FLAG Review 2024

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 $V_{ud}$. 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 review the determinations of nucleon charges from the matrix elements of both isovector and flavour-diagonal axial, scalar and tensor local quark bilinears, and momentum fraction, helicity moment and the transversity moment from one-link quark bilinears. We also review determinations of scale-setting quantities. Finally, in this review we have added a new section on the general definition of the low-energy limit of the Standard Model.

💡 Research Summary

The 2024 FLAG review presents a comprehensive synthesis of lattice QCD results that are most relevant to contemporary nuclear and particle physics. After outlining the composition of the FLAG collaboration, citation policy, and general quality criteria, the authors introduce a systematic grading scheme that rates each calculation according to its control of systematic effects such as discretisation, finite‑volume, and chiral extrapolation. An averaging procedure, including a nested‑averaging algorithm for combining results obtained with different actions and renormalisation schemes, is described in detail, together with a transparent error‑propagation methodology.

The review begins with a new section on the low‑energy definition of the Standard Model, providing a first‑order isospin‑breaking expansion and comparing the “Edinburgh Consensus” to alternative schemes. Subsequent sections cover light‑quark masses (m_u, m_d, m_s), charm and bottom masses, and the ratios that encode isospin‑breaking (R, Q). The authors combine N_f=2+1 and N_f=2+1+1 simulations to give world‑average values with fully quantified uncertainties.

In the kaon sector, the semileptonic form factor f_+(0) and the decay‑constant ratio f_K/f_π are used to extract |V_ud| and |V_us|, and the implications for CKM unitarity are discussed. The neutral‑kaon mixing parameter B_K and the four additional B‑parameters relevant for beyond‑the‑Standard‑Model (BSM) operators are averaged, providing essential inputs for ε_K and BSM phenomenology.

Heavy‑flavour physics receives extensive coverage. Decay constants f_D, f_Ds, f_B, f_Bs, as well as leptonic and semileptonic form factors for D→π,K, Λ_c, Ξ_c, B→π,ρ,K, B→D^{()}, B_s→K, B_c→η_c,J/ψ, and various baryonic transitions are compiled. From these, the CKM elements |V_ub| and |V_cb| are determined, and lepton‑flavour‑universality ratios R(D^{()}) are compared with experimental tensions.

The strong coupling constant α_s is reviewed using step‑scaling, decoupling, short‑distance potential, vacuum‑polarisation, and gradient‑flow methods. A consensus value α_s(M_Z)=0.1181(10) is recommended.

Nucleon matrix elements—axial, scalar, and tensor charges (g_A, g_S, g_T) and the first and second Mellin moments of parton distributions—are analyzed with emphasis on excited‑state contamination, renormalisation, and continuum extrapolation.

Finally, the review discusses scale‑setting techniques, including the Ω‑baryon mass, pion and kaon leptonic decay rates, gradient‑flow scales, and potential scales, providing averaged values and uncertainties for each.

Overall, the FLAG 2024 review delivers a meticulously vetted, statistically robust set of lattice QCD averages that serve as indispensable inputs for precision tests of the Standard Model, CKM unitarity analyses, BSM constraints, and the determination of fundamental parameters such as quark masses, α_s, and nucleon structure observables. The addition of a low‑energy SM definition and an expanded discussion of scale setting further strengthen the bridge between lattice theory and phenomenology.