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

Abstract

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 $D^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}\ddot{I}{€}^{+}\ddot{I}{€}^{-}$ and $D^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}\ddot{I}{€}^0\ddot{I}{€}^0$(non-$\hat{I}\cdot$). The fit fractions of individual components are obtained, and large interferences among the dominant components of $D^0\to a_1(1260)\ddot{I}€$, $D^0\to \ddot{I}€(1300)\ddot{I}€$, $D^0\to \ddot{I}(770)\ddot{I}(770)$ and $D^0\to 2(\ddot{I}€\ddot{I}€{)}_S$ are found in both channels. With the obtained amplitude model, the $CP$-even fractions of $D^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}\ddot{I}{€}^{+}\ddot{I}{€}^{-}$ and $D^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}\ddot{I}{€}^0\ddot{I}{€}^0$(non-$\hat{I}\cdot$) are determined to be $(75.2\pm 1.1_{stat.}\pm 1.5_{syst.})\%$ and $(68.9\pm 1.5_{stat.}\pm 2.4_{syst.})\%$, respectively. The branching fractions of $D^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}\ddot{I}{€}^{+}\ddot{I}{€}^{-}$ and $D^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}\ddot{I}{€}^0\ddot{I}{€}^0$(non-$\hat{I}\cdot$) are measured to be $(0.688\pm 0.010_{stat.}\pm 0.010_{syst.})\%$ and $(0.951\pm 0.025_{stat.}\pm 0.021_{syst.})\%$, respectively. The amplitude analysis provides an important model for binning strategy in the measurements of the strong phase parameters of $D^0\to 4\ddot{I}€$ when used to determine the CKM angle ${\hat{I}}^3(\ddot{I}{•}_3)$ via the $B^{-}\to D{K}^{-}$ decay.