The Higgs Mechanism

Goldstone's theorem says a spontaneously broken global symmetry gives a massless boson. When the broken symmetry is instead gauged, something more useful happens: the would-be Goldstone boson is absorbed by the gauge field, which becomes massive. This is the Higgs mechanism — the only known way to give gauge bosons mass without wrecking renormalizability, and the origin of the $W$, $Z$, and fermion masses in the electroweak theory.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Why gauge boson masses are a problem

A naive mass term $m^2{A}_{\mu }{A}^{\mu }$ for a gauge field is not gauge-invariant (under $A_{\mu }\to A_{\mu }+{\partial }_{\mu }\alpha$ it is not invariant), so it cannot simply be added to the Lagrangian. Worse, the massive-vector propagator has a $p_{\mu }{p}_{\nu }/m^2$ term that grows at high energy, spoiling renormalizability. The Higgs mechanism generates the mass dynamically, evading both problems.

Abelian Higgs model

Gauge the $U(1)$ Goldstone model: replace ${\partial }_{\mu }\to D_{\mu }={\partial }_{\mu }-ie{A}_{\mu }$ and add the photon kinetic term:

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+|D_{\mu }\varphi {|}^2-V(\varphi ),V=-{\mu }^2|\varphi {|}^2+\lambda |\varphi {|}^4.$$

As before the potential drives a vev $⟨\varphi ⟩=v/\sqrt{2}$. Expand $\varphi =\frac{1}{\sqrt{2}}(v+\sigma )e^{i\pi /v}$ as in the Goldstone analysis. The key new term is the covariant-derivative kinetic piece evaluated at the vev:

$$|D_{\mu }\varphi {|}^2\supset \frac{1}{2}{e}^2{v}^2{A}_{\mu }{A}^{\mu }.$$

A gauge-boson mass $m_A=ev$ has appeared — generated by the vev, fully gauge invariant. Counting degrees of freedom confirms the bookkeeping:

"Eating" the Goldstone boson

The Goldstone mode $\pi$ has not vanished — it has been absorbed. In unitary gauge one uses the gauge freedom to set $\pi =0$ entirely, removing it from the spectrum. Its single degree of freedom becomes the longitudinal polarization the now-massive gauge boson needs (a massless vector has 2 polarizations, a massive one has 3). The slogan: "the gauge boson eats the Goldstone boson and grows heavy." No massless particle appears — Goldstone's theorem is evaded precisely because the current is now a gauge current.

Renormalizability: the $R_{\xi }$ gauges

Unitary gauge makes the physical spectrum manifest but hides renormalizability (the propagator grows at high energy). The resolution — 't Hooft's — is the $R_{\xi }$ gauges (the gauge-fixing family), in which the would-be Goldstone reappears as an unphysical field with a $\xi$-dependent mass, and the gauge-boson propagator is well-behaved:

$$D_{\mu \nu }(p)=\frac{-i}{p^2-m_A^2}\left[{\eta }_{\mu \nu }-(1-\xi )\frac{p_{\mu }{p}_{\nu }}{p^2-\xi m_A^2}\right].$$

Because this falls off as $1/p^2$ at large $p$, power counting works and the theory is renormalizable — 't Hooft's 1971 proof (using BRST/Slavnov–Taylor identities) that established the electroweak theory as a consistent quantum theory. Physical amplitudes are $\xi$-independent, matching the unitary-gauge ($\xi \to \infty$) spectrum.

Non-abelian case and the electroweak theory

Gauging a non-abelian broken symmetry $G\to H$ gives mass to the gauge bosons of the broken generators (one longitudinal mode each, eaten from the $\dim G-\dim H$ would-be Goldstones), while the unbroken $H$ gauge bosons stay massless. In the electroweak theory:

$$SU(2{)}_L\times U(1{)}_Y\stackrel{⟨\Phi ⟩}{\to }U(1{)}_{\text{EM}},$$

with a scalar doublet $\Phi$ (four real fields). Three would-be Goldstones are eaten to give the massive $W^{\pm }$ and $Z^0$; the fourth is the physical Higgs boson (observed at 125 GeV, 2012). The unbroken $U(1{)}_{\text{EM}}$ keeps the photon massless. Fermion masses arise separately from Yukawa couplings $y{\bar{\psi }}_L\Phi {\psi }_R$, which become $m=yv/\sqrt{2}$ at the vev — the same vev doing all the mass generation.

Summary

• Gauging a spontaneously broken symmetry: the gauge boson acquires mass $m_A=ev$ from the scalar vev, gauge-invariantly.
• The Goldstone boson is eaten — it becomes the longitudinal polarization of the massive gauge boson (dof conserved).
• Unitary gauge shows the spectrum; $R_{\xi }$ gauges show renormalizability (’t Hooft).
• Electroweak: $SU(2)\times U(1)\to U(1{)}_{\text{EM}}$ gives $W,Z$ mass and the Higgs boson; fermion masses from Yukawa couplings.

Where this leads

• The full theory: electroweak theory and the Standard Model.
• Radiatively induced breaking: the effective potential.
• Why it is renormalizable: BRST and Slavnov–Taylor.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 20.1.
• Higgs, Phys. Rev. Lett. 13, 508 (1964); Englert & Brout, ibid. 321.
• 't Hooft, Nucl. Phys. B 35, 167 (1971).
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 21.