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

A measurement of the $C\!P$-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 $\textrm{fb}^{-1}$. Using a combination of $C\!P$ eigenstates, $D \to \pi^+\pi^-\pi^0$ and $D \to K_{S,L}^0\pi^{+}\pi^{-}$ as tagging modes, the $C\!P$-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 $C\!P$ violation and mixing in the $D^{0}$-$\bar{D}^{0}$ system.