Backward Compton Scattering Sum Rule and theπ0→2γ,η0→2γDecay Rates

Abstract

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 ($\ensuremath{\gamma}N\ensuremath{\rightarrow}\ensuremath{\gamma}N$) contribution contains the well-known nucleon pole terms and the continuum, which we replace by just the $\ensuremath{\pi}\ensuremath{-}N$ intermediate states; we then feed in photoproduction data. The $t$-channel ($\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}N\overline{N}$) contribution consists of the $\ensuremath{\pi}$, $\ensuremath{\eta}$ 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 ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ and ${\ensuremath{\eta}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ 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.