The B_c decays to a P-wave charmonium by the improved Bethe–Salpeter approach

The meson B_c is the ground state of the double heavy (both of the components are heavy) quark–antiquark binding system ($\bar{b}c$). In the standard model (SM), it is a unique meson which carries two different heavy flavors explicitly; thus, it decays weakly, that is, very different from the ground states of flavor-hidden double heavy mesons, such as η_c and η_b. Namely, B_c decays via weak interaction (via virtual W emission or annihilation) only, while the ground states of flavor-hidden double heavy mesons decay predominantly by annihilating to gluons (strong interaction) or/and photons (electronic interaction). The meson B_c has very rich and experimentally accessible decay channels, so studying the decays of the B_c meson is especially interesting. By comparing the experimental and theoretical results of the decays of the B_c meson, we can also reach some insight into the binding effects of the heavy quark–antiquark system, which are of QCD nature, besides knowledge of the weak interaction such as the CKM matrix elements, etc.

The meson B_c was first experimentally discovered by the CDF collaboration at Fermilab through the semileptonic decay B_c → J/ψ + l + ν [1], and soon it was confirmed not only by CDF itself via another decay channel B_c → J/ψ + π [2], but also by the other collaboration D0 at Fermilab [3]. The latest experimental report for its lifetime and mass in PDG [4] is $M_{B_c}=6.277\pm 0.006$ GeV and ${\tau }_{B_c}=(0.453\pm 0.041)\times 10^{-12}$ s. Because the cross section of B_c production is comparatively small, so to discover it is quite difficult in experiment. On the other hand, according to the estimates [5–7] that LHC will produce about 5 × 10¹⁰B_c events per year, it is expected that more measurements of decays and production of the meson B_c will be available soon at the LHC (LHCb, CMS, ATLAS), and it must push more studies of the decays of the B_c meson forward. So both experimental and theoretical studies on the B_c meson now become more interesting.

In fact, the decays of the B_c meson can be divided into three categories: (i) the anti-bottom quark $\bar{b}$ decays into $\bar{c}$ (or $\bar{u}$) with the c-quark being a spectator; (ii) the charm quark c decays into s (or d) with the $\bar{b}$-quark being a spectator; (iii) the two components, $\bar{b}$ and c, annihilate weakly. According to the decay products, we may realize which one, two or even three of the categories play roles in a concerned decay; thus, one can measure the CKM elements such as V_bc, V_ub, V_cs, V_cd through the decays. In this paper, we highlight the decays of the B_c meson to a P-wave charmonium, and one may easily realize that the decays being considered here belong to category (i). Since the lepton spectrum and the weak form factors, which relate to the binding effects (wavefunctions) precisely, may be measurable in semilepton decays as long as there are enough experimental data, we will shed quite a lot of light on them.

In fact, one may find a lot of theoretical methods to treat the semileptonic and nonleptonic decays of the B_c meson, such as the varieties of relativistic constituent quark models [8–15], QCD sum rules [16, 17], etc in the literature. Moreover, one may realize that among the relativistic constituent quark models, the method presented in [18] and adopted in [9, 10] is based on the instantaneous version [19] of the Bethe–Salpeter (B-S) equation [20], and the 'instantaneous treatment' is also extended to the weak-current matrix elements using the Mandelstam formulation [21], while the adopted approach in [8] is different from the one presented in [18] only in the kernel of the B-S equation and the 'instantaneous treatment', etc. Recently, in [22] an improvement to that of [18] has been proposed, and the relativistic effects in the binding systems and decays between the systems may be considered by the new development more properly, especially considering the fact that, of the new development, the part (factor) for dealing with the binding effects has been applied to study (test) the spectra of a positronium (a QED binding system) [23] and double heavy flavor binding systems (QCD binding systems) [24] and quite satisfactory results are obtained (see [23, 24]). So to test the new development [22] when experimental data are available in the foreseeable future, in this paper we try to apply the development to the decays of the B_c meson to a P-wave charmonium and to compare the obtained results with those obtained by the old method in [18] and obtained by other theoretical approaches. Since we suspect that the decays of the B_c meson to a P-wave charmonium might be more sensitive in testing the effects caused by the improvement than the decays of the B_c meson to an S-wave charmonium, here we focus our attention on the decays of the B_c meson to a P-wave charmonium.

The new development [22] contains two factors: one is about relativistic wavefunctions which describe bound states with definite quantum numbers, i.e. a relativistic form of wavefunctions (see appendix C) which are solutions of the full Salpeter equation (see appendix B). Note that here we solve the full equations (B9)–(B11), not only the first one (equation (B9)) as other authors did. The other factor of the improvement is about computing the weak-current matrix elements for the decays with the obtained relativistic wavefunctions as input. It is more 'complete' than that done in [9, 10, 18], i.e. the 'complete' formula in equation (15).

The paper is organized as follows. The formulations of the exclusive semileptonic and nonleptonic decays are outlined in section 2. The newly developed formulations, mainly for the matrix elements of the hadron weak decays, are presented in section 3. In section 4, numerical calculations for the exclusive semileptonic decays and nonleptonic decays are described, and the results and comparisons among the various approaches are presented. Section 5 is attributed to discussions. In the appendices, the formulations as necessary pieces for the calculations of the decays are given.

Let us now derive the formulations for the exclusive semileptonic and nonleptonic decays precisely (mainly quoted from [22]) for numerical calculations later on.

In the following subsections, we will focus on the matrix elements of weak currents and show how to present the amplitudes of the semileptonic or nonleptonic decays via the matrix elements of weak currents precisely. In fact, one may see that the newly developed method mainly is about the matrix elements of weak currents.

2.1. The semileptonic decays of the B_c meson

Figure 1 is a typical Feynman diagram responsible for a semileptonic decay of the B_c meson to a charmonium. The corresponding amplitude for the decay can be written as

$$\begin{equation} T=\frac{G_F}{\sqrt{2}}V_{bc}\bar{u}_{{\nu }_l}(p_\nu ){\gamma }_{\mu } (1-{\gamma }_5){\upsilon }_l(p_l)\langle \chi _c(h_{c})(P_f)|J^{\mu }|B_c(P)\rangle , \end{equation} \tag{ 1 }$$

where V_bc is the CKM matrix element, 〈χ_c(h_c)(P_f)|J^μ|B_c(P)〉 is the hadronic weak-current matrix element responsible for the decay, and P, P_f, p_ν and p_l are the momenta of initial state B_c, the finial P-wave state of $(c\bar{c})$ (i.e. h_c, χ_c0, χ_c1, χ_c2 and their excited states), the neutrino and the charged lepton, respectively.

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Generally, the form factors are defined in terms of the matrix elements of weak current responsible for the decays appearing in equation (1). Namely for the decay of the B_c meson to the scalar charmonium χ_c0, the form factors s₊ and s₋ are defined as follows:

$$\begin{eqnarray} &&\langle {\chi }_{c0}(P_f)|V^{\mu }|B_c(P)\rangle =0,\nonumber \\ && \langle {\chi }_{c0}(P_f)|A^{\mu }|B_c(P)\rangle =s_+(P+P_f)^{\mu }+s_-(P-P_f)^{\mu }. \end{eqnarray} \tag{ 2 }$$

For the decay of the B_c meson to the vector charmonium χ_c1, the relevant form factors f, u₁, u₂ and g are defined as follows:

$$\begin{eqnarray} && \langle {\chi }_{c1}(P_f)|V^{\mu }|B_c(P)\rangle =f(M+M_f){\varepsilon }^{\mu }+\big[u_1P^{\mu }+u_2P_f^{\mu }\big]\frac{{\varepsilon }\cdot P}{M},\nonumber \\ && \langle {\chi }_{c1}(P_f)|A^{\mu }|B_c(P)\rangle =\frac{2g}{M+M_f}{\rm i}{\epsilon }^{\mu \nu \rho \sigma } {\varepsilon }_{\nu }P_{\rho }{P_f}_{\sigma }. \end{eqnarray} \tag{ 3 }$$

For the decay of the B_c meson to the vector charmonium h_c, the relevant form factors V₀, V₁, V₂ and V₃ are defined as follows:

$$\begin{eqnarray} && \langle h_c(P_f)|V^{\mu }|B_c(P)\rangle =V_0(M+M_f){\varepsilon }^{\mu }+\big[V_1P^{\mu }+V_2P_f^{\mu }\big]\frac{{\varepsilon }\cdot P}{M},\nonumber \\ && \langle h_c(P_f)|A^{\mu }|B_c(P)\rangle =\frac{2V_3}{M+M_f}{\rm i}{\epsilon }^{\mu \nu \rho \sigma } {\varepsilon }_{\nu }P_{\rho }{P_f}_{\sigma }. \end{eqnarray} \tag{ 4 }$$

For the decay of the B_c meson to the tenser charmonium χ_c2, the relevant form factors k, c₁, c₂ and h are defined as follows:

$$\begin{eqnarray} &&\langle {\chi }_{c2}(P_f)|A^{\mu }|B_c(P)\rangle =k(M+M_f){\varepsilon }^{\mu \alpha }\frac{P_{\alpha }}{M} +{\varepsilon }_{\alpha \beta }\frac{P^{\alpha }P^{\beta }}{M^2} \big(c_1P^{\mu }+c_2P_f^{\mu }\big),\nonumber \\ && \langle {\chi }_{c2}(P_f)|V^{\mu }|B_c(P)\rangle = \frac{2h}{M+M_f}{\rm i}{\varepsilon }_{\alpha \beta }\frac{P^{\alpha }}{M} {\epsilon }^{\mu \beta \rho \sigma }P_{\rho }{P_f}_{\sigma }. \end{eqnarray} \tag{ 5 }$$

In the case without considering polarization, we have the squared decay-amplitude with the polarizations in final states being summed:

$$\begin{equation} \Sigma _{s_\nu ,s_l,S_{\chi _c(h_c)}}|T|^2=\frac{G_F^2}{2}|V_{bc}|^2l_{\mu \nu }h^{\mu \nu }, \end{equation} \tag{ 6 }$$

where l_μν is the leptonic tensor:

$$$ l_{\mu \nu }=\Sigma _{s_\nu ,s_l}\bar{{\upsilon }_l}(p_l){\gamma }_{\mu }(1-{\gamma }_5) {u}_{{\nu }_l}(p_\nu )\bar{u}_{{\nu }_l}(p_\nu ){\gamma }_{\nu } (1-{\gamma }_5){\upsilon }_l(p_l), $$$

and the hadronic tensor relating to the weak current in equation (1) is

$$\begin{eqnarray} &&\fl h^{\mu \nu } \equiv \Sigma _{S_{\chi _c(h_c)}}\langle B_c(P)|J^{\mu +}|\chi _c(h_{c})(P_f)\rangle \langle \chi _c(h_{c})(P_f)|J^{\nu }|B_c(P)\rangle \nonumber \\ &&\fl \quad \;\;\, =-{\alpha }g^{\mu \nu }+{\beta }_{++}(P+P_f)^{\mu }(P+P_f)^{\nu } +{\beta }_{+-}(P+P_f)^{\mu }(P-P_f)^{\nu }\nonumber \\ &&+\, {\beta }_{-+}(P-P_f)^{\mu }(P+P_f)^{\nu }+{\beta }_{--}(P-P_f)^{\mu }(P-P_f)^{\nu }\nonumber \\ &&+\, i\gamma {\epsilon }^{\mu \nu \rho \sigma }(P+P_f)_{\rho }(P-P_f)_{\sigma }, \end{eqnarray} \tag{ 7 }$$

where the functions α, β₊₊, β₊₋, β₋₊ and β₋₋, γ are related to the form factors and we put the relations in appendix A precisely.

The total decay width Γ can be written as

$$\begin{eqnarray} &&\fl \Gamma = \frac{1}{2M(2\pi )^9}\int \frac{{\rm d}^3\vec{P}_f}{2E_f} \frac{{\rm d}^3\vec{p}_l}{2E_l}\frac{{\rm d}^3\vec{p}_{\nu }}{2E_{\nu }} (2\pi )^4{\delta }^4(P-P_f-p_l-p_{\nu })\Sigma _{s_\nu ,s_l,S_{\chi _c(h_c)}}|T|^2, \end{eqnarray} \tag{ 8 }$$

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/M², the differential width of the decay can be reduced to

$$\begin{eqnarray} &&\fl \frac{{\rm d}^2\Gamma }{{\rm d}x\,{\rm d}y}=|V_{bc}|^2\frac{G_F^2M^5}{64{\pi }^3}\left\lbrace \frac{2\alpha }{M^2}\left(y-\frac{m_l^2}{M^2}\right)\right.\nonumber \\ &&+\, {\beta }_{++}\left[4\left(2x\left(1-\frac{M_f^2}{M^2}+y\right)-4x^2-y\right) +\frac{m_l^2}{M^2}\left(8x+4\frac{M_f^2}{M^2}-3y-\frac{m_l^2}{M^2}\right)\right]\nonumber \\ &&+\, ({\beta }_{+-}+{\beta }_{-+})\frac{m_l^2}{M^2} \left(2-4x+y-2\frac{M_f^2}{M^2}+\frac{m_l^2}{M^2}\right) +{\beta }_{--}\frac{m_l^2}{M^2}\left(y-\frac{m_l^2}{M^2}\right)\nonumber \\ &&\left.-\left[2{\gamma }y\left(1-\frac{M_f^2}{M^2}-4x+y+\frac{M_l^2}{M^2}\right)+ 2\gamma \frac{M_l^2}{M^2}\left(1-\frac{M_f^2}{M^2}\right)\right]\right\rbrace , \end{eqnarray} \tag{ 9 }$$

where M is the mass of the meson B_c, M_f is the mass of the charmonium in the final state and the total width of the decay is just an integration of the differential width, i.e. $\Gamma =\int {\rm d}x\int {\rm d}y\frac{{\rm d}^2\Gamma }{{\rm d}x\,{\rm d}y}$.

Thus, the key problem for calculating the semileptonic decays is turned to calculating the hadronic weak-current matrix elements.

2.2. The nonleptonic decays of the B_c meson

In this subsection, we mainly consider the nonleptonic two-body decays to a P-wave charmonium, i.e. decays B_c → M₁M₂ where M₁ is a P-wave charmonium and M₂ is a common meson. Figure 2 represents the Feynman diagram for the decays via the relevant effective Hamiltonian H_eff [25, 26]:

$$\begin{eqnarray} &&\fl H_{\rm eff}=\frac{G_F}{\sqrt{2}}\left\lbrace V_{cb}\big[c_1(\mu )O_1^{cb}+c_2(\mu )O_2^{cb}\big] -V_{tb}V_{tq}^*\left(\sum _{i=3}^{10}C_i(\mu )O_i\right)\right\rbrace +h.c., \end{eqnarray} \tag{ 10 }$$

where G_F is the Fermi constant, q = d, s, V_ij are the CKM matrix elements and c_i(μ) are the scale-dependent Wilson coefficients. O_i are the operators constructed by four quark fields and have the J^μJ_μ structure as follows:

$$\begin{eqnarray} &&\fl O_1^{cb}=[V_{ud}(\bar{d}_{\alpha }u_{\alpha })_{V-A}+V_{us}(\bar{s}_{\alpha }u_{\alpha })_{V-A}\ +V_{cd}(\bar{d}_{\alpha }c_{\alpha })_{V-A}+V_{cs}(\bar{s}_{\alpha }c_{\alpha })_{V-A}] (\bar{c}_{\beta }b_{\beta })_{V-A},\nonumber \\ &&\fl O_2^{cb}=[V_{ud}(\bar{d}_{\alpha }u_{\beta })_{V-A}+V_{us}(\bar{s}_{\alpha }u_{\beta })_{V-A}\ +V_{cd}(\bar{d}_{\alpha }c_{\beta })_{V-A}+V_{cs}(\bar{s}_{\alpha }c_{\beta })_{V-A}] (\bar{c}_{\beta }b_{\alpha })_{V-A},\nonumber \\ &&\fl O_3=(\bar{q}_\alpha b_\alpha )_{V-A}\sum \limits _{q^{\prime }}(\bar{q}^{\prime }_\beta q^{\prime }_\beta )_{V-A},\quad \quad \quad O_4=(\bar{q}_\beta b_\alpha )_{V-A}\sum \limits _{q^{\prime }}(\bar{q}^{\prime }_\alpha q^{\prime }_\beta )_{V-A}, \nonumber \\ &&\fl O_5=(\bar{q}_\alpha b_\alpha )_{V-A}\sum \limits _{q^{\prime }}(\bar{q}^{\prime }_\beta q^{\prime }_\beta )_{V+A}, \quad \quad \quad O_6=(\bar{q}_\beta b_\alpha )_{V-A}\sum \limits _{q^{\prime }}(\bar{q}^{\prime }_\alpha q^{\prime }_\beta )_{V+A}, \nonumber \\ &&\fl O_7=\frac{3}{2}(\bar{q}_\alpha b_\alpha )_{V-A}\sum \limits _{q^{\prime }}e_{q^{\prime }}(\bar{q}^{\prime }_\beta q^{\prime }_\beta )_{V+A},\quad \quad \quad O_8=\frac{3}{2}(\bar{q}_\beta b_\alpha )_{V-A}\sum \limits _{q^{\prime }}e_{q^{\prime }}(\bar{q}^{\prime }_\alpha q^{\prime }_\beta )_{V+A}, \nonumber \\ &&\fl O_9=\frac{3}{2}(\bar{q}_\alpha b_\alpha )_{V-A}\sum \limits _{q^{\prime }}e_{q^{\prime }}(\bar{q}^{\prime }_\beta q^{\prime }_\beta )_{V-A},\quad \quad \quad O_{10}=\frac{3}{2}(\bar{q}_\beta b_\alpha )_{V-A}\sum \limits _{q^{\prime }}e_{q^{\prime }}(\bar{q}^{\prime }_\alpha q^{\prime }_\beta )_{V-A}, \end{eqnarray} \tag{ 11 }$$

where $(\bar{q}_1q_2)_{V-A}=\bar{q}_1{\gamma }^{\mu }(1-{\gamma }_5)q_2$. The operators O₁ and O₂ are the current–current (tree) operators, O₃, ..., O₆ are the QCD-penguin operators and O₇, ..., O₁₀ are the electroweak penguin operators. Since we calculate the decay up to leading order, we just consider the contribution of O₁ and O₂.

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Here we apply the so-called naive factorization to H_eff, i.e. the operators O_i [27], so the nonleptonic two-body decay amplitude T can be reduced to a product of a transition matrix element of a weak current 〈M₁|J^μ|B_c〉 and an annihilation matrix element of another weak current 〈M₂|J_μ|0〉:

$$\begin{eqnarray} &&T=\langle M_1M_2|H_{\rm eff}|B_c\rangle \approx \langle M_1|J^{\mu }|B_c\rangle \langle M_2|J_{\mu }|0\rangle , \end{eqnarray} \tag{ 12 }$$

while the annihilation matrix element relates to a decay constant directly. The reason why we adopt the naive factorization here is that it works well enough due to the fact that all the decays concerned in this paper are 'constrained' to those in which the quark c as a 'spectator' goes from the initial B_c meson into the final meson M₁ always; thus, as pointed out by the authors of [28, 29], in the concerned cases, the corrections to the naive factorization are suppressed.

Since M₁ = χ_c(h_c), the matrix element 〈M₁|J^μ|B_c〉 is just the hadronic weak-current matrix element appearing in the previous subsection, the only difference being that the momentum transfer is fixed for the matrix element here (owing to the decays of one- to two-body). The annihilation matrix element 〈M₂|J_μ|0〉 with $J^{\mu }=(\bar{q}_1q_2)_{V-A}$ is related to the decay constant of a 'common meson' M₂ and can be measured via proper processes generally.

Precisely, let us now 'restrict ourselves' to analyze the B_c nonleptonic decays to the P-wave charmonium and the π⁺, ρ⁺, etc, which are governed by the weak decay $\bar{b}\rightarrow \bar{c}u\bar{d}$, or to the P-wave charmonium and K⁺, K*, etc, which are governed by the weak decay $\bar{b}\rightarrow \bar{c}u\bar{s}$. As an example, under naive factorization, we have the decay amplitude of B_c → χ_c0ρ⁺ as follows:

$$\begin{eqnarray} &&T(B_c \rightarrow {\chi }_{c0}{\rho }^+)=\frac{G_F}{\sqrt{2}}V_{cb}V^*_{ud} a_1(\mu )\langle {\chi }_{c0}|J^{\mu }|B_c\rangle \langle {\rho }^ +|J_{\mu }|0\rangle , \end{eqnarray} \tag{ 13 }$$

where $a_1=c_1+\frac{1}{N_c}c_2$ and N_c = 3 is the number of colors.

Since 〈M₂|J_μ|0〉 relates to the decay constant of the meson M₂ directly, calculating the widths of the nonleptonic decays is straightforward when the weak-current transition matrix elements 〈M₁|J^μ|B_c(P)〉 are well calculated. Thus, one may see that the problem of calculating the nonleptonic decays is essentially attributed to calculating the hadronic weak-current matrix elements 〈M₁|V^μ|B_c(P)〉 and 〈M₁|A^μ|B_c(P)〉 appearing in the above subsection for semileptonic decays.

From the previous section, we can see that calculating the weak-current matrix elements 〈M₁|J^μ|B_c(P)〉 is the key problem for the concerned semileptonic and nonleptonic decays, so let us now explain the reason why and show how to apply the newly developed method [22] to calculate the matrix elements. In fact it is also to prepare necessary formulas for final numerical calculations.

Here the weak-current matrix elements are for 'transitions' from a state of a double heavy meson to another double heavy meson. Due to the mass difference of the two states, the relativistic effects for the transitions are great enough that a proper formulation to deal with the relativistic effects is desired. It is known that the approach of the relativistic B-S equation for the bound states and Mandelstam formulation for the transition matrix elements may be taken into account quite well, and furthermore the B-S equation and Mandelstam formulation still work, even under the 'instantaneous approximation', because here the involved mesons are double heavy while the newly developed method [22], which applies the 'instantaneous approximation' to the current matrix elements and the B-S equation completely, should be better than the original one in [18], where the 'instantaneous approximation' is applied incompletely. The 'completeness' here means to apply it to the B-S equation, the solutions (B-S wavefunctions) and the transition matrix element (under Mandelstam formulation) properly, and let us outline it below.

According to the Mandelstam formulation [21], the corresponding hadronic matrix elements of weak current between the double heavy meson B_c in the initial state and the double heavy meson χ_c(h_c) in the final state, appearing in equation (1), equations (12) and (13), can be written as

$$\begin{eqnarray} &&\fl \langle \chi _c(h_c)(P_f)|J^{\mu }|B_c(P)\rangle \nonumber \\ &&={\rm i}\int \frac{{\rm d}^{4}q{\rm d}^4q^{\prime }}{(2\pi )^{4}} {\rm Tr}[\overline{\chi }_{\chi _c(h_c)}(P^{\prime },q^{\prime }) (\not\!{p_{1}}-m_{1})\chi _{_{B_c}}(P,q)V_{cb}\gamma ^{\mu } (1-\gamma _5)\delta (p_1-p^{\prime }_1)]\nonumber \\ &&={\rm i}\int \frac{{\rm d}^{4}q}{(2\pi )^{4}}{\rm Tr} \left[\overline{\chi }_{\chi _c(h_c)}(P^{\prime },q^{\prime }) (\alpha _1\not\!{P}+\not\!q-m_{1})\chi _{_{B_c}}(P,q)V_{cb} \gamma ^{\mu }(1-\gamma _5)\right], \end{eqnarray} \tag{ 14 }$$

where p₁ = α₁P + q ($\alpha _1 \equiv \frac{m_1}{ m_1 + m_2}$), p₂ = α₂P − q ($\alpha _2\equiv \frac{ m_2}{ m_1 + m_2}$) are the momenta of c-quark and $\bar{b}$-quark, respectively, inside the B_c meson; p'₁ = α'₁P_f + q' ($\alpha ^{\prime }_1 \equiv \frac{m^{\prime }_1}{ m^{\prime }_1 + m^{\prime }_2}$) and p'₂ = α'₂P_f − q' ($\alpha ^{\prime }_2\equiv \frac{ m^{\prime }_2}{ m^{\prime }_1 + m^{\prime }_2}$) are the momenta of the c-quark and $\bar{c}$-quark, respectively, inside the P-wave charmonium χ_c(h_c); moreover, for the final result (the last line of equation (14)), we have P = P_f + p_l + p_ν and q' = α₁P + q − α'₁P_f.

The newly developed method [22] essentially is to apply the 'instantaneous approximation' to the current matrix elements and the B-S equation completely, to outline it and for 'applying the instantaneous approximation' in a covariant way, we need to decompose the relative momentum q into two components: the time-like one q^μ_∥ and the space-like one q^μ_⊥ as follows:

$$\begin{eqnarray*} &&q^\mu =q^\mu _\parallel +q^\mu _\perp ,\quad q^\mu _\parallel \equiv \frac{ P\cdot q}{M^2}P^\mu ,\quad q^\mu _\perp \equiv q^\mu -q^\mu _\parallel , \end{eqnarray*}$$

$$\begin{eqnarray*} &&P^{\prime }{^{\mu }}=P^{\prime }{^{\mu }}_{\parallel }+P^{\prime }{^{\mu }}_{\perp } ,\quad P^{\prime }{^{\mu }}_{\parallel }\equiv \frac{(P\cdot P^{\prime })}{M^{2}}P^{\mu } ,\quad P^{\prime }{^{\mu }}_{\perp }\equiv P^{\prime }{^{\mu }}-P^{\prime }{^{\mu }}_{\parallel } ; \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&q^{\prime \mu }=q^{\prime \mu }_{\parallel }+q^{\prime \mu }_{\perp }, \;\;\;\; q^{\prime \mu }_{\parallel }\equiv (P\cdot q^{\prime }/M^{2})P^{\mu }, \;\;\;\; q^{\prime \mu }_{\perp }\equiv q^{\prime \mu }-q^{\prime \mu }_{\parallel }, \end{eqnarray*}$$

where M is the mass of the meson B_c, and we may further have two Lorentz invariant variables $q_P\equiv \frac{P\cdot q}{M}$ and $q_T\equiv \sqrt{-q_\perp ^2}$.

The 'instantaneous approximation' being applied to the matrix element is just to carry out the integration of dq^μ_∥ by a contour one in equation (14) precisely and to obtain the result as follows:

$$\begin{eqnarray} &&\fl \langle \chi _c(h_c)(P_f)|J^{\mu }|B_c(P)\rangle\nonumber\\ &&= {\rm i}\int \frac{{\rm d}^{4}q}{(2\pi )^{4}}{\rm Tr}[\overline{\chi }_{\chi _c(h_c)}(P^{\prime },q^{\prime }) (\alpha _1\not \!{P} \,+\not\! q-m_{1}) \chi _{_{B_c}}(P,q)V_{cb}\gamma ^{\mu }(1{-}\gamma _5)] \nonumber \\ &&=\int \frac{{\rm d}^3q_{\bot }}{(2\pi )^3}{\rm Tr} \Bigg \lbrace \Bigg [\bar{\varphi }^{\prime ++}(q_{\bot }^{\prime })\frac{\not\!P}{M}{\varphi }^{++} (q_{\bot })+\bar{\varphi }^{\prime ++}(q_{\bot }^{\prime })\frac{\not\!P}{M}{\psi }^{+-} (q_{\bot })\nonumber \\ &&-\,\bar{\psi }^{\prime -+}(q_{\bot }^{\prime })\frac{\not\!P}{M}{\varphi }^{++} (q_{\bot })-\bar{\psi }^{\prime +-}(q_{\bot }^{\prime })\frac{\not\!P}{M}{\varphi }^{--} (q_{\bot })+\bar{\varphi }^{\prime --}(q_{\bot }^{\prime })\frac{\not\!P}{M}{\psi }^{-+} (q_{\bot })\nonumber \\ &&-\,\bar{\varphi }^{\prime --}(q_{\bot }^{\prime })\frac{\not\!P}{M}{\varphi }^{--} (q_{\bot })\Bigg ]{\gamma }^{\mu }(1-{\gamma }_5)\Bigg \rbrace , \end{eqnarray} \tag{ 15 }$$

where

$$\begin{eqnarray} &&\fl \varphi ^{++}(q_\perp )=\frac{\Lambda _{1}^{+}(q_\perp )\eta (q_\perp ) \Lambda _{2}^{+}(q_\perp )}{M-\omega _{1}-\omega _{2}}, \quad \bar{\varphi }^{\prime ++}(q^{\prime }_{P\perp })=\frac{\Lambda _{2}^{\prime +}(q^{\prime }_{P\perp })\bar{\eta }(q^{\prime }_{P\perp }) \Lambda _{1}^{\prime +}(q^{\prime }_{P\perp })}{E_f-\omega ^{\prime }_{1}-\omega ^{\prime }_{2}},\nonumber \\ &&\fl \varphi ^{--}(q_\perp )=-\frac{\Lambda _{1}^{-}(q_\perp )\eta (q_\perp ) \Lambda _{2}^{-}(q_\perp )}{M+\omega _{1}+\omega _{2}}, \quad \bar{\varphi }^{\prime --}(q^{\prime }_{P\perp })=-\frac{\Lambda _{2}^{\prime -}(q^{\prime }_{P\perp })\bar{\eta }(q^{\prime }_{P\perp }) \Lambda _{1}^{\prime -}(q^{\prime }_{P\perp })}{E_f+\omega ^{\prime }_{1}+\omega ^{\prime }_{2}},\nonumber \\ &&\fl {\psi }^{-+}(q_\perp )= \frac{ \Lambda ^{-}_{1}(q_{P\perp }) \eta (q_\perp )\Lambda ^{+}_{2}(q_{P\perp }) }{M-\omega _{2}-\omega ^{\prime }_{2}-E_f}, \quad \bar{\psi }^{\prime -+}(q^{\prime }_{P\perp })= \frac{\Lambda ^{\prime -}_2(q^{\prime }_{P\perp }) \bar{\eta }^{\prime }(q^{\prime }_{P\perp })\Lambda ^{\prime +}_1(q^{\prime }_{P\perp })}{M-\omega _{2}-\omega ^{\prime }_{2}-E_f},\nonumber \\ &&\fl {\psi }^{+-}(q_\perp )= \frac{\Lambda ^{+}_{1}(q_{P\perp })\eta (q_\perp )\Lambda ^{-}_{2}(q_{P\perp }) }{M+\omega _{2}+\omega ^{\prime }_{2}-E_f}, \quad \bar{\psi }^{\prime +-}(q^{\prime }_{P\perp })= \frac{\Lambda ^{\prime +}_2(q^{\prime }_{P\perp }) \bar{\eta ^{\prime }}(q^{\prime }_{P\perp })\Lambda ^{\prime -}_1(q^{\prime }_{P\perp }) }{M+\omega _{2}+\omega ^{\prime }_{2}-E_f}. \end{eqnarray} \tag{ 16 }$$

φ^ij(q_⊥), ψ^ij(q_⊥) and ${\bar{\varphi }}^{\prime {ij}}(q_{P\bot }^{\prime }),\,\bar{\psi }^{\prime {ij}}(q_{P\bot }^{\prime })$ are the B-S wavefunctions as the B-S equation solutions under the 'complete instantaneous approximation' [23, 24] and with 'energy projection' Λ^± of the mesons in initial and finial states properly. The precise definitions of the 'energy projection' and the B-S 'vertex' η_P, $\bar{\eta }_{P}$ (η'_P, $\bar{\eta ^{\prime }}_{P}$) are presented in appendix B. One may also see that the three equations, equations (B.9)–(B.11), are B-S equations under the complete instantaneous approximation, instead of the incomplete instantaneous approximation which considers only equation (B.9).

Namely the 'improvements' from the 'newly development method' are attributed to the following: (i) with the complete instantaneous approximation to the current matrix element, as a result, six terms appeared in the square bracket of equation (15) while in earlier studies only the first term appeared,

$$\begin{eqnarray} &&\fl \langle \chi _c(h_c)(P_f)|J^{\mu }|B_c(P)\rangle =\int \frac{{\rm d}^3q_{\bot }}{(2\pi )^3}{\rm Tr} \left \lbrace \bar{\varphi }^{\prime ++}(q_{\bot }^{\prime })\frac{P}{M}{\varphi }^{++} (q_{\bot }){\gamma }^{\mu }(1-{\gamma }_5)\right \rbrace; \end{eqnarray} \tag{ 17 }$$

(ii) the B-S wavefunctions hidden in φ^ij(q_⊥), ψ^ij(q_⊥) and ${\bar{\varphi }}^{\prime {ij}}(q_{P\bot }^{\prime }),\, \bar{\psi }^{\prime {ij}}(q_{P\bot }^{\prime })$ are solved under complete instantaneous approximation to the B-S equation. For point (i), since the considered double heavy meson, B_c or χ_c(h_c), is the weak binding system, i.e. the binding energy ε ≡ M − ω₁ − ω₂ (or ε ≡ E_f − ω'₁ − ω'₂) is small ($\frac{\varepsilon }{M}\ll O(1)$), from equation (16) we are sure that φ⁺⁺(q_⊥) and ${\bar{\varphi }}^{\prime ++}(q_{P\bot }^{\prime })$ are much greater than the others φ^ij(q_⊥), ψ^ij(q_⊥) and ${\bar{\varphi }}^{\prime {ij}}(q_{P\bot }^{\prime }), \,\bar{\psi }^{\prime {ij}}(q_{P\bot }^{\prime })$, so that using equation (17) instead of equation (15) is a very good approximation, which we have precisely examined by considering the decay B_c → χ_c0lν_l as an example: in fact, the contributions of the second term and third term of equation (15) to the form factor are less than that of the first term roughly by a factor of 10⁻²–10⁻³. If the first three terms are considered, the decay width is 1.85 × 10⁻¹⁵ GeV, while if only the first term is considered, the decay width is 1.87 × 10⁻¹⁵ GeV, i.e. the two results are very similar. So the approximation is very good and we may use equation (17) instead of equation (15) to compute the weak-current matrix elements safely.

In this section, based on the formulations obtained in the paper, we evaluate the widths for semileptonic and nonleptonic decays and some interesting quantities for semileptonic decays, such as form factors, charged lepton spectrum, etc, and then discuss them briefly.

First of all, we need to fix the parameters appearing in the framework. We adjust the parameters a = e = 2.7183, λ = 0.21 GeV², Λ_QCD = 0.27 GeV, m_b = 4.96 GeV, m_c = 1.62 GeV and V₀ for the B-S kernel as those in [24, 31, 30], which is the best input for spectroscopy; then the spectra of the mesons and the masses $M_{B_c}=6.276$, $M_{{\chi }_{c0}}=3.414$, $M_{{\chi }_{c1}}=3.510$, $M_{{\chi }_{c2}}=3.555,$ $M_{{h}_{c}}=3.526$ GeV, etc [24], which are used in this paper, are obtained; moreover, the decay constants, average energies as well as annihilations of quarkonia are fitted [30–32].

With the obtained B-S wavefunctions (under the formulation defined in appendix B) and as a next step, we substitute the functions into φ⁺⁺(q_⊥) and ${\bar{\varphi }}^{\prime ++}(q_{P\bot }^{\prime })$, so that they are related to the components of the B-S wavefunctions precisely as depicted in appendix C. With formula (17), finally we represent the hadronic transition weak-current matrix elements as proper integrations of the components of the B-S wavefunctions. As the final results of this paper, the decay widths for the semileptonic and nonleptonic decays and some interesting quantities for the semileptonic decays, such as form factors, charged lepton spectrum, etc, are straightforwardly calculated numerically. In the following subsections, we present the results for the semileptonic decays and nonleptonic decays separately.

4.1. The semileptonic decays

When the weak-current transition matrix element for a definite semileptonic decay is calculated precisely and the values of the CKM matrix elements |V_ud| = 0.974, |V_us| = 0.225, |V_bc| = 0.0406 [4] are given, not only the decay width can be calculated straightforwardly, but also the form factors may be extracted. Moreover, as 'semifinished product', the spectrum of the charged lepton which may be measurable experimentally can also be acquired. Namely the functions α, β₊₊, β₊₋, β₋₊, β₋₋, γ appearing in the spectrum of the charged lepton (see equation (9)) are related to the form factors directly as shown in appendix A precisely. Therefore, when we calculate and present the results for semileptonic decays, not only the decay widths but also the spectra of the charged lepton in the decays are considered. Since the τ lepton is quite massive and m_μ ≃ m_e is quite a good approximation for the B_c meson decays, thus when we calculate and present the widths and the spectra of the charged lepton for the decays, only the cases where the lepton is an electron or τ are considered.

Note that since the input B-S wavefunctions obtained by solving the B-S equation for the double heavy mesons which are involved in the transition matrix elements of weak current have uncertainties, due to the parameters being fitted to fix the B-S kernel and quark masses, the way to solve the B-S equation numerically, and the approximation from equations (15) to (17) for the transition matrix elements of the weak currents, is considered, so we finally obtain that the numerical results have certain errors. To consider the uncertainties caused by the input parameters, we change all the input parameters simultaneously within 5% of the center values; then we obtain the uncertainties of numerical results for the semileptonic decays and the nonleptonic decays shown in table 1. We find that the uncertainties of the decays B_c → h_c(χ_c) + e + ν_e vary up to 30% of center values, while the uncertainties of B_c → h_c(χ_c) + τ + ν_τ are up to 60% in table 1; the reason is that the phase spaces for B_c → h_c(χ_c) + τ + ν_τ are smaller than the ones for B_c → h_c(χ_c) + e + ν_e because of the heavy τ lepton, and the uncertainties for the former are more sensitive to the changes of the phase space than the latter.

Table 1. The semileptonic decay widths (in units 10⁻¹⁵ GeV).

Mode	This work	[12]	[13]	[15]	[10]	[16]	[17]
B+c → χc0eν	1.87 ± 0.46	1.27	2.52	1.55	1.69	2.60 ± 0.73
B+c → χc0τν	0.23 ± 0.12	0.11	0.26	0.19	0.25	0.7 ± 0.23
B+c → χc1eν	1.52 ± 0.45	1.18	1.40	0.94	2.21	2.09 ± 0.60
B+c → χc1τν	0.14 ± 0.10	0.13	0.17	0.10	0.35	0.21 ± 0.06
B+c → χc2eν	1.50 ± 0.39	2.27	2.92	1.89	2.73
B+c → χc2τν	0.12 ± 0.07	0.13	0.20	0.13	0.42
B+c → hceν	3.98 ± 1.10	1.38	4.42	2.4	2.51	2.03 ± 0.57	4.2 ± 2.1
B+c → hcτν	0.28 ± 0.20	0.11	0.38	0.21	0.36	0.20 ± 0.05	0.53 ± 0.26

To compare with the results obtained by the other approaches, we present the decay widths calculated in this work with error bar and the results obtained by the other approaches by putting them together in a table, i.e. table 1.

In addition, we also present the obtained form factors and the spectra of the charged lepton in the decays in figures 3 and 4, respectively. To compare with the results of the previous work [10], we draw the curves of the spectra of the charged lepton obtained in this work and [10] in figure 5. However, in order to see the tendency of the form factors and the lepton spectrum clearly, we suspect that at the present stage it is enough, so in the figures, we draw the curves with the center values but do not involve the errors precisely.

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4.2. The nonleptonic decays

In section 4, the form factors (figure 3), energy spectra of the charge leptons (figures 4 and 5), decay widths (table 1) for the semileptonic decays and the decay widths for nonleptonic decays (table 2) are presented. Specially in the tables some comparisons with other approaches are also given. Thus, one may already read off a lot of interesting results.

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Since the form factors for the semileptonic decays, which are directly related to overlapping integrations of the components of the B-S wavefunctions of the initial and final states as shown in appendix C, are comparatively difficult to measure, in figure 3 we show the behavior of the form factors briefly (without errors). However, the measurement of the energy spectra of the charged lepton in the decays may not be so difficult, as long as the event example is great enough and the abilities of the detector are strong enough, and to see the differences between the spectra of the electron and the τ lepton clearly in figure 4, we plot the curves with center values without theoretical uncertainties. Moreover, to see the differences between this work and [10], in figure 5 we plot the spectra of the electron obtained in this work versus the ones [10] obtained by the previous approach, and for both of them only center values without theoretical uncertainties are taken. Since the spectra of muon (μ) are very similar to those of the electron in exclusive semileptonic decays, we do not present the spectra of muon at all. From figure 4, we can see the difference in the energy spectra among the B_c decays to different P-wave charmonia clearly, although the results for the electron are better than those for the τ lepton. From figure 5, we can see that the difference in the energy spectra of the electron due to different approaches, the difference caused by the newly improved approach and by the previous approach, can be quite large and hence can be tested experimentally in future. For the widths of the decays, from tables 1 and 2, both the semileptonic decays and the nonleptonic decays, one may see that in general the results of this work fall into the region of the predictions by various models, but the distribution of the predictions is quite wide, so future experimental data will be critical and may conclude which of the predictions is more reliable.

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Considering the fact that substantial tests of the B_c-meson decays have not been started yet, although the meson B_c has been observed at Tevatron for years and the LHC is running now, according to the estimates of the production at the LHC, one may believe reasonably that the tests of the predictions on the B_c decays will be started with the LHC and more measurements will be available. From the theoretical point of view, we think that the newly improved approach works better than the previous one; this needs to be tested by experiments. We would also like to note here that according to the estimates [33–36] of the production at an e⁻e⁺ collider running at the CM energy $\sqrt{S}\simeq m_Z$ (m_Z is the Z-boson mass) with very high luminosity (L = 10^{34 − 36} cm⁻² s⁻¹), i.e. a 'super-Z-factory', and considering the advantages, it may be more suitable to test the approaches by measuring the decays precisely than to test them at hadronic colliders such as Tevatron or the LHC, because at such a super-Z-factory, numerous B_c mesons may be produced and the energy–momentum of the produced B_c meson, as the e⁻e⁺ one of the collider, is precisely known in an e⁺–e⁻ collider environment.

This work was supported in part by the Natural Science Foundation of China (NSFC) under grant nos 10875032, 10805082, 10875155 and 10847001. This research was also supported in part by the Project of Knowledge Innovation Program (PKIP) of Chinese Academy of Sciences, grant no KJCX2.YW.W10.

Here according to the P-wave charmonium appearing in the final state, we present the useful functions α, β₊₊, β₊₋, β₋₊, β₋₋, γ precisely to relate to the form factors in turn.

• (a)  
  When B_c decays to χ_c0. Since the matrix elements of weak currents are described in terms of two form factors (s₊, s₋),

  $$\begin{eqnarray*} &\langle {\chi }_{c0}(P_f)|V^{\mu }|B_c(P)\rangle =0,\nonumber \\ & \langle {\chi }_{c0}(P_f)|A^{\mu }|B_c(P)\rangle =s_+(P+P_f)^{\mu }+s_-(P-P_f)^{\mu },\nonumber \end{eqnarray*}$$

  the functions read

  $$\begin{eqnarray} &&{\beta }_{++}=s^2_+,\quad {\beta }_{+-}={\beta }_{-+}=s_+s_-,\quad {\beta }_{--}=s^2_-. \end{eqnarray} \tag{ A.1 }$$
• (b)  
  When B_c decays to χ_c1. Since the matrix elements of weak currents can be described in terms of four form factors (f, u₁, u₂, g),

  $$\begin{eqnarray*} &\displaystyle \langle {\chi }_{c1}(P_f)|V^{\mu }|B_c(P)\rangle =f(M+M_f){\varepsilon }^{\mu }+\big[u_1P^{\mu }+u_2P_f^{\mu }\big]\frac{{\varepsilon }\cdot P}{M},\nonumber \\ &\displaystyle \langle {\chi }_{c1}(P_f)|A^{\mu }|B_c(P)\rangle =\frac{2g}{M+M_f}{\rm i}{\epsilon }^{\mu \nu \rho \sigma }{\varepsilon }_{\nu }P_{\rho }{P_f}_{\sigma },\nonumber \end{eqnarray*}$$

  the functions read

  $$\begin{eqnarray} &&\fl \alpha =f_1^2+4M^2g_1^2\vec{p}_f{\!\,}^2,\nonumber \\ &&\fl {\beta }_{++}=\frac{f_1^2}{4M_f^2}-M^2g_1^2y+\frac{1}{2} \left[\frac{M^2}{M_f^2}(1-y)-1\right]f_1u_+ +\frac{M^2\vec{p}_f{\!\,}^2}{M_f^2}u^2_+,\nonumber \\ &&\fl \beta _{+-}\,{=}\beta _{-+}{=}\,g_1^2\big(M^2-M^2_f\big){-}\frac{f_1^2}{4M^{2}_f} {-}\frac{1}{2}f_1(u_{+}+u_{-})-\frac{1}{2}\frac{ME_f}{M^2_f}f_1(u_{+}{-}u_{-}) {+}u_{+}u_{-}\frac{M^2{\vec{p}_f }{\!\,}^2}{M^2_f},\nonumber \\ &&\fl \beta _{--}=-g_1^2\big(M^2+2ME_f+M^2_f\big)+\frac{f_1^2}{4M^{2}_f} -\left(\frac{ME_f}{M^2_f}+1\right)f_1u_{-} +u^2_{-}\frac{M^2{\vec{p_f}^2}}{M^2_f},\nonumber \\ &&\fl \gamma =-2f_1g_1\, \end{eqnarray} \tag{ A.2 }$$

  when setting $f_1=f(M+M_f), u_+=\frac{(u_1+u_2)}{2M}, u_-=\frac{(u_1-u_2)}{2M}\;{\rm and }\;g_1=\frac{g}{M+M_f}$.
• (c)  
  When B_c decays to h_c. Since the matrix elements of weak currents can be described in terms of four invariant form factors (V₀, V₁, V₂, V₃),

  $$\begin{eqnarray*} &&\langle h_c(P_f)|V^{\mu }|B_c(P)\rangle =V_0(M+M_f){\varepsilon }^{\mu }+\big[V_1P^{\mu }+V_2P_f^{\mu }\big]\frac{{\varepsilon }\cdot P}{M},\nonumber \\ &&[-1pt] \langle h_c(P_f)|A^{\mu }|B_c(P)\rangle =\frac{2V_3}{M+M_f}{\rm i}{\epsilon }^{\mu \nu \rho \sigma }{\varepsilon }_{\nu }P_{\rho }{P_f}_{\sigma },\nonumber \end{eqnarray*}$$

  the functions read

  $$\begin{eqnarray} &&\fl \alpha =f_1^2+4M^2g_1^2\vec{p}_f{\!\,}^2,\nonumber \\ &&[-1.2pt] \fl {\beta }_{++}=\frac{f_1^2}{4M_f^2}-M^2g_1^2y+ \frac{1}{2}\left[\frac{M^2}{M_f^2}(1-y)-1\right]f_1a_+ +\frac{M^2\vec{p}_f{\!\,}^2}{M_f^2}a^2_+,\nonumber \\ &&[-1.2pt] \fl \beta _{+-}=\beta _{-+}\nonumber \\ &&\fl \quad \quad =g_1^2(M^2-M^2_f)-\frac{f_1^2}{4M^{2}_f} -\frac{1}{2}f_1(a_{+}+a_{-})-\frac{1}{2}\frac{ME_f}{M^2_f}f_1(a_{+}-a_{-}) +a_{+}a_{-}\frac{M^2{\vec{p}_f }{\!\,}^2}{M^2_f},\nonumber\\ &&[-1.2pt] \fl \beta _{--}=-g_1^2(M^2+2ME_f+M^2_f)+\frac{f_1^2}{4M^{2}_f} -\left(\frac{ME_f}{M^2_f}+1\right)f_1a_{-}+a^2_{-}\frac{M^2{\vec{p_f}{\!\,}^2}}{M^2_f},\nonumber \\ &&[-1.2pt] \fl \gamma =-2f_1g_1, \end{eqnarray} \tag{ A.3 }$$

  when setting $f_1=V_0(M+M_f), a_+=\frac{(V_1+V_2)}{2M}, a_-=\frac{(V_1-V_2)}{2M}\;{\rm and }\;g_1=\frac{V_3}{M+M_f}$.
• (d)  
  When B_c decays to χ_c2. Since the matrix elements of weak currents can be described in terms of four form factors (k, c₁, c₂, h),

  $$\begin{eqnarray*} &&\langle {\chi }_{c2}(P_f)|A^{\mu }|B_c(P)\rangle =k(M+M_f){\varepsilon }^{\alpha \mu }\frac{P_{\alpha }}{M} +{\varepsilon }_{\alpha \beta }\frac{P^{\alpha }P^{\beta }}{M^2}\big(c_1P^{\mu }+c_2P_f^{\mu }\big),\nonumber \\ &&[-1.2pt] \langle {\chi }_{c2}(P_f)|V^{\mu }|B_c(P)\rangle = \frac{2h}{M+M_f}{\rm i}{\varepsilon }_{\alpha \beta }\frac{P^{\alpha }}{M} {\epsilon }^{\mu \beta \rho \sigma }P_{\rho }{P_f}_{\sigma },\nonumber \end{eqnarray*}$$

  where ε_αβ(ε^αμ) is the polarization tensor of the tensor meson, the functions read

  $$\begin{eqnarray} &&\fl \alpha =\frac{c}{2}\big(k_1^2+4Mh_1^2\vec{p}_f{\!\,}^{2}\big),\nonumber \\ &&[-1.2pt] \fl {\beta }_{++}=\frac{ck_1^2}{8M_f^2}-\frac{ch_1^2}{2}M^2y+\frac{2}{3}c^2c_+^2\nonumber \\ &&[-1.2pt] +\frac{4}{3}ck_1c_+\left(\frac{M^2(1-y)+M_f^2}{4M_f^2}-\frac{1}{2}\right) +\frac{k_1^2}{6}\left(\frac{M^2(1-y)+M_f^2}{4M_f^2}-\frac{1}{2}\right)^2,\nonumber \\ &&[-1.2pt] \fl {\beta }_{+-}\,{=}\,{\beta }_{-+}\,{=}\,-\frac{ck_1^2}{8M_f^2}{+}\frac{ch_1^2}{2}(M^2-M_f^2) +\frac{k_1^2}{6}\left[\left(\frac{M^2(1-y)+M_f^2}{4M_f^2}\right)^2-\frac{1}{4}\right] +\frac{2}{3}c^2c_+c_-\nonumber \\ &&[-1.2pt] -\, \frac{2}{3}ck_1c_+\left(\frac{M^2(1{-}y){+}M_f^2}{4M_f^2}{+}\frac{1}{2}\right) +\frac{2}{3}ck_1c_-\left(\frac{M^2(1-y)+M_f^2}{4M_f^2}-\frac{1}{2}\right),\nonumber \\ &&[-1.2pt] \fl {\beta }_{--}=\frac{ck_1^2}{8M_f^2}-\frac{ch_1^2}{2}(2(M^2+M_f^2)-M^2y) +\frac{2}{3}c^2c_-^2\nonumber \\ &&[-1.2pt] +\frac{4}{3}ck_1c_-\left(-\frac{M^2(1-y)+M_f^2}{4M_f^2}-\frac{1}{2}\right) +\frac{k_1^2}{6}\left(\frac{M^2(1-y)+M_f^2}{4M_f^2}+\frac{1}{2}\right)^2,\nonumber \\ &&\fl \gamma =-ch_1k_1, \nonumber\\ &&[-21pt] \end{eqnarray} \tag{ A.4 }$$

  when setting $c=\frac{M^2{\vec{p}_f}{\!\,}^{2}}{M_f^2},\, k_1=k(1+\frac{M_f}{M}),\,c_+=\frac{c_1+c_2}{2M^2},\, c_-=\frac{c_1-c_2}{2M^2}\;{\rm and }\;h_1=\frac{h}{M(M+M_f)}$.

In this appendix, we outline the 'complete instantaneous approximation' onto the B-S equation when it has an instantaneous kernel, which describes a double heavy meson quite well.

The B-S equation [20] reads

$$\begin{equation} (\not\!p_{1}-m_{1})\chi _{p}(q)(\not\!p_{2}+m_{2})={\rm i}\int \frac{{\rm d}^{4}k}{(2\pi )^{4}}V(P,k,q)\chi _{p}(k), \end{equation} \tag{ B.1 }$$

where χ_p(q) is the B-S wavefunction of the relevant bound state, P is the four-momentum of the meson state, and p₁, p₂, m₁ and m₂ are the momenta and constituent masses of the quark and antiquark, respectively. From the definition, they relate to the total momentum P and relative momentum q as follows:

$$$ p_1 = \alpha _1P + q,\quad \ \alpha _1 \equiv \frac{m_1}{ m_1 + m_2} , $$$

$$$ p_2 = \alpha _2P-q,\quad \ \alpha _2\equiv \frac{ m_2}{ m_1 + m_2}. $$$

The interaction kernel V(P, k, q) for a double heavy system, being approximately instantaneous, can be treated as a potential after performing instantaneous approximation, i.e. the kernel takes the simple form (in the rest frame) [19]

$$$ V(P,k,q)\quad \Rightarrow \quad V(|\vec{k}-\vec{q}|) .\\[-15pt] $$$

For various usages, we divide the relative momentum q into two parts,

$$$ q^\mu =q^\mu _\parallel +q^\mu _\perp ,\quad \ \ q^\mu _\parallel \equiv \frac{ P\cdot q}{M^2}P^\mu ,\qquad \ \ q^\mu _\perp \equiv q^\mu -q^\mu _\parallel ,\\[-5pt] $$$

where M is the mass of the meson, and we may have two Lorentz invariant variables:

$$$ q_P\equiv \frac{P\cdot q}{M},\quad q_T\equiv \sqrt{-q_\perp ^2} .\\[-15pt] $$$

For convenience below, let us introduce the definitions

$$\begin{equation} \varphi _{p}\big(q_\perp ^{\mu }\big)\equiv {\rm i}\int \frac{\mathrm{d}q_{p}}{2\pi }\chi _{p}\big(q_\parallel ^{\mu },q_\perp ^{\mu }\big),\quad \eta \big(q_\perp ^{\mu }\big)\equiv \int \frac{\mathrm{d}k_\perp ^{3}}{(2\pi )^{3}}V\big(k_\perp ,q_\perp \big)\varphi _{p}\big(k_\bot ^{\mu }\big); \end{equation} \tag{ B.2 }$$

then the B-S equation can be rewritten as

$$\begin{equation} \chi (q_{\Vert },q_{\bot })=S_1(p_1)\eta (q_{\bot })S_2(p_2). \end{equation} \tag{ B.3 }$$

Owing to equations (B.2) and (B.3), it is reasonable and for convenience we may call η(q_⊥) the 'instantaneous B-S vertex'. The propagator of quark or antiquark may be decomposed as

$$$ S_i(p_i)=\frac{\Lambda _{i}^{+}(q_\perp )}{J(i)q_P+{\alpha }_iM-w_i+{\rm i}\epsilon } +\frac{\Lambda _{i}^{-}(q_\perp )}{J(i)q_P+{\alpha }_iM-w_i+{\rm i}\epsilon } , $$$

where i = 1, 2 for quark and antiquark, respectively, and J(i) = ( − 1)^{i + 1}, $\omega _{1}=\sqrt{m_{1}^{2}+q_T^{2}}$, $\omega _{2}=\sqrt{m_{2}^{2}+q_T^{2}}$ and Λ^±₁ and Λ₂^± are the generalized energy projection operators,

$$\begin{eqnarray} &&\fl \Lambda _{1}^{\pm }(q_\perp )\equiv \frac{1}{2\omega _{1}}\left[\frac{\not\!{P}}{M}\omega _{1}\pm (m_{1}+\not\!q_\perp )\right],\ \ \ \ \Lambda _{2}^{\pm }(q_\perp )\equiv \frac{1}{2\omega _{2}}\left[\frac{\not\!{P}}{M}\omega _{2}\mp (m_{2}+\not\!q_\perp )\right], \end{eqnarray} \tag{ B.4 }$$

and have the properties

$$\begin{eqnarray} &&\Lambda ^{+}_{iP}\big(q^\mu _{P_\perp }\big)+ \Lambda ^{-}_{iP}\big(q^\mu _{P_\perp }\big)= \frac{\not\!{P}}{M},\;\;\;\;\;\;\;\;\;\;\; \Lambda ^{\pm }_{iP}\big(q^\mu _{P_\perp }\big)\frac{\not\!{P}}{M} \Lambda ^{\mp }_{iP}\big(q^\mu _{P_\perp }\big)=0 ,\nonumber \\ &&\Lambda ^{\pm }_{iP}\big(q^\mu _{P_\perp }\big)\frac{\not\!{P}}{M} \Lambda ^{\pm }_{iP}\big(q^\mu _{P_\perp }\big) =\Lambda ^{\pm }_{iP}\big(q^\mu _{P_\perp }\big). \end{eqnarray} \tag{ B.5 }$$

The instantaneous approximation to the B-S equation is to do contour integration over q_P on both sides of equation (B.3) and obtain

$$\begin{equation} \varphi _{p}(q_\perp )=\frac{\Lambda _{1}^{+}(q_\perp )\eta (q_\perp ) \Lambda _{2}^{+}(q_\perp )}{M-\omega _{1}-\omega _{2}}-\frac{\Lambda _{1}^{-}(q_\perp )\eta (q_\perp ) \Lambda _{2}^{-}(q_\perp )}{M+\omega _{1}+\omega _{2}}. \end{equation} \tag{ B.6 }$$

If we introduce the notations

$$\begin{equation} \varphi _{p}^{\pm \pm }(q_\perp )\equiv \Lambda _{1}^{\pm }(q_\perp ) \frac{\not\!{P}}{M}\varphi _{p}(q_\perp )\frac{\not\!{P}}{M}\Lambda _{2}^{\pm }(q_\perp ), \end{equation} \tag{ B.7 }$$

we have

$$\begin{equation} \varphi _{p}(q_\perp )=\varphi _{p}^{++}(q_\perp )+\varphi _{p}^{+-}(q_\perp ) +\varphi _{p}^{-+}(q_\perp )+\varphi _{p}^{--}(q_\perp ). \end{equation} \tag{ B.8 }$$

With properties (B.5) and notations (B.7), the full Salpeter equation (B.6) can be written as

$$\begin{equation} (M-\omega _{1}-\omega _{2})\varphi _{p}^{++}(q_\perp )=\Lambda _{1}^{+}(q_\perp )\eta (q_\perp ) \Lambda _{2}^{+}(q_\perp ), \end{equation}\vspace*{-12pt} \tag{ B.9 }$$

$$\begin{equation} (M+\omega _{1}+\omega _{2})\varphi _{p}^{--}(q_\perp )=-\Lambda _{1}^{-}(q_\perp )\eta (q_\perp ) \Lambda _{2}^{-}(q_\perp ), \end{equation}\vspace*{-12pt} \tag{ B.10 }$$

$$\begin{equation} \varphi _{p}^{+-}(q_\perp )=\varphi _{p}^{-+}(q_\perp )=0. \end{equation} \tag{ B.11 }$$

The normalization condition for the B-S equations now reads

$$\begin{equation} \int \frac{q^2_T\,{\rm d}q_T}{2{\pi }^2}{\rm Tr}[\bar{\varphi }^{++}\frac{\not\!{P}}{M} {\varphi }^{++}\frac{\not\!{P}}{M} -\bar{\varphi }^{--}\frac{\not\!{P}}{M} {\varphi }^{--}\frac{\not\!{P}}{M}]=2P_0. \end{equation} \tag{ B.12 }$$

The coupled equations (B.9)–(B.11) with the normalization condition (B.12) are the final B-S (Salpeter) equation under the 'complete instantaneous approximation' versus the previous one, i.e. the Salpeter equation [19] where only equation (B.9) is considered.

In addition, note that in the model used here for the double heavy quark–antiquark systems, the QCD-inspired interaction kernel V, being instantaneous approximately and dictating the Cornell potential which is composed of a linear scalar interaction plus a vector interaction, reads

$$\begin{eqnarray} &&V(\vec{q})=V_s(\vec{q})+V_v(\vec{q})\gamma ^0\otimes \gamma ^0,\nonumber \\ &&V_s(\vec{q})=-\left(\frac{\lambda }{\alpha }+V_0\right)\delta ^3(\vec{q}) +\frac{\lambda }{\pi ^2}\frac{1}{{(\vec{q}\,^{2}}+\alpha ^2)^2},\nonumber \\ &&V_v(\vec{q})=-\frac{2}{3\pi ^2}\frac{\alpha _s(\vec{q})}{(\vec{q}\,^2+\alpha ^2)}, \end{eqnarray} \tag{ B.13 }$$

where the QCD running coupling constant $\alpha _s(\vec{q})=\frac{12\pi }{33-2N_f}\frac{1}{\mathrm{log}(a+\vec{q}^2/\Lambda _{{\rm QCD}}^2)}$; the constants λ, α, a, V₀ and Λ_QCD are the parameters characterizing the potential.

In the appendix, we present the reduced wavefunctions ${\varphi }^{++}(\vec{q})$ (and ${\psi }^{+-}(\vec{q})$) which directly relate to the solutions by newly solving the obtained coupled equations (B.9)–(B.11) under a new approach. The key point of the new approach is to solve the B-S equation according to the quantum numbers of the concerned bound states, respectively [24, 30, 31], i.e. to solve the equation under the new approach we need to give the most general formulation for the wavefunction first. Therefore, for the present usage, in this appendix, we precisely quote the solutions for the low-lying bound state B_c meson with the quantum numbers J^P = 0⁻, χ_c0; J^PC = 0⁺⁺, χ_c1; J^PC = 1⁺⁺, χ_c2; J^PC = 2⁺⁺, and h_c with the quantum numbers J^PC = 1⁺⁻ from [24, 30, 31], and then we write down the reduced wavefunctions ${\varphi }^{++}(\vec{q})$ and the form factors accordingly.

When the weak-current matrix elements are computed precisely, as an intermediate step, the form factors can be represented as overlapping integrations of the components appearing in the B-S solutions; thus, in this appendix we also give the formulas of the form factors in terms of the 'overlapping integrations'.

• (a)  
  For the B_c meson with the quantum numbers J^P = 0⁻. The B-S wavefunction (solution of equations (B.9)–(B.11)) of the B_c meson with J^P = 0⁻ reads

  $$\begin{equation} \fl {\varphi }_{_{B_c}}(\vec{q})=M\left [\frac{\not\!{P}}{M}{f}_{1}(\vec{q})\left \lbrace 1- \frac{\not\!{q_{\bot }}(w_1+w_2)}{m_2w_1+m_1w_2}\right \rbrace +{f}_{2}(\vec{q})\left \lbrace 1 +\frac{\not\!q_{\bot }(w_2-w_1)}{m_1w_2+m_2w_1}\right \rbrace \right ]{\gamma }_5, \end{equation} \tag{ C.1 }$$

  where M and P are the mass and the total momentum of the meson B_c, respectively, $q_{\bot }=(0,\vec{q})$, $\vec{q}$ is the relative momentum of quark and antiquark in the meson, so $q_{\bot }^2=-{\vec{q}}{\,\,}^{2}$.Then we can rewrite the reduced wavefunction as

  $$\begin{eqnarray} &&{\varphi }^{++}_{_{B_c}}(\vec{q})=b_1 \left[b_2+\frac{\not\!{P}}{M}+b_3\not\!{q_{\bot }} +b_4\frac{\not\!{q_{\bot }}\not\!{P}}{M}\right]{\gamma }_5, \end{eqnarray} \tag{ C.2 }$$

  where

  $$\begin{eqnarray*} &&b_1=\frac{M}{2}\left({f}_{1}(\vec{q}) +{f}_{2}(\vec{q})\frac{m_1+m_2}{w_1+w_2}\right),\;\;\;\; b_2=\frac{w_1+w_2}{m_1+m_2}, \end{eqnarray*}$$

  $$\begin{eqnarray*} &&b_3=-\frac{(m_1-m_2)}{m_1w_2+m_2w_1},\;\;\;\; b_4=\frac{(w_1+w_2)}{(m_1w_2+m_2w_1)}. \end{eqnarray*}$$

  In appendix B, in equation (B.2), we have

  $$\begin{eqnarray} &&\fl \eta (q_{\bot })=\int {\rm d}^3k\,V(\vec{k})M\Bigg [\frac{\not\!{P}}{M}{f}_{1}(\vec{k})\left \lbrace 1- \frac{\not\!{k_{\bot }}(w_{11}+w_{21})}{m_2w_{11}+m_1w_{21}}\right \rbrace \nonumber \\ &&+{f}_{2}(\vec{k})\left \lbrace 1 +\frac{\not\!k_{\bot }(w_{21}-w_{11})}{m_1w_{21}+m_2w_{11}}\right \rbrace \Bigg ]{\gamma }_5, \end{eqnarray} \tag{ C.3 }$$

  where $w_{11}=\sqrt{m_1^2-k_{\bot }^2}$, $w_{21}=\sqrt{m_2^2-k_{\bot }^2}$, $V(\vec{k})=V_s(\vec{k})+V_v(\vec{k})\gamma ^0\otimes \gamma ^0$.According to equation (C.1),

  $$\begin{eqnarray} &&\fl \eta (q_{\bot })=\int {\rm d}^3k(V_s(\vec{k})+V_v(\vec{k})\gamma ^0\otimes \gamma ^0)\nonumber \\ &&\fl M\Bigg [\frac{\not\!{P}}{M}{f}_{1}(\vec{k})\Bigg \lbrace 1- \frac{\not\!{k_{\bot }}(w_{11}+w_{21})}{m_2w_{11}+m_1w_{21}}\Bigg \rbrace +{f}_{2}(\vec{k})\Bigg \lbrace 1 +\frac{\not\!k_{\bot }(w_{21}-w_{11})}{m_1w_{21}+m_2w_{11}}\Bigg \rbrace \Bigg ]{\gamma }_5\nonumber \\ &&=M\left[g_1\frac{\not\!{P}}{M}+g_2+g_3\frac{\not\!{q_{\bot }}}{M}+g_4\frac{\not\!{P\not\!{q_{\bot }}}}{M^2}\right]\gamma _5; \end{eqnarray} \tag{ C.4 }$$

  $$\begin{eqnarray*} &&g_1=\int {\rm d}^3k[V_s-V_v]f_1(\vec{k}),\quad g_2=\int {\rm d}^3k[V_s-V_v]f_2(\vec{k}), \end{eqnarray*}$$

  $$\begin{eqnarray*} &&g_3=\int {\rm d}^3k[V_s+V_v]\frac{\vec{k}\cdot \vec{q}}{|\vec{q}|^2}f_2(\vec{k}) \frac{(w_{21}-w_{11})}{m_1w_{21}+m_2w_{11}},\nonumber\\ &&g_4=\int {\rm d}^3k[V_s+V_v]f_1(\vec{k})\frac{(w_{11}+w_{21})}{m_1w_{21}+m_2w_{11}}. \end{eqnarray*}$$

  So we can also write down the wavefunction of ψ⁺⁻(q_⊥) as

  $$\begin{eqnarray} &&\fl {\psi }^{+-}(q_\perp )= \frac{\Lambda ^{+}_{1}(q_{P\perp })\eta (q_\perp )\Lambda ^{-}_{2}(q_{P\perp }) }{M+\omega _{2}+\omega ^{\prime }_{2}-E_f} =\left[n_1\frac{\not\!{P}}{M}+n_2+n_3\not\!{q_{\bot }} +n_4\frac{\not\!{q_{\bot }}\not\!{P}}{M}\right]{\gamma }_5. \end{eqnarray} \tag{ C.5 }$$

  Set $tt=\frac{1}{4w_1w_2(M+\omega _{2}+\omega ^{\prime }_{2}-E_f)}$, where the symbol ' denotes of the final state, and

  $$$ n_1=tt[g_1M(-q^2+m_1m_2-w_1w_2)+g_2M(m_2w_1-m_1w_2)+g_3(w_1+w_2)q^2+g_4(m_1+m_2)q^2], $$$

  $$$ n_2=tt[g_1M(m_2w_1-m_1w_2)+g_2M(q^2+m_1m_2-w_1w_2)+g_3(m_1-m_2)q^2+g_4(w_1-w_2)q^2], $$$

  $$$ n_3=tt[-g_1M(w_1+w_2)-g_2M(m_1-m_2)+g_3(q^2+m_1m_2+w_1w_2)+g_4(m_2w_1+m_1w_2)], $$$

  $$$ n_4=tt[g_1M(m_1+m_2)+g_2M(w_1-w_2)-g_3(m_2w_1+m_1w_2)-g_4(-q^2+m_1m_2+w_1w_2)]. $$$
• (b)  
  For the charmonium χ_c0(J^PC = 0⁺⁺) and the form factors s₊ and s₋. The B-S wavefunction (solution of equations (B.9)–(B.11) under the new method to solve the coupled equations) of χ_c0 reads

  $$\begin{equation} {\varphi }_{\chi _{c0}}(\vec{q^{\prime }})={f}^{\prime }_{1}(\vec{q^{\prime }})\not\!{q^{\prime }_{\bot }}+{f}^{\prime }_{2}(\vec{q^{\prime }})\frac{\not\!{P_f}\not\!{q^{\prime }_{\bot }}}{M_f} +{f}^{\prime }_{3}(\vec{q^{\prime }})M_f+{f}^{\prime }_{4}(\vec{q^{\prime }})\not\!{P_f}, \end{equation} \tag{ C.6 }$$

  with constraints on the components of the wavefunction; for the charmonium, m'₁ = m'₂, w'₁ = w'₂, we obtain

  $$\begin{eqnarray*} &&f^{\prime }_3(\vec{q^{\prime }})=\frac{f^{\prime }_1(\vec{q^{\prime }})q_{\bot }^{\prime 2}}{M_fm^{\prime }_1},\;\;\; f^{\prime }_4(\vec{q^{\prime }})=0 , \end{eqnarray*}$$

  where M_f and P_f are the mass and the total momentum of the final meson χ_c0, respectively, $q^{\prime }_{\bot }=(0,\vec{q^{\prime }})$, $\vec{q^{\prime }}$ is the relative momentum of quark and antiquark in the meson, so $q_{\bot }^{\prime 2}=-{\vec{q^{\prime }}}^{ 2}$. Then the reduced wavefunction ${\varphi }_{^3P_0}^{++}(\vec{q^{\prime }})$ reads

  $$\begin{equation} {\varphi }_{\chi _{c0}}^{++}(\vec{q^{\prime }})=a_1\left[\not\!{q^{\prime }_{\bot }}+a_2\frac{\not\!{P_f}\not\!{q^{\prime }_{\bot }}}{M_f} +a_3+a_4\frac{\not\!{P_f}}{M_f}\right], \end{equation} \tag{ C.7 }$$

  with

  $$\begin{eqnarray*} &&a_1=\frac{1}{2}\left({f}^{\prime }_{1}(\vec{q^{\prime }}) +{f}^{\prime }_{2}(\vec{q^{\prime }})\frac{m^{\prime }_1}{w^{\prime }_1}\right),\;\; a_2=\frac{w^{\prime }_1}{m^{\prime }_1},\ \ \ a_3=\frac{q_{\bot }^{\prime 2}}{m^{\prime }_1},\;\;a_4=0. \end{eqnarray*}$$

  The wavefunction of $\bar{\psi }^{\prime -+}(q^{\prime }_{P\perp })$ is

  $$\begin{eqnarray} &&\fl \bar{\psi }^{\prime -+}(q^{\prime }_{P\perp })= \frac{\Lambda ^{\prime -}_2(q^{\prime }_{P\perp }) \bar{\eta }^{\prime }(q^{\prime }_{P\perp })\Lambda ^{\prime +}_1(q^{\prime }_{P\perp })}{M-\omega _{2}-\omega ^{\prime }_{2}-E_f}= n^{\prime }_1\not\!{q^{\prime }_{\bot }}+n^{\prime }_2\frac{\not\!{q^{\prime }_{\bot }}\not\!{P_f}}{M_f} +n^{\prime }_3+n^{\prime }_4\frac{\not\!{P_f}}{M_f}. \end{eqnarray} \tag{ C.8 }$$

  Set $tt^{\prime }=\frac{1}{4w_1^{\prime 2}(M-\omega _{2}-\omega ^{\prime }_{2}-E_f)}$, where

  $$\begin{eqnarray*} &&\fl n^{\prime }_1=tt^{\prime }[-2g^{\prime }_1q^{\prime 2}+2g^{\prime }_3M_fm^{\prime }_1],\ \ \ n^{\prime }_2=0, \end{eqnarray*}$$

  $$\begin{eqnarray*} &&\fl n^{\prime }_3=tt^{\prime }[-2g^{\prime }_1m^{\prime }_1q^{\prime 2}+2g^{\prime }_3M_fm_1^{\prime 2}], \ \ \ n^{\prime }_4=tt^{\prime }[-2g^{\prime }_1w^{\prime }_1q^{\prime 2}+2g^{\prime }_3M_fm^{\prime }_2w^{\prime }_1], \end{eqnarray*}$$

  and

  $$\begin{eqnarray*} &&\fl g^{\prime }_1=\int {\rm d}^3k^{\prime }[V_s-V_v]\frac{\vec{k^{\prime }}\cdot \vec{q^{\prime }}}{|\vec{q^{\prime }}|^{2}}f^{\prime }_1(\vec{k^{\prime }}), \ \ g^{\prime }_2=\int {\rm d}^3k^{\prime }[V_s-V_v]\frac{\vec{k^{\prime }}\cdot \vec{q^{\prime }}}{|\vec{q^{\prime }}|^2}f^{\prime }_2(\vec{k^{\prime }}), \end{eqnarray*}$$

  $$\begin{eqnarray*} &&\fl g^{\prime }_3=\int {\rm d}^3k^{\prime }[V_s+V_v]\frac{f^{\prime }_1(\vec{k^{\prime }})k_{\bot }^{\prime 2}}{M_fm^{\prime }_1}, \ \ g^{\prime }_4=0. \end{eqnarray*}$$

  With equation (17), the form factors may be presented by overlapping integrations:

  $$\begin{eqnarray} &&\fl s_+=\frac{1}{2}\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4a_1b_1}{MM_f} \left[a_3b_2M_f+{\alpha }_{11}E_f(a_2b_2E_f+M_f+a_2b_4\vec{q}\cdot \vec{P_f})\right.\nonumber \\ &&+\,b_3(M_fq^2+{\alpha }_{11}M_f\vec{q}\cdot \vec{P_f}) +M(a_2b_4q^2-{\alpha }_{11}a_2b_2E_f-{\alpha }_{11}M_f)\nonumber \\ &&\left.+\,M\frac{q\cos \theta }{|\vec{P_f}|} \left(1-\frac{E_f}{M}\right)(a_2b_2E_f-a_3b_4M_f+M_f+a_2b_4\vec{q}\cdot \vec{P_f})\right], \end{eqnarray} \tag{ C.9 }$$

  $$\begin{eqnarray} &&\fl s_-=\frac{1}{2}\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4a_1b_1}{MM_f} \left[a_3b_2M_f+{\alpha }_{11}E_f(a_2b_2E_f+M_f+a_2b_4\vec{q}\cdot \vec{P_f})\right.\nonumber \\ &&+\,b_3(M_fq^2+{\alpha }_{11}M_f\vec{q}\cdot \vec{P_f}) -M(a_2b_4q^2-{\alpha }_{11}a_2b_2E_f-{\alpha }_{11}M_f)\nonumber \\ &&\left.-\,M\frac{q\cos \theta }{|\vec{P_f}|} \left(1+\frac{E_f}{M}\right)(a_2b_2E_f-a_3b_4M_f+M_f+a_2b_4\vec{q}\cdot \vec{P_f})\right], \end{eqnarray} \tag{ C.10 }$$

  where ${\alpha }_{11}=\alpha ^{\prime }_1=\frac{m^{\prime }_1}{m^{\prime }_1+m^{\prime }_2}.$
• (c)  
  For the charmonium χ_c1(J^PC = 1⁺⁺) and the form factors f, u₁, u₂, g. The B-S wavefunction (solution of equations (B.9)–(B.11) under the new method to solve the coupled equations) of χ_c1 reads

  $$\begin{eqnarray} &&\fl {\varphi }_{\chi _{c1}}(\vec{q^{\prime }})={\rm i}{\epsilon }_{\mu \nu \alpha \beta }P_f^{\nu }q^{\prime \alpha }_{\bot }{\varepsilon }^{\beta } [f^{\prime }_1(\vec{q^{\prime }})M_f{\gamma }^{\mu }+f^{\prime }_2(\vec{q^{\prime }})\not\!{P_f}{\gamma }^{\mu }+f^{\prime }_3(\vec{q^{\prime }})\not\!{q^{\prime }_{\bot }}{\gamma }^{\mu } \nonumber\\ &&+\, {\rm i}f^{\prime }_4(\vec{q^{\prime }}){\epsilon }^{\mu \rho \sigma \delta }P_{f\sigma }q^{\prime }_{\bot \rho }{\gamma }_{\delta }{\gamma }_5/M_f]/M_f^2, \end{eqnarray} \tag{ C.11 }$$

  where ε is the polarization vector of the axial vector meson and with the constraint on the components,

  $$\begin{eqnarray*} &&f^{\prime }_3(\vec{q^{\prime }})=0,\;\;\; f^{\prime }_4(\vec{q^{\prime }})=\frac{f^{\prime }_2(\vec{q^{\prime }})M_f}{m^{\prime }_1}. \end{eqnarray*}$$

  Then the reduced wavefunction ${\varphi }_{^3P_1}^{++}(\vec{q^{\prime }})$ reads

  $$\begin{equation} \fl {\varphi}_{\chi _{c1}}^{++}(\vec{q^{\prime }})={\rm i}{\epsilon }_{\mu \nu \alpha \beta }P_f^{\nu }q^{\prime \alpha }_{\bot }{\varepsilon }^{\beta } a_1[M_f{\gamma }^{\mu }+a_2{\gamma }^{\mu }\not\!{P_f}+a_3{\gamma }^{\mu }\not\!{q^{\prime }_{\bot }} +a_4{\gamma }^{\mu }\not\!{P_f}\not\!{q^{\prime }_{\bot }}]/M_f^2, \end{equation} \tag{ C.12 }$$

  with

  $$\begin{eqnarray*} &&a_1=\frac{1}{2}\left(f^{\prime }_1(\vec{q^{\prime }})+f^{\prime }_2(\vec{q^{\prime }})\frac{w^{\prime }_1}{m^{\prime }_1}\right),\;\;\;\; a_2=-\frac{m^{\prime }_1}{w^{\prime }_1},\ \ \ a_3=0,\;\;\;\; a_4=-\frac{1}{w^{\prime }_1}. \end{eqnarray*}$$

  With equation (17), the form factors may be presented by overlapping integrations:

  $$\begin{eqnarray} &&\fl f=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4a_1b_1}{M_f^2(M+M_f)} \left[\left(a_4M_f^2q^2-a_2b_4M_f^2q^2 -a_2b_2E_f\vec{q}\cdot \vec{P_f}\right.\right.\nonumber \\ &&\hphantom{\fl f=} +(a_4-a_2b_4)(\vec{q}\cdot \vec{P_f})^2+{\alpha }_{11}^2a_4E_f^2\vec{P_f}^2+a_2b_2{\alpha }_{11}(E_f^3-E_fM_f^2) -2{\alpha }_{11}a_4E_f^2\vec{q}\cdot \vec{P_f} \nonumber \\ &&\hphantom{\fl f=} +{\alpha }_{11}a_2b_4E_f^2\vec{q}\cdot \vec{P_f}+b_3E_fM_f(q^2-{\alpha }_{11}\vec{q}\cdot \vec{P_f}) \left.\right) +\frac{q^2}{2}(\cos ^2\theta -1)\left(M_f^2(a_4-a_2b_4)\right. \nonumber \\ &&\hphantom{\fl f=} \left.\left.+\,b_3E_f(M_f+a_4\vec{q}\cdot \vec{P_f} -{\alpha }_{11}a_4\vec{P_f}^2)\right)\right], \end{eqnarray} \tag{ C.13 }$$

  $$\begin{eqnarray} &&\fl u_1=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4a_1b_1M}{M_f^2}\left[\frac{{\alpha }_{11}E_f}{M^2}\left(a_2b_2M_f^2 +M_fb_3\vec{q}\cdot \vec{P_f}+a_4({\alpha }_{11}E_fM_f^2+b_3(M_f^2q^2 \right.\right.\nonumber \\ &&\hphantom{\fl u_1=} \left.+\,(\vec{q}\cdot \vec{P_f})^2-{\alpha }_{11}\vec{q}\cdot \vec{P_f}E_f^2))\right)-{\alpha }_{11}E_f^2\frac{q\cos \theta }{M^2|\vec{P_f}|} \left((a_4-a_2b_4)M_f^2+b_3E_f(-{\alpha }_{11}a_4E_f^2\right.\nonumber \\ &&\hphantom{\fl u_1=} \left. +\,M_f+a_4\vec{q}\cdot \vec{P_f}+{\alpha }_{11}a_4M_f^2)\right)-\frac{E_f}{M^2}\frac{q\cos \theta }{|\vec{P_f}|}\left(a_2b_2M_f^2 \right.\nonumber \\ &&\hphantom{\fl u_1=} \left.+\,M_fb_3\vec{q}\cdot \vec{P_f}\right. \left.+\,\,a_4({\alpha }_{11}E_fM_f^2+b_3(M_f^2q^2 +(\vec{q}\cdot \vec{P_f})^2-{\alpha }_{11}\vec{q}\cdot \vec{P_f}E_f^2))\right)\nonumber \\ &&\hphantom{\fl u_1=} +\,\frac{q^2}{2M^2|\vec{P_f}|^2}(-M_f^2+(2E_f^2+M_f^2)\cos ^2\theta )\left(M_f^2(a_4-a_2b_4)\right. \nonumber \\ &&\hphantom{\fl u_1=} \left.\left.+\,b_3E_f(M_f+a_4\vec{q}\cdot \vec{P_f} -{\alpha }_{11}a_4\vec{P_f}^2)\right)\right], \end{eqnarray} \tag{ C.14 }$$

  $$\begin{eqnarray} &&\fl u_2=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4a_1b_1M}{M_f^2}\left[-\frac{1}{M}\left(b_3M_fq^2 +(a_2b_2+{\alpha }_{11}a_4E_f)({\alpha }_{11}E_f^2-\vec{q}\cdot \vec{P_f})\right)\right. \nonumber \\ &&\hphantom{\fl u_2=} +\frac{E_fq\cos \theta }{M|\vec{P_f}|}\left({\alpha }_{11}M_fb_3E_f +(a_2b_4-a_4)(\vec{q}\cdot \vec{P_f}-{\alpha }_{11}E_f^2)\right)\nonumber \\ &&\hphantom{\fl u_2=} +\frac{1}{M}\frac{q\cos \theta }{|\vec{P_f}|}\left(a_2b_2M_f^2 +M_f(b_3\vec{q}\cdot \vec{P_f}+{\alpha }_{11}b_4E_f\vec{q}\cdot \vec{P_f})\right.\nonumber \\ &&\hphantom{\fl u_2=} \left.+\,a_4({\alpha }_{11}E_fM_f^2+b_3(M_f^2q^2 +(\vec{q}\cdot \vec{P_f})^2-{\alpha }_{11}\vec{q}\cdot \vec{P_f}E_f^2))\right)\nonumber \\ &&\hphantom{\fl u_2=} -\frac{E_fq^2}{2M|\vec{P_f}|^2}(3\cos ^2\theta -1)\left(M_f^2(a_4-a_2b_4) \left.+b_3E_f(M_f+a_4\vec{q}\cdot \vec{P_f} -{\alpha }_{11}a_4\vec{P_f}^2)\right)\right], \nonumber\\ \end{eqnarray} \tag{ C.15 }$$

  $$\begin{eqnarray} &&\fl g=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4a_1b_1(M+M_f)}{M_f^2} \left[\frac{E_f}{M}\left(\frac{q\cos \theta }{|\vec{P_f}|}-{\alpha }_{11}\right)\left(a_2(b_2E_f+b_4\vec{q}\cdot \vec{P_f})+M_f \right.\right.\nonumber \\ &&\hphantom{\fl g=} \left.\left.+\,a_4b_3E_f(q^2-{\alpha }_{11}\vec{q}\cdot \vec{P_f}) \right)+\frac{q^2}{2M}(\cos ^2\theta -1) (b_3(M_f+a_4(\vec{q}\cdot \vec{P_f}-{\alpha }_{11}E_f^2)))\right]. \nonumber\\ \end{eqnarray} \tag{ C.16 }$$
• (d)  
  For the charmonium h_c(J^PC = 1⁺⁻) and the form factors V₀, V₁, V₂, V₃. The B-S wavefunction (solution of equations (B.9)–(B.11) under the new method to solve the coupled equations) of h_c reads

  $$\begin{equation} \fl {\varphi }_{h_c}(\vec{q^{\prime }})=q^{\prime }_{\bot }\cdot \varepsilon \left[f^{\prime }_1(\vec{q^{\prime }})+f^{\prime }_2(\vec{q^{\prime }})\frac{\not\!P_f}{M_f} +f^{\prime }_3(\vec{q^{\prime }})\not\!{q^{\prime }_{\bot }}+f^{\prime }_4(\vec{q^{\prime }})\frac{\not\!P_f\not\!{q^{\prime }_{\bot }}}{M_f^2}\right]{\gamma }_5, \end{equation} \tag{ C.17 }$$

  with the constraint on the components of the wavefunction,

  $$\begin{eqnarray*} &&f^{\prime }_3(\vec{q^{\prime }})=0,\;\;\; f^{\prime }_4(\vec{q^{\prime }})=-\frac{f^{\prime }_2(\vec{q^{\prime }})M_f}{m^{\prime }_1}, \end{eqnarray*}$$

  Then we have the reduced wavefunction ${\varphi }_{h_c}^{++}(\vec{q^{\prime }})$,

  $$\begin{equation} {\varphi }_{h_c}^{++}(\vec{q^{\prime }})=q^{\prime }_{\bot }\cdot \varepsilon a_1\left[1+a_2\frac{\not\!P_f}{M_f} +a_3\not\!{q^{\prime }_{\bot }}+a_4\frac{\not\!{q^{\prime }_{\bot }}\not\!P_f}{M_f}\right]{\gamma }_5, \end{equation} \tag{ C.18 }$$

  $$\begin{eqnarray*} &&a_1=\frac{1}{2}\left(f^{\prime }_1(\vec{q^{\prime }})+f^{\prime }_2(\vec{q^{\prime }})\frac{w^{\prime }_1}{m^{\prime }_1})\right) ,\;\; a_2=\frac{m^{\prime }_1}{w^{\prime }_1} ,\ \ a_3=0 ,\;\;\; a_4=\frac{1}{w^{\prime }_1} . \end{eqnarray*}$$

  With equation (17), the form factors may be presented by overlapping integrations:

  $$\begin{eqnarray} &&\fl V_0=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4a_1b_1}{M_f(M+M_f)}\frac{q^2}{2}(\cos ^2\theta -1)\left[a_4b_2E_f+a_2b_3E_f-b_4M_f+a_4b_4\vec{q}\cdot \vec{P_f}\right], \nonumber \\ \end{eqnarray} \tag{ C.19 }$$

  $$\begin{eqnarray} &&\fl V_1=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4a_1b_1}{M_f}\left[\frac{E_f}{M}\left({\alpha }_{11}-\frac{q\cos \theta }{|\vec{P_f}|}\right) ({\alpha }_{11}a_4b_2E_f^2+{\alpha }_{11}a_4b_4E_f\vec{q}\cdot \vec{P_f}\right.\nonumber\\ &&+\,b_2M_f+a_2b_3\vec{q}\cdot \vec{P_f}) \left(\frac{q^2}{2M|\vec{P_f}|^2}(-M_f^2+(2E_f^2+M_f^2)\cos ^2\theta )\right.\nonumber\\ &&\left. -{\alpha }_{11}\frac{E_f^2q\cos \theta }{M|\vec{P_f}|}\right) \left.(a_4b_2E_f+a_2b_3E_f-b_4M_f+a_4b_4\vec{q}\cdot \vec{P_f})\vphantom{\frac{P^f_f}{[p_f]}}\right], \end{eqnarray} \tag{ C.20 }$$

  $$\begin{eqnarray} &&\fl V_2=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4Ma_1b_1}{M_f}\left[\frac{E_f}{M} \left({\alpha }_{11}-\frac{q\cos \theta }{|\vec{P_f}|}\right) \left(-a_4b_4q^2+a_2-a_4b_2E_f{\alpha }_{11}\right)\right.\nonumber \\ &&\hphantom{\fl V_2=}\!\!\! +\left({\alpha }_{11}\frac{E_fq\cos \theta }{M|\vec{P_f}|}{+}\frac{E_fq^2}{2M|\vec{P_f}|^2}(3\cos ^2\theta {-}1)\right) \left.(a_4b_2E_f{+}a_2b_3E_f{-}b_4M_f{+}a_4b_4\vec{q}\cdot \vec{P_f})\vphantom{\frac{P^f_f}{[p_f]}}\right], \nonumber\\ \end{eqnarray} \tag{ C.21 }$$

  $$\begin{eqnarray} &&\fl V_3=-\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4a_1b_1(M+M_f)}{MM_f}\frac{q^2}{2}(\cos ^2\theta -1) [a_4(b_2+b_4E_f{\alpha }_{11})+b_3a_2]. \end{eqnarray} \tag{ C.22 }$$
• (e)  
  For the charmonium χ_c2(J^PC = 2⁺⁺) and the form factors k, c₁, c₂, h. The B-S wavefunction (solution of equations (B.9)–(B.11) under the new method to solve the coupled equations) of χ_c2 reads

  $$\begin{eqnarray} &&\fl {\varphi }_{\chi _{c2}}(\vec{q^{\prime }})={\varepsilon }_{\mu \nu }q^{\prime \nu }_{\bot }\left\lbrace q^{\prime \mu }_{\bot } \left[f^{\prime }_1(\vec{q^{\prime }})+\frac{\not\!P_f}{M_f}f^{\prime }_2(\vec{q^{\prime }}) +\frac{\not\!q^{\prime }_{\bot }}{M_f}f^{\prime }_3(\vec{q^{\prime }}) +\frac{\not\!P_f\not\!q^{\prime }_{\bot }}{M_f^2}f^{\prime }_4(\vec{q^{\prime }})\right]\right.\nonumber \\ &&\left.\hphantom{\fl {\varphi }_{\chi _{c2}}(\vec{q^{\prime }})} +{\gamma }^{\mu }[M_ff^{\prime }_5(\vec{q^{\prime }})+\not\!P_ff^{\prime }_6(\vec{q^{\prime }})+\not\!q^{\prime }_{\bot }f^{\prime }_7(\vec{q^{\prime }})]+ \frac{\rm i}{M_f}f^{\prime }_8(\vec{q^{\prime }}){\epsilon }^{\mu \alpha \beta \delta } P_{f\alpha }q^{\prime }_{\bot \beta }{\gamma }_{\delta }{\gamma }_5\vphantom{\frac{P^f_f}{[p_f]}}\right\rbrace , \end{eqnarray} \tag{ C.23 }$$

  with the constraint on the components of the wavefunction,

  $$$ f^{\prime }_1(\vec{q^{\prime }})=\frac{[q_{\bot }^{\prime 2}f^{\prime }_3(\vec{q^{\prime }})+M_f^2f^{\prime }_5(\vec{q^{\prime }})]}{M_fm^{\prime }_1} ,\ \ f^{\prime }_2(\vec{q^{\prime }})=0 ,\;\;\; f^{\prime }_7=0 ,\;\;\; f^{\prime }_8=\frac{f^{\prime }_6(\vec{q^{\prime }})M_f}{m^{\prime }_1} , $$$

  where ε_μν is a tensor for J = 2. Then we have the reduced wavefunction ${\varphi }_{\chi _{c2}}^{++}(\vec{q})$ as

  $$\begin{eqnarray} &&\fl {\varphi }^{++}_{\chi _{c2}}(\vec{q^{\prime }})={\varepsilon }_{\mu \nu }q^{\prime \nu }_{\bot }\left\lbrace q^{\prime \mu }_{\bot }\left[a_1+a_2\frac{\not\!P_f}{M_f} +a_3\frac{{\not\!q^{\prime }_{\bot }}}{M_f}\right.\right.\nonumber \\ &&\left.\left. +\,a_4\frac{\not\!q^{\prime }_{\bot }\not\!P_f}{M_f^2}\right] +{\gamma }^{\mu }\left[a_5+a_6\frac{\not\!P_f}{M_f}+a_7\frac{\not\!q^{\prime }_{\bot }}{M_f}+ a_8\frac{\not\!P_f\not\!q^{\prime }_{\bot }}{M_f^2}\right]\right\rbrace , \end{eqnarray} \tag{ C.24 }$$

  with

  $$\begin{eqnarray*} &&\fl a_1=\frac{q^{\prime 2}_{\bot }}{2M_fm^{\prime }_1}n_1 +\frac{(f^{\prime }_5(\vec{q^{\prime }})w^{\prime }_2-f^{\prime }_6(\vec{q^{\prime }})m^{\prime }_2)M_f}{2m^{\prime }_1w^{\prime }_2},\ \ \displaystyle \quad \quad \quad a_2=\frac{(f^{\prime }_6(\vec{q^{\prime }})w^{\prime }_2-f^{\prime }_5(\vec{q^{\prime }})m^{\prime }_2)M_f}{2m^{\prime }_1w^{\prime }_2},\nonumber \\ &&\fl a_3=\frac{1}{2}n_1+\frac{f^{\prime }_6(\vec{q^{\prime }})M_f^2}{2m^{\prime }_1w^{\prime }_2},\;\;\;\; a_4=\frac{1}{2}\left(-\frac{w^{\prime }_1}{m^{\prime }_1}\right)n_1+\frac{f^{\prime }_5(\vec{q^{\prime }})M_f^2}{2m^{\prime }_1w^{\prime }_2},\nonumber \\ &&\fl a_5=\frac{M_f}{2}n_2,\quad a_6=\frac{M_fm^{\prime }_1}{2w^{\prime }_1}n_2,\;\; \quad \quad a_7=0,\;\; \quad \quad a_8=\frac{M_f^2}{2w^{\prime }_1}n_2,\nonumber \\ &&\fl n_1=\frac{1}{2}\left(f^{\prime }_3(\vec{q^{\prime }})+f^{\prime }_4(\vec{q^{\prime }})\frac{m^{\prime }_1}{w^{\prime }_1}\right),\;\;\; n_2=\frac{1}{2}\left(f^{\prime }_5(\vec{q^{\prime }})-f^{\prime }_6(\vec{q^{\prime }})\frac{w^{\prime }_1}{m^{\prime }_1}\right).\nonumber \end{eqnarray*}$$

  With equation (17), the form factors may be presented by overlapping integrations:

  $$\begin{eqnarray} &&\fl k=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4b_1}{M_f^2(M+M_f)}\nonumber \\ &&\hphantom{\fl k=}\times\,\bigg[ \frac{q^2}{2}(\cos ^2\theta -1)(-{\alpha }_{11}E_f(M_f(-a_3-a_2b_3E_f+a_1b_4M_f) +a_4(b_2E_f+b_4\vec{q}\cdot \vec{P_f}))\nonumber \\ &&\hphantom{\fl k=}-\,({\alpha }_{11}a_8b_3E_f^2 -a_5b_3M_f^2+a_8b_3\vec{q}\cdot \vec{P_f})-{\alpha }_{11}E_f(M_f(-a_3-a_2b_3E_f+a_1b_4M_f)\nonumber \\ &&\hphantom{\fl k=}+\,a_4(b_2E_f+b_4\vec{q}\cdot \vec{P_f}) +2a_8b_3E_f))\nonumber \\ &&\hphantom{\fl k=}-E_f\bigg({\alpha }_{11}-\frac{q\cos \theta }{|\vec{P_f}|}\bigg) (-a_5M_f^2+a_8b_3E_fq^2+a_6b_2E_fM_{f}\nonumber\\ &&\hphantom{\fl k=}+a_6b_4M_f\vec{q}\cdot \vec{P_f} -{\alpha }_{11}a_8b_3E_f\vec{q}\cdot \vec{P_f})\nonumber \\ &&\hphantom{\fl k=} -\frac{E_fq^3\cos \theta }{|\vec{P_f}|}(1-\cos ^2\theta ) ((M_f(-a_3-a_2b_3E_f+a_1b_4M_f)\nonumber \\ &&\hphantom{\fl k=}+\,a_4(b_2E_f+b_4\vec{q}\cdot \vec{P_f})+2a_8b_3E_f))) \bigg], \end{eqnarray} \tag{ C.25 }$$

  $$\begin{eqnarray} &&\fl c_1=\int \frac{{\rm d}^3q}{(2\pi )^3} \frac{4b_1M}{M_f^2}\left[{\alpha }_{11}\frac{E_f}{M} \left({\alpha }_{11}\frac{E_f}{M}(-{\alpha }_{11}a_4b_2E_f^2 -{\alpha }_{11}a_4b_4E_f\vec{q}\cdot \vec{P_f} +a_1b_2M_f^2\right.\right.\nonumber \\ &&\hphantom{\fl c_1=}+a_2b_3M_f\vec{q}\cdot \vec{P_f} +a_3M_f(b_3q^2+{\alpha }_{11}E_f-{\alpha }_{11}b_3\vec{q}\cdot \vec{P_f})) \left.-2{\alpha }_{11}a_8b_3E_f\vec{q}\cdot \vec{P_f}/M\right)\nonumber \\ &&\hphantom{\fl c_1=} -\frac{E_fq\cos \theta }{M|\vec{P_f}|} \left(-M\left({\alpha }_{11}\frac{E_f}{M}\right)^2 (M_f(-a_3-a_2b_3E_f+a_1b_4M_f)\right.\nonumber \\ &&\hphantom{\fl c_1=}+\,a_4(b_2E_f +b_4\vec{q}\cdot \vec{P_f}))-{\alpha }_{11}\frac{E_f}{M}({\alpha }_{11}a_8b_3E_f^2 -a_5b_3M_f^2+a_8b_3\vec{q}\cdot \vec{P_f})\nonumber \\ &&\hphantom{\fl c_1=} +2{\alpha }_{11}\frac{E_f}{M}(-{\alpha }_{11}a_4b_2E_f^2-{\alpha }_{11}a_4b_4E_f\vec{q}\cdot \vec{P_f} +a_1b_2M_f^2+a_2b_3M_f\vec{q}\cdot \vec{P_f}\nonumber \\ &&\hphantom{\fl c_1=} +a_3M_f(b_3q^2+{\alpha }_{11}E_f-{\alpha }_{11}b_3\vec{q}\cdot \vec{P_f})) -{\alpha }_{11}2a_8b_3E_f\vec{q}\cdot \vec{P_f}/M\nonumber \\ &&\hphantom{\fl c_1=} \left.-{\alpha }_{11}\frac{E_f}{M}({\alpha }_{11}a_8b_3E_f^2 +a_5b_3M_f^2+a_8b_3\vec{q}\cdot \vec{P_f})\right)\nonumber \\ &&\hphantom{\fl c_1=} +\frac{q^2}{2M^2|\vec{P_f}|^2}(-M_f^2+(2E_f^2+M_f^2)\cos ^2\theta )\left(M(a_4b_4q^2+a_2M_f+{\alpha }_{11}a_4b_2E_f\right.\nonumber \\ &&\hphantom{\fl c_1=} \left.-\,{\alpha }_{11}a_3M_f -a_8-a_6b_4M_f+{\alpha }_{11}a_8b_3E_f)\right.\nonumber \\ &&\hphantom{\fl c_1=} -{\alpha }_{11}E_f(M_f(-a_3-a_2b_3E_f+a_1b_4M_f)+a_4(b_2E_f+b_4\vec{q}\cdot \vec{P_f}))\nonumber \\ &&\hphantom{\fl c_1=} -({\alpha }_{11}a_8b_3E_f^2 -a_5b_3M_f^2+a_8b_3\vec{q}\cdot \vec{P_f})\nonumber \\ &&\hphantom{\fl c_1=} -{\alpha }_{11}E_f (M_f(-a_3-a_2b_3E_f+a_1b_4M_f)+a_4(b_2E_f+b_4\vec{q}\cdot \vec{P_f})+2a_8b_3E_f)))\nonumber \\ &&\hphantom{\fl c_1=} -\frac{q^3E_f\cos \theta }{2M^2|\vec{P_f}|^3}(3M_f^2-(2E_f^2+3M_f^2)\cos ^2\theta )\left((M_f(-a_3-a_2b_3E_f+a_1b_4M_f)\right.\nonumber \\ &&\hphantom{\fl c_1=} \left.\left.+\,a_4(b_2E_f+b_4\vec{q}\cdot \vec{P_f})+2a_8b_3E_f)\right)\right], \end{eqnarray} \tag{ C.26 }$$

  $$\begin{eqnarray} &&\fl c_2=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4b_1M}{M_f^2} \left[{\alpha }_{11}\frac{E_f}{M}\left({\alpha }_{11}E_f (a_4b_4q^2+a_2M_f+{\alpha }_{11}a_4b_2E_f-{\alpha }_{11}a_3M_f)\right.\right.\nonumber \\ &&\hphantom{\fl c_2=} \left.-(-a_8b_3q^2-a_6b_2M_f+{\alpha }_{11}a_8E_f)\right)\nonumber \\ &&\hphantom{\fl c_2=}+\,\frac{E_fq\cos \theta }{M|\vec{P_f}|} \left((-{\alpha }_{11}^2E_f)(M_f(-a_3-a_2b_3E_f+a_1b_4M_f)+a_4(b_2E_f+b_4\vec{q}\cdot \vec{P_f}))\right.\nonumber \\ &&\hphantom{\fl c_2=}-{\alpha }_{11}({\alpha }_{11}a_8b_3E_f^2 -a_5b_3M_f^2a_8b_3\vec{q}\cdot \vec{P_f})-{\alpha }_{11}E_f (a_4b_4q^2+a_2M_f\nonumber \\ &&\hphantom{\fl c_2=} +{\alpha }_{11}a_4b_2E_f-{\alpha }_{11}a_3M_f)+(-a_8b_3q^2-a_6b_2M_f+{\alpha }_{11}a_8E_f)\nonumber \\ &&\hphantom{\fl c_2=} -{\alpha }_{11}E_f(a_4b_4q^2+a_2M_f+{\alpha }_{11}a_4b_2E_f-{\alpha }_{11}a_3M_f \left.-a_8-a_6b_4M_f+{\alpha }_{11}a_8b_3E_f)\right)\nonumber \\ &&\hphantom{\fl c_2=} +\frac{q^2}{2M|\vec{P_f}|^2}(-M_f^2+(2E_f^2+M_f^2)\cos ^2\theta )\left(a_4b_4q^2+a_2M_f+{\alpha }_{11}a_4b_2E_f\right.\nonumber \\ &&\hphantom{\fl c_2=} \left.-{\alpha }_{11}a_3M_f-a_8-a_6b_4M_f+{\alpha }_{11}a_8b_3E_f\right) -\frac{q^2E_f}{2M|\vec{P_f}|^2}(3\cos ^2\theta -1)\nonumber \\ &&\hphantom{\fl c_2=} \left(-{\alpha }_{11}E_f(M_f(-a_3-a_2b_3E_f+a_1b_4M_f)+a_4(b_2E_f+b_4\vec{q}\cdot \vec{P_f}))\right.\nonumber \\ &&\hphantom{\fl c_2=} -({\alpha }_{11}a_8b_3E_f^2 -a_5b_3M_f^2+a_8b_3\vec{q}\cdot \vec{P_f})-{\alpha }_{11}E_f(M_f(-a_3-a_2b_3E_f+a_1b_4M_f)\nonumber \\ &&\hphantom{\fl c_2=} \left.+a_4(b_2E_f{+}b_4\vec{q}\cdot \vec{P_f}){+}2a_8b_3E_f))\right) -\frac{q^3\cos \theta }{2M|\vec{P_f}|^3}[(4E_f^2{+}M_f^2)\cos ^2\theta -(2E_f^2{+}M_f^2)]\nonumber \\ &&\hphantom{\fl c_2=} \times\left(\right.\!\! (M_f(-a_3-a_2b_3E_f+a_1b_4M_f)+a_4(b_2E_f+b_4\vec{q}\cdot \vec{P_f})+2a_8b_3E_f))\left.\!\!\right)\left.\!\!\right], \end{eqnarray} \tag{ C.27 }$$

  $$\begin{eqnarray} &&\fl h=\int \frac{{\rm d}^3q}{(2\pi )^3}\frac{4b_1(M+M_f)}{M_f^2} \left[\left(\frac{q\cos \theta }{|\vec{P_f}|}-{\alpha }_{11}\right)\right. \frac{E_f}{M}(a_8b_3q^2+a_6b_2M_f+{\alpha }_{11}a_8E_f)\nonumber \\ &&\hphantom{\fl h=} +{\alpha }_{11}\frac{E_f^2q\cos \theta }{M|\vec{P_f}|} (a_8-a_6b_4M_f+{\alpha }_{11}a_8b_3E_f)-{\alpha }_{11}\frac{E_fq\cos \theta }{M|\vec{P_f}|} b_3({\alpha }_{11}a_8E_f^2+a_5M_f^2\nonumber \\ &&\hphantom{\fl h=} -a_8\vec{q}\cdot \vec{P_f})-\frac{q^2}{2M\vec{P_f}^2}(-M_f^2+(2E_f^2+M_f^2)\cos ^2\theta ) (a_8-a_6b_4M_f+{\alpha }_{11}a_8b_3E_f)\nonumber \\ &&\hphantom{\fl h=} +\frac{q^2E_f}{2M\vec{P_f}^2}(3\cos ^2\theta -1) (b_3({\alpha }_{11}a_8E_f^2+a_5M_f^2-a_8\vec{q}\cdot \vec{P_f}))\nonumber \\ &&\hphantom{\fl h=}\times\frac{q}{2}(\cos ^2\theta -1) \left.\left(-{\alpha }_{11}\frac{E_f}{M}(2(b_3M_f(a_3{\alpha }_{11}-a_2)+a_4(b_2 +{\alpha }_{11}b_4E_f))+4a_8b_3)\right)\right.\nonumber \\ &&\hphantom{\fl h=} \left.+\frac{E_fq^3\cos \theta }{2M|\vec{P_f}|^2}(1-\cos ^2\theta ) (2(b_3M_f(a_3{\alpha }_{11}-a_2)+a_4(b_2+{\alpha }_{11}b_4E_f))+4a_8b_3)\right]. \nonumber\\ \end{eqnarray} \tag{ C.28 }$$