Backward Compton Scattering Sum Rule and the ${\pi }^0\to 2\gamma$, ${\eta }^0\to 2\gamma$ Decay Rates

We use the superconvergence of certain Compton scattering ($s$-channel) helicity amplitudes for fixed $s$ and large $t$, to derive a sum rule for $t$- and $u$-channel processes. The $u$-channel ($\gamma N\to \gamma N$) contribution contains the well-known nucleon pole terms and the continuum, which we replace by just the $\pi -N$ intermediate states; we then feed in photoproduction data. The $t$-channel ($\gamma \gamma \to N\overline{N}$) contribution consists of the $\pi$, $\eta$ poles and the continuum. We choose a suitable combination of superconvergent amplitudes such that the effects of $0^{+}$, $2^{+}$, and $1^{+}$ resonances in the $t$ channel are eliminated. Assuming that this takes care of most of the $t$-channel continuum, we get a sum rule for the ${\pi }^0\to 2\gamma$ and ${\eta }^0\to 2\gamma$ widths, or alternatively, by using the experimental widths, we can check the consistency of the superconvergence in question. A brief comparison is made with related work by Goldberger and Abarbanel, and by Pagels.