Glueball–flavour-singlet-meson mixing in $N_f=1$ QCD

In $N_f=1$ QCD — SU(3) gauge theory with a single dynamical quark flavour — every meson is a flavour singlet, so glueballs and $q\bar{q}$ mesons share the same $J^{PC}$ channels and can mix through quark-annihilation (disconnected) diagrams. The $N_f=1$ glueball spectrum extracted to date used purely gluonic operators, which cannot resolve this mixing, and the one modern mixing calculation covers only the scalar channel at a strange-mass quark on a single lattice spacing, finding a large effect (mixing angle about $41^\circ$ , mixing energy $239(24)$ MeV). The open problem is to determine the full role of glueball–meson mixing in the $N_f=1$ spectrum: the coupled-basis spectrum, mixing angles and mixing matrix elements in the low-lying $0^{++}$ , $2^{++}$ and $0^{-+}$ channels, their dependence on the quark mass down to light masses, and their continuum limit, establishing where mixing produces detectable mass shifts or additional states.

Background

A glueball is a colour-singlet bound state made of gluons, predicted by QCD in the pure-gauge sector with masses starting near $1.6$ GeV for the lightest ( $0^{++}$ ) state (Athenodorou and Teper 2020). In QCD with $N_f=1$ — one dynamical quark flavour — there are no pions: every quark–antiquark ( $q\bar{q}$ ) meson is a flavour singlet, including a scalar ( $\sigma$ -like) and an anomalously heavy pseudoscalar ( $\eta$ -like) meson shifted up by the $U(1)_A$ anomaly (Farchioni et al. 2007 measured this one-flavour meson spectrum). Because glueballs and these singlet mesons carry identical quantum numbers $J^{PC}$ (spin, parity, charge conjugation), the physical states are mixtures: a scalar glueball $G$ and a scalar $q\bar{q}$ meson couple through a mixing energy, producing two eigenstates displaced from the unmixed masses with a mixing angle $\theta$ ; in the $2\times2$ case, $\tan 2\theta = 2x/(m_G-m_{q\bar{q}})$ with $x$ the off-diagonal mixing matrix element. This mixing is a key systematic in identifying glueball candidates among the experimental isoscalar scalar mesons. On the lattice, masses are extracted from Euclidean correlation functions of interpolating operators; a variational Generalized Eigenvalue Problem (GEVP) over a basis of operators isolates the eigenstates level by level. Glueball correlators have a very poor signal-to-noise ratio, and flavour-singlet meson operators require quark–antiquark annihilation (disconnected) contributions, which are computable with all-to-all techniques such as distillation — smearing quark fields gauge-covariantly and propagating them between all points. Lattice rotational symmetry is only the octahedral group, whose irreducible representations (e.g. $A_1$ , $E$ , $T_2$ with parity and charge-conjugation superscripts such as $A_1^{++}$ ) label the measured states, tending to continuum $J^{PC}$ as the lattice spacing $a$ goes to zero. Athenodorou, Bergner, Teper and Wenger (2023) computed the $N_f=1$ glueball spectrum versus the quark mass (parameterised by the PCAC quark mass — the mass defined through the partially conserved axial-current Ward identity, the standard lattice definition) using $\mathcal{O}(10\mathrm{K})$ configurations but a purely gluonic operator basis, finding the $0^{++}$ , $2^{++}$ and $0^{-+}$ glueball masses consistent with pure SU(3) gauge theory; they left the mixing with flavour-singlet mesons — and whether it shifts masses or introduces additional states at higher precision or lighter quark masses — open, stating only that mesonic operators would be considered in future work, since their purely gluonic operator basis could not detect mixing. A companion study with $N_f=4$ light quarks (Athenodorou, Finkenrath, Lantos, Teper 2023), again using a purely gluonic operator basis, found the tensor and pseudoscalar ground states near their pure-gauge values but an additional light scalar state carrying large quark content. In 2026, Gui, Sun, Chen, Gong, Li and Liu performed the first $N_f=1$ coupled-basis mixing computation, but only in the scalar channel, on one anisotropic $16^3\times128$ ensemble at lattice spacing $a_s\approx0.148$ fm with the quark mass tuned to the strange point: solving the GEVP with glueball and $s\bar{s}$ operators (distillation for the annihilation diagrams) they found two eigenstates at $1.290(20)$ and $1.777(49)$ GeV, a mixing angle $|\theta|\approx40.7(2.7)^\circ$ and mixing energy $x_s=239(24)$ MeV — a strong effect, with the eigenstate masses shifted by hundreds of MeV from the unmixed $m_{s\bar{s}}=1.486(2)$ GeV and $m_G\approx1.581(53)$ GeV. They themselves flag the continuum limit of $x_s$ and its quark-mass dependence as uninvestigated. Related mixing physics is known in neighbouring theories: $\eta$ –glueball mixing has been computed in $N_f=2$ lattice QCD (Jiang et al. 2022), and quenched-era studies of scalar glueball–singlet mixing exist, but neither addresses the single-flavour theory.

Problem Statement

Determine the role of glueball–flavour-singlet-meson mixing in the low-energy spectrum of $N_f=1$ QCD. Specifically, extract (i) the low-lying spectrum of the channels in which glueballs are measured — $0^{++}$ , $2^{++}$ and $0^{-+}$ — as eigenstates of the coupled glueball–meson system; (ii) the mixing angles and mixing matrix elements between the glueball and the corresponding flavour-singlet meson in each channel; and (iii) their dependence on the dynamical quark mass, from the strange-mass regime down to lighter quark masses where the singlet scalar and pseudoscalar mesons enter the glueball mass region, together with the continuum-limit extrapolation of the mixing parameters. Establish thereby whether mixing shifts the glueball masses detectably or introduces additional low-lying states, and reconcile the result with the pure-gauge-like spectrum obtained from purely gluonic operators. The scalar channel at a strange-mass quark on a single lattice spacing has been measured (mixing angle $|\theta|\approx41^\circ$ , mixing energy $239(24)$ MeV); the quark-mass dependence, the continuum limit, and the non-scalar channels remain to be determined.

References

Andreas Athenodorou, Georg Bergner, Michael Teper, Urs Wenger, "Glueballs in $N_f=1$ QCD", PoS LATTICE2023 (2024) 141, DOI 10.22323/1.453.0141, arXiv:2312.00470, https://arxiv.org/abs/2312.00470
Long-Cheng Gui, Wei Sun, Ying Chen, Ming Gong, Geng Li, Zhaofeng Liu, "Scalar glueball- $s\bar{s}$ mixing in one flavor lattice QCD", arXiv:2607.17294 (2026), https://arxiv.org/abs/2607.17294
F. Farchioni, I. Montvay, G. Münster, E.E. Scholz, T. Sudmann, J. Wuilloud, "Hadron masses in QCD with one quark flavour", Eur. Phys. J. C 52 (2007) 305, DOI 10.1140/epjc/s10052-007-0394-4, arXiv:0706.1131, https://arxiv.org/abs/0706.1131
Andreas Athenodorou, Jacob Finkenrath, Adam Lantos, Michael Teper, "Glueball Spectrum with four light dynamical fermions", arXiv:2308.10054 (2023), https://arxiv.org/abs/2308.10054
Andreas Athenodorou, Michael Teper, "The glueball spectrum of SU(3) gauge theory in 3+1 dimensions", JHEP 11 (2020) 172, DOI 10.1007/JHEP11(2020)172, arXiv:2007.06422, https://arxiv.org/abs/2007.06422
Xiangyu Jiang, Wei Sun, Feiyu Chen, Ying Chen, Ming Gong, Zhaofeng Liu, Renqiang Zhang, " $\eta$ -glueball mixing from $N_f=2$ lattice QCD", arXiv:2205.12541 (2022), https://arxiv.org/abs/2205.12541