The vacuum of the Standard Model is known to be unstable for the measured
values of the top and Higgs masses. Here we show how vacuum stability can be
achieved naturally if lepton number is violated spontaneously at the TeV scale.
More precise Higgs measurements in the next LHC run should provide a crucial
test of our symmetry breaking scenario. In addition, these schemes typically
lead to enhanced rates for processes involving lepton flavour violation .Comment: 9 pages, 4+2 figures; some references added, some textual
modifications: 2 figures added, appendices added. Results unchanged. Matches
published versio