High-energy dipole scattering amplitude from evolution of low-energy proton light-cone wave functions

Abstract

The forward scattering amplitude of a small dipole at high energies is given in the mean field approximation by the Balitsky-Kovchegov (BK) evolution equation. It requires an initial condition $N(r;{x}_{0})$ describing the scattering of a dipole with size $r$ off the target that is probed at momentum fraction ${x}_{0}$. Rather than using ad hoc parametrizations tuned to high-energy data at $x\ensuremath{\ll}{x}_{0}$, here we attempt to construct an initial scattering amplitude that is consistent with low-energy, large-$x$ properties of the proton. We start from a nonperturbative three quark light-cone model wave function from the literature. We add $\mathcal{O}(g)$ corrections due to the emission of a gluon, and $\mathcal{O}({g}^{2})$ virtual corrections due to the exchange of a gluon, computed in light-cone perturbation theory with exact kinematics. We provide numerical data as well as analytic parametrizations of the resulting $N(r;{x}_{0})$ for ${x}_{0}=0.01--0.05$. Solving the BK equation in the leading logarithmic approximation towards lower $x$, we obtain a fair description of the charm cross section in deeply inelastic scattering measured at HERA by fitting one parameter, the coupling constant ${\ensuremath{\alpha}}_{s}\ensuremath{\simeq}0.2$. However, without the option to tune the initial amplitude at ${x}_{0}$, the fit of the high precision data results in ${\ensuremath{\chi}}^{2}/{N}_{\mathrm{dof}}=2.3$ at ${N}_{\mathrm{dof}}=38$, providing clear statistical evidence for the need of systematic improvement, e.g., of the photon wave function, evolution equation, and initial condition.

Topics & Concepts