Possible H -like dibaryon states with heavy quarks
aa r X i v : . [ h e p - ph ] N ov Possible H -like dibaryon states with heavy quarks ∗ Hongxia Huang a , Jialun Ping a , and Fan Wang b a Department of Physics, Nanjing Normal University, Nanjing 210097, P.R. China and b Department of Physics, Nanjing University, Nanjing 210093, P.R. China
Possible H -like dibaryon states Λ c Λ c and Λ b Λ b are investigated within the framework of quarkdelocalization color screening model. The results show that the interaction between two Λ c ’s isrepulsive, so it cannot be bound state by itself. However, the strong attraction in Σ c Σ c and Σ ∗ c Σ ∗ c channels and the strong channel coupling, due to the central interaction of one-gluon-exchange andone-pion-exchange, among Λ c Λ c , Σ c Σ c and Σ ∗ c Σ ∗ c push the energy of system below the threshold ofΛ c Λ c by 22 MeV. The corresponding system Λ b Λ b has the similar properties as that of Λ c Λ c system,and a bound state is also possible in Λ b Λ b system. PACS numbers: 13.75.Cs, 12.39.Pn, 12.39.Jh
I. INTRODUCTION
The H dibaryon, a six quark ( uuddss ) state corre-sponding asymptotically to a bound ΛΛ system, was firstproposed by Jaffe in 1977 [1]. This hypothesis initiateda worldwide activity of theoretical studies and experi-mental searches for dibaryon states [2]. In 1987, M. Oka et al. claimed that a sharp resonance appears in S ΛΛscattering at E c.m. = 26 . H dibaryon state [3]. Moreover, M. Oka also pro-posed several J P = 2 + dibaryons in the quark clustermodel without meson exchange [4]. Despite numerousclaims, no dibaryon candidate has been confirmed exper-imentally so far. Recntly, the interest in the H dibaryonhave been revived by lattice QCD calculations of differentcollaborations, NPLQCD [5] and HALQCD [6]. Thesetwo groups reported that the H particle is indeed a boundstate at pion masses larger than the physical ones. Then,Carames and Valcarce examined the H dibaryon withina chiral constituent quark model and obtained a bound H dibaryon with binding energy B H = 7 MeV [7].Understanding the hadron-hadron interactions andsearching exotic quark states are important topics intemporary hadron physics. Recently observed manynear-threshold charmonium-like states, such as X (3872), Y (3940), and Z + (4430), triggered lots of studies onthe molecule-like bound states containing heavy quarkhadrons. Such a study may help us to understand furtherthe hadron-hadron interactions. In the heavy quark sec-tor, the large masses of the heavy baryons reduce the ki-netic of the system, which makes it easier to form boundstates. One may wonder whether a H -like dibaryon stateΛ c Λ c exist or not.In particular, the deuteron is a loosely bound stateof a proton and a neutron, which may be regarded asa hadronic molecular state. The possibility of existingdeuteron-like states, such as N Σ c , N Ξ ′ c , N Ξ cc , ΞΞ cc andso on, were investigated by several realistic phenomeno- ∗ Corresponding author: [email protected] (J.L. Ping) logical nucleon-nucleon interaction models [8, 9]. The N Λ c system and relevant coupled channel effects wereboth studied on hadron level [10] and on quark level [11].However, some different results were obtained by thesetwo methods. On hadron level [10], it is found thatmolecular bound states of N Λ c are plausible in both theone-pion-exchange potential model and the one-boson-exchange potential model. On quark level [11], our groupfound the attraction between N and Λ c is not strongenough to form any N Λ c bound state within our quarkdelocalization color screening model (QDCSM). Whereasthe attraction between N and Σ c is strong enough to forma bound state N Σ c ( S ), it becomes a resonance state bycoupling to the open N Λ c D − wave channels. We also ex-plored the corresponding states N Λ b , N Σ b and the simi-lar properties as that of states N Λ c , N Σ c were obtained.Recently, the possible Λ c Λ c molecular state was studiedin the one-boson-exchange potential model [12] and in theone-pion-exchange potential model [13] on hadron level.Different results were obtained by these two models. TheΛ c Λ c does not exist in the former model, whereas themolecular bound state of Λ c Λ c is possible in the latermodel. So the quark level study of the Λ c Λ c system isinteresting and necessary.The quark delocalization color screening model (QD-CSM) was developed in the 1990s with the aim of ex-plaining the similarities between nuclear and molecularforces [14]. The model gives a good description of N N and
Y N interactions and the properties of deuteron [15].It is also employed to calculate the baryon-baryon scat-tering phase shifts in the framework of the resonatinggroup method (RGM), and the dibaryon candidates arealso studied with this model [16, 17]. Recent study alsoshow that the intermediate-range attraction mechanismin the QDCSM, quark delocalization and color screen-ing, is an alternative mechanism for the σ -meson ex-change in the most common quark model, the chiralquark model [16, 17]. In the frame of QDCSM, the H dibaryon were also obtained [18]. Therefore, it is very in-teresting to investigate whether a H -like dibaryon stateΛ c Λ c exist or not in QDCSM.In present work, QDCSM is employed to study theproperties of Λ c Λ c systems. the channel-coupling effect ofΣ c Σ c , Σ c Σ ∗ c , Σ ∗ c Σ ∗ c and N Ξ cc are included. Our purposeis to understand the interaction properties of the Λ c Λ c system and to see whether an H -like dibaryon state Λ c Λ c exist or not. Extension of the study to the bottom caseis also interesting and is performed here. The structureof this paper is as follows. After the introduction, wepresent a brief introduction of the quark models used insection II. Section III devotes to the numerical results anddiscussions. The summary is shown in the last section. II. THE QUARK DELOCALIZATION COLORSCREENING MODEL (QDCSM)
The detail of QDCSM used in the present work can befound in the references [14–17]. Here, we just present thesalient features of the model. The model Hamiltonian is: H = X i =1 (cid:18) m i + p i m i (cid:19) − T c + X i 08, satisfying the re-lation µ us = µ uu ∗ µ ss [17]. When extending to the heavyquark case, there is no experimental data available, so wetake it as a common parameter. In the present work, wetake µ cc = 0 . 001 and µ uc = 0 . µ uc = µ uu ∗ µ cc . All the other parameters are takenfrom [11].The quark delocalization in QDCSM is realized byspecifying the single particle orbital wave function of QD-CSM as a linear combination of left and right Gaussians,the single particle orbital wave functions used in the or-dinary quark cluster model, ψ α ( s i , ǫ ) = ( φ α ( s i ) + ǫφ α ( − s i )) /N ( ǫ ) ,ψ β ( − s i , ǫ ) = ( φ β ( − s i ) + ǫφ β ( s i )) /N ( ǫ ) , N ( ǫ ) = p ǫ + 2 ǫe − s i / b . (2) φ α ( s i ) = (cid:18) πb (cid:19) / e − b ( r α − s i / φ β ( − s i ) = (cid:18) πb (cid:19) / e − b ( r β + s i / . Here s i , i = 1 , , ..., n are the generating coordinates,which are introduced to expand the relative motion wave-function [15]. The mixing parameter ǫ ( s i ) is not an ad-justed one but determined variationally by the dynamicsof the multi-quark system itself. This assumption allowsthe multi-quark system to choose its favorable configu-ration in the interacting process. It has been used toexplain the cross-over transition between hadron phaseand quark-gluon plasma phase [20]. III. THE RESULTS AND DISCUSSIONS Here, we perform a dynamical investigation of theΛ c Λ c system with IJ P = 00 + in the QDCSM. The chan-nel coupling effects are also considered. The labels of allcoupled channels are listed in Table I. TABLE I: The Λ c Λ c and Λ b Λ b states and the channels coupled to them.Channels 1 2 3 4 5 6 7 J P = 0 + Σ c Σ c ( S ) N Ξ cc ( S ) Λ c Λ c ( S ) Σ ∗ c Σ ∗ c ( S ) N Ξ ∗ cc ( D ) Σ c Σ ∗ c ( D ) Σ ∗ c Σ ∗ c ( D ) J P = 0 + Σ b Σ b ( S ) N Ξ bb ( S ) Λ b Λ b ( S ) Σ ∗ b Σ ∗ b ( S ) N Ξ ∗ bb ( D ) Σ b Σ ∗ b ( D ) Σ ∗ b Σ ∗ b ( D ) Because an attractive potential is necessary for formingbound state or resonance, we first calculate the effectivepotentials of all the channels listed in Table I. The effec-tive potential between two colorless clusters is defined as, V ( s ) = E ( s ) − E ( ∞ ), where E ( s ) is the diagonal matrixelement of the Hamiltonian of the system in the gener-ating coordinate. The effective potentials of the S -waveand D -wave channels are shown in Fig. 1(a) and (b)respectively. From Fig. 1(a), we can see that the poten-tials are attractive for the S channels Σ c Σ c , N Ξ cc andΣ ∗ c Σ ∗ c . While for the channel Λ c Λ c , the potential is repul-sive and so no bound state can be formed in this singlechannel. However, the attractions of Σ c Σ c channel andΣ ∗ c Σ ∗ c channel are very large, the channel coupling effectsof Σ c Σ c and Σ ∗ c Σ ∗ c to Λ c Λ c will push the energy of Λ c Λ c downward, it is possible to from a bound state. For the D channels shown in Fig. 1(b), the potentials are allrepulsive.In order to see whether or not there is any bound state,a dynamic calculation is needed. Here the RGM equa-tion is employed. Expanding the relative motion wave-function between two clusters in the RGM equation bygaussians, the integro-differential equation of RGM canbe reduced to algebraic equation, the generalized eigen-equation. The energy of the system can be obtainedby solving the eigen-equation. In the calculation, thebaryon-baryon separation ( | s n | ) is taken to be less than6 fm (to keep the matrix dimension manageably small).The single channel calculation shows that the energyof Λ c Λ c is above its threshold, the sum of masses of twoΛ c ’s. It is reasonable, because the interaction betweenthe two Λ c ’s is repulsive as mentioned above. For N Ξ cc channel, the attraction is too weak to tie the two particlestogether, so it is also unbound. At the same time, due tothe stronger attraction, the energies of Σ c Σ c and Σ ∗ c Σ ∗ c are below their corresponding thresholds. The bindingenergy of Σ c Σ c and Σ ∗ c Σ ∗ c states are listed in Table II, inwhich ’ ub ’ means unbound. For the D channels, theyare all unbound since the potentials are all repulsive, sowe leave them out from Table II. We also do a channel-coupling calculation and a bound state, which energy isbelow the threshold of Λ c Λ c , is obtained. The bindingenergy is also shown in Table II under the head ’ c.c. ’.There are several features which are discussed below.First, the individual S -wave Λ c Λ c channel is unboundin our quark level calculation, which is consistent withthe conclusion on the hadron level [12, 13]. For otherindividual channels, there are some different results. In TABLE II: The binding energy of every S channels of Λ c Λ c system and with channel coupling ( c.c. ).Channels Σ c Σ c N Ξ cc Λ c Λ c Σ ∗ c Σ ∗ c c.c. B.E.(MeV) − ub ub − − Ref. [13], the calculation shows all the individual channelsare unbound and in Ref. [12], Σ c Σ c is bound. Whilein our quark level calculation, the individual Σ c Σ c andΣ ∗ c Σ ∗ c are deeply bound.Secondly, by taking into account the channel-couplingeffect, a bound state is obtained for the Λ c Λ c system,which is also consistent with the conclusion on the hadronlevel [13]. However, the channel-coupling effect is differ-ent between our quark level calculation and their hadronlevel calculation. In Ref. [13], the coupling of Λ c Λ c tothe D -wave channels Σ c Σ ∗ c and Σ ∗ c Σ ∗ c are crucial in bind-ing two Λ c ’s. This indicates the importance of the tensorforce. This conclusion is the same as their calculationof N Λ c system [10]. While in our quark level calcula-tion, the coupling between Λ c Λ c , N Ξ cc , Σ c Σ c and Σ ∗ c Σ ∗ c channels is through the central force. The transition po-tentials of these four channels are shown in Fig. 2(a). Itis the strong coupling among these channels that makesthe Λ c Λ c ( S ) be bound state. The transition potentialsof three D -wave channels are shown in Fig. 2(b). Tosee the effects of tensor force, the transition potentialsfor S and D wave channels are shown in Fig. 3(a) and(b). From which one can see that the effects of tensorforce are much small compared with that of the centralforce. Thus the S and D wave channel-coupling effectis small in out quark model calculation. This conclusionis consistent with our calculation of N Λ c system [11], inwhich the effect of the N Σ ∗ c ( D ) channel coupling to N Λ c ( S ) is very small.Thirdly, the properties of the Λ c Λ c system in our quarkmodel is similar to that of the ΛΛ system. Our group hascalculated the H -dibaryon before [18], in which the singlechannel ΛΛ is unbound, when coupled to the channels N Ξ and ΣΣ, it becomes a bound state. Here, we extendour model to study the heavy flavor dibaryons, we findit is possible to form a bound state in the Λ c Λ c system,which it is a H -like dibaryon state.In the previous discussion, the Λ c Λ c system is investi-gated and a H -like dibaryon state is found. Because ofthe heavy flavor symmetry, we also extend the study tothe bottom case of Λ b Λ b system. The numerical results V ( s ) ( M e V ) s (fm) (a) S-wave (11) (22) (33) (44) 0 1 2 3-1000100200300400 V ( s ) ( M e V ) s (fm) (55) (66) (77)(b) D-wave FIG. 1: The potentials of different channels for the J P = 0 + case of the Λ c Λ c system. V ( s ) ( M e V ) s (fm) (12) (13) (14) (23) (24) (34) (a) s (fm) (56) (57) (67)(b) V ( s ) ( M e V ) FIG. 2: The transition potentials of (a): S -wave channels and (b): D -wave channels for the J P = 0 + case of the Λ c Λ c system. for the N Λ b system are listed in Figs. 4 and Table III.The results are similar to the Λ c Λ c system. From TableIII, we also find there is also a H -like dibaryon state inthe Λ b Λ b system in our quark model. IV. SUMMARY In this work, we perform a dynamical calculation of theΛ c Λ c system with IJ P = 00 + in the framework of QD-CSM. Our results show that the interaction between twoΛ c ’s is repulsive, so it cannot be a bound state by itself.The attractions of Σ c Σ c and Σ ∗ c Σ ∗ c channels are strong V ( s ) ( M e V ) s (fm) (15) (16) (17) (25) (26) (27)(a) V ( s ) ( M e V ) (35) (36) (37) (45) (46) (47)(b) s (fm) FIG. 3: The transition potentials of S − D wave channels for the J P = 0 + case of the Λ c Λ c system. V ( s ) ( M e V ) (a) S-wave (11) (22) (33) (44) s (fm) V ( s ) ( M e V ) (b) D-wave (55) (66) (77) s (fm) FIG. 4: The potentials of different channels for the J P = 0 + case of the Λ b Λ b system. enough to bind two Σ c ’s and two Σ ∗ c ’s together. It is pos-sible to form a H -like dibaryon state in the Λ c Λ c systemwith the binding energy 22 MeV in our quark model byincluding the channel-coupling effect. This result is con-sistent with the result of the calculation on the hadronlevel [13]. However, the effect of the channel couplingis different between these two approaches. The role ofthe central force is much more important than the tensor force in our quark level calculation, while in the calcula-tion on the hadron level [13], the tensor force is shownto be important and the D -wave channels are crucial inbinding two Λ c ’s. Further investigation should be doneto understand the difference between the approaches onthe hadron level and the quark level. It will help us tounderstand the quark-duality and exotic quark states.Extension of the study to the bottom case has also TABLE III: The binding energy of every S channels of Λ b Λ b system and with channel coupling ( c.c. ).Channels Σ b Σ b N Ξ bb Λ b Λ b Σ ∗ b Σ ∗ b c.c. B.E.(MeV) − ub ub − − been done. The results of Λ b Λ b system is similar to theΛ c Λ c system, and there exits a H -like dibaryon state inthe Λ b Λ b system with a binding energy of 19 MeV in ourquark model. On the experimental side, finding the H -like dibaryon states Λ c Λ c and Λ b Λ b will be a challengingsubject. [1] R. L. Jaffe, Phys. Rev. Lett. , 195 (1977).[2] R. L. Jaffe, Phys. Rep. , 1 (2005) and referencetherein.[3] M. Oka, K. Shimizu and K. Yazaki, Nucl. Phys. A ,700 (1987).[4] M. Oka, Phys. Rev. D , 298 (1988).[5] S. R. Beane, E. Chang, W. Detmold, et al. , Phys. Rev.Lett. , 162001 (2011).[6] T. Inone, N. Ishii, S. Aoki, et al. , Phys. Rev. Lett. ,162002 (2011).[7] T. F. Carames and A. Valcarce, Phys. Rev. C , 045202(2012).[8] F. Fromel, B. Julia-Diaz and D. O. Riska, Nucl. Phys. A , 337 (2005).[9] B. Julia-Diaz and D. O. Riska, Nucl. Phys. A , 431(2005).[10] Y. R. Liu and M. Oka, Phys. Rev. D , 014015 (2012).[11] H. X. Huang, J. L. Ping and F. Wang, Phys. Rev. C ,034002 (2013).[12] N. Lee, Z. G. Luo, X. L. Chen and S. L. Zhu, Phys. Rev.D , 014031 (20111).[13] W. Meguro, Y. R. Liu and M. Oka, Phys. Lett. B ,547 (2011).[14] F. Wang, G. H. Wu, L. J. Deng and T. Goldman, Phys.Rev. Lett. , 2901 (1992); G. H. Wu, L. J. Teng, J. L. Ping et al. , Phys. Rev. C , 1161 (1996).[15] J. L. Ping, F. Wang and T. Goldman, Nucl. Phys. A ,95 (1999); G. H. Wu, J. L. Ping, L. J. Teng et al. , Nucl.Phys. A , 279 (2000); H. R. Pang, J. L. Ping, F. wangand T. Goldman, Phys. Rev. C , 014003 (2001); J. L.Ping, F. Wang and T. Goldman, Nucl. Phys. A , 871(2001); J. L. Ping, H. R. Pang, F. Wang and T. Goldman,Phys. Rev. C , 044003 (2002).[16] L. Z. Chen, H. R. Pang, H. X. Huang, J. L. Ping andF. Wang, Phys. Rev. C , 014001 (2007); J. L. Ping,H. X. Huang, H. R. Pang, F. Wang and C. W. Wong,Phys. Rev. C , 024001 (2009); H. X. Huang, P. Xu, J.L. Ping and F. Wang, Phys. Rev. C , 064001 (2011).[17] M. Chen, H. X. Huang, J. L. Ping and F. Wang, Phys.Rev. C , 015202 (2011).[18] F. wang J. L. Ping, G. H. Wu, L. J. Teng and T. Gold-man, Phys. Rev. C , 3411 (1995); H. R. Pang, J. L.Ping, F. wang, T. Goldman and E. G. Zhao, Phys. Rev.C , 065207 (2004).[19] A. Valcarce, H. garcilazo, F. fernandez and P. Gonzalez,Rep. prog. Phys. , 965 (2005) and reference there in.[20] M. M. Xu, Y. M. Liu and L. S. Liu, Phys. Rev. Lett.100