and interactions and the charged charmonium-like state ††thanks: Supported by the Major State Basic Research Development Program in China (No. 2014CB845405), the National Natural Science Foundation of China (Grants No. 11275235)
The and interactions are studied in a one-boson-exchange model. Isovector bound state solutions with spin parity are found from the interaction, which may be related to the observed charged charmonium-like state . There is no bound state solution found from the interaction.
xotic state, charmed meson interaction, one-boson-exchange model, Bethe-Salpeter equation
4.40.Rt, 21.30.Fe, 11.10.St
A resonant structure near 4.43 GeV in the invariant mass distribution was first observed by the Belle Collaboration , and is the first evidence of the existence of charged charmonium-like states. The mass MeV and width MeV were extracted by using a Breit-Wigner resonance shape. A higher mass MeV and a larger width MeV were reported by the Belle Collaboration through a full amplitude analysis of decay and a spin parity of was favored over other hypotheses . Recently, the LHCb Collaboration released their new result on the decay, which confirmed the existence of the resonant structure with a mass MeV and a width MeV .
The was observed in the invariant mass spectrum, which suggests that it should be an exotic state beyond the conventional picture, which has a neutral charge. Many theoretical efforts have been made to understand the internal structure of the and a number of explanations have been offered. Since the carries charge, the hybrid interpretation is excluded . It is natural to explain the charge carrier as a multiquark system in which, as well as , there exist other light quarks. The first type of multiquark explanation is the excited tetraquark [5, 6, 7, 8, 9, 10] where four quarks are in a color singlet. Another type of multiquark explanation is a loosely bound state composed of two charmed mesons [11, 12], or charmed baryons. There also exist several nonresonant explanations, such as the threshold cusp effect  and a cusp in the channel .
The mass measured by the Belle Collaboration , MeV, is close to the threshold, so it has been popular to explain the as a -wave molecular state with in the one-boson-exchange (OBE) model [16, 17]. A calculation in the context of the QCD sum rule also favors the bound state explanation with spin-parity . The new Belle and LHCb results suggest the spin-parity of is , however. With such an assignment of spin parity, a new calculation by Barnes suggests that the is either a state dominated by long-range exchange, or a state with short-range components . It has also been suggested that the may be from -wave interaction because the mass is very close to the threshold .
In this paper, the and interactions will be studied by solving the Bethe-Salpeter equation combined with the one-boson-exchange model. The mass is close to the threshold of four configurations, , , , and . The large width of the , however, MeV , which means a very short lifetime, makes it difficult to bind it and the together to form a bound state with width about MeV. The configuration has also been related to the in the literature. However, its threshold is about 100 MeV higher than the mass. In this work, the constituents will be treated as stable particles as in the OBE model [16, 17]. However, the physical widths of (2420) and (2600) are about 27 MeV and 93 MeV, respectively. Form factors will be introduced to compensate the self energy effects. The non-zero width will also introduce the three-body effect, which is not included in the current work considering that the thresholds of the three-body channels, such as , and , are much lower than the mass of the . It is also the reason why the configuration (2500) is excluded. Since only loosely bound states are considered, only two configurations, and , will be included in the current calculation.
The paper is organized as follows. In the next section a theoretical frame will be developed to study the and interactions (we omit the numbers for the masses, and respectively, here and hereafter) by solving the Bethe-Salpeter equation. In Section 3, the potential is derived with the help of effective Lagrangians from the heavy quark effective theory. The numerical results are given in Section 4. A summary is given in the last section.
2 Bethe-Salpeter equation for vertices
The Bethe-Salpeter equation is a powerful tool to study bound state problems such as the deuteron . A Bethe-Salpeter formalism was developed and applied to study the and its decay pattern [23, 24], the as system  and the as system . In Refs. [27, 28, 29], the system was also studied by solving the Bethe-Salpeter equation with boson exchange mechanism to explore the possible relationship between the recently observed / and the interaction. We start from the Bethe-Salpeter equation for vertex ,
where and are the potential kernel and the propagator for the two constituents of the system. The vertex function of the system with two configurations can be written as
where and are the vertex functions after separating out the flavor parts and . In this paper SU(2) symmetry is considered, so the same vertex function is used for both configurations.
The explicit flavor structures for isovectors () or isoscalars () are 
where corresponds to -parity . The flavor structure for configuration is analogous to that of the configuration.
The vertex function is rewritten as
with or 2 for configuration or , and stands for the different components in a configuration. is the factor for in Eq. (3). After multiplying on both sides, the Bethe-Salpeter equation becomes
The above equation is a coupled-channel equation for the two channels and involved.
The Bethe-Salpeter equation is a 4-dimensional integral equation. It is popular to reduce it to a 3-dimensional equation by quasipotential approximation, and in principle there exist infinite choices to make the quasipotential approximation. As in Ref. , we adopt the covariant spectator theory [30, 31] to make the 3-dimensional reduction. With the help of the onshellness of the heavier constituent 2, , the numerator of the propagator with being the polarization vector with helicity . Different from Ref. , where the off-shell constituent is a pseudoscalar particle , constituent 1 here is a vector meson . So we will make an approximation with polarization on shell. Such an approximation will introduce an uncertainty of about several percent in the numerator of the propagator, which will be further smeared by the introduction of form factors which will be given in the next section. Now, the equation for the vertex is of a form
Written down in the center of mass frame where , the propagator is
where , with .
The integral equation can be written explicitly as
where the reduced potential kernel
with a factor as The normalized wave function can be related to the vertex as with the normalization factor
A partial wave expansion can reduce the 3-dimensional integral equation to a one-dimensional equation,
where is the number of wave functions with a certain spin-parity.
3 Lagrangians and potential
For a loosely bound system, long-range interaction through the exchange should be more important than short-range interaction through heavier mesons. Moreover, in the isovector sector the isospin factors are and for and mesons, respectively . The cancelation between the contributions from these two mesons introduces further suppression of the short-range interaction. Hence, the heavier mesons, and , are not considered in this paper. The exchange which mediates the medium range interaction is included as in Ref. . We will find that the exchange is negligible compared with exchange.
The effective Lagrangians describing the interaction between the light pseudoscalar meson and heavy flavor mesons are constructed with the help of the chiral symmetry and heavy quark symmetry [32, 33],
which corresponds to and . The coupling constant can be extracted from the experimental width with a value . Falk and Luke obtained an approximate relation in quark model . With the available experimental information, Casalbuoni and coworkers extracted GeV . The coupling constant for decaying into and can be extracted from the decay widths obtained in quark model as MeV and MeV . The values are and GeV. The relative phases between the Lagrangians are not fixed, which will be discussed later.
The exchange which mediates the medium range interaction is included as in Ref. . The Lagrangians for the scalar meson read,
The coupling constant with .
With the above Lagrangians, we can obtain the potential for direct and cross diagrams,
where is the initial (final) momentum for constituent 1 or 2. The flavor factor is listed in Table 3.\tabcaption
The flavor factors and for direct and cross diagrams and different exchange mesons.
The form factor is introduced to compensate the off-shell effect of heavy mesons, and is also required by the convergence of the equation. It is also convenient to interpret the form factors as self-energies, which is important in this work due to the large decay width of the heavier constituent, . In this work, we adopt
Here is adopted to make the equation convergent. We will present the results with , that is, an exponential type of form factor , also to show the sensitivity of results to . In the propagator of the exchange meson we make a replacement to remove the singularities as in Ref. . The form factor for the light meson is chosen as a monopole type . The cut-off can be related to the radius of the hadron , which is about fm for a meson. The cut-off is about GeV for exponential type or GeV for monopole type. Such an estimation is very rough, so in this work we choose the cut-off as a free parameter from 0.8-2 GeV.
4 Numerical results
To search for the bound state from the interactions, the integral equation will be solved following the procedure in Ref. . After discretizing and by Gaussian quadrature, the recursion method in Refs.  is adopted to solve the nonlinear spectral problem. The numerical results are presented in Fig. 4. To show the sensitivity of the results to parameter in the form factor in Eq. (20), the results with are also presented as solid bands. The results suggest the binding energies are not sensitive to . In this work, all quantum number will be considered in the range of cut-offs GeV.
The binding energies for the system (patterns (a) and (c)) and system (patterns (b) and (d)) with the variation of cut-off . The lines are for the results with in form factor in Eq. (20) and the bands for results with .
In Fig. 4(b) and (d), the coupled-channel results with both configurations, , are presented, and are almost the same as these with the configuration only (Fig. 4(a) and Fig. 4(c)), which suggests that the interaction is much weaker than the interaction and transitions between and are negligible. The wave system carries spin-parity which is consistent with the new experimental results and the threshold is very close to the mass measured in the new LHCb experiment . However, in our calculation, no bound state solution is found from the interaction with a coupling constant GeV. In this work, the coupling constant is determined from the decay width predicted in the quark model . So, we increase the value of to check if the results are sensitive to , and find that even with there is no bound state produced from the interaction.
Different from Ref. , the exchange is dominant in the interaction in our model, and the effect of exchange is negligible. In the exchange, the contributions from diagram, , the cross diagram, is much more important than the contribution from the direct diagram . Hence, the contribution from the cross diagram of the exchange is dominant in the coupled interaction. Since diagram is composed of two vertices, the phase of the Lagrangian will be canceled. Hence, its dominance guarantees that the results are not sensitive to the relative phases of the Lagrangians, which are not well fixed.
There exists a bound solution with quantum number with cut-off about 1.8 GeV (see Fig. 4(a) and Fig. 4(b)). Such an wave molecular state has been related to the with the assumption that it carries spin parity . However, the new experimental results favor quantum number , which corresponds to a wave bound state. In the isovector sector, only two bound states are produced from the interaction. One of them has quantum number which is consistent with the experimental observed quantum number of the , .
For the coupled system, the cross diagram contribution from the exchange for channel is dominant. So the results are only sensitive to the square of coupling constant, , for . The value GeV in Ref.  is extracted from the old experimental data, which corresponds to decay width MeV . Compared with the new suggested value in the PDG, MeV , the largest possible value of is about 1 GeV. It is of interest to check the variation of results, especially for bound states with the quantum numbers, with the variation of coupling constant . The results are presented in Fig. 4.
The binding energies for coupled system with the variation of cut-off . The lines are for the results with in form factor in Eq. (20) and the bands for results with .
With larger , bound states are generated from the interactions with smaller cut-offs. For example, with a coupling constant , the isovector bound states with and are generated with cut-offs about 1.3 GeV and 1.5 GeV, respectively.
The new experimental results released by the LHCb Collaboration exclude the wave molecular state interpretation with quantum number for the . In this paper we discuss the possibility to interpret the as or molecular state with quantum number .
Isovector bound state solutions with spin-parity are found from the interaction, which may be related to the charged charmonium-like state . Different from the Belle experiment , the new observed mass of is above the threshold. However, considering the current large uncertainties and broad width, further more precise measurements are expected. The is still a candidate for the molecular state. On the theoretical side, it is interesting to consider the possibility of interpreting the as a resonance from the interaction , which can provide a mass above the threshold and is still consistent with the conclusion in this work.
There is no bound state solution found from the interaction. A calculation with the coupled interaction is also performed, and it is found that the results are almost the same as those obtained from the configuration only.
The current work is performed with the assumption that only channels with thresholds close to the mass of the are important. A more comprehensive study with more coupled channels and more sophisticated treatment of the non-zero width of the excited meson will be helpful to further understand the internal structure of the and the molecular states with two excited mesons.
In this work many molecular states are found from the interaction, but only one of them can be related to the observed . This is not surprising because those states are not obtained with the same cut-off, which should be the same for a given interaction. Hence, the states obtained in this work do not exist simultaneously. Besides, the effect of some molecular state predicted states may be too small to be observed in current experiments. Further more precise experiments are expected to check their existence.
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