Unitary Coupled-Channel Hidden Gauge Formalism

• The unitary coupled-channel hidden gauge formalism is a nonperturbative framework that uses effective vector exchange interactions and hidden local gauge symmetry to dynamically generate hadronic molecules.
• It reduces complex meson-baryon and meson-meson dynamics to effective contact interactions, commonly resembling Weinberg–Tomozawa-type terms in the low-energy limit.
• Coupled-channel unitarization via the Bethe–Salpeter equation reveals resonance poles that are interpreted as molecular states, challenging traditional quark-model assignments.

The unitary coupled-channel hidden gauge formalism is a nonperturbative hadron-spectroscopy framework in which effective interaction kernels, derived from hidden local symmetry or local hidden gauge Lagrangians and often reducible in the low-energy limit to Weinberg–Tomozawa-type contact terms, are iterated in a coupled-channel scattering equation so that poles of the resulting amplitude are interpreted as dynamically generated hadronic molecules. In the literature represented here, the formalism has been applied to hidden-charm and hidden-beauty baryons, open-charm baryons, light and heavy meson-meson systems, bottom-strange molecules, and triple-heavy pentaquark candidates, with the common structure that resonances emerge from meson-baryon or meson-meson dynamics rather than being inserted as elementary states (Wu et al., 2010, Xiao et al., 2013, Wang et al., 2024).

1. Lagrangian basis and symmetry structure

The formalism is rooted in hidden local gauge symmetry. In the meson-baryon sector, the standard interaction terms are

$${\cal L}_{VVV}=ig\langle V^\mu[V^{\nu},\partial_\mu V_{\nu}]\rangle ,$$

$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$

$${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$

with $g=M_V/(2f)$. In vector-vector applications, the formalism also uses the four-vector contact term

$${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$

and the three-vector interaction

$${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$

The $\langle\cdots\rangle$ notation denotes flavor traces, and the field content is organized in meson and baryon multiplets $P$, $V$, and $B$ (Wu et al., 2010, Molina et al., 2010).

The symmetry realization depends on the sector. Light-hadron applications are formulated in $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$0, whereas hidden-charm, hidden-beauty, open-heavy, and triple-heavy applications extend the bookkeeping to $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$1 or $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$2-like matrices. These extensions are not treated as exact flavor symmetries. One hidden-charm formulation states explicitly that $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$3 is not exact, but is used as a working symmetry for the vertices, while mass differences of exchanged mesons and kinematics break the symmetry dynamically. The triple-heavy construction likewise notes that the derivation can be understood from quark content without imposing exact $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$4 symmetry (Wu et al., 2010, Wang et al., 2024).

Heavy-quark spin symmetry enters in the heavy sector through the heavy-quark spectator picture and through HQSS-adapted coupled-channel bases. In the hidden-charm baryon problem, the inclusion of both $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$5 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$6 channels, together with spin-$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$7 and spin-$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$8 charmed baryons, is demanded by HQSS. The 2013 hidden-charm analysis shows that the $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$9-extended hidden-gauge interaction is consistent with HQSS at leading order: $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$0 has no spin dependence, $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$1 carries only the trivial $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$2 factor, $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$3 transitions are suppressed, and transitions between spin-$${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$4 and spin-$${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$5 charmed baryons are subleading (Xiao et al., 2013).

2. Effective kernels from vector exchange

A defining step is the reduction of $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$6-channel vector exchange to an effective contact-like kernel in the low-energy regime. The approximation keeps only the $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$7 component of the baryon current, neglects external three-momenta relative to hadron masses, and approximates the exchanged vector propagator by its static limit. Under these assumptions, the interaction becomes Weinberg–Tomozawa-like. For vector-baryon channels, one representative expression is

$${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$8

and the pseudoscalar-baryon kernel has the same structure without the polarization factor. The coefficients $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$9 encode the coupled-channel dynamics and are given by flavor algebra or explicit wave-function overlaps (Wu et al., 2010, Xiao et al., 2013).

This reduction is used across sectors with different kinematic realizations. In hidden-charm baryons, the channels include $g=M_V/(2f)$0, $g=M_V/(2f)$1, $g=M_V/(2f)$2, $g=M_V/(2f)$3, $g=M_V/(2f)$4, $g=M_V/(2f)$5, and their vector analogs such as $g=M_V/(2f)$6, $g=M_V/(2f)$7, and $g=M_V/(2f)$8. In open charm, the same mechanism produces the generalized Weinberg–Tomozawa kernel for $g=M_V/(2f)$9, $${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$0, $${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$1, $${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$2, and related channels. In meson-meson systems, the same hidden-gauge logic leads to analogous exchange kernels; for $${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$3, for example, the dominant contribution comes from $${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$4-channel exchange of $${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$5, $${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$6, and $${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$7, with

$${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$8

The dominant attraction or repulsion is then controlled by the channel coefficients and the isospin projection (Liang et al., 2014, Wan et al., 2018).

Extended local hidden gauge formulations add mechanisms beyond the leading vector-exchange kernel. In hidden-charm $${\cal L}^{(c)}_{III}=\frac{g^2}{2}\langle V_\mu V_\nu V^\mu V^\nu-V_\nu V_\mu V^\mu V^\nu\rangle$$9-like systems, pion-exchange box diagrams mix pseudoscalar-baryon and vector-baryon sectors, and anomalous $${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$0 boxes add attraction in vector-baryon channels. Gauge invariance requires the Kroll–Ruderman contact term. The resulting box contributions are decomposed as

$${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$1

and their $${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$2-wave pieces are used to define effective off-diagonal transition potentials,

$${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$3

This construction is central to the explicit $${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$4 mixing used in the extended approach (Uchino et al., 2015).

3. Unitarization, loop functions, and pole analysis

The nonperturbative core of the formalism is coupled-channel unitarization. In most implementations, the amplitude is obtained from the on-shell factorized Bethe–Salpeter equation

$${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$5

or equivalently $${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$6. Here $${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$7 is the interaction kernel and $${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$8 is the diagonal two-body loop function. This algebraic resummation is the standard form used in the chiral unitary and hidden-gauge approach (Wu et al., 2010, Sun et al., 2019).

For meson-baryon scattering, a representative loop integral is

$${\cal L}^{(3V)}_{III}=ig\langle (\partial_\mu V_\nu -\partial_\nu V_\mu) V^\mu V^\nu\rangle .$$9

while meson-meson applications use the corresponding two-meson loop. Two regularization schemes recur. Dimensional regularization introduces a subtraction constant and scale, such as $\langle\cdots\rangle$0 MeV and $\langle\cdots\rangle$1 in hidden-charm baryons, or $\langle\cdots\rangle$2 and $\langle\cdots\rangle$3 MeV in the $\langle\cdots\rangle$4 problem. Cutoff schemes use three-momentum cutoffs, often around $\langle\cdots\rangle$5 GeV in hidden-charm baryons, and may be matched to dimensional regularization near threshold (Wu et al., 2010, Wan et al., 2018).

The recent bottom-strange $\langle\cdots\rangle$6 study uses a cutoff-regularized Lippmann–Schwinger equation rather than the on-shell algebraic form,

$\langle\cdots\rangle$7

with the two-body Green’s function

$\langle\cdots\rangle$8

In that formulation, the only free parameter is the cutoff $\langle\cdots\rangle$9, fixed to the LHCb state $P$0 (Sánchez-Illana et al., 30 Mar 2026).

Spectroscopy is extracted from the analytic structure of $P$1. Poles below threshold on the first sheet are interpreted as bound states; poles on unphysical sheets above threshold are interpreted as resonances. In multichannel situations, higher sheets also appear: the $P$2-like pole in the $P$3, $P$4 pseudoscalar-baryon system lies on the third Riemann sheet. Near a pole,

$P$5

so residues determine channel couplings. Some applications also quantify molecular content through a generalized compositeness relation,

$P$6

with $P$7 close to $P$8 indicating a predominantly molecular state (Sun et al., 2019, Wang et al., 2024).

4. Hidden-charm and hidden-beauty baryon spectroscopy

The formalism became particularly visible through hidden-charm baryon predictions. One 2010 hidden-charm calculation found six bound states in the basic coupled-channel problem and, after including decay mechanisms, quoted two $P$9 and four $V$0 hidden-charm resonances. The reported masses and total widths are $V$1 MeV with $V$2 MeV, $V$3 MeV with $V$4 MeV, $V$5 MeV with $V$6 MeV, $V$7 MeV with $V$8 MeV, $V$9 MeV with $B$0 MeV, and $B$1 MeV with $B$2 MeV. These states lie above $B$3 GeV and have widths below $B$4 MeV. The same study states that they definitely cannot be accommodated by quark models with three constituent quarks and interprets the dominant couplings as signaling a molecular structure (Wu et al., 2010).

A closely related hidden-charm analysis around $B$5–$B$6 GeV identified poles dominated by $B$7, $B$8, $B$9, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$00, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$01, and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$02. It also estimated PANDA production channels. For $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$03, the total cross section was estimated as $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$04, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$05, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$06, and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$07, depending on exchange model and form factors; for $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$08, the corresponding estimate was $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$09–$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$10 nb with $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$11-exchange (Wu et al., 2010).

When HQSS is imposed explicitly, the spectrum is reorganized into multiplets. The 2013 hidden-charm HQSS study finds seven $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$12 states but interprets them as four basic molecular structures: $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$13 in $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$14, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$15 in $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$16, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$17 nearly degenerate in $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$18, and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$19 nearly degenerate in $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$20. All are bound by about $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$21–$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$22 MeV with respect to their thresholds, and the $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$23 $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$24 pole has exactly zero width in the chosen channel space. No acceptable physical states are retained in $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$25 (Xiao et al., 2013).

The hidden-beauty extension yields a parallel spectrum at much higher mass. The 2010 hidden-beauty study predicts two hidden-beauty $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$26 states near $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$27 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$28 MeV and four hidden-beauty $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$29 states near $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$30, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$31, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$32, and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$33 MeV, with total widths between about $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$34 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$35 MeV. These are interpreted as dynamically generated hadronic molecules dominated by channels such as $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$36, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$37, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$38, and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$39. The same work gives production estimates of $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$40–$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$41 nb and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$42 nb for $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$43 GeV, together with an event-rate estimate of more than $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$44 events/day at luminosity $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$45 (Wu et al., 2010).

5. Open-charm baryons and explicit $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$46–$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$47 mixing

In the open-charm sector, the extended local hidden gauge approach combines local hidden gauge dynamics, HQSS implemented through the heavy-quark spectator picture, and coupled-channel unitarization. The central claim is that the negative-parity charmed baryons, especially $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$48 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$49, can be generated dynamically as meson-baryon molecules. The calculation includes $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$50, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$51, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$52, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$53, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$54, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$55, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$56, and the special $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$57 channel. The study finds two states with nearly zero width associated to $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$58 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$59: a lower $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$60 state coupled to $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$61 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$62, and a $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$63 state dominantly coupled to $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$64. It also predicts additional $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$65 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$66 states (Liang et al., 2014).

The mechanism that breaks the naive $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$67–$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$68 degeneracy is $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$69 mixing through pion-exchange box diagrams. In this formulation, the $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$70-wave piece of the box is absorbed into an effective $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$71 transition potential, while the $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$72-wave piece corrects diagonal interactions. The same paper emphasizes that the vector-baryon interaction in $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$73-wave, because it contains $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$74, would otherwise generate degenerate $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$75 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$76 states (Liang et al., 2014).

A later hidden-charm $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$77-like study uses the extended local hidden gauge approach to include both the Weinberg–Tomozawa term and pion-exchange box diagrams as box potentials. In the $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$78 sector it reports six states around $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$79–$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$80 GeV: two $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$81 admixture states dominated by

$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$82

one $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$83 $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$84 resonance, one spin-degenerate $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$85 bound state with $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$86, and two $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$87 bound states dominated by

$${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$88

That work states explicitly that including pion-exchange box diagrams is crucial, since without them the $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$89 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$90 sectors would be much less mixed and the $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$91 states would appear significantly higher in energy (Uchino et al., 2015).

6. Meson-meson, bottom-strange, and triple-heavy generalizations

The same formal architecture extends beyond baryons. In vector-vector scattering, the $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$92 and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$93 system is treated with the four-vector contact term, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$94- and $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$95-channel vector exchange, and Bethe–Salpeter unitarization. In $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$96, the threshold interaction is strongly attractive, with projected $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$97 potentials $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$98, $${\cal L}_{PPV}=-ig\langle V^\mu[P,\partial_\mu P]\rangle ,$$99, and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$00 for $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$01, whereas $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$02 is repulsive. The model generates one resonance for each spin $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$03; the $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$04 and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$05 states are associated with $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$06 and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$07, while the $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$08 state near $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$09 MeV is a prediction. The $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$10 box contributes only to $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$11 and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$12, which explains why the $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$13 state is narrow (Molina et al., 2010).

In hidden-beauty meson-meson dynamics, the formalism is combined explicitly with HQSS. The relevant channels are $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$14 and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$15, classified by $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$16, $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$17, and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$18-parity. The analysis finds six robust $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$19 bound states and six additional weakly bound hidden-strange states that depend on coupled-channel effects. A major conclusion is that the leading hidden-gauge interaction produces $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$20, so no $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$21 bound states are found within the framework (Ozpineci et al., 2013).

Light meson-meson applications illustrate the same mechanism in a simplified setting. The $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$22 study, effectively a single-channel $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$23 unitarization with hidden-gauge interaction kernel, finds an $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$24 pole at $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$25 MeV on the second sheet, which moves to $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$26 MeV after folding in the $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$27 width, and interprets it as a dynamically generated $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$28 molecular resonance that might correspond to $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$29. In the $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$30 sector it finds a much broader pole at $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$31 MeV, or $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$32 MeV after the width folding, with no established PDG counterpart (Wan et al., 2018).

The bottom-strange $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$33 system provides a recent heavy-flavor realization. With one cutoff fixed by identifying $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$34 as a $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$35 molecule, the on-shell scheme predicts $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$36 at $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$37 MeV, $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$38 at $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$39 MeV, $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$40 at $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$41 MeV, and nearly degenerate $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$42 states at $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$43 MeV in $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$44. The off-shell solution gives closely similar numbers. The shallow $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$45 and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$46 poles are interpreted as bottom-flavor partners of $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$47 and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$48 (Sánchez-Illana et al., 30 Mar 2026).

The triple-heavy extension applies the same logic to open-heavy meson-baryon systems in $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$49, using vector exchange, suppressed heavy-vector corrections, on-shell Bethe–Salpeter unitarization, and compositeness analysis. It predicts four $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$50-like states, four $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$51-like states, fourteen $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$52-like states, and ten $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$53-like states. Their binding energies are typically of order $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$54–$${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$55 MeV, the widths are usually small, and the compositeness values are used to argue that they are largely molecular (Wang et al., 2024).

7. Approximations, interpretive issues, and relation to other unitary methods

Several technical assumptions recur throughout the formalism. Most implementations work in $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$56-wave, neglect external three-momenta compared with hadron masses, approximate $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$57 near threshold, and employ on-shell factorization. Many models retain only vector exchange at tree level, treating heavier exchanged vectors as suppressed corrections; the triple-heavy study makes this explicit through $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$58, $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$59, and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$60, while heavier exchanges such as $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$61 and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$62 are neglected. Some implementations neglect anomalous $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$63 transitions or $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$64 mixing when they are argued to be very small; others, by contrast, make those mechanisms central through box diagrams (Wang et al., 2024, Wu et al., 2010).

Regularization is a persistent source of model dependence. Hidden-charm studies state that their conclusions are stable against reasonable changes in $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$65, but the hidden-beauty baryon analysis emphasizes that the beauty sector is more sensitive to regularization than the strange sector because the loop function varies more strongly with energy near heavy thresholds. The $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$66 analysis is also explicit that its calculation is highly simplified: it is dominated by $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$67-channel exchange, uses a single-channel approximation, and neglects most coupled channels by argument rather than explicit coupled-channel numerics (Wu et al., 2010, Wan et al., 2018).

The formalism is closely related to chiral unitary dynamics. A chiral $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$68 study of pseudoscalar meson-baryon octet scattering states that hidden-gauge methods often generate meson-baryon interactions via vector-meson exchange which, in the low-momentum limit, reduce to contact-like interactions closely related to the Weinberg–Tomozawa term. That work situates itself within the standard chiral unitary literature associated with Oller and Oset, Kaiser, Siegel, Weise, Inoue, Oset, Vicente Vacas, Döring, Nieves, and others. In its $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$69, $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$70 sector, the pole at $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$71 MeV couples much more strongly to $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$72, $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$73, and $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$74 than to $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$75, which it uses to argue that hidden-strangeness channels are essential to the dynamical generation of the $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$76-like resonance; in $${\cal L}_{BBV}=g \left(\langle\bar{B}\gamma_\mu [V^\mu,B]\rangle+\langle\bar{B}\gamma_\mu B\rangle\langle V^\mu\rangle\right),$$77, the interaction is repulsive and no resonance is generated (Sun et al., 2019).

The dominant physical interpretation across these applications is molecular rather than compact quark-model structure. Hidden-charm and hidden-beauty baryon papers state that the resulting states are not compatible with simple three-quark assignments, because they are dynamically generated from hadron-hadron interactions and contain hidden heavy flavor. At the same time, one hidden-charm study notes that distinguishing such states from possible five-quark interpretations would require further study. This suggests that the formalism is best understood as a precise dynamical framework for near-threshold hadron-hadron states, rather than as a unique ontological classification scheme (Wu et al., 2010).