Measurement of the ratio of branching fractions<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">B</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi mathvariant="script">B</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>using the CDF II detector

We present a measurement of ${R}_{\mathcal{B}}$, the ratio of the branching fraction for the rare decay ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ to that for the Cabibbo-favored decay ${D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$. Charge-conjugate decays are implicitly included. A signal of $2005\ifmmode\pm\else\textpm\fi{}104$ events for the decay ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ is obtained using the CDF II detector at the Fermilab Tevatron collider. The data set corresponds to an integrated luminosity of $0.35\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ produced in $\overline{p}p$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$. Assuming no mixing, we find ${R}_{\mathcal{B}}=[4.05\ifmmode\pm\else\textpm\fi{}0.21(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.11(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. This measurement is consistent with the world average, and comparable in accuracy with the best measurements from other experiments.

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