Exotics with gluonic excitations

Abstract: That the gluons of quantum chromodynamics self-interact with a strong coupling suggests the possibility of hadrons in which the gluonic field plays an essential role: Glueballs in which gluons stick together to produce a hadron in which no quarks are required, and hybrids in which an excited gluonic field couples to quarks. This chapter presents our contemporary theoretical understanding of such states, and the challenges associated with definitively identifying their presence within the experimental spectrum of hadrons.

• The paper synthesizes lattice QCD and experimental evidence, identifying a chromomagnetic gluonic excitation about 1.2 GeV above conventional hadrons and a hybrid supermultiplet containing an exotic 1−+ state.
• The paper argues that coupled-channel analyses favor one broad π1 resonance near 1564 MeV rather than separate π1(1400) and π1(1600) states, while predicting substantial decay through the πb1 channel.
• The paper finds glueball identification inconclusive because strong mixing with ordinary mesons and inconsistent resonance counts obscure the scalar, tensor, and pseudoscalar spectra, underscoring the need for multi-hadron lattice methods and coordinated amplitude analyses.

Motivation and scope

This pedagogic chapter addresses a structural question in hadron physics: whether the self-interacting gluonic field of QCD manifests as identifiable excitations of the hadron spectrum, either as glueballs (bound states of gluons alone) or hybrids (quark systems coupled to an excited gluonic field). The author, Jozef Dudek, frames the central difficulty precisely: because QCD is strongly coupled at low energies, physical eigenstates are plausibly admixtures of $q\bar{q}$, $qqq$, glueball, and hybrid basis states, and the observables available from experiment—quantum numbers, masses, widths, branching fractions, production couplings—are only indirectly sensitive to internal structure. The chapter surveys the theoretical status of each class of state, drawing heavily on lattice QCD results, and confronts them with the experimental record.

Glueballs

The cleanest evidence for bound gluon states comes from SU(3) Yang-Mills theory, where lattice calculations exhibit a detailed spectrum of color-singlet glueballs with the lightest having $0^{++}$, $2^{++}$, and $0^{-+}$ quantum numbers. In the quarkless theory these states are stable below roughly 3.5 GeV, but their translation into full QCD is the crux of the problem: since glueballs carry no flavor and are isoscalar, they share quantum numbers with ordinary mesons, and strong coupling gives no reason to expect suppression of glueball–$q\bar{q}$ mixing.

The phenomenological consequences are laid out candidly. The standard interpretive tools—the OZI rule and "ideal" flavor mixing—are empirical regularities whose dynamical origin remains poorly understood, and which demonstrably fail in some channels (the $\eta$, $\eta'$ are not ideally mixed). Symmetry arguments for glueball character are shown to be weak: SU(3) singlet behavior is equally consistent with a $u\bar{u}+d\bar{d}+s\bar{s}$ configuration. Even two-photon coupling, often used as a filter against pure-glue nature, is undermined by rescattering through charged meson pairs.

The scalar sector illustrates the impasse. Three resonances—$f_0(1370)$, $qqq$0, $qqq$1—are commonly taken as one more state than the $qqq$2 model predicts, with admixture schemes assigning the largest glueball component variously to the $qqq$3 or elsewhere. But the resonance content itself is contested: high-statistics BESIII radiative $qqq$4 data do not peak at most PDG-listed resonance masses, one coupled-channel fit finds fewer poles than the PDG lists (no $qqq$5 pole; only three tensor states), while other analyses of overlapping data claim eight scalars between 1 and 2.5 GeV. The author's conclusion that no universally accepted spectrum below 2.5 GeV exists in the scalar, tensor, or pseudoscalar sectors is a pointed statement about the field's current state. The recent proposal that $qqq$6 is the pseudoscalar glueball is explicitly judged speculative, resting on mass arguments and lattice-inferred branching limits whose rigor is questionable given the unclear relation between Yang-Mills and full-QCD glueballs.

Hybrid mesons

The hybrid meson discussion is the strongest part of the chapter, because here theory and experiment have begun to converge. Lattice QCD calculations at unphysical quark masses ($qqq$7 MeV) identify a lightest hybrid supermultiplet—exotic $qqq$8 plus non-exotic $qqq$9, $0^{++}$0, $0^{++}$1 partners—all sharing large overlap onto chromomagnetic operators built from commutators of gauge-covariant derivatives, interpretable as $0^{++}$2 in an $0^{++}$3-wave coupled to a $0^{++}$4 gluonic excitation. Notably, this excitation matches neither the constituent-gluon nor flux-tube model predictions, but corresponds to the lightest bag-model mode. The excitation energy above the $0^{++}$5 (or nucleon) baseline is approximately 1.2 GeV and largely quark-mass independent—a robust, quantitative prediction—and charmonium calculations extend the pattern to positive-parity exotics with a $0^{++}$6-wave $0^{++}$7.

The experimental resolution of the long-standing $0^{++}$8/$0^{++}$9 ambiguity is presented as a significant development. Coupled-channel analyses of COMPASS data describe both $2^{++}$0 and $2^{++}$1 peaks with a single broad resonance of mass $2^{++}$2 MeV and width $2^{++}$3 MeV, consistent with the single light $2^{++}$4 state anticipated by lattice QCD. This directly contradicts the decades-old PDG listing of two exotic isovectors and undermines tetraquark multiplet interpretations, none of whose predicted flavor-exotic partners were ever found.

On decays, the first-principles calculation of Woss et al., performed in the exact SU(3) flavor limit where the $2^{++}$5 state lies only above two-meson thresholds, finds a narrow resonance with small couplings to open channels but large coupling to the closed $2^{++}$6 channel—an emergent confirmation of the old model-based lore favoring $2^{++}$7-wave meson partners. Extrapolation to physical kinematics suggests a broad state dominated by $2^{++}$8 decay, broadly compatible with the measured total width, though the dominance of $2^{++}$9 remains experimentally untested. The author is appropriately explicit about the limitations: the extrapolation scheme assumes mass-independent scaled couplings, and the experimental $0^{-+}$0 lies above many-hadron thresholds inaccessible to current finite-volume formalism.

The recently claimed $0^{-+}$1 in BESIII's $0^{-+}$2 is treated with instructive skepticism. If it is the isoscalar partner of the $0^{-+}$3, flavor-singlet production through a gluonic intermediate state combined with the near-octet $0^{-+}$4/near-singlet $0^{-+}$5 composition forces it to be a flavor admixture—so the absence of the orthogonal second $0^{-+}$6 state in the same analysis is a genuine puzzle the paper leaves open.

Hybrid baryons

Hybrid baryons receive comparatively brief treatment, reflecting both the absence of exotic quantum numbers (all half-integral $0^{-+}$7 are reachable with $0^{-+}$8 configurations) and the compounding "missing resonances" problem. Lattice results show candidate hybrid baryons—$0^{-+}$9 in $q\bar{q}$0-wave coupled to the same chromomagnetic excitation—with an excitation energy matching the meson case at roughly 1.2 GeV, suggesting a common gluonic mode across sectors. Experimentally, however, established resonances do not yet even fill the expected conventional slots below the first predicted hybrids, making identification currently impractical. Electroproduction transition form factors are noted as a potentially discriminating observable, though no first-principles expectation for their $q\bar{q}$1 dependence yet exists.

Limitations and open questions

The paper is unusually forthright about limitations, and several bear directly on its conclusions. The lattice spectra underpinning the hybrid supermultiplets are computed at unphysically heavy quark masses, ignore decay effects, may miss broad resonances, and should be read as approximations to narrow-resonance masses rather than definitive spectra. Resonance extraction from finite-volume spectra is limited to two-hadron channels; three-body decays remain at the frontier, which blocks rigorous study of the physical $q\bar{q}$2 and most glueball candidates. On the experimental side, the resonance content of the isoscalar scalar, tensor, and pseudoscalar sectors is unsettled, with competing analyses of the same data yielding incompatible state counts. Open questions include: whether any of the scalar resonances between 1 and 2 GeV is dominantly glueball; whether the $q\bar{q}$3 is confirmed and why its flavor partner is absent; whether the predicted $q\bar{q}$4 dominance of $q\bar{q}$5 decay holds; and how Yang-Mills glueballs relate to states in full QCD.

Conclusion

The chapter argues that gluonic excitations are a natural consequence of non-Abelian QCD, that lattice QCD now provides concrete, quantitative expectations for them—notably the ~1.2 GeV chromomagnetic excitation energy and the single-broad-$q\bar{q}$6 picture—and that experimental evidence has begun to align with these expectations in the exotic $q\bar{q}$7 channel while remaining inconclusive for glueballs. Its methodological thesis is that progress requires converged coupled-channel amplitude analyses of multi-process data on the experimental side, and extension of finite-volume resonance formalism to multi-hadron decays on the theoretical side, before reliable statements about internal structure can be made.