Continuum-extrapolated glueball spectra

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The records below distinguish what the authors state, what this site examined, and what remains unestablished. Reading a source does not certify its proof.

Continuum-extrapolated pure SU(N) glueball spectra

numerical evidence · numerical support

Monte Carlo calculations estimate low-lying glueball masses at sampled values of N, with continuum extrapolations at fixed N and a subsequent large-N analysis.

Model: 3+1D; SU(N), N = 2, 3, 4, 5, 6, 8, 10, 12; Pure gauge
Geometry and regulator: Finite periodic lattice volumes with volume studies; Lattice with numerical continuum extrapolation. Boundary details: see the exact source conditions.
Assumptions: Wilson plaquette action; finite operator bases, statistical sampling and extrapolation assumptions; separate finite-volume and cutoff analyses.
Exact source: Sections 5.3-5.7; continuum extrapolation equation (27)
arXiv:2106.00364v3, revised 2021-12-10; JHEP 12 (2021) 082; DOI 10.1007/JHEP12(2021)082
Tables 23-30 (sampled groups)
arXiv:2106.00364v3, revised 2021-12-10; JHEP 12 (2021) 082; DOI 10.1007/JHEP12(2021)082
Table 24, printed page 74, A1++ ground state
arXiv:2106.00364v3, revised 2021-12-10; JHEP 12 (2021) 082; DOI 10.1007/JHEP12(2021)082
Table 34, printed page 83, 0++ gs SU(3)
arXiv:2106.00364v3, revised 2021-12-10; JHEP 12 (2021) 082; DOI 10.1007/JHEP12(2021)082
Table 34 PDF page 84 visually inspected; 0++ gs SU(3) cell 3.405(21), no star
arXiv:2106.00364v3, revised 2021-12-10; JHEP 12 (2021) 082; DOI 10.1007/JHEP12(2021)082
Publication: published; source date 2021-12-10 (day). First submitted: 2021-06-01.
What remains: Numerical estimates depend on statistical and systematic control. They do not prove a lower bound on the complete physical spectrum or construct the continuum theory.
Quoted numerical uncertainty is not a rigorous spectral lower bound or a guarantee that every systematic effect is covered. Spin assignment, poor continuum fits and multiparticle/ditorelon contamination require the paper's caveats. No simulation reproduced.
Limits and uniformity: a² sigma: Numerical continuum extrapolation via equation (27). Fixed: N; dimensionless observable definition. Finite-volume and fit-systematic checks are numerical controls, not a rigorous uniform spectral bound. Fixed-N continuum extrapolations precede the separate large-N fits.
1/N²: Separate large-N extrapolation where reported. Fixed: continuum-extrapolated dimensionless normalization. Sampled finite ranks and fit assumptions; no theorem for every compact simple group.
Proof-obligation relation: gap — numerical-support: Continuum-extrapolated pure SU(N) glueball spectra: Numerical estimates depend on statistical and systematic control. They do not prove a lower bound on the complete physical spectrum or construct the continuum theory.
Author claim: A scoped numerical evidence in the model above; no full four-dimensional solution is asserted by this record.
Technical assessment: selected sections; latest recorded action scope checked on 2026-10-24. No independent proof verification is recorded.
Community acceptance: No consensus assessment is inferred from publication or this site’s reading. External assessments, if documented, are separate records.
Clay recognition: These are partial-result records, not prize-recognition claims. See the separately sourced problem status.

Assessment history and unresolved checks

2026-10-24: scope checked

selected sections · Codex editorial source inspection; inherited and current actions remain distinct.

Examined: Sections 5.3-5.7; continuum extrapolation equation (27); Tables 23-30 (sampled groups); Table 24, printed page 74, A1++ ground state; Table 34, printed page 83, 0++ gs SU(3); Table 34 PDF page 84 visually inspected; 0++ gs SU(3) cell 3.405(21), no star.

Pure Wilson-action SU(N) lattice spectroscopy in 3+1 dimensions for N=2,3,4,5,6,8,10,12. Finite-spacing variational masses are extrapolated in a^2 sigma and further in 1/N^2 where reported.

Quoted numerical uncertainty is not a rigorous spectral lower bound or a guarantee that every systematic effect is covered. Spin assignment, poor continuum fits and multiparticle/ditorelon contamination require the paper's caveats. No simulation reproduced.
No independent proof certification or numerical simulation reproduction.

2026-10-23: scope checked

abstract · Inherited webmaster-kit planning inspection; not a new reading on import.

Examined: Abstract; Version history and publication metadata.

The abstract reports Wilson-action lattice simulations of SU(N) theories for sampled values of N between 2 and 12, with continuum extrapolations of low-lying spectra and further large-N extrapolations.

The data support numerical spectroscopy, not a rigorous lower bound on the entire spectrum. Verify exact sampled values of N, tables, scale setting, and statistical/systematic errors before transcribing benchmark values.
No independent expert proof verification or computational reproduction is recorded.

Editorial history

2026-10-24 · scope correction

Separated provenance and claim type for Continuum-extrapolated pure SU(N) glueball spectra.

Replace the inherited undifferentiated “source scope checked” badge with the actual recorded read depth and explicit limitations. Midnight UTC is a date-only editorial convention, not a source publication time.

Previous statement: “Source scope checked, 22 October 2026 (inherited undifferentiated label).”

2026-10-22 · new result

Inherited dossier: Continuum-extrapolated glueball spectra.

Original editorial snapshot, retained during migration. This is not the date the source result first appeared.

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Athenodorou and Teper’s SU(N) gauge theories in 3+1 dimensions: glueball spectrum, string tensions and topology studies pure gauge theories at the sampled values $N=2,3,4,5,6,8,10,12$ (the rank of $SU(N)$ is $N-1$ ). It estimates spectra using lattice simulations, extrapolates to the continuum at fixed $N$ , and then studies large $N$ . It is a quantitative benchmark, not a theorem constructing the physical Hilbert space.

What a glueball calculation measures

A glueball is a gauge-invariant excitation built from the gauge field. To search for it, choose operators with desired symmetry properties and subtract any vacuum expectation where necessary. Their Euclidean correlation matrix has, under the relevant spectral assumptions, the schematic form

C_{ij}(t) =\sum_{n\neq0} \langle0|O_i|n\rangle \langle n|O_j^\dagger|0\rangle e^{-E_nt}.

Here the vacuum energy is set to zero. This forward-propagation formula is most transparent at zero temperature with a sufficiently long Euclidean time direction. A periodic finite temporal extent adds backward-propagating contributions and must be modeled accordingly.

Large Euclidean time suppresses higher energies. The difficulty is that the signal also becomes small. A basis of several operators helps distinguish states and improve overlap with the levels of interest. This is why operator design and statistical precision both matter.

For a single clean exponential, an effective mass is

m_{\mathrm{eff}}(t) =\frac1a\log\frac{C(t)}{C(t+a)}.

If $C(t)=Ae^{-mt}$ exactly, this returns $m$ . With several exponentials, it varies with $t$ . A plateau is therefore a diagnostic to examine, not an algebraic guarantee that all unwanted states have disappeared. Explore this directly in the correlator lab.

Two limits that deserve separate plots

The box must be large enough to approximate infinite spatial volume. Independently, the lattice spacing must be small enough to approach the continuum. Holding the number of sites fixed while reducing $a$ shrinks the physical box, so it does not perform both operations together.

A dimensionless mass ratio, such as $m/\sqrt{\sigma}$ with string tension $\sigma$ , avoids introducing an arbitrary external unit. The measured lattice quantities are $am$ and $a^2\sigma$ . A finite physical mass is compatible with $am$ tending to zero as the regulator is removed.

This is a useful sanity check: a positive lower bound on $am$ independent of $a$ would imply an increasing physical mass in fixed units, not the desired finite continuum scale. The continuum-limit lesson develops this distinction.

How to read the benchmark

In the source, start with §2.2 on energies and correlators and §2.3 on systematic errors. For the central extrapolation, compare §5.2 on finite-volume effects with §5.5 on continuum masses; §5.6 treats large $N$ . Keep those three extrapolations separate in your notes.

A useful reading sheet has columns for operator channel, lattice spacing, physical volume, fit window, and extrapolation assumptions. Error bars on a final number cannot communicate every modeling decision without that supporting context.

Why strong evidence is not the complete gap statement

An operator sees only states with nonzero overlap. Even an accurately measured positive mass in several channels does not logically exclude a lighter state missed by the chosen basis. The Clay target concerns the complete physical spectrum of the constructed continuum theory.

Numerical evidence therefore answers an essential question—what spectrum the theory appears to have—while rigorous construction and a universal lower spectral bound remain distinct obligations. This distinction preserves the value of the calculation and clarifies what a proof would add.

Continue with operator overlap, glueball spectroscopy, and the proof-obligation map. The per-claim assessment history above records source scope, not an independent audit of simulation data.