FLAG Review 2019

TL;DR

FLAG 2019 consolidates lattice QCD determinations of low-energy constants in chiral perturbation theory, emphasizing SU(2) and SU(3) LECs, finite-volume regimes, and alternative extraction methods. It analyzes how lattice data constrain F, Sigma, and NLO LECs like bar_L3 and bar_L4, and discusses the interplay between x- and xi-expansions, discretization, and scale setting. The report highlights general consistency between SU(2) and SU(3) analyses, notes small Zweig-rule violations for L4 and L6, and calls for more diverse ensembles and lighter m_s to sharpen convergence and reduce uncertainties. Overall, it provides a comprehensive benchmark of current lattice QCD capabilities for low-energy hadronic physics and guides future computational efforts.

Abstract

We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to π$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_π$ and its consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $SU(2)_L\times SU(2)_R$ and $SU(3)_L\times SU(3)_R$ Chiral Perturbation Theory. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for $D$- and $B$-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $α_s$. Finally, in this review we have added a new section reviewing results for nucleon matrix elements of the axial, scalar and tensor bilinears, both isovector and flavor diagonal.

Figures (6)

• Figure 1: Cubic root of the $SU(2)$ quark condensate $\Sigma\equiv-\langle\ubar u\rangle|_{m_u,m_d\to0}$ in the $\overline{\rm MS}$-scheme, at the renormalization scale $\mu=2\GeV$. Green and red squares indicate determinations from correlators in the $p$-regime. Up triangles refer to extractions from the topological susceptibility, diamonds to determinations from the pion form factor, and star symbols refer to the spectral density method.
• Figure 2: Comparison of the results for the ratio of the physical pion decay constant $\Fpi$ and the leading-order $SU(2)$ low-energy constant $F$. The meaning of the symbols is the same as in Fig. \ref{['fig:sigma']}.
• Figure 3: Effective coupling constants $\lbar_3$, $\lbar_4$ and $\lbar_6$. Squares indicate determinations from correlators in the $p$-regime, diamonds refer to determinations from the pion form factor.
• Figure 4: Summary of the pion form factors $\<r^2\>_V^\pi$ (top) and $\<r^2\>_S^\pi$ (bottom).
• Figure 5: Summary of the $\pi$-$\pi$ scattering lengths $a_0^0\Mpi$ (top) and $a_0^2\Mpi$ (bottom).