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

which is introduced for mass generation. To generate the boson masses, the Higgs potential of Equation 2.25
is used, along with µ 2 > 0 and λ > 0. The appropriate vacuum expectation value is the one given in
Equation 2.26. This choice of vacuum expectation value ensures that the vacuum remains invariant under
U(1)em transformations, the photon remains massless, and massive gauge bosons are generated along. From
Equation 2.28 it can be shown that the masses of the bosons are:
MW = 1
2 vg and MZ = 1
2 v g 2 + g ′2 .
From there, it follows that MW
MZ = cos θW, with cos θW = g √
g2g ′2 . θW is the weak mixing angle.
(2.29)
The same Higgs doublet used above is also sufficient to give masses to the leptons and quarks. This is
accomplished by adding an interaction term, which involves Yukawa couplings of the scalars to the fermions.
This interaction term is symmetric under SU(2)L×U(1)Y and is a Lorentz scalar. The Higgs field couples the
left- and right-handed fermions. The mass of the fermions can be expressed in terms of the corresponding
Yukawa coupling, mf = 1
√ 2 λfv, where v is the constant in the vacuum expectation value. In contrast to the
boson masses, the fermion masses depend on a free parameter, the Yukawa coupling, not constrained by the
model. It is clear, and interesting, that given the current experimental values of the quark masses the top
quark Yukawa coupling is much larger than the Yukawa couplings for the rest of the quarks.
Cabibbo Kobayashi Maskawa Mixing Matrix
As mentioned above, the masses and mixing of quarks arise from the Yukawa interactions of the quarks with
the Higgs field. When the Higgs acquires a vacuum expectation value, quark mass terms are generated. An
important feature is that, as a consequence of the Higgs mechanism, the mass eigenstates of the physical
quarks are rotated with respect to the electroweak eigenstates. The mass states are related to the electroweak
states by the CKM matrix. As a result, the weak interactions couple to the physical up and down-type quarks
with couplings given by the CKM matrix, VCKM ≡ V u L V d† L :
⎛
⎜
VCKM = ⎜
⎝
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
12
⎞
⎟
⎠
(2.30)