Parton distribution function for quarks in an s-channel approach

TL;DR

This paper develops an $s$-channel formulation of the quark distribution at small $x$, showing that $x f_{q/p}(x,\mu)$ can be written as a dipole cross section weighted by a perturbative $1/\Delta^4$ lightcone wave function. A central result is the relation $x f_{q/p}(x,\mu) = \frac{N_c}{3\pi^4} \int d^2{\bm b} \int d^2{\bm \Delta} \Theta(\Delta^2\mu^2 > a^2) \frac{\Xi({\bm b},{\bm \Delta})}{\Delta^4}$, with UV renormalization handled in ${\overline{MS}}$ and a clear connection to the gluon distribution via RG matching. The dipole amplitude $\Xi$ is modeled using a saturation-inspired form $\Xi({\bm b},{\bm \Delta}) = 1 - \exp[-{\bm\Delta}^2 Q_s^2({\bm b})/4]$ linked to the local gluon density, and the same framework yields a dipole representation of the structure function $F_T(x,Q^2)$, establishing a direct link between the small-$x$ quark distribution and the gluon content of the proton. Overall, the work provides a unified perspective that connects the $s$-channel picture with the conventional parton model and highlights how saturation physics enters the quark sector at small $x$.

Abstract

We use an s-channel picture of hard hadronic collisions to investigate the parton distribution function for quarks at small momentum fraction x, which corresponds to very high energy scattering. We study the renormalized quark distribution at one loop in this approach. In the high-energy picture, the quark distribution function is expressed in terms of a Wilson-line correlator that represents the cross section for a color dipole to scatter from the proton. We model this Wilson-line correlator in a saturation model. We relate this representation of the quark distribution function to the corresponding representation of the structure function F_T(x,Q^2) for deeply inelastic scattering.

Figures (1)

• Figure 1: Quark distribution in the $s$-channel picture. The black circles at the top represent the insertion of the operators in Eq. (\ref{['eq:fqasforwardscattering']}). The double straight line represents the eikonal. Any number of slow gluons couple the fast parton and the eikonal to the proton.