Correlations of \(C\) and \(CP\) violation in \(\eta\to \pi^0\ell^+\ell^-\) and \(\eta'\to \eta\ell^+\ell^-\)

Based on recent progress in the systematic analysis of \(C\) and \(CP\) violation in the light-meson sector, we calculate the \(C\)-odd transition amplitudes \(\eta\to\pi^0\ell^+\ell^-\) and \(\eta'\to\eta\ell^+\ell^-\). Focusing on long-distance contributions driven by the lowest-lying hadronic intermediate states, we work out the correlations between these beyond-the-Standard-Model signals and the Dalitz-plot asymmetries in \(\eta \rightarrow \pi^0 \pi^+ \pi^-\) and \(\eta' \rightarrow \eta \pi^+ \pi^- \), using dispersion theory.