First Measurement of the Decay Dynamics in the Semileptonic Transition of the $D^{+(0)}$ into the Axial-vector Meson $\bar K_1(1270)$

Using $e^+e^-$ data taken at the center-of-mass energy of 3.773 GeV with the BESIII detector, corresponding to an integrated luminosity of 20.3 fb$^{-1}$, we report the first measurement of the decay dynamics of the semileptonic decays $D^{+(0)}\to K^-\pi^+\pi^{0(-)} e^+\nu_e$. The amplitude analysis gives the hadronic form factors of the semileptonic $D$ transitions into the axial-vector meson $\bar{K}_1(1270)$ to be $r_A=(-11.2\pm1.0_{\rm stat}\pm0.9_{\rm syst})\times10^{-2}$ and $r_V = (-4.3\pm 1.0_{\rm stat}\pm2.5_{\rm syst})\times 10^{-2}$. This is the first in the semileptonic decays of heavy mesons into axial-vector mesons. The angular analysis yields an up-down asymmetry $\mathcal{A}^\prime_{ud} = 0.01\pm0.11$, which is consistent with the Standard Model prediction. In addition, the branching fractions of $D^+\to \bar K_1(1270)^0 e^+\nu_e$ and $D^0\to K_1(1270)^- e^+\nu_e$ are determined with improved precision to be $(2.27\pm0.11_{\rm stat}\pm0.07_{\rm syst}\pm0.07_{\rm input})\times10^{-3}$ and $(1.02\pm0.06_{\rm stat}\pm0.06_{\rm syst}\pm0.03_{\rm input})\times10^{-3}$, respectively. No significant signals of $D^+\to \bar K_1(1400)^0 e^+\nu_e$ and $D^0\to K_1(1400)^- e^+\nu_e$ are observed and their branching fraction upper limits are set as $1.4\times10^{-4}$ and $0.7\times10^{-4}$ at 90\% confidence level, respectively.