Aspects of holographic Langevin diffusion in the presence of anisotropic magnetic field

TL;DR

This work addresses heavy quark diffusion in a strongly coupled, anisotropic plasma under a uniform magnetic field using holography. By computing all five Langevin diffusion coefficients with the membrane paradigm in the magnetic-branes background, it reveals how velocity and directions of motion and diffusion shape momentum broadening, including a violation of the isotropic universal relation $\\kappa_{\\parallel} > \\\kappa_{\\perp}$ in this anisotropic setup. The study provides both numerical results across parameter space and analytic results in the strong-field limit $\\mathcal{B} \gg T^2$, showing distinct $\\mathcal{B}$-scaling: $\\kappa^{v\\parallel B}_{\\parallel}$ is $B$-independent, $\\kappa^{v\\parallel B}_{\\perp} \propto B$, and $\\kappa^{v\\perp B}$ components scale as $B^{3/2}$ with velocity-dependent prefactors. These findings advance the understanding of heavy quark transport in magnetized, anisotropic plasmas and offer guidance for transport simulations in holographic QGP models.

Abstract

We study the holographic Langevin diffusion coefficients of a heavy quark, when traveling through a strongly coupled anisotropic plasma in the presence of magnetic field~$\mathcal{B}$. In particular, we clarified how motion velocities shape momentum broadening and its directional dependence. It is observed that the transverse Langevin diffusion coefficients depend more on the direction of motion rather than the directions of momentum diffusion at the ultra-fast limit, while an opposite conclusion is found when the moving speed is sufficiently low. For the Longitudinal Langevin diffusion coefficient, we find that motion perpendicular to~$\mathcal{B}$ affects the Langevin coefficients stronger at any fixed velocity. We should also emphasize that all five Langevin coefficients are becoming larger with increasing velocity. We find that the universal relation~$κ^{\parallel}>κ^{\perp}$ in the isotropic background, is broken in a different new case that a quark moving paralleled to~$\mathcal{B}$. This is one more particular example where the violation of the universal relation occurs for the anisotropic background. Further, we find the critical velocity of the violation will become larger with increasing~$\mathcal{B}$.

Figures (5)

• Figure 1: $v(b)$ (dash curve) and $w(b)$ (solid curve) against $b$.
• Figure 2: The transverse LGV-coefficients $\kappa^{v\parallel B}_{\perp}$, $\kappa^{v\perp B}_{(\parallel,\perp)}$ and $\kappa^{v\perp B}_{(\perp,\perp)}$, at $v=0.99$ (Upper panel) and at $v=0.1$ (Lower panel), as a function of $\mathcal{B}/T^2$ are normalized by the conformal limit.
• Figure 3: The longitudinal LGV-coefficients $\kappa^{v\parallel B}_{\parallel}$ and $\kappa^{v\perp B}_{\parallel}$, at $v=0.99$ (Upper panel) and at $v=0.1$ (Lower panel), as a function of $\mathcal{B}/T^2$ are normalized by the conformal limit.
• Figure 4: The transverse LGV-coefficients $\kappa_{\perp}$ are shown in the upper panel, while the longitudinal LGV-coefficients $\kappa_{\parallel}$ are shown in the lower panel. The corresponding conformal limit result is also included as a function of the motion velocity $v$.
• Figure 5: The ratios of $\kappa_{\parallel}/\kappa_{\perp}$ as a function of quark motion velocity $v$ in different case. The only violation is $\kappa^{v\perp B}_{\parallel}/\kappa^{v\perp B}_{(\perp,\perp)}$ at both $\mathcal{B}/T^2=9.6$ (Upper panel) and $\mathcal{B}/T^2=99.9$ (Lower panel).