Study of the axial-vector and tensor resonant contributions to the $D \to VP\ell^+\nu_\ell$ decays based on SU(3) flavor analysis

Semileptonic three-body $D \to M\ell^+\nu_\ell$ decays, non-leptonic $M \to VP$ decays, and semileptonic four-body $D \to M(M \to VP)\ell^+\nu_\ell$ decays are analyzed using the SU(3) flavor symmetry/breaking approach, where $\ell=e/\mu$, $M=A/T$, and $A/T/V/P$ denote the axial-vector/tensor/vector/pseudoscalar mesons, respectively. In terms of SU(3) flavor symmetry/breaking, the decay amplitudes of the $D \to M \ell^+ \nu_\ell$ decays and the vertex coefficients of the $M \to VP$ decays are related. The relevant non-perturbative parameters of the $D \to A\ell^+\nu_\ell$, $A \to VP$ and $T \to VP$ decays are constrained by the present experimental data, and the non-perturbative parameters of $D \to T\ell^+\nu_\ell$ decays are taken from the results in the light-front quark model since no experimental data are available at present. The branching ratios of the $D \to M \ell^+ \nu_\ell $, $M \to VP$, and $D\to M(M\to VP)\ell^+\nu_\ell$ decays are then predicted. We find that some processes receive both tensor and axial-vector resonant contributions, while other processes receive only axial-vector resonant contributions. In cases where both kinds of resonant contributions exist, the axial-vector contributions are dominant. Some branching ratios with axial-vector resonance states are large, and they may be measured experimentally in the near future. In addition, the sensitivities of the branching ratios of $D \to A \ell^+ \nu_\ell $ and $A \to VP$ decays to the parameters are also investigated, and some decay branching ratios are found to be sensitive to the non-perturbative parameters.