Analysis of radial scaling in single-particle inclusive reactions

An analysis of an extensive sample of the world's data has been performed to test the hypothesis of radial scaling. We have studied the inclusive reactions $p+p\ensuremath{\rightarrow}({\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{},0} or {K}^{\ifmmode\pm\else\textpm\fi{}} or p or \overline{p})+\mathrm{anything}$ to determine the behavior of the invariant cross section as a function of ${p}_{\ensuremath{\perp}}$, ${x}_{R}=\frac{{E}^{*}}{{{E}^{*}}_{max}}$, the radial scaling variable, and $s$. The data cover a range in ${p}_{\ensuremath{\perp}}$ from 0.25 to \ensuremath{\sim}6.0 GeV/c and a range in $\ensuremath{\surd}s$ from 3.0 to 63 GeV. For small ${x}_{R}$ and all available ${p}_{\ensuremath{\perp}}$ the single-particle inclusive cross sections for the reactions studied scale to a good approximation for all $\ensuremath{\surd}s$, even down to the kinematic threshold. For large ${x}_{R}$, the single-particle inclusive cross sections for increasing $\ensuremath{\surd}s$ show a rapid approach to the scaling limit from above. In these cases the scaling limit is always approached by $\ensuremath{\surd}s\ensuremath{\approx}10$ GeV. Thus, data for all particles to a good approximation exhibit radial scaling at all available ${p}_{\ensuremath{\perp}}$ and ${x}_{R}$ over the CERN ISR energy range. A comparison of radial scaling with Feynman scaling is given. It is shown that in the Feynman case the cross sections for small ${x}_{\ensuremath{\parallel}}$ (${x}_{\ensuremath{\parallel}}=\frac{{{p}^{*}}_{\ensuremath{\parallel}}}{{{p}^{*}}_{max}}$) approach their scaling limit from below, and that the approach to the scaling limit is slower than is exhibited for the case of small ${x}_{R}$. The systematic differences among the inclusive cross sections of various particles are discussed in the range of $\ensuremath{\surd}s$ where radial scaling has been shown to be valid. In particular, the ${p}_{\ensuremath{\perp}}$ and ${x}_{R}$ distributions of $E\frac{d\ensuremath{\sigma}}{d{p}^{3}}$ are examined.