Bulk Minimal & Gyromagnetic Couplings

• Bulk minimal and gyromagnetic couplings are key interactions defining how charged, spinning particles and black holes respond to background electromagnetic fields, with minimal coupling enforcing g=2 at the point-particle level.
• Non-minimal (Pauli) couplings introduce higher-dimensional operators that capture electromagnetic multipole moments and structure effects, allowing deviations from the universal gyromagnetic factor.
• The interplay between these couplings in effective field theories governs scattering amplitudes and ensures algebraic consistency, enabling accurate phenomenological predictions in both particle physics and black hole models.

Bulk minimal and gyromagnetic couplings govern the interaction of charged, spinning particles and black holes with background electromagnetic fields, constraining both the structure of the low-energy effective action and the physical observable known as the gyromagnetic factor, $\mathfrak{g}$ (or $g$ for point particles). Minimal coupling refers to the standard gauge-invariant interaction, while non-minimal (Pauli) couplings account for higher-multipole and structure effects, especially in higher-dimensional or composite systems. The gyromagnetic factor encodes the proportionality between spin and magnetic moment, and its value is dictated by both symmetry constraints and the allowed coupling operators. The precise structure of these couplings controls both the consistency of the underlying theory (degrees of freedom, causality, ghost-freedom) and its phenomenological predictions for systems ranging from fundamental particles to higher-dimensional charged black holes.

1. Minimal Coupling and Universal Gyromagnetic Ratio

Minimal coupling is realized by replacing partial derivatives with gauge-covariant derivatives in the kinetic term of a charged field. For massive Dirac or higher-spin fields in $(d+1)$-dimensional spacetime, minimal coupling alone generically yields a universal gyromagnetic factor $g=2$ at the “point-particle” level. This result holds for all integer and half-integer spins, as shown by enforcing algebraic constraint propagation (no extra degrees of freedom) and tree-level unitarity. For instance, a neutral spin-2 field described by the Fierz–Pauli system or a spin-$3/2$ Rarita–Schwinger system both require $g=2$ for consistent propagation in a background electromagnetic field. Any deviation from $g=2$ at the minimal-coupling level immediately causes unphysical modes or ambiguities in the constraint structure, signaling problems such as superluminal propagation or ghost states (Benakli, 10 Dec 2025).

2. Non-Minimal (Pauli) Couplings and Effective Field Theory

Non-minimal or Pauli couplings, typically dimension-5 operators, are necessary to encode the electromagnetic multipole moments and structure-dependent corrections of realistic systems. The canonical Pauli interaction for a Dirac fermion is $-i\kappa\,\bar\psi\sigma^{\mu\nu}F_{\mu\nu}\psi$, where $\kappa$ governs the strength of the anomalous magnetic dipole term. In higher spins, analogous dipole-type operators (of the form $F\Phi-\Phi F$ for spin-2 or $F_{mn}\chi^n$ and $\sigma^{ab} F_{ab}\chi_m$ for spin-$3/2$) generalize the Pauli coupling. For composite resonances, long-lived hadronic states, or when modeling black holes in higher dimensions, these non-minimal terms are essential to account for $g\ne2$. Their systematic inclusion is achieved via an Effective Field Theory (EFT) expansion in powers of the background field strength and derivatives, extending the validity of the theory up to the compositeness or cutoff scale (Benakli, 10 Dec 2025).

3. Bulk Couplings, Scattering Amplitudes, and Black Hole Gyromagnetic Ratio

The relationship between bulk coupling structure and observable gyromagnetic factors is probed by classical limits of scattering amplitudes in low-energy gravitational EFTs. Charged, rotating black holes in $(d+1)$-dimensional spacetime can be modeled as the classical outcome of scattering processes involving massive, charged fermionic sources with minimal and non-minimal (Pauli) couplings. The amplitude-based computation yields, at the leading (dipole) order, an asymptotic magnetic potential

$$A_i^{\rm dip}(x) = \frac{\mathfrak{g}}{2} \frac{1}{\Omega_{d-1} r^d} Q S_{ik} x^k,$$

where $S_{ik}$ is the spin-density tensor and $\Omega_{d-1}$ the volume of the unit $(d-1)$-sphere. The direct relation

$\mathfrak{g} = 2 + \frac{4m\kappa}{Q}$

emerges by matching the coefficient of the dipole term in the electromagnetic potential to the Pauli coupling parameter $\kappa$. In $3+1$ dimensions, the Myers–Perry–Kerr–Newman solution is reproduced for $\mathfrak{g}=2$ with $\kappa=0$ (minimal coupling); for $d>3$, the black-hole gyromagnetic ratio is

$\mathfrak{g}_{\rm BH} = \frac{d-1}{d-2}$

and minimal coupling is insufficient, requiring a nonzero $\kappa$ fixed by matching to this value (Gambino et al., 10 Sep 2025).

Dimension $(d+1)$	Minimal coupling sufficient?	Black-hole $\mathfrak{g}_{\rm BH}$	Pauli term required?
$4$	Yes	$2$	No
$>4$	No	$(d-1)/(d-2)$	Yes

4. Algebraic Consistency, Constraints, and Ghost-Freedom

Generalizing to arbitrary $g$ or $\mathfrak{g}$ via Pauli-type terms introduces potential pathologies unless the operator structure is carefully engineered. For spin-2 and spin-$3/2$, deviation from the universal value modifies the algebraic constraint chain. In the EFT framework, new higher-dimensional “completion” operators, such as $\beta F_{(m}{}^r\Delta\Phi_{n)r}/M^2$ for spin-2 and analogous ones for spin-$3/2$, restore constraint closure and preserve the correct count of propagating degrees of freedom. Consistency conditions fix the coefficients of these terms (e.g., $c_{\Delta,3/2}=2-g$ for spin-$3/2$) such that all algebraic and differential constraints for the field remain preserved at each order in the background field, maintaining ghost-freedom and strong hyperbolicity as long as the EFT expansion is valid (i.e., $|e F|/M^2\ll 1$) (Benakli, 10 Dec 2025).

5. Higher-Order Operators, Extensions, and Phenomenology

Beyond leading dipole order, the EFT provides a systematic expansion in higher-dimension operators (quadrupole, octupole, …), governing static polarizabilities and multipole moments. The structure and coefficients of these higher-order terms are constrained by positivity bounds, causality, and dispersive sum rules. The methodology extends straightforwardly to arbitrary spin, using Young-projected tensors and auxiliary constraint operators. Phenomenologically, these frameworks underpin first-principles modeling of hadronic resonances, dark-sector bound states, and black holes, enabling predictions for spectroscopy, electromagnetic response, and the extraction of gyromagnetic ratios from microscopic amplitudes or classical black-hole solutions. The projection to curved and inhomogeneous backgrounds further enriches the operator basis, introducing invariants like $\nabla_\ell F_{mn}$ and $R\Phi$, with EFT power counting dictating their relevance and ensuring the overall consistency of the description (Benakli, 10 Dec 2025).

6. Comparative Role of Minimal and Non-Minimal Couplings in Bulk Gravity

The interplay between bulk (gravitational and electromagnetic) minimal and non-minimal couplings dictates whether classical solutions, such as Myers–Perry-type black holes, can be derived from the underlying amplitude-based EFT. Only in $3+1$ dimensions does minimal coupling fully account for the correct gyromagnetic factor, unifying the amplitude-based and metric-based reconstruction of charged rotating black holes. For $d>3$, the required matching of the bulk asymptotics and black-hole gyromagnetic ratio necessitates precise tuning of the Pauli term, reflecting an intrinsic dimensional dependence of the physical multipole response in classical gravitational systems (Gambino et al., 10 Sep 2025). This reveals a deep connection between the algebraic structure of the EFT, classical black-hole parameters, and experimentally accessible observables such as magnetic moments and spin–field couplings.