The Strong Decays of X(3940) and X(4160)

The new mesons $X(3940)$ and $X(4160)$ have been found by Belle Collaboration in the processes $e^+e^-\to J/\psi D^{(*)}\bar D^{(*)}$. Considering $X(3940)$ and $X(4160)$ as $\eta_c(3S)$ and $\eta_c(4S)$ states, the two-body open charm OZI-allowed strong decay of $\eta_c(3S)$ and $\eta_c(4S)$ are studied by the improved Bethe-Salpeter method combine with the $^3P_0$ model. The strong decay width of $\eta_c(3S)$ is $\Gamma_{\eta_c(3S)}=(33.5^{+18.4}_{-15.3})$ MeV, which is closed to the result of $X(3940)$, therefore, $\eta_c(3S)$ is a good candidate of $X(3940)$. The strong decay width of $\eta_c(4S)$ is $\Gamma_{\eta_c(4S)}=(69.9^{+22.4}_{-21.1})$ MeV, considering the errors of the results, it's closed to the lower limit of $X(4160)$. But the ratio of the decay width $\frac{\Gamma(D\bar D^*)}{\Gamma (D^*\bar D^*)}$ of $\eta_c(4S)$ is larger than the experimental data of $X(4160)$. According to the above analysis, $\eta_c(4S)$ is not the candidate of $X(4160)$, and more investigations of $X(4160)$ is needed.


I. INTRODUCTION
In the past few years, many more new charmonium-like states, so-called XY Z states, have been observed by the Belle, BABAR and BESIII Collaborations [1]. The discovery of these states not only enriched the spectroscopy of charmonium-like states but also provided us an opportunity to research the properties of charmonium-like states. For example, the X(3940) state was observed from the inclusive process e + e − → J/ψX(3940) and had the decay mode X(3940) → D * D by the Belle Collaboration at a mass of (3943 ± 6 ± 6) MeV.
The mesons can be described by the B-S equation. Ref. [21] took the B-S equation to describe the light mesons π and K, then they calculated the mass and decay constant of π by the B-S amplitudes [22], they also studied the weak decays [23] and the strong decays [24] combine with the Dyson-Schwinger equation.
We will use the B-S equation to study the properties of heavy mesons. In Ref. [25], we had calculate the Spectrum of heavy quarkonia the improved Bethe-Salpeter(B-S) method, for the the charmonium state with the quantum numbers J P C = 0 −+ , the mass of 3 1 S 0 (η c (3S)) is M=3948.8 MeV which is closed to the mass of X(3940) with error, the mass of 4 1 S 0 (η c (4S)) was M=4224.6 MeV which was larger than the center mass of X(4160) about 70 MeV. In this paper, to check if the X(4160) is the charmonium η c (4S), we calculate the strong decay of η c (4S), but assign the mass of η c (4S) as 4156 MeV by varying the parameter V 0 in interaction potential, where in potential model the parameter V 0 is added to move the theoretical mass spectra parallel to match the experimental data.
Using the the improved B-S method, we calculated the weak decay of B c to η c (1S) and η c (2S) [26], and the weak decay of B c to η c (3S) and η c (4S) [27]. There is nobody to calculate B c to η c (4S) , but the results of B c to η c (1S), η c (2S), η c (3S) were close to the other theoretical results. We also studied the properties of some XY Z states, such as radiative E1 decay of X(3872) [9,10], two-body strong decay of Z(3930) which was χ c2 (2P ) state combine with the 3 P 0 model [11], the strong decay of X(3915) as χ c0 (2P ) state [28], and the strong decay of Υ [29]. All the theoretical results consist with experimental data or other theoretical results. Because the higher excited states have larger relativistic correction than the corresponding ground state, a relativistic model is needed in a careful study. The improved B-S method is a relativistic model that describe bound states with definite quantum number, the corresponding relativistic form of wavefunctions are solutions of the full Salpeter equations. So the improved B-S method is good method to describe the properties and decays of the radial high excited states, In this paper, we focus on the strong decays of X(3940) and X(4160) as radial high excited states η c (3S) and η c (4S) by the improved B-S method.
In our method, we study the natures of heavy mesons by the coupling of L + S for the quark and anti-quark in mesons. According to the L+S coupling, we show the wavefunctions of the heavy mesons in term of the quantum number J P (or J P C ) which are very good to describe the equal mass systems in heavy mesons. The quantum numbers J P C of η c (3S) and η c (4S) both are 0 −+ , the C parities are even which agree with the results of Ref. [4] and Ref. [5]. The corresponding Okubo-Zweig-Iizuka (OZI) [30][31][32] rule allowed two-body open charm strong decay modes are: 0 − → 0 − 1 − and 0 − → 1 − 1 − , while other strong decays in the final state are ruled out by the kinematic possible mass region. In order to calculate the two-body open charm strong decay, we adopt the 3 P 0 model which assumes that a quark-antiquark pairs is created with vacuum quantum numbers, J P C = 0 ++ [33][34][35]. The 3 P 0 model was proposed in Ref. [33], then Ref. [34] and Ref. [35] applied the 3 P 0 model to study the open-flavor strong decays of the light mesons. Now, People have extended this model to study the natures of heavy-light mesons [36,37] and heavy quarkonia [29,38,39].
In Ref. [11] and Ref. [29], we have calculated the OZI allowed two-body strong decays of charmonium and bottomonium in the 3 P 0 model with the relativistic B-S wavefunctions.
The results were good according with experimental data and the other theoretical results.
Furthermore, the strong decay widths are related to the parameter γ, but the ratio of the decay width Γ(ηc(4S)→DD * Γ(ηc(4S)→D * D * and Γ(ηc(4S)→DD Γ(ηc(4S)→D * D * were independent of the parameter γ, so the results of the ratios are more reliable than the decay widths. In this paper, we take the same method as Ref. [11] and Ref. [29] to study strong decays of η c (3S) and η c (4S) states.
The paper is organized as follows. In Sec. II, we introduce the instantaneous B-S equation; We show the relativistic wavefunctions of initial mesons and final mesons in Section. III; In Sec. IV, we give the formulation of two-body open charm strong decays; The corresponding results and conclusions are present in Sec. V.

II. INSTANTANEOUS BETHE-SALPETER EQUATION
In this section, we briefly review the Bethe-Salpeter equation and its instantaneous one, the Salpeter equation.
The BS equation is read as [40]: where χ(q) is the B-S wave function, P is the total momentum of the meson, q is relative quantum between quark and anti-quark, V (P, k, q) is the interaction kernel between the quark and anti-quark, p 1 , p 2 and m 1 , m 2 are the momentum and mass of the quark 1 and anti-quark 2, respectively.
We divide the relative momentum q into two parts, q and q ⊥ , Correspondingly, we have two Lorentz invariants: When → P = 0, q p = q 0 and q T = | q|, respectively.
In instantaneous approach, the kernel V (P, k, q) takes the simple form [41]: Let us introduce the notations ϕ p (q µ ⊥ ) and η(q µ ⊥ ) for three dimensional wave function as follows: Then the BS equation can be rewritten as: The propagators of the two constituents can be decomposed as: with where i = 1, 2 for quark and anti-quark, respectively, and J(i) = (−1) i+1 .
Introducing the notations ϕ ±± p (q ⊥ ) as: With contour integration over q p on both sides of Eq. (3), we obtain: , and the full Salpeter equation: For the different J P C (or J P ) states, we give the general form of wave functions. Reducing the wave functions by the last equation of Eq. (7), then solving the first and second equations in Eq. (7) to get the wave functions and mass spectrum. We have discussed the solution of the Salpeter equation in detail in Ref. [25,42].
The normalization condition for BS wave function is: In our model, the instantaneous interaction kernel V is Cornell potential, which is the sum of a linear scalar interaction and a vector interaction: where λ is the string constant and α s ( q) is the running coupling constant. In order to fit the data of heavy quarkonia, a constant V 0 is often added to confine potential. To avoid the infrared divergence V v ( q) at q = 0 in the momentum space, we introduce a factor e −αr to avoid the divergence: It is easy to know that when αr ≪ 1, the potential becomes to Eq. (9). In the momentum space and the C.M.S of the bound state, the potential reads : where the running coupling constant α s ( q) is : .
We introduce a small parameter a to avoid the divergence in the denominator. The constants λ, α, V 0 and Λ QCD are the parameters that characterize the potential. N f = 3 forbq (and cq) system.

III. THE RELATIVISTIC WAVEFUNCTIONS
In this paper, we focus on the two-body open charm strong decay of X(3940) and X(4160) which are considered as η c (3S) η c (4S) states. η c (3S) η c (4S) states have two decay modes: So we only discuss the relativistic wavefunctions of J P equal to 0 − ( 1 S 0 ) and 1 − ( 3 S 1 ) states.
A. For pseudoscalar meson with quantum numbers J P = 0 − The general form for the relativistic wavefunction of pseudoscalar meson can be written as [42]: where M is the mass of the pseudoscalar meson, and f i ( q) are functions of | q| 2 . Due to the last two equations of Eq. (7): where m 1 , m 2 and ω 1 = m 2 1 + q 2 , ω 2 = m 2 2 + q 2 are the masses and the energies of quark and anti-quark in mesons, q 2 ⊥ = −| q| 2 . The numerical values of radial wavefunctions f 1 , f 2 and eigenvalue M can be obtained by solving the first two Salpeter equations in Eq. (7). In Ref. [27], we have plot the wavefunctions of X(3940) and X(4160) which are considered as η c (3S) and η c (4S), respectively.
According to the Eq. (6) the relativistic positive wavefunction of pseudoscalar meson in C.M.S can be written as [42]: where the b i s (i = 1, 2, 3, 4) are related to the original radial wavefunctions f 1 , f 2 , quark masses m 1 , m 2 , quark energy w 1 , w 2 , and meson mass M: B. For vector meson with quantum numbers J P = 1 − The general form for the relativistic wavefunctions of vector state J P = 1 − (or J P C = 1 −− for quarkonium) can be written as eight terms, which are constructed by P f 1 , q f 1⊥ , ǫ 1 and gamma matrices [43], where ǫ 1 is the polarization vector of the vector meson in the final state.
Due to the last two equations of Eq. (7): ϕ +− 0 − = ϕ −+ 0 − = 0, we have [44]: The relativistic positive wavefunctions of 3 S 1 state can be written as [45]: where we first define the parameter n i which are the functions of f ′ i ( 3 S 1 wave functions): , , , then we define the parameters b i which are the functions of f ′ i and n i : .

IV. THE FORMULATION OF TWO-BODY OPEN CHARM STRONG DECAYS
For the two-body OZI-allowed open charm strong decays, such as η c (3S) → DD * , we adopt the 3 P 0 model to calculate the strong decay amplitude. The non-relativistic 3 P 0 model describe the decay matrix elements by the qq pair-production Hamiltonian: H = g d 3 xψψ [38]. According to the improved B-S method which is a relativistic model, we can extend the non-relativistic 3 P 0 model to the relativistic form: H = −ig d 4 xψψ [11,29].
Here ψ is the dirac quark field, g = 2m q γ, m q is the quark mass of the light quark-pairs, γ is a dimensionless constant which describe the pair-production strength and can be obtained by fitting the experimental data. In this paper, we choose γ = 0.483 [39] which give reasonable calculation of η c (3S), then we use the same value to η c (4S). Using the qq pair-production Hamiltonian, the amplitude of two-body OZI-allowed open charm strong decays A → B + C in Fig. 1, can be written as [11,29], where ϕ ++ P ( q), ϕ ++ P f 1 ( q f 1 ) and ϕ ++ P f 2 ( q f 2 ) are the relativistic positive wavefunctions of initial meson A, finial meson B and C, respectively.φ = γ 0 ϕ † γ 0 . We have given the detailed form of wavefunctions in Sec. III. P , P f 1 , P f 2 and q, q f 1 , q f 2 are the momentum and three dimension relative momentum between quark and anti-quark of initial meson A, finial meson B and C, respectively.  Finally, using Eq. (18) the two-body open charm strong decay width can be written as, where    Table. I, where DD * means D 0D * 0 +D + D * − , and D * D * means D * 0D * 0 +D * − D * + . for D 0D * 0 , D + D * − and D − s D * + s , we have considered the isospin conservation of the final mesons. In Table. II, we have presented the total widths with different theoretical model and the experimental data for convenience. We also consider the uncertainties by varying all the input parameters simultaneously within ±5% of the central values in Table. I and Table. II. In Table. I, we find that the dominant strong decay channels of η c (3S) is DD * , and agree  For η c (4S) state, the main strong decay channels are DD * and D * D * , η c (4S) → D − s D * + s is very small with the small phase space, and the decay η c (4S) → DD is forbidden. In Table. II, the total two-body open charm strong decay widths of η c (4S) is Γ ηc(4S) = (69.9 +22.4 −21.1 ) MeV. Our result is larger than the result of Ref. [17], but considering the uncertainties of the results, our result is closed to the lower limit of X(4160) for experimental data [3]. In our calculation, the ratio of the decay width Γ(ηc(4S)→DD) Γ(ηc(4S)→D * D * ) = 0, which is consistent with the experimental data Γ(X(4160)→DD) Γ(X(4160)→D * D * ) < 0.09 [3]. There is another ratio of the decay width: Γ(ηc(4S)→DD * ) Γ(ηc(4S)→D * D * ) = 3.67, which is much larger than the upper limit of the experimental data Γ(X(4160)→DD * ) Γ(X(4160)→D * D * ) < 0.22 which is reported by Belle [3]. In order to find out the relation of the decay width to the mass of η c (4S), we plot the relation of different decay width and decay ratio to the mass of η c (4S) in Fig. 3 and Fig. 4. Especially in Fig. 4, the decay ratio is decreased with the increased mass of η c (4S), but the decay ratio is larger than the experimental data at large mass, so η c (4S) is not the candidate of X(4160), and more investigations of X(4160) is needed in future.