Pseudoscalar transition form factors and the hadronic light-by-light contribution to the muon $g-2$

We report on our progress toward the computation of the ${\pi }^0$, $\eta$ and ${\eta }^{\mathrm{\prime }}$ transition form factors using staggered quarks on $N_f=2+1+1$ gauge ensembles generated by the Budapest-Marseille-Wuppertal collaboration. These form factors are essential ingredients to evaluate the pseudoscalar-pole contributions to the hadronic light-by-light scattering in the muon $g-2$. Preliminary results for the pseudoscalar-pole contributions are presented, at finite lattice spacing, for all three light mesons.