K Condensation in Neutron Star Matter with Quartet

Commun. Theor. Phys. (eijing, China) 54 (2010) pp. 500 508 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 3, September 15, 2010 K Condensation in Neutron Star Matter with Quartet DING Wen-o (ò ), 1,2, LIU Guang-Zhou ( ¾ ), 2, ZHU Ming-Feng (ý ò), 2, YU Zi (â ), 2 XU Yan (Æ ), 2 and ZHAO En-Guang ( ¾) 3 1 Physical Department, ohai University, Jinzhou 121000, China 2 Center for Theoretical Physics, Jilin University, Changchun 130023, China 3 Institute of Theoretical Physics, Chinese Academy of Sciences, eijing 100190, China (Received November 5, 2009; revised manuscript received June 25, 2010) Abstract In the framework of relativistic mean field theory, the condensations of K and K 0 in neutron star matter including baryon octet and quartet are studied. We find that in this case K and K 0 condensations can occur at relative shallow optical potential depth of K from 80 MeV to 160 MeV. oth K and K 0 condensations favor the appearances of resonances. With K condensations all the quartet can appear well inside the maximum mass stars. The appearances of resonances change the composition and distribution of particles at high densities. The populations of resonances can enhance K condensation. It is found that in the core of massive neutron stars, neutron star matter includes rich particle species, such as antikaons, baryon octet, and quartet. In the presence of resonances and K condensation, the EOS becomes softer and results in smaller maximum mass stars. Furthermore the impact of antikaon condensations, hyperons, and resonances on direct Urca process with nucleons is also discussed briefly. PACS numbers: 26.60.-c, 21.65.-f, 97.60.Jd, 95.30.Cq Key words: antikaon condensation, neutron stars, delta resonances 1 Introduction Neutron stars, as terrestrial laboratories to investigate the properties of dense matter, recently became the center of attention. At the core of neutron stars, with the increasing density, the chemical potentials of baryons and leptons increase rapidly. Consequently exotic forms of matter may appear, such as strange baryons, [1] condensation of antikaons, [2 5] deconfined quark matter [6 7] and so on. In 1986, Kaplan and Nelson [8 9] within a chiral SU(3) L SU(3) R model, first demonstrated that K mesons may undergo ose Einstein condensation in dense matter formed in heavy-ion collisions. Later, many articles [2,10 11] have predicted that K condensation may form in the core of neutron stars. Though the condition of the super high density and the very low temperature (only about 0.1 MeV for cold neutron stars) can not be simulated in laboratories, the data of antikaons in heavy ion collisions [3,12] lend strong support to the condensation of antikaons. However, it is found that antikaon condensation is very sensitive to the equation of state (EOS) and the optical potential depth of antikaon. Recent works on antikaon condensation give relatively shallow values of optical potential, which is supported by theoretical and experimental research. [13 15] Since Dashen and Tajaraman [16] have pointed out that quartet could be treated as a separate elementary particle in neutron stars, in this paper we are interested in the interaction of K condensation and quartet and the effects on the property of neutron stars. One will find that in neutron star matter with K and K 0 condensations quartet can appear and populate well inside the maximum mass star in the paper. And the presence of resonances can influence the distribution of particles, the maximum mass of neutron stars, the maximal central density, and so on. The composition and structure of neutron stars primarily depend on the nature of strong interaction. The threshold densities for baryon formation are strongly dependent on the EOS and the parameter used. In fact, we have to choose stiff EOSs, which favor the presences of antikaons and quartet. In neutron star matter, if there is no K condensation, quartet will hardly appear. In dense matter with a very stiff EOS and in the absence of K condensation,, 0 may appear at high densities, which are close to the maximal central density of neutron stars, so the influence of them on the gross properties of neutron stars is negligible. However, with K condensation, the abundance of K increases rapidly with baryon density, and the particles with positive charge, such as ++ and + will appear at lower densities to keep the charge neutrality of the star. Moreover, with the existence of K and K 0 condensations, the interaction potentials are more favorable to the quartet formation. On the other hand, resonances can also accelerate the increase of antikaons. Supported in part by National Natural Science Foundation of China under Grant Nos. 10275029 and 10675054 E-mail: dingwenbo1980@yahoo.cn E-mail: lgz@jlu.edu.cn E-mail: mfzhu@jlu.edu.cn

No. 3 K Condensation in Neutron Star Matter with Quartet 501 As we know, neutron stars are born after the supernova explosions, and the evolution of the remnant protoneutron stars is driven by the thermal neutrino diffusion from the dense core. However, how antikaon condensations, hyperons, and resonances influence this evolution is yet indeterminate and complicated. The neutrino scatting with regard to kaons includes the thermal kaon reactions, [17] kaon-induced Urca, [18 19] and modified Urca reactions. It is found that the formation of K condensation might not affect the delayed collapse of neutron stars. Thus far, the most effective cooling mechanism is the direct Urca process for nucleons, with a magnitude of emissivity 10 27 erg s 1 cm 3. Though the direct Urca process with hyperons or antikaons belongs to the enhanced cooling, the magnitude is lower, about 10 24 10 25 erg s 1 cm 3. Therefore, direct Urca process with nucleons plays the leading role in the neutrino emission, consequently the effect of antikaon condensations, hyperons, and resonances on this process is quite important, which will be discussed in the paper. Especially, by investigating the decay process of resonances, urgion and Link [20] proposed that young and rapidly-rotating neutron stars could be intense neutrino sources. It is noted that the effect of high temperatures is neglected in the paper, for the particle distribution is insensitive to the temperature. The paper is structured in the following way. In Sec. 2, we describe the relativistic mean field theory (RMFT) and different phases of matter. In Sec. 3, the parameters of the model are discussed and the results of our calculations are explained. A summary and conclusion are given in Sec. 4. 2 Formalism In this calculation, we adopt the RMFT to describe neutron star matter. [2] The interaction between baryons is mediated by the exchange of scalar σ, isoscalar vector ω, and vector isovector ρ mesons. This picture is extended to include the (anti) kaons. Therefore the Lagrangian density contains baryonic, kaonic, and leptonic parts. The baryon Lagrangian density is given by L = Ψ (iγ µ µ m + g σ σ g ω γ µ ω µ 1 2 g ργ µ τ ρ µ) Ψ + 1 2 µσ µ σ 1 2 m2 σ σ2 U(σ) 1 4 ω µνω µν + 1 2 m2 ωω µ ω µ 1 4 ρ µν ρ µν + 1 2 m2 ρ ρ µ ρ µ. (1) Here Ψ represents the Dirac spinor for baryon with vacuum mass m and isospin operator τ. denotes not only n, p, Λ, Σ +, Σ 0, Σ, Ξ 0, and Ξ of the baryon octet, but also ++, +, 0, of the quartet. g σ, g ω and g ρ denote the coupling constant to σ, ω, and ρ meson field, respectively. ω µν and ρ µν are field strength tensors. The scalar self-interaction term [21] is U(σ) = 1 3 bm N(g σn σ) 3 + 1 4 c(g σnσ) 4, (2) The (anti) kaon is coupled to the meson fields using minimal coupling L K = D µ KD µ K m 2 K KK, (3) where the covariant derivative is the standard form D µ = µ + ig ωk ω µ + ig ρk τ K ρ µ. (4) K = (K +, K 0 ), K = (K, K 0 ) denote the isospin doublet for kaon and untikaon. The effective mass of kaon is given by m K = m K g σk σ, (5) where m K is the bare kaon mass. From Lagrangian Eq. (3), the equation of motion for kaon can be derived by solving Euler Lagrange equation as [D µ D µ + m 2 ]K = 0. (6) K In the mean field approximation (MFA) adopted here, the meson fields are replaced by their expectation values, which are denoted by σ, ω 0, ρ 03. In cold neutron stars, for S-wave condensation of antikaons, the in-medium energies of K is given by w K = m K g σk σ g ωk ω 0 1 2 g ρkρ 03. (7) Since the σ and ω fields generally increase with density, both the terms containing σ and ω fields in Eq. (7) are attractive for antikaons and the in-medium energies of K are lowered. On the other hand, because of the negative ρ 03, the isovector ρ field lowers the effective energies of K0, consequently, it favors K 0 condensation over K condensation. The total Lagrangian density is with L l = l L = L + L l + L K, (8) Ψ l (iγ µ µ m l )Ψ l, and l = (e, µ). So the meson field equations in presence of baryons and antikaon condensation are given by m 2 σ σ = g σ ρ S + g σk ρ K U σ, K m 2 ωω 0 = g ω ρ g ωk ρ K, K m 2 ρρ 03 = g ρ I 3 ρ + g ρk I 3 Kρ K. (9) The scalar density and baryon number density are ρ S = 2J k + 1 F m 2π 2 k 2 dk, (10) 0 k2 + m 2 ρ = (2J + 1) k3 F 6π 2, (11) K

502 DING Wen-o, LIU Guang-Zhou, ZHU Ming-Feng, YU Zi, XU Yan, and ZHAO En-Guang Vol. 54 with Fermi momentum k F, spin J, and isospin projection I 3. In nucleon-only star matter, nucleons and electrons undergo β-decay processes n p + e + ν e. And when the chemical potential for an electron exceeds the mass of a muon, the reaction e µ + ν µ + ν e occurs. Considering cold neutron stars, we assume the neutrinos left freely. Therefore the chemical potentials of nucleons and leptons are governed by the equilibrium conditions µ n µ p = µ e = µ µ. (12) Here µ n, µ p, µ e, and µ µ denote the chemical potentials of neutrons, protons, electrons, and muons, respectively. When the Fermi momenta of nucleons are large enough, the strange changing processes, such as N N + K, e K + ν e, may occur, where N = (n, p) denotes the isospin doublets for nucleons. The threshold conditions for K condensation can be expressed as µ n µ p = µ K = µ e, (13) µ K 0 = 0. (14) Therefore, K will form a condensation if the effective energy w K equals to its chemical potential µ K, i.e. w K = µ K = µ e. (15) Similarly, K 0 condensation occurs when w K 0 = µ K 0 = 0. (16) The discuss above may extend to the case for neutron star matter including hyperons and isobars. When other baryons in the form of hyperons and isobars are present in neutron star matter, the β-decay processes for nucleons generalize to the form 1 2 + l + ν l and 2 + l 1 + ν l, where 1 and 2 are baryons. In general, for an arbitrary baryon, chemical equilibrium in a star may be expressed by a generic equation µ = µ n q µ e, (17) where µ and q are, respectively, the chemical potential and the electric charge of baryon. From Eq. (7), with the increase of baryon density, w K and w K 0 decrease, while µ e increases. So K condensation can occur easily. While the formation of K0 condensation is relatively difficult, because w K 0 has to decrease to zero. However, since the σ, ω, and ρ fields are all helpful to lower w K 0, K 0 meson condensation may set in at a higher density in all probability. For neutron star matter, we also need to include the baryon number conservation and charge neutrality condition ρ = n, (18) q ρ ρ e ρ µ ρ K = 0. (19) Here n is the total baryon number density. As soon as n is chosen, we can obtain the input-values of the EOS by solving the nonlinear equations, which include Eqs. (9), (13), (14), (17), (18), and (19). In the presence of K condensation, the total energy density consists of three parts, ε = ε + ε l + ε K, i.e., ε = 1 2 m2 σ σ2 + b 3 m N(g σn σ) 3 + c 4 (g σnσ) 4 + 1 2 m2 ω ω2 0 + 1 2 m2 ρ ρ2 03 + k 2J + 1 F k2 2π 2 + m 2 k2 dk 0 + k 1 l π 2 k 2 + m 2 l k2 dk l 0 The pressure is given by + m ρ K K + m K 0ρ K 0. (20) P = 1 2 m2 σ σ2 b 3 m N(g σn σ) 3 c 4 (g σnσ) 4 + 1 2 m2 ω ω2 0 + 1 2 m2 ρ ρ2 03 + 1 k 2J + 1 F 3 2π 2 0 + 1 3 l k 1 l π 2 0 k 4 dk k2 + m 2 k 4 dk. (21) k2 + m 2 l Here we treat both K and K 0 condensations as second-order phase transitions. [5] For the direct Urca process with nucleons of electrons, the energy per unit volume and time released in one cycle can be expressed as [22] I = 457π 10080 g2 F cos2 θ C (1 + 3g 2 )T 6 A 9 m p m n µ eθ npe, (22) where m p and m n are the effective masses of protons and neutrons respectively. Θ npe is a step function corresponding to the triangle condition, and g F, θ C, and g A are the constants in weak interactions. The situation for muons is similar to that for electrons. Through the neutrino emissivity, the neutrino luminosity of the neutron star can be calculated for a given mass of the neutron star. The relation between the interior temperature and the evolving time of a neutron star can be obtained by solving the cooling equation, then the cooling history can be described. 3 Results and Discussion In the model we adapt here, four distinct sets of coupling constants, meson-nucleon, meson- resonance, meson-hyperon, and meson-kaon are required. The meson-nucleon coupling constants are generated by reproducing the nuclear matter saturation properties. Here we exploit the GL91 parameter set, which is listed in Ref. [23]. Since resonances have the same quark composition with nucleons, we assume that meson- resonance coupling constants are equal to the ones for nucleons. According to Ref. [23], the well-depth for Λ hyperon is 28 MeV, and we

No. 3 K Condensation in Neutron Star Matter with Quartet 503 assume all the hyperons have the same couplings as Λ s, i.e., χ MH = χ M, where χ MH = g MH /g MN, H = {Λ, Σ, Ξ}, and M = {σ, ω, ρ}. Here we choose χ σ = χ ω = χ ρ = 1, which in fact leads to a stiff EOS. The vector meson-kaon coupling constants are derived from the quark and isospin counting rule, they are g ωk = 1 3 g ωn, g ρk = g ρn. (23) The scalar coupling constant is estimated from the real part of K optical potential depth in the normal nuclear matter density U K(n 0 ) = g σk σ g ωk ω 0. (24) There are experimental evidences that antikaons experience an attractive potential, whereas kaons feel a repulsive interaction in nuclear matter. [12,24] Recent calculations of K optical potential at the normal nuclear matter density give relatively shallow values compared with previous calculations, [25] for example, Ramos and Schaffner predicted a potential about 40 MeV, a combined chiral analysis of K atomic and K P scattering data lead to a potential of 55 MeV. [13] Most recent studies give rise to 80 MeV [15] and the values 70 120 MeV. [14] So we are inclined to believe that K condensation with a relatively shallow optical potential can occur. In our calculation, we select a set of optical potential depths starting from 80 MeV to 160 MeV. The scalar coupling constants for these values of optical potential depths, g σk, are listed in Table 1. Table 1 The coupling constants for the antikaon to σ- meson, g σk in the GL91 set, for various values of K optical potential depths, U K(n 0), at the saturation density. U K(n 0 )/MeV 80 100 120 140 160 GL91 0.252 73 0.880 28 1.507 83 2.135 38 2.762 94 It was shown in various calculations that hyperons, such as Λ and Σ always first appeared in neutron star matter around (2 3)n 0, which made the EOS soft. Therefore, the formation of K condensate is postponed to a higher density, compared with that in nucleon-only (np) matter. If a relatively deep optical potential depth of K ( 160 or 180 MeV) [4,26] had been used, the critical density of K condensate was about (2 4)n 0, and K 0 condensate might occur at a higher density, about (6 8)n 0. Some calculations even drew a conclusion that K 0 condensate could not occur. [7] However, in our calculation, it is found that the stiff EOS makes the K condensation occur easily, even for shallow optical potential depths. We demonstrate our results with the GL91 set. In Fig. 1, the mean meson potentials g ωn ω 0, g σn σ, and g ρn ρ 03 for neutron star matter including nucleons, hyperons, resonances, and leptons (nph ) (solid lines) and for matter further including antikaon, K, and K 0 (nph K K0 ) (dotted lines) as a function of baryon density, n /n 0 for K optical potential depth of U K(n 0 ) = 100 MeV are shown. For comparison, neutron star matter with K and K 0 condensations in the absent of quartet (nphk K0 ) (dashed lines) is also shown in Fig. 1. It is found that with antikaon condensation all the mean meson potentials including σ, ω, and ρ fields decrease compared with nph matter. However, with the occurrence of quartet (dotted lines), the vector ω meson potential decreases and scalar σ meson potential increases, whereas the isovector ρ meson potential does not change. These changes in meson fields may be attributed to the variations of the souce terms due to antikaon condensation and the occurrence of quartet in field equations Eq. (9). Fig. 1 The meson potentials versus normalized baryon density in the GL91 set for neutron star matter nph (solid lines), nphk K0 (dashed lines) and nph K K0 (dotted lines). The K optical potential at normal nuclear matter density is 100 MeV. The populations of particles as a function of baryon density for neutron star matter nph K and for matter excluding resonances nphk are shown in Fig. 2. It is seen that K condensation occurs at 4.29n 0, and then increases rapidly with baryon density. It is because K mesons as bosons in the lowest energy state are more favorable to maintain charge neutrality than any other negatively charged particles. It is noted that ++ appear at 5.67n 0, and then +, 0, and appear one by one. With the appearances of resonances, the abundance of K mesons is enhanced drastically (dotted lines). The arrows at the transverse axis indicate the central densities of the maximum neutron stars (see Tables 2 and 3). For comparison, the populations of particles in neutron star matter without antikaon condensation, i.e., nph are shown in Fig. 3. From Fig. 3, one can find that only and 0 can appear and their threshold densities are smaller than the central density of the maximum mass star. The large density of K condensation may be responsible for the appearances of ++ and +. Compareing with Fig. 3, one can find that in Fig. 2 the presence of K condensation

504 DING Wen-o, LIU Guang-Zhou, ZHU Ming-Feng, YU Zi, XU Yan, and ZHAO En-Guang Vol. 54 reverses the occurrence order of quartet. It is worthy to emphasize that K condensation favors the appearances and populations of resonances. In Fig. 2 all the threshold densities of quartet are smaller than the central density of the maximum mass star, which expresses the presence of resonances in the core of maximum mass stars. Fig. 2 The number density n i of various particles in neutron star matter nph K (dotted lines) and nphk (solid lines) at U K(n 0) = 100 MeV as a function of normalized baryon density. The arrows at the transverse axis indicate the central densities of the maximum neutron stars. density. In Fig. 4, we also noticed that the densitiy of neutrally charged hyperons Ξ 0 decreases drastically to zero, as soon as K 0 condensation forms. Due to the different expectation values for the isovector field, K0 condensate favors the occurrence of 0 rather than Ξ 0. In a word, the total effect of both K and K 0 condensates on resonances is to help the quartet appear early. The appearances of resonances further impact on the condensation of K and the distribution of baryons. Compared with the solid and dotted curves in Fig. 4, we find the effect of the suppression of K mesons by K 0 condensation may be attributed to the fact that K 0 condensation can make all baryons appeared in neutron star matter including nucleons, hyperons, and resonances become identical leading to an isospin saturated symmetric matter. In order to demonstrate the main role of K0 condensation, we show iosvector potentials for neutron star matter nph, nph K, and nph K K0 as a function of baryon density in Fig. 5. From Fig. 5, one finds that K 0 further reduces the iosvector potential with the increase of baryon density. At about 7n 0 the iosvector potential is close to zero. Fig. 4 Same as Fig. 2 but in neutron star matter further including K 0 condensation, i.e., nph K K0 and nphk K0 matter. Fig. 3 Same as Fig. 2 but excluds K condensation. The populations of neutron star matter nph K K0 and nphk K0 are displayed in Fig. 4. Comparing with Fig. 2, we find that once the K 0 condensation occurs at 5.68n 0, the distribution of particles will be changed (solid curves). With the increase of baryon density, K mesons decease and K 0 mesons increase rapidly, the neutron and proton abundances become almost identical. With K 0 condensation, we note that the occurrence of ++ and + is delayed, whereas the occurrence of 0 and negatively charged is brought forward. Furthermore, the threshold densities of quartet are very close to the same Fig. 5 Same as Fig. 1 but only isovector potential is shown.

No. 3 K Condensation in Neutron Star Matter with Quartet 505 The critical densities for antikaon mesons and all baryons in neutron star matter nph K K0 for different values of U K(n 0 ) are collected in Table 2. It is found that K condensation occurs earlier for a deeper optical potential of K and K mesons always condensate before the condensation of K0. In fact, K condensate delays the appearance of K0 condensation to a higher density. These results are consistent with the previous calculations. [27 28] However, it is worthy to note that both K and K 0 condensations can occur well at densities lower than the central densities corresponding to the maximum mass stars (see Table 3) for a wide range of 100 MeV U K(n 0 ) 160 MeV, though only K condensation can occur at U K(n 0 ) = 80 MeV. K condensate suppresses the abundances of negatively charged hyperons, Σ and Ξ, but favors the occurrence of the positively charged particles, such as Σ +, ++ and +. However, the effect is accentuated for a larger U K(n 0 ). The existence of antikaon condensations and resonances will impact on the EOS. Figure 6 shows the EOS, pressure P versus energy density ε for neutron star matter including K, nphk (dashed line), for matter including both K and K 0 condensations, nphk K0 (dashed-dotted line) and for matter further including, nph K K0 with antikaon optical potential depth at normal density of U K(n 0 ) = 100 MeV. For comparison, the EOS of neutron star matter including nucleons and hyperons, nph (dotted line) is also shown in Fig. 6. From Fig. 6 one finds that the EOS becomes softer in the presence of K condensation. After the onset of K0 condensation the EOS is further softened. The softest one is the EOS for neutron star matter nph K K0. The EOS for neutron star matter with antikaon condensation for different values of U K(n 0 ) are displayed in Fig. 7. In a line, the first (at low density) and second kinks correspond to the onsets of K and K 0 condensations, respectively. The third one refers to the appearances of resonances. From Fig. 7 we find that the softness in the EOS is sensitive to antikaon optical potential depth of K at the normal density. The stronger the attractive antikaon interaction, the softer the corresponding EOS. A large U K(n 0 ) corresponds to a low threshold density of quartet (see Table 2) and with the increase of U K(n 0 ), the central density, u cent (see Table 3), decreases in a neutron star. Static structures of spherically symmetric neutron stars are obtained by solving Tolman Oppenheimer Volkoff (TOV) equations. [29] With the EOS shown in Fig. 6 as input, we calculate the mass-radius relations of these neutron stars which are shown in Fig. 8. These calculation results are also listed in Table 3. From Fig. 8 one finds that the softer the EOS, the smaller the corresponding maximum mass. From Tables 2 and 3 one can notice that for all the values of U K(n 0 ), except for U K(n 0 ) = 160 MeV the threshold densities of quartet are lower than the central densities of the maximum mass stars. This implies that these maximum mass stars will contain quartet. Fig. 6 Pressure P as a function of energy density ε in the GL91 set for the neutron star matter nph, nphk, nphk K0, and nph K K0 at U K(n 0) = 100 MeV. In a line the first (at lower energy density) and second kinks correspond to the appearance of K and K 0 condensations, while the third one corresponds to the appearances of resonances. Table 2 The critical densities for K mesons and baryons in neutron star matter nph and nph K K0 of GL91 set. Here u cr = ρ cr/n 0. nph 80/MeV 100/MeV 120/MeV 140/MeV 160/MeV u cr(k ) 5.19 4.29 3.53 2.98 2.56 u cr( K 0 ) 5.68 4.92 4.23 3.60 u cr(λ) 1.89 1.89 1.89 1.89 1.89 1.89 u cr(σ + ) 4.32 4.32 4.31 4.02 3.75 3.47 u cr(σ 0 ) 3.38 3.38 3.38 3.38 3.37 3.42 u cr(σ ) 1.80 1.80 1.80 1.80 1.80 1.80 u cr(ξ 0 ) 4.78 4.78 5.09 u cr(ξ ) 3.53 3.53 3.53 u cr( ++ ) 5.67 5.30 4.89 u cr( + ) 6.54 5.74 5.36 4.94 u cr( 0 ) 5.97 5.96 5.82 5.43 5.00 u cr( ) 4.92 4.92 5.89 5.49 5.06

506 DING Wen-o, LIU Guang-Zhou, ZHU Ming-Feng, YU Zi, XU Yan, and ZHAO En-Guang Vol. 54 Table 3 The maximum mass, M max, the corresponding central densities, u cent = ρ cent/n 0, and the radii, R, of neutron stars for neutron star matter nph, nph K and nph K K0 in GL91 set. The U K(n 0) is taken from 80 MeV to 160 MeV. nph K nph K K 0 nph U K(n 0 )/MeV 80 100 120 140 160 100 120 140 160 M max/m 2.1110 2.0473 1.9201 1.7051 1.4571 2.0469 1.9164 1.6922 1.4561 2.1318 u cent 6.351 6.181 5.593 4.773 4.423 7.021 5.498 5.034 3.603 6.619 R/km 11.539 11.890 12.291 12.708 13.406 11.918 12.339 12.852 13.475 11.188 Fig. 7 The equation of state, pressure P vs. energy density ε in the GL91 set. The solid lines refer to the results for neutron star matter nph and nphk K0, and the dashed ones correspond to nph and nph K K0. nph K K0 matter with NL-SH set are shown in Fig. 9. It is seen that the distribution of baryons is similar to the one with GL91 set (see Fig. 4). K condensation occurs at 3.02n 0, and K 0 at 4.05n 0. It is found that u cr ( ++ ) = 4.02n 0, u cr ( + ) = 4.06n 0, u cr ( 0 ) = 4.11n 0, and u cr ( ) = 4.16n 0, the central density u cent is equal to 4.18n 0. The EOS with the NL-SH parameter set for nph, nphk K0, and nph K K0 neutron star matter for U K(n 0 ) = 140, 160, 180 MeV are shown in Fig. 10. The kinks in dotted lines at high densities corresponding to the appearances of resonances are seen obviously. All the results support the conclusions made from GL91 set. Moreover, many parameter sets which can lead to stiff EOSs may give the similar conclusions, such as NL0, NL3 and so on. Fig. 9 Same as Fig. 4 but for NL-SH set at U K(n 0) = 140 MeV. Fig. 8 The mass-radius relation for neutron star sequences near the limiting mass in the GL91 set, for the nph matter (dotted line)and the nph K K0 matter (solid lines) at different values of U K(n 0). The arrows correspond to the maximum masses. As mentioned above, the threshold densities for antikaon condensations are quite sensitive to the EOS we used. A stiffer EOS is more efficient for antikaon condensations. However, the EOS for neutron star matter at the high density region is uncertainty. Apart from the GL91 parameter set, the NL-SH [30] set are also exploited. The populations of particles in nphk K0 and The static properties of neutron stars have been studied above. Now we will discuss the effect of antikaon condensations, hyperons and resonances on the neutrino scatting. Since the direct Urca (durca) process plays the important role, here we present the impact on it in Fig. 11. It is found that in np matter, the neutrino emissivity for durca process I durca is the highest in most area of the star (dashed line). However, once hyperons and antikaon condensations are included (nphk K0 matter), I durca decreases obviously (solid line). The first fall occurs at roughly R = 8 km because of the appearance of hyperons, especially Σ and Λ hyperons. The second fall of I durca

No. 3 K Condensation in Neutron Star Matter with Quartet 507 is located at R = 4.3 km, where K condensation sets in. The third fall occurs at R = 2.1 km visibly, corresponding to the ceaseing of the durca process for muons, whose neutrino emissivity is equal to that for electrons. In fact, the low neutrino emissivity in nphk K0 matter, according to Eq. (22), is mainly attributed to the decrease of leptons and the scalar σ meson potential, even at the core of the neutron star, the triangle condition for muons can not be fulfilled. resonances make a further fall of I durca at the core (see dotted line). However for small area resonances existed in this effect is feeble. Fig. 10 Same as Fig. 7 but for NL-SH set at U K(n 0) = 140, 160, 180 MeV. Fig. 11 The neutrino emissivity of durca process with nucleons in neutron stars with the mass of 2.0469M for GL91 parameter set. U K(n 0) = 100 MeV. Noted that antikaon condensations, hyperons and resonances can provide neutrino scatting, but for the low magnitude, the trend of the total neutrino emissivity is the same to that of durca process. Neutrino luminosity can be calculated through the total neutrino emissivity, and then the cooling curves can be obtained by the cooling equation. Here we conjecture that hyperons, antikaon condensations, and resonances can hardly make the neutron star cool faster. 4 Summary and Conclusions We have studied K and K 0 condensations in neutron stars in the framework of the RMFT with GL91 set. esides baryon octet, we have also considered resonances. Here we treat both K and K 0 condensations as second-order phase transitions. The results for different neutron star matter compositions, such as nph, nph K, nph K K0, and nphk K0 matter for various depths of optical potential of K from 80 MeV to 160 MeV have been studied. In GL91 set it is found that both K and K 0 mesons can condensate inside the maximum mass neutron stars at the values of 160 MeV U K(n 0 ) 100 MeV. K condensation is chiefly responsible for the softing of the corresponding EOS, which leads to the reduction in the maximum mass of neutron stars. K0 condensation suppresses the density of K condensation, makes the EOS further softer and results in a further reduction in the maximum mass of neutron stars. However, the chief role of K 0 condensation is to reduce the isovector potential. This reduction leads to the formation of an isospin saturated symmetric matter. oth K and K 0 condensations favor the appearances of resonances. K condensation makes the occurrences of resonances earlier, whereas K 0 condensation makes the quartet be close to an isospin symmetry state. With antikaon condensations for U K(n 0 ) 140 MeV, all the quartet can appear well inside the maximum mass stars. The appearances of resonances change the composition and the distribution of particles at high densities. The populations of resonances can enhance K condensation. With the increase of baryon density, the K meson number density increases. Moreover, resonances soften the EOS, and further reduce the maximum mass of neutron stars. However, as the critical densities for resonances are close to the central densities of the maximum mass neutron stars, the reduction is not obvious. With antikaon condensations, the gross properties of neutron stars are very sensitive to the optical potential depth at the normal density. A deeper optical potential depth of K corresponds to a lower critical density of K condensation. On the other hand, the larger the value of U K(n 0 ), the fewer the species of baryon. With the increase of U K(n 0 ), the EOS becomes softer and the corresponding mass of the maximum mass star gets smaller. Considering all the cases we studied, we can conclude that in the cores of some large mass neutron stars, neutron star matter includes rich particle species, such as baryon octet and quartet and antikaon mesons. For example 4U0900 40 is reported to have mass of 1.85±0.3M. [31] It is also found in recent discovery [32] that the mass of neutron stars should be less than 2.2M, and the radius less than 17 km. The results we calculated for U K(n 0 ) 140 MeV are all consistent with those data. On the other hand, a neutron star with the canonical mass

508 DING Wen-o, LIU Guang-Zhou, ZHU Ming-Feng, YU Zi, XU Yan, and ZHAO En-Guang Vol. 54 of about 1.44M, for example PSR 1934 + 16 [33] may include both the K condensation with large U K(n 0 ) and the hyperon degrees of freedom. The results for nph K K0 matter with U K(n 0 ) = 160 MeV are very close to these data. However, the smaller mass stars of 1.35±0.04M [32] were given recently. Maybe in the star besides hyperons and K condensate, a phase transition from baryon to quark may exist. Though the impact of antikaon condensations, hyperons and resonances on the neutrino emission is quite complicated, by the calculation, we find that they can lower the neutrino emissivity of the most effective durca process with nucleons. Thus we conjecture that hyperons, antikaon condensations, and resonances may hardly help the neutron star cool faster. Acknowledgments We would like to thank Lie-Wen Chen for useful correspondence. References [1] N.K. Glendenning, Astrophys. J. 293 (1985) 470. [2] N.K. Glendenning and J. Schaffner-ielich, Phys. Rev. C 60 (1999) 025803. [3] C.H. Lee, G.E. rown, and D.P. Min, Nucl. Phys. A 585 (1995) 401. [4] S. anik and D. andyopadhyay, Phys. Rev. C 66 (2002) 065801. [5] E.E. Kolomeistev and D.N. Voskresensky, Phys. Rev. C 68 (2003) 015803. [6] N.K. Glendenning, Phys. Rep. 342 (2001) 393. [7] J.F. Gu, H. Guo, and X. Li, Phys. Rev. C 73 (2006) 055803. [8] D.. Kaplan and A.E. Nelson, Phys. Lett. 175 (1986) 57. [9] A.E. Nelson and D.. Kaplan, Phys. Lett. 192 (1987) 193. [10] G.E. rown, K. Kubodera, and M. Rho, Phys. Lett. 291 (1992) 355. [11] P.J. Ellis, R. Knorren, and M. Prakash, Phys. Lett. 349 (1995) 11. [12] S. Pal, C.M. Ko, and Z. Lin, Phys. Rev. C 62 (2000) 061903. [13] A. Cieply, E. Friedman, and A. Gal, Nucl. Phys. A 696 (2001) 173. [14] E.E. Kolomeister, C. Hartnack, and H.W. arz, J. Phys. G 31 (2005) s741. [15] W. Scheinast, I. öttcher, and M. Debowsk, Phys. Rev. Lett. 96 (2006) 072301. [16] R.F. Dashen and R. Rajaraman, Phys. Rev. D 10 (1974) 708. [17] T. Muto, T. Tatsumi, and N. Iwamoto, Phys. Rev. D 61 (2000) 083002. [18] S. Kubis, Phys. Rev. C 73 (2006) 015805. [19] W.. Ding, G.Z. Liu, and M.F. Zhu, Astron. & Astrophys. 506 (2009) L13. [20] G.F. urgio and. Link, Nucl. Phys. Proc. Suppl. 165 (2007) 231. [21] J. oguta and A.R. odmer, Nucl. Phys. A 292 (1977) 413. [22] J.M. Lattimer, C.J. Pethick, and M. Prakash, Phys. Rev. Lett. 66 (1991) 2701. [23] N.K. Glendenning and S.A. Moszkowski, Phys. Rev. Lett. 67 (1991) 2414. [24] G.Q. Li, C.H. Lee, and G.E. rown, Phys. Rev. Lett. 79 (1997) 5214. [25] E. Friedman, A. Gal, and J. Mares, Phys. Rev. C 60 (1999) 024314. [26] S. anik and D. andyopadhyay, Phys. Rev. C 64 (2001) 055805. [27] S. Pal, D. andyopadhyay, and W. Greiner, Nucl. Phys. A 674 (2000) 553. [28] J.F. Gu, H. Guo, and R. Zhou, Astrophys. J. 622 (2005) 549. [29] J.R. Oppenheimer and G.M. Volkoff, Phys. Rev. 55 (1939) 347. [30] G.A. Lalazissis, J. König, and P. Ring, Phys. Rev. C 55 (1997) 540. [31] P.C. Joss and S.A. Rappaport, Annu. Rev. Astron. Astrophys. 22 (1984) 537. [32] M.C. Miller, F.K. Lamb, and D. Psaltis, Astrophys. J. 508 (1998) 791. [33] J.H. Taylor and J.M. Weisberg, Astrophys. J. 345 (1989) 434.