Measurement of time-dependent $CP$ violation parameters in $B^{0} \to K_{S}^{0} π^{0} γ$ decays at Belle and Belle II

We perform a measurement of time-dependent $CP$ violation parameters in $B^{0} \to K_{S}^{0} π^{0} γ$ decays using a dataset of approximately $772 \times 10^6$ and $521 \times 10^6$ $Υ(4S)$ decays collected by the Belle and Belle II experiments, respectively. The measured parameters for the combined dataset in the $K^{*0}(892)$ dominated region ($M_{K_{S}^{0} π^{0}} \in [0.8,1.0] \mathrm{GeV}/c^2$) are $S = 0.09 \pm 0.16 \pm 0.02$ and $C = -0.09 \pm 0.08 \pm 0.04$. For the non-$K^{*0}(892)$ region ($M_{K_{S}^{0} π^{0}} \in [1.0,1.8] \mathrm{GeV}/c^2$), the corresponding values are $S = -0.32 \pm 0.33 \pm 0.09$ and $C = -0.07 \pm 0.17 \pm 0.08$. The first quoted uncertainties are statistical, while the second ones are systematic. These results are consistent with Standard Model predictions and more precise than previous measurements.