Shear viscosity of a pion gas resulting from $\rho \pi \pi$ and $\sigma \pi \pi$ interactions

We have evaluated the shear viscosity of pion gas with vanishing chemical potential taking into account its scattering with the low mass resonances $\sigma$ and $\rho$ during propagation in the medium. The thermal width (or collisional rate) of pions is calculated from $\pi \sigma$ and $\pi \rho$ loop diagrams using effective interactions in the real-time formulation of finite temperature field theory. A very small value of shear viscosity by entropy density ratio ($\eta /s$), close to the quantum bound, is obtained which approximately matches the range of values of $\eta /s$ used by Niemi et al. [H. Niemi, G. S. Denicol, P. Huovinen, E. Molnár, and D. H. Rischke, Phys. Rev. Lett. 106, 212302 (2011); Phys. Rev. C 86, 014909 (2012)] to fit the elliptic flow data from the Relativistic Heavy Ion Collider.

Figure 1

One-loop self-energy diagram of a pion.

Figure 2

The imaginary part of pion self-energy from (a) $\pi \rho$ and (b) $\pi \sigma$ loops as a function of the invariant mass $m=\sqrt{k_0^2-{\left|\mathbf{k}\right|}^2}$ for fixed values of temperature $T=0.150$ GeV and three-momentum $\left|\mathbf{k}\right|=0.300$ GeV. The vertical dotted line indicates the on-shell value $m=m_{\pi }$. Parameters are as follows: $m_{\pi }=0.140$ GeV, $m_{\rho }=0.770$ GeV, ${\Gamma }_{\rho }^0=0.150$ GeV, $g_{\rho }=6$, and Set 1 in Table 1 for parameters of $\sigma$ resonance.

Figure 3

Momentum dependence of (a) the thermal width and (b) of the mean free path for a fixed value of temperature $T=0.150$ GeV. Parameters are the same as in Fig. 2.

Figure 4

Temperature dependence of (a) the thermal width and (b) of the mean free path for a fixed value of momentum $\left|\mathbf{k}\right|=0.300$ GeV. Parameters are the same as in Fig. 2.

Figure 5

Temperature dependence of $\eta$ from the $\pi \sigma$ (dashed lines) and $\pi \rho$ (dotted lines) loops. The results (a) without and (b) with folding.

Figure 6

Results of $\eta$ versus $T$ obtained in this work compared to some other results by Lang et al. [39], Fraile et al. [31], and Mitra et al. [40].

Figure 7

The band of uncertainty of $\eta$ in the low temperature domain for different sets of $m_{\sigma }$, $\Gamma \left(m_{\sigma }\right)$ and $g_{\sigma }$ from Table 1.

Figure 8

(a) $\eta /s$ versus $T$ compared to other results by Lang et al. [39], Fraile et al. [31], and Chakraborty et al. [21] and also the KSS bound (dotted line). (b) Dependence of the effective pion mass $m_{\pi }^{*}$ with temperature $T$ at two different values of three-momentum $\left|\mathbf{k}\right|$.