{{Quantum field theory|cTopic=Some models}}
In

particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, the

**Higgs mechanism** is the process in which gauge bosons in a

gauge theoryIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

can acquire non-vanishing

massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

es through absorption of

Nambu-Goldstone bosonIn particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...

s arising in

spontaneous symmetry breakingSpontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....

.
The simplest implementation of the mechanism adds an extra

**Higgs field** to the gauge theory. The spontaneous symmetry breaking of the underlying local symmetry triggers conversion of components of this Higgs field to Goldstone bosons which interact with (at least some of) the other fields in the theory, so as to produce mass terms for (at least some of) the gauge bosons. This mechanism may also leave behind elementary scalar (

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

-0) particles, known as

Higgs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

s.
In the

standard modelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

, the phrase "Higgs mechanism" refers specifically to the generation of masses for the

W^{±}, and ZThe W and Z bosons are the elementary particles that mediate the weak interaction; their symbols are , and . The W bosons have a positive and negative electric charge of 1 elementary charge respectively and are each other's antiparticle. The Z boson is electrically neutral and its own...

weak gauge bosons through

electroweakIn particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different...

symmetry breaking. Although the evidence for the electroweak Higgs mechanism is overwhelming, experiments have yet to discover the single Higgs boson predicted by the standard model. The

Large Hadron ColliderThe Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....

at

CERNThe European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...

is currently searching for Higgs bosons, and attempting to understand the electroweak Higgs mechanism.

## In the standard model

The Higgs mechanism was incorporated into modern particle physics by

Steven WeinbergSteven Weinberg is an American theoretical physicist and Nobel laureate in Physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interaction between elementary particles....

and

Abdus SalamMohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk (Urdu: محمد عبد السلام, pronounced , (January 29, 1926– November 21, 1996) was a Pakistani theoretical physicist and Nobel laureate in Physics for his work on the electroweak unification of the...

, and is an essential part of the

standard modelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

.
In the standard model, at temperatures high enough so that electroweak symmetry is unbroken, all elementary particles are massless. At a critical temperature, the symmetry is spontaneously broken, and the

W and Z bosonsThe W and Z bosons are the elementary particles that mediate the weak interaction; their symbols are , and . The W bosons have a positive and negative electric charge of 1 elementary charge respectively and are each other's antiparticle. The Z boson is electrically neutral and its own...

acquire masses. (EWSB, ElectroWeak Symmetry Breaking, is an abbreviation used for this).
Fermions, such as the

leptonA lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...

s and

quarkA quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

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

**SU**(2)The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...

doublet, a complex spinor with four real components (or equivalently with two complex components), with a Standard Model

**U**(1) charge of −1.
It transforms as a spinor under

**SU**(2). Under

**U**(1) rotations, it gets multiplied by a phase; this mixes the real and imaginary part 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).
The Higgs field, through the interactions specified by its potential, induces spontaneous breaking of three out of the four generators ("directions") of the
gauge group

*SU(2)*×

*U(1)*, and three out of its four components would
ordinarily amount to Goldstone bosons, 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 W and Z bosons, while the one remaining degree of freedom becomes the

Higgs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

– a new scalar particle.

### The part that remains massless

The gauge group of the electroweak part of the standard model is

**SU**(2) ×

**U**(1).
The group

**SU**(2) is all unitary matrices, all the orthonormal changes of coordinates in a complex two dimensional vector space. Rotating the coordinates so that the first basis vector in the direction of

*H* makes the vacuum expectation value of

*H* the spinor

*(A, 0)*. The generators for rotations about the

*x*,

*y*, and

*z* axes are by half the Pauli matrices

$\backslash sigma\_x,\; \backslash sigma\_y,\; \backslash sigma\_z$, so that a rotation of angle θ about the

*z*-axis takes the vacuum to:NEWLINE

NEWLINE- NEWLINE
NEWLINE- $(A\; e^\{\backslash frac\{1\}\{2\}i\backslash theta\},0)\backslash ,$

NEWLINE

NEWLINE
While the X and Y generators mix up the top and bottom components of the spinor, the Z rotations only multiply by a phase. This phase can be undone by a

**U**(1) rotation of angle ½θ, which multiplies by the opposite phase, since the Higgs has charge −1. Under both an

**SU**(2)

*z*-rotation and a

**U**(1) rotation by an amount ½θ, the vacuum is invariant. This combination of generators:NEWLINE

NEWLINE- NEWLINE
NEWLINE- $Q\; =\; W\_z\; +\; \backslash frac\{1\}\{2\}V$

NEWLINE

NEWLINE
defines the unbroken gauge group, where

*W*_{z} is the generator of rotations around the z-axis in the

**SU**(2) and

*V* is the 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. The part of the gauge field in this direction stays massless, and this gauge field is the actual photon.
The phase that a field acquires under this combination of generators is its electric charge, and this is the formula for the electric charge in the standard model. In this convention, all the

*V* charges in the standard model are multiples of ⅓. To make all the

*V*-charges in the standard model integers, you can rescale the

*V* part of the formula by tripling all the

*V*-charges if you like, and rewrite the charge formula as:NEWLINE

NEWLINE- NEWLINE
NEWLINE- $Q\; =\; W\_z\; +\; \backslash frac\{1\}\{6\}V$

NEWLINE

NEWLINE
but the normalization with ½

*V* is the universal standard.

### Consequences for fermions

In spite of the introduction of spontaneous symmetry-breaking, also for

fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s the mass terms oppose the chiral gauge invariance. Therefore, also for these fields the mass terms should be replaced by a gauge-invariant "Higgs" mechanism. An obvious possibility is some kind of "Yukawa coupling" (see below) between the fermion field

*ψ* and the Higgs field

*Φ*, with unknown couplings

*$G\_\{\backslash psi\}$*, 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

$\backslash mathcal\{L\}\_\{\backslash mathrm\{Fermion\}\}(\backslash phi\; ,\; A,\; \backslash psi\; )\; =\; \backslash overline\{\backslash psi\}\; \backslash gamma^\{\backslash mu\}\; D\_\{\backslash mu\}\; \backslash psi\; +\; G\_\{\backslash psi\}\; \backslash overline\{\backslash psi\}\; \backslash phi\; \backslash psi\; \backslash ,,$
where again the gauge field

*A* only enters

$D\_\backslash mu$ (i.e., it is only indirectly visible). The quantities

*$\backslash gamma^\{\backslash mu\}$* are the Dirac matrices, and

$G\_\{\backslash psi\}$ 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

$|\backslash langle\backslash phi\backslash rangle\; |$, as described above. Again, this is crucial for the existence of the property "mass".

### Background

Spontaneous symmetry breakingSpontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....

offered a framework to introduce bosons into relativistic quantum field theories. However, according to Goldstone's theorem, these bosons should be massless. The only observed particles which could be approximately interpreted as Goldstone bosons were the

pionIn particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....

s, which

Yoichiro Nambuis a Japanese-born American physicist, currently a professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded a one-half share of the Nobel Prize in Physics in 2008 for the discovery of the mechanism of spontaneous broken symmetry in...

related to

chiral symmetryIn quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts of Dirac fields transform independently...

breaking.
A similar problem arises with Yang–Mills theory (also known as nonabelian

gauge theoryIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

), which predicts massless

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

-1

gauge bosonIn particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...

s. Massless weakly interacting gauge bosons lead to long-range forces, which are only observed for electromagnetism and the corresponding massless

photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

. Gauge theories of the weak force needed a way to describe massive gauge bosons in order to be consistent.

### Discovery

{{Wikinews|Prospective Nobel Prize for Higgs boson work disputed}}
The Higgs mechanism is also called the

**Brout–Englert–Higgs mechanism**, or

**Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism**, or

**Anderson–Higgs mechanism**. The mechanism was proposed in 1962 by

Philip Warren AndersonPhilip Warren Anderson is an American physicist and Nobel laureate. Anderson has made contributions to the theories of localization, antiferromagnetism and high-temperature superconductivity.- Biography :...

, who discussed its consequences for particle physics but did not work out an explicit relativistic model. The relativistic model was developed in 1964 by

Peter HiggsPeter Ware Higgs, FRS, FRSE, FKC , is an English theoretical physicist and an emeritus professor at the University of Edinburgh....

, and independently by

Robert BroutRobert Brout was an American-Belgian theoretical physicist who has made significant contributions in elementary particle physics...

and

Francois EnglertFrançois Englert is a Belgian theoretical physicist. He was awarded the 2010 J. J. Sakurai Prize for Theoretical Particle Physics , the Wolf Prize in Physics in 2004 and the High Energy and Particle Prize of the European Physical Society François Englert (born 6 November 1932) is a...

, and

Gerald GuralnikGerald Stanford Guralnik is the Chancellor’s Professor of Physics at Brown University. He is most famous for his co-discovery of the Higgs mechanism and Higgs Boson with C. R. Hagen and Tom Kibble...

,

C. R. HagenCarl Richard Hagen is a professor of particle physics at the University of Rochester. He is most noted for his contributions to the Standard Model and Symmetry breaking as well as the co-discovery of the Higgs mechanism and Higgs boson with Gerald Guralnik and Tom Kibble...

, and

Tom KibbleThomas Walter Bannerman Kibble, FRS, is a British scientist and senior research investigator at The Blackett Laboratory, at Imperial College London, UK. His research interests are in quantum field theory, especially the interface between high-energy particle physics and cosmology...

, who worked out the results by the spring of 1963. The mechanism is closely analogous to phenomena previously discovered by

Yoichiro Nambuis a Japanese-born American physicist, currently a professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded a one-half share of the Nobel Prize in Physics in 2008 for the discovery of the mechanism of spontaneous broken symmetry in...

involving the "vacuum structure" of quantum fields in

superconductivitySuperconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...

. A similar but distinct effect, known as the Stueckelberg mechanism, had previously been studied by

Ernst StueckelbergErnst Carl Gerlach Stueckelberg was a Swiss mathematician and physicist.- Career :In 1927 Stueckelberg got his Ph. D. at the University of Basel under August Hagenbach...

.
These physicists discovered that when a gauge theory is combined with an additional field breaking spontaneously the symmetry group, the gauge bosons can consistently acquire a finite 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 WeinbergSteven Weinberg is an American theoretical physicist and Nobel laureate in Physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interaction between elementary particles....

and

Abdus SalamMohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk (Urdu: محمد عبد السلام, pronounced , (January 29, 1926– November 21, 1996) was a Pakistani theoretical physicist and Nobel laureate in Physics for his work on the electroweak unification of the...

in 1967. Higgs's original article presenting the model was rejected by

Physics LettersPhysics Letters was a scientific journal published from 1962 to 1966, when it split in two series now published by Elsevier:*Physics Letters A: condensed matter physics, theoretical physics, nonlinear science, statistical physics, mathematical and computational physics, general and...

. When revising the article before resubmitting it to

Physical Review LettersPhysical Review Letters , established in 1958, is a peer reviewed, scientific journal that is published 52 times per year by the American Physical Society...

, he added a sentence at the end, mentioning that it implies the existence of one or more new, massive scalar bosons, which do not form complete

representationsIn the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

of the symmetry group; these are the

Higgs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

s.
The three papers by Guralnik, Hagen, and Kibble; Higgs; and Brout and Englert were each recognized as "milestone letters" by

*Physical Review Letters* in 2008. 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

J. J. Sakurai Prize for Theoretical Particle PhysicsThe J. J. Sakurai Prize for Theoretical Particle Physics, is presented by the American Physical Society at its annual "April Meeting", and honors outstanding achievement in particle physics theory...

for this work.

Benjamin W. LeeBenjamin Whisoh Lee or Ben Lee, was a Korean-American theoretical physicist...

is often credited with first naming the "Higgs-like" mechanism, although there is debate around when this first occurred. One of the first times the

*Higgs* name appeared in print was in 1972 when

Gerardus 't HooftGerardus 't Hooft is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G...

and

Martinus J. G. VeltmanMartinus Justinus Godefriedus Veltman is a Dutch theoretical physicist. He shared the 1999 Nobel Prize in physics with his former student Gerardus 't Hooft for their work on particle theory.-Biography:...

referred to it as the "Higgs-Kibble mechanism" in their Nobel winning paper.

## Examples of Higgs mechanism

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.

### Landau Model

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 (see Landau model of superconductivity).
A superconductor expels all magnetic fields from its interior, a phenomenon known as the

Meissner effectThe Meissner effect is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. The German physicists Walther Meissner and Robert Ochsenfeld discovered the phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin...

. 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 depthIn superconductors, the London penetration depth characterizes the distance to which a magnetic field penetrates into a superconductor and becomes equal to 1/e times that of the magnetic field at the surface of the superconductor...

, can be calculated from a simple model.
This simple model, due to

Lev LandauLev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...

and

Vitaly GinzburgVitaly Lazarevich Ginzburg ForMemRS was a Soviet theoretical physicist, astrophysicist, Nobel laureate, a member of the Russian Academy of Sciences and one of the fathers of Soviet hydrogen bomb...

, treats superconductivity as a charged

Bose–Einstein condensateA Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...

. Suppose that a superconductor contains bosons with charge

*q*. The wavefunction of the bosons can be described by introducing a

quantum fieldQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

, ψ, which obeys the

Schrödinger equation as a field equationIn quantum mechanics and quantum field theory, a Schrödinger field, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation...

(in units where

$\backslash hbar$, the Planck quantum divided by 2π, is replaced by 1):

$i\{\backslash partial\; \backslash over\; \backslash partial\; t\}\; \backslash psi\; =\; \{(\backslash nabla\; -\; iqA)^2\; \backslash over\; 2m\}\; \backslash psi$
The operator

*ψ(x)* annihilates a boson at the point

*x*, while its adjoint

$\backslash psi^\backslash dagger$ 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.
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.

$\backslash psi\; \backslash rightarrow\; e^\{iq\backslash phi(x)\}\; \backslash psi$$A\; \backslash rightarrow\; A\; +\; \backslash nabla\; \backslash phi$
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.
The condensate wave function can be written as

$\backslash psi(x)\; =\; \backslash rho(x)\backslash ,\; e^\{i\backslash theta(x)\},$
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.
The energy of slow changes of phase can be calculated from the Schrödinger kinetic energy,

$H=\; \{1\backslash over\; 2m\}\; |\{(qA+\backslash nabla\; )\backslash psi|^2\},$
and taking the density of the condensate ρ to be constant,

$H\; \backslash approx\; \{\backslash rho^2\; \backslash over\; 2m\}\; (qA\; +\; \backslash nabla\; \backslash theta)^2.$
Fixing the choice of gauge so that the condensate has the same phase everywhere, the electromagnetic field energy has an extra term,

$\{q^2\; \backslash rho^2\; \backslash over\; 2m\}\; A^2.$
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,

$E\backslash approx\; \{\{\backslash dot\; A\}^2\backslash over\; 2\}\; +\; \{q^2\; \backslash rho^2\; \backslash over\; 2m\}\; A^2.$
This is a harmonic oscillator with frequency:

$\backslash sqrt\{\backslash frac\{1\}\{m\}\; q^2\; \backslash rho^2\}.$
The quantity |ψ|

^{2} (=ρ

^{2}) is the density of the condensate of superconducting particles.
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 pairIn condensed matter physics, a Cooper pair or BCS pair is two electrons that are bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper...

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

Bardeen, Cooper and SchriefferBCS theory — proposed by Bardeen, Cooper, and Schrieffer in 1957 — is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a boson-like state...

in the famous BCS theory.

### Abelian Higgs Mechanism

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.
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.
The condensate value is described by a quantum field with an expectation value, just as in the Landau–Ginzburg model.
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.
The only

renormalizableIn quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

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.

$S(\backslash phi\; )\; =\; \backslash int\; \{1\backslash over\; 2\}\; |\backslash partial\; \backslash phi|^2\; -\; \backslash lambda\backslash cdot\; (|\backslash phi|^2\; -\; \backslash Phi^2)^2$
This defines the following Hamiltonian:

$H(\backslash phi\; )\; =\; \{1\backslash over\; 2\}\; |\backslash dot\backslash phi|^2\; +\; |\backslash nabla\; \backslash phi|^2\; +\; V(|\backslash phi|)$
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.
This potential energy

*V(z, Φ) = λ • (|z|*^{2} - Φ^{2})^{2} has a graph which looks like a

Mexican hatIn general, a Mexican hat is a sombrero – a broad-brimmed and high-crowned hat formerly used in rural areas of Mexico and still common today among mariachi musicians and foreign tourists.Mexican hat may also refer to:...

, 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.

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

$\backslash phi(x)\; =\; \backslash Phi\; e^\{i\backslash theta(x)\}$
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.
The Abelian Higgs model is the Mexican-hat model coupled to electromagnetism:

$S(\backslash phi\; ,A)\; =\; \backslash int\; \{1\backslash over\; 4\}\; F^\{\backslash mu\backslash nu\}\; F\_\{\backslash mu\backslash nu\}\; +\; |(\backslash partial\; -\; i\; q\; A)\backslash phi|^2\; +\; \backslash lambda\backslash cdot\; (|\backslash phi|^2\; -\; \backslash Phi^2)^2.$
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.
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:

$E\; =\; \{1\; \backslash over\; 2\}\; \backslash left\; |\backslash partial\; \backslash left\; (\backslash Phi\; e^\{iqAx\}\; \backslash right\; )\; \backslash right\; |^2\; =\; \{1\; \backslash over\; 2\}\; q^2\backslash Phi^2\; A^2$
And this energy is the same as a mass term ½m

^{2}A

^{2} where m = qΦ.

### Nonabelian Higgs mechanism

The Nonabelian Higgs model has the following action:

$S(\backslash phi\; ,\backslash mathbf\; A)\; =\; \backslash int\; \{1\backslash over\; 4g^2\}\; \backslash mathop\{\backslash textrm\{tr\}\}(F^\{\backslash mu\backslash nu\}F\_\{\backslash mu\backslash nu\})\; +\; |D\backslash phi|^2\; +\; V(|\backslash phi|)$
where now the nonabelian field

**A** is contained in

*D* and in the tensor components

$F^\{\backslash mu\; \backslash nu\}$ and

$F\_\{\backslash mu\; \backslash nu\}$ (the relation between

**A** and those components is well-known from the Yang–Mills theory).
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.

$D\backslash phi\; =\; \backslash partial\; \backslash phi\; -\; i\; A^k\; t\_k\; \backslash phi$
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.
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 SchwingerJulian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...

. 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

^{a} 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 (

$c\; =\; \backslash hbar\; =\; 1$).
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.
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.

### Affine Higgs mechanism

Ernst StueckelbergErnst Carl Gerlach Stueckelberg was a Swiss mathematician and physicist.- Career :In 1927 Stueckelberg got his Ph. D. at the University of Basel under August Hagenbach...

discovered a version of the Higgs mechanism by analyzing the theory of quantum electrodynamics with a massive photon. 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 disappears. The vector meson mass is equal to the product

*eH*, and stays finite.
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:

$\backslash theta\; \backslash rightarrow\; \backslash theta\; +\; e\backslash alpha\backslash ,$$A\; \backslash rightarrow\; A\; +\; \backslash alpha\; \backslash ,$
The gauge covariant derivative for the angle (which is actually gauge invariant) is:

$D\backslash theta\; =\; \backslash partial\; \backslash theta\; -\; e\; A\backslash ,$
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

$\backslash phi\; \backslash ;=\backslash ;\; He^\{\backslash frac\{1\}\{H\}i\backslash theta\}$.

$S\; =\; \backslash int\; \{1\; \backslash over\; 4\}F^2\; +\; \{1\; \backslash over\; 2\}(D\backslash theta)^2\; =\; \backslash int\; \{1\; \backslash over\; 4\}F^2\; +\; \{1\; \backslash over\; 2\}(\backslash partial\; \backslash theta\; -\; He\; A)^2\; =\; \backslash int\; \{1\; \backslash over\; 4\}F^2\; +\; \{1\; \backslash over\; 2\}(\backslash partial\; \backslash theta\; -\; m\; A)^2$
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:

$S=\; \backslash int\; \{1\; \backslash over\; 4\}\; F^2\; +\; \{1\; \backslash over\; 2\}\; m^2\; A^2\backslash ,$
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.
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 monopoleA magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...

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.

## See also

NEWLINE

NEWLINE- Higgs boson
The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

NEWLINE- Goldstone boson
In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...

NEWLINE- 1964 PRL symmetry breaking papers
NEWLINE- QCD vacuum
The QCD vacuum is the vacuum state of quantum chromodynamics . It is an example of a non-perturbative vacuum state, characterized by many non-vanishing condensates such as the gluon condensate or the quark condensate...

NEWLINE- Quantum triviality
In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. Ifthe only allowed value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting...

NEWLINE- Symmetry breaking
Symmetry breaking in physics describes a phenomenon where small fluctuations acting on a system which is crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations , the choice will appear arbitrary...

NEWLINE- Tachyon condensation
In particle physics, theoretical processes that eliminate or resolve particles or fields into better understood phenomena are called, by extension and metaphor with the macroscopic process, "condensation"...

NEWLINE- Top quark condensate
In particle physics, the top quark condensate theory is an alternative to the Standard Model in which a fundamental scalar Higgs field is replaced by a composite field composed of the top quark and its antiquark. These are bound by a four-fermion interaction, analogous to Cooper pairs in a BCS...

NEWLINE- Yang–Mills–Higgs equations
In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle...

NEWLINE

## Further reading

NEWLINE

NEWLINE- Schumm, Bruce A. (2004)
*Deep Down Things*. Johns Hopkins Univ. Press. Chpt. 9.

NEWLINE

## External links

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NEWLINE
{{DEFAULTSORT:Higgs Mechanism}}