Isgur-Wise function in Bc decays to charmonium with the Bethe-Salpeter method

  • The heavy quark effective theory vastly reduces the weak-decay form factors of hadrons containing one heavy quark. Many works attempt to apply this theory to the multiple heavy quarks hadrons directly. In this paper, we examine this confusing application by the instantaneous Bethe-Salpeter method from phenomenological respect, and give the numerical results for the $B_c$ decays to charmonium where the final states including $1S$, $1P$, $2S$ and $2P$. Our results indicate that the form factors parameterized by a single Isgur-Wise function deviate seriously from the full ones, especially involving the excited states. The relativistic corrections ($1/m_Q$ corrections) require the introduction of more non-perturbative universal functions, similarly to the Isgur-Wise function, which are the overlapping integrals of the wave functions with the relative momentum between the quark and antiquark.
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Zi-Kan Geng, Yue Jiang, Tianhong Wang, Hui-Wen Zheng and Guo-Li Wang. Isgur-Wise function in Bc decays to charmonium with the Bethe-Salpeter method[J]. Chinese Physics C.
Zi-Kan Geng, Yue Jiang, Tianhong Wang, Hui-Wen Zheng and Guo-Li Wang. Isgur-Wise function in Bc decays to charmonium with the Bethe-Salpeter method[J]. Chinese Physics C. shu
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Isgur-Wise function in Bc decays to charmonium with the Bethe-Salpeter method

    Corresponding author: Yue Jiang, jiangure@hit.edu.cn
  • Department of Physics, Harbin Institute of Technology, Harbin, 150001, China

Abstract: The heavy quark effective theory vastly reduces the weak-decay form factors of hadrons containing one heavy quark. Many works attempt to apply this theory to the multiple heavy quarks hadrons directly. In this paper, we examine this confusing application by the instantaneous Bethe-Salpeter method from phenomenological respect, and give the numerical results for the $B_c$ decays to charmonium where the final states including $1S$, $1P$, $2S$ and $2P$. Our results indicate that the form factors parameterized by a single Isgur-Wise function deviate seriously from the full ones, especially involving the excited states. The relativistic corrections ($1/m_Q$ corrections) require the introduction of more non-perturbative universal functions, similarly to the Isgur-Wise function, which are the overlapping integrals of the wave functions with the relative momentum between the quark and antiquark.

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    I.   INTRODUCTION
    • Under the heavy quark effective theory (HQET), a semileptonic decay process can be related to a rotation of the heavy quark flavor or spin [1, 2]. In the limit $ m_Q\to\infty $ (Q denotes the heavy quark or anti-quark), this rotation is a symmetry transformation. The form factors depend only on the Lorentz boost $ \gamma = v\cdot v' $ which connects the rest frames of the initial state and the final state. The transition can be described by a dimensionless function $ \xi(v\cdot v') $. Heavy-quark symmetry reduces the weak-decay form factors of heavy hadrons to this universal function. These relations were derived by Isgur and Wise firstly [3, 4], so called Isgur-Wise function (IWF).

      HQET vastly simplifies the calculations, thus it plays a crucial role in extracting the values of $ |V_{cb}| $ and $ |V_{ub}| $. For example, the differential semileptonic decay rate for $ B\to D $ in the heavy-quark limit can be model-independently described by [2]

      $ \frac{ {\rm{d}}\Gamma(\bar B\to D\ell\bar\nu)}{ {\rm{d}}\omega} = \frac{G_F^2}{48\pi^3}|V_{cb}|^2(m_B+m_D)^2m_D^3(\omega^2-1)^{3/2}\xi^2(\omega). $

      (1)

      The decay rate depends on only two quantities, $ |V_{cb}| $ and $ \xi(\omega) $. If the differential semileptonic decay rate is measured by experiments, one can obtain the value of $ |V_{cb}|\xi(\omega) $. Exploiting the normalization of Isgur-Wise function $ \xi(1) = 1 $, the value of $ |V_{cb}| $ can be extracted. Conversely exploiting the given value of $ |V_{cb}| $, the differential semileptonic decay rate can be obtained by calculating the Isgur-Wise function. But Isgur-Wise function cannot be calculated by perturbation theory in principle, and can only be obtained by various phenomenological models. Obviously the latter way is model-dependent. Then a great deal of efforts were directed to study the Isgur-Wise function and its applications in different frameworks [516]. In a phenomenological model, the Isgur-Wise function usually corresponds to the overlapping integral of the wave functions.

      However the leading order result is not accurate enough due to the heavy quark approximation. The symmetry-breaking corrections are needed when the study becomes more precise, since the masses of the heavy quarks or anti-quarks are not infinite actually. In addition the radiation correction cannot be ignored. The HQET provides a systematic framework to analyze these corrections. For example, Luke analyzed the $ 1/m_Q $ corrections for a more complicated case of weak decay form factors [17]. Falk et al. analyzed the structure of $ 1/m^2_Q $ corrections for weak decay form factors of both meson and baryon, and calculated the leading QCD radiative corrections [18, 19]. Other efforts of complements are too many to be listed here [2022].

      The flavor-spin symmetry can be used to weak decays involving not only ground hardons but also orbitally and radially excited states containing one heavy (anti-) quark. For example, Isgur and Wise exploited the flavor-spin symmetry to obtain model-independent predictions in weak decays from pseudoscalar meson of a heavy quark $ Q_i $ to P wave excited states of another heavy quark $ Q_j $ in terms of two Isgur-Wise functions [23], and so on [2427]. The HQET is not adopted here, so we do not introduce it much in order to avoid misunderstanding.

      When the systems contain two or more heavy degrees of freedom, the validity of HQET is suspectable, but they are still best described in the nonrelativistic QCD (NRQCD). However, some works have explored the application of the heavy quark symmetry to describe the weak decays of hadrons containing two heavy quarks. Sanchis-Lozano estimated that the flavor symmetry losts, while the spin symmetry holds for the double-heavy meson [28]. As for the baryon containing two heavy quarks, which constitute a bosonic diquark whose mass could be regarded as infinite, and therefore HQET still holds true for the diquark-light quark system. But HQET is not suitable for dealing with the diquark subsystem. Ebert et al. exploited the relativistic quark model to obtain the diquark wave functions, and then the transition amplitudes of heavy diquarks bb and bc going respectively to bc and cc are expressed through the overlap integrals of corresponding diquark wave functions [29]. Pathak et al. treated $ B_c $ meson as a typical heavy-light meson like B or D within a QCD potential model, and the semileptonic decay rates of $ B_c $ meson into $ \eta_c $, $ J/\psi $ are exploited [30]. Das et al. computed the slopes and curvatures of the form factors of semileptonic decays of heavy-light mesons including $ B_c $ [31]. The error of the result for $ B_c $ is very large. Wang et al. obtained form factors for the $ B_c $ into S-wave and P-wave charmonium by three universal Isgur-Wise functions [32]. Almost all of these studies only adopt the formulas in HQET directly, but the usefulness of the heavy quark symmetry to describe double heavy hadrons has little been reported.

      The aim of this paper is to investigate the heavy quark symmetry in the double-heavy mesons from phenomenological respect free of heavy quark limit. The heavy quark symmetry, if holds roughly, must be reflected in the results obtained by a suitable phenomenological model without the employment of heavy quark limit. The instantaneous Bethe-Salpeter (BS) equation is just a very effective method to deal with double-heavy mesons. This method has a comparatively solid foundation because both the BS equation and the Mandelstam formula are established on relativistic quantum field theory. Meanwhile the instantaneous approximation is reasonable, since the quark and antiquark in double-heavy mesons are both heavy. This method gives an analytical expression, so the symmetry can be found intuitively though the accuracy may not be as good as Lattice QCD. We choose the semileptonic $ B_c $ decays to charmonium, and the final mesons involve the orbitally and radially excited states. It is concluded that the flavor symmetry breaks, while the spin symmetry holds for the double-heavy meson, as Sanchis-Lozano estimated [28]. The HQET is applicable to these decays in this paper from phenomenological respect, because the spectator charm quark is not involved in the hard scattering in a short time at order of $ 1/m_W $ [32]. But the form factors parameterized by a single Isgur-Wise function deviate seriously from the full ones, especially involving the excited states. The relativistic corrections require the introduction of more non-perturbative universal functions, similarly to the Isgur-Wise function.

      Actually, some works have been done on IWF with BS equation. Kugo et al. expanded BS equation in orders of the inverse heavy quark mass and defined the leading term in the expansion of the first form factor as IWF [33]. El-Hady et al. pointed out that the IWF can be related to the overlap integral of normalized meson wave functions in the infinite momentum frame and it should be possible to calculate the form factors directly without using the heavy quark limit [34]. Zoller et al. calculated the numerical IWF by multiplying quark masses with a large factor directly [35]. Chang et al. obtained two universal functions in $ B_c\to h_c,\chi_{c} $ with the instantaneous BS method, but the wave functions they used are nonrelativistic [36, 37]. Nowadays the instantaneous BS method has developed to be quite covariant, and the full Salpeter equations are solved for different $ J^{P(C)} $ states [3841]. So the relativistic correction which equates to the symmetry-breaking correction has better been taken into account. Note that in this paper we do not use HQET, but attempt to examine the validity of HQET on double heavy mesons by the instantaneous BS method from phenomenological respect. The heavy quark limit is also not adopted here.

      The paper is organized as follows. In section II, we give the relativistic wave function and Mandelstam formalism in the instantaneous BS method. In section III, we extract the IWF and give the analytical results. In section IV, we give the numerical results and discussions. We summarize and conclude in section V, and put the Salpeter equation and some wave functions in the appendix A.

    II.   RELATIVISTIC WAVE FUNCTION AND FORM FACTORS
    • Usually, the nonrelativistic wave function for a pseudoscalar is written as [36]

      $ \Psi_P(\vec{q}\:) = ({\not\!\! P}+M)\gamma_5 f(\vec{q}\:) , $

      (2)

      where M and P are the mass and momentum of the meson, respectively; $ \vec{q} $ is the relative momentum between the quark and antiquark, and the radial wave function $ f(\vec{q}) $ can be obtained by solving the Schrodinger equation. But in our method, we solve the full Salpeter equation. The form of wave function is relativistic and should depend on the $ J^{P(C)} $ quantum number of the corresponding meson. The relativistic wave function of a pesudosclar can be constructed by P, $ q_{\perp} $ and $ \gamma $-matrices [42]

      $ \varphi_{0^-}(q_{\perp}) \!=\! M\!\left[\frac{{\not \!\!P}}{M}f_1(q_{\perp}) \!+\!f_2(q_{\perp})\!+\!\frac{{\not\!\! q}_{\perp}}{M}f_3(q_{\perp})\!+\!\frac{{\not\!\! P}{\not q}_{\perp}}{M^2}f_4(q_{\perp})\!\right]\!\gamma_5, $

      (3)

      where $ q = p_1-\alpha_1 P = \alpha_2 P-p_2 $ is the relative momentum between quark (with momentum $ p_1 $ and mass $ m_1 $) and antiquark (momentum $ p_2 $ and mass $ m_2 $), $ \alpha_1 = \displaystyle\frac{m_1}{m_1+m_2} $, $ \alpha_2 = \displaystyle\frac{m_2}{m_1+m_2} $; $ q_{\perp} = q-\displaystyle\frac{P\cdot q}{M^2}P $, in the rest frame of the meson, $ q_{\perp} = (0,\vec{q}) $.

      The quantum numbers of all items in Eq. (3) are $ 0^- $. This wave function is a general relativistic form for a pseudoscalar with the instantaneous approximation. If we drop the $ {\cal{O}}(\frac{{\not q}_{\perp}}{M}) $ items, and ignore the difference between $ f_1 $ and $ f_2 $, Eq. (3) would be reduced to the Schrodinger wave function Eq. (2).

      The procedure to solve the full Salpeter equation is put in the appendix A. The last two equations in Eq. (A9) are constraint equations, and therefore only two wave functions in Eq. (3) are independent. We retain $ f_1 $ and $ f_2 $, and derive the normalization condition as

      $ \!\int\!\!\!\frac{ {\rm{d}}\vec q}{(2\pi)^3}4f_1f_2M^2\!\left\{\!\frac{m_1\!\!+\!\!m_2}{\omega_1\!\!+\!\!\omega_2}\!+\!\frac{\omega_1\!\!+\!\!\omega_2}{m_1\!\!+\!\!m_2}\!+\!\frac{2\vec q^{\:2}(m_1\omega_1\!\!+\!\!m_2\omega_2)}{(m_2\omega_1\!\!+\!\!m_1\omega_2)^2}\!\right\} \!\!= \!2M. $

      (4)

      where the quark energy $ \omega_i = \sqrt{m^2_i-q_{\perp}^2} = \sqrt{m^2_i+\vec{q}^{\:2}} $ ($ i = 1,2 $). The projection operators are defined as

      $ \Lambda_{i}^{\pm}(q_{\perp}^{\mu})\equiv\frac{1}{2\omega_{i}}\left[\frac{{\not\!\! P}}{M}\omega_{i}\pm (-1)^{i+1}({\not q}_{\perp}+m_i)\right], $

      (5)

      and the positive energy wave function is

      $ \begin{aligned}[b] \varphi_{0^-}^{++}(q_{\perp}) \equiv & \Lambda_1^+\frac{{\not\!\! P}}{M}\varphi\frac{{\not P}}{M}\Lambda_2^+ = \left[A_1(q_{\perp})+\frac{{\not \!\!P}}{M}A_2(q_{\perp})\right.\\ &\left.+\frac{{\not\!\! q}_{\perp}}{M}A_3(q_{\perp})+\frac{{\not\!\! P}{\not \!\!q}_{\perp}}{M^2}A_4(q_{\perp})\right]\gamma^5 , \end{aligned} $

      (6)

      where

      $ \begin{aligned}[b] A_1 =& \displaystyle\frac{M}{2}\left[\frac{\omega_1+\omega_2}{m_1+m_2}f_1+f_2\right], \quad A_2 = \displaystyle\frac{M}{2}\left[f_1+\frac{m_1+m_2}{\omega_1+\omega_2}f_2\right],\\ A_3 =& -\displaystyle\frac{M(\omega_1-\omega_2)}{m_1\omega_2+m_2\omega_1}A_1,\quad A_4 = -\displaystyle\frac{M(m_1+m_2)}{m_1\omega_2+m_2\omega_1}A_1 . \end{aligned} $

      (7)

      In this paper, besides $ 0^- $ state, we also need to construct the wave functions for $ 1^{--} $ ($ J/\psi $), $ 1^{+-} $ ($ h_c $), $ 0^{++} $ ($ \chi_{c0} $), $ 1^{++} $ ($ \chi_{c1} $) and $ 2^{++} $ ($ \chi_{c2} $) states. We put the $ 2^{++} $ state wave function in the appendix A and the others can be referred to [43].

      The transition amplitude element for $ B_c^+\to (c\bar c)\ell^+\nu_{\ell} $ shown in figure 1, reads

      Figure 1.  Feynman diagram corresponding to the semileptonic decays $B_c^+\to (c\bar c)\ell^+\nu_{\ell}$.

      $ T = \frac{G_F}{\sqrt 2}V_{cb}\bar{u}_{\nu_{\ell}}\gamma^{\mu}(1-\gamma_5)v_{\ell}\left\langle (c\bar c)(P_f)|J_{\mu}|B_c^+(P)\right\rangle, $

      (8)

      where $ (c\bar c) $ denotes charmonium, and $ J_{\mu}\equiv V_{\mu}-A_{\mu} $ is the charged current responsible for this decay. The hadronic part can be calculated by the overlapping integral over the initial and final wave functions, which we call the Mandelstam formalism. The wave function should be solved from the full relativistic BS equation, but it is hard to do. Instead we solve the instantaneous one, namely, the full Salpeter equation. The hadronic transition element should be approximated accordingly [44]

      $ \langle(c\bar c)| \bar b\gamma^{\mu}(1-\gamma^5)c|B_c^+\rangle = \int\frac{ {\rm{d}}\vec q\:'}{(2\pi)^3} {\rm Tr}\bigg[\overline\varphi_{P_f}^{++}(\vec q\:')\frac{{\not P}}{M}\varphi_P^{++}(\vec q\:)\gamma^{\mu}(1-\gamma^5)\bigg], $

      (9)

      where $ \varphi_P^{++} $ denotes the positive energy wave function of the initial state, and $ \overline\varphi_{P_f}^{++}\equiv\gamma^0(\varphi_{P_f}^{++})^{\dagger}\gamma^0 $ for the final state; $ \vec q\:' $ is the relative momentum between quark (with mass $ m'_1 $) and antiquark (mass $ m'_2 $) in the final state, and $ \vec q = \vec q\:'+\displaystyle\frac{m'_1}{m'_1+m'_2} \vec P_f $. In this paper, we drop $ \varphi^{+-},\varphi^{-+},\varphi^{--} $ and keep only the positive energy component $ \varphi^{++} $, because those contributions are much smaller than 1% in transition of $ B_c \to (c\bar c) $ [45]. The integral argument $ \vec q\:' $ in Eq. (9) is convenient for P-wave final state [37].

      For $ B_c^+\to P\ell^+\nu_{\ell} $ (here P denotes $ \eta_c $ or $ \chi_{c0} $), the hadronic matrix element can be factorized as

      $ \left\langle P|\bar b\gamma^{\mu}(1-\gamma^5)c|B_c^+\right\rangle = S_+(P+P_f)^{\mu}+S_-(P-P_f)^{\mu} , $

      (10)

      where $ S_{+} $ and $ S_{-} $ are the form factors. For $ B_c^+\to V\ell^+\nu_{\ell} $ (here V denotes $ J/\psi $, $ h_c $ or $ \chi_{c1} $),

      $\begin{aligned}[b] \langle V|\bar b\gamma^{\mu}(1-\gamma^5)c|B_c^+\rangle =& (t_1 P^{\mu}+ t_2 P_f^{\mu})\frac{\epsilon\cdot P}{M}+t_3(M+M_f)\epsilon^{\mu}\\&+\frac{2t_4}{M+M_f}{\rm{i}}\varepsilon^{\mu\nu\sigma\delta}\epsilon_{\nu} P_{\sigma} P_{f\delta}, \end{aligned}$

      (11)

      where $ \epsilon^{\mu} $ is the polarization vector of the final vector meson; $ t_1 $, $ t_2 $, $ t_3 $ and $ t_4 $ are the form factors. For $ B_c^+\to T\ell^+\nu_{\ell} $ (here T denotes $ \chi_{c2} $),

      $ \begin{aligned}[b] \langle T|\bar b\gamma^{\mu}(1\!-\!\gamma^5)c|B_c^+\rangle \!= & (t_1 P^{\mu}\!+\! t_2 P_f^{\mu})\epsilon_{\alpha\beta}\frac{P^{\alpha} P^{\beta}}{M^2}\!+\!t_3(M\!+\!M_f)\\ & \times\epsilon^{\mu\alpha}\frac{P_{\alpha}}{M} \!+\!\frac{2t_4}{M+M_f}{\rm{i}}\varepsilon^{\mu\beta\sigma\delta}\epsilon_{\alpha\beta}\frac{P^{\alpha}}{M} P_{\sigma} P_{f\delta}, \end{aligned} $

      (12)

      where $ \epsilon_{\alpha\beta} $ is the polarization tensor of the final tensor meson; $ t_1 $, $ t_2 $, $ t_3 $ and $ t_4 $ are the form factors. In this paper, we focus only on the form factors but not the decay widths.

    III.   ISGUR-WISE FUNCTION
    • Since the binding energy is of the order of $ \Lambda_{QCD} $, which is smaller than the constituent quark mass in $ B_c $ and charmonium, the approximation

      $ \omega_i\equiv\sqrt{m_i^2+\vec{q}\:^2}\approx m_i+\frac{\vec{q}\:^2}{2m_i} $

      (13)

      is taken. In the numerical calculation, the large $ |\vec q| $ contribution will be suppressed by the wave function $ f_i(\vec q) $. After performing this approximation and the trace on the matrix element Eq. (9), all the form factors depend only on the overlapping integrals of the wave functions for the initial state and the final state. For instance, one type of overlapping integrals are

      $\begin{aligned}[b]& \int\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}f_1(|\vec q|)f'_1(|\vec q\:'|),\; \; \int\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}f_1(|\vec q|)f'_2(|\vec q\:'|),\end{aligned}$

      $\begin{aligned}[b]\int\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}f_2(|\vec q|)f'_1(|\vec q\:'|),\; \; \int\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}f_2(|\vec q|)f'_2(|\vec q\:'|), \end{aligned}$

      (14)

      where $ f_i $ denotes the wave function of initial state, and $ f'_i $ denotes the wave function of final state. Two wave functions from the same meson is very close numerically, i.e., $ f_1\approx f_2 $ and $ f'_1\approx f'_2 $. So the four overlapping integrals in Eq. (14) are approximately equal, and for convenience they are replaced by their average which is denoted as

      $ \xi_{00}(v\cdot v') = C\int\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}\overline{ff'}, $

      (15)

      where C is the normalization coefficient; $ v,v' $ are the four dimensional velocities of the initial state and final state respectively, and $ \overline{ff'} = \displaystyle\frac{f_1f'_1+f_1f'_2+f_2f'_1+f_2f'_2}{4} $. There are other overlapping integrals with the relative momentum $ \vec q\:' $ being inserted. One will see at once that they are the relativistic ($ 1/m_Q $) corrections to the leading order form factors which parameterized by a single function $ \xi_{00} $. They are denoted as $ \xi_{qx} $, where subscript q denotes the power of the relative momentum $ \vec q\:' $, subscript x denotes the power of $ \cos\theta $, and $ \theta $ is the angle between $ \vec q\:' $ and $ \vec P_f $, i.e.,

      $ \begin{aligned}[b] \xi_{11} = & C\int\displaystyle\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}\overline{ff'}\displaystyle\frac{|\vec q\:'|\cos\theta}{\sqrt{MM'}},\\ \xi_{20} = & C\int\displaystyle\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}\overline{ff'}\frac{\vec q\:'^2}{MM'},\\ \xi_{22} = & C\int\displaystyle\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}\overline{ff'}\displaystyle\frac{\vec q\:'^2\cos^2\theta}{MM'},\\ \xi_{31} = & C\int\displaystyle\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}\overline{ff'}\displaystyle\frac{|\vec q\:'|^3\cos\theta}{\sqrt{(MM')^3}},\\ \xi_{33} = & C\int\displaystyle\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}\overline{ff'}\displaystyle\frac{|\vec q\:'|^3\cos^3\theta}{\sqrt{(MM')^3}},\\ \xi_{40} = & C\int\displaystyle\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}\overline{ff'}\displaystyle\frac{\vec q\:'^4}{(MM')^2},\\ \xi_{42} = & C\int\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}\overline{ff'}\displaystyle\frac{\vec q\:'^4\cos^2\theta}{(MM')^2}, \end{aligned} $

      (16)

      and so on. When the final state is S-wave meson, we keep the first six functions and drop the higher order $ {\cal{O}}(q^4) $; When the final state is P-wave meson whose wave function includes a factor $ \vec q $, $ \xi_{00} $ disappears, thus we reserve the first eight functions and drop the higher order $ {\cal{O}}(q^5) $. The normalization coefficients C based on the normalized formulas are shown in Table. I for each process. Taking the process $ B_c\to\eta_c $ as an example, the initial and final states are both $ 0^- $ state. With the approximations $ \vec q = 0 $, $ f_1 = f_2 $ and $ \omega_i = m_i $, Eq.(4) is deduced as

      final state$\eta_c$$J/\psi$$h_c$$\chi_{c0}$$\chi_{c1}$$\chi_{c2}$
      $C$$4\sqrt{MM'}$$4\sqrt{MM'}$$\frac{4M}{\sqrt{3}} $$ 4M $$4\sqrt{\frac{2}{3}}M$$\frac{4MM'}{\sqrt{3}}$

      Table 1.  The normalization coefficients of different processes.

      $ \int\frac{ {\rm{d}}\vec q}{(2\pi)^3}4Mf^2 = 1. $

      (17)

      So the normalized wave function of $ 0^- $ state is $ 2\sqrt{M}f $, and the normalization coefficient is $ 4\sqrt{MM'} $ for the process $ B_c\to\eta_c $. The above approximation is only used to determine the normalization coefficient but not elsewhere.

      The form factors of semileptonic decay $ B_c\to\eta_c\ell\nu_{\ell} $ can be written as

      $ \begin{aligned}[b] S_+ = & -\frac{M+M'}{2\sqrt{MM'}}\xi_{00}+\frac{1}{4\sqrt{MM'}}\left[b_1\frac{1}{m_1}+b_2\frac{1}{m_2}\right]\alpha\xi_{00}\\ & +\frac{1}{4P'}\left[-b_1\frac{1}{m_1}-b_2\frac{1}{m_2}+a_1\frac{1}{m'_1}+a_2\frac{1}{m'_2}\right]\xi_{11}+\frac{(M-M')P'^2}{8\sqrt{MM'}}\frac{1}{m_1m_2}\alpha^2\xi_{00}\\ & +\frac{P'}{8}\left[(E'+M')\left(\frac{1}{m_1m'_1}+\frac{1}{m_2m'_2}\right)+(E'-M')\left(\frac{1}{m_1m'_2}+\frac{1}{m_2m'_1}\right)-(M-M')\frac{1}{m_1m_2}\right]\alpha\xi_{11}\\ & +\frac{\sqrt{MM'}}{8}\left[(M-M')\left(\frac{1}{m_1m_2}-\frac{1}{m_1m'_2}-\frac{1}{m_2m'_1}+\frac{1}{m'_1m'_2}\right)-(M+M')\left(\frac{1}{m_1m'_1}+\frac{1}{m_2m'_2}\right)\right]\xi_{20}\\ & +\frac{\sqrt{MM'}}{8}(M-E')\left[\frac{1}{m_1m'_1}+\frac{1}{m_1m'_2}+\frac{1}{m_2m'_1}+\frac{1}{m_2m'_2}\right]\xi_{22}\\ & -\frac{P'^2}{16\sqrt{MM'}}\left[b_1\frac{1}{m_1^3}+b_2\frac{1}{m_2^3}\right]\alpha^3\xi_{00}+\frac{P'}{16}\left[b_1\frac{3}{m_1^3}+b_2\frac{3}{m_2^3}+a_1\frac{1}{m_1m_2m'_2}+a_2\frac{1}{m_1m_2m'_1}\right]\alpha^2\xi_{11}\\ & -\frac{\sqrt{MM'}}{16}\left[b_1\left(\frac{1}{m_1^3}-\frac{1}{m_2m'_1m'_2}\right)+b_2\left(\frac{1}{m_2^3}-\frac{1}{m_1m'_1m'_2}\right)\right]\alpha\xi_{20}\\ &-\frac{\sqrt{MM'}}{8}\left[b_1\frac{1}{m_1^3}+b_2\frac{1}{m_2^3}+a_1\frac{1}{m_1m_2m'_2}+a_2\frac{1}{m_1m_2m'_1}\right]\alpha\xi_{22}\\ & +\frac{MM'}{16P'}\left[b_1\left(\frac{1}{m_1^3}-\frac{1}{m_2m'_1m'_2}\right)+b_2\left(\frac{1}{m_2^3}-\frac{1}{m_1m'_1m'_2}\right)\right.\\ & \left.-a_1\left(\frac{1}{m_1^{'3}}-\frac{1}{m_1m_2m'_2}\right)-a_2\left(\frac{1}{m_2^{'3}}-\frac{1}{m_1m_2m'_1}\right)\right]\xi_{31} \end{aligned} $

      (18)

      where $ a_1 = E'^2-E'M+E'M'-MM' = M'(E'-M)(\omega+1) $, $ a_2 = E'^2-E'M-E'M'+MM' = M'(E'-M)(\omega-1) $, $ b_1 = MM'-E'M'+E'M-M'^2 = M'(M-M')(1+\omega) $, $ b_2 = MM'-E'M'-E'M+M'^2 = M'(M+M')(1-\omega) $, $ b_1 = \vec P_f^2-a_1 $, $ b_2 = a_2-\vec P_f^2 $.

      $ \begin{aligned}[b] S_- = & \frac{M-M'}{2\sqrt{MM'}}\xi_{00}+\frac{1}{4\sqrt{MM'}}\left[-c_1\frac{1}{m_1}+c_2\frac{1}{m_2}\right]\alpha\xi_{00}\\ & +\frac{1}{4P'}\left[c_1\frac{1}{m_1}-c_2\frac{1}{m_2}+d_1\frac{1}{m'_1}+d_2\frac{1}{m'_2}\right]\xi_{11}-\frac{(M+M')P'^2}{8\sqrt{MM'}}\frac{1}{m_1m_2}\alpha^2\xi_{00}\\ & +\frac{P'}{8}\left[(M'+E')\left(\frac{1}{m_1m'_1}+\frac{1}{m_2m'_2}\right)+(E'-M')\left(\frac{1}{m_1m'_2}+\frac{1}{m_2m'_1}\right)+2(M+M')\frac{1}{m_1m_2}\right]\alpha\xi_{11}\\ & +\frac{\sqrt{MM'}}{8}\left[(M+M')\left(-\frac{1}{m_1m_2}+\frac{1}{m_1m'_2}+\frac{1}{m_2m'_1}-\frac{1}{m'_1m'_2}\right)+(M-M')\left(\frac{1}{m_1m'_1}+\frac{1}{m_2m'_2}\right)\right]\xi_{20}\\ & -\frac{\sqrt{MM'}}{8}(M+E')\left[\frac{1}{m_1m'_1}+\frac{1}{m_1m'_2}+\frac{1}{m_2m'_1}+\frac{1}{m_2m'_2}\right]\xi_{22} \end{aligned} $

      $ \begin{aligned}[b] & +\frac{P'^2}{16\sqrt{MM'}}\left[c_1\frac{1}{m_1^3}-c_2\frac{1}{m_2^3}\right]\alpha^3\xi_{00}+\frac{P'}{16}\left[-c_1\frac{3}{m_1^3}+c_2\frac{3}{m_2^3}+d_1\frac{1}{m_1m_2m'_2}+d_2\frac{1}{m_1m_2m'_1}\right]\alpha^2\xi_{11}\\ & +\frac{\sqrt{MM'}}{16}\left[c_1\left(\frac{1}{m_1^3}-\frac{1}{m_2m'_1m'_2}\right)-c_2\left(\frac{1}{m_2^3}-\frac{1}{m_1m'_1m'_2}\right)\right]\alpha\xi_{20}\\ & -\frac{\sqrt{MM'}}{8}\left[c_1\frac{1}{m_1^3}-c_2\frac{1}{m_2^3}-d_1\frac{1}{m_1m_2m'_2}-d_2\frac{1}{m_1m_2m'_1}\right]\alpha\xi_{22}\\ & +\frac{MM'}{16P'}\left[-c_1\left(\frac{1}{m_1^3}-\frac{1}{m_2m'_1m'_2}\right)+c_2\left(\frac{1}{m_2^3}-\frac{1}{m_1m'_1m'_2}\right)\right.\\ & \left.-d_1\left(\frac{1}{m_1^{'3}}-\frac{1}{m_1m_2m'_2}\right)-d_2\left(\frac{1}{m_2^{'3}}-\frac{1}{m_1m_2m'_1}\right)\right]\xi_{31} \end{aligned} $

      (19)

      where $ c_1 = E'M+E'M'+MM'+M'^2 $, $ c_2 = E'M-E'M'- MM'+M'^2 $, $ d_1 = E'^2+E'M+E'M'+MM' $, $ d_2 = E'^2+E'M- E'M'-MM' $, $ c_1 = d_1-\vec P_f^2 $, $ c_2 = d_2-\vec P_f^2 $.

      The function $ \xi_{00} $ is just the Isgur-Wise function appearing in HQET for $ 0^-\to 0^- $ decays. Because the form factors of this process will degenerate into those under the nonrelativistic limit if only the function $ \xi_{00} $ is considered [2],

      $ \begin{aligned}[b] \langle \eta_c|\bar b\gamma^{\mu}(1-\gamma^5)c|B_c^+\rangle =& -\sqrt{MM_f}\left[v^{\mu}+v_f^{\mu}\right]\xi_{00},\\ S_{\pm} =& \mp\frac{M\pm M'}{2\sqrt{MM'}}\xi_{00}. \end{aligned} $

      (20)

      The Eq. (18) and Eq. (19) clearly show that the other functions $ \xi_{qx}\; (q\neq 0) $ are the relativistic corrections ($ 1/m_i $ corrections) to the leading order form factors which parameterized by a single IWF $ \xi_{00} $, where i denotes a quark or anti-quark in the initial and the final mesons. The number of $ \vec q\:' $ contained in the function $ \xi_{qx} $ (subscript q) corresponds to the order of the correction. Note that there should have been another type of overlapping integrals with the relative momentum $ \vec q $ in the initial state. For example,

      $ C\int\frac{ {\rm{d}} \vec q\:'}{(2\pi)^3}\overline{ff'}\frac{|\vec q\:|\cos\beta}{\sqrt{MM'}}, $

      (21)

      where $ \beta $ is the angle between $ \vec q $ and $ \vec P_f $. Due to the relation $ \vec q = \vec q\:'+\alpha \vec P_f, \alpha = \frac{m'_1}{m'_1+m'_2} $, this overlapping integral Eq. (21) is decomposed into $ \xi_{11}+\alpha|\vec P_f\:|\xi_{00} $. So the item involving $ \alpha\xi_{00} $ should be considered as relativistic correction of the same order as $ \xi_{11} $. Generally, the item involving $ \alpha^n\xi_{qx} $ is the $ q+n $ order relativistic correction ($ 1/m_i^{q+n} $ correction) which can be confirmed in Eq. (18) and (19). The process $ 0^-\to 1^{--} $ is the same as above the case. The leading order result agrees with HQET [2], i.e.,

      $ \begin{aligned}[b] \langle J/\psi|\bar b\gamma^{\mu}(1-\gamma^5)c|B_c^+\rangle = & \sqrt{MM_f}\left[\epsilon\cdot v v_f^{\mu}\\ &-(v\cdot v_f+1)\epsilon^{\mu}+ {\rm{i}}\varepsilon^{\mu\nu\sigma\delta}\epsilon_{\nu} v_{\sigma} v_{f\delta}\right]\xi_{00}. \end{aligned}$

      (22)

      It is very natural that the leading order analytical results, Eq. (20) and Eq. (22), in this paper are entirely consistent with HQET for $ 0^-\to 0^- $ or $ 1^{--} $ processes. Because for the leading order results, the terms involving $ {\not q} $ disappear and $ \omega_i = m_i $, so that the BS wave functions degenerate into the nonrelativistic case, i.e.,

      $ 0^-:\frac{M+{\not P}}{2\sqrt M}\gamma^5\Psi,\quad 1^{--}:\frac{M+{\not P}}{2\sqrt M}{\not \epsilon} \Psi.$

      (23)

      A pseudoscalar meson and its corresponding vector have the same radial wave function $ \Psi $ in the nonrelativistic limit. But in this paper, the radial wave functions are obtained by solving BS equation numerically, and the $ \Psi $ in Eq. (23) corresponds to the normalized wave function $ 2\sqrt{M}f_i $, where $ f_i $ are the independent components of the BS wave function, just as $ f_1 $ and $ f_2 $ in Eq. (3). Because we do not use the heavy quark limit, the normalized radial wave functions of a pseudoscalar meson and its corresponding vector are not exactly the same numerically, see figures 2(b) and 2(c). Further, the normalized radial wave functions of $ B_c $ and $ \eta_c $ differ greatly, see figures 2(a) and 2(b). And then, the IWF $ \xi_{00} $ in this paper is not the leading order result in HQET, but the corrected IWF in HQET which contains the relativistic corrections ($ 1/m_Q $ corrections). However, our results in section 4 show that the form factors parameterized by this corrected IWF $ \xi_{00} $ still deviate seriously from the full ones, especially involving the excited states, even if the relativistic ($ 1/m_Q $ corrections) correction to $ \xi_{00} $ has been taken into account. Ones need to introduce more non-perturbative universal functions $ \xi_{qx} (q\neq 0) $ to get more accurate form factors. We call them the high order correction functions below.

      Figure 2.  The normalized radial wave functions of $ B_c $ and charmonium ($ n = 1 $).

      For P-wave meson as the final state, the nonrelativistic wave functions are usually written as

      $ \begin{aligned}[b] &\quad {{0}^{++}}:\frac{{{{{\not\!\! q}}}_{\bot }}}{|\vec{q}\ |}\frac{M+\not{P}}{2\sqrt{M}}\Phi ,\\& \quad{{1}^{++}}:{\rm{i}}{{\varepsilon }_{\mu \nu \alpha \beta }}\sqrt{\frac{3}{2}}\frac{{{P}^{\nu }}}{M}\frac{q_{\bot }^{\alpha }}{|\vec{q}\ |}{{\epsilon }^{\beta }}\frac{M+{\not\!\!P}}{2\sqrt{M}}{{\gamma }^{\mu }}\Phi , \\ &\quad {{2}^{++}}:\sqrt{3}{{\epsilon }_{\mu \nu }}{{\gamma }^{\mu }}\frac{q_{\bot }^{\nu }}{|\vec{q}\ |}\frac{M+\not{P}}{2\sqrt{M}}\Phi ,\\&\quad {{1}^{+-}}:\sqrt{3}\frac{{{q}_{\bot }}\cdot \epsilon }{|\vec{q}\ |}\frac{M+{\not\!\!P}}{2\sqrt{M}}{{\gamma }^{5}}\Phi . \end{aligned}$

      (24)

      and these states have the same radial wave function $ \Phi $. Similarly in this paper, the radial wave functions are obtained by solving BS equation numerically, and the $ \Phi $ in Eq. (24) corresponds to the normalized wave function. In these cases, $ \xi_{00} $ disappears and $ \xi_{11} $ is just the corrected Isgur-Wise function appearing in HQET for $ S \to P $ wave decays. We give the leading order results in the case that only the function $ \xi_{11} $ is considered,

      $ \begin{aligned}[b] \langle h_c|\bar b\gamma^{\mu}(1-\gamma^5)c|B_c^+\rangle =& \sqrt{3MM_f}\frac{v\cdot v_f}{|\vec v_f|}(\epsilon\cdot v)\left[v^{\mu}+v_f^{\mu}\right]\xi_{11},\\ \langle \chi_{c0}|\bar b\gamma^{\mu}(1-\gamma^5)c|B_c^+\rangle =& -\sqrt{MM_f}\frac{v\cdot v_f+1}{|\vec v_f|}\left[v\cdot v_fv^{\mu}-v_f^{\mu}\right]\xi_{11},\\ \langle \chi_{c1}|\bar b\gamma^{\mu}(1-\gamma^5)c|B_c^+\rangle =& \sqrt{\frac{3MM_f}{2}}\frac{v\cdot v_f}{|\vec v_f|}\left[\epsilon\cdot v(v^{\mu}-v\cdot v_fv_f^{\mu})\right.\\ & \left.+\vec v_f^2\epsilon^{\mu}+{\rm{i}}(v\cdot v_f+1)\varepsilon^{\mu\nu\sigma\delta}\epsilon_{\nu} v_{\sigma} v_{f\delta}\right]\xi_{11},\\ \langle\chi_{c2}|\bar b\gamma^{\mu}(1-\gamma^5)c|B_c^+\rangle =& -\sqrt{3MM_f}\frac{v\cdot v_f}{|\vec v_f|}\left[\epsilon_{\alpha\beta}v^{\alpha} v^{\beta} v_f^{\mu}\right.\\ & \left.-(v\cdot v_f+1)\epsilon^{\mu\alpha}v_{\alpha}+{\rm{i}}\epsilon_{\alpha\beta}v^{\alpha}\varepsilon^{\mu\beta\sigma\delta}v_{\sigma} v_{f\delta}\right]\xi_{11}. \end{aligned} $

      (25)

      These results do not agree with Ref. [32]. The latter analyzes the reduction of form factors in the heavy quark limit, and there are two IWFs $ \xi_E,\xi_Fv_{\alpha} $ for $ B_c $ to P-wave charmonium. While in this paper the leading order form factors depend only on the IWF $ \xi_{11} $. Ref. [32] does not further describe the used IWFs $ \xi_E,\xi_Fv_{\alpha} $. The disagreement needs further examination. Note that the above analytical results are not confined to the processes of $ B_c $ to charmonium, but hold true for each possible process whose initial and final mesons have the same $ J^{PC} $ with Eq. (20), (22) and (25). In the next section, we will give the numerical results and discussions on the specific processes.

    IV.   RESULTS AND DISCUSSIONS
    • The parameters used in this paper: $ \Gamma_{B_c} = 1.298\times $ $ 10^{-12}\; {\rm{GeV}}, G_F = 1.166\times 10^{-5}\; {\rm{GeV^{-2}}}, \,m_b = 4.96\; {\rm{GeV}},\, m_c = $$1.62\; {\rm{GeV}} $, $ M_{h_c(2P)} \!=\! 3.887\; {\rm{GeV}}, M_{\chi_{c0}(2P)}\! =\! 3.862\; {\rm{GeV}}, $$ M_{\chi_{c1}(2P)} \!=3.872\; {\rm{GeV}}, M_{\chi_{c2}(2P)} = 3.927\; {\rm{GeV}} $.

      After solving the corresponding full Salpeter equations, the numerical wave functions for different mesons are obtained and shown in figures 2-3. When $ |\vec q\:| $ is large, the wave functions will decrease rapidly. So the approximation Eq. (13) can be taken here, and the error from large $ |\vec q\:| $ will be suppressed by the wave function. The numerical values of two dominate (independent) components of BS wave function are almost equivalent for each meson, so the approximation that the four overlapping integrals in Eq. (14) are replaced by their average is reasonable. For $ 1^{--} $ or $ 2^{++} $ state, there are two other minor (also independent) components of BS wave function $ g_3,g_4 $, and $ g_3\approx -g_4 $. Taking the approximation $ g_3 = -g_4 $ and approximation Eq. (13), $ g_3,g_4 $ only appear in the $ {\cal{O}}(q^4) $ or higher order in the $ 1^{--} $ state BS wave function. Within the precision $ {\cal{O}}(q^3) $ of this study for process $ 0^-\to 1^{--} $, these two minor wave functions $ g_3,g_4 $ disappear. It is similar for $ 2^{++} $ state wave function. These are consistent with Eq. (23) and (24), in which there is only one radial wave function.

      Figure 3.  The normalized radial wave functions of the charmonium ($ n = 2 $).

      The behaviors of the Isgur-Wise function $ \xi_{00} $ and high-order corrections $ \xi_{qx} $, i.e., the overlapping integrals of the wave functions of the initial and final bound states, are computed numerically and plotted in Fig. 4-5, where $ \omega = v\cdot v_f = \frac{P\cdot P_f}{MM_f} $. These IWFs can be classified into four categories according to the configurations $ nL $ of initial and final states. They belong to the modes $ 1S\to 1S $, $ 1S\to 1P $, $ 1S\to 2S $ and $ 1S\to 2P $ respectively. In each mode, for example, in the processes $ B_c\to\eta_c $ and $ B_c\to J/\psi $, the behaviors of IWFs are virtually identical except their argument $ \omega = v\cdot v_f $. Because these decay processes are just related by a rotation of the heavy-quark spin or the meson spin, and this rotation is a symmetry transformation in the infinite-mass limit. Note that the infinite-mass limit is not used in this paper, but this spin-symmetry reflected in the results automatically, as Fig. 4-5 shows. This indicates that the spin-symmetry still holds though the initial and final states are both the double-heavy mesons. Comparing the different modes, for example, the final $ \eta_c $ turns into $ \eta_c(2S) $, the behaviors of IWFs become significantly different from before. Next we will discuss these four modes one by one.

      Figure 4.  The IWF and high-order corrections $\xi_{qx}$ vs $\omega$ for $B_c$ to charmonium ($n=1$), where $\omega=v\cdot v_f=\frac{P\cdot P_f}{MM_f}$. The solid line is the Isgur-Wise function, the dash and dot-dash one are the first order correction functions, the dot one is the second order correction function, and so on, in every subfigure.

      Figure 5.  The IWF and high-order corrections $\xi_{qx}$ vs $\omega$ for $B_c$ to charmonium ($n=2$), where $\omega=v\cdot v_f=\frac{P\cdot P_f}{MM_f}$. The meaning of each type line is the same as that in Fig. 4.

      The mode $ 1S\to 1S $ has been extensively studied in HQET. $ B_c $ and $ \eta_c $ are related by the replacement $ b\to c $, while $ \eta_c $ and $ J/\psi $ are related by the transformation $ c^{\Uparrow}\to c^{\Downarrow} $ here. These two rotations (flavour and spin rotations) are symmetry transformations in the infinite-mass limit. So the radial wave functions of these mesons will be identical in this limit, and the corresponding IWF at zero recoil, i.e. $ \xi(1) $ will be the same as the normalization formula of wave functions. It is very natural that $ \xi(1) = 1 $ in HQET. In this paper we solve the full Salpeter equations without the infinite-mass limit, and the normalized radial wave function is approximated to $ 2\sqrt{M}f $ for $ 1S $ state. The normalized wave functions have little difference for $ \eta_c $ and $ J/\psi $ (two dominate wave functions), which is consistent with Eq. (23). But the discrepancy between $ B_c $ and the former two is in the order of 30% (peak value), as Fig. 2(a)-2(c) shows. This indicates that in the double-heavy system the spin-symmetry holds, while the flavour-symmetry breaks. The masses of quark and antiquark are in the same order of magnitude, and therefore the change of flavour will lead to a great impact. Although the analytical expressions (Eq. (20) and Eq. (22)) of the form factors parameterized by a single IWF $ \xi_{00} $ are the same as HQET, these IWFs $ \xi_{00} $ are not strict unity at zero recoil in this paper, as figures 4(a)-4(b) show. The relativistic correction ($ 1/m_Q $ correction) reflected in IWF $ \xi_{00} $ is around 10% at zero recoil, which is consistent with the estimate from HQET [2]. In the mode $ 1S\to 1S $, it is convenient to fit the IWF as

      $ \xi_{00}(\omega) = \xi_{00}(1)\left[1-\rho^2(\omega-1)+c(\omega-1)^2\right], $

      (26)

      where $ \rho^2 $ is the slope parameter and c is the curvature parameter which characterizes the shape of the IWF. The slope by fitting are 2.25 in $ B_c\to\eta_c $, and it is 2.38 in $ B_c\to J/\psi $. The result agrees with the rule that the slope is bigger as the (reduced) mass is heavier [11, 31]. Note that the fitting Eq. (26) is only used to compare our slopes with other literatures, but not elsewhere. The other functions $ \xi_{qx} $ are the relativistic corrections to the form factors parameterized by a single IWF $ \xi_{00} $. The more $ \vec q\:' $ the correction function contains, the less contribution it makes. We may call the correction function with one relative momentum $ \vec q\:' $ as the first order correction, the correction function with two $ \vec q\:' $ as the second order correction, and so on. The values of $ \xi_{11} $ is about 1/20 of $ \xi_{00} $, as figures 4(a)-4(b) show. Because the decay width is proportional to modular square of amplitude, the first order correction to the decay width may reach 1/10 of the leading order result, which is important for accurate calculation. Our previous study shows that the higher order relativistic corrections also have considerable contributions, and the total relativistic correction can reach around 20% at the level of decay width [43].

      In the mode $ 1S\to 1P $, the configuration of initial state is $ 1S $, while the configuration of final state is $ 1P $. Their orbital angular momenta are different, so the symmetry transformations exist only between the final states, i.e., spin rotations. $ \chi_{c0} $, $ \chi_{c1} $ and $ \chi_{c2} $ are spin triplet states that are related by the rotation of total spin component (the component of total spin in the direction of orbital angular momentum), while $ h_c $ and the former three are related by the transformation $ c^{\Uparrow}\to c^{\Downarrow} $. These two spin rotations are symmetry transformations in the infinite-mass limit, so the normalized radial wave functions of these mesons will be identical. The infinite-mass limit is not adopted here, and the normalized radial wave functions of these mesons are approximated to $ \frac{2|\vec q\:|h}{\sqrt{3M}},\frac{2|\vec q\:|\phi}{\sqrt{M}},\frac{2\sqrt{2}|\vec q\:|\psi}{\sqrt{3M}} $ and $ \frac{2\sqrt{M}|\vec q\:|\zeta}{\sqrt 3} $ respectively, where $ h, \phi, \psi $ and $ \zeta $ are the independent components of BS wave functions. Their numerical results are almost the same, as figures 2(d)-2(f) shows, which is consistent with Eq. (24). This indicates that the spin-symmetry holds in the P-wave charmonium though the quark and anti-quark have the same masses. Because P-wave function contains a $ \vec q\: $, $ \xi_{00} $ disappears, and IWF is $ \xi_{11} $ whose behavior is obvious different from $ \xi_{00} $. Due to the presence of $ \cos\theta $, see Eq. (16), the IWF $ \xi_{11} $ is zero at zero recoi, and it is enhanced kinematically, as Figs. 4(c)-4(e) shows. This behavior agrees with Ref. [37]. There is a kinematically suppressed factor $ 1/|\vec v_f| $ in the form factors Eq. (25), and therefore the behaviors of the leading order form factors are not purely dependent on $ \xi_{11} $. The $ \xi_{20} $ and $ \xi_{22} $ are comparable to the leading order $ \xi_{11} $, especially at zero recoil. $ \xi_{22} $ is smaller than $ \xi_{20} $ due to the factor $ \cos^2\theta $. They decrease slowly when the momentum recoil increases and therefore the relativistic corrections may be comparable to the nonrelativistic results in this mode. Although the other correction functions seem to be very small, they are still important for accurate calculation, just as the mode $ 1S\to 1S $. For the final states as $ h_c $, $ \chi_{c0} $, $ \chi_{c1} $ and $ \chi_{c2} $, the total relativistic corrections are 50%, 64%, 34% and 14% at the level of decay width respectively[43]. The total relativistic correction of $ B_c\to\chi_{c2} $ is unusually small, because the different orders corrections cancel each other out. This can be seen in the following analysis of form factors.

      In the mode $ 1S\to 2S $, the configuration of initial state is different from the final state. Similarly, the only symmetry transformation is the spin rotation $ c^{\Uparrow}\to c^{\Downarrow} $ which relates $ \eta_c(2S) $ with $ \psi(2S) $. Their normalized wave functions are almost the same, as Fig. 3(a)-3(b) shows, which is consistent with Eq. (23). The numerical results of IWFs in this mode is negative. Though the signs of IWFs do not affect the result of width, the negative values of IWFs indicate that the negative parts of $ 2S $-wave functions play a primary role. The overlapping integral of wave functions $ \int {\rm{d}}\vec q = \int\vec q\:^2\sin\theta {\rm{d}}|\vec q\:| {\rm{d}}\theta {\rm{d}}\phi $ contains a factor $ \vec q\:^2 $. It is suppressed when $ |\vec q\:|<1 $ while is enhanced when $ |\vec q\:|>1 $. The negative parts of $ 2S $-wave functions are mainly in the range of $ |\vec q\:|>1 $, so the negative parts play a primary role in the overlapping integral. The IWF $ \xi_{00} $ is increasing together with the momentum recoil, as Figs. 5(a)-5(b) shows, and the leading order form factors have the same behaviors due to Eq. (20) and (22). The behaviors of other correction functions in this mode are similar to $ 1S\to 1S $, but they make more contributions here. For example, $ \xi_{11} $ and $ \xi_{20} $ are about one fifth and one eighth of $ \xi_{00} $ at the maximum recoil, respectively. So the relativistic corrections become greater, and our previous study shows they are about 19%–28% larger than those in the mode $ 1S\to 1S $ [43].

      Comparing to the mode $ 1S\to 1P $, the analysis about the symmetry and normalized wave functions is the same in $ 1S\to 2P $, as Figs. 5(c)-5(e) shows. The IWF $ \xi_{11} $ is also zero at zero recoil, but increases more slowly. The $ \xi_{20} $ and $ \xi_{22} $ are comparable to IWF $ \xi_{11} $, and are no more decreasing but increasing as the momentum recoil is increasing. So the relativistic corrections may be more significant in this mode. They are about 10%–16% larger than those in the mode $ 1S\to 1P $ [43].

      The above analysis of relativistic corrections is qualitative, because the kinematic factors multiplied by IWFs are different and complex. In order to discuss these relativistic corrections precisely, the form factors in these processes are calculated by different order corrections in turn. Their numerical results are compared with those calculated by instantaneous Bethe-Salpeter method directly, as Figs. 6-13 shows. In these Figs, $ t\equiv(P-P_f)^2 $ is the momentum transfer, and $ t_m $ is the maximum of t, so $ t_m-t = 2MM_f(v\cdot v'-1) $. The circle-solid line (BSE) denotes the form factor calculated by instantaneous Bethe-Salpeter method directly, and we regard it as the more precise result because this method is almost covariant; the solid line denotes the leading order (LO) of form factor calculated only by IWF; the dash line denotes the result with IWF and first order (1st) correction; the dot-dash line denotes the result with IWF, the first and second order (2nd) corrections; the dot line denotes the result with IWF, the first, second and third order (3rd) corrections.

      Figure 6.  The form factors of $B_c\to\eta_c,J/\psi$ calculated by IWFs and instantaneous BS method, where $t\equiv(P-P_f)^2$ is the momentum transfer, and $t_m-t=2MM_f(v\cdot v'-1)$. The circle-solid line denotes the form factor calculated by instantaneous Bethe-Salpeter method directly; the solid line denotes the leading order of form factor calculated only by IWF; the dash line denotes the result with IWF and first order correction; the dot-dash line denotes the result with IWF, the first and second order corrections; the dot line denotes the result with IWF, the first, second and third order corrections.

      Figure 7.  The form factors of $B_c\to h_c,\chi_{c0}$ calculated by IWFs and instantaneous Bethe-Salpeter method, where $t\equiv(P-P_f)^2$ is the momentum transfer, and $t_m-t=2MM_f(v\cdot v'-1)$. The meaning of each type line is the same as that in Fig. 6.

      Figure 8.  The form factors of $B_c\to \chi_{c1}$ calculated by IWFs and instantaneous Bethe-Salpeter method, where $t\equiv(P-P_f)^2$ is the momentum transfer, and $t_m-t=2MM_f(v\cdot v'-1)$. The meaning of each type line is the same as that in Fig. 6.

      Figure 9.  The form factors of $B_c\to \chi_{c2}$ calculated by IWFs and instantaneous Bethe-Salpeter method, where $t\equiv(P-P_f)^2$ is the momentum transfer, and $t_m-t=2MM_f(v\cdot v'-1)$. The meaning of each type line is the same as that in Fig. 6.

      Figure 10.  The form factors of $B_c\to\eta_c(2S),\psi(2S)$ calculated by IWFs and instantaneous Bethe-Salpeter method, where $t\equiv(P-P_f)^2$ is the momentum transfer, and $t_m-t=2MM_f(v\cdot v'-1)$. The meaning of each type line is the same as that in Fig. 6.

      Figure 13.  The form factors of $B_c\to \chi_{c2}(2P)$ calculated by IWFs and instantaneous Bethe-Salpeter method, where $t\equiv(P-P_f)^2$ is the momentum transfer, and $t_m-t=2MM_f(v\cdot v'-1)$. The meaning of each type line is the same as that in Fig. 6.

      For the process $ B_c\to\eta_c $, as Figs. 6(a)6(b) shows, there is some gap between the Leading order $ S_+ $ and the result from BSE. The $ S_+ $ with 1st correction is close to BSE. When the 3rd correction is taken into account, the result becomes very accurate. The difference between the LO $ S_- $ and BSE is slightly larger, but due to the small contribution of $ S_- $ to the decay width, the leading order results may be approximate. Though the high order corrections do not make $ S_- $ and BSE exactly the same, they become closer. For the process $ B_c\to J/\psi $, as Figs. 6(c)6(f) shows, the form factor $ t_3 $ makes main contribution to the decay width. The LO $ t_3 $ has a little gap with BSE, and the result with high order corrections is more accurate. The LO $ t_1 $ is zero that agrees with HQET, see Eq. (22), but far from BSE. The LO $ t_2 $ and $ t_4 $ are also different from BSE. The high order corrections bring them closer to BSE, of which 1st correction is the most important one. In the mode $ 1S\to 1S $, the leading order form factors which parameterized by a single IWF $ \xi_{00} $ may be approximate, but the high order corrections can make the result more precise. Note that, the accurate result of $ t_1 $ cannot be obtained by correcting $ \xi_{00} $, so we need to introduce new high-order correction functions.

      For the process $ B_c\to h_c $, as Figs. 7(a)7(d) shows, the form factor $ t_2 $ makes the main contribution to the decay width. The LO $ t_2 $ is slightly different from BSE. The 1st, 2nd and 3rd corrections are not small but almost cancel each other out. These corrections make $ t_2 $ closer to BSE. The LO $ t_3 $ and $ t_4 $ is zero which can be used to examine our method. The $ t_1 $, $ t_2 $ and $ t_4 $ with 1st correction are still a lot different from BSE, and therefore the higher order corrections are necessary. For the process $ B_c\to \chi_{c0} $, as Figs. 7(e)7(f) shows, the difference between the LO $ S_+ $ and BSE is large. At least the 1st correction needs to be considered in order to reach an approximate result, though the $ S_- $ with 1st correction is not accurate enough. The higher order corrections can make the results more accurate. For the process $ B_c\to \chi_{c1} $, as Figs. 8(a)8(d) shows, the form factor $ t_2 $ makes main contribution to the decay width. The 1st corrections are great except $ t_4 $. The higher order corrections are small but still important for accurate calculation. For the process $ B_c\to \chi_{c2} $, as Figs. 9(a)9(d) shows, the form factor $ t_3 $ makes main contribution to the decay width. The LO $ t_3 $ is close to BSE, and the high order corrections almost cancel each other out. It leads to the unusually small result of the total relativistic correction. The 1st correction makes $ t_1 $, $ t_2 $ and $ t_4 $ closer to BSE, but makes the main form factor $ t_3 $ farther from BSE. The result may be more imprecise if only the IWF and 1st correction are considered, so the higher-order corrections are very important. At zero recoil, the IWF $ \xi_{11} $ is zero and the kinematical factor $ (v\cdot v_f)/|\vec v_f| $ will lead to the divergence, see Eq. (25). However the most LO form factors are limited values at zero recoil. In general, the relativistic corrections are large, and the 1st corrections can only derive the approximate results in the mode $ 1S\to 1P $.

      In the modes $ 1S\to 2S $ and $ 1S\to 2P $, the form factors are no longer depressed but a little enhanced. The relativistic corrections are similar to those discussed above, but are greater and more complicated, as Figs. 10-13 shows. Generally, there are big gaps between the leading order form factors and those from BSE directly. The newly introduced high-order correction functions make significant contributions in these relativistic corrections.

      Figure 11.  The form factors of $B_c\to h_c(2P),\chi_{c0}(2P)$ calculated by IWFs and instantaneous Bethe-Salpeter method, where $t\equiv(P-P_f)^2$ is the momentum transfer, and $t_m-t=2MM_f(v\cdot v'-1)$. The meaning of each type line is the same as that in Fig. 6.

      Figure 12.  The form factors of $B_c\to \chi_{c1}(2P)$ calculated by IWFs and instantaneous Bethe-Salpeter method, where $t\equiv(P-P_f)^2$ is the momentum transfer, and $t_m-t=2MM_f(v\cdot v'-1)$. The meaning of each type line is the same as that in Fig. 6.

    V.   CONCLUSION
    • In this work, we examine the validity of the heavy quark effective theory in double heavy mesons by the instantaneous Bethe-Salpeter method from phenomenological respect. With some approximations, all form factors are parameterized by some universal functions $ \xi_{qx} $. These functions are calculated by the overlapping integrals of the phenomenological BS wave functions for the initial state and the final state. We reproduce the classical formulas of the HQET in which the leading order form factors are parameterized by a single Isgur-Wise function $ \xi_{00} $. The heavy quark limit is not adopted here, and therefore the IWF $ \xi_{00} $ in this paper is the corrected IWF in HQET which contains the relativistic correction ($ 1/m_Q $ correction). The IWF $ \xi_{00} $ is not strict unity numerically at zero recoil, i.e. $ \xi_{00}(1)\neq 1 $.

      We choose the semileptonic $ B_c $ decays to charmonium to calculate the numerical results of the Isgur-Wise functions and form factors, where the final states include $ 1S $, $ 1P $, $ 2S $ and $ 2P $. The form factors parameterized by a single corrected IWF $ \xi_{00} $ deviate seriously from the full ones, especially involving the excited states. The deviation requires the introduction of more non-perturbative universal functions $ \xi_{qx} (q\neq 0) $. These functions $ \xi_{qx} (q\neq 0) $ are the relativistic corrections ($ 1/m_Q $ corrections) to the leading order results. These functions $ \xi_{qx} $ mainly depend on the configurations $ nL $ of the initial and final states, so they are universal for each mode. They can simplify the calculations of form factors. This simplification can be generalized to other modes, including but not limited to $ 1S\to 1P $, $ 1S\to 2S $ and $ 1S\to 2P $ which this paper studies. We conclude that the HQET is applicable to these decays in this paper from phenomenological respect, but the higher order correction functions $ \xi_{qx}(q\neq 0) $ provide great relativistic corrections ($ 1/m_Q $ corrections), and must be introduced.

    APPENDIX A: EQUATION AND SOLUTION FOR HEAVY MESONS
    • BS equation for a quark-antiquark bound state generally is written as [46]

      $\tag{A1} ({\not p}_1-m_1)\chi_P(q)({\not\!\! p}_2+m_2) = {\rm{i}}\int\frac{{\rm{d}}^4k}{(2\pi)^4}V(P,k,q)\chi_P(k), $

      where $ p_1,p_2;m_1,m_2 $ are the momenta and masses of the quark and antiquark, respectively; $ \chi_P(q) $ is the BS wave function with the total momentum P and relative momentum q; $ V(P,k,q) $ is the kernel between the quark-antiquark in the bound state. P and q are defined as

      $ \tag{A2} \begin{aligned}[b]& \vec p_1 = \alpha_1\vec P+\vec q,\quad\alpha_1 = \frac{m_1}{m_1+m_2},\\& \vec p_2 = \alpha_2\vec P-\vec q,\quad\alpha_2 = \frac{m_2}{m_1+m_2}. \end{aligned} $

      We divide the relative momentum q into two parts, $ q_{P_{||}} $ and $ q_{P_{\perp}} $, a parallel part and an orthogonal one to P, respectively

      $ \tag{A3}q^{\mu} = q_{P_{||}}^{\mu}+q_{P_{\perp}}^{\mu}, $

      where $ q_{P_{||}}^{\mu}\equiv(P\cdot q/M^2)P^{\mu},\; q_{P_{\perp}}^{\mu}\equiv q^{\mu}-q_{P_{||}}^{\mu} $, and M is the mass of the relevant meson. Correspondingly, we have two Lorentz-invariant variables

      $\tag{A4} q_P = \frac{P\cdot q}{M},\; q_{P_T} = \sqrt{q_P^2-q^2} = \sqrt{-q_{P_{\perp}}^2}. $

      If we introduce two notations as below

      $ \tag{A5} \begin{aligned}[b] \eta(q_{P_{\perp}}^{\mu})\equiv & \int\frac{k_{P_T}^2 {\rm{d}} k_{P_T} {\rm{d}} s}{(2\pi)^2}V(k_{P_{\perp}},s,q_{P_{\perp}})\varphi(k_{p_{\perp}}^{\mu}),\\ \varphi(q_{p_{\perp}}^{\mu})\equiv & {\rm{i}}\int\frac{{\rm{d}}q_P}{2\pi}\chi_P(q_{P_{||}}^{\mu},q_{P_{\perp}}^{\mu}). \end{aligned} $

      Then the BS equation can take the form as follow

      $\tag{A6} \chi_P(q_{P_{||}}^{\mu},q_{P_{\perp}}^{\mu}) = S_1(p_1^{\mu})\eta(q_{P_{\perp}}^{\mu})S_2(p_2^{\mu}). $

      The propagators of the relevant particles with masses $ m_1 $ and $ m_2 $ can be decomposed as

      $\tag{A7} S_i(p_i^{\mu}) = \frac{\Lambda_{i_P}^+(q_{P_{\perp}}^{\mu})}{J(i)q_P+\alpha_iM-\omega_{i_P}+{\rm{i}}\varepsilon}+\frac{\Lambda_{i_P}^-(q_{P_{\perp}}^{\mu})}{J(i)q_P+\alpha_iM+\omega_{i_P}-{\rm{i}}\varepsilon}, $

      with

      $ \tag{A8} \begin{aligned}[b] \omega_{i_P} =& \sqrt{m_i^2+q_{P_T}^2},\\ \Lambda_{i_P}^{\pm}(q_{P_{\perp}}^{\mu}) =& \frac{1}{2\omega_{i_P}}\left[\frac{{\not P}}{M}\omega_{i_P}\pm J(i)({\not q}_{P_{\perp}}+m_i)\right], \end{aligned} $

      where $ i = 1,2 $ for quark and antiquark, respectively, and $ J(i) = (-1)^{i+1} $.

      Then the instantaneous Bethe-Salpeter equation can be decomposed into the coupled equations

      $ \tag{A9} \begin{aligned}[b]& (M-\omega_{1p}-\omega_{2p})\varphi^{++}(q_{P_{\perp}}) = \Lambda_1^+(P_{1p_{\perp}})\eta(q_{P_{\perp}})\Lambda_2^+(P_{2p_{\perp}}),\\& (M+\omega_{1p}+\omega_{2p})\varphi^{--}(q_{P_{\perp}}) = -\Lambda_1^-(P_{1p_{\perp}})\eta(q_{P_{\perp}})\Lambda_2^-(P_{2p_{\perp}}),\\ & \varphi^{+-}(q_{P_{\perp}}) = 0,\quad \qquad\varphi^{-+}(q_{P_{\perp}}) = 0. \end{aligned} $

      The instantaneous kernel has the following form

      $\tag{A10} V(P,k,q)\sim V(|k-q|), $

      especially when the two constituents of meson are very heavy. The kernel we used contains a linear scalar interaction for color-confinement, a vector interaction for one-gluon exchange and a constant $ V_0 $ which as a `zero-point', i.e.

      $\tag{A11} I(r) = \lambda r+V_0-\gamma_0\otimes\gamma^0\frac{4}{3}\frac{\alpha_s(r)}{r}, $

      where $ \lambda $ is the `string constant', $ \alpha_s(r) $ is the running coupling constant. In order to avoid the infrared divergence, a factor $ e^{-\alpha r} $ is introduced, i.e.

      $\tag{A12} \begin{aligned}[b] V_s(r) = &\frac{\lambda}{\alpha}(1-e^{-\alpha r}),\\ V_v(r) =& -\frac{4}{3}\frac{\alpha_s(r)}{r}e^{-\alpha r}. \end{aligned} $

      In momentum space the kernel reads:

      $\tag{A13} I(\vec q\,) = V_s(\vec q\,)+\gamma_0\otimes\gamma^0V_v(\vec q\,), $

      where

      $ \tag{A14} \begin{aligned}[b] 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},\\ V_v(\vec q\,) = &-\frac{2}{3\pi^2}\frac{\alpha_s(\vec q\,)}{\vec q\,^2+\alpha^2},\\ \alpha_s(\vec q\,) = &\frac{12\pi}{27}\frac{1}{{\rm{In}}(a+\vec q\,^2/\Lambda_{QCD}^2)}. \end{aligned} $

      The fitted parameters are $ a = e = 2.7183 $, $ \alpha = 0.06 $ GeV, $ \lambda = 0.21 $ $ {\rm GeV}^2 $, $ \Lambda_{QCD} = 0.27 $ GeV; $ V_0 $ is fixed by fitting the mass of the ground state. With these parameters, the mass spectrums, decay constants and some branching fractions of the double heavy mesons, including $ B_c $, charmonium and bottomium, can be obtained. These results are in good accord with the experimental data. More details can be referred to in the literatures [47, 48].

      The instantaneous Bethe-Salpeter wave function for $ 2^{++} $ states mesons have the general form[41]

      $ \tag{A15} \begin{aligned}[b] \varphi_{2^{++}}(q_{\perp}) =& \epsilon_{\mu\nu}q_{\perp}^{\mu} q_{\perp}^{\nu}\left[\zeta_1(q_{\perp})+\frac{{\not\!\! P}}{M}\zeta_2(q_{\perp}) +\frac{{\not \!\!q}_{\perp}}{M}\zeta_3(q_{\perp})\right.\\&\left.+\frac{{\not\!\! P}{\not q}_{\perp}}{M^2}\zeta_4(q_{\perp})\right]+M\epsilon_{\mu\nu}\gamma^{\mu} q_{\perp}^{\nu}\left[\zeta_5(q_{\perp})+\frac{{\not\!\! P}}{M}\zeta_6(q_{\perp}) \right.\\&\left.+\frac{{\not \!\!q}_{\perp}}{M}\zeta_7(q_{\perp})+\frac{{\not\!\! P}{\not \!\!q}_{\perp}}{M^2}\zeta_8(q_{\perp})\right] \end{aligned} $

      with

      $ \tag{A16} \begin{aligned}[b] \zeta_1(q_{\perp}) =& \frac{q_{\perp}^2\zeta_3(\omega_1+\omega_2)+2M^2\zeta_5\omega_2}{M(m_1\omega_2+m_2\omega_1)}\\ \zeta_2(q_{\perp}) =& \frac{q_{\perp}^2\zeta_4(\omega_1-\omega_2)+2M^2\zeta_6\omega_2}{M(m_1\omega_2+m_2\omega_1)}\\ \zeta_7(q_{\perp}) = &\frac{M(\omega_1-\omega_2)}{m_1\omega_2+m_2\omega_1}\zeta_5\\ \zeta_8(q_{\perp}) =& \frac{M(\omega_1+\omega_2)}{m_1\omega_2+m_2\omega_1}\zeta_6 \end{aligned} $

      The wave function corresponding to the positive projection has the form

      $ \tag{A17} \begin{aligned}[b] \varphi_{2^{++}}^{++}(q_{\perp}) =& \epsilon_{\mu\nu}q_{\perp}^{\mu} q_{\perp}^{\nu}\left[B_1(q_{\perp})+\frac{ {\not \!\!P}}{M}B_2(q_{\perp})+\frac{ {\not\!\! q}_{\perp}}{M}B_3(q_{\perp})\right.\\&\left.+\frac{ {\not\!\! P} {\not \!\!q}_{\perp}}{M^2}B_4(q_{\perp})\right]+M\epsilon_{\mu\nu}\gamma^{\mu} q_{\perp}^{\nu}\left[B_5(q_{\perp})+\frac{ {\not\!\! P}}{M}B_6(q_{\perp})\right.\\&\left.+\frac{ {\not \!\!q}_{\perp}}{M}B_7(q_{\perp})+\frac{ {\not\!\! P} {\not\!\! q}_{\perp}}{M^2}B_8(q_{\perp})\right] \end{aligned} $

      where

      $ \begin{aligned}[b] B_1 =& \frac{1}{2M(m_1\omega_2+m_2\omega_1)}[(\omega_1+\omega_2)q_{\perp}^2\zeta_3\\&+(m_1+m_2)q_{\perp}^2\zeta_4+2M^2\omega_2\zeta_5-2M^2m_2\zeta_6]\\ B_2 =& \frac{1}{2M(m_1\omega_2+m_2\omega_1)}[(m_1-m_2)q_{\perp}^2\zeta_3\\&+(\omega_1-\omega_2)q_{\perp}^2\zeta_4+2M^2\omega_2\zeta_6-2M^2m_2\zeta_5]\\ B_3 =& \frac{1}{2}\left[\zeta_3+\frac{m_1+m_2}{\omega_1+\omega_2}\zeta_4-\frac{2M^2}{m_1\omega_2+m_2\omega_1}\zeta_6\right]\\ B_4 =& \frac{1}{2}\left[\frac{\omega_1+\omega_2}{m_1+m_2}\zeta_3+\zeta_4-\frac{2M^2}{m_1\omega_2+m_2\omega_1}\zeta_5\right]\\ B_5 = &\frac{1}{2}\left[\zeta_5-\frac{\omega_1+\omega_2}{m_1+m_2}\zeta_6\right],\qquad A_6 = \frac{1}{2}\left[-\frac{m_1+m_2}{\omega_1+\omega_2}\zeta_5+\zeta_6\right] \end{aligned} $

      $ \tag{A18} \begin{aligned}[b] B_7 = &\frac{M}{2}\frac{\omega_1-\omega_2}{m_1\omega_2+m_2\omega_1}\left[\zeta_5-\frac{\omega_1+\omega_2}{m_1+m_2}\zeta_6\right]\\ B_8 =& \frac{M}{2}\frac{m_1+m_2}{m_1\omega_2+m_2\omega_1}\left[-\zeta_5+\frac{\omega_1+\omega_2}{m_1+m_2}\zeta_6\right] \end{aligned} $

      If the masses of the quark and antiquark are equal, the normalization condition reads as

      $\tag{A19} \int\!\!\!\frac{ {\rm{d}}\vec q}{(2\pi)^3}\frac{8\omega_1\vec q\:^2}{15m_1}\left[5\zeta_5\zeta_6M^2\!+\!2\zeta_4\zeta_5\vec q\:^2\!-\!2\vec q\:^2\zeta_3\left(\zeta_4\frac{\vec q\:^2}{M^2}\!+\!\zeta_6\right)\right] \!=\! 2M. $

Reference (48)

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