Clarifying the puzzling mass shift of the $ψ(4160)$ via a reanalysis of $R$-value data with unquenched charmonium spectroscopy
The long-standing upward shift of the extracted $ψ(4160)$ 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 $B^+\to K^+μ^+μ^-$ process within an unquenched charmonium framework, where the lower-mass $ψ(4160)$ 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 $ψ(4160)$ mass, and provide an independent examination. In contrast to the conventional quenched picture with $ψ(4040)$, $ψ(4160)$, and $ψ(4415)$, the unquenched vector-charmonium spectrum contains six states: $ψ(4040)$, $ψ(4160)$, $ψ(4220)$, $ψ(4380)$, $ψ(4415)$, and $ψ(4500)$. Including these states together with the near-threshold $ψ(3770)$, we find that the BESII $R$-value line shape can be well reproduced over the full energy range while retaining the lower-mass $ψ(4160)$ assignment. The enhancement around $4.19$~GeV then arises from the coherent interplay among the nearby $ψ(4040)$, $ψ(4160)$, and $ψ(4220)$ amplitudes, rather than requiring an upward shift of the $ψ(4160)$ 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.