Quantum nucleation of up-down quark matter and astrophysical implications

Quark matter with only $u$ and $d$ quarks ($ud$QM) might be the ground state of baryonic matter at large baryon number $A>A_{\rm min}$. With $A_{\rm min}\gtrsim 300$, this has no direct conflict with the stability of ordinary nuclei. An intriguing test of this scenario is to look for quantum nucleation of $ud$QM inside neutron stars due to their large baryon densities. In this paper, we study the transition rate of cold neutron stars to $ud$ quark stars ($ud$QSs) and the astrophysical implications, considering the relevant theoretical uncertainties and observational constraints. It turns out that a large portion of parameter space predicts an instantaneous transition, and so the observed neutron stars are mostly $ud$QSs. We find this possibility still viable under the recent gravitational wave and pulsar observations, although there are debates on its compatibility with some observations that involve complicated structure of quark matter. The tension could be partially relieved in the two-families scenario, where the high-mass stars ($M\gtrsim2 M_{\odot}$) are all $ud$QSs and the low-mass ones ($M\sim1.4\, M_{\odot}$) are mostly hadronic stars. In this case, the slow transition of the low-mass hadronic stars points to a very specific class of hadronic models with moderately stiff EOSs, and $ud$QM properties are also strongly constrained.

discuss these two possibilities and their astrophysical observations in detail, where different information on udQM and hadronic matter properties can be inferred. We conclude in Sec. V.
In Appendix A, we present the detailed calculations for the tidal deformability constraints with updated results for the recent event GW190425 from LIGO/Virgo [22]. In the rest of the paper, we use the natural unit with c = ħ h = k B = 1.

II. PROPERTIES OF udQM AND udQS
The novel possibility that udQM is actually the ground state of baryonic matter was explored in an effective theory of sub-GeV mesons in our recent paper [4]. Assuming a linear signal model, we fixed the free parameters in the meson potential by the masses and decay widths of mesons. In the presence of finite quark densities, the meson fields are pushed away from the vacuum along the least steep direction. As a result, the constituent quark masses are reduced and quark matter becomes energetically favorable. Due to the badly broken flavor symmetry in QCD, the potential shape around the vacuum is much stiffer along the strange direction than the non-strange one. The u, d quark mass then drops first as the Fermi momentum gradually increases from small values. Within the viable parameter space, an intermediate Fermi momentum is found to minimize the bulk energy per baryon ≡ E/A, where u, d quark mass already becomes negligible and the strange fraction remains zero. Thus, in contrast to the naive expectation from the bag model, udQM is more stable than SQM after taking into account the flavor symmetry breaking in the potential energy.
Around the Fermi momentum that is minimized, the energy per baryon for udQM in the bulk limit (large baryon number A 1) can be well approximated by contributions from a relativistic quark gas and from a spatially constant potential energy, where ρ is the energy density and n is the baryon number density. N C = 3 is the color factor for quarks, χ = i f 4/3 i is the flavor factor with the fraction f i = n i /(N C n), and p F = (3π 2 n) 1/3 is the Fermi momentum. The effective bag constant B eff denotes the potential difference along the valley oriented close to the non-strange direction. It is rather insensitive to p F and is closely related to the lightest meson mass. The minimum energy per baryon is then at the Fermi momentum p F,0 ≈ √ 2π χ −1/4 0 B 1/4 eff , with χ 0 = (2/3) 4/3 + (1/3) 4/3 for a charge neutral u, d gas. This shows the direct connection between the energy per baryon and the effective bag constant. For a large part of parameter space, we find udQM in the bulk limit more stable than the most stable nuclei 56 Fe, i.e. min 930 MeV, and so it is the ground state of baryonic matter with zero pressure.
Small udQM becomes less stable due to the finite size effects and the Coulomb energy contribution. For the baryon number A not too small, the former can be well approximated by a surface-tension term for the quark-vacuum interface. From the numerical fit, the surface tension is found to be quite insensitive to the variation of relevant parameters, with a robust value σ s0 ≈ 20 MeV fm −2 . To not ruin the stability of ordinary nuclei, it is safe to have the minimum baryon number A min of udQM larger than 300, corresponding to min 900 MeV.
Therefore, with Eq. (2), the scenario of stable udQM predicts the range of the effective bag constant to be 50 MeV fm −3 B eff 57 MeV fm −3 .
The stable udQM scenario could be realized in a more general setup. Going beyond the simple model in [4], the upper bound on B eff remains intact as it is directly related to the stability condition in the bulk limit. The lower bound derived from the condition A min 300, on the other hand, could be relaxed if the effective surface tension is larger, as predicted in some other models [23]. Assuming the same Coulomb contribution in the analytical approximation for the energy, a more general lower bound is σ s0 /(MeV fm −2 ) + 2 B eff /(MeV fm −3 ) 120, e.g. σ s0 30 MeV fm −2 is required for B eff ≈ 45 MeV fm −3 . As for SQM, a large perturbative QCD effect or a color superconducting phase could reduce B eff to a smaller range for the same stability condition of min [16,17,26].
The physics of compact stars relies on the properties of hadronic matter and udQM at certain temperature and pressure [15,24,25]. In this paper, we restrict to cold stars with zero temperature as a good approximation for mature neutron stars being formed after some time.
The approximation Eq. (1) ceases to apply at large pressure when a nonzero strange fraction becomes favored. It turns out that the pressure within reach for stable udQSs remains small and the strange fraction can be safely ignored, as we will show at the end of this section. Thus, in the rest of the paper, we stick with the effective bag constant range (3) inferred from the udQM properties. Quantum nucleation of udQM in hadronic matter phase relies on quark matter and hadronic matter properties at the same pressure. The flavor composition of udQM is then determined by that of the hadronic matter in chemical equilibrium due to the conservation of baryon and lepton numbers, and differs from udQM in equilibrium. Depending on the models, there could be a considerable fraction of electrons and muons in a neutron star interior, corresponding to an increasing number of protons.
To derive udQM properties as functions of the pressure, we start from udQM with relativistic electrons. The muon contribution is corrected by the non-negligible mass, and it will be discussed later. The energy density ρ as a function of n and f i can be found from Eq. (1) with (4) f e , f p denote the electron and proton fractions of the hadronic matter, and f e = f p when muons are absent. With Eq. (4), the pressure can be found through the thermodynamic relation where ρ 0 = min n 0 ≈ 4B eff is the surface density with zero pressure. Combining these two equations, we obtain the baryon number density The chemical potential µ can be found by substituting Eq. (5) and Eq. (6) into another thermodynamics relation with µ(0) = mim at the surface. This expression is equivalent to µ = i=u,d,e N C f i µ i , where µ i = p F i for relativistic particles. For increasing electron fraction f e , the flavor factor χ is larger, and udQM formed via transition has a smaller n(P) and a larger µ(P) in comparison to that in β-equilibrium. For f e ∼ O(10%), the chemical potential can change by the order of 10 MeV.
Including muons, instead of a mere redefinition of χ, there are non-negligible mass corrections to the energy density in Eq. (4) with the muon mass comparable to p F . For thermodynamic quantities relevant to quantum tunneling, the major change is for µ(P) with the additional contribution f µ µ µ in Eq. (7). Due to the chemical equilibrium, µ µ = µ e , f µ < f e , and the muon contribution is bounded from above by that from electrons.
As a useful approximation, if the P dependence of the flavor factor χ is mild, the thermodynamic relation in Eq. (5) can be rewritten as P ≈ n 2 d(ρ/n)/dn. Together with Eq. (7), this leads to a simple relation dµ/d P ≈ 1/n for either hadronic matter or quark matter. In the integral form, it becomes As we will show in Sec. III, this relation is crucial in understanding the general feature of hadronic matter to udQM transition.
Next, we discuss the properties of udQSs. The crucial quantity is the EOS of udQM in Eq. (5). It takes the same form as SQM in the bag model if ignoring the effect of strange quark mass, with the coefficient for ρ expected for a relativistic gas and a non-vanishing surface density ρ 0 = 4B eff . Referring to Eq. (2), udQM with the same minimum energy par baryon min as SQM has a smaller surface density ρ 0 due to a larger value of the flavor factor χ. For quark stars with an enormous A, gravitational interaction becomes important, and the density profile can be found by solving the Tolman-Oppenheimer-Volkoff (TOV) equation [27,28] with the udQM EOS in (5). As in the case for strange stars, this linear form of udQM EOS enables one to rewrite the TOV equation in terms of the following dimensionless variables [29,30],ρ and the ρ 0 or B eff dependence in the TOV equation is fully absorbed into the rescaled solution.   [18]. We can see that the B eff range (3) in the stable udQM scenario remains consistent with this upper bound. Due to uncertainties related to the hadronic matter and udQM properties, the lower 3 More massive pulsars have been suggested based on the optical spectroscopic and photometric observations [31]. But, as we will show later in Fig. 6, this imposes less stronger constraints due to large uncertainties of this method in comparison to the Shapiro delay measurements through radio timing. mass neutron stars with M ∼ 1.4M could be either hadronic stars or quark stars. For the latter case, i.e. all compact star being quark stars scenario, the radii of udQSs with M ∼ 1.4M can be compared with observations. This then provides additional constraints on B eff , as we will discuss in Sec. IV A. Fig. 1 (b) shows the tidal deformability Λ of udQSs in terms of the rescaled mass. As a useful quantity to characterize the tidal properties of udQS, the tidal deformability is determined by the Love number k 2 and the compactness C = GM /R =M /R with Λ = 2k 2 /(3C 5 ). As discussed in Appendix A, the dimensionless rescaling Eq. (10) can be extended to equations for k 2 so that the Love number for udQSs is determined only by C and is independent of B eff .
We find k 2 ranging from 0.7 to 0.06 as the compactness increases. Given theM −R relation, we can present Λ as a function ofM . Since the compactness increases with the mass, the tidal deformability becomes small for heavy stars with strong gravitational interactions, and it reaches the minimum value Λ min ≈ 23 at the maximum mass. The possibility that gravitational wave observations of coalescing neutron stars involve udQSs has been discussed in [18,21] for GW170817 from LIGO/Virgo [32]. In this paper, we extend the discussion to a newer event GW190425 [22], and constrain B eff together with other observations considering the transition rate estimation of neutron stars.
As a final remark, we justify the earlier assumption of ignoring the strange fraction. The central pressure for the maximum mass udQSs has the rescaling relation: P max ≈ 1. is larger only for about ten percent of models from our parameter scan [4]. 4 Thus, for most of the parameter space, it is a reasonable assumption to ignore the strangeness at the pressure accessible from a stable quark star.

III. QUANTUM NUCLEATION OF udQM IN COLD NEUTRON STAR MATTER
We start by reviewing the calculation framework for the quark matter nucleation rate. Inside a cold neutron star, a droplet of more stable udQM nucleates in the metastable hadronic phase through quantum tunneling. In the semiclassical approximation, the virtual droplet can be described by a sphere with radius R(t). The potential energy for such a fluctuation can be represented as [34,35] 4 The strange fraction will turn on above the special Fermi momentum p (s) F , when it is energetically favorable to produce non-relativistic or relativistic strange quarks. This gives p (s) where C P ≡ 4πn Q (µ H −µ Q )/3. µ H , µ Q are the chemical potentials of hadron and quark matter, and n Q is the baryon number density of the later. σ s is the surface tension for the quark-hadron interface, and it differs from σ s0 for the quark-vacuum interface in general. As σ s may suffer more from the theoretical uncertainties, i.e. σ s ∼ 10-150 MeV fm −2 found for different models in the literature, 5 we treat σ s as a free parameter in this paper. For this potential, the first term denotes the negative volume contribution that favors the quark matter with µ Q < µ H , and the second term denotes the positive surface contribution that prevents nucleation at smaller radii.
A potential barrier forms due to competition of the two contributions, as shown in Fig. 2. It is useful to characterize the potential by its peak value and a special radius that denotes the typical size of a droplet, The potential energy U(R) for a stable phase droplet with a radius R. U max denotes the potential peak value. R c is the nontrivial zero of the potential and denotes the typical size of a droplet. R ± denote the classical turning points in the tunneling rate calculation of a state of energy E.
The kinetic energy of a droplet results from a flow in the medium around the droplet when there is a density discontinuity between the two phases. For a general case, the Lagrangian for the fluctuation can be written as [35] whereṘ is the growth rate and M (R) is the effective mass for the droplet, where C M ≡ 4πρ H (1 − n Q /n H ) 2 and ρ H is the energy density for the hadronic phase. The kinetic term incorporates the relativistic effects. WhenṘ 1, it takes the non-relativistic form 1 2 M (R)Ṙ 2 as in the Lifshitz-Kagan theory [36].
For the quantum tunneling problem, a state of energy E satisfies the Schrodinger equation, With the standard semiclassical (WKB) approximation, the tunnelling probability for one droplet is In the non-relativistic limit, A(E) is roughly the action under the potential barrier. Taking into account relativistic effects, it takes the form R ± denote the classical turning points as given by U(R ± ) = E. The ground state energy E 0 is determined from Bohr's quantization condition where I(E) is the action for the zero-point oscillation, the minimum allowed energy for E 0 when the relativistic effects are large, i.e. U(R) > 2M (R). The transition time for one droplet is then, where Inside a hadronic star, the transition time Eq. (20) for a udQM droplet at the radius r is determined by the properties of udQM and hadronic matter evaluated at the pressure P(r), through the coefficients C P , σ s in the potential energy Eq. (11) and C M in the effective mass Eq. (14). After its quantum formation, the first droplet quickly expands by eating up nucleons and the whole star will be converted almost instantaneously. The transition time for a hadronic star τ s can then be approximated by formation time of the first droplet, where τ min denotes the minimum transition time a droplet could have inside the star and N s 1 denotes the number of such droplets. Given that the neutron star radius is around 10 km and the typical size of a droplet is in the order of fm, we have roughly N s ∼ (km/fm) 3 ∼ 10 54 .
A more careful estimate for transition at the core gives N s ∼ 10 48 [35]. To account for the related uncertainties, we assume N s ∼ 10 45 -10 55 in the rest of the paper.  quantities. The intersection of n(p) curves can be avoided for a stiff hadronic EOS such as GM1, but the chemical potential difference and then C P also become larger for this case. The final result for the transition time depends on the competition between C P and C M . This competition is expected due to the relation between the density difference and the chemical potential difference as from the thermodynamic relation in Eq. (8), Therefore, for the case that n H is bounded from above by n Q , both C M and C P become larger for a stiffer hadronic matter EOS. At certain point, a too large C P dominates the transition time and the increasing stiffness would not help to slow down the transition.

IV. CONVERSION OF NEUTRON STARS AND ASTROPHYSICAL OBSERVATIONS
For a more comprehensive understanding of the conversion of neutron stars, we start from a brief review of the hadronic matter EOSs in compact stars. A typical neutron star has an atmosphere and an interior. Fig. 5 (a) summarizes our current understanding of the hadronic matter EOSs in the interior. Below 0.5ρ nuc is a curst consisting of ions and electrons (and free neutrons when the density is above the neutron drip density). The curst EOS is testable in Astrophysical observations for neutron stars provide important clue to the EOS in this region.
Comparing with the joined constraints from recent observations, some models for example are disfavored at ρ 4ρ nuc . Perturbative QCD (pQCD) applies at an ultrahigh density, i.e. ρ 100ρ nuc . Although this region is far from accessible in a neutron star, its prediction may serve as an asymptotical limit for any model of the intermediate density region. Phenomenologically, an EOS needs to satisfy the monotony and causality conditions, i.e. 0 ≤ d P/dρ ≤ 1.
As highlighted in Sec. III, the spacing between the hadronic matter and udQM n(p) curves is crucial in determining the transition rate. Here we present the comparison of various n(p) curves in Fig. 5 (b). Given the theoretical range of B eff in Eq. (3), the udQM prediction is a quite narrow band as approximated by n(p) ≈ 0.003MeV −1 P + n(0). Interestingly, most of the hadronic models considered before are quite soft, and their n(p) curves can easily intersect with the udQM band at some low pressure, i.e. below 30 MeV fm −3 , accessible from an astrophysical neutron star. Thus, newly formed hadronic stars described by these EOSs will experience an instantaneous transition, and observed compact stars with M 1.4M are most likely to be udQSs. On the other hand, the uncertainty range of the hadronic matter EOSs as from the low energy theory and astrophysical observations remain large, where the major part of the udQM band is covered. A slow transition is then possible for a special set of viable EOSs with the n H (P) curve sitting moderately below the udQM band. In the following, we discuss these two possibilities and their observational implications in detail.
As a side remark, for the case that a crossing of n(p) curves occurs, the chemical potential difference µ H − µ Q starts to decrease above the crossing point with n H > n Q , referring to Eq. (22). At some higher pressure, µ H may become smaller than µ Q , indicating that the hadronic matter becomes more stable again. If this pressure is accessible from udQSs, there will be a transition back to hadronic matter in the deep interior of quark stars. This points to a new type of hybrid stars, in contrast to the conventional ones with a quark matter core. We leave the detailed study for future work.

A. All compact stars being udQSs
Neutron stars described by a soft hadronic matter EOS is more likely to convert to udQSs, the maximum mass of which remains compatible with the observed heaviest pulsars. The possibility that all compact stars are udQSs then provides a natural solution to the hyperon puzzle. For this case, the main question is the consistency of udQS predictions with most of the other neutron star observations that involve objects considerably lighter than 2M . Note that the joined constraints found in [47] and other references rely on the nuclear theory input for hadronic matter at the low density, and they cannot be directly used for udQM and udQSs. with a heavier mass but a much larger uncertainty. Overall, the theoretical range is consistent with these 2M bounds at 90% C.L.. NICER measures the X-ray emission from a rotating neutron star and is expected to reach better sensitivity for the mass and radius measurements.
As the first target, J0030+0451 points to a star with the mass around 1.4M and the radius around 13 km. Given the rescaled mass and radius relation for udQSs in Fig. 1 (a), the inferred range on the M − R plane can be translated to a range for B eff . A relatively small B eff is favored by this observation, with the theoretical prediction disfavored at 68% C.L.. The tension nonetheless goes away at 90% C.L., and there is even less concern if considering the theoretical uncertainties associated with the surface tension σ s0 .
Gravitational wave observations provide a unique chance to measure the tidal properties for the binary system. The average tidal deformabilityΛ can be extracted from a waveform at the inspiral stage, and it is a function of the mass ratio and the rescaled chirp mass. The The most popular interpretation involves a superfluid component and a rigid structure [50,51], with glitches produced by their angular momentum transfer. In the standard scenario, the rigid structure is provided by a solid crust, and the crustal moment of inertia is bounded from below by the observations of "giant glitches". Although still under debate [30,52,53], the normal nuclear curst of a quark star below the neutron dip pressure [54] is probably too small to account for the demanded crustal momentum inertia. An alternative for quark stars is a crust consisting of small chunks of quark matter instead of ordinary nuclei. For strange stars, this new crust might be large enough with the energy density contributed mainly by the strangelets [55]. The two ingredients may also be related to the peculiar properties of quark matter. One example is the inhomogeneous crystalline color superconducting phase, which is rigid as well as superfluid and may provide an explanation without a crust [56]. For the stable udQM case, the nuclear crust for udQS would be larger than that for strange stars due to a larger positive charge for udQM and a stronger Coulomb support of the crust. Implications of other mechanisms deserve further studies.
A more recent example for such kind of observations is quasi-periodic oscillations for the highly magnetized compact stars. In the simplest model, they are associated with the seismic oscillations of the stellar crust, and the frequencies are determined mainly by the crust thickness. Quark stars are disfavored due to their much thinner crust and the much higher frequency, even considering a crust consisting of quark matter [57]. However, to infer the crust thickness, modes identification between the observation and theory is needed, and this may depend crucially on other unknown features of the stars [58]. All in all, the current observations in tension with the quark star explanation seem to involve complicated structure of quark matter, and further studies are required for a more conclusive analysis.
Mergers of quark stars may produce small chunks of udQM, which we name as udlets, in line with the strangelets in the SQM hypothesis. Normally, a udlet would not be absorbed by an ordinary nucleus due to the Coulomb repulsion of the positive charges. But those generated from mergers may acquire large kinetic energy to overcome the Coulomb barrier, and their encounter with smaller hadronic stars, e.g. white dwarfs, planets, may lead to fast conversion into small quark stars. The final results depend on the flux and spectrum of udlets.
A recent numerical simulation [59] shows that the strangelet flux from strange stars merger is negatively correlated with B eff through the mass-radius relation. Since udQM has a smaller B eff than SQM with the same min , the conversion rate induced by the udlets might be higher than that for strangelets.

B. Co-existence of hadronic stars and udQSs
In the two-families scenario, high-mass stars with M ∼ 2M are all udQSs, while low-mass ones remain hadronic with a slow enough transition rate. As mentioned before, this points to a special class of hadronic matter EOSs, which is a little fine-tuned. But in view of observations, it shows that the transition behavior in this scenario is extremely sensitive to the variations of hadronic matter EOSs, and could be used to provide information that is otherwise inaccessible.
Another advantage of this scenario is the possibility to avoid the long-time debate regarding the compatibility with observations such as pulsar glitches, given that these observations are consistent with lower mass stars within uncertainties.   Fig. 7. The blue line denotes an example (HM1) that smoothly interpolates the high density and low density regimes as described by the pQCD and ChPT. Matching to the crust at 0.5ρ nuc , its EOS at higher density is given by the analytical expression 6 n(P) = (a + bP) ( with the parameters (a, b, c, Fig. 8 (b) mainly comes from the distinct C P , as being proportional to the chemical potential difference in Fig. 8 (a). Since HM2 is softer at lower density, it has smaller µ H and then a smaller chemical potential difference.
A nontrivial P-dependence of the flavor composition can also be helpful by reducing the chemical potential difference at high pressure. For illustration, we focus on hadronic matter models with a negligible strangeness, and the flavor composition varies mainly with the lepton fractions. 7 For the case with negligible contribution from leptons, given the thermodynamic relation Eq. (22), the chemical potential difference µ H −µ Q increases with pressure as expected from the condition n H (P) < n Q (P) , and the transition at the center of stars is the fastest. In the presence of nonzero lepton fractions, the chemical potential of udQM becomes larger, and the difference µ H − µ Q can be significantly reduced at high pressure due to the nontrivial P-dependence of the lepton fractions, which reach up to 20% for this model.
It is then clear that a soft hadronic matter EOS at lower density and a nontrivial lepton fractions can help to slow down the transition. For instance, as from Fig. 8 (b), assuming star. For such cases, as shown in Fig. 11, the hadronic matter EOS plays an important role, with quite different results for the two benchmarks. HM1 is ruled out by GW170817 at 90% C.L. for either a udQS-HS system or a HS-HS system simply due to a too large tidal effect for the hadronic star. The situation for HM2 is better, where we find no constraints for the HS-HS case and B eff 50MeV fm −3 for the udQS-HS case.
In the two-families scenario, low mass hadronic stars not yet converted by the quantum nucleation may experience a fast transition by encounter with a udlet, which can be produced by binary mergers involving heavy udQSs. Recent numerical simulations for strange stars show that the merger product of quark stars tends to promptly collapse to a black hole with much less ejecta [59]. A binary with heavy udQSs either have a too small companion or a too large total mass. For the former case the merger is too mild to produce ejecta, while for the latter case matter is mostly swallowed by the promptly formed black hole. Thus, the udlet flux as coming only from binary mergers involving heavy udQSs would be much smaller than the flux in the "all compact stars being udQSs" case, and the chance of low-mass stars converted by udlets is expected to be small.

V. SUMMARY
We investigated astrophysical implications of the stable udQM scenario in this paper, taking into account both the transition rate estimation for hadronic stars and the observational constraints.
With the effective bag constant B eff range (3)  The main issue we addressed here is the nature of low-mass compact stars with M ∼ 1.4M .
As shown in Fig. 3, the transition time mainly depends on the chemical potential and the density difference for the hadronic matter and quark matter phases, as well as the surface tension σ s of their interface. We found it convenient to track the EOS dependence through comparison of the n(P) curve of the two phases. As a result, the transition rate only becomes significant when moving into the interior. A prominent feature is that when the two n(P) curves cross, the udQM droplets turn ultra-relativistic, and the transition is instantaneously fast regardless of the values for other quantities.
Most of the hadronic models do predict a n(P) curve intersecting with that of udQM at a pressure accessible from a 1.4M compact star. This then points to the unconventional possibility that the observed neutron stars are mostly udQSs. This possibility is often overlooked due to a long-time debate on its compatibility with some well-established observations. Yet, complicated structures of quark stars are likely to be involved. For a more direct probe of the basic properties of udQM, we consider constraints from the recent gravitational wave and pulsar observations. As shown in Fig. 6, different observations push B eff to the opposite directions with a small region left open. We found the theoretical prediction of udQM still viable at 90% A slow transition of low-mass hadronic stars is also possible if the hadronic matter n(P) curve happens to be moderately below the udQM one, which is still allowed given the uncertainties. For this case, the transition time is extremely sensitive to variations of the relevant quantities, as summarized in Fig. 8. We found that a softer hadronic matter EOS at low pressure and a nontrivial lepton fraction can help to slow down the transition, and a reasonable lower bound on B eff and σ s can be obtained to have 1.4M compact stars being hadronic at the present universe. Heavy hadronic stars with M ≈ 2M , on the other hand, can quickly convert to quark stars. Thus, the transition behavior in the two-families scenario provides useful information for both udQM and hadronic matter. A softer hadronic EOS is favored by the recent observations as well, in particular GW170817 from LIGO/Virgo.
There are more to explore in the future. On the theoretical side, further model development for stable udQM may help to limit the allowed ranges for B eff and σ s , which will lend to a more definite conclusion for the two-families scenario. On the observational side, a hadronic star conversion is a dramatic event, where a large amount of energy is expected to be released.
This may trigger a neutrino burst accompanied by emission of gravitational waves [15]. The implication for udQSs deserves further studies. In this section, we discuss in detail the tidal deformability constraints from the compact star merger events on udQSs, and we use the geometric unit with G = c = ħ h = k B = 1 here.
The GW170817 event detected by LIGO/Virgo [32] is the first confirmed merger event of compact stars, with the chirp mass M c = 1.186 +0.001 −0.001 M , and a 90% highest posterior density interval ofΛ = 300 +420 −230 with q = 0.73 − 1.00 for the low spin prior from the collaboration [32,33]. More recently, a new event GW190425 was identified [22] with M c = 1.44 M , q = 0.8 − 1.0 andΛ ≤ 600 for the low spin prior at 90% credible interval. Ref. [18] showed that the GW170817 event may be a binary system with at least one udQS. In the following, we update the constraints in the context of the neutron star conversion, and extend the discussion to GW190425.
The tidal deformability Λ, which characterizes the response of compact stars to an external disturbance, can be expressed as Λ = 2k 2 /(3C 5 ). The Love number k 2 is defined as [60][61][62][63] where the compactness C = M /R, and y R is y(r) evaluated at the surface, which can be obtained by solving the following equation [63]: with boundary condition y(0) = 2. Here and For quark stars with a finite surface density, a matching condition should be imposed at the boundary y ext R = y int R − 4πR 3 ρ 0 /M [64]. Note that we can also utilize Eq. (10) to transform Eq. (A2) into a fully dimensionless form, withρ andP obtained from the rescaled TOV equation for quark stars as introduced in Sec. II. The solution then is in the form of y(r) with r = r √ 4B eff , and the variable y(R) evaluated at the surface can be converted further into the y(C) form with theM −R relation in Fig.1 (a). Therefore, for udQSs, the Love number k 2 and the tidal deformability Λ are only functions of the compactness C, as shown in Fig. 9 (a), (b) respectively, with the dependence on ρ 0 or B eff fully absorbed. This feature crucially relies on the linear form of quark matter EOSs. For a binary system, the average tidal deformability is defined as Λ = 16 13 (1 + 12q) (1 + q) 5 Λ(M 1 ) + 16 13 where q = M 2 /M 1 ≤ 1. For an equal mass binary with q = 1,Λ is simply Λ(M i ). In the other limit that q → 0,Λ is dominated by the massive component contribution. For the two-families scenario, a binary with at least one low-mass star could be either a udQS-HS system or a HS-HS system. Here we use the two benchmarks of hadron matter EOSs introduced earlier in Fig. 7, i.e. HM1 and HM2, which are proposed to realize a slow transition of the low-mass hadronic stars. This is in contrast to the previous study in Ref. [18], where Bsk19, SLy, Bsk21 models are chosen for a more general representation of the hadronic matter EOSs.
The corresponding results ofΛ for GW170817 and GW190425 are shown in Fig. 11. We can see that for GW190425 either a udQS-HS merger or a HS-HS merger is well compatible with the current constraint. For GW170817, the observations favor a relatively soft hadronic EOS and B eff 50MeV fm −3 for udQM, which match the expectation of Ref. [18].