deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 38af672..01b3b37 100644  --- a/untitled.tex  +++ b/untitled.tex 
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M^>(\omega)=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k}^{*}V_{\gamma q}\int \frac{d\epsilon}{2\pi} [G^{h,>}_{kq}(\epsilon) G^{<}_{\gamma\beta}(\omega+\epsilon)   \end{eqnarray}  We replace now  \begin{eqnarray}  &&G_{kq}^{h,>}(\omega+\epsilon) \begin{align*}  &G_{kq}^{h,>}(\epsilon)  = \sum_{\alpha\delta} [g_{k}^{h,r}(\omega+\epsilon) [g_{k}^{h,r}(\epsilon)  V_{\alpha k} G^{r}_{\alpha\delta}(\omega+\epsilon)V_{\delta G^{r}_{\alpha\delta}(\epsilon)V_{\delta  q}^* g_{q}^{h,>}(\omega) +g_{k}^{h,r}(\omega+\epsilon) g_{q}^{h,>}(\epsilon) +g_{k}^{h,r}(\epsilon)  V_{\alpha k} G^{>}_{\alpha\delta}(\omega+\epsilon)V_{\delta G^{>}_{\alpha\delta}(\epsilon)V_{\delta  q}^* g_{q}^{h,a}(\omega+\epsilon)] g_{q}^{h,a}(\epsilon)]  \\  \nonumber  &&g_{k}^{h,>}(\omega+\epsilon) &+g_{k}^{h,>}(\epsilon)  V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta G^{a}_{\gamma\beta}(\epsilon)V_{\beta  q}^* g_{q}^{h,a}(\omega) g_{q}^{h,a}(\epsilon)  ]\,, \end{eqnarray} \end{align*}  Then we get  \begin{eqnarray}  &&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}(\omega+\epsilon) g_{k}^{h,r}(\epsilon)  V_{\alpha k} G^{r}_{\alpha\delta}(\omega+\epsilon) G^{r}_{\alpha\delta}(\epsilon)  V_{\delta q}^* g_{q}^{h,>}(\omega+\epsilon) g_{q}^{h,>}(\epsilon)  V_{\gamma q} G^{r}_{\gamma\nu}(\epsilon) G^{r}_{\gamma\nu}(\omega+\epsilon)  \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\epsilon)(f_e(\epsilon)+f_h(\epsilon))+ G^{a}_{\gamma\nu}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))+  \\ \nonumber &&  V_{\beta k}^{*} g_{k}^{h,r}(\omega+\epsilon) g_{k}^{h,r}(\epsilon)  V_{\alpha k} G^{>}_{\alpha\delta}(\omega+\epsilon) G^{>}_{\alpha\delta}(\epsilon)  V_{\delta q}^* g_{q}^{h,<}(\omega+\epsilon) g_{q}^{h,<}(\epsilon)  V_{\gamma q} G^{r}_{\gamma\nu}(\epsilon) G^{r}_{\gamma\nu}(\omega+\epsilon)  \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\epsilon)(f_e(\epsilon)+f_h(\epsilon))\\ G^{a}_{\gamma\nu}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))\\  \nonumber &&  V_{\beta k}^{*} g_{k}^{h,>}(\omega+\epsilon) g_{k}^{h,>}(\epsilon)  V_{\alpha k} G^{a}_{\alpha\delta}(\omega+\epsilon) G^{a}_{\alpha\delta}(\epsilon)  V_{\delta q}^* g_{q}^{h,a}(\omega+\epsilon) g_{q}^{h,a}(\epsilon)  V_{\gamma q} G^{r}_{\gamma\nu}(\epsilon) \Gamma_{\nu\mu} G^{a}_{\gamma\nu}(\epsilon) (f_e(\epsilon)+f_h(\epsilon))\Biggr\} G^{a}_{\gamma\nu}(\omega+\epsilon) (f_e(\omega+\epsilon)+f_h(\epsilon))\Biggr\}  \end{eqnarray}  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}(\omega+\epsilon) G^{r}_{\alpha\delta}(\omega+\epsilon) [\Sigma^{r,h}_{0,\beta\alpha}(\epsilon) G^{r}_{\alpha\delta}(\epsilon)  \Sigma^{h,>}_{0,\delta\gamma} G^{r}_{\gamma\nu}(\epsilon)\Gamma_{\nu\mu}](f_{e}(\epsilon)+f_h(\epsilon)) 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) 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))+