Effects of the Pion-Pion Resonance and the Three-Pion Resonance or Bound State on Neutral-Pion Decay

We have applied the dispersion method to the problem of neutral-pion decay. It is shown that a pion-pion $P$-wave resonance can produce a large effect in the decay matrix element. The contribution is related to the semifundamental constant which determines the rate of photoproduction of pions from pions. The contribution of a strong three-pion state is also considered.