Measurement of the doubly Cabibbo-suppressed decay $D^+\to K^+π^+π^-π^0$ with semileptonic tags
Using an $e^+e^-$ annihilation data sample corresponding to an integrated luminosity of $2.93\,\rm fb^{-1}$ collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, the doubly Cabibbo-suppressed decay $D^+\to K^+π^+π^-π^0$ is studied with a semileptonic tag method. After removing the decays containing narrow intermediate resonances, $D^+\to K^+η$, $D^+\to K^+ω$, and $D^+\to K^+ϕ$, the branching fraction for the decay $D^+\to K^+π^+π^-π^0$ is determined to be $(1.03 \pm 0.12_{\rm stat} \pm 0.06_{\rm syst})\times 10^{-3}$. The ratio of the branching fraction for $D^+\to K^+π^+π^-π^0$ to its Cabibbo-favored counterpart $D^+\to K^-π^+π^+π^0$ is measured to be $ (1.65\pm0.21)\%$, corresponding to $(5.73\pm0.73)\tan^4θ_C$, where $θ_C$ 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.