Spontaneous Breakings of Chiral Symmetries

We analyze in detail the spontaneous breakings of chiral $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ and $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(2\right)\mathrm{}$. We determine the directions along which the two groups may break spontaneously. We discuss also the physical implications of these group-theoretical results, as the appearance of Goldstone particles, the particle mixings, and the consequences of the residual invariance.