Electric conductivity with the magnetic field and the chiral anomaly in a holographic QCD model

We calculate the electric conductivity $\sigma$ in deconfined QCD matter using a holographic QCD model, i.e., the Sakai-Sugimoto Model with varying magnetic field $B$ and chiral anomaly strength. After confirming that our estimated $\sigma$ for $B=0$ is consistent with the lattice-QCD results, we study the case with $B\neq 0$ in which the coefficient $\alpha$ in the Chern-Simons term controls the chiral anomaly strength. Our results imply that the transverse conductivity, $\sigma_\perp$, is suppressed to be $\lesssim 70\%$ at $B\sim 1\,\mathrm{GeV}^2$ as compared to the $B=0$ case when the temperature is fixed as $T= 0.2\,\mathrm{GeV}$. Since the Sakai-Sugimoto Model has massless fermions, the longitudinal conductivity, $\sigma_\parallel$, with $B\neq 0$ should diverge due to production of the matter chirality. Yet, it is possible to extract a regulated part out from $\sigma_\parallel$ with an extra condition to neutralize the matter chirality. This regulated quantity is interpreted as an Ohmic part of $\sigma_\parallel$. We show that the longitudinal Ohmic conductivity increases with increasing $B$ for small $\alpha$, while it is suppressed with larger $B$ for physical $\alpha=3/4$ due to anomaly induced interactions.