Convergent polynomial expansion and scaling in diffraction scattering I.ppscattering

Using Mandelstam analyticity of the $s$ and $cos\theta$ planes and conformal mapping, a variable $\chi$ is constructed which has the potentialities of reproducing Regge behavior and/or some known scaling variables. The role of the physical region in the mapped plane for the optimized polynomial expansion (OPE) is emphasized. Ambiguities in using the OPE in terms of Laguerre polynomials at finite energies are pointed out. However, at finite energies there exists a convergent polynomial expansion (CPE) for which the nature of polynomials and the rate of convergence vary with energy. The first term in the expansion gives a good fit to the world data on forward slopes for $\mathrm{pp}$ scattering for all energies with effective shapes of spectral function, but yields a good fit to the high-energy data for $s>\mathrm{}35\mathrm{}$ ${\mathrm{GeV}}^2$ with the theoretical boundaries. The possible existence of a scaling function at asymptotic energies as a series in Laguerre polynomials in the new variable $\chi$ is pointed out. Available high-energy data on the $\mathrm{pp}$ cross-section ratio for $p_{\mathrm{lab}}\ge 50$ GeV/c and all angles exhibit scaling in this variable. It is found that at high energies scaling occurs even for larger-$\mathrm{}\left|t\right|\mathrm{}$ data lying well outside the diffraction peak. The implication of this type of scaling in the data analysis at high energies using OPE is pointed out. The energy dependence of dip position at high energies is predicted to be $\left|t_d\mathrm{}\right(s\left)\right|=4\mathrm{}{m_{\pi }}^2{\left\{sinh{\left[\frac{(4.35\mathrm{}\pm 0.05)}{4{m_{\pi }}^{2}b\left(s\right)\mathrm{}}\right]}^{\frac{1}{2}}\right\}}^2$, which is in very good agreement with the existing data.