Longitudinal conductivity in strong magnetic field in perturbative QCD: Complete leading order

We compute the longitudinal electrical conductivity in the presence of a strong background magnetic field in complete leading order of perturbative QCD, based on the assumed hierarchy of scales alpha(s)eB << (m(q)(2), t(2)) << eB. We formulate an effective kinetic theory of lowest Landau level quarks with the leading order QCD collision term arising from 1-to-2 processes that become possible due to 1 + 1 dimensional Landau level kinematics. In the small m(q)/T << 1 regime, the longitudinal conductivity behaves as sigma(zz) similar to e(2) (eB)T/(alpha(s)m(q)(2) log(T/m(q))), where the quark mass dependence can be understood from the chiral anomaly with the axial charge relaxation provided by a finite quark mass m(q). We also present parametric estimates for the longitudinal and transverse color conductivities in the presence of the strong magnetic field, by computing dominant damping rates for quarks and gluons that are responsible for color charge transportation. We observe that the longitudinal color conductivity is enhanced by the strong magnetic field, which implies that the sphaleron transition rate in perturbative QCD is suppressed by the strong magnetic field due to the enhanced Lenz's law in color field dynamics.