The time-dependent $CP$ asymmetry in $B^0\to K_{\mathrm{res}}{\hat{I}}^3\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}{K}_S^0{\hat{I}}^3$ decays

Abstract

The time-dependent $CP$ asymmetry in $B^0\to K_{\mathrm{res}}{\hat{I}}^3\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}{K}_S^0{\hat{I}}^3$ is sensitive to the photon polarisation in the quark level process $b\to s{\hat{I}}^3$. 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 $B^0\to K_{\mathrm{res}}{\hat{I}}^3\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}{K}_S^0{\hat{I}}^3$ including the most dominant intermediate states. We propose a new observable that could be measured in a time-dependent amplitude analysis of $B^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}{K}_S^0{\hat{I}}^3$ 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.

24 pages, 7 figures