CERN-THESIS-2012-153 26/07/2012 - CERN Document Server

particles. Because of the introduction of h, this is known as the “Higgs mechanism”.
Figure 2.1: The potential V (φ) for a complex scalar field φ = (φ1 + iφ2)/ √ 2 for µ 2 < 0 and λ > 0. The
circle of minima has a radius v.
The Higgs mechanism described above corresponds to the spontaneous symmetry breaking of a U(1)
gauge symmetry. For the case of weak boson masses, this must be done in SU(2) gauge symmetry. Thus, a
similar derivation should be carried out in the Lagrangian:
where φ is an SU(2) doublet of complex scalar fields:
L = (∂µφ) † (∂ µ φ) − µ 2 φ † φ − λ(φ † φ) 2
φ =
⎛
⎜
⎝ φα
φβ
⎞
⎟
⎠ =
1
2
In this case, ∂µ is replaced by the covariant derivative:
⎛
⎜
⎝ φ1 + iφ2
φ3 + iφ4
⎞
(2.21)
⎟
⎠ . (2.22)
Dµ = ∂µ + ig τa
2 W a µ, (2.23)
and introducing three gauge fields W a µ(x) with a = 1,2,3. With the appropriate gauge field transformation,
the gauge invariant Lagrangian is:
L = ∂µφ + ig 1
†
τ · Wµφ
2
∂ µ + ig 1
2 τ · W µ φ
10
− V (φ) − 1 µν
WµνW
4
(2.24)