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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} \int \frac{d\omega}{2\pi} G_{\beta G^r_{\beta  \alpha} [V^*_{k\alpha} g^<_{k}(\omega)V_{k\delta} + V_{k\alpha} g^{h,<}_{k}(\omega) V^*_{k\delta}]G_{\delta V^*_{k\delta}]G^a_{\delta  \gamma}] \end{eqnarray}