Time-dependent analysis of B0 → KS0 π-π+γ decays and studies of the K+π-π+ system in B+ →k+π-π+γ decays

We measure the time-dependent CP asymmetry in the radiative-penguin decay B^0→K^0_Sπ^−π^+γ, using a sample of 471×10^6 Υ(4S)→BB¯ events recorded with the BABAR detector at the PEP-II e^+e^− storage ring at SLAC. Using events with mKππ<1.8  GeV/c^2, we measure the branching fractions of B^+→K^+π^−π^+γ and B^0→K^0π^−π^+γ, the branching fractions of the kaonic resonances decaying to K^+π^−π^+, as well as the overall branching fractions of the B^+→ρ^0K^+γ, B^+→K^(*0)π^+γ and S-wave B^+→(Kπ)^(*0)_0π^+γ components. For events from the ρ mass band, we measure the CP-violating parameters SK^0_Sπ^+π^−γ=0.14±0.25±0.03 and CK^0_Sπ^+π^−γ=−0.39±0.20+0.03−0.02, where the first uncertainties are statistical and the second are systematic. We extract from this measurement the time-dependent CP asymmetry related to the CP eigenstate ρ^0K^0_S and obtain SK^0_(Sργ)=−0.18±0.32^(+0.06)_(−0.05), which provides information on the photon polarization in the underlying b→sγ transition.