3.2 Feynman rules and calculations in QED

Feynman rules are the building blocks of quantum electrodynamics calculations. They provide a visual and mathematical framework for understanding particle interactions, allowing us to compute scattering amplitudes and cross-sections for various processes.

These rules connect the abstract world of quantum field theory to observable phenomena. By mastering Feynman diagrams and their associated mathematical expressions, we gain powerful tools for predicting and interpreting experimental results in particle physics.

Feynman Rules for QED

Fundamental Elements and Principles

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• Feynman rules serve as diagrammatic and mathematical tools for calculating scattering amplitudes in quantum electrodynamics (QED)
• Fundamental vertices in QED represent interactions between electrons, positrons, and photons with a coupling constant of $-ie\gamma^\mu$
• External lines in Feynman diagrams depict incoming or outgoing particles
• Internal lines represent virtual particles propagating between vertices
• Each Feynman diagram element corresponds to a specific mathematical factor (external lines, internal lines, vertices)
• Calculate overall scattering amplitude by multiplying all factors from the Feynman diagram and integrating over undetermined internal momenta
• Feynman diagrams must conserve energy, momentum, and charge at each vertex and for the overall process
• Determine order of perturbation theory in QED calculations by counting number of vertices in the Feynman diagram
  • Higher-order terms generally contribute less to the overall amplitude

Mathematical Representations and Calculations

• External fermion lines represented by spinors ($u(p)$ for particles, $v(p)$ for antiparticles)
• Internal fermion propagators given by $\frac{i(\not p + m)}{p^2 - m^2 + i\epsilon}$
• Photon propagator in Feynman gauge expressed as $\frac{-ig_{\mu\nu}}{q^2 + i\epsilon}$
• Vertex factor for electron-photon interaction $-ie\gamma^\mu$
• Integrate over loop momenta $\int \frac{d^4k}{(2\pi)^4}$ for each closed loop in the diagram
• Apply a factor of $(-1)$ for each closed fermion loop
• Include symmetry factors for identical particles in the final state

Tree-Level Feynman Diagrams

Fundamental QED Processes

• Tree-level diagrams represent simplest Feynman diagrams with no closed loops
• Electron-positron annihilation ($e^-e^+ \rightarrow \gamma\gamma$) serves as a fundamental QED process
  • Involves t-channel and u-channel diagrams
• Compton scattering ($e^-\gamma \rightarrow e^-\gamma$) forms another basis for complex interactions
  • Includes s-channel and u-channel diagrams
• Bhabha scattering ($e^-e^+ \rightarrow e^-e^+$) involves two tree-level diagrams
  • One diagram with virtual photon in s-channel
  • Another diagram with virtual photon in t-channel
• Møller scattering ($e^-e^- \rightarrow e^-e^-$) example of process with only t-channel diagrams at tree-level
• Electron-muon scattering ($e^-\mu^- \rightarrow e^-\mu^-$) another example of t-channel only process

Advanced Processes and Calculations

• Calculate cross-section for pair production ($\gamma\gamma \rightarrow e^-e^+$) using tree-level diagrams with virtual electron propagators
• Processes involving real photon emission (bremsstrahlung) require inclusion of external photon lines
• Estimate relative contribution of different tree-level diagrams to a process by comparing propagator denominators and coupling constants
• Apply crossing symmetry to relate different processes (annihilation, pair production, Compton scattering)
• Calculate interference terms between different tree-level diagrams for processes with multiple contributing diagrams
• Implement Ward identity to verify gauge invariance of calculated amplitudes

Virtual Particles in QED

Characteristics and Properties

• Virtual particles appear as internal lines in Feynman diagrams representing intermediate states in quantum interactions
• Energy-momentum relation for virtual particles does not obey usual mass-shell condition ($E^2 = p^2 + m^2$)
  • Allows for "off-shell" four-momenta
• Heisenberg's uncertainty principle permits temporary violation of energy conservation in virtual particle exchanges
  • Magnitude of violation inversely proportional to interaction time
• Virtual photons mediate electromagnetic force between charged particles in QED
• Propagator for virtual particles in Feynman diagrams represents amplitude for particle to travel between two spacetime points
• Concept of virtual particles extends beyond QED to other quantum field theories (QCD, weak interactions)

Role in QED Interactions and Corrections

• Virtual particle exchanges explain nature of electromagnetic interactions at quantum level
• Higher-order corrections in perturbation theory involve additional virtual particle exchanges
  • Lead to phenomena such as vacuum polarization
  • Contribute to running coupling constant
• Virtual electron-positron pairs in vacuum cause screening of electric charges
  • Results in distance-dependent effective charge
• Vacuum fluctuations involving virtual particles contribute to Lamb shift in atomic spectra
• Self-energy diagrams with virtual photon loops lead to mass renormalization of charged particles
• Virtual particle effects explain Casimir force between uncharged conducting plates

Cross-Sections and Decay Rates

Calculation Methods and Principles

• Calculate cross-sections and decay rates in QED using Fermi's Golden Rule
  • Relates transition probability to square of matrix element
• Compute matrix element from sum of all relevant Feynman diagrams for the process
  • Typically use perturbation theory to a specific order
• Differential cross-section proportional to square of matrix element
  • Integrate over final state phase space
  • Average over initial state spins
• Calculate decay rates similarly to cross-sections
  • Consider unstable particles transitioning to final states without incoming particles
• Optical theorem relates total cross-section to imaginary part of forward scattering amplitude
  • Provides useful check for calculations
• Apply spin summing and averaging techniques for unpolarized cross-sections
• Implement Cutkosky rules for calculating imaginary parts of amplitudes

Advanced Techniques and Corrections

• Higher-order corrections in perturbation theory (loop diagrams) contribute to more precise calculations
• Employ renormalization techniques for handling infinities in higher-order perturbative calculations
  • Ensures finite and physically meaningful results
• Utilize dimensional regularization to handle ultraviolet divergences in loop integrals
• Apply on-shell renormalization scheme for QED calculations
• Implement running coupling constant in calculations to account for scale dependence of interactions
• Consider radiative corrections (real and virtual) for precision calculations of cross-sections and decay rates
• Use parton distribution functions for calculations involving composite particles (proton structure in electron-proton scattering)