7.1 Feynman rules for QED

Quantum Electrodynamics (QED) is the theory of how light and matter interact. In this section, we'll learn about Feynman rules, which are like a recipe for calculating how particles behave in QED.

Feynman diagrams are pictures that show particle interactions. We'll see how to draw these diagrams and use them to figure out the math behind particle behavior. This is key to understanding how QED works in practice.

Feynman Diagrams for QED Processes

Constructing Feynman Diagrams

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• Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior and interaction of subatomic particles in perturbative quantum field theory
• QED describes the interactions between charged particles (electrons, positrons) and photons, the quanta of light
• The basic components of a Feynman diagram in QED are:
  • External lines representing initial and final state particles
  • Internal lines representing virtual particles
  • Vertices representing interactions
• Feynman rules associate each element of a Feynman diagram with a specific mathematical expression
• To construct a Feynman diagram for a QED process:
  • Identify all possible ways the initial state particles can interact to produce the final state particles, consistent with conservation laws
  • Draw all topologically distinct diagrams connecting the initial and final states
  • Assign appropriate labels (momenta, spins, etc.) to each line and vertex

Perturbative Expansion and Diagram Order

• The order of a Feynman diagram is determined by the number of vertices it contains
• Higher-order diagrams generally contribute less to the overall amplitude due to the smallness of the coupling constant (fine-structure constant $\alpha \approx 1/137$)
• The perturbative expansion of a QED process involves summing the contributions of diagrams at increasing orders:
  • Leading order (LO) diagrams have the minimum number of vertices required by the process
  • Next-to-leading order (NLO) diagrams have one additional vertex compared to LO
  • Next-to-next-to-leading order (NNLO) diagrams have two additional vertices, and so on
• The complexity of calculations increases rapidly with the diagram order, but higher-order corrections are essential for precise predictions

Interpretation of QED Feynman Diagrams

Mathematical Expressions for Diagram Elements

• Each external line representing an incoming or outgoing fermion (electron, positron) is associated with:
  • A spinor ($u$ for particle, $v$ for antiparticle)
  • A momentum factor ($\sqrt{2E}$)
• Each external line representing an incoming or outgoing photon is associated with:
  • A polarization vector ($\varepsilon^{\mu}$)
  • A momentum factor ($\sqrt{2\omega}$)
• Each internal line representing a virtual fermion is associated with a propagator factor:
  • $\frac{i}{\not{p} - m}$, where $p$ is the fermion's momentum and $m$ is its mass
  • $\not{p} \equiv \gamma^{\mu}p_{\mu}$ is the Feynman slash notation
• Each internal line representing a virtual photon is associated with a propagator factor:
  • $\frac{-ig_{\mu\nu}}{k^2}$, where $k$ is the photon's momentum and $g_{\mu\nu}$ is the metric tensor
• Each vertex where a photon interacts with a fermion is associated with a factor:
  • $-ie\gamma^{\mu}$, where $e$ is the electron charge and $\gamma^{\mu}$ are the Dirac gamma matrices

Evaluating Amplitudes from Diagrams

• The overall amplitude for a Feynman diagram is obtained by:
  • Multiplying all the factors associated with its elements (external lines, internal lines, vertices)
  • Integrating over the momenta of the virtual particles (internal lines)
  • Applying conservation of momentum at each vertex
• The amplitude is a complex number that depends on the momenta and spins of the external particles
• Feynman diagrams provide a visual way to organize the terms in the perturbative expansion of the amplitude
• The interpretation of Feynman diagrams is based on a set of rigorous mathematical rules derived from the underlying quantum field theory (QED)

Matrix Elements for QED Processes

Definition and Significance

• The matrix element (or amplitude) for a QED process is a complex number that encodes the probability of the process occurring
• It is a fundamental quantity that connects the theoretical predictions of QED with experimental observables
• The matrix element depends on the momenta and spins of the initial and final state particles
• To evaluate the matrix element, one must sum the amplitudes of all the Feynman diagrams contributing to the process at a given order in perturbation theory

Simple QED Processes and Leading-Order Diagrams

• Examples of simple QED processes include:
  • Electron-positron annihilation into photons ($e^+ e^- \to \gamma\gamma$)
  • Compton scattering ($e^- \gamma \to e^- \gamma$)
  • Møller scattering ($e^- e^- \to e^- e^-$)
• The leading-order Feynman diagrams for these processes involve just one or two vertices
• For $e^+ e^- \to \gamma\gamma$, there are two LO diagrams related by crossing symmetry
• For Compton scattering, there are also two LO diagrams (s-channel and u-channel)
• Møller scattering has a single LO diagram with a t-channel photon exchange

Calculating Cross Sections from Matrix Elements

• The spin-averaged squared matrix element $\overline{|\mathcal{M}|^2}$ is proportional to the cross section $\sigma$ of the process:
  • $\sigma = \frac{1}{2E_1 2E_2 |v_1 - v_2|} \int \frac{d^3p_3}{(2\pi)^3 2E_3} \frac{d^3p_4}{(2\pi)^3 2E_4} \overline{|\mathcal{M}|^2} (2\pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)$
  • The delta function enforces conservation of 4-momentum
• Evaluating matrix elements requires performing spinor and tensor algebra in 4-dimensional spacetime
• One must also handle singularities that may arise from the propagators when virtual particles go on-shell (e.g., in the forward or backward scattering limits)
• Techniques such as dimensional regularization and renormalization are used to obtain finite, physically meaningful results

Virtual Particles in QED Interactions

Properties and Interpretation

• Virtual particles are not directly observable and do not satisfy the usual energy-momentum dispersion relation $E^2 = \vec{p}^2 + m^2$
• They can be thought of as short-lived quantum fluctuations that mediate the interactions between real particles
• In QED, virtual photons mediate the electromagnetic interaction between charged particles
• Virtual fermions (electrons, positrons) appear in loop diagrams that contribute to higher-order corrections
• The presence of virtual particles is a consequence of the uncertainty principle, which allows energy conservation to be violated for short times

Role in Mediating Electromagnetic Interactions

• The range of the electromagnetic interaction is infinite because the photon is massless, allowing virtual photons to propagate over arbitrary distances
• The strength of the interaction decreases with distance due to the $1/k^2$ factor in the photon propagator, where $k$ is the photon momentum
• Virtual photons can also mediate interactions between charged particles and external electromagnetic fields (electric, magnetic)
• In higher-order diagrams, virtual photons can split into virtual fermion-antifermion pairs, which then recombine into photons
• These fermion loops contribute to the screening of electric charge and the running of the coupling constant $\alpha$

Renormalization and Divergences

• Virtual particles play a crucial role in renormalization, the process of absorbing divergent loop contributions into the definitions of physical quantities like charge and mass
• Naive calculations of loop diagrams often yield infinite results due to the integration over unbounded virtual momenta
• Renormalization provides a consistent way to handle these divergences by:
  • Regularizing the integrals (e.g., dimensional regularization)
  • Redefining the bare parameters (charge, mass) in terms of renormalized, physically measurable quantities
  • Absorbing the divergences into counterterms that cancel the infinities
• The renormalized perturbation theory is well-defined and yields finite, testable predictions for QED observables
• The success of renormalization in QED was a major triumph of quantum field theory and paved the way for the development of other gauge theories (weak and strong interactions)