Equation of state for hybrid stars with strangeness
EEquation of state for hybrid stars with strangeness
Tsuyoshi Miyatsu ∗ , Takahide Kambe, and Koichi Saito Department of Physics, Faculty of Science and Technology, Tokyo University of Science, Noda278-8510, JapanE-mail: [email protected] , [email protected] Considering the mass constraint from the resent pulsar observations, we study the properties ofneutron stars including hyperons and quarks explicitly. Using the chiral quark-meson couplingmodel with relativistic Hartree-Fock approximation, the equation of state (EoS) for hadronic mat-ter is calculated by taking into account the strange ( σ ∗ and φ ) mesons as well as the light non-strange ( σ , ω , ρρρ , and πππ ) mesons in SU(3) flavor symmetry. On the other hand, the EoS for quarkmatter is constructed with the simple MIT bag or the flavor-SU(3) Nambu-Jona-Lasinio model,and we investigate the effect of the hadron-quark coexistence on the neutron-star properties, im-posing smooth crossover or Gibbs criterion for chemical equilibrium. The mass-radius relationof a neutron star, as well as physical quantities such as EoSs, particle fractions, and the speed ofsound in matter are presented. We find that, in order to prevent the quark appearance at very lowdensities, the stiff hadronic EoS should be required under both of the hadron-quark crossover andthe first-order phase transition. The 26th International Nuclear Physics Conference11-16 September, 2016Adelaide, Australia ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ nu c l - t h ] F e b quation of state for hybrid stars with strangeness Tsuyoshi Miyatsu
1. Introduction
Neutron stars may be believed to be cosmological laboratories for nuclear matter at extremelylow temperature and high density, because the central density of neutron stars can reach severaltimes higher than the normal nuclear density. The neutron-star properties, especially the mass andradius, are particularly important for determining the equations of state (EoSs) in the core of aneutron star. Thus, the astrophysical observations of pulsars can provide some constraints on theEoS for dense nuclear matter [1–3]. The possibility of the other exotic degrees of freedom arealso expected in the core of a neutron star, such as hyperons [4, 5], quark matter [6, 7], mesoncondensations [8, 9], and/or dark matter [10].Since the precise observations of massive neutron stars with the mass around 2 M (cid:12) [11, 12],it is quite difficult to explain them theoretically, because the new degrees of freedom, especiallyhyperons, soften the EoS for neutron stars, and thus, the possible maximum mass of a neutronstar is drastically reduced. Hence, it seems to be important to solve the problem by proposing theneutron-star EoS which can satisfy the nuclear properties and the astrophysical constraints.In the present study, the EoS for neutron stars with strangeness is presented. We first showthe hadronic EoS with hyperons, which can support 2 M (cid:12) neutron stars, using the relativistic mean-field (Hartree) or Hartree-Fock (RHF) approximation in SU(3) flavor symmetry [13–20]. The EoSfor quark matter is then constructed with the simple MIT bag or the flavor-SU(3) Nambu–Jona-Lasinio (NJL) model, and we investigate the effect of the transition between hadron and quarkphases on the neutron-star properties, imposing smooth crossover or Gibbs criterion for chemicalequilibrium [21–23].
2. Hadronic EoS for neutron stars with hyperons
Lagrangian density for uniform hadronic matter is given by L H = L B + L M + L int , (2.1)with the baryon, meson, and interaction terms [17, 18]. We here consider the octet baryons ( B ):proton ( p ), neutron ( n ), Λ , Σ + − , and Ξ − . In addition, the mesons, which are composed of lightquarks ( σ , ω , ρρρ , and πππ ) and the strange quarks ( σ ∗ and φ ), are taken into account. The interactionLagrangian density is given by L int = ∑ B ¯ ψ B (cid:20) g σ B ( σ ) σ + g σ ∗ B ( σ ∗ ) σ ∗ − g ω B γ µ ω µ + f ω B M σ µν ∂ ν ω µ − g φ B γ µ φ µ + f φ B M σ µν ∂ ν φ µ − g ρ B γ µ ρρρ µ · III B + f ρ B M σ µν ∂ ν ρρρ µ · III B − f π B m γ γ µ ∂ µ πππ · III B (cid:21) ψ B , (2.2)where the common mass scale mass, M , is taken to be the free nucleon mass, and III B is the isospinmatrix for baryon B . The σ -, σ ∗ -, ω -, φ -, ρ -, π - B coupling constants are respectively denotedby g σ B ( σ ) , g σ ∗ B ( σ ∗ ) , g ω B , g φ B , g ρ B , and f π B , while f ω B , f φ B , and f ρ B are the tensor couplingconstants for the vector mesons.In order to consider the baryon structure variation in nuclear matter, we adopt the followingsimple parametrizations for the σ - and σ ∗ - B coupling constants, which are based on the chiral1 quation of state for hybrid stars with strangeness Tsuyoshi Miyatsu P ( M e V /f m ) ε (MeV/fm ) P = ε CQMC (RHF-SU3, 269)CQMC (RH-SU3, 309)CQMC (RH, 309)QMC (286)QHD+NL (286) M / M O • Radius (km)
CQMC (RHF-SU3)CQMC (RH-SU3)CQMC (RH)QMCQHD+NL
Figure 1:
EoS for neutron stars with hyperons (left panel) and mass-radius relation (right panel). The labeldenoted by “SU3” is for the case in SU(3) flavor symmetry with relativistic Hartree (RH) or Hartree-Fock(RHF) approximation, while the lines without the label are calculated in SU(6) spin-flavor symmetry withRH approximation. The filled circle shows the point at which a neutron star reaches the maximum mass.The incompressibility of symmetric nuclear matter at ρ is also shown by the last number in each label (leftpanel). quark-meson coupling (chiral QMC, CQMC) model [24–26]: g σ B ( σ ) = g σ B b B (cid:104) − a B ( g σ N σ ) (cid:105) , g σ ∗ B ( σ ∗ ) = g σ ∗ B b (cid:48) B (cid:20) − a (cid:48) B ( g σ ∗ Λ σ ∗ ) (cid:21) , (2.3)where a B , a (cid:48) B , b B , and b (cid:48) B are parameters listed in Refs [13, 14, 19, 21]. In contrast, the nonlinear(NL) self-interaction terms for the σ field are introduced in quantum hadrodynamics (QHD) [27],where the baryons are treated as point-like objects, and the coupling constants for the scalar mesonsare fixed as g σ B ( σ = ) and g σ ∗ B ( σ ∗ = ) in Eq. (2.2).The nucleon coupling constants are determined so as to reproduce the same saturation condi-tion: the binding energy per nucleon ( − . . ρ = .
155 fm − . In addition, the coupling constants for hyperons arealso fixed to simulate the experimental data of hypernuclei in SU(3) flavor symmetry or SU(6)spin-flavor symmetry [13–16].In Fig. 1, we show the EoS for neutron stars with hyperons and the mass-radius relation of aneutron star by solving the Tolman-Oppenheimer-Volkoff (TOV) equation, where leptons ( (cid:96) ) areintroduced to impose the charge neutrality and β equilibrium conditions. We find that, due tothe variation of the quark substructure of baryon in matter, the EoS in the QMC model is slightlystiffer than that in the QHD+NL model at the high energy densities. In addition, the effect ofgluon and pion exchanges between quarks in the CQMC model makes the EoS harder. However,it is impossible to support 2 M (cid:12) neutron stars even using the CQMC model in SU(6) spin-flavorsymmetry. With relativistic Hartree (RH) or Hartree-Fock (RHF) approximation in SU(3) flavorsymmetry, the maximum mass can exceed the astrophysical mass constraint, because the additionalrepulsive force due to the φ meson makes the EoS stiff, even if hyperons are taken into accountin the core. We note that, though the EoSs in SU(3) symmetry can satisfy the 2 M (cid:12) constraint2 quation of state for hybrid stars with strangeness Tsuyoshi Miyatsu from the astrophysical observations, the incompressibility of symmetric nuclear matter at ρ withthe RHF approximation shows a more reasonable value, K =
269 MeV, than that with the RHapproximation, K =
309 MeV. Furthermore, we find that the Fock contribution reduces the radiusof a neutron star maximally by about 1 km.
3. Quark matter description and hadron-quark coexistence
For a description of quark matter, we use two models: one is the simple MIT bag model withthe density-dependent bag constant, which is assumed to be given by a Gaussian parametrization, B ( ρ ) = B ∞ + ( B − B ∞ ) exp (cid:34) − β (cid:18) ρρ (cid:19) (cid:35) , (3.1)with ρ and β being the total baryon density and a adjustable parameter, respectively [21]. Theother is the flavor-SU(3) Nambu–Jona-Lasinio (NJL) model, L NJL = ¯ q (cid:0) i γ µ ∂ µ − (cid:98) m (cid:1) q + G S ∑ a = (cid:104) ( ¯ q λ a q ) + ( ¯ qi γ λ a q ) (cid:105) − G D [ det f ( ¯q ( + γ ) q ) + det f ( ¯q ( − γ ) q )] − g V ( ¯ q γ µ q ) , (3.2)where the quark field, q i ( i = u , d , s ), has three colors and three flavors with the current quark mass, m i , in the mass matrix, (cid:98) m . We here introduce a phenomenological vector-type interaction, and theparameter set given in Ref. [28] is used.In order to describe the hadron-quark mixture in the core, we consider the first-order phasetransition under β -equilibrium with the so-called Gibbs criterion for chemical equilibrium [21], orthe crossover phenomenon [22]. In addition, we use the following three cases for the hadron-quarkcrossover [29–31]: the energy density-baryon density ( ε - ρ ) interpolation, the pressure-baryon den-sity ( P - ρ ) interpolation, and the pressure-energy density ( P - ε ) interpolation. In all cases, we em-ploy the common interpolation functions: f ± ( ρ ) = (cid:20) ± tanh (cid:18) ρ − ¯ ρ Γ (cid:19)(cid:21) or f ± ( ε ) = (cid:20) ± tanh (cid:18) ε − ¯ ε Γ (cid:19)(cid:21) , (3.3)with ¯ ρ ( ¯ ε ) and Γ being the central density (energy density) and width for the crossover region,respectively.
4. Hybrid-star properties
The particle fraction, Y i = ρ i / ρ ( i = B , (cid:96), q ), for hybrid-star matter is presented in Fig. 2. In caseof the first-order phase transition shown in the left panel, the quarks appear at 0.6 fm − , followingthe hyperon creation. Considering the 2 M (cid:12) constraint from the astrophysical observations, theparameter in the density-dependent bag constant in Eq. (3.1) should be β ≤ . quation of state for hybrid stars with strangeness Tsuyoshi Miyatsu -3 -2 -1 -3 -2 -1 ρ (fm -3 ) P a r t i c l e F r a c t i on Y i np Λ Ξ − e − µ − u d s 10 -3 -2 -1 -3 -2 -1 ρ (fm -3 ) P a r t i c l e F r a c t i on Y i n p Λ Ξ − e − µ − ud s Figure 2:
Particle fraction, Y i , for hybrid-star matter. The left panel is for the case of the first-order phasetransition with Gibbs criterion ( β = . P - ρ interpolation with ¯ ρ = ρ NJL0 and Γ = ρ NJL0 . The hadronic EoS in the CQMC model with RH-SU3, as in Fig. 1, is used in both cases. For the quark EoS, we employ the MIT bag (NJL) model in the left(right) panel. The hatched area presents the mixed phase of hadrons and quarks, and the shaded area is thecrossover region. -3 -2 -1 -3 -2 -1 -3 -2 -1 -3 -2 -1 ρ (fm -3 ) P a r t i c l e F r a c t i on Y i n p Λ Ξ − Ξ e − µ − u ds M / M O • Radius (km) only hadron core1st order phase transition ( β = 0.025) ε - = 1100 (MeV fm -3 ), Γ = 110 (MeV fm -3 ) ε - = 1430 (MeV fm -3 ), Γ = 190 (MeV fm -3 ) ε - = 1790 (MeV fm -3 ), Γ = 270 (MeV fm -3 ) Figure 3:
Particle fraction, Y i , for hybrid-star matter under crossover phenomenon in the P - ε interpolation(left panel), and mass-radius relation for a hybrid star (right panel). We here adopt the CQMC model withRH-SU3 for the hadronic EoS and the flavor-SU(3) NJL model with g V = . G S for the quark EoS in bothpanels. The top (middle) [bottom] figure in the left panel is for the case with ¯ ε = and Γ =
110 (190) [270] MeV/fm . densities, using the typical range for the crossover phenomenon in the P - ρ interpolation, namely¯ ρ = ρ NJL0 and Γ = ρ NJL0 ( ρ NJL0 = .
17 fm − ) [29, 30]. Here the number density of particle species, ρ i , is defined as ρ i = f − ( ρ ) ρ HM i + f + ( ρ ) ρ QM i with ρ HM i ( ρ QM i ) being that in hadronic (quark)matter. Even if we use the soft hadronic EoS, the 2 M (cid:12) constraint can be easily satisfied by usingthe NJL model for quark matter, because of the strong repulsive force due to the phenomenological4 quation of state for hybrid stars with strangeness Tsuyoshi Miyatsu d P / d ε ρ (fm -3 ) ε - = 1100 (MeV fm -3 ), Γ = 110 (MeV fm -3 ) ε - = 1430 (MeV fm -3 ), Γ = 190 (MeV fm -3 ) ε - = 1790 (MeV fm -3 ), Γ = 270 (MeV fm -3 ) Figure 4:
Speed of sound as a function of baryon density in the P - ε interpolation with g V = . G S . vector-type interaction.We next study how the crossover region affects the hybrid-star properties. In order to preventthe quark creation at low densities, the crossover window should be moved to higher densities. Inthe left panel of Fig. 3, the particle fraction for hybrid-star matter in the P - ε interpolation with g V = . G S is presented. The central energy density and width for the crossover window are heredetermined so as to achieve the appearance of quarks at 0 . − . As the crossover region goes tohigh densities, the amount of strangeness increases and the Ξ appears at around 1.0 fm − in themiddle and bottom figures of the left panel. It means that the hadronic EoS dominates, and thenthe EoS for hybrid-star matter becomes soft. Thus, the possible maximum mass of a hybrid star isslightly reduced as shown in the right panel of Fig. 3. We note that, in the case with g V ≤ G S , themaximum mass of a hybrid star is smaller than that of a neutron star without quarks.The speed of sound in the P - ε interpolation with g V = . G S is given in Fig. 4. In thiscase, causality and thermodynamics stability are fully satisfied in any crossover window, namely0 < dP / d ε <
1. In addition, we find that these conditions can be satisfied in the P - ε and P - ρ interpolations, if the crossover region is located at high density. However, in the P - ρ interpolation,it is impossible to explain the 2 M (cid:12) constraint form the astrophysical observations.
5. Summary
We have calculated the equation of state (EoS) for neutron stars with hyperons and quarks ex-plicitly, where the nuclear properties and the astrophysical constraints are considered. The hadronicEoS for neutron-star matter has been calculated using the chiral quark-meson coupling (CQMC)model in order to include the effect of internal structure variation of baryons in matter. In addition,not only the light non-strange ( σ , ω , ρρρ , and πππ ) mesons but also the strange ( σ ∗ and φ ) mesonshave been taken into account with relativistic Hartree-Fock (RHF) approximation. The couplingconstants for baryons are also determined so as to reproduce the saturation properties and the ex-perimental data of hypernuclei in SU(3) flavor symmetry or SU(6) spin-flavor symmetry. We havefound that the baryon structure variation in matter, the Fock contribution, and the repulsive force5 quation of state for hybrid stars with strangeness Tsuyoshi Miyatsu due to the φ meson stiffen the EoS for neutron stars, and thus, the possible maximum mass of aneutron star can reach the 2 M (cid:12) constraint from the astrophysical observations, even if we considerthe hyperons in the core.On the other hand, the EoS for quark matter has been constructed with the simple MIT bagmodel with the density-dependent bag constant or the flavor-SU(3) Nambu–Jona-Lasinio (NJL)model with phenomenological vector-type interaction. The effect of hadron-quark coexistence onthe hybrid-star properties have been investigated by imposing the smooth crossover and the first-order phase transition for chemical equilibrium. It has been found that, using the MIT bag modelwith Gibbs criterion, the possible maximum mass of a hybrid star can reach 2 M (cid:12) , if we employthe stiff hadronic EoS. Furthermore, we have also found that, using the hadron-quark crossoverwith the three kinds of interpolations in the NJL model, both of the astrophysical constraints andthe speed of sound in hybrid-star matter can be satisfied only in the pressure-energy density ( P - ε )interpolation, if the crossover window is pushed upwards in order to prevent the creation of muchamount of quarks at very low densities. References [1] F. Weber,
Pulsars as astrophysical laboratories for nuclear and particle physics , IOP, Bristol (1999).[2] N. K. Glendenning,
Compact Stars: Nuclear physics, particle physics, and general relativity , 2nd ed.,Springer-Verlag, New York (2000).[3] J. M. Lattimer and M. Prakash,
Neutron star observations: Prognosis for equation of stateconstraints , Phys. Rept. (2007) 109 [ astro-ph/0612440 ].[4] N. K. Glendenning and S. A. Moszkowski,
Reconciliation of neutron star masses and binding of thelambda in hypernuclei , Phys. Rev. Lett. (1991) 2414.[5] J. Schaffner and I. N. Mishustin, Hyperon rich matter in neutron stars , Phys. Rev. C (1996) 1416[ nucl-th/9506011 ].[6] C. Kettner, F. Weber, M. K. Weigel, and N. K. Glendenning, Structure and stability of strange andcharm stars at finite temperatures , Phys. Rev. D (1995) 1440.[7] F. Weber, Strange quark matter and compact stars , Prog. Part. Nucl. Phys. (2005) 193[ astro-ph/0407155 ].[8] T. Takatsuka, K. Tamiya, T. Tatsumi, and R. Tamagaki, Solidification and Pion Condensation inNuclear Medium , Prog. Theor. Phys. (1978) 1933.[9] N. K. Glendenning and J. Schaffner-Bielich, Kaon condensation and dynamical nucleons in neutronstars , Phys. Rev. Lett. (1998) 4564 [ astro-ph/9810284 ].[10] M. A. Perez-Garcia, J. Silk, and J. R. Stone, Dark matter, neutron stars and strange quark matter , Phys. Rev. Lett. (2010) 141101 [ arXiv:1007.1421 ].[11] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels,
A two-solar-mass neutron starmeasured using Shapiro delay , Nature (2010) 1081 [ arXiv:1010.5788 ].[12] J. Antoniadis et al. , A Massive Pulsar in a Compact Relativistic Binary , Science (2013) 6131[ arXiv:1304.6875 ].[13] T. Miyatsu, M. K. Cheoun, and K. Saito,
Equation of state for neutron stars in SU(3) flavor symmetry , Phys. Rev. C (2013) 015802 [ arXiv:1304.2121 ]. quation of state for hybrid stars with strangeness Tsuyoshi Miyatsu[14] T. Miyatsu, M. K. Cheoun, and K. Saito,
Equation of state for neutron stars: Hyperon mixing inSU(3) flavor symmetry , JPS Conf. Proc. (2014) 013080 [ arXiv:1404.2428 ].[15] S. Weissenborn, D. Chatterjee, and J. Schaffner-Bielich, Hyperons and massive neutron stars: vectorrepulsion and SU(3) symmetry , Phys. Rev. C (2012) 065802; (2014) 019904(E)[ arXiv:1112.0234 ].[16] L. L. Lopes and D. P. Menezes, Hypernuclear matter in a complete SU(3) symmetry group , Phys. Rev. C (2014) 025805 [ arXiv:1309.4173 ].[17] T. Miyatsu, T. Katayama, and K. Saito, Effects of Fock term, tensor coupling and baryon structurevariation on a neutron star , Phys. Lett. B (2012) 242 [ arXiv:1110.3868 ].[18] T. Katayama, T. Miyatsu, and K. Saito,
EoS for massive neutron stars , Astrophys. J. Suppl. (2012)22 [ arXiv:1207.1554 ].[19] T. Miyatsu, S. Yamamuro, and K. Nakazato,
A new equation of state for neutron star matter withnuclei in the crust and hyperons in the core , Astrophys. J. (2013) 4 [ arXiv:1308.6121 ].[20] D. L. Whittenbury, J. D. Carroll, A. W. Thomas, K. Tsushima, and J. R. Stone,
Quark-MesonCoupling Model, Nuclear Matter Constraints and Neutron Star Properties , Phys. Rev. C (2014)065801 [ arXiv:1307.4166 ].[21] T. Miyatsu, M. K. Cheoun, and K. Saito, Equation of state for neutron stars with hyperons and quarksin the relativistic Hartree-Fock approximation , Astrophys. J. (2015) 135[ arXiv:1506.05552 ].[22] T. Kambe, T. Katayama, and K. Saito,
Equation of state for neutron star matter with NJL model andDirac-Brueckner-Hartree-Fock approximation , in proceedings of
NIC Symposium 2016 [ arXiv:1608.06449 ].[23] D. L. Whittenbury, H. H. Matevosyan, and A. W. Thomas, Hybrid stars using the quark-mesoncoupling and proper-time Nambu–Jona-Lasinio models , Phys. Rev. C (2016) 035807[ arXiv:1511.08561 ].[24] S. Nagai, T. Miyatsu, K. Saito, and K. Tsushima, Quark-meson coupling model with the cloudy bag , Phys. Lett. B (2008) 239 [ arXiv:0803.4362 ].[25] T. Miyatsu and K. Saito,
Effect of gluon and pion exchanges on hyperons in nuclear matter , Prog. Theor. Phys. (2010) 1035 [ arXiv:0903.1893 ].[26] K. Saito,
The quark-meson coupling model and chiral symmetry , AIP Conf. Proc. (2010) 238[ arXiv:1004.2763 ].[27] J. Boguta and A. R. Bodmer,
Relativistic Calculation of Nuclear Matter and the Nuclear Surface , Nucl. Phys. A (1977) 413.[28] T. Hatsuda and T. Kunihiro,
QCD phenomenology based on a chiral effective Lagrangian , Phys. Rept. (1994) 221 [ hep-ph/9401310 ].[29] K. Masuda, T. Hatsuda, and T. Takatsuka,
Hadron-Quark Crossover and Massive Hybrid Stars withStrangeness , Astrophys. J. (2013) 12 [ arXiv:1205.3621 ].[30] K. Masuda, T. Hatsuda, and T. Takatsuka,
Hadron-quark crossover and massive hybrid stars , Prog. Theor. Exp. Phys. (2013) 073D01 [ arXiv:1212.6803 ].[31] T. Hell and W. Weise,
Dense baryonic matter: constraints from recent neutron star observations , Phys. Rev. C (2014) 045801 [ arXiv:1402.4098 ].].