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

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.