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...
\end{eqnarray}
On the other hand we have for $G^>_{\beta\gamma}(\epsilon)$ (accordingly with J.S note)
\begin{eqnarray}
G^>_{\beta\gamma}(\epsilon) = \sum_{k\alpha\delta} G^r_{\beta \alpha}(\epsilon)
[V^*_{k\alpha} g^<_{k}(\epsilon)V_{k\delta} [V^*_{\alpha k} g^<_{k}(\epsilon)V_{\delta k} +
V_{k\alpha} V_{\alpha k} g^{h,<}_{k}(\epsilon)
V^*_{k\delta}]G^a_{\delta V^*_{\delta k}]G^a_{\delta \gamma}(\epsilon)
\end{eqnarray}
We need to compute the following product of Green functions: $G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$
\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_{k\delta} g^<_{k}(\omega)V_{\delta k} +
V_{k\alpha} V_{\alpha k} g^{h,<}_{k}(\omega)
V^*_{k\delta}]G^a_{\delta 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^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta} [V^*_{\alpha p} g^<_{p}(\epsilon)V_{ \delta}p +
V_{p\alpha} V_{\alpha p} g^{h,<}_{p}(\epsilon)
V^*_{p\delta}]G^a_{\delta V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,<}(\omega) \nonumber
\\
&&+G^r_{\beta \alpha}(\epsilon)
[V^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta} [V^*_{\alpha p} g^<_{p}(\epsilon)V_{ \delta}p +
V_{p\alpha} V_{\alpha p} g^{h,<}_{p}(\epsilon)
V^*_{p\delta}]G^a_{\delta V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)]\\ \nonumber
&&+G^r_{\beta \alpha}(\epsilon)
[V^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta} [V^*_{\alpha p} g^<_{p}(\epsilon)V_{ \delta}p +
V_{p\alpha} V_{\alpha p} g^{h,<}_{p}(\epsilon)
V^*_{p\delta}]G^a_{\delta V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon)[g_{k}^{h,<}(\omega+\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)]\,,
\end{eqnarray}
We now compute separately the different parts of the previous expression for the ac noise
\begin{eqnarray}
S^{>,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^*_{p\alpha} g^<_{p}(\omega)V_{p\delta} [V^*_{\alpha p} g^<_{p}(\epsilon)V_{ \delta}p +
V_{p\alpha} g^{h,<}_{p}(\omega) V^*_{k\delta}]G^a_{\delta 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)= \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^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta} [V^*_{\alpha p} g^<_{p}(\epsilon)V_{ \delta}p +
V_{p\alpha} V_{\alpha p} g^{h,<}_{p}(\epsilon)
V^*_{p\delta}]G^a_{\delta V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [V^*_{\beta k} g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)V_{\gamma q}]
\end{eqnarray}
\begin{eqnarray}
S^{>,3}(\omega)= \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \sum_{k,q,p\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon)
[V^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta} [V^*_{\alpha p} g^<_{p}(\epsilon)V_{ \delta}p +
V_{p\alpha} V_{\alpha p} g^{h,<}_{p}(\epsilon)
V^*_{p\delta}]G^a_{\delta V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [V^*_{\beta k} g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)V_{\gamma q}]
\end{eqnarray}
\begin{eqnarray}
S^{>,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^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta} \alpha}(\epsilon)[V^*_{\alpha p} g^<_{p}(\epsilon)V_{ \delta}p +
V_{p\alpha} V_{\alpha p} g^{h,<}_{p}(\epsilon)
V^*_{p\delta}]G^a_{\delta V^*_{\delta p}] G^a_{\delta \gamma}(\epsilon)[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}
