4Spontaneous symmetry breaking
III The Standard Model
4.5 The Higgs mechanism
Recall that we had a few conditions for our previous theorem to hold. In the case
of gauge theories, these typically don’t hold. For example, in QED, imposing
a Lorentz invariant gauge condition (e.g. the Lorentz gauge) gives us negative
norm states. On the other hand, if we fix the gauge condition so that we don’t
have negative norm states, this breaks Lorentz invariance. What happens in this
case is known as the Higgs mechanism.
Let’s consider the case of scalar electrodynamics. This involves two fields:
– A complex scalar field φ(x) ∈ C.
– A 4-vector field A(x) ∈ R
1,3
.
As usual the components of
A
(
x
) are denoted
A
µ
(
x
). From this we define the
electromagnetic field tensor
F
µν
= ∂
µ
A
ν
− ∂
ν
A
µ
,
and we have a covariant derivative
D
µ
= ∂
µ
+ iqA
µ
.
As usual, the Lagrangian is given by
L = −
1
4
F
µν
F
µν
+ (D
µ
φ)
∗
(D
µ
φ) − V (|φ|
2
)
for some potential V .
A U(1) gauge transformation is specified by some
α
(
x
)
∈ R
. Then the fields
transform as
φ(x) 7→ e
iα(x)
φ(x)
A
µ
(x) 7→ A
µ
(x) −
1
q
∂
µ
α(x).
We will consider a φ
4
theory, so that the potential is
V (|φ|
2
) = µ
2
|φ|
2
+ λ|φ|
4
.
As usual, we require
λ >
0, and if
µ
2
>
0, then this is boring with a unique
vacuum at φ = 0. In this case, A
µ
is massless and φ is massive.
If instead µ
2
< 0, then this is the interesting case. We have a minima at
|φ
0
|
2
= −
µ
2
2λ
≡
v
2
2
.
Without loss of generality, we expand around a real φ
0
, and write
φ(x) =
1
√
2
e
iθ(x)/v
(v + η(x)),
where η, θ are real fields.
Now we notice that
θ
isn’t a “genuine” field (despite being real). Our theory
is invariant under gauge transformations. Thus, by picking
α(x) = −
1
v
θ(x),
we can get rid of the θ(x) term, and be left with
φ(x) =
1
√
2
(v + η(x)).
This is called the unitary gauge. Of course, once we have made this choice, we
no longer have the gauge freedom, but on the other hand everything else going
on becomes much clearer.
In this gauge, the Lagrangian can be written as
L =
1
2
(∂
µ
η∂
µ
η + 2µ
2
η
2
) −
1
4
F
µν
F
µν
+
q
2
v
2
2
A
µ
A
µ
+ L
int
,
where L
int
is the interaction piece that involves more than two fields.
We can now just read off what is going on in here.
– The η field is massive with mass
m
2
µ
= −2µ
2
= 2λv
2
> 0.
– The photon now gains a mass
m
2
A
= q
2
v
2
.
–
What would be the Goldstone boson, namely the
θ
field, has been “eaten”
to become the longitudinal polarization of
A
µ
and
A
µ
has gained a degree
of freedom (or rather, the gauge non-degree-of-freedom became a genuine
degree of freedom).
One can check that the interaction piece becomes
L
int
=
q
2
2
A
µ
A
µ
η
2
+ qm
A
A
µ
A
µ
η −
λ
4
η
4
− m
η
r
λ
2
η
3
.
So we have interactions that look like
This η field is the “Higgs boson” in our toy theory.