Unitary Symmetry and the Transformationη→π0

The transformation $\eta \to {\pi }^0$, which is thought to play a dominant role in the decay mode $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$, is examined in the light of unitary symmetry and several different models for $\eta$ decay. Beside the models of Gell-Mann, Sharp, and Wagner (GSW), and of Barrett and Barton (BB), these include the strong coupling of pseudoscalar and vector meson octets with the unitary singlet vector meson $\varphi$, and the electromagnetic coupling of pseudoscalar and vector mesons. Group-theoretical methods are used to confirm that the contributions of all but one of the lowest order diagrams on the GSW model cancel one another. The same methods show that the cancellations in lowest order are not as serious for the BB model, and are nonexistent for models involving the $\varphi$ meson.