deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 95b6eb3..bde8166 100644  --- a/untitled.tex  +++ b/untitled.tex 
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Then we get  \begin{align*}  &M^>(\omega)=\frac{2 i e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta,\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{  V_{\beta k}^{*} g_{k}^{h,r}(\epsilon) V_{\alpha k} G^{r}_{\alpha\delta}(\epsilon) V_{\delta q}^* g_{q}^{h,>}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon)) G^{a}_{\nu\beta}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))  \\  &+{\beta &+V_{\beta  k}^{*} g_{k}^{h,r}(\epsilon) V_{\alpha k} G^{>}_{\alpha\delta}(\epsilon) V_{\delta q}^* g_{q}^{h,<}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon)) G^{a}_{\nu\beta}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))  \\   &+V_{\beta k}^{*} g_{k}^{h,>}(\epsilon) V_{\alpha k} G^{a}_{\alpha\delta}(\epsilon) V_{\delta q}^* g_{q}^{h,a}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\epsilon) G^{r}_{\gamma\nu}(\omega+\epsilon)  \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\omega+\epsilon) (f_e(\omega+\epsilon)+f_h(\epsilon))\Biggr\} G^{a}_{\nu\beta}(\omega+\epsilon) (f_e(\omega+\epsilon)+f_h(\omega+\epsilon))\Biggr\}  \end{align*}  We now use the explicit expressions for the self-energies  \begin{eqnarray}  &&M^>(\omega)=\frac{2i \begin{align*}  &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) 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)) \Sigma^{h,>}_{0,\delta\gamma}(\omega+\epsilon) G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\nu\beta}(\omega+\epsilon)](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon))  + \\   \nonumber  &[\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}(\omega+\epsilon)+f_h(\omega+\epsilon))+ (\omega+\epsilon)G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\nu\beta}(\omega+\epsilon)](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}(\omega+\epsilon)+f_h(\omega+\epsilon)) [\Sigma^{>,h}_{0,\beta\alpha}(\epsilon) G^{a}_{\alpha\delta}(\epsilon) \Sigma^{h,a}_{0,\delta\gamma}(\omega+\epsilon) G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}G^{a}_{\nu\beta}(\omega+\epsilon)](f_{e}(\omega+\epsilon)+f_h(\omega+\epsilon))  \Biggr\} \end{eqnarray} \end{align*}  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}(\omega+\epsilon)+f_h(\omega+\epsilon))(1-f_h(\omega+\epsilon)) +