QCD vacuum


The QCD vacuum is the quantum vacuum state of quantum chromodynamics. It is an example of a non-perturbative vacuum state, characterized by non-vanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the confined phase of quark matter.

Symmetries and symmetry breaking

Symmetries of the QCD Lagrangian

Like any relativistic quantum field theory, QCD enjoys Poincaré symmetry including the discrete symmetries CPT. Apart from these space-time symmetries, it also has internal symmetries. Since QCD is an SU gauge theory, it has local SU 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 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 symmetries are broken in the QCD Lagrangian:
When the Hamiltonian of a system has a certain symmetry, but the vacuum does not, then one says that spontaneous symmetry breaking has taken place.
A familiar example of SSB is in ferromagnetic materials. Microscopically, the material consists of atoms 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 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.

Symmetries of the QCD vacuum

The SU × SU 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 part of the chiral group. The diagnostic for this is the formation of a non-vanishing chiral condensate, where is the quark field operator, and the flavour index is summed. The Goldstone bosons of the symmetry breaking are the pseudoscalar mesons.
When, i.e., only the up and down quarks are treated as massless, the three pions are the Goldstone bosons. When the strange quark is also treated as massless, i.e.,, 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 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.

Experimental evidence

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.

Motivating results

Gradient coupling

In the 1950s, there were many attempts to produce a field theory to describe the interactions of pions and nucleons. The obvious renormalizable interaction between the two objects is the Yukawa coupling to a pseudoscalar:
And this is theoretically correct, since it is leading order and it takes all the symmetries into account. But it doesn't match experiment in isolation. When the nonrelativistic limit of this coupling is taken, the gradient-coupling model is obtained. To lowest order, the nonrelativistic pion field interacts by derivatives. This is not obvious in the relativistic form. A gradient 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, at least not by itself. The way nature fixes this in the pseudoscalar model is by simultaneous rotation of the proton-neutron and shift of the pion field. This, when the proper axial SU symmetry is included, is the Gell-Mann Levy σ-model, discussed below.
The modern explanation for the shift symmetry is now understood to be the Nambu-Goldstone non-linear symmetry realization mode, due to Yoichiro Nambu and Jeffrey Goldstone.
The pion field is a Goldstone boson, while the shift symmetry is a manifestation of a degenerate vacuum.

Goldberger–Treiman relation

There is a surprising relationship between the strong interaction coupling of the pions to the nucleons, the coefficient in the nucleon-pion-gradient coupling model, and the axial vector current coefficient of the nucleon, which determines the weak decay rate of the neutron. The relation is
and it is obeyed to 2.5% 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 non-Abelian gauge theory. The Goldberger–Treiman relation suggests that the pions, by dint of chiral symmetry breaking, interact as surrogates of sorts of the axial weak currents.

Partially conserved axial current

The structure which gives rise to the Goldberger–Treiman relation was called the partially conserved axial current hypothesis, spelled out in the pioneering σ-model paper. Partially conserved describes the modification of a spontaneously-broken symmetry current by an explicit breaking correction preventing its conservation. The axial current in question is also often called the chiral symmetry current.
The basic idea of SSB is that the symmetry current which performs axial rotations on the fundamental fields does not preserve the vacuum: This means that the current applied to the vacuum produces particles. The particles must be spinless, otherwise the vacuum wouldn't be Lorentz invariant. By index matching, the matrix element must be
where is the momentum carried by the created pion.
When the divergence of the axial current operator is zero, we must have
Hence these pions are massless,, in accordance with Goldstone's theorem.
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.
But if a small explicit breaking of the chiral symmetry is introduced, as in real life, the above divergence does not vanish, and the right hand side involves the mass of the pion, now a Pseudo-Goldstone boson.

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. 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; i.e., the octet of vector mesons. 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 which 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.

Eta prime meson

This pattern of SSB solves one of the earlier "mysteries" of the quark model, where all the pseudoscalar mesons should have been of nearly the same mass. Since, there should have been nine of these. However, one has quite a larger mass than the SU octet. In the quark model, this has no natural explanation – a mystery named the η−η′ mass splitting.
In QCD, one realizes that the η′ is associated with the axial UA which is explicitly broken through the chiral anomaly, and thus its mass is not "protected" to be small, like that of the η. The η–η′ mass splitting can be explained
through the 't Hooft instanton mechanism,
whose realization is also known as Witten–Veneziano mechanism.

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 come from such analysis.
Another method of analysis of correlation functions in QCD is through an operator product expansion. 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 refers to the gluon field tensor, to the quark field, and 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.