Pole analysis on the hadron spectroscopy of Λb→J/ΨpK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b\rightarrow J/\Psi p K^-$$\end{document}

In this paper we study the J/Ψp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\Psi p$$\end{document} spectroscopy in the process of Λb→J/ΨpK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b\rightarrow J/\Psi p K^-$$\end{document}. The final state interactions of coupled channel J/Ψp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\Psi p$$\end{document}–D¯Σc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{D} \Sigma _c$$\end{document}–D¯∗Σc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{D}^{*} \Sigma _c$$\end{document} are constructed based on K-matrix with the Chew–Mandelstam function. We build the Λb→J/ΨpK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b\rightarrow J/\Psi p K^-$$\end{document} amplitude according to the Au–Morgan–Pennington method. The event shape is fitted and the decay width of Λb→J/ΨpK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b\rightarrow J/\Psi p K^-$$\end{document} 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 Pc(4312)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_c(4312)$$\end{document} should be D¯Σc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{D}\Sigma _c$$\end{document} molecule, the Pc(4440)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_c(4440)$$\end{document} could be an S-wave compact pentaquark state, and the structure around Pc(4457)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_c(4457)$$\end{document} is caused by the cusp effect. The future experimental measurement of the decays of Λb→D¯ΣcK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b\rightarrow \bar{D}\Sigma _c K^-$$\end{document} and Λb→D¯∗ΣcK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b\rightarrow \bar{D}^*\Sigma _c K^-$$\end{document} would further help to study the nature of these resonances.


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/ p spectrum of b → J/ pK − : 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][22]. Among them, Ref. [12] uses a coupled channel K-matrix formalism to fit to the data around the P c (4312). They found the attractive effect of the + cD 0 channel, but it is not strong enough to form a bound state. In Ref. [21], the Lippmann-Schwinger equations have been used and the one pion exchange and short range scattering potential have been considered. Thay found the three resonances and also a narrow * cD state (P c (4380)). In addition, three more * cD * molecules have been seen in the analysis. All these poles are hadronic molecules of ( * ) cD ( * ) channels. Since lots of the paper support the molecule picture of these P c states, it would be rather interesting to distinguish the molecule and non-molecule structure. The pole counting rule [23,24] is a right way to do such study. Indeed the shadow poles (accompanying the ones being closest to the physical sheet) are also important to discuss the inner structure. 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 above (P c (4312), P c (4440), and P c (4457)). Our amplitudes describe the data around the P c (4440) and P c (4457) region well 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 b → J/ pK − . 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. [25][26][27][28][29][30][31][32][33][34][35]. We will use the Au-Morgan-Pennington (AMP) method [25,26] to include the FSI of J/ p-D c -D * 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. 2 we use Kmatrix to build the hadronic scattering amplitudes of J/ p-D c -D * c triple channels. And the b → J/ pK − amplitude is constructed by the AMP methods. In Sect. 3 we fit to the event shape and decay width of b → J/ pK − 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. 4 we discuss the fits to other datasets given by the LHCb. We end with a brief summary.

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 [36,37], there could beD c andD * c hadronic molecule states with quantum number I J P = 1 2 1 2 − at 4261 + i28.5 and 4412 + i23. 6 MeV of each channel. This is later studied in [38] by a coupled channel unitary approach. In Ref. [5], the P + c (4312) is found to be underD c threshold and more like an S-wave resonance. The other two resonances are proximate to theD * c threshold. We thus take J/ p-D c -D * c as the coupled channels. The helicity has been ignored as that the heavy quark spin symmetry ensures the spin-dependent interactions related to the heavy quark are of the order of 1/m Q [39] 1 . These assumptions are consistent with the analysis of Ref. [5], where it is also found that including P-wave factors in the Breit-Wigner amplitudes has negligible effect on the results.
In addition, the thresholds of * + cD 0 ( * ++ c D − ) and * + cD * 0 ( * ++ c D * − ) are 4382.33 (4388.06) MeV and 4524.35 (4528.67) MeV, respectively. They are far away from the three resonances we studied here. Besides, from the LHCb experiment measurement [5], there is no obvious structure around these thresholds, we thus do not consider their contribution here. For the + cD ( * )0 channels, the mass and width of the P c (4312) are M = 4311.9 ± 0.7 +6.8 −0.6 MeV and = 9.8 ± 2.7 +3.7 −4.5 MeV. Taking into account that the + cD ( * )0 threshold is 19 MeV below the peak of the P c (4312), twice as much as the width, and the + cD 0 is even lower, they won't contribute a lot to the structure. It should also be pointed out that their interactions are expected to be repulsive and can not form the hadronic bound state [5]. Thus all the channels, except for the J/ p-D c -D * c , are not included in our model. 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].
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 and it could be parameterized as To reduce the model dependence, we include as less as possible parameters. According to practice we set c i j n≥2 = 0 and s l≥2 = 0. C(s) is the diagonal matrix of the canonical definition of Chew-Mandelstam function [40,41], and it could be written in once subtracted dispersion relation: where i( j) = 1, 2, 3 represent for J/ p,D c ,D * 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/ p line shape (in the process of b → J/ pK − ) from LHCb [5] help us to constrain the hadronic scattering amplitude. To describe it, the rescattering of inelastic channels (D c ,D * c ) needs to be considered. We implement the AMP formalism [25,26] to include the FSI: where 'i' and 'k' have the same meaning as explained after Eq. (3). The α k (s) are polynomials of s, absorbing all the contributions of left hand cut and distant right hand cut. For simplicity we set them to be constant α 1,2,3 and ignore higher order terms. It is easy to check that Eq. (6) 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 d i d √ s with a normalization factor 'N '. The decay width given by the PDG [42] 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  8)) are P S 2 /P S 1 0.6, 0.4, respectively. This suggests that the branching ratios of Br 2 ( 0 b →D c K − ) and Br 3 ( 0 b →D * c K − ) could be the same order as that of Br 1 ( 0 b → J/ pK − ). 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.

Fitting strategy and pole analysis
For the 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 P-wave scattering amplitude we adopt the Blatt-Weisskopf barrier factor representation [29,30], see the supplement for details. The following fits are performed: (1) Fit 1: We do not include any poles in the K matrix.
(2) Fit 2: As in Fit 1 but we include one pole in the K matrix.
(3) Fit 3: As in Fit 1 we do not include poles in the K matrix, but add a P-wave instead.
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 P c weighted. It is worth to point out that this dataset has removed much of the interfering of the * , which is in the K − p channel and most populated at cos θ P c > 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  Fig. 1. For the P c (4312), our amplitude fits 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 theD 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 * c threshold effect. These will be discussed in next sections. We also separate the individual contribution of each channel to the J/ p spectroscopy, that is, we use F 1 = α i T i1 to replace of Eq.(6). It is found that theD c → J/ p channel dominates the contribution around the P c (4312), which is consistent with theD c molecule picture. All the three channels contributes like a peak or dip around the P c (4440), sug- gesting it to be a Breit-Wigner particle. Around the P c (4457) all the channels have non-ignorable structure aroundD * c threshold, while that of theD * c → J/ p channel dominates. What is more, theD * c contribution behaves like either a threshold effect or a bound state below the threshold. These supports that the P c (4457) could either be caused by cusp effect or a component ofD * c molecule. We will discuss it 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 [43]. Here we use the following definition [44] as shown in Table 1. The pole s R and its coupling/residue of the RS-n in the triple channel are defined as: where the subscript 'i, j' denote the hadronic channels as before. The poles in different RSs for all the fits are given in Table 2. In Fit 1, we only find poles of P c (4312). 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. [23,44]: 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 aD 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 aboveD c threshold, respectively. RS-II and RS-V are connected along the unitary cut above theD 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 theD c is strong and it is typical way how a virtual state (with weak interaction to theD c ) changes into a molecule. For the strength of the couplings please see the |g 2 | of  In the region aroundD * 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 theD * c threshold, see the graph in the topright corner of Fig. 2. Indeed, this structure is very similar to that of the η π + π − line shape aroundp p threshold, see Fig. 4 of [46] and Fig. 4 of [47].

Analysis of other datasets
We also check the case that we fit to other datasets given in Ref. [5], the 'm K p all' and the 'm K p > 1.9 GeV' ones. Indeed in all these datasets the treatment with the background is different. Note that the poles are model independent and in all the processes and models their locations should be the same. Thus the poles extracted from different Fits could be used to check the reliability of our analysis. The following extra fits are performed: (1) Fit A: As in Fit.2 but we fit to the data of 'm K p all' case (without requiring m K p > 1.9 GeV). (2) Fit B: As in Fit.2 but we fit to the data with requiring m K p > 1.9 GeV. (3) Fit C: As in Fit.2 we fit to the data with requiring cos θ P c weighted, with the isospin symmetry violation included. That is, the mass difference between D ( * )0 + c and D ( * )− ++ c is taken into account, see Appendix A for details.
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 * c threshold in Fits A and C. And in Fit B ours exhibits a pronounced cusp-like structure around theD * c threshold. These confirm the cusp effect origin of the P c (4457), but a component ofD * c molecule can't be excluded. This component could be generated by a virtual state in theD * c single channel scattering nearby the threshold, see Ref. [48] and discussions of the pole trajectory in the supplement in Appendix B.

Summary
In this paper we perform an amplitude analysis in the process of b → J/ pK − . The J/ p-D 0 + c -D * 0 + c triple channel scattering amplitude is constructed by K-matrix, within Chew-Mandelstam formalism. Based on it we apply the Au-Morgan-Pennington method to study the process of b → J/ pK − , taking into account the final state interactions. Qualified fits to the invariant mass spectroscopy [5] is obtained from J/ p threshold up to √ s = 4600 MeV. We extract out the poles (Fit 2) and find that the P c (4312), with the pole location 4314.  Fit of the J/ p spectroscopy of b → J/ pK − . All the datasets are from Ref. [5]. In Fit A the data is from the 'm K p all' case, where the red open circle is from Fig. 1 and green square is from Fig. 3 of Ref. [5]. In Fit B the data is from the 'm K p > 1.9 GeV' case. In Fit C the data is from the cos θ Pc -weighted one. The vertical lines denote theD

Data Availability Statement
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Appendix A: K matrix formalism
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 . See Refs. [29,30] for similar discussions on isospin symmetry breaking in K K .
The parameters and χ 2 /d.o. f of all our fits are given in Table 3.
For all the fits, the masses of the resonances are given by the PDG [36], To include the P-wave scattering amplitude we adopt the Blatt-Weisskopf barrier factor representation [25] with Q i (s) = 1 + q 2 /(s − (m i + M i ) 2 ) and q is chosen to be 1 GeV. Here M P and P→i are the input mass and width (decaying to the i-th channel) of the P-wave resonance in order. We have the scattering amplitudes and the relative 0 b decay amplitudes . (12) Note that the partial wave decomposition factor and the coupling γ P 0 b K + is absorbed into the β P . The branching ratios of all the fits are shown in Table 4. In all the fits our Br 1 ( 0 b → J/ pK − ) is exactly the same as that of PDG, while in most of fits Br 2 ( 0 b →D c K − ) is roughly 1/3 to 2/3 of Br 1 , and the Br 3 ( 0 b →D * c K − ) is of 3.7321 ± 1.5880 · · · · · · · · · P→3 (MeV) · · · · · · 1.0234 ± 0.4089 · · · · · · · · · β P · · · · · · 1.23 Table 4 The Br 1,2,3 denotes the branching ratios of 0 b → J/ pK − , 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 K p > 1.9 GeV reduces the event, and the cut out one should also contribute to the branching ratio. In contrast, the cos θ P c -weighted data has also cut out the * contibution, but the event shape of it is quite the same as that of the 'm K p all' data by multiplying a normalization factor. The latter data does not miss such contribution.
The poles and couplings are shown in Tables 5 and 6, respectively. For the P c (4312), we only find one pole in RS-V of Fits A and B. Roughly, in each Fit the residue coupling toD c (g 2 ) is much larger than that coupling to J/ p (g 1 ). This supports the 'D 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 * c (g 3 ). It reveals the strong transition between thē D c andD * 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 P c -weighted 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 notD * 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 * 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 * c threshold. However, a component ofD * c molecule can't be excluded, see the g 3 (P c (4457))  Table 6 The residues of each poles given by our fits. The unit is GeV. We only show the one is closest to the physical region. 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 i j (s) → λK i j (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 (4312) Except for the pole in RS-V, there are three other shadow poles in RS-III, IV and VII. These shadow poles are dragged far away from the pole of RS-V, due to the strong interaction between the pole and theD c channel. Finally all of them merge into one 'destination pole', which is a resonance ( √ s p = 4428.56 − i112.57 MeV) in RS-II ofD single channel scattering. Since all the poles in different RSs are originated fromD scattering, it supports theD molecular picture. This conclusion is consistent with the 'pole counting' rule. P c (4440) There are four poles nearby √ s = 4440 MeV in RS-III, IV, V, VII, supporting the non-molecular origin. The poles in RS-III and V merge together at the destination pole  Table 6 for the P c (4440). When the inelastic channels are shut down, the coupled channel scattering is spitted into several single channel scattering, and the destination pole in different single channel behaves differently. The trajectory supports the elementary particle picture (compact pentaquark). 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 * c threshold ( √ s th3 = 4459.75 MeV) and it is the virtual state in RS-II of theD * 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 * 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 the P c (4457).