Next-to-leading order QCD corrections to the top quark decay via the Flavor-Changing Neutral-Current operators with mixing effects

In this paper detailed calculations of the complete $\mathcal{O}({\alpha }_s)$ corrections to top quark decay widths $\mathrm{\Gamma }(t\to q+V)$ are presented ($V=g,\gamma ,Z$). Besides describing in detail the calculations in our previous paper (arXiv:0810.3889), we also include the mixing effects of the Flavor-Changing Neutral-Current (FCNC) operators for $t\to q+\gamma$ and $t\to q+Z$,whichwerenotconsideredinourpreviouspaper\mathrm{.}Theresultsfor$t\to q+g$ are the same as in our previous paper. But the mixing effects can either be large or small, and increase or decrease the branching ratios for $t\to q+\gamma$and$t\to q+Z$, depending on the values of the anomalous couplings (${\kappa }_{\mathrm{t}\mathrm{q}}^{g,\gamma ,Z}\mathrm{/}\mathrm{\Lambda }$, $f_{\mathrm{t}\mathrm{q}}^{g,\gamma ,Z}$ and $h_{\mathrm{t}\mathrm{q}}^{g,\gamma ,Z}$).