Neutron stars with Bogoliubov quark-meson coupling model
Aziz Rabhi, Constança Providência, Steven A. Moszkowski, João da Providência, Henrik Bohr
aa r X i v : . [ nu c l - t h ] S e p Neutron stars with Bogoliubov quark-meson coupling model
Aziz Rabhi ∗ University of Carthage, Avenue de la R´epublique BP 77 -1054 Amilcar, Tunisia, andCFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal.
Constan¸ca Providˆencia † and Jo˜ao da Providˆencia ‡ CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal.
Steven A. Moszkowski § UCLA, Los Angeles, CA 90095, USA
Henrik Bohr ¶ Department of Physics, B.307, Danish Technical University, DK-2800 Lyngby, Denmark (Dated: September 14, 2020)A quark-meson coupling model based on the quark model proposed by Bogoliubovfor the description of the quark dynamics is developed and applied to the descriptionof neutron stars. Starting from a su(3) symmetry approach, it is shown that thissymmetry has to be broken in order to satisfy the constraints set by the hypernucleiand by neutron stars. The model is able to describe observations such as two solarmass stars or the radius of canonical neutron stars within the uncertainties presentlyaccepted. If the optical potentials for Λ and Ξ hyperons in symmetric nuclear matterat saturation obtained from laboratory measurements of hypernuclei properties areimposed the model predicts no strangeness inside neutron stars. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] I. INTRODUTION
The study of nuclear matter properties has received, in the recent few decades, much atten-tion. Such investigations are particularly important in connection with nuclear-astrophysics.The recent detection of the gravitational waves GW170817 and the follow-up of the electromag-netic counterpart from a neutron star merger [1–3], together with the simulataneous measure-ment of the radius and mass of the pulsar PSR J0030-0451 by NICER [4, 5] are very importantobservations to constrain the equation of state (EoS) of dense matter. Besides, the two solarmass pulsars PSR J1614-2230 [6, 7], PSR J0348+0432 [8] and MSP J0740+6620 [9] are alsosetting important constraints on the nuclear matter EoS at high densities. In particular, thesemasses put some difficulties on the possible existence of non-nucleonic degrees of freedom, suchas hyperons or quark matter, in the inner core of the NS. In [6], it was even suggested that PSRJ1614-2230 would rule out the appearance of these degrees of freedom inside pulsars. Sincethen many works have shown that the present existing constraints on the high density EoS aresuffciently weak to still allow for the onset of hyperons, quarks or other non-nucleonic degreesof freedom inside two solar mass neutron stars [10–16].In relativistic mean field (RMF) models [17–20], the nucleon-nucleon interaction is describedin terms of the coupling of nucleons, assumed to be point particles, with isoscalar scalar mesons,isoscalar vector mesons, and isovector vector mesons. In order to describe adequately nuclearmatter properties RMF models include either self- and cross-interactions [18, 19] among thesemesons or density dependent couplings [20].There have been attempts, based on the MIT bag model [21], and on the Nambu-Jona-Lasinio (NJL) model [22], to take into account the quark structure of the nucleon, in order toincorporate the meson couplings at a more basic level. Along these lines, the nuclear equationof state (EoS) has been obtained and the properties of nuclear matter have been determined byGuichon, Saito and Thomas [21, 23, 24], and by others [25–28], in the framework of quark-mesoncoupling (QMC) models. Still within the same model, in [29, 30] the authors have studied theeffect of strong magnetic fields on kaon condensation and hyperonic matter, respectively.Recently, nuclear matter has also been investigated in the context of a modified QMC modelbased on the replacement of the nucleon bag by an independent quark potential [31–33]. Moti-vated by the idea of the string tension, Bogoliubov proposed an independent quark model forthe description of the quark dynamics [34]. The phenomenological description of hadronic mat-ter in the spirit of the QMC approach, combined with Bogoliubov’s interesting quark model,has been considered in [35], for non-strange matter, and in [36], for strange matter. We willrefer to the model considered in [35, 36] as the Bogoliubov-QMC model.In the present study, we consider a generalization of the model proposed in [36], where thecouplings of the quarks s to vector bosons have not been explicitly considered. Instead, in [36]it is postulated that the couplings of hyperons to the vector mesons are well constrained by thephenomenological hyperon potentials in nuclear matter. Here, the consequences of consideringthe coupling of the quarks u, d, s to appropriate vector bosons are explicitly investigated. Wediscuss under which conditions it is possible to describe two solar mass stars with a non zerostrangeness content and determine their chemical content.In section II we briefly present the model, in section III the description of hadronic matterwith strangeness is introduced and the β -equilirium equation of state is built. In the sectionIV, we obtain the structure and properties of neutron stars described by the present modelsand discuss the results. Finally some concluding remarks are drawn in the last section. II. THE MODEL
We consider the Hamiltonian h D = − i α · ∇ + β ( κ | r | + m − g qσ σ ) . (1)Here, m is the current quark mass, β and the components α x , α y , α z of α are Dirac matrices, σ denotes the external scalar field, g qσ denotes the coupling of the quark to the σ field and κ denotes the string tension, β = " I − I , α x = " σ x σ x , α y = " σ y σ y , α z = " σ z σ z , where σ x , σ y , σ z are the Pauli matrices. The current quark mass m is taken to be m = 0 for u, d quarks because their constituent mass is assumed to be determined exclusively by the valueof κ . The eigenvalues of h D are obtained by a scale transformation from the eigenvalues of h D = − i α · ∇ + β ( | r | − a ) . We need the lowest positive eigenvalue of h D . We cannot apply the variational principle to h D , because its eigenvalues are not bounded from below, but we can apply the variationalprinciple to the square of the Hamiltonian, h D = −∇ + ( | r | − a ) + iβ α · r | r | . (2)We wish to determine variationally the lowest positive eigenvalue of h D versus a . The varia-tional ansatz should take into account the Dirac structure of the quark wave-function, so thatwe consider the following ansatz,Ψ b,λ = " χiλ ( σ · r ) χ e − ( | r |− a − b ) / , (3)where b, λ are variational parameters, and χ is a 2-spinor. Minimizing the expectation value of h D for Ψ b,λ , the following expression for the quark mass is found, m ( κ, a ) κ = min λ,b h ψ b,λ | h D | ψ b,λ i κ h ψ b,λ | ψ b,λ i = min λ,b K + V + V λ + ( K + V ) λ N + N λ , (4)where N , V , K , N , V , and K are all given in [35].Minimization of Eq. (4) with respect to λ is readily performed, so that m ( κ, a ) κ = 12 min b K + V N + K + V N − s(cid:18) K + V N − K + V N (cid:19) + (cid:18) V √N N (cid:19) . (5)Minimization of the r.h.s. of Eq. (5) with respect to b may be easily implemented. We havefound that in the interval − . < a < .
4, that covers the range of densities we will consider,we may express the groundstate energy, m ( κ, a ) , of h D , with sufficient accuracy, as m ( κ, a ) κ = 2 . − . a + 0 . a − . a − . a + 0 . a + 0 . a . (6)Taking a = g qσ σ/ √ κ for quarks u, d , we get, in the vacuum, the constituent mass of thesequarks equal to 313 MeV, with a = 0 and κ = 37106 . . For the quark s , a = a s = − . g qσ σ/ √ κ reproduces the vacuum constituent mass 504 MeV of this quark.Consequently, the mass M ∗ B of the baryon B is given as follows M ∗ N = M ∗ P = 3 m ( κ, a ) , M ∗ Λ = 2 m ( κ, a ) + m ( κ, a s ) , M ∗ Ξ = m ( κ, a ) + 2 m ( κ, a s ) . As we will discuss in the following, the Σ-hyperons will not be considered, because experimentaldata seem to indicate that the potential of the Σ-hyperon in nuclear matter is quite repulsive[37], so that their appearance is disfavored.
III. HADRONIC MATTER
In order to describe hadronic matter, we introduce the vector-isoscalar ω meson, the vector-isovector b meson and use nuclear matter properties to fix the couplings of these mesons tonucleons.In the present model, the field ω is replaced by a vector field of the η type, in the spirit ofthe reference [38], with structure (¯ uu + ¯ dd + (1 + δ ) ¯ ss ) / p δ ) , where 1 + δ > , sothat the coupling of the ω -meson to the quark s is equal to the coupling to the quarks u, d multiplied by 1 + δ . The parameter δ will be fixed by the potencial U Λ of the Λ-hyperon insymmetric nuclear matter at saturation.In this framework, the energy density is given by E = γ (2 π ) X B, ( B =Σ) Z k FB d k q k + M ∗ B + X l Z k Fl d k p k + M l + 12 m σ σ + 12 m ω ω + 12 m b b , (7)and the thermodynamical potential is given byΦ = γ π X B, ( B =Σ) Z k FB k d k (cid:18)q k + M ∗ B − ( µ − q B λ ) (cid:19) + Z k Fl k d k (cid:16)p k + M l − λ (cid:17) + 12 m σ σ + 12 m ω ω + 12 m b b , (8)where the Lagrange multiplier µ controls the baryon density and λ the electrical charge. Thesources of the fields ω and b respectively ρ , and ρ are given by ρ = γ (2 π ) X B, ( B =Σ) ζ B Z k F B d k, ρ = γ (2 π ) X B, ( B =Σ) η B Z k F B d k, (9)with ζ P = ζ N = 1 , ζ Λ = 1 + δ, ζ Ξ = ζ Ξ − = 1 + 2 δ, (10) η P = 1 , η N = − , η Λ = 0 , η Ξ = 1 , η Ξ − = − . The relation between the fields and the respective sources is given by ω = 3 g qω ρ m ω , b = g qb ρ m b . (11)We start by fixing the free parameter κ of the Bogoliubov model. This is obtained byfitting the nucleon mass M = 939 MeV. Next, the desired values of the neutron effectivemass M ∗ /M = 0 . E B = ǫ/ρ B − M ∗ N = − . K = 315 . R B = 0 . ρ B = 0 . − , are obtained by µ Ν P ( M e V / f m ) δ = 0.0 µ ρ Β /ρ ε ( M e V / f m ) a) b) Figure 1. Pressure versus potential comparing neutron, proton, leptonic matter with hyperonic matterfor δ = 0 . , . , . , . , . δ (panel right), for β -equilibrium nucleonic and hyperonic matter.All EoS obtained for the Bogoliubov-QMC model. setting g qσ = 4 . g qω = g qωN = 9 . g qb = 3 . a = 29MeV and the symmetry energyslope L = 79 .
45 MeV, at saturation density. The value we consider for L is well inside therange of values obtained in [40] from a huge number of experimental data and astrophysicalobservations, L = 58 . ± . K .There is an appreciable mass difference between the hyperons Λ and Σ, which, according to[41–43] is due to an hyperfine splitting. Moreover, it should be kept in mind that the SU (2)symmetry is a very important one. The hyperon Λ is an isosinglet; the nucleon and the Ξare isodoublets; the Σ is an isotriplet. Besides, it is known that the Σ-nucleus potential insymmetric nuclear matter seems to be repulsive [37, 44]. Our bag model does not take intoaccount the mechanism responsible for the above mentioned hyperfine splitting, the Σ and Λhyperons are degenerate, and besides also leads to an attractive optical potential for the Σ. Weovercome this problem by omitting the Σ in sums over B , as explicitly indicated in (9), andin analogous sums in the sequel. We are only performing sums over baryons which are eitherisosinglets or isodoublets. The omission of the Σ-hyperon is in accordance with the generalresult obtained when a repulsive Σ-potential in symmetric matter at saturation density of theorder of 30 MeV is considered [11, 12, 16]: Σ-hyperons are not present inside neutron stars. Y i Y i Y i Y i ρ B (fm -3 )0.0010.010.11 Y i ρ B (fm -3 ) 0.0010.010.11 Y i δ=0.0 δ=0.05δ=0.15δ=0.1δ=0.2 δ=0.25 np Λ Ξ Ξ − µ ea) b)c) d)e) f) Figure 2. Baryonic and leptonic particle fractions as a function of the baryonic density, for severalvalues of the parameter δ . For δ = 0 .
25 the onset of hyperons is shifted to densities above 1.2 fm − .The central baryonic density lies between 0.9 and 1.1 fm − depending on the hyperonic content.Table I. Properties of the stable neutron star with maximum mass, for several values of δ , M max , M bmax , R , E , ρ c , R . , R . , U Λ ( ρ ) annd U Ξ ( ρ ) are respectively, the gravitational and baryonicmasses, the star radius, the central energy density, the central baryonic density, the radius of neutronsstar calculated for 1 . M ⊙ and 1 . M ⊙ , and the optical potentials for a Λ and Ξ-hyperon in symmetricnuclear matter at saturation. δ M max M bmax R E u c = ρ c /ρ R . R . U Λ ( ρ ) U Ξ ( ρ )[ M ⊙ ] [ M ⊙ ] [km] [fm − ] [km] [km] [MeV] [MeV]0 . .
02 2.02 2.34 11.16 6.85 7.293 13.740 13.624 -72.23 -87.770 .
05 2.08 2.43 11.42 6.43 6.882 13.752 13.680 -67.57 -78.450 . .
15 2.20 2.58 11.83 5.84 6.285 13.746 13.698 -52.03 -47.370 . .
25 2.21 2.60 11.84 5.82 6.255 13.746 13.696 -36.49 -16.29npe µ
10 11 12 13 14 15 16R[km]00.20.40.60.811.21.41.61.822.2 M [ M ] δ = 0.0 µ Figure 3. Mass-radius curves obtained from the integration of the TOV equations, for different valuesof the δ parameter. The curves stop at the maximum mass configuration. The family of stars fornucleonic stars constituted by npeµ matter is also represented. Minimization of Φ with respect to k F B leads to q k F B + M ∗ B + 3 g qω ωζ B + g qb b η B = µ − q B λ. (12)The quantity µ − q B λ is usually referred to as the chemical potential of baryon B . Minimizationof Φ with respect to k F e leads to q k F e + M e = λ, (13)so the Lagrange multiplier λ is usually called the electron Fermi energy.Explicitly, for N, Λ , Ξ , (12) reduces to q k F N + M ∗ N + 3 g qω ω + g qb b η N = µ − q N λ, q k F Λ + M ∗ + 3 g qω (1 + δ ) ω = µ, q k F Ξ + M ∗ + 3 g qω (1 + 2 δ ) ω + g qb b η Ξ = µ − q Ξ λ. Then, according to the prescription of [45], we have U Λ := M ∗ Λ − M Λ + 3 g qω (1 + δ ) ω,U Ξ := M ∗ Ξ − M Ξ + 3 g qω (1 + 2 δ ) ω, and it is possible to fix the coupling to the quark s in such a way that a reasonable U Λ , isobtained. We find that a small change of δ leads to big changes of U Λ and U Ξ . However, theEoS is almost insensitive to the value of δ for a wide range of values of U Λ around the properone. This model predicts a competition between negatively charged hyperons and leptons. Thisis natural in view of Bodmer-Witten’s Conjecture [46, 47], according to which the groundstateof baryonic matter at high densities should involve only quarks u, d, s, without leptons.In order to study the structure of neutron stars described by the present model we haveintegrated the Tolman-Oppenheimer-Volkov equations for spherical stars in equilibrium [48, 49].The complete EoS was obtained matching the Baym-Pethcik-Sutherland EoS for the outer crust[50], and the inner crust obtained within a Thomas Fermi description of the non-homogeneousmatter for the NL3 ωρ model with the symmetry energy slope at saturation equal to 77 MeV[51], to the core EoS.For δ = 0, we find that the EoS is too soft. However, a mass of 1.92 solar masses and a radiusof about 11 km are reached in the present model with K = 315 MeV. For δ ≥ .
2, the EoS andthe curve mass vs. radius are almost insensitive to the value of δ . The onset of hyperons occursat a density above ∼ . − and the hyperon fraction is too small. Let us point out that weobtain reasonable values for the hyperon-potentials in symmetric nuclear matter for δ ∼ . δ no hyperons will appear inside neutron stars. A similar conclusion wasobtained by [52] within a microscopic approach that includes three body contributions of theform N N Y . Under these results, two solar mass stars will not contain hyperons because theywill set in at densities of the order of the neturon star central density or above.The canonical star with a mass 1.4 M ⊙ has a mass of the order of 13.7 km, well within thevalues obtained by NICER [4, 5] and other observations [53], and within or just slightly abovethe predition obtained from terrestrial data [54], or the gravitational wave GW170817 [1, 2]detected by LIGO/Virgo from a neutron neutron star merger [55, 56, 58]. We have calculatedthe tidal deformability of a canonical star with a mass 1.4 M ⊙ according to [59]. The resultobtained was Λ . = 936 −
954 depending on the hyperon content, well above the prediction of[2] 70 < Λ . < . may indicate that the symmetry energyis too stiff as discussed in [56, 57], and the inclusion of a non-linear ω − ρ term will soften thesymmetry energy at high densities, and decrease the value of Λ . .Within the present model we are able to describe NS as massive as the pulsar MSPJ0740+6620 [9], in particular, if we constraint the optical potential of the Λ-hyperon in sym-0metric matter to experimental values. Only the hyperon fraction of baryonic matter and thevalue of the optical potential are sensitive to the precise value of δ , for δ ≥ . IV. CONCLUSIONS
In the present study we have developed a QMC model based in the Boguliubov quarkmodel. The nucleons interact via the exchange of a scalar-isoscalar meson, a vector-isoscalarmeson and a vector-isovector meson. The nucleon mass is derived from the energy of the bagwhich includes u , d and s quarks. The parameters introduced at this level are chosen so that thevacuum constituent quarks masses are reproduced. Hadronic matter is described by introducinga vector-isoscalar ω -meson, which also includes a ¯ ss content, and a vector-isovector b -meson.In order to satisfy constraints imposed by neutron stars and hypernuclei it is shown that thecoupling of the ω -meson to the s -quark must be more repulsive than its coupling to the u , and d -quarks, and a parameter that takes this aspect into account has to be introduced, so thatsu(3) symmetry is broken.The couplings of the mesons to the nucleons were fixed so that nuclear matter properties,binding energy, saturation density, incompressibility, symmetry energy and its slope at satu-ration, are adequately described. Once these parameters are fixed, only the parameter thatdefines how repulsive is the coupling of the ω -meson to hyperons, remains to be fixed. Takingthe optical potential of the Λ-hyperon of the order of −
30 MeV as discussed in [37, 60, 61], nohyperons will be present inside a two-solar mass. A similar conclusion has been drawn in [52]where, within an auxiliary field diffusion Monte Carlo algorithm, it was shown that the three-body hyperon-nucleon interaction has an important role in softening the EoS at large densities.Using experimental separation energies of medium-light hypernuclei to constraint the Λ
N N force, they have shown that the onset of hyperons will occur above 0.56 fm − , and concludedthat with the presently available experimental energies of Λ-hypernuclei it is not possible todraw a conclusive statement concerning the presence of hyperons inside neutron stars.The present model predicts for the canonical neutron star a radius that is compatible withobservations and predictionns from the analysis of the GW170817 detection. The tidal de-formability, is however, too large, and this may indicate that the symmetry energy is too stiff.A softer symmetry energy may be generated with the inclusion of a non-linear ω − ρ term in1the model [62]. [1] B. P. Abbott et al. (The Virgo, The LIGO Scientific Collaborations), Phys. Rev. Lett. 119, 161101(2017).[2] B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 121, 161101(2018).[3] B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSatCadmium Zinc Telluride Imager Team, IPN, Insight-Hxmt, ANTARES, Swift, AGILE Team,1M2H Team, Dark Energy Camera GW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA,ASKAP, Las Cumbres Observatory Group, OzGrav, DWF (Deeper Wider Faster Program),AST3, CAASTRO, VINROUGE, MASTER, J-GEM, GROWTH, JAGWAR, CaltechNRAO,TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team, TZAC Consortium, KU, Nordic OpticalTelescope, ePESSTO, GROND, Texas Tech University, SALT Group, TOROS, BOOTES, MWA,CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA, HAWC, Pierre Auger, ALMA, Euro VLBITeam, Pi of Sky, Chandra Team at McGill University, DFN, ATLAS Telescopes, High Time Res-olution Universe Survey, RIMAS, RATIR, SKA South Africa/MeerKAT), Astrophys. J. 848, L12(2017).[4] T. E. Riley, A. L. Watts, et al. Astrophys. J. 887, L21 (2019).[5] M. C. Miller, et al. , Astrophys. J. 887, L21 (2019).[6] P.B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, J. W. T. Hessels, Nature 467, 1081(2010).[7] Z. Arzoumanian et al. , The Astrophysical Journal Supplement Series 235, 37 (2018).[8] J. Antoniadis et al. , Science 340, 6131 (2013).[9] H. T. Cromartie, E. Fonseca, S. M. Ransom, and et al. , Nat. Astron. 4, 72 (2020).[10] I. Bednarek, P. Haensel, J. L. Zdunik, M. Bejger, and R. M´anka, Astron. Astrophys. 543, A157(2012).[11] S. Weissenborn, D. Chatterjee, and J. Schaffner-Bielich, Phys. Rev. C 85, 065802 (2012); 90,019904(E) (2014).[12] S. Weissenborn, D. Chatterjee, and J. Schaffner-Bielich, Nucl. Phys. A914, 421 (2013).[13] G. Colucci and A. Sedrakian, Phys. Rev. C 87, 055806 (2013).[14] C. Providencia and A. Rabhi, Phys. Rev. C 87, 055801 (2013). [15] E. N. E. van Dalen, G. Colucci, and A. Sedrakian, Phys. Lett. B 734, 383 (2014).[16] M. Fortin, C. Providencia, A. R. Raduta, F. Gulminelli, J. L. Zdunik, P. Haensel, and M. Bejger,Phys. Rev. C 94, 035804 (2016).[17] J. D. Walecka, Ann. Phys. 83, 491 (1974).[18] J. Boguta and A. R. Bodmer, Nucl. Phys. A292, 413 (1977).[19] H. M¨uller and B. D. Serot, Nucl. Phys. A606, 508 (1996)[20] S. Typel and H. H. Wolter, Nucl. Phys. A656, 331 (1999)[21] P.A.M. Guichon, Phys. Lett. 200B(1988) 235.[22] W. Bentz, A. W. Thomas, Nucl. Phys. A696, 138 (2001).[23] P.A.M. Guichon, K. Saito, E. Kodionov and A.W. Thomas, Nucl. Phys. A601 (1996) 349.[24] P. A. M. Guichon and A. W. Thomas, Phys. Rev. Lett. 93 (2004) 132502.[25] P. K. Panda, A. Mishra, J. M. Eisenberg, and W. Greiner, Phys. Rev. C 56, (1997) 3134.[26] P.K. Panda, D.P. Menezes and C. Providencia, Phys.Rev. C69 (2004) 025207.[27] P. K. Panda, G. Krein, D. P. Menezes, and C. Providencia, Phys. Rev. C68, (2003) 015201.[28] P.K. Panda, D.P. Menezes and C. Providencia, Phys.Rev. C69 (2004) 025207.[29] P. Yue and H. Shen, Phys. Rev. C 77, 045804 (2008).[30] P. Yue, F. Yang, and H. Shen, Phys. Rev. C 79, 025803 (2009).[31] E.F. Batista, B.V. Carlson, T. Frederico, Nucl. Phys. A 697 (2002) 469.[32] N. Barik, R.N. Mishra, D.K. Mohanty, P.K. Panda and T. Frederico, Phys. Rev. C88 (2013)015206.[33] R.N. Mishra, H. S. Sahoo, P.K. Panda, N. Barik and T. Frederico, Phys. Rev. C92 (2015) 045203.[34] P.N. Bogolyoubov, Ann. Inst. Henri Poincar´e 8 (1968) 163.[35] H. Bohr, S. A. Moszkowski, P. K. Panda, C. Providˆencia and J. da Providˆencia, Int. J. Mod.Phys. E 25, (2016) 1650007.[36] P. K. Panda, C. Providˆencia, S. Moszkowski and H. Bohr, Int. J. Mod. Phys. E 28, (2019) 1950034.[37] A. Gal, E. V. Hungerford, and D. J. Millener, Rev. Mod. Phys. 88, 035004 (2016).[38] S. L. Glashow, Phys. Rev. Lett. 11 (1963) 48.[39] J. R. Stone, N. J. Stone, and S. A. Moszkowski Phys. Rev. C 89, 044316 (2014).[40] M. Oertel, M. Hempel, T. Kl¨ahn, and S.Typel Rev. Mod. Phys. 89, 015007 (2017).[41] A. Le Yaouanc, L. Oliver, O. P`ene, and J. C. Raynal, “Phenomenological SU (6) breaking ofbaryon wave functions and the chromodynamic spin-spin force”, Phys. Rev. D 18 (1978) 1591.[42] N. Isgur and G. Karl, “Ground State Baryons in a Quark Model with Hyperfine Interactions”3