Measurement of the strong-phase difference between $D^0$ and $\overline{D^0}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ in bins of phase space

A first determination of the strong-phase difference between $D^0$ and $\overline{D^0}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ is performed using $e^{+}{e}^{-}\to \psi (3770)\to D\bar{D}$ data collected by the BESIII detector corresponding to an integrated luminosity of $20.3$ fb${}^{-1}$. The measurements are made in four pairs of bins in phase space, which are chosen to provide optimal sensitivity to the angle $\gamma$ of the Unitarity Triangle in $B^{\pm }\to D{K}^{\pm }$ decays. From these measurements, it follows that the $C\text{\hspace{0.17em}⁣}P$-even fraction of the decay is $F_{+}=0.754\pm {0.010}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.008}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}}$. In addition, the branching fraction of $D^0\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ is measured to be $(2.863\pm {0.028}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.045}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\times {10}^{-3}$, which is twice as precise as previous results obtained at other experiments.