G_F

Fermi constant as effective coupling

$G_F$ controls weak-interaction strength at low energy. Its relation to fundamental parameters:

$$ G_F = \frac{1}{\sqrt{2}\, v^2}, $$

makes $G_F$ a consequence of the Higgs VEV $v$.

Self-consistency of v in PT

The VEV is itself fixed self-consistently in the PT cascade (chapter 10):

$$ v = m_e / \alpha_{\rm therm}^{\,2 - 3\alpha_{\rm EM}}, $$

linking three fundamental scales: $m_e$ (measured input), $\alpha_{\rm therm}$ (thermal coupling on q_-), and $\alpha_{\rm EM}$ (electroweak feedback).

Numerically: $v = 246.2196$ GeV (PT) vs $246.22$ GeV (PDG, empirically derived from $G_F$).

Computing G_F

$$ G_F = \frac{1}{\sqrt{2} \cdot (246.2196)^2} = 1.16638 \times 10^{-5}\ \text{GeV}^{-2}. $$

PT: 1.16638 × 10⁻⁵ vs PDG: 1.16638 × 10⁻⁵. Gap: 0.0001% — self-consistent at PDG precision.

Status

Classed [SC] (self-consistency): the consistency $v$ ↔ $\alpha_{\rm therm}$ ↔ $G_F$ is forced by PT structure, not by a fit.