Study of $CP$ violation in $B^0\rightarrow \pi^0\pi^0 K_{\rm S}^0$ decay at BaBar

We present the measurement of the time-dependent $CP$ asymmetry for the neutral $B$-meson decay into the $CP=+1$ final state $K_ {\rm S}^0 \pi^0 \pi^0$. We use a sample of approximately 227 million $B$-meson pairs recorded at the $\Upsilon(4S)$ resonance by the BaBar detector at the PeP-II $B$-Factory at SLAC. From a maximum likelihood fit we extract the mixing-induced $CP$-violation parameter $S = 0.84\pm 0.71\pm 0.08$ and the direct $CP$-violation parameter $C = 0.27\pm 0.52\pm 0.13$, where the first uncertainty is statistical and the second systematic. We compare intensities of subdecay modes with expectations from related $B$-decay modes