Effective field theory and classical equations of motion

TL;DR

This paper develops a rigorous approach to effective field theories (EFTs) derived from UV theories with heavy and light fields, ensuring well-posedness and approximation accuracy of classical solutions.

Contribution

It introduces a method to make EFT equations well-posed and demonstrates their effectiveness in approximating classical UV solutions, including a modified EFT for non-approximable solutions.

Findings

EFT equations can be made well-posed with the proposed approach.

Classical solutions of UV theories are well approximated by EFT solutions.

Non-approximable solutions are close to solutions of a modified EFT.

Abstract

Given a theory containing both heavy and light fields (the UV theory), a standard procedure is to integrate out the heavy field to obtain an effective field theory (EFT) for the light fields. Typically the EFT equations of motion consist of an expansion involving higher and higher derivatives of the fields, whose truncation at any finite order may not be well-posed. In this paper we address the question of how to make sense of the EFT equations of motion, and whether they provide a good approximation to the classical UV theory. We propose an approach to solving EFTs which leads to a well-posedness statement. For a particular choice of UV theory we rigorously derive the corresponding EFT and show that a large class of classical solutions to the UV theory are well approximated by EFT solutions. We also consider solutions of the UV theory which are not well approximated by EFT solutions and demonstrate that these are close, in an averaged sense, to solutions of a modified EFT.