3.3 Renormalization and running coupling

Renormalization and running coupling are crucial concepts in Quantum Electrodynamics. They address the problem of infinite results in calculations and explain how the strength of interactions changes with energy scale.

These ideas are fundamental to making sense of quantum field theories. They allow us to extract meaningful predictions from QED and understand how the theory behaves across different energy ranges, connecting theory with experiment.

Divergences in QED

Types of Divergences

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• Quantum Electrodynamics (QED) describes interactions between charged particles and photons using perturbation theory involving Feynman diagram calculations
• Higher-order loop diagrams in QED lead to infinite results when integrated over all possible momenta of virtual particles
  • Vacuum polarization and electron self-energy diagrams exhibit this behavior
• Ultraviolet divergences arise from very high-energy (short-distance) processes in the theory
• Infrared divergences occur in calculations involving massless particles (photons) stemming from very low-energy (long-wavelength) photon emission
• These divergences contradict finite results observed in experiments, necessitating a mathematical procedure to handle them

Need for Renormalization

• Renormalization absorbs infinities into the definition of physical parameters, allowing for meaningful predictions in QED
• Systematic method introduced to handle divergences in quantum field theory calculations
• Resolves the contradiction between infinite theoretical results and finite experimental observations
• Enables extraction of physically meaningful and finite predictions from the theory
• Provides a framework for dealing with quantum corrections to classical parameters (mass, charge)
• Establishes a connection between bare (unobservable) parameters and physical (measurable) quantities

Regularization and Renormalization

Regularization Techniques

• Regularization makes divergent integrals finite by introducing a cutoff or dimensional parameter
• Common regularization methods:
  • Momentum cutoff: Limits integration to a finite momentum range
  • Dimensional regularization: Performs calculations in $d = 4 - \epsilon$ dimensions
  • Pauli-Villars regularization: Introduces fictitious heavy particles
• After regularization, theory contains divergent terms dependent on the regularization parameter
• Each method has advantages and applications in different contexts (gauge theories, supersymmetry)

Renormalization Process

• Redefines bare parameters of the theory in terms of physical, measurable quantities
• Absorbs divergences by introducing counterterms in the Lagrangian
• Counterterms cancel divergences order by order in perturbation theory
• Renormalization schemes provide specific prescriptions for removing divergences:
  • On-shell renormalization: Defines parameters at physical mass shell
  • Minimal subtraction (MS): Subtracts only divergent parts of regularized expressions
• Renormalization group describes how renormalized parameters change with energy scale
• Leads to the concept of running coupling constants, essential for understanding scale-dependent behavior

Running Coupling in QED

Concept and Origin

• Running coupling refers to the variation of effective coupling constant with interaction energy scale
• Arises from quantum corrections to the interaction vertex, which screen or anti-screen the bare charge
• In QED, vacuum polarization leads to charge screening, increasing effective coupling at higher energies
• Described by the renormalization group equation, relating coupling change to energy scale
• Crucial for understanding theory behavior at different energy scales
• Enables comparison of theoretical predictions with experimental measurements across energy ranges

Implications and Applications

• Running couplings are a general feature of quantum field theories (QED, QCD, Standard Model)
• Behavior at high energies has important implications for fundamental force unification
• Determines validity of the theory at extreme scales (Planck scale, grand unification scale)
• Essential for precision calculations in particle physics experiments (LHC, electron-positron colliders)
• Provides insights into the structure of vacuum and nature of fundamental interactions
• Connects low-energy effective theories with high-energy fundamental theories

Beta Function and Asymptotic Behavior

Beta Function Calculation

• Beta function $\beta(g)$ describes rate of change of coupling constant $g$ with energy scale $\mu$
• Defined by renormalization group equation: $\mu\frac{dg}{d\mu} = \beta(g)$
• For QED, lowest-order contribution calculated from vacuum polarization diagram
• QED beta function: $\beta(\alpha) = \frac{2\alpha^2}{3\pi} + O(\alpha^3)$, where $\alpha$ is fine structure constant
• Positive sign indicates coupling strength increases with energy (Landau pole behavior)
• Calculation involves evaluating loop diagrams and applying renormalization techniques
• Higher-order corrections can be computed for more precise results

Asymptotic Behavior Analysis

• Solving renormalization group equation with QED beta function determines running coupling $\alpha(\mu)$
• QED coupling increases logarithmically with energy scale
• Asymptotic behavior suggests theory becomes strongly coupled at very high energies
• Potential breakdown of perturbation theory at extreme scales (Landau pole)
• Contrasts with theories like QCD, which have negative beta functions (asymptotic freedom)
• Analysis of beta functions essential for understanding high-energy behavior of quantum field theories
• Provides insights into possible unification scenarios and limitations of current theoretical frameworks