Electric conductivity of hot and dense quark matter in a magnetic field with Landau level resummation via kinetic equations

We compute the electric conductivity of quark matter at finite temperature $T$ and quark chemical potential $\mu$ under a magnetic field $B$ beyond the Lowest Landau level approximation. The electric conductivity transverse to $B$ is dominated by the Hall conductivity $\sigma_H$. For the longitudinal conductivity $\sigma_\parallel$, we need to solve kinetic equations. Then, we numerically find that $\sigma_\parallel$ has only mild dependence on $\mu$ and the quark mass $m_q$. Moreover, $\sigma_\parallel$ first decreases and then linearly increases as a function of $B$, leading to an intermediate $B$ region which looks consistent with the experimental signature for the chiral magnetic effect. We also point out that $\sigma_\parallel$ at nonzero $B$ remains within the range of the lattice-QCD estimate at $B=0$.