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On powercounting in perturbative quantum gravity theories through color-kinematic duality

Rutger H. Boels, Reinke Sven Isermann

TL;DR

The paper investigates perturbative quantum gravity UV behavior through color-kinematic duality and the BCJ double-copy construction. By reformulating the duality as a linear-map problem with a generalized inverse, it derives the large-$z$ scaling of gravity tree and loop integrands from gauge theory data, arguing that cancellations are loop-order independent under plausible assumptions. It then applies the same linear-map framework to estimate UV divergences, reproducing known critical dimensions (e.g., for $ ext{N}=8$ supergravity) and clarifying how tadpoles and multi-trace structures influence the results. The work provides a unifying perspective that links on-shell BCFW behavior, off-shell numerator scaling, and UV power counting via a common linear-algebraic structure, while highlighting key open questions about loop-level color-kinematic duality and the full role of vanishing integrand contributions.

Abstract

The standard argument why gravity is not renormalisable relies on direct powercounting of Feynman graphs to estimate the degree of UV divergence. This analysis has in several (highly) supersymmetric examples be shown to overestimate divergences considerably. In these examples the main improvements arise from a conjectured duality between color and kinematics. In this paper we initiate the systematic study of quite general powercounting under the assumption that color-kinematic duality exists. The main technical tool is a reformulation of the duality in terms of linear maps, modulo subtleties at loop level mostly inherent to the duality. This tool may have wider applications in both gauge and gravity theories, up to resolution of the subtleties. Here it is first applied to the large Britto-Cachazo-Feng-Witten (BCFW) shift behavior of gravity integrands constructed through the duality. Assuming color-kinematic duality and reasonable technical requirements hold these shifts are shown to be independent of loop order, which would imply massive cancellations with respect to the Feynman graph expression. More speculatively, the same approach is then applied to provide estimates of the overall degree of UV divergence in quite general gravity theories, assuming the duality exists. The cancellations obtained in these estimates depends on the exact implementation of the duality at loop level, especially on graph topology. Finally, some evidence for the duality to all loop orders is provided from an analysis of BCFW shifts of gauge theory integrands through Feynman graphs.

On powercounting in perturbative quantum gravity theories through color-kinematic duality

TL;DR

The paper investigates perturbative quantum gravity UV behavior through color-kinematic duality and the BCJ double-copy construction. By reformulating the duality as a linear-map problem with a generalized inverse, it derives the large- scaling of gravity tree and loop integrands from gauge theory data, arguing that cancellations are loop-order independent under plausible assumptions. It then applies the same linear-map framework to estimate UV divergences, reproducing known critical dimensions (e.g., for supergravity) and clarifying how tadpoles and multi-trace structures influence the results. The work provides a unifying perspective that links on-shell BCFW behavior, off-shell numerator scaling, and UV power counting via a common linear-algebraic structure, while highlighting key open questions about loop-level color-kinematic duality and the full role of vanishing integrand contributions.

Abstract

The standard argument why gravity is not renormalisable relies on direct powercounting of Feynman graphs to estimate the degree of UV divergence. This analysis has in several (highly) supersymmetric examples be shown to overestimate divergences considerably. In these examples the main improvements arise from a conjectured duality between color and kinematics. In this paper we initiate the systematic study of quite general powercounting under the assumption that color-kinematic duality exists. The main technical tool is a reformulation of the duality in terms of linear maps, modulo subtleties at loop level mostly inherent to the duality. This tool may have wider applications in both gauge and gravity theories, up to resolution of the subtleties. Here it is first applied to the large Britto-Cachazo-Feng-Witten (BCFW) shift behavior of gravity integrands constructed through the duality. Assuming color-kinematic duality and reasonable technical requirements hold these shifts are shown to be independent of loop order, which would imply massive cancellations with respect to the Feynman graph expression. More speculatively, the same approach is then applied to provide estimates of the overall degree of UV divergence in quite general gravity theories, assuming the duality exists. The cancellations obtained in these estimates depends on the exact implementation of the duality at loop level, especially on graph topology. Finally, some evidence for the duality to all loop orders is provided from an analysis of BCFW shifts of gauge theory integrands through Feynman graphs.

Paper Structure

This paper contains 21 sections, 2 theorems, 157 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Review of concepts
  3. Color-kinematic duality
  4. Solving Jacobi relations: the $\textrm{D}^3\textrm{M}$ basis
  5. BCFW shifts and on-shell recursion
  6. Generalized inverses
  7. BCFW shifts of gravity amplitudes from gauge theory
  8. Warmup: four-point amplitudes revisited
  9. Rephrasing tree level color-kinematic duality as linear algebra
  10. BCFW shifts of gravity amplitudes constructed by double copy
  11. BCFW shifts of gravity integrands from gauge theory
  12. BCFW shifts of kinematic numerators using generalized inverses
  13. BCFW shifts of gravity integrands constructed by double copy
  14. Estimates of UV divergences assuming color-kinematic duality
  15. Towards an off-shell understanding of numerator scaling
  16. ...and 6 more sections

Key Result

Theorem 1

For every tree level amplitude in a theory with adjoint matter only one can find numerators $n_j$ such that they satisfy the Jacobi relations of equation eq:colorkinematicduality and reproduce the amplitudes through eq:BCJtree.

Figures (8)

  • Figure 1: A graphical illustration of the Jacobi relations.
  • Figure 2: Schematic application of the Jacobi relations (red ellipsis) to maximize the distance between two singled out legs (denoted by the hat) in a given graph.
  • Figure 3: Contact graphs appearing at three loops (left) and at four loops (right).
  • Figure 4: Possible four loop tadpole graph made of boxes.
  • Figure 5: The bubble graph divergence at one loop
  • ...and 3 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2