X(3872) as a molecular $D\bar{D}^*$ state in the Bethe-Salpeter equation approach

We discuss the possibility that the X(3872) can be a $D\bar{D}^*$ molecular bound state in the Bethe-Salpeter equation approach in the ladder and instantaneous approximations. We show that the $D\bar{D}^*$ bound state with quantum numbers $J^{PC}=1^{++}$ exists. We also calculate the decay width of $X(3872) \rightarrow \gamma J/\psi$ channel and compare our result with those from previous calculations.

In the theory aspect, the nature of the X(3872) is still a puzzle, and many theoretical models were proposed to explain the X(3872) state. The X(3872) is analyzed as a D 0D * 0 /D 0 D * 0 bound state [8][9][10][11][12][13][14][15][16][17], a tetraquark state [18][19][20][21][22][23], a hybrid charmonium (ccg gluonic hadrons) [24,25], and a charmonium (cc) [26][27][28], it has also been considered as a mixture of a charmonium with a D 0D * 0 /D 0 D * 0 component [29][30][31]. Among the above models, the molecular state provides a plausible explanation since the X(3872) can be identified as a weakly bound hadronic molecule which constituents are D and D * . The reason for this natural interpretation is that the mass of X(3872) is very close to the D 0D * 0 threshold and hence is in analogy to the deuteron-a weakly bound state of the proton and the neutron.
The radiative decay of the X(3872) into γJ/ψ is sensitive to its internal structure [26,32], and this decay channel has been studied in lots of literatures. The first observation of the X(3872) → γJ/ψ decay mode was reported by Belle collaboration [33]. Later on, this decay mode was confirmed by the BABAR collaboration [34] and again observed by Belle collaboration [35]. This decay mode received some early attention and was studied in Refs. [26,32,[36][37][38], assuming a charmonium state, a molecular state, or a mixture of a molecular state with a charmonium state.
The Bethe-Salpeter (BS) equation is a formally exact equation to describe the relativistic bound state [39][40][41] and has been applied to many theoretical studies concerning heavy mesons and heavy baryons [42][43][44][45][46][47][48][49][50]. In this paper, we will work in the BS equation approach which can automatically include relativistic corrections comparing with the potential model which was applied in Ref. [51] to investigate the possible states of KK, DK, and BK in the framework of the nonrelativistic Schrödinger equation with the potential between pseudoscalar mesons being derived from the relevant Lagrangian. We will try to investigate the possibility of X(3872) as the DD * molecular state with quantum numbers J P C = 1 ++ .
We will also study the decay of X(3872) to γJ/ψ in this picture.
The paper is organized as follows. In Sec. II, we establish the BS equation for the bound state of a vector meson and a pseudoscalar one. Then we discuss the interaction kernel. In Sec. III, we discuss the normalization condition of the BS wave function and obtain the numerical results of the BS wave 2 function. In Sec. IV, the decay of the DD * bound state to γJ/ψ final state is discussed and we give numerical results. Finally, Sec. V is devoted to summary and conclusion.

II. THE BETHE-SALPETER FORMALISM FOR DD * SYSTEM
In this section, we will review the general formalism of the BS equation and derive the BS equation for the system D and D * mesons. We will also derive the normalization condition for the BS wave function.
Let us start by defining the BS wave function for the bound state |P of a vector and a pseudoscalar mesons as the following: where D * α (x 1 ) and D(x 2 ) are the field operators of the vector meson D * and the pseudoscalar meson D at space coordinates x 1 and x 2 , respectively, P denotes the total momentum of the bound state with mass M and velocity v, and the relative coordinate x and the center-of-mass coordinate X are defined by or inversely, where η i = m i /(m 1 + m 2 ), m i (i = 1, 2) is the mass of the i-th constituent particle. In momentum space, the BS wave function can be defined as where p represents the relative momentum of the two constituents and p = η 2 p 1 − η 1 p 2 (or p 1 = η 1 P + p, The BS equation for the bound state of X(3872) can be written in the following form: where S αλ (p 1 ) and S(p 2 ) are the propagators of D * and D, respectively, and K λτ (P, p, q) is the kernel which contains two-particle-irreducible diagrams. For convenience, we define p l (= p·v) and p µ t (= p µ −p l v µ ) to be the longitudinal and transverse projections of the relative momentum (p) along the bound state momentum (P ). Then the D * propagator has the form and the propagator of D meson has the form 3 where ω 1(2) = m 2 1(2) − p 2 t . In general, for DD * system, χ α P (p) can be written as where ǫ β (P ) represents the polarization vector of the bound state and f i (i = 1, 2, 3, 4) are Lorentz-scalar functions. With the constraints imposed by parity and Lorentz transformations, it is easily to prove that χ α P (p) can be simplified as where the function φ P (p) contains all the dynamics and is a Lorentz-scalar function of p.
As discussed in the introduction, we will study the X(3872) as an S-wave bound state of the DD * system. We use the field doublets (D * 0 ,D * 0 ), (D * + , D * − ), (D 0 ,D 0 ), and (D + , D − ), which correspond to the following expansions: Since the isospin quantum number of X(3872) is zero, if we assume that it is composed of DD * , the flavor wave function of the X(3872) can be represented as in Refs. [54,55] Let us now project the bound states on the field operators D * 1 , D * 2 , D 3 and D 4 . From Eq. (10) we have where C ij (I,I 3 ) (i, j =1, 2, 3, 4) is the isospin coefficient. The coefficients C ij (I,I 3 ) for the isoscalar state are The kernel of the BS equation for the X(3872) can be derived from the meson exchange Feynman diagrams for the DD * system at the tree level which are shown in Fig. 1 and Fig. 2. Based on the chiral symmetry [56,63], the Lagrangians for the interactions among D(D * ) mesons and light pseudoscalar, scalar or vector mesons are where a, b denote the light quark flavour indices, The octet pseudoscalar P and the nonet vector V meson matrices are defined as respectively, and the coupling constants are given as From the above observations, at the tree level, in the t-channel we have the following kernel for the BS equation in the so-called lader approximation (see Figs. 1 and 2 for direct and crossed channels, where m σ , m P and m V represent the masses of the exchanged σ, pseudoscalar light meson and vector light meson, respectively. ∆ µν represents the propagator for a vector meson and ∆ represents pseudoscalar or scalar meson propagator, and they have the following forms: In order to describe the phenomena in the real world, we should include a form factor at each interacting vertex of hadrons to include the finite-size effects of these hadrons. For the meson-exchange case, the form factor is assumed to take the following form: where Λ, m and k represent the cutoff parameter, mass of the exchanged meson and momentum of the exchanged meson, respectively. 6 From Eqs. (5-7) and Eqs. (18)(19)(20)(21), we have where F mσ (k), F mη (k) and so on represent the form factors for different exchanged mesons.
Defineφ P (|p t |) = dp l 2π φ P (p l , p t ),φ P (|p t |) depends only on the norm of p t , |p t |. Therefore, after completing the azimuthal integration, the above BS equation becomes a one dimensional integral equation, which readsφ the expressions for V 1 (|p t |, |q t |) and V 2 (|p t |, |q t |) are given in Appendix A.

III. SOLUTION OF THE BS EQUATION FOR X(3872)
In this part, we will solve the BS equation numerically. To find out the bound state of the DD * system, one only needs to solve the homogeneous BS equation. However, when we want to calculate physical quantities such as the decay width we have to face the problem of the normalization of the BS wave function. In the following we will discuss the normalization of the BS wave function χ P (p).
Following Ref. [41] one can write down the normalization condition as where have the following form: Inserting Eqs. (27) and (28) into Eq. (26), the normalization condition can be written in the following form: Substituting Eq. (24) into Eq. (29) and completing the azimuthal integration we have It can be seen from Eq. (24) that there is one parameter in our model, the cutoff Λ, which contains the information about the non-point interaction due to the structure of hadrons at the interaction vertices.
Although the value of Λ cannot be exactly determined and depends on the specific process, it should be typically the scale of low-energy physics, which is about 1 GeV. In Ref. radiative decay X(3872) → γJ/ψ is [65] : where F µν = ∂ µ A ν − ∂ ν A µ and M µν = ∂ µ M ν − ∂ ν M µ is the stress tensor of the vector mesons with M = D * 0 , J ψ (in the Lagrangian we denote J/ψ by J ψ ), ǫ µναβ is levi-civita symbol. In the present calculation we will use the following values of the coupling constants [65]: The differential decay width of the bound state can be written as where |q| is the norm of the three-momentum of the particles in the final state in the rest of the initial bound state. M is the Lorentz-invariant decay amplitude of the process.
According to the above interactions, the decay X(3872) → γJ/ψ induced by D * exchange is shown in Fig. 4(a). We can write down the amplitude as where q 1 (q 2 ) is the momentum of γ(J ψ ) and q = η 2 q 1 − η 1 q 2 which is not the relative momentum of particles in the final state (note that η 1 and η 2 are defined as η i = m i /(m 1 + m 2 ), and m 1 and m 2 are the masses of the component particles of the initial bound state but not the final states), ǫ ρ (P ), ǫ λ (q 1 ) and ǫ γ (q 2 ) are the polarization vectors of X(3872), γ and J ψ , respectively. Similarly, the diagram for X(3872) → γJ/ψ through exchanging the D meson is shown in Fig. 4(b).
One can write the amplitude as In general, we can write the amplitude in the form where G 1 and G 2 are Lorentz invariant form factors.
In the calculation we stay in the rest frame of the initial bound state and hence P = (M, 0). We The decay width of X(3872) → γJ/ψ has been studied by several groups with X(3872) in different structures. For comparison, these results are dispalyed in Table 1 together with ours. In the table, X(3872) are considered as a cc state [26,32,66], a molecule state [32,38,65,67], a mixture of a charmonium with a D 0D * 0 /D 0 D * 0 component [68,69], a tetraquark state (ccqq) [52], and a mixture of cc and ccqq state [70], respectively.

V. SUMMARY AND CONCLUSION
In this paper, in order to investigate the structure of the observed state X(3872) with the quantum numbers J P = 1 ++ , we use the BS equation which has been successfully applied in many theoretical studies concerning heavy mesons and baryons and automatically includes relativistic corrections. We work in the picture that X(3872) is an S-wave DD * molecular bound state because it is very close to the DD * threshold. We establish the BS equation for the system composed of a vector meson and a pseudoscalar meson. Then we derive the BS equation for the DD * system using the kernel which is 11 induced by σ, π, η, ρ and ω exchange diagrams. In our model, we have used the ladder approximation which can considerably simplify the formalism. In addition, based on the fact that the DD * system is very weekly bound, we have used the instantaneous approximation in the BS equation, in which the energy exchange between the constituent particles is neglected. Since the constituent particles and the exchanged particles in the DD * system are not pointlike, we introduce form factors including a cutoff Λ which reflects the effects of structure of these particles. Since Λ is controlled by nonperturbative QCD and cannot be determined at present, we let it vary in a reasonable range to find its values with which X(3872) can be a DD * molecular bound state.