Unitary Symmetry and theK→2πDecay

Abstract

It is shown that all the $K\ensuremath{\rightarrow}2\ensuremath{\pi}$ decays are forbidden in the limit of complete S${\mathrm{U}}_{3}$ symmetry of strong interactions, provided only (a) that both $K$ and $\ensuremath{\pi}$ mesons belong to the same octet and (b) that the nonleptonic decay interactions belong to the $\mathrm{CP}$-invariant representation (the irreducible representation whose $Y=Q=0$ members are $\mathrm{CP}$-invariant). This is more general than the conclusion of Cabibbo and Gell-Mann. In their proof, the nonleptonic decay Lagrangian is assumed to belong to a $\mathrm{CP}$-invariant octet. This result suggests that the observed large decay rate of ${{K}_{1}}^{0}\ensuremath{\rightarrow}2\ensuremath{\pi}$ decay is due to the symmetry-breaking interactions and its large $Q$ value. Finally, a brief comment is given concerning the $({{K}_{1}}^{0}\ensuremath{\rightarrow}2\ensuremath{\pi})\ensuremath{-}({K}^{+}\ensuremath{\rightarrow}2\ensuremath{\pi})$ puzzle in the $\ensuremath{\Delta}I=\frac{1}{2}$ rule. It is shown that, if one assumes the $\ensuremath{\Delta}I=\frac{1}{2}$ rule in addition to the above hypothesis (b), the ${K}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}2\ensuremath{\pi}$ decay can occur only through the electromagnetic corrections and its effective decay Lagrangian should belong to the representations 10 and $\overline{1}\overline{0}$.