Chiral symmetry breaking, color superconductivity, and equation of state for magnetized strange quark matter

We investigate the vacuum structure of dense quark matter in strong magnetic fields in a three-flavor Nambu Jona Lasinio (NJL) model including the Kobayashi-Maskawa-t’Hooft (KMT) determinant term using a variational method. The method uses an explicit construct for the “ground” state in terms of quark-antiquark condensates as well as diquark condensates in the background of a constant magnetic field. The coupled mass gap equations and the superconducting gap equation are solved self-consistently and are used to compute the thermodynamic potential along with charge neutrality conditions imposed for bulk matter. Within the model, we observe inverse magnetic catalysis for chiral symmetry breaking for moderate magnetic fields. Further, we observe gapless modes in the presence of the magnetic field when charge neutrality conditions are imposed. The equation of state for charge neutral magnetized strange quark matter is derived, and found to be stiffer compared to the vanishing magnetic field counterpart. This could be relevant for gross structural properties of neutron stars.

Figure 1

Constituent quark masses and superconducting gap when charge neutrality conditions are not imposed. Part (a) shows the ${\mathrm{M}}_u$ at zero temperature as a function of quark chemical potential for different values of the magnetic field. Part (b) shows the same for the strange quark mass ${\mathrm{M}}_s$ and the superconducting gap.

Figure 2

Baryon number density in units of nuclear matter density as a function of chemical potential for different strengths of magnetic field at zero temperature.

Figure 3

Critical chemical potential for chiral transition at zero temperature as a function of magnetic field.

Figure 4

Gaps without determinant interaction at zero temperature as a function of quark chemical potential. The solid curve refers to masses of u-d quarks, the dashed curve refers to the mass of strange quark and the dotted curve corresponds to the superconducting gap.

Figure 5

Constituent quark masses as a function of magnetic field for $\mathrm{T}=0$. Part (a) shows the masses of the three quarks below the chiral transition for $\mu =200\mathrm{MeV}$. Part (b) shows the same for the masses along with the superconducting gap above the chiral transition for ${\mu }_q=400\mathrm{MeV}$.

Figure 6

Axial current density for $\tilde{e}B=5{\mathrm{m}}_{\pi }^2$ (black solid line) and $\tilde{e}B=10{\mathrm{m}}_{\pi }^2$ (red dashed line).

Figure 7

Constituent quark masses and superconducting gap when charge neutrality conditions are imposed. Part (a) shows the masses and superconducting gap at zero temperature as a function of quark chemical potential for magnetic field $\tilde{e}B=0.1{\mathrm{m}}_{\pi }^2$. Part (b) shows the same for $\tilde{e}B=10{\mathrm{m}}_{\pi }^2$.

Figure 8

Dispersion relation and the occupation number for condensing quarks at $\mathrm{T}=0$, ${\mu }_q=340\mathrm{MeV}$. Part (a) shows the dispersion relation for the condensing quarks for zeroth Landau level. The upper curve is for u quark and the lower curve corresponds to d quark dispersion relation. Part (b) shows the occupation number as a function of momentum for $\tilde{e}B=10{\mathrm{m}}_{\pi }^2$.

Figure 9

Number densities of up and down quarks participating in the superconductivity for $\tilde{e}B=0.1{\mathrm{m}}_{\pi }^2$ (dashed line) and $\tilde{e}B=10{\mathrm{m}}_{\pi }^2$ (solid line).

Figure 10

Chemical potential ${\mu }_E$ and ${\mu }_8$ for charge neutral quark matter. $|{\mu }_E|$ is plotted as a function of quark chemical potential ${\mu }_q$ for magnetic field $\tilde{e}B=0.1{\mathrm{m}}_{\pi }^2$ (a) and for $\tilde{e}B=10{\mathrm{m}}_{\pi }^2$ (b). In (a) and (b) we have also plotted the mass of strange quarks and superconducting gap as a function of quark chemical potential to highlight the dependence of charge chemical potential on these two parameters. In the lower two plots, the color chemical potential ${\mu }_8$ is plotted as a function ${\mu }_q$ for $\tilde{e}B=0.1{\mathrm{m}}_{\pi }^2$ (c) and for $\tilde{e}B=10{\mathrm{m}}_{\pi }^2$ (d).

Figure 11

Population of different species for charge neutral quark matter for $\tilde{e}B=0.1{\mathrm{m}}_{\pi }^2$ (a) and for $\tilde{e}B=10{\mathrm{m}}_{\pi }^2$ (b).

Figure 12

Equation of state for $\tilde{e}B=0.1{\mathrm{m}}_{\pi }^2$ (dashed line) and $\tilde{e}B=10{\mathrm{m}}_{\pi }^2$ (solid line).