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 $B eq 0$ 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 $Bsim 1,mathrm{GeV}^2$ 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 $B eq 0$ 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.