19
Spontaneously Broken Local Symmetry
In earlier parts of this book we have briefly indicated why we might want to
search for a gauge theory of the weak interactions. The reasons include: (i)
the goal of unification (e.g. with the U(1) gauge theory QED), as mentioned in
section 1.3.5; and (ii) certain ‘universality’ phenomena (to be discussed more
fully in chapter 20), which are reminiscent of a similar situation in QED (see
comment (ii) in section 2.6, and also section 11.6), and which are particularly
characteristic of a non-Abelian gauge theory, as pointed out in section 13.1
after equation (13.44). However, we also know from section 1.3.5 that weak
interactions are short-ranged, so that their mediating quanta must be massive.
At first sight, this seems to rule out the possibility of a gauge theory of weak
interactions, since a simple gauge boson mass violates gauge invariance, as we
pointed out for the photon in section 11.3 and for non-Abelian gauge quanta
in section 13.3.1, and will review again in the following section. Nevertheless,
there is a way of giving gauge field quanta a mass, which is by ‘spontaneously
breaking’ the gauge (i.e. local) symmetry. This is the topic of the present
chapter. The detailed application to the electroweak theory will be made in
chapter 21.
19.1 Massive and massless vector particles
Let us begin by noting an elementary (classical) argument for why a gauge
field quantum cannot have mass. The electromagnetic potential satisfies the
Maxwell equation (cf (2.22))
❗A
ν
− ∂
ν (∂ μ A
μ ) = j
ν
(19.1)
em
which, as discussed in section 2.3, is invariant under the gauge transformation
A
μ
→ A
′μ = A
μ
− ∂
μ χ.
(19.2)
However, if A
μ were to represent a massive field, the relevant wave equation
would be
(❗ + M
2 )A
ν
− ∂
ν (∂ μ A
μ ) = j
ν .
(19.3)
em
To get this, we have simply replaced the massless ‘Klein–Gordon’ operator ❗
by the corresponding massive one, ❗ + M
2 (compare sections 3.1 and 5.3).
255
Spontaneously Broken Local Symmetry
In earlier parts of this book we have briefly indicated why we might want to
search for a gauge theory of the weak interactions. The reasons include: (i)
the goal of unification (e.g. with the U(1) gauge theory QED), as mentioned in
section 1.3.5; and (ii) certain ‘universality’ phenomena (to be discussed more
fully in chapter 20), which are reminiscent of a similar situation in QED (see
comment (ii) in section 2.6, and also section 11.6), and which are particularly
characteristic of a non-Abelian gauge theory, as pointed out in section 13.1
after equation (13.44). However, we also know from section 1.3.5 that weak
interactions are short-ranged, so that their mediating quanta must be massive.
At first sight, this seems to rule out the possibility of a gauge theory of weak
interactions, since a simple gauge boson mass violates gauge invariance, as we
pointed out for the photon in section 11.3 and for non-Abelian gauge quanta
in section 13.3.1, and will review again in the following section. Nevertheless,
there is a way of giving gauge field quanta a mass, which is by ‘spontaneously
breaking’ the gauge (i.e. local) symmetry. This is the topic of the present
chapter. The detailed application to the electroweak theory will be made in
chapter 21.
19.1 Massive and massless vector particles
Let us begin by noting an elementary (classical) argument for why a gauge
field quantum cannot have mass. The electromagnetic potential satisfies the
Maxwell equation (cf (2.22))
❗A
ν
− ∂
ν (∂ μ A
μ ) = j
ν
(19.1)
em
which, as discussed in section 2.3, is invariant under the gauge transformation
A
μ
→ A
′μ = A
μ
− ∂
μ χ.
(19.2)
However, if A
μ were to represent a massive field, the relevant wave equation
would be
(❗ + M
2 )A
ν
− ∂
ν (∂ μ A
μ ) = j
ν .
(19.3)
em
To get this, we have simply replaced the massless ‘Klein–Gordon’ operator ❗
by the corresponding massive one, ❗ + M
2 (compare sections 3.1 and 5.3).
255