In-medium properties of the low-lying strange, charm, and bottom baryons in the quark-meson coupling model
aa r X i v : . [ h e p - ph ] J a n LFTC-18-13/34In-medium properties of the low-lying strange, charm, and bottom baryonsin the quark-meson coupling model
K. Tsushima Laborat´orio de F´ısica Te´orica e Computacional-LFTC,Universidade Cruzeiro do Sul, 01506-000, S˜ao Paulo, SP, Brazil
In-medium properties of the low-lying strange, charm, and bottom baryons in symmetric nuclearmatter are studied in the quark-meson coupling (QMC) model. Results for the Lorentz-scalareffective masses, mean field potentials felt by the light quarks in the baryons, in-medium bag radii,and the lowest mode bag eigenvalues are presented for those calculated using the updated data. Thisstudy completes the in-medium properties of the low-lying baryons in symmetric nuclear matter inthe QMC model, for the strange, charm and bottom baryons which contain one or two strange,one charm or one bottom quarks, as well as at least one light quark. Highlight is the predictionof the bottom baryon Lorentz-scalar effective masses, namely, the Lorentz-scalar effective mass ofΣ b becomes smaller than that of Ξ b at moderate nuclear matter density, m ∗ Σ b < m ∗ Ξ b , although invacuum m Σ b > m Ξ b . We study further the effects of the repulsive Lorentz-vector potentials on theexcitation (total) energies of these bottom baryons. I. INTRODUCTION
The study of baryon properties in a nuclear medium, especially for the baryons which contain charmand/or bottom quarks is very interesting [1–8], due to the emergence of heavy-quark symmetry also inthe baryon sector [9–11]. The existence of heavy quarks in hadrons makes it simpler to treat them inmany cases, e.g., one can treat them in a nonrelativistic framework with effective potentials such asnonrelativistic QCD [12, 13]. In particular, in-medium properties of heavy baryons which contain at leastone light u or d quarks, can provide us with important information on the dynamical chiral symmetrybreaking, and the roles of light quarks in partial restoration of chiral symmetry [14–16]. Despite ofthe importance, theoretical studies for the in-medium properties of heavy baryons do not seem to existmany [16–18], probably because the lack of models and/or methods which are simple enough to handleeasily.To study the in-medium properties of heavy baryons, we rely here on the quark-meson coupling (QMC)model, a quark-based model of nuclear matter, finite nuclei and hadron properties in a nuclear medium.The model was invented by Guichon [19]. (For other variants of the QMC model, see Ref. [14].) The QMCmodel has successfully been applied for various studies of the properties of finite (hyper)nuclei [20–30],hadron properties in a nuclear medium [31–36], reactions involving nuclear targets [37–45], and neutronstar structure [46, 47]. Self-consistent exchange of the Lorentz-scalar-isoscalar σ -, Lorentz-vector-isoscalar ω -, and Lorentz-vector-isovector ρ -mean fields, directly couple to the light quarks u and d , is the keyfeature of the model to be able to achieve the novel saturation properties of nuclear matter with a simpleand systematic treatment. All the relevant coupling constants of the σ -light-quark, ω -light-quark, and ρ -light-quark in any hadrons, are the same as those in nucleon, those fixed by the nuclear matter saturationproperties. The physics behind of this simple picture may be supported by the fact that the light-quarkcondensates reduces/changes faster than those of the strange and heavier quarks in finite density as thenuclear density increases [48, 49]. Or, partial restoration of chiral symmetry in nuclear medium is mainlydriven by the decrease in the magnitude of the light quark condensates. This is modeled in the QMCmodel by the fact that the scalar-isoscalar σ -, vector-isoscalar ω -, and vector-isovector ρ -mean fieldscouple directly only to the light quarks, but not to the strange nor heavier quarks.The present article completes the studies for the low-lying baryon properties in symmetric nuclearmatter in the QMC model with some updates. In particular, highlight is on the bottom baryon Lorentz-scalar effective masses in nuclear medium. Detailed results are presented explicitly, where many of themhave not been presented before [14, 15].We predict that the Lorentz-scalar effective mass of Σ b becomes smaller than that of Ξ b at moderatenuclear matter density, namely, m ∗ Σ b < m ∗ Ξ b , although m Σ b > m Ξ b in vacuum. We study further the effectsof the repulsive Lorentz-vector potentials on the excitation (total) energies of these bottom baryons,by considering two different possibilities for the vector potentials, one is extracted by the Λ and Σhypernucear experimental observation, the one which includes effective Pauli potentials based on thePauli-principle at the quark level, and the other is the vector potentials that are predicted by the QMCmodel without the effective
Pauli potentials . II. FINITE (HYPER)NUCLEUS IN THE QMC MODEL
In order to make this article self-contained, we briefly review the QMC model following Ref. [14, 15]with minor improvements for better understanding.Although Hartree-Fock treatment is possible within the QMC model [50], the main features of theresults, especially the density dependence of total energy per nucleon (nuclear matter energy density)is nearly identical as that of the Hartree approximation. Then, it is sufficient to rely on the Hartreeapproximation in this study. (See Ref. [46] for a detailed study made for the neutron star structure basedon the QMC model with the Hartree-Fock treatment.)Before discussing the heavy baryon properties in symmetric nuclear matter, we start by the case offinite (hyper)nucleus. Using the Born-Oppenheimer approximation, a relativistic Lagrangian densitywhich gives the same mean-field equations of motion for a nucleus or a hypernucleus, may be given in theQMC model [14, 15, 25] below, where the quasi-particles moving in single-particle orbits are three-quarkclusters with the quantum numbers of a nucleon, strange, charm or bottom hyperon when expanded tothe same order in velocity [20, 21, 25, 28, 30, 36]: L YQMC = L NQMC + L YQMC , (1) L NQMC ≡ ψ N ( ~r ) (cid:20) iγ · ∂ − m ∗ N ( σ ) − ( g ω ω ( ~r ) + g ρ τ N b ( ~r ) + e τ N ) A ( ~r ) ) γ (cid:21) ψ N ( ~r ) −
12 [( ∇ σ ( ~r )) + m σ σ ( ~r ) ] + 12 [( ∇ ω ( ~r )) + m ω ω ( ~r ) ]+ 12 [( ∇ b ( ~r )) + m ρ b ( ~r ) ] + 12 ( ∇ A ( ~r )) , (2) L YQMC ≡ ψ Y ( ~r ) (cid:2) iγ · ∂ − m ∗ Y ( σ ) − ( g Yω ω ( ~r ) + g Yρ I Y b ( ~r ) + eQ Y A ( ~r ) ) γ (cid:3) ψ Y ( ~r ) , ( Y = Λ , Σ , ± , Ξ , − , Λ + c , Σ , + , ++ c , Ξ , + c , Λ b , Σ , ± b , Ξ , − b ) , (3)where, for a normal nucleus, L YQMC in Eq. (1), namely Eq. (3) is not needed, but for the following studywe do need this. In the above ψ N ( ~r ) and ψ Y ( ~r ) are respectively the nucleon and hyperon (strange,charm or bottom baryon) fields. The mean-meson fields represented by, σ, ω and b which directly coupleto the light quarks self-consistently, are the Lorentz-scalar-isoscalar, Lorentz-vector-isoscalar and thirdcomponent of Lorentz-vector-isovector fields, respectively, while A stands for the Coulomb field.In an approximation where the σ -, ω - and ρ -mean fields couple only to the u and d light quarks, thecoupling constants for the hyperon appearing in Eq. (3) are obtained/identified as g Yω = ( n q / g ω , and g Yρ ≡ g ρ = g qρ , with n q being the total number of valence light quarks in the hyperon Y , where g ω and g ρ are the ω - N and ρ - N coupling constants. I Y and Q Y are the third component of the hyperon isospinoperator and its electric charge in units of the proton charge, e , respectively.As mentioned already, the approximation adopted in the QMC model, that the meson fields couple onlyto the light quarks, reflects the fact that the magnitudes of the light-quark condensates decrease fasteras increasing the nuclear density than those of the strange and heavy flavor quarks. This is associatedwith partial restoration of chiral symmetry in a nuclear medium (dynamically symmetry breaking and itspartial restoration). The dynamical symmetry breaking and its restoration can provide us with importantinformation on the origin of the (dynamical) masses of hadrons which we observe in our universe.The field dependent σ - N and σ - Y coupling strengths respectively for the nucleon N and hyperon Y , g σ ( σ ) ≡ g Nσ ( σ ) and g Yσ ( σ ) appearing in Eqs. (2) and (3), are defined by m ∗ N ( σ ) ≡ m N − g σ ( σ ) σ ( ~r ) , (4) m ∗ Y ( σ ) ≡ m Y − g Yσ ( σ ) σ ( ~r ) ( Y = Λ , Σ , Ξ , Λ c , Σ c , Ξ c , Λ b , Σ b , Ξ b ) , (5)where m N ( m Y ) is the free nucleon (hyperon) mass. Note that the dependence of these coupling strengthson the applied scalar field ( σ ) must be calculated self-consistently within the quark model [19, 20, 25, 28,29, 36]. Hence, unlike quantum hadrodynamics (QHD) [51, 52], even though g Yσ ( σ ) /g σ ( σ ) may be 2/3or 1/3 depending on the number of light quarks n q in the hyperon in free space, σ = 0 (even this is trueonly when their bag radii in free space are exactly the same in the standard QMC model with the MITbag), this will not necessarily be the case in a nuclear medium.The Lagrangian density Eq. (1) [or (2) and (3)] leads [lead] to a set of equations of motion for the finite(hyper)nuclear system:[ iγ · ∂ − m ∗ N ( σ ) − ( g ω ω ( ~r ) + g ρ τ N b ( ~r ) + e τ N ) A ( ~r ) ) γ ] ψ N ( ~r ) = 0 , (6)[ iγ · ∂ − m ∗ Y ( σ ) − ( g Yω ω ( ~r ) + g ρ I Y b ( ~r ) + eQ Y A ( ~r ) ) γ ] ψ Y ( ~r ) = 0 , (7)( −∇ r + m σ ) σ ( ~r ) = − (cid:20) ∂m ∗ N ( σ ) ∂σ (cid:21) ρ s ( ~r ) − (cid:20) ∂m ∗ Y ( σ ) ∂σ (cid:21) ρ Ys ( ~r ) , ≡ g σ C N ( σ ) ρ s ( ~r ) + g Yσ C Y ( σ ) ρ Ys ( ~r ) , (8)( −∇ r + m ω ) ω ( ~r ) = g ω ρ B ( ~r ) + g Yω ρ YB ( ~r ) , (9)( −∇ r + m ρ ) b ( ~r ) = g ρ ρ ( ~r ) + g Yρ I Y ρ YB ( ~r ) , (10)( −∇ r ) A ( ~r ) = eρ p ( ~r ) + eQ Y ρ YB ( ~r ) , (11)where, ρ s ( ~r ) ( ρ Ys ( ~r )), ρ B ( ~r ) = ρ p ( ~r )+ ρ n ( ~r ) ( ρ YB ( ~r )), ρ ( ~r ) = ρ p ( ~r ) − ρ n ( ~r ), ρ p ( ~r ) and ρ n ( ~r ) are the nucleon(hyperon) scalar, nucleon (hyperon) baryon, third component of isovector, proton and neutron densitiesat the position ~r in the (hyper)nucleus. On the right hand side of Eq. (8), − [ ∂m ∗ N ( σ ) /∂σ ] ≡ g σ C N ( σ )and − [ ∂m ∗ Y ( σ ) /∂σ ] ≡ g Yσ C Y ( σ ), where g σ ≡ g σ ( σ = 0) and g Yσ ≡ g Yσ ( σ = 0) hereafter all in this article,are the key ingredients of the QMC model. Note that, when there is σ -dependence, they will be explicitlywritten by g σ ( σ ) and g Yσ ( σ ) to avoid confusion. At the hadronic level, the entire information of the quarkdynamics is condensed in the effective couplings C N,Y ( σ ) of Eq. (8), which characterize the features ofthe QMC model, namely, scalar polarisability . Furthermore, when C N,Y ( σ ) = 1, which corresponds to astructureless nucleon or hyperon, the equations of motion given by Eqs. (6)-(11) can be identified withthose derived from naive QHD [51, 52].We note that, for the Dirac equation Eq. (7) for the hyperon Y , we include the effects due to thePauli blocking at the quark level by adding repulsive potentials based on the study made for the strangehyperons Λ , Σ, and Ξ. The net, repulsive “Pauli potentials”, which may be interpreted as also includingthe Λ N − Σ N channel coupling effect, was extracted by the fit to the Λ- and Σ-hypernuclei taking intoaccount the Σ N − Λ N channel coupling [25]. Of course, the effects of the channel coupling are expectedto be smaller for the corresponding charm and bottom baryons, since the corresponding mass differencesfor these cases are larger than that for the Λ and Σ hyperons. Thus, for the interesting case of theΣ b − Ξ b baryon system focused on later, we study two possibilities of the vector potentials, with andwithout including the effective Pauli potentials. The modified Dirac equation for Y = Λ , Σ , Ξ , Λ c,b , Σ c,b and Ξ c,b is, [ iγ · ∂ − M Y ( σ ) − ( λ Y ρ B ( ~r ) + g Yω ω ( ~r ) + g ρ I Y b ( ~r ) + eQ Y A ( ~r ) ) γ ] ψ Y ( ~r ) = 0 , (12)where λ Y ρ B ( ~r ) is the effective Pauli potential for the hyperon Y , with ρ B ( ~r ) being the baryon densityat the position ~r in the corresponding hypernucleus. The values of λ Y for Y = (Λ , Λ c,b ), and (Σ , Σ c,b )are respectively 60.25 MeV (fm) and 110.6 MeV (fm) , while for Y = Ξ and Ξ c,b , λ Y is (1 / × . based on the valence light-quark number. For the details of the effective Pauli potentials atthe quark level, see Ref. [25].The effective masses of the nucleon N ( m ∗ N ) and hyperon Y ( m ∗ Y ) are calculated later by Eq. (25)(by replacing h → N , and h → Y , respectively there). The explicit expressions for C N,Y ( σ ) ≡ S N,Y ( σ ) /S N,Y ( σ = 0) ( S N,Y ( σ ) to be defined next) and the effective masses m ∗ N,Y are related by, ∂m ∗ N,Y ( σ ) ∂σ = − n q g qσ Z bag d yψ q ( ~y ) ψ q ( ~y ) ≡ − n q g qσ S N,Y ( σ ) = − [ n q g qσ S N,Y ( σ = 0)] C N,Y ( σ ) = − ∂∂σ (cid:2) g N,Yσ ( σ ) σ (cid:3) , (13)where g qσ is the light-quark- σ coupling constant, and ψ q is the light quark wave function in the nucleon N or hyperon Y immersed in a nuclear medium. By the above relation, we define the σ - N and σ - Y coupling constants, g N,Yσ ≡ n q g qσ S N,Y ( σ = 0) , (14)where g Nσ ≡ g σ = g σ ( σ = 0) appeared already. Note that, as in the case of C N,Y ( σ ), the values of S N ( σ = 0) and S Y ( σ = 0) are different, because the light-quark wave functions in the nucleon N andhyperon Y are different in vacuum as well as in medium; that is, the bag radii of the N and Y aredifferent in both vacuum and medium.The parameters appearing at the nucleon, hyperon and meson Lagrangian level used for the study ofinfinite nuclear matter and finite nuclei [20, 21] are: m ω = 783 MeV, m ρ = 770 MeV, m σ = 550 MeV and e / π = 1 / . III. BARYON PROPERTIES IN SYMMETRIC NUCLEAR MATTER
We consider the rest frame of infinitely large, symmetric nuclear matter, a spin and isospin saturatedsystem with only strong interaction (Coulomb force is dropped as usual). One first keeps only L NQMC in Eq. (1), or correspondingly drops all the quantities with the super- and under-scripts Y , and sets theCoulomb field A ( ~r ) = 0 in Eqs. (6)-(11). Next one sets all the terms with any derivatives of the fieldsto be zero. Then, within the Hartree mean-field approximation, the nuclear (baryon) ρ B , and scalar ρ s densities are respectively given by, ρ B = 4(2 π ) Z d k θ ( k F − | ~k | ) = 2 k F π , (15) ρ s = 4(2 π ) Z d k θ ( k F − | ~k | ) m ∗ N ( σ ) q m ∗ N ( σ ) + ~k . (16)Here, m ∗ N ( σ ) is the value (constant) of the Lorentz-scalar effective nucleon mass at a given nuclear(baryon) density (see also Eq. (4)) and k F the Fermi momentum. In the standard QMC model [19], theMIT bag model is used for describing nucleons and hyperons (hadrons). The use of this quark model isan essential ingredient for the QMC model, namely the use of the relativistic, confined quarks.The Dirac equations for the quarks and antiquarks in nuclear matter, in a bag of a hadron, h , ( q = u or d , and Q = s, c or b , hereafter) neglecting the Coulomb force, are given by ( x = ( t, ~x ) and for | ~x | ≤ bag radius) [32, 34–37], (cid:20) iγ · ∂ x − ( m q − V qσ ) ∓ γ (cid:18) V qω + 12 V qρ (cid:19)(cid:21) (cid:18) ψ u ( x ) ψ ¯ u ( x ) (cid:19) = 0 , (17) (cid:20) iγ · ∂ x − ( m q − V qσ ) ∓ γ (cid:18) V qω − V qρ (cid:19)(cid:21) (cid:18) ψ d ( x ) ψ ¯ d ( x ) (cid:19) = 0 , (18)[ iγ · ∂ x − m Q ] ψ Q ( x ) = 0 , [ iγ · ∂ x − m Q ] ψ Q ( x ) = 0 , (19)where, the (constant) mean fields for a bag in nuclear matter are defined by V qσ ≡ g qσ σ , V qω ≡ g qω ω and V qρ ≡ g qρ b , with g qσ , g qω and g qρ being the corresponding quark-meson coupling constants. We assume SU(2)symmetry, m u, ¯ u = m d, ¯ d ≡ m q, ¯ q . The corresponding Lorentz-scalar effective quark masses are definedby, m ∗ u, ¯ u = m ∗ d, ¯ d = m ∗ q, ¯ q ≡ m q, ¯ q − V qσ . Since the ρ -meson mean field becomes zero, V qρ = 0 in Eqs. (17)and (18) in symmetric nuclear matter in the Hartree approximation, we will ignore it. (This is not true ina finite nucleus with equal and more than two protons even with equal numbers of protons and neutrons,since the Coulomb interactions among the protons induce an asymmetry between the proton and neutrondensity distributions to give ρ ( ~r ) = ρ p ( ~r ) − ρ n ( ~r ) = 0.)The same meson-mean fields σ and ω for the quarks in Eqs. (17) and (18), satisfy self-consistently thefollowing equations at the nucleon level (together with the Lorentz-scalar effective nucleon mass m ∗ N ( σ )of Eq. (4) to be calculated by Eq. (25)): ω = g ω m ω ρ B , (20) σ = g σ m σ C N ( σ ) 4(2 π ) Z d k θ ( k F − | ~k | ) m ∗ N ( σ ) q m ∗ N ( σ ) + ~k = g σ m σ C N ( σ ) ρ s , (21)where C N ( σ ) ≡ − g σ ( σ = 0) [ ∂m ∗ N ( σ ) /∂σ ] . (22)Because of the underlying quark structure of the nucleon used to calculate m ∗ N ( σ ) in nuclear medium, C N ( σ ) decreases as σ increases, whereas in the usual point-like nucleon-based models it is constant, C N ( σ ) = 1. As will be discussed later it can be parametrized in the QMC model as C N ( σ ) = 1 − a N × ( g σ σ ) ( a N > C N ( σ ) (or equivalently dependence of the scalar coupling ondensity, or σ , g σ ( σ )) that yields a novel saturation mechanism for nuclear matter in the QMC model, andcontains the important dynamics which originates from the quark structure of the nucleons and hadrons.It is the variation of this C N ( σ ), which yields three-body or density dependent effective forces, as hasbeen demonstrated by constructing an equivalent energy density functional [24, 53]. As a consequence ofthe derived , nonlinear couplings of the meson fields in the Lagrangian density at the nucleon (hyperon)and meson level, the standard QMC model yields the nuclear incompressibility of K ≃
280 MeV with m q = 5 MeV. This is in contrast to a naive version of QHD [51, 52] (the point-like nucleon model ofnuclear matter), results in the much larger value, K ≃
500 MeV; the empirically extracted value falls inthe range K = 200 −
300 MeV. (See Ref. [54] for an extensive analysis on this issue.)
TABLE I: Current quark mass values (inputs), quark-meson coupling constants and the bag pressure, B p . Notethat the m c value is updated from Refs. [14, 15] based on the data [55]. m u,d g qσ m s
250 MeV g qω m c g qρ m b B / p
170 MeVOnce the self-consistency equation for the σ field Eq. (21) is solved, one can evaluate the total energyper nucleon: E tot /A = 4(2 π ) ρ B Z d k θ ( k F − | ~k | ) q m ∗ N ( σ ) + ~k + m σ σ ρ B + g ω ρ B m ω . (23)We then determine the coupling constants, g σ and g ω at the nucleon level (see also Eq. (14)), by thefit to the binding energy of 15.7 MeV at the saturation density ρ = 0.15 fm − ( k F = 1.305 fm − ) forsymmetric nuclear matter, as well as g ρ to the symmetry energy of 35 MeV. The determined quark-mesoncoupling constants, and the current quark mass values are listed in table I. The coupling constants at thenucleon level are g σ / π = 3 . g ω / π = 5 .
31 and g ρ / π = 6 .
93. (See Eq. (14) for g σ = g Nσ .)We show in Fig. 1 the density dependence of the total energy per nucleon E tot /A − m N (left panel)and the Lorentz-scalar effective quark mass m ∗ q , vector ( V qω ) and scalar ( − V qσ ) potentials felt by the lightquarks (right panel) calculated using the quark-meson coupling constants determined. ρ Β /ρ ( ρ =0.15 fm -3 ) -20-100102030 ( E t o t / A ) - m N ( M e V ) ρ Β /ρ (ρ =0.15 fm -3 ) -300-200-1000100 m * q a nd po t e n ti a l s ( M e V ) V q ω m* q (m q =5 MeV)-V q σ FIG. 1: Total energy per nucleon E tot /A − m N (left panel), and the light-quark Lorentz-scalar effective mass m ∗ q ,vector ( V qω ) and scalar ( − V qσ ) potentials felt by the light quarks. In the following, let us consider the situation that a hadron h (or a hyperon Y ) is immersed in nuclearmatter. The normalized, static solution for the ground state quarks or antiquarks with flavor f in thehadron h may be written, ψ f ( x ) = N f exp − iǫ f t/R ∗ h ψ f ( ~r ), where N f and ψ f ( ~r ) are the normalizationfactor and corresponding spin and spatial part of the wave function. The bag radius in medium for thehadron h , denoted by R ∗ h , is determined through the stability condition for the mass of the hadron againstthe variation of the bag radius [19, 26] (see Eq. (25)). The eigenenergies in units of 1 /R ∗ h are given by, ǫ u ǫ ¯ u ! = Ω ∗ q ± R ∗ h (cid:18) V qω + 12 V qρ (cid:19) , ǫ d ǫ ¯ d ! = Ω ∗ q ± R ∗ h (cid:18) V qω − V qρ (cid:19) , ǫ Q = ǫ Q = Ω Q . (24)The hadron mass in a nuclear medium, m ∗ h (free mass is denoted by m h ), is calculated by m ∗ h = X j = q, ¯ q,Q,Q n j Ω ∗ j − z h R ∗ h + 43 πR ∗ h B p , ∂m ∗ h ∂R h (cid:12)(cid:12)(cid:12)(cid:12) R h = R ∗ h = 0 , (25)where Ω ∗ q = Ω ∗ ¯ q = [ x q + ( R ∗ h m ∗ q ) ] / ( q = u, d ), with m ∗ q = m q − g qσ σ = m q − V qσ , Ω ∗ Q = Ω ∗ Q =[ x Q + ( R ∗ h m Q ) ] / ( Q = s, c, b ), and x q,Q are the lowest mode bag eigenvalues. B p is the bag pres-sure (constant), n q ( n ¯ q ) and n Q ( n Q ) are the lowest mode valence quark (antiquark) numbers for thequark flavors q and Q in the hadron h , respectively, while z h parametrizes the sum of the center-of-massand gluon fluctuation effects, which are assumed to be independent of density [20]. The bag pressure B p = (170 MeV) (density independent) is determined by the free nucleon mass m N = 939 MeV withthe bag radius in vacuum R N = 0 . m q = 5 MeV as inputs, which are considered to be standardvalues in the QMC model [14]. (See also table I.) Concerning the Lorentz-scalar effective mass m ∗ q innuclear medium, it reflects nothing but the strength of the attractive scalar potential as in Eqs. (17)and (18), and thus naive interpretation of the mass for a (physical) particle, which is positive, should notbe applied. The model parameters are determined to reproduce the corresponding masses in free space.The quark-meson coupling constants, g qσ , g qω and g qρ , have already been determined by the nuclear mattersaturation properties. Exactly the same coupling constants, g qσ , g qω and g qρ , will be used for the lightquarks in all the hadrons as in the nucleon. These values are fixed, and will not be changed dependingon the hadrons.In table II we present the inputs, vacuum masses of baryons B , m B , the parameters z B , the calculatedlowest mode bag eigenvalues ( x , x , x ) of the corresponding valence quarks ( q , q , q ) in the baryon B , and the bag radii calculated in vacuum R B , as well as the corresponding quantities at ρ = 0 . − , namely, the Lorentz-scalar effective masses m ∗ B , in-medium bag radii R ∗ B , and the lowest mode TABLE II: The parameters related with the zero-point energy z B , baryon masses and the bag radii in free space[at normal nuclear matter density, ρ = 0 .
15 fm − ] m B (MeV), R B (fm) [ m ∗ B , R ∗ B ], the lowest mode bag eigenvalues x , x , x [ x ∗ , x ∗ , x ∗ ] of baryon B ( q , q , q ) with the corresponding valence quarks q , q , q in the baryon B , where z B s are kept the same as those in vacuum, i.e., density independent. Free space mass values m B for the heavybaryons are from Ref. [55], and those for the strange hyperons are from Ref. [14], as well as the nucleon bagradius R N = 0 . m q = 5 MeV), are inputs. The light quarks are indicated by q = u or d . Note that, thebaryons containing at least one light quark q , are modified in medium in the QMC model, but Ω , Ω c , and Ω b arenot modified in the QMC model. We remind that some inputs are updated from those in Refs. [14, 15] based onthe data [55]. For the recent data for Σ b , see Ref. [56], which give the averaged mass of m Σ b = 5813 . B ( q , q , q ) z B m B R B x x x m ∗ B R ∗ B x ∗ x ∗ x ∗ N ( qqq ) 3.295 939.0 0.800 2.052 2.052 2.052 754.5 0.786 1.724 1.724 1.724Λ( uds ) 3.131 1115.7 0.806 2.053 2.053 2.402 992.7 0.803 1.716 1.716 2.401Σ( qqs ) 2.810 1193.1 0.827 2.053 2.053 2.409 1070.4 0.824 1.705 1.705 2.408Ξ( qss ) 2.860 1318.1 0.820 2.053 2.406 2.406 1256.7 0.818 1.708 2.406 2.406Ω( sss ) 1.930 1672.5 0.869 2.422 2.422 2.422 — — — — —Λ c ( udc ) 1.642 2286.5 0.854 2.053 2.053 2.879 2164.2 0.851 1.691 1.691 2.878Σ c ( qqc ) 0.903 2453.5 0.892 2.054 2.054 2.889 2331.8 0.889 1.671 1.671 2.888Ξ c ( qsc ) 1.445 2469.4 0.860 2.053 2.419 2.880 2408.3 0.859 1.687 2.418 2.880Ω c ( ssc ) 1.057 2695.2 0.876 2.424 2.424 2.884 — — — — —Λ b ( udb ) -0.622 5619.6 0.930 2.054 2.054 3.063 5498.5 0.927 1.651 1.651 3.063Σ b ( qqb ) -1.554 5813.4 0.968 2.054 2.054 3.066 5692.8 0.966 1.630 1.630 3.066Ξ b ( qsb ) -0.785 5793.2 0.933 2.054 2.441 3.063 5732.7 0.931 1.649 2.440 3.063Ω b ( ssb ) -1.327 6046.1 0.951 2.446 2.446 3.065 — — — — —bag eigenvalues, ( x ∗ , x ∗ , x ∗ ). Note that in the QMC model, Ω( sss ) , Ω c ( ssc ) and Ω b ( ssb ) properties arenot modified in medium.One can notice a few things easily in table II: (i) the parameter z B decreases as the vacuum massof the baryon increases, (ii) the in-medium bag radius R ∗ B of the baryon B at ρ decreases than thecorresponding vacuum value, and the decreasing ratio becomes smaller as the vacuum baryon mass valueincreases, and (iii) the lowest mode bag eigenvalues decrease at ρ , and the decreasing magnitude islarger for the light quarks, but tiny for the heavier quarks. Note that, the bag radius is not the physicalobservable, and one must calculate the baryon radius using the corresponding quark wave function. Infact, such calculation shows that the slight increase of the in-medium radius. (See table 2 in Ref. [21].)In Figs. 2, 3, and 4 we show respectively the density dependence of the Lorentz-scalar effective baryonmasses, in-medium bag radii, and the lowest mode bag eigenvalues. In Figs. 2 and 4 each panel respectivelyshows for nucleon and strange baryons (top panel), for charm baryons (bottom-left panel), and for bottombaryons (bottom-right panel).For the Lorentz-scalar effective masses shown in Fig. 2, one can notice a very interesting feature forthe bottom baryons (bottom-right panel). The Lorentz-scalar effective mass of Σ b becomes smaller thanthat of Ξ b , namely m ∗ Σ b < m ∗ Ξ b at baryon density range larger than about 0 . ρ , although vacuum massessatisfy m Σ b > m Ξ b [55] (see table II). This is indeed interesting, and can be understood as follows. The Σ b baryon contains two light quarks, while the Ξ b baryon contains one. Because the light quark condensatesare much more sensitive to the nuclear density change than those of the strange, charm and bottom quarkones, one can expect that the partial restoration of chiral symmetry to take place faster for Σ b than Ξ b as increasing the nuclear density. Or in the QMC model picture, since the scalar potential is roughlyproportional to the number of the valence light quarks [14, 25, 36], the Lorentz-scalar effective mass ofΣ b decreases faster than that of Ξ b as increasing the nuclear matter density.The result of the reverse in the Lorentz-scalar effective masses of Σ b and Ξ b is one of the main predictionsof this article. We must seek how this interesting prediction possibly be connected with experimentalobservables. This would give very important information on the dynamical symmetry breaking and ρ B / ρ (ρ =0.15 fm -3 ) m * B ( M e V ) ΞΣΛ N ρ B / ρ (ρ =0.15 fm -3 ) m * B ( M e V ) Ξ c Σ c Λ c ρ B / ρ (ρ =0.15 fm -3 ) m * B ( M e V ) Ξ b Σ b Λ b FIG. 2: Density dependence of Lorentz-scalar baryon effective masses in symmetric nuclear matter. ρ B / ρ (ρ =0.15 fm -3 ) B a g r a d i u s R * B (f m ) N ΛΞΣΛ c Ξ c Σ c Λ b Ξ b Σ b FIG. 3: Density dependence of in-medium bag radii in symmetric nuclear matter. ρ B / ρ (ρ =0.15 fm -3 ) B a g e i g e nv a l u e x q , Q x u (N)x u,s (Λ) x u,s (Σ) x u,s (Ξ) x s x u ρ B / ρ (ρ =0.15 fm -3 ) B a g e i g e nv a l u e x q , Q x u,c ( Λ c )x u,c ( Σ c )x u,s,c ( Ξ c ) x c x u x s ρ B / ρ (ρ =0.15 fm -3 ) B a g e i g e nv a l u e x q , Q x u,b ( Λ b )x u,b ( Σ b )x u,s,b ( Ξ b ) x b x u x s FIG. 4: Density dependence of the lowest mode bag eigenvalues in symmetric nuclear matter. the partial restoration of chiral (dynamical) symmetry. However, the story is not that straightforward,since the baryons (light quarks) also feel repulsive Lorentz-vector potentials in addition to the attractiveLorentz-scalar potentials. Thus, we must take into account the effects of the repulsive vector potentialsfor considering more realistic/practical experimental situations, and we will study this later.Concerning the in-medium bag radii shown in Fig. 3, one can notice that all the in-medium bag radiidecrease as increasing the nuclear matter density. In particular, the decrease for the nucleon case is thelargest.As for the lowest mode bag eigenvalues shown in Fig. 4, they also decrease as increasing the nuclearmatter density, particularly noticeable for the light quarks, but tiny decreases for the heavier quarks.In connection with the Lorentz-scalar effective baryon masses shown in Fig. 2, it has been found thatthe function C B ( σ ) ( B = N, Λ , Σ , Ξ , Λ c , Σ c , Ξ c , Λ b , Σ b , Ξ b ) (see Eq. (13) and above), can be parameterizedas a linear form in the σ field, g σ σ , for a practical use [20, 21, 25]: C B ( σ ) = 1 − a B × ( g σ σ ) , ( B = N, Λ , Σ , Ξ , Λ c , Σ c , Ξ c , Λ b , Σ b , Ξ b ) . (26)The values obtained for a B are listed in table III. This parameterization works very well up to about threetimes of normal nuclear matter density 3 ρ . Then, the effective mass of baryons B in nuclear matter iswell approximated by: m ∗ B ≃ m B − n q g σ h − a B g σ σ ) i σ, ( B = N, Λ , Σ , Ξ , Λ c , Σ c , Ξ c , Λ b , Σ b , Ξ b ) , (27)with n q being the valence light-quark number in the baryon B . See Eqs. (4) and (5) to compare with g N,Y ( σ ) and the above expression. For the Σ b and Ξ b baryons, n q are respectively two and one in Eq. (27)with a Σ b ≃ a Ξ b from table III. Then, one can confirm that the decrease in the Lorentz-scalar effective0 TABLE III: Slope parameters, a B ( B = N, Λ , Σ , Ξ , Λ c , Σ c , Ξ c , Λ b , Σ b , Ξ b ). Note that the tiny differences in valuesof a B from those in Refs. [14, 15], are due to the differences in the number of data points for calculating a B , butsuch differences in a B give negligible effects. a B × − MeV − a B × − MeV − a B × − MeV − a N a Λ a Λ c a Λ b a Σ a Σ c a Σ b a Ξ a Ξ c a Ξ b b is larger than that for Ξ b as increasing the nuclear matter density, or as increasing the σ mean field. ρ B / ρ (ρ =0.15 fm -3 ) -400-300-200-1000100200300400500600700800 S ca l a r a nd V ec t o r po t o s . ( M e V ) N-vector Σ -vector(+Pauli) Λ -vector(+Pauli) Ξ -vector(+Pauli) Ξ -scalar Σ -scalar Λ -scalarN-scalar ρ B / ρ (ρ =0.15 fm -3 ) -400-300-200-1000100200300400500600700800 S ca l a r a nd V ec t o r po t o s . ( M e V ) Σ c -vector(+Pauli) Λ c -vector(+Pauli) Ξ c -vector(+Pauli) Ξ c -scalar Σ c -scalar Λ c -scalar ρ B / ρ (ρ =0.15 fm -3 ) -400-300-200-1000100200300400500600700800 S ca l a r a nd V ec t o r po t o s . ( M e V ) Σ b -vector(+Pauli) Λ b -vector(+Pauli) Ξ b -vector(+Pauli) Ξ b -scalar Σ b -scalar Λ b -scalar FIG. 5: Attractive Lorentz-scalar and repulsive Lorentz-vector potentials of baryons in symmetric nuclear matter.In figures “vector+Pauli” denote that the effective Pauli potentials are added, and “potos.” in each vertical axisis the abbreviation for “potentials”.
To analyze more carefully the interesting findings for the Σ b and Ξ b baryon Lorentz-scalar effectivemasses, we next discuss the “excitation energies” of baryons, to study the total energies (potentials) in anonrelativistic sense, the Lorentz-scalar plus Lorentz-vector potentials focusing on the Σ b and Ξ b baryons.First, results for the attractive scalar and repulsive vector potentials separately, are shown in Fig. 5,for nucleon and strange baryons (top panel), charm baryons (bottom-left panel), and bottom baryons(bottom-right panel). For the repulsive vector potentials, we show here only one case, the one including1 ρ B / ρ (ρ =0.15 fm -3 ) m * a nd [ m * + V v ] ( M e V ) Σ b (m*+V V ) Ξ b (m*+V V ) Ξ b (m*) Σ b (m*) ρ B / ρ (ρ =0.15 fm -3 ) m * a nd [ m * + V v Q M C ] ( M e V ) Σ b (m*+V VQMC ) Ξ b (m*+V VQMC ) Ξ b (m*) Σ b (m*) FIG. 6: Effective masses and excitation energies (total potentials) of Σ b and Ξ b baryons for the two cases of thevector potentials, with including the Pauli potentials (left panel) and without (right panel). the effective “Pauli potentials” introduced in Eq. (12), denoted by “vector(+Pauli)”. One can see thesimilarity in the amounts of the scalar and vector(+Pauli) potentials among the corresponding strange,charm and bottom sector baryons, namely among those three baryons in each brackets, (Λ , Λ c , Λ b ),(Σ , Σ c , Σ b ) and (Ξ , Ξ c , Ξ b ).Now we show in Fig. 6 Lorentz-scalar effective masses and excitation energies (total energies), Lorentz-scalar effective masses plus vector potentials for the two cases of the vector potentials focusing on Σ b and Ξ b . The left panel is the case with the Pauli potentials, while the right panel is without the Paulipotentials. Recall that, because the mass difference between the Λ b − Σ b system is much larger than thatfor the Λ − Σ and Λ c − Σ c systems, it is expected that the effective Pauli potentials should be smallerfor the Λ b , Σ b and Ξ b baryons than the corresponding strange and charm sector baryons. Thus, one canregard the more realistic case when we consider without the Pauli potentials, shown in the right panel ofFig. 6.We discuss separately the two cases of the vector potentials. First, for the case with the Pauli potentialsshown in the left panel of Fig. 6, the excitation energies (total potentials) for the Σ b and Ξ b never reversein magnitudes, and always the excitation energy of Σ b is larger than that for Ξ b . The smallest excitationenergy difference is about a few tens of MeV, and it is larger for Σ b . For the nuclear matter density largerthan around ρ , the difference in the excitation energies increases.Next, for the case without the Pauli potentials, which may be expected to be more realistic, is shown inthe right panel of Fig. 6. Interestingly, in the nuclear matter density range 0 . ρ < ρ B < . ρ , the twoexcitation energies for Σ b and Ξ b are nearly degenerate. This means that Σ b and Ξ b can be produced atrest with the nearly same costs of energies. This may imply the emergence of many interesting phenomena,for example, in heavy ion reactions and reactions in the systems of dense nuclear medium, such as in adeep core of neutron (compact) star.The results shown in Fig. 6 suggest that the two different types of the vector potentials may possibly bedistinguished, and give important information on the dynamical symmetry breaking and partial restora-tion of chiral symmetry, by studying the heavy bottom baryon properties in medium. For proving thesesuggestions, we have to seek what kind of experiments can be made to get a clue, in particular, for theLorentz-scalar effective masses of Σ b and Ξ b . It might be very interesting to measure the valence quark(parton) distributions of Σ b and Ξ b in medium, since the supports of the parton distributions of thesebaryons reflect their excitation energies. Other possibility may be to measure the strangeness-changingsemi-leptonic weak decay of Ξ b → Σ b in medium, which again reflects the excitation energy difference ofthem in medium.2 IV. SUMMARY AND DISCUSSION
In this article we have completed the study of baryon properties in symmetric nuclear matter in thequark-meson coupling model, for the low-lying strange, charm, and bottom baryons which contain atleast one light quark. We have presented the density dependence of the Lorentz-scalar effective masses,bag radii, the lowest mode bag eigenvalues, and vector potentials for the baryons.We predict that the Lorentz-scalar effective mass of Σ b becomes smaller than that of Ξ b in the nuclearmatter density range larger than ≃ . ρ ( ρ = 0 .
15 fm − ), while in vacuum the mass of Σ b is larger thanthat of Ξ b . We also give parametrization for the Lorentz-scalar effective masses of the baryons treated inthis article as a function of the scalar mean field for a convenient use.We have further studied the effects of the two different repulsive Lorentz-vector potentials to estimatethe excitation (total) energies focusing on Σ b and Ξ b baryons. In the case without the effective Paulipotentials, which is expected to be more realistic, the excitation energies for the Σ b and Ξ b baryons arepredicted to be nearly degenerate in the nuclear matter density range about [0 . ρ , . ρ ]. Thus, theproduction of Σ b and Ξ b baryon cost nearly the same energies at rest in this nuclear matter densityrange, and this may imply many interesting phenomena in heavy ion collisions, and reactions involvingthem in a deep core of neutron (compact) star.To make possible connections of the findings for the Lorentz-scalar effective masses and/or excitationenergies of Σ b and Ξ b baryons with experimental observables, we need to seek relevant experimentalmethods and situations. It might be very interesting to measure the valence quark (parton) distributionsof Σ b and Ξ b in medium, since the supports of the parton distributions of these baryons reflect theirexcitation energies. Other possibility may be to measure the strangeness-changing semi-leptonic weakdecay of Ξ b → Σ b in medium, which again reflects the excitation energy difference of them in medium.In conclusion, studies of heavy baryon properties, in particular Σ b and Ξ b baryons in nuclear medium,can provide us with very interesting and important information on the dynamical symmetry breakingand partial restoration of chiral symmetry, as well as the roles of the light quarks in medium. Acknowledgments
This work was partially supported by the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico- CNPq Grants, No. 400826/2014-3 and No. 308088/2015-8, and was also part of the projects, InstitutoNacional de Ciˆencia e Tecnologia - Nuclear Physics and Applications (INCT-FNA), Brazil, Process No.464898/2014-5, and FAPESP Tem´atico, Brazil, Process No. 2017/05660-0. [1] E. E. Jenkins and A. V. Manohar, Phys. Lett. B , 558 (1991).[2] M. B. Wise, Phys. Rev. D , no.7, R2188 (1992)[3] For a review, J. G. Korner, M. Kramer and D. Pirjol, Prog. Part. Nucl. Phys. , 787 (1994).[4] C. Albertus, J. E. Amaro, E. Hernandez and J. Nieves, Nucl. Phys. A , 333 (2004).[5] X. Liu, H. X. Chen, Y. R. Liu, A. Hosaka and S. L. Zhu, Phys. Rev. D , 014031 (2008).[6] E. E. Jenkins, Phys. Rev. D , 034012 (2008).[7] Z. S. Brown, W. Detmold, S. Meinel and K. Orginos, Phys. Rev. D , no.9, 094507 (2014).[8] V. Simonis, arXiv:1803.01809 [hep-ph].[9] N. Isgur and M. B. Wise, Phys. Rev. Lett. , 1130 (1991).[10] T. Mannel, W. Roberts and Z. Ryzak, Nucl. Phys. B , 38 (1991).[11] For a review, M. Neubert, Phys. Rept. , 259 (1994).[12] G. P. Lepage, L. Magnea, C. Nakhleh, U. Magnea and K. Hornbostel, Phys. Rev. D , 4052 (1992).[13] N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl. Phys. B , 275 (2000).[14] For a review, K. Saito, K. Tsushima and A. W. Thomas, Prog. Part. Nucl. Phys. , 1 (2007).[15] For a review, G. Krein, A. W. Thomas and K. Tsushima, Prog. Part. Nucl. Phys. , 161 (2018).[16] For a review, A. Hosaka, T. Hyodo, K. Sudoh, Y. Yamaguchi and S. Yasui, Prog. Part. Nucl. Phys. , 147 (2017); Erratum: [Nucl. Phys. A , 122 (2017)].[18] K. Azizi and N. Er, Nucl. Phys. A , 422 (2018).[19] P. A. M. Guichon, Phys. Lett. B (1988) 235.[20] P. A. M. Guichon, K. Saito, E. N. Rodionov and A. W. Thomas, Nucl. Phys. A , 349 (1996).[21] K. Saito, K. Tsushima and A. W. Thomas, Nucl. Phys. A , 339 (1996).[22] K. Saito, K. Tsushima and A. W. Thomas, Phys. Rev. C , 2637 (1997). [23] J. R. Stone, P. A. M. Guichon, P. G. Reinhard and A. W. Thomas, Phys. Rev. Lett. , no.9, 092501(2016).[24] P. A. M. Guichon, J. R. Stone and A. W. Thomas, Prog. Part. Nucl. Phys. , 262 (2018).[25] K. Tsushima, K. Saito, J. Haidenbauer and A. W. Thomas, Nucl. Phys. A , 691 (1998).[26] K. Tsushima, K. Saito and A. W. Thomas, Phys. Lett. B (1997) 9 Erratum: [Phys. Lett. B , 413(1998).[27] P. A. M. Guichon, A. W. Thomas and K. Tsushima, Nucl. Phys. A , 66 (2008).[28] K. Tsushima and F. C. Khanna, Phys. Rev. C , 015211 (2003).[29] K. Tsushima and F. C. Khanna, Prog. Theor. Phys. Suppl. , 160 (2003).[30] K. Tsushima and F. C. Khanna, J. Phys. G , 1765 (2004).[31] K. Saito, K. Tsushima and A. W. Thomas, Phys. Rev. C , 566 (1997).[32] K. Tsushima, K. Saito, A. W. Thomas and S. V. Wright, Phys. Lett. B , 239 (1998); Erratum: [Phys.Lett. B , 4531 (1998).[33] K. Tsushima, D. H. Lu, A. W. Thomas and K. Saito, Phys. Lett. B , 26 (1998).[34] K. Tsushima, D. H. Lu, A. W. Thomas, K. Saito and R. H. Landau, Phys. Rev. C , 2824 (1999).[35] A. Sibirtsev, K. Tsushima, K. Saito and A. W. Thomas, Phys. Lett. B , 23 (2000).[36] K. Tsushima and F. C. Khanna, Phys. Lett. B , 138 (2003).[37] A. Sibirtsev, K. Tsushima and A. W. Thomas, Eur. Phys. J. A , 351 (1999).[38] R. Shyam, K. Tsushima and A. W. Thomas, Phys. Lett. B , 51 (2009).[39] K. Tsushima, P. A. M. Guichon, R. Shyam and A. W. Thomas, Int. J. Mod. Phys. E , 2546 (2010).[40] R. Shyam, K. Tsushima and A. W. Thomas, Nucl. Phys. A , 255 (2012).[41] R. Chatterjee, R. Shyam, K. Tsushima and A. W. Thomas, Nucl. Phys. A , 116 (2013).[42] K. Tsushima, R. Shyam and A. W. Thomas, Few Body Syst. , 1271 (2013).[43] R. Shyam and K. Tsushima, Phys. Rev. D , no.7, 074041 (2016).[44] R. Shyam and K. Tsushima, Phys. Lett. B , 236 (2017).[45] R. Shyam and K. Tsushima, Few Body Syst. , no.3, 18 (2018).[46] D. L. Whittenbury, J. D. Carroll, A. W. Thomas, K. Tsushima and J. R. Stone, Phys. Rev. C , 065801(2014).[47] A. W. Thomas, D. L. Whittenbury, J. D. Carroll, K. Tsushima and J. R. Stone, EPJ Web Conf. , 03004(2013).[48] K. Tsushima, T. Maruyama and A. Faessler, Nucl. Phys. A , 497 (1991).[49] T. Maruyama, K. Tsushima and A. Faessler, Nucl. Phys. A , 303 (1992).[50] G. Krein, A. W. Thomas and K. Tsushima, Nucl. Phys. A , 313 (1999).[51] J. D. Walecka, Annals Phys. , 491 (1974).[52] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. , 1 (1986).[53] P. A. M. Guichon, H. H. Matevosyan, N. Sandulescu and A. W. Thomas, Nucl. Phys. A , 1 (2006).[54] M. Dutra, O. Lourenco, J. S. Sa Martins, A. Delfino, J. R. Stone and P. D. Stevenson, Phys. Rev. C ,035201 (2012).[55] C. Patrignani et al. (Particle Data Group), Chin. Phys. C, , 100001 (2016) and 2017 update.[56] R. Aaij et al.et al.