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\end{align*}  We now use the explicit expressions for the self-energies  \begin{eqnarray}  &&M^>(\omega)=\frac{2i e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{ [\Sigma^{r,h}_{0,\beta\alpha}(\epsilon) \\  &[\Sigma^{r,h}_{0,\beta\alpha}(\epsilon)  G^{r}_{\alpha\delta}(\epsilon) \Sigma^{h,>}_{0,\delta\gamma} G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon)) + \\   \nonumber  &&[\Sigma^{r,h}_{0,\beta\alpha}(\omega+\epsilon) &[\Sigma^{r,h}_{0,\beta\alpha}(\omega+\epsilon)  G^{>}_{\alpha\delta}(\omega+\epsilon) \Sigma^{h,a}_{0,\delta\gamma} G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\epsilon)+f_h(\epsilon))+ G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon))+  \\   \nonumber  && &  [\Sigma^{>,h}_{0,\beta\alpha}(\omega+\epsilon) G^{a}_{\alpha\delta}(\omega+\epsilon) \Sigma^{h,a}_{0,\delta\gamma} G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\epsilon)+f_h(\epsilon)) G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon))  \Biggr\} \end{eqnarray}  Then we obtain,   \begin{eqnarray}  &&M^>(\omega)=\frac{4 e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{ [-i\Gamma_{\beta\alpha}] G^{r}_{\alpha\delta}(\omega+\epsilon) \Gamma_{\delta\gamma} G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\epsilon)+f_h(\epsilon))(1-f_h(\omega+\epsilon)) G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon))(1-f_h(\omega+\epsilon))  + \\   \nonumber  &&[\sum_{\theta\tau} [-i\Gamma_{\beta\alpha}] G^{r}_{\alpha\theta}(\omega+\epsilon)\Gamma_{\theta\tau}G^{a}_{\tau\delta}(\omega+\epsilon) [i\Gamma_{\delta\gamma}] G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\epsilon)+f_h(\epsilon)) (1-f_{e}(\epsilon+\omega)+1-f_h(\epsilon+\omega)) G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}G ^{a}_{\mu\beta}(\epsilon)](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon)) (1-f_{e}(\epsilon)+1-f_h(\epsilon))  + \\   \nonumber  && [\Gamma_{\beta\alpha}(\omega+\epsilon) G^{a}_{\alpha\delta}(\omega+\epsilon) [i\Gamma_{\delta\gamma}] G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\epsilon)+f_h(\epsilon)(1-f_h(\omega+\epsilon))) G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon)(1-f_h(\epsilon)))  \Biggr\} \end{eqnarray}  Again, the "lesser" term for $M(t,t')$ is obtained by exchanging $1-f$ by $f$ and viceversa.  The last term that we need to compute is $Q>(t,t')+Q<(t,t')= G^{h,>}_{k\gamma}(t,t')G^{<}_{q \beta}(t',t)+ G^{h,<}_{k\gamma}(t,t')G^{>}_{q \beta}(t',t)$. We only calculate $Q^>(t,t')$. For such calculation we employ