Study of $\mathbf{B^{0}\rightarrow\rho^{+}\rho^{-}}$ decays and implications for the CKM angle $\mathbf{\phi_2}$

We present a measurement of the branching fraction and the longitudinal polarization fraction of $B^{0}\rightarrow\rho^+\rho^-$ decays, as well as the time-dependent $CP$ violating parameters in decays into longitudinally polarized $\rho^{+}\rho^{-}$ pairs with Belle's final data set of $772\times 10^6$ $B\bar{B}$ pairs, at the $\Upsilon(4S)$ resonance, collected at the asymmetric-energy $e^+ e^-$ collider KEKB. We obtain ${\cal B}(B^0 \rightarrow \rho^+ \rho^-)=(28.3\pm 1.5\;(\rm stat) \pm 1.5\;(\rm syst))\times 10^{-6}$, $f_L = 0.988\pm 0.012\;(\rm stat ) \pm0.023\;(\rm syst)$, ${\cal A}_{CP} = 0.00\pm0.10\;(\rm stat ) \pm0.06\;(\rm syst)$, ${\cal S}_{CP} = -0.13\pm0.15\;(\rm stat ) \pm0.05\;(\rm syst).$ We perform an isospin analysis to constrain the CKM angle $\phi_2$ and obtain two solutions with \begin{eqnarray} \phi_{2} = (93.7\pm10.6)^{\circ},\nonumber \end{eqnarray} being most compatible with other Standard-Model based fits to the data.