QCD vacuum
Encyclopedia
The QCD vacuum is the vacuum state
of quantum chromodynamics
(QCD). 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. These condensates characterize the normal phase or the confined phase of quark matter.
quantum field theory
, QCD
enjoys Poincaré symmetry
including the discrete symmetries CPT
(each of which is realized). Apart from these space-time symmetries, it also has internal symmetries. Since QCD is an SU(3) gauge theory
, it has local SU(3) gauge symmetry.
Since it has many flavour
s of quarks, it has approximate flavour and chiral symmetry
. This approximation is said to involve the chiral limit of QCD. Of these chiral symmetries, the baryon number symmetry is exact. Some of the broken symmetries include the axial U(1) symmetry of the flavour group. This is broken by the chiral anomaly
. The presence of instanton
s implied by this anomaly also breaks CP symmetry.
In summary, the QCD Lagrangian has the following symmetries:
The following classical symmetries are broken in the QCD Lagrangian:
of a system (or the Lagrangian
) has a certain symmetry, but the ground state
(i.e., the vacuum) does not, then one says that spontaneous symmetry breaking (SSB) has taken place.
A familiar example of SSB is in ferromagnetic materials
. Microscopically, the material consists of atom
s with a non-vanishing spin, each of which acts like a tiny bar magnet, i.e., a magnetic dipole
. The Hamiltonian of the material, describing the interaction of neighbouring dipoles, is invariant under rotation
s. At high temperature, there is no magnet
ization of a large sample of the material. Then one says that the symmetry of the Hamiltonian is realized by the system. However, at low temperature, there could be an overall magnetization. This magnetization has a preferred direction, since one can tell the north magnetic pole of the sample from the south magnetic pole. In this case, there is spontaneous symmetry breaking of the rotational symmetry of the Hamiltonian.
When a continuous symmetry
is spontaneously broken, massless boson
s appear, corresponding to the remaining symmetry. This is called the Goldstone phenomenon and the bosons are called Goldstone boson
s.
of the theory. The symmetry of the vacuum state is the diagonal SU(Nf) part of the chiral group. The diagnostic for this is the formation of a non-vanishing chiral condensate
, where ψi is the quark field operator, and the flavour index i is summed. The Goldstone boson
s of the symmetry breaking are the pseudoscalar meson
s.
When Nf=2, i.e., only the u and d quarks are treated as massless, the three pion
s are the Goldstone boson
s. When the s quark is also treated as massless, i.e., Nf=3, all eight pseudoscalar meson
s of the quark model
become Goldstone boson
s. The actual masses of these meson
s are obtained in chiral perturbation theory
through an expansion in the (small) actual masses of the quarks.
In other phases of quark matter the full chiral flavour
symmetry may be recovered, or broken in completely different ways.
s and nucleons. The obvious renormalizable interaction between the two objects is the Yukawa coupling to a pseudoscalar:
And this is clearly theoretically correct, since it is leading order and it takes all the symmetries into account. But it doesn't match experiment. The interaction that does couples the nucleons to the gradient of the pion field.
This is the gradient-coupling model. This interaction has a very different dependence on the energy of the pion—it vanishes at zero momentum.
This type of coupling means that a coherent state of low momentum pions barely interacts at all. This is a manifestation of an approximate symmetry, a shift symmetry of the pion field. The replacement
leaves the gradient coupling alone, but not the pseudoscalar coupling.
The modern explanation for the shift symmetry was first proposed by Yoichiro Nambu
. The pion field is a Goldstone boson
, and the shift symmetry is the lowest order approximation to moving along the flat directions.
and it is obeyed to 1% accuracy.
The constant
is the coefficient that determines the neutron decay rate. It gives the normalization of the weak interaction matrix elements for the nucleon. On the other hand, the pion-nucleon coupling is a phenomenological constant describing the scattering of bound states of quarks and gluons.
The weak interactions are current-current interactions ultimately because they come from a nonabelian gauge theory. The Goldberger Treiman relation suggests that the pions for some reason interact as if they are related to the same symmetry current.
The phenomena which gives rise to the Goldberger Treiman relation was called the "Partially Conserved Axial Current" hypothesis, or PCAC. Partially conserved is an archaic term for spontaneously broken, and the axial current is now called the chiral symmetry current.
The idea is that the symmetry current which performs axial rotations on the fundamental fields does not preserve the vacuum. This means that the current J applied to the vacuum produces particles. The particles must be scalars, otherwise the vacuum wouldn't be Lorentz invariant. By index matching, the matrix element is:
where
is the momentum carried by the created pion.
Since the divergence of the axial current operator is zero, we must have
Hence the pions are massless,
, in accordance with Goldstone's theorem.
Now if the scattering matrix element is considered, we have
Up to a momentum factor, which is the gradient in the coupling, it takes the same form as the axial current turning a neutron into a proton in the current-current form of the weak interaction.
to calculate the amplitudes for collisions which emit low energy pions from the amplitude for the same process with no pions. The amplitudes are those given by acting with symmetry currents on the external particles of the collision.
These successes established the basic properties of the strong interaction vacuum well before QCD.
of pseudoscalar mesons is very much lighter than the next lightest states; i.e., the octet of vector mesons (such as the rho
). The most convincing evidence for SSB of the chiral flavour
symmetry of QCD is the appearance of these pseudo-Goldstone boson
s. These would have been strictly massless in the chiral limit. There is convincing demonstration that the observed masses are compatible with chiral perturbation theory
. The internal consistency of this argument is further checked by lattice QCD
computations allow one to vary the quark mass and check that the variation of the pseudoscalar masses with the quark mass is as required by chiral perturbation theory
.
where all the pseudoscalar mesons should have been of nearly the same mass. Since Nf=3, there should have been nine of these. However, one (the SU(3) singlet η') has quite a different mass from the SU(3) octet. In the quark model this has no natural explanation— a mystery named the η-η' mass splitting (the η is one member of the octet which should have been degenerate in mass with the η'). In QCD one realizes that the η' is associated with the axial U(1) which is broken through the chiral anomaly
and not by SSB. One says therefore, that instanton
s cause the η-η' mass splitting.
also provide evidence for this pattern of SSB. Direct estimates of the chiral condensate also comes from such analysis.
Another method of analysis of correlation function
s in QCD is through an operator product expansion
(OPE). This writes the vacuum expectation value
of a non-local operator as a sum over VEVs of local operators, i.e., condensates. The value of the correlation function then dictates the values of the condensates. Analysis of many separate correlation functions gives consistent results for several condensates, including the gluon condensate
, the quark condensate, and many mixed and higher order condensates. In particular one obtains
Here G refers to the gluon
field
tensor
, ψ to the quark
field
, and g to the QCD coupling.
These analyses are being refined further through improved sum rule estimates and direct estimates in lattice QCD
. They provide the raw data which must be explained by models of the QCD vacuum.
and the hadron
spectrum. Lattice QCD
is making rapid progress towards providing the solution as a systematically improvable numerical computation. However, approximate models of the QCD vacuum remain useful in more restricted domains. The purpose of these models is to make quantitative sense of some set of condensates and hadron
properties such as masses and form factor
s.
This section is devoted to models. Opposed to these are systematically improvable computational procedures such as large N QCD and lattice QCD
, which are described in their own articles.
. However, Stanley Mandelstam
showed that a homogeneous vacuum field is also unstable. The instability of a homogeneous gluon field was argued by Niels Kjær Nielsen and Poul Olesen in their 1978 paper. These arguments suggest that the scalar condensates are an effective long-distance description of the vacuum, and at short distances, below the QCD scale, the vacuum may have structure.
s condense into Cooper pair
s. As a result magnetic flux
is squeezed into tubes. In the dual superconductor
picture of the QCD vacuum, chromomagnetic monopoles condense into dual Cooper pairs, causing chromoelectric flux to be squeezed into tubes. As a result, confinement
and the string picture of hadrons follows. This dual superconductor picture is due to Gerard 't Hooft and Stanley Mandelstam
. 't Hooft showed further that an Abelian projection of a non-Abelian gauge theory
contains magnetic monopole
s.
While the vortices in a type II superconductor are neatly arranged into a hexagonal or occasionally square lattice, as is reviewed in Olesen's 1980 seminar one may expect a much more complicated and possibly dynamical structure in QCD. For example, nonabelian Abrikosov-
Nielsen-Olesen vortices
may vibrate wildly or be knotted.
and hadron
s have a long history. They were first invented to explain certain aspects of crossing symmetry
in the scattering of two meson
s. They were also found to be useful in the description of certain properties of the Regge trajectory of the hadron
s. These early developments took on a life of their own called the dual resonance model
(later renamed string theory
). However, even after the development of QCD string models continued to play a role in the physics of strong interaction
s. These models are called non-fundamental strings or QCD string
s, since they should be derived from QCD, as they are, in certain approximations such as the strong coupling limit of lattice QCD
.
The model states that the colour electric flux between a quark and an antiquark collapses into a string, rather than spreading out into a Coulomb field as the normal electric flux does. This string also obeys a different force law. It behaves as if the string had constant tension, so that separating out the ends (quarks) would give a potential energy increasing linearly with the separation. When the energy is higher than that of a meson, the string breaks and the two new ends become a quark-antiquark pair, thus describing the creation of a meson. Thus confinement is incorporated naturally into the model.
In the form of the Lund model Monte Carlo program, this picture has had remarkable success in explaining experimental data collected in electron-electron and hadron-hadron collisions.
s. The model consists of putting some version of a quark model in a perturbative vacuum inside a volume of space called a bag. Outside this bag is the real QCD vacuum, whose effect is taken into account through boundary conditions on the quark wave functions. The hadron
spectrum is obtained by solving the Dirac equation
for quarks with the bag boundary conditions.
The chiral bag model couples the axial vector current
of the quarks at the bag boundary to a pionic field
outside of the bag. In the most common formulation, the chiral bag model basically replaces the interior of the skyrmion
with the bag of quarks. Very curiously, most physical properties of the nucleon become mostly insensitive to the bag radius. Prototypically, the baryon number of the chiral bag remains an integer, independent of bag radius: the exterior baryon number is identified with the topological winding number
density of the Skyrme soliton
, while the interior baryon number consists of the valence quarks (totaling to one) plus the spectral asymmetry
of the quark eigenstates in the bag. The spectral asymmetry is just the vacuum expectation value
summed over all of the quark eigenstates in the bag. Other values, such as the total mass and the axial coupling constant 
, are not precisely invariant like the baryon number, but are mostly insensitive to the bag radius, as long as the bag radius is kept below the nucleon diameter. Because the quarks are treated as free quarks inside the bag, the radius-independence in a sense validates the idea of asymptotic freedom
.
-like instanton
s play an important role in the vacuum structure of QCD. These instantons were discovered in 1975 by Belavin
, Polyakov, Schwartz and Tyupkin as topologically
stable solutions to the Yang-Mills field equations. They represent tunnelling
transitions from one vacuum state to another. These instantons are indeed found in lattice
calculations. The first computations performed with instantons used the dilute gas approximation. The results obtained did not solve the infrared problem of QCD, making many physicists turn away from instanton physics. Later, though, an instanton liquid model
was proposed, turning out to be more promising an approach.
The dilute instanton gas model departs from the supposition that the QCD vacuum consists of a gas of BPST-like instantons. Although only the solutions with one or few instantons (or anti-instantons) are known exactly, a dilute gas of instantons and anti-instantons can be approximated by considering a superposition of one-instanton solutions at great distances from one another. 't Hooft calculated the effective action for such an ensemble, and he found an infrared divergence
for big instantons, meaning that an infinite amount of infinitely big instantons would populate the vacuum.
Later, an instanton liquid model
was studied. This model starts from the assumption that an ensemble of instantons cannot be described by a mere sum of separate instantons. Various models have been proposed, introducing interactions between instantons or using variational methods (like the "valley approximation") endeavoring to approximate the exact multi-instanton solution as closely as possible. Many phenomenological successes have been reached. Whether an instanton liquid can explain confinement in 3+1 dimensional QCD is not known, but many physicists think that it is unlikely.
play an important role. These vortices are topological defect
s carrying a center
element as charge. These vortices are usually studied using lattice simulations
, and it has been found that the behavior of the vortices is closely linked with the confinement
-deconfinement
phase transition: in the confining phase vortices percolate and fill the space-time volume, in the deconfining phase they are much suppressed. Also it has been shown that the string tension vanished upon removal of center vortices from the simulations, hinting at an important role for center vortices.
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
of quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
(QCD). It is an example of a non-perturbative vacuum state, characterized by many non-vanishing condensates such as the gluon condensate
Gluon condensate
In Quantum chromodynamics , the gluon condensate is a non-perturbative property of the QCD vacuum which could be partly responsible for giving masses to certain hadrons.If the gluon field tensor is represented as Gμν, then...
or the quark condensate. These condensates characterize the normal phase or the confined phase of quark matter.
Symmetries of the QCD Lagrangian
Like any relativisticTheory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
quantum field theory
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...
, QCD
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
enjoys Poincaré symmetry
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
including the discrete symmetries CPT
CPT symmetry
CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...
(each of which is realized). Apart from these space-time symmetries, it also has internal symmetries. Since QCD is an SU(3) 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...
, it has local SU(3) gauge symmetry.
Since it has many flavour
Flavour (particle physics)
In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...
s of quarks, it has approximate flavour and 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...
. This approximation is said to involve the chiral limit of QCD. Of these chiral symmetries, the baryon number symmetry is exact. Some of the broken symmetries include the axial U(1) symmetry of the flavour group. This is broken by the chiral anomaly
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
. The presence of instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
s implied by this anomaly also breaks CP symmetry.
In summary, the QCD Lagrangian has the following symmetries:
- Poincare symmetry and CPTCPT symmetryCPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...
invariance - SU(3) local gauge symmetry
- approximate global SU(Nf)XSU(Nf) flavourFlavour (particle physics)In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...
chiral symmetryChiral 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...
and the U(1) baryon number symmetry
The following classical symmetries are broken in the QCD Lagrangian:
- scale, i.e., conformal symmetryConformal symmetryIn theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
(through the scale anomaly), giving rise to asymptotic freedomAsymptotic freedomIn physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum... - the axial part of the U(1) flavourFlavour (particle physics)In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...
chiral symmetry (through the chiral anomalyChiral anomalyA chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
), giving rise to the strong CP problem.
Spontaneous symmetry breaking
When the HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of a system (or the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
) has a certain symmetry, but the ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
(i.e., the vacuum) does not, then one says that spontaneous symmetry breaking (SSB) has taken place.
A familiar example of SSB is in ferromagnetic materials
Magnet
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.A permanent magnet is an object...
. Microscopically, the material consists of atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s with a non-vanishing spin, each of which acts like a tiny bar magnet, i.e., a magnetic dipole
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...
. The Hamiltonian of the material, describing the interaction of neighbouring dipoles, is invariant under rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s. At high temperature, there is no magnet
Magnet
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.A permanent magnet is an object...
ization of a large sample of the material. Then one says that the symmetry of the Hamiltonian is realized by the system. However, at low temperature, there could be an overall magnetization. This magnetization has a preferred direction, since one can tell the north magnetic pole of the sample from the south magnetic pole. In this case, there is spontaneous symmetry breaking of the rotational symmetry of the Hamiltonian.
When a continuous symmetry
Continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the...
is spontaneously broken, massless boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
s appear, corresponding to the remaining symmetry. This is called the Goldstone phenomenon and the bosons are called 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.
Symmetries of the QCD vacuum
The SU(Nf) × SU(Nf) chiral flavour symmetry of the QCD Lagrangian is broken in the vacuum stateVacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
of the theory. The symmetry of the vacuum state is the diagonal SU(Nf) part of the chiral group. The diagnostic for this is the formation of a non-vanishing chiral condensate
, where ψi is the quark field operator, and the flavour index i is summed. The Goldstone bosonGoldstone 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 of the symmetry breaking are the pseudoscalar meson
Meson
In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...
s.
When Nf=2, i.e., only the u and d quarks are treated as massless, the three 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 are the 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. When the s quark is also treated as massless, i.e., Nf=3, all eight pseudoscalar meson
Meson
In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...
s of the quark model
Quark model
In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons....
become 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. The actual masses of these meson
Meson
In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...
s are obtained in chiral perturbation theory
Chiral perturbation theory
Chiral perturbation theory is an effective field theory constructed with a Lagrangian consistent with the chiral symmetry of quantum chromodynamics , as well as the other symmetries of parity and charge conjugation. ChPT is a theory which allows one to study the low-energy dynamics of QCD...
through an expansion in the (small) actual masses of the quarks.
In other phases of quark matter the full chiral flavour
Flavour (particle physics)
In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...
symmetry may be recovered, or broken in completely different ways.
Evidence: experimental consequences
The evidence for QCD condensates comes from two eras, the pre-QCD era 1950-1973 and the post-QCD era, after 1974. The pre-QCD results established that the strong interactions vacuum contains a quark chiral condensate, while the post-QCD results established that the vacuum also contains a gluon condensate.Pre-QCD: gradient coupling
In the 1950s, there were many attempts to produce a field theory to describe the interactions of pionPion
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 and nucleons. The obvious renormalizable interaction between the two objects is the Yukawa coupling to a pseudoscalar:
And this is clearly theoretically correct, since it is leading order and it takes all the symmetries into account. But it doesn't match experiment. The interaction that does couples the nucleons to the gradient of the pion field.
This is the gradient-coupling model. This interaction has a very different dependence on the energy of the pion—it vanishes at zero momentum.
This type of coupling means that a coherent state of low momentum pions barely interacts at all. This is a manifestation of an approximate symmetry, a shift symmetry of the pion field. The replacement
leaves the gradient coupling alone, but not the pseudoscalar coupling.
The modern explanation for the shift symmetry was first proposed 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...
. The pion field is a 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...
, and the shift symmetry is the lowest order approximation to moving along the flat directions.
Pre-QCD: Goldberger-Treiman relation
There is a mysterious relationship between the strong interaction coupling of the pions to the nucleons, the coefficient g in the gradient coupling model, and the axial vector current coefficient of the nucleon which determines the weak decay rate of the neutron. The relation isand it is obeyed to 1% accuracy.
The constant
is the coefficient that determines the neutron decay rate. It gives the normalization of the weak interaction matrix elements for the nucleon. On the other hand, the pion-nucleon coupling is a phenomenological constant describing the scattering of bound states of quarks and gluons.The weak interactions are current-current interactions ultimately because they come from a nonabelian gauge theory. The Goldberger Treiman relation suggests that the pions for some reason interact as if they are related to the same symmetry current.
PCAC
The phenomena which gives rise to the Goldberger Treiman relation was called the "Partially Conserved Axial Current" hypothesis, or PCAC. Partially conserved is an archaic term for spontaneously broken, and the axial current is now called the chiral symmetry current.
The idea is that the symmetry current which performs axial rotations on the fundamental fields does not preserve the vacuum. This means that the current J applied to the vacuum produces particles. The particles must be scalars, otherwise the vacuum wouldn't be Lorentz invariant. By index matching, the matrix element is:
where
is the momentum carried by the created pion.Since the divergence of the axial current operator is zero, we must have
Hence the pions are massless,
, in accordance with Goldstone's theorem.Now if the scattering matrix element is considered, we have
Up to a momentum factor, which is the gradient in the coupling, it takes the same form as the axial current turning a neutron into a proton in the current-current form of the weak interaction.
Pre-QCD: soft pion emission
Extensions of the PCAC ideas allowed Steven WeinbergSteven 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....
to calculate the amplitudes for collisions which emit low energy pions from the amplitude for the same process with no pions. The amplitudes are those given by acting with symmetry currents on the external particles of the collision.
These successes established the basic properties of the strong interaction vacuum well before QCD.
Pseudo-Goldstone bosons
Experimentally it is seen that the masses of the octetAdjoint representation
In mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
of pseudoscalar mesons is very much lighter than the next lightest states; i.e., the octet of vector mesons (such as the rho
Rho
Rho is the 17th letter of the Greek alphabet. In the system of Greek numerals, it has a value of 100. It is derived from Semitic resh "head"...
). The most convincing evidence for SSB of the chiral flavour
Flavour (particle physics)
In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...
symmetry of QCD is the appearance of these pseudo-Goldstone boson
Pseudo-Goldstone boson
Pseudo-Goldstone bosons arise in a quantum field theory with both spontaneous and explicit symmetry breaking. The controlling approximate symmetries, if they were exact, would be spontaneously broken , and would thus engender massless Goldstone bosons. The additional explicit symmetry breaking...
s. These would have been strictly massless in the chiral limit. There is convincing demonstration that the observed masses are compatible with chiral perturbation theory
Chiral perturbation theory
Chiral perturbation theory is an effective field theory constructed with a Lagrangian consistent with the chiral symmetry of quantum chromodynamics , as well as the other symmetries of parity and charge conjugation. ChPT is a theory which allows one to study the low-energy dynamics of QCD...
. The internal consistency of this argument is further checked by lattice QCD
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....
computations allow one to vary the quark mass and check that the variation of the pseudoscalar masses with the quark mass is as required by chiral perturbation theory
Chiral perturbation theory
Chiral perturbation theory is an effective field theory constructed with a Lagrangian consistent with the chiral symmetry of quantum chromodynamics , as well as the other symmetries of parity and charge conjugation. ChPT is a theory which allows one to study the low-energy dynamics of QCD...
.
The η'
This pattern of SSB solves one of the mysteries of the quark modelQuark model
In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons....
where all the pseudoscalar mesons should have been of nearly the same mass. Since Nf=3, there should have been nine of these. However, one (the SU(3) singlet η') has quite a different mass from the SU(3) octet. In the quark model this has no natural explanation— a mystery named the η-η' mass splitting (the η is one member of the octet which should have been degenerate in mass with the η'). In QCD one realizes that the η' is associated with the axial U(1) which is broken through the chiral anomaly
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
and not by SSB. One says therefore, that instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
s cause the η-η' mass splitting.
Current algebra and QCD sum rules
PCAC and current algebraCurrent algebra
Current algebra is a mathematical framework in quantum field theory where the fields form a Lie algebra under their commutation relations.For instance, in a non-Abelian Yang–Mills symmetry, where ρ is the charge density,...
also provide evidence for this pattern of SSB. Direct estimates of the chiral condensate also comes from such analysis.
Another method of analysis of correlation function
Correlation function (quantum field theory)
In quantum field theory, the matrix element computed by inserting a product of operators between two states, usually the vacuum states, is called a correlation function....
s in QCD is through an operator product expansion
Operator product expansion
- 2D Euclidean quantum field theory :In quantum field theory, the operator product expansion is a Laurent series expansion of two operators...
(OPE). This writes the vacuum expectation value
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
of a non-local operator as a sum over VEVs of local operators, i.e., condensates. The value of the correlation function then dictates the values of the condensates. Analysis of many separate correlation functions gives consistent results for several condensates, including the gluon condensate
Gluon condensate
In Quantum chromodynamics , the gluon condensate is a non-perturbative property of the QCD vacuum which could be partly responsible for giving masses to certain hadrons.If the gluon field tensor is represented as Gμν, then...
, the quark condensate, and many mixed and higher order condensates. In particular one obtains
Here G refers to the gluon
Gluon
Gluons are elementary particles which act as the exchange particles for the color force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles....
field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
, ψ to the 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...
field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
, and g to the QCD coupling.
These analyses are being refined further through improved sum rule estimates and direct estimates in lattice QCD
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....
. They provide the raw data which must be explained by models of the QCD vacuum.
Models of the QCD vacuum
A full solution of QCD would automatically give a full description of the vacuum, confinementColour confinement
Color confinement, often simply called confinement, is the physics phenomenon that color charged particles cannot be isolated singularly, and therefore cannot be directly observed. Quarks, by default, clump together to form groups, or hadrons. The two types of hadrons are the mesons and the baryons...
and the hadron
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
spectrum. Lattice QCD
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....
is making rapid progress towards providing the solution as a systematically improvable numerical computation. However, approximate models of the QCD vacuum remain useful in more restricted domains. The purpose of these models is to make quantitative sense of some set of condensates and hadron
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
properties such as masses and form factor
Form factor
Form factor may refer to:*Form factor or emissivity, the proportion of energy transmitted by that object which can be transferred to another object...
s.
This section is devoted to models. Opposed to these are systematically improvable computational procedures such as large N QCD and lattice QCD
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....
, which are described in their own articles.
The Savvidy vacuum, instabilities and structure
This is a model of the QCD vacuum which at a basic level is a statement that it cannot be the conventional Fock vacuum empty of particles and fields. In 1977, George Savvidy showed that the QCD vacuum with zero field strength is unstable, and decays into a state with a calculable non vanishing value of the field. Since condensates are scalar, it seems like a good first approximation that the vacuum contains some non-zero but homogeneous field which gives rise to these condensates. This would then be a more complicated version of the Higgs mechanismHiggs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
. However, Stanley Mandelstam
Stanley Mandelstam
Stanley Mandelstam is a South African-born theoretical physicist. He introduced the relativistically invariant Mandelstam variables into particle physics in 1958 as a convenient coordinate system for formulating his double dispersion relations...
showed that a homogeneous vacuum field is also unstable. The instability of a homogeneous gluon field was argued by Niels Kjær Nielsen and Poul Olesen in their 1978 paper. These arguments suggest that the scalar condensates are an effective long-distance description of the vacuum, and at short distances, below the QCD scale, the vacuum may have structure.
The dual superconducting model
In a type II superconductor, electric chargeElectric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
s condense 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. As a result magnetic flux
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...
is squeezed into tubes. In the dual superconductor
Dual superconducting model
In the theory of quantum chromodynamics, dual superconductor models attempt to explain confinement of quarks in terms of an electromagnetic dual theory of superconductivity....
picture of the QCD vacuum, chromomagnetic monopoles condense into dual Cooper pairs, causing chromoelectric flux to be squeezed into tubes. As a result, confinement
Colour confinement
Color confinement, often simply called confinement, is the physics phenomenon that color charged particles cannot be isolated singularly, and therefore cannot be directly observed. Quarks, by default, clump together to form groups, or hadrons. The two types of hadrons are the mesons and the baryons...
and the string picture of hadrons follows. This dual superconductor picture is due to Gerard 't Hooft and Stanley Mandelstam
Stanley Mandelstam
Stanley Mandelstam is a South African-born theoretical physicist. He introduced the relativistically invariant Mandelstam variables into particle physics in 1958 as a convenient coordinate system for formulating his double dispersion relations...
. 't Hooft showed further that an Abelian projection of a non-Abelian 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...
contains 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.
While the vortices in a type II superconductor are neatly arranged into a hexagonal or occasionally square lattice, as is reviewed in Olesen's 1980 seminar one may expect a much more complicated and possibly dynamical structure in QCD. For example, nonabelian Abrikosov-
Abrikosov vortex
In superconductivity, an Abrikosov vortex is a vortex of supercurrent in a type-II superconductor. The supercurrent circulates around the normal core of the vortex. The core has a size \sim\xi — the superconducting coherence length...
Nielsen-Olesen vortices
Nielsen-Olesen vortex
In theoretical physics, a Nielsen–Olesen vortex is a point-like object localized in two spatial dimensions or, equivalently, a classical solution of field theory with the same property. This particular solution occurs if the configuration space of scalar fields contains non-contractible circles...
may vibrate wildly or be knotted.
String models
String models of confinementColour confinement
Color confinement, often simply called confinement, is the physics phenomenon that color charged particles cannot be isolated singularly, and therefore cannot be directly observed. Quarks, by default, clump together to form groups, or hadrons. The two types of hadrons are the mesons and the baryons...
and hadron
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
s have a long history. They were first invented to explain certain aspects of crossing symmetry
Crossing symmetry
In quantum field theory, a branch of theoretical physics, crossing is the property of scattering amplitudes that allows antiparticles to be interpreted as particles going backwards in time....
in the scattering of two meson
Meson
In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...
s. They were also found to be useful in the description of certain properties of the Regge trajectory of the hadron
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
s. These early developments took on a life of their own called the dual resonance model
Dual resonance model
In theoretical physics, a dual resonance model arose the early investigation of string theory as an S-matrix theory of the strong interaction....
(later renamed string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
). However, even after the development of QCD string models continued to play a role in the physics of strong interaction
Strong interaction
In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...
s. These models are called non-fundamental strings or QCD string
QCD string
In quantum chromodynamics, , if a connection which is colour confining occurs, it is possible for stringlike degrees of freedom called QCD strings or QCD flux tubes to form...
s, since they should be derived from QCD, as they are, in certain approximations such as the strong coupling limit of lattice QCD
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....
.
The model states that the colour electric flux between a quark and an antiquark collapses into a string, rather than spreading out into a Coulomb field as the normal electric flux does. This string also obeys a different force law. It behaves as if the string had constant tension, so that separating out the ends (quarks) would give a potential energy increasing linearly with the separation. When the energy is higher than that of a meson, the string breaks and the two new ends become a quark-antiquark pair, thus describing the creation of a meson. Thus confinement is incorporated naturally into the model.
In the form of the Lund model Monte Carlo program, this picture has had remarkable success in explaining experimental data collected in electron-electron and hadron-hadron collisions.
Bag models
Strictly, these models are not models of the QCD vacuum, but of physical single particle quantum states — the hadronHadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
s. The model consists of putting some version of a quark model in a perturbative vacuum inside a volume of space called a bag. Outside this bag is the real QCD vacuum, whose effect is taken into account through boundary conditions on the quark wave functions. The hadron
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
spectrum is obtained by solving the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
for quarks with the bag boundary conditions.
The chiral bag model couples the axial vector current
of the quarks at the bag boundary to a pionic fieldPion
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....
outside of the bag. In the most common formulation, the chiral bag model basically replaces the interior of the skyrmion
Skyrmion
In theoretical physics, a skyrmion is a mathematical model used to model baryons . It was conceived by Tony Skyrme.-Overview:...
with the bag of quarks. Very curiously, most physical properties of the nucleon become mostly insensitive to the bag radius. Prototypically, the baryon number of the chiral bag remains an integer, independent of bag radius: the exterior baryon number is identified with the topological winding number
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
density of the Skyrme soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
, while the interior baryon number consists of the valence quarks (totaling to one) plus the spectral asymmetry
Spectral asymmetry
In mathematics and physics, the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalues of an operator. In mathematics, the spectral asymmetry arises in the study of elliptic operators on compact manifolds, and is given a deep meaning by the Atiyah-Singer index theorem...
of the quark eigenstates in the bag. The spectral asymmetry is just the vacuum expectation value
summed over all of the quark eigenstates in the bag. Other values, such as the total mass and the axial coupling constant , are not precisely invariant like the baryon number, but are mostly insensitive to the bag radius, as long as the bag radius is kept below the nucleon diameter. Because the quarks are treated as free quarks inside the bag, the radius-independence in a sense validates the idea of asymptotic freedomAsymptotic freedom
In physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...
.
Instanton ensemble
Another view states that BPSTBPST instanton
The BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU Yang-Mills theory in Euclidean space-time , meaning it describes a transition between two...
-like instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
s play an important role in the vacuum structure of QCD. These instantons were discovered in 1975 by Belavin
Alexander Belavin
Alexander "Sasha" Abramovich Belavin is a Russian physicist, known for his contributions to string theory.He is a professor at the Independent University of Moscow and is researcher at the Landau Institute of Theoretical Physics....
, Polyakov, Schwartz and Tyupkin as topologically
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
stable solutions to the Yang-Mills field equations. They represent tunnelling
Quantum tunnelling
Quantum tunnelling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the sun, and has important...
transitions from one vacuum state to another. These instantons are indeed found in lattice
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....
calculations. The first computations performed with instantons used the dilute gas approximation. The results obtained did not solve the infrared problem of QCD, making many physicists turn away from instanton physics. Later, though, an instanton liquid model
Instanton fluid
In quantum chromodynamics, the instanton fluid model is a model of Wick rotated Euclidean quantum chromodynamics. If we examine the path integral of the action, we find that it has infinitely many local minima, corresponding to varying instanton numbers...
was proposed, turning out to be more promising an approach.
The dilute instanton gas model departs from the supposition that the QCD vacuum consists of a gas of BPST-like instantons. Although only the solutions with one or few instantons (or anti-instantons) are known exactly, a dilute gas of instantons and anti-instantons can be approximated by considering a superposition of one-instanton solutions at great distances from one another. 't Hooft calculated the effective action for such an ensemble, and he found an infrared divergence
Infrared divergence
In physics, an infrared divergence or infrared catastrophe is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very small energy approaching zero, or, equivalently, because of physical phenomena at very long distances.The infrared ...
for big instantons, meaning that an infinite amount of infinitely big instantons would populate the vacuum.
Later, an instanton liquid model
Instanton fluid
In quantum chromodynamics, the instanton fluid model is a model of Wick rotated Euclidean quantum chromodynamics. If we examine the path integral of the action, we find that it has infinitely many local minima, corresponding to varying instanton numbers...
was studied. This model starts from the assumption that an ensemble of instantons cannot be described by a mere sum of separate instantons. Various models have been proposed, introducing interactions between instantons or using variational methods (like the "valley approximation") endeavoring to approximate the exact multi-instanton solution as closely as possible. Many phenomenological successes have been reached. Whether an instanton liquid can explain confinement in 3+1 dimensional QCD is not known, but many physicists think that it is unlikely.
Center vortex picture
A more recent picture of the QCD vacuum is one in which center vorticesCenter vortex
Center vortices are line-like topological defects that exist in the vacuum of Yang–Mills theory and QCD. They seem to play an important role in the confinement of quarks.-In SU theories:...
play an important role. These vortices are topological defect
Topological defect
In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of...
s carrying a center
Center (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
element as charge. These vortices are usually studied using lattice simulations
Lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum electrodynamics, quantum chromodynamics and the Standard...
, and it has been found that the behavior of the vortices is closely linked with the confinement
Confinement
Confinement may refer to either* Civil confinement for psychiatric patients* Color confinement, the physical principle explaining the non-observation of color charged particles like free quarks* Solitary confinement, a strict form of imprisonment...
-deconfinement
Deconfinement
In physics, deconfinement is the property of a phase in which certain particles are allowed to exist as free excitations, rather than only within bound states...
phase transition: in the confining phase vortices percolate and fill the space-time volume, in the deconfining phase they are much suppressed. Also it has been shown that the string tension vanished upon removal of center vortices from the simulations, hinting at an important role for center vortices.
See also
- Vacuum stateVacuum stateIn quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
and vacuum - Spontaneous symmetry breakingSpontaneous symmetry breakingSpontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
- Quantum chromodynamicsQuantum chromodynamicsIn theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
and flavourFlavour (particle physics)In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry... - Top quark condensateTop quark condensateIn 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...
- Goldstone bosonGoldstone 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...
- Symmetry breakingSymmetry breakingSymmetry 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...
- Higgs mechanismHiggs mechanismIn particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....