Studying the $\bar{D}_1K$ molecule in the Bethe-Salpeter equation approach

We interpret the $X_1(2900)$ as an $S$-wave $\bar{D}_1K$ molecular state in the Bethe-Salpeter equation approach with the ladder and instantaneous approximations for the kernel. By solving the Bethe-Salpeter equation numerically with the kernel containing one-particle-exchange diagrams and introducing three different form factors (monopole, dipole, and exponential form factors) in the verties, we find the bound state exists. We also study the decay width of the decay $X_1(2900)$ to $D^-K^+$.

In the past decades, a growing number of good candidates of exotic states have been observed, with lots of them containing cc or bb quarks [6,7]. Thus, the discovery of X 0 (2900) and X 1 (2900) have drawn a lot of attentions. The X 0 (2900) can be interpreted as a csūd compact tetraquark both in the universal quark mass picture [8] and in the quark model [9], but not within an extended relativized quark model [10]. In Ref. [11], the authors used the two-body chromomagnetic interactions to find that the X 0 (2900) can be interpreted as a radial excited tetraquark and the X 1 (2900) can be an orbitally excited tetraquark. It was also suggested the X 0 (2900) can be interpreted as the S-wave D * − K * + molecule state and the X 1 (2900) as the P -wavecsud compact tetraquark state [12]. In the chiral constituent quark model, it was shown show that no candidate of X(2900) was founded in the IJ P = 00 + and IJ P = 01 + csqq system, while there were two states in the P -wave excited csqq system, D 1K and D JK , which could be candidates of X(2900) [13]. From the QCD Sum Rules, the X 0 (2900) and X 1 (2900) were studied in molecular and diquark-antidiquark tetraquark pictures, respectively, and the results for masses are in good agreement with the observed masses in the experiment [14]. Investigations bases the one-boson exchange model [15] and the phenomenological Lagrangian approach [16], showed that the X 0 (2900) can be a D * K * molecule, but the X 1 (2900) can not. In Ref. [17], the decay width for X 0 (2900) →DK process was found to be in agreement with the experimental data with the S-waveD * K * scenario for X 0 (2900) in the effective lagrangian approach. The study in Ref. [18] showed that the X 1 (2900) as aD 1 K is disfavored within the meson exchange model. In Ref. [19], in the quasipotential Bethe-Salpeter (BS) equation approach, the authors supported the assignment of X 0 (2900) as a D * K * molecular state and X 1 (2900) as aD 1 K virtual state.
Considering the mass of X 1 (2900) is about 10 MeV below theD 1 K threshold, it is natural to explore the existence of the S-waveD 1 K molecule. In this work, we will focus on the X 1 (2900) in the BS equation approach, investigating whether the X 1 (2900) can be an S-waveD 1 K bound state. We will also to study the decay width of X 1 (2900) → D − K + .
In the rest of the manuscript we will proceed as follows. In Sec. II, we will establish the BS equation for the bound state of an axial-vector meson (D 1 ) and a pseudoscalar meson (K). Then we will discuss the interaction kernel of the BS equation and calculate numerical results of the Lorentz scalar functions in the normalized BS wave function in Sec. III. In Sec. IV, the decay width of the X 1 (2900) to D − K + final state will be calculated. In Sec. V, we will present a summary of our results.

II. THE BS FORMALISM FORD 1 K SYSTEM
For the molecule composed of an axial-vector meson (D 1 ) and a pseudoscalar meson (K), its BS wave function is defined as whereD 1 (x 1 ) and K(x 2 ) are the field operators of the axial-vector mesonD 1 and the pseudoscalar meson K at space coordinates x 1 and x 2 , respectively, P = M v is the total momentum of bound state and v is its velocity. Let mD 1 and m K be the masses of theD 1 and K mesons, respectively, p be the relative momentum of the two constituents, and define λ 1 =mD 1 /(mD 1 + m K ), λ 2 =m K /(mD 1 + m K ). The BS wave function in momentum space is defined as where X = λ 1 x 1 + λ 2 x 2 is the coordinate of the center of mass and x = x 1 − x 2 . The momentum of thē D 1 meson is p 1 = λ 1 P + p and that of the K meson is p 2 = λ 2 P − p.
It can be shown that the BS wave function of theD 1 K system satisfies the following BS equation: where S µν D 1 (p 1 ) and S K (p 2 ) are the propagators ofD 1 and K mesons, respectively, and K νλ (P, p, q) is the kernel, which is defined as the sum of all the two particle irreducible diagrams with respect to D 1 and K mesons. For convenience, in the following we use the variables p l (= p · v) and p t (= p − p l v) as the longitudinal and transverse projections of the relative momentum (p) along the bound state momentum (P ), respectively. Then, in the heavy quark limit the propagator of D 1 is and the propagator of the K meson is In the BS equation approach, the interaction betweenD 1 and K mesons arises from the light vectormeson (ρ and ω) exchange. Based on the heavy quark symmetry and the chiral symmetry, the relevant effective Lagrangian used in this work is shown in the following [20]: where a and b represent the light flavor quark (u and d), V µ is a 3 × 3 Hermitian matrix containing ρ, ω, K * , and φ: The coupling constants involved in Eq. (6) are related to each other as follows [20]: where the parameters β 2 g V and λ 2 g V are given by 2g ρN N and 3 10m N (g ρN N + f ρN N ), respectively, with g 2 ρN N /4π = 0.84 and f ρN N /g ρN N = 6.10 [21]. The parameter g V = 5.8 is determined by the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin relations [20]. Then, at the tree level, in the t-channel the kernel for the BS equation of theD 1 K system in the lader FIG. 1: One-particle exchange diagrams induced by vector mesons ρ and ω.
approximation includes the following term (see Fig. 1): where m V (V = ρ, ω) represents the mass of the exchanged light vector meson ρ or ω , c I is the isospin coefficient: c 0 = 3, 1 and c 1 = −1, 1 for ρ and ω, respectively, ∆ µν represents the propagator for the light vector meson.
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 forms: in the monopole (M ), dipole (D), and exponential (E) models, respectively, where Λ, m and k represent the cutoff parameter, the mass of the exchanged meson and the momentum of the exchanged meson, respectively. The value of Λ is near 1 GeV which is the typical chiral symmetry breaking scale.
In general, for an axial-vector meson (D 1 ) and a pseudoscalar meson (K) bound state, the BS wave function χ µ P (p) has the following form: where f i (p) (i = 0, 1, 2, 3) are Lorentz-scalar functions and ǫ µ represents the polarization vector of the bound state. After considering the constraints imposed by parity and Lorentz transformations, it is easy to prove that χ µ P (p) can be simplified as where the scalar function f (p) contains all the dynamics.
In the following derivation of the BS equation, we will apply the instantaneous approximation, in which the energy exchanged between the constituent particles of the binding system is neglected. In our calculation we choose the absolute value of the binding energy E b of theD 1 K system (which is defined where k t = p t −q t is the momentum of the exchanged meson in the covariant instantaneous approximation. In Eq. (13) there are poles in the plane of p l at −λ 1 M + ω 1 − iǫ, λ 2 M + ω 2 − iǫ and λ 2 M − ω 2 + iǫ.
By choosing the appropriate contour, we integrate over p l on both sides of Eq. (13) in the rest frame of the bound state, then we obtain the following equation: wheref (p t ) ≡ dp l f (p). To find out the possible molecular bound states, one only needs to solve the homogeneous BS equation. One numerical solution of the homogeneous BS equation corresponds to a possible bound state. The integration region in each integral is discretized into n pieces, with n being sufficiently large. In this way, the integral equation is converted into an n × n nmatrix equation, and the scalar wave function will now be regarded as an n-dimensional vector. Then, the integral equation can be illustrated as is an n-dimensional vector, and A (n×n) (p t , q t ) is an n×n matrix, which corresponds to the matrix labeled by p t and q t in each integral equation. Generally, p t (q t ) varies from 0 to +∞. Here, p t (q t ) is transformed into a new variable t that varies from −1 to 1 based on the Gaussian integration method, where µ is a parameter introduced to avoid divergence in numerical calculations, w and y are parameters  Table I.

III. THE NORMALIZATION CONDITION OF THE BS WAVE FUNCTION
To find out whether the bound state of theD 1 K system exists or not, 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 µ P (p).
In the heavy quark limit, the normalization of the BS wave function of theD 1 K system can be written as [23] i d 4 pd 4 q (2π) 8χ µ P (p) where I P µν (p, q) = (2π) 4 δ 4 (p − q)S −1 µν (p 1 )S −1 (p 2 ). In the rest frame, the normalization condition can be written in the following form: From Eqs. (13) and (14), we obtain Then, one can recast the normalization condition for the BS wave function into the form The wave function obtained in the previous section (which is calculated numerically from Eq.(14)) can be normalized by Eq. (19). IV. THE DECAY OF X 1 (2900) → D − K + Besides investigating whether the bound state of theD 1 K system can be X 1 (2900) or not, we can also study the decay of the X 1 (2900) as the S-waveD 1 K bound state. The X 1 (2900) can decay to D − K +  6)) as the following [20]: where the coupling constants are given as [20], with the two parameters ζ 1 and µ 1 being involved in the coupling constants, about which the information is very scarce leading them undetermined. However, in the heavy quark limit, we can roughly assume that the coupling constants g DD 1 V and g ′ , respectively. The parameters µ = 0.1 GeV −1 and ζ = 0.1 are taken in Ref. [24].
According to the above interactions, we can write down the amplitude for the decay X 1 (2900) → FIG. 4: The diagrams contributing to the X 1 (2900) → D − K + decay process induced by ρ and ω.
D − K + induced by light vector meson (ρ and ω) exchanges as shown in Fig. 4, as the following: In the rest frame, we define p ′ 1 = (E ′ 1 , −p ′ 1 ) and p ′ 2 = (E ′ 2 , p ′ 2 ) to be the momenta of D and K, respectively. According to the kinematics of the two-body decay of the initial state in the rest frame, one and dΓ = 1 where |p ′ 1 | and |p ′ 2 | are the norm of the 3-momentum of the particles in the final states in the rest frame of the initial bound state and M is the Lorentz-invariant decay amplitude of the process. and 28.14 MeV with monopole form factor, 18.13 MeV with dipole form factor, 12.78 MeV with exponential form factor.
(25) From our calculation results, we can see that different form factors have a great influence on the decay width, and different cutoff Λ for the same form factor also have a great influence on the decay width.

V. SUMMARY
In this paper, we studied the X 1 (2900) with the hadronic molecule interpretation by regarding it as a bound state ofD 1 and K mesons in the BS equation approach. In our model, we applied the ladder and instantaneous approximations to obtain the kernel containing one-particle-exchange diagrams and introduced three different form factors (the monopole form factor, the dipole form factor, and the exponential form factor) at the interaction vertices. From the calculating results we find that there exist bound states of theD 1 K system. The binding energy depends on the value of the cutoff Λ. For the I = 0D 1 K system, we find the cutoff regions in which the solutions (with the binding energy E b ∈ (-5, with the monopole, dipole, and exponential form factors, respectively.) to calculate the decay widths of X 1 (2900) → D − K + induced by ρ and ω exchanges. We predict the decay widths are 70.73, 98.75, and 60.38 MeV and 28.14, 18.13, and 12.78 MeV for X 1 (2900) as I = 0 and I = 1D 1 K molecules with the corresponding cutoff in the decay process, respectively. From our study, the X 1 (2900) is suitable as I = 0 D 1 K molecular state. There are two uncertain factors in the calculation of the decay width, one is that the parameters ξ 1 and µ 1 have not been determined since the information about them is very scarce, the other is that we can not give the definite value of the cutoff Λ.