378
22. The GSW Gauge Theory of Electroweak Interactions
group of local phase transformations, i.e. the relevant weak gauge group? Several possibilities were suggested, but it is now very well established that the
one originally proposed by Glashow (1961), subsequently treated as a spontaneously broken gauge symmetry by Weinberg (1967) and by Salam (1968),
and later extended by other authors, produces a theory which is in remarkable
agreement with currently known data. We shall not give a critical review of
all the experimental evidence, but instead proceed directly to an outline of
the GSW theory, introducing elements of the data at illustrative points.
An important clue to the symmetry group involved in the weak interactions is provided by considering the transitions induced by these interactions.
This is somewhat analogous to discovering the multiplet structure of atomic
levels and hence the representations of the rotation group, a prominent symmetry of the Schr¨ odinger equation, by studying electromagnetic transitions.
However, there is one very important difference between the ‘weak multiplets’
we shall be considering, and those associated with symmetries which are not
spontaneously broken. We saw in chapter 12 how an unbroken non-Abelian
symmetry leads to multiplets of states which are degenerate in mass, but in
section 17.1 we learned that that result only holds provided the vacuum is
left invariant under the symmetry transformation. When the symmetry is
spontaneously broken, the vacuum is not invariant, and we must expect that
the degenerate multiplet structure will then, in general, disappear completely.
This is precisely the situation in the electroweak theory.
Nevertheless, as we shall see, essential consequences of the weak symmetry group – specifically, the relations it requires between otherwise unrelated
masses and couplings – are accessible to experiment. Moreover, despite the
fact that members of a multiplet of a global symmetry which is spontaneously
broken will, in general, no longer have even approximately the same mass,
the concept of a multiplet is still useful. This is because when the symmetry
is made a local one, we shall find (in sections 22.2.2 and 22.2.3) that the associated gauge quanta still mediate interactions between members of a given
symmetry multiplet, just as in the manifest local non-Abelian symmetry example of QCD. Now, the leptonic transitions associated with the weak charged
currents are, as we saw in chapter 20, ν e ↔ e, ν μ ↔ μ etc. This suggests that
these pairs should be regarded as doublets under some group. Further we
saw in section 20.7 how weak transitions involving charged quarks suggested
a similar doublet structure for them also. The simplest possibility is therefore to suppose that, in both cases, a ‘weak SU(2) group’ is involved, called
‘weak isospin’. We emphasize once more that this weak isospin is distinct
from the hadronic isospin of chapter 12, which is part of SU(3) f . We use the
symbols t, t 3 for the quantum numbers of weak isospin, and make the specific
assignments for the leptonic fields
(
(
)
(
)
(
)
1
t 3 = +1/2
ˆ
ν e
′
ν ˆ
′
t = ,
,
μ
ν ˆ
′
,
τ
2
t 3 = −1/2
e ˆ
−
μ ˆ
−
τ ˆ
L
L
−
L
(22.24)
22. The GSW Gauge Theory of Electroweak Interactions
group of local phase transformations, i.e. the relevant weak gauge group? Several possibilities were suggested, but it is now very well established that the
one originally proposed by Glashow (1961), subsequently treated as a spontaneously broken gauge symmetry by Weinberg (1967) and by Salam (1968),
and later extended by other authors, produces a theory which is in remarkable
agreement with currently known data. We shall not give a critical review of
all the experimental evidence, but instead proceed directly to an outline of
the GSW theory, introducing elements of the data at illustrative points.
An important clue to the symmetry group involved in the weak interactions is provided by considering the transitions induced by these interactions.
This is somewhat analogous to discovering the multiplet structure of atomic
levels and hence the representations of the rotation group, a prominent symmetry of the Schr¨ odinger equation, by studying electromagnetic transitions.
However, there is one very important difference between the ‘weak multiplets’
we shall be considering, and those associated with symmetries which are not
spontaneously broken. We saw in chapter 12 how an unbroken non-Abelian
symmetry leads to multiplets of states which are degenerate in mass, but in
section 17.1 we learned that that result only holds provided the vacuum is
left invariant under the symmetry transformation. When the symmetry is
spontaneously broken, the vacuum is not invariant, and we must expect that
the degenerate multiplet structure will then, in general, disappear completely.
This is precisely the situation in the electroweak theory.
Nevertheless, as we shall see, essential consequences of the weak symmetry group – specifically, the relations it requires between otherwise unrelated
masses and couplings – are accessible to experiment. Moreover, despite the
fact that members of a multiplet of a global symmetry which is spontaneously
broken will, in general, no longer have even approximately the same mass,
the concept of a multiplet is still useful. This is because when the symmetry
is made a local one, we shall find (in sections 22.2.2 and 22.2.3) that the associated gauge quanta still mediate interactions between members of a given
symmetry multiplet, just as in the manifest local non-Abelian symmetry example of QCD. Now, the leptonic transitions associated with the weak charged
currents are, as we saw in chapter 20, ν e ↔ e, ν μ ↔ μ etc. This suggests that
these pairs should be regarded as doublets under some group. Further we
saw in section 20.7 how weak transitions involving charged quarks suggested
a similar doublet structure for them also. The simplest possibility is therefore to suppose that, in both cases, a ‘weak SU(2) group’ is involved, called
‘weak isospin’. We emphasize once more that this weak isospin is distinct
from the hadronic isospin of chapter 12, which is part of SU(3) f . We use the
symbols t, t 3 for the quantum numbers of weak isospin, and make the specific
assignments for the leptonic fields
(
(
)
(
)
(
)
1
t 3 = +1/2
ˆ
ν e
′
ν ˆ
′
t = ,
,
μ
ν ˆ
′
,
τ
2
t 3 = −1/2
e ˆ
−
μ ˆ
−
τ ˆ
L
L
−
L
(22.24)