Hyperbolic form factors for Yukawa interactions, and applications to the Earth
Abstract
We define the hyperbolic form factor of a density distribution as its bilateral Laplace transform, related by duality or analytic continuation to its form factor. For a sphere it is given by $Φ(x = kR) =\langle \cosh \vec k.\vec r\rangle=\langle\sinh kr /kr\rangle $, expanded as $\sum \frac{x^{2n}}{(2n+1)!} \frac{\langle r^{2n}\rangle}{R^{2n}} $, and similarly for the form factor $ \langle \sin kr/kr\rangle$. It is also obtained from the bilateral Laplace transform of 2πr ρ(|r|), and enters in the determination of the outside Yukawa potential induced by a new charge for a mediator of mass m=k=1/λ. Φ(x) may be expressed as 3/x3 (x cosh x − sinh x) ρ̅(x)/ρ0, where ρ̅(x) is an effective density decreasing, for dρ/dr<0, from the average ρ0 at small x, down to ρ(R). An inversion formula allows one to recover ρ(r) from an analytic continuation of Φ(x), as ρ(r) =ρ0 (2R/3πr)∫Φ(ix) sin (xr/R) x dx. Φ(x) for the Earth is essential to determine limits on a new force, as tested by MICROSCOPE, depending on the density distribution within the Earth. Quite remarkably, much simplified density profiles, such as $ρ= ρ_0\ 2R/3r $ or ρ= ρ0 (5/4−r/R+ R/3r), provide analytic expressions of Φ(x) and ρ̅(x) giving almost the same values as in a 5-shell model. Φ(x)=(sinhx/2/x/2)2 is valid to within ≃ 1 % up to x=4. Φ(x)= [7x2cosh x−24cosh x+9xsinh x −4x2+24]/(4x4) is valid to within 1 % for $λ> $ 100 km (or m< 2× 10−12 eV/c2). For m=10−12 eV/c2 the coupling limits are increased by 34 as compared to a massless mediator, to $|g_{B-L}|< 3.6 \times 10^{-24} $ and |gB|<2.6× 10−23 for a spin-1 mediator, with slightly different limits in the spin-0 case.
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BibTeX
@article{fayet2026hyperbolic,
title = {{Hyperbolic form factors for Yukawa interactions, and applications to the Earth}},
author = {Pierre Fayet},
journal = {European Physical Journal C: Particles and Fields},
year = {2026},
month = oct,
doi = {10.1140/epjc/s10052-026-15933-4},
eprint = {2603.21275},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05568906v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Hyperbolic form factors for Yukawa interactions, and applications to the Earth
AU - Pierre Fayet
PY - 2026
DA - 2026/10/20
JO - European Physical Journal C: Particles and Fields
DO - 10.1140/epjc/s10052-026-15933-4
UR - https://hal.science/hal-05568906v1
AB - We define the hyperbolic form factor of a density distribution as its bilateral Laplace transform, related by duality or analytic continuation to its form factor. For a sphere it is given by $Φ(x = kR) =\langle \cosh \vec k.\vec r\rangle=\langle\sinh kr /kr\rangle $, expanded as $\sum \frac{x^{2n}}{(2n+1)!} \frac{\langle r^{2n}\rangle}{R^{2n}} $, and similarly for the form factor $ \langle \sin kr/kr\rangle$. It is also obtained from the bilateral Laplace transform of 2πr ρ(|r|), and enters in the determination of the outside Yukawa potential induced by a new charge for a mediator of mass m=k=1/λ. Φ(x) may be expressed as 3/x^3 (x cosh x − sinh x) ρ̅(x)/ρ_0, where ρ̅(x) is an effective density decreasing, for dρ/dr<0, from the average ρ_0 at small x, down to ρ(R). An inversion formula allows one to recover ρ(r) from an analytic continuation of Φ(x), as ρ(r) =ρ_0 (2R/3πr)∫Φ(ix) sin (xr/R) x dx. Φ(x) for the Earth is essential to determine limits on a new force, as tested by MICROSCOPE, depending on the density distribution within the Earth. Quite remarkably, much simplified density profiles, such as $ρ= ρ_0\ 2R/3r $ or ρ= ρ_0 (5/4−r/R+ R/3r), provide analytic expressions of Φ(x) and ρ̅(x) giving almost the same values as in a 5-shell model. Φ(x)=(sinhx/2/x/2)^2 is valid to within ≃ 1 % up to x=4. Φ(x)= [7x^2cosh x−24cosh x+9xsinh x −4x^2+24]/(4x^4) is valid to within 1 % for $λ> $ 100 km (or m< 2× 10^−12 eV/c^2). For m=10^−12 eV/c^2 the coupling limits are increased by 34 as compared to a massless mediator, to $|g_{B-L}|< 3.6 \times 10^{-24} $ and |g_B|<2.6× 10^−23 for a spin-1 mediator, with slightly different limits in the spin-0 case.
SN - 1434-6044
SN - 1434-6052
ER -