PDF at small $x$ in the non-perturbative region

TL;DR

The paper develops a non-perturbative framework for parton distributions at small $x$ by extending the parton model to include splitting and fusion processes. A single-type parton cascade is analyzed, linking generation depth to $x$ via $x_n=1/2^n$, and yielding a power-law PDF $x f(x)\u00A0\u22131$ in the moderately small-$x$ region with exponent $lta_w=log(2w)/log 2$, and saturation at very small $x$ when nonlinear fusion terms become important. Saturation manifests as a finite, high-density parton medium with $x f(x)$ and rapidity density $dN/dy$ approaching constant values determined by the parameters $w$ and $v$, corresponding to a dense gluon medium (color glass condensate) without relying on perturbative scales. The results qualitatively agree with perturbative saturation pictures while offering a non-perturbative mechanism, providing a foundation for quantitative extensions and cross-checks with PQCD approaches such as BFKL, GLR, and BK.

Abstract

Parton distribution function (PDF) at small $x$ in a fast-moving proton is investigated within an upgraded parton model that includes parton splitting and fusion. In the region of moderately small $x$, we obtain a power-law behavior of the parton density $x f(x)$ with an exponent proportional to the logarithm of the probability of parton splitting. Taking into account parton fusion leads to a nonlinear equation for the PDF. In the region of very small $x$, the phenomenon of saturation of the parton density is detected and a model estimate of its value in this regime is obtained. The results are compared with those obtained previously based on the analysis of equations in logarithmic approximations of perturbative QCD.