Weak four-fermion interaction in terms of SU(3) invariants. I

Abstract

The four-fermion weak interaction is formulated in terms of invariants of the groups G₂=L⊗SU (2) and G₃=L⊗SU(3). The four-fermion Hamiltonian is constructed as a four-fermion invariant of the group G₃ on the basis of the spin tensorsψ ^lmn _#x03B1;βγ , whereα, β, γ=1, 2, 3, 4 are the spinor indices andl, m, n=1, 2, 3 are the unitary indices of the representation of G₃. It is shown that in the case of one multipletψ ^lmn _#x03B1;βγ one can construct one nontrivial invariant and in the case of two multipletsψ ^lmn _#x03B1;βγ andϕ ^lmn _#x03B1;βγ , nine nontrivial invariants. Of these, only in the case of two of the invariants, which contain two multiplets, are the lepton and baryon numbers conserved independently. One of these invariants is considered in detail. In the case of SU(2) there is no fundamental difficulty and a number of relations are obtained for the constants of the weak interactions and the probabilities of processes. In the case of SU(3) it is shown that the known breakings of SU(3) in weak interactions can be localized in the lepton octet by choosing it in a special way.