Production of $D_{s0}(2590)^+$ in B nonleptonic decays

Abstract

In 2021, a new charm-strange meson, $D_{s0}(2590)^+$, was discovered, which is believed to be the $D_s^+(2^1S_0)$. However, its low mass and wide width are challenged by theoretical results. Given the small branching ratio of the current production channel, resulting in a small number of events and large errors, we suggest searching for the $D_{s0}(2590)^+$ in the B meson nonleptonic decays, $B_q\rightarrow D^{(*)}_qD_{s0}(2590)^+$ ($q=u,d$), followed by $D_{s0}(2590)^+\rightarrow D^*K$. We find that $Br(B_q\rightarrow D^{(*)}_qD_{s0}(2590)^+)\times Br(D_{s0}(2590)^+\rightarrow D^{*}K)=(2.16\sim 2.82)\times 10^{-3}$ is very large, and the result is not sensitive to the mass of $D_{s0}(2590)^+$. Due to the large branching ratio, numerous $D_{s0}(2590)^+$ events are expected. This study is based on the framework of the instantaneous Bethe–Salpeter equation, and the relativistic wave functions used for mesons contain different partial waves. The contributions of the different partial waves are also studied.

1 Introduction

In 2021, the Large Hadron Collider beauty (LHCb) Collaboration discovered a new charm-strange meson $D_{s0}(2590)^+$ in the decay process of $B^0\rightarrow D^-D^+K^+\pi ^-$ [1]. Its mass, width, and spin-parity were measured, and their values are $m=2591\pm 6\pm 7$ MeV, $\Gamma =89\pm 16\pm 12$ MeV, and $J^P=0^-$, respectively. This new particle is believed to be the $D_s^+(2^1S_0)$ state, the first radial excitation of the pseudoscalar ground-state $D_s^+$ meson.

Since its discovery, the $D_{s0}(2590)^+$ has attracted considerable attention because, as the radial excited state $D_s^+(2^1S_0)$, its mass is much lower than all the theoretical predictions, at least several tens of MeV [2,3,4,5,6,7]. Another issue is its width. On the surface, some theoretical results are consistent with experimental data. However, we pointed out in Ref. [8] that due to its main Okubo–Zweig–Iizuka (OZI)-allowed two-body strong decays occurring near the threshold, and because $\Gamma \propto {|{\textbf{P}}_f}|^3$ (${\textbf{P}}_f$ is the three-dimensional recoil momentum), its width will strongly depend on its mass. Therefore, with different $D_{s0}(2590)^+$ masses, the calculated widths should not be directly compared to each other. To address this issue, we introduced an almost mass-independent quantity. And our calculations showed that unlike surface phenomena, all the theoretically predicted widths are much smaller than the experimental data [5, 9,10,11].

Reference [12] confirms our results. Using a semi-relativistic model, they find that the mass of $D_s^+(2^1S_0)$ is about 60 MeV larger than the data and predict a decay width of $\Gamma (D_{s0}(2590)^+)\simeq 19$ MeV, consistent with our result of 19.7 MeV [8]. Applying the $3P^0$ model, Ref. [13] also estimates that the width of $D_s^+(2^1S_0)$ is about 20 MeV. To understand the $D_{s0}(2590)^+$, they take into account the coupling of $D^*K$, which reduces the mass by about 88 MeV [13]. In a coupled-channel framework, Ref. [14] confirms that to meet the assignment of the $D_{s0}(2590)^+$ as the $D_s^+(2^1S_0)$, the nearby meson–meson thresholds must be taken into account. From the current status, it can be seen that $D_{s0}(2590)^+$ is not well understood, so more careful theoretical and experimental studies are still needed. Further comments on $D_{s0}(2590)^+$ can be found in the review of Ref. [15].

Currently, the discovered channel of $D_{s0}(2590)^+$ is $B^0\rightarrow D^-D^+K^+\pi ^-$, and $D_{s0}(2590)^+$ is reconstructed in the $D^+K^+\pi ^-$ final state [1], which is a three-body strong decay of $D_{s0}(2590)^+$ and has a very small branching ratio, resulting in very limited $D_{s0}(2590)^+$ events with large uncertainties [16]. As the OZI-allowed two-body strong decay $D_{s0}(2590)^+\rightarrow D^*K$ has a dominant branching ratio, instead of $B^0\rightarrow D^-D^+K^+\pi ^-$, a large number of events and small errors are expected in $B^0\rightarrow D^-D^*K$. Thus, in this paper, we will further study the properties of $D_{s0}(2590)^+$ as the $D_s^+(2^1S_0)$ meson, and calculate its production rate in B meson weak decays of $B_q\rightarrow D_q^{(*)} D_{s0}(2590)^+$ with $q=u,d$, followed by $D_{s0}(2590)^+\rightarrow D^*K$.

The nonleptonic decays $B_q\rightarrow D^{(*)}_q D_{s0}(2590)^+$ are calculated using the factorization assumption. In addition, we consider the contribution not only of the leading color-allowed tree diagram, but also others such as the penguin diagram. When calculating the decay amplitude, the instantaneous Bethe–Salpeter equation [17] method, also called the Salpeter equation [18] method, is chosen. The advantage of this approach is that it is a relativistic method [19, 20], and the relativistic wave function contains rich information. For example, in addition to the common nonrelativistic main wave, there are other relativistic partial waves. Besides the decay branching ratio, we will also study the behavior of different partial waves in the hadronic transition.

This paper is organized as follows. In Sect. 2, taking $B^+\rightarrow {\bar{D}}^0D_{s0}(2590)^+$ as an example, we show how to calculate the transition matrix element and give the wave functions with different partial waves. In Sect. 3, we provide our results and some discussion.

2 Theoretical method

2.1 Decay amplitude

According to the effective Hamiltonian [21], the amplitude of the nonleptonic decay $B^+\rightarrow {\bar{D}}^0D_{s0}(2590)^+$ can be written as [22]

$$\begin{aligned} {\mathcal {M}}= & {} \frac{G_F}{\sqrt{2}}\bigg \{{V_{cb}V_{cs}^*}{a_1}\nonumber \\{} & {} +\sum _{q'=u,c}V_{q'b}V_{q's}^*\Big [a_{4}^{q'}+a_{10}^{q'}+\xi (a_{6}^{q'}+a_{8}^{q'})\Big ]\bigg \}A, \end{aligned}$$

(1)

where $G_F$ is the Fermi constant; $V_{cb}$, $V_{cs}$, $V_{ub}$ and $V_{us}$ are the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements [16]; $a_i$ ($i=1$,4,6,8,10) are the combinations of Willson coefficients (their formulas and the expression of coefficient $\xi$ are shown in the Appendix); and A is the hadronic matrix element. In the factorization assumption, it is written as

$$\begin{aligned} A= \bigg \langle {\bar{D}}^0{\mid }J_{\mu }{\mid }B^+\bigg \rangle \bigg \langle D_{s0}(2590)^+{\mid }J^{\mu }{\mid }0\bigg \rangle , \end{aligned}$$

(2)

where $\langle D_{s0}(2590)^+{\mid }J^{\mu }{\mid }0 \rangle =iF_PP_{f_2}^{\mu }$, and $F_P$ and $P_{f_2}$ are the decay constant and momentum of $D_{s0}(2590)^+$, respectively.

In the instantaneous approximation, the hadronic transition matrix element can be written as an overlapping integral with respect to the meson wave functions [20]:

(3)

where $\varphi _{P}^{++}$ and ${\overline{\varphi }}_{P_f}^{++}=\gamma _0(\varphi _{P_f}^{++})^\dag \gamma ^0$ are the positive-energy wave functions of the $B^+$ and ${\bar{D}}^0$ mesons, respectively; P and $P_f$ are the momenta of $B^+$ and ${\bar{D}}^0$, respectively; M is the mass of $B^+$; $\vec {q}$ and $\vec {q}_f$ are the relative momenta between quarks in $B^+$ and ${\bar{D}}^0$, respectively, and their relation is $\vec {q}_f=\vec {q}-\alpha '_1{\vec {P}_f}$, where $\alpha '_1=m_u/(m_u+m_c)$, $m_u$ and $m_c$ are the quark masses.

The formula for the decay width of the two-body is

$$\begin{aligned} \Gamma =\frac{\mid {\vec {P_f}}\mid }{8\pi {M^2}}\sum |{\mathcal {M}}|^2. \end{aligned}$$

(4)

2.2 Relativistic wave functions and their partial waves

In our method, the relativistic wave function of a meson is provided according to its $J^P$, and the numerical values of radial parts are obtained by solving the Salpeter equation. The wave function obtained in this way contains different partial waves.

2.2.1 $0^-$ state

In the instantaneous approximation $(P\cdot {q}=0)$, the relativistic wave function of a $0^-$ meson is written as [23]

(5)

where $q_\perp =q-\frac{P\cdot q}{M^2}P=(0,\vec {q})$, and the radial wave functions $f_i(i=1,2,3,4)$ are functions of $\vec {q}^2$. The corresponding positive-energy wave function is

(6)

where $A_{i}(i=1,2,3,4)$ are related to the radial wave functions $f_i$, and are shown in the Appendix. It can be verified that each term in Eq. (5) or Eq. (6) has a quantum number of $0^-$ [24]. We note that are S-waves, and the $A_3$, $A_4$ terms are P-waves [24]. In a nonrelativistic limit, only S-waves exist, so the $A_3$ and $A_4$ terms are relativistic corrections.

2.2.2 $1^-$ state

The relativistic wave function of the vector $1^-$ state is written as [25]

(7)

where $\epsilon _f$ is the polarization vector, and $g_i(i=1,2,...)$ are functions of ${\vec q}_{f}^2$. The positive-energy wave function is written as

(8)

where $C_i(i=1,2,3,\ldots )$ are functions of $g_i$, and their expressions are shown in the Appendix.

In Eq. (8), the $C_1$ and $C_2$ terms are nonrelativistic S-waves, and all others are relativistic corrections; $C_3$, $C_4$, $C_5$, and $C_6$ are P-waves; $C_7$ and $C_8$ terms are a D-waves mixed with S-waves. The pure D-wave is [26]

(9)

and the corresponding S-wave is

(10)

2.3 Form factors

Using the positive-energy wave functions, the transition matrix element in Eq. (3) is calculated straightforwardly. The transition matrix elements for the transitions $0^{-}\rightarrow 0^{-}$ and $0^{-}\rightarrow 1^{-}$ can then be expressed as functions of form factors

$$\begin{aligned} <{\bar{D}}^0{\mid }J_{\mu }{\mid }B^+>= & {} t_1P_{\mu }+t_2P_{f_{\mu }}, \end{aligned}$$

(11)

$$\begin{aligned} <{\bar{D}}^{*0}{\mid }J_{\mu }{\mid }B^+>= & {} id_1\varepsilon _{{\mu }\epsilon _{f}PP_{f}}\nonumber \\{} & {} +\epsilon _f\cdot {P}(d_2P_{\mu }+d_3P_{f_{\mu }})+d_4\epsilon _{f_{\mu }},\nonumber \\ \end{aligned}$$

(12)

where $t_1$, $t_2$, $d_{1}$, $d_{2}$, $d_{3}$, and $d_{4}$ are the form factors. In the upper formula, we have used the following shorthand notation: $\varepsilon _{{\mu }{\nu }\alpha \beta }\epsilon _f^\nu P^\alpha P_f^\beta =\varepsilon _{{\mu }\epsilon _{f}PP_{f}}$.

3 Numerical results and discussion

In this work, the quark masses $m_b=4.96$ GeV, $m_c=1.62$ GeV, $m_u=0.305$ GeV, $m_d=0.311$ GeV, and $m_s=0.5$ GeV were chosen. For the decay constant of $D_s^+(1968)$, we take $f_{D_s^+(1968)}=0.25$ GeV from the Particle Data Group (PDG) [16], while the decay constant of $D_{s0}(2590)^+$, $f_{D_{s0}(2590)^+}=0.20$ GeV is calculated by our model.

3.1 $B_q\rightarrow D^{(*)}_qD_s^+(1968)$

To validate our method, we first study the processes $B_q\rightarrow D^{(*)}_qD_s^+(1968)$ and show the decay branching ratios in Table 1.

Table 1 The branching ratios $(10^{-3})$ of $B_q\rightarrow D_q^{(*)}D_s^+(1968)$

Our results are very close to experimental data, which confirms that our method is feasible. Therefore, we apply it to $B_q\rightarrow D^{(*)}_qD_{s0}(2590)^+$.

3.2 $B_q\rightarrow D^{(*)}_qD_{s0}(2590)^+$

The calculated branching ratios of decays $B_q\rightarrow D_q^{(*)}D_{s0}$ $(2590)^+$ are shown in Table 2. Compared with the results in Table 1, we can see that the branching ratios of $B_q\rightarrow D_q^{(*)}D_{s0}(2590)^+$ are very large, about half of that of $B_q\rightarrow D_q^{(*)}D_s^+(1968)$. The differences mainly come from the differences in phase spaces and decay constants.

Table 2 The branching ratios $(10^{-3})$ of $B_q\rightarrow D_q^{(*)}D_{s0}(2590)^+$

Table 3 The branching ratios $(10^{-3})$ vary with the mass of $D_{s0}(2590)^+$

In Ref. [8], we obtained

$$\begin{aligned}{} & {} \Gamma (D_{s0}(2590)^+\rightarrow D^*(2007)^0K^+)=10.4 ~\textrm{MeV}, \\{} & {} \Gamma (D_{s0}(2590)^+\rightarrow D^*(2010)^+K^0)=9.29 ~\textrm{MeV}. \end{aligned}$$

Ignoring other secondary contributions, the total width can be estimated as their sum, $\Gamma (D_{s0}(2590)^+) \simeq 19.7 ~\textrm{MeV}$. Then we obtain the following results:

$$\begin{aligned}{} & {} Br(B^+\rightarrow {\bar{D}}^{0}D_{s0}(2590)^+)\times Br(D_{s0}(2590)^+\rightarrow D^{*0}K^+)\nonumber \\{} & {} \quad =2.62\times 10^{-3}, \nonumber \\{} & {} Br(B^+\rightarrow {\bar{D}}^{0}D_{s0}(2590)^+)\times Br(D_{s0}(2590)^+\rightarrow D^{*+}K^0)\nonumber \\{} & {} \quad =2.34\times 10^{-3}, \nonumber \\{} & {} Br(B^+\rightarrow {\bar{D}}^{*0}D_{s0}(2590)^+)\times Br(D_{s0}(2590)^+\rightarrow D^{*0}K^+)\nonumber \\{} & {} \quad =2.82\times 10^{-3}, \nonumber \\{} & {} Br(B^+\rightarrow {\bar{D}}^{*0}D_{s0}(2590)^+)\times Br(D_{s0}(2590)^+\rightarrow D^{*+}K^0)\nonumber \\{} & {} \quad =2.53\times 10^{-3},\nonumber \\ \end{aligned}$$

(13)

$$\begin{aligned}{} & {} Br(B^0\rightarrow {D}^{-}D_{s0}(2590)^+)\times Br(D_{s0}(2590)^+\rightarrow D^{*0}K^+)\nonumber \\{} & {} \quad =2.42\times 10^{-3}, \nonumber \\{} & {} Br(B^0\rightarrow {D}^{-}D_{s0}(2590)^+)\times Br(D_{s0}(2590)^+\rightarrow D^{*+}K^0)\nonumber \\{} & {} \quad =2.16\times 10^{-3}, \nonumber \\{} & {} Br(B^0\rightarrow {D}^{*-}D_{s0}(2590)^+)\times Br(D_{s0}(2590)^+\rightarrow D^{*0}K^+)\nonumber \\{} & {} \quad =2.59\times 10^{-3}, \nonumber \\{} & {} Br(B^0\rightarrow {D}^{*-}D_{s0}(2590)^+)\times Br(D_{s0}(2590)^+\rightarrow D^{*+}K^0)\nonumber \\{} & {} \quad =2.31\times 10^{-3}.\nonumber \\ \end{aligned}$$

(14)

Since the mass of $D_{s0}(2590)^+$ is lower than all the theoretical predictions, we adjust its mass in the range of $2600\sim 2670$ MeV, and show the corresponding branching ratio of $B_q\rightarrow D_q^{(*)}D_{s0}(2590)^+$ in Table 3. It can be seen that as the $D_{s0}(2590)^+$ mass increases, the branching ratio decreases. But due to the large phase space, the decay branching ratio is not very sensitive to the mass of $D_{s0}(2590)^+$.

For the strong decays $D_{s0}(2590)^+\rightarrow D^{*0}K^+$ and $D_{s0}(2590)^+\rightarrow D^{*+}K^0$, they occur near the threshold of $D_{s0}(2590)^+$, so the decay widths strongly depend on the mass of $D_{s0}(2590)^+$ [8]. However, the corresponding branching ratios are not sensitive to the mass [8], so we conclude that the results in Eqs. (13) and (14) are not sensitive to the mass of $D_{s0}(2590)^+$.

3.3 Contributions of different partial waves in hadron transition

The wave functions we provided contain different partial waves, so it is interesting to see their behaviors in the decays. We take the processes $B^+\rightarrow {\bar{D}}^{(*)0}D_{s0}(2590)^+$ as examples to provide the details. Since the $D_{s0}(2590)^+$ only contributes to the decay constant, we ignore the discussion of its partial waves and only consider the contribution of partial waves in the transition $B^+ \rightarrow {\bar{D}}^{(*)0}$.

3.3.1 $B^+ \rightarrow {\bar{D}}^0D_{s0}(2590)^+$

We show the results in Table 4, where the “whole” refers to the contribution using the complete wave function, while the “S-wave” indicates the result obtained using only the S-wave and ignoring others. We use the “prime” to represent the final state.

Table 4 Contributions of partial waves to the branching ratio of $B^+ \rightarrow {\bar{D}}^{0} D_{s0}(2590)^+$ ($10^{-3}$)

Table 4 shows that the $S\times S'$ contribution to the branching ratio is the largest, and the others are small. In the nonrelativistic limit, only $S\times S'$ exists, and its contribution is $Br_\mathrm{{non}}=4.02\times 10^{-3}$. All others provide the relativistic corrections. Our relativistic result is $Br_\mathrm{{rel}}=4.96\times 10^{-3}$, so the relativistic effect in transition $B^+ \rightarrow {\bar{D}}^0$ can be estimated as

$$\begin{aligned} \frac{Br_\mathrm{{rel}}-Br_\mathrm{{non}}}{Br_\mathrm{{rel}}}=19.0\%. \end{aligned}$$

(15)

3.3.2 $B^+ \rightarrow {\bar{D}}^{*0}D_{s0}(2590)^+$

Table 5 Contributions of partial waves to the branching ratio of $B^+ \rightarrow {\bar{D}}^{*0} D_{s0}(2590)^+$ ($10^{-3})$

Table 5 shows the contribution of partial waves in $B^+$ and ${\bar{D}}^{*0}$ to the branching ratio of $B^+$ $\rightarrow$ ${\bar{D}}^{*0}$ $D_{s0}(2590)^+$. The $S\times S'$ has the largest contribution, $3.59\times 10^{-3}$, but this is not the nonrelativistic result. When calculating the nonrelativistic result, we should delete the contribution of the $S'$-wave from $C_7$ and $C_8$ terms in Eq.(10), and we obtain $Br_\mathrm{{non}}=3.83\times 10^{-3}$. Then the relativistic effect is

$$\begin{aligned} \frac{Br_\mathrm{{rel}}-Br_\mathrm{{non}}}{Br_\mathrm{{rel}}}=28.4\%. \end{aligned}$$

(16)

Comparing Eqs. (15) and (16), the relativistic effects are slightly different, but comparable. However, there are significant differences in the details of relativistic corrections in Tables 4 and 5. In Table 4, the relativistic contributions of $P\times P'$, $S\times P'$, and $P\times S'$ are comparable, while the corresponding ones in Table 5 are very different, with the following relationship: $P\times P' \gg S\times P'\gg P\times S'$. The contribution of the $D'$-wave in ${\bar{D}}^{*0}$ is very small.

4 Conclusion

In the framework of the Salpeter equation, we study the production rates of $D_{s0}(2590)^+$ in the non-leptonic decays of B mesons, and the branching ratios are $Br(B^+\rightarrow {\bar{D}}^{0}D_{s0}(2590)^+){=}4.96{\times }10^{-3}$, $Br(B^+\rightarrow {\bar{D}}^{*0}D_{s0}(2590)^+)$ $=5.35\times 10^{-3}$, $Br(B^0\rightarrow D^-D_{s0}(2590)^+)=4.58\times 10^{-3}$, and $Br(B^0\rightarrow D^{*-}D_{s0}(2590)^+)=4.90\times 10^{-3}$. The $D_{s0}(2590)^+$ can be reconstructed in the $D^{*}K$ invariant mass spectrum. For these cascading decays, we have $Br(D_{s0}(2590)^+\rightarrow D^{*0}K^+)=0.528$ and $Br(D_{s0}(2590)^+$ $\rightarrow D^{*+}K^0)=0.472$. Thus, given the large number of B-meson events in Belle and BaBar [27, 28], etc., it is expected that $D_{s0}(2590)^+$ will be carefully studied through these channels.

The quantum number $J^P$ of a meson determines that its wave function is not a pure wave, but contains different partial waves. We studied the contributions of the partial waves in the transitions of $B^+\rightarrow {\bar{D}}^{0}$ and $B^+\rightarrow {\bar{D}}^{*0}$, and found that the main waves contribute to the nonrelativistic results, while other waves provide the relativistic corrections.

Appendices

Appendix A: Coefficients of the positive-energy wave function

For the positive-energy wave function of the $0^-$ state, we have the following expression: $A_{1}=\frac{1}{2}(f_{1}+f_{2}\frac{m_{1}+m_{2}}{\omega _{1}+\omega _{2}})$, $A_{2}=\frac{1}{2}(f_{2}+f_{1}\frac{\omega _{1}+\omega _{2}}{m_{1}+m_{2}})$, $A_{3}=-A_{1}\frac{m_{1}-m_{2}}{m_{2}\omega _{1}+m_{1}\omega _{2}}$, $A_{4}=-A_{1}\frac{\omega _{1}+\omega _{2}}{m_{2}\omega _{1}+m_{1}\omega _{2}}$, where $m_1$, $m_2$, $\omega _1=\sqrt{m_1^2+\vec {q}^2}$, and $\omega _2=\sqrt{m_2^2+\vec {q}^2}$ are the masses and energies of quark 1 and antiquark 2, respectively.

For the $1^-$ state, we have

• $C_{1}=\frac{1}{2}\left[ g_5-\frac{\omega _1+\omega _2}{m_1+m_2}g_6\right]$, $C_2=\frac{1}{2}\left[ -\frac{m_1+m_2}{\omega _1+\omega _2}g_5+g_6\right]$,

• $C_3=\frac{1}{2M_f(m_1\omega _2+m_2\omega _1)}[-(\omega _1+\omega _2)q^2_{f_\perp }g_3-(m_1+m_2)q^2_{f_\perp }g_4+2M_f^2\omega _2g_5-2M_f^2m_2g_6]$,

• $C_4=\frac{1}{2M_f(m_1\omega _2+m_2\omega _1)}[-(m_1-m_2)q^2_{f_\perp }g_3-(\omega _1-\omega _2)q^2_{f_\perp }g_4-2M_f^2m_2g_5+2M_f^2\omega _2g_6]$,

• $C_{5}=\frac{M_f}{2}\frac{\omega _1-\omega _2}{m_1\omega _2+m_2\omega _1}\left[ g_5-\frac{\omega _1+\omega _2}{m_1+m_2}g_6\right]$, $C_{6}=\frac{M_f}{2}\frac{m_1+m_2}{m_1\omega _2+m_2\omega _1}\left[ -g_5+\frac{\omega _1+\omega _2}{m_1+m_2}g_6\right]$,

• $C_7=\frac{1}{2}\left[ g_3+\frac{m_1+m_2}{\omega _1+\omega _2}g_4-\frac{2M_f^2}{m_1\omega _2+m_2\omega _1}g_6\right]$, $C_8=\frac{1}{2}\left[ g_4+\frac{\omega _1+\omega _2}{m_1+m_2}g_3-\frac{2M_f^2}{m_1\omega _2+m_2\omega _1}g_5\right]$.

Appendix B: Some related formulas

In Eq. (1), for decay $B^+\rightarrow {\bar{D}}^0D_s^+$, we have $\xi ={2M_{D_s^+}^2}/(m_b-m_c)/(m_c+m_s)$, while for $B^+\rightarrow {\bar{D}}^{*0}D_s^+$, $\xi =-{2M_{D_s^+}^2}/(m_b+m_c)/(m_c+m_s)$. We also have $a_{1}=c_{1}+{c_{2}}/{N_c}$, $a_{2i}=c_{2i}+{c_{2i}}/{N_c}$, where $i=2,3,4,5$, and $N_c=3$, and the values of Willson coefficients $c_i$ are the same as in Ref. [29]. The expression for $a_i^{q'}$ ($i=4,6,8,10$) is $a_i^{q'}=a_i+I_i^{q'}$ with $I_4^{q'}=I_6^{q'}=\frac{\alpha _s}{9\pi }\left\{ c_1\left[ \frac{10}{9}-G(m_{q'},k^2)\right] \right\}$, $I_8^{q'}=I_{10}^{q'}=\frac{\alpha _e}{9\pi }\frac{1}{N_c}\left\{ (c_1+c_2N_c)\left[ \frac{10}{9}-G(m_{q'},k^2)\right] \right\}$, where the coupling constants $\alpha _s=0.216$ and $\alpha _e={1}/{128}$; $G(m_{q'},k^2)=-4\int _0^1x(1-x)\ln \frac{m_{q'}^2-k^2x(1-x)}{m_b^2}dx$, and the momentum k is taken from Ref. [30].