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Scattering Amplitudes, Black Holes and Leading Singularities in Cubic Theories of Gravity

William T Emond, Nathan Moynihan

TL;DR

This work investigates cubic theories of gravity through modern scattering amplitudes to derive a semi-classical potential and black-hole corrections. By computing the one-loop amplitude both via unitarity cuts and via the leading singularity, the authors extract the classical gravitational effects of higher-derivative terms, showing consistency between methods and connecting the potential to a corrected Schwarzschild-like metric. The results encompass ECG and Gauss-Bonnet cubic invariants, revealing quantum corrections to the classical metric and providing explicit expressions for the potential in various string-inspired and $G_3$-inclusive scenarios. The study demonstrates a practical amplitude-based route to nontrivial gravitational corrections, with implications for understanding higher-derivative gravity and its phenomenology in the strong-field regime.

Abstract

We compute the semi-classical potential arising from a generic theory of cubic gravity, a higher derivative theory of spin-2 particles, in the framework of modern amplitude techniques. We show that there are several interesting aspects of the potential, including some non-dispersive terms that lead to black hole solutions (include quantum corrections) that agree with those derived in Einsteinian cubic gravity (ECG). We show that these non-dispersive terms could be obtained from theories that include the Gauss-Bonnet cubic invariant $G_3$. In addition, we derive the one-loop scattering amplitudes using both unitarity cuts and via the leading singularity, showing that the classical effects of higher derivative gravity can be easily obtained directly from the leading singularity with far less computational cost.

Scattering Amplitudes, Black Holes and Leading Singularities in Cubic Theories of Gravity

TL;DR

This work investigates cubic theories of gravity through modern scattering amplitudes to derive a semi-classical potential and black-hole corrections. By computing the one-loop amplitude both via unitarity cuts and via the leading singularity, the authors extract the classical gravitational effects of higher-derivative terms, showing consistency between methods and connecting the potential to a corrected Schwarzschild-like metric. The results encompass ECG and Gauss-Bonnet cubic invariants, revealing quantum corrections to the classical metric and providing explicit expressions for the potential in various string-inspired and -inclusive scenarios. The study demonstrates a practical amplitude-based route to nontrivial gravitational corrections, with implications for understanding higher-derivative gravity and its phenomenology in the strong-field regime.

Abstract

We compute the semi-classical potential arising from a generic theory of cubic gravity, a higher derivative theory of spin-2 particles, in the framework of modern amplitude techniques. We show that there are several interesting aspects of the potential, including some non-dispersive terms that lead to black hole solutions (include quantum corrections) that agree with those derived in Einsteinian cubic gravity (ECG). We show that these non-dispersive terms could be obtained from theories that include the Gauss-Bonnet cubic invariant . In addition, we derive the one-loop scattering amplitudes using both unitarity cuts and via the leading singularity, showing that the classical effects of higher derivative gravity can be easily obtained directly from the leading singularity with far less computational cost.

Paper Structure

This paper contains 11 sections, 78 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Cubic Theories of Gravity
  3. Scattering Amplitudes and the Effective Potential
  4. One Loop Amplitude
  5. Unitarity Cuts
  6. Leading Singularity
  7. Potential and Black Hole Solutions
  8. Discussion
  9. Acknowledgements
  10. Tree-Level 4-point Amplitude
  11. Fourier Transforms

Figures (4)

  • Figure 1: The kinematic setup, where particles 1 and 4 are incoming and 2 and 3 outgoing. In the center of mass frame, we consider the exchanged momentum $q = P_1+P_2 = (0,\textbf{q})$ (in the all outgoing convention) and $P_1 = (E_A,\textbf{q}/2)$, $P_2 = -(E_A,-\textbf{q}/2)$ with $E_A = \sqrt{m_A^2 + \textbf{p}^2 + \frac{\textbf{q}^2}{4}}$. $P_3$ and $P_4$ are defined similarly with $A\leftrightarrow B$.
  • Figure 2: Double Cut Diagram
  • Figure 3: Tree Diagrams
  • Figure 4: LS Triangle Diagram