Coupled-channel effects of the $\Sigma_c^{(*)}\bar{D}^{(*)}$-$\Lambda_c(2595)\bar{D}$ system and molecular nature of the $P_c$ pentaquark states from one-boson exchange model

The effects of the $\Sigma_c\bar{D}^*$-$\Lambda_{c}(2595)\bar{D}$ coupled-channel dynamics and various one-boson-exchange (OBE) forces for the LHCb pentaquark states, $P_c(4440)$ and $P_c(4457)$, are reinvestigated. Both the pion and $\rho$-meson exchanges are considered for the $\Sigma_c\bar{D}^*$-$\Lambda_{c}(2595)\bar{D}$ coupled-channel dynamics. It is found that the role of the $\Lambda_{c}(2595)\bar{D}$ channel in the descriptions of the $P_c(4440)$ and $P_c(4457)$ states is not significant with the OBE parameters constrained by other experimental sources. The naive OBE models with the short-distance $\delta(\vec{r})$ term of the one-pion exchange (OPE) keep failing to reproduce the $P_c(4440)$ and $P_c(4457)$ states simultaneously. The OPE potential with the full $\delta(\vec{r})$ term results in a too large mass splitting for the $J^P=1/2^-$ and $3/2^-$ $\Sigma_c\bar{D}^*$ bound states with total isospin $I=1/2$. The OBE model with only the OPE $\delta(\vec{r})$ term dropped may fit the splitting much better but somewhat underestimates the splitting. Since the $\delta(\vec r)$ potential is from short-distance physics, which also contains contributions from the exchange of mesons heavier than those considered explicitly, we vary the strength of the $\delta(\vec r)$ potential and find that the masses of the $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ can be reproduced simultaneously with the $\delta(\vec r)$ term in the OBE model reduced by about 80\%. While two different spin assignments are possible to produce their masses, in the preferred description, the spin parities of the $P_c(4440)$ and $P_c(4457)$ are $3/2^-$ and $1/2^-$, respectively.


I. INTRODUCTION
The study of multiquark states began even before the birth of QCD and was accelerated with the development of QCD. It is speculated that, apart from well-known qqq baryons and qq mesons, there would be multiquark states, glueballs and quark-gluon hybrids in the quark model notation, which are collectively called exotic hadrons. Multiquark states can be categorized into tetraquark states (qqqq), pentaquark states (qqqqq) and so on. The study of multiquark states and how the quarks are grouped inside (i.e., in a compact multiquark state or as a hadronic molecule) plays a crucial role for understanding the low-energy QCD, and it is very important to search for them in experiments.
In this century, many candidates of exotic tetraquark and pentaquark states in the charm sector have been observed. A great intriguing fact is that most of them are located near hadron-hadron thresholds. This property can be understood as there is an S wave attraction between the relevant hadron pair [1], and naturally leads to the hadronic molecular interpretation for them (as reviewed in Refs. [2][3][4][5][6]). The validity of the hadronic molecular picture is also reflected by the successful quantitative predictions of some exotic states in early theoretical works based on the hadron-hadron interaction dynamics [7][8][9][10][11][12][13][14][15].
A lot of theoretical works have been done to identify the spin parity of these three states. Chen et al. [82] study them with the one-boson-exchange (OBE) model assisted with heavy quark spin symmetry (HQSS). In their work, by considering the coupled-channel effects and the S-D wave parameters for the exchange of different light mesons (Λ σ , Λ π , Λ η , Λ ρ , and Λ ω ) as scenario I where the spin parities of the P c (4440) and P c (4457) are interchanged compared to scenario I. It may imply that the δ( r) term in the OBE model plays an important role in the hadronic molecular descriptions of the P c (4312), P c (4380), P c (4440) and P c (4457) states.
In principle, an effective field theory framework, which introduces counterterms to parametrize additional short-distance contributions, is needed for a consistent treatment of the δ( r) term. In this work, we rather follow a phenomenological path, and investigate the role of the δ( r) term by adjusting its strength in the OBE model. We will take the OBE model with the potential derived from the Lagrangian with HQSS as given in Refs. [88][89][90][91]. A percentage parameter a is introduced to represent how much the δ( r) term is reduced in the potential of our OBE model, and thus mimics the variation of a constant contact term in a nonrelativistic effective field theory framework at leading order. The parameter a is varied in the range of [0, 1], that is, a = 0(1) denotes fully including (excluding) the δ( r) term.
This paper is organized as follows. The details of the OBE model in the Σ ( * ) cD ( * ) -Λ c1D coupledchannel system are introduced in Sec. II. Numerical results and discussions on the role of Λ c1D channel and the δ( r) term in the OBE potential are given in Sec. III. Finally, we draw our conclusion in Sec. IV.

II. FORMALISM
In this section, the phenomenological heavy hadron chiral Lagrangian is reviewed, and the potentials for the Σ ( * ) cD ( * ) -Λ c1D system are constructed within the OBE framework.

A. Lagrangian
To investigate the interactions between a charmed baryon or an anticharmed meson with light scalar, pseudoscalar and vector mesons, we employ the effective Lagrangian taking into account HQSS which has been developed in Refs. [88][89][90][91][92][93][94]. The light chiral and heavy quark spin symmetric effective Lagrangian, which describes the low-energy interactions between the Qqq baryons/Qq mesons and the light bosons, can be written as where flavor indices are denoted by a, b and c and σ is the lightest scalar meson, which is taken to be an SU(3) flavor singlet, MeV is the pion decay constant. The symbols P and V α denote the light pseudoscalar octet and the vector nonet, respectively, Field operators for the S-(positive parity) and P wave (negative parity) heavy baryons Qqq are denoted as interpolating fields S µ ab and R µ ab , respectively. HQ a annihilates the antiheavy meson Qq. They are defined as where the anticharmed pseudoscalarP a and vectorP * aµ fields 1 are defined in flavor/isospin space as (D 0 , D − , D − s ) and (D * 0 , D * − , D * − s ), respectively, with the subscript a is the light flavor index, and v µ = (1, 0, 0, 0) is the 4-velocity of the heavy hadron. The charmed baryon fields B 6c and B 3c in the SU(3) flavor space are written as B * 6c and B * 3c are the spin excited states of B 6c and B 3c , respectively, and have the same flavor matrix forms as above; that is, B * 6c and B * 3c are spin-3/2 fields, and B 6c and B 3c are spin-1/2 ones.

B. Partial wave representation
In our analysis, we consider three possible spin-parity states, J P = 1/2 − , 3/2 − and 1/2 + for the Σ ( * ) cD ( * ) -Λ c1D coupled-channel system. The corresponding partial wave basis is listed in Table I where we use the notation 2S+1 L J to identify various partial waves. S, L and J stand for the total spin, orbital and total angular momenta, respectively.
1 Note that herePa andP * aµ are the heavy meson fields satisfying the normalization relations 0|P Q q(0 − ) = MP and 0|P * µ Q q(1 − ) = µ MP * [90]. With this convention, all physical effective couplings for the heavy meson pair (h1, h2) interacting with light mesons are related to the couplings in the Lagrangian (1) by multiplying an additional factor M h 1 M h 2 on the latter.
The partial wave function | 2S+1 L J is explicitly written in the standard form as where C Jm Lm l ,Sms is the Clebsch-Gordan (CG) coefficient, |Sm s is the spin wave function and |Lm l is the spherical harmonics.
One critical point in the partial wave implementation is the spin-orbital ordering convention which is hardly discussed in earlier works. Note that a change of the ordering of spin and orbital angular momenta that converts C Jm Lm l ,Sms into C Jm Sms,Lm l will lead to an additional sign on the matrix elements of the spin-orbital operators, such as S(r, σ, i † 4 × 2 ) and 2 ·r, which are obtained after being sandwiched between the | 2S+1 L J states. A detailed illustration can be found in Appendix A.
However, as long as the same convention is used throughout the calculation, the derived potentials would not depend on the convention. In this paper, all matrix elements are obtained with the C Jm Lm l ,Sms convention.

C. Potentials
To get the OBE potentials for the Σ ( * ) cD ( * ) -Λ c1D system, we need to derive the t-channel scattering amplitude M in the center-of-mass frame first. Note that the nonrelativistic approximation for the charmed hadrons is implemented in our calculation. Potentials in the momentum space can be obtained from the t-channel scattering amplitudes with the Breit approximation, that is, are the masses of particles in the final (initial) states.
It is convenient to label the five channels considered in our work for the Σ ( * ) cD ( * ) -Λ c1D system, i.e., Σ cD , Σ * cD , Λ c1D , Σ cD * , Σ * cD * , as the first, second, third, fourth, and fifth channels, respectively. They are sorted simply by their thresholds. The OBE potential in the momentum space for this five-channel coupled system can be written as where all components are given as, Here i, j and k are the channel indies with i, j = 1, 2, 4, 5 and k = 4, 5. O 1 , · · · , O 6 are the spin operators in the t-channel transition. They are given explicitly in Appendix A. q and µ ij = m 2 ex − ω 2 ij are the 3-momentum and effective mass for the exchanged meson, where ω ij is the energy of exchanged meson in the t-channel transition i → j. Note that there is no energy exchange in the elastic transition (i = j); that is, ω ii = 0. A, B, C, D, E and F denote symmetric constant matrices consisting of the coupling constants and the flavor factors. All nonzero elements are listed below, where {T π , T η , T ρ , T ω } = {−1, 1/6, −1, 1/2} are the flavor factors and τ π/ρ = 3/2 for the isospin-1/2 system.
After implementing the Fourier transformation, we can obtain potentials in the coordinate space. With the dipole form factors included, they read as with Λ ij = Λ 2 − ω 2 ij . For the σ meson exchange, one gets where Y (r, Λ ij , µ ij ) is the attractive Yukawa potential. Before implementing the Fourier integral of the pseudoscalar-exchange potential, we decompose it into two terms as usually done in the literature, Note that the constant 1 inside the parentheses in the first term leads to a short-range δ potential [δ( r) term in the coordinate space]. As discussed in the Introduction, the short-distance contribution cannot be fully captured by the OBE model, which may be viewed as there can be contributions from exchanging heavier particles. Thus, we introduce a parameter a to adjust the strength of the δ( r) potential. It is introduced as Thus, a = 0 corresponds to the case with the full δ( r) potential of the exchanged meson, and a = 1 corresponds to the case without it. Then, the V ij π/η (r) can be obtained as where With the same procedure, we can obtain the r-space potentials for Eqs. (12), (13) and (14), The δ( r) term in the central potential C(r) appears not only in the pseudoscalar exchange potentials but also in the vector meson case. Figure 1 shows the contribution of the δ( r) term to the central potential, where a = 0(1) means a full inclusion (exclusion) of the δ( r) term. If the δ( r) term is fully removed, the central potential becomes very weak and repulsive.
For the numerical analysis, we need some input parameters, such as masses of particles and coupling constants in Lagrangian Eq. (1). All the masses of particles are referred to the isospinaveraged values of the experimental masses listed in the Review of Particle Physics [86] and are collected in Table II. The coupling constants are given in Table III. It should be mentioned that the scalar meson coupling constant g S = 0.76 for the DDσ vertex is different from the values used in Refs. [68,91,99] by a factor of 2m D due to the different conventions as introduced in Sec. II A.
The coupling l S in the Σ c σ vertex is determined in Refs. [91,95] with the chiral multiplet assumption [100]. The pseudoscalar meson couplings g and g 1 are determined in Refs. [91,96]   relate g 1 to the latter process). The vector meson couplings β, λ, β S , λ S and g V are determined in Refs. [91,97] with the vector-meson dominance assumption. The coupling of the vector meson with the P -and S wave baryons h 3 may be roughly estimated from the Λ c1 → Σ c γ decay via the vector meson dominance assumption [101,102]. At the tree level, the radiative decay width of Λ c1 → Σ c γ can be calculated as where α EM = 1/137 is the fine structure constant. The radiative decay width Γ(Λ c1 → Σ c γ) has not been measured, but the prediction of it may help us to estimate the value of h 3 . The decay [103] with HQSS and predicted to be 71 keV. One can infer the coupling constant h 3 from this to be 0.24.

III. NUMERICAL RESULTS AND DISCUSSION
A. Role of the Λ c1D channel The Λ c1D channel has its threshold very close to the mass of the P c (4457), and couples to J P = 1/2 + in an S wave due to the negative parity of the Λ c1 baryon. It is interesting to study whether the Λ c1D channel can trigger the formation of some J P = 1/2 + molecular candidate near its threshold. In Ref. [84], the J P = 1/2 + system was investigated with the Σ cD * -Λ c1D coupled channels considering OPE with couplings from quark model. They demonstrated that if the nondiagonal potential of the Σ cD * -Λ c1D coupled channel is strong enough it is possible to reproduce the P c (4457) and P c (4440) as J P = 1/2 + and 3/2 − molecules simultaneously. Now, we revisit the role of Λ c1D channel by including the vector meson exchange potential in the coupled All potentials needed in our work are formulated in Sec. II C. Here we use potentials of the Λ c1D -Σ cD * -Σ * cD * coupled channels to solve the Schrödinger equation to probe the molecular states with J P = 1/2 + and 3/2 − . To be consistent with Sect. II C, we enumerate the three channels Λ c1D , Σ cD * , and Σ * cD * , with channel labels 3, 4 and 5, which are ordered according to the channel thresholds W 3 , W 4 and W 5 . The coupled-channel Schrödinger equation for spherically symmetric potentials can be written as where ∇ 2 i , µ i and R i (r) are the Laplacian, reduced mass and radial wave function of the ith channel (i = 3, 4, 5); ∆ 4 = W 4 − W 3 and ∆ 5 = W 5 − W 3 are the mass differences; and E is the binding energy. In the spherical coordinate, the Laplacian and radial wave function may be written as r , respectively. The ground-state binding energies of the J P = 1/2 + and 3/2 − systems are obtained by solving the Schrödinger equation with the help of the Gaussian-expansion method [104].
It is useful to take a look at the coupled-channel potentials. We plot the potentials with the coupling constants in Table III for the J P = 1/2 + and 3/2 − system with total isospin I = 1/2 in Fig. 2, in which we fully remove the δ( r) term by setting a = 1 as is the case in Ref. [84]. The diagonal Λ c1D potential is neglected in our formalism. This is because our motivation is to check the mechanism proposed in Ref. [84] where the Λ c1D -Σ ( * ) cD ( * ) coupling forms a bound state lying below the Λ c1D threshold. 2 The potentials for other exchanged mesons absent from the potential plots are forbidden due to the HQSS, isospin, and parity conservations. The exchange of the isoscalar mesons σ, ω and η for the Λ c1D → Σ cD * is forbidden considering isospin symmetry, which is the case here, and since the Λ c1 couples to Σ * c π in te D wave the pion exchange for Λ c1D → Σ * cD * is also not considered. We can see that in the 1/2 + system the diagonal potentials V 44 and V 55 are attractive, but they couple in P waves and are largely canceled by the strong repulsive centrifugal potential.
The off-diagonal elements involving the S wave Λ c1D channel contributes to the J P = 1/2 + system and may trigger the 1/2 + system to form a bound state. The quark model calculation in Ref. [84] indicates that the P c (4457) and P c (4440) can be simultaneously reproduced as J P = 1/2 + , 3/2 − states within the Λ c1D -Σ cD * coupled channel as long as the off-diagonal elements of the Λ c1D potential are strong enough. Here, we further investigate such a scenario within the OBE model.
The combined coupling of the OPE potential for the Λ c1D → Σ cD * channel is defined in Ref. [84] asĝ = where the values of g and h 2 are given in Table III, and the pion energy ω in the t-channel transition The physical value ofĝ is 0.17(4) GeV −1 , as mentioned in Ref. [84], with coupling constants in Table III. In Ref. [84], it is required to beĝ = 0.52 GeV −1 in order to set the lower 3/2 − state match the P c (4440). Note that quark model predictions of the OPE potential for the elastic channel Σ cD * which was studied in Ref. [84] is roughly 1.7 times stronger than the one derived in our work. Here, we take a simple ratio of the OPE potentials, where the quark model potential V q π is given in Ref. [84] with g q = 0.59 and V 44 π is obtained in our work. The ratio arises due to different determinations of the coupling constants at the quark and hadronic levels. As given in Table III, the pseudoscalar couplings g and g 1 are determined from the experimental data on the decays of D * → Dπ and Σ * c → Λ c π, respectively, and used in our work.
For the Λ c1D -Σ cD * -Σ * cD * coupled-channel system, we remove the δ( r) term in the whole potentials by setting a = 1. In this case, the J P = 1/2 − state, which was a deep bound state when a = 0, cannot be bound in this system with a value of Λ up to almost 5 GeV. In the rest of this subsection, we will keep a = 1 as the potential for the Λ c1D → Σ cD * , which triggers the J P = 1/2 + state to be bound is independent of the a parameter. We mainly focus on the states below Λ c1D and Σ cD * thresholds with spin parities J P = 1/2 + and J P = 3/2 − , respectively, in order to discuss the effect of the Λ c1D channel on these states, corresponding to the P c (4457) and the P c (4440). The other P c states at the resonance region above the Σ cD * threshold will be discussed in Sec. III B.
In our manipulation of the coupled-channel equation with threshold differences, the lowest threshold mass is always chosen to be the origin of energy and the S-D mixing is also considered. Table IV shows the binding energies of J P = 1/2 + , 3/2 − systems by varying the cutoff Λ, with the coupling constants given in Table III,  [MeV] [MeV] [MeV] [MeV] cD * coupled channels together with the OBE potentials, the results are given in the last column labeled case IV. One sees that the J P = 1/2 + system, which couples to the S wave Λ c1D and the P wave Σ cD * , does not form a bound state because the Λ c1D → Σ cD ( * ) transition potential is not strong enough, and there is only one bound state with J P = 3/2 − in this system, which is mainly formed by the S wave Σ cD * .
As shown in Ref. [84], ifĝ = 0.52 GeV −1 , then a J P = 1/2 + bound state emerges, and masses of both P c (4457) and P c (4440) would be reproduced with the same cutoff Λ = 1.42 GeV. The results withĝ = 0.52 GeV −1 are given in Another S wave channel for the 1/2 + state is the Λ c1D * . One may expect a sizeable role it would play, although its threshold is around 140 MeV above the Λ c1D one. However, its S wave elastic potential is repulsive since the repulsive ω-exchange force contributes dominantly. And it is found that the nondiagonal dynamics of the Λ c1D * channel is also negligible due to the experimental constraints on the Λ c1 couplings similarly to the Λ c1D case. An explicit inclusion of this channel into case III would only slightly increase the numerical values of the absolute values of the binding energies by less than 0.5 MeV for the cutoff in the range listed in Table IV. All other S wave channels for the 1/2 + state with higher thresholds are expect to be irrelevant.

B. Role of the δ( r) term in the OBE model
In the hadronic molecular picture, the masses of the latest observed pentaquarks P c (4312), P c (4440) and P c (4457) are claimed to be well reproduced as the Σ  [68]. A recent analysis using an effective field theory framework shows that the LHCb data can be well described with both quantum number assignments, while the latter is preferred because of its insensitivity on the cutoff values used in regularizing the coupled-channel Lippmann-Schwinger equation [81]. In this section, first, we use the OBE potentials derived in Sec. II to simultaneously reproduce all the observed P c states by varying the cutoff Λ and the magnitude of the δ( r) term without coupled channels. Then, we include the Σ ( * ) cD ( * ) coupled-channel effects and try to distinguish the two spin-parity assignments for P c (4440) and P c (4457) with help of the δ( r) term. Here, we consider four channels Σ cD , Σ * cD , Σ cD * and Σ * cD * and exclude the Λ c1D channel due to the fact that its contributions are negligible for negative parity states with the physical value forĝ.
To be consistent with Sec. II C, we enumerate the channels Σ cD , Σ * cD , Σ cD * , and Σ * cD * with 1, 2, 4 and 5, respectively, which are ordered according to the channel thresholds W 1 , W 2 , W 4 and W 5 . First, we take a look at the potentials for the J P = 1/2 − and 3/2 − systems. Figure 3 shows the diagonal S wave potentials, where we compare the potentials with the δ( r) term (a = 0) and those without it (a = 1) using a cutoff Λ = 1 GeV. The where W i is the threshold energy of the ith channel. Here, we directly solve the coupled-channel Schrödinger equation, where i is the channel index, u i is defined by u i (r) = rR i (r) with the radial wave function R i (r) for the ith channel, and µ i is the corresponding reduced mass. The eigenmomentum for channel i is given as q i = 2µ i (E − ∆ i ), where ∆ i is the threshold difference with respect to the threshold of the lowest channel. By solving Eq. (32), we obtain the coupled-channel wave function, which is normalized to satisfy the boundary condition for the jth channel given as [106] u (j) First, we show the results of the single channels Σ cD , Σ * cD , Σ cD * , and Σ * cD * with the S-D wave mixing effects considered. 3 The quantum numbers can be J P = 1/2 − , 3/2 − , and 5/2 − . Table VI   shows the binding energy of each single channel with various cutoffs. We list the results for two different δ( r)-term contributions, that is, a = 0 and a = 1. In the single-channel case, the binding energies of the 1/2 − (Σ cD ) and 3/2 − (Σ * cD ) states are independent of the δ( r) term, and the two systems are loosely bound. The corresponding potentials for these two systems are identical as given in Fig. 3(a). The small difference between the binding energies is completely caused by the different reduced masses. However, for all the other states, the δ( r) term has an impressive effect on the binding energies. And the binding energies are heavily dependent on the cutoff Λ when the δ( r) term is included because of the short-distance nature of the δ( r) term. The single-channel results show that we cannot reproduce the P c (4312), P c (4380), P c (4440) and P c (4457) simultaneously by including or excluding fully the δ( r) term.
Then, let us move to the cases with the value of the reduction parameter a taking a value somewhere between 0 and 1. The results with a = 0.78 and Λ = 1.6 GeV are obtained as Finally, we consider the Σ cD -Σ * cD -Σ cD * -Σ * cD * coupled-channel system with the S-D wave mixing effects. We try to reproduce the P c (4312), P c (4380), P c (4440), and P c (4457) states by varying the cutoff Λ and the reduction parameter a. As mentioned at the beginning of this section, the masses of P c (4380), P c (4440), and P c (4457) lie above the threshold of the Σ cD channel. Then, these three P c states should be solved as resonances in the current coupled-channel system, that is, the eigenenergies of P c (4380), P c (4440), and P c (4457) will take complex values. Going to the appropriate Riemann sheets, one can find the complex poles of the S matrix, which can be interpreted as resonances. We interpret the real and imaginary parts of the pole position as the mass and half width of the corresponding resonance. 4 Two sets of solutions are found that can reproduce the P c (4312), P c (4380), P c (4440), and P c (4457) masses simultaneously. They are marked by different reduction values, a = 0.55 and a = 0.79. Figure 5 shows the masses (upper panel) and widths (lower panel) of the 1/2 − (Σ cD ), 1/2 − (Σ cD * ) and 3/2 − (Σ cD * ) states as functions of Λ for each value of a. The horizontal gray bands represent the experimental uncertainties of P c masses [53], and the vertical gray bands stand for the cutoff range where masses of all P c states can be simultaneously reproduced. In this figure, we do not include the curves of the 3/2 − (Σ * cD ) molecule since its mass is always in line with the P c (4380) within the whole cutoff range covered by the plot (and thus higher than the P c (4337) structure reported recently in Ref. [107]). The vertical dashed lines are the best-fit solutions with Λ = 1.23 GeV for a = 0.55 and Λ = 1.40 GeV for a = 0.79, which are obtained by minimizing the χ 2 that represents the deviation between our solved P c masses and the LHCb measurements.
The masses of P c states for the best-fit solutions are listed in Table VII. Note that the state with spin parity J P = 1/2 − near the Σ cD threshold does not have decay width since it emerges as a bound state in our calculation where the channel coupling to the lower channels is omitted. As we can see from Table VII, the P c (4312) can be interpreted as the 1/2 − (Σ cD ) molecule in both solutions, and it is consistent with the single-channel result. The spin-parity assignments for the P c (4440) and P c (4457) states are interchanged between these two solutions. In the solution with a = 0.55, the masses of P c (4440) and P c (4457) can be reproduced well by the 1/2 − and 3/2 − Σ cD * molecules, respectively. However, the P c (4440) and P c (4457) are described as the 3/2 − and 1/2 − 4 Since only the Σ  Such spin-parity interchange can be understood as the dependence behavior of 1/2 − and 3/2 − Σ cD * elastic potential on the parameter a. We plot the impact of the parameter a on the Σ cD * elastic potential for J P = 1/2 − and 3/2 − in Fig. 6. As one can see, the 1/2 − potential gets shallower as a increases, leading to a smaller binding energy (absolute value of E B ) of the 1/2 − bound state, while the situation is reversed for the 3/2 − potential-the 3/2 − potential becomes deeper as a increases, and the 3/2 − bound state will have a larger binding energy. It results in 3/2 − -P c (4440) with larger a and 1/2 − -P c (4440) with smaller a. This behavior is originated from the sign difference of the δ(r) term in the 1/2 − and 3/2 − Σ cD * elastic potentials; see the value of the spin operator O 44 3 · O 44 4 in Appendix A. In our model, we may distinguish the two possible solutions by the decays of the two states into the subdominant channels Σ cD and Σ * cD , which behave differently in these two spin-parity assignments. In the model calculation of Ref. [108], the dominant decay channel for both P c  [53]. Future data with higher statistics will be able to resolve the Σ * cD * states.

IV. CONCLUSION
We investigate the coupled-channel dynamics of the Λ c1D and Σ ( * ) cD ( * ) channels within the OBE model to test the mechanism of a J P = 1/2 + state being triggered by the Λ c1D channel. It is found that the J P = 1/2 + system cannot be bound with the OBE parameters constrained by  We further investigate the role of the δ( r) term in the coupled-channel dynamics. Here, the δ( r) term comes from the constant term of the t-channel scattering amplitudes in the momentum space. Such a δ( r)-term contribution is of a short-distance nature and needs to be regularized.
Here the regularization is performed by introducing dipole form factors, the effects of which may be recognized as the short-range interactions derived by the exchange of mesons heavier than ρ and ω mesons. In this work, as a phenomenological study, the δ( r) term is corrected by introducing a reduction factor a that quantifies how much the δ( r) potential is reduced in the OBE potential. a varies in the range of [0, 1] in our analysis. Two set of solutions for the parameters, the cutoff Λ and the reduction parameter a, are found to be able to reproduce the masses of the observed P c states in the hadronic molecular picture. In the first solution, called scenario I, the best description of the P c masses is given by the parameters {Λ = 1.23 GeV, a = 0.55}, where P c (4312), P c (4440), and P c (4457) are interpreted as the 1/2 − (Σ cD ), 1/2 − (Σ cD * ), and 3/2 − (Σ cD * ) molecules, respectively.
The second solution, called scenario II, has {Λ = 1.4 GeV, a = 0.79}, and the spin-parity quantum numbers of the P c (4440) and P c (4457) states are 3/2 − and 1/2 − , respectively. Scenario II is favored since the partial decay width of the P c (4457) in scenario I is larger than the central value of the experimental value. This is consistent with previous analysis from an effective field theory point of view [81]. In this preferred scenario, the P c (4312), P c (4380), P c (4440), and P c (4457) states are the In the following, we only show the operators at the upper triangle of the coupled-channel potential matrix; the others can be obtained with the Hermitian condition of the potential matrix, and The other operators are zero. The partial wave projection of operators in the J P = 1/2 − , 3/2 − , 1/2 + system is shown in the Table VIII, which is calculated by sandwiching the operators given above between the partial waves of the initial and final states [109]. Every element of the spin operators is replaced by the corresponding partial wave projections in the actual calculation. Here, we calculate the spin projection for the transition 1 → 5 (Σ cD → Σ * cD * ) as an example. For J P = 1/2 − , the partial waves for the initial and final states are Σ cD : 2 S 1/2 , Σ * cD * : 2 S 1/2 , 4 D 1/2 , 6 D 1/2 , From Eq. (35) together with Eqs. (21) and (24), we know that the spin operators, which are universal for the pseudoscalar and vector exchange potentials, are O 15 3 · O 15 4 = (i χ † 3 × σχ 1 ) · * 4 for the spin-spin coupling and S( O 15 3 , O 15 4 ,r) = 3(i χ † 3 × σχ 1 ) ·r * 4 ·r − (i χ † 3 × σχ 1 ) · * 4 for the tensor coupling. Then it can be calculated as 2S +1 L J |(i χ † 3 × σχ 1 ) · * 4 | 2S+1 L J = iC J,m ml ,ms C J,m ml,ms C S ,ms m3,m4 C S,ms m1,m2 C s3,m3 h1,h2 × 3 i,j,k=1 ε ijk * i 3 (h2)χ † 3 (h1)σ j χ 1 (m1) * k 4 (m4) L , ml |L, ml , ε ijk * i 3 (h2)χ † 3 (h1)σ j χ 1 (m1) * k 4 (m4) L , ml |L, ml }, (38) where the lower indices of Clebsch-Gordan coefficients which represent the magnetic quantum numbers should be summed. 2S +1 L J | and | 2S+1 L J stand for the partial waves for the final and initial states, respectively. The spherical harmonics are integrated as L , ml |r k r o |L, ml = dΩY * L ,ml (θ, φ)r k r o Y L,ml (θ, φ), where Y L,ml (θ, φ) is the spherical harmonics and r k and r o are the components of the unit vector r in the spherical coordinate. After having calculated Eqs. (37) and (38) with the partial waves given Eq. (36), we get (− √ 2, 0, 0) and (0, 2/5, 3 2/5), respectively. One should be careful about the convention. If the ordering of the angular momenta in the Clebsch-Gordan coefficient C J,m ml ,ms changes to C J,m ms ,ml , the result of Eq. (38) will change to (0, − 2/5, 3 2/5). All partial waves in our work are calculated with the convention of C J,m ml ,ms , and the results are collected in Table VIII.