  We consider theories where the Standard Model (SM) neutrinos acquire masses
through the seesaw mechanism at the weak scale. We show that in such a
scenario, the requirement that any pre-existing baryon asymmetry, regardless of
its origin, not be washed out leads to correlations between the pattern of SM
neutrino masses and the spectrum of new particles at the weak scale, leading to
definite predictions for the LHC. For type I seesaw models with a TeV scale Z'
coupled to SM neutrinos, we find that for a normal neutrino mass hierarchy, at
least one of the right-handed neutrinos must be `electrophobic', decaying with
a strong preference into final states with muons and tauons rather than
electrons. For inverted or quasi-degenerate mass patterns, on the other hand,
we find upper bounds on the mass of at least one right-handed neutrino. In
particular, for an inverted mass hierarchy, this bound is 1 TeV, while the
corresponding upper limit in the quasi-degenerate case is 300 GeV. Similar
results hold in type III seesaw models, albeit with somewhat more stringent
bounds. For the Type II seesaw case with a weak scale SU(2) triplet Higgs, we
again find that an interesting range of Higgs triplet masses is disallowed by
these considerations.
