$η$ and $η'$ mesons from $N_f = 2+1$ lattice QCD at the physical point using topological charge operators

TL;DR

This work addresses the challenge of determining η and η' masses and their SU(3) flavor mixing in lattice QCD by using topological charge density operators on two 2+1-flavor ensembles at the physical point. The authors compute 4D topological-charge correlators, apply Wilson-flow smearing, and fit the resulting spectra with large-$\hat{r}$ single-state and constrained two-state forms to extract $m_\eta$, $m_{\eta'}$, and $\theta_1$, arriving at $m_\eta = 0.505(72)(75)$ GeV, $m_{\eta'} = 0.952(47)(40)$ GeV, and $\theta_1 = -8.9(2.1)(1.8)^{\circ}$. The analysis demonstrates that topological charge operators can effectively probe flavor-singlet mesons, offering a cross-check against quark-bilinear methods and a path toward higher-precision results with more configurations. The study also provides insight into the interplay between topology, the U_A(1) anomaly, and η–η' mixing in QCD.

Abstract

By fitting the two-point correlation functions of topological charge density operators calculated on two $2+1$-flavor gauge ensembles with physical pion mass, we determine both the $η$ and $η'$ masses and also the mixing angle to be $m_η= 0.505(72)(75)$ GeV, $m_{η'}=0.952(47)(40)$ GeV, and $θ_1 = -8.9(2.1)(1.8)^\circ$, respectively, where the first error is the statistical uncertainty and the second one is the systematic uncertainty. This is the first extraction of both $η/η'$ masses and the mixing angle $θ_1$ using topological charge operators. Compared with previous studies using quark bilinear operators, the error of the $η$ mass is relatively large, but the mixing angle has comparable precision. This demonstrates that the topological charge operators are well suited to study the $η$ and $η'$ mesons.

Figures (8)

• Figure 1: The two-point correlation function of the topological charge density operator $C_2(r)$ as a function of $r$ on the 48I ensemble. The zoomed in figure shows the negative correlation at large $r$.
• Figure 2: The correlation functions of topological charge density operators on the 64I ensembles. Additional minus sign is included and the data points with different flow times are shifted vertically for clarity.
• Figure 3: The topological charges of several different gauge configurations as a function of flow time on the 64I lattice. The horizontal dashed lines indicate the nearest integers.
• Figure 4: The effective masses and fitting results of the $\eta$ mass. The bands also indicate the fitting ranges.
• Figure 5: Left panel: single-state fitting results for the $\eta$ mass on the 64I ensemble as a function of the fitting starting point $\hat{r}_{\rm start}$. The corresponding $\chi^2/\text{d.o.f.}$ are also presented. The bands are to indicate the picked fitting and to show more clearly the difference of the fitting results between different $\hat{r}_{\rm start}$. Right panel: the same plot but for the 48I ensemble.