Shear viscosity to entropy density ratio of QCD below the deconfinement temperature

Using chiral perturbation theory we investigate the QCD shear viscosity (η) to entropy density (s) ratio below the deconfinement temperature (∼170MeV) with zero baryon number density. It is found that η/s of QCD is monotonically decreasing in temperature (T) and reaches 0.6 with estimated ∼50% uncertainty at T=120MeV. A naive extrapolation of the leading order result shows that η/s reaches the 1/4π minimum bound proposed by Kovtun, Son, and Starinets using string theory methods at T∼200MeV. This suggests a phase transition or cross over might occur at T≲200MeV in order for the bound to remain valid. Also, it is natural for η/s to stay close to the minimum bound around the phase transition temperature as was recently found in heavy ion collisions.