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

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 $\Phi(x = kR) =\langle \cosh \vec k.\vec r\rangle=\langle\sinh kr /kr\rangle$, 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 $2\pi r\,\rho(|r|)$, and enters in the determination of the outside Yukawa potential induced by a new charge for a mediator of mass $m=k=1/\lambda$. $\Phi(x)$ may be expressed as $\frac{3}{x^3}\,(x \cosh x - \sinh x)\ \bar\rho(x)/\rho_0$, where $\bar\rho(x)$ is an effective density decreasing, for $d\rho/dr<0$, from the average $\rho_0$ at small $x$, down to $\rho(R)$. An inversion formula allows one to recover $\rho(r)$ from an analytic continuation of $\Phi(x)$, as $\rho(r) =\rho_0\,(2R/3\pi r)\int\Phi(ix) \sin (x\frac{r}{R})\,x\,dx$. $\Phi(x)$ 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 $\rho = \rho_0\ 2R/3r$ or $\rho = \rho_0\, (\frac54-\frac{r}{R}+ \frac{R}{3r})$, provide analytic expressions of $\Phi(x)$ and $\bar\rho(x)$ giving almost the same values as in a 5-shell model. $\,\Phi(x)=(\sinh\frac{x}{2}/\frac{x}{2})^2$ is valid to within $\simeq 1\,\%$ up to $x=4$. $\,\Phi(x)= [7x^2\cosh x-24\cosh x+9x\sinh x -4x^2+24]/(4x^4)$ is valid to within 1 % for $\lambda >$ 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.