Abstract
In this work, we study the molecule in the Bethe-Salpeter (BS) equation approach. With the kernel containing one-particle-exchange diagrams and introducing two different form factors (monopole form factor and dipole form factor) in the vertex, we solve the BS equation numerically in the covariant instantaneous approximation. We investigate the isoscalar and isovector systems, and we find that cannot be a molecule.
1. Introduction
The physics of exotic multiquark states has been a subject of intense interest in recent years. One reason for this is that the experimental data are being accumulated on charmonium-like states and pentaquark states (see the review papers [1–3] for details) and more and more experimental data will be found in near future.
In 2016, the D0 Collaboration announced a new enhancement structure with the statistical significance of 5.1σ in the invariant mass spectrum, which has the mass (syst) MeV and width (syst) MeV [4]. The observed channel indicates that the isospin of the is 1 and if it decays into via a S-wave, the quantum numbers of the should be . Subsequent analyses by the LHCb [5], CMS [6], and ATLAS [7] Collaborations have not found evidence for the in proton-proton interactions at = 7 and 8 TeV. The CDF Collaboration has recently reported no evidence for in proton-antiproton collisions at = 1.96 TeV [8] with different kinematic. Recently, the D0 Collaboration reported a further evidence about this state in the decay of with a significance of 6.7σ [9] which is consistent with their previous measurement in the hadronic decay mode [4]. Therefore, the experimental status of the resonance remains unclear and controversial.
No matter whether the structure exists or not, it has been attracting a lot of attention from both experimental and theoretical sides. Many theoretical groups have studied possible ways to explain as a tetraquark state, a molecular state, etc., within various models, and they obtained different results. In Refs. [10–18], the authors based on QCD sum rules obtained the mass and/or decay width which are in agreement with the experimental data. In Refs. [19, 20], the authors showed that or could not be assigned to be an or molecular state. is also disfavored as a -wave coupled-channel scattering molecule involving the states , , , and in Ref. [21]. The authors of Ref. [22] pointed out that the and interactions were weak and could not be a -wave and molecular state. Based on the lattice QCD, there is no candidate for with [23]. The authors found that threshold, cusp, molecular, and tetraquark models were all unfavoured for [24]. as molecule and diquark-diquark model are considered in Ref. [25] using QCD two-point and light-cone sum rules, and their results strengthen the diquark-antidiquark picture for the state rather than a meson molecule structure. But the authors of Ref. [26] found that the signal can be reproduced by using coupled channel analysis, if the corresponding cutoff value was larger than a natural value GeV. In Ref. [27], the authors demonstrated that could be a kinematic reflection and explained the absence of in LHCb and CMS Collaborations. Based on the quark model, could exist as a mixture of a tetraquark and hadronic molecule [28].
By this chance, we will systematically study the molecular state in the BS equation approach. We investigate the -wave systems with both isospins being considered. We will vary in a much wider range and search for all the possible solutions. In this process, we naturally check whether can exist as -wave molecular state, or not.
The remainder of this paper is organized as follows. In Section 2, we discuss the BS equation for two pseudoscalar mesons and establish the one-dimensional BS function for this system. The numerical results of the systems are presented in Section 3. In the last section, we give a summary and some discussions.
2. The Bethe-Salpeter Formalism for System
In this section, we will review the general formalism of the BS equation and establish the BS equation for the system of two pseudoscalar mesons. Let us start by defining the BS wave function for the bound state as the following:where and are the field operators of the and mesons at space coordinates and , respectively, and denotes the total momentum of the bound state with mass and velocity . The BS wave function in momentum space is defined aswhere represents the relative momentum of the two constituents and (or , ). The relative coordinate and the center-of-mass coordinate are defined byor inversely, where and , and and are the masses of and mesons.
It can be shown that the BS wave function of bound state satisfies the following BS equation [29]:where and are the propagators of and , respectively, and is the kernel, which is defined as the sum of all the two particle irreducible diagrams with respect to and mesons. For convenience, in the following we use the variables and to be the longitudinal and transverse projections of the relative momentum () along the bound state momentum (). Then, the propagator of mesons can be expressed asand the propagator of the iswhere (we have defined ).
As discussed in the introduction, we will study the -wave bound state of system. The field doublets , , , and have the following expansions in momentum space:where is the energy of the particle.
The isospin of can be 0 or 1 for system, and the flavor wave function for the isoscalar bound state can be written as and the flavor wave functions of the isovector states for system are
Let us now project the bound states on the field operators , , , and . Then we havewhere is the common BS wave function for the bound state with isospin which depends only on but not of the state . The isospin coefficients for the isoscalar state areand for the isovector states we have
Now considering the kernel, Eq. (5) can be written down schematically, Then, from Eq. (12), for the isoscalar case, we have (take as an example) Similarly, for the isovector case, taking the component as an example, we have
In the BS equation approach, the interaction between and mesons can be due to the light vector-meson ( and ) exchanges. The corresponding effective Lagrangians describing the couplings of [30, 31] and [32, 33] arewhere the nonet vector meson matrix reads asIn addition, the coupling constants involved in Eq. (17) are taken as with , , while the coupling constants satisfy the relations in the limit, and [32].
From the above observations, at the tree level, in the -channel the kernel for the BS equation of the interaction between and in the so-called ladder approximation is taken to have the following form:where represent the masses of the exchanged light vector mesons and , and is the isospin coefficient: and for , , and represents the propagator for vector meson.
In order to manipulate the off shell effect of the exchanged mesons and and finite size effect of the interacting hadrons, we introduce a form factor at each vertex. Generally, the form factor has the monopole form and dipole form as shown in Ref. [34]:where , , and represent the cutoff parameter, mass of the exchanged meson, and momentum of the exchanged meson, respectively. These two kinds of form factors are normalized at the on shell momentum of . On the other hand, if were taken to be infinitely large (), the form factors, which can be expressed as the overlap integral of the wave functions of the hadrons at the vertex, would approach zero.
For the system, substituting Eqs. (6), (7), and (19) and aforementioned form factors Eqs. (20) and (21) into Eq. (5) and using the so-called covariant instantaneous approximation [35], (which ensures that the BS equation is still covariant after this approximation). Then one obtains the expression
In Eq. (22) there are poles in , , , and . By choosing the appropriate contour, we integrate over on both sides of Eq. (22) in the rest frame, and we will obtain the following equation:where .
3. Numerical Results
In this part, we will solve the BS equation numerically and study whether the S-wave bound state exists or not. It can be seen from Eq. (23) that there is only one free parameter in our model, the cutoff , it cannot be uniquely determined, and various forms and cutoff are chosen phenomenologically. It contains the information about the nonpoint interaction due to the structures of hadrons. The value of is near 1 GeV which is the typical scale of nonperturbative QCD interaction. In this work, we shall treat the cutoff in the form factors as a parameter varying in a much wider range 0.8-4.8 GeV, in which we will try to search for all the possible solutions of the bound states. For each pair of trial values of the cutoff and the binding energy of the system (which is defined as ), we will obtain all the eigenvalues of this eigenvalue equation. The eigenvalue closest to 1.0 for a pair of and will be selected out and called the trial eigenvalue. Fixing a value of the cutoff and varying the binding energy (from 0 to -220 MeV) we will obtain a series of the trial eigenvalues.
Since the BS wave function for the ground state is in fact rotationally invariant, depends only on . Generally, varies from 0 to + and would decrease to zero when . We replace by the variable, : where is a parameter introduced to avoid divergence in numerical calculations, and are parameters used in controlling the slope of wave functions and finding the proper solutions for these functions, and varies from -1 to 1. We then discretize Eq. (23) into pieces ( is large enough) through the Gauss quadrature rule. The BS wave function can be written as -dimension vectors, . The coupled integral equation becomes a matrix equation ( corresponding to the coefficients in Eq. (23)). Similar methods are also adopted in solving d Lippmann-Schwinger equation for [36–41] and [42].
In our calculation, we choose to work in the rest frame of the bound state in which . We take the averaged masses of the mesons from the PDG [43], , , , and 782.65 MeV. With the above preparation, we try to search for the all the possible solutions by solving the BS equation. The relations between and for the with are depicted in Figures 1 and 2, respectively.
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(b)
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4. Summary
Stimulated by , which is recently discovered by the D0 Collaboration, we carried out a study of the interaction of system with isospin =0, 1 in the Bethe-Salpeter equation approach. In order to solve the BS equation, we have used the ladder approximation and the instantaneous approximation. The value of is near 1 GeV which is the typical scale of nonperturbative QCD interaction. Thus, if strictly considering this criterion of the value of , we conclude that there does not exist isovector molecular state. And the cannot be the molecular state. The relations between and for the with are depicted in Figures 1 and 2, respectively.
Data Availability
The numerical data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
One of the authors (Z.-Y. Wang) thanks Dr. Xian-Wei Kang for a very careful reading of this manuscript. This work was supported by National Natural Science Foundation of China (Projects No. 11275025, No. 11775024, and No. 11605150) and K.C.Wong Magna Fund in Ningbo University.