\begin{document}$ t\rightarrow c\gamma $\end{document}, \begin{document}$ t\rightarrow cg $\end{document}, \begin{document}$ t\rightarrow cZ $\end{document}, and \begin{document}$ t\rightarrow ch $\end{document}. They are extremely rare processes in the standard model (SM). As the \begin{document}$ U(1) $\end{document} extension of the minimal supersymmetric standard model (MSSM), the \begin{document}$ U(1)_X $\end{document} SSM features new superfields such as the right-handed neutrinos and three Higgs singlets. We analyzed the effects of different sensitive parameters on the results and made reasonable theoretical predictions, thereby providing a useful reference for future experimental development. Considering the constraint set by the updated experimental data, the numerical results show that the branching ratios of the four processes, i.e., \begin{document}$ t\rightarrow c\gamma,\; cg,\; cZ,\; ch $\end{document}, can reach the same order of magnitude as their experimental upper limits. Among them, \begin{document}$ \tan\beta $\end{document} has the most evident effect on these processes and is the main parameter; \begin{document}$ g_X $\end{document}, \begin{document}$ g_{YX} $\end{document}, \begin{document}$ \mu $\end{document}, \begin{document}$ M_2 $\end{document}, \begin{document}$ \lambda_H $\end{document}, \begin{document}$ M_{U23}^2 $\end{document}, and \begin{document}$ M_{Q23}^2 $\end{document} are also important parameters for the processes, and have effects on the numerical results."> Top-quark rare decays with flavor violation -
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