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We compute the longitudinal electrical conductivity in the presence of strongbackground magnetic field in complete leading order of perturbative QCD, basedon the assumed hierarchy of scales $\alpha_s eB\ll (m_q^2,T^2)\ll eB$. Weformulate an effective kinetic theory of lowest Landau level quarks with theleading order QCD collision term arising from 1-to-2 processes that becomepossible due to 1+1 dimensional Landau level kinematics. In small $m_q/T\ll 1$regime, the longitudinal conductivity behaves as $\sigma_{zz}\sime^2(eB)T/(\alpha_s m_q^2\log(m_q/T))$, where the quark mass dependence can beunderstood from the chiral anomaly with the axial charge relaxation provided bya finite quark mass $m_q$. We also present parametric estimates for thelongitudinal and transverse "color conductivities" in the presence of strongmagnetic field, by computing dominant damping rates for quarks and gluons thatare responsible for color charge transportation. We observe that thelongitudinal color conductivity is enhanced by strong magnetic field, whichimplies that the sphaleron transition rate in perturbative QCD is suppressed bystrong magnetic field due to the enhanced Lenz's law in color field dynamics.

Longitudinal Conductivity in Strong Magnetic Field in Perturbative QCD: Complete Leading Order