# 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:
The Standard Model Higgs - University of Chicago
Higgs boson - Wikipedia
Fermi coupling constant - Wikipedia
Higgs vacuum expectation value - Wikipedia

## Answers and Replies

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

ℳ = GF/√2 [uγμ(1-γ5)u][uγμ(1-γ5)u]

ℳ = [g/√2 uγμ(1-γ5)u](1/(MW2 - q2))[g/√2 uγμ(1-γ5)u]

Comparing these shows that GF/√2 = g2/MW2.

On the other hand, look at the Higgs Lagrangian:

L = |(∂μ -g/2τ·Wμ - g'Y/2 Bμ)φ|2

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

MW2 = (½ vg)2

Equating these two results:

MW2 = (½ vg)2 = √2 g2/GF

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

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