QCD vacuum - Citizendia

The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). In Quantum field theory, the vacuum state (also called the vacuum) is the Quantum state with the lowest possible Energy. Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the 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. In Quantum field theory the vacuum expectation value (also called condensate) of an operator is its average Expected value in the vacuum In Quantum chromodynamics (QCD the gluon condensate is a Non-perturbative property of the QCD vacuum which could be partly responsible for giving masses A fermionic condensate is a Superfluid phase formed by Fermionic particles at low Temperatures It is closely related to the Bose-Einstein These condensates characterize the normal phase or the confined phase of quark matter. Quark matter or QCD matter (see QCD) refers to any of a number of theorized phases of matter whose degrees of freedom include Quarks and Gluons

Unsolved problems in physics: QCD in the non-perturbative regime: confinement. This is a list of some of the major unsolved problems in Physics. In Quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system Color confinement, often called just confinement, is the Physics phenomenon that Color charged particles (such as Quarks cannot be isolated singularly The equations of QCD remain unsolved at energy scales relevant for describing atomic nuclei. In Physics, length scale is a particular Length or Distance determined with the precision of one order (or a few orders of magnitude The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom How does QCD give rise to the physics of nuclei and nuclear constituents?

Symmetries and symmetry breaking

Symmetries of the QCD Lagrangian

Like any relativistic quantum field theory, QCD enjoys Poincare symmetry including the discrete symmetries CPT (each of which is realized). The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. In quantum field theory (QFT the forces between particles are mediated by other particles Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime 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. Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations

Since it has many flavours of quarks, it has approximate flavour and chiral symmetry. In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their In Quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts This approximation is said to involve the chiral limit of QCD. Of these chiral symmetries, the baryon number symmetry is exact. In Particle physics, the baryon number is an approximate conserved Quantum number of a system Some of the broken symmetries include the axial U(1) symmetry of the flavour group. This is broken by the chiral anomaly. A chiral anomaly is the anomalous Nonconservation of a chiral current The presence of instantons implied by this anomaly, also breaks CP symmetry. An instanton or pseudoparticle is a notion appearing in theoretical and Mathematical physics. In Particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics

In summary, the QCD Lagrangian has the following symmetries:

Poincare symmetry and CPT invariance
SU(3) local gauge symmetry
approximate global SU(N_f)XSU(N_f) flavour chiral symmetry and the U(1) baryon number symmetry

The following classical symmetries are broken in the QCD Lagrangian:

scale, ie, conformal symmetry (through the scale anomaly), giving rise to asymptotic freedom
the axial part of the U(1) flavour chiral symmetry (through the chiral anomaly), giving rise to the strong CP problem. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their In Quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts In Particle physics, the baryon number is an approximate conserved Quantum number of a system In theoretical physics conformal symmetry is a Symmetry under dilatation ( Scale invariance) and under the special conformal transformations. Conformal anomaly is an anomaly ie a quantum phenomenon that breaks the Conformal symmetry of the Classical theory. In Physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles such as Quarks, becomes arbitrarily In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their A chiral anomaly is the anomalous Nonconservation of a chiral current In Particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics

Spontaneous symmetry breaking

When the Hamiltonian of a system (or the Lagrangian) has a certain symmetry, but the ground state (ie, the vacuum) does not, then one says that spontaneous symmetry breaking (SSB) has taken place. In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system In Quantum mechanics, a stationary state is an Eigenstate of a Hamiltonian, or in other words a state of definite energy This vacuum means "absence of matter" or "an empty area or space" for the cleaning appliance see Vacuum cleaner.

A familiar example of SSB is in magnetic materials. A magnet (from Greek grc μαγνήτης λίθος " Magnesian stone" is a material or object that produces a Magnetic field. Microscopically, the material consists of atoms with a non-vanishing spin, each of which acts like a tiny bar magnet, ie, a magnetic dipole. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In physics there are two kinds of dipoles ( Hellènic: di(s- = two- and pòla = pivot hinge An electric dipole is a The Hamiltonian of the material, describing the interaction of neighbouring dipoles, is invariant under rotations. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation At high temperature, there is no magnetization of a large sample of the material. A magnet (from Greek grc μαγνήτης λίθος " Magnesian stone" is a material or object that produces a Magnetic field. 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 bosons appear, corresponding to the remaining symmetry. In Mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetries as motions as opposed to e In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein This is called the Goldstone phenomenon and the bosons are called Goldstone bosons. In particle and Condensed matter physics, Goldstone bosons (also known as Nambu -Goldstone bosons) are Bosons that appear in models

(For more details, see the page on spontaneous symmetry breaking). In Physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some Symmetry group goes into a Vacuum state

Symmetries of the QCD vacuum

The SU(N_f) × SU(N_f) chiral flavour symmetry of the QCD Lagrangian is broken in the vacuum state of the theory. In Quantum field theory, the vacuum state (also called the vacuum) is the Quantum state with the lowest possible Energy. The symmetry of the vacuum state is the diagonal SU(N_f) part of the chiral group. The diagnostic for this is the formation of a non-vanishing chiral condensate $\langle\overline\psi_i\psi_i\rangle$ , where ψ_i is the quark field operator, and the flavour index i is summed. A fermionic condensate is a Superfluid phase formed by Fermionic particles at low Temperatures It is closely related to the Bose-Einstein The Goldstone bosons of the symmetry breaking are the pseudoscalar mesons. In particle and Condensed matter physics, Goldstone bosons (also known as Nambu -Goldstone bosons) are Bosons that appear in models In Physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as Improper rotations In Particle physics, a meson is a strongly interacting Boson &mdashthat is a Hadron with integer spin.

When N_f=2, ie, only the u and d quarks are treated as massless, the three pions are the Goldstone bosons. In Particle physics, pion (short for pi meson) is the collective name for three Subatomic particles, and. In particle and Condensed matter physics, Goldstone bosons (also known as Nambu -Goldstone bosons) are Bosons that appear in models When the s quark is also treated as massless, ie, N_f=3, all eight pseudoscalar mesons of the quark model become Goldstone bosons. In Particle physics, a meson is a strongly interacting Boson &mdashthat is a Hadron with integer spin. In Physics, the quark model is a classification scheme for Hadrons in terms of their valence quarks, i In particle and Condensed matter physics, Goldstone bosons (also known as Nambu -Goldstone bosons) are Bosons that appear in models The actual masses of these mesons are obtained in chiral perturbation theory through an expansion in the (small) actual masses of the quarks. In Particle physics, a meson is a strongly interacting Boson &mdashthat is a Hadron with integer spin. Chiral perturbation theory (ChPT is an Effective field theory constructed with a Lagrangian consistent with the (approximate Chiral symmetry of Quantum

In other phases of quark matter the full chiral flavour symmetry may be recovered, or broken in completely different ways. Quark matter or QCD matter (see QCD) refers to any of a number of theorized phases of matter whose degrees of freedom include Quarks and Gluons In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their

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 pions and nucleons. In Particle physics, pion (short for pi meson) is the collective name for three Subatomic particles, and. The obvious renormalizable interaction between the two objects is the Yukawa coupling to a pseudoscalar:

$L_I= \bar{N}\gamma_5 \pi N \,$

And this is clearly theoretically correct, since it is leading order and it takes all the symmetries into account. In Particle physics, Yukawa's interaction, named after Hideki Yukawa, is an interaction between a scalar field \phi and a Dirac field But it doesn't match experiment. The interaction that does couples the nucleons to the gradient of the pion field.

$g \bar{N}\gamma^\mu \partial_\mu \pi N \,$

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

$\pi \rightarrow \pi+C \,$

leaves the gradient coupling alone, but not the pseudoscalar coupling.

The modern explanation for the shift symmetry was first proposed by Yoichiro Nambu. is a Japan -born American Physicist, currently a Professor at the University of Chicago. The pion field is a Goldstone boson, and the shift symmetry is the lowest order approximation to moving along the flat directions. In particle and Condensed matter physics, Goldstone bosons (also known as Nambu -Goldstone bosons) are Bosons that appear in models

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 is

$g_{\pi NN} F_\pi = G_A M_N \,$

and it is obeyed to 1% accuracy.

The constant $G A$ 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 nonablian 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:

$J_\mu |0\rangle = k_\mu |\pi\rangle \,,$

where $k μ$ is the momentum carried by the created pion. Since the divergence of the axial current operator is zero, we must have

$\partial_\mu J^\mu |0\rangle = k^\mu k_\mu |\pi\rangle = m_\pi^2|\pi\rangle = 0 \,.$

Hence the pions are massless, $m_\pi^2=0$ , in accordance with Goldstone's theorem. In particle and Condensed matter physics, Goldstone bosons (also known as Nambu -Goldstone bosons) are Bosons that appear in models

Now if the scattering matrix element is considered, we have

$k_\mu \langle N(p) |\pi(k) N(p') \rangle = \langle N(p) | J_\mu |N(p')\rangle \,.$

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.

$\langle N |J^\mu |N\rangle \langle e| J_\mu |\nu\rangle \,$

Pre-QCD: soft pion emission

Extensions of the PCAC ideas allowed Steven Weinberg to calculate the amplitudes for collisions which emit low energy pions from the amplitude for the same process with no pions. Steven Weinberg (born May 3, 1933) is an American Physicist, and Nobel laureate in Physics for his contributions with Abdus Salam 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 octet of pseudoscalar mesons is very much lighter than the next lightest states, ie, the octet of vector mesons (such as the rho). In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its The most convincing evidence for SSB of the chiral flavour symmetry of QCD is the appearance of these pseudo-Goldstone bosons. In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their Pseudo-Goldstone bosons arise in a Quantum field theory with an approximate Symmetry such that if the symmetry were exact then there would be Spontaneous symmetry 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 (ChPT is an Effective field theory constructed with a Lagrangian consistent with the (approximate Chiral symmetry of Quantum 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. In Physics, lattice quantum chromodynamics (lattice QCD is a theory of Quarks and Gluons formulated on a space-time lattice. Chiral perturbation theory (ChPT is an Effective field theory constructed with a Lagrangian consistent with the (approximate Chiral symmetry of Quantum

The η'

This pattern of SSB solves one of the mysteries of the quark model where all the pseudoscalar mesons should have been of nearly the same mass. In Physics, the quark model is a classification scheme for Hadrons in terms of their valence quarks, i Since N_f=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. A chiral anomaly is the anomalous Nonconservation of a chiral current One says therefore, that instantons cause the η-η' mass splitting. An instanton or pseudoparticle is a notion appearing in theoretical and Mathematical physics.

Current algebra and QCD sum rules

PCAC and current algebra also provide evidence for this pattern of SSB. Current algebra is a mathematical framework in Quantum field theory where the fields form a Lie algebra under their commutation relations Direct estimates of the chiral condensate also comes from such analysis.

Another method of analysis of correlation functions in QCD is through an operator product expansion (OPE). In Quantum field theory, correlation functions generalize the concept of Correlation functions in statistics In Quantum field theory, the operator product expansion ( OPE) is a Laurent series expansion of two operators This writes the vacuum expectation value of a non-local operator as a sum over VEVs of local operators, ie, condensates. In Quantum field theory the vacuum expectation value (also called condensate) of an operator is its average Expected value in the vacuum In Quantum field theory the vacuum expectation value (also called condensate) of an operator is its average Expected value in the vacuum 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 Quantum chromodynamics (QCD the gluon condensate is a Non-perturbative property of the QCD vacuum which could be partly responsible for giving masses A fermionic condensate is a Superfluid phase formed by Fermionic particles at low Temperatures It is closely related to the Bose-Einstein In particular one obtains—

$\langle (gG)^2\rangle\ \stackrel{\mathrm{def}}{=}\ \langle g^2 G_{\mu\nu}G^{\mu\nu}\rangle \simeq 0.5 {\rm\ GeV}^4$

$\langle \overline\psi\psi\rangle \simeq (-0.23)^3 {\rm\ GeV}^3$

$\langle (gG)^4\rangle\simeq 5:10\langle (gG)^2\rangle^2$

Here G refers to the gluon field tensor, ψ to the quark field and g to the QCD coupling. Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the In Physics, a field is a Physical quantity associated to each point of Spacetime. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. In Physics, a field is a Physical quantity associated to each point of Spacetime.

These analyses are being refined further through improved sum rule estimates and direct estimates in lattice QCD. In Physics, lattice quantum chromodynamics (lattice QCD is a theory of Quarks and Gluons formulated on a space-time lattice. 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, confinement and the hadron spectrum. Color confinement, often called just confinement, is the Physics phenomenon that Color charged particles (such as Quarks cannot be isolated singularly In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( Lattice QCD is making rapid progress towards providing the solution as a systematically improvable numerical computation. In Physics, lattice quantum chromodynamics (lattice QCD is a theory of Quarks and Gluons formulated on a space-time lattice. 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 factors. In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " (

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. In Physics, lattice quantum chromodynamics (lattice QCD is a theory of Quarks and Gluons formulated on a space-time lattice.

The Savvidy vacuum

This is not so much a model of the QCD vacuum as a statement of what it is not. In 1977, George Savvidy showed that the QCD vacuum with zero field strength is unstable, and decays into a state with a 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. In Quantum field theory the vacuum expectation value (also called condensate) of an operator is its average Expected value in the vacuum This would then be a more complicated version of the Higgs mechanism. The Higgs mechanism is Spontaneous symmetry breaking in a Gauge theory. However, Stanley Mandelstam showed that a homogeneous vacuum field is also unstable. Stanley Mandelstam (b 1928 Johannesburg) is a South African born theoretical physicist It seems 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 charges condense into Cooper pairs. Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Condensed matter physics, a Cooper pair is the name given to electrons that are bound together at low temperatures in a certain manner first described in 1956 by As a result magnetic flux is squeezed into tubes. Magnetic flux, represented by the Greek letter Φ ( Phi) is a measure of quantity of Magnetism, taking into account the strength and the extent of a Magnetic In the dual superconductor picture of the QCD vacuum, chromomagnetic monopoles condense into dual Cooper pairs, causing chromoelectric flux to be squeezed into tubes. In Quantum gauge theory, the dual superconducting model is a proposed explanation of confinement as the dual of a Superconductor. As a result, confinement and the string picture of hadrons follows. Color confinement, often called just confinement, is the Physics phenomenon that Color charged particles (such as Quarks cannot be isolated singularly This dual superconductor picture is due to Gerard 't Hooft and Stanley Mandelstam. Gerardus 't Hooft (xeːrɑrt ət hoːft (born July 5, 1946, Den Helder) is a professor in Theoretical physics at Utrecht University Stanley Mandelstam (b 1928 Johannesburg) is a South African born theoretical physicist 't Hooft showed further that an Abelian projection of a non-Abelian gauge theory contains magnetic monopoles. Gerardus 't Hooft (xeːrɑrt ət hoːft (born July 5, 1946, Den Helder) is a professor in Theoretical physics at Utrecht University Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Physics, a magnetic monopole is a hypothetical particle that is a Magnet with only one pole (see Maxwell's equations for more on magnetic There is continuing interest in checking whether further parts of this picture hold.

String models

String models of confinement and hadrons have a long history. Color confinement, often called just confinement, is the Physics phenomenon that Color charged particles (such as Quarks cannot be isolated singularly In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( They were first invented to explain certain aspects of crossing symmetry in the scattering of two mesons. In Quantum field theory, a branch of theoretical physics crossing symmetry is a symmetry that relates S-matrix elements In Particle physics, a meson is a strongly interacting Boson &mdashthat is a Hadron with integer spin. They were also found to be useful in the description of certain properties of the Regge trajectory of the hadrons. In Quantum physics, Regge theory is the study of the analytic properties of scattering as a function of angular momentum where the angular momentum is not restricted to be an In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( These early developments took on a life of their own called the dual resonance model (later renamed string theory). A dual resonance model is a term used in Theoretical physics which refers to the early investigation (1968-1974 on Strong interactions of string theory String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings However, even after the development of QCD string models continued to play a role in the physics of strong interactions. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and These models are called non-fundamental strings or QCD strings, since they should be derived from QCD, as they are, in certain approximations such as the strong coupling limit of lattice QCD. In Quantum chromodynamics, or more generally Quantum gauge theories with a connection which are confining, stringlike degrees of freedom called In Physics, lattice quantum chromodynamics (lattice QCD is a theory of Quarks and Gluons formulated on a space-time lattice.

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. See also QCD string In Particle physics, the Lund string model is a phenomenological model of Hadronization.

Bag models

Strictly, these models are not models of the QCD vacuum, but of physical single particle quantum states — the hadrons. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( 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. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The hadron spectrum is obtained by solving the Dirac equation for quarks with the bag boundary conditions. In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides

The chiral bag model couples the axial vector current $\overline\psi \gamma_5 \gamma_\mu\psi$ of the quarks at the bag boundary to a pionic field outside of the bag. In Particle physics, pion (short for pi meson) is the collective name for three Subatomic particles, and. In the most common formulation, the chiral bag model basically replaces the interior of the skyrmion with the bag of quarks. In Theoretical physics, a skyrmion, conceived by Tony Skyrme, is a mathematical model used to model Baryons (a Subatomic particle) 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. In Particle physics, the baryon number is an approximate conserved Quantum number of a system The term winding number may also refer to the Rotation number of an Iterated map. In Mathematics and Physics, a soliton is a self-reinforcing solitary wave (a wave packet or pulse that maintains its shape while it travels at constant speed In Mathematics and Physics, the spectral asymmetry is the asymmetry in the distribution of the Spectrum of Eigenvalues of an Operator The spectral asymmetry is just the vacuum expectation value $\langle \overline\psi \gamma_0\psi\rangle$ summed over all of the quark eigenstates in the bag. Other values, such as the total mass and the axial coupling constant $g A$ , 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. In Physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles such as Quarks, becomes arbitrarily

Instanton ensemble

Another view states that BPST-like instantons play an important role in the vacuum structure of QCD. The BPST instanton is the Instanton with Winding number 1 found by Belavin, Polyakov, Schwartz and Tyupkin. An instanton or pseudoparticle is a notion appearing in theoretical and Mathematical physics. These instantons were discovered in 1975 by Belavin, Polyakov, Schwartz and Tyupkin^[1] as topologically stable solutions to the Yang-Mills field equations. Year 1975 ( MCMLXXV) was a Common year starting on Wednesday (link will display full calendar of the Gregorian calendar. Alexander M Polyakov (born 27 September 1945) is a theoretical Physicist, formerly at the Landau Institute in Moscow, currently Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of They represent tunnelling transitions from one vacuum state to another. These instantons are indeed found in lattice calculations. In Physics, lattice quantum chromodynamics (lattice QCD is a theory of Quarks and Gluons formulated on a space-time lattice. 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. ^[2]

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,^[3] and he found an infrared divergence for big instantons, meaning that an infinite amount of infinitely big instantons would populate the vacuum. Gerardus 't Hooft (xeːrɑrt ət hoːft (born July 5, 1946, Den Helder) is a professor in Theoretical physics at Utrecht University In Physics, an infrared divergence is a situation in which an Integral, for example a Feynman diagram, diverges because of contributions of objects with

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. ^[2] Confinement seems to be the biggest issue in Yang-Mills theory for which instantons have no answer whatsoever.

References and external links

^ Belavin, A. In Quantum field theory, the vacuum state (also called the vacuum) is the Quantum state with the lowest possible Energy. This vacuum means "absence of matter" or "an empty area or space" for the cleaning appliance see Vacuum cleaner. In Physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some Symmetry group goes into a Vacuum state Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their In Particle physics, the top quark condensate theory is an alternative to the Standard Model in which a fundamental Scalar Higgs field is replaced In particle and Condensed matter physics, Goldstone bosons (also known as Nambu -Goldstone bosons) are Bosons that appear in models The Higgs mechanism is Spontaneous symmetry breaking in a Gauge theory. A. ; A.M. Polyakov, A. Alexander M Polyakov (born 27 September 1945) is a theoretical Physicist, formerly at the Landau Institute in Moscow, currently S. Schwartz and Yu. S. Tyupkin (1975). "Pseudoparticle solutions of the Yang-Mills equations". Phys. Lett. 59B: 85–87. doi:10.1016/0370-2693(75)90163-X. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ ^a ^b Hutter, Marcus (1995). "Instantons in QCD: Theory and application of the instanton liquid model". Ph. D. thesis. Retrieved on 2008-01-18. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 350 - Generallus Magnentius deposes Roman Emperor Constans and proclaims himself Emperor
^ 't Hooft, Gerard (1976). Gerardus 't Hooft (xeːrɑrt ət hoːft (born July 5, 1946, Den Helder) is a professor in Theoretical physics at Utrecht University "Computation of the quantum effects due to a four-dimensional pseudoparticle" (abstract). Phys. Rev. D14: 3432–3450. doi:10.1103/PhysRevD.14.3432. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.

The quantum quark, by Andrew Watson ISBN 0-521-82907-0
Handbook of QCD, by M. A. Shifman ISBN 981-238-028-0
The QCD vacuum, hadrons and superdense matter, by E. V. Shuryak ISBN 981-238-574-6

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