Spontaneous breaking of chiral symmetry, and eventually of parity, in a $\sigma$-model with two Mexican hats

A sigma-model with two linked Mexican hats is discussed. This scenario could be realized in low-energy QCD when the ground state and the first excited (pseudo)scalar mesons are included, and where not only in the subspace of the ground states, but also in that of the first excited states, a Mexican hat potential is present. This possibility can change some basic features of a low-energy hadronic theory of QCD. It is also shown that spontaneous breaking of parity can occur in the vacuum for some parameter choice of the model.