As Scales Become Separated: Lectures on Effective Field Theory

TL;DR

This paper provides a comprehensive pedagogical introduction to Effective Field Theory (EFT), emphasizing scale separation, renormalization group techniques, and applications to scalar theories and soft collinear effective theory (SCET).

Contribution

It introduces a detailed, step-by-step approach to EFT, including scale separation, matching, RG running, and the application of SCET to IR divergences and gauge theories.

Findings

Demonstrates scale separation and matching in scalar EFTs.

Shows how RG techniques sum large logarithms.

Builds a scalar toy version of SCET for IR divergence analysis.

Abstract

These lectures aim to provide a pedagogical introduction to the philosophical underpinnings and technical features of Effective Field Theory (EFT). Improving control of $S$-matrix elements in the presence of a large hierarchy of physical scales $m \ll M$ is emphasized. Utilizing $\lambda \sim m/M$ as a power counting expansion parameter, we show how matching an ultraviolet (UV) model onto an EFT makes manifest the notion of separating scales. Renormalization Group (RG) techniques are used to run the EFT couplings from the UV to the infrared (IR), thereby summing large logarithms that would otherwise reduce the efficacy of perturbation theory. A variety of scalar field theory based toy examples are worked out in detail. An approach to consistently evolving a coupling across a heavy particle mass threshold is demonstrated. Applying the same method to the scalar mass term forces us to confront the hierarchy problem. The summation of a logarithm that lacks explicit dependence on an RG scale is performed. After reviewing the physics of IR divergences, we build a scalar toy version of Soft Collinear Effective Theory (SCET), highlighting many subtle aspects of these constructions. We show how SCET can be used to sum the soft and collinear IR Sudakov double logarithms that often appear for processes involving external interacting light-like particles. We conclude with the generalization of SCET to theories of gauge bosons coupled to charged fermions. These lectures were presented at TASI 2018.