Yang–Mills Theory

Yang–Mills theory generalizes electromagnetism to a non-Abelian gauge group. Its gauge fields interact with one another even without matter. In four dimensions the pure theory is asymptotically free: short-distance perturbation theory can be controlled while long-distance dynamics becomes strongly coupled.

Non-Abelian field strength

Take gauge group $SU(N_c)$, with Hermitian fundamental generators normalized by $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$, and no matter fields. Define

$$\begin{aligned} D_\mu&=\partial_\mu-igA_\mu^aT^a,\\ F_{\mu\nu}^a&=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a +gf^{abc}A_\mu^bA_\nu^c,\\ \mathcal L&=-\frac14F_{\mu\nu}^aF^{a\mu\nu}. \end{aligned}$$

This profile takes the theta angle to be zero. The commutator $[D_\mu,D_\nu]=-igF_{\mu\nu}^aT^a$ checks the relative sign of the quadratic term. Because $F$ already contains two powers of $A$, the action contains three- and four-gauge-field interactions.

The equation is $(D_\mu F^{\mu\nu})^a=0$. Unlike a global color rotation of independent photons, the local non-Abelian structure ties the self-interactions to the gauge constraints. See Srednicki 2007, §§69–72.

From a dimensionless coupling to a scale

For pure $SU(N_c)$ theory, the one-loop beta function is

$$\mu\frac{\mathrm dg}{\mathrm d\mu} =-\frac{b_0g^3}{16\pi^2}+O(g^5), \qquad b_0=\frac{11N_c}{3}.$$

Integrating at this order gives

$$\frac1{g^2(\mu)} =\frac{b_0}{8\pi^2}\log\frac{\mu}{\Lambda}.$$

The integration constant $\Lambda$ has dimensions of mass even though $g$ is dimensionless. This is dimensional transmutation. The formula is useful at $\mu\gg\Lambda$; its apparent singularity near $\Lambda$ marks the failure of this perturbative approximation, not a derivation of the infrared spectrum. See Srednicki 2007, §73.

Gauge-invariant questions

Wilson loops, glueball correlation functions, and the response to temperature probe the theory beyond individual gauge-dependent fields. An area-law regime for an appropriate Wilson loop diagnoses a flux-tube energy in pure gauge theory; adding dynamical matter changes screening and string-breaking questions.

Perturbative gluons are useful short-distance variables. They do not themselves establish asymptotic colored particles, a mass gap, or confinement. Those conclusions require nonperturbative reasoning and evidence. Global gauge-group choices and allowed line operators also matter: the Lie algebra alone does not specify every version of the quantum theory.

Develop the calculation

The Yang–Mills Action and Gauge Self-Interaction develops the action. Non-Abelian Screening and Asymptotic Freedom develops the running. Add fundamental quarks to reach QCD, or compare a topological gauge action in Abelian Chern–Simons Theory.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§69–73 and 82. DOI.