Hidden-charm pentaquark states through current algebra: from their production to decay

$ \begin{aligned}[b] P_c(4380)^+: M=& 4380 \pm 8 \pm 29 \rm{ MeV} \, ,\\ \Gamma=& 205 \pm 18 \pm 86 \rm{ MeV} \, , \end{aligned} $

$ \begin{aligned}[b] P_c(4450)^+: M=& 4449.8 \pm 1.7 \pm 2.5 \rm{ MeV} \, , \\ \Gamma=& 39 \pm 5 \pm 19 \rm{ MeV} \, . \end{aligned} $

$ \begin{aligned}[b] P_c(4312)^+: M=& 4311.9 \pm 0.7 ^{+6.8}_{-0.6} \rm{ MeV} \, , \\ \Gamma=& 9.8 \pm 2.7 ^{+3.7}_{-4.5} \rm{ MeV} \, , \end{aligned} $

$ \begin{aligned}[b] P_c(4440)^+: M=& 4440.3 \pm 1.3 ^{+4.1}_{-4.7} \rm{ MeV} \, ,\\ \Gamma=& 20.6 \pm 4.9 ^{+8.7}_{-10.1} \rm{ MeV} \, , \end{aligned} $

$ \begin{aligned}[b] P_c(4457)^+: M=& 4457.3 \pm 0.6 ^{+4.1}_{-1.7} \rm{ MeV} \, ,\\ \Gamma=& 6.4 \pm 2.0 ^{+5.7}_{-1.9} \rm{ MeV} \, .\\ \end{aligned} $

$ \begin{aligned}[b] P_{cs}(4459)^0: M=& 4458.8 \pm 2.9^{+4.7}_{-1.1} \rm{ MeV} \, , \\ \Gamma=& 17.3 \pm 6.5^{+8.0}_{-5.7} \rm{ MeV} \, . \end{aligned} $

$ \begin{aligned}[b] \sqrt2 \xi_1(x) =& [\delta^{ab} \bar c_a(x) \gamma_5 d_b(x)] \\ & \times [\epsilon^{cde} u_c^T(x) \mathbb{C} \gamma_\mu u_d(x) \gamma^\mu \gamma_5 c_e(x)] \, , \end{aligned} $

$ \langle 0 | \xi_1 | D^- \Sigma_c^{++}; 1/2^-(q) \rangle = f_1 u(q) \, , $

$ \sqrt2 \xi_1 \rightarrow {1\over6} \; [\bar c_a \gamma_5 c_a]\; N - {1\over12} \; [\bar c_a \gamma_\mu c_a]\; \gamma^\mu \gamma_5 N + \cdots \, , $

$ \begin{aligned}[b] \langle 0 | \xi_1 | \eta_c p \rangle \approx& {\sqrt2\over12} \langle 0 | \bar c_{a} \gamma_5 c_a | \eta_c \rangle\; \langle 0 | N | p \rangle + \cdots \, , \\ \langle 0 | \xi_1 | \psi p \rangle \approx & -{\sqrt2\over24} \langle 0 | \bar c_{a} \gamma_\mu c_a | \psi \rangle\; \gamma^\mu \gamma_5 \langle 0 | N | p \rangle + \cdots \, . \end{aligned} $

$ \begin{aligned}[b] \theta(x) =& [\bar c_a(x) \Gamma^\theta_1 c_b(x)] \; \Big[[q^T_c(x) \mathbb{C} \Gamma^\theta_2 q_d(x)] \; \Gamma^\theta_3 q_e(x)\Big] \, , \\ \eta(x) =& [\bar c_a(x) \Gamma^\eta_1 u_b(x)] \; \Big[[u^T_c(x) \mathbb{C} \Gamma^\eta_2 d_d(x)] \; \Gamma^\eta_3 c_e(x) \Big] \, ,\\ \xi(x) =& [\bar c_a(x) \Gamma^\xi_1 d_b(x)] \; \Big[[u^T_c(x) \mathbb{C} \Gamma^\xi_2 u_d(x)] \; \Gamma^\xi_3 c_e(x) \Big] \, , \end{aligned} $

$ \delta^{ab} \epsilon^{cde} = \delta^{ac} \epsilon^{bde} + \delta^{ad} \epsilon^{cbe} + \delta^{ae} \epsilon^{cdb} \, . $

$\begin{aligned}[b]& \quad\quad\quad\quad\quad \bar c_a q_a \, [0^+] \, , \quad \, \bar c_a \gamma_5 q_a \, [0^-] \, ,\quad\\&\bar c_a \gamma_\mu q_a \, [1^-] \, ,\quad \, \bar c_a \gamma_\mu \gamma_5 q_a \, [1^+] \, , \,\quad \bar c_a \sigma_{\mu\nu} q_a \, [1^\pm] \, .\end{aligned} $

$ \sigma_{\mu\nu} \gamma_5 = {{\rm i}\over2} \epsilon_{\mu\nu\rho\sigma} \sigma^{\rho\sigma} \, . $

$ J_{D} = \bar c_a \gamma_5 q_a \, , \quad J_{D^{*}} = \bar c_a \gamma_\mu q_a \, . $

$ \begin{aligned}[b] J_{\Lambda_c^+} =& \epsilon^{abc} [u_a^T \mathbb{C} \gamma_{5} d_b] c_c \, , \\ \sqrt2 J_{\Sigma_c^{++}} =& \epsilon^{abc} [u_a^T \mathbb{C} \gamma_{\mu} u_b] \gamma^{\mu}\gamma_{5} c_c \, , \\ J_{\Sigma_c^+} =& \epsilon^{abc} [u_a^T \mathbb{C} \gamma_{\mu} d_b] \gamma^{\mu}\gamma_{5} c_c \, , \\ \sqrt2 J_{\Sigma_c^{0}} =& \epsilon^{abc} [d_a^T \mathbb{C} \gamma_{\mu} d_b] \gamma^{\mu}\gamma_{5} c_c \, , \\ \sqrt2 J^\alpha_{\Sigma_c^{*++}} =& \epsilon^{abc} P_{3/2}^{\alpha\mu} [u_a^T \mathbb{C} \gamma_{\mu} u_b] c_c \, , \\ J^\alpha_{\Sigma_c^{*+}} =& \epsilon^{abc} P_{3/2}^{\alpha\mu} [u_a^T \mathbb{C} \gamma_{\mu} d_b] c_c \, , \\ \sqrt2 J^\alpha_{\Sigma_c^{*0}} =& \epsilon^{abc} P_{3/2}^{\alpha\mu} [d_a^T \mathbb{C} \gamma_{\mu} d_b] c_c \, . \end{aligned} $

$ P_{3/2}^{\mu\nu} = g^{\mu\nu} - {1 \over 4} \gamma^\mu\gamma^\nu \, . $

$\begin{aligned}[b] | \bar D \Sigma_c; {1/2}^- ; \theta \rangle =& \cos\theta\; | \bar D^0 \Sigma_c^+;1/2^- \rangle\\& + \sin\theta\; | D^- \Sigma_c^{++};1/2^- \rangle \, , \end{aligned}$

$\begin{aligned}[b] | \bar D^* \Sigma_c; {1/2}^- ; \theta \rangle =& \cos\theta\; | \bar D^{*0} \Sigma_c^+;1/2^- \rangle \\&+ \sin\theta\; | D^{*-} \Sigma_c^{++};1/2^- \rangle \, , \end{aligned}$

$ \begin{aligned}[b] | \bar D^* \Sigma_c; {3/2}^- ; \theta \rangle =& \cos\theta\; | \bar D^{*0} \Sigma_c^+;3/2^- \rangle \\&+ \sin\theta\; | D^{*-} \Sigma_c^{++};3/2^- \rangle \, , \end{aligned} $

$\begin{aligned}[b] | \bar D \Sigma_c^*; {3/2}^- ; \theta \rangle =& \cos\theta\; | \bar D^{0} \Sigma_c^{*+};3/2^- \rangle \\&+ \sin\theta\; | D^- \Sigma_c^{*++};3/2^- \rangle \, , \end{aligned} $

$\begin{aligned}[b] | \bar D^* \Sigma_c^*; {1/2}^- ; \theta \rangle =& \cos\theta\; | \bar D^{*0} \Sigma_c^{*+};1/2^- \rangle \\&+ \sin\theta\; | D^{*-} \Sigma_c^{*++};1/2^- \rangle \, ,\end{aligned}$

$\begin{aligned}[b] | \bar D^* \Sigma_c^*; {3/2}^- ; \theta \rangle =& \cos\theta\; | \bar D^{*0} \Sigma_c^{*+};3/2^- \rangle \\&+ \sin\theta\; | D^{*-} \Sigma_c^{*++};3/2^- \rangle \, , \end{aligned} $

$ \begin{aligned}[b] | \bar D^* \Sigma_c^*; {5/2}^- ; \theta \rangle =& \cos\theta\; | \bar D^{*0} \Sigma_c^{*+};5/2^- \rangle\\& + \sin\theta\; | D^{*-} \Sigma_c^{*++};5/2^- \rangle \, , \end{aligned} $

$ \begin{aligned}[b] | \bar D^{(*)} \Sigma_c^{(*)}; J^P \rangle =& {\sqrt{1/3}}\; | \bar D^{(*)0} \Sigma_c^{(*)+}; J^P \rangle \\&- {\sqrt{2/3}}\; | D^{(*)-} \Sigma_c^{(*)++}; J^P \rangle . \end{aligned} $

$ J_i = \cos\theta\; \eta_i + \sin\theta\; \xi_i \, , $

$ \begin{aligned}[b] \eta_1 =& [\delta^{ab} \bar c_a \gamma_5 u_b] \; [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu d_d \gamma^\mu \gamma_5 c_e] \\ =& \bar D^0 \; \Sigma_c^+ \, , \end{aligned} $

$ \begin{aligned}[b] \eta_2 =& [\delta^{ab} \bar c_a \gamma_\nu u_b] \; \gamma^\nu \gamma_5 \; [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu d_d \gamma^\mu \gamma_5 c_e] \\ =& \bar D^{*0}_\nu \; \gamma^\nu \gamma_5 \; \Sigma_c^+ \, , \end{aligned} $

$ \begin{aligned}[b] \eta_3^\alpha =& P_{3/2}^{\alpha\nu} \; [\delta^{ab} \bar c_a \gamma_\nu u_b] \; [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu d_d \gamma^\mu \gamma_5 c_e] \\ =& P_{3/2}^{\alpha\nu} \; \bar D^{*0}_\nu \; \Sigma_c^+ \, , \end{aligned} $

$ \begin{aligned}[b] \eta_4^\alpha =& [\delta^{ab} \bar c_a \gamma_5 u_b] \; P_{3/2}^{\alpha\mu} [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu d_d c_e] \\ =& \bar D^0 \; \Sigma_c^{*+;\alpha} \, , \end{aligned} $

$ \begin{aligned}[b] \eta_5 =& [\delta^{ab} \bar c_a \gamma_\nu u_b] \; P_{3/2}^{\nu\mu} [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu d_d c_e] \\ =& \bar D^{*0}_\nu \; \Sigma_c^{*+;\nu} \, , \end{aligned} $

$ \begin{aligned}[b] \eta_6^{\alpha} =& [\delta^{ab} \bar c_a \gamma_\nu u_b] \; P_{3/2}^{\alpha\rho} \; \gamma^\nu \gamma_5 \; P^{3/2}_{\rho\mu} [\epsilon^{cde} u_c^T \mathbb{C} \gamma^\mu d_d c_e] \\ =& \bar D^{*0}_\nu \; P_{3/2}^{\alpha\rho} \; \gamma^\nu \gamma_5 \; \Sigma_{c;\rho}^{*+} \, , \end{aligned} $

$ \begin{aligned}[b] \eta_7^{\alpha\beta} =& P^{\alpha\beta,\nu\rho}_{5/2} \; [\delta^{ab} \bar c_a \gamma_\nu u_b] \; P^{3/2}_{\rho\mu} [\epsilon^{cde} u_c^T \mathbb{C} \gamma^\mu d_d c_e] \\ =& P^{\alpha\beta,\nu\rho}_{5/2} \; \bar D^{*0}_\nu \; \Sigma_{c;\rho}^{*+} \, , \end{aligned} $

$ \begin{aligned}[b] \xi_1 =& {1\over\sqrt2} \; [\delta^{ab} \bar c_a \gamma_5 d_b] \; [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu u_d \gamma^\mu \gamma_5 c_e] \\ =& D^- \; \Sigma_c^{++} \, , \end{aligned} $

$ \begin{aligned}[b] \xi_2 =& {1\over\sqrt2} \; [\delta^{ab} \bar c_a \gamma_\nu d_b] \; \gamma^\nu \gamma_5 \; [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu u_d \gamma^\mu \gamma_5 c_e] \\ =& D^{*-}_\nu \; \gamma^\nu \gamma_5 \; \Sigma_c^{++} \, , \end{aligned} $

$ \begin{aligned}[b] \xi_3^\alpha =& {1\over\sqrt2} \; P_{3/2}^{\alpha\nu} \; [\delta^{ab} \bar c_a \gamma_\nu d_b] \; [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu u_d \gamma^\mu \gamma_5 c_e] \\ =& P_{3/2}^{\alpha\nu} \; D^{*-}_\nu \; \Sigma_c^{++} \, , \end{aligned} $

$ \begin{aligned}[b] \xi_4^\alpha =& {1\over\sqrt2} \; [\delta^{ab} \bar c_a \gamma_5 d_b] \; P_{3/2}^{\alpha\mu} [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu u_d c_e] \\ =& D^- \; \Sigma_c^{*++;\alpha} \, , \end{aligned} $

$ \begin{aligned}[b] \xi_5 =& {1\over\sqrt2} \; [\delta^{ab} \bar c_a \gamma_\nu d_b] \; P_{3/2}^{\nu\mu} [\epsilon^{cde} u_c^T \mathbb{C} \gamma_\mu u_d c_e] \\ =& D^{*-}_\nu \; \Sigma_c^{*++;\nu} \, , \end{aligned} $

$ \begin{aligned}[b] \xi_6^{\alpha} =& {1\over\sqrt2} \; [\delta^{ab} \bar c_a \gamma_\nu d_b] \; P_{3/2}^{\alpha\rho} \gamma^\nu \gamma_5 \; P^{3/2}_{\rho\mu} [\epsilon^{cde} u_c^T \mathbb{C} \gamma^\mu u_d c_e] \\ =& D^{*-}_\nu \; P_{3/2}^{\alpha\rho} \; \gamma^\nu \gamma_5 \; \Sigma_{c;\rho}^{*++} \, ,\\[-10pt] \end{aligned} $

$ \begin{aligned}[b] \xi_7^{\alpha\beta} =& {1\over\sqrt2} \; P^{\alpha\beta,\nu\rho}_{5/2} \; [\delta^{ab} \bar c_a \gamma_\nu d_b] \; P^{3/2}_{\rho\mu} [\epsilon^{cde} u_c^T \mathbb{C} \gamma^\mu u_d c_e] \\ =& P^{\alpha\beta,\nu\rho}_{5/2} \; D^{*-}_\nu \; \Sigma_{c;\rho}^{*++} \, . \\[-10pt]\end{aligned} $

$ \begin{aligned}[b] P_{5/2}^{\mu\nu,\rho\sigma} =& {1\over2} g^{\mu\rho} g^{\nu\sigma} + {1\over2} g^{\mu\sigma} g^{\nu\rho} - {1 \over 6} g^{\mu\nu} g^{\rho\sigma} - {1 \over 12} g^{\mu\rho} \gamma^{\nu}\gamma^{\sigma}\\& - {1 \over 12} g^{\mu\sigma} \gamma^{\nu}\gamma^{\rho} - {1 \over 12} g^{\nu\sigma} \gamma^{\mu}\gamma^{\rho} - {1 \over 12} g^{\nu\rho} \gamma^{\mu}\gamma^{\sigma} \, . \end{aligned} $

$ \begin{aligned}[b] \langle 0 | J_{1,2,5} | X_{1,2,5}; 1/2^- \rangle =& f_{X_{1,2,5}} u (p) \, , \\ \langle 0 | J_{3,4,6}^\alpha | X_{3,4,6}; 3/2^- \rangle =& f_{X_{3,4,6}} u^\alpha (p) \, , \\ \langle 0 | J_{7}^{\alpha\beta} | X_{7}; 5/2^- \rangle =& f_{X_7} u^{\alpha\beta} (p) \, , \end{aligned} $

$ \begin{aligned}[b] \Pi_{1,2,5}\left(q^2\right) =& {\rm i} \int {\rm d}^4x {\rm e}^{{\rm i}q\cdot x} \langle 0 | T\left[J_{1,2,5}(x) \bar J_{1,2,5}(0)\right] | 0 \rangle \\ =& (\not q + M_{X_{1,2,5}}) \; \Pi_{1,2,5}\left(q^2\right) \, ,\\[-10pt] \end{aligned} $

$ \begin{aligned}[b] \Pi^{\alpha \alpha^\prime}_{3,4,6}\left(q^2\right) =& {\rm i} \int {\rm d}^4x {\rm e}^{{\rm i}q\cdot x} \langle 0 | T\left[J^{\alpha}_{3,4,6}(x) \bar J^{\alpha^\prime}_{3,4,6}(0)\right] | 0 \rangle \\ =& {\cal{G}}_{3/2}^{\alpha \alpha^\prime} (\not q + M_{X_{3,4,6}})\; \Pi_{3,4,6}\left(q^2\right) \, , \\[-8pt]\end{aligned} $

$ \begin{aligned}[b] \Pi^{\alpha \beta,\alpha^\prime \beta^\prime}_7\left(q^2\right) =& {\rm i} \int {\rm d}^4x {\rm e}^{{\rm i}q\cdot x} \langle 0 | T\left[J^{\alpha \beta}_7(x) \bar J^{\alpha^\prime \beta^\prime}_7(0)\right] | 0 \rangle \\ =& {\cal{G}}_{5/2}^{\alpha \beta,\alpha^\prime \beta^\prime} (\not q + M_{X_7})\; \Pi_7\left(q^2\right) \, , \end{aligned} $

$ {\cal{G}}_{3/2}^{\mu\nu}(p) = g^{\mu\nu} - {1\over3} \gamma^\mu \gamma^\nu - {p^\mu\gamma^\nu - p^\nu\gamma^{\mu} \over 3m} - {2p^{\mu}p^\nu \over 3m^2} \, , $

$ \begin{aligned}[b] {\cal{G}}_{5/2}^{\mu \nu,\rho \sigma}(p) =& \frac{1}{2}(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho}) - \frac{1}{5}g^{\mu\nu}g^{\rho\sigma} \\ &- \frac{1}{10}(g^{\mu\rho}\gamma^\nu\gamma^\sigma + g^{\mu\sigma}\gamma^\nu\gamma^\rho + g^{\nu\rho}\gamma^\mu\gamma^\sigma + g^{\nu\sigma}\gamma^\mu\gamma^\rho) \end{aligned} $

$ \begin{aligned}[b] &+ \frac{1}{10m}\Bigl(g^{\mu\rho}(p^\nu\gamma^\sigma - p^\sigma\gamma^\nu) + g^{\mu\sigma}(p^\nu\gamma^\rho - p^\rho\gamma^\nu) \\ & + g^{\nu\rho}(p^\mu\gamma^\sigma - p^\sigma\gamma^\mu) + g^{\nu\sigma}(p^\mu\gamma^\rho - p^\rho\gamma^\mu)\Bigr) \\ &+ \frac{1}{5m^2}(g^{\mu\nu}p^\rho p^\sigma + g^{\rho\sigma}p^\mu p^\nu) \\ &- \frac{2}{5m^2}(g^{\mu\rho}p^\nu p^\sigma + g^{\mu\sigma}p^\nu p^\rho + g^{\nu\rho}p^\mu p^\sigma + g^{\nu\sigma}p^\mu p^\rho) \\ &+ \frac{1}{10m^2}\Bigl(\gamma^\mu p^\nu (\gamma^\rho p^\sigma + \gamma^\sigma p^\rho) + \gamma^\nu p^\mu (\gamma^\rho p^\sigma + \gamma^\sigma p^\rho)\Bigl) \\ &+ \frac{1}{5m^3}\Bigl(p^\rho p^\sigma (\gamma^\mu p^\nu + \gamma^\nu p^\mu ) - p^\mu p^\nu (\gamma^\rho p^\sigma + \gamma^\sigma p^\rho ) \Bigl) \\ &+ \frac{2}{5m^4}p^\mu p^\nu p^\rho p^\sigma \, . \end{aligned} $

$ \langle 0 | J_{1\cdots7} | X_{1\cdots7}^\prime \rangle = f_{X^\prime_{1\cdots7}} \gamma_5 u (p) \, , $

$ \langle 0 | \gamma_5 J_{1\cdots7} | X_{1\cdots7} \rangle = f_{X_{1\cdots7}} \gamma_5 u (p) \, . $

$ \langle 0 | \gamma_5 J_{1\cdots7} | X_{1\cdots7}^\prime \rangle = f_{X^\prime_{1\cdots7}} u (p) \, . $

$ \Pi(q^2)={\frac{1}{\pi}}\int^\infty_{s_<}\frac{{\rm{Im}} \Pi(s)}{s-q^2-{\rm i}\varepsilon}{\rm d}s \, , $

$ \begin{aligned}[b] \rho_{\rm{phen}}(s) \equiv& {\rm{Im}}\Pi(s)/\pi \\ =& \sum_n\delta(s-M^2_n)\langle 0|\eta|n\rangle\langle n|{\eta^\dagger}|0\rangle \\ =& f_X^2\delta(s-m_X^2)+ \rm{continuum}. \end{aligned} $

$ \Pi(s_0, M_{\rm B}^2) \equiv f^2_X {\rm e}^{-M_X^2/M_{\rm B}^2} = \int^{s_0}_{s_<} {\rm e}^{-s/M_{\rm B}^2}\rho_{\rm{OPE}}(s){\rm d}s \, . $

$ M^2_X(s_0, M_{\rm B}) = \frac{\displaystyle\int^{s_0}_{s_<} {\rm e}^{-s/M_{\rm B}^2}s\rho_{\rm{OPE}}(s){\rm d}s}{\displaystyle\int^{s_0}_{s_<} {\rm e}^{-s/M_{\rm B}^2}\rho_{\rm{OPE}}(s){\rm d}s} \, , $

$ f_X^2(s_0, M_{\rm B}) = {\rm e}^{(M_X^2(s_0, M_{\rm B}))/ M_{\rm B}^2} \int^{s_0}_{s_<} {\rm e}^{-s/M_{\rm B}^2}\rho_{\rm{OPE}}(s){\rm d}s \, . $

$ \begin{aligned}[b] m_c=& 1.275 ^{+0.025}_{-0.035} \rm{ GeV} \, , \\ \langle \bar qq \rangle =& - (0.24 \pm 0.01)^3 \rm{ GeV}^3 \, , \\ \langle g_s^2GG\rangle =& (0.48 \pm 0.14) \rm{ GeV}^4\, , \\ \langle g_s \bar q \sigma G q \rangle =& M_0^2 \times \langle \bar qq \rangle\, , \\ M_0^2 =& (0.8 \pm 0.2) \rm{ GeV}^2 \, , \end{aligned} $

$ \rm{Convergence} \equiv \left|\frac{ \Pi^{D=10}(\infty, M_{\rm B}) }{ \Pi(\infty, M_{\rm B}) }\right| \leq 10\% \, . $

$ \rm{Pole-Contribution} \equiv \frac{ \Pi(s_0, M_{\rm B}) }{ \Pi(\infty, M_{\rm B}) } \geq 40\% \, . $

$ \begin{aligned}[b] M_{X_1} =& 4.30^{+0.10}_{-0.10} \rm{ GeV} \, , \\ f_{X_1} =& \left(1.19^{+0.19}_{-0.18}\right) \times 10^{-3} \rm{ GeV}^6 \, . \end{aligned} $

$ \begin{aligned}[b] M_{X_6} =& 4.64^{+0.10}_{-0.10} \rm{ GeV} \, , \quad f_{X_6} = \left(1.01^{+0.15}_{-0.14}\right) \times 10^{-3} \rm{ GeV}^6 \, , \\ M_{X_7} =& 4.64^{+0.14}_{-0.12} \rm{ GeV} \, , \quad f_{X_7} = \left(0.77^{+0.12}_{-0.11}\right) \times 10^{-3} \rm{ GeV}^6 \, . \end{aligned} $

$ \rm{Convergence} \equiv \left|\frac{ \Pi^{D=10}(\infty, M_{\rm B}) }{ \Pi(\infty, M_{\rm B}) }\right| \leq 15\% \, . $

$ \begin{aligned}[]b M^\prime_{X_6} =& 4.52^{+0.11}_{-0.11} \rm{ GeV} \, , \quad f^\prime_{X_6} = \left(0.85^{+0.14}_{-0.13}\right) \times 10^{-3} \rm{ GeV}^6 \, , \\ M^\prime_{X_7} =& 4.55^{+0.15}_{-0.13} \rm{ GeV} \, , \quad f^\prime_{X_7} = \left(0.65^{+0.11}_{-0.10}\right) \times 10^{-3} \rm{ GeV}^6 \, . \end{aligned} $

$ \Lambda_b^0 = udb \to ud c\; \bar c s \to udc\; \bar c s\; \bar u u \to J/\psi p K^- \, . $

$ \begin{aligned}[b] \Lambda_b^0 \to& ud c\; \bar c s\; (\bar u u + \bar d d) \\ \to& \sqrt{1\over3} D^{(*)-} \Sigma_c^{(*)++} K^- + \sqrt{1\over3} \bar D^{(*)0} \Sigma_c^{(*)0} \bar K^0 \\ & - \sqrt{1\over6} D^{(*)-} \Sigma_c^{(*)+} \bar K^0 - \sqrt{1\over6} \bar D^{(*)0} \Sigma_c^{(*)+} K^-\, . \end{aligned} $

$ \begin{aligned}[b]\\ \epsilon^{abc} \delta^{de} \delta^{fg} =& \left( \epsilon^{ebc} \delta^{da} + \epsilon^{aec} \delta^{db} + \epsilon^{abe} \delta^{dc} \right) \times \delta^{fg} = \epsilon^{gbc} \delta^{da} \delta^{fe} + \epsilon^{egc} \delta^{da} \delta^{fb} + \epsilon^{ebg} \delta^{da} \delta^{fc} \\ &+ \epsilon^{gec} \delta^{db} \delta^{fa} + \epsilon^{agc} \delta^{db} \delta^{fe} + \epsilon^{aeg} \delta^{db} \delta^{fc} + \epsilon^{gbe} \delta^{dc} \delta^{fa} + \epsilon^{age} \delta^{dc} \delta^{fb} + \epsilon^{abg} \delta^{dc} \delta^{fe} \, . \end{aligned} $

$ [\epsilon^{abc} u_a d_b c_c][\delta^{de}\bar c_d s_e][\delta^{fg}\bar u_f u_g] $

$ [\epsilon^{agc}u_a u_g c_c] [\delta^{db} \bar c_d d_b] [\delta^{fe}\bar u_f s_e] $

$ \begin{array}{*{20}{l}} \Lambda_b^0 &\xrightarrow{\; \; \; \; \; \; \; \; \; \; \; \; \; }& J_{\Lambda_b^0} = [\epsilon^{abc}u_a^T \mathbb{C} \gamma_5 d_b b_c], \end{array} $

$ \begin{array}{*{20}{l}}\quad\quad \xrightarrow{\; \; \; \; \rm weak\; \; \; \; } [\epsilon^{abc}u_a^T \mathbb{C} \gamma_5 d_b \gamma_\rho(1 - \gamma_5) c_c] \; \times\; [\delta^{de}\bar c_d \gamma^\rho(1 - \gamma_5) s_e], \end{array} $

$ \begin{array}{*{20}{l}} \quad\quad\xrightarrow{\; \; \; \; \rm QPC\; \; \; \; } [\epsilon^{abc}u_a^T \mathbb{C} \gamma_5 d_b \gamma_\rho(1 - \gamma_5) c_c] \; \times\; [\delta^{de}\bar c_d \gamma^\rho(1 - \gamma_5) s_e] \; \times\; [\delta^{fg}\bar u_f u_g], \end{array} $

$ \begin{array}{*{20}{l}} \quad\quad\underline{\underline {{\rm{\;\;color\;\;}}}} \epsilon^{agc}\delta^{db}\delta^{fe} \times u_a^T \mathbb{C} \gamma_5 d_b \gamma_\rho(1 - \gamma_5) c_c \times \bar c_d \gamma^\rho(1 - \gamma_5) s_e \times \bar u_f u_g + \cdots , \end{array} $

$ \begin{array}{*{20}{l}} \quad\quad\underline{\underline {{\rm{Fierz:}}d_b \leftrightarrow u_g}} -{\delta^{db}\delta^{fe}\over4} \times [\epsilon^{agc}u_a^T \mathbb{C} \gamma_\mu u_g \gamma_\rho(1 - \gamma_5) c_c] \times \bar c_d \gamma^\rho(1 - \gamma_5) s_e \times \bar u_f \gamma^\mu \gamma_5 d_b + \cdots, \end{array} $

$ \begin{aligned}[b]\quad\quad \underline{\underline {{\rm{Fierz:}}d_b \leftrightarrow s_e}} +& {1 + \gamma_5 \over16} \times [\epsilon^{agc}u_a^T \mathbb{C} \gamma_\mu u_g \gamma^\mu \gamma_5 c_c] \times [\delta^{db} \bar c_d \gamma_5 d_b] \times [\delta^{fe}\bar u_f \gamma_5 s_e] \\& + {(1 + \gamma_5) (g^{\nu\rho} - {\rm i} \sigma^{\nu\rho}) \over 32} \times [\epsilon^{agc}u_a^T \mathbb{C} \gamma_\mu u_g \gamma^\mu \gamma_5 c_c] \times [\delta^{db} \bar c_d \gamma_\nu d_b] \times [\delta^{fe}\bar u_f \gamma_\rho \gamma_5 s_e] \\ & + {(1 + \gamma_5) (g^{\alpha\nu} \gamma^\rho + g^{\alpha\rho} \gamma^\nu) \over 16} \times [P_{\alpha \mu}^{3/2}\epsilon^{agc}u_a^T \mathbb{C} \gamma^\mu u_g c_c] \times [\delta^{db} \bar c_d \gamma_\nu d_b] \times [\delta^{fe}\bar u_f \gamma_\rho \gamma_5 s_e] + \cdots, \end{aligned} $

$ \begin{aligned}[b]\quad\quad \underline{\underline { \; \; \; \; \; \; \; \; \; \; \; \; \; \;}} & + {1 + \gamma_5 \over8\sqrt2} \times \xi_1 \times [\bar u_a \gamma_5 s_a] + {(1 + \gamma_5) (g_{\nu\rho} - {\rm i} \sigma_{\nu\rho}) \over 16\sqrt2} \left(\xi_3^\nu - {1\over4}\gamma^\nu\gamma_5\xi_2\right) [\bar u_a \gamma^\rho \gamma_5 s_a] \\ & + {(1 + \gamma_5) (g_{\alpha\nu} \gamma_\rho + g_{\alpha\rho} \gamma_\nu) \over 8\sqrt2} \left(\xi_7^{\alpha\nu} - {1\over9}\gamma^\alpha\gamma_5\xi_6^\nu - {1\over9}\gamma^\nu\gamma_5\xi_6^\alpha + {2\over9} g^{\alpha\nu}\xi_5 \right) [\bar u_a \gamma^\rho \gamma_5 s_a] + \cdots \, . \end{aligned} $

$ \begin{aligned}[b] \xi_6^{\prime \alpha\beta} =& {1\over\sqrt2}\; P^{\alpha\beta,\nu\rho}_{3/2}\; [\delta^{ab} \bar c_a \gamma_\nu d_b] \; P^{3/2}_{\rho\mu} [\epsilon^{cde} u_c^T \mathbb{C} \gamma^\mu u_d c_e] \\ =& P^{\alpha\beta,\nu\rho}_{3/2}\; D^{*-}_\nu \; \; \Sigma_{c;\rho}^{*++} \, , \end{aligned} $

$ \begin{aligned}[b] P_{3/2}^{\mu\nu,\,\rho\sigma} =& {1\over2} g^{\mu\rho}g^{\nu\sigma} - {1\over2} g^{\mu\sigma}g^{\nu\rho} + {1\over6} \sigma^{\mu\nu}\sigma^{\rho\sigma} \\ & - {1\over4}g^{\mu\rho}\gamma^\nu\gamma^\sigma + {1\over4}g^{\mu\sigma}\gamma^\nu\gamma^\rho \\ & - {1\over4}g^{\nu\sigma}\gamma^\mu\gamma^\rho + {1\over4}g^{\nu\rho}\gamma^\mu\gamma^\sigma \, . \end{aligned} $

$ \langle 0| \eta_6^{\prime \alpha\beta} | D^{*-} \Sigma_c^{*++}; 3/2^- \rangle = i f^T_{6^\prime} (p^\alpha u^\beta - p^\beta u^\alpha) \, , $

$ \begin{aligned}[b] \langle 0 | \bar u_a \gamma_5 s_a | K^-(q) \rangle =& \lambda_{K} \, , \\ \langle 0 | \bar u_a \gamma_\mu \gamma_5 s_a | K^-(q) \rangle =& {\rm i} q_\mu f_{K} \, , \end{aligned} $

$ \begin{aligned}[b] & \langle \Lambda_b^0(q) \; |\; \bar D \Sigma_c; 1/2^-(q_1)\; K^-(q_2) \rangle \\ \approx& -c\; {\rm i} \lambda_K f_{|\bar D \Sigma_c; 1/2^- \rangle} \; \bar u_{\Lambda_b^0} \left( {1 + \gamma_5 \over16} \right) u \, , \end{aligned} $

$ \begin{aligned}[b] & \langle \Lambda_b^0(q) \; |\; \bar D^* \Sigma_c; 1/2^-(q_1)\; K^-(q_2) \rangle \\ \approx& c\; {\rm i} f_K f_{|\bar D^* \Sigma_c; 1/2^- \rangle} q_2^\rho \\ &\times \bar u_{\Lambda_b^0} \left( {(1 + \gamma_5) (g_{\nu\rho} - {\rm i} \sigma_{\nu\rho}) \over 32} \cdot \left(- {1\over4}\gamma^\nu\gamma_5\right) \right) u \, , \end{aligned} $

$ \begin{aligned}[b] & \langle \Lambda_b^0(q) \; |\; \bar D^* \Sigma_c; 3/2^-(q_1)\; K^-(q_2) \rangle \\ \approx& c\; {\rm i} f_K f_{|\bar D^* \Sigma_c; 3/2^- \rangle} q_2^\rho \\ &\times \bar u_{\Lambda_b^0} \left( {(1 + \gamma_5) (g_{\nu\rho} - {\rm i} \sigma_{\nu\rho}) \over 32} \right) u^\nu \, , \end{aligned} $

$ \begin{aligned}[b] & \langle \Lambda_b^0(q) \; |\; \bar D^* \Sigma_c^*; 1/2^-(q_1)\; K^-(q_2) \rangle \\ \approx& c\; {\rm i} f_K f_{|\bar D^* \Sigma_c^*; 1/2^- \rangle} q_2^\rho \\ &\times \bar u_{\Lambda_b^0} \left( {(1 + \gamma_5) (g_{\alpha\nu} \gamma_\rho + g_{\alpha\rho} \gamma_\nu) \over 16} \cdot {2\over9} g^{\alpha\nu} \right) u \, , \end{aligned} $

$ \begin{aligned}[b]& \langle \Lambda_b^0(q) \; |\; \bar D^* \Sigma_c^*; 3/2^-(q_1)\; K^-(q_2) \rangle \\ \approx& c\; {\rm i} f_K f_{|\bar D^* \Sigma_c^*; 3/2^- \rangle} q_2^\rho \\ & \times \bar u_{\Lambda_b^0} \Bigg( {(1 + \gamma_5) (g_{\alpha\nu} \gamma_\rho + g_{\alpha\rho} \gamma_\nu) \over 16} \\ & \times \left( - {1\over9}\gamma^\alpha\gamma_5 g^{\nu\beta} - {1\over9}\gamma^\nu\gamma_5 g^{\alpha\beta} \right) \Bigg) u_\beta \, , \end{aligned} $

$ \begin{aligned}[b] & \langle \Lambda_b^0(q) \; |\; \bar D^* \Sigma_c^*; 5/2^-(q_1)\; K^-(q_2) \rangle \\ \approx& c\; {\rm i} f_K f_{|\bar D^* \Sigma_c^*; 5/2^- \rangle} q_2^\rho \\ &\times \bar u_{\Lambda_b^0} \left( {(1 + \gamma_5) (g_{\alpha\nu} \gamma_\rho + g_{\alpha\rho} \gamma_\nu) \over 16} \right) u^{\alpha\nu} \, , \end{aligned} $

$ \begin{aligned}[b] M_{|\bar D \Sigma_c; 1/2^- \rangle} =& M_{P_c(4312)^+} = 4311.9\; {\rm{MeV}} \, , \\ M_{|\bar D^{*} \Sigma_c; 1/2^- \rangle} =& M_{P_c(4457)^+} = 4457.3\; {\rm{MeV}} \, , \\ M_{|\bar D^{*} \Sigma_c; 3/2^- \rangle} =& M_{P_c(4440)^+} = 4440.3\; {\rm{MeV}} \, , \\ M_{|\bar D \Sigma_c^{*}; 3/2^- \rangle} \approx& M_{D} + M_{\Sigma_c^*} = 4385\; {\rm{MeV}} \, , \\ M_{|\bar D^{*} \Sigma_c^*; 1/2^- \rangle} \approx& M_{D^*} + M_{\Sigma_c^*} = 4527\; {\rm{MeV}} \, , \\ M_{|\bar D^{*} \Sigma_c^*; 3/2^- \rangle} \approx& M_{D^*} + M_{\Sigma_c^*} = 4527\; {\rm{MeV}} \, , \\ M_{|\bar D^{*} \Sigma_c^*; 5/2^- \rangle} \approx& M_{D^*} + M_{\Sigma_c^*} = 4527\; {\rm{MeV}} \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(\Lambda_b^0 \to |\bar D \Sigma_c \rangle_{1/2^-} K^-) =& c^2\; 6.15 \times 10^{-11}\; {\rm{GeV}}^{17} \, , \\ \Gamma(\Lambda_b^0 \to |\bar D^* \Sigma_c \rangle_{1/2^-} K^-) =& c^2\; 8.76 \times 10^{-12}\; {\rm{GeV}}^{17} \, , \\ \Gamma(\Lambda_b^0 \to |\bar D^* \Sigma_c \rangle_{3/2^-} K^-) =& c^2\; 7.52 \times 10^{-12}\; {\rm{GeV}}^{17} \, , \\ \Gamma(\Lambda_b^0 \to |\bar D \Sigma_c^* \rangle_{3/2^-} K^-) =& 0 \\ \Gamma(\Lambda_b^0 \to |\bar D^* \Sigma_c^* \rangle_{1/2^-} K^-) =& c^2\; 3.57 \times 10^{-11}\; {\rm{GeV}}^{17} \, , \\ \Gamma(\Lambda_b^0 \to |\bar D^* \Sigma_c^* \rangle_{3/2^-} K^-) =& c^2\; 1.38 \times 10^{-12}\; {\rm{GeV}}^{17} \, , \\ \Gamma(\Lambda_b^0 \to |\bar D^* \Sigma_c^* \rangle_{5/2^-} K^-) =& 0 \, . \end{aligned} $

$ \begin{aligned}[b]\\[-5pt] & {{\cal{B}}\Bigg(\Lambda_b^0 \rightarrow K^-\Big( |\bar D \Sigma_c \rangle_{1/2^-} :|\bar D^* \Sigma_c \rangle_{1/2^-} :|\bar D^* \Sigma_c \rangle_{3/2^-} :|\bar D \Sigma_c^* \rangle_{3/2^-} :|\bar D^* \Sigma_c^* \rangle_{1/2^-} :|\bar D^* \Sigma_c^* \rangle_{3/2^-} :|\bar D^* \Sigma_c^* \rangle_{5/2^-} \Big)\Bigg) \over {\cal{B}}\left(\Lambda_b^0 \rightarrow |\bar D^* \Sigma_c \rangle_{3/2^-} K^- \right)} \\ \approx & \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;\quad\quad 8.2 \; \; \; \; : \; \; \; \; \; \,1.2\,\; \; \; \; \; : \; \; \; \; \; \; \; {\bf1}\;\;\; \; \; \; \; \; \, : \; \; \; \; \; \; 0\; \;\;\;\; \; \; \; \; : \; \; \; \; \; \,4.8\,\; \; \; \; \;\; : \; \; \; \; \; 0.18\; \; \; \; \; : \; \; \; \; \; 0\; \, . \end{aligned} $

$ \begin{aligned}[b] N = N_1 - N_2 = \epsilon^{abc} (u_a^T \mathbb{C} d_b) \gamma_5 u_c - \epsilon^{abc} (u_a^T \mathbb{C} \gamma_5 d_b) u_c \, . \end{aligned} $

$ \langle 0 | N | p \rangle = f_p u_p \, , $

$ f_p = 0.011 {\rm\; GeV}^3 \, . $

$ \langle 0 | J_{{\cal{B}}} | {\cal{B}} \rangle = f_{{\cal{B}}} u_{{\cal{B}}} \, . $

$ \begin{aligned}[b] f_{\Lambda_c} = 0.015 {\rm\; GeV}^3 \, , \quad f_{\Sigma_c} = 0.036 {\rm\; GeV}^3 \, . \end{aligned} $

$ \delta^{ab} \epsilon^{cde} = {1\over3}\; \delta^{ae} \epsilon^{bcd} - {1\over2}\; \lambda^{ae}_n \epsilon^{bcf} \lambda^{fd}_n + {1\over2}\; \lambda^{ae}_n \epsilon^{bdf} \lambda^{fc}_n \, , $

$ \begin{aligned}[b] \eta_4^\alpha \rightarrow & [\bar c_a \gamma_\mu c_a] \left( - {1\over32} g^{\alpha\mu} - {{\rm i}\over96} \sigma^{\alpha\mu} \right) N \\ & + \; [\bar c_a \gamma_\mu \gamma_5 c_a] \left( - {1\over32} g^{\alpha\mu} \gamma_5 - {{\rm i}\over96} \sigma^{\alpha\mu} \gamma_5 \right) N \\ & + \; [\bar c_a \sigma_{\mu\nu} c_a] \left( {{\rm i}\over48} g^{\alpha\mu}\gamma^\nu + {1\over96} \epsilon^{\alpha\mu\nu\rho} \gamma_\rho \gamma_5 \right) N \\ & + \; \cdots \, , \end{aligned} $

$ \begin{aligned}[b] \eta_5 \rightarrow & + {1\over8} \; [\bar c_a c_a] \; \gamma_5 N + {1\over8} \; [\bar c_a \gamma_5 c_a] \; N \\ & + {1\over16} \; [\bar c_a \gamma_\mu c_a] \; \gamma^\mu \gamma_5 N - {1\over16} \; [\bar c_a \gamma_\mu \gamma_5 c_a] \; \gamma^\mu N \\ & + {1\over48} \; [\bar c_a \sigma_{\mu\nu} c_a] \; \sigma^{\mu\nu} \gamma_5 N + \cdots \, , \\[-10pt]\end{aligned} $

$ \begin{aligned} [b] \eta_6^\alpha \rightarrow & [\bar c_a \gamma_\mu c_a] \left( {3\over32} g^{\alpha\mu} + {{\rm i}\over32} \sigma^{\alpha\mu} \right) N \\ & + \; [\bar c_a \gamma_\mu \gamma_5 c_a] \left( - {3\over32} g^{\alpha\mu} \gamma_5 - {{\rm i}\over32} \sigma^{\alpha\mu} \gamma_5 \right) N + \; \cdots \, , \end{aligned} $

$ \begin{aligned}[b] \eta_7^{\alpha\beta}\rightarrow & \Big( {{\rm i}\over144} \sigma^{\alpha\rho}\epsilon^{\beta\mu\nu\rho} + {1\over72} g^{\alpha\mu}\sigma^{\beta\nu}\gamma_5 - {1\over144} g^{\alpha\beta}\sigma^{\mu\nu}\gamma_5 \Big) \\ & \times \; [\bar c_a \sigma_{\mu\nu} c_a] \; N + \cdots \, . \end{aligned} $

$ \begin{aligned}[b] \sqrt2 \xi_4^\alpha \rightarrow & [\bar c_a \gamma_\mu c_a] \left( {1\over16} g^{\alpha\mu} + {{\rm i}\over48} \sigma^{\alpha\mu} \right) N \\ & + \; [\bar c_a \gamma_\mu \gamma_5 c_a] \left( {1\over16} g^{\alpha\mu} \gamma_5 + {{\rm i}\over48} \sigma^{\alpha\mu} \gamma_5 \right) N \\ & + \; [\bar c_a \sigma_{\mu\nu} c_a] \left( - {i\over24} g^{\alpha\mu}\gamma^\nu - {1\over48} \epsilon^{\alpha\mu\nu\rho} \gamma_\rho \gamma_5 \right) N \\ & + \; \cdots \, , \\[-5pt]\end{aligned} $

$ \begin{aligned}[b] \sqrt2 \xi_5 \rightarrow & - {1\over4} \; [\bar c_a c_a] \; \gamma_5 N - {1\over4} \; [\bar c_a \gamma_5 c_a] \; N \\ & - {1\over8} \; [\bar c_a \gamma_\mu c_a] \; \gamma^\mu \gamma_5 N + {1\over8} \; [\bar c_a \gamma_\mu \gamma_5 c_a] \; \gamma^\mu N \\ & - {1\over24} \; [\bar c_a \sigma_{\mu\nu} c_a] \; \sigma^{\mu\nu} \gamma_5 N + \cdots \, , \end{aligned} $

$ \begin{aligned}[b] \sqrt2 \xi_6^\alpha \rightarrow & [\bar c_a \gamma_\mu c_a] \left( - {3\over16} g^{\alpha\mu} - {{\rm i}\over16} \sigma^{\alpha\mu} \right) N \\ & + \; [\bar c_a \gamma_\mu \gamma_5 c_a] \left( {3\over16} g^{\alpha\mu} \gamma_5 + {{\rm i}\over16} \sigma^{\alpha\mu} \gamma_5 \right) N \\ & + \; \cdots \, , \end{aligned} $

$ \begin{aligned}[b] \sqrt2 \xi_7^{\alpha\beta} \rightarrow & \Big( - {{\rm i}\over72} \sigma^{\alpha\rho}\epsilon^{\beta\mu\nu\rho} - {1\over36} g^{\alpha\mu}\sigma^{\beta\nu}\gamma_5 \\ & + {1\over72} g^{\alpha\beta}\sigma^{\mu\nu}\gamma_5 \Big) \; [\bar c_a \sigma_{\mu\nu} c_a] \; N + \cdots \, . \end{aligned} $

$ \delta^{ab} \epsilon^{cde} = {1\over3}\; \delta^{ac} \epsilon^{bde} - {1\over2}\; \lambda^{ac}_n \epsilon^{bdf} \lambda^{fe}_n + {1\over2}\; \lambda^{ac}_n \epsilon^{bef} \lambda^{fd}_n \, , $

$\begin{aligned}[b] \delta^{ab} \epsilon^{cde} =& {1\over3}\; \delta^{ad} \epsilon^{cbe} + {1\over2}\; \lambda^{ad}_n \epsilon^{bcf} \lambda^{fe}_n \\&- {1\over2}\; \lambda^{ad}_n \epsilon^{bef} \lambda^{fc}_n \, , \end{aligned} $

$ \begin{aligned}[b] \eta_4^\alpha \rightarrow & \left( {1\over16}g^{\alpha\mu} + {{\rm i}\over48}\sigma^{\alpha\mu} \right) \; [\bar c_a \gamma_\mu u_a] \; \Lambda_c^+ \\ &+ \left( {{\rm i}\over384}\sigma^{\alpha\sigma} \epsilon^{\mu\nu\rho\sigma} - {1\over128}\epsilon^{\alpha\mu\nu\rho} \right) [\bar c_a \sigma_{\mu\nu} u_a] \gamma_\rho \gamma_5 \Sigma_c^+ \\ &+ \left( {{\rm i}\sqrt2\over384}\sigma^{\alpha\sigma} \epsilon^{\mu\nu\rho\sigma} - {\sqrt2\over128}\epsilon^{\alpha\mu\nu\rho} \right) \\ & \times \; [\bar c_a \sigma_{\mu\nu} d_a] \; \gamma_\rho \gamma_5 \Sigma_c^{++} + \cdots \, , \end{aligned} $

$ \begin{aligned}[b] \eta_5 \rightarrow & - {1\over4} \; [\bar c_a \gamma_5 u_a] \; \Lambda_c^+ - {1\over48} \; [\bar c_a \sigma_{\mu\nu} u_a] \; \sigma^{\mu\nu}\gamma_5 \Lambda_c^+ \\ & - {1\over32} \; [\bar c_a \gamma_\mu u_a] \; \gamma^\mu \gamma_5 \Sigma_c^+ + {1\over32} \; [\bar c_a \gamma_\mu \gamma_5 u_a] \; \gamma^\mu \Sigma_c^+ \\ & - {\sqrt2\over32} [\bar c_a \gamma_\mu d_a] \gamma^\mu \gamma_5 \Sigma_c^{++} + {\sqrt2\over32} [\bar c_a \gamma_\mu \gamma_5 d_a] \gamma^\mu \Sigma_c^{++} \\ & + \; \cdots \, , \end{aligned} $

$ \begin{aligned}[b] \eta_6^\alpha \rightarrow & \left( {{\rm i}\over16}g^{\alpha\mu}\gamma^\nu + {1\over32}\epsilon^{\alpha\mu\nu\rho}\gamma_\rho\gamma_5 \right) \; [\bar c_a \sigma_{\mu\nu} u_a] \; \Lambda_c^+ \\ &+ \left( {1\over96}g^{\alpha\mu}\gamma^\nu\gamma_5 + {1\over96}g^{\alpha\nu}\gamma^\mu\gamma_5 - {1\over192}g^{\mu\nu}\gamma^\alpha\gamma_5 \right) \\ & \times \; [\bar c_a \gamma_\mu u_a] \; \gamma_\nu \gamma_5 \Sigma_c^+ \\ &+ \left( {1\over64}g^{\alpha\mu}\gamma^\nu - {1\over64}g^{\alpha\nu}\gamma^\mu - {{\rm i}\over64}\epsilon^{\alpha \mu \nu \rho}\gamma_\rho\gamma_5 \right) \\ & \times \; [\bar c_a \gamma_\mu \gamma_5 u_a] \; \gamma_\nu \gamma_5 \Sigma_c^+ \\ &+ \left( {\sqrt2\over96}g^{\alpha\mu}\gamma^\nu\gamma_5 + {\sqrt2\over96}g^{\alpha\nu}\gamma^\mu\gamma_5 - {\sqrt2\over192}g^{\mu\nu}\gamma^\alpha\gamma_5 \right) \\ & \times \; [\bar c_a \gamma_\mu d_a] \; \gamma_\nu \gamma_5 \Sigma_c^{++} \\ &+ \left( {\sqrt2\over64}g^{\alpha\mu}\gamma^\nu - {\sqrt2\over64}g^{\alpha\nu}\gamma^\mu - {{\rm i}\sqrt2\over64}\epsilon^{\alpha \mu \nu \rho}\gamma_\rho\gamma_5 \right) \\ &\times \; [\bar c_a \gamma_\mu \gamma_5 d_a] \; \gamma_\nu \gamma_5 \Sigma_c^{++} + \cdots \, , \end{aligned} $

$ \begin{aligned}[b] \eta_7^{\alpha\beta} \rightarrow & \Big( {1\over36}g^{\alpha\mu}g^{\beta\nu} - {1\over144}g^{\alpha\beta}g^{\mu\nu} + {{\rm i}\over144}g^{\alpha\mu}\sigma^{\beta\nu} \\ & + {{\rm i}\over144}g^{\alpha\nu}\sigma^{\beta\mu} \Big) \; [\bar c_a \gamma_\mu u_a] \; \gamma_\nu \gamma_5 \Sigma_c^+ \\ &+ \Big( {\sqrt2\over36}g^{\alpha\mu}g^{\beta\nu} - {\sqrt2\over144}g^{\alpha\beta}g^{\mu\nu} + {{\rm i}\sqrt2\over144}g^{\alpha\mu}\sigma^{\beta\nu} \\ &+ {{\rm i}\sqrt2\over144}g^{\alpha\nu}\sigma^{\beta\mu} \Big) \; [\bar c_a \gamma_\mu d_a] \; \gamma_\nu \gamma_5 \Sigma_c^{++} + \cdots \, . \end{aligned} $

$ \begin{aligned}[b] \sqrt2 \xi_4^\alpha \rightarrow & \left( - {1\over8}g^{\alpha\mu} - {{\rm i}\over24}\sigma^{\alpha\mu} \right) \; [\bar c_a \gamma_\mu u_a] \; \Lambda_c^+ \\ &+ \left( {{\rm i}\over192}\sigma^{\alpha\sigma} \epsilon^{\mu\nu\rho\sigma} - {1\over64}\epsilon^{\alpha\mu\nu\rho} \right) \\ & \times \; [\bar c_a \sigma_{\mu\nu} u_a] \; \gamma_\rho \gamma_5 \Sigma_c^+ + \cdots \, , \end{aligned} $

$ \begin{aligned}[b] \sqrt2 \xi_5 \rightarrow & + {1\over2} \; [\bar c_a \gamma_5 u_a] \; \Lambda_c^+ + {1\over24} \; [\bar c_a \sigma_{\mu\nu} u_a] \; \sigma^{\mu\nu}\gamma_5 \Lambda_c^+ \\ & - {1\over16} [\bar c_a \gamma_\mu u_a] \gamma^\mu \gamma_5 \Sigma_c^+ + {1\over16} [\bar c_a \gamma_\mu \gamma_5 u_a] \gamma^\mu \Sigma_c^+ \\ & + \; \cdots \, , \\[-10pt]\end{aligned} $

$ \begin{aligned}[b] \sqrt2 \xi_6^\alpha \rightarrow & \left( - {{\rm i}\over8}g^{\alpha\mu}\gamma^\nu - {1\over16}\epsilon^{\alpha\mu\nu\rho}\gamma_\rho\gamma_5 \right) \; [\bar c_a \sigma_{\mu\nu} u_a] \; \Lambda_c^+ \\ &+ \left( {1\over48}g^{\alpha\mu}\gamma^\nu\gamma_5 + {1\over48}g^{\alpha\nu}\gamma^\mu\gamma_5 - {1\over96}g^{\mu\nu}\gamma^\alpha\gamma_5 \right) \\ & \times \; [\bar c_a \gamma_\mu u_a] \; \gamma_\nu \gamma_5 \Sigma_c^+ \\ &+ \left( {1\over32}g^{\alpha\mu}\gamma^\nu - {1\over32}g^{\alpha\nu}\gamma^\mu - {{\rm i}\over32}\epsilon^{\alpha \mu \nu \rho}\gamma_\rho\gamma_5 \right) \\ & \times \; [\bar c_a \gamma_\mu \gamma_5 u_a] \; \gamma_\nu \gamma_5 \Sigma_c^+ + \cdots \, , \end{aligned} $

$ \begin{aligned}[b] \sqrt2 \xi_7^{\alpha\beta} \rightarrow & \Big( {1\over18}g^{\alpha\mu}g^{\beta\nu} - {1\over72}g^{\alpha\beta}g^{\mu\nu} + {{\rm i}\over72}g^{\alpha\mu}\sigma^{\beta\nu} \\ & + {{\rm i}\over72}g^{\alpha\nu}\sigma^{\beta\mu} \Big) \; [\bar c_a \gamma_\mu u_a] \; \gamma_\nu \gamma_5 \Sigma_c^+ + \cdots \, . \end{aligned} $

$ \begin{aligned}[b] &\;\; \left[\delta^{ab} \bar c_a u_b\right] \; \left[\epsilon^{cde} u_c d_d c_e\right] \\ \underline{\underline {{\rm{color}}}} &\;\; {1\over3}\delta^{ae} \epsilon^{bcd} \; \bar c_au_b \; u_c d_d c_e + \cdots \\ \underline{\underline {{\rm{Fierz}}}} &\;\; {1\over3} \; \left[\delta^{ae}\bar c_a c_e\right] \; \left[\epsilon^{bcd} u_c d_d u_b\right] + \cdots . \end{aligned} $

$ \begin{aligned}[b] & \langle \bar D^0 \Sigma_c^{*+}; 3/2^-(q) \; |\; \eta_c(q_1)\; p(q_2) \rangle \\ \approx& {a_4}\; {\rm i} f_{\eta_c} f_p q_1^\mu \; \bar u^\alpha \left( - {1\over32} g_{\alpha\mu} \gamma_5 - {{\rm i}\over96} \sigma_{\alpha\mu} \gamma_5 \right) u_p \, , \end{aligned} $

$ \begin{aligned}[b] & \langle \bar D^0 \Sigma_c^{*+}; 3/2^-(q) | J/\psi(q_1,\epsilon_1)\; p(q_2) \rangle \\ \approx& {a_4}\; m_{J/\psi} f_{J/\psi} f_p \epsilon_1^\mu \; \bar u^\alpha \left( - {1\over32} g_{\alpha\mu} - {{\rm i}\over96} \sigma_{\alpha\mu} \right) u_p \\ &+ {a_4}\; {\rm i}f^T_{J/\psi} f_p \; \left(q_1^\mu \epsilon_1^\nu - q_1^\nu \epsilon_1^\mu \right) \\ &\times \bar u^\alpha \left( {{\rm i}\over48} g_{\alpha\mu}\gamma_\nu + {1\over96} \epsilon_{\alpha\mu\nu\rho} \gamma^\rho \gamma_5 \right) u_p \, . \end{aligned} $

$ \begin{aligned}[b] & \langle \bar D^0 \Sigma_c^{*+}; 3/2^-(q) \; |\; \bar D^{*0}(q_1,\epsilon_1)\; \Lambda_c^+(q_2) \rangle \\ \approx& {b_4}\; m_{D^*} f_{D^*} f_{\Lambda_c} \epsilon_1^\mu \; \bar u^\alpha \left( {1\over16}g_{\alpha\mu} + {{\rm i}\over48}\sigma_{\alpha\mu} \right) u_{\Lambda_c} \, , \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^0 \Sigma_c^{*+}; 3/2^- \rangle \to \eta_c p ) =& a_4^2 \; 42 \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^0 \Sigma_c^{*+}; 3/2^- \rangle \to J/\psi p ) =& a_4^2 \; 60 \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^0 \Sigma_c^{*+}; 3/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_4^2 \; 1.5 \times 10^{4}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^0 \Sigma_c^{*+}; 3/2^- \rangle \to J/\psi p )\big|_A =& a_4^2 \; 1.0 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^0 \Sigma_c^{*+}; 3/2^- \rangle \to J/\psi p )\big|_B =& a_4^2 \; 1.1 \times 10^{4}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|D^- \Sigma_c^{*++}; 3/2^- \rangle \to \eta_c p ) =& a_4^2 \; 84 \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^- \Sigma_c^{*++}; 3/2^- \rangle \to J/\psi p ) =& a_4^2 \; 120 \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^- \Sigma_c^{*++}; 3/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_4^2 \; 3.0 \times 10^{4}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to \eta_c p ) =& a_5^2 \; 3.3 \times 10^{5} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to J/\psi p ) =& a_5^2 \; 1.0 \times 10^{4} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to \chi_{c0} p ) =& a_5^2 \; 3.2 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to \chi_{c1} p ) =& a_5^2 \; 1.1 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to h_{c} p ) =& a_5^2 \; 220 \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to \bar D^{0} \Lambda_c^+ ) =& b_5^2 \; 3.5 \times 10^{5}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_5^2 \; 1.6 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to \bar D^{0} \Sigma_c^+ )=& b_5^2 \; 1.4 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to D^- \Sigma_c^{++} ) =& b_5^2 \; 2.9 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to \bar D^{*0} \Sigma_c^+ ) =& b_5^2 \; 3.3 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 1/2^- \rangle \to D^{*-} \Sigma_c^{++} ) =& b_5^2 \; 6.6 \times 10^{4}\; {\rm{GeV}}^7 \, , \end{aligned} $

$ \begin{aligned}[b] \Gamma(|D^{*-} \Sigma_c^{*++}; 1/2^- \rangle \to \eta_c p ) =& a_5^2 \; 6.5 \times 10^{5} \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 1/2^- \rangle \to J/\psi p ) =& a_5^2 \; 2.1 \times 10^{4} \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 1/2^- \rangle \to \chi_{c0} p ) =& a_5^2 \; 6.4 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 1/2^- \rangle \to \chi_{c1} p ) =& a_5^2 \; 2.1 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 1/2^- \rangle \to h_{c} p ) =& a_5^2 \; 450 \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 1/2^- \rangle \to \bar D^{0} \Lambda_c^+ ) =& b_5^2 \; 7.0 \times 10^{5}\; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 1/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_5^2 \; 3.1 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 1/2^- \rangle \to \bar D^{0} \Sigma_c^+ ) =& b_5^2 \; 2.9 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 1/2^- \rangle \to \bar D^{*0} \Sigma_c^+ ) =& b_5^2 \; 6.6 \times 10^{4}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 3/2^- \rangle \to \eta_c p ) =& a_6^2 \; 750 \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 3/2^- \rangle \to J/\psi p ) =& a_6^2 \; 1.2 \times 10^{5} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 3/2^- \rangle \to \chi_{c0} p ) =& a_6^2 \; 960 \; {\rm{GeV}}^7 \, ,\\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 3/2^- \rangle \to \bar D^{*0} \Lambda_c^+ )=& b_6^2 \; 4.5 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 3/2^- \rangle \to \bar D^{0} \Sigma_c^+ ) =& b_6^2 \; 36\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 3/2^- \rangle \to D^- \Sigma_c^{++} ) =& b_6^2 \; 71\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 3/2^- \rangle \to \bar D^{*0} \Sigma_c^+ ) =& b_6^2 \; 4.3 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*0} \Sigma_c^{*+}; 3/2^- \rangle \to D^{*-} \Sigma_c^{++} ) =& b_6^2 \; 8.7 \times 10^{4}\; {\rm{GeV}}^7 \, , \end{aligned} $

$ \begin{aligned}[b] \Gamma(|D^{*-} \Sigma_c^{*++}; 3/2^- \rangle \to \eta_c p ) =& a_6^2 \; 1.5 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 3/2^- \rangle \to J/\psi p ) =& a_6^2 \; 2.3 \times 10^{5} \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 3/2^- \rangle \to \chi_{c0} p ) =& a_6^2 \; 1.9 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 3/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_6^2 \; 9.1 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 3/2^- \rangle \to \bar D^{0} \Sigma_c^+ ) =& b_6^2 \; 71\; {\rm{GeV}}^7 \, , \\ \Gamma(|D^{*-} \Sigma_c^{*++}; 3/2^- \rangle \to \bar D^{*0} \Sigma_c^+ ) =& b_6^2 \; 8.7 \times 10^{4}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D \Sigma_c^{*}; 3/2^- \rangle \to \eta_c p ) =& a_4^2 \; 130 \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D \Sigma_c^{*}; 3/2^- \rangle \to J/\psi p ) =& a_4^2 \; 180 \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D \Sigma_c^{*}; 3/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_4^2 \; 4.5 \times 10^{4}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to \eta_c p ) =a_5^2 \; 9.8 \times 10^{5} \; {\rm{GeV}}^7 \, , \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to J/\psi p ) =& a_5^2 \; 3.1 \times 10^{4} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to \chi_{c0} p ) =& a_5^2 \; 9.5 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to \chi_{c1} p ) =& a_5^2 \; 3.2 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to h_{c} p ) =& a_5^2 \; 670 \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to \bar D^{0} \Lambda_c^+ ) =& b_5^2 \; 1.1 \times 10^{6}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_5^2 \; 4.7 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to \bar D^{0} \Sigma_c^+ ) =& b_5^2 \; 4.8 \times 10^{3}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to D^- \Sigma_c^{++} ) =& b_5^2 \; 9.6 \times 10^{3}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to \bar D^{*0} \Sigma_c^+ ) =& b_5^2 \; 1.1 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 1/2^- \rangle \to D^{*-} \Sigma_c^{++} ) =& b_5^2 \; 2.2 \times 10^{4}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^{*} \Sigma_c^{*}; 3/2^- \rangle \to \eta_c p ) =& a_6^2 \; 2.2 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 3/2^- \rangle \to J/\psi p ) =& a_6^2 \; 3.5 \times 10^{5} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 3/2^- \rangle \to \chi_{c0} p ) =& a_6^2 \; 2.9 \times 10^{3} \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 3/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_6^2 \; 1.4 \times 10^{5}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 3/2^- \rangle \to \bar D^{0} \Sigma_c^+ ) =& b_6^2 \; 12\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 3/2^- \rangle \to D^- \Sigma_c^{++} ) =& b_6^2 \; 24\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 3/2^- \rangle \to \bar D^{*0} \Sigma_c^+ ) =& b_6^2 \; 1.4 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c^{*}; 3/2^- \rangle \to D^{*-} \Sigma_c^{++} ) =& b_6^2 \; 2.9 \times 10^{4}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D \Sigma_c; 1/2^- \rangle \to \eta_c p ) =a_1^2 \; 3.2 \times 10^{5}\; {\rm{GeV}}^7 \, , \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D \Sigma_c; 1/2^- \rangle \to J/\psi p ) =& a_1^2 \; 8.5 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D \Sigma_c; 1/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_1^2 \; 5.9 \times 10^{4}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^{*} \Sigma_c; 1/2^- \rangle \to \eta_c p ) =& a_2^2 \; 1.8 \times 10^{5}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 1/2^- \rangle \to J/\psi p ) =& a_2^2 \; 5.1 \times 10^{5}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 1/2^- \rangle \to \chi_{c0} p ) =& a_2^2 \; 8.0 \times 10^{3}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 1/2^- \rangle \to \chi_{c1} p ) =& a_2^2 \; 200\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 1/2^- \rangle \to \bar D^{0} \Lambda_c^+ ) =& b_2^2 \; 1.7 \times 10^{6}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 1/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_2^2 \; 6.0 \times 10^{5}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 1/2^- \rangle \to \bar D^{0} \Sigma_c^+ ) =& b_2^2 \; 5.9 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 1/2^- \rangle \to D^- \Sigma_c^{++} ) =& b_2^2 \; 1.2 \times 10^{5}\; {\rm{GeV}}^7 \, . \end{aligned} $

$ \begin{aligned}[b] \Gamma(|\bar D^{*} \Sigma_c; 3/2^- \rangle \to \eta_c p ) =& a_3^2 \; 670\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 3/2^- \rangle \to J/\psi p ) =& a_3^2 \; 1.4 \times 10^{5}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 3/2^- \rangle \to \bar D^{*0} \Lambda_c^+ ) =& b_3^2 \; 4.6 \times 10^{4}\; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 3/2^- \rangle \to \bar D^{0} \Sigma_c^+ ) =& b_3^2 \; 1.4 \; {\rm{GeV}}^7 \, , \\ \Gamma(|\bar D^{*} \Sigma_c; 3/2^- \rangle \to D^- \Sigma_c^{++} ) =& b_3^2 \; 2.7 \; {\rm{GeV}}^7 \, . \end{aligned} $

$ {\cal{R}}_1(P_c) \equiv {{\cal{B}}\left(\Lambda_b^0 \rightarrow P_c K^- \right) \over {\cal{B}}\left(\Lambda_b^0 \rightarrow |\bar D^* \Sigma_c \rangle_{3/2^-} K^- \right)} \, , $

$ {\cal{R}}_2(P_c) \equiv { {\cal{B}}(\Lambda_b^0 \to P_c K^- \to J/\psi p K^-) \over {\cal{B}}(\Lambda_b^0 \to |\bar D^* \Sigma_c \rangle_{3/2^-} K^- \to J/\psi p K^-) } \, . $

$ \begin{aligned}[b] {\cal{R}}(P_c(4312)^+) =& 0.30 \pm 0.07 ^{+0.34}_{-0.09} \% \, , \\ {\cal{R}}(P_c(4440)^+) =& 1.11 \pm 0.33 ^{+0.22}_{-0.10} \% \, , \\ {\cal{R}}(P_c(4457)^+) =& 0.53 \pm 0.16 ^{+0.15}_{-0.13} \% \, , \end{aligned} $

$ \begin{aligned}[b] {{\cal{R}}(P_c(4312)^+) \over {\cal{R}}(P_c(4440)^+)} =& 0.27^{+0.32}_{-0.14} \, , \\ {{\cal{R}}(P_c(4457)^+) \over {\cal{R}}(P_c(4440)^+)} =& 0.48^{+0.25}_{-0.25} \, . \end{aligned} $

$ \begin{aligned}[b] {\cal{R}}_2(|\bar D \Sigma_c; 1/2^- \rangle) =& {{\cal{R}}_2(|\bar D \Sigma_c; 1/2^- \rangle) \over {\cal{R}}_2(|\bar D^* \Sigma_c; 3/2^- \rangle)} \approx 2.0 \, , \\ {\cal{R}}_2(|\bar D^* \Sigma_c; 1/2^- \rangle) =& {{\cal{R}}_2(|\bar D^* \Sigma_c; 1/2^- \rangle) \over {\cal{R}}_2(|\bar D^* \Sigma_c; 3/2^- \rangle)} \approx 0.25 \, , \end{aligned} $

$ \begin{aligned}[b] { {\cal{B}}(\Lambda_b^0 \to |\bar D^* \Sigma_c^* \rangle_{1/2^-} K^- \to J/\psi p K^-) \over {\cal{B}}(\Lambda^0_b \to J/\psi p K^-) } \approx 0.1{\text{%}} \, , \end{aligned} $

$ \begin{aligned}[b] { {\cal{B}}(\Lambda_b^0 \to |\bar D^* \Sigma_c^* \rangle_{3/2^-} K^- \to J/\psi p K^-) \over {\cal{B}}(\Lambda^0_b \to J/\psi p K^-) } \approx 0.2{\text{%}} \, . \end{aligned} $

$ \begin{aligned}[b]\\[-6pt] \rho_{1}(s) =& m_c \left( \rho^{\rm pert}_{1a}(s) + \rho^{{\langle\bar qq\rangle}}_{1a}(s) + \rho^{{\langle GG\rangle}}_{1a}(s)+ \rho^{{\langle\bar qGq\rangle}}_{1a}(s) + \rho^{{\langle\bar qq\rangle}^2}_{1a}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{1a}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{1a}(s) + \rho^{{\langle\bar qq\rangle}^3}_{1a}(s) \right) \\ &+ \not q \; \; \left( \rho^{\rm pert}_{1b}(s) + \rho^{{\langle\bar qq\rangle}}_{1b}(s) + \rho^{{\langle GG\rangle}}_{1b}(s)+ \rho^{{\langle\bar qGq\rangle}}_{1b}(s) + \rho^{{\langle\bar qq\rangle}^2}_{1b}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{1b}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{1b}(s) + \rho^{{\langle\bar qq\rangle}^3}_{1b}(s) \right) \, , \end{aligned}\tag{A1} $

$ \begin{aligned}[b] & \rho^{\rm pert}_{1a}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{13 (1 - \alpha - \beta)^3}{983040 \pi ^8 \alpha ^5 \beta ^4} \Bigg\} \, , \\ & \rho^{{\langle\bar qq\rangle}}_{1a}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-(1 - \alpha - \beta)^2}{768 \pi ^6 \alpha ^3 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle GG\rangle}}_{1a}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{13 (1 - \alpha - \beta)^3 \left(\alpha ^3+\beta ^3\right)}{1179648 \pi ^8 \alpha ^5 \beta ^4} \\ & \quad\quad\quad\quad + {\cal{F}}(s)^3 \times \frac{(\alpha +\beta -1) \left(80 \alpha ^3+\alpha ^2 (206 \beta -79)+\alpha \left(28 \beta ^2-27 \beta -1\right)-26 (\beta -1)^2 \beta \right)}{2359296 \pi ^8 \alpha ^5 \beta ^3} \Bigg\} \, , \end{aligned} $

$\begin{aligned}[b] & \rho^{{\langle\bar qGq\rangle}}_{1a}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{ (1 - \alpha - \beta) \left(14 \alpha ^2+2 \alpha (15 \beta -7)+(\beta -1) \beta \right)}{8192 \pi ^6 \alpha ^3 \beta ^3} \Bigg\} \, , \\ & \rho^{{\langle\bar qq\rangle}^2}_{1a}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-29}{1536 \pi ^4 \alpha ^2 \beta } \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{1a}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ {\cal{F}}(s) \times \frac{-6 \alpha -29 \beta}{3072 \pi ^4 \alpha ^2 \beta } \Bigg\} + {\cal{H}}(s) \times \frac{55 }{3072 \pi ^4 \alpha } \Bigg\} \, , \end{aligned} $

$\begin{aligned}[b] &\rho^{{\langle\bar qGq\rangle}^2}_{1a}(s)= {{\langle g_s\bar q\sigma Gq\rangle}^2 } \Bigg\{\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{52 \alpha ^2-75 \alpha +29}{12288 \pi ^4 \alpha } \Bigg\} + \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-13}{6144 \pi ^4 \alpha } \Bigg\}\Bigg\} \, , \\ &\rho^{{\langle\bar qq\rangle}^3}_{1a}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{13 }{288 \pi ^2} \Bigg\} \, , \\& \rho^{\rm pert}_{1b}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{13 (1 - \alpha - \beta)^3}{491520 \pi ^8 \alpha ^4 \beta ^4} \Bigg\} \, , \end{aligned} $

$\begin{aligned}[b] &\rho^{{\langle\bar qq\rangle}}_{1b}(s) ={m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-29 (1 - \alpha - \beta)^2}{12288 \pi ^6 \alpha ^2 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle GG\rangle}}_{1b}(s) ={{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{13 (1 - \alpha - \beta)^3 \left(\alpha ^3+\beta ^3\right)}{589824 \pi ^8 \alpha ^4 \beta ^4} \\ &\quad\quad\quad\quad + {\cal{F}}(s)^3 \times \frac{(\alpha +\beta -1) \left(167 \alpha ^2+\alpha (223 \beta -166)+80 \beta ^2-79 \beta -1\right)}{2359296 \pi ^8 \alpha ^3 \beta ^3} \Bigg\} \, , \\ & \rho^{{\langle\bar qGq\rangle}}_{1b}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{ (1 - \alpha - \beta) \left(110 \alpha ^2+\alpha (217 \beta -110)+3 (\beta -1) \beta \right)}{32768 \pi ^6 \alpha ^2 \beta ^3} \Bigg\} \, , \end{aligned} $

$\begin{aligned}[b]& \rho^{{\langle\bar qq\rangle}^2}_{1b}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-1}{96 \pi ^4 \alpha \beta } \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{1b}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ {\cal{F}}(s) \times \frac{-5 \alpha -15 \beta }{3072 \pi ^4 \alpha \beta } \Bigg\} + {\cal{H}}(s) \times \frac{31 }{3072 \pi ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}^2}_{1b}(s)= {{\langle g_s\bar q\sigma Gq\rangle}^2 } \Bigg\{ \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{30 \alpha ^2-40 \alpha +15}{12288 \pi ^4} \Bigg\} + \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-5 }{4096 \pi ^4} \Bigg\} \Bigg\} \, , \\ & \rho^{{\langle\bar qq\rangle}^3}_{1b}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{13 \alpha }{576 \pi ^2} \Bigg\} \, . \end{aligned} $

$ \begin{aligned}[b] \rho_{2}(s)=& m_c \left( \rho^{\rm pert}_{2a}(s) + \rho^{{\langle\bar qq\rangle}}_{2a}(s) + \rho^{{\langle GG\rangle}}_{2a}(s)+ \rho^{{\langle\bar qGq\rangle}}_{2a}(s) + \rho^{{\langle\bar qq\rangle}^2}_{2a}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{2a}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{2a}(s) + \rho^{{\langle\bar qq\rangle}^3}_{2a}(s) \right) \\ &+ \not q \; \; \left( \rho^{pert}_{2b}(s) + \rho^{{\langle\bar qq\rangle}}_{2b}(s) + \rho^{{\langle GG\rangle}}_{2b}(s)+ \rho^{{\langle\bar qGq\rangle}}_{2b}(s) + \rho^{{\langle\bar qq\rangle}^2}_{2b}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{2b}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{2b}(s) + \rho^{{\langle\bar qq\rangle}^3}_{2b}(s) \right) \, , \\ \end{aligned}\tag{A2} $

$ \begin{aligned}[b] &\rho^{\rm pert}_{2a}(s)= {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{(1 -\alpha -\beta)^3}{49152 \pi ^8 \alpha ^5 \beta ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}}_{2a}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-13 (1 -\alpha -\beta)^2}{3072 \pi ^6 \alpha ^3 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle GG\rangle}}_{2a}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{5 (1 - \alpha - \beta)^3 \left(\alpha ^3+\beta ^3\right)}{294912 \pi ^8 \alpha ^5 \beta ^4} \\ & \quad\quad\quad\quad + {\cal{F}}(s)^3 \times \frac{(1 - \alpha - \beta) \left(32 \alpha ^3-\alpha ^2 (16 \beta +31)+\alpha \left(-14 \beta ^2+15 \beta -1\right)+10 (\beta -1)^2 \beta \right)}{589824 \pi ^8 \alpha ^5 \beta ^3} \Bigg\} \, ,\\& \rho^{{\langle\bar qGq\rangle}}_{2a}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{ (1 - \alpha - \beta) (23 \alpha +2 \beta -2)}{4096 \pi ^6 \alpha ^3 \beta ^2} \Bigg\} \, , \\ & \rho^{{\langle\bar qq\rangle}^2}_{2a}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-5}{192 \pi ^4 \alpha ^2 \beta } \Bigg\} \, , \\ & \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{2a}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ {\cal{F}}(s) \times \frac{7 \alpha -20 \beta}{1536 \pi ^4 \alpha ^2 \beta } \Bigg\} + {\cal{H}}(s) \times \frac{11 }{512 \pi ^4 \alpha } \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}^2}_{2a}(s)= {{\langle g_s\bar q\sigma Gq\rangle}^2 } \Bigg\{ \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{26 \alpha ^2-53 \alpha +20}{6144 \pi ^4 \alpha } \Bigg\} + \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-13}{6144 \pi ^4 \alpha } \Bigg\} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] \rho^{{\langle\bar qq\rangle}^3}_{2a}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{23 }{144 \pi ^2} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{\rm pert}_{2b}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{23 (1 -\alpha -\beta)^3}{245760 \pi ^8 \alpha ^4 \beta ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}}_{2b}(s) ={m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-5(1 -\alpha -\beta)^2}{1536 \pi ^6 \alpha ^2 \beta ^3} \Bigg\} \, , \\ & \rho^{{\langle GG\rangle}}_{2b}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{23(1 - \alpha - \beta)^3 \left(\alpha ^3+\beta ^3\right)}{294912 \pi ^8 \alpha ^4 \beta ^4} \\ &\quad\quad\quad\quad + {\cal{F}}(s)^3 \times \frac{(1 - \alpha - \beta) \left(\alpha ^2-\alpha (11 \beta +1)-24 (\beta -1) \beta \right)}{196608 \pi ^8 \alpha ^3 \beta ^3} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] &\rho^{{\langle\bar qGq\rangle}}_{2b}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{13 (1 - \alpha - \beta)}{4096 \pi ^6 \alpha \beta ^2} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^2}_{2b}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-13}{384 \pi ^4 \alpha \beta } \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{2b}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ {\cal{F}}(s) \times \frac{ - 5 \alpha - 24 \beta}{1536 \pi ^4 \alpha \beta } \Bigg\} + {\cal{H}}(s) \times \frac{47 }{1536 \pi ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}^2}_{2b}(s)= {{\langle g_s\bar q\sigma Gq\rangle}^2 } \Bigg\{ \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{42 \alpha ^2-61 \alpha +24}{6144 \pi ^4} \Bigg\} +\int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-7}{2048 \pi ^4} \Bigg\} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^3}_{2b}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{5 \alpha }{144 \pi ^2} \Bigg\} \, . \end{aligned} $

$ \begin{aligned}[b] \rho_{3}(s) =& m_c \left( \rho^{\rm pert}_{3a}(s) + \rho^{{\langle\bar qq\rangle}}_{3a}(s) + \rho^{{\langle GG\rangle}}_{3a}(s)+ \rho^{{\langle\bar qGq\rangle}}_{3a}(s) + \rho^{{\langle\bar qq\rangle}^2}_{3a}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{3a}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{3a}(s) + \rho^{{\langle\bar qq\rangle}^3}_{3a}(s) \right) \\ &+ \not q \; \; \left( \rho^{\rm pert}_{3b}(s) + \rho^{{\langle\bar qq\rangle}}_{3b}(s) + \rho^{{\langle GG\rangle}}_{3b}(s)+ \rho^{{\langle\bar qGq\rangle}}_{3b}(s) + \rho^{{\langle\bar qq\rangle}^2}_{3b}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{3b}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{3b}(s) + \rho^{{\langle\bar qq\rangle}^3}_{3b}(s) \right) \, ,\ \end{aligned}\tag{A3} $

$ \begin{aligned}[b] & \rho^{\rm pert}_{3a}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{7 (1 - \alpha - \beta)^3 (\alpha +\beta +4)}{3932160 \pi ^8 \alpha ^5 \beta ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}}_{3a}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{- (1 - \alpha - \beta)^2 (8 \alpha +8 \beta +157)}{147456 \pi ^6 \alpha ^3 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle GG\rangle}}_{3a}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{7 (1 - \alpha - \beta)^3 (\alpha +\beta +4) \left(\alpha ^3+\beta ^3\right)}{4718592 \pi ^8 \alpha ^5 \beta ^4} \\ &\quad\quad\quad\quad + {\cal{F}}(s)^3 \times \Bigg( \frac{53 \alpha ^5+\alpha ^4 (530 \beta -464)+45 \alpha ^3 \left(22 \beta ^2-16 \beta +17\right)+70 \alpha ^2 \left(8 \beta ^3-3 \beta ^2-5\right)}{28311552 \pi ^8 \alpha ^5 \beta ^3} \\ &\quad\quad\quad\quad+ \frac{\alpha (\beta -1)^2 \left(5 \beta ^2+14 \beta -4\right)-42 (\beta -1)^3 \beta (\beta +4)}{28311552 \pi ^8 \alpha ^5 \beta ^3} \Bigg) \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qGq\rangle}}_{3a}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \; \frac{(1 - \alpha - \beta) \left(42 \alpha ^2+\alpha (50 \beta +311)+8 \beta ^2+14 \beta -22\right)}{196608 \pi ^6 \alpha ^3 \beta ^2} \Bigg\} \, , \\ &\rho^{{\langle\bar qq\rangle}^2}_{3a}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-4 \alpha -4 \beta -1}{384 \pi ^4 \alpha ^2 \beta } \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{3a}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s) \times \frac{4 \alpha ^2+\alpha (49-1128 \beta )-96 \beta (4 \beta +1)}{73728 \pi ^4 \alpha ^2 \beta } \Bigg\} \\ &\quad\quad\quad\;\;\quad\quad+ {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ {\cal{H}}(s) \times \frac{935 }{73728 \pi ^4 \alpha } \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}^2}_{3a}(s)= {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ \frac{\alpha -96 \beta}{73728 \pi ^4 \alpha } \Bigg\} + \frac{546 \alpha ^2-1079 \alpha +480}{294912 \pi ^4 \alpha } \Bigg\} \\ &\quad\quad\quad\;\;\quad+ {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-455 }{294912 \pi ^4 \alpha } \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^3}_{3a}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{9}{256 \pi ^2} \Bigg\} \, , \\ & \rho^{\rm pert}_{3b}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{9 (1 - \alpha - \beta)^3 (\alpha +\beta +2)}{1310720 \pi ^8 \alpha ^4 \beta ^4} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}}_{3b}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-5 (1 - \alpha - \beta)^2}{3072 \pi ^6 \alpha ^2 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle GG\rangle}}_{3b}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{3 (1 - \alpha - \beta)^3 (\alpha +\beta +2) \left(\alpha ^3+\beta ^3\right)}{524288 \pi ^8 \alpha ^4 \beta ^4} + {\cal{F}}(s)^3 \\ &\quad\quad\quad\quad\times \frac{(\alpha +\beta -1) \left(243 \alpha ^3+\alpha ^2 (673 \beta -834)+\alpha \left(761 \beta ^2-743 \beta +588\right)+331 \beta ^3+103 \beta ^2-437 \beta +3\right)}{28311552 \pi ^8 \alpha ^3 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}}_{3b}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{5 (1 - \alpha - \beta) (94 \alpha +3 \beta -3)}{196608 \pi ^6 \alpha ^2 \beta ^2} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^2}_{3b}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-5 (12 \alpha +12 \beta -1)}{6144 \pi ^4 \alpha \beta } \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] \rho_{4}(s) =& m_c \left( \rho^{pert}_{4a}(s) + \rho^{{\langle\bar qq\rangle}}_{4a}(s) + \rho^{{\langle GG\rangle}}_{4a}(s)+ \rho^{{\langle\bar qGq\rangle}}_{4a}(s) + \rho^{{\langle\bar qq\rangle}^2}_{4a}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{4a}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{4a}(s) + \rho^{{\langle\bar qq\rangle}^3}_{4a}(s) \right) \\ &+ \not q \; \; \left( \rho^{pert}_{4b}(s) + \rho^{{\langle\bar qq\rangle}}_{4b}(s) + \rho^{{\langle GG\rangle}}_{4b}(s)+ \rho^{{\langle\bar qGq\rangle}}_{4b}(s) + \rho^{{\langle\bar qq\rangle}^2}_{4b}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{4b}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{4b}(s) + \rho^{{\langle\bar qq\rangle}^3}_{4b}(s) \right) \, , \end{aligned}\tag{A4} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{3b}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s) \times \frac{- 24 \alpha ^2 - \alpha (1088 \beta -61) - 4 \beta (94 \beta -21)}{73728 \pi ^4 \alpha \beta } \Bigg\} \\ &\quad\quad\quad\quad\;\;\quad+ {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ {\cal{H}}(s) \times \frac{661 }{73728 \pi ^4} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] &\rho^{{\langle\bar qGq\rangle}^2}_{3b}(s)= {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ \frac{ - 3 \alpha - 47 \beta }{36864 \pi ^4} \Bigg\} + \frac{334 \alpha ^2-663 \alpha +292}{294912 \pi ^4} \Bigg\} \\ &\quad\;\;\quad\quad\quad+ {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-331}{294912 \pi ^4} \Bigg\} \, , \\ &\rho^{{\langle\bar qq\rangle}^3}_{3b}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{35 \alpha }{2304 \pi ^2} \Bigg\} \, .\\ & \rho^{\rm pert}_{4a}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{13 (1 - \alpha - \beta)^3 (\alpha +\beta +4)}{15728640 \pi ^8 \alpha ^5 \beta ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}}_{4a}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-(1 - \alpha -\beta)^2 (14 \alpha +14 \beta +43)}{147456 \pi ^6 \alpha ^3 \beta ^3} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle GG\rangle}}_{4a}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{13 (1 - \alpha - \beta)^3 (\alpha +\beta +4) \left(\alpha ^3+\beta ^3\right)}{18874368 \pi ^8 \alpha ^5 \beta ^4} \\ &\quad\quad\quad\quad + {\cal{F}}(s)^3 \times \Bigg\{ \frac{341 \alpha ^5+\alpha ^4 (598 \beta +220)-9 \alpha ^3 \left(10 \beta ^2-130 \beta +163\right)}{113246208 \pi ^8 \alpha ^5 \beta ^3} \\ & \quad\quad\quad\quad + \frac{\alpha ^2 \left(-688 \beta ^3+714 \beta ^2-936 \beta +910\right)-\alpha (\beta -1)^2 \left(419 \beta ^2+1152 \beta +4\right)-78 (\beta -1)^3 \beta (\beta +4)}{113246208 \pi ^8 \alpha ^5 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}}_{4a}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \\ &\quad\quad\quad\quad \times \frac{(1 - \alpha - \beta) \left(164 \alpha ^3+4 \alpha ^2 (137 \beta +74)+\alpha \left(382 \beta ^2+693 \beta -460\right)-2 \beta \left(\beta ^2+4 \beta -5\right)\right)}{1179648 \pi ^6 \alpha ^3 \beta ^3} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}^2}_{4a}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{- 4 \alpha - 4 \beta -85 }{24576 \pi ^4 \alpha ^2 \beta } \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{4a}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s) \times \frac{- 16 \alpha ^2 - \alpha (40 \beta +46) + \beta (4 \beta +85)}{147456 \pi ^4 \alpha ^2 \beta } \Bigg\} \\ &\quad\quad\;\quad\quad\quad+ {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ {\cal{H}}(s) \times \frac{509}{147456 \pi ^4 \alpha } \Bigg\} \, , \\ &\rho^{{\langle\bar qGq\rangle}^2}_{4a}(s)= {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ \frac{- 4 \alpha + \beta }{147456 \pi ^4 \alpha } \Bigg\} + \frac{468 \alpha ^2-317 \alpha -89}{589824 \pi ^4 \alpha } \Bigg\} \\&\quad\quad\quad\;\;\quad+ {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-121 }{294912 \pi ^4 \alpha } \Bigg\} \, , \\&\rho^{{\langle\bar qq\rangle}^3}_{4a}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{13 }{1536 \pi ^2} \Bigg\} \, , \\& \rho^{\rm pert}_{4b}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{13 (1 - \alpha - \beta)^3 (\alpha +\beta +2)}{7864320 \pi ^8 \alpha ^4 \beta ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}}_{4b}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{- (1 - \alpha - \beta)^2 (112 \alpha +112 \beta +155)}{589824 \pi ^6 \alpha ^2 \beta ^3} \Bigg\} \, , \\ & \rho^{{\langle GG\rangle}}_{4b}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{13 (1 - \alpha - \beta)^3 \left(\alpha ^4+\alpha ^3 (\beta +2)+\alpha \beta ^3+\beta ^3 (\beta +2)\right)}{9437184 \pi ^8 \alpha ^4 \beta ^4} \end{aligned} $

$ \begin{aligned}[b] & \quad\quad\quad\quad + {\cal{F}}(s)^3 \times \frac{5 (\alpha +\beta -1) \left(136 \alpha ^3+\alpha ^2 (176 \beta +29)+\alpha \left(40 \beta ^2+69 \beta -166\right)-80 \beta ^2+79 \beta +1\right)}{113246208 \pi ^8 \alpha ^3 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}}_{4b}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \\ &\quad\quad\quad\quad\times \frac{(1 - \alpha - \beta) \left(1312 \alpha ^3+\alpha ^2 (4400 \beta +358)+\alpha \left(3088 \beta ^2+1441 \beta -1670\right)-45 (\beta -1) \beta \right)}{4718592 \pi ^6 \alpha ^2 \beta ^3} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] &\rho^{{\langle\bar qq\rangle}^2}_{4b}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{- 2 \alpha - 2 \beta - 17}{6144 \pi ^4 \alpha \beta } \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{4b}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s) \times \frac{- 32 \alpha ^2 - \alpha (88 \beta +27) + 75 \beta}{147456 \pi ^4 \alpha \beta } \Bigg\} \\ &\quad\quad\quad\quad\;\;\quad+{{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ {\cal{H}}(s) \times \frac{437 }{147456 \pi ^4} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qGq\rangle}^2}_{4b}(s)={{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ \frac{-\alpha }{18432 \pi ^4} \Bigg\} + \frac{386 \alpha ^2-252 \alpha -75}{589824 \pi ^4} \Bigg\} \\ &\quad\;\;\quad\quad\quad+ {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-209 }{589824 \pi ^4}\Bigg\} \, , \\ & \rho^{{\langle\bar qq\rangle}^3}_{4b}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{65 \alpha }{9216 \pi ^2} \Bigg\} \, . \end{aligned} $

$ \begin{aligned}[b] \rho_{5}(s) =& m_c \left( \rho^{\rm pert}_{5a}(s) + \rho^{{\langle\bar qq\rangle}}_{5a}(s) + \rho^{{\langle GG\rangle}}_{5a}(s)+ \rho^{{\langle\bar qGq\rangle}}_{5a}(s) + \rho^{{\langle\bar qq\rangle}^2}_{5a}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{5a}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{5a}(s) + \rho^{{\langle\bar qq\rangle}^3}_{5a}(s) \right) \\ &+ \not q \; \; \left( \rho^{\rm pert}_{5b}(s) + \rho^{{\langle\bar qq\rangle}}_{5b}(s) + \rho^{{\langle GG\rangle}}_{5b}(s)+ \rho^{{\langle\bar qGq\rangle}}_{5b}(s) + \rho^{{\langle\bar qq\rangle}^2}_{5b}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{5b}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{5b}(s) + \rho^{{\langle\bar qq\rangle}^3}_{5b}(s) \right) \, , \\ \end{aligned}\tag{A5} $

$ \begin{aligned}[b] & \rho^{\rm pert}_{5a}(s) ={\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{3 (1 - \alpha - \beta)^3}{262144 \pi ^8 \alpha ^5 \beta ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}}_{5a}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-11(1-\alpha -\beta)^2}{16384 \pi ^6 \alpha ^3 \beta ^3} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle GG\rangle}}_{5a}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{5 (1 -\alpha -\beta)^3 \left(\alpha ^3+\beta ^3\right)}{524288 \pi ^8 \alpha ^5 \beta ^4} \\ &\quad\quad\quad\quad + {\cal{F}}(s)^3 \times \frac{ (1 - \alpha -\beta) \left(4 \alpha ^3+\alpha ^2 (92 \beta -5)+\alpha \left(94 \beta ^2-95 \beta +1\right)+30 (\beta -1)^2 \beta \right)}{3145728 \pi ^8 \alpha ^5 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}}_{5a}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{45 (1 - \alpha - \beta)}{65536 \pi ^6 \alpha ^2 \beta ^2} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^2}_{5a}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-5}{1024 \pi ^4 \alpha ^2 \beta } \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{5a}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ {\cal{F}}(s) \times \frac{-\alpha +20 \beta}{24576 \pi ^4 \alpha ^2 \beta } \Bigg\} + {\cal{H}}(s) \times \frac{97 }{24576 \pi ^4 \alpha } \Bigg\} \, , \\ & \rho^{{\langle\bar qGq\rangle}^2}_{5a}(s)={{\langle g_s\bar q\sigma Gq\rangle}^2 } \Bigg\{ \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{74 \alpha ^2-53 \alpha -20}{98304 \pi ^4 \alpha } \Bigg\} + \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-37}{98304 \pi ^4 \alpha } \Bigg\}\Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^3}_{5a}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{7}{256 \pi ^2} \Bigg\} \, , \\& \rho^{\rm pert}_{5b}(s) ={\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{21 (1-\alpha -\beta)^3}{1310720 \pi ^8 \alpha ^4 \beta ^4} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}}_{5b}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-5(1-\alpha -\beta)^2}{8192 \pi ^6 \alpha ^2 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle GG\rangle}}_{5b}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{7(1 - \alpha -\beta)^3 \left(\alpha ^3+\beta ^3\right)}{524288 \pi ^8 \alpha ^4 \beta ^4} \\ & \quad\quad\;\;\quad\quad+ {\cal{F}}(s)^3 \times \frac{(\alpha +\beta -1) \left(17 \alpha ^2-\alpha (179 \beta +13)-64 \beta ^2+68 \beta -4\right)}{9437184 \pi ^8 \alpha ^3 \beta ^3} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] &\rho^{{\langle\bar qGq\rangle}}_{5b}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{(35 \alpha -2 \beta +2) (1-\alpha -\beta)}{65536 \pi ^6 \alpha ^2 \beta ^2} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^2}_{5b}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-11}{2048 \pi ^4 \alpha \beta } \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{5b}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ {\cal{F}}(s) \times \frac{-13 \alpha +16 \beta }{24576 \pi ^4 \alpha \beta } \Bigg\} + {\cal{H}}(s) \times \frac{37 }{8192 \pi ^4} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qGq\rangle}^2}_{5b}(s)= {{\langle g_s\bar q\sigma Gq\rangle}^2 } \Bigg\{ \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{90 \alpha ^2-61 \alpha -16}{98304 \pi ^4} \Bigg\} + \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-15}{32768 \pi ^4} \Bigg\} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^3}_{5b}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{5 \alpha }{256 \pi ^2} \Bigg\} \, . \end{aligned} $

$ \begin{aligned}[b] \rho_{6}(s) =& m_c \left( \rho^{\rm pert}_{6a}(s) + \rho^{{\langle\bar qq\rangle}}_{6a}(s) + \rho^{{\langle GG\rangle}}_{6a}(s)+ \rho^{{\langle\bar qGq\rangle}}_{6a}(s) + \rho^{{\langle\bar qq\rangle}^2}_{6a}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{6a}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{6a}(s) + \rho^{{\langle\bar qq\rangle}^3}_{6a}(s) \right) \\ &+\not q \; \; \left( \rho^{\rm pert}_{6b}(s) + \rho^{{\langle\bar qq\rangle}}_{6b}(s) + \rho^{{\langle GG\rangle}}_{6b}(s)+ \rho^{{\langle\bar qGq\rangle}}_{6b}(s) + \rho^{{\langle\bar qq\rangle}^2}_{6b}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{6b}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{6b}(s) + \rho^{{\langle\bar qq\rangle}^3}_{6b}(s) \right) \, , \end{aligned}\tag{A6} $

$ \begin{aligned}[b]& \rho^{\rm pert}_{6a}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{7 (1 - \alpha - \beta)^3 (\alpha +\beta +4)}{15728640 \pi ^8 \alpha ^5 \beta ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}}_{6a}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-5(1 - \alpha - \beta)^2 (8 \alpha +8 \beta +31)}{196608 \pi ^6 \alpha ^3 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle GG\rangle}}_{6a}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{7(1 - \alpha - \beta)^3 (\alpha +\beta +4) \left(\alpha ^3+\beta ^3\right)}{18874368 \pi ^8 \alpha ^5 \beta ^4} \end{aligned} $

$ \begin{aligned}[b] &\quad\quad\quad\quad+ {\cal{F}}(s)^3 \times \Bigg\{ \frac{-391 \alpha^5 - 2 \alpha ^4 (839 \beta +326) - 3 \alpha ^3 \left(766 \beta ^2-96 \beta -819\right) - 42 (\beta -1)^3 \beta (\beta +4)}{113246208 \pi ^8 \alpha ^5 \beta ^3} \\ &\quad\quad\quad\quad+ \frac{-2 \alpha ^2 \left(584 \beta ^3-417 \beta ^2-864 \beta +697\right) - \alpha (\beta -1)^2 \left(199 \beta ^2+546 \beta +20\right)}{113246208 \pi ^8 \alpha ^5 \beta ^3} \Bigg\} \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}}_{6a}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{(1 - \alpha - \beta) (110 \alpha +110 \beta +243)}{262144 \pi ^6 \alpha ^2 \beta ^2} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^2}_{6a}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{4 \alpha +4 \beta -11}{2048 \pi ^4 \alpha ^2 \beta } \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] &\rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{6a}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s) \times \frac{28 \alpha ^2+\alpha (168 \beta +67)+8 \beta (11-4 \beta )}{98304 \pi ^4 \alpha ^2 \beta } \Bigg\} \\ &\quad\quad\quad\;\;\quad\quad+ {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ {\cal{H}}(s) \times \frac{301 }{98304 \pi ^4 \alpha } \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}^2}_{6a}(s)= {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ \frac{7 \alpha -8 \beta}{98304 \pi ^4 \alpha } \Bigg\} + \frac{342 \alpha ^2-381 \alpha -56}{393216 \pi ^4 \alpha } \Bigg\} \\&\quad\;\;\quad\quad\quad+ {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-133}{393216 \pi ^4 \alpha } \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}^3}_{6a}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{25 }{1024 \pi ^2} \Bigg\} \, , \\& \rho^{\rm pert}_{6b}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{5(1 - \alpha - \beta)^3 (\alpha +\beta +2)}{1048576 \pi ^8 \alpha ^4 \beta ^4} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}}_{6b}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-(1 - \alpha - \beta)^2 (16 \alpha +16 \beta +5)}{49152 \pi ^6 \alpha ^2 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle GG\rangle}}_{6b}(s) = {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{ m_c^2 {\cal{F}}(s)^2 \times \frac{25 (1 - \alpha - \beta)^3 \left(\alpha ^4+\alpha ^3 (\beta +2)+\alpha \beta ^3+\beta ^3 (\beta +2)\right)}{6291456 \pi ^8 \alpha ^4 \beta ^4} \\ &\quad\quad\quad\quad + {\cal{F}}(s)^3 \times \Bigg\{ \frac{(1 - \alpha - \beta) \left(261 \alpha ^3+\alpha ^2 (1799 \beta -738)+\alpha \left(1615 \beta ^2-1117 \beta +516\right)\right)}{113246208 \pi ^8 \alpha ^3 \beta ^3} \\ & \quad\quad\quad\quad + \frac{(1 - \alpha - \beta) \left(77 \beta ^3+761 \beta ^2-799 \beta -39\right)}{113246208 \pi ^8 \alpha ^3 \beta ^3} \Bigg\} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] &\rho^{{\langle\bar qGq\rangle}}_{6b}(s) = {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \; \frac{ (1 - \alpha - \beta) \left(472 \alpha ^2+\alpha (488 \beta -68)+16 \beta ^2-11 \beta -5\right)}{786432 \pi ^6 \alpha ^2 \beta ^2} \Bigg\} , \\& \rho^{{\langle\bar qq\rangle}^2}_{6b}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-20 \alpha -20 \beta -45}{8192 \pi ^4 \alpha \beta } \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{6b}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s) \times \frac{- 24 \alpha ^2 - \alpha (224 \beta +9) + 12 \beta (2 \beta +7)}{98304 \pi ^4 \alpha \beta } \Bigg\} \\ &\quad\quad\;\;\quad\quad\quad+ {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ {\cal{H}}(s) \times \frac{743 }{98304 \pi ^4} \Bigg\} \, , \\ & \rho^{{\langle\bar qGq\rangle}^2}_{6b}(s)={{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ \frac{ (\beta -\alpha )}{16384 \pi ^4} \Bigg\} + \frac{602 \alpha ^2-461 \alpha -108}{393216 \pi ^4} \Bigg\} \end{aligned} $

$ \begin{aligned}[b] &\quad\quad\quad\quad+ {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-353}{393216 \pi ^4} \Bigg\} \, , \\ &\rho^{{\langle\bar qq\rangle}^3}_{6b}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{35 \alpha }{9216 \pi ^2} \Bigg\} \, . \end{aligned} $

$ \begin{aligned}[b] \rho_{7}(s) =& m_c \left( \rho^{\rm pert}_{7a}(s) + \rho^{{\langle\bar qq\rangle}}_{7a}(s) + \rho^{{\langle GG\rangle}}_{7a}(s)+ \rho^{{\langle\bar qGq\rangle}}_{7a}(s) + \rho^{{\langle\bar qq\rangle}^2}_{7a}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{7a}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{7a}(s) + \rho^{{\langle\bar qq\rangle}^3}_{7a}(s) \right) \\ &+ \not q \; \; \left( \rho^{\rm pert}_{7b}(s) + \rho^{{\langle\bar qq\rangle}}_{7b}(s) + \rho^{{\langle GG\rangle}}_{7b}(s)+ \rho^{{\langle\bar qGq\rangle}}_{7b}(s) + \rho^{{\langle\bar qq\rangle}^2}_{7b}(s) + \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{7b}(s)+ \rho^{{\langle\bar qGq\rangle}^2}_{7b}(s) + \rho^{{\langle\bar qq\rangle}^3}_{7b}(s) \right) \, , \end{aligned}\tag{A7} $

$ \begin{aligned}[b] \rho^{\rm pert}_{7a}(s) =& {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{7 (1 - \alpha - \beta)^3 \left(3 \alpha ^2+2 \alpha (3 \beta +7)+3 \beta ^2+14 \beta +33\right)}{88473600 \pi ^8 \alpha ^5 \beta ^4} \Bigg\} \, , \\ \rho^{{\langle\bar qq\rangle}}_{7a}(s) =& {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{-(1 - \alpha - \beta)^2 (10 \alpha +10 \beta +23)}{73728 \pi ^6 \alpha ^3 \beta ^3} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] \rho^{{\langle GG\rangle}}_{7a}(s) =& {{\langle g_s^2GG\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta}\Bigg\{m_c^2 {\cal{F}}(s)^2 \times \Bigg\{ \frac{7(1 - \alpha - \beta)^3 \left(3 \alpha ^5+2 \alpha ^4 (3 \beta +7)+\alpha ^3 \left(3 \beta ^2+14 \beta +33\right)+3 \alpha ^2 \beta ^3\right)}{106168320 \pi ^8 \alpha ^5 \beta ^4} \\ & + \frac{7(1 - \alpha - \beta)^3 \left(2 \alpha \beta ^3 (3 \beta +7)+\beta ^3 \left(3 \beta ^2+14 \beta +33\right)\right)}{106168320 \pi ^8 \alpha ^5 \beta ^4} \Bigg\} \\ &+ {\cal{F}}(s)^3 \times \Bigg\{ \frac{(\alpha +\beta -1) \left(252 \alpha ^5-\alpha ^4 (324 \beta +1273)-\alpha ^3 \left(2136 \beta ^2+3771 \beta +3733\right)\right)}{1274019840 \pi ^8 \alpha ^5 \beta ^3} \\ & + \frac{(\alpha +\beta -1) \left(\alpha ^2 \left(2544 \beta ^3+5595 \beta ^2+818 \beta -4817\right)-84 (\beta -1)^2 \beta \left(3 \beta ^2+14 \beta +33\right)\right)}{1274019840 \pi ^8 \alpha ^5 \beta ^3} \\ & + \frac{(\alpha +\beta -1) \left(-\alpha \left(1236 \beta ^4+3769 \beta ^3+1717 \beta ^2-6785 \beta +63\right)\right)}{1274019840 \pi ^8 \alpha ^5 \beta ^3} \Bigg\} \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] \rho^{{\langle\bar qGq\rangle}}_{7a}(s) =& {m_c{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{-530 \alpha ^3 - 105 \alpha ^2 (10 \beta +1) - 102 \alpha \left(5 \beta ^2+\beta -6\right) + (\beta -1)^2 (10 \beta +23)}{1769472 \pi ^6 \alpha ^3 \beta ^2} \Bigg\} \, , \\ \rho^{{\langle\bar qq\rangle}^2}_{7a}(s)=& {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{- 10 \alpha - 10 \beta - 1}{3072 \pi ^4 \alpha ^2 \beta } \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}{\langle\bar qGq\rangle}}_{7a}(s)= {{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s) \times \frac{ 10 \alpha ^2-140 \alpha \beta +\alpha +3 \beta (10 \beta +1) }{55296 \pi ^4 \alpha ^2 \beta } \Bigg\} \\ &\quad\;\quad\quad\quad\quad+{{\langle\bar qq\rangle}{\langle g_s\bar q\sigma Gq\rangle} } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ {\cal{H}}(s) \times \frac{11 }{3072 \pi ^4 \alpha } \Bigg\} \, , \\& \rho^{{\langle\bar qGq\rangle}^2}_{7a}(s)={{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta \Bigg\{ \frac{5 \alpha + 15 \beta}{110592 \pi ^4 \alpha } \Bigg\} + \frac{108 \alpha ^2-86 \alpha -33}{221184 \pi ^4 \alpha } \Bigg\} \\ &\quad\quad\;\;\quad\quad+ {{\langle g_s\bar q\sigma Gq\rangle}^2 } \int^{1}_{0}{\rm d}\alpha \Bigg\{ m_c^2 \delta\left(s - {m_c^2 \over \alpha(1-\alpha)}\right) \times \frac{-11}{24576 \pi ^4 \alpha } \Bigg\} \, , \end{aligned} $

$ \begin{aligned}[b] & \rho^{{\langle\bar qq\rangle}^3}_{7a}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{7 }{864 \pi ^2} \Bigg\} \, , \\ & \rho^{\rm pert}_{7b}(s) = {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^5 \times \frac{7 (1 - \alpha - \beta)^3 \left(6 \alpha ^2+\alpha (12 \beta +13)+6 \beta ^2+13 \beta +21\right)}{58982400 \pi ^8 \alpha ^4 \beta ^4} \Bigg\} \, , \\ &\rho^{{\langle\bar qq\rangle}}_{7b}(s) = {m_c {\langle\bar qq\rangle} } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^3 \times \frac{- (1 - \alpha - \beta)^2 (13 \alpha +13 \beta +20)}{73728 \pi ^6 \alpha ^2 \beta ^3} \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^2}_{7b}(s)= {{\langle\bar qq\rangle}^2 } {\int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha\int^{\beta_{\rm max}}_{\beta_{\rm min}}{\rm d}\beta} \Bigg\{ {\cal{F}}(s)^2 \times \frac{- 13 \alpha - 13 \beta + 2}{3072 \pi ^4 \alpha \beta } \Bigg\} \, , \\& \rho^{{\langle\bar qq\rangle}^3}_{7b}(s)= {m_c{\langle\bar qq\rangle}^3 } \int^{\alpha_{\rm max}}_{\alpha_{\rm min}}{\rm d}\alpha \Bigg\{ \frac{35 \alpha}{5184 \pi ^2} \Bigg\} \, . \end{aligned} $

$ {\cal{L}}^A_{\psi p} = g_A\; \bar P_c^\alpha \left( t_1 g_{\alpha\mu} + t_2 \sigma_{\alpha\mu} \right) N\; \psi^\mu \, , \tag{B1} $

${\cal{L}}^B_{\psi p} = g_B\; \bar P_c^\alpha \left( t_3 g_{\alpha\mu}\gamma_\nu + t_4 \epsilon_{\alpha\mu\nu\rho} \gamma^\rho \gamma_5 \right) N\; \partial^\mu\psi^\nu \, , \tag{B2} $

$ {\cal{L}}^C_{\psi p} = g_C\; \bar P_c \gamma_\mu \gamma_5 N\; \psi^\mu \, ,\tag{B3} $

$ {\cal{L}}^D_{\psi p}= g_D\; \bar P_c \sigma_{\mu\nu} \gamma_5 N\; \partial^\mu\psi^\nu \, . \tag{B4} $

$ {\cal{L}}^E_{\eta_c p}=g_E\; \bar P_c N\; \eta_c \, , \tag{B5} $

$ {\cal{L}}^F_{\eta_c p} = g_F\; \bar P_c \gamma_\mu N\; \partial^\mu\eta_c \, . \tag{B6}$