7.5 Renormalization of QED

Renormalization in QED tackles the tricky infinities that pop up in calculations. It's like cleaning up messy math to get real-world answers. By tweaking the theory, we can predict stuff like how electrons behave and interact with light.

This process is crucial for making QED work in practice. It helps explain cool phenomena like the electron's magnetic moment and the Lamb shift. Without renormalization, QED would be a hot mess of infinities instead of the super-accurate theory it is.

Renormalization in QED

Concept and Motivation for Renormalization

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• Renormalization systematically removes infinities and divergences that arise in perturbative calculations of quantum field theory
• In QED, the bare electron mass and charge, which appear in the original Lagrangian, are not directly observable due to the presence of divergent self-energy and vacuum polarization corrections
• Renormalization absorbs these infinities into the definitions of the physical (renormalized) electron mass and charge, which are the observable quantities in experiments (electron g-factor, Lamb shift)

Renormalization Procedure and Counterterms

• The renormalization procedure introduces counterterms in the Lagrangian, which cancel the divergent parts of the perturbative expansions
• Counterterms are added to the bare parameters (mass, charge) to define the renormalized parameters
• Renormalization ensures that the perturbative calculations in QED yield finite and well-defined results for physical observables (scattering cross-sections, decay rates)
• The renormalization procedure is carried out order by order in the perturbative expansion, ensuring consistency and finiteness at each order

Regularization Techniques for QED Divergences

Dimensional Regularization

• Dimensional regularization is a commonly used technique in QED, where the spacetime dimension is analytically continued from 4 to 4-ε dimensions, with ε being a small parameter
  • Divergent integrals become finite functions of ε in the regularized theory, allowing for a systematic expansion in powers of ε
  • The divergences appear as poles (1/ε terms) in the regularized expressions, which can be isolated and removed by renormalization
• Dimensional regularization preserves gauge invariance and Lorentz invariance, making it a preferred choice in gauge theories like QED

Other Regularization Methods

• Pauli-Villars regularization introduces fictitious heavy particles with opposite statistics to cancel the high-energy behavior of the divergent integrals
  • The fictitious particles act as a regulator, providing a cutoff for the divergent integrals
  • The Pauli-Villars regulator masses are taken to infinity after renormalization, removing their effect on physical observables
• Lattice regularization discretizes spacetime on a lattice, providing a natural cutoff for high-energy modes and making the theory finite
  • The lattice spacing acts as a regulator, and the continuum limit is recovered by taking the lattice spacing to zero after renormalization
  • Lattice regularization is particularly useful for non-perturbative studies of QED and other quantum field theories

Renormalization of Electron Mass and Charge

Electron Mass Renormalization

• The renormalization of the electron mass involves the self-energy correction, which arises from the interaction of the electron with its own electromagnetic field
  • The bare electron mass (m₀) is split into the physical (renormalized) mass (m) and a counterterm (δm), such that m₀ = m + δm
  • The counterterm (δm) is chosen to cancel the divergent part of the self-energy correction, rendering the renormalized mass (m) finite and observable
• The renormalized electron mass (m) corresponds to the pole of the renormalized electron propagator, which includes the effects of the self-energy correction

Electron Charge Renormalization

• The renormalization of the electron charge involves the vacuum polarization correction, which arises from the creation and annihilation of virtual electron-positron pairs in the presence of an electromagnetic field
  • The bare electron charge (e₀) is split into the physical (renormalized) charge (e) and a counterterm (δe), such that e₀ = e + δe
  • The counterterm (δe) is chosen to cancel the divergent part of the vacuum polarization correction, rendering the renormalized charge (e) finite and observable
• The renormalized electron charge (e) is defined by the renormalization condition imposed on the photon propagator, ensuring that the effective charge at a specific energy scale (renormalization point) matches the experimentally measured value

Physical Interpretation of Renormalized Quantities

Renormalized Electron Mass

• The renormalized electron mass (m) represents the physical mass of the electron, which includes the effects of virtual particle interactions
  • The bare mass (m₀) is an unobservable quantity that diverges when the cutoff is removed, while the renormalized mass (m) remains finite and corresponds to the experimentally measured value (0.511 MeV)
• The renormalized mass is scale-dependent, meaning it varies with the energy scale at which it is measured, as described by the renormalization group equations

Renormalized Electron Charge and Running Coupling

• The renormalized electron charge (e) represents the physical charge of the electron, which takes into account the screening effects due to vacuum polarization
  • The bare charge (e₀) is an unobservable quantity that diverges when the cutoff is removed, while the renormalized charge (e) remains finite and corresponds to the experimentally measured value (1.602 × 10⁻¹⁹ C)
• The renormalized charge depends on the choice of the renormalization scale, leading to the concept of running coupling constant (fine-structure constant, α)
  • The running of the coupling constant with energy scale is described by the renormalization group equations, which provide a framework for understanding the behavior of QED at different energy scales
  • The fine-structure constant increases with increasing energy scale, indicating that the electromagnetic interaction becomes stronger at higher energies (asymptotic freedom)

Renormalized Perturbative Expansions for QED Processes

Renormalized Propagators and Vertices

• The renormalized perturbative expansions for QED processes involve Feynman diagrams with renormalized propagators and vertices
• The electron self-energy correction modifies the electron propagator, leading to a shift in the electron mass and a wave function renormalization factor
  • The renormalized electron propagator includes the effects of the self-energy correction and ensures that the pole of the propagator corresponds to the physical electron mass
• The vacuum polarization correction modifies the photon propagator, leading to a running of the electromagnetic coupling constant (fine-structure constant)
  • The renormalized photon propagator includes the effects of the vacuum polarization correction and describes the effective strength of the electromagnetic interaction at different energy scales

Calculations of Simple QED Processes

• Simple QED processes, such as electron-electron scattering (Møller scattering) or electron-muon scattering, can be calculated using the renormalized perturbative expansions
  • The leading-order (tree-level) diagrams are evaluated using the renormalized propagators and vertices, yielding finite and well-defined results for the scattering amplitudes and cross-sections
  • Higher-order corrections, such as one-loop diagrams (vertex correction, box diagram), are systematically included in the renormalized perturbative expansions, providing more accurate predictions for the observables
• The renormalized perturbative expansions have been successfully applied to calculate various QED processes, such as:
  • Bhabha scattering (e⁺e⁻ → e⁺e⁻)
  • Compton scattering (e⁻γ → e⁻γ)
  • Pair annihilation (e⁺e⁻ → γγ)
• The agreement between the theoretical predictions based on renormalized QED and the experimental measurements provides strong evidence for the validity and success of the renormalization procedure in QED