Realization of chiral symmetry in the domain model of QCD

The domain model for the QCD vacuum has previously been developed and shown to exhibit confinement of quarks, a strong correlation of the local chirality of quark modes, and a duality of the background domainlike gluon field. Quark fluctuations satisfy a chirality violating boundary condition parametrized by a random chiral angle ${\alpha }_j$ on the jth domain. The free energy of an ensemble of $\overrightarrow{N}\infty$ domains depends on $\{{\alpha }_j,j=1,\dots ,N\}$ through the logarithm of the quark determinant. Its parity odd part is given by the axial anomaly. The anomaly contribution to the free energy suppresses continuous axial $U\left(1\right)\mathrm{}$ degeneracy in the ground state, leaving only a residual axial $Z\left(2\right)\mathrm{}$ symmetry. This discrete symmetry and the flavor ${SU(N}_f{)}_L\times {SU(N}_f{)}_R$ chiral symmetry in turn are spontaneously broken with a quark condensate arising due to the asymmetry of the spectrum of the Dirac operator. In order to illustrate the splitting between the ${\eta }^{\prime }$ from octet pseudoscalar mesons realized in the domain model, we estimate the masses of the light pseudoscalar and vector mesons.