APPENDIX C: RENORMALIZATION GROUP (RG) FLOW

This appendix details the mathematical mechanism that bridges the 67-order-of-magnitude gap between the Planck-scale stiffness ($T_P$) and the observed QCD string tension ($\kappa_{QCD}$).

C.1 The Running Coupling Ansatz

We define the dimensionless stiffness coupling $\alpha(L)$ at length scale $L$ as:

$$\alpha(L) = \kappa(L) \cdot L^2$$

The evolution of this coupling is governed by the Beta function:

$$\beta(\alpha) = \frac{d\alpha}{d\ln L} = \alpha^2$$

C.2 Physical Justification for $\beta = \alpha^2$

In a granular force network, stress transmission is not continuous. It relies on the probability of grains forming a stable chain.

• Reconnection Probability: For a chain to extend or reconnect, it requires the simultaneous alignment of two contact points.

• Scaling: The probability of two independent events scales as the square of the single-event probability ($P_{joint} \sim P^2$).

• Result: The effective stiffness "runs" quadratically with scale, leading to the $\beta = \alpha^2$ form.

C.3 The Solution

Integrating the Beta function yields the running coupling equation:

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

C.4 Calculation Check

• Input (Planck Scale): $\alpha(L_P) \approx 0.0215$ (derived from vacuum yield stress).

• Scale Factor: $\ln(1 \text{ fm} / L_P) \approx \ln(10^{20}) \approx 46.0$.

• Denominator: $1 - (0.0215 \times 46.0) = 1 - 0.989 = 0.011$.

• Result (QCD Scale): $\alpha(1 \text{ fm}) \approx 0.0215 / 0.011 \approx 1.95$.

• String Tension: $\kappa = \alpha / L^2 \approx 0.2 \text{ GeV}^2$.

This derivation confirms that the "weak" force of QCD confinement is actually the "strong" Planck tension diluted by the probability of maintaining force chains over $10^{20}$ grain diameters.