Measurement of time-dependent $CP$ asymmetries in $B^0\to K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-}\gamma$ decays at Belle and Belle II

We present a measurement of the time-dependent $CP$ asymmetry in $B^0\to K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-}\gamma$ decays using a data set of 365 fb${}^{-1}$ recorded by the Belle II experiment and the final data set of 711 fb${}^{-1}$ recorded by the Belle experiment at the $\mathrm{{\rm Y}}(4\mathrm{S})$ resonance. The direct and mixing-induced time-dependent $CP$ violation parameters $C$ and $S$ are determined along with two additional quantities, $S^{+}$ and $S^{-}$, defined in the two halves of the $m^2(K_{\mathrm{S}}^0{\pi }^{+})-m^2(K_{\mathrm{S}}^0{\pi }^{-})$ plane. The measured values are $C=-0.17\pm 0.09\pm 0.04$, $S=-0.29\pm 0.11\pm 0.05$, $S^{+}=-0.57\pm 0.23\pm 0.10$ and $S^{-}=0.31\pm 0.24\pm 0.05$, where the first uncertainty is statistical and the second systematic.