\begin{document}$ D\to\gamma \,\ell \,\nu $\end{document} decay. For the first time, we provide the analytic expressions of next-to-leading power contributions and the error estimation associated with the power expansion of \begin{document}$ {\cal O}(\Lambda_{\rm QCD}/m_c) $\end{document}. In our calculation, we adopt two different models of the D-meson distribution amplitudes \begin{document}$ \phi_{D,\rm I}^+ $\end{document} and \begin{document}$ \phi_{D,\rm II}^+ $\end{document}. Within the framework of QCD factorization as well as the dispersion relation, we evaluate the soft contribution up to the next-to-leading logarithmic accuracy and also consider the higher-twist contribution from the two-particle and three-particle distribution amplitudes. Finally, we find that all the sub-leading power contributions are significant at \begin{document}$ \lambda_D(\mu_0) = 354 $\end{document} MeV, and the next-to-leading power contributions lead to 143% in \begin{document}$ \phi_{D,\rm I}^+ $\end{document} and 120% in \begin{document}$ \phi_{D,\rm II}^+ $\end{document} corrections to leading power vector form factors with \begin{document}$ E_{\gamma} = 0.5 $\end{document} GeV. As the corrections from the higher-twist and local sub-leading power contributions are enhanced with increasing inverse moment, it is difficult to extract an appropriate inverse moment of the D-meson distribution amplitude. The predicted branching fractions are \begin{document}$ (1.88_{-0.29}^{+0.36})\times10^{-5} $\end{document} for \begin{document}$ \phi_{D,\rm I}^+ $\end{document} and \begin{document}$ (2.31_{-0.54}^{+0.65})\times10^{-5} $\end{document} for \begin{document}$ \phi_{D,\rm II}^+ $\end{document}."> Factorization of radiative leptonic <i>D</i>-meson decay with sub-leading power corrections -
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