Unpolarized GPDs at small $x$ and non-zero skewness

TL;DR

This work extends the small-$x$ description of unpolarized GPDs and GTMDs to non-zero skewness by expressing these distributions through the dipole amplitude $N_{10}(Y)$ and the odderon amplitude $\mathcal{O}_{10}(Y)$ within the LCOT/shock-wave framework. A key result is that non-zero skewness modifies the evolution rapidity, replacing $Y$ by $Y = \ln \big( \min \{ 1/|x|, 1/|\xi| \} \big)$, derived from lifetime/plus-momentum ordering of the gluon cascade in LCPT and valid for linear and non-linear small-$x$ evolution (BFKL/BK/JIMWLK). Consequently, the gluon and quark GTMDs and their GPD counterparts acquire skewness-dependent evolution, altering predictions for exclusive high-energy processes and the GPD-to-PDF ratio, with notable ERBL-region behavior where the odderon contributes to quark—and not gluon—GPDs. The results provide a consistent prescription to include skewness in small-$x$ evolution and open avenues for studying all leading-twist GPDs/GTMDs in this regime, including potential sub-eikonal corrections and spin-dependent extensions.

Abstract

We study the small-$x$ asymptotics of unpolarized generalized parton distributions (GPDs) and generalized transverse momentum distributions (GTMDs). Unlike the previous works in the literature, we consider the case of non-zero (but small) skewness while allowing for non-linear contributions to the evolution equations. We show that unpolarized GPDs and GTMDs at small $x$ are related to the eikonal dipole amplitude $N$, whose small-$x$ evolution is given by the BK/JIMWLK evolution equations, and to the odderon amplitude $\cal O$, whose evolution is also known in the literature. We show that the effect of non-zero skewness $ξ\neq 0$ is to modify the value of the evolution parameter (rapidity) in the arguments for the dipole amplitudes $N$ and $\cal O$ from $Y = \ln (1/x)$ to $Y = \ln \min \left\{ 1/|x| , 1/|ξ| \right\}$.

Figures (2)

• Figure 1: A ladder diagram contributing to the small-$x$ evolution of a quark and anti-quark dipole in the general non-forward case for an elastic scattering process.
• Figure 2: A gluon cascade in the non-forward elastic scattering case. The rectangle represents the shock wave, comprising subsequent evolution and interaction with the target.