SU(3)⊗SU(3), Symmetry Breaking, and Cabibbo Angle

Abstract

Making use of bootstrap equations for symmetry-breaking vectors in $(3^{*},3)\oplus (3,3^{*})$ and $(1,8)\oplus (8,1)$ representations of $\mathrm{SU}\mathrm{}\left(3\right)\mathrm{}\otimes \mathrm{SU}\mathrm{}\left(3\right)\mathrm{}$ with both octet and 27-plet electromagnetic and weak driving terms, and requiring partial conservation of vector current and the Cabibbo angle ${\theta }_c\ne 0$, we obtain a solution of the form of Gell-Mann, Oakes, and Renner for semistrong breaking interaction. Knowing the deviation from $\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\otimes \mathrm{SU}\mathrm{}\left(2\right)\mathrm{}$ in their work, we get ${\theta }_c\approx 0.22$. The nonleptonic $\Delta S=1\mathrm{}$ interaction cannot be rotated away as it is in $(1,8)\oplus (8,1)$. Also the electromagnetic tadpole cannot be in $(3^{*},3)\oplus (3,3^{*})$ but can be in $(1,8)\oplus (8,1)$.