Geometric QCD III: Exact transition amplitudes and the glueball spectrum

Abstract

We complete the analysis of planar Makeenko--Migdal loop equations in the Lorentz-invariant continuum limit. Using our confining twistor-string representation, we compute the quantum fluctuation determinant, which in Minkowski space reduces to a discrete product of finite-dimensional matrix quadratures. The $ζ$-regularized weight is independent of winding number $w$. Near the mass shell, the pole singularity is generated by $w \to \infty$, suppressing fluctuation variance as $1/w$. The path integral localizes on the classical trajectory, rendering the pole spectrum and transition residues parametrically exact in the large-winding WKB limit. For the open-string meson sector, we fit 40 observed states across five topological boundary sectors ($h=0, \pm 1, \pm 2$). The holonomy shift $h$ accounts for exact geometric degeneracies between parity families, reproducing mass splittings without phenomenological spin-orbit parameters. Evaluated one-loop residues yield theoretical transition cross-sections capturing heavy-mass quenching and phase-space enhancement for high-spin light states. Applying this framework to the pure Yang--Mills closed string, we demonstrate the dynamical stability of the pure-gauge minimal surface: the conformal Liouville anomaly drives the string strictly to the trigonometric minimum ($q=0$). The complex elliptic geometry analytically collapses, yielding linear Regge trajectories. The translation zero-mode measure dynamically nullifies the transition amplitude of the massless scalar ghost, providing an analytic mechanism for a purely gluonic mass gap. Anchoring parameter-free glueball trajectories to the open-string tension natively recovers the exact L"uscher intercept $α(0)=1/12$, perfectly matching established PDG unassigned isoscalar candidates and large-$N_c$ lattice QCD extrapolations.