  The vacuum stability condition of the Standard Model Higgs potential with
mass in the range of 124-127 GeV puts an upper bound on the Dirac mass of the
neutrinos. We study this constraint with the right-handed neutrino masses upto
TeV scale. The heavy neutrinos contribute to $\Delta L=2$ processes like
neutrinoless double beta decay and same-sign-dilepton production in the
colliders. The vacuum stability criterion also restricts the light-heavy
neutrino mixing and constrains the branching ratio of lepton flavour violating
process, like $\mu \to e \gamma$ mediated by the heavy neutrinos. We show that
neutrinoless double beta decay with a lifetime $\sim 10^{25}$ years can be
observed if the the lightest heavy neutrino mass is $<$ 4.5 TeV. We show that
the vacuum stability condition and the experimental bound on $\mu \rightarrow e
\gamma$ together put a constrain on heavy neutrino mass $M_R >$ 3.3 TeV.
Finally we show that the observation of same-sign-dileptons (SSD) associated
with jets at the LHC needs much larger luminosity than available at present. We
have estimated the possible maximum cross-section for this process at the LHC
and show that with an integrated luminosity 100 $fb^{-1}$ it may be possible to
observe the SSD signals as long as $M_R <$ 400 GeV.
