Convergent polynomial expansion and scaling in inelastic charge-exchange scattering processes

Abstract

A method of approach developed in a previous paper for elastic diffraction scattering process at high energies using Mandelstam analyticity and convergent polynomial expansion (CPE) is applied in this paper for several inelastic charge-exchange processes. A conformal mapping $Z$ of the unsymmetrically cut $cos\ensuremath{\theta}$ plane, which does not develop any spurious cut in the mapped plane or require any knowledge of zeros, is combined with that of the $s$ plane to construct a variable $\ensuremath{\chi}(s,t)$ which has the potentialities to reproduce some known scaling variables and Regge behavior and to provide information on the asymptotic behavior of the slope parameter of the type ${(\mathrm{ln}s)}^{m}$, with $m=0, 1, 2, \dots{}$. At high energies and away from the peak region the variable becomes $\ensuremath{\sim}b(s){(\mathrm{ln}t)}^{2}$. Because of the absence of any spurious cut in the mapped plane, it is possible to obtain information on the existence of entire function for the differential-cross-section ratio $f(s,t)$ at asymptotic energies. However, at finite energies, neither the rate of convergence nor the nature of the polynomials in the CPE in $Z$ or $\ensuremath{\chi}$ is uniquely fixed. At asymptotic energies the polynomials are uniquely the Laguerre polynomials and the CPE goes over to the optimized polynomial expansion (OPE). The approach from the CPE to the OPE is faster in the $\ensuremath{\chi}$ plane for physical values of $s$ if $m>0$. The possible existence of the scaling function at asymptotic energies as a series in Laguerre polynomials in $\ensuremath{\chi}$ is pointed out. The first term in the CPE gives a good fit to the high-energy data on the slope parameter for each of the six processes ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}n$, ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}\ensuremath{\eta}n$, ${K}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\overline{K}}^{0}n$, ${K}^{+}n\ensuremath{\rightarrow}{K}^{0}p$, ${K}^{+}p\ensuremath{\rightarrow}{K}^{0}{\ensuremath{\Delta}}^{++}$, and ${K}^{\ensuremath{-}}n\ensuremath{\rightarrow}{\overline{K}}^{0}{\ensuremath{\Delta}}^{\ensuremath{-}}$. The asymptotic behavior of the slope parameter for every process is $\ensuremath{\sim}\mathrm{ln}s$. This suggests that $f(s,t)$, for every process, is an entire function of $\ensuremath{\chi}$, as $s\ensuremath{\rightarrow}\ensuremath{\infty}$. For every process all the available data on $f(s,t)$ at high energies exhibit scaling in a remarkable manner. Our analysis shows definite evidence of early onset of scaling in the energy scale for several processes.