Axial U(1) and spontaneous chiral symmetry breaking within a confining model of the QCD vacuum

A model for the QCD vacuum based on domain-like background gluon fields has previously been developed and demonstrated to exhibit confinement of quarks and strong correlation of the local chirality of quark modes and duality of the background gluon field. On the boundary of the j-th domain, quark fluctuations satisfy chirality violating boundary conditions parametrized by a chiral angle ${\alpha }_j$ which is treated as a collective variable. The dependence of the free energy of an ensemble of $N\to \mathrm{\infty }$ domains on $\{{\alpha }_j,j=\mathrm{1...}N\}$ is studied. For one massless flavour there are only two degenerate minima (${\alpha }_j=0,\pi$ $\mathrm{\forall }j$) due to the contribution of the anomaly to the free energy, which suppresses the continuous U(1) degeneracy. The residual discrete symmetry is spontaneously broken with a quark condensate of $-(238 {\rm{MeV}})^3.$For$N_f$masslessflavoursacontinuous$SU(N_f)_L\times SU(N_f)_R$degeneracyappears,whilethe$U_A(1)$ direction remains suppressed.