Excluded-volume model for quarkyonic matter II: Three-flavor shell-like distribution of baryons in phase space
IINT-PUB-20-028
Excluded Volume Model for Quarkyonic Matter II: Three-flavor Shell-likeDistribution of Baryons in Phase Space
Dyana C. Duarte, ∗ Saul Hernandez-Ortiz, † and Kie Sang Jeong ‡ Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA (Dated: July 28, 2020)We extend the excluded volume model of iso-spin symmetric two-flavor dense Quarkyonic mat-ter [1] including strange baryons and quarks and address its implications for neutron stars. Theeffective size of baryons are defined from the diverging hard-core potentials in the short interdistanceregime. Around the hard-core density, the repulsive core between baryons at short-distances leadsto a saturation in the number density of baryons and genarates the perturbative quarks from thelower phase space which leads to the shell-like distribution of baryons by Pauli’s exclusion principle.The strange quark Fermi sea always appears in the high densities but Λ hyperon shell only appearswhen the effective size of Λ hyperon is smaller than the effective size of nucleons. We find that thepressure of strange quarkyonic matter can be large enough to support neutron stars with two-timesolar mass and can have a large sound speed c s (cid:39) .
7. The fraction of the baryon number carriedby perturbative quarks is about 30% at the inner core of most massive neutron stars. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ nu c l - t h ] J u l I. INTRODUCTION
Observation of GW170817 [2, 3] provided important information for understanding dense nuclear matter. Thepossible range of tidal deformability is confined in 90% confidence level [2, 3] and the subsequent analyses constrainedthe corresponding radius R . ≤ . R M ≥ . ≤
15 km including higher mass states. Meanwhile, it has been required for the hard enough equation of state(EoS) to support two-times solar mass ( M (cid:12) ) states, which usually leads to larger radius state [13–15]. To reconcilethese observations, the EoS should be soft enough for the low density regime and hard enough for the higher densityregime, so that the strong pressure of the inner core in higher densities can support larger mass state and the weakerpressure of the outer core in lower densities can satisfy R . ≤ . c s > . ρ ) [16–24]. Beyond the density regime of inner core where the hard EoS is supported, the softened EoS is expectedunder the causality and conformal limit constraints [5, 6, 18–25].However, it is hard to reconcile both constraints from fundamental principles. If one considers mean-field potentialsbetween baryons, certain universal repulsive contributions are expected for EoS [26, 27] at high densities as the newlygenerated degrees of freedom lead to the soft EoS through various decay channels into the low energy states [28–31].Even if the stiff evolution is obtained by some kind of model, it is hard to explain the expected softening evolution atthe high density limit by these same first principles. Some kind of phase transition to quark matter can be introduced.A phase transition to quark matter attenuates the hard nature of the EoS. There is much literature that debates aboutthe signals of such a hypothetical phase transition [16–18, 25, 32–36]. As an alternative candidate for a solution, it isworthwhile to consider Quarkyonic-like model [1, 37–41] which naturally generates the hard-soft evolution of EoS.The Quarkyonic matter concept is based on large N c quantum chromodynamics (QCD) [42, 43]. If one supposes alarge Fermi sphere ( T →
0) in the large N c limit, the quasi-quark state around the Fermi surface whose momenta aredistributed in the range of confinement ( | (cid:126)k Q i − (cid:126)k Q j | < Λ QCD , | (cid:126)k Q i | (cid:39) k QF where i, j = { , · · · N c } ) will be confined inthe baryon-like state. The confinement mechanism is expected to be similar to the mechanism for the baryon statein vacuum as the Debye screening due to the quark loop is suppressed by 1 /N c [37]. In this circumstance, as theconfined quark momenta are correlated within | (cid:126)k Q i − (cid:126)k Q j | < Λ QCD , the baryon-like state has the minimum momentum k bF (cid:39) N c k QF . Thus, one can expect the shell-like distribution of the baryons and the almost free quarks occupyingthe lower phase sphere. The transition from the ordinary nuclear matter to Quarkyonic matter may occur at fewtimes of ρ where the soft-hard evolution of EoS is expected. Whence k bF ∼ O (Λ QCD ), the lower phase sphere will besaturated by the free quarks. Then, by Pauli’s principle, the momenta of confined quarks should become larger thanthe saturated momenta [1, 39], which leads to the sudden enhancement of chemical potential of the baryon-like state( k bF ∼ O (Λ QCD ) → N c k QF ). This is not a usual first-order phase transition as the pressure is not fixed but suddenlyand smoothly enhanced by the enhanced chemical potential and there is no discontinuity for the increment of energydensity and the baryon number density [37]. From this point, most of baryon number increment is taken by thesaturated quarks and eventually, the shell-like baryon distribution will disappear and the perturbative QCD matterwill appear at extreme density limit ( k QF ∼ O ( √ N c Λ QCD )), as the Debye screening begins to block the confinementprocess ( r Debye ∼ O ( N c )).This concept was introduced to describe the hard-soft evolution of EoS in the previous literature [1, 39–41]. Themodel construction with the explicit shell-like distribution reproduced the plausible result satisfying the aforemen-tioned constraints [39]. The 2-flavor generalization of the aforementioned model [39] is studied under the β -equilibriumcondition [40]. In the phenomenological model construction, one can consider the hard-core repulsive interaction whosescale can be regarded as the effective size of the baryon [44–63]. In the single flavor excluded volume model [1], therepulsive core dynamically generates the shell-like phase structure of baryon which reproduces the stiff evolution ofEoS with c s (cid:39) . II. EXPLICIT STRUCTURE OF SHELL-LIKE DISTRIBUTION OF BARYONS
In dense Quarkyonic matter [37], the quark wave functions distributed around the quark Fermi surface are clearlyconfined in baryon-like states because Debye screening is suppressed in large N c limit [42, 43]. The matter looks like thenormal nuclear matter in the low density regime as the momentum of a quark is distributed in the confinement range.However, when the matter density reaches few times of ρ where k bF ∼ O (Λ QCD ) so that lowest momentum statesbecome distributed away from the clear confinement range, the quark Fermi sea is formed from the low momentumphase space. Whence the Fermi sea is saturated, the confined quark should take larger momentum than the fullyoccupied lower phase by Pauli’s exclusion principle (shell-like momentum distribution of baryons) [1, 37, 39]. Aroundthe onset moment, the pressure of the system will be continuously and stiffly increasing as the chemical potential shouldshow stiffness and continuity ( k bF (cid:39) N c k QF ), contrary to the expected evolution in a first-order phase transition. Thephenomenological configuration strongly depends on how one defines the lower boundary of the distribution becausethe dynamical equilibrium constraints are related through the shell-like distribution. In this section, we will brieflyintroduce the excluded volume model approach, and present the explicit structure of the shell-like baryon distributionformed by the dynamically saturated quark Fermi sea. We will use the following abbreviations to denote the baryonsand quarks: B represents the total baryons including quarks, b represents the baryon (hadron) and the Q representsthe saturated quarks. The abbreviations appearing with the lowcase romans, b i represents the baryon flavor flavors { n, p, Λ } and Q i represents the quark flavors { u, d, s } . A. Brief summary of excluded volume model for Quarkyonic-like model
As introduced in the previous literature [1, 41], one can simplify the baryon-baryon central potential whose strong re-pulsive core is expected at high density regime [65–71] by supposing the infinite-well shaped potential whose hard-coreradius is around r c (cid:39) . n , p , and Λ) system can be suggested as a simplest multi-flavor extensionwhere the effective size of particle is understood from the hard-core repulsion around n B ∼ n [65–71]. If only thequasi-baryons are assumed, the number density in excluded volume can be defined as follows [1, 41]: n exb i = n b i − ˜ n b /n = 2(2 π ) (cid:90) K biF d k, (1)˜ n b = n n + n p + (1 + α ) n Λ , (2)where K b i F represents the enhanced Fermi momentum due to the reduced available volume and α determines thestrength of hard core repulsive interaction between surrounding baryon and Λ hyperon in range of | α | < .
2. Inthe context of the presumed effective size of particle, this approach could be understood as the cold-dense limit ofthe Van der Waals (VdW) EoS in Fermi-Dirac statistics [49–63]. As a simple example, K bF can be obtained from µ ∗ = µ id ( n exb , T →
0) without attraction term if one derives the intensive number from the VdW EoS in Fermi-Diracstatistics [55]. However, as we focus on the high density regime where the interdistance of particles becomes order ofthe hard-core radius, n > .
65 fm − (cid:39) ρ will be considered, which is a different order of magnitude from the sizeused in Refs. [55–63]. One may adopt a well constructed model [18, 20, 24, 26, 27, 61] and use a Maxwell constructionto accommodate the low density properties of nuclear matter. The variation range | α | < . ε b = (cid:18) − ˜ n b n (cid:19) π { n,p, Λ } (cid:88) i (cid:90) K biF dkk (cid:0) k + m b i (cid:1) + (3 π ) π n e , (3)where the electron mass is suppressed. If one takes non-relativistic limit, baryon chemical potential can be obtainedas follows: µ i (cid:39) m b i + (3 π ) π m i n exb i + ω i { n,p, Λ } (cid:88) j (3 π ) π m j n n exb j + · · · , (4)where ω i = ∂ ˜ n b /∂n i ( ω n,p = 1, ω Λ = 1 + α ). As one can find from the third term, the chemical potential of a specificflavor (4) can be enhanced without having large n exb i if the some part of system volume is occupied by the other finitesize particles. Thus, to accommodate a heavier baryon (denote flavor h ), its effective size should be small so that thecontribution from the third term be suppressed ( ω h (cid:28) n . Due to the intrinsic divergence around n B ∼ n ,this system cannot accommodate n B > n and contains unphysical configuration ( v s (cid:29) k bF = N c k QF . Even if the asymmetric configurationis considered, the scale can be estimated around k bF ∼ N c max. (cid:2) k uF , k dF , k sF (cid:3) as the quarks confined in a baryon shouldhave a common scale of momentum. The detailed argument for the k b i F in the iso-spin asymmetric configuration willbe given in next subsection. The baryon number in excluded volume density within the explicit shell-like structurecan be written from k b i F as ¯ n exb i = n b i − ˜ n b /n = 2(2 π ) (cid:90) [ k F +∆] bi k biF d k, (5)where the upper boundary of the baryon distribution has been defined by assuming the fully occupied phase space:[ k F + ∆] b i = (cid:16) π ¯ n exb i + k b i F (cid:17) , (6)where the ∆ is the width of the baryon distribution [1].In Quarkyonic matter, the quark Fermi sea would be continuously saturated without any signature of a first-order phase transition according to large N c gauge dynamics [1, 37, 41]. Thus, smooth interpolation of a quark’senergy should be possible in both directions around the Fermi surface. In this excluded volume model approach, theenergy interpolation is continuous by analytic definition but an unphysical divergence appears at the onset moment ofsaturation. Huge energy enhancement due to the sudden formation of the shell-like baryon distribution leads to theunphysical energy dispersion relation corresponding to ∂n B /∂n ˜ Q (cid:29) , ∂n B /∂n b (cid:28) n B = n b + n ˜ Q ). To attenuatethe unphysical divergence, an enhanced phase measure M i ( k ) for the saturated quarks can be introduced. Themodified measure M i ( k ) > k effectively enhances the free quark density around the saturation moment of freequarks and converges to the ideal gas limit ( M i ( k ) → k ) at high density regime ( k Q i F (cid:29) Λ QCD ). Then the quarknumber density can be written in baryon number unit as follows: n ˜ Q i ≡ π (cid:90) k QiF dk M i ( k ) , (7)where the tilde in the subscript denotes the number density in baryon unit. The relatively rapid growth of quarkdensity at the onset moment ( n b (cid:39) n ) makes an effective barrier for δn ˜ Q in the variation of the baryon numberdensity ( n b < n − δn ˜ Q ), which prevents the unphysical divergence and leads to the gradual formation of the shell-likedistribution. The energy density with explicit shell-like baryon distribution can be written as follows: ε qy. = 2 (cid:18) − ˜ n b n (cid:19) { n,p, Λ } (cid:88) i (cid:90) [ k F +∆] bi k biF d k (2 π ) (cid:0) k + m b i (cid:1) + N c π { u,d,s } (cid:88) j (cid:90) k QjF dk M j ( k ) (cid:16) k + m Q j (cid:17) + (3 π ) π n e . (8)Corresponding baryon ( n, p , and Λ) chemical potential can be obtained as µ b i = ∂ε qy. ∂n b i = (cid:18) − ˜ n b n (cid:19) [ k F + ∆] b i π (cid:16) [ k F + ∆] b i + m b i (cid:17) ∂ [ k F + ∆] b i ∂n b i + { n,p, Λ } (cid:88) j (cid:54) = i [ k F + ∆] b j π (cid:16) [ k F + ∆] b j + m b j (cid:17) ∂ [ k F + ∆] b j ∂n b i − ω i n { n,p, Λ } (cid:88) k π (cid:90) [ k F +∆] bk k bkF dkk (cid:0) k + m b k (cid:1) , = (cid:18) n − (˜ n b − ω i n b i ) n − ˜ n b (cid:19) (cid:16) [ k F + ∆] b i + m b i (cid:17) + ω i n { n,p, Λ } (cid:88) j (cid:54) = i ¯ n exb j (cid:16) [ k F + ∆] b j + m b j (cid:17) − { n,p, Λ } (cid:88) k π (cid:90) [ k F +∆] bk k bkF dkk (cid:0) k + m b k (cid:1) , (9)where the partial derivatives are calculated as ∂ [ k F + ∆] b i ∂n b i = π [ k F + ∆] b i (cid:18) − ˜ n b /n (cid:19) (cid:18) − ˜ n b − ω i n b i n (cid:19) , (10) ∂ [ k F + ∆] b j ∂n b i = π [ k F + ∆] b j (cid:18) − ˜ n b /n (cid:19) (cid:18) ω i n b j n (cid:19) , (11)with ω n,p = 1, ω Λ = 1 + α . Again, the characteristic feature of excluded volume model can be found from the ω i dependent terms of chemical potential (9). Even if there exist only few numbers of a specific flavor of baryon, thecorresponding chemical potential can be enhanced if the space is taken by the other baryons. By the same reason,we only consider 3-flavors for the baryon side as n , p , and Λ are expected to have similar order of n . The quarkchemical potential in baryon units can be obtained in a similar way: µ ˜ Q i = ∂ε qy. ∂n ˜ Q i = (cid:18) − ˜ n b n (cid:19) { u,d,s } (cid:88) k (cid:40) [ k F + ∆] b k π (cid:16) [ k F + ∆] b k + m b k (cid:17) ∂ [ k F + ∆] b k ∂n ˜ Q i − k b k F π (cid:16) k b k F + m b k (cid:17) ∂k b k F ∂n ˜ Q i (cid:41) + N c (cid:18) m Q i + (cid:16) k Q i F (cid:17) (cid:19) = (cid:18) − ˜ n b n (cid:19) { u,d,s } (cid:88) k ∂k b k F ∂k Q i F k b k F M i (cid:16) k Q i F (cid:17) (cid:26)(cid:16) [ k F + ∆] b k + m b k (cid:17) − (cid:16) k b k F + m b k (cid:17) (cid:27) + N c (cid:18)(cid:16) k Q i F (cid:17) + m Q i (cid:19) , (12)where the partial derivatives are calculated as ∂ [ k F + ∆] b k ∂n ˜ Q i = π [ k F + ∆] b k k b k F M i (cid:16) k Q i F (cid:17) ∂k b k F ∂k Q i F , (13) ∂k Q i F ∂n ˜ Q i = π M i (cid:16) k Q i F (cid:17) . (14)As a consequence of Pauli’s exclusion principle, the chemical potential of saturated quark gets contributions fromthe baryon distribution as well because k b k F emerges as a consequence of the saturated quark Fermi sea. Also, one A detailed argument for the possible emergence of ∆(1232) is given in Appendix A. k [ MeV ] n Q ( k ) ( a ) n d ( k ) n u ( k ) k [ MeV ] n Q ( k ) ( b ) n d ( k ) n u ( k ) FIG. 1. Illustration of quark momentum correlation in the confined state ( k dF > k uF ). Blue (red) dot-dashed (dashed) linerepresents d ( u ) quark distribution. The states whose momentum is distributed in not fully occupied phase ( n Q ( k ) <
1) areunderstood as confined quark waves in the baryon-like state. The quark distribution depicted in (a) and (b) represents theconfiguration under the strong and weak correlation assumption, respectively. In the strongly correlated configuration (a) , theconfined quarks have almost same size of momenta so that the distribution of u quark is concentrated around k dF . However,in the weakly correlated configuration (b) , k u conf. is broadly distributed as the confined quark momenta can be deviated fromeach other. can anticipate another singularity possibly arising in the iso-spin asymmetric configuration. Suppose the shell-likedistribution ( k b k F >
0) formed by ahead saturation of d quark Fermi sea ( k dF >
0) and u Fermi quark sea about toappear ( k uF (cid:39) M u (cid:0) k uF (cid:1) → k uF → M i (cid:0) k (cid:1) = k + Λ Q i which leads to followingconfigurations: n ˜ Q i = 1 π (cid:90) k QiF dk (cid:0) k + Λ Q i (cid:1) = k Q i F π (cid:18) (cid:16) Λ Q i /k Q i F (cid:17) (cid:19) , (15) ε qy. = 2 (cid:18) − ˜ n b n (cid:19) { n,p, Λ } (cid:88) i (cid:90) [ k F +∆] bi k biF d k (2 π ) (cid:0) k + m b i (cid:1) + N c π { u,d,s } (cid:88) j (cid:90) k QjF dk (cid:16) k + Λ Q j (cid:17) (cid:16) k + m Q j (cid:17) + (3 π ) π n e , (16)where the criteria for M i (cid:0) k (cid:1) are satisfied in the both limits. The regulator Λ Q i could be understood as an a priorinon-perturbative contribution remaining on the saturated quark Fermi surface. B. Explicit structure of shell-like distribution of baryon
The iso-spin asymmetry naturally appears under consideration of electro-weak interactions and subsequent equi-librium conditions. This asymmetric configuration can arise in either of the baryons and quarks. The details will bestrongly dependent on k bF as the physical constraints between the baryons and quarks are related through the shell-likebaryon distribution. k bF can be supposed differently depending on the assumption about the confined quark statedistributed slightly above the saturated quark Fermi surface: if the confined quark momenta around the saturatedquark Fermi surface are strongly correlated, k bF should show weak dependence on the flavor asymmetry while it candepend strongly on the asymmetry if the confined quark momenta are weakly correlated. Following we propose twophenomenological assumptions for the two different scenarios.
1. Assumption I: strongly correlated momentum of confined quark
In the large N c limit, one may assume a strong correlation between the momenta of confined quarks as the confine-ment mechanism should be very similar to the one of hadron state in vacuum. This clear confinement should occureven for the quarks whose momentum is distributed just above the Fermi surface, where the occupation number isalmost one. In a simplest guess for the constituent quarks of baryon, one can imagine the confined quarks sharinga same size of momentum k Q conf. = k b / | k q i conf. − k q j conf. | (cid:28) Λ QCD , k q i conf. > k q i F ,and | k q i F − k q j F | > Λ QCD where i, j denotes the quark flavor). For example, one may imagine iso-spin asymmetric con-figuration where d quarks are saturated first as total baryon density increases and u quarks follow after (Fig. 1(a)).Then, the momentum of u quark confined in the lower boundary of the baryon shell should be closely distributedaround k dF as follows: k u conf. = k dF + r sqq w s (cid:0) k dF − k uF (cid:1) , (17)where r sqq determines the correlation strength and the strong correlation weight function w s ( x ) is assumed to slowlyconverge to 1 in the x = | k dF − k uF | > Λ QCD limit: w s ( x ) = 1 − exp (cid:0) −| x | /δ (cid:1) , (18)where δ determines the non-trivial range ( w s ( x < Λ QCD ) < r sqq , the minimaldifference | k u conf. − k d conf. | (cid:28) Λ QCD will be guaranteed. As illustrated in Fig. 2(a), k u conf. → k dF + r sqq in | k dF − k uF | > Λ QCD limit and k u conf. slowly reduces even when a large | r sqq | is assigned, which implies k n,pF (cid:39) N c max. (cid:2) k uF , k dF (cid:3) . Under thisassumption, one can anticipate the minimal flavor asymmetry in quark Fermi sea: populating a specific flavor of quarkleads to the large shift of the shell-like distribution k bF (cid:39) N c max. (cid:2) k Q i F (cid:3) . In the strong correlation assumption, thelower boundary of baryon distribution can be defined as follows: k nF = Θ( k dF − k uF ) (cid:0) k dF + r sqq w s (cid:0) k dF − k uF (cid:1)(cid:1) + Θ( k uF − k dF ) (cid:0) k uF + 2 r sqq w s (cid:0) k uF − k dF (cid:1)(cid:1) , (19) k pF = Θ( k dF − k uF ) (cid:0) k dF + 2 r sqq w s (cid:0) k dF − k uF (cid:1)(cid:1) + Θ( k uF − k dF ) (cid:0) k uF + r sqq w s (cid:0) k uF − k dF (cid:1)(cid:1) , (20) k Λ F = Θ( k dF − k sF ) (cid:0) k dF + r sqq w s (cid:0) k dF − k uF (cid:1) + r sqs w s (cid:0) k dF − k sF (cid:1)(cid:1) + Θ( k sF − k dF ) (cid:0) k sF + r sqs w s (cid:0) k sF − k dF (cid:1) + r sqs w s ( k sF − k uF ) (cid:1) , (21)where Θ( x ) represents the unit step function and r sqs determines the correlation strength between the light and strangequark momentum. Around the saturation moment of quark Fermi sea, k bF will be determined by the largest quarkFermi momentum as k bF (cid:39) k dF . Also, k dF ≥ k uF , k sF ≥ k uF conditions in the high density regime are understood fromthe weak decay channel of d and s quarks.
2. Assumption II: weakly correlated momentum of confined quark
On the other hand, one can imagine the weakly correlated momenta of the confined quarks in the confinement range | k q i conf. − k q j conf. | (cid:46) Λ QCD . If one considers the non-zero chiral condensate in the confined baryon phase and the symmetryrestoration at high density regime [77–83], the confinement mechanism of the quarks distributed slightly above thesaturated Fermi surface would be quite different from the one of vacuum case where the symmetry is broken. Theconfined state would rather look like the correlated state of three non-perturbative quarks whose ground energy scaleis m b (cid:39) k q conf. > k qF ) so that the flavor asymmetrybecomes large, the other confined quarks can take some lower unoccupied phase space ( k qF > k q i conf. > k qF − Λ QCD ) tominimize the ground state energy (Fig. 1(b)). If we assume the similar condition where d quark Fermi sea is saturatedfirst, then confined u quark momentum can be suggested as k u conf. = k dF + r wqq w w (cid:0) k dF − k uF (cid:1) , (22)where r wqq determines the weak correlation strength and the weak correlation weight function w w ( x ) is assumed torapidly converge to 1 in the x = | k dF − k uF | > Λ QCD limit: w w ( x ) = erf( −| x | /δ ) , (23) k Fd - k Fu [ GeV ] k c on f . u k Fd k Fd - Λ QCD ( a ) k Fd - k Fu [ GeV ] k c on f . u k Fd k Fd - Λ QCD ( b ) FIG. 2. Illustration of k u conf. within k dF > k uF condition. Left (a) : k u conf. under the strong correlation assumption (17) is plottedwith black solid (red dashed) line and r sqq = −
30 MeV ( r sqq = −
60 MeV). The confined quark momenta are closely locatedaround k dF and weakly dependent on | k dF − k uF | . Right (b) : k u conf. under the weak correlation assumption (22) is plotted withblack solid (red dashed) line and r wqq = −
100 MeV ( r wqq = −
140 MeV). The confined quark momenta rapidly deviate away from k dF as | k dF − k uF | becomes large. where erf( x ) denotes the error function and δ has the same role given in Eq. (17). With a large negative r wqq , the non-negligible difference | k u conf. − k d conf. | < Λ QCD can be obtained. As one can find in Fig. 2(b), the error function makesrelatively fast reduction of k u conf. even at small | k dF − k uF | . Comparing to the case of strongly correlated assumption,relatively larger flavor asymmetry is anticipated among the saturated quarks: if k u conf. is distributed away from k dF , k n,pF can have smaller magnitude than k n,pF (cid:39) N c k dF for large | k dF − k uF | . Under the same conditions, k bF can be definedas follows: k nF = Θ( k dF − k uF ) (cid:0) k dF + r wqq w w (cid:0) k dF − k uF (cid:1)(cid:1) + Θ( k uF − k dF ) (cid:0) k uF + 2 r wqq w w (cid:0) k uF − k dF (cid:1)(cid:1) , (24) k pF = Θ( k dF − k uF ) (cid:0) k dF + 2 r wqq w w (cid:0) k dF − k uF (cid:1)(cid:1) + Θ( k uF − k dF ) (cid:0) k uF + r wqq w w (cid:0) k uF − k dF (cid:1)(cid:1) , (25) k Λ F = Θ( k dF − k sF ) (cid:0) k dF + r wqq w w (cid:0) k dF − k uF (cid:1) + r wqs w w (cid:0) k dF − k sF (cid:1)(cid:1) + Θ( k sF − k dF ) (cid:0) k sF + r wqs w w (cid:0) k sF − k dF (cid:1) + r wqs w w ( k sF − k uF ) (cid:1) , (26)where r sqs determines the correlation strength between the light and strange quark momentum. III. EQUATION OF STATE FOR THE QUARKYONIC-LIKE MATTER
Equilibrium constraints and parameter set: in the 3-flavor system with electron clouds, the physical configu-ration should be constrained by the baryon number conservation, charge neutrality, and possible weak interactions.As summarized in Ref. [41], the weak interactions leads to following constraints: µ n = µ p + µ e , (27) µ ˜ d = µ ˜ u + N c µ e , (28) µ n = µ Λ (when n Λ (cid:54) = 0 . n Λ = 0 if µ Λ < m Λ ) , (29) µ ˜ d = µ ˜ s (when n ˜ s (cid:54) = 0 . n ˜ s = 0 if µ ˜ s < N c m s ) , (30)where µ ˜ Q i = N c µ Q i denotes the quark chemical potential in the unit of baryon number. The saturated quark onthe Fermi surface are allowed to decay onto the other Fermi surface of different flavor. Under the baryon numberconservation and charge neutrality, these constraints leads to the dynamical equilibrium condition:if n Λ = 0 , n s = 0 , µ n = N c µ d − µ e = µ p + µ e , (31)if n Λ (cid:54) = 0 , n s (cid:54) = 0 , µ n = N c µ d − µ e = µ Λ = µ p + µ e = N c µ s − µ e , (32)if n Λ = 0 , n s (cid:54) = 0 , µ n = N c µ d − µ e = µ Λ = µ p + µ e , (33)if n Λ (cid:54) = 0 , n s = 0 , µ n = N c µ d − µ e = µ p + µ e = N c µ s − µ e , (34)where µ n = N c µ d − µ e is the generalization of the dynamical equilibrium constraint µ N = N c µ q in the iso-spinsymmetric configuration [1]. Hereafter, we will calculate all the physical quantities under the constraints. Followingnumbers will be used as the representative parameter set: N c = 3 for the number of colors, { m n,p = 1 GeV, m Λ = 1 . m q = 0 .
333 GeV, and m s = 0 .
533 GeV } for the fermion masses, { r sqq = −
30 MeV and r sqs = −
60 MeV } for the strong correlation assumption, { r wqq = −
100 MeV and r wqs = −
140 MeV } for the weak correlation assumption, n = 6 ρ for the hard-core density ( r c (cid:39) . Q = 180 MeV which attenuate theunphysical noise. A. Density profile of particles in the excluded volume model with shell-like baryon distribution
We present the density profile of particles to understand the complicated dynamical properties from the shell-likedistribution. The density profiles of particles are plotted in Fig. 3 under n = 6 ρ , and Λ Q = 180 MeV conditions.As one can find in the profile (a) and (c) of Fig. 3, the stronger repulsive core ( α = 0 .
2) for Λ hyperon suppressesthe emergence of Λ degree of freedom even in the high density regime while the weaker repulsive core ( α = − . n Λ > ω Λ dependentterms of the baryon chemical potential (9): it becomes hard to satisfy the equilibrium constraint (29) with the otherconstraints in simultaneous way because µ Λ is enhanced by ω Λ > s quark takes relatively large portion than the one of the caseswhere n Λ > α = − . µ ˜ s for α = 0 . n ˜ s can beaccommodated satisfying the constraint (30) where µ ˜ d has the contribution from the n, p shell. By the same reason,one can understand the difference between the profiles from the strong and weak correlation assumptions. Underthe strong correlation of the confined quark momenta, the large iso-spin asymmetry in the quark Fermi sea enhancesthe lower boundary of nucleon momentum as k n,pF (cid:39) k dF , by which the nucleons in the shell obtain huge energyenhancement. Thus, it is dynamically favored for the iso-spin symmetric configuration of the light quark Fermi seaby the constraint (33) (Fig. 3(a)). If there is the Λ shell ( n Λ > k n,pF ≤ k dF and k Λ F ≤ k sF . Therefore, the constraints (32) and (33) can be satisfied in that asymmetricconfiguration.In all the cases, the quark Fermi sea is saturated in order of d , u , and s quark flavor. After the saturation,( n B ≥ ρ ), each baryon density profile looks converging to the asymptotic number and the quark Fermi sea takes allthe increment of the baryon number density ( dn B (cid:39) dn ˜ Q ). By definition, this model does not contain the essentialattractive and repulsive potentials required to reproduce the low density properties of nuclear matter. The saturationmoment of d quark Fermi sea can differ by the proper modifications to acquire the low density properties. The expectedpossible configurations at the low density regime is denoted as shaded area in Fig. 3. The qualitative behavior of thedensity profile does not change when different hard-core density n = 5 ρ is assigned. B. Equation of state and speed of sound
In the zero-temperature limit , the pressure and corresponding sound velocity can be found as p qy. = − ε qy. + µ B n B , (35) c s = ∂p qy. ∂ε qy. = n B µ B ∂n B ∂µ B . (36)0 np Λ uds n B / ρ n i / ρ ( a ) n = ρ , α = np Λ uds n B / ρ n i / ρ ( b ) n = ρ , α =- np Λ uds n B / ρ n i / ρ ( c ) n = ρ , α = np Λ uds n B / ρ n i / ρ ( d ) n = ρ , α =- FIG. 3. Density profiles of fermions ( n = 6 ρ , and Λ Q = 0 .
18 GeV). The profiles in the upper (a, b) and lower (c, d) sidesare obtained under the strong and weak correlation assumption, respectively. The profiles in the left (a, c) and right (b,d) sides are obtained under α = 0 . α = − . d quark Fermi sea which may rely on the proper EoS which covers the low densityregime. As can be found in the EoS plots (Figs. 4(a, c)), the EoS stiffly increases around the saturation moment of the quarkFermi sea. The evolution becomes more stiff when the stronger repulsive core ( n = 5 ρ ) is considered. Because eachquark flavor can be separately saturated in this system, there are several stiffly increasing segments in the evolutioncurve of EoS. This tendency does not appear when all the quark flavors simultaneously saturate [1, 40]. This evolutionlook similar to the results presented in Refs. [34, 35] where the first-order phase transition is implied via the hybridquark-meson-nucleon model [32–35]. However, as one can find in the sound velocity plots (Figs. 4(b, d)), c s > c s = 1 /
3) as one can anticipate from the definition of the model (16).Although the stiffness converges to the ideal limit, the weaker repulsive core ( n = 6 ρ ) leads to harder EoS in thehigh density regime because the system can accommodate more baryons under the weaker repulsion.The details of the stiff increments of EoS can be understood from the corresponding sound velocity plots and thedensity profiles. Under the strong correlation assumption for the confined quark momenta, one can read the overallstiffness of EoS from the the peak value of sound velocity as max.[ c s ] (cid:39) . c s ] (cid:39) . s quark saturation1 n = ρ , α = n = ρ , α = - n = ρ , α = n = ρ , α = - ℰ qy. [ MeV / fm ] p qy . [ M e V / f m ] ( a ) n = ρ , α = n = ρ , α = - n = ρ , α = n = ρ , α = - n B / ρ c s ( b ) n = ρ , α = n = ρ , α = - n = ρ , α = n = ρ , α = - ℰ qy. [ MeV / fm ] p qy . [ M e V / f m ] ( c ) n = ρ , α = n = ρ , α = - n = ρ , α = n = ρ , α = - n B / ρ c s ( d ) FIG. 4. Evolution of EoS (left) and corresponding sound velocity (right) . The plots in the upper (a, b) and lower (c, d) sides are obtained under the strong and weak correlation assumption, respectively. The stiffly increasing segments in (a, c) correspond to the onset moments of the new degrees of freedom and subsequently expanding shell-like phase structure of thebaryons. The peaks in (b, d) corresponds to the stiff behavior of the EoS in (a, c) . is determined by the existence of the Λ shell-like distribution. After the saturation of all the quark Fermi sea, theshell-like distributions of the baryons rapidly expand as the saturated quarks take almost all of the total baryonnumber density increment ∂n B /∂n ˜ Q (cid:39)
1, so that k bF (cid:39) N c k QF . If n Λ > α = − . α = 0 .
2) circumstance. The locations of early appearing peakcan be altered by the phenomenological modification of EoS for the low density regime as the saturation moment of d quark Fermi sea may depend on the modification. C. Mass-Radius relation of Quarkyonic neutron star
Now we can explore the possible Quarkyonic configuration in the compact stellar state by solving Tolman-Oppenheimer-Volkof (TOV) equations [84, 85]: dp ( r ) dr = G ( (cid:15) ( r ) + p ( r ))( M ( r ) + 4 πr p ( r )) r ( r − GM ( r )) , (37) dM ( r ) dr = 4 πr (cid:15) ( r ) , (38)2 Nucl. Matt. α = ( s ) α = - ( s ) α = ( w ) α = - ( w ) ℰ [ MeV / fm ] p [ M e V / f m ] ( a ) n = ρ Urbana IX Nucl. Matt. α = ( s ) α = - ( s ) α = ( w ) α = - ( w ) R [ km ] M [ M ⊙ ] ( b ) n = ρ PSR J0740 + Nucl. Matt. α = ( s ) α = - ( s ) α = ( w ) α = - ( w ) ℰ [ MeV / fm ] p [ M e V / f m ] ( c ) n = ρ V π PW + V μ = R Nucl. Matt. α = ( s ) α = - ( s ) α = ( w ) α = - ( w ) R [ km ] M [ M ⊙ ] ( d ) n = ρ PSR J0740 + FIG. 5. Maxwell constructions of EoS (left) and mass-radius relations from the solution of TOV equations (right) . Abbrevi-ations (s) and (w) represent the strong and weak correlation assumptions for the confined quark momenta, respectively. Thecrust EoS [86, 87] is used for the lower density regime of 0 ≤ ρ B ≤ . ρ . In the density range of 0 . ρ ≤ ρ B ≤ ρ M , twodifferent parameter sets [27] are used for the nuclear EoS: { Urbana IX force: ˜ a = − . , ˜ b = 10 . } for the plots in (a, b) and { V PW π + V Rµ =150 : ˜ a = − . , ˜ b = 13 . } for the plots in (c, d) . The EoS of the excluded volume model isused for the higher density regime beyond ρ M . Stiffer evolution with n = 5 ρ (a, b) : M max. = 2 . M (cid:12) and R . = 12 . n = 6 ρ (c, d) : M max. = 1 . M (cid:12) and R . = 11 . M max. estimated from Ref. [15]. where G is the gravitational constant and the boundary conditions p ( R star ) = 0 and M ( R star ) = M star is assumed.To obtain physically reasonable mass-radius relation, the low density part of our model needs modification as it doesnot contain the essential attractive and repulsive contribution required to describe the low density nuclear matterproperties. The EoS of our model will be kept from the intermediate regime to high density limit because we focus onthe role of the hard-core repulsive interaction and the dynamically generated shell-like distribution of baryons at thehigh density regime. Below a critical density (say n B ≤ ρ M ), some proper EoS can be adopted instead of introducingadditional mean-field potentials. At the extremely low density regime (0 ≤ n B ≤ . ρ ), the EoS of outer crust [86, 87]will be used. Since the low density configuration of our model can be simply regarded as the neutron matter (see theprofiles in Fig. 3), the nuclear EoS developed for neutron rich matter [27] can be used for the intermediate densityregime (0 . ρ ≤ n B ≤ ρ M ) as E/A = (cid:16) p nF + m n (cid:17) − m n + ˜ a (cid:18) n n ρ (cid:19) + ˜ b (cid:18) n n ρ (cid:19) , (39)3where p nF = (3 π n n ) / represents the neutron Fermi momentum in the ideal gas limit. The attractive (˜ a ) andrepulsive (˜ b ) coefficients have been determined by the possible 3-body nucleon forces. The parameter sets { Urbana IXforce: ˜ a = − . , ˜ b = 10 . } and { V P W π + V Rµ =150 : ˜ a = − . , ˜ b = 13 . } are used in the stifferand softer nuclear EoS, respectively. The stiffer (softer) nuclear EoS is interpolated with our high density EoS with n = 5 ρ ( n = 6 ρ ), requiring minimal Maxwell construction interval ( P nucl. ( µ M ) = P qy. ( µ M ) where µ M = µ B ( ρ M )).As one can find in the Maxwell constructions of the EoS obtained under n = 5 ρ condition (Figs. 5(a, b)), themoment of interpolation differs by the assumptions on the high density EoS. Among the interpolated curves plottedin Fig. 5(a), the α = 0 . α = − . α = − . { M max. = 2 . M (cid:12) , R M max. = 11 . } where theweaker repulsive core of Λ ( α = − .
2) and the weakly correlated confined quark momenta are assumed. The othercurves are barely located in the possible range estimated from the recent observation [15] ( M max. = 2 . +0 . − . M (cid:12) ).The constraints from the GW observations, R . = 12 . < . R . = 12 . <
15 km [12], aresatisfied.If the n = 6 ρ condition is considered (Figs. 5(c, d)), the interpolation interval minimally appears in all the casesand the Maxwell construction is done before the saturation of d quark Fermi sea. The low mass stage is governed by thenuclear EoS with V P W π + V Rµ =150 potential. While the interpolated EoS can be regarded as the Quarkyonic-like model, M max. = 2 . +0 . − . M (cid:12) cannot be reproduced from the EoS. The highest mass state appears as { M max. = 1 . M (cid:12) , R M max. = 10 . } in both cases of the weaker repulsive core of Λ ( α = − . M max. (cid:39) . M (cid:12) and the corresponding radius in the range of 10 km ≤ R M max. ≤ . { M star = 1 . M (cid:12) , R M = 12 km } and from that moment,the saturated quarks begin to take most of the total baryon density increment ( ∂n B /∂n ˜ Q (cid:39) M max. = 2 . M (cid:12) state (on the curve of α = − . n ˜ Q (cid:39) . n B and ε ˜ Q (cid:39) . ε qy. . For M max. = 2 . M (cid:12) state (on the curve of α = 0 . n ˜ Q (cid:39) . n B and ε ˜ Q (cid:39) . ε qy. . The scale of sound velocity appears as max.[ c s ] (cid:39) . IV. SUMMARY AND DISCUSSION
In this paper, the single flavor excluded volume model [1] is extended to the three-flavor model under considerationof Pauli’s exclusion principle which leads to the shell-like distribution of baryons. In perspective of the presumedfinite effective size of the particles, the baryon part of this excluded volume model could be understood as the zero-temperature and hard-core density limit of the multi-flavor VdW model in Fermi-Dirac statistics [49, 53, 55, 58–60].If one considers the quark degrees of freedom in the system, the quark Fermi sea is dynamically saturated by the hard-core repulsive nature of the baryon system [1, 41, 61]. Since the presence of the shell-like distribution increases thesaturated quark chemical potential by Pauli’s principle, the multi-flavor configuration of the shell-like distribution ofbaryon is determined by the repulsive core size of Λ hyperon and the correlation strength of confined quark momenta.If the repulsive core size of Λ hyperon is larger than the size of nucleons ( α > s quark number increases as there is no shell-like Λ distribution whose presenceincreases the chemical potential of s quark. In the opposite case where α <
0, the saturated s quark density becomessimilar order to the saturated d quark density. Similarly, when one assumes the strong correlation among the confinedquark momenta, the flavor asymmetry in the saturated quark Fermi sea leads to the large enhancement of quarkchemical potential as the minimum momenta of the baryons in the shell-like distribution get enhanced as k BF (cid:39) k QF by Pauli’s principle. Thus, the saturated quarks prefer the flavor symmetric configuration. When the weak correlationis assumed for the confined quark momenta, the flavor asymmetry of the saturated quark side becomes larger than4the strong correlation case.As a consequence of the saturation of the quark Fermi sea, the pressure increases stiffly by two or three stepswith emergence of the shell-like distributions, which is a different feature from the result of other literatures whereall the quark degrees of freedom appears simultaneously [1, 39, 40]. The corresponding sound velocity shows itspeak value as max.[ c s ] (cid:39) . c s ] (cid:39) .
7) for the strong (weak) correlation assumption for the confined quarkmomenta. Although the Quarkyonic matter would not have the evident first-order phase transition nature, the EoSlooks similar to the result of the previously reported works where the phase transition is implied [34, 35, 61]. Beyondthe saturation moments of the quark Fermi sea, the stiffness of EoS becomes moderate and converges to the conformallimit at the high density regime. Using the nuclear EoS introduced for neutron rich matter [27] for the low densities,we investigated the mass-radius relation of neutron star. The higher mass states tail of the mass-radius relation curveis determined by the EoS of the excluded volume model. The maximum mass state appears as { M max. = 2 . M (cid:12) , R M max. = 11 . } under the condition of n = 5 ρ , α = − .
2, and the weak correlation of the confined quarkmomenta. At the core of the M star = 2 . M (cid:12) state, the portions taken by the saturated quarks are estimated as n ˜ Q (cid:39) . n B and ε ˜ Q (cid:39) . ε qy. , comparable numbers with the results of Refs. [25, 34, 35, 61].In comparison with the previous study [41] where the stiff evolution was not evident enough, this excluded volumemodel approach reproduced the required stiff evolution of EoS even for the 3-flavor circumstance. It would berather required for the existence of Λ degree of freedom to support 2 M (cid:12) state in the high densities. At least, wedemonstrated that the repulsive hard-core of baryons and the dynamically generated Quarkyonic-like configurationcan be an alternative approach for understanding dense nuclear matter via fundamental principle. However, it wouldneed more improvements as the current model cannot cover all the possible range of the massive neutron star [15] andaccommodate the matter properties at low densities. If one keeps the physical scale of the hard-core radius [64–69],various types of potential [26, 27] can be referred to for the low density regime and the model can be refined to satisfythe low density matter constraints. Meanwhile, one can improve the current model in the VdW EoS framework [44–63].For example, the baryon part of the current model can be improved by following the treatment of Carnahan-Starlingmodification [46, 59, 62, 63] where the additional repulsive contribution is reflected in the larger repulsive core sizethan the estimated scale in Refs. [64–69]. In either approaches, the required soft nature for the low densities andstiffer nature for the high densities can be achieved by the additional attractive and repulsive contributions to theEoS.In microscopic aspect, there would be debates about the baryon-like state located on the lower boundary of theshell-like distribution. In this model, the baryon like state is clearly distinguished from the saturated quark statesand the non-perturbative regulator Λ Q is introduced for the quark phase measure. However, depending on the chiralsymmetry restoration [77–83] and the quark confinement mechanism around the quark Fermi sea [88–91], the baryonlike-state can be differently understood. It would be still the baryon state with restored chiral symmetry or thecorrelated state of quarks under non-perturbative dynamics. The similarity and discrepancy between the Quarkyonicmatter concept and the other approaches which involve the quark degrees of freedom would be understood via furtherstudies about the possible baryon-like states since the phase transition nature should also be related with the strongcorrelation patterns of quarks on the surface. ACKNOWLEDGMENTS
Authors acknowledge useful discussions with Larry McLerran and Sanjay Reddy during development of this work.Authors also thank Toru Kojo and Gerald Miller for inspiring discussions. Authors acknowledge the support ofthe Simons Foundation under the Multifarious Minds Program grant 557037. The work of Dyana C. Duarte, SaulHernandez-Ortiz and Kie Sang Jeong was supported by the U.S. DOE under Grant No. DE-FG02-00ER41132
Appendix A: Possible emergence of ∆(1232) isobar
The ∆(1232) isobar may emerge via the energetic collisions or in the dense neutron rich matter. In this work,the low density configuration ( n B ≤ ρ ) appears as the neutron rich matter (see the profiles plotted in Fig. 3). Ifone assumes similar size of the repulsive core for the baryons ( ω n,p = ω ∆ = 1, n = 5 ρ ), the chemical potentials of5baryon (9) can be written as follows: µ n = (cid:18) n n − n n (cid:19) (cid:16) K nF + m n (cid:17) − π n (cid:90) K nF dkk (cid:0) k + m n (cid:1) , (A1) µ p = m p + 1 n (cid:40) ¯ n exn (cid:16) K nF + m n (cid:17) − π (cid:90) K nF dkk (cid:0) k + m n (cid:1) (cid:41) , (A2) µ ∆ = m ∆ + 1 n (cid:40) ¯ n exn (cid:16) K nF + m n (cid:17) − π (cid:90) K nF dkk (cid:0) k + m n (cid:1) (cid:41) , (A3)where m n = 1 GeV, m ∆ = 1 . n B (cid:39) n n , n p , n ∆ (cid:39)
0) is understood. Firstly, µ ∆ > µ n in all the relevant density regime ( n B ≤ ρ ). If one considers the possible emergence via the scattering nn → p ∆ − , the energy relation is satisfied in the relevant densities ( n B (cid:39) . ρ ). However, this scattering barelyhappens as the momentum conservation is not always matched. Another possibility can be imagined in our modelas n + d → ∆ − + u after the saturation of d quark Fermi sea. In this scenario, the liberated u quark falls down tothe lower phases space but the emerging ∆ − should fill the phase space from the lower shell boundary ( k ∆ F (cid:39) k dF ).Under the simplified configuration where n B (cid:39) n n + n ˜ d , the baryon (12) and quark (9) chemical potentials can bewritten as follows: µ n = (cid:18) n n − n n (cid:19) (cid:16) [ k F + ∆] n + m n (cid:17) − π n (cid:90) [ k F +∆] n k nF dkk (cid:0) k + m n (cid:1) , (A4) µ ∆ = (cid:16) k ∆ F + m (cid:17) + 1 n (cid:40) ¯ n exn (cid:16) [ k F + ∆] n + m n (cid:17) − π (cid:90) [ k F +∆] n k nF dkk (cid:0) k + m n (cid:1) (cid:41) , (A5) µ d = 2 (cid:18) − n n n (cid:19) k nF k dF + Λ d (cid:26)(cid:16) [ k F + ∆] n + m n (cid:17) − (cid:16) k nF + m n (cid:17) (cid:27) + (cid:16)(cid:0) k dF (cid:1) + m d (cid:17) , (A6) µ u = (cid:18) − n n n (cid:19) k nF Λ u (cid:26)(cid:16) [ k F + ∆] n + m n (cid:17) − (cid:16) k nF + m n (cid:17) (cid:27) + m u , (A7)where k nF = k ∆ F = 3 k dF is assumed in small k dF limit. In this case, µ n + µ d < µ − ∆ + µ u around the expected saturationmoments of the quark Fermi sea (3 ρ ≤ n B ≤ ρ ) but the energy relation can be satisfied when the iso-spinasymmetry of the saturated quarks is large. However, it is unlikely to accommodate ∆ isobar degrees of freedom inthe Quarkyonic-like system as that large flavor asymmetry of saturated quark is not appearing under the physicalcontraints. [1] K. S. Jeong, L. McLerran and S. Sen, Phys. Rev. C , no.3, 035201 (2020) doi:10.1103/PhysRevC.101.035201[arXiv:1908.04799 [nucl-th]].[2] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett. , no. 16, 161101 (2017)doi:10.1103/PhysRevLett.119.161101 [arXiv:1710.05832 [gr-qc]].[3] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett. , no. 16, 161101 (2018)doi:10.1103/PhysRevLett.121.161101 [arXiv:1805.11581 [gr-qc]].[4] F. J. Fattoyev, J. Piekarewicz and C. J. Horowitz, Phys. Rev. Lett. , no. 17, 172702 (2018)doi:10.1103/PhysRevLett.120.172702 [arXiv:1711.06615 [nucl-th]].[5] E. Annala, T. Gorda, A. Kurkela and A. Vuorinen, Phys. Rev. Lett. , no. 17, 172703 (2018)doi:10.1103/PhysRevLett.120.172703 [arXiv:1711.02644 [astro-ph.HE]].[6] A. Vuorinen, Nucl. Phys. A , 36 (2019) doi:10.1016/j.nuclphysa.2018.10.011 [arXiv:1807.04480 [nucl-th]].[7] C. Raithel, F. zel and D. Psaltis, Astrophys. J. , no. 2, L23 (2018) doi:10.3847/2041-8213/aabcbf [arXiv:1803.07687[astro-ph.HE]].[8] E. R. Most, L. R. Weih, L. Rezzolla and J. Schaffner-Bielich, Phys. Rev. Lett. , no. 26, 261103 (2018)doi:10.1103/PhysRevLett.120.261103 [arXiv:1803.00549 [gr-qc]].[9] I. Tews, J. Margueron and S. Reddy, Eur. Phys. J. A , no. 6, 97 (2019) doi:10.1140/epja/i2019-12774-6 [arXiv:1901.09874[nucl-th]].[10] I. Tews, J. Margueron and S. Reddy, AIP Conf. Proc. , no. 1, 020009 (2019) doi:10.1063/1.5117799 [arXiv:1905.11212[nucl-th]].[11] C. D. Capano, I. Tews, S. M. Brown, B. Margalit, S. De, S. Kumar, D. A. Brown, B. Krishnan and S. Reddy, Nat Astron. , 625632 (2020) doi:10.1038/s41550-020-1014-6 [arXiv:1908.10352 [astro-ph.HE]]. [12] B. Abbott et al. [LIGO Scientific and Virgo], Astrophys. J. Lett. , L3 (2020) doi:10.3847/2041-8213/ab75f5[arXiv:2001.01761 [astro-ph.HE]].[13] P. Demorest, T. Pennucci, S. Ransom, M. Roberts and J. Hessels, Nature , 1081-1083 (2010) doi:10.1038/nature09466[arXiv:1010.5788 [astro-ph.HE]].[14] J. Antoniadis et al. Science , 6131 (2013) doi:10.1126/science.1233232 [arXiv:1304.6875 [astro-ph.HE]].[15] H. T. Cromartie et al.
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