Pole analysis on the hadron spectroscopy of $\Lambda_b\to J/\Psi p K^-$

In this paper we study the $J/\Psi p$ spectroscopy in the process of $\Lambda_b\to J/\Psi p K^-$. The final state interactions of coupled channel $J/\Psi p$ ~-~ $\bar{D} \Sigma_c$~-~$\bar{D}^{*} \Sigma_c$ are constructed based on K-matrix with the Chew-Mandelstam function. We build the $\Lambda_b\to J/\Psi p K^-$ amplitude according to the Au-Morgan-Pennington method. The event shape is fitted and the decay width of $\Lambda_b\to J/\Psi p K^-$ is used to constrain the parameters, too. With the amplitudes we extract out the poles and their residues. Our amplitude and pole analysis suggest that the $P_c(4312)$ should be $\bar{D}\Sigma_c$ molecule, the $P_c(4440)$ could be an S-wave compact pentaquark state, and the structure around $P_c(4457)$ is caused by the cusp effect. The future experimental measurement of the decays of $\Lambda_b\to \bar{D}\Sigma_c K^-$ and $\Lambda_b\to \bar{D}^*\Sigma_c K^-$ would further help to study the nature of these resonances.


I. INTRODUCTION
The discovery of hidden-charm hadrons P + c (4380) and P + c (4450) [1] started a new era of hadron physics as they obviously contain at least five quark componentccuud. Whether they are compact pentaquark states or hadronic molecules or generated by kinematic effect is still not clear. For recent review on the hadronic molecules and multiquark states, we refer to [2][3][4]. Recently the LHCb experiment made a great progress [5]. The decay events collected now by Run 1 and Run 2 are about nine times more than that of Run 1 analysis. As a result, the bin size has been decreased from 15 to 2 MeV. With the high statistics they found three structures in the J/Ψp spectrum of Λ b → J/ΨpK − : with all the units being MeV. Then the question is, what inner structure are they? A cornucopia of models have been done to study the property of these resonances [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Among them, Ref. [12] uses a coupled channel Kmatrix formalism to fit to the data around the P c (4312). In Ref. [21], the Lippmann-Schwinger equations have been used and the one pion exchange and short range scattering potential have been considered. They found a narrow Σ * cD state (P c (4380)). In addition, three more Σ * cD * molecules have been seen in the analysis. Here we use the Chew-Mandelstam formalism to write the unitary cut in once subtracted dispersion relation, and we fit up to 4.6 GeV to discuss the three resonances listed * Electronic address: dailingyun@hnu.edu.cn above (P c (4312), P c (4440), and P c (4457)). It is worth to point out that the pole counting rule [22,23] is useful to distinguish the molecule and non-molecule structure. Indeed the shadow poles (not only the ones being closest to the physical sheet) are also important to discuss the inner structure. Our amplitudes around P c (4440) and P c (4457) are in high quality and they are helpful for discussing the structure. These will be discussed in next sections.
To extract the information of these resonances, we need amplitude analysis to describe the invariant mass spectroscopy of Λ b → J/ΨpK − . The final state interactions (FSI) are important to be considered. There have been lots of papers that indicate the importance of the FSI, see e.g. [24][25][26][27][28][29][30][31][32]. We will use the Au-Morgan-Pennington (AMP) method [24] to include the FSI of J/Ψp -DΣ c -D * Σ c triple channels. From the amplitudes we extract out the pole information and discuss their property according to the pole counting rule. This paper is organized as follows: In Sect. II we use K-matrix to build the hadronic scattering amplitudes of J/Ψp -DΣ c -D * Σ c triple channels. The Λ b → J/ΨpK − amplitude is constructed by AMP method. In Sect. III we fit to the event shape and decay width of Λ b → J/ΨpK − and determine the parameters. The amplitudes are continued into un-physical Riemann sheets (RSs) and the poles in different RSs are extracted out. By pole analysis the origin of these poles are discussed. In Sect. IV we discuss the fits to other datasets given by LHCb. We end with a brief summary.

II. DECAY AMPLITUDE
To get the information of poles, we need an amplitude analysis to get accurate hadronic scattering amplitudes. The following problem is which channel should be included? As is predicted in [33,34], there could beDΣ c andD * Σ c hadronic molecule states with quantum number IJ P = 1 2 1 2 − at 4261 + i28.5 and 4412 + i23.6 MeV of each channel. This is later studied in [35] by a cou-pled channel unitary approach. In Ref. [5], the P + c (4312) is found to be underDΣ c threshold and more like an Swave resonance. The other two resonances are proximate to theD * Σ c threshold. We thus take J/Ψp -DΣ c -D * Σ c as the coupled channels. The helicity has been ignored as that the heavy quark spin symmetry ensures the spindependent interactions related to the heavy quark are of the order of 1/m Q [36] 1 . This assumption is consistent with the analysis of Ref. [5], where it is found that including P-wave factors in the Breit-Wigner amplitudes has negligible effect on the results. We construct our amplitude based on K-Matrix to keep unitarity, and have where √ s is the energy in the center of mass frame. K(s) is a real matrix. C(s) is the diagonal matrix of the canonical definition of Chew-Mandelstam functions [37,38], and it could be written in once subtracted dispersion relation: where i(j) = 1, 2, 3 represent for J/Ψp,DΣ c ,D * Σ c , respectively, with isospin 1/2. s thi = (M i + m i ) 2 is the threshold and M i (m i ) is the mass of meson (baryon) in the i-th channel. The phase space factor has only diagonal elements: The Chew-Mandelstam function could be expressed explicitly as The new high statistics results of J/Ψp line shape (in the process of Λ b → J/ΨpK − ) from LHCb [5] help us to constrain the hadronic scattering amplitude. To describe it, the rescattering of inelastic channels (DΣ c ,D * Σ c ) needs to be considered. We implement the AMP formalism [24] to include the FSI: where 'i' has the same meaning as explained after Eq. (2). The α i (s) are polynomials of s, absorbing all the contributions of left hand cut and distant right hand cut. For 1 We are aware of that the spin dependent interactions between the light quarks can not be ignored, and we refer readers to read Ref. [21].
simplicity we set them to be constant α 1,2,3 and ignore higher order terms. It is easy to check that Eq. (5) satisfies the final state interactions theorem: With this amplitude we fit to the invariant mass spectroscopy Here the Källén function λ is defined as λ(x, y, z) = (x − y − z) 2 − 4yz. To fit to the invariant mass spectroscopy one would need to time dΓi d √ s with a normalization factor 'N '. The decay width given by the PDG [39] could be used to constrain the α i , too. But still we lack adequate constraints on F 2 (s) and F 3 (s) amplitudes. Indeed in Ref. [17], the ratio of the coupling constants  7)) are P S 2 /P S 1 ≃ 0.6, 0.4, respectively. This suggests that the branching ratios of Br 2 (Λ 0 b →DΣ c K − ) and Br 3 (Λ 0 b →D * Σ c K − ) could be the same order as that of Br 1 We thus make the 'data' as Br 2,3 = Br 1 , and the uncertainty of Br 2,3 is set to be 10 times as the central values of them. We input these as constraints.

III. FITTING STRATEGY AND POLE ANALYSIS
For K matrix, we try to use as less parameters as possible. Only when more parameters are indispensable to reduce the χ 2 distinctly do we include them. A pole 2 in the K matrix is necessary to fit to the event shape around P c (4440). Adding a P-wave instead of inputting a K matrix pole is also checked. It is somehow helpful to distinguish the quantum number of P c (4440). For the Pwave scattering amplitude we adopt the Blatt-Weisskopf barrier factor representation [27], see the supplement for details. The following fits are performed: The parameters of all the fits are shown in the supplement. The fit results are shown in Fig. 1. Our amplitudes fit well to the high statistics LHCb data in 2019 [5], with cos θ Pc weighted. It is worth to point out that this dataset has removed much of the interfering of the Λ * , which is in the K − p channel and most populated at cos θ Pc > 0. Owing to this our K matrix fit without three body final state interactions is feasible. The branching ratios of Br 1 is exactly the same as that of PDG, and in most of the fits the Br 2 is of 10 −4 , and Br 3 is of 10 −5 . Notice that in Fit 1 it does not have the structure around the √ s = 4440 MeV. To study the resonances we enlarge the size of the plots around the structures, as shown in Fig to the data well. Though the amplitude is a bit lower than the data on the left side of the 'peak', it is within the margin of the data error. Note that we do not input a K-matrix pole around P c (4312) in the K-matrix formalism. This is why our 'peak' exactly locates at theDΣ c threshold ( √ s = 4.3177 GeV), shifting a bit to the right side of the peak of the data. For the P c (4440), our Fits 2 and 3 fit to the data well, while in Fit 1 one can not find such a structure. This is caused by that in Fit 1 we do not include the K matrix pole in K(s). As shown in Fig.2, the Fit 2 is better than Fit 3 in both the P c (4440) and the P c (4457) region. Our results prefer an S-wave P c (4440). For the P c (4457), our amplitude behaves more like a cusp caused by theD * Σ c threshold effect. These will be discussed in next sections.
With the amplitudes given by the Chew-Mandelstam formalism, the information of the poles can be extracted out. We continue the T (s) amplitude to the unphysical Riemann sheets based on unitarity and analyticity. The definition of the Riemann sheet (RS) could be found in [40]. Here we use the following definition [41] as shown in Table I. The pole s R and its coupling/residue of the RS-n in the triple channel are defined as: where the subscript 'i, j' denotes the hadronic channels as before. The poles in different RSs for all the fits are given in Table II  And in Fits 2 and 3 we find poles of all the three resonances. Fit 2 describes the data better with less assumptions, we choose it as our optimistic one. To classify the inner structure of the poles, we use the 'criteria' proposed in Ref. [22,41]: A triple channel Breit-Wigner resonance should appear as quadruplet poles in different RSs, while a molecule has less poles.
P c (4312) This resonance (pole) is rather stable in all the fits. We can find it without an input pole in the K(s). The poles locate at the RS-III and/or V. For each fit we can find only one or two poles. According to the 'pole counting', it is aDΣ c molecule. The masses of the poles are a bit below the threshold √ s th2 = 4317.73 MeV, and their widths (2 times of the imaginary part of the pole) are only a few MeV. This supports the molecule picture. As is known, RS-II and III are the closest sheet to the physical one below and aboveDΣ c threshold, respectively. RS-II and RS-V are connected along the unitary cut above theDΣ c threshold. The shadow poles appear in RS-V but not in RS-II suggests that there is a strong dynamics to drag the pole from RS-II to RS-V. The pole in RS-III confirms such observation. This is consistent with the molecule picture, as our interaction between the resonance and theDΣ c is strong, this is typical way how a virtual state (with weak interaction to theDΣ c ) changes into a molecule. For the strength of the couplings please see the |g 2 | of Table A.4, in the Appendix.
The masses of the poles are quite close to the input K matrix pole 4443.60 MeV in Fit 2. These poles are found in four sheets: RS-III, IV, V, VII/VIII, being close to each other. The widths are quite narrow and the poles are quite close to the real axis. According to 'pole counting', it is an elementary particle. Since it decays into J/Ψp and have five quark component, this implies a compact pentaquark picture. Note that our widths of the P c (4440) are small. Indeed in all our fits except for Fit 1, including Fits A, B and C, all the P c (4440) have small widths, see also Table A.3 in the Appendix. And all of our fits describe the blow-up of the data in the P c (4440) region well. If neither the pole in K matrix nor the higher partial wave resonance is input, one can not obtain such a blow-up (Fit 1). Obviously the kinematic behaviour can not supply such a structure. This supports the compact pentaquark picture. For its quantum number, in Refs. [8,[33][34][35], they suggest P c (4440) to be 1/2 − , while in Refs. [20,42], they suggest 3/2 − . In our case, Fit 3 is worse than Fit 2 and we prefer the P c (4440) to be Swave, but it is not possible to distinguish the quantum number.
P c (4457) We do not find poles in Fit 1, and find poles in RS-III and VII in Fit 2 and a pole in RS-VIII in Fit 3. For RS-VII and VIII they are faraway from the physical region. Absence of poles close to the physical region means that the structure is caused by cusp effect. The pole in RS-III (Fit 2) is 7 MeV above the threshold √ s th3 = 4459.75 MeV, barely close to the physical region. From the 'pole counting' it does not support the molecule or 'Breit-Wigner' types, but a molecular component can not be entirely excluded in Fit 2. In the region aroundD * Σ c , our amplitude behaves more like a cusp but not a normal Breit-Wigner structure. In Fits 1 and 3 they are caused by cusp effect and in Fit 2 there is a sharp decline near thē D * Σ c threshold, see the graph in the top-right corner of Fig.2. Indeed, this structure is very similar to that of the η ′ π + π − line shape aroundpp threshold, see Fig.4 of [43] and Fig.4 of [44].

IV. ANALYSIS OF OTHER DATASETS
We also check the case that we fit to other datasets given in Ref. [5], the 'm Kp all' and the 'm Kp > 1.9 GeV' ones. In all these fits the treatment with the background is different, but the pole information should be model independent. Following it, these fits could be used to check the reliability of our amplitude analysis. The following extra fits are performed: 1) Fit A: As in Fit.2 but we fit to the data of 'm Kp all' case (without requiring m Kp > 1.9 GeV).
2) Fit B: As in Fit.2 but we fit to the data with requiring m Kp > 1.9 GeV.
3) Fit C: As in Fit.2 we fit to the data with requiring cos θ Pc weighted, with the isospin symmetry violation included.
The invariant mass spectrum is given in Fig.3. Notice that in Fit C, with isospin violation included, it fits better to the data near thresholds.
The discussion of the event shape structure of these fits are consistent with that given by Fits 1, 2, and 3. In each fit, there are only one or two poles in RS-III and/or V of the P c (4312). It supports the molecular component. And there are four poles of the P c (4440) in RS-III, IV, V, VII/VIII. They are close to each other and the widths are narrow. This supports the compact pentaquark component. For the P c (4457) we do not find poles nearby in Fits A and B but find two poles in RS-III and VII in Fit C. Our line shape falls down obviously near theD * Σ c threshold in Fits A and C. And in Fit B ours exhibits a pronounced cusp-like structure around thē D * Σ c threshold. These confirm the cusp effect origin of the P c (4457), but a component ofD * Σ c molecule can't be excluded. This component could be generated by a virtual state in theD * Σ c single channel scattering nearby the threshold, see Ref. [45] and discussions of the pole trajectory in the supplement.

V. SUMMARY
In this paper we perform an amplitude analysis in the process   Fig.1 and green square is from Fig.3 of Ref. [5]. In Fit B the data is from the 'mKp > 1.9 GeV' case. In Fit C the data is from the cos θP c -weighted one. The vertical lines denote theD * Σc thresholds.
in RS-IV, prefers to be a compact S-wave pentaquark. The P c (4457) is most likely to be caused by cusp effect, while a component ofD * Σ c molecule can't be excluded. We also predict the branching ratios (Fit 2) of the decay of Λ 0 b →DΣ c K − and Λ 0 b →D * Σ c K − as (1.49 ± 0.26) × 10 −4 and (0.30 ± 0.08) × 10 −4 , respectively. The future LHCb measurement of the decays, for  The real K matrix K(s) could be parametrized as To reduce the model dependence, we include as less as possible parameters. According to practice we set c ij n≥2 = 0 and s l≥2 = 0. In Fits A and B we apply the same method as that of Fit 2. In Fit C we take into account the contribution of isospin violation, where the phase space factor ρ 2 (s) is replaced by 1 2 , and so on for ρ 3 (s). See Ref. [25] for similar discussions on isospin symmetry breaking in KK.
(A.4) Note that the partial wave decomposition factor and the coupling g P Λ 0 b K + is absorbed into the β P . The branching ratios of all the fits are shown in Table A.2. In all the fits our Br 1 (Λ 0 b → J/ΨpK − ) is ex- actly the same as that of PDG, while in most of fits Br 2 (Λ 0 b →DΣ c K − ) is roughly 1/3 to 2/3 of Br 1 , and the Br 3 (Λ 0 b →D * Σ c K − ) is of the order of 10 −5 . The branching ratio Br 3 of Fit 3 is much different from those of other fits. Notice that in Fit 3 we include the P-wave. This indicates that measuring the Br 2,3 would be rather important for the amplitude analysis. Nevertheless, all the Br 2,3 are of the order of 10 −4 − 10 −5 . The uncertainty of the branching ratios is collected by the uncertainty from MINUIT and the statistics of a dozen of other solutions. We are aware that the amplitude above √ s = 4.6 GeV is not fitted to the data. However, as we have checked, using a polynomial to fit to the data above 4.6 GeV and input it in the integration of Eq. (7), the difference is only several percents, at most 11%. We thus use our K matrix amplitude to do the integration in the whole energy region, and input the difference discussed above as part of the uncertainty. Notice that the branching ratios in Fit B do not have the credibility as our other results. This is reflected by the italic type. The reason is that the cut condition m Kp > 1.9 GeV reduces the event, and the cut out one should also contribute to the branching ratio. In contrast, the cos θ Pc -weighted data has also cut out the Λ * contibution, but the event shape of it is quite the same as that of the 'm Kp all' data by multiplying a normalization factor. The latter data does not miss such contribution.
The poles and couplings are shown in Tables A.3 and A.4, respectively. For the P c (4312), we only find one pole in RS-V of Fits A and B. This convince the molecule picture of this resonance. Roughly, in each Fit the residue coupling toDΣ c (g 2 ) is much larger than that coupling to J/Ψp (g 1 ). This supports the 'DΣ c ' molecule picture,  being in compatible with that given by the 'pole counting' rule. While it has more or less the same order magnitude as that coupling toD * Σ c (g 3 ). This suggests the strong transition between theDΣ c andD * Σ c channels. For the P c (4440), we find poles in four RSs for all the fits, the same as that of Fits 1, 2 and 3, which fits to the cos θ Pcweighted dataset. As discussed above, it is most likely to be a compact pentaquark. The residue g 3 (P c (4440)), except for Fits 3 and A, is smaller than g 3 (P c (4312)) and g 3 (P c (4457)). This supports that the P c (4440) is not D * Σ c molecule origin. For the P c (4457), we do not find poles close to the 'strcuture' as indicated by the data in Fits A and B, while in Fit C the poles are barely close to the physical RS. Indeed, the event shapes of Fit A and C are similar to that of Fit 2, dropping rapidly around theD * Σ c threshold. And in Fit B the line shape has a very clear cusp around the threshold. It confirms that the P c (4457) is caused by cusp effect at theD * Σ c threshold. However, a component ofD * Σ c molecule can't be excluded, see the g 3 (P c (4457)) of Fits 2, 3 and C. It is much larger than g 1,2 (P c (4457)), though the poles are barely close to the physical RS.

Appendix B: Pole trajectory
In this section we track the trajectories of poles by reducing the magnitude of inelastic channels. That is, we change K ij (s) → λK ij (s), where i = j and vary λ from one to zero. The trajectories of all the poles of Fit 2 are shown in Fig.4. It should be noticed that the amplitude at λ = 0 is not physical and the pole could run into weird position. However, as discussed below, it can still give useful hints about the resonances. P c (4457) Only in Fits 2, 3 and C do we find poles close to the P c (4457) region, and all these poles are not close to the physical sheet. No pole close to the physical region suggests that the structure is generated by the cusp effect. The P c (4457) pole trajectory of RS-III will go across the cut and get into RS-VIII, finally it meets the pole coming from RS-VII at the real axis. The destination pole ( √ s p = 4458.42 MeV) lies closely below theD * Σ c threshold ( √ s th3 = 4459.75 MeV) and it is the virtual state in RS-II of theD * Σ c single channel. As discussed above, a virtual state close to threshold could also generates 'cusp' structure around threshold, see in Ref. [43]. This means that a component ofD * Σ c molecule is also possible, and it could be coming from a virtual state origin in single channel. However, our pole analysis reveals that most likely the threshold effect generates the structure of P c (4457) as shown in the data.