Editing Higgs mechanism (section)

== Standard Model ==
The Higgs mechanism was incorporated into modern particle physics by [[Steven Weinberg]] and [[Abdus Salam]], and is an essential part of the [[Standard Model]].

In the Standard Model, at temperatures high enough that electroweak symmetry is unbroken, all elementary particles are massless. At a critical temperature, the Higgs field develops a [[vacuum expectation value]]; some theories suggest the symmetry is spontaneously broken by [[tachyon condensation]], and the [[W and Z bosons]] acquire masses (also called "electroweak symmetry breaking", or ''EWSB''). In the history of the universe, this is believed to have happened about a [[picosecond]] {{nowrap|(10<sup>−12</sup> s)}} after the hot big bang, when the universe was at a temperature {{val|159.5|1.5|u=[[GeV]]/''k''<sub>B</sub>}}.<ref>{{cite journal  |doi = 10.1103/PhysRevD.93.025003 |arxiv=1508.07161|title=Standard model cross-over on the lattice|journal=Physical Review D|volume=93|issue=2|pages=025003|year=2016|last1=d'Onofrio|first1=Michela|last2=Rummukainen|first2=Kari|bibcode=2016PhRvD..93b5003D|s2cid=119261776}}</ref>

Fermions, such as the [[lepton]]s and [[quark]]s in the Standard Model, can also acquire mass as a result of their interaction with the Higgs field, but not in the same way as the gauge bosons.

=== Structure of the Higgs field ===
In the Standard Model, the Higgs field is an [[Special unitary group|SU(2)]] [[Doublet state|doublet]] (i.e. the standard representation with two complex components called isospin), which is a [[Scalar field theory|scalar]] under Lorentz transformations. Its electric charge is zero; its [[weak isospin]] is {{sfrac|1|2}} and the third component of weak isospin is −{{sfrac|1|2}}; and its [[weak hypercharge]] (the charge for the [[Unitary group|U(1)]] gauge group defined up to an arbitrary multiplicative constant) is 1. Under U(1) rotations, it is multiplied by a phase, which thus mixes the real and imaginary parts of the complex spinor into each other, combining to the standard two-component complex representation of the group U(2).

The Higgs field, through the interactions specified (summarized, represented, or even simulated) by its potential, induces spontaneous breaking of three out of the four generators ("directions") of the gauge group U(2). This is often written as {{nowrap|SU(2)<sub>L</sub> × U(1)<sub>Y</sub>}}, (which is strictly speaking only the same on the level of infinitesimal symmetries) because the diagonal phase factor also acts on other fields – [[quark]]s in particular. Three out of its four components would ordinarily resolve as [[Goldstone boson]]s, if they were not coupled to gauge fields.

However, after symmetry breaking, these three of the four degrees of freedom in the Higgs field mix with the three [[W and Z boson]]s ({{SubatomicParticle|W boson+}}, {{SubatomicParticle|W boson-}} and {{SubatomicParticle|Z boson0}}), and are only observable as components of these [[weak boson]]s, which are made massive by their inclusion; only the single remaining degree of freedom becomes a new scalar particle: the [[Higgs boson]]. The components that do not mix with Goldstone bosons form a massless photon.

=== The photon as the part that remains massless ===
The [[gauge group]] of the electroweak part of the standard model is {{nowrap|SU(2)<sub>L</sub> × U(1)<sub>Y</sub>}}. The group SU(2) is the group of all 2-by-2 unitary matrices with unit determinant; all the orthonormal changes of coordinates in a complex two dimensional vector space.

Rotating the coordinates so that the second basis vector points in the direction of the Higgs boson makes the [[vacuum expectation value]] of ''H'' the spinor&nbsp;{{math|(0, ''v'')}}. The generators for rotations about the x-, y-, and z-axes are by half the [[Pauli matrices]] {{mvar|σ}}{{sub|x}}, {{mvar|σ}}{{sub|y}}, and {{mvar|σ}}{{sub|z}}, so that a rotation of angle {{mvar|θ}} about the z-axis takes the vacuum to
: <math>\ \Bigl(\ 0\ ,\ v\ e^{-\tfrac{ 1 }{ 2 }\ i\ \theta}\ \Bigr) ~.</math>

While the  {{mvar|T}}{{sub|x}}  and  {{mvar|T}}{{sub|y}}  generators mix up the top and bottom components of the [[spinor]], the {{mvar|T}}{{sub|z}} rotations only multiply each by opposite phases. This phase can be undone by a U(1) rotation of angle {{nobr| {{sfrac| 1 | 2 }} {{mvar|θ}}}}. Consequently, under both an SU(2) {{mvar|T}}{{sub|z}}-rotation and a U(1) rotation by an amount {{nobr| {{sfrac| 1 | 2 }} {{mvar|θ}}}}, the vacuum is invariant.

This combination of generators
: <math>\ Q = T_3 + \tfrac{\ 1\ }{ 2 }\ Y_\mathsf{W}\ </math>
defines the unbroken part of the gauge group, where {{mvar|Q}} is the electric charge, {{mvar|T}}{{sub|3}}  is the generator of rotations around the 3-axis in the [[adjoint representation]] of SU(2) and {{mvar|Y}}{{sub|W}} is the [[weak hypercharge]] generator of the U(1). This combination of generators (a ''3'' rotation in the SU(2) and a simultaneous U(1) rotation by half the angle) preserves the vacuum, and defines the unbroken gauge group in the standard model, namely the electric charge group. The part of the gauge field in this direction stays massless, and amounts to the physical photon.
By contrast, the broken trace-orthogonal charge <math>\ T_3 - \tfrac{\ 1\ }{ 2 }\ Y_\mathsf{W} = 2\ T_3 - Q\ </math> couples to the massive {{SubatomicParticle|Z boson0|link}}&nbsp;boson.

=== Consequences for fermions ===
In spite of the introduction of spontaneous symmetry breaking, the mass terms preclude chiral gauge invariance. For these fields, the mass terms should always be replaced by a gauge-invariant "Higgs" mechanism. One possibility is some kind of [[Yukawa coupling]] (see below) between the fermion field {{mvar|ψ}} and the Higgs field {{math|''φ''}}, with unknown couplings {{mvar|G{{sub|ψ}}}}, which after symmetry breaking (more precisely: after expansion of the Lagrange density around a suitable ground state) again results in the original mass terms, which are now, however (i.e., by introduction of the Higgs field) written in a gauge-invariant way. The Lagrange density for the Yukawa interaction of a fermion field {{mvar|ψ}} and the Higgs field {{math|''φ''}} is
: <math>\ \mathcal{L}_{\mathrm{Fermion}}(\phi, A, \psi) ~=~ \overline{\psi}\ \gamma^{\mu}\ D_{\mu}\ \psi ~+~ G_{\psi}\ \overline{\psi}\ \phi\ \psi\ ,</math>
where again the gauge field {{mvar|A}} only enters via the gauge covariant derivative operator {{mvar|D{{sub|μ}}}} (i.e., it is only indirectly visible). The quantities {{math|''γ''{{sub|''μ''}}}} are the [[Dirac matrices]], and {{math|''G''{{sub|''ψ''}}}} is the already-mentioned Yukawa coupling parameter for {{mvar|ψ}}. Now the mass-generation follows the same principle as above, namely from the existence of a finite expectation value {{tmath|1= \vert\langle\phi\rangle\vert }}. Again, this is crucial for the existence of the property ''mass''.