Measurement of $D^0\mathrm{\text{−}}{\overline{D}}^0$ mixing using the ratio of lifetimes for the decays $D^0\to K^{-}{\pi }^{+}$, $K^{-}{K}^{+}$, and ${\pi }^{-}{\pi }^{+}$

We present a measurement of $D^0\mathrm{\text{−}}{\overline{D}}^0$ mixing parameters using the ratios of lifetimes extracted from a sample of $D^0$ mesons produced through the process $D^{*+}\to D^0{\pi }^{+}$, which decay to $K^{-}{\pi }^{+}$, $K^{-}{K}^{+}$, or ${\pi }^{-}{\pi }^{+}$. The lifetimes of the $CP$-even, Cabibbo-suppressed modes $K^{-}{K}^{+}$ and ${\pi }^{-}{\pi }^{+}$ are compared with that of the $CP$-mixed, Cabibbo-favored mode $K^{-}{\pi }^{+}$ to obtain a measurement of $y_{CP}$, which in the limit of $CP$ conservation corresponds to the mixing parameter $y$. The analysis is based on a data sample of $384{\mathrm{fb}}^{-1}$ collected by the BABAR detector at the PEP-II asymmetric-energy $e^{+}{e}^{-}$ collider. We obtain $y_{CP}=[1.24\pm 0.39(\mathrm{stat})\pm 0.13(\mathrm{syst})]\%$, which is evidence for $D^0\mathrm{\text{−}}{\overline{D}}^0$ mixing at the $3\sigma$ level, and $\Delta Y=[-0.26\pm 0.36(\mathrm{stat})\pm 0.08(\mathrm{syst})]\%$, where $\Delta Y$ constrains possible $CP$ violation. Combining this result with a previous BABAR measurement of $y_{CP}$ obtained from a separate sample of $D^0\to K^{-}{K}^{+}$ events, we obtain $y_{CP}=[1.03\pm 0.33(\mathrm{stat})\pm 0.19(\mathrm{syst})]\%$.