Convergent polynomial expansion and scaling in diffraction scattering. III. Conformal mapping without spurious cut and scaling for elastic diffraction scattering processes

In this and a following paper we generalize the method of approach by the convergent polynomial expansion (CPE) for both elastic diffractive and inelastic nondiffractive hadron-hadron collision processes at high energies. The presence of spurious cuts in some of the conformal mappings used earlier is pointed out. A conformal mapping of the unsymmetrically cut costheta plane, which does not develop any spurious cut or require any knowledge of zeros, is combined with that of the s plane to construct a variable chi(s,t) which has the potentialities to reproduce some known scaling variables and Regge behavior and to provide information about asymptotic behavior of slope parameters of the type approx. (lns)/sup n/, with n=0,1,2. Away from the diffraction peak the variable becomes b(s)(lnt)2. Because of the absence of spurious cuts in the mapped plane, the variable has the potentialities to provide information on the possible existence of entire functions for the differential-cross-section ratio f(s,t) at asymptotic energies. However, the rate of convergence and the nature of the polynomials in the proposed CPE are not uniquely fixed at finite energies. Only at asymptotic energies the polynomials are uniquely the Laguerre polynomials and the CPE goes over to the optimized polynomial expansion. The possible existence of a scaling function for the differential-cross-section ratio at asymptotic energies as a series in Laguerre polynomials in the variable chi is pointed out. The first term in the expansion in the CPE gives a good description of the energy dependence of the forward slopes for different processes without needing any effective shape of spectral function. From the asymptotic behaviors of slope parameters obtained from data analysis we find that qualitatively the forward slopes for π+-p and K+-p scattering grow at the some rate, like approx. lns, as s → infinity