Measurement of the absolute branching fraction of the inclusive decay $D_s^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$

Using an $e^{+}{e}^{-}$ collision data sample with a total integrated luminosity of $3.19$ fb${}^{-1}$ collected with the BESIII detector at a center-of-mass energy of 4.178 GeV, the branching fraction of the inclusive decay of the $D_s^{+}$ meson to final states including at least three charged pions is measured for the first time to be $\mathcal{B}(D_s^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X)=(32.81\pm {0.35}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.63}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\mathrm{\%}$. In this measurement the charged pions from $K_S^0$ meson decays are excluded. The partial branching fractions of $D_s^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}X$ are also measured as a function of the ${\pi }^{+}{\pi }^{+}{\pi }^{-}$ invariant mass.