Quantum Chromodynamics
The SU(3) gauge theory of the strong interaction
Video Lectures
For comprehensive video lectures on QCD:
- Tobias Osborne QFT 2016:YouTube Playlist
Covers non-Abelian gauge theories and QCD
- David Tong Gauge Theory:Lecture Notes (Cambridge)
Chapter 3 covers QCD in detail
- Peskin & Schroeder:Chapters 16-17 on QCD
QCD: The Theory of Strong Interactions
Quantum Chromodynamics is the SU(3) Yang-Mills theory describing the strong force between quarks and gluons. It is characterized by two remarkable phenomena: asymptotic freedom and confinement.
SU(3)color gauge symmetry: q → eiαaTaqColor Charge
Quarks carry color charge in three varieties:
r
Red
g
Green
b
Blue
Antiquarks carry anticolor: r̄, ḡ, b̄. Observable hadrons are color singlets:
- Mesons: qq̄ (color-anticolor pair)
- Baryons: qqq (rgb combination = white)
- Antibaryons: q̄q̄q̄ (r̄ḡb̄ combination)
The QCD Lagrangian
LQCD = -¼ GaμνGaμν + Σf q̄f(iD̸ - mf)qfwhere the gluon field strength tensor is:
Gaμν = ∂μAaν - ∂νAaμ + gsfabcAbμAcνThe covariant derivative: Dμ = ∂μ - igsTaAaμ
Gluons: The Force Carriers
Properties
- 8 gluons (adjoint of SU(3))
- Massless spin-1 bosons
- Carry color charge themselves
- Self-interacting (unlike photons)
Gluon Colors
- rb̄, rḡ, br̄, bḡ, gr̄, gb̄
- (rr̄ - bb̄)/√2
- (rr̄ + bb̄ - 2gḡ)/√6
Key Difference from QED
Unlike photons, gluons carry color charge and interact with each other. This leads to 3-gluon and 4-gluon vertices, which are responsible for the unique behavior of QCD.
Asymptotic Freedom
The QCD coupling constant decreases at high energies (short distances):
αs(Q²) = αs(μ²) / [1 + (αs(μ²)/12π)(33 - 2nf) ln(Q²/μ²)]The β-function for QCD:
β(g) = -g³/(16π²)(11 - 2nf/3) + O(g&sup5;)Nobel Prize 2004
Gross, Wilczek, and Politzer received the Nobel Prize for discovering asymptotic freedom, showing that QCD is the correct theory of strong interactions.
Confinement
At low energies (large distances), the QCD coupling becomes strong, leading to color confinement: quarks and gluons are never observed as free particles.
Linear Potential
V(r) ∼ σr at large r
String tension σ ≈ 1 GeV/fm
Flux Tubes
Color field lines squeeze into tubes between quarks, unlike spreading in QED
ΛQCD ≈ 200 MeV: The scale where αs becomes O(1) and perturbation theory breaks down.
QCD Vertices
Quark-Gluon Vertex
-igsγμTa
Like QED, but with color matrix
3-Gluon Vertex
gsfabc[gμν(k-p)ρ + ...]
Non-Abelian self-coupling
4-Gluon Vertex
-igs²[fabefcde + ...]
Required by gauge invariance
Ghost-Gluon Vertex
gsfabcpμ
From Faddeev-Popov procedure
Running Coupling
Experimental values of αs at different scales:
| Energy Scale | αs | Process |
|---|---|---|
| MZ = 91.2 GeV | 0.118 | Z boson decays |
| Mτ = 1.78 GeV | 0.33 | τ decays |
| 1 GeV | ∼0.5 | Lattice QCD |
Deep Inelastic Scattering & Parton Model
At high Q², hadrons can be probed as collections of quasi-free partons (quarks and gluons):
F2(x, Q²) = Σq eq² x[q(x, Q²) + q̄(x, Q²)]The parton distribution functions (PDFs) evolve with Q² according to the DGLAP equations:
dq(x, Q²)/d ln Q² = (αs/2π) ∫ (dy/y) Pqq(x/y) q(y, Q²)Non-Perturbative QCD
Lattice QCD
Discretize spacetime, compute path integrals numerically. Essential for hadron masses, form factors.
Chiral Symmetry Breaking
Spontaneous breaking of SU(2)L×SU(2)R generates most of nucleon mass.
Instantons
Tunneling between topologically distinct vacua. Solve U(1)A problem.
QCD Vacuum
Non-trivial vacuum with gluon and quark condensates. θ-vacuum structure.
Summary
- ✓ QCD: SU(3) Yang-Mills theory with quarks in fundamental representation
- ✓ Color charge: Three colors (r, g, b), eight gluons carry color
- ✓ Asymptotic freedom: Coupling decreases at high energies (β < 0)
- ✓ Confinement: Quarks bound into color-singlet hadrons
- ✓ Running coupling: αs(MZ) ≈ 0.118, increases at low energies
- ✓ DGLAP evolution: Parton distributions evolve with momentum transfer
- ✓ Non-perturbative: Lattice QCD, chiral symmetry breaking, instantons