Search for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math>Violation in the Decays<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><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>K</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>D</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:math>

We measure time-integrated $CP$-violating asymmetries of neutral charmed mesons in the modes ${D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{K}^{+}$ and ${D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ with the highest precision to date by using ${D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ decays to correct detector asymmetries. An analysis of $385.8\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data collected with the BABAR detector yields values of ${a}_{CP}^{KK}=\mathbf{(}0.00\ifmmode\pm\else\textpm\fi{}0.34(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.13(\mathrm{syst})\mathbf{)}%$ and ${a}_{CP}^{\ensuremath{\pi}\ensuremath{\pi}}=\mathbf{(}\ensuremath{-}0.24\ifmmode\pm\else\textpm\fi{}0.52(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.22(\mathrm{syst})\mathbf{)}%$, which agree with standard model predictions.

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