The time-dependent $CP$ asymmetry in $B^0 \to K_{\rm res} \gamma \to \pi^+ \pi^- K^0_{\scriptscriptstyle S} \gamma$ decays

The time-dependent CP asymmetry in B$^{0}$ → K$_{res}$γ → π$^{+}$π$^{−}$K$^{S}$γ is sensitive to the photon polarisation in the quark level process b → sγ. While this polarisation is predominantly left-handed in the standard model, it could be modified by the existence of new physics contributions that may possess different CP properties. In this paper, we derive the CP violation formulae for B$^{0}$ → K$_{res}$γ → π$^{+}$π$^{−}$K$^{S}$γ including the most dominant intermediate states. We propose a new observable that could be measured in a time-dependent amplitude analysis of B$^{0}$ → π$^{+}$π$^{−}$K$^{S}$γ decays, providing a stringent contraint on the photon polarisation. We discuss the future prospects for obtaining such constraints from measurements at Belle II and LHCb.