We exploit the quantum coherence between pair-produced $D^0$ and ${\overline{D}}^0$ in $\psi (3770)$ decays to study charm mixing, which is characterized by the parameters $x$ and $y$, and to make a first determination of the relative strong phase $\delta$ between doubly Cabibbo-suppressed $D^0\to K^{+}{\pi }^{-}$ and Cabibbo-favored ${\overline{D}}^0\to K^{+}{\pi }^{-}$. We analyze a sample of $1.0\times {10}^6$ $D^0{\overline{D}}^0$ pairs from $281{\mathrm{pb}}^{-1}$ of $e^{+}{e}^{-}$ collision data collected with the CLEO-c detector at $E_{\mathrm{cm}}=3.77\mathrm{GeV}$. By combining CLEO-c measurements with branching fraction input and time-integrated measurements of $R_{\mathrm{M}}\equiv (x^2+y^2)/2$ and $R_{\mathrm{WS}}\equiv \Gamma (D^0\to K^{+}{\pi }^{-})/\Gamma ({\overline{D}}^0\to K^{+}{\pi }^{-})$ from other experiments, we find $\cos \delta ={1.03}_{-0.17}^{+0.31}\pm 0.06$, where the uncertainties are statistical and systematic, respectively. In addition, by further including external measurements of charm mixing parameters, we obtain an alternate measurement of $\cos \delta =1.10\pm 0.35\pm 0.07$, as well as $x\sin \delta =({4.4}_{-1.8}^{+2.7}\pm 2.9)\times {10}^{-3}$ and $\delta =({22}_{-12-11}^{+11+9})°$.

• Received 18 February 2008

DOI:https://doi.org/10.1103/PhysRevD.78.012001