Factorization of radiative leptonic D-meson decay with sub-leading power corrections

Abstract

In this work, we calculate the sub-leading power contributions to the radiative leptonic $D\to \gamma \ell \nu$ 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 $\mathcal{O}({\Lambda }_{\mathrm{QCD}}/m_c)$. In our calculation, we adopt two different models of the $D$-meson distribution amplitudes ${\varphi }_{D,\mathrm{I}}^{+}$ and ${\varphi }_{D,\mathrm{II}}^{+}$. Within the framework of the QCD factorization as well as the dispersion relation, we evaluate the soft contribution up to the next-to-leading logarithmic accuracy, and the higher-twist contribution from the two-particle and three-particle distribution amplitudes is also considered. Finally, we find that all the sub-leading power contributions are significant at ${\lambda }_D({\mu }_0)=354\mathrm{MeV}$, and the next-to-leading power contributions will lead to $143\%$ in ${\varphi }_{D,\mathrm{I}}^{+}$ and $120\%$ in ${\varphi }_{D,\mathrm{II}}^{+}$ corrections to leading power vector form factors with $E_{\gamma }=0.5\mathrm{GeV}$. As the corrections from the higher-twist and local sub-leading power contributions will be enhanced with the growing of the inverse moment, it is difficult to extract an appropriate inverse moment of the $D$-meson distribution amplitude. The predicted branching fractions are $(1.88_{-0.29}^{+0.36})\times 10^{-5}$ for ${\varphi }_{D,\mathrm{I}}^{+}$ and $(2.31_{-0.54}^{+0.65})\times 10^{-5}$ for ${\varphi }_{D,\mathrm{II}}^{+}$, respectively.

Comments: 19 pages, 7 figures, to be published in Chinese Physics C