The Yukawa potential of a non-homogeneous sphere, with new limits on an ultralight boson

Abstract

Extremely weak long-range forces may lead to apparent violations of the Equivalence Principle. The final MICROSCOPE result, leading at 95 % c.l. to $|\delta |<4.5\times 10^{-15}$ or $6.5\times 10^{-15}$ for a positive or negative Eötvös parameter $\delta$, requires taking into account the spin of the mediator, and the sign of $\Delta (Q/A_r{)}_{\mathrm{Ti}-\mathrm{Pt}}$ ($Q$ denoting the new charge involved). A coupling to $B-L$ or $B$ should verify $|g_{B-L}|<1.1\times 10^{-25}$ or $|g_B|<8\times 10^{-25}$, for a spin-1 mediator of mass $m<10^{-14}$ eV$/c^2$, with slightly different limits of $1.3\times 10^{-25}$ or $6.6\times 10^{-25}$ in the spin-0 case. The limits increase with $m$, in a way which depends on the density distribution within the Earth. This involves an hyperbolic form factor, expressed through a bilateral Laplace transform as $\Phi (x=mR)=⟨\sinh mr/mr⟩$, related by analytic continuation to the Earth form factor $\Phi (ix)=⟨\sin mr/mr⟩$. It may be expressed as $\Phi (x)=\frac{3}{x^2}(\cosh x-\frac{\sinh x}{x})\times \bar{\rho }(x)/{\rho }_0$, where $\bar{\rho }(x)$ is an effective density, decreasing from the average ${\rho }_0$ at $m=0$ down to the density at the periphery. We give general integral or multishell expressions of $\Phi (x)$, evaluating it, and $\bar{\rho }(x)$, in a simplified 5-shell model. $\Phi (x)$ may be expanded as $\sum \frac{x^{2n}}{(2n+1)!}\frac{⟨r^{2n}⟩}{R^{2n}}\simeq 1+.0827x^2+.00271x^4+4.78\times 10^{-5}{x}^6+5.26\times 10^{-7}{x}^8+...$, absolutely convergent for all $x$ and potentially useful up to $x\approx 5$. The coupling limits increase at large $x$ like $mR{e}^{mz/2}/\sqrt{1+mr}$ ($z=r-R$ being the satellite altitude), getting multiplied by $\simeq 1.9,34$, or $1.2\times 10^9$, for $m=10^{-13},10^{-12}$ or $10^{-11}$ eV$/c^2$, respectively.

25 pages, 4 figures; v2, published version

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