Electrical conductivity of hadronic matter from different possible mesonic and baryonic loops

Electromagnetic current-current correlators in pionic and nucleonic medium have been evaluated in the static limit to obtain electrical conductivities for pion and nucleon components, respectively, where the former decreases and the latter increases with the variation of temperature $T$ and baryon chemical potential ${\mu }_N$. Therefore, total electrical conductivity of pion and nucleon system exhibits a valley structure in the $T\text{−}{\mu }_N$ plane. To get nondivergent and finite values of correlators, and finite thermal widths of medium constituents, the pion and nucleon have been considered, where the thermal widths have been determined from the in-medium scattering probabilities of the pion and nucleon with other mesonic and baryonic resonances, based on the effective hadronic model. At ${\mu }_N=0$, the results of the present work more or less agree with the results of earlier works and their finite ${\mu }_N$ extension shows a decreasing nature of electrical conductivity for the hadronic medium.

Figure 1

The diagram (a) is a schematic one-loop representation of the electromagnetic current-current correlator for the medium with pionic constituents. The external photon lines are coupled with double dashed internal lines of pions, which have some finite thermal width. The thermal width of the pion can be derived from its self-energy diagrams (b)–(d), where (b) represents pion self-energy for mesonic ($\pi M$) loops, whereas diagrams (c) and (d) are direct cross diagrams of pion self-energy for $NB$ loops.

Figure 2

The diagram (a) is a schematic one-loop representation of the electromagnetic current-current correlator for the medium with nucleonic constituents. Similar to the double dashed lines of pions in Fig. (1), here double solid lines of the nucleon indicate that they have finite thermal width, which can be obtained from the nucleon self-energy diagram (b) for $\pi B$ loops.

Figure 3

Temperature dependence of electrical conductivity for the pionic medium due to its different mesonic loops, $\pi \sigma$ (dotted line), $\pi \rho$ (dashed line) loops and their total (solid line). With and without the folding effect of resonances $M=\sigma$, $\rho$ are taken in the upper (a) and lower (b) panels, respectively.

Figure 4

Effect of baryonic loops ($N\mathrm{\Delta }$ loop, dashed line; $NB$ loops, solid line) after adding with mesonic loops ($\pi M$ loops, dotted line) of the pion on ${\sigma }_{\pi }(T)$ at ${\mu }_N=0$ (a) and ${\mu }_N=0.300\mathrm{GeV}$ (b).

Figure 5

The same as Fig. 4 for ${\sigma }_{\pi }({\mu }_N)$ at $T=0.120$ (a) and $T=0.150\mathrm{GeV}$ (b).

Figure 6

Temperature dependence of electrical conductivity for pion (dotted line) and nucleon (dashed line) components and their total (solid line) at ${\mu }_N=0.500$ (a) and ${\mu }_N=0.300\mathrm{GeV}$.

Figure 7

${\mu }_N$ dependence of electrical conductivity for pion (dotted line) and nucleon (dashed line) components and their total (solid line) at $T=0.120$ (a) and $T=0.150\mathrm{GeV}$.

Figure 8

Total electrical conductivity ${\sigma }_T$ in the $T-{\mu }_N$ plane.

Figure 9

Our results of $\sigma (T,{\mu }_N=0)/T$ are compared with the results of Refs. [11, 12, 18, 19] (a). The valley structure of $\sigma (T)$ at ${\mu }_N=0.400$, 0.500, and 0.600 GeV is shown by solid, dotted, and dashed lines, respectively, in panel (b).

Figure 10

(a) Valley structure of $\sigma ({\mu }_N)$ at three different $T$. (b) The points of minima (solid circles) and the freeze-out line [40] (solid line) are shown in the $T-{\mu }_N$ plane.

Figure 11

One-loop and two-loop contributions of electrical conductivity of pionic medium for $\sigma \pi \pi$ interaction are shown by the dotted line (a) and dash-dotted line (b). The solid line (a) is their total.