The origin of Bjorken-$x$ dependence in DIS: a case for a $z$-dependent weight functional in the CGC

TL;DR

This paper proposes a modification to the CGC framework for DIS that incorporates a $z$-dependent weight functional, addressing an ambiguity in how Bjorken-$x$ dependence is modeled and aligning better with physical expectations.

Contribution

It introduces a $z$-dependent weight functional in the CGC description of DIS, resolving an ambiguity and improving compatibility with $k_t$-factorization.

Findings

The $x_b$ dependence is not solely from small-$x$ evolution.

A $z$-dependent weight functional can fit DIS data well.

The modified framework aligns with physical expectations and $k_t$-factorization.

Abstract

We discuss what is, at best, an ambiguity, and possibly an inconsistency of the eikonal Color Glass Condensate (CGC) description of Deep Inelastic Scattering (DIS). In this framework, the Bjorken-$x$ dependence enters the cross section solely through the rapidity cutoff $\Lambda=x_b$, leading to an all-order cross section independent of $x_b$. To address this issue, we explore a natural modification in which the weight functional depends explicitly on the light-cone momentum fraction $z$, with integration limits determined by $x_b$. This modification is consistent with the physical expectation that the observed non-perturbative structure depends on the probe energy. Our analysis implies that the $x_b$ variation of the cross section is not solely driven by small-$x$ evolution equations. We support this conclusion through an analysis of existing DIS fits and by demonstrating that a similarly good description of the data can be obtained within the modified framework. Finally, we show that the modified formulation is compatible with $k_t$-factorization, unlike the standard one.