Auxiliary counterterms and their role in effective field theory

TL;DR

This paper discusses the role of contact-range interactions and auxiliary counterterms in effective field theories, emphasizing their necessity, redundancy, and utility in ensuring cutoff independence and improving convergence.

Contribution

It clarifies the distinction between physical and auxiliary counterterms, highlighting their roles in renormalization and effective field theory improvements.

Findings

Cutoff independence naturally generates counterterms encoding physical information.

Auxiliary counterterms can resolve renormalization inconsistencies and enhance convergence.

Residual cutoff dependence is typically negligible compared to EFT uncertainties.

Abstract

Effective field theories include contact-range interactions (or counterterms) for two reasons: representing the unknown short-range physics in a model independent manner and ensuring the cutoff independence of observables. Both are intertwined: cutoff independence alone (modulo truncation errors) already generates counterterms encoding physical information not present in the known long-range physics. Yet, there is also residual cutoff dependence, which is smaller than the uncertainties that are achievable within the effective field theory description and thus can be safely neglected in most settings. If one insists on exact cutoff independence though, new counterterms will be required, but they encode no new physical information and are thus what one could call redundant, or auxiliary, counterterms. It happens that auxiliary counterterms are still useful for solving certain inconsistencies that appear during renormalization or for improving the convergence of the effective field theory expansion. Examples of these use cases are discussed, including the interpretation of the improved actions or the relation between perturbative and non-perturbative renormalization.