A SU(3) amplitudes for a {\cal B} or a {\cal C} decays into an octet pentaquark.

# Some Predictions of Diquark Model for Hidden Charm Pentaquark Discovered at the LHCb

## Abstract

The LHCb has discovered two new states with preferred quantum numbers and from decays. These new states can be interpreted as hidden charm pentaquarks. It has been argued that the main features of these pentaquarks can be described by diquark model. The diquark model predicts that the and are in two separate octet multiplets of flavor and there is also an additional decuplet pentaquark multiplet. Finding the states in these multiplets can provide crucial evidence for this model. The weak decays of b-baryon to a light meson and a pentaquark can have Cabibbo allowed and suppressed decay channels. We find that in the limit, for -spin related decay modes the ratio of the decay rates of Cabibbo suppressed to Cabibbo allowed decay channels is given by . There are also other testable relations for b-baryon weak decays into a pentaquark and a light pseudoscalar. These relations can be used as tests for the diquark model for pentaquark.

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Introduction

The LHCb collaboration has recently discovered two new states(1) which can be interpreted as two different pentaquarks from , followed by . This has generated a lot of theoretical investigations(2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15). The quark contents can be identified as . Although the states contain charm quarks, they are hidden charm pentaquarks because and appear together and the net charm quantum number is zero. The best fit quantum numbers and their masses are

 JP=3/2−with a mass of 4380 MeV,andJP=5/2+with a mass of 4450 MeV. (1)

Experimentally quantum numbers and for these two states are not ruled out.

The existence of these states and need to be further confirmed as there may be some other effects which can minic similar effects(8); (11); (13). If these states are genuine pentaquarks, one may ask whether they are molecular states of two hadrons or composite hadron systems(2); (3); (4); (5); (7); (10); (14), or a tightly bound five quark system , or the quarks bound in other forms(9); (12); (15). It is intriguing that the in the S-wave state has a mass very close to the . Such a molecular state was actually studied just before the experimental discovery(2). With and bound sates one can obtain both and states(3). These prompt the speculation that the pentaquarks might be molecular states. On the other hand, it has been argued that the two pentaquarks from the LHCb are five quark systems organized in and diquarks(9), and the , the diquark model for pentaquark. The diquark model also has supports from tetraquark studies(9). At present, with limited data it is not possible to distinguish whether the pentaquarks are molecular states or more tightly bound quark states or even mixture of these states. The diquark model has simple structure to analysis. We also find it very predictive. Therefore in this work we choose to study some properties of diquark model for the pentaquarks discovered at the LHCb.

Diquark Model for Pentaquarks

In the diquark model the two pentaquarks from the LHCb are five quark systems organized in and diquarks(9), and the . More explicitly indicated as the following

 P=ϵαβγ{¯cα[cq]β,s=0,1[q′q′′]γ,s=0,1,L} (2)

where the greek letters are color indices and indicates the spin.

Under flavor symmetry, the diquark transforms as and and the diquark transforms as a . Therefore the pentaquarks can have and multiplets. We indicate the pentaquarks with and by and , respectively. Assuming that the two pentaquarks have and , these two fields are component fields in octet multiplets with the following spin diquark combinations fit the picture well(9)

 PS(3/2−)=ϵαβγ{¯cα[cq]β,s=1[q′q′′]γ,s=1,L=0} PA(5/2+)=ϵαβγ{¯cα[cq]β,s=1[q′q′′]γ,s=0,L=1} (3)

We denote the octet pentaquark component fields as

 (Pji(JP))=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Σ08√2+Λ08√6Σ+8p8Σ−8−Σ08√2+Λ08√6n8Ξ−8Ξ08−2Λ08√6⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (4)

For , there should also be a decuplet (totally symmetric in sub-indices) multiplet. The component pentaquark fields are

 P111=Δ++10,P112=1√3Δ+10,P122=1√3Δ010,P222=Δ−10, P113=1√3Σ+10,P123=1√6Σ010,P223=1√3Σ−10, P133=1√3Ξ010,P233=1√3Ξ−10, P333=Ω−10. (5)

The two observed pentaquarks are identified as and , respectively. It is clear that there are other members of pentaquarks. Similar to the decay channels , the pentaquarks in the octet and decuplet will be able to decay into a plus baryons in the low-lying octet and decuplet, respectively. For , there should be a companion singlet pentaquark . However, there is no low-lying baryon singlet, will not be able to decay into a plus an ordinary low-lying baryon, but may be an ordinary baryon and multi-mesons forming a singlet. The masses of the pentaquarks are degenerate in the limit. But may be different due to s-quark mass being much larger than u- and d-quarks. Estimate of mass differences for some of the pentaquarks in the diquark model has been carried out in Ref.(15). Discoverying these additional pentaquarks identified above is one of the way to verify the diquark model for pentaquark which can in principle be carried at the LHCb.

Pentaquark Weak Decays

We now discuss weak decay modes of b-baryon into an octet or a decuplet pentaquark and a light pseudoscalar octet meson. The leading effective Hamiltonian inducing decays in the SM has both parity conserving and violating parts given by

 Heff(q)=4GF√2[VcbV∗cq(c1O1+c2O2), (6)

where can be or . is the CKM matrix element. The coefficients are the Wilson Coefficients (WC) which WC have been studied by several groups and can be found in Ref. (16). The operators are given by

 O1=(¯qicj)V−A(¯cibj)V−A,O2=(¯qc)V−A(¯cb)V−A, (7)

where . In the above, we have neglected contributions from penguin diagrams since they are significantly smaller than the tree contributions given above.

The operators transfer under the flavor as a . We indicate it as . The non-zero entries of the matrices are given as the following,

 H(¯¯¯3)2=1,for ΔS=0,q=d% , H(¯¯¯3)3=1,for ΔS=−1,q=s. (8)

For , the decay amplitude is proportional to , which we refer to as Cabibbo allowed interaction. For , the decay amplitude is proportional to , which is Cabibbo suppressed interaction.

The low-lying b-baryons are made up of a quark and two light quarks. Here the light quark is one of the , or quarks. Under the flavor symmetry, the quark is a singlet and the light quark is a member in the fundamental representation . The b-baryons then have representations under flavor as , that is, the b-baryons contain an anti-triplet and a sextet in the flavor space(17). The anti-triplet and the sextet b-baryons will be indicated by

 (Bij)=⎛⎜ ⎜⎝0Λ0bΞ0b−Λ0b0Ξ−b−Ξ0b−Ξ−b0⎞⎟ ⎟⎠,(Cij)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Σ+bΣ0b√2Ξ′0b√2Σ0b√2Σ−bΞ′−b√2Ξ′0b√2Ξ′−b√2Ω−b⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (9)

The pseudoscalar octet mesons will be indicated by . They are

 (Mji)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝π0√2+η8√6π+K+π−−π0√2+η8√6K0K−¯K0−2η8√6⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (10)

At the hadron level, the decay amplitude can be generically written as

 A=⟨PM|Heff(q)|Bor% C⟩=VcbV∗cqT(q). (11)

To obtain the invariant decay amplitude for a b-baryon, one first uses the Hamiltonian to annihilate the b-quark in (or ) and then contract indices in an appropriate way with final states and . Taking the anti-triplet tree amplitude and sextet tree amplitude as examples, following the procedures for b-baryon charmless two-body decays in Ref.(18), we have

 Tt8(q) = at(¯¯¯3)⟨PklMlk|H(¯¯¯3)i|Bi′i′′⟩ϵii′i′′+bt(¯¯¯3)⟨PkjMik|H(¯¯¯3)j|Bi′i′′⟩ϵii′i′′ (12) + ct(¯¯¯3)⟨PikMkj|H(¯¯¯3)j|Bi′i′′⟩ϵii′i′′ + dt(¯¯¯3)⟨Pi′j′Mij|H(¯¯¯3)i′′|Bjj′⟩ϵii′i′′+et(¯¯¯3)⟨Pi′j′Mij|H(¯¯¯3)j|Bi′′j′⟩ϵii′i′′ + ft(¯¯¯3)⟨Pi′jMij′|H(¯¯¯3)j|Bi′′j′⟩ϵii′i′′.

and

 Ts8(q) = ds(¯¯¯3)⟨Pi′j′Mij|H(¯¯¯3)i′′|Cjj′⟩ϵii′i′′+es(¯¯¯3)⟨Pi′j′Mij|H(¯¯¯3)j|Ci′′j′⟩ϵii′i′′ (13) + fs(¯¯¯3)⟨Pi′jMij′|H(¯¯¯3)j|Ci′′j′⟩ϵii′i′′.

The invariant decay amplitude involve decuplet pentaquarks can be written as

 Tt10(q) = at10⟨PkjlMki|H(¯¯¯3)l|Bij⟩, Ts10(q) = as10⟨PkjlMki|H(¯¯¯3)l|Cij⟩+bs10⟨PkjiMkl|H(¯3)l|Cij⟩. (14)

For baryons decay into , there are six possible terms as far as properties are concerned. The dominant contributions are from terms with coefficients , , and . In figure 1, we show diagrams correspond to these terms. In these figures, the quarks , and are contracted by the totally antisymmetric tensor or . This property plays an important role in determining whether or sextet diquarks are allowed to form. The two terms with coefficients and are allowed, but it turns out that for these two terms the state is actually a higher order fork state with the same quantum numbers but with more quark contents as can be seen in figure 2. Therefore, one expects that the coefficients and to be smaller than , , and . We will include all in our later discussions. For the properties we emphasis will not be affected whether to include and or not.

For baryons decay into , only three possible terms, terms with coefficients, , and . The corresponding diagrams are shown in figure 1.b, 1.c, and 1.d. Because the two light quarks in are symmetric, similar terms to those with coefficients , , for decays are identically zero.

For and baryons decay into decuplet, the final light quarks need to be in totally symmetric state. There are less possibilities. In figure 3 we show the corresponding diagrams for the allowed terms.

In the above, we have suppressed the Lorentz indices and spinor forms, but concentrated in flavor indices. The results apply to and also multiplets. Expanding the above amplitudes, one can obtain the individual decay amplitude. The full expansions are given in the appendices. From these results, one can read off many properties concerning weak decays of b-baryons to a pentaquark and a light pseudoscalar. We present some of the interesting properties in the following.

Discussions and conclusions

We begin with discussing b-baryons decay into octet pentaquarks. The two pentaquarks discovery at the LHCb belong to this category. Without a detailed dynamic model, one can not calculate the size of the various amplitudes, but the flavor symmetry can relate to different decay modes and can be tested experimentally. Since the and belong to different octet, there are no relations among the decay amplitudes related to these two pentaquarks. But within each multiplet with the same , there are relations which can be tested experimentally.

We find the -spin related amplitudes ( and ) for anti-triplet satisfy the following relations

 Tt(Ξ−b→K−n8)=Tt(Ξ−b→π−Ξ08),Tt(Ξ0b→¯K0n8)=−Tt(Λ0b→K0Ξ08), Tt(Ξ−b→K0Ξ−8)=Tt(Ξ−b→¯K0Σ−8),Tt(Ξ0b→K0Ξ08)=−Tt(Λ0b→¯K0n8), Tt(Ξ0b→π−Σ+8)=−Tt(Λ0b→K−p8),Tt(Λ0b→π−p8)=−Tt(Ξ0b→K−Σ+8), Tt(Ξ0b→π+Σ−8)=−Tt(Λ0b→K+Ξ−8),Tt(Λ0b→K+Σ−8)=−Tt(Ξ0b→π+Ξ−8), Tt(Ξ0b→K−p8)=−T(Λ0b→π−Σ+8),Tt(Ξ0b→K+Ξ−8)=−Tt(Λ0b→π+Σ−8). (15)

While the -spin related amplitudes for sextet satisfy

 Ts(Σ+b→n8π+)=−Ts(Σ+b→Ξ08K+),Ts(Σ+b→Σ+8K0)=−Ts(Σ+b→p8¯K0), Ts(Σ−b→n8π−)=−Ts(Ω−b→Ξ08K−),Ts(Σ−b→Σ−8K0)=−Ts(Ω−b→Ξ−8¯K0), Ts(Ω−b→Ξ08π−)=−Ts(Σ−b→n8K−),Ts(Ω−b→Σ−8¯K0)=−Ts(Σ−b→Ξ−8K0), Ts(Σ0b→Σ−8K+)=−Ts(Ξ′0b→Ξ−8π+),Ts(Σ0b→p8π−)=−Ts(Ξ′0b→Σ+8K−), Ts(Ξ′0b→Ξ−8K+)=−Ts(Σ0b→Σ−8π+),Ts(Ξ′0b→Σ−8π+)=−Ts(Σ0b→Ξ−8K+), Ts(Ξ′0b→p8K−)=−Ts(Σ0b→Σ+8π−),Ts(Ξ′0b→Σ+8π−)=−Ts(Σ0b→p8K−), Ts(Ξ′0b→Ξ08K0)=−Ts(Σ0b→n8¯K0),Ts(Ξ′0b→n8¯K0)=−Ts(Σ0b→Ξ08K0), Ts(Ξ′−b→n8K−)=−Ts(Ξ′−b→Ξ08π−),Ts(Ξ′−b→Ξ−8K0)=−Ts(Ξ′−b→Σ−8¯K0). (16)

The above relations can be directly read off from the diagrams shown in figures 1 and 2 when specific quarks are assigned for each decay modes. For illustrations we take the pairs a). and for decay, and b). and for decay as examples to provide some details. From tables in Appendix A, we find that for the pair in a), there are contributions from and terms. We focus on the diagram corresponds to . For the pair in b), there is only contribution. Specifying quark contents, we have

 a) ForΞ0b→K+Ξ−8,(qj′,qj,qi,qi′′)=(s,u,s,d), ForΛ0b→π+Σ−8,(qj′,qj,qi,qi′′)=(d,u,d,s). b) ForΩ−b→π−Ξ08,(qj′,qj,qi,qi′′)=(s,d,u,s), (17) ForΣ−b→K−n8,(qj′,qj,qi,qi′′)=(d,s,u,d).

Note that for each pairs one just needs to exchange all s quarks with all d quarks to go from one to another within a given pair. If U spin is a good symmetry, the amplitudes defined in eq.15 and eq.16 are therefore equal in magnitude. The relative minus sign for the amplitude in each pair in eq.15 and eq.16 comes from the fact that each diagram is contracted by and for and decays, respectively. For the pair in a), and for and decays, and for the pair in b), and for and decays, respectively. These specific values for , and explain the relative minus sign in the relations.

The above relations also hold even one include small penguin contributions(18). These relations apply to both octets with and . Due to mixing between and , the decay modes with in the final sates is not as clean as those with and in the final state to study. We have not listed processes involve above.

The above relations lead to the following relations for each pairs above,

 A(B→MP,ΔS=0)=VcbV∗cdT,A(B→MP,ΔS=−1)=VcbV∗csT, (18)

and

 Γ(B→MP,ΔS=0)Γ(B→MP,ΔS=−1)=|Vcd|2|Vcs|2≈4.5%. (19)

When more data become available, with more pentaquarks discovered, the above relations can be tested. To study the Cabbibo suppressed decays, 20 times more data are needed.

For the b-baryons decay into decuplet pentaquark, let us focus on b-baryons which undergo visible weak decays, namely , , , and decays. The full lists are given in the appendix. The b-baryons , , belong to the anti-triplet . Expanding the first equation in eq.(14), we have for amplitudes

 Λ0b: at10(π+Δ−10+1√3K+Σ−10+√2√3π0Δ010−1√3π−Δ+10−1√6K0Σ010) Ξ0b: at10(12√3π0Σ010+12η8Σ010+1√3π+Σ−10+1√3K+Ξ−10−1√3K−Δ+10−1√3¯K0Δ010), Ξ−b: at10(1√6π−Σ010−1√6π0Σ−10+1√2η8Σ−10+1√3K0Ξ−10−1√3K−Δ010−¯K0Δ−10), (20)

and for amplitudes, we have

 Λ0b: at10(1√3π0Σ010+1√3π+Σ−10+1√3K+Ξ−10−1√3π−Σ