Chiral susceptibility in the Nambu–Jona-Lasinio model: A Wigner function approach

We estimate here chiral susceptibility at finite temperature within the framework of the Nambu–Jona-Lasinio model (NJL) using the Wigner function approach. We also estimate it in the presence of chiral chemical potential (${\mu }_5$) as well as a nonvanishing magnetic field ($B$). We use a medium separation regularization scheme (MSS) in the precence of magnetic field to calculate the chiral condensate and corresponding susceptibility. It is observed that for a fixed value of chiral chemical potential (${\mu }_5$), transition temperature increases with the magnetic field. While for the fixed value of the magnetic field, transition temperature decreases with chiral chemical potential. For a strong magnetic field, we observe nondegeneracy in susceptibility for up and down type quarks.

Figure 1

Left plot: variation of constituent quark mass $M_u=M_d$ with temperature ($T$) for zero magnetic field but for various values of chiral chemical potential. Right plot: variation of chiral susceptibility ${\chi }_c$ with temperature ($T$) for zero magnetic field but with different values of chiral chemical potential. Prominent peak in the chiral susceptibility plot shows the chiral transition temperature. From the left plot, it is clear that with increasing chiral chemical potential (${\mu }_5$) constituent mass decreases. From the susceptibility plot, it is clear that transition temperature decreases with chiral chemical potential.

Figure 2

Variation of constituent quark masses $M_u$ and $M_d$ with temperature for vanishing chiral chemical potential but with finite magnetic field for different values of $\alpha$. For vanishing magnetic field, $M_u$ and $M_d$ are same. Note that in the presence of magnetic field, for $\alpha =0.5$, although $⟨{\overline{\psi }}_u{\psi }_u⟩\ne ⟨{\overline{\psi }}_d{\psi }_d⟩$, but the constituent quark masses $M_u=M_d$. However, for $\alpha \ne 0.5$, the constituent quark masses $M_u\ne M_d$ in the presence of magnetic field. $\alpha =0.0$ corresponds to the case when there is no flavor mixing interaction, and $\alpha =0.5$ corresponds to maximal flavor mixing.

Figure 3

Left plot: variation of constituent quark mass $M_u$ and $M_d$, with temperature for vanishing chiral chemical potential, but with different values of magnetic field for $\alpha =0.5$. Right plot: variation of chiral susceptibility ${\chi }_c$ with temperature ($T$) for vanishing chiral chemical potential, but with different values of magnetic field for $\alpha =0.5$. From the left plot, it is clear that with increasing magnetic field constituent mass $M_u=M_d$ increases. From the susceptibility plot, it is clear that transition temperature increases with magnetic field.

Figure 4

Left plot: variation of constituent quark mass $M_u$ and $M_d$, with temperature for vanishing chiral chemical potential, but with different values of magnetic field for $\alpha =0.0$. Right plot: variation of chiral susceptibility ${\chi }_c$ with temperature ($T$) for vanishing chiral chemical potential, but with different values of magnetic field for $\alpha =0.0$. From the left plot, it is clear that with increasing magnetic field constituent mass increases. From the susceptibility plot, it is clear that transition temperature increases with magnetic field. In the right plot, we can observe two distinct peaks at relatively large magnetic fields.

Figure 5

Left plot: variation of ${\chi }_{cu}$, ${\chi }_{cd}$, and ${\chi }_c$ with temperature at vanishing chiral chemical potential for $eB=0.4{\mathrm{GeV}}^2$ and $\alpha =0.0$. Right plot: variation of ${\chi }_{cu}$, ${\chi }_{cd}$, and ${\chi }_c$ with temperature at vanishing chiral chemical potential for $eB=0.4{\mathrm{GeV}}^2$ and $\alpha =0.5$. From the left plot, it is clear that chiral susceptibility shows two distinct peaks at large magnetic field. This is due to the fact that at large magnetic field, difference between $M_u$ and $M_d$ is large. On the other hand, the right plot shows that for $\alpha =0.5$, $⟨{\overline{\psi }}_u{\psi }_u⟩\ne ⟨{\overline{\psi }}_d{\psi }_d⟩$, ${\chi }_{cu}$ and ${\chi }_{cd}$ show peak at same temperature. Hence, for $\alpha =0.5$, at finite magnetic field, chiral transition temperature for $u$ and $d$ quarks is the same.

Figure 6

Left plot: variation of constituent quark mass $M_u=M_d$, with temperature for finite magnetic field and finite chiral chemical potential. Right plot: variation of chiral susceptibility ${\chi }_c$ with temperature for finite magnetic field and finite chiral chemical potential. From this figure, it is clear that with increasing chiral chemical potential quark mass as well as the chiral transition temperature decreases.