Low-Energy Pion-Photon Interaction: The (2π, 2γ) Vertex

In the ($2\pi , 2\gamma$) problem, the Mandelstam representation is written for the two independent gauge-invariant amplitudes. On the basis of unitarity limitations on the asymptotic behavior of these amplitudes, only a $j=1\mathrm{}$ subtraction in the $\gamma +\pi \to \gamma +\pi$ channel and a $j=0\mathrm{}$ subtraction in the $\gamma +\gamma \to \pi +\pi$ channel are allowed. No over-all subtraction constants are required and the Thomson limit is automatically maintained. Only the effect of $2\pi$ intermediate states is considered. The odd-$j$ $\pi \pi$ contribution involves the amplitude for the process $\gamma +\pi \to 2\pi$ analyzed by Wong and shown to be proportional to a pseudo-elementary constant $\Lambda$. Even with a $\pi \pi$ $P$ resonance, the correction is negligible (≲1%) if we use the value of $\Lambda$ estimated by Wong on the basis of ${\pi }^0$ decay and confirmed by Ball in connection with photopion production on nucleons. A moderately important contribution comes from the $S$-wave interaction if we use a recent estimate of $\pi \pi$ $S$-wave phase shifts obtained from crossing relations. For the pion-pion coupling constant $\lambda$ of order -0.20, this effect is ∼10% in $\gamma +\pi \to \gamma +\pi$ scattering. For $\gamma +\gamma \to \pi +\pi$, the correction for the $I=0\mathrm{}$ state at threshold is positive and ∼100% of the Born approximation. However, as the energy is increased, the correction quickly changes sign.