Chinese Physics C Vol., No. 9 (07) 09303 Spectra of charmed and ottom aryons with hyperfine interaction * Zhen-Yang Wang( ) ;) Jing-Juan Qi( ) ;) Xin-Heng Guo( ) ;3) Ke-Wei Wei( ) ;) College of Nuclear Science and Technology, Beijing Normal University, Beijing 0075, China College of Physics and Electrical Engineering, Anyang Normal University, Anyang 55000, China Astract: Up to now, the excited charmed and ottom aryon states have still not een well studied experimentally or theoretically. In this paper, we predict the mass of Ω, the only L0 aryon state which has not een oserved, to e 6069. MeV. The spectra of charmed and ottom aryons with the orital angular momentum L are studied in two popular constituent quark models, the Goldstone oson exchange (GBE) model and the one gluon exchange (OGE) hyperfine interaction model. Inserting the latest experimental data from the Review of Particle Physics, we find that in the GBE model, there exist some multiplets (Σ c(), Ξ c() and Ω c()) in which the total spin of the three quarks in their lowest energy states is 3/, ut in the OGE model there is no such phenomenon. This is the most important difference etween the GBE and OGE models. These results can e tested in the near future. We suggest more efforts to study the excited charmed and ottom aryons oth theoretically and experimentally, not only for the aundance of aryon spectra, ut also for determining which hyperfine interaction model est descries nature. Keywords: charmed aryons, ottom aryons, mass spectra, fine and hyperfine structure PACS:.0.Lq,.0.Mr, 33.5.Ta DOI: 0.0/67-37//9/09303 Introduction The Goldstone oson exchange (GBE) and one gluon exchange (OGE) hyperfine interaction terms descrie quark interactions in the constituent quark model and are popular for studying aryon spectra [ 5]. These two different kinds of hyperfine interactions have een used to descrie the oserved spectra of light aryons and ground state heavy aryons [, 5, 6]. The GBE model can correctly descrie the Roper resonance ut the OGE model cannot, as stated in Ref. [7]. This is a ig difference etween the GBE and OGE models in light aryons. With the ongoing development of experiments there should e more heavy aryons oserved experimentally in the near future, which will in turn guide theoretical studies in this area. The most important motivation of this paper is to compare the differences etween the numerical results for negative parity charmed and ottom aryons (with the orital angular momentum L) in these two models. Baryonic physics in the charmed and ottom sectors has experienced spectacular progress in recent years due to the experimental activities of the BaBar, CLEO, Belle, CDF, and LHC Collaorations and theoretical developments. Up to now, most of the charmed and ottom aryon ground states have een oserved experimentally, ut excited heavy aryon states are still poorly known []. Recently, the LHC Collaoration oserved four ottom aryon resonances, i.e. M Λ 0 (59) 59.97 MeV and M Λ 0 (590) 599.77 MeV, which are interpreted as the oritally excited states of Λ 0 [9], and 5935.0 MeV and M Ξ 5955.33 MeV, which M Ξ are expected in this mass region with spin-parity J P (/) + and J P (3/) +, respectively [0]. There are also some states which have een oserved, ut their J P numers have not een determined experimentally. For example, the charmed aryon Σ c (00) (Belle 005 []) was first reported in the decay modes Λ + c π, Λ + c π 0 and Λ + c π+, with the mass differences M(Σ c )M(Λ + ) c MeV for the neutral state, 6 +37+5 33 MeV for the charged state, and 75 ++ 3 MeV measured to e 6 ++ 33 for the douly charged state, and was also oserved y the BaBar Collaoration in 00 []. However, the J P numers of Σ c (00) have not een determined experi- Received 0 Feruary 07, Revised May 07 Supported y National Natural Science Foundation of China (7500, 57503, U05) ) E-mail: wangz-y@mail.nu.edu.cn ) E-mail: qijj@mail.nu.edu.cn 3) E-mail: xhguo@nu.edu.cn, Corresponding author ) E-mail: weikw@hotmail.com, Corresponding author Content from this work may e used under the terms of the Creative Commons Attriution 3.0 licence. Any further distriution of this work must maintain attriution to the author(s) and the title of the work, journal citation and DOI. Article funded y SCOAP 3 and pulished under licence y Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pulishing Ltd 09303-
Chinese Physics C Vol., No. 9 (07) 09303 mentally. There will e more heavy aryons to e oserved experimentally in the near future, which can help to examine which hyperfine interaction model can etter descrie the aryon spectra. In our paper, we will study the spectra of charmed and ottom aryon states with negative parity and oital angular momentum L. The constituent quark model is a simple and effective phenomenological model to study mass spectra [3, ]. With the data for the light aryons and ground state charmed and ottom aryons which are laeled with three or four stars in the Review of Particle Physics and hence have een well estalished experimentally, we can calculate the parameters in the constituent quark model. After that, masses of the oritally excited L charmed and ottom aryons can e calculated. Many calculations ased on the quark flavor group SU(3) are remarkaly consistent with experiments. When we add another degree of freedom in flavor space, known as charm or eauty, a natural generalisation is to extend the flavor group to SU() (which means the flavor space we consider is u, d, s, c or u, d, s, of SU()), which is however actually adly roken. So in the calculation we introduce a perturation term which contains two parts. One is the mass difference etween the light quark (u,d) and the heavy quark (c,), and the other is s, which is from the mass difference etween the light quarks (u and d) and the s quark. We will neglect the quark mass difference etween the light quarks (u and d) in the calculation. The remainder of this paper is organized as follows. In Section, we present the theoretical framework, which includes explicit forms of the employed hyperfine interactions etween quarks. Numerical results for the spectra of L charmed and ottom aryons are presented in Section 3. Finally, Section contains a rief conclusion. Theoretical framework In the constituent quark model, the non-relativistic Hamiltonian for a three-quark system can e expressed as [, 3] H H 0 +H hyp + 3 m i, () where m i denotes the constituent mass of the ith quark, H hyp is the hyperfine interaction etween quarks, which is often treated as a perturation, and H 0 is the Hamiltonian concerning orital motions of the quarks, which should contain two parts, namely the kinetic term and the confining potential etween quarks. Both the orital Hamiltonian H 0 and the hyperfine interaction H hyp for the three-quark system have een discussed intensively in the literature [,, 5, 3]. i. H 0 and the wave-function for a aryon system The form of H 0 employed for the non-relativistic harmonic oscillator potential in the three-quark system is as follows [, 3]: 3 p 3 i H 0 + V conf ( r ij ), () m i i where p i and m i denote the momentum and mass of the ith quark, respectively. The quantity r ij r i r j is the relative position of the (ij) pair of quarks, and V conf ( r ij ) is the confinement potential. The harmonic oscillator, as one of the most commonly used quark confinement potentials, has een successfully applied to the spectroscopy of nonstrange and strange aryon ground states and excitations [, 3]. So we take V conf ( r ij ) to e the harmonic oscillator form as follows: i<j V conf ( r ij ) mω r ij+v 0, (3) where m is the constituent mass of the light quarks (u, d), ω is the angular frequency of the oscillator interaction and V 0 represents the unharmonic part of V conf, which is treated as a constant in this paper. For charmed or ottom aryons, compared with a system including three light quarks, we can rewrite H 0 for a system including one heavy quark (c or ) as the following (here we do not consider the mass difference etween the light quarks (u, d) and s quark): 3 p 3 i H 0 m + mω ( r i r j ) +3V 0 +H 0, () i i<j where H 0 represents the corrections due to the mass difference etween the light quark and the heavy quark. If we neglect the contriution from the perturation term H 0, the exact eigenvalue of H 0 should e E 0 (N+3)ω+3V 0, (5) where N is the quantum numer of the excited state. ω can e determined from the mass difference etween the nucleon and N(00), as pointed out in Ref. []. For the perturation term H 0 of charmed or ottom aryons which comes from the heavy quark (c or ) mass difference with the light quark, as shown in Ref. [3], we take it to e flavour-dependent, H 0 m hm m 3 p i δ ih, (6) m h where m and m h represent the constituent masses of the light and heavy quarks, respectively, and the Kronecker symol δ ih is a flavor dependent parameter with value for a heavy quark and 0 for a light quark. In the constituent quark model which is governed y the aove non-relativistic Hamiltonian, we introduce i 09303-
Chinese Physics C Vol., No. 9 (07) 09303 the three-quark wave function, which is factorized into orital colour flavour spin. In this paper, the wave function is descried y the Young pattern [f], where f is a sequence of integers that indicate the numer of oxes in the successive rows of the corresponding Young patterns. The pattern [3] represents a completely symmetric state, [] is the mixed symmetric state, and [] is the completely antisymmetric one. Due to the Pauli principle, the wave function of a three-quark system must e totally antisymmetric under the exchange of any quark pair, so it can e written as [] XCF S with the suscripts X, C, F and S (we use S to represent the total spin of the three quarks in the following) representing orital, colour, flavour and spin degrees of freedom, respectively. For the L 0 aryon state all the quarks are in the orital ground state with [3] X configuration, and for the L state two quarks are in the orital ground state and the other in the P state with the [] X configuration. Because of colour confinement, the colour wave function must e [] C. There are three possile flavour wave functions for a aryon system: [3] F, [] F and [] F in the Weyl taleaux of the SU(3) group [5 7]. The total spins could e S/ and S3/, corresponding to [] S and [3] S configuration, respectively. The explicit wave functions ased on the orital-colour-flavour-spin configurations can easily e derived from the Clesch- Gordan coefficients. In the case of N 0, all the three quarks are in their ground state, so the matrix elements of H 0 are the same for all the N 0 configurations [3]: <g.s. H g.s.> 0 δ, (7) where g.s. > represents the ground state and δ ( m/m h )ω. The perturation (6) is flavor dependent and its matrix elements etween different P shell multiplets of Λ + c take the following values [3]: <Λ + c H 0 Λ + c > []F S [] F [] S 3 δ, <Λ + c H 0 Λ + c > []F S [] F [] S 3 δ, <Λ + H c 0 Λ+ > c [] F S [] F [3] S 7 δ, () where the suscript [] F S [] F [] S, for example, means the configuration with [] F S flavour-spin symmetry, [] F flavour wave-function and [] S spin wavefunction, and similarly for the other suscripts. For the P shell excitations of Σ c, the matrix elements of H 0 are [3] <Σ + c H 0 Σ + c > []F S [] F [] S 3 δ, <Σ + c H 0 Σ+ c > [] F S [3] F [] S 3 δ, <Σ + H c 0 Σ+ > c [] F S [] F [3] S 3 δ. (9) For the L negative parity excitations of Ξ c, corrections arising from H 0 are the same as those of, and for the negative parity excitations of Ξ c and Ω c, the corresponding corrections are equal to those of Σ c.. Hyperfine interactions etween quarks To calculate the mass splittings of the degenerate configurations, explicit perturative hyperfine interactions are needed. Since the GBE [, 3] and OGE [, 3, 0] interactions etween quark pairs have een discussed intensively efore, we just present a very rief review and apply them to charmed and ottom aryons. For charmed and ottom aryons, the GBE hyperfine Hamiltonian can e written in the following form, as in Ref. [, ]: H GBE i<j V M λ a i λa j σ i σ j, (0) where λ a i (a,,) are the SU() extension of the SU(3) Gell-Mann matrices in flavour space, σ i are Pauli spin matrices (the suscrie i and j represent the ith and jth quarks, respectively), and V M is a flavor dependent parameter to descrie the strength of the exchange of a meson M (M contains π, K, η, D, D s, B and B s mesons). Because η c and J/ψ are purely c c mesons, we do not need to consider the fifteenth Gell-Mann matrix λ 5 of SU(). Explicitly, the hyperfine interaction Eq. (0) etween two quarks has the following form for the GBE interaction in the case of the SU() extension: H GBE { 3 7 V π λ a i λa+ j V K λ a i λa+v j ηλ i λ j i<j a a } + V D λ a i λa+ j V Ds λ a i λa j σ i σ j. () a9 a3 For the OGE interaction [], the commonly used hyperfine interaction can e written as: H OGE i,j C i,j λ C i λc j σ i σ j, () where the λ C i and σ i are Gell-Mann SU(3) matrices in colour space and Pauli spin matrices, respectively, and C i,j are the colormagnetic interaction strengths. 09303-3
Chinese Physics C Vol., No. 9 (07) 09303 3 Numerical results In this section, we present the numerical results for the L charmed and ottom aryon spectra using the hyperfine interactions given in the GBE and OGE models. Before that we should fix the parameters in these models. For the constituent quark masses we take the values from Refs. [,, 5], which are determined y fitting the experimental aryon masses, i.e. m u m d 360 MeV, m s 530 MeV, m c 700 MeV, m 503 MeV. The angular frequency is determined from the mass difference etween the nucleon and N(00) [, 3], ω 57.3 MeV. All other parameters in these two different hyperfine interaction models will e otained from the ground aryon state splittings, which will e discussed in the following. 3. Fine structure corrections of the light, charmed and ottom ground aryons Generally, the fine structure corrections (δm) contain three parts, the hyperfine interaction, the difference s etween the constituent masses of the light quarks (u and d) and s quark, and the energy shift in Eq. (6) which is caused y the heavy quark mass difference. For L0 states, the energy shift is δ. For the GBE and OGE models, all these corrections (δm) are presented in Tale and Tale, where all the masses of aryons are taken from the Review of Particle Physics [], except for Ω, which has not een oserved experimentally. We calculate the mass of Ω elow. In the GBE hyperfine interaction, we assume that V qq η in qs pair state are equal to the in qq pair state and V qs η exchange potential from π and K as in Ref. [], respectively. In the OGE hyperfine interaction, C qq is a flavor dependent strength parameter. The parameters V π, V K, C qs and C qq can e otained from the N(939) (3) and Σ(93)Σ (35) mass splittings: Therefore, M (3) M N(939) 0V π 6C qq, M Σ (35)M Σ(93) 0V K 6C qs. (3) V π 9.3 MeV, C qq.3 MeV, V K 0.3 MeV, C qs.96 MeV. () To determine the parameters V D, V Ds, C qc and C sc, we consider the Σ c Σ c and Ω c Ω c mass splittings: Therefore, M Σ c M Σc 6V D 6C qc, M Ω c M Ωc 6V Ds 6C sc. (5) V D 0.77 MeV, C qc.0 MeV, V Ds.7 MeV, C sc. MeV. (6) The mass splitting etween Σ and Σ is: Therefore, M Σ M Σ 6V B 6C q. (7) V B 3.37MeV, C q.6mev. () Then we consider the Ξ Ξ mass splitting, M Ξ M Ξ 3V B +3V Bs C q +C s. (9) Sustituting V B 3.37 MeV, C q.6 MeV, and the masses of Ξ and Ξ into Eq. (9), we can get V Bs 3.0 MeV and C s. MeV. In Refs. [, 3], it is pointed out that V K ( mu m s )V π, V ss ( mu m s )V K. Therefore, V ss V K V π.0 MeV. All the parameters in the two different hyperfine interaction models are summarized in Tale 3. Tale. Fine structure corrections (δm) to the masses (in MeV) of the light ground state aryons (L0) from the GBE interaction and OGE interaction. The experimental values are from Ref. []. [f] C [f] F S [f] F [f] S state δm (GBE) δm (OGE) Exp. mass [] C [3] F S [] F [] S N(939) V π δ Cqq δ 93.9 [] C [3] F S [3] F [3] S (3) V π δ Cqq δ 3 [] C [3] F S [] F [] S Λ(6) V π6v K + s δ Cqq+ s δ 5.6 [] C [3] F S [] F [] S Σ(93) [] C [3] F S [3] F [3] S Σ (35) 3 Vπ 3 3 V K+ s δ 3 Cqq+ 3 3 Cqs+ s δ 93.5 3 Vπ 3 V K+ s δ 3 Cqq+ 6 3 Cqs+ s δ 3.57 [] C [3] F S [] F [] S Ξ(3) 3 3 V K 3 V ss η + s δ 3 3 Cqs+ 3 Css+ s δ 3.9 [] C [3] F S [3] F [3] S Ξ (530) 3 V K 3 Vss+ s δ 6 3 Cqs+ 3 Css+ s δ 533. [] C [3] F S [3] F [3] S Ω (67) V ss η +3 s δ Css+3 s δ 67.5 09303-
Chinese Physics C Vol., No. 9 (07) 09303 Tale. Fine structure corrections (δm) to the masses (in MeV) of the charmed and ottom ground aryons (L0) from the GBE interaction and the OGE interaction. The experimental values are from Ref. []. [f] C [f] F S [f] F [f] S state δm (GBE) δm (OGE) Exp. mass [] C [3] F S [] F [] S V π6v D δ Cqq δ 6.6 [] C [3] F S [] F [] S Σ c 3 Vπ0V D δ 3 Cqq 3 3 Cqc δ 5.90 [] C [3] F S [3] F [3] S Σ c 3 VπV D δ 3 Cqq+ 6 3 Cqc δ 57.50 [] C [3] F S [] F [] S Ξ c V K 3V D 3V Ds + s δ Cqs+ s δ 67.0 [] C [3] F S [] F [] S Ξ c 3 V K5V D 5V Ds + s δ 6 6 Cqs Cqc 3 3 3 Csc+ s δ 575.60 [] C [3] F S [3] F [3] S Ξ c 3 V KV D V Ds + s δ 3 Cqs+ 3 Cqc+ 3 Csc+ s δ 65.90 [] C [3] F S [] F [] S Ω c 3 V ss η 0V D s + s δ 3 Css 3 3 C sc+ s δ 695.0 [] C [3] F S [3] F [3] S Ω c 3 V ss η V Ds + s δ 3 Css+ 6 3 Csc+ s δ 765.90 [] C [3] F S [] F [] S Λ V π6v B δ Cqq δ 569.50 [] C [3] F S [] F [] S Σ 3 Vπ0V B δ 3 Cqq 3 3 C q δ 53. [] C [3] F S [3] F [3] S Σ 3 VπV B δ 3 Cqq+ 6 3 C q δ 533.6 [] C [3] F S [] F [] S Ξ V K 3V B 3V Bs + s δ Cqs+ s δ 579.9 [] C [3] F S [] F [] S Ξ 3 V K5V B 5V Bs + s δ 3 Cqs 6 3 C q 6 3 C s+ s δ 535.0 [] C [3] F S [3] F [3] S Ξ 3 V KV B V Bs + s δ 3 Cqs+ 3 C q+ 3 C s+ s δ 5955.33 [] C [3] F S [] F [] S Ω 3 V ss η 0V Bs + s δ 3 Css 3 3 C s+ s δ 60. [] C [3] F S [3] F [3] S Ω 3 V ss η V B s + s δ 3 Css+ 6 3 C s+ s δ Tale 3. Parameters (in MeV) of the two hyperfine interaction models. GBE V π 9.3 V K 0.3 V ss.0 V D 0.77 V Ds.7 V B 3.37 V Bs 3.0 OGE C qq.3 C qs.96 C ss 7. C qc.0 C sc. C q.6 C s. For the L0 aryon state, Ω is the only state which has not een oserved experimentally. From Tale, for oth GBE and OGE models, we find that (M Ω M Ω )+(M Σ M Σ )(M Ξ M Ξ ). (0) Note that there is no difference for the relations in Eq. (0) etween these two different hyperfine interaction models. We predict M Ω 6069. MeV from Eq. (0). We will discuss the L charmed and ottom aryons in the GBE and OGE models in the next susection. 3. Masses of charmed and ottom aryon states with L With all the fixed hyperfine interactions parameters and the configurations of the negative parity charmed and ottom aryon systems with L outlined in Tale, the masses of the charmed and ottom aryon states can e calculated. There are three steps to otain numerical results in our models. First, one has to calculate the fine structure corrections of charmed and ottom aryon configurations with L from these two different kinds of hyperfine interactions. These can e otained y calculating the matrix elements of the hyperfine interations in Eq. (0) and Eq. (), s and the energy shift in Eq. (6). Second, one should calculate the mass of the configurations from the mass splittings etween charmed and ottom aryon states with L0. Finally, y diagonalization of the matrices, we can get the masses of the aryon states. 09303-5
Chinese Physics C Vol., No. 9 (07) 09303 Tale. Flavor-spin configurations for the charmed and ottom aryon systems with L. configuration multiplet () [] X [] c[] F S [] F [] S, 3,() () [] X [] c[] F S [] F [] S, 3,() () 3 [] X [] c[] F S [] F [3] S Σ c() [] X [] c[] F S [] F [] S, 3 Σ c() [] X [] c[] F S [3] F [] S Σ c() 3 [] X [] c[] F S [] F [3] S, 3 Ξ c() [] X [] c[] F S [] F [] S Ξ c() [] X [] c[] F S [] F [] S Ξ c() 3 [] X [] c[] F S [] F [3] S, 3 Ξ c() [] X [] c[] F S [] F [] S Ξ c() [] X [] c[] F S [3] F [] S Ξ c() 3 [] X [] c[] F S [] F [3] S, 3 Ω c() [] X [] c[] F S [] F [] S Ω c() [] X [] c[] F S [3] F [] S Ω c() 3 [] X [] c[] F S [] F [3] S, 3, 5,(), 3,Σ c(), 3,Σ c(), 5,Σ c(), 3,Ξ c(), 3,Ξ c(), 5,Ξ c(), 3,Ξ c(), 3,Ξ c(), 5,Ξ c(), 3,Ω c(), 3,Ω c(), 5,Ω c() For the Λ + c multiplet, (595) + and (65) +, with J P and J P 3, respectively, M Λc(595) + 59.5 MeV, M Λc(65) + 6. MeV from the latest Review of Particle Physics []. Then we can easily get the mass splitting M Λc(65) + M (595) + 35.6 MeV. Similarly, for the states Ξ c (790) + (J P ) and Ξ c (5) + (J P 3 ), the mass splitting is 7.5 MeV. There are also two oritally excited singly ottom aryons measured experimentally: Λ (59) and Λ (590) with J P and J P 3, respectively, and M Λ (59) 59. MeV, M Λ (590) 599. MeV. The mass difference of these two states is small (< MeV). The spin-orital interaction has a smaller influence on the mass of the charmed and ottom aryons with the increase of constituent quark masses. As Capstick and Isgur pointed out in Ref. [3], the spin-orit terms are quite small. So we can safely neglect the impact of spin-orit coupling on our calculation. Since we have neglected spin-orital effects, S ecomes a good quantum numer. Because and other states have two configurations with the same total spin S/ as listed in Tale, we need to consider the mixing of these two configurations. However, for the total spin S / and S 3/ states the mixing is zero ecause [] S is orthogonal to [3] S. Then after explicit derivation, the matrices of fine structure corrections in these two models are: H GBE H GBE Σ c H GBE Ξ c H GBE Ξ c 3 V πv D 3 δ 9 V π+ 3 V D 0 9 V π+ 3 V D 3 V π+v D 39.3.9 0 3 V πv D 7.9 03.9 0, δ 0.5 3 V πv D 3 δ 9 V π+ 3 V D 0 9 V π+ 3 V D 3 V π+v D 65..7 0 3 V π+v D 3.7 0.5 0, δ 0.5 3 V KV D V Ds + s 3 δ 9 V K+ 3 V D+ 3 V D s 0 9 V K+ 3 V D+ 3 V D s 3 V K+V D +V Ds + s.9.5 0 3 V KV D V Ds + s 7.5 55.7 0, δ 9.3 3 V KV D V Ds + s 3 δ 9 V K+ 3 V D+ 3 V D s 0 9 V K+ 3 V D+ 3 V D s 3 V K+V D +V Ds + s 9.9 3. 3 V K+V D +V Ds + s 3 3.0 59.5 0, δ 7.5 09303-6
H GBE Ω c 3 V ss η V D s + s 3 δ 9 V ss Chinese Physics C Vol., No. 9 (07) 09303 + η 3 V D s 0 9 V ss η + 3 V D s 3 V ss η +V Ds + s 3 V ss η 53. 9. +V Ds + s 3 9.0 3.5 0, () δ 50.5 where, for example, the matrix (H GBE ) ij is the element of the matrix of i H GBE +H 0+n s Λ j, and n is the numer of s quarks in the aryon state. H OGE H OGE Σ c H OGE Ξ c H OGE Ξ c H OGE Ω c 3 (C qq+c qc ) 3 δ 3 (C qqc qc ) 0 3 (C qqc qc ) 3 (C qq+c qc ) 53.0 3. 0 3 (C qq+c qc ) 7 3. 53., δ.9 3 (C qc+c qq ) 3 δ 6 3 (C qqc qc ) 0 6 3 (C qqc qc ) 3 (C qc+c qq ) 53.0 76. 0 3 (C qc+c qq ) 3 76. 53., δ.6 3 (C us+c uc +C sc )+ s 3 δ 3 (C qsc qc C sc ) 0 3 (C qsc qc C sc ) 3 (C us+c uc +C sc )+ s 3 (C us+c uc +C sc )+ s 7 δ 3.9 0.6 0 0.6 3.9 0, 5. 3 (C us+c uc +C sc )+ s 3 δ 3 (C qsc qc C sc ) 0 3 (C qsc qc C sc ) 3 (C us+c uc +C sc )+ s 3.9. 0. 3.9 0, 3.5 3 (C sc+c ss )+ s 3 δ 6 3 (C ssc sc ) 0 6 3 (C ssc sc ) 3 (C sc+c ss )+ s 3 (C us+c uc +C sc )+ s 3 δ 3 (C sc+c ss )+ s 3 δ 0. 3.5 0 3.5 0. 0. 9. () From Eqs. () to (6) and the corresponding calculation aove, we can find that the eigenvalues for the configurations are: 3 E m i +(N+3)ω+3V 0 + H GBE(OGE) + H +n s, (3) i where m i denotes the constituent mass of the ith quark, and N and n represent the quantum numer of the excited state and the numer of s quarks in the aryon state, respectively. Then, the expressions for the mass splittings etween the L and L0 states are: M L M L0 6 3 V π+v D 6 δ+ω, 6 3 C qq 6 3 C qc 6 δ+ω, () 09303-7
Chinese Physics C Vol., No. 9 (07) 09303 M L M L0 M L 3 M L0 6 3 V π+v D 6 δ+ω, 6 3 C qq 6 3 C qc (5) 6 δ+ω, 3 3 V π+v D δ+ω, 3 3 C qq+ 6 3 C qc (6) δ+ω, where the first (second) line in each equation is the result in the GBE (OGE) model. Inserting the parameters listed in Tale 3 and the values for M L0 into the aove expressions, one can easily get the masses M L,,3. Similarly, we can also otain the masses of Σ c, Ξ c, Ξ c, Ω c states with L y considering the mass splittings etween them and their corresponding ground states, which are listed in Tale. These masses just represent the energies of the configurations and are listed in the diagonal terms of matrices (7) and (), ut are not the real charmed aryons physical masses, which will e calculated later. All the input masses of corresponding ground states are taken from the latest Review of Particle Physics []. Then we can get matrices H (H represents the nonrelativistic Hamiltonian for a three-quark system) for every multiplet configuration, and the numerical values are listed in the following matrices: Ξ c Ξ c 600.9.9 0 753..7 0.9 665.6 0, Σ c.7.5 0, 79. 70. 735..5 0 56.6 3..5 03., Ξ 3.0 9.3 0, c 93.9 3.5 99. 3. 0 53. 76. 0 3. 99. 0, Σ c 76. 53. 0, 650.3 63. 65.7 0.6 0 67.0. 0 0.6 65.7 0, Ξ. 67., c 96.7 39. Ω c Ω c 966. 9. 9.0 3036. 0, (7) 96. 0. 3.5 0 3.5 0. 0. () 96. In the GBE model, from matrix (7), one can see the lowest energy states of three multiplets (Σ c, Ξ c and Ω c) have spin 3/. According to our analysis, this is ecause the contriutions from fine structure corrections to Σ c, Ξ c and Ω c states with spin 3/ are smaller than those states with spin / in the GBE model from the matrix (). However, in the OGE model there is no such phenomenon. This is the most important difference etween the GBE and OGE models. By diagonalizing matrices (7) and (), we can get the energies for the physical charmed aryon states as shown in Tale 5, in which we also show these states as the linear cominations of the configurations given in Tale, with the corresponding coefficients for the cominations listed in Tale 5. From Tale 5 we can also see the spin of the states with the lowest energy is also S3/ for Σ c, Ξ c and Ω c states only in the GBE model, ut not in the OGE model. As shown in Tale 5, in the GBE model, the mixing of the configurations in the GBE model is much weaker than that in the OGE model. For instance, for the states, the mixing coefficient etween the configurations () and () is aout 0.6, ut in the OGE model it is aout 0.7. According to our calculation results in matrix (), the diagonal matrix elements and nondiagonal matrix elements have the same results when the spin is / in the OGE model. So the mixing coefficients etween the configurations with spin / should e the same and the mixing is stronger than in the GBE model,as shown in Tale 5. The asolute values of the nondiagonal matrix elements in the OGE model are larger than those in the GBE model. For the ottom aryon states, the fine structure correction matrices are analogous to the expressions in () and (); we just need change the c quark, D and D s mesons to quark, B and B s mesons, respectively. The expressions for the mass splittings etween the negative parity ottom aryon states with L and the corresponding states of L 0 are similar to Eqs. (), (5) and (6). We will not give the explicit expressions for these mass splittings here. The numerical results for the ottom aryon configurations with L are listed in matrices (9) and (30), which are otained in the same way as for the charmed aryons in the GBE and OGE models. We also consider the ottom aryon configuration mixing, and the masses and corresponding coef- 09303-
Chinese Physics C Vol., No. 9 (07) 09303 ficients for the mixing are listed in Tale 6. There are two differences, however, etween ottom aryon states and charmed aryon states. The first is that the larger constituent mass of the quark leads to the increase of the mass difference correction in Eq. (6), and the second is that the hyperfine interaction contriutions from the GBE interaction Eq. (0) and the OGE interaction Eq. () to the cases of the ottom aryon states should e less important than those for the charmed aryon states in these two models. This is ecause the parameters as listed in Tale 3 for ottom aryon states are smaller than for charmed aryon states. Intriguingly, from Tale 6, we can also see the spins of the lowest energy states in the GBE model for Σ, Ξ and Ω are S 3/. The same phenomenon has een found for charmed aryon states. This is the most special aspect of the GBE model compared with the OGE model. Tale 5. Energies and coefficients for mixing etween the configurations with S / and S 3/ for the charmed aryon states with L in the GBE and OGE models. GBE 3 Σ c Σ c Σ c 3 Ξ c Ξ c Ξ c 3 595. 0.965 0.6 0 7.7 0.965-0.6 0 735.3 0.999 0. 670.7-0.6 0.965 0 3.6 0.6 0.965 3.0-0..999 0 79. 70. 93.9 Ξ c Ξ c Ξ c 3 Ω c Ω c Ω c 3 56.5 0.999 0.05 0 96.9 0.99 0.5 0 9. -0.05 0.999 0 3037.9 0.5 0.99 0 3.5 96. OGE 3 Σ c Σ c Σ c 3 Ξ c Ξ c Ξ c 3 6. 0.707-0.707 0 37.3 0.707-0.707 0 69.5 0.707-0.707 0 537. 0.707 0.707 0 59.5 0.707 0.707 0 660.7 0.707 0.707 0 650.3 63. 96.7 Ξ c Ξ c Ξ c 3 Ω c Ω c Ω c 3 69. 0.707-0.707 0 7.9 0.707-0.707 0 7. 0.707 0.707 0 3. 0.707 0.707 0 39. 96. Λ Ξ Ξ c 595.5. 0. 5935.7 0 6090.7 60.9 6. 0 6. 6063. 0 6079. 53. 5.5 0 5.5 53. 0 5966.6 59.9.5 0.5 59.9 0 600. Σ Ξ Σ c Ξ c 605.5 3. 0 3. 607.7 0 597.6 69. 3.6 0 3.6 669.5 0 6093.6 555. 90.9 0 90.9 555. 0 595. 600. 57. 0 57. 600. 0 6096. Ω Ω c 69.0. 0. 669. 0. (9) 607.6 6.5 0. 0 0. 6.5 0. (30) 66.9 09303-9
Chinese Physics C Vol., No. 9 (07) 09303 Tale 6. Energies and coefficients for mixing etween the configurations with S states with L. and S3/ for ottom aryon Λ Λ Λ 3 Σ Σ Σ 3 Ξ Ξ Ξ 3 GBE 599.7 0.3 0.55 0 6035.7 0.3-0.55 0 600. 0.957 0.9 595.5 0.55 0.3 0 607. -0.55 0.3 0 605. 0.9.957 0 6090.7 597.6 6079. Ξ Ξ Ξ 3 Ω Ω Ω 3 6. 0.9-0.7 0 6. 0.99-0.03 0 676.3-0.7 0.9 0 669.6-0.03 0.99 0 6093.6 607.6 Λ Λ Λ 3 Σ Σ Σ 3 Ξ Ξ Ξ 3 OGE 5797.9 0.707-0.707 0 576.5 0.707-0.707 0 5956. 0.707-0.707 0 5. 0.707 0.707 0 596.3 0.707 0.707 0 603.5 0.707 0.707 0 5966.6 595. 600. Ξ Ξ Ξ 3 Ω Ω Ω 3 5953.7 0.707-0.707 0 60. 0.707-0.707 0 6067.9 0.707 0.707 0 6.3 0.707 0.707 0 6096. 66.9 Conclusions In this paper, we studied the difference etween two hyperfine interactions including the Goldstone oson exchange (GBE) and the one gluon exchange (OGE) hyperfine interaction models, y predicting the masses of charmed and ottom aryons with L negative parity. The results for the L negative parity charmed and ottom aryon masses were otained from the mass splittings etween these states and their corresponding ground states, and the input parameters were determined y fitting the experimental aryon masses. With these two models, we first expressed the fine structure correction parts (δm) for the light, charmed and ottom ground aryons as listed in Tale and Tale. Then, with the mass splittings etween ground aryon states, the parameters for these two kinds of hyperfine interaction models were extracted and listed in Tale 3. We predicted the mass of Ω, the only L 0 aryon state which has not e oserved, to e 6069. MeV. After that, the masses of the negative parity charmed and ottom aryon configurations with L were estimated from the splittings etween their corresponding charmed and ottom aryon states with L0. In our calculations, mixing etween the configurations with same spin quantum numers was also taken into account. Then the physical masses of the negative parity charmed and ottom aryon states with L were predicted after diagonalizing the matrices (7), (), (9) and (30) and we gave the corresponding coefficients for the mixing etween these configurations in the two hyperfine interaction models. From the latest Review of Particle Physics [], the splitting etween (595) + with J P and Λ c (65) + with J P 3 is 35.6 MeV, and that etween Ξ c (790) + with J P and Ξ c (5) + with J P 3 is 7.5 MeV. For Λ (59) and Λ (590) with J P and J P 3, respectively, M Λ (59) 59. MeV, M Λ (590) 599. MeV, and the mass difference etween these two states is small (< MeV). This indicates that the spin-orital interaction has a smaller impact on the masses of the charmed and ottom aryons with increase of the constituent quark masses. So in our calculation we neglected the contriution from the spin-orital interaction. It is very interesting that in the GBE model, there exist three multiplets (Σ c(), Ξ and Ω c() c()), of which the spins of their lowest energy states are 3/. However, in the OGE model there is no such phenomenon. There are also no such phenomenon for singly heavy aryons in QCD-motivated relativistic quark model [6] and hypercentral constituent quark model [7, ], and for Ω c states in chiral quark model [9] and nonrelativistic constituent quark model [30]. This is the most ovious difference etween the GBE and OGE models. According to our analysis, we find the contriutions from the diagonal matrix elements of fine structure corrections to the energies of spin / states are larger than the contriutions to the energies of the S 3/ states only in the GBE model, as listed in matrices () and (). Another ovious feature is that the mixing in the case of the ottom aryon states is stronger than that in the charmed aryon states oth in the GBE and OGE models, as listed in Tales 5 and 6. This is ecause the larger constituent mass of the quark reduces the hyperfine 09303-0
Chinese Physics C Vol., No. 9 (07) 09303 interaction contriutions to ottom aryon states compared with the charmed aryon states. We expect that our results for these two different models in this work can e tested at the LHC and other experiments in the near future. The predicted masses in this paper may e also useful for the discovery of the unoserved charmed and ottom aryon states and the J P assignment of these aryon states when they are oserved in the near future. It will also allow us to compare these two different hyperfine interaction models from their results and examine which phenomenological model can etter descrie the spectra. Therefore, more efforts should e given to study charmed and ottom aryons oth theoretically and experimentally. References L. A. Copley, N. Isgur, and G. Karl, Phys. Rev. D, 0: 76 (979) L. Y. Glozman and D. O. Riska, Phys. Rept., 6: 63 (996) 3 L. Y. Glozman and D. O. Riska, Nucl. Phys. A, 603: 36 (996) N. Isgur and G. Karl, Phys. Rev. D, 9: 653 (979) 5 N. Isgur and G. Karl, Phys. Rev. D, : 7 (97) 6 K. Maltman and N. Isgur, Phys. Rev. D, : 70 (90) 7 N. Mathur, Y. Chen, S. J. Dong, T. Draper, I. Horvath, F. X. Lee, K. F. Liu, and J. B. Zhang, Phys. Lett. B, 605: 37 (005) K. A. Olive et al (Particle Data Group Collaoration), Chin. Phys. C, 3: 09000 (0) 9 R. Aaij et al (LHC Collaoration), Phys. Rev. Lett., 09: 7003 (0) 0 R. Aaij et al (LHC Collaoration), Phys. Rev. Lett., : 0600 (05) R. Mizuk et al (Belle Collaoration), Phys. Rev. Lett., 9: 00 (005) B. Auert et al (BaBar Collaoration), Phys. Rev. D, 7: 003 (00) 3 A. De Rujula, H. Georgi, and S. L. Glashow, Phys. Rev. D, : 7 (975) S. Godfrey and N. Isgur, Phys. Rev. D, 3: 9 (95) 5 J. Q. Chen, Group Representation Theory for Physicists, (Singapore: World Scientific, 99) 6 Zhong-Qi Ma, Group Theory in Physics (in Chinese) (Beijing: Science Press, 99) 7 F. E. Close, An Introduction to Quarks and Partons, (New York: Academic Press 979) N. Isgur and G. Karl, Phys. Lett. B 7: 09 (977) 9 N. Isgur and G. Karl, Phys. Rev. D, 0: 9 (979) 0 S. Capstick and W. Roerts, Prog. Part. Nucl. Phys., 5: S (000) V. Borka Jovanovic, S. R. Ignjatovic, D. Borka, and P. Jovanovic, Phys. Rev. D, : 750 (00) F. Buisseret, N. Matagne, and C. Semay, Phys. Rev. D, 5: 03600 (0) 3 S. Capstick and N. Isgur, Phys. Rev. D, 3: 09 (96) V. Borka Jovanovic and D. Borka, Rom. J. Phys., 57: 03 (0) 5 M. Karliner and J. L. Rosner, Phys. Rev. D, 90(9): 09007 (0) 6 D. Eert, R. N. Faustov, and V. O. Galkin, Phys. Rev. D, : 005 (0) 7 Z. Shah, K. Thakkar, A. Kumar Rai, and P. C. Vinodkumar, Eur. Phys. J. A, 5(0): 33 (06) Z. Shah, K. Thakkar, A. K. Rai, and P. C. Vinodkumar, Chin. Phys. C, 0(): 30 (06) 9 K. L. Wang, L. Y. Xiao, X. H. Zhong, and Q. Zhao, arxiv: 703.0930 [hep-ph] 30 B. Chen and X. Liu, arxiv:70.053 [hep-ph] 09303-