Using an amplitude analysis to measure the photon polarisation in $B\to K\pi \pi \gamma$ decays

Abstract

A method is proposed to measure the photon polarisation parameter ${\lambda }_{\gamma }$ in $b\to s\gamma$ transitions using an amplitude analysis of $B\to K\pi \pi \gamma$ decays. Simplified models of the $K\pi \pi$ i system are used to simulate $B^{+}\to K^{+}{\pi }^{-}{\pi }^{+}\gamma$ and $B^0\to K^{+}{\pi }^{-}{\pi }^0$ decays, validate the amplitude analysis method, and demonstrate the feasibility of a measurement of the ${\lambda }_{\gamma }$ parameter irrespective of the model parameters. Similar sensitivities to ${\lambda }_{\gamma }$ 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 $B^{+}\to K^{+}{\pi }^{-}{\pi }^{+}\gamma$ 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 $B^0\to K^{+}{\pi }^{-}{\pi }^0$ decays in a Belle II data sample corresponding to an integrated luminosity of $5$ab${}^{-1}$ would lead to a statistical uncertainty of $0.018$