Quantitative decay phenomenology of the lightest vector glueball: absolute widths, quarkonium mixing, and robustness of the predicted branching-ratio hierarchy

The lightest glueball with quantum numbers $J^{PC}=1^{--}$ (vector glueball) is predicted by quenched lattice QCD near $3.8\ \mathrm{GeV}$ but has never been identified experimentally. An effective-Lagrangian study (Giacosa, Sammet, Janowski 2017) within the extended Linear Sigma Model (eLSM) fixed three chirally invariant interaction terms of a bare vector glueball with light mesons and obtained parameter-free branching-ratio hierarchies, with $\omega\pi\pi$ (via $b_1\pi$ ) and $\pi KK^*(892)$ as golden channels; however, the overall coupling constants $\lambda_{\mathcal{O},1}$ , $\lambda_{\mathcal{O},2}$ , $\alpha$ are unknown, glueball-quarkonium mixing, unquenching effects on the mass, the validity of the eLSM extrapolation from $\sim1.7\ \mathrm{GeV}$ to $\sim3.8\ \mathrm{GeV}$ , additional chirally allowed operators, and the physical- $N_c$ width scale were all left open. The literature status through 2025 shows that none of these inputs has been determined; a 2025 proof-of-concept shows $\psi(3686)$ -glueball mixing can address the $\rho\pi$ puzzle and estimate the total width, while a 2024 holographic study predicts a very broad spin-1 glueball, sharpening the width question. The problem of converting the predicted hierarchies into quantitative, testable identification criteria for a $\sim3.8\ \mathrm{GeV}$ vector glueball remains open.

Background

A glueball is a hadron made of gluons with no valence quarks, predicted by QCD because gluons self-interact; none has been unambiguously identified experimentally. Lattice QCD (discrete numerical simulation of the QCD path integral) in the quenched approximation — gluons evolved without dynamical quark-antiquark loops — predicts the lightest vector glueball, with quantum numbers $J^{PC}=1^{--}$ (spin 1, parity $-$ , charge conjugation $-$ ), at about $3.81\ \mathrm{GeV}$ (Chen et al. 2006), with an estimated $10$ - $15\%$ uncertainty; an unquenched ( $N_f=2+1$ ) lattice study of glueball masses found them broadly consistent with quenched values (Gregory et al. 2012). Because a $1^{--}$ state couples to a virtual photon, it could be directly formed in $e^+e^-$ annihilation at BESIII (the Beijing electron-positron collider experiment, in a $3.5$ - $4\ \mathrm{GeV}$ energy scan) and, if narrow enough, in proton-antiproton fusion at the PANDA experiment at FAIR (the Facility for Antiproton and Ion Research, Darmstadt). What experiments need from theory are the dominant decay channels and, ultimately, absolute widths. The extended Linear Sigma Model (eLSM) is a chiral effective hadron model: its Lagrangian is built from nonets of light $(\bar q q)$ -like (pseudo)scalar and (axial-)vector meson fields and a scalar dilaton field representing the gluon condensate (the nonperturbative expectation value of gluon fields in the QCD vacuum), constrained by chiral symmetry $U(3)_L\times U(3)_R$ (the approximate flavor symmetry of QCD with massless light quarks), by dilatation invariance (the approximate symmetry associated with scale transformations, broken by the gluon condensate and quark masses), and by parity and charge conjugation. The eLSM describes meson masses and decay widths up to roughly $1.7\ \mathrm{GeV}$ . In the source paper (Giacosa, Sammet, Janowski, Phys. Rev. D 95, 114004 (2017)), three chirally invariant effective interaction terms coupling a bare vector glueball field $\mathcal{O}_\mu$ to light mesons were constructed (two of them also dilatation invariant); since their overall coupling constants $\lambda_{\mathcal{O},1}$ , $\lambda_{\mathcal{O},2}$ (dimensionless) and $\alpha$ (dimension $\mathrm{energy}^{-2}$ ) cannot be fixed within the model, the authors computed parameter-free branching-ratio hierarchies for a glueball mass of $3.81\ \mathrm{GeV}$ . They found the golden channels $\mathcal{O}\to b_1\pi\to\omega\pi\pi$ , $\mathcal{O}\to\omega\pi\pi$ , and $\mathcal{O}\to\pi KK^*(892)$ (with $b_1(1230)$ an axial-vector meson and $K^*(892)$ a vector kaon), and subleading but observable $\rho\pi$ and $K^*(892)K$ modes that could bear on the $\rho\pi$ puzzle of $\psi(2S)$ — the longstanding violation of the pQCD '13% rule' relating $\psi(2S)$ and $J/\psi$ hadronic branching fractions, for which earlier perturbative estimates quoted in that paper find a glueball- $\psi(2S)$ mixing angle $|\theta|\lesssim2^\circ$ . The width scale itself was argued only qualitatively from large- $N_c$ counting (the $N_c\to\infty$ limit of QCD with $N_c$ color charges, in which glueball decays are doubly OZI-suppressed, where the OZI rule suppresses decays requiring quark-antiquark annihilation), suggesting possibly narrow ( $\lesssim100\ \mathrm{MeV}$ ) glueballs; a companion large- $N_c$ estimate gave roughly $10\ \mathrm{MeV}$ for heavy glueballs, with the physical case $N_c=3$ explicitly uncontrolled.

Problem Statement

Can the decay phenomenology of the lightest ( $J^{PC}=1^{--}$ ) vector glueball near $3.8\ \mathrm{GeV}$ be made quantitative and testable — i.e., can the missing inputs of the chiral effective-Lagrangian description be determined and its predictions shown robust — so that experimental vector-glueball searches can definitively identify or exclude a candidate? Concretely, building on the parameter-free branching-ratio hierarchies of Giacosa, Sammet, Janowski (2017): (i) determine the overall effective couplings ( $\lambda_{\mathcal{O},1}$ , $\lambda_{\mathcal{O},2}$ , $\alpha$ , and their analogues in any extended operator basis) — and hence the absolute partial and total decay widths; (ii) quantify glueball-quarkonium mixing (in particular with $\psi(2S)$ ) and its phenomenological impact on the $\rho\pi$ puzzle; (iii) establish the vector-glueball mass in full (unquenched) QCD, including hadronic-loop mass shifts, to quantify deviations from the quenched $3.81\ \mathrm{GeV}$ input; (iv) test the validity of extrapolating the eLSM interaction structures from $\sim1.7\ \mathrm{GeV}$ to $\sim3.8\ \mathrm{GeV}$ ; (v) establish how the predicted branching-ratio hierarchy and total width (including whether the state is narrow or very broad at $N_c=3$ ) change under the full set of chirally allowed operators, symmetry-breaking terms, and mixing. These items jointly serve one objective: a quantitatively grounded, experimentally usable identification (or exclusion) criterion — predicted dominant channels, branching ratios, widths, and mixing signatures for a $\sim3.8\ \mathrm{GeV}$ vector glueball; they are routes toward that single objective rather than independent obligations.

References

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