Low-Energy Theorem forγ+γ→π+π+π

We use the hypothesis of the partially conserved axial-vector current (PCAC) to show that the matrix elements for $\gamma +\gamma \to {\pi }^0+{\pi }^0+{\pi }^0$ and $\gamma +\gamma \to {\pi }^0+{\pi }^{+}+{\pi }^{-}$ vanish in the soft-${\pi }^0$ limit. This, combined with photon gauge invariance, implies low-energy theorems relating these matrix elements to the matrix elements for $\gamma +\gamma \to {\pi }^0$ and $\gamma \to {\pi }^0+{\pi }^{+}+{\pi }^{-}$. Since the magnitude of the former is determined by the ${\pi }^0$ lifetime, while the ratio of the latter to the former is determined in a model-independent way by isospin and low-energy-theorem arguments, a model-independent prediction for the $\gamma +\gamma \to \pi +\pi +\pi$ amplitude can be given. Our results agree with those of Aviv, Hari Dass, and Sawyer in the neutral case, but not in the charged case. We give a diagrammatic and effective-Lagrangian interpretation of our formulas which explains the discrepancy.