Measurement of the doubly Cabibbo-suppressed decay ${D}^{+}\to {K}^{+}{\pi }^{+}{\pi }^{-}{\pi }^{0}$ with semileptonic tags

We propose a new semileptonic tag method to study the doubly Cabibbo-suppressed $D$ decay in $\psi \left(3770\right)\to D\overline{D}$ reaction. Utilizing the dataset corresponding to an integrated luminosity of $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ at a center-of-mass energy of 3.773 GeV collected by the BESIII detector, we determine the branching fraction for ${D}^{+}\to {K}^{+}{\pi }^{+}{\pi }^{-}{\pi }^{0}$ to be $\left(1.03±0.1{2}_{\mathrm{stat}}±0.0{6}_{\mathrm{syst}}\right)×{10}^{-3}$, in which the contributions from narrow intermediate resonances, ${D}^{+}\to {K}^{+}\eta $, ${D}^{+}\to {K}^{+}\omega $, and ${D}^{+}\to {K}^{+}\varphi $ have been excluded. Combining the world average of the branching fraction of ${D}^{+}\to {K}^{-}{\pi }^{+}{\pi }^{+}{\pi }^{0}$, we determine $B\left({D}^{+}\to {K}^{+}{\pi }^{+}{\pi }^{-}{\pi }^{0}\right)/B\left({D}^{+}\to {K}^{-}{\pi }^{+}{\pi }^{+}{\pi }^{0}\right)=\left(1.65±0.21\right)%$, corresponding to $\left(5.73±0.73\right){\mathrm{tan}}^{4}{\theta }_{c}$, where ${\theta }_{c}$ is the Cabibbo mixing angle. These results are consistent with our previous measurement with hadronic tags but are significantly larger than other doubly Cabibbo-suppressed decays in the charm sector.