Predictions of the hidden-charm molecular states with four-quark component

In this work, we study the $T\bar{T}$-type molecular systems systematically via one pion exchange model, where $T$ denotes the narrow $J^P=1^+$ $D_1$ meson or $2^+$ $D_2^*$ meson and $\bar{T}$ is its antiparticle. With the effective potentials, we try to find the bound state solutions of the corresponding systems, which provide crucial information of whether there exist the $T\bar{T}$-type molecular states. According to our analysis, we predict some $T\bar{T}$-type molecular states which may be accessible at future experiments like LHCb and forthcoming BelleII.


I. INTRODUCTION
Searching for exotic states is a promising research topic full of opportunities and challenges in hadron physics. In 2013, the observation of the charged charmonium-like structure Z c (3900) from the BESIII [1] and Belle [2] collaborations have inspired extensive discussions on the four-quark matter. Very recently, the LHCb Collaboration announced two hidden-charm pentaquarks P c (4380) and P c (4450) [3], which have aroused theorists' interest in the five-quark matter again [4][5][6][7][8].
As one of the possible exotic hadron configurations, the multiquark state is a new hadronic matter beyond the conventional qq meson and qqq baryon. Among the multiquark states, the molecular state is a very popular configuration, which has been applied to understand some of the experimental observations of the near-threshold charmonium-like states, which are also named as the XYZ states.
With further experimental progress, more candidates of exotic states will be reported. For theorists, it is time to make reliable predictions of the hidden-charm molecular states. In this work, we focus on the hidden-charm molecular states with * Electronic address: chenr2012@lzu.edu.cn † Electronic address: xiangliu@lzu.edu.cn ‡ Electronic address: yrliu@sdu.edu.cn § Electronic address: zhusl@pku.edu.cn four-quark component, which are composed of the charmed and anti-charmed mesons. Their properties are determined by the corresponding components (charmed and anti-charmed mesons).
In the heavy quark limit [27], the S-wave and P-wave charmed mesons can be grouped into three doublets H = (0 − , 1 − ), S = (0 + , 1 + ), and T = (1 + , 2 + ). In Ref. [15], the hidden-charm molecular states composed of the charmed/anticharmed mesons in the H doublet were studied, which are abbreviated as the HH-type hidden-charm molecular states. Later, the possible SS -type hidden-charm molecular states are predicted [28], which are constructed by P-wave charmstrange meson D s0 (2317) and D s1 (2460) and their corresponding anti-particles, where D s0 (2317) and D s1 (2460) have very narrow widths. In addition, the authors of Ref. [29] analyzed the molecular systems composed of the charmed mesons in the H doublet and S doublet carefully, which are also named as HS -type hidden-charm molecular states.
Along this line, in this work, we continue to carry out the study of the TT -type hidden-charm molecular states, which are composed of the charmed and anti-charmed mesons in the T doublet. The charmed mesons in the T doublet have narrow widths. In Ref. [30], it is argued that the broad width of an open-charm meson probably results in unobservable bound states containing this meson. Thus, the charmed mesons in the T doublet are the suitable building blocks of the molecular states.
In order to investigate the TT -type hidden-charm molecular states, we adopt the one pion exchange (OPE) model [12,13,31] to deduce the effective potentials. With the obtained potentials and by solving Schrödinger equation, we finally get the bound state solution and can judge whether there exists the bound states. We need to specify that the S-D mixing effect [21,22,32,33] and coupled-channel effect [34,35] will be included in our calculation. The detailed deductions of the OPE potentials will be given in the next section. We hope that the present study may stimulate interest in searching for the TT -type hidden-charm molecular states in experiments like LHCb and forthcoming BelleII.
A hadron-hadron bound state such as the deuteron is stable since its constituents do not decay. In the present case, both D 1 and D * 2 have a width of tens of MeV. Although a binding energy of several MeV for a possible molecule is smaller than the width of its constituent, some molecular type resonance may still result from the TT interaction. This character may be approximately studied by treating the mesons as stable states and encoding the width effects into some parameters. Such a point was discussed in Ref. [36] for the NN * (1440) interaction.
This paper is organized as follows. After introduction, we present the deduction of the OPE potentials in Sec. II. In Sec. III, we present the corresponding numerical results. We summarize our results in Sec. IV.

II. DEDUCTION OF THE EFFECTIVE POTENTIALS
In this section, we will derive the one-pion-exchange interaction potential between the charmed mesons in the T doublet and their antiparticles. These charmed-anticharmed meson pairs can form three S -wave systems D 1D1 , D 1D 2 − D * 2D 1 . The meson-antimeson molecular system is similar to the well-known deuteron [12,13] and the tensor force might be important. We also include the coupled channel effects due to the D-wave interaction between the constituents. The allowed quantum numbers and possible coupled channels for the systems are where the notation | 2S +1 L J is used. Contributions from the higher partial wave channels are significantly suppressed and we do not include such channels here. The P-parity is always +1. The C-parity for the D 1D1 and D * 2D * 2 systems is determined with C = (−1) L+S while the C-parity for the D 1D * 2 system is not constrained by symmetry. Since the orbital angular momentum L = 0, 2, one observes that the total spin S of the system is always odd (even) for the D 1D1 and D * 2D * 2 states with negative (positive) C-parity while both odd and even S are allowed for the D 1D * 2 states. The molecular system composed of two mesons AB is defined as Here, C S ,m S s A ,m A ;s B ,m B and C J,M S ,m S ;L,m L denote the CG coefficients, |Y L,m L is the spherical harmonic function, and ǫ m is the polarization vector ε m for the axial-vector meson or the polarization tensor ξ m for the tensor meson. In the static limit, the special expressions for the polarization vector are ε ±1 = 1 √ 2 (0, ±1, i, 0) and ε 0 = (0, 0, 0, −1). The polarization tensor ξ m can be constructed from the polarization vectors, ξ m = m 1 ,m 2 C 2,m 1,m 1 ;1,m 2 ε m 1 ε m 2 [40]. One may use each z-component (M = 0, ±1, · · · , ±J) to derive the potential.

B. The effective Lagrangian
We will study the meson-antimeson interaction in a onepion-exchange (OPE) model. The effective potential is derived using the effective Lagrangian with the heavy quark symmetry and chiral symmetry, which can be written in a compact form [41], Here, the axial field A µ is constructed with the pion field where f π = 132 MeV is the pion decay constant. The T doublet now is expressed with a form of the superfield where the four velocity v = (1, 0) in the static approximation. The normalization for the involved mesons are 0|P µ 1 |Qq(1 + ) = ε µ M P 1 and 0|P * µν 2 |Qq(2 + ) = ξ µν M P * 2 . The antimeson doublet containing a heavy anti-quarkQ is obtained from the charge conjugate operation, where the transpose is for the γ matrices, C is the charge conjugate operator and the matrix C = iγ 2 γ 0 . The convention for the C-parity transformation for meson and antimeson fields is and the hermitian conjugate fields are defined bȳ In the compact Lagrangian, ... denotes trace in the spin and flavor space. Expanding Eq. (7), one obtains the interaction terms of the D 1 (D * 2 ) andD 1 (D * 2 ) mesons with the pion, The lowest charmed meson doublet is H = (0 − , 1 − ). The coupling constant g for the H − H − π interaction can be extracted from the strong decay D * → Dπ: g = 0.59±0.07±0.01 [42]. In this work, the coupling constant k is taken as the same value of g [24,35,43].

C. The OPE effective potential
With the above effective Lagrangians, we can write down the amplitudes for the t-channel scattering processes, which are related to the required potentials. The involved processes Generally, the effective potential in the momentum space is related to the obtained scattering amplitude through the Breit approximation. Here, we take the process D 1D1 → D 1D1 as an example. The relation between the effective potential V and scattering amplitude M is where M i and M f denote the masses of the initial and final states, respectively. We calculate the scattering amplitude M(D 1D1 → D 1D1 ) in the one-pion-exchange approximation.
In addition, one obtains the effective potential in the coordinate space by performing the Fourier transformation, is the monopole type form factor with m E the mass of the exchanged meson. The monopole form factor is introduced at each interaction vertex to compensate the off shell effect of the exchanged meson and describe the structure effect of the vertex. In practice, the phenomenological cutoff Λ is around one to several GeV.
Following the above procedure, we obtain the OPE potentials for the D 1D1 , D 1D * 2 and D * 2D * 2 systems. The form of the final potentials for these systems may be summarized into a compact form Here, I is the isospin of the state and G(I) is defined as an isospin factor, with the value G(0) = 3/2 and G(1) = −1/2. One gets these potentials by sandwiching the above (I, J)independent potentials between the states in Eqs. (1) and (2). We take the D 1D1 system as an example to illustrate the structure of the potentials. Now Here, E 1 (S 1 ) is the spin-spin (tensor) operator and f (r) (g(r)) is a potential function depending on the cutoff Λ, the pion mass m π , and the radial coordinate r. Obviously, V 1 is a n × n matrix, where n is the number of the coupled channels in Eq.
(3) which depends on the total angular momentum J. The explicit matrix elements for the operator E 1 are and those for S 1 are When the quantum numbers are the same, the transitions between the states D 1D1 , D 1D * 2 , and D * 2D * 2 may happen. We also consider such coupled channel effects in this paper. The Relevant transition potentials areV 2,3,4,7,9 . We will show the form of the final potentials in the next section. We collect all the defined subpotentialsV k (k = 1, 2, · · · , 11) in Eqs. (A3)-(A13) in Appendix A. The related matrix elements are given there in Tables VII and VIII.

III. NUMERICAL RESULTS
With the effective potentials (12)-(14), we can search for the bound state solutions by solving the coupled-channel Schrödinger equation, where , and µ is the reduced mass of the system. The wave function ψ(r) is a column matrix depending on V(r).
Then we adopt the FESSDE program [44,45] to calculate the energy eigenvalue E in units of MeV. If a bound state solution exists, the binding energy E is defined as the mass difference M system − M constituents . We further calculate the corresponding root-mean-square (RMS) radius r RMS in units of fm. E and r RMS depend on the cutoff parameter Λ which varies from 0.5 to 5 GeV. In this subsection, we present three sets of numerical results for the pure D 1D1 , D 1D * 2 and D * 2D * 2 systems in Tables I and  II. There are fourteen isoscalar and four isovector bound state solutions in the TT systems. The Λ dependence of the obtained binding energy is shown in FIG. 1. The results are very sensitive to the cutoff parameter, which is a common feature in these types of investigations [9, 15, 21, 22, 24, 28, 29, 32-35, 38, 41, 46-49]. At present, we cannot determine its value for the present systems from experimental observables. One should be cautious that the binding solution for the mesonantimeson systems are possible only when the cutoff falls into a reasonable range. A repulsive potential or attractive potential with an inappropriate cutoff does not result in bound states. systems. E, r RMS , and Λ are in units of MeV, fm, and GeV, respectively. The notation · · · means that no binding solution is found.   system. E, r RMS , and Λ are in units of MeV, fm, and GeV, respectively. The notation · · · means that no binding solution is found. If we compare all the bound state solutions system by system, we notice an inequality Λ I=0 if the binding energies are fixed at roughly the same value. To some extent, the existence possibility of one molecule is related to the value of the cutoff Λ. We notice P(0 + (0 ++ )) > P(0 − (1 +− )) > P(0 + (2 ++ )), where P(0 + (0 ++ )) denotes the existence possibility for the D 1D1 with I G (J PC ) = 0 + (0 ++ ). In other words, the D 1D1 with I G (J PC ) = 0 + (0 ++ ) is the most stable loosely bound structure, followed by 0 − (1 +− ) and 0 + (2 ++ ). This binding feature results certainly from the potentials. Since the main contribution comes from the S-wave interaction, the matrix elements for the operator E 1 in the diagonal S-wave channels, 1 S 0 |E 1 | 1 S 0 = 2, 3 S 1 |E 1 | 3 S 1 = 1, and 5 S 2 |E 1 | 5 S 2 = −1, affect dominantly the attraction strength. Therefore, the 0 ++ state is the most easily bound. The isoscalar scalar D 1D1 system may be easiest to detect in future experiments. Moreover, we also notice the same trend in the isoscalar D 1D * 2 and isoscalar D * 2D * 2 cases.
In terms of the isovector states, there are only four binding solutions, which are the D 1D1 system with 1 − (2 ++ ), D 1D * 2 with 1 + (3 +− ), 1 − (3 ++ ) and D * 2D * 2 with 1 − (4 ++ ). All these TT systems could be easily observed if they do exist since the charged state is easy to identify experimentally. Moreover, their total angular momentum J always takes the largest value for these systems. The reason is that the sign for the potential in the isovector case is opposite to that in the isoscalar case. A smaller matrix element for the operator E 1 in the diagonal S-wave channel leads to a stronger attraction. From the matrix elements in Table VII, the smallest value usually corresponds to the largest J. Compared to the isoscalar states, the values of the cutoff Λ in the isovector case are all around 4 to 5 GeV. From the experience of the deuteron (Λ ∼ 1 GeV), this large cutoff means that the attraction is not so strong. It is not difficult to understand this observation from the potentials, where an isospin factor G(I) always exists. The magnitude of the factor (and thus the potential) in the isovector case is smaller than that in the isoscalar case by a factor of three. As a result, one usually finds more binding solutions in the isoscalar case. Further studies of the existence possibility for the isovector states are required.
When we ignore the channel couplings among D 1D1 , D 1D * Searching for these prime molecular candidates is very interesting in the future. In the following we also discuss their decay behavior. We present all the allowed S-wave two-body decay modes. The decay channels of the D 1D1 molecular state with 0 + (0 ++ ) include DD, D * D * , η c (nS )η (n = 1, 2) and ψ(nS )ω (n = 1, 2, 3). For the D 1D *

B. The coupled system
Since the charmed meson masses in the T doublet (D 1 , D * 2 ) are very close, it is very necessary to consider the coupled channel effects for various systems. In Table III, we present the possible channels to be considered in this work. The states with J PC = 1 ++ , 2 +− , 3 ++ , and 4 ++ appear only in either D 1D * 2 or D * 2D * 2 system, which have been discussed in the above subsection.
Compared with Eqs. (3)-(5), several D-wave channels are omitted in Table III since their contributions are small in our calculation. For illustration, we present the contributions from each S-wave and D-wave channel to the pure D 1D * 2 and D *

2D
* 2 systems for some cases In Table IV. It is obvious that the contributions from the neglected channels such as | 5 D 1 , | 7 D 1 , | 7 D 2 , | 9 D 2 , and | 5 D 3 are small (less than 0.6%) in the J = 1 case. In the coupled channel cases, the contributions of the above neglected D-wave D 1D * 2 and D * 2D * 2 channels are even smaller than those for the pure systems. We have confirmed that the omission of these tiny D-wave channels do not affect the main results after solving the coupled Schrodinger equation. Now, the binding energy is defined relative to the D 1D1 threshold. One has to add (M D * 2 − M D 1 ) to the kinetic term in Eq. To get the final potentials, one repeats the procedure to obtain Eq. (15) with the relevant transition amplitudes. For the J PC = 0 ++ case, we have where V 4 is a 2 by 2 matrix. The base of the whole 4 × 4 matrix is the column matrix composed of the four scalar states in Table III. Similarly, the final potentials for the cases J PC = 1 +− , 2 ++ , and 3 +− are V(I, J = 2, r) and respectively. The bases of these potential matrices (also the order of elements) are completely determined by the channels in Table III. Since we ignore small contributions from several channels, the dimension of V 5,6,10 here is smaller than that in Eqs. (13) and (14). Note we do not give V 8 and V 11 explicitly as in Eq. (13), since they are equal to V 5 and V 6 , respectively. One may find the expressions of these eleven potentials V k (k = 1, 2, · · · , 11) in the Appendix A. Following the same procedure to solve the bound state problem, one gets the possible binding energies and RMS radius. In Table V, we present the numerical results for the coupled TT systems. Probabilities (p i (%)) for each channel are also given. There are six bound state solutions with the quantum numbers I G (J PC ) = 0 + (0 ++ ), 0 − (1 +− ), 0 + (2 ++ ), 1 − (2 ++ ), 0 − (3 +− ) and 1 + (3 +− ). In Fig. 2, we illustrate the cutoff dependence for the binding energy E and the RMS radius r RMS .
The coupled system with J = 0 includes four channels . It is a good molecular candidate since the cutoff parameter Λ is around 1 GeV and consistent with the experience from the deuteron. The mass of the candidate is expected to be below the D 1D1 threshold (around 4840 MeV). Moreover, it is a typical S -wave bound state as the channel D 1D1 (| 1 S 0 ) dominates exclusively and the D * TABLE IV: Probabilities of each channel for the selected pure D 1D * 2 and D * 2D * 2 systems. The notation · · · means that no binding solution is found.

IV. SUMMARY
In this work, we have studied the interactions of the TTtype molecular systems within the framework of the one-pionexchange model. We have found the bound state solutions for some TT -type molecular systems. There exist possible TTtype molecular states, which are summarized in Table VI for readers' convenience.
In Table VI, we adopt the same criterion as that in Ref. [33] and show the existence possibility of these bound-state solutions with asterisks. A state with more ⋆ implies higher possibility to find this molecular state. Here, the four-star, threestar, two-star and one-star are applied to mark these states, which have bound state solutions with cutoff Λ < 1.5 GeV, 1.5 < Λ < 2.5 GeV, 2.5 < Λ < 3.5 GeV and 3.5 < Λ < 5 GeV, respectively. Thus, we suggest experiments to focus on these four-star states, which include the D 1D1 state with I G (J PC ) = 0 + (0 ++ ), the D 1D * 2 states with 0 ± (1) +± , the D *  V: The binding solutions (binding energy E and root-mean-square radius r RMS ) after the coupled channel effects are considered. p i (%) denotes the probability for the i-th channel. E, r RMS , and Λ are in units of MeV, fm, and GeV, respectively. The notation · · · means that no binding solution is found.
around the D 1D1 threshold are highly probable. In the present model we adopt a phenomenological form factor at each interaction vertex with the adjustable cutoff parameter Λ. The numerical results are sensitive to its value. Because of lack of available experimental data, we restrict its reasonable range from the experience with the deuteron. The numerical results are preliminary. In principle, the results should be stable with the variation of the cutoff. Probably the consideration of the other meson exchange forces may reduce the sensitivity once the coupling constants can be appropriately determined. Hopefully one can improve the model and make more reliable predictions in the future.
Searching for the exotic multiquark states continues to be a very interesting issue of hadron physics. The present predictions of the hidden-charm molecular states, which are composed of anti-charmed and charmed meson in the T doublet, provide useful information for future experimental exploration of them. The present work is only the starting point for the study of the TT -type molecular systems, which need to be investigated further by other approaches.
Recent experimental observation of the two hidden-charm pentaquarks P c (4380) and P c (4450) [3] enhances our confidence in the existence of the multiquark states. With experimental progresses, especially from LHCb and forthcoming BelleII, we expect that more candidates of the multiquark states will be announced. This field is full of challenges and opportunities for both theorists and experimentalists.