Shear viscosity to entropy density ratio of a relativistic Hagedorn resonance gas
Abstract
The new state of matter produced at Relativistic Heavy Ion Collider reveals a strongly coupled quarkgluon plasma with an extremely small shear viscosity to entropy density ratio . We calculate the of an equilibrated hadron matter characterized by a relativistic hadron resonance gas with a Hagedorn mass spectrum that grows exponentially with the hadron mass. We find with increase in temperature of the system the value decreases due to rapid increase in the multiplicity of massive resonances. In the vicinity of the critical temperature for deconfinement transition, the minimum value of in the Hagedorn resonance gas is found to be consistent with the current estimates for a strongly coupled quarkgluon plasma.
pacs:
12.38.Mh, 24.85.+p, 25.75.qHeavy ion collisions at the BNL Relativistic Heavy Ion Collider (RHIC) have revealed a new state of matter BRAHMS ; PHOBOS ; STAR ; PHENIX comprising of strongly interacting quarks and gluons (sQGP) DIS . This conclusion is based on viscous hydrodynamic model analysis of elliptic flow that requires a very small shear viscosity to entropy density ratio of Mike ; Paul ; Song . The main uncertainty in these estimates stems from the equation of state and the initial conditions employed. This value is remarkably similar to the lower bound obtained by KovtunSonStarinets (KSS) KSS for infinitely coupled supersymmetric YangMills gauge theory based on the AdS/CFT duality conjecture. A recent lattice calculation Meyer of gluonic plasma do support the current estimates. While leading order perturbative calculations result in a significantly large value of for Arnold .
It has been also argued Csernai ; Lacey that with increasing temperature in the hadronic phase decreases and reaches a minimum at or near the critical temperature and increases thereafter in the deconfined phase. Indeed this behavior has been observed in several substances in nature all of which satisfy the KSS bound suggesting the bound could be universal. However, no consensus have yet been reached Blaizot on the physical mechanism that accounts for the thermodynamic and transport properties of the system that leads to the minimal viscosity. Understanding the transport coefficients of the matter formed at RHIC is thus very challenging. Since in a heavy ion collision the system evolves from a QGP phase to the confined hadrons (low temperature phase of QCD), it is important to understand systematically the transport properties of a hadronic system in order to ascertain the sQGP properties with minimal uncertainty. In this letter, we investigate the transport properties of a infinite equilibrated hadronic matter comprising of massive hadronic resonances the Hagedorn states Hagedorn ; Kapusta ; Venu and all the lowlying observed hadrons within a Monte Carlo sequential binary emission model PalPD ; Pal .
Several calculations of in the hadron phase have been performed with various techniques such as the chiral perturbation theory, linearized Boltzmann equation Dobado ; Chen ; Itakura , and microscopic transport theory Muronga . However, most of these calculations employ at most two component system or mesonic gas with a wide variation in the estimate of at . A recent microscopic transport calculation Bass within the UrQMD model accounted all the measured hadron species of mass GeV yields a minimum at zero baryon chemical potential. This is significantly higher than in the viscous hydrodynamic estimate. Clearly it indicates the importance of massive resonances on the transport properties of hadron gas in the strong interaction domain of QCD matter especially near the critical temperature MeV predicted in lattice simulations Cheng .
On the other hand, it was proposed Hagedorn that the density of hadronic states grows exponentially with resonance mass , , where MeV is the Hagedorn temperature. The measured hadronic states up to GeV are indeed qualitatively consistent with the Hagedorn mass spectrum Bron . In absence of hadronic interaction, the energy density for such a system will diverge leading to a maximum/limiting temperature for the hadronic matter. In the infinite mass limit, the thermodynamical quantities for the Hagedorn resonance gas would thus show critical behavior as it crosses which may be associated with deconfining transition Castorina . It has been also argued that the idea of Hagedorn states (HS) arise naturally in QCD at large Cohen and their indirect evidence can be found in lattice studies of thermodynamic properties of hadron matter above the deconfinement temperature Bringoltz . For a hadron resonance gas with Hagedorn states the reduction of in the vicinity of has been demonstrated Noronha within the rate equation approach.
In the present study of the properties of an equilibrated Hagedorn gas in the binary emission model PalPD ; Pal , we include the Hagedorn states (mass GeV) and all the lowlying measured hadrons available in the Particle Data Book. Detail description of the decay widths of HS and other resonances and their formation cross section in binary collisions during dynamic evolution of the system can be found in Ref. PalPD ; Pal . For brevity, we mention that depending on the mass, a HS may undergo binary decay into one of the three possible channels (i) two observed discrete hadrons (DH) of GeV, (ii) a DH and a HS, (iii) two HSs. Based on Hagedorn hypothesis the density of massive states are assumed to be
(1) 
in the usual notation PalPD ; Pal . Here GeV, and a HS of mass is characterized by its baryon, strangeness, spin and isospin quantum numbers with a ground state mass ; the parameters () are determined empirically from the measured smallest masses. This model was found to successfully describe the stable and resonance yield ratios and their spectra at RHIC. We consider here the Hagedorn temperature MeV consistent with the critical temperature in lattice QCD prediction of a crossover transition at vanishing baryon chemical potential Cheng . A resulting exponent is then estimated by comparing the theoretical and experimental cumulants of the spectrum Bron ; PalPD .
To generate the equilibrated matter we initiate a system of Hagedorn states in a box with periodic boundary conditions in configuration space. Subsequent binary decay and their regeneration in binary collision during time evolution enforce the system to thermodynamic equilibrium. Chemical equilibrium is determined from the saturation of various particle densities when their production and annihilation rates become identical. While kinetic equilibration occurs when the system approaches momentum isotropization. By fitting the particle energy spectra with a Boltzmann distribution , thermal equilibration is verified when all the particle species at later times can be described by the same slope temperature . For pointparticles employed in our simulation, the pressure can be evaluated from the virial theorem as
(2) 
where and are the momenta and energy of the th particle. While the inclusion of Hagedorn states in the system provides an attractive interaction, use of point particles in the simulation ignores the repulsive interactions among the finite size hadrons especially the massive HS Hagedorn ; Kapusta ; Venu that would drive the system to deconfinement at . The effects of repulsive interaction can be included via excludedvolume approach Kapusta where the volume occupied by th hadron in the relativistic approximation is , where is the effective MIT bag constant. The excludedvolume temperature and pressure are related to their pointparticle analogs as Kapusta
(3) 
The other thermodynamic quantities can be obtained using the thermodynamic identities. We consider GeV that results in with a corresponding limiting temperature Kapusta as evident from Eq. (3). We restrict our calculations to in the confined hadronic system.
Figure 1 shows the time evolution of various particle abundances at zero net baryon and strangeness densities. At a temperature MeV, the initial distribution of massive Hagedorn states ( GeV) decay and regenerate within few fm/c. Subsequently the HS multiplicity decreases rapidly to a small value with simultaneous production of lighter ( GeV) hadrons. In contrast, at MeV each massive HS decays dominantly to a lighter HS and a discrete hadron resulting in a continuous increase in the density of Hagedorn states and other hadron resonances. At about fm/c the particle densities saturate suggesting chemical equilibration has been achieved.
In Fig. 2 (left panel) we present the equation of state, i.e. pressure versus energy density, of an equilibrated hadron matter at zero net baryon density. With increasing energy density when the temperature , enhanced particle production of especially heavy resonances and HS causes softening of the equation of state. Consequently the speed of sound in the medium drops gradually from at MeV to about 0.09 at . The estimated minimum speed of sound is consistent with the lattice data Cheng . In contrast, the UrQMD model hadron resonance gas (without HS) yields a constant value of Bass .
To gauge the dynamics of hadronhadron interaction, we also investigate the thermodynamic properties in an independent statistical model for an hadron resonance gas in the grand canonical ensemble. The particle densities in the statistical model is given by Belkacem
(4) 
where the sign refers to fermions/bosons. For the th species, is the spinisospin degeneracy and is the chemical potential with and the baryon and strangeness chemical potentials, respectively. For the observed resonances, is taken as BreitWigner mass distribution while for HS it is replaced by of Eq. (1). The results are found to be rather insensitive to the choice of upper mass limit of integration for HS GeV. In the limit , as the estimated exponent in Eq. (1), the pointparticle energy density, pressure and entropy density of an ideal classical (Boltzmann) Hagedorn gas remain finite at and diverge at exhibiting critical behavior Castorina ; albeit in the excludedvolume approach, the corresponding quantities evaluated at would remain finite.
In Fig. 2 (left panel) we also show the equation of state obtained in the statistical model. Compared to a pion gas (stars), inclusion of measured discrete resonances up to mass GeV (triangles) results in a considerable decrease in the system pressure. The interaction measure, , then gradually increases as . Further inclusion of Hagedorn states in the statistical model opens up additional degrees of freedom and provides an attractive interaction that slows the pressure increase at GeV/fm. The trace anomaly then increases dramatically near . The speed of sound in the statistical model with HS turns out to be rather small at . Comparison of this curve with sequential binary emission model result provides a measure of the dynamics of repulsive interaction in the medium. The interaction effect starts at temperature above the pion mass and becomes significant at high energies where massive resonances are produced.
The pointparticle entropy density of the system can be computed using the Gibbs formula
(5)  
where is the net baryon density and the (initial) net strangeness density is set to zero in our calculation. The baryon chemical potential is evaluated from the particle yield ratios at equilibrium. Figure 2 (right panel) shows the excludedvolume entropy density Kapusta as a function of temperature at zero baryon chemical potential. In the statistical model, compared to the pion gas, larger degrees of freedom in the hadron (and Hagedorn) resonance gas at the same energy density lowers its temperature which results in higher entropy PalPratt . In the sequential emission model the increase (or slope) of entropy density with temperature near is somewhat less dramatic and captures the trend associated with that of a crossover transition obtained in the lattice data Cheng .
We now extract the shear viscosity for the dynamically evolving system of an equilibrated hadron resonance gas with Hagedorn states using the Kubo relation Kubo . The transport coefficients determine the dynamics of fluctuations of dissipative fluxes about the equilibrated state. The Kubo formalism employs the linear response theory to relate the transport coefficients as correlations of dissipative fluxes by considering the dissipative fluxes as perturbations to local thermodynamic equilibrium Kubo ; Hosoya ; Paech . The GreenKubo relation for shear viscosity is
(6) 
Here refers to time after the system equilibrates which is set at , and is the shear component (traceless part) of the energy momentum tensor :
(7) 
The equivalence accounts for the point particles used in our simulation. The averaging in Eq. (6) is over the ensemble of events generated in the simulation.
Figure 3 displays the correlation function of the shearstress tensor as a function of time. The correlation functions are found to damp exponentially with time, i.e. . Note at small temperatures (right panel) the correlations sustain for long times however show a power law tail. The relaxation times for the shear fluxes so obtained by fitting the correlation functions are found to decrease with increasing temperature. The shear viscosity is estimated from Eq. (6) by integrating the correlation function over time using the exponential decay found empirically.
The shear viscosity to entropy density ratio for the Hagedorn resonance gas in equilibrium at zero baryon chemical potential is presented in Fig. 4. Also shown in the figure are the values for hadron resonance gas without Hagedorn states in the UrQMD model Bass . Our results for are comparable to the UrQMD estimates at temperatures MeV. At higher temperatures, massive resonances contribute dominantly resulting in gradual reduction of with temperature Noronha and reaches a minimum of at MeV. In contrast, the UrQMD model calculation shows almost a constant value for at MeV. The reduction of with Hagedorn mass spectrum can be realized as primarily due to enhancement of massive resonances that leads to a decrease in the mean free path of a particle and a corresponding increase in the average binary collision cross section . In classical transport theory, as (where is the average momentum of the particle), implies a reduction of shear viscosity. The at zero baryon chemical potential for the equilibrated Hagedorn resonance gas is significantly above the KSS bound of KSS while somewhat close to the upper bound of 0.24 obtained from viscous hydrodynamic analysis Paul ; Song of elliptic flow; albeit is assumed constant throughout the hydrodynamic evolution of the system. Note however, for Hagedorn resonance gas treated within the rate equation approach Noronha , an upper limit of estimated in the kinetic theory is found to be as small as the KSS bound near .
The measured antibaryontobaryon ratio at midrapidity for central Au+Au collision at RHIC BRAHMS ; PHOBOS ; STAR ; PHENIX suggests the hadronic matter formed possess, though small, but a finite net baryon number density . During the late hadronic stage of evolution essentially from chemical equilibrium to thermal equilibrium the system could acquire nonequilibrium chemical composition Hirano . In fact, reproduction of the measured particle yield ratios at RHIC requires a nonzero baryon chemical potential in the sequential binary emission model PalPD ; Pal , while thermal model fits Braun ; Rafelski to the Au+Au collision data at the RHIC energy of GeV yields a chemical freezeout temperature of MeV and baryon chemical potential MeV. In Fig. 4 we show as a function of temperature at a finite baryon chemical potential () created by inducing a finite in the initial distribution of Hagedorn states. At a given , the energy density of the system is larger for nonzero baryon chemical potential leading to further enhancement of particularly massive particle abundances. In the region explored, the shear viscosity increases compared to that at . This increment can be understood classically substantially large particle densities though increase and thereby reduces of each species , however the total contribution, , from these species is greatly enhanced Itakura . Since the entropy density of the system also grows with , the resulting at is found to be similar to . In Fig. 4 we also illustrate the effect of high associated with heavy ion collisions at the top AGS energy of GeV and at the lowest CERN/SPS energy of GeV Braun ; Rafelski ; albeit the temperatures reached in these reactions are much smaller at MeV. We find that only for such large the extracted decreases further. The present study also suggests the need to incorporate, in the viscous hydrodynamics calculations, the Hagedorn states in the afterburning hadronic phase to quantify the dissipation and thus ratio.
While weakly coupled perturbative QCD estimate of at , the QGP formed at RHIC in the region is thought to be strongly interacting. The current estimate of Mike ; Paul ; Song ; Meyer ; PalAMPT at RHIC may be related to a minimum reached in the deconfined phase just above . Our estimate of for the Hagedorn resonance gas suggests that, as the sQGP cools, the in deconfined phase makes a relatively continuous/smooth transition into the confined phase near without any discontinuity or a sharp increase proposed Bass in absence of Hagedorn states.
In summary, we have studied the thermodynamic and transport properties of an infinite equilibrated matter composed of interacting hadron resonance gas plus an exponentially increasing Hagedorn mass spectrum. At temperatures close to the Hagedorn temperature, MeV, the particle densities of the massive resonances are enhanced significantly. This leads to a sharp rise in the trace anomaly and the entropy density of the system which captures the trend seen in the lattice data at vanishing baryon chemical potential. The shear viscosity to entropy density ratio of the Hagedorn resonance gas, both in and out of chemical equilibrium, decreases with temperature and baryon chemical potential and reaches a minimum of at about . The extracted minimum value is near the upper bound of current estimates for a strongly coupled quarkgluon plasma formed in ultrarelativistic heavy ion collisions at RHIC.
I would like to thank Steffen Bass, Rajeev Bhalerao and Pawel Danielewicz for many helpful discussions.
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