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Scale without Conformal Invariance: An Example

Jean-François Fortin, Benjamín Grinstein, Andreas Stergiou

TL;DR

The paper establishes that scale invariance can coexist with non-conformal behavior in a relativistic quantum field theory, realized through limit-cycle RG trajectories in D=4−ε. It provides an explicit example with two Weyl spinors and two real scalars showing scale invariance on a cyclic RG flow, while also proving that scale implies conformal invariance for scalar-only models up to two loops and for models with one real scalar and any number of Weyl spinors to all orders. By analyzing the dilatation current and the associated Q and P obstructions, the authors connect the existence of limit cycles to the underlying symmetry structure and RG dynamics. These results open pathways to exploring scale without conformal invariance in other dimensions and setups, with potential implications for fundamental QFT and beyond.

Abstract

We give an explicit example of a model in D=4-epsilon space-time dimensions that is scale but not conformally invariant, is unitary, and has finite correlators. The invariance is associated with a limit cycle renormalization group (RG) trajectory. We also prove, to second order in the loop expansion, in D=4-epsilon, that scale implies conformal invariance for models of any number of real scalars. For models with one real scalar and any number of Weyl spinors we show that scale implies conformal invariance to all orders in perturbation theory.

Scale without Conformal Invariance: An Example

TL;DR

The paper establishes that scale invariance can coexist with non-conformal behavior in a relativistic quantum field theory, realized through limit-cycle RG trajectories in D=4−ε. It provides an explicit example with two Weyl spinors and two real scalars showing scale invariance on a cyclic RG flow, while also proving that scale implies conformal invariance for scalar-only models up to two loops and for models with one real scalar and any number of Weyl spinors to all orders. By analyzing the dilatation current and the associated Q and P obstructions, the authors connect the existence of limit cycles to the underlying symmetry structure and RG dynamics. These results open pathways to exploring scale without conformal invariance in other dimensions and setups, with potential implications for fundamental QFT and beyond.

Abstract

We give an explicit example of a model in D=4-epsilon space-time dimensions that is scale but not conformally invariant, is unitary, and has finite correlators. The invariance is associated with a limit cycle renormalization group (RG) trajectory. We also prove, to second order in the loop expansion, in D=4-epsilon, that scale implies conformal invariance for models of any number of real scalars. For models with one real scalar and any number of Weyl spinors we show that scale implies conformal invariance to all orders in perturbation theory.

Paper Structure

This paper contains 16 sections, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Introduction
  3. Preliminaries
  4. Preliminaries
  5. Trajectories and enhanced-symmetry fixed points
  6. Trajectories and enhanced-symmetry fixed points
  7. Scale implies conformal: two classes of models in 4-epsilon
  8. Scale implies conformal: two classes of models in 4-epsilon
  9. Models of scalars
  10. Models of scalars
  11. Models of Weyl spinors and one real scalar
  12. Models of Weyl spinors and one real scalar
  13. The simplest example
  14. The simplest example
  15. Discussion and conclusion
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The couplings of the model with one Weyl spinor and two real scalars on an RG cycle, as a function of RG time. Here $\epsilon=0.01$ and the starting conditions are the solutions to given in the text, Eqs. \ref{['ScaleInvPoint']}.
  • Figure 2: Artistic rendition of scale-invariant trajectories as a function of $\epsilon$.