Charmless two-body B meson decays in the perturbative QCD factorization approach

$ \begin{aligned}[b] {{\cal{H}}}_{{\rm{eff}}}(\Delta b = - 1) =& \frac{{G_\rm{F}}}{\sqrt{2}} \Big[ V_{U q}^\ast \, V_{Ub} \left( C_1(\mu) \, O_1(\mu) + C_2(\mu) \, O_2(\mu) \right) \\ & - V_{tq}^\ast \, V_{tb} \, \sum_{i=3}^{10} \, C_{i}(\mu) \, O_{i}(\mu) \\&- V_{tq}^\ast \, V_{tb} C_{8g}(\mu) \, O_{8g}(\mu) \Big] , \end{aligned} $

$ \begin{aligned}[b] O_1 =& \left( \bar{q}_\alpha \, U_\beta \right)_{{\rm{V-A}}} \left( \bar{U}_\beta \, b_\alpha \right)_{{\rm{V-A}}} \,,\\ O_2 =& \left( \bar{q}_\alpha \, U_\alpha \right)_{{\rm{V-A}}} \left( \bar{U}_\beta \, b_\beta \right)_{{\rm{V-A}}} \,, \end{aligned} $

$ \begin{aligned}[b]O_3 =& \left( \bar{q}_\alpha \, b_\alpha \right)_{{\rm{V-A}}} \sum_{q^\prime} \left( \bar{q}^\prime_\beta \, q^\prime_\beta \right)_{{\rm{V-A}}} \,, \\ O_4 =& \left( \bar{q}_\alpha \, b_\beta \right)_{{\rm{V-A}}} \sum_{q^\prime} \left( \bar{q}^\prime_\beta \, q^\prime_\alpha \right)_{{\rm{V-A}}} \,, \\ O_5 =& \left( \bar{q}_\alpha \, b_\alpha \right)_{{\rm{V-A}}} \sum_{q^\prime} \left( \bar{q}^\prime_\beta \, q^\prime_\beta \right)_{{\rm{V+A}}} \,, \\ O_6 =&\left( \bar{q}_\alpha \, b_\beta \right)_{{\rm{V-A}}} \sum_{q^\prime} \left( \bar{q}^\prime_\beta \, q^\prime_\alpha \right)_{{\rm{V+A}}} \,, \end{aligned} $

$ \begin{aligned}[b] O_7 = &\frac{3}{2} \left( \bar{q}_\alpha \, b_\alpha \right)_{{\rm{V-A}}} \sum_{q^\prime} e_{q^\prime} \left( \bar{q}^\prime_\beta \, q^\prime_\beta \right)_{{\rm{V+A}}} \,, \\ O_8 =& \frac{3}{2} \left( \bar{q}_\alpha \, b_\beta \right)_{{\rm{V-A}}} \sum_{q^\prime} e_{q^\prime} \left( \bar{q}^\prime_\beta \, q^\prime_\alpha \right)_{{\rm{V+A}}} \,, \\O_9 =& \frac{3}{2} \left( \bar{q}_\alpha \, b_\alpha \right)_{{\rm{V-A}}} \sum_{q^\prime} e_{q^\prime} \left( \bar{q}^\prime_\beta \, q^\prime_\beta \right)_{{\rm{V-A}}} \,, \\ O_{10} = &\frac{3}{2} \left( \bar{q}_\alpha \, b_\beta \right)_{{\rm{V-A}}} \sum_{q^\prime} e_{q^\prime} \left( \bar{q}^\prime_\beta \, q^\prime_\alpha \right)_{{\rm{V-A}}} \,, \end{aligned} $

$ O_{8 g} = \frac{g_s}{8 \pi^2} \, m_b \, \bar{q}_\alpha \, \sigma^{\mu\nu} \left( 1 + \gamma_5 \right) T_{\alpha\beta}^a \, G_{\mu\nu}^a \, b_\beta \,. $

$ \begin{eqnarray} p_3 = \frac{1}{\sqrt{2}} \left( 0, m_B, {\bf{0}}_T \right) . \end{eqnarray} $

$ \begin{eqnarray} \frac{1}{k_T^2 - xm_B^2 - {\rm i} \epsilon} = {{\cal{P}}} \left( \frac{1}{k_T^2 - xm_B^2} \right) + {\rm i} \pi \delta(k_T^2 - xm_B^2). \end{eqnarray} $

$ \begin{aligned}[b] {{\cal{M}}}(B \to M_2M_3) =& H_{r}(M_W, t ) \otimes H(t,\mu) \otimes \phi(x,P^+,b,\mu) \\ =& \sum_{i} C_i(M_W,t) \otimes H_i(t,b) \otimes \phi(x,b,1/b) \\& \cdot \exp \Bigg[ -s(P^+,b) - \int_{1/b}^{t} \frac{{\rm d}\overline{\mu}}{\overline{\mu}} \gamma_\phi(\alpha_s(\overline{\mu}))\Bigg]. \end{aligned} $

$ \begin{aligned}[b] &\int {\rm d}^4z_1 \, {\rm e}^{{\rm i} \bar{k}_1 \cdot z_1} \big\langle 0 \vert \bar{d}_\sigma(z_1) \, b_{ \beta}(0) \vert \bar{B}^0(p_1) \big\rangle \\ =& \frac{{\rm i} f_B}{\sqrt {2 N_c}} \Bigg\{ ( \not p_1 + m_B) \gamma_5 \Bigg[ \frac{ \not n_+}{\sqrt{2}} \varphi_+(\bar{x}_1, b_1) \\&+ \left( \frac{ \not n_-}{\sqrt{2}} - k_1^+ \gamma_\perp^\nu \frac{\partial}{\partial {\bf{k}}^\nu_{1 T}}\right) \varphi_-(\bar{x}_1, b_1) \Bigg] \Bigg\}_{\beta\sigma} \, \\ =& \frac{-{\rm i} }{\sqrt {2 N_c}} \left\{ ( \not p_1 + m_B) \gamma_5 \left[ \varphi_B(x_1, b_1) - \frac{ \not n_+ - \not n_-}{\sqrt{2}} \bar{\varphi}_B(x_1, b_1) \right] \right\} _{\beta\sigma} \,. \end{aligned} $

$ \begin{aligned}[b] \varphi_B(x_1, b_1) =& \frac{1}{2} \left[ \varphi_+(x_1, b_1) + \varphi_-(x_1, b_1) \right] \,, \\ \bar{\varphi}_B(x_1, b_2) =& \frac{1}{2} \left[ \varphi_-(x_1, b_1) - \varphi_+(x_1, b_1) \right] \,. \end{aligned} $

$ \begin{eqnarray} \varphi_B(x_1,b_1) = N_B \,x_1^2 \, (1-x_1)^2 \, {\rm{exp}} \left[ - \frac{x_1^2m_B^2}{2 \omega_B^2} - \frac{(\omega_B b_1)^2}{2} \right] , \end{eqnarray} $

$ \begin{eqnarray} \int_0^1 {\rm d} x_1 \, \varphi_B(x_1, b_1=0) = \frac{f_B}{2 \sqrt{2N_c}} \,. \end{eqnarray} $

$ \begin{aligned}[b] & \int {\rm d}^4z \, {\rm e}^{-{\rm i} \bar{k} \cdot z} \,\big\langle \pi^+(p) \big\vert \bar{u}_\delta(0) \, d_\alpha(z) \big\vert 0 \big\rangle \\ =& \frac{-\rm i }{\sqrt {2N_c}} \, \Big\{ \gamma_5 \Big[ \, \not p \, \varphi_\pi^{\rm{a}}( x, b) + m_0^\pi \, \varphi_\pi^{\rm{p}}(x, b) \\&-m_0^\pi \left( \not n_+ \not n_- - 1\right) \, \varphi_\pi^{\rm{t}}(x, b) \Big] \Big\}_{\alpha\delta} . \end{aligned} $

$ \begin{eqnarray} \varphi_P^{\rm{a}}(x, \mu) = \frac{f_{\pi}}{2\sqrt{2N_c}}6x\bar{x} \sum_{n=0} \, a^P_n(\mu) \, C_n^{3/2}(2x-1) \,. \end{eqnarray} $

$ \begin{aligned}[b] \varphi^{\rm{p}}_P(x, \mu) =& \frac{f_{\pi}}{2\sqrt{2N_c}} \Bigg[1 + 3 \rho^P \Big( 1-3a_1^P+6a_2^P\Big)(1+\ln x) \\&- \frac{\rho^P}{2} \Big( 3-27a_1^P+54a_2^P\Big) \, C_1^{1/2}(2x-1) \\ &+ 3 \Big(10 \eta_{3P} - \rho^P (a_1^P - 5 a_2^P) \Big) \, C_2^{1/2}(2x-1) \\&+ \Big(10 \eta_{3P} \lambda_{3P} - \frac{9}{2}\rho^Pa_2^P \Big) \, C_3^{1/2}(2x-1) \\ &- 3 \eta_{3P} \omega_{3P} \, C_4^{1/2}(2x-1) \Bigg] \, , \end{aligned} $

$ \begin{aligned}[b] \varphi_P^{\rm{t}}(x, \mu)=& \frac{1}{6} \, \frac{\rm d}{{\rm d} x }\varphi_P^\sigma(x, \mu), \\ \varphi_P^\sigma(x, \mu) =& \frac{f_{\pi}}{2\sqrt{2N_c}} 6x(1-x) \Bigg\{ 1 + \frac{\rho^P}{2} \Big(2 - 15 a_1^P + 30 a_2^P\Big) \\&+ \rho^P \Big(3a_1^P - \frac{15}{2}a_2^P\Big) \, C_1^{3/2}(2x-1) \\ &+ \frac{1}{2} \Big( \eta_{3P}(10-\omega_{3P}) + 3\rho^P a_2^P \Big) \, C_2^{3/2}(2x-1) \\&+ \eta_{3P} \lambda_{3P} \, C_3^{3/2}(2x-1) \\ &+ 3 \rho^P \Big( 1- 3a_1^P + 6 a_2^P \Big) \, \ln x \Bigg\} \,. \end{aligned} $

$ \begin{aligned}[b] \big\langle \rho^+(p, \epsilon^\lambda) \big\vert \bar{u}(0) \gamma_\tau d(0) \big\vert 0 \big\rangle =& - {\rm i} f^\parallel_\rho m_\rho \epsilon^\lambda_\tau \,, \\ \big\langle \rho^+(p, \epsilon^\lambda) \big\vert \bar{u}(0) \sigma_{\tau,\tau^\prime} d(0) \big\vert 0 \big\rangle =& - {\rm i} f^\perp_\rho \left( \epsilon^\lambda_\tau p_{\tau^\prime} - \epsilon^\lambda_{\tau^\prime} p_\tau \right) \,. \end{aligned} $

$ \begin{aligned}[b] &\int {\rm d}^4 z \, {\rm e}^{-{\rm i} \bar{k} \cdot z} \, \big\langle \rho^+(p, \epsilon^\parallel) \big\vert \bar{u}_{\delta}(0) d_\alpha(z) \big\vert 0 \big\rangle \\ =&\frac{-{\rm i} }{\sqrt {2N_c}} \left\{m_\rho \not \epsilon \,^\parallel \, \varphi_{\rho}^\parallel( x) + \not \epsilon \,^\parallel \not p \, \varphi_{\rho}^{t,\parallel}( x) - m_\rho \, \bar{\psi}_{\rho}^{s,\parallel}( x) \right\}_{\alpha\delta} \,, \end{aligned} $

$ \begin{aligned}[b] &\int {\rm d}^4z \, {\rm e}^{-{\rm i} \bar{k} \cdot z} \, \big\langle \rho^+(p, \epsilon^\perp) \big\vert \bar{u}_{\delta}(0) d_\alpha(z) \big\vert 0 \big\rangle \\ =&\frac{-{\rm i} }{\sqrt {2N_c}} \Bigg\{ \not \epsilon \,^\perp \not p \, \varphi_{\rho}^\perp( x) + m_\rho \not \epsilon\,^\perp\, \varphi_{\rho}^{t,\perp}( x) \\&- \frac{{\rm i} \, m_\rho}{(p \cdot n_-)} \varepsilon_{\tau\tau'\kappa\kappa'} \gamma_5 \gamma^\tau \epsilon^{\perp \tau'} p^\kappa n_-^{\kappa'} \, \bar{\psi}_{\rho}^{s,\perp}(x) \Bigg\}_{\alpha\delta} \,. \end{aligned} $

$ \begin{aligned}[b] \int_0^1 {\rm d} x \, \varphi^{\parallel(\perp)}_\rho(x) =&\frac{f^{\parallel(\perp)}_{\rho}}{2\sqrt{2N_c}} \,,\\ \int_0^1 {\rm d} x \, \varphi_{\rho}^{t,\parallel(\perp)}(x) =& \frac{1}{2\sqrt{2N_c}} \left(f^{\perp(\parallel)}_{\rho} - f^{\parallel(\perp)}_{\rho} \frac{m_u+m_d}{m_\rho} \right)\,. \end{aligned} $

$ \varphi^\parallel_{V}(x, \mu) =\frac{3 f_\rho^\parallel}{\sqrt{6}} x\bar{x} \sum_{n=0} \, a^{V, \parallel}_n(\mu) \, C_n^{3/2}(2x-1) \,, $

$ \begin{aligned}[b] \varphi^{t,\parallel}_{V}(x, \mu) =& \frac{3 f_\rho^\perp}{2\sqrt{6}} (2x-1) \Big\{C_1^{1/2}(2x-1) \\&+ \, a_{V, 1}^\perp \, C_2^{1/2}(2x-1) + \, a_{V, 2}^\perp \, C_3^{1/2}(2x-1) \Big\} \,, \end{aligned} $

$ \begin{eqnarray} \psi^{s,\parallel}_{V}(x, \mu) = \frac{3 f_\rho^\perp}{2\sqrt{6}} x\bar{x} \left\{1 + \frac{a_{V,1}^\perp}{3} \, C_1^{3/2}(2x - 1) + \frac{a_{V, 2}^\perp}{6} \, C_2^{3/2}(2x - 1) \right\} \,. \end{eqnarray} $

$ \begin{eqnarray} \varphi^\perp_{V}(x, \mu) &=& \frac {3 f_\rho^\perp}{\sqrt{6}} x\bar{x} \sum_{n=0} \, a^{V, \perp}_n(\mu) \, C_n^{3/2}(2x-1) \,, \end{eqnarray} $

$ \begin{aligned}[b] \varphi^{t,\perp}_{V}(x, \mu) =& \frac{3 f_\rho^\parallel}{8 \sqrt{6}} \big\{ [1 + (2x-1)^2] + 2\, a_{V,1}^{\parallel} \, (2x-1)^3\\& +8 \, a_{V, 2}^\parallel \, C_2^{1/2}(2x-1) + \frac{6}{7} \, a_{V, 2}^\parallel \, C_4^{1/2}(2x-1) \big\} \,, \end{aligned} $

$ \begin{aligned}[b] \psi^{s,\perp}_{V}(x, \mu) =& \frac{3 f_\rho^\parallel}{4 \sqrt{6}} x\bar{x} \Bigg\{1 + \frac{a_{V,1}^\parallel}{3} \, C_1^{3/2}(2x-1) \\&+ \frac{a_{V, 2}^\parallel}{6} \, C_2^{3/2}(2x-1) \Bigg\} \,. \end{aligned} $

$ \begin{aligned}[b] \left( \begin{array}{c} \big\vert \eta \rangle \\ \big\vert \eta^\prime \rangle \end{array} \right) =& U(\theta) \, \left( \begin{array}{c} \big\vert \eta_8 \rangle \\ \big\vert \eta_1 \rangle \end{array} \right) \, = U(\phi) \, \left( \begin{array}{c} \big\vert \eta_q \rangle \\ \big\vert \eta_s \rangle \end{array} \right) \,\\ =& \left( \begin{array}{ccc} \cos \phi & - \sin \phi \\ \sin \phi & \cos \phi \end{array} \right) \, \left( \begin{array}{c} \big\vert \eta_q \rangle \\ \big\vert \eta_s \rangle \end{array} \right) \, . \end{aligned} $

$ \begin{aligned}[b] \big\langle 0 \big\vert \, \bar{q} \gamma_\tau \gamma_5 q \, \big\vert \eta (\eta') \big\rangle =& {\rm i} f_{\eta (\eta')}^{q} p_\tau \,, \\\big\langle 0 \big\vert \, \bar{s} \gamma_\tau \gamma_5 s \, \big\vert \eta (\eta') \big\rangle =& {\rm i} f_{\eta (\eta')}^{s} p_\tau \,, \end{aligned} $

$ \begin{eqnarray} &&\big\langle 0 \big\vert \, \bar{q} \gamma_\tau \gamma_5 q \, \big\vert \eta_q (p) \big\rangle = \frac{\rm i}{\sqrt{2}} \, p_\tau \, f_q \,, \,\,\,\,\,\, \big\langle 0 \big\vert \, \bar{s} \gamma_\tau \gamma_5 s\, \big\vert \eta_s (p) \big\rangle = {\rm i} \, p_\tau \, f_s \, \end{eqnarray} $

$ \begin{eqnarray} \left( \begin{array}{c c } f_\eta^q & f_\eta^s \\ f_{\eta^\prime}^q & f_{\eta^\prime}^s \end{array} \right) = U(\phi) \, \left( \begin{array}{c c} f_q & 0 \\ 0 & f_s \end{array} \right) \,. \end{eqnarray} $

$ \begin{aligned}[b] \partial^\tau \left(J_{\tau, 5}^{q} \right) =& \sqrt{2} \left( m_u \bar{u} \gamma_5 u + m_d \bar{d} \gamma_5 d + \frac{\alpha_s}{4\pi} G \tilde{G} \right) \,, \\ \partial^\tau \left(J_{\tau, 5}^{s} \right) =& 2 m_s \bar{s} \gamma_5 s + \frac{\alpha_s}{4\pi} G \tilde{G} \,, \end{aligned} $

$ \begin{aligned}[b] &\left\langle 0 \left\vert \partial^\tau \left( \begin{array}{c c}J_{\tau, 5}^{q} & 0 \\ 0 & J_{\tau, 5}^{s}\end{array} \right)\right\vert \left(\begin{array}{c c} \eta & 0 \\ 0 & \eta^\prime \end{array} \right) \right\rangle \\=& \left(\begin{array}{c c } m_{\eta}^2 & 0 \\ 0 & m_{\eta^\prime}^2 \end{array} \right) \left(\begin{array}{c c } f_\eta^q & f_\eta^s \\ f_{\eta^\prime}^q & f_{\eta^\prime}^s \end{array} \right) = {\cal{M}}^2 U(\phi) \left( \begin{array}{c c} f_q & 0 \\ 0 & f_s \end{array} \right) \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{M}}_{qs}^2 =& U^\dagger(\phi) {\cal{M}}^2 U(\phi) \\=& \left( \begin{array}{c c} m_{qq}^2+\dfrac{\sqrt{2}}{f_q}\big\langle 0 \big\vert \dfrac{\alpha_s}{4\pi} G \tilde{G} \big\vert \eta_q \big\rangle & \dfrac{1}{f_s}\big\langle 0 \big\vert \dfrac{\alpha_s}{4\pi} G \tilde{G} \big\vert \eta_q \big\rangle \\ \dfrac{\sqrt{2}}{f_q}\big\langle 0 \big\vert \dfrac{\alpha_s}{4\pi} G \tilde{G} \big\vert \eta_s \big\rangle & m_{ss}^2+\dfrac{1}{f_s}\big\langle 0 \big\vert \dfrac{\alpha_s}{4\pi} G \tilde{G} \big\vert \eta_s \big\rangle \end{array} \right) \, \end{aligned} $

$ \begin{aligned}[b] m_{qq}^2 \equiv& {\rm i} \frac{\sqrt{2}}{f_q} \big\langle 0 \big\vert m_u \bar{u} \gamma_5 u + m_d \bar{d} \gamma_5 d \big\vert \eta_q \big\rangle \\ =& m_\eta^2 \cos^2 \phi + m^2_{\eta^\prime} \sin^2\phi- \frac{\sqrt{2}f_s}{f_q} \left( m^2_{\eta^\prime} - m^2_\eta \right) \cos\phi \sin\phi \,, \\ m_{ss}^2 \equiv& \frac{2}{f_s} \big\langle 0 \big\vert m_s \bar{s} \gamma_5 s \big\vert \eta_s \big\rangle = m_\eta^2 \sin^2 \phi + m^2_{\eta^\prime} \cos^2\phi \\&- \frac{f_q}{\sqrt{2}f_s} \left( m^2_{\eta^\prime} - m^2_\eta \right) \cos\phi \sin\phi\,. \end{aligned} $

$ \begin{eqnarray} \alpha_s(\mu) = \frac{\pi}{2\,\beta_1\, {\rm{log}}(\mu/\Lambda^{(n_f)})} \left[ 1 - \frac{\beta_2}{\beta_1^2} \, \frac{ {\rm{log}}(2\, {\rm{log}}(\mu/\Lambda^{(n_f)}))}{2\, {\rm{log}}(\mu/\Lambda^{(n_f)})} \right] \,, \end{eqnarray} $

$ \begin{aligned}[b]& \overline{m}_c(\overline{m}_c) = 1.28 \, {\rm{GeV}}\,, \,\,\,\,\,\, \overline{m}_b(\overline{m}_b)= 4.18 \, \mathrm{GeV}\,, \\&\overline{m}_t(\overline{m}_t) = 165 \, \mathrm{GeV} \,. \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}_{M_3}^{{\bf{LL}}} =& -{\cal{E}}_{M_3}^{{\bf{LR}}}= 8 \pi C_F m_B^4 f_{M_2} \int_0^1 {\rm d}x_1 \, {\rm d}x_3 \,\\&\times \int_0^{1/\Lambda} b_1 {\rm d} b_1 \, b_3 {\rm d} b_3 \varphi_B(x_1,b_1)\, \\ &\cdot \Big\{ h_{e}(x_1,x_3,b_1,b_3) E_e(\mu_e) \Big[ (2\,r_b-\bar{x}_3)\varphi_{\pi}^a(x_3)\\&-r_3\left(r_b-2\bar{x}_3 \right)\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right) \Big] \\ & + h_{e}(x_3,x_1,b_3,b_1)E_{e^{\prime}}(\mu_{e^{\prime}}) \,2\,r_3\varphi_{\pi}^p(x_3) \Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}_{M_3}^{{\bf{SP}}} =& 16 \pi r_2 C_F m_B^4 f_{M_2} \int_0^1 {\rm d} x_1 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{e}(x_1,x_3,b_1,b_3) E_e(\mu_e) \\ &\times\Big[ \left(2-r_b \right)\varphi_{\pi}^a(x_3) +r_3\left(4r_b+x_3-2\right)\varphi_{\pi}^p(x_3)-r_3x_3\varphi_{\pi}^t(x_3) \Big] + h_{e}(x_3,x_1,b_3,b_1) E_{e^{\prime}}(\mu_{e^{\prime}})\, \left(x_1\varphi_{\pi}^a(x_3)+2r_3 \bar{x}_1 \varphi_{\pi}^p(x_3)\right) \Big\} \,. \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}_{NF,M_3}^{{\bf{LL}}} =& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\,\varphi_\pi(x_2) \\ &\cdot \Big\{ h_{ne}(x_1,{\bar x}_2,x_3,b_1,b_2)E_{ne}(\mu_{ne}) \Big[ (\bar{x}_2-x_1)\varphi_{\pi}^a(x_3)-r_3x_3\left(\varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_3)\right) \Big] \\ & + h_{ne}(x_1,x_2,x_3,b_1,b_2) E_{ne}(\mu_{ne^{\prime}}) \Big[\left(x_1-x_2-x_3\right)\varphi_{\pi}^a(x_3) + r_3x_3\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right)\Big] \Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}_{NF,M_3}^{{\bf{LR}}} =& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 r_2 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\, \\ &\cdot \Big\{ h_{ne}(x_1,{\bar x}_2,x_3,b_1,b_2)E_{ne}(\mu_{ne}) \Big[ \left(\bar{x}_2-x_1\right)\varphi_{\pi}^a(x_3)\left(\varphi_{\pi}^p(x_2) +\varphi_{\pi}^t(x_2) \right) + r_3x_3 \left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2) \right)\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right) \\ & + r_3(\bar{x}_2- x_1) \left(\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2) \right)\left(\varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_3)\right) \Big] + h_{ne}(x_1,x_2,x_3,b_1,b_2) E_{ne}(\mu_{ne^{\prime}}) \Big[ -x_2\varphi_{\pi}^a(x_3)\left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right) \\ & - r_3x_2\left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_3)\right) - r_2x_3\left(\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right) \\ & + x_1\left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right) \left(\varphi_{\pi}^a(x_3)+r_3(\varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_3)) \right) \Big]\Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}_{NF,M_3}^{{\bf{SP}}} =& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\,\varphi_\pi(x_2) \\ &\cdot \Big\{ h_{ne}(x_1,{\bar x}_2,x_3,b_1,b_2)E_{ne}(\mu_{ne}) \Big[\left(x_1-\bar{x}_2-x_3\right)\varphi_{\pi}^a(x_3) +r_3x_3\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3) \right) \Big] \\ & + h_{ne}(x_1,x_2,x_3,b_1,b_2)E_{ne}(\mu_{ne^{\prime}}) \Big[\left( x_2 -x_1\right) \varphi_{\pi}^a(x_3)- r_3x_3\left(\varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_3) \right) \Big]\Big\} \,. \end{aligned} $

$ \begin{aligned}[b]\\[-12pt] {\cal{A}}^{{\bf{LL}}}_{M_3}=&{\cal{A}}^{{\bf{LR}}}_{M_3}= 8 \pi C_F m_B^4 f_B \int_0^1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \Big\{ h_{a}(x_2,{\bar x}_3,b_2,b_3)E_a({\mu_a}) \Big[ -\bar{x}_3 \varphi_{\pi}^a(x_2)\varphi_{\pi}^a(x_3)+ 2r_2r_3 \varphi_{\pi}^p(x_2) \\ & \cdot \Big[ - \left( \varphi_{\pi}^p(x_3) + \varphi_{\pi}^t(x_3) \right) + {\bar x}_3 \left( \varphi_{\pi}^t(x_3) - \varphi_{\pi}^p(x_3) \right) \Big] \Big] + h_{a}({\bar x}_3,x_2,b_3,b_2) E_{a^\prime}({\mu_{a^\prime}}) \Big[ x_2 \varphi_{\pi}^a(x_2)\varphi_{\pi}^a(x_3) \\ &+ 2r_2r_3 \varphi_{\pi}^p(x_3) \Big[ \left( \varphi_{\pi}^p(x_2) - \varphi_{\pi}^t(x_2) \right) + x_2 \left( \varphi_{\pi}^p(x_2) + \varphi_{\pi}^t(x_2) \right) \Big] \Big]\Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{SP}}}_{M_3}=& 16 \pi C_F m_B^4 f_B \int_0^1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \cdot \Big\{ h_{a}(x_2,{\bar x}_3,b_2,b_3)E_a({\mu_a}) \Big[ r_3\bar{x}_3\varphi_{\pi}^a(x_2) \left( \varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right)\\ &+2r_2\varphi_{\pi}^p(x_2)\varphi_{\pi}^a(x_3) \Big] + h_{a}({\bar x}_3,x_2,b_3,b_2)E_{a^\prime}({\mu_{a^\prime}}) \Big[r_2x_2\varphi_{\pi}^a(x_3)\left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right) + 2r_3\varphi_{\pi}^a(x_2)\varphi_{\pi}^p(x_3) \Big]\Big\} \,. \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{LL}}}_{NF,M_3}=& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{na}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na}) \\&\times\Big[ \left(\bar x_1-r_b-x_2\right) \varphi_{\pi}^a(x_2)\varphi_{\pi}^a(x_3)-4r_br_2r_3\varphi_{\pi}^p(x_2)\varphi_{\pi}^p(x_3) \\ & - r_2r_3\left(x_1-\bar{x}_2 \right)\left(\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_3)\right) + r_2r_3x_3 \left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right)\Big] \\ & + h_{na^\prime}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na^{\prime}}) \Big[ \bar{x}_3\varphi_{\pi}^a(x_2)\varphi_{\pi}^a(x_3) + \bar{x}_3r_2r_3\left(\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_3) \right) \\ & + r_2r_3 (x_2-x_1) \left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right) \Big]\Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{LR}}}_{NF,M_3}=& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{na}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na}) \\&\times\Big[ r_3\left(r_b+x_3\right) \varphi_{\pi}^a(x_2)\left(\varphi_{\pi}^t(x_3)-\varphi_{\pi}^p(x_3)\right) + r_2\left(r_b-x_1+\bar{x}_2 \right)\varphi_{\pi}^a(x_3) \left(\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2) \right)\Big] \\ & + h_{na^\prime}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na^{\prime}}) \Big[ r_3\bar{x}_3\varphi_{\pi}^a(x_2)\left(\varphi_{\pi}^t(x_3)-\varphi_{\pi}^p(x_3)\right) + r_2\left(x_2-x_1 \right)\varphi_{\pi}^a(x_3) \left(\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2) \right) \Big]\Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{SP}}}_{NF,M_3}=& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{na}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na}) \\&\times\Big[ (x_3-r_b) \varphi_{\pi}^a(x_2)\varphi_{\pi}^a(x_3)-4r_2 r_3 \varphi_{\pi}^p(x_2)\varphi_{\pi}^p(x_3) + r_2r_3\left(\bar{x}_2-x_1 \right) \left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right) \\ & + r_2r_3x_3 \left(\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2)\right) \big( \varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_2) \Big) \Big] + h_{na^\prime}(x_1,x_2,x_3,b_1,b_2) E_{na}(\mu_{na^{\prime}}) \Big[ \left(x_2-x_1\right)\varphi_{\pi}^a(x_2)\varphi_{\pi}^a(x_3) \\ & + r_2r_3\left(x_2- x_1\right)\left(\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_3)\right) + r_2r_3\bar{x}_3\left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right)\Big]\Big\} \,. \end{aligned} $

$ \begin{eqnarray} &&R1=\Big\{ \varphi_{\pi}^a(x_3) \to \varphi^{\parallel}_{\rho}(x_3) \,, \; \varphi_{\pi}^p(x_3) \to \bar{\psi}_{\rho}^{s,\parallel}(x_3) \,, \; \varphi_{\pi}^t(x_3) \to \varphi_{\rho}^{t,\parallel}(x_3) \Big\} \,, \end{eqnarray} $

$ \begin{eqnarray} &&R2=\Big\{ \varphi_{\pi}^a(x_3) \to \varphi^{\parallel}_{\rho}(x_3) \,, \; \varphi_{\pi}^p(x_3) \to - \bar{\psi}_{\rho}^{s,\parallel}(x_3) \,, \; \varphi_{\pi}^t(x_3) \to - \varphi_{\rho}^{t,\parallel}(x_3) \Big\} \,. \end{eqnarray} $

$ \begin{aligned}[b]&{\cal{E}}^{{\bf LL}}_{{V_3}}=-{\cal{E}}^{{\bf LR}}_{{V_3}} \stackrel{R1}\Longleftarrow {\cal{E}}^{{\bf{LL}}}_{{M_3}} \,, \quad\ {\cal{E}}^{{\bf SP}}_{{V_3}} \stackrel{R1}\Longleftarrow -{\cal{E}}^{{\bf{SP}}}_{{M_3}} \,, \\ &{\cal{E}}^{{\bf LL}}_{{NF,V_3}} \stackrel{R1}\Longleftarrow {\cal{E}}^{{\bf{LL}}}_{{NF,M_3}} \,, \quad {\cal{E}}^{{\bf LR}}_{{NF,V_3}} \stackrel{R1}\Longleftarrow -{\cal{E}}^{{\bf{LR}}}_{{NF,M_3}} \,, \\& {\cal{E}}^{{\bf SP}}_{{NF,V_3}} \stackrel{R1}\Longleftarrow {\cal{E}}^{{\bf{SP}}}_{{NF,M_3}} \,, \\&{\cal{A}}^{{\bf LL}}_{{V_3}}=-{\cal{A}}^{{\bf LR}}_{{V_3}} \stackrel{R2}\Longleftarrow {\cal{A}}^{{\bf{LL}}}_{{M_3}} \,, \quad {\cal{A}}^{{\bf SP}}_{{V_3}} \stackrel{R2}\Longleftarrow -{\cal{A}}^{{\bf{SP}}}_{{M_3}} \,, \\ &{\cal{A}}^{{\bf LL}}_{{NF,V_3}} \stackrel{R2}\Longleftarrow {\cal{A}}^{{\bf{LL}}}_{{NF,M_3}} \,, \quad {\cal{A}}^{{\bf LR}}_{{NF,V_3}} \stackrel{R2}\Longleftarrow -{\cal{A}}^{{\bf{LR}}}_{{NF,M_3}} \,, \\& {\cal{A}}^{{\bf SP}}_{{NF,V_3}} \stackrel{R2}\Longleftarrow -{\cal{A}}^{{\bf{SP}}}_{{NF,M_3}} \,. \end{aligned} $

$ \begin{eqnarray} R=\Big\{ \varphi_{\pi}^a(x_2) \to \varphi^{\parallel}_{\rho}(x_2) \,, \; \varphi_{\pi}^p(x_2) \to \bar{\psi}_{\rho}^{s,\parallel}(x_2) \,, \; \varphi_{\pi}^t(x_2) \to \varphi_{\rho}^{t,\parallel}(x_2) \Big\} \,. \end{eqnarray} $

$ \begin{aligned}[b]{\cal{E}}^{{\bf LL}}_{{P_3}} ={\cal{E}}^{{\bf LR}}_{{P_3}} \stackrel{R}\Longleftarrow {\cal{E}}^{{\bf{LL}}}_{{M_3}} \,, \quad {\cal{E}}^{{\bf SP}}_{{P_3}} =0 \,, \quad{\cal{E}}^{{\bf LL}}_{{NF,P_3}} \stackrel{R}\Longleftarrow {\cal{E}}^{{\bf{LL}}}_{{NF,M_3}} \,, \end{aligned} $

$ \begin{aligned}[b] & {\cal{E}}^{{\bf LR}}_{{NF,P_3}} \stackrel{R}\Longleftarrow {\cal{E}}^{{\bf{LR}}}_{{NF,M_3}} \,, \quad {\cal{E}}^{{\bf SP}}_{{NF,P_3}} \stackrel{R}\Longleftarrow -{\cal{E}}^{{\bf{SP}}}_{{NF,M_3}} \,,\\&{\cal{A}}^{{\bf LL}}_{{P_3}}= -{\cal{A}}^{{\bf LR}}_{{P_3}} \stackrel{R}\Longleftarrow {\cal{A}}^{{\bf{LL}}}_{{M_3}} \,, \quad {\cal{A}}^{{\bf SP}}_{{P_3}} \stackrel{R}\Longleftarrow {\cal{A}}^{{\bf{SP}}}_{{M_3}} \,, \\ &{\cal{A}}^{{\bf LL}}_{{NF,P_3}} \stackrel{R}\Longleftarrow {\cal{A}}^{{\bf{LL}}}_{{NF,P_3}} \,, \quad {\cal{A}}^{{\bf LR}}_{{NF,P_3}} \stackrel{R}\Longleftarrow {\cal{A}}^{{\bf{LR}}}_{{NF,P_3}} \,, \\& {\cal{A}}^{{\bf SP}}_{{NF,P_3}} \stackrel{R}\Longleftarrow -{\cal{A}}^{{\bf{SP}}}_{{NF,M_3}} \,. \end{aligned} $

$ \begin{aligned}[b]&{\cal{E}}^{{\bf LL}}_{{V_3}}= {\cal{E}}^{{\bf LR}}_{{V_3}} \stackrel{R3} \Longleftarrow {\cal{E}}^{{\bf{LL}}}_{{M_3}} \,, \quad {\cal{E}}^{{\bf SP}}_{{V_3}} =0 \,, \quad{\cal{E}}^{{\bf LL}}_{{NF,V_3}} \stackrel{R3}\Longleftarrow {\cal{E}}^{{\bf{LL}}}_{{NF,M_3}} \,, \\& {\cal{E}}^{{\bf LR}}_{{NF,V_3}} \stackrel{R3}\Longleftarrow -{\cal{E}}^{{\bf{LR}}}_{{NF,M_3}} \,, \quad {\cal{E}}^{{\bf SP}}_{{NF,V_3}} \stackrel{R3}\Longleftarrow -{\cal{E}}^{{\bf{SP}}}_{{NF,M_3}} \,, \\ &{\cal{A}}^{{\bf LL}}_{{V_3}} = {\cal{A}}^{{\bf LR}}_{{V_3}} \stackrel{R4}\Longleftarrow {\cal{A}}^{{\bf{LL}}}_{{M_3}} \,, \quad {\cal{A}}^{{\bf SP}}_{{V_3}} \stackrel{R4}\Longleftarrow -{\cal{A}}^{{\bf{SP}}}_{{M_3}} \,, \\ &{\cal{A}}^{{\bf LL}}_{{NF,V_3}} \stackrel{R4}\Longleftarrow {\cal{A}}^{{\bf{LL}}}_{{NF,M_3}} \,, \quad {\cal{A}}^{{\bf LR}}_{{NF,V_3}} \stackrel{R4}\Longleftarrow -{\cal{A}}^{{\bf{LR}}}_{{NF,M_3}} \,, \\& {\cal{A}}^{{\bf SP}}_{{NF,V_3}} \stackrel{R4}\Longleftarrow {\cal{A}}^{{\bf{SP}}}_{{NF,M_3}} \,. \end{aligned} $

$ \begin{aligned}[b]\\[-5pt] R3=\Big\{ \varphi^a_{\pi}(x_{2(3)}) \to \varphi^\parallel_{\rho}(x_{2(3)}) \,, \; \varphi_{\pi}^p(x_{2(3)}) \to \bar{\psi}_{\rho}^{s,\parallel}(x_{2(3)}) \,, \; \varphi_{\pi}^t(x_{2(3)}) \to \varphi_{\rho}^{t,\parallel}(x_{2(3)}) \Big\} \,, \end{aligned} $

$ \begin{eqnarray} &&R4=\Big\{ \varphi^a_{\pi}(x_{2(3)}) \to \varphi^\parallel_{\rho}(x_{2(3)}) \,, \; \varphi_{\pi}^p(x_{2(3)}) \to (-) \bar{\psi}_{\rho}^{s,\parallel}(x_{2(3)}) \,, \; \varphi_{\pi}^t(x_{2(3)}) \to (-) \varphi_{\rho}^{t,\parallel}(x_{2(3)}) \Big\} \,. \end{eqnarray} $

$ \begin{aligned}[b] {\cal{E}}^{{\bf{LL,N}}}_{V_3}=& 8 \pi C_F m_B^4 f_{V_2}^\parallel r_2 \int_0^1 {\rm d}x_1 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{e}(x_1,x_3,b_1,b_3) E_e(\mu_e) \Big[ \left(2-r_b \right)\varphi_{\rho}^{\perp}(x_3) - r_3x_3\bar{\psi}_{\rho}^{s,\perp}(x_3) \\ & +r_3\left(4r_b+x_3-2\right)\varphi_{\rho}^{t,\perp}(x_3) \Big] + h_{e}(x_3,x_1,b_3,b_1)E_{e^{\prime}}(\mu_{e^{\prime}}) \Big[r_3 (x_1+1)\bar{\psi}_{\rho}^{s,\perp}(x_3)+r_3 \bar x_1 \varphi_{\rho}^{t,\perp}(x_3)\Big]\Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}^{{\bf{LL,T}}}_{V_3}=& 8 \pi C_F m_B^4 f_{V_2}^\parallel r_2 \int_0^1 {\rm d}x_1 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{e}(x_1,x_3,b_1,b_3) E_e(\mu_e) \\&\times\Big[ \left(2-r_b \right)\varphi_{\rho}^{\perp}(x_3) + r_3\left(4r_b+x_3-2\right)\bar{\psi}_{\rho}^{s,\perp}(x_3) -r_3x_3\varphi_{\rho}^{t,\perp}(x_3) \Big] + h_{e}(x_3,x_1,b_3,b_1)E_{e^{\prime}}(\mu_{e^{\prime}}) \Big[r_3 \bar{x}_1\bar{\psi}_{\rho}^{s,\perp}(x_3)\\&+r_3(1+x_1)\varphi_{\rho}^{t,\perp}(x_3) \Big]\Big\} \,, \end{aligned} $

$ \begin{eqnarray} {\cal{E}}^{{\bf{LR,N}}}_{V_3}&=&{\cal{E}}^{{\bf{LL,N}}}_{V_3}\,, \quad {\cal{E}}^{{\bf{LR,T}}}_{V_3}={\cal{E}}^{{\bf{LL,T}}}_{V_3}\,, \quad {\cal{E}}^{{\bf{SP,N}}}_{V_3}={\cal{E}}^{{\bf{SP,T}}}_{V_3}=0\,. \end{eqnarray} $

$ \begin{aligned}[b] {\cal{E}}^{{\bf{LL,N}}}_{NF,V_3}=& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 r_2 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{ne}(x_1,{\bar x}_2,x_3,b_1,b_2)E_{ne}(\mu_{ne}) \\&\times \Big[ (\bar x_2-x_1)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right) \varphi_{\rho}^{\perp}(x_3) \Big] + h_{ne}(x_1,x_2,x_3,b_1,b_2) E_{ne}(\mu_{ne^{\prime}}) \Big[(x_2-x_1)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right)\varphi_{\rho}^{\perp}(x_3) \\ & + 2r_3(x_1-x_2-x_3)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3) +\varphi_{\rho}^{t,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)\right)\Big] \Big\} \,, \\[-12pt]\end{aligned} $

$ \begin{aligned}[b] {\cal{E}}^{{\bf{LL,T}}}_{NF,V_3}=& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 r_2 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{ne}(x_1,{\bar x}_2,x_3,b_1,b_2)E_{ne}(\mu_{ne}) \\&\times \Big[ (\bar x_2-x_1 )\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right) \varphi_{\rho}^{\perp}(x_3) \Big] + h_{ne}(x_1,x_2,x_3,b_1,b_2) E_{ne}(\mu_{ne^{\prime}}) \Big[(x_2-x_1)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right)\varphi_{\rho}^{\perp}(x_3) \\ & + 2r_3(x_1-x_2-x_3)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3) +\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)\right)\Big] \Big\} \,,\\[-12pt] \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}^{{\bf{LR,N}}}_{NF,V_3}=&{\cal{E}}^{{\bf{LR,T}}}_{NF,V_3}= \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 r_3 x_3 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \cdot \varphi_B(x_1,b_1)\, \varphi_{\rho}^{\perp}(x_2) \Big[\bar{\psi}_{\rho}^{s,\perp}(x_3) - \varphi_{\rho}^{t,\perp}(x_3) \Big] \\ &\cdot \Big\{ h_{ne}(x_1,{\bar x}_2,x_3,b_1,b_2)E_{ne}(\mu_{ne}) + h_{ne}(x_1,x_2,x_3,b_1,b_2) E_{ne}(\mu_{ne^{\prime}}) \Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}^{{\bf{SP,N}}}_{NF,V_3}=&\frac{16 \sqrt {6}}{3} \pi C_F m_B^4 r_2 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1) \cdot \Big\{ h_{ne}(x_1,{\bar x}_2,x_3,b_1,b_2)E_{ne}(\mu_{ne}) \\&\times \Big[ ( x_1 -\bar{x}_2 )\left(\varphi_{\rho}^{t,\perp}(x_2)-\bar{\psi}_{\rho}^{s,\perp}(x_2)\right)\varphi_{\rho}^{\perp}(x_3) + 2r_3(x_1+\bar{x}_2+x_3)\left(\varphi_{\rho}^{t,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)-\bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)\right)\Big] \\ & + h_{ne}(x_1,x_2,x_3,b_1,b_2) E_{ne}(\mu_{ne^{\prime}}) \Big[(x_2-x_1)\left( \bar{\psi}_{\rho}^{s,\perp}(x_2)-\varphi_{\rho}^{t,\perp}(x_2)\right) \varphi_\rho ^\perp (x_3) \Big] \Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}^{{\bf{SP,T}}}_{NF,V_3}=& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 r_2 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{ne}(x_1,{\bar x}_2,x_3,b_1,b_2) E_{ne}(\mu_{ne}) \\&\times \Big[ (x_1-\bar{x}_2)\left(\varphi_{\rho}^{t,\perp}(x_2)-\bar{\psi}_{\rho}^{s,\perp}(x_2)\right)\varphi_{\rho}^{\perp}(x_3) + 2r_3(x_1-\bar{x}_2-x_3)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)-\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)\right) \Big] \\ & + h_{ne}(x_1,x_2,x_3,b_1,b_2) E_{ne}(\mu_{ne^{\prime}}) \Big[(x_2-x_1)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)-\varphi_{\rho}^{t,\perp}(x_2) \right) \varphi_\rho ^\perp (x_3)\Big] \Big\} \,. \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{LL,N}}}_{V_3}=&{\cal{A}}^{{\bf{LR,N}}}_{V_3}= 8 \pi C_F m_B^4 f_B r_2 r_3 \int_0^1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \cdot \Big\{ h_{a}(x_2,{\bar x}_3,b_2,b_3)E_a({\mu_a}) \\&\times\Big[(x_3-2)(\bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)) - x_3(\bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3) )\Big] \\ &+ h_{a}({\bar x}_3,x_2,b_3,b_2) E_{a^\prime}({\mu_{a^\prime}}) \Big[(x_2+1)\left (\varphi_{\rho}^{t,\perp}(x_2) \varphi_{\rho}^{t,\perp}(x_3)+\bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3) \right) \\& - \bar x_2 \left(\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3) + \bar{\psi}_{\rho}^{s,\perp}(x_2) \varphi_{\rho}^{t,\perp}(x_3) \right) \Big]\Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{LL,T}}}_{V_3}=& -{\cal{A}}^{{\bf{LR,T}}}_{V_3}= -8 \pi C_F m_B^4 f_B r_2 r_3 \int_0^1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \cdot\Big\{h_{a}(x_2, {\bar x}_3,b_2,b_3)E_a({\mu_a}) \\&\times\Big[2\left(\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)+\bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)\right) + x_3\left(\varphi_{\rho}^{t,\perp}(x_2)-\bar{\psi}_{\rho}^{s,\perp}(x_2)\right)\left(\varphi_{\rho}^{t,\perp}(x_3)-\bar{\psi}_{\rho}^{s,\perp}(x_3)\right) \Big] \end{aligned} $

$ \begin{aligned}[b]\quad + h_{a}({\bar x}_3,x_2,b_3,b_2) E_{a^\prime}({\mu_{a^\prime}}) \Big[\bar x_2 \left (\varphi_{\rho}^{t,\perp}(x_2) \varphi_{\rho}^{t,\perp}(x_3)+\bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3) \right)- (1+x_2) \left(\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3) + \bar{\psi}_{\rho}^{s,\perp}(x_2) \varphi_{\rho}^{t,\perp}(x_3) \right) \Big]\Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{SP,N}}}_{V_3} =& {\cal{A}}^{{\bf{SP,T}}}_{V_3}= 16 \pi C_F m_B^4 f_B \int_0^1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \cdot \Big\{ h_{a}(x_2,{\bar x}_3,b_2,b_3) E_a({\mu_a}) \, r_2 \varphi_\rho ^\perp (x_3) \Big[\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\Big] \\ & + h_{a}({\bar x}_3,x_2,b_3,b_2) E_{a^\prime}({\mu_{a^\prime}}) \, r_3 \varphi_\rho ^\perp (x_2) \Big[\varphi_{\rho}^{t,\perp}(x_3)-\bar{\psi}_{\rho}^{s,\perp}(x_3) \Big]\Big\} \, \,. \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{LL,N}}}_{NF,V_3} =& -\frac{16 \sqrt {6}}{3} \pi C_F m_B^4 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\,\\ & \cdot \Big\{ h_{na}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na}) 2r_2r_3r_b \Big[ \varphi_{\rho}^{t,\perp}(x_2)\varphi_{\rho}^{t,\perp}+\bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)\Big] \\ & + h_{na^\prime}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na^{\prime}}) \Big[ \varphi_\rho^\perp (x_2) \varphi_\rho^\perp (x_3) [r_3^2(x_3-1)-r_2^2x_2] \big] \Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{LL,T}}}_{NF,V_3}=& -\frac{16 \sqrt {6}}{3} \pi C_F m_B^4 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\, \\ &\cdot \Big\{h_{na}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na}) 2r_2r_3r_b \Big[ \varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)+\bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)\Big] \\ & + h_{na^\prime}(x_1,x_2,x_3,b_1,b_2) E_{na}(\mu_{na^{\prime}}) \Big[ \varphi_\rho^\perp (x_2) \varphi_\rho^\perp (x_3) [r_3^2(x_3-1) + r_2^2x_2] \big] \Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{LR,N}}}_{NF,V_3}=& \frac{16 \sqrt {6}}{3} \pi C_F m_B^4 \int_0^1 {\rm d}x_1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_2 {\rm d}b_2 \varphi_B(x_1,b_1)\, \cdot \Big\{ h_{na}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na}) \\&\times \Big[ r_2(r_b-x_1+\bar{x}_2)\varphi_{\rho}^{\perp}(x_3)\left(\varphi_{\rho}^{t,\perp}(x_2)+\bar{\psi}_{\rho}^{s,\perp}(x_2)\right) + r_3(r_b+x_3)\varphi_{\rho}^{\perp}(x_2)\left(\bar{\psi}_{\rho}^{s,\perp}(x_3)-\varphi_{\rho}^{t,\perp}(x_3)\right)\Big] \\ & + h_{na^\prime}(x_1,x_2,x_3,b_1,b_2)E_{na}(\mu_{na^{\prime}}) \Big[ r_3 \bar x_3 \varphi_{\rho}^{\perp}(x_2)\left(\bar{\psi}_{\rho}^{s,\perp}(x_3)-\varphi_{\rho}^{t,\perp}(x_3)\right) - r_2(x_1-x_2)\varphi_{\rho}^{\perp}(x_3)\left(\varphi_{\rho}^{t,\perp}(x_2)+\bar{\psi}_{\rho}^{s,\perp}(x_2)\right)\Big]\Big\} \,, \end{aligned} $

$ \begin{eqnarray} {\cal{A}}^{{\bf{LR,T}}}_{NF,V_3}&=&{\cal{A}}^{{\bf{LR,N}}}_{NF,V_3} \,, \quad {\cal{A}}^{{\bf{SP,N}}}_{NF,V_3}={\cal{A}}^{{\bf{LL,N}}}_{NF,V_3} \,, \quad {\cal{A}}^{{\bf{SP,T}}}_{NF,V_3}=-{\cal{A}}^{{\bf{LL,T}}}_{NF,V_3} \,. \end{eqnarray} $

$ \begin{aligned}[b] a_{1,2}(\mu)\to& a_{1,2}(\mu) + \frac{\alpha_s(\mu)C_F}{4\pi} \frac{C_{1,2}(\mu)}{N_C} \, V_{1,2}(M_2) \,, \\ a_{i,i+1}(\mu) \to& a_{i,i+1}(\mu) + \frac{\alpha_s(\mu)C_F}{4\pi} \frac{C_{i+1,i}}{N_C} \, V_{i,i+1}(M_2) \\& {\rm{with}} \; i = 3,5,7,9 \,. \end{aligned} $

$ \begin{aligned}[b] V_i(P) =& 12 \ln\frac{m_b}{\mu} - 18 + \int_0^1 {\rm d} x \, \varphi_{P}^a(x) \, g(x) ~ {\rm{with}} ~ i = 1-4,9,10 \,, \\ V_i(P) =& - 12 \ln\frac{m_b}{\mu} + 6 - \int_0^1 {\rm d}x \, \varphi_{P}^a(x) \, g(1-x) ~ {\rm{with}}~ i = 5,7 \,, \end{aligned} $

$ \begin{aligned}[b]V_i(P) = - 6 + \int_0^1 {\rm d} x \, \varphi^{{\rm{p}}}_{P}(x) \, h(x) ~ {\rm{with}} ~ i = 6,8 \,, \end{aligned} $

$ \begin{aligned}[b] g(x) =& 3 \left( \frac{1-2x}{1-x} \ln x - {\rm i} \pi \right) + \Bigg[ 2 \, {\rm{Li_2}}(x) - \ln^2 x \\&+ \frac{2 \, \ln x}{1-x} + (3+2{\rm i}\pi) \, \ln x - \{ x \leftrightarrow 1-x\} \Bigg] \,, \\ h(x) =& 2 \, {\rm{Li_2}}(x) - \ln^2 x - (1+2{\rm i} \pi) \, \ln x - \{ x \leftrightarrow 1-x \} \,. \end{aligned} $

$ \begin{eqnarray} {{\cal{C}}}^{(u,c)}(\mu, l^2) = \left[ {{\cal{G}}}^{(u,c)}(\mu, l^2) - \frac{2}{3} \right] C_2(\mu) \,, \end{eqnarray} $

$ \begin{aligned}[b] {{\cal{C}}}^{(t)} (\mu, l^2)=& \left[ {{\cal{G}}}^{(s)}(\mu, l^2) - \frac{2}{3} \right] C_3(\mu) \\&+ \sum_{q^{\prime\prime}}^{u,d,s,c} {{\cal{G}}}^{(q^{\prime\prime})}(\mu, l^2) \left[ C_4(\mu) + C_6(\mu) \right] \,, \end{aligned} $

$ \begin{eqnarray} {{\cal{G}}}^{(c)}(\mu, l^2) = - 4 \int_0^1 \, {\rm d}x \, x(1-x) \ln \left[ \frac{m_c^2 - x(1-x) l^2}{\mu^2} \right] \,, \end{eqnarray} $

$ \begin{aligned}[b] {\rm{Re}} \left[ {{\cal{G}}}^{(c)}(\mu, l^2) \right] =& \frac{2}{3} \left( \frac{5}{3} + \frac{4 m_c^2}{l^2} - \ln \frac{m_c^2}{\mu^2} \right) + \frac{2}{3} \left( 1 + \frac{2 m_c^2}{l^2} \right) \\&\times\left\{ \begin{array}{*{20}{l}} \sqrt{\beta} \ln \dfrac{\sqrt{\beta} - 1}{\sqrt{\beta} + 1}\,, & l^2\in (- \infty, 0) \,, \\ -2 \sqrt{- \beta} \cot^{-1} \sqrt{- \beta} \,, & l^2\in [0, 4m_c^2) \,, \\ -2 \sqrt{\beta} \,, & l^2 = 4 m_c^2 \,,\\ \sqrt{\beta} \ln \dfrac{1 - \sqrt{\beta}}{1 + \sqrt{\beta}} \,, & l^2\in (4 m_c^2, \infty) \,, \end{array} \right. \, \end{aligned} $

$ \begin{eqnarray} {\rm{Im}} \left[ {{\cal{G}}}^{(c)}(\mu, l^2) \right] = \frac{2 \pi}{3} \left( 1 + \frac{2 m_c^2}{l^2} \right) \sqrt{\beta} \, \Theta[\beta] \,, \end{eqnarray} $

$ \begin{aligned}[b] {{\cal{M}}}^{({\rm{ql}})}_{B \to P_2 P_3} =& - \frac{8}{\sqrt{6}} C_F^2 m_B^4 \int_0^1 {\rm d}x_1{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1) \, \Big\{h_{e}(x_1,x_3,b_1,b_3) E_q(\mu_q) \\ &\times\Big[ \varphi_{\pi}^a(x_2)\big[ \left(2r_b-\bar{x}_3\right)\varphi_{\pi}^a(x_3) - r_3\left(r_b-2\bar{x}_3\right)\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right)\big] \\ & - 2r_2\varphi_{\pi}^p(x_2)\big[\left(r_b-2\right)\varphi_{\pi}^a(x_3) - r_3\left(4r_b+x_3-2\right)\varphi_{\pi}^p(x_3) +r_3x_3\varphi_{\pi}^t(x_3)\big] \Big] \\ & + h_{e}(x_3,x_1,b_3,b_1)E_{q^{\prime}}(\mu_{q^{\prime}}) \, 2 \big[ r_3\varphi_{\pi}^a(x_2)\varphi_{\pi}^p(x_3) + r_2\varphi_{\pi}^p(x_2) (x_1\varphi_{\pi}^a(x_3)+2r_3\bar{x}_1\varphi_{\pi}^p(x_3) )\big] \Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}^{({\rm{ql}})}_{ B\to P_2V_3} =& \frac{8}{\sqrt{6}} C_F^2 m_B^4 \int_0^1 {\rm d}x_1 {\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1) \, \Big\{ \, h_{e}(x_1,x_3,b_1,b_3) E_q(\mu_q) \\ &\times \Big[ \varphi_{\pi}^a(x_2) \big[ \left(\bar{x}_3- 2 r_b\right)\varphi_\rho^\parallel(x_3)+r_3\left(r_b-2\bar{x}_3\right) \left(\bar{\psi}_{\rho}^{s,\parallel}(x_3)+\varphi_{\rho}^{t,\parallel}(x_3)\right) \big] \\ & - 2r_2\varphi_{\pi}^p(x_2)\big[\left(r_b-2\right)\varphi_\rho^\parallel(x_3)-r_3\left(4r_b+x_3-2\right)\bar{\psi}_{\rho}^{s,\parallel}(x_3)+r_3x_3\varphi_{\rho}^{t,\parallel}(x_3) \big] \Big] \\ &+ h_{e}(x_3,x_1,b_3,b_1)E_{q^{\prime}}(\mu_{q^{\prime}}) \, 2 \big[ -r_3 \varphi_{\pi}^a(x_2) \bar{\psi}_{\rho}^{s,\parallel}(x_3) + r_2 \varphi_{\pi}^p(x_2) (x_1 \varphi_{\rho}^{\parallel}(x_3) +2 r_3 \bar x_1 \bar{\psi}_{\rho}^{s,\parallel}(x_3) ) \big] \Big\} \,, \end{aligned} $

$ \begin{eqnarray} {{\cal{M}}}^{({\rm{ql}})}_{B\to V_2P_3} &\stackrel{R}\Longleftarrow& {{\cal{M}}}^{({\rm{ql}})}_{B \to P_2 P_3} , \quad\quad\quad\quad {{\cal{M}}}^{({\rm{ql}})}_{B \to V_2V_3} \stackrel{R}\Longleftarrow {{\cal{M}}}^{({\rm{ql}})}_{ B\to P_2V_3} \,, \end{eqnarray} $

$ \begin{aligned}[b] {{\cal{M}}}^{({\rm{ql}}), {\bf N}}_{B\to V_2V_3} =& -\frac{8}{\sqrt{6}} C_F^2 m_B^4 r_2 \int_0^1 {\rm d}x_1 {\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1)\, \Big\{ h_{e}(x_1,x_3,b_1,b_3) E_q(\mu_q) \\ &\times\Big[ r_3\big[(4r_b+x_3-2)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3) +\varphi_{\rho}^{t,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)\right) - x_3\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)\right) \\ & - (r_b-2)\varphi_{\rho}^{\perp}(x_3)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right)\big]\Big] + h_{e}(x_3,x_1,b_3,b_1)E_{q^{\prime}}(\mu_{q^{\prime}}) \, r_3\Big[\bar{\psi}_{\rho}^{s,\perp}(x_2)\left(\bar{x}_1\bar{\psi}_{\rho}^{s,\perp}(x_3)+(1+x_1)\varphi_{\rho}^{t,\perp}(x_3)\right) \\& + \varphi_{\rho}^{t,\perp}(x_2)\left((1+x_1)\bar{\psi}_{\rho}^{s,\perp}(x_3)+\bar{x}_1\varphi_{\rho}^{t,\perp}(x_3)\right)\Big]\Big\} \,, \\[-12pt]\end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}^{({\rm{ql}}), {\bf T}}_{B\to V_2V_3} =& -\frac{8}{\sqrt{6}} C_F^2 m_B^4 r_2 \int_0^1 {\rm d}x_1{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1)\, \Big\{ \, h_{e}(x_1,x_3,b_1,b_3) E_q(\mu_q) \\ &\times \Big[r_3 \big[ x_3\left(\varphi_{\rho}^{t,\perp}(x_2)-\bar{\psi}_{\rho}^{s,\perp}(x_2)\right)\left(\bar{\psi}_{\rho}^{s,\perp}(x_3)-\varphi_{\rho}^{t,\perp}(x_3)\right) + 2(2 r_b-1)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)\right) \big] \\ & - (r_b-2)\varphi_{\rho}^{\perp}(x_3)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right) \Big] + h_{e}(x_3,x_1,b_3,b_1)E_{q^{\prime}}(\mu_{q^{\prime}}) \,r_3 \Big[ \bar{\psi}_{\rho}^{s,\perp}(x_2) \big[ (1+x_1)\bar{\psi}_{\rho}^{s,\perp}(x_3)+\bar{x}_1\varphi_{\rho}^{t,\perp}(x_3) \big] \\ & + \varphi_{\rho}^{t,\perp}(x_2) \big[ \bar{x}_1\bar{\psi}_{\rho}^{s,\perp}(x_3)+(1+x_1)\varphi_{\rho}^{t,\perp}(x_3) \big] \Big] \Big\} \,. \\[-12pt] \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}^{({\rm{mp}})}_{B\to P_2P_3} =& -\frac{8}{\sqrt{6}} C_F^2 m_B^6 \int_0^1 {\rm d}x_1 {\rm d}x_2\, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1) \cdot\Big\{ h_g(\alpha_g,\beta_g,\gamma_g,b_1,b_2,b_3) E_{8g}(\mu_g) \\&\times \Big[-\bar{x}_3\varphi_{\pi}^a(x_2)\Big(2(r_b-2)\varphi_{\pi}^a(x_3) +r_3 \big[ \varphi_{\pi}^t(x_3)(-2r_b+x_3+1)- \varphi_{\pi}^p(x_3)\left(6r_b-\bar{x}_3\right) \big] \Big) \\ & +r_2 \big[ x_2\left(3\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right) \big(\varphi_{\pi}^a(x_3)(2r_b-\bar{x}_3)-r_3(r_b-2\bar{x}_3) \left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right)\Big) \\& - r_3(r_b-2)\bar{x}_3\left(3\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2)\right)\left(\varphi_{\pi}^p(x_3)-\varphi_{\pi}^t(x_3)\right) \big] \Big] \\ & + h_g(\alpha_{g^\prime},\beta_{g^\prime},\gamma_{g^\prime},b_1,b_2,b_3) E_{8g^{\prime}}(\mu_{g^{\prime}}) \Big[x_1\varphi_{\pi}^a(x_3)\big[ \varphi_{\pi}^a(x_2)+r_2\left(3\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2)\right)\Big] \end{aligned} $

$ \begin{aligned}[b] \quad\quad\quad\quad- 2 r_3\varphi_{\pi}^p(x_3) \big[(x_1-2)\varphi_{\pi}^a(x_2)+r_2\left(6x_1\varphi_{\pi}^p(x_2)+x_2\left(\varphi_{\pi}^t(x_2)-3\varphi_{\pi}^p(x_2)\right)\right)\big] \Big]\Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}^{({\rm{mp}})}_{B\to P_2V_3} =& \frac{8}{\sqrt{6}} C_F^2 m_B^6 \int_0^1 {\rm d}x_1 {\rm d}x_2\, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1)\, \Big\{ h_g(\alpha_g,\beta_g,\gamma_g,b_1,b_2,b_3) E_{8g}(\mu_g)\\&\times\Big[ \bar{x}_3\varphi_{\pi}^a(x_2)\Big(2(r_b-2)\varphi_\rho^\parallel(x_3) + r_3\big[ \varphi_{\rho}^{t,\parallel}(x_3)(-2r_b+x_3+1) - \bar{\psi}_\rho^{s,\parallel}(x_2)\left(6r_b-\bar x_3 \right) \Big) \big] \\ & + r_2 \big[ x_2\left(3\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right)\big(\varphi_\rho^\parallel(x_3)(2r_b-\bar{x}_3) - r_3(r_b-2\bar{x}_3) (\bar{\psi}_\rho^{s,\parallel}(x_2)+\varphi_{\rho}^{t,\parallel}(x_3) )\big) \\ &+ r_3(r_b-2)\bar{x}_3\left(3\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2)\right)\left( \varphi_{\rho}^{t,\parallel}(x_3)-\bar{\psi}_\rho^{s,\parallel}(x_3)\right) \big] \Big] \\ & + h_g(\alpha_{g^\prime},\beta_{g^\prime},\gamma_{g^\prime},b_1,b_2,b_3)E_{8g^{\prime}}(\mu_{g^{\prime}}) \Big[x_1\varphi_\rho^\parallel(x_3)\big[ r_2\left(3\varphi_{\pi}^p(x_2)+\varphi_{\pi}^t(x_2)\right)-\varphi_{\pi}^a(x_2) \big] \\ & + 2 r_3\bar{\psi}_\rho^{s,\parallel}(x_3)\big[ (x_1-2)\varphi_{\pi}^a(x_2)-r_2\left(6x_1\varphi_{\pi}^p(x_2)+x_2\left(\varphi_{\pi}^t(x_2)-3\varphi_{\pi}^p(x_2)\right)\right)\big] \Big]\Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}^{({\rm{mp}}), {\bf N}}_{B\to V_2V_3}=& -\frac{8}{\sqrt{6}} C_F^2 m_B^6 \int_0^1 {\rm d}x_1{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1)\,\cdot \Big\{ h_g(\alpha_g,\beta_g,\gamma_g,b_1,b_2,b_3) E_{8g}(\mu_g) \\ & \Big[r_2 r_3\Big( \big[ 2\left(1+x_2\right)\bar{x}_3-r_b\left(x_2+\bar{x}_3\right) \big] \big[ \bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3) + \varphi_{\rho}^{t,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)\big] \\ &- \big[ 2\bar{x}_2\bar{x}_3+r_b\left(x_2-\bar{x}_3\right)\big] \big[\bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)\big]\Big) \\ & + \left( \bar{x}_3-2r_b \right) \big[ r_3\bar{x}_3\varphi_{\rho}^{\perp}(x_2)\left(\bar{\psi}_{\rho}^{s,\perp}(x_3)-\varphi_{\rho}^{t,\perp}(x_3)\right) - r_2x_2\varphi_{\rho}^{\perp}(x_3)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right) \big] \Big] \\ & + h_g(\alpha_{g^\prime},\beta_{g^\prime},\gamma_{g^\prime},b_1,b_2,b_3) E_{8g^{\prime}}(\mu_{g^{\prime}}) \, r_3 \Big[ x_1\varphi_{\rho}^{\perp}(x_2)\left(\bar{\psi}_{\rho}^{s,\perp}(x_3)-\varphi_{\rho}^{t,\perp}(x_3)\right) \\ & + r_2\big[ x_2\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right) \left(\bar{\psi}_{\rho}^{s,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_3)\right) - 2x_1\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)\right)\big] \Big] \Big\} \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}^{({\rm{mp}}), {\bf T}}_{B\to V_2V_3} =& -\frac{8}{\sqrt{6}} C_F^2 m_B^6 \int_0^1 {\rm d}x_1{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_1 {\rm d}b_1 b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1)\, \cdot \Big\{h_g(\alpha_g,\beta_g,\gamma_g,b_1,b_2,b_3) E_{8g}(\mu_g) \\ &\times \Big[r_2 r_3\Big(-[ 2\bar{x}_2\bar{x}_3+r_b\left(x_2-\bar{x}_3\right) \big] \big[ \bar{\psi}_{\rho}^{s,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3) + \varphi_{\rho}^{t,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3) \big] \\ & + \big[ 2\left(1+x_2 \right)\bar{x}_3-r_b\left(x_2+\bar{x}_3\right)\big] \big[ \bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)\big] \Big) \\ & + \left(\bar{x}_3-2r_b\right) \big[ r_3\bar{x}_3\varphi_{\rho}^{\perp}(x_2)\left(\bar{\psi}_{\rho}^{s,\perp}(x_3)-\varphi_{\rho}^{t,\perp}(x_3)\right) - r_2x_2\varphi_{\rho}^{\perp}(x_3)\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right) \big] \Big] \\ & + h_g(\alpha_{g^\prime},\beta_{g^\prime},\gamma_{g^\prime},b_1,b_2,b_3) E_{8g^{\prime}}(\mu_{g^{\prime}}) \, r_3 \Big[ x_1\varphi_{\rho}^{\perp}(x_2)\left(\bar{\psi}_{\rho}^{s,\perp}(x_3)-\varphi_{\rho}^{t,\perp}(x_3)\right) \\ &+ r_2\big[ x_2\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)+\varphi_{\rho}^{t,\perp}(x_2)\right) \left(\bar{\psi}_{\rho}^{s,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_3)\right) - 2x_1\left(\bar{\psi}_{\rho}^{s,\perp}(x_2)\varphi_{\rho}^{t,\perp}(x_3)+\varphi_{\rho}^{t,\perp}(x_2)\bar{\psi}_{\rho}^{s,\perp}(x_3)\right)\big] \Big] \Big\} \,, \end{aligned} $

$ \begin{eqnarray} {{\cal{M}}}^{({\rm{mp}})}_{B\to V_2P_3} &\stackrel{R}\Longleftarrow& {{\cal{M}}}^{({\rm{mp}})}_{B\to P_2P_3} \,, \quad {{\cal{M}}}^{({\rm{mp}})}_{B\to V_2V_3} \stackrel{R}\Longleftarrow {{\cal{M}}}^{({\rm{mp}})}_{B\to P_2V_3} \,. \end{eqnarray} $

$ \begin{aligned}[b] {\rm{F_{T2}}^{(1)}}(x_i, \zeta_3) =& \frac{\alpha_s(\mu_f)C_F}{4\pi} \Bigg[ \frac{21}{4}\ln \frac{\mu^2}{m_B^2} - \left( \ln \frac{m_B^2}{\zeta_1^2} + \frac{13}{2} \right) \ln \frac{\mu_f^2}{m_B^2} + \frac{7}{16} \ln^2(x_1x_3) + \frac{1}{8} \ln^2 x_1 + \frac{1}{4} \ln x_1 \ln x_3 \\&+ \left( 2 \ln \frac{m_B^2}{\zeta_1^2} - \frac{1}{4} \right) \ln x_1 - \frac{3}{2} \ln x_3 - \left( \frac{3}{2} \ln \frac{m_B^2}{\zeta_1^2} + 1 \right) \ln \frac{m_B^2}{\zeta_1^2} + \frac{101}{48} \pi^2 + \frac{219}{16} \Bigg] \,, \end{aligned} $

$ \begin{aligned}[b] {\rm{F_{T3}}^{(1)}}(x_i, \zeta_3 )=& \frac{\alpha_s(\mu_f)C_F}{4\pi} \Bigg[ \frac{21}{4}\ln \frac{\mu^{\prime 2}}{m_B^2} - \left( \frac{1}{2} \ln \frac{m_B^2}{\zeta_1^2} + 3 \right) \ln \frac{\mu_f^2}{m_B^2} + \frac{7}{16} \ln^2 x_1 - \frac{3}{8} \ln^2 x_3 + \frac{9}{8} \ln x_1 \ln x_3 \\&+ \left( \ln \frac{m_B^2}{\zeta_1^2} - \frac{29}{8} \right) \ln x_1 + \ln \frac{m_B^2}{\zeta_3^2} + \left( \ln \frac{m_B^2}{\zeta_3^2} - \frac{25}{16} \right) \ln x_3 - \left( \frac{1}{4} \ln \frac{m_B^2}{\zeta_1^2} - \frac{1}{2} \right) \ln \frac{m_B^2}{\zeta_1^2} + \frac{37}{32} \pi^2 + \frac{91}{32} \Bigg] \, . \end{aligned} $

$ \begin{eqnarray} \mu \equiv \mu(\mu_f) =\Big\{ {\rm{Exp}} \Big[ c_1 + \left( \ln \frac{m_B^2}{\zeta_1^2} + \frac{5}{4} \right) \ln \frac{\mu_f^2}{m_B^2} \Big] x_1^{c_2} x_2^{c_3}\Big\}^{2/21} \mu_f \,, \end{eqnarray} $

$ \begin{eqnarray} \mu^\prime \equiv \mu^\prime(\mu_f) = \Big\{ {\rm{Exp}} \Big[ c^\prime_1 + \left( \frac{1}{2}\ln \frac{m_B^2}{\zeta_1^2} - \frac{9}{4} \right) \ln \frac{\mu_f^2}{m_B^2} \Big] x_1^{c_2^\prime} x_2^{c_3^\prime}\Big\}^{2/21} \mu_f \,. \end{eqnarray} $

$ \begin{aligned}[b] c_1 =& \left( \frac{3}{2} \ln \frac{m_B^2}{\zeta_1^2} + 1 \right) \ln \frac{m_B^2}{\zeta_1^2} - \frac{101}{48} \pi^2 - \frac{219}{16} \,, \\ c_2 =& - \left( 2 \ln \frac{m_B^2}{\zeta_1^2} - \frac{1}{4} \right) \,, \quad c_3 = + \frac{3}{2} \,, \end{aligned} $

$ \begin{aligned}[b] c_1^\prime =& \left( \frac{1}{4} \ln \frac{m_B^2}{\zeta_1^2} - \frac{1}{2} \right) \ln \frac{m_B^2}{\zeta_1^2} - \frac{37}{32} \pi^2 - \frac{91}{32} \,, \\ c_2^\prime =& - \left( \ln \frac{m_B^2}{\zeta_1^2} - \frac{29}{8} \right) \,, \quad c_3^\prime = + \frac{25}{16} \,, \end{aligned} $

$ \begin{aligned}[b] {\cal{E}}_{M_3}^{{\bf{LL}}, {\rm{NLO}}} =& -{\cal{E}}_{M_3}^{{\bf{LR}}, {\rm{NLO}}}= 8 \pi C_F m_B^4 f_{M_2} \int_0^1 {\rm d}x_1 \, {\rm d}x_3 \, \\&\times\int_0^{1/\Lambda} b_1 {\rm d}b_1 \, b_3 {\rm d}b_3 \varphi_B(x_1,b_1)\, \\ &\cdot \Big\{ h_{e}(x_1,x_3,b_1,b_3) E_e(\mu_e) \Big[ (2\,r_b-\bar{x}_3)\\&\times \left[ 1 + {\rm{F_{T2}}^{(1)}}(x_i, \zeta_3) \right] \varphi_{\pi}^a(x_3) \\ & - r_3\left(r_b-2\bar{x}_3 \right)\left(\varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right) \Big] \\ & + h_{e}(x_3,x_1,b_3,b_1) E_{e^{\prime}}(\mu_{e^{\prime}}) \,2\,r_3\\&\times \left[ 1 + {\rm{F_{T3}}^{(1)}}(x_i, \zeta_3) \right] \varphi_{\pi}^p(x_3) \Big\} \,. \end{aligned} $

$ \begin{aligned}[b] {\rm{G}}_{\pi, {\rm{T2}}}^{(1)} =& \frac{\alpha_s(\mu) C_F}{4 \pi} \Big[ -\frac{3}{4} \ln \frac{\mu^2}{m_B^2} - \frac{1}{4} \ln^2 \left(\frac{4x_2x_3}{m_B^2b_2^2}\right) - \frac{17}{4} \ln^2 x_3 + \frac{27}{8} \ln x_2 \ln x_3 + \left( \frac{17}{8} \ln x_3 + \frac{23}{16} + \gamma_E + {\rm i} \frac{\pi}{2} \right) \ln \left( \frac{4x_2x_3}{m_B^2b_2^2} \right)\\& - \left( \frac{13}{8} + \frac{17}{4} \gamma_E - {\rm i} \frac{17}{8} \pi \right) \ln x_3 + \frac{31}{16} \ln x_2 - \frac{\pi^2}{2} + (1 - 2 \gamma_E) \pi + \frac{\ln 2}{2} + \frac{53}{4} - \frac{23 \gamma_E}{8} - \gamma_E^2 + {\rm i} \pi \left( \frac{171}{16} + \gamma_E \right) \Big] \,, \end{aligned} $

$ \begin{eqnarray} {\rm{G}}_{\pi, {\rm{T3, 1}}}^{(1)} = \frac{\alpha_s(\mu) C_F}{4 \pi} \Big[ \frac{9}{4} \ln \frac{\mu^2}{m_B^2} - \frac{53}{32} \ln \left(\frac{4x_2x_3}{m_B^2b_2^2} \right) - \frac{23}{32} \ln \left( \frac{4x_3}{m_B^2b_3^2}\right) - \frac{1}{8} \ln^2 x_2 - \frac{9}{8} \ln x_2 - \frac{137}{96} \pi^2 + \frac{19}{4} \gamma_E + \frac{337}{64} + {\rm i} \pi \frac{39}{8} \Big] \,, \end{eqnarray} $

$ \begin{aligned}[b] {\rm{G}}_{\pi, {\rm{T3, 2}}}^{(1)} =& \frac{\alpha_s(\mu) C_F}{4 \pi} \Bigg[ \frac{9}{4} \ln \frac{\mu^2}{m_B^2} - 2 \ln \left(\frac{4x_2x_3}{m_B^2b_2^2} \right) - \frac{1}{8} \ln^2 \left( \frac{4x_3}{m_B^2b_3^2}\right) + \left( \frac{\gamma_E}{2} + \frac{3}{4} {\rm i} \pi \right) \ln \left( \frac{4x_3}{m_B^2b_3^2}\right) \\ &+ 2\ln x_2 - \frac{\pi^2}{4} - \frac{\gamma_E^2}{2} + 4 \gamma_E + \frac{\ln 2}{4} + \frac{11}{2} + {\rm i} \pi \left( \frac{15}{4} - \frac{3}{2} \gamma_E \right)\Bigg]\,. \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{LL}}, {\rm{NLO}}}_{M_3}=&{\cal{A}}^{{\bf{LR}}, {\rm{NLO}}}_{M_3}= 8 \pi C_F m_B^4 f_B \int_0^1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/ \Lambda} b_2 {\rm d}b_2 \, b_3 {\rm d}b_3 \\ &\cdot \Big\{ h_{a}(x_2,{\bar x}_3,b_2,b_3)E_a({\mu_a}) \Big[ -\bar{x}_3 \left( 1 + {\rm{G}}_{\pi, {\rm{T2}}}^{(1)} \right) \varphi_{\pi}^a(x_2)\varphi_{\pi}^a(x_3) + 2r_2r_3 \varphi_{\pi}^p(x_2) \\ & \cdot \big[ - \left( 1 + {\rm{G}}_{\pi, {\rm{T3, 2}}}^{(1)} \right) \left( \varphi_{\pi}^p(x_3) + \varphi_{\pi}^t(x_3) \right) + {\bar x}_3 \left( 1 + {\rm{G}}_{\pi, {\rm{T3, 1}}}^{(1)} \right) \left( \varphi_{\pi}^t(x_3) - \varphi_{\pi}^p(x_3) \right) \big] \Big] \\ & + h_{a}({\bar x}_3,x_2,b_3,b_2) E_{a^\prime}({\mu_{a^\prime}}) \Big[ x_2 \left( 1 + {\rm{G}}_{\pi, {\rm{T2}}}^{(1)} \right) \varphi_{\pi}^a(x_2)\varphi_{\pi}^a(x_3) + 2r_2r_3 \varphi_{\pi}^p(x_3) \\ & \cdot \big[\left( 1 + {\rm{G}}_{\pi, {\rm{T3, 1}}}^{(1)} \right) \left( \varphi_{\pi}^p(x_2) - \varphi_{\pi}^t(x_2) \right) + x_2 \left( 1 + {\rm{G}}_{\pi, {\rm{T3, 2}}}^{(1)} \right) \left( \varphi_{\pi}^p(x_2) + \varphi_{\pi}^t(x_2) \right) \big] \Big]\Big\} \,. \end{aligned} $

$ \begin{aligned}[b] {\cal{A}}^{{\bf{SP}}}_{M_3}=& 16 \pi C_F m_B^4 f_B \int_0^1 \,{\rm d}x_2 \, {\rm d}x_3 \, \int_0^{1/\Lambda} b_2 {\rm d}b_2 \, b_3 {\rm d} b_3\cdot \Big\{ h_{a}(x_2,{\bar x}_3,b_2,b_3)E_a({\mu_a}) \Big[ r_3\bar{x}_3\varphi_{\pi}^a(x_2) \left( \varphi_{\pi}^p(x_3)+\varphi_{\pi}^t(x_3)\right) \\&+ 2 r_2 \left( 1 + \Gamma^{(1)}(Q^2)\right) \varphi_{\pi}^p(x_2)\varphi_{\pi}^a(x_3) \Big] + h_{a}({\bar x}_3,x_2,b_3,b_2)E_{a^\prime}({\mu_{a^\prime}}) \Big[ r_2 x_2\varphi_{\pi}^a(x_3)\left(\varphi_{\pi}^p(x_2)-\varphi_{\pi}^t(x_2)\right) \\&+ 2 r_3 \left( 1 + \Gamma^{(1)}(Q^2)\right) \varphi_{\pi}^a(x_2)\varphi_{\pi}^p(x_2) \Big]\Big\} \, \end{aligned} $

$ \begin{aligned}[b] \Gamma^{(1)}=& \frac{\alpha_s C_F}{8\pi} \Big\{ \frac{5}{2} \ln \frac{\mu^2}{m_B^2} - \frac{1}{16} \ln^2 \left( \frac{4 x_2 (1-x_3)}{m_B^2 b_3^2} \right) + \left( \frac{33}{8} - \frac{\gamma_E}{4} \right) \ln \left( \frac{4 x_2 (1-x_3)}{m_B^2 b_3^2} \right) - \ln^2 (1-x_3) - \frac{3}{8} \ln^2 x_2 \\&+ \frac{1}{2} \ln x_2 \ln (1-x_3) - \frac{71}{4} \ln (1-x_3) - \frac{7}{4} \ln x_2 - \frac{105}{48} \pi^2 + \frac{41}{2} - \frac{\gamma_E^2}{2} - \frac{33 \gamma_E}{4} + {\rm i} \pi \frac{5}{2} \Big\} \,. \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}(B^+ \to \pi^+ K^0) =& \frac{G_F}{\sqrt{2}} V^{\ast}_{ub} V_{us} \Big[ a_1 {{\cal{A}}}_{\pi}^{{\bf LL}} + C_1 {{\cal{A}}}_{NF, \pi}^{{\bf LL}} +{{\cal{M}}}^{({\rm{ql,u}})}_{B \to K^0 \pi^+} \Big]+\frac{G_F}{\sqrt{2}} V^{\ast}_{cb} V_{cs}{{\cal{M}}}^{({\rm{ql,c}})}_{B \to K^0 \pi^+} - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{ts} \Bigg[ \left( a_4-\frac{a_{10}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf LL}}\\& + \left( a_6 - \frac{a_{8}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf SP}} + \left( C_3 - \frac{C _{9}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf LL}} + \left( C_5 - \frac{C_{7}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf LR}} + \left( a_4 + a_{10} \right) {{\cal{A}}}_{\pi}^{{\bf LL}} + \left( a_6 + a_{8} \right) {{\cal{A}}}_{\pi}^{{\bf SP}} \\ &+ \left( C_3 + C_{9} \right) {{\cal{A}}}_{NF, \pi}^{{\bf LL}} + \left( C_5 + C_{7} \right) {{\cal{A}}}_{NF, \pi}^{{\bf LR}} + {{\cal{M}}}^{({\rm{ql,t }})}_{B \to K^0 \pi^+}+ {{\cal{M}}}^{({\rm{mp}})}_{B \to K^0 \pi^+} \Bigg] \,, \end{aligned} $

$ \begin{aligned}[b] \sqrt{2}{{\cal{M}}}(B^+ \to \pi^0 K^+) =& \frac{G_F}{\sqrt{2}} V^{\ast}_{ub }V_{us} \Big[ a_1 \left( {{\cal{E}}}_{\pi}^{{\bf LL}} + {{\cal{A}}}_{\pi}^{{\bf LL}} \right) + a_2 {{\cal{E}}}_{K}^{{\bf LL}} + C_1 \left( {{\cal{E}}}_{NF \pi}^{{\bf LL}} + {{\cal{A}}}_{NF, \pi}^{{\bf LL}} \right) + C_2 {{\cal{E}}}_{NF, K}^{{\bf LL}} +{{\cal{M}}}^{({\rm{ql,u}})}_{B \to K^+ \pi^0} \Big] \\ &+ \frac{G_F}{\sqrt{2}} V^{\ast}_{cb }V_{cs} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to K^+ \pi^0} - \frac{G_F}{\sqrt{2}} V^{\ast}_{tb}V_{ts} \Big[ \left( a_4+a_{10} \right) {{\cal{E}}}_{\pi}^{{\bf LL}} + \left( a_6 + a_{8} \right) {{\cal{E}}}_{\pi}^{{\bf SP}} +\frac{3a_{9}}{2} {{\cal{E}}}_{K}^{{\bf LL}} + \frac{3a_{7}}{2} {{\cal{E}}}_{K}^{{\bf LR}} \, \\ & + \frac{3C_{10}}{2} {{\cal{E}}}_{NF, K}^{{\bf LL}} + \frac{3C_{8}}{2} {{\cal{E}}}_{NF, K}^{{\bf SP}} + \left( C_3 + C_{9} \right) {{\cal{E}}}_{NF, \pi}^{{\bf LL}} + \left( C_5 + C_{7} \right) {{\cal{E}}}_{NF, \pi}^{{\bf LR}} \\ & + \left( a_4 + a_{10} \right) {{\cal{A}}}_{\pi}^{{\bf LL}} + \left( a_6 + a_{8} \right) {{\cal{A}}}_{\pi}^{{\bf SP}} + \left( C_3 + C_{9} \right) {{\cal{A}}}_{NF, \pi}^{{\bf LL}} + \left( C_5 + C_{7} \right) {{\cal{A}}}_{NF, \pi}^{{\bf LR}} + {{\cal{M}}}^{({\rm{ql,t}})}_{B \to K^+ \pi^0} + {{\cal{M}}}^{({\rm{mp}})}_{B \to K^+ \pi^0} \Big] \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}(B^+ \to \eta_s K^+) = & \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{us} \Big[ a_1 {{\cal{A}}}_{K}^{{\bf{LL}}} + C_1 {{\cal{A}}}_{NF, K}^{{\bf{LL}}} + {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \eta_s K^+} \Big] + \frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cs} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \eta_s K^+} \\ & - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{ts} \Big[ \left( a_3 + a_4-\frac{a_{9}+a_{10}}{2} \right) {{\cal{E}}}_{K}^{{\bf{LL}}} + \left( a_{5}-\frac{a_{7}}{2} \right) {{\cal{E}}}_{K}^{{\bf{LR}}} + \left( a_6 - \frac{a_{8}}{2} \right) {{\cal{E}}}_{K}^{{\bf{SP}}} \end{aligned} $

$ \begin{aligned}[b] \quad\quad & + \left( C_3 + C_4 - \frac{C_{9}+C_{10}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{LL}}} + \left( C_5 - \frac{C_{7}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{LR}}} + \left( C_6 - \frac{C_{8}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{SP}}} \\ & + \left( a_4 + a_{10} \right) {{\cal{A}}}_{K}^{{\bf{LL}}} + \left( a_6 + a_{8} \right) {{\cal{A}}}_{K}^{{\bf{SP}}} + \left( C_3 + C_{9} \right) {{\cal{A}}}_{NF, K}^{{\bf{LL}}} + \left( C_5 + C_{7} \right) {{\cal{A}}}_{NF, K}^{{\bf{LR}}} +{{\cal{M}}}^{({\rm{ql,t}})}_{B \to \eta_s K^+} +{{\cal{M}}}^{({\rm{mp}})}_{B \to \eta_s K^+} \Big] \,, \end{aligned} $

$ \begin{aligned}[b] \sqrt{2}{{\cal{M}}}(B^+ \to \eta_q K^+) = & \frac{G_F}{\sqrt{2}} V^{\ast}_{ub} V_{us} \Big[ a_1 \left( {{\cal{E}}}_{\eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{\eta_q}^{{\bf{LL}}} \right) + a_2 {{\cal{E}}}_{K}^{{\bf{LL}}} + C_1 \left( {{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{NF, \eta_q}^{{\bf{LL}}} \right) + C_2 {{\cal{E}}}_{NF, K}^{{\bf{LL}}} + {{\cal{M}}}^{({\rm{ql,u}})}_{B \to K^+ \eta_q } \Big] \\&+ \frac{G_F}{\sqrt{2}} V^{\ast}_{cb} V_{cs} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to K^+ \eta_q } - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb}\, V_{ts} \Big[ \left( a_4 + a_{10} \right) {{\cal{E}}}_{\eta_q}^{{\bf{LL}}} + \left( a_6 + a_{8} \right) {{\cal{E}}}_{\eta_q}^{{\bf{SP}}} + \left( C_3 + C_{9} \right) {{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}}\\ & + \left( C_5 + C_{7} \right) {{\cal{E}}}_{NF, \eta_q}^{{\bf{LR}}} + \left( 2 a_{3} + \frac{a_{9}}{2} \right) {{\cal{E}}}_{K}^{{\bf{LL}}} + \left( 2a_{5} +\frac{a_{7}}{2} \right) {{\cal{E}}}_{K}^{{\bf{LR}}} + \left( 2C_4 + \frac{C_{10}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{LL}}} \\ &+ \left( 2C_6 + \frac{C_{8}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{SP}}} + \left( a_4 + a_{10} \right) {{\cal{A}}}_{\eta_q}^{{\bf{LL}}} + \left( a_6 + a_{8} \right) {{\cal{A}}}_{\eta_q}^{{\bf{SP}}} + \left( C_3 + C_{9} \right) {{\cal{A}}}_{NF, \eta_q}^{{\bf{LL}}} \\ &+ \left( C_5 + C_{7} \right) {{\cal{A}}}_{NF,\eta_q}^{{\bf{LR}}} + {{\cal{M}}}^{({\rm{ql,t}})}_{B \to K^+ \eta_q } + {{\cal{M}}}^{({\rm{mp}})}_{B \to K^+ \eta_q } \Big] \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}(B^+ \toK^+ \bar K^0) =& \frac{G_F}{\sqrt{2}} V^{\ast}_{ub} V_{ud} \Big[ a_1 {{\cal{A}}}_{K^+}^{{\bf{LL}}} + C_1 {{\cal{A}}}_{NF, K^+}^{{\bf{LL}}} + {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \bar K^0 K^+ } \Big] + \frac{G_F}{\sqrt{2}} V^{\ast}_{cb} V_{cd} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \bar K^0 K^+ } - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left( a_4 - \frac{a_{10}}{2} \right) {{\cal{E}}}_{K^+}^{{\bf{LL}}} \\ & + \left( a_6 - \frac{a_{8}}{2} \right) {{\cal{E}}}_{K^+}^{{\bf{SP}}} + \left( C_3 - \frac{C_{9}}{2} \right) {{\cal{E}}}_{NF, K^+}^{{\bf{LL}}} + \left( C_5 - \frac{C_{7}}{2} \right) {{\cal{E}}}_{NF, K^+}^{{\bf{LR}}} + \left( a_4 + a_{10} \right) {{\cal{A}}}_{K^+}^{{\bf{LL}}} + \left( a_6 + a_{8} \right) {{\cal{A}}}_{K^+}^{{\bf{SP}}} + \left( C_3 + C_{9} \right) {{\cal{A}}}_{NF, K^+}^{{\bf{LL}}} \\ & + \left( C_5 + C_{7} \right) {{\cal{A}}}_{NF, K^+}^{{\bf{LR}}} + {{\cal{M}}}^{({\rm{ql,t}})}_{B \to \bar K^0 K^+ } +{{\cal{M}}}^{({\rm{mp}})}_{B \to \bar K^0 K^+ } \Big] \,, \end{aligned} $

$ \begin{aligned}[b] \sqrt{2}{{\cal{M}}}(B^+ \to \pi^0 \pi^+ ) =& \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{ud} \Big[ a_2 {{\cal{E}}}_{\pi^+}^{{\bf{LL}}} + C_2 {{\cal{E}}}_{NF, \pi^+}^{{\bf{LL}}} + a_1 {{\cal{E}}}_{\pi^0}^{{\bf{LL}}} + C_1 {{\cal{E}}}_{NF, \pi^0}^{{\bf{LL}}} + {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \pi^+ \pi^0} - {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \pi^0\pi^+ } \Big] \\& + \frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cd} ( {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \pi^+ \pi^0} - {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \pi^0\pi^+ } ) - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left( \frac{3a_{9}+a_{10}}{2} -a_4 \right) {{\cal{E}}}_{\pi^+}^{{\bf{LL}}} + \frac{3a_7}{2} {{\cal{E}}}_{\pi^+}^{{\bf{LR}}} +(\frac{a_8}{2}-a_6) {{\cal{E}}}_{\pi^+}^{{\bf{SP}}} \\ & + \left(\frac{ (3 C_{10} + C_9 )}{2}-C_3\right) {{\cal{E}}}_{NF, \pi^+}^{{\bf{LL}}} + \left(\frac{C_7}{2}-C_5\right) {{\cal{E}}}_{NF, \pi^+}^{{\bf{LR}}} +\frac{3C_8}{2} {{\cal{E}}}_{NF, \pi^+}^{{\bf{SP}}} + (a_4+a_{10}){{\cal{E}}}_{\pi^0}^{{\bf{LL}}} + (a_6+a_8){{\cal{E}}}_{\pi^0}^{{\bf{SP}}} \\&+(C_3+C_9) {{\cal{E}}}_{NF, \pi^0}^{{\bf{LL}}} + (C_5+C_7) {{\cal{E}}}_{NF, \pi^0}^{{\bf{LR}}} + {{\cal{M}}}^{({\rm{ql,t}})}_{B \to \pi^+ \pi^0} - {{\cal{M}}}^{({\rm{ql,t}})}_{B \to \pi^0\pi^+ }+ {{\cal{M}}}^{({\rm{mp}})}_{B \to \pi^+ \pi^0} - {{\cal{M}}}^{({\rm{mp}})}_{B \to \pi^0\pi^+ } \Big] \,, \end{aligned} $

$ {{\cal{M}}}(B^+ \to \pi^+ \eta_s) = -\frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left( a_3-\frac{a_{9}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LL}}} + \left(a_5-\frac{a_{7}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LR}}} + \left( C_4-\frac{C_{10}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} + \left( C_6-\frac{C_{8}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{SP}}} \Big] \,, $

$ \begin{aligned}[b] \sqrt{2}{{\cal{M}}}(B^+ \to \pi^+ \eta_q) =& \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{ud} \Big[ a_1\left( {{\cal{E}}}_{\eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{\eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{\pi}^{{\bf{LL}}} \right) + a_2 {{\cal{E}}}_{\pi}^{{\bf{LL}}} + C_2 {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} \\ & + C_1 \left( {{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{NF, \eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{NF, \pi}^{{\bf{LL}}} \right) + {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \pi^+\eta_q } +{{\cal{M}}}^{({\rm{ql,u}})}_{B \to \eta_q \pi^+ } \Big] +\frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cd} ({{\cal{M}}}^{({\rm{ql,c}})}_{B \to \pi^+\eta_q } +{{\cal{M}}}^{({\rm{ql,c}})}_{B \to \eta_q \pi^+ }) \\ & - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left( 2a_3 + a_4 + \frac{a_{9}-a_{10} }{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LL}}} + \left( 2a_{5}+\frac{a_{7}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LR}}} + \left(a_6-\frac{a_{8}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{SP}}} \\ &+ \left( C_3+2C_4 - \frac{C_{9}-C_{10}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} + \left( C_5-\frac{C_{7}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LR}}} + \left(2 C_6 + \frac{C_{8}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{SP}}} \\ & + \left( a_4+a_{10} \right) {{\cal{E}}}_{\eta_q}^{{\bf{LL}}} + \left(a_6+a_{8}\right) {{\cal{E}}}_{\eta_q}^{{\bf{SP}}} + \left( C_3+C_{9} \right) {{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}} + \left( C_5+C_{7} \right) {{\cal{E}}}_{NF, \eta_q}^{{\bf{LR}}} \\ & + \left( a_4+a_{10} \right) \left({{\cal{A}}}_{\pi}^{{\bf{LL}}} + {{\cal{A}}}_{\eta_q}^{{\bf{LL}}} \right) + \left(a_6+a_{8}\right) \left({{\cal{A}}}_{\pi}^{{\bf{SP}}} + {{\cal{A}}}_{\eta_q}^{{\bf{SP}}} \right) \\ & + \left(C_3+C_{9} \right) \left({{\cal{A}}}_{NF, \pi}^{{\bf{LL}}} + {{\cal{A}}}_{NF, \eta_q}^{{\bf{LL}}} \right) + \left( C_5+C_{7} \right) \left( {{\cal{A}}}_{NF, \pi}^{{\bf{LR}}} + {{\cal{A}}}_{NF, \eta_q}^{{\bf{LR}}} \right) \\ &+{{\cal{M}}}^{({\rm{ql,t}})}_{B \to \pi^+\eta_q } +{{\cal{M}}}^{({\rm{ql,t}})}_{B \to \eta_q \pi^+ } + {{\cal{M}}}^{({\rm{mp}})}_{B \to \pi^+\eta_q} +{{\cal{M}}}^{({\rm{mp}})}_{B \to \eta_q \pi^+ } \Big] \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}(B^0 \to \pi^- K^+) =& \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{us} \Big[ a_1 {{\cal{E}}}_{\pi}^{{\bf{LL}}} + C_1 {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} + {{\cal{M}}}^{({\rm{ql,u}})}_{B \to K^+ \pi^- } \Big]+ \frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cs} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to K^+ \pi^- } \\ & - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{ts} \Big[ \left(a_4+a_{10}\right) {{\cal{E}}}_{\pi}^{{\bf{LL}}} + \left(a_6+a_{8}\right) {{\cal{E}}}_{\pi}^{{\bf{SP}}} + \left( C_3+C_{9} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} + \left( C_5+C_{7} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LR}}} + \left(a_4-\frac{a_{10}}{2} \right) {{\cal{A}}}_{\pi}^{{\bf{LL}}} \\ &+ \left(a_6-\frac{a_{8}}{2}\right) {{\cal{A}}}_{\pi}^{{\bf{SP}}} + \left(C_3-\frac{C_{9}}{2}\right) {{\cal{A}}}_{NF, \pi}^{{\bf{LL}}} + \left(C_5-\frac{C_{7}}{2} \right) {{\cal{A}}}_{NF, \pi}^{{\bf{LR}}}+ {{\cal{M}}}^{({\rm{ql,t}})}_{B \to K^+ \pi^- } +{{\cal{M}}}^{({\rm{mp}})}_{B \to K^+ \pi^-} \Big] \,, \end{aligned} $

$ \begin{aligned}[b] \sqrt{2}{{\cal{M}}}(B^0 \to \pi^0 K^0) =& \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{us} \Big[ a_2 {{\cal{E}}}_{K}^{{\bf{LL}}} + C_2 {{\cal{E}}}_{NF, K}^{{\bf{LL}}} -{{\cal{M}}}^{({\rm{ql,u}})}_{B \to K^0 \pi^0 } \Big] -\frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cs} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to K^0 \pi^0 } - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{ts} \Big[ \left(-a_4+\frac{a_{10}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LL}}} \\ & + \left(-a_6+\frac{a_{8}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{SP}}} +\frac{3a_{9}}{2} {{\cal{E}}}_{K}^{{\bf{LL}}} + \frac{3a_{7}}{2} {{\cal{E}}}_{K}^{{\bf{LR}}} + \frac{3C_{10}}{2} {{\cal{E}}}_{NF, K}^{{\bf{LL}}} + \frac{3C_{8}}{2} {{\cal{E}}}_{NF, K}^{{\bf{SP}}} + \left(-C_3+\frac{C_{9}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} \\ &+ \left(-C_5+\frac{C_{7}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LR}}} + \left(-a_4+\frac{a_{10}}{2} \right) {{\cal{A}}}_{\pi}^{{\bf{LL}}} + \left(-a_6+\frac{a_{8}}{2} \right) {{\cal{A}}}_{\pi}^{{\bf{SP}}} + \left(-C_3+\frac{C_{9}}{2} \right) {{\cal{A}}}_{NF, \pi}^{{\bf{LL}}} \\ & + \left(-C_5+\frac{C_{7}}{2} \right) {{\cal{A}}}_{NF, \pi}^{{\bf{LR}}}-{{\cal{M}}}^{({\rm{ql,t}})}_{B \to K^0 \pi^0 } - {{\cal{M}}}^{({\rm{mp}})}_{B \to K^0 \pi^0}\Big] \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}(B^0\to \eta_s K^0) = &\frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{us} {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \eta_s K^0 } +\frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cs} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \eta_s K^0 } -\frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{ts} \Big[ \left(a_3+a_4-\frac{a_{9}+a_{10}}{2} \right) {{\cal{E}}}_{K}^{{\bf{LL}}}\\ & + \left( a_{5}-\frac{a_{7}}{2} \right) {{\cal{E}}}_{K}^{{\bf{LR}}} + \left(a_6-\frac{a_{8}}{2} \right) {{\cal{E}}}_{K}^{{\bf{SP}}} + \left(C_3+C_4-\frac{C_{9}+C_{10}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{LL}}} + \left(C_5-\frac{C_{7}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{LR}}} \\ & + \left(C_6-\frac{C_{8}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{SP}}} + \left(a_4-\frac{a_{10}}{2} \right) {{\cal{A}}}_{K}^{{\bf{LL}}} + \left(a_6-\frac{a_{8}}{2} \right) {{\cal{A}}}_{K}^{{\bf{SP}}} + \left(C_3-\frac{C_{9}}{2} \right) {{\cal{A}}}_{NF, K}^{{\bf{LL}}} \\ & + \left(C_5-\frac{C_{7}}{2} \right) {{\cal{A}}}_{NF, K}^{{\bf{LR}}} + {{\cal{M}}}^{({\rm{ql,t}})}_{B \to \eta_s K^0 }+ {{\cal{M}}}^{({\rm{mp}})}_{B \to \eta_s K^0 } \Big] \,, \end{aligned} $

$ \begin{aligned}[b] \sqrt{2}{{\cal{M}}}(B^0 \to \eta_q K^0) =& \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{us} \Big[a_2 {{\cal{E}}}_{K}^{{\bf{LL}}} + C_2 {{\cal{E}}}_{NF, K}^{{\bf{LL}}} + {{\cal{M}}}^{({\rm{ql,u}})}_{B \to K^0 \eta_q } \Big] + \frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cs} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to K^0 \eta_q } \\ &-\frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{ts} \Big[ \left(a_4-\frac{a_{10}}{2} \right) {{\cal{E}}}_{\eta_q}^{{\bf{LL}}} + \left( a_6-\frac{a_{8}}{2} \right) {{\cal{E}}}_{\eta_q}^{{\bf{SP}}} + \left( 2a_{3}+\frac{a_{9}}{2} \right) {{\cal{E}}}_{K}^{{\bf{LL}}} \\ & +\left(2a_{5}+\frac{a_{7}}{2} \right) {{\cal{E}}}_{K}^{{\bf{LR}}} + \left(2C_4+\frac{C_{10}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{LL}}} + \left( 2C_6+\frac{C_{8}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{SP}}} \\ & + \left( C_3-\frac{C_{9}}{2} \right) {{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}} + \left(C_5-\frac{C_{7}}{2} \right) {{\cal{E}}}_{NF, \eta_q}^{{\bf{LR}}} + \left(a_4-\frac{a_{10}}{2} \right) {{\cal{A}}}_{\eta_q}^{{\bf{LL}}} + \left(a_6-\frac{a_{8}}{2} \right) {{\cal{A}}}_{\eta_q}^{{\bf{SP}}} \\ & + \left(C_3-\frac{C_{9}}{2} \right) {{\cal{A}}}_{NF, \eta_q}^{{\bf{LL}}} + \left(C_5-\frac{C_{7}}{2} \right) {{\cal{A}}}_{NF, \eta_q}^{{\bf{LR}}} + {{\cal{M}}}^{({\rm{ql,t}})}_{B \to K^0 \eta_q }+ {{\cal{M}}}^{({\rm{mp}})}_{B \to K^0 \eta_q }\Big] \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}(B^0 \to K^0 \bar K^0) =&\frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{ud} {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \bar K^0 K^0 } +\frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cd} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \bar K^0 K^0 } \\ & -\frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left(a_4-\frac{a_{10}}{2}\right) {{\cal{E}}}_{K}^{{\bf{LL}}} + \left(a_6-\frac{a_{8}}{2} \right) {{\cal{E}}}_{K}^{{\bf{SP}}} + \left(C_3-\frac{C_{9}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{LL}}} \\ & \left(C_5-\frac{C_{7}}{2} \right) {{\cal{E}}}_{NF, K}^{{\bf{LR}}} + \left(a_3+a_4-\frac{a_{9}+a_{10}}{2} \right) {{\cal{A}}}_{K}^{{\bf{LL}}} + \left(a_{5}-\frac{a_{7}}{2} \right) {{\cal{A}}}_{K}^{{\bf{LR}}} \\ & + \left(a_6-\frac{a_{8}}{2} \right) {{\cal{A}}}_{K}^{{\bf{SP}}} + \left(C_3+C_4-\frac{C_{9}+C_{10}}{2} \right) {{\cal{A}}}_{NF, K}^{{\bf{LL}}} + \left(C_5-\frac{C_{7}}{2} \right) {{\cal{A}}}_{NF, K}^{{\bf{LR}}} \\ & + \left(C_6-\frac{C_{8}}{2} \right) {{\cal{A}}}_{NF, K}^{{\bf{SP}}} + \left( a_3 - \frac{a_9}{2} \right) {{\cal{A}}}_{\bar{K}}^{{\bf{LL}}} + \left( a_5 - \frac{a_7}{2} \right) {{\cal{A}}}_{\bar{K}}^{{\bf{LR}}} + \left( C_4^\prime - \frac{C_{10}^\prime}{2} \right) {{\cal{A}}}_{NF, \bar{K}}^{{\bf{LL}}} \\ & + \left( C_6^\prime - \frac{C_8^\prime}{2} \right) {{\cal{A}}}_{NF, \bar{K}}^{{\bf{SP}}} +{{\cal{M}}}^{({\rm{ql,t}})}_{B \to \bar K^0 K^0 } + {{\cal{M}}}^{({\rm{mp}})}_{B \to \bar K^0 K^0 } \Big] \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}(B^0 \to K^+ K^-) = & \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{ud} \Big[ a_2 {{\cal{A}}}_{K^+}^{{\bf{LL}}} + C_2 {{\cal{A}}}_{NF, K^+}^{{\bf{LL}}} \Big] - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left(a_3-\frac{a_{9}}{2} \right) {{\cal{A}}}_{K^+}^{{\bf{LL}}} + \left(a_5-\frac{a_{7}}{2} \right) {{\cal{A}}}_{K^+}^{{\bf{LR}}} + \left(C_4-\frac{C_{10}}{2} \right) {{\cal{A}}}_{NF, K^+}^{{\bf{LL}}} \\ & + \left(C_6-\frac{C_8}{2} \right) {{\cal{A}}}_{NF, K^+}^{{\bf{SP}}} + \left(a_3+a_{9} \right) {{\cal{A}}}_{K^-}^{{\bf{LL}}} + \left(a_5+a_7 \right) {{\cal{A}}}_{K^-}^{{\bf{LR}}} + \left(C_4+C_{10} \right) {{\cal{A}}}_{NF, K^-}^{{\bf{LL}}} + \left(C_6+C_{8}\right) {{\cal{A}}}_{NF, K^-}^{{\bf{SP}}} \Big] \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}(B^0 \to \pi^+\pi^-) =&\frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{ud} \Big[ a_1 {{\cal{E}}}_{\pi^-}^{{\bf{LL}}} + a_2 {{\cal{A}}}_{\pi^+}^{{\bf{LL}}} + C_1 {{\cal{E}}}_{NF, \pi^-}^{{\bf{LL}}} + C_2 {{\cal{A}}}_{NF, \pi^+}^{{\bf{LL}}} + {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \pi^+ \pi^-} \Big] +\frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cd} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \pi^+ \pi^-} \\ & - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left(a_4+a_{10} \right) {{\cal{E}}}_{\pi^-}^{{\bf{LL}}} + \left(a_6+a_{8}\right) {{\cal{E}}}_{\pi^-}^{{\bf{SP}}} + \left(C_3+C_{9} \right) {{\cal{E}}}_{NF, \pi^-}^{{\bf{LL}}} + \left(C_5+C_{7} \right) {{\cal{E}}}_{NF, \pi^-}^{{\bf{LR}}} \\ & + \left(a_3+a_4-\frac{a_{9}+a_{10}}{2} \right) {{\cal{A}}}_{\pi^-}^{{\bf{LL}}} + \left(a_{5}-\frac{a_{7}}{2} \right) {{\cal{A}}}_{\pi^-}^{{\bf{LR}}} + \left(a_6-\frac{a_{8}}{2} \right) {{\cal{A}}}_{\pi^-}^{{\bf{SP}}} + \left(C_3+C_4-\frac{C_{9}+C_{10}}{2} \right) {{\cal{A}}}_{NF, \pi^-}^{{\bf{LL}}} \\ & + \left(C_5-\frac{C_{7}}{2} \right) {{\cal{A}}}_{NF, \pi^-}^{{\bf{LR}}} + \left(C_6-\frac{C_{8}}{2} \right) {{\cal{A}}}_{NF, \pi^-}^{{\bf{SP}}} + \left(a_3+a_9 \right) {{\cal{A}}}_{\pi^+}^{{\bf{LL}}} + \left(a_{5}+ a_{7} \right) {{\cal{A}}}_{\pi^+}^{{\bf{LR}}} \\ & + \left(C_4+C_{10} \right) {{\cal{A}}}_{NF, \pi^+}^{{\bf{LL}}}+\left(C_6+C_{8} \right) {{\cal{A}}}_{NF, \pi^+}^{{\bf{SP}}} + {{\cal{M}}}^{({\rm{ql,t}})}_{B \to \pi^+ \pi^-} +{{\cal{M}}}^{({\rm{mp}})}_{B \to \pi^+ \pi^- } \Big] \,, \end{aligned} $

$ \sqrt{2}{{\cal{M}}}(B^0 \to \pi^0\eta_s) = -\frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb}V_{td} \Big[ \left(-a_3+\frac{a_{9}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LL}}} + \left(-a_5+\frac{a_{7}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LR}}} + \left(-C_4+\frac{C_{10}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} + \left(-C_6+\frac{C_{8}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{SP}}} \Big] \,, $

$ {{\cal{M}}}(B^0 \to \eta_s\eta_s) = - \frac{G_F} {\sqrt{2}} \, V^{\ast}_{tb}V_{td} \, 2 \Big[ \left(a_3-\frac{a_{9}}{2}\right) {{\cal{A}}}_{\eta_s}^{{\bf{LL}}} + \left(a_5-\frac{a_{7}}{2} \right) {{\cal{A}}}_{\eta_s}^{{\bf{LR}}} + \left(C_4-\frac{C_{10}}{2} \right) {{\cal{A}}}_{NF, \eta_s}^{{\bf{LL}}} + \left(C_6-\frac{C_{8}}{2} \right) {{\cal{A}}}_{NF, \eta_s}^{{\bf{SP}}} \Big] \,, $

$ \begin{aligned}[b] \sqrt{2} {{\cal{M}}}(B^0 \to \pi^0\pi^0) =& \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub}V_{ud} \Big[a_2 \left( {{\cal{A}}}_{\pi}^{{\bf{LL}}} - {{\cal{E}}}_{\pi}^{{\bf{LL}}} \right) + C_2 \left( {{\cal{A}}}_{NF, \pi}^{{\bf{LL}}} - {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} \right) + {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \pi^0 \pi^0} \Big] \\ & +\frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb}V_{cd} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \pi^0 \pi^0} - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left(a_4-\frac{3a_{9}}{2}-\frac{a_{10}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LL}}} + \left(-\frac{3a_{7}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LR}}} \\ & + \left(a_6-\frac{a_{8}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{SP}}} + \left(C_3-\frac{C_{9}}{2}-\frac{3C_{10}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} + \left(C_5-\frac{C_{7}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LR}}} \\ & - \left(\frac{3C_{8}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{SP}}} + \left(2a_3+a_4+\frac{a_{9}-a_{10}}{2} \right) {{\cal{A}}}_{\pi}^{{\bf{LL}}} + \left(2a_{5}+\frac{a_{7}}{2} \right) {{\cal{A}}}_{\pi}^{{\bf{LR}}} \\ &+ \left(a_6-\frac{a_{8}}{2} \right) {{\cal{A}}}_{\pi}^{{\bf{SP}}} + \left(C_3+2C_4+\frac{-C_{9}+C_{10}}{2} \right) {{\cal{A}}}_{NF, \pi}^{{\bf{LL}}} + \left(C_5-\frac{C_{7}}{2} \right) {{\cal{A}}}_{NF, \pi}^{{\bf{LR}}} \\ & + \left(2C_6+\frac{C_{8}}{2} \right) {{\cal{A}}}_{NF, \pi}^{{\bf{SP}}} + {{\cal{M}}}^{({\rm{ql,t}})}_{B \to \pi^0 \pi^0} + {{\cal{M}}}^{({\rm{mp}})}_{B \to \pi^0 \pi^0 } \Big] \,, \end{aligned} $

$ \begin{aligned}[b] 2{{\cal{M}}}(B^0 \to\pi^0\eta_q) = & \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub} V_{ud} \Big[a_2 \left({{\cal{E}}}_{\eta_q}^{{\bf{LL}}} - {{\cal{E}}}_{\pi}^{{\bf{LL}}} + {{\cal{A}}}_{\eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{\pi}^{{\bf{LL}}} \right) + C_2 \left( {{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}} - {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} + {{\cal{A}}}_{NF, \pi}^{{\bf{LL}}} + {{\cal{A}}}_{NF, \eta_q}^{{\bf{LL}}} \right) - {{\cal{M}}}^{({\rm{ql,u}})}_{B \to \eta_q \pi^0 }-{{\cal{M}}}^{({\rm{ql,u}})}_{B \to \pi^0 \eta_q } \Big] \\ & - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb} V_{cd} ( {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \eta_q \pi^0 }+{{\cal{M}}}^{({\rm{ql,c}})}_{B \to \pi^0 \eta_q } ) - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb}V_{td} \Big[ \left(-a_4+\frac{3a_{9}}{2}+\frac{a_{10}}{2} \right) \left( {{\cal{E}}}_{\eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{\eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{\pi}^{{\bf{LL}}} \right) \\ & + \left(\frac{3a_{7}}{2} \right) \left( {{\cal{E}}}_{\eta_q}^{{\bf{LR}}} + {{\cal{A}}}_{\eta_q}^{{\bf{LR}}} + {{\cal{A}}}_{\pi}^{{\bf{LR}}} \right) + \left(-a_6+ \frac{a_{8}}{2} \right) \left( {{\cal{E}}}_{\pi}^{{\bf{SP}}} + {{\cal{E}}}_{\eta_q}^{{\bf{SP}}} + {{\cal{A}}}_{\pi}^{{\bf{SP}}} + {{\cal{A}}}_{\eta_q}^{{\bf{SP}}} \right) + \left(-2a_3-a_4-\frac{a_{9}}{2}+\frac{a_{10}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LL}}} \\ &+ \left(-2a_5-\frac{a_{7}}{2} \right) {{\cal{E}}}_{\pi}^{{\bf{LR}}} + \left(-C_3+\frac{C_{9}}{2}+\frac{3C_{10}}{2} \right) \left({{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{NF, \eta_q}^{{\bf{LL}}} + {{\cal{A}}}_{NF, \pi}^{{\bf{LL}}} \right) \\ & + \left(-C_5+\frac{C_{7}}{2} \right) \left( {{\cal{E}}}_{NF, \pi}^{{\bf{LR}}} + {{\cal{E}}}_{NF, \eta_q}^{{\bf{LR}}} + {{\cal{A}}}_{NF, \pi}^{{\bf{LR}}} + {{\cal{A}}}_{NF, \eta_q}^{{\bf{LR}}} \right) + \left(\frac{3C_{8}}{2} \right) \left( {{\cal{E}}}_{NF, \eta_q}^{{\bf{SP}}} + {{\cal{A}}}_{NF, \pi}^{{\bf{SP}}} + {{\cal{A}}}_{NF, \eta_q}^{{\bf{SP}}} \right) \\ &+ \left(-C_3-2C_4+\frac{C_{9}-C_{10}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{LL}}} + \left(-2C_6-\frac{C_{8}}{2} \right) {{\cal{E}}}_{NF, \pi}^{{\bf{SP}}}-{{\cal{M}}}^{({\rm{ql,t}})}_{B \to \eta_q \pi^0 }- {{\cal{M}}}^{({\rm{ql,t}})}_{B \to \eta_q \pi^0 }-{{\cal{M}}}^{({\rm{mp}})}_{B \to \eta_q \pi^0 }- {{\cal{M}}}^{({\rm{mp}})}_{B \to \eta_q \pi^0 }\Big] \,, \end{aligned} $

$ \begin{aligned}[b] {{\cal{M}}}(B^0 \to \eta_q\eta_q) =& \frac{G_F}{\sqrt{2}} \, V^{\ast}_{ub}V_{ud} \Big[a_2 \left( {{\cal{A}}}_{\eta_q}^{{\bf{LL}}} + {{\cal{E}}}_{\eta_q}^{{\bf{LL}}} \right) + C_2 \left({{\cal{A}}}_{NF, \eta_q}^{{\bf{LL}}} + {{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}} \right) +{{\cal{M}}}^{({\rm{ql,u}})}_{B \to \eta_q \eta_q} \Big] \\ & + \frac{G_F}{\sqrt{2}} \, V^{\ast}_{cb}V_{cd} {{\cal{M}}}^{({\rm{ql,c}})}_{B \to \eta_q \eta_q} - \frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left(2a_3+a_4+\frac{a_{9}-a_{10}}{2} \right) \left( {{\cal{A}}}_{\eta_q}^{{\bf{LL}}} + {{\cal{E}}}_{\eta_q}^{{\bf{LL}}} \right) \\ & + \left(2a_{5}+\frac{a_{7}}{2} \right)\left({{\cal{A}}}_{\eta_q}^{{\bf{LR}}}+{{\cal{E}}}_{\eta_q}^{{\bf{LR}}} \right) + \left(a_6-\frac{a_{8}}{2} \right) \left( {{\cal{A}}}_{\eta_q}^{{\bf{SP}}} + {{\cal{E}}}_{\eta_q}^{{\bf{SP}}} \right) + \left(C_3+2C_4+\frac{-C_{9}+C_{10}}{2}\right) \left( {{\cal{A}}}_{NF, \eta_q}^{{\bf{LL}}} + {{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}} \right) \\ & + \left(C_5-\frac{C_{7}}{2} \right) \left({{\cal{A}}}_{NF, \eta_q}^{{\bf{LR}}} + {{\cal{E}}}_{NF, \eta_q}^{{\bf{LR}}} \right) + \left(2C_6+\frac{C_{8}}{2} \right) \left({{\cal{A}}}_{NF, \eta_q}^{{\bf{SP}}} + {{\cal{E}}}_{NF, \eta_q}^{{\bf{SP}}} \right) + {{\cal{M}}}^{({\rm{ql,t}})}_{B \to \eta_q \eta_q} +{{\cal{M}}}^{({\rm{mp}})}_{B \to \eta_q \eta_q } \Big] \,, \end{aligned} $

$ \begin{eqnarray} \sqrt{2}{{\cal{M}}}(B^0 \to \eta_q\eta_s) = -\frac{G_F}{\sqrt{2}} \, V^{\ast}_{tb} V_{td} \Big[ \left(a_3-\frac{a_{9}}{2} \right) {{\cal{E}}}_{\eta_q}^{{\bf{LL}}} + \left(a_5-\frac{a_{7}}{2} \right) {{\cal{E}}}_{\eta_q}^{{\bf{LR}}} + \left(C_4-\frac{C_{10}}{2} \right) {{\cal{E}}}_{NF, \eta_q}^{{\bf{LL}}} + \left(C_6-\frac{C_{8}}{2} \right) {{\cal{E}}}_{NF, \eta_q}^{{\bf{SP}}} \Big] \,. \end{eqnarray} $

$ \begin{aligned}[b] a_1=& \frac{C_1}{3} + C_2\,, \quad a_2 =\frac{C_2}{3}+C_1 \,, \\ a_i =& \frac{C_{i+1}}{3} + C_i \quad {\rm with}\quad i=3,5,7,9 \,, \\ a_i = & \frac{C_{i-1}}{3} + C_i \quad {\rm with}\quad i=4,6,8,10 \,. \end{aligned} $