Hidden-heavy mesons are unusually valuable laboratories for quantum chromodynamics, or QCD, because their heavy constituents make the systems simpler than ordinary hadrons containing light quarks. Their masses and splittings still reflect confinement, the short-distance exchange of gluons, relativistic motion and spin-dependent forces, however. In charmonium, these effects are especially visible because the charm quark is comparatively light and the bound states are more vulnerable to open-charm thresholds and coupled-channel mixing. Bottomonium is more compact and closer to the nonrelativistic limit, making it a sharper test of short-distance dynamics while displaying smaller relative fine-structure and spin-flip effects.

To calculate the spectrum, the researchers used a semi-relativistic Hamiltonian containing the kinetic energies of the quark and antiquark plus an effective interaction. Its central component combines a Coulomb-like term, representing one-gluon exchange at short distances, with a confining term that rises approximately linearly at moderate separation but becomes screened at larger distances. Screening is important for highly excited states because a quark-antiquark pair with a large spatial extent can begin to feel the influence of open-flavour thresholds and other configurations. The kinetic energy was expanded beyond the leading nonrelativistic term, retaining successive powers of momentum to incorporate relativistic corrections. Gaussian variational wave functions were then optimized separately for different radial and orbital states.

The central potential determines the broad organization of the spectrum, but the physical quantum states emerge only after spin-dependent terms are included. The calculation accounts for spin-orbit interactions, the contact hyperfine interaction and tensor forces. The contact term is particularly important for S-wave states because it depends on the wave function at the origin, where the quark and antiquark overlap most strongly. That interaction generates the splitting between pseudoscalar and vector partners such as the ηc and J/ψ, or the ηb and Υ. Spin-orbit and tensor terms instead resolve the fine structure of P-, D- and higher-wave multiplets, allowing the model to assign individual states with quantum numbers denoted by JPC, where J is total angular momentum and P and C represent parity and charge conjugation.

The resulting low-lying spectra follow the established experimental pattern in both sectors. For charmonium, the calculated 1S and 1P spin-averaged masses lie close to experimental averages, while the first D-wave levels occupy the region associated with observed states including the ψ2(3823) and ψ3(3842). The higher charmonium region is less straightforward. Resonances near open-charm thresholds, including χc1(3872), may not be pure conventional quark-antiquark states and could contain molecular or other multiquark components. The researchers therefore present their high-lying results primarily as a conventional cc̄ reference spectrum, rather than claiming that every observed structure has a simple quarkonium interpretation.

Bottomonium displays the same broad ordering but with a noticeably more compressed structure. Its larger bottom-quark mass localizes the wave functions and reduces the relative importance of relativistic corrections. The calculated 1S, 1P and 2P centers remain close to the known experimental pattern, while the higher states extend smoothly upward. The smaller ηb–Υ separation and reduced χb fine structure are natural consequences of the stronger mass suppression of spin-dependent interactions. Even here, caution is required near open-bottom thresholds, where coupled-channel effects and exotic structures such as the Zb states can complicate the interpretation of nominal quarkonium levels.

The study tests its spectrum in several independent ways. In Regge analysis, the researchers plot angular momentum or radial excitation number against the squared meson mass, looking for approximately linear trajectories. These patterns do not add new information to the mass calculation; instead, they reveal whether the same potential arranges orbital and radial excitations coherently over a broad range. Charmonium produces steeper trajectories because its excitation gaps are larger, whereas bottomonium trajectories are flatter and more compressed. Spin-averaged trajectories are especially useful because they suppress fine-structure details and expose the underlying behavior of the central interaction. Deviations from perfect linearity can signal the effects of screening, thresholds or different dynamics in radial and orbital excitation.

Radiative decays provide a more demanding test because they probe the detailed shapes of the wave functions, not just their energies. Electric-dipole, or E1, transitions generally change orbital angular momentum by one unit while preserving the total spin. Their widths depend on the cube of the photon energy, angular momentum coefficients and a radial matrix element of the form ⟨f|r|i⟩. Transitions such as 1P to 1S or 1D to 1P can be strong when the initial and final wave functions overlap substantially. Radially excited states are more revealing: nodes in their wave functions can produce cancellations, so small changes in the predicted structure may lead to large differences in the calculated width.

Magnetic-dipole, or M1, transitions mainly flip the spin while preserving orbital angular momentum. Their amplitudes contain a spherical-Bessel overlap that accounts for the finite photon momentum. Allowed transitions between corresponding radial levels, such as J/ψ to ηc plus a photon, are controlled largely by the hyperfine splitting and the heavy-quark magnetic moment. Hindered transitions between different radial levels are much more sensitive to orthogonality, node cancellations and relativistic corrections. The predicted hierarchy is clear: charmonium M1 widths are generally larger, while bottomonium spin-flip transitions are strongly suppressed by the heavier quark mass. Together, the E1 and M1 results connect observed or future radiative signals directly to the spatial and spin structure of the calculated states.

The researchers also used two-point and three-point QCD sum rules to calculate observables tied to the currents that create the mesons from the vacuum. In a two-point sum rule, a current correlation function is expanded using perturbative contributions and nonperturbative condensates. A Borel transformation suppresses poorly known higher states, while a continuum threshold separates the ground-state pole from the rest of the spectrum. The extracted decay constants were 392 ± 25 MeV for ηc, 403 ± 36 MeV for J/ψ, 642 ± 81 MeV for ηb and 722 ± 105 MeV for Υ. These values reflect the stronger localization of bottomonium, although its electromagnetic annihilation rates are reduced by the smaller electric charge of the bottom quark.

Those decay constants feed into estimates of annihilation widths, including pseudoscalar decays to two photons or two gluons and vector decays to lepton pairs or three gluons. The formulas make the uncertainties transparent: widths proportional to a decay constant squared inherit twice its fractional uncertainty, while gluonic channels carry additional sensitivity to the strong coupling αs. Three-point sum rules were used separately to obtain electromagnetic transition form factors, which describe how amplitudes change when one photon carries spacelike momentum. The study represents pseudoscalar two-photon form factors with a monopole dependence and vector-to-pseudoscalar transitions with a dipole form, providing compact predictions that can be compared with future measurements.

Finally, the team constructed finite-spectrum thermodynamic indicators from the predicted discrete levels. Each state contributes to a partition function according to its excitation energy above the calculated ground state and its spin degeneracy, 2J + 1. From this sum, the researchers derive a free energy, mean excitation energy, entropy and specific heat. The calculation includes 74 charmonium states between 2.988 and 4.940 GeV and 72 bottomonium states between 9.425 and 11.353 GeV. These are not properties of a complete hot-QCD medium: they omit continuum states, explicit thermal mass shifts and the full dynamics of deconfinement. Instead, they show how level spacing and multiplicity alone shape a finite quarkonium spectrum as temperature rises. Across the observables, the same physical message recurs: bottomonium is more compact and spin-suppressed, while charmonium is more responsive to fine structure, radial nodes, thresholds and medium-related effects. The framework’s main limitation is equally clear, and points toward future work incorporating coupled channels, state mixing and genuine finite-temperature dynamics.