Using simple considerations of Mandelstam analyticity of the $s$ plane along with that of the $cos\theta$ plane by conformal mapping and the convergent polynomial expansion (CPE), a variable $\chi \mathrm{}(s,t)\mathrm{}$ is constructed which has the potentialities of reproducing some known scaling variables, Regge behavior, and providing information about asymptotic behavior of slope parameters in diffraction scattering of the type $\sim {\left(\ln s\right)}^p$, where $p=0,1,2\mathrm{}$. CPE in terms of Laguerre polynomials in the proposed variable is possible for all energies, but maximum convergence of the series is possible only at asymptotic energies. Use of the first one or two terms in the CPE in $\chi$ provides improved fits to the forward slope data at all energies and reasonably good fits to the high-energy slope-parameter data at $\mathrm{}\left|t\right|=0.2\mathrm{}$ ${\mathrm{GeV}}^2$, for $\mathrm{pp}$, $\overline{p}p$, $K^{\pm }p$ and ${\pi }^{\pm }p$ scattering. Information on the asymptotic behavior of slope parameters for these processes is obtained. Data analysis reveals that the strong behaviors in the amplitudes for all these processes cannot be present inside the corresponding figures of convergence in the mapped planes for all energies except near the threshold. A fit to the slope-parameter data for any process determines unknown parameters in the corresponding $\chi$. When the cross-section-ratio data for different processes are plotted against the corresponding $\chi$'s all the data for every process starting from $P_{\mathrm{lab}}=3\mathrm{}$ GeV/c up to the highest available energy and within the range $\mathrm{}\left|t\right|\mathrm{}\le {\mathrm{}\left|t\right|\mathrm{}}_{max}$, where ${\mathrm{}\left|t\right|\mathrm{}}_{max}=1.25,0.5,2.0,1.0,\mathrm{and} 1.25$ ${\mathrm{GeV}}^2$ for $\mathrm{pp}$, $\overline{p}p$, $K^{+}p$, $K^{-}p$, and ${\pi }^{\pm }p$ scattering, respectively, lie on a scaling curve. For $p_{\mathrm{lab}}>3\mathrm{}$ GeV/c data points for $\mathrm{}\left|t\right|>{\mathrm{}\left|t\right|\mathrm{}}_{max}$ also approach the scaling curve, and the data for all available values of $\mathrm{}\left|t\right|\mathrm{}$ with $P_{\mathrm{lab}}\ge 50$ GeV/c lie on it. For $\mathrm{pp}$ scattering it is found that at high energies scaling in the variable $\chi$ occurs for larger-$\mathrm{}\left|t\right|\mathrm{}$ data lying well outside the diffraction-peak region. The scaling curves for $\mathrm{pp}$ and $\overline{p}p$ scattering are found to be different and those for ${\pi }^{+}p$ and ${\pi }^{-}p$ scattering are found to be very much the same. The scaling curves for $K^{+}p$ and $K^{-}p$ scattering are almost the same. Our analysis implies that the data lying on the scaling curves can be represented by CPE in terms of Laguerre polynomials in the variable $\chi$, with the coefficients of expansion independent of energy. The implication of such type of scaling in the data analysis at high energies using CPE is pointed out.

• Received 29 November 1977

DOI:https://doi.org/10.1103/PhysRevD.19.164