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Shear and Bulk Viscosities of a Weakly Coupled Quark Gluon Plasma with Finite Chemical Potential and Temperature---Leading-Log Results

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 contrary, 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 \geq 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.