Massive Δ-resonance admixed hypernuclear stars with anti-kaon condensations
Vivek Baruah Thapa, Monika Sinha, Jia Jie Li, Armen Sedrakian
MMassive ∆ -resonance admixed hypernuclear stars with anti-kaon condensations Vivek Baruah Thapa, ∗ Monika Sinha, † Jia Jie Li,
2, 3, 4, ‡ and Armen Sedrakian
5, 6, § Indian Institute of Technology Jodhpur, Jodhpur 342037, India School of Physical Science and Technology, Southwest University, Chongqing 400700, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute for Theoretical Physics, Goethe University,Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße, 1, 60438 Frankfurt am Main, Germany Institute of Theoretical Physics, University of Wrocław, pl. M. Borna 9, 50-204 Wrocław, Poland
In this work, we study the effect of (anti)kaon condensation on the properties of compact starsthat develop hypernuclear cores with and without an admixture of ∆ -resonances. We work withinthe covariant density functional theory with the parameters adjusted to K -atomic and kaon-nucleonscattering data in the kaonic sector. The density-dependent parameters in the hyperonic sector areadjusted to the data on Λ and Ξ − hypernuclei data. The ∆ -resonance couplings are tuned to thedata obtained from their scattering off nuclei and heavy-ion collision experiments. We find that(anti)kaon condensate leads to a softening of the equation of state and lower maximum masses ofcompact stars than in the absence of the condensate. Both the K − and ¯ K -condensations occurthrough a second-order phase transition, which implies no mixed-phase formation. For large valuesof (anti)kaon and ∆ -resonance potentials in symmetric nuclear matter, we observe that condensationleads to an extinction of Ξ − , hyperons. We also investigate the influence of inclusion of additionalhidden-strangeness σ ∗ meson in the functional and find that it leads to a substantial softening ofthe equation of state and delay in the onset of (anti)kaons. I. INTRODUCTION
Born in supernova explosions neutron (or compact)stars (NSs) are the densest cosmic bodies in the mod-ern Universe. They provide a unique domain of densityrange to study the novel states of matter. Indeed, mat-ter in compact stars is compressed by the gravitationalforce to densities a few times nuclear saturation density, n [1–4].During the last decade electromagnetic as well as gravi-tational wave observations placed a number of constraintson the global properties of compact stars (masses, radii,deformabilities, etc.) which significantly constrain therange of admissible equation of state (EoS) models ofdense matter. We briefly list below the most importantobservational results. The masses of massive M ∼ M (cid:12) compact star (millisecond pulsars) in binaries with whitedwarfs were determined for J − ( M = 1 . ± . M (cid:12) ) [5, 6], J ( . ± . M (cid:12) ) [7] andJ ( . +0 . − . M (cid:12) with 95 % credibility) [8]. Theradius of a canonical . M (cid:12) compact stars was inferredfrom low-mass X-ray binaries in globular clusters in therange ≤ R ≤ km [9]. The mass-radius measure-ments of PSR J by the NICER experiment de-termined M = 1 . +0 . − . M (cid:12) , R = 13 . +1 . − . km [10] and M = 1 . +0 . − . M (cid:12) , R = 12 . +1 . − . km (with . cred-ibility) [11]. The first multimessenger gravitational-waveevent GW170817 observed by the LIGO-Virgo collabora- ∗ [email protected] † [email protected] ‡ [email protected] § sedrakian@fias.uni-frankfurt.de tion (LVC) [12–14] set constraints on the tidal deforma-bilities of involved stars which through a tight correla-tion with the radii predict a radius ≤ R . ≤ kmfor the canonical-mass star M = 1 . M (cid:12) . The LVC ob-servation of the GW190425 event in gravitational wavesdetermined the component masses range . − . M (cid:12) [15]. Another event GW190814 suggests a binary with alight component with a mass . +0 . − . M (cid:12) [16] which fallsin the “mass-gap” ( . M (cid:12) ≤ M ≤ M (cid:12) ). The nature ofthe lighter companion is still not resolved [17–21], but theneutron star interpretation appears to be in tension withformation of heavy baryons (hyperons, ∆ -resonances) incompact stars [22–24].Due to large densities reached in the core region ofcompact stars, new hadronic degrees of freedom are ex-pected to nucleate in addition to the nucleons. One suchpossibility is the onset of hyperons, as initially suggestedin Ref. [25]. This occurs in the inner core of compact starsat about (2 − n . Even though the presence of hyperonsin compact stars may seem to be unavoidable, it leads toan incompatibility of the theory with the observationsof massive pulsars mentioned above, as is evidenced bymany studies which used either phenomenological [26–30]or microscopic [31–35] approaches. Specifically, hyperonslead to a softening of the EoS and imply a low value ofthe maximum mass of compact stars, below those ob-served. This problem is known as the “hyperon puzzle”.The studies prior to the discovery of massive pulsars,the work during the last decade focused mainly on mod-els which provide sufficient repulsion among the hadronicinteractions which guarantees stiffer EoS and larger max-imum masses of hypernuclear stars; these have been car-ried out mostly within the covariant density functionaltheory [36–52]. But microscopic models have also been a r X i v : . [ a s t r o - ph . H E ] F e b employed [53, 54].Another fascinating possibility of the onset of non-nucleonic degrees of freedom is the appearance of stable ∆ -resonances in the matter. Whether ∆ -resonances playany role in the NSs is still a matter of debate [49, 55].Early work [26, 56] indicated that the threshold densityfor the appearance of ∆ -resonances could be as high as (9 − n . More recent works [49, 57–59] have shownthat indeed these non-strange baryons may appear in nu-clear matter at density in the range (1 − n . In par-ticular, the recent work which included both hyperonsand ∆ -resonance [49, 60] showed that the inclusion of ∆ -resonances into the NS matter composition reduces theradius of a canonical . M (cid:12) mass compact star, whereasthe maximum mass implied by the EoS does not changesignificantly. The onset of ∆ -resonances also shifts theonset of hyperons to higher densities [49, 57, 60].Yet another possibility of a new non-nucleonic degreeof freedom at high densities is the appearance of vari-ous meson (pion, kaon, ρ -meson) condensates. Initially,pion-condensation and its implications for neutron starphysics was studied [61–63]. Later, the focus shiftedtowards the strangeness-carrying (anti)kaons ( ¯ K ) con-densate, initially suggested within a chiral perturbativemodel in Refs. [64, 65]; for further models and develop-ments see [28, 66, 67]. It has been then realized that therepulsive optical potential developed by the K + mesonsin the nuclear matter disfavors the presence of kaons inneutron star matter. Several authors [68–74] have stud-ied the (anti)kaon condensation in nuclear as well ashypernuclear matter. The onset of (anti)kaons in thecompact star matter is very sensitive to the K − opti-cal potentials as well as the presence of hyperons. Inthe latter case, it is observed that the threshold densityof (anti)kaons is shifted to even higher matter densities[75]. A generic feature of the onset of the condensates isthe softening of the EoS and the reduction of the maxi-mum masses of compact stars, which could become po-tentially incompatible with the observations of massivepulsars. The onset of (anti)kaon condensation affectsmany properties of compact stars beyond the equationof state, such as superfluidity [76], neutrino emission viadirect Urca processes [77, 78], and bulk viscosity [79].This is a direct consequence of the changes in the single-particle spectrum of fermions, e.g., the Fermi momenta,effective masses, etc.In the present work, we explore the possibility of(anti)kaon condensation in β -equilibrated ∆ -resonanceadmixed hypernuclear matter in the core region of com-pact stars within the framework of covariant densityfunctional (CDF) model. To construct the EoS, we im-plement the DD-ME2 parametrization of density func-tional with density-dependent couplings [80]. This modelhas been extended previously to the ∆ -resonance ad-mixed hypernuclear matter without (anti)kaon conden-sation [49, 60], showing that the resulting EoS is broadlycompatible with the available astrophysical constraints.It has been also extended to include the effect of strong magnetic fields [81]. This work, therefore, will focus onthe novel aspects that are introduced by the (anti)kaoncondensation.The paper is arranged as follows. In Sec. II we brieflydescribe the density-dependent CDF formalism and itsextension to (anti)kaons condensation in ∆ -resonancesadmixed hypernuclear matter. Sec. III is devoted to nu-merical results and their discussions. The conclusionsand future perspectives are given in Sec. IV. We use nat-ural units (cid:126) = c = 1 throughout. II. FORMALISMA. Density Dependent CDF Model
In this work, we consider the density dependent CDFmodel to study the transition of matter from hadronicto (anti)kaon condensed phase in β -equilibrated ∆ -resonance admixed hypernuclear matter. The mattercomposition is considered to be of the baryon octet( b ≡ N, Λ , Σ , Ξ ), ∆ -resonances ( d ≡ ∆ ++ , ∆ + , ∆ , ∆ − ),(anti)kaons ( ¯ K ≡ K − , ¯ K ) alongside leptons ( l ) such aselectrons and muons. The strong interactions betweenthe baryons as well as the (anti)kaons are mediated bythe isoscalar-scalar σ , σ ∗ , isoscalar-vector ω µ , φ µ andisovector-vector ρ µ meson fields. The additional hid-den strangeness mesons ( σ ∗ , φ µ ) are considered to medi-ate the hyperon-hyperon as well as (anti)kaon-hyperoninteractions. The total Lagrangian density consistingof the baryonic, leptonic and kaonic parts is given by[49, 68, 69, 82] L = (cid:88) b ¯ ψ b ( iγ µ D µ ( b ) − m ∗ b ) ψ b + (cid:88) d ¯ ψ dν ( iγ µ D µ ( d ) − m ∗ d ) ψ νd + (cid:88) l ¯ ψ l ( iγ µ ∂ µ − m l ) ψ l + D ( ¯ K ) ∗ µ ¯ KD µ ( ¯ K ) K − m ∗ K ¯ KK + 12 ( ∂ µ σ∂ µ σ − m σ σ ) + 12 ( ∂ µ σ ∗ ∂ µ σ ∗ − m σ ∗ σ ∗ ) − ω µν ω µν + 12 m ω ω µ ω µ − ρ µν · ρ µν + 12 m ρ ρ µ · ρ µ − φ µν φ µν + 12 m φ φ µ φ µ (1)where the fields ψ b , ψ νd , ψ l correspond to the baryonoctet, ∆ -baryon and lepton fields. m b , m d , m K and m l represent the bare masses of members of baryon octet, ∆ -quartet, isospin doublet for (anti)kaons and leptonsrespectively. The covariant derivative in Eq.(1) is D µ ( j ) = ∂ µ + ig ωj ω µ + ig ρj τ j · ρ µ + ig φj φ µ (2)with j denoting the baryons ( b, d ) and (anti)kaons ( ¯ K ).The density-dependent coupling constants are denotedby g pj where ‘ p ’ index labels the mesons. The isospin op-erator for the isovector-vector meson fields is representedby τ j . The gauge field strength tensors for the vectormeson fields are given by ω µν = ∂ ν ω µ − ∂ µ ω ν , ρ µν = ∂ ν ρ µ − ∂ µ ρ ν ,φ µν = ∂ ν φ µ − ∂ µ φ ν . (3)The Dirac effective baryon and (anti)kaon masses in Eq.(1) are given by m ∗ b = m b − g σb σ − g σ ∗ b σ ∗ , m ∗ d = m d − g σd σ,m ∗ K = m K − g σK σ − g σ ∗ K σ ∗ (4)In the relativistic mean-field approximation, the mesonfields obtain expectation values which are given by σ = (cid:88) b m σ g σb n sb + (cid:88) d m σ g σd n sd + (cid:88) ¯ K m σ g σK n s ¯ K , σ ∗ = (cid:88) b m σ ∗ g σ ∗ b n sb + (cid:88) ¯ K m σ ∗ g σ ∗ K n s ¯ K ,ω = (cid:88) b m ω g ωb n b + (cid:88) d m ω g ωd n d − (cid:88) ¯ K m ω g ωK n ¯ K , φ = (cid:88) b m φ g φb n b − (cid:88) ¯ K m φ g φK n ¯ K ρ = (cid:88) b m ρ g ρb τ b n b + (cid:88) d m ρ g ρd τ d n d + (cid:88) ¯ K m ρ g ρK τ ¯ K n ¯ K (5)where n s = (cid:104) ¯ ψψ (cid:105) and n = (cid:104) ¯ ψγ ψ (cid:105) denote the scalar andvector (number) densities respectively. The explicit formof scalar and vector density of baryons in the T = 0 limitis n sj = 2 J j + 14 π m ∗ j (cid:34) p F j E F j − m ∗ j ln (cid:32) p F j + E F j m ∗ j (cid:33)(cid:35) ,n j = 2 J j + 16 π p F j (6)respectively with J j , p F j and E F j being the spin, Fermimomentum and Fermi energy of the j -th baryon. For thecase of s -wave (anti)kaons, the number density is givenas n K − , ¯ K = 2 (cid:18) ω ¯ K + g ωK ω + g φK φ ± g ρK ρ (cid:19) = 2 m ∗ K ¯ KK. (7)Here, ω ¯ K represents the in-medium energies of(anti)kaons and are given by (considering isospin pro-jection as ∓ / for K − , ¯ K ) ω K − , ¯ K = m ∗ K − g ωK − g φK φ ∓ g ρK ρ . (8)In case of leptons ( l ), the number density is given by n l = p F l / π . The chemical potential of the j -th baryonis µ j = (cid:113) p F j + m ∗ j + Σ B , (9)where Σ B = Σ + Σ r denotes the vector self-energy with Σ = g ωj ω + g φj φ + g ρj τ j ρ , (10) Σ r = (cid:88) b (cid:20) ∂g ωb ∂n ω n b − ∂g σb ∂n σn sb + ∂g ρb ∂n ρ τ b n b + ∂g φb ∂n φ n b (cid:21) + (cid:88) d ( ψ b −→ ψ νd ) . (11)Eq.(11) is the rearrangement term which is required incase of density-dependent meson-baryon coupling modelsto maintain the thermodynamic consistency [83]. Here, n = (cid:80) j n j represents the total baryon number density.The threshold condition for the onset of j -th baryoninto the nuclear matter is given by [69] µ j = µ n − q j µ e (12)where q j is the charge of the j -th baryon. µ e = µ n − µ p is the chemical potential of electron with µ n , µ p denot-ing the same for neutron and proton. With increasingdensity, the Fermi energy of electrons increases and onceit reaches the rest mass of muons i.e. µ e = m µ , muonsstart to appear in the nuclear matter.In case of (anti)kaons, the threshold conditions aregoverned by the strangeness changing processes such as, N (cid:10) N + ¯ K and e − (cid:10) K − [1, 70] and are given by µ n − µ p = ω K − = µ e , ω ¯ K = 0 . (13)The total energy density due to the fermionic part isgiven by ε f = 12 m σ σ + 12 m ω ω + 12 m ρ ρ + (cid:88) j ≡ b,d J j + 12 π (cid:34) p F j E F j − m ∗ j (cid:32) p F j E F j + m ∗ j ln (cid:32) p F j + E F j m ∗ j (cid:33)(cid:33)(cid:35) + 12 m σ ∗ σ ∗ + 12 m φ φ + 1 π (cid:88) l (cid:20) p F l E F l − m l (cid:18) p F l E F l + m l ln (cid:18) p F l + E F l m l (cid:19)(cid:19)(cid:21) . (14)And the energy density contribution from the kaonic mat-ter is ε ¯ K = m ∗ K ( n K − + n ¯ K ) (15)giving the total energy density as ε = ε ¯ K + ε f . Now,because (anti)kaons being bosons are in the condensedphase at T = 0 , the matter pressure is provided only bythe baryons and leptons and is given by the Gibbs-Duhemrelation p m = (cid:88) j ≡ b,d µ j n j + (cid:88) l µ l n l − ε f . (16)The rearrangement term in Eq. (11) contributes explic-itly to the matter pressure term only through the vectorself-energy term.Two additional constraints – the charge neutrality andglobal baryon number conservation – should be takeninto account to calculate the equation of state self-consistently. The charge neutrality condition is given by (cid:88) b q b n b + (cid:88) d q d n d − n K − − n e − n µ = 0 . (17) B. Coupling parameters
In the density dependent CDF model implemented inthis work, DD-ME2 [80] coupling parametrization is in-corporated. The coupling functional dependence of thescalar σ and vector ω -meson on density is given by g iN ( n ) = g iN ( n ) f i ( x ) , for i = σ, ω, (18)where, x = n/n , n , n being the total baryon numberdensity and nuclear saturation density respectively with f i ( x ) = a i b i ( x + d i ) c i ( x + d i ) (19)For the case with ρ -meson, the density dependence cou-pling functional is defined as g ρN ( n ) = g ρN ( n ) e − a ρ ( x − (20)The parameters of the meson-nucleon couplings in DD-ME2 parametrization model is given in Table I. The co-efficients associated with DD-ME2 model are fitted toreproduce nuclear phenomenology; the details of whichcan be found in Ref. [80]. Since the nucleons do notcouple to the strange mesons, g σ ∗ N = g φN = 0 . The TABLE I. The meson masses and parameters of the DD-ME2parametrization used in Eq. (18) and (19).Meson ( i ) m i (MeV) a i b i c i d i g iN σ ω
783 1.3892 0.9240 1.4620 0.4775 13.0189 ρ
763 0.5647 7.3672 masses of the additional hidden strangeness mesons aretaken as m σ ∗ = 975 MeV and m φ = 1019 . MeV. Thenuclear saturation properties are provided in Table II.The parameters E , K , Q denote the saturation en-ergy, incompressibility, and skewness in isoscalar sector,and E sym , L sym for symmetry energy coefficient and itsslope in isoscalar sector, all evaluated at the saturationdensity. It should be noted, that the experimentally ob-tained values of some of these parameters have an un-certainty range given by n ∈ [0 . − . fm − [84], − E ∈ [15 − MeV [84], K ∈ [220 − MeV [85, 86], E sym ∈ [28 . − . MeV [87, 88]. Once the parameters ofthe model are fixed to particular values within the rangeindicated above, the EoS is obtained by straightforwardextrapolation to the high-density regime. At present, thehigh-density properties of dense matter are constrainedby astrophysics of compact stars and modeling of heavy-ion collision experiments, both of which carry uncertain-ties of their own.
TABLE II. The nuclear properties of the density-dependentCDF model (DD-ME2) at n . n E K Q E sym L sym m ∗ N /m N (fm − ) (MeV) (MeV) (MeV) (MeV) (MeV)0.152 − . The bare masses of the members of the baryon octet, ∆ -quartet and (anti)-kaons considered in this work are, m Λ = 1115 . MeV, m Ξ = 1314 . MeV, m Ξ − =1321 . MeV, m Σ + = 1189 . MeV, m Σ = 1192 . MeV, m Σ + = 1197 . MeV, m ∆ = 1232 MeV, m K = 493 . MeV.For the meson-hyperon vector coupling parameters, weincorporated the SU(6) symmetry and quark countingrule [72, 89] as g ω Λ = 12 g ω Σ = g ω Ξ = 13 g ωN , g φ Λ = 2 g φ Σ = g φ Ξ = − √ g ωN , g ρ Σ = g ρ Ξ = g ρN , g ρ Λ = 0 . (21)The scalar meson-hyperon couplings are calculated byconsidering the optical potentials of Λ , Σ , Ξ as − MeV, +30
MeV and − MeV respectively [81]. Furthermore,the scalar strange meson σ ∗ -hyperon coupling is evalu-ated from the measurements on light double- Λ nuclei andfitted to the optical potential depth U ΛΛ ( n /
5) = − . MeV [90]. The scalar meson-hyperon couplings for theother strange baryons can be obtained from the relation-ship, g σ ∗ Y g φY = g σ ∗ Λ g φ Λ , Y ∈ { Ξ , Σ } . (22)Table III provides the numerical values of the meson-hyperon couplings at nuclear saturation density, where R σY = g σY /g σN , R σ ∗ Y = g σ ∗ Y /g σN denote the scal-ing factors for non-strange and strange scalar mesonscoupling to hyperons respectively. Because experimen- TABLE III. Scalar meson-hyperon coupling constants for DD-ME2 parametrization.
Λ Ξ Σ R σY R σ ∗ Y tal information on the ∆ -resonance is scarce, the meson- ∆ baryon couplings are treated as parameters. In thesubsequent discussion we consider g ωd = 1 . g ωN and g ρd = g ρN for vector-meson couplings [49, 91]. For thescalar meson- ∆ baryon couplings we use two values of theisoscalar potentials viz. V ∆ = V N and / V N with V N being the nucleon potential [23, 49]. These values wereextracted from the studies of electron and pion scatteringoff nuclei studies as well as studies of heavy-ion-collisionswhich involved ∆ -resonance production.The numerical values of scalar meson- ∆ -baryon cou-pling parameters with V ∆ = V N is R σd = 1 . and thatwith V ∆ = 5 / V N is R σd = 1 . , where R σd = g σd /g σN denotes the non-strange scalar meson coupling to ∆ -resonances. Similar to the nucleons, ∆ -resonances do notcouple to σ ∗ , φ -mesons, i.e, g σ ∗ d = g φd = 0 .The meson-(anti)kaon couplings are fixed according toRefs. [75, 92] and are taken as density indepedent. Thevector meson-(anti)kaon coupling parameters are evalu-ated from the isospin counting rule [75] and are givenas g ωK = 13 g ωN , g ρK = g ρN . (23) And in case of the additional hidden strange force medi-ating mesons, the couplings are given as [72] g σ ∗ K = 2 . , g φK = 4 . . (24)The scalar meson-(anti)kaon coupling constants arecalculated by fitting to the real part of K − optical po-tential at nuclear saturation density. The readers mayrefer to Ref. [74] for details. Refs. [63, 93, 94] show thatthe (anti)kaons experience an attractive potential in nu-clear matter whereas the opposite is true for the casewith kaons in nuclear matter [95, 96]. Different modelcalculations [93, 94, 97–99] have provided the K − opti-cal potential in normal nuclear matter to be in the rangefrom − MeV to − MeV. We have chosen a K − op-tical potential range of − ≤ U ¯ K ≤ − MeV in thiswork and numerical values of g σK for the mentioned op-tical potential range is provided in Table IV. TABLE IV. Scalar σ meson-(anti)kaon coupling parametervalues in DD-ME2 parametrization at n . U ¯ K (MeV) − − − − g σK III. RESULTS
In this section we report our numerical results for mat-ter composition with (anti)kaons and (a) Nucleons + Hy-perons (NY), (b) Nucleons + Hyperons + ∆ -resonances(NY ∆ ) for varying values of (anti)kaon optical poten-tials. The case of pure nuclear matter with (anti)kaonswas considered already in Ref. [74] and the reader is re-ferred to that work. From calculations, it is found thatthe phase transition to (anti)kaon condensed phase is ofthe second-order for both NY and NY ∆ compositions. Inall the calculations the K − -meson is observed to appearbefore the onset of ¯ K . Table V provides the thresholddensities of (anti)kaons for different values of ∆ -baryonas well as U ¯ K potentials for two matter compositions.It is observed that the (anti)kaons do not appear atall in case of U ¯ K = − MeV for all matter compo-sitions. (Anti)kaons are observed to appear only after U ¯ K = − MeV with V ∆ = 5 / V N . This happens asthe higher ∆ -potential shifts the onset of hyperons tohigher densities making the way for the (anti)kaons. Inall the cases considered, it is observed that with the in-clusion of ∆ -resonances into the composition of matterthe threshold densities of onset of (anti)kaon is shifted tohigher densities.Figure 1 shows the in-medium (effective) energies of ¯ K mesons as a function of baryon (vector) number den-sity in NY ∆ matter described by the DD-ME2 CDF. Theonset of K − mesons condense in the compact star mat-ter occurs when the respective effective energy crossesthe electron chemical potential, which then marks the n/n K i n - m e d i u m e n e r g i e s ( M e V ) U Κ = −130 MeVU Κ = −140 MeVU Κ = −150 MeV ω K NY ∆Κ (V ∆ = V N ) ω K - ω K - ω K DD-ME2 µ e - µ e - NY ∆Κ (V ∆ = 5/3 V N ) FIG. 1. The effective energy of (anti)kaons as a function ofbaryon number density in NY ∆ matter for ∆ -potential val-ues V ∆ = 1 (top panels) and / V N (bottom panels). Leftand right panels show the energies of K − and ¯ K respectively.The chemical potential of electron for the same matter compo-sition is depicted by the dashed curve. The solid, dash-dotted,dotted lines represent the U ¯ K values of − , − , − MeVrespectively.TABLE V. Threshold densities, n u for (anti)kaon conden-sation in NY and NY ∆ matter for different values of ∆ -potentials and K − optical potential depths U ¯ K ( n ) .Config. NY ¯ K NY ∆ ¯ KV ∆ = V N V ∆ = 5 / V N U ¯ K n u ( K − ) n u ( ¯ K ) n u ( K − ) n u ( ¯ K ) n u ( K − ) n u ( ¯ K )(MeV) ( n ) ( n ) ( n ) ( n ) ( n ) ( n ) − − − − − − −− − − − − − − threshold density. In the case ¯ K mesons, the conden-sate appears when their in-medium energy value reacheszero. With the increase in the values of U ¯ K , the densitythreshold for the onset of the (anti)kaons is shifted tolower densities.The EoSs with NY and NY ∆ matter compositions inthe absence as well as in presence of (anti)kaon degreesof freedom are shown in Fig. 2. In the case with no(anti)kaons in the matter, the EoSs of NY ∆ matter isstiffer than the EoS of NY matter in the high-densityregime and the opposite is true in the low-density regime.This is consistent with the results of Ref. [49] foundwithin the same DD-ME2 parametrization.The middle and right panels of Fig. 2 include(anti)kaons with potential values U ¯ K = − , − MeVrespectively. It is seen that the onset of (anti)kaon con-densation softens the EoS, which is marked by a changein the slope of EoSs beyond the condensation thresh- old. Furthermore, the softening is more pronounced inthe case of NY ∆ composition, which reverses the high-density behavior seen in the left panel: the EoS withNY ∆ composition is now the softest among all consid-ered cases. It is further seen that the higher the value of U ¯ K the more pronounced is the softening of the EoSs.The mass-radius ( M - R ) relations corresponding to theEoSs in Fig. 2 were obtained by solving the Tolman-Oppenheimer-Volkoff (TOV) equations for static non-rotating spherical stars [1] and are shown in Fig. 3. Forthe crust region, the BPS EoS [100] is used. The in-clusion of additional exotic degrees of freedom reducesthe maximum mass of NSs in comparison to nucleonicmatter from . M (cid:12) to ∼ M (cid:12) . The compactness isalso observed to be enhanced due to the appearance of ∆ − -resonance at lower densities. The parameter valuesof the maximum mass stars are provided in a tabulatedform in Table VI. From Tables V and VI it can be in-ferred that K − meson appears in all the EoS models with U ¯ K = − , − MeV. But ¯ K meson does not appearin the hypernuclear star with U ¯ K = − MeV and ∆ -baryon admixed hypernuclear star with V ∆ = V N and U ¯ K = − MeV. Consistent with the (anti)kaon soften-ing of the EoS seen in Fig. 2 the maximum masses of thestars with NY ∆ composition and (anti)kaon condensa-tion lie below those without ∆ resonances, which is thereverse of what is observed when (anti)kaon condensationis absent.From the analysis above, we conclude that compactstars containing (anti)kaons are consistent with the as-trophysical constraints set by the observations of massivepulsars, the NICER measurements of parameters of PSRJ , the low-mass X-ray binaries in a globularcluster, and the gravitational wave event GW190425, seeSec. I. Although we do not provide here the deforma-bilities of our models, from the values of the radii ob-tained it is clear that our models are also consistent withthe GW170817 event. Finally, our models are inconsis-tent with the interpretation of the light companion of theGW190814 binary as a compact star. Including the ro-tation even at its maximal mass-shedding limit will notbe sufficient to produce a ∼ . M (cid:12) mass compact star,see Refs. [22, 23].Figure 4 shows the particle composition in NY matterwith (anti)kaons as a function of baryon number den-sity and for U ¯ K = − , − MeV. At low densities,before the onset of strange particles, the charge neu-trality is maintained among the protons, electrons andmuons. At somewhat higher density ( ≥ n ) Λ and Ξ − appear in the matter (because of the repulsive natureof Σ -potential in dense nuclear matter, Σ -baryons do notappear in the composition). Finally, the (anti) kaons and Ξ appear in the high-density regime ( ≥ n ). Compar-ing the upper and lower panels of the figure, we observethat the higher U ¯ K value implies a lower density thresh-old of the onset of (anti)kaon, as expected. The onsetof (anti)kaons also affects the population of leptons; K − are efficient in replacing electrons and muons once they ε (MeV/fm ) P ( M e V / f m ) NY Κ NY ∆Κ (V ∆ = V N )NY ∆Κ (V ∆ = 5/3 V N )300 600 900 1200 300 600 900 1200 DD-ME2U Κ = −140 MeV DD-ME2U Κ = −150 MeVDD-ME2U Κ = 0 FIG. 2. Pressure as a function of energy density (EoS) for zero-temperature, charge-neutral NY matter (solid lines), NY ∆ matter with ∆ -potential V ∆ = V N (dashed lines) and V ∆ = 5 / V N (dash-dotted lines). The three panels correspond to differentvalues of (anti)kaon potential: U ¯ K = 0 (left panel), U ¯ K = − (middle panel), and U ¯ K = − MeV (right panel).
R (km) M / M s o l a r
10 12 14 16
10 12 14 16
GW190814 MSP J0740+6620 NY Κ NY ∆Κ (V N )NY ∆Κ (5/3 V N )PSR J0030+0451 Low-massX-ray binaries DD-ME2 DD-ME2U Κ = 0 U Κ = −140 GW190425
DD-ME2U Κ = −150 FIG. 3. The mass-radius relationships for EoS shown in Fig. 2 for NY matter (solid lines), NY ∆ matter with ∆ -potential V ∆ = V N (dashed lines) and V ∆ = 5 / V N (dash-dotted lines). The three panels correspond to different values of (anti)kaonpotential: U ¯ K = 0 , i.e., no (anti)kaon condensation, (left panel), U ¯ K = − (middle panel), and U ¯ K = − MeV (right panel).The astrophysical constraints from GW190425 [15], GW190814 [16], MSP J [8], PSR J [10, 11], low-massX-ray binaries [9] are shown by dot-double dashed, dotted, long-, short-dashed boxes and horizontal solid line respectively. appear, thus they contribute to the extinction of leptons,which occurs at lower densities for higher values of U ¯ K .In the case of U ¯ K = − MeV, the Ξ − fraction is seen tobe strongly affected with the appearance of K − mesons.This is expected as K − being bosons are more energeti-cally favorable for maintaining the charge neutrality com-pared to fermionic Ξ − . The composition in the case of U ¯ K = − MeV, does have ¯ K mesons ( n u ∼ . n )whereas for U ¯ K = − MeV, ¯ K appears at onset den-sity n u ∼ . n which leads to an additional softeningof the EoS. Figure 5, which is analogous to Fig. 4, showsthe particle population in N Y ∆ -matter as a function ofbaryon number density for U ¯ K = − MeV. It is ob-served that for V ∆ = V N only ∆ − resonance appears, TABLE VI. Properties of maximum mass stars for various compositions and values of (anti)kaon potential U ¯ K ( n ) . For eachcomposition/potential value the enteries include: maximum mass (in units of M (cid:12) ) the radius (in units of km), and centralnumber density (in units of n ).Configuration NY ¯ K NY ∆ ¯ KV ∆ = V N V ∆ = 5 / V N U ¯ K (MeV) M max ( M (cid:12) ) R (km) n c ( n ) M max ( M (cid:12) ) R (km) n c ( n ) M max ( M (cid:12) ) R (km) n c ( n ) .
008 11 .
651 6 .
107 2 .
021 11 .
565 6 .
160 2 .
049 11 .
226 6 . −
140 2 .
005 11 .
652 6 .
096 2 .
019 11 .
566 6 .
151 2 .
032 11 .
343 6 . −
150 1 .
994 11 .
664 6 .
13 2 .
006 11 .
61 6 .
143 1 .
973 11 .
448 6 . n/n n i / n npe − µ − ΛΞ − Ξ Κ − Κ DD-ME2U Κ = −140 MeV NY Κ p ne − µ − Λ Ξ − Ξ Κ − Κ U Κ = −150 MeV
FIG. 4. Particle abundances n i (in units of n ) as a functionof normalized baryon number density in NY matter for valuesof U ¯ K − MeV (top panel) and − MeV (bottom panel). whereas for V ∆ = 5 / V N the onset of the entire quartetof ∆ -resonances is possible. It seen that in general the ∆ -resonances effectively shift the threshold densities ofhyperons to higher densities, thus diminishing their role.This concerns both the neutral Λ as well as Ξ − -hyperon.This shift is stronger for larger values of V ∆ . Resonancesalso suppress the lepton fraction by lowering the den-sity at which they disappear in NY ∆ -matter, this effectbeing magnified for larger values of V ∆ . In the high-density regime the negative charge is provided by ∆ − – Ξ − – K − mixture and it is seen that the rapid increase in n/n n i / n npe − µ − ∆ − Λ Ξ − Ξ Κ − Κ DD-ME2U Κ = −140 MeV NY ∆Κ (V N )p ne − µ − ∆ − Λ ∆ ∆ + ∆ ++ Ξ − Ξ Κ − Κ NY ∆Κ (5/3 V N ) FIG. 5. Same as Fig. 4 but for NY ∆ matter for V ∆ = V N (top panel) and V ∆ = 5 / V N (bottom panel) and fixed valueof U ¯ K = − MeV. the K − population suppresses the ∆ − - Ξ − abundancesfor V ∆ = 5 / V N , as kaons are energetically more favor-able than the heavy-baryons. Note also that the onsetof ¯ K meson abruptly decreases the abundance of Ξ − ,as seen in the lower panel; (in the upper panel, i.e. for U ¯ K = − MeV and V ∆ = V N , the ¯ K mesons do notappear). There is some qualitative differences betweenthe two cases V ∆ = 1 and / V N : (a) the ∆ − baryondisappears at higher matter densities for V ∆ = 1 but itsabundance is almost constant in for V ∆ = 5 / V N ; (b)the Λ hyperon dominates over the neutron fraction athigher density for ∼ . n in case of V ∆ = V N comparedto ∼ . n in case of V ∆ = 5 / V N . n/n n i / n npe − µ − ∆ − Λ Ξ − Ξ Κ − Κ DD-ME2U Κ = −150 MeV NY ∆Κ (V N )p ne − µ − ∆ − Λ∆ ∆ + ∆ ++ Κ − Κ NY ∆Κ (5/3 V N ) ∆ ∆ + FIG. 6. Same as Fig. 5 but for a larger (absolute) value ofpotential U ¯ K = − MeV.
Figure 6 shows the same as in Fig. 5 but for U ¯ K = − MeV. The particle fractions show identical trends as inFig. 5 until the appearance of (anti)kaons. The largerpotential favors earlier onset of (anti)kaons in matter;for example, the K − sets in before the Ξ − and it is nowthe dominant negatively charged component shortly afterthe density increases beyond the onset value. The effectof the onset of ¯ K on the Ξ − and ∆ − , which is medi-ated via changes in the abundances of K − , is seen clearlyagain. As before, for a large value of V ∆ = 5 / V N , allthe members of the quartet of ∆ -resonances are presentin the matter composition. Another notable fact is thecomplete extinction of Ξ − , baryons, which is consistentwith the trends seen in Figs. 4 and 5. Interestingly,in the case V ∆ = 5 / V N the (anti)kaons abundancesare the largest among all particles in the high-densityregime, which leads also to the softening of the EoS ob-served above. Figure 7 shows the (anti)kaon effectivemass as a function of normalized baryon number den-sity for various strengths of U ¯ K with different mattercompositions. The effective mass of (anti)kaons tendsto decrease rather steeply in case of higher strengths of n/n m * K / m K NY Κ , U Κ = −140 MeVNY Κ , U Κ = −150 MeVNY ∆Κ (V N ), U Κ = −140 MeVNY ∆Κ (V N ), U Κ = −150 MeVNY ∆Κ (5/3 V N ), U Κ = −140 MeVNY ∆Κ (5/3 V N ), U Κ = −150 MeV
DD-ME2
FIG. 7. Effective (anti)kaon mass (in units of its bare mass, m ¯ K ) as a function of baryon number density for NY and NY ∆ matter compositions and two values of (anti)kaon potentialdepth. U ¯ K . It is observed that in the low-density regime, the(anti)kaon effective mass decreases relatively quickly inthe case of ∆ -resonances admixed matter compared tothat with the only hyperonic matter. The reason is thelarger scalar potential values arising from the onset ofadditional non-strange baryons at lower densities. Andat higher densities, the (anti)kaon effective mass valuesare observed to be larger in the former case than the lat-ter one. This may be attributed to the delayed onset ofhyperons because of the ∆ -resonances appearance. P ( M e V / f m ) σωρφσωρσ ∗ φ
300 600 900 1200 ε (MeV/fm ) DD-ME2U Κ = −120 MeVNY Κ NY ∆Κ (V ∆ = 5/3 V N )no Κ no Κ FIG. 8. The EoS of NY matter (left panel) and NY ∆ matter(right panel) with (anti)kaon potential U ¯ K = − MeV in-cluding σ ∗ meson (dashed lines) and without it (solid lines).The ∆ -potential value is fixed at / V N . The matter pressure as a function of energy density0for different matter compositions with and without σ ∗ meson for the hyperon-hyperon interactions is shown inFig. 8. Being a scalar, σ ∗ meson makes the EoS softeras is evident from the figure. It is observed that incor-porating σ ∗ meson rules out the possibility of (anti)kaonphase transition with U ¯ K = − MeV. This is becausethis scalar meson further reduces the effective mass of(anti)kaons halting their onset in the matter. The phasetransition from the purely hadronic to (anti)kaon con-densed phase is second-order. The results of mass-radius
R (km) M / M s o l a r σωρφσωρσ ∗ φ GW190814 MSP J0740+6620 NY Κ NY ∆Κ (5/3 V N ) PSR J0030+0451
Low-massX-ray binaries
DD-ME2U Κ = −120 MeVno Κ no Κ GW190425
FIG. 9. The M - R relations corresponding to the EoSs inFig. 8 are shown for NY matter (left panel) and NY ∆ matter(right panel) with (anti)kaon potential U ¯ K = − MeV in-cluding σ ∗ meson (dashed lines) and without it (solid lines).The ∆ -potential value is fixed at / V N . The astrophysicalobservables (constraints) are similar as in Fig. 3. ( M - R ) relationship obtained by solving the TOV equa-tions for non-rotating spherical stars corresponding tothe EoSs in Fig. 8 are presented in Fig. 9. It is observedthat in both cases of NY for NY ∆ matter the inclusion of σ ∗ meson leads to lower maximum mass. It is also seenthat the addition of ∆ ’s reduces the radius of the of thestars and mildly increases the maximum, which consis-tent with the findings without (anti)kaon condensation.Table VII provides the stellear maximum masses, radiiand corresponding central densities evaluated from theEoSs in Fig. 8 with U ¯ K = − MeV.Figure 10 shows the particle abundances in case of hy-pernuclear matter with U ¯ K = − MeV with and with-out σ ∗ meson. The main qualitative difference is that K − appears for n ≥ . n in the first case and it does notappear up to n ∼ n in the second case. Consequently,the charge neutrality is maintained between e − Ξ − + K − and protons in the first case and only e − Ξ − and pro-tons in the second case. Given by more than one orderof magnitude smaller abundance of electrons, the abun-dances of Ξ − and protons almost coincide in the secondcase. Another feature seen in Fig. 10 is that the electron TABLE VII. Properties of maximum mass stars for variouscompositions, U ¯ K = − MeV, V ∆ = 5 / V N in the caseswith σ ∗ meson and without. In both cases we list the maxi-mum mass (in units of M (cid:12) ) the radius (in units of km), andcentral number density (in units of n ).Config. NY ¯ K NY ∆ ¯ K ( V ∆ = 5 / V N ) M max R n c M max R n c ( M (cid:12) ) (km) ( n ) ( M (cid:12) ) (km) ( n ) σωρφ .
124 11 .
673 5 .
973 2 .
137 11 .
023 6 . σωρσ ∗ φ .
008 11 .
651 6 .
107 2 .
049 11 .
226 6 . n/n n i / n npe − µ − Κ − Λ Ξ − (no Κ ) Ξ − Ξ DD-ME2U Κ = −120 MeV σωρφ p ne − µ − U Κ = −120 MeV
Λσωρσ ∗ φ (NY Κ ) FIG. 10. Particle abundances n i (in units of n ) as a functionof normalized baryon number density in NY matter for valueof U ¯ K = − in the case of σωρφ exchange (top panel) and σωρσ ∗ φ (bottom panel). (Anti)kaons are absent in the secondcase. and muon populations disappear faster with increasingdensity in the case where the σ ∗ meson is included.Figure 11, which is similar to Fig. 10, shows the com-position of particles in NY ∆ matter and for U ¯ K = − MeV. In this case also, (anti)kaons are observed to appearonly in the EoS where σ ∗ meson is excluded. It is seen,that the main difference between the two cases is that σ ∗ driven interactions prefer lower threshold density of Ξ and their larger fraction, which effectively leads to anexclusions of (anti)kaons in the density range considered.1 n/n n i / n np e − µ − ∆ − Λ Ξ − Ξ Κ − DD-ME2U Κ = −120 MeV NY ∆Κ (5/3 V N )p ne − µ − ∆ − Λ ∆ ∆ + ∆ ++ Ξ − Ξ NY ∆Κ (5/3 V N ) (no Κ)∆ ∆ + ∆ ++ σωρφσωρσ ∗ φ FIG. 11. Same as Fig. 10 but for NY ∆ matter with V ∆ =5 / V N . Unlike the case with only hypernuclear matter, in thiscase the lepton fractions are unaffected by the exclusionor inclusion of σ ∗ meson, because of the negative chargeis supplied by the ∆ − -resonance. IV. SUMMARY AND CONCLUSIONS
In this work, we discussed the second-order phase tran-sition to Bose-Einstein condensation of (anti)kaons in hy-pernuclear matter with and without an admixture of ∆ -resonances within the framework of density-dependentCDF theory. The resulting EoS, matter composition,and the structure of the associated static, sphericallysymmetrical star models were presented. The strong in-teractions viz. baryon-baryon and (anti)kaon-baryon arehandled on the same footing. The mediators consideredin this work are σ , ω , ρ for the non-strange baryons andtwo strange particle interaction mediating mesons- σ ∗ , φ .The K − optical potentials ( − ≤ U ¯ K ≤ − MeV)at nuclear saturation density are considered in a rangewhich fulfills the observational compact star maximummass constraint ( ∼ M (cid:12) ).We find that the (anti)kaon condensates cannot appearin the hypernuclear matter, within our parametrization, if U ¯ K ≤ − MeV. ¯ K condensation is absent in max-imum mass compact stars with U ¯ K = − MeV. Theinclusion of hyperons into the matter composition shiftsthe onset of (anti)kaons to higher density regimes in com-parison to the case without hyperons, i.e. only nuclearmatter, c.f. to Ref. [74]. For higher U ¯ K values, the ap-pearance of both the (anti)kaons becomes possible in themaximum mass models. The K − meson fraction is seento dominate over the Ξ − baryon for high U ¯ K strengths.This can be attributed to the fact that the K − parti-cle being bosons is more favored over the fermionic Ξ − -particles.Next, in the case of ∆ baryon admixed hypernuclearmatter, the onset of (anti)kaons is shifted to even higherdensities compared to only hyperonic matter. (Anti)kaoncondensation is absent with U ¯ K ≤ − MeV. The con-densed phase is observed to appear in matter with U ¯ K = − MeV and V ∆ = V N . However, ¯ K condensationis absent for this particular U ¯ K strength. Larger valuesof ∆ -potentials V ∆ imply that the entire ∆ -resonancesquartet is present in matter. It is also observed thatin a particular matter composition ( U ¯ K = − MeV, V ∆ = V N ), the onset of K − occurs even before thatof Ξ − particles. Moreover, for higher strengths of U ¯ K and V ∆ , the ∆ -baryons and (anti)kaons take over the Ξ − , particles leading to their complete suppression inthe matter. Lepton populations are suppressed with in-creasing density more quickly in case of higher strengthsof V ∆ . We find that the effective mass of (anti)kaonsis weakly dependent on the composition of matter anddecreases almost linearly in the relevant density range ≤ n/n ≤ , which reflects the density dependence ofthe scalar potential.The influence of the strange scalar interaction mediat-ing meson σ ∗ on the composition and EoS are twofold:firstly, including the σ ∗ meson softens the EoS sig-nificantly leading to lower maximum masses of com-pact stars. Secondly, exclusion of σ ∗ meson allows for(anti)kaon K − to appear for weakly attractive potentialstrength U ¯ K ∼ − MeV in both the hyperonic as wellas ∆ admix hypernuclear matter.As indicated in the discussion (Sec. III) the presentmodel with a suitable choice of parameters characterizingthe (anti)kaon condensate is consistent with the currentlyavailable astrophysical constraints listed in Sec. I. Thepresent model can, therefore, be used to model physicalprocesses in (anti)kaon condensate featuring ∆ -admixedhypernuclear star. Examples include cooling processes,bulk viscosity, thermal conductivity, to list a few. ACKNOWLEDGMENTS
The authors thank the anonymous referee for the con-structive comments which have bestowed to enhance thequality of the manuscript notably. VBT and MS ac-knowledge the financial support from the Science andEngineering Research Board, Department of Science and2Technology, Government of India through Project No.EMR/2016/006577 and Ministry of Education, Govern-ment of India. They are also thankful to SarmisthaBanik and Debades Bandyopadhyay for vital and fruit-ful discussions. M.S. also thanks Alexander von Hum-boldt Foundation for the support of a visit to GoetheUniversity, Frankfurt am Main. J. J. Li acknowledges the A. von Humboldt Foundation for support in the ini-tial stages of this work. A. S. acknowledges the sup-port by the Deutsche Forschungsgemeinschaft (Grant No.SE 1836/5-1) and the European COST Action CA16214PHAROS “The multi-messenger physics and astrophysicsof neutron stars”. [1] N. K. Glendenning,
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