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V(ϕ)=−μ2 ϕ†ϕ+λ (ϕ†ϕ)2,μ2>0, λ>0V(\phi) = -\mu^2\,\phi^\dagger\phi + \lambda\,(\phi^\dagger\phi)^2, \qquad \mu^2 > 0,\ \lambda > 0

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The simplest version of the mechanism starts from a single complex scalar field ϕ\phi with a potential energy density of the form V(ϕ)=−μ2 ϕ†ϕ+λ (ϕ†ϕ)2,μ2>0, λ>0V(\phi) = -\mu^2\,\phi^\dagger\phi + \lambda\,(\phi^\dagger\phi)^2, \qquad \mu^2 > 0,\ \lambda > 0. Plotted against the two real components of ϕ\phi , this is the shape popularly called the Mexican hat: a local maximum at ϕ\phi = 0 , surrounded by a circular trough of degenerate minima at |ϕ\phi| = μ\mu/2λ\sqrt{2\lambda} . The sign of the mass term is the entire mechanism. With μ2\mu^2 < 0 the origin would be the unique minimum and there would be nothing to discuss; it is the negative mass-squared term, an assumption built into the Standard Model’s Higgs sector rather than derived from a deeper principle, that manufactures the trough in the first…

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ϕ\phi

Symbol phi

phi is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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ϕ†\phi^\dagger

Symbol phi^dagger

phidi^dagger is one of the signed contributions combined to compute the quantity on the left.

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Published contexts (1)

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V(ϕ)=−μ2 ϕ†ϕ+λ (ϕ†ϕ)2,μ2>0, λ>0.V(\phi) = -\mu^2\,\phi^\dagger\phi + \lambda\,(\phi^\dagger\phi)^2, \qquad \mu^2 > 0,\ \lambda > 0.

Equation 2 · Evolutionary Physics

The Vacuum Has a History: Symmetry Breaking as Speciation

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

The simplest version of the mechanism starts from a single complex scalar field ϕ\phi with a potential energy density of the form V(ϕ)=−μ2 ϕ†ϕ+λ (ϕ†ϕ)2,μ2>0, λ>0V(\phi) = -\mu^2\,\phi^\dagger\phi + \lambda\,(\phi^\dagger\phi)^2, \qquad \mu^2 > 0,\ \lambda > 0. Plotted against the two real components of ϕ\phi , this is the shape popularly called the Mexican hat: a local maximum at ϕ\phi = 0 , surrounded by a circular trough of degenerate minima at |ϕ\phi| = μ\mu/2λ\sqrt{2\lambda} . The sign of the mass term is the entire mechanism. With μ2\mu^2 < 0 the origin would be the unique minimum and there would be nothing to discuss; it is the negative mass-squared term, an assumption built into the Standard Model’s Higgs sector rather than derived from a deeper principle, that manufactures the trough in the first…

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