Understanding $P_{cs}(4459)$ as a hadronic molecule in the $\Xi_b^-\to J/\psi \Lambda K^-$ decay

Recently, the LHCb Collaboration reported on the evidence for a hidden charm pentaquark state with strangeness, i.e., $P_{cs}(4459)$, in the $J/\psi\Lambda$ invariant mass distribution of the $\Xi_b^-\to J/\psi \Lambda K^-$ decay. In this work, assuming that $P_{cs}(4459)$ is a $\bar{D}^*\Xi_c$ molecular state, we study this decay via triangle diagrams $\Xi_b\rightarrow \bar{D}_s^{(*)}\Xi_c\to (\bar{D}^{(*)}\bar{K})\Xi_c\to P_{cs} \bar{K}\to (J/\psi\Lambda) \bar{K}$. Our study shows that the production yield of a spin 3/2 $\bar{D}^*\Xi_c$ state is approximately one order of magnitude larger than that of a spin $1/2$ state due to the interference of $\bar{D}_s\Xi_c$ and $\bar{D}_s^*\Xi_c$ intermediate states. We obtain a model independent constraint on the product of couplings $g_{P_{cs}\bar{D}^*\Xi_c}$ and $g_{P_{cs}J/\psi\Lambda}$. With the predictions of two particular molecular models as inputs, we calculate the branching ratio of $\Xi_b^-\to (P_{cs}\to)J/\psi\Lambda K^- $ and compare it with the experimental measurement. We further predict the lineshape of this decay which could be useful to future experimental studies.

Most recently, the LHCb Collaboration reported on the first evidence for a structure in the J/ψΛ invariant mass distribution of the Ξ − b → J/ψΛK − decay [45], hinting at the existence of a pentaquark state with strangeness, i.e., P cs (4459). It should be noted that the existence of pentaquark states with strangeness was predicted together with their non-strange counterparts in the molecular picture [2,3]. The P cs (4459) state is located close to theD * Ξ c threshold, leading naturally to a molecular interpretation [46][47][48][49][50][51]. One interesting point to be noted is that in addition to the four P c 's discovered experimentally, there may be three more candidates which strongly couple toD * Σ * c with J P = 1 2 − , 3 2 − , 5 2 − , as dictated by the heavy quark spin symmetry (HQSS) [30,35,36,39]. As for pentaquark states with strangeness, one expects 10 of them [47,50,52,53].
In addition to the masses and widths of the pentaquark states, the LHCb Collaboration also reported the production yields of P c (4312), P c (4380), P c (4440), P c (4457), and P cs (4459), which are collected in Table I. One notes that the production yield for P c (4380) is one order of magnitude larger than that of P c (4312), which provides an explanation why P c (4312) was not observed in 2015. However, we note that the sum of the production yields of P c (4440) and P c (4457) is only half of that of P c (4450), which may indicate that something is missing, maybe a new resonance as suggested in several works [19,[54][55][56]. Clearly, understanding the production yields, particularly, the pattern shown in Table I, will greatly improve our understanding of the pentaquark states.
In the present work, we study the branching ratio of Ξ − b → P cs K − . 1 The present work differs  [45] from those of Refs. [58,59] in two ways. First, the weak production formalism is different from that of Ref. [58] (see also Ref. [60]), which allows for a prediction of the absolute branching ratio of the Ξ − b → P cs K − → J/ψΛK − decay, within a molecular model. Compared to Ref. [59], we use different parameterizations of form factors and predict the lineshape of the Ξ − b → P cs K − → J/ψΛK − decay. Taking two molecular models for the P cs (4459) state [47], we compare the soobtained branching ratios with the experimental data.
The present work is organized as follows. In Sec. II, we explain in detail the mechanism for the P cs (4459) production in the Ξ − b decay, which involves a weak interaction part and a strong interaction part. In Sec. III, we present the numerical results and compare with the experimental data, followed by a short summary in Sec. IV.

II. THEORETICAL FRAMEWORK
Assuming that P cs (4459) is a molecule mainly composed ofD * Ξ c , the Ξ − b → P cs K − decay can proceed as shown in diagram (a) of Fig. 1. The Ξ b state first decays into Ξ c by emitting a W − boson which is then converted into a pair ofcs, which after hadronization turns into a D ( * ) s . Next the D ( * ) s meson emits a kaon and aD * . The final state interaction ofD * Ξ c dynamically generates the P cs (4459) state which then decays into J/ψΛ, as shown in Fig. 2.
In addition to the W -emission diagram discussed above, the Ξ − b decay can also proceed via the branching ratio and the π +(0) η lineshape are well described. In particular, the large branching ratio of D + s → a + 0 π 0 (a 0 0 π + ) is naturally explained, while for a pure W -annihilation process one would expect a much smaller FIG. 1. External W-emission (a) and internal W-conversion (b) mechanism for the Ξ b decay.
the internal W -exchange mechanism shown in diagram (b) of Fig. 1. The ssd cluster can either directly hadronize into a Ξ − or, by picking up a pair of qq from the vacuum, hadronizes into ΛK − . The former is indeed a main decay channel of Ξ − b [61], while the latter has been studied in Ref. [58].
A. Branching ratio of Ξ b → P cs K − In the following, we describe how to calculate the diagrams of Fig. 2. The effective Lagrangian s decay reads The A 1 , A 2 , B 1 , B 2 , A, and B can be expressed with the six form factors describing the 2 Here we adopt the convention for the form factors of Ref. [63] in which there exists an extra minus in front of f A 2 and f V 2 .  Following the double-pole parametrization proposed in Ref. [65], one can rewrite the form factors as Fitting to the results of the relativistic quark-diquark model [63], we can obtain the values of F (0), Λ 1 , and Λ 2 , which are tabulated in Table II.
The effective Lagrangians for the P cs →D * Ξ c andD ( * ) where P cs1 and P cs2 denote the P cs (4459) state with J P = 1 2 − and 3 2 − , respectively. The g KDsD * and g KD * s D * are the kaon meson couplings to D s D * and D * s D * , respectively. We take g KDsD * = 5.0 and g KD * s D * = 7.0 GeV −1 in the present work, which are extracted from Ref. [66]. The g P cs1 ΞcD * and g P cs2 ΞcD * are the couplings between P cs and its components, whose values are not known a priori, but can be computed with the compisteness conditions [67][68][69] or in molecular models, e.g., Refs. [47], or in lattice QCD.
With the effective Lagrangians above, the decay amplitudes for The decay amplitudes of We follow Ref. [37] and introduce a monopole form factor to depict the off-shell effect of the exchangedD * mesons, where Λ = m + αΛ QCD with Λ QCD = 220 MeV, and α is a model parameter. In this way, the triangle diagrams are free of any ultraviolet divergence. Collecting all the pieces together, the decay width for Ξ b → P csK could be expressed as where | p| denotes the momentum ofK or P cs in the rest frame of Ξ b .
With the weak decay vertices described in Eq. (1) and Eq. (2), we can further work out the invariant mass distribution of the Ξ − b → J/ψΛK − decay. Parameterizing the intermediate P cs (4459) state with a Breit-Wigner resonance, the amplitudes of Fig. 3 read where (p 4 + p 5 ) 2 = p 2 45 = M 2 45 denotes the invariant mass of the J/ψΛ final state and The partial decay rate for Ξ b → J/ψΛK as a function of the invariant mass M J/ψΛ then reads with

III. RESULTS AND DISCUSSIONS
In this section, we explore the decay mechanism proposed in this work. We divide our discussions into two categories, those only depend on the decay mechanism explored and those depend on a particular molecular model.
In our framework, the parameter α is not known, though its value is often assumed to be about 1 [70][71][72][73]. Therefore, we first study how the calculated branching ratios depend on the value of α.
Varying α from 0.8 to 1.2, we plot the values of Br[Ξ b → P cs1 (P cs2 )K]/g 2 P cs1/2 ΞcD * in Fig. 4. One can see that the branching ratios for P cs1 and P cs2 are moderately sensitive to the value of α in the range studied. As a consequence, in the following, we will take α = 1.0 ± 0.1 to take into account the uncertainties from α.

A. Model independent predictions
To compute the absolute branching ratio Br[Ξ b → P csK ], we need to know the coupling constants g P cs1 ΞcD * and g P cs2 ΞcD * . They can be determined model independently with the compositeness conditions [67][68][69], as was done in, e.g., Ref. [32] for the pentaquark states. With the experimental mass of P cs , the couplings read g P cs1 ΞcD * = 1.59 and g P cs2 ΞcD * = 2.76, corresponding to a cutoff Λ = 1.0 GeV (more details can be found in the Appendix). With these couplings, we find, surprisingly, that the branching ratio for the P cs state with J P = 3/2 − is approximately one order of magnitude larger than that for the P cs state with J P = 1/2 − , which are In addition, using the experimental branching ratio Br[Ξ b → J/ψΛK] = (2.31 ± 1.37) × 10 −4 (see the Appendix how to derive this), the mass, width, and branching ratio R of the P cs state given in Table I as inputs, we can provide a model independent constraint on the product of the two couplings in Eq. (4), g PcsD * Ξc and g PcsJ/ψΛ , within the decay mechanism studied in the present work. The experimental branching ratio given in Table I The above product can be used to constrain molecular models.

B. Comparison with models
In order to produce the branching ratio R defined in the introduction, in addition to the information derived above, we need to know the partial decay width of P cs into J/ψΛ. For this, we turn to specific molecular models. In the following, we study the unitary approach of Ref. [47] and the one-boson-exchange (OBE) model of Ref. [51], calculate the branching ratio R, and compare with the LHCb measurement.
First, we focus on Ref. [47]. Note that the difference between the definition of their couplings and ours (see the Appendix for details) and with the branching ratios Br[P cs → J/ψΛ] = 3.31% for P cs1 and 14.68% for P cs2 from Ref. [47], we obtain the couplings as g P cs1 J/ψΛ = 0.07 and g P cs2 J/ψΛ = 0.27. The branching ratios R for the spin-parity assignment 1/2 − and 3/2 − are found to be In the one-boson-exchange (OBE) model of Ref. [51], the P cs (4459) state is interpreted as a J P = 3/2 − molecular state and the partial decay width of P cs → J/ψΛ is estimated to be 0.06 ∼ 0.2 MeV. The main decay mode is found to be P cs → K * Ξ(ωΛ), which accounts for 80% of the total decay width. These numbers lead to an even smaller branching ratio Br[P cs → J/ψΛ] = 0.6% − 0.8% corresponding to the total decay width ranging from 10 to 25 MeV. For the coupling between the P cs state with J P = 3/2 − and its components, we adopt the value g P cs2 ΞcD * = 2.76 obtained from the compositeness condition. Using 0.7% as the central value for Br[P cs → J/ψΛ] and 0.1% as its error, we obtain All these numbers are compared with the LHCb measurement in Fig. 5. It is clear that the result of the OBE model seems to agree with the experimental measurement, as well as the J P = 1/2  Finally, in Fig. 6, we show the J/ψΛ invariant mass distribution of the Ξ − b → J/ψΛK − decay with all the relevant couplings provided by the unitary approach of Ref. [47] (see the Appendix for more details). They might be useful for future experimentla searches.

IV. SUMMARY
In this work, we studied the decay of Ξ − b → P cs K − → J/ψΛK − via a triangle mechanism. The decay consists of three steps. First, Ξ − b decays weakly intoD s state then emits a kaon and aD * . TheD * and Ξ c interact with each other to dynamically generate the P cs (4459) state, which then decays into J/ψΛ. From such a decay mechanism, we derived a constraint on the product of couplings of the P cs (4459) state to theD * Ξ c and J/ψΛ channels. Determining the coupling between P cs and theD * Ξ c channel using the compositeness condition, we predicted the branching ratio Br[Ξ − b → P cs K − ]. These can be useful to understanding the nature of P cs (4459) as a molecular state.
Using the predicted couplings by the unitary approach [47] and the one-boson exchange model of Ref. [51], we calculated the branching ratios Br[Ξ − b → (P cs →)J/ψΛK − ]. We found that in the unitary approach, the J P = 1/2 assignment is prefered, while the J P = 3/2 assignment gives a branching ratio much larger than the experimental measurement. On the other hand, the 3/2 assignment in the one-boson-exchange model of Ref. [51] yields a branching ratio in agreement with the LHCb data. This can be traced back to the drastically different partial decay width of P cs → J/ψΛ.
In principle, the present formalism can also be utilized to study the Λ b → J/ψpK − decay, where the four pentaquark states, P c (4312), P c (4380), P c (4440), and P c (4457), were discovered.
This has been explored in Ref. [37], which, however, suffers from the fact that the weak decay s Σ c is suppressed because the ud quark pair in Λ b has spin 0, but that in Σ c has spin 1. As a result, the relevant transition form factors are not known and therefore one could not arrive at a quantitative determination of the branching ratios. In addition, compared to the present case, the suppression of the Λ b →D ( * ) s Σ c transition indicates that other mechanisms may play a role than the external W -emission studied in the present work, which complicates the study a lot. With the assumption that the P cs state observed by the LHCb Collaboration can be interpreted as a molecular state ofD * Ξ c with J P = 1/2 − or J P = 3/2 − , we can calculate the couplings between the P cs state and its components with the compositeness condition, which is quite similar to what was done in Refs. [32,74].D * (k 2 ) Ξ c (k 1 ) P cs (k 0 ) P cs (k 0 )

FIG. 7. Mass operators of the P c
According to the compositeness rule [67][68][69], the coupling constant g P cs1/2 ΞcD * can be determined from the fact that the renormalization constant of the wave function of a composite particle should be zero. That is, where Σ Pcs denotes the self-energy of P cs1 and P cs2 . Applying the effective Lagrangians listed in Eq. (4), the self-energy Σ P cs1/2 reads with ω Ξc = m Ξc m Ξc + m D * , The Φ[−p 2 ] = exp(p 2 /Λ 2 ) is the Fourier transformation of the correlation in the Gaussian form with Λ being the size parameter which characterizes the distribution of components inside the molecule. With all the formula above and taking Λ = 1.0 GeV, we obtain the couplings between the P cs states andD * Ξ c , which are g P cs1 ΞcD * = 1.59 for J P = 1/2 − and g P cs2 ΞcD * = 2.76 for Experimentally, the branching ratio of Ξ b → J/ψΛK has been measured to be [75] f where f Ξ b and f Λ b refer to the b quark fragmentation fractions into Ξ − b and Λ 0 b , the ratio of which is [76] f Ξ b f Λ b = (6.7 ± 0.5 ± 0.5 ± 2.0) × 10 −2 , while the branching ratio of Λ b → J/ψΛ has been measured by the CDF Collaboration [77] Br[Λ b → J/ψΛ] = (3.7 ± 1.7 ± 0.7) × 10 −4 .
With all the ratios given above, one can compute the branching ratio of Ξ b → J/ψΛK The large uncertainty can be traced back to the experimental uncertainty in the braching ratio Br[Λ b → J/ψΛ], which accounts for about 50%, and the large uncertainty in the ratio of fragmentation fractions coming from the estimation of SU(3) breaking effects [76].

C. Couplings from the unitary approach
In our convention, the J/ψΛ partial decay widths of the P cs state with J P = 1/2 − and 3/2 − are expressed as Γ P cs1 →J/ψΛ = 1 2 Pcs |q| |A P cs1 | 2 , Γ P cs2 →J/ψΛ = 1 4 where the modules of amplitude squared are in which q denotes the momentum of J/ψ in the rest frame of the P cs state. Using the partial decay widths from Ref. [47], we obtain g P cs1 J/ψΛ = 0.07 and g P cs2 J/ψΛ = 0.27. Similarly, we obtain g P cs1 ΞcD * = 1.25 and g P cs2 ΞcD * = 2.17.