369
22.1. Difficulties with the current–current and ‘naive’ IVB models
where f J is the partial wave amplitude for angular momentum J and k is the
CM momentum. It is a consequence of unitarity, or flux conservation (see, for
example, Merzbacher 1998, chapter 13), that the partial wave amplitude may
be written in terms of a phase shift δ J :
f J = e
iδJ sin δ J
(22.6)
so that
|f J | ≤ 1.
(22.7)
Thus the cross section in each partial wave is bounded by
σ J ≤ 4π(2J + 1)/k
2
(22.8)
which falls as the CM energy rises. By contrast, in (22.3) we have a cross
section that rises with CM energy:
σ ∼ E
2 .
(22.9)
Moreover, since the amplitude (equation (22.2)) only involves (cos θ)
0 and
(cos θ)
1 contributions, it is clear that this rise in σ is associated with only a
few partial waves, and is not due to more and more partial waves contributing
to the sum in σ. Therefore, at some energy E, the unitarity bound will be
violated by this lowest-order (Born approximation) expression for σ.
This is the essence of the ‘unitarity disease’ of the current–current model.
To fill in all the details, however, involves a careful treatment of the appropriate partial wave analysis for the case when all particles carry spin. We shall
avoid those details. Instead we argue, again on dimensional grounds, that the
dimensionless partial wave amplitude f J (note the 1/k
2 factor in (22.5)) must
be proportional to G F E
2 , which violates the bound (22.7) for CM energies
−1/2
E ≥ G
∼ 300GeV.
(22.10)
F
At this point the reader may recall a very similar-sounding argument made
in section 11.8, which led to the same estimate of the ‘dangerous’ energy scale
(22.10). In that case, the discussion referred to a hypothetical ‘4-fermi’ interaction without the V–A structure, and it was concerned with renormalization
rather than unitarity. The gamma-matrix structure is irrelevant to these issues, which ultimately have to do with the dimensionality of the coupling
constant, in both cases. In fact, as we shall see, unitarity and renormalizability are actually rather closely related.
Faced with this unitarity difficulty, we appeal to the most successful theory
we have, and ask: what happens in QED? We consider an apparently quite
+
similar process, namely e e
−
→ μ
+ μ
− in lowest order (figure 22.2). In chapter
8 the total cross section for this process, neglecting lepton masses, was found
to be (see problem 8.18 and equation (9.87))
σ = 4πα
2 /3E
2
(22.11)
22.1. Difficulties with the current–current and ‘naive’ IVB models
where f J is the partial wave amplitude for angular momentum J and k is the
CM momentum. It is a consequence of unitarity, or flux conservation (see, for
example, Merzbacher 1998, chapter 13), that the partial wave amplitude may
be written in terms of a phase shift δ J :
f J = e
iδJ sin δ J
(22.6)
so that
|f J | ≤ 1.
(22.7)
Thus the cross section in each partial wave is bounded by
σ J ≤ 4π(2J + 1)/k
2
(22.8)
which falls as the CM energy rises. By contrast, in (22.3) we have a cross
section that rises with CM energy:
σ ∼ E
2 .
(22.9)
Moreover, since the amplitude (equation (22.2)) only involves (cos θ)
0 and
(cos θ)
1 contributions, it is clear that this rise in σ is associated with only a
few partial waves, and is not due to more and more partial waves contributing
to the sum in σ. Therefore, at some energy E, the unitarity bound will be
violated by this lowest-order (Born approximation) expression for σ.
This is the essence of the ‘unitarity disease’ of the current–current model.
To fill in all the details, however, involves a careful treatment of the appropriate partial wave analysis for the case when all particles carry spin. We shall
avoid those details. Instead we argue, again on dimensional grounds, that the
dimensionless partial wave amplitude f J (note the 1/k
2 factor in (22.5)) must
be proportional to G F E
2 , which violates the bound (22.7) for CM energies
−1/2
E ≥ G
∼ 300GeV.
(22.10)
F
At this point the reader may recall a very similar-sounding argument made
in section 11.8, which led to the same estimate of the ‘dangerous’ energy scale
(22.10). In that case, the discussion referred to a hypothetical ‘4-fermi’ interaction without the V–A structure, and it was concerned with renormalization
rather than unitarity. The gamma-matrix structure is irrelevant to these issues, which ultimately have to do with the dimensionality of the coupling
constant, in both cases. In fact, as we shall see, unitarity and renormalizability are actually rather closely related.
Faced with this unitarity difficulty, we appeal to the most successful theory
we have, and ask: what happens in QED? We consider an apparently quite
+
similar process, namely e e
−
→ μ
+ μ
− in lowest order (figure 22.2). In chapter
8 the total cross section for this process, neglecting lepton masses, was found
to be (see problem 8.18 and equation (9.87))
σ = 4πα
2 /3E
2
(22.11)