Dispersion relation for hadronic light-by-light scattering: $\eta$ and $\eta'$ poles

TL;DR

This paper develops a dispersive approach to determine the $ ext{eta}$ and $ ext{eta'}$ transition form factors, enabling precise calculation of their contributions to the muon's anomalous magnetic moment with improved theoretical constraints.

Contribution

It introduces a dispersive analysis of $ ext{eta}$ and $ ext{eta'}$ TFFs, incorporating multiple experimental and theoretical constraints, including resonance effects and asymptotic behavior.

Findings

Calculated $ ext{eta}$-pole contribution to muon $g-2$ as $14.7(9) imes 10^{-11}$.

Calculated $ ext{eta'}$-pole contribution to muon $g-2$ as $13.5(7) imes 10^{-11}$.

Provided a comprehensive dispersive framework for pseudoscalar contributions to hadronic light-by-light scattering.

Abstract

The pseudoscalar-pole contributions to hadronic light-by-light scattering are determined by the respective transition form factors (TFFs) into two virtual photons. These TFFs constitute complicated functions of the photon virtualities that, in turn, can be reconstructed in a dispersive approach from their discontinuities. In this work, we present such an analysis for the $\eta^{(\prime)}$ TFFs, implementing a number of constraints from both experiment and theory: normalizations from the $\eta^{(\prime)}\to\gamma\gamma$ decay widths, unitarity constraints from the $\eta^{(\prime)}\to\pi^+\pi^-\gamma$ spectra, chiral symmetry for the $\eta^{(\prime)}\to 2(\pi^+\pi^-)$ amplitudes, vector-meson couplings, singly-virtual data from $e^+e^-\to e^+e^-\eta^{(\prime)}$, and the asymptotic behavior predicted by the light-cone expansion. In particular, we account for the leading left-hand-cut singularity by including effects from the $a_2$ resonance, necessitating the solution of an inhomogeneous Muskhelishvili-Omn\`es problem via a carefully chosen path deformation. The resulting TFFs allow us to evaluate the $\eta^{(\prime)}$-pole contributions to the anomalous magnetic moment of the muon, $a_\mu^{\eta\text{-pole}}=14.7(9)\times 10^{-11}$ and $a_\mu^{\eta'\text{-pole}}=13.5(7)\times 10^{-11}$, completing a dedicated program for the lowest-lying pseudoscalar intermediate states in a dispersive approach to hadronic light-by-light scattering, $a_\mu^{\text{PS-poles}}=91.2^{+2.9}_{-2.4}\times 10^{-11}$.