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Higgs Mechanism

The Higgs mechanism explains the origin of mass for fundamental particles through the interaction with the Higgs field, a cornerstone of the Standard Model.
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The statement of the theorem

Consider the Higgs doublet Φ\Phi with potential V(Φ)=μ2Φ†Φ+λ(Φ†Φ)2V(\Phi) = \mu^2 \Phi^{\dagger}\Phi + \lambda (\Phi^{\dagger}\Phi)^2. If μ2<0\mu^2 < 0 and λ>0\lambda > 0, the potential minimum occurs at ⟨Φ†Φ⟩=v2/2\langle \Phi^{\dagger}\Phi \rangle = v^2/2, where v=−μ2/λv = \sqrt{-\mu^2 / \lambda}. Expanding Φ\Phi around its vacuum expectation value ⟨Φ⟩=12(0 v)\langle \Phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \ v \end{pmatrix}, the gauge boson kinetic terms Lkin=12∑A=W,Z(DμΦ)†(DμΦ)\mathcal{L}_{kin} = \frac{1}{2} \sum_{A=W,Z} (D_{\mu} \Phi)^{\dagger} (D^{\mu} \Phi) yield mass terms for the WW and ZZ bosons: \nLmass=12MW2Wμ ⁣Wμ+12MZ2ZμZμ\mathcal{L}_{mass} = \frac{1}{2} M_W^2 W_{\mu}\!W^{\mu} + \frac{1}{2} M_Z^2 Z_{\mu}Z^{\mu} \nwhere MW=12gvM_W = \frac{1}{2} g v and MZ=12g2+g′2vM_Z = \frac{1}{2} \sqrt{g^2 + g'^2} v.