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 \(z\tau\)-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. The \(z\tau\)-face of the prism. The two-dimensional \(F_{z\tau}\) flux splits orthogonally into its sourced part, the confining string with a quark and antiquark at its ends, and its closed part, the circulation around the face, which carries no boundary and no charge. The glueball is the closed part.

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, \((d_0^{\mathsf T}b_g)_v=0\), which is exactly the kernel of the edge operator: since \(L_1=d_0 d_0^{\mathsf T}\), \(L_1 b=0\iff d_0^{\mathsf T}b=0\). 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 \(z\tau\)-face can carry a closed colour contour, the circulation \(b_g\).

Sourcedness fixes the trace

The representation factor \(\Theta\) 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, \(\Theta_{b_s}=3/8\). 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, \(\pi\pi\), \(K\bar K\), \(\eta\eta\), and \(\eta\eta'\), so \(\Theta_{b_g}=1/4\). Through the master equation with the spectral weight squared,

Scalar glueball mass
\(m_{0^{++}}=m_P\,\Theta_{b_g}\,\Pi_{b_g}^{2}\,\exp(-1/\xi^{*})=1732~\text{MeV}\)   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.

in agreement with the quenched lattice scalar glueball \(m(0^{++})=1730(50)(80)\) MeV from the anisotropic Wilson action.

The note

The full derivation is set out in a short companion note.

The Glueball (PDF)