Vacuum symmetry structure in quantum chromodynamics

The quantum-chromodynamic vacuum is described in terms of additional angles associated with chiral $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ vacuum degeneracy and the phases of the quark mass matrix. This description helps to remove certain ambiguities associated with instantons, chiral $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ breaking, and the U(1) problem. It is verified that the chiral Ward identities are consistently satisfied in the presence of chiral $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ breaking without the need of a "ninth" massless pseudoscalar meson.