Measurement of the $CP$-even fraction of ${D}^{0}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$

A measurement of the $CP$-even fraction of the decay ${D}^{0}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ is performed with a quantum-correlated $\psi \left(3770\right)\to D\overline{D}$ data sample collected by the BESIII experiment, corresponding to an integrated luminosity of $2.93\text{}\text{}{\mathrm{fb}}^{-1}$. Using a combination of $CP$ eigenstates, $D\to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ and $D\to {K}_{S,L}^{0}{\pi }^{+}{\pi }^{-}$ as tagging modes, the $CP$-even fraction is measured to be ${F}_{+}^{4\pi }=0.735±0.015±0.005$, where the first uncertainty is statistical and the second is systematic. This is the most precise determination of this quantity to date. It provides valuable model-independent input for the measurement of the angle $\gamma $ of the Cabibbo-Kobayashi-Maskawa matrix with ${B}^{±}\to D{K}^{±}$ decays, and for time-dependent studies of $CP$ violation and mixing in the ${D}^{0}\text{−}{\overline{D}}^{0}$ system.