Axial-vector and scalar contributions to hadronic light-by-light scattering

We present results for single axial-vector and scalar meson pole contributions to the hadronic light-by-light scattering (HLbL) part of the muon's anomalous magnetic moment. In the dispersive approach to these quantities (in narrow width approximation) the central inputs are the corresponding space-like electromagnetic transition form factors. We determine these directly using a functional approach to QCD by Dyson-Schwinger and Bethe-Salpeter equations in the very same setup we used previously to determine pseudo-scalar meson exchange ($\pi$, $\eta$ and $\eta'$) as well as meson ($\pi$ and $K$) box contributions. Particular care is taken to preserve gauge invariance and to comply with short distance constraints in both the form factors and the HLbL tensor. Our result for the contributions from a tower of axial-vector states including short distance constraints is { $a_\mu^{\text{HLbL}}[\text{AV-tower+SDC}] = 24.8 \,(6.1) \times 10^{-11}$}. For the combined contributions from $f_0(980), a_0(980), f_0(1370)$ and $a_0(1450)$ we find $a_\mu^{\text{HLbL}}[\text{scalar}] = -1.6 \,(5) \times 10^{-11}$.