Coupled-Channel Quark Cluster Model
- The Coupled-Channel Quark Cluster Model is a hadron-structure framework that treats observed hadrons as mixtures of compact quark cores and explicit hadronic channels.
- It employs coupling mechanisms such as ³P₀ pair creation, quark exchange, and dispersion relations to induce mass shifts, continuum admixtures, and threshold effects.
- The framework is applied to heavy-quarkonium, multiquark, and hexaquark systems, providing insights into resonance formation and decay patterns near thresholds.
The coupled-channel quark cluster model is a class of hadron-structure frameworks in which an observed hadron is treated not as a single isolated quark-model eigenstate but as a superposition of compact quark configurations and explicit hadronic channels. In the formulations surveyed here, the coupled basis may contain a bare or core plus open-flavor meson-meson continua, meson-meson and diquark-antidiquark cluster rearrangements, or six-quark subamplitudes associated with different pair clusterings. The common organizing principle is that channel coupling—implemented through pair creation, string breaking, quark exchange, or dispersion-relation kernels—modifies masses, induces continuum admixtures, generates widths above threshold, and can produce threshold-dominated molecular or hybridized states rather than pure valence configurations (Anwar et al., 2021, Cardoso et al., 2014, Terashima et al., 2022, He et al., 2023, Gerasyuta et al., 2010).
1. Conceptual scope and channel content
Across the cited literature, the term denotes a family of related constructions rather than a single canonical Hamiltonian. Some realizations are “unquenched” quark models for heavy quarkonia, where a confined state is dressed by open-charm loops; others are multichannel cluster calculations in which meson-meson, hidden-color, and diquark-antidiquark partitions are solved simultaneously; still others are relativistic six-quark coupled-subamplitude systems formulated through dispersion relations (Man et al., 2024, He et al., 2023, Gerasyuta et al., 2010).
| Realization | Channel basis | Characteristic coupling |
|---|---|---|
| Unquenched quarkonium model | bare + open-charm continuum | pair creation |
| Multichannel Schrödinger model | confinement + meson-meson channels | delta-shell string breaking |
| Multiquark cluster calculation | , , , 0 | channel mixing in full Hamiltonian |
| Relativistic hexaquark formalism | pair-cluster subamplitudes 1 | coupled dispersion kernels |
In the quarkonium versions, the physical state is explicitly decomposed into a compact core and hadronic continua. In the multiquark versions, the “cluster” language refers to alternative quark rearrangements with the same total quantum numbers. In the six-quark case, the cluster concept is promoted to a relativistic coupled-subamplitude hierarchy, formally analogous to a relativistic Faddeev–Yakubovsky construction (Anwar et al., 2021, Cardoso et al., 2014, Gerasyuta et al., 2010).
A recurring physical theme is threshold sensitivity. When a bare state lies near an 2-wave or 3-wave–4-wave threshold, the continuum component can become numerically important, alter spectroscopy by tens of MeV, and control decay or production patterns. This is the central mechanism behind the heavy-quarkonium and multiquark applications discussed below (Anwar et al., 2021, Man et al., 2024, Abe et al., 25 Apr 2026).
2. Hilbert-space construction and coupling mechanisms
A standard unquenched formulation expands the physical hadron as
5
with 6 a bare charmonium state from a quenched potential model and 7 charmed-meson continuum states. The Hamiltonian is written as
8
where 9 is a Cornell-model charmonium Hamiltonian, 0 is the free two-meson continuum, and 1 is a pair-creation interaction. In these implementations the interaction is commonly taken in the standard 2 form
3
so that light 4 creation couples the compact core to open-flavor channels (Anwar et al., 2021).
A closely related multichannel Schrödinger realization writes
5
and solves a coupled radial equation in which the confined 6 sector is governed by a harmonic-oscillator potential,
7
while the meson-meson sector is treated as free relative motion and the two are linked by a delta-shell transition potential,
8
In that model the same string-breaking radius 9 is used for 0, 1, and 2, with 3, 4, 5, and 6 after fitting the 7 and 8 masses (Cardoso et al., 2014).
In multiquark cluster calculations, the basis is enlarged further. For 9, for example, the coupled-channel basis contains eight channels: six meson-meson channels and two diquark-antidiquark channels,
0
The coupled problem is then reduced to a generalized eigenvalue equation of the form
1
implemented with the Gaussian expansion method (He et al., 2023).
The six-quark dispersion-relation formalism adopts a different but related decomposition: 2 so that the full six-body amplitude is represented by 15 pair subamplitudes. For the illustrative 3 system, a representative pair amplitude is further decomposed into three subamplitudes 4, 5, and 6 depending on progressively larger cluster invariants. This cluster hierarchy is the dynamical backbone of the coupled-channel construction in the hexaquark sector (Gerasyuta et al., 2010).
3. Self-energies, effective kernels, and analytic structure
In unquenched quarkonium models, the coupled-channel effect enters spectroscopy through self-energy integrals. A typical mass shift is
7
and the continuum diagnostic
8
is used to quantify how strongly a state couples to a given intermediate channel. In one implementation, because 9 is not fixed from a full spectrum fit, these channel “probabilities” are explicitly described as unnormalized coupling strengths rather than absolute probabilities (Anwar et al., 2021).
A systematic charmonium model uses the same 0 amplitudes to generate masses, widths, and core fractions simultaneously. For states below threshold the pole condition is
1
with
2
For states above threshold, 3 acquires an imaginary part; the real mass is extracted from the principal value of the integral, while the total open-charm width is
4
In that framework the approximate 5 fraction is
6
though the paper states that for broad resonances it is not a strict probability (Man et al., 2024).
The Feshbach viewpoint makes the effective interaction structure explicit. For a two-channel 7-8 system,
9
eliminating one channel yields an effective Hamiltonian in the other. Eliminating the hadron channel gives
0
The resulting effective potential is therefore generically non-local and energy dependent, irrespective of the properties of the transition potential. When the eliminated channel contains scattering states, the effective interaction acquires an imaginary part above threshold, encoding decay into the open hadronic channel. A local derivative expansion,
1
is possible formally, but the cited analysis stresses that such a truncation loses the imaginary part above threshold and can therefore distort the analytic structure (Terashima et al., 2022).
The multichannel Schrödinger approach with a delta-shell transition potential exhibits the same logic in a different representation. There the channel wave functions are continuous at the string-breaking radius 2, while their derivatives have discontinuities fixed by the delta interaction; the matching conditions determine the secular equation for the eigenenergy and the normalized channel composition (Cardoso et al., 2014).
4. Heavy-quarkonium realizations
The heavy-quarkonium sector is the most developed arena for coupled-channel quark-cluster methods. In the charmoniumlike vector region, one coupled-channel quark model treats 3 as predominantly a 4 molecule mixed with a short-distance charmonium core, and then uses heavy-quark spin symmetry (HQSS) to identify partner states. The relevant thresholds are 5 at 6 MeV, 7 at 8 MeV, 9 at 0 MeV, and 1 at 2 MeV. Within that analysis, the dominant coupled channel near the 3 region is 4, while for 5 the strongest coupling is 6. The resulting HQSS multiplet picture is 7 as mostly 8, 9 as mostly 0, and 1 as mostly 2 (Anwar et al., 2021).
A broader charmonium coupled-channel study uses the same 3 mechanism to analyze masses, open-charm widths, and 4-5 mixing. There the vector assignments are 6 as 7-dominated, 8 as 9-dominated, 0 as 1-dominated, 2 as 3-dominated, and 4 as 5-dominated. The extracted mixing angles are 6 for 7, 8 for 9, 00 for 01, 02 for 03, and 04 for 05. This illustrates a central feature of the formalism: loop-induced mixing can be as important as the mass shifts themselves (Man et al., 2024).
The 06 is a benchmark case for channel hybridization. In the multichannel Schrödinger model with HO confinement and a delta-shell transition potential, the fitted wave function contains 07 08, 09 10, 11 12, and 13 14. In the same model the radiative widths are 15 and 16, giving 17 (Cardoso et al., 2014). A different charmonium coupled-channel calculation instead treats 18 as a 19 state with significant continuum contribution 20, a physical mass 21, and a dominant 22 self-energy contribution 23 (Man et al., 2024). This suggests that the coupled-channel framework accommodates strongly mixed core-plus-continuum interpretations while leaving the precise partition between “quarkonium” and “molecule” implementation dependent.
Coupled-channel effects are equally important in bottomonium transitions. For hindered M1 processes between 24-wave bottomonia, virtual 25 loops can dominate the quenched spin-flip amplitude. The triangle-loop power counting gives
26
so the amplitude is enhanced near open-bottom thresholds. The same study emphasizes that two-loop pion-exchange contributions can be order one relative to the triangle contribution, which limits precision but reinforces the conclusion that the coupled-channel mechanism can overwhelm the direct quenched amplitude (Guo et al., 2016).
5. Multiquark and hadron-level cluster realizations
The multiquark sector extends the coupled-channel quark cluster concept beyond quarkonium dressing. In the hidden-local-symmetry extension of the chiral quark model, the 27 calculation combines confinement, one-gluon exchange, 28, 29, 30, and 31 exchange and solves an eight-channel coupled problem. In a single-channel calculation 32 is not bound, but after channel coupling the binding becomes
33
relative to the threshold 34. The same work reports RMS distances
35
consistent with a loose molecular state. For 36, by contrast, full channel coupling gives 37 relative to 38, with much smaller RMS distances, indicating a more compact configuration (He et al., 2023).
The 39 channel provides a hadron-plus-quark realization of the same logic. The coupled basis contains
40
for 41, 42. In this model the diagonal meson-exchange potentials are weak, the off-diagonal one-meson-exchange transitions are also weak, but short-distance quark exchange generates strong off-diagonal transitions, particularly 43. The quark-exchange interaction is effectively only off-diagonal at leading order, and after a global reduction factor 44 the coupled-channel amplitudes reproduce the qualitative HAL QCD pattern: weak diagonal 45, strong open-charm to hidden-charm mixing, cusp-like threshold structures when more channels are added, and no bound state or resonance pole in the main coupled-channel analysis (Abe et al., 25 Apr 2026).
The relativistic hexaquark program pushes the cluster idea to six-body spectroscopy. There the total amplitude is built from 15 pair subamplitudes, the quark-quark interaction is modeled by an effective four-fermion interaction with gluon quantum numbers, and two-body diquark amplitudes are written in 46-type form
47
By extracting leading singularities and evaluating reduced amplitudes at the central Dalitz point
48
the integral system is reduced to algebraic equations with a common denominator 49, and the hexaquark masses are obtained from the pole condition
50
Using 51, 52, 53, and two fitted gluon-coupling sets, that study reports 39 dibaryons, including a deuteron-like state near 54 and an 55, 56 57-type state near 58 (Gerasyuta et al., 2010).
6. Observables, diagnostics, and methodological limits
A notable strength of the coupled-channel quark cluster framework is that it connects spectroscopy to channel-specific observables. In the HQSS analysis of the 59 and 60 states, decay modes are proposed as discriminants of long- and short-distance structure. If 61 is a 62 molecule, the dominant 63 decay implies a natural 64 final state but not 65. If 66 is dominantly 67, then both 68 and 69 should be populated, and a broad enhancement around 70 GeV in 71 is noted as consistent with that assignment (Anwar et al., 2021).
In the charmonium applications, the same transition amplitudes that produce continuum dressing also determine open-charm widths and leptonic observables. For vector states the loop-induced 72-73 admixture is essential because pure 74-wave vectors would otherwise have too-small 75. For 76, a quasi-three-body treatment using the finite width of 77 gives 78, consistent with the observed width 79 MeV within that model’s assumptions (Man et al., 2024). For the multichannel 80 model, unquenching strongly suppresses the quenched prediction 81 and yields the more balanced radiative pattern quoted above (Cardoso et al., 2014).
The formalism, however, has persistent methodological limits. One coupled-channel quark model states explicitly that its 82 values are unnormalized coupling strengths because the 83 strength 84 is not fitted globally (Anwar et al., 2021). Another notes that 85 is only an approximate measure for broad resonances (Man et al., 2024). The 86 multichannel Schrödinger calculation assumes no direct meson-meson interaction in the 87 sector (Cardoso et al., 2014). The Feshbach analysis shows that eliminating channels produces non-local and energy-dependent kernels and that a naive local approximation can remove the imaginary part above threshold, thereby misrepresenting decay physics (Terashima et al., 2022). In bottomonium hindered M1 transitions, absolute widths depend on unknown couplings 88, 89, and 90, and some two-loop effects are estimated to be order one, so ratios of widths are identified as more robust discriminants than absolute partial widths (Guo et al., 2016).
These limitations do not weaken the central conclusion shared by the cited work. Rather, they define the technical frontier of the field: coupled-channel quark cluster models are most informative when the basis captures the relevant nearby thresholds, the transition operators preserve the correct analytic structure, and spectroscopy is constrained simultaneously by masses, widths, and channel-specific decay or scattering observables (Terashima et al., 2022, Man et al., 2024, Abe et al., 25 Apr 2026).
