$η'(η)\toγγ$: A Tale of Two Anomalies

Abstract

The radiative decays of the pseudoscalar mesons $P=(π^0,η,η')$ are of special interest as they provide an experimental window on the electromagnetic and colour chiral $U_A(1)$ anomalies. $π^0\toγγ$ is well described by the electromagnetic $U_A(1)$ anomaly under the assumption that $π^0$ is a Goldstone boson for spontaneously broken chiral symmetry, but the analogous results for $η'(η)\toγγ$ are complicated by the colour $U_A(1)$ anomaly in QCD. This paper reviews the theory of $η'(η)\toγγ$ 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 $η'$ and the QCD topological susceptibility is derived. The implications for the phenomenological analysis of $P\toγγ$ are described and a proposal is made for a comprehensive re-analysis of existing data on $P\toγγ$, $P\to Vγ$, where $V=(ρ,ω, ϕ)$ is a light $1^-$ meson, $η'(η)\to ππγ$, $ψ\toη'(η)γ$, etc. Other applications, to the muon anomalous magnetic moment $g_\m-2$ and the determination of the photon structure function $g_1^γ$ in polarised deep inelastic scattering, are also briefly discussed.