We calculate the shear ($\eta$) and bulk ($\zeta$) viscosities of a weakly coupled quark gluon plasma at the leading-log order with finite temperature $T$ and quark chemical potential $\mu$. We find that the shear viscosity to entropy density ratio $\eta /s$ increases monotonically with $\mu$ and eventually scales as $(\mu /T{)}^2$ at large $\mu$. In contrast, $\zeta /s$ is insensitive to $\mu$. Both $\eta /s$ and $\zeta /s$ are monotonically decreasing functions of the quark flavor number $N_f$ when $N_f\ge 2$. This property is also observed in pion gas systems. Our perturbative calculation suggests that QCD becomes the most perfect (i.e. with the smallest $\eta /s$) at $\mu =0$ and $N_f=16$ (the maximum $N_f$ with asymptotic freedom). It would be interesting to test whether the currently smallest $\eta /s$ computed close to the phase transition with $\mu =0$ and $N_f=0$ can be further reduced by increasing $N_f$.

• Received 23 December 2012

DOI:https://doi.org/10.1103/PhysRevD.87.036002