on between strange quark and light quark are expected to appear in the correlations of final particles in experiments, if the correlations in QGP are not washed out by the processes afterwards. Because the correlations of carry on the correlation of strange quark, the balance functions of are supposed to have less dependence on the centrality than that of the balance functions of . This is consistent with the findings on balance functions in RHIC experiments [24].
4. Summary
In this paper we have calculated the evolution of chemical composition of the quark-gluon system with 2 + 1 staggered quark flavors, using diagonal quark number susceptibilities and the EOS calculated by Wuppertal-Budapest group. Our results indicate that the number of the light quark increases by 75% while for strange quark it drops by 18% at the considered temperature region.
In our work a continuing charge production picture was used in the calculations. This continuing, late production mechanism will place a significant impact on conserved charge correlation. The correlations of strange particles have less dependence on centrality than that of hadrons comprised of light quarks. This can be tested by investigating the balance functions between particle species.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Y.-H. Pan would like to thank Dr. S. Pratt for his kind hospitality and very useful discussion when visited at Michigan State University during the year 2011. This work was supported by the National Natural Science Foundation of China under Grant no. 11275037.