# The simplest massive S-matrix: from minimal coupling to Black Holes

## TL;DR

This paper investigates the properties of massive higher spin states in electromagnetism and gravity, establishing that minimal coupling leads to g=2 and connecting these states to Kerr black holes, with implications for classical potentials.

## Contribution

It constructs the simplest massive S-matrix amplitudes, shows their relation to minimal coupling and black holes, and derives spin-dependent potentials at 2PM order.

## Key findings

- Minimal coupling yields g=2 for arbitrary charged spins.
- Deformations away from minimal coupling are inconsistent in gravity.
- Large spin states correspond to Kerr black holes.

## Abstract

In this paper, we explore the physics of electromagnetically and gravitationally coupled massive higher spin states from the on-shell point of view. Starting with the three-point amplitude, we focus on the simplest amplitude which is characterized by matching to minimal coupling in the UV. In the IR such amplitude leads to g = 2 for arbitrary charged spin states, and the best high energy behavior for a given spin. We proceed to construct the (gravitational) Compton amplitude for generic spins. We find that the leading deformation away from minimal coupling, in the gravitation sector, will lead to inconsistent factorizations and are thus forbidden. As the corresponding deformation in the gauge sector encodes the anomalous magnetic dipole moment, this leads to the prediction that for systems with gauge2 =gravity relations, such as perturbative string theory, all charged states must have g = 2. It is then natural to ask for generic spin, what is the theory that yields such minimal coupling. By matching to the one body effective action, remarkably we verify that for large spins, the answer is Kerr black holes. This identification is then an on-shell avatar of the no hair theorem. Finally using this identification as well as the newly constructed Compton amplitudes, we proceed to compute the spin dependent pieces for the classical potential at 2PM order up to degree four in spin operator of either black holes.