Spontaneous symmetry breaking and the Higgs mechanism

Take a complex scalar with canonical kinetic term $|\partial\phi|^2$ and potential $V=\lambda(|\phi|^2-v^2/2)^2$, where $\lambda>0$ and $v>0$. For a global U(1), minima satisfy $|\phi|=v/\sqrt2$. In a neighborhood of a selected vacuum write $\phi=(v+h)e^{i\pi/v}/\sqrt2$. The radial coordinate $h$ and angular coordinate $\pi$ have canonical quadratic kinetic terms, although higher interactions make the polar-coordinate metric field dependent.

Expand the potential: $V=\lambda v^2h^2+\lambda vh^3+\lambda h^4/4$. Compare its quadratic term with $m_h^2h^2/2$, yielding $m_h^2=2\lambda v^2$. There is no angular potential, so $\pi$ is massless at this classical level. For a genuine spontaneously broken continuous global symmetry, Goldstone's theorem protects the corresponding gapless behavior under its assumptions, including a suitable infinite-volume limit and Lorentz-invariant setting.

Now use $D_\mu=\partial_\mu-igA_\mu$. The scalar kinetic term contains $(v+h)^2(\partial_\mu\pi/v-gA_\mu)^2/2$. In a suitable local gauge the angular coordinate is removed from that expression, leaving a vector mass term $g^2v^2A_\mu A^\mu/2$. Compare with the Proca normalization to obtain $m_A=gv$. The original massless vector's two physical polarizations and the scalar angular degree of freedom reorganize into three massive-vector polarizations.

This is the Higgs mechanism, not the insertion of an arbitrary gauge-breaking mass by hand. Covariant gauges retain an angular field and ghosts with gauge-dependent masses; their contributions cancel appropriately in physical quantities. The radial excitation remains a physical mode in the weakly coupled expansion. The phrase that a vector 'eats' a scalar is only a mnemonic for the changed polarization count, not a literal process in which a particle is swallowed.

Without gauge fixing, a local gauge-variant field average cannot serve as an ordinary physical order parameter. Elitzur's theorem in its standard gauge-invariant lattice setting rules out spontaneous breaking of the local gauge redundancy. A nonzero scalar expectation value in a chosen gauge remains a useful perturbative coordinate, but it is not a direct gauge-invariant measurement. The Higgs regime is characterized through physical spectra, correlation functions, and suitable gauge-invariant diagnostics.

Global symmetry breaking also requires attention to limits. A finite symmetric system ordinarily has a symmetric expectation value unless an external source selects a direction. One takes an infinite-volume limit before removing an infinitesimal source to define a broken phase. A classical Mexican-hat plot does not by itself prove a quantum phase transition, determine finite-temperature behavior, or guarantee a sharp distinction between every Higgs and confining regime for every matter representation. Those questions depend on the full theory and its observables.