Kaon Condensation in Neutron Star Using Relativistic Mean Field Models

Kaon condensation in neutron star using relativistic mean ﬁeld models

S.W. Hong1, C.H. Hyun1, and C.Y. Ryu2 1 Department of Physics and Basic Atomic Energy Research Institute, Sungkyunkwan University, Suwon 440-746, Republic of Korea 2 Research Center for Nuclear Physics (RCNP), Ibaraki, Osaka 567-0047, Japan

c Societ`a Italiana di Fisica / Springer-Verlag 2007

Abstract. We use the modiﬁed quark-meson coupling and the quantum hadrodynamics models to study the properties of neutron star. Coupling constants of both models are adjusted to reproduce the same saturation properties. The onset of kaon condensation in neutron star matter is studied in detail over a wide range of kaon optical potential values. Once the kaon condensation takes place, the population of kaons increases very rapidly, and kaons become the dominant component, possibly making the neutron star matter a kaonic matter if the kaon optical potential is large.

PACS. 97.60.Jd Neutron stars – 14.20.Jn Strange particles

1 Introduction 2Models Recently, the magnitudes of the kaon-nucleus potential The model Lagrangian consists of several terms for the in matter have attracted much attention. Some calcu- octet baryons, exchange mesons, leptons and kaons; Ltot = − lations [1,2] show that the real part of the K -nucleus LB +LM +Ll +LK . In the mean-field approximation, each optical potential UK− is shallow (UK− ≈−50 MeV), but term reads some other calculations suggest that UK− can be as large ∗ ∗ LB = ψ¯B iγ · ∂ − m (σ, σ ) as about −120 MeV [3,4] or even close to −200 MeV [5]. If B B we consider the strangeness degrees of freedom in neutron 0 1 star matter by taking into account hyperon creation −γ gωBω0 + gφBφ0 + gρBτzρ03 ψB, (1) and kaon condensation [6–8], the information on the 2 interaction of hyperons and kaons in matter is essential in 1 2 2 1 2 ∗2 1 2 2 1 2 2 LM = − m σ − m ∗ σ + m ω + m φ determining the equation of state (EoS) of dense nuclear 2 σ 2 σ 2 ω 0 2 φ 0 matter. In view of the above-mentioned ambiguity in the 1 − + m2ρ2 , (2) K -nucleus interaction, we consider in this work the pos- 2 ρ 03 sibility of deep optical potential of kaons in nuclei and ex- L ¯ · − plore the consequences in the composition of neutron star l = ψl(iγ ∂ ml)ψl,, (3) matter and the mass-radius relation of the neutron star. l ∗ ∗ µ ∗ 2 ∗ To treat the dense nuclear matter we employ two dif- LK = DµK D K − mK K K, (4) ferent relativistic mean field models; the modified quark- where B denotes the sum over the octet baryons and l meson coupling (MQMC) model [9] and the quantum stands for the sum over the free electrons and muons. The hadrodynamics (QHD) model [10]. In the MQMC model interactions between the non-strange light quarks (u and the nucleons and hyperons in the baryon octet are treated d) are mediated by σ, ω and ρ mesons, and σ∗ and φ as MIT bags, whereas in QHD they are assumed to be mesons [11] are introduced to take into account the inter- point particles. The parameters of the two models are cali- actions between s quarks. One of the most important dif- brated to produce exactly the same saturation properties. ferences between MQMC and QHD lies in the treatment By comparing the results from the two different models of baryons; the former treats baryons as composite sys- we can investigate the model dependence at high densi- tems of quarks, and the latter assumes baryons as point ties. We find the onset densities of the kaon condensation particles. As a result, the analytic form of the effective and the compositions of matter at high densities are non- mass of a baryon differs greatly as follows. negligibly model dependent. The mass-radius relation of In MQMC, the effective mass of a baryon in matter the neutron star can significantly differ depending on the ∗ ∗ ∗ 2 xq 2 interaction of the kaon in nuclear matter. mB(σ, σ ) can be written as mB = EB − q R , S.W. Hong et al.: Kaon condensation in neutron star using relativistic mean field models

Table 1. Non-strange meson coupling constants for the Table 2. gσK for several UK− values in MQMC and QHD. MQMC (left four columns) and for the QHD (right four q − − − columns) models. gρN in QHD is the same as gρ in MQMC. UK− (MeV) 120 140 160

q q B q −1 MQMC 2.75 3.50 4.25 gσ gω gσ gρ gσN gωN g2(fm ) g3 QHD 2.83 3.61 4.39 1.02.712.277.888.06 8.19 12.1 48.4

We have five meson-kaon coupling constants, gσK, ∗ where EB is the bag energy of a baryon, R the bag radius gωK, gρK , gσ K and gφK . The quark counting rule is em- ployed for gωK and gρK . gσ∗K can be fixed from f0(980) and xq the eigenvalue of the quarks in the bag. The bag en- √ Ωq ZB 4 3 decay, and gφK from the SU(6) relation 2gφK = gππρ. ergy of a baryon is given by EB = q R − R + 3 πR BB, gσ∗K and gφK thus fixed are 2.65 and 4.27, respectively. where BB and ZB are the bag constant and a phe- The remaining coupling constant, gσK, can be related to nomenological constant for the zero-point motion of a the real part of the optical potential of a kaon at the sat- 2 ∗ 2 baryon B, respectively, and Ωq = xq +(Rmq ) ,where uration density through UK− = −(gσKσ + gωKω0). The ∗ q q ∗ mq (= mq − gσσ − gσ∗ σ ) is the effective mass of a quark. resulting gσK values for several UK− are given in Table 2. In the MQMC model, the bag constant BB is assumed Once the coupling constants are determined, one can to depend on density, and we use the extended form of obtain the EoS of neutron star matter by solving self- Ref. [8]. consistently the equation of motion of exchange mesons, In QHD, the effective mass of a baryon is written as charge neutrality condition, baryon number conservation, ∗ ∗ mB = mB − gσBσ − gσ∗Bσ . We require MQMC and and β-equilibrium conditions for baryons and kaons. The QHD to satisfy identically the same saturation proper- mass-radius relation of the neutron star can be obtained ties. To produce the compression modulus within a rea- by inserting the EoS into the Tolman-Oppenheimer- sonable range one needs to include self-interaction terms Volkoff (TOV) equation. QHD 1 3 1 4 of the σ-field represented by Uσ = 3 g2 σ + 4 g3 σ in the Lagrangian for QHD. The meson term in the QHD QHD QHD thus reads LM = LM − Uσ . 3 Results The kaon is treated as a point particle in both MQMC and QHD models in this work, and its effective mass is A particle fraction is defined by the density of a parti- ∗ ∗ cle divided by the baryon density. Numerical results of given as mK = mK −gσKσ−gσ∗K σ . Covariant derivatives in Eq. (4) include interactions with vector mesons through the particle fraction is shown in Fig. 1. The onset den- 1 Dµ = ∂µ + igωKωµ − igφK φµ + i 2 gρK τ · ρµ. A plane wave solution for the kaon field equation gives the dispersion MQMC QHD relation of the anti-kaon 1 1 n n

1 p p ∗ - ωK = mK − gωKω0 + gφK φ0 − gρK ρ03. (5) 0.1 K Σ+ 0.1 2 e- e- |UK |=120 Σ+ µ- µ- Nuclear saturation properties are relatively well 0.01 0.01 Σ0 K- Σ0 Σ- Λ Ξ- Σ- Λ Ξ- Ξ0 known, and the values quoted in the literatures lie in the Ξ0

−3 Relative populations ∼ ∼ 0.001 0.001 range 0.15 0.17 fm for the saturation density ρ0,15 0 2 4 6 8 10 0 2 4 6 8 10 ∼ 1 1 16.3 MeV for the binding energy per nucleon Eb,30 35 n n ∼ MeV for the symmetry energy asym, 200 300 MeV for p p K- + the compression modulus K,and(0.7 ∼ 0.8)mN for the 0.1 0.1 Σ e- e- |UK|=140 0 eﬀective mass of the nucleon. In this work we assume the µ- µ- Σ −3 Σ+ saturation properties ρ0 =0.17 fm , Eb =16.0MeV, 0.01 0.01 ∗ - - 0 - - asym =32.5MeV,K = 285 MeV, and mN =0.78mN . Σ Λ K Σ Σ Λ Ξ Relative populations 0.001 0.001 These conditions determine the coupling constants of non- 0 2 4 6 8 10 0 2 4 6 8 10 1 1 strange mesons to u and d quarks in the MQMC, and to n

n nucleons in the QHD. These coupling constants are sum- p p - marized in Table 1. 0.1 K 0.1 e- e- |UK|=160 For the coupling constants between the non-strange µ- µ- Σ+ Σ+ mesons (σ, ω,andρ) and hyperons we use the quark count- 0.01 0.01 ing rule assuming that the s-quark does not couple to u Σ- Λ Σ- Λ K- Σ0

s s s Relative populations 0.001 0.001 and d quarks. This gives us gσ = gω = gρ =0, and these 0 2 4 6 8 10 0 2 4 6 8 10 relations determine the coupling constants of non-strange ρ / ρ0 ρ / ρ0 mesons to hyperons in both MQMC and QHD. For the Fig. 1. The relative populations of particles in nuclear matter coupling constants of strange meson to hyperons,√ we as- calculated from the MQMC and QHD models are displayed s u,d sume√ SU(6) symmetry, which gives us gσ∗ = 2gσ and in the left and right columns, respectively, for UK− = −120, s u,d − − gφ = 2gω . 140 and 160 MeV.

394 S.W. Hong et al.: Kaon condensation in neutron star using relativistic mean ﬁeld models

sity of the kaon condensation ρcrit decreases as |UK− | in- 400 Without K 400 Without K ∗ 350 |UK| = 120 MeV 350 |UK| = 120 MeV creases. This is expected because a larger gσK makes mK ) ) 3 300 |UK| = 140 MeV 3 300 |UK| = 140 MeV smaller, which then causes the chemical equilibrium of 250 |UK| = 160 MeV 250 |UK| = 160 MeV kaons, ωK = µn − µp fulfilled at lower densities, where 200 200 150 150 µn(p) is the chemical potential of the neutron (proton). 100 100 P ( MeV / fm P ( MeV / fm An interestring feature is that once the kaon is created, 50 50 − 0 0 the density of K increases very rapidly and dominates 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 the particle population over the hyperons and even the H ( MeV / fm3 ) H ( MeV / fm3 ) nucleons. This can be partly attributed to the fact that Fig. 2. The EoS without using any equilibrium condition in the contribution from the ω-meson to the octet baryons the mixed phase. The left figure is from MQMC, and the right is repulsive whereas the contribution from the ω-meson figure is from QHD. The unstable region where the pressure to the kaon is attractive. The ω-meson term in the en- decreases with increasing density has to be remedied by either ergy of K− in Eq. (5) has a negative sign and is thus Maxwell or Gibbs condition. attractive, but it is repulsive for the chemical potential of 2 2 the octet baryons. The ω-meson enhances the population MQMC QHD − of K but suppresses baryons, and thus the kaon den- 1.5 1.5 sity increases rapidly. In addition, due to the competition sun sun − 1 |UK| = 120 MeV 1 |UK| = 120 MeV between the negatively charged hyperons and K in the / M / M N |UK| = 140 MeV N |UK| = 140 MeV

M |UK| = 160 MeV M |UK| = 160 MeV charge neutrality condition, the negatively charged hyper- 0.5 0.5 ons are highly suppressed and in some cases are not even 0 0 created at all as soon as the kaon condensation sets in. 8 9 10 11 12 13 14 8 9 10 11 12 13 14 Positively charged hyperons, on the other hand, receive Radius (km) Radius (km) the opposite effects from the kaon condensation, and Σ+ Fig. 3. The mass-radius relation from the MQMC (left) and is created at lower densities as |UK− | increases. The pro- QHD (right) models. Both figures are obtained with Maxwell ton density is also enhanced by large abundance of K−, construction. which facilitates in turn the enhancement of Σ+ population through the chemical equilibrium condition of the positively charged hyperons. plotted in Fig. 2 to avoid too many lines in one figure but should be obvious.) From Fig. 2 we can see that kaon con- We find that the onset density of kaon condensation densation begins at a point where the EoS curve deviates ρcrit from the MQMC model is lower than that from from the solid curve obtained without kaons. The kaon − − − − QHD. For UK = 120, 140, and 160 MeV, ρcrit from condensation makes the EoS softer than the one without MQMC are 5.9ρ0,3.8ρ0 and 3.0ρ0, respectively, while they it. As |UK− | becomes larger, the EoS becomes softer. By are 9.8ρ0,4.3ρ0 and 3.3ρ0 in QHD. The model dependence comparing the two solid curves in the left and in the right | − | of ρcrit becomes less pronounced for a larger UK value. panels we see that the EoS without kaons is softer with Kaon condensation leads to a phase transition from the QHD model, but the effect of kaon condensation is a pure hadronic phase to a kaon condensed one with a more significant in the MQMC model. mixed state of hadron and kaon phases. To deal with such Once the EoS is determined, it is straightforward to a mixed phase one needs to use Gibbs conditions [12]. solve the TOV equation to obtain the mass-radius rela- However, for simplicity, we employ Maxwell condition for tion of a neutron star. Fig. 3 shows the mass-radius curves the mixed phase, which is often used as an approximation calculated by the MQMC (left) and QHD (right) mod- to the Gibbs condition. Though the Maxwell construction els. For UK− = −120 MeV, the effect of kaon conden- may not give us very accurate results particularly for the sation is rather small and negligible particularly for the EoS, it can still provdie us with reasonable results for the QHD case. The maximum mass of a neutron star with mass-radius relation of the neutron star. (The comparison only nucleons is about 2M, as mentioned in the Intro- of the results from the Maxwell construction with those duction. When hyperons are created, the maximum mass from the Gibbs condition will be given elsewhere [13].) of a neutron star decreases about 0.5M. As the depth − Fig. 2 shows the EoS including kaon condensation, but of optical potential for K increases, the effect of kaon calculated without using either Maxwell or Gibbs con- condensation becomes more significant. If UK− = −160 dition. There is an instability region where pressure de- MeV, the maximum mass decreases by about 10 ∼ 12% creases as the density increases. If such an instability exists from the maximum mass without kaon condensation. In in the interior of a neutron star, there is a high pressure re- the case of QHD the maximum mass with UK− = −160 gion on top of a low pressure one. The low pressure region MeV becomes so small as 1.32M that most of the ob- cannot sustain the high pressure region, and the unsta- served masses in Ref. [14] exceeds this (maximum) value. ble region will collapse, and the matter will be rearranged This raises a question whether a very deep optical poten- − to fix the instability. Including such a rearrangement in tial of K can be comptible with the existing observations the calculation of the EoS is equivalent to implmenting of the neutron star mass. We will discuss this again in the Maxwell or Gibbs condition. With Maxwell condition, the next section. unstable part is replaced by a flat line, which is deter- One of the unsettled issues in hypernuclear physics mined by the equal area method. (The flat lines are not is the interaction of Σ hyperon in nuclear medium. Use

395 S.W. Hong et al.: Kaon condensation in neutron star using relativistic mean ﬁeld models

1 1 n n between nucleons can make the EoS stiff at high densi- p p ties. The effect of hard core has been considered in the 0.1 0.1 e- e- high density/temperature limit region [18–20], and it is UK =-120 UK =-160 µ- µ- shown that it makes the deconfined quark phase the most UΣ =+30 UΣ =+30 0.01 0.01 stable state of matter at high density and/or temperature. Λ Ξ- Ξ0 Λ K- Ξ0 Σ+

Relative populations Relative populations One can take into account such effects in the neutron star 0.001 0.001 0 2 4 6 8 10 0 2 4 6 8 10 matter together with exotic states. The investigation is in ρ ρ ρ ρ / 0 / 0 progress. Fig. 4. The particle fraction with repulsive in-medium interaction of Σ hyperonintheQHDmodel.UΣ is fixed as +30 This work was supported by grant No.R01-2005-000-10050-0 MeV, and UK− is chosen as −120 (left) and −160 (right) MeV. from the Basic Research Program of the Korea Science & En- gineering Foundation. Work of CHH was supported by Ko- rea Research Foundation Grant funded by Korea Government of plain quark counting rule for the hyperon-meson cou- (MOEHRD, Basic Research Promotion Fund) (KRF-2005-206- pling constants gives us attractive optical potentials of C00007). Σ hyperons in the range −40 ∼−30 MeV at the saturation density. Some studies of Σ-hypernuclei, however, indicates that the interaction of Σ hyperon in medium is References repulsive and its optical potential at the saturation density takes a positive value [15]. To take into account this 1. A. Ramos and E. Oset, Nucl. Phys. A671, 481 (2000). possibility, we have repeated the calculations using the 2. A. Cieply, E. Friedman, A. Gal and J. Mares, Nucl. Phys. QHD model with the repulsive Σ-hyperon optical poten- A696, 173 (2001). tial of +30 MeV at the saturation density and show the 3. E. Friedman, A. Gal and C. J. Batty, Nucl. Phys. A579, results in Fig. 4. The particle compositions are displayed 578 (1994). 4. N. Kaiser, P.B. Siegel and W. Weise, Nucl. Phys. A594, for UK− = −120 and −160 MeV. By comparing these two figures with the figures on the right panels of Fig. 1, one 325 (1995). can see that the repulsive Σ-hyperon interaction makes 5. C. J. Batty, E. Friedman, A. Gal, Phys. Rep. 287, 385 (1997). big changes in the population of hyperons, but the onset 6. R. Knorren, M. Prakash and P. J. Ellis, Phys. Rev. C 52, densities and the population of kaons are little affected. 3470 (1995). The maximum masses of a neutron star with UΣ =+30 7. G.-Q. Li, C.-H. Lee and G. E. Brown, Nucl. Phys. A625, − − MeV are 1.52M,1.48M and 1.32M for UK = 120, 372 (1997); Phys. Rev. Lett. 79, 5214 (1997). − − 140 and 160 MeV, respectively. These results deviate 8. S. Pal, M. Hanauske, I. Zakout, H. Stöcker and W. Greiner, from those with attractive Σ-hyperon interaction only by Phys. Rev. C 60, 015802 (1999). about 1.4% at most. 9. X. Jin and B. K. Jennings, Phys. Lett. B374, 13 (1996); Phys. Rev. C 54, 1427 (1996). 10. B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16,1 4 Discussion (1986). 11. J. Schaffner and I. N. Mishustin, Phys. Rev. C 53, 1416 Masses of some pulsars [16,17] that are more recently (1996). measured are reported to lie much above the canonical 12. N. K. Glendenning and J. Schaffner-Bielich, Phys. Rev. value of Ref. [14]. Some of the large masses are in the range Lett. 81, 4564 (1998); Phys. Rev. C 60, 025803 (1999). 13. C. Y. Ryu, C. H. Hyun, S. W. Hong, and B. T. Kim, to of (1.7−2.1) M, though the results depend on the binary be published in Phys. Rev. C. systems. These values are even larger than the maximum 14. S. E. Thorsett and D. Chakrabarty, Astrophys. J. 512, 288 mass of a neutron star composed of only nucleons and (1999). hyperons (in short, a baryon star). If kaon condensate 15. P. K. Saha et al., Phys. Rev. C 70, 044613 (2004). states are added to the baryonic state, it will soften the 16. D. J. Nice, E. M. Splaver, I. H. Stairs, O. Löhmer, A. EoS and reduce the maximum mass further down. Then Jessner, M. Kramer, and J. M. Cordes, Astrophys. J. 634, one confronts the question: Are exotic states such as kaon 1242 (2005). condensation ruled out by large mass neutron stars? To 17. D. Page and S. Reddy, Ann. Rev. Nucl. Part. Sci. 56, 327 answer the question, we have to understand better the (2006). interaction of particles and state of matter at densities 18. D. H. Rischke, M. I. Gorenstein, H. Stöcker and W. above the nuclear saturation. For instance, we don’t know Greiner, Z. Phys. C 51, 485 (1991). yet to what densities our mean field lagrangian models 19. S. Kagiyama, A. Nakamura and T. Omodaka, Z. Phys. C are valid. If the large mass neutron star does exist, we 53, 163 (1992). have to have a mechanism which makes the EoS at high 20. J. Cleymans, J. Stalnacke and E. Suhonen, Z. Phys. C 55, densities stiffer than now. Introducing hard core repulsion 317 (1992).

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