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diff --git a/untitled.tex b/untitled.tex
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--- a/untitled.tex
+++ b/untitled.tex
...
We need to compute the following product of Green functions: $P^>(t,t')=e^2/\hbar^2\sum_{k\beta,q\gamma} V_{\beta k}V^*_{\gamma q} G^>_{\beta\gamma}(\epsilon)G_{kq}^{<,h}(\omega+\epsilon)$
\begin{align*}
& \sum_{k q\beta\gamma} V_{\gamma q}^* V_{\beta k} G^>_{\beta\gamma}(\epsilon) G_{kq}^{<,h}(\omega+\epsilon) = \sum_{k,p,q \beta \alpha\delta \gamma} \Biggr\{ V_{\gamma q}^* V_{\beta k} G^r_{\beta \alpha}(\epsilon) [V^*_{p\alpha} g^>_{p}(\epsilon)V_{\delta p} + V_{\alpha p} g^{>,h}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) g_{q}^{<,h}(\omega+\epsilon)\delta_{kq} \\ \nonumber
&+ G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p}
g^>_{p}(\epsilon)V_{ \delta}p g^>_{p}(\epsilon)V_{\delta p} + V_{\alpha p} g^{>,h}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [ V_{\gamma q}^* g_{q}^{r,h}(\omega+\epsilon) V_{\tau q} G^{r}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{<,h}(\omega) V_{\beta k} \nonumber
\\
& + G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p} g^>_{p}(\epsilon)V_{
\delta}p \delta p} + V_{\alpha p} g^{>,h}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [ V_{\gamma q}^* g_{q}^{r,h}(\omega+\epsilon) V_{\tau q} G^{<}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{a,h}(\omega) V_{\beta k}
\nonumber
\\
&+ \sum_{p \alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p} g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p} g^{>,h}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [ V_{\gamma q}^* g_{q}^{<,h}(\omega+\epsilon) V_{\tau q} G^{a}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{a,h}(\omega) V_{\beta k} \Biggr\}
...
\begin{eqnarray}
P^{>,4}(\omega) =\frac{e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{k,q,p\beta\gamma\alpha\delta\tau\theta} G^r_{\beta \alpha}(\epsilon) (\Sigma_{0,\alpha\delta}^{>,e}+ \Sigma_{0,\alpha\delta}^{>,h}) G^a_{\delta \gamma}(\epsilon)\Sigma_{\gamma\tau}^{<,h}(\omega+\epsilon) G_{\tau\theta}^a(\omega+\epsilon)\Sigma_{\theta\beta}^{a,h}(\omega+\epsilon)
\end{eqnarray}
Now we
explicitely explicitly write down the expressions for the self-energies
\begin{equation}
\Sigma^>_{0,\alpha\delta}(\epsilon) = \Sigma^{<,e}_{0,\alpha\delta}(\epsilon)+\Sigma^{<,h}_{0,\alpha\delta}(\epsilon)
\end{equation}
...
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{align*}
&P^{>,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{4e^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))] f_{h}(\omega+\epsilon) \Gamma_{\alpha\delta}(-(\omega+\epsilon))]
\end{align*}
In the particle-hole case we take $\Gamma(\epsilon)=\Gamma(-\epsilon)$. Besides we consider the WBL and take $\Gamma$ as constants, then
\begin{align*}
& P^{>,1}(\omega)= \frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha} G^r_{\beta \alpha}(\epsilon) \Sigma^>_{0,\alpha\delta}(\epsilon) G^a_{\delta
\gamma}(\epsilon)\Sigma^{h,<}_{0,\alpha\delta}(\epsilon+\omega)= \gamma}(\epsilon)\Sigma^{h,<}_{0,\gamma\beta}(\omega+\epsilon)= \frac{4e^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} \Gamma_{\gamma\beta}
\\ \nonumber
&[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]
\end{align*}
...
\begin{eqnarray}
& P^{>,3}(\omega)= \frac{-2i e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta\tau\theta} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) [-i\Gamma_{\gamma\tau}] G^{<}_{\tau\theta}(\omega+\epsilon) [i\Gamma_{\theta\beta}][(1-f_{e}(\epsilon))+(1-f_{h}(\epsilon))]
\end{eqnarray}
We replace
$G^{<}_{\gamma\beta}(\omega+\epsilon) $G^{<}_{\theta\tay}(\omega+\epsilon) = 2i \sum_{\nu\mu}
G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}(f_{e}(\omega+\epsilon)+f_{h}(\omega+\epsilon))G^{a}_{\mu\beta}(\omega+\epsilon)$, G^{r}_{\theta\nu}(\omega+\epsilon)\Gamma_{\nu\mu}(f_{e}(\omega+\epsilon)+f_{h}(\omega+\epsilon))G^{a}_{\mu\tau}(\omega+\epsilon)$, then
\begin{align*}
& P^{>,3}(\omega)= \frac{4 e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta\tau\theta\nu\mu} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) [-i\Gamma_{\gamma\tau}] G^{r}_{\tau\nu}(\omega+\epsilon)\Gamma_{\nu\mu} G^{a}_{\mu\theta}(\omega+\epsilon) [i\Gamma_{\theta\beta}] \\ & [(1-f_{e}(\epsilon))+(1-f_{h}(\epsilon))] [f_{e}(\epsilon+\omega)+f_{h}(\epsilon+\omega)]
\end{align*}
...
P^{>,2}(\omega)+ P^{>,4}(\omega) = -i\frac{4e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta}(\epsilon) G^a_{\delta \gamma}(\epsilon)\Gamma_{\gamma\tau}
\\ [G^{r}_{\tau\theta}(\omega+\epsilon)-G^{a}_{\tau\theta}(\omega+\epsilon)] \Gamma_{\theta\beta}[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]
\end{multline}
Now we replace
$[G^{r}_{\gamma\tau}(\omega+\epsilon)-G^{a}_{\gamma\beta}(\omega+\epsilon)]= -4iG^r_{\gamma\nu}\Gamma_{\nu\mu}(\omega+\epsilon)G^{a}_{\mu\beta}(\omega+\epsilon)$, $[G^{r}_{\tau\theta}(\omega+\epsilon)-G^{a}_{\tau\theta}(\omega+\epsilon)]= -4iG^r_{\tau\nu}\Gamma_{\nu\mu}(\omega+\epsilon)G^{a}_{\theta\beta}(\omega+\epsilon)$, then
\begin{align*}
&P^{>,2}(\omega)+ P^{>,4}(\omega) = \frac{-16e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta\mu\nu} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta}(\epsilon) \\
& G^a_{\delta \gamma}(\epsilon)\Gamma_{\gamma\tau}[G^r_{\tau\nu}(\omega+\epsilon)\Gamma_{\nu\mu}(\omega+\epsilon)G^{a}_{\mu\theta}(\omega+\epsilon)]
\Gamma_{\theta\beta}[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) \Gamma_{\theta\beta}[f_{e}(\epsilon)(1-f_{h}(\epsilon+\omega))+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]
\end{align*}
We now define $F_{\tau\tau'}
=f_\tau(\epsilon)(1-f_\tau'(\epsilon+\omega))+f_\tau(\epsilon+\omega)(1-f_\tau'(\epsilon))$ =f_\tau(\epsilon)(1-f_{\tau'}(\epsilon+\omega))+f_\tau(\epsilon+\omega)(1-f_{\tau'}(\epsilon))$ with $\tau=e,h$. Then collecting all the terms for $P$ (including the two pieces $P^>$ and $P^<$ we have)
\begin{eqnarray}
P^{A}(\omega)=
\frac{4 P^3(\omega)=\frac{4 e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta\nu\mu} G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) [\Gamma_{\gamma\tau}] G^{r}_{\tau\nu}(\omega+\epsilon)\Gamma_{\nu\mu} G^{a}_{\mu\theta}(\omega+\epsilon) [\Gamma_{\theta\beta}][F_{ee}+F_{hh}+F_{eh}+F_{he}]
\end{eqnarray}
and
\begin{eqnarray}
...
\end{eqnarray}
and
\begin{eqnarray}
P^{>,1}(\omega)= P^C=P^{1}(\omega)= \frac{4e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta\tau\theta} [G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) \Gamma_{\alpha\delta}
[F_{eh}+F_{hh}] [F_{he}+F_{hh}]
\end{eqnarray}
with the total contribution is $P=P^A+P^B+P^C$.
...
\begin{equation}
N^>(t,t')=(e^2/h)\sum_{k\beta,q\gamma} V_{\beta k} V^*_{\gamma q}G_{\beta q}^>(t,t') G^{h,<}_{\gamma k}(t',t),\quad\, N^<(t,t')=(e^2/h)\sum_{k\beta,q\gamma} V_{\beta k} V^*_{\gamma q}G_{\beta q}^<(t,t') G^{h,>}_{\gamma k}(t',t).
\end{equation}
The total one is then $N=N^>+N^<$. We start with $N^>$
that reads then
\begin{eqnarray}
G^>_{\beta q}(t,t') = \frac{1}{h} \sum_\gamma \int dt_1 [G_{\beta\gamma}^r(t,t_1) V_{\gamma q} g^{>}_{q}(t_1,t')+ G_{\beta\gamma}^>(t,t_1) V_{\gamma q} g^{a}_{q}(t_1,t')]\,,
\end{eqnarray}
...
Then, we have
\begin{align*}
&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} V_{\beta k} V^*_{\gamma q}
[G_{\beta\nu}^r(\omega+\epsilon) [G_{\beta\nu}^r(\epsilon) V_{\nu q}
g^{>}_{q}(\omega+\epsilon)+ G_{\beta\nu}^>(\omega+\epsilon) g^{>}_{q}(\epsilon)+ G_{\beta\nu}^>(+\epsilon) V_{\nu q} g^{a}_{q}(\omega+\epsilon)]
\\ \nonumber
&[G_{\gamma\mu}^r(\epsilon) &[G_{\gamma\mu}^r(\omega+\epsilon) V^*_{\mu k}
g^{<,h}_{k}(\epsilon)+ G_{\gamma\mu}^<(\epsilon) g^{<,h}_{k}(\omega+\epsilon)+ G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k}
g^{a,h}_{k}(\epsilon)] g^{a,h}_{k}(\omega+\epsilon)]
\end{align*}
\begin{align*}
...
Inserting the expressions for the self-energies we get
\begin{align*}
&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}
[G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon)
G_{\gamma\mu}^r(\epsilon+\omega) \Sigma_{0,\mu\beta}^{<,h}(\epsilon+\omega)]+[G_{\beta\nu}^r(\epsilon) G_{\gamma\mu}^r(\omega+\epsilon+) \Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon+)]+[G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon)
G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)] G_{\gamma\mu}^<(\omega+\epsilon) \Sigma_{0,\mu\beta}^{a,h}(\omega+\epsilon)]
\\
&+[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon) G_{\gamma\mu}^r(\omega+\epsilon)\Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon)]+[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon)
G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)] G_{\gamma\mu}^<(\omega+\epsilon+) \Sigma_{0,\mu\beta}^{a,h}(\omega+\epsilon)]
\end{align*}
\begin{align*}
