Possible S -wave N D (∗) and N B ¯ (∗) bound states in a chiral quark model
aa r X i v : . [ nu c l - t h ] M a r Possible S -wave N D ( ∗ ) and N ¯ B ( ∗ ) bound states in a chiral quark model Dan Zhang,
1, 2, ∗ Dan Yang, † Xiao-Fei Wang, ‡ and Kanzo Nakayama § School of Physical Science and Technology, Inner Mongolia University, Hohhot 010021, People’s Republic of China Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA S -wave bound-states composed of a nucleon( N ) and a heavy meson ( D , D ∗ , ¯ B or ¯ B ∗ ) are investi-gated in both the chiral SU(3) quark model and the extended chiral SU(3) quark model by solvingthe resonating group method equation. The results reveal that the ND and ND ∗ interactions inthe corresponding relative S -wave states are attractive, arising mainly from one boson exchangeprocesses between light quarks. It is shown that these attractions are strong enough to form six ND and ND ∗ S -wave bound states in the extended chiral SU(3) quark model with the binding energiesin the range of 3 −
45 MeV, and three S -wave bound states within the chiral SU(3) quark model withbinding energies of 2 − c (2800) is interpreted tobe most likely an S − wave ND state with the total isopsin I = 0 and spin-parity J P = 1 / − , whileΛ c (2940) + as an S − wave ND ∗ state with I = 0 and J P = 3 / − . Further information on the ND and ND ∗ interactions in the (unbound) scattering kinematics are obtained from the corresponding S -wave phase shifts. The N ¯ B and N ¯ B ∗ systems are also investigated within the present two modelsand some S − wave bound states with binding energies in the range of 1 −
60 MeV are predicted inthese systems: six (in total) within the extended chiral SU(3) quark model and, four, within thechiral SU(3) quark model.
PACS numbers: 12.39.Jh, 12.39.Pn, 14.20.Pt, 21.10.Dr
I. INTRODUCTION
In the past decade, many new charmed hadrons havebeen detected experimentally (see the review literature[1] and references therein). Of particular interest amongthese hadronic states is the two charmed baryonic states:the Σ c (2800) state observed by the Belle Collabora-tion [2] and the Λ c (2940) + state reported by the BabarCollaboration [3]. In 2005, an isospin triplet charmedbaryon Σ c (2800) decaying into Λ + c π was first reportedby the Belle Collaboration [2], who tentatively assignedthe quantum numbers J P = 3 / − . Later, the BabarCollaboration [4] proposed a possible confirmation of theneutral state Σ c (2800) with a weak evidence of J = 1 / ± ±
10 MeV ishigher than the value quoted by Belle [2]. In addition, theBabar Collaboration [3] observed a new charmed baryon,Λ c (2940) + , with a mass of 2939 . ± . ± . c (2455) π decay with measuredmass of 2938 . ± . +2 . − . MeV.Since both the Σ c (2800) and Λ c (2940) + states are justbelow the N D and
N D ∗ thresholds, respectively, theyare likely explained as the N D and
N D ∗ molecular states[1], respectively. Dong et al. [6] considered the isotripletΣ c (2800) as a hadronic molecule composed of a nucleonand a D meson. Their widths of the strong two-bodydecay Σ c → Λ c π for the spin-parity J P = 3 / − and ∗ Electronic address: [email protected] † Corresponding author. Email: [email protected] ‡ Corresponding author. Email: [email protected] § Corresponding author. Email: [email protected] J P = 1 / + assignments are consistent with the currentdata. In Ref.[7], based on a coupled channels unitaryapproach, it is suggested the Σ c (2800) is a dynamicallygenerated resonance with a dominant N D configurationhaving J P = 1 / − . The author of Ref.[8] proposedΣ c (2800) to be an S -wave N D state with J P = 1 / − within the framework of QCD sum rules. Although itscalculated mass is somewhat larger than the correspond-ing experimental value, the possibility of Σ c (2800) tobe a molecular state can not be arbitrarily excluded.Wang et al. [9] assumed that the observed Σ c (2800) is an S − wave N D molecular state with J P = 1 / − in the Bethe-Salpeter equation approach. Concerningthe Λ c (2940) + state, Zhang[8] suggested it to be an S -wave N D ∗ state with J P = 3 / − in the framework ofQCD sum rules. The S -wave pD ∗ molecular state with J P = 1 / − was suggested in Ref.[10]. Dong et al. [11, 12]studied Λ c (2940) + , suggesting it to be an N D ∗ molecu-lar state with J P = 1 / ± ; their results also suggest thatthe spin-parity J P = 1 / − should be ruled out. He etal. [13] indicated the existence of the N D ∗ systems with J P = 1 / ± , / ± which, not only provided valuable in-formation to underlying the structure of Λ c (2940) + , butalso improves our knowledge of the interaction of nu-cleon and D ∗ . A possible molecular candidate for theΛ c (2940) + with J P = 3 / − was obtained in a chiral con-stituent quark model[14, 15].On the other hand, there is an alternative way of the-oretical study to consider Σ c (2800) and Λ c (2940) + asconventional charmed baryons[16–20]. In a relativizedpotential model, the masses of Σ c (2800) and Λ c (2940) + were close to theoretical values of Σ ∗ c with J P = 3 / − or J P = 5 / − and Λ ∗ c with J P = 5 / − or J P = 3 / + ,respectively[16]. In the relativistic quark-diquark pic-ture, the Σ c (2800) state has been suggested as one ofthe orbital excitations(1 P ) of the ground state Σ c with J P = 1 / − , 3 / − or 5 / − . The Λ c (2940) + has been pro-posed as the first radial excitation of Σ c with J P = 3 / + [17]. In Ref.[18], based on the Faddeev method, it is in-dicated that the Σ c (2800) state would correspond to anorbital excitation with J P = 1 / − or 3 / − and that theΛ c (2940) + state may constitute the second orbital exci-tation of Λ c . In a mass loaded flux tube model, Chen etal. [19] suggested that Λ c (2940) + could be the orbitallyexcited Λ + c with J P = 5 / − . He et al. [20] evaluatedthe production rate of Λ c (2940) + as a charmed baryonin view of future experiments at PANDA.Despite considerable efforts spent in the study of theΣ c (2800) and Λ c (2940) + states, they are not yet fullyunderstood. Thus, it is timely to make further efforts tostudy these states to reveal their properties.Earlier investigations indicate that the chiral SU(3)quark model [21] and the extended chiral SU(3) quarkmodel [22] are successful in studying hadronic systemswith light flavors, such as N N , N Y [21, 22],
N K [23, 24],∆ K , Σ K [25], N ¯Ω[26] interactions, and the structuresof pentaquark states [27, 28]. Recently, these twoquark models were applied to the heavy flavor sectorsand valuable results were obtained, which include themasses of the singly heavy ground-state baryons [29], thetetraquark states including heavy quarks [30–32], inter-actions of DK [33], D ¯ D and B ¯ B [34], Σ c ¯ D and Λ c ¯ D [35],and structures of X (3872) [36], Z b (10610) and Z b (10650)[37]. In this paper, we shall extend the application ofthese two successful models [21, 22, 25–28] to explore the S -wave N D and
N D ∗ as well as N ¯ B and N ¯ B ∗ heavyquark systems with all possible quantum numbers. Wesolve the resonating group method (RGM) equation toobtain the pertinent solutions. In particular, we give aninterpretation of the structures of both the Σ c (2800) andΛ c (2940) + states within these models. The present mod-els are also applied to the S -wave N ¯ B and N ¯ B ∗ systemsto predict some new bound states. In addition, some use-ful information on the N D ( ∗ ) and N ¯ B ( ∗ ) interactions willbe obtained from the corresponding scattering processes.Throughout this paper, we use the notation N D ( ∗ ) toindicate both N D and
N D ∗ systems. Likewise, N ¯ B ( ∗ ) indicates both N ¯ B and N ¯ B ∗ systems.We mention that the current models are different fromthe chiral constituent quark model of Refs.[14, 15] in thedetails of the assumed Hamiltonian. In particular, weconsider the vector-meson exchange contributions explic-itly (extended chiral SU(3) quark model), while this isabsent in Refs.[14, 15].The paper is organized as follows: Sec. II presents abrief description of the chiral and extended chiral SU(3)quark models and associated parameter values used inthe present work, as well as the resonating group method.In Sec. III, we discuss the numerical results. Finally, Sec.IV gives a summary and conclusion. II. FORMALISMA. Model
The details of the chiral and extended chiral SU(3)quark models considered in the present work have beendescribed in Refs. [21, 22]. Thus, here, we only summa-rize the most relevant aspects of the model to the presentwork. The Hamiltonian of the baryon( qqq )-meson( Q ¯ q ) ( q stands for light quark and, Q , for heavy quark) systemcan be written as H = X i T i − T cm + X i 8) denotes the low-lying scalar-(pseudoscalar-)meson nonet fields. In the above equa-tion, V σ a ( r ij ) = − C ( g ch , m σ a , Λ) X ( m σ a , Λ , r ij ) × [ λ a ( i ) λ a ( j )] , (5)and V π a ( r ij ) = C ( g ch , m π a , Λ) m π a m i m j X ( m π a , Λ , r ij ) × ( σ i · σ j )[ λ a ( i ) λ a ( j )] , (6)where C ( g ch , m, Λ) = g ch π Λ Λ − m m, (7) X ( m, Λ , r ) = Y ( mr ) − Λ m Y (Λ r ) , (8) X ( m, Λ , r ) = Y ( mr ) − (cid:18) Λ m (cid:19) Y (Λ r ) , (9) Y ( x ) = 1 x e − x , (10)with m σ a ( m π a ) denoting the mass of thescalar(pseudoscalar) meson, and Λ the cutoff massfor mesons. g ch represents the coupling constantfor the scalar and pseudoscalar chiral field couplings.[ λ a ( i ) λ a ( j )] is the operator in the flavor space. Note thatthe flavor SU(3) classification of the low-lying scalarmesons is not well established, since the underlyingstructures of these mesons are still an open issue. Nev-ertheless, considering these mesons as members of theSU(3) nonet seems to work rather well [21–24, 26–35, 37].The extended chiral SU(3) quark model includes thevector meson exchanges in addition, viz., V chqq ( ij ) = X a =0 V σ a ( r ij ) + X a =0 V π a ( r ij ) + X a =0 V ρ a ( r ij ) . (11)where ρ a ( a = 0 , .., 8) denotes the low-lying vector-mesonnonet fields. In the above equation, V ρ a ( r ij ) = C ( g chv , m ρ a , Λ) { X ( m ρ a , Λ , r ij )+ m ρ a m i m j (cid:18) f chv g chv m i + m j M P + f chv g chv m i m j M P (cid:19) X ( m ρ a , Λ , r ij ) · ( σ i · σ j ) } [ λ a ( i ) λ a ( j )] , (12)with m ρ a denoting the mass of the vector meson and M P is a mass scale, which is taken to be the proton mass. g chv and f chv are the coupling constants associated with thevector and tensor couplings of the vector meson fields,respectively.The flavor singlet-octet mixing ( π - π ) of the pseu-doscalar mesons is accounted for according to η = π cos θ ps − π sin θ ps ,η ′ = π sin θ ps + π cos θ ps , (13) with the mixing angle of θ ps = − ◦ [22–24, 28, 33].Analogously, the mixing angle θ s corresponding to thesinglet-octet ( σ - σ ) mixing of the scalar mesons is takento be θ s = 0 ◦ [21, 22, 26, 33], i.e., the σ meson is assumedto be a pure singlet ( σ ) and f , a pure octet ( σ ) meson,respectively. Since the vector mesons ω and φ are nearlyideally mixed states of ρ and ρ , they are approximatedto be pure ( u ¯ u + d ¯ d ) and s ¯ s states, respectively [23, 24,26], i.e., θ v ≈ θ videal = − . ◦ . V conf in Eq. (2) stands for the confinement potential,taken as the linear form in this work following Refs. [29–35], V confqq ( ij ) = − ( λ ci · λ cj )( a ij r ij + a ij ) , (14)with a ij denoting the confinement strength and a ij thezero-point energy.The quark-anti-quark interaction V q ¯ q can be obtainedfrom the quark-quark interaction V qq specified above bysimple transformations [23–28]. For V OGEq ¯ q and V confq ¯ q ,the transformation is given by the replacement ( λ ci · λ cj ) → ( − λ ci · λ c ∗ j ) in V OGEqq and V confqq given by Eq. (3) and (14),respectively, while for V chq ¯ q , we have V chq ¯ q = X j ( − G j V ch,jqq . (15)Here ( − G j represents the G parity of the j th meson.In the heavy quark sector, chiral symmetry is explic-itly broken and, therefore, V ch is not considered in theinteractions involving heavy quarks [14, 29–37]. Hence, V Qq ( ij ) = V conf ( ij ) + V OGE ( ij ) , (16)and likewise for V Q ¯ q . The confinement and one-gluon-exchange potentials, V conf ( ij ) and V OGE ( ij ), in theabove equation can be obtained from those in V qq and V q ¯ q , respectively, by replacing the mass of a light quarkby that of a heavy quark.Note that, since we confine ourselves to the S partialwaves in this work, the spin-orbit and tensor forces - inprinciple present in Eq.(3), (5), (6), and (12)- do not playany role. B. The framework of resonating group method The resonating group method (RGM) is a well estab-lished method for learning about the interaction betweentwo clusters, which has been widely used in NuclearPhysics and in constituent quark models [15, 21–28, 33–40]. For the systems composed of the baryon N andmeson M Q , the wave function of the five-quark system istaken asΨ = A [ φ N ( ξ , ξ ) φ M Q ( ξ ) χ ( R NM Q )] α , (17)where ξ , ξ are the internal coordinates for the cluster N , and ξ is the internal coordinate for the cluster M Q , ξ = r − r , ξ = r − m r + m r m + m ξ = r − r . (18) R NM Q ≡ R N − R M Q is the relative coordinate betweenthe two clusters, N and M Q . α represents the set of allquantum numbers to specify a state of the baryon-mesonsystem. φ N ( ξ , ξ ) and φ M Q ( ξ ) in Eq.(17) are the internalcluster wave functions of N and M Q , respectively. Theyare given by φ N ( ξ , ξ ) = (cid:16) m ξ ωπ (cid:17) / exp (cid:16) − m ξ ω ξ (cid:17) · (cid:16) m ξ ωπ (cid:17) / exp (cid:16) − m ξ ω ξ (cid:17) , (19) φ M Q ( ξ ) = (cid:16) m ξ ωπ (cid:17) / exp (cid:16) − m ξ ω ξ (cid:17) , (20)with m ξ = m m m + m , m ξ = ( m + m ) m m + m + m m ξ = m m m + m , (21)and ω = 1 m u b u . (22) A in Eq.(17) is the antisymmetrization operator, de-fined as A ≡ − X i =1 P i = 1 − P , (23)where P i represents the permutation operator betweenthe three quarks in the cluster N and the heavy quark in M Q . Note that, since there is no quark exchange betweenthe two color-singlet clusters N and D ( ∗ ) or ¯ B ( ∗ ) , theantisymmetrization operator when acting on the quarksbetween these two clusters becomes A = 1, i.e., P = 0.The consequence of this is that the matrix elements ofthe color operator ( λ ci · λ cj ) in Eq.(3) and Eq.(14) vanishidentically, which lead to the absence of the one-gluon-exchange and confinement potentials between the clus-ters N and D ( ∗ ) or ¯ B ( ∗ ) . These potentials, however,act between the quarks within the individual clusters N and D ( ∗ ) or ¯ B ( ∗ ) . Hence, the OGE and confinement po-tentials affect the binding energies (cf. Eq. (30)) of thebound N D ( ∗ ) and N ¯ B ( ∗ ) states only indirectly throughthe calculated masses of the individual clusters N , D ( ∗ ) and ¯ B ( ∗ ) . Moreover, to the extent that the parameters of the OGE and confining potentials are adjusted to repro-duce the masses of these individual clusters as explainedlater in this section, they have no effect on the calculatedbinding energies of the N D ( ∗ ) and N ¯ B ( ∗ ) bound states.Finally, χ ( R NM Q ) in Eq.(17) is the relative wave func-tion of the two clusters, which can be obtained by solvingthe following equation Z φ + N ( ξ , ξ ) φ + M Q ( ξ )( H − E )Ψd ξ d ξ d ξ = 0 . (24)With Ψ, φ N ( ξ , ξ ) and φ M Q ( ξ ) given by Eqs.(16,18,19), the RGM equation becomes Z L ( R ′ , R ) χ ( R )d R = 0 , (25)with L ( R ′ , R ) ≡ H ( R ′ , R ) − E N ( R ′ , R ) , (26) H ( R ′ , R ) = Z φ + N ( ξ , ξ ) φ + M Q ( ξ ) δ ( R ′ − R NM Q ) · H · φ N ( ξ , ξ ) φ M Q ( ξ ) δ ( R − R NM Q ) · d ξ d ξ d ξ d R NM Q , (27)and N ( R ′ , R ) = Z φ + N ( ξ , ξ ) φ + M Q ( ξ ) δ ( R ′ − R NM Q ) · · φ N ( ξ , ξ ) φ M Q ( ξ ) δ ( R − R NM Q ) · d ξ d ξ d ξ d R NM Q . (28)By solving the RGM equation (25), we can obtain thebinding energies or scattering phase-shifts and cross sec-tions for the two-cluster systems. The details of solvingthe RGM equation can be found in Refs. [38–41]. C. Model parameters For light quarks, all the model parameters are takenfrom our previous work [21–28], which can give a satis-factory description of the energies of the baryon groundstates, the binding energy of deuteron, and the N N scat-tering phase-shifts. The harmonic-oscillator width pa-rameter b u in Eq.(22) is taken with different values forthe two models: b u = 0 . 50 fm in the chiral SU(3) quarkmodel and b u = 0 . 45 fm in the extended chiral SU(3)quark model. The up (down) quark mass, m u ( m d ), istaken to be the usual value of m u = m d = 313 MeV.The coupling constant for the scalar and pseudoscalarchiral field couplings, g ch , is determined according to therelation g ch π = (cid:18) (cid:19) g NNπ π m u M N , (29) TABLE I: Model parameters for the light quarks. The mesonmasses: m a = 980 MeV, m f = 980 MeV, m π = 138 MeV, m η = 549 MeV, m η ′ = 957 MeV, m ρ = 770 MeV, m ω = 782MeV, m φ = 1020 MeV. The cutoff masses: Λ = 1100 MeV forall mesons. The mixing angles between the flavor singlet andoctet mesons: θ ps = − ◦ , θ s = 0 ◦ , θ v = θ videal = − . ◦ . χ -SU(3)QM Ex. χ -SU(3) QMI II III f chv /g chv = 0 f chv /g chv = 2 / b u (fm) 0.5 0.45 0.45 m u (MeV) 313 313 313 g u (= g d ) 0.875 0.236 0.363 g ch g chv − m σ (MeV) 595 535 547 a uu (MeV/fm) 87.5 75.3 66.2 a uu (MeV) − − − with the empirical value g NNπ / π = 13 . g chv in Eq.(12) istaken to be g chv = 2 . 351 and g chv = 1 . 973 correspondingto the ratio f chv /g chv = 0 and f chv /g chv = 2 / 3, respec-tively. The OGE coupling constants, g u (= g d ) can bedetermined by the mass splits between N and ∆. Theconfinement strengths a uu is fixed by the stability con-ditions of N , and the zero-point energies a uu by fittingthe masses of N . The masses of the mesons are takento be those determined experimentally, except for the σ meson, whose value is adjusted to fit the binding energyof the deuteron. The cutoff radius Λ − (inverse of thecutoff mass Λ) is taken to be the value close to the chi-ral symmetry breaking scale[42–45], and common to allmesons.All the parameter values of the present models associ-ated with the light quarks are displayed in Table I, wherethe first set corresponds to the chiral SU(3) quark model(I), the second and third sets are for the extended chiralSU(3) quark model by taking the ratio f chv /g chv to be 0(II) and 2 / m c = 1430 MeV[30–35, 37], m c =1550 MeV[33, 35, 46], m c = 1870 MeV[33, 35, 47] and, m b = 4720 MeV[34, 37], m b = 5100 MeV[48], m b = 5259MeV[47]. Our numerical results indicate that the heavy-quark-mass dependence is small and, therefore, we onlyreport the results corresponding to m c = 1430 MeV and m b = 4720 MeV, which are the values used in the earlierwork[29–37].The OGE coupling constants g Q and the confinementstrengths a Qq , and the zero-point energies a Qq can be de-termined by fitting the masses of the mesons with singleheavy quark[29–33]. Thus, in the present study, follow-ing Refs. [29, 32], the OGE coupling constants g c and g b are taken to be g c = 0 . 58 and g b = 0 . 52, respectively.The confinement parameters a Qq and a Qq are adjusted such that the calculated heavy-meson masses be closeto the corresponding experimental values. The result-ing model parameters for the c quark, a cu and a cu (to-gether with the OGE coupling constant g c ), in the chiralSU(3) quark model(set I) with m c = 1430 MeV are dis-played in Table II. The three sets of adjusted values of a cu and a cu shown there illustrate the sensitivity of the D and D ∗ meson masses to these parameter values. Sim-ilar results/sensitivity are found for the correspondingparameters in the extended chiral SU(3) quark model.We refrain from showing them here. The above findingsconcerning the confining potential involving the param-eters a bu and a bu hold for the case of b quarks as well.As pointed out before, although the OGE and confin-ing potentials enter in the calculations of the individualclusters N , D ( ∗ ) and ¯ B ( ∗ ) , they do not play any rolein the interaction between the clusters N and D ( ∗ ) or¯ B ( ∗ ) , since these potential contributions vanish identi-cally. This means that the interaction between N and D ( ∗ ) or ¯ B ( ∗ ) is not sensitive to the light-heavy-quark pa-rameters g Q , a Qq , and a Qq ; therefore, we only discuss theresults of one set of them, namely, that of the first rowin Table II. TABLE II: Model parameters for the c quark in the chiralSU(3) quark model(set I) with m c = 1430 MeV. The corre-sponding calculated and experimental[49] values of the massesof the D and D ∗ mesons are listed as well. g c a cu (MeV/fm) a cu (MeV) D (MeV) D ∗ (MeV)0.58 275.6 -169.0 1869.6 1974.8275.5 -162.9 1901.8 2007.0275.2 -165.0 1889.6 1994.8Exp. data[49] 1869.6 2006.9 III. RESULTS AND DISCUSSIONS In this work we investigate the S -wave N D ( ∗ ) and N ¯ B ( ∗ ) systems within the chiral and extended chiralSU(3) quark models. We consider these systems in boththe bound and unbound (scattering) kinematics. Thespin of N is S = 1 / 2, of D and ¯ B is S = 0, and of D ∗ and ¯ B ∗ is S = 1. The isospin of all these five cluster par-ticles is I = 1 / 2. Therefore, the N D and N ¯ B systems inthe S partial-waves can form the states with total spin( J ) and isospin ( I ), JI , with J = 1 / I = 0 , 1. Like-wise, the N D ∗ and N ¯ B ∗ systems can form the JI stateswith J = 1 / , / I = 0 , N D ( ∗ ) and N ¯ B ( ∗ ) sys-tems separately. -240-180-120-60060-240-180-120-600600.0 0.5 1.0 1.5-240-180-120-60060 0.0 0.5 1.0 1.5 [ND] J=1/2 V ( s ) ( M e V ) s (fm) I=1 I=0 scalar vector total -240-180-120-60060-240-180-120-600600.0 0.5 1.0 1.5-240-180-120-60060 0.0 0.5 1.0 1.5 scalar pseudo vector total V ( s )( M e V ) s (fm) I=0 I=1 [ND * ] J=3/2 -240-180-120-60060-240-180-120-600600.0 0.5 1.0 1.5-240-180-120-60060 0.0 0.5 1.0 1.5 scalar pseudo vector total s (fm) V ( s )( M e V ) [ND * ] J=1/2 I=1 I=0 FIG. 1: The effective potentials between the two clusters N and D and, N and D ∗ , corresponding to the S -wave [ ND ] J =1 / (left two columns), [ ND ∗ ] J =3 / (middle two columns) and [ ND ∗ ] J =1 / (right two columns) states as functions of the generatorcoordinate s . The two columns within a given state in the ND or ND ∗ system as indicated, correspond to the total isospin I = 0 (left column) and I = 1 (right column). The panels in the upper rows correspond to the chiral SU(3) quark model (setI). The panels in the middle and lower rows correspond to the extended chiral SU(3) quark model with the parameter sets IIand III as given in Table. I, respectively. The dotted (red) lines represent the effective potentials due to the σ -meson exchange( σ ). The dashed (green) lines are the sum of the contributions from the scalar-meson( a , f ) exchanges (scalar), and thedash-double-dotted (magenta) lines are the sum of the contributions from the pseudoscalar-meson( π, η, η ′ ) exchanges(pseudo).The dash-dotted (blue) lines are those from the vector-meson( ω , ρ ) exchanges (vector) in the extended chiral SU(3) quarkmodel(sets II and III). The solid (black) lines present the sum of all the individual contributions to the effective potentials(total). A. ND ( ∗ ) systems First we discuss the N D ( ∗ ) systems ( N D and N D ∗ ) inthe bound kinematics and then in the unbound (scatter-ing) kinematics.Since neither the OGE nor the confinement potentialsbetween the two color-singlet clusters N and D ( ∗ ) arepresent, as mentioned in Sec. II, the effective potentialbetween the two clusters arises only from the one-meson-exchange interactions, V chqq , between the light quarks.The OGE and confinement potentials are present in thecalculations of the individual clusters N and D ( ∗ ) (inparticular, their masses), however. Figure 1 displays theeffective potentials between the N and D , and, N and D ∗ clusters in the S -wave [ N D ] J =1 / , [ N D ∗ ] J =3 / and[ N D ∗ ] J =1 / states, respectively, as functions of s , thegenerator coordinate which describes, qualitatively, thedistance between the two clusters in the generator co-ordinate method (GCM) calculation[38–41]. In Fig. 1, the dotted (red) curves correspond to the scalar σ -mesonexchange contribution, while the dashed (green) curvesrepresent the sum of the scalar a - and f -meson contri-butions (no σ meson contribution included here). Thedash-double-dotted (magenta) curves correspond to thesum of the pseudoscalar π -, η - and η ′ -meson and thedash-dotted (blue) curves to the sum of the vector ρ -and ω -meson contributions. Note that the φ -meson con-tribution vanishes identically because, as mentioned inSec. IIA, it is taken to be a pure s ¯ s state in the presentwork. The net contribution of all mesons is representedby the solid (black) curves. We mention that the con-tribution of the pseudoscalar meson η ′ is negligible com-pared to the other meson contributions, hence, in thefollowing, we shall make no reference to this meson con-tribution.As can be seen, the net (total) effective potentialis attractive in all the cases. Except for the case of[ N D ∗ ] J =1 / with I = 1, the bulk of the attraction isprovided by the isoscalar σ -meson exchange. Comparedto the chiral SU(3) quark model (set I), the net effec-tive potential is more attractive in the extended chiralSU(3) quark model (sets II and III) due to the presence ofthe vector-meson ( ρ plus ω ) exchange contribution whichprovides an additional attraction. Note, in particular,that the vector-meson contribution is comparable to thatof the σ -meson in the case of [ N D ∗ ] J =1 / with I = 1; thisis due to the isovector ρ -meson which provides a muchstronger attraction compared to the cases of [ N D ] J =1 / and [ N D ∗ ] J =3 / . The pseudoscalar-meson exchange con-tribution is largest in the case of [ N D ∗ ] J =1 / with I = 1due to the π -meson providing the bulk of the repulsion,comparable in magnitude to those of the σ - and vector-mesons. It cancels partly the otherwise very strong at-traction arising from the σ - and vector-meson contribu-tions in the extended chiral SU(3) quark model. In thechiral SU(3) quark model, due to the absence of the at-tractive vector-meson potential, the resulting (attractive)net potential is much shallower than that in the formermodel. As we shall show later, this has a direct con-sequence in the formation of bound states in the N D ∗ system with quantum numbers J = 1 / I = 1.Note that the pseudoscalar-meson exchanges are absentin the case of [ N D ] J =1 / because of the antisymmetricspin wave function of the D cluster.The aspects of the effective potentials between the N and D ( ∗ ) clusters discussed above are the dominant fea-tures exhibited by these potentials. We now look at someof the more detailed aspects of the underlying dynam-ics exhibited by our effective potentials. Our analysisindicates that, for the isospin I = 0 states, the scalar-isoscalar meson f contributes a weak attraction whilethe scalar-isovector meson a , a repulsion comparable inmagnitude to the attractive f -meson potential. Theseresult in nearly vanishing contribution of the sum ofthe scalar-meson exchanges excluding the σ -meson, asshown in Fig. 1 (dashed (green) lines). For the I = 1states, both f and a provide weak attractions, leadingto slightly attractive potentials, also shown as dashed(green) curves in Fig. 1. Note that the potential from anisovector meson contribution for the I = 1 state differ bya factor of -3 compared to the corresponding contribu-tion for the I = 0 state. The effective potentials arisingfrom the pseudoscalar mesons exhibit quite differentlybehavior depending on the JI quantum numbers as wellas on the systems N D or N D ∗ . First of all, as has beenmentioned before, for the [ N D ] J =1 / state with I = 0 , N D ∗ ] J =3 / , I = 1state, their overall contribution is weakly attractive dueto the attraction from the π meson being a bit largerthan the repulsion from the η meson. Here, the relativelysmall attractive individual contributions arising from thepseudoscalar and scalar ( a plus f ) mesons add up to asignificant attraction. For the corresponding I = 0 state,both the η and π mesons contribute a repulsion, addingup to a still relatively small net repulsion. The situationin the case of the [ N D ∗ ] J =1 / states is reversed from the case of [ N D ∗ ] J =3 / in the chiral SU(3) quark model(setI) as far as the total isospin I dependence is concerned.Furthermore, the contribution is enhanced by more thana factor of 2 in the I = 1 state as compared to the case ofthe [ N D ∗ ] J =3 / , I = 0 state. Here, for the I = 0 state,the overall contribution of the pseudoscalar mesons is at-tractive due to the attraction from π being larger thanthe repulsion from η . However, for the I = 1, η and π contribute a relatively strong repulsion, comparable inmagnitude to the strong attraction arising from the σ meson. In the extended chiral SU(3) quark model (setsII and III), we have the additional contributions to theeffective potential from the vector mesons. The ω mesoncontributes an attraction for all the states [ N D ] J =1 / ,[ N D ∗ ] J =1 / and [ N D ∗ ] J =3 / , The ρ -meson contributesa weaker repulsion for the [ N D ] J =1 / and [ N D ∗ ] J =1 / states with I = 0, while an attraction for the correspond-ing states with I = 1. This leads to a net attractivevector-meson contribution for I = 0 that is weaker thanthe corresponding attraction for I = 1. Note, in addi-tion, that the attraction for the [ N D ∗ ] J =1 / , I = 1 stateis much larger than that for the [ N D ] J =1 / , I = 1 state.For [ N D ∗ ] J =3 / , the vector mesons contribute a net at-tractive potential for both I = 0 and I = 1. However,the attraction for I = 1 is weaker than for I = 0, which isopposite to what is observed for the corresponding statesin [ N D ∗ ] J =1 / as far as the isospin dependence is con-cerned. Finally, we point out that the relatively small dif-ference in the corresponding vector-meson contributionsbetween set II and set III, is sorely due to the ω -mesontensor coupling as shown in Table. I.The features of the effective potentials exhibited inFig. 1 and discussed above are reflected in the calcu-lated binding energies of the possible bound states in the S -wave N D and N D ∗ systems displayed in Table.III, cor-responding to the three sets of model parameters givenin Table. I. TABLE III: Binding energies E b (MeV) of possible S -wave ND ( ∗ ) . The last column is the results from Ref.[15]. χ -SU(3)QM Ex. χ -SU(3) QM J P Isospin I II III Ref.[15] ND − − ND ∗ − − − − The binding energy E b of the nucleon-meson ( N M Q )system is defined as E b = − (cid:2) M NM Q − ( M N + M M Q ) (cid:3) . (30) M NM Q , M N , and M M Q are the calculated masses of thefive-quark system [ N M Q ], nucleon N , and heavy meson M Q , respectively. If E b is positive, the system is boundas a molecular state.Table III displays all the bound states predicted inthe present work with the corresponding binding ener-gies. As can be seen, in the extended chiral SU(3) quarkmodel(set II and III), we obtain two bound states inthe [ N D ] system, namely, [ N D ] J =1 / with I = 0 and I = 1, and four bound states in the N D ∗ system, twowith spin-parity J P = 1 / − and two with J P = 3 / − ,i.e., [ N D ∗ ] J =1 / with I = 0 and I = 1, and [ N D ∗ ] J =3 / with I = 0 and I = 1. In contrast, in the chiral SU(3)quark model (set I), no bound states [ N D ] J =1 / and[ N D ∗ ] J =3 / with I = 0 and [ N D ∗ ] J =1 / with I = 1 arefound. This is a direct consequence of the absence of theattractive vector-meson exchange potential in the latermodel, which makes the resulting total effective poten-tials much shallower than in the extended SU(3) quarkmodel, as can be seen in Fig. 1. Note, in particular,that the binding energies of the predicted states in thechiral SU(3) quark model are much smaller than the cor-responding binding energies predicted in the extendedchiral SU(3) quark model. Furthermore, the binding en-ergies corresponding to set II are larger than those of setIII, in accordance with the stronger attractive potentialin set II than in set III as can also be seen in Fig. 1.We know that the experimentally observed Σ c (2800)is about 8 MeV below the N D threshold. Therefore, thepredicted [ N D ] J =1 / with I = 0 molecular state with10 . . c (2800) state. Its spin-parity J P = 1 / − is consistent with the findings from other independentcalculations [7–9], as well as with the weak evidence of J = 1 / c (2800) candidate is I = 0,which is at odds with the experimentally inferred valueof I = 1[1, 2]. We expect further experiments to confirmor dismiss our results.Similarly, the experimentally observed Λ c (2940) + isabout 6 MeV below the N D ∗ threshold. Both the presentchiral and extended chiral SU(3) quark models predict an N D ∗ bound state with J P = 3 / − in this energy region.The former model yields an [ N D ∗ ] J P =3 / − state with I = 1 at about 8.0 MeV (set I) binding energy, while thelatter model gives an [ N D ∗ ] J P =3 / − state with I = 0 atabout 6.6 MeV (set II) or 3.2 MeV (set III) binding en-ergy. The recent experiment [50] assigns the spin-parity J P = 3 / − while the other experiments [1, 3, 5, 49] assignthe isospin I = 0 to the Λ c (2940) + . This rules out thechiral SU(3) quark model, indicating that the extendedchiral SU(3) quark model is more effective in predict-ing the S -wave bound states in the N D ∗ system, whichmeans that the vector-meson exchange interactions playa significant role.At this point, we mention that the authors of Ref.[15]have also investigated the possible N D and N D ∗ molecules in a chiral constituent quark model, and theirresults are listed in the last column of Table III. They suggest the [ N D ∗ ] J =3 / with I = 0 with 8 . 02 MeV bind-ing energy as the observed Λ c (2940) + , and don’t findthe candidate for the Σ c (2800). We corroborate thesefindings. Our binding energies for the [ N D ] J =1 / with I = 0 and [ N D ∗ ] J =1 / with I = 1 states are larger thantheirs. In contrast to the present work, where we haveincluded explicitly the low-lying vector-meson nonet intothe calculation (extended chiral SU(3) quark model), inRef. [15], the vector-mesons are not considered based onthe argument to avoid double counting because the vec-tor mesons provide the short-range interaction, which istaken over by the OGE potential [51]. As we have pointedout, however, the OGE potential does not act betweenthe N and D ( ∗ ) or ¯ B ( ∗ ) clusters. Note that the possibledouble-counting issue that arises in the light quark flavorsector, where both the OGE and vector-meson exchangecontribute is avoided to the extent that the OGE cou-pling is reduced when the vector mesons are also includedto always reproduce the relevant physical quantities (Seethe value of the coupling constant g u in Table. I which isreduced by nearly a factor of 4 in going from the chiralto extended chiral SU(3) quark model.)We now turn our attention to the issue of the flavorsingle-octet mixing of the scalar mesons σ and σ . Asmentioned in Sec. II.A, both the chiral and extended chi-ral SU(3) quark models applied in the present work ig-nore the σ - σ mixing, i.e., θ s = 0 ◦ . These models werebuilt to describe successfully many hadronic systems inthe light flavor sector [21, 22, 26, 33]. The present anal-ysis of the N D ( ∗ ) systems reveals that the σ meson hasa very important contribution to the effective potentialbetween the N and D ( ∗ ) cluster, providing the necessaryattractions to form bound states (cf. Fig. 1). It happensthat the resulting attraction in the effective potentialand, consequently, the binding energies are rather sen-sitive to the mixing angle θ s . Unfortunately, the mixingangle θ s is not as well established as the correspondingangles for the pseudoscalar and vector mesons. The anal-ysis of Ref. [52], based on Unitarized Chiral PerturbationTheory, have shown that singlet-octet mixing holds in thescalar-meson sector and quotes a value of θ s = 19 ◦ ± ◦ .The positive sign of θ s is preferred over the negative sign,since the latter sign-choice leads to a large U(3) sym-metry breaking. On the other hand, in Ref. [53], basedon an approximate chiral symmetry effective Lagrangian,the mixing angle of θ s = − ◦ is quoted. These very dis-tinct values of θ s reveal that the issue of σ - σ mixingis far from being settled. We emphasize that a properassessment of the sensitivity of the present results to θ s should be performed in a way that all the other param-eter values of the model are consistent with the chosenvalue of θ s . This requires refitting those parameters todescribe the relevant hadronic systems in the light flavorsector which is beyond the scope of the present paper.We reserve this for a future work. In this regard, it isnoteworthy that, in a recent chiral SU(3) quark modelstudy by Huang and Wang[54], the masses of the octetand decuplet baryon ground states, the binding energy ofdeuteron, and the N N scattering phase shifts have beensimultaneously reproduced in a rather consistent man-ner. Specifically, the harmonic-oscillator size parametersfor constituent quarks were determined by a variationalmethod instead of being treated as predetermined pa-rameters and taken to be the same for all single baryons.This ensures that all single baryons are minima of theHamiltonian. It would be interesting to consider theirmodel as a basis for including the N D ( ∗ ) and N ¯ B ( ∗ ) sys-tems. I=1 I=0 [ND] J=1/2 [ND * ] J=1/2 ph a s e s h i f t ( d e g ) [ND * ] J=3/2 E cm (MeV) [ND] J=1/2 [ND * ] J=1/2 [ND * ] J=3/2 FIG. 2: ND ( ∗ ) S -wave phase-shifts as functions of the center-of-mass energy. The solid lines represent the results in thechiral SU(3) quark model (set I), and the dash-dotted anddashed lines represent the results in the extended chiral SU(3)quark model (sets II and III). Panels on the left are for I = 0and those on the right are for I = 1. In the present work, we also study the S partial-wave N D ( ∗ ) elastic scattering processes by solving the RGMequation in order to obtain information on the underly-ing N D ( ∗ ) interactions in the scattering kinematics. Thecalculated phase-shifts are illustrated in Fig.2 as a func-tion of c.m. energy E cm . As can be seen in Fig.2, the S -wave phase-shifts corresponding to the parameter setsI(chiral SU(3) quark model), II, and III(extended chi-ral quark model) are all positive, which means that thecorresponding underlying interactions are all attractive.They decrease rapidly as functions of energy. The mag-nitudes of the S -wave phase-shifts in the extended chi- ral SU(3) quark model are larger than that of the chiralSU(3) quark model, especially that corresponding to theparameter set II. Also, magnitudes for I = 1 are largerthan the corresponding ones with I = 0, except for thecase of [ N D ∗ ] J =1 / corresponding to set I. The larger isthe phase-shift, the greater is the attraction. B. N ¯ B ( ∗ ) systems The analysis of the N D ( ∗ ) systems in the precedingsubsection can be carried over to the N ¯ B ( ∗ ) systems, byreplacing the c quark by the b quark. The behaviors ofthe resulting effective interactions between N and ¯ B ( ∗ ) are similar to those exhibited by the N D ( ∗ ) interactionsshown in Fig. 1, except for the fact that the attractionsbetween N and ¯ B ( ∗ ) are a bit larger than those between N and D ( ∗ ) . The binding energies of all possible S -wave N ¯ B ( ∗ ) states obtained are tabulated in Table IV. The[ N ¯ B ] J =1 / states with I = 0 and I = 1 are both boundwith the binding energies in the range of 1 − 53 MeV de-pending on the models and parameter sets (I, II, and III)considered. The [ N ¯ B ∗ ] J =3 / bound state with I = 0 isfound only in the extended chiral SU(3) quark model withbinding energy of 13 . . N ¯ B ∗ ] J =3 / bound state with I = 1 isfound in both the chiral and extended chiral SU(3) quarkmodels with the binding energies in the range of 16 − N ¯ B ∗ ] J =1 / states with I = 0 and I = 1 arebound with the binding energy in the range of 7 − N ¯ B ∗ ] J =1 / state with I = 1 which is absent in the chiral SU(3) quark model(set I). One sees that the binding energies of the N ¯ B ( ∗ ) states are larger than those of the corresponding N D ( ∗ ) states.Also, as shown in Table IV, Ref.[15] found four N ¯ B ( ∗ ) bound states. Our binding energies of [ N ¯ B ] J =1 / and[ N ¯ B ∗ ] J =3 / with I = 0 are close to their correspond-ing results, while those of [ N ¯ B ] J =1 / with I = 1 and[ N ¯ B ∗ ] J =1 / with I = 0 are larger than theirs. TABLE IV: Binding energies E b (MeV) of possible S -wave N ¯ B ( ∗ ) (MeV). The last column is the results from Ref.[15]. χ -SU(3)QM Ex. χ -SU(3) QM J P Isospin I II III Ref.[15] N ¯ B − N ¯ B ∗ − − − − The results of scattering process of N ¯ B ( ∗ ) and N D ( ∗ ) S -wave phase-shifts in N ¯ B ( ∗ ) scat-tering have similar behavior to those corresponding onesin N D ( ∗ ) shown in Fig.2 with slightly larger magnitudes.The difference is no more than 10 degrees. This fea-ture indicates that the attraction between N and ¯ B ( ∗ ) isgreater than that between N and D ( ∗ ) , which is also thefeature seen in the binding energies in Tables.III and IVin the bound state kinematics. IV. SUMMARY AND CONCLUSION We have explored some of the properties of the N D ( ∗ ) and N ¯ B ( ∗ ) systems in the S partial waves by solvingthe RGM equation in both the chiral and extended chi-ral SU(3) quark models, including the bound-state andelastic scattering processes. We have found that the ef-fective potential between the two clusters, N and D ( ∗ ) or N and ¯ B ( ∗ ) , is attractive, and this attraction in theextended chiral SU(3) quark model (specially in set II) isstronger than that in the chiral SU(3) quark model. Also,the attraction is stronger for the I = 1 states than forthe I = 0 states, except for the case of the [ N D ∗ ] J =1 / and [ N ¯ B ∗ ] J =1 / states in the chiral SU(3) quark model,where the attraction is stronger for the I = 0 states thanfor the I = 1 states. The attractive nature of the ef-fective potential has been shown to arise primarily fromthe σ -meson exchange in both the chiral and extendedchiral SU(3) quark models for [ N D ] J =1 / , [ N D ∗ ] J =3 / with both I = 0 , N D ∗ ] J =1 / with I = 0. For[ N D ∗ ] J =1 / with I = 1 there is a sizable repulsive con-tribution from the pion exchange and, in the extendedchiral SU(3) quark model, an attractive vector mesoncontribution comparable to that of the σ -meson. In thelater model, the vector meson ( ρ , ω ) exchanges providefurther attraction. This extra attraction from the vec-tor mesons suffices to form the bound states [ N D ] J =1 / with I = 0, [ N D ∗ ] J =1 / with I = 1 and [ N D ∗ ] J =3 / with I = 0, which are absent in the chiral SU(3) quarkmodel. Analogously, the bound states [ N ¯ B ∗ ] J =1 / with I = 1 and [ N ¯ B ∗ ] J =3 / with I = 0 are only formed in theextended chiral SU(3) quark model.According to the present model analysis, the observedΣ c (2800) and Λ c (2940) + may be interpreted, respec- tively, as the S -wave N D molecular state with I = 0, J P = 1 / − and the S -wave N D ∗ molecular state with I = 0, J P = 3 / − . This finding indicates the extendedchiral SU(3) quark model is more effective in describ-ing the S -wave N D ( ∗ ) system as compared to the chiralSU(3) quark model. Although more in depth studies arerequired, we would like to point out that the basic qual-itative features of all possible S -wave N D ( ∗ ) and N ¯ B ( ∗ ) states obtained in this study are reasonable when com-pared to the currently available experimental and theo-retical information. Also, the future experimental searchfor the N ¯ B ( ∗ ) molecular states is an interesting topic.The calculated S -wave phase-shifts corresponding tothe N D ( ∗ ) and N ¯ B ( ∗ ) scattering processes reveal thatthe corresponding interactions are also quite attractive ,decreasing rapidly as the energy increases.Finally, we mention that we have found that the S -wave bound states in the N D ( ∗ ) and N ¯ B ( ∗ ) systems aresensitive to the flavor singlet-octet mixing angle of thescalar meson nonet. Unfortunately the correspondingmixing angle θ s is not well known. This imposes a limi-tation on the predictive power of the present type model,where this parameter value is required. A proper assess-ment of the sensitivity of the calculated binding energiesto θ s requires a re-analysis of the relevant hadronic sys-tems in the light flavor sector to consistently determineall the parameters of the model as a function of θ s . 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