Measurements of the branching fractions of the inclusive decays $D^0(D^{+})\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$

Using $e^{+}{e}^{-}$ annihilation data corresponding to an integrated luminosity of $2.93{\mathrm{fb}}^{-1}$ taken at a center-of-mass energy of 3.773 GeV with the BESIII detector, we report the first measurements of the branching fractions of the inclusive decays $D^0\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$ and $D^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$, where pions from $K_S^0$ decays have been excluded from the ${\pi }^{+}{\pi }^{+}{\pi }^{-}$ system and $X$ denotes any possible particle combination. The branching fractions of $D^0(D^{+})\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$ are determined to be $\mathcal{B}(D^0\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X)=(17.60\pm 0.11\pm 0.22)\%$ and $\mathcal{B}(D^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X)=(15.25\pm 0.09\pm 0.18)\%$, where the first uncertainties are statistical and the second systematic.

Figure 1

Fits to the $M_{\mathrm{BC}}$ distributions of the ST candidates for ${\overline{D}}^0\to K^{+}{\pi }^{-}$ (left) and $D^{-}\to K^{+}{\pi }^{-}{\pi }^{-}$ (right), where the points with error bars are data, the blue solid curves are the fit results, and the red dashed curves are the fitted combinatorial backgrounds.

Figure 2

Comparisons of the $M_{3\pi }$ (left) and $M_{\mathrm{miss}}$ (right) distributions of the DT candidates for $D^0\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$ (top) and $D^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$ (bottom), where the points with error bars are data and the color filled histograms are the inclusive MC sample. To ensure a $\overline{D}$ in the ST side, events must satisfy the requirements mentioned in the text and an additional requirement of $|M_{\mathrm{BC}}-M_D|<0.005\mathrm{GeV}/c^2$, where $M_D$ is the known $D$ mass [5].

Figure 3

Comparisons of the momentum distributions of the selected pions of the DT candidates for $D^0\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$ (top) and $D^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$ (bottom), where the points with error bars are data and the color filled histograms are the inclusive MC sample. To ensure a $\overline{D}$ in the ST side, events must satisfy the requirements mentioned in the text and an additional requirement of $|M_{\mathrm{BC}}-M_D|<0.005\mathrm{GeV}/c^2$, where $M_D$ is the known $D$ mass [5], and ${\pi }_1^{+}$ and ${\pi }_2^{+}$ denote the higher and lower momentum ${\pi }^{+}\mathrm{s}$, respectively.

Figure 4

Fits to the $M_{\mathrm{BC}}$ distributions of the tag side when the candidates for (a) $D^0\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$ and (b) $D^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$ are found in various reconstructed $M_{3\pi }$ intervals (in unit of $\mathrm{GeV}/c^2$) indicated on each plot. The points with error bars are data, the blue solid curves are the fit results, the pink dashed curves are the peaking background summing over Mis-ID $K$ BKG, Mis-ID $\mu$ BKG, Mis-ID $e$ BKG, and $K_S^0$ BKG, and the black dashed curves are the fitted combinatorial backgrounds.