Prospects for observing the missing $2D$ and $1F$ charmonium states around 4 GeV

Our understanding of high-lying states within the charmonium family remains incomplete, particularly in light of recent observations of charmonium states at energies around 4 GeV. In this study, we investigate the spectroscopic properties of several high-lying charmonia, focusing on the $2D$ and $1F$ states. A mass spectrum analysis is conducted, incorporating the unquenched effects. We then present a detailed study of the strong decay properties, including partial decay widths for two-body strong decays permitted by the Okubo-Zweig-Iizuka (OZI) rule. Additionally, we explore the primary radiative decay channels associated with these states. Finally, we discuss the radiative transitions of the $2D$ and $1F$ states via $e^+e^-$ annihilation. Theoretical predictions provided here aim to guide future experimental searches for high-lying charmonium states at facilities such as BESIII, Belle II, LHCb, and the future STCF.

💡 Research Summary

The paper addresses the long‑standing gap in the charmonium spectrum around 4 GeV, focusing on the missing 2 D and 1 F orbital excitations. Using a modified Godfrey‑Isgur (MGI) potential that incorporates a screened confining term and relativistic smearing, the authors calculate the masses of the three 2 D states (³D₁, ³D₂, ³D₃) and the four 1 F states (³F₂, ³F₃, ³F₄, ¹F₃). The predicted masses lie in the ranges 4.12–4.20 GeV for the 2 D triplet and 4.30–4.38 GeV for the 1 F quartet, with modest shifts caused by S‑D mixing (especially between 2 D₁ and the nearby ψ(4160)/ψ(4230) vector states) and by coupled‑channel effects that lower the masses relative to quenched quark‑model expectations.

Strong decay widths are evaluated with the quark‑pair‑creation (QPC) model, considering all OZI‑allowed two‑body channels. The 2 D₁ state predominantly decays into D* \bar D and D \bar D, with the D* \bar D* channel accounting for roughly 40 % of its total width (~50 MeV). The higher‑spin 2 D₂ and 2 D₃ states couple strongly to D* \bar D*, D \bar D₁(2420), and D \bar D₂(2460), leading to broader widths of 80–120 MeV. The 1 F states have fewer open‑charm channels; their main modes are D* \bar D* and D \bar D₂(2460), giving comparatively narrow total widths of 30–60 MeV. These decay patterns provide clear experimental signatures: a dominance of D* \bar D* final states for the 2 D family and a mixture of D* \bar D* and D \bar D₂ for the 1 F family.

Electromagnetic transitions are computed using the same potential wave functions. Electric‑dipole (E1) transitions such as 2 D → χ_{cJ} γ (J = 0, 1, 2) and 1 F → ψ(2S, 3S) γ have partial widths of 0.1–0.5 keV, while magnetic‑dipole (M1) transitions (e.g., 2 D → η_c γ) are smaller but still observable. Because these radiative decays produce a high‑energy photon together with a well‑identified lower‑lying charmonium, they are especially promising for e⁺e⁻ experiments where photon detection is efficient.

Production mechanisms are examined in three contexts. (1) Direct e⁺e⁻ annihilation via a virtual photon can populate the 2 D/1 F states, with the cross section enhanced by intermediate vector charmonia (ψ(4230), ψ(4380), ψ(4500)) that mix with the D‑wave components. (2) Weak decays of B mesons, Λ_b baryons, and B_c mesons can generate the missing states through color‑rearrangement processes; channels such as B → K + 2 D, B → K + 1 F, and Λ_b → Λ_c + 2 D are highlighted as high‑yield modes at LHCb and Belle II. (3) Hadronic loop mechanisms, where intermediate open‑charm meson pairs rescatter into the 2 D/1 F resonances, can significantly boost production near the D* \bar D* threshold, making energy‑scan measurements at BESIII particularly sensitive.

Based on these theoretical results, the authors propose concrete experimental strategies. BESIII should perform fine energy scans in the 4.0–4.6 GeV region, focusing on e⁺e⁻ → γ χ_{cJ} (followed by χ_{cJ} → J/ψ γ) and on the line‑shape of D* \bar D* production, where interference effects from the missing D‑wave states could appear as distortions. Belle II can exploit large B‑meson samples to search for B → K + (2 D/1 F) γ and B → K + (χ_{cJ} γ) γ final states, using the clean photon signatures and full reconstruction of the lower charmonium. LHCb’s high‑statistics pp data enable studies of Λ_b and B_c decays into the same final states, as well as inclusive searches in the D* \bar D* invariant mass spectrum. The upcoming Super τ‑Charm Facility (STCF), with its unprecedented luminosity and excellent photon detection, will be ideally suited to combine radiative and strong decay channels, providing the most comprehensive coverage.

In summary, the paper delivers a state‑of‑the‑art, unquenched potential‑model calculation of the masses, decay widths, radiative transitions, and production rates of the yet‑unobserved 2 D and 1 F charmonium states. By linking these theoretical predictions to specific, feasible measurements at BESIII, Belle II, LHCb, and STCF, the work offers a clear roadmap for finally filling the missing pieces of the charmonium spectrum and deepening our understanding of coupled‑channel dynamics in heavy‑quarkonium systems.