Measurement of the doubly Cabibbo suppressed decay $D^0 \to K^+\pi^-$ and a search for charm mixing

We present an analysis of the decay $D^0 \to K^+\pi^-$ based on FOCUS data. From a sample of 234 signal events, we find a branching ratio of $\frac{\Gamma(D^0 \to K^+\pi^-)}{\Gamma(D^0 \to K^-\pi^+)} = (0.429^{+0.063}_{-0.061}\pm 0.027)%$ under the assumptions of no mixing and no CP violation. Allowing for CP violation, we find a branching ratio of $(0.429 \pm 0.063 \pm 0.028)%$ and a \textit{CP} asymmetry of $0.18 \pm 0.14 \pm 0.04$. The branching ratio for the case of mixing with no CP violation is $(0.381^{+0.167}_{-0.163} \pm 0.092)%$. We also present limits on charm mixing.