Hadron Resonance Gas EoS and the Fluidity of Matter Produced in HIC

Kadam, Guruprasad; Pawar, Swapnali

doi:https://doi.org/10.1155/2019/6795041

Advances in High Energy Physics

On this page

Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 6795041 | https://doi.org/10.1155/2019/6795041

Hadron Resonance Gas EoS and the Fluidity of Matter Produced in HIC

Guruprasad Kadam¹and Swapnali Pawar²

Academic Editor: Sally Seidel

Received24 Nov 2018

Revised22 Feb 2019

Accepted06 Mar 2019

Published26 Mar 2019

Abstract

We study the equation of state (EoS) of hot and dense hadron gas by incorporating the excluded volume corrections into the ideal hadron resonance gas (HRG) model. The total hadron mass spectrum of the model is the sum of the discrete mass spectrum consisting of all the experimentally known hadrons and the exponentially rising continuous Hagedorn states. We confront the EoS of the model with lattice quantum chromodynamics (LQCD) results at finite baryon chemical potential. We find that this modified HRG model reproduces the LQCD results up to MeV at zero as well as finite baryon chemical potential. We further estimate the shear viscosity within the ambit of this model in the context of heavy-ion collision experiments.

1. Introduction

The phase diagram of quantum chromodynamics (QCD) is under intense theoretical investigation especially in the context of heavy-ion collision (HIC) experiments where two heavy nuclei are accelerated to very high energies and then they collide to produce very hot and dense QCD matter. These experiments can probe part of the QCD phase diagram where the most interesting nonperturbative aspects of QCD lie. The most reliable theoretical tool at zero baryon chemical potential is lattice quantum chromodynamics (LQCD). One of the most important prediction of LQCD simulation is that the phase transition from hadronic to quark-gluon-plasma (QGP) is an analytic crossover [1]. However, at finite baryon chemical potential QCD has been plagued with the so-called sign problem and one has to resort to certain approximation schemes to fetch the reliable results [2]. One can adopt an alternative approach where the effective model of QCD can be constructed which not only preserves certain important symmetries of QCD but is tractable at finite chemical potential as well. The hadron resonance gas model is the statistical model of QCD describing the low temperature hadronic phase of quantum chromodynamics. The essential starting point of the model is the Dashen-Ma-Bernstein theorem [3] which allows us to compute the partition function of the interacting system in terms of scattering matrix. Using this theorem it can be shown that if the dynamics of the system is dominated by narrow-resonance formation it behaves like a noninteracting system [4–6]. Thus the thermodynamics of interacting gas of hadrons through formation of resonances can be well approximated by the noninteracting gas of hadrons and resonances. The HRG model has been very successful in describing the hadron multiplicities in HICs [7–16]. Recently, the interacting HRG model with multicomponent hard-core repulsion has been successful in describing the heavy-ion collision data with the unprecedented accuracy [17–20].

One possible improvement in the HRG model is to include the exponentially rising Hagedorn density of states apart from the known hadrons and resonances included in the form of discrete mass spectrum. This exponential mass spectrum arises in the string picture [21–25] or the glueball picture of the hadrons [26]. A typical form of the continuous mass spectrum is which satisfies the statistical bootstrap condition [27, 28]. With the proper choice of the parameters and with all the experimentally found hadrons fit in this exponential mass spectrum [29, 30]. Finite temperature LQCD simulations provide strong evidence of the existence of Hagedorn states in hot and dense matter created in heavy-ion collision experiments. The mass spectrum of these states is found to be of the form ( is the Hagedorn temperature) [31].

Another possible improvement in the ideal HRG model is to take into account short range repulsive interactions between hadrons. There are many different ways of incorporating the repulsive interactions in the ideal HRG model without spoiling the thermodynamical consistency of the model [32–34]. We shall consider the excluded volume correction scheme of [32] where the repulsive interactions are accounted for through the excluded volume correction in the ideal gas partition function.

In the past few decades the LQCD results of the equation of state at zero baryon chemical potential have been analyzed within the HRG model and its extensions [35–40]. Recently, in [41] the authors have observed that the hadron resonance gas model with the discrete mass spectrum augmented with the continuous Hagedorn mass spectrum is not sufficient to explain the recent lattice QCD results. Further the excluded volume corrections to the ideal HRG model also fail to do the same. But if both these physical effects, , the Hagedorn states as well as excluded volume corrections, are included in the ideal HRG model then the model reproduces the lattice QCD data all the way up to MeV.

In the context of relativistic heavy-ion collision (RHIC) experiments the shear viscosity coefficient governs the evolution of the nonequilibrium system towards the equilibrium state. In the off-central nuclear collision the spatial anisotropy of the produced matter gets converted into momentum anisotropy and the equilibration of this momentum anisotropy is governed by the shear viscosity coefficient. The produced matter in the fireball after the collision, with quarks and gluons degrees of freedom, behaves like a strongly interacting liquid with very small shear viscosity. Assuming that this liquid of quarks and gluons is in thermal equilibrium, it expands due to pressure difference and cools and finally undergoes a phase transition to hadronic degrees of freedom which finally free stream to the detector. One of the successful descriptions of such an evolution is through dissipative relativistic hydrodynamics [48–56] and transport simulations [57–64]. In the hydrodynamic description of the heavy-ion collision experiments finite but the small ratio of shear viscosity () to entropy () is necessary to explain the flow data [65, 66]. The smallness of this ratio and its connection to the conjectured Kovtun-Son-Starinets (KSS) bound of obtained using AdS/CFT correspondence [67] have motivated many theoretical investigations of this ratio to understand and derive it rigorously from a microscopic theory [43–46, 68–79].

In this work we confront the equation of state of the excluded volume HRG model which includes the discrete hadron states and continuum Hagedorn states in the density of states at finite baryon chemical potential. We further attempt to make rough estimates of the shear viscosity coefficient within the ambit of this extended HRG model in the context of heavy-ion collision experiments. Throughout the discussion we shall adopt the Boltzmann approximation (i.e., Boltzmann classical statistics) since it is a rather excellent approximation in the region of the QCD phase diagram in which we are interested (i.e., MeV).

We organize the paper as follows. In Section 2 we briefly describe the thermodynamics of the hadron resonance gas model. In Section 3 we give brief derivation of the shear viscosity coefficient for the multicomponent hadronic matter using the relativistic Boltzmann equation in relaxation time approximation. In Section 4 we present the results and discuss their implications in the context of relativistic heavy-ion collision experiments. Finally we summarize and conclude in Section 5.

2. The Hadron Resonance Gas Model

Thermodynamical properties of the hadron resonance gas model can be deduced from the grand canonical partition function defined as where is the baryon chemical potential and and are the mass spectrum of the bosons and fermions, respectively. We assume that the hadron mass spectrum is a combination of discrete (HG) and continuous (HS) Hagedorn states given by where

This discrete mass spectrum consists of all the experimentally known hadrons with cut-off . One can set different cut-off values for baryons and mesons. The Hagedorn density of states is assumed to bewhere is constant and is the Hagedorn temperature. One can physically interpret as QCD phase transition temperature. It is important to note that one cannot define baryon chemical potential for the Hagedorn states. Hagedorn states depend on the trend in the increase of the number of states as the hadron mass increases. Since this is not a thermodynamic statement, concepts such as the baryon density or chemical potential would not apply to Hagedorn states. Further the Hagedorn states are defined by the density of states defined by (4) satisfying the bootstrap condition. Since the quark content of these states is unknown it is not legitimate to define baryon number and hence baryon chemical potential to these states. Thus for all practical purposes we set for Hagedorn states in our calculation. It is to be noted that the heavy Hagedorn states have finite width and large width of Hagedorn states is of great importance to describe the lattice QCD thermodynamics and to explain the chemical equilibrium of hadronic matter born from these states [80, 81]. But we will work in narrow width approximation for Hagedorn states.

Repulsive interaction can be accounted for in the ideal HRG model via excluded volume correction () to the partition function. The pressure of the HRG model with the discrete mass spectrum and excluded volume correction (which we shall call EHRG) turns out to be [32]where , is the ideal gas pressure, and is the excluded volume parameter of the hadron with hard-core radius . Note that we assume a uniform hard-core radius to all the hadrons. The excluded volume models can be extended to multicomponent gas of a different hard-core radius [82]. These models can further be extended to take into account the Lorentz contraction of hadrons due to their relativistic motion [83]. But we will neglect these effects for the sake of simplicity. In the Boltzmann approximation the contribution to the ideal gas pressure due to hadronic species is where is the degeneracy factor and is the modified Bessel function of the second kind. In the Boltzmann approximation, (5) is simplified to In the case of the continuum Hagedorn spectrum the sum in (7) is replaced by the integration and the discrete mass spectrum given by the delta function is replaced by the Hagedorn mass spectrum given by (4). The other thermodynamical quantities, , energy density (), entropy density (), number density (), and speed of sound (), can be obtained from the pressure by taking appropriate derivatives as per thermodynamical identities.

3. Transport Coefficients: A Relaxation Time Approximation

In the relativistic kinetic theory the evolution of distribution function is determined by the Boltzmann equation [84]where and is called the collision term. Solving this equation, which is in general an “integrodifferential” equation, for is rather a very difficult task, if not impossible. So it is customary to resort to certain approximations so that solving (8) becomes feasible. We assume that the system is only slightly away from the equilibrium; i.e., with . is an equilibrium distribution function. If we further assume that the collisions bring the system towards equilibrium with the time scale , then the collision term can be approximated as

In this so-called relaxation time approximation, Boltzmann equation (8) becomes

The equilibrium distribution function in the Boltzmann approximation is given by where is the fluid velocity.

In the theory of fluid dynamics shear () and bulk () viscosities enter as coefficients of the space-space component of the energy-momentum tensor away from equilibrium as where is the ideal part of the stress tensor.

In the local Lorentz frame the dissipative part of the stress energy tensor can be written as

In the kinetic theory is defined as where , and is the degeneracy. The space-space component of the above equation is where

In the local rest frame, (11) is simplified to Assuming steady flow of the form and space-time independent temperature, (14) is simplified to

From (17) and (18) we get (using (12) with )

Comparing the coefficients of gradients in (19) and (20) we get the coefficient of shear viscosity

Thus for multicomponent hadron gas at finite chemical potential the shear viscosity coefficient is [68]

The relaxation time is in general energy dependent. But for simplicity we use averaged relaxation time () which is rather a good approximation as energy dependent relaxation time [85]. Thus the averaged partial relaxation time is defined by where is the number density of hadronic species and is the thermal average of the cross section given by [86]where is a center of mass energy and is the modified Bessel function of the second kind. Note that we have assumed the cross section to be constant for all the species in the system.

The massive Hagedorn states cannot be described rigorously using the Boltzmann equation. Thus it is very difficult to compute the contribution of these massive and highly unstable hadrons to the shear viscosity of low temperature QCD matter. But since these states contribute significantly to the thermodynamics of hadronic matter close to QCD transition temperature their contribution cannot be ignored. In fact they decay so rapidly that it is legitimate to assume that their presence will affect the relaxation time of the system. In [87] the authors studied the effect of Hagedorn states on the shear viscosity of QCD matter near . They assumed that the relaxation time is inversely related to the decay width through the relation . The decay width of Hagedorn states can be obtained from the linear fit to the decay widths of all the resonances in the particle data book. This prescription corresponds to the decay cross section and neglects the collisional cross section for the momentum transport that might contribute the shear viscosity as per the Boltzmann equation. Such approximation gives only a rough estimate of the shear viscosity. Since we are also interested in the rough estimate of shear viscosity we assume that the Hagedorn states contribute to the shear viscosity through collisions with other hadrons. We further assume that these states are hard sphere particles of radius . Thus the thermodynamics of the Hagedorn states can be estimated using the excluded volume HRG model. It can be shown that in the excluded volume approximation the shear viscosity of the Hagedorn gas is [88]where is the number density of the Hagedorn gas. Viscosity coefficients of the pure Hagedorn fluid in the relaxation time approximation of the Boltzmann equation have also been studied in [77]. We again mention here that the heavy Hagedorn states have finite width but we are working in the narrow width approximation of these states.

4. Results and Discussion

We include all the hadrons and resonances up to 2.225 GeV listed in [89]. More specifically we choose cut-offs GeV for mesons and GeV for the baryons. Note here that in HRGM usually one has to include all hadrons and resonances up to GeV in order to get a good description of the experimental data. But the mass spectrum of hadrons with masses between 1 GeV and 2.2 GeV is approximately Hagedorn-like. We set uniform hard-core radius fm to all the hadrons. Note here that the excluded volume corrections to the ideal HRG model are consistent only if we choose a uniform hard-core radius to all the hadrons. To specify the Hagedorn mass spectrum we choose Ge. Finally we set the Hagedorn temperatures MeV for and MeV for MeV. This specific choice is made to get the best fit with the LQCD data at zero as well as finite chemical potential.

Figure 1 shows scaled pressure () and the scaled interaction measure () at zero chemical potential. The black dots (with error bars) correspond to lattice QCD data taken from [42]. The blue curve corresponds to the ideal HRG model with only the discrete mass spectrum while the green dashed curve corresponds to the ideal HRG with both the discrete and continuous Hagedorn mass spectrum. The brown curve corresponds to the excluded volume HRG model with only experimentally known hadrons included. The solid magenta curve corresponds to excluded volume corrections to the ideal HRG model in which both experimentally known hadrons and continuous Hagedorn states are included. We shall call HRG with excluded volume corrections and Hagedorn spectrum MEHRG (modified excluded volume HRG) for brevity. It can be noted that the HRG model alone cannot reproduce lattice data for the pressure as well as the interaction measure. The HRG estimates for the pressure are merely within the error bars. In case of interaction measure, while the LQCD predict the rapid rise till MeV, HRG estimates do not show such rapid rise. Similar points can be noted at finite chemical potential as shown in Figure 2. Further, the inclusion of the Hagedorn mass spectrum or the excluded volume corrections to the ideal HRG model does not improve the results. But if excluded volume corrections are made to the ideal HRG model with the discrete as well as continuous Hagedorn mass spectrum then the resulting model (MEHRG) reproduces the LQCD results up to temperature MeV. This observation has already been made at zero chemical potential in [41] where the authors confronted the LQCD data at with the HRG model with the Hagedorn mass spectrum and excluded volume correction. We observe that a similar conclusion can be drawn at finite chemical potential as well. While the ideal HRG does not satisfactorily reproduce all the features of LQCD data, MEHRG reproduces the lattice data all the way up to MeV. It is to be noted in passing that at finite chemical potential QCD phase transition occurs at lower temperature than that at zero chemical potential.

(a)

(b)

(a)

(b)

Figure 3 shows estimated in MEHRG with the Hagedorn mass spectrum of the form . The values of the parameters are the same as in [41]. We note that our conclusion, i.e., the ideal HRG needing to be improved with excluded volume corrections as well as Hagedorn density of states, does not depend on the choice of the Hagedorn mass spectrum.

Figure 4 shows comparison of the ratio estimated within our model with that of various other methods [43–46]. The red dashed curve corresponds to the Chapman-Enskog method with constant cross sections [43]. The dashed green curve corresponds to the relativistic Boltzmann equation in relaxation time approximation. The thermodynamical quantities in this model have been estimated using the scaled hadron masses and coupling (SHMC) model [44]. The brown dashed curve corresponds to estimations made using the relativistic Boltzmann equation in RTA. The thermodynamical quantities are estimated within the EHRG model [45]. The dot-dashed orchid curve corresponds to of meson gas estimated using the chiral perturbation theory [46]. While the ratio in our model is relatively large at low temperature as compared to other models it rapidly falls and approaches closer to the Kovtun-Son-Starinets (KSS) bound [67], at high temperature. This rapid fall may be attributed to the rapidly rising entropy density due to Hagedorn states which are absent in the other models.

Figure 4

Comparison of the ratio estimated within our model, i.e., MEHRG (solid magenta curve), with various other methods. The red dashed curve corresponds to the Chapman-Enskog method [43] with constant cross sections. The dashed green curve corresponds to estimations made within the SHMC model of the hadronic matter [44]. The brown dashed curve corresponds to estimations made within the EHRG model [45]. The dot-dashed orchid curve corresponds to of meson gas estimated using the chiral perturbation theory [46].

Figure 5(a) shows the ratio of shear viscosity to entropy density at two different chemical potentials. The ratio decreases with increase in temperature and approaches the KSS bound at high temperature. The rapid fall in estimated within MEHRG can again be attributed to the rapid rise in the entropy density due to exponetially rising Hagedorn states. At finite this ratio approaches the KSS bound more closely as compared to the zero chemical potential case.

(a)

(b)

It has been argued in [90] that at finite chemical potential the correct fluidity measure is not but the quantity . At zero chemical potential the basic thermodynamical identity implies that two fluidity measures and are the same. However, at finite chemical potential they may differ. Figure 5(b) shows the fluidity measure at zero as well as at finite chemical potential. It can be noted that shows behavior similar to that of .

One can make estimations of in the context of heavy-ion collision experiments by finding the beam energy () dependence of the temperature and chemical potential. This is extracted from a statistical thermal model description of the particle yield at various [47, 91–96]. The freeze-out curve is parametrized by [47]where MeV, MeVGeV, GeV MeV, MeV, MeV, and Ge. The functions and smoothly connect the different behaviors of . Figure 6 shows fluidity measures, and , along the chemical freeze-out line. It can be noted that the fluidity measures decrease as the center of mass energy increases and attains the minimum, and then it does not vary much as increases. This indicates that the fluid behavior of hadronic matter does not change much for the wide range of higher values of collision energies. Thus the matter produced in heavy-ion collision experiments with the wide range of high collision energies can exhibit substantial elliptic flow.

(a)

(b)

5. Summary and Conclusion

In this work we confronted the HRG equation of state with the LQCD results at finite baryon chemical potential. We noted that the ideal HRG model along with its extended forms, , HRG with excluded volume corrections and HRG with discrete as well as continuous Hagedorn states, cannot explain the lattice data simultaneously at zero as well as finite chemical potential. We found that the LQCD data can be accurately reproduced up to MeV at zero as well as finite baryon chemical potential if both the excluded volume corrections and Hagedorn states are simultaneously included in the ideal HRG model. We noted that while the excluded volume corrections suppress the thermodynamical quantities as compared to ideal HRG results, the Hagedorn states provide necessary rapid rise in the trace anomaly observed in LQCD simulation results. With these observations we conclude that the unobserved heavy Hagedorn states play an important role in the thermodynamics of hadronic matter especially near QCD transition temperature.

We also estimated the shear viscosity coefficient within the ambit of the HRG model augmented with excluded volume corrections and Hagedorn states. We found that the behaviors of the fluidity measures and are in agreement with the existing results. We noted that both fluidity measures fall rapidly at high temperature. This fall may be attributed to the rapid rise in entropy density, , and the quantity which are thermodynamically related to each other. We further noted that the fluidity measures do not change much at high collision energies. This indicates that the matter produced in heavy-ion collision experiments with the wide range of higher collision energies can exhibit substantial elliptic flow.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

Guruprasad Kadam is financially supported by the DST-INSPIRE faculty award under Grant No. DST/INSPIRE/04/2017/002293.

References

Y. Aoki, G. Endrődi, Z. Fodor, S. D. Katz, and K. K. Szabó, “The order of the quantum chromodynamics transition predicted by the standard model of particle physics,” Nature, vol. 443, no. 7112, pp. 675–678, 2006.
View at: Publisher Site | Google Scholar
M. P. Lombardo, “Lattice QCD at Finite Temperature and Density,” Modern Physics Letters A, vol. 22, no. 07n10, pp. 457–471, 2007.
View at: Publisher Site | Google Scholar
R. Dashen, S.-K. Ma, and H. J. Bernstein, “S-matrix formulation of statistical mechanics,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 187, no. 1, pp. 345–370, 1969.
View at: Publisher Site | Google Scholar
R. F. Dashen and R. Rajaraman, “Narrow resonances in statistical mechanics,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 10, no. 2, pp. 694–708, 1974.
View at: Publisher Site | Google Scholar
G. Welke, R. Venugopalan, and M. Prakash, “The speed of sound in an interacting pion gas,” Physics Letters B, vol. 245, no. 2, pp. 137–141, 1990.
View at: Publisher Site | Google Scholar
R. Venugopalan and M. Prakash, “Thermal properties of interacting hadrons,” Nuclear Physics A, vol. 546, no. 4, pp. 718–760, 1992.
View at: Publisher Site | Google Scholar
P. Braun-Munzinger, J. Stachel, J. Wessels, and N. Xu, “Thermal and hadrochemical equilibration in nucleus-nucleus collisions at the SPS,” Physics Letters B, vol. 365, pp. 1–6, 1996.
View at: Publisher Site | Google Scholar
G. D. Yen and M. I. Gorenstein, “Analysis of particle multiplicities in Pb+Pb collisions at 158A GeV/c within hadron gas models,” Physical Review C: Nuclear Physics, vol. 59, no. 5, pp. 2788–2791, 1999.
View at: Publisher Site | Google Scholar
F. Becattini, J. Cleymans, A. Keranen, E. Suhonen, and K. Redlich, “Features of particle multiplicities and strangeness production in central heavy ion collisions between 1.7A and 158A GeV/c,” Physical Review C: Nuclear Physics, vol. 64, article no. 024901, 2001.
View at: Publisher Site | Google Scholar
J. Cleymans and H. Satz, “Thermal hadron production in high energy heavy ion collisions,” Zeitschrift für Physik C Particles and Fields, vol. 57, no. 1, pp. 135–147, 1993.
View at: Publisher Site | Google Scholar
P. Braun-Munzinger, D. Magestro, K. Redlich, and J. Stachel, “Hadron production in Au-Au collisions at RHIC,” Physics Letters B, vol. 518, p. 41, 2001.
View at: Publisher Site | Google Scholar
J. Rafelski and J. Letessier, “Testing limits of statistical hadronization,” Nuclear Physics A, vol. 715, p. 98, 2003.
View at: Publisher Site | Google Scholar
A. Andronic, P. Braun-Munzinger, and J. Stachel, “Hadron production in central nucleus-nucleus collisions at chemical freeze-out,” Nuclear Physics A, vol. 772, no. 3-4, pp. 167–199, 2006.
View at: Publisher Site | Google Scholar
S. Chatterjee, R. M. Godbole, and S. Gupta, “Strange freezeout,” Physics Letters B, vol. 727, pp. 554–557, 2013.
View at: Publisher Site | Google Scholar
S. Chatterjee, S. Das, L. Kumar et al., “Freeze-out parameters in heavy-ion collisions at AGS, SPS, RHIC, and LHC energies,” Advances in High Energy Physics, vol. 2015, Article ID 349013, 20 pages, 2015.
View at: Publisher Site | Google Scholar
A. K. Dash, R. Singh, S. Chatterjee, C. Jena, and B. Mohanty, “Role of system size in freeze-out conditions extracted from transverse momentum spectra of hadrons,” Physical Review C, vol. 89, no. 6, article no. 064902, 2018.
View at: Publisher Site | Google Scholar
K. Bugaev, V. Sagun, A. Ivanytskyi et al., “Going beyond the second virial coefficient in the hadron resonance gas model,” Nuclear Physics A, vol. 970, pp. 133–155, 2018.
View at: Publisher Site | Google Scholar
V. V. Sagun, K. A. Bugaev, and A. I. Ivanytskyi, “Hadron resonance gas model with induced surface tension,” The European Physical Journal A, vol. 54, no. 6, p. 100, 2018.
View at: Publisher Site | Google Scholar
K. A. Bugaev, D. R. Oliinychenko, J. Cleymans et al., “Chemical freeze-out of strange particles and possible root of strangeness suppression,” EPL, Europhysics Letters, Physics Journal, vol. 104, no. 2, article no. 064902, 2013.
View at: Publisher Site | Google Scholar
K. A. Bugaev, D. R. Oliinychenko, A. S. Sorin, and G. M. Zinovjev, “Simple solution to the Strangeness Horn description puzzle,” The European Physical Journal A, vol. 49, no. 30, 2013.
View at: Publisher Site | Google Scholar
S. Fubini and G. Veneziano, “Duality in operator formalism,” Il Nuovo Cimento A, vol. 67, no. 1, pp. 29–47, 1970.
View at: Publisher Site | Google Scholar
K. Bardakci and S. Mandelstam, “Analytic solution of the linear-trajectory bootstrap,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 184, no. 5, pp. 1640–1644, 1969.
View at: Publisher Site | Google Scholar
S. Fubini, D. Gordon, and G. Veneziano, “A general treatment of factorization in dual resonance models,” Physics Letters B, vol. 29, no. 10, pp. 679–682, 1969.
View at: Publisher Site | Google Scholar
K. Huang and S. Weinberg, “Ultimate temperature and the early universe,” Physical Review Letters, vol. 25, no. 13, pp. 895–897, 1970.
View at: Publisher Site | Google Scholar
T. D. Cohen, “QCD strings and the thermodynamics of the metastable phase of QCD at large N_c,” Physics Letters B, vol. 637, no. 1-2, pp. 81–84, 2006.
View at: Publisher Site | Google Scholar
C. B. Thorn, “Infinite Nc QCD at finite temperature: Is there an ultimate temperature?” Physics Letters B, vol. 99, no. 6, pp. 458–462, 1981.
View at: Publisher Site | Google Scholar
R. Hagedorn, “Statistical thermodynamics of strong interactions at high energies,” Nuovo Cimento. Supplemento, vol. 3, pp. 147–186, 1965.
View at: Google Scholar
S. Frautschi, “Statistical bootstrap model of hadrons,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 3, no. 11, pp. 2821–2834, 1971.
View at: Publisher Site | Google Scholar
W. Broniowski and W. Florkowski, “Different Hagedorn temperatures for mesons and baryons from experimental mass spectra,” Physics Letters B, vol. 490, no. 3-4, pp. 223–227, 2000.
View at: Publisher Site | Google Scholar
W. Broniowski, W. Florkowski, and L. Y. Glozman, “Update of the Hagedorn mass spectrum,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 70, no. 11, 2004.
View at: Publisher Site | Google Scholar
S. Borsányi, G. Endrodi, Z. Fodor et al., “The QCD equation of state with dynamical quarks,” Journal of High Energy Physics, vol. 2010, no. 11, article 77, pp. 1–31, 2010.
View at: Publisher Site | Google Scholar
D. H. Rischke, J. Schaffner, M. I. Gorenstein, A. Schaefer, H. Stoecker, and W. Greiner, “Quasi-confinement in the SU(3)-gluon plasma,” Zeitschrift für Physik C Particles and Fields, vol. 56, pp. 325–337, 1992.
View at: Publisher Site | Google Scholar
J. I. Kapusta and K. A. Olive, “Thermodynamics of hadrons: Delimiting the temperature,” Nuclear Physics A, vol. 408, no. 3, pp. 478–494, 1983.
View at: Publisher Site | Google Scholar
C. Singh, B. Patra, and K. Singh, “Thermodynamically consistent EOS for hot dense hadron gas,” Physics Letters B, vol. 387, no. 4, pp. 680–684, 1996.
View at: Publisher Site | Google Scholar
F. Karsch, K. Redlich, and A. Tawfik, “Hadron resonance mass spectrum and lattice QCD thermodynamics,” The European Physical Journal C, vol. 29, pp. 549–556, 2003.
View at: Publisher Site | Google Scholar
F. Karsch, K. Redlich, and A. Tawfik, “Thermodynamics at non-zero Baryon number density: a comparison of lattice and Hadron resonance gas model calculations,” Physics Letters B, vol. 571, pp. 67–74, 2003.
View at: Publisher Site | Google Scholar
K. Redlich, F. Karsch, and A. Tawfik, “Heavy-ion collisions and lattice QCD at finite baryon density,” Journal of Physics G: Nuclear and Particle Physics, vol. 30, p. S1271, 2004.
View at: Google Scholar
M. Albright, J. Kapusta, and C. Young, “Matching excluded-volume hadron-resonance gas models and perturbative QCD to lattice calculations,” Physical Review C: Nuclear Physics, vol. 90, no. 2, article no. 024915, 2014.
View at: Publisher Site | Google Scholar
A. Andronic, P. Braun-Munzinger, J. Stachel, and M. Winn, “Interacting hadron resonance gas meets lattice QCD,” Physics Letters B, vol. 718, no. 1, pp. 80–85, 2012.
View at: Publisher Site | Google Scholar
G. P. Kadam and H. Mishra, “Bulk and shear viscosities of hot and dense hadron gas,” Nuclear Physics A, vol. 934, pp. 133–147, 2014.
View at: Publisher Site | Google Scholar
V. Vovchenko, D. V. Anchishkin, and M. I. Gorenstein, “Hadron resonance gas equation of state from lattice QCD,” Physical Review C: Nuclear Physics, vol. 91, no. 2, article no. 024905, 2015.
View at: Publisher Site | Google Scholar
S. Borsanyi, G. Endrodi, Z. Fodor et al., “QCD equation of state at nonzero chemical potential: continuum results with physical quark masses at order μ₂,” Journal of High Energy Physics, vol. 1208, no. 053, 2012.
View at: Publisher Site | Google Scholar
G. S. Denicol, C. Gale, S. Jeon, and J. Noronha, “Fluid behavior of a baryon-rich hadron resonance gas,” Physical Review C: Nuclear Physics, vol. 88, no. 6, article no. 201000056, 2013.
View at: Publisher Site | Google Scholar
A. S. Khvorostukhin, V. D. Toneev, and D. N. Voskresensky, “Viscosity coefficients for hadron and quark–gluon phases,” Nuclear Physics A, vol. 845, no. 1-4, pp. 106–146, 2010.
View at: Publisher Site | Google Scholar
G. P. Kadam and H. Mishra, “Dissipative properties of hot and dense hadronic matter in an excluded-volume hadron resonance gas model,” Physical Review C, vol. 92, no. 3, article no. 035203, 2015.
View at: Publisher Site | Google Scholar
D. Fernández-Fraile and A. Gómez Nicola, “Transport coefficients and resonances for a meson gas in chiral perturbation theory,” The European Physical Journal C, vol. 62, no. 1, pp. 37–54, 2009.
View at: Publisher Site | Google Scholar
K. A. Bugaev, A. I. Ivanytskyi, D. R. Oliinychenko, E. G. Nikonov, V. V. Sagun, and G. M. Zinovjev, “Non-smooth chemical freeze-out and apparent width of wide resonances and quark gluon bags in a thermal environment,” https://arxiv.org/abs/1312.4367.
View at: Google Scholar
C. Gale, S. Jeon, and B. Schenke, “Hydrodynamic modeling of heavy-ion collisions,” International Journal of Modern Physics A, vol. 28, no. 11, article no. 1340011, 2013.
View at: Publisher Site | Google Scholar
B. Schenke, “Flow in heavy-ion collisions—theory perspective,” Journal of Physics G: Nuclear and Particle Physics, vol. 38, article no. 124009, 2011.
View at: Publisher Site | Google Scholar
C. Shen and U. Heinz, “Collision energy dependence of viscous hydrodynamic flow in relativistic heavy-ion collisions,” Physical Review C: Nuclear Physics, vol. 85, no. 5, article no. 054902, 12 pages, 2012.
View at: Publisher Site | Google Scholar
P. F. Kolb and U. W. Heinz, “Quark Gluon Plasma,” R. C. Hwa, Ed., pp. 634–714.
View at: Google Scholar
D. Teaney, J. Lauret, and E. V. Shuryak, “Flow at the SPS and RHIC as a quark-gluon plasma signature,” Physical Review Letters, vol. 86, no. 21, pp. 4783–4786, 2001.
View at: Publisher Site | Google Scholar
L. Del Zanna, V. Chandra, G. Inghirami et al., “Relativistic viscous hydrodynamics for heavy-ion collisions with ECHO-QGP,” The European Physical Journal C, vol. 73, no. 2524, 2013.
View at: Publisher Site | Google Scholar
I. Karpenko, P. Huovinen, and M. Bleicher, “A 3 + 1 dimensional viscous hydrodynamic code for relativistic heavy ion collisions,” Computer Physics Communications, vol. 185, no. 11, pp. 3016–3027, 2014.
View at: Publisher Site | Google Scholar
H. Holopainen, H. Niemi, and K. J. Eskola, “Elliptic flow from event-by-event hydrodynamics,” Journal of Physics G: Nuclear and Particle Physics, vol. 38, no. 12, 2011.
View at: Google Scholar
A. Jaiswal, B. Friman, and K. Redlich, “Relativistic second-order dissipative hydrodynamics at finite chemical potential,” Physics Letters B, vol. 751, pp. 548–552, 2015.
View at: Google Scholar
Z. Xu and C. Greiner, “Thermalization of gluons in ultrarelativistic heavy ion collisions by including three-body interactions in a parton cascade,” Physical Review C, vol. 71, article no. 064901, 2005.
View at: Google Scholar
I. Bouras, E. Molnar, H. Niemi et al., “Investigation of shock waves in the relativistic Riemann problem: A comparison of viscous fluid dynamics to kinetic theory,” Physical Review C, vol. 82, article no. 064901, 2010.
View at: Google Scholar
I. Bouras, A. El, O. Fochler, H. Niemi, Z. Xu, and C. Greiner, “Transition from ideal to viscous Mach cones in a kinetic transport approach,” Physics Letters B, vol. 710, no. 4-5, pp. 641–646, 2012.
View at: Google Scholar
O. Fochler, Z. Xu, and C. Greiner, “Energy loss in a partonic transport model including bremsstrahlung processes,” Physical Review C: Nuclear Physics, vol. 82, no. 2, article no. 024907, 2010.
View at: Publisher Site | Google Scholar
C. Wesp, A. El, F. Reining, Z. Xu, I. Bouras, and C. Greiner, “Calculation of shear viscosity using Green-Kubo relations within a parton cascade,” Physical Review C: Nuclear Physics, vol. 84, no. 5, article no. 054911, 2011.
View at: Publisher Site | Google Scholar
J. Uphoff, O. Fochler, Z. Xu, and C. Greiner, “Open heavy flavor in Pb + Pb collisions at √s=2.76 TeV within a transport model,” Physics Letters B, vol. 717, no. 4-5, pp. 430–435, 2012.
View at: Google Scholar
M. Greif, F. Reining, I. Bouras, G. S. Denicol, Z. Xu, and C. Greiner, “Heat conductivity in relativistic systems investigated using a partonic cascade,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 87, no. 3, article no. 033019, 2013.
View at: Publisher Site | Google Scholar
P. Danielewicz and M. Gyulassy, “Dissipative phenomena in quark-gluon plasmas,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 31, no. 1, pp. 53–62, 1985.
View at: Publisher Site | Google Scholar
M. Gyulassy and L. McLerran, “New forms of QCD matter discovered at RHIC,” Nuclear Physics A, vol. 750, no. 1, pp. 30–33, 2005.
View at: Publisher Site | Google Scholar
L. P. Csernai, J. I. Kapusta, and L. D. McLerran, “Strongly interacting low-viscosity matter created in relativistic nuclear collisions,” Physical Review Letters, vol. 97, no. 15, article no. 033019, 2006.
View at: Publisher Site | Google Scholar
P. K. Kovtun, D. T. Son, and A. O. Starinets, “Viscosity in strongly interacting quantum field theories from black hole physics,” Physical Review Letters, vol. 94, no. 11, article no. 111601, 2005.
View at: Publisher Site | Google Scholar
S. Gavin, “Transport coefficients in ultra-relativistic heavy-ion collisions,” Nuclear Physics A, vol. 435, no. 3-4, pp. 826–843, 1985.
View at: Publisher Site | Google Scholar
M. Prakash, M. Prakash, R. Venugopalan, and G. Welke, “Non-equilibrium properties of hadronic mixtures,” Physics Reports, vol. 227, no. 6, pp. 321–366, 1993.
View at: Publisher Site | Google Scholar
A. Dobado and F. J. Llanes-Estrada, “Viscosity of meson matter,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 69, no. 11, article no. 116004, 2004.
View at: Publisher Site | Google Scholar
J. W. Chen, Y. H. Li, Y. F. Liu, and E. Nakano, “QCD viscosity to entropy density ratio in the hadronic phase,” Physical Review D, vol. 76, article no. 114011, 2007.
View at: Publisher Site | Google Scholar
K. Itakura, O. Morimatsu, and H. Otomo, “Shear viscosity of a hadronic gas mixture,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 77, no. 1, article no. 014014, 2008.
View at: Publisher Site | Google Scholar
A. Dobado, F. J. Llanes-Estrada, and J. M. Torres-Rincon, “η/s and phase transitions,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 79, no. 1, article no. 014014, 2009.
View at: Publisher Site | Google Scholar
N. Demir and S. A. Bass, “Shear-viscosity to entropy-density ratio of a relativistic hadron gas,” Physical Review Letters, vol. 102, no. 17, article no. 172302, 2009.
View at: Publisher Site | Google Scholar
A. Puglisi, S. Plumari, and V. Greco, “Shear viscosity η to electric conductivity σ el ratio for the quark–gluon plasma,” Physics Letters B, vol. 751, pp. 326–330, 2015.
View at: Publisher Site | Google Scholar
L. Thakur, P. Srivastava, G. P. Kadam, M. George, and H. Mishra, “Shear viscosity η to electrical conductivity σ el ratio for an anisotropic QGP,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 95, no. 9, article no. 096009, 2017.
View at: Publisher Site | Google Scholar
A. Tawfik and M. Wahba, “Bulk and shear viscosity in Hagedorn fluid,” Annalen Phys, vol. 522, article no. 201000056, p. 849, 2010.
View at: Publisher Site | Google Scholar
G. Kadam, S. Pawar, and H. Mishra, “Estimating transport coefficients of interacting pion gas with K-matrix cross sections,” Journal of Physics G: Nuclear and Particle Physics, vol. 46, no. 1, 2019.
View at: Publisher Site | Google Scholar
G. P. Kadam, H. Mishra, and L. Thakur, “Electrical and thermal conductivities of hot and dense hadronic matter,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 98, no. 11, article no. 114001, 2018.
View at: Publisher Site | Google Scholar
K. A. Bugaev, V. K. Petrov, and G. M. Zinovjev, “Quark gluon bags as reggeons,” Physical Review C: Nuclear Physics, vol. 79, no. 5, article no. 054913, 2009.
View at: Publisher Site | Google Scholar
M. Beitel, K. Gallmeister, and C. Greiner, “Thermalization of hadrons via Hagedorn states,” Physical Review C: Nuclear Physics, vol. 90, no. 4, 2014.
View at: Publisher Site | Google Scholar
M. I. Gorenstein, A. P. Kostyuk, and Y. D. Krivenko, “Van der Waals excluded-volume model of multicomponent hadron gas,” Journal of Physics G: Nuclear and Particle Physics, vol. 25, no. 9, pp. L75–L83, 1999.
View at: Publisher Site | Google Scholar
K. Bugaev, M. Gorenstein, H. Stöcker, and W. Greiner, “Van der Waals excluded volume model for Lorentz contracted rigid spheres,” Physics Letters B, vol. 485, no. 1-3, pp. 121–125, 2000.
View at: Publisher Site | Google Scholar
L. D. Landau and E. M. Lifshitz, Physical Kinetics, Pergamon, Oxford, 1981.
O. Moroz, “Shear and bulk viscosities of a hadron gas within relaxation time approximation and its test,” Ukrainian Journal of Physics, vol. 58, no. 12, p. 1127, 2013.
View at: Google Scholar
P. Gondolo and G. Gelmini, “Cosmic abundances of stable particles: improved analysis,” Nuclear Physics B, vol. 360, no. 1, pp. 145–179, 1991.
View at: Publisher Site | Google Scholar
J. Noronha-Hostler, J. Noronha, and C. Greiner, “Transport coefficients of hadronic matter near T_c,” Physical Review Letters, vol. 103, no. 17, 2009.
View at: Publisher Site | Google Scholar
J. Noronha-Hostler, J. Noronha, and C. Greiner, “Hadron mass spectrum and the shear viscosity to entropy density ratio of hot hadronic matter,” Physical Review C: Nuclear Physics, vol. 86, no. 2, article no. 024913, 2012.
View at: Publisher Site | Google Scholar
C. Amsler, T. DeGrand, and B. Krusche, “Review of particle physics,” Physics Letters B, vol. 667, no. 1–5, pp. 1–6, 2008.
View at: Publisher Site | Google Scholar
S. Kumar, S. Kumar, and R. K. Puri, “Fluidity and supercriticality of the QCD matter created in relativistic heavy ion collisions,” Physical Review C, vol. 81, article no. 014611, 2010.
View at: Publisher Site | Google Scholar
J. Cleymans, H. Oeschler, K. Redlich, and S. Wheaton, “Comparison of chemical freeze-out criteria in heavy-ion collisions,” Physical Review C: Nuclear Physics, vol. 73, article no. 034905, 2006.
View at: Publisher Site | Google Scholar
A. Tawfik, “On the higher moments of particle multiplicity, chemical freeze-out, and QCD critical endpoint,” Advances in High Energy Physics, vol. 2013, Article ID 574871, 22 pages, 2013.
View at: Publisher Site | Google Scholar
A. Tawfik, “Chemical freeze-out and higher order multiplicity moments,” Nuclear Physics A, vol. 922, pp. 225–236, 2014.
View at: Publisher Site | Google Scholar
A. Tawfik, “Constant-trace anomaly as a universal condition for the chemical freeze-out,” Physical Review C, vol. 88, article no. 035203, 2013.
View at: Publisher Site | Google Scholar
A. Nasser Tawfik, L. I. Abou-Salem, A. G. Shalaby et al., “Particle production and chemical freezeout from the hybrid UrQMD approach at NICA energies,” The European Physical Journal A, vol. 52, no. 10, 2016.
View at: Publisher Site | Google Scholar
A. N. Tawfik, H. Yassin, and E. R. Abo Elyazeed, “Chemical freezeout parameters within generic nonextensive statistics,” Indian Journal of Physics, vol. 92, no. 10, p. 1325, 2018.
View at: Google Scholar

Copyright

Copyright © 2019 Guruprasad Kadam and Swapnali Pawar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

PDF Download Citation

Download other formats

Order printed copies

Views

1556

Downloads

1038

Citations