Comparison of Finitely and Infinitely Composite Models for High-Energy Proton-Proton Scattering

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We have extended the Glauber theory of high-energy scattering to the collision of two clusters with $N=2, 3, 4, \mathrm{and} \mathrm{infinitely}$ many constitutents each. For the special case of Gaussian scattering amplitudes and form factors, a convenient matrix formula is derived which reduces the evaluation of the multiple Gaussian integrals to the diagonalization of a symmetric matrix which can be read off from each graph and allows complete summation (by computer) of the multiple-scattering series. The convergence of the Glauber series is investigated and found to be excellent for most of the parameter values of interest. These results have been derived with the intention of comparing the hadronic differential cross sections in different quark-type models. We find that all the models discussed lead to excellent fits to the experimental proton-proton differential cross section. While the $N=3$ quark model or the $N=\ensuremath{\infty}$ droplet model may be preferred on other grounds, we find that the limited freedom one has in deciding to what degree the shape of the scattering amplitude is due to the form factors and to what degree to the elementary scattering amplitudes makes it impossible to differentiate between the two models on the basis of the structure of the differential cross section alone.