Appendix
The QCD spectral densities \(\rho ^{11,1}_{sss}(s)\), \(\widetilde{\rho }^{11,0}_{sss}(s)\), \(\rho ^{11,1}_{uss}(s)\), \(\widetilde{\rho }^{11,0}_{uss}(s)\), \(\rho ^{11,1}_{uus}(s)\), \(\widetilde{\rho }^{11,0}_{uus}(s)\), \(\rho ^{11,1}_{uuu}(s)\), \(\widetilde{\rho }^{11,0}_{uuu}(s)\), \(\rho ^{10,1}_{sss}(s)\), \(\widetilde{\rho }^{10,0}_{sss}(s)\), \(\rho ^{10,1}_{uss}(s)\), \(\widetilde{\rho }^{10,0}_{uss}(s)\), \(\rho ^{10,1}_{uus}(s)\), \(\widetilde{\rho }^{10,0}_{uus}(s)\), \(\rho ^{10,1}_{uuu}(s)\), and \(\widetilde{\rho }^{10,0}_{uuu}(s)\) of the pentaquark states,
$$\begin{aligned}&\rho ^{11,1}_{sss}(s)=\frac{1}{61440\pi ^8}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^4(s-\overline{m}_c^2)^4(8s-3\overline{m}_c^2 ) \nonumber \\&\quad +\frac{m_s m_c}{12288\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad - \frac{m_c\langle \bar{s}s\rangle }{768\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad + \frac{m_s\langle \bar{s}s\rangle }{128\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^2(s-\overline{m}_c^2)^2 (2s-\overline{m}_c^2 ) \nonumber \\&\quad + \frac{3m_c\langle \bar{s}g_s\sigma Gs\rangle }{1024\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \\&\quad + \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{512\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad - \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{1024\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad +\frac{\langle \bar{s}s\rangle ^2}{48\pi ^4}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)(s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad +\frac{5 m_s m_c\langle \bar{s}s\rangle ^2}{48\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(s-\overline{m}_c^2 ) \nonumber \\&\quad -\frac{5\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{192\pi ^4}\int \mathrm{d}y\mathrm{d}z \,yz (4s-3\overline{m}_c^2)\nonumber \\&\quad +\frac{\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{192\pi ^4}\int \mathrm{d}y\mathrm{d}z \,(y+z)(1-y-z) (4s-3\overline{m}_c^2)\nonumber \\&\quad -\frac{121m_s m_c \langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{2304\pi ^4}\int \mathrm{d}y \nonumber \\&\quad +\frac{m_s m_c\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{48\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) -\frac{m_c\langle \bar{s}s\rangle ^3}{18\pi ^2}\int \mathrm{d}y\nonumber \\&\quad +\frac{m_s\langle \bar{s}s\rangle ^3}{12\pi ^2} \int \mathrm{d}y \,y(1-y) \left[ 1+\frac{s}{3}\delta \left( s-\widetilde{m}_c^2 \right) \right] \nonumber \end{aligned}$$
$$\begin{aligned}&\quad -\frac{3\langle \bar{s}g_s\sigma Gs\rangle ^2}{512\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad +\frac{3\langle \bar{s}g_s\sigma Gs\rangle ^2}{256\pi ^4}\int \mathrm{d}y \, y(1-y)\left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{192\pi ^4}\int \mathrm{d}y \left( \frac{1-y}{y}+\frac{y}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad +\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{72\pi ^4}\int \mathrm{d}y \, y(1-y)\left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{2304\pi ^4}\int \mathrm{d}y \, \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ), \end{aligned}$$
(55)
$$\begin{aligned}&\widetilde{\rho }^{11,0}_{sss}(s)=\frac{1}{122880\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^4(s-\overline{m}_c^2)^4\nonumber \\&\quad \quad \times (7s-2\overline{m}_c^2 ) +\frac{m_s m_c}{6144\pi ^8}\int \mathrm{d}y\mathrm{d}z \,(1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad - \frac{m_c\langle \bar{s}s\rangle }{384\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad + \frac{m_s\langle \bar{s}s\rangle }{768\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2(s-\overline{m}_c^2)^2 (5s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{3m_c\langle \bar{s}g_s\sigma Gs\rangle }{512\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad - \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{256\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2 (s-\overline{m}_c^2 ) (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{512\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 ) (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{\langle \bar{s}s\rangle ^2}{48\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)(s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{5 m_s m_c\langle \bar{s}s\rangle ^2}{24\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (s-\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{5\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{384\pi ^4}\int \mathrm{d}y\mathrm{d}z \,(y+z) (3s-2\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{96\pi ^4}\int \mathrm{d}y\mathrm{d}z \,(1-y-z) (3s-2\overline{m}_c^2)\nonumber \\&\quad \quad -\frac{121m_s m_c\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{1152\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{48\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) \nonumber \\&\quad \quad -\frac{m_c\langle \bar{s}s\rangle ^3}{9\pi ^2}\int \mathrm{d}y +\frac{m_s\langle \bar{s}s\rangle ^3}{36\pi ^2}\int \mathrm{d}y \, \left[ 1+\frac{s}{2}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{\langle \bar{s}g_s\sigma Gs\rangle ^2}{256\pi ^4}\int \mathrm{d}y\mathrm{d}z \, \left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{192\pi ^4}\int \mathrm{d}y \, \left( \frac{1}{y}+\frac{1}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{31m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{2304\pi ^4}\int \mathrm{d}y \, \left( 1+\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ) , \end{aligned}$$
(56)
$$\begin{aligned}&\rho ^{11,1}_{uss}(s) =\frac{1}{61440\pi ^8}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^4(s-\overline{m}_c^2)^4(8s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c}{18432\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad - \frac{m_c\left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{2304\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad + \frac{m_s\left[ 2\langle \bar{s}s\rangle -\langle \bar{q}q\rangle \right] }{192\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^2\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^2 (2s-\overline{m}_c^2 ) + \frac{m_c\left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{1024\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad + \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{384\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{1536\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad - \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{3072\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{\langle \bar{s}s\rangle \left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)(s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{ m_s m_c\langle \bar{s}s\rangle \left[ 11\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(s-\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{5\left[ \langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{576\pi ^4}\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,yz (4s-3\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{576\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z)(1-y-z) (4s-3\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{m_s m_c\left[ \langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) \nonumber \\&\quad \quad -\frac{m_s m_c \left[ 34\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +33\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle -5\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\int \mathrm{d}y \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{m_s m_c \langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{2304\pi ^4}\int \mathrm{d}y-\frac{m_c\langle \bar{q}q\rangle \langle \bar{s}s\rangle ^2}{18\pi ^2}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{m_s\langle \bar{q}q\rangle \langle \bar{s}s\rangle ^2}{18\pi ^2}\int \mathrm{d}y \,y(1-y) \left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{\langle \bar{s}g_s\sigma Gs\rangle \left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{512\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{\langle \bar{s}g_s\sigma Gs\rangle \left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{256\pi ^4}\end{aligned}$$
$$\begin{aligned}&\quad \quad \times \int \mathrm{d}y \, y(1-y)\left[ 1+\frac{s}{3}\delta \left( s-\widetilde{m}_c^2 \right) \right] \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{288\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( \frac{1-y}{y}+\frac{y}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ 17\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{6912\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ), \end{aligned}$$
(57)
$$\begin{aligned}&\widetilde{\rho }^{11,0}_{uss}(s)=\frac{1}{122880\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^4(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad \times (7s-2\overline{m}_c^2 )+\frac{m_s m_c}{9216\pi ^8}\int \mathrm{d}y\mathrm{d}z \,(1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad - \frac{m_c\left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{1152\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad + \frac{m_s\left[ 2\langle \bar{s}s\rangle -\langle \bar{q}q\rangle \right] }{1152\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^2 (5s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_c\left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{512\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad - \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{768\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2 \nonumber \\&\quad \quad \times (s-\overline{m}_c^2 ) (2s-\overline{m}_c^2 ) + \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{384\pi ^6}\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z) (1-y-z) (s-\overline{m}_c^2 ) (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{1536\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) \nonumber \\&\quad \quad \times (s-\overline{m}_c^2 ) (2s-\overline{m}_c^2 ) +\frac{\langle \bar{s}s\rangle \left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{144\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)(s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{ m_s m_c\langle \bar{s}s\rangle \left[ 11 \langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{72\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (s-\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{5\left[ \langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{1152\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z) (3s-2\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{\left[ \langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{288\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(1-y-z) (3s-2\overline{m}_c^2)\nonumber \\ \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{m_s m_c\left[ \langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) \nonumber \\&\quad \quad -\frac{m_s m_c\left[ 34\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +33\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle -5\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{864\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{1152\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad -\frac{m_c\langle \bar{q}q\rangle \langle \bar{s}s\rangle ^2}{9\pi ^2}\int \mathrm{d}y +\frac{m_s\langle \bar{q}q\rangle \langle \bar{s}s\rangle ^2}{54\pi ^2}\int \mathrm{d}y \, \left[ 1+\frac{s}{2}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{\langle \bar{s}g_s\sigma Gs\rangle \left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{768\pi ^4}\int \mathrm{d}y\mathrm{d}z \, \left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{288\pi ^4}\int \mathrm{d}y \, \left( \frac{1}{y}+\frac{1}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ 17\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( 1\!+\!\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ) -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{6912\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( 1+\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ) , \end{aligned}$$
(58)
$$\begin{aligned}&\rho ^{11,1}_{uus}(s)=\frac{1}{61440\pi ^8}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^4(s-\overline{m}_c^2)^4(8s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c}{36864\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad - \frac{m_c\left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{2304\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad + \frac{m_s\left[ 3\langle \bar{s}s\rangle -2\langle \bar{q}q\rangle \right] }{384\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^2\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^2 (2s-\overline{m}_c^2 ) + \frac{m_c\left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{1024\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad - \frac{m_s\left[ 3\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{12288\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{384\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{q}g_s\sigma Gq\rangle }{1536\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \,\quad +\frac{\langle \bar{q}q\rangle \left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)(s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{ m_s m_c\langle \bar{q}q\rangle \left[ 6\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(s-\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{5\left[ \langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{576\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,yz (4s-3\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{\langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{576\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z)(1-y-z) (4s-3\overline{m}_c^2)\nonumber \\ \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{m_s m_c\langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) \nonumber \\&\quad \quad -\frac{m_s m_c \left[ 36\langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle -2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle -3\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{1728\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{m_s m_c \langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{2304\pi ^4}\int \mathrm{d}y -\frac{m_c\langle \bar{q}q\rangle ^2\langle \bar{s}s\rangle }{18\pi ^2}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{m_s\langle \bar{q}q\rangle ^2\langle \bar{s}s\rangle }{36\pi ^2}\int \mathrm{d}y \,y(1-y) \left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{\langle \bar{q}g_s\sigma Gq\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{512\pi ^4}\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{\langle \bar{q}g_s\sigma Gq\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{256\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, y(1-y)\left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle ^2}{576\pi ^4}\int \mathrm{d}y \left( \frac{1-y}{y}+\frac{y}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \left[ 9\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{6912\pi ^4}\int \mathrm{d}y \, \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ), \end{aligned}$$
(59)
$$\begin{aligned}&\widetilde{\rho }^{11,0}_{uus}(s)=\frac{1}{122880\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^4(s-\overline{m}_c^2)^4(7s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c}{18432\pi ^8}\int \mathrm{d}y\mathrm{d}z \,(1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad - \frac{m_c\left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{1152\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2 (s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad + \frac{m_s\left[ 3\langle \bar{s}s\rangle -2\langle \bar{q}q\rangle \right] }{2304\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^2 (5s-2\overline{m}_c^2 ) + \frac{m_c\left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{512\pi ^6}\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (1-y-z) (s-\overline{m}_c^2 )^2 - \frac{m_s\left[ 3\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{3072\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2 (s-\overline{m}_c^2 ) (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{384\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 ) (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{q}g_s\sigma Gq\rangle }{1536\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)(s-\overline{m}_c^2 ) (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{\langle \bar{q}q\rangle \left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)\nonumber \\&\quad \quad \times (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{ m_s m_c\langle \bar{q}q\rangle \left[ 6\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{72\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (s-\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{5\left[ \langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{1152\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z) (3s-2\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{\langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{288\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(1-y-z) (3s-2\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{1152\pi ^4}\int \mathrm{d}y \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{m_s m_c\left[ 36\langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle -2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle -3\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{864\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad -\frac{m_c\langle \bar{q}q\rangle ^2\langle \bar{s}s\rangle }{9\pi ^2}\int \mathrm{d}y +\frac{m_s\langle \bar{q}q\rangle ^2\langle \bar{s}s\rangle }{108\pi ^2}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left[ 1+\frac{s}{2}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{\langle \bar{q}g_s\sigma Gq\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{768\pi ^4}\int \mathrm{d}y\mathrm{d}z \, \left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle ^2}{576\pi ^4}\int \mathrm{d}y \, \left( \frac{1}{y}+\frac{1}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \left[ 9\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( 1+\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{6912\pi ^4}\int \mathrm{d}y \, \left( 1+\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ) , \end{aligned}$$
(60)
$$\begin{aligned} \rho ^{11,1}_{uuu}(s)= & {} \rho ^{11,1}_{sss}(s)\mid _{m_s \rightarrow 0,\,\,\langle \bar{s}s\rangle \rightarrow \langle \bar{q}q\rangle ,\,\,\langle \bar{s}g_s\sigma Gs\rangle \rightarrow \langle \bar{q}g_s\sigma Gq\rangle } , \nonumber \\\end{aligned}$$
(61)
$$\begin{aligned} \widetilde{\rho }^{11,0}_{uuu}(s)= & {} \widetilde{\rho }^{11,0}_{sss}(s)\mid _{m_s \rightarrow 0,\,\,\langle \bar{s}s\rangle \rightarrow \langle \bar{q}q\rangle ,\,\,\langle \bar{s}g_s\sigma Gs\rangle \rightarrow \langle \bar{q}g_s\sigma Gq\rangle } ,\nonumber \\ \end{aligned}$$
(62)
$$\begin{aligned}&\rho ^{10,1}_{sss}(s)=\frac{1}{122880\pi ^8}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^4(s-\overline{m}_c^2)^4(8s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{1}{153600\pi ^8}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^5\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^3(18s^2-16s\overline{m}_c^2+3\overline{m}_c^4 ) \nonumber \\&\quad \quad +\frac{m_s m_c}{12288\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad +\frac{m_s m_c}{49152\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^4(s-\overline{m}_c^2)^3 (7s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{m_c\langle \bar{s}s\rangle }{768\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad - \frac{m_c\langle \bar{s}s\rangle }{768\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2)^2(2s-\overline{m}_c^2) \nonumber \\&\quad \quad - \frac{m_s\langle \bar{s}s\rangle }{256\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^2(s-\overline{m}_c^2)^2 (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{s}s\rangle }{128\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^3(s-\overline{m}_c^2)\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times (7s^2-8s\overline{m}_c^2+2\overline{m}_c^4 ) \nonumber \\&\quad \quad + \frac{11m_c \langle \bar{s}g_s\sigma Gs\rangle }{4096\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad + \frac{11m_c \langle \bar{s}g_s\sigma Gs\rangle }{8192\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 \nonumber \\&\quad \quad \times (s-\overline{m}_c^2 ) (5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{7m_c \langle \bar{s}g_s\sigma Gs\rangle }{8192\pi ^6}\int \mathrm{d}y\mathrm{d}z \, \left( \frac{z}{y}+\frac{y}{z}\right) (1-y-z)^2 (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad - \frac{m_c \langle \bar{s}g_s\sigma Gs\rangle }{4096\pi ^6}\int \mathrm{d}y\mathrm{d}z \, \left( \frac{z}{y}+\frac{y}{z}\right) (1-y-z)^3\nonumber \\&\quad \quad \times (s-\overline{m}_c^2 ) (5s-3\overline{m}_c^2 ) \end{aligned}$$
$$\begin{aligned}&\quad \quad + \frac{11m_s\langle \bar{s}g_s\sigma Gs\rangle }{2048\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{256\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^2 (15s^2-20s\overline{m}_c^2+6\overline{m}_c^4 ) \nonumber \\&\quad \quad + \frac{3m_s\langle \bar{s}g_s\sigma Gs\rangle }{8192\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 \nonumber \\&\quad \quad \times (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{\langle \bar{s}s\rangle ^2}{48\pi ^4}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)(s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{ m_s m_c\langle \bar{s}s\rangle ^2}{24\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(s-\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{ m_s m_c\langle \bar{s}s\rangle ^2}{48\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)(4s-3\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{19\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{768\pi ^4}\int \mathrm{d}y\mathrm{d}z \,yz (4s-3\overline{m}_c^2)\nonumber \\&\quad \quad -\frac{\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{512\pi ^4}\int \mathrm{d}y\mathrm{d}z \,(y+z)(1-y-z) (4s-3\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{89m_s m_c \langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{3072\pi ^4}\int \mathrm{d}y\mathrm{d}z \,(y+z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{181m_s m_c \langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{9216\pi ^4}\int \mathrm{d}y +\frac{3m_s m_c\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{512\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) -\frac{3m_s m_c\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{256\pi ^4}\nonumber \\&\quad \quad \int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) (1-y-z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{m_c\langle \bar{s}s\rangle ^3}{36\pi ^2}\int \mathrm{d}y +\frac{m_s\langle \bar{s}s\rangle ^3}{12\pi ^2}\int \mathrm{d}y \,y(1-y) \left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{43\langle \bar{s}g_s\sigma Gs\rangle ^2}{18432\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{11\langle \bar{s}g_s\sigma Gs\rangle ^2}{1024\pi ^4}\int \mathrm{d}y \, y(1-y)\left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{768\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) \left( 1+\frac{s}{2M^2}\right) \delta (s-\overline{m}_c^2 )\nonumber \\&\quad \quad -\frac{17m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{9216\pi ^4}\int \mathrm{d}y \, \left( \frac{1-y}{y}+\frac{y}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{17m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{4608\pi ^4}\int \mathrm{d}y \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ) , \end{aligned}$$
(63)
$$\begin{aligned}&\widetilde{\rho }^{10,0}_{sss}(s)=\frac{1}{122880\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^4\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^4(7s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{1}{614400\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^5(s-\overline{m}_c^2)^3\nonumber \\&\quad \quad \times (28s^2-21s\overline{m}_c^2+3\overline{m}_c^4 ) \nonumber \\&\quad \quad +\frac{m_s m_c}{3072\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad -\frac{m_s m_c}{12288\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^4(s-\overline{m}_c^2)^3 (3s-\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{m_c\langle \bar{s}s\rangle }{192\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad + \frac{m_c\langle \bar{s}s\rangle }{1152\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^3(s-\overline{m}_c^2)^2(5s-2\overline{m}_c^2) \nonumber \\&\quad \quad - \frac{m_s\langle \bar{s}s\rangle }{768\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2(s-\overline{m}_c^2)^2 (5s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{m_s\langle \bar{s}s\rangle }{256\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2)\nonumber \\&\quad \quad \times (5s^2-5s\overline{m}_c^2+\overline{m}_c^4 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad + \frac{11m_c\langle \bar{s}g_s\sigma Gs\rangle }{1024\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad - \frac{11m_c\langle \bar{s}g_s\sigma Gs\rangle }{2048\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2 (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{7m_c\langle \bar{s}g_s\sigma Gs\rangle }{4096\pi ^6}\int \mathrm{d}y\mathrm{d}z \, \left( \frac{1}{y}+\frac{1}{z}\right) (1-y-z)^2 (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad + \frac{m_c\langle \bar{s}g_s\sigma Gs\rangle }{2048\pi ^6}\int \mathrm{d}y\mathrm{d}z \, \left( \frac{1}{y}+\frac{1}{z}\right) (1-y-z)^3\nonumber \\&\quad \quad \times (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{512\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 \nonumber \\&\quad \quad \times (10s^2-12s\overline{m}_c^2+3\overline{m}_c^4 ) \nonumber \\&\quad \quad + \frac{11m_s\langle \bar{s}g_s\sigma Gs\rangle }{1024\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) \nonumber \\&\quad \quad \times (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{\langle \bar{s}s\rangle ^2}{24\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)(s-\overline{m}_c^2)(2s-\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{ m_s m_c\langle \bar{s}s\rangle ^2}{6\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{ m_s m_c\langle \bar{s}s\rangle ^2}{24\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)(3s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{19\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{768\pi ^4}\int \mathrm{d}y\mathrm{d}z \,(y+z) (3s-2\overline{m}_c^2)\nonumber \\&\quad \quad -\frac{89m_s m_c \langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{2304\pi ^4}\int \mathrm{d}y\mathrm{d}z \left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{199m_s m_c \langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{2304\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{3m_s m_c\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{256\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{128\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) (1-y-z)\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times \left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 ) \right] \nonumber \\&\quad \quad -\frac{m_c\langle \bar{s}s\rangle ^3}{9\pi ^2}\int \mathrm{d}y +\frac{m_s\langle \bar{s}s\rangle ^3}{18\pi ^2}\int \mathrm{d}y \left[ 1+\frac{s}{2}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{11\langle \bar{s}g_s\sigma Gs\rangle ^2}{1536\pi ^4}\int \mathrm{d}y \left[ 1+\frac{s}{2}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{1536\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) \left( 1+\frac{s}{M^2} \right) \delta (s-\overline{m}_c^2 )\nonumber \\&\quad \quad -\frac{17m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{4608\pi ^4}\int \mathrm{d}y \left( \frac{1}{y}+\frac{1}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{125m_s m_c\langle \bar{s}g_s\sigma Gs\rangle ^2}{9216\pi ^4}\int \mathrm{d}y \left( 1+\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ) , \end{aligned}$$
(64)
$$\begin{aligned}&\rho ^{10,1}_{uss}(s)=\frac{1}{122880\pi ^8}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^4(s-\overline{m}_c^2)^4(8s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{1}{153600\pi ^8}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^5\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^3(18s^2-16s\overline{m}_c^2+3\overline{m}_c^4 ) \nonumber \\&\quad \quad +\frac{m_s m_c}{18432\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad +\frac{m_s m_c}{73728\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^4\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times (s-\overline{m}_c^2)^3 (7s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{m_c\left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{2304\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 (s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad - \frac{m_c\left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{2304\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^2(2s-\overline{m}_c^2) \nonumber \\&\quad \quad - \frac{m_s\left[ 2\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{384\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^2\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^2 (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{s}s\rangle }{192\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^3(s-\overline{m}_c^2)(7s^2-8s\overline{m}_c^2+2\overline{m}_c^4 ) \nonumber \\&\quad \quad + \frac{11m_c\left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{12288\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad + \frac{11m_c \left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{24576\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 (s-\overline{m}_c^2 ) (5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{7m_c \left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{24576\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, \left( \frac{z}{y}+\frac{y}{z}\right) (1-y-z)^2 (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad - \frac{m_c \left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{12288\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, \left( \frac{z}{y}+\frac{y}{z}\right) (1-y-z)^3 (s-\overline{m}_c^2 ) (5s-3\overline{m}_c^2 ) \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\quad \quad - \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{384\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^2 (15s^2-20s\overline{m}_c^2+6\overline{m}_c^4 ) \nonumber \\&\quad \quad + \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{8192\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{q}g_s\sigma Gq\rangle }{384\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{2048\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) +\frac{\langle \bar{s}s\rangle \left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{144\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)(s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{ m_s m_c\langle \bar{s}s\rangle \left[ 5\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(s-\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{ m_s m_c\langle \bar{s}s\rangle \left[ \langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{144\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)(4s-3\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{19\left[ \langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{2304\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,yz (4s-3\overline{m}_c^2)\nonumber \\&\quad \quad -\frac{\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle }{1536\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z)(1-y-z) (4s-3\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{m_s m_c \left[ 2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +3\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle +5\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{576\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{m_s m_c \langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{1024\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z ( y+z) \left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{m_s m_c \langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{3072\pi ^4}\int \mathrm{d}y \nonumber \\&\quad -\frac{m_s m_c \left[ 16\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +15\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle -5\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{m_s m_c\left[ 2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{576\pi ^4}\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) +\frac{m_s m_c \langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{4608\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) -\frac{m_s m_c\langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{256\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) (1-y-z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{m_c\langle \bar{q}q\rangle \langle \bar{s}s\rangle ^2}{36\pi ^2} \int \mathrm{d}y +\frac{m_s\langle \bar{q}q\rangle \langle \bar{s}s\rangle ^2}{18\pi ^2}\nonumber \\&\quad \quad \times \int \mathrm{d}y \,y(1-y) \left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{13\langle \bar{s}g_s\sigma Gs\rangle \left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{18432\pi ^4}\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{11\langle \bar{s}g_s\sigma Gs\rangle \left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{3072\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, y(1-y)\left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{2304\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) \left( 1+\frac{s}{2M^2}\right) \delta (s-\overline{m}_c^2 )\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ 5\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{3456\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( \frac{1-y}{y}+\frac{y}{1-y}\right) \delta (s-\widetilde{m}_c^2 )\nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{27648\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( \frac{1-y}{y}+\frac{y}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ 7\langle \bar{q}g_s\sigma Gq\rangle -2\langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{4608\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ), \end{aligned}$$
(65)
$$\begin{aligned}&\widetilde{\rho }^{10,0}_{uss}(s)=\frac{1}{122880\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^4(s-\overline{m}_c^2)^4(7s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{1}{614400\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^5(s-\overline{m}_c^2)^3\nonumber \\&\quad \quad \times (28s^2-21s\overline{m}_c^2+3\overline{m}_c^4 ) \nonumber \\&\quad \quad +\frac{m_s m_c}{4608\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad -\frac{m_s m_c}{18432\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^4(s-\overline{m}_c^2)^3 (3s-\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{m_c\left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{576\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad + \frac{m_c\left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{3456\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^3(s-\overline{m}_c^2)^2(5s-2\overline{m}_c^2) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad - \frac{m_s\left[ 2\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{1152\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2(s-\overline{m}_c^2)^2 (5s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{m_s\langle \bar{s}s\rangle }{384\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2) (5s^2-5s\overline{m}_c^2+\overline{m}_c^4 ) \nonumber \\&\quad \quad + \frac{11m_c\left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{3072\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad - \frac{11m_c\left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{6144\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2 (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{7m_c\left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{12288\pi ^6}\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, \left( \frac{1}{y}+\frac{1}{z}\right) (1-y-z)^2 (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad + \frac{m_c\left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{6144\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, \left( \frac{1}{y}+\frac{1}{z}\right) (1-y-z)^3 (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) + \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{768\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 (10s^2-12s\overline{m}_c^2+3\overline{m}_c^4 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{q}g_s\sigma Gq\rangle }{192\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad + \frac{m_s\left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{1024\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{\langle \bar{s}s\rangle \left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{72\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)(s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{ m_s m_c\langle \bar{s}s\rangle \left[ 5\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{36\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{ m_s m_c\langle \bar{s}s\rangle \left[ \langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{72\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)(3s-2\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{19\left[ \langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{2304\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z) (3s-2\overline{m}_ c^2)\nonumber \\&\quad \quad -\frac{m_s m_c \left[ 2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +3\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle +5\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{432\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 )\right] -\frac{m_s m_c\langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{768\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 ) \right] \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{m_s m_c\langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{768\pi ^4} \int \mathrm{d}y \nonumber \\&\quad \quad -\frac{m_s m_c \left[ 16\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +15\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle -5\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{432\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{2304\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) \nonumber \\&\quad \quad +\frac{m_s m_c\left[ 2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}s\rangle \langle \bar{s}g_s\sigma Gs\rangle \right] }{288\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) +\frac{m_s m_c\langle \bar{s}s\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{384\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) (1-y-z)\left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 ) \right] \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{m_c\langle \bar{q}q\rangle \langle \bar{s}s\rangle ^2}{9\pi ^2}\int \mathrm{d}y +\frac{m_s\langle \bar{q}q\rangle \langle \bar{s}s\rangle ^2}{27\pi ^2}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left[ 1+\frac{s}{2}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{11\langle \bar{s}g_s\sigma Gs\rangle \left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{4608\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left[ 1+\frac{s}{2}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{4608\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) \left( 1+\frac{s}{M^2} \right) \delta (s-\overline{m}_c^2 )\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ 5\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( \frac{1}{y}+\frac{1}{1-y}\right) \delta (s-\widetilde{m}_c^2 )\nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{13824\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( \frac{1}{y}+\frac{1}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ 15\langle \bar{q}g_s\sigma Gq\rangle -3\langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( 1+\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{s}g_s\sigma Gs\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{9216\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( 1+\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ), \end{aligned}$$
(66)
$$\begin{aligned}&\rho ^{10,1}_{uus}(s)=\frac{1}{122880\pi ^8}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^4(s-\overline{m}_c^2)^4(8s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{1}{153600\pi ^8}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^5(s-\overline{m}_c^2)^3\nonumber \\&\quad \quad \times (18s^2-16s\overline{m}_c^2+3\overline{m}_c^4 ) \nonumber \\&\quad \quad +\frac{m_s m_c}{36864\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2)^4 \nonumber \\&\quad \quad +\frac{m_s m_c}{147456\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^4\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^3 (7s-3\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad - \frac{m_c\left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{2304\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad - \frac{m_c\left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{2304\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^2(2s-\overline{m}_c^2) \nonumber \\&\quad \quad - \frac{m_s\left[ 4\langle \bar{q}q\rangle -3\langle \bar{s}s\rangle \right] }{768\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^2\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^2 (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{s}s\rangle }{384\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^3(s-\overline{m}_c^2)\nonumber \\&\quad \quad \times (7s^2-8s\overline{m}_c^2+2\overline{m}_c^4 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad + \frac{11m_c\left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{12288\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad + \frac{11m_c \left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{24576\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 (s-\overline{m}_c^2 ) (5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{7m_c \left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{24576\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, \left( \frac{z}{y}+\frac{y}{z}\right) (1-y-z)^2 (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad - \frac{m_c \left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{12288\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, \left( \frac{z}{y}+\frac{y}{z}\right) (1-y-z)^3 (s-\overline{m}_c^2 ) (5s-3\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad - \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{768\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)^2 \nonumber \\&\quad \quad \times (15s^2-20s\overline{m}_c^2+6\overline{m}_c^4 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{q}g_s\sigma Gq\rangle }{8192\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 \nonumber \\&\quad \quad \times (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\left[ 2\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{768\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{q}g_s\sigma Gq\rangle }{2048\pi ^6}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z) \nonumber \\&\quad \quad \times (s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{\langle \bar{q}q\rangle \left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \, yz(1-y-z)(s-\overline{m}_c^2 )(5s-3\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{ m_s m_c\langle \bar{q}q\rangle \left[ 3\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(s-\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{ m_s m_c\langle \bar{q}q\rangle \langle \bar{s}s\rangle }{144\pi ^4}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)(4s-3\overline{m}_c^2 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{19\left[ \langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{2304\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,yz (4s-3\overline{m}_c^2)\nonumber \\&\quad \quad -\frac{\langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{1536\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z)(1-y-z) (4s-3\overline{m}_c^2)\nonumber \\&\quad \quad +\frac{m_s m_c \left[ 2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +3\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{576\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{m_s m_c \langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{1024\pi ^4}\int \mathrm{d}y\mathrm{d}z ( y+z) \left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{m_s m_c \langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{3072\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad -\frac{m_s m_c \left[ 18\langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle -2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle -3\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{1728\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{m_s m_c\left[ 2 \langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] \langle \bar{q}g_s\sigma Gq\rangle }{576\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) \nonumber \\&\quad \quad +\frac{m_s m_c \langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{4608\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{256\pi ^4}\int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) (1-y-z)\nonumber \\&\quad \quad \times \left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] -\frac{m_c\langle \bar{q}q\rangle ^2\langle \bar{s}s\rangle }{36\pi ^2}\int \mathrm{d}y +\frac{m_s\langle \bar{q}q\rangle ^2\langle \bar{s}s\rangle }{36\pi ^2}\nonumber \\&\quad \quad \times \int \mathrm{d}y \,y(1-y) \left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{13\langle \bar{q}g_s\sigma Gq\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{18432\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)\left[ 1+\frac{s}{3}\delta (s-\overline{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{11\langle \bar{q}g_s\sigma Gq\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{3072\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, y(1-y)\left[ 1+\frac{s}{3}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{2304\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{z}{y}+\frac{y}{z}\right) \left( 1+\frac{s}{2M^2}\right) \delta (s-\overline{m}_c^2 )\nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \left[ 3\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{3456\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( \frac{1-y}{y}+\frac{y}{1-y}\right) \delta (s-\widetilde{m}_c^2 )\nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{27648\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( \frac{1-y}{y}+\frac{y}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \left[ 9\langle \bar{q}g_s\sigma Gq\rangle -4\langle \bar{s}g_s\sigma Gs\rangle \right] }{3456\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{4608\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( 1+\frac{s}{2M^2}\right) \delta (s-\widetilde{m}_c^2 ), \end{aligned}$$
(67)
$$\begin{aligned}&\widetilde{\rho }^{10,0}_{uus}(s)=\frac{1}{122880\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^4(s-\overline{m}_c^2)^4(7s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{1}{614400\pi ^8}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^5(s-\overline{m}_c^2)^3\nonumber \\&\quad \quad \times (28s^2-21s\overline{m}_c^2+3\overline{m}_c^4 )+\frac{m_s m_c}{9216\pi ^8}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (1-y-z)^3(s-\overline{m}_c^2)^4-\frac{m_s m_c}{36864\pi ^8}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (1-y-z)^4(s-\overline{m}_c^2)^3 (3s-\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{m_c\left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{576\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2(s-\overline{m}_c^2)^3 \nonumber \\&\quad \quad + \frac{m_c\left[ 2\langle \bar{q}q\rangle +\langle \bar{s}s\rangle \right] }{3456\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (1-y-z)^3(s-\overline{m}_c^2)^2(5s-2\overline{m}_c^2) \nonumber \\&\quad \quad - \frac{m_s\left[ 4\langle \bar{q}q\rangle -3\langle \bar{s}s\rangle \right] }{2304\pi ^6}\int \mathrm{d}y\mathrm{d}z \,(y+z)(1-y-z)^2\nonumber \\&\quad \quad \times (s-\overline{m}_c^2)^2 (5s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{m_s\langle \bar{s}s\rangle }{768\pi ^6}\int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^3(s-\overline{m}_c^2) (5s^2-5s\overline{m}_c^2+\overline{m}_c^4 ) \end{aligned}$$
$$\begin{aligned}&\quad \quad + \frac{11m_c\left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{3072\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (1-y-z) (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad - \frac{11m_c\left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{6144\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (1-y-z)^2 (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad - \frac{7m_c\left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{12288\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, \left( \frac{1}{y}+\frac{1}{z}\right) (1-y-z)^2 (s-\overline{m}_c^2 )^2 \nonumber \\&\quad \quad + \frac{m_c\left[ 2\langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{s}g_s\sigma Gs\rangle \right] }{6144\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, \left( \frac{1}{y}+\frac{1}{z}\right) (1-y-z)^3 \left( s-\overline{m}_c^2 \right) (2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{s}g_s\sigma Gs\rangle }{1536\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)^2 (10s^2-12s\overline{m}_c^2+3\overline{m}_c^4 ) \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad + \frac{m_s\left[ 2\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{384\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad + \frac{m_s\langle \bar{q}g_s\sigma Gq\rangle }{1024\pi ^6}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z) (s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{\langle \bar{q}q\rangle \left[ \langle \bar{q}q\rangle +2\langle \bar{s}s\rangle \right] }{72\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (y+z)(1-y-z)(s-\overline{m}_c^2 )(2s-\overline{m}_c^2 ) \nonumber \\&\quad \quad +\frac{ m_s m_c\langle \bar{q}q\rangle \left[ 3\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] }{36\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (s-\overline{m}_c^2 ) +\frac{ m_s m_c\langle \bar{q}q\rangle \langle \bar{s}s\rangle }{72\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \, (1-y-z)(3s-2\overline{m}_c^2 ) \nonumber \\&\quad \quad -\frac{19\left[ \langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle +\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{2304\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \,(y+z) (3s-2\overline{m}_c^2)\nonumber \\ \end{aligned}$$
$$\begin{aligned}&\quad \quad -\frac{m_s m_c \left[ 2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle +3\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{432\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 )\right] -\frac{m_s m_c\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{768\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 ) \right] +\frac{m_s m_c\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{768\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad -\frac{m_s m_c \left[ 18\langle \bar{q}q\rangle \langle \bar{q}g_s\sigma Gq\rangle -2\langle \bar{q}q\rangle \langle \bar{s}g_s\sigma Gs\rangle -3\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle \right] }{432\pi ^4}\int \mathrm{d}y \nonumber \\&\quad \quad +\frac{m_s m_c\left[ 2\langle \bar{q}q\rangle -\langle \bar{s}s\rangle \right] \langle \bar{q}g_s\sigma Gq\rangle }{288\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) +\frac{m_s m_c\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{2304\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) +\frac{m_s m_c\langle \bar{s}s\rangle \langle \bar{q}g_s\sigma Gq\rangle }{384\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) (1-y-z)\left[ 1+\frac{s}{2}\delta (s-\overline{m}_c^2 ) \right] \nonumber \\&\quad \quad -\frac{m_c\langle \bar{q}q\rangle ^2\langle \bar{s}s\rangle }{9\pi ^2}\int \mathrm{d}y +\frac{m_s\langle \bar{q}q\rangle ^2\langle \bar{s}s\rangle }{54\pi ^2} \int \mathrm{d}y \left[ 1+\frac{s}{2}\delta (s-\widetilde{m}_c^2 )\right] \nonumber \end{aligned}$$
$$\begin{aligned}&\quad \quad +\frac{11\langle \bar{q}g_s\sigma Gq\rangle \left[ \langle \bar{q}g_s\sigma Gq\rangle +2\langle \bar{s}g_s\sigma Gs\rangle \right] }{4608\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left[ 1+\frac{s}{2}\delta (s-\widetilde{m}_c^2 )\right] -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{4608\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y\mathrm{d}z \left( \frac{1}{y}+\frac{1}{z}\right) \left( 1+\frac{s}{M^2} \right) \delta (s-\overline{m}_c^2 )\nonumber \\&\quad \quad -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \left[ 3\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( \frac{1}{y}+\frac{1}{1-y}\right) \delta (s-\widetilde{m}_c^2 )-\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{13824\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( \frac{1}{y}+\frac{1}{1-y}\right) \delta (s-\widetilde{m}_c^2 ) \nonumber \\&\quad \quad +\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \left[ 9\langle \bar{q}g_s\sigma Gq\rangle -\langle \bar{s}g_s\sigma Gs\rangle \right] }{1728\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \left( 1+\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ) -\frac{m_s m_c\langle \bar{q}g_s\sigma Gq\rangle \langle \bar{s}g_s\sigma Gs\rangle }{9216\pi ^4}\nonumber \\&\quad \quad \times \int \mathrm{d}y \, \left( 1+\frac{s}{M^2}\right) \delta (s-\widetilde{m}_c^2 ), \end{aligned}$$
(68)
$$\begin{aligned} \rho ^{10,1}_{uuu}(s)= & {} \rho ^{10,1}_{sss}(s)\mid _{m_s \rightarrow 0,\,\,\langle \bar{s}s\rangle \rightarrow \langle \bar{q}q\rangle ,\,\,\langle \bar{s}g_s\sigma Gs\rangle \rightarrow \langle \bar{q}g_s\sigma Gq\rangle } , \nonumber \\\end{aligned}$$
(69)
$$\begin{aligned} \widetilde{\rho }^{10,0}_{uuu}(s)= & {} \widetilde{\rho }^{10,0}_{sss}(s)\mid _{m_s \rightarrow 0,\,\,\langle \bar{s}s\rangle \rightarrow \langle \bar{q}q\rangle ,\,\,\langle \bar{s}g_s\sigma Gs\rangle \rightarrow \langle \bar{q}g_s\sigma Gq\rangle } ,\nonumber \\ \end{aligned}$$
(70)
where \(\int \mathrm{d}y\mathrm{d}z=\int _{y_i}^{y_f}\mathrm{d}y \int _{z_i}^{1-y}\mathrm{d}z\), \(\int \mathrm{d}y=\int _{y_i}^{y_f}\mathrm{d}y\), \(y_{f}=\frac{1+\sqrt{1-4m_c^2/s}}{2}\), \(y_{i}=\frac{1-\sqrt{1-4m_c^2/s}}{2}\), \(z_{i}=\frac{y m_c^2}{y s -m_c^2}\), \(\overline{m}_c^2=\frac{(y+z)m_c^2}{yz}\), \( \widetilde{m}_c^2=\frac{m_c^2}{y(1-y)}\), \(\int _{y_i}^{y_f}\mathrm{d}y \rightarrow \int _{0}^{1}\mathrm{d}y\), \(\int _{z_i}^{1-y}\mathrm{d}z \rightarrow \int _{0}^{1-y}\mathrm{d}z\) when the \(\delta \) functions \(\delta (s-\overline{m}_c^2)\) and \(\delta \left( s-\widetilde{m}_c^2\right) \) appear.