The results of renormalized perturbation theory, in QCD and other quantum field theories, are ambiguous at any finite order, due to renormalization-scheme dependence. The perturbative results depend upon extraneous scheme variables, including the renormalization scale, that the exact result cannot depend on. Such 'non-invariant approximations' occur in many other areas of physics, too. The sensible strategy is to find where the approximant is stationary under small variations of the extraneous variables. This general principle is explained and illustrated with various examples. Also dimensional transmutation, RG equations, the essence of renormalization and the origin of its ambiguities are explained in simple terms, assuming little or no background in quantum field theory. The minimal-sensitivity approach leads to 'optimized perturbation theory,' which is developed in detail. Applications to Re⁺e⁻, the infrared limit, and to the optimization of factorized quantities, are also discussed thoroughly.

Contents:

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  Acknowledgments

  
  


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  Introduction

  
  


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  Renormalization and Its Ambiguities:

  
  
  
  
  • Dimensional Analysis in Quantum Field Theory
  
  
  • Renormalization as Reparametrization
  
  
  • Non-invariant Approximations and the Principle of Minimal Sensitivity
  
  
  • Induced Convergence
  
  
  
  


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  Optimized Perturbation Theory:

  
  
  
  
  • Preliminaries: RG invariance, int-β Equation, Λ Definition, and CG Relation
  
  
  • Parametrization of RS Dependence and the ρn Invariants
  
  
  • Finite Orders and Optimization
  
  
  • Solution for the Optimized rₘ Coefficients and Optimization Algorithm
  
  
  • Numerical Examples for Re⁺e⁻ in QCD
  
  
  
  


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  Special Topics:

  
  
  
  
  • Infrared Limit: Fixed and Unfixed Points
  
  
  • Optimization of Factorized Quantities
  
  
  • Exploring All-Orders OPT in the Small-b (BZ) Limit
  
  
  
  


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  Bibliography

  
  

Readership: Researchers and graduate students in perturbative quantum field theory, and especially QCD (Quantum Chromodynamics). Physicists in all fields where 'non-invariant approximations' arise. Mathematicians interested in approximations and expansions.

Key Features:

• First book on this topic — crucial to obtaining numerical predictions from perturbative quantum field theory — written by its originator


• Unique pedagogical approach that de-mystifies renormalization, renormalization group equations, and dimensional transmutation


• Thorough explanation of the Principle of Minimal Sensitivity and its ramifications, with detailed examples from quantum mechanics and elsewhere


• Insights into the "induced convergence" phenomenon whereby convergent results are naturally obtained from otherwise divergent series


• Analysis of the infrared limit, predicting "freezing" of the QCD coupling and revealing the possibility of infrared limits not governed by the usual "fixed-point" lore


• Elegant resolution of the factorization-scheme-dependence problem