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Yang–Mills Theory

Yang–Mills theory is the local relativistic theory of a connection for a non-Abelian gauge group. Its curvature contains the gauge potential quadratically, so the same principle that removes redundant polarizations also fixes cubic and quartic gauge-boson interactions. This chapter follows that fact from the classical action and Gauss constraint through ghosts, color vertices, asymptotic freedom, the background-field method, and the BRST statement of perturbative consistency.

Enter this chapter

Section titled “Enter this chapter”

Begin with The Yang–Mills Action and Gauge Self-Interaction to derive non-Abelian self-interactions from the field strength.

Topics

Section titled “Topics”
  1. The Yang–Mills Action and Gauge Self-Interaction — Derives cubic and quartic interactions from the curvature.
  2. Yang–Mills Equations, Constraints, and Observables — Separates dynamical equations from the Gauss constraint.
  3. Gauge-Fixed Yang–Mills Action and Ghost Sector — Builds the ghost action accompanying covariant gauge fixing.
  4. Yang–Mills Color Algebra and Perturbative Vertices — Combines Lorentz and group factors in perturbative vertices.
  5. Non-Abelian Screening and Asymptotic Freedom — Determines when gauge fluctuations produce ultraviolet antiscreening.
  6. Background-Field Yang–Mills Effective Action — Preserves background gauge covariance while integrating fluctuations.
  7. Consistency of Perturbative Yang–Mills Theory — Connects symmetry identities to physical unitarity.

How the ideas fit together

Section titled “How the ideas fit together”

The nonlinear curvature fixes both the classical equations and the relative strengths of gauge-boson vertices. Gauge fixing changes how that structure is represented in perturbation theory; its field-dependent Jacobian introduces ghosts. The resulting identities tie gauge, matter, and ghost counterterms together, so diagram-by-diagram gauge dependence is not a physical inconsistency. The physical test concerns the complete amplitude and the states selected by the gauge constraints. A perturbative consistency argument does not establish the nonperturbative mass gap. Schwartz 2014, chs. 25–26.

Where to go next

Section titled “Where to go next”

Continue to Perturbative QCD and Partons to identify the scales on which a QCD calculation is perturbative.

References

Section titled “References”
  • Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.