The motivation comes from one of the deepest puzzles in modern physics. The standard model of particle physics, built on the symmetry group SU(3) x SU(2) x U(1), describes the strong, weak and electromagnetic interactions with spectacular success, while general relativity handles gravity. Yet many theorists suspect that this edifice is incomplete. One popular possibility is an extra U(1) symmetry, often called a dark U(1), whose mediator would be an extremely light spin-1 boson resembling a generalized dark photon. Such a particle would generate an extraordinarily feeble new force whose couplings to ordinary matter would involve combinations of baryon number B, lepton number L, or their difference B minus L. A spin-0 mediator could produce a similar effect. Because these forces would couple differently to different materials, they could produce apparent violations of the equivalence principle, the exact identity between inertial and gravitational mass that underlies general relativity and was famously tested by Baron Eötvös more than a century ago.

The sharpest current probe of such effects is the MICROSCOPE satellite mission, which monitored the relative acceleration of two freely falling test masses, one of titanium and one of platinum, aboard a drag-free spacecraft orbiting at an average altitude of roughly 710 kilometers. The mission found no deviation, constraining the Eötvös parameter, which measures the relative difference in accelerations, to a level of a few parts in ten to the fifteen. Translating this null result into limits on the strength of a hypothetical new force, however, requires knowing exactly how the Earth’s extended mass distribution sources that force, and this is where the new mathematics enters.

For a new interaction mediated by a particle of mass m, the force has a finite range lambda equal to the inverse mass in natural units. If the range greatly exceeds the Earth’s radius, the potential outside the planet is essentially the same as that of a point charge at the center. But as the range shrinks toward the satellite altitude, the potential at a given point is increasingly generated by the parts of the Earth closest to it, down to a minimum distance equal to the altitude itself. Fayet shows that the outside potential can always be written as the Yukawa potential of a pointlike source multiplied by a correction factor Phi(x), where x is the ratio of the Earth’s radius to the force range. This factor, always greater than or equal to one, is what he calls the hyperbolic form factor.

The definition is elegantly simple: the hyperbolic form factor is the bilateral Laplace transform of the density distribution, expressible as the average of the hyperbolic cosine of the dot product of the wave vector and position vector, or equivalently as the average of the hyperbolic sine of kr divided by kr over the body. Remarkably, this quantity is related by a duality transformation, or by analytic continuation through the substitution of imaginary for real wave numbers, to the ordinary form factor familiar from scattering theory, which involves the ordinary sine and cosine. The two are two faces of the same underlying function, and the connection extends to a compact formula in which the product of the wave number and the form factor appears as the bilateral Laplace transform of the quantity two-pi times r times the density. An inversion formula even allows one to reconstruct the density profile itself from an analytic continuation of the form factor, closing the loop between structure and potential.

Fayet also introduces a companion concept, the effective density. For each force range, one can ask what uniform sphere would generate the same outside potential as the real, layered Earth. The answer is a density that decreases as the range shortens, sliding from the global average density for very long-range forces down toward the density near the surface for short ranges. This makes physical sense: a short-ranged force is sourced preferentially by the outer shells of the planet, which are less dense than the iron core. The effective density thus encodes, in a single curve, how the sensitivity of an experiment migrates inward or outward as the hypothesized mediator mass changes.

The practical payoff comes from a surprising discovery about the Earth itself. Rather than needing a detailed numerical model with many layers, Fayet finds that remarkably simple analytic density profiles reproduce the form factor of a realistic five-shell Earth model, which distinguishes inner and outer cores, inner and outer mantles, and crust, with shell boundaries at radii of 1121, 3480, 5701, 6341 and 6371 kilometers. A density falling off as one over the radius yields a form factor equal to the square of the hyperbolic sine of x over 2 divided by x over 2, accurate to within about one percent up to x equal to 4. Even better, a hybrid profile combining the one-over-r behavior with a linear decrease gives a closed-form expression that matches the five-shell calculation to within 0.7 percent all the way up to x equal to 64, corresponding to force ranges above 100 kilometers or mediator masses below roughly two times ten to the minus twelve electron-volts. The power series expansion of this analytic expression, beginning with one plus 0.0833 x squared, sits extraordinarily close to the five-shell result, and the tiny differences that do exist are in fact mathematically mandatory, since identical form factors would imply identical densities through the inversion formula.

These approximations matter because the experimental limits on new-force couplings scale with the form factor. For a mediator mass of ten to the minus twelve electron-volts, the coupling limits derived from MICROSCOPE are weakened by a factor of about 34 compared with the massless case. Concretely, the analysis yields upper limits of roughly 3.6 times ten to the minus twenty-four on the magnitude of a coupling to B minus L for a spin-1 mediator, and about 2.6 times ten to the minus twenty-three for a coupling to baryon number alone, with slightly different numbers, differing by a factor of about 1.2, in the spin-0 case. The limits grow rapidly with mediator mass, and the analytic and numerical approaches agree to better than 0.4 percent across the relevant mass range, meaning future analyses can replace layered numerical models with a single compact formula without loss of accuracy. An even simpler approximation, in which the square root of the form factor is just the hyperbolic sine of x over 2 divided by x over 2, suffices to within about 2.3 percent for ranges down to a tenth of the Earth’s radius.

Beyond the immediate application to equivalence-principle tests, the framework has broad reach. The same hyperbolic form factors can be computed for Gaussian density distributions, where they reduce to simple exponentials, and for the electron cloud of a hydrogen atom in its ground state, where the ordinary form factor reproduces the well-known result governing electron scattering. The formalism also illuminates curious resonant behavior: for certain density profiles, the ordinary form factor vanishes at specific values of the wave number, corresponding to internal solutions of the field equations with vanishing outside potential, the static analogue of a stationary wave confined within the sphere. And because the coupling limits depend only weakly on the fine details of the Earth’s deep interior, further refinements of the internal structure are unlikely to change the conclusions. What began as an exercise in redefining a form factor may thus sharpen one of the most sensitive searches for physics beyond the standard model, turning the whole planet into a precisely calibrated instrument for detecting forces too faint to imagine.

Subject of Research: Hyperbolic form factors describing Yukawa potentials of extended bodies and their application to constraining new finite-range forces using Earth's density distribution

Article Title: Hyperbolic form factors for Yukawa interactions, and applications to the Earth

Article References: Fayet, P. (2026). Hyperbolic form factors for Yukawa interactions, and applications to the Earth. The European Physical Journal C, 86(9), Article 1125. https://doi.org/10.1140/epjc/s10052-026-15933-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-15933-4

Keywords: Yukawa potential, hyperbolic form factor, dark photon, equivalence principle, MICROSCOPE mission, new forces, Earth density, dark sector, fifth force, Laplace transform, particle physics, geophysics