Dalitz plot binning optimization for the model-independent measurement of the CKM angle β with B⁰ → Dπ⁺π⁻, D → K_S⁰π⁺π⁻ decays

The paper does what it says: it adapts the Fisher-information binning optimization (previously used for γ) to the time-dependent β measurement in B0→Dπ+π-, D→KSπ+π- and obtains a ~10% sensitivity gain over equal-phase-difference binning. The derivation of Q2 from Fisher information is correct, the simulated annealing implementation is reasonable, and the toy MC validates the predicted improvement under the stated conditions. The robustness checks against decay time acceptance, efficiency, resolution, and background are thoughtful and show the optimization is not fragile to those effects.

The soft spot, and it is a real one, is that the optimization and the validation both use the same B0→D0π+π- amplitude model [23] as their sole input for the hadronic parameters kj, cj, sj. No alternative model or perturbed version is tested to see whether the 10% improvement survives a different strong-phase structure. The authors acknowledge the reliance on the LHCb model, but they do not propagate any uncertainty from the model into the improvement estimate. If the true strong-phase variation differs from [23] in a way that the equal-phase-difference and optimized binnings are affected differently, the 10% could be an overestimate or could vanish. This is a load-bearing concern for any quantitative claim about the size of the improvement.

That said, the paper is clear about its assumptions and does not oversell. It is a methodological contribution aimed at experimental groups who will have their own amplitude model (or will eventually fit the parameters from data). The 10% number is a placeholder based on one model; the value of the paper is the optimization framework itself, not the precise number. I would support sending this to peer review. The authors should be asked to either quote the result with a caveat about model dependence stronger than the current one, or, ideally, to test the optimization with a second model or with systematic variations of the amplitude parameters.

For my own work I wouldn't cite this directly because I don't work on β measurements, but the paper is a competent, incremental contribution that belongs in the literature.

The central claim of a ~10% sensitivity improvement is entirely derived from and validated using the B0→D0π+π- amplitude model of Ref. [23]. The binning optimization (Sec. 3.3) computes the hadronic parameters kj, cj, sj from this model, and the toy Monte Carlo verification (Sec. 4.3) generates pseudoexperiments from the same model and fixes those parameters at their model-predicted values. No test is performed using an alternative model or a perturbed version of [23] to assess whether the relative improvement changes significantly when the strong-phase distribution across the B Dalitz plot is different. The authors acknowledge that 'the optimization relies on the accuracy of the B0→D0π+π- amplitude model from LHCb' (reader's weakest_assumption), but they do not propagate any model uncertainty into the improvement estimate. If the true strong-phase structure deviates from [23] in a way that the equal-phase-difference baseline and the optimized binning are differently affected (e.g., because the optimized binning exploits model-specific features), the claimed 10% could be an overestimate or even vanish. This is a load-bearing concern because the entire quantitative claim hangs on a single unverified input.

The reader identified the reliance on the amplitude model as the weakest assumption. Our stress-test concurs and sharpens the concern by noting that the improvement is computed and validated within the same model, without any cross-check. This does not invalidate the paper's methodology—the optimization framework is sound and the robustness checks on experimental effects are thorough—but it means the 10% figure should be understood as conditional on the model. Because the paper is otherwise strong and the authors acknowledge the limitation in passing (Sec. 3.3: 'the publicly available amplitude model of B0→D0π+π- from Ref. [23]'), we do not recommend changing the ACCEPT verdict. However, adding a quantitative test of model dependence would strengthen the claim significantly. The agreement with the reader is complete.

The work provides a concrete, data-driven method to improve the sensitivity of a theoretically clean β measurement at LHCb and Belle II. The ~10% gain is modest but valuable, especially when combined with the model-independence of the final analysis. The optimization methodology (Fisher information + simulated annealing) is clearly presented and could be applied to other binned analyses. The robustness checks against experimental effects strengthen confidence in the result.

This paper shows that the statistical sensitivity to the CKM angle β can be improved by approximately 10% by optimizing how the B⁰ → Dπ⁺π⁻ Dalitz plot is divided into bins. The optimization uses a quality factor constructed from the Fisher information and a simulated annealing search, targeting the time-dependent likelihood that will be used in a real analysis. The improvement is validated with toy Monte Carlo experiments, and the optimized binning is shown to be robust against realistic detector effects such as decay-time acceptance, Dalitz-plot efficiency, mass resolution, and background contamination. The result matters because a cleaner measurement of β tests the Standard Model of CP violation and resolves ambiguities left by sin 2β measurements alone.

What carries the argument. The central object is a binning quality factor Q² = I_binned / I_unbinned, where I is the Fisher information for β computed from the time‑dependent probability density of the double‑Dalitz decay chain. Simulated annealing is used to search the space of symmetric 2M‑bin partitions of the B⁰ → Dπ⁺π⁻ Dalitz plot, evaluating Q² for each candidate. The B‑decay hadronic parameters kⱼ, cⱼ, sⱼ that enter Q² are pre‑computed from the amplitude model of Ref. [23]; the D‑decay parameters are fixed from external charm‑factory measurements.

Core claim. The authors demonstrate that a binning scheme for the B⁰ → Dπ⁺π⁻ Dalitz plot, selected by maximizing the ratio of binned to unbinned Fisher information for β, yields a 9.7 % reduction in the expected statistical uncertainty on β relative to the conventional equal‑phase‑difference binning. The time‑dependent quality factor (Q²_TD) accounts for B⁰–B̄⁰ oscillations, decay‑time dependence, and flavour tagging. Toy Monte Carlo studies confirm that the Q² prediction matches the actual improvement seen in fits where β is the only free parameter. For the recommended 16‑bin (M = 8) scheme, the sensitivity gain persists when realistic detector effects (efficiency, resolution, background) are introduced, with the relative improvement rising to 11.6 % when Dalitz‑plot efficiency is included.

The premise it rests on. The optimization depends on the accuracy of the LHCb amplitude model for B⁰ → D⁰π⁺π⁻ [23]; if that model misrepresents the variation of the strong‑phase difference across the B Dalitz plot, the computed hadronic parameters (and thus the optimized binning) may not give the predicted sensitivity improvement.

The recommended M = 8 binning scheme can be adopted by LHCb and Belle II for a model‑independent β measurement with B⁰ → Dπ⁺π⁻, D → K_S⁰π⁺π⁻ decays, directly improving the statistical power of future datasets.

The optimization methodology—Fisher information plus simulated annealing—is transferable to other binned Dalitz‑plot analyses where the binning choice affects sensitivity, such as γ measurements in B⁺ → DK⁺ decays.

The robustness studies confirm that the 10 % gain is not eroded by typical experimental effects (efficiency, resolution, background), so the scheme can be used without re‑optimizing under realistic conditions.

If the B‑decay amplitude model used in the optimization is later refined, the same procedure can produce an updated binning, ensuring the scheme evolves with improved theoretical or experimental input.

The paper fixes the B‑decay hadronic parameters to values from a single amplitude model; a complete systematic uncertainty assessment would require propagating the model's uncertainties (or variations across alternative models) into the optimized binning and the final β sensitivity.

The toy validation assumes a simplified fit where β is the only free parameter and the hadronic parameters are known. In a real analysis, β and the hadronic parameters are fitted simultaneously, which may alter the effective sensitivity gain. Extending the Fisher‑information formalism to the full parameter set would test whether the 10 % improvement holds.

The method could be adapted to optimize the D‑decay binning jointly with the B‑decay binning, rather than fixing it to the established scheme, potentially yielding further gains if the D‑decay strong‑phase parameters are also subject to refinement.

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