SU(3) Polyakov linear-$\sigma$ model in an external magnetic field

In the present work, we analyze the effects of an external magnetic field on the chiral critical temperature $T_c$ of strongly interacting matter. In doing this, we can characterize the magnetic properties of the quantum chromodynamics (QCD) strongly interacting matter, the quark-gluon plasma (QGP). We investigate this in the framework of the SU(3) Polyakov linear sigma model (PLSM). To this end, we implement two approaches representing two systems, in which the Polyakov-loop potential added to PLSM is either renormalized or non-normalized. The effects of Landau quantization on the strongly interacting matter are conjectured to reduce the electromagnetic interactions between quarks. In this case, the color interactions will be dominant and increasing, which in turn can be achieved by increasing the Polyakov-loop fields. Obviously, each of them equips us with a different understanding about the critical temperature under the effect of an external magnetic field. In both systems, we obtain a paramagnetic response. In one system, we find that $T_c$ increases with increasing magnetic field. In the other one, $T_c$ significantly decreases with increasing magnetic field.

Figure 1

Left-hand panel (a): the strange and nonstrange condensates of the renormalized approach, Eq. (19), at magnetic fields $eB=0.019$ ${\mathrm{GeV}}^2$ (double-dotted curve), $eB=0.2$ ${\mathrm{GeV}}^2$ (dotted curve), and $eB=0.4$ ${\mathrm{GeV}}^2$ (dashed curve). Right-hand panel (b): the same as in the left-hand panel (a) but for the nonrenormalized approach, Eq. (20).

Figure 2

Left-hand panel (a): the Polyakov loop field $\varphi$ of the renormalized approach, Eq. (20), at magnetic fields $eB=0.019$ ${\mathrm{GeV}}^2$ (double-dotted curve), $eB=0.2$ ${\mathrm{GeV}}^2$ (dotted curve), and $eB=0.4$ ${\mathrm{GeV}}^2$ (dashed curve). Right-hand panel (b): the same as in the left-hand panel (a) but for the nonrenormalized approach, Eq. (19).

Figure 3

Left-hand panel (a): the derivative of strange and nonstrange condensates $\partial {\sigma }_f/\partial T$ of the renormalized approach, Eq. (20), in the magnetic fields $eB=0.019$ ${\mathrm{GeV}}^2$ (double-dotted curve), $eB=0.2$ ${\mathrm{GeV}}^2$ (dotted curve), and $eB=0.4$ ${\mathrm{GeV}}^2$ (dashed curve). Right-hand panel (b): the same as in the left-hand panel (a) but for the nonrenormalized approach, Eq. (19).

Figure 4

Left-hand panel (a): the Polyakov-loop potential $\varphi$ of the renormalized approach, Eq. (20) at magnetic fields $eB=0.019$ ${\mathrm{GeV}}^2$ (double-dotted curve), $eB=0.2$ ${\mathrm{GeV}}^2$ (dotted curve), and $eB=0.4$ ${\mathrm{GeV}}^2$ (dashed curve). Right-hand panel (b): the same as in the left-hand panel (a) but for the nonrenormalized approach, Eq. (19).

Figure 5

Left-hand panel (a): the subtracted chiral-condensate ${\Delta }_{ls}$ of the renormalized approach, Eq. (20), in external magnetic fields $eB=0.019$ ${\mathrm{GeV}}^2$ (double-dotted curve), $eB=0.2$ ${\mathrm{GeV}}^2$, (dotted curve) and $eB=0.4$ ${\mathrm{GeV}}^2$ (dashed curve). Right-hand panel (b): the same as in the left-hand panel (a) but for the nonrenormalized approach, Eq. (19).

Figure 6

The thermal evolution of the mesonic potential of LSM is studied at vanishing chemical potential. Accordingly, this part of potential can be excluded, especially at high temperatures.

Figure 7

Left-hand panel (a): thermodynamic pressure of the renormalized approach, Eq. (20), at magnetic fields $eB=0.019$ ${\mathrm{GeV}}^2$ (double-dotted curve), $eB=0.2$ ${\mathrm{GeV}}^2$ (dotted curve), and $eB=0.4$ ${\mathrm{GeV}}^2$ (dashed curve). Right-hand panel (b): the same as in the left-hand panel (a) but for the nonrenormalized approach, Eq. (19).

Figure 8

Left-hand panel (a): thermodynamic trace anomaly of the renormalized approach, Eq. (20), at various values of the external magnetic field $eB=0.019$ ${\mathrm{GeV}}^2$ (double-dotted curve), $eB=0.2$ ${\mathrm{GeV}}^2$ (dotted curve), and $eB=0.4$ ${\mathrm{GeV}}^2$ (dashed curve). Right-hand panel (b): the same as in the left-hand panel (a) but for nonrenormalized approach, Eq. (19).

Figure 9

The magnetic susceptibility $\chi$ of the renormalized approach, Eq. (20) (dotted curve), and for the nonrenormalized approach, Eq. (19) (dashed curve). The symbols represent the lattice QCD results at different lattice spacing $a$ and sizes $L_s$ [17].

Figure 10

Left-hand panel (a): the magnetic chiral phase diagram in an external magnetic field. The critical temperature $T_c$ is given in dependence on $eB$ from the renormalized approach, Eq. (20), for light chiral condensate (double-dotted curve) and for strange chiral condensate (dotted curve). Right-hand panel (b): the same as in the left-hand panel (a) but for the nonrenormalized approach, Eq. (19).

Figure 11

The magnetic chiral phase diagram, $T_c$ vs $eB$, in an external magnetic field from the renormalized approach, Eq. (20), for strange (dotted curve) and light quark chiral phase transition (double dotted curve), compared with lattice QCD calculations [39] for light (open boxes) and strange quark (open circles) chiral phase transition [39] (dotted curve).

Figure 12

The magnetic chiral phase diagram, $T$ vs $eB$, in an external magnetic field from the non-normalized approach, Eq. (19), for light quark chiral phase transition of PLSM (double dotted curve), compared with PNJL (dash-dotted curve) and EPNJL (dotted curve) models [68].