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arXiv:2106.03449v2 [nucl-th] 23 Aug 2021 nuht ecnitn ihrcn bevtoso massive Keywords: of observations recent with consistent be to enough ssonta otnn fteeuto fsaesemn fro stemming e state repulsive of equation the the of softening that mo shown string-junction is the in force repulsive three- sal fmsienurnsasa ag s2 as large as stars observati The massive : dense of various highly on of properties information and important phases with star provided neutron have with nomena associated waves gravitational and rays, h eeto fgaiainlwvsfo eto-trmer neutron-star 19 from 18, waves 17, gravitational [16, of EOS detection the The on constraint stringent put have mass) the er 1 ] ancnesto otlkl cuswe the when occurs likely most condensation lowest Kaon 2]. [1, metry in 2 2 3 4 15]. emi They 14, 13, 12, enhanced 11]. [2, via sions 10, processes [8, radius cooling max- extraordinary its the a and reducing also star stars neutron would well. compact of as more mass soft to imum EOS lead the make phases would These of existence The n[,8.I a lobe ugse htmxn fhyperons of mixing that suggested been also has se It ( condensation kaon 8]. (EOS) once state [4, of matter, in equation dense the in that shown softened been largely has It 4]. [3, tial rpitsbitdt ora fL of Journal to submitted Preprint den saturation at energy binding the to force three- eet h est-eedneo clradvco eo m vector and scalar of density-dependence the reflects aeko ( kaon wave ,3 ,5 ,7 ] rvn oc fko odnaini the is condensation kaon of force Driving 8]. 7, 6, 5, 4, 3, physi 2, matter condensed par and of astrophysics, fields physics, interdisciplinary nuclear the in attention long much has ceived condensation kaon matter, interacting strongly in Introduction 1. relativis the with combined matt being -mixed interactions, kaon-kaon in phase kaon-condensed of Possibility Abstract [email protected] [email protected] Y cusi h rudsaeo eto trmte 9 0 11] 10, [9, matter star neutron of state ground the in occurs ) eetmlimsegrosrain ihrdowaves, radio with observations multi-messenger Recent sanvlfr ihmcocpcapaac fstrangeness of appearance macroscopic with form novel a As ∗ mi addresses: Email author Corresponding s -wave K − E nry hc erae ihbro est u to due density baryon with decreases which energy, ff K ancnesto,hprnmxn,euto fsae uni state, of equation hyperon-mixing, condensation, kaon K ff - cso he-aynfre nko odnaini hyperon in condensation kaon on forces three-baryon of ects -ulo ( )-nucleon N ff c hra vlto fnurnsasthrough stars neutron of evolution thermal ect c ftetrebro oc n h eaiitce relativistic the and force three-baryon the of ect trcin et h lcrnceia poten- chemical the meets attraction, [email protected] a eateto hsc,CiaIsiueo ehooy 2-1- Technology, of Institute Chiba Physics, of Department N c neato pcfidb hrlsym- chiral by specified interaction ) A nttt fEuain sk agoUiest,3-1-1Nak University, Sangyo Osaka Education, of Institute T b E TsiaaTatsumi) (Toshitaka dacdSineRsac etr aa tmcEeg Agen Energy Atomic Japan Center, Research Science Advanced Templates X TsiiMaruyama), (Toshiki auiMuto Takumi M ⊙ ( M ⊙ Tkm Muto), (Takumi en h solar the being a, ∗ ohk Maruyama Toshiki , e n hnmnlgcltrencenatatv oc a force attractive three-nucleon phenomenological and del i enfil hoyfrtobd aynitrcin nad In interaction. baryon two-body for theory mean-field tic s[1, cs yre- ly ticle- phe- gers iy h ancnesdpaei yeo-ie atrbec matter hyperon-mixed in phase kaon-condensed The sity. ons ohko odnainadmxn fhprn scompensat is hyperons of mixing and condensation kaon both m eto stars. neutron X ri osdrdo h ai fcia ymtyfrkaon-bar for symmetry chiral of basis the on considered is er s- s is ts ]. - - a-ed,wihi osrie ytecnrbto ftea the of contribution the by constrained is which ean-fields, . ff oeo xhneptnil[7 8 and 28] multi- as [27, such potential exchange between forces multi-body th and account into body taking works several appeared there quently ( three-nucleon as l 2] h di The [29]. els e Λ mn yeosadncen ( (UTBR repulsion and three-baryon hyperons universal among the necess the introducing puzzle, of hyperon the calle been resolve To has EOS puzzle”. the “hyperon of softening dramatic such from nating stars. neutron massive of observations with compatible aso eto tr ihte( the with stars neutron of mass uso thg este by densities high at pulsion h yeo uze .g h U3 ymtymdlfrvec- for model symmetry SU(3) the meson- , resolvi tor g. for e. models other puzzle, some hyperon are the There 34]. 33, 32, [31, tion assadculns[6,etc. [36], couplings and masses oc o ne est fK n h O fkaon-condensed of EOS the and KC of density onset for force itda ( as viated ohK and KC both sacneuneo h obnde the combined by the of consequence a as hs ihko odnae K)and (KC) condensates kaon with phase matter. light dense shed of have EOS the 23] constraining [22, on (NICER) ExploreR Composition rior of radius through and stars mass of neutron measurements and 21] [20, (GW170817) MMM c o w-oybro-aynitrcin h atre latter The interaction. baryon-baryon two-body for ect ff NN hbzn,Nrsio hb 7-03 Japan 275-0023, Chiba Narashino, Shibazono, 1 ntecs fpr yeo-ie atr h rbe origi- problem the matter, hyperon-mixed pure of case the In sfrko odnain motneo ula three-body nuclear of importance condensation, kaon for As nasre forwrs ehv tde osbecoexistent possible studied have we works, our of series a In ciefil hoyhsbe ple o the for applied been has theory field ective esltrebro force three-baryon versal b ohtk Tatsumi Toshitaka , gio at,Oaa5483,Japan 574-8530, Osaka Daito, agaito, neatoshsbe oe[0,adrcnl h chiral the recently and [30], done been has interactions s yedarm nterltvsi enfil RF mod- (RMF) mean-field relativistic the in diagrams type -wave Y Y y,Iaai3919,Japan 319-1195, Ibaraki , cy + opig 3] h oe ihsaigo hyperon of scaling with model the [35], couplings Y K mxn edt infiatsfeigo h EOS the of softening significant to lead -mixing K bro ( -baryon ff hs][4 5 6.I a ensonthat shown been has It 26]. 25, [24, phase] ) so ot al td including study Carlo Monte usion NNN X oc a one u 1] Subse- [10]. out pointed was force ) ryosrainb eto trInte- star Neutron by observation -ray B c trcinadaodn the avoiding and attraction ) Y mxn 1] hstemaximum the Thus [10]. -mixing Y + mxdmatter -mixed YYY K ff hs ok o o obe to low too looks phase ) cso eraigenergy decreasing of ects , Y YYN mxdmte [abbre- matter -mixed B , eitoue.It introduced. re msn( -meson YNN iin univer- dition, YNN uut2,2021 24, August mssti omes swell as ) Λ ttractive o and yon N interac- dwith ed N - M N ff ree- ) and ect M re- ity ng ff d ) , phase in neutron-star matter has been discussed [37, 38]. In the each coefficient in front of the number density of baryon, ρb, is (Y+K) phase, it may also be legitimate to assume that three- specified as the V- charge, and f the meson decay constant, body repulsions among baryons (Y and N) at high densities for which we simply take the decay constant ( 93 MeV) should work on an equal footing as three-nucleon repulsion, in lowest-order in chiral perturbation. The classical≃ kaon field although there is few empirical information on the UTBR. is represented as K = ( f / √2)θ exp( iµK t), where θ is the chi- ± ± In the present work, in order to circumvent the significant ral angle and µK the kaon chemical potential. The Lagrangian softening of the EOS in the case of the (Y+K) phase, we take density for the classical kaon field reads [26] into account the repulsion by the UTBR. In the (Y + K) phase, 2 1 2 2 both bosonic (KC) and fermionic (hyperons) degrees of free- K = f (µK sin θ) mK (1 cos θ)+2µK X0(1 cos θ) , (1) dom take part in the realization of in the ground L h 2 − − − i state of matter. In particular, we clarify how the former is af- where, mK is the free kaon mass, and the last term in the bracket fected by the UTBR for stiffening the EOS, while the latter is stands for the s-wave K-B vector interaction. The s-wave K- directly affected by receiving the repulsion from the UTBR. B scalar interaction is absorbed into the effective baryon mass We adopt the RMF model for two-body B-B interaction Mb∗. It is to be noted that the K-K nonlinear self-interaction is mediated by meson-exchange, discarding the nonlinear self- naturally incorporated through the terms proportional to sin2 θ interacting σ, ω, or ω ρ meson-couplingpotentials [9]. We call − and cos θ as a consequence of the nonlinear representation of this model a minimal RMF (MRMF) throughout this paper. We the K-field in the effective chiral Lagrangian. introduce the density-dependent effective two-body potentials for the UTBR, which has been derived from the string-junction model by Tamagaki [39] and originally applied to Y-mixed mat- 3. Baryon interactions ter by Tamagaki, Takatsuka and Nishizaki[40]. Together with 3.1. Minimal RMF for two-body baryon interaction the UTBR, phenomenological three-nucleon attraction (TNA) The Lagrangian density for baryons and which de- is taken into account, and we construct the baryon interaction scribes the two-body interaction is given by model that reproduces saturation properties of symmetric nu- clear matter (SNM) and empirical values of incompressibility, µ (b) 1 µ 2 2 BM = ψb iγ Dµ Mb∗ ψb + ∂ σ∂µσ mσσ symmetry energy, and its slope at the nuclear saturation density L X − 2 − 3 b     ρ0 (= 0.16 fm− ). Then we consider the (Y+K) phase based e 1 µ 2 2 1 µν 1 2 µ + ∂ σ∗∂ σ∗ m σ∗ ω ω + m ω ω upon the effective chiral Lagrangian coupled with the present µ σ∗ µν ω µ 2  −  − 4 2 baryon interaction model (MRMF+UTBR+TNA). Effects of 1 µν a 1 2 µ a 1 µν 1 2 µ R R + m R R φ φµν + m φ φµ , (2) the UTBR and TNA on the whole EOS with the (Y+K) phase − 4 a µν 2 ρ a µ − 4 2 φ are clarified. As implications for thermal evolution of neutron where ψb stands for baryon field b, and M∗ is the effective stars, rapid cooling mechanisms in the presence of the (Y+K) b baryonmass defined by M∗ Mb gσbσ gσ bσ∗ ΣKb(1 cos θ) phase are briefly mentioned. b ≡ − − ∗ − − with Mb being the free bayon mass and gσb, gσ∗b being the 2. Kaon condensation on the basis of chiral symmetry -baryon coupling constants. The fields for the ω, ρ, φ mesons are denoted as ωµ, Rµ with the The s-wave K-B scalar and vector interactions relevant to a component a, and φµ ( s¯γµ s), respectively. The ki- kaon condensation are embodied in the effective chiral La- netic terms of the vector mesons∼ are given in terms of ωµν grangian [1]. The former is simulated by the “K-baryon sigma µν µ ∂µων ∂νωµ, R ∂µRν ∂νR , and φµν ∂µφν ∂νφµ. The≡ terms” which explicitly break chiral symmetry, i. e., Σ a a a Kb vector− meson-baryon≡ couplings− are introduced≡ through− the co- (m + m )/2 b uu¯ + ss¯ b with b qq¯ b being the content≡ u s variant derivative, D(b) ∂ + ig ω + ig Iˆ (b)(R ) + ig φ , in the baryon· hb.| The value| i of theh |Σ | iis taken to be (300 400) µ µ ωb µ ρb 3 3 0 φb µ Kn where g is the vector≡ meson-baryon coupling constant and MeV as standard values [3, 26]. Σ is closely related− to the mb Kn Iˆ (b) is a sign of the third componentof the isospin for baryon b. πN sigma term, Σ , for which we adopt the phenomenological 3 πN The vector meson couplings for hyperons(Y) are here related value, Σ = 45 MeV [41]. The upper value for the Σ (= 400 πN Kn with those for the nucleon N by SU(6) symmetry [43] as g = MeV) is obtained at leading order in chiral perturbation theory ωΛ g Σ = 2g Ξ = (2/3)g , g Λ = 0, g Σ = 2g Ξ = 2g , so as to be consistent with the octet baryon mass splitting. In ω − ω − ωN ρ ρ − ρ − ρN gφΛ = gφΣ = (1/2)gφΞ = ( √2/3)gωN. this case, one has a large strangeness content in the nucleon, − − − The scalar (σ, σ∗) mesons-hyperoncouplings are determined yN 2 N ss¯ N / N uu¯ + dd¯ N = 0.44. The lower value cor- ≡ h | | i h | | i from the phenomenologicalanalyses of recent hypernuclear ex- responds to the case yN 0 [42]. In this case, ΣKn is related ≃ periments. The σ-Y coupling constant, gσY , is related with the to ΣπN by ΣKn = Σπn(mu + ms)/[2(mu + md)] on the assumption potential depth of the hyperon Y (Y =Λ, Σ−, Ξ−) at ρB = ρ0 in n uu¯ n n dd¯ n . With Σπn = 45 MeV, mu = 6 MeV, md = 12 h | | i ∼ h | | i SNM, VN , which is written in the RMF as MeV, and ms = 240 MeV [1], one obtains ΣKn 300 MeV. (Re- Y ∼ cent result for the quark masses from the lattice QCD, mu=2.2 N VY = gσY σ 0 + gωY ω0 0 + ∂ UTBR/∂ρY , (3) MeV, md = 4.7 MeV,and ms = 95 MeV, little changes the result. − h i h i E ) The latter vector interaction, corresponding to the Tomozawa- where σ 0 and ω0 0 are the meson mean fields at ρB = ρ0 in Weinberg term for the meson-N scattering amplitude, is propor- SNM, andh i the lasth termi comes from the energy density contri- X ρ + 1 ρ 1 ρ ρ / f 2 tional to the term: 0 p 2 n 2 Σ− Ξ− (2 ), where bution from the UTBR, UTBR, which is derived from Eq. (6) in ≡  − −  E 2 N N N Y+K Sec. 3.2. By setting VΛ = 27 MeV, VΣ = 23.5 MeV, and VΞ 4. Energy density for the ( ) phase and saturation prop- − − − = 14 MeV in Eq. (3) [44], one obtains gσΛ, gσΣ , and gσΞ . erties in SNM − − − For the σ∗-Y coupling constant, gσ∗Y , we determined gσ∗Λ to be 7.2 so as to reproduce the empirical values of the separation en- The energy density for the (Y+K) phase is given as the sum 11 ergy BΛΛ(ΛΛ Be), with use of the B-B interaction model in the of the KC, baryon and mesons for two-body baryon interaction, RMF extended to finite nuclei [45, 46]. Within our B-B interac- UTBR and TNA for three-body interaction, and : = + +( + )+ . The groundstate energyforE the tion model, gσ∗Ξ− is taken to be 4.0, for which one obtains the K B,M UTBR TNA e th 15 (EY +EK) phaseE is obtainedE underE the charge neutrality condition theoretical values of the separation energies BΞ ( Ξ(s)C) = 8.1 th 12 and the β-equilibrium condition at a given density ρB. MeV and BΞ ( Ξ(s)Be) = 5.1 MeV, which are consistent with the 14 The coupling constants, gσN , gωN , and the meson mean- empirical values deduced from the “Kiso” event, Ξ− + N 15 10 5 → fields, σ 0, ω0 0 are determined so as to reproduce the prop- ΞC ΛBe + ΛHe [47, 48]. The remaining unknown coupling h i h i → erties of the SNM with saturation density ρ0 and the binding constant, gσ∗Σ− , is simply set to be zero. energy B0 (=16.3 MeV), together with the equations of motion σ ω 2 σ = ρs 3.2. Three-baryon repulsive force for the and mean-fields, mσ 0 gσN N with the nuclear scalar density ρs , and m2 ω =h gi ρ . Further, the coupling ff U N ω 0 0 ωN 0 The e ective two-body potential SJM is obtained from constant g and the parametersh i γ , η in TNA associated with W r , r , r ρN a a the three-body baryon interaction ( 1 2 3) in the string- isospin-dependence are obtained to meet empirical values of junction model by integrating out variables of the third baryon the incompressibility K=240 MeV [51], the symmetry energy multiplying the short-range correlation (s.r.c.) function squared 2 S 0 (=31.5 MeV) [52] for a given value of the slope L, which is fsrc(r): ρ (∂ /∂ρ ) = ρ /ρ defined as L 3 0 S B ρB=ρ0,x=1/2 3Pneutron matter( 0) 0. There is controversy≡ about the empirical value of L, ranging 3 2 2 USJM(1, 2; ρB) = ρB d r3W(r1, r2; r3) fsrc(r1 r3) fsrc(r2 r3) from 30 MeV to 90 MeV [53, 54, 55]. For instance, there Z − − (4) are several models with the smaller L (. 40 MeV), satisfy- with W(r1, r2; r3) = W0g(r1 r3)g(r2 r3), where W0 ( 2 ing the constraints from compact star observations and HIC ex- GeV) is the strength of the order− of B-antibaryon− (B¯) excitation≃ periments [36, 56]. On the other hand, there are rather larger energy [39], and g(ri r j) is the wavefunction between Bi and values estimated from the observations associated with X-ray − B j. Taking the wavefunction g(r) as the Gaussian form, g(r) = bursters [57]. Recent PREX-II experiments on measurement of 2 208 exp[ (r/ηc) ] with r = ri r j and ηc [= (0.45 0.50) fm] being the neutron skin thickness of Pb have also reported a large the range− of the repulsive| − core| for baryon forces,∼ one obtains value of L = (73 146) MeV [58]. Further detailed analyses of experimental and− observational information will be needed to ρBW0 ∞ L U (r; ρ ) = dqq2 j (qr) (G (q))2 , (5) constrain the precise value of the . We take the lower values SJM B 2π2 Z 0 src L = (60 70) MeV so that the density-dependence of the energy 0 − contributions around ρ0 in SNM does not deviate much from where Gsrc(q) is the Fourier transform of the B-B wavefunction those obtained by the standard variational calculation in terms with the s.r.c. and j0(qr)(=sin(qr)/(qr)) is the spherical Bessel of the V14 two-body potential with addition of the phenomeno- function. Here the approximate form of USJM is used as logical TNI by Lagaris-Pandahripande [LP (1981)] [50] [e.g., 2 see Fig. 1 for L = 65 MeV]. In Table 1, the relevant quanti- U (r; ρ ) = Vrρ (1 + crρ /ρ ) exp[ (r/λr) )] , (6) SJM2 B B B 0 − ties with the MRMF+UTBR+TNA model are listed for typical 3 = where Vr=95 MeV fm , cr=0.024, and λr=0.86 fm correspond- cases of L (60, 65, 70) MeV. · η = U In Fig. 1, the total energy per baryon, E (total) (= /ρB), ing to c 0.50 fm for SJM2 [40]. The SJM grows almost E linearly with ρB. Finally one obtains the effective two-body po- and each energy contribution from the three-nucleon-repulsion tential, USJM(r : ρB) = fsrc(r)USJM(r; ρB). [E (TNR)], the three-nucleon attraction [E (TNA)], and the sum of kinetic and two-body interaction energies [E (two-body)] in e 3.3. Three-nucleon attractive force SNM are shown as functions of ρB obtained by the present L = As for the TNA, we adopt the density-dependent effective model in the case of 65 MeV by the solid lines. For two-body potential by Nishizaki, Takatsuka and Hiura [49], comparison, those of LP (1981) are shown by the dotted lines, E E E which was phenomenologically introduced and the direct term where the (total), (two-body) and (TNR) are read from of which agrees with the expression by Lagaris and Pandhari- Fig. 2, Tables 4 and 5 in [50], and the parametersin TNA in the case of LP (1981) are set to be γ = 700 MeV fm6 and η = pande [50] [we later call it LP (1981)]: a a 13.6 fm3 [50]. − · 2 2 UTNA(r; ρB) = VaρB exp( ηaρB) exp[ (r/λa) ](~τ1 ~τ2) , (7) One finds that both the TNR and TNA have substantial con- − − · tributions to the binding energy at ρ0; E (TNR)= 4.1 MeV and where the range parameter λa is fixed to be 2.0 fm. The E (TNA)= 6.6 MeV, which are similar to those of LP (1981); U (r; ρ ) depends upon not only density but also isospin E (TNR)=−3.5 MeV and E (TNA)= 6.1 MeV. It should be TNA B − τ1 τ2. The parameters Va and ηa are determined together with noted that the energy contribution from the three-B force is de- other· parameters to reproduce the saturation properties of the termined by the volume-integral of the effective two-body V 3 SNM for the allowable values of L. potential, i. e. (UTBR) U (0; ρ ) (λr) in the case of V ∼ UTBR B · 3 Table 1: The coupling constants, gσN , gωN , gρN , and the meson mean-fields, σ 0, ω0 0 in SNM in case of the MRMF+ UTBR+TNA model and the parameters h i h i γa, ηa for TNA in case of L=60, 65, and 70 MeV. The effective mass ratio for the nucleon, (MN∗ /MN )0, in SNM at ρB = ρ0 is also listed.

γa ηa gσN gωN gρN σ 0 ω0 0 (MN∗ /MN )0 (MeV fm6) (fm3) (MeV) (MeV)h i (MeV)h i · SJM2+TNA-L60 1662.63 17.18 5.27 8.16 3.29 39.06 16.37 0.78 SJM2+TNA-L65 −1597.67 18.25 5.71 9.07 3.35 42.16 18.18 0.74 SJM2+TNA-L70 −1585.48 19.82 6.07 9.77 3.41 44.62 19.59 0.71 − (MeV)

Minimal RMF +UTBR+TNA σ 0, ω0 0 are modified to keep the saturation properties of the SNM.h i h Asi a result, for a larger L, a contribution to the binding LP(1981) energy from the two-body B-B interaction through the σ and ω meson-exchange in the RMF framework gets larger, and so E(TNR) are the coupling constants and meson mean-fields at ρ0 (see Ta- ble 1). At high densities beyond ρ0, where attraction from the σ-exchange is saturated, the remaining repulsion from the ω- E (TNA) exchange is more marked for larger ω-mean field, so that the E (two−body) stiffness of the EOS stands out as the L increases from 60 MeV E (total) to 70 MeV. Hence, in our model, the slope L controls the stiff- ness of the EOS in SNM not only around ρ , but also at high ρ 0 0 ρ − 3 B (fm ) densities. This feature is applied also to hadronic matter with the (Y+K) phase. (see Sec. 5 and Sec. 6). As for the experimental constraints of the EOS of the SNM, Figure 1: The total energy per baryon, E (total) (= /ρB), and each energy con- the flow of matter in heavy ion collisions was analyzed to de- E tribution to it from the three-nucleon-repulsion [E (TNR)], the three-nucleon termine the pressures at density region ρ = (2 5)ρ [59]. attraction [E(TNA)], and the sum of kinetic and two-body interaction energies B 0 Our results in the present model show that the pressure-dens∼ ity [E(two-body)] in SNM are shown as functions of ρB by the solid lines in the case of L = 65 MeV. For comparison, those obtained from LP (1981)[50] are curve in the SNM passes slightly above the upper limit of the shown by the dotted lines. allowable region constrained from the experimental data, which UTBR with UUTBR(0; ρB) being the repulsive-core height and implies some softening may occur in the SNM for the relevant λr the range of the potential, not by UUTBR(0; ρB) or λr sepa- densities. rately. Actually, there is a large difference of both the height and the range of the potentials between SJM2 and LP (1981) 5. Onset density of KC and EOS for the (Y+K) phase : USJM2(0; ρ0) =15.6 MeV and λr (SJM2)=0.86 fm, while U (0; ρ )= 2.89 MeV and λ (LP) = 1.40 fm. Nevertheless TNR,LP 0 r In Fig. 2, the energy per baryon /ρ with the (Y+K) phase (SJM2) coincides with (LP) at ρ =ρ , and both potentials B B 0 measured from the free nucleon massE is shown as functions contributeV to almost the sameV amount of the E (TNR) at ρ =ρ . B 0 of baryon number density ρ for (a) L= 60 MeV, (b) L = 65 One can see, in Fig. 1, the difference of the E (TNR) between B MeV, and (c) L = 70 MeV. In each figure, the bold solid line SJM2 and LP(1981) is tiny (0.6 MeV) at ρ = ρ , but it be- B 0 is for Σ = 300 MeV and the thin solid line is for Σ = 400 comes large at high densities ρ & 0.40 fm 3 due to the sensi- Kn Kn B − MeV, respectively. For comparison, the energy per baryon for tive density-dependenceof the E (TNR) ( ρ2 ). The stiffness of B pure hyperon-mixedmatter, where KC is switched off by setting the EOS in the present model comes partially∝ from such strong θ = 0, is shown by the green dashed line. The onset densities repulsion of the E (TNR) at high densities. In addition, for of KC [ρc (K)], Λ [ρc (Λ)], and Ξ [ρc (Ξ )] in the case of Σ two-body interaction in the RMF picture, attraction by the σ- B B − B − Kn = 300 MeV are denoted by the filled circle, filled triangle, and meson exchange is saturated at some density, while repulsion filled inverted triangle, respectively. The onset densities for KC by the ω-meson exchange increases steadily. This relativistic and Ξ in the case of Σ = 400 MeV are denoted by the open effect also renders stiff EOS as compared with the nonrelativis- − Kn circle and open inverted triangle, respectively. Note that the Λ tic variational method of LP (1981). c hyperons always precede KC for (a), (b), (c), so that ρB(Λ) is The value of γa in E (TNA) is correlated with a choice of 2 common to both cases of ΣKn = 300 MeV and 400 MeV. For L the slope L through the relation, ∆L (TNA)= 6γaρ0(ηaρ0 − = 65 MeV and 70 MeV with ΣKn = 300 MeV, even Ξ hyper- 2)e ηaρ0 (< 0), where ∆L (TNA) is the contribution to L from − − ons appear at lower density than KC. Mixing of the negatively the E (TNA), stemming from the isospin-dependence of the charged hyperons Ξ pushes the onset of KC to high densities, E (TNA). Therefore a larger L corresponds to a smaller γ , − a so that ρc (K) is delayed to higher densities for larger L. Con- where the TNA has a less contribution to the binding energy| | B versely, in case KC onsets at a lower density than the Ξ− hy- at ρB = ρ0, ∆B0 (TNA). By adjusting to the ∆B0 (TNA), the perons (for L = 60 MeV with ΣKn = 300 MeV and for L = (60, coupling constants gσN , gωN , gρN and meson mean-fields at ρ0, 65, 70) MeV with ΣKn = 400 MeV), mixing of the Ξ− hyperons 4 400 400 400 (a) L = 60 MeV (b) L = 65 MeV (c) L = 70 MeV

300 (MeV) 300 (MeV) 300 (MeV) N N N M M M − − − B B 200 200 B 200

ρ ρ ρ

/ / /

ε ε ε

100 100 100

0 0 0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 ρ − 3 ρ − 3 ρ − 3 B (fm ) B (fm ) B (fm )

Figure 2: The energy per baryon /ρB with the (Y+K) phase measured from the free nucleon mass as functions of baryon number density ρB for (a) L= 60 MeV, E (b) L = 65 MeV, and (c) L = 70 MeV, obtained with the MRMF+UTBR+TNA. In each figure, the bold solid line is for ΣKn = 300 MeV and the thin solid line is for ΣKn = 400 MeV, respectively. For comparison, the energy per baryon for pure hyperon-mixed matter, where KC is switched off by setting θ = 0, is shown by the c c c green dashed line. The onset densities of KC [ρB (K)], Λ [ρB(Λ)], and Ξ− [ρB(Ξ−)] in the case of ΣKn = 300 MeV are denoted by the filled circle, filled triangle, and filled inverted triangle, respectively. The onset densities for KC and Ξ− in the case of ΣKn = 400 MeV are denoted by the open circle and open inverted triangle, respectively. See the text for details. is pushed up to high densities, or even does not occur over the 65, 70) MeV, obtained with the MRMF+UTBR+TNA. The relevant densities: KC and Ξ− hyperons compete against each branches including KC in the core are denoted as the black other through the repulsive K-Ξ− vector interaction term in X0, bold solid lines (blue thin solid lines) for ΣKn = 300 MeV (400 the form of which is dictated by chiral symmetry. MeV). For comparison, the branch including pure hyperon- c From Fig. 2, the onsetdensity of KC is readas ρB(K) = (0.56, mixed matter, where KC is switched off by setting θ = 0, is 3 3 0.62, 0.70) fm− [ (0.46, 0.48, 0.50) fm− ] for L = (60, 65, 70) shown by the green dashed line for each case of L. The green MeV in the case of ΣKn = 300 MeV (400 MeV). The appear- filled triangle [N] stands for the branch point where the Λ hy- ance of KC in the hyperon (Λ)-mixed matter leads to further perons appear from nuclear matter in the center of the star. The decrease in energy of the system due to the s-wave K-B attrac- branch point at which KC appears in the center of the star is tion from that due to the Λ-mixing. As a result, the EOS for indicated by the filled circle [ ] (open circle [ ]) in the case • ◦ the (Y+K) phase is further softened in comparison with that in of ΣKn = 300 MeV (400 MeV). The maximum mass point for the pure Y-mixed matter (the dashed lines in Fig. 2). There is each branch including the (Y+K) phase is indicated by the open a clear difference in energy for ΣKn = 400 MeV from the case square []. The cross point [ ] corresponds to the causal limit × of the pure Y-mixed matter, while the difference is tiny for ΣKn at which the sound velocity exceeds the . One can = 300 MeV, in particular, in the case of L = 70 MeV. It should see, in Fig. 2, that the EOS becomes stiffer at high densities for be noted that the stronger three-baryon repulsion as a net effect larger L. This fact is reflected in that the maximum mass and its of the UTBR and TNA leads to more remarkable saturation and radius shift to larger values for larger L, as seen in Fig. 3. Also s subsequent reduction of the nuclear scalar density ρp,n as a rel- the radius of neutron stars for a given mass in the stable branch ativistic effect. As a result, a part of the attractive energy from increases with L. KC, which comes from the effective baryon mass term in (2) The maximum masses for L =(65, 70) MeV are consistent with s being proportional to ρN ΣKN , is suppressed more, so that the recent observations of massive neutron stars in both cases of decrease in energy due to KC is moderated by the introduction ΣKn = 300 MeV and 400 MeV, while the masses within the of the three-baryon force, leading to suppression of the signifi- causal limit for L = 60 MeV do not reach the range allow- cant softening of the EOS even in the presence of KC. able from the observations of most massive neutron stars to date [18, 19] (the green and yellow bands in Fig. 3). The ra- 6. Mass-radius relations of kaon-condensed neutron stars dius R in the stable branches is consistent with observational constraints from gravitational waves of the binary Based on the EOS including the (Y+K) phase, we discuss mergers GW170817 [20, 21]. Also R for M 1.3M lie within ≃ ⊙ the effects of KC on the structure of compact stars. In Fig. 3, the range of the mass and radius deduced from NICER observa- the gravitational mass M - radius R relations after solving the tions of PSR J0030+0451 [22, 23]. The observation of a com- Tolman-Oppenheimer-Volkoff equation are shown for L = (60, pact object with a mass of (2.50 2.67) M in the GW190814 − ⊙ 5 reach it fonly of sists rnsasfrtevle fteslope, neu the in of hyperon-mixing values the of for those stars to tron similar densities at mixed esol osdrhwsseai eaigo h ouein- volume the of relaxing systematic how consider should We stars. neutron massive of observations recent with and mass resulting the and oee,teei ag miut bu h empirical the about ambiguity large is there However, h eino ais38k 48k) nteohrhn,for hand, other the On km). (4.8 with km stars 3.8 neutron radius of region the arncpcueacmayn h ( the accompanying picture hadronic .Smayadcnldn remarks concluding and Summary 7. matter. hyperon-mixed the in KC of mechanisms o h aeo rvt,w o’ osdrpsiiiyo mix of possibility consider the don’t of we brevity, of sake the For the and constants coupling oecmoe fKC, of composed core of e ( MeV rnsaswith stars tron ih( with with h ( the Tolman-Oppenheimer-Volko iue3 h rvttoa mass gravitational The 3: Figure ssice o switched is oe stebakbl oi ie bu hnsldlns fo lines) solid thin (blue lines solid bold black the as noted MRMF of e,atog ute ealdaaye r eddt obtai to needed are mat case analyses conclusion. dense definite former detailed of EOS The further the although on [60]. ter, constraint heaviest hole stringent the black the provide lightest is will the companion or second star the neutron if question a provoked 40MV.Frcmaio,tebac ihpure with branch the comparison, For MeV). (400 L eety thsbe one u htthe that out pointed been has it Recently, nti ok ehv xdteUB oteSM model. SJM2 the to UTBR the fixed have we work, this In ehv hw htte( the that shown have We ti xetdta ev eto tr with stars neutron heavy that expected is It L . Y [ M

= M/M + Y ρ + ∆ L UTBR(SJM2) B c + K − (60 = 0.0 0.5 1.0 1.5 2.0 2.5 ( = K n ecnetaeo aigcertesuppression the clear making on concentrate we and , Λ oe ag oto ftecr a eoccupied be may core the of portion large A core. ) ff 2.0 hs o asv eto tr:Frnurnstars neutron For stars: neutron massive for phase ) ) 0MVand MeV 70 9 8 − ysetting by n 0 e]and MeV] 70) ∼ , M M p 0 Maximum mass n etn ( leptons and , . ⊙ fm 4 & + Kaon condensatesappear Kaon condensatesappear ntecs of case the in M N.Tebace nldn Ci h oeaede- are core the in KC including branches The TNA. θ 1.7 Σ hyperons( Σ pure hyperon−mixedmatter = Kn − Kn ff Λ . 3 ,i hw ytegendse iefrec case each for line dashed green the by shown is 0, M n h rudsaei h oecon- core the in state ground the and , qainfor equation = 400MeV Σ 1.4 = 300MeV 10 and ⊙ Kn eedn ntealwbevalues allowable the on depending , Y / Σ rrdu fcmatsaswithin stars compact of radius or ∆ M + M Kn = Ξ Λ e K sbrptnili atr[56]. matter in potential isobar − oradius to ⊙ R ) appear − 0 e) n a h ( the has one MeV), 400 [ mxdbroi atrwithin matter baryonic -mixed 11 h eta est osnot does density central the , hs a eraie nneu- in realized be can phase ) ). = L causal limit (km) L (300 = = Y 60 MeV L L 6,6,7)MVotie with obtained MeV 70) 65, (60, + 5MVand MeV 65 = R − 21 14 13 12 K = Y 0)MV.TeEOS The MeV]. 400) eain fe ovn the solving after relations hs r consistent are phase ) (40 mxdmte,weeKC where matter, -mixed ∆ 65 M − − PSR J1810+1744 PSR J0740+6620 sbr a be may isobars & r 0 e [61]. MeV 60) Σ 1 Kn 70 . Σ 7 = Kn ∆ M 0 MeV 300 -meson ⊙ = Y have + 300 a n ing K ) - - 6 h ( the ndb oprn ihohrrslsi ur oesincludi e models Pauli quark quark in the results other with comparing by ined erlfrteUB a UTBR the for tegral te stedrc ra(U rcs nKC, in process (DU) Urca direct the is other µ h classical the otn oeo hra vlto fnurnsas n sth is One stars: process, neutron (KU) of Urca evolution kaon-induced thermal on role portant ν For h rtnmxn ai sudrtrsodfrteD proces DU the for threshold ρ under is ratio -mixing the oiatcoigpoes o h asv eto stars neutron massive the For process. ( cooling dominant a Cfrsmlct.I h rsneo yeos the hyperons, of presence the In simplicity. for KC modify structure. t pasta may KC which forces, the order, of three-baryon second aspects the of various with becomes KC result to firs present transition of the transition the In to lead order. pasta KC to the According accompanying significa ening forces, [67]. three-baryon phase without result condensed previous kaon the of structure pasta a indispensable. is 6 matter of [65, wit position reactions processes these for emissivities cooling of description rapid Unified temperatu extraordinary low requires anomalously have that stars neutron Several cess. Λ ok ancoigpoesi iie ytemass presen the the by divided in is results process cooling the main to th work, According of density-dependence [13]. the energy upon symmetry depending met is reaction aro h atcehl olciemdswith modes collective -hole the of pair ehnssascae ihthe with associated mechanisms fthe of a may forces three-baryon The u ubr ( numbers tum .Tktuafrhsclaoaino h rsn subject. Prof present the late on the collaboration thank his We for Technology. Takatsuka T. of Institute Chiba by port Acknowledgement at freedom works. of future degrees in considered quark be will with densities co phase high The hadronic 70]. of [69, massi nection observations obtain with to compatible qua transition stars to crossover neutron connected the was by matter smoothly matter hadronic ext particular, studied In been have sively. transition phase -quark and ter iswhere ties N KNY B ¯ p M K p msin a eknmtclypsil,wihpasa im- an plays which possible, kinematically be may emissions neato.I a ensonta pnaeu raino creation spontaneous a that shown been has It interaction. + /ρ ntepeec fK,rpdcoigmcaim through mechanisms cooling rapid KC, of presence the In hogotti ok ehv ocnrtdo the on concentrated have we work, this Throughout considered Tatsumi) T. and Maruyama (T. authors the of Two saohrpcuefrsti for picture another As n fteatos(.Mt)akolde h nnilsup- financial the acknowledges Muto) (T. authors the of One + → omk h ecinknmtclypsil 2 2.The 12]. [2, possible kinematically reaction the make to & M e Y B e − neato eesrl rssi diint the to addition in arises necessarily interaction − + p . 1 . p → K . → + wv ancnesdpaebtas ai cooling rapid also but phase condensed kaon -wave 7 hs.Vldt fteUB hudas eexam- be also should UTBR the of Validity phase. ) 1 1 e . n M Λ 4 / − N .For 9. M + ⊙ K + yeosaemr bnatta rtn [68]. than abundant more are hyperons ,teK rcs eoe ancoigpro- cooling main a becomes process KU the ), + p ⊙ − ν ν wv ancnesto)myocra densi- at occur may condensation) kaon -wave ti ie ytemdfidUc rcs since process Urca modified the by given is it , ¯ e h ff e sln steknmtclcniinfrthe for condition kinematical the as long as , edwihsple h ytmwt energy with system the supplies which field K , cs[2 3 n atc C eut [64]. results QCD lattice and 63] [62, ects − p M i + + ff cstesti the ects > e ν − e 1 ff → ( . N nn h O,srneqakmat- quark strange EOS, the ening 4 M ff + Λ = c o nyteostadEOS and onset the only not ect ⊙ p yeo ( Hyperon , wv ancondensation. kaon -wave p ff N , ν eso h O including EOS the of ness n e ,where ), + 1]sat n becomes and starts [14] h K − → i n Λ K h raprocess, Urca ) → + K and − N M p i + s + tnsfor stands ∼ -wave K e e − p 1 tsoft- nt − s com- h − -wave -wave . quan- 4 + + M en- ν 6]. ν K ¯ ¯ ng a f he ve n- rk re ⊙ ν s, e e e e - t t , , . , . References [41] J. Gasser, H. Leutwyler, and M. E. Sainio, Phys. Lett. B 253, 252 (1991). [42] H. Ohki et al.(JLQCD Collaboration), Phys. Rev. D 78, 054502 (2008). References [43] J. Schaffner, C. B. Dover, A. Gal, C. Greiner, D. J. Millener, and H. St¨ocker, Ann. Phys. 235, 35 (1994). [44] A. Gal, E. V.Hungerford, and D. J. Millener, Rev. Mod. Phys. 88, [1] D. B. Kaplan and A. E. Nelson, Phys. Lett. B 175, 57 (1986). 035004 (2016), and references therein. [2] T. Tatsumi, Prog. Theor. Phys. 80, 22 (1988). [45] T. Muto, T. Maruyama, and T. Tatsumi, Phys. Rev. C 79, 035207 (2009). [3] T. Muto and T. Tatsumi, Phys. Lett. B 283, 165 (1992). [46] T. Muto, T. Maruyama, and T. Tatsumi, JPS Conf. Proc. 1, 013081 (2014); [4] V. Thorsson, M. Prakash, and J. M. Lattimer, Nucl. Phys. A EPJ Web of Conferences 73, 05007 (2014). 572, 693 (1994); Nucl. Phys. A 574, 851 (1994) (E). [47] K. Nakazawa et al., Prog. Theor. Exp. Phys. (2015) 033D02. [5] E. E. Kolomeitsev, D. N. Voskresensky, B. K¨ampfer, Nucl. Phys. A [48] S. H. Hayakawa et al., Phys. Rev. Lett. 126, 062501 (2021). 588 (1995) 889. [49] S. Nishizaki, T. Takatsuka and J. Hiura, Prog. Theor. Phys. 92, 93 (1994). [6] C. -H. Lee, G. E. Brown, D. -P. Min, and M. Rho, Nucl. Phys. A [50] I. E. Lagaris and V. R. Pandharipande, Nucl. Phys. A 359 (1981),349. 585, 401 (1995). [51] U. Garg and G. Col`o, Prog. Part. Nucl. Phys. 101, 55 (2018). [7] K. Tsushima, K. Saito, A. W. Thomas, and S. V. Wright, Phys. Lett. B [52] Bao-An Li and X.Han, Phys. Lett. B 727 (2013), 276. 429, 239 (1998); ibid. 436, 453 (1998) (E). [53] J. M. Lattimer and M. Prakash, Phys. Rep. 621, 127 (2016). [8] H. Fujii, T. Maruyama, T. Muto, and T. Tatsumi, Nucl. Phys. A [54] M. Oertel, M. Hempel, T. K¨ahn, and S. Typel, 597, 645 (1996). Rev. Mod. Phys. 89, 015007 (2017). [9] N. K. Glendenning, Astrophys. J. 293, 470 (1985) ; N. K. Glendenning, [55] C. -J. Xia, T. Maruyama, N. Yasutake, T. Tatsumi, and Y. -X. Zhang, and S. A. Moszkowski, Phys. Rev. Lett. 67, 2414 (1991). Phys. Rev. C 103, 055812 (2021). [10] S. Nishizaki, Y. Yamamoto, and T. Takatsuka, [56] E. E. Kolomeitsev, K. A. Maslov, and D. N. Voskresensky, Nucl. Phys. A Prog. Theor. Phys. 108, 703 (2002). 961, 106 (2017). [11] G. F. Burgio, H. -J. Schulze, I. Vida˜na, and J. -B. Wei, arXiv: [57] H. Sotani, K. Iida, and K. Oyamatsu, Phys. Rev. C 91, 015805 (2015). 2105.03747 [nucl-th]. [58] B. T. Reed, F. J. Fattoyev, C. J. Horowitz, and J. Piekarewicz, [12] G. E. Brown, K. Kubodera, D. Page, and P. Pizzecherro, Phys. Rev. D37 Phys. Rev. Lett. 126, 172503 (2021); arXiv: 2101. 03193v1 [nucl-th]. (1988) 2042 . [59] P. Danielewicz, R. Lacy, and W. G. Lynch, Science 298, 1592 (2002). [13] H. Fujii, T. Muto, T. Tatsumi and R. Tamagaki, [60] R. Abbott et al., Astrophys. J.Lett. 896, L44 (2020). Nucl. Phys. A571, 758 (1994); Phys. Rev. C50, 3140 (1994). [61] A. Drago, A. Lavagno, G. Pagliara, and D. Pigato, Phys. Rev C [14] M. Prakash, J. M. Lattimer, and C. J. Pethick, Astro- 90, 065809 (2014). phys. J. 390, L77 (1992). [62] M. Oka, Nucl. Phys. A 881, 6 (2012). [15] H. Grigorian, D. N. Voskresensky, and K. A. Maslov, [63] C. Nakamoto, Y. Suzuki, Phys. Rev. C 94, 035803 (2016). Nucl. Phys. 980, 105 (2018). [64] T. Inoue for Hal QCD, AIP Conf. Proc. 2130, 020002 (2019). [16] P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, and [65] S. Tsuruta, Phys. Rep. 292, 1 (1998). J. W. T. Hessels, Nature 467, 1081 (2010); E. Fonseca, T. T. Pennucci [66] A. Dohi, K. Nakazato, M. Hashimoto, Y. Matsuo, and T. Noda, et al., Astrophys. J 832, 167 (2016). Prog. Theor. Exp. Phys. 2019, 113E01 (2019). [17] J. Antoniadis et al., Science 340, 6131 (2013). [67] T. Maruyama, T. Tatsumi, D. N. Voskresensky, T. Tanigawa, T. Endo, and [18] H. T. Cromartie et al., Nat. Astron. 4, 72 (2020); E. Fonseca, H. T. Cro- S. Chiba, Phys. Rev. C 73, 035802 (2006). martie, T. T. Pennucci, et al., arXiv:2104.00880 [astro-ph.HE]. [68] T. Muto, Nucl. Phys. A 697, 225 (2002). [19] R. W. Romani, D. Kandel, A. V. Filippenko et al., Astro- [69] K. Masuda, T. Hatsuda, and T. Takatsuka, Astrophys. J. 764, 12 (2013); phys. J. L. 908, L46 (2021). PTEP 2016. No. 2, 021D01 (2016). [20] B. P. Abbott et al., Phys. Rev. Lett. 121, 161101 (2018). [70] G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song, and T. Takatsuka, [21] F. J. Fattoyev, J. Piekarewicz, and C. J. Horowitz, Rept. Prog. Phys. 81, 056902 (2018). Phys. Rev. Lett. 120, 172702 (2018). [22] T. E. Riley et al., Astrophys. J. 887, L21 (2019). [23] M. C. Miller et al., Astrophys. J. 887, L24 (2019). [24] T. Muto, Phys. Rev. C 77, 015810 (2008), and references cited therein. [25] T. Muto, T. Maruyama, T. Tatsumi, and T. Takatsuka, JPS Conf. Proc. 26, 024019 (2019). [26] T. Muto, T. Maruyama, and T. Tatsumi, to be submitted. [27] Y. Yamamoto, T. Furumoto, N. Yasutake, and Th. A. Rijken, Phys. Rev. C 90, 045805 (2014). [28] Y. Yamamoto, H. Togashi, T. Tamagawa, T. Furumoto, N. Yasutake, and Th. A. Rijken, Phys. Rev. C 96, 065804 (2017). [29] K. Tsubakihara and A. Ohnishi, Nucl. Phys. A 914, 438 (2013). [30] D. Lonardoni, A. Lovato, S. Gandolfi, and F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015). [31] M. Kohno, Phys. Rev. C 97, 035206 (2018). [32] D. Logoteta, I. Vida˜na, and I. Bombaci, Eur. Phys. J. A 55, 207 (2019). [33] D. Gerstung, N. Kaiser, and W. Weise, Eur. Phys. J. A 56, 175 (2020). [34] S. Petschauer, J. Haidenbauer, N. Kaiser, Ulf-G. Meißner, and W. Weise, Frontiers in Physics 8, 12 (2020), and references cited therein. [35] S. Weissenborn, D. Chatterjee, and J. Schaffner-Bielich, Phys. Rev. C 85, 065802 (2012). [36] K. A. Maslov, E. E. Kolomeitsev, and D. N. Voskresensky, Phys. Lett. B 748, 369 (2015). [37] W. Zuo, A. Li, Z. H. Li, and U. Lombardo, Phys. Rev. C 70, 055802 (2004). [38] A. Li, G. F. Burgio, U. Lombardo, and W. Zuo, Phys. Rev. C 74, 055801 (2006). [39] R. Tamagaki, Prog. Theor. Phys. 119, 965 (2008). [40] T. Takatsuka, S. Nishizaki, and R. Tamagaki, AIP Conf. Proc. 1011, 209 (2008). 7