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Shear viscosity $η$ to electric conductivity $σ_{el}$ ratio for the Quark-Gluon Plasma

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, $η/s$, and the electric conductivity, $σ_{el}$. Once the relaxation time is tuned to have a minimum value of $η/s=1/4π$ near the critical temperature $T_c$, one simultaneously predicts $σ_{el}/T$ very close to recent lQCD data. More generally, we discuss why the ratio of $(η/s)/(σ_{el}/T)$ 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 $(η/s)/(σ_{el}/T)$, independently on the running coupling $α_s(T)$, should increase up to about $\sim 20$ for $T \rightarrow T_c$, while it goes down to a nearly flat behavior around $\simeq 4$ for $T \geq 4\, T_c$. Therefore we in general predict a stronger T dependence of $σ_{el}/T$ with respect to $η/s$ that in a quasiparticle approach is constrained by lQCD thermodynamics. A conformal theory, instead, predicts a similar T dependence of $η/s$ and $σ_{el}/T$.