Measurement of time-dependent $CP$ asymmetries in B⁰ → KS⁰ \: π⁺ π⁻ γ decays at Belle and Belle II

We present a measurement of the time-dependent $CP$ asymmetry in B⁰ → KS⁰ \: π⁺ π⁻ γ 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 Υ(4S) 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 m²(KS⁰ π⁺)-m²(KS⁰ π⁻) plane. The measured values are C = -0.17 ± 0.09 ± 0.04, S = -0.29 ± 0.11 ± 0.05, S⁺ = -0.57 ± 0.23 ± 0.10 and S⁻ = 0.31 ± 0.24 ± 0.05, where the first uncertainty is statistical and the second systematic.