PLB34606S0370-2693(19)30286-210.1016/j.physletb.2019.04.053PhenomenologyTable 1The values of the scattering lengths and effective ranges of the S-wave amplitudes for different channels. The uncertainties for a and r are determined by adding in quadrature the resulting ones from the systematic and statistical errors of the masses and widths of the Pc states. The errors of the different thresholds are negligible in comparison with the uncertainties of the masses and widths of the Pc states.Table 1ResonanceMassWidthThresholdar

(MeV)(MeV)(MeV)(fm)(fm)

Pc(4312)4311.9 ± 6.89.8 ± 5.2Σc+D¯0 (4317.7)−2.9 ± 0.8−1.7 ± 0.7

Σc++D− (4323.6)−2.4 ± 0.6−1.2 ± 0.3

Pc(4440)4440.3 ± 4.920.6 ± 11.2Σc+D¯⁎0 (4459.8)−1.7 ± 0.2−0.9 ± 0.1

Σc++D⁎− (4464.2)−1.6 ± 0.2−0.8 ± 0.1

Pc(4457)4457.3 ± 4.16.4 ± 6.0Σc+D¯⁎0 (4459.8)−3.8 ± 1.6−2.3 ± 1.3

Σc++D⁎− (4464.2)−3.0 ± 0.7−1.6 ± 0.4

Table 2Results obtained with X = X1 + X2 = 1. The J/ψp and ΣcD¯(⁎) channels, which are labeled as 1 and 2 respectively, are included.Table 2Resonance|g1||g2|Γ1Γ2X1X2

(GeV)(GeV)(MeV)(MeV)

Pc(4312)

mΣc++mD¯02.1−2.1+0.810.9−2.9+2.16.5−6.5+4.93.3−3.3+10.50.006−0.006+0.0050.994−0.005+0.006

mΣc+++mD−2.5−0.9+0.612.6−2.6+1.68.5−4.6+4.71.3−1.3+6.10.008−0.005+0.0050.992−0.005+0.005

Pc(4440)

mΣc++mD¯⁎03.2−0.9+0.614.9−1.4+1.216.3−7.4+6.74.3−4.3+9.20.010−0.004+0.0050.990−0.005+0.004

mΣc+++mD⁎−3.3−0.9+0.615.6−1.1+1.017.7−8.2+6.92.9−2.9+8.30.011−0.005+0.0050.989−0.005+0.005

Pc(4457)

mΣc++mD¯⁎01.5−1.0+0.79.5−5.1+2.23.5−3.5+4.22.9−2.9+9.50.002−0.002+0.0030.998−0.003+0.002

mΣc+++mD⁎−1.8−0.9+0.611.2−2.5+1.65.4−4.0+4.21.0−1.0+6.10.003−0.002+0.0030.997−0.003+0.002

Table 3Results obtained for X = 0.8 and X = 0.5 by including the J/ψp (labeled as 1) and ΣcD¯(⁎) (labeled as 2) channels. The values in the table are calculated by using the masses Σc+ and D¯(⁎)0.Table 3Resonance|g1||g2|Γ1Γ2X1X2

(GeV)(GeV)(MeV)(MeV)

Pc(4312)

X = 0.82.3−1.8+0.79.8−2.5+1.87.1−6.8+5.02.7−2.7+7.30.007−0.007+0.0050.793−0.005+0.007

X = 0.52.4−1.2+0.77.7−2.0+1.58.1−6.2+5.11.7−1.7+5.10.008−0.006+0.0050.492−0.005+0.006

Pc(4440)

X = 0.83.2−0.9+0.713.3−1.3+1.017.2−8.2+7.63.4−3.4+7.40.011−0.005+0.0050.789−0.005+0.005

X = 0.53.4−1.0+0.710.5−1.0+0.718.5−9.3+9.02.1−2.1+4.50.012−0.006+0.0060.488−0.006+0.006

Pc(4457)

X = 0.81.6−1.5+0.78.5−4.5+2.04.1−4.1+4.62.3−2.3+7.90.002−0.002+0.0030.798−0.003+0.003

X = 0.51.7−1.6+0.86.7−3.3+1.55.0−5.0+5.11.4−1.4+5.00.003−0.003+0.0030.497−0.003+0.003

Table 4Results obtained when including the Λc+D¯⁎0 (labeled as 1) and Σc+D¯⁎0 (labeled as 2) channels for X = 1.0, 0.8 and 0.5.Table 4Resonance|g1||g2|Γ1Γ2X1X2

(GeV)(GeV)(MeV)(MeV)

Pc(4312)

X = 1.04.0−3.8+2.010.5−2.5+1.36.8−6.8+5.43.0−3.0+10.60.09−0.09+0.160.91−0.16+0.09

X = 0.84.2−3.4+2.09.2−2.0+1.27.5−7.2+5.52.3−2.3+8.10.10−0.10+0.160.70−0.16+0.10

X = 0.54.5−2.5+2.06.8−1.2+0.98.5−6.5+5.71.3−1.3+4.30.11−0.09+0.170.39−0.17+0.09

Pc(4440)

X = 1.03.8−1.0+0.714.8−1.3+1.016.4−7.5+6.84.2−4.2+9.10.03−0.02+0.010.97−0.01+0.02

X = 0.83.9−1.1+0.813.1−1.1+0.917.3−8.3+7.73.3−3.3+7.20.03−0.02+0.010.77−0.01+0.02

X = 0.54.0−1.2+1.010.2−0.8+0.618.6−9.4+9.22.0−2.0+4.30.03−0.01+0.020.47−0.02+0.01

Pc(4457)

X = 1.01.7−1.6+0.99.4−5.0+2.33.5−3.5+3.72.9−2.9+9.50.005−0.005+0.0070.995−0.007+0.005

X = 0.81.9−1.9+0.88.4−4.4+2.04.1−4.1+4.62.3−2.3+7.90.006−0.006+0.0080.794−0.008+0.006

X = 0.52.0−2.0+0.96.6−3.2+1.65.0−5.0+5.11.4−1.4+4.90.008−0.008+0.0080.492−0.008+0.008

Anatomy of the newly observed hidden-charm pentaquark states: Pc(4312), Pc(4440) and Pc(4457)Zhi-HuiGuoa⁎zhguo@hebtu.edu.cnJ.A.Ollerboller@um.esaDepartment of Physics and Hebei Advanced Thin Films Laboratory, Hebei Normal University, Shijiazhuang 050024, ChinaDepartment of PhysicsHebei Advanced Thin Films LaboratoryHebei Normal UniversityShijiazhuang050024ChinabDepartamento de Física, Universidad de Murcia, E-30071 Murcia, SpainDepartamento de FísicaUniversidad de MurciaMurciaE-30071Spain⁎Corresponding author.Editor: B. GrinsteinAbstractWe study the newly reported hidden-charm pentaquark candidates Pc(4312), Pc(4440) and Pc(4457) from the LHCb Collaboration, in the framework of the effective-range expansion and resonance compositeness relations. The scattering lengths and effective ranges from the S-wave ΣcD¯ and ΣcD¯⁎ scattering are calculated by using the experimental results of the masses and widths of the Pc(4312), Pc(4440) and Pc(4457). Then we calculate the couplings between the J/ψp,ΣcD¯ channels and the pentaquark candidate Pc(4312), with which we further estimate the probabilities of finding the J/ψp and ΣcD¯ components inside Pc(4312). The partial decay widths and compositeness coefficients are calculated for the Pc(4440) and Pc(4457) states by including the J/ψp and ΣcD¯⁎ channels. Similar studies are also carried out for the three Pc states by including the ΛcD¯⁎ and ΣcD¯(⁎) channels.1IntroductionThe first discovery of the hidden-charm pentaquark states Pc(4380) and Pc(4450) [1] has triggered a plethora of in-depth theoretical studies [2]. Very recently, the LHCb Collaboration has reported updated results on the pentaquark states based on the combinations of the Run 1 + Run 2 data [3]. The first notable finding from the updated measurements is that a new hidden-charm pentaquark state Pc(4312) is observed with the mass 4311.9±0.7−0.6+6.8 MeV and the width 9.8±2.7+3.7−4.5 MeV. The second notable and intriguing observation is that the previous single state Pc(4450) is superseded by two nearby states Pc(4440) and Pc(4457), with their masses 4440.3±1.3+4.1−4.7 MeV and 4457.3±0.6−1.7+4.1 MeV, respectively, and their widths 20.6±4.9−10.1+8.7 MeV and 6.4±2.0+5.7−1.9 MeV, respectively. The previous peak around the Pc(4380) state now becomes less clear and its existence needs to be confirmed further by the experimental analysis. The new measurements have already attracted attention from many groups [4–13].All of the three new states are observed in the J/ψp invariant mass distributions from the Λb→J/ψpK− decay. One of the common features of the newly measured pentaquark states is that they all have small widths. Another important common feature is that they lie quite close to the thresholds of two underlying hadrons. In the following discussions we take a conservative way to estimate the experimental values of the masses and widths for the Pc(4312),Pc(4440) and Pc(4457) [3]. To be more specific, we take the larger values in magnitude of the upper or lower limits for the systematic uncertainties, and add them quadratically to the statistical ones as the total uncertainties. The resulting masses and widths are summarized in the second and third columns of Table 1. The differences between the mass of the Pc(4312) and the Σc+D¯0 and Σc++D− thresholds are 5.8±6.8 MeV and 11.7±6.8 MeV, respectively. The mass of Pc(4440) lie 19.5±4.9 MeV and 23.9±4.9 MeV below the Σc+D¯⁎0 and Σc++D⁎− thresholds, respectively. For the Pc(4457), the differences between its mass and the Σc+D¯⁎0 and Σc++D⁎− thresholds are 2.5±4.1 MeV and 6.9±4.1 MeV, respectively. Taking into account the uncertainties of the experimental measurements of the Pc(4312), we notice that its mass can be either below or above the Σc+D¯0 threshold, but it is always below the Σc++D− threshold. For the mass of the Pc(4457) a similar situation occurs, so that, within the present experimental uncertainties [3], its mass can be also either below or above the Σc+D¯⁎0 threshold, but it is always below the Σc++D⁎− threshold. As a result one would expect that the isospin breaking effects could be visible [6]. In order to quantify the possible isospin breaking effects, we shall distinguish the elastic scattering with different thresholds involving Σc+ or Σc++ in a later study.In this work, our key aim is to quantify the possibilities of the Pc(4312) as the S-wave ΣcD¯, and the Pc(4440) and Pc(4457) as the S-wave ΣcD¯⁎ molecular states. The effective range expansion (ERE) approach offers a reliable tool to analyze the dynamics around the threshold energy region. The combinations of the analyticity, unitarity and ERE have been demonstrated to be successful in analyzing the heavy-flavor exotic hadrons near thresholds [14–17]. Another powerful tool that can help to reveal the inner structures of the hadrons is the Weinberg's compositeness relation [18], which is extended to the resonance case in Refs. [19–21]. Other forms of generalization for other compositeness relation to address the resonances can be also found in Refs. [22–26]. In the current work we combine analyticity, unitarity, the ERE and the resonance compositeness relation to study the three newly measured pentaquark states.2Effective-range-expansion study of the pentaquark statesThe ERE approach relies on the power series expansion of the K-matrix V(k) at around threshold(1)V(k)=−1a+12rk2+O(k2), where a is the scattering length, r denotes the effective range and k stands for the magnitude of three-momentum in the center of mass (CM) frame. For a two-particle system with masses m1 and m2, in the non-relativistic limit the three-momentum k is related to the CM energy E through(2)k=2μ(E−mth), with the threshold mth=m1+m2 and the reduced mass μ=m1m2m1+m2.For the ΣcD¯ scattering near the Pc(4312) and the ΣcD¯⁎ scattering near the Pc(4440) and Pc(4457) energy regions, the magnitudes of the three-momenta of the two-particle systems can range from 0 to 250 MeV, after taking into account the experimental uncertainties of the masses of the Pc states [3]. For the scattering of two heavy-flavor hadrons, it is plausible that the pion exchanges can be treated perturbatively [27–31]. For the heavier vector-resonance exchanges, their contributions can be effectively included via contact interactions, since their masses are clearly larger than the scale of the relevant three-momenta. Therefore we take the point of view from the pionless effective field theory, which only needs to include the local contact interactions [32].Under these circumstances only the unitarity/right-hand cut enters and there is no crossed-channel dynamics. The elastic S-wave scattering amplitude around threshold that results from Eq. (1) (without the crossed-channel cuts) can be written as(3)T(E)=1−1a+12rk2−ik, which satisfies the unitarity condition(4)ImT(E)−1=−k,(E>mth).The formula T(E) in Eq. (3) generally works well in the energy region near threshold even when resonances appear, except in the special situation that an underlying Castillejo-Dalitz-Dyson (CDD) pole sits on top of the threshold. In the latter case, one has to explicitly include the CDD pole in Eq. (3) and we refer to Ref. [17] for further details. It is difficult to know whether there is a CDD pole near threshold a priori. Nevertheless in Refs. [15,17] it is proved that when a CDD pole approaches to the threshold one has the following behaviors for the scattering length and effective range(5)a→−mth−MCDDgCDD,r→−gCDDμ(mth−MCDD)2, with MCDD the bare CDD pole mass and gCDD the residue. According to Eq. (5), one can infer that there exists a CDD pole near the threshold only for the situations with |a|≪1 fm and |r|≫1 fm. In this situation, one should use the formalism developed in Ref. [17] to proceed, instead of Eq. (3).In the present work we first blindly use the ERE formalism in Eq. (3). If the resulting a and r have natural values of the long-range hadronic scale at 1/mπ∼1 fm, one could then safely conclude that the formalism in Eq. (3) is applicable in our study (with no indication of a near-threshold CDD pole). We demonstrate below that the resulting values of a and r from the S-wave ΣcD¯ scattering around Pc(4312) and the S-wave ΣcD¯⁎ scattering around the Pc(4440) and Pc(4457), indeed have typical long-range hadronic scale around 1 fm. Another issue that needs to be clarified is that we implicitly assume a definite isospin number for the Pc states (although we do not need to specify it), otherwise we had to use a coupled-channel scattering formalism in the ERE study. Regarding the quantum numbers of JP, the negative P parity can be uniquely fixed in the S-wave ΣcD¯(⁎) scattering. Similarly, the total angular momentum is J=1/2 for ΣcD¯ S-wave scattering, while for the analogous ΣcD¯⁎ case there are two possibilities, J=1/2 or 3/2, which can not be pinned down from our study.For a resonance pole, its position ER is denoted as(6)ER=MR−iΓR/2, where MR is the resonance mass and ΓR denotes its width. The resonance poles lie on the second Riemann sheet (RS) of the scattering amplitude TII(E), which is given by(7)TII(E)=1−1a+12rk2+ik. We mention that the convention Imk>0 should be taken in Eqs. (3) and (7). Given the mass and width of the resonance, we can determine the scattering length a and effective range r by requiring that TII(ER)−1=0, i.e.(8)−1a+12rkR2+ikR=0, where kR is the corresponding three-momentum at the pole position(9)kR=μ(ER−mth).By solving Eq. (8), it is straightforward to determine the values of a and r once the masses and widths of the resonances are given, with the result [15](10)a=−2ki|kR|2,r=−1ki, where kr=RekR and ki=ImkR. As mentioned above in the Introduction, we distinguish the different charged states Σc+ and Σc++, in order to quantify the isospin breaking effects. The thresholds of the different charged states are explicitly given in the fourth column of Table 1. The results for the scattering lengths a and effective ranges r with uncertainties are collected in the fifth and sixth columns of Table 1, respectively.According to the values obtained for a and r in Table 1, although we see some discrepancies in the central values for the channels with different charged states, they are compatible after taking into account the uncertainties. It implies that the isospin breaking effects in the three Pc states seem mild and further experimental reduction of the uncertainties could help to identify the roles of the isospin breaking.All of the resulting scattering lengths a and effective ranges r in Table 1 have natural values of the order of 1 fm, indicating that indeed there is no need for introducing CDD poles near the thresholds. Let us notice that this outcome is consistent with the application of Eq. (3) in our study. Furthermore, the natural values of the a and r allow us to qualitatively conclude that the Pc(4312) can be described as an S-wave ΣcD¯ molecular state, and the Pc(4440) and Pc(4457) are S-wave ΣcD¯⁎ composite states. Nevertheless, in the ERE approach we can not use the prescription in Ref. [19] to give a quantitative estimate of the probabilities of the ΣcD¯ component in the Pc(4312) and of the ΣcD¯⁎ component in the Pc(4440) and Pc(4457) resonances. In Ref. [19] it has been demonstrated that one can only give the probabilistic interpretation of the compositeness coefficients when the resonance pole sR=ER2 lies in an unphysical RS that is directly connected to the physical one in the region sk<s<sk+1, such that sk<ResR<sk+1, with sk and sk+1 the two nearby thresholds. In the single-channel scattering case, it requires that the resonance pole mass should lie above the threshold in the second RS. However in most of cases the pole positions of the Pc states in Table 1 are below the thresholds. This fact refrains us from discussing the probabilities of finding the two-particle components in the Pc states in the ERE approach.In the present formalism we are assuming that the whole width of a resonance is due to the corresponding ΣcD¯(⁎) channel, and the resulting a and r are real. On general grounds, because of the presence of the inelastic channels below threshold, like the J/ψp one to which these resonances decay, the ERE parameters a and r are complex. One possible way to proceed is to include explicitly the inelastic channels below the ΣcD¯(⁎) channel, such as the aforementioned J/ψp. However, in the coupled-channel scattering case, there would be needed extra scattering input which is beyond the scope of the present study. In order to give quantitative information of the inner structures of the Pc states, and take into account at least one inelastic channel, we proceed the study by relating the compositeness coefficients with the partial decay widths in next section.3Compositeness relations and the partial widthsAs mentioned previously, we can not access the quantitative information of the constituents inside the Pc(4312) in the elastic scattering ΣcD¯ and the Pc(4440) and Pc(4457) from the elastic ΣcD¯⁎ scattering. A straightforward extension is to include the additional J/ψp channel, in which invariant-mass distribution the different Pc resonances are actually detected [3]. For the two-channel J/ψp and ΣcD¯(⁎) systems, it is natural to assume that the Pc resonances lie in the second RS, which now allows us to exploit the formalism in Ref. [19] to calculate the probabilities of the two-particle components in the Pc. Analogous study has been carried out for the obsolete Pc(4450) state by including the J/ψp and χc1p channels in Ref. [21].The essential prescription of Ref. [19] to calculate the partial compositeness coefficient Xj of a resonance R contributed by the jth channel is given by(11)Xj=|gj|2|∂Gj(sR)∂s|, where gj denotes the coupling between the two-particle state and the resonance R, and the one-loop two-point function G(s) is given by(12)G(s)=i∫d4q(2π)41(q2−m12+iϵ)[(P−q)2−m22+iϵ],s≡P2. This expression can be explicitly integrated out by using a once-subtracted dispersion relation or dimensional regularization (replacing the divergence by a constant), which then reads [33](13)G(s)=116π2{a(μg)+lnm12μg2+s−m12+m222slnm22m12+σ2s[ln(s−m22+m12+σ)−ln(−s+m22−m12+σ)+ln(s+m22−m12+σ)−ln(−s−m22+m12+σ)]}, where(14)σ(s)=[s−(m1+m2)2][s−(m1−m2)2]. The evaluation of Gj(s) for the jth channel in Eq. (11) requires to use the proper masses m1 and m2 in that channel. In this equation ∂Gj(sR)/∂s denotes the partial derivative evaluated at the resonance pole position sR=ER2=(MR−iΓR/2)2. Notice that ∂Gj(sR)/∂s is independent on the subtraction constant a(μg) and the regularization scale μg in Eq. (13).In order to fix the two couplings gi=1,2, we impose that the decay widths ΓR of the Pc states are saturated by the two channels J/ψp and ΣcD¯(⁎). The partial decay width Γ1 to J/ψp takes the standard form [34](15)Γ1=|g1|2q(MR2)8πMR2, where the relativistic three-momentum q(MR2) is(16)q(MR2)=[MR2−(m1+m2)2][MR2−(m1−m2)2]2MR. Since in many cases the masses of the Pc resonances are below the thresholds of ΣcD¯(⁎), we introduce a Lorentzian mass distribution to calculate the partial width Γ2 to the ΣcD¯(⁎) channel as(17)Γ2=|g2|2∫mthMR+2ΓRdwq(w2)16π2w2ΓR(MR−w)2+ΓR2/4. To restrict the discussion to the resonance energy region, we set the upper integration limit at MR+2ΓR in Eq. (17), as in Ref. [21]. After taking into account Eqs. (15) and (17), the saturation condition of the Pc decay widths by the J/ψp and ΣcD¯(⁎) channels gives(18)|g1|2q1(MR2)8πMR2+|g2|2∫mthMR+2ΓRdwq2(w2)16π2w2ΓR(MR−w)2+ΓR2/4=ΓR, with q1 and q2 the three-momenta of the J/ψp and ΣcD¯(⁎) channels, respectively.For the resonance poles in the second RS in the coupled-channel J/ψp and ΣcD¯(⁎) scattering, one can identify the compositeness coefficient Xj in Eq. (11) as the probability to find the two-particle state from the jth channel in the considered resonance. We mention that within the uncertainties of the masses of the Pc(4312) and Pc(4457), a tiny portion of their poles lies in the third RS (in which the three-momenta of the two channels flip sign) so that they are continuously connected with the physical RS above the ΣcD¯(⁎) threshold. Nevertheless, due to their closeness to the thresholds, their effects can be covered by the large uncertainties of the Pc masses. Therefore we shall only focus on the poles on the second RS in the following.As a clarification remark, let us notice that in Eq. (11) the coupling is taken constant in the range of masses of the resonance along its Lorentzian mass distribution because of the finite width of the resonance, cf. Eq. (17). In this way, there is a smooth transition in the calculation of X2 as the value of the nominal resonance pole mass MR varies from above to below the threshold. This allows us some flexibility in order to bypass the strict requirement that the resonance mass should lie above the thresholds of the channels for which Xj is calculated. However, in the elastic ERE approach discussed in Sec. 2, the whole width is accounted for only by the channel explicitly taken into account (the second one in the present coupled-channel study), and the situation is more stringent in this respect [15].The total compositeness X is the sum of X1 and X2, with X1 the partial compositeness coefficient of the J/ψp and X2 the coefficient of ΣcD¯(⁎). By using Eq. (11), we can obtain(19)|g1|2|∂G1II(sR)∂s|+|g2|2|∂G2(sR)∂s|=X, where G1II(s) stands for the G(s) function on the second RS and it is related to the expression in Eq. (13) through GII(s)=G(s)+iσ(s)/(8πs).For a given value of the total compositeness X contributed by the J/ψp and ΣcD¯(⁎) channels, we can determine the couplings |g1| and |g2| by combining Eqs. (18) and (19). In this way, we can further calculate the partial compositeness coefficients X1,2 using Eq. (11) and the partial decay widths Γ1,2 via Eqs. (15) and (17). In principle the partial widths Γ2 consist of combinations of the Σc+D¯(⁎)0 and Σc++D(⁎)− channels, depending on the isospin of the pentaquark states Pc. Nevertheless, we point out that the method employed is not sensitive to whether we assume a definite isospin for the Pc states or not, as long as the same masses of the ΣcD¯(⁎) are taken in Eqs. (17) and (18). The reason is because the couplings squared of the different charged states simply add together in these equations. In order to check the isospin breaking effects, we separately solve Eqs. (18) and (19) by using either the masses of Σc+D¯0(D¯⁎0) or Σc++D−(D⁎−).Concerning the value of X in Eq. (19), we distinguish three different scenarios. In the first scenario, we assume that the compositeness of the Pc states is completely saturated by the J/ψp and ΣcD¯(⁎)channels, that is, we first assume that X=1. For each Pc state, we separately perform the calculations by using either the masses of Σc+D¯0(D¯⁎0) or Σc++D−(D⁎−). The resulting values of the couplings |g1| and |g2|, the partial widths Γ1 and Γ2, and the partial compositeness coefficients X1 and X2 are summarized in Table 2. The first lesson we learn from Table 2 is that the Pc couplings |g1| to the J/ψp channel are much smaller than the couplings |g2| to the ΣcD¯(⁎) channel. The situation for the partial decay widths becomes less clear, since many of them have large uncertainties. In all the cases, the overwhelmingly dominant components of the Pc states are found to be the ΣcD¯(⁎), in agreement with our qualitative understanding in Sec. 2 from the values of a and r given in Table 1.In the next two scenarios, we set the compositeness X=0.8 and 0.5 in Eq. (19). In order not to overload the table, we only show the values obtained by using the masses of Σc+ and D¯(⁎)0 in Table 3. The results by using the masses of Σc++ and D(⁎)− are quantitatively similar. All the values in Table 3 show quite similar trends as those in Table 2, with X2≫X1.The previous discussions rely on the assumption that the decay widths of the Pc states are saturated by the J/ψp and ΣcD¯(⁎) channels. Other decay patterns are also predicted, such as those in Refs. [35,36], which suggest that the partial decay widths of the pentaquark states to the ΛcD¯(⁎) channels could be more important than to the J/ψp. In order to check the robustness of our conclusion, we include the ΛcD¯(⁎) and ΣcD¯(⁎) channels to perform a similar study. To be specific, we give the results in Table 4 by using the masses of Λc+D¯⁎0 and Σc+D¯⁎0. It is verified that to use the masses of other charged states leads to quantitatively similar results. Since to replace the ΛcD¯⁎ channel by the ΛcD¯ does not lead to qualitatively new trends, we do not explicitly show the corresponding results. Comparing the numbers in Tables 2, 3 and those in Table 4, not only the partial decay widths of the two different sets of dynamical channels are quite similar, but also the compositeness coefficients in the different cases are compatible within uncertainties.Summarizing, we have studied the newly discovered hidden-charm exotic states Pc(4312), Pc(4440) and Pc(4457) by the LHCb Collaboration [3]. We have first applied elastic effective-range expansion in the ΣcD¯(⁎) channel with the scattering length and the effective range fixed by reproducing the mass and width of every resonance separately. In all the cases one obtains values for these parameters of O(1) fm, which supports their interpretation as composite resonances of ΣcD¯(⁎). We have also employed another coupled-channel approach involving the two channels J/ψp and ΣcD¯(⁎) for each resonance, so that we require the saturation of the total width of the resonance. By assuming some values for the total compositeness coefficients for these two channels, ranging from 0.5 to 1, we conclude that the weight of the ΣcD¯(⁎) channel is much larger than the one for J/ψp, in agreement with the ERE approach. We have also performed similar studies by including alternatively the ΛcD¯⁎ and ΣcD¯(⁎) as dynamical channels. The conclusions are basically the same as those obtained in the J/ψp and ΣcD¯(⁎) channels. 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