QCD vacuum

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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.

Unsolved problems in physics: QCD in the non-perturbative regime: confinement. The equations of QCD remain unsolved at energy scales relevant for describing atomic nuclei. How does QCD give rise to the physics of nuclei and nuclear constituents?

Contents

[edit] Symmetries and symmetry breaking

[edit] 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). 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 flavours 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 instantons implied by this anomaly, also breaks CP symmetry.

In summary, the QCD Lagrangian has the following symmetries:

The following classical symmtries are broken in the QCD Lagrangian:

[edit] 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.

A familiar example of SSB is in magnetic materials. Microscopically, the material consists of atoms with a non-vanishing spin, each of which acts like a tiny bar magnet, ie, a magnetic dipole. The Hamiltonian of the material, describing the interaction of neighbouring dipoles, is invariant under rotations. At high temperature, there is no magnetization 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 bosons appear, corresponding to the remaining symmetry. This is called the Goldstone phenomenon and the bosons are called Goldstone bosons.

(For more details, see the page on spontaneous symmetry breaking).

[edit] Symmetries of the QCD vacuum

The SU(Nf)XSU(Nf) chiral flavour symmetry of the QCD Lagrangian is broken in the vacuum state 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 \langle\overline\psi_i\psi_i\rangle, where ψi is the quark field operator, and the flavour index i is summed. The Goldstone bosons of the symmetry breaking are the pseudoscalar mesons.

When Nf=2, ie, only the u and d quarks are treated as massless, the three pions are the Goldstone bosons. When the s quark is also treated as massless, ie, Nf=3, all eight pseudoscalar mesons of the quark model become Goldstone bosons. The actual masses of these mesons 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.

[edit] Evidence: experimental consequences

[edit] Pseudo-Goldstone bosons

Experimentally it is seen that the masses of the octet of pseudoscalar mesons is very much lighter than the next heaviest states, ie, 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 bosons. 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.

[edit] 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. 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 as 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 instantons cause the η-η' mass splitting.

[edit] Current algebra and QCD sum rules

PCAC and current algebra 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 functions 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, ie, 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—

\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.

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.

[edit] Models of the QCD vacuum

A full solution of QCD would automatically give a full description of the vacuum, confinement 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 factors.

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.

[edit] 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. This would then be a more complicated version of the Higgs mechanism. However, Stanley Mandelstam showed that a homogeneous vacuum field is also unstable. 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.

[edit] The dual superconducting model

In a type II superconductor, electric charges condense into Cooper pairs. 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 monopoles. There is continuing interest in checking whether further parts of this picture hold.

[edit] String models

String models of confinement and hadrons have a long history. They were first invented to explain certain aspects of crossing symmetry in the scattering of two mesons. They were also found to be useful in the description of certain properties of the Regge trajectory of the hadrons. 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 interactions. 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.

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.

[edit] Bag models

Strictly, these models are not models of the QCD vacuum, but of physical single particle quantum states — the hadrons. 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 \overline\psi \gamma_5 \gamma_\mu\psi 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 \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 gA, 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.

[edit] Instanton gas and liquid

[edit] See also

[edit] References and external links

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