Time-dependent amplitude analysis of B0→KS0π+π−

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We perform a time-dependent amplitude analysis of ${B}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays to extract the $CP$ violation parameters of ${f}_{0}(980){K}_{S}^{0}$ and ${\ensuremath{\rho}}^{0}(770){K}_{S}^{0}$ and the direct $CP$ asymmetry of ${K}^{*+}(892){\ensuremath{\pi}}^{\ensuremath{-}}$. The results are obtained from a data sample of $(383\ifmmode\pm\else\textpm\fi{}3)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ decays, collected with the BABAR detector at the PEP-II asymmetric--energy $B$ factory at SLAC. We find two solutions, with an equivalent goodness-of-fit. Including systematic and Dalitz plot model uncertainties, the combined confidence interval for values of the $CP$ parameter ${\ensuremath{\beta}}_{\mathrm{eff}}$ in ${B}^{0}$ decays to ${f}_{0}(980){K}_{S}^{0}$ is $18\ifmmode^\circ\else\textdegree\fi{}<{\ensuremath{\beta}}_{\mathrm{eff}}<76\ifmmode^\circ\else\textdegree\fi{}$ at 95% confidence level (C.L). $CP$ conservation in ${B}^{0}$ decays to ${f}_{0}(980){K}_{S}^{0}$ is excluded at 3.5 standard deviations including systematic uncertainties. For ${B}^{0}$ decays to ${\ensuremath{\rho}}^{0}(770){K}_{S}^{0}$, the combined confidence interval is $\ensuremath{-}9\ifmmode^\circ\else\textdegree\fi{}<{\ensuremath{\beta}}_{\mathrm{eff}}<57\ifmmode^\circ\else\textdegree\fi{}$ at 95% C.L. In decays to ${K}^{*+}(892){\ensuremath{\pi}}^{\ensuremath{-}}$ we measure the direct $CP$ asymmetry to be ${A}_{CP}=\ensuremath{-}0.20\ifmmode\pm\else\textpm\fi{}0.10\ifmmode\pm\else\textpm\fi{}0.01\ifmmode\pm\else\textpm\fi{}0.02$. The measured phase difference (including ${B}^{0}{\overline{B}}^{0}$ mixing) between decay amplitudes of ${B}^{0}\ensuremath{\rightarrow}{K}^{*+}(892){\ensuremath{\pi}}^{\ensuremath{-}}$ and ${\overline{B}}^{0}\ensuremath{\rightarrow}{K}^{*\ensuremath{-}}(892){\ensuremath{\pi}}^{+}$, excludes the interval $\ensuremath{-}137\ifmmode^\circ\else\textdegree\fi{}<\ensuremath{\Delta}\ensuremath{\Phi}({K}^{*+}(892){\ensuremath{\pi}}^{\ensuremath{-}})<\ensuremath{-}5\ifmmode^\circ\else\textdegree\fi{}$ at 95% C.L.

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