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.
In other phases of quark matter the full chiral flavour symmetry may be recovered, or broken in completely different ways.
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.
This type of coupling means that a coherent state of low momentum pions barely interacts at all.
The way nature fixes this in the pseudoscalar model is by simultaneous rotation of the proton-neutron and shift of the pion field.
The pion field is a Goldstone boson, while the shift symmetry is a manifestation of a degenerate vacuum.
The constant GA 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 (strong) scattering of bound states of quarks and gluons.
The Goldberger–Treiman relation suggests that the pions, by dint of chiral symmetry breaking, interact as surrogates of sorts of the axial weak currents.
The structure which gives rise to the Goldberger–Treiman relation was called the partially conserved axial current (PCAC) hypothesis, spelled out in the pioneering σ-model paper.
By index matching, the matrix element must be where kμ 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, m2π = 0, in accordance with Goldstone's theorem.
The most convincing evidence for SSB of the chiral flavour symmetry of QCD is the appearance of these pseudo-Goldstone bosons.
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.
In QCD, one realizes that the η′ is associated with the axial UA(1) which is explicitly broken through the chiral anomaly, and thus its mass is not "protected" to be small, like that of the η.
Another method of analysis of correlation functions in QCD is through an operator product expansion (OPE).
Lattice QCD is making rapid progress towards providing the solution as a systematically improvable numerical computation.
The purpose of these models is to make quantitative sense of some set of condensates and hadron properties such as masses and form factors.
The instability of a homogeneous gluon field was argued by Niels Kjær Nielsen and Poul Olesen in their 1978 paper.
't Hooft showed further that an Abelian projection of a non-Abelian gauge theory contains magnetic monopoles.
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[14] one may expect a much more complicated and possibly dynamical structure in QCD.
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.
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.
The wave functions of the quarks satisfy the boundary conditions of a fermion in an infinitely deep potential well of scalar type with respect to the Lorentz group.
The spectral asymmetry is just the vacuum expectation value ⟨ψγ0ψ⟩ summed over all of the quark eigenstates in the bag.
Another view states that BPST-like instantons play an important role in the vacuum structure of QCD.
The results obtained did not solve the infrared problem of QCD, making many physicists turn away from instanton physics.
A more recent picture of the QCD vacuum is one in which center vortices play an important role.