Appendix

Calculation of the Electromagnetic Contribution to

A.1. The Absorptive Part (Y)

This may be obtained from the unitarity relation of fig. 9 [39]. We assume . For , this is completely justified, but for , may need to be considered (it has been calculated [4] as ). It may be even more significant in and decays. For , we obtain

Numerically, this becomes

We note that a large direct (weak) decay rate would contribute to the nominally pure electromagnetic Y, but we do not expect this to be significant.

A.2. The Dispersive Part (X)

The simplest assumption possible in this calculation is that P is pointlike. This is not satisfactory, however, since for large . We must thus find some way to cut off this integration; we give structure to the meson. Measurements of the pionform factor [41] indicate that it conforms well to vector-meson dominance predictions. Although this has no direct physical connection with the process under discussion, it suggests that vector-meson dominance should be used there.

[ Figure 9 ] The unitarity relation for the imaginary part of the amplitude for .

[ Figure 10 ] Diagram for the process , which allows the form factor of P to be probed.

A number of estimates have been made for . We give some for :

We have used model 3, in which one photon is saturated with , the other with . Model 4 saturates only one photon with a meson. Model 1 takes P to be a uniform sphere of charge in momentum space; it is not at all clear that its radius should be . Other models for include taking as a virtual state [40] (obtaining ), and a very general one [44] whose results vary by an order of magnitude around those of model 3.

The cut-off factor in can, in principle, be deduced from measurements on [45], which proceeds by the diagram of fig. 10. , overtaking for GeV. However, is difficult experimentally, because the final state particles come out with small . Nevertheless, should be known from experiment within a few years, allowing as accurate value for to be obtained.

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