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 $|{\overline{\alpha }}_g|<3.2\times {10}^{-11}$, $2.3\times {10}^{-13}$, $2.2\times {10}^{-13}$, $6.7\times {10}^{-13}$, and $1.5\times {10}^{-12}$ at $2\sigma$, for a coupling to $B$, $L$, $B-L$, $B+L$, or $3B+L$; or, for a coupling to isospin, $|{\alpha }_g|<8.4\times {10}^{-12}$. 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 ${\overline{\alpha }}_g$. 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 ${\overline{\alpha }}_g={\alpha }_g/4$ for an average nucleon. A force coupled to $B+L=2Z+N$ would act twice as much on $p$ as on $n$, getting enhanced from ${\alpha }_g$ for neutrons to about ${\overline{\alpha }}_g=\frac{9}{4}{\alpha }_g$ for an average nucleon. It is thus convenient to view such forces as acting on $\overline{Q}=B$, $2L$, $2(B-L)$, $2(B+L)/3$, or $2(3B+L)/7$ (normalized to 2 for $p+e+n$), leading to ${\overline{\alpha }}_g={\alpha }_g\times (1,1/4,1/4,9/4,\text{or}49/4)$. The sensitivity for a coupling to $L$ or $B-L$ is better than for $B$ by 2 orders of magnitude [as $\mathrm{\Delta }(2L/A_r)\simeq 144\mathrm{\Delta }(B/A_r)$ for Ti-Pt], and about 3 or 7 times better than for $B+L$ or $3B+L$. A coupling to $({\epsilon }_{B}B+{\epsilon }_{Q_{\mathrm{el}}}{Q}_{\mathrm{el}})e$ should verify $|{\epsilon }_B|<5\times {10}^{-24}$; similarly $|{\epsilon }_L|$ or $|{\epsilon }_{B-L}|<0.9\times {10}^{-24}$, $|{\epsilon }_{B+L}|<0.5\times {10}^{-24}$, $|{\epsilon }_{3B+L}|<0.32\times {10}^{-24}$, and $|{\epsilon }_{B-2L}|<2.6\times {10}^{-24}$, implying a new interaction weaker than electromagnetism by more than $10^{46}$ to $10^{48}$. The resulting hierarchy between couplings, typically by $\gtrsim {10}^{24}$, may be related within supersymmetry with a large hierarchy in energy scales by $\gtrsim {10}^{12}$. This points to a $\sqrt{\xi }\approx {10}^{16}\mathrm{GeV}$ scale, associated with a huge vacuum energy density that may be responsible for the inflation of the early universe.

• Received 13 September 2018

DOI:https://doi.org/10.1103/PhysRevD.99.055043

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Published by the American Physical Society