Critical Correlations of Wilson Lines in SU(3) and the High Energy $\gamma^*p$ Cross Section

Abstract

We discuss deep inelastic scattering at high energies as a critical phenomenon in 2+1 space - time dimensions. In the limit of Bjorken $x \to 0$, $QCD$ SU(3) with quark fields becomes a critical theory with a diverging correlation length $\xi(x) \propto x^{-\frac{1}{2 \lambda_2}}$ where the exponent $\lambda_2=2.52$ is obtained from the center group Z(3) of SU(3). We conjecture that the dipole wave function of the virtual photon for transverse sizes $1/Q<x_{\bot}<\xi$ obeys correlation scaling $\Psi \propto (x_{\bot})^{-(1+n)}$ before exponentially decaying for distances larger than the correlation length. Using this behavior combined with different $x$ -independent dipole proton cross sections we calculate the proton structure function and compare with the experimental data. We take the good agreement with the measured proton structure function F$_2(x,Q^2)$ as an indication that at high energies dimensional reduction to an effective three dimensional theory with a critical point occurs.