Bulk viscosity for pion and nucleon thermal fluctuation in the hadron resonance gas model

We have calculated microscopically bulk viscosity of hadronic matter, where equilibrium thermodynamics for all hadrons in medium are described by the hadron resonance gas (HRG) model. Considering pions and nucleons as abundant medium constituents, we have calculated their thermal widths, which inversely control the strength of bulk viscosities for respective components and represent their in-medium scattering probabilities with other mesonic and baryonic resonances, present in the medium. Our calculations show that bulk viscosity increases with both temperature and baryon chemical potential, whereas viscosity to entropy density ratio decreases with temperature and with baryon chemical potential, the ratio increases first and then decreases. The decreasing nature of the ratio with temperature has been observed in most of the earlier investigations with few exceptions. We find that the temperature dependence of bulk viscosity crucially depends on the structure of the relaxation time. Along the chemical freeze-out line in nucleus-nucleus collisions with increasing collision energy, bulk viscosity as well as the bulk viscosity to entropy density ratio decreases, which also agrees with earlier references. Our results indicate the picture of a strongly coupled hadronic medium.

Figure 1

Pion self-energy diagram with mesonic loops (a) and baryonic loops [(b) and (c) are direct and cross diagrams] and nucleon self-energy diagram (d).

Figure 2

Momentum dependence of pion thermal width for mesonic (dash-dotted line), baryonic loops (dotted line), and their total (solid line) and nucleon thermal width (dashed line) at three different medium parameters: (a) $(T,{\mu }_B)=(0.130$ GeV, $0)$, (b) $(0.170$ GeV, $0)$, and (c) $(0.130$ GeV, 0.300 GeV).

Figure 3

$\zeta \left(T\right)$ due to pion thermal width for $\pi \sigma$ (dotted line), $\pi \rho$ (dashed line) loops, and their total (solid line) at $v_n^2=0$ (a), $v_n^2=1/3$ (b), and $v_n^2\left(T\right)$ from HRG (c). Blue circles represents the results of Ref. [37].

Figure 4

(a) Temperature dependence of bulk viscosity for pion thermal width with mesonic loops (dash-dotted line), meson + baryon loops (dotted line), for nucleon thermal width (dashed line), and their total ${\zeta }_T={\zeta }_{\pi }+{\zeta }_N$ (solid line). (b) $\zeta \left(T\right)$ for $v_n^2\left(T\right)$ from HRG and two constant values of $v_n^2$ ($v_n^2=0.15$: dotted line and $v_n^2=0.25$: dashed line).

Figure 5

$\zeta \left(T\right)$ due to nucleon thermal width (a), pion thermal width for meson loops (b), and meson + baryon loops (c) at ${\mu }_B=0.250$ GeV (dotted line) and 0.500 GeV (dashed line).

Figure 6

Same as Fig. 5 along ${\mu }_B$ axis at $T=0.050$ GeV (dotted line), 0.100 GeV (dashed, line) and 0.150 GeV (solid line).

Figure 7

(a) $v_n^2\left(T\right)$ at ${\mu }_B=0$ (solid line), 0.250 GeV (dotted line), and 0.500 GeV (dashed line), and LQCD results of $v_n^2(T,{\mu }_B=0)$ (circles) [3]; (b) ${\left(\frac{\partial P}{\partial n_B}\right)}_{\epsilon }$ vs $T$ at ${\mu }_B=0.250$ GeV (dotted line) and 0.500 GeV (dashed line); (c) $v_n^2\left({\mu }_B\right)$ at $T=0.050$ GeV (dotted line), 0.100 GeV (dashed line), and 0.150 GeV (solid line); (d) ${\left(\frac{\partial P}{\partial n_B}\right)}_{\epsilon }$ vs ${\mu }_B$ at $T=0.050$ GeV (dotted line), 0.100 GeV (dashed line), and 0.150 GeV (solid line).

Figure 8

$T$ dependence of total bulk viscosities (a), entropy densities from HRG (b), and their ratios $\zeta /s$ (c) at ${\mu }_B=0$ (solid line), 0.250 GeV (dotted line) and 0.500 GeV (dashed line).

Figure 9

${\mu }_B$ dependence of total bulk viscosities (a), entropy densities from HRG (b), and their ratios $\zeta /s$ (c) at $T=0.050$ GeV (dotted line), 0.100 GeV (dashed line), and 0.150 GeV (solid line).

Figure 10

Center of mass energy ($\sqrt{s}$) dependence of total bulk viscosity (a), entropy density from HRG (b), and their ratio $\zeta /s$ (c).

Figure 11

Our results of $\zeta$ (a) and $\zeta /s$ (b) vs $T$ at ${\mu }_B=0$ for $\pi$ component (dotted line), ($\pi +N$) components (dashed line) and ($\pi +N$ + other resonances $R$) components are compared with the earlier results of Sasaki et al. (green triangles down [20]), Deb et al. (pink solid squares [25]), Chakraborty et al. (brown stars [28]), Marty et al. (open circles [21]), Kadam et al. (violet pluses [34]), Fraile et al. (blue solid circles [37]), Hostler et al. (open squares [36]), Mitra et al. (grey triangles up [32]).