Shear and bulk viscosities of quark matter from quark-meson fluctuations in the Nambu–Jona-Lasinio model

We have calculated the temperature dependence of shear $\eta$ and bulk $\zeta$ viscosities of quark matter due to quark-meson fluctuations. The quark thermal width originating from quantum fluctuations of quark-$\pi$ and quark-$\sigma$ loops at finite temperature is calculated with the formalism of real-time thermal field theory. Temperature-dependent constituent-quark and meson masses and quark-meson couplings are obtained in the Nambu–Jona-Lasinio model. We found a nontrivial influence of the temperature-dependent masses and couplings on the Landau-cut structure of the quark self-energy. Our results for the ratios $\eta /s$ and $\zeta /s$, where $s$ is the entropy density (also determined in the Nambu–Jona-Lasinio model in the quasiparticle approximation), are in fair agreement with results of the literature obtained from different models and techniques. In particular, our result for $\eta /s$ has a minimum very close to the quantum lower bound, $\eta /s=1/4\pi$.

Figure 1

The diagram (a) represents a schematic one-loop diagram of the quark correlator, which can be obtained from the two-point function of the viscous-stress tensor for the quark constituents. The double-dashed lines for the quark propagators indicate that they have some finite thermal width which can be derived from the quark self-energy diagrams (b) for quark-meson loops (for $\pi ,\sigma$ meson).

Figure 2

(a) Temperature dependence of ${\mathrm{\Gamma }}_Q$ (solid line) for $k=0.5$ GeV and the individual contribution from $\pi$- (dashed line) and $\sigma$-quark loops (dotted line). (b) Temperature dependence of quark mass (solid line), the pion (dotted line), and the sigma (dashed line) meson masses and quark-meson couplings $(g_{QQ\pi }^2/4\pi$: dash-dotted line; $g_{QQ\sigma }^2/4\pi$: dash-double-dotted line). The straight vertical red line denotes the Mott temperature $(T_M=0.187$ GeV).

Figure 3

Invariant mass distribution of the quark thermal width from the $Q\pi$ (solid line) and $Q\sigma$ (dotted line) loops for $\left|\mathit{k}\right|=500$ MeV for temperatures (a) $T=0.180$ GeV, (b) $0.190$ GeV, and (c) $0.200$ GeV. The curve for ${\mathrm{\Gamma }}_{Q\left(Q\sigma \right)}\left(M\right)$ has been multiplied by $-1$.

Figure 4

(a) Temperature dependence of the quark thermal width ${\mathrm{\Gamma }}_Q$ and (b) collisional time ${\tau }_Q=1/{\mathrm{\Gamma }}_Q$ for three different values of momentum: $\mathit{k}=0$ (dashed line), $0.5$ GeV (solid line), and 1 GeV (dotted line). The vertical red line indicates the Mott temperature.

Figure 5

(a) Temperature dependence of $\eta$ and (b) $\eta /s$ for different values of constant collisional time: ${\tau }_Q=5$ fm (dashed line), 1 fm (solid line), and $0.2$ fm (dotted line). The straight horizontal red line indicates the KSS bound, $\eta /s=1/4\pi$.

Figure 6

(a) The temperature dependence of entropy density, normalized by Stephan-Boltzmann (SB) limiting value $s_{SB}$, given in Eq. (30). (b) The squared speed of sound $c_s^2$ as a function of the temperature. Straight horizontal dotted lines stand for the corresponding SB limits.

Figure 7

(a) Temperature dependence of $\zeta$ and (b) $\zeta /s$ for different values of constant ${\tau }_Q$. In the lower panel, also shown are results from the literature: Ref. [40] (filled circles and squares), Ref. [53] (open circles), Ref. [52] (stars).

Figure 8

Temperature dependence of the auxiliary quantities defined in the text. (a) $A=1/3-c_s^2$ (solid line) and $B=d\left({\beta }^2{M}_Q^2\right)/d{\beta }^2$ (dashed line). (b) $C=A{\mathit{k}}^2$, for $\left|\mathit{k}\right|=0.5$ GeV (solid line) and $\left|\mathit{k}\right|=0.3$ GeV (dashed line), $D=c_s^{2}B$ (dotted line). (c) $E={(C-D)}^2$ for $\left|\mathit{k}\right|=0.5$ GeV (solid line) and $\left|\mathit{k}\right|=0.3$ GeV (dashed line).

Figure 9

(a) Temperature dependence of $\eta$ and (b) $\eta /s$ using the full momentum dependence of the quark thermal width ${\mathrm{\Gamma }}_Q\left(\mathit{k}\right)$. Comparison with results from the literature: Ref. [26] (filled circles), Ref. [44] (open circles), Ref. [34] (open squares), Ref. [28] (filled squares), Ref. [32] (triangles), Ref. [36] (stars).

Figure 10

(a) Temperature dependence of $\zeta$ and (b) $\zeta /s$ using the full momentum dependence of the quark thermal width ${\mathrm{\Gamma }}_Q\left(\mathit{k}\right)$. Comparison with results from the literature: Ref. [57] (circles), Ref. [28] (squares), Ref. [52] (triangles), Ref. [34] (stars).

Figure 11

$\eta /s$ (a) and $\zeta /s$ (b) in the high temperature domain. HTL (circles) and FRG (stars) estimations of $\eta /s$ by Arnold et al. [9] and Christiansen et al. [73] are compared with $\eta /s$ of Eq. (14) at ${\tau }_Q=4.5$ fm (dashed line) and ${\tau }_Q=0.75$ fm (solid line). The HTL (circles) estimation of $\eta /s$ by Arnold et al. [10] is compared with $\zeta /s$ of Eq. (15) at ${\tau }_Q=0.3$ fm (dashed line) and ${\tau }_Q=5$ fm (solid line).