Neutron star properties with careful parametrization in the vector and axial-vector meson extended linear sigma model

The existence of quark matter inside the cores of heavy neutron stars is a possibility which can be probed with modern astrophysical observations. We use a vector and axial-vector meson extended quark-meson model to describe quark matter in the core of neutron stars. We discover that an additional parameter constraint is necessary in the quark model to ensure chiral restoration at high densities. By investigating hybrid star sequences with various parameter sets, we show that low sigma meson masses are needed to fulfill the upper radius constraints and that the maximum mass of stable hybrid stars is only slightly dependent on the parameters of the crossover-type phase transition. Using this observation and results from recent astrophysical measurements, a constraint of $2.5<g_V<4.3$ is set for the constituent quark–vector meson coupling. The effect of a nonzero bag constant is also investigated, and we observe that its effect is small for values adopted in previous works.

Figure 1

The dependence of the ${\varphi }_{N/S}$ scalar condensates on the quark chemical potential with (solid lines) and without (dashed lines) the asymptotic condition of Eq. (30). The parameter sets for these results are shown in Table 1 and in Table IV of Ref. [45], and $g_V=5$ in both cases. This shows that a naive parametrization of the model results in an increase of the scalar condensates at higher densities and therefore the recurrence of chiral symmetry breaking.

Figure 2

The chemical potential dependence of the ${\varphi }_N$ (blue) and ${\varphi }_S$ (green) scalar condensates. The parameter set with $m_{\sigma }=290\mathrm{MeV}$ was used for each curve. Increasingly bright tones correspond to increasing vector couplings of 0 (solid), 2 (dashed), 4 (dot-dashed), and 6 (dotted).

Figure 3

Top: the EoS ($p$ as a function of $ϵ$) of the SFHo model (yellow solid line), as well as for quark (solid) and hybrid (dashed) stars using the eLSM with the parameter set corresponding to $m_{\sigma }=290\mathrm{MeV}$ and vector couplings $g_V=0$ (green) and $g_V=5$ (blue). The inset contains the same curves for the $n_B({\mu }_B)$ dependence, while the circles correspond to the central conditions inside the maximum mass NSs. For the hybrid EoSs, the energy density interpolation was used with ${\overline{n}}_B=3.5n_0$ and $\mathrm{\Gamma }=1.5n_0$. Bottom: the speed of sound squared for the same EoSs.

Figure 4

$M-R$ relations for the SFHo model (yellow) and for different quark (blue) and hybrid (green) EoSs. Increasingly bright tones correspond to increasing vector couplings of 0 (solid), 2 (dashed), 4 (dot-dashed), and 6 (dotted). The different shaded areas correspond to the lower and upper radius constraints of Bauswein et al. [87] and Annala et al. [77], respectively; different credibility limits of the upper mass constraint of Rezzolla et al. [88]; and the mass of PSR $\mathrm{J}0740+6620$ with the 95.4% lower bound denoted by the horizontal dot-dashed line [16]. The yellow shaded area corresponds to the region excluded by causality. The two NICER measurements are not included in this figure, since they do not provide additional hard constraints on the $M-R$ relations.

Figure 5

The same curves as in Fig. 4, but for the tidal deformabilities $\mathrm{\Lambda }$ as function of the compact star masses $M$. The red bar corresponds to the region excluded by the constraint $70<\mathrm{\Lambda }(1.4M_{\odot })<720$ deduced from the measurement of GW170817 [81].

Figure 6

The maximum mass of stable NS sequences as a function of the vector coupling $g_V$, for different sigma masses. The parameters of the concatenation are $\overline{n}=3.5n_0$ and $\mathrm{\Gamma }=1.5n_0$. For higher sigma masses, this results in EoSs that violate either thermodynamic stability or causality.

Figure 7

The chemical potential dependence of the ${\varphi }_N$ (blue solid) and ${\varphi }_S$ (green dotted) scalar condensates. Increasingly bright tones correspond to parameter sets with increasing sigma meson masses with $g_V=3$ for each curve. For larger $m_{\sigma }$ masses, the first-order phase transition [indicated by the slight back bending of the $\varphi ({\mu }_q)$ curves] disappears, and for $m_{\sigma }=700\mathrm{MeV}$, the vacuum expectation value of ${\varphi }_N$ becomes larger than the value of ${\varphi }_S$.

Figure 8

$M-R$ relations of hybrid stars for different vector couplings and sigma meson masses, using the SFHo hadronic EoS and the energy density interpolation with ${\overline{n}}_B=3.5n_0$ and $\mathrm{\Gamma }=2n_0$. Increasingly bright tones correspond to the five parameter sets with increasing $m_{\sigma }$ (the same as in Fig. 7), while the different types of lines correspond to different vector couplings of 1 (solid), 3 (dashed), and 5 (dot-dashed). The curves for $m_{\sigma }=700\mathrm{MeV}$ with $g_V=1$ and 3 are omitted, since the hybrid EoSs produced with these parameters are not stable.

Figure 9

$M-R$ relations of hybrid stars with the SFHo (green) and the DD2 (blue) hadronic EoSs and the energy density interpolation method with different values for ${\overline{n}}_B$ (solid) and for $\mathrm{\Gamma }$ (dashed). Brighter tones correspond to larger values in both cases. $g_V=3$ for each curve. For the DD2 case, only the $M-R$ curves of stable EoSs are included.

Figure 10

Top: hybrid EoSs produced by the two concatenation methods using the SFHo model (yellow solid line) and the eLSM with the parameter set corresponding to $m_{\sigma }=290\mathrm{MeV}$ and $g_V=3$ (green solid line). The black crosses correspond to $n_L\equiv n_{B,H}({\mu }_{\mathrm{BL}})$ and $n_U\equiv n_{B,Q}({\mu }_{\mathrm{BU}})$, while the circles correspond to the central conditions inside the maximum mass NSs. The parameters of the two types of interpolation methods were chosen so that both arrive to the hadronic and quark EoSs at approximately the same densities. Bottom: the speed of sound squared for the same EoSs.

Figure 11

$M-R$ relations of hybrid stars for different concatenations as well as for the SFHo (yellow) and DD2 (blue) hadronic EoSs. The parameters for the $ϵ(n)$ (dashed) and $p(\mu )$ (dotted) concatenations are the same as in Fig. 10.

Figure 12

$M-R$ relations in case of a nonzero bag constant for the $ϵ(n)$ (top) and the $p(\mu )$ (bottom) interpolation methods. In the $p(\mu )$ case, the interpolation limits were chosen so that larger values for $B$ could be accommodated as well without producing unstable EoSs.

Figure 13

$M-R$ relations with various bag constants in case of a polynomial interpolation in $ϵ(n)$. The interpolation parameters are $n_L=1.2n_0$ and $n_U=7n_0$. In this case, the problem with the incorrect low-density behavior is evaded.