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.