Revisiting $D$-meson twist-2, 3 distribution amplitudes

Due to the significant difference between the experimental measurements and the theoretical predictions of standard model (SM) for the value of $\mathcal{R}(D)$ of the semileptonic decay $B\to D\ell\bar{\nu}_{\ell}$, people speculate that it may be the evidence of new physics beyond the SM. Usually, the $D$-meson twist-2, 3 distribution amplitudes (DAs) $\phi_{2;D}(x,\mu)$, $\phi_{3;D}^p(x,\mu)$ and $\phi_{3;D}^\sigma(x,\mu)$ are the main error sources when using perturbative QCD factorization and light-cone QCD sum rules to study $B\to D\ell\bar{\nu}_{\ell}$. Therefore, it is important to get more reasonable and accurate behaviors for those DAs. Motivated by our previous work [Phys. Rev. D 104, no.1, 016021 (2021)] on pionic leading-twist DA, we revisit $D$-meson twist-2, 3 DAs $\phi_{2;D}(x,\mu)$, $\phi_{3;D}^p(x,\mu)$ and $\phi_{3;D}^\sigma(x,\mu)$. New sum rules formulae for the $\xi$-moments of these three DAs are suggested to obtain more accurate values. The light-cone harmonic oscillator models for those DAs are improved, and whose model parameters are determined by fitting the values of $\xi$-moments with the least squares method.