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

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.

💡 Research Summary

In this work the authors perform a systematic QCD‑sum‑rule (QCDSR) analysis of the D‑wave charmonium sector, i.e. the 1 D states with orbital angular momentum L = 2. The study focuses on the spin‑triplet states with quantum numbers J^{PC}=1^{–}, 2^{–}, 3^{–} (denoted ψ₁, ψ₂, ψ₃) and the spin‑singlet state with J^{PC}=2^{-+} (the η_{c2}). The motivation is twofold: (i) several D‑wave candidates—ψ(3770), ψ₂(3823) and ψ₃(3842)—have been observed in recent experiments (BESIII, Belle, LHCb), and (ii) the η_{c2} state has not yet been seen, so a theoretical prediction of its mass is valuable for future searches.

Theoretical framework
The authors construct interpolating currents for each state using two covariant derivatives ↔D_μ acting on the charm quark fields. This choice guarantees gauge invariance while encoding the L = 2 orbital structure. The currents are:

• J_μ for ψ₁ (1³D₁),
• J_{μν}^{(1)} and J_{μν}^{(2)} for ψ₂ (1³D₂) and η_{c2} (1¹D₂) respectively,
• J_{μνρ} for ψ₃ (1³D₃).
  Two‑point correlation functions Π(p) are defined in the usual way and saturated with a complete set of intermediate hadronic states. The pole contributions are parametrised by decay constants f and masses M, and the tensor decomposition isolates the scalar invariant Π(p²) that carries the information of interest.

On the QCD side the operator product expansion (OPE) is carried out up to dimension‑6 operators: the perturbative term, the gluon condensate ⟨α_s G²⟩ (dimension 4) and the three‑gluon condensate ⟨g_s³ G³⟩ (dimension 6). The authors present explicit analytic expressions for the spectral densities ρ₁(s), ρ_{12}(s), ρ_{22}(s) and ρ₃(s) corresponding to the four currents. The spectral densities contain rational functions of s and the charm‑quark mass m_c, as well as logarithmic terms arising from loop integrals.

Sum‑rule construction
A Borel transform is applied to suppress higher‑state contributions and improve OPE convergence. The Borel parameter T² and the continuum threshold s₀ are chosen such that the pole contribution accounts for roughly 55 % of the total dispersion integral—a standard criterion in QCDSR analyses. The authors adopt s₀ ≈ M + (0.45–0.65) GeV, reflecting the typical gap between the ground D‑wave state and its first radial excitation. Two sets of vacuum‑condensate values are used to test the stability of the results:

• Set I: ⟨α_s G²⟩ = 0.012 GeV⁴, ⟨g_s³ G³⟩ = 0.045 GeV⁶,
• Set II: ⟨α_s G²⟩ = 0.022 GeV⁴, ⟨g_s³ G³⟩ = 0.616 GeV⁶.
  The charm‑quark mass is taken in the \overline{MS} scheme, m_c(m_c)=1.275 ± 0.025 GeV, and evolved to the scale μ = m_c using the four‑loop renormalisation‑group equation.

Numerical analysis
For each state the authors scan a reasonable Borel window (T² ≈ 2.9–3.8 GeV²) and adjust s₀ (≈ 4.30–4.40 GeV) to achieve stability of the extracted mass with respect to T². The resulting masses are:

• ψ₁(1³D₁): M = 3.77 ± 0.09 GeV,
• ψ₂(1³D₂): M = 3.82 ± 0.09 GeV,
• η_{c2}(1¹D₂): M = 3.83 ± 0.09 GeV (prediction),
• ψ₃(1³D₃): M = 3.84 ± 0.08 GeV.
  The quoted uncertainties combine the variation of T², the choice of s₀, and the two condensate sets. Decay constants are also extracted (f ≈ 13–15 GeV⁴).

The analysis shows that the perturbative term dominates the OPE (≈ 90 % of the total), while the gluon condensate contributes only a few percent and the three‑gluon condensate is negligible. This pattern is typical for heavy‑quark systems at relatively high masses, where non‑perturbative vacuum effects are suppressed.

Comparison with experiment and other models
The predicted masses for ψ₁, ψ₂ and ψ₃ agree within uncertainties with the experimentally measured ψ(3770), ψ₂(3823) and ψ₃(3842), respectively. The η_{c2} prediction lies at 3.83 GeV, very close to the ψ₂ and ψ₃ masses, suggesting that it may be accessible in radiative transitions from ψ(3770) or in B‑meson decays (e.g. B → K η_{c2}). The authors note that potential‑model calculations (Cornell, screened, Godfrey‑Isgur, etc.) give similar values, typically within 10–30 MeV, confirming the reliability of the QCDSR approach.

Conclusions and outlook
The paper demonstrates that QCD sum rules provide a robust, model‑independent framework for extracting the masses of D‑wave charmonium states. The agreement with existing data validates the method, and the prediction for the yet‑unobserved η_{c2} offers a concrete target for upcoming experiments at BESIII, Belle II and LHCb. The authors suggest that extending the analysis to higher‑dimensional condensates, incorporating radiative corrections, and studying decay widths or transition form factors would further enrich the understanding of the D‑wave sector and its interplay with open‑charm thresholds.