Measurement of the $C\text{\hspace{0.17em}⁣}P$-even fraction of $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$

A measurement of the $C\text{\hspace{0.17em}⁣}P$-even fraction of the decay $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ is performed with a quantum-correlated $\psi (3770)\to D\bar{D}$ data sample collected by the BESIII experiment, corresponding to an integrated luminosity of 2.93 ${\text{fb}}^{-1}$. Using a combination of $C\text{\hspace{0.17em}⁣}P$ eigenstates, $D\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and $D\to K_{S,L}^0{\pi }^{+}{\pi }^{-}$ as tagging modes, the $C\text{\hspace{0.17em}⁣}P$-even fraction is measured to be $F_{+}^{4\pi }=0.735\pm 0.015\pm 0.005$, 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 CKM angle $\gamma$ with $B^{\pm }\to D{K}^{\pm }$ decays, and for time-dependent studies of $C\text{\hspace{0.17em}⁣}P$ violation and mixing in the $D^0$-${\bar{D}}^0$ system.