Crystalline chiral condensates as a component of compact stars

We investigate the influence of spatially inhomogeneous chiral symmetry-breaking condensates in a magnetic field background on the equation of state for compact stellar objects. After building a hybrid star composed of nuclear and quark matter using the Maxwell construction, we find, by solving the Tolman-Oppenheimer-Volkoff equations for stellar equilibrium, that our equation of state supports stars with masses around $2M_{\odot }$ for values of the magnetic field that are in accordance with those inferred from magnetar data. The inclusion of a weak vector interaction term in the quark part allows one to reach two solar masses for relatively small central magnetic fields, making this composition a viable possibility for describing the internal degrees of freedom of this class of astrophysical objects.

Figure 1

Amplitudes ${\mathrm{\Delta }}_f$ ($f\in (u,d))$ and wave numbers $b_f$ for the CDW modulation [Eq. (8)] as a function of chemical potential in the presence of a constant external magnetic field.

Figure 2

Magnetic field profiles corresponding to Eq. (49) for different parametrizations ($\kappa ,\gamma$) in the chemical potential region under consideration.

Figure 3

Magnetization contribution compared to the matter thermodynamic potential [Eq. (12)] as a function of the quark chemical potential for a two-flavor system in a fixed external magnetic field. The Haas-van Alphen oscillations are richer in the isospin-asymmetric matter here considered due to the difference in the maximum Landau levels reachable for the up and down quark.

Figure 4

Equation of state (considering $P_{\perp }$ on the left panel and $P_{\parallel }$ in the right one) for hybrid stars for different values of the vector repulsion as indicated. The nuclear EoS does not include hyperons when considering quark matter with only quarks up and down [$\mathrm{GM}1\mathrm{n}+\mathrm{SU}(2)$ case] and includes hyperons when the strange quark is included in the quark phase as well [$\mathrm{GM}1\mathrm{nh}+\mathrm{SU}(3)$ case]. In the first case, the phase transition must happen at lower values of ${\mu }_B$. The magnetic field follows Eq. (49) and is taken with $H_{\mathrm{S}}=1\times {10}^{15}\mathrm{G}$, $H_C=2.5\times {10}^{18}\mathrm{G}$, $\gamma =2.5$, and $\kappa =12$. The horizontal lines joining the jumps in the energy density produced by the first-order phase transitions are only graphical artifacts to connect the curves before and after the phase transitions. No shift in the vacuum value of the quark phase was introduced.

Figure 5

Parallel (lower curves) and perpendicular (upper curves) pressures for a two-flavor hybrid star as a function of the baryonic chemical potential. The surface magnetic field is taken as $1\times {10}^{15}\mathrm{G}$ and the central field is $2.5\times {10}^{18}\mathrm{G}$ for the full lines and $6.8\times {10}^{18}\mathrm{G}$ for the dashed ones with $\gamma =2.5$ and $\kappa =12$. The curves end at the maximum value of the baryon chemical potential achieved in the interior of each configuration.

Figure 6

The mass-radius relations for a two-flavor (left) and three-flavor (right) inhomogeneous hybrid star considering $H_S=1\times {10}^{15}\mathrm{G}$, $H_C=2.5\times {10}^{18}\mathrm{G}$, $\gamma =2.5$, and $\kappa =12$. We employed the perpendicular pressure and the shifted vacuum pressure in units of $\mathrm{MeV}/{\mathrm{fm}}^3$ (see text for details) is indicated. The curves for stars composed entirely of nuclear matter in the GM1 parametrization with no magnetic field are also shown for comparison. The mass constraints ($\pm \sigma$) of pulsars PSR $\mathrm{J}1614-2230$ and PSR $\mathrm{J}0348+0432$ are shown as shaded regions.

Figure 7

Comparison of the mass-radius relations for a two-flavor hybrid star obtained when using the parallel and perpendicular pressures. The magnetic field is modeled with $H_S=1\times {10}^{15}\mathrm{G}$, $H_C=2.5\times {10}^{18}\mathrm{G}$, $\gamma =2.5$, and $\kappa =12$. The values of $G_V$ and shifted vacuum pressure in units of $\mathrm{MeV}/{\mathrm{fm}}^3$ are indicated. We note that the decrease in the maximum mass attainable is less than 5% when using the parallel pressure over the perpendicular one.