Transverse Momentum Dependent Parton Distributions with self-similarity at small $x$ and models of proton structure function

TL;DR

This paper re-analyzes a self-similarity based model of the proton structure function at small x, ensuring it is free of singularities and compatible with QCD, thereby extending its validity range and extrapolating to large x.

Contribution

It introduces a singularity-free version of the self-similarity model and generalizes TMDs to align with QCD, expanding the model's applicability across a wider kinematic range.

Findings

Singularity-free model valid in a restricted Q^2 range

Proper TMD generalizations remove singularities and match QCD growth

Model extrapolated to large x without additional parameters

Abstract

In this paper we make re-analysis of a self-similarity based model of the proton structure function at small $x$ pursued in recent years. The additional assumption is that it should be singularity free in the entire kinematic range $0\leq \textit{x}\leq 1$. Our analysis indicates that the singularity free version of the model is valid in a more restrictive range of $Q^{2}$. We then analyse the defining Transverse Momentum Dependent Parton Distributions (TMD) occurred in the models and show that the proper generalizations and initial conditions on them not only remove the undesired singularity but also results in a QCD compatible structure function with logarithmic growth in $Q^2$. The phenomenological range of validity is then found to be much larger than the earlier versions. We also extrapolate the models to large $x$ in a parameter free way.