**4.** *ud***QM Star**

By fulfilling simultaneously the local charge neutrality condition <sup>2</sup> 3 *<sup>n</sup><sup>u</sup>* <sup>−</sup> <sup>1</sup> 3 *n<sup>d</sup>* = *n<sup>e</sup>* + *n<sup>µ</sup>* and *β*-stability condition *µ<sup>u</sup>* + *µ<sup>e</sup>* = *µ<sup>u</sup>* + *µ<sup>µ</sup>* = *µ<sup>d</sup>* , the equation of state (EOS) for *ud*QM star matter can be fixed based on the equivparticle model illustrated in Section 2. Then the structure of *ud*QM star can be obtained by solving the TOV equation

$$\frac{\mathrm{d}P}{\mathrm{d}r} = -\frac{\mathrm{GME}}{r^2} \frac{(1 + P/E)(1 + 4\pi r^3 P/M)}{1 - 2GM/r},\tag{17}$$

$$\frac{\text{d}M}{\text{d}r} = 4\pi E r^2,\tag{18}$$

where the gravity constant is taken as *<sup>G</sup>* <sup>=</sup> 6.707 <sup>×</sup> <sup>10</sup>−<sup>45</sup> MeV−<sup>2</sup> . The tidal deformability can also be estimated with perturbation method, where a special boundary treatment on the surface is required for bare *ud*QM star [83–86].

The energy per baryon of *ud*QM nuggets is decreasing monotonously with baryon number for certain choices of parameters in Figure 2, which excludes a crust of *ud*QM nuggets. The corresponding mass-radius relations of bare *ud*QM stars are then presented in Figure 4. Note that the third term in the quark mass scaling in Equation (3) generally introduces an repulsive interaction among quarks, so that larger masses and radii are obtained for larger symmetry energy (larger *C<sup>I</sup>* ). The obtained maximum masses for the parameter sets (*C*, √ *D* in MeV): (−0.5, 176), (−0.3, 167), (0.1, 149), and (0.5, 135) reach the observational mass (2.14+0.20 <sup>−</sup>0.18*<sup>M</sup>*, 95.4% credibility) of PSR J0740+6620 [87], while the radii of the two-solar-mass stars are consistent with that of PSR J0740+6620 (12.39+1.30 <sup>−</sup>0.98 km and 2.072+0.067 <sup>−</sup>0.066*<sup>M</sup>*) only for *<sup>C</sup>* <sup>=</sup> 0.1 and <sup>√</sup> *D* = 149 MeV [88]. Nevertheless, if we examine the tidal deformation of those stars, only the *ud*QM stars obtained with the parameter sets (−0.5, 176) and (−0.3, 167) meet the constraints 70 ≤ Λ1.4 ≤ 580 from the binary neutron star merger event GW170817 [89].

**Figure 4.** Mass-radius relation of *ud*QM stars without crusts, where *C<sup>I</sup>* = 0. For the two cases with larger radii, *C<sup>I</sup>* = 40 MeV/fm<sup>3</sup> are adopted.

For the cases where *ud*QM nuggets at certain sizes (*A* ≈ 1000) are more stable than others, the surface of *ud*QM star will fragment into a crystalline crust. To obtain the EOSs of the *ud*QM star crust, as an rough estimation, we neglect the effects of charge screening and assume vanishing surface tension, i.e., Gibbs construction. By equating the pressures of the quark phase [*P*QM(*µ*b, *µe*) + *Pe*(*µe*)] and electrons [*Pe*(*µe*)], we find that the pressure of pure quark matter should vanish, i.e.,

$$P\_{\rm QM}(\mu\_{\rm b}, \mu\_{\varepsilon}) = 0.\tag{19}$$

Then the pressure of the nonuniform phase is exactly the pressure of electrons *P* = *Pe*(*µe*). The volume fraction *χ* of the quark phase is fixed according to the global charge neutrality condition, i.e.,

$$
\chi \left( \frac{2}{3} n\_{\iota} - \frac{1}{3} n\_{d} \right) = n\_{\iota}. \tag{20}
$$

Combining both the quark phase and electron phase, the energy density is determined by

$$E = \chi E\_{\text{QM}}(\mu\_{\text{b}}, \mu\_{\text{e}}) + E\_{\text{e}}(\mu\_{\text{e}}).\tag{21}$$

The obtained EOSs are presented in Figure 5, where the nonuniform phase takes place at *E* . 200 MeV/fm<sup>3</sup> . Note that the nonuniform phase in *ud*QM star is similar to that of the outer crust of neutron stars. Since the surface tension is nonzero and the energy per baryon of the most stable *ud*QM nugget does not reach *ε*0, we expect the formation of various geometrical structures in *ud*QM stars. A detailed investigation is thus necessary to obtain the realistic structures and EOSs [50,52], which is intended in our future works. Additionally, we should mention that the possible strong attractive interactions among quark clusters could result in very interesting conclusions [90–92], where the formations of strangeon matter and strangeon stars are expected [93,94].

**Figure 5.** Equation of state for *ud*QM star matter, which includes both the uniform phase at large densities and nonuniform phase at small densities.

Based on the EOSs presented in Figure 5, we solve the TOV Equation (17) and obtain the structures of *ud*QM stars with crusts, where the mass-radius relations are presented in Figure 6. Due to the presence of a crust, the mass-radius relations are similar to those of neutron stars, where the radius is increasing as we decrease the density in the center. This is essentially different from that of hybrid stars with unstable quark matter, where the deconfinement phase transition would reduce the radius and even lead to high-mass twins in case of a strong first-order phase transition [95]. To show this explicitly, the

mass-radius relation for hybrid stars is presented in Figure 5, which is obtained with the combination of a density functional PKDD for nuclear matter, *p*QCD with *C*<sup>1</sup> = 2.5 and ∆*µ* = 770 MeV for quark matter, and a surface tension value *σ* = 5 MeV/fm<sup>2</sup> as indicated in Reference [96]. For *ud*QM stars, it is found that the maximum masses for the parameter sets (*C*, √ *D* in MeV): (−0.3, 167), (0.1, 149), and (0.5, 135) reach the observational mass (2.14+0.20 −0.18*<sup>M</sup>*, 95.4% credibility) of PSR J0740+6620 [87], while the radii of the twosolar-mass stars coincide with that of PSR J0740+6620 (12.391.30 <sup>−</sup>0.98 km and 2.072+0.067 <sup>−</sup>0.066*<sup>M</sup>*) only for *<sup>C</sup>* <sup>=</sup> <sup>−</sup>0.3 and <sup>√</sup> *D* = 167 MeV. Similar as the cases of bare *ud*QM stars, the GW170817 constraint 70 ≤ Λ1.4 ≤ 580 is consistent with the parameter sets (−0.5, 176) and ( <sup>√</sup> <sup>−</sup>0.3, 167) [89]. Combined with all these constraints, only the case of *<sup>C</sup>* <sup>=</sup> <sup>−</sup>0.3 and *D* = 167 MeV is consistent with various pulsar observations.

**Figure 6.** Mass-radius relations (left) and tidal deformabilities (right) of *ud*QM stars with crusts, where the EOSs presented in Figure 5 were adopted. The corresponding values for typical hybrid stars from Reference [96] are presented as well.

In addition to the pulsar-like objects discussed so far, it was realized that for SQM there exists low-mass large-radius strangelet dwarfs if the surface tension is small enough [97]. A strangelet dwarf is comprised of a charge separated phase, which is energetically favorable to form crystalline structures with strangelets and electrons. Adopting the EOSs with *P*QM = 0 in Figure 5 and solving the TOV Equation (17), we have observed similar objects comprised of only *ud*QM nuggets and electrons as indicated in Figure 7. At sufficiently low temperatures, the *ud*QM nuggets and electrons form crystalline structures inside the star, which is in analogy with white dwarfs comprised of nuclei and electrons. We thus refer to them as *ud*QM dwarfs. Comparing with strangelet dwarfs [97], the masses of *ud*QM dwarfs are much larger and approaching to the Chandrasekhar limit (∼1.4 *M*), which is mainly due to the large charge-to-mass ratio of *ud*QM nuggets. In such cases, it is likely that some of the observed white dwarfs may in fact be *ud*QM dwarfs. Nevertheless, it is worth mentioning that the results indicated in Figure 7 should be considered as an upper limit, where the emergence of geometrical structures with various surface tension values is expected to play an important role [97]. A detailed investigation is thus necessary and intended for our future study.

**Figure 7.** Same as Figure 6 but for the mass-radius relations of *ud*QM dwarfs.
