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LI Ang(o[)1∗∗, ZUO Wei()2, Schulze H. J.3, Lombardo U.3 1Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005 2 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000 3 INFN-LNS, Via Santa Soﬁa 44, I-95123 Catania, and Department of Physics and Astrophysics, Catania University, Via Santa Soﬁa 64, I-95123, Italy
(Received 7 June 2008) We perform a systematic calculation of the equation of state of asymmetric nuclear matter at ﬁnite temperature within the framework of the Brueckner–Hartree–Fock approach with a microscopic three-body force. When applying it to the study of hot kaon condensed matter, we ﬁnd that the thermal eﬀect is more profound in comparison with normal matter, in particular around the threshold density. Also, the increase of temperature makes the equation of state slightly stiﬀer through suppression of kaon condensation.
PACS: 21. 65. +f, 26. 60. +c, 24. 10. Cn, 97. 60. Jd, 13. 75. Cs
One of the challenging problems in nuclear physics the strangeness content of the proton, which is still is to elucidate the behaviour of nuclear matter under quite controversial.[3−5] Neutrino-trapping may shift high-density and/or high-temperature environments. the kaon threshold to higher densities as compared to Despite the importance of the structure and properties the neutrino-free matter. Also, the presence of other of β-stable matter at extreme densities of several times strange particles (for instance, hyperons or more gen- −3 of normal nuclear matter density (ρ0 ∼ 0.17 fm ), as erally strange-quark degrees of freedom) was found yet, its internal constitution and the equation of state to push the onset of kaon condensation to higher (EoS) are not yet known with certainty. Strangeness densities, even out of the physically relevant density may occur in the form of hadrons (such as hyperons or regime. a K− meson condensation) or in the form of strange Overall most of the previous investigations have quarks. The existence of these strange phases may been carried out for cold matter, and have underesti- have important consequences for the structure of com- mated the dependence of kaon condensation on tem- pact stars, for the cooling dynamic of a remnant star perature. That is not perfectly justiﬁed because ther- after supernova explosion and the subsequent forma- mal eﬀects are found to signiﬁcantly aﬀect the prop- tion of a neutron star. erties of proto-neutron stars (PNS). Therefore in With respect to kaons, Kaplan and Nelson this Letter we extend our previous work[15,16] to hot  showed in tree order with SU(3)L × SU(3)R chi- matter. Our objective here is to investigate the im- ral Lagrangian that at a density around 3ρ0, the pact of a kaon condensation on NS matter at ﬁnite ground state of baryonic matter might contain a Bose– temperature. Einstein condensation of negatively charged kaons. In the kaon condensed phase of neutron star mat- Following that, kaon condensation has prompted ex- ter, the energy density consists of three parts of con- tensive investigations and discussions[2−16] on its im- tributions, i.e. plications for astrophysical phenomena in a neutron ε = ε + ε + ε , (1) star (NS). An improved mechanism for kaon conden- NN lep KN sations going beyond the tree order which is consis- where εNN is the baryonic part, εlep denotes the con- tent with kaon–nuclear interactions was proposed by tribution of leptons, and KN is the contribution from Brown, Kubodera, Rho and Thorsson. This work the kaon–nucleon interaction. as well as others that followed it conﬁrmed that the In the present work, we employ the Brueckner– K K critical density ρc lies in the range 2ρ0 < ρc < Hartree–Fock (BHF) approach for asymmetric nuclear [18−21] 4ρ0, more or less in accord with the ﬁrst prediction matter at ﬁnite temperature to calculate the of Ref. . After a kaon condensation is present, baryonic contribution to the EoS of stellar matter. proton abundance increases dramatically, and even The essential ingredient of the BHF approach is the in- antileptons are allowed to exist. Nevertheless, the teraction matrix G, which satisﬁes the self-consistent critical density depends sensitively on the value of equations
∗Supported by the National Natural Science Foundation of China under Grant No 10575119, the Knowledge Innovation Project of Chinese Academy of Sciences under Grant No KJCX3-SYW-N2, and the Asia-Europe Link project (CN/ASIA-LINK/008(94791)) of the European Commission. ∗∗Email: [email protected] c 2008 Chinese Physical Society and IOP Publishing Ltd 4234 LI Ang et al. Vol. 25 the correlations at T 6= 0 are assumed to be essen- tially the same as at T = 0. This means that the s.p. X 1 G(ρ, xp; ω) = vNN + vNN potential Ui(k) at ﬁnite temperature for the compo- ω − (k1) − (k2) k1k2 nent i is approximated by the one calculated at T = 0.
· |k1k2i[1 − n(k1)][1 − n(k2)]hk1k2| Then the free energy density has the following simpli- ﬁed expression: · G(ρ, xp; ω), (2) X X k2 1 and f = n (k) + U (k) − T s , (6) i 2m 2 i i i i X 0 0 0 0 k U(k; ρ)=Re n(k )hkk |G[ρ; e(k)+e(k )]|kk ia, (3) 0 k ≤kF where the entropy density si for component i is ob- tained from the free gas model with a spectrum where ω is the starting energy; xp = ρp/ρ is the of i(k) (see, for example, Ref. ). It turns out proton fraction, i.e. ρp and ρ the proton and the that the assumed independence is valid to a good total baryon density, respectively; (k) is the single accuracy,[18,19] at least for not too high temperature. particle energy given by (k) = ¯h2k2/(2m) + U(k). The realistic nucleon-nucleon (NN) interaction vNN adopted in the present calculation is the Argonne v18 two-body force supplemented with the microscopic three-body force (TBF) given in the following. At ﬁnite temperature, n(k) is a Fermi distribution. For a given density and temperature, Eqs. (2) and (3) have to be solved self-consistently with the following equation for the auxiliary chemical potentialµ ˜,
X X h (k) − µ˜ i−1 ρ = n(k, T ) = exp + 1 . (4) T k k The microscopic TBF adopted in the present calcula- tions is constructed from the meson-exchange current approach, which has been reduced to an eﬀective two- Fig. 1. Finite temperature EoS for symmetric (left-hand body interaction in r-space as panel) and purely neutron (right-hand panel) matter. The upper panels show the internal energy and the lower pan- 0 0 eﬀ hr1r2|V3 (T )|r1r2i els the free energy per particle as a function of the nucleon density. The temperatures vary from 0 to 50 MeV in steps 1 X Z = Tr n(k, T ) dr dr0 φ∗ (r0 ) of 10 MeV. 4 3 3 n 3 k In Fig.1 we display the EoSs obtained following 0 0 × (1 − η(r13,T ))(1 − η(r23,T )) the above-discussed procedure, for symmetric nuclear 0 0 0 × W3(r1r2r3|r1r2r3)φn(r3) matter and purely neutron matter. In the upper pan- els we display the internal energy per particle, whereas × (1 − η(r13,T ))(1 − η(r23,T )), (5) the lower panels show the free energy as a function of where the trace is taken with respect to the spin and the baryon density, for several values of temperature isospin of the third nucleon. The function η(r, T ) between 0 and 50 MeV. We notice that the free en- is the defect function. Since the defect function is ergy of symmetric matter is a monotonically decreas- directly determined by the solution of the Bethe– ing function of temperature. At T = 0 the free energy Goldstone (BG) equation, i.e. Eq. (2), and depends coincides with the internal energy and the correspond- on temperature. The transformation of the TBF to ing curve is just the usual nuclear matter saturation the above eﬀective interaction entails a self-consistent curve. In contrast, the internal energy is an increasing coupling between the TBF and the Brueckner proce- function of temperature. The eﬀect is less pronounced dure of solving the BG equation. It is obvious from for pure neutron matter due to the larger Fermi en- eﬀ Eq. (5) that the temperature dependence of V3 (T ) ergy of the neutrons at given density. We notice the comes from the Fermi distribution n(k, T ) and the de- results of this microscopic TBF are always stiﬀer than fect function η(r, T ), therefore it depends implicitly those of the phenomenological Urbana TBF. on temperature via density. As to the thermal kaon-nucleon part, we apply the To save computational time and simplify the nu- new formulation constructed by Tatsumi et al. and merical procedure, in the following we introduce the treat ﬂuctuations around the condensate within the so-called frozen correlations approximation, i.e., framework of chiral symmetry. A detailed description No. 12 LI Ang et al. 4235 of the procedure is given in Refs. [27,28]. In addition and charge neutrality conditions leads to the following to the classic kaon contribution, the thermodynamic three coupled equations:[4,5,27,28] potential of the thermal kaon contribution is intro- ρ duced as follows: 0 = f 2 sin θ m2 + (a x + a + 2a )m K 1 p 2 3 s f 2 Z d3p th th −βE+(p) −βE−(p) ρ(1 + xp) 2 ∂(ΩK /V ) ΩK =TV ln 1−e 1−e , (7) − 2µ − µ cos θ + , (2π)3 K 4f 2 K ∂θ (11) and the kaon–nucleon energy density obtained in this way is (θ is the amplitude of the condensation) and θ θ µ = µ − µ = 4(1 − 2x )F sec2 ε = f 2 − µ2 sin2 θ/2 + 2m2 sin2 K n p p sym 2 KN K K 2 2 θ Z d3p − 2a1ms tan , (12) + [E (p)f (E (p)) 2 (2π)3 − B −
+ E+(p)fB(E+(p))] + µK QK , (8) QK + Qe + Qµ = Qp(= xpρ), (13) [4,5] where E± are the zero-momentum kaonic energies where µ represents the chemical potential of the neg- and the kaonic charge density QK is given by atively charged particles. Fsym is a parameter simi- lar with the symmetry energy Esym, and to a good 2 2 2 QK = µK f sin θ + (1 + xp) sin (θ/2)ρ + cos θ accuracy can be deﬁned as Fsym = F/A(xp = 0) − Z 3 d p F/A(xp = 0.5). Also we employ Esym = E/A(xp = · fB(E−(p)) − fB(E+(p)) . (9) (2π)3 0) − E/A(xp = 0.5) as usual. The last two equa- tions are from the chemical equilibrium and charge In the above equations, mK is the mass of kaons, and neutrality conditions, respectively. The EoS and the µK is the chemical potential of kaons; f = 93 MeV is composition of the kaon condensed phase in the chemi- the pion decay constant. We adopt a1ms = −67 MeV, cally equilibrated neutron star matter can be obtained  a2ms = 134 MeV and a3ms = −134, −204, by solving the coupled equations (11), (12), and (13). −310 MeV to perform our numerical calculations. The The critical density for kaon condensation is deter- lepton number density is given by (i = e, µ) mined as the point above which a real solution for the coupled equations can be found. Z d3p Q = 2 [f ( (p)−µ )−f (¯(p)+µ )], (10) i (2π)3 F i i F i i
p 2 2 with i(p) = ¯i(p) = mi + p .
Fig. 3. Energy per baryon as a function of the baryon density for a3ms = −134, −204, −310 MeV at the temper- ature of T = 0 and 30 MeV.
Fig. 2. Relative population (a) and the resulting equa- In the following we present some numerical calcu- tions of state (b) as a function of the baryon density for lations to see how our formalism works and how large a3ms = −204 MeV at the temperature of T = 0, 10, 30, the thermal eﬀects are. In Fig.2, left panel, we display and 50 MeV. the relative particle fractions (including neutrons, pro- One can determine the ground state by minimizing tons, electrons, muons and kaons) as a function of the the total energy density with respect to the conden- baryon density for several values of temperature T = sate amplitude θ while keeping all densities ﬁxed. This 0, 10, 30, and 50 MeV with a3ms = −204. We notice minimization together with the chemical equilibrium that temperature eﬀects inﬂuence the population not 4236 LI Ang et al. Vol. 25 only in the low-density region but in the high-density References region. 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