Spectrum of the strange hidden charm molecular pentaquarks in chiral effective field theory

We calculate the effective potentials of the $\Xi_c\bar{D}^{(\ast)}$, $\Xi_c^\prime\bar{D}^{(\ast)}$ and $\Xi_c^\ast\bar{D}^{(\ast)}$ systems with the chiral effective field theory up to the next-to-leading order. We simultaneously consider the short-, intermediate- and long-range interactions. With the newly observed $P_c$ spectra as inputs, we construct the quark-level contact Lagrangians to relate the low energy constants to those of $\Sigma_c\bar{D}^{(\ast)}$ with the help of quark model. Our calculation indicates there are seven bound states in the $I=0$ strange hidden charm $[\Xi_c^\prime\bar{D}^{(\ast)}]_J~(J=\frac{1}{2},\frac{3}{2})$ and $[\Xi_c^\ast\bar{D}^{(\ast)}]_J~(J=\frac{1}{2},\frac{3}{2},\frac{5}{2})$ systems. Our analyses also disfavor the $\Lambda_c\bar{D}^{(\ast)}$ bound states. However, we obtain three new hadronic molecules in the isoscalar $[\Xi_c\bar{D}^{(\ast)}]_J~(J=\frac{1}{2},\frac{3}{2})$ systems. The masses of $[\Xi_c\bar{D}]_{1/2}$, $[\Xi_c\bar{D}^{\ast}]_{1/2}$ and $[\Xi_c\bar{D}^{\ast}]_{3/2}$ are predicted to be $4319.4^{+2.8}_{-3.0}$ MeV, $4456.9^{+3.2}_{-3.3}$ MeV and $4463.0^{+2.8}_{-3.0}$ MeV, respectively. We also notice the one-eta-exchange influence is rather feeble. Binding solutions in the $I=1$ channels are nonexistent. We hope the future analyses at LHCb can seek for these new $P_{cs}$s in the $J\psi\Lambda$ final states, especially near the thresholds of $\Xi_c\bar{D}^{(\ast)}$.


I. INTRODUCTION
In the past decades, the renaissance of hadron physics was witnessed. Mesons and baryons, which have the internal configurations qq and qqq respectively, have been extensively studied with lattice QCD and various QCD inspired models. The abundant conventional hadrons in the Reviews of Particle Physics [1] reflect the great victory of the quark model. Other more complicated quark configurations, such as qqqq, qqqqq, and qqqqqq, etc., are not forbidden by QCD. Thus hunting for these type states is a long standing problem for theorists and experimenters. X(3872) is the poster child that opened a new era for hadron physics [2]. After that, more and more XY Z states were discovered. The multiquark matter becomes one of the hottest topics in recent years [3][4][5][6][7][8].
Very recently, the LHCb Collaboration reported the observation of three pentaquark states P c (4312), P c (4440) and P c (4457) [9]. Their masses lie several to tens MeV below the Σ cD and Σ cD * thresholds, thus the molecular explanation is naturally proposed in many works [10][11][12][13][14][15][16][17][18][19][20][21]. The J P quantum numbers are undetermined yet, but the theoretically favored ones in the molecular scenario for these three P c s are 1 2 − , 1 2 − and 3 2 − , respectively. Therefore, if the forthcoming measurements for the J P s indeed meet the predictions from the molecular pictures, which would give more robust support for the molecular interpretations.
The XY Z and P c states are the candidates of the hiddencharm multiquark states with inner quark components QQqq and QQqqq, respectively. As indicated in Refs. [22][23][24], it seems the heavy quark core plays an important role in stabilizing the exotic clusters. This is indeed the case in the atomic physics. For example, the hydrogen molecule consists of two protons and two electrons, which stably exists in the nature.
In the hadronic molecular scenario, the interaction between two color singlets (e.g., Σ c andD) is very similar to that between electroneutral atoms (e.g., H and H). The "covalent bond" in the former is attributed to the residual strong interactions, which is equivalently described by the pion-exchange in chiral effective theory or the meson-exchange (e.g., π, ρ, σ, ...) in one-boson-exchange model. Therefore, one could actually anticipate the existence of more hadronic molecules in the charmed baryon-anticharmed meson systems when the flavor symmetry group is enlarged to SU(3).
Starting from the deuteron (an I = 0 loosely bound np molecule), one can notice that the interactions between two heavy matter fields tend to form the bound states in the lowest isospin channels. X(3872) is another example, which is a good candidate of the D 0D * 0 molecule with isospin I = 0. The newly reported P c s are widely accepted as the Σ cD ( * ) bound states with isospins I = 1 2 . Some investigations on the DD * andB ( * )B( * ) systems also demonstrated the existence of bound states in I = 0 channels [25,26].
In our previous work [11], we have systematically investigated the interactions of the Σ cD , Σ cD * , Σ * cD and Σ * cD * systems in chiral effective field theory. We simultaneously reproduced the newly observed three P c s as the I = 1 2 hidden-charm Σ cD and Σ cD * molecules by introducing the Λ c contribution in the two-pion-exchange loop diagrams. In this work, we extend our study to the Ξ cD ( * ) , Ξ cD ( * ) and Ξ * cD ( * ) systems to see whether there exist the bound states in the lowest isospin, i.e., I = 0 channels. Likewise, these strange hidden charm molecular states might be reconstructed in the J/ψΛ channel at the LHCb experiment. Some investigations suggest searching for these states in the decay modes [27][28][29].
Based on Ref. [11], we further study the effective potentials of six systems, i.e., Ξ cD ( * ) , Ξ cD ( * ) and Ξ * cD ( * ) . They all contain one strange quark. The short-range contact interaction, long-range one-pion-exchange contribution and intermediate-range two-pion-exchange diagrams are all included in the framework of chiral effective field theory (For arXiv:1912.12592v2 [hep-ph] 3 Feb 2020 the reviews of chiral effective field theory, we refer to [30][31][32][33][34]). Considering the hadronic molecules are shallowly bound states, the strange quark dynamics are freezed in the present calculations, which contribution is partially involved in the contact terms. We ignore the η and K meson contributions in the loops. The low energy constants (LECs) are well determined by fitting the P c spectra. In this way, we predict the possible strange hidden charm molecular pentaquarks.
This paper is organized as follows. In Sec. II, we give the effective Lagrangians constructed with the chiral and heavy quark symmetries. In Sec. III, we show the expressions of effective potentials. In Sec. IV, we give the numerical results and discussions. In Sec. V, we conclude this work with a short summary. In Appendix A, we bridge the LECs to those of In addition, the chiral connection Γ µ and axial-vector current u µ read respectively where and f π = 92.4 MeV is the pion decay constant. Expanding Eq. (2) one can get the coupling terms among B3, B 6 and B * µ 6 . The detailed forms and the corresponding axial couplings can be found in Refs. [11,[39][40][41].
The leading order Lagrangians that delineate the interactions between the anticharmed mesons and light Goldstones read [42,43] where · · · represents the trace in spinor space. The covariant derivative D µ = ∂ µ + Γ µ . δ b is defined as δ b = mD * − mD. g −0.59 stands for the axial coupling, which can be extracted from the partial decay width of D * + → D 0 π + [1]. TheH is the super-field for the anticharmed mesons, which reads The contact Lagrangians that describe the short distance interactions between the charmed baryon sextets and an-ticharmed mesons have been constructed in Refs. [10,11] with the super-field representations, which read where D a , D b , E a and E b are the LECs. They can be determined by fitting the P c spectra. λ i denotes the Gell-Mann matrices. Besides, we also need the Lagrangians to depict the contact interactions of the charmed baryon antitriplet and anticharmed mesons. They can be analogously constructed as follows, whereD a ,D b ,Ẽ a andẼ b are another sets of the LECs. These LECs are different from the ones in Eq. (7), since the B3 and B ( * ) 6 are not the partner states under heavy quark spin symmetry. But we can establish the corresponding relationship with the D a , D b , E a and E b with the help of quark model. We show this operation in the Appendix A.

III. EXPRESSIONS OF THE EFFECTIVE POTENTIALS
There exists a simple relation between the effective potential and scattering amplitude in momentum space under the Breit approximation, where m i and m f are the masses of the initial and final states. q denotes the transferred momentum between two scattering particles. Then the effective potential in coordinate space can be easily obtained by Fourier transformation, which yields where a Gauss regulator F(q) = exp(−q 2n /Λ 2n ) is introduced to suppress the high momentum contribution [44,45]. n = 2 is used in this work [46,47]. Considering the ρ meson mass m ρ is treated as the typical hard scale in chiral effective theory, thus the cutoff Λ should be smaller than m ρ (A detailed discussion on the range of Λ can be found in Ref. [11]). The Λ is chosen to be around 0.5 GeV to perform fittings and give predictions [10,11,32,46]. The topological diagrams are shown in Fig. 1. There are three types of Feynman diagrams in our calculations, i.e., the leading order contact interaction, one-pion-exchange diagram, and the next-to-leading order two-pion-exchange diagrams. The Feynman diagrams for the Ξ cD ( * ) (Ξ cD ( * ) ) and Ξ * cD ( * ) systems are totally the same as the Σ cD ( * ) and Σ * cD ( * ) , respectively. In other words, one can build the following correspondence, So we do not explicitly show the detailed graphs for each systems at each order. One can find them in figures 2−7 of Ref. [11].
In the following, we write down the leading order contact potential of each system, which can be easily obtained by expanding the Lagrangians in Eqs. (7) and (8), respectively.
The expressions of the one-pion-exchange diagrams for Ξ cD * and Ξ * cD * are the same as the Σ cD * and Σ * cD * in Ref. [11] up to different matrix elements of I 1 ·I 2 operator. For Ξ cD * , the coupling between Ξ c and π vanishes since the pion does not couple to the scalar isoscalar light diquark within Ξ c because of the parity and angular momentum conservation. Thus the one-pion-exchange does not contribute to the effective potential of the Ξ cD * system. For the two-pion-exchange diagrams, the graph (F i.j ) is governed by the chiral connection term, thus all the systems share one single expression, i.e., For graphs (T i.j ), (B i.j ) and (R i.j ), their analytical expressions generally have the following structures, V Bi.j V Ri.j where the subscript "Sys." denotes the corresponding system, such as Ξ cD ( * ) and so on. The superscript "T i.j ", "B i.j " and "R i.j " represent the labels of the Feynman diagrams. Various J functions, such as J T x , J B x and J R x are the scalar loop functions, which are defined and given in Refs. [10,11,26]. The coefficients C  [11]. According to the correspondence in Eq. (11), one can easily get them by matching with the results in Ref. [11]. The coefficients C Ti.j Sys. and C Bi.j Sys. (C Ri.j Sys. ) for each system are given in Tables I and II.  In order to get the numerical results, we have to determine the eight LECs in Eqs. (7) and (8). As in Ref. [49], we also propose a SU(3) quark model to estimate the LECs. One can find the detailed derivations in Appendix A.
The effective potentials of some representative I = 0 channels are given in Fig. 2. We notice that the leading order contact interaction supplies a strong attractive potential for all the considered channels. This situation is the same as those of the Σ ( * ) cD ( * ) systems [11]. For the Ξ cD and Ξ cD * systems, one can find the two-pion-exchange contributions are also significant because of the accidental degeneration between Ξ cD and Ξ cD * in the loops. However, the behavior of the two-pionexchange potentials for Ξ cD and Σ cD are totally different due to the opposite sign of the mass differences. For example, the mass difference between Σ cD and Λ cD * is about 28 MeV, while that for Ξ cD and Ξ cD * is about −32 MeV. By solving the Schrödinger equation, we find the bound states in the Ξ cD ( * ) and Ξ * cD ( * ) systems, likewise. The predicted binding energies and masses are given in Table III.
Theoretically, the existence of the bound states in the Ξ cD ( * ) and Ξ * cD ( * ) systems is not a surprise, because Ξ c and Ξ * c belong to the same flavor multiplets with the Σ c and Σ * c , respectively in the SU(3) flavor symmetry [48]. Nevertheless, things become interesting when we go to the Ξ cD ( * ) systems. On the one hand, no bound states are experimentally observed near the Λ cD ( * ) thresholds up to now [9]. On the other hand, some theoretical calculations also do not support the existence  For the Λ cD ( * ) and Ξ cD ( * ) systems, the long-range onepion-exchange vanishes, thus only the contact term and twopion-exchange contribute to their potentials. The isospin I and spin J (J denotes the total spin of the light degrees of freedom in Λ c ) of the Λ c are both zero, so there are no isospin-isospin and spin-spin interactions for the Λ cD ( * ) systems 1 . Ignoring the η and K meson contributions in the loops, one can roughly get the two-pion-exchange potential of the Λ cD ( * ) from the Ξ cD ( * ) expressions by setting the matrix element I 1 · I 2 to be zero. Although the two-pion-exchange potential of the Λ cD ( * ) (or say the contribution from couplechannel effect) is attractive, the one from the leading orderD a term [e.g., see Eq. (8)] is repulsive. Their contributions almost cancel each other, thus there are no binding solutions for the Λ cD ( * ) systems. The J of Ξ c is also zero, thus there are no spin-spin interactions for the Ξ cD ( * ) systems as well. But I = 1 2 for Ξ c , i.e., the isospin-isospin interaction in the Ξ cD ( * ) systems is the main reason that leads to the different scenarios for the Λ cD ( * ) and Ξ cD ( * ) systems. The leading order isospin re-latedẼ a term provides a very strong attractive potential, as well as the attractive two-pion-exchange potential. Their contributions together yield three bound states in the Ξ cD ( * ) systems (see Table III). A recent study based on the local hidden gauge approach also gives a similar conclusion [52].
We also tried to include the one-eta-exchange contribution for the Ξ cD * and Ξ * cD * systems at the leading order, where we use the experimental values of f η and m η as inputs [39][40][41]53]. We notice the one-eta-exchange contribution is marginal, which only introduces about 1% and even much smaller than 1% corrections to the Ξ cD * and Ξ * cD * binding energies, respectively. Additionally, we also calculate the potentials of these six systems in the I = 1 channel. However, the total potentials 1 At the leading order of the heavy quark expansion, the spin-spin interaction between Λc andD * are represented by their light degrees of freedom, i.e., J Λc · J D * . Its matrix element vanishes for the ΛcD * system. are all repulsive, i.e., no bound states exist for the Ξ cD ( * ) , Ξ cD ( * ) and Ξ * cD ( * ) systems in the isovector channels.
These I = 0 molecular pentaquarks with a strange quark can be reconstructed at the J/ψΛ final states. We hope LHCb collaborations may search for these new P cs s near the Ξ cD ( * ) thresholds. Some discussions on the Λ cD

V. SUMMARY
In this work, we have systematically calculated the effective potentials of Ξ cD ( * ) , Ξ cD ( * ) and Ξ * cD ( * ) systems with the chiral effective field theory up to the next-to-leading order. The contact interaction, one-pion-exchange contribution and two-pion-exchange diagrams are considered. By fitting the newly observed P c spectra, we relate the LECs to those of the Σ cD * systems with the quark model (see Appendix A).
As the partners of Σ cD ( * ) and Σ * cD ( * ) , we also find seven bound states in the isoscalar [Ξ cD ( * ) ] J and [Ξ * cD ( * ) ] J systems. The contact terms provide the attractive potentials, which are dominant for these systems. The two-pionexchange interactions are important to the Ξ cD and Ξ cD * systems due to the accidental degeneration of the intermediate states in the loops.
With the estimated LECs, we also obtain three bound states in the isoscalar [Ξ cD ( * ) ] J systems. This is very different from their partners Λ cD ( * ) . Our analyses do not support the existence of any molecular pentaquarks in the Λ cD ( * ) systems. The difference between Λ cD ( * ) and [Ξ cD ( * ) ] J arises from the isospin-isospin interaction, which vanishes for the Λ cD ( * ) .
We considered the influence of one-eta-exchange interaction. Its contribution only gives rise to 1% corrections to the magnitude of the binding energies. Thus the tiny effect is neglected in our numerical results. Our calculation indicates that the potentials of the I = 1 channels are all strongly repulsive and no bound states exist in these I = 1 channels.
In summary, we obtain ten molecular pentaquarks P cs in the isoscalar Ξ cD ( * ) , Ξ cD ( * ) and Ξ * cD ( * ) systems. Their signals can be reconstructed in the J/ψΛ final states at LHCb experiment.  In this part, we estimate the eight LECs in Eqs. (7) and (8) from the viewpoints of quark model. We assume the shortrange contact interaction stems from some heavy particle exchanges, which is analogous to the resonance saturation model [54]. However, we do not specify the exchange particles (such as ρ, ω, f 0 , etc..) as in the one-boson-exchange scheme, because their contributions are partially mimicked by the two-pion-exchange diagrams. Rather than calculating the contact potential at the hadron level, we construct the quark-level Lagrangians to depict the short-range interaction by borrowing some concepts from the quark-hadron duality and quark model. Generally, one can formulate the quarklevel Lagrangians as follows, where q = (u, d, s), g s and g a are two independent coupling constants. The S and A µ are two fictitious fields, which create (annihilate) the scalar and axial-vector spurions respectively. We assume the S and A µ are flavor octets and have the similar matrix form as that in the second term of Eq. (5). They are introduced to produce the central potential and spin-spin interaction between two quarks, respectively. With the quark-level Lagrangians, the Σ cD * effective potential can be expressed as where the superscript "Q.L." is the abbreviation of quark level.
We take the SU(3) flavor symmetry and ignore the mass differences in the S and A µ multiplets, respectively. In addition, the assumption that q 2 m 2 S (m 2 A ) is used. Thus if we know the square of the "charge-to-mass ratios" g 2 s /m 2 S and g 2 a /m 2 A , we could obtain the potentials of the other systems that contain the light quarks. The contact potential of the Σ cD * in the I = 1 2 channel is given as [11] V ΣcD * = −D 1 − 2 3 D 2 (σ · T).
By fitting the P c spectra with the cutoff Λ = 0.4 GeV, we get D 1 = 63.1 GeV −2 , D 2 = 6.5 GeV −2 . Defining C 1 = g 2 s /m 2 S , C 2 = g 2 a /m 2 A , and comparing Eq. (A2) and Eq. (A3), one easily gets Following the same procedure, we can also calculate the Ξ cD * contact potential with the quark-level Lagrangians, which yield V Q.L.
Matching Eq. (13) and Eq. (A5), one can obtain the LECs in Eq. (7) under the SU(3) symmetry, which read Similarly, one can also calculate the contact potential of Because the exchange particles we considered in Eq. (A1) are only octets, thus the final results show that the LECs D a , D b andD a are all zero. Their values should be contributed by the singlets exchange. We could estimate the D a , D b and D a by replacing the octets in S and A with the nonet. An alternative way is to expand the Gell-Mann matrices in Eqs. (7) and (8) with λ 0 = 2/3 diag{1, 1, 1}. The extra λ 0 terms can be matched to the D a , D b andD a terms. The relations read We attempt to include the influences of D a , D b andD a on the numerical results in the SU(3) case. Considering the masses of the singlets are heavier than those of the octets, we adopt the half values in Eq. (A9) as their limits to give a conservative estimation. Finally, the values of the these LECs are given in Table IV. DaD bẼaẼb 0 ± 3.2 0 18.9 0