MICROSCOPE limits for new long-range forces and implications for unified theories

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 - .61Q, 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 Eotvos parameter delta, and to almost the same limits for epsilon(B-L) or epsilon(L), as long as no indication for delta not equal 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 vertical bar epsilon(B-L)vertical bar or vertical bar epsilon(L)vertical bar < .8 10(-24), and a force coupled to [epsilon(Q)Q + epsilon B-B]e, vertical bar epsilon(B)vertical bar < 5 10(-24). We also discuss, within supersymmetric theories, how such extremely small gauge couplings g '', typically less than or similar to 10(-24), may be related to a correspondingly large xi"D" term associated with a huge initial vacuum energy density, proportional to 1/g ''(2). The corresponding hierarchy between energy scales, by a factor. 1/root g '' greater than or similar to 10(12), involves a very large scale similar to 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.