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

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.