Observation of the semileptonic decays ${D}^{0}\to {K}_{S}^{0}{\pi }^{-}{\pi }^{0}{e}^{+}{\nu }_{e}$ and ${D}^{+}\to {K}_{S}^{0}{\pi }^{+}{\pi }^{-}{e}^{+}{\nu }_{e}$

By analyzing e + e − annihilation data corresponding to an integrated luminosity of 2.93 fb −1 collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, the first observation of the semileptonic decays ${D}^{0}\to {K}_{S}^{0}{\pi }^{-}{\pi }^{0}{e}^{+}{\nu }_{e}$ and ${D}^{+}\to {K}_{S}^{0}{\pi }^{+}{\pi }^{-}{e}^{+}{\nu }_{e}$ is reported. In the hypothesis that all events correspond to K 1(1270) decays, the branching fractions are measured to be $B\left({D}^{0}\to {K}_{1}{\left(1270\right)}^{-}\left(\to {K}_{S}^{0}{\pi }^{-}{\pi }^{0}\right){e}^{+}{\nu }_{e}\right)=\left({1.69}_{-0.46}^{+0.53}±0.15\right)×{10}^{-4}$ and $B\left({D}^{+}\to {\overline{K}}_{1}{\left(1270\right)}^{0}\left(\to {K}_{S}^{0}{\pi }^{+}{\pi }^{-}\right){e}^{+}{\nu }^{e}\right)=\left({1.47}_{-0.40}^{+0.45}±0.14\right)×{10}^{-4}$ with statistical significance of 5.4σ and 5.6σ, respectively. When combined with measurements of the K 1(1270) → K + π − π decays, the absolute branching fractions are determined to be $B\left({D}^{0}\to {K}_{1}{\left(1270\right)}^{-}{e}^{+}{\nu }_{e}\right)=\left({1.08}_{-0.13-0.10}^{+0.14+0.08}±0.21\right)×{10}^{-3}$ and $B\left({D}^{+}\to {\overline{K}}_{1}{\left(1270\right)}^{0}{e}^{+}{\nu }_{e}\right)=\left({1.70}_{-0.23}^{+0.26}±0.13±0.35\right)×{10}^{-3}$ . The first and second uncertainties are statistical and systematic, respectively, and the third uncertainties originate from the assumed branching fractions of the K 1(1270) → Kππ decays.