$$\phi = 0$$

Equation 4 · Evolutionary Physics

The Vacuum Has a History: Symmetry Breaking as Speciation

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Plotted against the two real components of $\phi$ , this is the shape popularly called the Mexican hat: a local maximum at $\phi$ = 0 , surrounded by a circular trough of degenerate minima at |$\phi$| = $\mu$/$\sqrt{2\lambda}$ . The sign of the mass term is the entire mechanism. With $\mu^2$ < 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 place. Nothing in the Lagrangian singles out any point on that trough. A system placed at the unstable summit will roll downhill in a direction fixed…

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