Measurement of angular and CP asymmetries in $D^0\to {\pi }^{+}{\pi }^{-}{\mu }^{+}{\mu }^{-}$ and $D^0\to K^{+}{K}^{-}{\mu }^{+}{\mu }^{-}$ decays

The first measurements of the forward-backward asymmetry of the dimuon pair ($A_{FB}$), the triple-product asymmetry ($A_{2\varphi }$), and the charge-parity-conjugation asymmetry ($A_{CP}$), in $D^0\to {\pi }^{+}{\pi }^{-}{\mu }^{+}{\mu }^{-}$ and $D^0\to K^{+}{K}^{-}{\mu }^{+}{\mu }^{-}$ decays are reported. They are performed using data from proton-proton collisions collected with the LHCb experiment from 2011 to 2016, corresponding to a total integrated luminosity of 5 fb${}^{-1}$. The asymmetries are measured to be \begin{align*} A_{FB}(D^0\to\pi^+\pi^-\mu^+\mu^-) &= (\phantom{-}3.3\pm3.7\pm0.6)%,\ A_{2\phi}(D^0\to\pi^+\pi^-\mu^+\mu^-)&= (-0.6\pm3.7\pm0.6)%,\ A_{CP}(D^0\to\pi^+\pi^-\mu^+\mu^-) &= (\phantom{-}4.9\pm3.8\pm0.7)%,\ A_{FB}(D^0\to K^+K^-\mu^+\mu^-) &= (0\pm11\pm2)%,\ A_{2\phi}(D^0\to K^+K^-\mu^+\mu^-)&= (9\pm11\pm1)%,\ A_{CP}(D^0\to K^+K^-\mu^+\mu^-) &= (0\pm11\pm2)%, \end{align*} where the first uncertainty is statistical and the second systematic. The asymmetries are also measured as a function of the dimuon invariant mass. The results are consistent with the Standard Model predictions.