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

Semileptonic three-body \(D \rightarrow M\ell ^+\nu _\ell \) decays, non-leptonic \(M \rightarrow VP\) decays, and semileptonic four-body \(D \rightarrow M(M \rightarrow VP)\ell ^+\nu _\ell \) 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 \(D \rightarrow M \ell ^+ \nu _\ell \) decays and the vertex coefficients of the \(M \rightarrow VP\) decays are related. The relevant non-perturbative parameters of the \(D \rightarrow A\ell ^+\nu _\ell \), \(A \rightarrow VP\), and \(T \rightarrow VP\) decays are constrained by the present experimental data, and the non-perturbative parameters of \(D \rightarrow T\ell ^+\nu _\ell \) 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 \(D \rightarrow M \ell ^+ \nu _\ell \), \(M \rightarrow VP\), and \(D\rightarrow M(M\rightarrow VP)\ell ^+\nu _\ell \) 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 \(D \rightarrow A \ell ^+ \nu _\ell \) and \(A \rightarrow VP\) decays to the parameters are also investigated, and some decay branching ratios are found to be sensitive to the non-perturbative parameters.