3 The Standard Model of Electroweak Interactions
47
It is also evident that for Higgs doublets
ρ 0 =
m 2
W
m 2
Z cos 2 θ W
= 1 .
(3.55)
This relation is typical of one or more Higgs doublets and would be spoiled by the
existence of Higgs triplets etc. In general,
ρ 0 =
i ((t i ) 2 − (t
3
i ) 2 + t i )v 2
i
i 2(t 3
i ) 2 v 2
i
(3.56)
for several Higgs bosons with VEVs v i , weak isospin t i , and z-component t 3
i .
These results are valid at the tree level and are modified by calculable EW radiative
corrections, as discussed in Sect. (3.7).
The measured values of the W and Z masses are [5, 8] (see Chap. 6):
m W = 80.398 ± 0.025 GeV,
m Z = 91.1875 ± 0.0021 GeV.
(3.57)
In the minimal version of the SM only one Higgs doublet is present. Then
the fermion–Higgs couplings are in proportion to the fermion masses. In fact,
from the Yukawa couplings g φ ¯
f f ( ¯
f L φf R + h.c.), the mass m f is obtained by
replacing φ by v, so that m f = g φ ¯
f f v. In the minimal SM three out of the four
Hermitian fields are removed from the physical spectrum by the Higgs mechanism
and become the longitudinal modes of W + , W − , and Z. The fourth neutral
Higgs is physical and should be found. If more doublets are present, two more
charged and two more neutral Higgs scalars should be around for each additional
doublet.
The couplings of the physical Higgs H can be simply obtained from L Higgs , by
the replacement (the remaining three hermitian fields correspond to the would be
Goldstone bosons that become the longitudinal modes of W ± and Z):
φ(x) =
φ + (x)
φ 0 (x)
→
0
v + (H /
√
2)
,
(3.58)
[so that (D μ φ) † (D μ φ) = 1/2(∂ μ H ) 2 + . . .], with the results
L[H, W, Z] = g
2 v
√
2
W
+
μ W
−μ H +
g 2
4
W
+
μ W
−μ H
2
+
+ g
2
v
2
√
2 cos 2 θ W
Z μ Z
μ H +
g 2
8 cos 2 θ W
Z μ Z
μ H
2 . (3.59)
Note that the trilinear couplings are nominally of order g 2 , but the adimensional
coupling constant is actually of order g if we express the couplings in terms of the
47
It is also evident that for Higgs doublets
ρ 0 =
m 2
W
m 2
Z cos 2 θ W
= 1 .
(3.55)
This relation is typical of one or more Higgs doublets and would be spoiled by the
existence of Higgs triplets etc. In general,
ρ 0 =
i ((t i ) 2 − (t
3
i ) 2 + t i )v 2
i
i 2(t 3
i ) 2 v 2
i
(3.56)
for several Higgs bosons with VEVs v i , weak isospin t i , and z-component t 3
i .
These results are valid at the tree level and are modified by calculable EW radiative
corrections, as discussed in Sect. (3.7).
The measured values of the W and Z masses are [5, 8] (see Chap. 6):
m W = 80.398 ± 0.025 GeV,
m Z = 91.1875 ± 0.0021 GeV.
(3.57)
In the minimal version of the SM only one Higgs doublet is present. Then
the fermion–Higgs couplings are in proportion to the fermion masses. In fact,
from the Yukawa couplings g φ ¯
f f ( ¯
f L φf R + h.c.), the mass m f is obtained by
replacing φ by v, so that m f = g φ ¯
f f v. In the minimal SM three out of the four
Hermitian fields are removed from the physical spectrum by the Higgs mechanism
and become the longitudinal modes of W + , W − , and Z. The fourth neutral
Higgs is physical and should be found. If more doublets are present, two more
charged and two more neutral Higgs scalars should be around for each additional
doublet.
The couplings of the physical Higgs H can be simply obtained from L Higgs , by
the replacement (the remaining three hermitian fields correspond to the would be
Goldstone bosons that become the longitudinal modes of W ± and Z):
φ(x) =
φ + (x)
φ 0 (x)
→
0
v + (H /
√
2)
,
(3.58)
[so that (D μ φ) † (D μ φ) = 1/2(∂ μ H ) 2 + . . .], with the results
L[H, W, Z] = g
2 v
√
2
W
+
μ W
−μ H +
g 2
4
W
+
μ W
−μ H
2
+
+ g
2
v
2
√
2 cos 2 θ W
Z μ Z
μ H +
g 2
8 cos 2 θ W
Z μ Z
μ H
2 . (3.59)
Note that the trilinear couplings are nominally of order g 2 , but the adimensional
coupling constant is actually of order g if we express the couplings in terms of the