5Electroweak theory
III The Standard Model
5.1 Electroweak gauge theory
We start by understanding the gauge and Higgs part of the theory. As mentioned,
the gauge group is
G = SU(2)
L
× U(1)
Y
.
We will see that this is broken by the Higgs mechanism.
It is convenient to pick a basis for su(2), which we will denote by
τ
a
=
σ
a
2
.
Note that these are not genuinely elements of
su
(2). We need to multiply these
by
i
to actually get members of
su
(2) (thus, they form a complex basis for the
complexification of
su
(2)). As we will later see, these act on fields as
e
iτ
a
instead
of e
τ
a
. Under this basis, the structure constants are f
abc
= ε
abc
.
We start by describing how our gauge group acts on the Higgs field.
Definition
(Higgs field)
.
The Higgs field
φ
is a complex scalar field with
two components,
φ
(
x
)
∈ C
2
. The
SU
(2) action is given by the fundamental
representation on C
2
, and the hypercharge is Y =
1
2
.
Explicitly, an (infinitesimal) gauge transformation can be represented by
elements
α
a
(
x
)
, β
(
x
)
∈ R
, corresponding to the elements
α
a
(
x
)
τ
a
∈ su
(2) and
β(x) ∈ u(1)
∼
=
R. Then the Higgs field transform as
φ(x) 7→ e
iα
a
(x)τ
a
e
i
1
2
β(x)
φ(x),
where the
1
2
factor of β(x) comes from the hypercharge being
1
2
.
Note that when we say
φ
is a scalar field, we mean it transforms trivially via
Lorentz transformations. It still takes values in the vector space C
2
.
The gauge fields corresponding to
SU
(2) and U(1) are denoted
W
a
µ
and
B
µ
,
where again
a
runs through
a
= 1
,
2
,
3. The covariant derivative associated to
these gauge fields is
D
µ
= ∂
µ
+ igW
a
µ
τ
a
+
1
2
ig
0
B
µ
for some coupling constants g and g
0
.
The part of the Lagrangian relating the gauge and Higgs field is
L
gauge,φ
= −
1
2
Tr(F
W
µν
F
W,µν
) −
1
4
F
B
µν
F
B,µν
+ (D
µ
φ)
†
(D
µ
φ) − µ
2
|φ|
2
− λ|φ|
4
,
where the field strengths of W and B respectively are given by
F
W,a
µν
= ∂
µ
W
a
ν
− ∂
ν
W
a
µ
− gε
abc
W
b
µ
W
c
ν
F
B
µν
= ∂
µ
B
ν
− ∂
ν
V
µ
.
In the case
µ
2
<
0, the Higgs field acquires a VEV, and we wlog shall choose
the vacuum to be
φ
0
=
1
√
2
0
v
,
where
µ
2
= −λv
2
< 0.
As in the case of U(1) symmetry breaking, the gauge term gives us new things
with mass. After spending hours working out the computations, we find that
(D
µ
φ)
†
(D
µ
φ) contains mass terms
1
2
v
2
4
g
2
(W
1
)
2
+ g
2
(W
2
)
2
+ (−gW
3
+ g
0
B)
2
.
This suggests we define the following new fields:
W
±
µ
=
1
√
2
(W
1
µ
∓ iW
2
µ
)
Z
0
µ
A
µ
=
cos θ
W
−sin θ
W
sin θ
W
cos θ
W
W
3
µ
B
µ
,
where we pick θ
W
such that
cos θ
W
=
g
p
g
2
+ g
02
, sin θ
W
=
g
0
p
g
2
+ g
02
.
This θ
W
is called the Weinberg angle. Then the mass term above becomes
1
2
v
2
g
2
4
((W
+
)
2
+ (W
−
)
2
) +
v
2
(g
2
+ g
02
)
4
Z
2
.
Thus, our particles have masses
m
W
=
vg
2
m
Z
=
v
2
p
g
2
+ g
02
m
γ
= 0,
where
γ
is the
A
µ
particle, which is the photon. Thus, our original
SU
(2)
×
U(1)
Y
breaks down to a U(1)
EM
symmetry. In terms of the Weinberg angle,
m
W
= m
Z
cos θ
W
.
Thus, through the Higgs mechanism, we find that the
W
±
and
Z
bosons gained
mass, but the photon does not. This agrees with what we find experimentally.
In practice, we find that the masses are
m
W
≈ 80 GeV
m
Z
≈ 91 GeV
m
γ
< 10
−18
GeV.
Also, the Higgs boson gets mass, as what we saw previously. It is given by
m
H
=
p
−2µ
2
=
√
2λv
2
.
Note that the Higgs mass depends on the constant
λ
, which we haven’t seen
anywhere else so far. So we can’t tell
m
H
from what we know about
W
and
Z
.
Thus, until the Higgs was discovered recently, we didn’t know about the mass of
the Higgs boson. We now know that
m
H
≈ 125 GeV.
Note that we didn’t write out all terms in the Lagrangian, but as we did before,
we are going to get W
±
, Z-Higgs and Higgs-Higgs interactions.
One might find it a bit pointless to define the
W
+
and
W
−
fields as we did,
as the kinetic terms looked just as good with
W
1
and
W
2
. The advantage of
this new definition is that
W
+
µ
is now the complex conjugate
W
−
µ
, so we can
instead view the
W
bosons as given by a single complex scalar field, and when
we quantize, W
+
µ
will be the anti-particle of W
−
µ
.