Eur. Phys. J. A (2025) 61: 169
https://doi.org/10.1140/epja/s10050-025-01635-0

Regular Article

$\eta$, ${\eta }^{\prime }$ mesons from lattice QCD in fully physical conditions

¹ PRISMA+ Cluster of Excellence and Institut für Kernphysik, Johannes Gutenberg-Universität Mainz, 55099, Mainz, Germany
² Computation-Based Science and Technology Research Center, The Cyprus Institute, Nicosia, Cyprus
³ Theoretical Physics Department, Esplanade des Particules 1, 1211, Geneva 23, Switzerland
⁴ Helmholtz-Institut für Strahlen- und Kernphysik, University of Bonn, Bonn, Germany
⁵ Bethe Center for Theoretical Physics, University of Bonn, Bonn, Germany
⁶ Albert Einstein Center, Institute for Theoretical Physics, University of Bern, Bern, Switzerland

^a kottnad@uni-mainz.de

Received: 19 March 2025
Accepted: 3 July 2025
Published online: 22 July 2025

Abstract

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 $a=0.057\text{fm},\dots ,0.092\text{fm}$ 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 $M_{\eta }=551(16)\text{MeV}$ and $M_{{\eta }^{\prime }}=972(20)\text{MeV}$, 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 ${\varphi }^{\text{phys}}=39.3{(2.0)}^{\circ }$ and our results for the decay-constant parameters are given by $f_l^{\text{phys}}=138.6(4.4)\text{MeV}$ and $f_s^{\text{phys}}=170.7(3.3)\text{MeV}$.