Heavy quark transport in a viscous semi-QGP

We study the effect of shear and bulk viscosities on the heavy quark transport coefficient within the matrix model of semi-QGP. Dissipative effects are incorporated through the first-order viscous correction in the quark/antiquark and gluon distribution function. It is observed that while the shear viscosity effects reduce the drag of heavy quark the bulk viscosity effects increase the drag and the diffusion coefficients of heavy quark. For finite values of $\eta /s$ and $\xi /s$, the Polyakov loop further decreases the drag coefficient and enhances the momentum diffusion coefficient as compared to perturbative QCD. It is also observed that with increases in $\eta /s$ and $\xi /s$ spatial diffusion coefficient decreases.

Figure 1

Coulomb scattering (left) of HQ (bold solid line) and light quark/antiquark (thin solid line). t-channel Compton scattering (right). The curly line represent a gluon.

Figure 2

s-channel Compton scattering (left). u-channel Compton scattering (right).

Figure 3

Left panel: The ratio of the drag coefficient i.e., $A(\eta )/A(\eta =0)$ is shown as a function of temperature for $p=1\mathrm{GeV}$, $\eta /s=0.1$, $\tau =0.3{\mathrm{fm}}^{-1}$ and $\xi /s=0$. The blue curve represents the perturbative QCD result while the red curve corresponds to including Polyakov loop within the matrix model. Right panel: The ratio $A(\xi )/A(\xi =0)$ is shown as a function of temperature for $\xi /s=0.03$, $p=1\mathrm{GeV}$, $\tau =0.3{\mathrm{fm}}^{-1}$ and $\eta /s=0$. Similar to the left panel, the blue curve corresponds to pQCD results and the red curve corresponds to including Polyakov loop within the matrix model.

Figure 4

Left panel: The ratio $A(\eta )/A(\eta =0)$ as a function of temperature for HQ momentum $p=1\mathrm{GeV}$ and $\xi /s=0$. The topmost curve is for $\eta /s=0.1$, $\tau =0.3{\mathrm{fm}}^{-1}$, bottom-most, i.e., dashed black curve is for $\eta /s=0.17$, $\tau =0.3{\mathrm{fm}}^{-1}$ and the blue curve is for $\eta /s=0.1$, $\tau =0.5{\mathrm{fm}}^{-1}$. Right panel: The ratio of the drag coefficient $A(\xi )/A(\xi =0)$ as a function of temperature for HQ momentum $p=1\mathrm{GeV}$ and $\eta /s=0$. The blue curve corresponds to $\xi /s=0.03$, $\tau =0.3{\mathrm{fm}}^{-1}$, the red curve corresponds to $\xi /s=0.03$, $\tau =0.5{\mathrm{fm}}^{-1}$ and the dashed black curve corresponds to $\xi /s=0.015$, $\tau =0.3{\mathrm{fm}}^{-1}$.

Figure 5

Left panel: Variation of $A(\eta )/A(\eta =0)$ as a function of momentum for $\xi /s=0$ and $T=220\mathrm{MeV}$. The red curve corresponds to $\eta /s=0.1$, $\tau =0.5{\mathrm{fm}}^{-1}$, the blue curve corresponds to $\eta /s=0.1$, $\tau =0.3{\mathrm{fm}}^{-1}$ and the dashed black curve corresponds to $\eta /s=0.17$, $\tau =0.3{\mathrm{fm}}^{-1}$. Right panel: Variation of $A(\xi )/A(\xi =0)$ as a function of momentum for $\eta /s=0$ and $T=220\mathrm{MeV}$. The blue curve corresponds to $\xi /s=0.03$, $\tau =0.3{\mathrm{fm}}^{-1}$, the red curve corresponds to $\xi /s=0.03$, $\tau =0.5{\mathrm{fm}}^{-1}$ and the dashed black curve corresponds to $\xi /s=0.01$, $\tau =0.3{\mathrm{fm}}^{-1}$.

Figure 6

Left panel: The ratio $A(\eta ,\xi )/A(\eta =0,\xi =0)$ as a function of temperature for HQ momentum $p=1\mathrm{GeV}$ and $\tau =0.3{\mathrm{fm}}^{-1}$. The blue curve corresponds to $\eta /s=0.2$, $\xi /s=0.035$, the red curve corresponds to $\eta /s=0.15$, $\xi /s=0.020$, and the black curve corresponds to $\eta /s=0.1$, $\xi /s=0.01$. Right panel: The ratio $A(\eta ,\xi )/A(\eta =0,\xi =0)$ as a function of heavy quark momentum for $T=220\mathrm{MeV}$ and $\tau =0.3{\mathrm{fm}}^{-1}$. The blue curve corresponds to $\eta /s=0.1$, $\xi /s=0.01$, the red curve corresponds to $\eta /s=0.15$, $\xi /s=0.020$, and the black curve corresponds to $\eta /s=0.22$, $\xi /s=0.04$.

Figure 7

Left panel: The ratio $B_0(\eta ,\xi )/B_0(\eta =0,\xi =0)$ of diffusion coefficient as a function of temperature for HQ momentum $p=1\mathrm{GeV}$ and $\tau =0.3{\mathrm{fm}}^{-1}$. The black curve corresponds to $\eta /s=0.1$, $\xi /s=0.01$, the red curve corresponds to $\eta /s=0.15$, $\xi /s=0.03$, and the blue curve corresponds to $\eta /s=0.2$, $\xi /s=0.05$. Right panel: The ratio $B_0(\eta ,\xi )/B_0(\eta =0,\xi =0)$ of diffusion coefficient as a function of HQ momentum for temperature $T=220\mathrm{MeV}$ and $\tau =0.3{\mathrm{fm}}^{-1}$. The black curve corresponds to $\eta /s=0.1$, $\xi /s=0.01$, the red curve corresponds to $\eta /s=0.15$, $\xi /s=0.03$, and the blue curve corresponds to $\eta /s=0.2$, $\xi /s=0.05$.

Figure 8

The spatial diffusion $2\pi D_{x}T$ as a function of temperature scaled by $T_c$ and $\tau =0.3{\mathrm{fm}}^{-1}$. The blue curve (dashed dotted) corresponds to $\eta /s=0.0$, $\xi /s=0.0$, the red curve (dashed) corresponds to $\eta /s=0.1$, $\xi /s=0.01$, and the black curve (solid) corresponds to $\eta /s=0.2$, $\xi /s=0.04$. Here, the lattice data for Polyakov loop is taken from Ref. [44] and the QCD running coupling constant ${\alpha }_s$ as defined in Eq. (60). The brown curve (dotted) are LO pQCD results for constant coupling ${\alpha }_s=0.4$ [43] and the green (dots) are lattice results taken from the Ref. [18].