${\eta }^{\prime }(\eta )\to \gamma \gamma$: A Tale of Two Anomalies

Abstract

The radiative decays of the pseudoscalar mesons $P=({\pi }^0,\eta ,{\eta }^{\prime })$ are of special interest as they provide an experimental window on the electromagnetic and colour chiral $U_A(1)$ anomalies. ${\pi }^0\to \gamma \gamma$ is well described by the electromagnetic $U_A(1)$ anomaly under the assumption that ${\pi }^0$ is a Goldstone boson for spontaneously broken chiral symmetry, but the analogous results for ${\eta }^{\prime }(\eta )\to \gamma \gamma$ are complicated by the colour $U_A(1)$ anomaly in QCD. This paper reviews the theory of ${\eta }^{\prime }(\eta )\to \gamma \gamma$ decays, emphasising the role of the colour $U_A(1)$ anomaly and the renormalisation group. The relation to the Witten-Veneziano mass formula for the ${\eta }^{\prime }$ and the QCD topological susceptibility is derived. The implications for the phenomenological analysis of $P\to \gamma \gamma$ are described and a proposal is made for a comprehensive re-analysis of existing data on $P\to \gamma \gamma$, $P\to V\gamma$, where $V=(\rho ,\omega ,\varphi )$ is a light $1^{-}$ meson, ${\eta }^{\prime }(\eta )\to \pi \pi \gamma$, $\psi \to {\eta }^{\prime }(\eta )\gamma$, etc. Other applications, to the muon anomalous magnetic moment $g_{\text{m}}-2$ and the determination of the photon structure function $g_1^{\gamma }$ in polarised deep inelastic scattering, are also briefly discussed.

Comments: 18 pages, 0 figures, LaTeX with espcrc2.sty