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Perturbative Quantum Gravity and its Relation to Gauge Theory

Zvi Bern

TL;DR

The paper articulates a principled link between perturbative gauge theories and gravity through KLT relations, enabling gravity amplitudes to be constructed from gauge-theory data. It then leverages D-dimensional unitarity to compute loop amplitudes in gravity from tree-level inputs, demonstrating substantial improvements in the ultraviolet behavior of maximal N=8 supergravity. Concrete results include one- and two-loop graviton amplitudes derived from gauge theory, the all-plus helicity sequences, and a general finiteness bound l < 10/(D−2). The work highlights both the practical power of this approach and the open theoretical challenge of deriving KLT directly from the Einstein–Hilbert Lagrangian, as well as the broader implications for gravity as a gauge-theory cousin in perturbation theory.

Abstract

In this review we describe a non-trivial relationship between perturbative gauge theory and gravity scattering amplitudes. At the semi-classical or tree level, the scattering amplitudes of gravity theories in flat space can be expressed as a sum of products of well defined pieces of gauge theory amplitudes. These relationships were first discovered by Kawai, Lewellen and Tye in the context of string theory, but hold more generally. In particular, they hold for standard Einstein gravity. A method based on D-dimensional unitarity can then be used to systematically construct all quantum loop corrections order-by-order in perturbation theory using as input the gravity tree amplitudes expressed in terms of gauge theory ones. More generally, the unitarity method provides a means for perturbatively quantizing massless gravity theories without the usual formal apparatus associated with the quantization of constrained systems. As one application, this method was used to demonstrate that maximally supersymmetric gravity is less divergent in the ultraviolet than previously thought.

Perturbative Quantum Gravity and its Relation to Gauge Theory

TL;DR

The paper articulates a principled link between perturbative gauge theories and gravity through KLT relations, enabling gravity amplitudes to be constructed from gauge-theory data. It then leverages D-dimensional unitarity to compute loop amplitudes in gravity from tree-level inputs, demonstrating substantial improvements in the ultraviolet behavior of maximal N=8 supergravity. Concrete results include one- and two-loop graviton amplitudes derived from gauge theory, the all-plus helicity sequences, and a general finiteness bound l < 10/(D−2). The work highlights both the practical power of this approach and the open theoretical challenge of deriving KLT directly from the Einstein–Hilbert Lagrangian, as well as the broader implications for gravity as a gauge-theory cousin in perturbation theory.

Abstract

In this review we describe a non-trivial relationship between perturbative gauge theory and gravity scattering amplitudes. At the semi-classical or tree level, the scattering amplitudes of gravity theories in flat space can be expressed as a sum of products of well defined pieces of gauge theory amplitudes. These relationships were first discovered by Kawai, Lewellen and Tye in the context of string theory, but hold more generally. In particular, they hold for standard Einstein gravity. A method based on D-dimensional unitarity can then be used to systematically construct all quantum loop corrections order-by-order in perturbation theory using as input the gravity tree amplitudes expressed in terms of gauge theory ones. More generally, the unitarity method provides a means for perturbatively quantizing massless gravity theories without the usual formal apparatus associated with the quantization of constrained systems. As one application, this method was used to demonstrate that maximally supersymmetric gravity is less divergent in the ultraviolet than previously thought.

Paper Structure

This paper contains 21 sections, 60 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Outline of Traditional Approach to Perturbative Quantum Gravity
  3. Overview of Gravity Feynman Rules
  4. Divergences in Quantum Gravity
  5. Gravity and Gauge Theory Feynman Rules
  6. The Kawai-Lewellen-Tye Relations
  7. The KLT Relations in String Theory
  8. The KLT Relations in Field Theory
  9. Tree-level Applications
  10. Soft and Collinear Properties of Gravity Amplitudes from Gauge Theory
  11. The Einstein-Hilbert Lagrangian and Gauge Theory
  12. From Trees to Loops
  13. Gravity Loop Amplitudes from Gauge Theory
  14. One-loop Four-point Example
  15. Arbitrary Numbers of Legs at One Loop
  16. ...and 6 more sections

Figures (13)

  • Figure 1: The Feynman propagator and three- and four-point vertices in Einstein gravity.
  • Figure 2: Sample gravity tree-level Feynman diagrams. The lines represent any particles in a gravity theory.
  • Figure 3: Sample loop-level Feynman diagrams. Each additional loop represents an extra power of Planck's constant.
  • Figure 4: An example of a five-loop diagram.
  • Figure 5: String theory suggests that the three-graviton vertex can be expressed in terms of products of three-gluon vertices.
  • ...and 8 more figures