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The gravitational cusp anomalous dimension from AdS space

D. J. Miller, C. D. White

TL;DR

This work extends the radial AdS Wilson line framework from gauge theories to perturbative gravity, showing that the gravitational cusp anomalous dimension $\,\Gamma\,$ can be obtained from the Newtonian limit in Euclidean AdS space. It demonstrates a unified, interpolating formulation between QED and gravity through a general operator ${\cal W}_n$ with a continuous parameter $n$, and analyzes both the massive (nonrelativistic) and lightlike limits, including the cancellation of soft collinear singularities in gravity. The authors derive the gravity cusp anomalous dimension from a static-energy calculation in radial space and provide a general method to construct such quantities via modified current densities to enforce proper boundary conditions. These results illuminate structural similarities between gauge and gravity IR behavior and offer a framework for exploring gauge–gravity connections and potential conformal gauge approaches in Wilson-line calculations.

Abstract

Recently a new picture has been developed for examining Wilson lines, and the corresponding anomalous dimensions which govern their renormalization properties. By making a particular coordinate transform, the calculation of the cusp anomalous dimension in QED or QCD can be related to the energy of a pair of static charges in Euclidean Anti-de-Sitter (AdS) space. This paper shows how the same picture can be used to describe Wilson lines in quantum gravity. We show how the relevant cusp anomalous dimension (which has recently been shown to be one loop exact) can be obtained using the Newtonian limit of General Relativity. We also show how both the QED and gravity cases emerge as special cases of a general formulation, and that a continuous parameter exists which interpolates between them. The results may be useful in examining the relations between gauge and gravity theories.

The gravitational cusp anomalous dimension from AdS space

TL;DR

This work extends the radial AdS Wilson line framework from gauge theories to perturbative gravity, showing that the gravitational cusp anomalous dimension can be obtained from the Newtonian limit in Euclidean AdS space. It demonstrates a unified, interpolating formulation between QED and gravity through a general operator with a continuous parameter , and analyzes both the massive (nonrelativistic) and lightlike limits, including the cancellation of soft collinear singularities in gravity. The authors derive the gravity cusp anomalous dimension from a static-energy calculation in radial space and provide a general method to construct such quantities via modified current densities to enforce proper boundary conditions. These results illuminate structural similarities between gauge and gravity IR behavior and offer a framework for exploring gauge–gravity connections and potential conformal gauge approaches in Wilson-line calculations.

Abstract

Recently a new picture has been developed for examining Wilson lines, and the corresponding anomalous dimensions which govern their renormalization properties. By making a particular coordinate transform, the calculation of the cusp anomalous dimension in QED or QCD can be related to the energy of a pair of static charges in Euclidean Anti-de-Sitter (AdS) space. This paper shows how the same picture can be used to describe Wilson lines in quantum gravity. We show how the relevant cusp anomalous dimension (which has recently been shown to be one loop exact) can be obtained using the Newtonian limit of General Relativity. We also show how both the QED and gravity cases emerge as special cases of a general formulation, and that a continuous parameter exists which interpolates between them. The results may be useful in examining the relations between gauge and gravity theories.

Paper Structure

This paper contains 11 sections, 120 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of necessary concepts
  3. The radial picture for Wilson lines
  4. Infrared singularities in Quantum GR
  5. Gravitational Wilson lines in the radial picture
  6. Gravitational Wilson lines in the lightlike limit
  7. General formulation
  8. Conclusion
  9. Newton's law from the gravitational Wilson line
  10. Normalisation of the solution to Newton's equation
  11. Field theory calculation of the gravitational cusp anomalous dimension

Figures (4)

  • Figure 1: The emission of a graviton from an external scalar line.
  • Figure 2: (a) Wilson loop contour in Minkowski space used for the calculation of the Newtonian potential; (b) the relevant one loop diagram.
  • Figure 3: The 3-surface used to normalise the coefficient $A_1$ in the general solution to the homogeneous Newton equation. The horizontal plane is representative of the spacelike directions, where the radial distance in this plane represents $\beta$.
  • Figure 4: Diagram entering the calculation of the cusp anomalous dimension.