Spontaneous symmetry breaking, gauge hierarchy and electroweak vacuum metastability

The so-called metastability bound asserts that an unnaturally small Higgs mass is a necessary condition for electroweak vacuum metastability, offering a new approach towards solving the hierarchy problem. So far, this result relies on the assumption of a negative Higgs mass parameter, or equivalently, on electroweak spontaneous symmetry breaking. We derive a new, corresponding bound for the case of a positive mass parameter. When the new bound is significantly more restrictive than or comparable to its established counterpart, it may offer an explanation for the sign of the Higgs mass parameter, and thus, spontaneous symmetry breaking itself. New physics at scales $\mathcal{O} (1-10)$ TeV can lower these bounds as far as the TeV-scale. As an illustration, we consider vacuum stability in the presence of additional TeV-scale fermions with Yukawa couplings to the Higgs, as well as a dimension-six term parameterizing new physics in the UV. This scenario requires new physics that couples strongly to the Higgs, and can potentially be probed at future colliders. Finally, to allow for comparison with concrete mechanisms predicting metastability, we provide the mass-dependent lifetime of the electroweak vacuum for this model.