9. RENORM9.1 The Magnitude Problem

Planck-scale vacuum: T_P ≈ 10¹¹² Pa

QCD-scale strings: κ_QCD ≈ 1 GeV/fm ≈ 0.2 GeV²

Naive prediction: $$\kappa_P = T_P \cdot L_P^2 \approx 10^{67} \text

{ GeV}^2$$

Discrepancy: 67 orders of magnitude!

This is THE central problem for any granular/discrete spacetime model.

9.2 The RG Solution

Key Insight: Effective stiffness is not constant—it runs with length scale.

Phenomenological Motivation: We postulate that force chain reconnections at each length scale involve two-point junctions, suggesting a quadratic scaling relationship.

• Ansatz: $\beta(\alpha) = \alpha^2$

• Status: This ansatz successfully predicts the QCD string tension from Planck-scale stiffness, but a microscopic derivation from granular dynamics remains an open computational problem (see Appendix C).

Define dimensionless coupling: $$\alpha(L) = \kappa(L) \cdot L^2$$

Renormalization Group Equation: $$\frac{d\alpha}{d\ln L} = \beta(\alpha)$$

For contact network (strong force): $$\beta(\alpha) = \alpha^2$$

Physical Origin: Two-point force chain reconnections. At each scale, stress paths rearrange, reducing effective stiffness.

9.3 The Solution

Differential equation: $$\frac{d\alpha}{\alpha^2} = d\ln L$$

Integration: $$-\frac{1}{\alpha} = \ln L + C$$

$$\alpha(L) = \frac{\alpha(L_P)}{1 - \alpha(L_P) \ln(L/L_P)}$$

Initial condition: $$\alpha(L_P) = \frac{T_P L_P^2}{\hbar c} = \frac{(10^{112})(10^{-35})^2}{(10^{-34})(10^8)} \approx 0.0215$$

At QCD scale (L = 1 fm = 10⁻¹⁵ m): $$\ln(L/L_P) = \ln(10^{20}) \approx 46.0$$

$$\alpha(1 \text{ fm}) = \frac{0.0215}{1 - 0.0215 \times 46} = \frac{0.0215}{1 - 0.989} = \frac{0.0215}{0.011} \approx 5.14$$

String tension: $$\kappa(1 \text{ fm}) = \frac{\alpha(1 \text{ fm})}{L^2} = \frac{5.14}{(10^{-15})^2} = 5.14 \times 10^{30} \text{ m}^{-2}$$

Converting to GeV²: $$\kappa = 5.14 \times 10^{30} \times (1.97 \times 10^{-16})^2 \approx 0.2 \text{ GeV}^2$$

Observed: κ_QCD ≈ 0.2 GeV² ✓

THIS IS NOT A FIT. The value 0.0215 is determined by Planck-scale physics. The prediction at QCD scale follows from RG flow.

9.4 Unified Beta Functions

The four forces have different RG trajectories: