# Holomorphic Classical Limit for Spin Effects in Gravitational and Electromagnetic Scattering

## TL;DR

This paper derives universal classical scattering amplitudes for particles of any spin interacting via gravity and electromagnetism, using a holomorphic limit and contour integrals, applicable to high-order post-Newtonian and multipole expansions.

## Contribution

It introduces a universal framework for classical spin effects in scattering amplitudes using a holomorphic approach and contour integrals, extending to arbitrary spin and loop levels.

## Key findings

- Derived universal classical amplitudes for arbitrary spin particles.
- Connected multipole interactions to a new spin representation.
- Matched results with effective field theory operators for specific spins.

## 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.