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 ordinary form factor.  For a sphere it is given by \(\Phi (x \!= \! kR) =\langle \,\cosh \,\vec k.\vec r\,\rangle = \langle \,\frac{\sinh kr}{kr}\,\rangle \), expanded as \(\,\sum _0^\infty \frac{x^{2n}}{(2n+1)! }\, \frac{\langle r^{2n}\rangle }{R^{2n}}\), and similarly for the form factor \( \langle \,\frac{\sin kr}{kr}\,\rangle \). 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 \(\frac{3}{x^3}\,(x\,\cosh x - \sinh x) \times {{\bar{\rho }}} (x)/\rho _0\), where \({{\bar{\rho }}}(x)\) is an effective density decreasing (for \(d\rho /dr <0)\) from the average \(\rho _0\) at small x, down to the density \(\rho (R)\) near the surface. An inversion formula allows one to recover the density distribution \(\rho (r)\) from an analytic continuation of \(\Phi (x)\), as \(\rho (r) =\rho _0\, (2R/3\pi r)\int _0^\infty \Phi (ix) \, \sin (x\frac{r}{R})\ x\,dx\,\).