  We recall how a new light gauge boson emerged in Supersymmetric extensions of
the Standard Model with an extra singlet superfield, and how it could behave
very much as a light pseudoscalar, with the corresponding symmetry broken at a
scale higher than electroweak.
  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 it be found directly in e+e- annihilations ? Not so
easily, due to various constraints on its couplings, especially axial ones,
leading to an axionlike behavior or extra parity-violation effects. Searches
for the decay Upsilon -> gamma + invisible U may be used to constrain severely
its axial coupling to the electron, f_eA = f_bA, to be less than about 10^-6
m_U(MeV).
  The vector coupling to the electron may in principle be larger. We find from
g_mu-2, under reasonable assumptions, that this coupling can be at most as
large as ~ 1.3 10^-3, for m_U < m_mu. Such a coupling of the order of 10^-3
could also be responsible for the somewhat large value of g_mu-2, compared to
Standard Model expectations.
  The U couplings to electrons are otherwise likely to be smaller, e.g. \lsim 3
10^-6 m_U(MeV), if the couplings to neutrinos and electrons are similar. This
restricts significantly the possibility of detecting a light U boson in e+e- ->
gamma U. Despite that, 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.
