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Higher Derivative Sigma Models

John F. Donoghue, Gabriel Menezes

TL;DR

The paper argues that in higher-derivative sigma models, formal renormalization-group running does not always translate into physical running in scattering amplitudes. It develops a framework combining heat-kernel analysis and Passarino-Veltman reduction to separate kinematic (logarithmic) running from mass-scale effects, revealing region-dependent behavior with an intrinsic scale $m$. A key result is that the fundamental coupling $f$ in the HD SU(N) nonlinear sigma model does not run at one loop in physical processes, contrary to previous claims of asymptotic freedom; other couplings exhibit running tied to the low-energy EFT or the high-energy full theory. The work highlights limitations of relying solely on divergences or FRG for physical predictions in theories with multiple scales and nonlocal log structures, and offers a practical roadmap for extracting physical beta functions in such theories.

Abstract

We explore the nature of running couplings in the higher derivative linear and nonlinear sigma models and show that the results in dimensional regularization for the physical running couplings do not always match the values quoted in the literature. Heat kernel methods identify divergences correctly, but not all of these divergences are related to physical running couplings. Likewise the running found using the Functional Renormalization Group does not always appear as running couplings in physical processes, even for the case of logarithmic running. The basic coupling of the higher derivative SU(N) nonlinear sigma model does not run at all at one loop, in contrast to published claims for asymptotic freedom. At one loop we describe how to properly identify the physical running couplings in these theories, and provide revised numbers for the higher derivative nonlinear sigma model.

Higher Derivative Sigma Models

TL;DR

The paper argues that in higher-derivative sigma models, formal renormalization-group running does not always translate into physical running in scattering amplitudes. It develops a framework combining heat-kernel analysis and Passarino-Veltman reduction to separate kinematic (logarithmic) running from mass-scale effects, revealing region-dependent behavior with an intrinsic scale . A key result is that the fundamental coupling in the HD SU(N) nonlinear sigma model does not run at one loop in physical processes, contrary to previous claims of asymptotic freedom; other couplings exhibit running tied to the low-energy EFT or the high-energy full theory. The work highlights limitations of relying solely on divergences or FRG for physical predictions in theories with multiple scales and nonlocal log structures, and offers a practical roadmap for extracting physical beta functions in such theories.

Abstract

We explore the nature of running couplings in the higher derivative linear and nonlinear sigma models and show that the results in dimensional regularization for the physical running couplings do not always match the values quoted in the literature. Heat kernel methods identify divergences correctly, but not all of these divergences are related to physical running couplings. Likewise the running found using the Functional Renormalization Group does not always appear as running couplings in physical processes, even for the case of logarithmic running. The basic coupling of the higher derivative SU(N) nonlinear sigma model does not run at all at one loop, in contrast to published claims for asymptotic freedom. At one loop we describe how to properly identify the physical running couplings in these theories, and provide revised numbers for the higher derivative nonlinear sigma model.
Paper Structure (14 sections, 171 equations, 1 figure)

This paper contains 14 sections, 171 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Identifying running couplings
  3. Theories with higher derivatives
  4. Renormalization group techniques and physical running
  5. Structure for the paper
  6. The higher derivative linear sigma model
  7. Renormalization without running
  8. Running without renormalization
  9. The EFT divide
  10. The U(1) non-linear sigma model
  11. The SU(N) non-linear sigma model
  12. Identifying physical running
  13. One-loop effective action for the SU(N) non-linear sigma model
  14. Discussion and conclusions

Figures (1)

  • Figure 1: Diagrams (a) and (b) are referred to as tadpole diagrams, with any number of external legs. Diagram (c) is a bubble diagram. The important distinction is whether the external momentum flows through the loop or not.