Scaling laws for large-momentum-transfer processes

Abstract

Dimensional scaling laws are developed as an approach to understanding the energy dependence of high-energy scattering processes at fixed center-of-mass angle. Given a reasonable assumption on the short-distance behavior of bound states, and the absence of an internal mass scale, we show that at large $s$ and $t$, $\frac{d\sigma }{\mathrm{dt}}(AB\to CD)\sim s^{-n+2\mathrm{}}f\left(\frac{t}{s}\right)$; $n$ is the total number of fields in $A$, $B$, $C$, and $D$ which carry a finite fraction of the momentum. A similar scaling law is obtained for large-$p_{\perp }$ inclusive scattering. When the quark model is used to specify $n$, we find good agreement with experiments. For instance, this accounts naturally for the ${\left(q^2\right)}^{-2\mathrm{}}$ asymptotic behavior of the proton form factor. We examine in detail the field-theoretic foundations of the scaling laws and the assumption which needs to be made about the short-distance and infrared behavior of a bound state.