Flavor singlet phenomena in lattice QCD

TL;DR

Flavor singlet phenomena in lattice QCD interrogates how nonperturbative vacuum structure and disconnected quark loops shape low-energy observables. The paper surveys lattice methodologies for computing the pion–nucleon sigma term $\sigma_{\pi N}$, the flavor-singlet axial coupling $G_A^1$, and the $\eta'$ mass, contrasting quenched and full QCD, and contrasting direct and topological approaches. Key findings show substantial disconnected contributions in $\sigma_{\pi N}$ within quenched QCD and a suppressed $G_A^1$ consistent with axial anomaly effects, while the $\eta'$ mass can be understood through topological vacuum structure. The results highlight significant statistical and systematic challenges, especially for disconnected amplitudes, and argue that high-statistics full-QCD simulations with nonperturbative renormalization are essential to achieve precise, continuum-accurate predictions that illuminate the QCD vacuum.

Abstract

Flavor singlet combinations of quark operators ${\cal{O}}_S^Γ = \bar{u}Γu + \bar{d}Γd + \bar{s}Γs$ contribute to many important physical observables in the low energy region of QCD. Experimentally one finds the values of some of these observables to be in sharp contrast to the naive (perturbative) theoretical expectations. This indicates that non perturbative vacuum properties might play a major role in the comprehension of these phenomena. An example of such a vacuum contribution is the axial anomaly, which appears in the divergence of the flavor singlet axial current and which is connected to the topological properties of QCD. From a field theoretical point of view flavor singlet matrix elements differ from non singlet amplitudes in the occurrence of so called disconnected insertions. These are correlations of hadron propagators with quark-antiquark loops or correlations between quark-antiquark loops, which are mediated by vacuum fluctuations. According to their respective flavor composition, the disconnected insertions cancel largely in non singlet processes, but add in flavor singlet amplitudes. The lattice approach provides an ideal tool to study flavor singlet phenomena. Being a first principle method it should be capable to uncover non perturbative vacuum contributions and to yield, on the long run, reliable results for the size of such contributions in QCD. The present article reviews the status of flavor singlet matrix element calculations in lattice QCD with respect to methods, results and reliability. Special emphasis is paid to the discussion of state of the art calculations of the pion nucleon sigma term $σ_{πN}$, the flavor singlet axial coupling of the proton $G_A^1$, and the $η'$ mass.

Figures (21)

• Figure 1: Connected (a) and disconnected (b) insertions of a proton interacting with a current $j$. Please note that all quark lines, including the quark loop, are connected by infinitely many gluon lines and virtual quark loops.
• Figure 2: Connected (a) and disconnected (b) contributions to the $\eta'$ propagator.
• Figure 3: Ref. japan_nsigma: The ratio $R^{SUM}(t)$ for the connected and disconnected contributions (u+d insertion) to the scalar density amplitude of a proton $\langle P|\bar{u}u+\bar{d}d|P\rangle$, at a quark mass corresponding to $m_{\pi}/m_{\rho}=0.604$.
• Figure 4: Connected and disconnected contributions to the scalar density amplitude of a proton, $\langle P|\bar{u}u+\bar{d}d|P\rangle$. The data has been taken from Fukugita et al.japan_nsigma and from Dong et al.liu_nsigma. To compare the results of different simulations we have renormalized the amplitudes consistently with the tadpole improved renormalization factor $Z_s$sw_tadpole_imp.
• Figure 5: Disconnected contribution to $\sigma_{\pi N }$. Comparison of wall source + summation method with $Z_{2}$ stochastic estimator + PAM at a sea quark mass corresponding to $m_{\pi}/m_{\rho} = 0.833$. The solid lines indicate the region of ground state behavior of the signal.