FLAG Review 2021

TL;DR

FLAG 2021 synthesizes lattice QCD results for low-energy constants within SU(2) and SU(3) ChPT, tying meson observables and scattering lengths (ππ, πK, KK) to fundamental LECs and their scale dependence. The report highlights new LO and NLO SU(2) determinations (F, Σ^{1/3}, ℓ̄3, ℓ̄4) and LO SU(3 results (F0, Σ0, Σ/Σ0), plus select NLO SU(3) constants, while detailing how ππ and πK/KK scattering constrain these parameters through Lüscher-type finite-volume methods. It discusses the implications for Zweig-rule violations and the consistency (or tension) with FLAG19 averages, stressing the need for correlated extractions of multiple LECs. Overall, the edition advances the lattice-ChPT program by expanding the lattice input to LECs, clarifying the reliability of SU(2) versus SU(3) expansions, and informing CKM-related phenomenology via updated low-energy constants.

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 the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. 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 $α_s$. We consider nucleon matrix elements, and review the determinations of the axial, scalar and tensor bilinears, both isovector and flavor diagonal. Finally, in this review we have added a new section reviewing determinations of scale-setting quantities.

Figures (6)

• Figure 1: Cubic root of the $SU(2)$ quark condensate $\Sigma\equiv-\lim_{m_u,m_d\to0}\langle\ubar u\rangle$ in the $\overline{\rm MS}$-scheme, at the renormalization scale $\mu=2\GeV$. Square symbols 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 bullet points 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 $F$. Square symbols indicate determinations from correlators in the $p$-regime, and diamonds from the pion form factor.
• Figure 3: Effective coupling constant $\lbar_4$. 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 factor $\<r^2\>_V^\pi$. The publication status of $\chi$QCD 20 Wang:2020nbf changed from "preprint" to "accepted" after our closing date.
• Figure 5: Summary of the $\pi\pi$ scattering lengths $a_0^0\Mpi$ (top) and $a_0^2\Mpi$ (bottom). Results in Tab. \ref{['tab:pipi']} with statistical error only are not shown.