4Spontaneous symmetry breaking
III The Standard Model
4.1 Discrete symmetry
We begin with a toy example, namely that of a discrete symmetry. Consider a
real scalar field φ(x) with a symmetric potential V (φ), so that
V (−φ) = V (φ).
This gives a discrete Z/2Z symmetry φ ↔ −φ.
We will consider the case of a φ
4
theory, with Lagrangian
L =
1
2
∂
µ
φ∂
µ
φ −
1
2
m
2
φ
2
−
λ
4!
φ
4
for some λ.
We want the potential to
→ ∞
as
φ → ∞
, so we necessarily require
λ >
0.
However, since the
φ
4
term dominates for large
φ
, we are now free to pick the
sign of m
2
, and still get a sensible theory.
Usually, this theory has m
2
> 0, and thus V (φ) has a minimum at φ = 0:
φ
S(φ)
However, we could imagine a scenario where
m
2
<
0, where we have “imaginary
mass”. In this case, the potential looks like
φ
V (φ)
v
To understand this potential better, we complete the square, and write it as
V (φ) =
λ
4
(φ
2
− v
2
)
2
+ constant,
where
v =
r
−
m
2
λ
.
We see that now
φ
= 0 becomes a local maximum, and there are two (global)
minima at
φ
=
±v
. In particular,
φ
has acquired a non-zero vacuum expectation
value (VEV ).
We shall wlog consider small excitations around φ = v. Let’s write
φ(x) = v + cf(x).
Then we can write the Lagrangian as
L =
1
2
∂
µ
f∂
µ
f − λ
v
2
f
2
+ +vf
3
+
1
4
f
4
,
plus some constants. Therefore f is a scalar field with mass
m
2
f
= 2v
2
λ.
This
L
is not invariant under
f → −f
. The symmetry of the original Lagrangian
has been broken by the VEV of φ.