Clarifying the puzzling mass shift of the $ψ(4160)$ via a reanalysis of $R$-value data with unquenched charmonium spectroscopy

RESUMO

The long-standing upward shift of the extracted $\psi(4160)$ mass, from about $4.16$~GeV to $4.19$~GeV in later analyses, remains a puzzling issue in charmonium spectroscopy. In our previous study, this problem was investigated through the $B^+\to K^+\mu^+\mu^-$ process within an unquenched charmonium framework, where the lower-mass $\psi(4160)$ assignment was found to be compatible with the data. Here we revisit the BESII $R$-value data, which played an important role in the historical extraction of the higher $\psi(4160)$ mass, and provide an independent examination. In contrast to the conventional quenched picture with $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$, the unquenched vector-charmonium spectrum contains six states: $\psi(4040)$, $\psi(4160)$, $\psi(4220)$, $\psi(4380)$, $\psi(4415)$, and $\psi(4500)$. Including these states together with the near-threshold $\psi(3770)$, we find that the BESII $R$-value line shape can be well reproduced over the full energy range while retaining the lower-mass $\psi(4160)$ assignment. The enhancement around $4.19$~GeV then arises from the coherent interplay among the nearby $\psi(4040)$, $\psi(4160)$, and $\psi(4220)$ amplitudes, rather than requiring an upward shift of the $\psi(4160)$ mass itself. The additional higher states also naturally describe the line-shape structure in the $4.4$~GeV region. We further show that the seven-resonance coherent amplitude contains six complex zeros, yielding $2^6=64$ mathematically equivalent solutions with identical line shapes but substantially different di-electron widths and relative phases. Comparing these solutions with available experimental information and representative unquenched charmonium predictions, we provide a qualitative assessment of their phenomenological consistency and highlight several solutions that appear more compatible with present information.