Shear Viscosity of Quark-Gluon Plasma in Weak Magnetic Field in Perturbative QCD: Leading Log

We compute the shear viscosity of two-flavor QCD plasma in an external magnetic field in perturbative QCD at leading log order, assuming that the magnetic field is weak or soft: $eB\sim g^4\log (1\mathrm{/}g)T^2$. We show that the shear viscosity takes a form $\eta =\bar{\eta }(\bar{B})T^3\mathrm{/}(g^4\log (1\mathrm{/}g))$ with a dimensionless function $\bar{\eta }(\bar{B})$ in terms of a dimensionless variable $\bar{B}=(eB)\mathrm{/}(g^4\log (1\mathrm{/}g)T^2)$. The variable $\bar{B}$ corresponds to the relative strength of the effect of cyclotron motions compared to the QCD collisions: $\bar{B}\sim l_{mfp}\mathrm{/}{l}_{cyclo}$. We provide a full numerical result for the scaled shear viscosity $\bar{\eta }(\bar{B})$.