II. Non-commuting Matrix Solution of DGLAP; $F_2 {p,d}$ Data Leading to Partons Directly without Parameterization

TL;DR

This paper introduces a non-parametric, direct method for determining quark and gluon distribution functions from data, avoiding traditional parameter assumptions by solving singular systems of equations at exact data points.

Contribution

It develops a novel non-commuting matrix solution for DGLAP equations that directly extracts parton distributions from experimental data without parameterization.

Findings

Achieves direct estimation of parton distributions at data points.

Solves singular linear systems for quark and gluon components.

Provides a non-commuting evolution solution on natural data points.

Abstract

Dominant present path for determination of quarks and gluon distribution functions from data is based on pre-assumed form of parameters. Here, an alternative direct, or non-parametric method is spelled out. As the main task, least square estimates of the central values are obtained at the exact $x$ points of the analysed ${{F_2}^{p,d}}$ data points, at a chosen $Q^2$. In the process, numerically singular system of weighted linear combination of LO decomposition equations of the data points, each at a given $(x_k, {Q_{kl}}^2), l=1, ..., n_k$, obtained from a respective $\chi^2$, together with the equations of Zero Mass Variable Flavour Number constraints, are solved. In each data equation, the corresponding data points are decomposed into their quarks and gluon components, evolved from a set of unknowns at $(x_k, Q^2), k=1, ..., n$. A similar evolution is done in the constraints. As a complementary task, the constrained discrete $x$ set, required for the commuting solution of evolution equation, \cite{I}, is relaxed, and a non-commuting solution on a more natural set of exact $x$ points of the data is developed.