The production of charm pentaquark from B meson within SU(3) analysis

We study the masses and production modes of pentaquark with the quark constituent cq¯qqq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\bar{q}qqq$$\end{document}, by the tridiquark-diquark model and systematical light flavor quark symmetry SU(3) analysis. The mass spectrums show that the S-wave singly charm pentaquark cn¯nnn,cs¯ssn,cn¯ssn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c{\bar{n}}nnn,~c{\bar{s}}ssn,~c{\bar{n}}ssn$$\end{document} and cn¯snn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c{\bar{n}}snn$$\end{document}(n=u,d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n=u,d)$$\end{document} are above their strong decay thresholds, while cs¯nnn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c{\bar{s}} nnn$$\end{document} and cs¯snn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\bar{s} snn$$\end{document} with parity 12-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}^-$$\end{document} are below their strong decay thresholds, which imply the possibility of stable states. Furthermore, we discuss the production of the concerned 15 states from B meson, the analysis within SU(3) symmetry yields the golden channels of production process, B¯s0→Fs¯udd+p¯,B-→Fu¯dds-Λ¯0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{B}}^0_s\rightarrow F_{\bar{s}udd}^{+} {\overline{p}},\ B^-\rightarrow F_{\bar{u}dds}^{-} {\overline{\Lambda }}^0$$\end{document}, which expected to be work worthily in b-factory.

In the letter, we wound consider the charm pentaquark within the triquark-diquark model.The model is a suitable appropriate [30,31] to deal with the many quarks system, in this framework, the diquark with spin 0, color anti-triplet, known as good diquark |(q ′ q ′′ ) S(0) C( 3) >, and triquark with spin 1 2 , color triplet, builded as |(cq q) S( 12 ) C(3) >.The lowest-lying pentaquark states {cq q} − {q ′ q ′′ }, whose orbital angular momentum L = 0 and parity J P = 1 2 − , accordingly are divided into two non-singlet color clusters, which combining through the color-triplet binding mechanism [32].To investigate the mass splitting of singly charm pentaquark, we adopt the effective Hamiltonian approach, which apart from the constituent quark and diquark masses, including dominant spin-spin and spin-orbit interactions [33].
Note that the light quark symmetry have been successfully adopted in generic hadronic system [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48], which provides a general insight for the decays and productions of hadron, then to be an useful approach to discuss the pentaquark production from B meson.The study is based on the representations of pentaquark, final hadrons and weak transition operator.After constructing the corresponding production Hamiltonian, naturally expanding into transition matrix element, production channels can be attained.A large branching ratios and relatively clear final products wound be better for the choosing golden production channels.We prefer the production channels, which are expected to be observed in b-factory.
The paper is organized as follows.In Sec.II, we present the discussion of the mass splitting of the charm pentaquark.In Sec.III, we analysis the production of the pentaquark states from B meson decays.Some further comments are proposed in Sec.IV.

II. THE MASS SPLITTING OF THE CHARM PENTAQUARK STATES
We study the mass spectrum of S-wave pentaquark states cqqqq(q = u, d, s) in the framework of non-relativity doubly heavy triquark-diquark model.Under the diquark-triquark picture, the pentaquark states can be labeled as |((cq)3(q)3) C=3 (q ′ q ′′ ) C= 3 , with two light quarks q ′ q ′′ into color 3 state and singly heavy diquark cq in color 3 plus a light anti-quark q in color 3. Following with effective Hamiltonian approach [33], the mass spectrum of charm pentaquark with hyperfine structure from spin-spin interactions, can be written as H ld = m ld + 2(K qq ′ )3(S q • S q ′ ) + 2(K cq ′ )3(S c • S ′ q ) + 2(K qq ′′ )3(S q • S q ′′ ) + 2(K cq ′′ )3(S c • S q ′′ ) + 2(K qq ′ )3(S q • S q ′ ) + 2(K q ′ q)(S ′ q • S q) + 2(K q ′′ q)(S q ′′ • S q). ( The Hamiltonian H t is related with the color triquark, where m q and m hd are the constituent masses of the antiquark and singly charm diquark, respectively.The remanent terms of H t describe the spin-spin interactions in the singly charm diquark and between the diquark constituents and the antiquark.Among the three spin-spin couplings (K cq )3, (K cq ) and (K q q ), the spin-spin interaction inside the diquark (K cq )3 is argued to be the dominant one.The values of the spin-spin couplings and the masses of quarks and diquarks [33,50].The H ld term in the Hamiltonian contains the operators responsible for the spin-spin interaction in the light diquark and its interaction with the triquark.In the singly heavy triquarkdiquark system, the suggested spin of singly heavy charm diquark is S cq = 0, 1. Similarily, the spin of "good" light diquark in the S-wave pentaquark state cqqqq is chosen as S q ′ q ′′ = 0 [49].
Accordingly, we can write directly the possible configuration of S-wave pentaquark cq qqq, signed as |S cq , S t , L t ; S q ′ q ′′ , L q ′ q ′′ ; S, L .Here the light diquark q ′ q ′′ can be any one of the constituents (ud, du, us, su, ds, sd).Except for the spin of singly heavy diquark S cq and light diquark S q ′ q ′′ , the S-wave states with orbital angular momentum L t = L qq ′ = L = 0, the spin of triquark ((cq)q) S t turns out to be 1/2 or 3/2.
The states with parity J P = 1 2 − and J P = 3 2 − , sandwiching the effective mass Hamiltonian Eq. ( 1), then yield the mass spectrum matrix of S-wave pentaquark cqqqq, In particular, the determination of spin-spin interaction between three spins inside the triquark, i.e., S c • S q and S q • S q, can be drawn by the wigner 6j-symbols.Further more, for the interaction between triquark and light diquark, such as S q • S q ′ and S q • S q ′′ , it is convenient to utilize the wigner 9j-symbols to describe the recouplings.The remaining step is the numerical calculation.In this work, the spin-spin coupling and the mass of quark and diquark can be taken as in Tab.II.
Certainly, one should consider the uncertainties from these couplings and masses, we take 10% as the error in this work.
We  Light quarks satisfy the SU(3) flavor symmetry, and behave well at the level of hadrons.We can use group representations to describe the hadrons, in consideration of individual spin or orbital quantum number.We can transform the singly charm pentaquark cqqqq under the SU(3) symmetry, After group decomposition above, we get the irreducible representations of new combination states 6, 15 and 24.By tensor reduction, the irreducible representations can be expressed as different tensor forms that labeled with T jkl i , and the coefficients of irreducible representations can be taken as follows, The new combination states 6, 15 and 24 can be expressed as irreducible representations T 6 , T 15 and T 24 .Here the anti-symmetry index can be identified as [ij], and the symmetry indexes can be signed with {ij}.The coefficients consist of tensor δ and antisymmetric tensor ε.We can get the quark components of the states in flavor space by expanding the tensor representations which have been deduced above, and the nonzero components are listed in Tab III.We also give the weight in the 15 state, do not contain q q pair, so these six states are relatively stable [23] and wound be the concerned states in the work.
FIG. 2: The weight diagrams(a-c) show the 6, 15, 24 multiple states of singly charm pentaquark, signed as Immediately, the masses of

B. The Production by SU(3) analysis
According to the analysis of SU(3) light quark flavor symmetry, we will discuss the possible production and decay modes of the pentaquark states cqqqq in this subsection.This production can be realized by studying the weak decay of B meson.At the stage of decay modes, we focus on the explores of stable pentaquark candidates, which give priority to the weak decays similarly.The 15 state -1 2 we offer the nonzero SU(3) tensor components of Cabibbo allowed transition given as consistently.
In the quark level, there are two production processes of the pentaquark states from B meson decay.
The operator of the transition b → ccd/s form an triplet, with (H 3 ) 2 = V * cd , (H 3 ) 3 = V * cs .And the operator of the transition b → cūd/s can form an octet 8, whose nonzero composition followed as Among the calculation of the production with SU(3) symmetry analysis, the representations of initial and final states are essential inputs.The initial states, B meson, including a bottom quark and one light quark, B i = B − , B 0 , B 0 s .The representations of light anti-baryons can be decomposed into an octet and an anti-decuplet [42].The decuplet can be referred to [50], the octet can be written as The representation of singly anti-charm anti-baryons can be decomposed into one triple states and anther anti-sextet, which can be written respectively as The possible Hamiltonian for the production of concerned pentaquark ground 15 states from one B meson, induced by the transition b → ccd/s and b → cūd/s in the quark level, can be writen directly as, The parameters a i (b i or c i ) with i = 1, 2, 3, are the non-perturbative coefficients.
It can be found that for the concerned pentaquark T 15 states, the difference between the decay widths of different production processes from B mesons is relatively large, but they are also interrelated each other.Once anyone decay channel will be detected in the future, we can give other decay widths.And also if two decay widths can be measured, our predictions can also be tested.
Therefor, the golden channels producing the singly charm pentaquark ground states are selected in Tab.IV, from which, we screen out several finest processes for the concerned pentaquark F 15 , One can make a rough estimate about the branching ratios of the processes, by assuming the weak coupling constants for the B coupling with final pentaquark and baryon [48].Immediately, we find that the branching ratios can reach to the order of 10 −7 .
Here, the weak couplings constants are near 1.0 KeV order, and one monopole function is introduced to describe the inner structure effect of the interaction vertices, F (q 2 ) = Λ 2 Λ 2 +q 2 .The parameter Λ = 300 MeV, and q 2 is the anti-baryon three-momentum in the rest frame of the B meson.

IV. CONCLUSIONS
In this work, we study the mass splitting of the S-wave singly charm pentaquark state cqqqq(q = u, d, s) in the framework of non-relativity doubly heavy triquark-diquark model, in which the hyperfine structure from spin-spin and spin-orbit interaction.We find that cūsdd, c dsuu, c duud and csudd with parity 1 2 − are below their strong decay thresholds, which imply that they are the stable states.It should be checked in future experiments.
Within the SU(3) flavor symmetry, we discuss the production of the ground pentaquark state from B meson, several golden channels are selected, in particular, the estimation of branching ratios can reach to a sizeable order of 10 −7 .We further intend to consider the processes with more effective means, such as the effective Lagrangian method, expecting to acquire more convincing results for the experimental detection.

Appendix A: tensor decompositon
We list the tensor representations of pentaquark cqqqq with 6, 15 and 24 respectively.The 6 states T 6 can be obtained from the decomposition of 1 ⊗ 6, 8 ⊗ 3 or 8 ⊗ 6.In addition, the tensor T 24 is derived from 8 ⊗ 6.
(T 24 ) The possible Hamiltonian for the production of pentaquark ground 6 states from one B meson, induced by the transition b → ccd/s and b → cūd/s in the quark level, can be writen directly as, The production channels directly collected into Tab.VIII, in addition, we show the fully production processes about all 15 states in the Tab.VI and Tab.VII.The relations between different production widths can then be deduced, the complete results for sextet given as, The production width relations of 15 states are gathered as,

2 − 2 −
diagonalize the mass matrix and obtain the mass splittings of pentaquark cqqqq shown in Tab.II.The results from Chromomagnetic Interaction(CMI) model, QCD sum rules(QCDSR) and chiral effective field theory(ChEFT) are collected for comparison.Our study is similar with the calculation from CMI, but is different from the results of QCDSR and ChEFT, especially for the ground states with parity 1 .In our considering, both components cssnn, cnssn and cnsnn are lower than their strong thresholds Ξ c K, Ω c π and Ξ c π respectively, further more, the mass of the component csssn is close to the strong thresholds Ξ c η.It is interesting that the pentaquark with constituent cūdss, c duss with parity 1 are near their strong decay thresholds Ω c π, which may imply the possibility of stable states.The pentaquark with constituent cūdss, c duss near the corresponding threshold Ω c π, respectively below that about 178 MeV and 178 MeV, which wound

FIG. 1 :
FIG.1:The typical topology diagrams for the production of charm pentaquark P cqqqq from the bottom meson.The production depends on the weak decay of b quark, which leads to different topologies (a,b,c) including charm pentaquark, and light anti-baryon or anti-charm anti-baryon in final states.

graphs of states 6 ,
15 and 24 in Fig 2, whose flavor structures given in Appendix.A.The six states 308 MeV, 308 MeV, 319 MeV and 319 MeV.While the masses of F − −1 and F + 1 are both 2.835 GeV, nearing their strong thresholds Ω 0 c π − about 178 MeV, indicating that they may be stable pentaquark states.However the sizeable value of the inaccurate coming from the spin-spin coupling K coefficients and the mass of diquark m cq , are large enough to change the decay behaviour of the states.Accordingly the stability of the states F − −1 and F + 1 remains to be an open question.Nevertheless, the F − −3/2 , F ++ 3/2 , F ++ 1/2 and F + −1/2 with less controversial in our work, should be weakly decay, which expected to recognized in future experiments.
weak interaction of production for the pentaquark states, induced by the transition b → cqq/ccq, can be classified by the quantities of CKM matrix elements.For non-leptonic decays of b quark, we classify the transitions into two groups.b → ccs/cūd, b → ccd/cūs, which are Cabibbo allowed, and singly Cabibbo suppressed transitions respectively.The transition b → cqq can be decomposed as 3⊗ 3 = 1⊕8.The transition b → ccq can be decomposed as 3.Here

F
c3 and F c 6 are the triplet and anti-sextet of anti-charm anti-baryon, while F 8 and F 10 are octet and decuplet of the light anti-baryon.The concerned pentaquark ground states are noted with T 15 , B represents the bottom meson.In the quark level, the productions of singly charm pentaquark multiple states can be described with topological diagrams, as shown with Fig.1.We expand the Hamiltonian and collect the possible processes of concerned pentaquark 15 states, gathering into Tab.V, as well as the fully results shown in Tab.VI and Tab.VII of Appendix.B.Meanwhile, it is ready to reduce the relations of decay widths between different channels, which given as follows.

TABLE II :
The mass splitting of the S-wave pentaquark P cqqqq coming from the hyperfine structures of F ++ 1/2 and F + −1/2 are found respectively as 2.300 GeV, 2.300 GeV, 2.465 GeV and 2.465 GeV.Obviously they are lower than their corresponding strong thresholds

TABLE III :
The possible representations of pentaquark cqqqq with states 6, 15 and 24.Let S, F and T be the names of the states respectively.Meanwhile, we give the tensor representations T 6 /T 15 /T 24 , isospin I 3 and hyper-charge Y of the corresponding states.

TABLE IV :
The golden channels for the production of concerned ground pentaquark states from B mesons.

TABLE VI :
The productions of pentaquark F 15 and light anti-baryons (qqq) 8/10 from B mesons.