Probing hidden-charm decay properties of $P_c$ states in a molecular scenario

The $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ observed by the LHCb Collaboration are very likely to be $S$-wave $\Sigma_c\bar{D}^{(*)}$ molecular candidates due to their near-threshold character. In this work, we study the hidden-charm decay modes of these $P_c$ states, $P_c\to J/\psi p(\eta_cp)$, using a quark interchange model. The decay mechanism for the $P_c\to J/\psi p(\eta_cp)$ processes arises from the quark-quark interactions, where all parameters are determined by the mass spectra of mesons. We present our results in two scenarios. In scenario I, we perform the dynamical calculations and treat the $P_c$ states as pure $\Sigma_c \bar D^{(*)}$ molecules. In scenario II, after considering the coupled channel effect between different flavor configurations $\Sigma^{(*)}_c\bar D^{(*)}$, we calculate these partial decay widths again. The decay patterns in these two scenarios can help us to explore the molecular assignment and the inner flavor configurations for the $P_c$ states. In particular, the decay widths of $\Gamma(P_c(4312)\to\eta_cp)$ are comparable to the $J/\psi p$ decay widths in both of these two scenarios. Future experiments like LHCb may confirm the existence of the $P_c(4312)$ in the $\eta_cp$ channel.


I. INTRODUCTION
In 2015, the LHCb collaboration reported two pentaquark states P c (4380) and P c (4450) in the J/ψp invariant mass distribution of the decay Λ 0 b → J/ψpK − [1]. Very recently, with Run I and Run II data, the LHCb Collaboration found that that P c (4450) + should contain two substructures P c (4440) + and P c (4457) + [2]. In addition, another new narrow state P c (4312) + is observed. Resonance parameters for these observed P c states are [2] P + c (4312) : M P + c (4312) = 4311.9 ± 0.7 +6.8 −0. 6 MeV, Γ P + c (4312) = 9.8 ± 2.7 +3.7 Before the LHCb's observation, the molecular pentaquark states have been predicted in Refs. [3][4][5][6][7]. The observation of P c states by the LHCb Collaboration in 2015 inspired theorists' great enthusiasm on the study of hidden-charm pentaquark states. Various interpretations have been proposed, such as the loosely bound meson-baryon molecular states , the tightly bound pentaquark states [30][31][32][33][34][35][36], and the hadrocharmonium states [37]. A recent review is referred to * wgj@pku.edu.cn † lyxiao@ustb.edu.cn ‡ chen rui@pku.edu.cn § xiaohai.liu@tju.edu.cn ¶ xiangliu@lzu.edu.cn * * zhusl@pku.edu.cn Refs. [38][39][40][41][42][43]. Since P c (4312) and P c (4440)/P c (4457) locate several MeV below the thresholds of the Σ cD system and Σ cD * systems, respectively, the meson-baryon molecule scheme is a more natural explanation. More precise data from LHCb plays an important role in distinguishing different hadronic configurations for the P c states. It is necessary to provide a clear and consistent interpretation of their inner structures for these P c states. And we should pay more attentions to the other properties of P c states, like their decay behaviors, productions, and various reactions, and so on.
In particular, the decay patterns can provide a golden platform for probing their inner dynamics. After the observation of the P c (4312) and P c (4440)/P c (4457), there are several phenomenological investigations on the decay properties of the P c states as Σ ( * ) cD ( * ) molecules by using the heavy quark symmetry [18,22], effective Lagrangian approach [20,24], the QCD sum rule [44] and other methods [23,45]. In this work, we will explore the hidden-charm decay properties of the P c states in the meson-baryon molecular scheme with the quark interchange model [46][47][48][49][50][51]. Within the model, the decay widths are related to the scattering process of the hadrons at Born order. The scattering Hamiltonian is then approximated by the well-established quark-quark interactions, with all coupling constants determined by the mass spectra of the hadrons. This method has been adopted to study the decay patterns of the exotic states [52][53][54]. In this work, we extend the approach to give a quantitative estimate of the decay patterns of the P c states.
Although these P c states are probably hidden-charm molecules composed of a charmed baryon and a charmed meson, we have to find out whether they are pure Σ ( * ) cD ( * ) molecular states or not. In general, the coupled channel effect may play an important role for the systems with the same quantum number and small mass splitting. In the heavy quark limit, the (D,D * ) and (Σ c , Σ * c ) doublets are degenerate, re-molecules and the admixtures of different flavor configurations. This paper is organized as follows. After the introduction, we introduce the formalisms which relate the decay width of the pentaquark state to the effective potentials between the Σ ( * ) cD ( * ) and the J/ψN(η c N) channels in Sec. II. In Sec. III, we derive these effective potentials using the quark interchange model. In Sec. IV and Sec. V, we perform the numerical calculations in two scenarios, respectively. In scenario I, the P c states are treated as pure Σ ( * ) cD ( * ) molecules, while in scenario II, they are the admixtures of different flavor configurations. The paper ends with a summary in Sec. VI. The details of the calculations are illustrated in the Appendix.

II. DECAY WIDTH
For a decay process from a pentaquark state into a two-body finial state P c → C + D, its decay width reads where M is the mass of the pentaquark state and p c is the three momentum of the meson C in the final state. The decay amplitude M is related to the T −matrix as follows, where E C,D is the energy of hadrons in the final state. The ψ CD is the relative wave function in the final state. In the molecular model, ψ P c (p) is the normalized relative wave function between the constituent meson A and baryon B in the P c state. V eff (p c , p) is the effective potential between the AB and CD channels, which is generally a function of the initial momentum p and the final momentum p c . At Bonn order, it is derived by the amplitude of the two-body scattering process, where the numbers 1 − 5 stand for the inner quarks. The detailed derivation is illustrated in Sec. III. In general, a pentaquark may be the superposition of the components with different orbital angular momentum, and the relative wave function in the momentum space can be expressed as The T − matrix is decomposed as with where u = cosθ, and θ is the angle between the p and p c . P l is the Legendre polynomial. In this work, we do not include the spin-orbital and tensor interactions. Then, the orbital angular momentum is kept unchanged in the decay process. T -matrix is diagonal in l. Then, the decay width is

III. EFFECTIVE POTENTIAL
In the quark interchange model, the hadron-hadron scattering process is approximated by the interaction between the inner quarks [47][48][49][50][51]. In the process from two heavy hadrons scattering into a heavy quarkonium plus a nucleon, the shortrange interactions dominant the scattering process. We adopt the V i j for the quark-quark interaction [46,55]. The effective potential V i j in the momentum space can be expressed as, where λ i (−λ T i ) is the color factor for the quark (antiquark). q is the transferred momentum. The three terms in the V i j correspond to the Coulomb, linear confinement, and hyperfine potentials, respectively. α s is the running coupling constant and a function of the Q 2 , which is the square of the invariant mass of the interacting quarks. We perform our calculation in the momentum space. The V i j contains a constant potential in the spatial space. In addition, the Fourier transformation of the Coulomb and linear confinement potentials induce the divergent terms. The constant term and the divergence vanish due to the exact cancellation of the color factors as illustrated in the following. The parameters in Eq. (9) are determined by fitting the mass spectra of the mesons. Their values are listed in Table I. In the quark model, the wave function of a hadron is factorized as where χ c, f,s and φ(p) are the wave functions in the color, flavor, spin and momentum space, respectively. Correspondingly, the T −matrix can be factorized as where the factors I with the subscripts color, flavor-spin, and space stand for the overlap of the wave functions in the corresponding space. The notation= means that the spin and spatial factors can be separated for the S -wave scattering process. The so-called "prior-post" ambiguity in the scattering process arises due to different decompositions of the Hamiltonian [49]. The Hamiltonian is separated as where H 0 is the Hamiltonian for a free hadron and V AB(CD) is the residual potential between two color-singlet hadrons. These two decomposition methods result in the "Prior" and "Post" formalisms as illustrated in Fig. 1. In the quark interchange model, the V AB(CD) is approximated as the sum of the two-body interactions between the constituent quarks in hadrons AB (CD). In the baryon, we mark the light quarks with definite symmetry as the fourth and the fifth quarks. Then, the potentials V AB(CD) lead to the four diagrams in the "Prior" and "Post" formalisms, respectively. The "prior-post" ambiguity disappears if the exact solutions of the wave functions are adopted and the hadrons are on-shell 1 [49,56]. In this work, we adopt the averaged scattering amplitudes to reduce the "prior-post" ambiguity.
With the quark interchange model, we first study the decay processes where p and n denote the proton and the neutron, respectively. Then, the scripts 1, 2, 3, 4, and 5 in Fig.1 denote thec, d (u), c, u (d), and u (d) quarks, respectively. The forth and the fifth quarks have the same flavor. Their spin and isospin are equal to one constrained by the Fermi statistics, which simplifies the calculation of the spin-flavor factor I flavor-spin .
In the quark interchange process, the color factor reads In Table II, we collect the numerical results of the color factors I color . Since the sum of the color factors are zero, the constant V cons in the quark-quark potential and the divergence induced by the Fourier transformation of the Coulomb and linear confinement potentials in the momentum space cancel out exactly. The spin factor is where S ( ′ ) is the total spin of the initial (final) state, s denotes the spin of the four hadrons.V s is defined as the spin-spin interaction operator. In the quark model, theV s is unitary for the Coulomb and linear confinement interactions, and S i · S j for the hyperfine potential. The derivation of the spin factor I spin is illustrated in Appendix B. Numerical results for the color-spin-flavor factors are collected in Table III.  Additionally, the explicit forms of spatial factors I space for different diagrams can be written as where the p 3(4) is defined as the momentum of the third (fourth) quark. f A and f C are expressed as with m i being the mass of the ith quark. φ(p) is the spatial wave function of the hadron presented in Appendix A. The integral of the linear confinement potential is divergent at q = 0. The divergency cancels out exactly with each other due to the color factors. According to the isospin symmetry, one obtains the following relations,

A. Heavy quark symmetry
Before the numerical analysis, we would like to discuss the hidden-charm strong decay behaviors of Σ cD ( * ) states with the heavy quark spin symmetry. The corresponding interaction amplitudes of Σ ( * ) cD ( * ) → J/ψ(η c )p processes are collected in Table IV. For the Σ cD ( * ) states with J P = 1 2 − , one obtains For the Σ cD * state with J P = 3 2 − , its decay into the η c p and J/ψp processes occur via D−wave and S −wave interactions, respectively. The decay width of Σ cD suppressed by a (p c /M) 4 factor, which is of O(10 −3 ). Thus, we do not consider this decay process. Anyway, one can still find that The above conclusions can also be applied to their partner P b states as the heavy quark flavor symmetry. According to Eqs. (20)- (21), we want to emphasis that the η c p final state is a very The T-matrix of the scattering process Σ ( * ) cD ( * ) → J/ψp or η c p in the heavy quark limit. All the spatial information is included in the matrix element H h,l . The subscripts h and l denote the heavy and light degrees of freedom in the initial and final states. The near threshold behavior of P c states provides an intuitive explanation that they are good candidates for the hiddencharm meson-baryon molecules. The different Σ ( * ) cD ( * ) configurations with the same J P may couple with one another and contribute to the same pentaquark state. In the following, we present the numerical results in two scenarios, which corresponds to P c states as pure Σ ( * ) In scenario I, we study the decay patterns of the P c states as pure Σ ( * ) cD ( * ) molecules. According to their mass spectra, the lowest P c (4312) is probability the Σ cD molecule with J P = in Refs. [13][14][15]. Both of these two spin assignments will be discussed in this section. Here, we firstly adopt an S-wave Gaussian function in Eq. (A1) to estimate the relative wave function for the P c states in the meson-baryon picture. The wave function contains an undetermined oscillating parameter β P c , which is related to the root mean square radius of the P c state. For an S −wave loosely bound molecule composed of two hadrons, the typical molecular size can be estimated as r ∼ 1/ 2µ(M A + M B − M) with its reduced mass µ = M A M B M A +M B [41,57,58]. With this input, β P c can be related to the mass of the P c state, Moreover, we still allow 10% uncertainty for this relation in the following numerical calculations. In Fig. 2, we present the mass dependence of the partial decay width for the Σ cD ( * ) molecules decaying into the J/ψp and η c p channels. The binding energies for these Σ cD ( * ) molecules vary from −50 to −1 MeV. With a smaller binding energy, the P c state has a larger mass, which results in a larger relative momentum in the final state. As illustrated in Eq. (7), the decay width depends on both of the final momentum and the potential V eff (p c , p, µ). With the increasing initial and final relative momenta, the spatial factor I space suffers an exponential suppression and the effective potential decreases. In the limit of large relative momentum of the final states, the I space and the decay width vanish. As shown in Fig. 2, when the binding energy is taken as −50 MeV, the decay widths of P c → J/ψp(η c p) become the largest with their smaller phase space.
In Fig. 3, we present the decay ratio Γ(P c → η c p)/Γ(P c → J/ψp). We find that the decay ratios decrease with the larger P c mass because the relative momentum in the η c p channel is larger than that in the J/ψp channel. The effective potential decreases faster for the η c p channel.
As shown in Fig. 3, the loosely bound Σ cD molecule prefers to decay into the η c p channel rather than the J/ψp channel with its binding energy in the range of −50 ∼ −1 MeV. If the P c (4312) is the Σ cD molecule, its partial widths decaying into the η c p and J/ψp channels are 0.89±0.25 MeV and 0.32±0.08 MeV, respectively. Here, the errors come from the uncertainty of the relative molecular wave function. The decay ratio R 1 is 2.84 ± 0.03. Thus, the η c p should be the other promising decay channel to observe the P c (4312) molecular state.
For the Σ cD * molecules with J P = 1 2 − and J P = 3 2 − , as shown in Fig. 2, the J/ψp decay channel is remarkably more important than the η c p channel. Even with a larger phase space in the η c p channel, the Σ cD * state decays much more easily into J/ψp. The Σ cD * couples more strongly with the J/ψp channel. For the Σ cD * molecule with J P = 1 2 − , the ratios The interaction between Σ cD * and J/ψp is sensitive to the total angular momentum. And the interaction between the Σ cD * the upper limits are determined to be 2.3% and 3.8% at 90% confidence level in GlueX [59].
The numerical results show that 1. If the P c (4380) is the Σ * cD molecule, the partial decay width for the P c (4380) → J/ψp is 0.49 ± 0.14 MeV.
2. The Σ * cD * state with J P = 1/2 − prefers to decay into the η c p channel rather than the J/ψp channel. In addition, we further extend our framework to explore decay patterns of the hidden-bottom Σ ( * ) b B ( * ) molecular pentaquarks assuming their existence. Their partial decay widths and the branching fraction ratios versus the masses are presented in Fig. 6 and Fig. 7 in the Appendix C. 2 The central value of the P c decay width is used to estimate the branching fractions.

CONFIGURATIONS
In scenario II, we improve our results with the exact molecular wave functions obtained by solving the coupled channel Schödinger equation in the scheme of the one-bosonexchange (OBE) model. The explicit details of the calculations are referred to Ref. [8]. The relative molecular wave functions between the constituent meson and baryon are pre-sented in Fig. 5. In Ref. [8], the P c states observed by the LHCb collaboration can coexist as the admixtures of Σ ( * ) cD ( * ) molecular states as illustrated in Table V. The P c (4312) and P c (4440) are the J P = 1 2 − molecular states mainly composed of Σ cD and Σ cD * channels, while the P c (4457) and P c (4380) are the J P = 3 2 − molecules mainly composed of Σ cD * and Σ * cD channels, respectively. In particular, the other flavor configurations also provide important contributions to reproduce the P c states. For instance, the Σ * cD * channel couples with the Σ cD * and contributes to the P c (4457) state with its probability around 21%. Since the D−wave components contribute a tiny proportion to the P c states with their probabilities less than 5%, the decay from the D−wave components into the J/ψp (η c p) channels will be suppressed. Here, we only consider the S −wave flavor configurations in calculating the partial decay widths.
In Table VI, we collect the T −matrices for the possible flavor configurations decaying into the J/ψp(η c p) channels. Their sum contributes to the partial decay widths. It is interesting that the coupled channel effect significantly changes the decay pattern of the P c (4312). As illustrated in section IV B, taking the P c (4312) as the pure Σ cD molecule, the partial decay width ratios between the η c p and J/ψp is nearly three. Considering the coupled channel effect, the S −wave component Σ cD * is considerable and occupies around 11%. However, its contribution to the J/ψp decay mode is larger than that from the dominant Σ cD configuration and enlarges the partial decay width. These results indicate that the interaction between the Σ cD * and J/ψp is much stronger than that in the Σ cD system with J P = 1/2 − . This is also consistent with the prediction in the heavy quark limit, in which the V eff (Σ cD * − J/ψp) is the largest while the V eff (Σ cD − J/ψp) is the smallest for the channels with J P = 1 2 − . At present, R 1 becomes 0.53, this is strongly contrasted with the single channel ratio 2.84. When we exclude the contribution from the Σ cD * configuration, the decay ratio R 1 of the P c (4312) becomes around 3, which is similar to the value in scenario I.
In Ref. [8], the coupled channel effect is also considerably large for the P c (4457). It contains 21% Σ * cD * configuration, which strongly couples with the J/ψp channel as illustrated in section IV B. The interference effect between the contributions from the Σ * cD * and Σ cD * is constructive and enlarges the partial decay width. Here, the decay width of Γ(P c (4457) → J/ψp) is 0.90 MeV, which is several times larger than that of the pure Σ cD * molecular state with J P = 3/2 − . For the P c (4440) molecular state mainly composed of the Σ cD * channel with J P = 1/2 − , as listed in Table VI, the signs of the scattering amplitudes for the Σ cD * and Σ * cD * with the η c p channel are opposite. The contributions partly cancel with each other. Finally, this cancelation leads to a quiet small partial decay width of the decay P c (4440) → η c p, which is suppressed by several orders compared with the J/ψp decay. In other words, it may be a little hard to observe the P c (4440) in the η c p channel.
Since a loosely bound molecular state P c (4380) mainly composed of an S −wave Σ * cD component can be reproduced using the same set of parameters in our previous work [8], we also obtain the decay width of P c (4380) → J/ψp, Γ = 2.34 MeV. The partial decay width of P c (4380) → η c p is suppressed by O(10 −3 ).

VI. SUMMARY
Inspired by the observations from the LHCb collaboration [2], we have calculated the partial decay widths for the P c states as the Σ ( * ) cD ( * ) molecules decaying into the J/ψp and η c p channels in the quark interchange model. Their partial decay widths are related with the scattering amplitudes between the Σ ( * ) cD ( * ) and J/ψp (η c p) channels, which are derived by the quark interchange model. In the quark level, the interactions between the hadrons are equivalently represented in terms of the interactions between the quarks. All the parameters in the quark model are determined by the mass spectra of the mesons.
In our calculations, we discuss the hidden-charm partial decay behaviors of the P c states as the pure (scenario I) or the coupled Σ ( * ) cD ( * ) molecules (scenario II). The corresponding results are summarized in Table VII.
As a pure Σ cD molecule with J P = 1/2 − , the P c (4312) has a larger decay width for the η c p decay mode than the J/ψp mode. Thus, one can expect the observation of the P c (4312) in the η c p decay channel. For the P c (4440)/P c (4457) states in scenario I, the decay widths of P c → J/ψp are larger than that of P c → η c p because of the stronger interaction between the Σ cD * state and the J/ψp channel.
With the coupled channel effect between different flavor configurations taken into account, one can reproduce the masses of P c (4312), P c (4440), P c (4457), and P c (4380) states simultaneously in Ref. [8]. They are the molecules mainly composed of Σ cD with J P = 1 Our results indicate that the P c (4312) is easier to decay into the J/ψp than the η c p channel. Since the partial decay widths into the two channels are not so different, we expect P c (4312) can be observed in the η c p channel in the near future experiment, like the LHCb Collaboration.
In contrast to the prediction in scenario I, the P c (4440) may be a little hard to be observed in the η c p channel in scenario II due to the destructive interference of the configurations Σ cD * and Σ * cD * . The tensions between the two scenarios will help to probe the molecular components in the P c (4312) and P c (4440) states.
In Table VII, we find that the predicted partial decay widths in scenario II are generally larger than those in scenario I except that of P c (4440) → η c p. This may come from the different relative molecular wave functions in the two scenarios. In scenario I, the radial wave function in the momentum space |p|R 0 (p) (R 0 (p) is the molecular wave function in Eq. (5) with p being the relative momentum) is zero at |p| = 0, while in scenario II the wave function obtained in the OBE model is not zero after Fourier transformation of the wave functions in Fig. 5. In the overlap of the wave functions, the region with small |p| may play an important role.
The obtained decay widths and branching fraction ratios are useful to explore the molecular assignment for the hiddencharm pentaquak states, which can be also examined by the experiments and the Lattice QCD in the coming years. The investigation of the decay properties will help to explore the reasonability of the molecular assignment and understand the inner dynamics of the exotic states. V. The masses (in unit of MeV) of the hidden-charm P c state and probabilities for different flavor configurations in the state [8]. The "∼ 0%" denotes that the proportion of the corresponding configuration is tiny and negligible.
where S ( ′ ) and I ( ′ ) are the spin and isospin of the initial (final) state, respectively. s a,b,c,d and I a,b,c,d denote the spin and isospin for the different hadrons.V s represents the spin operator.
We use the decayD −( * ) Σ ( * )++ c → J/ψ(η c )p as an example to illustrate the calculation of the I flavor-spin . In the scattering process, the flavor factor I flavor = 1. For the spin factor I spin , whenV s = 1, it reads withX = √ 2X + 1. WhenV s = S i · S j , the I spin for different diagrams in Fig. 1  With these method, one can easily calculate I spin for the post diagrams.