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

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