Harmonic Entanglement Theory
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Confinement

The glueball

The glueball is the sourceless counterpart of the confining string, the closed part of the same colour flux on the same zτ-face. A contour with no ends carries no external colour charge; read through the master equation with the singlet representation factor, it gives the scalar glueball mass.

The z-tau face of the prism, with the colour flux split into its sourced (string) and closed (glueball) parts.
Figure 4. The two orthogonal parts of the colour flux on the zτ-face at (x,y)=(0,0). In (a) the sourced part, the confining string, whose boundary is −2,+2,+2,−2 at the four face vertices, a quark and an antiquark standing at the ends of the flux. In (b) the closed part, the glueball contour, whose boundary vanishes at every vertex, so that it carries no external colour charge. The two exhaust the Fzτ parity class.

Closed flux on the same face

Where the string is the sourced part of the colour flux, the glueball is the closed part on the same face. Closure is the vanishing of the net flow at every vertex, \((d_{0}^{T}b_{g})_{v}=0,\) which is exactly the kernel of the edge operator: since \(L_{1}=d_{0}\) d0T, L1 b=0iff \(d_{0}^{T}b=0\). The edge space splits into the seventeen-dimensional kernel of closed contours and the sourced eigenspaces \(\lambda=2,4,6,8\). On the prism only the zτ-face can carry a closed colour contour, the circulation bg.

Sourcedness fixes the trace

The representation factor Θ is a ratio of state-space dimensions, and confinement decides which channels are open. The confining string carries fundamental colour, so it propagates through the octet of eight gluons, Θbs=3/8. The glueball is a colour singlet, and a singlet does not propagate through gluons, which confinement removes from the physical spectrum; its channels are hadronic. For the scalar there are four, the two-pseudoscalar channels of the light nonet, ππ, Kbar K, ηη, and ηη', so Θbg=1/4. Through the master equation with the spectral weight squared,

Scalar glueball mass
m0++=mP Θbg Πbg2 exp(−1/ξ*)=1732.34 MeV   lattice 1730(50)(80) MeV

The octet is internal to the construction, the colour assigned to the z-edges of the gauge sectors; the four hadronic channels are those of the light quark nonet and are not derived here.