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Perturbative Gravity from QCD Amplitudes

Z. Bern, A. K. Grant

TL;DR

This paper establishes that perturbative gravity amplitudes can be constructed from QCD gluon amplitudes using KLT relations, enabling a gravity Lagrangian to be derived directly from gauge-theory data. It provides an explicit construction up to five gravitons and shows how field redefinitions and a suitable gauge choice map the resulting Lagrangian to the Einstein–Hilbert action, yielding simpler gravity Feynman rules. The approach also supports unitarity-based loop techniques by recycling gauge-theory results and demonstrates concrete applications such as gravity soft factors derived from QCD. Overall, it deepens the connection between gauge theory and gravity and points to broader extensions to supergravity and curved spacetimes.

Abstract

We demonstrate that QCD gluon amplitudes can be used to construct a Lagrangian for gravity. This procedure makes use of perturbative `squaring' relations between gravity and gauge theory that follow from string theory. We explicitly carry out the construction for up to five-point interactions and discuss a set of field variables in the Einstein-Hilbert Lagrangian for interpreting the Lagrangian obtained from QCD. A spin-off from our analysis is that it can be used to provide simpler tree-level gravity Feynman rules than for conventional gauges.

Perturbative Gravity from QCD Amplitudes

TL;DR

This paper establishes that perturbative gravity amplitudes can be constructed from QCD gluon amplitudes using KLT relations, enabling a gravity Lagrangian to be derived directly from gauge-theory data. It provides an explicit construction up to five gravitons and shows how field redefinitions and a suitable gauge choice map the resulting Lagrangian to the Einstein–Hilbert action, yielding simpler gravity Feynman rules. The approach also supports unitarity-based loop techniques by recycling gauge-theory results and demonstrates concrete applications such as gravity soft factors derived from QCD. Overall, it deepens the connection between gauge theory and gravity and points to broader extensions to supergravity and curved spacetimes.

Abstract

We demonstrate that QCD gluon amplitudes can be used to construct a Lagrangian for gravity. This procedure makes use of perturbative `squaring' relations between gravity and gauge theory that follow from string theory. We explicitly carry out the construction for up to five-point interactions and discuss a set of field variables in the Einstein-Hilbert Lagrangian for interpreting the Lagrangian obtained from QCD. A spin-off from our analysis is that it can be used to provide simpler tree-level gravity Feynman rules than for conventional gauges.

Paper Structure

This paper contains 5 sections, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of the KLT relations and applications
  3. Construction of a gravity Lagrangian from QCD
  4. Sample applications
  5. Discussion

Figures (3)

  • Figure 1: One can recycle tree amplitudes into loop amplitudes via $D$-dimensional unitarity cuts.
  • Figure 2: A sample diagram for showing that we may use propagator (\ref{['NewPropagator']}) instead of (\ref{['SymmetricPropagator']}).
  • Figure 3: One can obtain a four vertex with the left-right factorization property by subtracting the diagrams containing the kinematic poles from the $S$-matrix.