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

Recent estimates for the electromagnetic fields produced in the early stages of noncentral ultrarelativistic heavy ion collisions indicate the presence of magnetic fields $B\sim \mathcal{O}(0.1-15m_{\pi }^2)$, 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 paper we study the anisotropy in the shear viscosity induced by an external magnetic field in a strongly coupled $\mathcal{N}=4$ super Yang-Mills (SYM) plasma. Due to the spatial anisotropy created by the magnetic field, the most general viscosity tensor of a magnetized plasma has five shear viscosity coefficients and two 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 $B\ne 0$ 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.

Figure 1

The rescaling parameters $v$ (solid blue curve) and $w$ (dashed black curve) as a function of $b/\sqrt{3}$.

Figure 2

The normalized entropy density $s/(N^2{\mathcal{B}}^{3/2})$ as a function of the dimensionless combination $T/\sqrt{\mathcal{B}}$.

Figure 3

The ratio of shear viscosities $(\eta /s{)}_{\parallel }/(\eta /s{)}_{\perp }$ as a function of $\mathcal{B}/T^2$. The solid blue line is the numerical result from $(\eta /s{)}_{\parallel }/(\eta /s{)}_{\perp }=w/v$; the dashed red line is the asymptotic result valid only when $\mathcal{B}\gg T^2$. (35).