Amplitude Analysis of the Decays $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ and ${\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$

Using $e^{+}{e}^{-}$ annihilation data corresponding to an integrated luminosity of 2.93 $\mathrm{f}{\mathrm{b}}^{-1}$ taken at the center-of-mass energy $\sqrt{s}=3.773$~GeV with the BESIII detector, a joint amplitude analysis is performed on the decays $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ and $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$(non-$\eta$). The fit fractions of individual components are obtained, and large interferences among the dominant components of $D^0\to a_1(1260)\pi$, $D^0\to \pi (1300)\pi$, $D^0\to \rho (770)\rho (770)$ and $D^0\to 2(\pi \pi {)}_S$ are found in both channels. With the obtained amplitude model, the $CP$-even fractions of $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ and $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$(non-$\eta$) are determined to be $(75.2\pm {1.1}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {1.5}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$ and $(68.9\pm {1.5}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {2.4}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$, respectively. The branching fractions of $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ and $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$(non-$\eta$) are measured to be $(0.688\pm {0.010}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.010}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$ and $(0.951\pm {0.025}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.021}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}})\mathrm{\%}$, respectively. The amplitude analysis provides an important model for binning strategy in the measurements of the strong phase parameters of $D^0\to 4\pi$ when used to determine the CKM angle $\gamma ({\phi }_3)$ via the $B^{-}\to D{K}^{-}$ decay.