The ratio of viscosity to entropy density in a pion gas satisfies the KSS holographic bound

We evaluate the ratio of shear viscosity to entropy density in a pion gas employing the UehlingUehlenbeck equation and experimental phase-shifts parameterized by means of the SU(2) Inverse Amplitude Method. We find that the ratio for this monocomponent gas stays well above the KSS 1/(4π) bound. We find similar results with other sets of phase shifts and conclude the bound is nowhere violated. PACS. 05.20.Dd – 51.20.+d Recently, Kovtun, Son and Starinets [1] have conjectured a universal bound for the viscosity to entropy ratio of any one-component dilute gas based on gravity duality arguments and the Heisenberg uncertainty principle. In natural units the conjecture is that the quotient of the shear viscosity to the entropy density η/s is greater or equal than 4π, for any quantum field theory with one field alone. A general proof does not exist but no counterexample is yet known either. This has caused considerable stir in the heavy-ion collision community as RHIC experiments are providing us with a picture of strongly interacting matter [2] that is close to a perfect fluid and features low viscosity. Prospects for an indirect measurement of viscosity in heavy-ion collisions through sufficiently precise hydrodynamic codes remain appealing [3] Follow-up studies are under preparation by several groups from the hadron and from the quark-gluon phases in RHICtheory to see the effect of the phase transition on the viscosity. We have presented a comprehensive study of the shear viscosity of a hadronic gas at low and moderate temperature [4]. Other transport coefficients are under active study, for example there is a recent paper featuring the electrical conductivity of the pion gas in chiral perturbation theory [5] and a recalculation of the thermal conductivity is under preparation. In this brief report we employ our published results to address the viscosity to entropy density ratio. We employ the same notation and conventions as in our earlier publication [4]. In particular we employ the SU(2) Inverse Amplitude Method parametrization of the pion-pion scattering experimental phase shifts (as well as alternative parametrizations to check the sensitivity of the calculation presented). To show a consistent ratio where both numerator and denominator, η and s = S/V are computed with the same approximation, we need to realize that the viscosity of the pion gas is defined hydrodynamically near equilibrium, that is, for infinitesimally small gradients of the gas velocity field. Therefore the same approximation can be taken in the calculation of the entropy, and this has to be evaluated at equilibrium employing the Bose-Einstein distribution function f0. Further, the Uehling-Uehlenbeck (quantum Boltzmann) equation is decoupled from the BGKY hierarchy by assuming low density, that is, between two consecutive interactions, well separated by a large mean free path, particles decorrelate and behave as if the density of available states corresponded to a free gas. Therefore it is also fair to employ the free gas approximation in the computation of the entropy density. Hence we are entitled to write