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Crucial inputs for a variety of $CP$-violation studies can be determinedthrough the analysis of pairs of quantum-entangled neutral $D$ mesons, whichare produced in the decay of the $\psi(3770)$ resonance. The relativestrong-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 bythe BESIII detector at a center-of-mass energy of 3.773 GeV. Results arepresented in regions of the phase space of the decay. These are the mostprecise measurements to date of the strong-phase parameters in $D \toK_{S,L}^0\pi^+\pi^-$ decays. Using these parameters, the associated uncertaintyon the Cabibbo-Kobayashi-Maskawa angle $\gamma/\phi_3$ is expected to bebetween $0.7^\circ$ 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 ofthree smaller than that achievable with previous measurements. Furthermore,these results provide valuable input for charm-mixing studies, othermeasurements of $CP$ violation, and the measurement of strong-phase parametersfor other $D$-decay modes.

physical review. d/physical review. d.

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 $\gamma/\phi_3$