2.2 Particle interactions and Feynman diagrams

Feynman diagrams are like secret codes that reveal how tiny particles interact. They use lines and squiggles to show electrons, photons, and other subatomic particles dancing together. These diagrams help us visualize the invisible world of quantum physics.

By following special rules, we can turn these diagrams into math equations. This lets us calculate the chances of different particle interactions happening. It's like predicting the outcome of a cosmic game of chance!

Feynman diagrams for particle interactions

Graphical representation and basic elements

• Feynman diagrams graphically represent mathematical expressions describing subatomic particle behavior
• Time flows left to right with incoming particles on left and outgoing particles on right
• Particles represented by specific line types
  • Straight lines for fermions (electrons, quarks)
  • Wavy lines for photons
  • Curly lines for gluons
• Antiparticles depicted by arrows pointing backwards in time
• Vertices signify particle interactions where conservation laws must be obeyed
• Closed loops represent virtual particle-antiparticle pairs briefly coming into existence

Interpretation and complexity

• Diagram complexity relates to perturbation theory order
• Higher-order diagrams generally contribute less to overall interaction probability
• Analyze diagrams to determine:
  • Particles involved in interaction
  • Nature of force mediating the interaction (electromagnetic, strong, weak)
  • Conservation of quantum numbers (charge, lepton number, baryon number)
• Examples of common interactions:
  • Electron-positron annihilation: $e^- + e^+ \rightarrow \gamma$
  • Compton scattering: $e^- + \gamma \rightarrow e^- + \gamma$

Fundamental vertices and propagators

Basic interaction vertices

• Fundamental vertices represent basic interactions allowed by Standard Model
• Electromagnetic vertex connects fermion line to photon line (charged particle-photon interaction)
• Strong interaction vertex joins quark lines to gluon lines (quark-gluon interaction)
• Weak interaction vertex links fermion lines to W or Z boson lines (weak force interactions)
• Examples of fundamental vertices:
  • QED vertex: $e^- \rightarrow e^- + \gamma$
  • QCD vertex: $q \rightarrow q + g$

Propagators and internal lines

• Propagators represent internal lines in Feynman diagrams (virtual particle exchange)
• Different propagator types include:
  • Fermion propagators (straight lines)
  • Photon propagators (wavy lines)
  • Gluon propagators (curly lines)
• Propagator line direction indicates momentum and quantum number flow
• Mathematical expressions for propagators:
  • Fermion propagator: $\frac{i(\gamma^\mu p_\mu + m)}{p^2 - m^2}$
  • Photon propagator: $\frac{-ig_{\mu\nu}}{q^2}$

Probabilities and cross-sections using Feynman rules

Amplitude calculation

• Feynman rules systematically translate diagrams into mathematical expressions
• Each diagram element corresponds to specific mathematical factor
• Calculate overall interaction amplitude by multiplying all factors from Feynman rules
• Higher-order corrections included by incorporating more complex diagrams
• Coupling constants determine interaction vertex strength in probability calculations
• Example calculation for electron-muon scattering:
  • Amplitude $\mathcal{M} = \bar{u}(p_3)(-ie\gamma^\mu)u(p_1) \cdot \frac{-ig_{\mu\nu}}{q^2} \cdot \bar{u}(p_4)(-ie\gamma^\nu)u(p_2)$

Cross-section determination

• Calculate cross-sections by squaring amplitude and integrating over final state phase space
• Cross-section formula: $\sigma = \int |\mathcal{M}|^2 d\Phi_n$
• Renormalization techniques handle infinities in higher-order calculations
• Define physical parameters in terms of measurable quantities
• Example cross-section calculation:
  • $e^+e^- \rightarrow \mu^+\mu^-$ cross-section: $\sigma = \frac{4\pi\alpha^2}{3s}$

Virtual particles and internal lines

Characteristics of virtual particles

• Virtual particles represented by internal lines in Feynman diagrams
• Do not appear in initial or final interaction states
• Can violate energy-momentum conservation briefly (allowed by uncertainty principle)
• Mediate forces between real particles explaining action at a distance in quantum field theory
• Examples of virtual particle effects:
  • Coulomb force mediated by virtual photons
  • Nuclear force mediated by virtual pions

Role in quantum processes

• Virtual particle propagation described by propagators (probability amplitude for particle travel between points)
• Sum over all possible virtual particle exchanges contributes to overall interaction probability
• Higher-order diagrams with multiple exchanges typically contribute less but necessary for precise calculations
• Lead to observable phenomena:
  • Vacuum polarization (virtual electron-positron pairs around charged particles)
  • Casimir effect (attractive force between uncharged conducting plates)