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

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.

• The paper demonstrates that leading singularity methods fundamentally reproduce classical long-range gravitational corrections in cubic gravity.
• It employs unitarity cuts and one-loop amplitude computations to isolate non-analytic corrections to the static potential and black hole metrics.
• Explicit modifications for Einsteinian Cubic Gravity and Gauss-Bonnet terms highlight significant quantum and classical implications for gravitational phenomenology.

Scattering Amplitudes and Black Holes in Cubic Gravity: Amplitude Techniques and Singularities

Overview

This paper provides a comprehensive analysis of the classical and quantum corrections to gravitational potentials and black hole solutions within generic cubic (six-derivative) theories of gravity. By employing on-shell amplitude methods, the authors derive semi-classical corrections to the Newtonian potential and the corresponding modifications to black hole metrics. A significant emphasis is placed on contrasting the computational efficiency and structural insights offered by unitarity cut-based techniques and the leading singularity method within perturbative quantum gravity.

Formulation of Cubic Gravity and Amplitude Structure

Generic cubic gravity in four dimensions is parametrized by an action incorporating all possible contractions of three Riemann or Ricci tensors, extending the Einstein-Hilbert action by terms cubic in curvature invariants. The corresponding Lagrangian is controlled by coefficients $\beta_i$ and a mass-dimension $-2$ coupling $\lambda$. Notably, Einsteinian cubic gravity (ECG) propagates only the massless spin-2 sector, maintaining the on-shell spectrum of GR, and is captured by the specific choices of $\beta_i$.

At the amplitude level, cubic terms alter only the graviton self-interaction vertices—modifying the three-point amplitudes among identical helicity gravitons—while leaving matter-graviton couplings unaltered at leading order. This feature ensures that, at tree-level, the Newtonian potential and leading classical effects coincide with GR; modifications emerge at one-loop and higher.

The Gauss-Bonnet cubic invariant $G_3$, though topological in pure gravity, induces non-trivial effects when coupled to matter, giving rise to non-dispersive, non-analytic contributions that can survive in the static potential and black hole metrics.

Computation of One-Loop Amplitudes: Unitarity Cuts and Leading Singularities

The authors detail a full field-theoretic computation of the $2 \rightarrow 2$ amplitude for massive scalars. At one-loop, the relevant diagrams include triangle and bubble integrals, with only diagrams containing modified graviton three-point vertices sensitive to the cubic terms. Quantum and classical corrections to the potential are isolated by retaining the non-analytic (long-distance) parts of the amplitude, which are unaffected by regularization ambiguities.

A careful Passarino-Veltman reduction is performed, extracting leading terms in small momentum transfer $t$, facilitating straightforward identification of corrections to the static potential and, by extension, the black hole metric.

Importantly, the leading singularity method is introduced as an efficient alternative, employing complex analysis to localize loop integrals on maximal codimension singularities. This approach computes the relevant classical terms from triangle leading singularities (not box or bubble), significantly simplifying calculations and providing direct contact with the physically relevant residue structure of the amplitude. For cubic gravity, the authors demonstrate that the leading singularity fully reproduces the classical long-range contributions obtained via unitarity cuts, with substantially reduced computational effort.

Static Potential and Black Hole Metric Corrections

By Fourier transforming the non-relativistic limit of the amplitude, explicit corrections to the effective potential between two massive scalars are obtained. These corrections are organized into classical terms, scaling as $1/r^6$, and quantum terms, scaling as $\hbar/r^7$, with coefficients set by the underlying cubic couplings and masses.

For ECG ($\beta_1=12$, $\beta_2=1$) and Gauss-Bonnet $G_3$, the paper presents closed-form expressions:

• Einsteinian Cubic Gravity: The leading classical correction to the Schwarzschild solution is proportional to $G^4 \lambda m_A^2/r^6$, in precise agreement with black hole perturbations computed directly in the modified field equations. Quantum terms generate further, subleading corrections with explicit coefficients.
• Cubic Gauss-Bonnet: Non-dispersive corrections, proportional to invariants in $G_3$, contribute at the same order and are present even if GR is supplemented solely by $G_3$.

This analysis unifies the computation of quantum and classical corrections, showing that amplitude methods can efficiently recover results previously derived via considerably more complex perturbative solutions to higher-derivative field equations.

Implications for Theoretical and Practical Developments

The explicit demonstration that leading singularities capture the relevant classical physics has several key implications:

• Efficiency and Universality: Modern amplitude methods, particularly the leading singularity approach, provide a general and computationally tractable framework for deriving post-Minkowskian potentials and metric corrections in higher-derivative gravity. This method brings new efficiency to the calculation of terms appearing in the effective action after integrating out high-energy modes, with direct relevance for gravitational wave physics in the LIGO era.
• Black Hole Solutions and Non-dispersive Terms: The presence of non-dispersive terms in the potential, especially those connected to the Gauss-Bonnet cubic invariant, may have non-trivial implications for the spectrum, stability, and thermodynamics of black hole solutions in string-inspired and higher-curvature gravity theories.
• Quantum Corrections to Classical Gravity: The explicit quantum corrections calculated here provide concrete input for future studies of quantum gravitational effects in phenomenological models, particularly in regimes with strong gravity or at sub-Planckian distances.

Conclusion

The paper establishes that for a broad class of cubic and higher-derivative gravity theories, amplitude-based techniques, notably the leading singularity approach, are not only computationally advantageous but also provide direct and transparent access to both classical and quantum corrections to gravitational potentials and black hole metrics. The resulting corrections align with perturbative solutions known from the field-theoretic literature in ECG and Gauss-Bonnet gravity, but are obtained with significantly enhanced rigor and conceptual clarity. These results pave the way for systematic explorations of gravitational dynamics in modified gravity and effective field theory frameworks, with direct applications to gravitational phenomenology and quantum gravity research.