Hadronic Axial-Vector Molecule Overview

• Hadronic axial-vector molecules are near-threshold bound or virtual states formed predominantly by meson pairs, where at least one meson has axial-vector (J^P = 1^+) properties.
• They are classified within SU(3) flavor and HQSS multiplets and analyzed using unitarized chiral perturbation theory, lattice QCD, and QCD sum rules to extract mass spectra and decay patterns.
• Their predicted mass, width, and decay observables provide experimentally testable signatures in charmed, light-quark, and heavy-quark sectors, distinguishing them from compact multiquark states.

A hadronic axial-vector molecule is a near-threshold bound or virtual state composed primarily of two mesons, at least one of which is itself an axial-vector ($J^P=1^+$) state, whose binding and observable properties are dominantly the result of residual hadron–hadron interactions rather than a compact multiquark (diquark–antidiquark) structure. This concept is essential to the phenomenology of positive-parity open-charm mesons, light-quark axial-vector resonances, and multiquark candidates in the heavy quarkonium sector, where chiral symmetry, unitarity, and heavy-quark spin symmetry (HQSS) play a pivotal role in model construction and interpretation.

1. SU(3), Spin, and Flavor Decomposition

Dynamically generated axial-vector hadronic molecules can be systematically classified within SU(3)%%%%1%%%% and HQSS multiplets. As shown in the context of charmed-light systems, the direct product of a ground-state $D$ or $D^*$ heavy meson ($[\overline{3}]_c$) with the light-meson octet ($[8]_\phi$) yields

$$[\overline{3}]_c \otimes [8]_\phi = [\overline{3}] \oplus [6] \oplus [\overline{15}].$$

The Weinberg–Tomozawa leading-order term dictates that the $[\overline{3}]$ is most attractive, the $[6]$ is moderately attractive, and the $[\overline{15}]$ is repulsive. In the hadronic-molecule scenario, only the attractive channels—$[\overline{3}]$ and $[6]$—are candidates for harboring bound or virtual states, while the repulsive $[\overline{15}]$ will not support near-threshold poles. These patterns are expected to be identical in both scalar and axial-vector sectors due to HQSS (Gregory et al., 31 Mar 2025).

2. Dynamical Generation and Scattering Properties

The dynamical origin of hadronic axial-vector molecules is grounded in non-perturbative resummation techniques such as unitarized chiral perturbation theory (UChPT) and Bethe–Salpeter-equation approaches. The typical workflow is as follows:

• Construct effective chiral Lagrangians, including contact and exchange terms relevant for vector–pseudoscalar (VP) or vector–vector (VV) channels.
• Compute the on-shell VP$\to$VP potential $V_{ij}(s)$, project onto $S$-wave, and encode leading-order flavor coefficients $C_{ij}$.
• Unitarize via $t(s) = [1 - V(s) G(s)]^{-1} V(s)$, where $G(s)$ is the meson–meson loop function.
• Identify poles in $t_{ij}(s)$ in the complex $s$-plane as resonance states, with residues $g_{R,i}$ quantifying channel couplings.

When implemented for light axial-vectors, this strategy yields poles corresponding to $a_1(1260)$, $b_1(1235)$, $f_1(1285)$, $h_1(1170)$, $h_1(1380)$, and the two-pole structure of $K_1(1270)$ (Dai et al., 2018). Compositeness analyses via the Weinberg criterion typically show large molecular probabilities ($X\sim 0.6-0.9$), implying dominant meson–meson content.

3. Lattice QCD and Model Discrimination

Ab initio lattice QCD calculations, such as the Bonn–Jülich–Beihang study (Gregory et al., 31 Mar 2025), are crucial for discriminating between hadronic-molecular and tetraquark hypotheses. Key features include:

• Construction of four-quark interpolators projected onto $[6]$ and $[\overline{15}]$ SU(3) irreps, with Dirac structures appropriate for $J^P=1^+$.
• Computation of correlators with all relevant contraction topologies (direct, exchange).
• Extraction of ground-state energies and formation of energy shifts:

$\Delta E_{[d]} = E_0([d]) - [M_{D^{*}} + M_\pi].$

• Empirical findings: for both scalar ($0^+$) and axial-vector ($1^+$) sectors, $[6]$ is attractive ($\Delta E < 0$), $[\overline{15}]$ is repulsive ($\Delta E > 0$), with near-equal shifts in both parity channels.

These observations are fully consistent with the hadronic molecular scenario and exclude compact tetraquark models predicting a low-lying $[\overline{15}]$ only in the axial-vector sector. The methodology leverages the Lüscher quantization condition for relating finite-volume spectrum to infinite-volume scattering parameters, thereby connecting lattice QCD to physical compositeness and binding criteria.

4. QCD Sum Rules and Heavy Axial-Vector Molecules

QCD sum rules provide an alternative means of quantifying mass spectra, decay constants, and transition amplitudes of putative hadronic molecules. The construction for an axial-vector molecule, for instance a $(B_c^{*\pm} B_c^\mp)$ system ($J^{PC}=1^{++}$), proceeds as follows (Agaev et al., 24 Jul 2025):

• Build an interpolating current with the correct quantum numbers, e.g.,

$$J_\mu(x) = \tfrac{1}{2} \Bigl( [\bar b_a i\gamma_5 c_a][\bar c_b \gamma_\mu b_b] + [\bar b_a \gamma_\mu c_a][\bar c_b i\gamma_5 b_b] \Bigr).$$

• Evaluate the two-point correlator and isolate the ground-state pole.
• Compute the OPE up to dimension-4 operators (condensate terms), perform a Borel transform, and match correlators.
• Extract mass and coupling by standard ratio sum-rule techniques:

$$m^2 = \frac{d}{d(-1/M^2)}\Pi(M^2, s_0)/\Pi(M^2, s_0).$$

• Use three-point sum rules to obtain hadronic couplings for dominant decay channels (e.g., $J/\psi\eta_b$, $\Upsilon\eta_c$, $B_c^*B_c$), and compute partial widths.

Quantitative results for $B_c^* B_c$ molecules are

$$m = 12770 \pm 60\,\text{MeV},\quad \Gamma = 93 \pm 14\,\text{MeV},$$

with largest branching fractions to $J/\psi\eta_b$ and $\Upsilon\eta_c$. The molecule and the tetraquark alternative are nearly degenerate in mass but differ in predicted widths and decay patterns (Agaev et al., 24 Jul 2025).

5. Experimental and Phenomenological Implications

The hadronic axial-vector molecule hypothesis has several experimentally accessible consequences:

1. Mass and width systematics: Molecules are expected near two-meson thresholds, with relatively narrow widths in cases where S-wave decays dominate (if phase space allows), but potentially much broader if several fall-apart channels are open.
2. Decay patterns: Branching ratios are predictable from three-point sum rules and compositeness; extensive decay to two-meson final states is expected.
3. Production in weak processes: Chiral-unitary models predict specific invariant mass distributions in weak decays, such as $\tau^- \to \nu_\tau P^- A$, providing additional direct tests (Dai et al., 2018).
4. Interpretation of heavy states: For example, X(7300) is found to be compatible with a dominantly $\chi_{c1}\chi_{c1}$ molecule with a sizable tetraquark admixture, and the coupling of both types of interpolating current to the physical state is $O(1)$ in the ratio, suggesting non-pure composition (Agaev et al., 2023).

6. Distinctions from Compact Multiquark States

Key evidence for the molecular nature of hadronic axial-vector states, as opposed to compact diquark–antidiquark configurations, includes:

• Absence of low-lying $[\overline{15}]$ axial-vector states in both lattice and sum-rule spectra, contra tetraquark expectations (Gregory et al., 31 Mar 2025).
• Parity and spin sector universality (scalar and axial-vector sectors behave identically), a hallmark of the molecular UChPT scenario due to HQSS.
• Large compositeness $X \gtrsim 0.6$ for light-quark axial-vectors in unitarized models (Dai et al., 2018).
• Direct experimental branching ratios and invariant mass spectra are explainable within the triangle diagram mechanism, sensitive to the molecular content.

7. Applications and Open Problems

Axial-vector hadronic molecules constitute a key paradigm for interpreting a wide range of exotic meson spectroscopy:

• They provide a natural explanation for the proliferation of positive-parity states near two-meson thresholds in both light and heavy sectors.
• Explicit lattice and QCD sum rule treatments supply quantitative targets for mass, width, and coupling, relevant for experimental searches in $e^+e^-$, $B$-decay, and heavy-ion environments.
• Uncertainty remains for the detailed composition (pure molecule vs. molecule–tetraquark mixing), especially for fully heavy systems or highly excited states (Agaev et al., 2023).
• Future work includes higher-statistics lattice calculations with explicit multi-hadron interpolators, parallel QCD-SR analyses, and experimental mapping of decay spectra and pole structures.

In summary, the hadronic axial-vector molecule framework is rigorously tested by contemporary lattice QCD, unitarized effective field theory, and sum rule methods, and constitutes the leading explanation for the lightest positive-parity charmed mesons, axial-vector exotics, and analogous heavy tetraquark-like states (Gregory et al., 31 Mar 2025, Agaev et al., 2023, Dai et al., 2018, Agaev et al., 24 Jul 2025).