$CP$ violation in $\eta^{(\prime)}\to\pi^+\pi^-\mu^+\mu^-$ decays

It has been pointed out recently that a certain set of dimension-6 scalar $P$- and $CP$-violating light-quark-muon operators may be tested in $\eta$ and $\eta'$ decays to various final states involving $\mu^+\mu^-$ pairs, at a level not yet excluded by constraints from electric dipole moments. We here work out the hadronic matrix elements required for the predictions for the decays $\eta, \eta' \to\pi^+\pi^-\mu^+\mu^-$. We relate a new asymmetry in the angular distribution between the dipion and dimuon decay planes to the corresponding Wilson coefficients. Despite the advantage of not requiring the measurement of muon polarization, the projected sensitivities are shown to be moderate due to a combination of small phase space and chiral suppression. Such sensitivity studies are most timely in view of upcoming or planned high-statistics $\eta^{(\prime)}$ experiments such as the JLab eta factory or REDTOP.