Glossary

3.2 Feynman rules and calculations in QED

4 min read•august 1, 2024

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 of −ieγμ-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
    • generally contribute less to the overall amplitude

Mathematical Representations and Calculations

  • External fermion lines represented by spinors (u(p)u(p) for particles, v(p)v(p) for antiparticles)
  • Internal fermion given by i(p̸+m)p2−m2+iϵ\frac{i(\not p + m)}{p^2 - m^2 + i\epsilon}
  • propagator in Feynman gauge expressed as −igμνq2+iϵ\frac{-ig_{\mu\nu}}{q^2 + i\epsilon}
  • for -photon interaction −ieγμ-ie\gamma^\mu
  • Integrate over loop momenta ∫d4k(2π)4\int \frac{d^4k}{(2\pi)^4} for each closed loop in the diagram
  • Apply a factor of (−1)(-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
  • (e−e+→γγe^-e^+ \rightarrow \gamma\gamma) serves as a fundamental QED process
    • Involves t-channel and u-channel diagrams
  • (e−γ→e−γe^-\gamma \rightarrow e^-\gamma) forms another basis for complex interactions
    • Includes s-channel and u-channel diagrams
  • Bhabha scattering (e−e+→e−e+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−→e−e−e^-e^- \rightarrow e^-e^-) example of process with only t-channel diagrams at tree-level
  • Electron-muon scattering (e−μ−→e−μ−e^-\mu^- \rightarrow e^-\mu^-) another example of t-channel only process

Advanced Processes and Calculations

  • Calculate cross-section for pair production (γγ→e−e+\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 (E2=p2+m2E^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- 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 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 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)

Key Terms to Review (19)

Compton Scattering: Compton scattering refers to the phenomenon where X-rays or gamma rays collide with matter, typically electrons, resulting in a change in the direction and energy of the incoming photons. This interaction highlights the particle-like behavior of light and is a key example of how electromagnetic radiation interacts with charged particles, showcasing the principles underlying particle interactions and represented through Feynman diagrams. It is fundamental to understanding how quantum electrodynamics (QED) describes these processes and applies Feynman rules to calculate outcomes in such interactions.
Coupling Constant: A coupling constant is a number that quantifies the strength of interaction between particles in quantum field theories. It plays a crucial role in determining the probability of a given interaction occurring, such as those mediated by force carriers in various fundamental forces. The value of the coupling constant can vary depending on the energy scale of the interactions, highlighting its significance in processes described by both quantum electrodynamics and electroweak theory.
Electron: An electron is a subatomic particle with a negative electric charge, found in the outer regions of atoms and playing a crucial role in chemical bonding and electricity. Electrons are fundamental components of atoms, which make up matter, and are classified as leptons in the framework of particle physics, being part of the Standard Model that describes fundamental particles and their interactions.
Electron-positron annihilation: Electron-positron annihilation is a process in which an electron and its antimatter counterpart, the positron, collide and annihilate each other, resulting in the production of energy typically in the form of gamma-ray photons. This phenomenon illustrates key principles of particle interactions and can be represented using Feynman diagrams, which depict the exchange of virtual particles during these interactions. The annihilation process is significant in quantum electrodynamics (QED), where it serves as a fundamental example of particle interactions governed by electromagnetic forces.
External Leg Rule: The external leg rule is a guideline used in quantum electrodynamics (QED) that pertains to the assignment of momenta to external legs in Feynman diagrams. This rule helps ensure that momentum conservation is accurately represented in calculations, allowing for proper representation of incoming and outgoing particles in the interactions depicted in the diagrams. By applying this rule, physicists can systematically analyze particle interactions, ensuring that all physical constraints are met.
Fine-structure constant: The fine-structure constant, often denoted as $$\alpha$$, is a fundamental physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. It plays a crucial role in quantum electrodynamics (QED), influencing the calculations of particle interactions and the precision of predictions made within the theory. Its approximate value of $$\alpha \approx \frac{1}{137}$$ highlights its importance in determining the scale of electromagnetic effects compared to other fundamental forces.
First-order correction: First-order correction refers to the adjustment made to a quantity in quantum mechanics or quantum field theory that accounts for the first level of perturbation or deviation from an unperturbed state. This concept is critical in calculations involving interactions, where the corrections help improve the accuracy of predictions for physical processes, especially in Quantum Electrodynamics (QED). By applying Feynman rules, one can systematically compute these corrections, providing deeper insights into particle interactions.
Green's Functions: Green's functions are mathematical constructs used to solve inhomogeneous differential equations, particularly in the context of quantum field theory and many-body physics. They provide a way to express the response of a system to external perturbations, and they are essential in calculating propagators in quantum electrodynamics, linking source terms to field values at different points in space and time.
Higher-Order Terms: Higher-order terms refer to contributions in a series expansion or perturbative calculation that go beyond the leading order terms, capturing more complex interactions and effects in quantum field theory calculations. These terms become increasingly significant in precise calculations and can affect the overall outcome, especially in processes described by perturbation theory, such as in quantum electrodynamics (QED). The inclusion of higher-order terms often leads to corrections that refine predictions made by simpler models.
Julian Schwinger: Julian Schwinger was a prominent American theoretical physicist known for his foundational contributions to quantum electrodynamics (QED), particularly the development of Feynman rules and the formulation of a more formal approach to particle interactions. His work, alongside Richard Feynman and Sin-Itiro Tomonaga, led to the establishment of QED as a well-defined theory that describes how light and matter interact, influencing the rules used in calculations within this framework.
Loop diagram: A loop diagram is a graphical representation used in quantum field theory, specifically in quantum electrodynamics (QED), to depict the interactions and processes involving virtual particles within a Feynman diagram. These diagrams illustrate the contributions of virtual particles to scattering amplitudes and other physical quantities, playing a crucial role in calculating loop corrections that arise from quantum fluctuations.
Photon: A photon is a quantum of electromagnetic radiation, characterized by its energy, momentum, and its ability to exhibit both particle-like and wave-like behavior. Photons are the fundamental carriers of electromagnetic force in the context of particle interactions, playing a crucial role in various phenomena such as light emission, absorption, and scattering. They are essential to understanding how particles interact through electromagnetic forces, particularly in quantum electrodynamics and the broader framework of quantum field theory.
Positron: A positron is the antimatter counterpart of an electron, possessing the same mass as an electron but with a positive charge. The discovery of the positron was a significant milestone in particle physics, leading to the understanding of particle-antiparticle pairs and their role in quantum field theories. This concept plays a crucial role in modern physics, especially in quantum electrodynamics, where interactions between particles and their antiparticles are fundamental.
Propagators: Propagators are mathematical functions used in quantum field theory that describe how particles propagate from one point to another in spacetime. They are essential for calculating the probabilities of various particle interactions and are often represented graphically in Feynman diagrams. Understanding propagators is key to applying Feynman rules and performing calculations in quantum electrodynamics (QED) and other quantum field theories.
Regularization: Regularization is a mathematical technique used to deal with infinities and divergences that arise in quantum field theory calculations, particularly in particle physics. It introduces parameters or modifications to the calculations, allowing for the treatment of otherwise ill-defined integrals and ensuring that the results remain meaningful and manageable. This technique is essential when deriving Feynman rules and during renormalization processes, helping to stabilize the calculations and link them with physical predictions.
Renormalization: Renormalization is a mathematical process used in quantum field theory to remove infinities that arise in calculations, allowing for the extraction of meaningful physical predictions. This process involves redefining certain parameters within a theory, such as mass and charge, to account for interactions at different energy scales. Renormalization is essential for ensuring that theoretical predictions match experimental results, particularly in quantum electrodynamics.
Richard Feynman: Richard Feynman was a prominent American theoretical physicist known for his work in quantum mechanics and particle physics, particularly for his contributions to quantum electrodynamics (QED). His innovative approaches and ideas not only advanced the understanding of fundamental particles and forces but also shaped modern physics education and interdisciplinary connections.
Tree Diagram: A tree diagram is a graphical representation used to illustrate the possible outcomes of a process or interaction, particularly in particle physics calculations. It serves as a visual tool to depict the relationships between initial and final states of particles, helping to simplify the calculation of probabilities for various processes in quantum electrodynamics (QED). Each branch of the diagram represents a different interaction or decay process, allowing physicists to systematically analyze complex interactions.
Vertex factor: In quantum electrodynamics (QED), a vertex factor is a mathematical expression that describes the interaction between particles at a vertex in a Feynman diagram. This factor encodes information about the coupling constants, particle types, and the conservation laws that govern particle interactions, making it essential for calculating scattering amplitudes and cross-sections in particle physics.
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