Calculation of shear viscosity in Au+Au collisions at NICA energies

Shear viscosity of hot and dense nuclear matter, produced in the central zone of central gold-gold collisions at energies of NICA, is calculated within the UrQMD model. Besides the microscopic simulations of heavy ion collisions, the procedure assumes the application of statistical model to determine the temperature and chemical potentials in the system, and study of the relaxation process within the UrQMD box with periodic boundary conditions. The latter is used for calculation of the correlator which enters the Green-Kubo formula for shear viscosity. The fluctuations at early and late stages of the system evolution are studied. Results are compared to predictions of other models.


Introduction
Experiments on heavy-ion collisions at relativistic and ultrarelativistic energies are the only means to study the properties of hot and dense nuclear matter in the laboratory. Analyzing the data obtained at RHIC (BNL) physicists came to the conclusion that the created substance possessed the properties of perfect fluid [1][2][3][4]. However, it became clear soon that one had to employ the nonzero shear viscosity η for the correct description of differential elliptic flow v 2 of hadrons as function of transverse momentum p T within the framework of hydrodynamics. Recall, that the absolute minimum for the ratio of η to the entropy density s, h s, estimated in the AdS-CFT formalism, equals p 1 4 [5] in system of natural units, . Hydrodynamic calculations use a bit higher values to describe the experimental data, i.e. h = s 0.12 at RHIC for Au+Au collisions at = s 200GeV, and 0.20 at LHC (CERN) for Pb+Pb collisions at = s 2.76 TeV [6]. As was pointed out in [7], the ratio h s reaches minimum in the vicinity of tricritical point for all known substances. It appears that at energies of LHC the phase transition between the quark-gluon plasma (QGP) and hadrons is a smooth crossover. The phase transition QGP−hadrons is expected to be of the first order at much lower energies accessible for beam energy scan (BES) at RHIC and future facilities NICA (JINR) and FAIR (GSI). The search for the tricritical point, where the first order phase transition becomes the second order one, is in the agenda of all experiments planned at the aforementioned accelerators. In the present work we would like to study the shear viscosity and its ratio to entropy density in the midrapidity range of central heavy-ion collisions at energies between = E 10 lab and 40AGeV generated by the microscopic transport model UrQMD [8,9]. Note, that except of [10], h s ratio was studied in various models for closed systems with fixed values of energy density and baryon density [11][12][13][14][15][16][17][18][19][20][21][22]. We are going to extend the results of [10], obtained within the equilibrium approach for calculation Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of the entropy density, to the nonequilibrium case. The paper is organized as follows. Determination of shear viscosity by means of Green-Kubo approach is described in section 2. Section 3 presents the basic ingredients of model calculations including the UrQMD model, statistical model (SM) of an ideal hadron gas, and UrQMD box, which is a closed system. Results of our study obtained both with equilibrium and nonequilibrium entropy are discussed in section 4. Conclusions are drawn in section 5.

Determination of shear viscosity. Green-Kubo formalism
To determine the shear viscosity one usually employs the Green-Kubo method [23,24]. It assumes the exponential damping of fluctuations whereas the closed system relaxes to equilibrium. The shear viscosity for the system with volume V and temperature T reads where t 0 and t is the initial time and final time, respectively. The correlator in the integrand is It contains the traceless parts of the energy-momentum tensor T ij and E k (t) denote the i( j)-th components of momentum and energy of k-th particle. The sum in Eq.(3) runs over all particles. The correlator (2) should drop exponentially with time in the vicinity of equilibrium, therefore, it can be approximated by the exponential containing the effective relaxation time τ. Thus, to find the shear viscosity one has to determine both T and τ, because Eq.(1) is reduced to Calculation of these parameters is explained in the next section.

Model setup
From the description of the method of shear viscosity extraction it becomes clear that we have to organize several steps to complete our task. First, while running the transport string model for heavy-ion collisions at given energy, we should determine the volume to search for local equilibrium.
Previous studies reveal [25][26][27][28] that central cubic cell with volume V=5×5×5=125fm 3 is well suited for our analysis. But the cell is an open system and particles can leave it freely. The initially hot and dense fireball quickly expands, and its bulk characteristics are promptly changing. In order to investigate how far is the matter in the cell from local equilibrium one has to extract three basic parameters, namely, energy density ε, net baryon density r B , and net strangeness density r S , and insert it as an input to the statistical model of an ideal hadron gas. If the abundances of hadronic species and their energy spectra in the microscopic model calculations are close to those given by the SM, we can conclude that the matter is in the vicinity of chemical and thermal equilibrium. This procedure enables us to determine the temperature T, baryochemical potential m B , and the strangeness chemical potential m S . Finally, the behavior of the correlator should be studied. This can be done with the help of box with periodic boundary conditions to keep key system parameters constant. At this stage we get the value of shear viscosity at given m m T , , B S . The basic principles of the three stages are presented below.

Microscopic transport model
The UrQMD is formulated as Monte-Carlo event generator allowing to perform various analyzes of the measurable quantities by introducing all necessary experimental cuts. The model is designed to describe hadronic, hadron-nucleus, and nucleus-nucleus collisions in a broad energy range. In the hadronic sector UrQMD treats the production of new particles via formation and fragmentation of specific colored objects, strings. Strings are uniformly stretched, with constant string tension k » 1 GeV/fm, between the quarks, diquarks and their antistates. The excited string is fragmenting into pieces via the Schwinger-like mechanism ofqq-pair production, and the produced hadrons are uniformly distributed in the rapidity space.
In contrast to models which rely on the color exchange mechanism of string excitation, like QGSM [29,30] or NEXUS [31], the UrQMD model employs the longitudinal excitation of strings. Here the string masses arise from the momentum transfer. Tables of the experimentally available information, like hadron cross sections, resonance widths, their decay modes, and so forth are implemented. In case of lacking the information, the model employs the detailed balance considerations, the one-boson exchange model, and isospin symmetry conditions. The propagation of particles is governed by Hamilton equation of motion. Newly produced hadrons can interact further only after a certain formation time. The Pauli principle is taken into account via the blocking of the final state, if the outgoing phase space is occupied.

Statistical model of an ideal hadron gas
If the system is in thermal and chemical equilibrium, its macroscopic characteristics are fully determined by particle distribution functions where p and m i is the momentum and the mass of the hadron species i, respectively. Sign − stands for bosons and + for fermions. One has to know just three parameters, namely, temperature T and chemical potentials assigned to the conserved charges, i.e. baryon chemical potential m B and strangeness chemical potential m S . Chemical potential of i-th hadron depends on its baryon and strangeness content, The dependence on chemical potential m Q associated with electric charge is disregarded here, because m Q is usually an order of magnitude weaker compared to m B and m S . Then, the expressions for particle number density n i , energy density e i and pressure P read with g i being the spin-isospin degeneracy factor. The entropy density s i can be calculated either from the Gibbs thermodynamic identity The total energy density ε, baryon density r B and strangeness density r S calculated microscopically within the cell at time t are inserted into the set of nonlinear equations to determine temperature T, baryon chemical potential m B and strangeness chemical potential m S . After that all characteristics of the system in equilibrium are known and particle spectra can be compared with those obtained from microscopic model calculations.

Box for simulation of infinite nuclear matter
The standard choice for such a simulation is a cubic box with periodic boundary conditions [32][33][34] to ensure the energy and momentum conservation. If one particle leaves the box, another particle, fully identical to the first one, enters the box from the opposite side. The box should be neither too big nor too small, e.g. =´´= V 10 10 10 1000 fm 3 [32] or =´´= V 5 5 5 125 fm 3 [33]. The energy density ε, the net baryon density r B , and the net strangeness density r S are fixed at the initial stage. Recall, that for the infinite nuclear matter the net strangeness is zero, but for the central cell in relativistic heavy-ion collisions, which is an open system, r S can differ from zero [25,27]. In case of r = 0 S the initial configuration in the box consists of protons and neutrons uniformly distributed in the configuration space. Their momenta are then rescaled to get the required energy density. If r ¹ 0 S certain admixture of kaons can be added. When the system is prepared, hadrons start to interact, and one can follow the microscopic model calculations to trace the system evolution and study the relaxation of hadron-string matter in the box to equilibrium.

Results
A bit more than 50.000 Au+Au collisions were generated at each of four bombarding energies. Figures 1(a)-(c) shows energy density, net baryon density, and net strangeness density obtained from the microscopic calculations in the central cell at times between 1 and 20fm/c. Within first few fermiseconds the colliding nuclei fully overlap. This moment corresponds to highest energy density and baryon density in the fireball. The fireball expands, and both characteristics drop. Strangeness demonstrates another behavior. It is negative for all four energies. After reaching minimum, r S is relaxing to zero at late times. This behavior is explained by domination of baryons over antibaryons in the cell. Therefore, according to [25][26][27][28] We are ready now to start the box calculations.    The behavior of t fit , obtained after fitting our results to equation (3), is presented in figure 4. It is very similar to that of t int , but it has no stochastic oscillations. Note that the distributions demonstrates some slope in the plateau region at Results, obtained after averaging over the plateau, are displayed in figure 5. Note that the statistical errors are smaller than the symbol sizes. One can see that shear viscosity distributions for all four energies converge to each other at » t 6 fm/c. The matter in the cell becomes very dilute at the late times, and η drops almost to zero. This behavior is almost identical to the decrease of the cell temperatures seen in figure 1(d). Recall that the chemical potentials of net baryon charge and net strangeness are different, as shown in figure 1(e)-(f). It means that shear viscosity is predominantly determined by the temperature and not by m B and m S . The entropy density, however, does depend on both temperature and chemical potentials.
To show this we present in figure 6 the dependencies of ratio h s on (a) time, (b) temperature, (c) baryon chemical potential, and (d) strangeness chemical potential for all four reactions in question.
The nonequilibrium stages of the system evolution are shown by dashed lines. We see that this ratio decreases with  increase of the bombarding energy. Since the shear viscosity is almost the same, this circumstance implies the drop of entropy density with rising E lab . It can be explained by faster loss of energy density in the central cell in case of more energetic collisions. Also, no distinct minima are observed. Recall, however, that we used the entropy density, calculated for the hadron gas in thermal and chemical equilibrium, even for the early stages of the collision when the matter in the cell was far from equilibrium. Obviously, the entropy density of nonequilibrated system should be lower than that of the equilibrated one. To estimate the nonequilibrium entropy, we insert in Eq. (11) instead of the equilibrium distribution functions (6) another ones provided by the hadron momentum distributions It is worth noting that the temperature at nonequilibrium stage is determined as average of partial temperatures of hadron species. Time evolution of h s noneq in the cell together with its dependencies on temperature, on baryon chemical potential, and on strangeness chemical potential are shown in figure 7. Here the minima for all four energies are clearly seen. This occurs at time »t 5 6 fm/c corresponding to maximum baryon density in the system. The lower the  bombarding energy, the deeper the minimum in of h s distribution. The study should be extended to lower energies in order to see where the ratio of shear viscosity to entropy density will stop to decrease. We are in the energy region where the equation of state (EOS) is expected to be changed because of the formation of non-hadronic objects, quarkgluon strings.

Conclusions
The self-consistent procedure for determination of shear viscosity and its ratio to entropy density within the microscopic model calculations is developed. It includes three steps. First, we define the volume in A+A collisions to look for the local equilibrium. This is a very important condition, because the Green-Kubo formalism employed for determination of shear viscosity implies the relaxation of out-of-equilibrium matter to the equilibrated state. The central cubic cell with volume V=125fm 3 is well suited for our analysis. Second, we extract the energy density, net baryon density, and net strangeness density out of the tested volume. The procedure is repeated with the time step D = t 1 fm/c. The extracted values are inserted into a system of non-linear equations of the statistical model of an ideal hadron gas with essentially the same number of degrees of freedom, as in the microscopic model. This allows one to determine temperature, baryon chemical potential, and strangeness chemical potential in the tested volume in case the hadron yields and energy spectra, calculated microand macroscopically, are close to each other. Finally, the values of m m T , , B S are used to initialize the box with periodic boundary conditions in the framework of the same microscopic transport model. Here the Green-Kubo formalism is applied to determine η, ratio h s, and so forth.
The developed procedure was used to study the shear viscosity of hot and dense nuclear matter in the central zone of central gold-gold collisions at energies = E 10, 20, 30 lab and 40AGeV, accessible for BES, NICA, and FAIR facilities. For calculations we employ the UrQMD model. We found that, for all four tested energies, the temperatures and shear viscosities are very close to each other after 5-6fm/c. Both T, η and s in the cell drop with time, whereas the ratios h s, however, reach minima at » t 5 fm/c, irrespective of the bombarding energy. Then the ratios rise until the very late stages of the system evolution. The minima in h s become more pronounced if the nonequilibrium entropy density is used. The rise of h s is accompanied by the simultaneous increase of baryon chemical potential and decrease of both temperature and strangeness chemical potential in the cell.