An Effective Model for the Constructive Realization of the Gluon and the Mass Gap via β-Vacuum

We present a non-perturbative construction of gluon-like states from an infrared effective model valid for momentum scales Q ≤ m_gap ≈ 1.5 GeV. The β-Vacuum is defined by the Lagrangian Lβ = β/4 (∂μΦᵃ ∂ᵐΦᵃ)² + α/2 Φᵃ □² Φᵃ + m₀²/2 Φᵃ Φᵃ with a = 1,...,8. We prove the strict monotonicity of the associated Euler-Lagrange operator in W¹˙⁴ ∩ H² and deduce existence and uniqueness of weak solutions. From the spectral analysis of the linearized operator we establish a mass gap m_gap = (m₀²/α)¹ᐟ⁴ > 0. After canonical quantization with the Gupta-Bleuler condition, the projection Aμᵃ = κ ∂μ Φᵃ yields 8 massive vector fields of spin 1, transforming in the adjoint representation of SU(3) and possessing 3-point and 4-point vertices with coefficients given by the structure constants fᵃᵇᶜ. The effective coupling g_eff is fixed by matching to perturbative QCD at the scale m_gap. These states are identified with effective gluons. Their color-singlet bound states are interpreted as glueballs. The model predicts m_gap ≈ 1.5 GeV and a linear static potential V(r) = σ r with σ ≈ 0.18 GeV². An experimental procedure for the detection of the gap is proposed. This model is not derived from Yang-Mills but is constructed to reproduce its IR properties: mass gap, adjoint representation, and confinement. It should be viewed as an effective Lagrangian in the spirit of chiral perturbation theory.