218
17. Spontaneously Broken Global Symmetry
ˆ
(vacuum) condition (17.98). That is, we need to define ‘‹0|φ|0›’ and expand
about it, as in (17.84). In the present case, however, the situation is more
complicated than (17.84) since the complex doublet (17.95) contains four real
fields as indicated in (17.95), and (17.98) becomes
‹0| ˆ
φ
2
ˆ
+ φ
2
ˆ
+ φ
2
ˆ
1
2
3 + φ
2
4 |0› = v
2 .
(17.102)
ˆ
It is evident that we have a lot of freedom in choosing the ‹0|φ i |0› so that
(17.102) holds, and it is not at first obvious what an appropriate generalization
of (17.84) and (17.85) might be.
Furthermore, in this more complicated (non-Abelian) situation a qualitatively new feature can arise: it may happen that the chosen condition
‹ ˆ
0|φ i |0› = 0 is invariant under some subset of the allowed symmetry transformations. This would effectively mean that this particular choice of the
vacuum state respected that subset of symmetries, which would therefore not
be ‘spontaneously broken’ after all. Since each broken symmetry is associated
with a massless Goldstone boson, we would then get fewer of these bosons
than expected. Just this happens (by design) in the present case.
ˆ
Suppose, then, that we could choose the ‹0|φ i |0› so as to break this SU(2)
× U(1) symmetry completely: we would then expect four massless fields.
Actually, however, it is not possible to make such a choice. An analogy may
make this point clearer. Suppose we were considering just SU(2), and the field
ˆ
ˆ
ˆ
‘φ’ was an SU(2)-triplet, φ. Then we could always write ‹0|φ|0› = vn where
n is a unit vector; but this form is invariant under rotations about the n-axis,
irrespective of where that points. In the present case, by using the freedom
of global SU(2) × U(1) phase changes, an arbitrary ‹ ˆ
0|φ|0› can be brought to
the form
(
)
0
‹0| ˆ
φ|0› =
v/
√
.
(17.103)
2
In considering what symmetries are respected or broken by (17.103), it is easiest to look at infinitesimal transformations. It is then clear that the particular
transformation
ˆ
δφ = −
ˆ
i∈(1 + τ 3 )φ
(17.104)
(which is a combination of (17.101) and the ‘third component’ of (17.100)) is
still a symmetry of (17.103) since
(
) (
)
0
(1 + τ 3 ) v/
√
0
=
,
(17.105)
2
0
so that
‹0|φ|0› = ‹0|φ + δφ|0›;
(17.106)
we say that ‘the vacuum is invariant under (17.104)’, and when we look at
the spectrum of oscillations about that vacuum we expect to find only three
massless bosons, not four.
/
17. Spontaneously Broken Global Symmetry
ˆ
(vacuum) condition (17.98). That is, we need to define ‘‹0|φ|0›’ and expand
about it, as in (17.84). In the present case, however, the situation is more
complicated than (17.84) since the complex doublet (17.95) contains four real
fields as indicated in (17.95), and (17.98) becomes
‹0| ˆ
φ
2
ˆ
+ φ
2
ˆ
+ φ
2
ˆ
1
2
3 + φ
2
4 |0› = v
2 .
(17.102)
ˆ
It is evident that we have a lot of freedom in choosing the ‹0|φ i |0› so that
(17.102) holds, and it is not at first obvious what an appropriate generalization
of (17.84) and (17.85) might be.
Furthermore, in this more complicated (non-Abelian) situation a qualitatively new feature can arise: it may happen that the chosen condition
‹ ˆ
0|φ i |0› = 0 is invariant under some subset of the allowed symmetry transformations. This would effectively mean that this particular choice of the
vacuum state respected that subset of symmetries, which would therefore not
be ‘spontaneously broken’ after all. Since each broken symmetry is associated
with a massless Goldstone boson, we would then get fewer of these bosons
than expected. Just this happens (by design) in the present case.
ˆ
Suppose, then, that we could choose the ‹0|φ i |0› so as to break this SU(2)
× U(1) symmetry completely: we would then expect four massless fields.
Actually, however, it is not possible to make such a choice. An analogy may
make this point clearer. Suppose we were considering just SU(2), and the field
ˆ
ˆ
ˆ
‘φ’ was an SU(2)-triplet, φ. Then we could always write ‹0|φ|0› = vn where
n is a unit vector; but this form is invariant under rotations about the n-axis,
irrespective of where that points. In the present case, by using the freedom
of global SU(2) × U(1) phase changes, an arbitrary ‹ ˆ
0|φ|0› can be brought to
the form
(
)
0
‹0| ˆ
φ|0› =
v/
√
.
(17.103)
2
In considering what symmetries are respected or broken by (17.103), it is easiest to look at infinitesimal transformations. It is then clear that the particular
transformation
ˆ
δφ = −
ˆ
i∈(1 + τ 3 )φ
(17.104)
(which is a combination of (17.101) and the ‘third component’ of (17.100)) is
still a symmetry of (17.103) since
(
) (
)
0
(1 + τ 3 ) v/
√
0
=
,
(17.105)
2
0
so that
‹0|φ|0› = ‹0|φ + δφ|0›;
(17.106)
we say that ‘the vacuum is invariant under (17.104)’, and when we look at
the spectrum of oscillations about that vacuum we expect to find only three
massless bosons, not four.
/