373
22.1. Difficulties with the current–current and ‘naive’ IVB models
where E is the CM energy and θ the CM scattering angle. We see that the
(unsquared) amplitude must behave essentially as g
2 E
2 /M
2 , the quantity
W
g
2 /M
2 effectively replacing G F of the current–current model. The unitarity
W
bound is violated for E ≥ M W /g ∼ 300 GeV, taking g ∼ e.
Other unitarity-violating processes can easily be invented, and we have to
conclude that the IVB model is, in this respect, no more fitted to be called a
theory than was the four-fermion model. In the case of the latter, we argued
that the root of the disease lay in the fact that G F was not dimensionless, yet
somehow this was not a good enough cure after all: perhaps (it is indeed so)
‘dimensionlessness’ is necessary but not sufficient (see the following section).
Why is this? Returning to M λ1 ,λ2 for νν ¯ → W
+ W
− (equation (22.16)) and
setting ∈ = k μ /M for the longitudinal polarization vectors, we see that we are
involved with an effective amplitude
2
g
/ p 1 − / k 1
v ¯(p 2 ) / k 2 (1 − γ 5 )
/ k 1 (1 − γ 5 )u(p 1 ).
(22.21)
M 2
(p 1 − k 1 ) 2
W
2
Using the Dirac equation / p 1 u(p 1 ) = 0 and p = 0, this can be reduced to
1
2
g
−
v ¯(p 2 ) / k 2 (1 − γ 5 )u(p 1 ).
(22.22)
M 2
W
We see that the longitudinal ∈’s have brought in the factors M
−2 , which are
W
‘compensated’ by the factor / k 2 , and it is this latter factor which causes the rise
with energy. The longitudinal polarization states have effectively reintroduced
a dimensional coupling constant g/M W .
What happens in QED? We learnt in section 7.3 that, for real photons,
the longitudinal state of polarization is absent altogether. We might well
suspect, therefore, that since it was the longitudinal W’s that caused the ‘bad’
high-energy behaviour of the IVB model, the ‘good’ high-energy behaviour of
QED might have its origin in the absence of such states for photons. And
this circumstance can, in its turn, be traced (cf section 7.3.1 ) to the gauge
invariance property of QED.
Indeed, in section 8.6.3 we saw that in the analogue of (22.17) for photons
(this time involving only the two transverse polarization states), the righthand side could be taken to be just -g μν , provided that the Ward identity
(8.166) held, a condition directly following from gauge invariance.
We have arrived here at an important theoretical indication that what we
really need is a gauge theory of the weak interactions, in which the W’s are
gauge quanta. It must, however, be a peculiar kind of gauge theory, since
normally gauge invariance requires the gauge field quanta to be massless.
However, we have already seen how this ‘peculiarity’ can indeed arise, if the
local symmetry is spontaneously broken (chapter 19). But before proceeding
to implement that idea, in the GSW theory, we discuss one further disease
(related to the unitarity one) possessed by both current–current and IVB
models – that of non-renormalizability.
22.1. Difficulties with the current–current and ‘naive’ IVB models
where E is the CM energy and θ the CM scattering angle. We see that the
(unsquared) amplitude must behave essentially as g
2 E
2 /M
2 , the quantity
W
g
2 /M
2 effectively replacing G F of the current–current model. The unitarity
W
bound is violated for E ≥ M W /g ∼ 300 GeV, taking g ∼ e.
Other unitarity-violating processes can easily be invented, and we have to
conclude that the IVB model is, in this respect, no more fitted to be called a
theory than was the four-fermion model. In the case of the latter, we argued
that the root of the disease lay in the fact that G F was not dimensionless, yet
somehow this was not a good enough cure after all: perhaps (it is indeed so)
‘dimensionlessness’ is necessary but not sufficient (see the following section).
Why is this? Returning to M λ1 ,λ2 for νν ¯ → W
+ W
− (equation (22.16)) and
setting ∈ = k μ /M for the longitudinal polarization vectors, we see that we are
involved with an effective amplitude
2
g
/ p 1 − / k 1
v ¯(p 2 ) / k 2 (1 − γ 5 )
/ k 1 (1 − γ 5 )u(p 1 ).
(22.21)
M 2
(p 1 − k 1 ) 2
W
2
Using the Dirac equation / p 1 u(p 1 ) = 0 and p = 0, this can be reduced to
1
2
g
−
v ¯(p 2 ) / k 2 (1 − γ 5 )u(p 1 ).
(22.22)
M 2
W
We see that the longitudinal ∈’s have brought in the factors M
−2 , which are
W
‘compensated’ by the factor / k 2 , and it is this latter factor which causes the rise
with energy. The longitudinal polarization states have effectively reintroduced
a dimensional coupling constant g/M W .
What happens in QED? We learnt in section 7.3 that, for real photons,
the longitudinal state of polarization is absent altogether. We might well
suspect, therefore, that since it was the longitudinal W’s that caused the ‘bad’
high-energy behaviour of the IVB model, the ‘good’ high-energy behaviour of
QED might have its origin in the absence of such states for photons. And
this circumstance can, in its turn, be traced (cf section 7.3.1 ) to the gauge
invariance property of QED.
Indeed, in section 8.6.3 we saw that in the analogue of (22.17) for photons
(this time involving only the two transverse polarization states), the righthand side could be taken to be just -g μν , provided that the Ward identity
(8.166) held, a condition directly following from gauge invariance.
We have arrived here at an important theoretical indication that what we
really need is a gauge theory of the weak interactions, in which the W’s are
gauge quanta. It must, however, be a peculiar kind of gauge theory, since
normally gauge invariance requires the gauge field quanta to be massless.
However, we have already seen how this ‘peculiarity’ can indeed arise, if the
local symmetry is spontaneously broken (chapter 19). But before proceeding
to implement that idea, in the GSW theory, we discuss one further disease
(related to the unitarity one) possessed by both current–current and IVB
models – that of non-renormalizability.