η , η ' mesons from lattice QCD in fully physical conditions

We determine masses and mixing parameters of the $\eta$ and ${\eta }^{\text{'}}$ meson in lattice QCD. The calculations are carried out on a set of 13 ETMC gauge ensembles with $N_f=2+1+1$ (maximally) twisted-mass Clover-improved quarks. These ensemble cover four values of the lattice spacing $a=0.057\text{fm},\dots ,0.092\text{fm}$ and pion masses from 140 to $360\text{MeV}$ , including three ensembles at physical quark masses and six ensembles with $M_{\pi }<200\text{MeV}$ . The strange-quark contribution is treated in a mixed-action approach using Osterwalder-Seiler fermions to avoid complications due to flavor mixing in the heavy quark sector and to enable the use of the one-end trick in the computation of strange quark-disconnected diagrams. With the strange-quark mass tuned to its physical value and several ensembles having close-to-physical light-quark mass, uncertainties related to the chiral extrapolations are reduced significantly compared to earlier studies. Physical results are computed with fully controlled systematics from a combined chiral, continuum and infinite-volume extrapolation, and a full error budget is obtained from model averages over of various fit ansätze and data cuts. Our results for the masses are given by $M_{\eta }=551\left(16\right)\text{MeV}$ and $M_{{\eta }^{\text{'}}}=972\left(20\right)\text{MeV}$ , respectively, where statistical and systematic errors have been added in quadrature. For the mixing angle and decay-constant parameters the Feldmann-Kroll-Stech scheme is employed to compute them from pseudoscalar matrix elements in the quark-flavor basis. For the mixing angle we obtain ${\varphi }^{\text{phys}}=39.3{\left(2.0\right)}^{\circ }$ and our results for the decay-constant parameters are given by $f_l^{\text{phys}}=138.6\left(4.4\right)\text{MeV}$ and $f_s^{\text{phys}}=170.7\left(3.3\right)\text{MeV}$ .