We present measurements of the branching fractions for the singly Cabibbo-suppressed decays $D^{+}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^0$ and $D_s^{+}\to K^{+}{\pi }^{-}{\pi }^{+}{\pi }^0$, and the doubly Cabibbo-suppressed decay $D^{+}\to K^{+}{\pi }^{-}{\pi }^{+}{\pi }^0$, based on $980{\mathrm{fb}}^{-1}$ of data recorded by the Belle experiment at the KEKB $e^{+}{e}^{-}$ collider. We measure these modes relative to the Cabibbo-favored modes $D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^0$ and $D_s^{+}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^0$. Our results for the ratios of branching fractions are $\mathcal{B}(D^{+}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^0)/\mathcal{B}(D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^0)=(11.32\pm 0.13\pm 0.26)\%$, $\mathcal{B}(D^{+}\to K^{+}{\pi }^{-}{\pi }^{+}{\pi }^0)/\mathcal{B}(D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^0)=(1.68\pm 0.11\pm 0.03)\%$, and $\mathcal{B}(D_s^{+}\to K^{+}{\pi }^{-}{\pi }^{+}{\pi }^0)/\mathcal{B}(D_s^{+}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^0)=(17.13\pm 0.62\pm 0.51)\%$, where the uncertainties are statistical and systematic, respectively. The second value corresponds to $(5.83\pm 0.42)\times {\tan }^4{\theta }_C$, where ${\theta }_C$ is the Cabibbo angle; this value is larger than other measured ratios of branching fractions for a doubly Cabibbo-suppressed charm decay to a Cabibbo-favored decay. Multiplying these results by world average values for $\mathcal{B}(D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^0)$ and $\mathcal{B}(D_s^{+}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^0)$ yields $\mathcal{B}(D^{+}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^0)=(7.08\pm 0.08\pm 0.16\pm 0.20)\times {10}^{-3}$, $\mathcal{B}(D^{+}\to K^{+}{\pi }^{-}{\pi }^{+}{\pi }^0)=(1.05\pm 0.07\pm 0.02\pm 0.03)\times {10}^{-3}$, and $\mathcal{B}(D_s^{+}\to K^{+}{\pi }^{-}{\pi }^{+}{\pi }^0)=(9.44\pm 0.34\pm 0.28\pm 0.32)\times {10}^{-3}$, where the third uncertainty is due to the branching fraction of the normalization mode. The first two results are consistent with, but more precise than, the current world averages. The last result is the first measurement of this branching fraction.

• Received 3 May 2022
• Accepted 28 September 2022

DOI:https://doi.org/10.1103/PhysRevD.107.033003

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Published by the American Physical Society