= Spontaneous breaking of a Z2 scalar symmetry
{title2=$V(\phi)=\lambda(\phi^2-v^2)^2/4,\quad m_h^2=2\lambda v^2$}
A <real scalar field> with the displayed potential is invariant under the <Z2 symmetry> $\phi\mapsto-\phi$, but a chosen broken-phase pure vacuum has nonzero <vacuum expectation value> of one sign. Expanding about $\phi=v$ gives $V=\lambda v^2h^2+\lambda vh^3+\lambda h^4/4$. The excitation is massive, and no continuous broken generator requires a <Goldstone boson>. In a finite-volume theory with symmetry-preserving conditions, tunneling can produce a symmetric ground state. A broken pure vacuum is selected by taking the infinite-volume limit before removing an infinitesimal source, or by specifying appropriate boundary conditions. The two signs are related by the symmetry, although the individual pure vacua are not invariant under it.
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