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 $Φ(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 $2πr\,ρ(|r|)$, and enters in the determination of the outside Yukawa potential induced by a new charge for a mediator of mass $m=k=1/λ$. $Φ(x)$ may be expressed as $\frac{3}{x^3}\,(x \cosh x - \sinh x)\ \barρ(x)/ρ_0$, where $\barρ(x)$ is an effective density decreasing, for $dρ/dr<0$, from the average $ρ_0$ at small $x$, down to $ρ(R)$. An inversion formula allows one to recover $ρ(r)$ from an analytic continuation of $Φ(x)$, as $ρ(r) =ρ_0\,(2R/3πr)\intΦ(ix) \sin (x\frac{r}{R})\,x\,dx$. $Φ(x)$ 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 $ρ= ρ_0\ 2R/3r $ or $ρ= ρ_0\, (\frac54-\frac{r}{R}+ \frac{R}{3r})$, provide analytic expressions of $Φ(x)$ and $\barρ(x)$ giving almost the same values as in a 5-shell model. $\,Φ(x)=(\sinh\frac{x}{2}/\frac{x}{2})^2$ is valid to within $\simeq 1\,\%$ up to $x=4$. $\,Φ(x)= [7x^2\cosh x-24\cosh x+9x\sinh x -4x^2+24]/(4x^4)$ is valid to within 1 % for $λ> $ 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.

Figures (3)

• Figure 1: $\Phi(x)$ as a function of $x=R/\lambda$, for four density distributions: 1) an homogeneous $\rho_0$, with $\phi(x) =$$\frac{3}{x^3}\,(x\cosh x-\sinh x)$ (dotted green); 2) $\rho(r)=\rho_0 \,\frac{2R}{3r}$, with $\Phi(x) = \frac{2}{x^2}\,(\cosh x-1)$ (dotted red); 3) $\rho'(r)=$$\rho_0 \,(\frac{5}{4} - \frac{r}{R}+\frac{R}{3r})$, with $\Phi'(x) =\frac{1}{4x^4}\,[\,7x^2 \cosh x-24 \cosh x+ 9x\sinh x -4x^2+24\,]$ (cf. Sec. \ref{['sec:earth']}, in blue); 4) $\rho(r)$ in a 5-shell Earth model (also in blue) yuk. The last two curves are superposed, coinciding to within $\simeq .5\,\%$ for $x <50\,$. $\,\Phi(x) = \frac{2}{x^2}\,(\cosh x-1)$, in dotted green, slightly overestimates this result, by $\simeq 1\,\%$ for $x= 4$, and 4 % for $x = 8$. $\phi(x)$ for an homogenous density, in dotted green, overestimates it by $\simeq 36\,\%$ for $x=8$.
• Figure 2: The effective density $\bar{\rho}(x)=\rho_0\, \Phi(x)/\phi(x)$ as a function of $x=R/\lambda$ : 1) for $\bar{\rho} =\rho_0$, in the homogeneous case (dotted green); 2) for $\rho(r) = \rho_0 \,2R/3r$ with $\bar{\rho}(x) = \rho_0\ [\,2x\,(\cosh x -1)\,]/[\,3\,(x\cosh x-\sinh x)\,]$, decreasing to $\frac{2}{3}\,\rho_0$ at large $x$ (dotted red); 3) for $\rho'(r) = \rho_0\,(\frac{5}{4}-\frac{r}{R}+\frac{R}{3r})$ with $\bar{\rho}\,'(x) = [ \,7x^2\,\cosh x -24\,\cosh x + 9\,x\sinh x -4\,x^2+24\,]\,/ \,[\,12x\,(x\,\cosh x-\sinh x)\,]$, decreasing to $\frac{7}{12}\,\rho_0$ (in blue); 4) in a 5-shell model, also in blue. The last two curves almost exactly coincide, to within .7 % for $x < 64$ i.e. $\lambda > 100$ km. The $1/r$ density profile also provides a good approximation of $\Phi(x)$ and $\bar{\rho}(x)$ (in dotted red) up to $x\simeq 4\,$.
• Figure 3: Upper limits on $|g_{B-L}|$ or $|g_L|$ (blue), and $|g_B |$ (orange), at the 95 % CL. The limits for an Eötvös parameter $\delta < 0$ are larger than for $\delta > 0$ by $\simeq 1.2$. For $\lambda\gg R$ they are $1.1 \times 10^{-25}$ for $|g_{B-L}|$ (spin-1) and $|g_L|$ (spin-0), solid blue line ; and $1.3 \times 10^{-25}$ for $|g_L|$ (spin-1) and $|g_{B-L}|$ (spin-0), dashed blue line. For $|g_B |$ they are $7.7\times 10^{-25}$ (spin 1, solid orange) and $6.4\times 10^{-25}$ (spin-0, dashed orange). For $m= 10^{-13},\, 10^{-12}$ or $10^{-11}$ eV/$c^2$ the limits are multiplied by $\simeq 1.9, \ 34$ or $1.2\times 10^9$, respectively, as compared to the nearly massless case. The limits obtained from $\Phi(x)$ in the 5-shell model or from the analytic expression $\Phi'(x)$ in eq. (\ref{['phin']}) differ by less than .4 % for $m < 2\times 10^{-12}$ eV/$c^2$, the corresponding curves being superposed.