F₂ Glueball Mass Gap: Multi-β CFN-Frame Mechanism on the SU(2) Wilson Lattice

Title: F$_2$ Glueball Mass Gap: Multi-$\beta$ CFN-Frame Mechanism on the SU(2) Wilson Lattice

Author: Alexander Novickis (alex.novickis@gmail.com)

We study pure SU(2) lattice gauge theory through the Cho–Faddeev–Niemi (CFN) decomposition of the link variables, and report a numerical mechanism result on standard SU(2) Wilson configurations. On heatbath-thermalised configurations we measure the equal-time Path-D observables $\langle |X|\rangle$ (off-diagonal-field norm) and $\langle |\hat n|\rangle$ (colour-direction-field norm). Across a ten-point coupling scan (nominal $\beta \in \{4,\dots,20\}$ at $L = 18$, spanning a factor $\sim 4$ in lattice spacing) these are stable to $\sim 10^{-4}$: $\langle |X|\rangle = 0.9487 \pm 0.0001$ and $\langle |\hat n|\rangle = 0.1098 \pm 0.0001$, so ~94.9% of the link content sits in the off-diagonal $X$ sector and ~11.0% in the colour-direction $\hat n$ sector. This X-dominance pattern is a robust frame fingerprint of thermalised SU(2) Wilson configurations, essentially independent of lattice spacing. The result is an existence/mechanism claim — the F$_2$-CFN frame decomposition is numerically realised and stable — not a continuum glueball-mass extraction. The single-slice effective mass of the unsmeared scalar operator is a UV-scale diagnostic (it tracks the lattice cutoff rather than decreasing with $a$); a continuum $m_{0^{++}}$ requires a smeared multi-operator GEVP, the Wilson-loop string tension, and a universal-ratio Symanzik fit, which are not performed here. The programme's analytical glueball mass (Paper CLIX, $m_{0^{++}} = 1.729$ GeV) and external lattice QCD (1.71–1.74 GeV) supply the value; this paper supplies the mechanism.

Key results:

• The F$_2$-CFN frame decomposition is numerically realised on heatbath-thermalised SU(2) Wilson configurations, with the off-diagonal $X$ sector carrying ~94.9% of the link content and the colour-direction $\hat n$ sector ~11.0%.
• The X-dominance fingerprint is stable across the coupling scan ($\beta_{\rm nom} \in \{4,\dots,20\}$, $L = 18$): $\langle |X|\rangle = 0.9487 \pm 0.0001$ and $\langle |\hat n|\rangle = 0.1098 \pm 0.0001$, constant to ~$10^{-4}$ over a factor ~4 in lattice spacing.
• The finite-volume systematic is measured via a companion L-scan at $\beta_{\rm nom}=8$ ($L = 18/20/24$, identical plaquette to five decimals): the X-dominance observables vary by only $\sim 10^{-3}$ across the three volumes, with no monotone drift.
• No continuum glueball-mass value is claimed. The single-slice effective mass of the unsmeared scalar operator is a UV-scale diagnostic; a continuum $m_{0^{++}}$ requires a smeared multi-operator GEVP, the Wilson-loop string tension, and a universal-ratio Symanzik fit against the Teper SU(2) benchmark $m_{0^{++}}/\sqrt{\sigma} = 3.74 \pm 0.12$.
• The value is supplied independently: Paper CLIX §6.4 (analytical, 1.729 GeV) and external lattice QCD (1.71–1.74 GeV). This paper contributes the mechanism, not the value.

Keywords: [physics, QCD, glueball, mass gap, Yang-Mills, SU(2), Wilson lattice, lattice gauge theory, CFN decomposition, Cho-Faddeev-Niemi, flag manifold, color-direction field, monopole, Abelian dominance, heatbath, Kennedy-Pendleton, finite-volume systematic, finite-size scaling, plaquette, string tension, GEVP, Symanzik improvement, Teper benchmark, continuum extrapolation, Hopf soliton, F2, topological soliton]

DOI: 10.5281/zenodo.20519108

Series: Paper CXVIII in the Hopf Soliton Programme