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 $$\Phi (x \!= \! kR) =\langle \,\cosh \,\vec k.\vec r\,\rangle = \langle \,\frac{\sinh kr}{kr}\,\rangle $$ Φ ( x = k R ) = ⟨ cosh k → . r → ⟩ = ⟨ sinh k r kr ⟩ , expanded as $$\,\sum _0^\infty \frac{x^{2n}}{(2n+1)! }\, \frac{\langle r^{2n}\rangle }{R^{2n}}$$ ∑ 0 ∞ x 2 n ( 2 n + 1 ) ! ⟨ r 2 n ⟩ R 2 n , and similarly for the form factor $$ \langle \,\frac{\sin kr}{kr}\,\rangle $$ ⟨ sin k r kr ⟩ . It is also obtained from the bilateral Laplace transform of $$2\pi r\,\rho (|r|)$$ 2 π r ρ ( | r | ) , and enters in the determination of the outside Yukawa potential induced by a new charge for a mediator of mass $$m= k=$$ m = k = $$1/\lambda $$ 1 / λ . $$\Phi (x)$$ Φ ( x ) may be expressed as $$\frac{3}{x^3}\,(x\,\cosh x - \sinh x) \times {{\bar{\rho }}} (x)/\rho _0$$