Possible strange hidden-charm pentaquarks from $\Sigma_c^{(*)}\bar{D}_s^*$ and $\Xi^{(',*)}_c\bar{D}^*$ interactions

Using the one-boson-exchange model, we investigate the $\Lambda_c\bar{D}_s^*$, $\Sigma_c\bar{D}_s^*$, $\Sigma_c^*\bar{D}_s^*$, $\Xi_c\bar{D}^*$, $\Xi_c'\bar{D}^*$, and $\Xi_c^*\bar{D}^*$ interactions by considering the one-eta-exchange and/or one-pion-exchange contributions. We further predict the existence of hidden-charm molecular pentaquarks. Promising candidates for hidden-charm molecular pentaquarks include a $\Xi_c'\bar{D}^*$ state with $0(\frac{1}{2}^-)$ and the $\Xi_c^*\bar{D}^*$ states with $0(\frac{1}{2}^-)$ and $0(\frac{3}{2}^-)$. Experimental searches for these predicted hidden-charm molecular pentaquarks are an interesting future research topic for experiments like LHCb.


Introduction
As a hot research issue, studying exotic hadronic states is attractive for both experimentalists and theorists. In the past 14 years, more and more charmonium-like and bottomoniumlike states and states with open-charm and open-bottom quantum numbers have been reported in experiments. This gives us a good chance to identify possible candidates for exotic states (see the comprehensive reviews in Refs. [1,2] for recent progress in this field).
Under the molecular state assignment to P c (4380) and P c (4450) [29,30,[38][39][40][41][42][43][44], the interactions of charmed baryons with anti-charmed mesons were studied. If P c (4380) and P c (4450) are molecular pentaquarks composed of a charmed baryon and an anti-charmed meson, we have reason to believe that their partners should exist. For example, in Ref. [30], possible charm-strange molecular pentaquarks were studied by the OBE model. The authors suggested Λ 0 b →D 0 D 0 Λ 0 as an appropriate channel to search for two predicted pentaquarks P cs (3340) and P cs (3400), which correspond to the Σ cK * configuration with I(J P ) = 1 2 ( 3 2 − ) and the Σ * cK * configuration with 1 2 ( 5 charm molecular pentaquarks with a strange quark, which have the concrete quark component [ccsqq]. These molecular pentaquarks, which are also called strange hidden-charm molecular pentaquarks in this work, are closely related to the Λ cD * cD * interactions. There are some previous theoretical studies of strange hidden-charm molecular pentaquarks [32][33][34]. Very recently, Karliner and Rosner [57] proposed that Λ b → J/ψΛ(π + π − /η) is a promising channel to find a possible strange hidden-charm molecular pentaquark composed of Λ c andD * s . Different from the former studies of strange hiddencharm molecular pentaquark in Refs. [32][33][34]57], in this work we carry out a comprehensive investigation of the cD * interactions under the OBE model, by which we further predict the strange hidden-charm molecular pentaquarks. Here, two kinds of molecular configurations will be taken into consideration: molecular systems composed of a charmed baryon and an anti-charmed-strange meson (molecular pentaquarks with components Λ cD * s , Σ cD * s , and Σ * cD * s ), and systems composed of a charmed-strange baryon and an anti-charmed meson (molecular pentaquarks with components Ξ cD * , Ξ ′ cD * , and Ξ * cD * ). In the following section, we will give a detailed illustration of deducing the effective potentials involved in the study of strange hiddencharm molecular pentaquarks. We hope that the present work may provide valuable information about strange hiddencharm molecular pentaquarks, which will be helpful for further experimental searches for them.
The paper is organized as follows. We present the deduction of the effective potentials in Section 2. In Section 3, the corresponding numerical results for the strange hidden-charm pentaquarks are given. A summary is then given in Section 4.
2 Effective potentials related to the Λ cD * cD * systems First, we need to illustrate the details of deducing the effective potential. For the Λ cD * cD * systems, their total wave functions are constructed by including color, flavor, spin-orbit, and spatial wave functions. For colorless molecular states, the color wave function is simply taken as 1.
In addition, we adopt the notation | 2S +1 L J to define the spinorbit wave function. The total angular momentum J can be 1 2 and 3 2 for the Λ cD * s , Σ cD * s , Ξ ′ cD * and Ξ cD * systems, and 1 2 , 3 2 and 5 2 for the Σ * cD * s and Ξ * cD * systems. The spin-orbit wave function | 2S +1 L J can be explicitly expressed as where S and D denote the corresponding systems with orbit angular momentum L = 0 and L = 2, respectively. We need to specify that in our calculation we consider the mixing of S -wave and D-wave, which is the lesson learned from deuteron studies. Here, S-D mixing contributes to the tensor force, which is crucial to form the shallow deuteron. The explicit expressions for the spin-orbit wave function are categorized into two typical groups by the spin S B of the baryon in the system, i.e., where Y L,m L is the spherical harmonics function, and the constants C J,M S m S ,Lm L , C S ,m S cD * : where Σ ( * ) c indicates the charmed baryons Σ c and Σ * c , and the charmed-strange baryons Ξ c , Ξ ′ c and Ξ * c , are denoted by Ξ (′, * ) c . In the following, we continue to deduce the effective potentials of the Λ cD * cD * systems. In general, the effective potential in momentum space is related to the scattering amplitude, i.e., where M(ab → cd) denotes the scattering amplitude of a process ab → cd, and m i (i = a, b, c, d) is the mass of particle a/b/c/d. At the hadronic level, we can write out the expression of M(ab → cd) by the effective Lagrangian approach.
Then, an effective potential in momentum space V E (q) can be transferred into an effective potential in coordinate space by performing the Fourier transformation In the above Fourier transformation, the form factor F (q 2 , m 2 E ) should be introduced at each interaction vertex to compensate ******-2 the off-shell effect of the exchanged meson and reflect the inner structure of each interaction vertex. Usually, a monopole form like F (q 2 , m 2 [5,6] * † , where m E and q denote the mass and four-momentum of the exchanged particle, respectively. In addition, the cutoff Λ is a model parameter in our calculation. Later, we will discuss the dependence of the numerical result on Λ. With the obtained effective potential, we try to find bound state solutions by solving the Schrödinger equation. In this way, we can further predict the mass spectrum of the possible molecular states.
When writing out the scattering amplitude, we adopt the effective Lagrangian approach. Due to both heavy quark symmetry and chiral symmetry [58][59][60][61][62][63], the relevant Lagrangians can be constructed as where H (Q) b and S µ are defined as field operators.
The superfield operator S µ is related to baryons B 6 with J P = 1/2 + and B * 6 with J P = 3/2 + in the 6 F flavor representation. The expression of S µ reads as Additionally, the four velocity has the form v = (1, 0). The axial current satisfies A µ = 1 2 (ξ † ∂ µ ξ − ξ∂ µ ξ † ) with ξ = exp(iP/ f π ) and the pion decay constant is taken as f π = 132 MeV. The concrete expressions of matrices P, B3, and B ( * ) 6 are By further expanding Eqs. (7)(8)(9), the concrete expressions of the effective Lagrangians adopted in our calculation can be obtained, i,e., In the above Lagrangians, there are three coupling constants, g, g 1 , and g 2 , to be determined. Based on the D * decay width (Γ(D * + ) = 96±4±22 keV) [64], the coupling constant g in Eq.
In the following, we continue to the deduction of the effective potential. The normalization relations for vector meson P * , baryon B3(B 6 ) with spin-1/2, and baryon B * µ 6 with spin- * In Refs. [5,6], Törnqvist studied the one-pion exchange potential contribution to NN systems. Especially, the deuteron was discussed with the introduced monopole form factor, where an expression for a spherical pion source was given, R = √ 10 fm. According to this relation, the Λ value was estimated to be 0.8 − 1.5 GeV for the NN interaction. For the discussed hidden-charm molecular pentaquarks, a smaller R should be expected, which results in a larger Λ. Due to this reason, in this work we choose Λ = 0.8 − 5 GeV to present our numerical results. Although the bound state solution by scanning this wide Λ range can be found, we still should be careful to make a definite conclusion of the existence of the corresponding hidden-charm molecular pentaquarks. Thus, in this work we take a stricter criterion, i.e., if the bound state solutions appear when taking Λ = 1−1.5 GeV, the existence of the corresponding molecular state becomes more possible. In Section 3, we give more detailed discussions.
† Indeed we can take other forms for the form factor, such as the dipole form factor F (q 2 , m 2 E ) = (Λ 2 −m 2 E ) 2 /(Λ 2 −q 2 ) 2 , to regularize the potential. If taking this form of form factor in the calculation, we need to fix the possible range of Λ by restudying the NN interaction.
‡ In Ref. [63], the coupling constant g 1 is related to another coupling constant g 4 by the relation [63] for more details). ******-3 respectively. Here, M b (b = B3, B 6 ,P * ) denotes the corresponding mass of vector meson P * /baryon B3/baryon B * µ 6 . The σ and p are the Pauli matrix and the momentum of the corresponding heavy hadron, respectively.
With the above preparation, we get the general expressions of the one-pion-exchange and one-eta-exchange effective potentials for the molecular systems considered in this work, i.e., Here, the subscript of V ΛcD * s →ΛcD * s η (r) denotes that the exchanged meson is η. We use the same notation for the other effective potentials listed in Eqs. (15)- (22). At first sight, the Λ cD * s and Ξ cD * interactions are forbidden because of the constraint from heavy quark symmetry. For the Σ cD * s and Σ * cD * s interactions, the one pion exchange is suppressed according to the OZI rule and vanishes under the symmetries considered in the current work. In the one-pion-exchange effective potentials listed in Eq. (19) and Eq. (21), an isospin factor G(I) is introduced, which is taken as G = 1 for the isovector sector with I = 1, and G = −3 for the isoscalar sector with I = 0. For the convenience of the reader, two auxiliary potential functions V 1 and V 2 are given here, i.e., where S (r, x, y) = 3(r·x)(r·y)−x·y, and the functions Y(Λ, m, r), Z(Λ, m, r), and T (Λ, m, r) have the definitions T (Λ, m, r) = r ∂ ∂r The values of the angular momentum operators in Eqs. (23)- (24) sandwiched between the wave functions can be read from Table 1, which will be used in the calculation.
With the effective potentials obtained, the bound state solutions (binding energy E and corresponding root-meansquare radius r RMS ) can be obtained by solving the coupledchannel Schrödinger equation. The corresponding kinetic terms for the systems investigated read as where ∇ 2 = 1 r 2 ∂ ∂r r 2 ∂ ∂r , ∇ 2 1 = ∇ 2 − 6/r 2 , and M α is the reduced mass with the subscript α standing for the different systems Σ cD * s , Σ * cD * s , Ξ ′ cD * , and Ξ * cD * . Recall that the Λ cD * s and Ξ cD * interactions are forbidden under heavy quark symmetry and chiral symmetry. In Section 3.1 and Section 3.2, we will present our results for two different types of molecular system, i.e., the Σ ( * ) cD * s systems, with the strange quark in the constituent meson, and the Ξ ( ′ , * ) cD * systems, with the strange quark in the constituent baryon, respectively.

Numerical results
In Table 2, we list the masses and quantum numbers of charmed hadrons involved in our calculation. In this work, special attention will be paid to the roles of the one-pionexchange (OPE) potential and the one-eta-exchange (OEE) potential in forming a hadronic molecular state. Table 2. Properties of hadrons involved in this work [66].
The Σ cD * s and Σ * cD * s systems For the Σ cD * s and Σ * cD * s systems, the OPE potential does not exist, and only the contribution from the OEE should be considered. In Table 3, the corresponding bound state solutions are listed. In addition, we also present the Λ dependence of the bound state solutions in Fig. 1.
by which experiments like LHCb may search for these three strange hidden-charm molecular pentaquarks in future.

The Ξ ′
cD * and Ξ * cD * systems For the Ξ ′ cD * and Ξ * cD * systems, the OPE is not suppressed, so it works with the OEE to provide the interaction force. The numerical results of bound state solutions for the Ξ ′ cD * and Ξ * cD * systems are collected in Table 4 and Table 5, respectively. Here, we still scan the cutoff Λ from 0.6 GeV to 5 GeV. Table 4. Typical values of the bound state solutions (binding energy E and root-mean-square radius r RMS ) for the Ξ ′ cD * system. Here, E, r RMS , and Λ are in units of MeV, fm, and GeV, respectively.  Table 4, we cannot find a bound state solution for the Ξ ′ To show the contributions from the OPE and the OEE more clearly, the effective potentials for the Ξ ′ cD * interaction with 0( 1 2 − ) when taking the cutoff Λ = 1.32 GeV are shown in Fig. 2. The OPE effective potentials are much larger than the OEE effective potentials and dominate the total potentials for the V 11 , V 12 , and V 22 cases. The curves for the OPE potentials nearly overlap with these for the total effective potentials. Besides, as indicated by the results for the bound state solution, the OEE behavior is similar to that of the OPE but with a smaller contribution to the total effective potential. ******-6 In Table 5, we present the numerical results for the Ξ * tive, which makes the value of the cutoff Λ become smaller when considering both OPE and OEE contributions if reproducing the same binding energy as when only considering the OPE contribution. Different from the above three cases, the OEE contribution provides a repulsive potential for the Ξ * cD * system with 1( 5 2 − ). Thus, we can naturally understand why the cutoff for the OPE&OEE mode is larger than for the OPE mode, as shown in Table 5. The effective potentials for the Ξ * cD * state with 0( 3 2 − ) are presented in Fig. 3. A similar conclusion to that of the Ξ ′ cD * state with 0( 3 2 − ) can be reached. The OEE contribution is much smaller than the OPE contribution for all V i j potentials considered here, which is consistent with the observation from these results in Table 5.  Additionally, we also provide the allowed two-body decay channels for the Ξ ( ′ , * ) cD * molecular pentaquarks with different quantum numbers in Table 6, which may be useful for the further experimental study of these molecular pentaquarks. Table 6. Allowed decay channels for Ξ ′ cD * and Ξ * cD * with different quantum numbers.

Summary
Searching for exotic hadronic states is a research field full of challenges and opportunities. With recent experimen-tal progress, more and more novel phenomena have been revealed in experiments, which has stimulated theorists' extensive interest in studying exotic states. Interested readers may read about the relevant progress in the review papers in Refs. [1,2].
In 2015, the observation of two P c states at LHCb [31] inspired many new investigations of hidden-charm pentaquarks, and molecular assignments to the two P c states are a popular explanation [29,30,[38][39][40][41][42][43][44]. In this situation, we have reason to believe that there should exist partners of the two P c states. Thus, we need to perform dynamical studies relevant to their partners. In this work, we mainly focus on the Λ cD * s /Σ ( * ) cD * s /Ξ (′, * ) cD * interactions and predict the existence of some strange hidden-charm molecular pentaquarks, as described in Section 3.
Our numerical results show that the most promising strange hidden-charm molecular pentaquarks are a Ξ ′ cD * state with 0( 1 2 − ) and the Ξ * cD * states with 0( 1 2 − ) and 0( 3 2 − ). Thus, we strongly suggest that experimentalists search for these.
In summary, the two observed P c states from LHCb have opened fascinating new avenues of research. In future, theorists and experimentalists should make more effort to study hidden-charm pentaquarks, especially the partners of P c (4380) and P c (4450). As we face this research field full of challenges, more opportunities are waiting for us.