MICROSCOPE limits on the strength of a new force, with comparisons to gravity and electromagnetism

Abstract

Extremely weak new forces could lead to apparent violations of the Equivalence Principle. The MICROSCOPE experiment implies that the relative strength of a new long-range force, compared with gravity, is constrained to $|\bar\alpha_g|<3.2\ 10^{-11},2.3\ 10^{-13},2.2\ 10^{-13},6.7\ 10^{-13}$ and $1.5\ 10^{-12}$ at $2\sigma$, for a coupling to $B,\ L,\ B-L,\ B+L$ or $3B+L$; or, for a coupling to isospin, $|\alpha_g|<8.4\ 10^{-12}$. This is a gain in sensitivity $\simeq 3$ for a coupling to $B$, to $\approx$ 15 in the other cases, including $B-L$ as suggested by grand unification. This requires paying attention to the definition of $\bar\alpha_g$. A force coupled to $L$ (or $B-L$) would act effectively on protons (or neutrons) only, its relative intensity being reduced from $\alpha_g$ to about $\bar\alpha_g=\alpha_g/4$ for an average nucleon. It is thus convenient to view such forces as acting on $\bar Q =B,\ 2L,\ 2(B-L),2(B+L)/3$ or $2(3B+L)/7$, leading to $\bar\alpha_g=\alpha_g\times(1,1/4,1/4,9/4$ or $49/4$). The sensitivity for a coupling to $L$ or $B-L$ is better than for $B$ by two orders of magnitude (as $\Delta (2L/A_r)\simeq 144\ \Delta (B/A_r)$ for Ti-Pt); and about 3 or 7 times better than for $B+L$ or $3B+L$. A coupling to $(\epsilon_BB+\epsilon_{Q_{el}}Q_{el})e$ should verify $|\epsilon_B|<5\ 10^{-24}$; similarly $|\epsilon_L|$ or $|\epsilon_{B-L}|<.9\ 10^{-24}$, $|\epsilon_{B+L}|<.5\ 10^{-24},|\epsilon_{3B+L}|<.32\ 10^{-24}$ and $|\epsilon_{B-2L}|<2.6\ 10^{-24}$, implying a new interaction weaker than electromagnetism by more than $10^{46}$ to $10^{48}$. The resulting hierarchy between couplings, typically by $>10^{24}$, may be related within supersymmetry with a large hierarchy in energy scales by $>10^{12}$. This points to a $\sqrt\xi\approx 10^{16}$ GeV scale, associated with a huge vacuum energy density that may be responsible for the inflation of the early Universe.