\(\eta \), \(\eta ^\prime \) mesons from lattice QCD in fully physical conditions

We determine masses and mixing parameters of the \(\eta \) and \(\eta ^\prime \) 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 \(a=0.057\,\textrm{fm},\ldots ,0.092\,\textrm{fm}\) and pion masses from 140 to \(360\,\textrm{MeV}\), including three ensembles at physical quark masses and six ensembles with \(M_\pi <200\,\textrm{MeV}\). 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 \(M_\eta =551(16)\,\textrm{MeV}\) and \(M_{\eta ^\prime }=972(20)\,\textrm{MeV}\), 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 \(\phi ^\textrm{phys}=39.3(2.0)^\circ \) and our results for the decay-constant parameters are given by \(f_l^\textrm{phys}=138.6(4.4)\,\textrm{MeV}\) and \(f_s^\textrm{phys}=170.7(3.3)\,\textrm{MeV}\).