Decaysπ0→2γ,η→2γand Sum Rules for Nucleon Compton Scattering

Abstract

Motivated by the success of the Goldberger—Treiman calculation of charged-pion decay and the assumption of pion-pole dominance of the divergence of the axial current subsequently introduced, we study the decays of the neutral pion and eta in the same way. Examining Compton scattering of real photons from the nucleons presents a fruitful analogy to the nuclear $\beta$-decay process. Identifying the dominant nucleon and meson-pole-term contributions to Compton scattering and utilizing the content of the exact low-energy theorem on Compton scattering, a no-subtraction hypothesis, supported by the Regge-pole phenomenology, then enables us to establish exact sum rules for the lifetimes ${\tau }_{{\pi }^0\to 2\gamma }$ and ${\tau }_{\eta \to 2\gamma }$ and also the Drell—Hearn sum rules for $\kappa _p^{}{}_{}{}^2\pm \kappa _n^{}{}_{}{}^2$. We also consider in detail the relation of our sum rule to the Goldberger—Treiman calculation of ${\pi }^0$ decay and forward-Compton-scattering sum rules for systems of spin=1. Neglecting continuum contributions, which are small, we find from our sum rules $\kappa _p^{}{}_{}{}^2=\kappa _n^{}{}_{}{}^2$ and $\tau _{{\pi }^0\to 2\gamma }^{}{}_{}{}^{-1\mathrm{}}=\frac{\pi {\alpha }^{2}m_{\pi }^{}{}_{}{}^3\kappa _p^{}{}_{}{}^2}{4g_{\pi N^2}{M}_{N^2}}\simeq 3.1$ eV or ${\tau }_{{\pi }^0\to 2\gamma }\simeq 2.2\times {10}^{-16\mathrm{}}$ sec in approximate agreement with the experimental value ${\tau }_{{\pi }^0\to 2\gamma }=(1.0\mathrm{}\pm 0.5)\times {10}^{-16\mathrm{}}$ sec. Better photopion-production data for the nonresonant multipoles would enable us to accurately estimate the continuum contributions. Including the dominant contributions to the photopion production continuum, we find that the Drell-Hearn sum rule for $\kappa _p^{}{}_{}{}^2+\kappa _n^{}{}_{}{}^2$ is well satisfied. We can estimate the $\eta \to 2\gamma$ lifetime but our result depends on a knowledge of the eta-nucleon coupling and on nonresonant background contributions to the photopion multipoles. Once better photopion production data for the nonresonant multipoles $E_{l\pm }$, $M_{l\pm }$, $l<~2$ for energies ${\pi }^0\to 2\gamma$, $\eta \to 2\gamma$.