**1. Introduction**

At increasing density of baryonic matter, it is expected that a deconfinement phase transition will take place and form quark matter. The properties of quark matter is of particular interest to us since its absolute stability would permit an explanation of dark matter within the framework of the standard model [1]. In the beginning of the 1970s, it was suggested that strange quark matter (SQM) comprised of *u*, *d*, and *s* quarks may be more stable than nuclear matter [1–3], which can exist in various forms, e.g., strangelets [4–7], nuclearites [8,9], meteorlike compact ultradense objects [10], and strange stars [11–13]. Nevertheless, the absolute stability of SQM was challenged by chiral models due to a too large strange quark mass with dynamical chiral symmetry breaking [14,15]. Then SQM only exists in extreme conditions such as the center of compact stars [16–22] and heavy-ion collisions [23,24]. In recent years, an interesting proposition was raised suggesting that quark matter comprised of only *u* and *d* quarks (*ud*QM) may be more stable [25], so that *ud*QM nuggets and *ud*QM stars can exist in the Universe. Due to a much smaller surface tension, the ordinary nuclei would not decay into *ud*QM nuggets [25,26]. In fact, it was shown that in a large parameter space the energy per baryon of *ud*QM nuggets is larger than 930 MeV at *A* . 300 [25]. The properties of nonstrange quark stars and their astrophysical implications are then examined extensively in recent years, e.g., those in References [27–30]. In particular, the merger of binary quark stars would eject *ud*QM nuggets into space. If those objects become supercritically charged, the *e* +*e* − pair production would inevitably start and release a large amount of energy. The positron emission of the supercritically charged objects are thus expected to play important roles in the short *γ*-ray burst during

**Citation:** Wang, L.; Hu, J.; Xia, C.-J.; Xu, J.-F.; Peng, G.-X.; Xu, R.-X. Stable Up-Down Quark Matter Nuggets, Quark Star Crusts, and a New Family of White Dwarfs. *Galaxies* **2021**, *9*, 70. https://doi.org/10.3390/ galaxies9040070

Academic Editors: Elena Moretti and Francesco Longo

Received: 30 July 2021 Accepted: 23 September 2021 Published: 28 September 2021

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the merger of binary quark stars, the 511 keV continuum emission, and the narrow faint emission lines in X-ray spectra from galaxies and galaxy clusters [26].

Based on various investigations, it was shown that the interface effects of quark matter play key roles in the properties of strangelets, *ud*QM nuggets, compact stars, and the processes of quark-hadron transition [26,31,32]. The energy contribution due to the interface effects is often taken into account with a surface tension *σ*, while its exact value is still veiled in mystery. Adopting the bag model [33], linear sigma model [34–36], NJL model [37,38], three-flavor Polyakov-quark-meson model [39], Dyson-Schwinger equation approach [40], equivparticle model [41], nucleon-meson model [42], and Fermi gas approximations [43,44], recent estimations indicate a small value with *σ* . 30 MeV/fm<sup>2</sup> , while larger values were obtained in previous studies [45–47].

In the framework of the bag model, it was shown that a small strangelet can be destabilized substantially if *σ* 1/3 <sup>≈</sup> *<sup>B</sup>* 1/4 with *B* being the bag constant [4], while the minimum baryon number for metastable strangelets *A*min ∝ *σ* 3 [5,48]. Depending on the values of surface tension, large strangelets and strange stars will face very different fates. On the one hand, if a moderate value for *σ* is adopted, larger strangelets are more stable than smaller ones and strange stars' surfaces are likely bare [49]. On the other hand, if *σ* is smaller than a critical value *σ*crit, large strangelets will decay via fission [50] and strange stars' surfaces may fragment into crystalline crusts [51]. Adopting linearization for the charge density, it was shown that the critical surface tension can be obtained with [50]

$$
\sigma\_{\rm crit} = 0.1325 n\_Q^2 \lambda\_D / \chi\_{Q'} \tag{1}
$$

where *n<sup>Q</sup>* is the charge density, *λ<sup>D</sup>* = 1/ p 4*παχ<sup>Q</sup>* the Debye screening length, and *χ<sup>Q</sup>* = ∑*<sup>i</sup> qi ∂n<sup>Q</sup> ∂µ<sup>i</sup>* the electric charge susceptibility of quark matter at zero electric charge chemical potential *µ<sup>e</sup>* = 0. Assuming noninteracting SQM, Equation (1) suggests *σ*crit ∝ *m*<sup>4</sup> *s* with *m<sup>s</sup>* being the strange quark mass [51,52]. As we increase *m<sup>s</sup>* , the strangeness per baryon *f<sup>s</sup>* for *β*-stable SQM decreases and eventually reaches *f<sup>s</sup>* = 0, where SQM is converted into *ud*QM. We thus expect that the critical surface tension of *ud*QM is much larger than that of SQM, so it is more likely that there exist *ud*QM nuggets at certain size that are more stable than others. Additionally, varying the symmetry energy of quark matter will alter the values of *nQ*, *λD*, *χQ*, and consequently *σ*crit according to Equation (1). Since it was shown that the symmetry energy of quark matter plays an important role on the structures of quark stars [53–57], in this work we investigate its impact on the properties of small objects such as *ud*QM nuggets.

The purpose of our current study is thus twofold, i.e., investigate the properties of *ud*QM nuggets with various symmetry energies and discuss their implications on *ud*QM stars' structures. The paper is organized as follows. In Section 2, we discuss briefly the equivparticle model and present the corresponding Lagrangian density. To investigate the impact of symmetry energy, an isospin dependent term is added to the quark mass scaling. Then the properties of *ud*QM nuggets are investigated adopting the method discussed in our previous publications [58–61], where the stability window for *ud*QM is obtained according to the binding energy of the heaviest *β*-stable nucleus <sup>266</sup>Hs. The properties of *ud*QM stars with and without crusts are then examined in Section 4 according to the stability of *ud*QM nuggets. We draw our conclusion in Section 5.
