Using an amplitude analysis to measure the photon polarisation in $B \to Kππγ$ decays

A method is proposed to measure the photon polarisation parameter $λ_γ$ in $b\to sγ$ transitions using an amplitude analysis of $B\to Kππγ$ decays. Simplified models of the $Kππ$ i system are used to simulate $B^+ \to K^+π^-π^+γ$ and $B^0\to K^+π^-π^0$ decays, validate the amplitude analysis method, and demonstrate the feasibility of a measurement of the $λ_γ$ parameter irrespective of the model parameters. Similar sensitivities to $λ_γ$ are obtained with both the charged and neutral hadronic systems. In the absence of any background and distortion due to experimental effects, the statistical uncertainty expected from an analysis of $B^+ \to K^+π^-π^+γ$ decays in an LHCb data set corresponding to an integrated luminosity of $9\,$fb$^{-1}$ is estimated to be $0.009$. A similar measurement using $B^0\to K^+π^-π^0$ decays in a Belle II data sample corresponding to an integrated luminosity of $5\,$ab$^{-1}$ would lead to a statistical uncertainty of $0.018$