Appendix 1
The coefficients \( \delta_{j} \) and \( \Delta_{j} (j = 1,2, \ldots ,16) \) which are used in Eqs. (7) and (8), as follows:
$$ \begin{aligned} & \delta_{1} = c_{12} ,\quad \delta_{2} = c_{11} - 4c_{12} - c_{22} ,\quad \delta_{3} = 5c_{12} + 3c_{22} - 3c_{11} - c_{21} ,\quad \delta_{4} = 2\left( {c_{11} - c_{22} - c_{12} + c_{21} } \right), \\ & \delta_{5} = c_{21} ,\quad \delta_{6} = c_{11} - 2c_{31} + c_{22} ,\quad \delta_{7} = 4c_{31} - 3c_{11} - c_{22} ,\quad \delta_{8} = 2\left( {c_{11} - c_{31} + c_{21} } \right);\quad \delta_{9} = c_{24} , \\ & \delta_{10} = c_{14} + c_{23} + 2c_{32} ,\quad \delta_{11} = 3c_{14} + c_{23} + 4c_{32} ,\quad \delta_{12} = 2\left( {c_{14} + c_{32} + c_{24} } \right),\quad \delta_{13} = c_{13} , \\ & \delta_{14} = c_{23} - c_{14} + 4c_{13} ,\quad \delta_{15} = c_{24} - 3c_{23} + 3c_{14} - 5c_{13} ,\quad \delta_{16} = 2\left( {c_{23} - c_{14} - c_{24} + c_{13} } \right), \\ & \Delta_{1} = b_{11} ,\quad \Delta_{2} = 2b_{31} + b_{21} + b_{12} ,\quad \Delta_{3} = 4b_{31} + 3b_{21} + b_{12} ,\quad \Delta_{4} = 2(b_{31} + b_{21} + b_{11} ), \\ & \Delta_{5} = b_{22} ,\quad \Delta_{6} = b_{21} - 4b_{22} - b_{12} ,\quad \Delta_{7} = 5b_{22} + 3b_{12} - b_{11} - 3b_{21} , \\ & & & \Delta_{8} = 2b_{21} - 2b_{22} - 2b_{12} + 2b_{11} ,\quad \Delta_{9} = b_{14} ,\quad \Delta_{10} = 2b_{32} - b_{13} - b_{24} , \\ & \Delta_{11} = b_{13} + 3b_{24} - 4b_{32} ,\quad \Delta_{12} = 2b_{32} - 2b_{24} - 2b_{14} ,\quad \Delta_{13} = b_{23} ,\quad \Delta_{14} = b_{13} - b_{24} + 4b_{23} , \\ & \Delta_{15} = b_{14} - 3b_{13} + 3b_{24} - 5b_{23} ,\quad \Delta_{16} = 2b_{13} - 2b_{24} + 2b_{23} - 2b_{14} . \\ \end{aligned} $$
(24)
where the following definitions apply:
$$ \begin{aligned} c_{11} = a_{11}^{1} b_{11} + a_{12}^{1} b_{21} ,\quad c_{12} = a_{11}^{1} b_{12} + a_{12}^{1} b_{22} ,\quad c_{13} = a_{11}^{1} b_{13} + a_{12}^{1} b_{23} + a_{11}^{2} , \hfill \\ c_{14} = a_{11}^{1} b_{14} + a_{12}^{1} b_{24} + a_{12}^{2} ,\quad c_{21} = a_{21}^{1} b_{11} + a_{22}^{1} b_{21} ,\quad c_{22} = a_{21}^{1} b_{12} + a_{22}^{1} b_{22} , \hfill \\ c_{23} = a_{21}^{1} b_{13} + a_{22}^{1} b_{14} + a_{21}^{2} ,\quad c_{24} = a_{21}^{1} b_{14} + a_{22}^{1} b_{13} + a_{22}^{2} ,\quad c_{31} = a_{66}^{1} b_{31} ,\quad c_{32} = a_{66}^{1} b_{32} + a_{66}^{2} , \hfill \\ b_{11} = a_{22}^{0} /L_{0} ,\quad b_{12} = - a_{12}^{0} /L_{0} ,\quad b_{13} = \left( {a_{12}^{0} a_{21}^{1} - a_{11}^{1} a_{22}^{0} } \right)/L_{0} ,\quad b_{14} = \left( {a_{12}^{0} a_{22}^{1} - a_{12}^{1} a_{22}^{0} } \right)/L_{0} , \hfill \\ b_{21} = - a_{21}^{0} /L_{0} ,\quad b_{22} = a_{11}^{0} /L_{0} ,\quad b_{23} = \left( {a_{21}^{0} a_{11}^{1} - a_{21}^{1} a_{11}^{0} } \right)/L_{0} ,\quad b_{24} = \left( {a_{21}^{0} a_{12}^{1} - a_{22}^{1} a_{11}^{0} } \right)/L_{0} , \hfill \\ b_{31} = 1/a_{66}^{0} ,\quad b_{32} = - a_{66}^{1} /a_{66}^{0} ,\quad L_{0} = a_{11}^{0} a_{22}^{0} - a_{12}^{0} a_{21}^{0} . \hfill \\ \end{aligned} $$
(25)
in which
$$ \begin{aligned} a_{11}^{k} & = \frac{{E_{01} h^{k + 1} }}{{1 - \nu_{1} \nu_{2} }}\int\limits_{ - 1/2}^{1/2} {Z^{k} e^{\eta (Z - 0.5)} \, \text{d}Z} ,\quad a_{22}^{k} = \frac{{E_{01} h^{k + 1} }}{{1 - \nu_{1} \nu_{2} }}\int\limits_{ - 1/2}^{1/2} {Z^{k} e^{\eta (Z - 0.5)} \, \text{d}Z{ ,}} \\ a_{12}^{k} & = \nu_{2} a_{11}^{k} = \nu_{1} a_{22}^{k} = a_{21}^{k} ,\quad a_{66}^{k} = 2G_{0} h^{k + 1} \int\limits_{ - 1/2}^{1/2} {Z^{k} e^{\eta (Z - 0.5)} \, \text{d}Z,} \quad k = 0,1,2. \\ \end{aligned} $$
(26)
Appendix 2
The coefficients \( T_{j} (j = 1,2, \ldots ,5) \) and \( P_{jj} (j = 1,2) \), which are contained in Eqs. (11) and (12) are:
$$ \begin{aligned} T_{1} & = \frac{{S_{2} e^{{ - 2x_{0} \lambda }} }}{{32x_{0}^{4} \lambda \left( {2\lambda - 1} \right)\left[ {\left( {2\lambda - 1} \right)^{2} x_{0}^{2} + m^{2} \pi^{2} } \right]}}\left( {m^{2} \pi^{2} e^{{2x_{0} \lambda }} \left\{ {\lambda \left\langle {8\left[ { - 6\lambda^{4} \delta_{1} - 4\left( {\delta_{2} + 3\delta_{1} } \right)\lambda^{3} } \right.} \right.} \right.} \right. \\ & \quad \left. { - \left( {3\delta_{2} + 2\delta_{3} } \right)\lambda^{2} + 2\left( {2\delta_{2} + 3\delta_{1} + \delta_{3} } \right)\lambda + 6\delta_{1} + 3\delta_{4} + 5\delta_{2} + 4\delta_{3} } \right]x_{0}^{4} + \pi^{4} m^{4} \delta_{1} - 2m^{2} \pi^{2} x_{0}^{2} \\ & \quad \left. { \times \left[ {4\lambda^{2} \delta_{1} + 4\left( {\delta_{2} + 5\delta_{1} } \right)\lambda + 2\delta_{3} + 7\delta_{2} + 16\delta_{1} } \right]} \right\rangle \left. { - 2x_{0}^{2} \left( {2\lambda - 1} \right)S_{2} \left[ {\left( {2\lambda - 1} \right)^{2} x_{0}^{2} + m^{2} \pi^{2} } \right]\cot \gamma } \right\} \\ & \quad - \lambda e^{{x_{0} }} \left\langle {32\left[ {\lambda^{3} \delta_{1} + \left( {3\delta_{1} + \delta_{2} } \right)\lambda^{2} } \right.} \right.\left. { + \left( {\delta_{3} + 2\delta_{2} + 3\delta_{1} } \right)\lambda + \delta_{2} + \delta_{3} + \delta_{4} + \delta_{1} } \right]\left( {\lambda + 1} \right)\left( {2\lambda - 1} \right)^{2} x_{0}^{6} \\ & \quad - 8m^{2} \pi^{2} x_{0}^{4} \left[ {14\lambda^{4} \delta_{1} + 4\left( {\delta_{2} - \delta_{1} } \right)} \right.\lambda^{3} - \left( {15\delta_{2} + 2\delta_{3} + 42\delta_{1} } \right)\lambda^{2} - \left( {10\delta_{3} + 4\delta_{4} + 16\delta_{1} + 15\delta_{2} } \right)\lambda \\ & \quad + 4\delta_{2} + 8\delta_{1} + \delta_{3} \left. { - \delta_{4} } \right]\left. { - 2m^{4} \pi^{4} x_{0}^{2} \left[ {\lambda^{2} \delta_{1} + 8\left( {\delta_{2} + 4\delta_{1} } \right)\lambda + 5\delta_{2} + 2\delta_{3} + 7\delta_{1} } \right] + \pi^{6} \delta_{1} m^{6} } \right\rangle \\ & \quad \left. { - 2x_{0}^{2} \cot \gamma \left( {2\lambda - 1} \right)S_{2} \left[ {\left( {2\lambda - 1} \right)^{2} x_{0}^{2} + m^{2} \pi^{2} } \right]\left[ {8\lambda \left( {\lambda + 1} \right)x_{0}^{2} + m^{2} \pi^{2} } \right]} \right) \\ \end{aligned} $$
$$ \begin{aligned} T_{2} & = \frac{{n^{2} S_{2} e^{{ - 2\lambda x_{0} }} }}{{64\left( {2\lambda - 1} \right)x_{0}^{2} \sin^{4} \gamma \left[ {\left( {2\lambda - 1} \right)^{2} x_{0}^{2} + m^{2} \pi^{2} } \right]}}\left( {\pi^{2} m^{2} e^{{2\lambda x_{0} }} \left\{ {\left[ {\left( {3\delta_{7} + 2\delta_{8} } \right.} \right.} \right.} \right.\left. { + 2\lambda \delta_{6} - 2\lambda^{2} \delta_{6} + 4\delta_{6} } \right)x_{0}^{2} \\ & \quad \left. { - 0.5\delta_{6} m^{2} \pi^{2} } \right]\cos^{2} \gamma + x_{0}^{2} \left( {2n^{2} \delta_{5} + 2\lambda^{2} \delta_{6} - 3\delta_{7} - 4\delta_{6} - 2\lambda \delta_{6} - 2\delta_{8} } \right)\left. { + 0.5\delta_{6} m^{2} \pi^{2} } \right\} \\ & \quad - e^{{x_{0} }} \left\langle {\left\{ {4\left( {2\lambda - 1} \right)^{2} x_{0}^{4} \left[ {\lambda^{2} \delta_{6} + \left( {2\delta_{6} + \delta_{7} } \right)\lambda + \delta_{6} + \delta_{8} + \delta_{7} } \right]} \right.} \right. + m^{2} x_{0}^{2} \pi^{2} \left[ {2\lambda^{2} \delta_{6} + 2\left( {5\delta_{6} + 2\delta_{7} } \right)\lambda } \right. \\ & \quad \left. {\left. { + \delta_{7} + 2\delta_{8} - \delta_{6} } \right] - 0.5\delta_{6} m^{4} \pi^{4} } \right\}\cos^{2} \gamma + 4\left( {2\lambda - 1} \right)^{2} x_{0}^{4} \left[ {n^{2} \delta_{5} - \lambda^{2} \delta_{6} } \right. - \left( {2\delta_{6} + \delta_{7} } \right)\lambda \,\left. { - \delta_{7} - \delta_{8} - \delta_{6} } \right] \\ & \quad + \pi^{2} m^{2} x_{0}^{2} \left[ {2n^{2} \delta_{5} - 2\lambda^{2} \delta_{6} - 2\left( {2\delta_{7} + 5\delta_{6} } \right)} \right.\lambda - \delta_{7} - \delta_{8} \,\left. {\left. {\left. { + \delta_{6} } \right] + 0.5\delta_{6} m^{4} \pi^{4} } \right\rangle } \right) \\ \end{aligned} $$
$$ \begin{aligned} T_{3} & = - \frac{{e^{{ - 2\lambda x_{0} }} }}{{2x_{0}^{4} \left( {\lambda - 1} \right)\left[ {4\left( {\lambda - 1} \right)^{2} x_{0}^{2} + m^{2} \pi^{2} } \right]\sin^{4} \gamma }}\left[ \frac{1}{8} \right.\pi^{2} m^{2} \left\langle {\left\{ {\left[ {2\left( {2\delta_{12} + \delta_{13} } \right)\lambda^{3} - 3\delta_{12} \lambda^{4} + \left( {\delta_{14} - 3\delta_{13} } \right)\lambda^{2} } \right.} \right.} \right. \\ & \quad \left. { - 2\delta_{14} \lambda + \delta_{15} } \right]x_{0}^{4} + 0.25\pi^{2} m^{2} \left[ {2\left( {\delta_{13} - 2\delta_{12} } \right)\lambda } \right. - 2\delta_{12} \lambda^{2} \left. { + \delta_{14} + \delta_{13} } \right]x_{0}^{2} \left. { + \frac{1}{16}\delta_{12} m^{4} \pi^{4} } \right\}\cos^{4} \gamma \\ & \quad + \left\{ {x_{0}^{4} \left[ 6 \right.\delta_{12} \lambda^{4} - 4\left( {\delta_{13} + 2\delta_{12} } \right)} \right.\lambda^{3} + \left( {6\delta_{13} - n^{2} \delta_{10} } \right.\left. { - 2\delta_{14} } \right)\lambda^{2} \left. { + 2\left( {n^{2} \delta_{10} + 2\delta_{14} } \right)\lambda + \left( {\delta_{16} - \delta_{11} } \right)n^{2} - 2\delta_{15} } \right] \\ & \quad - 0.25\pi^{2} m^{2} x_{0}^{2} \left[ {4\left( {\delta_{13} - 2\delta_{12} } \right)\lambda - 4\delta_{12} \lambda^{2} \left. {\left. { + n^{2} \delta_{10} + 2\delta_{13} + 2\delta_{14} } \right] - 0.125\delta_{12} m^{4} \pi^{4} } \right\}\cos^{2} \gamma } \right. \\ & \quad + x_{0}^{4} \left[ {2\left( {2\delta_{12} + \delta_{13} } \right)} \right.\lambda^{3} - 3\delta_{12} \lambda^{4} + \left( {n^{2} \delta_{10} - 3\delta_{13} } \right.\,\left. {\left. { + \delta_{14} } \right)\lambda^{2} - 2\left( {\delta_{14} + n^{2} \delta_{10} } \right)\lambda + n^{4} \delta_{9} + \left( {\delta_{11} - \delta_{16} } \right)n^{2} + \delta_{15} } \right] \\ & \quad \left. { + 0.25\pi^{2} m^{2} x_{0}^{2} \left[ {2\left( {\delta_{13} - 2\delta_{12} } \right)\lambda - 2\delta_{12} \lambda^{2} + \delta_{13} + \delta_{14} + n^{2} \delta_{10} } \right] + 0.125\delta_{12} m^{4} \pi^{4} } \right\rangle e^{{2\lambda x_{0} }} \\ & \quad + \left\langle {\left\{ {\lambda \left( {\lambda - 1} \right)} \right.^{2} \left( {\delta_{13} \lambda^{2} - \delta_{12} \lambda^{3} + \delta_{14} \lambda - \delta_{15} } \right)x_{0}^{6} - 0.125\pi^{2} m^{2} x_{0}^{4} \left[ {2\left( {\delta_{13} + 10\delta_{12} } \right)\lambda^{3} - 7\delta_{12} \lambda^{4} } \right.} \right. + 2\delta_{14} - \delta_{15} \\ & \quad \left. { - \left( {9\delta_{13} + 12\delta_{12} + \delta_{14} } \right)\lambda^{2} + 2\left( {\delta_{15} - \delta_{14} + 3\delta_{13} } \right)\lambda \,\,} \right] - \frac{1}{32}\pi^{4} m^{4} \left( {2\delta_{12} - 8\delta_{12} \lambda^{2} } \right. + \delta_{14} \left. { - \delta_{13} + 4\delta_{13} \lambda } \right)x_{0}^{2} \\ & \quad \left. { - \frac{1}{128}\delta_{12} m^{6} \pi^{6} } \right\}\cos^{4} \gamma + x_{0}^{6} \left\{ {\left( {\lambda - 1} \right)^{2} \left[ {2\delta_{12} \lambda^{4} - 2\delta_{13} \lambda^{3} - \left( {n^{2} \delta_{10} } \right.} \right.} \right.\left. { + 2\delta_{14} } \right)\lambda^{2} + \left( {n^{2} \delta_{11} + 2\delta_{15} } \right)\lambda \left. { - n^{2} \delta_{16} } \right] \\ & \quad - 0.125\pi^{2} m^{2} x_{0}^{4} \left\{ {4\left( {\delta_{13} + 10\delta_{12} } \right)\lambda^{3} } \right. - \left( {18\delta_{13} + n^{2} \delta_{10} + 24\delta_{12} + 2\delta_{14} } \right)\lambda^{2} - 14\delta_{12} \lambda^{4} + 4\delta_{14} - 2\delta_{15} \\ & \quad + 2\left[ {\left( {\delta_{11} - \delta_{10} } \right)n^{2} + 2\delta_{15} - 2\delta_{14} + 6\delta_{13} } \right]\lambda \left. { + \left( {2\delta_{10} - \delta_{16} - \delta_{11} } \right)n^{2} } \right\} \\ & \quad - \frac{1}{4}\pi^{4} m^{4} x_{0}^{2} \left( {2\delta_{12} - 8\delta_{12} \lambda^{2} + \delta_{14} } \right.\left. { - \delta_{13} + 0.5n^{2} \delta_{10} + 4\delta_{13} \lambda } \right)\left. { - \frac{1}{16}\delta_{12} m^{6} \pi^{6} } \right\}\cos^{2} \gamma \\ & \quad + \left( {\lambda - 1} \right)^{2} x_{0}^{6} \left[ {\delta_{13} \lambda^{3} - \delta_{12} \lambda^{4} + } \right.\left( {\delta_{14} + n^{2} \delta_{10} } \right)\lambda^{2} \,\left. { - \left( {n^{2} \delta_{11} + \delta_{15} } \right)\lambda + n^{2} \left( {\delta_{16} - \delta_{9} n^{2} } \right)} \right] \\ & \quad - 0.125\pi^{2} m^{2} x_{0}^{4} \left\{ { - 7\delta_{12} \lambda^{4} + \left( {\delta_{13} + 10\delta_{12} } \right)\lambda^{3} } \right.\, + \left( { - 12\delta_{12} - 9\delta_{13} - n^{2} \delta_{10} - \delta_{14} } \right)\lambda^{2} + n^{4} \delta_{9} \\ & \quad \left. { + 2\left[ {\left( {\delta_{11} - \delta_{10} } \right)n^{2} + \delta_{15} - \delta_{14} + 3\delta_{13} } \right]\lambda + \left( {2\delta_{10} - \delta_{16} - \delta_{11} } \right)n^{2} + 2\delta_{14} - \delta_{15} } \right\} \\ & \quad - \frac{1}{32}\pi^{4} m^{4} x_{0}^{2} \left( {4\delta_{13} \lambda + 2\delta_{12} + n^{2} \delta_{10} - 8\delta_{12} \lambda^{2} + \delta_{14} } \right.\left. {\left. {\left. { - \delta_{13} } \right) - \frac{1}{128}\delta_{12} m^{6} \pi^{6} } \right\rangle e^{{2x_{0} }} } \right] \\ \end{aligned} $$
$$ \begin{aligned} T_{4} & = - \frac{{3S_{2}^{2} e^{{ - 2x_{0} \lambda }} }}{{4\lambda x_{0}^{4} \sin^{4} \gamma \left( {4\lambda^{2} x_{0}^{2} + m^{2} \pi^{2} } \right)}}\left( {\left\langle {\left\{ {\left[ {3\Delta_{8} \lambda^{4} + \left( {8\Delta_{8} + 2\Delta_{7} } \right)\lambda^{3} } \right.} \right.} \right.} \right. + \left( {3\Delta_{7} + 6\Delta_{8} + \Delta_{6} } \right)\lambda^{2} - \Delta_{7} - \Delta_{8} \\ & \quad \left. { - \Delta_{6} - \Delta_{5} } \right]x_{0}^{4} + 0.25m^{2} \pi^{2} x_{0}^{2} \left[ {2\Delta_{8} \lambda^{2} + 2\left( {\Delta_{7} + 4\Delta_{8} } \right)\lambda } \right.\left. { + \Delta_{6} + 3\Delta_{7} + 6\Delta_{8} } \right]\left. { - \frac{1}{16}\Delta_{8} m^{4} \pi^{4} } \right\}\cos^{4} \gamma \\ & \quad + \left\{ {x_{0}^{4} \left[ { - 6\Delta_{8} \lambda^{4} - 4\left( {4\Delta_{8} + \Delta_{7} } \right)} \right.} \right.\lambda^{3} + \left( {\Delta_{2} n^{2} - \Delta_{6} - 12\Delta_{8} - 6\Delta_{7} } \right)\lambda^{2} + 2\Delta_{7} + \left( {\Delta_{3} - 1.5\Delta_{2} - \Delta_{4} } \right)n^{2} \\ & \quad + 2\Delta_{8} + 2\Delta_{6} \left. { + 2\Delta_{5} } \right] - 0.25\left[ {4\Delta_{8} \lambda^{2} + 4\left( {\Delta_{7} + 4\Delta_{8} } \right)\lambda + 12\Delta_{8} - \Delta_{2} n^{2} + 2\Delta_{6} + 6\Delta_{7} } \right]m^{2} \pi^{2} x_{0}^{2} \\ & \quad \left. { + 0.125\Delta_{8} m^{4} \pi^{4} } \right\}\cos^{2} \gamma + \left[ {3\Delta_{8} \lambda^{4} + \left( {8\Delta_{8} + 2\Delta_{7} } \right)} \right.\lambda^{3} + \left( { - \Delta_{2} n^{2} + \Delta_{6} + 3\Delta_{7} + 6\Delta_{8} } \right)\lambda^{2} - \Delta_{8} - \Delta_{7} \\ & \quad \left. { - n^{4} \Delta_{1} + \left( {\Delta_{2} + \Delta_{4} - \Delta_{3} } \right)n^{2} - \Delta_{6} - \Delta_{5} } \right]x_{0}^{4} + 0.25\left[ {2\Delta_{8} \lambda^{2} } \right. + 2\left( {\Delta_{7} + 4\Delta_{8} } \right)\lambda + 3\Delta_{7} - \Delta_{2} n^{2} \\ & \quad \left. {\left. { + 6\Delta_{8} + \Delta_{6} } \right]m^{2} \pi^{2} x_{0}^{2} - \frac{1}{16}\Delta_{8} m^{4} \pi^{4} } \right\rangle m^{2} \pi^{2} e^{{2x_{0} \lambda }} \\ & \quad + \left\{ {8\lambda^{2} \left( {\lambda + 1} \right)x_{0}^{6} \left[ {\Delta_{8} \lambda^{3} + \left( {\Delta_{7} + 3\Delta_{8} } \right)\lambda^{2} + \left( {\Delta_{6} + 2\Delta_{7} + 3\Delta_{8} } \right)\lambda + \Delta_{5} + \Delta_{7} + \Delta_{8} + \Delta_{6} } \right]} \right. \\ & \quad - m^{2} \pi^{2} x_{0}^{4} \left[ {7\Delta_{8} \lambda^{4} + \left( {2\Delta_{7} + 9\Delta_{8} } \right)} \right.\lambda^{3} - \left( {3\Delta_{7} + \Delta_{6} + 6\Delta_{8} } \right)\lambda^{2} - 2\left( {2\Delta_{6} + 3\Delta_{7} + \Delta_{5} + 4\Delta_{8} } \right)\lambda - \Delta_{8} \\ & \quad \left. { - \Delta_{6} - \Delta_{5} - \Delta_{7} } \right] - 0.25m^{4} \pi^{4} x_{0}^{2} \left[ {8\Delta_{8} \lambda^{2} + 4\left( {\Delta_{7} + 4\Delta_{8} } \right)\lambda + \Delta_{6} + 6\Delta_{8} + 3\Delta_{7} } \right]\left. { + \frac{1}{48}\Delta_{8} m^{6} \pi^{6} } \right\}\cos^{4} \gamma \\ & \quad - \left\langle 8 \right.\lambda^{2} x_{0}^{6} \left\{ {2\Delta_{8} \lambda^{4} + 2\left( {\Delta_{7} + 4\Delta_{8} } \right)\lambda^{3} + \left( {2\Delta_{6} + 6\Delta_{7} - \Delta_{2} n^{2} + 12\Delta_{8} } \right)} \right.\lambda^{2} + \left( {\Delta_{3} - \Delta_{4} - \Delta_{2} } \right)n^{2} + 2\Delta_{6} \\ & \quad + \left[ {8\Delta_{8} + 6\Delta_{7} + \left( {\Delta_{3} - 2\Delta_{2} } \right)n^{2} + 2\Delta_{5} + 4\Delta_{6} } \right]\lambda + 2\Delta_{8} + 2\Delta_{7} \left. { + 2\Delta_{5} } \right\} + m^{2} \pi^{2} \left\{ {\left( {4\Delta_{7} + 16\Delta_{8} } \right)\lambda^{3} } \right. \\ & \quad + 14\Delta_{8} \lambda^{4} \, + \left( {\Delta_{2} n^{2} - 12\Delta_{8} - 6\Delta_{7} - 2\Delta_{6} } \right)\lambda^{2} + \left[ {\left( {4\Delta_{2} - 2\Delta_{3} } \right)n^{2} - 16\lambda^{3} \left. { - 12\Delta_{7} - 8\Delta_{6} - 4\Delta_{5} } \right]\lambda } \right. \\ & \quad - 2\Delta_{8} - 2\Delta_{7} + \left( {\Delta_{4} - \Delta_{3} + \Delta_{2} } \right)n^{2} \left. { - 2\Delta_{6} - 2\Delta_{5} } \right\} - 0.25\left[ {16\Delta_{8} \lambda^{2} + 8\left( {\Delta_{7} + 4\Delta_{8} } \right)\lambda + 12\Delta_{8} } \right. + 2\Delta_{6} \\ & \quad \left. { + 6\Delta_{7} - \Delta_{2} n^{2} } \right]m^{4} \pi^{4} x_{0}^{2} \left. { + 0.125\Delta_{8} m^{6} \pi^{6} } \right\rangle \cos^{2} \gamma + 8\lambda^{2} \left\{ {\Delta_{8} \lambda^{4} + \left( {\Delta_{7} + 4\Delta_{8} } \right)} \right.\lambda^{3} \left( {\Delta_{6} + 3\Delta_{7} + 6\Delta_{8} } \right. \\ & \quad \left. { - \Delta_{2} n^{2} } \right)\lambda^{2} + \left[ {4\Delta_{8} } \right. + 3\Delta_{7} + \left( {\Delta_{3} - 2\Delta_{2} } \right)n^{2} \left. { + \Delta_{5} + 2\Delta_{6} } \right]\lambda + \Delta_{8} + \Delta_{7} + n^{4} \Delta_{1} + \left( {\Delta_{3} - \Delta_{4} - \Delta_{2} } \right)n^{2} \\ & \quad \left. { + \Delta_{6} + \Delta_{5} } \right\}x_{0}^{6} - m^{2} \pi^{2} x_{0}^{4} \left\{ {7\Delta_{8} \lambda^{4} + \left( {2\Delta_{7} + 8\Delta_{8} } \right)\lambda^{3} + \left( {\Delta_{2} n^{2} - \Delta_{6} - 6\Delta_{8} - 3\Delta_{7} } \right)} \right.\lambda^{2} - 2\left[ 4 \right.\Delta_{8} + 3\Delta_{7} \\ & \quad \left. { - \left( {2\Delta_{2} - \Delta_{3} } \right)n^{2} + \Delta_{5} + 2\Delta_{6} } \right]\lambda - \Delta_{8} - \Delta_{7} - n^{4} \Delta_{1} + \left( {\Delta_{2} + \Delta_{4} - \Delta_{3} } \right)n^{2} \left. { - \Delta_{6} - \Delta_{5} } \right\} \\ & \quad - 0.25m^{4} \pi^{4} x_{0}^{2} \left[ {8\Delta_{8} \lambda^{2} + 4\left( {\Delta_{7} + 4\Delta_{8} } \right)\lambda + 6\Delta_{8} + \Delta_{6} - \Delta_{2} n^{2} + 3\Delta_{7} } \right]\left. { + \frac{1}{16}\Delta_{8} m^{6} \pi^{6} } \right) \\ \end{aligned} $$
$$ \begin{aligned} T_{5} &= - \frac{{S_{2} e^{{ - 2\lambda x_{0} }} }}{{2x_{0}^{4} \left( {2\lambda - 1} \right)\left[ {\left( {2\lambda - 1} \right)^{2} x_{0}^{2} + m^{2} \pi^{2} } \right]\sin^{4} \gamma }}\left[ {m^{2} \left\langle {\left\{ {x_{0}^{4} \left[ { - 3\Delta_{16} \lambda^{4} + \left( {2\Delta_{15} } \right.} \right.} \right.} \right.} \right.\left. { + 2\Delta_{16} } \right)\lambda^{3} - \Delta_{14} \lambda \\ & \quad \left. { - 0.5\Delta_{13} + \left( {\Delta_{14} - 1.5\Delta_{15} } \right)\lambda^{2} } \right] + 0.25m^{2} \pi^{2} x_{0}^{2} \left[ { - 2\lambda^{2} \Delta_{16} + \left( {2\Delta_{15} - 2\Delta_{16} } \right)\lambda + \Delta_{14} } \right.\left. { + 0.5\Delta_{15} } \right] \\ & \quad \left. { + \frac{1}{16}\Delta_{16} m^{4} \pi^{4} } \right\}\cos^{4} \gamma + \left\{ {x_{0}^{4} \left[ {6\Delta_{16} \lambda^{4} - 4\left( {\Delta_{15} + \Delta_{16} } \right)\lambda^{3} + } \right.} \right.\left( {3\Delta_{15} - 2\Delta_{14} + n^{2} \Delta_{10} } \right)\lambda^{2} + \Delta_{13} \\ & \quad \left. { + \left( {2\Delta_{14} - n^{2} \Delta_{10} } \right)\lambda - \left( {\Delta_{12} + 0.5\Delta_{11} } \right)n^{2} } \right] - m^{2} \pi^{2} x_{0}^{2} \left[ {\left( {\Delta_{15} - \Delta_{16} } \right)\lambda - \lambda^{2} \Delta_{16} - 0.25n^{2} \Delta_{10} } \right. \\ & \quad \left. {\left. { + 0.5\Delta_{14} + 0.5\Delta_{15} } \right] - 0.125\Delta_{16} m^{4} \pi^{4} } \right\}\cos^{2} \gamma + x_{0}^{4} \left[ {2\left( {\Delta_{15} + \Delta_{16} } \right)\lambda^{3} - 3\Delta_{16} \lambda^{4} - \left( {1.5\Delta_{15} } \right.} \right. \\ & \quad \left. {\left. { + n^{2} \Delta_{10} - \Delta_{14} } \right)\lambda^{2} + \left( {n^{2} \Delta_{10} - \Delta_{14} } \right)\lambda + n^{4} \Delta_{9} + \left( {0.5\Delta_{11} + \Delta_{12} } \right)n^{2} - 0.5\Delta_{13} } \right] + 0.25m^{2} \pi^{2} x_{0}^{2} \\ & \quad \left. {\left. { \times \left[ {\Delta_{14} - 2\lambda^{2} \Delta_{16} } \right. + \left( {2\Delta_{15} - 2\Delta_{16} } \right)\lambda - n^{2} \Delta_{10} + 0.5\Delta_{15} } \right] + \Delta_{16} m^{4} \pi^{4} /16} \right\rangle \pi^{2} e^{{2\lambda x_{0} }} \\ & \quad + e^{{x_{0} }} \left\langle {\left\{ {2\left( {2\lambda - 1} \right)^{2} \lambda x_{0}^{6} \left. {\left( {\Delta_{15} \lambda^{2} - \Delta_{16} \lambda^{3} } \right. + \Delta_{14} \lambda + \Delta_{13} } \right) + m^{2} \pi^{2} x_{0}^{4} } \right.} \right.\left[ {7\Delta_{16} \lambda^{4} - 2\left( {5\Delta_{16} + \Delta_{15} } \right)\lambda^{3} } \right. \\ & \quad \left. { + \lambda^{2} \left( {3\Delta_{16} + \Delta_{14} + 4.5\Delta_{15} } \right) + \lambda \left( {2\Delta_{13} + \Delta_{14} - 1.5\Delta_{15} } \right) - 0.5\Delta_{14} - 0.5\Delta_{13} } \right] - 0.25m^{4} \pi^{4} x_{0}^{2} \\ & \quad \times \left( {\Delta_{14} - 0.5\Delta_{15} + 4\lambda \Delta_{15} - 8\lambda^{2} \Delta_{16} } \right.\left. {\left. { + 0.5\Delta_{16} } \right) - \Delta_{16} m^{6} \pi^{6} /16} \right\}\cos^{4} \gamma \\ & \quad + \left\langle {2\left( {2\lambda - 1} \right)^{2} x_{0}^{6} \left[ {2\Delta_{16} \lambda^{4} - 2\Delta_{15} \lambda^{3} - \left( {2\Delta_{14} - n^{2} \Delta_{10} } \right)} \right.} \right.\lambda^{2} \left. { - \left( {2\Delta_{13} - n^{2} \Delta_{11} } \right)\lambda + n^{2} \Delta_{12} } \right] \\ & \quad - m^{2} \pi^{2} x_{0}^{4} \left\{ {14\Delta_{16} \lambda^{4} - 4\left( {5\Delta_{16} + \Delta_{15} } \right)\lambda^{3} + \left( {9\Delta_{15} - n^{2} \Delta_{10} } \right.} \right.\left. {\, + 3\Delta_{16} + \Delta_{14} } \right)\lambda^{2} \\ & \quad + n^{2} \left( {0.5\Delta_{11} - \Delta_{12} } \right.\left. { + 0.5\Delta_{10} } \right) + \left[ {4\Delta_{13} + 2\Delta_{14} - 3\Delta_{15} - \left( {2\Delta_{11} + \Delta_{10} } \right)n^{2} } \right]\lambda \left. { - \Delta_{13} - \Delta_{14} } \right\} \\ & \quad + 0.25m^{4} \pi^{4} x_{0}^{2} \left( {2\Delta_{14} - 16\lambda^{2} \Delta_{16} - n^{2} \Delta_{10} + 8\lambda \Delta_{15} + \Delta_{16} - \Delta_{15} } \right)\left. { + 0.125\Delta_{16} m^{6} \pi^{6} } \right\rangle \cos^{2} \gamma \\ & \quad + 2\left( {2\lambda - 1} \right)^{2} \left[ {\Delta_{15} \lambda^{3} - \Delta_{16} \lambda^{4} + \left( {\Delta_{14} - n^{2} \Delta_{10} } \right)\lambda^{2} + \left( { - n^{2} \Delta_{11} + \Delta_{13} } \right)\lambda \left. { - n^{2} \Delta_{12} - n^{2} \Delta_{9} } \right]x_{0}^{6} } \right. \\ & \quad + m^{2} \pi^{2} x_{0}^{4} \left\{ {7\Delta_{16} \lambda^{4} - 2\left( {5\Delta_{16} + \Delta_{15} } \right)\lambda^{3} + \left( {4.5\Delta_{15} + \Delta_{14} - n^{2} \Delta_{10} + 3\Delta_{16} } \right)\lambda^{2} } \right. - 0.5\Delta_{14} \\ & \quad + \left[ {2\Delta_{13} - \left( {2\Delta_{11} + \Delta_{10} } \right)n^{2} + \Delta_{14} - 1.5\Delta_{15} } \right]\lambda - n^{4} \Delta_{9} + 0.5\left( {\Delta_{10} - 2\Delta_{12} + \Delta_{11} } \right)n^{2} \left. { - 0.5\Delta_{13} } \right\} \\ & \quad \left. { - 0.25m^{4} \pi^{4} x_{0}^{2} \left( {4\lambda \Delta_{15} - 0.5\Delta_{15} - 8\lambda^{2} \Delta_{16} - n^{2} \Delta_{10} + \Delta_{14} + 0.5\Delta_{16} } \right) - \left. {\Delta_{16} m^{6} \pi^{6} /16} \right\rangle } \right] \\ & \quad - S_{2}^{2} \cot \gamma \left[ {\pi^{2} m^{2} \left( {e^{{2x_{0} \lambda }} - 1} \right) + 8x_{0}^{2} \lambda \left( {\lambda - 1} \right)} \right]e^{{ - 2x_{0} \lambda }} /(16x_{0}^{2} \lambda ) \\ \end{aligned} $$
(27)
$$ \begin{aligned} P_{11} & = - \frac{{e^{{ - x_{0} (1 + 2\lambda )}} S_{2}^{3} \tan \gamma }}{{16x_{0}^{2} \left( {2\lambda + } \right)\left[ {\left( {2\lambda + 1} \right)^{2} x_{0}^{2} + m^{2} \pi^{2} } \right]}}\left\{ {m^{2} \pi^{2} \left[ {2\left( {2\lambda^{2} + 2\lambda - 1} \right)x_{0}^{2} + m^{2} \pi^{2} } \right]e^{{x_{0} (1 + 2\lambda )}} } \right. \\ & \quad + 8x_{0}^{4} \left( {4\lambda^{3} - 3\lambda - 1} \right)\lambda + 4m^{2} \pi^{2} x_{0}^{2} \left( {\lambda^{2} - 3\lambda - 1} \right)\left. { - m^{4} \pi^{4} } \right\} \\ P_{22} & = \frac{{e^{{ - x_{0} (1 + 2\lambda )}} S_{2}^{3} \tan \gamma }}{{4\left( {2\lambda + 1} \right)\left( {x_{0}^{2} (2\lambda + 1)^{2} + m^{2} \pi^{2} } \right)}}\left\langle { - \left\{ {m^{2} \pi^{2} \left[ {1 + e^{{x_{0} (1 + 2\lambda )}} } \right] + 4x_{0}^{2} \lambda (2\lambda + 1)^{2} + 4m^{2} \pi^{2} \lambda } \right\}} \right. \\ & \quad \left. { + 2\frac{{n^{2} }}{{\sin^{2} \gamma }}\left\{ {m^{2} \pi^{2} \left[ {1 - e^{{x_{0} (1 + 2\lambda )}} } \right] + 2\left( {2\lambda + 1} \right)^{2} x_{0}^{2} } \right\}} \right\rangle \\ \end{aligned} $$
(28)
