Invariant properties of weak interactions and the Cabibbo angle in chiral dynamics

It is shown that if the Cabibbo angle is introduced by an axial (and not vector) rotation about the seventh axis in $SU_3$-space, and it is assumed that the pion is massless before the rotation, the well-known contradiction between the allowed regions of values of the parameters of the broken $SU_3\otimes SU_3$ chiral symmetry can be eliminated. Different models are discussed and an investigation made of the Hamiltonian of the $SU_3\otimes SU_3$ chiral symmetry breaking before the Cabibbo rotation. The results are generalized to $SU_4\otimes SU_4$. The invariance properties of the weak interactions under permutation of the quark isodoublets in the Glashow–Iliopoulos–Maiani model are used. Corresponding unitary transformations are constructed and a theorem proved about the structure of the Hamiltonian of the $SU_4\otimes SU_4$ chiral symmetry breaking.