Systematic analysis of the D-wave charmonium states with the QCD sum rules

TL;DR

This work uses QCD sum rules to systematically study D-wave charmonium states, deriving four two-point sum rules for the 1^3D1, 1^3D2, 1^1D2, and 1^3D3 channels. The analysis includes the OPE up to gluon and three-gluon condensates and employs quark-hadron duality with a Borel transform to extract masses from the spectral densities ρ_1(s), ρ_2^1(s), ρ_2^2(s), ρ_3(s). The predicted masses are M ≈ 3.77, 3.82, 3.83, 3.84 GeV for ψ1, ψ2, η_c2, ψ3, respectively, aligning ψ1 with ψ(3770), ψ2 with ψ2(3823) and ψ3 with ψ3(3842), and predicting η_c2 near 3.83 GeV awaiting observation. The results remain robust across different condensate parameter sets, highlighting the robustness of the QCD sum-rule approach for charmonia near open-charm thresholds and guiding future experimental searches.

Abstract

We systematically study the 1D charmonium spin-triplet (with the $J^{PC}=1^{--}, 2^{--}, 3^{--}$) and spin-singlet (with the $J^{PC}=2^{-+}$) via the QCD sum rules in comparison with the recent experimental results. The predicted mass $M_{ψ_1}=3.77\pm{0.09}\,\rm {GeV}$ supports identifying the $ψ_1$ as the $ψ(3770)$, the value $M_{ψ_2}=3.82\pm{0.09}\,\rm {GeV}$ is consistent with the reported observation of the $ψ_2(3823)$, the prediction $M_{ψ_3}=3.84\pm{0.08}\,\rm {GeV}$ supports identifying the $ψ_3$ as the $ψ_3(3842)$. Additionally, we estimate the unobserved $η_{c2}$ state lies at $3.83\pm{0.09}\,\rm {GeV}$, and suggest detection prospects in the future. More experimental data will help us to unravel the mass spectrum of the charmonium states near the open-charm thresholds.

Figures (4)

• Figure 1: The Feynman diagrams contribute to the gluon condensate $\langle \frac{\alpha_sGG}{\pi}\rangle$.
• Figure 2: The Feynman diagrams contribute to the three-gluon condensate $\langle g_s^3 GGG\rangle$.
• Figure 3: The contributions of the vacuum condensates of dimension $n$ with variations of the Borel parameters $T^2$, where the $A$, $B$, $C$ and $D$ denote $\psi_1$, $\psi_2$, $\eta_{c2}$ and $\psi_3$ charmonium states, respectively.
• Figure 4: The masses of the charmonium states with variations of the Borel parameters $T^2$, where the $A$, $B$, $C$ and $D$ denote $\psi_1$, $\psi_2$, $\eta_{c2}$ and $\psi_3$ charmonium states, respectively.