Derivation of the Higgs mass equation?

[Orion1]

Higgs vacuum expectation value for the Standard Model: (ref. 1 pg. 14)
$$v_h = \sqrt{\frac{(\hbar c)^3}{\sqrt{2} G_F}}$$

Higgs mass equation for the Standard Model: (ref. 1 pg. 14)
$$m_H = \sqrt{2 \lambda_h} v_h$$
$\lambda_h$ - Higgs self-coupling parameter.

Integration via substitution:
$$m_H = \sqrt{2 \lambda_h} v_h = \sqrt{\frac{(\hbar c)^3 \lambda_h}{G_F}}$$

Higgs mass:
$$\boxed{m_H = \sqrt{\frac{(\hbar c)^3 \lambda_h}{G_F}}}$$

How was the Higgs mass equation listed in reference 1 page 14 derived?

Reference:
http://hep.uchicago.edu/~pilcher/p463/Old/Lecture08%20Higgs.bw.pdf
Higgs boson - Wikipedia
Fermi coupling constant - Wikipedia
Higgs vacuum expectation value - Wikipedia

 

[Bill_K]

The invariant amplitude for a weak interaction, say muon decay, can be expressed two ways: in terms of the weak coupling constant G_F, and in terms of the W-meson:

ℳ = G_F/√2 [uγ^μ(1-γ⁵)u][uγ^μ(1-γ⁵)u]

ℳ = [g/√2 uγ^μ(1-γ⁵)u](1/(M_W² - q²))[g/√2 uγ^μ(1-γ⁵)u]

Comparing these shows that G_F/√2 = g²/M_W².

On the other hand, look at the Higgs Lagrangian:

L = |(∂_μ -g/2τ·W_μ - g'Y/2 B_μ)φ|²

set φ = v, and pull out of this the M_W mass term:

M_W² = (½ vg)²

Equating these two results:

M_W² = (½ vg)² = √2 g²/G_F

The g's cancel, and you get a relationship between v and G_F.