Extended Local Hidden Gauge Approach

• The extended local hidden gauge approach is a framework that uses t-channel vector-meson exchange and unitarization to model low-energy hadron interactions while treating heavy quarks as spectators.
• It applies heavy-flavor extensions to classic hidden gauge methods, generating molecular states in systems like hidden-charm/beauty baryons, meson molecules, and multiquark pentaquarks with near-degenerate HQSS multiplets.
• The approach employs Weinberg–Tomozawa-type kernels and various regularization schemes to yield binding energies and decay widths that depend on the dynamical input and channel coupling.

The extended local hidden gauge approach is a heavy-flavor extension of the local hidden gauge or hidden local symmetry framework in which low-energy hadron interactions are driven primarily by $t$-channel vector-meson exchange, reduced in $S$ wave to Weinberg–Tomozawa-type contact interactions, and then unitarized in coupled channels. In its characteristic implementations, heavy quarks act as spectators in the dominant light-vector-exchange processes, which makes the leading interaction largely independent of heavy spin and flavor and therefore naturally compatible with heavy-quark spin symmetry (HQSS). The framework has been used to generate hidden-charm and hidden-beauty baryons, open-charm and open-beauty baryons, hidden-beauty meson molecules, and more recent double-heavy, triple-heavy, and five-flavor molecular pentaquarks (Xiao et al., 2013, Xiao et al., 2013, Liang et al., 2014).

1. Formation and scope of the framework

A characteristic formulation of the extended local hidden gauge approach appeared in 2013 in coupled-channel studies of hidden-charm and hidden-beauty meson–baryon systems, where local hidden gauge dynamics was combined explicitly with HQSS to generate molecular baryons from $\bar D^{(*)}\Sigma_c^{(*)}$ and $B^{(*)}\Sigma_b^{(*)}$ dynamics (Xiao et al., 2013, Xiao et al., 2013). It was then developed further in open-beauty and open-charm sectors, where $\bar B^{(*)}N$ and $D^{(*)}N$ channels were coupled to light channels and pion-exchange box diagrams were included to break the leading spin degeneracies of the vector–baryon sector (Liang et al., 2014, Liang et al., 2014).

The same logic was extended to hidden-beauty meson–meson systems, where the formalism predicts a robust $I=0$ sextet of bound $B\bar B$, $B\bar B^*$, and $B^*\bar B^*$ molecules, while the leading $S$0 interaction vanishes because of $S$1–$S$2 cancellation (Ozpineci et al., 2013). It was also used in a two-body heavy-meson application to the $S$3 and $S$4 systems, where a bound state slightly lower than the $S$5 threshold was generated in the isospin-zero sector and interpreted as a possible $S$6 candidate (Sun et al., 2017).

More recent work has pushed the same formal structure into multiquark-heavy sectors. These applications include $S$7 molecular pentaquarks (Wang et al., 2023), triple-heavy molecular pentaquarks of $S$8-, $S$9-, $\bar D^{(*)}\Sigma_c^{(*)}$0-, and $\bar D^{(*)}\Sigma_c^{(*)}$1-type (Wang et al., 2024), $\bar D^{(*)}\Sigma_c^{(*)}$2 molecules (Wang et al., 29 Aug 2025), and five-flavor $\bar D^{(*)}\Sigma_c^{(*)}$3 molecular pentaquarks built from local hidden gauge symmetry combined with HQSS and heavy-quark flavor symmetry (HQFS) (Suntharawirat et al., 8 Jun 2026). Across these sectors, the approach keeps the same conceptual core: light-vector exchange fixes the kernel, heavy-vector exchange is suppressed, and unitarization generates poles interpreted as hadronic molecules.

2. Interaction structure and Weinberg–Tomozawa kernels

The dynamical input is supplied by the hidden local symmetry interaction vertices. A standard set is

$\bar D^{(*)}\Sigma_c^{(*)}$4

$\bar D^{(*)}\Sigma_c^{(*)}$5

with

$\bar D^{(*)}\Sigma_c^{(*)}$6

where $\bar D^{(*)}\Sigma_c^{(*)}$7 MeV and $\bar D^{(*)}\Sigma_c^{(*)}$8 is taken as a light-vector mass such as $\bar D^{(*)}\Sigma_c^{(*)}$9 (Xiao et al., 2013). Equivalent forms are used throughout the later heavy-sector applications, often written as $B^{(*)}\Sigma_b^{(*)}$0, $B^{(*)}\Sigma_b^{(*)}$1, and $B^{(*)}\Sigma_b^{(*)}$2 with the same universal coupling (Wang et al., 2023, Wang et al., 29 Aug 2025).

In the meson–baryon sector, projecting light-vector exchange into $B^{(*)}\Sigma_b^{(*)}$3 wave gives the standard Weinberg–Tomozawa-type kernel

$B^{(*)}\Sigma_b^{(*)}$4

where the coefficients $B^{(*)}\Sigma_b^{(*)}$5 are fixed by flavor and channel quantum numbers (Xiao et al., 2013). In open-charm and open-beauty implementations, the same interaction is often written in the relativistic form

$B^{(*)}\Sigma_b^{(*)}$6

which reduces to the familiar Weinberg–Tomozawa structure near threshold (Liang et al., 2014, Liang et al., 2014).

For vector–baryon channels, the same scalar strength appears with the polarization factor $B^{(*)}\Sigma_b^{(*)}$7, and in heavy sectors temporal polarizations are neglected, so the leading kernel is spin independent at the quark level (Xiao et al., 2013, Wang et al., 29 Aug 2025). Heavy-vector exchange, by contrast, is systematically treated as subleading. In hidden-beauty baryons it enters only through small off-diagonal low-energy constants such as $B^{(*)}\Sigma_b^{(*)}$8 and $B^{(*)}\Sigma_b^{(*)}$9 (Xiao et al., 2013). In later multiquark applications it is parameterized by suppression factors such as $\bar B^{(*)}N$0, $\bar B^{(*)}N$1, $\bar B^{(*)}N$2, and $\bar B^{(*)}N$3 (Wang et al., 2023, Wang et al., 29 Aug 2025).

3. Heavy-flavor extension, spectator heavy quarks, and HQSS

What makes the framework “extended” is not merely the enlargement of field multiplets to include charm and bottom, but the specific dynamical assumption that the heavy quarks act as spectators in the dominant interactions. Light-vector exchange then proceeds between light quarks only, so the heavy-quark spin is untouched at leading order (Liang et al., 2014, Liang et al., 2014). Several later papers stress that SU(4) or SU(5) matrices are used mainly as bookkeeping devices, while the actual dynamics is governed by light-flavor exchange and physical hadron masses rather than exact heavy-flavor symmetry (Wang et al., 2023, Wang et al., 29 Aug 2025).

In the hidden-charm and hidden-beauty baryon studies, the channel space is reorganized into an HQSS basis labeled by the heavy-pair spin $\bar B^{(*)}N$4, the spin of the light degrees of freedom $\bar B^{(*)}N$5, and the total $\bar B^{(*)}N$6. HQSS implies that the QCD Hamiltonian is diagonal in $\bar B^{(*)}N$7, $\bar B^{(*)}N$8, and $\bar B^{(*)}N$9 and independent of $D^{(*)}N$0, which leads to interaction matrices expressed through a finite set of reduced low-energy constants (Xiao et al., 2013, Xiao et al., 2013). In hidden beauty, this produces nine low-energy constants, $D^{(*)}N$1 and $D^{(*)}N$2, which are isospin dependent but $D^{(*)}N$3 independent. Matching them to the local hidden gauge potential yields, for example, $D^{(*)}N$4, $D^{(*)}N$5, $D^{(*)}N$6, and $D^{(*)}N$7 in the $D^{(*)}N$8 sector (Xiao et al., 2013).

In hidden-beauty meson–meson systems the same mechanism appears in a different notation. There the HQSS low-energy constants satisfy $D^{(*)}N$9, $I=0$0, and $I=0$1, making the $I=0$2, $I=0$3, and $I=0$4 kernels identical across $I=0$5 in the heavy limit (Ozpineci et al., 2013). A related five-flavor formulation reconstructs the physical channel matrices as

$I=0$6

with the heavy-vector suppression encoded in

$I=0$7

thereby making the HQSS and HQFS content of the approach explicit (Suntharawirat et al., 8 Jun 2026).

A direct physical consequence is the recurrent appearance of near-degenerate spin partners. In hidden beauty baryons, $I=0$8 appears in $I=0$9 and $B\bar B$0 in $B\bar B$1 (Xiao et al., 2013). In open beauty, $B\bar B$2 states are degenerate under the Weinberg–Tomozawa interaction before pion exchange is included (Liang et al., 2014). In more recent multiquark calculations, the same pattern reappears as $B\bar B$3 doublets and $B\bar B$4 triplets (Wang et al., 2023, Wang et al., 2024, Suntharawirat et al., 8 Jun 2026).

4. Unitarization and regularization in the heavy sector

The interaction kernels are resummed with the on-shell Bethe–Salpeter equation

$B\bar B$5

or equivalently $B\bar B$6 in later notation (Xiao et al., 2013, Wang et al., 2023). The loop function for a meson–baryon channel is

$B\bar B$7

with $B\bar B$8 (Xiao et al., 2013).

A recurring technical issue is that the heavy sector is sensitive to regularization. Several works note that dimensional regularization matched to a cutoff near threshold can lead to unphysical behavior below threshold, including cases where $B\bar B$9 (Xiao et al., 2013, Liang et al., 2014). For this reason, different implementations coexist.

One strategy is a sharp three-momentum cutoff,

$B\bar B^*$0

used extensively in the open-charm and open-beauty baryon calculations, often with separate cutoff scales for pseudoscalar–baryon loops, vector–baryon loops, and box diagrams (Liang et al., 2014, Liang et al., 2014, Uchino et al., 2015).

A second strategy derives a physical form factor directly from the exchanged light-vector propagator,

$B\bar B^*$1

so that

$B\bar B^*$2

This was emphasized in hidden-beauty baryons and meson molecules as a more physical way to retain the large loop momenta relevant near heavy thresholds (Xiao et al., 2013, Ozpineci et al., 2013).

A third strategy, prominent in later pentaquark work, uses dimensional regularization with subtraction constants fixed channel by channel by matching the dimensional loop to a cutoff expression at threshold. Representative choices are $B\bar B^*$3 MeV in the $B\bar B^*$4 calculation, $B\bar B^*$5 MeV in the $B\bar B^*$6 and triple-heavy calculations, and $B\bar B^*$7 GeV with $B\bar B^*$8 in the five-flavor $B\bar B^*$9 study (Wang et al., 2023, Wang et al., 2024, Wang et al., 29 Aug 2025, Suntharawirat et al., 8 Jun 2026).

Pole searches above threshold require analytic continuation to the second Riemann sheet. Typical formulae are

$B^*\bar B^*$0

in cutoff-based implementations (Liang et al., 2014), or

$B^*\bar B^*$1

in dimensional regularization (Suntharawirat et al., 8 Jun 2026). Residues then define the channel couplings through

$B^*\bar B^*$2

or the equivalent expression in $B^*\bar B^*$3 (Xiao et al., 2013, Wang et al., 2023).

5. Dynamically generated spectra across sectors

The framework has generated a broad, internally coherent molecular spectrum across heavy hadron sectors.

Sector	Representative output	Source
Hidden charm baryons	Four $B^*\bar B^*$4 molecular families from $B^*\bar B^*$5	(Xiao et al., 2013)
Hidden beauty baryons	Four basic $B^*\bar B^*$6 states around 11 GeV; no $B^*\bar B^*$7 poles	(Xiao et al., 2013)
Open charm and beauty baryons	$B^*\bar B^*$8, $B^*\bar B^*$9, $S$00, $S$01 plus extra multiplets	(Liang et al., 2014, Liang et al., 2014)
Hidden beauty meson molecules	Six robust $S$02 bound states; leading $S$03 interaction too weak to bind	(Ozpineci et al., 2013)
Two-body heavy meson molecules	Isoscalar $S$04 and $S$05 bound states	(Sun et al., 2017)
Heavy pentaquarks	$S$06, triple-heavy, $S$07, and $S$08 molecular states	(Wang et al., 2023, Wang et al., 2024, Wang et al., 29 Aug 2025, Suntharawirat et al., 8 Jun 2026)

In hidden charm, the approach produces four molecular families in $S$09: $S$10 with $S$11, $S$12 with $S$13, a near-degenerate $S$14 doublet in $S$15, and a near-degenerate $S$16 triplet in $S$17. These states lie about $S$18–$S$19 MeV below their dominant thresholds, decay mostly into $S$20 and $S$21, and have widths of order $S$22–$S$23 MeV except for the $S$24 state, whose width is zero within the chosen space (Xiao et al., 2013). Hidden beauty baryons reproduce the same HQSS architecture at higher masses: $S$25, $S$26, $S$27, and $S$28 appear as $S$29 bound states around 11 GeV, with binding energies about $S$30–$S$31 MeV and widths from $S$32 to $S$33 MeV, while no $S$34 bound states or resonances are found (Xiao et al., 2013).

In the open-heavy sectors, pion-exchange box diagrams become quantitatively important because they mix pseudoscalar–baryon and vector–baryon channels. In open beauty this mechanism yields two nearly zero-width states identified with $S$35 and $S$36, both dominated by $S$37, with a $S$38–$S$39 splitting of about $S$40 MeV generated by pion exchange through intermediate $S$41 states (Liang et al., 2014). In open charm, the same pattern gives a narrow $S$42 with mixed $S$43 and $S$44 character, a narrow $S$45 dominated by $S$46, and additional $S$47 and $S$48 states, including nearly degenerate $S$49 partners around 2990 MeV in $S$50 (Liang et al., 2014). A later hidden-charm baryon study added pion-exchange boxes and anomalous $S$51 terms to the local hidden gauge kernel and obtained six states, including two admixture states in the $S$52–$S$53 sector and a spin-degenerate $S$54 bound state with $S$55 (Uchino et al., 2015).

In hidden-beauty meson–meson dynamics, the approach predicts six robust $S$56 bound states, all three spin sectors being degenerate in binding because of HQSS. For $S$57 MeV, the representative masses are about 10523 MeV for $S$58, 10568 MeV for $S$59, and 10613 MeV for $S$60, each roughly 37 MeV below threshold. Weakly bound hidden-strange partners can appear in single-channel approximations but disappear in the full coupled-channel calculation (Ozpineci et al., 2013). In a related two-body application, the $S$61 interaction in $S$62 generates a bound state near 3872 MeV, interpreted as a possible $S$63, and the same formalism predicts an isoscalar $S$64 bound state with no current Particle Data Group counterpart (Sun et al., 2017).

The later pentaquark literature extends the approach into systematically heavier systems. The $S$65 calculation predicts isoscalar states with $S$66, $S$67, and $S$68, binding energies of order $S$69–$S$70 MeV, and widths below $S$71 MeV (Wang et al., 2023). The triple-heavy study reports four $S$72-like states, four $S$73-like states, fourteen $S$74-like states, and ten $S$75-like states, together with couplings and compositeness consistent with dominant molecular configurations (Wang et al., 2024). The $S$76 work finds a total of fourteen molecular states with quantum numbers $S$77, $S$78, and $S$79 and binding energies about $S$80–$S$81 MeV, the range depending on the free parameter $S$82 (Wang et al., 29 Aug 2025). The five-flavor $S$83 analysis predicts ten threshold-associated isoscalar poles with $S$84 in the range $S$85 to $S$86 GeV, organized into HQSS multiplets, and also two more deeply bound $S$87 and $S$88 poles generated by strong inter-channel coupling (Suntharawirat et al., 8 Jun 2026).

6. Assumptions, limitations, and recurrent points of debate

The approach is not an exact SU(4) or SU(5) theory. Several papers explicitly state that heavy-flavor matrices are introduced for bookkeeping, while the actual dynamics uses physical masses and relies on light-vector exchange, with heavy quarks treated as spectators (Wang et al., 2023, Wang et al., 29 Aug 2025). A common misconception is therefore to identify the framework with naive heavy-flavor symmetry; the calculations instead combine light-sector hidden local symmetry with controlled heavy-sector symmetry breaking.

A second recurring issue is regulator dependence. In hidden-beauty baryons, varying the upper momentum limit between 1.5 and 3 GeV/c shifts binding energies by about 20 MeV, while imposing an additional sharp cutoff near 800 MeV reduces the binding to about 50 MeV (Xiao et al., 2013). In open beauty, the absolute masses move with the cutoff, although the $S$89 MeV splitting between the $S$90 and $S$91 analogues remains stable (Liang et al., 2014). In the $S$92 and five-flavor pentaquark studies, changing the subtraction scale shifts pole positions by order $S$93–$S$94 MeV while leaving the qualitative multiplet pattern intact (Wang et al., 29 Aug 2025, Suntharawirat et al., 8 Jun 2026).

A third issue is the treatment of pion exchange. In open-charm and open-beauty baryons, pion-exchange box diagrams are a key quantitative ingredient because they mix pseudoscalar–baryon and vector–baryon channels and lift HQSS-induced degeneracies (Liang et al., 2014, Uchino et al., 2015). By contrast, the $S$95 study concluded that one-pion exchange need not be considered when solving the Bethe–Salpeter equation, because the $S$96 exchange kernel already generates the bound state (Sun et al., 2017). These are not contradictory formalisms so much as sector-dependent implementations with different dominant mechanisms.

The predicted widths also require careful interpretation. Several states have zero width only within a restricted basis. The hidden-charm $S$97 state with $S$98, the hidden-beauty $S$99 state with $\bar D^{(*)}\Sigma_c^{(*)}$00, and some open-beauty bound states are widthless only because no open decay channel is included in the selected coupled-channel space (Xiao et al., 2013, Xiao et al., 2013, Liang et al., 2014). Later refinements that add further channels, pion boxes, or anomalous terms generally broaden at least part of the spectrum (Uchino et al., 2015).

Finally, the leading-order ELHG kernel does not universally bind every phenomenologically interesting threshold system. In hidden-beauty meson–meson dynamics, the $\bar D^{(*)}\Sigma_c^{(*)}$01 interaction cancels at leading order because $\bar D^{(*)}\Sigma_c^{(*)}$02, so the observed $\bar D^{(*)}\Sigma_c^{(*)}$03 states are interpreted there as requiring subleading HQSS-breaking terms, long-range one-pion exchange, or explicit coupling to bottomonium-plus-pion channels (Ozpineci et al., 2013). This illustrates a general feature of the method: its most robust predictions are usually the HQSS multiplet structure, the dominant molecular channels, and the existence of attraction in specific isospin sectors, whereas exact pole positions and the fate of marginal states depend more strongly on the regulator, channel completeness, and subleading dynamics.