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 parameterizations tuned to high-energy data at $x\ll x_0$, 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 non-perturbative 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 parameterizations of the resulting $N(r;x_0)$ for $x_0=0.01-0.05$. Solving the BK equation in the leading logarithmic (LL) 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 ${\alpha }_s\simeq 0.2$. However, without the option to tune the initial amplitude at $x_0$, the fit of the high precision data results in ${\chi }^2/N_{\text{dof}}=2.3$ at $N_{\text{dof}}=38$, providing clear statistical evidence for the need of systematic improvement e.g. of the photon wave function, evolution equation, and initial condition.

11 pages, 10 figures. Obtained dipole-proton amplitudes at different x are included in the ancillary files, v2 matches published version

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• Global Bayesian Analysis of $\mathrm{J}/\psi$ Photoproduction on Proton and Lead Targets
• Complete Next-to-Leading Order Calculation of Single Inclusive ${\pi }^0$ Production in Forward Proton-Nucleus Collisions
• Determination of the initial condition for the Balitsky-Kovchegov equation with transformers
• Directed flow from parton spin-orbit coupling in $pp$ and $pA$ collisions
• Proton spin from small-$x$ with constraints from the valence quark model