Shear viscosity $\eta$ to electric conductivity ${\sigma }_{el}$ ratio for the Quark-Gluon Plasma

Abstract

The transport coefficients of strongly interacting matter are currently subject of intense theoretical and phenomenological studies due to their relevance for the characterization of the quark-gluon plasma produced in ultra relativistic heavy-ion collisions (uRHIC). We discuss the connection between the shear viscosity to entropy density ratio, $\eta /s$, and the electric conductivity, ${\sigma }_{el}$. Once the relaxation time is tuned to have a minimum value of $\eta /s=1/4\pi$ near the critical temperature $T_c$, one simultaneously predicts ${\sigma }_{el}/T$ very close to recent lQCD data. More generally, we discuss why the ratio of $(\eta /s)/({\sigma }_{el}/T)$ supplies a measure of the quark to gluon scattering rates whose knowledge would allow to significantly advance in the understanding of the QGP phase. We also predict that $(\eta /s)/({\sigma }_{el}/T)$, independently on the running coupling ${\alpha }_s(T)$, should increase up to about $\sim 20$ for $T\to T_c$, while it goes down to a nearly flat behavior around $\simeq 4$ for $T\ge 4T_c$. Therefore we in general predict a stronger T dependence of ${\sigma }_{el}/T$ with respect to $\eta /s$ that in a quasiparticle approach is constrained by lQCD thermodynamics. A conformal theory, instead, predicts a similar T dependence of $\eta /s$ and ${\sigma }_{el}/T$.

Comments: 5 pages, 3 figures. Revised for publication