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\begin{eqnarray}  &&N(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{  [G_{\beta\nu}^r(\omega+\epsilon) \Gamma_{\nu\gamma} G_{\gamma\mu}^r(\omega) \Gamma_{\mu\beta} (1-f_e(\omega+\epsilon) f_{h}(\omega)] \\ \nonumber  &&\sum_{\lambda\delta}[G_{\beta\nu}^r(\omega+\epsilon) \Gamma_{\nu\gamma} G_{\gamma\lambda}^a(\omega) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_e(\omega+\epsilon)f_{h}(\omega))] \Gamma_{\lambda\delta}G^a_{\delta\mu}(\omega+\epsilon)[i\Gamma_{\mu\beta}](1-f_e(\omega+\epsilon)(f_{h}(\omega)+f_e(\omega))]  \\ \nonumber &&\sum_{\lambda\delta} G_{\beta\lambda}^r(\omega+\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\omega+\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\mu}^r(\omega)(1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))f_h(\omega+\epsilon)\\ \nonumber  &&\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\omega+\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\omega+\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\theta}^r(\omega)\Gamma_{\theta\tau} G_{\tau\mu}^a(\omega)[i\Gamma_{\mu\beta}](1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))(f_{e}(\omega+\epsilon)+f_{h}(\omega+\epsilon))] \Biggr\}  \end{eqnarray}