Performance study of a novel time-dependent CP analysis of B $\to$ ${\pi }^0{\pi }^0$ and projection of the measurement of the unitarity angle ${\varphi }_2$/$\alpha$ at the Belle II Experiment

The measurement of the time-dependent CP violation parameters of the decay channel B $\to$ ${\pi }^0{\pi }^0$ is crucial for the determination of the unitarity angle ${\varphi }_2$. At former B-factories, only the direct CP violation parameter A${}_{CP}$ could be measured in time-integrated studies. For the mixing-induced CP violation parameter S${}_{CP}$, a time-dependent analysis is required, a highly challenging task. This analysis requires a precise determination of the B${}^0$-decay vertex, which cannot be achieved in the dominant four-photons final state. Only rare events with a so-called Dalitz decay ${\pi }^0$ $\to$ $e^{+}{e}^{-}$$\gamma$ or with converted photons provide information for the B${}^0$ vertex reconstruction. Thus, large samples of B${}^0$ $\to$ ${\pi }^0{\pi }^0$ decays are needed and, at present, the size of the current B-factories data samples is insufficient. Exploiting the capabilities of the new Belle II pixel vertex detector and the full Belle II data sample at 50 ab${}^{-1}$, the presented study shows that the expected uncertainty on the first measurement of S${}_{{\pi }^0{\pi }^0}$ is estimated to be $\Delta$S${}_{{\pi }^0{\pi }^0}$ = 0.29. This results in a reduction of the current 8-fold ${\varphi }_2$/$\alpha$-ambiguity to a 2-fold considering B $\to$ $\pi \pi$ decays. Considering also B $\to$ $\rho \rho$ decays, the final uncertainty on ${\varphi }_2$/$\alpha$ is projected to be about five times smaller than the current world average.