Higgs mechanism

# Higgs mechanism

Discussion

Encyclopedia
{{Quantum field theory|cTopic=Some models}} In particle physics
Particle physics
Particle 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 theory
Gauge theory
In 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 mass
Mass
Mass 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 boson
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...

s arising in spontaneous symmetry breaking
Spontaneous symmetry breaking
Spontaneous 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 (spin
Spin (physics)
In 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 boson
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...

s. In the standard model
Standard Model
The 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 Z
W and Z bosons
The 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 electroweak
Electroweak interaction
In 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 Collider
The 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 CERN
CERN
The 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 Weinberg
Steven Weinberg
Steven 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 Salam
Abdus Salam
Mohammad 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 model
Standard Model
The 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 bosons
W and Z bosons
The 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 lepton
Lepton
A 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 quark
Quark
A 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)
Special unitary group
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 boson
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...

– 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 $\sigma_x, \sigma_y, \sigma_z$, so that a rotation of angle θ about the z-axis takes the vacuum to:NEWLINE
NEWLINE
NEWLINE
NEWLINE
$\left(A e^\left\{\frac\left\{1\right\}\left\{2\right\}i\theta\right\},0\right)\,$
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
$Q = W_z + \frac\left\{1\right\}\left\{2\right\}V$
NEWLINE
NEWLINE defines the unbroken gauge group, where Wz 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
$Q = W_z + \frac\left\{1\right\}\left\{6\right\}V$
NEWLINE
NEWLINE but the normalization with ½V is the universal standard.

### Consequences for fermions

In spite of the introduction of spontaneous symmetry-breaking, also for fermion
Fermion
In 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 $G_\left\{\psi\right\}$, 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 $\mathcal\left\{L\right\}_\left\{\mathrm\left\{Fermion\right\}\right\}\left(\phi , A, \psi \right) = \overline\left\{\psi\right\} \gamma^\left\{\mu\right\} D_\left\{\mu\right\} \psi + G_\left\{\psi\right\} \overline\left\{\psi\right\} \phi \psi \,,$ where again the gauge field A only enters $D_\mu$ (i.e., it is only indirectly visible). The quantities $\gamma^\left\{\mu\right\}$ are the Dirac matrices, and $G_\left\{\psi\right\}$ 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 $|\langle\phi\rangle |$, as described above. Again, this is crucial for the existence of the property "mass".

### Background

Spontaneous symmetry breaking
Spontaneous symmetry breaking
Spontaneous 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 pion
Pion
In 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 Nambu
Yoichiro Nambu
is 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 symmetry
Chiral symmetry
In 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 theory
Gauge theory
In 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 spin
Spin (physics)
In 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 boson
Gauge boson
In 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 photon
Photon
In 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 Anderson
Philip Warren Anderson
Philip 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 Higgs
Peter Higgs
Peter Ware Higgs, FRS, FRSE, FKC , is an English theoretical physicist and an emeritus professor at the University of Edinburgh....

, and independently by Robert Brout
Robert Brout
Robert Brout was an American-Belgian theoretical physicist who has made significant contributions in elementary particle physics...

and Francois Englert
François Englert
François 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 Guralnik
Gerald Guralnik
Gerald 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. Hagen
C. R. Hagen
Carl 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 Kibble
Tom W. B. Kibble
Thomas 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 Nambu
Yoichiro Nambu
is 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 superconductivity
Superconductivity
Superconductivity 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 Stueckelberg
Ernst Stueckelberg
Ernst 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 Weinberg
Steven Weinberg
Steven 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 Salam
Abdus Salam
Mohammad 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 Letters
Physics Letters
Physics 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 Letters
Physical Review Letters
Physical 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 representations
Group representation
In 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 boson
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...

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 Physics
Sakurai Prize
The 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. Lee
Benjamin W. Lee
Benjamin 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 Hooft
Gerardus 't Hooft
Gerardus '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. Veltman
Martinus J. G. Veltman
Martinus 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 effect
Meissner effect
The 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 depth
London penetration depth
In 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 Landau
Lev Landau
Lev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...

and Vitaly Ginzburg
Vitaly Ginzburg
Vitaly 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 condensate
Bose–Einstein condensate
A 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 field
Quantum field theory
Quantum 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 equation
Schrödinger field
In 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 $\hbar$, the Planck quantum divided by 2π, is replaced by 1): $i\left\{\partial \over \partial t\right\} \psi = \left\{\left(\nabla - iqA\right)^2 \over 2m\right\} \psi$ The operator ψ(x) annihilates a boson at the point x, while its adjoint $\psi^\dagger$ 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. $\psi \rightarrow e^\left\{iq\phi\left(x\right)\right\} \psi$$A \rightarrow A + \nabla \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 $\psi\left(x\right) = \rho\left(x\right)\, e^\left\{i\theta\left(x\right)\right\},$ 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, $H= \left\{1\over 2m\right\} |\left\{\left(qA+\nabla \right)\psi|^2\right\},$ and taking the density of the condensate ρ to be constant, $H \approx \left\{\rho^2 \over 2m\right\} \left(qA + \nabla \theta\right)^2.$ Fixing the choice of gauge so that the condensate has the same phase everywhere, the electromagnetic field energy has an extra term, $\left\{q^2 \rho^2 \over 2m\right\} A^2.$ 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, $E\approx \left\{\left\{\dot A\right\}^2\over 2\right\} + \left\{q^2 \rho^2 \over 2m\right\} A^2.$ This is a harmonic oscillator with frequency: $\sqrt\left\{\frac\left\{1\right\}\left\{m\right\} q^2 \rho^2\right\}.$ The quantity |ψ|2 (=ρ2) 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 pair
Cooper pair
In 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 Schrieffer
BCS theory
BCS 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 renormalizable
Renormalization
In 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. $S\left(\phi \right) = \int \left\{1\over 2\right\} |\partial \phi|^2 - \lambda\cdot \left(|\phi|^2 - \Phi^2\right)^2$ This defines the following Hamiltonian: $H\left(\phi \right) = \left\{1\over 2\right\} |\dot\phi|^2 + |\nabla \phi|^2 + V\left(|\phi|\right)$ 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 - Φ2)2 has a graph which looks like a Mexican hat
Mexican hat
In 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 $\phi\left(x\right) = \Phi e^\left\{i\theta\left(x\right)\right\}$ 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: $S\left(\phi ,A\right) = \int \left\{1\over 4\right\} F^\left\{\mu\nu\right\} F_\left\{\mu\nu\right\} + |\left(\partial - i q A\right)\phi|^2 + \lambda\cdot \left(|\phi|^2 - \Phi^2\right)^2.$ 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: $E = \left\{1 \over 2\right\} \left |\partial \left \left(\Phi e^\left\{iqAx\right\} \right \right) \right |^2 = \left\{1 \over 2\right\} q^2\Phi^2 A^2$ And this energy is the same as a mass term ½m2A2 where m = qΦ.

### Nonabelian Higgs mechanism

The Nonabelian Higgs model has the following action: $S\left(\phi ,\mathbf A\right) = \int \left\{1\over 4g^2\right\} \mathop\left\{\textrm\left\{tr\right\}\right\}\left(F^\left\{\mu\nu\right\}F_\left\{\mu\nu\right\}\right) + |D\phi|^2 + V\left(|\phi|\right)$ where now the nonabelian field A is contained in D and in the tensor components $F^\left\{\mu \nu\right\}$ and $F_\left\{\mu \nu\right\}$ (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. $D\phi = \partial \phi - i A^k t_k \phi$ 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 Schwinger
Julian Schwinger
Julian 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 φa 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 ($c = \hbar = 1$). 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 Stueckelberg
Ernst Stueckelberg
Ernst 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:$\theta \rightarrow \theta + e\alpha\,$$A \rightarrow A + \alpha \,$ The gauge covariant derivative for the angle (which is actually gauge invariant) is:$D\theta = \partial \theta - e A\,$ 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 $\phi \;=\; He^\left\{\frac\left\{1\right\}\left\{H\right\}i\theta\right\}$. $S = \int \left\{1 \over 4\right\}F^2 + \left\{1 \over 2\right\}\left(D\theta\right)^2 = \int \left\{1 \over 4\right\}F^2 + \left\{1 \over 2\right\}\left(\partial \theta - He A\right)^2 = \int \left\{1 \over 4\right\}F^2 + \left\{1 \over 2\right\}\left(\partial \theta - m A\right)^2$ 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: $S= \int \left\{1 \over 4\right\} F^2 + \left\{1 \over 2\right\} m^2 A^2\,$ 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 monopole
Magnetic monopole
A 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.

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• Higgs boson
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...

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• Goldstone boson
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...

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• 1964 PRL symmetry breaking papers
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• QCD vacuum
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...

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• Quantum triviality
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...

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• Symmetry breaking
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...

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• Tachyon condensation
Tachyon condensation
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• Top quark condensate
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...

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• Yang–Mills–Higgs equations
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...

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