Part V: Gauge Field Theories • Chapter 4

Yang-Mills Theory

The complete non-Abelian gauge theory structure

📺 Video Lectures

For video lectures on Yang-Mills theory and gluon self-interactions, see:

• Susskind's Theoretical Minimum:Lectures 3-5: Group Theory & Gauge Fields
Yang-Mills Lagrangian and 3-gluon, 4-gluon vertices
• Tobias Osborne QFT 2016:YouTube Playlist
Complete derivation of Yang-Mills action and equations of motion
• David Tong Gauge Theory:Lecture Notes (Cambridge)
Chapters 1-2 cover Yang-Mills action and classical solutions (instantons)

The Yang-Mills Action

The complete pure Yang-Mills Lagrangian:

ℒ_YM = -(1/4) F_μν^a F^{μν a}

where:

F_μν^a = ∂_μA_ν^a - ∂_νA_μ^a + gf^abcA_μ^bA_ν^c

Self-Interaction Vertices

Expanding the Yang-Mills Lagrangian reveals interaction terms:

Kinetic term (2-point):

-(1/2) (∂_μA_ν^a)(∂^μA^{ν a})

3-gluon vertex:

g f^abc (∂_μA_ν^a) A^μb A^νc

Cubic self-interaction - unique to non-Abelian theories

4-gluon vertex:

-(g²/4) f^abe f^cde A_μ^aA_ν^bA^μcA^νd

Quartic self-interaction from (A×A)² terms

Physical Meaning

These 3- and 4-gluon vertices mean gluons can split into gluons, merge together, and scatter off each other. This is fundamentally different from QED where photons pass through each other unchanged.

Yang-Mills Equations of Motion

The field equation from varying the action:

D_μ F^{μν a} = 0

Expanding:

∂_μF^{μν a} + gf^abc A_μ^b F^{μν c} = 0

These are highly nonlinear partial differential equations. Unlike Maxwell's equations, they have no general analytic solution and exhibit rich classical dynamics including solitons and instantons.

Classical Solutions

Important classical solutions:

• Instantons (BPST solution):

Finite-action solutions in Euclidean space, important for tunneling and θ-vacua

• Monopoles (in spontaneously broken theories):

't Hooft-Polyakov monopoles in theories with Higgs mechanism

• Sphalerons:

Unstable static solutions at the barrier between topological sectors

Summary

✓ Yang-Mills action: ℒ = -(1/4)F^a_μνF^{μν a} with F containing gf^abcA^bA^c term
✓ 3-gluon vertex: Cubic self-interaction ∝ gf^abc
✓ 4-gluon vertex: Quartic self-interaction ∝ g²f^abef^cde
✓ Nonlinear equations: Much richer than linear Maxwell theory
✓ Classical solutions: Instantons, monopoles, sphalerons

Quantum Field Theory Course