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# The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries

DOI link for The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries

The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries book

# The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries

DOI link for The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries

The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries book

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## ABSTRACT

There are actually several situations involving one, or more in the case of the supersymmetric version, Higgs boson. These can be presented in a variety of ways and in different gauges. We shall present one of these to make the basic idea as simple as possible. It is the first one studied by Peter Higgs [1] himself. We already know what happens when a gauge theory, such as the Goldstone model [2], is spontaneously broken. Massless scalars or pseudoscalars appear as Goldstone bosons corresponding to the broken generators. Then the mass-less gauge vector or pseudovector bosons absorb the Goldstone bosons and the result is that they develop masses. So now Higgs started by making the Goldstone model locally U(1) invariant in the, by now, familiar way. We note from the start that we begin with four degrees of freedom, one each from the two scalars and two from the massless electromagnetic field potential A^{µ}
(x) corresponding to the transverse waves of electromagnetism or equivalently the independent senses of polarization of the photons. To fully exhibit the particle interpretation of this theory, it is best to change to a “radial and phase” field description. Then we expect it to be simple when the symmetry becomes local, because it is exactly local phase variations that are being considered. In fact, the theory, now with A^{μ}
present, is invariant under local phase changes. Indeed we can get rid of the phase field altogether. We start from
ϕ
=
1
2
[
f
+
ρ
(
x
)
]
exp
[
−
i
θ
(
x
)
f
]
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/1f26c7ae-535e-4efd-af5a-60cf7c5e5cf4/content/math11_1_B.tif"/>
where fλ = μ. Under a local phase (or gauge) transformation
ϕ
→
exp
[
−
i
e
χ
(
x
)
]
ϕ
(
x
)
,
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/1f26c7ae-535e-4efd-af5a-60cf7c5e5cf4/content/math11_2_B.tif"/>
A
μ
→
A
μ
+
∂
μ
χ
(
x
)
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/1f26c7ae-535e-4efd-af5a-60cf7c5e5cf4/content/math11_3_B.tif"/>
and we see that
ρ
=
ρ
′
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/1f26c7ae-535e-4efd-af5a-60cf7c5e5cf4/content/math11_4_B.tif"/>
θ
′
=
θ
+
e
f
χ
.
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429184550/1f26c7ae-535e-4efd-af5a-60cf7c5e5cf4/content/math11_5_B.tif"/>