We recall how a new light gauge boson emerged in Supersymmetric extensions of the Standard Model with an extra singlet chiral superfield, and how it could often behave very much as a light pseudoscalar, with the corresponding symmetry broken at a scale higher than electroweak.The possible existence of such a new gauge boson U , light and very weakly coupled, allows for Light Dark Matter particles, which could be at the origin of the 511 keV line from the galactic bulge. Could such a light gauge boson be found directly in e + e − annihilations ? Not so easily, in fact, due to various constraints limiting the size of its couplings, especially the axial ones, leading to an axionlike behavior or extra parity-violation effects. In particular, searches for the decay Υ → γ + invisible U may be used to constrain severely the axial coupling of the U to the electron, fe A = f b A , to be less than about 10 −6 mU (MeV), 50 times smaller than the ≃ 5 10 −5 mU (MeV) that could otherwise have been allowed from ge − 2.The vector coupling of the U to the electron may in principle be larger, but is also limited in size. Even under favorable circumstances (no axial couplings to quarks and charged leptons, and very small couplings to neutrinos), taking also into account possible Z-U mixing effects, we find from gµ − 2 , under reasonable assumptions (no cancellation effect, lepton universality), that the vector coupling of the U to the electron can be at most as large as ≃ 1.3 10 −3 , for mU < mµ . Such a coupling to the muon of the order of 10 −3 could also be responsible for the somewhat large value of the measured gµ − 2 , as compared to Standard Model expectations, should this effect turn out to be real.The U couplings to electrons are otherwise likely to be smaller, e.g. < ∼ 3 10 −6 mU (MeV), if the couplings to neutrinos and electrons are similar. This restricts significantly the possibility of detecting a light U boson in e + e − → γ U , making this search quite challenging. Despite the smallness of these couplings, U exchanges can provide annihilation cross sections of LDM particles of the appropriate size, even if this may require that light dark matter be relatively strongly self-interacting.