46
G. Altarelli and S. Forte
contribute to gauge-boson masses. The condition that the photon remains massless
is equivalent to the condition that the vacuum is electrically neutral:
Q|v = (t
3
+
1
2
Y )|v = 0 .
(3.47)
We now explicitlly consider the case of a single Higgs doublet:
φ =
φ +
φ 0
, v =
0
v
,
(3.48)
The charged W mass is given by the quadratic terms in the W field arising from
L Higgs , when φ(x) is replaced by v in Eq. (3.41). By recalling Eq. (3.10), we obtain
m
2
W W
+
μ W
−μ
= g
2
|(t
+ v/
√
2)|
2 W
+
μ W
−μ ,
(3.49)
whilst for the Z mass we get [recalling Eqs. (3.12–3.14)]
1
2
m
2
Z Z μ Z
μ
= |[g cos θ W t
3
− g
sin θ W (Y/2)]v|
2 Z μ Z
μ ,
(3.50)
where the factor of 1/2 on the left-hand side is the correct normalization for the
definition of the mass of a neutral field. By using Eq. (3.47), relating the action of t 3
and Y/2 on the vacuum v, and Eqs. (3.16), we obtain
1
2
m
2
Z = (g cos θ W + g
sin θ W )
2
|t
3 v|
2
= (g
2 / cos
2 θ W )|t
3 v|
2 .
(3.51)
For a Higgs doublet, as in Eq. (3.48), we have
|t
+ v|
2
= v
2 , |t
3 v|
2
= 1/4v
2 ,
(3.52)
so that
m
2
W = 1/2g
2 v
2 , m
2
Z = 1/2g
2 v
2 / cos
2 θ W .
(3.53)
Note that by using Eq. (3.19) we obtain
v = 2
−3/4 G
−1/2
F
= 174.1 GeV .
(3.54)