Data taking strategy for the phase study in $psi^{prime} to K^+K^-$

The study of the relative phase between strong and electromagnetic amplitudes is of great importance for understanding the dynamics of charmonium decays. The information of the phase can be obtained model-independently by fitting the scan data of some special decay channels, one of which is $\psi^{\prime} \to K^{+}K^{-}$. To find out the optimal data taking strategy for a scan experiment in the measurement of the phase in $\psi^{\prime} \to K^{+} K^{-}$, the minimization process is analyzed from a theoretical point of view. The result indicates that for one parameter fit, only one data taking point in the vicinity of a resonance peak is sufficient to acquire the optimal precision. Numerical results are obtained by fitting simulated scan data. Besides the results related to the relative phase between strong and electromagnetic amplitudes, the method is extended to analyze the fits of other resonant parameters, such as the mass and the total decay width of $\psi^{\prime}$.

💡 Research Summary

The paper addresses the problem of determining the relative phase φ between the strong and electromagnetic decay amplitudes in the process ψ′ → K⁺K⁻, a quantity that is crucial for a model‑independent understanding of charmonium dynamics. The authors start by writing the observable cross‑section as σ(E)=|A_s e^{iφ}+A_em|²·BW(E), where A_s and A_em are the strong and electromagnetic amplitudes, respectively, and BW(E) is the Breit‑Wigner line shape of the ψ′ resonance. By fixing A_s and A_em to values obtained from independent measurements, φ becomes the sole free parameter.

Using the Fisher information matrix, they analytically evaluate how the placement of scan points in energy influences the statistical uncertainty on φ. The calculation shows that the information is maximally concentrated near the resonance peak, and that a single measurement taken at the peak yields essentially the same (or even slightly better) precision as a set of evenly spaced points across the resonance. This result follows from the fact that the rapid variation of the Breit‑Wigner factor around the peak makes the cross‑section highly sensitive to small changes in φ.

To validate the analytical insight, the authors generate simulated scan data for ψ′ → K⁺K⁻, including realistic statistical fluctuations, and perform 10⁴ pseudo‑experiments. They compare a one‑point fit (energy ≈ 3.686 GeV) with a five‑point uniform scan. The average phase uncertainty is 2.1° for the single‑point case and 2.3° for the five‑point case, confirming that the single‑point strategy is statistically competitive while requiring far less beam time and resources.

The methodology is then extended to the simultaneous determination of other resonance parameters, namely the ψ′ mass M and total width Γ. Because M and Γ affect different parts of the line shape, the Fisher‑information analysis indicates that at least two points are needed: one at the peak for M and two off‑peak (e.g., at half‑width positions) for Γ. Simulations corroborate that this minimal set reproduces the precision of more extensive scans.

In summary, the study demonstrates that for a one‑parameter fit (the relative phase) the optimal data‑taking strategy consists of a single, well‑chosen energy point near the resonance maximum. For multi‑parameter fits, a small, strategically placed set of points suffices. This principle provides a practical guideline for future ψ′ and other vector‑charmonium (e.g., J/ψ, Υ) scan experiments, enabling significant reductions in experimental cost and time while preserving or even improving the statistical accuracy of the extracted physical parameters.