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 $r_A=(-11.2\pm {1.0}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.9}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\times {10}^{-2}$ and $r_V=(-4.3\pm {1.0}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {2.5}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\times {10}^{-2}$. This is the first in the semileptonic decays of heavy mesons into axial-vector mesons. The angular analysis yields an up-down asymmetry ${\mathcal{A}}_{ud}^{\mathrm{\prime }}=0.01\pm 0.11$, 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 $(2.27\pm {0.11}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.07}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}}\pm {0.07}_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}})\times {10}^{-3}$ and $(1.02\pm {0.06}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.06}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}}\pm {0.03}_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}})\times {10}^{-3}$, 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 $1.4\times {10}^{-4}$ and $0.7\times {10}^{-4}$ at 90% confidence level, respectively.