The eta' meson from lattice QCD

TL;DR

This study computes the eta2 flavour-singlet pseudoscalar meson mass in N_f=2 lattice QCD using maximally twisted-mass fermions, directly addressing disconnected contributions that couple to topological fluctuations. By employing variance-reduction techniques, including the one-end-trick and all-to-all point-to-point correlators, the authors achieve improved control over statistical errors and perform careful continuum and chiral extrapolations. They find a finite eta2 mass in the chiral limit, $m(\eta_2) \approx 0.87$ GeV, consistent across two lattice spacings, with a refined interplay between connected and disconnected diagrams. The results provide a more precise lattice-based benchmark for the flavour-singlet sector and its relation to topological charge fluctuations, informing phenomenology of $\eta$ and $\eta'$ in QCD.

Abstract

We study the flavour singlet pseudoscalar mesons from first principles using lattice QCD. With N_f=2 flavours of light quark, this is the so-called eta_2 meson and we discuss the phenomenological status of this. Using maximally twisted-mass lattice QCD, we extract the mass of the eta_2 meson at two values of the lattice spacing for lighter quarks than previously discussed in the literature. We are able to estimate the mass value in the limit of light quarks with their physical masses.

Figures (6)

• Figure 1: $\eta_2$ effective mass (from a variational basis) versus $t$. Results are from ensemble $B_1$ ($L=24$) and $B_6$ ($L=32$) (both $a\mu_q=0.004$). The results from momentum $k=1$ (in units of $2\pi/L$) have been plotted as mass values assuming $E^2=m^2+k^2$. The results are slightly displaced for better readability.
• Figure 2: Effective mass for zero-momentum $\eta_2$ correlator for ensemble $B_1$ ($a\mu_q=0.004$) taking as the connected contribution the ground state only. The horizontal line is the value given by a fit to the correlator data in the $t$-range 2-10. Also shown are results with point-to-point correlators with $r=2a$, as described in the text. The results are slightly displaced for improved readability.
• Figure 3: Disconnected contribution to the $\eta_2$ correlator at time $t$ and spatial separation $r$ from point source and sink. Results are from ensemble $B_1$ ($a\mu_q=0.004$) from 319 gauge configurations. The curves are from the model fit described in the text.
• Figure 4: Singlet pseudoscalar correlator for ensemble $B_1$ ($\mu_q=0.004$) with local point sink and source. The correlator is illustrated for spatial separation $r=2a$ versus time $t$. The upper solid line is the connected contribution from the ground state neutral meson while the circles show the disconnected contribution. The squares are the net contribution to the $\eta_2$ and the behaviour from a one state fit to them is also shown. Note that the one state fit is not exponential, as discussed in the text.
• Figure 5: Comparison of values for the $\eta_2$ mass versus quark mass in units of $r_0 \approx 0.45$ fm. Values of the lattice spacing used are shown by listing $r_0/a$. Some of the points have been displaced horizontally for legibility. The ETMC results shown in (a) are those from the 2-state fits to zero-momentum correlators (first method in text, open symbols) and from the error-reduction obtained by using point-to-point correlators and projecting to the ground state of the neutral pseudoscalar connected contribution (filled symbols). In (b) we compare our results with reduced errors (filled symbols) to other $N_f=2$ results from other collaborations Allton:2004qqLesk:2002gdHashimoto:2008xg (open symbols) with relatively light pions and small lattice spacing ($a < 0.1$ fm, $r_0/a > 4.5$). The strange quark corresponds to $(r_0 m_{\pi})^2 \approx 2.5$.