Precision evaluation of the $η$- and $η'$-pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon

Abstract

Next to the $π^0$ pole, $η$ and $η'$ intermediate states give rise to the leading singularities of the hadronic light-by-light tensor, resulting in sizable contributions to the anomalous magnetic moment of the muon $a_μ$. The strength of the poles is determined by the respective transition form factors (TFFs) to two (virtual) photons. We present a calculation of these TFFs that implements a number of low- and high-energy constraints, including the $η^{(\prime)}\toγγ$ decay widths, $η^{(\prime)}\toπ^+π^-γ$ spectra, chiral symmetry for the $η^{(\prime)}\to 2(π^+π^-)$ amplitudes, vector-meson couplings, and asymptotic limits. Crucially, we investigate the role of the leading left-hand singularity generated by the exchange of the $a_2$ tensor meson, yielding, for the first time, an estimate of the associated factorization-breaking corrections. Our final results, $a_μ^{η\text{-pole}}=14.7(9)\times 10^{-11}$ and $a_μ^{η'\text{-pole}}=13.5(7)\times 10^{-11}$, conclude a dedicated effort to evaluate the pseudoscalar-pole contributions to hadronic light-by-light scattering using dispersion relations, amounting to a combined $a_μ^{\text{PS-poles}}=91.2^{+2.9}_{-2.4}\times 10^{-11}$.