Is the $3S$-$2D$ mixing strong for the charmonia $ψ(4040)$ and $ψ(4160)$?

TL;DR

The study investigates whether the strong $S$-$D$ mixing between the charmonia $\psi(4040)$ and $\psi(4160)$ can be explained by a coupled-channel mechanism beyond the conventional tensor interaction. Using a $3S$-$2D$ mixing framework for $\psi(3^3S_1)$ and $\psi(2^3D_1)$, the authors incorporate open-charm channel couplings via a once-subtracted dispersion relation and compute transition amplitudes with the quark-pair-creation model, yielding mixing angles $\theta_1=(7.3^{+0.1}_{-0.4})^{\circ}$ and $\theta_2=(10.4^{+2.0}_{-3.5})^{\circ}$ and physical masses around $M(\psi^{\prime})\approx 4006$ MeV and $M(\psi^{\prime\prime})\approx 4089$ MeV. These results are notably larger than the tensor-only prediction (about $0.3^{\circ}$) but still smaller than some prior extractions that relied on dileptonic widths, highlighting a major sensitivity to $\Gamma_{e^+e^-}$ inputs. The authors show that the predicted dielectronic widths and the ratio of dileptonic widths, $\Gamma^{\psi^{\prime}}_{e^+e^-}/\Gamma^{\psi^{\prime\prime}}_{e^+e^-}$, conflict with PDG values, emphasizing the need for high-precision, exclusive measurements to resolve the $S$-$D$ mixing puzzle in these states. The work underlines the importance of unquenched, coupled-channel dynamics in charmonium spectroscopy and points to future experimental data from BESIII as crucial to clarifying the mixing mechanism.

Abstract

In this work, we revisit the $3S$-$2D$ mixing scheme for the charmonia $ψ(4040)$ and $ψ(4160)$. We introduce a coupled-channel mechanism-distinct from the tensor-force contribution in potential models, which alone is insufficient to induce significant mixing-to describe the mixing between these states. Our analysis yields mixing angles of $θ_1=7^\circ$ and $θ_2=10^\circ$, inconsistent with the larger angle inferred from experimental data, such as the dilectronic widths of the $ψ(4040)$ and $ψ(4160)$. We discuss possible origins of this discrepancy and emphasize the need for future experiments to resolve it. Precise measurements of the resonance parameters and dilectronic decay widths, via both inclusive and exclusive processes, will be crucial in clarifying this issue.

Figures (1)

• Figure 1: The ratio of dielectronic widths $\Gamma^{\psi^{\prime}_{3S\text{-}2D}}_{e^+e^-}/\Gamma^{\psi^{\prime\prime}_{3S\text{-}2D}}_{e^+e^-}$ as a function of the $3S\text{-}2D$ mixing angles with our parameters from Eqs. (\ref{['eq:Gamma1']})and (\ref{['eq:Gamma2']}). The red pentagram represents the prediction of this work ($\theta \approx 10^\circ$ and ratio $\Gamma^{\psi^{\prime}_{3S\text{-}2D}}_{e^+e^-}/\Gamma^{\psi^{\prime\prime}_{3S\text{-}2D}}_{e^+e^-}\approx 4.1$). The light green and dark green shaded bands indicate the experimental values $1.79 \pm 0.83$ and $1.04 \pm 0.12$, respectively, from the PDG ParticleDataGroup:2024cfk. The triangle and square indicate the mixing angles $20^{\circ}$ and $34.8^{\circ}$, as obtained in Refs. Wang:2022jxj and Badalian:2008dv, respectively.