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 $\Phi (x=kR)=⟨\cosh \vec{k}.\vec{r}⟩=⟨\sinh kr/kr⟩$, expanded as $\sum \frac{x^{2n}}{(2n+1)!}\frac{⟨r^{2n}⟩}{R^{2n}}$, and similarly for the form factor $⟨\sin kr/kr⟩$. 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 $\rho (r)={\rho }_0(2R/3\pi r)\int \Phi (ix)\sin (x\frac{r}{R})xdx$. $\Phi (x)$ 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 $\rho ={\rho }_0(\frac{5}{4}-\frac{r}{R}+\frac{R}{3r})$, 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.

Comments: 23 pages, 3 figures