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

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 ${\gamma }^{\ast }{\gamma }^{\ast }\to \pi \pi /\bar{K}K$ 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 $a_0(980)$: $a_{\mu }^{\text{HLbL}}[\text{scalars}]=-9(1)\times 10^{-11}$.

9 pages, 2 figures; journal version

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