A Dynamical Calculation of theη→3πDecay Rate

We formulate dispersion relations for calculating the strength of the virtual $\eta \to {\pi }^0$ transition and include contributions from all possible intermediate baryon-antibaryon states. In our approximation, the change in isospin is derived from the baryon electromagnetic mass splittings, other contributory causes being neglected. The $\eta \to 3\pi$ decay rate is governed by the strength of this transition if, as is believed, the decay proceeds via virtual single pseudoscalar boson intermediate states. We recall that the contributions from the two sequences $\eta \to \left({\pi }^0\right)\to {\pi }^{+}+{\pi }^{-}+{\pi }^0$ and $\eta \to {\pi }^{+}+{\pi }^{-}+\left(\eta \right)\to {\pi }^{+}+{\pi }^{-}+{\pi }^0$ cancel each other exactly if the eightfold way interaction $4\pi \lambda {(\pi ·\pi +\eta \eta )}^2$ is used, and if the strength of the $\eta -{\pi }^0$ transition is taken to be independent of which particle is on its mass shell. Here this difficulty is resolved since our dynamical calculation allows for the mass dependence of the $\eta -{\pi }^0$ "black box." Comparison of our provisionally evaluated result with that of a previous calculation on $\eta \to 2\gamma$ gives an estimate for the $\eta \to \frac{2\gamma }{\eta }\to {\pi }^{+}{\pi }^{-}{\pi }^0$ branching ratio which is in satisfactory agreement with experiment.