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Similar equations are hold for $\Sigma^{>,e }(\epsilon)= 2i [1-f_{e}(\epsilon)] \Gamma_{\alpha\delta}(\epsilon)$ and$\Sigma^{>,h}(\epsilon)= 2i [1-f_{h}(\epsilon)] \Gamma_{\alpha\delta}(-\epsilon)$. Then,  \begin{eqnarray}  &&S^{>,1}(\omega)= \frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Sigma^>_{0,\alpha\delta}(\epsilon) G^a_{\delta \gamma}(\epsilon)\Sigma^{h,<}_{0,\alpha\delta}(\epsilon+\omega) \\ \nonumber  && = 2 i -4  \frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) [ (1-f_{e}(\epsilon)) \Gamma_{\alpha\delta}(\epsilon) + (1-f_{h}(\epsilon)) \Gamma_{\alpha\delta}(-\epsilon)] G^a_{\delta \gamma}(\epsilon) f_{h}(\epsilon) \Gamma_{\alpha\delta}(-(\epsilon+\omega))] \end{eqnarray}  In the particle-hole case we take $\Gamma(\epsilon)=\Gamma(-\epsilon)$. Besides we consider the WBL and take $\Gamma$ as constants, then  \begin{eqnarray}  &&S^{>,1}(\omega)= \frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Sigma^>_{0,\alpha\delta}(\epsilon) G^a_{\delta \gamma}(\epsilon)\Sigma^{h,<}_{0,\alpha\delta}(\epsilon+\omega) \\ \nonumber  && = -4\frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} [G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) \Gamma_{\alpha\delta} [(1-f_{e}(\epsilon))f_{h}(\epsilon)+(1-f_{h}(\epsilon)) f_{h}(\epsilon)]  \end{eqnarray}