Measurement of $CP$-violating asymmetries in $B^0 \rightarrow (\rho \pi)^0$ using a time-dependent Dalitz-plot analysis of the {\sc BaBar}

A measurement of $CP$-violating asymmetries is performed in the mode $B^0 \rightarrow \pi^+ \pi^- \pi^0$ where the decay mode is dominated by intermediate $\rho \rightarrow \pi \pi$ resonances. The use of a full time-dependent Dalitz plot analysis allows sensitivity to the interference effects caused by the relative strong and weak phases in the regions where the $\rho^+$, $\rho^-$, and $\rho^0$ resonances overlap. This feature of the analysis makes possible the unambiguous extraction of the strong and weak phases as well as the $CP$-violating parameter $\alpha \equiv {\rm arg}[-V_{\rm td}V^{*}_{\rm tb}/V_{\rm ud}V^{*}_{\rm ub}]$ (where the $V_{q q^\prime}$ are components of the CKM quark mixing matrix). A precision measurement of $\alpha$ serves to further test the Standard Model and constrain new physics. The analysis is performed using a sample of 431 ${\rm fb}^{-1}$ of data corresponding to $471 \times 10^6$ $B{\overline B}$ meson pairs collected at the $\Upsilon(4S)$ resonances by the {\sc BaBar} experiment using the PEP-II asymmetric-energy $e^+e^-$ collider at the SLAC National Accelerator Laboratory.