Diabatic Dynamical Diquark Model

• The diabatic dynamical diquark model is a coupled-channel framework that integrates compact tetraquark configurations with meson-meson thresholds using a Born–Oppenheimer approach.
• It employs a diabatic basis to replace nonadiabatic derivative couplings with explicit potential matrices, enabling a unified description of molecular and compact states.
• Numerical implementations reveal manifold state compositions and threshold effects, providing insights into heavy-quark phenomenology and exotic-hadron structure.

The diabatic dynamical diquark model is a coupled-channel extension of the dynamical diquark description of exotic hadrons in which compact diquark–antidiquark configurations and nearby di-hadron thresholds are treated on an equal dynamical footing within a Born–Oppenheimer framework. In place of a purely adiabatic description by a single diquark–antidiquark potential, it employs a diabatic basis containing one compact $\delta\bar\delta$ configuration together with explicit hadron–hadron channels, so that compact tetraquark, threshold-dressed, and, in some regions of parameter space, purely molecular behavior can be described within the same Hamiltonian scheme (Lebed et al., 2022).

1. Origins in the dynamical diquark program

The underlying dynamical diquark model treats exotic hadrons as systems built from compact colored clusters: for tetraquarks, a color–$\bar{\mathbf 3}$ diquark $\delta=(Qq)$ and a color–$\mathbf 3$ antidiquark $\bar\delta=(\bar Q\bar q')$; for pentaquarks, a triquark–diquark analogue is used. These clusters are regarded as heavy color sources connected by a gluonic flux tube, and the resulting spectrum is organized by Born–Oppenheimer potentials labeled by the quantum numbers of the light fields. The first numerical implementation of this program used lattice-calculated Born–Oppenheimer potentials to generate $\Sigma_g^+(1S)$, $\Sigma_g^+(1P)$, $\Sigma_g^+(2S)$, and higher bands for tetraquarks and pentaquarks, while treating spin and isospin splittings as subleading structure on top of BO multiplet averages (Giron et al., 2019).

In that adiabatic version, the lowest tetraquark spin multiplet in $\Sigma_g^+(1S)$ is built from the standard diquark-spin basis

$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$

and analogous constructions were then extended to hidden-bottom, hidden-charm/strange, fully heavy, and doubly charmed sectors. Several later studies explicitly noted that these applications remained adiabatic even when coupled BO equations or threshold effects were discussed, thereby establishing the BO dynamical diquark framework as the baseline to be generalized diabatically (Mutuk, 2022).

2. Adiabatic and diabatic formulations

The heavy-source Hamiltonian is written as

$\bar{\mathbf 3}$0

with $\bar{\mathbf 3}$1 the reduced mass of the heavy sources. In the adiabatic Born–Oppenheimer picture one first solves, at fixed inter-source separation $\bar{\mathbf 3}$2,

$\bar{\mathbf 3}$3

and then expands the full state in the $\bar{\mathbf 3}$4-dependent basis $\bar{\mathbf 3}$5. The price of this diagonal light-field basis is the appearance of nonadiabatic couplings

$\bar{\mathbf 3}$6

which become important near threshold crossings and avoided crossings (Lebed et al., 2022).

The diabatic formalism trades those derivative couplings for an explicit potential matrix in a fixed basis. One expands instead as

$\bar{\mathbf 3}$7

with $\bar{\mathbf 3}$8 chosen away from strong mixing regions, and defines

$\bar{\mathbf 3}$9

The resulting coupled-channel Schrödinger system is

$\delta=(Qq)$0

Operationally, the diabatic basis is chosen so that one channel is a compact $\delta=(Qq)$1 configuration and the others are explicit threshold states $\delta=(Qq)$2, baryon–antibaryon channels, or analogous hadron pairs. In the hidden-strangeness implementation, this is described as a refinement of the adiabatic model “in which each state is described solely by a diquark-antidiquark potential” into a form that “incorporates effects of di-hadron thresholds upon the states” (Jafarzade et al., 17 Oct 2025).

3. Effective Hamiltonian and mixing potentials

In the standard diabatic implementation, the channel-space potential matrix is

$\delta=(Qq)$3

with no direct off-diagonal couplings among distinct di-hadron thresholds in the first instance. The compact channel is usually modeled by a Cornell-like BO potential,

$\delta=(Qq)$4

or, when the constituent masses are written explicitly,

$\delta=(Qq)$5

while each threshold channel is approximated by a flat potential equal to the threshold energy,

$\delta=(Qq)$6

A widely used diabatic mixing ansatz is

$\delta=(Qq)$7

so that mixing is strongest near the radius where the compact BO potential crosses the threshold (Lebed et al., 2022).

The numerical values of $\delta=(Qq)$8, $\delta=(Qq)$9, $\mathbf 3$0, $\mathbf 3$1, and $\mathbf 3$2 are sector-dependent in published implementations. In the hidden-charm tetraquark realization, the parameters are

$\mathbf 3$3

with

$\mathbf 3$4

In the hidden-strangeness diabatic calculation, the same $\mathbf 3$5 and $\mathbf 3$6 are used but

$\mathbf 3$7

These numbers are not presented as universal constants of the model but as choices tied to particular sectoral fits and phenomenological assumptions (Jafarzade et al., 17 Oct 2025).

A later extension introduced explicit meson–meson molecular binding potentials instead of purely flat thresholds. Two simple forms were studied: $\mathbf 3$8 and

$\mathbf 3$9

That modification was used to interpolate continuously between mixed compact–molecular states and purely molecular states (Lebed et al., 27 Feb 2025).

4. State composition, observables, and numerical implementation

For a normalized coupled-channel solution with radial components $\bar\delta=(\bar Q\bar q')$0, the probability of channel $\bar\delta=(\bar Q\bar q')$1 is

$\bar\delta=(\bar Q\bar q')$2

This definition is used throughout the diabatic literature to quantify the compact $\bar\delta=(\bar Q\bar q')$3 content and the individual threshold fractions of a state. In the first hidden-charm implementation, the $\bar\delta=(\bar Q\bar q')$4 state identified with $\bar\delta=(\bar Q\bar q')$5 was found at $\bar\delta=(\bar Q\bar q')$6 with composition $\bar\delta=(\bar Q\bar q')$7 and $\bar\delta=(\bar Q\bar q')$8, whereas the corresponding $\bar\delta=(\bar Q\bar q')$9 and $\Sigma_g^+(1S)$0 states remained predominantly compact, at $\Sigma_g^+(1S)$1 and $\Sigma_g^+(1S)$2 $\Sigma_g^+(1S)$3, respectively (Lebed et al., 2022).

The bound-state problem is solved numerically with standard coupled-channel methods such as renormalized Numerov and log-derivative techniques. A separate scattering formulation later re-expressed the large-$\Sigma_g^+(1S)$4 solutions in terms of a K matrix and an S matrix,

$\Sigma_g^+(1S)$5

allowing elastic open-charm di-meson cross sections to be computed as functions of center-of-mass energy. Within that framework the model yields true resonances, near resonances, and threshold cusp effects in channels with zero, open, and hidden strangeness (Lebed et al., 2023).

A complementary Hamiltonian treatment of open thresholds computed self-energy mass shifts and partial widths for states lying above open channels. In that implementation the total predicted widths range from $\Sigma_g^+(1S)$6 MeV for a $\Sigma_g^+(1S)$7 example to $\Sigma_g^+(1S)$8 MeV for a $\Sigma_g^+(1S)$9 example, with the general width formula

$\Sigma_g^+(1P)$0

derived from the overlap of the compact radial wavefunction with the diabatic mixing potential and the open-channel continuum (Lebed et al., 2024).

5. Heavy-quark phenomenology and the compact–molecular interplay

The most studied application remains hidden charm. In the original diabatic bound-state analysis, $\Sigma_g^+(1P)$1 emerged with a dominant $\Sigma_g^+(1P)$2 component but also a considerable diquark–antidiquark component, and that compact component was argued to be relevant for short-distance observables such as radiative decays (Lebed et al., 2022). The later scattering calculation sharpened that picture by showing that the $\Sigma_g^+(1P)$3 channel can generate a near-threshold resonance enhanced by the $\Sigma_g^+(1P)$4 threshold, either with or without an additional contribution from a conventional $\Sigma_g^+(1P)$5 channel (Lebed et al., 2023).

The same framework predicts that not all exotic states near thresholds are predominantly molecular. In the hidden-charm sectors studied so far, the $\Sigma_g^+(1P)$6, $\Sigma_g^+(1P)$7, and many open-strange states remain mainly compact $\Sigma_g^+(1P)$8 configurations even after threshold mixing is included. This supports a non-dichotomous interpretation: the diabatic dynamical diquark model is designed precisely to describe coherent mixtures whose compact and threshold components vary strongly from channel to channel rather than to force a universal “tetraquark” or “molecule” label (Lebed et al., 2024).

The model has also been applied to $\Sigma_g^+(1P)$9. In that case the coupled basis consists of an elementary $\Sigma_g^+(2S)$0 channel and the two charged $\Sigma_g^+(2S)$1 thresholds, $\Sigma_g^+(2S)$2 and $\Sigma_g^+(2S)$3. The published analysis found that the influence of $\Sigma_g^+(2S)$4 is larger than that of $\Sigma_g^+(2S)$5 but not overwhelmingly so, and that $\Sigma_g^+(2S)$6 contains an $\Sigma_g^+(2S)$7 $\Sigma_g^+(2S)$8 component (Lebed et al., 2024). A plausible implication is that the same formalism can interpolate smoothly between compact-core and threshold-dominated behavior even in systems with explicit isospin breaking at the threshold level.

6. Hidden-strangeness realization

The hidden-strangeness sector provides a particularly stringent test because the strange quark is only marginally heavy in the BO sense. The adiabatic precursor study treated $\Sigma_g^+(2S)$9 and $\Sigma_g^+(1S)$0 tetraquarks in the dynamical diquark model and argued that states near $\Sigma_g^+(1S)$1 GeV with peculiar properties, such as $\Sigma_g^+(1S)$2, $\Sigma_g^+(1S)$3, and $\Sigma_g^+(1S)$4, are plausible tetraquark candidates (Jafarzade et al., 21 May 2025). The subsequent diabatic generalization tabulated all relevant thresholds, computed the di-hadron content of each predicted state, and found no hidden-strange tetraquark candidate whose structure is dominated by di-hadron content; this was explicitly contrasted with the charm sector, where many exotic states are strongly associated with thresholds (Jafarzade et al., 17 Oct 2025).

Representative hidden-strangeness results are summarized below.

State	Diabatic mass	Dominant content
$\Sigma_g^+(1S)$5	$\Sigma_g^+(1S)$6	$\Sigma_g^+(1S)$7
$\Sigma_g^+(1S)$8	$\Sigma_g^+(1S)$9	$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$0
$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$1	$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$2	$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$3
$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$4	$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$5	$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$6
$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$7 fully strange near $$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$8	$$\begin{aligned} J^{PC}=0^{++}:&\quad X_0,\ X_0',\ J^{PC}=1^{++}:&\quad X_1,\ J^{PC}=1^{+-}:&\quad Z,\ Z',\ J^{PC}=2^{++}:&\quad X_2, \end{aligned}$$9	$\bar{\mathbf 3}$00

The broader hidden-strangeness pattern is that threshold admixtures are present but typically modest. For the spin-averaged $\bar{\mathbf 3}$01 multiplet, the published diabatic masses are

$\bar{\mathbf 3}$02

with corresponding compact fractions $\bar{\mathbf 3}$03 and $\bar{\mathbf 3}$04. Even the largest single hidden-strangeness threshold admixture reported so far, the $\bar{\mathbf 3}$05 fraction in the fully strange $\bar{\mathbf 3}$06 state near $\bar{\mathbf 3}$07, does not overturn compact-core dominance. On this basis, $\bar{\mathbf 3}$08, $\bar{\mathbf 3}$09, and $\bar{\mathbf 3}$10 remain strong hidden-strange tetraquark candidates in the diabatic dynamical diquark model.

7. Interpretation, controversies, and open problems

A common misconception is that the diabatic dynamical diquark model is simply a compact-tetraquark model with perturbative threshold corrections. Published work does not support that reading. By construction, the framework allows states to range from predominantly compact to predominantly molecular, and with additional meson–meson binding potentials it develops parameter regions in which one may produce mass eigenstates exactly matching the specific examples of $\bar{\mathbf 3}$11 and $\bar{\mathbf 3}$12, as well as regions in which a pure di-meson molecular state emerges (Lebed et al., 27 Feb 2025). The model is therefore best regarded as a coupled-channel BO scheme rather than as a single-structure ansatz.

Its limitations are equally explicit. The Gaussian mixing potential with universal $\bar{\mathbf 3}$13 and $\bar{\mathbf 3}$14 is phenomenological; direct meson–meson interactions are often replaced by flat thresholds; quarkonium mixing is omitted in most channels; fine structure is not yet treated fully diabatically in all sectors; and channel truncation is unavoidable in current calculations (Lebed et al., 2024). In the strange sector an additional conceptual caveat appears: $\bar{\mathbf 3}$15, so the BO separation of scales is substantially less controlled than in charm or bottom, and stronger configuration mixing with ordinary mesons, hybrids, and glueballs is expected (Jafarzade et al., 21 May 2025).

The published future directions are correspondingly clear. They include spin- and flavor-dependent diabatic couplings, multi-threshold scattering, refined meson–meson molecular potentials, lattice input on string breaking and static tetraquark potentials, and extensions to open-flavor tetraquarks and pentaquarks (Lebed et al., 2023). Taken together, these developments suggest that the diabatic dynamical diquark model is evolving from a threshold-dressed BO spectroscopy scheme into a more general multichannel framework for exotic-hadron structure, line shapes, and decays.