{
  "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 $\\frac{3}{x^3}\\,(x \\cosh x - \\sinh x)\\ \\barρ(x)/ρ_0$, where $\\barρ(x)$ 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 $ρ(r) =ρ_0\\,(2R/3πr)\\intΦ(ix) \\sin (x\\frac{r}{R})\\,x\\,dx$. $Φ(x)$ 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 $ρ= ρ_0\\, (\\frac54-\\frac{r}{R}+ \\frac{R}{3r})$, provide analytic expressions of $Φ(x)$ and $\\barρ(x)$ giving almost the same values as in a 5-shell model. $\\,Φ(x)=(\\sinh\\frac{x}{2}/\\frac{x}{2})^2$ is valid to within $\\simeq 1\\,\\%$ up to $x=4$. $\\,Φ(x)= [7x^2\\cosh x-24\\cosh x+9x\\sinh x -4x^2+24]/(4x^4)$ is valid to within 1 % for $λ> $ 100 km (or $m< 2\\times 10^{-12}$ 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 $|g_B|<2.6\\times 10^{-23}$ for a spin-1 mediator, with slightly different limits in the spin-0 case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2603.21275",
  "author_total": 1,
  "authors": [
    {
      "name": "Pierre Fayet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0s26g62ffgv77.bib",
  "cited_by": 0,
  "doi": "10.1140/epjc/s10052-026-15933-4",
  "fulltext_url": null,
  "id": "0s26g62ffgv77",
  "issns": [
    "1434-6044",
    "1434-6052"
  ],
  "landing_url": "https://hal.science/hal-05568906v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.phys"
    ],
    [
      "hal",
      "1.phys.hexp"
    ],
    [
      "hal",
      "1.phys.hphe"
    ],
    [
      "openalex",
      "3103|Relativity and Gravitational Theory"
    ]
  ],
  "openalex": "W7140180681",
  "pdf_url": "https://hal.science/hal-05568906/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": "Springer Verlag (Germany)",
  "ris": "https://research.draconitic.com/paper/0s26g62ffgv77.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:59Z",
      "native_id": "hal-05568906",
      "source": "hal",
      "url": "https://hal.science/hal-05568906v1"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "hal:1.phys.hexp",
      "subfield": "high-energy"
    }
  ],
  "title": "Hyperbolic form factors for Yukawa interactions, and applications to the Earth",
  "title_text": "Hyperbolic form factors for Yukawa interactions, and applications to the Earth",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:17:59Z",
  "url": "https://research.draconitic.com/paper/0s26g62ffgv77",
  "venue": "European Physical Journal C: Particles and Fields",
  "versions": []
}