Holomorphic Classical Limit for Spin Effects in Gravitational and Electromagnetic Scattering

TL;DR

The paper presents a covariant, spin-agnostic framework to extract the classical piece of gravitational and electromagnetic scattering amplitudes for arbitrary spin, using a holomorphic classical limit and Leading Singularity techniques to capture all multipole contributions. The method yields universal spin-multipole structures and enables explicit matching to standard EFT operators for low spins, while also handling massless probes to reproduce light-bending results. By introducing a massive spinor-helicity representation, the authors extend the analysis to spinning external particles and show that the resulting spin-dependent interactions are universal and align with known 1-loop and EFT results. This approach provides a streamlined, relativistic path to PN-like corrections and suggests deeper connections to the double-copy structure between gravity and gauge theory, with potential extensions to higher loops and finite-size effects.

Abstract

We provide universal expressions for the classical piece of the amplitude given by the graviton/photon exchange between massive particles of arbitrary spin, at both tree and one loop level. In the gravitational case this leads to higher order terms in the post-Newtonian expansion, which have been previously used in the binary inspiral problem. The expressions are obtained in terms of a contour integral that computes the Leading Singularity, which was recently shown to encode the relevant information up to one loop. The classical limit is performed along a holomorphic trajectory in the space of kinematics, such that the leading order is enough to extract arbitrarily high multipole corrections. These multipole interactions are given in terms of a recently proposed representation for massive particles of any spin by Arkani-Hamed et al. This explicitly shows universality of the multipole interactions in the effective potential with respect to the spin of the scattered particles. We perform the explicit match to standard EFT operators for $S=\frac{1}{2}$ and $S=1$. As a natural byproduct we obtain the classical pieces up to one loop for the bending of light.

Figures (4)

• Figure 1: A typical scattering process contributing to the effective potential. In this case two massive particles represented by $P_{1}^{2}=P_{2}^{2}=m_{a}^{2}$ and $P_{3}^{2}=P_{4}^{2}=m_{b}^{2}$ exchange several gravitons. The momentum transfer is given by $K=P_{1}-P_{2}=(0,\vec{q})$ in the COM frame.
• Figure 2: A one photon/graviton exchange process. In the HCL the internal particle is on-shell and the two polarizations need to be considered.
• Figure 3: The triangle diagram used to compute the leading singularity, corresponding to the $b$- topology. The $a$-topology is obtained by reflection, i.e. by appropriately exchanging the external particles.
• Figure 4: The matrix element of the stress-energy tensor $\langle T_{\mu\nu}(K)\rangle$ corresponds to the 3 point function associated to a pair of massive particles and an external off-shell graviton. The coupling to internal gravitons emanating from the massive source yields, in the long-range behavior, higher corrections in $G$.