We study a “toy” model which describes the bound state of two scalar particles. We parameterize the “soft” part of the wave function of the bound state using results obtained in the study of a nontopological soliton model. (This part of the wave function has predominantly lowmomentum components.) We improve this wave function by considering a single “hard” scattering, which we treat in perturbation theory. Thus, we have a model in which the wave function has both “soft” and “hard” parts, in the language used when performing perturbative QCD studies. (The relation of these results to those obtained for electromagnetic form factors in perturbative QCD studies requires further analysis.) We find that the form factor calculated using the “soft” part of the wave function behaves as F(Q2)=F(0)/(1+λ2Q2), where Q2=−q2>0. The individual hard scattering terms yield form factors which have the same form, except for a change of the scale factor, λ. We discuss the approach to the asymptotic form, F(Q2)~Q−2, for the various amplitudes which are summed to obtain the complete form factor. We also discuss the modification of the form factor under a change in mass scale. This modification is relatively simple to study since, in our model, there is only a single dimensionful parameter, κ, which sets the scale for all other dimensional parameters.
Hard scattering amplitude for the higher helicity components of the pion form factor