Model-independent determination of the relative strong-phase difference between ${D}^{0}$ and ${\overline{D}}^{0}\to {K}_{S,L}^{0}{\pi }^{+}{\pi }^{-}$ and its impact on the measurement of the CKM angle $\gamma /{\varphi }_{3}$

Crucial inputs for a variety of $CP$-violation studies can be determined through the analysis of pairs of quantum-entangled neutral $D$ mesons, which are produced in the decay of the $\psi \left(3770\right)$ resonance. The relative strong-phase parameters between ${D}^{0}$ and ${\overline{D}}^{0}$ in the decays ${D}^{0}\to {K}_{S,L}^{0}{\pi }^{+}{\pi }^{-}$ are studied using $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ annihilation data delivered by the BEPCII collider and collected by the BESIII detector at a center-of-mass energy of 3.773 GeV. Results are presented in regions of the phase space of the decay. These are the most precise measurements to date of the strong-phase parameters in $D\to {K}_{S,L}^{0}{\pi }^{+}{\pi }^{-}$ decays. Using these parameters, the associated uncertainty on the Cabibbo-Kobayashi-Maskawa angle $\gamma /{\varphi }_{3}$ is expected to be between 0.7° and 1.2° for an analysis using the decay ${B}^{±}\to D{K}^{±}$, $D\to {K}_{S}^{0}{\pi }^{+}{\pi }^{-}$, where $D$ represents a superposition of ${D}^{0}$ and ${\overline{D}}^{0}$ states. This is a factor of 3 smaller than that achievable with previous measurements. Furthermore, these results provide valuable input for charm-mixing studies, other measurements of $CP$ violation, and the measurement of strong-phase parameters for other $D$-decay modes.