First Measurement of the Decay Dynamics in the Semileptonic Transition of $ {\mathrm{D}}^{+(0)}$ into the Axial-Vector Meson $ {\overline{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\text{}\text{}{\mathrm{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 $ {\overline{K}}_{1}(1270)$ to be $ {r}_{A}=(-11.2\pm {1.0}_{\mathrm{stat}}\pm {0.9}_{\mathrm{syst}})\times {10}^{-2}$ and $ {r}_{V}=(-4.3\pm {1.0}_{\mathrm{stat}}\pm {2.5}_{\mathrm{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}}_{\mathrm{ud}}^{\prime }=0.01\pm 0.11$, which is consistent with the standard model prediction. In addition, the branching fractions of $ {D}^{+}\to {\overline{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 0.1{1}_{\mathrm{stat}}\pm 0.0{7}_{\mathrm{syst}}\pm 0.0{7}_{\text{input}})\times {10}^{-3}$ and $ (1.02\pm 0.0{6}_{\mathrm{stat}}\pm 0.0{6}_{\mathrm{syst}}\pm 0.0{3}_{\text{input}})\times {10}^{-3}$, respectively. No significant signals of $ {D}^{+}\to {\overline{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.