Visualisation of $CP$-violation effects in decay-time-dependent analyses of multibody $B$-meson decays

Decay-time-dependent $CP$-violation effects in transitions of neutral $B$ mesons to $CP$-eigenstates can be visualised by oscillations in the asymmetry, as a function of decay time, between decay yields from mesons tagged as initially having $\overline{B}$ or $B$ flavour. Such images, for example for $B^0 \to J/\psi K^0_{\rm S}$ decays where the magnitude of the oscillation is proportional to $\sin(2\beta)$ with $\beta$ being an angle of the Cabibbo--Kobayashi--Maskawa Unitarity Triangle, provide a straightforward illustration of the underlying physics. Until now there has been no comparable method to provide visualisation for the case of decays to multibody final states that are not $CP$-eigenstates, where interference between $CP$-even and -odd amplitudes provides additional physics sensitivity. A method is proposed to weight the data so that the terms of interest can be projected out and used to obtain asymmetries that visualise the relevant effects. Application of the weighting to $B^0_s$ decays, where effects due to non-zero width difference are not negligible, provides a novel method to observe $CP$ violation in interference between mixing and decay without tagging the production flavour.