17
Spontaneously Broken Global Symmetry
Previous chapters have introduced the non-Abelian symmetries SU(2) and
SU(3) in both global and local forms, and we have seen how they may be
applied to describe such typical physical phenomena as particle multiplets,
and massless gauge fields. Remarkably enough, however, these symmetries
are also applied, in the Standard Model, in two cases where the physical
phenomena appear to be very different. Consider the following two questions:
(i) Why are there no signs in the baryonic spectrum, such as parity doublets
in particular, of the global chiral symmetry introduced in section 12.3.2? (ii)
How can weak interactions be described by a local non-Abelian gauge theory
when we know the mediating gauge field quanta are not massless? The answers
to these questions each involve the same fundamental idea, which is a crucial
component of the Standard Model, and perhaps also of theories which go
beyond it. This is the idea that a symmetry can be ‘spontaneously broken’,
or ‘hidden’. By contrast, the symmetries considered hitherto may be termed
‘manifest symmetries’.
The physical consequences of spontaneous symmetry breaking turn out to
be rather different in the global and local cases. However, the essentials for
a theoretical understanding of the phenomenon are contained in the simpler
global case, which we consider in this chapter. The application to spontaneously broken chiral symmetry will be treated in chapter 18, and spontaneously broken local symmetry will be discussed in chapter 19, and applied in
chapter 22.
17.1 Introduction
We begin by considering, in response to question (i) above, what could go
wrong with the argument for symmetry multiplets that we gave in chapter 12.
To understand this, we must use the field theory formulation of section 12.3,
in which the generators of the symmetry are Hermitian field operators, and
the states are created by operators acting on the vacuum. Thus consider two
1
states |A›, |B› :
†
†
ˆ
ˆ
|A› = φ |0›,
|B› = φ |0›
(17.1)
A
B
1 We now revert to the ordinary notation |0> for the vacuum state, rather than |Ω>, but
it must be borne in mind that |0> is the full (interacting) vacuum.
195
Spontaneously Broken Global Symmetry
Previous chapters have introduced the non-Abelian symmetries SU(2) and
SU(3) in both global and local forms, and we have seen how they may be
applied to describe such typical physical phenomena as particle multiplets,
and massless gauge fields. Remarkably enough, however, these symmetries
are also applied, in the Standard Model, in two cases where the physical
phenomena appear to be very different. Consider the following two questions:
(i) Why are there no signs in the baryonic spectrum, such as parity doublets
in particular, of the global chiral symmetry introduced in section 12.3.2? (ii)
How can weak interactions be described by a local non-Abelian gauge theory
when we know the mediating gauge field quanta are not massless? The answers
to these questions each involve the same fundamental idea, which is a crucial
component of the Standard Model, and perhaps also of theories which go
beyond it. This is the idea that a symmetry can be ‘spontaneously broken’,
or ‘hidden’. By contrast, the symmetries considered hitherto may be termed
‘manifest symmetries’.
The physical consequences of spontaneous symmetry breaking turn out to
be rather different in the global and local cases. However, the essentials for
a theoretical understanding of the phenomenon are contained in the simpler
global case, which we consider in this chapter. The application to spontaneously broken chiral symmetry will be treated in chapter 18, and spontaneously broken local symmetry will be discussed in chapter 19, and applied in
chapter 22.
17.1 Introduction
We begin by considering, in response to question (i) above, what could go
wrong with the argument for symmetry multiplets that we gave in chapter 12.
To understand this, we must use the field theory formulation of section 12.3,
in which the generators of the symmetry are Hermitian field operators, and
the states are created by operators acting on the vacuum. Thus consider two
1
states |A›, |B› :
†
†
ˆ
ˆ
|A› = φ |0›,
|B› = φ |0›
(17.1)
A
B
1 We now revert to the ordinary notation |0> for the vacuum state, rather than |Ω>, but
it must be borne in mind that |0> is the full (interacting) vacuum.
195