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Topic 19 of 22

Quantum Chromodynamics

Quantum Chromodynamics is the SU(3) gauge theory of the strong force. Quarks carry one of three "color" charges and interact by exchanging 8 gluons. QCD has two defining and seemingly contradictory properties: asymptotic freedom (quarks behave as free particles at short distances, enabling perturbative QCD) and confinement (colored objects are never observed in isolation at long distances). Together they explain why perturbative methods work in high-energy collisions but fail for low-energy hadron physics.

Key Concepts

  • SU(3) color: quarks in fundamental (3) representation, gluons in adjoint (8)
  • QCD Lagrangian: ℒ = Σ_f q̄_f(iD̸−m_f)q_f − ¼F^a_μν F^{aμν} (sum over 6 quark flavors)
  • Running coupling: αₛ(M_Z) ≈ 0.118, αₛ(1 GeV) ≈ 0.4 — decreases at high energy
  • ΛQCD ≈ 200 MeV — energy scale where αₛ becomes O(1), marking boundary of perturbative QCD
  • Confinement: quark-antiquark potential V(r) ≈ κr for large r (string tension κ ≈ 0.18 GeV²)
  • Parton model: at Q² ≫ ΛQCD², nucleon = incoherent beam of quarks and gluons (partons)

Key Equations

QCD Lagrangian
LQCD=∑fqˉf(i̸ ⁣D−mf)qf−14FμνaFaμν\mathcal{L}_{\rm QCD}=\sum_f\bar{q}_f(i\not\!D-m_f)q_f-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}
QCD running coupling
αs(μ)≈2πβ0ln⁡(μ/ΛQCD),β0=11−23Nf\alpha_s(\mu)\approx\frac{2\pi}{\beta_0\ln(\mu/\Lambda_{\rm QCD})},\quad\beta_0=11-\tfrac{2}{3}N_f
DIS structure function
F2(x,Q2)=∑fef2 xff(x,Q2)F_2(x,Q^2)=\sum_f e_f^2\,x f_f(x,Q^2)
Color factors
CF=43,  CA=3,  TF=12C_F=\tfrac{4}{3},\;C_A=3,\;T_F=\tfrac{1}{2}
Worked Example

Example Problem

Problem

At M_Z = 91 GeV, αₛ = 0.118. Estimate αₛ at μ = 1000 GeV using 1/αₛ(μ) = 1/αₛ(μ₀) + (β₀/2π)ln(μ/μ₀) with N_f=6, β₀=7.

Solution

1/αₛ(1000) = 1/0.118 + (7/2π)ln(1000/91) = 8.475 + 1.114×2.397 = 8.475 + 2.670 = 11.145. So αₛ(1000 GeV) ≈ 0.0897.

Key Takeaways

  • QCD is SU(3) Yang-Mills + 6 quark flavors in the fundamental representation; the 8 gluons carry color and self-interact
  • Asymptotic freedom: αₛ decreases at high energy → perturbative QCD valid for hard processes (jet production, DIS, top quark)
  • Confinement: at low energies, colored objects cannot be isolated; quarks are bound in colorless hadrons by a linear potential
  • The R-ratio and DIS structure functions at high Q² directly measure the number of colors (N_c=3) and quark charges
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