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A Non-Relativistic Weyl Anomaly

Ido Adam, Ilarion V. Melnikov, Stefan Theisen

TL;DR

This work analyzes Weyl anomalies for a $z=3$ Lifshitz scalar in a four-dimensional Horava–Lifshitz gravity background. It develops a position-space, ε-regularization technique and validates it against relativistic cases before applying it to the Lifshitz system, extracting anomaly data from two-point functions of conserved quantities. The main finding is a type-B Weyl anomaly at one-loop in the Lifshitz setting, with no type-A contribution at this order, and the authors discuss implications for the UV structure of HL gravity and possible higher-point/type-A insights via consistency relations. The paper also outlines a roadmap for extending the analysis to higher-point correlators and more general backgrounds, which could sharpen constraints on Lifshitz theories with matter.

Abstract

We examine the Weyl anomaly for a four-dimensional z=3 Lifshitz scalar coupled to Horava's theory of anisotropic gravity. We find a one-loop break-down of scale-invariance at second order in the gravitational background.

A Non-Relativistic Weyl Anomaly

TL;DR

This work analyzes Weyl anomalies for a Lifshitz scalar in a four-dimensional Horava–Lifshitz gravity background. It develops a position-space, ε-regularization technique and validates it against relativistic cases before applying it to the Lifshitz system, extracting anomaly data from two-point functions of conserved quantities. The main finding is a type-B Weyl anomaly at one-loop in the Lifshitz setting, with no type-A contribution at this order, and the authors discuss implications for the UV structure of HL gravity and possible higher-point/type-A insights via consistency relations. The paper also outlines a roadmap for extending the analysis to higher-point correlators and more general backgrounds, which could sharpen constraints on Lifshitz theories with matter.

Abstract

We examine the Weyl anomaly for a four-dimensional z=3 Lifshitz scalar coupled to Horava's theory of anisotropic gravity. We find a one-loop break-down of scale-invariance at second order in the gravitational background.

Paper Structure

This paper contains 16 sections, 98 equations.

Table of Contents

  1. Introduction
  2. The Lifshitz scalar in a gravitational background
  3. The free scalar in flat space
  4. The scalar in a curved HLG background
  5. Conservation laws in flat space
  6. Quantum corrections and the anomaly
  7. A regularization and position space computations
  8. The regularization scheme
  9. The relativistic scalar in two dimensions
  10. The relativistic scalar in four dimensions
  11. The Weyl anomaly
  12. Counter-term analysis
  13. Scale dependence
  14. Discussion
  15. The regularized anisotropic $\delta$-function
  16. ...and 1 more sections