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 $p+p\to ({\pi }^{\pm ,0\mathrm{}} or K^{\pm } or p or \overline{p})+\mathrm{anything}$ to determine the behavior of the invariant cross section as a function of $p_{\perp }$, $x_R=\frac{E^{*}}{{E^{*}}_{max}}$, the radial scaling variable, and $s$. The data cover a range in $p_{\perp }$ from 0.25 to ∼6.0 GeV/c and a range in $\surd s$ from 3.0 to 63 GeV. For small $x_R$ and all available $p_{\perp }$ the single-particle inclusive cross sections for the reactions studied scale to a good approximation for all $\surd s$, even down to the kinematic threshold. For large $x_R$, the single-particle inclusive cross sections for increasing $\surd s$ show a rapid approach to the scaling limit from above. In these cases the scaling limit is always approached by $\surd s\approx 10$ GeV. Thus, data for all particles to a good approximation exhibit radial scaling at all available $p_{\perp }$ 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 $x_{\parallel }$ ($x_{\parallel }=\frac{{p^{*}}_{\parallel }}{{p^{*}}_{max}}$) 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 $\surd s$ where radial scaling has been shown to be valid. In particular, the $p_{\perp }$ and $x_R$ distributions of $E\frac{d\sigma }{d{p}^3}$ are examined.