The QCD vacuum is the quantum vacuum state of quantum chromodynamics (QCD). 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 (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:
Poincaré symmetry and CPT invariance SU(3) local gauge symmetry approximate global SU(Nf) × SU(Nf) flavour chiral symmetry and the U(1) baryon number symmetry The following classical symmetries are broken in the QCD Lagrangian:
scale, i.e., 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.
Spontaneous symmetry breaking
When the Hamiltonian of a system (or the Lagrangian) has a certain symmetry, but 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 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(Nf) × SU(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 ⟨ψiψi⟩, 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, 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., 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.
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 ( π {\displaystyle \pi } ) and nucleons ( N {\displaystyle N} ). The obvious renormalizable interaction between the two objects is the Yukawa coupling to a pseudoscalar boson:
L I = N ¯ γ 5 π N {\displaystyle L_{I}={\bar {N}}\gamma _{5}\pi N}
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
π → π + C {\displaystyle \pi \rightarrow \pi +C}
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(2) 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.
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