Dalitz plot binning optimization for the model-independent measurement of the CKM angle $\beta$ with $B^0\to D{\pi }^{+}{\pi }^{-}$, $D\to K_S^0{\pi }^{+}{\pi }^{-}$ decays

A model-independent measurement of the CKM angle $\beta$ can be performed using a time-dependent analysis of $B^0\to D{\pi }^{+}{\pi }^{-}$, $D\to K_S^0{\pi }^{+}{\pi }^{-}$ decays, where the Dalitz plots of both the $B$ and $D$ decays are divided into bins. The choice of binning scheme for the $B^0\to D{\pi }^{+}{\pi }^{-}$ Dalitz plot directly impacts the statistical sensitivity to $\beta$. This paper presents an optimization of this binning using a Fisher information-based quality factor $Q^2$ and simulated annealing. The optimized binning improves the sensitivity to $\beta$ by approximately $10\mathrm{\%}$ relative to the equal-phase-difference baseline, as validated by toy Monte Carlo studies without experimental effects. The robustness of this improvement against decay time acceptance, Dalitz plot efficiency, Dalitz plot resolution, and background contributions is also assessed.