Search for D0 - anti-D0 mixing and a measurement of the doubly Cabibbo suppressed decay rate in D0 ---> K pi decays

We present results of a search for $D^0$-$\kern 0.2em\bar{\kern -0.2em D}{}^0$~mixing and a measurement of $R_{\rm D}$, the ratio of doubly Cabibbo-suppressed decays to Cabibbo-favored decays, based on an analysis of $D^0 \to K^{+}\pi^{-}$ decays in 57.1~${fb}^{-1}$ of data collected at or just below the $\Upsilon{(4S)}$ resonance with the {\slshape BABAR} detector at the PEP-II collider. Our results are compatible with no mixing and no $CP$ violation. At the 95% C.L., allowing for CP violation, we find the mixing parameters ${x^{\prime}}^2<0.0022$ and $-0.056<y^{\prime}<0.039$, and the mixing rate $R_{\rm M} < 0.16%$. In the limit of no mixing, $R_{\rm D} = (0.357 \pm 0.022 \hbox{(stat.)} \pm 0.027 \hbox{(syst.)})%$ and the $CP$~violating asymmetry \mbox{$A_{\rm D} = 0.095 \pm 0.061 \hbox{(stat.)} \pm 0.083 \hbox{(syst.)}$.}