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We present preliminary results from our calculation of the pseudoscalartransition form factors of the $\eta$ and $\eta'$ mesons using staggered quarkson $N_f=2+1+1$ gauge ensembles generated by the Budapest-Marseille-Wuppertalcollaboration. These transition form factors are an important input for thehadronic light-by-light contribution to the muon $(g-2)$. We first elaborate onthe extraction of the masses of the $\eta$ and $\eta'$ mesons, that mix underthe dynamics of QCD. Thereafter, we show our preliminary results for thepseudoscalar transition form factors, focusing on the $\eta$ meson in theabsence of mixing.

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