Estimating the production rates of D-wave charmed mesons via the semileptonic decays of bottom mesons

In this work, using the covariant light front approach with conventional vertex functions, we estimate the production rates of $D$-wave charmed/charmed-strange mesons via $B_{(s)}$ semileptonic decays. Due to these calculated considerable production rates, it is possible to experimentally search for $D$-wave charmed/charmed-strange mesons via the semileptonic decays, which may provide extra approach to explore $D$-wave charmed/charmed-strange mesons.

When checking the production processes involving the D-wave charmed and charmed-strange mesons, we notice that these states are mainly produced via the nonleptonic weak decays of bottom/bottom-strange mesons. However, as an important decay mode, the semileptonic decays of B/B s mesons are an ideal platform to produce D-wave D/D s mesons because they can be estimated more accurately than nonleptonic ones. For estimating the branching ratios of these processes, we need to perform a serious theoretical study on the production of D-wave D/D s mesons via the semileptonic decays of B/B s mesons, which is a main task of the present work.
Until now, there has been no work on the study of the productions of D-wave D/D s mesons through the semileptonic decays of B/B s mesons in the covariant light-front approach, which makes the present work be the first paper on this issue. As illustrated in the following sections, the technical details of deduction relevant to the above processes are far more complicated than those of S -and P-wave mesons. Thus, our work is not only an application of the LFQM but also development of this research field since the formula presented in this work can be helpful to the study of other processes involving D-wave mesons. Because we consider it is valuable to readers, we provide more details of deduction.
Finally, we still hope that the present study can stimulate experimentalists' interest in searching for D-wave D/D s mesons by the semi-leptonic decays of B/B s mesons. It will open another window to explore D-wave D/D s mesons, on which the experimental information will become more abundant.
This paper is organized as follows. In Sec. II, we introduce the covariant light-front approach for D-wave mesons and their corresponding form factors. In Sec. III, we list our numerical results including the form factors as well as the decay branching ratios. The final section is devoted to a summary of our work. Appendices A through D will describe the algebraic details related to the D-wave mesons in the LFQM.

II. COVARIANT LIGHT-FRONT QUARK MODEL
In the conventional light-front quark model, quark and antiquark inside a meson are required to be on their mass shells. Then, one can extract physical quantities by calculating the plus component of the corresponding matrix element. However, as discussed in Ref. [35], this treatment may result in missing the so-called Z-diagram contribution and make the matrix element non-covariant. A systematic way of incorporating the zero mode effects was proposed in Ref. [49] to maintain the associated current matrix element covariance.
In this work, we will apply the covariant light-front approach to investigate the production of D * * (s) mesons via the semileptonic decays of B/B s mesons (see Fig. 1), where D * * (s) denotes a general D-wave D/D s meson. Firstly, we briefly introduce how to deal with the transition amplitude. According to Ref. [49], the relevant form factors are calculated in terms of Feynman loop integrals which are manifestly covariant. The constituent quarks within a hadron are off-shell, i.e., the incoming (outgoing) meson has the momentum P ′(′′) = p ′(′′) 1 + p 2 , where p ′(′′) 1 and p 2 are the off-shell momenta of quark and antiquark, respectively. These momenta can be expressed in terms of the appropriate internal variables, (x i , p ′ ⊥ ), defined by, with x 1 + x 2 = 1. In the light-front coordinate, P ′ = (P ′− , P ′+ , P ′ ⊥ ) with P ′± = P ′0 ± P ′3 , which has the relation P ′2 = P ′+ P ′− − P ′2 ⊥ . One needs to specify that there exist different conventions for the momentum conservation under the covariant light-front and conventional light-front approaches. In the covariant light-front approach, four components of a momentum are conserved at each vertex, where the quark and antiquark are off-shell. In the conventional light-front approach, the plus and transverse components of a momentum are conserved quantities, where quark and antiquark are required to be on their mass shells. Thus, it would be useful to define some internal quantities for on-shell quarks 3) where M ′2 0 is the kinetic invariant mass squared of the incoming meson. e (′) i denotes the energy of quark i, while m ′ 1 and m 2 are the masses of quark and antiquark, respectively.

A. General definition and BSW definition of form factors
In Ref. [35], the form factors for the semileptonic decays of bottom mesons into S -wave and P-wave charmed mesons were obtained within the framework of the covariant light-front quark model. In the following, we adopt the same approach to deduce the form factors of the production of D-wave charmed/charmed-strange mesons through the semileptonic decays of bottom/bottom-strange mesons. Here, the D-wave D/D s mesons with notations D * * (s)1 , D * * (s)2 , D * * ′ (s)2 , and D * * (s)3 have the quantum numbers 2S +1 L J = 3 D 1 , 1 D 2 , 3 D 2 , and 3 D 3 , respectively. In the following deduction, we will use such notation for simplification.
In the heavy quark limit m Q → ∞, the heavy quark spin s Q decouples from the other degrees of freedom. Hence, a more convenient way to describe charmed/charmed-strange mesons is to use the |J, j ℓ basis, where J denotes the total spin and j ℓ denotes the total angular momentum of the light quark. There exists connection between physical states |J, j l and the states described by |J, 2S +1 L L for L = 2 [50,51], i.e., This relation shows that two physical states D (s)2 and D ′ (s)2 with J P = 2 − are linear combinations of the D * * (s)2 ( 1 D 2 ) and D * * ′ (s)2 ( 3 D 2 ) states. When dealing with the transition amplitudes of the production of the D 2 and D ′ 2 states, we need to consider the mixing of states shown in Eqs. (2.6) and (2.7).
One can write out the general definitions for the matrix elements of the production of D-wave D/D s mesons via the semileptonic decays of B/B s mesons, i.e., with P = P ′ + P ′′ , q = P ′ − P ′′ and ǫ 0123 = 1. We have assumed Lorentz covariance to define these form factors. One should notice that the B (s) → D * * (s) transition occurs through a V − A current, where D * * (s) denotes the general D-wave charmed (charmedstrange) meson. For the semileptonic decays involving the 3 D 1 and 3 D 3 states, the ǫ µναβ term arises in D * * (s)1 D * * (s)3 V µ B (s) , which corresponds to the contribution of the vector current. Different from the case of the 3 D 1 and 3 D 3 states, for the 1 D 2 and 3 D 2 states, ǫ µναβ term arises in the axial vector current. Here, a minus sign is added in front of this term so that we have D * * (′) (s)2 −A µ B (s) = ǫ µναβ ǫ ′′ * νλ P λ P α q β n (′) (q 2 ). When the sign of the ǫ µναβ term is fixed, the signs of other form factors can be also determined.
The form factors listed in Eq. (2.8) for the semileptonic decays relevant to the 3 D 1 state can be rewritten by inserting the Bauer-Stech-Wirbel (BSW) form factors [52], i.e., (2.14) where dependence on the external masses has been extracted by considering the Lorentz invariance so that resultant form factors, V B (s) →D * * (s)1 (q 2 ), A B (s) →D * * (s)1 1 (q 2 ), etc., become dimensionless. For the semileptonic decays related to the 1 D 2 and 3 D 2 states, the general form factors defined in Eqs. (2.9)-(2.10) can be rewritten by inserting the BSW form factors: We get the expressions for the semileptonic decay widths of the transitions under discussion in terms of the BSW form factors.
In this work, we will also use the BSW convention to present our numerical results of form factors. Analogy to the transition matrix element of the 3 D 1 state, we give the definition of the BSW form factors for the semileptonic decay relevant to the 3 D 3 state as, Following the calculation in Ref. [35], we first obtain the B (s) → D * * (s)1 transition form factors, and then continue to calculate the processes involving other D-wave charmed/charmed-strange states. Here, one needs to introduce the vertex wave functions to describe the B (s) and D * * (s) mesons. The expression of a wave function for an initial B (s) meson has been obtained in Ref. [35]. In the following, we will give detailed discussion for the covariant vertex function of the final state D * * (s)1 meson. The conventional D-wave vertex function has been studied in Ref. [53]. However, there exist some inconsistences in their deduction. Thus, in this work we present our analysis of D-wave vertex functions in the frame work of the conventional LFQM (see Appendix B for more details), which will be applied to the calculation of B (s) → D * * (s)1 hadronic matrix element. In the conventional LFQM, the p ′ 1 and p 2 are on their mass shell, while in the covariant [49] light-front approach, the quark and antiquark are off-shell, but the total momentum P ′ = p ′ 1 + p 2 is still the on-shell momentum of a meson, i.e., P ′2 = M ′2 with M ′ being the mass of an incoming meson. One needs to relate the vertex function deduced in the conventional LFQM to the vertex in the covariant light-front approach. A practical method for this process has been proposed in a covariant light-front approach in Ref. [49]. Since the covariant vertex has the same Lorentz structure as that of a conventional vertex function, following the method in Ref. [49] and adopting the approach similar to the conventional vertex function for 3 D 1 state in Appendix B, we obtain the corresponding covariant vertex function as  26) where N c is the number of colors, N ′(′′) is the vertex function of a pseudoscalar meson, and One can integrate over p ′− 1 via a contour integration with d 4 p ′ 1 = P ′+ d p ′− 1 dx 2 d 2 p ′ ⊥ /2 and the integration picks up a residue p 2 =p 2 , where the antiquark is set to be on-shell,p 2 2 = m 2 2 . The momentum of the quark is given by the momentum conservation, p ′ 1 = P ′ −p 2 . Consequently, after performing the p ′− 1 integration, we make the replacements: where the explicit trace expansion ofŜ 3 D 1 µν after integrating Eq. (2.26) over p ′− 1 is presented in Appendix A. Additionally, h ′ P has been given in Ref. [35] as, where ϕ is the solid harmonic oscillator for S -wave and describes the momentum distribution of an initial B (s) meson.
As noted in Ref. [53], after carrying out the contour integral over p − 1 , the quantities, H ′′ , and ǫ * ′′ are replaced by the corresponding h ′′ which is derived in Appendix B and with p ′′ ⊥ = p ′ ⊥ − x 2 q ⊥ . Furthermore, as pointed out in Ref. [35,49],p ′ 1 can be expressed in terms of three external vectors, P ′ , q, andω:p [35,49] is a light-like four vector, and theω dependent terms can not be canceled by a Lorentz transformation since the corresponding matrix elements are not covariant. Thisω dependence will also appear in the products of a couple of p ′ 1 's. Following the discussion in Ref. [35,49], to avoid theω dependence and to maintain the covariance, one needs to do the following replacements: where P = P ′ + P ′′ and A (i) j and Z 2 are functions of x 1 , p ′2 ⊥ , p ′ ⊥ · q ⊥ , and q 2 . One can refer to Appendix C for their explicit expressions. These functions have also been obtained in Ref. [49].
It is still necessary to give more details on the procedure of the above replacements. Let us take the result ofp ′ 1µ as an example to follow discussions on the tensor decomposition. As obtained in Ref. [49], the tensor decomposition ofp ′ 1µ iŝ where the coefficient that depends onω is obtained as [49] C (1) This term is associated with zero-mode contributions and this effect can be included by performing the replacementN 2 → Z 2 . This treatment will set C (1) 1 = 0 and the p ′ 1µ have the tensor form of Eq. (2.31), which is free ofω dependence and the covariance is maintained.
The same procedure should also be applied to the tensor decomposition ofp ′ 1µp ′ 1ν . The full tensor decomposition has also been obtained in Ref. [49]. By constraining our discussion in the leading order ofω, there are two tensor decomposition coefficients [49] left that are related toω, i.e., B (2) 1 , C (2) 1 . As indicated in [35,49], the B (2) 1 function is not associated with zeromode contributions, and should give very small contribution or just vanishes under a certain selection of vertex functions. The expression of C (2) 1 is related to the zero-mode contributions and has been given as [49] By using Eq. (2.38) and set C (2) 1 to 0, we find combining with the relation x 2N2 = 0 in Ref. [49], we can easily obtain Eq. (2.35). Again, we find the results of p ′ 1µ p ′ 1ν that is covariant and free ofω after p ′− 1 integration. Similar deduction should also be applied to analysis of more products of p ′ 1 's. We have performed this kind of analysis to the product of up to five p ′ 1 's and will present them in the following discussions.  [35,53], the Lorentz structure of a vertex function for 3 S 1 state is identical to that of 3 D 1 state. Hence, we can show the explicit form factors, g D (q 2 ), f D (q 2 ), a D+ (q 2 ) and a D− (q 2 ) here as, which are consistent with those obtained in Ref. [35]. The same procedure can also be applied to the transitions relevant to 1 D 2 , 3 D 2 as well as 3 D 3 states, whose results of form factors are given in Appendix A. In the following, we continue to discuss these states and focus on the new subjects that should be introduced when dealing with higher spin D-wave states. By analogy to the conventional vertex functions obtained in Appendix B, we write out the covariant vertex functions for 1 D 2 , 3 D 2 , and 3 D 3 in one loop Feynman diagrams as, respectively, where H2S+1 D J and W2S+1 D J are the functions of associated states in the momentum space. In order to obtain the B (s) → D * * (′) (s)2 , D * * (s)3 transition form factors, the matrix elements are denoted as It is straightforward to obtain the explicit expressions of the corresponding one loop integrals as By integrating over p ′− 1 as discussed in the case of B B (s) (D * * (s1) ) µ , the following replacements should be taken, where M appearing in the subscripts or superscripts denotes 1 D 2 , 3 D 2 , and 3 D 3 , by which these physical quantities corresponding to different transitions can be easily distinguished. The explict forms of h ′ M are given by Eqs. (B.26) in Appendix B. We also present the trace expansions ofŜ and ǫ ′′ , respectively. The next step is to maintain theω independence, which makes the whole expression covariant. Then, the zero mode effecst can be included.

C. Semileptonic decay widths
In this subsecton, we will give the explicit forms of decay widths for the semiloptonic decays, which are necessary for numerical calculations. In this work, we study the production of D-wave D * * (s) mesons and their partners through the semileptonic decay of B (s) mesons. The effective weak Hamiltonian involved in the where G F is the Fermi coupling constant and V cb denotes the Cabibbo-Kobayashi-Maskawa (CKM) matrix element.
We adopt the BSW [52] form factors defined in Sect. II A to deduce the decay widths of the semileptonic decays under discussion. The concrete expression of decay widths can be obtained by using the helicity amplitude as, and When it comes to production of charmed/charmed-strange mesons with J = 2 via the semileptonic decays of bottom/bottom-strange mesons, there exist some relations between the decay width involved in a 3 D 1 state and that relevant to the production of 1 D 2 and 3 D 2 states, i.e., where the factors 2 3 and 1 2 are due to the second order polarization tensors ǫ µν (L z ), the explicit forms of which are given in is multiplied to get the correct dimension for the decay width.
In a similar way, we can also obtain the decay width for 3 D 3 states, i.e., (2.65)

III. NUMERICAL RESULTS
In the framework of the light-front quark model [35,53,54], one usually adopts a single simple harmonic oscillator (SHO) wave function to approximately describe the corresponding spatial wave function of a meson, where the parameter β in the SHO wave function is extracted from the corresponding decay constant. Due to the limited information on the decay constants of D-wave charmed/charmed-strange mesons, we may consult another approach to get it.
In Refs. [26,27], the mass spectrum of D/D s mesons have been systematically studied in the framework of the modified Godfrey-Isgur (GI) model, by which their numerical spatial wave functions can be also obtained. As illustrated in Appendix B, we may adopt numerical spatial wave functions as an input in our calculation (see Appendix B for more details). Here, the numerical wave functions for the discussed D-wave D/D s mesons can be precisely described by the expansion in the twenty-one SHO bases, where the corresponding expansion coefficients form the eigenvectors. In Tables I and II, we collect the masses and the corresponding eigenvectors of the involved D-wave D/D s mesons. Although the observed D * (2760), D(2750) [2,3], D * s1 (2860) and D * s3 (2860) [4,5] can be good candidates of 1D states in charmed and charmed-strange meson families [6][7][8][9][10][11][12][13][14][15], in this work we still take the theoretical masses of D-wave D/D s mesons as an input when studying these semileptonic decays .
In our calculation, other input parameters include the constituent quark masses, m u,d = 220 MeV, m s = 419 MeV, m c = 1628 MeV and m b = 4977 MeV, which are consistent with those given in the modified GI model [26,27]. In order to determine the shape parameter β in Eq.(B.9) of Appendix B for initial pseudoscalar bottom and bottom-strange mesons, we use the direct results of the lattice QCD [55], where f B = 190 MeV and f B s = 231 MeV. Then, the parameter β can be extracted from these two decay constants by [35] where m ′ 1 and m 2 denote the constituent quark masses of b and light quark, respectively. Finally, we have β B = 0.567 GeV and β B s = 0.6263 GeV for bottom and bottom-strange mesons, respectively.
In Appendix A, we list the detailed expressions for the form factors relevant to the production of 3 D 1 , 1 D 2 , 3 D 2 , and 3 D 3 states. In the light-front quark model, q + = 0 is assumed. Due to the equality q 2 = q + q − − q 2 ⊥ , all the results obtained for the form factors are only effective in the q 2 ≤ 0 region. This means that we need to extrapolate our results of the form factors to the time-like region.
In this work, we introduce the so-called z-series parametrization used in [56][57][58] to obtain our form factors in the time-like region. This parametrization is suggested by including the general and analytical properties of form factors [57]. The explicit expression can be written as [58] F(q 2 ) = F(0) where the conformal transformation is introduced, 3)          [26,27].
Here, in order to give an accurate matching for D-wave B (s) → D * * (s) transition form factors, two parameters b 1 and b 2 are introduced. In Table III, the fitting parameters and form factors for D * * (s) |V − A| B (s) transitions are collected. We should emphasize that in our calculation the mixing between 1 D 2 and 3 D 2 states is considered. When presenting the form factors MA and MV for the processes of B (s) → D (s)2 (2D ′ (s)2 ) and B (s) → D (s)2 (2D ′ (s)2 ), such mixing is included in the BSW formalism, i.e., We present the form factors obtained for the B (s) → D * * (s) transitions in Table III. In addition, we also show the q 2 dependence of the form factors in Figs. 3-6. The form factors for the processes of producing D-wave D mesons have the similar behavior to those for the processes of producing D-wave D s mesons, which is due to identical vertex function structures and similar phase spaces belonging to the transitions under discussion. With the above preparation, we perform the numerical calculation of branching ratios for the B (s) semileptonic decays to D-wave D/D s mesons, which are listed in Table IV. Since the branching ratios obtained are considerably large, experimental search for the semileptonic decays relevant to the production of D-wave D/D s mesons will be an intriguing issue for future experiment.

IV. SUMMARY
In the past years, a great progress on observing D-wave D/D s mesons have been made in experiment [2][3][4][5]   For getting the numerical results of the discussed semileptonic decays, we adopt the light-front quark model, which has been extensively applied to study decay processes including semileptonic decays [35][36][37][38][39][40][41][42][43][44][45][46][47][48]. Our study under the framework of the LFQM shows that the whole deduction relevant to the D-wave D/D s mesons produced via B/B s mesons is much more complicated than that relevant to production of the S -wave and P-wave D/D s mesons [35][36][37]. In this paper, we have given detailed derivation of many formulas necessary for obtaining the final semileptonic decay widths. The numerical results obtained for the discussed semileptonic decays have shown that the semileptonic decays of B/B s mesons are suitable for finding the Dwave charmed and charmed-strange mesons.
Before the present work, there were some theoretical studies on the B (s) semi-leptonic decays to D-wave charmed mesons by the QCD sum rule [59,60] and the instantaneous Bethe-Salpeter method [51]. We have noticed that different theoretical groups have given different results for the B (s) semi-leptonic decays to D-wave charmed mesons. Thus, experimental search for our predicted semileptonic decays will provide a crucial test for the theoretical frameworks applied to study on the B (s) semi-leptonic decays. As indicated by our numerical results, the semileptonic decays of pseudoscalar B/B s mesons can be an ideal platform to carry out the investigation on D-wave charmed and charmed-strange mesons. With the running of LHCb at 13 TeV and forthcoming BelleII, we expect the experimental progress on this issue furthermore.   µαβν . Form factors associated with these expressions are also collected here.
When integrating over p ′− 1 , we need to do the following integrationŝ where the trace expansions ofŜ The corresponding form factors for a 1 D 2 state are The corresponding form factors for a 3 D 2 state are The corresponding form factors for a 3 D 3 state are In the conventional light-front approach, a meson with momentum P and spin J can be defined as where q 1 andq 2 denote the quark and antiquark within the meson, respectively, and p 1 and p 2 are the on-shell light-front momenta of quark and antiquark, respectively. The symbol "˜" means an operation on a momentum to extract only plus and transverse components, i.e.,p and In the light-front coordinate, the definition of the light-front relative momentum (x, p ⊥ ) reads The meson wave function Ψ J J z LS in momentum space is expressed as where ϕ LL z (x 2 , p ⊥ ) describes the momentum distribution of the constituent quarks inside a meson with the orbital angular momentum L, and LS ; L z S z |LS ; JJ z is the corresponding Clebsch-Gordan (CG) coefficient. For a meson with s total momentum J, it has 2J + 1 components in J z direction. In order to express all CG coefficients with the same J but different J z components in a unified formula, we need to reexpress the CG coefficients into a tensor contraction form, which is crucial to obtain the vertex function. We will discuss it more explicitly in the following. In Eq. (B.4), R S S z λ 1 λ 2 transforms a light-front helicity (λ 1 , λ 2 ) eigenstate into a state with spin (S , S z ) whereP is the momentum of meson in the rest frame,P = p 1 + p 2 , According to the spinor representation ofū and v in Appendix of Ref. [61], one has calculated the explicit expression of R S S z λ 1 λ 2 in Ref. [62]. In the framework of the LFQM [35,53,54], one usually uses a single SHO wave function to approximately depict the spatial wave function of a meson. For a D-wave state, the harmonic oscillator wave function is written as [53] where K = (p 2 − p 1 )/2 and ϕ is the harmonic oscillator wave function for an S wave, which has the form This harmonic oscillator wave function given by Eq. (B.6) describes the momentum distribution of a meson with n r = 0 and L = 2, where n r is the radial quantum number. Under this treatment, the value β in the wave function of a meson becomes a parameter determined by the decay constant of a meson. Due to the limited information of decay constants of D-wave charmed/charmed-strange mesons in experiment and theory, in this work we adopt another approach to get it, which will be illustrated in the following.
With the help of the potential model, for a definite meson state with quantum numbers n r 2S +1 L J , its mass and the corresponding numerical spatial wave function can be calculated. The obtained spatial wave function can be reproduced by the expansion of a list of SHO wave functions (the number of the SHO wave function bases taken is set to be N), where the expansion coefficients form the corresponding eigenvector. Practically, N = 20 is enough if precisely reproducing the numerical wave function, since the obtained masse value becomes stable when taking N > 20. Here, a general expression of an SHO wave function reads and Y lm p x , p y , p z = p l Y l,m (θ, ϕ) . (B.10) In the light-front coordinate, an SHO wave function corresponding to Eq. (B.8) is where the term e 1 e 2 x(1−x)M 0 is due to the Jacobi transformation between rectangular and light-front coordinates. Additionally, the normalization condition is To describe the wave function of a D-wave meson by the expansion in a list of SHO wave functions in the light-front coordinate, one has where 15 8π ǫ µν (L z )K µ K ν denotes the D-wave solid harmonic Y 2Lz (x, p ⊥ ). An overall factor 2 (2π) 3 is introduced to satisfy the normalization condition [35], By comparing Eq. (B.13) with Eq. (B.6), the numerical wave function can be introduced by the following replacement in Eq. (B.6), By using the replacement Eq. (B.15), the meson wave function Ψ J J z 2S (p 1 , p 2 , λ 1 , λ 2 ) in momentum space reads Here, ϕ N = N n β 2 √ 2 a n R ′ n2 (x, p ⊥ )π 30e 1 e 2 x(1−x)M 0 , and Γ ( 2S +1 D J ) denotes the corresponding vertex structure of D-wave mesons. Now, we focus on the vertex structure Γ ( 2S +1 D J ) in Eq. (B.16) for D-wave mesons. In order to derive the concise expression of a vertex function, the corresponding CG coefficient should be expressed in a tensor contraction form. For example, the CG coefficient relevant to the production of 3 D 1 state is given by, Other CG coefficients involved in our calculation are calculated to be, which are related to the productions of the 1 D 2 , 3 D 2 and 3 D 3 states, respectively. The structure of these CG coefficients can be constructed by the tensor algebra as well as the symmetry of CG coefficients. The detailed derivation for the transformation of the CG coefficient 21; L z S z |21; 1J z can be found in Appendix D. Additionally, readers can also refer to Refs. [53,63] for details 2 .
Substituting Eq. (B.17) and Eq. (B.18) into Eq. (B.16), we obtain the corresponding vertex structures for 3 D 1 , 1 D 2 , 3 D 2 and 3 D 3 states, i.e., When deriving the Γ ( 3 D 2 ) vertex function, there appears an anti-symmetric tensor ǫ µανβ introduced from the tensor contraction form of a CG coefficient, which has a relation, iǫ µανβ γ β = S µανβ γ β γ 5 − γ µ γ α γ ν γ 5 (B.23) with S µανβ = g µα g νβ + g µβ g να − g µν g αβ . In the following, we further check our calculation. From Eq. (B.16), we can derive the tensor contraction forms of CG coefficients in Eq. (B.17) and Eq. (B.18) to obtain the corresponding Γ ( 2S +1 D J ) vertex structures. The wave function Ψ J J z 2S (p 1 , p 2 , λ 1 , λ 2 ) has 2J + 1 components. Then, we can use the expression of a vertex function and the explicit expressions ofū and v in Ref. [61] to expand all the J z components of Ψ J J z 2S (p 1 , p 2 , λ 1 , λ 2 ). On the other hand, we can also expand all the J z components of Ψ J J z 2S (p 1 , p 2 , λ 1 , λ 2 ) in numerical CG coefficients. These two expansions must be identical to Ψ J J z 2S (p 1 , p 2 , λ 1 , λ 2 ) for each J z component. In this way, we examine all the vertex functions in the present work, and find that our calculation satisfies the above condition. We use these vertex functions to do the following calculations.