2023 Investigating Z cs (3985) and Z cs (4000) exotic states in Λ b → Z − cs p decays

,


I. INTRODUCTION
Exploring exotic states is an intriguing frontier in hadron physics that has seen significant progress in the past decade.A growing number of candidates for exotic states have been experimentally observed, as discussed in recent reviews [1][2][3][4][5][6].Among these states, the charmonium-like states, which consist of a cc pair, have formed a large family since the first observation of the X(3872) in 2003 [7].Recently, a Z cs (3985) state, as a strangeness-flavour partner of the Z c (3900) state, was observed by BESIII [8] in 2021 with a mass of 3982.5 +1. 8  −2.6 ± 2.1 MeV, a width of 12.8 +5.3  −4.4 ± 3.0 MeV, and spin-parity J P = 1 + .This experimental observation was anticipated in theoretical models, such as the hadronic molecular [1,[9][10][11][12][13][14][15], the compact tetraquark [16,17], etc.Following the observation of Z cs (3985), a Z cs (4000) state was discovered by LHCb [18] with a mass of 4003 ± 6 +4 −14 MeV, a width of 131 ± 15 ± 26 MeV, and J P = 1 + .Although LHCb claimed there is no evidence that Z cs (4000) is the same state as the Z cs (3985) state, Refs.[12,19] discussed the possibility that they may correspond to the same state.In particular, Ref. [19] demonstrates that both the BESIII and LHCb data can be fitted simultaneously treating they as the same state.This attracts significant attention to the molecular model, which naturally interprets Z cs (3985) and Z cs (4000) as two "C-parity partners" 1 [9,10,14,20,21]: On the other side, Ref. [22] proposes the classification of these two strange resonances as the strange constituents of two S-wave tetraquark nonets within the framework of an SU(3) quark model.
Moreover, the production of Z cs can occur through the decays of b-flavored baryons.Reference [23] predicts that branching fractions of b-flavored baryons two-body decays involving Z cs for short-distance annihilation mechanisms are ≤ O(10 −7 ).However, it is also possible for the Z cs states to be produced through long-distance annihilation mechanisms, which could significantly enhance the branching fractions [24,25].Specifically, the long-distance In this report, we will study the Λ b → Z − cs p decay with the interpretations of Z cs as the molecular states and via the triangle-rescattering diagrams.The size of the contribution from the triangle-rescattering effect heavily depends on the couplings of the involved intermediate interactions.Fortunately, the branching fractions of Λ b → Λ c D ( * )− s have been measured or estimated to be at the 10 −2 level, indicating sizable weak couplings of the baryon decays [26], and the strong couplings of Λ c → D ( * ) p have been found to be significant [27].In addition, the Z cs , as candidates for DD * − s /D * D − s molecular states, should strongly couple to DD * − s and D * D − s .Therefore, the Λ b → Z − cs p decays could be dominated by long-distance effect and study of these decays is crucial to pin down the nature of the Z cs .
the decay constant, and f 1,3 (g 1,3 ) the Λ b → Λ + c transition form factors, while a 1 results from the factorization.The second part is the amplitude of Λ c → pD ( * ) , given by [27] where p µ is the momentum of p.The same definitions and notation as in Ref. [27] are used for the coupling constants (g V , g T , g ΛcpD ) and the polarization four-vector ǫ µ .The third part is the amplitude of where g ZD * Ds and g ZDD * s are the coupling constants, and ǫ Zcs the polarization four-vector.Eventually, the amplitude of the triangle-rescattering processes for the Λ b → Z − cs p decay are given by with where q 2 = p Zcs − q 1 and q 3 = p Λ b − q 1 correspond to the momentum flows in Fig. 1; χ represents the phase difference between M a and M b .The monopole form factors 2 ) with the cutoff parameters λ a(b) are to avoid the overestimation with q 2 to ±∞.
Since Z cs has J P = 1 + , the amplitude M a(b) can be expressed in the general form [28]: To obtain , and B ′ 2a(b) , one needs to integrate over the phase space of the triangle loop in Eq. (7).For convenience, we define A ′ (B ′ ) ≡ A(B) − Ã( B) and factorize Eq. ( 7) as The first term of Eq. ( 9) corresponds to A and B, and the second term Ã and B. The integrations of the multi-point functions are discussed in App.A and Refs.[30][31][32][33].By comparing Eqs. ( 8) and ( 9 , A and B are obtained to be with The definitions of C 0 , C ij , D ij are the same as in Refs [30][31][32][33].In the same way, Ã( B) are obtained by replacing m D * and m D in Eq. ( 10) with λ a(b) .Following Ref. [28], the decay width is determined to be with where p c is the center-of-mass momentum.
The resonance states of Λ c can also contribute to the branching fractions of Λ b → Z − cs p.Here, we provide a rough estimate of the contribution of the lightest resonance, Λ c (2595).
One could expect that heavier resonance state will contribute less or at the same order.
with the strong coupling of Λ c → pD ( * ) in Eq. ( 4) and the strong coupling of Λ c (2595) → Using g c1 = 3.69 √ 2 , g c2 = 5.7 √ 2 quoted from Ref. [35] (Refs.[36,37] also provide similar values) and ignoring the mass difference between Λ c and Λ c (2595), one can estimate the contribution of Λ c (2595) by This indicates the contribution of Λ c (2595) is orders of magnitude smaller than that of the ground sate Λ c .Furthermore, using the branching fractions and couplings given in Refs [38,39] will lead to the same conclusion.Note that the information about M(Λ c (2595) → pD * ) and B(Λ b → Λ c (2595)D * − s ) is absent in Refs [38,39], but one could assume that them are of the same order of magnitude as M(Λ c (2595) → pD) and B(Λ b → Λ c (2595)D − s ).

IV. CONCLUSIONS
In conclusion, the study of the hidden-charm states Z cs is crucial in understanding its process for Λ b → Z − cs p starts with the Λ b → Λ c D ( * )− s decay followed by Λ c and 1 Although |D D * s ; 1 ++ and |D D * s ; 1 +− do not have the C-parity, their quantum numbers are still denoted as J P C = 1 +± here since they are considered as the strangeness partners of |D D * ; 1 ++ and |D D * ; 1 +− in the molecular model [10].D ( * )− s rescattering.(In the following of this report, Z cs denotes |D D * s J=1 ±|D * Ds J=1 .)The rescattering process transforms Λ c and D ( * )− s to Z − cs and p via D ( * ) exchange in the triangle loop, as depicted in Fig. 1.

FIG. 3 .
FIG. 3. The branching fractions of Λ b → Z − cs p related to χ.The dashed line represents Z cs (3985) − and the solid line Z cs (4000) − .
According to Ref[29], the branching fractions of the Λ b → Λ c D ( * )− s are as followsB(Λ b → Λ c D − s ) = 11.92 +7.69 −5.28 × 10 −3 , B(Λ b → Λ c D * − s ) = 17.49+10.60 −7.48 × 10 −3 ,(17)and these of the Λ b → Λ c (2595)D −( * ) s are given by B(Λ b → Λ c (2595)D − s ) ∼ = (1.72 +1.71 −1.01 ) × 10 −3 , B(Λ b → Λ c (2595)D * − s ) ∼ = (2.28 +2.21 −1.29 ) × 10 −3 , exotic nature.By employing different channels to produce Z cs , we can identify and confirm the molecular model or other models proposed for them.Our study has focused on the triangle-rescattering decays Λ b → Λ c D ( * )− s → Z − cs p, where Λ c and D − s (D * − s ) transform into Z − cs and p by exchanging a D * (D) in the triangle loop.We have proposed Λ b → Z − cs p with Z − cs → J/ψK ( * )− p as candidate decays to search for the Z cs exotic states, and predicted B(Λ b → Z − cs p) = (3.1 +1.4 −2.6 ) × 10 −4 in the molecular model.It is worth noting that Ref. [5] also provides the coupling values g ZDsD * and g ZD * s D under the assumption that Z cs is a tetraquark.These couplings are approximately one third of those in the molecular model.Consequently, the branching ratio of Z cs → Z − cs p based on the tetraquark model is an order of magnitude smaller than that of the molecular model.We are optimistic about the prospects of our proposal being tested by the LHCb Collaboration.With 3 fb −1 of pp collision data taken at √ s = 7 and 8 TeV and 6 fb −1 at √ s = 13 TeV, LHCb can achieve a sensitivity level of 10 −6 for branching fractions of the Λ b decays.Explicitly, the LHCb experiment has discovered exotic structures on the J/ψp spectrum of the Λ b → J/ψK − p events [40-42].Our proposal is based on the same final state particles, while on the J/ψK − spectrum, indicating an accessible opportunity to search for the Z cs exotic states via the Λ b → Z − cs p decay.