Non-Abelian gauge theory

Choose Hermitian generators $T^a$ with $[T^a,T^b]=if^{abc}T^c$ and $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$. Write $D_\mu=\partial_\mu-igA_\mu^aT^a$ and define the field strength by $[D_\mu,D_\nu]=-igF_{\mu\nu}$. Expanding this commutator produces derivative terms and a commutator of potentials. The latter vanishes for U(1), but not for SU(N). The transformation rule for the connection is chosen so that the derivative of a transformed matter field transforms covariantly.

The gauge kinetic term is $-F^a_{\mu\nu}F^{a\mu\nu}/4$ in Minkowski signature. Squaring a field strength that contains a term quadratic in $A$ produces both cubic and quartic gauge vertices. Their coefficients are related by gauge invariance; they are not independent couplings one can tune separately while preserving the same theory. Gauge-fixed perturbation theory also includes ghosts, whose internal diagrams are part of the symmetry-controlled result.

For an SU(N) theory with $n_f$ massless Dirac fundamentals, the one-loop coefficient is $b_0=(11N-2n_f)/3$. The beta function is negative when this coefficient is positive. Integrating the differential equation shows that the inverse squared coupling increases logarithmically with the renormalization scale. This is asymptotic freedom: sufficiently short-distance processes can become increasingly accessible to perturbation theory rather than increasingly strongly coupled.

This formula assumes active flavors whose masses are small relative to the scale and a weak enough coupling for one-loop accuracy. Heavy flavors require threshold matching rather than retaining the same $n_f$ everywhere. The gauge contribution and matter contribution oppose one another; the final coefficient includes all required gauge and ghost terms in a consistent calculation. At large $n_f$, loss of asymptotic freedom is a statement about ultraviolet running, not by itself a complete classification of the infrared phase.

A Wilson loop is a trace of parallel transport around a closed path. In pure gauge theory, an area-law falloff for a large rectangular loop corresponds to a linearly rising static-source potential under suitable limiting conditions. This supplies a gauge-invariant diagnostic that is more meaningful than a plot of a gauge potential. Dynamical fundamental matter can screen sources and break the long string, so the naive area-law criterion needs qualification in full QCD.

The one-loop coupling grows toward the infrared, telling us where perturbation theory becomes unreliable. It does not prove confinement, compute a hadron mass, or establish the Yang–Mills mass gap. Lattice calculations, symmetry constraints, and special analytic limits supply additional evidence and tools. The existence of a rigorous four-dimensional pure Yang–Mills continuum theory with a positive mass gap remains a mathematical problem; realistic hadron phenomenology and that foundational theorem are related but distinct achievements.