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

TL;DR

This paper introduces hyperbolic form factors related to density distributions and their Laplace transforms, providing analytic tools to analyze Yukawa interactions and apply them to Earth's density profile for testing new forces.

Contribution

It defines the hyperbolic form factor as a bilateral Laplace transform, derives analytic expressions for Earth's density profiles, and applies these to set limits on new Yukawa-mediated forces.

Findings

Analytic expressions for Earth's hyperbolic form factor (x)

Simplified density profiles yield accurate (x) approximations

Constraints on new force coupling constants for light mediators

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 $\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.