Review of lattice results concerning low energy particle physics

TL;DR

This FLAG review systematically compiles lattice QCD results for low-energy hadronic physics, extending beyond light-quark observables to heavy-quark sectors and the strong coupling. It introduces a formal quality framework, separates results by dynamical flavour (N_f), and provides conservative, consensus-based averages for quark masses, decay constants, form factors, mixing parameters, and α_s. The work highlights robust cross-checks across discretizations and actions, emphasizes nonperturbative renormalization where possible, and uses CKM unitarity and SM consistency tests to validate lattice inputs. Overall, lattice results now yield precise determinations of m_ud, m_s, f_K/f_π, B_K, and numerous heavy-quark quantities, with α_s rooted in multiple, complementary approaches, supporting the Standard Model picture while pointing to areas needing further refinement and cross-checks.

Abstract

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determination of the light-quark masses, the form factor f+(0), arising in semileptonic K -> pi transition at zero momentum transfer, as well as the decay constant ratio fK/fpi of decay constants and its consequences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)LxSU(2)R and SU(3)LxSU(3)R Chiral Perturbation Theory and review the determination of the BK parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, for this review, we focus on D- and B-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant alpha_s.

Figures (26)

• Figure 1: Mass of the strange quark (${\overline{\mathrm{MS}}}$ scheme , running scale 2 GeV). The central and top panels show the lattice results listed in tables \ref{['tab:masses2']} and \ref{['tab:masses3']}. For comparison, the bottom panel collects a few sum rule results and also indicates the current PDG estimate. Diamonds represent results based on perturbative renormalization, while squares indicate that, in the relation between the lattice regularized and renormalized ${\overline{\mathrm{MS}}}$ masses, nonperturbative effects are accounted for. The black squares and the grey bands represent our estimates (\ref{['eq:quark masses Nf=2']}) and (\ref{['eq:nf3msmud']}). The significance of the colours is explained in section \ref{['sec:qualcrit']}.
• Figure 2: Mean mass of the two lightest quarks, $m_{ud}=\frac{1}{2}(m_u+m_d)$ (for details see Fig. \ref{['fig:ms']}).
• Figure 3: Results for the ratio $m_s/m_{ud}$. The upper part indicates the lattice results listed in Table \ref{['tab:ratio_msmud']}. The lower part shows results obtained from $\chi$PT and sum rules, together with the current PDG estimate.
• Figure 4: Comparison of lattice results (squares) for $f_+(0)$ and $f_K/ f_\pi$ with various model estimates based on $\chi$PT (blue circles). The black squares and grey bands indicate our estimates. The significance of the colours is explained in section \ref{['sec:qualcrit']}.
• Figure 5: The plot compares the information for $|V_{ud}|$, $|V_{us}|$ obtained on the lattice with the experimental result extracted from nuclear $\beta$ transitions. The dotted arc indicates the correlation between $|V_{ud}|$ and $|V_{us}|$ that follows if the three-flavour CKM-matrix is unitary.