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November 7, 2013 233
There must, therefore, exist an angle θ W such that
Q W = √ 8 g W sin θ W , g WWZ = √ 8 g W cos θ W . (9.60)
In the following we shall use the notation s W = sin θ W and c W = cos θ W . This
angle is called the weak mixing angle, and it parametrizes essentially all of the
minimal model of electroweak interactions we are constructing here. In the first
place, we know that the charge of the W must be equal to the charge of the
electron (since neutrinos are neutral) and therefore we might prefer to write
g W =
Q W
√
8 sW
(9.61)
which leads to a parametrization of the W mass itself 15 :
or
(¯h c m W ) 2 =
π α
√
2 1.16 10
−5
1
s W
2 GeV2 , (9.62)
¯h c m W = 37.3
s W
GeV . (9.63)
As we see, the assumption of the existence of a single, neutral Z boson immediately
implies that the W has a mass of at least 37.3 GeV. Notice that no
prediction for the mass of the Z is obtained, however.
The other unknowns in our treatment can now be expressed in terms of θ W .
Adopting the usual convention of denoting by e the positive unit charge, we find
by straightforward algebra
Q W = −e , g WWZ = −e c W
,
s W
e
a U = −a D = ,
4s W c
(
W
2
v U = a U 1 − 4s Q )
U
W ,
e
(
2
v D = a D 1 + 4s Q )
D
W . (9.64)
e
We note here that θ W is defined at this stage as a relation between coupling
constants ; later on we shall encounter it in another guise !
9.3.3 W, Z and γ four-point interactions
The 2 → 2 processes involving either four fermions or two fermions and two
bosons have led us to postulate W and Z particles and their interactions with
fermions, as well as their mutual three-point vertices. Since as excercise ?? shows
15 To arrive at this experession we have used the definition (9.5) for G F , and the result
(7.28) of α.