  A large value of the cosmological constant (CC) is induced in the Standard
Model (SM) of Elementary Particle Physics because of Spontaneous Symmetry
Breaking. To provide a small value of the observable CC one has to introduce
the vacuum term which cancels the induced one at some point in the very far
infrared cosmic scale. Starting from this point we investigate whether the
cancellation is preserved at different energy scales. We find that the running
of the Higgs mass, couplings and the vacuum term inevitably result in a scaling
dependence of the observable CC value. As a consequence one meets a nonzero CC
at an energy scale comparable to the typical electron neutrino mass suggested
by some experiments, and the order of magnitude of this constant is roughly the
one derived from recent supernovae observations. However the sign of it is
negative -- opposite to what is suggested by these observations. This
discrepancy may be a hint of the existence of an extra very light scalar,
perhaps a Cosmon-like dilaton, which should essentially decouple from the SM
Lagrangian, but that it nevertheless could mediate new macroscopic forces in
the submillimeter range.
