48
G. Altarelli and S. Forte
masses according to Eqs. (3.53):
L[H, W, Z] = gm W W
+
μ W
−μ H +
g 2
4
W
+
μ W
−μ H
2
+
+
gm Z
2 cos 2 θ W
Z μ Z
μ H +
g 2
8 cos 2 θ W
Z μ Z
μ H
2 .
(3.60)
Thus the trilinear couplings of the Higgs to the gauge bosons are also proportional
to the masses. The quadrilinear couplings are genuinely of order g 2 . Recall that to
go from the lagrangian to the Feynman rules for the vertices the statistical factors
must be taken into account: for example, the Feynman rule for the ZZH H vertex
is ig μν g 2 /2 cos 2 θ W .
The generic coupling of H to a fermion of type f is given by (after diagonalization):
L[H, ¯
ψ, ψ] =
g f
√
2
¯
ψψH,
(3.61)
with
g f
√
2
=
m f
√
2v
= 2
1/4 G
1/2
F m f .
(3.62)
The Higgs self couplings are obtained from the potential in Eq. (3.40) by the
replacement in Eq. (3.58). Given that, from the minimum condition:
v =
μ 2
λ
(3.63)
one obtains:
V = −μ
2 (v +
H
√
2
)
2
+
μ 2
2v 2 (v +
H
√
2
)
4
= −
μ 2 v 2
2
+ μ
2 H
2
+
μ 2
√
2v
H
3
+
μ 2
8v 2 H
4
(3.64)
The constant term can be omitted in our context. We see that the Higgs mass is
positive (compare with Eq. (3.40)) and is given by:
m
2
H = 2μ
2
= 2λv
2
(3.65)
We see that for
√
λ ∼ o(1) the Higgs mass should be of the order of the weak scale.
The difficulty of the Higgs search is due to the fact that it is heavy and coupled
in proportion to mass: it is a heavy particle that must be radiated by another heavy
particle. So a lot of phase space and luminosity is needed. At LEP2 the main process
for Higgs production was the Higgs-strahlung process e + e − → ZH shown in