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Singularities in the RG flow

Antal Jakovac

Abstract

In this work, we show examples when a perturbatively irrelevant operator becomes relevant in the infrared because of the presence of an IR singularity (IR Landau pole). An example of this behavior is the four-fermion interaction that allows the formation of bound states. The reason of the appearance of the IR Landau pole is not the singular loop as in the purely perturbative case, but the infinite number of modes appearing in the RG flow.

Singularities in the RG flow

Abstract

In this work, we show examples when a perturbatively irrelevant operator becomes relevant in the infrared because of the presence of an IR singularity (IR Landau pole). An example of this behavior is the four-fermion interaction that allows the formation of bound states. The reason of the appearance of the IR Landau pole is not the singular loop as in the purely perturbative case, but the infinite number of modes appearing in the RG flow.
Paper Structure (12 sections, 70 equations, 1 figure)

This paper contains 12 sections, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The problem
  3. Scaling dimension and relevance
  4. Anomalies in scaling
  5. The failure of the scaling argumentation
  6. The RG equations of the extended QED
  7. The electron-proton scattering interaction
  8. Explicit contribution to the running of $\lambda$
  9. Self-consistent $\lambda$ evolution
  10. Discussion: bound state pole in the renormalization group
  11. Conclusion
  12. FRG calculation of the beta functions

Figures (1)

  • Figure 1: The qualitative behavior of the rescaled 4-fermion coupling. At high scale, even if it starts with a value around unity, it decreases with a power law, rendering it irrelevant at lower scales. At a certain $k=K$ scale, however, it reaches a pole, and it becomes relevant again.