[Paper Review] Les Houches Lectures on Renormalization Theory and Effective Field Theories

Matthias Neubert|arXiv (Cornell University)|Jan 19, 2019

Random Matrices and Applications3 references9 citations

TL;DR

This paper provides a comprehensive review of renormalization theory in quantum field theories, with a focus on effective field theories (EFTs), using QED and QCD as primary examples. It details the systematic removal of ultraviolet divergences via renormalization, the role of running couplings and anomalous dimensions, and the Wilsonian approach to EFTs, culminating in the derivation of the renormalization group equations and their solution for the running coupling, which remains reliable even in the large logarithmic regime due to perturbative suppression of higher-order corrections.

ABSTRACT

These lectures review the formalism of renormalization in quantum field theories with special regard to effective quantum field theories. While renormalization theory is part of every advanced course on quantum field theory, for effective theories some more advanced topics become particularly important. This includes the renormalization of composite operators, operator mixing under scale evolution, and the resummation of large logarithms of scale ratios. This course thus sets the basis for many of the more specialized lecture courses delivered at the 2017 Les Houches Summer School.

Motivation & Objective

• To provide a systematic review of renormalization formalism in quantum field theories, especially in the context of effective field theories (EFTs).
• To clarify the physical meaning of renormalization through the Wilsonian approach, emphasizing scale separation in EFTs.
• To derive and explain the renormalization group (RG) equations for running couplings and Wilson coefficients in QED and QCD.
• To demonstrate the robustness of the leading-order running coupling formula in QCD, even when logarithms become large, by analyzing higher-order corrections.

Proposed method

• Uses the on-shell and minimal subtraction renormalization schemes to handle ultraviolet divergences in QED and QCD.
• Applies the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula to connect Feynman diagrams to physical scattering amplitudes.
• Derives the beta function and anomalous dimensions via one-loop calculations of self-energy and vertex functions in QED and QCD.
• Constructs the renormalization group (RG) equations for coupling constants and Wilson coefficients using the Callan-Symanzik equation.
• Solves the RG equations perturbatively, showing that higher-order corrections are suppressed by powers of αs/4π, ensuring reliability in the perturbative regime.
• Applies Fierz identities and one-loop UV analysis to compute the anomalous dimension matrix for four-fermion operators in the effective weak Hamiltonian.

Experimental results

Research questions

• RQ1 How can ultraviolet divergences in quantum field theories be systematically removed using renormalization, and what is the role of counterterms in this process?
• RQ2 What is the physical significance of the running coupling in QCD, and how does it emerge from the renormalization group evolution?
• RQ3 How do anomalous dimensions of composite operators influence the scale dependence of Wilson coefficients in effective field theories?
• RQ4 Why is the leading-order formula for the running coupling in QCD reliable even when logarithmic terms become large?
• RQ5 What is the structure of the anomalous dimension matrix for dimension-6 four-fermion operators in the effective theory of nonleptonic B-meson decays?

Key findings

• The running coupling in QCD, αs(μ), is reliably described by the leading-order formula αs(μ) ≈ αs(Q)/(1 + β₀αs(Q)/(4π) ln(μ²/Q²)) for all μ ≫ ΛQCD.
• Higher-order corrections to the beta function do not spoil the leading-order formula because they are suppressed by powers of αs/4π, ensuring perturbative control.
• The anomalous dimension matrix for the two leading four-fermion operators in nonleptonic B decays is γ = (αs/4π) × [[−6/Nc, 6], [6, −6/Nc]] + O(αs²), derived from one-loop UV divergences and wave-function renormalization.
• The solution to the RG equation for Wilson coefficients is obtained iteratively, with the leading-order term dominating even in the large-logarithm regime.
• The Wilsonian approach provides a deeper physical understanding of renormalization by explicitly separating physics at different energy scales.
• The structure of infrared divergences is not explored in this work, though effective field theories have provided new insights into them elsewhere.