Derivation of the Higgs mass equation?

In summary, the Higgs vacuum expectation value (v_h) for the Standard Model is calculated using the following equation: v_h = \sqrt{\frac{(\hbar c)^3}{\sqrt{2} G_F}}. The Higgs mass (m_H) for the Standard Model is then determined using the Higgs mass equation: m_H = \sqrt{2 \lambda_h} v_h, where \lambda_h is the Higgs self-coupling parameter. This equation can also be derived using integration via substitution. The Higgs mass equation is given by \boxed{m_H = \sqrt{\frac{(\hbar c)^3 \lambda_h}{G_F}}}. The derivation of this equation can be
  • #1
Orion1
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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:
http://hep.uchicago.edu/~pilcher/p463/Old/Lecture08%20Higgs.bw.pdf
Higgs boson - Wikipedia
Fermi coupling constant - Wikipedia
Higgs vacuum expectation value - Wikipedia
 
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  • #2
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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What is the Higgs mass equation and why is it important?

The Higgs mass equation, also known as the Higgs mechanism, is a fundamental equation in particle physics that explains how particles acquire mass. It is a crucial component of the Standard Model, which describes the known particles and their interactions. The Higgs mass equation helps us understand the origin of mass and plays a significant role in our understanding of the universe.

Who developed the Higgs mass equation?

The Higgs mass equation was independently developed by three scientists: Peter Higgs, François Englert, and Robert Brout. Their work was published in 1964 and they were awarded the Nobel Prize in Physics in 2013 for their contributions to the theory of the Higgs mechanism.

How does the Higgs mass equation work?

The Higgs mass equation proposes the existence of a field, called the Higgs field, that permeates all of space. When particles interact with this field, they acquire mass. The more they interact, the more massive they become. The Higgs boson, a particle predicted by the equation, is the carrier of this field.

What evidence supports the Higgs mass equation?

One of the primary pieces of evidence for the Higgs mass equation is the discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012. The LHC is a particle accelerator that allows scientists to study the fundamental building blocks of matter at high energies. The detection of the Higgs boson confirmed the existence of the Higgs field and provided strong support for the theory.

What is the current status of the Higgs mass equation?

The Higgs mass equation is a well-established part of the Standard Model and has been extensively tested and validated through experiments at the LHC and other particle accelerators. However, there are still unanswered questions and ongoing research to further understand the Higgs field and its implications for our understanding of the universe. Scientists continue to explore the Higgs mass equation and its potential extensions to further our knowledge of the fundamental laws of nature.

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