Anisotropic shear viscosity of a strongly coupled non-Abelian plasma from magnetic branes

Abstract: Recent estimates for the electromagnetic fields produced in the early stages of non-central ultra-relativistic heavy ion collisions indicate the presence of magnetic fields $B\sim \mathcal{O}(0.1-15\,m_\pi^2)$, where $m_\pi$ is the pion mass. It is then of special interest to study the effects of strong (Abelian) magnetic fields on the transport coefficients of strongly coupled non-Abelian plasmas, such as the quark-gluon plasma formed in heavy ion collisions. In this work we study the anisotropy in the shear viscosity induced by an external magnetic field in a strongly coupled $\mathcal{N} = 4$ SYM plasma. Due to the spatial anisotropy created by the magnetic field, the most general viscosity tensor of a magnetized plasma has 5 shear viscosity coefficients and 2 bulk viscosities. We use the holographic correspondence to evaluate two of the shear viscosities, $\eta_{\perp} \equiv \eta_{xyxy}$ (perpendicular to the magnetic field) and $\eta_{\parallel} \equiv \eta_{xzxz}=\eta_{yzyz}$ (parallel to the field). When $B\neq 0$ the shear viscosity perpendicular to the field saturates the viscosity bound $\eta_{\perp}/s = 1/(4\pi)$ while in the direction parallel to the field the bound is violated since $\eta_{\parallel}/s < 1/(4\pi)$. However, the violation of the bound in the case of strongly coupled SYM is minimal even for the largest value of $B$ that can be reached in heavy ion collisions.