Transport coefficients from the QCD Kondo effect

We study the transport coefficients from the QCD Kondo effect in quark matter which contains heavy quarks as impurity particles. We estimate the coupling constant of the interaction between a light quark and a heavy quark at finite density and temperature by using the renormalization group equation up to two-loop order. We also estimate the coupling constant at zero temperature by using the mean-field approximation as nonperturbative treatment. To calculate the transport coefficients, we use the relativistic Boltzmann equation and apply the relaxation time approximation. We calculate the electric resistivity from the relativistic kinetic theory, and the viscosities from the relativistic hydrodynamics. We find that the electric resistivity is enhanced and the shear viscosity is suppressed due to the QCD Kondo effect at low temperature.

Figure 1

Diagrams for one loop ($i{\mathrm{\Gamma }}_4^{(1)p}$ (left) and $i{\mathrm{\Gamma }}_4^{(1)h}$ (right)). The thin lines represent the light-quark propagators, the thick lines represent the heavy-quark propagators.

Figure 2

Diagram for the four-point vertex with two-loop. The thin lines represent the light-quark propagator, the thick line represents the heavy-quark propagator. The dashed lines are the interaction.

Figure 3

Diagram for the heavy quark propagator with two-loop. The thin lines represent the light-quark propagator, the thick line represents the heavy-quark propagator.

Figure 4

The contour plot of the thermodynamic potential ${\tilde{\mathrm{\Omega }}}_0(\mu ;\lambda ,\delta )$ as a function of $\lambda$ and $\delta$, Eq. (56). The used parameter set is $N_c=3$, $G_c=2(9/2)/{\mathrm{\Lambda }}_{\mathrm{UV}}^2$, ${\mathrm{\Lambda }}_{\mathrm{UV}}=0.65\mathrm{GeV}$ and $\mu =0.5\mathrm{GeV}$. The dashed half circle and straight line indicate the plots of Eqs. (57) and (58), respectively.

Figure 5

The temperature dependence of the effective coupling constants. The black lines for $G_c^{*}(T)=G_c$, the red lines for $G_c^{*}(T)=G_c^{(1)}(T)$, the blue lines for $G_c^{*}(T)=G_c^{(2)}(T)$ and the black blobs for $G_c^{*}(T=0)=G_c^{\mathrm{gs}}$.

Figure 6

The temperature dependence of the relaxation time. The notations are the same as used in Fig. 5.

Figure 7

The temperature dependence of the resistivity. The notations are the same as used in Fig. 5.

Figure 8

The temperature dependence of the shear viscosity. The notations are the same as used in Fig. 5.