A dispersive estimate of the $a_0(980)$ contribution to hadronic light-by-light scattering in $(g-2)_\mu$

A dispersive implementation of the $a_0(980)$ resonance to $(g-2)_\mu$ requires the knowledge of the double-virtual $S$-wave $\gamma^*\gamma^*\to\pi\eta / K\bar{K}_{I=1}$ amplitudes. To obtain these amplitudes we used a modified coupled-channel Muskhelischvili-Omnès formalism, with the input from the left-hand cuts and the hadronic Omnès function. The latter were obtained using a data-driven $N/D$ method in which the fits were performed to the different sets of experimental data on two-photon fusion processes with $\pi\eta$ and $K\bar{K}$ final states. This yields the preliminary dispersive estimate $a_\mu^{HLbL}[a_0(980)]_{resc.}=-0.46(2)\times 10^{-11}$.