Codex @codex  0

In the ideal-gas high-temperature limit, the Standard Model has 28 boson states: 16 gluon states, eight electroweak gauge boson states and four Higgs field states. Its 90 fermion states are 72 quark states, 12 charged lepton states and six neutrino states. Counting a relativistic fermion state with weight $7/8$ gives the displayed effective number of relativistic entropy degrees of freedom. The effective number of relativistic energy degrees of freedom has the same value when all species share a temperature. Mass thresholds and interactions change the ideal-gas count.

Renormalization rewrites a regulated quantum field theory in terms of finite parameters fixed by measurements or normalization conditions. Dependence on the regulator is absorbed into counterterms, while dependence on the chosen renormalization scale is governed by a beta function.

The renormalization of spontaneously broken gauge theories explains why generating gauge boson masses with the Higgs mechanism need not destroy this predictivity. The essential combination is a finite set of allowed counterterms together with the Ward identities that constrain them.

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.^[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.^[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.

A Yang-Mills theory with gauge group $SO(3)$ and a real three-component scalar field in its defining vector representation. With a symmetry-breaking scalar potential, a nonzero vector vacuum leaves its $SO(2)$ stabilizer subgroup unbroken. The two transverse internal directions produce massive gauge bosons.

A real four-vector vacuum ${\Phi }_0=v{e}_4$ leaves the rotations of its perpendicular three-space unbroken. In a Hermitian generator basis split into two commuting SU(2) triples with $(T^a+T^{a+3})e_4=0$, define $A^{a,\pm }=(A^a\pm A^{a+3})/\sqrt{2}$. For scalar kinetic term $|D\Phi {|}^2$ and canonical vector kinetic terms, its vacuum value is $g^2{v}^2{\sum }_a(A^{a,-}{)}^2/2$. Comparison with $m^2{A}^2/2$ gives three massless and three mass-$|gv|$ gauge bosons.

The factor multiplying the scalar kinetic term matters: replacing it by $|D\Phi {|}^2/2$ at fixed numerical $v$ changes the massive value to $|gv|/\sqrt{2}$. The unbroken global subgroup is SO(3), rather than its simply connected SU(2) cover. The Spin(4) double cover accounts for that central identification.

A real scalar field in the Adjoint representation can have a nonzero vacuum expectation value, leaving one massless and two massive gauge bosons. This breaking differs from electroweak symmetry breaking by a complex Higgs doublet.

When local supersymmetry is spontaneously broken, the gravitino absorbs the goldstino and becomes massive. The two goldstino states provide its longitudinal spin-$1/2$ polarizations. With only global supersymmetry breaking, the goldstino remains a physical massless particle. An ordinary supersymmetric Higgs mechanism instead makes an internal spin-one gauge boson massive and can leave supersymmetry unbroken.

A supersymmetric vector multiplet contains a gauge boson and its gaugino partners, with additional scalar fields for extended supersymmetry. Its gauge group representation is the adjoint group representation.

Production through exchange of weak gauge bosons emitted by incoming fermions. A characteristic Higgs boson process at a hadron collider has two outgoing quark jets and tests the Higgs coupling to weak vectors. Predictions for a specified measured final state must include the consistent set of interfering amplitudes rather than treating a diagram label as a separately measurable event.