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

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 $Y = \ln \min \left{ 1/|x| , 1/|ξ| \right}$.

💡 Research Summary

This paper presents a significant advancement in the study of Generalized Parton Distributions (GPDs) within the high-energy limit of Quantum Chromodynamics (QCD). It focuses on deriving the small-x asymptotic behavior of unpolarized GPDs and their more fundamental counterparts, Generalized Transverse Momentum Distributions (GTMDs), while accounting for a previously overlooked variable: the skewness parameter (ξ).

The central achievement of the work is the systematic incorporation of non-zero (but small) skewness into the non-linear evolution equations that govern high-energy scattering. Traditionally, the small-x/saturation formalism describes scattering amplitudes in terms of the eikonal scattering amplitude of a color dipole off a target, evolving with rapidity Y ~ ln(1/x). This formalism often assumed ξ ≈ 0 for simplicity. The authors demonstrate that this assumption is not necessary and provide a precise prescription for including ξ.

Through a detailed analysis using Light Cone Operator Treatment (LCOT) and small-x approximations, the authors first show that the operator definitions for unpolarized gluon and quark GTMDs at small x and small ξ can be simplified. They are expressed as Fourier transforms of fundamental non-perturbative quantities: the C-even dipole scattering amplitude N and the C-odd odderon amplitude O (Eqs. 11, 17). The gluon GPD, obtained by integrating the GTMD over parton transverse momentum, depends solely on N (Eq. 14), while the quark GPD may retain an odderon contribution outside the ERBL region (|x| > |ξ|) (Eq. 18).

The most crucial result lies in determining the correct evolution rapidity for these dipole amplitudes. By analyzing ladder diagrams contributing to the dipole evolution, including those responsible for non-linear saturation effects (BK/JIMWLK evolution), the authors prove that the effective rapidity parameter Y is not simply ln(1/x). Instead, it is set by the smaller lifetime between the incoming and outgoing partons, leading to the modification: Y = ln(1/x) → Y = ln min{ 1/|x|, 1/|ξ| }. This finding generalizes a previous result in the linear (BFKL) evolution context to the full non-linear saturation regime. The physical interpretation is that the evolution is cut off by the larger of the two momentum fractions involved, |x| or |ξ|, which limits the phase space for gluon emission.

The implications of this work are substantial for phenomenology. It provides a rigorous framework for analyzing high-energy exclusive scattering processes (like DVCS) at future facilities such as the Electron-Ion Collider (EIC), where data will probe the small-x, non-zero ξ region with precision. Correctly accounting for skewness in the evolution is essential for extracting accurate 3D images of the nucleon, including its mass, angular momentum, and mechanical properties, from GPDs. The work also clarifies the potential role of odderon exchange in quark distributions at small x.