402
22. The GSW Gauge Theory of Electroweak Interactions
FIGURE 22.23
¯
One-Z
0 and one-γ annihilation contribution to f λ=−1 f λ=1 → W 0
+ W 0
− .
renormalizability. Consider, for example, a fermion–antifermion annihilation
process of the form
¯
W
−
f f → W
+
0 ,
(22.122)
0
where the subscript indicates the λ = 0 (longitudinal) polarization state of the
W
± . We studied such a reaction in section 22.1.1 in the context of unitarity violations (in lowest-order perturbation theory) for the IVB model. Appelquist
and Chanowitz (1987) considered first the case in which ‘f ’ is a lepton with
1
1
t = 2 , t 3 = − 2 coupling to W’s, Z
0 and γ with the usual SU(2) L × U(1)
couplings, but having an explicit (Dirac) mass m f . They found that in the
‘right’ helicity channels for the leptons (λ = +1 for ¯
1 for f ) the
f , λ = −
bad high energy behaviour associated with a fermion-exchange diagram of
the form of figure 22.4 was cancelled by that of the diagrams shown in figure
22.23. The sum of the amplitudes tends to a constant as s (or E
2 ) → ∞.
Such cancellations are a feature of gauge theories, as we indicated at the end
of section 22.1.2, and represent one aspect of the renormalizability of the theory. But suppose, following Appelquist and Chanowitz (1987), we examine
channels involving the ‘wrong’ helicity component, for example λ = +1 for
the fermion f. Then it is found that the cancellation no longer occurs, and we
shall ultimately have a ‘non-renormalizable’ problem on our hands, all over
again.
An estimate of the energy at which this will happen can be made by
recalling that the ‘wrong’ helicity state participates only by virtue of a factor √
(m f /energy) (recall section 20.2.2), which here we can take to be m f / s.
The typical bad high energy behaviour for an amplitude M was M ∼ G F s,
which we expect to be modified here to
√
√
M ∼ G F sm f / s ∼ G F m f s.
(22.123)
The estimate obtained by Appelquist and Chanowitz differs only by a factor of
√
2. Attending to all the factors in the partial wave expansion gives the result
22. The GSW Gauge Theory of Electroweak Interactions
FIGURE 22.23
¯
One-Z
0 and one-γ annihilation contribution to f λ=−1 f λ=1 → W 0
+ W 0
− .
renormalizability. Consider, for example, a fermion–antifermion annihilation
process of the form
¯
W
−
f f → W
+
0 ,
(22.122)
0
where the subscript indicates the λ = 0 (longitudinal) polarization state of the
W
± . We studied such a reaction in section 22.1.1 in the context of unitarity violations (in lowest-order perturbation theory) for the IVB model. Appelquist
and Chanowitz (1987) considered first the case in which ‘f ’ is a lepton with
1
1
t = 2 , t 3 = − 2 coupling to W’s, Z
0 and γ with the usual SU(2) L × U(1)
couplings, but having an explicit (Dirac) mass m f . They found that in the
‘right’ helicity channels for the leptons (λ = +1 for ¯
1 for f ) the
f , λ = −
bad high energy behaviour associated with a fermion-exchange diagram of
the form of figure 22.4 was cancelled by that of the diagrams shown in figure
22.23. The sum of the amplitudes tends to a constant as s (or E
2 ) → ∞.
Such cancellations are a feature of gauge theories, as we indicated at the end
of section 22.1.2, and represent one aspect of the renormalizability of the theory. But suppose, following Appelquist and Chanowitz (1987), we examine
channels involving the ‘wrong’ helicity component, for example λ = +1 for
the fermion f. Then it is found that the cancellation no longer occurs, and we
shall ultimately have a ‘non-renormalizable’ problem on our hands, all over
again.
An estimate of the energy at which this will happen can be made by
recalling that the ‘wrong’ helicity state participates only by virtue of a factor √
(m f /energy) (recall section 20.2.2), which here we can take to be m f / s.
The typical bad high energy behaviour for an amplitude M was M ∼ G F s,
which we expect to be modified here to
√
√
M ∼ G F sm f / s ∼ G F m f s.
(22.123)
The estimate obtained by Appelquist and Chanowitz differs only by a factor of
√
2. Attending to all the factors in the partial wave expansion gives the result