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Vacuum Energy

The principle that neutrino masses arise from the expectation value of the Dirac operator in the vacuum state, related to vacuum energy.
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The statement of the theorem

The vacuum energy density ρvac\rho_{vac} is related to the expectation value of the Lagrangian L\mathcal{L} in the vacuum state ∣0⟩|0\rangle. If neutrino masses arise from spontaneous symmetry breaking (SSB) via a mechanism like the seesaw, the effective mass matrix MνM_{\nu} is determined by the vacuum expectation values (VEVs) of the relevant scalar fields ⟨ϕ⟩\langle \phi \rangle. Specifically, the effective potential VeffV_{eff} must yield a non-zero minimum ⟨ϕ⟩≠0\langle \phi \rangle \neq 0, leading to the mass term: \nLmass=−12νˉMνν\mathcal{L}_{mass} = -\frac{1}{2} \bar{\nu} M_{\nu} \nu \nwhere Mν∝1⟨ϕ⟩⟨Higgs⟩M_{\nu} \propto \frac{1}{\langle \phi \rangle} \langle \text{Higgs} \rangle (schematically).