Determination of U-spin breaking parameters with an amplitude analysis of the decay $D^0\to K_{\mathrm{L}}^0{\pi }^{+}{\pi }^{-}$

We present a study of the resonant structure of the decay $D^0\to K_{\mathrm{L}}^0{\pi }^{+}{\pi }^{-}$, using quantum-correlated $D^0{\bar{D}}^0$ data produced at $\sqrt{s}=3.773$ GeV. The data sample was collected by the BESIII experiment and corresponds to an integrated luminosity of $2.93$ fb${}^{-1}$. This study is the first amplitude analysis of a decay mode involving a $K_{\mathrm{L}}^0$, which also results in the first measurement of the complex $\mathit{\text{U-spin}}\mathit{\text{breaking}}\mathit{\text{parameters}}$ ($\widehat{\rho }$) related to various $CP$-eigenstate resonant modes through which the three-body decay proceeds. The moduli of the $\widehat{\rho }$ parameters have central values in a wide range from $0.4$ to $12.1$, which indicates substantial U-spin symmetry breaking. We present the fractional resonant contributions and average strong-phase parameters over regions of phase space for both $K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-}$ and $K_{\mathrm{L}}^0{\pi }^{+}{\pi }^{-}$ modes. We also report the ratio of the branching fractions between $K_{\mathrm{L}}^0{\pi }^{+}{\pi }^{-}$ and $K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-}$ decay modes and the $CP$-even fraction of the $K_{\mathrm{L}}^0{\pi }^{+}{\pi }^{-}$ state calculated using the U-spin breaking parameters.