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Auxiliary counterterms and their role in effective field theory

Manuel Pavon Valderrama

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.

Auxiliary counterterms and their role in effective field theory

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.
Paper Structure (17 sections, 101 equations)

This paper contains 17 sections, 101 equations.

Table of Contents

  1. Introduction
  2. Exact versus EFT Cutoff Independence
  3. Pionless EFT with PDS
  4. Pionless EFT with a cutoff
  5. The cutoff as a light scale
  6. The cutoff as the EFT convergence dial
  7. Improved actions and their interpretation
  8. Convergence around a bound state
  9. Comparing the EFT and $\hbar$ expansions
  10. The inconsistency
  11. Solving the inconsistency
  12. Power counting and the $\hbar$ expansion
  13. A matter of interpretation?
  14. Perturbative and non-perturbative renormalization
  15. Convergence of the perturbative series
  16. ...and 2 more sections