Abstract
Many theories beyond the Standard Model involve an extra U(1) gauge group. The resulting gauge boson U, in general mixed with the Z and the photon, may be massless or very light, and very weakly coupled. It may be viewed as a generalized dark photon interacting with matter through a linear combination (\epsilon_Q Q + \epsilon_B B+\epsilon_L L) e, involving B-L in a grand-unified theory, presumably through B-L-.61 Q, inducing effectively a very small repulsive force between neutrons. This new force, if long-ranged, may manifest through apparent violations of the Equivalence Principle. They are approximately proportional to \epsilon_B+\epsilon_L/2, times a combination involving mostly \epsilon_L. New forces coupled to B-L or L should lead to nearly opposite values of the E\"otv\"os parameter \delta, and to almost the same limits for \epsilon_{B-L} or \epsilon_L, as long as no indication for \delta \neq 0 is found. We derive new limits from the first results of the MICROSCOPE experiment testing the Equivalence Principle in space. A long-range force coupled to (\epsilon_Q Q + \epsilon_{B-L} (B-L)) e or (\epsilon_Q Q + \epsilon_L L) e should verify |\epsilon_{B-L}| or |\epsilon_L| < .8 10^{-24}, and a force coupled to (\epsilon_Q Q + \epsilon_B B) e, |\epsilon_B| < 5 10^{-24}. We also discuss, within supersymmetric theories, how such extremely small gauge couplings g", typically \simle 10^{-24}, may be related to a correspondingly large \xi"D" term associated with a huge initial vacuum energy density, \propto 1/g"^2. The corresponding hierarchy between energy scales, by a factor \propto 1/\sqrt g" \simge 10^{12}, involves a very large scale ~ 10^{16} GeV, that may be associated with inflation, or supersymmetry breaking with a very heavy gravitino, leading to possible values of {\delta} within the experimentally accessible range.
Highlights
U, in general mixed with the Z and the photon, may be massless or very light and very weakly coupled
The − sign corresponds to the fact that, for a new force effectively coupled to B − L, the test mass richer in neutrons should undergo a stronger repulsive force from the neutrons in the Earth, leading to a smaller apparent “free-fall” acceleration
The resulting apparent violations of the equivalence principle will be typically ≲10−15 for a new force coupled to B, up to a few 10−14 for a coupling involving L, including most notably B − L
Summary
Are all fundamental particles and forces known? It would be presumptuous to think so. Approaches towards a consistent theory of quantum gravity, and most notably string theories, usually involve additional Uð1Þ symmetries and describe many new fields and particles, some of them extremely weakly coupled This may occur in supersymmetric theories, possibly in connection with grand-unification, supersymmetry breaking, supergravity and the vacuum energy density, which plays an important role in the evolution of the Universe. FÞ; ð6Þ in the visible sector, remains conserved at this stage, as long as no neutrino Majorana mass terms are considered, with the grand-unification gauge bosons XÆ4/3 and YÆ1/3 having Of this possibility of grand-unification, the vev of the spin-0 doublet φ, taken with F 1⁄4 Y 1⁄4 1 (for γ ≠ 0), induces the electroweak breaking in a way involving the extra-Uð1Þ gauge field C, according to. This may allow to distinguish the spin-1 and spin-0 induced cases, should such a force be found
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