- 1Theoretical Astrophysics, Eberhard-Karls Universität Tübingen, Tübingen, Germany
- 2Department of Theoretical Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece
Recent analyses on the properties of the central compact object in the HESS J1731-347 remnant and the PSR J1231-1411 pulsar indicated that these two compact objects are characterized by similar (low) masses and possibly different radii. This paper aims at reconciling the aforementioned measurements by utilizing the widely employed color-flavor locked (CFL) MIT bag model. The main objective is related to the examination of the acceptable values for the color superconducting gap
1 Introduction
One of the most important unresolved questions in theoretical astrophysics is related to the nature of matter in the cores of compact stars. Notably, compact stars could be composed solely by hadrons (nucleons and hyperons), but the extreme conditions that prevail in their interior may allow for the presence of exotic forms of matter, such as deconfined quarks (Witten, 1984; Annala et al., 2020). The latter opens up intriguing scenarios, such as the existence of strange stars (Weber, 2005), composed purely of quark matter, or hybrid stars (Heiselberg and Hjorth-Jensen, 2000), where a quark core is surrounded by a layer of hadrons. Interestingly, given that different hypotheses for the composition of stellar matter may predict distinct properties for the structure of compact stars (Lattimer and Prakash, 2001; Glendenning and Kettner, 2000), precisely inferred measurements on masses and radii of compact objects may provide important insight into their nature.
In 2022, the analysis of Doroshenko et al. (2022) provided puzzling values for the mass
The recent analysis of Salmi et al. (2024a) also provided intriguing values for the properties of the pulsar PSR J1231-1411. Notably, their results appeared to be sensitive to the selection of the radius prior they used. By limiting the radius to be consistent with previous observational constraints and nuclear theory, the authors indicated that
Given that color-flavor locked (CFL) quark matter (Alford et al., 1999; Alford, 2001; Alford et al., 2008) has been successfully employed in the reconciliation of HESS J1731-347, not only for its mass and radius but also for its thermal evolution (Di Clemente et al., 2024; Horvath et al., 2023), we aim to consider it in order to examine the possible simultaneous explanation of the PSR J1231-1411 properties. To do so, we consider two distinct hypotheses for strange quark matter (SQM): a) absolutely stable, b) energetically favored at high baryon density. Of utmost importance is to examine if the resulting EOSs satisfy other precisely inferred mass-radius measurements (NICER mission) (Choudhury et al., 2024; Salmi et al., 2024b; Vinciguerra et al., 2024; Riley et al., 2019; Miller et al., 2019) and also allow for the existence of stable massive stars beyond
When considering Case I (Salmi et al., 2024a), it is rather interesting that, while both the CCO in the HESS J1731-347 remant and the PSR J1231-1411 pulsar have similar masses, their radii do not overlap at the
In Case II (Qi et al., 2025), the radius of PSR J1231-1411 has an upper bound (at
This paper is structured as follows: Section 2 sets the theoretical framework of the present study. Section 3 presents the results for the two distinct hypotheses on the nature of compact stars (quark or hybrid stars) along with a detailed discussion on our findings. Lastly, Section 4 highlights the main insights and conclusions derived from this research.
2 Color-flavor locked equation of state
The equation of state for CFL quark matter can be formulated within the MIT bag model framework. To the order of
where
Equation 3 reduces to the one presented in Lugones and Horvath (2002) for
Conversely, a lower limit can be determined by requiring that two-flavor quark matter should be less stable than nuclear matter. In the context of the MIT bag model
This value decreases when considering lower values for
Finally, it is worth noting that in order to ensure that the CFL phase represents the favorable state of matter (compared to 2SC, unpaired or gapless CFL matter) the following condition needs to be met (Alford et al., 2005b):
Notably, Equation 6 holds for all of the quark EOSs constructed in the present study.
3 Results and discussion
3.1 Strange stars
In this section, we present our results derived under the assumption that SQM represents the true ground state of matter. However, a key question, that requires a proper discussion, immediately arises: if SQM is more stable than nuclear matter why does normal matter persist? One explanation may be that nuclear matter is metastable [see Horvath et al. (1992), Olesen and Madsen (1994), Iida and Sato (1998), Bombaci et al. (2008), Bombaci et al. (2009), Ren and Zhang (2020) and references therein], separated from the favorable SQM state by a significant energy barrier. Thus, at low densities, the conversion to SQM may be suppressed due to quantum tunneling limitations. However, the extreme neutron star environment, characterized by high densities, may enhance the possibility of a SQM droplet appearing via quantum fluctuations. In addition, extreme events such as supernovae and neutron star mergers may also facilitate SQM formation. Given that the appearance of a SQM seed could trigger the conversion of the entire hadronic star into a strange star, it has long been hypothesized that all compact stars may be of SQM nature.
Proceeding with our calculations, we solved the system of the TOV equations (Oppenheimer and Volkoff, 1939), using a wide range of
Notably, in the present study we work under the assumption of a universal EOS (a single EOS that accounts for all observations). Therefore, the derived EOSs should be compatible to all state-of-the-art multimessenger constraints on the mass and radius of compact stars. Thus, apart from HESS J1731-347 and PSR J1231-1411, we also utilized the corresponding data related to PSR J0952-0607 (Romani et al., 2022), PSR J0030+0451 (Miller et al., 2019) and PSR J0437-4715 (Choudhury et al., 2024). Lastly, to highlight the peculiar nature of the recent XTE J1814-338 measurement (Kini et al., 2024), we have included it in our analysis, although we did not attempt to interpret it simultaneously with the aforementioned constraints [for some recent works on its reconciliation see Zhou and Huang (2025), Yang et al. (2025), Laskos-Patkos and Moustakidis (2025), Lopes and Issifu (2025), M. Veselský et al. (2025)].
Figure 1 displays the

Figure 1.
Figures 1a,b indicate that HESS J1731-347 and PSR J1231-1411 (Case I), can both be reconciled within the range of parameters that are compatible to the latter, while also respecting the maximum-mass constraints imposed by PSR J0952-0607. This region is characterized by small values for the bag constant, close to the minimum possible one, and large values for the color superconducting gap. As a result, the CFL EOS is extremely stiff, leading to very high maximum masses for all

Figure 2. (a) Mass-radius curves that are compatible with PSR J1231-1411 Case I (solid lines) and Case II (dashed lines), with
Another interesting point from Figures 1a,b is related to the fact that the parameter spaces which are compatible with PSR J0437-4715 and PSR J1231-1411 (Case I) do not overlap at the
Considering Case II, Figures 1c,d show that there is a small range of parameters which is compatible with all measurements at the
The parameter space in which all measurements can be explained by pure CFL matter (Case II) is shown in Figure 2c, alongside the corresponding maximum mass predictions (for the
A final remark needs to be made with regards to the possible explanation of the thermal evolution for the CCO in the HESS J1731-347 remnant. Notably, the CCO has a rather high temperature for its estimated age which suggests slow cooling (Di Clemente et al., 2024), analogous to those of purely hadronic stars. Previous works (Di Clemente et al., 2024; Horvath et al., 2023) have qualitatively suggested that superconductivity may suppress the rapid cooling processes that are expected in unpaired SQM and therefore the temperature of the CCO could be explained. Interestingly, according to Horvath et al. (2023) this could occur only under the consideration of vanishingly small gap values
3.2 Hybrid stars
In the previous section, we considered parametrizations that support the absolute stability of CFL quark matter. At this point, we aim to examine the scenario of explaining all current multimessenger constraints by considering a first-order phase transition, from hadronic to CFL quark matter, in the stellar interior.
Notably, the simultaneous reconciliation of both HESS J1731-347 and PSR J1231-1431 (in Case I) measurements would be rather difficult assuming a purely hadronic EOS. In particular, the hadronic model should be rather soft, at low densities, to support the low radius associated with HESS J1731-347 in the sub-solar mass region, and then it should rapidly stiffen to achieve the higher radius of PSR J1231-1431 at slightly larger masses. Interestingly, the low-density domain of the nuclear EOS can be effectively constrained through parity-violating electron scattering experiments, which aim to measure the neutron skin thickness of different nuclei. In particular, the PREX-II collaboration provided a measurement for the neutron skin thickness of lead, which pointed to stiff EOS behavior at low densities (Adhikari et al., 2021; Reed et al., 2021). Then, the subsequent CREX experiment extracted a puzzling value for the neutron skin thickness of calcium (Adhikari et al., 2022), supporting softer models. More precisely, the simultaneous explanation of both CREX and PREX-II was not possible with the use of traditional energy density functionals (Reinhard et al., 2023; Tagami et al., 2022; Miyatsu et al., 2023; Kumar et al., 2023; Burgio et al., 2024). However, recent attempts have proposed sophisticated modifications to the Lagrangians which are used to describe nuclear matter, and they achieved an explanation of both experimental values (Reed et al., 2024; Kumar et al., 2024). Nonetheless, the predicted EOSs are characterized as extremely stiff. Notably, the latter issue was at some extent resolved in Salinas and Piekarewicz (2024).
Considering that current experimental constraints might point to stiff behavior for the hadronic EOS at low density, in the present study, we are going to employ a nuclear model which is sufficiently stiff, so that it crosses the PSR J1231-1411 contour (Case I), and, therefore, potentially incompatible to the HESS J1731-347 constraints. In particular, we are going to use a widely employed Skyrme model, namely Ska (Köhler, 1976; Typel et al., 2022; Gulminelli and Raduta, 2015; Danielewicz and Lee, 2009; Baym et al., 1971). Thus, in our attempt for explaining all current astronomical measurements, by considering a first-order phase transition to CFL quark matter, we will rely on the hybrid branch to account for the existence of the central compact object in the HESS J1731-347 remnant. It is important to comment that, the employed nuclear model is of nucleonic composition. In principle, as density increases, hyperonic degrees of freedom may appear, altering the properties of the EOS. Given the uncertainties related to hyperon-hyperon and hyperon-nucleon interactions, in the present study we work under a simplified framework that neglects their existence.
In Figure 3a, we depicted the mass-radius dependence for hybrid EOSs constructed by varying the values of

Figure 3. Mass-radius diagrams for hybrid EOSs combining the Ska hadronic EOS and the CFL MIT bag model. The gray contour regions denote mass and radius measurements related to PSR J0740 + 6620 (Salmi et al., 2024b), PSR J0030+0451 (Miller et al., 2019), HESS J1731-347 (Doroshenko et al., 2022), PSR J0437-4715 (Choudhury et al., 2024), PSR J1231-1411 (Salmi et al., 2024a), XTE J1814-338 (Kini et al., 2024) and GW170817 (Abbot et al., 2017). The solid (dashed) contours correspond to
Up to this moment, we have only considered Case I for PSR J1231-1411. However, it is clear that the consideration of Case II would not alter any conclusion about our inability of reproducing all observational constraints when we combine the CFL model with a stiff hadronic EOS (similar to the employed one). Nevertheless, future refinement of theoretical models or nuclear experiments that point towards a softer nuclear model may alter the current picture.
Interestingly, we can attempt to reconcile all of the considered astronomical constraints by allowing the hadronic branch to reach the two solar masses and then induce an extremely strong phase transition to make the mass-radius diagram drop to cross the HESS J1731-347 contour. Notably, hybrid stars may be stable even at a descending branch of a mass-radius diagram, assuming that the phase conversion is sufficiently slow (Pereira et al., 2018). The characterization slow refers to the magnitude of the phase conversion timescale
In Figure 4a, we depict the mass-radius dependence for hybrid EOSs constructed by varying the color superconducting gap in the range [0,600] MeV. The bag constant was derived by considering that the phase transition occurs when the hadronic mass reaches the

Figure 4. (a) Mass-radius diagrams for hybrid EOSs combining the Ska and the CFL bag model with
Notably, the central baryon chemical potential range that is being considered, in order to achieve the reconciliation of the HESS J1731-347 constraints, reaches such high values that it may cross the regime where the results of pQCD, for the EOS of strongly interacting quark matter, are potentially reliable. Typically, pQCD is considered to be credible at densities around
A final remark is appropriate regarding the fact that while the existence of slow stable hybrid stars is theoretically intriguing it is not clear how such objects are born. In that sense, future work on possible formation scenarios of such objects would be of utmost importance. Some recent progress on the astrophysical paths that may lead to the existence of twin stars (but for a hybrid branch respecting the turning point criterion) has been made in the work of Naseri et al. (2024). In that direction, we expect that future research will hopefully shed light on the possible existence of slow stable hybrid star branches.
4 Conclusion
In this paper, we have presented a systematic study of the CFL MIT bag model in light of recent observations of low-mass compact stars. We have shown that the intriguing measurements of HESS J1731-347 and PSR J1231-1411 can be simultaneously explained within the framework of pure (absolutely stable) CFL matter, while also satisfying the maximum mass constraint set by PSR J0952-0607 and the latest multimessenger constraints on compact star masses and radii (PSR J0030+0451 and PSR J0437-4715). The parameter space consistent with all these measurements yields
Notably, in the case of absolutely stable SQM, we worked under the assumption that only one state of matter may appear in the stellar interior. However, it is worth mentioning that a new exotic scenario has been reported in the literature suggesting the possible reappearance of hadrons, at large densities, even if SQM represents the true ground state of matter (Zhang and Ren, 2023; Negreiros et al., 2025; Zhang et al., 2024). Thus, an interesting direction for future work would be to investigate how such a consideration might alter the results reported in the present study.
When considering the framework of hybrid stars, we found that although a hybrid branch originating in the PSR J1231-1411 mass-radius region [as provided by Salmi et al. (2024a)] can also accommodate the HESS J1731-347 measurement, the resulting maximum masses remain well below the well-established threshold of
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author. The data associated with this study are available from the authors upon reasonable request.
Author contributions
KK: Writing – original draft, Writing – review and editing. PL-P: Writing – original draft, Writing – review and editing. CM: Writing – original draft, Writing – review and editing.
Funding
The author(s) declare that financial support was received for the research and/or publication of this article. PL-P. acknowledges that the research work was supported by the Hellenic Foundation for Research and Innovation (HFRI) under the fifth Call for HFRI PhD Fellowships (Fellowship Number: 19175).
Acknowledgments
The authors would like to thank Y. Kini for providing the data for the contour regions for XTE J1814-338.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.
Generative AI statement
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Keywords: neutron stars, quark stars, color-flavor locked matter, hadron-quark phase transition, hybrid stars, equation of state
Citation: Kourmpetis K, Laskos-Patkos P and Moustakidis CC (2025) Confronting recent light compact star observations with color-flavor locked quark matter. Front. Astron. Space Sci. 12:1600563. doi: 10.3389/fspas.2025.1600563
Received: 26 March 2025; Accepted: 11 June 2025;
Published: 22 July 2025.
Edited by:
Armen Sedrakian, University of Wrocław, PolandReviewed by:
Malte Albrecht, Jefferson Lab (DOE), United StatesGiuseppe Pagliara, University of Ferrara, Italy
Milva Orsaria, National Scientific and Technical Research Council (CONICET), Argentina
Copyright © 2025 Kourmpetis, Laskos-Patkos and Moustakidis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ch. C. Moustakidis, bW91c3Rha2lAYXV0aC5ncg==