Universal fit top−pelastic diffraction scattering from the Lorentz-contracted geometrical model

We examine the prediction of the Lorentz-contracted geometrical model for proton-proton elastic scattering at small angles. The model assumes that when two high-energy particles collide, each behaves as a geometrical object which has a Gaussian density and is spherically symmetric except for the Lorentz contraction in the incident direction. It predicted that $\frac{d\sigma }{\mathrm{dt}}$ should be independent of energy when plotted against the variable $\frac{{\beta }^2{P_{\perp }}^2{\sigma }_{\mathrm{tot}}\mathrm{}\left(s\right)\mathrm{}}{\left(38.3\mathrm{} \mathrm{mb}\right)}$. Thus the energy dependence of the diffraction-peak slope ($b$ in an $e^{-b\left|t\right|\mathrm{}}$ plot) is given by $b\left(s\right)=\frac{A^2{\beta }^2{\sigma }_{\mathrm{tot}}\mathrm{}\left(s\right)\mathrm{}}{\left(38.3\mathrm{} \mathrm{mb}\right)}$, where $\beta$ is the proton's c.m. velocity and $A$ is its radius. We used recently measured values of ${\sigma }_{\mathrm{tot}}\mathrm{}\left(s\right)\mathrm{}$, and obtained an excellent fit to the elastic slope in both $t$ regions [$-t<\mathrm{}0.1\mathrm{}$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$ and $0.1<-t<\mathrm{}0.4\mathrm{}$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$] at all energies from $s=6\mathrm{} \mathrm{to} 4000$ ${\mathrm{}\left(\mathrm{GeV}\right)\mathrm{}}^2$.