Observation of the Doubly Cabibbo-Suppressed Decay ${D}^{+}\to {K}^{+}{\pi }^{+}{\pi }^{-}{\pi }^{0}$ and Evidence for ${D}^{+}\to {K}^{+}\omega $

Using $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ collision data collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, the first observation of the doubly Cabibbo-suppressed decay ${D}^{+}\to {K}^{+}{\pi }^{+}{\pi }^{-}{\pi }^{0}$ is reported. After removing decays that contain narrow intermediate resonances, including ${D}^{+}\to {K}^{+}\eta $, ${D}^{+}\to {K}^{+}\omega $, and ${D}^{+}\to {K}^{+}\varphi $, the branching fraction of the decay ${D}^{+}\to {K}^{+}{\pi }^{+}{\pi }^{-}{\pi }^{0}$ is measured to be $\left(1.13±0.0{8}_{\mathrm{stat}}±0.0{3}_{\mathrm{syst}}\right)×{10}^{-3}$. The ratio of branching fractions of ${D}^{+}\to {K}^{+}{\pi }^{+}{\pi }^{-}{\pi }^{0}$ over ${D}^{+}\to {K}^{-}{\pi }^{+}{\pi }^{+}{\pi }^{0}$ is found to be $\left(1.81±0.15\right)%$, which corresponds to $\left(6.28±0.52\right){\mathrm{tan}}^{4}{\theta }_{C}$, where ${\theta }_{C}$ is the Cabibbo mixing angle. This ratio is significantly larger than the corresponding ratios for other doubly Cabibbo-suppressed decays. The asymmetry of the branching fractions of charge-conjugated decays ${D}^{±}\to {K}^{±}{\pi }^{±}{\pi }^{\mp }{\pi }^{0}$ is also determined, and no evidence for $CP$ violation is found. In addition, the first evidence for the ${D}^{+}\to {K}^{+}\omega $ decay, with a statistical significance of $3.3\sigma $, is presented and the branching fraction is measured to be $B\left({D}^{+}\to {K}^{+}\omega \right)=\left({5.{7}_{-2.1}^{+2.5}}_{\mathrm{stat}}±0.{2}_{\mathrm{syst}}\right)×{10}^{-5}$.