A Mixed Action Analysis of $\eta$ and $\eta'$ Mesons

We study $\eta$ and $\eta'$ mesons and their mixing angle in a mixed action approach with so-called Osterwalder-Seiler valence quarks on a Wilson twisted mass sea. The gauge configurations have been generated by ETMC for $N_f=2+1+1$ dynamical quark flavours and for three values of the lattice spacing. The main results are that differences in between the mixed action and the unitary approach vanish towards the continuum limit with the expected rate of $\mathcal{O}(a^2)$. The individual size of the lattice artifacts depends, however, strongly on the observable used to match unitary and valence actions. Moreover, we show that for the $\eta$ mass valence and valence plus sea quark mass dependence differ significantly. Hence, in this case a re-tuning of the simulation parameters in the valence sector only is not sufficient to compensate for mismatches in the original quark masses.