deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 194fc1a..8d6d555 100644  --- a/untitled.tex  +++ b/untitled.tex 
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\end{eqnarray}  Then  \begin{align*}  \frac{ &Q^>(\omega)\frac{  e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha} \int \frac{d\epsilon}{2\pi}\\ &[g^{r,h}_{k}(\epsilon) &V_{\beta k}^* V_{\gamma q}[g^{r,h}_{k}(\epsilon)  V_{\alpha k} G_{\alpha\gamma}^>(\epsilon)+g^{>,h}_{k}(\epsilon) V_{\alpha k} G_{\alpha\gamma}^a(\epsilon)] [ V_{\gamma q} g^{r}_{q}(\omega+\epsilon) V^*_{\alpha q} G_{\alpha\beta}^<(\omega+\epsilon)+ V_{\gamma q} g^{<}_{q}(\epsilon) V^*_{\alpha q} G_{\alpha\beta}^a(\omega+\epsilon) ] \end{align*}  Then we have  \begin{align*}  &Q^>(\omega) = \frac{ e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha} \int \frac{d\epsilon}{2\pi}[\Sigma_{0,\beta\alpha}^r (\omega+\epsilon)G_{\alpha\gamma}^>(\omega+\epsilon)+ (\epsilon)G_{\alpha\gamma}^>(\epsilon)+  \Sigma_{0,\beta\alpha}^> (\omega+\epsilon) G_{\alpha\gamma}^a(\omega+\epsilon)] (\epsilon) G_{\alpha\gamma}^a(\epsilon)]  \\  &[\Sigma^r_{0,\gamma \alpha}(\epsilon) G_{\alpha\beta}^<(\epsilon)+ \alpha}(\omega+\epsilon) G_{\alpha\beta}^<(\omega+\epsilon)+  \Sigma^<_{0,\gamma \alpha}(\epsilon)G_{\alpha\beta}^a(\omega)] \alpha}(\omega+\epsilon)G_{\alpha\beta}^a(\omega+\epsilon)]  \end{align*}  We can split $Q^>(\omega)$ in four contributions  \begin{eqnarray}  Q^{>,1}(\omega) = \frac{ 4e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\mu\nu\tau,\delta\lambda} \int \frac{d\epsilon}{2\pi}[-i\Gamma_{\beta\alpha}]G_{\alpha\delta}^r(\omega+\epsilon)\Gamma_{\delta\lambda} G_{\lambda\gamma}^a(\omega+\epsilon) [-i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\epsilon)\Gamma_{\mu\nu} G_{\nu\beta}^a(\epsilon)((1-f_h(\omega+\epsilon)+(1-f_e(\omega+\epsilon)))(f_e(\epsilon)+f_h(\epsilon)) \frac{d\epsilon}{2\pi}[-i\Gamma_{\beta\alpha}]G_{\alpha\delta}^r(\epsilon)\Gamma_{\delta\lambda} G_{\lambda\gamma}^a(\epsilon) [-i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\omega+\epsilon)\Gamma_{\mu\nu} G_{\nu\beta}^a(\omega+\epsilon)((1-f_h(\epsilon)+(1-f_e(\epsilon)))(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))  \end{eqnarray}  \begin{eqnarray}  Q^{>,2}(\omega) = \frac{ 4e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\mu\nu\tau} \int \frac{d\epsilon}{2\pi}[-i\Gamma_{\beta\alpha}]G_{\alpha\gamma}^a(\omega+\epsilon) [-i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\epsilon)\Gamma_{\mu\nu} G_{\nu\beta}^a(\epsilon) (1-f_h(\omega+\epsilon))(f_e(\epsilon)+f_h(\epsilon)) \frac{d\epsilon}{2\pi}[-i\Gamma_{\beta\alpha}]G_{\alpha\gamma}^a(\epsilon) [-i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\omega+\epsilon)\Gamma_{\mu\nu} G_{\nu\beta}^a(\omega+\epsilon) (1-f_h(\epsilon))(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))  \end{eqnarray}  \begin{eqnarray}  Q^{>,3}(\omega) = \frac{ 4e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\mu\nu\tau} \int \frac{d\epsilon}{2\pi} [-i\Gamma_{\beta\alpha}]G_{\alpha\delta}^r(\omega+\epsilon)\Gamma_{\delta\lambda} G_{\lambda\gamma}^a(\omega+\epsilon) \Gamma_{\gamma\tau}G^a_{\tau\beta}(\epsilon)((1-f_h(\omega+\epsilon)+(1-f_e(\omega+\epsilon)))f_e(\epsilon) [-i\Gamma_{\beta\alpha}]G_{\alpha\delta}^r(\epsilon)\Gamma_{\delta\lambda} G_{\lambda\gamma}^a(\epsilon) \Gamma_{\gamma\tau}G^a_{\tau\beta}(\omega+\epsilon)((1-f_h(\epsilon)+(1-f_e(\epsilon)))f_e(\omega+\epsilon)  \end{eqnarray}  \begin{eqnarray}  Q^{>,4}(\omega) = \frac{ 4e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\mu\nu\tau } \int \frac{d\epsilon}{2\pi}[\Gamma_{\beta\alpha}]G_{\alpha\gamma}^a(\omega+\epsilon) [-i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\epsilon)\Gamma_{\mu\nu} G_{\nu\beta}^a(\epsilon)(f_h(\omega+\epsilon)+f_e(\omega+\epsilon))(1-f_h(\epsilon)) \frac{d\epsilon}{2\pi}[\Gamma_{\beta\alpha}]G_{\alpha\gamma}^a(\epsilon) [-i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\omega+\epsilon)\Gamma_{\mu\nu} G_{\nu\beta}^a(\omega+\epsilon)(f_h(\omega+\epsilon)+f_e(\omega+\epsilon))(1-f_h(\epsilon))  \end{eqnarray}