Topological susceptibility and chiral condensate with $N_f=2+1+1$ dynamical flavors of maximally twisted mass fermions

We study the ‘spectral projector’ method for the computation of the chiral condensate and the topological susceptibility, using $N_f=2+1+1$ dynamical flavors of maximally twisted mass Wilson fermions. In particular, we perform a study of the quark mass dependence of the chiral condensate $\mathrm{\Sigma }$ and topological susceptibility ${\chi }_{top}$ in the range $270 MeV < m_{\pi} < 500 MeV$andcompareourdatawithanalyticalpredictions\mathrm{.}Inaddition,wecompute$\chi_{top}$ in the quenched approximation where we match the lattice spacing to the $N_f=2+1+1$ dynamical simulations. Using the Kaon, $\eta$ and ${\eta }^{\mathrm{\prime }}$ meson masses computed on the $N_f=2+1+1$ ensembles, we then perform a preliminary test of the Witten-Veneziano relation.