Revisting the boiling of quark nuggets at nonzero chemical potential
aa r X i v : . [ a s t r o - ph . C O ] A ug Revisiting the boiling of primordial quarknuggets at nonzero chemical potential
Ang Li a , b , , Tong Liu a , b , c , , Philipp Gubler d , ,and Ren-Xin Xu e , a Department of Astronomy and Institute of Theoretical Physics and Astrophysics,Xiamen University, Xiamen 361005, China b State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China c Key Laboratory for the Structure and Evolution of Celestial Objects, ChineseAcademy of Sciences, Kunming, Yunnan 650011, China d RIKEN Nishina Center, RIKEN, Wako 351-0198, Japan e School of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, China
Abstract
The boiling of possible quark nuggets during the quark-hadron phase transitionof the Universe at nonzero chemical potential is revisited within the microscopicBrueckner-Hartree-Fock approach employed for the hadron phase, using two kindsof baryon interactions as fundamental inputs. To describe the deconfined phase ofquark matter, we use a recently developed quark mass density-dependent modelwith a fully self-consistent thermodynamic treatment of confinement. We studythe baryon number limit A boil (above which boiling may be important) with threetypical values for the confinement parameter D . It is firstly found that the baryoninteraction with a softer equation of state for the hadron phase would only lead toa small increase of A boil . However, results depend sensitively on the confinementparameter in the quark model. Specifically, boiling might be important during theUniverse cooling for a limited parameter range around D / = 170 MeV, a valuesatisfying recent lattice QCD calculations of the vacuum chiral condensate, while forother choices of this parameter, boiling might not happen and cosmological quarknuggets of 10 < A < could survive. PACS:
Key words:
Equation of state; Quark nuggets
Preprint submitted to Elsevier 15 May 2018
Introduction
It has long been proposed that much of the baryon number ( A ) of the Universeis condensed into the quark phase (usually called quark nuggets, QNs) duringthe quark-hadron phase transition [1]. To survive in the hot QCD medium ( ∼
150 MeV), a QN of a certain size must outlive two decay processes, namelysurface evaporation [2] and boiling [nucleation of hadronic bubbles (HBs)] [3,4].The former is generally very efficient when the environment is transparent toneutrinos, and the details mainly depend [2] on the dynamic properties of theneutrino-driven cooling, for example the neutrino opacity. Our interest lies inthe latter case, i.e., the boiling of QNs into hadrons, which is closely relatedto the underlying microscopic physics of the quark-hadron phase transition.In one of the earliest studies, Alcock & Olinto [3] described nucleons in termsof an ideal gas, and assumed that the pressure in the strange-quark matterwould be contributed entirely by the thermal spectrum of light particles (elec-trons, neutrinos, and photons). Based on the idea that if the total surface areaof HBs exceeds the QN surface area, boiling would be inefficient, they founda baryon number minimum A boil above which boiling is important, and con-cluded that this limit must be as high as 10 - 10 . They have furthermoretreated the surface tension of QNs, σ , as a free parameter and obtained for itan unusually large lower limit, namely (178 MeV) , which would mean thatalmost all QNs could not survive boiling. Later Madsen & Olesen [4] treatedthe hadron phase as a Walecka-type interacting neutron-proton-electron ( npe )gas and also introduced the fermion pressure in the quark phase using the MITbag-like model [5]. They found a rapid dependence of A boil on the parameters( σ, B ), where B is the bag constant. They argued that QNs may survive boil-ing for some choice of ( σ, B ), and that therefore for such a case boiling is notthe dominant decay process for QNs, compared to the evaporation mentionedabove. In the recent work of Lugones & Horvath [6], quark pairing and thecurvature energy were introduced in the quark phase and it was concludedthat both boiling and surface evaporation would be suppressed by the pairinggap. The authors also argued that boiling might be unlikely for intermediatetemperatures ( T < T ∆ ∼ . [email protected] [email protected] [email protected] [email protected] .
17 fm − , and also fulfill the recent 2-solar-mass neutron star mass measure-ment [31,32]. They, however, give a very different high-density EOS ( > . − ) [14]. In particular, the microscopic TBF turns out to be more repulsivethan the Urbana model at high densities, and the discrepancy between thetwo predictions becomes increasingly large as the density increases. Since thethreshold of the quark-hadron transition is essentially determined by the stiff-ness of different hadron EOSs, we should keep in mind that the EOS from themicroscopic TBF is stiffer than that of the phenomenological one. Hereafter,we refer to “stiff EOS” as the one with the microscopic TBF, and to “softEOS” as the one with the phenomenological TBF. Calculations are mainlydone using the microscopic TBF, and results with the phenomenological TBFare presented as well in several cases for comparison.Here we have neglected the possible appearance of hyperons and pion or kaon3ondensates in the hadron phase, which in general might soften the high-density EOS. How to confront them with the high-mass neutron stars is animportant topic discussed frequently in recent papers [17,33,34]. It would bestraightforward to include the strangeness in the hadron phase in a subsequentstudy, once the controversial high-density EOS is clarified.Let us also mention here that the subject of study of the boiling of QNs(into hadrons), in principle demands that the QCD phase transition is of firstorder (see for instance [6] and references therein). Lattice QCD studies overthe past years have however reached the conclusion that, for physical quarkmasses and a vanishing baryon chemical potential µ , this transition is rather asmooth crossover than a first order phase transition [35,36,37]. If the Universefollows the “standard” scenario and undergoes the QCD phase transition withonly a very small µ , this would mean that QNs could not have been createdand the discussion of their properties would thus be irrelevant for the Universethat we live in. It should however be stressed here that there is room for analternative scenario, which has been discussed in the literature [38,39,40,41] aslittle inflation. In this case, the Universe follows a path with larger µ and cantherefore undergo a first order phase transition as the QCD phase diagramis expected to have a critical endpoint at some finite value of µ , above whichthe the quark-gluon plasma and hadron gas phases are separated by a firstorder phase transition line. The creation of QNs can hence not be ruled outand studying their properties may still be of relevance for the physics of ourUniverse.The paper is organized as follows. In section 2, we establish our physical modeland describe in details the numerical methods for the calculation. In section3, numerical results are discussed. We present our main conclusions in section4. In the hot QCD medium, the hadron gas may be energetically favored inthermal fluctuations, and bubbles of hadronic gas would nucleate throughoutthe volume of the produced nuggets of strange matter. This process is called“boiling of QNs”. If boiling happens, the QN would dissolute into hadrons anddisappear in the Universe.Following the estimation using classical nucleation theory by Alcock & Olinto [3],the work done to form a bubble of radius r composed by the hadronic phase4 " / e e (cid:80) (cid:74) (cid:81) (cid:81) (cid:16) (cid:14) (cid:32) / (cid:80) (cid:74) (cid:81) (cid:81) (cid:32) / n n p p e e (cid:80)(cid:16) (cid:14) (cid:122) / u u d d s s e e (cid:80)(cid:16) (cid:14) (cid:122) / (cid:80) (cid:74) (cid:81) (cid:81) (cid:32) Fig. 1. Pictorial representation of the model. Zero chemical potential ( µ = 0) rep-resents that the relevant particles are thermally produced. inside the quark phase is W = − πr ∆ P + 4 πσr , (1)where σ is the QN surface tension. We follow Madsen & Olesen [4] and self-consistently calculate the surface tension from all fermion species ( i = u, d, s, e )as: σ i = 3 T π Z ∞ (1 − π arctan km i ) ln[1 + exp( − e i ( k ) − µ i T )] kdk. (2)where the energy e ( k ) is given as e ( k ) = q k + m i with m i ( µ i ) being the mass(chemical potential) of component i . T is the temperature. ∆ P = P H − P Q isthe pressure difference between the hadron phase (with a pressure of P H ) andthe quark phase (with a pressure of P Q ). Assuming that the phase transitionis first order, its properties are calculated from the the pressure differencebetween the two phases based on the chemical equilibrium condition: µ Q ( P Q ) = µ H ( P H ) ≡ µ (3)where µ Q and µ H are the baryon chemical potentials for the hadron and quarkphases, respectively. 5pecifically, as illustrated in Fig. 1, besides the common pressures from ther-mal photons and neutrinos, P H is contributed by hadrons and e ± pairs whichwill be dealt as accurately as possible here, P Q constitutes a nonthermal pres-sure from u, d, s quarks and e ± pairs. Since the pressure of the quark phase, P Q , is equal to the pressure in the Universe (mainly contributed by thermalphotons, neutrinos, electrons, and positrons), the nonthermal contributionsof quarks and e ± pairs in the quark phase balance exactly the thermal elec-trons and positrons in the Universe. This constrain is used to determine theindependent baryon chemical potential, µ . Clearly, ∆ P comes from the pres-sure difference apart from the common contribution from thermal photonsand neutrinos, and can be computed from the pressure of hadron phase, P H ,subtracted by the pressure from thermally produced electrons and positrons:∆ P = P H − π π ~ ) Z ∞ (f e − + f e + ) k /e ( k ) dk | µ e =0 (4)with f i the Fermi–Dirac distribution written asf e − ( k, T ) = 1exp[( e ( k ) − µ e ) /k B T ] + 1 , (5)f e + ( k, T ) = 1exp[( e ( k ) + µ e ) /k B T ] + 1 , (6)where k B is the Boltzmann constant and the energy e ( k ) = q k + m e with m e ( µ e ) being the mass (chemical potential) of the electrons or positrons.Whether boiling is important or not depends on the formation rate of criticalbubbles, since only those bubbles with a radius greater than the critical radiuswill be able to grow. By maximizing W in Eq. (1) w.r.t. r , the work of acritical-size bubble can be obtained as W c = 16 π σ ∆ P . (7)The rate at which critical bubbles appear is then p ( T, µ ) ∼ T exp( − W c /T ) . (8)If the number of bubbles becomes so small that the total surface area of thebubbles is smaller than the bounding surface area of a QN, boiling would beinefficient. That gives a minimum baryon number A boil above which boiling isefficient [4]: A boil ≈ . × − ∆ P T σ exp(16 π σ T ∆ P ) . (9)Much of the microscopic physics introduced in the present work, including thequark-quark interaction, would enter in ∆ P and consequently influence the6 [MeV]= 158 170 225 / / M e V T /MeV
20 40 60 80 100020040060080010001200 D [MeV] = 158 170 225 / M e V T /MeV
Fig. 2. (Left) surface tensions σ and (right) baryon chemical potentials µ are shownas a function of temperature T for three values of the quark confining parameter D / = 158 MeV, 170 MeV, 225 MeV in the quark phase. The microscopic TBF isemployed for the hadron phase. physics of boiling. As will be shown later, the high powers and exponentialterm in Eq. (9) would result in a rapid dependence on parameters (∆ P , σ , T ).The remaining part of this section is devoted to the calculation of ∆ P . Asmentioned above, assuming the phase transition is first order, calculations canbe done separately for two phases. They are related by the constraint of thecommon chemical potentials µ of Eq. (3). In the following, theoretical modelsfor the the hadron phase and the quark phase are illustrated, respectively. Let us first study the hadron phase, that is nuclear matter consisting of nu-cleons and e ± pairs in β -equilibrium of the following weak reactions: n ⇋ p + e − + ¯ ν e , (10) e + + e − ⇋ ν e + ¯ ν e (11)Under the condition of neutrino escape, this equilibrium can be written as µ n − µ p = µ e − = − µ e + . (12)The requirement of charge neutrality implies n p = n e − − n e + , (13)where n i is the number density of component i .7he chemical potentials of the non-interacting leptons e ± are obtained bysolving numerically the free Fermi gas model at finite temperature. The nu-cleonic chemical potentials required in Eq. (12) are derived from the free en-ergy density of nuclear matter, based on the finite-temperature BHF nuclearmany-body approach discussed below.The BHF approach [15] is one of the most advanced microscopic approachesto the EOS of nuclear matter. Recently, this model was extended to the finite-temperature regime within the Bloch-De Dominicis formalism [42,43,44]. Thecentral quantity of the BHF formalism is the G -matrix, which in the finite-temperature extension [15,16,42,43,44] is determined by solving numericallythe Bethe-Goldstone equation, and can be written in operatorial form as G ab [ W ] = V ab + X c X p,p ′ V ac (cid:12)(cid:12)(cid:12) pp ′ E Q c W − E c + iǫ D pp ′ (cid:12)(cid:12)(cid:12) G cb [ W ] , (14)where the indices a, b, c indicate pairs of nucleons and the Pauli operator Q and energy E determine the propagation of intermediate nucleon pairs. In agiven nucleon-nucleon channel c = (12) one has Q (12) = [1 − f ( k )][1 − f ( k )] , (15) E (12) = m + m + e ( k ) + e ( k ) , (16)with the single-particle (s.p.) energy e i ( k ) = k / m i + U i ( k ), the Dirac-Fermidistribution f i ( k ) = (cid:16) e [ e i ( k ) − ˜ µ i ] /T + 1 (cid:17) − , the starting energy W , and the above-mentioned baryon interaction V as fundamental input. The various single-particle (s.p.) potentials within the continuous choice are given by U ( k ) = Re X n,p X k n ( k ) D k k (cid:12)(cid:12)(cid:12) G (12)(12) h E (12) i (cid:12)(cid:12)(cid:12) k k E A , (17)where k i generally denote momentum and spin. For given partial densities n i ( i = n, p ) and temperature T , Eqs. (14-17) have to be solved self-consistentlyalong with the equations for the auxiliary chemical potentials ˜ µ i , n i = R k f i ( k ).Once the different s.p. potentials for the species i = n, p are known, the freeenergy density of nuclear matter can be obtained using the following simplifiedexpression f N = X i "X k f i ( k ) k m i + 12 U i ( k ) ! − T s i , (18)8here s i = − X k (cid:18) f i ( k ) ln f i ( k ) + [1 − f i ( k )] ln[1 − f i ( k )] (cid:19) (19)is the entropy density for component i treated as a free gas with s.p. spectrum e i ( k ) [16,15]. All thermodynamic quantities of interest can then be computedfrom the free energy density, Eq. (18); namely, the “true” chemical potentials µ i ( i = n, p ), internal energy density ǫ N , and pressure P N are µ i = ∂f N ∂n i , (20) s N = − ∂f N ∂T , (21) ǫ N = f N + T s N , (22) P N = n B ∂ ( f H /n B ) ∂n B = X i µ i n i − f N . (23)One can then proceed to calculate the composition of the hot β -equilibriummatter by solving Eqs. (12) and (13), together with the conservation of thebaryon number, n n + n p = n B . Then the total energy density ǫ and the totalpressure P of the system are ǫ H = ǫ l + ǫ N , (24) P H = P l + P N , (25)where ǫ l and P l are the standard contributions of the leptons: ǫ l = 8 π (2 π ~ ) Z ∞ (f e − + f e + ) e ( k ) k dk , (26) P l = 8 π π ~ ) Z ∞ (f e − + f e + ) k /e ( k ) dk . (27) We consider the quark phase as a mixture of interacting u , d , s quarks,electrons and positrons. In the QMDD model, the mass of the quarks m i ( i = u, d, s ) is parameterized by the baryon number density n B as follows9 i ≡ m i + m I = m i + Dn zB , (28)The density-dependent mass m i includes two parts: one is the original mass orcurrent mass m i , the other is the interacting part m I . The light-quark massesare very small, and we simply take m u = m d = 0. As to the uncertain strangequark mass, a modest value of m s = 95 MeV is chosen. In principle, the quarkmass scaling should be determined from QCD, which is obviously impossiblepresently. As mentioned before, we use the cubic scaling z = 1/3 [45], based onthe linear confinement and in-medium chiral condensates. It was first derivedat zero temperature [45] and then expanded to finite temperature [8]. D is the confinement parameter. From our previous work of Peng et al.2008, we know that it has a lower bound of D / = 156 MeV, and an up-per bound of D / = 270 MeV. The lower bound comes from the nuclearphysics constraint, demanding that at P = 0, non-strange nuclear mattershould be stable against decay to ( ud ) quark matter. This leads to the con-dition E/A > M F e c /
56 = 930 MeV for ( ud ) quark matter, which gives theabove-mentioned lower bound. The upper bound can be derived from a rela-tion between D and the quark-condensate [9] and the known range of valuesfor this condensate. The upper boundary of 270 MeV is in fact a very conser-vative one. According to the updated quark condensate determined nowadaysvery precisely by lattice QCD [46], a range of (161 MeV, 195 MeV) can be ob-tained. Therefore in this work, we take three typical values of the confinementparameter as D / = 158 MeV, 170 MeV, 225 MeV. The lower one (158 MeV)is chosen to account for the realizable case of absolutely stable strange-quarkmatter [9], the middle one (170 MeV) because it satisfies the constraints fromthe newest lattice QCD results [46], and the large one (225 MeV) to study thesituation in the region of the upper boundary.The relevant chemical potentials µ u , µ d , µ s , and µ e ± satisfy the weak-equilibriumcondition (we again assume that neutrinos leave the system freely): µ u − µ d = µ e − = − µ e + , µ d = µ s . (29)The baryon number density and the charge density can be given as n B = 13 ( n u + n d + n s ) , (30) q Q = 23 n u − n d − n s + n e + − n e − . (31)The charge-neutrality condition requires q Q = 0.10n the present model, the single-particle energies depend on density and tem-perature via the quark masses. Based on the quasiparticle assumption, thequark energy density can be written as ( i = u, d, s ) [9] ǫ Q = g X i X k q k + m i f i ( k, T ) . (32)where the statistical weight is g = 6 for quarks. From the Landau definitionof the single-particle energy extended to finite temperature, we have ε i ( k ) = δǫ Q δ f i ( k, T )= q k + m i + g X j m j f j ( k, T ) q k + m j ∂m j ∂n i ≡ e i ( k ) − µ I , (33)where e i ( k ) ≡ q k + m i is the usual dispersion relation of free particles. Theextra term µ I can be added to the chemical potential, so defining µ ∗ i ≡ µ i + µ I . (34)Accordingly, the net density of the particle type i is n i = g P k [f i ( k, T ) − f ¯ i ( k, T )] , or, explicitly, we have n i = g Z ∞ (cid:26)
11 + e [ ε i ( k ) − µ ∗ i ] /T −
11 + e [ ε i ( k )+ µ ∗ i ] /T (cid:27) p d k π . (35)Inverting this equation, one determines µ ∗ i as a function of n i so that the freeenergy density of quarks can be given as f q = X i h f + i + f − i i (36)with f ± i = g Z ∞ ( − T ln (cid:20) e − ( √ k + m i ∓ µ ∗ i ) /T (cid:21) ± µ ∗ i e ( √ k + m i ∓ µ ∗ i ) /T ) k d k π . (37)One can then determine the real chemical potentials and pressure, accordingto the well-known relations ( i = u, d, s ):11 i = ∂f q ∂n i , P q = − f q + X i µ i n i . (38)Solving Eqs. (29), (30) and (31), the total pressure of the system can beobtained: P Q = P l + P q , (39)after adding the contribution of the leptons P l (Eq. (27)). We begin in Fig. 2 with the surface tensions of QNs (left) and the baryonchemical potentials (right) as a function of temperature for three values ofthe quark confining parameter in the quark phase, and the microscopic TBFchosen for the hadron phase. During the cooling of the Universe, σ increases,which means that QNs tend to be more bound. Also, the larger the confiningparameter, the larger the surface tension, which is reasonable because quarksinteract more strongly. The baryon chemical potentials are decreasing func-tions with the temperature. With the increase of the confining parameter D ,the equilibrium chemical potential µ of the two phases increases due to a largerpressure difference ∆ P as shown in Fig. 3.To see the influence of the hadron EOS, results from the the phenomenologicalTBF (labelled as “soft EOS”) is also shown for the case of D / = 170 MeV.It is clearly observed that a soft hadron EOS will result in a slight decrease ofthe pressure difference of the two phases. It has a similar decreasing effect onthe equilibrium chemical potential µ , but the difference is very small and cannot be distinguished in Fig. 2.More microscopically, we show in the left panel of Fig. 4 the baryon numberdensities of the two phases at chemical equilibrium. The densities in QNsalways decrease with the temperature T , but the densities in HBs first increasewith T then decrease with it. Soft EOS in the hadron phase will in generallower the baryon densities of HBs at chemical equilibrium only slightly foreach temperature T . Compared with that of the hadron EOS, the effect of thequark confining parameter D is more evident. A larger D value will result inincreasing baryon densities both in QNs and HBs, and also lead to a largerdensity gap between the two phases, as shown in the right panel of Fig. 4.One can also notice a small decreasing effect of the soft hadron EOS on thedensity gap.For a smaller density gap between the two phases, the boiling of a QN to12 [MeV] = 158 170 225 P / M e V f m - T /MeV soft EOS
Fig. 3. Pressure differences ∆ P are shown as a function of temperature T for threevalues of the quark confining parameter D / = 158 MeV, 170 MeV, 225 MeV inthe quark phase. The microscopic TBF is employed for the hadron phase. Resultsfrom the the phenomenological TBF (labelled as “soft EOS”) is also shown for thecase of D / = 170 MeV. HBs becomes more difficult, as it is clearly demonstrated in Fig. 5, where thebaryon number limits A boil , above which boiling could happen, are shown as afunction of the temperature T . The calculation is done with D / = 170 MeVin the quark phase, for both the microscopic TBF and the phenomenologicalTBF for the hadron phase. A slightly smaller pressure difference (as shownin Fig. 3), or a smaller density gap (as shown in Fig. 4) in the soft EOS case(namely the case of the phenomenological TBF) will result in a larger workfor a critical-size bubble, and finally lead to a lower probability of bubblesnucleation, namely a larger value of the limit A boil .More importantly, it is found that the limit A boil is a sharp decreasing functionof the temperature T , as indicated by Eq. (9). With the chosen D parameter,large amounts of cosmological quark nuggets between 10 < A < wouldboil around T = 90 MeV. A small change of the D value from 170 MeV wouldresult in a sensitively sharp increase of the A boil value, which is far from thebaryon number of a cosmological QN. This is what happens in the cases of158 MeV and 225 MeV. It means that only for a small parameter range of theconfining parameter D , there is a chance that the QN boiling could happenvery efficiently during the early stage ( T ∼
90 MeV) of the cooling. Otherwise,the baryon limit is irrelevant and no QNs could boil and might survive until thepresent time. We mention here that our calculation demonstrates an amazing13 D en s i t y gap /f m - H ad r on bubb l e s Q ua r k nugge t s D [MeV] = 158 170 170 (soft EOS) B a r y on den s i t y /f m - T /MeV
T /MeV
Fig. 4. Baryon densities of the two phases (left panel) and density gaps betweenthe two phases (right panel) at chemical equilibrium are shown as a function ofthe temperature T , for two values of the quark confining parameter D / = 158MeV (solid lines) and 170 MeV (dashed lines) in the quark phase. The microscopicTBF is employed for the hadron phase. Results from the the phenomenological TBF(labelled as “soft EOS”) is also shown for the case of D / = 170 MeV (dash-dottedlines). coincidence of the lattice results [46] just falling into the range where boilingcould be important. In this paper, we have renewed the study of the boiling of possible QNs duringthe quark-hadron phase transition of the Universe at nonzero chemical poten-tial. For this purpose, a parameter-free microscopic BHF model extended tofinite temperature is employed for the hadron phase to describe HBs, and aquark model with self-consistent thermodynamic treatment of the confinementis used to deal with the quark phase for the description of QNs. Both phasesare in beta-stable equilibrium through weak processes. The phase transitionis regarded as a first-order one, in order to get the pressure difference betweenthe two phases. Also, the important parameter of the QN surface tension iscalculated self-consistently at each sets of parameters. Detailed presentationsof the QN surface tension, the baryon chemical potential, as well as the pres-sures and the baryon densities for the two phases are shown.We found a larger effect of the confinement parameter, than that of the hadron14
Log [ A B o il ] T /MeV
Soft EOSStiff EOS
Fig. 5. Baryon number limits A boil , above which boiling could happen, are shown asa function of the temperature T . The calculation is done with D / = 170 MeV inthe quark phase, for both the microscopic TBF (solid line, labelled as “stiff EOS”)and the phenomenological TBF (dashed line, labelled as “soft EOS”) for the hadronphase. The cases of D / = 158 MeV and D / = 225 MeV would result in A boil values far from the baryon number of a cosmological QN, therefore they are notshown here. EOS. The baryon number limits are found above which boiling could be ef-ficient. It turns out that only a limited range of the confinement parameterallow boiling to happen around a temperature of 90 MeV. This allowed rangeof the confinement parameter falls comfortably into the range provided by lat-tice QCD. This therefore appears to be the favoured scenario at present. Largenumbers of QNs would boil to HNs in this case and QNs may not exist in thepresent Universe. For other values of this parameter, boiling is impossible tohappen, and QNs with a cosmological baryon number of 10 < A < couldbe possibly found. Future experiments of detecting dark matter can provide acrucial cross-check to this problem.Furthermore, the importance of quark pairing gap was demonstrated in thework of Lugones & Horvath [6] within the MIT model, taking into accountthe BCS-type of pairing for quarks. Therefore, a more proper treatment ofpairing, for example using the Nambu-Jona-Lasinio (NJL), or the Polyakov-NJL model (PNJL), should be interesting. Since the NJL (or PNJL) model, notlike the MIT + BCS scheme, incorporates all the symmetries known from thefundamental theory of strong interactions (i.e., quantum chromodynamics), itcould improve our understanding of the boiling problem and its relation withthe phase diagram of the high-density strong interaction matter.15 Acknowledgments
Two of us (AL, TL) would like to thank Dr. H. Mao for his suggestions for themanuscript and his hospitality during our stay in Hangzhou. This work wassupported by the National Basic Research Program (973 Program) of ChinaGrant (Nos 2012CB821800, 2014CB845800), the Fundamental Research Fundsfor the Central Universities, the National Natural Science Foundation of China(Grant Nos 11078015, 11103015, 11225314, 11233006, U1331101), and the CASOpen Research Program of Key Laboratory for the Structure and Evolutionof Celestial Objects under grant OP201305.
References [1] Witten, E. 1984, Phys. Rev. D, 30, 272[2] Alcock C., & Farhi, E. 1985, Phys. Rev. D, 32, 1273[3] Alcock C., & Olinto, A. 1989, Phys. Rev. D, 39, 1233[4] Madsen J., & Olesen, M. 1991, Phys. Rev. D, 43, 1069[5] Farhi E., & Jaffe, 1984, R. L. Phys. Rev. D, 30, 2379[6] Lugones, G., & Horvath, J. E. 2004, Phys. Rev. D, 69, 063509[7] Peng, G. X., Chiang, H. Q., Zou, B. S., Ning, P. Z., & Luo, S. J. 1999, Phys.Rev. C, 62, 025801[8] Wen, X. J., Zhong, X. H., Peng, G. X., Shen, P. N., & Ning, P. Z. 2005 Phys.Rev. C, 72, 015204[9] Peng, G.-X., Li, A., & Lombardo, U. 2008, Phys. Rev. C, 77, 065807[10] Lugones, G. & Horvath,J. E. 2003, Int. J. Mod. Phys. D, 12, 495[11] Zheng, X. P., Liu, X. W., Kang, M., & Yang, S. H. 2004, Phys. Rev. C, 70,015803[12] Li, A., Xu, R. X., & Lu, J.-F. 2010, MNRAS, 402, 2715[13] Li, A., Peng, G.-X., & Lu, J.-F. 2011, RAA, 11, 482[14] Li, A., Burgio, G. F., Lombardo, U., & Zuo, W., 2006, Phys. Rev. C, 74, 055801[15] Baldo, M. 1999, in Nuclear Methods and the Nuclear Equation of State, ed. M.Baldo (Singapore: World Scientific), 1[16] Baldo, M., & Ferreira, L. S. 1999, Phys. Rev. C, 59, 682[17] Burgio, G. F., Schulze, H. J., & Li, A. 2011, Phys. Rev. C, 83, 025804
18] Baldo, M., Bombaci, I., & Burgio, G. F. 1997, A&A, 328, 274[19] Sahu, P. K., Burgio, G. F., & Baldo, M. 2002, ApJ, 566, L89[20] Burgio, G. F., Schulze, H. J., & Weber, F. 2003, A&A, 408, 675[21] Zuo, W., Li, A., Li, Z.-H., & Lombardo, U., 2004, Phys. Rev. C, 70, 055802[22] Nicotra, O. E., Baldo, M., Burgio, F., & Schulze, H. J. 2006, A&A, 451, 213[23] Li, A., Zhou, X. R., Burgio, F., & Schulze, H. J. 2010, Phys. Rev. C, 81, 025806[24] Baldo, M., & Burgio, G. F. 2012, Rep. Prog. Phys., 75, 026301[25] Bonanno, A., Baldo, M,. Burgio, G. F., & Urpin, V. 2013, A&A accepted,preprint(astro-ph/13112153)[26] Chen, H., Burgio, G. F., Schulze, H. J., & Yasutake, N. 2013, A&A, 551, A13[27] Wiringa, R. B., Stoks, V. G. J., & Schiavilla, R. 1995, Phys. Rev. C, 51, 38[28] Pudliner, B. S., Pandharipande, V. R., Carlson, J., & Wiringa, R. B. 1995,Phys. Rev. Lett., 74, 4396[29] Pudliner, B. S., Pandharipande, V. R., Carlson, J., Pieper, S. C., & Wiringa,R. B. 1997, Phys. Rev. C, 56, 1720[30] Grang´e, P., Lejeune, A., Martzolff, M., & Mathiot, J.-F. 1989, Phys. Rev. C,40, 1040[31] Demorest, P., Pennucci, T., Ransom, S., Roberts, M., Hessels, J. 2010, Nature,467, 1081[32] Antoniadis J. et al., 2013, Science, 340, 6131[33] Li, A. Huang, F., & Xu, R. X. 2012, Astropart. Phys., 37, 70[34] Miyatsu, T., Yamamuro, S., & Nakazato K. 2013 ApJ, 777, 4[35] F. R. Brown et al. , Phys. Rev. Lett. , 2491 (1990).[36] S. Aoki et al. (JLQCD Collaboration), Nucl. Phys. Proc. Suppl. B73 , 459(1999).[37] Y. Aoki, G. Endrodi, Z. Fodor, S.D. Katz, K.K. Szabo, Nature , 675 (2006).[38] B. Kaempfer, Astron. Nachr. , 231 (1986).[39] N. Borghini, W.N. Cottingham and R.V. Mau, J. Phys. G , 771 (2000).[40] T. Boeckel and J. Schaffner-Bielich, Phys. Rev. Lett. , 041301 (2010).[41] T. Boeckel and J. Schaffner-Bielich, Phys. Rev. D , 103506 (2012).[42] Bloch, C., & De Dominicis, C. 1958, Nucl. Phys., 7, 459[43] Bloch, C., & De Dominicis, C. 1959a, Nucl. Phys., 10, 181
44] Bloch, C., & De Dominicis, C. 1959b, Nucl. Phys., 10, 509[45] Peng, G. X., Chiang, H. Q., Yang, J. J., Li, L., & Liu, B. 1999, Phys. Rev. C,61, 015201[46] Aoki, S. et al. 2013, preprint(hep-lat/13108555)44] Bloch, C., & De Dominicis, C. 1959b, Nucl. Phys., 10, 509[45] Peng, G. X., Chiang, H. Q., Yang, J. J., Li, L., & Liu, B. 1999, Phys. Rev. C,61, 015201[46] Aoki, S. et al. 2013, preprint(hep-lat/13108555)