Shear viscosity to entropy density ratio of a relativistic Hagedorn resonance gas Subrata Pal Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India The new state of matter produced at Relativistic Heavy Ion Collider reveals a strongly coupled quark-gluon plasma with an extremely small shear viscosity to entropy density ratio η/s . We calculate the η/s 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 η/s 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 η/s in the Hagedorn resonance gas is found to be consistent with the current estimates for a strongly coupled quark-gluon plasma. PACS numbers: 12.38.Mh, 24.85.+p, 25.75.-q It has been also argued [12, 13] that with increasing temperature in the hadronic phase η/s decreases and reaches a minimum at or near the critical temperature T$_{c}$ 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 [14] on the physical mechanism that ac- counts for the thermodynamic and transport properties of the system that leads to the minimal viscosity. Under- standing 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 equi- librated hadronic matter comprising of massive hadronic resonances − the Hagedorn states [15–17] and all the low- lying observed hadrons within a Monte Carlo sequential binary emission model [18, 19]. Several calculations of η/s in the hadron phase have been performed with various techniques such as the chi- In the present study of the properties of an equilibrated Hagedorn gas in the binary emission model [18, 19], we include the Hagedorn states (mass > ∼ 2 GeV) and all the low-lying 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 On the other hand, it was proposed [15] that the den- sity of hadronic states grows exponentially with reso- nance mass m , ρ$_{HS}$ ( m ) ∼ m − $^{α}$exp( m/T$_{H}$ ), where T$_{H}$ ∼ 150 − 200 MeV is the Hagedorn temperature. The mea- sured hadronic states up to m ∼ 2 GeV are indeed qual- itatively consistent with the Hagedorn mass spectrum [26]. In absence of hadronic interaction, the energy den- sity for such a system will diverge leading to a maxi- mum/limiting temperature T$_{H}$ for the hadronic matter. In the infinite mass limit, the thermodynamical quan- tities for the Hagedorn resonance gas would thus show critical behavior as it crosses T$_{H}$ ≈ T$_{c}$ which may be as- sociated with deconfining transition [27]. It has been also argued that the idea of Hagedorn states (HS) arise natu- rally in QCD at large N$_{c}$ [28] and their indirect evidence can be found in lattice studies of thermodynamic proper- ties of hadron matter above the deconfinement temper- ature [29]. For a hadron resonance gas with Hagedorn states the reduction of η/s in the vicinity of T$_{c}$ has been demonstrated [30] within the rate equation approach. ral perturbation theory, linearized Boltzmann equation [20–22], and microscopic transport theory [23]. However, most of these calculations employ at most two compo- nent system or mesonic gas with a wide variation in the estimate of η/s ∼ 0 . 08 − 1 at ∼ T$_{c}$ . A recent microscopic transport calculation [24] within the UrQMD model ac- counted all the measured hadron species of mass m < ∼ 2 GeV yields a minimum η/s ≈ 0 . 9 at zero baryon chemical potential. This is significantly higher than in the viscous hydrodynamic estimate. Clearly it indicates the impor- tance of massive resonances on the transport properties of hadron gas in the strong interaction domain of QCD matter especially near the critical temperature T$_{c}$ = 196 MeV predicted in lattice simulations [25]. arXiv:1001.1585v1 [nucl-th] 11 Jan 2010 Heavy ion collisions at the BNL Relativistic Heavy Ion Collider (RHIC) have revealed a new state of matter [1– 4] comprising of strongly interacting quarks and gluons (sQGP) [5]. This conclusion is based on viscous hydro- dynamic model analysis of elliptic flow that requires a very small shear viscosity to entropy density ratio η/s of 0 . 08 − 0 . 24 [6–8]. The main uncertainty in these es- timates stems from the equation of state and the initial conditions employed. This value is remarkably similar to the lower bound η/s ≥ 1 / 4 π obtained by Kovtun-Son- Starinets (KSS) [9] for infinitely coupled supersymmetric Yang-Mills gauge theory based on the AdS/CFT duality conjecture. A recent lattice calculation [10] of gluonic plasma do support the current estimates. While leading order perturbative calculations result in a significantly large value of η/s ≈ 0 . 8 for α$_{s}$ = 0 . 3 [11].