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| {{Standard model of particle physics}}
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| {{Quantum field theory}}
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| In [[particle physics]], the '''Higgs mechanism''' is essential to explain the [[mass generation mechanism|generation mechanism]] of the property "[[mass]]". This is a most-important property of almost all [[elementary particles]].<ref>An exception is the "[[photon]]", the quantum of "light radiation", which is massless.</ref>
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| According to this theory, particles gain mass by interacting with the so-called [[Introduction to the Higgs field|Higgs field]] that permeates all space. More precisely, the Higgs mechanism endows the three so-called [[gauge boson]]s Z, W<sup>+</sup> and W<sup>−</sup> with mass. These particles would otherwise be massless; but actually they are very heavy, with values around 80 GeV/c<sup>2</sup>. But also more common particles are endowed with mass by this mechanism, e.g. the simple [[electron]], or the [[quark]] constituents of, e.g., [[proton]]s, technically through [[spontaneous symmetry breaking]], where, however, due to the specific form of the symmetry breaking, instead of the usual transverse [[Goldstone boson|Nambu–Goldstone boson]]s a longitudinal boson, named the [[Higgs boson]], appears.
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| The simplest implementation of the mechanism adds an extra Higgs field to the gauge theory. The specific spontaneous [[symmetry breaking]] of the underlying local symmetry, which is similar to that one appearing in the theory of [[superconductivity]], triggers conversion of the longitudinal field component to the Higgs boson, which interacts with itself and (at least a part of) the other fields in the theory, so as to produce mass terms for the three gauge bosons, and also to [[fermion]]s.
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| In the [[Standard Model]], the phrase "Higgs mechanism" refers specifically to the generation of masses for the [[W and Z bosons|W<sup>±</sup>, and Z]] [[Weak force|weak]] gauge bosons through [[Electroweak interaction|electroweak]] symmetry breaking.<ref name=PDG>G. Bernardi, M. Carena, and T. Junk: "Higgs bosons: theory and searches", Reviews of Particle Data Group: Hypothetical particles and Concepts, 2007, http://pdg.lbl.gov/2008/reviews/higgs_s055.pdf</ref> The [[Large Hadron Collider]] at [[CERN]] announced results consistent with the Higgs particle on July 4, 2012 but stressed that further testing is needed to confirm the Standard Model.
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| The mechanism was proposed in 1962 by [[Philip Warren Anderson]]. The relativistic model was developed in 1964 by three independent groups: by [[Robert Brout]] and [[François Englert]]; by [[Peter Higgs]]; and by [[Gerald Guralnik]], [[C. R. Hagen]], and [[Tom W. B. Kibble|Tom Kibble]].
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| The Higgs mechanism is therefore also called the '''Brout–Englert–Higgs mechanism''' or '''Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism''',<ref>{{cite web|url=http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism |title=Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia |publisher=Scholarpedia.org |date= |accessdate=2012-06-16}}</ref> '''Anderson–Higgs mechanism''',<ref>{{cite doi|10.1103/PhysRevB.65.132513|noedit}}</ref> '''Anderson–Higgs-Kibble mechanism''',<ref>{{cite doi|10.1016/0550-3213(75)90215-1|noedit}}</ref> '''Higgs–Kibble mechanism''' by Abdus Salam<ref name="frank_close_infinity_puzzle">{{cite book|last=Close|first=Frank|title=The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe|year=2011|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-959350-7}}</ref> and '''ABEGHHK'tH mechanism''' [for Anderson, Brout, Englert, Guralnik, Hagen, Higgs, Kibble and [[Gerard 't Hooft|'t Hooft]]] by Peter Higgs.<ref name="frank_close_infinity_puzzle"/>
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| On October 8, 2013, it was announced that [[Peter Higgs]] and [[François Englert]] had been awarded the 2013 [[Nobel Prize in Physics]] "for the theoretical discovery of a mechanism that contributes to our understanding of the origin of the mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS and CMS experiments at CERN’s Large Hadron Collider".<ref>{{cite web |url=http://www.nobelprize.org/nobel_prizes/physics/laureates/2013/press.pdf |title=Press release from Royal Swedish Academy of Sciences |date=8 October 2013 |accessdate=8 October 2013}}</ref>
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| == Standard model ==
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| The Higgs mechanism was incorporated into modern particle physics by [[Steven Weinberg]] and [[Abdus Salam]], and is an essential part of the [[standard model]].
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| 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 becomes [[Tachyonic field|tachyonic]], the symmetry is spontaneously broken by [[Tachyon condensation|condensation]], and the [[W and Z bosons]] acquire masses. (EWSB, ElectroWeak Symmetry Breaking, is an abbreviation used for this.)
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| 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.
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| === Structure of the Higgs field ===
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| In the standard model, the Higgs field is an [[Special unitary group|'''SU'''(2)]] doublet, a complex [[spinor]] with four real components (or equivalently with two complex components).
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| Its ([[weak hypercharge]]) '''U'''(1) charge is 1. That means that it transforms as a spinor under '''SU'''(2). 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—so this is ''not the same'' as two complex spinors mixing under '''U'''(1) (which would have eight real components between them), but instead is the spinor representation of the group '''U'''(2).
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| 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 '''SU'''(2) × '''U'''(1): three out of its four components would ordinarily amount to [[Goldstone bosons]], if they were not coupled to gauge fields.
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| However, after symmetry breaking, these three of the four degrees of freedom in the Higgs field mix with the three W and Z bosons ({{SubatomicParticle|W boson+}}, {{SubatomicParticle|W boson-}} and {{SubatomicParticle|Z boson}}), and are only observable as spin components of these [[weak boson]]s, which are now massive; while the one remaining degree of freedom becomes the [[Higgs boson]]—a new scalar particle.
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| === The photon as the part that remains massless ===
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| The gauge group of the electroweak part of the standard model is '''SU'''(2) × '''U'''(1). The group '''SU'''(2) is all 2-by-2 unitary matrices; all the orthonormal changes of coordinates in a complex two dimensional vector space.
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| 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 (0, ''v''). The generators for rotations about the ''x'', ''y'', and ''z'' axes are by half the Pauli matrices σ<sub>''x''</sub>, σ<sub>''y''</sub>, and σ<sub>''z''</sub>, so that a rotation of angle θ about the ''z''-axis takes the vacuum to
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| ::<math> (0, v e^{-i\theta/2}).</math>
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| While the ''T''<sub>x</sub> and ''T''<sub>y</sub> generators mix up the top and bottom components of the spinor, the ''T''<sub>z</sub> rotations only multiply each by opposite phases. This phase can be undone by a '''U'''(1) rotation of angle {{sfrac|1|2}}θ. Consequently, under both an '''SU'''(2) ''T''<sub>z</sub>-rotation and a '''U'''(1) rotation by an amount {{sfrac|1|2}}θ, ''the vacuum is invariant''.
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| This combination of generators
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| ::<math> Q = T_z + \frac{Y}{2}</math>
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| defines the unbroken part of the gauge group, where ''T''<sub>z</sub> is the generator of rotations around the z-axis in the '''SU'''(2) and ''Y'' is the hypercharge generator of the '''U'''(1). This combination of generators (a ''z'' 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.
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| === Consequences for fermions ===
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| In spite of the introduction of spontaneous symmetry breaking, the mass terms oppose the 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 ψ and the Higgs field Φ, with unknown couplings ''G''<sub>ψ</sub>, 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 ψ and the Higgs field Φ is
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| :<math> \mathcal{L}_{\mathrm{Fermion}}(\phi, A, \psi) = \overline{\psi} \gamma^{\mu} D_{\mu} \psi + G_{\psi} \overline{\psi} \phi \psi,</math>
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| where again the gauge field ''A'' only enters ''D''<sub>μ</sub> (i.e., it is only indirectly visible). The quantities γ<sup>μ</sup> are the [[Dirac matrices]], and ''G''<sub>ψ</sub> is the already-mentioned "Yukawa" coupling parameter. Already now the mass-generation follows the same principle as above, namely from the existence of a finite expectation value <math>|\langle\phi\rangle |</math>, as described above. Again, this is crucial for the existence of the property "mass".
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| == History of research ==
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| === Background ===
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| [[Spontaneous symmetry breaking]] offered a framework to introduce bosons into relativistic quantum field theories. However, according to [[Goldstone's theorem]], these bosons should be massless.<ref>{{cite web|url=http://www.datafilehost.com/download-7d512618.html |title=Guralnik, G S; Hagen, C R and Kibble, T W B (1967). Broken Symmetries and the Goldstone Theorem. Advances in Physics, vol. 2 |publisher=Datafilehost.com |date= |accessdate=2012-06-16}}</ref> The only observed particles which could be approximately interpreted as Goldstone bosons were the [[pion]]s, which [[Yoichiro Nambu]] related to [[chiral symmetry]] breaking.
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| A similar problem arises with [[Yang–Mills theory]] (also known as [[nonabelian]] [[gauge theory]]), which predicts massless [[Spin (physics)|spin]]-1 [[gauge boson]]s. Massless weakly interacting gauge bosons lead to long-range forces, which are only observed for electromagnetism and the corresponding massless [[photon]]. Gauge theories of the [[weak force]] needed a way to describe massive gauge bosons in order to be consistent.
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| === Discovery ===
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| [[File:AIP-Sakurai-best.JPG|thumb|Five of the six 2010 APS [[Sakurai Prize]] Winners – (L to R) Tom Kibble, Gerald Guralnik, Carl Richard Hagen, François Englert, and Robert Brout]]
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| [[File:Higgs, Peter (1929).jpg|thumb|Number six: Peter Higgs 2009]]
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| {{Wikinews|Prospective Nobel Prize for Higgs boson work disputed}}
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| The mechanism was proposed in 1962 by [[Philip Warren Anderson]],<ref>{{cite journal | author=P. W. Anderson | title=Plasmons, Gauge Invariance, and Mass | journal=Physical Review | volume=130 | year=1962 | issue=1 | pages=439–442 | doi=10.1103/PhysRev.130.439 |bibcode = 1963PhRv..130..439A }}</ref> who discussed its consequences for particle physics but did not work out an explicit relativistic model. The relativistic model was developed in 1964 by three independent groups – [[Robert Brout]] and [[François Englert]];<ref>{{cite journal | author=F. Englert and R. Brout | title=Broken Symmetry and the Mass of Gauge Vector Mesons | journal=Physical Review Letters | volume=13 | year=1964 | issue=9 | pages=321–323 | doi=10.1103/PhysRevLett.13.321 | bibcode=1964PhRvL..13..321E}}</ref> [[Peter Higgs]];<ref>{{cite journal | author=Peter W. Higgs | title=Broken Symmetries and the Masses of Gauge Bosons | journal=Physical Review Letters | volume=13 | year=1964 | issue=16 | pages=508–509 | doi=10.1103/PhysRevLett.13.508 | bibcode=1964PhRvL..13..508H}}</ref> and [[Gerald Guralnik]], [[C. R. Hagen|Carl Richard Hagen]], and [[Tom W. B. Kibble|Tom Kibble]].<ref>{{cite journal | author=G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble | title=Global Conservation Laws and Massless Particles | journal=Physical Review Letters | volume=13 | year=1964 | issue=20 | pages=585–587 | doi=10.1103/PhysRevLett.13.585 | bibcode=1964PhRvL..13..585G}}</ref><ref>{{cite journal | author=Gerald S. Guralnik | title=The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles | journal=International Journal of Modern Physics | volume=A24 | issue=14 | year=2009 | pages=2601–2627 | arxiv=0907.3466 | doi=10.1142/S0217751X09045431|bibcode = 2009IJMPA..24.2601G }}</ref><ref>[http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism_%28history%29 History of Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia.]</ref> The mechanism is closely analogous to phenomena previously discovered by [[Yoichiro Nambu]] involving the "vacuum structure" of quantum fields in [[superconductivity]].<ref>{{cite journal | last =Nambu | first = Y | year = 1960 | title = Quasiparticles and Gauge Invariance in the Theory of Superconductivity | journal = Physical Review | volume = 117 | issue =3 | pages = 648–663| doi = 10.1103/PhysRev.117.648|bibcode = 1960PhRv..117..648N }}</ref> A similar but distinct effect (involving an affine realization of what is now recognized as the Higgs field), known as the [[Stueckelberg action|Stueckelberg mechanism]], had previously been studied by [[Ernst Stueckelberg]].
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| These physicists discovered that when a gauge theory is combined with an additional field that spontaneously breaks the symmetry group, the gauge bosons can consistently acquire a nonzero mass. In spite of the large values involved (see below) this permits a gauge theory description of the weak force, which was independently developed by [[Steven Weinberg]] and [[Abdus Salam]] in 1967. Higgs's original article presenting the model was rejected by [[Physics Letters]]. When revising the article before resubmitting it to [[Physical Review Letters]], he added a sentence at the end,<ref>{{Cite journal
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| |first=Peter |last=Higgs
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| |year=2007
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| |title=Prehistory of the Higgs boson
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| |journal=Comptes Rendus Physique
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| |volume=8
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| |issue=9 |pages=970–972
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| |doi=10.1016/j.crhy.2006.12.006
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| |postscript=<!--None-->
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| |bibcode = 2007CRPhy...8..970H }}</ref> mentioning that it implies the existence of one or more new, massive scalar bosons, which do not form complete [[group representation|representations]] of the symmetry group; these are the Higgs bosons.
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| The three papers by Brout and Englert; Higgs; and Guralnik, Hagen, and Kibble were each recognized as "milestone letters" by ''Physical Review Letters'' in 2008.<ref>{{cite web|url=http://prl.aps.org/50years/milestones#1964 |title=Physical Review Letters – 50th Anniversary Milestone Papers |publisher=Prl.aps.org |date= |accessdate=2012-06-16}}</ref> While each of these seminal papers took similar approaches, the contributions and differences among the [[1964 PRL symmetry breaking papers]] are noteworthy. All six physicists were jointly awarded the 2010 [[Sakurai Prize|J. J. Sakurai Prize for Theoretical Particle Physics]] for this work.<ref>{{cite web|url=http://www.aps.org/units/dpf/awards/sakurai.cfm |title=American Physical Society – J. J. Sakurai Prize Winners |publisher=Aps.org |date= |accessdate=2012-06-16}}</ref>
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| [[Benjamin W. Lee]] is often credited with first naming the "Higgs-like" mechanism, although there is debate around when this first occurred.<ref>{{cite web|author=Department of Physics and Astronomy |url=http://www.pas.rochester.edu/urpas/news/Hagen_030708 |title=Rochester's Hagen Sakurai Prize Announcement |publisher=Pas.rochester.edu |date= |accessdate=2012-06-16}}</ref><ref>{{cite web|author=FermiFred |url=http://www.youtube.com/watch?v=QrCPrwRBi7E&feature=PlayList&p=BDA16F52CA3C9B1D&playnext_from=PL&index=9 |title=C.R. Hagen discusses naming of Higgs Boson in 2010 Sakurai Prize Talk |publisher=Youtube.com |date=2010-02-15 |accessdate=2012-06-16}}</ref><ref>{{cite web|last=Sample |first=Ian |url=http://www.guardian.co.uk/science/blog/2009/may/29/why-call-it-the-god-particle-higgs-boson-cern-lhc |title=Anything but the God particle by Ian Sample |publisher=Guardian |date=2009-05-29 |accessdate=2012-06-16}}</ref> One of the first times the ''Higgs'' name appeared in print was in 1972 when [[Gerardus 't Hooft]] and [[Martinus J. G. Veltman]] referred to it as the "Higgs–Kibble mechanism" in their Nobel winning paper.<ref>{{cite journal | author=G. 't Hooft and M. Veltman | title=Regularization and Renormalization of Gauge Fields | journal=Nuclear Physics B | volume=44 | year=1972 | issue=1 | pages=189–219 | doi=10.1016/0550-3213(72)90279-9 |bibcode = 1972NuPhB..44..189T }}</ref><ref>{{cite web|url=http://igitur-archive.library.uu.nl/phys/2005-0622-155148/13877.pdf |title=Regularization and Renormalization of Gauge Fields by t'Hooft and Veltman (PDF) |format=PDF |date= |accessdate=2012-06-16}}</ref>
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| == Examples ==
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| The Higgs mechanism occurs whenever a charged field has a vacuum expectation value. In the nonrelativistic context, this is the Landau model of a charged Bose–Einstein condensate, also known as a superconductor. In the relativistic condensate, the condensate is a scalar field, and is relativistically invariant.
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| === Landau model ===
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| The Higgs mechanism is a type of [[superconductivity]] which occurs in the vacuum. It occurs when all of space is filled with a sea of particles which are charged, or, in field language, when a charged field has a nonzero vacuum expectation value. Interaction with the quantum fluid filling the space prevents certain forces from propagating over long distances (as it does in a superconducting medium; e.g., in the [[Ginzburg–Landau theory]]).
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| A superconductor expels all magnetic fields from its interior, a phenomenon known as the [[Meissner effect]]. This was mysterious for a long time, because it implies that electromagnetic forces somehow become short-range inside the superconductor. Contrast this with the behavior of an ordinary metal. In a metal, the conductivity shields electric fields by rearranging charges on the surface until the total field cancels in the interior. But magnetic fields can penetrate to any distance, and if a magnetic monopole (an isolated magnetic pole) is surrounded by a metal the field can escape without collimating into a string. In a superconductor, however, electric charges move with no dissipation, and this allows for permanent surface currents, not just surface charges. When magnetic fields are introduced at the boundary of a superconductor, they produce surface currents which exactly neutralize them. The Meissner effect is due to currents in a thin surface layer, whose thickness, the [[London penetration depth]], can be calculated from a simple model
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| (the [[Ginzburg–Landau theory]]).
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| This simple model treats superconductivity as a charged [[Bose–Einstein condensate]]. Suppose that a superconductor contains bosons with charge ''q''. The wavefunction of the bosons can be described by introducing a [[quantum field theory|quantum field]], ψ, which obeys the [[Schrödinger field|Schrödinger equation as a field equation]] (in units where the [[reduced Planck constant]], ''ħ'', is set to 1):
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| :<math>i{\partial \over \partial t} \psi = {(\nabla - iqA)^2 \over 2m} \psi.</math>
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| The operator ψ(''x'') annihilates a boson at the point ''x'', while its adjoint ψ<sup>†</sup> creates a new boson at the same point. The wavefunction of the Bose–Einstein condensate is then the [[expectation value]] ψ of ψ(''x''), which is a classical function that obeys the same equation. The interpretation of the expectation value is that it is the phase that one should give to a newly created boson so that it will coherently superpose with all the other bosons already in the condensate.
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| When there is a charged condensate, the electromagnetic interactions are screened. To see this, consider the effect of a [[gauge transformation]] on the field. A gauge transformation rotates the phase of the condensate by an amount which changes from point to point, and shifts the vector potential by a gradient:
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| :<math>\begin{align}
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| \psi &\rightarrow e^{iq\phi(x)} \psi \\
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| A &\rightarrow A + \nabla \phi.
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| \end{align}</math>
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| When there is no condensate, this transformation only changes the definition of the phase of ψ at every point. But when there is a condensate, the phase of the condensate defines a preferred choice of phase.
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| The condensate wave function can be written as
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| :<math>\psi(x) = \rho(x)\, e^{i\theta(x)},</math>
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| where ρ is real amplitude, which determines the local density of the condensate. If the condensate were neutral, the flow would be along the gradients of θ, the direction in which the phase of the Schrödinger field changes. If the phase θ changes slowly, the flow is slow and has very little energy. But now θ can be made equal to zero just by making a gauge transformation to rotate the phase of the field.
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| The energy of slow changes of phase can be calculated from the Schrödinger kinetic energy,
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| :<math>H= {1\over 2m} |{(qA+\nabla)\psi|^2},</math>
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| and taking the density of the condensate ρ to be constant,
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| :<math>H \approx {\rho^2 \over 2m} (qA + \nabla \theta)^2. </math>
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| Fixing the choice of gauge so that the condensate has the same phase everywhere, the electromagnetic field energy has an extra term,
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| :<math>{q^2 \rho^2 \over 2m} A^2.</math>
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| When this term is present, electromagnetic interactions become short-ranged. Every field mode, no matter how long the wavelength, oscillates with a nonzero frequency. The lowest frequency can be read off from the energy of a long wavelength A mode,
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| :<math> E \approx {{\dot A}^2\over 2} + {q^2 \rho^2 \over 2m} A^2. </math>
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| This is a harmonic oscillator with frequency
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| :<math>\sqrt{\frac{1}{m} q^2 \rho^2}.</math>
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| The quantity |ψ|<sup>2</sup> (=ρ<sup>2</sup>) is the density of the condensate of superconducting particles.
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| In an actual superconductor, the charged particles are electrons, which are fermions not bosons. So in order to have superconductivity, the electrons need to somehow bind into [[Cooper pair]]s. The charge of the condensate ''q'' is therefore twice the electron charge ''e''. The pairing in a normal superconductor is due to lattice vibrations, and is in fact very weak; this means that the pairs are very loosely bound. The description of a Bose–Einstein condensate of loosely bound pairs is actually more difficult than the description of a condensate of elementary particles, and was only worked out in 1957 by [[BCS theory|Bardeen, Cooper and Schrieffer]] in the famous BCS theory.
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| === Abelian Higgs mechanism ===
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| Gauge invariance means that certain transformations of the gauge field do not change the energy at all. If an arbitrary gradient is added to ''A'', the energy of the field is exactly the same. This makes it difficult to add a mass term, because a mass term tends to push the field toward the value zero. But the zero value of the vector potential is not a gauge invariant idea. What is zero in one gauge is nonzero in another.
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| So in order to give mass to a gauge theory, the gauge invariance must be broken by a condensate. The condensate will then define a preferred phase, and the phase of the condensate will define the zero value of the field in a gauge invariant way. The gauge invariant definition is that a gauge field is zero when the phase change along any path from parallel transport is equal to the phase difference in the condensate wavefunction.
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| The condensate value is described by a quantum field with an expectation value, just as in the [[Ginzburg–Landau theory|Landau–Ginzburg model]].
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| In order for the phase of the vacuum to define a gauge, the field must have a phase (also referred to as 'to be charged'). In order for a scalar field Φ to have a phase, it must be complex, or (equivalently) it should contain two fields with a symmetry which rotates them into each other. The vector potential changes the phase of the quanta produced by the field when they move from point to point. In terms of fields, it defines how much to rotate the real and imaginary parts of the fields into each other when comparing field values at nearby points.
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| The only [[renormalization|renormalizable]] model where a complex scalar field Φ acquires a nonzero value is the Mexican-hat model, where the field energy has a minimum away from zero. The action for this model is
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| :<math>S(\phi) = \int {1\over 2} |\partial \phi|^2 - \lambda \left(|\phi|^2 - \Phi^2\right)^2,</math>
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| which results in the Hamiltonian
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| :<math>H(\phi) = {1\over 2} |\dot\phi|^2 + |\nabla \phi|^2 + V(|\phi|).</math>
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| The first term is the kinetic energy of the field. The second term is the extra potential energy when the field varies from point to point. The third term is the potential energy when the field has any given magnitude.
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| This potential energy, ''V''(''z'', Φ) = λ(|''z''|<sup>2</sup> − Φ<sup>2</sup>)<sup>2</sup>,<ref>{{cite doi|10.1007/BF02812722|noedit}}</ref> has a graph which looks like a [[Mexican hat]], which gives the model its name. In particular, the minimum energy value is not at ''z'' = 0, but on the circle of points where the magnitude of ''z'' is Φ.
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| [[Image:Mecanismo de Higgs PH.png|thumb|Higgs potential ''V''. For a fixed value of λ the potential is presented upwards against the real and imaginary parts of Φ. The ''Mexican-hat'' or ''champagne-bottle profile'' at the ground should be noted.]]
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| When the field Φ(''x'') is not coupled to electromagnetism, the Mexican-hat potential has flat directions. Starting in any one of the circle of vacua and changing the phase of the field from point to point costs very little energy. Mathematically, if
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| :<math>\phi(x) = \Phi e^{i\theta(x)}</math>
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| with a constant prefactor, then the action for the field θ(''x''), i.e., the "phase" of the Higgs field Φ(x), has only derivative terms. This is not a surprise. Adding a constant to θ(''x'') is a symmetry of the original theory, so different values of θ(''x'') cannot have different energies. This is an example of [[Goldstone's theorem]]: spontaneously broken continuous symmetries normally produce massless excitations.
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| The Abelian Higgs model is the Mexican-hat model coupled to [[Maxwell's theory|electromagnetism]]:
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| :<math>S(\phi, A) = \int - {1\over 4} F^{\mu\nu} F_{\mu\nu} + |(\partial - i q A)\phi|^2 - \lambda (|\phi|^2 - \Phi^2)^2.</math>
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| The classical vacuum is again at the minimum of the potential, where the magnitude of the complex field φ is equal to Φ. But now the phase of the field is arbitrary, because gauge transformations change it. This means that the field θ(''x'') can be set to zero by a gauge transformation, and does not represent any actual degrees of freedom at all.
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| Furthermore, choosing a gauge where the phase of the vacuum is fixed, the potential energy for fluctuations of the vector field is nonzero. So in the abelian Higgs model, the gauge field acquires a mass. To calculate the magnitude of the mass, consider a constant value of the vector potential A in the x direction in the gauge where the condensate has constant phase. This is the same as a sinusoidally varying condensate in the gauge where the vector potential is zero. In the gauge where A is zero, the potential energy density in the condensate is the scalar gradient energy:
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| :<math>E = {1 \over 2} \left |\partial \left (\Phi e^{iqAx} \right) \right |^2 = {1 \over 2} q^2\Phi^2 A^2.</math>
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| This energy is the same as a mass term {{sfrac|1|2}}''m''<sup>2</sup>''A''<sup>2</sup> where ''m'' = ''q''Φ.
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| === Nonabelian Higgs mechanism ===
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| The Nonabelian Higgs model has the following action:
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| :<math>S(\phi,\mathbf A) = \int {1\over 4g^2} \mathop{\textrm{tr}}(F^{\mu\nu}F_{\mu\nu}) + |D\phi|^2 + V(|\phi|)</math>
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| where now the nonabelian field '''A''' is contained in ''D'' and in the tensor components <math>F^{\mu \nu}</math> and <math>F_{\mu \nu}</math> (the relation between '''A''' and those components is well-known from the [[Yang–Mills theory]]).
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| It is exactly analogous to the Abelian Higgs model. Now the field φ is in a representation of the gauge group, and the gauge covariant derivative is defined by the rate of change of the field minus the rate of change from parallel transport using the gauge field A as a connection.
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| :<math>D\phi = \partial \phi - i A^k t_k \phi</math>
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| Again, the expectation value of Φ defines a preferred gauge where the vacuum is constant, and fixing this gauge, fluctuations in the gauge field A come with a nonzero energy cost.
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| Depending on the representation of the scalar field, not every gauge field acquires a mass. A simple example is in the renormalizable version of an early electroweak model due to [[Julian Schwinger]]. In this model, the gauge group is '''SO'''(3) (or '''SU'''(2) − there are no spinor representations in the model), and the gauge invariance is broken down to '''U'''(1) or '''SO'''(2) at long distances. To make a consistent renormalizable version using the Higgs mechanism, introduce a scalar field φ<sup>a</sup> which transforms as a vector (a triplet) of '''SO'''(3). If this field has a vacuum expectation value, it points in some direction in field space. Without loss of generality, one can choose the ''z''-axis in field space to be the direction that φ is pointing, and then the vacuum expectation value of φ is (0, 0, ''A''), where ''A'' is a constant with dimensions of mass (<math>c = \hbar = 1</math>).
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| Rotations around the ''z''-axis form a '''U'''(1) subgroup of '''SO'''(3) which preserves the vacuum expectation value of φ, and this is the unbroken gauge group. Rotations around the ''x'' and ''y''-axis do not preserve the vacuum, and the components of the '''SO'''(3) gauge field which generate these rotations become massive vector mesons. There are two massive W mesons in the Schwinger model, with a mass set by the mass scale ''A'', and one massless '''U'''(1) gauge boson, similar to the photon.
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| The Schwinger model predicts [[magnetic monopoles]] at the electroweak unification scale, and does not predict the Z meson. It doesn't break electroweak symmetry properly as in nature. But historically, a model similar to this (but not using the Higgs mechanism) was the first in which the weak force and the electromagnetic force were unified.
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| === Affine Higgs mechanism ===
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| [[Ernst Stueckelberg]] discovered<ref>Stueckelberg, E. C. G. (1938), "Die Wechselwirkungskräfte in der Elektrodynamik und in der Feldtheorie der Kräfte", ''Helv. Phys. Acta.'' '''11:''' 225</ref> a version of the Higgs mechanism by analyzing the theory of quantum electrodynamics with a massive photon. Effectively, [[Stueckelberg action|Stueckelberg's model]] is a limit of the regular Mexican hat Abelian Higgs model, where the vacuum expectation value ''H'' goes to infinity and the charge of the Higgs field goes to zero in such a way that their product stays fixed. The mass of the Higgs boson is proportional to ''H'', so the Higgs boson becomes infinitely massive and decouples, so is not present in the discussion. The vector meson mass, however, equals to the product ''eH'', and stays finite.
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| The interpretation is that when a '''U'''(1) gauge field does not require quantized charges, it is possible to keep only the angular part of the Higgs oscillations, and discard the radial part. The angular part of the Higgs field θ has the following gauge transformation law:
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| :<math> \theta \rightarrow \theta + e\alpha\,</math>
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| :<math> A \rightarrow A + \alpha. \,</math> | |
| The gauge covariant derivative for the angle (which is actually gauge invariant) is:
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| :<math> D\theta = \partial \theta - e A.\,</math>
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| In order to keep θ fluctuations finite and nonzero in this limit, θ should be rescaled by H, so that its kinetic term in the action stays normalized. The action for the theta field is read off from the Mexican hat action by substituting <math>\phi \;=\; He^{\frac{1}{H}i\theta}</math>.
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| :<math> S = \int {1 \over 4}F^2 + {1 \over 2}(D\theta)^2 = \int {1 \over 4}F^2 + {1 \over 2}(\partial \theta - He A)^2 = \int {1 \over 4}F^2 + {1 \over 2}(\partial \theta - m A)^2</math>
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| since ''eH'' is the gauge boson mass. By making a gauge transformation to set θ = 0, the gauge freedom in the action is eliminated, and the action becomes that of a massive vector field:
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| :<math> S= \int {1 \over 4} F^2 + {1 \over 2} m^2 A^2.\,</math>
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| To have arbitrarily small charges requires that the '''U'''(1) is not the circle of unit complex numbers under multiplication, but the real numbers '''R''' under addition, which is only different in the global topology. Such a '''U'''(1) group is ''non-compact''. The field θ transforms as an affine representation of the gauge group. Among the allowed gauge groups, only non-compact '''U'''(1) admits affine representations, and the '''U'''(1) of electromagnetism is experimentally known to be compact, since charge quantization holds to extremely high accuracy.
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| The Higgs condensate in this model has infinitesimal charge, so interactions with the Higgs boson do not violate charge conservation. The theory of quantum electrodynamics with a massive photon is still a renormalizable theory, one in which electric charge is still conserved, but [[magnetic monopole]]s are not allowed. For nonabelian gauge theory, there is no affine limit, and the Higgs oscillations cannot be too much more massive than the vectors.
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| == See also ==
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| * [[Electromagnetic mass]]
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| * [[Higgs bundle]]
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| * [[Mass generation]]
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| * [[QCD vacuum]]
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| * [[Quantum triviality]]
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| * [[Top quark condensate]]
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| * [[Yang–Mills–Higgs equations]]
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| == References ==
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| <!--This article uses the Cite.php citation mechanism. If you would like more information on how to add footnotes to this article, please see http://meta.wikimedia.org/wiki/Cite/Cite.php -->
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| {{Reflist|30em}}
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| == Further reading ==
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| * Schumm, Bruce A. (2004) ''Deep Down Things''. Johns Hopkins Univ. Press. Chpt. 9.
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| == External links ==
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| *{{cite journal | last1 = Guralnik | first1 = G.S. | last2 = Hagen | first2 = C.R. | last3 = Kibble | first3 = T.W.B. | year = 1964 | title = Global Conservation Laws and Massless Particles | journal = [[Physical Review Letters]] | volume = 13 | issue = 20 | pages = 585–87 |bibcode = 1964PhRvL..13..585G |doi = 10.1103/PhysRevLett.13.585 }}
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| *Mark D. Roberts (1999) "[http://www.arXiv.org/abs/hep-th/9904080 A Generalized Higgs Model]"
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| *[http://www.youtube.com/view_play_list?p=BDA16F52CA3C9B1D 2010 Sakurai Prize - All Events - YouTube]
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| *[http://cerncourier.com/cws/article/cern/32522 From BCS to the LHC - CERN Courier] Jan 21, 2008, Steven Weinberg, University of Texas at Austin.
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| *{{YouTube|Zl4W3DYTIKw|Higgs, dark matter and supersymmetry: What the Large Hadron Collider will tell us (Steven Weinberg) - YouTube}} 06-11-2009
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| *[http://www.youtube.com/watch?v=WLZ78gwWQI0 Gerry Guralnik speaks at Brown University about the 1964 PRL papers]
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| *Guralnik, Gerald (2013). [http://www.sps.ch/en/articles/milestones_in_physics/heretical_ideas_that_provided_the_cornerstone_for_the_standard_model_of_particle_physics_1/ "Heretical Ideas that Provided the Cornerstone for the Standard Model of Particle Physics".] SPG MITTEILUNGEN March 2013, No. 39, (p. 14)
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| *[http://www.pas.rochester.edu/urpas/news/Hagen_030708 Steven Weinberg Praises Teams for Higgs Boson Theory]
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| *[http://prl.aps.org/50years/milestones#1964 Physical Review Letters – 50th Anniversary Milestone Papers]
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| *[http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_13-6-2008-12-42-20?newsid=38514 Imperial College London on PRL 50th Anniversary Milestone Papers]
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| *[http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia]
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| *[http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism_%28history%29 History of Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia]
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| *[http://apps3.aps.org/aps/meetings/april10/roser.pdf The Hunt for the Higgs at Tevatron]
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| *{{YouTube|Y-vKh_jKX7Q|The Mystery of Empty Space}}. A lecture with [[UCSD]] physicist Kim Griest (43 minutes)
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| {{Standard model of physics}}
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| {{DEFAULTSORT:Higgs Mechanism}}
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| [[Category:Electroweak theory]]
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| [[Category:Phase transitions]]
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| [[Category:Quantum field theory]]
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| [[Category:Standard Model]]
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| [[Category:Symmetry]]
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