Model-independent determination of the relative strong-phase difference between $D^0$ and $\bar{D}^0\rightarrow K^0_{S,L}\pi^+\pi^-$ and its impact on the measurement of the CKM angle $\gamma/\phi_3$

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 $\psi(3770)$ resonance. The relative strong-phase parameters between $D^0$ and $\bar{D}^0$ in the decays $D^0\rightarrow K^0_{S,L}\pi^+\pi^-$ are studied using 2.93~${\rm fb}^{-1}$ 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 $\gamma/\phi_3$ is expected to be between $0.7^\circ$ and $1.2^\circ$, for an analysis using the decay $B^{\pm}\rightarrow DK^{\pm}$, $D\rightarrow K^0_S\pi^+\pi^-$, where $D$ represents a superposition of $D^0$ and $\bar{D^0}$ states. This is a factor of three 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.