= Renormalization of spontaneously broken gauge theories
<Renormalization of spontaneously broken gauge theories> means that the <Higgs mechanism> can generate <gauge boson> <masses> while a fixed finite set of <renormalized couplings>, <masses> and field normalizations still suffices to absorb <ultraviolet divergences>. The symmetry constrains the <counterterms> even when the chosen <vacuum expectation value> hides it. This makes <radiative corrections> to the <electroweak interaction> calculable after inputs have been fixed by <renormalization conditions>.
A useful distinction is between choosing variables that display the physical <particles> and choosing variables that display good <power counting>. <Unitary gauge> removes the would-be <Goldstone bosons> from the explicit fields. <R-xi gauge> keeps them and the required <Faddeev-Popov ghosts> in intermediate <Feynman diagrams>, exposing simpler large-<momentum> behavior. The physical <scattering amplitude> must agree after all contributions and <counterterms> have been included consistently.
In Renormalizable Lagrangians for massive Yang-Mills fields (1971), <Gerard 't Hooft> constructed spontaneously broken <gauge theories> with massive <gauge bosons>. His worked <SU(2) gauge theory with an adjoint Higgs field> leaves a massless <photon>; it is a demonstration model, not the full chiral <Standard Model>. The <gauge fixing>, <Faddeev-Popov ghosts> and <Ward identities> connect <renormalizability> with physical <unitarity>.<Renormalizable Lagrangians for massive Yang-Mills fields by Gerardus 't Hooft>[§§3–5, pp. 171–177; Eqs. (3.8)–(3.14)]{ref}
Regularization and renormalization of gauge fields (1972), by <Gerard 't Hooft> and <Martinus J. G. Veltman>, developed <dimensional regularization> with recursive subtraction of <ultraviolet subdivergences>. It continued <Feynman integrals> in the dimension, keeping the relevant <Ward identities> in non-anomalous cases; its final section examined <quantum anomalies>. The procedure does not make an inconsistent chiral <gauge theory> consistent: <gauge anomaly> cancellation remains essential.<Regularization and renormalization of gauge fields by Gerardus 't Hooft and Martinus Justinus Godefridus Veltman>[§1, pp. 189–191; §4, pp. 198–205; §6, pp. 206–208]{ref}
The organizing proof outline is to use good large-<momentum> <propagators>, subtract every divergent subgraph, and constrain the remaining local divergences by the <Ward identities>. A finite family of allowed <counterterms> must close under this operation at every <loop order>. The same identities must remove unphysical intermediate contributions from physical amplitudes. <Power counting> alone establishes neither closure nor these cancellations. Modern <BRST symmetry> packages these restrictions; the scalar subtraction below illustrates only locality.
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