MICROSCOPE limits on the strength of a new force, compared 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 \(|\bar\alpha_g|<2.7\ 10^{-11},1.9 \ 10^{-13},1.9\ 10^{-13},6\ 10^{-13}\) and \(1.3\ 10^{-12}\) at \(2\sigma\), for a coupling to \(B,\ L,\ B-L,\ B+L\) or \(3B+L\); or, for a coupling to isospin, \(|\alpha_g|<7\ 10^{-12}\). This is a gain in sensitivity \(\simeq 5\) for a coupling to \(B\), to \(\approx\) 20 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 \(|\epsilon_{B-L}|<.8\ 10^{-24}\), \(\epsilon_{B+L}<.5\ 10^{-24},|\epsilon_{3B+L}|<.3\ 10^{-24}\) and \(\epsilon_{B-2L}<2.4\ 10^{-24}\), 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.