Semileptonic and nonleptonic decays of $D$ into tensor mesons with light-cone sum rule

Form factors of $D$ decays into $J^{PC}=2^{++}$ tensor mesons are calculated in the light-cone sum rules approach up to twist-4 distribution amplitudes of the tensor meson. The masses of the tensor mesons are comparable to that of the charm quark mass $m_c$; therefore all terms including powers of $m_T/m_c$ are kept out in the expansion of the two-particle distribution amplitude $\langle T|\bar q_{1\alpha}(x) \, q_{2\delta}(0)|0\rangle$. Branching ratios of the semileptonic $D \to T\,\mu \,\bar{\nu}_{\mu}$ decays and nonleptonic $D\to T~ P ~(P=K, \pi)$ decays are taken into consideration. A comparison is also made between our results and predictions of other methods and the existing experimental values for the nonleptonic case. The semileptonic branching ratios are typically of the order of $10^{-5}$, and the nonleptonic ones show better agreement with the experimental data in comparison to the Isgur-Scora-Grinstein-Wise predictions.