Exploring the first-order phase transition in neutron stars using the parity doublet model and NJL-type quark model

Quantum Chromodynamics (QCD) is believed to be the fundamental theory that describes the strong interactions between quarks and gluons. Under normal conditions, quarks and gluons are confined within hadrons, such as protons, neutrons, and pions. These composite particles serve as effective degrees of freedom (d.o.f.) for describing the physics in vacuum and low density/temperature. However, the situation changes with the increasing of temperature or the density as hadrons begin to overlap and quarks and gluons d.o.f. will come into play. Such extreme forms of matter are believed to have existed in the early universe and are present in astrophysical objects like neutron stars (NSs) (see, e.g., Refs. Fukushima and Hatsuda (2011); Fukushima and Sasaki (2013); Baym et al. (2018); Ma and Rho (2020); Annala et al. (2020); Lovato et al. (2022); Brandes and Weise (2024) and references therein).

Along the temperature axis in hot QCD, a heated hadron resonance gas smoothly transforms into a quark-gluon plasma, despite the apparent differences between these two phases. Experiments have successfully created such hot matter through high-energy heavy ion collisions, where the colliding energy is converted into heat. A coherent picture of hot QCD matter has emerged through the combined efforts of experimental analyses, ab-initio lattice simulations of QCD, and model calculations Fukushima and Hatsuda (2011); Lovato et al. (2022); Fukushima (2012); Petreczky (2012).

However, the situation is much different for cold dense matter, especially about the exact nature of the transition from nuclear matter to quark matter. Theoretically, this cannot be accessed due to the lack of solid approach anchored on nonperturbative QCD. Moreover, the model dependent predictions cannot be diagnosed by terrestrial experiments since such high density cannot be produced or Lattice QCD simulation due to the sign problem. The only constraints on nuclear matter properties are from the experimental data around the saturation density $n_{0}$ ($n_{0}=0.16$ fm³) and predictions from the perturbative QCD at extreme high density $\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}50n_{0}$.

Since the detection of the gravitational waves (GWs) from a binary NS merger GW170817 Abbott (2017); Abbott et al. (2017, 2018), astrophysics enters the multi-messenger era. In combination with the observation of massive pulsars around two-solar mass Miller et al. (2021); Riley et al. (2021); Vinciguerra et al. (2024), NS has become a natural laboratory for investigating the properties of matter under extreme conditions, particularly at ultra-high densities up to $\sim 10n_{0}$. The characteristics of neutron stars are determined by the equation of state (EOS) of the matter composing them. Therefore, the theoretically obtained EOSs can be constrained by observational data, such as the mass-radius ($M\text{-}R$) relation of a NS, the tidal deformability, and the signals of gravitational waves stemming from binary NS mergers Annala et al. (2020); Brandes and Weise (2024); Hotokezaka et al. (2011, 2013); De Pietri et al. (2016); Maione et al. (2016); Moustakidis et al. (2017); Radice et al. (2017); Paschalidis et al. (2018); Annala et al. (2018); Torres-Rivas et al. (2019); Most et al. (2019); Tews et al. (2018); Tsang et al. (2018); Bauswein et al. (2019); Orsaria et al. (2019); Most et al. (2020); Yang et al. (2021); Sekiguchi et al. (2011); Choi et al. (2021); Annala et al. (2022); Raaijmakers et al. (2021); Prakash et al. (2021, 2021); Kedia et al. (2022); Fujimoto et al. (2023); Guo et al. (2023).

One of the key questions in the study of NS EOSs is whether there is a possible transition from hadronic matter to quark matter—either a phase transition or a crossover. Such a transition could have significant implications for the structure and properties of NSs (see, e.g., Baym et al. (2018); Lenzi and Lugones (2012); Kojo (2021); Benic et al. (2015)). When the transition is a first-order phase transition, there can be a sudden release or absorption of energy, leading to changes in the equilibrium state of the star. This can manifest as a sudden change in the radius of the neutron star, as the internal pressure and composition adjust to the new equilibrium state.

In previous works, by assuming the transition is a cossover transition, some of us constructed unified EOSs by interpolating the hadronic EOS from a hadronic model based on the parity doublet structure and the quark EOS from an Nambu-Jona-Lasinio (NJL)-type quark model Minamikawa et al. (2021a, b); Gao et al. (2022); Kong et al. (2023); Gao et al. (2024a). In this paper, we devote ourselves to investigate the possibility and the impacts of the first-order phase transition in dense matter using the parity doublet model (PDM) for the hadronic phase and the NJL-type model for the quark phase.

The PDM offers a unique perspective of baryon structure by incorporating chiral symmetry and its restoration, allowing the parity partners to have a degenerate chiral invariant mass $m_{0}$ at high densities when chiral symmetry is restored. The notion of chiral invariant mass was first introduced in DeTar and Kunihiro (1989); Jido et al. (2001) and is consistent with the recent lattice QCD simulation Aarts et al. (2015, 2017, 2019) as well as the skyrmion crystal approach to nuclear physics Ma et al. (2013, 2014). Effective models based on parity doublet structure is widely used for studying the hadron structure and nuclear/NS physicsNishihara and Harada (2015); Minamikawa et al. (2023a); Hatsuda and Prakash (1989); Zschiesche et al. (2007); Dexheimer et al. (2008); Sasaki and Mishustin (2010); Sasaki et al. (2011); Gallas et al. (2011); Steinheimer et al. (2011); Paeng et al. (2012); Dexheimer et al. (2013); Paeng et al. (2013); Mukherjee et al. (2017a); Suenaga (2018); Takeda et al. (2018); Abuki et al. (2018); Marczenko et al. (2019a); Motohiro et al. (2015); Yamazaki and Harada (2019); Mukherjee et al. (2017b); Gao et al. (2022); Marczenko et al. (2022); Minamikawa et al. (2023b); Baym et al. (2018); Minamikawa et al. (2021a, b); Marczenko et al. (2019b); Kong et al. (2023); Gao et al. (2024b) . The key feature of the PDM is that a larger $m_{0}$ leads to a weaker $\sigma$ coupling to nucleon since nucleon does not have to acquire its mass entirely from the $\sigma$ field. The corresponding $\omega$ field also becomes smaller due to the equilibrium state at the saturation density $n_{0}$. Then at density larger than $n_{0}$, the $\sigma$ field decreases while the $\omega$ field increases. As a result, a larger $m_{0}$ would lead to a softer EOS.

By combining the PDM and the NJL-type model, we aim to construct a hybrid EOS that captures the transition from hadronic matter to quark matter. We will explore the parameter space of both models to identify the conditions under which a first-order phase transition can occur. The resulting EOSs will be compared with observational constraints from NS observation to assess their compatibility with the latest astronomical data.

This paper is organized as follows. In Sec. II, we present the EOSs used in our study. We describe the hadronic matter EOS based on the PDM and the quark matter EOS derived from the NJL-type model. We also discuss the construction of the hybrid EOS incorporating a first-order phase transition. In Sec.III, we investigate the neutron star properties using the EOSs constructed in Sec.II, focusing on the mass-radius relation and the constraints on the onset density of the phase transition. Finally, we summarize our findings and discuss their implications in Sec.IV.

In this section, we develop EOSs for neutron star matter in the hadron and quark phases separately, and then explore the unified EOS with a first-order phase transition by integrating them.

In this section, we study the NS properties by using the EOSs constructed in the previous section. By solving the Tolman-Oppenheimer-Volkoff (TOV) equation for spherically symmetric and static stars, we obtain the NS mass-radius (M-R) relation.

We first show the M-R relation in pure hadronic PDM in Fig. 4. The green and red contours represent the NS observational constraints from LIGO-VIRGO Abbott (2017); Abbott et al. (2017, 2018) and the Neutron Star Interior Composition Explorer (NICER) Miller et al. (2021); Riley et al. (2021); Vinciguerra et al. (2024), respectively. We observe that for $m_{0}$ = 700 and 800 MeV, the maximum mass of the NS is smaller than 2$M_{\odot}$ constraint. Since the first-order phase transition soften the EOS and lower the maximum mass of the NS, it is impossible to realize the NS with $M/M_{\odot}\geq 2$ for $m_{0}\geq 700$ MeV. Consequently, we focus on the cases with $m_{0}$ = 500 and 600 MeV in the subsequent analysis.

Next, we examine the impact of the first-order phase transition on the M-R relation by combining the PDM with the NJL-type model. We show some typical example in Fig. 5 and Fig. 6, with the black line representing the pure PDM and different colors corresponding to various NJL parameter combinations $(H,g_{V})$. For the purpose to investigate the effects of parameters $H$ and $g_{V}$, we change one values with keeping the others fixed, as shown in Fig. 5(a) and Fig. 5(b). The cross marks indicate the points of maximum mass for each combination.

From Fig. 5, one can conclude that the increasing of $g_{V}$ or the decreasing of $H$ increases the onset density of the first-order phase transition. When the phase transition occurs at a very low density, as indicated by the orange line in Fig. 5, the magnitude of chiral symmetry breaking is suppressed. As the onset density of the first-order phase transition is pushed to higher densities by increasing $g_{V}$ or decreasing $H$, the chiral restoration around 2$n_{0}$ for $m_{0}=500$ MeV begins to manifest. In the figure, the orange, red, and blue lines represent stable NS configurations, with the maximum mass $\sim 2m_{\odot}$ appearing after the first-order phase transition. However, the brown line and the magenta line in Fig. 5(b) exhibit a significant reduction in radius and mass following the phase transition, implying an NS with a unstable quark core since the ${\rm d}M/{\rm d}n_{B}<0$ after the phase transition Alford and Sedrakian (2017); Glendenning and Kettner (2000).

We also show the case for $m_{0}=600$ MeV in Fig. 6 with some typical choices of the NJL parameters. In this case, the onset of $N(1535)$ point is shifted to around $3n_{0}$ and a softer quark EOS is required to form a stable NS compared with the case in $m_{0}=500$ MeV. In the two figures in Fig. 6, we find that the onset density of the phase transition, governed by the interplay between hadronic and quark matter parameters, significantly influences the stability of the NS configuration. A phase transition occurring at lower densities allows for stable NS configurations, while a phase transition at excessively high densities can lead to an unstable NS, leads to ${\rm d}M/{\rm d}n_{B}<0$ after the phase transition. This is because if the onset density is very high, only NSs with very high central densities will undergo this phase transition. These stars are typically very massive and close to the TOV limit (the maximum mass limit for a stable neutron star). At such high densities, the EoS change due to the phase transition can make the core more compressible or softer, which can destabilize the star. This destabilization may lead to the collapse of the NS star into a black hole if the pressure support is insufficient to counteract the gravitational pull.

Finally, by fine-tuning the NJL parameters, we can identify the suitable parameter space and constrain the onset density of the first-order phase transition. In Fig. 7, we show all possible combinations of $(H,g_{V})$ for $m_{0}$ = 500, 600 and 700 MeV. The color of each point indicates the maximum mass of the NS obtained from the corresponding parameters, as shown by the vertical bar on the right side of each figure. A cross mark indicates that the EOS constructed from the combination of $(H,g_{V})$ does not have an intersection point with the EOS constructed from the PDM, while a colored indicates that the combination is allowed. For $m_{0}$=700 MeV, all the combination of NJL parameters lead to a NS maximum mass smaller than 2$M_{\odot}$, leading to the conclusion that the $m_{0}$ should be smaller than 700 MeV while considering the first-order phase transition.

After considering NS with a stable quark core, we further constrain the parameter space to the region surrounded by red lines which represents the combinations that can simultaneously satisfy the maximum mass constraints and the stability condition. Using these parameters, we can determine the possible range of densities at which the first-order phase transition occurs.

We choose $m_{0}$ to be 500, 550, and 600 MeV and examine the density range of the first-order phase transition as shown in Fig. 8. The blue line in the figure represents the density range at which the phase transition begins, while the red line represents the density range where the phase transition ends. As the value of $m_{0}$ increases, we find that both the onset density and the ending density of the phase transition increase. For $m_{0}$ = 500 MeV, the phase transition occurs between $1.9n_{0}$ and $2.95n_{0}$ and ends between $2.1n_{0}$ and $3.6n_{0}$. When $m_{0}$ = 600 MeV, the phase transition takes place between $2.9n_{0}$ and $4.1n_{0}$ and concludes between $3.4n_{0}$ and $4.6n_{0}$.

In this paper, we have investigated the possibility and impacts of a first-order phase transition from hadronic matter to quark matter in NSs using the PDM for the hadronic phase and the NJL-type model for the quark phase. By constructing hybrid EOSs and analyzing the M-R relations, we have explored the role of key parameters in both models, such as the chiral invariant mass ($m_{0}$) in the PDM and the attractive diquark correlation strength ($H$) and vector repulsion strength ($g_{V}$) in the NJL-type quark model.

Our findings show that the onset density of the first-order phase transition is strongly influenced by the interplay between hadronic and quark matter parameters. A smaller chiral invariant mass $m_{0}$ in the PDM leads to an earlier onset of the chiral phase transition associated with the emergence of the negative parity baryon. The stability of the NS configuration is sensitive to the onset density of the first-order phase transition, with transitions occurring at lower densities allowing for stable configurations, while transitions at excessively high densities can lead to NS with an unstable quark core. By fine-tuning the parameters $(H,g_{V})$ and considering the stability of the NS, we have identified suitable parameter spaces and constrained the onset density of the first-order phase transition. For $m_{0}$ = 500 MeV, the phase transition occurs between 1.9$n_{0}$ and 2.95$n_{0}$ and ends between 2.1$n_{0}$ and 3.6$n_{0}$. For $m_{0}$ = 600 MeV, the phase transition occurs between 2.9$n_{0}$ and 4.1$n_{0}$ and ends between 3.4$n_{0}$ and 4.6$n_{0}$.

Finally, we compare cases assuming a crossover transition with those involving a first-order phase transition, as illustrated in Fig. 9. For each $m_{0}$ value, we select NJL parameters from the region surrounded by red lines in Fig. 7 that yield both the largest and smallest maximum masses. Solid lines represent cases with first-order phase transitions, while dotted lines of the same color indicate cases assuming crossover transitions. Our analysis reveals several key differences between these two scenarios. We observe that the crossover assumption leads to a smooth change in the $M$-$R$ curve, while the first-order phase transition results in a more abrupt change, often characterized by a ”kink” in the M-R relation. Furthermore, the maximum neutron star mass in the first-order phase transition scenario is typically smaller than in the crossover case, which can be attributed to the sudden softening of the equation of state during a first-order transition.

The work of B.G., and M.H. are supported in part by JSPS KAKENHI Grant Nos.  23H05439 and 24K07045. B.G. is also supported by JST SPRING, Grant No. JPMJSP2125. B.G. would like to take this opportunity to thank the “Interdisciplinary Frontier Next-Generation Researcher Program of the Tokai Higher Education and Research System.” The work of Y. L. M. was supported in part by the National Key R&D Program of China under Grant No. 2021YFC2202900 and the National Science Foundation of China (NSFC) under Grant No. 12347103 and No. 11875147.