This comprehensive study aims to address a fundamental inquiry: how can we attain a precise measurement of the gluon Parton Distribution Function (PDF) to enhance our comprehension of the intricate quark-gluon composition within nucleons? Employing a dual methodology that integrates Quantum Chromodynamics (QCD) and high-energy particle collisions, our research leverages computational tools and experimental techniques, with a specific focus on probing protons and neutrons. Through scattering experiments and advanced detectors, we delve into the intricate structure of nucleons.

The investigation encompasses a synergy of theoretical frameworks, notably utilizing QCD equations and the Momentum Sum Rule to scrutinize Parton Distribution Functions at distinct momentum fractions and energy scales. Our primary objective is to achieve an unparalleled precision in measuring the gluon PDF, a paramount undertaking crucial for unraveling the complexities associated with the strong force and internal dynamics of nucleons.

 Incorporating theoretical plots and equations, our study vividly illustrates the hypothetical behaviors of gluon distribution, quark distribution, and splitting functions. These representations contribute significantly to a profound understanding of fundamental particles, shedding light on the underlying dynamics and interactions that govern their behavior. By combining computational simulations with real-world experimental data, this research offers a holistic approach, advancing our knowledge of the intricate interplay between quarks and gluons within nucleons, ultimately contributing to the broader understanding of the fabric of matter.

Keywords: Quantum Chromo Dynamics (QCD), Parton Distribution Functions (PDFs), DGLAP Evolution Equation,
Quarks and Gluons, Monte Carlo Simulations, Uncertainties in αs (MZ)