Glueball-quark-meson mixing and glueball decay widths from unquenched lattice QCD, and their confrontation with experiment: an unresolved question posed by Politzer's 1982 chiral analysis of the quenched approximation

Politzer's 1982 Phys. Lett. B 116, 171 analysis proposed that spontaneous chiral symmetry breaking acts at shorter distances than confinement, giving quarks a soft dynamical mass of roughly 300 MeV at confinement momenta, and organized the flavor-singlet corrections to the then-standard quenched lattice QCD approximation through a singlet mass-shift parameter $K$, a quark-antiquark-glueball mixing parameter $C$, a one-loop quenched-to-unquenched scale factor of $3.0\pm0.3$, and O(1/15)-type sea-quark suppression estimates. The open problem is the unquenched part of the 1982 glueball request: a quantitative first-principles determination of glueball-quark-antiquark (and multi-meson) mixing amplitudes and glueball decay widths in dynamical QCD, the size of the sea-quark corrections relative to the quenched determinations, and the confrontation of these results with the experimental spectrum to identify or exclude glueball-dominant states. The quenched determinations exist (Sexton et al., Morningstar and Peardon), the dynamical ones remain exploratory (Brett et al., Morningstar's 2024 review), and no experimental glueball is established.

Background

Lattice QCD is the discretization of quantum chromodynamics (QCD, the theory of quarks and gluons) on a Euclidean space-time grid so that path integrals can be evaluated by Monte Carlo simulation. In the early 1980s only the quenched or valence approximation was numerically affordable: the fermion determinant, which represents virtual quark-antiquark pairs in the gauge-field average, is replaced by its zero-loop value, so only the explicitly created valence quarks propagate. Hamber and Parisi and Weingarten produced the first quenched estimates of the hadron spectrum in this way, while Kogut, Stone, Wyld, Shigemitsu, Shenker and Sinclair reported evidence from finite-temperature lattice simulations that the force responsible for chiral symmetry breaking is short-ranged and independent of confinement, obtaining chiral-symmetry restoration temperatures for quarks in both the fundamental and the adjoint representations of color (adjoint quarks do not screen fundamental color charges, so chiral restoration there cannot be a deconfinement effect), which suggests the two phenomena are dynamically distinct.

Chiral symmetry here means the approximate invariance of QCD with massless light quarks under independent rotations of left- and right-handed quark fields; it is spontaneously broken by a condensate of quark-antiquark pairs, which endows a massless quark with a momentum-dependent constituent or soft mass $m_{\text{soft}}(Q^{2})$ defined through the quark propagator. Politzer's 1982 paper proposed that this chiral breaking occurs at shorter distances and weaker coupling than confinement, conjectured that $m_{\text{soft}}(Q^{2})$ becomes roughly 300 MeV on the scale relevant to confinement, and argued that this renders most light hadrons effectively non-relativistic (explaining the success of the non-relativistic quark model) and suppresses quark-loop corrections to valence observables by powers of $Q^{2}/m_{\text{soft}}^{2}$. For decay widths he estimated the effect of one additional quark-antiquark pair as the ratio $\Gamma_{\omega}/\Gamma_{\rho}\sim1/15$ of the omega to rho meson widths, i.e. an O(1/15) suppression per pair.

The suppression fails in the flavor-singlet sector (states invariant under flavor rotations, such as the eta and eta-prime mesons) because the axial anomaly, the quantum effect that gives the eta-prime its large mass, is independent of the quark mass. Politzer therefore organized the singlet dynamics in a meson-glueball self-energy matrix $\pi_{ij}(Q^{2})$ acting on a basis of interpolating fields (the local operators used to create and probe the states of interest), including the would-be Goldstone boson of mass $m_{0}$ and a glueball of mass $m_{G}$. Glueballs are bound states of gluons predicted by the self-interacting non-Abelian gauge field alone; they are well-defined states of pure-gauge (quenched) QCD, with the lightest scalar ($0^{++}$) near 1.7 GeV. Two parameters were left to be determined numerically: an additive singlet mass-shift $K$, arising because the dominant quark-loop contributions form a disconnected series that exponentiates (so that $m_{0}+K$ estimates the singlet meson mass, and no finite-loop Monte Carlo calculation improves the estimate), and a mixing parameter $C$ controlling quark-antiquark-glueball mixing. Politzer flagged the extrapolation of $\pi_{ij}(Q^{2})$ from the point $Q^{2}=m_{0}^{2}$, where disconnected correlators are evaluated, to on-shell and glueball momenta (neglecting cuts and nearby poles) as a serious possible source of error, especially as $m_{0}\to0$; estimated a one-loop renormalization factor $3.0\pm0.3$ (at bare coupling $g_{0}=1$) between masses computed with and without quark loops, implying that a quenched determination of the QCD scale parameter $\Lambda$ (the intrinsic mass scale of QCD) underestimates the true $\Lambda$ by about a factor of three; and predicted that physical glueballs, whose unmixed widths should be comparable to the omega width (about 10 MeV), may be strongly broadened by mass-mixing with neutral quark-antiquark states, with all mixings estimable from quenched Monte Carlo correlators.

Later work reformulated much of this program. Quenched chiral perturbation theory (the effective field theory, i.e. the most general local Lagrangian organized in powers of small momenta and quark masses, of the quenched theory), developed by Sharpe and systematized by Bernard and Golterman, makes the exponentiating disconnected series an eta-prime hairpin vertex (a two-vertex singlet insertion connecting otherwise disconnected quark loops) that generates double poles in quenched flavor-singlet correlators; its strength is a low-energy constant fit directly to lattice data. The unquenched era then replaced the quenched extrapolation program: dynamical simulations with light sea quarks became standard, and the eta and eta-prime masses are now reproduced from first principles at the physical quark masses. Glueball physics, however, remains split: the pure-gauge spectrum and the quenched mixing and decay computations exist, but the corresponding dynamical-quark determinations and the experimental identification of glueball-dominant states do not.

Problem Statement

The open problem is the unquenched half of the 1982 glueball program: determine, from dynamical-quark lattice QCD at (near-)physical quark masses with controlled continuum, finite-volume, and resonance systematics,

(1) the mixing of gluonic interpolating fields with quark-antiquark and multi-meson interpolating fields in the isoscalar (flavor-singlet) channels, i.e. the mixing amplitudes generalizing the 1982 parameter $C$ and the quantitative gluonic content of the physical scalar and pseudoscalar isoscalar states;

(2) the decay widths of glueball-rich states, including the amount by which physical glueballs are broadened by mixing with neutral quark-antiquark states, generalizing the 1982 estimate of unmixed widths of order 10 MeV;

(3) the numerical size of the sea-quark (O(1/15)-type) corrections to the quenched determinations of (1) and (2), i.e. the resolution of the 1982 suppression estimates into measured differences between quenched and dynamical results;

and confront these first-principles results with the experimental meson spectrum so as to identify, or exclude, glueball-dominant resonances. The three determinations and the experimental confrontation form one coherent objective: they jointly decide the same underlying question, whether and how the gluonic bound states predicted by QCD manifest in the physical spectrum.

References

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