A study of structure functions with the DGLAP: equations at small $x$ with $O(x)$ and $O(x ^2 )$

TL;DR

This paper derives and solves second order differential equations from the DGLAP equations at small x, incorporating terms up to O(x) and O(x^2), and compares solutions to analyze structure functions in QCD.

Contribution

It introduces a novel approach to approximate DGLAP equations at small x using Taylor expansion up to O(x^2) and compares solutions obtained by different methods.

Findings

Solutions for O(x) and O(x^2) are generally not identical.

The analysis is consistent with recent HERA data.

The methods provide insights into the behavior of structure functions at small x.

Abstract

We obtain a pair of second order differential equations in two variables $x$ and $t$ from the coupled DGLAP QCD evolution equations at small $x$ using the standard Taylor series expansion method.To that end we keep terms upto $O(x^2 )$.We use the standard assumption about the relationship between the singlet Structure Function and the gluon distributions available in current literature. We solve the taylor approximated $O(x)$ DGLAP equations by Lagrange's auxiliary method and $O(x^2)$ equation by Method of Separation of Variables and then show that the two solutions obtained in each for $O(x)$ and $O(x^2)$ are not identical in general.Analysis of the results obtained are done in the range of the recent HERA data.