Leading Singularities and Classical Gravitational Scattering

TL;DR

The paper develops leading singularities (LS) as a principled on-shell tool to extract classical gravitational scattering information for two massive bodies. It shows that at one loop the double discontinuity in the $t$-channel, realized by triangle-type LS, contains the full classical piece, while box-type LS do not contribute to the classical part; a compact fully relativistic one-loop formula for the classical amplitude is obtained and reconciled with the post-Newtonian expansion in the non-relativistic limit. The authors then extend the LS framework to higher loops, demonstrating an infinite ladder family and detailing how multiple $t$-channel dispersions project out quantum pieces, suggesting a path to all-loop classical results. The work argues that classical gravitational physics can be built from on-shell building blocks, offering a potentially scalable route to PN corrections and insights into the analytic structure of gravity amplitudes. Overall, it provides a novel, on-shell-centric method to derive classical potentials from quantum amplitudes with clear connections to known PN results and future multi-loop avenues.

Abstract

In this work we propose to use leading singularities to obtain the classical pieces of amplitudes of two massive particles whose only interaction is gravitational. Leading singularities are generalizations of unitarity cuts. At one-loop we find that leading singularities obtained by multiple discontinuities in the t-channel contain all the classical information. As the main example, we show how to obtain a compact formula for the fully relativistic classical one-loop contribution to the scattering of two particles with different masses. The non-relativistic limit of the leading singularity agrees with known results in the post-Newtonian expansion. We also compute a variety of higher loop leading singularities including some all-loop families and study some of their properties.

Figures (7)

• Figure 1: Scattering process of massive scalars. Throughout the text we will extensively discuss the case $m_1=m_2=m_{\textsc{a}}$ and $m_3=m_4=m_{\textsc{b}}$. Note that all momenta are incoming.
• Figure 2: a) Box with internal and external massless particles. b) Triangle with massive external particles and massless internal particles. All lines in this work represent on-shell particles.
• Figure 3: The box and triangle leading singularities can be computed from the triple-cut diagram.
• Figure 4: Alternative parametrization of the loop momenta in the scalar triangle diagram.
• Figure 5: $2$-loop nested triangle. The solid lines represent massive particles exchanging massless states, represented by dashed lines.