Yang–Mills Theory as the Mathematical Framework of the Strong Nuclear Interaction

The strong nuclear interaction is described fundamentally by quantum chromodynamics (QCD), a non-Abelian gauge theory constructed according to the Yang–Millsformalism. In QCD, the local gauge symmetry is represented by the group SU(3)c, whose generators act in the internal color space of quarks. Unlike the Abelian U(1) gauge theory of electromagnetism, the gluon fields of QCD carry color charge and therefore interact directly with one another. This self-interaction produces two central properties of the strong interaction: asymptotic freedom at short distances and color confinement at large distances.This paper presents the mathematical and physical structure of Yang–Mills theory as applied to the strong nuclear force. The formulation begins with gauge symmetry, covariant derivatives, field-strength tensors, and the QCD Lagrangian.The renormalization-group equation, Wilson loop, lattice formulation, and nonperturbative methods are then introduced as essential tools for studying the highenergy and low-energy regimes of the theory. The analysis demonstrates how Yang–Mills theory provides the conceptual basis for understanding the internal structure of protons and neutrons, hadron formation, nuclear binding, high-energyscattering, and strongly interacting matter.