Scaling Laws at Large Transverse Momentum

The application of simple dimensional counting to bound states of pointlike particles enables us to derive scaling laws for the asymptotic energy dependence of electromagnetic and hadronic scattering at fixed c.m. angle which only depend on the number of constituent fields of the hadrons. Assuming quark constituents, some of the $s\ensuremath{\rightarrow}\ensuremath{\infty}$, fixed-$\frac{t}{s}$ predictions are ${(\frac{d\ensuremath{\sigma}}{\mathrm{dt}})}_{\ensuremath{\pi}p\ensuremath{\rightarrow}\ensuremath{\pi}p}\ensuremath{\sim}{s}^{\ensuremath{-}8}$, ${(\frac{d\ensuremath{\sigma}}{\mathrm{dt}})}_{\mathrm{pp}\ensuremath{\rightarrow}\mathrm{pp}}\ensuremath{\sim}{s}^{\ensuremath{-}10}$, ${(\frac{d\ensuremath{\sigma}}{\mathrm{dt}})}_{\ensuremath{\gamma}p\ensuremath{\rightarrow}\ensuremath{\pi}p}\ensuremath{\sim}{s}^{\ensuremath{-}7}$, ${(\frac{d\ensuremath{\sigma}}{\mathrm{dt}})}_{\ensuremath{\gamma}p\ensuremath{\rightarrow}\ensuremath{\gamma}p}\ensuremath{\sim}{s}^{\ensuremath{-}6}$, ${F}_{\ensuremath{\pi}}({q}^{2})\ensuremath{\sim}{({q}^{2})}^{\ensuremath{-}1}$, and ${F}_{1p}({q}^{2})\ensuremath{\sim}{({q}^{2})}^{\ensuremath{-}2}$. We show that such scaling laws are characteristic of renormalizable field theories satisfying certain conditions.