Rates of $D^{*}_{0}(2400)$, $ D_J^*(3000) $ as the $D^{*}_{0}(2P)$ and $D^{*}_{0}(3P)$ in $B$ Decays

In this paper, we use the instantaneous Bethe-Salpeter method to calculate the semi-leptonic and non-leptonic production of the orbitally excited scalar $ D_0^* $ in $ B $ meson decays. When the final state is $ 1P $ state $ D_0^*(2400) $, our theoretical decay rate is consistent with experimental data. For $ D_J^*(3000) $ final state, which was observed by LHCb collaboration recently and here treated as the orbitally excited scalar $D^{*}_{0}(2P)$, its rate is in the order of $ 10^{-4} \sim 10^{-6}$. We find the special node structure of $D^{*}_{0}(2P)$ wave function possibly results in the suppression of its branching ratio and the abnormal uncertainty. The $3P$ states production rate is in the order of $ 10^{-5}$.


I. INTRODUCTION
The semi-leptonic and non-leptonic decays of B mesons are the frequently studied decays and also the dominant production channels of charmed mesons. During the last decades, for many important cases such as providing precise value of CKM element V cb , the channels of B decays to S-wave ground states of D mesons have been extensively measured and studied by the ALEPH, CLEO, OPAL, BABAR and Belle Collaborations [1][2][3][4][5][6][7][8] besides the theoretical studies.
In recent years, many collaborations reported several charmed resonances including some orbitally excited D mesons, which attracts lots of attention. The Belle and BABAR Collaborations reported the semi-leptonic B decays to P-wave D * mesons by using fully reconstructed B tags [9,10] and the Belle, BABAR and LHCb Collaborations reported the non-leptonic decays B → D * 0 π(K) [11][12][13][14][15]. They inspired many theoretical studies on the excited charmed states using different models, for example, the light-front quark model [16], the constituent quark model [17], as well as the Bethe-Salpeter method [18], etc.
In 2013, the LHCb Collaboration reported several resonances around 3000 MeV, D J (3000) 0 and D * J (3000) +,0 [19]. The D J (3000) and D * J (3000) were observed in the D * π and Dπ invariant mass spectrum respectively. Their quantum numbers J P are still undetermined and many theoretical studies give different assignments [20][21][22][23]. In our previous works [24,25], we calculated the strong decays and the leptonic productions of D J (3000) and we favoured it as the excited 2P (1 + ) broad state . For D * J (3000), we calculated its strong decays and our results favoured it as the excited scalar 2P (0 + ) state [26].
We notice that, in current experiments and theories, the knowledge of B semi-leptonic and non-leptonic decays to orbitally excited D * 0 meson is still rather poor. Thus this work will focus on the leptonic decays B → D * 0 − ν and non-leptonic decays B → D * 0 X, where the initial state could be B − or B 0 , the final state D * 0 is the excited scalar D * 0 (nP ) 0 or D * 0 (nP ) + ( n = 1, 2, 3), and X is a light meson. Currently, the 1P states D * 0 (2400) 0 and D * 0 (2400) + have been well studied, while 2P and 3P states haven't. The newly detected D * J (3000) +,0 are treated as the 2P scalars in this paper and our results will help to determine their quantum numbers. The processes of B leptonic decays to them could be their important production ways.
In our previous study [27], we found that large relativistic corrections exist in the processes where a heavy-light excited state is involved. We also found that the highly excited state has larger relativistic effect than its corresponding ground state. Thus when a process includes an excited state, a relativistic method or model is needed. In this paper, we use the Bethe-Salper (BS) method based on the relativistic BS equation. The relativistic effect is well concerned by solving the BS equation and applying the BS wave function.
The rest contents of this paper are organized as follows: in section 2, we present the formalism of semi-leptonic production process, including the leptonic and hadronic matrix elements by using the BS method. Then the factorization approach is used to derive the formalism of non-leptonic process in section 3. In section 4, we show our numerical results and comparison with the results of other model. Finally, discussions and short summary are given in section 4.

II. FORMALISM OF SEMI-LEPTONIC DECAYS
We take B − (B 0 ) → D * 0 0 (D * + 0 ) − ν as an example to show the calculation details of semi-leptonic process. The Feynman diagram is shows in Fig. 1.
The transition amplitude T can be expressed as : where G F is the Fermi weak coupling constant, V cb is the CKM matrix element, J ξ is the charged weak current and l ξ is the leptonic matrix.
Then the square of amplitude can be expressed by the function of hadronic and leptonic tensor, where the leptonic tensor can be written as following form: We derive the hadronic matrix element by using the relativistic BS method where q f = q − mu m b +mu P f , ϕ P and ϕ P f are the instantaneous BS wave functions of the initial state B meson and final state D * 0 mesons. They are obtained by completely solving the BS equation. The processes of solving the Salpeter equation and obtaining the wave functions are not shown here. More details can be found in our previous works [28][29][30]. We just give a brief review in the appendix. n i are the form factors whose results are shown in next section.
Then the decay width can be given by the phase-space integral After the simplification, it can be rewritten as
The effective Hamiltonian H ef f can be expressed as [31,32]: c i is the Wilson coefficient which depends on the renormalization scale µ.
By using the factorization approach, the transition matrix elements D * 0 X|H ef f |B involving the 4-quark operators can be split into the product of two matrix elements Then the transition amplitude can be expressed as where, q denotes the d or s quark; a 1 = c 1 + 1 Nc c 2 is the effective Wilson coefficient, where N c = 3 is the number of colors. We choose the scale µ ≈ m b for B decays and adopt the effective Wilson coefficient a 1 = 1.14 [35]. The annihilation matrix element can be written as where X P means pseudoscalar (π, K) and X V means vector mesons (ρ, K * ); f X is the corresponding decay constant; ε µ is the polarization vector of X V and it satisfies the completeness Same as the semi-leptonic case, we write the square of amplitude by the hadronic and light meson tensor where h µν is same as Eq. 7, and the light meson tensor is Then the decay width can be obtained by

IV. RESULTS AND DISCUSSION
In our calculations, we adopt the same parameters as what we used before [26]: The CKM matrix elements [36] and the involved mesons' decay constants are [19,36,37]:

A. Semi-leptonic decays
The form factors relevant to the hadronic transition matrix elements of B → D * 0 (1P − 3P ) ν are shown in Fig. 3, where t = (P − P f ) 2 and t m is the momentum transfer at the zero recoil(the maximum value of t). Table I shows the decay widths and branching ratios of semi-leptonic production of the ground state D * 0 (2400). With varying the parameters by ±5%, we can obtain the uncertainty of the results. Because there is almost no difference between the results of l = e and l = µ, only the values of l = e are given below.
For comparison, we also give the results of cascade decays to Dπ which are shown in Table II. Recently, Ref. [16] used covariant light-front quark model to calculate the channel B + → D * 0 (2400) 0 l + ν l → D − π + , whose result of branching ratio is 2.31 ± 0.25 × 10 −3 . And Ref. [17] shows that B(B + → D * 0 with the constituent quark model. Considering the uncertainty, our results are consistent with the experimental and other models' results. The form factors for the semi-leptonic production process of D * 0 (2400) , D * J (3000) and Then we use the same method to calculate the 2P state D J (3000), which is shown in given in Table IV and get smaller uncertainty.  For 3P states, the wave function has two nodes, and the value change from negative to positive after the second node. The cancellation gets smaller than the case of the 2P state.
Thus, the final branching ratio seems to be fine and the uncertainty is not very large. Considering the masses of these states have errors, the branching ratio of their semi-leptonic production changing with their masses are given in Fig. 5. For 1P state D * 0 (2400), the results have small changes. But the branching ratio of 2P state D * J (3000) dramatically decrease to nearly zero with increase of mass. The mass changing will also cause the wave function shift. That means the overlapping integral cancellation will increase as the mass increasing. For 3P state, it can be seen that the curves of l = e have minimum points around m = 3.175 GeV, which means the overlapping integrals have the maximum cancellation at that mass value. After that, the values increase again. For l = τ , because of the small phase space, the branching ratios have the downtrend from the beginning to the end.
Then, the normalized lepton spectra of the semi-leptonic production are presented in Fig.   6. Because there are almost no difference between l = e and l = µ, only the channels of B − → D * 0 0 e − ν e and B − → D * 0 0 τ − ν τ are given here. The spectrum peaks of 2P and 3P states move left because phase space decreases, especially for l = τ .

B. Non-leptonic production
The non-leptonic production results of 1P state D * 0 (2400) are shown in Table V. The D * 0 π and D * 0 ρ channels get the order of 10 −3 , while D * 0 K and D * 0 K * modes are in the order of 10 −5 and 10 −4 , respectively.    (7.7 ± 0.5 ± 0.3 ± 0.3 ± 0.4) × 10 −5 [14], respectively, which are much lower than the branching ratio of the previous channel. For D * 0 K channel, the LHCb collaboration shows the result [38], which is also lower than our calculation. From the perspective of symmetry, the non-leptonic results of these channels should be similar. But different experimental results show marked discrepancy, which need more experimental data accumulations and theoretical attentions.

Mass GeV
Then, like the previous section, the non-leptonic productions of 2P and 3P states are also considered and the results are shown in Table VII and VIII, respectively.
Similar to the semi-leptonic occasion, the results of 2P state D * 0 (3000) have a large uncertainty , which can also be explained by the node structure of BS wave function. Because the phase spaces of 2P and 3P state are close, their branching ratios reach similar magnitude.
We also draw the branching ratios changing with the mass of D * 0 in Fig. 7. Like the semi-  There are also minimum points of 3P states' curves at 3.175 GeV, where the maximum cancellation of overlapping integral occurs.

V. SUMMARY
Based on the instantaneous BS framework, we calculate the semi-leptonic and nonleptonic productions of several excited D * 0 states from B mesons. For 1P state D * 0 (2400),

Mass(GeV)
Branching Ratio (c)Braching ratio vers the mass of 2P state 3.10 3.12 3.14 3. the branching ratios of B → D * 0 e − ν e are in the order of 10 −3 , which is consistent with the results of present experiments and other models. For non-leptonic channels, the experiments didn't get quite consistent results while our calculating consists with parts of present experimental results. For 2P states D * J (3000), we get suppressed branching ratios in the order of 10 −5 ∼ 10 −6 and large uncertainty in both semi-leptonic and non-leptonic channels. The cancellation in overlapping integral, which is caused by its one-nodes structure of the BS wave functions, could explain the abnormal results. For 3P states, their ratios are of the same order of magnitude as 2P states' results because they have similar phase spaces. The two-nodes structure of 3P states wave functions makes the cancellation smaller than that of 2P states and get the minimum branching ratios if their masses are around 3.175 GeV.
Our work could give some inspiration to future experiment and we expect more attention on these production processes of the orbitally excited D mesons.