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Measurement of the time-dependent<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>-violating asymmetry in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mi>γ</mml:mi></mml:math>decays
We present a measurement of the time-dependent $CP$-violating asymmetry in ${B}^{0}\ensuremath{\rightarrow}{K}^{*0}\ensuremath{\gamma}$ decays with ${K}^{*0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{0}$ based on $232\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $\ensuremath{\Upsilon}(4S)\ensuremath{\rightarrow}B\overline{B}$ decays collected with the BABAR detector at the PEP-II asymmetric-energy ${e}^{+}{e}^{\ensuremath{-}}$ collider at SLAC. In a sample containing $157\ifmmode\pm\else\textpm\fi{}16$ signal decays, we measure ${S}_{{K}^{*0}\ensuremath{\gamma}}=\ensuremath{-}0.21\ifmmode\pm\else\textpm\fi{}0.40\ifmmode\pm\else\textpm\fi{}0.05$ and ${C}_{{K}^{*0}\ensuremath{\gamma}}=\ensuremath{-}0.40\ifmmode\pm\else\textpm\fi{}0.23\ifmmode\pm\else\textpm\fi{}0.03$, where the first error is statistical …