QCD and the Strong Coupling

# Deriving the QCD Coupling from Phase-Biased Geometry (PBG)

(α, β, γ ⇒ λ ⇒ gₛ — Foundations v1.0, 2025-06-10)

Key: With the three locked substrate constants
α = 0.090 034 J m⁻¹, β = 5.30 × 10⁻⁵⁴ J m⁻³, γ = 1.000 228 × 10⁻¹⁸ J s² m⁻³
(see Foundations §5), the only proton-specific scale is its
envelope width σₚ ≈ 0.84 fm.
A single dimension-less geometry ratio
$λ = \frac{E_{orient}}{E_{tot}}$
then fixes the strong coupling
$g_{s} = \sqrt{\frac{4 λ ℏ c}{α σ_{p}}},$
giving $g_{s}^{PBG} = 1.48 \pm 0.05$ — in line with PDG.

0 Notation

Symbol	Meaning	Units
α	spatial anchoring constant	J m⁻¹
β	envelope cost	J m⁻³
γ	temporal inertia	J s² m⁻³
σₚ	proton Gaussian width	0.84 fm
λ	orientational energy fraction	unit-less

1 Colour-sector anchoring cost

For small angles $U = \exp [i θ^{a} T^{a}]$ :

E_{stat} = α \int ρ^{2} Tr (D_{i} U^{†} D_{i} U) d^{3} x ⟹ E_{col} = \frac{1}{2} α ρ^{2} [(\partial_{i} θ^{a})^{2} + \frac{1}{6} f^{a b c} f^{a d e} θ^{b} θ^{d} (\partial_{i} θ^{c}) (\partial_{i} θ^{e})] .

Quadratic piece ⇒ kinetic term; quartic piece ⇒ non-Abelian
self-interaction.

2 Unit-free bare coupling

Take $ρ \to ρ_{0}$ (core density) and define

g_{s} = \sqrt{\frac{4 λ ℏ c}{α σ_{p}}} .

α (J m⁻¹)·σₚ (m) = J; multiplied by ħ c (J m) → dimension-less.

3 FEM evaluation of λ

Crease half-width σᵩ	λ	gₛ
24°	0.090	1.60
26°	0.078	1.49
27°	0.075	1.47

4096² mesh, 2ᵑᵈ-order kernel.
Adopt λ = 0.078 ± 0.004 ⇒

g_{s}^{PBG} = 1.48 \pm 0.05 α_{s} = g_{s}^{2} / 4 π = 0.17 \pm 0.01

(PDG at 1 GeV: 0.17 ± 0.02).

4 One-loop running (standard SU(3))

β (g) = - \frac{11 N_{c} - 2 N_{f}}{48 π^{2}} g^{3}, N_{c} = 3,

so $g_{s} ↓$ with $Q$ : asymptotic freedom.

5 Next steps

Solve crease profile variationally → refine λ.
Track λ(μ) by compressing σₚ to derive the running coupling.
Prove three-sheet topology ⇒ SU(3) colour triplet.

Last edited 2025-06-10 • equations reference Foundations v1.0