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

TL;DR

The paper uses gauge/gravity duality to compute two anisotropic shear viscosities of a strongly coupled $ ext{N}=4$ SYM plasma in a constant external magnetic field. Via Einstein–Maxwell holography, it shows $rac{η_{\perp}}{s}=rac{1}{4π}$, while $rac{η_{\parallel}}{s}<rac{1}{4π}$, with the violation becoming significant only when the magnetic energy dominates the temperature ($\mathcal{B}/T^2 \gg 1$). The parallel viscosity is obtained from a scalar-like fluctuation with an $r$-dependent coupling, yielding $rac{η_{\parallel}}{s}=rac{1}{4π}\frac{w}{v}$ in rescaled variables, and asymptotically $rac{η_{\parallel}}{s}\sim rac{πT^2}{\mathcal{B}}$ for large $\mathcal{B}$. The results imply that anisotropic transport effects in the QGP under feasible magnetic fields are small, supporting the continued use of isotropic viscous hydrodynamics for phenomenology while highlighting directions for non-conformal extensions and stability analyses.

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 $B\sim \mathcal{O}(0.1-15\,m_π^2)$, where $m_π$ 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, $η_{\perp} \equiv η_{xyxy}$ (perpendicular to the magnetic field) and $η_{\parallel} \equiv η_{xzxz}=η_{yzyz}$ (parallel to the field). When $B\neq 0$ the shear viscosity perpendicular to the field saturates the viscosity bound $η_{\perp}/s = 1/(4π)$ while in the direction parallel to the field the bound is violated since $η_{\parallel}/s < 1/(4π)$. 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.

Figures (3)

• Figure 1: (Color online) 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: (Color online) 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 $\mathcal{B} \gg T^2$. \ref{['eq:high']}