Four-body Semileptonic Charm Decays $D\to P_1P_2\ell^+\nu_\ell $ Based on SU(3) Flavor Analysis

Motivated by the significant experimental progress in probing semileptonic decays $D\to P_1P_2\ell^+\nu_\ell~(\ell=\mu,e)$, we analyze the branching ratios of the $D\to P_1P_2\ell^+\nu_\ell $ decays with the non-resonant, the light scalar meson resonant and the vector meson resonant contributions in this work. We obtain the hadronic amplitude relations between different decay modes by the SU(3) flavor analysis, and then predict relevant branching ratios of the $D\to P_1P_2\ell^+\nu_\ell$ decays by the present experimental data with $2 \sigma$ errors. Most of our predicted branching ratios are consistent with present experimental data within $2\sigma$ error bars, and others are consistent with the data within $3\sigma$ error bars. We find that some branching ratios of the non-resonant decays are on the order of $\mathcal{O}(10^{-3}-10^{-4})$, the vector meson resonant contributions are dominant in many decays, the non-resonant, the vector meson resonant and the scalar resonant contributions are all important in the $D^0\to\eta\pi^-\ell^+\nu_\ell$ decays, and both the non-resonant and the scalar resonant contributions are important in the $D^0\to K^-K^0\ell^+\nu_\ell,\eta'\pi^-\ell^+\nu_\ell$ and $D^+\to \overline{K}^0K^0\ell^+\nu_\ell,\pi^0\pi^0\ell^+\nu_\ell,\eta\pi^0\ell^+\nu_\ell,\eta'\pi^0\ell^+\nu_\ell$ decays. According to our predictions, many decay modes could be observed in the experiments like BESIII, LHCb and BelleII, and some decay modes might be measured in these experiments in near future.