Spatial Wilson Loops and Energy Loss for Heavy Quarks in Magnetized HQCD Model

TL;DR

This work uses an anisotropic holographic HQCD model with an external magnetic field to study spatial Wilson loops of heavy quarks. A Born–Infeld type action yields DW and horizon string configurations, producing a DW–horizon transition whose temperature $T_{cr}$ exhibits magnetic catalysis and dependence on the anisotropy parameter $\nu$; the phase structure is largely independent of dilaton boundary conditions. In the isotropic limit $\nu=1$, SWL tensions display the expected $\sigma \propto T^2$ behavior, while for $\nu=4.5$ the $T$-dependence deviates, signaling pronounced anisotropic effects on energy loss. The results connect SWL tensions to drag forces and show qualitative agreement with lattice observations, offering a holographic route to explore heavy-quark dynamics in magnetized, anisotropic QGP and guiding future lattice comparisons.

Abstract

We investigate the effective potential and the string tension for the spatial Wilson loop (SWL) in hot dense QGP with two types of anisotropy, i.e. external magnetic field and spatial anisotropy, employing a holographic approach for the heavy quark model. In this approach, the string is extended in the 5th, holographic direction and has a turning point either on a dynamical wall (DW) configuration or on the horizon configuration in the 5th direction. We obtain the magnetic catalysis behavior for a phase transition between DW and horizon configuration of the string. The structure of the phase diagram does not depend on the boundary conditions choice for the dilaton field. Inclusion of the external magnetic field and spatial anisotropy enhance the string tension in the horizon configuration, namely drag force. For the spatially isotropic case $ν= 1$ at different magnetic field values the string tension is proportional to $T^2$ and is qualitatively consistent with lattice results. However, for the anisotropic case, $ν= 4.5$, it deviates from the quadratic term.

Figures (26)

• Figure 1: Temperature as a function of horizon $T(z_h)$ for (A) different $\mu$ at fixed $\nu=1$, (B) $\nu = 4.5$, and (C) different $\nu$ and $c_B$ at fixed $\mu=0$, (D) $\mu=0.2$ GeV; $q_3 = 5$, $R_{gg} = 1.16$ GeV${}^2$, $p = 0.273$ GeV${}^4$; $[T]=[\mu]= [z_h]^{-1} = [c_B]^{\frac{1}{2}}=$ GeV.
• Figure 2: Effective potential ${\cal V}_1(z)$ in the first orientation ${\cal{W}}_{x Y_{1}}$ at (A) $\nu = 1$, and (B) $\nu = 4.5$ for different $c_B$ considering zero-boundary condition \ref{['zerobc']}; $[z]^{-1} = [c_B]^{\frac{1}{2}} =$ GeV.
• Figure 3: Spatial string tension $\sigma_1$ in the first orientation ${\cal{W}}_{x Y_{1}}$ as a function of temperature $T$ for isotropic case $\nu=1$ at (A) $\mu = 0$, (B) $\mu = 0.5$ GeV, and (C) $\mu = 1$ GeV with different $c_B$ considering zero-boundary condition \ref{['zerobc']}. The cyan line and green curve show DW and horizon configuration, respectively; $[\sigma]^{\frac{1}{2}}=[T]=[\mu] = [c_B]^{\frac{1}{2}} =$ GeV.
• Figure 4: Spatial string tension $\sigma_1$ in the first orientation ${\cal{W}}_{x Y_{1}}$ as a function of temperature $T$ for anisotropic case $\nu=4.5$ at (A) $\mu = 0$, (B) $\mu = 0.5$ GeV, and (C) $\mu = 1$ GeV with different $c_B$ considering zero-boundary condition \ref{['zerobc']}. The cyan line and blue curve show DW and horizon configuration, respectively; $[\sigma]^{\frac{1}{2}}=[T]=[\mu] = [c_B]^{\frac{1}{2}} =$ GeV.
• Figure 5: Phase transition diagrams in $(\mu,T)$-plane in the first orientation ${\cal{W}}_{x Y_{1}}$ at (A) $\nu = 1$, (B) $\nu = 4.5$, and (C) comparison between $\nu =1$ and $\nu = 4.5$ at fixed $c_B=-\,1$ GeV$^2$ corresponding to the transition from the DW configuration to the horizon configuration of $\sigma_1$ considering zero-boundary condition \ref{['zerobc']}; $[T]=[\mu] = [c_B]^{\frac{1}{2}} =$ GeV.