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Time-dependent analysis of $B^0 \to {K^0_{S}} π^- π^+ γ$ decays and studies of the $K^+π^-π^+$ system in $B^+ \to K^+ π^- π^+ γ$ decays

We measure the time-dependent $CP$ asymmetry in the radiative-penguin decay $B^0 \to {K^0_{S}} π^- π^+ γ$, using a sample of $471$ million $Υ(4S) \to B{\kern 0.18em\overline{\kern -0.18em B}}$ events recorded with the BaBar detector at the PEP-II $e^+e^-$ storage ring at SLAC. Using events with $m_{Kππ}<1.8{\mathrm{\,Ge\kern -0.1em V/}c^2}$, we measure the branching fractions of $B^+ \to K^+ π^- π^+ γ$ and $B^0 \to K^0 π^+ π^- γ$, the branching fractions of the kaonic resonances decaying to $K^+ π^- π^+$, as well as the overall branching fractions of the $B^+ \to ρ^0 K^+ γ$, $B^+ \to K^{*0} π^+ γ$ and S-wave $B^+ \to (K π)^{*0}_0 π^+ γ$ components. For events from the $ρ$ mass band, we measure the $CP$-violating parameters ${\mathcal{S}_{K^0_{S}π^+π^-γ}} = 0.14 \pm 0.25 \pm 0.03$ and ${\mathcal{C}_{K^0_{S}π^+π^-γ}} = -0.39 \pm 0.20^{+0.03}_{-0.02}$, 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 $ρ^0 {K^0_{S}}$ and obtain ${\mathcal{S}_{K^0_{S}ργ}} = -0.18 \pm 0.32^{+ 0.06}_{-0.05}$, which provides information on the photon polarization in the underlying $b \to s γ$ transition.