Measurements of Branching Fractions and 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 Asymmetries in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mi>K</mml:mi></mml:math>Decays

We present measurements of the $B\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}K$ branching fractions; for ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{K}^{+}$ we measure also the time-integrated charge asymmetry ${\mathcal{A}}_{\mathrm{ch}}$, and for ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{K}_{S}^{0}$ the time-dependent $CP$-violation parameters $S$ and $C$. The data sample corresponds to $232\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ pairs produced by ${e}^{+}{e}^{\ensuremath{-}}$ annihilation at the $\ensuremath{\Upsilon}(4S)$. The results are $\mathcal{B}(B\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{K}^{+})=(68.9\ifmmode\pm\else\textpm\fi{}2.0\ifmmode\pm\else\textpm\fi{}3.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{K}^{0})=(67.4\ifmmode\pm\else\textpm\fi{}3.3\ifmmode\pm\else\textpm\fi{}3.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, ${\mathcal{A}}_{\mathrm{ch}}=0.033\ifmmode\pm\else\textpm\fi{}0.028\ifmmode\pm\else\textpm\fi{}0.005$, $S=0.30\ifmmode\pm\else\textpm\fi{}0.14\ifmmode\pm\else\textpm\fi{}0.02$, and $C=\ensuremath{-}0.21\ifmmode\pm\else\textpm\fi{}0.10\ifmmode\pm\else\textpm\fi{}0.02$, where the first error quoted is statistical and the second is systematic.

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