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Spontaneous Symmetry Breaking

Spontaneous symmetry breaking is the process by which the electroweak theory transitions from a symmetric state to a broken state, giving rise to mass.
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The statement of the theorem

Consider a scalar field ϕ\phi governed by a potential V(ϕ)V(\phi) with a Mexican hat shape, V(ϕ)=μ2ϕ†ϕ+λ(ϕ†ϕ)2V(\phi) = \mu^2 \phi^{\dagger} \phi + \lambda (\phi^{\dagger} \phi)^2, where μ2<0\mu^2 < 0. The system minimizes its energy by acquiring a non-zero vacuum expectation value (VEV): \n⟨ϕ⟩=v/2, where v=−μ2/λ\langle \phi \rangle = v / \sqrt{2}, \text{ where } v = \sqrt{-\mu^2 / \lambda} \nThis process breaks the symmetry G→HG \to H (e.g., SU(2)L×U(1)Y→U(1)EMSU(2)_L \times U(1)_Y \to U(1)_{EM}), resulting in mass generation for the gauge bosons and fermions.