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Derivation of the Higgs mass equation?

  1. Sep 23, 2012 #1

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

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

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

    Higgs mass:
    [tex]\boxed{m_H = \sqrt{\frac{(\hbar c)^3 \lambda_h}{G_F}}}[/tex]

    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
     
  2. jcsd
  3. Sep 23, 2012 #2

    Bill_K

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    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.
     
    Last edited: Sep 23, 2012
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