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\begin{eqnarray}
G^>_{\beta\gamma}(\epsilon) = \sum_{k\alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha k} g^<_{k}(\epsilon)V_{\delta k} + V_{\alpha k} g^{h,<}_{k}(\epsilon) V^*_{\delta k}]G^a_{\delta \gamma}(\epsilon)
\end{eqnarray}
We need to compute the following product of Green functions:
$G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$ $P^>(t,t')=G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$
\begin{eqnarray}
&&G^>_{\beta\gamma}(\epsilon) G_{kq}^{h,<}(\omega+\epsilon) = \sum_{k\alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{k\alpha} g^>_{k}(\omega)V_{\delta k} + V_{\alpha k} g^{h,>}_{k}(\omega) V^*_{\delta k}]G^a_{\delta \gamma}(\epsilon) g_{q}^{h,<}(\omega+\epsilon)\delta_{kq} \\ \nonumber
&+& \sum_{p\beta\gamma\alpha\gamma} 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) [g_{q}^{h,r}(\omega+\epsilon) V_{\gamma q} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta k}^* g_{k}^{h,<}(\omega) \nonumber
...
\end{eqnarray}
We now compute separately the different parts of the previous expression for the ac noise
\begin{eqnarray}
S^{>,1}(\omega)= P^{>,1}(\omega)= \frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{k,q,p\beta\gamma\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_{\beta k} g_{q}^{h,<}(\omega+\epsilon)V^*_{\gamma q}\delta_{kq}
\end{eqnarray}
\begin{eqnarray}
&&S^{>,2}(\omega) &&P^{>,2}(\omega) = \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{k,q,p\beta\gamma\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) \\ \nonumber
&& [V_{\gamma q}^* g_{q}^{h,r}(\omega+\epsilon) V_{\gamma q} G^{<}_{\gamma\beta}(\omega+\epsilon) V_{\beta k} g_{k}^{h,a}(\omega+\epsilon) V_{\beta k}^*]
\end{eqnarray}
\begin{eqnarray}
&&S^{>,3}(\omega)= &&P^{>,3}(\omega)= \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \sum_{k,q,p\beta\gamma\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) \\ \nonumber
&& [V_{\gamma q}^* g_{k}^{h,r}(\omega+\epsilon) V^*_{\gamma q} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\gamma q} g_{q}^{h,a}(\omega+\epsilon) V^*_{\gamma q}]
\end{eqnarray}
\begin{eqnarray}
&&S^{>,4}(\omega)= &&P^{>,4}(\omega)= \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{k,q,p\beta\gamma\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)\\ \nonumber
&& [V_{\beta k} g_{k}^{h,<}(\omega+\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)V^*_{\gamma q}]
\end{eqnarray}
They can be reformulated in terms of self-energies as
\begin{eqnarray}
S^{>,1}(\omega)= 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)
\end{eqnarray}
\begin{eqnarray}
S^{>,2}(\omega) P^{>,2}(\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,r }_{0,\gamma\gamma}(\epsilon+\omega) G^{r}_{\gamma\beta}(\omega+\epsilon) \Sigma^{h,<}_{0,\beta\beta}(\epsilon+\omega) ]
\end{eqnarray}
\begin{eqnarray}
S^{>,3}(\omega)= P^{>,3}(\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,r }_{0,\gamma\gamma}(\epsilon+\omega) G^{<}_{\gamma\beta}(\omega+\epsilon) \Sigma^{h,a}_{0,\beta\beta}(\epsilon+\omega) ]
\end{eqnarray}
\begin{eqnarray}
S^{>,4}(\omega)= P^{>,4}(\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,\gamma\gamma}(\epsilon+\omega) G^{a}_{\gamma\beta}(\omega+\epsilon) \Sigma^{h,a}_{0,\beta\beta}(\epsilon+\omega) ]
\end{eqnarray}
Now we explicitely write down the expressions for the self-energies
\begin{equation}
...
\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{eqnarray}
&&S^{>,1}(\omega)= &&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{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))]
\end{eqnarray}
In the particle-hole case we take $\Gamma(\epsilon)=\Gamma(-\epsilon)$. Besides we consider the WBL and take $\Gamma$ as constants, then
\begin{eqnarray}
&&
S^{>,1}(\omega)= 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
&=& \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}
\\ \nonumber
&& [(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]
\end{eqnarray}
\begin{eqnarray}
&&
S^{>,2}(\omega) P^{>,2}(\omega) = \frac{4e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} \\ \nonumber
&& G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta}(\epsilon) G^a_{\delta \gamma}(\epsilon) [-i\Gamma_{\gamma\gamma}] G^{r}_{\gamma\beta}(\omega+\epsilon) \Gamma_{\beta\beta}[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]
\end{eqnarray}
\begin{eqnarray}
&&
S^{>,3}(\omega)= P^{>,3}(\omega)= \frac{-2i e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} \\ \nonumber
&& G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) [-i\Gamma_{\gamma\gamma}] G^{<}_{\gamma\beta}(\omega+\epsilon) [i\Gamma_{\beta\beta}][(1-f_{e}(\epsilon))+(1-f_{h}(\epsilon))]
\end{eqnarray}
We replace $G^{<}_{\gamma\beta}(\omega+\epsilon) = 2i \sum_{\nu\mu} G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}(f_{e}(\omega+\epsilon)+f_{h}(\omega+\epsilon))G^{a}_{\mu\beta}(\omega+\epsilon)$, then
\begin{eqnarray}
&&
S^{>,3}(\omega)= 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) [-i\Gamma_{\gamma\gamma}] G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon) [i\Gamma_{\beta\beta}]\\ \nonumber && [(1-f_{e}(\epsilon))+(1-f_{h}(\epsilon))] [f_{e}(\epsilon+\omega)+f_{h}(\epsilon+\omega)]
\end{eqnarray}
\begin{eqnarray}
&&S^{>,4}(\omega) &&P^{>,4}(\omega) = \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_{\gamma\gamma} G^{a}_{\gamma\beta}(\omega+\epsilon) [i\Gamma_{\beta\beta}]\\ \nonumber
&&\times[(1-f_{e}(\epsilon))+(1-f_{h}(\epsilon))] f_{h}(\epsilon+\omega)
\end{eqnarray}
Now we collect $S^{>,2}(\omega)+S^{>,4}(\omega)$
\begin{eqnarray}
&&S^{>,2}(\omega)+ &&P^{>,2}(\omega)+ S^{>,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\gamma} \\ \nonumber
&& [G^{r}_{\gamma\beta}(\omega+\epsilon)-G^{a}_{\gamma\beta}(\omega+\epsilon)] \Gamma_{\beta\beta}[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]
\end{eqnarray}
Now we replace $[G^{r}_{\gamma\beta}(\omega+\epsilon)-G^{a}_{\gamma\beta}(\omega+\epsilon)]= -4iG^r_{\gamma\nu}\Gamma_{\nu\mu}(\omega+\epsilon)G^{a}_{\mu\beta}(\omega+\epsilon)$, then
\begin{eqnarray}
&&S^{>,2}(\omega)+ S^{>,4}(\omega) &&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\gamma}[G^r_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu}(\omega+\epsilon)G^{a}_{\mu\beta}(\omega+\epsilon)] \Gamma_{\beta\beta}
\\ \nonumber
&&[(1-f_{e}(\epsilon))f_{h}(\epsilon+\omega)+(1-f_{h}(\epsilon)) f_{h}(\epsilon+\omega)]
\end{eqnarray}
We now define $F_{\tau\tau'} =f_\tau(\epsilon)(1-f_\tau'(\epsilon+\omega))+f_\tau(\epsilon+\omega)(1-f_\tau'(\epsilon))$ with $\tau=e,h$. Then collecting all the terms for
the noise $P$ (including the two pieces
$S^>$ $P^>$ and
$S^<$ $P^<$ we have
\begin{eqnarray}
S^{A}(\omega)= P^{A}(\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\gamma}] G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon) [\Gamma_{\beta\beta}][F_{ee}+F_{hh}+F_{eh}+F_{he}]
\end{eqnarray}
and
\begin{eqnarray}
S^{B}(\omega)= P^{B}(\omega)= \frac{16 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\gamma}] G^{r}_{\gamma\nu}(\omega+\epsilon)\Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon) [\Gamma_{\beta\beta}][F_{eh}+F_{hh}]
\end{eqnarray}
and
\begin{eqnarray}
S^{>,1}(\omega)= P^{>,1}(\omega)= \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} [F_{eh}+F_{hh}]
\end{eqnarray}
with the total
noise as $S=S^A+S^B+S^C$ contribution is $P=P^A+P^B+P^C$
No we compute the second contribution to the noise from $N^>(t,t')=(e^2/h)\sum_{k\beta,q\gamma} V_{\beta k} V^*_{\gamma q}G_{\beta q}^>(t,t') G^{h,<}_{\gamma k}(t',t)$ and $N^<(t,t')=(e^2/h)\sum_{k\beta,q\gamma} V_{\beta k} V^*_{\gamma q}G_{\beta q}^<(t,t') G^{h,>}_{\gamma k}(t',t)$, the total one is then $N=N^>+N^<$. We start with $N^>$ that reads
\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')
...
&& [\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))) \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
$Q^>(t,t')$ $Q^>(t,t')$. For such calculation we employ
\begin{eqnarray}
G^{h,>}_{k\gamma}(\omega+\epsilon) = \sum_\alpha [g^{r,h}_{k}(\omega) V_{\alpha k} G_{\alpha\gamma}^>(\omega)+g^{>,h}_{k}(\omega) V_{\alpha k} G_{\alpha\gamma}^a(\omega)]
\end{eqnarray}
\begin{eqnarray}
G^{<,h}_{q \beta}(\omega) = \sum_\alpha [g^{r,h}_{q}(\omega) V^*_{\alpha q} G_{\alpha\beta}^r(\omega)+g^{r,<}_{q}(\omega) V^*_{\alpha q} G_{\alpha\beta}^a(\omega) ]
\end{eqnarray}
