Hadroquarkonia in QCD

• Hadroquarkonia are exotic states where a compact heavy quarkonium core (e.g., charmonium) is embedded in a light hadronic environment, interacting via soft gluon fields.
• The binding mechanism relies on chromoelectric dipole interactions and a QCD multipole expansion, distinctly separating it from hadronic molecules and tetraquark models.
• Lattice QCD studies reveal modest binding energy shifts (a few MeV) that depend on the quarkonium core’s polarizability and the host hadron’s energy-momentum distribution.

Hadroquarkonia, often called hadro-charmonium or hadro-bottomonium in the relevant heavy-flavor sectors, are exotic hadrons in which a compact heavy quarkonium core $Q\bar Q$ is embedded inside light hadronic matter and interacts with it primarily through soft gluonic fields. In the standard picture the heavy core remains approximately intact, acts as a nearly color-neutral but polarizable probe, and binds to the surrounding light hadron through chromoelectric interactions treated in a QCD multipole expansion. This framework is distinct from hadronic molecules, compact multiquarks, and hybrids, and it has been invoked in discussions of parts of the $XYZ$ spectrum and hidden-charm pentaquarks; at the same time, lattice and phenomenological studies indicate that the size, selectivity, and even viability of the mechanism are strongly channel dependent (Alberti et al., 2016, Polyakov et al., 2018, Guo et al., 28 Aug 2025, Knechtli et al., 2016).

1. Defining picture and physical scales

The defining structural assumption is a separation of scales between a compact heavy quarkonium core and a more extended light-hadron environment. For charmonium $S$-states, the core is characterized in the literature as a small-size quarkonium with radius $\lesssim 0.3$–$0.5\ \mathrm{fm}$, embedded in a larger light-hadron “gluonic cloud.” At long distances the $Q\bar Q$ pair is color neutral, but it remains polarizable, so the soft gluonic fields of the host hadron induce an effective dipole interaction. At leading order in the heavy-quark expansion the interaction is mainly spin-independent and governed by the chromoelectric field squared of the host hadron (Alberti et al., 2016).

This picture differs qualitatively from two other standard nonconventional assignments. In hadronic molecules, the heavy quark and antiquark are distributed inside open-heavy hadrons, and binding is tied to hadron-hadron forces, threshold proximity, pion exchange, and coupled channels. In compact multiquark models, the binding is instead attributed to short-distance color correlations, often organized through diquark substructures. Hadroquarkonium preserves a compact quarkonium core as a recognizable subsystem, with the light cloud providing the binding medium rather than dissolving the core into open-heavy constituents (Guo et al., 28 Aug 2025).

The expected strength of the mechanism depends on the chromoelectric polarizability of the core. In the Coulombic estimate summarized in the recent quarkonium review, the core size scales as $R_{Q\bar Q}\sim 1/(m_Q v)$, so $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$. Bottomonia are therefore more compact and much less polarizable than charmonia, which makes hadroquarkonium binding more plausible in the charmonium sector and suppressed in the bottomonium sector. Phenomenological expectations in that review range from a few$\times (10\ \mathrm{MeV})$ up to $O(100\ \mathrm{MeV})$ for charmonia with sizable $XYZ$0, whereas bottomonium binding is expected to be modest or marginal (Guo et al., 28 Aug 2025).

2. Multipole expansion, chromoelectric polarizability, and EMT formulation

The microscopic rationale is the QCD multipole expansion. When the quarkonium radius is much smaller than the scale over which the surrounding gluonic fields vary, the leading interaction is the chromoelectric dipole $XYZ$1. The first-order operator is written as

$XYZ$2

and the induced second-order interaction takes the form

$XYZ$3

Equivalently, the induced binding potential in a given light-hadron environment is

$XYZ$4

For a compact core one may write schematically

$XYZ$5

with the sum running over color-octet intermediate excitations. In this sense the binding mechanism is the hadroquarkonium analog of van der Waals binding (Guo et al., 28 Aug 2025).

A more structural formulation relates the same interaction to the host hadron’s energy-momentum tensor. In the EMT-based framework, the effective central potential is

$XYZ$6

where $XYZ$7 is the energy density, $XYZ$8 the pressure, $XYZ$9, and $S$0. The host hadron’s mechanical properties therefore enter directly the spectroscopy and decay problem. For spherically symmetric systems, the spatial stress tensor is decomposed as

$S$1

with shear-force density $S$2 and isotropic pressure $S$3. EMT conservation implies

$S$4

and the von Laue condition

$S$5

The associated D-term $S$6 encodes internal forces and is negative for stable systems. In the nucleon example reviewed in the EMT literature, model calculations give $S$7 and $S$8, with pressure positive in the inner region and negative beyond $S$9; those profiles then determine the range and shape of $\lesssim 0.3$0 for a hadroquarkonium core (Polyakov et al., 2018).

The same framework also constrains widths. Binding is governed by diagonal polarizabilities such as $\lesssim 0.3$1, while decays between quarkonium configurations depend on transition polarizabilities such as $\lesssim 0.3$2, which are typically much smaller. This difference is central to explanations of narrow hidden-charm states in hadroquarkonium models (Polyakov et al., 2018).

3. Distinction from molecules, compact multiquarks, and hybrids

The phenomenological distinction between hadroquarkonium and competing assignments is organized around threshold behavior, flavor selectivity, and decay topology. In the hadroquarkonium picture, one expects preferential hidden-heavy-flavor decays into a specific quarkonium plus soft pions or kaons, suppressed two-body open-flavor decays, and radiative transitions that preserve recognizable quarkonium-core selection patterns. By contrast, hadronic molecules are strongly tied to nearby $\lesssim 0.3$3-wave thresholds and typically exhibit large open-flavor branching fractions, while compact tetraquarks or pentaquarks can populate a broader set of flavor and spin multiplets and need not be threshold locked (Guo et al., 28 Aug 2025).

The experimental review of exotica from $\lesssim 0.3$4-factories and BESIII documents several of the empirical patterns that have motivated hadroquarkonium discussions without endorsing the framework as an interpretation. In particular, it emphasizes the “remarkable feature” that $\lesssim 0.3$5, much larger than for typical charmonium, and reports the BESIII Born cross section for $\lesssim 0.3$6 at $\lesssim 0.3$7 as $\lesssim 0.3$8. The same review stresses that near-threshold placement and open-bottom dominance make $\lesssim 0.3$9 and $0.5\ \mathrm{fm}$0 natural molecular candidates, and it explicitly notes that it does not formulate hadroquarkonium-specific HQSS multiplets, selection rules, or quantitative decay formulas (Choi, 2014).

The concise diagnostic checklist in the 2025 quarkonium review sharpens this separation. Strong threshold locking points toward molecules; hidden-flavor-dominant decays with core specificity, together with core-like E1/M1 radiative transitions, support hadroquarkonium; large charged isovector multiplets with strong open-flavor couplings favor molecule or tetraquark dynamics. This suggests that hadroquarkonium is not a generic label for all hidden-flavor exotica, but a specific dynamical hypothesis whose empirical footprint is comparatively restrictive (Guo et al., 28 Aug 2025).

4. Lattice QCD test in the static limit

A direct nonperturbative test was carried out on CLS $0.5\ \mathrm{fm}$1 lattices by measuring how the static $0.5\ \mathrm{fm}$2 potential changes in the background of a light hadron. The main ensemble, C101, has volume $0.5\ \mathrm{fm}$3, lattice spacing $0.5\ \mathrm{fm}$4, spatial extent $0.5\ \mathrm{fm}$5, and pion mass $0.5\ \mathrm{fm}$6–$0.5\ \mathrm{fm}$7; a second ensemble, S100, was used for finite-volume checks. Heavy quarks were treated as static sources, Wilson loops were measured at all positions and directions, and the hadron-modified potential was extracted from the ratio

$0.5\ \mathrm{fm}$8

with

$0.5\ \mathrm{fm}$9

Plateau stability was found for $Q\bar Q$0, and $Q\bar Q$1 was used as a good approximation to $Q\bar Q$2 (Alberti et al., 2016, Knechtli et al., 2016).

The qualitative result is uniform across the hadrons studied: $Q\bar Q$3, so the static potential becomes more attractive in a hadron background. For the nucleon, the shift is approximately $Q\bar Q$4 to $Q\bar Q$5 at $Q\bar Q$6 and $Q\bar Q$7 to $Q\bar Q$8 at $Q\bar Q$9. At the representative distance $R_{Q\bar Q}\sim 1/(m_Q v)$0, the effect is $R_{Q\bar Q}\sim 1/(m_Q v)$1 to $R_{Q\bar Q}\sim 1/(m_Q v)$2 for all hadrons investigated. The $R_{Q\bar Q}\sim 1/(m_Q v)$3-dependence is well described by a Cornell-like parametrization,

$R_{Q\bar Q}\sim 1/(m_Q v)$4

with a reduction of the linear slope, i.e. the effective string tension, as the dominant effect. A finite-volume comparison of $R_{Q\bar Q}\sim 1/(m_Q v)$5 between C101 and S100 found no significant finite-volume effects within errors (Alberti et al., 2016).

The phenomenological question is whether such a potential shift is sufficient to bind physical quarkonia. The lattice analysis propagated the effect to quarkonium spectroscopy by solving

$R_{Q\bar Q}\sim 1/(m_Q v)$6

or, in first-order perturbation theory,

$R_{Q\bar Q}\sim 1/(m_Q v)$7

Typical level shifts across the hadrons studied are $R_{Q\bar Q}\sim 1/(m_Q v)$8 to $R_{Q\bar Q}\sim 1/(m_Q v)$9 for $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$0, $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$1 to $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$2 for $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$3, and $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$4 to $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$5 for $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$6 charmonium. The attraction was therefore characterized as “deuteron-like,” namely only a few MeV, and substantially smaller than the tens-of-MeV binding often assumed in phenomenological hadro-charmonium scenarios (Alberti et al., 2016, Knechtli et al., 2016).

5. Phenomenological realizations and candidate states

Within the $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$7 sector, the most persistent hadroquarkonium candidates are the vector $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$8 states. The 2025 review highlights $$\alpha_{Q\bar Q}\sim R^3\sim 1/(m_Q^3 v^{3\text{--}4})$$9, $\times (10\ \mathrm{MeV})$0, and $\times (10\ \mathrm{MeV})$1 as the clearest arena for the idea, because they couple strongly to hidden charm with channel selectivity and do not show dominant two-body open-charm decays. A common narrative summarized there is that $\times (10\ \mathrm{MeV})$2 predominantly carries a $\times (10\ \mathrm{MeV})$3 core, while $\times (10\ \mathrm{MeV})$4 and $\times (10\ \mathrm{MeV})$5 couple most strongly to $\times (10\ \mathrm{MeV})$6. The same review also notes the observation of $\times (10\ \mathrm{MeV})$7 near $\times (10\ \mathrm{MeV})$8, which is compatible with production of a charmonium core from a vector environment. By contrast, $\times (10\ \mathrm{MeV})$9, $O(100\ \mathrm{MeV})$0, $O(100\ \mathrm{MeV})$1, $O(100\ \mathrm{MeV})$2, and the narrow $O(100\ \mathrm{MeV})$3 states are described there as better explained by molecular dynamics or compact multiquark admixtures because of their threshold alignment and strong open-flavor channel correlations (Guo et al., 28 Aug 2025).

Hidden-charm pentaquarks provide a more controversial application. One QCD-inspired hadroquarkonium model identifies $O(100\ \mathrm{MeV})$4 as a $O(100\ \mathrm{MeV})$5-nucleon $O(100\ \mathrm{MeV})$6-wave bound state with $O(100\ \mathrm{MeV})$7. In that approach the interaction is written in terms of the baryon energy density $O(100\ \mathrm{MeV})$8 and pressure $O(100\ \mathrm{MeV})$9,

$XYZ$00

and the partial width into $XYZ$01 is obtained from a two-channel transition potential:

$XYZ$02

Using $XYZ$03, that study finds $XYZ$04 and predicts two nearly degenerate hidden-charm baryon octets with $XYZ$05. For the $XYZ$06 octet it quotes $XYZ$07 at $XYZ$08, $XYZ$09 at $XYZ$10, $XYZ$11 at $XYZ$12, and $XYZ$13 at $XYZ$14, satisfying a Gell-Mann-Okubo relation (Eides et al., 2017).

A different line of analysis embeds hadroquarkonium inside a fully covariant four-body Bethe-Salpeter/Faddeev-Yakubovsky framework, where the hadroquarkonium topology is the color-singlet cluster $XYZ$15. In that calculation, hadroquarkonium dominates the $XYZ$16 channel across charm and bottom, contributes substantially in $XYZ$17, and is subleading for all states with $XYZ$18, which are instead dominated by heavy-light meson pairs. Concrete examples include a $XYZ$19 $XYZ$20 state in the $XYZ$21 region with $XYZ$22, $XYZ$23, and $XYZ$24–$XYZ$25 mixing $XYZ$26; a $XYZ$27 $XYZ$28 state in the $XYZ$29 region with $XYZ$30; and an overwhelmingly hadroquarkonium-dominated $XYZ$31 $XYZ$32 state at $XYZ$33 with $XYZ$34 in that truncation (Hoffer et al., 2024).

6. Systematics, limitations, and open directions

The central limitation of the lattice test is that it isolates only one mechanism: the static-potential shift produced by embedding a heavy $XYZ$35 pair in a single hadron background. The calculation was performed at one lattice spacing, with $XYZ$36, no dynamical charm, and in the heavy-quark static limit. It omits finite-$XYZ$37 and spin-dependent heavy-quark corrections, avoids the string-breaking region by restricting $XYZ$38, and in some channels faces sizable statistical uncertainties. The authors accordingly identify lighter pion masses, continuum extrapolation, dynamical charm, and treatments beyond the static limit, such as NRQCD/HQET with $XYZ$39 corrections, as necessary next steps (Alberti et al., 2016).

The EMT-based approach has a different set of systematics. Host-hadron EMT densities are still extracted from a mixture of models, lattice calculations at unphysical pion masses, and dispersive analyses; disconnected contributions are often omitted; quark/gluon decompositions are scheme dependent; and chromoelectric polarizabilities remain uncertain. The EMT review emphasizes open problems such as quantifying the gluon D-term experimentally, refining hadronic radii and EMT form factors, and clarifying how confinement-scale forces map onto the pressure and shear distributions that enter $XYZ$40 (Polyakov et al., 2018).

On the phenomenological side, the 2025 review stresses that single-channel Breit-Wigner fits are inadequate in much of the $XYZ$41 region and that decisive progress requires coupled-channel analyses, systematic measurements of hidden-flavor versus open-flavor branching ratios, radiative transition patterns, and isospin-partner structure. Taken together, the present literature supports a differentiated conclusion: hadroquarkonium remains a coherent and technically specific framework for a subset of hidden-flavor exotics, especially where hidden-heavy-flavor transitions are selective and threshold locking is weak, but the isolated chromoelectric attraction measured on the lattice is only of order a few MeV, and many near-threshold states are more naturally described by molecular or mixed dynamics than by a pure hadroquarkonium core (Guo et al., 28 Aug 2025).