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

We calculate the electric conductivity $σ$ 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 $σ$ for $B=0$ is consistent with the lattice-QCD results, we study the case with $B\neq 0$ in which the coefficient $α$ in the Chern-Simons term controls the chiral anomaly strength. Our results imply that the transverse conductivity, $σ_\perp$, 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, $σ_\parallel$, with $B\neq 0$ should diverge due to production of the matter chirality. Yet, it is possible to extract a regulated part out from $σ_\parallel$ with an extra condition to neutralize the matter chirality. This regulated quantity is interpreted as an Ohmic part of $σ_\parallel$. We show that the longitudinal Ohmic conductivity increases with increasing $B$ for small $α$, while it is suppressed with larger $B$ for physical $α=3/4$ due to anomaly induced interactions.