A dispersive estimate of scalar contributions to hadronic light-by-light scattering

TL;DR

This paper refines the estimate of scalar resonance contributions to hadronic light-by-light scattering in the muon's anomalous magnetic moment, using dispersion relations and coupled-channel analysis to improve accuracy.

Contribution

It introduces a dispersive approach to evaluate scalar resonance effects, including heavier scalars, improving upon previous hadronic model estimates.

Findings

Scalar contributions are estimated to be around -9(1)×10^{-11} in the muon's anomalous magnetic moment.

The $f_0(980)$ contribution is approximately -0.2(2)×10^{-11}.

Heavier scalar resonances contribute less than 1×10^{-11}.

Abstract

We consider the contribution of scalar resonances to hadronic light-by-light scattering in the anomalous magnetic moment of the muon. While the $f_0(500)$ has already been addressed in previous work using dispersion relations, heavier scalar resonances have only been estimated in hadronic models so far. Here, we compare an implementation of the $f_0(980)$ resonance in terms of the coupled-channel $S$-waves for $\gamma^*\gamma^*\to \pi\pi/\bar K K$ to a narrow-width approximation, which indicates $a_\mu^{\text{HLbL}}[f_0(980)]=-0.2(2)\times 10^{-11}$. With a similar estimate for the $a_0(980)$, the combined effect is thus well below $1\times 10^{-11}$ in absolute value. We also estimate the contribution of heavier scalar resonances. In view of the very uncertain situation concerning their two-photon couplings we suggest to treat them together with other resonances of similar mass when imposing the matching to short-distance constraints. Our final result is a refined estimate of the $S$-wave rescattering effects in the $\pi \pi$ and $\bar K K$ channel up to about $1.3$ GeV and including a narrow-width evaluation of the $a_0(980)$: $a_\mu^\text{HLbL}[\text{scalars}]=-9(1)\times 10^{-11}$.