The evolution-dominated hydrodynamics and thepseudorapidity
distributions in Au-Au collisions at BNL Relativistic Heavy Ion Collider
Z.J.Jiang*, Y. Huang, H. L. Zhang,Y. Zhangand H. P. Deng
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
*Correspondence authorE-mail:
The pseudorapidity rapidity distributions of charged particles produced in high energy heavy ion collisions are discussed in the framework of evolution-dominated hydrodynamics together with the effects of leading particles. A comparison is made between the theoretical predictions and the experimental measurements performed by PHOBOS Collaboration at BNL-RHIC in Au-Au collisions at = 62.4 and 130 GeV. The theoretical and experimental results match up well.
PACS numbers: 25.75.-q, 25.75.Ld, 25.75.Dw, 24.10.Nz
Keywords: Evolution-dominated hydrodynamics; pseudorapidity distributions; Khalatnikov potential;
leading particles; Gaussian distributions
I. Introduction
Since the pioneering work of Landau in 1953 [1], Relativistic hydrodynamics has been applied to calculate a large number of variables developed in the context of particle or nucleus collisions at high energy. Especially, along with the successful description of elliptic flow measured at the Relativistic Heavy-Ion Collider (RHIC)at Brookhaven National Laboratory (BNL) [2]and recently at the Large Hadron Collider(LHC) at CERN[3], the hydrodynamic research has entered into a more active phase. It has now been widely accepted as one of the best approaches for understanding the space-time evolution of the matter created in collisions[4-10].
Owing to the tremendous complexity of hydrodynamic equations, the most analytical investigations are, up till now, mainly limited to the hydrodynamics of 1+1 dimensions[11-18]. The 3+1 dimensional hydrodynamics is less developed,and no general exact solutionsare known so far.
A direct application of the 1+1 dimensional hydrodynamics is the analysis of the pseudorapidity distributions of charged particles produced in particle or nucleus collisions. In the presentpaper, by taking into account the effects of leading particles, we shall discuss such distributions in the framework of evolution-dominated hydrodynamics[14, 15].
II. A brief introduction to evolution-dominated hydrodynamics
The key ingredients of evolution-dominated hydrodynamics are as follows:
(1) The 1+1 expansion of a perfect fluid obeys equations
(1)
where,and are respectively the energy density, pressure and ordinary rapidity of fluid. is the light-cone coordinates, the proper time, and the space-time rapidity of fluid.
Eq. (1) is a complicated, non-linear and coupled one. In order to solve it, one introduces Khalatnikov potential , which relates to and by relations
(2)
where
(3)
is the temperature of fluid, and its initial scale. In terms of , Eq. (1) can be reduced to
(4)
where is the speed of sound. The above equation is now a linear second-order partial differential equation, which works for any form of .
(2) In case of constant, the solution of Eq. (4) takes the form as
(5)
where stands for the initial distribution of sources of hydrodynamic flow.
Experimental investigations have shown that the speed of sound changes very slowly with incident energies and centrality cuts[19-21]. For a given energy, it can be well taken as a constant.In nucleus-nucleus collisions at energies of BNL-RHIC scale, such a constant is about or [19].
(3) For evolution-dominated hydrodynamics, the initial distribution of source takes the form as [14, 22, 23]
, (6)
where is a constant, and the Heaviside step function. Inserting it into Eq. (5), it reads
(7)
where is the 0th order modified Bessel function of the first kind.
In heavy ion collisions at high energy, owing to the violent compression and Lorentz contraction of collision system along beam direction, the initial pressure gradient of created matter in this direction is very large. By contrast, theeffect of initial flow of sources is negligible. The motion of fluid is mainly dominated by the following evolution. Hence, the evolution-dominated hydrodynamics should reflect the reality of expansion of fluid.
(4) The freeze-out of fluid is assumed to take place at a space-like hypersurface with a fixed temperature of . From this assumption together with the direct proportional relation between the number of charged particles and entropy, we can get the rapidity distributions of the charged particles in terms of Khalatnikov potential
, (8)
where , which is related to the initial temperature of fluid and is therefore dependent on the incident energies and centrality cuts. Its specific value can be determined by comparing the theoretical predictions with experimental data.
Inserting Eq. (7) into above equation, it becomes
， (9)
where , independent of rapidity y, is an overall normalization constant, and the 1st order modified Bessel function of the first kind, the impact parameter, and the center-of-mass energy per pair of nucleons.
III. The rapidity distributions of leading particles
Apart from the freeze-out of fluid, leading particles also have certain contributions to the yields of charged particles.
Investigations have shown that the leading particles are believed to be formed outside the nucleus, i. e., outside the colliding region[24, 25]. The generation of them is therefore free from fluid evolution. Hence, the formulation of rapidity distributionsof them is beyondthe scope of hydrodynamics and should be treated separately.
In our previous work [5, 6, 15], we once argued that the rapidity distribution of leading particles takes the Gaussian form
, (10)
where, and are respectively the number of leading particles, central position and width of distribution. This conclusion comes from the consideration that, for a given incident energy, different leading particles resulting from each time of nucleus-nucleus collisions have approximately the same amount of energy or rapidity. Then, the central limit theorem [26, 27]ensures the reasonability of above argument. Actually, experimental investigations have shown that any kind of charged particles resulting in collisions forms a good Gaussian rapidity distribution[28-30].
in Eq. (10) is the average position of leading particles. It should increase with incident energies and centrality cuts. The value of relies on the relative energy or rapidity differences among leading particles. It should not, at least not apparently, depend on the incident energies, centrality cuts and even colliding systems. The concrete values of and can be determined by tuning the theoretical predictions to experimental data.
By definition, leading particles mean the particles which carry on the quantum numbers of colliding nucleons and take away most of incident energy. Then, the number of leading particles is equal to that of participants. For nucleon-nucleon, such as p-p collisions, there are only two leading particles. They are separately in projectile and target fragmentation region. For an identical nucleus-nucleus collision, the number of leading particles
, (11)
where is the total number of participants equaling[31-33]
, (12)
where the variable is the transverse coordinates in the overlap region with respect to the center of the projectile nucleus. The integrand is the total number of participants inthe flux tube with a unit bottom area, located at positionalong beam direction. It takes the form as
where is the inelastic nucleon-nucleon cross section.It increases slowly with energies,e. g., for =62.4 and 130 GeV, = 36 and 41mb [34], respectively. The subscripts A and B in the above equation represent the projectile and target nucleus. or is the nuclear thickness function with the value equaling the nucleon number in the flux as defined above. It is equal to
,
where
is Woods-Saxon distribution of nuclear density. and , taken somewhat different values in different papers [31], are respectively the skin depth and radius of nucleus. In this paper, they take the values of=0.54 fm and , where is the mass number of nucleus.
Known from the investigations in Ref. 15, Eq. (12) can give a correct result in different nucleus-nucleus collisions at different centrality cuts and energies from BNL-RHIC to CERN-LHC scale.
IV. The comparisons between theoretical predictions and experimental measurements
From rapidity distributions, we can get pseudorapidity distributions by relation[35]
, (13)
where is the transverse mass, and the transverse momentum. The first factor on the right-hand side of above equation is actually the Jacobian determinant. This transformation is closed by another relation
. (14)
Taking into account the contributions from both the freeze-out of fluid and leading particles, the rapidity distributions in Eq. 13 can be written as
. (15)
Substituting above equation or the sum of Eqs. (9) and (10) into (13), we can get the pseudorapidity distributions of charged particles. Figures 1-3 show such distributions in different centrality Au-Au collisions at = 62.4 and 130 GeV, respectively. The solid dots in figures denote the experimental measurements[36].The dashed curves are the results got from evolution-dominated hydrodynamics of Eq. (9). The dotted curves are the results obtained from leading particles of Eq. (10). The solid curves are the results achieved from Eq. (15), i. e., the sums of dashed and dotted curves.It can be seen that the theoretical results are well consistent with experimental measurements.
In calculations, for collisions at 62.4 and 130 GeV, the parameter in Eq. (9) takes the values of 1.71-1.87 and 2.30-2.65for centrality cuts from small to large. It can be seen that increases with incident energies and centrality cuts. It should, since in Eq. (9) determines the widths of distributions, which increase with energies and centrality cuts (c. f. Figs. 1-3). The width parameter in Eq. (10) takes a constant of 0.85 for collisionsat different energies and centrality cuts. As the analyses given above, is independent of incident energies and centrality cuts. For collisions from low to high energies, the center parameter in Eq. (10) takes the values of 2.62-2.85 and 2.62-3.15for centrality cuts from small to large. It can be seen that, although increases with energies and centrality cuts, the increasing speed is very slow.This is in agreement with the experimental observations that the rapidity loss of participants increases with energies[37].
Fig. 1. The pseudorapidity distributions of charged particles produced in different centrality Au-Au collisions at . The solid dots are the experimental measurements [36]. The dashed curves are the results from evolution-dominated hydrodynamics of Eq. (9). The dotted curves are the results from leading particles of Eq. (10). The solid curves are the sums of dashed and dotted ones.
V. Conclusions
The charged particles produced in heavy ion collisions are divided into two parts. One is from the hot and dense matter created in collisions, which is presumed to expand according to evolution-dominated hydrodynamics. The other is from leading particles, which is outside the scope of hydrodynamic description.
Compared with the tremendously high pressure gradient in beam direction, the effect of initial flow of the hot and dense matter in this direction is negligible. The motion of this matter is mainly governed by the evolution of fluid. This thus guarantees the rationality of evolution-dominated hydrodynamics. With the scheme of Khalatnikov potential, this theoretical model can be solved analytically.The exact solutions can then be used to formulate the rapidity distributions of charged particlefrozen out from fluid. If the freeze-out of fluid is assumed to occur at a space-like hypersurface with a fixed temperature of , the rapidity distributions can be expressed in a simple analytical form in terms of 0th and 1st order modified Bessel function of the first kind with only two parameters and . takes the value from experiments. is fixed by fitting the theoretical results with experimental data.
Fig. 2. The pseudorapidity distributions of charged particles produced in different centrality Au-Au collisions at . The solid dots are the experimental measurements [36]. The dashed curves are the results from evolution-dominated hydrodynamics of Eq. (9). The dotted curves are the results from leading particles of Eq. (10). The solid curves are the sums of dashed and dotted ones.
As for leading particles, we argue that they possess the Gaussian rapidity distribution normalized to the number of participants. This is the same as that proposed in Refs. 5, 6 and15. It is interested to notice that the investigations of the present paper once again show that, for a given colliding system, the central position of Gaussian rapidity distribution increases with incident energies and centrality cuts. While, the width of the distribution is irrelevant to them. These are consistent with the conclusions arrived at in our previous work[5, 6, 15].
Comparing with the experimental measurements carried out by PHOBOS Collaborationat BNL-RHIC in Au-Aucollisions at 62.4 and 130GeV, we can see that the total contributions from both evolution-dominated hydrodynamics and leading particles are well consistent with experimental data.
Acknowledgments
This work is partly supported by the Transformation Project of Science and Technology of Shanghai Baoshan District with Grant No. CXY-2012-25; the Cultivating Subject of National Project with Grant No. 15HJPY-MS04; the Hujiang Foundation of China with Grant No. B14004.
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