⚛️ high-energy theory

The gyromagnetic factor of charged rotating black holes in various dimensions from scattering amplitudes

This paper demonstrates that classical charged rotating black hole spacetimes in various dimensions can be reconstructed from scattering amplitudes, revealing that a non-minimal Pauli coupling is required to correctly describe the gyromagnetic factor in dimensions higher than four.

Original authors: Claudio Gambino, Fabio Riccioni, Victor Sanz Sanchis

Published 2026-02-10

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CC BY 4.0

Original authors: Claudio Gambino, Fabio Riccioni, Victor Sanz Sanchis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic Magnet: How Tiny Particles Tell Us About Giant Black Holes

Imagine you are trying to understand the shape and behavior of a massive, spinning, electrified ocean liner, but you are forbidden from ever getting close to it. You can’t sail near it, you can’t use sonar, and you can’t even see it through the fog.

How would you figure out its properties? You might try throwing tiny pebbles at it and watching how they bounce off. By studying the way those tiny pebbles scatter, you could eventually deduce if the ship is spinning, how much it weighs, and how much of a magnetic pull it has.

This is essentially what physicists are doing in this paper. Instead of "pebbles," they use quantum particles (like photons and gravitons). Instead of an "ocean liner," they are studying Black Holes.

1. The Core Idea: The "Mirror" Effect

The central premise of the paper is a beautiful concept in physics: The small reflects the large.

There is a mathematical bridge between the world of the incredibly tiny (quantum scattering amplitudes) and the world of the incredibly massive (classical black holes). The researchers found that if you calculate how a tiny, charged particle "scatters" (bounces) off gravity and light, the math you get is a perfect mirror image of the gravitational field surrounding a giant black hole.

2. The "Gyromagnetic Factor": The Black Hole's Personality

The researchers focused on a specific trait called the gyromagnetic factor ( $g$ ).

Think of this as a "personality score" for how a spinning object interacts with magnetism.

Some objects are "magnetic extroverts"—they have a huge magnetic field for their size.
Others are "magnetic introverts"—they spin wildly but barely affect the magnetic field around them.

In our familiar 3D world (plus time), black holes like the Kerr-Newman black hole have a very specific, "natural" personality score of $g = 2$ . This is the "standard" setting, and it happens automatically if you assume the simplest possible physics (called "minimal coupling").

3. The Twist: Higher Dimensions Change the Rules

The "Aha!" moment of this paper comes when the scientists moved beyond our 3D world into higher dimensions (like 4D or 5D space).

They discovered that in these extra dimensions, the "standard" settings don't work anymore. If you try to describe a higher-dimensional black hole using the same simple rules we use for our own universe, the math breaks. The black hole's "personality" changes.

They found a universal formula for this personality in higher dimensions:
$g = \frac{d-1}{d-2}$
(Where $d$ is the number of spatial dimensions.)

What does this mean in plain English?
It means that in higher dimensions, black holes are "magnetic introverts" compared to our world. They require a more complex set of rules—specifically, a "non-minimal coupling" (which you can think of as a special magnetic adjustment) to accurately describe them.

In our 3D world, the black hole is "simple." In a 4D or 5D world, the black hole is "extra"—it has extra layers of interaction that we have to account for to get the math right.

4. Why does this matter?

By finding this formula, the researchers have provided a "universal translator."

Whether a physicist is studying a complex, five-dimensional supergravity solution (like the CCLP solution mentioned in the paper) or a simpler rotating black hole, they now know exactly how much "magnetic adjustment" is needed to make the tiny quantum math match the giant cosmic reality.

Summary Table

Concept	Everyday Analogy	In the Paper
Scattering Amplitudes	Throwing pebbles to map a ship	Using tiny particles to map a black hole
Gyromagnetic Factor ( $g$ )	A "personality score" for magnetism	How much a spinning BH affects magnetic fields
Minimal Coupling	The "default" settings on a device	The simplest way particles interact with gravity
Higher Dimensions	Playing a game with extra rules/levels	Moving from 3D to 4D or 5D space

Technical Summary: The Gyromagnetic Factor of Charged Rotating Black Holes in Various Dimensions from Scattering Amplitudes

1. Problem Statement

The paper addresses the challenge of reconstructing the classical far-field properties (metric and electromagnetic potential) of charged, rotating black holes (BHs) in arbitrary spacetime dimensions ( $d+1$ ) using the framework of Effective Field Theory (EFT) and scattering amplitudes.

While the 4D Kerr-Newman solution is well-understood to emerge from the classical limit of minimally coupled charged particles, higher-dimensional black holes are more complex. In dimensions $d > 3$ , uniqueness theorems do not hold, and exact charged rotating solutions are often unknown or require additional interactions (like Chern-Simons terms in supergravity). The authors specifically aim to determine the gyromagnetic factor ( $g$ )—a parameter characterizing the magnetic dipole moment of a rotating charged object—for these higher-dimensional solutions and identify the necessary particle-level couplings required to reproduce them.

2. Methodology

The authors employ a Post-Minkowskian (PM) expansion approach, matching the classical limit ( $\hbar \to 0$ ) of quantum scattering amplitudes to the far-field expansion of classical BH metrics.

Source Modeling: The black hole is modeled as a massive, electrically charged spin-1/2 Dirac fermion field. This choice is strategic because the spin structure naturally encodes the angular momentum and mass required for a dipole-order analysis.
EFT Framework: To allow for a non-trivial gyromagnetic factor, the authors go beyond minimal coupling by introducing a Pauli non-minimal coupling term:
$\mathcal{L}_{nm} = -i \frac{\zeta Q}{2m} \bar{\psi} \Sigma_{\mu\nu} \psi F^{\mu\nu}$
where $\zeta$ is a dimensionless parameter that allows the gyromagnetic factor to vary.
Amplitude Computation: They compute the 1-loop (first PM order) scattering amplitudes for:
1. Graviton emission: To derive the metric $h_{\mu\nu}$ .
2. Photon emission: To derive the electromagnetic potential $A_\mu$ .
Chern-Simons Integration: In 5D ( $d=4$ ), they include a Chern-Simons (CS) term $\mathcal{L}_{CS} \propto \epsilon F F A$ to account for the specific interactions found in supergravity, which are known to produce the Chong-Cveticič-Lü-Pope (CCLP) solution.
Matching Procedure: The resulting amplitude-derived potentials and metrics are compared against the PM expansions of known classical solutions (Kerr-Newman and CCLP).

3. Key Contributions and Results

The paper provides three major technical results:

Reconstruction of 4D Kerr-Newman: The authors demonstrate that the 4D Kerr-Newman solution is exactly reproduced by a minimally coupled Dirac field ( $\zeta = 0$ ), which implies a gyromagnetic factor of $g = 2$ .
Reconstruction of 5D CCLP: The authors show that the 5D CCLP solution (from supergravity) cannot be recovered via minimal coupling. It requires both a specific Chern-Simons coupling ( $\lambda = 1$ ) and a non-minimal Pauli coupling with $\zeta = -1/4$ , resulting in a gyromagnetic factor of $g = 3/2$ .
Universal Gyromagnetic Factor Formula: By generalizing the analysis to $d+1$ dimensions, the authors derive a universal formula for the gyromagnetic factor of a charged Myers-Perry-like black hole:
$g_{BH} = \frac{d-1}{d-2}$
This formula correctly predicts $g=2$ for $d=3$ (4D) and $g=3/2$ for $d=4$ (5D).

4. Significance

The work has profound implications for the intersection of quantum field theory and general relativity:

EFT Requirements for Gravity: It establishes that minimal coupling is insufficient to describe charged rotating black holes in any dimension higher than four. In higher dimensions, the "particle" source must possess intrinsic non-minimal electromagnetic properties (a Pauli term) to match the macroscopic properties of a black hole.
Universality: The result $g = (d-1)/(d-2)$ is shown to be independent of the Chern-Simons term, as the CS term affects the electromagnetic potential but does not alter the dipole moment at the order considered. This suggests a universal relationship between spacetime dimensionality and the magnetic properties of black holes.
Methodological Validation: The paper validates the use of scattering amplitudes as a powerful tool for "reverse-engineering" classical spacetime geometries, providing a bridge between the particle physics of EFTs and the geometric descriptions of General Relativity.