4Spontaneous symmetry breaking
III The Standard Model
4.6 Non-abelian gauge theories
We’ll not actually talk about spontaneous symmetry in non-abelian gauge theories.
That is the story of the next chapter, where we start studying electroweak theory.
Instead, we’ll just briefly set up the general framework of non-abelian gauge
theory.
In general, we have a gauge group
G
, which is a compact Lie group with
Lie algebra
g
. We also have a representation of
G
on
C
n
, which we will assume
is unitary, i.e. each element of
G
is represented by a unitary matrix. Our field
ψ(x) ∈ C
n
takes values in this representation.
A gauge transformation is specified by giving a
g
(
x
)
∈ G
for each point
x
in
the universe, and then the field transforms as
ψ(x) 7→ g(x)ψ(x).
Alternatively, we can represent this transformation infinitesimally, by producing
some t(x) ∈ g, and then the transformation is specified by
ψ(x) 7→ exp(it(x))ψ(x).
Associated to our gauge theory is a gauge field
A
µ
(
x
)
∈ g
(i.e. we have an element
of
g
for each
µ
and
x
), which transforms under an infinitesimal transformation
t(x) ∈ g as
A
µ
(x) 7→ −∂
µ
t(x) + [t, A
µ
].
The gauge covariant derivative is again give by
D
µ
= ∂
µ
+ igA
µ
.
where all fields are, of course, implicitly evaluated at
x
. As before, we can define
F
µν
= ∂
µ
A
ν
− ∂
ν
A
µ
− g[A
µ
, A
ν
] ∈ g,
Alternatively, we can write this as
[D
µ
, D
ν
] = igF
µν
.
We will later on work in coordinates. We can pick a basis of
g
, say
t
1
, ··· , t
n
.
Then we can write an arbitrary element of
g
as
θ
a
(
x
)
t
a
. Then in this basis, we
write the components of the gauge field as
A
a
µ
, and similarly the field strength is
denoted F
a
µν
. We can define the structure constants f
abc
by
[t
a
, t
b
] = f
abc
t
c
.
Using this notation, the gauge part of the Lagrangian L is
L
g
= −
1
4
F
a
µν
F
aµν
= −
1
2
Tr(F
µν
F
µν
).
We now move on to look at some actual theories.