The weak decay $B_c$ to $Z(3930)$ and $X(4160)$ by Bethe-Salpeter method

Considering $Z(3930)$ and $X(4160)$ as $\chi_{c2}(2P)$ and $\chi_{c2}(3P)$ states, the semileptonic and nonleptonic of $B_c$ decays to $Z(3930)$ and $X(4160)$ are studied by the improved Bethe-Salpeter(B-S) Method. The decay form factors are calculated through the overlap integrals of the meson wave functions in the whole accessible kinematical range. And the influence of relativistic corrections are used in the exclusive decays. Branching ratios of $B_c$ weak decays to $Z(3930)$ and $X(4160)$ are predicted. Some of the branching ratios are: $Br(B_c^+\to Z(3930)e^+\nu_e)$$=3.03\times 10^{-4}$ and $Br(B_c^+\to X(4160)e^+\nu_e)$$=3.55\times 10^{-6}$. These results may provide the new channels to discover $Z(3930)$ and $X(4160)$ and the necessary information for the phenomenological study of $B_c$ meson physics.


I. INTRODUCTION
In this work, we do not talk the explanations of Z(3930) and X(4160), we only consider Z(3930) and X(4160) as charmonium states with the possible quantum numbers, then study their production in B c decays. we will consider the possibilities of Z(3930) and X(4160) as P −wave charmonium states χ c2 (2P ) and χ c2 (3P ), respectively. We focus on the productions of Z(3930) and X(4160) in exclusive weak decays of B c meson by the improved the Bethe-Salpeter(B-S) Method. On the one hand, the χ c2 (2P ) and χ c2 (3P ) have larger relativistic correction than the corresponding χ c2 (1P ), a relativistic model is needed in a careful study.
On the other hand, this study can improve the knowledge of B c meson, and B c meson only decay weakly which is an ideal particle to study the weak decays. The properties of B c meson have been studied by different relativistic constituent quark models [24][25][26][27][28][29][30][31][32], the covariant light-front quark model [33,34] and perturbative QCD factorization approach [35] and so on. We also discussed the properties of B c meson by the improved B-S method, include B c decays to P −wave mesons, the rare weak decays and rare radiative decays of B c , the nonleptonic charmless decays of B c , and so on [36][37][38][39][40][41][42]. In previous work, we only studied B c decays to χ c2 (1P ) state [36], because when the final states were χ c2 (2P ) and χ c2 (3P ) states, the corresponding branching ratios were very small, and there were only limited data of B c available. Now the Large Hadron Collider (LHC) will produce as much as 5 × 10 10 B c events per year [43,44]. The huge amount of B c events will provide us a chance to study B c decay to χ c2 (2P ) and χ c2 (3P ) states, some channels also provide an opportunities to discover the new particles in B c decay.
The paper is organized as follows. In Sec. II, we give the formulations of the exclusive semileptonic and nonleptonic decays; We show the hadronic weak-current matrix elements which is related to the wave functions of initial mesons and final mesons in Section. III; We show the wave functions of initial and final mesons in Sec. IV; The corresponding results and conclusions are present in Sec. V; Finally in Appendix, we introduce the instantaneous Bethe-Salpeter equation.

DECAY OF B c
In this section we present the formulations of semileptonic decay and nonleptonic decay of B c mesons to Z(3930) and X(4160) which are considered as χ c2 (2P ) and χ c2 (3P ) states.

A. Semileptonic decay of B c
The feynman diagram of B c semileptonic decay to Z(3930) or X(4160) is shown in Fig.   1. The corresponding amplitude for the decay can be written as where V bc is the CKM matrix element, G F is the the Fermi constant, J µ = V µ − A µ is the charged weak current, P and P f are the momentum of the initial meson B c and the final state, respectively. ε is the polarization tensor for final meson. The leptonic partū ν ℓ γ µ (1 − γ 5 )v ℓ is model independent and it's easy to calculate. The hadronic part X(P f , ε)|J µ |B c (P ) can be written as, where k, c 1 , c 2 , h are the Lorentz invariant form factors, M is the mass of the meson B c , M f is the mass of the charmonium in final state.
In the case without considering polarization, we have the squared decay-amplitude with the polarizations in final states being summed: where l µν is the leptonic tensor: and the hadronic tensor relating to the weak-current in Eq. (1) is where the functions α, β ++ , β +− , β −+ , β −− , γ are related to the form factors.
The total decay width Γ can be written as: where E f , E l and E ν are the energies of the charmonium, the charged lepton and the neutrino respectively. If we define x ≡ E l /M, y ≡ (P − P f ) 2 /M 2 , the differential width of the decay can be reduced to: The total width of the decay is just an integration of the differential width i.e. Γ = dx dy d 2 Γ dxdy .

B. Nonleptonic decay of B c
For the nonleptonic decay of B c → X + M 2 in Fig. 2, the relevant effective Hamiltonian H ef f is [45,46]: where G F is the Fermi constant, V bc is the CKM matrix element and c i (µ) are the scaledependent Wilson coefficients. O i are the operators responsible for the decays constructed by four quark fields and have the structure as follows: Since this is the primary study of these nonleptonic decays, we apply the naive factorization to H ef f [47], the nonleptonic two-body decay amplitude T can be reduce to a product of a transition matrix element of a weak current X|J µ |B c and an annihilation matrix element of another weak current M 2 |J µ |0 : a 1 = c 1 + 1 Nc c 2 and N c = 3 is the number of colors. While the annihilation matrix element M 2 |J µ |0 is related to the decay constant of M 2 . When M 2 is a pseudoscalar meson [48], In Eq. (6) and Eq. (9), we find that the most important things to get the decay width of the corresponding decay are to calculate hadronic weak-current matrix elements X(P f )|J µ |B c (P ) . We will give the detailed calculation of the hadronic weak-current matrix elements in the Section. III.

III. THE HADRONIC WEAK-CURRENT MATRIX ELEMENTS
The calculation of the hadronic weak-current matrix element are different for different model. In this paper, we combine the B-S method which is based on relativistic B-S equation with Mandelstam formalism [50] and relativistic wave functions to calculate the hadronic matrix element. The numerical values of wave functions have been obtained by solving the full Salpeter equation which we will introduce in Appendix. As an example, we consider the semileptonic decay B c → Xℓ + ν ℓ in Fig. 1. In this way, at the leading order the hadronic matrix element can be written as an overlapping integral over the wave functions of initial and final mesons [51], where q ( q f ) is the relative three-momentum between the quark and anti-quark in the initial We will show the Salpeter wave functions for the different mesons in next section.

IV. THE RELATIVISTIC WAVE FUNCTIONS OF MESON
A.
For B c meson with quantum numbers J P = 0 − The general form for the relativistic wave function of pseudoscalar meson B c can be written as [52]: 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. (A7): ϕ +− 0 − = ϕ −+ 0 − = 0, we have: 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 B c mesons, q ⊥ = q − (q · P/M 2 )P , and q 2 ⊥ = −| q| 2 . The numerical values of radial wave functions f 1 , f 2 and eigenvalue M can be obtained by solving the first two Salpeter equations in Eq. (A7). According to the Eq. (A6) the relativistic positive wave function of pseudoscalar meson B c in C.M.S can be written as [52]: where the b i s (i = 1, 2, 3, 4) are related to the original radial wave functions f 1 , f 2 , quark masses m 1 , m 2 , quark energy w 1 , w 2 , and meson mass M: B. For Z(3930) and X(4160) mesons with quantum numbers J P = 2 ++ Considering Z(3930) and X(4160) as χ c2 (2P ) and χ c2 (3P ), the general expression for the relativistic wave function of tensor J P = 2 ++ state can be written as [53] with the constraint on the components of the wave function: Then we have the reduced wave function ϕ 2 ++ ( q f ) as: with Where M f , P f , f ′ i ( q f ) are the mass, momentum and the radial wave functions of Z(3930) and X(4160), respectively. m ′ 1 , m ′ 2 and ω ′ 1 = m ′2 1 + q 2 f , ω ′ 2 = m ′2 2 + q 2 f are the masses and the energies of quark and anti-quark in Z(3930) and X(4160). To show the numerical results of wave functions explicitly, we plot the wave functions of Z(3930) and X(4160) states in f −2ME f which provides the kinematic range for the semileptonic decay of B c . It varies from t = 0 to t = 5.52 GeV 2 for the decays to Z(3930) and from t = 0 to t = 4.48 GeV 2 for the decays to X(4160). In Fig. 4 we give the relations of (t m − t)(t m = (M − M f ) 2 is the maximum of t) and the form factors. Taking the form factor to the Eq. (6), then we will get the leptonic energy spectra dΓ ΓdPe for semileptonic B c decay to Z(3930) and X(4160), the leptonic energy spectra are plotted in Fig. 5 which are related to the momentum of the final mesons.
Using the leptonic energy spectra, we calculate the decay widths of the semileptonic B c → Xℓ + ν ℓ (X = Z(3930) or X(4160), ℓ = e, µ, τ ) and give the results in Table. I. Since m τ is very large and m e ≃ m µ is quite a good approximation for the B c meson decays, thus only the cases where the lepton is an electron or τ are considered in Table. I. Because of the larger kinematic ranges and the different wave functions in Fig. 3, the corresponding decay   Table. II. The results of B c nonleptonic decay are affected by the CKM matrix elements, so the results of light mesons π, ρ are larger than the ones of light mesons Table. II, respectively.
In order to compare the numerical values with experimental measurements in the future, Taking the values a 1 = 1.14 for nonleptonic decays [45,46], combining the life time of B c meson, we calculate the branching ratios of the decays and list them in Table. III. Because of B c →Z(3930), X(4160) have small kinematic ranges and the wave functions have some minus parts in Z(3930), and X(4160), comparing our results with B c decays to χ c2 (1P ) in Ref. [36], the results are smaller than the results of B c decay to χ c2 (1P ).
In summary, many more heavy charmonium states, such as Z(3930) and X(4160) are observed by the experiments. In this work, considering Z(3930) and X(4160) as χ c2 (2P )  if we can observe the sufficient events, some channels will provide us a sizable ratios, the branching ratios of the order of (10 −6 ) could be measured precisely at the LHC, and may be they will detect the productions of Z(3930) and X(4160) in B c exclusive weak semileptonic and nonleptonic decay. Then our results will provide a new way to observe the Z(3930) and X(4160) and the necessary information for the study of B c meson.
In instantaneous approach, the kernel V (P, k, q) takes the simple form [59]: 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. (A3), 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. 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. One can see that V v (r) diverges at r = 0, we introduce a factor e −αr to avoid the divergence: It is easy to know that when αr ≪ 1, the potential becomes to Eq. (A9). 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.