Quark-nuclear hybrid star equation of state with excluded volume effects

A two-phase description of the quark–nuclear matter hybrid equation of state that takes into account the effect of excluded volume in both the hadronic and the quark-matter phases is introduced. The nuclear phase manifests a reduction of the available volume as density increases, leading to a stiffening of the matter. The quark-matter phase displays a reduction of the effective string tension in the confining density functional from available volume contributions. The nuclear equation of state is based upon the relativistic density-functional model DD2 with excluded volume. The quark-matter equation of state is based upon a quasiparticle model derived from a relativistic density-functional approach and will be discussed in greater detail. The interactions are decomposed into mean scalar and vector components. The scalar interaction is motivated by a string potential between quarks, whereas the vector interaction potential is motivated by higher-order interactions of quarks leading to an increased stiffening at high densities. As an application, we consider matter under compact star constraints of electric neutrality and $\beta$ equilibrium. We obtain mass-radius relations for hybrid stars that form a third family, disconnected from the purely hadronic star branch, and fulfill the $2M_{\odot }$ constraint.

Figure 1

Effective quark mass as a function of the baryon density. The effective confinement manifests itself by a divergence of the quasiparticle mass at low densities.

Figure 2

Illustration of the effective reduction of the string tension (density of color field lines) at high densities. At low densities, (a) the field lines are compressed to thin flux tubes by the dual Meissner effect, while at high densities, (b) this pressure is reduced, and consequently the effective string tension is lowered.

Figure 3

Pressure versus baryon chemical potential for the hadronic DD2 EoS with different excluded volume parameters and for a quark-matter EoS (QEoS) with parameters $\alpha =0.2{\mathrm{fm}}^6$, $a=-2.0\mathrm{MeV}{\mathrm{fm}}^3$, $b=2.0\mathrm{MeV}{\mathrm{fm}}^9$, and $c=0.036{\mathrm{fm}}^6$. As the excluded volume is increased, the slope of the hadronic DD2 is lowered and the onset of the hadron-to-quark-matter transition (the crossing of quark and hadron EoSs) is lowered.

Figure 4

Upper panel: Pressure versus energy density for the hybrid EoS that emerge from the Maxwell constructions corresponding to Fig. 3. As the excluded volume parameter is increased, the slope in the hadronic phase is raised, exhibiting the stiffening of the hadronic matter. Lower panel: Squared speed of sound as a function of energy density, verifying the causality condition ($c_s^2<1$).

Figure 5

M-R relations for the hybrid EoS shown in Fig. 4 with varying hadronic excluded volume parameters. The thin dotted lines represent the unstable configurations of hybrid stars between the second and third families of stable compact stars.

Figure 6

Same as Fig. 3, but for varying values of the eight-quark coupling parameter $b$, while the other parameters are kept fixed at $\alpha =0.2{\mathrm{fm}}^6$, $a=-2.0\mathrm{MeV}{\mathrm{fm}}^3$, and $c=0.036{\mathrm{fm}}^6$, and the hadronic phase is described by the DD2p40 EoS.

Figure 7

The same as Fig. 4 for the hybrid EoS cases following from the EoS shown in Fig. 6.

Figure 8

M-R relations for the hybrid EoS shown in Fig. 7, differing in the values of the eight-quark coupling parameter $b$. The thin dotted lines represent the unstable configurations of hybrid stars.

Figure 9

Same as Fig. 3, but for varying values of the 4-quark coupling $a$, while remaining parameters are fixed at $\alpha =0.2{\mathrm{fm}}^6$, $b=2.0\mathrm{MeV}{\mathrm{fm}}^9$, and $c=0.036{\mathrm{fm}}^6$.

Figure 10

The same as Fig. 7 for the hybrid EoS cases following from the EoS shown in Fig. 9.

Figure 11

M-R relations for the hybrid EoS shown in Fig. 10, differing in the values of the four-quark coupling parameter $a$. The thin dotted lines represent the unstable configurations of hybrid stars.

Figure 12

Same as Fig. 3 for varying values of the available volume parameter $\alpha$, while the remaining parameters are fixed at $a=-2.0\mathrm{MeV}{\mathrm{fm}}^3$, $b=2.0\mathrm{MeV}{\mathrm{fm}}^9$, and $c=0.036{\mathrm{fm}}^6$.

Figure 13

The same as Fig. 10 for the hybrid EoS cases following from the EoS shown in Fig. 12.

Figure 14

M-R relations for the hybrid EoS shown in Fig. 13, differing in the values of the available volume parameter $\alpha$. The thin dotted lines represent the unstable configurations of hybrid stars.

Figure 15

M-R relations for different values of the available volume $\alpha$, while $a=-1.0\mathrm{MeV}{\mathrm{fm}}^3$, $b=2.0\mathrm{MeV}{\mathrm{fm}}^9$, and $c=0.036{\mathrm{fm}}^6$ are kept fixed. The hadronic phase is described by the DD2p40 EoS. The thin dotted lines represent the unstable configurations of hybrid stars. The points on the plot indicate a possible twin solution for each branch.

Figure 16

The radial profiles of pressure (upper panel), energy density (middle panel), and baryon density (lower panel) for the twin solutions at $M=1.47M_{\odot }$ for $\alpha =0.2$, indicated in Fig. 15 by plus signs. Dashed lines correspond to the purely hadronic (NS) solution, while the solid lines are for the hybrid star case.

Figure 17

Same as Fig. 16, for the high-mass twin solutions with $M=2.02M_{\odot }$ obtained for $\alpha =0.3$ and indicated in Fig. 15 by a star sign.

Figure 18

The EoS shown in Fig. 13 compared to the EoS constraint region analyzed with a multi-polytrope ansatz for the high-density EoS by Ref. [50].